2$ we have $\\dim S_k^{+}(\\Gamma _0(2))^{\\rm new} - \\dim S_k^{-}(\\Gamma _0(2))^{\\rm new}={\\left\\lbrace \\begin{array}{ll} -1 & k\\equiv 2 \\, \\bmod \\, 8 \\\\0 & k \\equiv 4,6 \\, \\bmod \\, 8 \\\\1 & k \\equiv 0 \\, \\bmod \\, 8\\, .","\\\\\\end{array}\\right.","}$ We recall the notion of Yoshida type lifts.","Yoshida lifts are explained in [27]; see also [28], [26], [5], [21].","These are eigen forms associated to a pair of elliptic modular eigenforms whose spinor L-function is a product of the twisted L-functions of the elliptic modular cusp forms.","In [3] a number of conjectures on the existence of Yoshida lifts were made and these were proved by Rösner [20].","These conjectures deal with Siegel modular cusp forms of weight $(j,k)$ with $k\\ge 3$ .","It can be extended to the case of weight $(j,2)$ .","We denote the subspace of $S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}$ generated by Yoshida lifts by $YS_{j,2}^{s[\\varpi ]}$ .","Theorem 1.1 We have $YS_{j,2}^{s[w]} = 0$ unless we are in the following cases: (1) $w=[1^6]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ eigen-newforms of level $\\Gamma _0(2)$ of different sign.","In this case we have $\\dim YS_{j,2}^{s[w]} \\,= \\,\\dim \\,S^+_{j+2}(\\Gamma _0(2))^{\\rm new} \\otimes S^-_{j+2}(\\Gamma _0(2))^{\\rm new}.$ (2) $w=[2,1^4]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\Gamma _0(4)$ .","The multiplicity $\\mu (j)$ of $s[2,1^4]$ in $YS_{j,2}^{s[w]}$ is then $\\mu (j)\\, =\\, \\dim \\, \\Lambda ^2 S_{j+2}(\\Gamma _0(4))^{\\rm new}.$ (3) $w=[2^3]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\Gamma _0(2)$ with the same sign.","The multiplicity $\\nu (j)$ of $s[2^3]$ is $\\nu (j) \\,= \\,\\dim \\Lambda ^2 S^+_{j+2}(\\Gamma _0(2))^{\\rm new} \\oplus \\Lambda ^2 S^-_{j+2}(\\Gamma _0(2))^{\\rm new}.$ The proof of this theorem follows from results of Rösner and Weissauer, in a way very similar to Rösner's proof of the Bergström-Faber-van der Geer conjecture in weight $k\\ge 3$ [20].","In the second appendix Chenevier explains how to derive it.","We now formulate our conjecture.","Conjecture 1.2 The space $S_{j,2}(\\Gamma _2[2])$ is generated by Yoshida type lifts.","Note that this implies that $S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}=(0)$ unless $\\varpi = [1^6], [2,1^4]$ or $[2^3]$ .","In particular, it implies that $S_{2,j}(\\Gamma _2)=(0)$ .","The evidence we have for the latter is the following.","Theorem 1.3 We have $\\dim S_{j,2}(\\Gamma _2)=0$ for $j\\le 52$ .","For $j\\le 20$ the vanishing of $S_{j,2}(\\Gamma _2)$ was proved by Ibukiyama, Wakatsuki and Uchida [16], [17], and [25].","The evidence we have for the $s[1^6]$ -part of the conjecture is the following.","Theorem 1.4 The dimension of $S_{j,2}(\\Gamma _2,\\epsilon )$ is given by the coefficient of $t^j$ in the expansion of $t^{12}/(1-t^6)(1-t^8)(1-t^{12})$ for $j\\le 30$ .","Modular forms in $S_{j,2}(\\Gamma _2,\\epsilon )$ can be constructed explicitly using covariants as explained in [7].","We prove this theorem by constructing a basis of the space $S_{j,7}(\\Gamma _2)$ using covariants of binary sextics (see [7]) and then by checking which forms are divisible by the cusp form $\\chi _5 \\in S_{0,5}(\\Gamma _2,\\epsilon )$ .","We thus give generators for these spaces $S_{j,2}(\\Gamma _2[2])^{s[1^6]}$ for $j\\le 30$ and we can calculate Hecke eigenvalues for these.","For $j>30$ this becomes quite laborious.","For all irreducible representations we have the following vanishing result: Proposition 1.5 For any $\\varpi $ we have $\\dim S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}=0$ for $j<12$ .","We end with some remarks on other `small' weights.","The vanishing of $S_{j,1}(\\Gamma _2)$ follows from work of Skoruppa [23].","In an appendix to this paper besides providing a different proof of the vanishing of $S_{j,2}(\\Gamma _2)$ for $j\\le 38$ , Chenevier gives a proof of the vanishing of $S_{j,1}(\\Gamma _2[2])$ .","For $k=3$ one knows that $S_{j,3}(\\Gamma _2)=(0)$ for $j<36$ .","But $S_{36,3}$ is 1-dimensional and using covariants we can construct a form of this weight in a relatively easy manner; cf.","the remarks in [17] on the difficulty of constructing such a form.","Acknowledgement.","We thank Gaëtan Chenevier warmly for asking the question on the existence of modular forms of weight $(j,2)$ on the full group ${\\rm Sp}(4,{\\mathbb {Z}})$ and for agreeing to add his results in the form of an appendix.","He also pointed out an error in an earlier version of our conjecture and provided a proof of the extension of Rösner's result on Yoshida lifts.","We thank Mirko Rösner for correspondence.","We also thank the Max-Planck Institut für Mathematik in Bonn for excellent working conditions."],["Modular Forms of Degree Two","For the definitions of Siegel modular forms and elementary properties we refer to [14].","We denote the Siegel upper half space of degree $g$ by $\\mathfrak {H}_g$ .","The Siegel modular group $\\Gamma _g={\\rm Sp}(2g,{\\mathbb {Z}})$ acts on $\\mathfrak {H}_g$ by fractional linear transformations $\\tau \\mapsto (a\\tau +b)(c\\tau +d)^{-1} \\qquad \\text{\\rm for $\\left(\\begin{matrix}a & b\\cr c & d \\cr \\end{matrix} \\right) \\in {\\rm Sp}(2g,{\\mathbb {Z}})$and $\\tau \\in \\mathfrak {H}_g$.","}$ If $\\rho : {\\rm GL}(g,{\\mathbb {C}}) \\rightarrow {\\rm GL}(W)$ is a finite-dimensional complex representation then a holomorphic map $f: \\mathfrak {H}_g \\rightarrow W$ is called a Siegel modular form of weight $\\rho $ if $f(a\\tau +b)(c\\tau +d)^{-1})=\\rho (c\\tau +d)f(\\tau )$ for all $(a,b;c,d) \\in \\Gamma _g$ .","The space of modular forms of weight $\\rho $ is finite-dimensional and denoted by $M_{\\rho }(\\Gamma _g)$ .","If $g=2$ then an irreducible representation of ${\\rm GL}(2,{\\mathbb {C}})$ is of the form ${\\rm Sym}^j({\\rm St}) \\otimes {\\det }({\\rm St})^k$ with ${\\rm St}$ the standard representation of ${\\rm GL}(2,{\\mathbb {C}})$ for some $j \\in {\\mathbb {Z}}_{\\ge 0}$ and $k \\in {\\mathbb {Z}}$ .","For $\\rho ={\\rm Sym}^j({\\rm St}) \\otimes {\\det }({\\rm St})^k$ we denote $M_{\\rho }$ by $M_{j,k}$ and we call $(j,k)$ the weight.","If $j=0$ we are dealing with scalar-valued modular forms.","The space of Siegel modular forms of degree 2 and weight $(j,k)$ is denoted by $M_{j,k}(\\Gamma _2)$ .","There is the Siegel operator $\\Phi _g$ that maps Siegel modular forms of degree $g$ to Siegel modular forms of degree $g-1$ .","The kernel of $\\Phi _2$ in $M_{j,k}(\\Gamma _2)$ is called the space of cusp forms of weight $(j,k)$ and denoted by $S_{j,k}(\\Gamma _2)$ .","Note that for $k=2$ we have $M_{j,2}(\\Gamma _2)=S_{j,2}(\\Gamma _2)$ , see [16].","For a finite index subgroup $\\Gamma $ of ${\\rm Sp}(4,{\\mathbb {Z}})$ we have similar notions.","Here we deal with the groups $\\Gamma _2$ and $\\Gamma _2[2]$ .","The quotient group $\\Gamma _2/\\Gamma _2[2]\\cong {\\rm Sp}(4,{\\mathbb {Z}}/2{\\mathbb {Z}})$ is identified with the symmetric group $\\mathfrak {S}_6$ as in the Introduction.","This group acts in a natural way on the space of cusp forms $S_{j,k}(\\Gamma _2[2])$ and we can decompose this space in isotypical components $S_{j,k}(\\Gamma _2[2])^{s[\\varpi ]}$ corresponding to the irreducible representations $s[\\varpi ]$ of $\\mathfrak {S}_6$ which in turn correspond bijectively to the partitions $\\varpi $ of 6.","The ring $R$ of scalar-valued Siegel modular forms on $\\Gamma _2$ was determined by Igusa in the 1960s, see [18].","In the 1980s Tsushima gave in [24] formulas for the dimensions of the spaces of vector-valued cusp forms on a subgroup between $\\Gamma _2[2]$ and $\\Gamma _2$ .","Bergström extended this to $\\Gamma _2[2]$ with the action of $\\mathfrak {S}_6$ , see [2].","We thus know the dimension of $S_{j,k}(\\Gamma _2[2])^{s[\\varpi ]}$ for all $j$ and $k\\ge 3$ .","The vector-valued modular forms of degree 2 form a ring $M=\\oplus _{j,k} M_{j,k}(\\Gamma _2)$ .","It is also a module over the ring $R$ .","For level 2 similar things hold.","A vector-valued Siegel modular form $f$ of weight $(j,k)$ on $\\Gamma _2$ has a Fourier-Jacobi expansion $f(\\tau )= \\sum _{m\\ge 0} \\varphi _m(\\tau _1,z) \\, e^{2\\pi i m \\tau _2}\\qquad \\text{ where $\\tau = \\left( \\begin{matrix} \\tau _1 & z \\\\z & \\tau _2 \\\\ \\end{matrix} \\right)$ }$ with $\\varphi _m: \\mathfrak {H}_1 \\times {\\mathbb {C}} \\rightarrow {\\rm Sym}^j({\\mathbb {C}}^2)$ a holomorphic map that satisfies certain functional equations under the action of the so-called Jacobi group ${\\rm SL}(2,{\\mathbb {Z}}) \\ltimes {\\mathbb {Z}}^2$ .","This group is embedded in $\\Gamma _2$ via $\\left( \\begin{matrix} a & b \\\\ c & d \\\\ \\end{matrix}\\right) \\mapsto \\left( \\begin{matrix} a & 0 & b & 0 \\\\0 & 1 & 0 & 0 \\\\c & 0 & d & 0 \\\\0 & 0 & 0 & 1 \\\\ \\end{matrix} \\right), \\qquad (\\lambda , \\mu ) \\mapsto \\left( \\begin{matrix} 1 & 0 & 0 & \\mu \\\\\\lambda & 1 & \\mu & 0 \\\\0 & 0 & 1 & -\\lambda \\\\0 & 0 & 0 & 1 \\\\ \\end{matrix} \\right)$ with action $\\tau \\mapsto \\left( \\begin{matrix} (a\\tau _1+b)/(c\\tau _1+d) & z/(c\\tau _1+d) \\\\z/(c\\tau _1+d) & \\tau _2-cz^2/(c\\tau _1+d)\\\\ \\end{matrix} \\right)$ and $\\tau \\mapsto \\left( \\begin{matrix} \\tau _1 & z+\\lambda \\tau _1+\\mu \\\\z+\\lambda \\tau _1+\\mu & \\tau _2+ \\lambda ^2\\tau _1+2\\lambda z +\\lambda \\mu \\\\ \\end{matrix} \\right) \\, .$ The fact that $f$ is a modular form of weight $(j,k)$ implies the corresponding functional equations $\\varphi _m(\\frac{a \\tau _1+b}{c\\tau _1+d}, \\frac{z}{c\\tau _1+d})e^{-2\\pi i m \\frac{c z^2}{c\\tau _1+d}}= {\\rm Sym}^j\\left( \\begin{matrix} c\\tau _1+d & c z \\\\ 0 & 1 \\\\ \\end{matrix} \\right) (c\\tau _1+d)^k\\varphi _m(\\tau _1,z)$ and $\\varphi _m(\\tau _1,z+\\lambda \\tau _1+\\mu ) e^{2\\pi i m (\\lambda ^2 \\tau _1 +2\\lambda z +\\lambda \\mu )} ={\\rm Sym}^j \\left( \\begin{matrix} 1 & -\\lambda \\\\ 0 & 1 \\\\ \\end{matrix} \\right) \\varphi _m(\\tau _1,z) \\, ,$ where we write $\\varphi _m$ as the transpose of the row vector $(\\varphi _m^{(0)}, \\ldots , \\varphi _m^{(j)})$ .","Corollary 2.1 If $f \\in M_{j,k}(\\Gamma _2)$ (resp.","$f \\in S_{j,k}(\\Gamma _2)$ ) then the last coordinate $\\varphi _m^{(j)}$ of the coefficient $\\varphi _m$ of $e^{2\\pi i m \\tau _2}$ in the Fourier-Jacobi expansion of $f$ is a Jacobi form (resp.","Jacobi cusp form) of weight $k$ and index $m$ .","We note that Jacobi cusp forms of weight 2 and index $m$ are zero for $m< 37$ , see [11].","This imposes strong conditions on forms of weight $(j,2)$ on $\\Gamma _2$ .","Since we shall compute the action of Hecke operators later we now describe formulas for the action of Hecke operators on forms on $\\Gamma _2$ .","For forms without character we refer to [9], so we deal with the case of forms with a character.","For $f \\in M_{j,k}(\\Gamma _2,\\epsilon )$ we write its Fourier expansion as $f(\\tau )= \\sum _{n\\ge 0} a(n) \\, e^{\\pi i {\\rm Tr}(n\\tau )} \\, ,$ where $n$ runs over the positive semi-definite half-integral symmetric matrices.","We will write $[n_1,n_2,n_3]$ for $\\left( {\\begin{matrix}n_1 & n_2/2 \\\\ n_2/2 & n_3 \\end{matrix}} \\right)$ .","For an odd prime $p$ we denote by $T_p$ the Hecke operator for $\\Gamma _2$ at $p$ .","Then we write the transform of $f$ under $T_p$ as $T_p(f)(\\tau )= \\sum _{n\\ge 0} a_p(n) \\, e^{\\pi i {\\rm Tr} (n\\tau )}\\, .$ Here for $p \\lnot \\equiv 1 \\bmod \\, 3$ the coefficient $a_p([1,1,1])$ is given by $a([p,p,p])$ , and for $p=3$ by $a([3,3,3])-3^{k-2}{\\mathrm {Sym}}^j\\left({\\begin{matrix}3 & -1 \\\\0 & 1 \\\\\\end{matrix}}\\right)a([1,3,3]) \\, ,$ while for $p \\equiv 1 \\bmod \\, 3$ by $a([p,p,p])+p^{k-2}\\sum _{i=1}^{2}(-1)^{m_i}{\\mathrm {Sym}}^j\\left({\\begin{matrix}p & -m_i \\\\0 & 1 \\\\\\end{matrix}}\\right)a([\\frac{1+m_i+m_i^2}{p},1+2m_i, p]) \\, ,$ where in the latter case $m_1$ and $m_2$ are the two roots of the polynomial $1+X+X^2$ over $\\mathbb {F}_p$ , which we view here as the set $\\left\\lbrace 0,\\ldots ,p-1\\right\\rbrace $ .","Similarly, the coefficient $a_{p^2}([1,1,1])$ of the transform of $f$ under the Hecke operator $T_{p^2}$ is given for $p \\lnot \\equiv 1 \\bmod \\, 3$ by $a([p^2,p^2,p^2])$ , and for $p=3$ by $a([9,9,9])-3^{k-2}{\\mathrm {Sym}}^j\\left({\\begin{matrix}3 & -1 \\\\0 & 1 \\\\\\end{matrix}}\\right)a([3,9,9]) \\, .$ As an example, consider the modular form $\\chi _5 \\in S_{0,5}(\\Gamma _2,\\epsilon )\\,,$ the product of the ten even theta characteristics and the square root of Igusa's cusp form $\\chi _{10}$ , that will play an important role in this paper.","It provides a check on these formulas for the Hecke operators.","Indeed, one knows $\\lambda _p(\\chi _5)=p^3+a_p(f)+p^4, \\qquad \\lambda _{p^2}(\\chi _5)=\\lambda _p(\\chi _5)^2-(p^4+p^3)\\lambda _p(\\chi _5)+p^{8}\\, ,$ where $f=q-8\\, q^2+12\\, q^3+64\\, q^4-210\\, q^5 +\\cdots $ is the normalized Hecke eigenform in $S_{8}^+(\\Gamma _0(2))^{\\text{new}}$ , which illustrates that $\\chi _5$ is a Saito-Kurokawa lift.","One can check that the above formulas agree with this."],["Restriction to the diagonal","In order to put restrictions on the existence of Siegel modular forms we restrict these to the `diagonal' given by the embedding $i: \\mathfrak {H}_1\\times \\mathfrak {H}_1 \\rightarrow \\mathfrak {H}_2,\\qquad (z_1,z_2) \\mapsto \\left(\\begin{matrix}z_1 & 0 \\cr 0 & z_2\\cr \\end{matrix}\\right)\\, .$ The stabilizer of $i(\\mathfrak {H}_1\\times \\mathfrak {H}_1)$ in ${\\rm Sp}(4,{\\mathbb {R}})$ is an extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ of the image of ${\\rm SL}(2,{\\mathbb {R}})^2$ under the embedding $(\\left({\\begin{matrix}a_1 & b_1 \\\\ c_1 & d_1\\end{matrix}}\\right),\\left({\\begin{matrix}a_2 & b_2 \\\\ c_2 & d_2\\end{matrix}}\\right)) \\, \\mapsto \\,\\left({\\begin{matrix}a_1 & 0 & b_1 & 0\\\\0 & a_2 & 0 & b_2 \\\\c_1 & 0 & d_1 & 0\\\\0 & c_2 & 0 & d_2 \\\\\\end{matrix}}\\right)$ The extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ corresponds to the involution that interchanges $\\tau _1$ and $\\tau _2$ in $\\tau =\\left({\\begin{matrix}\\tau _1 & \\tau _{12}\\cr \\tau _{12} & \\tau _2 \\cr \\end{matrix}} \\right) \\in \\mathfrak {H}_2$ (and $z_1$ and $z_2$ on $\\mathfrak {H}_1^2$ ).","This corresponds to the element $\\iota =\\left({\\begin{matrix} a & 0 \\\\ 0 & d \\\\ \\end{matrix}}\\right)$ in $\\Gamma _2$ with $a=d= \\left({\\begin{matrix} 0 & 1 \\\\ 1 & 0 \\\\ \\end{matrix}}\\right)$ .","The stabilizer inside $\\Gamma _2$ (resp.","inside $\\Gamma _2[2]$ ) is an extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ of ${\\rm SL}(2,{\\mathbb {Z}})\\times {\\rm SL}(2,{\\mathbb {Z}})$ (resp.","of $\\Gamma _1[2]\\times \\Gamma _1[2]$ ).","If $F=(F_0,\\ldots ,F_{j})^t$ is a Siegel modular form of weight $(j,k)$ of level 2, then its pullback under $i$ to $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ gives rise to an element of $(f_0,\\ldots , f_j)^t$ with $f_l \\in M_{j+k-l}(\\Gamma _1[2])\\otimes M_{k+l}(\\Gamma _1[2])$ .","By restricting a cusp form of level 1 we get cusp forms of level 1.","The action of $\\iota $ is given by a map $S_{j+k-i}(\\Gamma _1[2])\\otimes S_{k+i}(\\Gamma _1[2]) \\rightarrow S_{k+i}(\\Gamma _1[2])\\otimes S_{j+k-i}(\\Gamma _1[2]), \\qquad a\\otimes b \\mapsto (-1)^k b\\otimes a$ for $\\Gamma _1[2]$ and a similar one for $\\Gamma _1$ .","So for a form of level 1 without character we loose no information by looking at $\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\Gamma _1)\\otimes S_{k+i}(\\Gamma _1)\\bigoplus {\\left\\lbrace \\begin{array}{ll}{\\wedge }^2 S_{j/2+k}(\\Gamma _1) & \\text{for $k$ odd} \\\\{\\rm Sym}^2 S_{j/2+k}(\\Gamma _1) & \\text{for $k$ even.}","\\\\\\end{array}\\right.","}$ By multiplying with $\\chi _5$ we get an injective map $S_{j,2}(\\Gamma _2,\\epsilon ) \\rightarrow S_{j,7}(\\Gamma _2)$ .","The generating series for the dimensions is $\\sum _{j=2}^{\\infty } \\dim S_{j,7}(\\Gamma _2) \\, t^j ={t^{12} \\over (1-t)(1-t^3)(1-t^4)(1-t^6)} \\, .$ We observe that our conjecture on $S_{j,2}(\\Gamma _2,\\epsilon )$ implies that $\\dim S_{j,7}(\\Gamma _2) -\\dim S_{j,2}(\\Gamma _2,\\epsilon ) =\\sum _{i=0}^{j/2-1} \\dim S_{j+7-i}(\\Gamma _1) \\dim S_{7+i}(\\Gamma _1)+ \\dim \\wedge ^2 S_{j/2+7}(\\Gamma _1) \\, ,$ or equivalently, that the restriction $\\rho $ to the diagonal fits in an exact sequence $0 \\rightarrow S_{j,2}(\\Gamma _2,\\epsilon ) {\\cdot \\chi _5 \\over \\longrightarrow }S_{j,7}(\\Gamma _2){\\rho \\over \\longrightarrow } \\oplus _{i=0}^{j/2-1}S_{j+7-i}(\\Gamma _1)\\otimes S_{7+i}(\\Gamma _1)\\oplus \\wedge ^2 S_{j/2+7}(\\Gamma _1)\\rightarrow 0 \\, .$ If $F\\in S_{j,k}(\\Gamma _2,\\epsilon )$ we find by using $\\iota $ that we can restrict to $\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\Gamma _1[2])^{s[1^3]}\\otimes S_{k+i}(\\Gamma _1[2])^{s[1^3]}\\bigoplus {\\rm Sym}^2(S_{j/2+k}(\\Gamma _1[2])^{s[1^3]})\\, .$ Indeed, the group $\\mathfrak {S}_3={\\rm SL}(2,{\\mathbb {Z}}/2{\\mathbb {Z}})$ acts on $S_k(\\Gamma _1[2])$ and for a form on $\\Gamma _2$ with a character the components $f_i, f_i^{\\prime }$ of the restriction to $i(\\mathfrak {H}_1\\times \\mathfrak {H}_1)$ are modular forms on $\\Gamma _1[2]$ with a character, i.e.","they lie in the $s[1^3]$ -isotypical part of $S_k(\\Gamma _1[2])$ .","The module $\\oplus _k S_k(\\Gamma _1[2])^{s[1^3]}$ is a module over the ring ${\\mathbb {C}}[e_4,e_6]$ of modular forms on $\\Gamma _1$ and is generated by the cusp form $\\delta =\\eta ^{12}$ , a square root of $\\Delta \\in S_{12}(\\Gamma _1)$ .","The generating series for the dimensions is now $\\sum _{j=2}^{\\infty } \\dim S_{j,7}(\\Gamma _2,\\epsilon )\\, t^j ={t^6 \\over (1-t)(1-t^3)(1-t^4)(1-t^6)}\\, .$ Conjecturally we now find an exact sequence $\\begin{aligned}0 \\rightarrow S_{j,7}(\\Gamma _2,\\epsilon ) {\\rho \\over \\longrightarrow }\\oplus _{i=0}^{j/2-1} S_{j+7-i}(\\Gamma _0(4))^{n}\\otimes S_{7+i}(\\Gamma _0(4))^{n} \\,\\oplus {\\rm Sym}^2 (S_{j/2+7}(\\Gamma _0(4))^{n}) &\\\\\\rightarrow K \\rightarrow 0\\, , &\\\\\\end{aligned}$ where $S_k(\\Gamma _0(4))^{n}=S_k(\\Gamma _0(4))^{\\rm new}$ and the dimension of the cokernel $K$ is now predicted by minus the extrapolation to $k=2$ of the algorithms used in [3] to calculate the dimension of $S_{j,k}(\\Gamma _2)$ and which give negative numbers here."],["Constructing Modular Forms Using Covariants","In the paper [7] we explained how to use invariant theory to construct Siegel modular forms.","In this paper we shall make extensive use of the procedure.","Let $V$ be the standard representation space of ${\\rm GL}(2,{\\mathbb {C}})$ with basis $x_1,x_2$ .","We consider the space ${\\rm Sym}^6(V)$ of binary sextics, where we write an element as $f= \\sum _{i=0}^6 a_i \\binom{6}{i} x_1^{6-i}x_2^i \\, .$ Sometimes we call this expression the universal binary sextic.","For a description of invariants and covariants for the action of ${\\rm GL}(2,{\\mathbb {C}})$ we refer to [7].","An invariant can be viewed as a polynomial in the coefficients $a_i$ that is invariant under the action of ${\\rm SL}(2,{\\mathbb {C}})$ , while a covariant of degree $(a,b)$ can be viewed as a form of degree $a$ in the $a_i$ and degree $b$ in $x_1,x_2$ .","If $A[\\lambda _1,\\lambda _2]$ is an irreducible representation of highest weight $(\\lambda _1 \\ge \\lambda _2)$ of ${\\rm GL}(2,{\\mathbb {C}})$ embedded equivariantly in ${\\rm Sym}^d({\\rm Sym}^6(V))$ this defines a covariant of degree $(d,\\lambda _1-\\lambda _2)$ and it is unique up to a multiplicative non-zero constant.","We denote the ring of covariants by ${\\mathcal {C}}$ .","Clebsch and others constructed in the 19th century generators for this ring.","There are 26 generators, 5 invariants and 21 covariants, satisfying many relations.","They can be found in the book of Grace and Young [15].","For the convenience of the reader we reproduce these here.","In the following table $C_{a,b}$ denotes a generator of degree $(a,b)$ .","height2pt$a \\backslash b$ 0 2 4 6 8 10 12 1 $C_{1,6}$ 2 $C_{2,0}$ $C_{2,4}$ $C_{2,8}$ 3 $C_{3,2}$ $C_{3,6}$ $C_{3,8}$ $C_{3,12}$ 4 $C_{4,0}$ $C_{4,4}$ $C_{4,6}$ $C_{4,10}$ 5 $C_{5,2}$ $C_{5,4}$ $C_{5,8}$ 6 $C_{6,0}$ $C_{6,6}^{(1)}$ $C_{6,6}^{(2)}$ 7 $C_{7,2}$ $C_{7,4}$ 8 $C_{8,2}$ 9 $C_{9,4}$ 10 $C_{10,0}$ $C_{10,2}$ 12 $C_{12,2}$ 15 $C_{15,0}$ A theorem of Gordan says that all these covariants can be constructed explicitly by using so-called transvectants from the universal binary sextic.","If ${\\rm Sym}^m(V)$ denotes the space of binary quantics of degree $m$ then we define the $k$ th transvectant as follows.","It is a map ${\\rm Sym}^m(V) \\times {\\rm Sym}^n(V) \\rightarrow {\\rm Sym}^{m+n-2k}(V)$ that sends a pair $(f,g)$ to $(f,g)_k=\\frac{(m-k)!(n-k)!}{m!n!","}\\sum _{j=0}^k (-1)^j\\binom{k}{j}\\frac{\\partial ^k f}{\\partial x_1^{k-j}\\partial x_2^j}\\frac{\\partial ^k g}{\\partial x_1^{j}\\partial x_2^{k-j}}\\, .$ When $k=1$ , we omit the index: $(f,g)=(f,g)_1$ .","The next table gives the construction of the covariants in the preceding table.","height2pt 1 $C_{1,6}=f$ 2 $C_{2,0}=(f,f)_6$ $C_{2,4}=(f,f)_4$ $C_{2,8}=(f,f)_2$ 3 $C_{3,2}=(C_{1,6},C_{2,4})_4$ $C_{3,6}=(f,C_{2,4})_2$ $C_{3,8}=(f,C_{2,4})$ $C_{3,12}=(f,C_{2,8})$ 4 $C_{4,0}=(C_{2,4},C_{2,4})_4$ $C_{4,4}=(f,C_{3,2})_2$ $C_{4,6}=(f,C_{3,2})$ $C_{4,10}=(C_{2,8},C_{2,4})$ 5 $C_{5,2}=(C_{2,4},C_{3,2})_2$ $C_{5,4}=(C_{2,4},C_{3,2})$ $C_{5,8}=(C_{2,8},C_{3,2})$ 6 $C_{6,0}=(C_{3,2},C_{3,2})_2$ $C_{6,6}^{(1)}=(C_{3,6},C_{3,2})$ $C_{6,6}^{(2)}=(C_{3,8},C_{3,2})_2$ 7 $C_{7,2}=(f,C_{3,2}^2)_4$ $C_{7,4}=(f,C_{3,2}^2)_3$ 8 $C_{8,2}=(C_{2,4},C_{3,2}^2)_3$ 9 $C_{9,4}=(C_{3,8},C_{3,2}^2)_4$ 10 $C_{10,0}=(f,C_{3,2}^3)_6$ $C_{10,2}=(f,C_{3,2}^3)_5$ 12 $C_{12,2}=(C_{3,8},C_{3,2}^3)_6$ 15 $C_{15,0}=(C_{3,8},C_{3,2}^4)_8$ Let $M$ be the ring of vector-valued Siegel modular forms of degree 2.","It is a module over the ring $R$ of scalar-valued Siegel modular forms of degree 2.","In [7] we defined maps $M \\longrightarrow {\\mathcal {C}} {\\nu \\over \\longrightarrow }M_{\\chi _{10}}\\, ,$ where $M_{\\chi _{10}}$ is the localization of $M$ at $\\chi _{10}$ .","A modular form of weight $(j,k)$ maps to a covariant of degree $(j/2+k,j)$ and a covariant of degree $(a,b)$ is sent to a meromorphic modular form of weight $(b,a-b/2)$ .","Under the map $\\nu $ the universal binary sextic $f$ is mapped to $\\chi _{6,3}/\\chi _5$ of weight $(6,-2)$ .","Here $\\chi _{6,3}$ is a holomorphic form in $S_{6,3}(\\Gamma _2,\\epsilon )$ .","The beginning of its Fourier expansion is given in [7].","In practice instead of $\\nu $ often we use a slightly modified map $\\mu : {\\mathcal {C}} \\longrightarrow M\\oplus M_{\\epsilon }\\, ,$ where $M_{\\epsilon }= \\oplus M_{j,k}(\\Gamma _2,\\epsilon )$ , is the $R$ -module of modular forms with a character.","Under $\\mu $ the universal sextic $f$ is mapped to $\\chi _{6,3}$ .","Then a covariant maps of degree $(a,b)$ maps to a holomorphic Siegel modular form of weight $(b,6a-b/2)$ and character $\\epsilon ^a$ .","Remark 4.1 Since $\\chi _{6,3}$ vanishes simply at infinity the definition of $\\mu $ implies that the image under $\\mu $ of a covariant of degree $(a,b)$ vanishes at infinity with order $\\ge a$ .","Recall that the order of vanishing of $\\chi _5$ at infinity is 1.","For example, Igusa's generators $E_4,E_6,\\chi _{10}, \\chi _{12}$ and $\\chi _{35}$ of $R$ are up to a non-zero multiplicative constant obtained as $\\begin{aligned}&E_4=\\mu (75\\, C_{4,0} -8\\, C_{2,0}^2)/\\chi _{10}^2, \\quad E_6= \\mu (224\\, C_{2,0}^3 -1425\\, C_{2,0}C_{4,0} -1125 \\, C_{6,0})/\\chi _{10}^3 \\\\&\\chi _{10}^6 = \\mu (C_{\\chi _{10}}), \\quad \\chi _{12}= \\mu (C_{2,0}), \\quad \\chi _{35}= \\mu (C_{15,0})/\\chi _5^{11}, \\\\\\end{aligned}$ with $C_{\\chi _{10}}$ , up to a multiplicative constant equal to the discriminant, given by $768\\, C_{2,0}^5-7625 \\,C_{4,0}C_{2,0}^3-1875\\left(7 \\, C_{6,0}C_{2,0}^2 -10 \\, C_{4,0}^2C_{2,0}-30 \\, C_{6,0}C_{4,0}- 13860 \\, C_{10,0}\\right) \\, .$ Remark 4.2 The first scalar-valued cusp form on $\\Gamma _2$ with character is of weight 30 and can be obtained by dividing $\\mu (C_{15,0})$ by $\\chi _5^{12}$ .","Note that we have $M_{j,k}(\\Gamma _2,\\epsilon )=S_{j,k}(\\Gamma _2,\\epsilon ).$ (see [17])."],["Cusp forms of weight $(j,2)$ on {{formula:3c24a2d5-8cca-456e-8c8a-10623d91c714}} with a character","Our conjecture says that $S_{j,2}(\\Gamma _2,\\epsilon )=(0)$ for $j<12$ .","We begin by showing this.","Lemma 5.1 For $j=0, 2, 4, 6, 8$ and 10, we have $S_{j,2}(\\Gamma _2,\\epsilon )=(0).$ We know that $\\dim S_{2j,4}(\\Gamma _2)=0$ for $j=0,2,4,6,8,10$ .","Assume that for one of these values of $j$ there is a non-zero element $f\\in S_{j,2}(\\Gamma _2,\\epsilon )$ .","Then ${\\rm Sym}^2(f)\\in S_{2j,4}(\\Gamma _2)=(0)$ must be zero.","Using the fact that the ring of holomorphic functions on $\\mathfrak {H}_2$ is an integral domain, we get a contradiction.","The first case where $S_{j,2}(\\Gamma _2,\\epsilon )$ is predicted to be non-zero is for $j=12$ .","In the paper [7] we constructed a modular form $\\chi _{12,2}$ in this space using the covariant $C_{3,12}$ associated to $A[15,3]$ occurring in ${\\rm Sym}^3({\\rm Sym}^6(V))$ , after dividing by the cusp form $\\chi _{10}$ .","Its Fourier expansion starts with $\\chi _{12,2}(\\tau )=\\left({\\begin{matrix}0\\\\0\\\\0\\\\2(R-R^{-1})\\\\9(R+R^{-1})\\\\12(R-R^{-1})\\\\0\\\\-12(R-R^{-1})\\\\-9(R+R^{-1})\\\\-2(R-R^{-1})\\\\0\\\\0\\\\0\\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,$ where $Q_1= e^{\\pi i \\tau _1}$ , $Q_2=e^{\\pi i \\tau _2}$ and $R=e^{\\pi i \\tau _{12}}$ for $\\tau =\\left({\\begin{matrix} \\tau _1 & \\tau _{12} \\\\ \\tau _{12} & \\tau _2\\\\\\end{matrix}} \\right)$ .","By multiplication by $\\chi _{6,3}$ we get an injective map $S_{12,2}(\\Gamma _2,\\epsilon ) \\rightarrow S_{18,5}(\\Gamma _2)$ and this latter space is 1-dimensional.","Corollary 5.2 We have $\\dim S_{12,2}(\\Gamma _2,\\epsilon )=1$ and it is generated by $\\chi _{12,2}$ .","We compute a few Hecke eigenvalues as described in Section .","To compute these Hecke eigenvalues, we used the following Fourier coefficients: $a([1,1,1])^t&=[0, 0, 0, 2, 9, 12, 0, -12, -9, -2, 0, 0, 0]\\\\a([1,3,3])^t&=[0, 0, 0, -2, -27, -156, -504, -996, -1233, -934, -396, -72, 0]\\\\a([3,3,3])^t&=[0, 216, 1188, 258, -7749, -12708, 0, 12708, 7749, -258, -1188, -216, 0]\\\\a([5,5,5])^t&=[0, 0, 0, -106920, -481140, -641520, 0, 641520, 481140, 106920, 0, 0, 0]\\\\a([1,5,7])^t&=[0, 0, 0, 2, 45, 444, 2520, 9060, 21375, 33046, 32220, 17928, 4320]\\\\a([7,7,7])^t&=[0, -8208, -45144, -542204, -2101338, -2711496, 0, 2711496, 2101338, 542204, 45144, 8208, 0]\\\\a([3,9,7])^t&=[0, -72, -1188, -8854, -39339, -115764, -236880, -343884,\\\\& \\qquad \\qquad -354141, -253514, -120132, -33912, -4320]\\\\$ These and a few more (too big to be written here) yield the following eigenvalues: height2pt $p$ 3 5 7 11 13 17 height2pt $\\lambda _p$ $-600$ $-53460$ $-369200$ 4084344 $-2845700$ 131681700 together with $\\lambda _9=-1090791$ .","We find that for the operator $T_p$ these are indeed of the form $\\lambda _p(f^{+})+\\lambda _p(f^{-})$ with $f^{\\pm }$ generators of $S^{\\pm }_{14}(\\Gamma _0(2))^{\\rm new}$ , with $f^{+}(\\tau )&=q - 64\\, q^2 - 1836\\, q^3 + 4096\\, q^4 + 3990\\, q^5 +117504\\, q^6 +\\cdots \\\\f^{-}(\\tau )&=q + 64\\, q^2 + 1236\\, q^3 + 4096\\, q^4 - 57450\\, q^5 +79104\\, q^6 +\\cdots \\, ,$ while for $T_{p^2}$ we find $\\lambda _{p}(f^{+})^2+\\lambda _p(f^{+})\\lambda _{p}(f^{-})+\\lambda _p(f^{-})^2-2\\, (p+1)p^j$ .","This fits with being a Yoshida lift.","Lemma 5.3 We have $ S_{14,2}(\\Gamma _2,\\epsilon )=(0)=S_{16,2}(\\Gamma _2,\\epsilon )$ .","To see that $S_{14,2}(\\Gamma _2,\\epsilon )=(0)$ we multiply a form in $S_{14,2}(\\Gamma _2,\\epsilon )$ with $\\chi _5$ and we end up in $S_{14,7}(\\Gamma _2)$ and this space is generated by a form associated to $C_{1,6}C_{3,8}$ after division by $\\chi _5^2$ .","Restricting this form $\\chi _{14,7}$ to $\\mathfrak {H}_1^2$ gives $\\sum _{i=0}^{7}(f_i\\otimes f_i^{\\prime })$ and only the term $f_5 \\otimes f_5^{\\prime }$ in $S_{16}(\\Gamma _1)\\otimes S_{12}(\\Gamma _1)$ can be non-zero and it is equal to $56\\, e_4\\Delta \\otimes \\Delta $ .","So it does not vanish along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ , hence $\\chi _{14,7}$ is not divisible by $\\chi _5$ .","We conclude $S_{14,2}(\\Gamma _2[2],\\epsilon )=(0)$ .","For $S_{16,2}(\\Gamma _2,\\epsilon )$ we multiply by $\\chi _{6,3}$ and land in $S_{22,5}(\\Gamma _2)$ and this space is zero.","Next we deal with the case of weight $(18,2)$ .","Proposition 5.4 The space $S_{18,2}(\\Gamma _2,\\epsilon )$ has dimension 1.","First we construct a non-zero element in this space by using the covariant $C=135\\,C_{1,6}^{2}C_{4,6}+56\\,C_{1,6}C_{2,0}C_{3,12}-270\\,C_{2,8}C_{4,10}-930\\,C_{3,6}C_{3,12}.$ It occurs in $\\rm Sym^6(\\rm Sym^6(V))$ and provides a cusp form, $F_C$ , of weight $(18,27)$ on $\\Gamma _2$ .","The order of vanishing of $F_C$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 5, so we can divide it by $\\chi _{5}^5$ and we get by Remark REF a cusp form, denoted $\\chi _{18,2}$ , of weight $(18,2)$ on $\\Gamma _2$ with character.","Again we multiply by $\\chi _{5}$ and land in $S_{18,7}(\\Gamma _2)$ .","This space is 2-dimensional and we can construct a basis using the following covariants $C_1=C_{3,12}(8\\, C_{1,6}C_{2,0}-75\\, C_{3,6})\\quad \\text{ and} \\quad C_2= C_{1,6}^2 C_{4,6}-2\\, C_{2,8}C_{4,10}-3\\, C_{3,6}C_{3,12}.$ They occur in $\\rm Sym^6(\\rm Sym^6(V))$ and provide two cusp forms, $F_{C_i}$ , of weight $(18,27)$ on $\\Gamma _2$ .","Each cusp form $F_{C_i}$ vanishes with order 4 along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ , so we can divide it by $\\chi _{5}^4$ and get a cusp form, $\\chi _{18,7}^{(i)}$ , of weight $(18,7)$ on $\\Gamma _2$ .","The cusp forms $\\chi _{18,7}^{(1)}$ and $\\chi _{18,7}^{(2)}$ are $\\mathbb {C}$ -linearly independent as can be read off from the first terms of their Fourier expansions and the pullbacks to $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ are of the form $\\sum _{r=0}^9 f_r \\otimes f_r^{\\prime }$ with only non-zero terms for $r=5$ and these are $216\\,e_4^2\\Delta \\otimes \\Delta $ and $48\\, e_4^2\\Delta \\otimes \\Delta $ .","Up to a non-zero scalar there is only one non-trivial linear combination, that vanishes along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ and that gives a non-zero form in $S_{18,2}(\\Gamma _2,\\epsilon )$ after division by $\\chi _5$ .","Proposition 5.5 The space $S_{20,2}(\\Gamma _2,\\epsilon )$ has dimension 1.","We construct a non-zero form in this space by taking the covariant $\\begin{aligned}C=&224\\,C_{1,6}^{2}C_{5,8}+312\\,C_{1,6}C_{2,4}C_{4,10}-560\\,C_{1,6}C_{2,8}C_{4,6}\\\\&-108\\,C_{1,6}C_{3,6}C_{3,8}+728\\,C_{2,0}C_{2,8}C_{3,12}-1235\\,C_{2,4}^{2}C_{3,12}.\\end{aligned}$ occurring in ${\\rm Sym}^7({\\rm Sym}^6(V))$ and providing a cusp form, $F_C$ , of weight $(20,32)$ on $\\Gamma _2$ with character.","The order of vanishing of $F_C$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 6, so we can divide it by $\\chi _{5}^6$ and we get a cusp form, $\\chi _{20,2}$ , of weight $(20,2)$ on $\\Gamma _2$ with character.","In a similar way we construct a basis of the space $S_{20,7}(\\Gamma _2)$ by taking the covariants $\\begin{aligned}C_1&=480\\,C_{1,6}^{2}C_{5,8}-180\\,C_{1,6}C_{3,6}C_{3,8}+728\\,C_{2,0}C_{2,8}C_{3,12}-1315\\,C_{2,4}^{2}C_{3,12},\\\\C_2&=80\\,C_{1,6}^{2}C_{5,8}-80\\,C_{1,6}C_{2,8}C_{4,6}+104\\,C_{2,0}C_{2,8}C_{3,12}-125\\,C_{2,4}^{2}C_{3,12},\\\\C_3&=80\\,C_{1,6}^{2}C_{5,8}-40\\,C_{1,6}C_{2,4}C_{4,10}+56\\,C_{2,0}C_{2,8}C_{3,12}-55\\,C_{2,4}^{2}C_{3,12}.\\end{aligned}$ which provide cusp forms with character of weight $(20,32)$ and these are divisible by $\\chi _5^5$ and thus give cusp forms of weight $(20,7)$ generating $S_{20,7}(\\Gamma _2)$ .","By restriction to the diagonal one sees that there is just a 1-dimensional space of forms vanishing on the diagonal.","The case of weight $(24,2)$ is dealt with in a similar way.","Proposition 5.6 We have $\\dim S_{24,2}(\\Gamma _2[2],\\epsilon )=2$ .","We know that $\\dim S_{24,7}(\\Gamma _2)=5$ and we can construct a basis using the procedure described in Section .","In the case at hand we have $A[39,15]$ occurring in ${\\rm Sym}^9({\\rm Sym}^6(V))$ with multiplicity 13 and this gives a subspace of $S_{24,42}(\\Gamma _2,\\epsilon )$ of dimension 13.","One checks that there is a 5-dimensional subspace of forms vanishing with multiplicity 7 along the diagonal and dividing by $\\chi _5^7$ gives a 5-dimensional subspace of $S_{24,7}(\\Gamma _2)$ , hence the whole space.","Again one checks that there is a 2-dimensional space of forms vanishing on the diagonal and we can divide these forms by $\\chi _5$ .","So the two generators of $S_{24,2}(\\Gamma _2,\\epsilon )$ are defined by the covariants $C_1$ and $C_2$ given respectively by $\\begin{aligned}&-499408\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}-1505385\\,C_{1,6}^{3}C_{6,6}^{(1)}-14727825\\,C_{1,6}^{2}C_{2,4}C_{5,8}+6916455\\,C_{1,6}^{2}C_{2,8}C_{5,4}\\\\&-5728590\\,C_{1,6}^{2}C_{3,12}C_{4,0}+6972210\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}+4257120\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}\\\\&+2182950\\,C_{2,8}^{2}C_{5,8}+11708550\\,C_{2,8}C_{3,6}C_{4,10}+595350\\,C_{2,8}C_{3,12}C_{4,4}+35171325\\,C_{3,6}^{2}C_{3,12}\\\\&-400950\\,C_{3,8}^{3}\\\\\\end{aligned}$ and $\\begin{aligned}&-42235648\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}+4434583545\\,C_{1,6}^{3}C_{6,6}^{(1)}+580982220\\,C_{1,6}^{3}C_{6,6}^{(2)}\\\\&+4919972400\\,C_{1,6}^{2}C_{2,4}C_{5,8}+4827362400\\,C_{1,6}^{2}C_{3,12}C_{4,0}-3504891600\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}\\\\&+1245336960\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}-4131252720\\,C_{2,8}^{2}C_{5,8}-24904998720\\,C_{2,8}C_{3,6}C_{4,10}\\\\&-281640240\\,C_{2,8}C_{3,12}C_{4,4}-58751907480\\,C_{3,6}^{2}C_{3,12}+1375354080\\,C_{3,8}^{3}\\, .\\end{aligned}$ The order of vanishing of $F_{C_i}$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 8, so we can divide it by $\\chi _{5}^8$ and we get two cusp forms, $\\chi _{24,2}^{(i)}$ , ($i=1,2$ ) of weight $(24,2)$ on $\\Gamma _2$ with character.","We set $\\chi _{24,2}^{(1)}=-12150\\, F_{C_1}/\\chi _5^8$ and $\\chi _{24,2}^{(2)}=-675(5368\\, F_{C_1} +5 \\, F_{C_2})/31528 \\chi _5^8$ .","Then their Fourier expansions are given by $\\chi _{24,2}^{(1)}=\\left({\\begin{matrix}0\\\\0\\\\0\\\\104(R-R^{-1})\\\\1092(R+R^{-1})\\\\3640(R-R^{-1})\\\\0\\\\-27678(R-R^{-1})\\\\-58905(R+R^{-1})\\\\-2916(R-R^{-1})\\\\148470(R+R^{-1})\\\\190778(R-R^{-1})\\\\0\\\\\\vdots \\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,\\quad \\chi _{24,2}^{2}(\\tau )=\\left({\\begin{matrix}0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\2(R-R^{-1})\\\\17(R+R^{-1})\\\\60(R-R^{-1})\\\\110(R+R^{-1})\\\\98(R-R^{-1})\\\\0\\\\\\vdots \\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,$ where $Q_1=e^{i\\pi \\tau _1}$ , $Q_2=e^{i\\pi \\tau _2}$ , $R=e^{i\\pi \\tau _{12}}$ .","The action of $\\iota =({\\begin{matrix} a & 0 \\\\ 0 & d\\\\ \\end{matrix}})\\in \\Gamma _2$ with $a=d= ({\\begin{matrix} 0 & 1 \\\\ 1 & 0\\\\ \\end{matrix}})$ implies that the $i$ th coordinate is equal to $(-1)^{k+1}$ times the $(j+1-i)$ th coordinate, which gives the non-displayed coordinates .","A Hecke eigenbasis of the space $S_{24,2}(\\Gamma _2,\\epsilon )$ is: $F_1=&\\,439 \\chi _{24,2}^{(1)} +(114847+650\\sqrt{106705})\\, \\chi _{24,2}^{(2)}\\\\F_2=& \\, 439\\chi _{24,2}^{(1)} +(114847-650\\sqrt{106705})\\, \\chi _{24,2}^{(2)}$ with eigenvalues height2pt $p$ $\\lambda _p(F_1)$ $\\lambda _{p^2}(F_1)$ height2pt 3 $287880-4800\\sqrt{106705}$ $545747143689-2293459200\\sqrt{106705}$ 5 $711981900+1555200\\sqrt{106705}$ – 7 $-41070905840+92534400\\sqrt{106705}$ – 11 $ -10344705071976 + 4819953600\\sqrt{106705}$ – in perfect agreement with the eigenforms being Yoshida lifts.","Indeed a basis of the space $S_{26}(\\Gamma _0[2])^{\\text{new}}$ is given by $f&=q - 4096\\, q^2 + 97956\\, q^3 + 16777216\\, q^4 + 341005350\\, q^5- 401227776\\, q^6 +\\cdots \\\\g&=q + 4096\\, q^2 + (2048-a/2)\\, q^3 + 16777216\\, q^4 +(431848374+162\\, a)\\, q^5 +\\cdots \\\\g^{\\prime }&=q + 4096\\, q^2 + (2048+a/2)\\, q^3 + 16777216\\, q^4 + (431848374-162\\, a)\\, q^5 +\\cdots \\, ,$ where $ a=-375752+9600\\sqrt{106705} $ , and $f,g^{\\prime } \\in S^{-}_{26}$ and $g \\in S_{26}^{+}$ .","Then we check for example that $\\lambda _5(F_1)&=711981900+1555200\\sqrt{106705}=a_5(f)+a_5(g)\\\\\\lambda _5(F_2)&=711981900-1555200\\sqrt{106705}=a_5(f)+a_5(g^{\\prime }).$ Proposition 5.7 One has $\\dim S_{26,2}(\\Gamma _2,\\epsilon )=1=\\dim S_{28,2}(\\Gamma _2,\\epsilon )$ and $\\dim S_{30,2}(\\Gamma _2,\\epsilon )=2$ .","The proof of this proposition is similar to the above.","For the first statement we consider the space $S_{26,7}(\\Gamma _2)$ which has dimension 6 and construct a basis of this space using covariants associated to $A[43,17]$ in ${\\rm Sym}^{10}({\\rm Sym}^6(V))$ which occurs with multiplicity 17, thus giving rise to a 17-dimensional subspace of $S_{26,47}(\\Gamma _2)$ .","By restricting along the diagonal one checks that there is a 6-dimensional subspace of cusp forms divisible by $\\chi _5^8$ leading to the construction of $S_{26,7}(\\Gamma _2)$ .","Again by restricting to the diagonal one sees that there is exactly a 1-dimensional subspace of this space that vanish along the diagonal.","By dividing by $\\chi _5$ we thus find the space $S_{26,2}(\\Gamma _2,\\epsilon )$ .","For weight $(28,2)$ we now use the representation $A[47,19]$ that occurs with multiplicity 23 in ${\\rm Sym}^{11}({\\rm Sym}^6(V))$ and leading to a 23-dimensional subspace of $S_{28,52}(\\Gamma _2)$ in which we find a 7-dimensional subspace of forms divisible by $\\chi _5^9$ and division gives forms that generate $S_{28,7}(\\Gamma _2)$ .","In this space the subspace of forms divisible by $\\chi _5$ is of dimension 1, proving our claim.","In the case of weight $(30,7)$ the 9-dimensional space $S_{30,7}$ is constructed using covariants resulting from $A[51,21]$ that occurs with multiplicity 31 in ${\\rm Sym}^{12}({\\rm Sym}^6(V))$ leading to a space of dimension 31 of cusp forms of weight $(30,57)$ .","There is a 9-dimensional subspace of cusp forms divisible by $\\chi _5^{10}$ and we thus generate $S_{30,7}(\\Gamma _2)$ .","It turns out that there is a 2-dimensional subspace of forms divisible by $\\chi _5$ and this proves the claim.","For all the cases treated we can check our construction by verifying that the Hecke eigenvalues for $p=3,5,7,11,13, 17$ agree with the forms being Yoshida lifts like we indicated for $j=12$ and $j=24$ .","In a forthcoming paper ([8]) we shall use the relation with covariants to describe modules of forms with a character."],["Modular Forms of Weight $(j,2)$ on {{formula:8ac6fb49-ab1c-4eac-b5ae-da9c6611aa41}}","In this section we explain how we checked that $S_{j,2}(\\Gamma _2)=(0)$ for $j\\le 52$ .","We begin with a simple lemma.","Recall that we have maps $M \\rightarrow {\\mathcal {C}} {\\mu \\over \\longrightarrow } M$ .","Lemma 6.1 Let $f \\in M_{j,k}(\\Gamma _2)$ .","Then there exists a covariant $c_f$ of degree $(d,j)$ with $d\\le j/2+k$ such that $f=\\nu (c_f)=\\mu (c_f)/\\chi _5^d$ .","If $f$ is a cusp form then there is a covariant $c_f^{\\prime }$ of degree $\\le j/2+k-10$ such that $f=\\mu (c_f^{\\prime })/\\chi _5^r$ for some $r$ .","The first statement follows directly from [7].","If $f$ is a cusp form then the covariant it defines vanishes on the discriminant locus.","But then the covariant $c_f$ is divisible by the discriminant, and $\\mu (c_f)$ by $\\chi _5^{d+2}$ .","This makes it possible to check the existence of a non-zero form $f \\in S_{j,2}(\\Gamma _2)$ by checking whether the forms of weight $(j,2+5d)$ provided via $\\mu $ by the non-zero covariants of degree $(d,j)$ with $d\\le j/2-8$ are divisible by $\\chi _5^{d}$ .","We applied this for values of $j\\le 52$ using the covariants of degree $d\\le 18$ .","For smaller values of $j$ other methods of showing that $S_{j,2}(\\Gamma _2)=(0)$ are available.","We sketch some methods below.","In this way we checked that $S_{j,2}(\\Gamma _2)=(0)$ for $j\\le 52$ .","Another method is to construct a basis of $S_{j,7}(\\Gamma _2,\\epsilon )$ by using covariants.","We then check the divisibility by $\\chi _5$ of elements in $S_{j,7}(\\Gamma _2,\\epsilon )$ by restricting the modular forms in this space to the diagonal.","As an illustration we give the proof for the case $j=24$ .","We construct a basis of the 9-dimensional space $S_{24,7}(\\Gamma _2,\\epsilon )$ .","For this we use the covariants associated to the $A[54,30]$ -isotypical component of ${\\rm Sym}^{14}({\\rm Sym}^6(V))$ .","The representation $A[54,30]$ occurs with multiplicity 65 and leads to a 65-dimensional subspace of modular forms of weight $(24,72)$ on $\\Gamma _2$ .","By restricting to $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ we can check that there exists a 9-dimensional subspace of cusp forms that are divisible by $\\chi _5^{13}$ .","This leads to a basis of $S_{24,7}(\\Gamma _2,\\epsilon )$ .","We then check by restriction to $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ again that there is no non-trivial element in this space that is divisible by $\\chi _5$ .","This proves the result for $j=24$ .","We carried this out for all the cases $j\\le 52$ and thus proved Theorem REF .","Sometimes there are other and easier ways to eliminate cases.","For example, by restricting a modular form of weight $(j,2)$ to the diagonal we get an element $\\sum _{i=0}^{j/2} f_i \\otimes f_i^{\\prime }\\in \\oplus _{i=0}^{j/2} S_{j+2-i}(\\Gamma _1)\\otimes S_{2+i}(\\Gamma _1)$ .","If $j<24$ , $j\\ne 20$ the spaces in question are zero.","Therefore a form $f\\in S_{j,2}(\\Gamma _2)$ will vanish on $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ .","But then $f/\\chi _5$ will be a holomorphic modular form of weight $(j,-3)$ and this has to be zero.","So $f=0$ for $j\\le 18$ and $j=22$ .","As yet another example of eliminating cases we give a somewhat different argument for $j=26$ .","We write elements of $F\\in S_{j,k}(\\Gamma _2)$ as vectors $F=(F_0,\\ldots ,F_j)^t$ with the $F_i$ holomorphic functions on $\\mathcal {H}_2$ , that is, in a module of rank 27 over the ring ${\\mathcal {F}}$ of holomorphic functions on $\\mathcal {H}_2$ .","Take a basis $s_1,\\ldots ,s_3$ of $S_{26,6}(\\Gamma _2)$ and a basis $s_{4},\\ldots ,s_{12}$ of $S_{26,8}(\\Gamma _2)$ .","If there exists a non-zero form $f$ of weight $(26,2)$ then the vectors $E_4\\, f$ and $E_6\\, f$ are linearly dependent and thus the exterior product $s_1\\wedge \\cdots \\wedge s_{12}$ must vanish.","By calculating bases of $S_{26,6}(\\Gamma _2)$ and $S_{26,8}(\\Gamma _2)$ one can check that this exterior product does not vanish.","So $S_{26,2}(\\Gamma _2)=(0)$ ."],["Other Small Weights","We begin with an elementary argument that shows that $S_{j,k}(\\Gamma _2[2])=(0)$ for $j\\le 8$ and $k\\le 2$ .","Proposition 7.1 For $j\\le 8$ and $k\\le 2$ we have $\\dim S_{j,k}(\\Gamma _2[2])=0$ .","We need to deal with the cases $j$ even and $k=1$ and $k=2$ only since for other values $S_{j,k}(\\Gamma _2[2])$ vanishes.","We restrict to the ten components of the Humbert surface $H_1$ in $\\Gamma _2[2]\\backslash \\mathfrak {H}_2$ , one component of which is given by the diagonal $\\tau _{12}=0$ .","The group $\\mathfrak {S}_6$ acts transitively on these ten components.","The stabilizer inside $\\mathfrak {S}_6$ of a component of $H_1$ is an extension of $\\mathfrak {S}_3\\times \\mathfrak {S}_3$ by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ .","By restricting a modular form $f \\in S_{j,1}(\\Gamma _2[2])$ to a component we get an element of $\\oplus _{r=0}^{j} S_{j+1-r}(\\Gamma _1[2])\\otimes S_{1+r}(\\Gamma _1[2])$ and for $f\\in S_{j,2}(\\Gamma _2[2])$ we get an element of $\\oplus _{r=0}^{j} S_{j+2-r}(\\Gamma _1[2])\\otimes S_{2+r}(\\Gamma _1[2]) \\, .$ For $j\\le 8$ and $k=1$ and for $j<8$ and $k=2$ these spaces are zero.","Thus a form $f\\in S_{j,2}(\\Gamma _2[2])$ restricts to zero on all irreducible components of $H_1$ , hence is divisible by $\\chi _{5}$ , and so $f$ must be zero.","For $j=8$ and $k=2$ the restriction to $H_1$ gives an injective $\\mathfrak {S}_6$ -equivariant map $S_{8,2}(\\Gamma _2[2]) \\rightarrow \\oplus _{i=1}^{10}\\, {\\rm Sym}^2 S_6(\\Gamma _1[2]) \\, ,$ where the action on the right is the induced representation from the extension of $\\mathfrak {S}_3 \\times \\mathfrak {S}_3$ by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ to $\\mathfrak {S}_6$ .","Now $S_6(\\Gamma _1[2])$ is 1-dimensional and of type $s[1^3]$ and we check that the representation of $\\mathfrak {S}_6$ on the 10-dimensional space $\\oplus _{i=1}^{10} \\rm Sym^2 S_6(\\Gamma _1[2])$ is of type s[6]+s[4,2].","Since $S_{8,2}(\\Gamma _2)=(0)$ , we conclude that only $S_{8,2}(\\Gamma _2[2])^{s[4,2]}$ can be non-zero.","If $S_{8,2}(\\Gamma _2[2])^{s[4,2]}$ is non-zero, then $S_{8,2}(\\Gamma _0[2])$ is non-zero (see [10]).","By restricting we get an element in a similar decomposition as before but with $\\Gamma _1[2]$ replaced by $\\Gamma _0[2]$ .","As we know that all theses spaces are zero, we can divide by $\\chi _5$ .","This contradiction concludes our claim.","More generally we have Proposition 7.2 For $j<12$ we have $S_{j,2}(\\Gamma _2[2])=(0)$ .","The space $S_{j,2}(\\Gamma _2[2])$ is defined over ${\\mathbb {Q}}$ .","All the cusps of $\\Gamma _2[2]$ are defined over ${\\mathbb {Q}}$ and the action of $\\mathfrak {S}_6$ is defined over ${\\mathbb {Q}}$ .","The q-expansion principle says that a modular form in $S_{j,k}(\\Gamma _2[2])$ with $k\\ge 3$ is defined over ${\\mathbb {Q}}$ if its Fourier coefficients at all cusps are defined over ${\\mathbb {Q}}$ , see [19] and [13].","We apply this to $f\\chi _{10}$ with $f \\in S_{j,2}(\\Gamma _2[2])$ defined over ${\\mathbb {Q}}$ and we conclude that the Fourier coefficients of $\\sigma (f\\chi _{10})$ with $\\sigma \\in \\mathfrak {S}_6$ are real, hence also those of $\\sigma (f)$ and if $f\\ne 0$ we find by looking at the `first' non-zero term in a Fourier expansion that $\\sum _{\\sigma \\in \\mathfrak {S}_6} \\sigma (f)^2$ is non-zero and because of $\\sigma (f^2)=\\sigma (f)^2$ also invariant under $\\mathfrak {S}_6$ .","Thus it defines a non-zero element of $S_{2j,4}(\\Gamma _2)$ .","So $S_{j,2}(\\Gamma _2[2])$ implies $S_{2j,4}(\\Gamma _2)\\ne (0)$ .","But we know that $S_{2j,4}(\\Gamma _2)=(0)$ for $j<12$ .","Remark 7.3 Note that the argument of the proof shows that our conjecture on the vanishing of $S_{j,2}(\\Gamma _2)$ for all $j$ implies the vanishing of $S_{j,1}(\\Gamma _2[2])$ for all $j$ .","In order to put our evidence for the vanishing of $S_{j,2}(\\Gamma _2)$ in perspective we show a small table that gives for each value of $k$ the smallest $j_0$ such that $\\dim S_{j_0,k}(\\Gamma _2)\\ne 0$ .","height2pt $k$ 3 4 5 6 7 8 height2pt $j_0$ 36 24 18 12 12 6 We can easily construct the generators of the corresponding spaces.","In [9] we constructed a generator $\\chi _{6,3}$ of $S_{6,3}(\\Gamma _2,\\epsilon )$ and above we gave the generator $\\chi _{12,2}$ of $S_{12,2}(\\Gamma _2,\\epsilon )$ .","The modular forms $\\chi _{12,2}^2$ , $\\chi _{6,3}\\chi _{12,2}$ , $\\chi _{6,3}^2$ , $\\chi _5\\chi _{12,2}$ and $\\chi _5\\chi _{6,3}$ give the generators for $k=4,\\ldots ,8$ .","We end by constructing a generator of $S_{36,3}(\\Gamma _2)$ ; the non-vanishing of this space plays a role in the appendix.","We look in ${\\rm Sym}^{11}({\\rm Sym}^6(V))$ and at $A[51,15]$ occuring with multiplicity 17 there.","We find the covariant $297 \\, C_{1,6}^2C_{3,8}^3 -8316 \\, C_{1,6}C_{3,8}C_{3,12}C_{4,10}+4116 \\, C_{1,6}C_{3,12}^2C_{4,6} -5488 \\, C_{2,0}C_{3,12}^3+9030 \\, C_{2,4}C_{3,8}C_{3,12}^2$ giving a form $f$ of weight $(36,48)$ that is divisible by $\\chi _5^{9}$ and $f/\\chi _5^9$ generates $S_{36,3}(\\Gamma _2)$ .","As a check we note that the Fourier coefficient at $n=[1,1,1]$ is of the form $[0, 0, 0, 0, 0, 0, 0, 32/6089428125, 464/6089428125, \\ldots ]$ and the coefficient at $n=[5,5,5]$ is $[0, 0, 0, 0, 0, 0, 0, -8687121398144/81192375, -125963260273088/81192375,\\ldots ]$ giving the eigenvalue for the Hecke operator $T_5$ as their ratio $ -20360440776900$ , in agreement with the value given in [2]."],["by Gaëtan Chenevier","Let $j$ and $k$ be integers with $j\\ge 0$ , and $\\Gamma \\subset {\\rm Sp}_4(\\mathbb {Z})$ a congruence subgroup.","Recall that $S_{j,k}(\\Gamma )$ denotes the space of cuspidal Siegel modular forms for the subgroup $\\Gamma $ with values in the representation ${\\rm Sym}^j\\, \\otimes \\,\\det ^k$ of ${\\rm GL}_2(\\mathbb {C})$ .","We first consider the full Siegel modular group $\\Gamma _2 = {\\rm Sp}_4(\\mathbb {Z})$ and provide alternative proofs of the following results : Proposition 1.1 We have ${S}_{j,1}(\\Gamma _2)=0$ for any $j$ , and ${S}_{j,2}(\\Gamma _2)=0$ for $j \\le 38$ .","The vanishing of ${S}_{j,2}(\\Gamma _2)$ for all $j\\le 52$ is also proved in this paper by Cléry and van der Geer (Theorem REF ).","The vanishing of ${S}_{j,1}(\\Gamma _2)$ was at least known to Ibukiyama, who asserts in [16] that it is a consequence of the vanishing of all Jacobi forms of weight 1 for ${\\rm SL}_2(\\mathbb {Z})$ proven by Skoruppa [23].","Here we shall rather use automorphic representation theoretic methods.","First we need to fix some notations and make some preliminary remarks.","We denote by $r : {\\rm Sp}_4(\\mathbb {C}) \\rightarrow {\\rm GL}_4(\\mathbb {C})$ the tautological inclusion.","For a real reductive Lie group $H$ we shall denote the infinitesimal character of the Harish-Chandra module $U$ by ${\\rm inf}\\, U$ .","In the case $H={\\rm GL}_n(\\mathbb {R})$ (resp.","$H={\\rm PGSp}_4(\\mathbb {R})$ ), and following Harish-Chandra and Langlands, ${\\rm inf} \\, U$ may be viewed in a canonical way as a semisimple conjugacy class in the Lie algebra $\\mathfrak {h}=\\mathfrak {gl}_n(\\mathbb {C})$ (resp.","$\\mathfrak {h}=\\mathfrak {sp}_4(\\mathbb {C})$ ).","In both cases we may and shall identify this conjugacy class with the multiset of its eigenvalues in the natural representation of $\\mathfrak {h}$ .","We denote by ${\\rm W}_\\mathbb {R}$ the Weil group of $\\mathbb {R}$ (a certain extension of $\\mathbb {Z}/2\\mathbb {Z}$ by $\\mathbb {C}^\\times $ ), and for any integer $w$ we define ${\\rm I}_{w}$ as the 2-dimensional representation of ${\\rm W}_{\\mathbb {R}}$ induced from the unitary character $z \\mapsto (z/|z|)^{w}$ of $\\mathbb {C}^\\times $ .","(a) As we have ${S}_{j,k}(\\Gamma _2)=0$ for any odd $j$ or for $k\\le 0$ , we may once and for all assume $j \\equiv 0 \\bmod 2$ and $k>0$ .","As is well-known, for any such $(j,k)$ there is an irreducible unitary Harish-Chandra module for ${\\rm PGSp}_4(\\mathbb {R})$ , unique up to isomorphism, generated by a highest-weight vector of $K$ -type ${\\rm Sym}^j \\,\\otimes \\det ^k$ .","This module, that we shall denote by ${\\rm U}_{j,k}$ , is a holomorphic discrete series for $k \\ge 3$ , a limit of holomorphic discrete series for $k=2$ , and non-tempered for $k=1$ ; it is non-generic in all cases.","We have ${\\rm inf}\\, {\\rm U}_{j,k} =\\lbrace \\frac{j+2k-3}{2}, \\,\\frac{j+1}{2}, \\,-\\frac{j+1}{2}, \\,-\\frac{j+2k-3}{2}\\rbrace $ .","More precisely, if $\\varphi : {\\rm W}_\\mathbb {R}\\rightarrow {\\rm Sp}_4(\\mathbb {C})$ denotes the Langlands parameter of ${\\rm U}_{j,k}$ , then we have $r \\circ \\varphi \\simeq {\\rm I}_{j+2k-3} \\oplus {\\rm I}_{j+1}$ for $k>1$ , and $r \\circ \\varphi \\simeq {\\rm I}_{j} \\otimes |.|^{1/2}\\oplus {\\rm I}_{j} \\otimes |.|^{-1/2}$ for $k=1$ (see e.g.","[22] for a survey of those properties, and the references therein).","The relevance of ${\\rm U}_{j,k}$ here is that if $\\pi $ is a cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by an element of ${S}_{j,k}(\\Gamma _2)$ , then the Archimedean component $\\pi _\\infty $ of $\\pi $ is isomorphic to ${\\rm U}_{j,k}$ .","The other important property of $\\pi $ is that $\\pi _p$ is unramified for each prime $p$ (i.e.","admits non-zero invariants under ${\\rm PGSp}_4(\\mathbb {Z}_p)$ ).","As ${\\rm PGSp}_4$ is isomorphic to the split classical group ${\\rm SO}_5$ over $\\mathbb {Z}$ , we may apply Arthur's theory [1] to such a $\\pi $ .","(b) One of the main results of Arthur [1] associates to any discrete automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ a unique isobaric automorphic representation $\\pi ^{\\rm GL}$ of ${\\rm GL}_4$ over $\\mathbb {Q}$ , characterized by the following property : for any prime $p$ such that $\\pi _p$ is unramified, then $(\\pi ^{\\rm GL})_p$ is unramified as well and its Satake parameter is the image of the one of $\\pi _p$ under the map $r$ .","The infinitesimal character of $(\\pi ^{\\rm GL})_\\infty $ is the image of ${\\rm inf}\\, \\pi _\\infty $ under the derivative of $r$ , namely $\\mathfrak {sp}_4(\\mathbb {C}) \\rightarrow \\mathfrak {gl}_4(\\mathbb {C})$ .","Moreover, there is a unique collection of distinct triplets $(d_i,n_i,\\pi _i)_{i \\in I}$ , with integers $d_i,n_i \\ge 1$ and cuspidal selfdual automorphic representations $\\pi _i$ of ${\\rm GL}_{n_i}$ with $\\pi ^{\\rm GL} \\,\\simeq \\,\\boxplus _{i \\in I} \\,(\\,\\boxplus _{l=0}^{d_i-1}\\,\\, \\,\\pi _i \\,\\otimes |.|^{\\frac{d_i-1}{2}-l}\\,)\\, \\, \\, \\, \\, {\\rm and} \\,\\,\\,\\,\\,4 = \\sum _{i \\in I} n_i d_i.$ The selfdual representation $\\pi _i$ is symplectic in Arthur's sense if, and only if, $d_i$ is odd.","All of this is included in [1].","(c) For $\\pi $ as in (b), then ${\\rm inf}\\, \\pi _\\infty $ is the union, over all $i \\in I$ and all $0 \\le l< d_i$ , of the multisets $\\frac{d_i-1}{2}-l\\,+\\,{\\rm inf}\\, (\\pi _i)_\\infty $ .","In particular, if we have $\\lambda \\in \\frac{1}{2}\\mathbb {Z}$ and $\\lambda -\\mu \\in \\mathbb {Z}$ for all $\\lambda ,\\mu \\in \\,{\\rm inf}\\, \\pi _\\infty $ , then ${\\rm inf}\\, (\\pi _i)_\\infty $ has the same property for each $i$ : such a $\\pi _i$ is called algebraic.","If $\\omega $ is a cuspidal selfdual algebraic automorphic representation of ${\\rm GL}_m$ over $\\mathbb {Q}$ , then $\\omega _\\infty $ is tempered by the Jaquet-Shalika estimates, and its Langlands parameter is trivial on the central subgroup $\\mathbb {R}_{>0}$ of ${\\rm W}_\\mathbb {R}$ (this is the so-called Clozel purity lemma, see e.g.","[6]).","(d) The only selfdual cuspidal automorphic representation $\\pi $ of ${\\rm GL}_1$ over $\\mathbb {Q}$ such that $\\pi _p$ is unramified for each prime $p$ is the trivial Hecke character 1 (which is of course selfdual orthogonal).","Moreover, for any integer $k\\ge 1$ , the number of cuspidal automorphic representations $\\pi $ of ${\\rm GL}_2$ over $\\mathbb {Q}$ such that $\\pi _p$ is unramified for each prime $p$ , and with ${\\rm inf}\\, \\pi _\\infty =\\lbrace -\\frac{k-1}{2},\\frac{k-1}{2}\\rbrace $ , is the dimension of the space ${S}_k(\\Gamma _1)$ of cuspidal modular forms of weight $k$ for $\\Gamma _1={\\rm SL}_2(\\mathbb {Z})$ .","Indeed, this is well-known for $k>1$ , and for $k=1$ it follows from the fact that there is no Maass form of eigenvalue $1/4$ for ${\\rm SL}_2(\\mathbb {Z})$ (a fact due to Selberg, see also [6] for a short proof).","(of the vanishing of ${S}_{j,1}(\\Gamma _2)$ for any $j$ ).","It is enough to show that there is no discrete automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ which is unramified at every prime and with $\\pi _\\infty \\simeq {\\rm U}_{j,1}$ .","For that we study $\\pi ^{\\rm GL}$ , and the associated collection $(d_i,n_i,\\pi _i)_{i \\in I}$ given by (b) above.","By (a), the infinitesimal character of $(\\pi ^{\\rm GL})_\\infty $ is ${\\rm inf}\\, {\\rm U}_{j,1} \\, =\\,\\lbrace \\,\\frac{j+1}{2}, \\,\\frac{j-1}{2}, \\,-\\frac{j-1}{2}, \\,-\\frac{j+1}{2}\\rbrace .$ If we have $d_i=1$ for each $i$ , then $(\\pi ^{\\rm GL})_\\infty $ is tempered by (c), hence so is $\\pi _\\infty $ by Arthur's local-global compatibility [1], a contradiction as ${\\rm U}_{j,1}$ is non-tempered by (a).","Fix $i \\in I$ with $d_i\\ge 2$ .","If we have $n_i \\ge 2$ then we must have $I=\\lbrace i\\rbrace $ and $n_i=d_i=2$ by the equality $4 = \\sum _i n_i d_i$ .","By (c) and the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ above, we necessarily have ${\\rm inf} \\, (\\pi _i)_\\infty = \\lbrace j/2,-j/2\\rbrace $ .","But this is absurd by (d) and the vanishing ${S}_{j+1}(\\Gamma _1)=0$ for any even integer $j \\ge 0$ .","We have thus $(d_i,n_i,\\pi _i)=(2,1,1)$ .","Choose $i^{\\prime } \\in I-\\lbrace i\\rbrace $ .","As we have $(d_{i^{\\prime }},n_{i^{\\prime }},\\pi _{i^{\\prime }}) \\ne (d_i,n_i,\\pi _i)$ , the previous argument shows $d_{i^{\\prime }}=1$ , so $\\pi _{i^{\\prime }}$ is symplectic by (b), which forces $n_{i^{\\prime }}$ to be even, hence the only possibility is $n_{i^{\\prime }}=2$ and $I=\\lbrace i,i^{\\prime }\\rbrace $ .","But then the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ and (c) show that we have either $j=0$ and ${\\rm inf} (\\pi _{i^{\\prime }})_\\infty = \\lbrace 1/2, -1/2 \\rbrace $ or $j=1$ and ${\\rm inf} (\\pi _{i^{\\prime }})_\\infty = \\lbrace 3/2, -3/2 \\rbrace $ .","Both cases are absurd by (d) as we have ${S}_4(\\Gamma _1)={S}_2(\\Gamma _1)=0$ , and we are done.","The second assertion of the proposition will be a consequence of the following two lemmas.","Lemma 1.2 Let $j \\ge 0$ be an even integer.","The integer $\\dim {S}_{j,2}(\\Gamma _2)$ is the number of cuspidal, selfdual symplectic, automorphic representations $\\Pi $ of ${\\rm GL_4}$ over $\\mathbb {Q}$ whose local components $\\Pi _p$ are unramified for each prime $p$ , and with ${\\rm inf} \\, \\Pi _\\infty \\, = \\,\\lbrace \\,\\frac{j+1}{2}, \\,\\frac{j+1}{2}, \\,-\\frac{j+1}{2}, \\,-\\frac{j+1}{2}\\rbrace $ .","Let $\\pi $ be a cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by an arbitrary Hecke eigenform $F$ in ${S}_{j,2}(\\Gamma _2)$ .","Consider its associated automorphic representation $\\pi ^{\\rm GL}$ of ${\\rm GL}_4$ and collection of $(d_i,n_i,\\pi _i)$ 's as in (b).","We claim that $\\pi ^{\\rm GL}$ is necessarily cuspidal, i.e.","$I=\\lbrace i\\rbrace $ is a singleton and $d_i=1$ , so that $\\Pi =\\pi ^{\\rm GL}$ satisfies all the assumptions of the statement by (a) and (b).","Let us show first that we have $d_i=1$ for each $i \\in I$ .","Otherwise, (c) shows that two elements of ${\\rm inf}\\, {\\rm U}_{j,2}$ must differ by 1, which only happens for $j=0$ .","But for $j=0$ an argument similar to the one in the previous proof shows that if $d_i>1$ then we have $I=\\lbrace i,i^{\\prime }\\rbrace $ with $(d_i,n_i,\\pi _i)=(2,1,1)$ , $n^{\\prime }_i=2$ and ${\\rm inf}\\, (\\pi _i)_{\\infty } = \\lbrace 1/2,-1/2\\rbrace $ , which is absurd by the vanishing ${S}_2(\\Gamma _1)=0$ and (d), and we are done.","As a consequence, the Langlands parameter of $(\\pi ^{\\rm GL})_\\infty $ is ${\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ by (c).","Let us denote by $\\psi $ Arthur's substitute for the global parameter of the representation $\\pi $ of ${\\rm PGSp}_{4}$ defined in [1].","We have just proved that $\\psi _\\infty $ is the tempered Langlands parameter of ${\\rm PGSp}_4(\\mathbb {R})$ with $r \\,\\circ \\,\\psi _\\infty \\simeq {\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ , i.e.","the Langlands parameter of ${\\rm U}_{j,2}$ by (a).","We have $\\mathcal {S}_{\\psi _\\infty } = \\mathbb {Z}/2\\mathbb {Z}$ : the corresponding $L$ -packet of ${\\rm PGSp}_4(\\mathbb {R})$ (limit of discrete series) has two elements, namely ${\\rm U}_{j,2}$ and a generic limit of discrete series with same infinitesimal character.","We now apply Arthur's multiplicity formula to the element $\\pi $ of the global packet $\\Pi _\\psi $ defined by Arthur.","As we have $d_i=1$ for all $i$ , either $\\pi ^{\\rm GL}$ is cuspidal or we have $I=\\lbrace i,i^{\\prime }\\rbrace $ with $n_i=n_{i^{\\prime }}=2$ and ${\\rm inf}\\, (\\pi _i)_\\infty ={\\rm inf}\\, (\\pi _{i^{\\prime }})_\\infty = \\lbrace \\frac{j+1}{2},-\\frac{j+1}{2}\\rbrace $ .","If $\\pi ^{\\rm GL}$ is not cuspidal, then according to Arthur's definitions the natural map $\\mathcal {S}_\\psi \\rightarrow \\mathcal {S}_{\\psi _\\infty }$ is an isomorphism of groups of order 2.","But then his multiplicity formula shows that $\\pi _\\infty $ has to be generic since $\\pi _p$ is unramified for each prime $p$ , a contradiction as ${\\rm U}_{j,2}$ is not generic.","(We have shown that $\\pi $ is not of “Yoshida type”.)","We have thus proved that $\\pi ^{\\rm GL}$ is cuspidal.","Note that in this case we have $\\mathcal {S}_\\psi =1$ , thus by the multiplicity formula again, the multiplicity of $\\pi $ in the automorphic discrete spectrum of ${\\rm PGSp}_4$ is 1; in particular, the Hecke eigenspace of the eigenform $F$ we started from has dimension 1.","It thus only remains to show that any $\\Pi $ as in the statement is in the image of the construction of the first paragraph above.","Let $\\Pi $ be as in the statement.","The Langlands parameter of $\\Pi _\\infty $ is the image under $r$ of the one of ${\\rm U}_{j,2}$ by (a) and (c).","A trivial application of Arthur's multiplicity formula shows the existence of a discrete automorphic $\\pi $ for ${\\rm PGSp}_4$ with $\\pi _\\infty \\simeq {\\rm U}_{j,2}$ , which is unramified at every prime, and satisfying $\\pi ^{\\rm GL} \\simeq \\Pi $ .","As ${\\rm U}_{j,2}$ is tempered, a classical result of Wallach ensures that $\\pi $ is actually cuspidal, hence generated by an element of ${S}_{j,2}(\\Gamma _2)$ : this concludes the proof.","Lemma 1.3 For any even integer $0 \\le j \\le 38$ there is no $\\Pi $ as in Lemma REF .","In order to contradict the existence of a $\\Pi $ as in Lemma REF for small $j$ , and following work of Odlyzko, Mestre, Fermigier, Miller and Chenevier-Lannes, we shall apply the so-called explicit formula “à la Riemann-Weil” to a suitable test function $F$ and to the complete Rankin-Selberg $L$ -function ${\\rm L}(s,\\Pi \\times \\Pi ^{\\prime })$ , first to $\\Pi ^{\\prime } = \\Pi ^{\\vee }$ (the contragredient of $\\Pi $ ) and then to some other well-chosen cuspidal automorphic representations $\\Pi ^{\\prime }$ .","Let us stress that the analytic properties of those Rankin-Selberg $L$ -functions (meromorphic continuation to $\\mathbb {C}$ , functional equation, determination of the poles, and boundedness in vertical strips away from the poles) which have been established by Gelbart, Jacquet, Shalika and Shahidi, will play a crucial role in the argument.","It will be convenient to follow the exposition of the explicit formula given in [6], which is designed for this kind of applications, and from which we shall borrow our notations.","In particular, we choose for the test function $F$ the scaling of Odlyzko's function which is denoted by ${\\rm F}_{\\lambda }$ in [6], and we denote by ${\\rm K}_{\\infty }$ the Grothendieck ring of finite dimensional complex representations of the quotient of the compact group ${\\rm W}_{\\mathbb {R}}$ by its central subgroup $\\mathbb {R}_{>0}$ , and by ${\\rm J}_{F} : {\\rm K}_{\\infty } \\rightarrow \\mathbb {R}$ the concrete linear form associated to $F$ defined in Proposition-Def.","3.7 loc.","cit.","Let $j\\ge 0$ be an even integer and let $\\Pi $ be as in Lemma REF .","As already explained, it follows from (c) that the Langlands parameter of $\\Pi _\\infty $ is ${\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ , whose square in the ring ${\\rm K}_\\infty $ is $4\\, ( {\\rm I}_{2j+2}+ {\\rm I}_{0})$ .","The explicit formula applied to $\\Pi \\times \\Pi ^\\vee $ leads to the inequality [6] : $ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{2j+2})+{\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0}) \\le \\frac{2}{\\pi ^{2}} \\lambda $ for all $\\lambda >0$ .","As ${\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{w})$ is a non-increasing function of $w$ , the truth of the proposition for $j \\le 34$ is a consequence of the following numerical computation for $\\lambda = 3.3$ ${\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{70})+ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0}) \\simeq 0.679 \\, \\, \\, \\, \\, \\, {\\rm and}\\, \\, \\, \\, \\, \\frac{2}{\\pi ^{2}}\\lambda \\simeq 0.669$ (the values are given up to $10^{-3}$ , and the left-hand side has been computed using the formula for ${\\rm J}_{F_{\\lambda }}$ given in [6]).","We now explain how to deal with the cases $j=36$ and 38.","By Tsushima's formula (proved for $k=3$ independently by Petersen and Taïbi), we know that the first value of $j$ such that ${S}_{j,3}(\\Gamma _2)$ is non-zero is $j=36$ , in which case it has dimension 1.","Let $\\pi $ be the cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by ${S}_{36,3}(\\Gamma _2)$ and set $\\Pi ^{\\prime } = \\pi ^{\\rm GL}$ .","This $\\Pi ^{\\prime }$ is a selfdual cuspidal representation by [6], and the Langlands parameter of $\\Pi ^{\\prime }_\\infty $ is ${\\rm I}_{39} \\oplus {\\rm I}_{37}$ (the existence of such a $\\Pi ^{\\prime }$ actually “explains” why the argument above breaks down at $j=36$ , as the Langlands parameter of $\\Pi ^{\\prime }_\\infty $ is “close” to ${\\rm I}_{37}\\oplus {\\rm I}_{37}$ ).","We now apply the explicit formula to $\\Pi \\times \\Pi ^\\vee $ , $\\Pi ^{\\prime } \\times {\\Pi ^{\\prime }}^\\vee $ and $\\Pi \\times {\\Pi ^{\\prime }}^\\vee $ .","It leads to a simple criterion, given in [6], for $\\Pi $ not to exist : the explicit quantity denoted there by ${\\rm t}(V,V^{\\prime },\\lambda )$ has to be $\\ge 0$ for all $\\lambda $ , where $V$ and $V^{\\prime }$ are the respective Langlands parameter of $\\Pi _\\infty $ and $\\Pi ^{\\prime }_\\infty $ .","But a computation gives ${\\rm t}({\\rm I}_{37} + {\\rm I}_{37},{\\rm I}_{39} + {\\rm I}_{37},4) \\simeq -0.429\\,\\,\\,\\,\\, {\\rm and} \\,\\,\\, \\, \\, {\\rm t}({\\rm I}_{39}+ {\\rm I}_{39},{\\rm I}_{39} + {\\rm I}_{37},4) \\simeq -0.039$ (these values are given up to $10^{-3}$ ) which are both $<0$ .","This concludes the proof.","Remark 1.4 (i) As we have ${S}_{j,3}(\\Gamma _2)=0$ for $j<36$ by Tsushima's formula, the vanishing of ${S}_{j,2}(\\Gamma _2)$ for $j \\le 38$ is a very mild evidence toward Conjecture 1.1 of Cléry and van der Geer.","(ii) Lemma REF and the explicit formula can also be used to obtain upper bounds on $\\dim {S}_{j,2}(\\Gamma _2)$ .","Indeed, keeping the notations in the above proof, and applying [6] (due to Taïbi), we get that the inequality $({\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{2j+2})+ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0})) \\dim {S}_{j,2}(\\Gamma _2) \\le \\frac{2}{\\pi ^{2}} \\lambda $ holds for all $\\lambda >0$ .","When the parenthesis on left hand side is $> 0$ , which happens (for big enough $\\lambda $ ) for all $j \\le 138$ , we obtain an explicit upper bound for $\\dim {S}_{j,2}(\\Gamma _2)$ .","For instance, we get $\\dim {S}_{j,2}(\\Gamma _2) \\le 1$ for all $j < 54$ and $\\dim {S}_{j,2}(\\Gamma _2) \\le 2$ for all $j < 66$ (choose respectively $\\lambda =5$ and $\\lambda =6$ ).","Our last and main result concerns the kernel $\\Gamma _2[2]$ of the reduction ${\\rm Sp}_4(\\mathbb {Z}) \\rightarrow {\\rm Sp}_4(\\mathbb {Z}/2\\mathbb {Z})$ .","Theorem 1.5 We have ${S}_{j,1}(\\Gamma _2[2])=0$ for any $j$ .","Our proof will be an elaboration of the one of Proposition REF .","We shall also use the vanishing ${S}_{j,1}(\\Gamma _2[2])=0$ for $j\\le 8$ , proved by Cléry and van der Geer in this paper (Proposition REF ).","As we have $-1 \\in \\Gamma _2[2]$ we may also assume $j$ is even.","Let us denote by $J$ the principal congruence subgroup of ${\\rm PGSp}_4(\\mathbb {Z}_2)$ and by $\\mathbb {A}$ the adele ring of $\\mathbb {Q}$ ; we easily check ${\\rm PGSp}_4(\\mathbb {A}) = {\\rm PGSp}_4(\\mathbb {Q}) \\cdot ({\\rm PGSp}_4(\\mathbb {R})^0 \\times J \\times \\prod _{p\\ne 2} {\\rm PGSp}_4(\\mathbb {Z}_p)).$ Moreover, classical arguments show that we have $S_{j,1}(\\Gamma _2[2])=0$ if, and only if, there is no cuspidal automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ which is unramified at every odd prime, such that $\\pi _2$ has a non-zero invariant under $J$ , and with $\\pi _\\infty \\simeq {\\rm U}_{j,1}$ .","Thus we fix such a $\\pi $ and consider $\\pi ^{\\rm GL}$ , as well as the associated collection $(d_i,n_i,\\pi _i)_{i \\in I}$ , given by (b) above.","By the same argument as in the case $\\Gamma =\\Gamma _2$ , there exists $i \\in I$ with $d_i \\ge 2$ .","If we have $n_i=1$ , which forces ${\\rm inf}\\, \\pi _i = \\lbrace 0\\rbrace $ , we must have $d_i=2$ and $j \\in \\lbrace 0,2\\rbrace $ by the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ .","But this is a contradiction as ${S}_{j,1}(\\Gamma _2[2])=0$ for $j=0,2$ .","So we must have $n_i=d_i=2$ , $I=\\lbrace i\\rbrace $ , and $\\pi _i$ is orthogonal with ${\\rm inf} (\\pi _i)_\\infty = \\lbrace -j/2,j/2\\rbrace $ .","The classification of orthogonal cuspidal automorphic representations of ${\\rm GL}_2$ over $\\mathbb {Q}$ , a very special case of Arthur's results, is well-known.","First of all, the central character of such a representation has order 2, hence corresponds to some uniquely defined quadratic extension $K$ of $\\mathbb {Q}$ .","Moreover, for any Hecke character $\\chi $ of $K$ which is trivial on the idele group of $\\mathbb {Q}$ , and with $\\chi ^2 \\ne 1$ , the automorphic induction of $\\chi $ to $\\mathbb {Q}$ is an orthogonal cuspidal automorphic representation of ${\\rm GL}_2$ over $\\mathbb {Q}$ that we shall denote by ${\\rm ind}(\\chi )$ .","It turns out that they all have this form, and that we have moreover ${\\rm ind}(\\chi ) \\simeq {\\rm ind}(\\chi ^{\\prime })$ if, and only if, we have $\\chi =\\chi ^{\\prime }$ or $\\chi ^{-1} = \\chi ^{\\prime }$ .","Last but not least, to any $\\chi $ as above Arthur associates a global packet $\\Pi (\\chi )=\\bigotimes ^{\\prime }_v \\Pi _v(\\chi _v)$ of irreducible admissible representations of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ , whose elements are exactly the discrete automorphic representations $\\omega $ which satisfy $\\omega ^{\\rm GL} = {\\rm ind}(\\chi ) \\otimes |.|^{1/2} \\boxplus {\\rm ind}(\\chi ) \\otimes |.|^{-1/2}$ (Soudry type); in this “stable case” any element of $\\Pi (\\chi )$ is automorphic by Arthur's multiplicity formula.","Going back to our specific situation, let $K$ and $\\chi $ be such that $\\pi _i \\simeq {\\rm ind}(\\chi )$ .","Since $\\pi _i$ is unramified outside 2, then so is $K$ and we necessarily have $K=\\mathbb {Q}(\\sqrt{d})\\, \\, \\, {\\rm with}\\, \\, \\, d\\,\\in \\lbrace -2,\\,-1,\\,2\\rbrace .$ We first claim $d \\ne 2$ .","Indeed, a Hecke character of real quadratic field has the form $|.|^{s_0} \\chi _0$ with $\\chi _0$ a finite order character and $s_0 \\in \\mathbb {C}$ .","We would thus have $\\lbrace s_0,s_0\\rbrace = {\\rm inf} (\\pi _i)_\\infty =\\lbrace j/2,-j/2\\rbrace $ , which implies $s_0=j=0$ , which is again absurd.","So $K$ is imaginary quadratic.","As $\\chi _\\infty $ is trivial on $\\mathbb {R}^\\times $ by assumption on $\\chi $ , and up to replacing $\\chi $ by $\\chi ^{-1}$ if necessary, the shape of ${\\rm inf}\\, (\\pi _i)_\\infty $ implies then $\\chi _\\infty (z) = (z/\\overline{z})^{j/2}$ .","Here comes the main trick.","Let $\\eta $ be the Hecke character of the statement of Lemma REF below, and set $w_K=|\\mathcal {O}_K^\\times |$ .","We may assume $j\\ge 2$ , so there are unique integers $r$ and $j^{\\prime }/2$ , with $r\\ge 0$ and $1 \\le j^{\\prime }/2 \\le w_K$ , such that $j/2 \\,=\\,r \\,w_K \\, +\\, j^{\\prime }/2$ .","Consider the Hecke character $\\chi ^{\\prime } = \\chi \\eta ^{-r}$ of $K$ .","It is obviously trivial on the idele group of $\\mathbb {Q}$ , and it satisfies $(\\chi ^{\\prime })^2 \\ne 1$ as we have $\\chi ^{\\prime }_\\infty (z)= (z/\\overline{z})^{j^{\\prime }/2}$ with $j^{\\prime }>0$ .","Consider now the packet $\\Pi (\\chi ^{\\prime })$ .","As we have $\\chi _2=\\chi ^{\\prime }_2$ , the local Arthur packets $\\Pi _2(\\chi _2)$ and $\\Pi _2(\\chi ^{\\prime }_2)$ do coincide.","As $\\pi $ belongs to $\\Pi (\\chi )$ , its local component $\\pi _2$ also belongs to $\\Pi _2(\\chi _2)$ .","We may thus consider a representation $\\pi ^{\\prime }=\\bigotimes ^{\\prime }_v \\pi ^{\\prime }_v$ in $\\Pi (\\chi ^{\\prime })$ with $\\pi ^{\\prime }_2\\simeq \\pi _2$ , with $\\pi ^{\\prime }_p$ unramified for all odd prime $p$ (since $\\chi ^{\\prime }_p$ is unramified for such a $p$ ), and with $\\pi ^{\\prime }_\\infty \\simeq {\\rm U}_{j^{\\prime },1}$ .","This last property holds because the Langlands packet associated to the Arthur packet $\\Pi _\\infty (\\chi ^{\\prime }_\\infty )$ , which is included in $\\Pi _\\infty (\\chi ^{\\prime }_\\infty )$ by [1], is the one of ${\\rm U}_{j^{\\prime },1}$ by Remark (a) above.","As already explained, the representation $\\pi ^{\\prime }$ is discrete automorphic by Arthur, and even cuspidal as its Archimedean component is tempered.","As $\\pi ^{\\prime }_2 \\simeq \\pi _2$ has non-zero invariants under the principal congruence subgroup $J$ of ${\\rm PGSp}_4(\\mathbb {Z}_2)$ , it follows that $\\pi ^{\\prime }$ is generated by an element in ${S}_{j^{\\prime },1}(\\Gamma _2[2])$ by the first paragraph above.","But now we have the inequality $j^{\\prime } \\,\\le \\,2 \\,w_K \\,\\le \\,8$ , a contradiction by the vanishing ${S}_{j^{\\prime },1}(\\Gamma _2[2])=0$ for $j^{\\prime }\\le 8$ .","Lemma 1.6 Let $K=\\mathbb {Q}(\\sqrt{d}) \\subset \\mathbb {C}$ with $d=-1,-2$ and set $w_K=|\\mathcal {O}_K^\\times |$ .","There is a Hecke character $\\eta $ of $K$ which is unramified outside $\\lbrace 2,\\infty \\rbrace $ and which satisfies $\\eta _2=1$ and $\\eta _\\infty (z)=(z/\\overline{z})^{w_K}$ for all $z \\in K_\\infty ^\\times $ .","Moreover, $\\eta $ is trivial on the idele group of $\\mathbb {Q}$ .","Denote by $\\mathbb {A}_F$ the adele ring of the number field $F$ .","As $\\mathcal {O}_K$ has class number 1 we have the decomposition $\\mathbb {A}_K^\\times = K^\\times \\cdot (K_\\infty ^\\times \\times K_2^\\times \\times \\prod _{v\\ne 2,\\infty } {\\mathcal {O}^\\times _{K_v}})$ .","This implies first the (unrequired) uniqueness of $\\eta $ , and shows that its existence is equivalent to the fact that the morphism $z \\mapsto (z/\\overline{z})^{w_K}, \\mathbb {C}^\\times \\rightarrow \\mathbb {C}^\\times $ , is trivial on the subgroup $(\\mathcal {O}_K[1/2])^\\times $ .","This is indeed the case as this latter group is generated by $\\mathcal {O}_K^\\times $ and by some element $\\pi \\in \\mathcal {O}_K$ with norm 2 and which satisfies $\\pi /\\overline{\\pi } \\in \\mathcal {O}_K^\\times $ .","The last assertion follows from the equality $\\mathbb {A}_\\mathbb {Q}^\\times = \\mathbb {Q}^\\times \\cdot (\\mathbb {R}^\\times \\times \\prod _p \\mathbb {Z}_p^\\times )$ and the properties of $\\eta $ ."],["Proof of Theorem ","In this second appendix, we explain how to deduce Theorem REF stated in the paper of Cléry and van der Geer from the works of Rösner [20] and Weissauer [26], for the convenience of the reader.","We first recall some results on Yoshida lifts taken from Weissauer's work [26].","Let us fix $f$ and $g$ two non-proportional elliptic eigen newforms of same even weight $j+2$ , and let us denote by $\\pi $ and $\\pi ^{\\prime }$ the (distinct) cuspidal automorphic representations of ${\\rm GL_2}(\\mathbb {A})$ that they generate, $\\mathbb {A}$ being the adele ring of $\\mathbb {Q}$ .","In particular, $\\pi _\\infty $ and $\\pi ^{\\prime }_\\infty $ are isomorphic discrete series, and we may assume that $\\pi $ and $\\pi ^{\\prime }$ are normalized such that this discrete series has trivial central character (as $j$ is even).","For Yoshida lifts $Y(f,g)$ of $f$ and $g$ to exist, we also need to assume that $f$ and $g$ have the same nebentypus, i.e.","that $\\pi $ and $\\pi ^{\\prime }$ have the same central character.","Let $\\Pi (\\pi ,\\pi ^{\\prime })=\\bigotimes ^{\\prime }_v \\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ be the restricted tensor product, over all the places $v$ of $\\mathbb {Q}$ , of the local $L$ -packet $\\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ of irreducible admissible representations of ${\\rm GSp}_4(\\mathbb {Q}_v)$ associated to the pair $\\lbrace \\pi _v,\\pi ^{\\prime }_v\\rbrace $ by Weissauer.","For each place $v$ of $\\mathbb {Q}$ , this local $L$ -packet has either 1 or 2 elements, including a unique generic element; it has another element if, and only if, both $\\pi _v$ and $\\pi ^{\\prime }_v$ are discrete series [26] [20].","In particular, the (Langlands) Archimedean packet $\\Pi _\\infty (\\pi _\\infty ,\\pi ^{\\prime }_\\infty )$ has two elements, the non-generic one being the holomorphic limit of discrete series ${\\rm U}_{j,2}$ recalled in appendix .","Also, if both $\\pi $ and $\\pi ^{\\prime }$ are unramified at the finite place $v$ then $\\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ is a singleton (thus $\\Pi (\\pi ,\\pi ^{\\prime })$ is finite).","The multiplicity formula proved by Weissauer [26] states that an element $\\varpi $ of $\\Pi (\\pi ,\\pi ^{\\prime })$ is discrete automorphic if, and only if, there is an even number of places $v$ such that $\\varpi _v$ is non-generic.","He also shows that such an element has multiplicity one in the discrete spectrum of ${\\rm GSp}_4$ ; it is necessarily cuspidal as the two elements of $\\Pi _\\infty (\\pi _\\infty ,\\pi ^{\\prime }_\\infty )$ are tempered.","Let us denote by $J \\subset {\\rm GSp}_4(\\mathbb {Z}_2)$ the principal congruence subgroup.","Some classical arguments show that the Yoshida lifts $Y(f,g)$ which belong to the space $YS_{j,2}^{s[w]}$ of the statement are in natural bijection with certain vectors of the finite part of the cuspidal automorphic representations $\\varpi $ in $\\Pi (\\pi ,\\pi ^{\\prime })$ having the following properties : (i) $\\varpi _\\infty \\simeq {\\rm U}_{j,2}$ , (ii) $\\varpi _p^{{\\rm GSp}_4(\\mathbb {Z}_p)} \\ne 0$ (in which case we have $\\, \\dim \\, \\varpi _p^{{\\rm GSp}_4(\\mathbb {Z}_p)}=1$ ), (iii) the $s[w]$ -isotypic component $\\varpi _2^{J}$ is non-zero.","More precisely, the Yoshida lifts $Y(f,g)$ corresponding to such a $\\varpi $ form a linear subspace $YS_{j,2}^{s[w]}[\\varpi ] \\subset YS_{j,2}^{s[w]}$ isomorphic to the $s[w]$ -isotypic component of $\\varpi _2^{J}$ as an $S_6$ -representation.","The space $YS_{j,2}^{s[w]}$ of the statement is then the direct sum of its subspaces $YS_{j,2}^{s[w]}[\\varpi ]$ where $f,g$ and $\\varpi $ vary, with $\\varpi $ cuspidal automorphic satisfying (i), (ii) and (iii).","We still fix elliptic newforms $f$ and $g$ as above, hence $\\pi $ and $\\pi ^{\\prime }$ as well.","By [20], we know first that $\\Pi _p(\\pi _p,\\pi ^{\\prime }_p)$ has an element with non-zero invariants under ${\\rm GSp}_4(\\mathbb {Z}_p)$ if, and only if, both $\\pi _p$ and $\\pi ^{\\prime }_p$ are unramified.","Moreover, the same corollary asserts that if $\\Pi _2(\\pi _2,\\pi ^{\\prime }_2)$ has an element with non-zero $J$ -invariants, then both $\\pi _2$ and $\\pi ^{\\prime }_2$ have non-zero invariants under the principal congruence subgroup of ${\\rm GL}_2(\\mathbb {Z}_2)$ .","As a first consequence, if $\\varpi \\in \\Pi (\\pi ,\\pi ^{\\prime })$ does satisfy (ii) and (iii) then $f$ and $g$ are newforms on $\\Gamma _0(N)$ with $N|4$ (and both $\\pi $ and $\\pi ^{\\prime }$ have a trivial central character).","Moreover, by the statement recalled above concerning the multiplicity formula (\"even parity of the number of non-generic places\"), there is at most one cuspidal automorphic $\\varpi \\in \\Pi (\\pi ,\\pi ^{\\prime })$ satisfying (i), (ii) and (iii) above, and it has the property that $\\varpi _2$ is the non-generic element of $\\Pi _2(\\pi _2,\\pi ^{\\prime }_2)$ .","In particular, this latter $L$ -packet has two elements and both $\\pi _2$ and $\\pi ^{\\prime }_2$ are discrete series of ${\\rm GL}_2(\\mathbb {Q}_2)$ (thus neither $f$ nor $g$ can have level 1).","By Rösner [20], there are only three possibilities for the isomorphism class of a representation of ${\\rm PGL}_2(\\mathbb {Q}_2)$ generated by an elliptic newform of level $\\Gamma _0(2)$ or $\\Gamma _0(4)$ , namely the Steinberg representation St and its unramified quadratic twist ${\\rm St}^{\\prime }$ in level $\\Gamma _0(2)$ , and a certain supercuspidal representation ${\\rm Sc}$ in level $\\Gamma _0(4)$ .","In particular, there are all discrete series.","Assume now that $\\sigma $ and $\\sigma ^{\\prime }$ are (possibly equal) elements of the set $\\lbrace {\\rm St}, {\\rm St^{\\prime }}, {\\rm Sc}\\rbrace $ and let $\\tau $ be the non-generic element of $\\Pi _2(\\sigma ,\\sigma ^{\\prime })$ .","View the finite dimensional vector space $\\tau ^J$ as a representation of $S_6$ .","In order to prove the theorem it only remains to show that either $\\tau ^J$ is 0 or we are in exactly one of the following situations : (a) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm St},{\\rm St^{\\prime }}\\rbrace $ and $\\tau ^J \\simeq s[1^6]$ , (b) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm Sc}\\rbrace $ and $\\tau ^J \\simeq s[2,1^4]$ , (c) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm St}\\rbrace $ or $\\lbrace {\\rm St^{\\prime }}\\rbrace $ and $\\tau ^J \\simeq s[2^3]$ .","This is a delicate analysis which fortunately has been carried out by Rösner.","Indeed, this is exactly the content of the left-bottom part of [20] with $q=2$ .","This table shows that there are just three non-zero possible representations for $\\tau ^J$ , denoted by $\\theta _2$ , $\\theta _5$ and $\\chi _9(1)$ there but which correspond respectively to the representations $s[2^3]$ , $s[1^6]$ and $s[2,1^4]$ by Rösner's other table [20], exactly according to the three cases above (read the table with $\\Pi _1={\\rm Sc}$ , $\\xi _\\mu {\\rm St} = {\\rm St}^{\\prime }$ , and take for $\\mu $ either the trivial character of $\\mathbb {Q}_2^\\times $ or its unramified quadratic character).","$\\square $ Acknowledgement.","Gaëtan Chenevier is supported by the C.N.R.S.","and by the project ANR-14-CE25 (PerCoLaTor).","He would like to thank Fabien Cléry and Gerard van der Geer for inviting him to include these two appendices to their paper.","Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.","E-mail address: gaetan.chenevier@math.cnrs.fr"]],"string":"[\n [\n \"On vector-valued Siegel modular forms of degree 2 and weight (j,2)\"\n ],\n [\n \"Abstract We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it.\",\n \"We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants.\",\n \"Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.\"\n ],\n [\n \"Introduction\",\n \"The usual methods for determining the dimensions of spaces of Siegel modular forms do not work for low weights.\",\n \"For Siegel modular forms of degree 2 this means that we do not have formulas for the dimensions of the spaces of Siegel modular forms of weight $(j,k)$ , that is, corresponding to ${\\\\rm Sym}^j \\\\otimes \\\\det {}^k$ , in case $k<3$ .\",\n \"In this paper we propose a description of the spaces of cusp forms of weight $(j,2)$ on the level 2 principal congruence subgroup $\\\\Gamma _2[2]=\\\\ker ({\\\\rm Sp}(4,{\\\\mathbb {Z}}) \\\\rightarrow {\\\\rm Sp}(4,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}}))$ of $\\\\Gamma _2={\\\\rm Sp}(4,{\\\\mathbb {Z}})$ and we provide some evidence for this conjectural description.\",\n \"Let $S_{j,k}(\\\\Gamma _2[2])$ be the space of cusp forms of weight $(j,k)$ , that is, corresponding to the factor of automorphy ${\\\\rm Sym}^j(c\\\\tau +d) \\\\det (c\\\\tau +d)^k$ on the group $ \\\\Gamma _2[2]$ .\",\n \"Recall that the group ${\\\\rm Sp}(4,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}})$ is isomorphic to the symmetric group $\\\\mathfrak {S}_6$ .\",\n \"We fix an explicit isomorphism by identifying the symplectic lattice over ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ with the subspace $\\\\lbrace (a_1,\\\\ldots ,a_6) \\\\in ({\\\\mathbb {Z}}/2{\\\\mathbb {Z}})^6: \\\\sum a_i=0\\\\rbrace $ modulo the diagonally embedded ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ with form $\\\\sum _i a_ib_i$ as in [3]; it is given explicitly on generators of $\\\\mathfrak {S}_6$ in [10].\",\n \"So $\\\\mathfrak {S}_6$ acts on the space of cusp forms $S_{j,k}(\\\\Gamma _2[2])$ and this space thus decomposes into isotypical components for the symmetric group $\\\\mathfrak {S}_6$ .\",\n \"The irreducible representations of $\\\\mathfrak {S}_6$ correspond to the partitions of 6 and we thus have for each such partition $\\\\varpi $ a subspace $S_{j,k}(\\\\Gamma _2[2])^{s[\\\\varpi ]}$ of $S_{j,k}(\\\\Gamma _2[2])$ where $\\\\mathfrak {S}_6$ acts as $s[\\\\varpi ]$ .\",\n \"Note that the case $s[6]$ corresponds to cusp forms on ${\\\\rm Sp}(4,{\\\\mathbb {Z}})$ , while the case $s[1^6]$ corresponds to modular forms of weight $(j,k)$ on ${\\\\rm Sp}(4,{\\\\mathbb {Z}})$ with a quadratic character: $S_{j,k}(\\\\Gamma _2[2])^{s[1^6]} = S_{j,k}({\\\\rm Sp}(4,{\\\\mathbb {Z}}),\\\\epsilon )$ with $\\\\epsilon $ the unique quadratic character of ${\\\\rm Sp}(4,{\\\\mathbb {Z}})$ .\",\n \"Before we formulate our conjecture we recall that the group ${\\\\rm SL}(2,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}})\\\\cong \\\\mathfrak {S}_3$ acts on the space $S_k(\\\\Gamma _1[2])$ of cusp forms on the principal congruence subgroup of level 2 $\\\\Gamma _1[2]=\\\\ker ({\\\\rm SL}(2,{\\\\mathbb {Z}})\\\\rightarrow {\\\\rm SL}(2,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}}))$ .\",\n \"We can thus decompose this space in isotypical components corresponding to the irreducible representations of $\\\\mathfrak {S}_3$ .\",\n \"The map $f(z) \\\\mapsto f(2z)$ defines an isomorphism $S_k(\\\\Gamma _1[2]){\\\\sim \\\\over \\\\longrightarrow }S_k(\\\\Gamma _0(4))$ with $\\\\Gamma _0(4)$ the usual congruence subgroup of $\\\\Gamma _1={\\\\rm SL}(2,{\\\\mathbb {Z}})$ .\",\n \"If we write its isotypical decomposition as $S_k(\\\\Gamma _1[2])= a_k\\\\, s[3]+b_k \\\\, s[2,1]+c_k \\\\, s[1^3]\\\\, ,$ then $\\\\dim S_k(\\\\Gamma _1)=a_k$ , $\\\\dim S_k(\\\\Gamma _0(2))^{\\\\rm new}=b_k-a_k$ and $\\\\dim S_k(\\\\Gamma _0(4))^{\\\\rm new}= c_k$ with their generating series given by $\\\\sum a_k t^k= t^{12}/(1-t^4)(1-t^6), \\\\quad \\\\sum b_k t^k=t^8/(1-t^2)(1-t^6)\\\\, ,\\\\quad \\\\sum c_k t^k=t^6/(1-t^4)(1-t^6)\\\\, .$ The Fricke involution $w_2: \\\\tau \\\\mapsto -1/2\\\\tau $ defines an involution on $S_k(\\\\Gamma _0(2))^{\\\\rm new}$ and this space splits into eigenspaces $S_k^{\\\\pm }(\\\\Gamma _0(2))^{\\\\rm new}$ and for $k>2$ we have $\\\\dim S_k^{+}(\\\\Gamma _0(2))^{\\\\rm new} - \\\\dim S_k^{-}(\\\\Gamma _0(2))^{\\\\rm new}={\\\\left\\\\lbrace \\\\begin{array}{ll} -1 & k\\\\equiv 2 \\\\, \\\\bmod \\\\, 8 \\\\\\\\0 & k \\\\equiv 4,6 \\\\, \\\\bmod \\\\, 8 \\\\\\\\1 & k \\\\equiv 0 \\\\, \\\\bmod \\\\, 8\\\\, .\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ We recall the notion of Yoshida type lifts.\",\n \"Yoshida lifts are explained in [27]; see also [28], [26], [5], [21].\",\n \"These are eigen forms associated to a pair of elliptic modular eigenforms whose spinor L-function is a product of the twisted L-functions of the elliptic modular cusp forms.\",\n \"In [3] a number of conjectures on the existence of Yoshida lifts were made and these were proved by Rösner [20].\",\n \"These conjectures deal with Siegel modular cusp forms of weight $(j,k)$ with $k\\\\ge 3$ .\",\n \"It can be extended to the case of weight $(j,2)$ .\",\n \"We denote the subspace of $S_{j,2}(\\\\Gamma _2[2])^{s[\\\\varpi ]}$ generated by Yoshida lifts by $YS_{j,2}^{s[\\\\varpi ]}$ .\",\n \"Theorem 1.1 We have $YS_{j,2}^{s[w]} = 0$ unless we are in the following cases: (1) $w=[1^6]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ eigen-newforms of level $\\\\Gamma _0(2)$ of different sign.\",\n \"In this case we have $\\\\dim YS_{j,2}^{s[w]} \\\\,= \\\\,\\\\dim \\\\,S^+_{j+2}(\\\\Gamma _0(2))^{\\\\rm new} \\\\otimes S^-_{j+2}(\\\\Gamma _0(2))^{\\\\rm new}.$ (2) $w=[2,1^4]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\\\Gamma _0(4)$ .\",\n \"The multiplicity $\\\\mu (j)$ of $s[2,1^4]$ in $YS_{j,2}^{s[w]}$ is then $\\\\mu (j)\\\\, =\\\\, \\\\dim \\\\, \\\\Lambda ^2 S_{j+2}(\\\\Gamma _0(4))^{\\\\rm new}.$ (3) $w=[2^3]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\\\Gamma _0(2)$ with the same sign.\",\n \"The multiplicity $\\\\nu (j)$ of $s[2^3]$ is $\\\\nu (j) \\\\,= \\\\,\\\\dim \\\\Lambda ^2 S^+_{j+2}(\\\\Gamma _0(2))^{\\\\rm new} \\\\oplus \\\\Lambda ^2 S^-_{j+2}(\\\\Gamma _0(2))^{\\\\rm new}.$ The proof of this theorem follows from results of Rösner and Weissauer, in a way very similar to Rösner's proof of the Bergström-Faber-van der Geer conjecture in weight $k\\\\ge 3$ [20].\",\n \"In the second appendix Chenevier explains how to derive it.\",\n \"We now formulate our conjecture.\",\n \"Conjecture 1.2 The space $S_{j,2}(\\\\Gamma _2[2])$ is generated by Yoshida type lifts.\",\n \"Note that this implies that $S_{j,2}(\\\\Gamma _2[2])^{s[\\\\varpi ]}=(0)$ unless $\\\\varpi = [1^6], [2,1^4]$ or $[2^3]$ .\",\n \"In particular, it implies that $S_{2,j}(\\\\Gamma _2)=(0)$ .\",\n \"The evidence we have for the latter is the following.\",\n \"Theorem 1.3 We have $\\\\dim S_{j,2}(\\\\Gamma _2)=0$ for $j\\\\le 52$ .\",\n \"For $j\\\\le 20$ the vanishing of $S_{j,2}(\\\\Gamma _2)$ was proved by Ibukiyama, Wakatsuki and Uchida [16], [17], and [25].\",\n \"The evidence we have for the $s[1^6]$ -part of the conjecture is the following.\",\n \"Theorem 1.4 The dimension of $S_{j,2}(\\\\Gamma _2,\\\\epsilon )$ is given by the coefficient of $t^j$ in the expansion of $t^{12}/(1-t^6)(1-t^8)(1-t^{12})$ for $j\\\\le 30$ .\",\n \"Modular forms in $S_{j,2}(\\\\Gamma _2,\\\\epsilon )$ can be constructed explicitly using covariants as explained in [7].\",\n \"We prove this theorem by constructing a basis of the space $S_{j,7}(\\\\Gamma _2)$ using covariants of binary sextics (see [7]) and then by checking which forms are divisible by the cusp form $\\\\chi _5 \\\\in S_{0,5}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"We thus give generators for these spaces $S_{j,2}(\\\\Gamma _2[2])^{s[1^6]}$ for $j\\\\le 30$ and we can calculate Hecke eigenvalues for these.\",\n \"For $j>30$ this becomes quite laborious.\",\n \"For all irreducible representations we have the following vanishing result: Proposition 1.5 For any $\\\\varpi $ we have $\\\\dim S_{j,2}(\\\\Gamma _2[2])^{s[\\\\varpi ]}=0$ for $j<12$ .\",\n \"We end with some remarks on other `small' weights.\",\n \"The vanishing of $S_{j,1}(\\\\Gamma _2)$ follows from work of Skoruppa [23].\",\n \"In an appendix to this paper besides providing a different proof of the vanishing of $S_{j,2}(\\\\Gamma _2)$ for $j\\\\le 38$ , Chenevier gives a proof of the vanishing of $S_{j,1}(\\\\Gamma _2[2])$ .\",\n \"For $k=3$ one knows that $S_{j,3}(\\\\Gamma _2)=(0)$ for $j<36$ .\",\n \"But $S_{36,3}$ is 1-dimensional and using covariants we can construct a form of this weight in a relatively easy manner; cf.\",\n \"the remarks in [17] on the difficulty of constructing such a form.\",\n \"Acknowledgement.\",\n \"We thank Gaëtan Chenevier warmly for asking the question on the existence of modular forms of weight $(j,2)$ on the full group ${\\\\rm Sp}(4,{\\\\mathbb {Z}})$ and for agreeing to add his results in the form of an appendix.\",\n \"He also pointed out an error in an earlier version of our conjecture and provided a proof of the extension of Rösner's result on Yoshida lifts.\",\n \"We thank Mirko Rösner for correspondence.\",\n \"We also thank the Max-Planck Institut für Mathematik in Bonn for excellent working conditions.\"\n ],\n [\n \"Modular Forms of Degree Two\",\n \"For the definitions of Siegel modular forms and elementary properties we refer to [14].\",\n \"We denote the Siegel upper half space of degree $g$ by $\\\\mathfrak {H}_g$ .\",\n \"The Siegel modular group $\\\\Gamma _g={\\\\rm Sp}(2g,{\\\\mathbb {Z}})$ acts on $\\\\mathfrak {H}_g$ by fractional linear transformations $\\\\tau \\\\mapsto (a\\\\tau +b)(c\\\\tau +d)^{-1} \\\\qquad \\\\text{\\\\rm for $\\\\left(\\\\begin{matrix}a & b\\\\cr c & d \\\\cr \\\\end{matrix} \\\\right) \\\\in {\\\\rm Sp}(2g,{\\\\mathbb {Z}})$and $\\\\tau \\\\in \\\\mathfrak {H}_g$.\",\n \"}$ If $\\\\rho : {\\\\rm GL}(g,{\\\\mathbb {C}}) \\\\rightarrow {\\\\rm GL}(W)$ is a finite-dimensional complex representation then a holomorphic map $f: \\\\mathfrak {H}_g \\\\rightarrow W$ is called a Siegel modular form of weight $\\\\rho $ if $f(a\\\\tau +b)(c\\\\tau +d)^{-1})=\\\\rho (c\\\\tau +d)f(\\\\tau )$ for all $(a,b;c,d) \\\\in \\\\Gamma _g$ .\",\n \"The space of modular forms of weight $\\\\rho $ is finite-dimensional and denoted by $M_{\\\\rho }(\\\\Gamma _g)$ .\",\n \"If $g=2$ then an irreducible representation of ${\\\\rm GL}(2,{\\\\mathbb {C}})$ is of the form ${\\\\rm Sym}^j({\\\\rm St}) \\\\otimes {\\\\det }({\\\\rm St})^k$ with ${\\\\rm St}$ the standard representation of ${\\\\rm GL}(2,{\\\\mathbb {C}})$ for some $j \\\\in {\\\\mathbb {Z}}_{\\\\ge 0}$ and $k \\\\in {\\\\mathbb {Z}}$ .\",\n \"For $\\\\rho ={\\\\rm Sym}^j({\\\\rm St}) \\\\otimes {\\\\det }({\\\\rm St})^k$ we denote $M_{\\\\rho }$ by $M_{j,k}$ and we call $(j,k)$ the weight.\",\n \"If $j=0$ we are dealing with scalar-valued modular forms.\",\n \"The space of Siegel modular forms of degree 2 and weight $(j,k)$ is denoted by $M_{j,k}(\\\\Gamma _2)$ .\",\n \"There is the Siegel operator $\\\\Phi _g$ that maps Siegel modular forms of degree $g$ to Siegel modular forms of degree $g-1$ .\",\n \"The kernel of $\\\\Phi _2$ in $M_{j,k}(\\\\Gamma _2)$ is called the space of cusp forms of weight $(j,k)$ and denoted by $S_{j,k}(\\\\Gamma _2)$ .\",\n \"Note that for $k=2$ we have $M_{j,2}(\\\\Gamma _2)=S_{j,2}(\\\\Gamma _2)$ , see [16].\",\n \"For a finite index subgroup $\\\\Gamma $ of ${\\\\rm Sp}(4,{\\\\mathbb {Z}})$ we have similar notions.\",\n \"Here we deal with the groups $\\\\Gamma _2$ and $\\\\Gamma _2[2]$ .\",\n \"The quotient group $\\\\Gamma _2/\\\\Gamma _2[2]\\\\cong {\\\\rm Sp}(4,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}})$ is identified with the symmetric group $\\\\mathfrak {S}_6$ as in the Introduction.\",\n \"This group acts in a natural way on the space of cusp forms $S_{j,k}(\\\\Gamma _2[2])$ and we can decompose this space in isotypical components $S_{j,k}(\\\\Gamma _2[2])^{s[\\\\varpi ]}$ corresponding to the irreducible representations $s[\\\\varpi ]$ of $\\\\mathfrak {S}_6$ which in turn correspond bijectively to the partitions $\\\\varpi $ of 6.\",\n \"The ring $R$ of scalar-valued Siegel modular forms on $\\\\Gamma _2$ was determined by Igusa in the 1960s, see [18].\",\n \"In the 1980s Tsushima gave in [24] formulas for the dimensions of the spaces of vector-valued cusp forms on a subgroup between $\\\\Gamma _2[2]$ and $\\\\Gamma _2$ .\",\n \"Bergström extended this to $\\\\Gamma _2[2]$ with the action of $\\\\mathfrak {S}_6$ , see [2].\",\n \"We thus know the dimension of $S_{j,k}(\\\\Gamma _2[2])^{s[\\\\varpi ]}$ for all $j$ and $k\\\\ge 3$ .\",\n \"The vector-valued modular forms of degree 2 form a ring $M=\\\\oplus _{j,k} M_{j,k}(\\\\Gamma _2)$ .\",\n \"It is also a module over the ring $R$ .\",\n \"For level 2 similar things hold.\",\n \"A vector-valued Siegel modular form $f$ of weight $(j,k)$ on $\\\\Gamma _2$ has a Fourier-Jacobi expansion $f(\\\\tau )= \\\\sum _{m\\\\ge 0} \\\\varphi _m(\\\\tau _1,z) \\\\, e^{2\\\\pi i m \\\\tau _2}\\\\qquad \\\\text{ where $\\\\tau = \\\\left( \\\\begin{matrix} \\\\tau _1 & z \\\\\\\\z & \\\\tau _2 \\\\\\\\ \\\\end{matrix} \\\\right)$ }$ with $\\\\varphi _m: \\\\mathfrak {H}_1 \\\\times {\\\\mathbb {C}} \\\\rightarrow {\\\\rm Sym}^j({\\\\mathbb {C}}^2)$ a holomorphic map that satisfies certain functional equations under the action of the so-called Jacobi group ${\\\\rm SL}(2,{\\\\mathbb {Z}}) \\\\ltimes {\\\\mathbb {Z}}^2$ .\",\n \"This group is embedded in $\\\\Gamma _2$ via $\\\\left( \\\\begin{matrix} a & b \\\\\\\\ c & d \\\\\\\\ \\\\end{matrix}\\\\right) \\\\mapsto \\\\left( \\\\begin{matrix} a & 0 & b & 0 \\\\\\\\0 & 1 & 0 & 0 \\\\\\\\c & 0 & d & 0 \\\\\\\\0 & 0 & 0 & 1 \\\\\\\\ \\\\end{matrix} \\\\right), \\\\qquad (\\\\lambda , \\\\mu ) \\\\mapsto \\\\left( \\\\begin{matrix} 1 & 0 & 0 & \\\\mu \\\\\\\\\\\\lambda & 1 & \\\\mu & 0 \\\\\\\\0 & 0 & 1 & -\\\\lambda \\\\\\\\0 & 0 & 0 & 1 \\\\\\\\ \\\\end{matrix} \\\\right)$ with action $\\\\tau \\\\mapsto \\\\left( \\\\begin{matrix} (a\\\\tau _1+b)/(c\\\\tau _1+d) & z/(c\\\\tau _1+d) \\\\\\\\z/(c\\\\tau _1+d) & \\\\tau _2-cz^2/(c\\\\tau _1+d)\\\\\\\\ \\\\end{matrix} \\\\right)$ and $\\\\tau \\\\mapsto \\\\left( \\\\begin{matrix} \\\\tau _1 & z+\\\\lambda \\\\tau _1+\\\\mu \\\\\\\\z+\\\\lambda \\\\tau _1+\\\\mu & \\\\tau _2+ \\\\lambda ^2\\\\tau _1+2\\\\lambda z +\\\\lambda \\\\mu \\\\\\\\ \\\\end{matrix} \\\\right) \\\\, .$ The fact that $f$ is a modular form of weight $(j,k)$ implies the corresponding functional equations $\\\\varphi _m(\\\\frac{a \\\\tau _1+b}{c\\\\tau _1+d}, \\\\frac{z}{c\\\\tau _1+d})e^{-2\\\\pi i m \\\\frac{c z^2}{c\\\\tau _1+d}}= {\\\\rm Sym}^j\\\\left( \\\\begin{matrix} c\\\\tau _1+d & c z \\\\\\\\ 0 & 1 \\\\\\\\ \\\\end{matrix} \\\\right) (c\\\\tau _1+d)^k\\\\varphi _m(\\\\tau _1,z)$ and $\\\\varphi _m(\\\\tau _1,z+\\\\lambda \\\\tau _1+\\\\mu ) e^{2\\\\pi i m (\\\\lambda ^2 \\\\tau _1 +2\\\\lambda z +\\\\lambda \\\\mu )} ={\\\\rm Sym}^j \\\\left( \\\\begin{matrix} 1 & -\\\\lambda \\\\\\\\ 0 & 1 \\\\\\\\ \\\\end{matrix} \\\\right) \\\\varphi _m(\\\\tau _1,z) \\\\, ,$ where we write $\\\\varphi _m$ as the transpose of the row vector $(\\\\varphi _m^{(0)}, \\\\ldots , \\\\varphi _m^{(j)})$ .\",\n \"Corollary 2.1 If $f \\\\in M_{j,k}(\\\\Gamma _2)$ (resp.\",\n \"$f \\\\in S_{j,k}(\\\\Gamma _2)$ ) then the last coordinate $\\\\varphi _m^{(j)}$ of the coefficient $\\\\varphi _m$ of $e^{2\\\\pi i m \\\\tau _2}$ in the Fourier-Jacobi expansion of $f$ is a Jacobi form (resp.\",\n \"Jacobi cusp form) of weight $k$ and index $m$ .\",\n \"We note that Jacobi cusp forms of weight 2 and index $m$ are zero for $m< 37$ , see [11].\",\n \"This imposes strong conditions on forms of weight $(j,2)$ on $\\\\Gamma _2$ .\",\n \"Since we shall compute the action of Hecke operators later we now describe formulas for the action of Hecke operators on forms on $\\\\Gamma _2$ .\",\n \"For forms without character we refer to [9], so we deal with the case of forms with a character.\",\n \"For $f \\\\in M_{j,k}(\\\\Gamma _2,\\\\epsilon )$ we write its Fourier expansion as $f(\\\\tau )= \\\\sum _{n\\\\ge 0} a(n) \\\\, e^{\\\\pi i {\\\\rm Tr}(n\\\\tau )} \\\\, ,$ where $n$ runs over the positive semi-definite half-integral symmetric matrices.\",\n \"We will write $[n_1,n_2,n_3]$ for $\\\\left( {\\\\begin{matrix}n_1 & n_2/2 \\\\\\\\ n_2/2 & n_3 \\\\end{matrix}} \\\\right)$ .\",\n \"For an odd prime $p$ we denote by $T_p$ the Hecke operator for $\\\\Gamma _2$ at $p$ .\",\n \"Then we write the transform of $f$ under $T_p$ as $T_p(f)(\\\\tau )= \\\\sum _{n\\\\ge 0} a_p(n) \\\\, e^{\\\\pi i {\\\\rm Tr} (n\\\\tau )}\\\\, .$ Here for $p \\\\lnot \\\\equiv 1 \\\\bmod \\\\, 3$ the coefficient $a_p([1,1,1])$ is given by $a([p,p,p])$ , and for $p=3$ by $a([3,3,3])-3^{k-2}{\\\\mathrm {Sym}}^j\\\\left({\\\\begin{matrix}3 & -1 \\\\\\\\0 & 1 \\\\\\\\\\\\end{matrix}}\\\\right)a([1,3,3]) \\\\, ,$ while for $p \\\\equiv 1 \\\\bmod \\\\, 3$ by $a([p,p,p])+p^{k-2}\\\\sum _{i=1}^{2}(-1)^{m_i}{\\\\mathrm {Sym}}^j\\\\left({\\\\begin{matrix}p & -m_i \\\\\\\\0 & 1 \\\\\\\\\\\\end{matrix}}\\\\right)a([\\\\frac{1+m_i+m_i^2}{p},1+2m_i, p]) \\\\, ,$ where in the latter case $m_1$ and $m_2$ are the two roots of the polynomial $1+X+X^2$ over $\\\\mathbb {F}_p$ , which we view here as the set $\\\\left\\\\lbrace 0,\\\\ldots ,p-1\\\\right\\\\rbrace $ .\",\n \"Similarly, the coefficient $a_{p^2}([1,1,1])$ of the transform of $f$ under the Hecke operator $T_{p^2}$ is given for $p \\\\lnot \\\\equiv 1 \\\\bmod \\\\, 3$ by $a([p^2,p^2,p^2])$ , and for $p=3$ by $a([9,9,9])-3^{k-2}{\\\\mathrm {Sym}}^j\\\\left({\\\\begin{matrix}3 & -1 \\\\\\\\0 & 1 \\\\\\\\\\\\end{matrix}}\\\\right)a([3,9,9]) \\\\, .$ As an example, consider the modular form $\\\\chi _5 \\\\in S_{0,5}(\\\\Gamma _2,\\\\epsilon )\\\\,,$ the product of the ten even theta characteristics and the square root of Igusa's cusp form $\\\\chi _{10}$ , that will play an important role in this paper.\",\n \"It provides a check on these formulas for the Hecke operators.\",\n \"Indeed, one knows $\\\\lambda _p(\\\\chi _5)=p^3+a_p(f)+p^4, \\\\qquad \\\\lambda _{p^2}(\\\\chi _5)=\\\\lambda _p(\\\\chi _5)^2-(p^4+p^3)\\\\lambda _p(\\\\chi _5)+p^{8}\\\\, ,$ where $f=q-8\\\\, q^2+12\\\\, q^3+64\\\\, q^4-210\\\\, q^5 +\\\\cdots $ is the normalized Hecke eigenform in $S_{8}^+(\\\\Gamma _0(2))^{\\\\text{new}}$ , which illustrates that $\\\\chi _5$ is a Saito-Kurokawa lift.\",\n \"One can check that the above formulas agree with this.\"\n ],\n [\n \"Restriction to the diagonal\",\n \"In order to put restrictions on the existence of Siegel modular forms we restrict these to the `diagonal' given by the embedding $i: \\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1 \\\\rightarrow \\\\mathfrak {H}_2,\\\\qquad (z_1,z_2) \\\\mapsto \\\\left(\\\\begin{matrix}z_1 & 0 \\\\cr 0 & z_2\\\\cr \\\\end{matrix}\\\\right)\\\\, .$ The stabilizer of $i(\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1)$ in ${\\\\rm Sp}(4,{\\\\mathbb {R}})$ is an extension by ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ of the image of ${\\\\rm SL}(2,{\\\\mathbb {R}})^2$ under the embedding $(\\\\left({\\\\begin{matrix}a_1 & b_1 \\\\\\\\ c_1 & d_1\\\\end{matrix}}\\\\right),\\\\left({\\\\begin{matrix}a_2 & b_2 \\\\\\\\ c_2 & d_2\\\\end{matrix}}\\\\right)) \\\\, \\\\mapsto \\\\,\\\\left({\\\\begin{matrix}a_1 & 0 & b_1 & 0\\\\\\\\0 & a_2 & 0 & b_2 \\\\\\\\c_1 & 0 & d_1 & 0\\\\\\\\0 & c_2 & 0 & d_2 \\\\\\\\\\\\end{matrix}}\\\\right)$ The extension by ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ corresponds to the involution that interchanges $\\\\tau _1$ and $\\\\tau _2$ in $\\\\tau =\\\\left({\\\\begin{matrix}\\\\tau _1 & \\\\tau _{12}\\\\cr \\\\tau _{12} & \\\\tau _2 \\\\cr \\\\end{matrix}} \\\\right) \\\\in \\\\mathfrak {H}_2$ (and $z_1$ and $z_2$ on $\\\\mathfrak {H}_1^2$ ).\",\n \"This corresponds to the element $\\\\iota =\\\\left({\\\\begin{matrix} a & 0 \\\\\\\\ 0 & d \\\\\\\\ \\\\end{matrix}}\\\\right)$ in $\\\\Gamma _2$ with $a=d= \\\\left({\\\\begin{matrix} 0 & 1 \\\\\\\\ 1 & 0 \\\\\\\\ \\\\end{matrix}}\\\\right)$ .\",\n \"The stabilizer inside $\\\\Gamma _2$ (resp.\",\n \"inside $\\\\Gamma _2[2]$ ) is an extension by ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ of ${\\\\rm SL}(2,{\\\\mathbb {Z}})\\\\times {\\\\rm SL}(2,{\\\\mathbb {Z}})$ (resp.\",\n \"of $\\\\Gamma _1[2]\\\\times \\\\Gamma _1[2]$ ).\",\n \"If $F=(F_0,\\\\ldots ,F_{j})^t$ is a Siegel modular form of weight $(j,k)$ of level 2, then its pullback under $i$ to $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ gives rise to an element of $(f_0,\\\\ldots , f_j)^t$ with $f_l \\\\in M_{j+k-l}(\\\\Gamma _1[2])\\\\otimes M_{k+l}(\\\\Gamma _1[2])$ .\",\n \"By restricting a cusp form of level 1 we get cusp forms of level 1.\",\n \"The action of $\\\\iota $ is given by a map $S_{j+k-i}(\\\\Gamma _1[2])\\\\otimes S_{k+i}(\\\\Gamma _1[2]) \\\\rightarrow S_{k+i}(\\\\Gamma _1[2])\\\\otimes S_{j+k-i}(\\\\Gamma _1[2]), \\\\qquad a\\\\otimes b \\\\mapsto (-1)^k b\\\\otimes a$ for $\\\\Gamma _1[2]$ and a similar one for $\\\\Gamma _1$ .\",\n \"So for a form of level 1 without character we loose no information by looking at $\\\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\\\Gamma _1)\\\\otimes S_{k+i}(\\\\Gamma _1)\\\\bigoplus {\\\\left\\\\lbrace \\\\begin{array}{ll}{\\\\wedge }^2 S_{j/2+k}(\\\\Gamma _1) & \\\\text{for $k$ odd} \\\\\\\\{\\\\rm Sym}^2 S_{j/2+k}(\\\\Gamma _1) & \\\\text{for $k$ even.}\",\n \"\\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ By multiplying with $\\\\chi _5$ we get an injective map $S_{j,2}(\\\\Gamma _2,\\\\epsilon ) \\\\rightarrow S_{j,7}(\\\\Gamma _2)$ .\",\n \"The generating series for the dimensions is $\\\\sum _{j=2}^{\\\\infty } \\\\dim S_{j,7}(\\\\Gamma _2) \\\\, t^j ={t^{12} \\\\over (1-t)(1-t^3)(1-t^4)(1-t^6)} \\\\, .$ We observe that our conjecture on $S_{j,2}(\\\\Gamma _2,\\\\epsilon )$ implies that $\\\\dim S_{j,7}(\\\\Gamma _2) -\\\\dim S_{j,2}(\\\\Gamma _2,\\\\epsilon ) =\\\\sum _{i=0}^{j/2-1} \\\\dim S_{j+7-i}(\\\\Gamma _1) \\\\dim S_{7+i}(\\\\Gamma _1)+ \\\\dim \\\\wedge ^2 S_{j/2+7}(\\\\Gamma _1) \\\\, ,$ or equivalently, that the restriction $\\\\rho $ to the diagonal fits in an exact sequence $0 \\\\rightarrow S_{j,2}(\\\\Gamma _2,\\\\epsilon ) {\\\\cdot \\\\chi _5 \\\\over \\\\longrightarrow }S_{j,7}(\\\\Gamma _2){\\\\rho \\\\over \\\\longrightarrow } \\\\oplus _{i=0}^{j/2-1}S_{j+7-i}(\\\\Gamma _1)\\\\otimes S_{7+i}(\\\\Gamma _1)\\\\oplus \\\\wedge ^2 S_{j/2+7}(\\\\Gamma _1)\\\\rightarrow 0 \\\\, .$ If $F\\\\in S_{j,k}(\\\\Gamma _2,\\\\epsilon )$ we find by using $\\\\iota $ that we can restrict to $\\\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\\\Gamma _1[2])^{s[1^3]}\\\\otimes S_{k+i}(\\\\Gamma _1[2])^{s[1^3]}\\\\bigoplus {\\\\rm Sym}^2(S_{j/2+k}(\\\\Gamma _1[2])^{s[1^3]})\\\\, .$ Indeed, the group $\\\\mathfrak {S}_3={\\\\rm SL}(2,{\\\\mathbb {Z}}/2{\\\\mathbb {Z}})$ acts on $S_k(\\\\Gamma _1[2])$ and for a form on $\\\\Gamma _2$ with a character the components $f_i, f_i^{\\\\prime }$ of the restriction to $i(\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1)$ are modular forms on $\\\\Gamma _1[2]$ with a character, i.e.\",\n \"they lie in the $s[1^3]$ -isotypical part of $S_k(\\\\Gamma _1[2])$ .\",\n \"The module $\\\\oplus _k S_k(\\\\Gamma _1[2])^{s[1^3]}$ is a module over the ring ${\\\\mathbb {C}}[e_4,e_6]$ of modular forms on $\\\\Gamma _1$ and is generated by the cusp form $\\\\delta =\\\\eta ^{12}$ , a square root of $\\\\Delta \\\\in S_{12}(\\\\Gamma _1)$ .\",\n \"The generating series for the dimensions is now $\\\\sum _{j=2}^{\\\\infty } \\\\dim S_{j,7}(\\\\Gamma _2,\\\\epsilon )\\\\, t^j ={t^6 \\\\over (1-t)(1-t^3)(1-t^4)(1-t^6)}\\\\, .$ Conjecturally we now find an exact sequence $\\\\begin{aligned}0 \\\\rightarrow S_{j,7}(\\\\Gamma _2,\\\\epsilon ) {\\\\rho \\\\over \\\\longrightarrow }\\\\oplus _{i=0}^{j/2-1} S_{j+7-i}(\\\\Gamma _0(4))^{n}\\\\otimes S_{7+i}(\\\\Gamma _0(4))^{n} \\\\,\\\\oplus {\\\\rm Sym}^2 (S_{j/2+7}(\\\\Gamma _0(4))^{n}) &\\\\\\\\\\\\rightarrow K \\\\rightarrow 0\\\\, , &\\\\\\\\\\\\end{aligned}$ where $S_k(\\\\Gamma _0(4))^{n}=S_k(\\\\Gamma _0(4))^{\\\\rm new}$ and the dimension of the cokernel $K$ is now predicted by minus the extrapolation to $k=2$ of the algorithms used in [3] to calculate the dimension of $S_{j,k}(\\\\Gamma _2)$ and which give negative numbers here.\"\n ],\n [\n \"Constructing Modular Forms Using Covariants\",\n \"In the paper [7] we explained how to use invariant theory to construct Siegel modular forms.\",\n \"In this paper we shall make extensive use of the procedure.\",\n \"Let $V$ be the standard representation space of ${\\\\rm GL}(2,{\\\\mathbb {C}})$ with basis $x_1,x_2$ .\",\n \"We consider the space ${\\\\rm Sym}^6(V)$ of binary sextics, where we write an element as $f= \\\\sum _{i=0}^6 a_i \\\\binom{6}{i} x_1^{6-i}x_2^i \\\\, .$ Sometimes we call this expression the universal binary sextic.\",\n \"For a description of invariants and covariants for the action of ${\\\\rm GL}(2,{\\\\mathbb {C}})$ we refer to [7].\",\n \"An invariant can be viewed as a polynomial in the coefficients $a_i$ that is invariant under the action of ${\\\\rm SL}(2,{\\\\mathbb {C}})$ , while a covariant of degree $(a,b)$ can be viewed as a form of degree $a$ in the $a_i$ and degree $b$ in $x_1,x_2$ .\",\n \"If $A[\\\\lambda _1,\\\\lambda _2]$ is an irreducible representation of highest weight $(\\\\lambda _1 \\\\ge \\\\lambda _2)$ of ${\\\\rm GL}(2,{\\\\mathbb {C}})$ embedded equivariantly in ${\\\\rm Sym}^d({\\\\rm Sym}^6(V))$ this defines a covariant of degree $(d,\\\\lambda _1-\\\\lambda _2)$ and it is unique up to a multiplicative non-zero constant.\",\n \"We denote the ring of covariants by ${\\\\mathcal {C}}$ .\",\n \"Clebsch and others constructed in the 19th century generators for this ring.\",\n \"There are 26 generators, 5 invariants and 21 covariants, satisfying many relations.\",\n \"They can be found in the book of Grace and Young [15].\",\n \"For the convenience of the reader we reproduce these here.\",\n \"In the following table $C_{a,b}$ denotes a generator of degree $(a,b)$ .\",\n \"height2pt$a \\\\backslash b$ 0 2 4 6 8 10 12 1 $C_{1,6}$ 2 $C_{2,0}$ $C_{2,4}$ $C_{2,8}$ 3 $C_{3,2}$ $C_{3,6}$ $C_{3,8}$ $C_{3,12}$ 4 $C_{4,0}$ $C_{4,4}$ $C_{4,6}$ $C_{4,10}$ 5 $C_{5,2}$ $C_{5,4}$ $C_{5,8}$ 6 $C_{6,0}$ $C_{6,6}^{(1)}$ $C_{6,6}^{(2)}$ 7 $C_{7,2}$ $C_{7,4}$ 8 $C_{8,2}$ 9 $C_{9,4}$ 10 $C_{10,0}$ $C_{10,2}$ 12 $C_{12,2}$ 15 $C_{15,0}$ A theorem of Gordan says that all these covariants can be constructed explicitly by using so-called transvectants from the universal binary sextic.\",\n \"If ${\\\\rm Sym}^m(V)$ denotes the space of binary quantics of degree $m$ then we define the $k$ th transvectant as follows.\",\n \"It is a map ${\\\\rm Sym}^m(V) \\\\times {\\\\rm Sym}^n(V) \\\\rightarrow {\\\\rm Sym}^{m+n-2k}(V)$ that sends a pair $(f,g)$ to $(f,g)_k=\\\\frac{(m-k)!(n-k)!}{m!n!\",\n \"}\\\\sum _{j=0}^k (-1)^j\\\\binom{k}{j}\\\\frac{\\\\partial ^k f}{\\\\partial x_1^{k-j}\\\\partial x_2^j}\\\\frac{\\\\partial ^k g}{\\\\partial x_1^{j}\\\\partial x_2^{k-j}}\\\\, .$ When $k=1$ , we omit the index: $(f,g)=(f,g)_1$ .\",\n \"The next table gives the construction of the covariants in the preceding table.\",\n \"height2pt 1 $C_{1,6}=f$ 2 $C_{2,0}=(f,f)_6$ $C_{2,4}=(f,f)_4$ $C_{2,8}=(f,f)_2$ 3 $C_{3,2}=(C_{1,6},C_{2,4})_4$ $C_{3,6}=(f,C_{2,4})_2$ $C_{3,8}=(f,C_{2,4})$ $C_{3,12}=(f,C_{2,8})$ 4 $C_{4,0}=(C_{2,4},C_{2,4})_4$ $C_{4,4}=(f,C_{3,2})_2$ $C_{4,6}=(f,C_{3,2})$ $C_{4,10}=(C_{2,8},C_{2,4})$ 5 $C_{5,2}=(C_{2,4},C_{3,2})_2$ $C_{5,4}=(C_{2,4},C_{3,2})$ $C_{5,8}=(C_{2,8},C_{3,2})$ 6 $C_{6,0}=(C_{3,2},C_{3,2})_2$ $C_{6,6}^{(1)}=(C_{3,6},C_{3,2})$ $C_{6,6}^{(2)}=(C_{3,8},C_{3,2})_2$ 7 $C_{7,2}=(f,C_{3,2}^2)_4$ $C_{7,4}=(f,C_{3,2}^2)_3$ 8 $C_{8,2}=(C_{2,4},C_{3,2}^2)_3$ 9 $C_{9,4}=(C_{3,8},C_{3,2}^2)_4$ 10 $C_{10,0}=(f,C_{3,2}^3)_6$ $C_{10,2}=(f,C_{3,2}^3)_5$ 12 $C_{12,2}=(C_{3,8},C_{3,2}^3)_6$ 15 $C_{15,0}=(C_{3,8},C_{3,2}^4)_8$ Let $M$ be the ring of vector-valued Siegel modular forms of degree 2.\",\n \"It is a module over the ring $R$ of scalar-valued Siegel modular forms of degree 2.\",\n \"In [7] we defined maps $M \\\\longrightarrow {\\\\mathcal {C}} {\\\\nu \\\\over \\\\longrightarrow }M_{\\\\chi _{10}}\\\\, ,$ where $M_{\\\\chi _{10}}$ is the localization of $M$ at $\\\\chi _{10}$ .\",\n \"A modular form of weight $(j,k)$ maps to a covariant of degree $(j/2+k,j)$ and a covariant of degree $(a,b)$ is sent to a meromorphic modular form of weight $(b,a-b/2)$ .\",\n \"Under the map $\\\\nu $ the universal binary sextic $f$ is mapped to $\\\\chi _{6,3}/\\\\chi _5$ of weight $(6,-2)$ .\",\n \"Here $\\\\chi _{6,3}$ is a holomorphic form in $S_{6,3}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"The beginning of its Fourier expansion is given in [7].\",\n \"In practice instead of $\\\\nu $ often we use a slightly modified map $\\\\mu : {\\\\mathcal {C}} \\\\longrightarrow M\\\\oplus M_{\\\\epsilon }\\\\, ,$ where $M_{\\\\epsilon }= \\\\oplus M_{j,k}(\\\\Gamma _2,\\\\epsilon )$ , is the $R$ -module of modular forms with a character.\",\n \"Under $\\\\mu $ the universal sextic $f$ is mapped to $\\\\chi _{6,3}$ .\",\n \"Then a covariant maps of degree $(a,b)$ maps to a holomorphic Siegel modular form of weight $(b,6a-b/2)$ and character $\\\\epsilon ^a$ .\",\n \"Remark 4.1 Since $\\\\chi _{6,3}$ vanishes simply at infinity the definition of $\\\\mu $ implies that the image under $\\\\mu $ of a covariant of degree $(a,b)$ vanishes at infinity with order $\\\\ge a$ .\",\n \"Recall that the order of vanishing of $\\\\chi _5$ at infinity is 1.\",\n \"For example, Igusa's generators $E_4,E_6,\\\\chi _{10}, \\\\chi _{12}$ and $\\\\chi _{35}$ of $R$ are up to a non-zero multiplicative constant obtained as $\\\\begin{aligned}&E_4=\\\\mu (75\\\\, C_{4,0} -8\\\\, C_{2,0}^2)/\\\\chi _{10}^2, \\\\quad E_6= \\\\mu (224\\\\, C_{2,0}^3 -1425\\\\, C_{2,0}C_{4,0} -1125 \\\\, C_{6,0})/\\\\chi _{10}^3 \\\\\\\\&\\\\chi _{10}^6 = \\\\mu (C_{\\\\chi _{10}}), \\\\quad \\\\chi _{12}= \\\\mu (C_{2,0}), \\\\quad \\\\chi _{35}= \\\\mu (C_{15,0})/\\\\chi _5^{11}, \\\\\\\\\\\\end{aligned}$ with $C_{\\\\chi _{10}}$ , up to a multiplicative constant equal to the discriminant, given by $768\\\\, C_{2,0}^5-7625 \\\\,C_{4,0}C_{2,0}^3-1875\\\\left(7 \\\\, C_{6,0}C_{2,0}^2 -10 \\\\, C_{4,0}^2C_{2,0}-30 \\\\, C_{6,0}C_{4,0}- 13860 \\\\, C_{10,0}\\\\right) \\\\, .$ Remark 4.2 The first scalar-valued cusp form on $\\\\Gamma _2$ with character is of weight 30 and can be obtained by dividing $\\\\mu (C_{15,0})$ by $\\\\chi _5^{12}$ .\",\n \"Note that we have $M_{j,k}(\\\\Gamma _2,\\\\epsilon )=S_{j,k}(\\\\Gamma _2,\\\\epsilon ).$ (see [17]).\"\n ],\n [\n \"Cusp forms of weight $(j,2)$ on {{formula:3c24a2d5-8cca-456e-8c8a-10623d91c714}} with a character\",\n \"Our conjecture says that $S_{j,2}(\\\\Gamma _2,\\\\epsilon )=(0)$ for $j<12$ .\",\n \"We begin by showing this.\",\n \"Lemma 5.1 For $j=0, 2, 4, 6, 8$ and 10, we have $S_{j,2}(\\\\Gamma _2,\\\\epsilon )=(0).$ We know that $\\\\dim S_{2j,4}(\\\\Gamma _2)=0$ for $j=0,2,4,6,8,10$ .\",\n \"Assume that for one of these values of $j$ there is a non-zero element $f\\\\in S_{j,2}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"Then ${\\\\rm Sym}^2(f)\\\\in S_{2j,4}(\\\\Gamma _2)=(0)$ must be zero.\",\n \"Using the fact that the ring of holomorphic functions on $\\\\mathfrak {H}_2$ is an integral domain, we get a contradiction.\",\n \"The first case where $S_{j,2}(\\\\Gamma _2,\\\\epsilon )$ is predicted to be non-zero is for $j=12$ .\",\n \"In the paper [7] we constructed a modular form $\\\\chi _{12,2}$ in this space using the covariant $C_{3,12}$ associated to $A[15,3]$ occurring in ${\\\\rm Sym}^3({\\\\rm Sym}^6(V))$ , after dividing by the cusp form $\\\\chi _{10}$ .\",\n \"Its Fourier expansion starts with $\\\\chi _{12,2}(\\\\tau )=\\\\left({\\\\begin{matrix}0\\\\\\\\0\\\\\\\\0\\\\\\\\2(R-R^{-1})\\\\\\\\9(R+R^{-1})\\\\\\\\12(R-R^{-1})\\\\\\\\0\\\\\\\\-12(R-R^{-1})\\\\\\\\-9(R+R^{-1})\\\\\\\\-2(R-R^{-1})\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\end{matrix}}\\\\right)Q_1Q_2+\\\\cdots \\\\, ,$ where $Q_1= e^{\\\\pi i \\\\tau _1}$ , $Q_2=e^{\\\\pi i \\\\tau _2}$ and $R=e^{\\\\pi i \\\\tau _{12}}$ for $\\\\tau =\\\\left({\\\\begin{matrix} \\\\tau _1 & \\\\tau _{12} \\\\\\\\ \\\\tau _{12} & \\\\tau _2\\\\\\\\\\\\end{matrix}} \\\\right)$ .\",\n \"By multiplication by $\\\\chi _{6,3}$ we get an injective map $S_{12,2}(\\\\Gamma _2,\\\\epsilon ) \\\\rightarrow S_{18,5}(\\\\Gamma _2)$ and this latter space is 1-dimensional.\",\n \"Corollary 5.2 We have $\\\\dim S_{12,2}(\\\\Gamma _2,\\\\epsilon )=1$ and it is generated by $\\\\chi _{12,2}$ .\",\n \"We compute a few Hecke eigenvalues as described in Section .\",\n \"To compute these Hecke eigenvalues, we used the following Fourier coefficients: $a([1,1,1])^t&=[0, 0, 0, 2, 9, 12, 0, -12, -9, -2, 0, 0, 0]\\\\\\\\a([1,3,3])^t&=[0, 0, 0, -2, -27, -156, -504, -996, -1233, -934, -396, -72, 0]\\\\\\\\a([3,3,3])^t&=[0, 216, 1188, 258, -7749, -12708, 0, 12708, 7749, -258, -1188, -216, 0]\\\\\\\\a([5,5,5])^t&=[0, 0, 0, -106920, -481140, -641520, 0, 641520, 481140, 106920, 0, 0, 0]\\\\\\\\a([1,5,7])^t&=[0, 0, 0, 2, 45, 444, 2520, 9060, 21375, 33046, 32220, 17928, 4320]\\\\\\\\a([7,7,7])^t&=[0, -8208, -45144, -542204, -2101338, -2711496, 0, 2711496, 2101338, 542204, 45144, 8208, 0]\\\\\\\\a([3,9,7])^t&=[0, -72, -1188, -8854, -39339, -115764, -236880, -343884,\\\\\\\\& \\\\qquad \\\\qquad -354141, -253514, -120132, -33912, -4320]\\\\\\\\$ These and a few more (too big to be written here) yield the following eigenvalues: height2pt $p$ 3 5 7 11 13 17 height2pt $\\\\lambda _p$ $-600$ $-53460$ $-369200$ 4084344 $-2845700$ 131681700 together with $\\\\lambda _9=-1090791$ .\",\n \"We find that for the operator $T_p$ these are indeed of the form $\\\\lambda _p(f^{+})+\\\\lambda _p(f^{-})$ with $f^{\\\\pm }$ generators of $S^{\\\\pm }_{14}(\\\\Gamma _0(2))^{\\\\rm new}$ , with $f^{+}(\\\\tau )&=q - 64\\\\, q^2 - 1836\\\\, q^3 + 4096\\\\, q^4 + 3990\\\\, q^5 +117504\\\\, q^6 +\\\\cdots \\\\\\\\f^{-}(\\\\tau )&=q + 64\\\\, q^2 + 1236\\\\, q^3 + 4096\\\\, q^4 - 57450\\\\, q^5 +79104\\\\, q^6 +\\\\cdots \\\\, ,$ while for $T_{p^2}$ we find $\\\\lambda _{p}(f^{+})^2+\\\\lambda _p(f^{+})\\\\lambda _{p}(f^{-})+\\\\lambda _p(f^{-})^2-2\\\\, (p+1)p^j$ .\",\n \"This fits with being a Yoshida lift.\",\n \"Lemma 5.3 We have $ S_{14,2}(\\\\Gamma _2,\\\\epsilon )=(0)=S_{16,2}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"To see that $S_{14,2}(\\\\Gamma _2,\\\\epsilon )=(0)$ we multiply a form in $S_{14,2}(\\\\Gamma _2,\\\\epsilon )$ with $\\\\chi _5$ and we end up in $S_{14,7}(\\\\Gamma _2)$ and this space is generated by a form associated to $C_{1,6}C_{3,8}$ after division by $\\\\chi _5^2$ .\",\n \"Restricting this form $\\\\chi _{14,7}$ to $\\\\mathfrak {H}_1^2$ gives $\\\\sum _{i=0}^{7}(f_i\\\\otimes f_i^{\\\\prime })$ and only the term $f_5 \\\\otimes f_5^{\\\\prime }$ in $S_{16}(\\\\Gamma _1)\\\\otimes S_{12}(\\\\Gamma _1)$ can be non-zero and it is equal to $56\\\\, e_4\\\\Delta \\\\otimes \\\\Delta $ .\",\n \"So it does not vanish along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ , hence $\\\\chi _{14,7}$ is not divisible by $\\\\chi _5$ .\",\n \"We conclude $S_{14,2}(\\\\Gamma _2[2],\\\\epsilon )=(0)$ .\",\n \"For $S_{16,2}(\\\\Gamma _2,\\\\epsilon )$ we multiply by $\\\\chi _{6,3}$ and land in $S_{22,5}(\\\\Gamma _2)$ and this space is zero.\",\n \"Next we deal with the case of weight $(18,2)$ .\",\n \"Proposition 5.4 The space $S_{18,2}(\\\\Gamma _2,\\\\epsilon )$ has dimension 1.\",\n \"First we construct a non-zero element in this space by using the covariant $C=135\\\\,C_{1,6}^{2}C_{4,6}+56\\\\,C_{1,6}C_{2,0}C_{3,12}-270\\\\,C_{2,8}C_{4,10}-930\\\\,C_{3,6}C_{3,12}.$ It occurs in $\\\\rm Sym^6(\\\\rm Sym^6(V))$ and provides a cusp form, $F_C$ , of weight $(18,27)$ on $\\\\Gamma _2$ .\",\n \"The order of vanishing of $F_C$ along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ is 5, so we can divide it by $\\\\chi _{5}^5$ and we get by Remark REF a cusp form, denoted $\\\\chi _{18,2}$ , of weight $(18,2)$ on $\\\\Gamma _2$ with character.\",\n \"Again we multiply by $\\\\chi _{5}$ and land in $S_{18,7}(\\\\Gamma _2)$ .\",\n \"This space is 2-dimensional and we can construct a basis using the following covariants $C_1=C_{3,12}(8\\\\, C_{1,6}C_{2,0}-75\\\\, C_{3,6})\\\\quad \\\\text{ and} \\\\quad C_2= C_{1,6}^2 C_{4,6}-2\\\\, C_{2,8}C_{4,10}-3\\\\, C_{3,6}C_{3,12}.$ They occur in $\\\\rm Sym^6(\\\\rm Sym^6(V))$ and provide two cusp forms, $F_{C_i}$ , of weight $(18,27)$ on $\\\\Gamma _2$ .\",\n \"Each cusp form $F_{C_i}$ vanishes with order 4 along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ , so we can divide it by $\\\\chi _{5}^4$ and get a cusp form, $\\\\chi _{18,7}^{(i)}$ , of weight $(18,7)$ on $\\\\Gamma _2$ .\",\n \"The cusp forms $\\\\chi _{18,7}^{(1)}$ and $\\\\chi _{18,7}^{(2)}$ are $\\\\mathbb {C}$ -linearly independent as can be read off from the first terms of their Fourier expansions and the pullbacks to $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ are of the form $\\\\sum _{r=0}^9 f_r \\\\otimes f_r^{\\\\prime }$ with only non-zero terms for $r=5$ and these are $216\\\\,e_4^2\\\\Delta \\\\otimes \\\\Delta $ and $48\\\\, e_4^2\\\\Delta \\\\otimes \\\\Delta $ .\",\n \"Up to a non-zero scalar there is only one non-trivial linear combination, that vanishes along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ and that gives a non-zero form in $S_{18,2}(\\\\Gamma _2,\\\\epsilon )$ after division by $\\\\chi _5$ .\",\n \"Proposition 5.5 The space $S_{20,2}(\\\\Gamma _2,\\\\epsilon )$ has dimension 1.\",\n \"We construct a non-zero form in this space by taking the covariant $\\\\begin{aligned}C=&224\\\\,C_{1,6}^{2}C_{5,8}+312\\\\,C_{1,6}C_{2,4}C_{4,10}-560\\\\,C_{1,6}C_{2,8}C_{4,6}\\\\\\\\&-108\\\\,C_{1,6}C_{3,6}C_{3,8}+728\\\\,C_{2,0}C_{2,8}C_{3,12}-1235\\\\,C_{2,4}^{2}C_{3,12}.\\\\end{aligned}$ occurring in ${\\\\rm Sym}^7({\\\\rm Sym}^6(V))$ and providing a cusp form, $F_C$ , of weight $(20,32)$ on $\\\\Gamma _2$ with character.\",\n \"The order of vanishing of $F_C$ along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ is 6, so we can divide it by $\\\\chi _{5}^6$ and we get a cusp form, $\\\\chi _{20,2}$ , of weight $(20,2)$ on $\\\\Gamma _2$ with character.\",\n \"In a similar way we construct a basis of the space $S_{20,7}(\\\\Gamma _2)$ by taking the covariants $\\\\begin{aligned}C_1&=480\\\\,C_{1,6}^{2}C_{5,8}-180\\\\,C_{1,6}C_{3,6}C_{3,8}+728\\\\,C_{2,0}C_{2,8}C_{3,12}-1315\\\\,C_{2,4}^{2}C_{3,12},\\\\\\\\C_2&=80\\\\,C_{1,6}^{2}C_{5,8}-80\\\\,C_{1,6}C_{2,8}C_{4,6}+104\\\\,C_{2,0}C_{2,8}C_{3,12}-125\\\\,C_{2,4}^{2}C_{3,12},\\\\\\\\C_3&=80\\\\,C_{1,6}^{2}C_{5,8}-40\\\\,C_{1,6}C_{2,4}C_{4,10}+56\\\\,C_{2,0}C_{2,8}C_{3,12}-55\\\\,C_{2,4}^{2}C_{3,12}.\\\\end{aligned}$ which provide cusp forms with character of weight $(20,32)$ and these are divisible by $\\\\chi _5^5$ and thus give cusp forms of weight $(20,7)$ generating $S_{20,7}(\\\\Gamma _2)$ .\",\n \"By restriction to the diagonal one sees that there is just a 1-dimensional space of forms vanishing on the diagonal.\",\n \"The case of weight $(24,2)$ is dealt with in a similar way.\",\n \"Proposition 5.6 We have $\\\\dim S_{24,2}(\\\\Gamma _2[2],\\\\epsilon )=2$ .\",\n \"We know that $\\\\dim S_{24,7}(\\\\Gamma _2)=5$ and we can construct a basis using the procedure described in Section .\",\n \"In the case at hand we have $A[39,15]$ occurring in ${\\\\rm Sym}^9({\\\\rm Sym}^6(V))$ with multiplicity 13 and this gives a subspace of $S_{24,42}(\\\\Gamma _2,\\\\epsilon )$ of dimension 13.\",\n \"One checks that there is a 5-dimensional subspace of forms vanishing with multiplicity 7 along the diagonal and dividing by $\\\\chi _5^7$ gives a 5-dimensional subspace of $S_{24,7}(\\\\Gamma _2)$ , hence the whole space.\",\n \"Again one checks that there is a 2-dimensional space of forms vanishing on the diagonal and we can divide these forms by $\\\\chi _5$ .\",\n \"So the two generators of $S_{24,2}(\\\\Gamma _2,\\\\epsilon )$ are defined by the covariants $C_1$ and $C_2$ given respectively by $\\\\begin{aligned}&-499408\\\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}-1505385\\\\,C_{1,6}^{3}C_{6,6}^{(1)}-14727825\\\\,C_{1,6}^{2}C_{2,4}C_{5,8}+6916455\\\\,C_{1,6}^{2}C_{2,8}C_{5,4}\\\\\\\\&-5728590\\\\,C_{1,6}^{2}C_{3,12}C_{4,0}+6972210\\\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}+4257120\\\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}\\\\\\\\&+2182950\\\\,C_{2,8}^{2}C_{5,8}+11708550\\\\,C_{2,8}C_{3,6}C_{4,10}+595350\\\\,C_{2,8}C_{3,12}C_{4,4}+35171325\\\\,C_{3,6}^{2}C_{3,12}\\\\\\\\&-400950\\\\,C_{3,8}^{3}\\\\\\\\\\\\end{aligned}$ and $\\\\begin{aligned}&-42235648\\\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}+4434583545\\\\,C_{1,6}^{3}C_{6,6}^{(1)}+580982220\\\\,C_{1,6}^{3}C_{6,6}^{(2)}\\\\\\\\&+4919972400\\\\,C_{1,6}^{2}C_{2,4}C_{5,8}+4827362400\\\\,C_{1,6}^{2}C_{3,12}C_{4,0}-3504891600\\\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}\\\\\\\\&+1245336960\\\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}-4131252720\\\\,C_{2,8}^{2}C_{5,8}-24904998720\\\\,C_{2,8}C_{3,6}C_{4,10}\\\\\\\\&-281640240\\\\,C_{2,8}C_{3,12}C_{4,4}-58751907480\\\\,C_{3,6}^{2}C_{3,12}+1375354080\\\\,C_{3,8}^{3}\\\\, .\\\\end{aligned}$ The order of vanishing of $F_{C_i}$ along $\\\\mathfrak {H}_1\\\\times \\\\mathfrak {H}_1$ is 8, so we can divide it by $\\\\chi _{5}^8$ and we get two cusp forms, $\\\\chi _{24,2}^{(i)}$ , ($i=1,2$ ) of weight $(24,2)$ on $\\\\Gamma _2$ with character.\",\n \"We set $\\\\chi _{24,2}^{(1)}=-12150\\\\, F_{C_1}/\\\\chi _5^8$ and $\\\\chi _{24,2}^{(2)}=-675(5368\\\\, F_{C_1} +5 \\\\, F_{C_2})/31528 \\\\chi _5^8$ .\",\n \"Then their Fourier expansions are given by $\\\\chi _{24,2}^{(1)}=\\\\left({\\\\begin{matrix}0\\\\\\\\0\\\\\\\\0\\\\\\\\104(R-R^{-1})\\\\\\\\1092(R+R^{-1})\\\\\\\\3640(R-R^{-1})\\\\\\\\0\\\\\\\\-27678(R-R^{-1})\\\\\\\\-58905(R+R^{-1})\\\\\\\\-2916(R-R^{-1})\\\\\\\\148470(R+R^{-1})\\\\\\\\190778(R-R^{-1})\\\\\\\\0\\\\\\\\\\\\vdots \\\\end{matrix}}\\\\right)Q_1Q_2+\\\\cdots \\\\, ,\\\\quad \\\\chi _{24,2}^{2}(\\\\tau )=\\\\left({\\\\begin{matrix}0\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\0\\\\\\\\2(R-R^{-1})\\\\\\\\17(R+R^{-1})\\\\\\\\60(R-R^{-1})\\\\\\\\110(R+R^{-1})\\\\\\\\98(R-R^{-1})\\\\\\\\0\\\\\\\\\\\\vdots \\\\end{matrix}}\\\\right)Q_1Q_2+\\\\cdots \\\\, ,$ where $Q_1=e^{i\\\\pi \\\\tau _1}$ , $Q_2=e^{i\\\\pi \\\\tau _2}$ , $R=e^{i\\\\pi \\\\tau _{12}}$ .\",\n \"The action of $\\\\iota =({\\\\begin{matrix} a & 0 \\\\\\\\ 0 & d\\\\\\\\ \\\\end{matrix}})\\\\in \\\\Gamma _2$ with $a=d= ({\\\\begin{matrix} 0 & 1 \\\\\\\\ 1 & 0\\\\\\\\ \\\\end{matrix}})$ implies that the $i$ th coordinate is equal to $(-1)^{k+1}$ times the $(j+1-i)$ th coordinate, which gives the non-displayed coordinates .\",\n \"A Hecke eigenbasis of the space $S_{24,2}(\\\\Gamma _2,\\\\epsilon )$ is: $F_1=&\\\\,439 \\\\chi _{24,2}^{(1)} +(114847+650\\\\sqrt{106705})\\\\, \\\\chi _{24,2}^{(2)}\\\\\\\\F_2=& \\\\, 439\\\\chi _{24,2}^{(1)} +(114847-650\\\\sqrt{106705})\\\\, \\\\chi _{24,2}^{(2)}$ with eigenvalues height2pt $p$ $\\\\lambda _p(F_1)$ $\\\\lambda _{p^2}(F_1)$ height2pt 3 $287880-4800\\\\sqrt{106705}$ $545747143689-2293459200\\\\sqrt{106705}$ 5 $711981900+1555200\\\\sqrt{106705}$ – 7 $-41070905840+92534400\\\\sqrt{106705}$ – 11 $ -10344705071976 + 4819953600\\\\sqrt{106705}$ – in perfect agreement with the eigenforms being Yoshida lifts.\",\n \"Indeed a basis of the space $S_{26}(\\\\Gamma _0[2])^{\\\\text{new}}$ is given by $f&=q - 4096\\\\, q^2 + 97956\\\\, q^3 + 16777216\\\\, q^4 + 341005350\\\\, q^5- 401227776\\\\, q^6 +\\\\cdots \\\\\\\\g&=q + 4096\\\\, q^2 + (2048-a/2)\\\\, q^3 + 16777216\\\\, q^4 +(431848374+162\\\\, a)\\\\, q^5 +\\\\cdots \\\\\\\\g^{\\\\prime }&=q + 4096\\\\, q^2 + (2048+a/2)\\\\, q^3 + 16777216\\\\, q^4 + (431848374-162\\\\, a)\\\\, q^5 +\\\\cdots \\\\, ,$ where $ a=-375752+9600\\\\sqrt{106705} $ , and $f,g^{\\\\prime } \\\\in S^{-}_{26}$ and $g \\\\in S_{26}^{+}$ .\",\n \"Then we check for example that $\\\\lambda _5(F_1)&=711981900+1555200\\\\sqrt{106705}=a_5(f)+a_5(g)\\\\\\\\\\\\lambda _5(F_2)&=711981900-1555200\\\\sqrt{106705}=a_5(f)+a_5(g^{\\\\prime }).$ Proposition 5.7 One has $\\\\dim S_{26,2}(\\\\Gamma _2,\\\\epsilon )=1=\\\\dim S_{28,2}(\\\\Gamma _2,\\\\epsilon )$ and $\\\\dim S_{30,2}(\\\\Gamma _2,\\\\epsilon )=2$ .\",\n \"The proof of this proposition is similar to the above.\",\n \"For the first statement we consider the space $S_{26,7}(\\\\Gamma _2)$ which has dimension 6 and construct a basis of this space using covariants associated to $A[43,17]$ in ${\\\\rm Sym}^{10}({\\\\rm Sym}^6(V))$ which occurs with multiplicity 17, thus giving rise to a 17-dimensional subspace of $S_{26,47}(\\\\Gamma _2)$ .\",\n \"By restricting along the diagonal one checks that there is a 6-dimensional subspace of cusp forms divisible by $\\\\chi _5^8$ leading to the construction of $S_{26,7}(\\\\Gamma _2)$ .\",\n \"Again by restricting to the diagonal one sees that there is exactly a 1-dimensional subspace of this space that vanish along the diagonal.\",\n \"By dividing by $\\\\chi _5$ we thus find the space $S_{26,2}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"For weight $(28,2)$ we now use the representation $A[47,19]$ that occurs with multiplicity 23 in ${\\\\rm Sym}^{11}({\\\\rm Sym}^6(V))$ and leading to a 23-dimensional subspace of $S_{28,52}(\\\\Gamma _2)$ in which we find a 7-dimensional subspace of forms divisible by $\\\\chi _5^9$ and division gives forms that generate $S_{28,7}(\\\\Gamma _2)$ .\",\n \"In this space the subspace of forms divisible by $\\\\chi _5$ is of dimension 1, proving our claim.\",\n \"In the case of weight $(30,7)$ the 9-dimensional space $S_{30,7}$ is constructed using covariants resulting from $A[51,21]$ that occurs with multiplicity 31 in ${\\\\rm Sym}^{12}({\\\\rm Sym}^6(V))$ leading to a space of dimension 31 of cusp forms of weight $(30,57)$ .\",\n \"There is a 9-dimensional subspace of cusp forms divisible by $\\\\chi _5^{10}$ and we thus generate $S_{30,7}(\\\\Gamma _2)$ .\",\n \"It turns out that there is a 2-dimensional subspace of forms divisible by $\\\\chi _5$ and this proves the claim.\",\n \"For all the cases treated we can check our construction by verifying that the Hecke eigenvalues for $p=3,5,7,11,13, 17$ agree with the forms being Yoshida lifts like we indicated for $j=12$ and $j=24$ .\",\n \"In a forthcoming paper ([8]) we shall use the relation with covariants to describe modules of forms with a character.\"\n ],\n [\n \"Modular Forms of Weight $(j,2)$ on {{formula:8ac6fb49-ab1c-4eac-b5ae-da9c6611aa41}}\",\n \"In this section we explain how we checked that $S_{j,2}(\\\\Gamma _2)=(0)$ for $j\\\\le 52$ .\",\n \"We begin with a simple lemma.\",\n \"Recall that we have maps $M \\\\rightarrow {\\\\mathcal {C}} {\\\\mu \\\\over \\\\longrightarrow } M$ .\",\n \"Lemma 6.1 Let $f \\\\in M_{j,k}(\\\\Gamma _2)$ .\",\n \"Then there exists a covariant $c_f$ of degree $(d,j)$ with $d\\\\le j/2+k$ such that $f=\\\\nu (c_f)=\\\\mu (c_f)/\\\\chi _5^d$ .\",\n \"If $f$ is a cusp form then there is a covariant $c_f^{\\\\prime }$ of degree $\\\\le j/2+k-10$ such that $f=\\\\mu (c_f^{\\\\prime })/\\\\chi _5^r$ for some $r$ .\",\n \"The first statement follows directly from [7].\",\n \"If $f$ is a cusp form then the covariant it defines vanishes on the discriminant locus.\",\n \"But then the covariant $c_f$ is divisible by the discriminant, and $\\\\mu (c_f)$ by $\\\\chi _5^{d+2}$ .\",\n \"This makes it possible to check the existence of a non-zero form $f \\\\in S_{j,2}(\\\\Gamma _2)$ by checking whether the forms of weight $(j,2+5d)$ provided via $\\\\mu $ by the non-zero covariants of degree $(d,j)$ with $d\\\\le j/2-8$ are divisible by $\\\\chi _5^{d}$ .\",\n \"We applied this for values of $j\\\\le 52$ using the covariants of degree $d\\\\le 18$ .\",\n \"For smaller values of $j$ other methods of showing that $S_{j,2}(\\\\Gamma _2)=(0)$ are available.\",\n \"We sketch some methods below.\",\n \"In this way we checked that $S_{j,2}(\\\\Gamma _2)=(0)$ for $j\\\\le 52$ .\",\n \"Another method is to construct a basis of $S_{j,7}(\\\\Gamma _2,\\\\epsilon )$ by using covariants.\",\n \"We then check the divisibility by $\\\\chi _5$ of elements in $S_{j,7}(\\\\Gamma _2,\\\\epsilon )$ by restricting the modular forms in this space to the diagonal.\",\n \"As an illustration we give the proof for the case $j=24$ .\",\n \"We construct a basis of the 9-dimensional space $S_{24,7}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"For this we use the covariants associated to the $A[54,30]$ -isotypical component of ${\\\\rm Sym}^{14}({\\\\rm Sym}^6(V))$ .\",\n \"The representation $A[54,30]$ occurs with multiplicity 65 and leads to a 65-dimensional subspace of modular forms of weight $(24,72)$ on $\\\\Gamma _2$ .\",\n \"By restricting to $\\\\mathfrak {H}_1 \\\\times \\\\mathfrak {H}_1$ we can check that there exists a 9-dimensional subspace of cusp forms that are divisible by $\\\\chi _5^{13}$ .\",\n \"This leads to a basis of $S_{24,7}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"We then check by restriction to $\\\\mathfrak {H}_1 \\\\times \\\\mathfrak {H}_1$ again that there is no non-trivial element in this space that is divisible by $\\\\chi _5$ .\",\n \"This proves the result for $j=24$ .\",\n \"We carried this out for all the cases $j\\\\le 52$ and thus proved Theorem REF .\",\n \"Sometimes there are other and easier ways to eliminate cases.\",\n \"For example, by restricting a modular form of weight $(j,2)$ to the diagonal we get an element $\\\\sum _{i=0}^{j/2} f_i \\\\otimes f_i^{\\\\prime }\\\\in \\\\oplus _{i=0}^{j/2} S_{j+2-i}(\\\\Gamma _1)\\\\otimes S_{2+i}(\\\\Gamma _1)$ .\",\n \"If $j<24$ , $j\\\\ne 20$ the spaces in question are zero.\",\n \"Therefore a form $f\\\\in S_{j,2}(\\\\Gamma _2)$ will vanish on $\\\\mathfrak {H}_1 \\\\times \\\\mathfrak {H}_1$ .\",\n \"But then $f/\\\\chi _5$ will be a holomorphic modular form of weight $(j,-3)$ and this has to be zero.\",\n \"So $f=0$ for $j\\\\le 18$ and $j=22$ .\",\n \"As yet another example of eliminating cases we give a somewhat different argument for $j=26$ .\",\n \"We write elements of $F\\\\in S_{j,k}(\\\\Gamma _2)$ as vectors $F=(F_0,\\\\ldots ,F_j)^t$ with the $F_i$ holomorphic functions on $\\\\mathcal {H}_2$ , that is, in a module of rank 27 over the ring ${\\\\mathcal {F}}$ of holomorphic functions on $\\\\mathcal {H}_2$ .\",\n \"Take a basis $s_1,\\\\ldots ,s_3$ of $S_{26,6}(\\\\Gamma _2)$ and a basis $s_{4},\\\\ldots ,s_{12}$ of $S_{26,8}(\\\\Gamma _2)$ .\",\n \"If there exists a non-zero form $f$ of weight $(26,2)$ then the vectors $E_4\\\\, f$ and $E_6\\\\, f$ are linearly dependent and thus the exterior product $s_1\\\\wedge \\\\cdots \\\\wedge s_{12}$ must vanish.\",\n \"By calculating bases of $S_{26,6}(\\\\Gamma _2)$ and $S_{26,8}(\\\\Gamma _2)$ one can check that this exterior product does not vanish.\",\n \"So $S_{26,2}(\\\\Gamma _2)=(0)$ .\"\n ],\n [\n \"Other Small Weights\",\n \"We begin with an elementary argument that shows that $S_{j,k}(\\\\Gamma _2[2])=(0)$ for $j\\\\le 8$ and $k\\\\le 2$ .\",\n \"Proposition 7.1 For $j\\\\le 8$ and $k\\\\le 2$ we have $\\\\dim S_{j,k}(\\\\Gamma _2[2])=0$ .\",\n \"We need to deal with the cases $j$ even and $k=1$ and $k=2$ only since for other values $S_{j,k}(\\\\Gamma _2[2])$ vanishes.\",\n \"We restrict to the ten components of the Humbert surface $H_1$ in $\\\\Gamma _2[2]\\\\backslash \\\\mathfrak {H}_2$ , one component of which is given by the diagonal $\\\\tau _{12}=0$ .\",\n \"The group $\\\\mathfrak {S}_6$ acts transitively on these ten components.\",\n \"The stabilizer inside $\\\\mathfrak {S}_6$ of a component of $H_1$ is an extension of $\\\\mathfrak {S}_3\\\\times \\\\mathfrak {S}_3$ by ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ .\",\n \"By restricting a modular form $f \\\\in S_{j,1}(\\\\Gamma _2[2])$ to a component we get an element of $\\\\oplus _{r=0}^{j} S_{j+1-r}(\\\\Gamma _1[2])\\\\otimes S_{1+r}(\\\\Gamma _1[2])$ and for $f\\\\in S_{j,2}(\\\\Gamma _2[2])$ we get an element of $\\\\oplus _{r=0}^{j} S_{j+2-r}(\\\\Gamma _1[2])\\\\otimes S_{2+r}(\\\\Gamma _1[2]) \\\\, .$ For $j\\\\le 8$ and $k=1$ and for $j<8$ and $k=2$ these spaces are zero.\",\n \"Thus a form $f\\\\in S_{j,2}(\\\\Gamma _2[2])$ restricts to zero on all irreducible components of $H_1$ , hence is divisible by $\\\\chi _{5}$ , and so $f$ must be zero.\",\n \"For $j=8$ and $k=2$ the restriction to $H_1$ gives an injective $\\\\mathfrak {S}_6$ -equivariant map $S_{8,2}(\\\\Gamma _2[2]) \\\\rightarrow \\\\oplus _{i=1}^{10}\\\\, {\\\\rm Sym}^2 S_6(\\\\Gamma _1[2]) \\\\, ,$ where the action on the right is the induced representation from the extension of $\\\\mathfrak {S}_3 \\\\times \\\\mathfrak {S}_3$ by ${\\\\mathbb {Z}}/2{\\\\mathbb {Z}}$ to $\\\\mathfrak {S}_6$ .\",\n \"Now $S_6(\\\\Gamma _1[2])$ is 1-dimensional and of type $s[1^3]$ and we check that the representation of $\\\\mathfrak {S}_6$ on the 10-dimensional space $\\\\oplus _{i=1}^{10} \\\\rm Sym^2 S_6(\\\\Gamma _1[2])$ is of type s[6]+s[4,2].\",\n \"Since $S_{8,2}(\\\\Gamma _2)=(0)$ , we conclude that only $S_{8,2}(\\\\Gamma _2[2])^{s[4,2]}$ can be non-zero.\",\n \"If $S_{8,2}(\\\\Gamma _2[2])^{s[4,2]}$ is non-zero, then $S_{8,2}(\\\\Gamma _0[2])$ is non-zero (see [10]).\",\n \"By restricting we get an element in a similar decomposition as before but with $\\\\Gamma _1[2]$ replaced by $\\\\Gamma _0[2]$ .\",\n \"As we know that all theses spaces are zero, we can divide by $\\\\chi _5$ .\",\n \"This contradiction concludes our claim.\",\n \"More generally we have Proposition 7.2 For $j<12$ we have $S_{j,2}(\\\\Gamma _2[2])=(0)$ .\",\n \"The space $S_{j,2}(\\\\Gamma _2[2])$ is defined over ${\\\\mathbb {Q}}$ .\",\n \"All the cusps of $\\\\Gamma _2[2]$ are defined over ${\\\\mathbb {Q}}$ and the action of $\\\\mathfrak {S}_6$ is defined over ${\\\\mathbb {Q}}$ .\",\n \"The q-expansion principle says that a modular form in $S_{j,k}(\\\\Gamma _2[2])$ with $k\\\\ge 3$ is defined over ${\\\\mathbb {Q}}$ if its Fourier coefficients at all cusps are defined over ${\\\\mathbb {Q}}$ , see [19] and [13].\",\n \"We apply this to $f\\\\chi _{10}$ with $f \\\\in S_{j,2}(\\\\Gamma _2[2])$ defined over ${\\\\mathbb {Q}}$ and we conclude that the Fourier coefficients of $\\\\sigma (f\\\\chi _{10})$ with $\\\\sigma \\\\in \\\\mathfrak {S}_6$ are real, hence also those of $\\\\sigma (f)$ and if $f\\\\ne 0$ we find by looking at the `first' non-zero term in a Fourier expansion that $\\\\sum _{\\\\sigma \\\\in \\\\mathfrak {S}_6} \\\\sigma (f)^2$ is non-zero and because of $\\\\sigma (f^2)=\\\\sigma (f)^2$ also invariant under $\\\\mathfrak {S}_6$ .\",\n \"Thus it defines a non-zero element of $S_{2j,4}(\\\\Gamma _2)$ .\",\n \"So $S_{j,2}(\\\\Gamma _2[2])$ implies $S_{2j,4}(\\\\Gamma _2)\\\\ne (0)$ .\",\n \"But we know that $S_{2j,4}(\\\\Gamma _2)=(0)$ for $j<12$ .\",\n \"Remark 7.3 Note that the argument of the proof shows that our conjecture on the vanishing of $S_{j,2}(\\\\Gamma _2)$ for all $j$ implies the vanishing of $S_{j,1}(\\\\Gamma _2[2])$ for all $j$ .\",\n \"In order to put our evidence for the vanishing of $S_{j,2}(\\\\Gamma _2)$ in perspective we show a small table that gives for each value of $k$ the smallest $j_0$ such that $\\\\dim S_{j_0,k}(\\\\Gamma _2)\\\\ne 0$ .\",\n \"height2pt $k$ 3 4 5 6 7 8 height2pt $j_0$ 36 24 18 12 12 6 We can easily construct the generators of the corresponding spaces.\",\n \"In [9] we constructed a generator $\\\\chi _{6,3}$ of $S_{6,3}(\\\\Gamma _2,\\\\epsilon )$ and above we gave the generator $\\\\chi _{12,2}$ of $S_{12,2}(\\\\Gamma _2,\\\\epsilon )$ .\",\n \"The modular forms $\\\\chi _{12,2}^2$ , $\\\\chi _{6,3}\\\\chi _{12,2}$ , $\\\\chi _{6,3}^2$ , $\\\\chi _5\\\\chi _{12,2}$ and $\\\\chi _5\\\\chi _{6,3}$ give the generators for $k=4,\\\\ldots ,8$ .\",\n \"We end by constructing a generator of $S_{36,3}(\\\\Gamma _2)$ ; the non-vanishing of this space plays a role in the appendix.\",\n \"We look in ${\\\\rm Sym}^{11}({\\\\rm Sym}^6(V))$ and at $A[51,15]$ occuring with multiplicity 17 there.\",\n \"We find the covariant $297 \\\\, C_{1,6}^2C_{3,8}^3 -8316 \\\\, C_{1,6}C_{3,8}C_{3,12}C_{4,10}+4116 \\\\, C_{1,6}C_{3,12}^2C_{4,6} -5488 \\\\, C_{2,0}C_{3,12}^3+9030 \\\\, C_{2,4}C_{3,8}C_{3,12}^2$ giving a form $f$ of weight $(36,48)$ that is divisible by $\\\\chi _5^{9}$ and $f/\\\\chi _5^9$ generates $S_{36,3}(\\\\Gamma _2)$ .\",\n \"As a check we note that the Fourier coefficient at $n=[1,1,1]$ is of the form $[0, 0, 0, 0, 0, 0, 0, 32/6089428125, 464/6089428125, \\\\ldots ]$ and the coefficient at $n=[5,5,5]$ is $[0, 0, 0, 0, 0, 0, 0, -8687121398144/81192375, -125963260273088/81192375,\\\\ldots ]$ giving the eigenvalue for the Hecke operator $T_5$ as their ratio $ -20360440776900$ , in agreement with the value given in [2].\"\n ],\n [\n \"by Gaëtan Chenevier\",\n \"Let $j$ and $k$ be integers with $j\\\\ge 0$ , and $\\\\Gamma \\\\subset {\\\\rm Sp}_4(\\\\mathbb {Z})$ a congruence subgroup.\",\n \"Recall that $S_{j,k}(\\\\Gamma )$ denotes the space of cuspidal Siegel modular forms for the subgroup $\\\\Gamma $ with values in the representation ${\\\\rm Sym}^j\\\\, \\\\otimes \\\\,\\\\det ^k$ of ${\\\\rm GL}_2(\\\\mathbb {C})$ .\",\n \"We first consider the full Siegel modular group $\\\\Gamma _2 = {\\\\rm Sp}_4(\\\\mathbb {Z})$ and provide alternative proofs of the following results : Proposition 1.1 We have ${S}_{j,1}(\\\\Gamma _2)=0$ for any $j$ , and ${S}_{j,2}(\\\\Gamma _2)=0$ for $j \\\\le 38$ .\",\n \"The vanishing of ${S}_{j,2}(\\\\Gamma _2)$ for all $j\\\\le 52$ is also proved in this paper by Cléry and van der Geer (Theorem REF ).\",\n \"The vanishing of ${S}_{j,1}(\\\\Gamma _2)$ was at least known to Ibukiyama, who asserts in [16] that it is a consequence of the vanishing of all Jacobi forms of weight 1 for ${\\\\rm SL}_2(\\\\mathbb {Z})$ proven by Skoruppa [23].\",\n \"Here we shall rather use automorphic representation theoretic methods.\",\n \"First we need to fix some notations and make some preliminary remarks.\",\n \"We denote by $r : {\\\\rm Sp}_4(\\\\mathbb {C}) \\\\rightarrow {\\\\rm GL}_4(\\\\mathbb {C})$ the tautological inclusion.\",\n \"For a real reductive Lie group $H$ we shall denote the infinitesimal character of the Harish-Chandra module $U$ by ${\\\\rm inf}\\\\, U$ .\",\n \"In the case $H={\\\\rm GL}_n(\\\\mathbb {R})$ (resp.\",\n \"$H={\\\\rm PGSp}_4(\\\\mathbb {R})$ ), and following Harish-Chandra and Langlands, ${\\\\rm inf} \\\\, U$ may be viewed in a canonical way as a semisimple conjugacy class in the Lie algebra $\\\\mathfrak {h}=\\\\mathfrak {gl}_n(\\\\mathbb {C})$ (resp.\",\n \"$\\\\mathfrak {h}=\\\\mathfrak {sp}_4(\\\\mathbb {C})$ ).\",\n \"In both cases we may and shall identify this conjugacy class with the multiset of its eigenvalues in the natural representation of $\\\\mathfrak {h}$ .\",\n \"We denote by ${\\\\rm W}_\\\\mathbb {R}$ the Weil group of $\\\\mathbb {R}$ (a certain extension of $\\\\mathbb {Z}/2\\\\mathbb {Z}$ by $\\\\mathbb {C}^\\\\times $ ), and for any integer $w$ we define ${\\\\rm I}_{w}$ as the 2-dimensional representation of ${\\\\rm W}_{\\\\mathbb {R}}$ induced from the unitary character $z \\\\mapsto (z/|z|)^{w}$ of $\\\\mathbb {C}^\\\\times $ .\",\n \"(a) As we have ${S}_{j,k}(\\\\Gamma _2)=0$ for any odd $j$ or for $k\\\\le 0$ , we may once and for all assume $j \\\\equiv 0 \\\\bmod 2$ and $k>0$ .\",\n \"As is well-known, for any such $(j,k)$ there is an irreducible unitary Harish-Chandra module for ${\\\\rm PGSp}_4(\\\\mathbb {R})$ , unique up to isomorphism, generated by a highest-weight vector of $K$ -type ${\\\\rm Sym}^j \\\\,\\\\otimes \\\\det ^k$ .\",\n \"This module, that we shall denote by ${\\\\rm U}_{j,k}$ , is a holomorphic discrete series for $k \\\\ge 3$ , a limit of holomorphic discrete series for $k=2$ , and non-tempered for $k=1$ ; it is non-generic in all cases.\",\n \"We have ${\\\\rm inf}\\\\, {\\\\rm U}_{j,k} =\\\\lbrace \\\\frac{j+2k-3}{2}, \\\\,\\\\frac{j+1}{2}, \\\\,-\\\\frac{j+1}{2}, \\\\,-\\\\frac{j+2k-3}{2}\\\\rbrace $ .\",\n \"More precisely, if $\\\\varphi : {\\\\rm W}_\\\\mathbb {R}\\\\rightarrow {\\\\rm Sp}_4(\\\\mathbb {C})$ denotes the Langlands parameter of ${\\\\rm U}_{j,k}$ , then we have $r \\\\circ \\\\varphi \\\\simeq {\\\\rm I}_{j+2k-3} \\\\oplus {\\\\rm I}_{j+1}$ for $k>1$ , and $r \\\\circ \\\\varphi \\\\simeq {\\\\rm I}_{j} \\\\otimes |.|^{1/2}\\\\oplus {\\\\rm I}_{j} \\\\otimes |.|^{-1/2}$ for $k=1$ (see e.g.\",\n \"[22] for a survey of those properties, and the references therein).\",\n \"The relevance of ${\\\\rm U}_{j,k}$ here is that if $\\\\pi $ is a cuspidal automorphic representation of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ generated by an element of ${S}_{j,k}(\\\\Gamma _2)$ , then the Archimedean component $\\\\pi _\\\\infty $ of $\\\\pi $ is isomorphic to ${\\\\rm U}_{j,k}$ .\",\n \"The other important property of $\\\\pi $ is that $\\\\pi _p$ is unramified for each prime $p$ (i.e.\",\n \"admits non-zero invariants under ${\\\\rm PGSp}_4(\\\\mathbb {Z}_p)$ ).\",\n \"As ${\\\\rm PGSp}_4$ is isomorphic to the split classical group ${\\\\rm SO}_5$ over $\\\\mathbb {Z}$ , we may apply Arthur's theory [1] to such a $\\\\pi $ .\",\n \"(b) One of the main results of Arthur [1] associates to any discrete automorphic representation $\\\\pi $ of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ a unique isobaric automorphic representation $\\\\pi ^{\\\\rm GL}$ of ${\\\\rm GL}_4$ over $\\\\mathbb {Q}$ , characterized by the following property : for any prime $p$ such that $\\\\pi _p$ is unramified, then $(\\\\pi ^{\\\\rm GL})_p$ is unramified as well and its Satake parameter is the image of the one of $\\\\pi _p$ under the map $r$ .\",\n \"The infinitesimal character of $(\\\\pi ^{\\\\rm GL})_\\\\infty $ is the image of ${\\\\rm inf}\\\\, \\\\pi _\\\\infty $ under the derivative of $r$ , namely $\\\\mathfrak {sp}_4(\\\\mathbb {C}) \\\\rightarrow \\\\mathfrak {gl}_4(\\\\mathbb {C})$ .\",\n \"Moreover, there is a unique collection of distinct triplets $(d_i,n_i,\\\\pi _i)_{i \\\\in I}$ , with integers $d_i,n_i \\\\ge 1$ and cuspidal selfdual automorphic representations $\\\\pi _i$ of ${\\\\rm GL}_{n_i}$ with $\\\\pi ^{\\\\rm GL} \\\\,\\\\simeq \\\\,\\\\boxplus _{i \\\\in I} \\\\,(\\\\,\\\\boxplus _{l=0}^{d_i-1}\\\\,\\\\, \\\\,\\\\pi _i \\\\,\\\\otimes |.|^{\\\\frac{d_i-1}{2}-l}\\\\,)\\\\, \\\\, \\\\, \\\\, \\\\, {\\\\rm and} \\\\,\\\\,\\\\,\\\\,\\\\,4 = \\\\sum _{i \\\\in I} n_i d_i.$ The selfdual representation $\\\\pi _i$ is symplectic in Arthur's sense if, and only if, $d_i$ is odd.\",\n \"All of this is included in [1].\",\n \"(c) For $\\\\pi $ as in (b), then ${\\\\rm inf}\\\\, \\\\pi _\\\\infty $ is the union, over all $i \\\\in I$ and all $0 \\\\le l< d_i$ , of the multisets $\\\\frac{d_i-1}{2}-l\\\\,+\\\\,{\\\\rm inf}\\\\, (\\\\pi _i)_\\\\infty $ .\",\n \"In particular, if we have $\\\\lambda \\\\in \\\\frac{1}{2}\\\\mathbb {Z}$ and $\\\\lambda -\\\\mu \\\\in \\\\mathbb {Z}$ for all $\\\\lambda ,\\\\mu \\\\in \\\\,{\\\\rm inf}\\\\, \\\\pi _\\\\infty $ , then ${\\\\rm inf}\\\\, (\\\\pi _i)_\\\\infty $ has the same property for each $i$ : such a $\\\\pi _i$ is called algebraic.\",\n \"If $\\\\omega $ is a cuspidal selfdual algebraic automorphic representation of ${\\\\rm GL}_m$ over $\\\\mathbb {Q}$ , then $\\\\omega _\\\\infty $ is tempered by the Jaquet-Shalika estimates, and its Langlands parameter is trivial on the central subgroup $\\\\mathbb {R}_{>0}$ of ${\\\\rm W}_\\\\mathbb {R}$ (this is the so-called Clozel purity lemma, see e.g.\",\n \"[6]).\",\n \"(d) The only selfdual cuspidal automorphic representation $\\\\pi $ of ${\\\\rm GL}_1$ over $\\\\mathbb {Q}$ such that $\\\\pi _p$ is unramified for each prime $p$ is the trivial Hecke character 1 (which is of course selfdual orthogonal).\",\n \"Moreover, for any integer $k\\\\ge 1$ , the number of cuspidal automorphic representations $\\\\pi $ of ${\\\\rm GL}_2$ over $\\\\mathbb {Q}$ such that $\\\\pi _p$ is unramified for each prime $p$ , and with ${\\\\rm inf}\\\\, \\\\pi _\\\\infty =\\\\lbrace -\\\\frac{k-1}{2},\\\\frac{k-1}{2}\\\\rbrace $ , is the dimension of the space ${S}_k(\\\\Gamma _1)$ of cuspidal modular forms of weight $k$ for $\\\\Gamma _1={\\\\rm SL}_2(\\\\mathbb {Z})$ .\",\n \"Indeed, this is well-known for $k>1$ , and for $k=1$ it follows from the fact that there is no Maass form of eigenvalue $1/4$ for ${\\\\rm SL}_2(\\\\mathbb {Z})$ (a fact due to Selberg, see also [6] for a short proof).\",\n \"(of the vanishing of ${S}_{j,1}(\\\\Gamma _2)$ for any $j$ ).\",\n \"It is enough to show that there is no discrete automorphic representation $\\\\pi $ of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ which is unramified at every prime and with $\\\\pi _\\\\infty \\\\simeq {\\\\rm U}_{j,1}$ .\",\n \"For that we study $\\\\pi ^{\\\\rm GL}$ , and the associated collection $(d_i,n_i,\\\\pi _i)_{i \\\\in I}$ given by (b) above.\",\n \"By (a), the infinitesimal character of $(\\\\pi ^{\\\\rm GL})_\\\\infty $ is ${\\\\rm inf}\\\\, {\\\\rm U}_{j,1} \\\\, =\\\\,\\\\lbrace \\\\,\\\\frac{j+1}{2}, \\\\,\\\\frac{j-1}{2}, \\\\,-\\\\frac{j-1}{2}, \\\\,-\\\\frac{j+1}{2}\\\\rbrace .$ If we have $d_i=1$ for each $i$ , then $(\\\\pi ^{\\\\rm GL})_\\\\infty $ is tempered by (c), hence so is $\\\\pi _\\\\infty $ by Arthur's local-global compatibility [1], a contradiction as ${\\\\rm U}_{j,1}$ is non-tempered by (a).\",\n \"Fix $i \\\\in I$ with $d_i\\\\ge 2$ .\",\n \"If we have $n_i \\\\ge 2$ then we must have $I=\\\\lbrace i\\\\rbrace $ and $n_i=d_i=2$ by the equality $4 = \\\\sum _i n_i d_i$ .\",\n \"By (c) and the shape of ${\\\\rm inf}\\\\, {\\\\rm U}_{j,1}$ above, we necessarily have ${\\\\rm inf} \\\\, (\\\\pi _i)_\\\\infty = \\\\lbrace j/2,-j/2\\\\rbrace $ .\",\n \"But this is absurd by (d) and the vanishing ${S}_{j+1}(\\\\Gamma _1)=0$ for any even integer $j \\\\ge 0$ .\",\n \"We have thus $(d_i,n_i,\\\\pi _i)=(2,1,1)$ .\",\n \"Choose $i^{\\\\prime } \\\\in I-\\\\lbrace i\\\\rbrace $ .\",\n \"As we have $(d_{i^{\\\\prime }},n_{i^{\\\\prime }},\\\\pi _{i^{\\\\prime }}) \\\\ne (d_i,n_i,\\\\pi _i)$ , the previous argument shows $d_{i^{\\\\prime }}=1$ , so $\\\\pi _{i^{\\\\prime }}$ is symplectic by (b), which forces $n_{i^{\\\\prime }}$ to be even, hence the only possibility is $n_{i^{\\\\prime }}=2$ and $I=\\\\lbrace i,i^{\\\\prime }\\\\rbrace $ .\",\n \"But then the shape of ${\\\\rm inf}\\\\, {\\\\rm U}_{j,1}$ and (c) show that we have either $j=0$ and ${\\\\rm inf} (\\\\pi _{i^{\\\\prime }})_\\\\infty = \\\\lbrace 1/2, -1/2 \\\\rbrace $ or $j=1$ and ${\\\\rm inf} (\\\\pi _{i^{\\\\prime }})_\\\\infty = \\\\lbrace 3/2, -3/2 \\\\rbrace $ .\",\n \"Both cases are absurd by (d) as we have ${S}_4(\\\\Gamma _1)={S}_2(\\\\Gamma _1)=0$ , and we are done.\",\n \"The second assertion of the proposition will be a consequence of the following two lemmas.\",\n \"Lemma 1.2 Let $j \\\\ge 0$ be an even integer.\",\n \"The integer $\\\\dim {S}_{j,2}(\\\\Gamma _2)$ is the number of cuspidal, selfdual symplectic, automorphic representations $\\\\Pi $ of ${\\\\rm GL_4}$ over $\\\\mathbb {Q}$ whose local components $\\\\Pi _p$ are unramified for each prime $p$ , and with ${\\\\rm inf} \\\\, \\\\Pi _\\\\infty \\\\, = \\\\,\\\\lbrace \\\\,\\\\frac{j+1}{2}, \\\\,\\\\frac{j+1}{2}, \\\\,-\\\\frac{j+1}{2}, \\\\,-\\\\frac{j+1}{2}\\\\rbrace $ .\",\n \"Let $\\\\pi $ be a cuspidal automorphic representation of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ generated by an arbitrary Hecke eigenform $F$ in ${S}_{j,2}(\\\\Gamma _2)$ .\",\n \"Consider its associated automorphic representation $\\\\pi ^{\\\\rm GL}$ of ${\\\\rm GL}_4$ and collection of $(d_i,n_i,\\\\pi _i)$ 's as in (b).\",\n \"We claim that $\\\\pi ^{\\\\rm GL}$ is necessarily cuspidal, i.e.\",\n \"$I=\\\\lbrace i\\\\rbrace $ is a singleton and $d_i=1$ , so that $\\\\Pi =\\\\pi ^{\\\\rm GL}$ satisfies all the assumptions of the statement by (a) and (b).\",\n \"Let us show first that we have $d_i=1$ for each $i \\\\in I$ .\",\n \"Otherwise, (c) shows that two elements of ${\\\\rm inf}\\\\, {\\\\rm U}_{j,2}$ must differ by 1, which only happens for $j=0$ .\",\n \"But for $j=0$ an argument similar to the one in the previous proof shows that if $d_i>1$ then we have $I=\\\\lbrace i,i^{\\\\prime }\\\\rbrace $ with $(d_i,n_i,\\\\pi _i)=(2,1,1)$ , $n^{\\\\prime }_i=2$ and ${\\\\rm inf}\\\\, (\\\\pi _i)_{\\\\infty } = \\\\lbrace 1/2,-1/2\\\\rbrace $ , which is absurd by the vanishing ${S}_2(\\\\Gamma _1)=0$ and (d), and we are done.\",\n \"As a consequence, the Langlands parameter of $(\\\\pi ^{\\\\rm GL})_\\\\infty $ is ${\\\\rm I}_{j+1} \\\\oplus {\\\\rm I}_{j+1}$ by (c).\",\n \"Let us denote by $\\\\psi $ Arthur's substitute for the global parameter of the representation $\\\\pi $ of ${\\\\rm PGSp}_{4}$ defined in [1].\",\n \"We have just proved that $\\\\psi _\\\\infty $ is the tempered Langlands parameter of ${\\\\rm PGSp}_4(\\\\mathbb {R})$ with $r \\\\,\\\\circ \\\\,\\\\psi _\\\\infty \\\\simeq {\\\\rm I}_{j+1} \\\\oplus {\\\\rm I}_{j+1}$ , i.e.\",\n \"the Langlands parameter of ${\\\\rm U}_{j,2}$ by (a).\",\n \"We have $\\\\mathcal {S}_{\\\\psi _\\\\infty } = \\\\mathbb {Z}/2\\\\mathbb {Z}$ : the corresponding $L$ -packet of ${\\\\rm PGSp}_4(\\\\mathbb {R})$ (limit of discrete series) has two elements, namely ${\\\\rm U}_{j,2}$ and a generic limit of discrete series with same infinitesimal character.\",\n \"We now apply Arthur's multiplicity formula to the element $\\\\pi $ of the global packet $\\\\Pi _\\\\psi $ defined by Arthur.\",\n \"As we have $d_i=1$ for all $i$ , either $\\\\pi ^{\\\\rm GL}$ is cuspidal or we have $I=\\\\lbrace i,i^{\\\\prime }\\\\rbrace $ with $n_i=n_{i^{\\\\prime }}=2$ and ${\\\\rm inf}\\\\, (\\\\pi _i)_\\\\infty ={\\\\rm inf}\\\\, (\\\\pi _{i^{\\\\prime }})_\\\\infty = \\\\lbrace \\\\frac{j+1}{2},-\\\\frac{j+1}{2}\\\\rbrace $ .\",\n \"If $\\\\pi ^{\\\\rm GL}$ is not cuspidal, then according to Arthur's definitions the natural map $\\\\mathcal {S}_\\\\psi \\\\rightarrow \\\\mathcal {S}_{\\\\psi _\\\\infty }$ is an isomorphism of groups of order 2.\",\n \"But then his multiplicity formula shows that $\\\\pi _\\\\infty $ has to be generic since $\\\\pi _p$ is unramified for each prime $p$ , a contradiction as ${\\\\rm U}_{j,2}$ is not generic.\",\n \"(We have shown that $\\\\pi $ is not of “Yoshida type”.)\",\n \"We have thus proved that $\\\\pi ^{\\\\rm GL}$ is cuspidal.\",\n \"Note that in this case we have $\\\\mathcal {S}_\\\\psi =1$ , thus by the multiplicity formula again, the multiplicity of $\\\\pi $ in the automorphic discrete spectrum of ${\\\\rm PGSp}_4$ is 1; in particular, the Hecke eigenspace of the eigenform $F$ we started from has dimension 1.\",\n \"It thus only remains to show that any $\\\\Pi $ as in the statement is in the image of the construction of the first paragraph above.\",\n \"Let $\\\\Pi $ be as in the statement.\",\n \"The Langlands parameter of $\\\\Pi _\\\\infty $ is the image under $r$ of the one of ${\\\\rm U}_{j,2}$ by (a) and (c).\",\n \"A trivial application of Arthur's multiplicity formula shows the existence of a discrete automorphic $\\\\pi $ for ${\\\\rm PGSp}_4$ with $\\\\pi _\\\\infty \\\\simeq {\\\\rm U}_{j,2}$ , which is unramified at every prime, and satisfying $\\\\pi ^{\\\\rm GL} \\\\simeq \\\\Pi $ .\",\n \"As ${\\\\rm U}_{j,2}$ is tempered, a classical result of Wallach ensures that $\\\\pi $ is actually cuspidal, hence generated by an element of ${S}_{j,2}(\\\\Gamma _2)$ : this concludes the proof.\",\n \"Lemma 1.3 For any even integer $0 \\\\le j \\\\le 38$ there is no $\\\\Pi $ as in Lemma REF .\",\n \"In order to contradict the existence of a $\\\\Pi $ as in Lemma REF for small $j$ , and following work of Odlyzko, Mestre, Fermigier, Miller and Chenevier-Lannes, we shall apply the so-called explicit formula “à la Riemann-Weil” to a suitable test function $F$ and to the complete Rankin-Selberg $L$ -function ${\\\\rm L}(s,\\\\Pi \\\\times \\\\Pi ^{\\\\prime })$ , first to $\\\\Pi ^{\\\\prime } = \\\\Pi ^{\\\\vee }$ (the contragredient of $\\\\Pi $ ) and then to some other well-chosen cuspidal automorphic representations $\\\\Pi ^{\\\\prime }$ .\",\n \"Let us stress that the analytic properties of those Rankin-Selberg $L$ -functions (meromorphic continuation to $\\\\mathbb {C}$ , functional equation, determination of the poles, and boundedness in vertical strips away from the poles) which have been established by Gelbart, Jacquet, Shalika and Shahidi, will play a crucial role in the argument.\",\n \"It will be convenient to follow the exposition of the explicit formula given in [6], which is designed for this kind of applications, and from which we shall borrow our notations.\",\n \"In particular, we choose for the test function $F$ the scaling of Odlyzko's function which is denoted by ${\\\\rm F}_{\\\\lambda }$ in [6], and we denote by ${\\\\rm K}_{\\\\infty }$ the Grothendieck ring of finite dimensional complex representations of the quotient of the compact group ${\\\\rm W}_{\\\\mathbb {R}}$ by its central subgroup $\\\\mathbb {R}_{>0}$ , and by ${\\\\rm J}_{F} : {\\\\rm K}_{\\\\infty } \\\\rightarrow \\\\mathbb {R}$ the concrete linear form associated to $F$ defined in Proposition-Def.\",\n \"3.7 loc.\",\n \"cit.\",\n \"Let $j\\\\ge 0$ be an even integer and let $\\\\Pi $ be as in Lemma REF .\",\n \"As already explained, it follows from (c) that the Langlands parameter of $\\\\Pi _\\\\infty $ is ${\\\\rm I}_{j+1} \\\\oplus {\\\\rm I}_{j+1}$ , whose square in the ring ${\\\\rm K}_\\\\infty $ is $4\\\\, ( {\\\\rm I}_{2j+2}+ {\\\\rm I}_{0})$ .\",\n \"The explicit formula applied to $\\\\Pi \\\\times \\\\Pi ^\\\\vee $ leads to the inequality [6] : $ {\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{2j+2})+{\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{0}) \\\\le \\\\frac{2}{\\\\pi ^{2}} \\\\lambda $ for all $\\\\lambda >0$ .\",\n \"As ${\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{w})$ is a non-increasing function of $w$ , the truth of the proposition for $j \\\\le 34$ is a consequence of the following numerical computation for $\\\\lambda = 3.3$ ${\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{70})+ {\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{0}) \\\\simeq 0.679 \\\\, \\\\, \\\\, \\\\, \\\\, \\\\, {\\\\rm and}\\\\, \\\\, \\\\, \\\\, \\\\, \\\\frac{2}{\\\\pi ^{2}}\\\\lambda \\\\simeq 0.669$ (the values are given up to $10^{-3}$ , and the left-hand side has been computed using the formula for ${\\\\rm J}_{F_{\\\\lambda }}$ given in [6]).\",\n \"We now explain how to deal with the cases $j=36$ and 38.\",\n \"By Tsushima's formula (proved for $k=3$ independently by Petersen and Taïbi), we know that the first value of $j$ such that ${S}_{j,3}(\\\\Gamma _2)$ is non-zero is $j=36$ , in which case it has dimension 1.\",\n \"Let $\\\\pi $ be the cuspidal automorphic representation of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ generated by ${S}_{36,3}(\\\\Gamma _2)$ and set $\\\\Pi ^{\\\\prime } = \\\\pi ^{\\\\rm GL}$ .\",\n \"This $\\\\Pi ^{\\\\prime }$ is a selfdual cuspidal representation by [6], and the Langlands parameter of $\\\\Pi ^{\\\\prime }_\\\\infty $ is ${\\\\rm I}_{39} \\\\oplus {\\\\rm I}_{37}$ (the existence of such a $\\\\Pi ^{\\\\prime }$ actually “explains” why the argument above breaks down at $j=36$ , as the Langlands parameter of $\\\\Pi ^{\\\\prime }_\\\\infty $ is “close” to ${\\\\rm I}_{37}\\\\oplus {\\\\rm I}_{37}$ ).\",\n \"We now apply the explicit formula to $\\\\Pi \\\\times \\\\Pi ^\\\\vee $ , $\\\\Pi ^{\\\\prime } \\\\times {\\\\Pi ^{\\\\prime }}^\\\\vee $ and $\\\\Pi \\\\times {\\\\Pi ^{\\\\prime }}^\\\\vee $ .\",\n \"It leads to a simple criterion, given in [6], for $\\\\Pi $ not to exist : the explicit quantity denoted there by ${\\\\rm t}(V,V^{\\\\prime },\\\\lambda )$ has to be $\\\\ge 0$ for all $\\\\lambda $ , where $V$ and $V^{\\\\prime }$ are the respective Langlands parameter of $\\\\Pi _\\\\infty $ and $\\\\Pi ^{\\\\prime }_\\\\infty $ .\",\n \"But a computation gives ${\\\\rm t}({\\\\rm I}_{37} + {\\\\rm I}_{37},{\\\\rm I}_{39} + {\\\\rm I}_{37},4) \\\\simeq -0.429\\\\,\\\\,\\\\,\\\\,\\\\, {\\\\rm and} \\\\,\\\\,\\\\, \\\\, \\\\, {\\\\rm t}({\\\\rm I}_{39}+ {\\\\rm I}_{39},{\\\\rm I}_{39} + {\\\\rm I}_{37},4) \\\\simeq -0.039$ (these values are given up to $10^{-3}$ ) which are both $<0$ .\",\n \"This concludes the proof.\",\n \"Remark 1.4 (i) As we have ${S}_{j,3}(\\\\Gamma _2)=0$ for $j<36$ by Tsushima's formula, the vanishing of ${S}_{j,2}(\\\\Gamma _2)$ for $j \\\\le 38$ is a very mild evidence toward Conjecture 1.1 of Cléry and van der Geer.\",\n \"(ii) Lemma REF and the explicit formula can also be used to obtain upper bounds on $\\\\dim {S}_{j,2}(\\\\Gamma _2)$ .\",\n \"Indeed, keeping the notations in the above proof, and applying [6] (due to Taïbi), we get that the inequality $({\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{2j+2})+ {\\\\rm J}_{{\\\\rm F}_{\\\\lambda }}({\\\\rm I}_{0})) \\\\dim {S}_{j,2}(\\\\Gamma _2) \\\\le \\\\frac{2}{\\\\pi ^{2}} \\\\lambda $ holds for all $\\\\lambda >0$ .\",\n \"When the parenthesis on left hand side is $> 0$ , which happens (for big enough $\\\\lambda $ ) for all $j \\\\le 138$ , we obtain an explicit upper bound for $\\\\dim {S}_{j,2}(\\\\Gamma _2)$ .\",\n \"For instance, we get $\\\\dim {S}_{j,2}(\\\\Gamma _2) \\\\le 1$ for all $j < 54$ and $\\\\dim {S}_{j,2}(\\\\Gamma _2) \\\\le 2$ for all $j < 66$ (choose respectively $\\\\lambda =5$ and $\\\\lambda =6$ ).\",\n \"Our last and main result concerns the kernel $\\\\Gamma _2[2]$ of the reduction ${\\\\rm Sp}_4(\\\\mathbb {Z}) \\\\rightarrow {\\\\rm Sp}_4(\\\\mathbb {Z}/2\\\\mathbb {Z})$ .\",\n \"Theorem 1.5 We have ${S}_{j,1}(\\\\Gamma _2[2])=0$ for any $j$ .\",\n \"Our proof will be an elaboration of the one of Proposition REF .\",\n \"We shall also use the vanishing ${S}_{j,1}(\\\\Gamma _2[2])=0$ for $j\\\\le 8$ , proved by Cléry and van der Geer in this paper (Proposition REF ).\",\n \"As we have $-1 \\\\in \\\\Gamma _2[2]$ we may also assume $j$ is even.\",\n \"Let us denote by $J$ the principal congruence subgroup of ${\\\\rm PGSp}_4(\\\\mathbb {Z}_2)$ and by $\\\\mathbb {A}$ the adele ring of $\\\\mathbb {Q}$ ; we easily check ${\\\\rm PGSp}_4(\\\\mathbb {A}) = {\\\\rm PGSp}_4(\\\\mathbb {Q}) \\\\cdot ({\\\\rm PGSp}_4(\\\\mathbb {R})^0 \\\\times J \\\\times \\\\prod _{p\\\\ne 2} {\\\\rm PGSp}_4(\\\\mathbb {Z}_p)).$ Moreover, classical arguments show that we have $S_{j,1}(\\\\Gamma _2[2])=0$ if, and only if, there is no cuspidal automorphic representation $\\\\pi $ of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ which is unramified at every odd prime, such that $\\\\pi _2$ has a non-zero invariant under $J$ , and with $\\\\pi _\\\\infty \\\\simeq {\\\\rm U}_{j,1}$ .\",\n \"Thus we fix such a $\\\\pi $ and consider $\\\\pi ^{\\\\rm GL}$ , as well as the associated collection $(d_i,n_i,\\\\pi _i)_{i \\\\in I}$ , given by (b) above.\",\n \"By the same argument as in the case $\\\\Gamma =\\\\Gamma _2$ , there exists $i \\\\in I$ with $d_i \\\\ge 2$ .\",\n \"If we have $n_i=1$ , which forces ${\\\\rm inf}\\\\, \\\\pi _i = \\\\lbrace 0\\\\rbrace $ , we must have $d_i=2$ and $j \\\\in \\\\lbrace 0,2\\\\rbrace $ by the shape of ${\\\\rm inf}\\\\, {\\\\rm U}_{j,1}$ .\",\n \"But this is a contradiction as ${S}_{j,1}(\\\\Gamma _2[2])=0$ for $j=0,2$ .\",\n \"So we must have $n_i=d_i=2$ , $I=\\\\lbrace i\\\\rbrace $ , and $\\\\pi _i$ is orthogonal with ${\\\\rm inf} (\\\\pi _i)_\\\\infty = \\\\lbrace -j/2,j/2\\\\rbrace $ .\",\n \"The classification of orthogonal cuspidal automorphic representations of ${\\\\rm GL}_2$ over $\\\\mathbb {Q}$ , a very special case of Arthur's results, is well-known.\",\n \"First of all, the central character of such a representation has order 2, hence corresponds to some uniquely defined quadratic extension $K$ of $\\\\mathbb {Q}$ .\",\n \"Moreover, for any Hecke character $\\\\chi $ of $K$ which is trivial on the idele group of $\\\\mathbb {Q}$ , and with $\\\\chi ^2 \\\\ne 1$ , the automorphic induction of $\\\\chi $ to $\\\\mathbb {Q}$ is an orthogonal cuspidal automorphic representation of ${\\\\rm GL}_2$ over $\\\\mathbb {Q}$ that we shall denote by ${\\\\rm ind}(\\\\chi )$ .\",\n \"It turns out that they all have this form, and that we have moreover ${\\\\rm ind}(\\\\chi ) \\\\simeq {\\\\rm ind}(\\\\chi ^{\\\\prime })$ if, and only if, we have $\\\\chi =\\\\chi ^{\\\\prime }$ or $\\\\chi ^{-1} = \\\\chi ^{\\\\prime }$ .\",\n \"Last but not least, to any $\\\\chi $ as above Arthur associates a global packet $\\\\Pi (\\\\chi )=\\\\bigotimes ^{\\\\prime }_v \\\\Pi _v(\\\\chi _v)$ of irreducible admissible representations of ${\\\\rm PGSp}_4$ over $\\\\mathbb {Q}$ , whose elements are exactly the discrete automorphic representations $\\\\omega $ which satisfy $\\\\omega ^{\\\\rm GL} = {\\\\rm ind}(\\\\chi ) \\\\otimes |.|^{1/2} \\\\boxplus {\\\\rm ind}(\\\\chi ) \\\\otimes |.|^{-1/2}$ (Soudry type); in this “stable case” any element of $\\\\Pi (\\\\chi )$ is automorphic by Arthur's multiplicity formula.\",\n \"Going back to our specific situation, let $K$ and $\\\\chi $ be such that $\\\\pi _i \\\\simeq {\\\\rm ind}(\\\\chi )$ .\",\n \"Since $\\\\pi _i$ is unramified outside 2, then so is $K$ and we necessarily have $K=\\\\mathbb {Q}(\\\\sqrt{d})\\\\, \\\\, \\\\, {\\\\rm with}\\\\, \\\\, \\\\, d\\\\,\\\\in \\\\lbrace -2,\\\\,-1,\\\\,2\\\\rbrace .$ We first claim $d \\\\ne 2$ .\",\n \"Indeed, a Hecke character of real quadratic field has the form $|.|^{s_0} \\\\chi _0$ with $\\\\chi _0$ a finite order character and $s_0 \\\\in \\\\mathbb {C}$ .\",\n \"We would thus have $\\\\lbrace s_0,s_0\\\\rbrace = {\\\\rm inf} (\\\\pi _i)_\\\\infty =\\\\lbrace j/2,-j/2\\\\rbrace $ , which implies $s_0=j=0$ , which is again absurd.\",\n \"So $K$ is imaginary quadratic.\",\n \"As $\\\\chi _\\\\infty $ is trivial on $\\\\mathbb {R}^\\\\times $ by assumption on $\\\\chi $ , and up to replacing $\\\\chi $ by $\\\\chi ^{-1}$ if necessary, the shape of ${\\\\rm inf}\\\\, (\\\\pi _i)_\\\\infty $ implies then $\\\\chi _\\\\infty (z) = (z/\\\\overline{z})^{j/2}$ .\",\n \"Here comes the main trick.\",\n \"Let $\\\\eta $ be the Hecke character of the statement of Lemma REF below, and set $w_K=|\\\\mathcal {O}_K^\\\\times |$ .\",\n \"We may assume $j\\\\ge 2$ , so there are unique integers $r$ and $j^{\\\\prime }/2$ , with $r\\\\ge 0$ and $1 \\\\le j^{\\\\prime }/2 \\\\le w_K$ , such that $j/2 \\\\,=\\\\,r \\\\,w_K \\\\, +\\\\, j^{\\\\prime }/2$ .\",\n \"Consider the Hecke character $\\\\chi ^{\\\\prime } = \\\\chi \\\\eta ^{-r}$ of $K$ .\",\n \"It is obviously trivial on the idele group of $\\\\mathbb {Q}$ , and it satisfies $(\\\\chi ^{\\\\prime })^2 \\\\ne 1$ as we have $\\\\chi ^{\\\\prime }_\\\\infty (z)= (z/\\\\overline{z})^{j^{\\\\prime }/2}$ with $j^{\\\\prime }>0$ .\",\n \"Consider now the packet $\\\\Pi (\\\\chi ^{\\\\prime })$ .\",\n \"As we have $\\\\chi _2=\\\\chi ^{\\\\prime }_2$ , the local Arthur packets $\\\\Pi _2(\\\\chi _2)$ and $\\\\Pi _2(\\\\chi ^{\\\\prime }_2)$ do coincide.\",\n \"As $\\\\pi $ belongs to $\\\\Pi (\\\\chi )$ , its local component $\\\\pi _2$ also belongs to $\\\\Pi _2(\\\\chi _2)$ .\",\n \"We may thus consider a representation $\\\\pi ^{\\\\prime }=\\\\bigotimes ^{\\\\prime }_v \\\\pi ^{\\\\prime }_v$ in $\\\\Pi (\\\\chi ^{\\\\prime })$ with $\\\\pi ^{\\\\prime }_2\\\\simeq \\\\pi _2$ , with $\\\\pi ^{\\\\prime }_p$ unramified for all odd prime $p$ (since $\\\\chi ^{\\\\prime }_p$ is unramified for such a $p$ ), and with $\\\\pi ^{\\\\prime }_\\\\infty \\\\simeq {\\\\rm U}_{j^{\\\\prime },1}$ .\",\n \"This last property holds because the Langlands packet associated to the Arthur packet $\\\\Pi _\\\\infty (\\\\chi ^{\\\\prime }_\\\\infty )$ , which is included in $\\\\Pi _\\\\infty (\\\\chi ^{\\\\prime }_\\\\infty )$ by [1], is the one of ${\\\\rm U}_{j^{\\\\prime },1}$ by Remark (a) above.\",\n \"As already explained, the representation $\\\\pi ^{\\\\prime }$ is discrete automorphic by Arthur, and even cuspidal as its Archimedean component is tempered.\",\n \"As $\\\\pi ^{\\\\prime }_2 \\\\simeq \\\\pi _2$ has non-zero invariants under the principal congruence subgroup $J$ of ${\\\\rm PGSp}_4(\\\\mathbb {Z}_2)$ , it follows that $\\\\pi ^{\\\\prime }$ is generated by an element in ${S}_{j^{\\\\prime },1}(\\\\Gamma _2[2])$ by the first paragraph above.\",\n \"But now we have the inequality $j^{\\\\prime } \\\\,\\\\le \\\\,2 \\\\,w_K \\\\,\\\\le \\\\,8$ , a contradiction by the vanishing ${S}_{j^{\\\\prime },1}(\\\\Gamma _2[2])=0$ for $j^{\\\\prime }\\\\le 8$ .\",\n \"Lemma 1.6 Let $K=\\\\mathbb {Q}(\\\\sqrt{d}) \\\\subset \\\\mathbb {C}$ with $d=-1,-2$ and set $w_K=|\\\\mathcal {O}_K^\\\\times |$ .\",\n \"There is a Hecke character $\\\\eta $ of $K$ which is unramified outside $\\\\lbrace 2,\\\\infty \\\\rbrace $ and which satisfies $\\\\eta _2=1$ and $\\\\eta _\\\\infty (z)=(z/\\\\overline{z})^{w_K}$ for all $z \\\\in K_\\\\infty ^\\\\times $ .\",\n \"Moreover, $\\\\eta $ is trivial on the idele group of $\\\\mathbb {Q}$ .\",\n \"Denote by $\\\\mathbb {A}_F$ the adele ring of the number field $F$ .\",\n \"As $\\\\mathcal {O}_K$ has class number 1 we have the decomposition $\\\\mathbb {A}_K^\\\\times = K^\\\\times \\\\cdot (K_\\\\infty ^\\\\times \\\\times K_2^\\\\times \\\\times \\\\prod _{v\\\\ne 2,\\\\infty } {\\\\mathcal {O}^\\\\times _{K_v}})$ .\",\n \"This implies first the (unrequired) uniqueness of $\\\\eta $ , and shows that its existence is equivalent to the fact that the morphism $z \\\\mapsto (z/\\\\overline{z})^{w_K}, \\\\mathbb {C}^\\\\times \\\\rightarrow \\\\mathbb {C}^\\\\times $ , is trivial on the subgroup $(\\\\mathcal {O}_K[1/2])^\\\\times $ .\",\n \"This is indeed the case as this latter group is generated by $\\\\mathcal {O}_K^\\\\times $ and by some element $\\\\pi \\\\in \\\\mathcal {O}_K$ with norm 2 and which satisfies $\\\\pi /\\\\overline{\\\\pi } \\\\in \\\\mathcal {O}_K^\\\\times $ .\",\n \"The last assertion follows from the equality $\\\\mathbb {A}_\\\\mathbb {Q}^\\\\times = \\\\mathbb {Q}^\\\\times \\\\cdot (\\\\mathbb {R}^\\\\times \\\\times \\\\prod _p \\\\mathbb {Z}_p^\\\\times )$ and the properties of $\\\\eta $ .\"\n ],\n [\n \"Proof of Theorem \",\n \"In this second appendix, we explain how to deduce Theorem REF stated in the paper of Cléry and van der Geer from the works of Rösner [20] and Weissauer [26], for the convenience of the reader.\",\n \"We first recall some results on Yoshida lifts taken from Weissauer's work [26].\",\n \"Let us fix $f$ and $g$ two non-proportional elliptic eigen newforms of same even weight $j+2$ , and let us denote by $\\\\pi $ and $\\\\pi ^{\\\\prime }$ the (distinct) cuspidal automorphic representations of ${\\\\rm GL_2}(\\\\mathbb {A})$ that they generate, $\\\\mathbb {A}$ being the adele ring of $\\\\mathbb {Q}$ .\",\n \"In particular, $\\\\pi _\\\\infty $ and $\\\\pi ^{\\\\prime }_\\\\infty $ are isomorphic discrete series, and we may assume that $\\\\pi $ and $\\\\pi ^{\\\\prime }$ are normalized such that this discrete series has trivial central character (as $j$ is even).\",\n \"For Yoshida lifts $Y(f,g)$ of $f$ and $g$ to exist, we also need to assume that $f$ and $g$ have the same nebentypus, i.e.\",\n \"that $\\\\pi $ and $\\\\pi ^{\\\\prime }$ have the same central character.\",\n \"Let $\\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })=\\\\bigotimes ^{\\\\prime }_v \\\\Pi _v(\\\\pi _v,\\\\pi ^{\\\\prime }_v)$ be the restricted tensor product, over all the places $v$ of $\\\\mathbb {Q}$ , of the local $L$ -packet $\\\\Pi _v(\\\\pi _v,\\\\pi ^{\\\\prime }_v)$ of irreducible admissible representations of ${\\\\rm GSp}_4(\\\\mathbb {Q}_v)$ associated to the pair $\\\\lbrace \\\\pi _v,\\\\pi ^{\\\\prime }_v\\\\rbrace $ by Weissauer.\",\n \"For each place $v$ of $\\\\mathbb {Q}$ , this local $L$ -packet has either 1 or 2 elements, including a unique generic element; it has another element if, and only if, both $\\\\pi _v$ and $\\\\pi ^{\\\\prime }_v$ are discrete series [26] [20].\",\n \"In particular, the (Langlands) Archimedean packet $\\\\Pi _\\\\infty (\\\\pi _\\\\infty ,\\\\pi ^{\\\\prime }_\\\\infty )$ has two elements, the non-generic one being the holomorphic limit of discrete series ${\\\\rm U}_{j,2}$ recalled in appendix .\",\n \"Also, if both $\\\\pi $ and $\\\\pi ^{\\\\prime }$ are unramified at the finite place $v$ then $\\\\Pi _v(\\\\pi _v,\\\\pi ^{\\\\prime }_v)$ is a singleton (thus $\\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })$ is finite).\",\n \"The multiplicity formula proved by Weissauer [26] states that an element $\\\\varpi $ of $\\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })$ is discrete automorphic if, and only if, there is an even number of places $v$ such that $\\\\varpi _v$ is non-generic.\",\n \"He also shows that such an element has multiplicity one in the discrete spectrum of ${\\\\rm GSp}_4$ ; it is necessarily cuspidal as the two elements of $\\\\Pi _\\\\infty (\\\\pi _\\\\infty ,\\\\pi ^{\\\\prime }_\\\\infty )$ are tempered.\",\n \"Let us denote by $J \\\\subset {\\\\rm GSp}_4(\\\\mathbb {Z}_2)$ the principal congruence subgroup.\",\n \"Some classical arguments show that the Yoshida lifts $Y(f,g)$ which belong to the space $YS_{j,2}^{s[w]}$ of the statement are in natural bijection with certain vectors of the finite part of the cuspidal automorphic representations $\\\\varpi $ in $\\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })$ having the following properties : (i) $\\\\varpi _\\\\infty \\\\simeq {\\\\rm U}_{j,2}$ , (ii) $\\\\varpi _p^{{\\\\rm GSp}_4(\\\\mathbb {Z}_p)} \\\\ne 0$ (in which case we have $\\\\, \\\\dim \\\\, \\\\varpi _p^{{\\\\rm GSp}_4(\\\\mathbb {Z}_p)}=1$ ), (iii) the $s[w]$ -isotypic component $\\\\varpi _2^{J}$ is non-zero.\",\n \"More precisely, the Yoshida lifts $Y(f,g)$ corresponding to such a $\\\\varpi $ form a linear subspace $YS_{j,2}^{s[w]}[\\\\varpi ] \\\\subset YS_{j,2}^{s[w]}$ isomorphic to the $s[w]$ -isotypic component of $\\\\varpi _2^{J}$ as an $S_6$ -representation.\",\n \"The space $YS_{j,2}^{s[w]}$ of the statement is then the direct sum of its subspaces $YS_{j,2}^{s[w]}[\\\\varpi ]$ where $f,g$ and $\\\\varpi $ vary, with $\\\\varpi $ cuspidal automorphic satisfying (i), (ii) and (iii).\",\n \"We still fix elliptic newforms $f$ and $g$ as above, hence $\\\\pi $ and $\\\\pi ^{\\\\prime }$ as well.\",\n \"By [20], we know first that $\\\\Pi _p(\\\\pi _p,\\\\pi ^{\\\\prime }_p)$ has an element with non-zero invariants under ${\\\\rm GSp}_4(\\\\mathbb {Z}_p)$ if, and only if, both $\\\\pi _p$ and $\\\\pi ^{\\\\prime }_p$ are unramified.\",\n \"Moreover, the same corollary asserts that if $\\\\Pi _2(\\\\pi _2,\\\\pi ^{\\\\prime }_2)$ has an element with non-zero $J$ -invariants, then both $\\\\pi _2$ and $\\\\pi ^{\\\\prime }_2$ have non-zero invariants under the principal congruence subgroup of ${\\\\rm GL}_2(\\\\mathbb {Z}_2)$ .\",\n \"As a first consequence, if $\\\\varpi \\\\in \\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })$ does satisfy (ii) and (iii) then $f$ and $g$ are newforms on $\\\\Gamma _0(N)$ with $N|4$ (and both $\\\\pi $ and $\\\\pi ^{\\\\prime }$ have a trivial central character).\",\n \"Moreover, by the statement recalled above concerning the multiplicity formula (\\\"even parity of the number of non-generic places\\\"), there is at most one cuspidal automorphic $\\\\varpi \\\\in \\\\Pi (\\\\pi ,\\\\pi ^{\\\\prime })$ satisfying (i), (ii) and (iii) above, and it has the property that $\\\\varpi _2$ is the non-generic element of $\\\\Pi _2(\\\\pi _2,\\\\pi ^{\\\\prime }_2)$ .\",\n \"In particular, this latter $L$ -packet has two elements and both $\\\\pi _2$ and $\\\\pi ^{\\\\prime }_2$ are discrete series of ${\\\\rm GL}_2(\\\\mathbb {Q}_2)$ (thus neither $f$ nor $g$ can have level 1).\",\n \"By Rösner [20], there are only three possibilities for the isomorphism class of a representation of ${\\\\rm PGL}_2(\\\\mathbb {Q}_2)$ generated by an elliptic newform of level $\\\\Gamma _0(2)$ or $\\\\Gamma _0(4)$ , namely the Steinberg representation St and its unramified quadratic twist ${\\\\rm St}^{\\\\prime }$ in level $\\\\Gamma _0(2)$ , and a certain supercuspidal representation ${\\\\rm Sc}$ in level $\\\\Gamma _0(4)$ .\",\n \"In particular, there are all discrete series.\",\n \"Assume now that $\\\\sigma $ and $\\\\sigma ^{\\\\prime }$ are (possibly equal) elements of the set $\\\\lbrace {\\\\rm St}, {\\\\rm St^{\\\\prime }}, {\\\\rm Sc}\\\\rbrace $ and let $\\\\tau $ be the non-generic element of $\\\\Pi _2(\\\\sigma ,\\\\sigma ^{\\\\prime })$ .\",\n \"View the finite dimensional vector space $\\\\tau ^J$ as a representation of $S_6$ .\",\n \"In order to prove the theorem it only remains to show that either $\\\\tau ^J$ is 0 or we are in exactly one of the following situations : (a) $\\\\lbrace \\\\sigma ,\\\\sigma ^{\\\\prime }\\\\rbrace =\\\\lbrace {\\\\rm St},{\\\\rm St^{\\\\prime }}\\\\rbrace $ and $\\\\tau ^J \\\\simeq s[1^6]$ , (b) $\\\\lbrace \\\\sigma ,\\\\sigma ^{\\\\prime }\\\\rbrace =\\\\lbrace {\\\\rm Sc}\\\\rbrace $ and $\\\\tau ^J \\\\simeq s[2,1^4]$ , (c) $\\\\lbrace \\\\sigma ,\\\\sigma ^{\\\\prime }\\\\rbrace =\\\\lbrace {\\\\rm St}\\\\rbrace $ or $\\\\lbrace {\\\\rm St^{\\\\prime }}\\\\rbrace $ and $\\\\tau ^J \\\\simeq s[2^3]$ .\",\n \"This is a delicate analysis which fortunately has been carried out by Rösner.\",\n \"Indeed, this is exactly the content of the left-bottom part of [20] with $q=2$ .\",\n \"This table shows that there are just three non-zero possible representations for $\\\\tau ^J$ , denoted by $\\\\theta _2$ , $\\\\theta _5$ and $\\\\chi _9(1)$ there but which correspond respectively to the representations $s[2^3]$ , $s[1^6]$ and $s[2,1^4]$ by Rösner's other table [20], exactly according to the three cases above (read the table with $\\\\Pi _1={\\\\rm Sc}$ , $\\\\xi _\\\\mu {\\\\rm St} = {\\\\rm St}^{\\\\prime }$ , and take for $\\\\mu $ either the trivial character of $\\\\mathbb {Q}_2^\\\\times $ or its unramified quadratic character).\",\n \"$\\\\square $ Acknowledgement.\",\n \"Gaëtan Chenevier is supported by the C.N.R.S.\",\n \"and by the project ANR-14-CE25 (PerCoLaTor).\",\n \"He would like to thank Fabien Cléry and Gerard van der Geer for inviting him to include these two appendices to their paper.\",\n \"Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.\",\n \"E-mail address: gaetan.chenevier@math.cnrs.fr\"\n ]\n]"},"id":{"kind":"string","value":"1709.01748"}}},{"rowIdx":983,"cells":{"text":{"kind":"list like","value":[["Index of Dirac operators and classification of topological insulators"],["Abstract Real and complex Clifford bundles and Dirac operators defined on them are considered.","By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard.","Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained.","This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems."],["Introduction","The recent discovery of new topological phases of matter has extended the connections between condensed matter physics and topology , , , .","The prominent examples of topological phases are topological insulators and they correspond to bulk insulating and edge conducting materials.","Topology of the bulk implies the edge conductivity and this differentiates topological insulators from trivial insulators.","Besides the classical integer quantum Hall effect, Chern insulators and quantum spin Hall effect are also realized experimentally as examples of topological insulators , .","Moreover, topological superconductors can also be described as bulk insulating and edge conducting materials that exemplifies the topological phases of matter .","This bulk/edge correspondence in topological materials is a manifestation of the holography principle in condensed matter physics.","The existence of topological phases depends on the symmetries of the Hamiltonian and the time-reversal (T) and charge conjugation (C) symmetries play prominent roles in the classification of topological materials.","Topological insulators and superconductors can be characterized by the relevant topological invariants defined from the eigenstates of the corresponding Hamiltonians.","There are two types of topological materials depending on the group that the topological invariants take values which are $\\mathbb {Z}$ -insulators (Chern insulators) and $\\mathbb {Z}_2$ -insulators.","$\\mathbb {Z}$ -invariants are generally characterized by Chern numbers or winding numbers defined from the Berry curvature of the system and $\\mathbb {Z}_2$ -invariants correspond to Kane-Mele invariants or Chern-Simons invariants , .","Classification of topological phases in different dimensions and symmetry classes has been obtained by using Clifford algebras and K-theory , , , , , , .","Because of the Bott periodicity of K-groups, the topological phases can be described by a finite periodic table.","The rows of the table correspond to the ten different symmetry classes of Hamiltonians which are classified in and the columns describe different dimensions.","The periodic table of topological insulators and superconductors shows that there are five different topological symmetry classes in each dimension and three of them are $\\mathbb {Z}$ -insulators and two of them are $\\mathbb {Z}_2$ -insulators.","Since the symmetry classes of Hamiltonians can be related to Cartan symmetric spaces, the periodic table can be divided into two parts that characterize the complex and real classes separately.","Besides Clifford algebras and K-theory, the periodic table can also be obtained by analyzing the stability of gapless boundary states against perturbations or by the dimensional reduction method of massive Dirac Hamiltonians .","Although the relations between Clifford algebras, K-groups and topological phases are known, the origin of the periodic table in mathematical terms is not discussed extensively in the literature.","In this paper, we show that the periodic table of topological insulators and superconductors can be obtained directly from the Clifford chessboard of Clifford algebras and the index theorems of Dirac operators.","We start by turning the Clifford chessboard to the table of Clifford bundles and defining relevant Dirac operators on these bundles.","Index theorems for Dirac operators relates the analytic index to the topological index , , and we find the table of indices of Dirac operators in that way.","From the relations between index theorems, K-theory groups, Grothendieck groups of Clifford algebra representations and symmetric spaces, we obtain the periodic table of topological phases for all dimensions and symmetry classes.","These steps are applied both to real and complex classes separately.","Instead of starting with the symmetry classes of free-fermion Hamiltonians and obtaining the topological classes by using Clifford algebras and K-theory which is done in the literature , , , , we start from the Clifford chessboard and find the symmetry classes of Hamiltonians and topological phases by using the index theorems of Dirac operators.","This approach gives the different topological classes in all dimensions and relates the topological invariants in these classes with the indices of Dirac operators.","The paper is organized as follows.","In Section 2, the mathematical concepts discussed in the subsequent sections are connected to the properties of the physical systems that describe topological materials.","Section 3 includes the periodicity relations of Clifford algebras and the construction of the Clifford chessboard.","In Section 4, we obtain the periodic table of real topological phases by starting with the table of Clifford bundles and by using the index theorems of Dirac operators and relations with K-groups, Grothendieck groups and symmetric spaces.","Section 5 deals with the problem of obtaining the periodic table of complex topological phases with the methods discussed in Section 4.","Section 6 concludes the paper."],["Dirac Hamiltonians of Topological Materials","In this section, the mathematical concepts used in the subsequent sections to obtain the classification table of topological insulators are connected to the properties of real physical systems via their Hamiltonians and eigenstates.","The free-fermion Hamiltonians are classified in ten different symmetry classes depending on the existence or non-existence of anti-unitary symmetries such as time-reversal and charge-conjugation .","These so-called Cartan classes represent the defining Hamiltonians of free-fermion systems.","The low energy effective Hamiltonians of topological insulators and superconductors in all dimensions and symmetry classes are characterized by Dirac-like Hamiltonians.","They can be written generally in terms of the momentum variables $\\bf {k}$ in the following form , $H(\\bf {k})=\\bf {d}(\\bf {k}).\\bf {\\sigma }$ where $\\bf {d}(\\bf {k})$ denotes the functions written in terms of $\\bf {k}$ and $\\bf {\\sigma }$ are Clifford algebra generators in relevant dimensions.","For example, the two dimensional Haldane model of graphene , which is a Chern insulator characterized by a $\\mathbb {Z}$ topological invariant, has the following Bloch Hamiltonian $H({\\bf {k}})=\\sum _{i=1}^3\\left\\lbrace 2t_2\\cos (\\phi )\\cos ({\\bf {k}}.","{\\bf {b}}_i)\\sigma _0+t_1\\left[\\cos ({\\bf {k}}.","{\\bf {a_i}})\\sigma _1+\\sin ({\\bf {k}}.","{\\bf {a}}_i)\\sigma _2\\right]+\\left[\\frac{M}{3}-2t_2\\sin (\\phi )\\sin ({\\bf {k}}.","{\\bf {b}}_i)\\right]\\sigma _3\\right\\rbrace $ where $t_1$ and $t_2$ are nearest neighbour and next nearest neighbour hopping parameters, $\\phi $ is the Haldane phase and ${\\bf {a}}_i$ and ${\\bf {b}}_i$ are nearest neighbour and next nearest neighbour displacement vectors.","$M$ is the on-site energy and $\\sigma _i$ are Pauli matrices that satisfy the Clifford algebra relations.","At the low energy limit around the ${\\bf {K}}$ point, this Hamiltonian reduces to the following continuum limit $H({\\bf {K}})=-3t_2\\cos (\\phi )\\sigma _0+\\frac{3}{2}t_1\\left(\\kappa _2\\sigma _1-\\kappa _1\\sigma _2\\right)+\\left(M-3\\sqrt{3}t_2\\sin (\\phi )\\right)\\sigma _3$ where $\\kappa _i$ are the momentum space Pauli matrices.","This Hamiltonian is in the form of a Dirac Hamiltonian which corresponds to a Dirac operator written in terms of the Clifford algebra basis $\\sigma _i$ .","Similarly, the two-dimensional time-reversal invariant $\\mathbb {Z}_2$ insulator which is characterized by the Kane-Mele model has the following form of Bloch Hamiltonian $H({\\bf {k}})=\\sum _{i=1}^5d_i({\\bf {k}})\\Gamma _i+\\sum _{ip$}\\end{array}\\right.$ The dimension of the Clifford algebra $Cl_{p,q}$ for $p+q=n$ is equal to $2^n$ .","As special cases, we denote $Cl_n\\equiv Cl_{n,0}\\nonumber \\\\Cl^*_n\\equiv Cl_{0,n}$ for the Clifford algebras with only negative and only positive generators, respectively.","An automorphism $\\eta :Cl_{p,q}\\rightarrow Cl_{p,q}$ can be defined on $Cl_{p,q}$ that gives the $\\mathbb {Z}_2$ -grading of the Clifford algebra; $Cl_{p,q}=Cl^0_{p,q}\\oplus Cl^1_{p,q}$ .","This corresponds to the superalgebra structure of the Clifford algebras and the even part $Cl^0_{p,q}$ constitute a subalgebra structure.","There is an algebra isomorphism between Clifford algebras and their even subalgebras which is written as $Cl_{p,q}\\cong Cl^0_{p+1,q}\\nonumber \\\\Cl_n\\cong Cl^0_{n+1}$ Other isomorphisms for Clifford algebras are $Cl_{p,q}\\cong Cl_{p-4,q+4}\\nonumber \\\\Cl_{p,q+1}\\cong Cl_{q,p+1}$ Moreover, some tensor product isomorphisms of Clifford algebras can also be stated as follows $Cl_{p,q+2}\\cong Cl_{q,p}\\otimes Cl_{0,2}\\nonumber \\\\Cl_{p+2,q}\\cong Cl_{q,p}\\otimes Cl_{2,0}\\\\Cl_{p+1,q+1}\\cong Cl_{p,q}\\otimes Cl_{1,1}\\nonumber $ On the other hand, if we choose $V=\\mathbb {C}^{p+q}$ , then we can define the complex Clifford algebras as the complexification of real Clifford algebras with complex quadratic form; $\\mathbb {C}l_{p,q}\\equiv Cl_{p,q}\\otimes _{\\mathbb {R}}\\mathbb {C}\\cong Cl(\\mathbb {C}^{p+q},Q\\otimes \\mathbb {C})$ .","Since there is no distinction between positive and negative generators in the complex case, we can simply denote the complex Clifford algebras as $\\mathbb {C}l_n$ .","The structure of real and complex Clifford algebras can be obtained from the following periodicity relations , $Cl_{p+8,q}&\\cong & Cl_{p,q}\\otimes Cl_{8,0}\\nonumber \\\\Cl_{p,q+8}&\\cong & Cl_{p,q}\\otimes Cl_{0,8}\\\\\\mathbb {C}l_{n+2}&\\cong & \\mathbb {C}l_n\\otimes _{\\mathbb {C}}\\mathbb {C}l_2\\nonumber $ As can be seen from the above isomorphisms, while the real Clifford algebras have eight-fold periodicity, the complex Clifford algebras have two-fold periodicity.","Indeed, Clifford algebras correspond to simple or semi-simple matrix algebras constructed from the division algebras $\\mathbb {R}$ , $\\mathbb {C}$ and $\\mathbb {H}$ .","From a simple observation of the Clifford algebra relation (11) satisfied by the generators of the Clifford algebra, the lower dimensional Clifford algebras can be obtained in terms of division algebras as follows $Cl_{1,0}&\\cong & \\mathbb {C}\\nonumber \\\\Cl_{0,1}&\\cong & \\mathbb {R}\\oplus \\mathbb {R}\\nonumber \\\\Cl_{1,1}&\\cong & \\mathbb {R}(2)\\nonumber \\\\Cl_{2,0}&\\cong & \\mathbb {H}\\nonumber \\\\Cl_{0,2}&\\cong & \\mathbb {R}(2)\\nonumber $ where $\\mathbb {R}(2)$ denotes the 2$\\times $ 2 matrix algebra with real elements.","These basic Clifford algebras and the periodicity relations in (14), (15) and (16) determine the corresponding division algebras of higher dimensional Clifford algebras as in the following table Table: NO_CAPTION Table: NO_CAPTION where $\\mathbb {R}(2^n)$ , $\\mathbb {C}(2^n)$ and $\\mathbb {H}(2^n)$ denotes the $2^n\\times 2^n$ matrix algebras that take values in $\\mathbb {R}$ , $\\mathbb {C}$ and $\\mathbb {H}$ , respectively.","Especially, we can write the Clifford algebras $Cl^*_n$ and $\\mathbb {C}l_n$ as follows Table: NO_CAPTIONFrom the considerations given above, one can construct a table of Clifford algebras which is called the Clifford chessboard in the following way Table: NO_CAPTION"],["Real Clifford bundles and real classes of periodic table","In this section, we consider vector bundles whose fibres correspond to the real Clifford algebras.","By considering Dirac operators and index theorems, we will obtain the periodic table of real classes in the classification of topological insulators and superconductors in several steps."],["$Cl_k$ -bundles and table of {{formula:7c30e545-f042-4f49-9a92-1549739b9d2f}}","On a manifold $M$ , the Clifford bundle $Cl(E)$ is defined as the bundle of Clifford algebras over $M$ .","If the fibers correspond to the real Clifford algebras, then we call it the real Clifford bundle.","The algebra structure of the space of sections of $Cl(E)$ arises from the fibrewise multiplication in $Cl(E)$ .","Moreover, a Dirac bundle over $M$ is a bundle $S$ of left modules of $Cl(M)$ with a Riemannian metric and a connection on it where $Cl(M)$ is the Clifford algebra defined on $M$ .","A special case of a Dirac bundle is the spinor bundle over $M$ whose fibres correspond to the spinor spaces as left modules of $Cl(M)$ .","The Clifford algebra $Cl_k$ denotes the real Clifford algebra with $k$ negative generators as defined in Section 3.","A $Cl_k$ -Dirac bundle over $M$ is a real Dirac bundle $S$ over $M$ with a right action $Cl_k\\hookrightarrow \\text{Aut}(S)$ which is parallel and commutes with multiplication by the elements of $Cl(M)$ .","So, a $Cl_k$ -Dirac bundle is a Dirac bundle with an extra multiplication by $Cl_k$ .","Because of the $\\mathbb {Z}_2$ -grading property of $Cl_k$ , the $Cl_k$ -Dirac bundle also carry a $\\mathbb {Z}_2$ -grading $S=S^0\\oplus S^1$ with a $\\mathbb {Z}_2$ -grading for the $Cl_k$ -action.","Similarly, one can define $Cl^*_k$ -Dirac bundles from the definition (4) whose fibres correspond to the left modules of Clifford algebras with $k$ positive generators.","Let us consider $Cl^*_k$ -Dirac bundles.","Since the real Clifford algebras have eight-fold periodicity as stated in (16), it is enough to consider $Cl^*_{k(\\text{mod }8)}$ .","We can construct a square table of $8\\times 8$ entries whose elements correspond to $Cl^*_{s-n (\\text{mod }8)}$ -Dirac bundles where $s,n=0,1,...,7$ Table: NO_CAPTIONThe table can be extended to bigger values of $s$ and $n$ , however, it repeats the pattern because of the eight-fold periodicity.","Moreover, it is equivalent to the table of Clifford chessboard because of the following isomorphism $Cl_{n,s}\\cong Cl^*_{s-n(\\text{mod }8)}.$ So, we obtain the table of $Cl^*_{s-n (\\text{mod }8)}$ -Dirac bundles from the Clifford chessboard of Clifford algebras."],["$Cl_k$ -Dirac operators","On a Dirac bundle $S$ on $M$ with sections $\\Gamma (S)$ , a first-order differential operator $D:\\Gamma (S)\\longrightarrow \\Gamma (S)$ called Dirac operator can be defined.","For the frame basis $\\lbrace X_a\\rbrace $ and co-frame basis $\\lbrace e^a\\rbrace $ , the Dirac operator can be written in terms of the connection $\\nabla $ and local coordinates as $D=e^a.\\nabla _{X_a}$ where $.$ denotes the Clifford product.","On a Riemannian manifold $M$ , the Dirac operator of any Dirac bundle is self-adjoint.","If the Dirac bundle is $\\mathbb {Z}_2$ -graded $S=S^0\\oplus S^1$ , then the Dirac operator can be written in the form $D=\\left(\\begin{array}{cc}0 & D^1 \\\\D^0 & 0 \\\\\\end{array}\\right)$ where $D^0:\\Gamma (S^0)\\longrightarrow \\Gamma (S^1)$ and $D^1:\\Gamma (S^1)\\longrightarrow \\Gamma (S^0)$ .","Since $D$ is self-adjoint, $D^0$ and $D^1$ are adjoints of each other and the index of $D^0$ is written as $\\text{ind}(D^0)=\\text{dim}(\\text{ker } D^0)-\\text{dim}(\\text{ker } D^1)$ where $\\text{ker }D^0$ denotes the kernel of $D^0$ .","For $Cl_k$ -Dirac bundles, we denote the Dirac operator as $\\displaystyle {\\lnot }D_k$ and it commutes with the $Cl_k$ -action which means that it is a $Cl_k$ -linear operator.","If the bundle is $\\mathbb {Z}_2$ -graded, then the $Cl_k$ -Dirac operator is $\\displaystyle {\\lnot }D_k=\\left(\\begin{array}{cc}0 & \\displaystyle {\\lnot }D^1_k \\\\\\displaystyle {\\lnot }D^0_k & 0 \\\\\\end{array}\\right)$ The index of the $Cl_k$ -Dirac operator is given by $\\text{ind}\\displaystyle {\\lnot }D^0_k=\\text{dim}(\\text{ker }\\displaystyle {\\lnot }D^0_k)-\\text{dim}(\\text{ker }\\displaystyle {\\lnot }D^1_k)$ The constructions for $Cl^*_k$ -Dirac bundles are similar.","By defining $Cl^*_k$ -Dirac operators on $Cl^*_k$ -Dirac bundles, we can construct the table of $Cl^*_{s-n}$ -Dirac operators from the table of $Cl^*_{s-n}$ -bundles defined in subsection 4.A as follows Table: NO_CAPTION"],["Index theorem for $Cl_k$ -Dirac operators","If the Dirac bundle $S$ is the Clifford bundle on $M$ , then the Dirac operator $D$ corresponds to the Hodge-de Rham operator and its square $D^2=\\Delta $ is the Hodge Laplacian.","The differential forms that are in the kernel of the Hodge Laplacian are called harmonic forms.","Similarly, if $S$ is the spinor bundle on $M$ , then $D$ is the usual Dirac operator on spinors and the spinors that are in the kernel of $D$ , namely the spinors that satisfy $D\\psi =0$ , are called harmonic spinors since $\\text{ker }D=\\text{ker }D^2$ for any compact Riemannian manifold $M$ .","These definitions can be extended to the case of $Cl_k$ -Dirac operators and we denote the kernel of $\\displaystyle {\\lnot }D_k$ , that is the harmonic space, as $\\textbf {H}_k=\\text{ker }\\displaystyle {\\lnot }D_k$ .","Now, we consider the index theorem for $Cl_k$ -Dirac operators.","The analytic index of $\\displaystyle {\\lnot }D_k$ defined in (21) can be written in terms of the dimension of the harmonic space $\\textbf {H}_k$ and characteristic classes of the bundle .","The Atiyah-Singer index theorem for $\\displaystyle {\\lnot }D_k$ gives the following equality $\\text{ind}(\\displaystyle {\\lnot }D_k)&=&\\left\\lbrace \\begin{array}{ll}\\text{dim}_{\\mathbb {C}}\\textbf {H}_k (\\text{mod }2), & \\hbox{ for $k\\equiv 1$ $(\\text{mod }8)$} \\\\\\text{dim}_{\\mathbb {H}}\\textbf {H}_k (\\text{mod }2), & \\hbox{ for $k\\equiv 2$ $(\\text{mod }8)$} \\\\\\frac{1}{2}\\widehat{A}(M), & \\hbox{ for $k\\equiv 4$ $(\\text{mod }8)$} \\\\\\widehat{A}(M), & \\hbox{ for $k\\equiv 0$ $(\\text{mod }8)$}\\end{array}\\right.$ where $\\text{dim}_{\\mathbb {C}}$ and $\\text{dim}_{\\mathbb {H}}$ denote the complex and quaternionic dimensions, respectively.","$\\widehat{A}(M)$ is the $\\widehat{A}$ -genus of the manifold $M$ and it is defined as a power series expansion $\\widehat{A}(M)=\\prod _{i=1}^n \\frac{x_i/2}{\\text{sinh}(x_i/2)}.$ It can be written in terms of the Pontrjagin classes $p_i$ as follows $\\widehat{A}(M)&=&1-\\frac{1}{24}p_1+\\frac{1}{5760}\\left(7p_1^2-4p_2\\right)+\\frac{1}{967680}\\left(-31p_1^3+44p_1p_2-16p_3\\right)+...$ $\\widehat{A}$ -genus is an integer number for compact manifolds and it is an even integer for the dimensions $4 (\\text{mod }8)$ .","So, the index of $\\displaystyle {\\lnot }D_k$ takes values in $\\mathbb {Z}_2$ for $k\\equiv 1 \\text{ and }2 (\\text{mod }8)$ and in $\\mathbb {Z}$ for $k\\equiv 0 \\text{ and }4 (\\text{mod}8)$ .","Then, we can write the table of $Cl^*_{s-n}$ -Dirac operators in subsection 4.B in terms of the index of $Cl^*_{s-n}$ -Dirac operators in the following way Table: NO_CAPTIONThe Atiyah-Singer index theorem in (22) attaches different topological invariants to $Cl^*_k$ -Dirac operators $\\displaystyle {\\lnot }D_k$ for different values of $k$ .","This means that we have non-trivial topological classes for different values of $s$ and $n$ which have non-zero index.","For the cases of zero index, there are only trivial classes.","In that way, we obtain a table of topological equivalence classes of $Cl^*_k$ -Dirac bundles from the Clifford chessboard by using the index theorems."],["Relation with $KO$ -groups","The Dirac operator $D$ on a real Dirac bundle is a self-adjoint operator with finite dimensional kernel and cokernel.","So, it is an element of the set of real Fredholm operators $\\mathfrak {F}_{\\mathbb {R}}$ .","The index of Fredholm operators is constant on connected components of $\\mathfrak {F}_{\\mathbb {R}}$ and we have the following isomorphism $\\text{ind}:\\pi _0(\\mathfrak {F}_{\\mathbb {R}})\\stackrel{\\simeq }{\\longrightarrow }\\mathbb {Z}$ Moreover, $\\mathfrak {F}_{\\mathbb {R}}$ is a classifying space for $KO$ -groups.","The stable equivalence classes of real vector bundles on a manifold $M$ are characterized by $KO$ -groups and they are real equivalences of the $K$ -groups of complex vector bundles.","For a Haussdorf space $X$ , index map defines an isomorphism $\\text{ind}:[X,\\mathfrak {F}_{\\mathbb {R}}]\\longrightarrow KO(X)$ where $[X,\\mathfrak {F}_{\\mathbb {R}}]$ denotes the homotopy classes of maps between $X$ and $\\mathfrak {F}_{\\mathbb {R}}$ and $KO(X)$ is the $KO$ -group on $X$ that classifies the stably equivalent real vector bundles on $X$ .","So, this gives an isomorphism between the homotopy groups of $\\mathfrak {F}_{\\mathbb {R}}$ and the $KO$ -groups at the point $\\text{ind}:\\pi _k(\\mathfrak {F}_{\\mathbb {R}})\\stackrel{\\simeq }{\\longrightarrow }KO^{-k}(\\text{pt})\\equiv \\widetilde{KO}(S^k)$ where $KO^{-k}(X)$ is equivalent to the reduced $KO$ -group of $k$ -fold suspension $\\Sigma ^kX$ of $X$ ; $KO^{-k}(X)\\equiv \\widetilde{KO}(\\Sigma ^kX)$ .","The reduced $KO$ -group $\\widetilde{KO}(X)$ is defined as the kernel of the map $KO(X)\\longrightarrow KO(\\text{pt})\\cong \\mathbb {Z}$ and $KO(\\text{pt})$ is the $KO$ -group at the point.","Since the $k$ -fold suspension of the point is equivalent to the $k$ -sphere $\\Sigma ^k(\\text{pt})\\equiv S^k$ , we have $KO^{-k}(\\text{pt})\\equiv \\widetilde{KO}(S^k)$ .","Let us denote the subset of Fredholm operators which are $\\mathbb {Z}_2$ -graded, $Cl_k$ -linear and self-adjoint as $\\mathfrak {F}_k$ .","Then, the index of the operators in $\\mathfrak {F}_k$ is constant on connected components of $\\mathfrak {F}_k$ $\\text{ind}:\\pi _0(\\mathfrak {F}_k)\\longrightarrow KO^{-k}(\\text{pt})$ So, $\\mathfrak {F}_k$ is a classifying space for $KO^{-k}$ and for a Haussdorf space $X$ and for any $k$ , there is an isomorphism $\\text{ind}:[X,\\mathfrak {F}_k]\\longrightarrow KO^{-k}(X)$ This means that the index of the $Cl_k$ -Dirac operator $\\displaystyle {\\lnot }D_k$ takes values in the group $KO^{-k}(\\text{pt})$ and we can write the table of the index of $Cl^*_{s-n}$ -Dirac operators in subsection 4.C as the table of $KO^{-(s-n)}(\\text{pt})$ groups Table: NO_CAPTIONSince the $KO$ -groups of the point is given as in the following table Table: NO_CAPTION we obtain the table of $KO^{-(s-n)}(\\text{pt})$ groups as Table: NO_CAPTIONand this is compatible with the fact that the index of $\\displaystyle {\\lnot }D_k$ takes values in $\\mathbb {Z}$ and $\\mathbb {Z}_2$ as stated in subsection 4.C."],["Isomorphism between $KO$ -groups and Grothendieck groups","Now, we consider the Atiyah-Bott-Shapiro (ABS) isomorphism which relates $KO$ -groups with the real representations of Clifford algebras .","Let us denote the group of equivalence classes of irreducible real representations of $Cl_k$ as $\\mathfrak {M}_k$ which is called the Grothendieck group.","This is the free abelian group generated by the distinct irreducible representations over $\\mathbb {R}$ .","For different values of $k$ , we have the Grothendieck groups as in the following table Table: NO_CAPTIONThe inclusion $i:\\mathbb {R}^k\\hookrightarrow \\mathbb {R}^{k+1}$ given by $i(x_1,...,x_k)=(x_1,...,x_k,0)$ induces an algebra homomorphism $i_*:Cl_k\\longrightarrow Cl_{k+1}$ .","By restricting the action from $Cl_{k+1}$ to $Cl_k$ , we obtain the homomorphism $i^*:\\mathfrak {M}_{k+1}\\longrightarrow \\mathfrak {M}_k$ .","So, we can define the groups $\\mathfrak {M}_k/i^*\\mathfrak {M}_{k+1}$ .","In this quotient group, a representation that can be obtained by restricting a representation of $Cl_{k+1}$ to $Cl_k$ is equivalent to zero.","Similarly, the Grothendieck group of real $\\mathbb {Z}_2$ -graded modules over $Cl_k$ is denoted by $\\widehat{\\mathfrak {M}}_k$ and there is the natural isomorphism $\\widehat{\\mathfrak {M}}_k=\\mathfrak {M}_{k-1}$ that comes from (13).","This implies the following isomorphism $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}\\cong \\mathfrak {M}_{k-1}/i^*\\mathfrak {M}_k$ From the groups $\\mathfrak {M}_k$ defined in the above table, we obtain $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}&\\cong &\\left\\lbrace \\begin{array}{ll}\\mathbb {Z}, & \\hbox{ for $k\\equiv 0$ \\text{or} $4(\\text{mod }8)$} \\\\\\mathbb {Z}_2, & \\hbox{ for $k\\equiv 1$ \\text{or} $2(\\text{mod }8)$} \\\\0, & \\hbox{ otherwise}\\end{array}\\right.$ where the $\\mathbb {Z}$ groups arise when $\\mathfrak {M}_{k-1}$ correspond to $\\mathbb {Z}\\oplus \\mathbb {Z}$ and the $\\mathbb {Z}_2$ groups arise when the dimension of the representation of $Cl_k$ is double of the dimension of the representation of $Cl_{k-1}$ .","The graded tensor product of modules gives a multiplication in $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}$ and this defines the graded ring $\\widehat{\\mathfrak {M}}_*/i^*\\widehat{\\mathfrak {M}}_{*+1}\\equiv \\bigoplus _{k\\ge 0}\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}$ .","ABS isomorphism defines an isomorphism between the $KO$ -groups at the point and the quotients of Grothendieck groups in the following way $\\phi :\\widehat{\\mathfrak {M}}_*/i^*\\widehat{\\mathfrak {M}}_{*+1}\\stackrel{\\simeq }{\\longrightarrow }KO^{-*}(\\text{pt})$ This can easily be seen from the definitions in this and previous subsections.","So, we can write the table of $KO^{-(s-n)}(\\text{pt})$ groups in subsection 4.D as the table of $\\widehat{\\mathfrak {M}}_{s-n}/i^*\\widehat{\\mathfrak {M}}_{s-n+1}$ groups as follows Table: NO_CAPTIONand the Clifford chessboard in section 3 turns into the table of quotients of Grothendieck groups."],["From Grothendieck groups to symmetric spaces","$KO$ -groups and Grothendieck groups are in relation with Cartan symmetric spaces through their connected components.","If we consider the following sequence of groups $O(16n)\\longrightarrow U(8n)\\longrightarrow Sp(4n)\\longrightarrow Sp(2n)\\times Sp(2n)\\longrightarrow Sp(2n)\\longrightarrow U(2n)\\longrightarrow O(2n)\\longrightarrow O(n)\\times O(n)\\longrightarrow O(n)$ and denote every element in the sequence by $G_i$ , then the coset spaces $R_i\\equiv G_i/G_{i+1}$ correspond to the eight of ten symmetric spaces of Cartan.","These symmetric spaces and the groups that give their connected components are given as follows Table: NO_CAPTIONAs can be seen from the last column of the table, the connected components of the symmetric spaces are in one-to-one correspondence with $KO$ -groups at the point and hence to the Grothendieck groups.","Indeed, Cartan symmetric spaces correspond to the classifying spaces of vector bundles and we have the following isomorphism for a Haussdorf space $X$ $KO^{-k}(X)\\cong [X,R_k]$ and so $KO^{-k}(\\text{pt})\\cong \\pi _0(R_k).$ From the ABS isomorphism in subsection 4.E, we also have $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}\\cong \\pi _0(R_k).$ Hence, we can write the table of the quotients of Grothendieck groups in subsection 4.E in terms of the connected components of the symmetric spaces as follows Table: NO_CAPTIONHowever, the homotopy groups of symmetric spaces have the property $\\pi _n(R_k)=\\pi _{n+1}(R_{k-1})$ , so we have $\\pi _0(R_{s-n(\\text{mod }8)})=\\pi _{8-n(\\text{mod }8)}(R_s)$ and the table turns into Table: NO_CAPTIONThis shows that every row in the table denoted by $s$ corresponds to a Cartan symmetric space $R_s$ and every column denoted by $n$ corresponds to the $(8-n)$ th homotopy group of the symmetric space."],["From symmetric spaces to periodic table","Eight symmetric spaces defined in the previous subsection are in one-to-one correspondence with the extensions of some real Clifford algebras.","By starting with a real Clifford algebra and answering the question how many different types of generators can be added to the existing ones gives the symmetric spaces .","The symmetric spaces and corresponding Clifford algebra extensions are given as follows Table: Conclusion"]],"string":"[\n [\n \"Index of Dirac operators and classification of topological insulators\"\n ],\n [\n \"Abstract Real and complex Clifford bundles and Dirac operators defined on them are considered.\",\n \"By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard.\",\n \"Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained.\",\n \"This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems.\"\n ],\n [\n \"Introduction\",\n \"The recent discovery of new topological phases of matter has extended the connections between condensed matter physics and topology , , , .\",\n \"The prominent examples of topological phases are topological insulators and they correspond to bulk insulating and edge conducting materials.\",\n \"Topology of the bulk implies the edge conductivity and this differentiates topological insulators from trivial insulators.\",\n \"Besides the classical integer quantum Hall effect, Chern insulators and quantum spin Hall effect are also realized experimentally as examples of topological insulators , .\",\n \"Moreover, topological superconductors can also be described as bulk insulating and edge conducting materials that exemplifies the topological phases of matter .\",\n \"This bulk/edge correspondence in topological materials is a manifestation of the holography principle in condensed matter physics.\",\n \"The existence of topological phases depends on the symmetries of the Hamiltonian and the time-reversal (T) and charge conjugation (C) symmetries play prominent roles in the classification of topological materials.\",\n \"Topological insulators and superconductors can be characterized by the relevant topological invariants defined from the eigenstates of the corresponding Hamiltonians.\",\n \"There are two types of topological materials depending on the group that the topological invariants take values which are $\\\\mathbb {Z}$ -insulators (Chern insulators) and $\\\\mathbb {Z}_2$ -insulators.\",\n \"$\\\\mathbb {Z}$ -invariants are generally characterized by Chern numbers or winding numbers defined from the Berry curvature of the system and $\\\\mathbb {Z}_2$ -invariants correspond to Kane-Mele invariants or Chern-Simons invariants , .\",\n \"Classification of topological phases in different dimensions and symmetry classes has been obtained by using Clifford algebras and K-theory , , , , , , .\",\n \"Because of the Bott periodicity of K-groups, the topological phases can be described by a finite periodic table.\",\n \"The rows of the table correspond to the ten different symmetry classes of Hamiltonians which are classified in and the columns describe different dimensions.\",\n \"The periodic table of topological insulators and superconductors shows that there are five different topological symmetry classes in each dimension and three of them are $\\\\mathbb {Z}$ -insulators and two of them are $\\\\mathbb {Z}_2$ -insulators.\",\n \"Since the symmetry classes of Hamiltonians can be related to Cartan symmetric spaces, the periodic table can be divided into two parts that characterize the complex and real classes separately.\",\n \"Besides Clifford algebras and K-theory, the periodic table can also be obtained by analyzing the stability of gapless boundary states against perturbations or by the dimensional reduction method of massive Dirac Hamiltonians .\",\n \"Although the relations between Clifford algebras, K-groups and topological phases are known, the origin of the periodic table in mathematical terms is not discussed extensively in the literature.\",\n \"In this paper, we show that the periodic table of topological insulators and superconductors can be obtained directly from the Clifford chessboard of Clifford algebras and the index theorems of Dirac operators.\",\n \"We start by turning the Clifford chessboard to the table of Clifford bundles and defining relevant Dirac operators on these bundles.\",\n \"Index theorems for Dirac operators relates the analytic index to the topological index , , and we find the table of indices of Dirac operators in that way.\",\n \"From the relations between index theorems, K-theory groups, Grothendieck groups of Clifford algebra representations and symmetric spaces, we obtain the periodic table of topological phases for all dimensions and symmetry classes.\",\n \"These steps are applied both to real and complex classes separately.\",\n \"Instead of starting with the symmetry classes of free-fermion Hamiltonians and obtaining the topological classes by using Clifford algebras and K-theory which is done in the literature , , , , we start from the Clifford chessboard and find the symmetry classes of Hamiltonians and topological phases by using the index theorems of Dirac operators.\",\n \"This approach gives the different topological classes in all dimensions and relates the topological invariants in these classes with the indices of Dirac operators.\",\n \"The paper is organized as follows.\",\n \"In Section 2, the mathematical concepts discussed in the subsequent sections are connected to the properties of the physical systems that describe topological materials.\",\n \"Section 3 includes the periodicity relations of Clifford algebras and the construction of the Clifford chessboard.\",\n \"In Section 4, we obtain the periodic table of real topological phases by starting with the table of Clifford bundles and by using the index theorems of Dirac operators and relations with K-groups, Grothendieck groups and symmetric spaces.\",\n \"Section 5 deals with the problem of obtaining the periodic table of complex topological phases with the methods discussed in Section 4.\",\n \"Section 6 concludes the paper.\"\n ],\n [\n \"Dirac Hamiltonians of Topological Materials\",\n \"In this section, the mathematical concepts used in the subsequent sections to obtain the classification table of topological insulators are connected to the properties of real physical systems via their Hamiltonians and eigenstates.\",\n \"The free-fermion Hamiltonians are classified in ten different symmetry classes depending on the existence or non-existence of anti-unitary symmetries such as time-reversal and charge-conjugation .\",\n \"These so-called Cartan classes represent the defining Hamiltonians of free-fermion systems.\",\n \"The low energy effective Hamiltonians of topological insulators and superconductors in all dimensions and symmetry classes are characterized by Dirac-like Hamiltonians.\",\n \"They can be written generally in terms of the momentum variables $\\\\bf {k}$ in the following form , $H(\\\\bf {k})=\\\\bf {d}(\\\\bf {k}).\\\\bf {\\\\sigma }$ where $\\\\bf {d}(\\\\bf {k})$ denotes the functions written in terms of $\\\\bf {k}$ and $\\\\bf {\\\\sigma }$ are Clifford algebra generators in relevant dimensions.\",\n \"For example, the two dimensional Haldane model of graphene , which is a Chern insulator characterized by a $\\\\mathbb {Z}$ topological invariant, has the following Bloch Hamiltonian $H({\\\\bf {k}})=\\\\sum _{i=1}^3\\\\left\\\\lbrace 2t_2\\\\cos (\\\\phi )\\\\cos ({\\\\bf {k}}.\",\n \"{\\\\bf {b}}_i)\\\\sigma _0+t_1\\\\left[\\\\cos ({\\\\bf {k}}.\",\n \"{\\\\bf {a_i}})\\\\sigma _1+\\\\sin ({\\\\bf {k}}.\",\n \"{\\\\bf {a}}_i)\\\\sigma _2\\\\right]+\\\\left[\\\\frac{M}{3}-2t_2\\\\sin (\\\\phi )\\\\sin ({\\\\bf {k}}.\",\n \"{\\\\bf {b}}_i)\\\\right]\\\\sigma _3\\\\right\\\\rbrace $ where $t_1$ and $t_2$ are nearest neighbour and next nearest neighbour hopping parameters, $\\\\phi $ is the Haldane phase and ${\\\\bf {a}}_i$ and ${\\\\bf {b}}_i$ are nearest neighbour and next nearest neighbour displacement vectors.\",\n \"$M$ is the on-site energy and $\\\\sigma _i$ are Pauli matrices that satisfy the Clifford algebra relations.\",\n \"At the low energy limit around the ${\\\\bf {K}}$ point, this Hamiltonian reduces to the following continuum limit $H({\\\\bf {K}})=-3t_2\\\\cos (\\\\phi )\\\\sigma _0+\\\\frac{3}{2}t_1\\\\left(\\\\kappa _2\\\\sigma _1-\\\\kappa _1\\\\sigma _2\\\\right)+\\\\left(M-3\\\\sqrt{3}t_2\\\\sin (\\\\phi )\\\\right)\\\\sigma _3$ where $\\\\kappa _i$ are the momentum space Pauli matrices.\",\n \"This Hamiltonian is in the form of a Dirac Hamiltonian which corresponds to a Dirac operator written in terms of the Clifford algebra basis $\\\\sigma _i$ .\",\n \"Similarly, the two-dimensional time-reversal invariant $\\\\mathbb {Z}_2$ insulator which is characterized by the Kane-Mele model has the following form of Bloch Hamiltonian $H({\\\\bf {k}})=\\\\sum _{i=1}^5d_i({\\\\bf {k}})\\\\Gamma _i+\\\\sum _{ip$}\\\\end{array}\\\\right.$ The dimension of the Clifford algebra $Cl_{p,q}$ for $p+q=n$ is equal to $2^n$ .\",\n \"As special cases, we denote $Cl_n\\\\equiv Cl_{n,0}\\\\nonumber \\\\\\\\Cl^*_n\\\\equiv Cl_{0,n}$ for the Clifford algebras with only negative and only positive generators, respectively.\",\n \"An automorphism $\\\\eta :Cl_{p,q}\\\\rightarrow Cl_{p,q}$ can be defined on $Cl_{p,q}$ that gives the $\\\\mathbb {Z}_2$ -grading of the Clifford algebra; $Cl_{p,q}=Cl^0_{p,q}\\\\oplus Cl^1_{p,q}$ .\",\n \"This corresponds to the superalgebra structure of the Clifford algebras and the even part $Cl^0_{p,q}$ constitute a subalgebra structure.\",\n \"There is an algebra isomorphism between Clifford algebras and their even subalgebras which is written as $Cl_{p,q}\\\\cong Cl^0_{p+1,q}\\\\nonumber \\\\\\\\Cl_n\\\\cong Cl^0_{n+1}$ Other isomorphisms for Clifford algebras are $Cl_{p,q}\\\\cong Cl_{p-4,q+4}\\\\nonumber \\\\\\\\Cl_{p,q+1}\\\\cong Cl_{q,p+1}$ Moreover, some tensor product isomorphisms of Clifford algebras can also be stated as follows $Cl_{p,q+2}\\\\cong Cl_{q,p}\\\\otimes Cl_{0,2}\\\\nonumber \\\\\\\\Cl_{p+2,q}\\\\cong Cl_{q,p}\\\\otimes Cl_{2,0}\\\\\\\\Cl_{p+1,q+1}\\\\cong Cl_{p,q}\\\\otimes Cl_{1,1}\\\\nonumber $ On the other hand, if we choose $V=\\\\mathbb {C}^{p+q}$ , then we can define the complex Clifford algebras as the complexification of real Clifford algebras with complex quadratic form; $\\\\mathbb {C}l_{p,q}\\\\equiv Cl_{p,q}\\\\otimes _{\\\\mathbb {R}}\\\\mathbb {C}\\\\cong Cl(\\\\mathbb {C}^{p+q},Q\\\\otimes \\\\mathbb {C})$ .\",\n \"Since there is no distinction between positive and negative generators in the complex case, we can simply denote the complex Clifford algebras as $\\\\mathbb {C}l_n$ .\",\n \"The structure of real and complex Clifford algebras can be obtained from the following periodicity relations , $Cl_{p+8,q}&\\\\cong & Cl_{p,q}\\\\otimes Cl_{8,0}\\\\nonumber \\\\\\\\Cl_{p,q+8}&\\\\cong & Cl_{p,q}\\\\otimes Cl_{0,8}\\\\\\\\\\\\mathbb {C}l_{n+2}&\\\\cong & \\\\mathbb {C}l_n\\\\otimes _{\\\\mathbb {C}}\\\\mathbb {C}l_2\\\\nonumber $ As can be seen from the above isomorphisms, while the real Clifford algebras have eight-fold periodicity, the complex Clifford algebras have two-fold periodicity.\",\n \"Indeed, Clifford algebras correspond to simple or semi-simple matrix algebras constructed from the division algebras $\\\\mathbb {R}$ , $\\\\mathbb {C}$ and $\\\\mathbb {H}$ .\",\n \"From a simple observation of the Clifford algebra relation (11) satisfied by the generators of the Clifford algebra, the lower dimensional Clifford algebras can be obtained in terms of division algebras as follows $Cl_{1,0}&\\\\cong & \\\\mathbb {C}\\\\nonumber \\\\\\\\Cl_{0,1}&\\\\cong & \\\\mathbb {R}\\\\oplus \\\\mathbb {R}\\\\nonumber \\\\\\\\Cl_{1,1}&\\\\cong & \\\\mathbb {R}(2)\\\\nonumber \\\\\\\\Cl_{2,0}&\\\\cong & \\\\mathbb {H}\\\\nonumber \\\\\\\\Cl_{0,2}&\\\\cong & \\\\mathbb {R}(2)\\\\nonumber $ where $\\\\mathbb {R}(2)$ denotes the 2$\\\\times $ 2 matrix algebra with real elements.\",\n \"These basic Clifford algebras and the periodicity relations in (14), (15) and (16) determine the corresponding division algebras of higher dimensional Clifford algebras as in the following table Table: NO_CAPTION Table: NO_CAPTION where $\\\\mathbb {R}(2^n)$ , $\\\\mathbb {C}(2^n)$ and $\\\\mathbb {H}(2^n)$ denotes the $2^n\\\\times 2^n$ matrix algebras that take values in $\\\\mathbb {R}$ , $\\\\mathbb {C}$ and $\\\\mathbb {H}$ , respectively.\",\n \"Especially, we can write the Clifford algebras $Cl^*_n$ and $\\\\mathbb {C}l_n$ as follows Table: NO_CAPTIONFrom the considerations given above, one can construct a table of Clifford algebras which is called the Clifford chessboard in the following way Table: NO_CAPTION\"\n ],\n [\n \"Real Clifford bundles and real classes of periodic table\",\n \"In this section, we consider vector bundles whose fibres correspond to the real Clifford algebras.\",\n \"By considering Dirac operators and index theorems, we will obtain the periodic table of real classes in the classification of topological insulators and superconductors in several steps.\"\n ],\n [\n \"$Cl_k$ -bundles and table of {{formula:7c30e545-f042-4f49-9a92-1549739b9d2f}}\",\n \"On a manifold $M$ , the Clifford bundle $Cl(E)$ is defined as the bundle of Clifford algebras over $M$ .\",\n \"If the fibers correspond to the real Clifford algebras, then we call it the real Clifford bundle.\",\n \"The algebra structure of the space of sections of $Cl(E)$ arises from the fibrewise multiplication in $Cl(E)$ .\",\n \"Moreover, a Dirac bundle over $M$ is a bundle $S$ of left modules of $Cl(M)$ with a Riemannian metric and a connection on it where $Cl(M)$ is the Clifford algebra defined on $M$ .\",\n \"A special case of a Dirac bundle is the spinor bundle over $M$ whose fibres correspond to the spinor spaces as left modules of $Cl(M)$ .\",\n \"The Clifford algebra $Cl_k$ denotes the real Clifford algebra with $k$ negative generators as defined in Section 3.\",\n \"A $Cl_k$ -Dirac bundle over $M$ is a real Dirac bundle $S$ over $M$ with a right action $Cl_k\\\\hookrightarrow \\\\text{Aut}(S)$ which is parallel and commutes with multiplication by the elements of $Cl(M)$ .\",\n \"So, a $Cl_k$ -Dirac bundle is a Dirac bundle with an extra multiplication by $Cl_k$ .\",\n \"Because of the $\\\\mathbb {Z}_2$ -grading property of $Cl_k$ , the $Cl_k$ -Dirac bundle also carry a $\\\\mathbb {Z}_2$ -grading $S=S^0\\\\oplus S^1$ with a $\\\\mathbb {Z}_2$ -grading for the $Cl_k$ -action.\",\n \"Similarly, one can define $Cl^*_k$ -Dirac bundles from the definition (4) whose fibres correspond to the left modules of Clifford algebras with $k$ positive generators.\",\n \"Let us consider $Cl^*_k$ -Dirac bundles.\",\n \"Since the real Clifford algebras have eight-fold periodicity as stated in (16), it is enough to consider $Cl^*_{k(\\\\text{mod }8)}$ .\",\n \"We can construct a square table of $8\\\\times 8$ entries whose elements correspond to $Cl^*_{s-n (\\\\text{mod }8)}$ -Dirac bundles where $s,n=0,1,...,7$ Table: NO_CAPTIONThe table can be extended to bigger values of $s$ and $n$ , however, it repeats the pattern because of the eight-fold periodicity.\",\n \"Moreover, it is equivalent to the table of Clifford chessboard because of the following isomorphism $Cl_{n,s}\\\\cong Cl^*_{s-n(\\\\text{mod }8)}.$ So, we obtain the table of $Cl^*_{s-n (\\\\text{mod }8)}$ -Dirac bundles from the Clifford chessboard of Clifford algebras.\"\n ],\n [\n \"$Cl_k$ -Dirac operators\",\n \"On a Dirac bundle $S$ on $M$ with sections $\\\\Gamma (S)$ , a first-order differential operator $D:\\\\Gamma (S)\\\\longrightarrow \\\\Gamma (S)$ called Dirac operator can be defined.\",\n \"For the frame basis $\\\\lbrace X_a\\\\rbrace $ and co-frame basis $\\\\lbrace e^a\\\\rbrace $ , the Dirac operator can be written in terms of the connection $\\\\nabla $ and local coordinates as $D=e^a.\\\\nabla _{X_a}$ where $.$ denotes the Clifford product.\",\n \"On a Riemannian manifold $M$ , the Dirac operator of any Dirac bundle is self-adjoint.\",\n \"If the Dirac bundle is $\\\\mathbb {Z}_2$ -graded $S=S^0\\\\oplus S^1$ , then the Dirac operator can be written in the form $D=\\\\left(\\\\begin{array}{cc}0 & D^1 \\\\\\\\D^0 & 0 \\\\\\\\\\\\end{array}\\\\right)$ where $D^0:\\\\Gamma (S^0)\\\\longrightarrow \\\\Gamma (S^1)$ and $D^1:\\\\Gamma (S^1)\\\\longrightarrow \\\\Gamma (S^0)$ .\",\n \"Since $D$ is self-adjoint, $D^0$ and $D^1$ are adjoints of each other and the index of $D^0$ is written as $\\\\text{ind}(D^0)=\\\\text{dim}(\\\\text{ker } D^0)-\\\\text{dim}(\\\\text{ker } D^1)$ where $\\\\text{ker }D^0$ denotes the kernel of $D^0$ .\",\n \"For $Cl_k$ -Dirac bundles, we denote the Dirac operator as $\\\\displaystyle {\\\\lnot }D_k$ and it commutes with the $Cl_k$ -action which means that it is a $Cl_k$ -linear operator.\",\n \"If the bundle is $\\\\mathbb {Z}_2$ -graded, then the $Cl_k$ -Dirac operator is $\\\\displaystyle {\\\\lnot }D_k=\\\\left(\\\\begin{array}{cc}0 & \\\\displaystyle {\\\\lnot }D^1_k \\\\\\\\\\\\displaystyle {\\\\lnot }D^0_k & 0 \\\\\\\\\\\\end{array}\\\\right)$ The index of the $Cl_k$ -Dirac operator is given by $\\\\text{ind}\\\\displaystyle {\\\\lnot }D^0_k=\\\\text{dim}(\\\\text{ker }\\\\displaystyle {\\\\lnot }D^0_k)-\\\\text{dim}(\\\\text{ker }\\\\displaystyle {\\\\lnot }D^1_k)$ The constructions for $Cl^*_k$ -Dirac bundles are similar.\",\n \"By defining $Cl^*_k$ -Dirac operators on $Cl^*_k$ -Dirac bundles, we can construct the table of $Cl^*_{s-n}$ -Dirac operators from the table of $Cl^*_{s-n}$ -bundles defined in subsection 4.A as follows Table: NO_CAPTION\"\n ],\n [\n \"Index theorem for $Cl_k$ -Dirac operators\",\n \"If the Dirac bundle $S$ is the Clifford bundle on $M$ , then the Dirac operator $D$ corresponds to the Hodge-de Rham operator and its square $D^2=\\\\Delta $ is the Hodge Laplacian.\",\n \"The differential forms that are in the kernel of the Hodge Laplacian are called harmonic forms.\",\n \"Similarly, if $S$ is the spinor bundle on $M$ , then $D$ is the usual Dirac operator on spinors and the spinors that are in the kernel of $D$ , namely the spinors that satisfy $D\\\\psi =0$ , are called harmonic spinors since $\\\\text{ker }D=\\\\text{ker }D^2$ for any compact Riemannian manifold $M$ .\",\n \"These definitions can be extended to the case of $Cl_k$ -Dirac operators and we denote the kernel of $\\\\displaystyle {\\\\lnot }D_k$ , that is the harmonic space, as $\\\\textbf {H}_k=\\\\text{ker }\\\\displaystyle {\\\\lnot }D_k$ .\",\n \"Now, we consider the index theorem for $Cl_k$ -Dirac operators.\",\n \"The analytic index of $\\\\displaystyle {\\\\lnot }D_k$ defined in (21) can be written in terms of the dimension of the harmonic space $\\\\textbf {H}_k$ and characteristic classes of the bundle .\",\n \"The Atiyah-Singer index theorem for $\\\\displaystyle {\\\\lnot }D_k$ gives the following equality $\\\\text{ind}(\\\\displaystyle {\\\\lnot }D_k)&=&\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\text{dim}_{\\\\mathbb {C}}\\\\textbf {H}_k (\\\\text{mod }2), & \\\\hbox{ for $k\\\\equiv 1$ $(\\\\text{mod }8)$} \\\\\\\\\\\\text{dim}_{\\\\mathbb {H}}\\\\textbf {H}_k (\\\\text{mod }2), & \\\\hbox{ for $k\\\\equiv 2$ $(\\\\text{mod }8)$} \\\\\\\\\\\\frac{1}{2}\\\\widehat{A}(M), & \\\\hbox{ for $k\\\\equiv 4$ $(\\\\text{mod }8)$} \\\\\\\\\\\\widehat{A}(M), & \\\\hbox{ for $k\\\\equiv 0$ $(\\\\text{mod }8)$}\\\\end{array}\\\\right.$ where $\\\\text{dim}_{\\\\mathbb {C}}$ and $\\\\text{dim}_{\\\\mathbb {H}}$ denote the complex and quaternionic dimensions, respectively.\",\n \"$\\\\widehat{A}(M)$ is the $\\\\widehat{A}$ -genus of the manifold $M$ and it is defined as a power series expansion $\\\\widehat{A}(M)=\\\\prod _{i=1}^n \\\\frac{x_i/2}{\\\\text{sinh}(x_i/2)}.$ It can be written in terms of the Pontrjagin classes $p_i$ as follows $\\\\widehat{A}(M)&=&1-\\\\frac{1}{24}p_1+\\\\frac{1}{5760}\\\\left(7p_1^2-4p_2\\\\right)+\\\\frac{1}{967680}\\\\left(-31p_1^3+44p_1p_2-16p_3\\\\right)+...$ $\\\\widehat{A}$ -genus is an integer number for compact manifolds and it is an even integer for the dimensions $4 (\\\\text{mod }8)$ .\",\n \"So, the index of $\\\\displaystyle {\\\\lnot }D_k$ takes values in $\\\\mathbb {Z}_2$ for $k\\\\equiv 1 \\\\text{ and }2 (\\\\text{mod }8)$ and in $\\\\mathbb {Z}$ for $k\\\\equiv 0 \\\\text{ and }4 (\\\\text{mod}8)$ .\",\n \"Then, we can write the table of $Cl^*_{s-n}$ -Dirac operators in subsection 4.B in terms of the index of $Cl^*_{s-n}$ -Dirac operators in the following way Table: NO_CAPTIONThe Atiyah-Singer index theorem in (22) attaches different topological invariants to $Cl^*_k$ -Dirac operators $\\\\displaystyle {\\\\lnot }D_k$ for different values of $k$ .\",\n \"This means that we have non-trivial topological classes for different values of $s$ and $n$ which have non-zero index.\",\n \"For the cases of zero index, there are only trivial classes.\",\n \"In that way, we obtain a table of topological equivalence classes of $Cl^*_k$ -Dirac bundles from the Clifford chessboard by using the index theorems.\"\n ],\n [\n \"Relation with $KO$ -groups\",\n \"The Dirac operator $D$ on a real Dirac bundle is a self-adjoint operator with finite dimensional kernel and cokernel.\",\n \"So, it is an element of the set of real Fredholm operators $\\\\mathfrak {F}_{\\\\mathbb {R}}$ .\",\n \"The index of Fredholm operators is constant on connected components of $\\\\mathfrak {F}_{\\\\mathbb {R}}$ and we have the following isomorphism $\\\\text{ind}:\\\\pi _0(\\\\mathfrak {F}_{\\\\mathbb {R}})\\\\stackrel{\\\\simeq }{\\\\longrightarrow }\\\\mathbb {Z}$ Moreover, $\\\\mathfrak {F}_{\\\\mathbb {R}}$ is a classifying space for $KO$ -groups.\",\n \"The stable equivalence classes of real vector bundles on a manifold $M$ are characterized by $KO$ -groups and they are real equivalences of the $K$ -groups of complex vector bundles.\",\n \"For a Haussdorf space $X$ , index map defines an isomorphism $\\\\text{ind}:[X,\\\\mathfrak {F}_{\\\\mathbb {R}}]\\\\longrightarrow KO(X)$ where $[X,\\\\mathfrak {F}_{\\\\mathbb {R}}]$ denotes the homotopy classes of maps between $X$ and $\\\\mathfrak {F}_{\\\\mathbb {R}}$ and $KO(X)$ is the $KO$ -group on $X$ that classifies the stably equivalent real vector bundles on $X$ .\",\n \"So, this gives an isomorphism between the homotopy groups of $\\\\mathfrak {F}_{\\\\mathbb {R}}$ and the $KO$ -groups at the point $\\\\text{ind}:\\\\pi _k(\\\\mathfrak {F}_{\\\\mathbb {R}})\\\\stackrel{\\\\simeq }{\\\\longrightarrow }KO^{-k}(\\\\text{pt})\\\\equiv \\\\widetilde{KO}(S^k)$ where $KO^{-k}(X)$ is equivalent to the reduced $KO$ -group of $k$ -fold suspension $\\\\Sigma ^kX$ of $X$ ; $KO^{-k}(X)\\\\equiv \\\\widetilde{KO}(\\\\Sigma ^kX)$ .\",\n \"The reduced $KO$ -group $\\\\widetilde{KO}(X)$ is defined as the kernel of the map $KO(X)\\\\longrightarrow KO(\\\\text{pt})\\\\cong \\\\mathbb {Z}$ and $KO(\\\\text{pt})$ is the $KO$ -group at the point.\",\n \"Since the $k$ -fold suspension of the point is equivalent to the $k$ -sphere $\\\\Sigma ^k(\\\\text{pt})\\\\equiv S^k$ , we have $KO^{-k}(\\\\text{pt})\\\\equiv \\\\widetilde{KO}(S^k)$ .\",\n \"Let us denote the subset of Fredholm operators which are $\\\\mathbb {Z}_2$ -graded, $Cl_k$ -linear and self-adjoint as $\\\\mathfrak {F}_k$ .\",\n \"Then, the index of the operators in $\\\\mathfrak {F}_k$ is constant on connected components of $\\\\mathfrak {F}_k$ $\\\\text{ind}:\\\\pi _0(\\\\mathfrak {F}_k)\\\\longrightarrow KO^{-k}(\\\\text{pt})$ So, $\\\\mathfrak {F}_k$ is a classifying space for $KO^{-k}$ and for a Haussdorf space $X$ and for any $k$ , there is an isomorphism $\\\\text{ind}:[X,\\\\mathfrak {F}_k]\\\\longrightarrow KO^{-k}(X)$ This means that the index of the $Cl_k$ -Dirac operator $\\\\displaystyle {\\\\lnot }D_k$ takes values in the group $KO^{-k}(\\\\text{pt})$ and we can write the table of the index of $Cl^*_{s-n}$ -Dirac operators in subsection 4.C as the table of $KO^{-(s-n)}(\\\\text{pt})$ groups Table: NO_CAPTIONSince the $KO$ -groups of the point is given as in the following table Table: NO_CAPTION we obtain the table of $KO^{-(s-n)}(\\\\text{pt})$ groups as Table: NO_CAPTIONand this is compatible with the fact that the index of $\\\\displaystyle {\\\\lnot }D_k$ takes values in $\\\\mathbb {Z}$ and $\\\\mathbb {Z}_2$ as stated in subsection 4.C.\"\n ],\n [\n \"Isomorphism between $KO$ -groups and Grothendieck groups\",\n \"Now, we consider the Atiyah-Bott-Shapiro (ABS) isomorphism which relates $KO$ -groups with the real representations of Clifford algebras .\",\n \"Let us denote the group of equivalence classes of irreducible real representations of $Cl_k$ as $\\\\mathfrak {M}_k$ which is called the Grothendieck group.\",\n \"This is the free abelian group generated by the distinct irreducible representations over $\\\\mathbb {R}$ .\",\n \"For different values of $k$ , we have the Grothendieck groups as in the following table Table: NO_CAPTIONThe inclusion $i:\\\\mathbb {R}^k\\\\hookrightarrow \\\\mathbb {R}^{k+1}$ given by $i(x_1,...,x_k)=(x_1,...,x_k,0)$ induces an algebra homomorphism $i_*:Cl_k\\\\longrightarrow Cl_{k+1}$ .\",\n \"By restricting the action from $Cl_{k+1}$ to $Cl_k$ , we obtain the homomorphism $i^*:\\\\mathfrak {M}_{k+1}\\\\longrightarrow \\\\mathfrak {M}_k$ .\",\n \"So, we can define the groups $\\\\mathfrak {M}_k/i^*\\\\mathfrak {M}_{k+1}$ .\",\n \"In this quotient group, a representation that can be obtained by restricting a representation of $Cl_{k+1}$ to $Cl_k$ is equivalent to zero.\",\n \"Similarly, the Grothendieck group of real $\\\\mathbb {Z}_2$ -graded modules over $Cl_k$ is denoted by $\\\\widehat{\\\\mathfrak {M}}_k$ and there is the natural isomorphism $\\\\widehat{\\\\mathfrak {M}}_k=\\\\mathfrak {M}_{k-1}$ that comes from (13).\",\n \"This implies the following isomorphism $\\\\widehat{\\\\mathfrak {M}}_k/i^*\\\\widehat{\\\\mathfrak {M}}_{k+1}\\\\cong \\\\mathfrak {M}_{k-1}/i^*\\\\mathfrak {M}_k$ From the groups $\\\\mathfrak {M}_k$ defined in the above table, we obtain $\\\\widehat{\\\\mathfrak {M}}_k/i^*\\\\widehat{\\\\mathfrak {M}}_{k+1}&\\\\cong &\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathbb {Z}, & \\\\hbox{ for $k\\\\equiv 0$ \\\\text{or} $4(\\\\text{mod }8)$} \\\\\\\\\\\\mathbb {Z}_2, & \\\\hbox{ for $k\\\\equiv 1$ \\\\text{or} $2(\\\\text{mod }8)$} \\\\\\\\0, & \\\\hbox{ otherwise}\\\\end{array}\\\\right.$ where the $\\\\mathbb {Z}$ groups arise when $\\\\mathfrak {M}_{k-1}$ correspond to $\\\\mathbb {Z}\\\\oplus \\\\mathbb {Z}$ and the $\\\\mathbb {Z}_2$ groups arise when the dimension of the representation of $Cl_k$ is double of the dimension of the representation of $Cl_{k-1}$ .\",\n \"The graded tensor product of modules gives a multiplication in $\\\\widehat{\\\\mathfrak {M}}_k/i^*\\\\widehat{\\\\mathfrak {M}}_{k+1}$ and this defines the graded ring $\\\\widehat{\\\\mathfrak {M}}_*/i^*\\\\widehat{\\\\mathfrak {M}}_{*+1}\\\\equiv \\\\bigoplus _{k\\\\ge 0}\\\\widehat{\\\\mathfrak {M}}_k/i^*\\\\widehat{\\\\mathfrak {M}}_{k+1}$ .\",\n \"ABS isomorphism defines an isomorphism between the $KO$ -groups at the point and the quotients of Grothendieck groups in the following way $\\\\phi :\\\\widehat{\\\\mathfrak {M}}_*/i^*\\\\widehat{\\\\mathfrak {M}}_{*+1}\\\\stackrel{\\\\simeq }{\\\\longrightarrow }KO^{-*}(\\\\text{pt})$ This can easily be seen from the definitions in this and previous subsections.\",\n \"So, we can write the table of $KO^{-(s-n)}(\\\\text{pt})$ groups in subsection 4.D as the table of $\\\\widehat{\\\\mathfrak {M}}_{s-n}/i^*\\\\widehat{\\\\mathfrak {M}}_{s-n+1}$ groups as follows Table: NO_CAPTIONand the Clifford chessboard in section 3 turns into the table of quotients of Grothendieck groups.\"\n ],\n [\n \"From Grothendieck groups to symmetric spaces\",\n \"$KO$ -groups and Grothendieck groups are in relation with Cartan symmetric spaces through their connected components.\",\n \"If we consider the following sequence of groups $O(16n)\\\\longrightarrow U(8n)\\\\longrightarrow Sp(4n)\\\\longrightarrow Sp(2n)\\\\times Sp(2n)\\\\longrightarrow Sp(2n)\\\\longrightarrow U(2n)\\\\longrightarrow O(2n)\\\\longrightarrow O(n)\\\\times O(n)\\\\longrightarrow O(n)$ and denote every element in the sequence by $G_i$ , then the coset spaces $R_i\\\\equiv G_i/G_{i+1}$ correspond to the eight of ten symmetric spaces of Cartan.\",\n \"These symmetric spaces and the groups that give their connected components are given as follows Table: NO_CAPTIONAs can be seen from the last column of the table, the connected components of the symmetric spaces are in one-to-one correspondence with $KO$ -groups at the point and hence to the Grothendieck groups.\",\n \"Indeed, Cartan symmetric spaces correspond to the classifying spaces of vector bundles and we have the following isomorphism for a Haussdorf space $X$ $KO^{-k}(X)\\\\cong [X,R_k]$ and so $KO^{-k}(\\\\text{pt})\\\\cong \\\\pi _0(R_k).$ From the ABS isomorphism in subsection 4.E, we also have $\\\\widehat{\\\\mathfrak {M}}_k/i^*\\\\widehat{\\\\mathfrak {M}}_{k+1}\\\\cong \\\\pi _0(R_k).$ Hence, we can write the table of the quotients of Grothendieck groups in subsection 4.E in terms of the connected components of the symmetric spaces as follows Table: NO_CAPTIONHowever, the homotopy groups of symmetric spaces have the property $\\\\pi _n(R_k)=\\\\pi _{n+1}(R_{k-1})$ , so we have $\\\\pi _0(R_{s-n(\\\\text{mod }8)})=\\\\pi _{8-n(\\\\text{mod }8)}(R_s)$ and the table turns into Table: NO_CAPTIONThis shows that every row in the table denoted by $s$ corresponds to a Cartan symmetric space $R_s$ and every column denoted by $n$ corresponds to the $(8-n)$ th homotopy group of the symmetric space.\"\n ],\n [\n \"From symmetric spaces to periodic table\",\n \"Eight symmetric spaces defined in the previous subsection are in one-to-one correspondence with the extensions of some real Clifford algebras.\",\n \"By starting with a real Clifford algebra and answering the question how many different types of generators can be added to the existing ones gives the symmetric spaces .\",\n \"The symmetric spaces and corresponding Clifford algebra extensions are given as follows Table: Conclusion\"\n ]\n]"},"id":{"kind":"string","value":"1709.01778"}}},{"rowIdx":984,"cells":{"text":{"kind":"list like","value":[["Ergodic Exploration using Binary Sensing for Non-Parametric Shape\n Estimation"],["Abstract Current methods to estimate object shape---using either vision or touch---generally depend on high-resolution sensing.","Here, we exploit ergodic exploration to demonstrate successful shape estimation when using a low-resolution binary contact sensor.","The measurement model is posed as a collision-based tactile measurement, and classification methods are used to discriminate between shape boundary regions in the search space.","Posterior likelihood estimates of the measurement model help the system actively seek out regions where the binary sensor is most likely to return informative measurements.","Results show successful shape estimation of various objects as well as the ability to identify multiple objects in an environment.","Interestingly, it is shown that ergodic exploration utilizes non-contact motion to gather significant information about shape.","The algorithm is extended in three dimensions in simulation and we present two dimensional experimental results using the Rethink Baxter robot."],["Introduction","Tactile sensing is often associated with shape estimation [1], [2], [3], [4] and mapping problems [5] in conditions where visual sensing may be limited.","In some instances, tactile sensing is used to supplement vision-based sensing to improve shape estimates [6], [7].","The richness of touch as a sensing modality is underscored by the development of novel tactile sensors [8], [9], [10], [11] for use in a myriad of applications ranging from robot-assisted tumor detection [7], to texture recognition, and feature localization [12], [13].","These advances in tactile sensor technology require corresponding advances in active exploration algorithms and the interpretation of tactile-based sensor data.","Recent approaches for active exploration use random sampling-based search algorithms for sensor motion planning that require a separate controller for path following [14].","Other methods use task-specific probabilistic spatial methods for shape estimation that are updated based on sensor poses that minimize measurement uncertainty [2], [15], [16], [17], [18].","A common approach in the literature is motion planning for sensing followed by feedback regulation of the generated plan [19], [20], [18], [17].","Moreover, most methods focus on one object at a time.","In contrast, the presented work integrates planning and control into a feedback law.","As a result, the method uses sensor motion to actively sense for time-varying spatial information.","We take the framework in [21] and use the feedback law developed here to enable real-time execution during shape estimation.","Lastly, the feedback in the active sensing algorithm compensates for low resolution sensors and the specification of the algorithm is independent of the number of objects.","We show that ergodic exploration with Sequential Action Control (eSAC) can be used for active exploration with respect to time-varying tactile-based distributed information [22], [23].","Previous applications of ergodic theory utilize parametric measurement models for localization tasks [24], [21].","The current work demonstrates localization and estimation of non-parametric shapes using a binary sensor model with classification methods.","In contrast to other methods of active sensing for shape estimation [19], [25], [26], [16], [18], the proposed algorithm automatically encodes dynamical constraints without any overhead spatial discretization or motion planning.","In addition, the algorithm incorporates sensor measurement information to actively adjust shape estimates and synthesize tactile-information based control actions.","As a result, the algorithm automatically adjusts the control synthesis for multiple objects in an environment.","Notably, ergodic exploration uses non-contact motion data (sensor motion not in contact with an object) [27], [28] to improve the shape estimate.","The idea of utilizing free space is often found in other related works of pose estimation and tracking [27], [28], [29], [30] and is emphasized in our work.","As a final contribution, we show the algorithm is modular with respect to the choice of shape representation and tactile information distribution.","The paper outline is as follows: Section motivates the use of a binary contact sensor and outlines the implications for tactile sensing.","Section describes the robot control algorithm for active exploration and motivates ergodicity as a measure for exploration.","Section explains how shape estimation is achieved and how measurements update the shape estimate and the control policy.","Experimental and simulated results and conclusion are shown in sections and respectively."],["Tactile Information","In biological systems, tactile sensing is generally an active process that incorporates feedback from multiple environmental cues [31], [32].","Humans, for instance, use fingers to grasp objects for manipulation and feature detection.","Many rodents use hair-like appendages (whiskers) for tactile sensing [33], [34], [35].","These biological sensors have a large set of actuators and sensor channels, thus the process control for active sensing is complex.","We investigate a lower resolution version of tactile sensing that is simple to control for active sensing.","Specifically, we show that a binary form of tactile sensing (i.e., collision detection) [10], [8], has enough information for shape estimation, when combined with an active exploration algorithm that automatically takes into account regions of shape information.","A binary tactile measurement model consists of a transition state from “no collision” to “collision” and vice versa.","We denote this measurement model as $\\Upsilon (x) ={\\left\\lbrace \\begin{array}{ll}\\hfill 1, \\hfill & \\phi (x) \\le 0 \\\\\\hfill 0, \\hfill & \\phi (x) > 0 \\\\\\end{array}\\right.","}$ where $x$ is the sensor state and $\\phi (x)$ is a boundary function that determines a transition state if $\\phi (x) \\le 0$ (output of 1).","The goal of shape estimation is to determine $\\phi (x)$ ."],["Motivation","Typical algorithms used in active exploration and information maximization cast the problem of information acquisition as “exploratory” (wide spread search for diffuse information such as localization) or “exploitative” (direct search for highly structured information such as shape contours) [36], [37].","Ergodic exploration [22], [38], [23], [24], [21] is responsive to both diffuse information densities and highly focused information.","Thus, ergodic control seamlessly encodes wide spread coverage for diffuse information and localized search for focused information for both needs."],["Ergodic Metric","A trajectory $x(t)$ of a sensor is ergodic with respect to a probability (“target”) distribution $\\Phi (x)$ when the fraction of time the trajectory spends in an area within the search space is equal to the spatial statistics across the search space [22], [39].","The ergodic metric that measures this characteristic is given as [22], [23], $ \\mathcal {E} (x(t)) = \\sum _{k \\in {\\mathbb {Z}}^n} \\Lambda _{k} [c_{k}(x(t)) - \\phi _{k}]^2$ where $c_{k} (x(t)) & = & \\frac{1}{t_f - t_0}\\int _{t_0}^{t_f} F_k (x(\\tau )) d\\tau , \\\\\\phi _{k} & = & \\int _{\\mathcal {X}} \\Phi ({\\bf x}) F_k ({\\bf x}) d{\\bf x} .$ Here, $F_k (x)$ is a Fourier basis function, and $\\Lambda _{k}$ are weights described in [22].","The target distribution $\\Phi (x)$ is what drives the control synthesis for active-exploration.","Comparison between robot trajectory and distribution is done with equation (REF ) which takes the Fourier power series decomposition using (3) and (4) and directly compares the statistics of a trajectory with that of the spatial statistics $\\Phi (x)$ .","As the ergodic metric is convex [22], convex objectives subject to linear affine constraints are still convex.","As a consequence, for linear affine dynamical systems, there are no local minimizers and must eventually search the whole space.","A path is “ergodic” with respect to a distribution if the fraction of time the trajectory spends in a region in space is equivalent to the measure associated with that region."],["Ergodic Sequential Action Control","Ergodic control determines a discrete set of control inputs that aims to reduce the ergodic metric over time.","Ergodic Sequential Action Control (eSAC) is a control scheme that generates ergodic control actions.","Although trajectory optimization has been used to generate control synthesis for ergodic exploration [23], [21], sequential action control provides a closed-form control that can immediately respond to changes in the ergodic metric in an infinite horizon setting [40].","In comparison with sample-based planners, SAC incorporates dynamical constraints.","In particular, non-linear feedback control is desired when dealing with collisions and contact information that must be incorporated at the time of measurement acquisition.","Thus, the motivating factor for using eSAC is a combination of compactness in its formulation and the ability to quickly react to measurements.","Assuming control-affine dynamics of the form $\\dot{x} = f(x(t), u) = g(x) + h(x) u$ where $x \\in \\mathbb {R}^n$ and $u \\in \\mathbb {R}^m$ , the trajectory cost is $J(x(t)) = q \\mathcal {E} (x(t)) + \\int _{t_0}^{t_f} \\frac{1}{2} u(\\tau )^T R u(\\tau ) d\\tau $ where $q \\in \\mathbb {R}$ weights the ergodic metric and $R \\in \\mathbb {R}^{m \\times m}$ is a positive definite matrix.","During each sampling instance, eSAC computes the optimal action to reduce the cost function $J(x(t))$ in the following steps:"],["Prediction","eSAC forward simulates the system dynamics starting from the sampled sensor state $x_0$ at time $t_0$ to a finite time horizon $t_0 + T$ assuming a default control $u_0$ similar to NMPC methods [41], [42].","Cost sensitivity is computed by backwards simulating an adjoint variable, $\\rho \\in \\mathbb {R}^n$ that satisfies the following differential equation $\\dot{\\rho } &=& 2 \\frac{q}{T} \\sum _{k \\in \\mathbb {Z}^n} \\Lambda _k (c_k (x(t)) - \\phi _k) \\frac{\\partial }{\\partial x} F_k (x(t)) \\nonumber \\\\& &\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ - \\frac{\\partial }{\\partial x} f(x, u_0)^T \\rho \\\\&&\\text{subject to } \\rho (t_0 + T) = \\vec{0} \\nonumber .$ The pair $x(t)$ and $\\rho (t)$ are used to compute the optimal control action.","Figure: (A) Simulated time sequence of tactile-based eSAC estimating a square (green) in 𝒳⊂ℝ 2 ⊂0,1×0,1\\mathcal {X} \\subset \\mathbb {R}^2 \\subset \\left[0,1 \\right] \\times \\left[0,1 \\right] .","Robot trajectory (magenta) translates in time from the previous final state (shown as a circle) to the current initial condition (shown as a magenta square).","The boundary of the surface estimate is shown as a black line and the posterior likelihood is shown in the density plot.","High likelihood of acquiring a collision is shown in red and low likelihood is shown in blue.","Bottom Row: Γ\\Gamma metric is defined as Γ=∫ x φ emp (x)-φ actual (x) 2 dx\\Gamma = \\int _x \\left(\\phi _{\\text{emp}}(x) - \\phi _{\\text{actual}} (x) \\right) ^2 dx the integrated difference of the shape and the estimated shape squared.","The Γ\\Gamma measure drops quickly after first contact and then remains at an equilibrium as the shape estimate is updated.","(B) Note the posterior at the end of the simulation (top subplot) resembles the Expected Information Density (EID) (see for derivation of EID) of the parametric model of the square shape (bottom subplot)."],["Calculate Optimal Control Actions","eSAC produces a schedule of control actions $u^* (t) \\in [t_0, t_0 + T]$ that optimizes $J_u = \\frac{1}{2} \\int _{t_0}^{t_0 + T} \\left[ \\frac{dJ_{\\text{t}rack}}{d \\lambda } - \\alpha _d \\right]^2 + \\Vert u(t)\\Vert _R^2 dt \\\\\\text{where } \\frac{dJ_{\\text{t}rack}}{d \\lambda } = \\rho (t)^T (f(x(t), u) - f(x(t), u_0)).$ The term $\\frac{dJ_{\\text{t}rack}}{d \\lambda }$ computes the rate of change of the cost with respect to a switch of infinitesimal duration $\\lambda $ [40], [43].","The value $\\alpha _d \\in \\mathbb {R}^-$ dictates the aggressiveness of the control (more negative values tend to saturate the control).","In this work a value for $\\alpha _d = -555$ was used.","The closed form solution of $u^*$ is then given as $u^* = (\\Lambda + R^T)^{-1} [\\gamma u_0 + h(x)^T \\rho \\alpha _d]$ where $\\gamma \\triangleq h(x)^T \\rho \\rho ^T h(x)$ .","Additional information on the derivation of SAC can be found in [40].","In this work, we used values of $q=30$ , $R=\\text{diag} ( [ 0.01, 0.01 ] )$ , $T=0.8$ for eSAC.","eSAC [1] given $x_0, c_{k,0}, \\phi _{k,0}, t_{curr}, T, t_s, i=0$ $(x(t), \\rho (t)) \\leftarrow $ prediction($x_0, t_{curr}, T$ ) $u^* (t) \\leftarrow $ calcControl($x(t), \\rho (t)$ ) return $u^*(t) \\in \\left[t_{curr}, t_{cuss}+t_s \\right]$"],["Shape Estimation","The process for shape estimation is described in Fig.","REF .","Shape estimates are obtained through repeated samples of the search space using directed motion of the sensor, as calculated by eSAC.","The samples are processed using kernel basis functions in order to create decision boundaries.","The decision boundaries are then processed into a posterior likelihood estimate using a common method in machine learning to generate statistics on a classification fit known as Platt Scaling [44], [45], [46], [47].","The returned spatial statistics on the fit is used for active sampling of the search space, thus updating the shape estimates.","The process iterates at a user-specified sampling frequency.","We further explain the process in the following sections.","Figure: Block diagram for shape estimation.","The measurements are processed by a Gaussian Kernel and made into a shape estimate.","With Platt Scaling, the shape estimate is transformed into a posterior estimate of the likelihood of acquiring a specific binary sensor measurement, which then is converted into control using eSAC."],["Binary Measurements for Shape Estimation","Given a set of measurements $y_k \\in \\left[0, 1 \\right]$ at time indexed by $k$ sampled at the corresponding set of sensor state $x_k$ , shape estimation is accomplished by probabilistic classification methods [18], [16], [25], [26].","Using the set of measurements, a decision boundary for the object is approximated as $\\phi (x) \\approx \\sum _{k} \\alpha _k y_k K(x_k, x) + b$ where $K(x_k, x)$ is the kernel basis function that determines the basis shape of the decision boundary and the parameters $\\alpha _k$ and $b$ are optimized parameters based on the set of $y_k$ and $x_k$ .","The choice of kernel basis function determines the shape of the decision function [48].","Arbitrary shape estimation is desired so a Gaussian kernel basis (also known as a radial basis function) is chosen [49].","However, any kernel basis function can be used as a design choice.","The Gaussian Kernel is given as $K(x_k,x) = e^{-\\frac{(x_k - x)^2}{\\sigma ^2}} .$ A kernel of this form maps data into infinite dimensional feature space which provides flexibility for decision boundaries [50].","(In this work, the python package Scikit-Learn [51] is used for optimizing the boundary fit.)","Figure: Time series shape estimation of diamond and clover shape.","The posterior likelihood estimate is shown at 0.1,1,2,6,110.1, 1, 2, 6, 11, and 30 seconds.","Final shape estimates at 30s30s is shown in the enclosed box.","Likelihood of measuring a collision measurement is denoted by the contours.","The sensor trajectory (shown in magenta) traverses from the previous time window onto the current posterior.","Shape estimates are depicted in the black line.","Actual continuous contact motion does not begin until around 6 seconds into the simulation.","Nonetheless, through non-contact motion, the algorithm is able to estimate the shapes.","After continuous contact occurs, the algorithm refines the shape estimate.","The final shape estimates are highlighted in the enclosed rectangular box.","Note that the shape estimate (denoted as the black line) for the square is underneath the actual shape.","With the clover, part of the shape estimate is outside the shape boundary."],["Posterior Likelihood Estimate (Platt Scaling)","As the measurement model is unknown, we design the controller to be ergodic with respect to the likelihood distribution of acquiring a collision measurement.","The distribution is updated for each measurement via Platt Scaling, defined as $P( y_k = 1 | x) = \\frac{1}{1 + e^{A \\phi (x) + B}} ,$ where $A$ and $B$ are solved through a regression fit and $\\phi (x)$ is the current shape estimate.","Figure: Time series shape estimation of a spatial sine wave.","High likelihood probabilities of a collision are shown in red and low probabilities in blue.","Start point of sensor shown as the magenta square.","By following the trajectory from 4.5s4.5s to 5.5s5.5s, the shape estimate depicted by the black line is updated by the sensor even without a collision.","At 6s6s the middle peak of the sine wave has been estimated and the sensor trajectory has not collided with the sine wave boundary.Figure: Modularity in the eSAC algorithm is shown with the example shapes used in and their approach for generating shape estimate and a target distribution.","(A) A Gaussian Process (GP) is used for shape representation and the uncertainty in the GP fit as the target distribution (this implies the robot will search near regions of high uncertainty in the fit).","(B) Support Vector Machine (SVM) approach for shape representation using the posterior likelihood as the target distribution.","Both methods using eSAC provide similar results in final shape estimates."],["Simulation Results","Active shape estimation in $\\mathbb {R}^2$ is done with double integrator dynamics given as $\\dot{{\\bf x}} = f({\\bf x},u) =\\begin{bmatrix}\\dot{x}, &\\dot{y}, &u_1, &u_2\\end{bmatrix}^T .$ The binary measurements are collisions detected during the crossing of the boundary function $\\phi (x) = 0$ .","Estimation in $\\mathbb {R}^3$ is done with a similar double integrator model with dynamics $\\dot{{\\bf x}} = f({\\bf x},u) =\\begin{bmatrix}\\dot{x}, &\\dot{y}, &\\dot{z}, &u_1, &u_2, &u_3\\end{bmatrix}^T .$ No assumptions are needed on the location of the object and we initialize with a uniform distribution as the target distribution.","Figure: Time series of eSAC exploring multiple objects.","The posterior is viewed at times 0.1,1,2,6,11,300.1, 1, 2, 6, 11, 30.","Trajectories start at the magenta square.","In each subplot, the sensor trajectory (magenta) starts from the endpoint of the previous time instant.","Although the robot has no prior knowledge of the number of objects in the sensor state, the algorithm is still able to correctly estimate all the objects.","Therefore, regardless of the number of shapes, the algorithm is still able to estimate all the shapes in the environment given a large enough time horizon.Figure: Baxter Robot used for experiments.","An object is pinned down to the table for shape estimation using the end-effector probe.","The probe itself has no sensors.","Thus, the measured joint torques are used to detect collisions based on rapidly changing measurements at specific joint locations near the end-effector.First, we demonstrate that the proposed algorithm's shape estimate converges to the actual shape.","Figure REF shows time evolution of the posterior as well as the resulting shape estimate.","The estimate depicted by the black line in Fig.","REF eventually converges to the shape, although much of the shape restructuring is accomplished from non-contact motion up until $6s$ .","Moreover, by comparing the posterior and the windowed sensor path, it is shown in Fig.","REF that the sensor path is drawn to high likelihood densities.","This ensures that the posterior estimate is verified and updated as new sensor information is acquired.","Note that the sensor is unable to access the interior of the shape, resulting in high likelihood probabilities within the shape.","The posterior likelihood of the square shape estimate shows high likelihood probabilities near the corners of the square.","As an aside, we take note of the similarities that the resultant likelihood has with the expected information density (EID) of the known shape measurement model (see [21] for EID derivation).","Specifically, high likelihood near the corners correspond to the similar large expectation of information for a square shape.","If we define the measurement model of the known square as $\\Upsilon ({x}, x)$ where $x$ is search space and ${x}$ is the robot's state space, then a measure of information is the Fisher Information Matrix [52] defined by $I ({x}, x) = \\frac{\\partial \\Upsilon }{\\partial x} ^T \\Sigma ^{-1} \\frac{\\partial \\Upsilon }{\\partial x},$ where $\\Sigma $ is the measurement covariance.","Assuming the measurement model is of the same form as in equation (REF ) then the region with the largest information is at the corners where the slopes of the edges collide.","Thus, large likelihood estimates should exist near corners if they have not been previously searched.","Figure: eSAC trajectory for a Torus 𝕊 2 \\mathbb {S}^2 in 𝒳⊂ℝ 3 ⊂0,1×0,1×0,1\\mathcal {X} \\subset \\mathbb {R}^3 \\subset \\left[0,1 \\right] \\times \\left[0,1 \\right] \\times \\left[0,1 \\right] search space.","Top: In magenta, the robot trajectory on a Torus.","Bottom: Density plot of the shape estimate.","Although the simulation was run for 40s40s, the use of eSAC is shown to capture the doughnut shape inner circle feature which defines the torus shape.","Utilizing an array of tactile sensors would reduce the error on the outer ring of the torus as larger coverage is needed for ℝ 3 \\mathbb {R}^3.Figure: Comparison of gEER and eSAC for detecting and refining multiple shape estimates.","We randomly selected 20 initial conditions with uniform distribution in the search space for both gEER and eSAC methods.","(A) Trial samples shown from both eSAC and gEER.","eSAC consistently detects all objects in the search space.","(B) Refined estimates of the shape after detection.","eSAC is shown to coarsely explore the whole region while spending a larger fraction of time around the shape estimates.","gEER is driven by larger values which reduce uncertainty which is susceptible to local minima.","(C) eSAC is shown to detect all shapes in the environment within 40s40s of simulation.","In all 20 trials, gEER does not successfully detect all shapes in the environment.Figure: Baxter experimental results for shape estimation.","(A) shows the shapes used for estimation.","(B) shows the location of Baxter's probe at the end of the estimation as well as the estimate of the shape as the dotted black line.","The posterior at the end of the experiment is shown in the contour map.","Note the prominent posterior likelihood in the triangle estimation around the edges are colored with high likelihood estimate, indicating that the estimated measurement model retains information about corners that is useful for localization.","Error in shape area is within 2.5%,7.5%,2.5 \\%, 7.5 \\%, and 10.0%10.0 \\% for the circle, triangle, and square respectively.","(Coarse discretization of the distribution was created to generate the figure to prevent run-time degradation on Baxter which results in an off-centered visual error of 1cm1cm in the distribution.",")The non-contact motion shown in Fig.","REF is beneficial for tactile-based exploration.","The sensor trajectory from time $4.5s$ to $6s$ updates the shape estimation of the sine wave boundary without measuring a collision for $t \\in [4.5,6s]$ .","Knowing that no contact has occurred indicates that the shape is not in the current sensor state, but likely in other unexplored regions of sensor state.","(Interestingly, tactile sensor arrays appear often in biological systems such as the rat whisker system [53]).","The algorithm is shown in Fig.","REF to estimate both a clover shape and a diamond shape.","Between times $2s$ to $6s$ in Fig.","REF , non-contact motion is shown to drive the sensor trajectory towards an enclosing path that estimates the location of the center of the shape.","After a few seconds, the sensor regularly is in contact with the shape as the estimate converges.","The final estimates of the diamond and clover are shown by black lines in the rectangular box in Fig.","REF .","The final time posterior shown in Fig.","REF and Fig.","REF show a resemblance in the posterior likelihood near the edges of the shape.","This similarity in posterior estimates shows the algorithm approximates regions where there is high expected tactile information.","Shown in Fig.","REF , multiple objects are allocated randomly in the sensor state.","The sensor has no prior knowledge of where the shapes are, nor the number of shapes.","The sensor contacts the first shape around $1s$ (Fig.","REF ), then between 2 and $6s$ the sensor comes into contact with the remaining two shapes.","Following the trajectory (magenta) traversed from time $11-30s$ shown in Fig.","REF , the sensor distributes the time spent amongst the three shapes.","Because the ergodic metric is convex with respect to the information distribution and thus convex with respect to the shape, unexplored regions of sensor state must be explored [22].","We further expand upon multiple shape detection with a comparison with a version of Greedy expected entropy reduction (gEER) exploration algorithm used in [21], [54], [55], [56], [57].","The algorithm samples nearby states centered at the robot probe's current location for regions with the largest probability of acquiring a collision measurement.","The algorithm then moves the probe to that location, sampling along the way to adjust the shape estimate.","In Fig.","REF , we run 20 trials of uniformly sampled initial conditions for both eSAC and gEER.","Figure REF (A) shows sample shape estimates for $80s$ simulations of both algorithms.","eSAC is shown to detect all the shapes and begin shape refinement while gEER algorithm at most detects two shapes, but refines only one.","We can see in (C) that eSAC detects all shapes within $40s$ of simulation time whereas the gEER detects two shapes at most.","The time it takes to detect objects does depend on the distance of the shape and the control saturation of the robot probe which we maintained constant with both algorithms.","In (B), we see the difference in the algorithm's area coverage.","In particular, eSAC covers most of the search space coarsely with densely collected collision measurements around the shape.","Modularity with respect to shape representation is demonstrated by comparing with the shapes used in [19] with the Gaussian Process (GP) for shape estimation (see Fig.","REF ).","Here, eSAC is shown to work with both a GP and SVM for shape estimates.","In particular, in Fig.","REF (A), eSAC uses the touch point selection metric used in [19] for control synthesis which is based on the covariance of the GP fit.","In comparison with eSAC, the work done in [19] uses an Markov Decision Process (MDP) in order to control a robot manipulator towards acquiring touch data for shape estimation.","In [19], a grid is defined and a path planner is used to traverse the grid.","The large overhead state discretization and motion planning code that is necessary for the MDP formulation is not required for eSAC.","Moreover, eSAC automatically encodes robot dynamic constraints.","Notably, eSAC can be used with other visual-based estimators that feed initial shape estimates such that tactile data is used to refine the estimate.","This algorithm can be trivially extended to $\\mathbb {R}^3$ as shown in simulation.","Figure REF shows the sensor trajectory in the top plot (magenta) actively sensing the torus shape.","Shape estimation of the torus is shown to be successful in the lower sub-plot of Fig.","REF resulting in scalability to objects in $\\mathbb {R}^3$ .","Performance for estimation in $\\mathbb {R}^3$ depends on the robot dynamical constraints as well as the sensor area coverage."],["Experimental Results","Experiments are done using the Baxter robot (Fig.","REF ).","Binary measurements are taken by the end-effector probe during a large change of joint torques, indicating a collision has occurred.","During collision, controller weight $R$ prevents the robot from dragging along the object.","In the case that the control weight $R$ is not significant, heuristics are used to prevent overexertion.","Baxter's end-effector is used as a probe for the $\\mathbb {R}^2$ search space.","No assumptions are needed on the location of the object or the height of the object.","Experimental results show that shape estimation can be accomplished with a robotic system with multiple sensors (Fig.","REF ).","Although there is odometry error within the robot's joint states, shape estimations are still visually seen to match.","Within the posterior regime seen in Fig.","REF , the probability densities for the shapes with corners are shown to have unique features.","In particular, the posterior for the triangle in Fig.","REF extends outwards along the three corners indicating high probability of obtaining a collision measurement.","Further measurements surrounding the edge regions reduce the likelihood of acquiring estimate which can be seen in with results from estimating the square shape."],["Conclusions and Future Work","In this paper, an algorithm is presented for shape estimation using a binary sensor combined with ergodic exploration.","It is shown that a binary measurement sensor can be treated as a lower resolution tactile sensor for shape estimation with active sensing.","In addition, using ergodic exploration as the principle for active sensing uses sensor motion to capture shape features during estimation.","The algorithm is shown to estimate shapes in $\\mathbb {R}^2$ and $\\mathbb {R}^3$ as well as an any number of shapes in the sensor domain.","The algorithm is shown to be modular with respect to choice of shape representation.","The algorithm is shown to have potential applications for active sensing with low resolution sensors with respect to shape estimation as well as applications in localization.","Future research directions include the use of multiple sensors for estimation.","Another direction is to augment the current algorithm to include localization using the estimated measurement model.","Thus, simultaneous localization and shape estimation should be possible."]],"string":"[\n [\n \"Ergodic Exploration using Binary Sensing for Non-Parametric Shape\\n Estimation\"\n ],\n [\n \"Abstract Current methods to estimate object shape---using either vision or touch---generally depend on high-resolution sensing.\",\n \"Here, we exploit ergodic exploration to demonstrate successful shape estimation when using a low-resolution binary contact sensor.\",\n \"The measurement model is posed as a collision-based tactile measurement, and classification methods are used to discriminate between shape boundary regions in the search space.\",\n \"Posterior likelihood estimates of the measurement model help the system actively seek out regions where the binary sensor is most likely to return informative measurements.\",\n \"Results show successful shape estimation of various objects as well as the ability to identify multiple objects in an environment.\",\n \"Interestingly, it is shown that ergodic exploration utilizes non-contact motion to gather significant information about shape.\",\n \"The algorithm is extended in three dimensions in simulation and we present two dimensional experimental results using the Rethink Baxter robot.\"\n ],\n [\n \"Introduction\",\n \"Tactile sensing is often associated with shape estimation [1], [2], [3], [4] and mapping problems [5] in conditions where visual sensing may be limited.\",\n \"In some instances, tactile sensing is used to supplement vision-based sensing to improve shape estimates [6], [7].\",\n \"The richness of touch as a sensing modality is underscored by the development of novel tactile sensors [8], [9], [10], [11] for use in a myriad of applications ranging from robot-assisted tumor detection [7], to texture recognition, and feature localization [12], [13].\",\n \"These advances in tactile sensor technology require corresponding advances in active exploration algorithms and the interpretation of tactile-based sensor data.\",\n \"Recent approaches for active exploration use random sampling-based search algorithms for sensor motion planning that require a separate controller for path following [14].\",\n \"Other methods use task-specific probabilistic spatial methods for shape estimation that are updated based on sensor poses that minimize measurement uncertainty [2], [15], [16], [17], [18].\",\n \"A common approach in the literature is motion planning for sensing followed by feedback regulation of the generated plan [19], [20], [18], [17].\",\n \"Moreover, most methods focus on one object at a time.\",\n \"In contrast, the presented work integrates planning and control into a feedback law.\",\n \"As a result, the method uses sensor motion to actively sense for time-varying spatial information.\",\n \"We take the framework in [21] and use the feedback law developed here to enable real-time execution during shape estimation.\",\n \"Lastly, the feedback in the active sensing algorithm compensates for low resolution sensors and the specification of the algorithm is independent of the number of objects.\",\n \"We show that ergodic exploration with Sequential Action Control (eSAC) can be used for active exploration with respect to time-varying tactile-based distributed information [22], [23].\",\n \"Previous applications of ergodic theory utilize parametric measurement models for localization tasks [24], [21].\",\n \"The current work demonstrates localization and estimation of non-parametric shapes using a binary sensor model with classification methods.\",\n \"In contrast to other methods of active sensing for shape estimation [19], [25], [26], [16], [18], the proposed algorithm automatically encodes dynamical constraints without any overhead spatial discretization or motion planning.\",\n \"In addition, the algorithm incorporates sensor measurement information to actively adjust shape estimates and synthesize tactile-information based control actions.\",\n \"As a result, the algorithm automatically adjusts the control synthesis for multiple objects in an environment.\",\n \"Notably, ergodic exploration uses non-contact motion data (sensor motion not in contact with an object) [27], [28] to improve the shape estimate.\",\n \"The idea of utilizing free space is often found in other related works of pose estimation and tracking [27], [28], [29], [30] and is emphasized in our work.\",\n \"As a final contribution, we show the algorithm is modular with respect to the choice of shape representation and tactile information distribution.\",\n \"The paper outline is as follows: Section motivates the use of a binary contact sensor and outlines the implications for tactile sensing.\",\n \"Section describes the robot control algorithm for active exploration and motivates ergodicity as a measure for exploration.\",\n \"Section explains how shape estimation is achieved and how measurements update the shape estimate and the control policy.\",\n \"Experimental and simulated results and conclusion are shown in sections and respectively.\"\n ],\n [\n \"Tactile Information\",\n \"In biological systems, tactile sensing is generally an active process that incorporates feedback from multiple environmental cues [31], [32].\",\n \"Humans, for instance, use fingers to grasp objects for manipulation and feature detection.\",\n \"Many rodents use hair-like appendages (whiskers) for tactile sensing [33], [34], [35].\",\n \"These biological sensors have a large set of actuators and sensor channels, thus the process control for active sensing is complex.\",\n \"We investigate a lower resolution version of tactile sensing that is simple to control for active sensing.\",\n \"Specifically, we show that a binary form of tactile sensing (i.e., collision detection) [10], [8], has enough information for shape estimation, when combined with an active exploration algorithm that automatically takes into account regions of shape information.\",\n \"A binary tactile measurement model consists of a transition state from “no collision” to “collision” and vice versa.\",\n \"We denote this measurement model as $\\\\Upsilon (x) ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\hfill 1, \\\\hfill & \\\\phi (x) \\\\le 0 \\\\\\\\\\\\hfill 0, \\\\hfill & \\\\phi (x) > 0 \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ where $x$ is the sensor state and $\\\\phi (x)$ is a boundary function that determines a transition state if $\\\\phi (x) \\\\le 0$ (output of 1).\",\n \"The goal of shape estimation is to determine $\\\\phi (x)$ .\"\n ],\n [\n \"Motivation\",\n \"Typical algorithms used in active exploration and information maximization cast the problem of information acquisition as “exploratory” (wide spread search for diffuse information such as localization) or “exploitative” (direct search for highly structured information such as shape contours) [36], [37].\",\n \"Ergodic exploration [22], [38], [23], [24], [21] is responsive to both diffuse information densities and highly focused information.\",\n \"Thus, ergodic control seamlessly encodes wide spread coverage for diffuse information and localized search for focused information for both needs.\"\n ],\n [\n \"Ergodic Metric\",\n \"A trajectory $x(t)$ of a sensor is ergodic with respect to a probability (“target”) distribution $\\\\Phi (x)$ when the fraction of time the trajectory spends in an area within the search space is equal to the spatial statistics across the search space [22], [39].\",\n \"The ergodic metric that measures this characteristic is given as [22], [23], $ \\\\mathcal {E} (x(t)) = \\\\sum _{k \\\\in {\\\\mathbb {Z}}^n} \\\\Lambda _{k} [c_{k}(x(t)) - \\\\phi _{k}]^2$ where $c_{k} (x(t)) & = & \\\\frac{1}{t_f - t_0}\\\\int _{t_0}^{t_f} F_k (x(\\\\tau )) d\\\\tau , \\\\\\\\\\\\phi _{k} & = & \\\\int _{\\\\mathcal {X}} \\\\Phi ({\\\\bf x}) F_k ({\\\\bf x}) d{\\\\bf x} .$ Here, $F_k (x)$ is a Fourier basis function, and $\\\\Lambda _{k}$ are weights described in [22].\",\n \"The target distribution $\\\\Phi (x)$ is what drives the control synthesis for active-exploration.\",\n \"Comparison between robot trajectory and distribution is done with equation (REF ) which takes the Fourier power series decomposition using (3) and (4) and directly compares the statistics of a trajectory with that of the spatial statistics $\\\\Phi (x)$ .\",\n \"As the ergodic metric is convex [22], convex objectives subject to linear affine constraints are still convex.\",\n \"As a consequence, for linear affine dynamical systems, there are no local minimizers and must eventually search the whole space.\",\n \"A path is “ergodic” with respect to a distribution if the fraction of time the trajectory spends in a region in space is equivalent to the measure associated with that region.\"\n ],\n [\n \"Ergodic Sequential Action Control\",\n \"Ergodic control determines a discrete set of control inputs that aims to reduce the ergodic metric over time.\",\n \"Ergodic Sequential Action Control (eSAC) is a control scheme that generates ergodic control actions.\",\n \"Although trajectory optimization has been used to generate control synthesis for ergodic exploration [23], [21], sequential action control provides a closed-form control that can immediately respond to changes in the ergodic metric in an infinite horizon setting [40].\",\n \"In comparison with sample-based planners, SAC incorporates dynamical constraints.\",\n \"In particular, non-linear feedback control is desired when dealing with collisions and contact information that must be incorporated at the time of measurement acquisition.\",\n \"Thus, the motivating factor for using eSAC is a combination of compactness in its formulation and the ability to quickly react to measurements.\",\n \"Assuming control-affine dynamics of the form $\\\\dot{x} = f(x(t), u) = g(x) + h(x) u$ where $x \\\\in \\\\mathbb {R}^n$ and $u \\\\in \\\\mathbb {R}^m$ , the trajectory cost is $J(x(t)) = q \\\\mathcal {E} (x(t)) + \\\\int _{t_0}^{t_f} \\\\frac{1}{2} u(\\\\tau )^T R u(\\\\tau ) d\\\\tau $ where $q \\\\in \\\\mathbb {R}$ weights the ergodic metric and $R \\\\in \\\\mathbb {R}^{m \\\\times m}$ is a positive definite matrix.\",\n \"During each sampling instance, eSAC computes the optimal action to reduce the cost function $J(x(t))$ in the following steps:\"\n ],\n [\n \"Prediction\",\n \"eSAC forward simulates the system dynamics starting from the sampled sensor state $x_0$ at time $t_0$ to a finite time horizon $t_0 + T$ assuming a default control $u_0$ similar to NMPC methods [41], [42].\",\n \"Cost sensitivity is computed by backwards simulating an adjoint variable, $\\\\rho \\\\in \\\\mathbb {R}^n$ that satisfies the following differential equation $\\\\dot{\\\\rho } &=& 2 \\\\frac{q}{T} \\\\sum _{k \\\\in \\\\mathbb {Z}^n} \\\\Lambda _k (c_k (x(t)) - \\\\phi _k) \\\\frac{\\\\partial }{\\\\partial x} F_k (x(t)) \\\\nonumber \\\\\\\\& &\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ - \\\\frac{\\\\partial }{\\\\partial x} f(x, u_0)^T \\\\rho \\\\\\\\&&\\\\text{subject to } \\\\rho (t_0 + T) = \\\\vec{0} \\\\nonumber .$ The pair $x(t)$ and $\\\\rho (t)$ are used to compute the optimal control action.\",\n \"Figure: (A) Simulated time sequence of tactile-based eSAC estimating a square (green) in 𝒳⊂ℝ 2 ⊂0,1×0,1\\\\mathcal {X} \\\\subset \\\\mathbb {R}^2 \\\\subset \\\\left[0,1 \\\\right] \\\\times \\\\left[0,1 \\\\right] .\",\n \"Robot trajectory (magenta) translates in time from the previous final state (shown as a circle) to the current initial condition (shown as a magenta square).\",\n \"The boundary of the surface estimate is shown as a black line and the posterior likelihood is shown in the density plot.\",\n \"High likelihood of acquiring a collision is shown in red and low likelihood is shown in blue.\",\n \"Bottom Row: Γ\\\\Gamma metric is defined as Γ=∫ x φ emp (x)-φ actual (x) 2 dx\\\\Gamma = \\\\int _x \\\\left(\\\\phi _{\\\\text{emp}}(x) - \\\\phi _{\\\\text{actual}} (x) \\\\right) ^2 dx the integrated difference of the shape and the estimated shape squared.\",\n \"The Γ\\\\Gamma measure drops quickly after first contact and then remains at an equilibrium as the shape estimate is updated.\",\n \"(B) Note the posterior at the end of the simulation (top subplot) resembles the Expected Information Density (EID) (see for derivation of EID) of the parametric model of the square shape (bottom subplot).\"\n ],\n [\n \"Calculate Optimal Control Actions\",\n \"eSAC produces a schedule of control actions $u^* (t) \\\\in [t_0, t_0 + T]$ that optimizes $J_u = \\\\frac{1}{2} \\\\int _{t_0}^{t_0 + T} \\\\left[ \\\\frac{dJ_{\\\\text{t}rack}}{d \\\\lambda } - \\\\alpha _d \\\\right]^2 + \\\\Vert u(t)\\\\Vert _R^2 dt \\\\\\\\\\\\text{where } \\\\frac{dJ_{\\\\text{t}rack}}{d \\\\lambda } = \\\\rho (t)^T (f(x(t), u) - f(x(t), u_0)).$ The term $\\\\frac{dJ_{\\\\text{t}rack}}{d \\\\lambda }$ computes the rate of change of the cost with respect to a switch of infinitesimal duration $\\\\lambda $ [40], [43].\",\n \"The value $\\\\alpha _d \\\\in \\\\mathbb {R}^-$ dictates the aggressiveness of the control (more negative values tend to saturate the control).\",\n \"In this work a value for $\\\\alpha _d = -555$ was used.\",\n \"The closed form solution of $u^*$ is then given as $u^* = (\\\\Lambda + R^T)^{-1} [\\\\gamma u_0 + h(x)^T \\\\rho \\\\alpha _d]$ where $\\\\gamma \\\\triangleq h(x)^T \\\\rho \\\\rho ^T h(x)$ .\",\n \"Additional information on the derivation of SAC can be found in [40].\",\n \"In this work, we used values of $q=30$ , $R=\\\\text{diag} ( [ 0.01, 0.01 ] )$ , $T=0.8$ for eSAC.\",\n \"eSAC [1] given $x_0, c_{k,0}, \\\\phi _{k,0}, t_{curr}, T, t_s, i=0$ $(x(t), \\\\rho (t)) \\\\leftarrow $ prediction($x_0, t_{curr}, T$ ) $u^* (t) \\\\leftarrow $ calcControl($x(t), \\\\rho (t)$ ) return $u^*(t) \\\\in \\\\left[t_{curr}, t_{cuss}+t_s \\\\right]$\"\n ],\n [\n \"Shape Estimation\",\n \"The process for shape estimation is described in Fig.\",\n \"REF .\",\n \"Shape estimates are obtained through repeated samples of the search space using directed motion of the sensor, as calculated by eSAC.\",\n \"The samples are processed using kernel basis functions in order to create decision boundaries.\",\n \"The decision boundaries are then processed into a posterior likelihood estimate using a common method in machine learning to generate statistics on a classification fit known as Platt Scaling [44], [45], [46], [47].\",\n \"The returned spatial statistics on the fit is used for active sampling of the search space, thus updating the shape estimates.\",\n \"The process iterates at a user-specified sampling frequency.\",\n \"We further explain the process in the following sections.\",\n \"Figure: Block diagram for shape estimation.\",\n \"The measurements are processed by a Gaussian Kernel and made into a shape estimate.\",\n \"With Platt Scaling, the shape estimate is transformed into a posterior estimate of the likelihood of acquiring a specific binary sensor measurement, which then is converted into control using eSAC.\"\n ],\n [\n \"Binary Measurements for Shape Estimation\",\n \"Given a set of measurements $y_k \\\\in \\\\left[0, 1 \\\\right]$ at time indexed by $k$ sampled at the corresponding set of sensor state $x_k$ , shape estimation is accomplished by probabilistic classification methods [18], [16], [25], [26].\",\n \"Using the set of measurements, a decision boundary for the object is approximated as $\\\\phi (x) \\\\approx \\\\sum _{k} \\\\alpha _k y_k K(x_k, x) + b$ where $K(x_k, x)$ is the kernel basis function that determines the basis shape of the decision boundary and the parameters $\\\\alpha _k$ and $b$ are optimized parameters based on the set of $y_k$ and $x_k$ .\",\n \"The choice of kernel basis function determines the shape of the decision function [48].\",\n \"Arbitrary shape estimation is desired so a Gaussian kernel basis (also known as a radial basis function) is chosen [49].\",\n \"However, any kernel basis function can be used as a design choice.\",\n \"The Gaussian Kernel is given as $K(x_k,x) = e^{-\\\\frac{(x_k - x)^2}{\\\\sigma ^2}} .$ A kernel of this form maps data into infinite dimensional feature space which provides flexibility for decision boundaries [50].\",\n \"(In this work, the python package Scikit-Learn [51] is used for optimizing the boundary fit.)\",\n \"Figure: Time series shape estimation of diamond and clover shape.\",\n \"The posterior likelihood estimate is shown at 0.1,1,2,6,110.1, 1, 2, 6, 11, and 30 seconds.\",\n \"Final shape estimates at 30s30s is shown in the enclosed box.\",\n \"Likelihood of measuring a collision measurement is denoted by the contours.\",\n \"The sensor trajectory (shown in magenta) traverses from the previous time window onto the current posterior.\",\n \"Shape estimates are depicted in the black line.\",\n \"Actual continuous contact motion does not begin until around 6 seconds into the simulation.\",\n \"Nonetheless, through non-contact motion, the algorithm is able to estimate the shapes.\",\n \"After continuous contact occurs, the algorithm refines the shape estimate.\",\n \"The final shape estimates are highlighted in the enclosed rectangular box.\",\n \"Note that the shape estimate (denoted as the black line) for the square is underneath the actual shape.\",\n \"With the clover, part of the shape estimate is outside the shape boundary.\"\n ],\n [\n \"Posterior Likelihood Estimate (Platt Scaling)\",\n \"As the measurement model is unknown, we design the controller to be ergodic with respect to the likelihood distribution of acquiring a collision measurement.\",\n \"The distribution is updated for each measurement via Platt Scaling, defined as $P( y_k = 1 | x) = \\\\frac{1}{1 + e^{A \\\\phi (x) + B}} ,$ where $A$ and $B$ are solved through a regression fit and $\\\\phi (x)$ is the current shape estimate.\",\n \"Figure: Time series shape estimation of a spatial sine wave.\",\n \"High likelihood probabilities of a collision are shown in red and low probabilities in blue.\",\n \"Start point of sensor shown as the magenta square.\",\n \"By following the trajectory from 4.5s4.5s to 5.5s5.5s, the shape estimate depicted by the black line is updated by the sensor even without a collision.\",\n \"At 6s6s the middle peak of the sine wave has been estimated and the sensor trajectory has not collided with the sine wave boundary.Figure: Modularity in the eSAC algorithm is shown with the example shapes used in and their approach for generating shape estimate and a target distribution.\",\n \"(A) A Gaussian Process (GP) is used for shape representation and the uncertainty in the GP fit as the target distribution (this implies the robot will search near regions of high uncertainty in the fit).\",\n \"(B) Support Vector Machine (SVM) approach for shape representation using the posterior likelihood as the target distribution.\",\n \"Both methods using eSAC provide similar results in final shape estimates.\"\n ],\n [\n \"Simulation Results\",\n \"Active shape estimation in $\\\\mathbb {R}^2$ is done with double integrator dynamics given as $\\\\dot{{\\\\bf x}} = f({\\\\bf x},u) =\\\\begin{bmatrix}\\\\dot{x}, &\\\\dot{y}, &u_1, &u_2\\\\end{bmatrix}^T .$ The binary measurements are collisions detected during the crossing of the boundary function $\\\\phi (x) = 0$ .\",\n \"Estimation in $\\\\mathbb {R}^3$ is done with a similar double integrator model with dynamics $\\\\dot{{\\\\bf x}} = f({\\\\bf x},u) =\\\\begin{bmatrix}\\\\dot{x}, &\\\\dot{y}, &\\\\dot{z}, &u_1, &u_2, &u_3\\\\end{bmatrix}^T .$ No assumptions are needed on the location of the object and we initialize with a uniform distribution as the target distribution.\",\n \"Figure: Time series of eSAC exploring multiple objects.\",\n \"The posterior is viewed at times 0.1,1,2,6,11,300.1, 1, 2, 6, 11, 30.\",\n \"Trajectories start at the magenta square.\",\n \"In each subplot, the sensor trajectory (magenta) starts from the endpoint of the previous time instant.\",\n \"Although the robot has no prior knowledge of the number of objects in the sensor state, the algorithm is still able to correctly estimate all the objects.\",\n \"Therefore, regardless of the number of shapes, the algorithm is still able to estimate all the shapes in the environment given a large enough time horizon.Figure: Baxter Robot used for experiments.\",\n \"An object is pinned down to the table for shape estimation using the end-effector probe.\",\n \"The probe itself has no sensors.\",\n \"Thus, the measured joint torques are used to detect collisions based on rapidly changing measurements at specific joint locations near the end-effector.First, we demonstrate that the proposed algorithm's shape estimate converges to the actual shape.\",\n \"Figure REF shows time evolution of the posterior as well as the resulting shape estimate.\",\n \"The estimate depicted by the black line in Fig.\",\n \"REF eventually converges to the shape, although much of the shape restructuring is accomplished from non-contact motion up until $6s$ .\",\n \"Moreover, by comparing the posterior and the windowed sensor path, it is shown in Fig.\",\n \"REF that the sensor path is drawn to high likelihood densities.\",\n \"This ensures that the posterior estimate is verified and updated as new sensor information is acquired.\",\n \"Note that the sensor is unable to access the interior of the shape, resulting in high likelihood probabilities within the shape.\",\n \"The posterior likelihood of the square shape estimate shows high likelihood probabilities near the corners of the square.\",\n \"As an aside, we take note of the similarities that the resultant likelihood has with the expected information density (EID) of the known shape measurement model (see [21] for EID derivation).\",\n \"Specifically, high likelihood near the corners correspond to the similar large expectation of information for a square shape.\",\n \"If we define the measurement model of the known square as $\\\\Upsilon ({x}, x)$ where $x$ is search space and ${x}$ is the robot's state space, then a measure of information is the Fisher Information Matrix [52] defined by $I ({x}, x) = \\\\frac{\\\\partial \\\\Upsilon }{\\\\partial x} ^T \\\\Sigma ^{-1} \\\\frac{\\\\partial \\\\Upsilon }{\\\\partial x},$ where $\\\\Sigma $ is the measurement covariance.\",\n \"Assuming the measurement model is of the same form as in equation (REF ) then the region with the largest information is at the corners where the slopes of the edges collide.\",\n \"Thus, large likelihood estimates should exist near corners if they have not been previously searched.\",\n \"Figure: eSAC trajectory for a Torus 𝕊 2 \\\\mathbb {S}^2 in 𝒳⊂ℝ 3 ⊂0,1×0,1×0,1\\\\mathcal {X} \\\\subset \\\\mathbb {R}^3 \\\\subset \\\\left[0,1 \\\\right] \\\\times \\\\left[0,1 \\\\right] \\\\times \\\\left[0,1 \\\\right] search space.\",\n \"Top: In magenta, the robot trajectory on a Torus.\",\n \"Bottom: Density plot of the shape estimate.\",\n \"Although the simulation was run for 40s40s, the use of eSAC is shown to capture the doughnut shape inner circle feature which defines the torus shape.\",\n \"Utilizing an array of tactile sensors would reduce the error on the outer ring of the torus as larger coverage is needed for ℝ 3 \\\\mathbb {R}^3.Figure: Comparison of gEER and eSAC for detecting and refining multiple shape estimates.\",\n \"We randomly selected 20 initial conditions with uniform distribution in the search space for both gEER and eSAC methods.\",\n \"(A) Trial samples shown from both eSAC and gEER.\",\n \"eSAC consistently detects all objects in the search space.\",\n \"(B) Refined estimates of the shape after detection.\",\n \"eSAC is shown to coarsely explore the whole region while spending a larger fraction of time around the shape estimates.\",\n \"gEER is driven by larger values which reduce uncertainty which is susceptible to local minima.\",\n \"(C) eSAC is shown to detect all shapes in the environment within 40s40s of simulation.\",\n \"In all 20 trials, gEER does not successfully detect all shapes in the environment.Figure: Baxter experimental results for shape estimation.\",\n \"(A) shows the shapes used for estimation.\",\n \"(B) shows the location of Baxter's probe at the end of the estimation as well as the estimate of the shape as the dotted black line.\",\n \"The posterior at the end of the experiment is shown in the contour map.\",\n \"Note the prominent posterior likelihood in the triangle estimation around the edges are colored with high likelihood estimate, indicating that the estimated measurement model retains information about corners that is useful for localization.\",\n \"Error in shape area is within 2.5%,7.5%,2.5 \\\\%, 7.5 \\\\%, and 10.0%10.0 \\\\% for the circle, triangle, and square respectively.\",\n \"(Coarse discretization of the distribution was created to generate the figure to prevent run-time degradation on Baxter which results in an off-centered visual error of 1cm1cm in the distribution.\",\n \")The non-contact motion shown in Fig.\",\n \"REF is beneficial for tactile-based exploration.\",\n \"The sensor trajectory from time $4.5s$ to $6s$ updates the shape estimation of the sine wave boundary without measuring a collision for $t \\\\in [4.5,6s]$ .\",\n \"Knowing that no contact has occurred indicates that the shape is not in the current sensor state, but likely in other unexplored regions of sensor state.\",\n \"(Interestingly, tactile sensor arrays appear often in biological systems such as the rat whisker system [53]).\",\n \"The algorithm is shown in Fig.\",\n \"REF to estimate both a clover shape and a diamond shape.\",\n \"Between times $2s$ to $6s$ in Fig.\",\n \"REF , non-contact motion is shown to drive the sensor trajectory towards an enclosing path that estimates the location of the center of the shape.\",\n \"After a few seconds, the sensor regularly is in contact with the shape as the estimate converges.\",\n \"The final estimates of the diamond and clover are shown by black lines in the rectangular box in Fig.\",\n \"REF .\",\n \"The final time posterior shown in Fig.\",\n \"REF and Fig.\",\n \"REF show a resemblance in the posterior likelihood near the edges of the shape.\",\n \"This similarity in posterior estimates shows the algorithm approximates regions where there is high expected tactile information.\",\n \"Shown in Fig.\",\n \"REF , multiple objects are allocated randomly in the sensor state.\",\n \"The sensor has no prior knowledge of where the shapes are, nor the number of shapes.\",\n \"The sensor contacts the first shape around $1s$ (Fig.\",\n \"REF ), then between 2 and $6s$ the sensor comes into contact with the remaining two shapes.\",\n \"Following the trajectory (magenta) traversed from time $11-30s$ shown in Fig.\",\n \"REF , the sensor distributes the time spent amongst the three shapes.\",\n \"Because the ergodic metric is convex with respect to the information distribution and thus convex with respect to the shape, unexplored regions of sensor state must be explored [22].\",\n \"We further expand upon multiple shape detection with a comparison with a version of Greedy expected entropy reduction (gEER) exploration algorithm used in [21], [54], [55], [56], [57].\",\n \"The algorithm samples nearby states centered at the robot probe's current location for regions with the largest probability of acquiring a collision measurement.\",\n \"The algorithm then moves the probe to that location, sampling along the way to adjust the shape estimate.\",\n \"In Fig.\",\n \"REF , we run 20 trials of uniformly sampled initial conditions for both eSAC and gEER.\",\n \"Figure REF (A) shows sample shape estimates for $80s$ simulations of both algorithms.\",\n \"eSAC is shown to detect all the shapes and begin shape refinement while gEER algorithm at most detects two shapes, but refines only one.\",\n \"We can see in (C) that eSAC detects all shapes within $40s$ of simulation time whereas the gEER detects two shapes at most.\",\n \"The time it takes to detect objects does depend on the distance of the shape and the control saturation of the robot probe which we maintained constant with both algorithms.\",\n \"In (B), we see the difference in the algorithm's area coverage.\",\n \"In particular, eSAC covers most of the search space coarsely with densely collected collision measurements around the shape.\",\n \"Modularity with respect to shape representation is demonstrated by comparing with the shapes used in [19] with the Gaussian Process (GP) for shape estimation (see Fig.\",\n \"REF ).\",\n \"Here, eSAC is shown to work with both a GP and SVM for shape estimates.\",\n \"In particular, in Fig.\",\n \"REF (A), eSAC uses the touch point selection metric used in [19] for control synthesis which is based on the covariance of the GP fit.\",\n \"In comparison with eSAC, the work done in [19] uses an Markov Decision Process (MDP) in order to control a robot manipulator towards acquiring touch data for shape estimation.\",\n \"In [19], a grid is defined and a path planner is used to traverse the grid.\",\n \"The large overhead state discretization and motion planning code that is necessary for the MDP formulation is not required for eSAC.\",\n \"Moreover, eSAC automatically encodes robot dynamic constraints.\",\n \"Notably, eSAC can be used with other visual-based estimators that feed initial shape estimates such that tactile data is used to refine the estimate.\",\n \"This algorithm can be trivially extended to $\\\\mathbb {R}^3$ as shown in simulation.\",\n \"Figure REF shows the sensor trajectory in the top plot (magenta) actively sensing the torus shape.\",\n \"Shape estimation of the torus is shown to be successful in the lower sub-plot of Fig.\",\n \"REF resulting in scalability to objects in $\\\\mathbb {R}^3$ .\",\n \"Performance for estimation in $\\\\mathbb {R}^3$ depends on the robot dynamical constraints as well as the sensor area coverage.\"\n ],\n [\n \"Experimental Results\",\n \"Experiments are done using the Baxter robot (Fig.\",\n \"REF ).\",\n \"Binary measurements are taken by the end-effector probe during a large change of joint torques, indicating a collision has occurred.\",\n \"During collision, controller weight $R$ prevents the robot from dragging along the object.\",\n \"In the case that the control weight $R$ is not significant, heuristics are used to prevent overexertion.\",\n \"Baxter's end-effector is used as a probe for the $\\\\mathbb {R}^2$ search space.\",\n \"No assumptions are needed on the location of the object or the height of the object.\",\n \"Experimental results show that shape estimation can be accomplished with a robotic system with multiple sensors (Fig.\",\n \"REF ).\",\n \"Although there is odometry error within the robot's joint states, shape estimations are still visually seen to match.\",\n \"Within the posterior regime seen in Fig.\",\n \"REF , the probability densities for the shapes with corners are shown to have unique features.\",\n \"In particular, the posterior for the triangle in Fig.\",\n \"REF extends outwards along the three corners indicating high probability of obtaining a collision measurement.\",\n \"Further measurements surrounding the edge regions reduce the likelihood of acquiring estimate which can be seen in with results from estimating the square shape.\"\n ],\n [\n \"Conclusions and Future Work\",\n \"In this paper, an algorithm is presented for shape estimation using a binary sensor combined with ergodic exploration.\",\n \"It is shown that a binary measurement sensor can be treated as a lower resolution tactile sensor for shape estimation with active sensing.\",\n \"In addition, using ergodic exploration as the principle for active sensing uses sensor motion to capture shape features during estimation.\",\n \"The algorithm is shown to estimate shapes in $\\\\mathbb {R}^2$ and $\\\\mathbb {R}^3$ as well as an any number of shapes in the sensor domain.\",\n \"The algorithm is shown to be modular with respect to choice of shape representation.\",\n \"The algorithm is shown to have potential applications for active sensing with low resolution sensors with respect to shape estimation as well as applications in localization.\",\n \"Future research directions include the use of multiple sensors for estimation.\",\n \"Another direction is to augment the current algorithm to include localization using the estimated measurement model.\",\n \"Thus, simultaneous localization and shape estimation should be possible.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01560"}}},{"rowIdx":985,"cells":{"text":{"kind":"list like","value":[["Pure radiation in space-time models that admit integration of the\n eikonal equation by the separation of variables method"],["Abstract We consider space-time models with pure radiation, which admit integration of the eikonal equation by the method of separation of variables.","For all types of these models, the equations of the energy-momentum conservation law were integrated.","The resulting form of metric, energy density and wave vectors of radiation as functions of metric for all types of spaces under consideration is presented.","The solutions obtained can be used for any metric theories of gravitation."],["Introduction","At present, activity has increased in the field of studying realistic cosmological models, which is due to observational data on the accelerated expansion of the universe, the presence of isotropy disturbances in background cosmological radiation, and so on.","If Einstein's theory of gravity is correct, then there must exist \"dark\" energy and \"dark\" matter, which interact gravitationally, but do not interact electromagnetically.","The study of the properties of these exotic substances is based on astrophysical observations and theoretical modeling sometimes leads to exotic equations of state.","There is another option implemented, in which GR is an approximation to a more realistic theory of gravity, the search for which is also actively pursued.","In any case, the situation needs new ideas, methods and tools for studying problems of cosmology and astrophysics.","In this research, we propose an approach to the modeling of space-time with pure radiation based on a combined approach with the following requirements: realistic theory of gravity is a metric theory, i.e.","Gravity is modeled by a metric tensor, and free bodies and radiation move along the geodesic lines of space-time; the law of conservation of energy-momentum of matter is satisfied; to construct integrable models, we will use spaces that allow the integration of the eikonal equation by the method of separation of variables.","As will be shown later, for the space-time models that meet these requirements, one can integrate the equations of the energy-momentum conservation law and obtain expressions for the energy density and the wave vector of pure radiation in the form of functional expressions in terms of functions included in the space-time metric.","The explicit form of the field equations of any gravitation theory is not required.","Solutions are written in coordinate systems that allow separation of variables in the eikonal equation.","At the same time, part of the solutions obtained includes the use of an isotropic (wave-like) variable, which is used in modeling wave processes similar to electromagnetic or gravitational radiation (massless fields).","In this paper, we find and enumerate all types of models considered.","The results obtained can be used when comparing similar models in Einstein's theory of gravity and in various modified theories of gravitation.","When considering metric theories of gravity, then the motion of the test particles and radiation along geodesic lines, just as like the law of conservation of energy-momentum of matter, remain unchanged theory fundamentals: $\\nabla ^iT_{ij}=0,\\qquad i,j,k=0...3,$ where $T_{ij}$ is the energy-momentum tensor of matter.","Therefore, to study and compare modified theories of gravity, the use of space-time models that allow the existence of coordinate systems, where the eikonal equation for massless test particles (null geodesic line for the metric $g^{ij}$ ) $g^{ij}S_{,i}S_{,j}=0,$ or the Hamilton-Jacobi equation for test uncharged mass test particles (an analogue of the geodesic equations) $g^{ij}S_{,i}S_{,j}=m^2,$ or the Hamilton-Jacobi equation for a charged test particle in an electromagnetic field (with a potential $A_i$ ) $g^{ij}(S_{,i}+A_i)(S_{,j}+A_j)=m^2,$ allowing integration by the method of complete separation of variables is of great interest (S is the action of the test particle in the Hamilton-Jacobi formalism, $S_{,i}=\\partial S/\\partial x^i$ ).","For these models, as it turns out, it is possible to integrate the differential equations of energy-momentum conservation law (REF ) for dust matter with the energy-momentum tensor of the form $&T_{ij}=\\rho \\, u_iu_j, &\\\\& u_iu^i=1,&$ where $\\rho $ – energy density of dust matter and $u_i$ – field of the four-velocity of dust matter.","The same situation and for pure radiation (in the high-frequency approximation of geometric optics) with the energy-momentum tensor of the form $& T_{ij}=\\varepsilon \\, {L}_i{L}_j, &\\\\&g^{ij} {L}_i{L}_i=0,&$ where $\\varepsilon $ – energy density and $L_i$ – wave vector of radiation.","As a result, for the energy density of dust matter $\\rho $ and radiation $\\varepsilon $ , for the field of four-velocity of matter $u_i$ and for the wave vector of radiation $L_i$ functional expressions can be obtained only through the space-time metric.","Thus, in these models, the use of field equations of specific theories of gravity can be reduced only to equations for the metric of space-time.","The spaces in which integration of the eikonal equation (REF ) is possible by the method of complete separation of variables are called conformally Stackel spaces (CSS), in contrast to the Stackel spaces, which admit complete separation of variables in the Hamilton-Jacobi equation for test uncharged masses (REF ).","In contrast to the case of the Hamilton-Jacobi equation for test mass particles, in the case of integration of the eikonal equation, the space-time metric admits an arbitrary conformal factor.","The task of this paper is to obtain a classification of functional expressions for the energy density of radiation and the wave vector by integrating the differential energy-momentum conservation law (REF ) for pure radiation for all possible space-time models, where the eikonal equation (REF ) admits integration by the method of separation of variables.","An analogous problem for the case of Stackel spaces and dust matter was considered earlier (see ref. 00).","The theory of Stackel spaces for the case of the Lorentz signature was considered in the works of various authors (see, for example, refs.","0-1) and was formulated in the final form in the works of V.N.","Shapovalov (see refs. 2-3).","We are based on the formulations of the theory and notation in (ref. 4).","The covariant condition of Stackel space is the presence of the so-called ”complete set” of a commuting Killing vector and tensor fields, which satisfy some additional algebraic relations (see details in ref. 4).","The Stackel space metric tensor in privileged coordinate systems (where separation of variables is allowed) is determined by a set of arbitrary functions where each function depends only on one variable.","Stackel space-times (SST) application in gravitation theories (refs.","5–13) is based on the fact that exact integrable models can be developed for these spaces.","The majority of well-known exact solutions is classified as SST (Schwarzschild, Kerr, Friedman-Robertson-Walker, NUT, etc.).","It is important to note that the other single-particle equations of motion - Klein-Gordon-Fock, Dirac, Weyl - admit a separation of variables in SST.","The same method can be used to obtain solutions in the theories of modified gravity (refs.","14–16)."],["Pure radiation in conformally Stackel space-times","The Stackel spaces type is defined by a set of two numbers $(N.N_0)$ , where $N$ – the number of commuting Killing vectors $Y^i_{(p)}$ ($p=1,N$ ) admitted by the ”complete set” (the dimension of the Abelian group of space-time motions), and $N_0=N-rank|Y^i_{(p)}g_{ij}Y^j_{(q)}|$ – is the number of null (\"wave-like\") variables in privileged coordinate systems.","For 4-dimensional Stackel spaces of Lorentz signature $N=0...3$ , $N_0=0,1$ .","Stackel spaces metrics are conveniently written in privileged coordinate systems in a contravariant form.","The coordinates of space-time will be numbered from 0 to 3 by indices $i$ , $ j$ , $k$ .","Ignored variables (on which the metric does not depend) will be numbered in indices $ p$ , $q$ , $r$ and non-ignored by Greek letters (indices) $ \\mu $ , $ \\nu $ , $ \\sigma $ .","The following notation will be used in the paper: $P=\\ln \\left|\\frac{{\\varepsilon }^2}{\\Delta ^2 \\det g^{ij}}\\right|,$ where $g^{ij}$ is the space-time metric, $\\varepsilon $ is the energy density of radiation and $\\Delta $ is the conformal factor of metric $g_{ij}$ .","For conformally Stackel spaces the conformal factor of metric is an arbitrary function of all variables.","In the privlieged coordinate system, the wave vector of the radiation has a \"separated\" form, i.e.","the corresponding covariant components of the wave vector depend only on one variable ${L}_i={L}_i(x^i)$ : $& L_0=L_0(x^0),\\quad L_1=L_1(x^1),&\\\\&L_2=L_2(x^2),\\quad L_3=L_3(x^3).&\\nonumber $ Later in the paper we obtain a functional form of the energy density and the wave vector of radiation for all types of conformally Stackel space-times in privileged coordinate systems (where the separation of variables in the eikonal equation (REF ) is allowed)."],["Conformally Stackel space-times (3.0) type","Conformally Stackel space-times (3.0) type admit three commuting Killing vectors.","In the privileged coordinate system, the metric of a conformally Stackel space of type (3.0) can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&a_0&b_0&c_0\\cr 0&b_0&d_0&e_0\\cr 0&c_0&e_0&f_0},$ where $\\Delta =\\Delta (x^0,x^1,x^2,x^3)$ and $ a_0, b_0, c_0, d_0, e_0, f_0$ are functions of a variable $x^0$ .","The wave vector of the radiation $L_i$ in the privileged coordinate system has the following \"separated\" form: $&{L}_0={L}_0(x^0),\\quad {L}_1=\\alpha ,\\quad {L}_2=\\beta ,\\quad {L}_3=\\gamma ,&\\\\&\\alpha , \\beta , \\gamma - const.&\\nonumber $ Wave vector norm condition () takes the form: $\\alpha ^2a_0+2\\alpha \\beta b_0+\\beta ^2d_0+2\\alpha \\gamma c_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=-{L}_0{}^2.$ From the equations of the energy-momentum conservation law (REF ) we obtain: $& {L}_0P_{,0}+(\\alpha a_0+\\beta b_0+\\gamma c_0)P_{,1}+(\\alpha b_0+\\beta d_0+\\gamma e_0)P_{,2} &\\nonumber \\\\&+\\mbox{} (\\alpha c_0+\\beta e_0+\\gamma f_0)P_{,3}+2{L}_0{}^{\\prime }=0,&$ here and in the sequel, a comma means a partial differentiation, and a prime means differentiation with respect to a single variable on which the function depends.","Using condition (REF ) and finding the integrals of equation (REF ), one can obtain expressions for the wave vector and the energy density of radiation.","Below are listed all the solutions we have obtained for conformally Stackel space-times (3.0) type.","The wave vector of the radiation has the form: $&{L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right),\\quad \\alpha , \\beta , \\gamma - const,&\\\\[1ex]&{L}_0=\\sqrt{\\alpha ^2a_0+2\\alpha \\beta b_0+\\beta ^2d_0+2\\alpha \\gamma c_0+2\\beta \\gamma e_0+\\gamma ^2 f_0}.&\\nonumber $ For the energy density of radiation we obtain the expression: $&{\\varepsilon }=F(X,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}/{{L}_0},&\\\\[1ex]&X=x^1-\\int {(\\alpha a_0+\\beta b_0+\\gamma c_0)}/{{L}_0}\\,dx^0,&\\nonumber \\\\&Y=x^2-\\int {(\\alpha b_0+\\beta d_0+\\gamma e_0)}/{{L}_0}\\,dx^0,&\\nonumber \\\\&Z=x^3-\\int {(\\alpha c_0+\\beta e_0+\\gamma f_0)}/{{L}_0}\\,dx^0,& \\nonumber $ where $ F (X, Y, Z) $ is an arbitrary function of its variables."],["CSS (3.0) type, case 2. (${L}_0= 0$ ).","For this degenerate case there is an additional condition for the functions of the metric (REF ): $&\\alpha ^2a_0+2\\alpha \\beta b_0+2\\alpha \\gamma c_0+\\beta ^2d_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=0,& \\nonumber \\\\&\\alpha , \\beta , \\gamma - const.&$ The wave vector takes the form: ${L}_i=(0,\\alpha , \\beta , \\gamma ).$ For the energy density of radiation we obtain the expression: $&{\\varepsilon }=F(x^0,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}&\\\\[1ex]&Y={x^1}/{(\\alpha a_0+\\beta b_0+\\gamma c_0)}-{x^2}/{(\\alpha b_0+\\beta d_0+\\gamma e_0)},& \\nonumber \\\\&Z={x^1}/{(\\alpha a_0+\\beta b_0+\\gamma c_0)}-{x^3}/{(\\alpha c_0+\\beta e_0+\\gamma f_0)},& \\nonumber $ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."],["CSS (3.0) type, case 3. (${L}_0=0$ , {{formula:70047d94-5e5d-40db-85a6-32afca0e0cfd}} ).","In this degenerate case, when one of the terms in (REF ) additionally vanishes together with $ {L}_0 $ (for example, the term with $ P_{, 1} $ ), we obtain additional conditions for the metric (REF ): $\\alpha a_0+\\beta b_0+\\gamma c_0=0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\alpha ^2a_0+2\\alpha \\beta b_0+2\\alpha \\gamma c_0+\\beta ^2d_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=0.$ The wave vector of the radiation has the form: ${L}_i=(0,\\alpha , \\beta , \\gamma ).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,x^1,Z){\\Delta \\sqrt{- \\det g^{ij}}}$ $Z=x^2 (\\alpha c_0+\\beta e_0+\\gamma f_0)- x^3 (\\alpha b_0+\\beta d_0+\\gamma e_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables."],["Conformally Stackel space-times (3.1) type","Conformally Stackel space-times (3.1) type admits 3 commuting Killing vectors $Y^i_{(p)}$ ($p=1,3$ ), but $rank\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=2$ .","In a privileged coordinate system the metric of a conformally Stackel space-times (3.1) type can be written in the following form, where the variable $ x^0 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{0&1&a_0&b_0\\cr 1&0&0&0\\cr a_0&0&c_0&f_0\\cr b_0&0&f_0&d_0},$ where $\\Delta =\\Delta (x^0,x^1,x^2,x^3)$ and $a_0, b_0, c_0, d_0, f_0$ are functions of a variable $ x^0$ .","The wave vector of the radiation has the form: $&{L}_0={L}_0(x^0),\\quad {L}_1=\\alpha ,\\quad {L}_2=\\beta ,\\quad {L}_3=\\gamma ,&\\\\&\\alpha , \\beta , \\gamma - const.\\nonumber $ The system of equations (REF ), () takes the form: $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0+2(\\alpha +\\beta a_0+\\gamma b_0){L}_0=0,$ $&(\\alpha +\\beta a_0+\\gamma b_0)P_{,0}+{L}_0P_{,1}+(a_0{L}_0+\\beta c_0+\\gamma f_0)P_{,2}&\\nonumber \\\\&\\mbox{}+(b_0{L}_0+\\beta f_0+\\gamma d_0)P_{,3}+2\\beta a_0^{\\prime }+2\\gamma b_0^{\\prime }=0.&$ From equations (REF )-(REF ) we can obtain functional expressions for the wave vector and the energy density of radiation.","Below all the solutions for conformally Stackel space-times (3.1) are listed."],["CSS (3.1) type, case 1. ($\\alpha +\\beta a_0+\\gamma b_0\\ne 0$ ).","The wave vector of the radiation has the form: $ {L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right),\\quad \\alpha , \\beta , \\gamma - const,$ $ {L}_0=\\frac{-(\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0)}{2\\,(\\alpha +\\beta a_0+\\gamma b_0)}.$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(X,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}/{(\\alpha +\\beta a_0+\\gamma b_0)},$ $X&=&x^1-\\int \\frac{{L}_0}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber \\\\Y&=&x^2-\\int \\frac{(a_0{L}_0+\\beta c_0+\\gamma f_0)}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber \\\\Z&=&x^3-\\int \\frac{(b_0{L}_0+\\beta f_0+\\gamma d_0)}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber $ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (3.1) type, case 2. ($L_0\\ne 0$ , {{formula:06fa1d48-9bb8-49b0-b333-846375975dad}} ). ","In this degenerate case there are additional conditions for the metric (REF ): $\\alpha +\\beta a_0+\\gamma b_0= 0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: $ {L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,Y,Z) \\Delta \\sqrt{- \\det g^{ij}}.$ $Y&=& x^1-\\frac{x^2(a_0{L}_0+\\beta c_0+\\gamma f_0) }{L_0},\\nonumber \\\\Z&=& x^1-\\frac{x^3(b_0{L}_0+\\beta f_0+\\gamma d_0) }{L_0},\\nonumber $ where $L_0(x^0)$ and $F(x^0,Y,Z)$ are arbitrary functions of their variables."],["CSS (3.1) type, case 3. ($L_0= 0$ , {{formula:88ed6080-40a1-42aa-a8a8-2b68967c0299}} ).","In this degenerate case there are additional conditions for the metric (REF ): $\\alpha +\\beta a_0+\\gamma b_0= 0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: ${L}_i=\\left( 0, \\alpha , \\beta , \\gamma \\right).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,x^1,Z) \\Delta \\sqrt{- \\det g^{ij}}.$ $Z=x^2(\\beta f_0+\\gamma d_0)-x^3(\\beta c_0+\\gamma f_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables.","Conformally Stackel space-times (2.0) type admit two commuting Killing vectors.","In the privileged coordinate system, the metric can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&{{\\epsilon }} &0&0\\cr 0&0&A&B\\cr 0&0&B&C},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),\\quad {\\epsilon } =\\pm 1,\\quad A=a_0(x^0)+a_1(x^1),&\\\\&B=b_0(x^0)+b_1(x^1),\\qquad C=c_0(x^0)+c_1(x^1) .&$ The wave vector of the radiation ${L}_i$ has the form: $&{L}_0={L}_0(x^0),\\qquad {L}_1={L}_1(x^1),&\\\\&{L}_2=\\alpha ,\\qquad {L}_3=\\beta , \\qquad \\alpha , \\beta - const.&\\nonumber $ From the norm condition () one can obtain: $&{{L}_0}^2+\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0-\\gamma =0,&\\\\&{\\epsilon } {{L}_1}^2+\\alpha ^2 a_1+2\\alpha \\beta b_1+\\beta ^2 c_1+\\gamma =0.&$ From the conservation law (REF ) one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+{\\epsilon } {L}_1P_{,1}+(\\alpha A+\\beta B)P_{,2}+(\\alpha B+\\beta C)P_{,3}&\\nonumber \\\\&\\mbox{}+2{L}_0^{\\prime }+2{\\epsilon } {L}_1^{\\prime }=0.&$ Below are listed all the solutions for conformally Stackel space-times (2.0) type."],["CSS (2.0) type, case 1. (${L}_0{L}_1\\ne 0$ ):","The wave vector of the radiation has the form: $&{L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta ,\\gamma - const,&\\nonumber \\\\[1ex]&{L}_0=\\sqrt{\\gamma -\\alpha ^2 a_0-2\\alpha \\beta b_0-\\beta ^2 c_0},&\\nonumber \\\\&{L}_1=\\sqrt{{\\epsilon } (-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}.&$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/({{L}_0\\,{L}_1}),$ $&X=\\int 1/{{L}_0}\\, dx^0- \\int {\\epsilon }/{{L}_1}\\, dx^1,&\\\\&Y=x^2-\\int {(\\alpha a_0+\\beta b_0)}/{{L}_0}\\, dx^0 - {\\epsilon } \\int {(\\alpha a_1+\\beta b_1)}/{{L}_1}\\, dx^1,&\\\\&Z=x^3-\\int {(\\alpha b_0+\\beta c_0)}/{{L}_0}\\, dx^0 - {\\epsilon } \\int {(\\alpha b_1+\\beta c_1)}/{{L}_1}\\, dx^1,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (2.0) type, case 2. (${L}_0=0, {L}_1\\ne 0$ ):","Let us consider the degenerate case when $ {L}_0=0$ or ${L}_1 = 0 $ and fo rcertainty we take $ {L}_0 = 0 $ and $ {L}_1 \\ne 0 $ , then the wave vector will take the following form: ${L}_i=\\left( 0, {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta , \\gamma - const,$ ${L}_1=\\sqrt{{\\epsilon } (-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}.$ In this degenerate case there are additional conditions for the metric (REF ): $\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0-\\gamma =0.$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_1},$ $Y=x^2-\\int {(\\alpha a_1+\\beta b_1)}/{{L}_1}\\, dx^1,$ $Z=x^3- {\\epsilon } \\int {(\\alpha b_1+\\beta c_1)}/{{L}_1}\\, dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."],["CSS (2.0) type, case 3. (${L}_0={L}_1= 0$ ):","In this degenerate case, there are additional conditions for the metric (REF ): $\\alpha ^2 A+2\\alpha \\beta B+\\beta ^2 C=0,\\qquad \\alpha , \\beta - const.$ The wave vector of the radiation has the form: ${L}_i=\\left( 0, 0, \\alpha , \\beta \\right).$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,x^1,Z)\\,\\Delta \\sqrt{-\\det g^{ij}},$ $Z=x^2 (\\alpha B+\\beta C) -x^3(\\alpha A+\\beta B),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables."],["Conformally Stackel space-times (2.1) type","Conformally Stackel space-times (2.1) type admits two commuting Killing vectors $Y^i_{(p)}$ ($p=1,2$ ), but $rank\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=1$ .","In a privileged coordinate system the metric of a conformally Stackel space-times (2.1) type can be written in the following form, where $ x^1 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&0&f_1(x^1)&1\\cr 0&f_1(x^1)&A&B\\cr 0&1&B&C},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3), \\quad A=a_0(x^0)+a_1(x^1),&\\\\&B=b_0(x^0)+b_1(x^1),\\quad C=c_0(x^0)+c_1(x^1).&$ The wave vector of radiation $L_i$ has the form: $&{L}_0={L}_0(x^0),\\quad {L}_1={L}_1(x^1),&\\nonumber \\\\&{L}_2=\\alpha ,\\quad {L}_3=\\beta , \\quad \\alpha , \\beta , \\gamma - const.&$ From the norm condition () one can obtain: $&\\gamma ={{L}_0}^2+\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0,&\\\\&-\\gamma =2(\\alpha f_1+\\beta ){L}_1+\\alpha ^2 a_1+2\\alpha \\beta b_1+\\beta ^2 c_1.&$ From the conservation law (REF ), one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+(\\alpha f_1+\\beta ) P_{,1}+(\\alpha A+\\beta B+f_1 {L}_1)P_{,2}&\\nonumber \\\\&\\mbox{}+(\\alpha B+\\beta C+{L}_1)P_{,3}+2\\alpha f_1^{\\prime }+2{{L}_0}^{\\prime }=0.&$ From the system of equations (REF )–(REF ) we can obtain functional expressions through the metric for the wave vector and the radiation energy density.","Below all the solutions for conformally Stackel space-times (2.1) type are listed."],["CSS (2.1) type, case 1. ${L}_0(\\alpha f_1+\\beta )\\ne 0$ :","The wave vector of radiation has the form: $&{L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta ,\\gamma - const,&\\nonumber \\\\[1ex]&{L}_0=\\sqrt{\\gamma -\\alpha ^2 a_0-2\\alpha \\beta b_0-\\beta ^2 c_0},&\\\\&{L}_1={(-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}/\\big ({2\\,(\\alpha f_1+\\beta )}\\big ).&\\nonumber $ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{\\big ({L}_0(\\alpha f_1+\\beta )\\big )},$ $X=\\int \\frac{dx^0}{{L}_0}- \\int \\frac{dx^1}{(\\alpha f_1+\\beta )},$ $Y=x^2-\\int \\frac{(\\alpha a_0+\\beta b_0)}{{L}_0}\\,dx^0 -\\int \\frac{(\\alpha a_1+\\beta b_1+f_1L_1)}{\\alpha f_1+\\beta }\\, dx^1,$ $Z=x^3-\\int \\frac{(\\alpha b_0+\\beta c_0)}{{L}_0}\\,dx^0 -\\int \\frac{(\\alpha b_1+\\beta c_1+L_1)}{\\alpha f_1+\\beta }\\, dx^1,$ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (2.1) type, case 2. $(\\alpha f_1+\\beta )= 0$ , {{formula:89e7bb2c-cabe-4b63-969f-80c8a732525b}} .","In this degenerate case, there are additional conditions for the metric (REF ): $a_1=f_1=0.$ The wave vector of radiation has the form ($\\beta =\\gamma =0$ ): ${L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , 0\\right),$ ${L}_0=\\alpha \\sqrt{- a_0},\\quad L_1=\\sigma -\\alpha b_1,\\quad \\alpha , \\sigma - const.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{L_0(x^0)},$ $Y=\\alpha x^2+\\int {L_0}\\,dx^0,\\quad Z= x^3 -\\int {(\\sigma +\\alpha b_0)}/{L_0}\\,dx^0,$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables."],["CSS (2.1) type, case 3. ${L}_0= 0$ , {{formula:b7686906-83ba-49e5-96c2-9923fa201a29}} .","In this degenerate case, there are additional conditions for the metric (REF ): $&\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0=\\gamma ,&\\\\&p(\\alpha a_0+\\beta b_0)+q (\\alpha b_0+\\beta c_0)=r,&\\\\&\\alpha , \\beta , \\gamma , p, q, r \\mbox{ -- const}.&\\nonumber $ The wave vector of radiation has the form: $&{L}_i=\\left( 0, {L}_1(x^1), \\alpha , \\beta \\right),&\\\\&{L}_1= {(-\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1-\\gamma )}/{(2\\,(\\alpha f_1+\\beta ))}.&\\nonumber $ The energy density of the radiation has the form: $ {\\varepsilon }=F(x^0,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{(\\alpha f_1+\\beta )},$ $Y=\\alpha x^2+\\beta x^3 +\\int L_1(x^1)\\,dx^1,$ $Z=px^2+qx^3$ $\\mbox{}-\\int \\frac{r+ p(\\alpha a_1+\\beta b_1+f_1L_1)+q(\\alpha b_1+\\beta c_1+L_1)}{(\\alpha f_1+\\beta )}\\,dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."],["CSS (2.1) type, case 4. ${L}_0= 0$ , {{formula:468ec54a-5663-4dcf-914e-e423945394f8}} , {{formula:fdcb860c-4b2a-4abd-a024-0e7e81517f01}} .","The wave vector of radiation has the form: ${L}_i=\\left( 0, {L}_1(x^1), 0, 0\\right),\\qquad \\alpha = \\beta =\\gamma =0.$ The energy density of the radiation has the form: $ {\\varepsilon }=F(x^0,x^1,Z)\\,\\Delta \\sqrt{-\\det g^{ij}},$ $Z=x^2 -x^3f_1(x^1),$ where $L_1(x^1)$ and $F(x^0,x^1,Z)$ are arbitrary functions of their variables."],["Conformally Stackel space-times (1.0) type","Conformally Stackel space-times (1.0) type admits one Killing vector, and in a privileged coordinate system the metric of this space-time can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{\\Omega &0&0&0\\cr 0&V^1&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^1=t_2(x^2)-t_3(x^3),\\qquad V^2=t_3(x^3)-t_1(x^1),&\\\\&V^3=t_1(x^1)-t_2(x^2),\\quad \\Omega =\\omega _\\nu (x^\\nu ) V^\\nu ,\\quad \\mu ,\\,\\nu =1...3.&$ The wave vector of radiation has the form: $ {L}_i=\\Big (\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ),\\qquad \\alpha =const.$ From equations (REF ), () we have: $ \\Omega \\alpha ^2 + V^\\mu {{L}_\\mu }^2=0,$ $ \\alpha \\Omega P_{,0}+ V^\\mu ({L}_\\mu P_{,\\mu }+2{L}_\\mu ^{\\prime })=0.","$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.","Below are listed all the solutions for conformally Stackel space-times (1.0) type."],["CSS (1.0) type, case 1. ${L}_1{L}_2{L}_3\\ne 0$ .","The wave vector of radiation has the form: $&{L}_i=\\left(\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\right),\\quad \\alpha ,\\beta ,\\gamma \\mbox{ -- const}.&\\nonumber \\\\[1ex]&{L}_\\mu =\\sqrt{-\\alpha ^2\\omega _\\mu +\\beta t_\\mu +\\gamma },\\quad \\mu , \\nu =1...3.&$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_1\\,{L}_2\\,{L}_3)},$ $&X=x^0 -\\alpha \\sum _\\mu \\int ({\\omega _\\mu }/{{L}_\\mu })\\, dx^\\mu ,&\\\\&Y=\\sum _\\mu \\int ({t_\\mu }/{{L}_\\mu })\\, dx^\\mu ,\\qquad Z= \\sum _\\mu \\int (1/{{L}_\\mu }){\\,dx^\\mu } ,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (1.0) type, case 2. ${L}_1=0, {L}_2{L}_3 \\ne 0$ .","In this degenerate case, when one of the components of the wave vector $ {L}_\\mu $ becomes zero (for definiteness, let $ L_1 = 0 $ , $ {L}_2 {L}_3 \\ne 0 $ ) we have an additional condition for the metric (REF ): $\\alpha ^2\\omega _1=\\beta t_1+\\gamma ,\\qquad \\mbox{$\\alpha ,\\beta ,\\gamma $ -- const}.$ The wave vector of radiation has the form: ${L}_i=\\Big (\\alpha ,0,{L}_2(x^2),{L}_3(x^3)\\Big ),$ ${L}_2=\\sqrt{-\\alpha ^2\\omega _2+\\beta t_2+\\gamma },\\quad {L}_3=\\sqrt{-\\alpha ^2\\omega _3+\\beta t_3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,\\Delta \\sqrt{-\\det g^{ij}}/\\prod _{\\mu \\ne 1}{L}_\\mu ,$ $Y=\\alpha x^0 -\\alpha ^2\\sum _{\\mu \\ne 1} \\int ({\\omega _\\mu }/{{L}_\\mu })\\, dx^\\mu +\\beta \\sum _{\\mu \\ne 1} \\int ({t_\\mu }/{{L}_\\mu })\\, dx^\\mu ,$ $Z=\\sum _{\\mu \\ne 1} \\int (1/{{L}_\\mu }){\\,dx^\\mu },$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables."],["CSS (1.0) type, case 3. $L_1\\ne 0, {L}_2=0, {L}_3= 0$ .","In this degenerate case, when two of the components of the wave vector $ {L}_\\mu $ becomes zero (for definiteness, let $ L_1 \\ne 0 $ , $ {L}_2 ={L}_3= 0 $ ) we have additional conditions for the metric (REF ): $\\alpha ^2\\omega _2=\\beta t_2+\\gamma ,\\quad \\alpha ^2\\omega _3=\\beta t_3+\\gamma ,$ where $\\alpha $ , $\\beta $ , $\\gamma $ – const.","The wave vector of radiation has the form: ${L}_i=\\left(\\alpha ,{L}_1(x^1),0,0\\right),\\quad {L}_1=\\sqrt{\\beta t_1-\\alpha ^2\\omega _1+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^2,x^3, Y)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_1(x^1)},$ $Y=\\alpha x^0+\\int L_1(x^1) \\,dx^1,$ where $F(x^2,x^3,Y)$ is an arbitrary function of its variables."],["Conformally Stackel space-times (1.1) type","Conformally Stackel space-times (1.1) type admits one Killing vector.","In a privileged coordinate system the metric of a conformally Stackel space-times (1.1) type can be written in the following form, where $ x^1 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{\\Omega &V^1&0&0\\cr V^1&0&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^1=t_2(x^2)-t_3(x^3),\\quad V^2=t_3(x^3)-t_1(x^1),&\\\\&V^3=t_1(x^1)-t_2(x^2),\\quad \\Omega =\\omega _\\mu (x^\\mu ) V^\\mu ,\\quad \\mu ,\\nu =1...3.$ The wave vector of radiation has the following \"separated\" form: $ {L}_i=\\Big (\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ),\\quad \\alpha - const.$ From equations (REF ), () we have: $ \\alpha V^1(2 {L}_1+\\alpha \\omega _1)+V^2({L}_2{}^2+\\alpha ^2\\omega _2)+V^3({L}_3{}^2+\\alpha ^2\\omega _3)=0,$ $(V^1{L}_1+\\alpha \\Omega )P_{,0}+ \\alpha V^1P_{,1}+ {L}_2V^2P_{,2}+ {L}_3V^3P_{,3}$ $\\mbox{}+ 2V^2{L}_2^{\\prime }+ 2V^3{L}_3^{\\prime } = 0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.","Below are listed all the solutions for conformally Stackel space-times (1.1) type."],["CSS (1.1) type, case 1. $\\alpha L_2 L_3\\ne 0$ .","The wave vector of radiation has the form: ${L}_0=\\alpha ,\\quad {L}_1=\\frac{1}{2\\alpha }(\\beta t_1 -\\alpha ^2\\omega _1+\\gamma ),\\quad \\alpha ,\\beta , \\gamma - const,$ ${L}_2=\\sqrt{\\beta t_2-\\alpha ^2\\omega _2+\\gamma },\\quad {L}_3=\\sqrt{\\beta t_3-\\alpha ^2\\omega _3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_2\\,{L}_3)},$ $X=x^0 -\\frac{1}{\\alpha } \\int ({L}_1+\\alpha \\omega _1)\\,dx^1 -\\alpha \\left( \\int \\frac{\\omega _2}{{L}_2}\\,dx^2 +\\int \\frac{\\omega _3}{{L}_3}\\,dx^3 \\right),$ $Y=-\\frac{1}{\\alpha }\\int t_1\\, dx^1 +\\int \\frac{t_2}{{L}_2}\\,dx^2+\\int \\frac{t_3}{{L}_3}\\,dx^3 ,$ $Z= \\frac{x^1}{\\alpha }+\\int \\frac{dx^2}{{L}_2}+\\int \\frac{dx^3}{{L}_3},$ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (1.1) type, case 2. $\\alpha = 0$ .","In this degenerate case, when $ {L}_0 = 0 $ , we obtain an additional condition for the metric (REF ): $t_1=0.$ The wave vector of the radiation $L_i$ has the following form ($\\gamma =0$ ): ${L}_0=0,\\quad {L}_1=L_1(x^1),\\quad {L}_2=\\sqrt{\\beta t_2},\\quad {L}_3=\\sqrt{\\beta t_3}.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_2\\,{L}_3)},$ $Y=\\int \\frac{t_2}{{L}_2}\\,dx^2+\\int \\frac{t_3}{{L}_3}\\,dx^3 ,\\quad Z=\\frac{x^0}{L_1}+ \\int \\frac{dx^2}{{L}_2}+\\int \\frac{dx^3}{{L}_3},$ where $F(x^1,Y,Z)$ and $L_1(x^1)$ are arbitrary functions of its variables."],["CSS (1.1) type, case 3. $\\alpha L_2\\ne 0, L_3=0$ .","In the degenerate case, when one of the components of the wave vector $L_3$ becomes zero (similarly for $ L_2 $ with the replacement of the indices 3 by 2) we have additional conditions for the metric (REF ): $\\alpha ^2\\omega _3=\\beta t_3+\\gamma , \\qquad \\alpha , \\beta , \\gamma - const,$ $(p\\alpha -q\\beta )\\,t_3=0,\\qquad p,q - const.$ The wave vector of the radiation $L_i$ has the following form: ${L}_0=\\alpha ,\\qquad {L}_1=\\frac{1}{2\\alpha }(\\beta t_1 -\\alpha ^2\\omega _1+\\gamma ),$ ${L}_2=\\sqrt{\\beta t_2-\\alpha ^2\\omega _2+\\gamma },\\qquad {L}_3=0.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_2},$ $Y=\\alpha x^0+\\int L_1\\,dx^1+\\int L_2\\,dx^2,$ $Z=qx^0+\\frac{1}{\\alpha }\\int \\Big ( q(\\gamma /\\alpha -\\alpha \\omega _1 -L_1)+pt_1\\Big )dx^1$ $\\mbox{}+\\int \\frac{q(\\gamma /\\alpha -\\alpha \\omega _2)+pt_2}{L_2}\\,dx^2,$ where $F(x^1,Y,Z)$ is an arbitrary function of their variables."],["Conformally Stackel space-times (0.0) type","The metric of conformally Stackel space-times (0.0) type in a privileged coordinate system can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{V^0 &0&0&0\\cr 0&V^1&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^0=a_1(b_2-b_3)+a_2(-b_1+b_3)+a_3(b_1-b_2),&\\\\&V^1= a_0(-b_2+b_3)+a_2(b_0-b_3)+a_3(-b_0+b_2),&\\\\&V^2=a_0(b_1-b_3)+a_1(-b_0+b_3)+a_3(b_0-b_1),&\\\\&V^3=a_0(-b_1+b_2)+a_1(b_0-b_2)+a_2(-b_0+b_1),&$ where functions $ a $ , $ b $ , $ c $ are functions of only one variable whose index corresponds to the lower index of the function.","For example $ a_0 = a_0 (x ^ 0) $ , $ b_1 = b_1 (x ^ 1) $ , etc.","The wave vector of the radiation in a privileged coordinate system has the following \"separated\" form: $ {L}_i=\\Big ({L}_0(x^0),{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ).$ From the law of energy-momentum conservation (REF ) and condition () we have: $V^i{L}_i{}^2=0,\\qquad i=0...3,$ $ V^i({L}_iP_{,i}+2{L}_i^{\\prime })=0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.","Below are listed all the solutions for conformally Stackel space-times (0.0) type."],["CSS (0.0) type, case 1. ${L}_0{L}_1{L}_2{L}_3\\ne 0$ .","The wave vector of the radiation $L_i$ has the following form ($\\alpha $ , $\\beta $ , $\\gamma $ – const): ${L}_i=\\sqrt{\\alpha a_i+\\beta b_i+\\gamma },\\qquad i,j \\mbox{ = 0...3.","}$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_0\\,{L}_1\\,{L}_2\\,{L}_3)},$ $X=\\sum _i \\int \\frac{dx^i}{{L}_i},\\quad Y=\\sum _i \\int \\frac{a_i}{{L}_i}\\,dx^i,\\quad Z=\\sum _i \\int \\frac{b_i}{{L}_i}\\,dx^i,$ where $F(X,Y,Z)$ is an arbitrary function of its variables."],["CSS (0.0) type, case 2. ${L}_0=0, \\quad {L}_1{L}_2{L}_3\\ne 0$ .","In the degenerate case, when one of the components of the wave vector turns to zero (for definiteness a component with the number $ i=0 $ , that is, $ {L}_0 = 0 $ ), then we obtain an additional condition for the metric (REF ): $\\alpha a_0+\\beta b_0+\\gamma =0,\\qquad \\mbox{$\\alpha $, $\\beta $, $\\gamma $ -- const.","}$ For the wave vector of radiation ${L}_i$ we obtain: ${L}_{0}=0,\\qquad {L}_i=\\sqrt{\\alpha a_i+\\beta b_i+\\gamma },\\qquad i\\ne 0.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,X,Y)\\,\\Delta \\sqrt{-\\det g^{ij}}/{\\prod _{i\\ne 0}{L}_i},$ $X=\\sum _{i\\ne 0} \\int \\frac{dx^i}{{L}_i},\\qquad Y=\\sum _{i\\ne 0} \\int {L}_i\\,dx^i,$ where $F(x^0,X,Y)$ is an arbitrary function of its variables."],["CSS (0.0) type, case 3. ${L}_0={L}_1= 0, \\quad {L}_2{L}_3\\ne 0$ .","In the degenerate case, when the two components of the wave vector vanish (let it be the components $ {L}_0 =0$ , $ {L}_1 = 0 $ ), then we obtain additional conditions for the metric ($\\alpha $ , $\\beta $ , $\\gamma $ – const): $\\alpha a_0+\\beta b_0+\\gamma =0,\\quad \\alpha a_1+\\beta b_1+\\gamma =0.$ For the wave vector of radiation we have: ${L}_i=\\Big (0,0,L_2(x^2), L_3(x^3)\\Big ),$ ${L}_2=\\sqrt{\\alpha a_2+\\beta b_2+\\gamma },\\quad {L}_3=\\sqrt{\\alpha a_3+\\beta b_3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,x^1,Y)\\,\\frac{\\Delta \\sqrt{-\\det g^{ij}}}{{L}_2{L}_3},$ $Y= \\int {L}_2\\,dx^2 + \\int {L}_3\\,dx^3,$ where $F(x^0,x^1,Y)$ is an arbitrary function of its variables.","Note that for Stackel space-times (0.0) type the three components of the wave vector of the radiation can not be zero, since this leads to a violation of the norm condition.","In the paper we obtain and enumerate all solutions of the equations of the energy-momentum conservation law of pure radiation for models of space-times that admit separation of variables in the eikonal equation.","The forms of the energy-momentum tensor of pure radiation (energy density and wave vector of radiation) for all types of space-times under consideration are obtained.","In privileged coordinate systems (where the separation of variables is admitted), the energy density and the wave vector of the radiation are determined through functions of the space-time metric.","The results obtained can be used to construct integrable models for various metric theories of gravitation, including for comparing similar models in the theory of gravity of Einstein and in modified theories of gravity.","Research partially supported by the Ministry of Education and Science of Russia under contract 3.1386.2017/4.6."]],"string":"[\n [\n \"Pure radiation in space-time models that admit integration of the\\n eikonal equation by the separation of variables method\"\n ],\n [\n \"Abstract We consider space-time models with pure radiation, which admit integration of the eikonal equation by the method of separation of variables.\",\n \"For all types of these models, the equations of the energy-momentum conservation law were integrated.\",\n \"The resulting form of metric, energy density and wave vectors of radiation as functions of metric for all types of spaces under consideration is presented.\",\n \"The solutions obtained can be used for any metric theories of gravitation.\"\n ],\n [\n \"Introduction\",\n \"At present, activity has increased in the field of studying realistic cosmological models, which is due to observational data on the accelerated expansion of the universe, the presence of isotropy disturbances in background cosmological radiation, and so on.\",\n \"If Einstein's theory of gravity is correct, then there must exist \\\"dark\\\" energy and \\\"dark\\\" matter, which interact gravitationally, but do not interact electromagnetically.\",\n \"The study of the properties of these exotic substances is based on astrophysical observations and theoretical modeling sometimes leads to exotic equations of state.\",\n \"There is another option implemented, in which GR is an approximation to a more realistic theory of gravity, the search for which is also actively pursued.\",\n \"In any case, the situation needs new ideas, methods and tools for studying problems of cosmology and astrophysics.\",\n \"In this research, we propose an approach to the modeling of space-time with pure radiation based on a combined approach with the following requirements: realistic theory of gravity is a metric theory, i.e.\",\n \"Gravity is modeled by a metric tensor, and free bodies and radiation move along the geodesic lines of space-time; the law of conservation of energy-momentum of matter is satisfied; to construct integrable models, we will use spaces that allow the integration of the eikonal equation by the method of separation of variables.\",\n \"As will be shown later, for the space-time models that meet these requirements, one can integrate the equations of the energy-momentum conservation law and obtain expressions for the energy density and the wave vector of pure radiation in the form of functional expressions in terms of functions included in the space-time metric.\",\n \"The explicit form of the field equations of any gravitation theory is not required.\",\n \"Solutions are written in coordinate systems that allow separation of variables in the eikonal equation.\",\n \"At the same time, part of the solutions obtained includes the use of an isotropic (wave-like) variable, which is used in modeling wave processes similar to electromagnetic or gravitational radiation (massless fields).\",\n \"In this paper, we find and enumerate all types of models considered.\",\n \"The results obtained can be used when comparing similar models in Einstein's theory of gravity and in various modified theories of gravitation.\",\n \"When considering metric theories of gravity, then the motion of the test particles and radiation along geodesic lines, just as like the law of conservation of energy-momentum of matter, remain unchanged theory fundamentals: $\\\\nabla ^iT_{ij}=0,\\\\qquad i,j,k=0...3,$ where $T_{ij}$ is the energy-momentum tensor of matter.\",\n \"Therefore, to study and compare modified theories of gravity, the use of space-time models that allow the existence of coordinate systems, where the eikonal equation for massless test particles (null geodesic line for the metric $g^{ij}$ ) $g^{ij}S_{,i}S_{,j}=0,$ or the Hamilton-Jacobi equation for test uncharged mass test particles (an analogue of the geodesic equations) $g^{ij}S_{,i}S_{,j}=m^2,$ or the Hamilton-Jacobi equation for a charged test particle in an electromagnetic field (with a potential $A_i$ ) $g^{ij}(S_{,i}+A_i)(S_{,j}+A_j)=m^2,$ allowing integration by the method of complete separation of variables is of great interest (S is the action of the test particle in the Hamilton-Jacobi formalism, $S_{,i}=\\\\partial S/\\\\partial x^i$ ).\",\n \"For these models, as it turns out, it is possible to integrate the differential equations of energy-momentum conservation law (REF ) for dust matter with the energy-momentum tensor of the form $&T_{ij}=\\\\rho \\\\, u_iu_j, &\\\\\\\\& u_iu^i=1,&$ where $\\\\rho $ – energy density of dust matter and $u_i$ – field of the four-velocity of dust matter.\",\n \"The same situation and for pure radiation (in the high-frequency approximation of geometric optics) with the energy-momentum tensor of the form $& T_{ij}=\\\\varepsilon \\\\, {L}_i{L}_j, &\\\\\\\\&g^{ij} {L}_i{L}_i=0,&$ where $\\\\varepsilon $ – energy density and $L_i$ – wave vector of radiation.\",\n \"As a result, for the energy density of dust matter $\\\\rho $ and radiation $\\\\varepsilon $ , for the field of four-velocity of matter $u_i$ and for the wave vector of radiation $L_i$ functional expressions can be obtained only through the space-time metric.\",\n \"Thus, in these models, the use of field equations of specific theories of gravity can be reduced only to equations for the metric of space-time.\",\n \"The spaces in which integration of the eikonal equation (REF ) is possible by the method of complete separation of variables are called conformally Stackel spaces (CSS), in contrast to the Stackel spaces, which admit complete separation of variables in the Hamilton-Jacobi equation for test uncharged masses (REF ).\",\n \"In contrast to the case of the Hamilton-Jacobi equation for test mass particles, in the case of integration of the eikonal equation, the space-time metric admits an arbitrary conformal factor.\",\n \"The task of this paper is to obtain a classification of functional expressions for the energy density of radiation and the wave vector by integrating the differential energy-momentum conservation law (REF ) for pure radiation for all possible space-time models, where the eikonal equation (REF ) admits integration by the method of separation of variables.\",\n \"An analogous problem for the case of Stackel spaces and dust matter was considered earlier (see ref. 00).\",\n \"The theory of Stackel spaces for the case of the Lorentz signature was considered in the works of various authors (see, for example, refs.\",\n \"0-1) and was formulated in the final form in the works of V.N.\",\n \"Shapovalov (see refs. 2-3).\",\n \"We are based on the formulations of the theory and notation in (ref. 4).\",\n \"The covariant condition of Stackel space is the presence of the so-called ”complete set” of a commuting Killing vector and tensor fields, which satisfy some additional algebraic relations (see details in ref. 4).\",\n \"The Stackel space metric tensor in privileged coordinate systems (where separation of variables is allowed) is determined by a set of arbitrary functions where each function depends only on one variable.\",\n \"Stackel space-times (SST) application in gravitation theories (refs.\",\n \"5–13) is based on the fact that exact integrable models can be developed for these spaces.\",\n \"The majority of well-known exact solutions is classified as SST (Schwarzschild, Kerr, Friedman-Robertson-Walker, NUT, etc.).\",\n \"It is important to note that the other single-particle equations of motion - Klein-Gordon-Fock, Dirac, Weyl - admit a separation of variables in SST.\",\n \"The same method can be used to obtain solutions in the theories of modified gravity (refs.\",\n \"14–16).\"\n ],\n [\n \"Pure radiation in conformally Stackel space-times\",\n \"The Stackel spaces type is defined by a set of two numbers $(N.N_0)$ , where $N$ – the number of commuting Killing vectors $Y^i_{(p)}$ ($p=1,N$ ) admitted by the ”complete set” (the dimension of the Abelian group of space-time motions), and $N_0=N-rank|Y^i_{(p)}g_{ij}Y^j_{(q)}|$ – is the number of null (\\\"wave-like\\\") variables in privileged coordinate systems.\",\n \"For 4-dimensional Stackel spaces of Lorentz signature $N=0...3$ , $N_0=0,1$ .\",\n \"Stackel spaces metrics are conveniently written in privileged coordinate systems in a contravariant form.\",\n \"The coordinates of space-time will be numbered from 0 to 3 by indices $i$ , $ j$ , $k$ .\",\n \"Ignored variables (on which the metric does not depend) will be numbered in indices $ p$ , $q$ , $r$ and non-ignored by Greek letters (indices) $ \\\\mu $ , $ \\\\nu $ , $ \\\\sigma $ .\",\n \"The following notation will be used in the paper: $P=\\\\ln \\\\left|\\\\frac{{\\\\varepsilon }^2}{\\\\Delta ^2 \\\\det g^{ij}}\\\\right|,$ where $g^{ij}$ is the space-time metric, $\\\\varepsilon $ is the energy density of radiation and $\\\\Delta $ is the conformal factor of metric $g_{ij}$ .\",\n \"For conformally Stackel spaces the conformal factor of metric is an arbitrary function of all variables.\",\n \"In the privlieged coordinate system, the wave vector of the radiation has a \\\"separated\\\" form, i.e.\",\n \"the corresponding covariant components of the wave vector depend only on one variable ${L}_i={L}_i(x^i)$ : $& L_0=L_0(x^0),\\\\quad L_1=L_1(x^1),&\\\\\\\\&L_2=L_2(x^2),\\\\quad L_3=L_3(x^3).&\\\\nonumber $ Later in the paper we obtain a functional form of the energy density and the wave vector of radiation for all types of conformally Stackel space-times in privileged coordinate systems (where the separation of variables in the eikonal equation (REF ) is allowed).\"\n ],\n [\n \"Conformally Stackel space-times (3.0) type\",\n \"Conformally Stackel space-times (3.0) type admit three commuting Killing vectors.\",\n \"In the privileged coordinate system, the metric of a conformally Stackel space of type (3.0) can be written in the following form: $g^{ij}=\\\\frac{1}{\\\\Delta }{1&0&0&0\\\\cr 0&a_0&b_0&c_0\\\\cr 0&b_0&d_0&e_0\\\\cr 0&c_0&e_0&f_0},$ where $\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3)$ and $ a_0, b_0, c_0, d_0, e_0, f_0$ are functions of a variable $x^0$ .\",\n \"The wave vector of the radiation $L_i$ in the privileged coordinate system has the following \\\"separated\\\" form: $&{L}_0={L}_0(x^0),\\\\quad {L}_1=\\\\alpha ,\\\\quad {L}_2=\\\\beta ,\\\\quad {L}_3=\\\\gamma ,&\\\\\\\\&\\\\alpha , \\\\beta , \\\\gamma - const.&\\\\nonumber $ Wave vector norm condition () takes the form: $\\\\alpha ^2a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2d_0+2\\\\alpha \\\\gamma c_0+2\\\\beta \\\\gamma e_0+\\\\gamma ^2 f_0=-{L}_0{}^2.$ From the equations of the energy-momentum conservation law (REF ) we obtain: $& {L}_0P_{,0}+(\\\\alpha a_0+\\\\beta b_0+\\\\gamma c_0)P_{,1}+(\\\\alpha b_0+\\\\beta d_0+\\\\gamma e_0)P_{,2} &\\\\nonumber \\\\\\\\&+\\\\mbox{} (\\\\alpha c_0+\\\\beta e_0+\\\\gamma f_0)P_{,3}+2{L}_0{}^{\\\\prime }=0,&$ here and in the sequel, a comma means a partial differentiation, and a prime means differentiation with respect to a single variable on which the function depends.\",\n \"Using condition (REF ) and finding the integrals of equation (REF ), one can obtain expressions for the wave vector and the energy density of radiation.\",\n \"Below are listed all the solutions we have obtained for conformally Stackel space-times (3.0) type.\",\n \"The wave vector of the radiation has the form: $&{L}_i=\\\\left( L_0(x^0), \\\\alpha , \\\\beta , \\\\gamma \\\\right),\\\\quad \\\\alpha , \\\\beta , \\\\gamma - const,&\\\\\\\\[1ex]&{L}_0=\\\\sqrt{\\\\alpha ^2a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2d_0+2\\\\alpha \\\\gamma c_0+2\\\\beta \\\\gamma e_0+\\\\gamma ^2 f_0}.&\\\\nonumber $ For the energy density of radiation we obtain the expression: $&{\\\\varepsilon }=F(X,Y,Z){\\\\Delta \\\\sqrt{- \\\\det g^{ij}}}/{{L}_0},&\\\\\\\\[1ex]&X=x^1-\\\\int {(\\\\alpha a_0+\\\\beta b_0+\\\\gamma c_0)}/{{L}_0}\\\\,dx^0,&\\\\nonumber \\\\\\\\&Y=x^2-\\\\int {(\\\\alpha b_0+\\\\beta d_0+\\\\gamma e_0)}/{{L}_0}\\\\,dx^0,&\\\\nonumber \\\\\\\\&Z=x^3-\\\\int {(\\\\alpha c_0+\\\\beta e_0+\\\\gamma f_0)}/{{L}_0}\\\\,dx^0,& \\\\nonumber $ where $ F (X, Y, Z) $ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (3.0) type, case 2. (${L}_0= 0$ ).\",\n \"For this degenerate case there is an additional condition for the functions of the metric (REF ): $&\\\\alpha ^2a_0+2\\\\alpha \\\\beta b_0+2\\\\alpha \\\\gamma c_0+\\\\beta ^2d_0+2\\\\beta \\\\gamma e_0+\\\\gamma ^2 f_0=0,& \\\\nonumber \\\\\\\\&\\\\alpha , \\\\beta , \\\\gamma - const.&$ The wave vector takes the form: ${L}_i=(0,\\\\alpha , \\\\beta , \\\\gamma ).$ For the energy density of radiation we obtain the expression: $&{\\\\varepsilon }=F(x^0,Y,Z){\\\\Delta \\\\sqrt{- \\\\det g^{ij}}}&\\\\\\\\[1ex]&Y={x^1}/{(\\\\alpha a_0+\\\\beta b_0+\\\\gamma c_0)}-{x^2}/{(\\\\alpha b_0+\\\\beta d_0+\\\\gamma e_0)},& \\\\nonumber \\\\\\\\&Z={x^1}/{(\\\\alpha a_0+\\\\beta b_0+\\\\gamma c_0)}-{x^3}/{(\\\\alpha c_0+\\\\beta e_0+\\\\gamma f_0)},& \\\\nonumber $ where $F(x^0,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (3.0) type, case 3. (${L}_0=0$ , {{formula:70047d94-5e5d-40db-85a6-32afca0e0cfd}} ).\",\n \"In this degenerate case, when one of the terms in (REF ) additionally vanishes together with $ {L}_0 $ (for example, the term with $ P_{, 1} $ ), we obtain additional conditions for the metric (REF ): $\\\\alpha a_0+\\\\beta b_0+\\\\gamma c_0=0,\\\\quad \\\\alpha , \\\\beta , \\\\gamma - const,$ $\\\\alpha ^2a_0+2\\\\alpha \\\\beta b_0+2\\\\alpha \\\\gamma c_0+\\\\beta ^2d_0+2\\\\beta \\\\gamma e_0+\\\\gamma ^2 f_0=0.$ The wave vector of the radiation has the form: ${L}_i=(0,\\\\alpha , \\\\beta , \\\\gamma ).$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(x^0,x^1,Z){\\\\Delta \\\\sqrt{- \\\\det g^{ij}}}$ $Z=x^2 (\\\\alpha c_0+\\\\beta e_0+\\\\gamma f_0)- x^3 (\\\\alpha b_0+\\\\beta d_0+\\\\gamma e_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"Conformally Stackel space-times (3.1) type\",\n \"Conformally Stackel space-times (3.1) type admits 3 commuting Killing vectors $Y^i_{(p)}$ ($p=1,3$ ), but $rank\\\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=2$ .\",\n \"In a privileged coordinate system the metric of a conformally Stackel space-times (3.1) type can be written in the following form, where the variable $ x^0 $ is a null (\\\"wave-like\\\") variable: $g^{ij}=\\\\frac{1}{\\\\Delta }{0&1&a_0&b_0\\\\cr 1&0&0&0\\\\cr a_0&0&c_0&f_0\\\\cr b_0&0&f_0&d_0},$ where $\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3)$ and $a_0, b_0, c_0, d_0, f_0$ are functions of a variable $ x^0$ .\",\n \"The wave vector of the radiation has the form: $&{L}_0={L}_0(x^0),\\\\quad {L}_1=\\\\alpha ,\\\\quad {L}_2=\\\\beta ,\\\\quad {L}_3=\\\\gamma ,&\\\\\\\\&\\\\alpha , \\\\beta , \\\\gamma - const.\\\\nonumber $ The system of equations (REF ), () takes the form: $\\\\beta ^2 c_0+2\\\\beta \\\\gamma f_0+\\\\gamma ^2 d_0+2(\\\\alpha +\\\\beta a_0+\\\\gamma b_0){L}_0=0,$ $&(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)P_{,0}+{L}_0P_{,1}+(a_0{L}_0+\\\\beta c_0+\\\\gamma f_0)P_{,2}&\\\\nonumber \\\\\\\\&\\\\mbox{}+(b_0{L}_0+\\\\beta f_0+\\\\gamma d_0)P_{,3}+2\\\\beta a_0^{\\\\prime }+2\\\\gamma b_0^{\\\\prime }=0.&$ From equations (REF )-(REF ) we can obtain functional expressions for the wave vector and the energy density of radiation.\",\n \"Below all the solutions for conformally Stackel space-times (3.1) are listed.\"\n ],\n [\n \"CSS (3.1) type, case 1. ($\\\\alpha +\\\\beta a_0+\\\\gamma b_0\\\\ne 0$ ).\",\n \"The wave vector of the radiation has the form: $ {L}_i=\\\\left( L_0(x^0), \\\\alpha , \\\\beta , \\\\gamma \\\\right),\\\\quad \\\\alpha , \\\\beta , \\\\gamma - const,$ $ {L}_0=\\\\frac{-(\\\\beta ^2 c_0+2\\\\beta \\\\gamma f_0+\\\\gamma ^2 d_0)}{2\\\\,(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)}.$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(X,Y,Z){\\\\Delta \\\\sqrt{- \\\\det g^{ij}}}/{(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)},$ $X&=&x^1-\\\\int \\\\frac{{L}_0}{(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)}\\\\,dx^0,\\\\nonumber \\\\\\\\Y&=&x^2-\\\\int \\\\frac{(a_0{L}_0+\\\\beta c_0+\\\\gamma f_0)}{(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)}\\\\,dx^0,\\\\nonumber \\\\\\\\Z&=&x^3-\\\\int \\\\frac{(b_0{L}_0+\\\\beta f_0+\\\\gamma d_0)}{(\\\\alpha +\\\\beta a_0+\\\\gamma b_0)}\\\\,dx^0,\\\\nonumber $ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (3.1) type, case 2. ($L_0\\\\ne 0$ , {{formula:06fa1d48-9bb8-49b0-b333-846375975dad}} ). \",\n \"In this degenerate case there are additional conditions for the metric (REF ): $\\\\alpha +\\\\beta a_0+\\\\gamma b_0= 0,\\\\quad \\\\alpha , \\\\beta , \\\\gamma - const,$ $\\\\beta ^2 c_0+2\\\\beta \\\\gamma f_0+\\\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: $ {L}_i=\\\\left( L_0(x^0), \\\\alpha , \\\\beta , \\\\gamma \\\\right).$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(x^0,Y,Z) \\\\Delta \\\\sqrt{- \\\\det g^{ij}}.$ $Y&=& x^1-\\\\frac{x^2(a_0{L}_0+\\\\beta c_0+\\\\gamma f_0) }{L_0},\\\\nonumber \\\\\\\\Z&=& x^1-\\\\frac{x^3(b_0{L}_0+\\\\beta f_0+\\\\gamma d_0) }{L_0},\\\\nonumber $ where $L_0(x^0)$ and $F(x^0,Y,Z)$ are arbitrary functions of their variables.\"\n ],\n [\n \"CSS (3.1) type, case 3. ($L_0= 0$ , {{formula:88ed6080-40a1-42aa-a8a8-2b68967c0299}} ).\",\n \"In this degenerate case there are additional conditions for the metric (REF ): $\\\\alpha +\\\\beta a_0+\\\\gamma b_0= 0,\\\\quad \\\\alpha , \\\\beta , \\\\gamma - const,$ $\\\\beta ^2 c_0+2\\\\beta \\\\gamma f_0+\\\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: ${L}_i=\\\\left( 0, \\\\alpha , \\\\beta , \\\\gamma \\\\right).$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(x^0,x^1,Z) \\\\Delta \\\\sqrt{- \\\\det g^{ij}}.$ $Z=x^2(\\\\beta f_0+\\\\gamma d_0)-x^3(\\\\beta c_0+\\\\gamma f_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables.\",\n \"Conformally Stackel space-times (2.0) type admit two commuting Killing vectors.\",\n \"In the privileged coordinate system, the metric can be written in the following form: $g^{ij}=\\\\frac{1}{\\\\Delta }{1&0&0&0\\\\cr 0&{{\\\\epsilon }} &0&0\\\\cr 0&0&A&B\\\\cr 0&0&B&C},$ $&\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3),\\\\quad {\\\\epsilon } =\\\\pm 1,\\\\quad A=a_0(x^0)+a_1(x^1),&\\\\\\\\&B=b_0(x^0)+b_1(x^1),\\\\qquad C=c_0(x^0)+c_1(x^1) .&$ The wave vector of the radiation ${L}_i$ has the form: $&{L}_0={L}_0(x^0),\\\\qquad {L}_1={L}_1(x^1),&\\\\\\\\&{L}_2=\\\\alpha ,\\\\qquad {L}_3=\\\\beta , \\\\qquad \\\\alpha , \\\\beta - const.&\\\\nonumber $ From the norm condition () one can obtain: $&{{L}_0}^2+\\\\alpha ^2 a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2 c_0-\\\\gamma =0,&\\\\\\\\&{\\\\epsilon } {{L}_1}^2+\\\\alpha ^2 a_1+2\\\\alpha \\\\beta b_1+\\\\beta ^2 c_1+\\\\gamma =0.&$ From the conservation law (REF ) one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+{\\\\epsilon } {L}_1P_{,1}+(\\\\alpha A+\\\\beta B)P_{,2}+(\\\\alpha B+\\\\beta C)P_{,3}&\\\\nonumber \\\\\\\\&\\\\mbox{}+2{L}_0^{\\\\prime }+2{\\\\epsilon } {L}_1^{\\\\prime }=0.&$ Below are listed all the solutions for conformally Stackel space-times (2.0) type.\"\n ],\n [\n \"CSS (2.0) type, case 1. (${L}_0{L}_1\\\\ne 0$ ):\",\n \"The wave vector of the radiation has the form: $&{L}_i=\\\\left( L_0(x^0), {L}_1(x^1), \\\\alpha , \\\\beta \\\\right), \\\\qquad \\\\alpha , \\\\beta ,\\\\gamma - const,&\\\\nonumber \\\\\\\\[1ex]&{L}_0=\\\\sqrt{\\\\gamma -\\\\alpha ^2 a_0-2\\\\alpha \\\\beta b_0-\\\\beta ^2 c_0},&\\\\nonumber \\\\\\\\&{L}_1=\\\\sqrt{{\\\\epsilon } (-\\\\gamma -\\\\alpha ^2 a_1-2\\\\alpha \\\\beta b_1-\\\\beta ^2 c_1)}.&$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(X,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/({{L}_0\\\\,{L}_1}),$ $&X=\\\\int 1/{{L}_0}\\\\, dx^0- \\\\int {\\\\epsilon }/{{L}_1}\\\\, dx^1,&\\\\\\\\&Y=x^2-\\\\int {(\\\\alpha a_0+\\\\beta b_0)}/{{L}_0}\\\\, dx^0 - {\\\\epsilon } \\\\int {(\\\\alpha a_1+\\\\beta b_1)}/{{L}_1}\\\\, dx^1,&\\\\\\\\&Z=x^3-\\\\int {(\\\\alpha b_0+\\\\beta c_0)}/{{L}_0}\\\\, dx^0 - {\\\\epsilon } \\\\int {(\\\\alpha b_1+\\\\beta c_1)}/{{L}_1}\\\\, dx^1,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (2.0) type, case 2. (${L}_0=0, {L}_1\\\\ne 0$ ):\",\n \"Let us consider the degenerate case when $ {L}_0=0$ or ${L}_1 = 0 $ and fo rcertainty we take $ {L}_0 = 0 $ and $ {L}_1 \\\\ne 0 $ , then the wave vector will take the following form: ${L}_i=\\\\left( 0, {L}_1(x^1), \\\\alpha , \\\\beta \\\\right), \\\\qquad \\\\alpha , \\\\beta , \\\\gamma - const,$ ${L}_1=\\\\sqrt{{\\\\epsilon } (-\\\\gamma -\\\\alpha ^2 a_1-2\\\\alpha \\\\beta b_1-\\\\beta ^2 c_1)}.$ In this degenerate case there are additional conditions for the metric (REF ): $\\\\alpha ^2 a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2 c_0-\\\\gamma =0.$ For the energy density of radiation we obtain the expression: ${\\\\varepsilon }=F(x^0,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{{L}_1},$ $Y=x^2-\\\\int {(\\\\alpha a_1+\\\\beta b_1)}/{{L}_1}\\\\, dx^1,$ $Z=x^3- {\\\\epsilon } \\\\int {(\\\\alpha b_1+\\\\beta c_1)}/{{L}_1}\\\\, dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (2.0) type, case 3. (${L}_0={L}_1= 0$ ):\",\n \"In this degenerate case, there are additional conditions for the metric (REF ): $\\\\alpha ^2 A+2\\\\alpha \\\\beta B+\\\\beta ^2 C=0,\\\\qquad \\\\alpha , \\\\beta - const.$ The wave vector of the radiation has the form: ${L}_i=\\\\left( 0, 0, \\\\alpha , \\\\beta \\\\right).$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^0,x^1,Z)\\\\,\\\\Delta \\\\sqrt{-\\\\det g^{ij}},$ $Z=x^2 (\\\\alpha B+\\\\beta C) -x^3(\\\\alpha A+\\\\beta B),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"Conformally Stackel space-times (2.1) type\",\n \"Conformally Stackel space-times (2.1) type admits two commuting Killing vectors $Y^i_{(p)}$ ($p=1,2$ ), but $rank\\\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=1$ .\",\n \"In a privileged coordinate system the metric of a conformally Stackel space-times (2.1) type can be written in the following form, where $ x^1 $ is a null (\\\"wave-like\\\") variable: $g^{ij}=\\\\frac{1}{\\\\Delta }{1&0&0&0\\\\cr 0&0&f_1(x^1)&1\\\\cr 0&f_1(x^1)&A&B\\\\cr 0&1&B&C},$ $&\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3), \\\\quad A=a_0(x^0)+a_1(x^1),&\\\\\\\\&B=b_0(x^0)+b_1(x^1),\\\\quad C=c_0(x^0)+c_1(x^1).&$ The wave vector of radiation $L_i$ has the form: $&{L}_0={L}_0(x^0),\\\\quad {L}_1={L}_1(x^1),&\\\\nonumber \\\\\\\\&{L}_2=\\\\alpha ,\\\\quad {L}_3=\\\\beta , \\\\quad \\\\alpha , \\\\beta , \\\\gamma - const.&$ From the norm condition () one can obtain: $&\\\\gamma ={{L}_0}^2+\\\\alpha ^2 a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2 c_0,&\\\\\\\\&-\\\\gamma =2(\\\\alpha f_1+\\\\beta ){L}_1+\\\\alpha ^2 a_1+2\\\\alpha \\\\beta b_1+\\\\beta ^2 c_1.&$ From the conservation law (REF ), one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+(\\\\alpha f_1+\\\\beta ) P_{,1}+(\\\\alpha A+\\\\beta B+f_1 {L}_1)P_{,2}&\\\\nonumber \\\\\\\\&\\\\mbox{}+(\\\\alpha B+\\\\beta C+{L}_1)P_{,3}+2\\\\alpha f_1^{\\\\prime }+2{{L}_0}^{\\\\prime }=0.&$ From the system of equations (REF )–(REF ) we can obtain functional expressions through the metric for the wave vector and the radiation energy density.\",\n \"Below all the solutions for conformally Stackel space-times (2.1) type are listed.\"\n ],\n [\n \"CSS (2.1) type, case 1. ${L}_0(\\\\alpha f_1+\\\\beta )\\\\ne 0$ :\",\n \"The wave vector of radiation has the form: $&{L}_i=\\\\left( L_0(x^0), {L}_1(x^1), \\\\alpha , \\\\beta \\\\right), \\\\qquad \\\\alpha , \\\\beta ,\\\\gamma - const,&\\\\nonumber \\\\\\\\[1ex]&{L}_0=\\\\sqrt{\\\\gamma -\\\\alpha ^2 a_0-2\\\\alpha \\\\beta b_0-\\\\beta ^2 c_0},&\\\\\\\\&{L}_1={(-\\\\gamma -\\\\alpha ^2 a_1-2\\\\alpha \\\\beta b_1-\\\\beta ^2 c_1)}/\\\\big ({2\\\\,(\\\\alpha f_1+\\\\beta )}\\\\big ).&\\\\nonumber $ The energy density of the radiation has the form: ${\\\\varepsilon }=F(X,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{\\\\big ({L}_0(\\\\alpha f_1+\\\\beta )\\\\big )},$ $X=\\\\int \\\\frac{dx^0}{{L}_0}- \\\\int \\\\frac{dx^1}{(\\\\alpha f_1+\\\\beta )},$ $Y=x^2-\\\\int \\\\frac{(\\\\alpha a_0+\\\\beta b_0)}{{L}_0}\\\\,dx^0 -\\\\int \\\\frac{(\\\\alpha a_1+\\\\beta b_1+f_1L_1)}{\\\\alpha f_1+\\\\beta }\\\\, dx^1,$ $Z=x^3-\\\\int \\\\frac{(\\\\alpha b_0+\\\\beta c_0)}{{L}_0}\\\\,dx^0 -\\\\int \\\\frac{(\\\\alpha b_1+\\\\beta c_1+L_1)}{\\\\alpha f_1+\\\\beta }\\\\, dx^1,$ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (2.1) type, case 2. $(\\\\alpha f_1+\\\\beta )= 0$ , {{formula:89e7bb2c-cabe-4b63-969f-80c8a732525b}} .\",\n \"In this degenerate case, there are additional conditions for the metric (REF ): $a_1=f_1=0.$ The wave vector of radiation has the form ($\\\\beta =\\\\gamma =0$ ): ${L}_i=\\\\left( L_0(x^0), {L}_1(x^1), \\\\alpha , 0\\\\right),$ ${L}_0=\\\\alpha \\\\sqrt{- a_0},\\\\quad L_1=\\\\sigma -\\\\alpha b_1,\\\\quad \\\\alpha , \\\\sigma - const.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^1,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{L_0(x^0)},$ $Y=\\\\alpha x^2+\\\\int {L_0}\\\\,dx^0,\\\\quad Z= x^3 -\\\\int {(\\\\sigma +\\\\alpha b_0)}/{L_0}\\\\,dx^0,$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (2.1) type, case 3. ${L}_0= 0$ , {{formula:b7686906-83ba-49e5-96c2-9923fa201a29}} .\",\n \"In this degenerate case, there are additional conditions for the metric (REF ): $&\\\\alpha ^2 a_0+2\\\\alpha \\\\beta b_0+\\\\beta ^2 c_0=\\\\gamma ,&\\\\\\\\&p(\\\\alpha a_0+\\\\beta b_0)+q (\\\\alpha b_0+\\\\beta c_0)=r,&\\\\\\\\&\\\\alpha , \\\\beta , \\\\gamma , p, q, r \\\\mbox{ -- const}.&\\\\nonumber $ The wave vector of radiation has the form: $&{L}_i=\\\\left( 0, {L}_1(x^1), \\\\alpha , \\\\beta \\\\right),&\\\\\\\\&{L}_1= {(-\\\\alpha ^2 a_1-2\\\\alpha \\\\beta b_1-\\\\beta ^2 c_1-\\\\gamma )}/{(2\\\\,(\\\\alpha f_1+\\\\beta ))}.&\\\\nonumber $ The energy density of the radiation has the form: $ {\\\\varepsilon }=F(x^0,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{(\\\\alpha f_1+\\\\beta )},$ $Y=\\\\alpha x^2+\\\\beta x^3 +\\\\int L_1(x^1)\\\\,dx^1,$ $Z=px^2+qx^3$ $\\\\mbox{}-\\\\int \\\\frac{r+ p(\\\\alpha a_1+\\\\beta b_1+f_1L_1)+q(\\\\alpha b_1+\\\\beta c_1+L_1)}{(\\\\alpha f_1+\\\\beta )}\\\\,dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (2.1) type, case 4. ${L}_0= 0$ , {{formula:468ec54a-5663-4dcf-914e-e423945394f8}} , {{formula:fdcb860c-4b2a-4abd-a024-0e7e81517f01}} .\",\n \"The wave vector of radiation has the form: ${L}_i=\\\\left( 0, {L}_1(x^1), 0, 0\\\\right),\\\\qquad \\\\alpha = \\\\beta =\\\\gamma =0.$ The energy density of the radiation has the form: $ {\\\\varepsilon }=F(x^0,x^1,Z)\\\\,\\\\Delta \\\\sqrt{-\\\\det g^{ij}},$ $Z=x^2 -x^3f_1(x^1),$ where $L_1(x^1)$ and $F(x^0,x^1,Z)$ are arbitrary functions of their variables.\"\n ],\n [\n \"Conformally Stackel space-times (1.0) type\",\n \"Conformally Stackel space-times (1.0) type admits one Killing vector, and in a privileged coordinate system the metric of this space-time can be written in the following form: $g^{ij}=\\\\frac{1}{\\\\Delta }{\\\\Omega &0&0&0\\\\cr 0&V^1&0&0\\\\cr 0&0&V^2&0\\\\cr 0&0&0&V^3},$ $&\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3),&\\\\\\\\&V^1=t_2(x^2)-t_3(x^3),\\\\qquad V^2=t_3(x^3)-t_1(x^1),&\\\\\\\\&V^3=t_1(x^1)-t_2(x^2),\\\\quad \\\\Omega =\\\\omega _\\\\nu (x^\\\\nu ) V^\\\\nu ,\\\\quad \\\\mu ,\\\\,\\\\nu =1...3.&$ The wave vector of radiation has the form: $ {L}_i=\\\\Big (\\\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\\\Big ),\\\\qquad \\\\alpha =const.$ From equations (REF ), () we have: $ \\\\Omega \\\\alpha ^2 + V^\\\\mu {{L}_\\\\mu }^2=0,$ $ \\\\alpha \\\\Omega P_{,0}+ V^\\\\mu ({L}_\\\\mu P_{,\\\\mu }+2{L}_\\\\mu ^{\\\\prime })=0.\",\n \"$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.\",\n \"Below are listed all the solutions for conformally Stackel space-times (1.0) type.\"\n ],\n [\n \"CSS (1.0) type, case 1. ${L}_1{L}_2{L}_3\\\\ne 0$ .\",\n \"The wave vector of radiation has the form: $&{L}_i=\\\\left(\\\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\\\right),\\\\quad \\\\alpha ,\\\\beta ,\\\\gamma \\\\mbox{ -- const}.&\\\\nonumber \\\\\\\\[1ex]&{L}_\\\\mu =\\\\sqrt{-\\\\alpha ^2\\\\omega _\\\\mu +\\\\beta t_\\\\mu +\\\\gamma },\\\\quad \\\\mu , \\\\nu =1...3.&$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(X,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{({L}_1\\\\,{L}_2\\\\,{L}_3)},$ $&X=x^0 -\\\\alpha \\\\sum _\\\\mu \\\\int ({\\\\omega _\\\\mu }/{{L}_\\\\mu })\\\\, dx^\\\\mu ,&\\\\\\\\&Y=\\\\sum _\\\\mu \\\\int ({t_\\\\mu }/{{L}_\\\\mu })\\\\, dx^\\\\mu ,\\\\qquad Z= \\\\sum _\\\\mu \\\\int (1/{{L}_\\\\mu }){\\\\,dx^\\\\mu } ,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (1.0) type, case 2. ${L}_1=0, {L}_2{L}_3 \\\\ne 0$ .\",\n \"In this degenerate case, when one of the components of the wave vector $ {L}_\\\\mu $ becomes zero (for definiteness, let $ L_1 = 0 $ , $ {L}_2 {L}_3 \\\\ne 0 $ ) we have an additional condition for the metric (REF ): $\\\\alpha ^2\\\\omega _1=\\\\beta t_1+\\\\gamma ,\\\\qquad \\\\mbox{$\\\\alpha ,\\\\beta ,\\\\gamma $ -- const}.$ The wave vector of radiation has the form: ${L}_i=\\\\Big (\\\\alpha ,0,{L}_2(x^2),{L}_3(x^3)\\\\Big ),$ ${L}_2=\\\\sqrt{-\\\\alpha ^2\\\\omega _2+\\\\beta t_2+\\\\gamma },\\\\quad {L}_3=\\\\sqrt{-\\\\alpha ^2\\\\omega _3+\\\\beta t_3+\\\\gamma }.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^1,Y,Z)\\\\,\\\\Delta \\\\sqrt{-\\\\det g^{ij}}/\\\\prod _{\\\\mu \\\\ne 1}{L}_\\\\mu ,$ $Y=\\\\alpha x^0 -\\\\alpha ^2\\\\sum _{\\\\mu \\\\ne 1} \\\\int ({\\\\omega _\\\\mu }/{{L}_\\\\mu })\\\\, dx^\\\\mu +\\\\beta \\\\sum _{\\\\mu \\\\ne 1} \\\\int ({t_\\\\mu }/{{L}_\\\\mu })\\\\, dx^\\\\mu ,$ $Z=\\\\sum _{\\\\mu \\\\ne 1} \\\\int (1/{{L}_\\\\mu }){\\\\,dx^\\\\mu },$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (1.0) type, case 3. $L_1\\\\ne 0, {L}_2=0, {L}_3= 0$ .\",\n \"In this degenerate case, when two of the components of the wave vector $ {L}_\\\\mu $ becomes zero (for definiteness, let $ L_1 \\\\ne 0 $ , $ {L}_2 ={L}_3= 0 $ ) we have additional conditions for the metric (REF ): $\\\\alpha ^2\\\\omega _2=\\\\beta t_2+\\\\gamma ,\\\\quad \\\\alpha ^2\\\\omega _3=\\\\beta t_3+\\\\gamma ,$ where $\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ – const.\",\n \"The wave vector of radiation has the form: ${L}_i=\\\\left(\\\\alpha ,{L}_1(x^1),0,0\\\\right),\\\\quad {L}_1=\\\\sqrt{\\\\beta t_1-\\\\alpha ^2\\\\omega _1+\\\\gamma }.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^2,x^3, Y)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{{L}_1(x^1)},$ $Y=\\\\alpha x^0+\\\\int L_1(x^1) \\\\,dx^1,$ where $F(x^2,x^3,Y)$ is an arbitrary function of its variables.\"\n ],\n [\n \"Conformally Stackel space-times (1.1) type\",\n \"Conformally Stackel space-times (1.1) type admits one Killing vector.\",\n \"In a privileged coordinate system the metric of a conformally Stackel space-times (1.1) type can be written in the following form, where $ x^1 $ is a null (\\\"wave-like\\\") variable: $g^{ij}=\\\\frac{1}{\\\\Delta }{\\\\Omega &V^1&0&0\\\\cr V^1&0&0&0\\\\cr 0&0&V^2&0\\\\cr 0&0&0&V^3},$ $&\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3),&\\\\\\\\&V^1=t_2(x^2)-t_3(x^3),\\\\quad V^2=t_3(x^3)-t_1(x^1),&\\\\\\\\&V^3=t_1(x^1)-t_2(x^2),\\\\quad \\\\Omega =\\\\omega _\\\\mu (x^\\\\mu ) V^\\\\mu ,\\\\quad \\\\mu ,\\\\nu =1...3.$ The wave vector of radiation has the following \\\"separated\\\" form: $ {L}_i=\\\\Big (\\\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\\\Big ),\\\\quad \\\\alpha - const.$ From equations (REF ), () we have: $ \\\\alpha V^1(2 {L}_1+\\\\alpha \\\\omega _1)+V^2({L}_2{}^2+\\\\alpha ^2\\\\omega _2)+V^3({L}_3{}^2+\\\\alpha ^2\\\\omega _3)=0,$ $(V^1{L}_1+\\\\alpha \\\\Omega )P_{,0}+ \\\\alpha V^1P_{,1}+ {L}_2V^2P_{,2}+ {L}_3V^3P_{,3}$ $\\\\mbox{}+ 2V^2{L}_2^{\\\\prime }+ 2V^3{L}_3^{\\\\prime } = 0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.\",\n \"Below are listed all the solutions for conformally Stackel space-times (1.1) type.\"\n ],\n [\n \"CSS (1.1) type, case 1. $\\\\alpha L_2 L_3\\\\ne 0$ .\",\n \"The wave vector of radiation has the form: ${L}_0=\\\\alpha ,\\\\quad {L}_1=\\\\frac{1}{2\\\\alpha }(\\\\beta t_1 -\\\\alpha ^2\\\\omega _1+\\\\gamma ),\\\\quad \\\\alpha ,\\\\beta , \\\\gamma - const,$ ${L}_2=\\\\sqrt{\\\\beta t_2-\\\\alpha ^2\\\\omega _2+\\\\gamma },\\\\quad {L}_3=\\\\sqrt{\\\\beta t_3-\\\\alpha ^2\\\\omega _3+\\\\gamma }.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(X,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{({L}_2\\\\,{L}_3)},$ $X=x^0 -\\\\frac{1}{\\\\alpha } \\\\int ({L}_1+\\\\alpha \\\\omega _1)\\\\,dx^1 -\\\\alpha \\\\left( \\\\int \\\\frac{\\\\omega _2}{{L}_2}\\\\,dx^2 +\\\\int \\\\frac{\\\\omega _3}{{L}_3}\\\\,dx^3 \\\\right),$ $Y=-\\\\frac{1}{\\\\alpha }\\\\int t_1\\\\, dx^1 +\\\\int \\\\frac{t_2}{{L}_2}\\\\,dx^2+\\\\int \\\\frac{t_3}{{L}_3}\\\\,dx^3 ,$ $Z= \\\\frac{x^1}{\\\\alpha }+\\\\int \\\\frac{dx^2}{{L}_2}+\\\\int \\\\frac{dx^3}{{L}_3},$ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (1.1) type, case 2. $\\\\alpha = 0$ .\",\n \"In this degenerate case, when $ {L}_0 = 0 $ , we obtain an additional condition for the metric (REF ): $t_1=0.$ The wave vector of the radiation $L_i$ has the following form ($\\\\gamma =0$ ): ${L}_0=0,\\\\quad {L}_1=L_1(x^1),\\\\quad {L}_2=\\\\sqrt{\\\\beta t_2},\\\\quad {L}_3=\\\\sqrt{\\\\beta t_3}.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^1,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{({L}_2\\\\,{L}_3)},$ $Y=\\\\int \\\\frac{t_2}{{L}_2}\\\\,dx^2+\\\\int \\\\frac{t_3}{{L}_3}\\\\,dx^3 ,\\\\quad Z=\\\\frac{x^0}{L_1}+ \\\\int \\\\frac{dx^2}{{L}_2}+\\\\int \\\\frac{dx^3}{{L}_3},$ where $F(x^1,Y,Z)$ and $L_1(x^1)$ are arbitrary functions of its variables.\"\n ],\n [\n \"CSS (1.1) type, case 3. $\\\\alpha L_2\\\\ne 0, L_3=0$ .\",\n \"In the degenerate case, when one of the components of the wave vector $L_3$ becomes zero (similarly for $ L_2 $ with the replacement of the indices 3 by 2) we have additional conditions for the metric (REF ): $\\\\alpha ^2\\\\omega _3=\\\\beta t_3+\\\\gamma , \\\\qquad \\\\alpha , \\\\beta , \\\\gamma - const,$ $(p\\\\alpha -q\\\\beta )\\\\,t_3=0,\\\\qquad p,q - const.$ The wave vector of the radiation $L_i$ has the following form: ${L}_0=\\\\alpha ,\\\\qquad {L}_1=\\\\frac{1}{2\\\\alpha }(\\\\beta t_1 -\\\\alpha ^2\\\\omega _1+\\\\gamma ),$ ${L}_2=\\\\sqrt{\\\\beta t_2-\\\\alpha ^2\\\\omega _2+\\\\gamma },\\\\qquad {L}_3=0.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^1,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{{L}_2},$ $Y=\\\\alpha x^0+\\\\int L_1\\\\,dx^1+\\\\int L_2\\\\,dx^2,$ $Z=qx^0+\\\\frac{1}{\\\\alpha }\\\\int \\\\Big ( q(\\\\gamma /\\\\alpha -\\\\alpha \\\\omega _1 -L_1)+pt_1\\\\Big )dx^1$ $\\\\mbox{}+\\\\int \\\\frac{q(\\\\gamma /\\\\alpha -\\\\alpha \\\\omega _2)+pt_2}{L_2}\\\\,dx^2,$ where $F(x^1,Y,Z)$ is an arbitrary function of their variables.\"\n ],\n [\n \"Conformally Stackel space-times (0.0) type\",\n \"The metric of conformally Stackel space-times (0.0) type in a privileged coordinate system can be written in the following form: $g^{ij}=\\\\frac{1}{\\\\Delta }{V^0 &0&0&0\\\\cr 0&V^1&0&0\\\\cr 0&0&V^2&0\\\\cr 0&0&0&V^3},$ $&\\\\Delta =\\\\Delta (x^0,x^1,x^2,x^3),&\\\\\\\\&V^0=a_1(b_2-b_3)+a_2(-b_1+b_3)+a_3(b_1-b_2),&\\\\\\\\&V^1= a_0(-b_2+b_3)+a_2(b_0-b_3)+a_3(-b_0+b_2),&\\\\\\\\&V^2=a_0(b_1-b_3)+a_1(-b_0+b_3)+a_3(b_0-b_1),&\\\\\\\\&V^3=a_0(-b_1+b_2)+a_1(b_0-b_2)+a_2(-b_0+b_1),&$ where functions $ a $ , $ b $ , $ c $ are functions of only one variable whose index corresponds to the lower index of the function.\",\n \"For example $ a_0 = a_0 (x ^ 0) $ , $ b_1 = b_1 (x ^ 1) $ , etc.\",\n \"The wave vector of the radiation in a privileged coordinate system has the following \\\"separated\\\" form: $ {L}_i=\\\\Big ({L}_0(x^0),{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\\\Big ).$ From the law of energy-momentum conservation (REF ) and condition () we have: $V^i{L}_i{}^2=0,\\\\qquad i=0...3,$ $ V^i({L}_iP_{,i}+2{L}_i^{\\\\prime })=0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.\",\n \"Below are listed all the solutions for conformally Stackel space-times (0.0) type.\"\n ],\n [\n \"CSS (0.0) type, case 1. ${L}_0{L}_1{L}_2{L}_3\\\\ne 0$ .\",\n \"The wave vector of the radiation $L_i$ has the following form ($\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ – const): ${L}_i=\\\\sqrt{\\\\alpha a_i+\\\\beta b_i+\\\\gamma },\\\\qquad i,j \\\\mbox{ = 0...3.\",\n \"}$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(X,Y,Z)\\\\,{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}/{({L}_0\\\\,{L}_1\\\\,{L}_2\\\\,{L}_3)},$ $X=\\\\sum _i \\\\int \\\\frac{dx^i}{{L}_i},\\\\quad Y=\\\\sum _i \\\\int \\\\frac{a_i}{{L}_i}\\\\,dx^i,\\\\quad Z=\\\\sum _i \\\\int \\\\frac{b_i}{{L}_i}\\\\,dx^i,$ where $F(X,Y,Z)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (0.0) type, case 2. ${L}_0=0, \\\\quad {L}_1{L}_2{L}_3\\\\ne 0$ .\",\n \"In the degenerate case, when one of the components of the wave vector turns to zero (for definiteness a component with the number $ i=0 $ , that is, $ {L}_0 = 0 $ ), then we obtain an additional condition for the metric (REF ): $\\\\alpha a_0+\\\\beta b_0+\\\\gamma =0,\\\\qquad \\\\mbox{$\\\\alpha $, $\\\\beta $, $\\\\gamma $ -- const.\",\n \"}$ For the wave vector of radiation ${L}_i$ we obtain: ${L}_{0}=0,\\\\qquad {L}_i=\\\\sqrt{\\\\alpha a_i+\\\\beta b_i+\\\\gamma },\\\\qquad i\\\\ne 0.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^0,X,Y)\\\\,\\\\Delta \\\\sqrt{-\\\\det g^{ij}}/{\\\\prod _{i\\\\ne 0}{L}_i},$ $X=\\\\sum _{i\\\\ne 0} \\\\int \\\\frac{dx^i}{{L}_i},\\\\qquad Y=\\\\sum _{i\\\\ne 0} \\\\int {L}_i\\\\,dx^i,$ where $F(x^0,X,Y)$ is an arbitrary function of its variables.\"\n ],\n [\n \"CSS (0.0) type, case 3. ${L}_0={L}_1= 0, \\\\quad {L}_2{L}_3\\\\ne 0$ .\",\n \"In the degenerate case, when the two components of the wave vector vanish (let it be the components $ {L}_0 =0$ , $ {L}_1 = 0 $ ), then we obtain additional conditions for the metric ($\\\\alpha $ , $\\\\beta $ , $\\\\gamma $ – const): $\\\\alpha a_0+\\\\beta b_0+\\\\gamma =0,\\\\quad \\\\alpha a_1+\\\\beta b_1+\\\\gamma =0.$ For the wave vector of radiation we have: ${L}_i=\\\\Big (0,0,L_2(x^2), L_3(x^3)\\\\Big ),$ ${L}_2=\\\\sqrt{\\\\alpha a_2+\\\\beta b_2+\\\\gamma },\\\\quad {L}_3=\\\\sqrt{\\\\alpha a_3+\\\\beta b_3+\\\\gamma }.$ The energy density of the radiation has the form: ${\\\\varepsilon }=F(x^0,x^1,Y)\\\\,\\\\frac{\\\\Delta \\\\sqrt{-\\\\det g^{ij}}}{{L}_2{L}_3},$ $Y= \\\\int {L}_2\\\\,dx^2 + \\\\int {L}_3\\\\,dx^3,$ where $F(x^0,x^1,Y)$ is an arbitrary function of its variables.\",\n \"Note that for Stackel space-times (0.0) type the three components of the wave vector of the radiation can not be zero, since this leads to a violation of the norm condition.\",\n \"In the paper we obtain and enumerate all solutions of the equations of the energy-momentum conservation law of pure radiation for models of space-times that admit separation of variables in the eikonal equation.\",\n \"The forms of the energy-momentum tensor of pure radiation (energy density and wave vector of radiation) for all types of space-times under consideration are obtained.\",\n \"In privileged coordinate systems (where the separation of variables is admitted), the energy density and the wave vector of the radiation are determined through functions of the space-time metric.\",\n \"The results obtained can be used to construct integrable models for various metric theories of gravitation, including for comparing similar models in the theory of gravity of Einstein and in modified theories of gravity.\",\n \"Research partially supported by the Ministry of Education and Science of Russia under contract 3.1386.2017/4.6.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01718"}}},{"rowIdx":986,"cells":{"text":{"kind":"list like","value":[["Distributed Deep Neural Networks over the Cloud, the Edge and End\n Devices"],["Abstract We propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and end devices.","While being able to accommodate inference of a deep neural network (DNN) in the cloud, a DDNN also allows fast and localized inference using shallow portions of the neural network at the edge and end devices.","When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.","Due to its distributed nature, DDNNs enhance sensor fusion, system fault tolerance and data privacy for DNN applications.","In implementing a DDNN, we map sections of a DNN onto a distributed computing hierarchy.","By jointly training these sections, we minimize communication and resource usage for devices and maximize usefulness of extracted features which are utilized in the cloud.","The resulting system has built-in support for automatic sensor fusion and fault tolerance.","As a proof of concept, we show a DDNN can exploit geographical diversity of sensors to improve object recognition accuracy and reduce communication cost.","In our experiment, compared with the traditional method of offloading raw sensor data to be processed in the cloud, DDNN locally processes most sensor data on end devices while achieving high accuracy and is able to reduce the communication cost by a factor of over 20x."],["Introduction","Neural networks (NNs), and deep neural networks (DNNs) in particular, have achieved great success in numerous applications in recent years.","For example, deep Convolutional Neural Networks (CNNs) continuously achieve state-of-the-art performances on various tasks in computer vision as shown in Figure REF .","At the same time, the number of end devices, including Internet of Things (IoT) devices, has increased dramatically.","These devices are appealing targets for machine learning applications as they are often directly connected to sensors (e.g., cameras, microphones, gyroscopes) that capture a large quantity of input data in a streaming fashion.","However, the current state of machine learning systems on end devices leaves an unsatisfactory choice: either (1) offload input sensor data to large NN models (e.g., DNNs) in the cloud, with the associated communication costs, latency issues and privacy concerns, or (2) perform classification directly on the end device using simple Machine Learning (ML) models e.g., linear Support Vector Machine (SVM), leading to reduced system accuracy.","To address these shortcomings, it is natural to consider the use of a distributed computing approach.","Hierarchically distributed computing structures consisting of the cloud, the edge and devices (see, e.g., [1], [2]) have inherent advantages, such as supporting coordinated central and local decisions, and providing system scalability, for large-scale intelligent tasks based on geographically distributed IoT devices.","An example of one such distributed approach is to combine a small NNThe term network layer may refer to either a layer in a NN or a layer in the distributed computing hierarchy (e.g., edge or cloud).","In order to remove ambiguity, when we refer to network layers for NN we explicitly use the term NN layers.","model (less number of parameters) on end devices and a larger NN model (more number of parameters) in the cloud.","The small model at an end device can quickly perform initial feature extraction, and also classification if the model is confident.","Otherwise, the end device can fall back to the large NN model in the cloud, which performs further processing and final classification.","This approach has the benefit of low communication costs compared to always offloading NN input to the cloud and can achieve higher accuracy compared to a simple model on device.","Additionally, since a summary based on extracted features from the end device model are sent instead of raw sensor data, the system could provide better privacy protection.","However, this kind of distributed approach over a computing hierarchy is challenging for a number of reasons, including: End devices such as embedded sensor nodes often have limited memory and battery budgets.","This makes it an issue to fit models on the devices that meet the required accuracy and energy constraints.","A straightforward partitioning of NN models over a computing hierarchy may incur prohibitively large communication costs in transferring intermediate results between computation nodes.","Incorporating geographically distributed end devices is generally beyond the scope of DNN literature.","When multiple sensor inputs on different end devices are used, they need to be aggregated together for a single classification objective.","A trained NN will need to support such sensor fusion.","Multiple models at the cloud, the edge and the device need to be learned jointly to allow coordinated decision making.","Computation already performed on end device models should be useful for further processing on edge or cloud models.","Usual layer-by-layer processing of a DNN from the NN's input layer all the way to the NN's output layer does not directly provide a mechanism for local and fast inference at earlier points in the neural networks (e.g., end devices).","A balance is needed between the accuracy of a model (with the associated model size) at a given distributed computing layer and the cost of communicating to the layer above it.","The solution must have reasonably good lower NN layers on the end devices capable of accurate local classification for some input while also providing useful features for classification in the cloud for other input.","To address these concerns under the same optimization framework, it is desirable that a system could train a single end-to-end model, such as a DNN, and partition it between end devices and the cloudFor presentation simplicity, we often just consider the device-cloud scenario.","Our methodology can similarly apply to general device-edge (fog)-cloud scenarios., in order to provide a simpler and more principled approach.","To this end, we propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and geographically distributed end devices.","In implementing a DDNN, we map sections of a single DNN onto a distributed computing hierarchy.","By jointly training these sections, we show that DDNNs can effectively address the aforementioned challenges.","Specifically, while being able to accommodate inference of a DNN in the cloud, a DDNN allows fast and localized inference using some shallow portions of the DNN at the edge and end devices.","Moreover, via distributed computing, DDNNs naturally enhance sensor fusion, data privacy and system fault tolerance for DNN applications.","When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.","DDNN leverages our earlier work on BranchyNet [3] which allows early exit points to be placed in a DNN.","Samples can be classified and exited locally when the system is confident and offloaded to the edge and the cloud when additional processing is required.","In addition, DDNN leverages the recent work of binary neural networks (BNNs) [4], which greatly reduce the required memory cost of neural network layers and enables multi-layer NNs to run on end devices with small memory footprints [5].","By training DDNN end-to-end, the network optimally configures lower NN layers to support local inference at end devices, and higher NN layers in the cloud to improve overall classification accuracy of the system.","As a proof of concept, we show a DDNN can exploit geographical diversity of sensors (on a multi-view multi-camera dataset) in sensor fusion to improve recognition accuracy.","The contributions of this paper include A novel DDNN framework and its implementation that maps sections of a DNN onto a distributed computing hierarchy.","A joint training method that minimizes communication and resource usage for devices and maximizes usefulness of extracted features which are utilized in the cloud, while allowing low-latency classification via early exit for a high percentage of input samples.","Aggregation schemes that allows automatic sensor fusion of multiple sensor inputs to improve the overall performance (accuracy and fault tolerance) of the system.","The DDNN codebase is open source and can be found here: https://github.com/kunglab/ddnn.","Figure: Progression towards deeper neural network structures in recent years (see, e.g., , , , , )."],["Related Work","In this section, we briefly review related work in distributed computing hierarchies and recent deep learning algorithms that enable our proposed method to run in a distributed fashion.","We then discuss other approaches involving distributed deep networks."],["Distributed Computing Hierarchy","The framework of a large-scale distributed computing hierarchy has assumed new significance in the emerging era of IoT.","It is widely expected that most of data generated by the massive number of IoT devices must be processed locally at the devices or at the edge, for otherwise the total amount of sensor data for a centralized cloud would overwhelm the communication network bandwidth.","In addition, a distributed computing hierarchy offers opportunities for system scalability, data security and privacy, as well as shorter response times (see, e.g., [2], [11]).","For example, in [11], a face recognition application shows a reduced response time is achieved when a smartphone's photos are proceeded by the edge (fog) as opposed to the cloud.","In this paper, we show that DDNN can systematically exploit the inherent advantages of a distributed computing hierarchy for DNN applications and achieve similar benefits."],["Deep Neural Network Extensions","Binarized neural networks (BNNs) are a recent type of neural networks, where the weights in linear and convolutional layers are constrained to $\\lbrace -1, 1\\rbrace $ (stored as 0 and 1 respectively).","This representation has been shown to achieve similar classification accuracy for some datasets such as MNIST and CIFAR-10 [12] when compared to a standard floating-point neural network while using less memory and reduced computation due to the binary format [4].","Embedded binarized neural networks (eBNNs) extends BNNs to allow the network to fit on embedded devices by reducing floating-point temporaries through reordering the operations in inference [5].","These compact models are especially attractive in end device settings, where memory can be a limiting factor and low power consumption is required.","In DDNN, we use BNNs, eBNNs and the alike to accommodate the end devices, so that they can be jointly trained with the NN layers in the edge and cloud.","BranchyNet proposed a solution of classifying samples at earlier points in a neural network, called early exit points, through the use of an entropy-based confidence criteria [3].","If at an early exit point a sample is deemed confident based on the entropy of the computed probability vector for target classes, then it is classified and no further computation is performed by the higher NN layers.","In DDNN, exit points are placed at physical boundaries (e.g., between the last NN layer on an end device and the first NN layer in the next higher layer of the distributed computing hierarchy such as the edge or the cloud).","Input samples that can already be classified early will exit locally, thereby achieving a lowered response latency and saving communication to the next physical boundary.","With similar objectives, SACT [13] allocates computation on a per region basis in an image, and exits each region independently when it is deemed to be of sufficient quality."],["Distributed Training of Deep Networks","Current research on distributing deep networks is mainly focused on improving the runtime of training the neural network.","In 2012, Dean et al.","proposed DistBelief, which maps large DNNs over thousands of CPU cores during training [14].","More recently, several methods have been proposed to scale up DNN training across GPU clusters [15], [16], which further reduces the runtime of network training.","Note that this form of distributing DNNs (over homogeneous computing units) is fundamentally different from the notion presented in this paper.","We proposes a way to train and perform feedforward inference over deep networks that can be deployed over a distributed computing hierarchy, rather than processed in parallel over bus- or switch-connected CPUs or GPUs in the cloud.","In this section we give an overview of the proposed distributed deep neural network (DDNN) architecture and describe how training and inference in DDNN is performed."],["DDNN Architecture","DDNN maps a trained DNN onto heterogeneous physical devices distributed locally, at the edge, and in the cloud.","Since DDNN relies on a jointly trained DNN framework at all parts in the neural network, for both training and inference, many of the difficult engineering decisions are greatly simplified.","Figure REF provides an overview of the DDNN architecture.","The configurations presented show how DDNN can scale the inference computation across different physical devices.","The cloud-based DDNN in (a) can be viewed as the standard DNN running in the cloud as described in the introduction.","In this case, sensor input captured on end devices is sent to the cloud in original format (raw input format), where all layers of DNN inference is performed.","We can extend this model to include a single end device, as shown in (b), by performing a portion of the DNN inference computation on the device rather than sending the raw input to the cloud.","Using an exit point after device inference, we may classify those samples which the local network is confident about, without sending any information to the cloud.","For more difficult cases, the intermediate DNN output (up to the local exit) is sent to the cloud, where further inference is performed using additional NN layers and a final classification decision is made.","Note that the intermediate output can be designed to be much smaller than the sensor input (e.g., a raw image from a video camera), and therefore drastically reduce the network communication required between the end device and the cloud.","The details of how communication is considered in the network is discussed in section REF .","DDNN can also be extended to multiple end devices which may be geographically distributed, shown in (c), that work together to make a classification decision.","Here, each end device performs local computation as in (b), but their output is aggregated together before the local exit point.","Since the entire DDNN is jointly trained across all end devices and exit points, the network automatically aggregates the input with the objective of achieving maximum classification accuracy.","This automatic data fusion (sensor fusion) simplifies runtime inference by avoiding the necessity of manually combining output from multiple end devices.","We will discuss the design of feature aggregation in detail in section REF .","As before, if the local exit point is not confident about the sample, each end devices sends intermediate output to the cloud, where another round of feature aggregation is performed before making a final classification decision.","DDNN scales vertically as well, by using an edge layer in the distributed computing hierarchy between the end devices and cloud, shown in (d) and (e).","The edge acts similarly to the cloud, by taking output from the end devices, performing aggregation and classification if possible, and forwarding its own intermediate output to the cloud if more processing is needed.","In this way, DDNN naturally adjusts the network communication and response time of the system on a per sample basis.","Samples that can be correctly classified locally are exiting without any communication to the edge or cloud.","Samples that require more feature extraction than can be provided locally are sent to the edge, and eventually the cloud if necessary.","Finally, DDNNs can also scale geographically across the edge layer as well, which is shown in (f).","Figure: Overview of the DDNN architecture.","The vertical lines represent the DNN pipeline, which connects the horizontal bars (NN layers).","(a) is the standard DNN (processed entirely in the cloud), (b) introduces end devices and a local exit point that may classify samples before the cloud, (c) extends (b) by adding multiple end devices which are aggregated together for classification, (d) and (e) extend (b) and (c) by adding edge layers between the cloud and end devices, and (f) shows how the edge can also be distributed like the end devices."],["DDNN Aggregation Methods","In DDNN configurations with multiple end devices (e.g., (c), (e), and (f) in Figure REF ), the output from each end device must be aggregated in order to perform classification.","We present several different schemes for aggregating the output.","Each aggregation method makes different assumptions about how the device output should be combined and therefore can result in different system accuracy.","We present three approaches: Max pooling (MP).","MP aggregates the input vectors by taking the max of each component.","Mathematically, max pooling can be written as $ {\\hat{v}}_{j} = \\max _{1 \\le i \\le n} v_{ij}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.","Average pooling (AP).","AP aggregates the input vectors by taking the average of each component.","This is written as $ {\\hat{v}}_{j} = \\sum _{i=1}^n \\frac{v_{ij}}{n}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.","Averaging may reduce noisy input presented in some end devices.","Concatenation (CC).","CC simply concatenates the input vectors together.","CC retains all information which is useful for higher layers (e.g., the cloud) that can use the full information to extract higher level features.","Note that this expands the dimension of the resulting vector.","To map this vector back to the same number of dimensions as input vectors, we add an additional linear layer.","We analyzes these aggregation methods in Section REF ."],["DDNN Training","While DDNN inference is distributed over the distributed computing hierarchy, the DDNN system can be trained on a single powerful server or in the cloud.","One aspect of DDNN that is different from most conventional DNN pipelines is the use of multiple exit points as shown in Figure REF .","At training time, the loss from each exit is combined during back-propagation so that the entire network can be jointly trained, and each exit point achieves good accuracy relative to its depth.","For this work, we follow joint training as described in GoogleNet [9] and BranchyNet [3].","For the system evaluation discussed in Section , we apply DDNNs to a classification task.","We use the softmax cross entropy loss function as the optimization objective.","We now describe formally how we train DDNNs.","Let $$ be a one-hot ground-truth label vector, $$ be an input sample and $\\mathcal {C}$ be the set of all possible labels.","For each exit, the softmax cross entropy objective function can be written as L(, ;) = -1|C| cC yc yc, where = softmax() = ()cC (zc), and = fexitn(;), where $f_{\\text{exit}_n}$ is a function representing the computation of the neural network layers from an entry point to the $n$ -th exit branch and $\\theta $ represents the network parameters such as weights and biases of those layers.","To train the DDNN we form a joint optimization problem as minimizing a weighted sum of the loss functions of each exit: L(, ;) = n=1N wn L(exitn, ;), where $N$ is the total number of exit points and $w_n$ is the associated weight of each exit.","Equal weights are used for the experimental results of this paper."],["DDNN Inference","Inference in DDNN is performed in several stages using multiple preconfigured exit thresholds $$ (one element $T$ at each exit point) as a measure of confidence in the prediction of the sample.","One way to define $$ is by searching over the ranges of $$ on a validation set and pick the one with the best accuracy.","We use a normalized entropy threshold as the confidence criteria (instead of unnormalized entropy as used in [3]) that determines whether to classify (exit) a sample at a particular exit point.","The normalized entropy is defined as $\\eta (\\mathbf {x}) =-\\sum _{i=1}^{|\\mathcal {C}|}{\\frac{x_{i}\\log x_{i}}{\\log |\\mathcal {C}|}},$ where $\\mathcal {C}$ is the set of all possible labels and $\\textbf {x}$ is a probability vector.","This normalized entropy $\\eta $ has values between 0 and 1 which allows easier interpretation and searching of its corresponding threshold $T$ .","For example, $\\eta $ close to 0 means that the DDNN is confident about the prediction of the sample; $\\eta $ close to 1 means it is not confident.","At each exit point, $\\eta $ is computed and compared against $T$ in order to determine if the sample should exit at that point.","At a given exit point, if the predictor is not confident in the result (i.e., $\\eta > T$ ), the system falls back to a higher exit point in the hierarchy until the last exit is reached which always performs classification.","We now provide an example of the inference procedure for a DDNN which has multiple end devices and three exit points (configuration (e) in Figure REF ): Each end device first sends summary information to local aggregator.","The local aggregator determines if the combined summary information is sufficient for accurate classification.","If so, the sample is classified (exited).","If not, each device sends more detailed information to the edge in order to perform further processing for classification.","If the edge believes it can correctly classify the sample it does so and no information is sent to the cloud.","Otherwise, the edge forwards intermediate computation to the cloud which makes the final classification."],["Communication Cost of DDNN Inference","The total communication cost for an end device with the local and cloud aggregator is calculated as $c = 4\\times |\\mathcal {C}| + (1-l)\\frac{f\\times o}{8}$ where $l$ is the percentage of samples exited locally, $\\mathcal {C}$ is the set of all possible labels (3 in our experiments), $f$ is the number of filters, and $o$ is the output size of a single filter for the final NN layer on the end-device.","The constant 4 corresponds to 4 bytes which are used to represent a floating-point number and the constant 8 corresponds to bits used to express a byte output.","The first term assumes a single floating-point per class, which conveys the probability that the sample to be transmitted from the end device to the local aggregator belongs to this class.","This step happens regardless of whether the sample is exited locally or at a later exit point.","The second term is the communication between end device and cloud which happens $(1-l)$ fraction of the time, when the sample is exited in the cloud rather than locally."],["Accuracy Measures","Throughout the evaluation in Section , we use different accuracy measures for the various exit points in a DDNN as follows: Local Accuracy is the accuracy when exiting 100% of samples at the local exit of a DDNN.","Edge Accuracy is the accuracy when exiting 100% of samples at the edge exit of a DDNN.","Cloud Accuracy is the accuracy when exiting 100% of samples at the cloud exit of a DDNN.","Overall Accuracy is the accuracy when exiting some percentage of samples at each exit point in the hierarchy.","The samples classified at each exit point are determined by the entropy threshold $T$ for that exit.","The impact of $T$ on classification accuracy and communication cost is discussed in Section REF .","Individual Accuracy is the accuracy of an end device NN model trained separately from DDNN.","The NN model for each end device consists of a ConvP block followed by a FC block (a single end device portion as shown in Figure REF ).","In the evaluation, individual accuracy for each device is computed by classifying all samples using the individual NN model and not relying on the local or cloud exit points of a DDNN."],["DDNN System Evaluation","In this section, we evaluate DDNN on a scenario with multiple end devices and demonstrate the following characteristics of the approach: DDNNs allow multiple end devices to work collaboratively in order to improve accuracy at both the local and cloud exit points.","DDNNs seamlessly extend the capability of end devices by offloading difficult samples to the cloud.","DDNNs have built-in fault tolerance.","We illustrate that missing any single end device does not dramatically affect the accuracy of the system.","Additionally, we show how performance gradually degrades as more end devices are lost.","DDNNs reduce communication costs for end devices compared to traditional system that offloads all input sensor data to the cloud.","We first introduce the DDNN architecture and dataset used in our evaluation."],["DDNN Evaluation Architecture","To accommodate the small memory size of the end devices, we use Binary Neural Network [4] blocksA block consists of one or more conventional NN layers.","We make use of two types of blocks in [5]: the fused binary fully connected (FC) block and fused binary convolution-pool (ConvP) block as shown in Figure REF .","FC blocks each consist of a fully connected layer with $m$ nodes for some $m$ , batch normalization and binary activation.","ConvP blocks each consist of a convolutional layer with $f$ filters for some $f$ , a pooling layer and batch normalization and binary activation.","A convolution layer has a kernel of size 3x3 with stride 1 and padding 1.","A pooling layer has a kernel of size 3x3 with stride 2 and padding 1.","For our experiments, we use version (c) from Figure REF , with six end devices.","The system presented can be generalized to a more elaborated structure which includes an edge layer, as shown in (d), (e) or (f) of Figure REF .","Figure REF depicts a detailed view of the DDNN system used in our experiments.","In this system, we have six end devices shown in red, a local aggregator, and a cloud aggregator.","During training, output from each device is aggregated together at each exit point using one of the aggregation schemes described in Section REF .","We provide detailed analysis on the impact of aggregation schemes at both the local and cloud exit points in Section REF .","All DDNNs in our experiments are trained with Adam [17] using the following hyper-parameter settings: $\\alpha $ of 0.001, $\\beta _1$ of 0.9, $\\beta _2$ of 0.999, and $\\epsilon $ of 1e-8.","We train each DDNN for 100 epochs.","When training the DDNN, we use equal weights for the local and cloud exit points.","We explored heavily weighting both the local exit and the cloud exit, but neither weighting scheme significantly changed the accuracy of the system.","This indicates that this solution to the dataset and the problem we are exploring is not sensitive to the weights, but this may not be true for other datasets and problemsIn GoogleNet [9], a less than 1% difference in accuracy was observed based on the values of the weight parameters.","Figure: Fused binary blocks consisting of one or more standard NN layers.","The fused binary fully connected (FC) block is a fully connected layer with nn nodes, batch normalization and binary activation.","The fused binary convolution-pool (ConvP) block consists of a convolutional layer with ff filters, a pooling layer, batch normalization and binary activation.","The convolution layer has a kernel of size 3x3 with stride 1 and padding 1.","The pooling layer has a kernel of size 3x3 with stride 2 and padding 1.","These blocks are used as they are presented in .Figure: The DDNN architecture used in the system evaluation.","The FC and ConvP blocks in red and blue correspond to layers run on end devices and the cloud respectively.","The dashed orange boxes represent the end devices and show which blocks of the DDNN are mapped onto each device.","The local aggregator shown in red combines the exit output (a short vector with length equal to the number of classes) from each end device in order to determine if local classification for the given input sample can be performed accurately.","If the local exit is not confident (i.e.","η(x)>T\\eta (x) > T), the activation output after the last convolutional layer from each end device is sent to the cloud aggregator (shown in blue), which aggregates the input from each device, performs further NN layer processing, and outputs a final classification result.","The aggregation of input for multiple end devices is discussed in Section ."],["Multi-view Multi-camera Dataset","We evaluate the proposed DDNN framework on a multi-view multi-camera dataset [18].","This dataset consists of images acquired at the same time from six cameras placed at different locations facing the same general area.","For the purpose of our evaluation, we assume that each camera is attached to an end device, which can transmit the captured images over a bandwidth-constraint wireless network to a physical endpoint connected to the cloud.","The dataset provides object bounding box annotations.","Multiple bounding boxes may exist in a single image, each of which corresponds to a different object in the frame.","In preparing the dataset, for each bounding box, we extract an image, and manually synchronizeIn practical object tracking systems, this synchronization step is typically automated [19].","the same object across the multiple devices that the object appears in for the given frame.","Examples of the extracted images are shown in Figure REF .","Each row corresponds to a single sample used for classification.","We resize each extracted sample to a 32x32 RGB pixel image.","For each device that a given object does not appear in, we use a blank image and assign a label of -1, meaning that the object is not present in the frame.","Labels 0, 1, and 2 correspond to car, bus and person, respectively.","Objects that are not present in a frame (i.e., label of -1) are not used during training.","We split the dataset into 680 training samples and 171 testing samples.","Figure REF shows the distribution of samples at each device.","Due to the imbalanced number of class samples in the dataset, the individual accuracy of each end device differs widely, as shown by the “Individual\" curve of Figure REF .","A full description of the training process for the individual NN models is provided in Section REF .","The processed dataset used in this paper is available at [20].","Figure: Example images of three objects (person, bus, car) from the multi-view multi-camera dataset.","The six devices (each with their own camera) capture the same object from different orientations.","An all grey image denotes that the object is not present in the frame.Figure: The distribution of class samples for each end device in the multi-view multi-camera dataset."],["Impact of Aggregation Schemes","In order to perform classification on the input from multiple end devices, we must aggregate the information from each end device.","We consider three aggregation methods (max pooling, average pooling, and concatenation) outlined in Section REF , at both the local and cloud exit points.","The accuracy of different aggregation schemes are shown in Table REF .","The first two letters identify the local aggregation scheme and the last two letters identify the scheme used by the cloud aggregator.","For example, MP-CC means the local aggregator uses max-pooling and the cloud uses concatenation.","Recall that each input to the local aggregator is a floating-point vector of length equal to the number of classes (corresponding to the output from the final FC block for a single device as shown in Figure REF ) and the device output sent to the cloud aggregator is the output from the final ConvP block.","Table: Accuracy of aggregation schemes.","The first two letters identify the local aggregation scheme, and the last two letters identify the cloud aggregation scheme.","For example, MP-CC means the local aggregator uses max-pooling and the cloud aggregator uses concatenation.","The accuracy of each exit point (either local or cloud) is computed using the entire test set.","In practice, we will exit a portion of samples locally based on the entropy threshold TT and send the remaining samples in the cloud.","Due to its high performance, MP-CC is used in the remaining experiments of this paper.The MP-MP scheme has good classification accuracy for the local aggregator but poor performance in the cloud.","The elements in the vectors at the local aggregator correspond to the same features (e.g., the first item is the likelihood that the input corresponds to that class).","Therefore, max pooling corresponds to taking the max response for each class over all end devices, and shows good performance.","On the other hand, since the information sent from the end devices to the cloud is the activation output from the filters at each device, which corresponds to different visual features in the input from the viewpoint of each individual end device, max pooling these features does not perform well.","Comparing MP-MP and MP-CC schemes, though both use MP for local aggregators, MP-CC increases the accuracy of the local classifier.","In the training phrase, during backpropagation the MP-MP scheme only passes gradients through a device that gives the highest response while MP-CC scheme passes gradients through all devices.","Therefore, using CC aggregator in the cloud allows all devices to learn better filters (filter weights) that give a stronger response for the local MP aggregator, resulting in a better classification accuracy.","The CC-CC scheme shows an opposite trend where the local accuracy is poor while the cloud accuracy is high.","Concatenating the local information (instead of a pooling scheme), does not enforce any relationship between output for the same class on multiple devices and therefore performs worse.","Concatenating the output for the cloud aggregator maintains the most information for NN layer processing in the cloud and therefore performs well.","Generally, for the local aggregator, average pooling performs worse than max pooling.","This is because some of the end devices do not have the object present in the given frame.","Average pooling take average of all outputs from end devices; this compromises the strong outputs from end devices in which the object is present.","Based on these results, we use the MP-CC aggregation scheme throughout the paper."],["Entropy Threshold","The entropy threshold for an exit point, $T$ , corresponds to the level of confidence that is required in order to exit a sample.","A threshold value of 0 would mean that no samples will exit and a value of 1 would mean that all samples exit at that point.","Figure REF shows the relationship between $T$ at the local aggregator and the overall accuracy of the DDNN.","We observe that as more samples are exited at the local exit, the overall accuracy decreases.","This is expected, as the accuracy of the local exit is typically lower than that of the cloud exit.","We need to set the threshold appropriately to achieve a balance between the communication cost, as defined in Section REF , latency and accuracy of the system.","In this case, we see that setting the threshold to $0.8$ results in the best overall accuracy with significantly reduced communication, i.e., 97% accuracy while exiting 60.82% of samples locally as shown in Table REF where in addition to local exit (%) and overall accuracy (%), communication cost in bytes is given.","We set $T=0.8$ for the remaining experiments in the system evaluation, unless noted otherwise.","The local classifier may do better than cloud for certain samples where low-level features are more robust in classification than higher-level features.","By setting an appropriate threshold $T$ , we can improve overall accuracy.","In this experiment, $T=0.8$ corresponds to that sweet spot where some samples which are incorrectly classified by the cloud classifier can actually be correctly classified by the local classifier.","Such a threshold indicates the optimal point where both local and cloud classifier work best together.","Table: Effects of different exit threshold (TT) settings for the local exit.","T=0.8T=0.8 is used in the remaining experiments.Figure: Overall accuracy of the system as the entropy threshold for the local exit is varied from 0 to 1.","For this experiment, 4 filters are used in the ConvP blocks on the end devices."],["Impact of Scaling Across End Devices","In order to scale DDNNs across multiple end devices, we distribute the lower sections of Figure REF , shown in red, over the corresponding devices, outlined in orange.","Figure REF shows how the accuracy of the system improves as additional end devices (each with its attached input cameras) are added.","The devices are added in order sorted by their individual accuracy from worst to best (i.e., the device with the lowest accuracy first and the device with the highest accuracy last).","The first observation is the large variation in the individual accuracy of the end devices, as noted earlier.","Due to the nature of the dataset, some devices are naturally better positioned and generally have clearer observations of the objects.","Looking at the viewpoints of each camera in Figure REF , we see that the selected examples for Device 6 have clear frontal views of each object.","This viewpoint gives Device 6 the highest individual accuracy at over 70%.","By comparison, Device 2 has the lowest individual accuracy at under 40%.","The “Local” and “Cloud” curves show the accuracy of the system at each exit point when all samples are exited at that point.","We observe that the cloud exit point outperforms the local exit point at all numbers of end devices.","The gap is widest when there are fewer devices.","This suggests that the additional NN layers in the cloud significantly improve the final classification result when the problem is more difficult due to limited labeled training data for an end device.","Once all six end devices are added, both the local and cloud aggregators have high accuracy.","The “Overall” curve represents the overall accuracy of the system when the threshold for the local exit point is set to $0.8$ .","We see that this curve is roughly equivalent to exiting all samples at the cloud (but at a much reduced communication cost as 60.82% of samples are exited locally).","Generally, these results show that by combining multiple viewpoints we can increase the classification accuracy at both the local and cloud level by a substantial margin when compared to the individual accuracy of any device.","The resulting accuracy of the DDNN system is superior to any individual device accuracy by over 20%.","Moreover, we note that the 60.82% of samples which exit locally enjoy lowered latency in response time.","Figure: Accuracy of the DDNN system as additional end devices are added.","The accuracy of “Overall” is obtained by exiting a percentage of the samples locally and the rest in the cloud.","The accuracy of “Cloud” and “Local” are computed by exiting all samples at each point, respectively.","The end devices are ordered by their “Individual” classification accuracy, sorted from worst to best."],["Impact of Cloud Offloading on Accuracy Improvements","DDNNs improve the overall accuracy of the system by offloading difficult samples to the cloud, which perform further NN layer processing and final classification.","Figure REF shows the accuracy and communication costs of DDNN as the number of filters on the end devices increases.","For all settings, the NN layers stored on an end device require under 2 KB of memory.","In this experiment, we configure the local exit threshold $T$ such that around $75\\%$ of samples are exited locally and around $25\\%$ of samples are offloaded to the cloud.","We see that DDNNs achieve around a $5\\%$ improvement in accuracy compared to using just the local aggregator.","This demonstrates the advantage for offloading to the cloud even when larger models (more filters) with improved local accuracy are used on the end devices.","Figure: Accuracy and communication cost (in bytes) for increasingly larger end device memory sizes that accommodate additional filters.","We notice that cloud offloading leads to improved accuracy."],["Fault Tolerance of DDNNs","A key motivation for distributed systems is fault tolerance.","Fault tolerance implies that the system still works well even when some parts are broken.","In order to test the fault tolerance of DDNN, we simulate end device failures and look at the resulting accuracy of the system.","Figure REF shows the accuracy of the system under the presence of individual device failures.","Regardless of the device that is missing, the system still achieves over a $95\\%$ overall classification accuracy.","Specifically, even when the device with the highest individual accuracy has failed, which is Device 6, the overall accuracy is reduced by only $3\\%$ .","This suggests that for this dataset, the automatic fault tolerance provided by DDNN makes the system reliable even in the presence of device failure.","We can also view figure REF from the perspective of providing fault tolerance for the system.","As we decrease the number of end devices from 6 to 4, we observe that the overall accuracy of the system drops only $4\\%$ .","This suggests that the system can also be robust to mutliple failing end devices.","Figure: The impact on DDNN system accuracy when any single end device has failed."],["Reducing Communication Costs","DDNNs significantly reduces the communication cost of inference compared to the standard method of offloading raw sensor input to the cloud.","Sending a 32x32 RGB pixel image (the input size of our dataset) to the cloud costs 3072 bytes per image sample.","By comparison, as shown in Table REF , the largest DDNN model used in our evaluation section requires only 140 bytes of communication per sample on average (an over 20x reduction in communication costs).","This communication reduction for an end device results from transmitting class-label related intermediate results to the local aggregator for all samples and binarized communication with the cloud when additional NN layer processing is required for classification with improved accuracy."],["DDNN Provision for Horizontal and Vertical Scaling","The evaluation in the previous section shows that DDNN is able to achieve high overall accuracy through provisioning the network to scale both horizontally, across end devices, and vertically, over the network hierarchy.","Specifically, we show that DDNN scales vertically, by exiting easier input samples locally for low-latency response and offloading difficult samples to the cloud for high overall recognition accuracy, while maintaining a small memory footprint on the end devices and incurring a low communication cost.","This result is not obvious, as we need sufficiently good feature representations from the lower parts of the DNN (running on the end devices with limited resources) in order for the upper parts of the neural network (running in the cloud) to achieve high accuracy under the low communication cost constraint.","Therefore, we show in a positive way that the proposed method of jointly training a single DNN with multiple exit points at each part of the distributed hierarchy allows us to meet this goal.","That is, DDNN optimizes the lower parts of the DNN to create a sufficiently good feature representations to support both samples exited locally and those processed further in the cloud.","To meet the goal of horizontal scaling, we provide a principled way of jointly training a DNN with inputs from multiple devices through feature pooling via local and cloud aggregators and demonstrate that by aggregating features from each device we can dramatically improve the accuracy of the system both at the local and cloud level.","Filters on each device are automatically tuned to process the geographically unique inputs and work together toward to the same overall objective leading to high overall accuracy.","Additionally, we show that DDNN provides built-in fault tolerance across the end devices and is still able to achieve high accuracy in the presence of failed devices."],["Conclusion","In this paper, we propose a novel distributed deep neural network architecture (DDNN) that is distributed across computing hierarchies, consisting of the cloud, the edge and end devices.","We demonstrate for a multi-view, multi-camera dataset that DDNN scales vertically from a few NN layers on end devices or the edge to many NN layers in the cloud and scales horizontally across multiple end devices.","The aggregation of information communicated from different devices is built into the joint training of DDNN and is handled automatically at inference time.","This approach simplifies the implementation and deployment of distributed cloud offloading and automates sensor fusion and system fault tolerance.","The experimental results suggest that with our DDNN framework, a single DNN properly trained can be mapped onto a distributed computing hierarchy to meet the accuracy, communication and latency requirements of a target application while gaining inherent benefits associated with distributed computing such as fault tolerance and privacy.","DDNNs reduce the required communication compared to a standard cloud offloading approach by exiting many samples at the local aggregator and sending a compact binary feature representation to the cloud when additional processing is required.","For our evaluation dataset, the communication cost of DDNN is reduced by a factor of over 20x compared to offloading raw sensor input to a DNN in the cloud which performs all of the inference computation.","DDNN provides a framework for further research in mapping DNN into a distributed computing hierarchy.","For future work, we will investigate the performance of DDNNs on applications with a larger dataset with multiple types of input modalities [21] and more end devices.","Currently, all layers in DDNN are binary.","While binary layers are a requirement for end devices due to the limited space on devices, it is not necessary in the cloud.","We will explore other types of aggregation schemes and mixed precisions schemes where the end devices use binary NN layers and the cloud uses mixed-precision or floating-point NN layers."],["Acknowledgment","This work is supported in part by gifts from the Intel Corporation and in part by the Naval Supply Systems Command award under the Naval Postgraduate School Agreements No.","N00244-15-0050 and No.","N00244-16-1-0018."]],"string":"[\n [\n \"Distributed Deep Neural Networks over the Cloud, the Edge and End\\n Devices\"\n ],\n [\n \"Abstract We propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and end devices.\",\n \"While being able to accommodate inference of a deep neural network (DNN) in the cloud, a DDNN also allows fast and localized inference using shallow portions of the neural network at the edge and end devices.\",\n \"When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.\",\n \"Due to its distributed nature, DDNNs enhance sensor fusion, system fault tolerance and data privacy for DNN applications.\",\n \"In implementing a DDNN, we map sections of a DNN onto a distributed computing hierarchy.\",\n \"By jointly training these sections, we minimize communication and resource usage for devices and maximize usefulness of extracted features which are utilized in the cloud.\",\n \"The resulting system has built-in support for automatic sensor fusion and fault tolerance.\",\n \"As a proof of concept, we show a DDNN can exploit geographical diversity of sensors to improve object recognition accuracy and reduce communication cost.\",\n \"In our experiment, compared with the traditional method of offloading raw sensor data to be processed in the cloud, DDNN locally processes most sensor data on end devices while achieving high accuracy and is able to reduce the communication cost by a factor of over 20x.\"\n ],\n [\n \"Introduction\",\n \"Neural networks (NNs), and deep neural networks (DNNs) in particular, have achieved great success in numerous applications in recent years.\",\n \"For example, deep Convolutional Neural Networks (CNNs) continuously achieve state-of-the-art performances on various tasks in computer vision as shown in Figure REF .\",\n \"At the same time, the number of end devices, including Internet of Things (IoT) devices, has increased dramatically.\",\n \"These devices are appealing targets for machine learning applications as they are often directly connected to sensors (e.g., cameras, microphones, gyroscopes) that capture a large quantity of input data in a streaming fashion.\",\n \"However, the current state of machine learning systems on end devices leaves an unsatisfactory choice: either (1) offload input sensor data to large NN models (e.g., DNNs) in the cloud, with the associated communication costs, latency issues and privacy concerns, or (2) perform classification directly on the end device using simple Machine Learning (ML) models e.g., linear Support Vector Machine (SVM), leading to reduced system accuracy.\",\n \"To address these shortcomings, it is natural to consider the use of a distributed computing approach.\",\n \"Hierarchically distributed computing structures consisting of the cloud, the edge and devices (see, e.g., [1], [2]) have inherent advantages, such as supporting coordinated central and local decisions, and providing system scalability, for large-scale intelligent tasks based on geographically distributed IoT devices.\",\n \"An example of one such distributed approach is to combine a small NNThe term network layer may refer to either a layer in a NN or a layer in the distributed computing hierarchy (e.g., edge or cloud).\",\n \"In order to remove ambiguity, when we refer to network layers for NN we explicitly use the term NN layers.\",\n \"model (less number of parameters) on end devices and a larger NN model (more number of parameters) in the cloud.\",\n \"The small model at an end device can quickly perform initial feature extraction, and also classification if the model is confident.\",\n \"Otherwise, the end device can fall back to the large NN model in the cloud, which performs further processing and final classification.\",\n \"This approach has the benefit of low communication costs compared to always offloading NN input to the cloud and can achieve higher accuracy compared to a simple model on device.\",\n \"Additionally, since a summary based on extracted features from the end device model are sent instead of raw sensor data, the system could provide better privacy protection.\",\n \"However, this kind of distributed approach over a computing hierarchy is challenging for a number of reasons, including: End devices such as embedded sensor nodes often have limited memory and battery budgets.\",\n \"This makes it an issue to fit models on the devices that meet the required accuracy and energy constraints.\",\n \"A straightforward partitioning of NN models over a computing hierarchy may incur prohibitively large communication costs in transferring intermediate results between computation nodes.\",\n \"Incorporating geographically distributed end devices is generally beyond the scope of DNN literature.\",\n \"When multiple sensor inputs on different end devices are used, they need to be aggregated together for a single classification objective.\",\n \"A trained NN will need to support such sensor fusion.\",\n \"Multiple models at the cloud, the edge and the device need to be learned jointly to allow coordinated decision making.\",\n \"Computation already performed on end device models should be useful for further processing on edge or cloud models.\",\n \"Usual layer-by-layer processing of a DNN from the NN's input layer all the way to the NN's output layer does not directly provide a mechanism for local and fast inference at earlier points in the neural networks (e.g., end devices).\",\n \"A balance is needed between the accuracy of a model (with the associated model size) at a given distributed computing layer and the cost of communicating to the layer above it.\",\n \"The solution must have reasonably good lower NN layers on the end devices capable of accurate local classification for some input while also providing useful features for classification in the cloud for other input.\",\n \"To address these concerns under the same optimization framework, it is desirable that a system could train a single end-to-end model, such as a DNN, and partition it between end devices and the cloudFor presentation simplicity, we often just consider the device-cloud scenario.\",\n \"Our methodology can similarly apply to general device-edge (fog)-cloud scenarios., in order to provide a simpler and more principled approach.\",\n \"To this end, we propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and geographically distributed end devices.\",\n \"In implementing a DDNN, we map sections of a single DNN onto a distributed computing hierarchy.\",\n \"By jointly training these sections, we show that DDNNs can effectively address the aforementioned challenges.\",\n \"Specifically, while being able to accommodate inference of a DNN in the cloud, a DDNN allows fast and localized inference using some shallow portions of the DNN at the edge and end devices.\",\n \"Moreover, via distributed computing, DDNNs naturally enhance sensor fusion, data privacy and system fault tolerance for DNN applications.\",\n \"When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.\",\n \"DDNN leverages our earlier work on BranchyNet [3] which allows early exit points to be placed in a DNN.\",\n \"Samples can be classified and exited locally when the system is confident and offloaded to the edge and the cloud when additional processing is required.\",\n \"In addition, DDNN leverages the recent work of binary neural networks (BNNs) [4], which greatly reduce the required memory cost of neural network layers and enables multi-layer NNs to run on end devices with small memory footprints [5].\",\n \"By training DDNN end-to-end, the network optimally configures lower NN layers to support local inference at end devices, and higher NN layers in the cloud to improve overall classification accuracy of the system.\",\n \"As a proof of concept, we show a DDNN can exploit geographical diversity of sensors (on a multi-view multi-camera dataset) in sensor fusion to improve recognition accuracy.\",\n \"The contributions of this paper include A novel DDNN framework and its implementation that maps sections of a DNN onto a distributed computing hierarchy.\",\n \"A joint training method that minimizes communication and resource usage for devices and maximizes usefulness of extracted features which are utilized in the cloud, while allowing low-latency classification via early exit for a high percentage of input samples.\",\n \"Aggregation schemes that allows automatic sensor fusion of multiple sensor inputs to improve the overall performance (accuracy and fault tolerance) of the system.\",\n \"The DDNN codebase is open source and can be found here: https://github.com/kunglab/ddnn.\",\n \"Figure: Progression towards deeper neural network structures in recent years (see, e.g., , , , , ).\"\n ],\n [\n \"Related Work\",\n \"In this section, we briefly review related work in distributed computing hierarchies and recent deep learning algorithms that enable our proposed method to run in a distributed fashion.\",\n \"We then discuss other approaches involving distributed deep networks.\"\n ],\n [\n \"Distributed Computing Hierarchy\",\n \"The framework of a large-scale distributed computing hierarchy has assumed new significance in the emerging era of IoT.\",\n \"It is widely expected that most of data generated by the massive number of IoT devices must be processed locally at the devices or at the edge, for otherwise the total amount of sensor data for a centralized cloud would overwhelm the communication network bandwidth.\",\n \"In addition, a distributed computing hierarchy offers opportunities for system scalability, data security and privacy, as well as shorter response times (see, e.g., [2], [11]).\",\n \"For example, in [11], a face recognition application shows a reduced response time is achieved when a smartphone's photos are proceeded by the edge (fog) as opposed to the cloud.\",\n \"In this paper, we show that DDNN can systematically exploit the inherent advantages of a distributed computing hierarchy for DNN applications and achieve similar benefits.\"\n ],\n [\n \"Deep Neural Network Extensions\",\n \"Binarized neural networks (BNNs) are a recent type of neural networks, where the weights in linear and convolutional layers are constrained to $\\\\lbrace -1, 1\\\\rbrace $ (stored as 0 and 1 respectively).\",\n \"This representation has been shown to achieve similar classification accuracy for some datasets such as MNIST and CIFAR-10 [12] when compared to a standard floating-point neural network while using less memory and reduced computation due to the binary format [4].\",\n \"Embedded binarized neural networks (eBNNs) extends BNNs to allow the network to fit on embedded devices by reducing floating-point temporaries through reordering the operations in inference [5].\",\n \"These compact models are especially attractive in end device settings, where memory can be a limiting factor and low power consumption is required.\",\n \"In DDNN, we use BNNs, eBNNs and the alike to accommodate the end devices, so that they can be jointly trained with the NN layers in the edge and cloud.\",\n \"BranchyNet proposed a solution of classifying samples at earlier points in a neural network, called early exit points, through the use of an entropy-based confidence criteria [3].\",\n \"If at an early exit point a sample is deemed confident based on the entropy of the computed probability vector for target classes, then it is classified and no further computation is performed by the higher NN layers.\",\n \"In DDNN, exit points are placed at physical boundaries (e.g., between the last NN layer on an end device and the first NN layer in the next higher layer of the distributed computing hierarchy such as the edge or the cloud).\",\n \"Input samples that can already be classified early will exit locally, thereby achieving a lowered response latency and saving communication to the next physical boundary.\",\n \"With similar objectives, SACT [13] allocates computation on a per region basis in an image, and exits each region independently when it is deemed to be of sufficient quality.\"\n ],\n [\n \"Distributed Training of Deep Networks\",\n \"Current research on distributing deep networks is mainly focused on improving the runtime of training the neural network.\",\n \"In 2012, Dean et al.\",\n \"proposed DistBelief, which maps large DNNs over thousands of CPU cores during training [14].\",\n \"More recently, several methods have been proposed to scale up DNN training across GPU clusters [15], [16], which further reduces the runtime of network training.\",\n \"Note that this form of distributing DNNs (over homogeneous computing units) is fundamentally different from the notion presented in this paper.\",\n \"We proposes a way to train and perform feedforward inference over deep networks that can be deployed over a distributed computing hierarchy, rather than processed in parallel over bus- or switch-connected CPUs or GPUs in the cloud.\",\n \"In this section we give an overview of the proposed distributed deep neural network (DDNN) architecture and describe how training and inference in DDNN is performed.\"\n ],\n [\n \"DDNN Architecture\",\n \"DDNN maps a trained DNN onto heterogeneous physical devices distributed locally, at the edge, and in the cloud.\",\n \"Since DDNN relies on a jointly trained DNN framework at all parts in the neural network, for both training and inference, many of the difficult engineering decisions are greatly simplified.\",\n \"Figure REF provides an overview of the DDNN architecture.\",\n \"The configurations presented show how DDNN can scale the inference computation across different physical devices.\",\n \"The cloud-based DDNN in (a) can be viewed as the standard DNN running in the cloud as described in the introduction.\",\n \"In this case, sensor input captured on end devices is sent to the cloud in original format (raw input format), where all layers of DNN inference is performed.\",\n \"We can extend this model to include a single end device, as shown in (b), by performing a portion of the DNN inference computation on the device rather than sending the raw input to the cloud.\",\n \"Using an exit point after device inference, we may classify those samples which the local network is confident about, without sending any information to the cloud.\",\n \"For more difficult cases, the intermediate DNN output (up to the local exit) is sent to the cloud, where further inference is performed using additional NN layers and a final classification decision is made.\",\n \"Note that the intermediate output can be designed to be much smaller than the sensor input (e.g., a raw image from a video camera), and therefore drastically reduce the network communication required between the end device and the cloud.\",\n \"The details of how communication is considered in the network is discussed in section REF .\",\n \"DDNN can also be extended to multiple end devices which may be geographically distributed, shown in (c), that work together to make a classification decision.\",\n \"Here, each end device performs local computation as in (b), but their output is aggregated together before the local exit point.\",\n \"Since the entire DDNN is jointly trained across all end devices and exit points, the network automatically aggregates the input with the objective of achieving maximum classification accuracy.\",\n \"This automatic data fusion (sensor fusion) simplifies runtime inference by avoiding the necessity of manually combining output from multiple end devices.\",\n \"We will discuss the design of feature aggregation in detail in section REF .\",\n \"As before, if the local exit point is not confident about the sample, each end devices sends intermediate output to the cloud, where another round of feature aggregation is performed before making a final classification decision.\",\n \"DDNN scales vertically as well, by using an edge layer in the distributed computing hierarchy between the end devices and cloud, shown in (d) and (e).\",\n \"The edge acts similarly to the cloud, by taking output from the end devices, performing aggregation and classification if possible, and forwarding its own intermediate output to the cloud if more processing is needed.\",\n \"In this way, DDNN naturally adjusts the network communication and response time of the system on a per sample basis.\",\n \"Samples that can be correctly classified locally are exiting without any communication to the edge or cloud.\",\n \"Samples that require more feature extraction than can be provided locally are sent to the edge, and eventually the cloud if necessary.\",\n \"Finally, DDNNs can also scale geographically across the edge layer as well, which is shown in (f).\",\n \"Figure: Overview of the DDNN architecture.\",\n \"The vertical lines represent the DNN pipeline, which connects the horizontal bars (NN layers).\",\n \"(a) is the standard DNN (processed entirely in the cloud), (b) introduces end devices and a local exit point that may classify samples before the cloud, (c) extends (b) by adding multiple end devices which are aggregated together for classification, (d) and (e) extend (b) and (c) by adding edge layers between the cloud and end devices, and (f) shows how the edge can also be distributed like the end devices.\"\n ],\n [\n \"DDNN Aggregation Methods\",\n \"In DDNN configurations with multiple end devices (e.g., (c), (e), and (f) in Figure REF ), the output from each end device must be aggregated in order to perform classification.\",\n \"We present several different schemes for aggregating the output.\",\n \"Each aggregation method makes different assumptions about how the device output should be combined and therefore can result in different system accuracy.\",\n \"We present three approaches: Max pooling (MP).\",\n \"MP aggregates the input vectors by taking the max of each component.\",\n \"Mathematically, max pooling can be written as $ {\\\\hat{v}}_{j} = \\\\max _{1 \\\\le i \\\\le n} v_{ij}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.\",\n \"Average pooling (AP).\",\n \"AP aggregates the input vectors by taking the average of each component.\",\n \"This is written as $ {\\\\hat{v}}_{j} = \\\\sum _{i=1}^n \\\\frac{v_{ij}}{n}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.\",\n \"Averaging may reduce noisy input presented in some end devices.\",\n \"Concatenation (CC).\",\n \"CC simply concatenates the input vectors together.\",\n \"CC retains all information which is useful for higher layers (e.g., the cloud) that can use the full information to extract higher level features.\",\n \"Note that this expands the dimension of the resulting vector.\",\n \"To map this vector back to the same number of dimensions as input vectors, we add an additional linear layer.\",\n \"We analyzes these aggregation methods in Section REF .\"\n ],\n [\n \"DDNN Training\",\n \"While DDNN inference is distributed over the distributed computing hierarchy, the DDNN system can be trained on a single powerful server or in the cloud.\",\n \"One aspect of DDNN that is different from most conventional DNN pipelines is the use of multiple exit points as shown in Figure REF .\",\n \"At training time, the loss from each exit is combined during back-propagation so that the entire network can be jointly trained, and each exit point achieves good accuracy relative to its depth.\",\n \"For this work, we follow joint training as described in GoogleNet [9] and BranchyNet [3].\",\n \"For the system evaluation discussed in Section , we apply DDNNs to a classification task.\",\n \"We use the softmax cross entropy loss function as the optimization objective.\",\n \"We now describe formally how we train DDNNs.\",\n \"Let $$ be a one-hot ground-truth label vector, $$ be an input sample and $\\\\mathcal {C}$ be the set of all possible labels.\",\n \"For each exit, the softmax cross entropy objective function can be written as L(, ;) = -1|C| cC yc yc, where = softmax() = ()cC (zc), and = fexitn(;), where $f_{\\\\text{exit}_n}$ is a function representing the computation of the neural network layers from an entry point to the $n$ -th exit branch and $\\\\theta $ represents the network parameters such as weights and biases of those layers.\",\n \"To train the DDNN we form a joint optimization problem as minimizing a weighted sum of the loss functions of each exit: L(, ;) = n=1N wn L(exitn, ;), where $N$ is the total number of exit points and $w_n$ is the associated weight of each exit.\",\n \"Equal weights are used for the experimental results of this paper.\"\n ],\n [\n \"DDNN Inference\",\n \"Inference in DDNN is performed in several stages using multiple preconfigured exit thresholds $$ (one element $T$ at each exit point) as a measure of confidence in the prediction of the sample.\",\n \"One way to define $$ is by searching over the ranges of $$ on a validation set and pick the one with the best accuracy.\",\n \"We use a normalized entropy threshold as the confidence criteria (instead of unnormalized entropy as used in [3]) that determines whether to classify (exit) a sample at a particular exit point.\",\n \"The normalized entropy is defined as $\\\\eta (\\\\mathbf {x}) =-\\\\sum _{i=1}^{|\\\\mathcal {C}|}{\\\\frac{x_{i}\\\\log x_{i}}{\\\\log |\\\\mathcal {C}|}},$ where $\\\\mathcal {C}$ is the set of all possible labels and $\\\\textbf {x}$ is a probability vector.\",\n \"This normalized entropy $\\\\eta $ has values between 0 and 1 which allows easier interpretation and searching of its corresponding threshold $T$ .\",\n \"For example, $\\\\eta $ close to 0 means that the DDNN is confident about the prediction of the sample; $\\\\eta $ close to 1 means it is not confident.\",\n \"At each exit point, $\\\\eta $ is computed and compared against $T$ in order to determine if the sample should exit at that point.\",\n \"At a given exit point, if the predictor is not confident in the result (i.e., $\\\\eta > T$ ), the system falls back to a higher exit point in the hierarchy until the last exit is reached which always performs classification.\",\n \"We now provide an example of the inference procedure for a DDNN which has multiple end devices and three exit points (configuration (e) in Figure REF ): Each end device first sends summary information to local aggregator.\",\n \"The local aggregator determines if the combined summary information is sufficient for accurate classification.\",\n \"If so, the sample is classified (exited).\",\n \"If not, each device sends more detailed information to the edge in order to perform further processing for classification.\",\n \"If the edge believes it can correctly classify the sample it does so and no information is sent to the cloud.\",\n \"Otherwise, the edge forwards intermediate computation to the cloud which makes the final classification.\"\n ],\n [\n \"Communication Cost of DDNN Inference\",\n \"The total communication cost for an end device with the local and cloud aggregator is calculated as $c = 4\\\\times |\\\\mathcal {C}| + (1-l)\\\\frac{f\\\\times o}{8}$ where $l$ is the percentage of samples exited locally, $\\\\mathcal {C}$ is the set of all possible labels (3 in our experiments), $f$ is the number of filters, and $o$ is the output size of a single filter for the final NN layer on the end-device.\",\n \"The constant 4 corresponds to 4 bytes which are used to represent a floating-point number and the constant 8 corresponds to bits used to express a byte output.\",\n \"The first term assumes a single floating-point per class, which conveys the probability that the sample to be transmitted from the end device to the local aggregator belongs to this class.\",\n \"This step happens regardless of whether the sample is exited locally or at a later exit point.\",\n \"The second term is the communication between end device and cloud which happens $(1-l)$ fraction of the time, when the sample is exited in the cloud rather than locally.\"\n ],\n [\n \"Accuracy Measures\",\n \"Throughout the evaluation in Section , we use different accuracy measures for the various exit points in a DDNN as follows: Local Accuracy is the accuracy when exiting 100% of samples at the local exit of a DDNN.\",\n \"Edge Accuracy is the accuracy when exiting 100% of samples at the edge exit of a DDNN.\",\n \"Cloud Accuracy is the accuracy when exiting 100% of samples at the cloud exit of a DDNN.\",\n \"Overall Accuracy is the accuracy when exiting some percentage of samples at each exit point in the hierarchy.\",\n \"The samples classified at each exit point are determined by the entropy threshold $T$ for that exit.\",\n \"The impact of $T$ on classification accuracy and communication cost is discussed in Section REF .\",\n \"Individual Accuracy is the accuracy of an end device NN model trained separately from DDNN.\",\n \"The NN model for each end device consists of a ConvP block followed by a FC block (a single end device portion as shown in Figure REF ).\",\n \"In the evaluation, individual accuracy for each device is computed by classifying all samples using the individual NN model and not relying on the local or cloud exit points of a DDNN.\"\n ],\n [\n \"DDNN System Evaluation\",\n \"In this section, we evaluate DDNN on a scenario with multiple end devices and demonstrate the following characteristics of the approach: DDNNs allow multiple end devices to work collaboratively in order to improve accuracy at both the local and cloud exit points.\",\n \"DDNNs seamlessly extend the capability of end devices by offloading difficult samples to the cloud.\",\n \"DDNNs have built-in fault tolerance.\",\n \"We illustrate that missing any single end device does not dramatically affect the accuracy of the system.\",\n \"Additionally, we show how performance gradually degrades as more end devices are lost.\",\n \"DDNNs reduce communication costs for end devices compared to traditional system that offloads all input sensor data to the cloud.\",\n \"We first introduce the DDNN architecture and dataset used in our evaluation.\"\n ],\n [\n \"DDNN Evaluation Architecture\",\n \"To accommodate the small memory size of the end devices, we use Binary Neural Network [4] blocksA block consists of one or more conventional NN layers.\",\n \"We make use of two types of blocks in [5]: the fused binary fully connected (FC) block and fused binary convolution-pool (ConvP) block as shown in Figure REF .\",\n \"FC blocks each consist of a fully connected layer with $m$ nodes for some $m$ , batch normalization and binary activation.\",\n \"ConvP blocks each consist of a convolutional layer with $f$ filters for some $f$ , a pooling layer and batch normalization and binary activation.\",\n \"A convolution layer has a kernel of size 3x3 with stride 1 and padding 1.\",\n \"A pooling layer has a kernel of size 3x3 with stride 2 and padding 1.\",\n \"For our experiments, we use version (c) from Figure REF , with six end devices.\",\n \"The system presented can be generalized to a more elaborated structure which includes an edge layer, as shown in (d), (e) or (f) of Figure REF .\",\n \"Figure REF depicts a detailed view of the DDNN system used in our experiments.\",\n \"In this system, we have six end devices shown in red, a local aggregator, and a cloud aggregator.\",\n \"During training, output from each device is aggregated together at each exit point using one of the aggregation schemes described in Section REF .\",\n \"We provide detailed analysis on the impact of aggregation schemes at both the local and cloud exit points in Section REF .\",\n \"All DDNNs in our experiments are trained with Adam [17] using the following hyper-parameter settings: $\\\\alpha $ of 0.001, $\\\\beta _1$ of 0.9, $\\\\beta _2$ of 0.999, and $\\\\epsilon $ of 1e-8.\",\n \"We train each DDNN for 100 epochs.\",\n \"When training the DDNN, we use equal weights for the local and cloud exit points.\",\n \"We explored heavily weighting both the local exit and the cloud exit, but neither weighting scheme significantly changed the accuracy of the system.\",\n \"This indicates that this solution to the dataset and the problem we are exploring is not sensitive to the weights, but this may not be true for other datasets and problemsIn GoogleNet [9], a less than 1% difference in accuracy was observed based on the values of the weight parameters.\",\n \"Figure: Fused binary blocks consisting of one or more standard NN layers.\",\n \"The fused binary fully connected (FC) block is a fully connected layer with nn nodes, batch normalization and binary activation.\",\n \"The fused binary convolution-pool (ConvP) block consists of a convolutional layer with ff filters, a pooling layer, batch normalization and binary activation.\",\n \"The convolution layer has a kernel of size 3x3 with stride 1 and padding 1.\",\n \"The pooling layer has a kernel of size 3x3 with stride 2 and padding 1.\",\n \"These blocks are used as they are presented in .Figure: The DDNN architecture used in the system evaluation.\",\n \"The FC and ConvP blocks in red and blue correspond to layers run on end devices and the cloud respectively.\",\n \"The dashed orange boxes represent the end devices and show which blocks of the DDNN are mapped onto each device.\",\n \"The local aggregator shown in red combines the exit output (a short vector with length equal to the number of classes) from each end device in order to determine if local classification for the given input sample can be performed accurately.\",\n \"If the local exit is not confident (i.e.\",\n \"η(x)>T\\\\eta (x) > T), the activation output after the last convolutional layer from each end device is sent to the cloud aggregator (shown in blue), which aggregates the input from each device, performs further NN layer processing, and outputs a final classification result.\",\n \"The aggregation of input for multiple end devices is discussed in Section .\"\n ],\n [\n \"Multi-view Multi-camera Dataset\",\n \"We evaluate the proposed DDNN framework on a multi-view multi-camera dataset [18].\",\n \"This dataset consists of images acquired at the same time from six cameras placed at different locations facing the same general area.\",\n \"For the purpose of our evaluation, we assume that each camera is attached to an end device, which can transmit the captured images over a bandwidth-constraint wireless network to a physical endpoint connected to the cloud.\",\n \"The dataset provides object bounding box annotations.\",\n \"Multiple bounding boxes may exist in a single image, each of which corresponds to a different object in the frame.\",\n \"In preparing the dataset, for each bounding box, we extract an image, and manually synchronizeIn practical object tracking systems, this synchronization step is typically automated [19].\",\n \"the same object across the multiple devices that the object appears in for the given frame.\",\n \"Examples of the extracted images are shown in Figure REF .\",\n \"Each row corresponds to a single sample used for classification.\",\n \"We resize each extracted sample to a 32x32 RGB pixel image.\",\n \"For each device that a given object does not appear in, we use a blank image and assign a label of -1, meaning that the object is not present in the frame.\",\n \"Labels 0, 1, and 2 correspond to car, bus and person, respectively.\",\n \"Objects that are not present in a frame (i.e., label of -1) are not used during training.\",\n \"We split the dataset into 680 training samples and 171 testing samples.\",\n \"Figure REF shows the distribution of samples at each device.\",\n \"Due to the imbalanced number of class samples in the dataset, the individual accuracy of each end device differs widely, as shown by the “Individual\\\" curve of Figure REF .\",\n \"A full description of the training process for the individual NN models is provided in Section REF .\",\n \"The processed dataset used in this paper is available at [20].\",\n \"Figure: Example images of three objects (person, bus, car) from the multi-view multi-camera dataset.\",\n \"The six devices (each with their own camera) capture the same object from different orientations.\",\n \"An all grey image denotes that the object is not present in the frame.Figure: The distribution of class samples for each end device in the multi-view multi-camera dataset.\"\n ],\n [\n \"Impact of Aggregation Schemes\",\n \"In order to perform classification on the input from multiple end devices, we must aggregate the information from each end device.\",\n \"We consider three aggregation methods (max pooling, average pooling, and concatenation) outlined in Section REF , at both the local and cloud exit points.\",\n \"The accuracy of different aggregation schemes are shown in Table REF .\",\n \"The first two letters identify the local aggregation scheme and the last two letters identify the scheme used by the cloud aggregator.\",\n \"For example, MP-CC means the local aggregator uses max-pooling and the cloud uses concatenation.\",\n \"Recall that each input to the local aggregator is a floating-point vector of length equal to the number of classes (corresponding to the output from the final FC block for a single device as shown in Figure REF ) and the device output sent to the cloud aggregator is the output from the final ConvP block.\",\n \"Table: Accuracy of aggregation schemes.\",\n \"The first two letters identify the local aggregation scheme, and the last two letters identify the cloud aggregation scheme.\",\n \"For example, MP-CC means the local aggregator uses max-pooling and the cloud aggregator uses concatenation.\",\n \"The accuracy of each exit point (either local or cloud) is computed using the entire test set.\",\n \"In practice, we will exit a portion of samples locally based on the entropy threshold TT and send the remaining samples in the cloud.\",\n \"Due to its high performance, MP-CC is used in the remaining experiments of this paper.The MP-MP scheme has good classification accuracy for the local aggregator but poor performance in the cloud.\",\n \"The elements in the vectors at the local aggregator correspond to the same features (e.g., the first item is the likelihood that the input corresponds to that class).\",\n \"Therefore, max pooling corresponds to taking the max response for each class over all end devices, and shows good performance.\",\n \"On the other hand, since the information sent from the end devices to the cloud is the activation output from the filters at each device, which corresponds to different visual features in the input from the viewpoint of each individual end device, max pooling these features does not perform well.\",\n \"Comparing MP-MP and MP-CC schemes, though both use MP for local aggregators, MP-CC increases the accuracy of the local classifier.\",\n \"In the training phrase, during backpropagation the MP-MP scheme only passes gradients through a device that gives the highest response while MP-CC scheme passes gradients through all devices.\",\n \"Therefore, using CC aggregator in the cloud allows all devices to learn better filters (filter weights) that give a stronger response for the local MP aggregator, resulting in a better classification accuracy.\",\n \"The CC-CC scheme shows an opposite trend where the local accuracy is poor while the cloud accuracy is high.\",\n \"Concatenating the local information (instead of a pooling scheme), does not enforce any relationship between output for the same class on multiple devices and therefore performs worse.\",\n \"Concatenating the output for the cloud aggregator maintains the most information for NN layer processing in the cloud and therefore performs well.\",\n \"Generally, for the local aggregator, average pooling performs worse than max pooling.\",\n \"This is because some of the end devices do not have the object present in the given frame.\",\n \"Average pooling take average of all outputs from end devices; this compromises the strong outputs from end devices in which the object is present.\",\n \"Based on these results, we use the MP-CC aggregation scheme throughout the paper.\"\n ],\n [\n \"Entropy Threshold\",\n \"The entropy threshold for an exit point, $T$ , corresponds to the level of confidence that is required in order to exit a sample.\",\n \"A threshold value of 0 would mean that no samples will exit and a value of 1 would mean that all samples exit at that point.\",\n \"Figure REF shows the relationship between $T$ at the local aggregator and the overall accuracy of the DDNN.\",\n \"We observe that as more samples are exited at the local exit, the overall accuracy decreases.\",\n \"This is expected, as the accuracy of the local exit is typically lower than that of the cloud exit.\",\n \"We need to set the threshold appropriately to achieve a balance between the communication cost, as defined in Section REF , latency and accuracy of the system.\",\n \"In this case, we see that setting the threshold to $0.8$ results in the best overall accuracy with significantly reduced communication, i.e., 97% accuracy while exiting 60.82% of samples locally as shown in Table REF where in addition to local exit (%) and overall accuracy (%), communication cost in bytes is given.\",\n \"We set $T=0.8$ for the remaining experiments in the system evaluation, unless noted otherwise.\",\n \"The local classifier may do better than cloud for certain samples where low-level features are more robust in classification than higher-level features.\",\n \"By setting an appropriate threshold $T$ , we can improve overall accuracy.\",\n \"In this experiment, $T=0.8$ corresponds to that sweet spot where some samples which are incorrectly classified by the cloud classifier can actually be correctly classified by the local classifier.\",\n \"Such a threshold indicates the optimal point where both local and cloud classifier work best together.\",\n \"Table: Effects of different exit threshold (TT) settings for the local exit.\",\n \"T=0.8T=0.8 is used in the remaining experiments.Figure: Overall accuracy of the system as the entropy threshold for the local exit is varied from 0 to 1.\",\n \"For this experiment, 4 filters are used in the ConvP blocks on the end devices.\"\n ],\n [\n \"Impact of Scaling Across End Devices\",\n \"In order to scale DDNNs across multiple end devices, we distribute the lower sections of Figure REF , shown in red, over the corresponding devices, outlined in orange.\",\n \"Figure REF shows how the accuracy of the system improves as additional end devices (each with its attached input cameras) are added.\",\n \"The devices are added in order sorted by their individual accuracy from worst to best (i.e., the device with the lowest accuracy first and the device with the highest accuracy last).\",\n \"The first observation is the large variation in the individual accuracy of the end devices, as noted earlier.\",\n \"Due to the nature of the dataset, some devices are naturally better positioned and generally have clearer observations of the objects.\",\n \"Looking at the viewpoints of each camera in Figure REF , we see that the selected examples for Device 6 have clear frontal views of each object.\",\n \"This viewpoint gives Device 6 the highest individual accuracy at over 70%.\",\n \"By comparison, Device 2 has the lowest individual accuracy at under 40%.\",\n \"The “Local” and “Cloud” curves show the accuracy of the system at each exit point when all samples are exited at that point.\",\n \"We observe that the cloud exit point outperforms the local exit point at all numbers of end devices.\",\n \"The gap is widest when there are fewer devices.\",\n \"This suggests that the additional NN layers in the cloud significantly improve the final classification result when the problem is more difficult due to limited labeled training data for an end device.\",\n \"Once all six end devices are added, both the local and cloud aggregators have high accuracy.\",\n \"The “Overall” curve represents the overall accuracy of the system when the threshold for the local exit point is set to $0.8$ .\",\n \"We see that this curve is roughly equivalent to exiting all samples at the cloud (but at a much reduced communication cost as 60.82% of samples are exited locally).\",\n \"Generally, these results show that by combining multiple viewpoints we can increase the classification accuracy at both the local and cloud level by a substantial margin when compared to the individual accuracy of any device.\",\n \"The resulting accuracy of the DDNN system is superior to any individual device accuracy by over 20%.\",\n \"Moreover, we note that the 60.82% of samples which exit locally enjoy lowered latency in response time.\",\n \"Figure: Accuracy of the DDNN system as additional end devices are added.\",\n \"The accuracy of “Overall” is obtained by exiting a percentage of the samples locally and the rest in the cloud.\",\n \"The accuracy of “Cloud” and “Local” are computed by exiting all samples at each point, respectively.\",\n \"The end devices are ordered by their “Individual” classification accuracy, sorted from worst to best.\"\n ],\n [\n \"Impact of Cloud Offloading on Accuracy Improvements\",\n \"DDNNs improve the overall accuracy of the system by offloading difficult samples to the cloud, which perform further NN layer processing and final classification.\",\n \"Figure REF shows the accuracy and communication costs of DDNN as the number of filters on the end devices increases.\",\n \"For all settings, the NN layers stored on an end device require under 2 KB of memory.\",\n \"In this experiment, we configure the local exit threshold $T$ such that around $75\\\\%$ of samples are exited locally and around $25\\\\%$ of samples are offloaded to the cloud.\",\n \"We see that DDNNs achieve around a $5\\\\%$ improvement in accuracy compared to using just the local aggregator.\",\n \"This demonstrates the advantage for offloading to the cloud even when larger models (more filters) with improved local accuracy are used on the end devices.\",\n \"Figure: Accuracy and communication cost (in bytes) for increasingly larger end device memory sizes that accommodate additional filters.\",\n \"We notice that cloud offloading leads to improved accuracy.\"\n ],\n [\n \"Fault Tolerance of DDNNs\",\n \"A key motivation for distributed systems is fault tolerance.\",\n \"Fault tolerance implies that the system still works well even when some parts are broken.\",\n \"In order to test the fault tolerance of DDNN, we simulate end device failures and look at the resulting accuracy of the system.\",\n \"Figure REF shows the accuracy of the system under the presence of individual device failures.\",\n \"Regardless of the device that is missing, the system still achieves over a $95\\\\%$ overall classification accuracy.\",\n \"Specifically, even when the device with the highest individual accuracy has failed, which is Device 6, the overall accuracy is reduced by only $3\\\\%$ .\",\n \"This suggests that for this dataset, the automatic fault tolerance provided by DDNN makes the system reliable even in the presence of device failure.\",\n \"We can also view figure REF from the perspective of providing fault tolerance for the system.\",\n \"As we decrease the number of end devices from 6 to 4, we observe that the overall accuracy of the system drops only $4\\\\%$ .\",\n \"This suggests that the system can also be robust to mutliple failing end devices.\",\n \"Figure: The impact on DDNN system accuracy when any single end device has failed.\"\n ],\n [\n \"Reducing Communication Costs\",\n \"DDNNs significantly reduces the communication cost of inference compared to the standard method of offloading raw sensor input to the cloud.\",\n \"Sending a 32x32 RGB pixel image (the input size of our dataset) to the cloud costs 3072 bytes per image sample.\",\n \"By comparison, as shown in Table REF , the largest DDNN model used in our evaluation section requires only 140 bytes of communication per sample on average (an over 20x reduction in communication costs).\",\n \"This communication reduction for an end device results from transmitting class-label related intermediate results to the local aggregator for all samples and binarized communication with the cloud when additional NN layer processing is required for classification with improved accuracy.\"\n ],\n [\n \"DDNN Provision for Horizontal and Vertical Scaling\",\n \"The evaluation in the previous section shows that DDNN is able to achieve high overall accuracy through provisioning the network to scale both horizontally, across end devices, and vertically, over the network hierarchy.\",\n \"Specifically, we show that DDNN scales vertically, by exiting easier input samples locally for low-latency response and offloading difficult samples to the cloud for high overall recognition accuracy, while maintaining a small memory footprint on the end devices and incurring a low communication cost.\",\n \"This result is not obvious, as we need sufficiently good feature representations from the lower parts of the DNN (running on the end devices with limited resources) in order for the upper parts of the neural network (running in the cloud) to achieve high accuracy under the low communication cost constraint.\",\n \"Therefore, we show in a positive way that the proposed method of jointly training a single DNN with multiple exit points at each part of the distributed hierarchy allows us to meet this goal.\",\n \"That is, DDNN optimizes the lower parts of the DNN to create a sufficiently good feature representations to support both samples exited locally and those processed further in the cloud.\",\n \"To meet the goal of horizontal scaling, we provide a principled way of jointly training a DNN with inputs from multiple devices through feature pooling via local and cloud aggregators and demonstrate that by aggregating features from each device we can dramatically improve the accuracy of the system both at the local and cloud level.\",\n \"Filters on each device are automatically tuned to process the geographically unique inputs and work together toward to the same overall objective leading to high overall accuracy.\",\n \"Additionally, we show that DDNN provides built-in fault tolerance across the end devices and is still able to achieve high accuracy in the presence of failed devices.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we propose a novel distributed deep neural network architecture (DDNN) that is distributed across computing hierarchies, consisting of the cloud, the edge and end devices.\",\n \"We demonstrate for a multi-view, multi-camera dataset that DDNN scales vertically from a few NN layers on end devices or the edge to many NN layers in the cloud and scales horizontally across multiple end devices.\",\n \"The aggregation of information communicated from different devices is built into the joint training of DDNN and is handled automatically at inference time.\",\n \"This approach simplifies the implementation and deployment of distributed cloud offloading and automates sensor fusion and system fault tolerance.\",\n \"The experimental results suggest that with our DDNN framework, a single DNN properly trained can be mapped onto a distributed computing hierarchy to meet the accuracy, communication and latency requirements of a target application while gaining inherent benefits associated with distributed computing such as fault tolerance and privacy.\",\n \"DDNNs reduce the required communication compared to a standard cloud offloading approach by exiting many samples at the local aggregator and sending a compact binary feature representation to the cloud when additional processing is required.\",\n \"For our evaluation dataset, the communication cost of DDNN is reduced by a factor of over 20x compared to offloading raw sensor input to a DNN in the cloud which performs all of the inference computation.\",\n \"DDNN provides a framework for further research in mapping DNN into a distributed computing hierarchy.\",\n \"For future work, we will investigate the performance of DDNNs on applications with a larger dataset with multiple types of input modalities [21] and more end devices.\",\n \"Currently, all layers in DDNN are binary.\",\n \"While binary layers are a requirement for end devices due to the limited space on devices, it is not necessary in the cloud.\",\n \"We will explore other types of aggregation schemes and mixed precisions schemes where the end devices use binary NN layers and the cloud uses mixed-precision or floating-point NN layers.\"\n ],\n [\n \"Acknowledgment\",\n \"This work is supported in part by gifts from the Intel Corporation and in part by the Naval Supply Systems Command award under the Naval Postgraduate School Agreements No.\",\n \"N00244-15-0050 and No.\",\n \"N00244-16-1-0018.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01921"}}},{"rowIdx":987,"cells":{"text":{"kind":"list like","value":[["A Stochastic Lagrangian particle system for the Navier-Stokes equations"],["Abstract This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process.","If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\to \\infty$.","In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \\to \\infty$.","The objective of this short note is to describe a resetting procedure that removes this deficiency.","We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy."],["Introduction","The Navier-Stokes equations $ \\partial _t u + (u \\cdot \\nabla )&u - \\nu \\Delta u+ \\nabla p= 0,\\\\ \\nabla &\\cdot u=0$ describe the evolution of a velocity field $u$ of an incompressible fluid with kinematic viscosity $ \\nu >0 $ , and where $p$ denotes the pressure.","When the viscosity vanishes, we end up with the incompressible Euler equation: $\\partial _t u + (u \\cdot \\nabla )&u+ \\nabla p= 0,\\\\\\nabla &\\cdot u=0$ which describe the motion of an ideal incompressible fluid.","The mathematical theory of these equations have been extensively studied and the existence of regular solutions is still an open problem in PDE theory C,F.","We are interested in developing probabilistic techniques, that could help solve this problem.","Probabilistic representations of solutions of partial differential equations as the expected value of functionals of stochastic processes date back to the work of Einstein, Feynman, Kac, and Kolmogorov in physics and mathematics.","The Feynman-Kac formula is the most well-known example, which has provided a link between linear parabolic partial differential equations and probability theory [13], [15].","These stochastic representation methods have provided in some cases tools to show existence and uniqueness of solutions to partial differential equations.","For nonlinear partial differential equations the earliest work was done by McKean [14], where a probabilistic representation of the solution for the nonlinear Kolmogorov-Petrovsky-Piskunov equation was given.","The theory for nonlinear partial differential equations, however, is far less understood.","The questions studied in this work are motivated by the development of a probabilistic formulation of (REF )-() proposed by P.Constantin and G.Iyer in [6].","There, the Navier-Stokes equation is interpreted as the average of a stochastic perturbation of the Euler equation.","More specifically, a Weber formula is used to represent the velocity of the inviscid equation in terms of the particle trajectories of the inviscid equation without including time derivatives, then the classical Lagrangian trajectories are replaced by stochastic flows.","Averaging these stochastic trajectories gives us the solution of (REF )-().","In [11] G.Iyer and J.Mattingly used a Monte-Carlo method to approximate the described probabilistic formulation.","They took $N$ independent copies of the Wiener process and replaced the expected value in the above formalism with the empirical mean, $\\frac{1}{N}$ times the sum over these $N$ independent copies (we review the details of this method in Section ).","By the law of the large numbers it is natural to expect that any average could be replaced by its empirical mean: $1/N\\sum _{i=1}^N X_i \\approx \\mathbb {E}(X)$ , where $X$ and $X_i$ , $i=1,\\dots , N$ are i.i.d.","It turns out that a straightforward approximation of this average by its empirical mean is not adequate here.","It was shown in [11] that in two dimensions the $N$ -particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\rightarrow \\infty $ .","In contrast, the solution of the corresponding Navier-Stokes equations does dissipate all of its energy as $t \\rightarrow \\infty $ .","The goal of this paper is to alleviate this deficiency of the particle system developed in [11] by modifying it, so that the modified particle system dissipates all of its energy.","Our modification is inspired by [12], where G.Iyer and A.Novikov studied a particle system formulation for the Burgers equation.","There we can find another example, where the particle system does not fully mimic properties of the corresponding PDE.","Namely, the viscous Burgers equation does not develop shocks, but the corresponding $N$ -particle system shocks almost surely in finite time.","In order to remove these shocks, the authors [12] considered a resetting procedure that prevented their formation.","In this paper we propose another resetting procedure, such that the particle system for the Navier-Stokes equations dissipate all its energy.","The particle system in [11] does not completely dissipate its energy, because, roughly speaking, the gradients of the velocities for the $N$ particles become decorrelated with time.","We reinforce correlation of these velocities and their gradients by resetting, and this allows complete dissipation of energy.","When the resetting condition holds, then the particle system dissipates its energy exponentially.","Once the resetting condition is not satisfied, we reset the particle system, and restart the procedure again.","In addition to the exponential dissipation of energy, the resetting procedure itself adds more dissipation each time we average our data.","Our theorem states that if we keep repeating the resetting procedure, the particle system will dissipate all its energy.","We now highlight briefly some other probabilistic formulations for the Navier-Stokes equations and related work.","The initial work on probabilistic representation of the Navier-Stokes equations was done by A.Chorin.","It was shown in [1] that in two dimensions vorticity evolves according to Fokker-Planck type equation.","Using this A.Chorin gave a probabilistic representation for the vorticity using random walks and a particle limit, and then related it to the velocity vector using the Biot-Savart Law.","Using a different approach Y.Le Jan and A.Sznitman in [17] developed a probabilistic representation of Navier-Stokes equations, where they used a backward-in-time branching process in Fourier space to express the velocity of the three dimensional viscous fluid as the average of a stochastic process.","These works do not allow to develop a self contained well-posedness theory.","Here our motivation is to develop new probabilistic techniques that may help us establish regularity theory for partial differential equations of Fluid Dynamics.","The stochastic representation in [6] allows such theory, and we, therefore, chose to work with it.","Building on the work done by P.Constantin and G.Iyer in [6], X.Zhang in [18] gave a stochastic representation for the backward incompressible Navier-Stokes equation using stochastic Lagrangian path.","Later, linking the work of X.Zhang in [18] with P.Constantin and G.Iyer in [6], F.Delbaen, J.Qiu, and S.Tang in [7] developed a coupled forward-backward stochastic differential system in the space of fields for the incompressible Navier-Stokes equation in the whole space.","Using probabilistic tools, they were able to obtain local uniqueness results for the forward-backward stochastic differential system.","In addition, they were able to show the existence of global solutions for the case with small Reynolds number or when the dimension is two.","We also mention [2], where F.Cipriano and A.Cruzeiro using the Brownian motions on the group of homeomorphisms on the torus, established a stochastic variational principle for the two dimensional Navier-Stokes equations.","Moreover, A.Cruzeiro and E.Shamarova in [3] formulated a connection between the Navier-Stokes equations and a system of forward-backward stochastic differential equations on the group of volume-preserving diffeomorphisms of a flat torus.","This paper is organized as follows.","In Section we describe the stochastic-Lagrangian representation of the Navier-Stokes equations and construct the particle system.","In addition, we explain the resetting scheme which will then be used in Section .","In Section , the main section of this paper, we study the energy of the Navier-Stokes's particle system and we use the resetting procedure and show that by repeating it often enough, the particle system for the Navier-Stokes equations dissipates all its energy."],["The particle system and the resetting","In this section we construct the particle system for the Navier-Stokes equations based on stochastic Lagrangian trajectories.","We begin by describing a stochastic Lagrangian formulation of the Navier-Stokes equations [11].","Let $B_t$ be a standard 2 or 3-dimensional Brownian motion on a torus $\\mathbb {T}$ , and let $u_0$ be some given periodic, and divergence free $C^{{k,\\alpha }}$ initial data.","Let $\\mathbb {E}$ denote the expected value with respect to the Wiener measure and $\\mathrm {P}$ be the Leray-Hodge projection onto divergence free vector fields.","Consider the system of equations $\\ dX_t(x) &= u_t ( X_t(x) ) \\, dt + \\sqrt{2\\nu }\\,dB_t, \\qquad X_0(x) = x, \\\\u_t &= \\mathbb {E}\\, \\mathrm {P}\\left[ (\\nabla ^*Y_t) (u_0 \\circ Y_t)\\right],\\qquad Y_t = X_t^{-1}.","$ Above $X_t$ is the stochastic flow of diffeomorphisms on $\\mathbb {T}$ , and we denote $Y_t = X_t^{-1}$ to be the spatial inverse of $X_t$ for any given time $t \\geqslant 0$ .","We denote $\\nabla ^*Y_t$ to be the transpose of the Jacobian of $Y_t$ .","It was shown in [6] that if the initial data $u_0 \\in C^{{k,\\alpha }}$ is a deterministic divergence free vector field with $k \\geqslant 2$ and if we impose periodic boundary conditions on $u_t$ and $X_t - \\mathbb {I}$ , then for a short time the system (REF )-() is equivalent to the incompressible Navier-Stokes equations, that is, $u_t$ satisfies $\\partial _t u_t + (u_t \\cdot \\nabla ) u_t - \\nu \\Delta u_t+ \\nabla p= 0, \\hspace{2.84544pt}\\nabla \\cdot u_t=0.$ In the case when the viscosity is zero $\\nu = 0$ , the equations (REF )-() are the Lagrangian formulation for the incompressible Euler equation developed in [4].","Note that we need the law of the entire flow $X$ in order to compute $u$ , this is due to the fact that the term $\\nabla ^{*}Y$ is present in ().","In order to approximate the system (REF )-(), we replace the flow $X_t$ with $N$ different copies $X^{i}_t$ where each one is driven independently by a Wiener process $B^i_t$ , $i=1,2, \\dots , N$ .","Fix a (sufficiently large) $N$ , and we end up with the following approximate system $dX^{i}_t &= u_t \\left( X^{i}_t \\right) \\, dt + \\sqrt{2\\nu }\\,dB^i_t,\\quad Y^{i}_t = \\left( X^{i}_t \\right)^{-1},\\\\u_t &= \\frac{1}{N} \\sum _{i=1}^N u^{i}_t, \\hspace{8.5359pt}u^{i}_t = \\mathrm {P}[ (\\nabla ^*Y^{i}_t) u_0\\circ Y^{i}_t], $ with initial data $X_0(x) = x$ .","We impose periodic boundary conditions on the initial data $u_0$ , and the displacement $\\lambda ^{i}_t(x) =X^{I}_t(x) - x$ .","The following Lemma describes the evolution of the velocity of the particle system () as SPDE.","Lemma 2.1 (Lemma 4.2 in [11]) Let $u^{i}_t = \\mathrm {P}[ (\\nabla ^*Y^{i}_t) u_{0} \\circ Y^{i}_t]$ be the $i^\\text{th}$ summand in ().","Then $u^{i}_t$ satisfies the SPDE $\\ d u^{i}_t + \\left[ (u_t \\cdot \\nabla ) u^{i}_t - \\nu \\triangle u^{i}_t + (\\nabla ^*u_t) u^{i}_t + \\nabla p^{i}_t\\right]dt + \\sqrt{2\\nu } \\nabla u^{i}_t dB^i_t = 0,$ and $u_t$ satisfies the SPDE $d u_t + \\left[ (u_t \\cdot \\nabla ) u_t - \\nu \\triangle u_t + \\nabla p_t \\right] \\, dt + \\frac{\\sqrt{2\\nu }}{N} \\sum _{i=1}^N \\nabla u^{i}_t dB^i_t = 0.$ In contrast to the true Navier-Stokes equations (REF )-() the particle system (REF )-(), for any finite $N$ , may not dissipate all of its energy as $t \\rightarrow \\infty $ .","In two dimensions this was proven in [11].","In this work, we propose to alleviate this deficiency by considering a resetting scheme, in which we start by solving the system (REF )-() on the interval $(t_0, t_1] $ , where the resetting time $t_1$ is going to be specified later according to our proposed resetting condition $(\\ref {times})$ below.","Next, we average our data, replace the original initial data with $u^{N}_{t_1}$ , and we restart the system (REF )-() for the next time interval using this new initial data.","We keep repeating this procedure on each interval $(t_m, t_{m+1}]$ for $m \\in \\mathbb {N}$ .","The resetting criterion comes from comparison of the rate of change of the energies of the true Navier-Stokes equation and the particle system.","Namely, the rate of change of the energy of our particle system (REF )-() is (see Theorem REF below) as follows.","$ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\frac{1}{N^2} \\hspace{4.26773pt} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^{j}_t, \\nabla u^{i}_t \\rangle \\right].$ Observe that the rate of change of energy depends on the average of inner products of the gradients of $N$ velocities.","In contrast, for the true Navier-Stokes equation the rate of change of the energy $\\frac{1}{2\\nu } \\partial _t \\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2.$ For large $N$ the right-hand sides of (REF ) and (REF ) are essentiallyIf $\\nabla u^{j}_t=\\nabla u^{i}_t$ , then the right-hand side of (REF ) is $ (N-1) N^{-1} \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\rightarrow \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ , as $N \\rightarrow \\infty $ .","the same, if $\\nabla u^{j}_t=\\nabla u^{i}_t$ for all $i$ and $j$ .","This observation motivates our approach.","We will use resetting to keep the sum of the expected value of the inner products $ \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^{j}_t, \\nabla u^{i}_t \\rangle \\right] \\geqslant \\hspace{1.42271pt} c N^2 \\hspace{1.42271pt} \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ on each interval $(t_m, t_{m+1}]$ where $m \\in \\mathbb {N}$ and for some constant $c >0$ that does not depend on $N$ .","This will make the inner products in (REF ) to be positive and thus the rate of energy dissipation will be negative on each interval $(t_m, t_{m+1}]$ .","Therefore we consider the following resetting system $dX^{i}_{t } &= u_t \\left( X^{i}_{t } \\right) \\, dt + \\sqrt{2\\nu }\\,dB^i_t, \\hspace{5.69046pt} X^{i}_{t_m }(x_0)=x_0,~Y^{i}_{t }= \\left( X^{i}_{t } \\right)^{-1},\\\\u_{t } &= \\frac{1}{N} \\sum _{i=1}^N u^{i}_t, \\hspace{5.69046pt} u^{i}_t = \\mathrm {P}\\left[ (\\nabla ^*Y^{i}_{t }) (u_{t_m} \\circ Y^{i}_{ t }) \\right], \\hspace{5.69046pt} \\text{for} \\hspace{5.69046pt}t \\in (t_m, t_{m+1}], $ where $m \\in \\mathbb {N}$ , the set of non-negative integers, $t_0=0$ , and the resetting times are defined recursively $ t_{m} = \\inf \\Big \\lbrace t > t_{m-1}: \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\nabla u^i_t, \\nabla u^j_t \\rangle \\right] < (1- \\varepsilon ) N(N-1) \\left\\Vert \\nabla u_{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\Big \\rbrace .$ for some positive fixed $\\varepsilon <1$ .","We say we reset the system at every $t = t_m$ , because we treat $u_{ t_m}$ as initial conditions for each of the intervals $t \\in [ t_m, t_{m+1} )$ .","Theorem 2.2 Suppose we are in two dimensions.","Let the initial condition $u_{0}$ be a $\\mathcal {F}_{0}$ -measurable, periodic mean zero function such that the norm $\\Vert u_{0}\\Vert _{{1,\\alpha }}$ , $\\alpha >0$ is almost surely bounded.","If we let $\\lbrace t_m\\rbrace _{m=0}^{\\infty }$ to be the sequence of resetting times defined in (REF ), then the particle system with resetting (REF )-() dissipates all its energy.","We remark that the particle system with resetting (REF )-() dissipates its energy using two mechanisms.","It dissipates energy exponentially anytime the inequality $(\\ref {EnergyCondition})$ holds, and it dissipates energy when we average our data each time we reset our system.","We also want to highlight that the manner in which we defined our resetting times in (REF ) causes the length of time increments $\\delta _m= t_m-t_{m-1}$ to vary among resetting intervals.","This gives rise to the case that if the sequence of time increments $\\delta _m $ decays to zero too fast, then the limit of the sequence of resetting times $t_m \\rightarrow T$ , for some constant time $T$ .","Thus, we have to consider two cases.","First case is when the limit of sequence of resetting times $t_m \\rightarrow \\infty $ , as $m \\rightarrow \\infty $ .","In this case we show that the energy of the particle system with resetting (REF )-() dissipates its energy mainly exponentially.","The second case is when the limit of the sequence of resetting times $t_m \\rightarrow T$ , for some finite time $T$ .","In this case, we show that the system (REF )-() dissipates its energy mainly by averaging each time we reset."],[" Energy decay by resetting","By Lemma REF , $u^{i}_t$ satisfies the SPDE $d u^{i}_t + \\left[ (u_t \\cdot \\nabla ) u^{i}_t - \\nu \\triangle u^{i}_t + (\\nabla ^*u_t) u^{i}_t + \\nabla p^{i}_t\\right]dt + \\sqrt{2\\nu } \\nabla u^{i}_t dB^i_t = 0,$ on each interval $t \\in § (t_m, t_{m+1}]$ .","In two dimensions, the vorticity $\\omega ^i_t = \\nabla \\times u^i_t$ solves $d \\omega ^i_t+ [ (u_t \\cdot \\nabla ) \\omega ^i -\\nu \\Delta \\omega ^i_t] dt +\\sqrt{2 \\nu } \\nabla \\omega ^i_t dB^i_t=0.$ By Itô's formula, we obtain $\\frac{1}{2} d \\left|\\omega ^i_t\\right|^2 + \\omega ^i_t \\cdot \\left[ (u_t \\cdot \\nabla ) \\omega ^i_t - \\nu \\triangle \\omega ^i_t \\right] \\, dt+ \\sqrt{2 \\nu } \\omega ^i_t \\cdot (\\nabla \\omega ^i_t\\, dB^i_t) -\\nu \\left|\\nabla \\omega ^i_t\\right|^2 \\, dt=0$ Integrating in space and using the fact that $u^i_t$ and $u_t$ are divergences free, we have $ d \\Vert \\omega ^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=0$ for all $ t \\in [t_m, t_{m+1})$ .","Thus the norm of all the vorticities is preserved on such time-intervals.","Since $\\Vert \\omega ^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 =\\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ we have $\\left\\Vert \\nabla u_{t_m}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 =\\left\\Vert \\nabla u^1_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=\\left\\Vert \\nabla u^2_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=\\dots =\\left\\Vert \\nabla u^N_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for $t \\in § (t_m, t_{m+1}]$ .","Using Lemma REF and Itô's formula, we also have $\\frac{1}{2} d \\left|u_t\\right|^2 + u_t \\cdot \\left[ (u_t \\cdot \\nabla ) u_t - \\nu \\triangle u_t + \\nabla p \\right] \\, dt+ \\frac{\\sqrt{2 \\nu }}{N} \\sum _{i=1}^N u_t \\cdot (\\nabla u^i_t\\, dB^i_t) \\\\=\\frac{\\nu }{N^2} \\sum _{i=1}^N \\vert \\nabla u^i_t\\vert ^2 \\, dt$ Integrating in space and taking expected values, we obtain $ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= \\mathbb {E}\\left[\\frac{1}{N^2} \\sum _{i=1}^N \\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 -\\Vert \\nabla u_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\right]\\\\ &= \\mathbb {E}\\left[ \\frac{1}{N^2} \\sum _{i=1}^N \\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 - \\frac{1}{N^2} \\bigg [ \\sum _{i=1}^N \\Vert \\nabla u^i_t \\Vert ^2_{L^2}+ \\sum _{i \\ne j }^N \\langle \\nabla u^j_t, \\nabla u^i_t \\rangle \\bigg ] \\right]$ This simplifies to $ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\frac{1}{N^2} \\hspace{4.26773pt} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^j_t, \\nabla u^i_t \\rangle \\right]$ Case I: $\\lim _{m \\rightarrow \\infty } t_m \\rightarrow \\infty $ .","In this case the sequence of resetting times goes to infinity.","Using resetting, we have $ \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\nabla u^i_t, \\nabla u^j_t \\rangle \\right] \\geqslant (1- \\varepsilon ) N(N-1) \\left\\Vert \\nabla u_{t_{m} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for all $t \\in [t_m, t_{m+1})$ .","Thus, using $(\\ref {ResettingCondtion}) $ we can obtain the following estimate on the rate of change of energy $(\\ref {RateEnergy3})$ $\\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant - (1- \\varepsilon )\\frac{N-1}{N} \\left\\Vert \\nabla u_{t_m }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ $\\leqslant -C(1- \\varepsilon )\\frac{N-1}{N} \\mathbb {E}\\left\\Vert u_{t_m }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant -C(1- \\varepsilon )\\frac{N-1}{N} \\mathbb {E}\\left\\Vert u_{t }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for some constant $C>0$ that arises from using Poincaré's inequality.","Using the Gronwall's inequality, we obtain exponential dissipation of energy.","Case II: $ \\lim _{m \\rightarrow \\infty } t_m \\rightarrow T $ .","In this case the sequence of resetting times converges to a finite time $T$ .","At every resetting time we have $\\omega _{t_{m} }= \\frac{1}{N} \\hspace{4.26773pt} \\sum _{i=1 }^N \\omega ^i_{t_{m}}.$ Thus, $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2= \\frac{1}{N^2} \\sum _{i=1 }^N \\mathbb {E}\\left\\Vert \\omega ^i_{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 + \\frac{1}{N^2} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}} , \\omega ^j_{t_{m}} \\rangle \\right]$ $= \\frac{1}{N} \\left\\Vert \\omega _{t_{m-1}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 + \\frac{1}{N^2} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}} , \\omega ^j_{t_{m}} \\rangle \\right].","$ Since $\\langle \\nabla u^i_{t_{m}}, \\nabla u^j_{t_{m}} \\rangle =\\langle \\omega ^i_{t_{m}}, \\omega ^j_{t_{m}} \\rangle $ , we have $ \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}}, \\omega ^j_{t_{m}} \\rangle \\right] \\leqslant (1- \\varepsilon ) N(N-1) \\left\\Vert \\omega _{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big ) \\left\\Vert \\omega _{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2, \\alpha =1 -\\varepsilon +\\frac{\\varepsilon }{N} <1.$ Iterating over $m$ , we have $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big )^m \\left\\Vert \\omega _{0 }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2, \\hspace{2.84544pt} \\text{where} \\hspace{2.84544pt} \\omega _0 = \\nabla \\times u_0.$ Thus, if $\\lim _{m \\rightarrow \\infty } t_m \\rightarrow T$ , for some finite time $T$ , this means we are going to reset our particle system a countable number of times.","Hence, $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big )^m \\left\\Vert \\omega _{0}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\rightarrow 0\\hspace{5.69046pt} \\text{as} \\hspace{5.69046pt} m \\rightarrow \\infty .$ Thus, the particle system dissipates all its energy in a finite time $T$ ."]],"string":"[\n [\n \"A Stochastic Lagrangian particle system for the Navier-Stokes equations\"\n ],\n [\n \"Abstract This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process.\",\n \"If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\\\to \\\\infty$.\",\n \"In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \\\\to \\\\infty$.\",\n \"The objective of this short note is to describe a resetting procedure that removes this deficiency.\",\n \"We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy.\"\n ],\n [\n \"Introduction\",\n \"The Navier-Stokes equations $ \\\\partial _t u + (u \\\\cdot \\\\nabla )&u - \\\\nu \\\\Delta u+ \\\\nabla p= 0,\\\\\\\\ \\\\nabla &\\\\cdot u=0$ describe the evolution of a velocity field $u$ of an incompressible fluid with kinematic viscosity $ \\\\nu >0 $ , and where $p$ denotes the pressure.\",\n \"When the viscosity vanishes, we end up with the incompressible Euler equation: $\\\\partial _t u + (u \\\\cdot \\\\nabla )&u+ \\\\nabla p= 0,\\\\\\\\\\\\nabla &\\\\cdot u=0$ which describe the motion of an ideal incompressible fluid.\",\n \"The mathematical theory of these equations have been extensively studied and the existence of regular solutions is still an open problem in PDE theory C,F.\",\n \"We are interested in developing probabilistic techniques, that could help solve this problem.\",\n \"Probabilistic representations of solutions of partial differential equations as the expected value of functionals of stochastic processes date back to the work of Einstein, Feynman, Kac, and Kolmogorov in physics and mathematics.\",\n \"The Feynman-Kac formula is the most well-known example, which has provided a link between linear parabolic partial differential equations and probability theory [13], [15].\",\n \"These stochastic representation methods have provided in some cases tools to show existence and uniqueness of solutions to partial differential equations.\",\n \"For nonlinear partial differential equations the earliest work was done by McKean [14], where a probabilistic representation of the solution for the nonlinear Kolmogorov-Petrovsky-Piskunov equation was given.\",\n \"The theory for nonlinear partial differential equations, however, is far less understood.\",\n \"The questions studied in this work are motivated by the development of a probabilistic formulation of (REF )-() proposed by P.Constantin and G.Iyer in [6].\",\n \"There, the Navier-Stokes equation is interpreted as the average of a stochastic perturbation of the Euler equation.\",\n \"More specifically, a Weber formula is used to represent the velocity of the inviscid equation in terms of the particle trajectories of the inviscid equation without including time derivatives, then the classical Lagrangian trajectories are replaced by stochastic flows.\",\n \"Averaging these stochastic trajectories gives us the solution of (REF )-().\",\n \"In [11] G.Iyer and J.Mattingly used a Monte-Carlo method to approximate the described probabilistic formulation.\",\n \"They took $N$ independent copies of the Wiener process and replaced the expected value in the above formalism with the empirical mean, $\\\\frac{1}{N}$ times the sum over these $N$ independent copies (we review the details of this method in Section ).\",\n \"By the law of the large numbers it is natural to expect that any average could be replaced by its empirical mean: $1/N\\\\sum _{i=1}^N X_i \\\\approx \\\\mathbb {E}(X)$ , where $X$ and $X_i$ , $i=1,\\\\dots , N$ are i.i.d.\",\n \"It turns out that a straightforward approximation of this average by its empirical mean is not adequate here.\",\n \"It was shown in [11] that in two dimensions the $N$ -particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\\\rightarrow \\\\infty $ .\",\n \"In contrast, the solution of the corresponding Navier-Stokes equations does dissipate all of its energy as $t \\\\rightarrow \\\\infty $ .\",\n \"The goal of this paper is to alleviate this deficiency of the particle system developed in [11] by modifying it, so that the modified particle system dissipates all of its energy.\",\n \"Our modification is inspired by [12], where G.Iyer and A.Novikov studied a particle system formulation for the Burgers equation.\",\n \"There we can find another example, where the particle system does not fully mimic properties of the corresponding PDE.\",\n \"Namely, the viscous Burgers equation does not develop shocks, but the corresponding $N$ -particle system shocks almost surely in finite time.\",\n \"In order to remove these shocks, the authors [12] considered a resetting procedure that prevented their formation.\",\n \"In this paper we propose another resetting procedure, such that the particle system for the Navier-Stokes equations dissipate all its energy.\",\n \"The particle system in [11] does not completely dissipate its energy, because, roughly speaking, the gradients of the velocities for the $N$ particles become decorrelated with time.\",\n \"We reinforce correlation of these velocities and their gradients by resetting, and this allows complete dissipation of energy.\",\n \"When the resetting condition holds, then the particle system dissipates its energy exponentially.\",\n \"Once the resetting condition is not satisfied, we reset the particle system, and restart the procedure again.\",\n \"In addition to the exponential dissipation of energy, the resetting procedure itself adds more dissipation each time we average our data.\",\n \"Our theorem states that if we keep repeating the resetting procedure, the particle system will dissipate all its energy.\",\n \"We now highlight briefly some other probabilistic formulations for the Navier-Stokes equations and related work.\",\n \"The initial work on probabilistic representation of the Navier-Stokes equations was done by A.Chorin.\",\n \"It was shown in [1] that in two dimensions vorticity evolves according to Fokker-Planck type equation.\",\n \"Using this A.Chorin gave a probabilistic representation for the vorticity using random walks and a particle limit, and then related it to the velocity vector using the Biot-Savart Law.\",\n \"Using a different approach Y.Le Jan and A.Sznitman in [17] developed a probabilistic representation of Navier-Stokes equations, where they used a backward-in-time branching process in Fourier space to express the velocity of the three dimensional viscous fluid as the average of a stochastic process.\",\n \"These works do not allow to develop a self contained well-posedness theory.\",\n \"Here our motivation is to develop new probabilistic techniques that may help us establish regularity theory for partial differential equations of Fluid Dynamics.\",\n \"The stochastic representation in [6] allows such theory, and we, therefore, chose to work with it.\",\n \"Building on the work done by P.Constantin and G.Iyer in [6], X.Zhang in [18] gave a stochastic representation for the backward incompressible Navier-Stokes equation using stochastic Lagrangian path.\",\n \"Later, linking the work of X.Zhang in [18] with P.Constantin and G.Iyer in [6], F.Delbaen, J.Qiu, and S.Tang in [7] developed a coupled forward-backward stochastic differential system in the space of fields for the incompressible Navier-Stokes equation in the whole space.\",\n \"Using probabilistic tools, they were able to obtain local uniqueness results for the forward-backward stochastic differential system.\",\n \"In addition, they were able to show the existence of global solutions for the case with small Reynolds number or when the dimension is two.\",\n \"We also mention [2], where F.Cipriano and A.Cruzeiro using the Brownian motions on the group of homeomorphisms on the torus, established a stochastic variational principle for the two dimensional Navier-Stokes equations.\",\n \"Moreover, A.Cruzeiro and E.Shamarova in [3] formulated a connection between the Navier-Stokes equations and a system of forward-backward stochastic differential equations on the group of volume-preserving diffeomorphisms of a flat torus.\",\n \"This paper is organized as follows.\",\n \"In Section we describe the stochastic-Lagrangian representation of the Navier-Stokes equations and construct the particle system.\",\n \"In addition, we explain the resetting scheme which will then be used in Section .\",\n \"In Section , the main section of this paper, we study the energy of the Navier-Stokes's particle system and we use the resetting procedure and show that by repeating it often enough, the particle system for the Navier-Stokes equations dissipates all its energy.\"\n ],\n [\n \"The particle system and the resetting\",\n \"In this section we construct the particle system for the Navier-Stokes equations based on stochastic Lagrangian trajectories.\",\n \"We begin by describing a stochastic Lagrangian formulation of the Navier-Stokes equations [11].\",\n \"Let $B_t$ be a standard 2 or 3-dimensional Brownian motion on a torus $\\\\mathbb {T}$ , and let $u_0$ be some given periodic, and divergence free $C^{{k,\\\\alpha }}$ initial data.\",\n \"Let $\\\\mathbb {E}$ denote the expected value with respect to the Wiener measure and $\\\\mathrm {P}$ be the Leray-Hodge projection onto divergence free vector fields.\",\n \"Consider the system of equations $\\\\ dX_t(x) &= u_t ( X_t(x) ) \\\\, dt + \\\\sqrt{2\\\\nu }\\\\,dB_t, \\\\qquad X_0(x) = x, \\\\\\\\u_t &= \\\\mathbb {E}\\\\, \\\\mathrm {P}\\\\left[ (\\\\nabla ^*Y_t) (u_0 \\\\circ Y_t)\\\\right],\\\\qquad Y_t = X_t^{-1}.\",\n \"$ Above $X_t$ is the stochastic flow of diffeomorphisms on $\\\\mathbb {T}$ , and we denote $Y_t = X_t^{-1}$ to be the spatial inverse of $X_t$ for any given time $t \\\\geqslant 0$ .\",\n \"We denote $\\\\nabla ^*Y_t$ to be the transpose of the Jacobian of $Y_t$ .\",\n \"It was shown in [6] that if the initial data $u_0 \\\\in C^{{k,\\\\alpha }}$ is a deterministic divergence free vector field with $k \\\\geqslant 2$ and if we impose periodic boundary conditions on $u_t$ and $X_t - \\\\mathbb {I}$ , then for a short time the system (REF )-() is equivalent to the incompressible Navier-Stokes equations, that is, $u_t$ satisfies $\\\\partial _t u_t + (u_t \\\\cdot \\\\nabla ) u_t - \\\\nu \\\\Delta u_t+ \\\\nabla p= 0, \\\\hspace{2.84544pt}\\\\nabla \\\\cdot u_t=0.$ In the case when the viscosity is zero $\\\\nu = 0$ , the equations (REF )-() are the Lagrangian formulation for the incompressible Euler equation developed in [4].\",\n \"Note that we need the law of the entire flow $X$ in order to compute $u$ , this is due to the fact that the term $\\\\nabla ^{*}Y$ is present in ().\",\n \"In order to approximate the system (REF )-(), we replace the flow $X_t$ with $N$ different copies $X^{i}_t$ where each one is driven independently by a Wiener process $B^i_t$ , $i=1,2, \\\\dots , N$ .\",\n \"Fix a (sufficiently large) $N$ , and we end up with the following approximate system $dX^{i}_t &= u_t \\\\left( X^{i}_t \\\\right) \\\\, dt + \\\\sqrt{2\\\\nu }\\\\,dB^i_t,\\\\quad Y^{i}_t = \\\\left( X^{i}_t \\\\right)^{-1},\\\\\\\\u_t &= \\\\frac{1}{N} \\\\sum _{i=1}^N u^{i}_t, \\\\hspace{8.5359pt}u^{i}_t = \\\\mathrm {P}[ (\\\\nabla ^*Y^{i}_t) u_0\\\\circ Y^{i}_t], $ with initial data $X_0(x) = x$ .\",\n \"We impose periodic boundary conditions on the initial data $u_0$ , and the displacement $\\\\lambda ^{i}_t(x) =X^{I}_t(x) - x$ .\",\n \"The following Lemma describes the evolution of the velocity of the particle system () as SPDE.\",\n \"Lemma 2.1 (Lemma 4.2 in [11]) Let $u^{i}_t = \\\\mathrm {P}[ (\\\\nabla ^*Y^{i}_t) u_{0} \\\\circ Y^{i}_t]$ be the $i^\\\\text{th}$ summand in ().\",\n \"Then $u^{i}_t$ satisfies the SPDE $\\\\ d u^{i}_t + \\\\left[ (u_t \\\\cdot \\\\nabla ) u^{i}_t - \\\\nu \\\\triangle u^{i}_t + (\\\\nabla ^*u_t) u^{i}_t + \\\\nabla p^{i}_t\\\\right]dt + \\\\sqrt{2\\\\nu } \\\\nabla u^{i}_t dB^i_t = 0,$ and $u_t$ satisfies the SPDE $d u_t + \\\\left[ (u_t \\\\cdot \\\\nabla ) u_t - \\\\nu \\\\triangle u_t + \\\\nabla p_t \\\\right] \\\\, dt + \\\\frac{\\\\sqrt{2\\\\nu }}{N} \\\\sum _{i=1}^N \\\\nabla u^{i}_t dB^i_t = 0.$ In contrast to the true Navier-Stokes equations (REF )-() the particle system (REF )-(), for any finite $N$ , may not dissipate all of its energy as $t \\\\rightarrow \\\\infty $ .\",\n \"In two dimensions this was proven in [11].\",\n \"In this work, we propose to alleviate this deficiency by considering a resetting scheme, in which we start by solving the system (REF )-() on the interval $(t_0, t_1] $ , where the resetting time $t_1$ is going to be specified later according to our proposed resetting condition $(\\\\ref {times})$ below.\",\n \"Next, we average our data, replace the original initial data with $u^{N}_{t_1}$ , and we restart the system (REF )-() for the next time interval using this new initial data.\",\n \"We keep repeating this procedure on each interval $(t_m, t_{m+1}]$ for $m \\\\in \\\\mathbb {N}$ .\",\n \"The resetting criterion comes from comparison of the rate of change of the energies of the true Navier-Stokes equation and the particle system.\",\n \"Namely, the rate of change of the energy of our particle system (REF )-() is (see Theorem REF below) as follows.\",\n \"$ \\\\frac{1}{2\\\\nu } \\\\partial _t \\\\mathbb {E}\\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 &= - \\\\frac{1}{N^2} \\\\hspace{4.26773pt} \\\\sum _{i \\\\ne j }^N \\\\mathbb {E}\\\\left[ \\\\langle \\\\nabla u^{j}_t, \\\\nabla u^{i}_t \\\\rangle \\\\right].$ Observe that the rate of change of energy depends on the average of inner products of the gradients of $N$ velocities.\",\n \"In contrast, for the true Navier-Stokes equation the rate of change of the energy $\\\\frac{1}{2\\\\nu } \\\\partial _t \\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 &= - \\\\Vert \\\\nabla u_t \\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2.$ For large $N$ the right-hand sides of (REF ) and (REF ) are essentiallyIf $\\\\nabla u^{j}_t=\\\\nabla u^{i}_t$ , then the right-hand side of (REF ) is $ (N-1) N^{-1} \\\\Vert \\\\nabla u_t \\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\rightarrow \\\\Vert \\\\nabla u_t \\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ , as $N \\\\rightarrow \\\\infty $ .\",\n \"the same, if $\\\\nabla u^{j}_t=\\\\nabla u^{i}_t$ for all $i$ and $j$ .\",\n \"This observation motivates our approach.\",\n \"We will use resetting to keep the sum of the expected value of the inner products $ \\\\sum _{i \\\\ne j }^N \\\\mathbb {E}\\\\left[ \\\\langle \\\\nabla u^{j}_t, \\\\nabla u^{i}_t \\\\rangle \\\\right] \\\\geqslant \\\\hspace{1.42271pt} c N^2 \\\\hspace{1.42271pt} \\\\mathbb {E}\\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ on each interval $(t_m, t_{m+1}]$ where $m \\\\in \\\\mathbb {N}$ and for some constant $c >0$ that does not depend on $N$ .\",\n \"This will make the inner products in (REF ) to be positive and thus the rate of energy dissipation will be negative on each interval $(t_m, t_{m+1}]$ .\",\n \"Therefore we consider the following resetting system $dX^{i}_{t } &= u_t \\\\left( X^{i}_{t } \\\\right) \\\\, dt + \\\\sqrt{2\\\\nu }\\\\,dB^i_t, \\\\hspace{5.69046pt} X^{i}_{t_m }(x_0)=x_0,~Y^{i}_{t }= \\\\left( X^{i}_{t } \\\\right)^{-1},\\\\\\\\u_{t } &= \\\\frac{1}{N} \\\\sum _{i=1}^N u^{i}_t, \\\\hspace{5.69046pt} u^{i}_t = \\\\mathrm {P}\\\\left[ (\\\\nabla ^*Y^{i}_{t }) (u_{t_m} \\\\circ Y^{i}_{ t }) \\\\right], \\\\hspace{5.69046pt} \\\\text{for} \\\\hspace{5.69046pt}t \\\\in (t_m, t_{m+1}], $ where $m \\\\in \\\\mathbb {N}$ , the set of non-negative integers, $t_0=0$ , and the resetting times are defined recursively $ t_{m} = \\\\inf \\\\Big \\\\lbrace t > t_{m-1}: \\\\sum _{i \\\\ne j }^N \\\\hspace{4.26773pt} \\\\mathbb {E}\\\\left[ \\\\langle \\\\nabla u^i_t, \\\\nabla u^j_t \\\\rangle \\\\right] < (1- \\\\varepsilon ) N(N-1) \\\\left\\\\Vert \\\\nabla u_{t_{m-1} }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\Big \\\\rbrace .$ for some positive fixed $\\\\varepsilon <1$ .\",\n \"We say we reset the system at every $t = t_m$ , because we treat $u_{ t_m}$ as initial conditions for each of the intervals $t \\\\in [ t_m, t_{m+1} )$ .\",\n \"Theorem 2.2 Suppose we are in two dimensions.\",\n \"Let the initial condition $u_{0}$ be a $\\\\mathcal {F}_{0}$ -measurable, periodic mean zero function such that the norm $\\\\Vert u_{0}\\\\Vert _{{1,\\\\alpha }}$ , $\\\\alpha >0$ is almost surely bounded.\",\n \"If we let $\\\\lbrace t_m\\\\rbrace _{m=0}^{\\\\infty }$ to be the sequence of resetting times defined in (REF ), then the particle system with resetting (REF )-() dissipates all its energy.\",\n \"We remark that the particle system with resetting (REF )-() dissipates its energy using two mechanisms.\",\n \"It dissipates energy exponentially anytime the inequality $(\\\\ref {EnergyCondition})$ holds, and it dissipates energy when we average our data each time we reset our system.\",\n \"We also want to highlight that the manner in which we defined our resetting times in (REF ) causes the length of time increments $\\\\delta _m= t_m-t_{m-1}$ to vary among resetting intervals.\",\n \"This gives rise to the case that if the sequence of time increments $\\\\delta _m $ decays to zero too fast, then the limit of the sequence of resetting times $t_m \\\\rightarrow T$ , for some constant time $T$ .\",\n \"Thus, we have to consider two cases.\",\n \"First case is when the limit of sequence of resetting times $t_m \\\\rightarrow \\\\infty $ , as $m \\\\rightarrow \\\\infty $ .\",\n \"In this case we show that the energy of the particle system with resetting (REF )-() dissipates its energy mainly exponentially.\",\n \"The second case is when the limit of the sequence of resetting times $t_m \\\\rightarrow T$ , for some finite time $T$ .\",\n \"In this case, we show that the system (REF )-() dissipates its energy mainly by averaging each time we reset.\"\n ],\n [\n \" Energy decay by resetting\",\n \"By Lemma REF , $u^{i}_t$ satisfies the SPDE $d u^{i}_t + \\\\left[ (u_t \\\\cdot \\\\nabla ) u^{i}_t - \\\\nu \\\\triangle u^{i}_t + (\\\\nabla ^*u_t) u^{i}_t + \\\\nabla p^{i}_t\\\\right]dt + \\\\sqrt{2\\\\nu } \\\\nabla u^{i}_t dB^i_t = 0,$ on each interval $t \\\\in § (t_m, t_{m+1}]$ .\",\n \"In two dimensions, the vorticity $\\\\omega ^i_t = \\\\nabla \\\\times u^i_t$ solves $d \\\\omega ^i_t+ [ (u_t \\\\cdot \\\\nabla ) \\\\omega ^i -\\\\nu \\\\Delta \\\\omega ^i_t] dt +\\\\sqrt{2 \\\\nu } \\\\nabla \\\\omega ^i_t dB^i_t=0.$ By Itô's formula, we obtain $\\\\frac{1}{2} d \\\\left|\\\\omega ^i_t\\\\right|^2 + \\\\omega ^i_t \\\\cdot \\\\left[ (u_t \\\\cdot \\\\nabla ) \\\\omega ^i_t - \\\\nu \\\\triangle \\\\omega ^i_t \\\\right] \\\\, dt+ \\\\sqrt{2 \\\\nu } \\\\omega ^i_t \\\\cdot (\\\\nabla \\\\omega ^i_t\\\\, dB^i_t) -\\\\nu \\\\left|\\\\nabla \\\\omega ^i_t\\\\right|^2 \\\\, dt=0$ Integrating in space and using the fact that $u^i_t$ and $u_t$ are divergences free, we have $ d \\\\Vert \\\\omega ^i_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2=0$ for all $ t \\\\in [t_m, t_{m+1})$ .\",\n \"Thus the norm of all the vorticities is preserved on such time-intervals.\",\n \"Since $\\\\Vert \\\\omega ^i_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 =\\\\Vert \\\\nabla u^i_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ we have $\\\\left\\\\Vert \\\\nabla u_{t_m}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 =\\\\left\\\\Vert \\\\nabla u^1_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2=\\\\left\\\\Vert \\\\nabla u^2_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2=\\\\dots =\\\\left\\\\Vert \\\\nabla u^N_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ for $t \\\\in § (t_m, t_{m+1}]$ .\",\n \"Using Lemma REF and Itô's formula, we also have $\\\\frac{1}{2} d \\\\left|u_t\\\\right|^2 + u_t \\\\cdot \\\\left[ (u_t \\\\cdot \\\\nabla ) u_t - \\\\nu \\\\triangle u_t + \\\\nabla p \\\\right] \\\\, dt+ \\\\frac{\\\\sqrt{2 \\\\nu }}{N} \\\\sum _{i=1}^N u_t \\\\cdot (\\\\nabla u^i_t\\\\, dB^i_t) \\\\\\\\=\\\\frac{\\\\nu }{N^2} \\\\sum _{i=1}^N \\\\vert \\\\nabla u^i_t\\\\vert ^2 \\\\, dt$ Integrating in space and taking expected values, we obtain $ \\\\frac{1}{2\\\\nu } \\\\partial _t \\\\mathbb {E}\\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 &= \\\\mathbb {E}\\\\left[\\\\frac{1}{N^2} \\\\sum _{i=1}^N \\\\Vert \\\\nabla u^i_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 -\\\\Vert \\\\nabla u_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\right]\\\\\\\\ &= \\\\mathbb {E}\\\\left[ \\\\frac{1}{N^2} \\\\sum _{i=1}^N \\\\Vert \\\\nabla u^i_t\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 - \\\\frac{1}{N^2} \\\\bigg [ \\\\sum _{i=1}^N \\\\Vert \\\\nabla u^i_t \\\\Vert ^2_{L^2}+ \\\\sum _{i \\\\ne j }^N \\\\langle \\\\nabla u^j_t, \\\\nabla u^i_t \\\\rangle \\\\bigg ] \\\\right]$ This simplifies to $ \\\\frac{1}{2\\\\nu } \\\\partial _t \\\\mathbb {E}\\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 &= - \\\\frac{1}{N^2} \\\\hspace{4.26773pt} \\\\sum _{i \\\\ne j }^N \\\\mathbb {E}\\\\left[ \\\\langle \\\\nabla u^j_t, \\\\nabla u^i_t \\\\rangle \\\\right]$ Case I: $\\\\lim _{m \\\\rightarrow \\\\infty } t_m \\\\rightarrow \\\\infty $ .\",\n \"In this case the sequence of resetting times goes to infinity.\",\n \"Using resetting, we have $ \\\\sum _{i \\\\ne j }^N \\\\hspace{4.26773pt} \\\\mathbb {E}\\\\left[ \\\\langle \\\\nabla u^i_t, \\\\nabla u^j_t \\\\rangle \\\\right] \\\\geqslant (1- \\\\varepsilon ) N(N-1) \\\\left\\\\Vert \\\\nabla u_{t_{m} }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ for all $t \\\\in [t_m, t_{m+1})$ .\",\n \"Thus, using $(\\\\ref {ResettingCondtion}) $ we can obtain the following estimate on the rate of change of energy $(\\\\ref {RateEnergy3})$ $\\\\frac{1}{2\\\\nu } \\\\partial _t \\\\mathbb {E}\\\\left\\\\Vert u_t\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\leqslant - (1- \\\\varepsilon )\\\\frac{N-1}{N} \\\\left\\\\Vert \\\\nabla u_{t_m }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ $\\\\leqslant -C(1- \\\\varepsilon )\\\\frac{N-1}{N} \\\\mathbb {E}\\\\left\\\\Vert u_{t_m }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\leqslant -C(1- \\\\varepsilon )\\\\frac{N-1}{N} \\\\mathbb {E}\\\\left\\\\Vert u_{t }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ for some constant $C>0$ that arises from using Poincaré's inequality.\",\n \"Using the Gronwall's inequality, we obtain exponential dissipation of energy.\",\n \"Case II: $ \\\\lim _{m \\\\rightarrow \\\\infty } t_m \\\\rightarrow T $ .\",\n \"In this case the sequence of resetting times converges to a finite time $T$ .\",\n \"At every resetting time we have $\\\\omega _{t_{m} }= \\\\frac{1}{N} \\\\hspace{4.26773pt} \\\\sum _{i=1 }^N \\\\omega ^i_{t_{m}}.$ Thus, $\\\\mathbb {E}\\\\left\\\\Vert \\\\omega _{t_{m}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2= \\\\frac{1}{N^2} \\\\sum _{i=1 }^N \\\\mathbb {E}\\\\left\\\\Vert \\\\omega ^i_{t_{m}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 + \\\\frac{1}{N^2} \\\\sum _{i \\\\ne j }^N \\\\mathbb {E}\\\\left[ \\\\langle \\\\omega ^i_{t_{m}} , \\\\omega ^j_{t_{m}} \\\\rangle \\\\right]$ $= \\\\frac{1}{N} \\\\left\\\\Vert \\\\omega _{t_{m-1}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 + \\\\frac{1}{N^2} \\\\sum _{i \\\\ne j }^N \\\\mathbb {E}\\\\left[ \\\\langle \\\\omega ^i_{t_{m}} , \\\\omega ^j_{t_{m}} \\\\rangle \\\\right].\",\n \"$ Since $\\\\langle \\\\nabla u^i_{t_{m}}, \\\\nabla u^j_{t_{m}} \\\\rangle =\\\\langle \\\\omega ^i_{t_{m}}, \\\\omega ^j_{t_{m}} \\\\rangle $ , we have $ \\\\sum _{i \\\\ne j }^N \\\\hspace{4.26773pt} \\\\mathbb {E}\\\\left[ \\\\langle \\\\omega ^i_{t_{m}}, \\\\omega ^j_{t_{m}} \\\\rangle \\\\right] \\\\leqslant (1- \\\\varepsilon ) N(N-1) \\\\left\\\\Vert \\\\omega _{t_{m-1} }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2$ $\\\\mathbb {E}\\\\left\\\\Vert \\\\omega _{t_{m}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\leqslant \\\\big ( 1 -\\\\alpha \\\\big ) \\\\left\\\\Vert \\\\omega _{t_{m-1} }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2, \\\\alpha =1 -\\\\varepsilon +\\\\frac{\\\\varepsilon }{N} <1.$ Iterating over $m$ , we have $\\\\mathbb {E}\\\\left\\\\Vert \\\\omega _{t_{m}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\leqslant \\\\big ( 1 -\\\\alpha \\\\big )^m \\\\left\\\\Vert \\\\omega _{0 }\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2, \\\\hspace{2.84544pt} \\\\text{where} \\\\hspace{2.84544pt} \\\\omega _0 = \\\\nabla \\\\times u_0.$ Thus, if $\\\\lim _{m \\\\rightarrow \\\\infty } t_m \\\\rightarrow T$ , for some finite time $T$ , this means we are going to reset our particle system a countable number of times.\",\n \"Hence, $\\\\mathbb {E}\\\\left\\\\Vert \\\\omega _{t_{m}}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\leqslant \\\\big ( 1 -\\\\alpha \\\\big )^m \\\\left\\\\Vert \\\\omega _{0}\\\\right\\\\Vert _{\\\\smash{L^{\\\\!2}_{\\\\vphantom{h}}}\\\\vphantom{L^{\\\\!2}}}^2 \\\\rightarrow 0\\\\hspace{5.69046pt} \\\\text{as} \\\\hspace{5.69046pt} m \\\\rightarrow \\\\infty .$ Thus, the particle system dissipates all its energy in a finite time $T$ .\"\n ]\n]"},"id":{"kind":"string","value":"1709.01536"}}},{"rowIdx":988,"cells":{"text":{"kind":"list like","value":[["Contact forms with large systolic ratio in dimension three"],["Abstract The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume.","We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio.","This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena."],["Introduction and main result","Given a Riemannian metric $g$ on a closed surface $S$ , we denote the length of a shortest non-constant closed geodesic by $\\ell _{\\min }(S,g)$ and consider the scaling invariant ratio $\\rho (S,g) := \\frac{\\ell _{\\min }(S,g)^2}{\\mathrm {area}(S,g)},$ where $\\mathrm {area}(S,g)$ denotes the Riemannian area.","It is a classical question in systolic geometry to find upper bounds, and possibly optimal ones, for this ratio.","When $S$ is not the 2-sphere, the fact that $S$ is not simply connected allows one to bound $\\rho (S,g)$ from above by the ratio $\\rho _{\\rm nc}(S,g) :=\\frac{\\mathrm {sys}_1(S,g)^2}{\\mathrm {area}(S^2,g)},$ where $\\mathrm {sys}_1(S,g)$ denotes the length of a shortest non-contractible curve on $(S,g)$ , which is of course a closed geodesic.","When $S$ is the 2-torus 2, a classical result of Loewner from the end of the 1940s states that $\\rho _{\\rm nc}(2,\\cdot )$ is maximised by the flat metric corresponding to the lattice generated by two sides of an equilateral triangle in $\\mathbb {R}^2$ .","The corresponding maximal value of $\\rho _{\\rm nc}$ is $2/\\sqrt{3}$ .","Shortly after, Pu [15] considered the case of the projective plane $\\mathbb {R}\\mathbb {P}^2$ and showed that $\\rho _{\\rm nc}(\\mathbb {R}\\mathbb {P}^2,\\cdot )$ is maximized by the standard constant curvature metric, with maximal value $\\pi /2$ .","In both cases, the metrics maximizing $\\rho _{\\rm nc}$ have no contractible closed geodesics, so these metrics maximize also the ratio $\\rho $ .","For an arbitrary closed non-simply connected surface $S$ , the ratio $\\rho _{\\rm nc}(S,\\cdot )$ is bounded from above by the non-optimal universal constant 2, as shown by Gromov in [12], but the supremum is in general not achieved.","Actually, in [12] Gromov proved a far reaching generalization of this result by showing that for any essential $n$ -dimensional closed Riemannian manifold $(M,g)$ , the quotient $\\rho _{\\rm nc}(M,g) := \\frac{\\mathrm {sys}_1(M,g)^n}{\\mathrm {vol}(M,g)}$ has an upper bound which depends only on the dimension $n$ .","Here, a non-simply connected closed oriented manifold $M$ is said to be essential if its fundamental class is non-zero in the homology of the Eilenberg-MacLane space $K(\\pi _1(M),1)$ .","This condition generalizes asphericity, that is the fact that all homotopy groups of degree larger than one vanish, and characterizes manifolds for which the ratio $\\rho _{\\rm nc}$ is bounded from above, see [6].","Going back to surfaces, from the fact mentioned above we deduce that the number 2 is an upper bound also for $\\rho (S,\\cdot )$ , when $S\\ne S^2$ .","In the case of the two-sphere $S^2$ , the first upper bound for $\\rho (S^2,\\cdot )$ was found by Croke in [7] and later improved by several authors, the best one so far being the bound 32 found by Rotman in [16].","We conclude that $\\rho (S,\\cdot )$ is always bounded from above when $S$ is a closed surface, and there is an upper bound which is independent of $S$ .","The ratios $\\rho $ and $\\rho _{\\rm nc}$ generalize to Finsler metrics, by replacing the Riemannian area by the Holmes-Thompson area.","Recent results of Àlvarez Paiva, Balacheff and Tzanev allow to extend the Riemannian bounds to the Finsler setting: The value of the supremum might get larger (and sometimes it does get larger, as in the case of 2, for which the Loewner metric does not maximize $\\rho $ and $\\rho _{\\rm nc}$ , even locally, among Finsler metrics) but is in any case finite.","See [5].","The next natural generalization of the ratio $\\rho $ is to the setting of contact geometry.","We recall that a contact form $\\alpha $ on a 3-manifold $M$ is a smooth 1-form such that $\\alpha \\wedge d\\alpha $ is a volume form.","The kernel of $\\alpha $ is a plane distribution and is called the contact structure defined by $\\alpha $ .","In this paper, contact structures are always assumed to be co-orientable, that is, defined by a global contact form.","If two contact forms $\\alpha _1,\\alpha _2$ on $M$ define the same contact structure then the volume forms $\\alpha _1\\wedge d\\alpha _1$ and $\\alpha _2\\wedge d\\alpha _2$ define the same orientation.","This orientation is said to be induced by the contact structure, and in this paper all contact 3-manifolds are oriented by the contact structure.","A contact form $\\alpha $ induces a nowhere vanishing vector field $R_{\\alpha }$ on $M$ , which is defined by the conditions $\\imath _{R_{\\alpha }} d\\alpha = 0 , \\qquad \\alpha (R_{\\alpha })=1,$ and is called the Reeb vector field for $\\alpha $ .","Any Reeb vector field on a closed 3-manifold $M$ admits periodic orbits, as proved by Taubes in [17], and we denote by $T_{\\min }(M,\\alpha )$ the minimum among the periods of all periodic orbits of $R_{\\alpha }$ .","Then it is natural to define $\\rho (M,\\alpha ):= \\frac{T_{\\min }(M,\\alpha )^2}{\\mathrm {vol}(M,\\alpha \\wedge d\\alpha )}.$ We will refer to this quantity as the systolic ratio of $(M,\\alpha )$ .","It is scaling invariant, because if we multiply $\\alpha $ by a non-zero real number $c$ then both $T_{\\min }^2$ and the volume get multiplied by $c^2$ .","The inverse of this quantity appears in [4] under the name of systolic volume of $(M,\\alpha )$ .","Let $F$ be a smooth Finsler metric on the closed surface $S$ .","The unit cotangent sphere bundle $S^*_FS$ is the space of cotangent vectors having norm 1 with respect to the dual Finsler metric $F^*$ .","The canonical Liouville 1-form $p\\, dq$ of $T^* S$ restricts to a contact form $\\alpha _F$ on $S_F^*S$ .","Two different Finsler metrics $F$ and $F^{\\prime }$ induce contact forms $\\alpha _F$ and $\\alpha _{F^{\\prime }}$ which correspond to the same contact structure, once $S_F^*S$ and $S_{F^{\\prime }}^*S$ are identified by means of the radial projection.","The flow of the corresponding Reeb vector field $R_{\\alpha _F}$ is precisely the geodesic flow of $F$ , once this is read on the cotangent bundle by the Legendre transform.","Therefore, $\\ell _{\\min }(S,F)=T_{\\min }(\\alpha _F)$ .","Moreover, the volume of $S^*_FS$ with respect to $\\alpha _F \\wedge d\\alpha _F$ is, essentially by definition, $2\\pi $ times the Holmes-Thompson area of $(S,F)$ .","We conclude that $\\rho (S^*_FS,\\alpha _F) = \\frac{\\rho (S,F)}{2\\pi },$ and the systolic ratio of a contact form is a genuine generalization of the corresponding notion from Riemannian and Finsler geometry.","All of this generalizes to higher dimensions, but here we restrict our attention to three-dimensional contact manifolds.","Seeing Riemannian and Finsler geometry in the larger contact setting is often fruitful.","On the one hand, results about existence and multiplicity of Riemannian and Finsler closed geodesics have often natural generalizations to Reeb flows, see e.g.","[13], and the same holds for some statements about the topological entropy of geodesic flows, see e.g.","[14], [9], [1].","On the other hand, techniques from contact geometry have been recently found to be useful to address systolic questions in Riemannian and Finlser geometry, such as the local systolic maximality of Zoll metrics, see [4], [2], [3].","It is therefore a natural question to ask whether the systolic ratio of contact forms inducing a given contact structure on a closed three-manifold also has uniform upper bounds.","The purpose of this paper is to give a negative answer to this question.","More precisely, we shall prove the following: Theorem Let $\\xi $ be a contact structure on a closed 3-manifold $M$ .","For every $c>0$ there exists a contact form $\\alpha $ satisfying $\\ker \\alpha =\\xi $ and $\\rho (M,\\alpha )\\ge c.$ When applied to the cotangent sphere bundle of an arbitrary closed surface $S$ , the above theorem has the following consequence: If $T$ is any positive number, then there exists a fiberwise starshaped domain $A\\subset T^*S$ (i.e.","a connected open neighborhood of the zero-section with smooth boundary $\\partial A$ such that for every $q\\in S$ the set $\\partial A \\cap T_q^* S$ is a closed curve in $T^*_q S\\setminus \\lbrace 0\\rbrace $ which is transverse to the radial direction) with volume 1 (with respect to the standard symplectic form $dp\\wedge dq$ of $T^*S$ ) and such that every closed orbit of the Reeb vector field on $\\partial A$ determined by the restriction of the Liouville form $p\\, dq$ has period larger than $T$ .","When $A$ is fiberwise convex then $\\partial A$ is the unit sphere bundle of some Finsler metric and the results of Àlvarez Paiva, Balacheff and Tzanev mentioned above prevent this phenomenon.","This shows that convexity plays an essential role in systolic inequalities.","In the special case of the tight three-sphere $(S^3,\\xi _{\\rm std})$ , that is the standard unit sphere in $\\mathbb {R}^4$ endowed with coordinaten $(x_1,y_1,x_2,y_2)$ and with the contact structure $\\xi _{\\rm std}$ which is given by the kernel of the contact form $\\alpha _0 = \\frac{1}{2} \\sum _{j=1}^2 (x_j \\, dy_j - y_j\\, dx_j) \\Big |_{S^3}$ the unboundedness of the systolic ratio was proved in [3].","The argument in [3] starts with the construction of a special symplectomorphism of the disk with a good lower bound on actions of periodic points and with a suitable negative value of the Calabi invariant (see Section and references therein for the definition of these notions).","This disk-map is then embedded as a return map to a disk-like global surface of section of some Reeb flow on $(S^3,\\xi _{\\rm std})$ in a way that there is a precise “dictionary”: Actions of periodic points correspond to periods of closed Reeb orbits and the Calabi invariant corresponds to the contact volume, up to explicit additive constants.","The tight three-sphere is very simple from a contact-topological point of view, since it admits the trivial supporting open book decomposition with unknotted binding, disk-like pages and trivial monodromy (see Section below and references therein for the definition of these notions).","Moreover, the construction of the disk-map in [3] uses the special symmetries of the disk.","In order to deal with a general contact structure $\\xi $ on a general orientable closed three-manifold $M$ we argue in the following way.","First we use the fact that by a theorem of Giroux [11] $\\xi $ is supported by an open book decomposition of $M$ to construct a contact form $\\alpha $ which is adapted to the open book decomposition and is such that the supports of the monodromy and return maps are contained in a region of small contact area, as visualized in Figure 1.","The construction of this special contact form is explained in Section , where the necessary background about open book decompositions is also recalled.","The next step is to construct suitable contact forms on a solid torus which are standard near the boundary, have a small contact volume and no closed Reeb orbits with small period.","These are constructed in Section , by building on results from [3].","Finally, the latter solid tori are used as plugs to modify the contact form $\\alpha $ constructed in Section .","Indeed, these plugs can be inserted in the region spanned by the large portions of the pages on which the monodromy and first return map are the identity: Their effect is to eat up volume without creating short periodic orbits.","In this way we can reduce the contact volume as much as we wish, while keeping $T_{\\min }$ bounded away from zero, and hence making the systolic ratio arbitrarily large.","This final step is performed in Section ."],["The research of A. Abbondandolo and B. Bramham is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the Deutsche Forschungsgemeinschaft.","U. Hryniewicz thanks the Floer Center for Geometry (Bochum) for its warm hospitality, and acknowledges the generous support of the Alexander von Humboldt Foundation.","U. Hryniewicz is also supported by CNPq grant 309966/2016-7.","P. Salomão is supported by FAPESP grant 2016/25053-8 and CNPq grant 306106/2016-7.","Figure: On the left a typical page as one would picture it in an open book decomposition.","On the right, the same page on an open book supporting a contact structure where the topology, and the supports of the monodromy and return maps are squeezed in a region with small contact area.","This leaves lots of space to embed solid tori with high systolic ratio."],["A special contact form","We recall that a global surface of section for a smooth flow $\\phi ^t$ on a 3-dimensional manifold $M$ is a compact surface $S$ smoothly embedded in $M$ whose boundary $\\partial S$ is $\\phi ^t$ -invariant and such that all orbits which are not contained in $\\partial S$ meet the surface transversally infinitely many times in the future and in the past.","The corresponding first return time function is the smooth function $\\tau : S \\setminus \\partial S \\rightarrow (0,+\\infty ), \\qquad \\tau (p) := \\inf \\lbrace t>0 \\mid \\phi ^t(p)\\in S\\rbrace ,$ and the corresponding first return map is the smooth diffeomorphism $\\varphi : S \\setminus \\partial S \\rightarrow S \\setminus \\partial S, \\qquad \\varphi (p) := \\phi ^{\\tau (p)}(p).$ When $\\phi ^t$ is the Reeb flow of a contact form $\\alpha $ , the transversality of the flow to $S\\setminus \\partial S$ implies that the restriction of $d\\alpha $ to $S\\setminus \\partial S$ is an area form.","The exactness of $d\\alpha $ implies that in the Reeb case the boundary of $S$ is necessarily non-empty.","Since Reeb vector fields are non-singular, the finitely many circles forming the boundary of $S$ are periodic orbits.","Moreover, it is well known that the contact volume of $M$ coincides with the integral of $\\tau $ on $S$ , once this surface is equipped with the area form given by the restriction of $\\alpha $ : $\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) = \\int _S \\tau \\, d\\alpha .$ See e.g.","[3] for the easy proof.","Proposition 1 Let $M$ be a closed connected 3-manifold with a contact structure $\\xi $ .","Then there is an embedded compact surface $S\\subset M$ with the following property: For every $\\epsilon >0$ there exists a contact form $\\alpha $ on $M$ satisfying $\\xi =\\ker \\alpha $ such that $S$ is a global surface of section for the Reeb flow of $\\alpha $ and: If we orient $S$ by $d\\alpha $ and give $\\partial S$ the boundary orientation, we have that $\\int _C \\alpha = 1$ for every connected component $C$ of $\\partial S$ .","In particular, $\\int _S d\\alpha = \\int _{\\partial S} \\alpha = \\ell ,$ where $\\ell \\ge 1$ denotes the number of circles forming the boundary of $S$ .","The first return time function $\\tau $ of the Reeb flow of $\\alpha $ extends to a smooth function on $S$ and the corresponding first return map $\\varphi $ extends to a smooth diffeomorphism of $S$ onto itself.","There exists an open tubular neighborhood $U$ of $\\partial S$ in $S$ such that the support of $\\varphi $ is contained in $S\\setminus U$ and $\\int _{S\\setminus U} d\\alpha < \\epsilon .$ $\\Vert \\tau -1\\Vert _{\\infty } < \\epsilon $ and $\\tau $ is constantly equal to 1 on the neighborhood $U$ from (iii).","In order to prove this proposition, we need to recall some facts about open book decompositions.","See [8] for more details.","An open book decomposition of an oriented closed 3-manifold $M$ is a pair $(\\Pi ,L)$ , where $L\\subset M$ is a smooth link and $\\Pi :M\\setminus L \\rightarrow \\mathbb {R}/2\\pi \\mathbb {Z}$ is a smooth (locally trivial) fibration which near $L$ is a normal angular coordinate.","More precisely, there exists a compact neighborhood $N$ of $L$ and a diffeomorphism $N \\simeq L \\times such that $ L L {0}$ and $ |N L$ is represented as$$(\\lambda ,r e^{ix})\\mapsto x,$$where $$ belongs to $ L$ and $ (r,x)(0,1]R/2Z$ are polar coordinates on the punctured closed unit disc $ {0}.","The link $L$ is called the binding and the fibers of $\\Pi $ are called the pages of the open book decomposition.","The closure of each page is a Seifert surface for $L$ , that is, a compact smoothly embedded surface with boundary $L$ .","The co-orientation of the pages given by the map $\\Pi $ and the ambient orientation induce an orientation of the pages.","The link $L$ inherits the boundary orientation.","Let $S$ be the closure in $M$ of the page $\\Pi ^{-1}(0)$ .","The coordinates introduced above induce coordinates $(\\lambda ,r) \\in \\partial S\\times [0,1]$ on the compact neighborhood $N\\cap S$ of $\\partial S$ in $S$ , where $[0,1]$ is seen as the intersection of $ with the positive real half-axis.$ The vector field $\\partial _x$ on $N \\simeq L \\times is smooth, vanishes at $ LL {0}$ and generates the group of rotations\\begin{equation}\\bigl (t,(\\lambda ,\\rho e^{ix}) \\bigr ) \\mapsto (\\lambda ,\\rho e^{i(x+t)}).\\end{equation}Let $ Z$ be a smooth vector field on $ M$ which coincides with $ x$ on $ N$ and is positively transverse to all the pages, meaning that $ dZ >0$ on $ ML$.","The set of such vector fields is obviously convex and in particular path connected.$ The surface $S$ is a global surface of section for the flow of $Z$ .","The form () of the flow of $Z$ on $N$ implies that the corresponding first return map is the identity on the set $N \\cap (S\\setminus \\partial S)$ , and hence it extends smoothly to the boundary and gives us an orientation preserving diffeomorphism $h: S \\rightarrow S,$ which is the identity on $N \\cap S$ .","The diffeomorphism $h$ is called the monodromy map of the open book decomposition which depends on the choice of the vector field $Z$ .","However as the set of such vector fields is path connected the isotopy class of $h$ within the space of diffeomorphisms of $S$ that are the identity in a neighborhood of $\\partial S$ is uniquely determined.","The mapping torus of $h$ is the smooth 3-manifold with boundary $S(h) := \\left([0,2\\pi ]\\times S\\right)/ \\sim \\qquad (2\\pi ,q) \\sim (0,h(q)).$ It is equipped with the fibration $x: S(h) \\rightarrow \\mathbb {R}/2\\pi \\mathbb {Z},$ given by the projection onto the first component.","The piece $([0,2\\pi ]\\times (S\\cap N))/\\sim $ is diffeomorphic to $\\mathbb {R}/2\\pi \\mathbb {Z}\\times (S\\cap N)$ , because $h$ is the identity on $S\\cap N$ , and is equipped with coordinates $(x,\\lambda ,r) \\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\partial S \\times [0,1].$ In particular, the boundary of $S(h)$ is the product $\\mathbb {R}/2\\pi \\mathbb {Z}\\times \\partial S$ , which is a union of finitely many 2-tori.","The flow of $Z$ preserves the family of leaves in the neighborhood $N$ and is transverse to the interiors of the leaves.","Thus we may renormalise $Z$ outside of $N$ so that its flow preserves the leaves also in $M\\backslash N$ .","Then the flow of $Z$ induces a diffeomorphism between the interior of $S(h)$ and $M\\setminus L$ , which identifies the fibrations $x$ and $\\Pi $ .","Moreover, $M$ is obtained from $S(h)$ by collapsing each boundary torus onto a circle.","More precisely, we can identify $M$ with the quotient $\\bigl (S(h) \\sqcup (\\partial S\\times \\bigr )/\\sim $ where $(x,\\lambda ,r) \\sim (\\lambda ,r e^{ix})$ .","Here, the smooth structure near the link is induced by the identification $\\partial S \\times N$ .","By a theorem of Giroux [11] (see also [8] for a detailed proof), the contact structure $\\xi $ is supported by some open book decomposition $(\\Pi ,L)$ of $M$ : This means that $\\xi =\\ker \\alpha $ , where $\\alpha $ is a contact form such that: the restriction of $d\\alpha $ to each page is a positive area form; $\\alpha $ is positive on $L$ .","As smoothly embedded compact surface $S$ we choose the closure of the page $\\Pi ^{-1}(0)$ .","We freely draw from the notation about open book decompositions established above; in particular, $N$ is the closed tubular neighborhood of $L=\\partial S$ with diffeomorphism $N\\simeq L\\times and $ h:S S$ is a choice of monodromy map which is supported in $ SN$.$ If $\\alpha $ and $\\alpha ^{\\prime }$ are contact forms satisfying (A1) and (A2), then $\\ker \\alpha $ and $\\ker \\alpha ^{\\prime }$ are isotopic (see [11] and [8]).","In this case, Gray's stability theorem (see [10]) implies the existence of a diffeomorphism bringing $\\ker \\alpha $ into $\\ker \\alpha ^{\\prime }$ .","Since the properties (i), (ii), (iii) and (iv) that we wish our contact form to have are preserved by the action of a diffeomorphism, it is enough to find a contact form $\\alpha $ which satisfies them together with the conditions (A1) and (A2) above, i.e.","without explicitly assuming that $\\ker \\alpha $ is the given contact structure.","Denote by $C_1,\\dots ,C_{\\ell }$ the connected components of $\\partial S$ .","The tubular neighborhood $N\\cap S$ of $\\partial S$ in $S$ is the union of $\\ell $ closed annuli $A_j$ carrying coordinates $(\\theta ,r)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,1]$ , where $\\theta $ is an orientation preserving parametrization of $C_j$ .","Here the boundary component $C_j$ corresponds to $r=0$ .","The fact that $C_j$ is given the boundary orientation from $S$ implies that $2r\\,d\\theta \\wedge dr$ is a positive 2-form on each half-open annulus $A_j\\setminus \\partial S$ .","Choose a 2-form $\\Omega $ on $S$ such that $\\Omega >0$ on $S\\setminus L$ , $\\int _S\\Omega = 2\\pi \\ell $ and $\\Omega = 2r\\, d\\theta \\wedge dr \\qquad \\mbox{on each closed annulus } A_j^{\\prime }:= \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,\\rho ]\\subset A_j,$ for a suitably small number $\\rho \\in (0,1]$ .","Perhaps after shrinking $\\rho $ , we can assume that there exists a primitive $\\eta $ of $\\Omega $ agreeing with $(1-r^2)d\\theta $ on each $A_j^{\\prime }$ .","To see this, choose any 1-form $\\sigma _0$ on $S$ which agrees with $(1-r^2)d\\theta $ near the $C_j$ 's.","Then $\\Omega -d\\sigma _0$ is a 2-form with support contained in $S\\setminus \\partial S$ .","Since $\\int _S \\bigl ( \\Omega - d\\sigma _0 \\bigr ) = 2\\pi \\ell - \\int _{\\partial S} \\sigma _0 = 2\\pi \\ell - 2\\pi \\ell = 0,$ we can invoke de Rham's theorem to find a 1-form $\\sigma _1$ supported in $S\\setminus \\partial S$ and such that $\\Omega -d\\sigma _0=d\\sigma _1$ .","Then $\\eta = \\sigma _0+\\sigma _1$ has the desired property.","Let $\\beta :\\mathbb {R}\\rightarrow [0,1]$ be a smooth function satisfying $\\beta (0)=0$ , $\\beta (2\\pi )=1$ and $\\mathrm {supp\\,}(\\beta ^{\\prime })\\subset (0,2\\pi )$ .","The family of 1-forms $\\tilde{\\alpha }_s := dx + s \\bigl ( (1-\\beta (x)) \\eta + \\beta (x) h^*\\eta \\bigr ), \\qquad s\\in (0,+\\infty ),$ is well defined on the mapping torus $S(h)$ of $h$ which is defined in (REF ).","Here $x$ denotes the fibration (REF ), which corresponds to the fibration $\\Pi $ under the identification of the interior of $S(h)$ with $M\\setminus L$ .","From now on, we shall identify $M$ with the manifold $\\left( S(h) \\sqcup \\bigsqcup _{j=1}^{\\ell } \\bigl ( \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho ) \\right) / \\sim $ as in (REF ), where a point in $\\mathbb {R}/2\\pi \\mathbb {Z}\\times A_j^{\\prime }\\subset S(h)$ with coordinates $(x,\\theta ,r)$ , is identified with the point $(\\theta ,r e^{ix})$ in the corresponding copy of $\\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho .","With this identification, the set$$V := \\bigsqcup _{j=1}^{\\ell } \\bigl ( \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho )$$is a compact tubular neighborhood of $ L$ in $ M$.$ We will think of $\\tilde{\\alpha }_s$ as defined in $M\\setminus L \\simeq S(h)\\setminus \\partial S(h)$ .","The fact that $h$ is the identity on $A_j$ and the form of $\\eta $ on $A_j^{\\prime }$ imply that $\\tilde{\\alpha }_s=dx + s (1-r^2)\\, d\\theta \\qquad \\mbox{on }V\\setminus L,$ with respect to the standard coordinates $(\\theta ,r,x)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times (0,\\rho ] \\times \\mathbb {R}/2\\pi \\mathbb {Z}$ on $V \\setminus L$ .","Fix $\\delta >0$ .","There exists $s_1>0$ , depending only on $\\rho $ and $\\delta $ , with the following properties: If $s\\in (0,s_1)$ then there are smooth functions $f,g:[0,\\rho ] \\rightarrow [0,+\\infty )$ defining a curve in the complex plane $\\gamma : [0,\\rho ] \\rightarrow \\qquad \\gamma (r) = f(r) + i g(r),$ satisfying: $\\gamma (r) = 1 + is(1-r^2)$ on $[r_1,\\rho ]$ , for some $r_1 \\in (0,\\rho )$ ; $g^{\\prime }<0$ on $(0,\\rho ]$ .","$\\gamma ([0,r_0]) \\subset \\lbrace x+iy \\mid x+y=1+\\delta \\rbrace $ for some $r_0\\in (0,r_1)$ such that $g(r_0)-g(\\rho )=g(r_0)-s(1-\\rho ^2) \\le 2\\delta .$ Moreover, $\\gamma (r) = r^2 + i (1+\\delta -r^2)$ when $r$ is close enough to 0.","The derivative of the argument of $\\gamma $ , that is the function $ \\frac{g^{\\prime }f-f^{\\prime }g}{f^2+g^2} $ is negative on $(0,\\rho ]$ .","The derivative of the argument of $\\gamma ^{\\prime }$ , that is the function $ \\frac{g^{\\prime \\prime }f^{\\prime }-f^{\\prime \\prime }g^{\\prime }}{(f^{\\prime })^2+(g^{\\prime })^2} $ is non-positive on $[0,\\rho ]$ .","Figure: Functions f,gf,g with the above properties exist.Finally, we define a smooth 1-form $\\alpha _s$ on $M$ by $\\alpha _s = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\frac{\\tilde{\\alpha }_s}{2\\pi (1+\\delta )}} & \\mbox{on } M\\setminus V, \\\\ \\\\\\displaystyle {\\frac{f(r)\\,dx + g(r)\\,d\\theta }{2\\pi (1+\\delta )}} & \\mbox{on } V. \\end{array} \\right.$ The smoothness of $\\alpha _{s}$ near the boundary of $V$ follows from (REF ) and (B1).","By (B3) we have the formula $\\alpha _{s} = \\frac{r^2 \\, dx + (1+\\delta -r^2)\\, d\\theta }{2\\pi (1+\\delta )} \\qquad \\mbox{near $L$},$ which shows that $\\alpha _{s}$ is smooth in a neighborhood of $L$ .","Therefore, $\\alpha _{s}$ is a smooth 1-form on $M$ for every $s\\in (0,s_1)$ .","We claim that there exists $s_2 \\in (0,s_1)$ , depending on $\\delta ,\\rho ,h,\\eta ,\\beta $ , such that if $s \\in (0,s_2)$ then $\\alpha _s$ is a contact form and satisfies conditions (A1) and (A2).","Let us prove this claim.","By (B3) the 1-form $\\alpha _{s}$ has the following expression on $L$ : $\\alpha _{s} = \\frac{1}{2\\pi } \\, d\\theta \\qquad \\mbox{on } L \\simeq \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\lbrace 0\\rbrace \\subset \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho $ which shows that condition (A2) holds for every $s\\in (0,s_1)$ .","On $V$ we have $d\\alpha _s = \\frac{f^{\\prime }dr \\wedge dx-g^{\\prime }d\\theta \\wedge dr}{2\\pi (1+\\delta )}.$ The restriction of $d\\alpha _{s}$ on the page $\\Pi ^{-1}(x_0) = x^{-1}(x_0)$ intersected with $V$ is $-\\frac{g^{\\prime }d\\theta \\wedge dr}{2\\pi (1+\\delta )},$ which is a positive area form thanks to (B2) and the positivity of $d\\theta \\wedge dr$ .","So the condition (A1) holds on $V$ for every $s\\in (0,s_1)$ .","Moreover, $\\alpha _s \\wedge d\\alpha _s = \\frac{f^{\\prime }g-fg^{\\prime }}{4\\pi ^2 (1+\\delta )^2} \\ dx\\wedge d\\theta \\wedge dr = \\frac{f^{\\prime }g-fg^{\\prime }}{4\\pi ^2 (1+\\delta )^2 r} \\ d\\theta \\wedge (r\\, dr \\wedge dx).$ By the last part of (B3), $\\frac{f^{\\prime }g-fg^{\\prime }}{r} =2(1+\\delta )$ when $r$ is close enough to 0.","Together with (B4), this implies that the smooth function $(f^{\\prime }g-fg^{\\prime })/r$ is strictly positive on $V$ .","This implies that $\\alpha _{s}$ is a contact form on $V$ for every $s\\in (0,s_1)$ .","Now we analyze $\\alpha _s$ on $M\\setminus V$ .","The formula $\\alpha _s=\\frac{dx+s \\bigl ((1-\\beta )\\eta +\\beta h^*\\eta \\bigr )}{2\\pi (1+\\delta )} \\qquad \\mbox{on }M\\setminus V$ gives $d\\alpha _s = s \\, \\frac{\\beta ^{\\prime }dx\\wedge (h^*\\eta -\\eta ) + \\omega }{2\\pi (1+\\delta )} \\qquad \\mbox{on }M\\setminus V,$ where $\\omega $ is the smooth 2-form on $M\\setminus L$ with kernel $\\partial _x$ and whose restriction to the page $\\Pi ^{-1}(x) \\simeq S\\setminus \\partial S$ is the positive area form $\\omega _x := (1-\\beta (x))d\\eta +\\beta (x) h^*d\\eta .$ The restriction of $d\\alpha _{s}$ to the page $\\Pi ^{-1}(x_0)=x^{-1}(x_0)\\simeq S\\setminus S$ intersected with $M\\setminus V$ is a positive multiple of the positive area form $\\omega _{x_0}$ , so $\\alpha _{s}$ satisfies (A1) on $M\\setminus V$ for every $s \\in (0,s_1)$ .","Moreover, $\\frac{4\\pi ^2 (1+\\delta )^2}{s} \\, \\alpha _s \\wedge d\\alpha _s = dx\\wedge \\omega + s \\ \\beta ^{\\prime } \\eta \\wedge dx \\wedge h^*\\eta \\qquad \\mbox{on } M\\setminus V.$ Since $dx\\wedge \\omega $ is a volume form on $M\\setminus V$ , the above formula shows that we can choose $s_2\\in (0,s_1]$ , depending on the data $h,\\eta ,\\beta $ , such that $\\alpha _s$ is a contact form on $M\\setminus V$ whenever $s\\in (0,s_2)$ .","We conclude that $\\alpha _s$ is a contact form satisfying conditions (A1) and (A2), for every $s\\in (0,s_2)$ .","The next task is to understand the Reeb flow of $\\alpha _s$ and to show it fullfills the requirements (i), (ii), (iii) and (iv) with respect to the surface $S$ , when $s$ is small enough.","Condition (i) is actually fulfilled for any $s\\in (0,s_1)$ , due to (REF ), so we need to focus only on (ii), (iii) and (iv).","Denote by $R_{s}$ the Reeb vector field of $\\alpha _{s}$ .","We start by analysing the flow of $R_{s}$ on $M\\setminus V$ .","Since $\\beta ^{\\prime }$ is supported in $(0,2\\pi )$ , $h^* \\eta -\\eta $ is compactly supported in $S\\setminus V$ and $\\omega $ restricts to an area form on each page, there is a unique smooth vector field $Y$ on $M$ which is supported in $M\\setminus V$ , is tangent to the pages and satisfies $\\omega (Y,\\cdot ) = -\\beta ^{\\prime }(x) (h^*\\eta -\\eta ) \\qquad \\mbox{on } \\Pi ^{-1}(x).$ By (REF ), we have the identity $\\begin{split}\\frac{2\\pi (1+\\delta )}{s} \\ d\\alpha _s(\\partial _x+Y,\\cdot ) &= \\beta ^{\\prime }(h^*\\eta -\\eta ) - \\beta ^{\\prime }(h^*\\eta -\\eta )(Y)\\, dx + \\omega (Y,\\cdot ) \\\\ &=\\beta ^{\\prime }(h^*\\eta -\\eta ) + \\omega (Y,Y)\\, dx - \\beta ^{\\prime }(h^*\\eta -\\eta ) = 0,\\end{split}$ which shows that the non-vanishing vector field $\\partial _x+Y$ is in the kernel of $d\\alpha _{s}$ .","Therefore, the Reeb vector field $R_{s}$ of $\\alpha _{s}$ has the form $R_s = \\frac{\\partial _x+Y}{\\alpha _s(\\partial _x+Y)} \\qquad \\mbox{on } M\\setminus V,$ for every $s\\in (0,s_2)$ .","The fact that $Y$ is supported in $M\\setminus V$ and the form of $V$ imply that $M\\setminus V$ is invariant for the Reeb flow, and hence the same is true for its complement $V$ .","Moreover, the above formula implies that $R_{s}$ is transverse to the portion of the pages $x^{-1}(x_0)$ lying in $M\\setminus V$ , and in particular to $S\\setminus V$ .","Next we study the Reeb flow on $V$ .","By the form of $\\alpha _{s}$ on $V$ , $R_{s}$ has the form $R_{s} = 2\\pi (1+\\delta ) \\frac{f^{\\prime } \\partial _{\\theta } - g^{\\prime } \\partial _x}{f^{\\prime }g-g^{\\prime }f} \\qquad \\mbox{on } V.$ The condition (B2) implies that $R_{s}$ is transverse the the portions of the pages $x^{-1}(x_0)$ which lie in $V$ , and in particular to $(S\\setminus \\partial S)\\cap V$ .","The expression for $R_{s}$ near $L$ becomes, thanks to (B3), $R_{s} = 2\\pi (\\partial _{\\theta } + \\partial _x) \\qquad \\mbox{for } (\\theta ,r,x)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,r_0] \\times \\mathbb {R}/2\\pi \\mathbb {Z}.$ In particular, since $\\partial _x$ vanishes on $L$ , $R_{s} = 2\\pi \\,\\partial _{\\theta }\\qquad \\mbox{on } L,$ and the link $L$ consists of periodic orbits of $R_{s}$ of period 1.","We conclude that the vector field $R_{s}$ is transverse to the pages $x^{-1}(x_0)$ and leaves the binding $L$ invariant.","In particular, in view of the form of $R_s$ near $L$ described above, $S$ is a global surface of section for the Reeb flow, and the first return time function and first return map $\\tau : S \\setminus \\partial S \\rightarrow (0,+\\infty ), \\qquad \\varphi : S \\setminus \\partial S \\rightarrow S \\setminus \\partial S$ are well defined.","By (REF ), on $(S\\setminus \\partial S)\\cap V$ the first return time function is explicitly given by the formula $\\tau = \\frac{g^{\\prime }f-f^{\\prime }g}{(1+\\delta )g^{\\prime }} \\qquad \\mbox{on } (S\\setminus \\partial S)\\cap V,$ and the first return time map by $\\varphi : (r,\\theta ) \\mapsto \\left( r,\\theta -2\\pi \\frac{f^{\\prime }(r)}{g^{\\prime }(r)} \\right) \\qquad \\mbox{on } (S\\setminus \\partial S) \\cap V.$ By (B3), we have $\\tau (r,\\theta ) = 1 \\qquad \\mbox{and} \\qquad \\varphi (r,\\theta ) = (r,\\theta -2\\pi ) \\qquad \\forall (r,\\theta )\\in (0,r_0]\\times \\mathbb {R}/2\\pi \\mathbb {Z}.$ This implies that $\\tau $ and $\\varphi $ extend smoothly to the boundary of $S$ and proves (ii).","By (REF ), the restriction of the first return time function $\\tau $ to $S\\cap V$ depends only on $r$ and we have $\\partial _r \\tau = \\frac{g(f^{\\prime }g^{\\prime \\prime }-g^{\\prime }f^{\\prime \\prime })}{(1+\\delta )(g^{\\prime })^2}.$ By (B5), this function is non-positive on $[0,\\rho ]$ .","Together with (REF ) and $\\tau (\\rho ,\\theta ) = \\frac{1}{1+\\delta },$ where we have used (REF ) and (B1), we find the bounds $\\frac{1}{1+\\delta } \\le \\tau \\le 1 \\qquad \\mbox{on } S\\cap V,$ from which $\\sup _{S\\cap V} |\\tau -1| \\le 1 - \\frac{1}{1+\\delta } = \\frac{\\delta }{1+\\delta } < \\delta .$ From (REF ) we have $dx(R_{s}) = \\frac{1}{\\alpha _{s}(\\partial _x + Y)} \\qquad \\mbox{on } M\\setminus V.$ Since the function $\\alpha _{s}(\\partial _x + Y) = \\frac{1}{2\\pi (1+\\delta )} \\tilde{\\alpha }_{s}(\\partial _x + Y) = \\frac{1}{2\\pi (1+\\delta )} \\bigl ( 1 + s ((1-\\beta ) \\eta (Y) + \\beta h^* \\eta (Y))\\bigr )$ converges to $1/(2\\pi (1+\\delta ))$ for $s\\rightarrow 0$ uniformly on $M\\setminus V$ , by (REF ) the function $dx(R_{s})$ converges to $2\\pi (1+\\delta )$ for $s\\rightarrow 0$ uniformly on $M\\setminus V$ .","This implies that $\\tau $ converges uniformly to $1/(1+\\delta )$ on $S\\setminus V$ .","Together with (REF ), this implies that there exists $s_3\\in (0,s_2]$ such that for every $s\\in (0,s_3)$ we have $\\Vert \\tau -1\\Vert _{\\infty } < 2\\delta .$ Define $U$ to be the open tubular neighborhood of $L=\\partial S$ in $S$ consisting of those $(r,\\theta )$ in $S\\cap V$ with $0\\le r < r_0$ .","By (REF ), $\\tau =1$ and $\\varphi =\\mathrm {id}$ on $U$ .","Together with (REF ), this proves that (iv) holds, by choosing $\\delta \\le \\epsilon /2$ .","The identity (REF ) gives us a constant $C$ , depending only on $\\beta $ , $\\eta $ and $h$ , such that $\\int _{S\\setminus V} d\\alpha _{s} \\le C s \\qquad \\forall s\\in (0,s_1).$ We now fix a $s_4\\in (0,s_3]$ such that $C s_4<\\delta $ .","By (REF ), the $d\\alpha _{s}$ -area of the annulus in $A_j^{\\prime }$ corresponding to the values of $r$ in the interval $[r_0,\\rho ]$ is $\\frac{1}{2\\pi (1+\\delta )} \\int _{\\mathbb {R}/2\\pi \\mathbb {Z}\\times [r_0,\\rho ]} (-g^{\\prime }) \\, d\\theta \\wedge dr = \\frac{1}{1+\\delta } \\bigl ( g(r_0) - g(\\rho ) \\bigr ) \\le \\frac{2\\delta }{1+\\delta }< 2\\delta ,$ where the first upper bound follows from (B3).","If $s$ is in the interval $(0,s_4)$ , the above two inequalities imply that $\\int _{S\\setminus U} d\\alpha _{s} = \\int _{S\\setminus V} d\\alpha _{s} + \\frac{1}{2\\pi (1+\\delta )} \\int _{\\mathbb {R}/2\\pi \\mathbb {Z}\\times [r_0,\\rho ]} (-g^{\\prime }) \\, d\\theta \\wedge dr < 3\\delta ,$ and the conclusion (iii) follows by choosing $\\delta =\\epsilon /3$ ."],["Construction of the plug","The aim of this section is to show how some results from [3] can be used in order to build a contact form on a solid torus such that the contact volume is small and all closed orbits of the corresponding Reeb flow have large period.","In the next section, this solid torus will be used as a plug to modify the special contact form which we constructed in the previous section.","Here is the statement which summarizes the properties of the plug.","Proposition 2 Fix positive numbers $r$ and $\\epsilon $ .","Let $\\lambda $ be a primitive of the standard area form $dx\\wedge dy$ on the closed disc $r\\mathbb {D}$ .","Then there exists a smooth contact form $\\beta $ on the solid torus $r\\mathbb {D} \\times \\mathbb {R}/\\mathbb {Z}$ with the following properties: $\\beta = \\lambda + ds$ in a neighborhood of $\\partial (r \\times \\mathbb {R}/\\mathbb {Z}$ , where $s$ denotes the coordinate on $\\mathbb {R}/ \\mathbb {Z}$ ; in particular, the Reeb vector field $R_{\\beta }$ of $\\beta $ coincides with $\\partial _s$ near the boundary of $r\\mathbb {R}/\\mathbb {Z}$ , and its flow is globally well-defined; the contact form $\\beta $ is smoothly isotopic to the contact form $\\lambda + ds$ on $r\\mathbb {R}/ \\mathbb {Z}$ through a path of contact forms which agree with $\\lambda + ds$ in a neighborhood of $\\partial (r \\times \\mathbb {R}/ \\mathbb {Z}$ ; all the closed orbits of $R_{\\beta }$ have period at least 1; $\\mathrm {vol}\\,(r\\mathbb {R}/ \\mathbb {Z}, \\beta \\wedge d\\beta ) < \\epsilon $ .","Before discussing the proof of the above proposition, we recall the definition of the Calabi invariant for compactly supported area-preserving diffeomorphisms of the plane.","Endow the plane $\\mathbb {R}^2$ with the standard area form $dx\\wedge dy$ and let $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ be the group of compactly supported area-preserving diffeomorphisms of $\\mathbb {R}^2$ .","Let $\\varphi $ be an element of $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ and let $\\lambda $ be a primitive of $dx\\wedge dy$ .","To $\\varphi $ and $\\lambda $ we can associate the unique compactly supported smooth function $\\sigma _{\\varphi ,\\lambda }: \\mathbb {R}^2 \\rightarrow \\mathbb {R}$ satisfying $\\varphi ^* \\lambda - \\lambda = d\\sigma _{\\varphi ,\\lambda }.$ This function is called the action of $\\varphi $ with respect to $\\lambda $ .","Its value at fixed points of $\\varphi $ is independent of the choice of the primitive $\\lambda $ , and so is its integral $\\mathrm {CAL}(\\varphi ) = \\int _{\\mathbb {R}^2} \\sigma _{\\varphi ,\\lambda } \\, dx\\wedge dy,$ which is called the Calabi invariant of $\\varphi $ .","The Calabi invariant defines a homomorphism from $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ onto $\\mathbb {R}$ .","Let $\\lambda _0$ be the following radially symmetric primitive of $dx\\wedge dy$ $\\lambda _0 := \\frac{1}{2} ( x\\, dy - y \\, dx).$ We shall make use of the following result: Proposition 3 Fix positive numbers $r$ and $L$ .","For every $\\epsilon >0$ there exists a positive integer $n$ and an area-preserving diffeomorphism $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ such that the following properties hold: $\\sigma _{\\varphi ,\\lambda _0} \\ge - L + L/n $ ; $\\mathrm {CAL}(\\varphi )< - L\\pi r^2 + \\epsilon $ ; all the fixed points of $\\varphi $ have non-negative action; all the periodic points of $\\varphi $ which are not fixed points have period at least $n$ .","This result is proved in [3] for $r=1$ .","The general case follows by a simple rescaling argument, using that if $\\varphi $ is in $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ then the rescaled map $\\varphi _r(z) := r \\varphi \\left( \\frac{z}{r} \\right)$ is also in $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ and $\\sigma _{\\varphi _r,\\lambda _0}(z) = r^2 \\sigma _{\\varphi ,\\lambda _0}\\left( \\frac{z}{r} \\right), \\qquad \\mathrm {CAL}(\\varphi _r)= r^4 \\, \\mathrm {CAL}(\\varphi ).$ The last ingredient which we need is the following result from [3][Proposition 3.1], which allows us to lift an area preserving diffeomorphism of a disc to a Reeb flow on a solid torus.","Proposition 4 Fix positive numbers $r$ and $L$ .","Let $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ and assume that the function $\\tau := \\sigma _{\\varphi ,\\lambda _0} + L$ is positive on $r.","Then there exists a smooth contact form $$ on the solid torus $ rR/L Z$ with the following properties:\\begin{enumerate}[({c}1)]\\item \\beta = \\lambda _0 + ds in a neighborhood of \\partial (r \\times \\mathbb {R}/L \\mathbb {Z}, where s denotes the coordinate on \\mathbb {R}/L \\mathbb {Z}; in particular, the Reeb vector field R_{\\beta } of \\beta coincides with \\partial _s near the boundary of r\\mathbb {R}/L \\mathbb {Z}, and its flow is globally well-defined;\\item the surface r\\lbrace 0\\rbrace is transverse to the flow of R_{\\beta }, and the orbit of every point in r\\mathbb {R}/L \\mathbb {Z} intersects r\\lbrace 0\\rbrace both in the future and in the past;\\item the first return map and the first return time of the flow of R_{\\beta } associated to the surface r\\lbrace 0\\rbrace \\cong r are the map \\varphi and the function \\tau ;\\item \\mathrm {vol}\\,(r\\mathbb {R}/L \\mathbb {Z}, \\beta \\wedge d\\beta ) = L \\,\\pi r^2+ \\mathrm {CAL}(\\varphi );\\item the contact form \\beta is smoothly isotopic to \\lambda _0 + ds on r\\mathbb {R}/L \\mathbb {Z} through a path of contact forms which agree with \\lambda _0 + ds in a neighborhood of \\partial (r \\times \\mathbb {R}/L \\mathbb {Z}.\\end{enumerate}$ Again, Proposition 3.1 in [3] is stated for $r=1$ , but the general case follows by rescaling.","In fact consider $r,L$ and $\\varphi $ as in the statement of Proposition REF .","Then, as explained above, we have $\\sigma _{\\varphi _{r^{-1}},\\lambda _0}(z) = r^{-2} \\sigma _{\\varphi ,\\lambda _0}\\left( rz \\right)$ .","Applying [3] to the map $\\varphi _{r^{-1}}$ with $Lr^{-2}$ in the place of $L$ we get a contact form $\\beta _{r^{-1}}$ on $\\mathbb {R}/Lr^{-2}\\mathbb {Z}$ satisfying the desired conclusions.","Consider now the diffeomorphism $\\Phi _{r^{-1}}:r\\mathbb {R}/L\\mathbb {Z}\\rightarrow \\mathbb {R}/Lr^{-2}\\mathbb {Z}$ defined as $(z,s) \\mapsto (r^{-1}z,r^{-2}s)$ .","Direct calculations reveal that $\\beta = r^2\\Phi _{r^{-1}}^*\\beta _{r^{-1}}$ is the desired contact form.","The statement of [3] is actually slightly more general, since it allows $\\lambda _0$ to be replaced by a more general primitive of $dx\\wedge dy$ , and gives more properties of the contact form $\\beta $ .","Building on the above two propositions, it is now easy to prove Proposition REF .","The first step is to prove the existence of a contact form $\\beta _0$ on $r\\mathbb {R}/\\mathbb {Z}$ which satisfies the required conditions (a1)-(a4), but where in (a1) and (a2) the primitive $\\lambda $ of $dx\\wedge dy$ is the radially symmetric primitive $\\lambda _0$ .","Let $n\\in \\mathbb {N}$ , $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ be given by an application of Proposition REF with $L=1$ .","By statement (b1) in this proposition, we have the lower bound $1 + \\sigma _{\\varphi ,\\lambda _0} \\ge \\frac{1}{n}.$ In particular, the function $\\tau := 1 + \\sigma _{\\varphi ,\\lambda _0}$ is everywhere positive.","Then Proposition REF implies the existence of a contact form $\\beta _0$ on the solid torus $r\\mathbb {D} \\times \\mathbb {R}/\\mathbb {Z}$ which satisfies the conditions (c1)-(c5) with $L=1$ .","We just need to check that $\\beta _0$ satisfies (a1)-(a4) with respect to $\\lambda _0$ .","Conditions (a1) and (a2) are precisely (c1) and (c5).","Condition (a4) follows from (c4) and (b2): $\\mathrm {vol}\\,(r\\mathbb {R}/ \\mathbb {Z}, \\beta _0\\wedge d\\beta _0) \\stackrel{(c4)}{=} \\pi r^2 + \\mathrm {CAL}(\\varphi ) \\stackrel{(b2)}{<} \\epsilon .$ There remains to prove that all closed orbits of the Reeb flow of $\\beta _0$ have period at least 1.","By (c2) and (c3), closed orbits of $R_{\\beta _0}$ are in one-to-one correspondence with periodic points of $\\varphi $ , and if $z\\in r\\mathbb {D}$ is a periodic point of $\\varphi $ with minimal period $k\\in \\mathbb {N}$ , then the corresponding closed orbit has period $T:= \\sum _{j=0}^{k-1} \\tau (\\varphi ^j(z)).$ When $k=1$ , $z$ is a fixed point of $\\varphi $ and by (b3) we have $T=\\tau (z) = 1 + \\sigma _{\\varphi ,\\lambda _0}(z) \\ge 1.$ By condition (b4), all periodic points of $\\varphi $ which are not fixed points have period $k\\ge n$ .","If $z$ is such a point, condition (b1) gives us $T = \\sum _{j=0}^{k-1} \\tau (\\varphi ^j(z)) = \\sum _{j=0}^{k-1} \\bigl (1+\\sigma _{\\varphi ,\\lambda _0}(\\varphi ^j(z))\\bigr ) \\ge \\sum _{j=0}^{k-1} \\frac{1}{n} = \\frac{k}{n} \\ge 1.$ This proves (a3) and concludes the first step.","Now we wish to modify $\\beta _0$ in a neighborhood of the boundary in order to obtain (a1) and (a2) with respect to the given primitive $\\lambda $ of $dx\\wedge dy$ , while keeping conditions (a3) and (a4).","The 1-form $\\lambda -\\lambda _0$ is closed and hence exact on $r.","Let $ u$ be a smooth function on $ r such that $\\lambda - \\lambda _0 = du.$ Let $\\lbrace \\beta _0^t\\rbrace _{t\\in [0,1]}$ be a smooth path of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ going from $\\beta _0$ to $\\lambda _0+ds$ and agreeing with $\\lambda _0+ds$ in a neighborhood $(rr^{\\prime }\\times \\mathbb {R}/\\mathbb {Z}$ of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ , where $r^{\\prime }\\in (0,r)$ .","Let $\\chi $ be a smooth function on $r such that $ =0$ on $ r' and $\\chi =1$ on a neighborhood of $\\partial (r$ .","Consider the following smooth contact form on $r\\mathbb {R}/\\mathbb {Z}$ : $\\beta := \\beta _0 + d(\\chi u).$ This contact form satisfies (a1) with respect to $\\lambda $ .","The formula $\\beta ^t := \\beta _0^t + d(\\chi u)$ defines a smooth path of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ going from $\\beta $ to the 1-form $\\lambda _0 + ds+d(\\chi u),$ and agreeing with $\\lambda +ds$ in a neighborhood of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ .","The latter contact form can be joined to $\\lambda + ds$ by the smooth isotopy $\\lambda _0 + t\\, du + (1-t) d(\\chi u) + ds,$ which consists of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ agreeing with $\\lambda +ds$ in a neighborhood of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ .","This proves that $\\beta $ satisfies (a2) with respect to $\\lambda $ .","The volume form induced by $\\beta $ is $\\beta \\wedge d\\beta = \\beta _0 \\wedge d\\beta _0 + d(\\chi u) \\wedge d\\beta _0.$ The 3-form $d(\\chi u) \\wedge d\\beta _0$ vanishes identically, because $d(\\chi u)$ is supported in the region in which $d\\beta _0=d\\lambda _0$ and the contractions of this 1-form and this 2-form along $\\partial _s$ are both zero.","We deduce that $\\beta \\wedge d\\beta = \\beta _0 \\wedge d\\beta _0$ on the whole $r\\mathbb {R}/\\mathbb {Z}$ , and the fact that $\\beta _0$ satisfies (a4) implies that also $\\beta $ does.","The fact that $\\beta $ differs from $\\beta _0$ by an exact 1-form which vanishes along the direction of $R_{\\beta _0}$ implies that the Reeb vector field of $\\beta $ coincides with the one of $\\beta _0$ .","Therefore, the fact that $\\beta _0$ satisfies (a3) implies that also $\\beta $ does."],["Proof of the main theorem","We are now ready to prove the theorem stated in the introduction.","Let $\\xi $ be a contact structure on the closed three-manifold $M$ .","Let $S$ be the smoothly embedded compact surface $S\\subset M$ given by Proposition REF and let $\\epsilon <1$ be a fixed positive number.","By Proposition REF we can find a contact form $\\tilde{\\alpha }$ on $M$ such that $\\ker \\tilde{\\alpha } = \\xi $ , $S$ is a global surface of section for the Reeb flow of $\\tilde{\\alpha }$ and statements (i)-(iv) hold, where $\\alpha $ is renamed as $\\tilde{\\alpha }$ .","In the following, $U$ , $\\varphi $ and $\\tau $ are the objects appearing in this proposition.","Statements labeled by a roman number refer to the statements of this proposition.","Denote by $C_1,\\dots ,C_{\\ell }$ the circles forming the boundary of $S$ .","Each connected component of $U$ is a half-open annulus $A_j$ and $d\\tilde{\\alpha }$ restricts to an area form on the interior of each $A_j$ .","Let $a_j>0$ be the $d\\tilde{\\alpha }$ -area of each $A_j$ .","By (i) and (iii) we have $\\ell = \\int _S d\\tilde{\\alpha } = \\int _{S\\setminus U} d\\tilde{\\alpha }+ \\sum _{j=1}^{\\ell } \\int _{A_j} d\\tilde{\\alpha } < \\epsilon + \\sum _{j=1}^{\\ell } a_j.$ For any $j\\in \\lbrace 1,\\dots ,\\ell \\rbrace $ , let $r_j>0$ be such that $\\pi r_j^2 = (1-\\epsilon ) a_j,$ and let $\\psi _j: r_j \\mathbb {D} \\hookrightarrow A_j \\setminus \\partial S$ be a smooth embedding such that $\\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.","Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\Psi _j: r_j \\mathbb {R}/\\mathbb {Z}\\hookrightarrow M$$such that\\begin{equation}\\Psi _j(\\cdot ,0) = \\psi _j \\qquad \\mbox{and} \\qquad \\Psi _j^*(R_{\\tilde{\\alpha }}) = \\partial _s,\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\begin{equation}\\Psi _j^*(\\tilde{\\alpha }) = \\lambda _j + ds\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.","Indeed, set for simplicity $\\tilde{\\alpha }_j:= \\Psi _j^*(\\tilde{\\alpha })$ .","The the second identity in () implies that the Reeb vector field of $\\tilde{\\alpha }_j$ is $\\partial _s$ .","Since any contact form is invariant by its Reeb flow, $\\tilde{\\alpha }_j$ is invariant under the translations $(x,y,s) \\mapsto (x,y,s+t)$ and hence has the form $\\tilde{\\alpha }_j = \\lambda _j + u_j(x,y)\\, ds,$ where $\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.","Then the condition $\\tilde{\\alpha }_j(\\partial _s)=1$ implies that $u_j=1$ .","Finally, the first identity in () implies that $d\\lambda _j = d\\tilde{\\alpha }_j|_{r_j \\lbrace 0\\rbrace } = \\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy,$ which concludes the proof of the claim.","Denote by $W_j$ the image of the embedding $\\Psi _j$ .","Its volume with respect to $\\tilde{\\alpha } \\wedge d\\tilde{\\alpha }$ is $\\begin{split}\\mathrm {vol}\\,( W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) &=\\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, \\tilde{\\alpha }_j \\wedge d\\tilde{\\alpha }_j ) = \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, d\\lambda _j\\wedge ds) \\\\ &= \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z},dx\\wedge dy\\wedge ds) = \\pi r_j^2 = (1-\\epsilon )a_j.\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }$ has the upper bound $\\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) = \\int _S \\tau \\, d\\tilde{\\alpha } \\le (1+\\epsilon ) \\int _S d\\tilde{\\alpha } = (1+\\epsilon ) \\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\begin{split}\\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } &W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) = \\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) - \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(W_j,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) \\\\&\\le (1+\\epsilon ) \\ell - (1-\\epsilon ) \\sum _{j=1}^{\\ell } a_j < (1+\\epsilon ) \\ell - (1-\\epsilon )(\\ell - \\epsilon ) \\\\ &= \\epsilon (2\\ell +1) - \\epsilon ^2 < \\epsilon (2\\ell +1),\\end{split}$ where the second inequality follows from (REF ).","Thanks to Proposition REF , on every solid torus $r_j \\mathbb {R}/\\mathbb {Z}$ there is a contact form $\\beta _j$ which agrees with $\\lambda _j+ds$ near the boundary, has total volume less than $\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.","Moreover, $\\beta _j$ is smoothly isotopic to $\\lambda _j+ds$ through contact forms which agree with $\\lambda _j+ds$ near the boundary.","Let $\\alpha $ be the contact form on $M$ which coincides with $\\tilde{\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\beta _j$ by the embedding $\\Psi _j$ .","This form is indeed smooth due to ().","It is smoothly isotopic to the contact form $\\tilde{\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\xi =\\ker \\tilde{\\alpha }$ .","By (REF ) and the fact that the volume of each $r_j \\mathbb {R}/\\mathbb {Z}$ with respect to $\\beta _j\\wedge d\\beta _j$ is smaller than $\\epsilon $ , the volume of $M$ with respect to $\\alpha \\wedge d\\alpha $ has the upper bound $\\begin{split}\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) &= \\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) + \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(r_j \\mathbb {R}/\\mathbb {Z},\\beta _j\\wedge d\\beta _j) \\\\ &< \\epsilon (2\\ell +1) + \\epsilon \\ell = \\epsilon (3\\ell +1).\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\alpha $ .","The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\beta _j$ 's.","The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .","Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\alpha $ have period larger than $1-\\epsilon $ .","We conclude that $T_{\\min }(\\alpha ) > 1-\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\alpha $ has the lower bound $\\rho _{\\mathrm {sys}}(\\alpha ) = \\frac{T_{\\min }(\\alpha )^2}{\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha )} > \\frac{(1-\\epsilon )^2}{\\epsilon (3\\ell +1)}.$ Since the latter quantity tends to $+\\infty $ for $\\epsilon \\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\xi $ on $M$ can be made arbitrarily large.","This concludes the proof of the theorem stated in the introduction.","We are now ready to prove the theorem stated in the introduction.","Let $\\xi $ be a contact structure on the closed three-manifold $M$ .","Let $S$ be the smoothly embedded compact surface $S\\subset M$ given by Proposition REF and let $\\epsilon <1$ be a fixed positive number.","By Proposition REF we can find a contact form $\\tilde{\\alpha }$ on $M$ such that $\\ker \\tilde{\\alpha } = \\xi $ , $S$ is a global surface of section for the Reeb flow of $\\tilde{\\alpha }$ and statements (i)-(iv) hold, where $\\alpha $ is renamed as $\\tilde{\\alpha }$ .","In the following, $U$ , $\\varphi $ and $\\tau $ are the objects appearing in this proposition.","Statements labeled by a roman number refer to the statements of this proposition.","Denote by $C_1,\\dots ,C_{\\ell }$ the circles forming the boundary of $S$ .","Each connected component of $U$ is a half-open annulus $A_j$ and $d\\tilde{\\alpha }$ restricts to an area form on the interior of each $A_j$ .","Let $a_j>0$ be the $d\\tilde{\\alpha }$ -area of each $A_j$ .","By (i) and (iii) we have $\\ell = \\int _S d\\tilde{\\alpha } = \\int _{S\\setminus U} d\\tilde{\\alpha }+ \\sum _{j=1}^{\\ell } \\int _{A_j} d\\tilde{\\alpha } < \\epsilon + \\sum _{j=1}^{\\ell } a_j.$ For any $j\\in \\lbrace 1,\\dots ,\\ell \\rbrace $ , let $r_j>0$ be such that $\\pi r_j^2 = (1-\\epsilon ) a_j,$ and let $\\psi _j: r_j \\mathbb {D} \\hookrightarrow A_j \\setminus \\partial S$ be a smooth embedding such that $\\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.","Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\Psi _j: r_j \\mathbb {R}/\\mathbb {Z}\\hookrightarrow M$$such that\\begin{equation}\\Psi _j(\\cdot ,0) = \\psi _j \\qquad \\mbox{and} \\qquad \\Psi _j^*(R_{\\tilde{\\alpha }}) = \\partial _s,\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\begin{equation}\\Psi _j^*(\\tilde{\\alpha }) = \\lambda _j + ds\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.","Indeed, set for simplicity $\\tilde{\\alpha }_j:= \\Psi _j^*(\\tilde{\\alpha })$ .","The the second identity in () implies that the Reeb vector field of $\\tilde{\\alpha }_j$ is $\\partial _s$ .","Since any contact form is invariant by its Reeb flow, $\\tilde{\\alpha }_j$ is invariant under the translations $(x,y,s) \\mapsto (x,y,s+t)$ and hence has the form $\\tilde{\\alpha }_j = \\lambda _j + u_j(x,y)\\, ds,$ where $\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.","Then the condition $\\tilde{\\alpha }_j(\\partial _s)=1$ implies that $u_j=1$ .","Finally, the first identity in () implies that $d\\lambda _j = d\\tilde{\\alpha }_j|_{r_j \\lbrace 0\\rbrace } = \\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy,$ which concludes the proof of the claim.","Denote by $W_j$ the image of the embedding $\\Psi _j$ .","Its volume with respect to $\\tilde{\\alpha } \\wedge d\\tilde{\\alpha }$ is $\\begin{split}\\mathrm {vol}\\,( W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) &=\\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, \\tilde{\\alpha }_j \\wedge d\\tilde{\\alpha }_j ) = \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, d\\lambda _j\\wedge ds) \\\\ &= \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z},dx\\wedge dy\\wedge ds) = \\pi r_j^2 = (1-\\epsilon )a_j.\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }$ has the upper bound $\\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) = \\int _S \\tau \\, d\\tilde{\\alpha } \\le (1+\\epsilon ) \\int _S d\\tilde{\\alpha } = (1+\\epsilon ) \\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\begin{split}\\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } &W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) = \\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) - \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(W_j,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) \\\\&\\le (1+\\epsilon ) \\ell - (1-\\epsilon ) \\sum _{j=1}^{\\ell } a_j < (1+\\epsilon ) \\ell - (1-\\epsilon )(\\ell - \\epsilon ) \\\\ &= \\epsilon (2\\ell +1) - \\epsilon ^2 < \\epsilon (2\\ell +1),\\end{split}$ where the second inequality follows from (REF ).","Thanks to Proposition REF , on every solid torus $r_j \\mathbb {R}/\\mathbb {Z}$ there is a contact form $\\beta _j$ which agrees with $\\lambda _j+ds$ near the boundary, has total volume less than $\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.","Moreover, $\\beta _j$ is smoothly isotopic to $\\lambda _j+ds$ through contact forms which agree with $\\lambda _j+ds$ near the boundary.","Let $\\alpha $ be the contact form on $M$ which coincides with $\\tilde{\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\beta _j$ by the embedding $\\Psi _j$ .","This form is indeed smooth due to ().","It is smoothly isotopic to the contact form $\\tilde{\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\xi =\\ker \\tilde{\\alpha }$ .","By (REF ) and the fact that the volume of each $r_j \\mathbb {R}/\\mathbb {Z}$ with respect to $\\beta _j\\wedge d\\beta _j$ is smaller than $\\epsilon $ , the volume of $M$ with respect to $\\alpha \\wedge d\\alpha $ has the upper bound $\\begin{split}\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) &= \\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) + \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(r_j \\mathbb {R}/\\mathbb {Z},\\beta _j\\wedge d\\beta _j) \\\\ &< \\epsilon (2\\ell +1) + \\epsilon \\ell = \\epsilon (3\\ell +1).\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\alpha $ .","The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\beta _j$ 's.","The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .","Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\alpha $ have period larger than $1-\\epsilon $ .","We conclude that $T_{\\min }(\\alpha ) > 1-\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\alpha $ has the lower bound $\\rho _{\\mathrm {sys}}(\\alpha ) = \\frac{T_{\\min }(\\alpha )^2}{\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha )} > \\frac{(1-\\epsilon )^2}{\\epsilon (3\\ell +1)}.$ Since the latter quantity tends to $+\\infty $ for $\\epsilon \\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\xi $ on $M$ can be made arbitrarily large.","This concludes the proof of the theorem stated in the introduction."]],"string":"[\n [\n \"Contact forms with large systolic ratio in dimension three\"\n ],\n [\n \"Abstract The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume.\",\n \"We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio.\",\n \"This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena.\"\n ],\n [\n \"Introduction and main result\",\n \"Given a Riemannian metric $g$ on a closed surface $S$ , we denote the length of a shortest non-constant closed geodesic by $\\\\ell _{\\\\min }(S,g)$ and consider the scaling invariant ratio $\\\\rho (S,g) := \\\\frac{\\\\ell _{\\\\min }(S,g)^2}{\\\\mathrm {area}(S,g)},$ where $\\\\mathrm {area}(S,g)$ denotes the Riemannian area.\",\n \"It is a classical question in systolic geometry to find upper bounds, and possibly optimal ones, for this ratio.\",\n \"When $S$ is not the 2-sphere, the fact that $S$ is not simply connected allows one to bound $\\\\rho (S,g)$ from above by the ratio $\\\\rho _{\\\\rm nc}(S,g) :=\\\\frac{\\\\mathrm {sys}_1(S,g)^2}{\\\\mathrm {area}(S^2,g)},$ where $\\\\mathrm {sys}_1(S,g)$ denotes the length of a shortest non-contractible curve on $(S,g)$ , which is of course a closed geodesic.\",\n \"When $S$ is the 2-torus 2, a classical result of Loewner from the end of the 1940s states that $\\\\rho _{\\\\rm nc}(2,\\\\cdot )$ is maximised by the flat metric corresponding to the lattice generated by two sides of an equilateral triangle in $\\\\mathbb {R}^2$ .\",\n \"The corresponding maximal value of $\\\\rho _{\\\\rm nc}$ is $2/\\\\sqrt{3}$ .\",\n \"Shortly after, Pu [15] considered the case of the projective plane $\\\\mathbb {R}\\\\mathbb {P}^2$ and showed that $\\\\rho _{\\\\rm nc}(\\\\mathbb {R}\\\\mathbb {P}^2,\\\\cdot )$ is maximized by the standard constant curvature metric, with maximal value $\\\\pi /2$ .\",\n \"In both cases, the metrics maximizing $\\\\rho _{\\\\rm nc}$ have no contractible closed geodesics, so these metrics maximize also the ratio $\\\\rho $ .\",\n \"For an arbitrary closed non-simply connected surface $S$ , the ratio $\\\\rho _{\\\\rm nc}(S,\\\\cdot )$ is bounded from above by the non-optimal universal constant 2, as shown by Gromov in [12], but the supremum is in general not achieved.\",\n \"Actually, in [12] Gromov proved a far reaching generalization of this result by showing that for any essential $n$ -dimensional closed Riemannian manifold $(M,g)$ , the quotient $\\\\rho _{\\\\rm nc}(M,g) := \\\\frac{\\\\mathrm {sys}_1(M,g)^n}{\\\\mathrm {vol}(M,g)}$ has an upper bound which depends only on the dimension $n$ .\",\n \"Here, a non-simply connected closed oriented manifold $M$ is said to be essential if its fundamental class is non-zero in the homology of the Eilenberg-MacLane space $K(\\\\pi _1(M),1)$ .\",\n \"This condition generalizes asphericity, that is the fact that all homotopy groups of degree larger than one vanish, and characterizes manifolds for which the ratio $\\\\rho _{\\\\rm nc}$ is bounded from above, see [6].\",\n \"Going back to surfaces, from the fact mentioned above we deduce that the number 2 is an upper bound also for $\\\\rho (S,\\\\cdot )$ , when $S\\\\ne S^2$ .\",\n \"In the case of the two-sphere $S^2$ , the first upper bound for $\\\\rho (S^2,\\\\cdot )$ was found by Croke in [7] and later improved by several authors, the best one so far being the bound 32 found by Rotman in [16].\",\n \"We conclude that $\\\\rho (S,\\\\cdot )$ is always bounded from above when $S$ is a closed surface, and there is an upper bound which is independent of $S$ .\",\n \"The ratios $\\\\rho $ and $\\\\rho _{\\\\rm nc}$ generalize to Finsler metrics, by replacing the Riemannian area by the Holmes-Thompson area.\",\n \"Recent results of Àlvarez Paiva, Balacheff and Tzanev allow to extend the Riemannian bounds to the Finsler setting: The value of the supremum might get larger (and sometimes it does get larger, as in the case of 2, for which the Loewner metric does not maximize $\\\\rho $ and $\\\\rho _{\\\\rm nc}$ , even locally, among Finsler metrics) but is in any case finite.\",\n \"See [5].\",\n \"The next natural generalization of the ratio $\\\\rho $ is to the setting of contact geometry.\",\n \"We recall that a contact form $\\\\alpha $ on a 3-manifold $M$ is a smooth 1-form such that $\\\\alpha \\\\wedge d\\\\alpha $ is a volume form.\",\n \"The kernel of $\\\\alpha $ is a plane distribution and is called the contact structure defined by $\\\\alpha $ .\",\n \"In this paper, contact structures are always assumed to be co-orientable, that is, defined by a global contact form.\",\n \"If two contact forms $\\\\alpha _1,\\\\alpha _2$ on $M$ define the same contact structure then the volume forms $\\\\alpha _1\\\\wedge d\\\\alpha _1$ and $\\\\alpha _2\\\\wedge d\\\\alpha _2$ define the same orientation.\",\n \"This orientation is said to be induced by the contact structure, and in this paper all contact 3-manifolds are oriented by the contact structure.\",\n \"A contact form $\\\\alpha $ induces a nowhere vanishing vector field $R_{\\\\alpha }$ on $M$ , which is defined by the conditions $\\\\imath _{R_{\\\\alpha }} d\\\\alpha = 0 , \\\\qquad \\\\alpha (R_{\\\\alpha })=1,$ and is called the Reeb vector field for $\\\\alpha $ .\",\n \"Any Reeb vector field on a closed 3-manifold $M$ admits periodic orbits, as proved by Taubes in [17], and we denote by $T_{\\\\min }(M,\\\\alpha )$ the minimum among the periods of all periodic orbits of $R_{\\\\alpha }$ .\",\n \"Then it is natural to define $\\\\rho (M,\\\\alpha ):= \\\\frac{T_{\\\\min }(M,\\\\alpha )^2}{\\\\mathrm {vol}(M,\\\\alpha \\\\wedge d\\\\alpha )}.$ We will refer to this quantity as the systolic ratio of $(M,\\\\alpha )$ .\",\n \"It is scaling invariant, because if we multiply $\\\\alpha $ by a non-zero real number $c$ then both $T_{\\\\min }^2$ and the volume get multiplied by $c^2$ .\",\n \"The inverse of this quantity appears in [4] under the name of systolic volume of $(M,\\\\alpha )$ .\",\n \"Let $F$ be a smooth Finsler metric on the closed surface $S$ .\",\n \"The unit cotangent sphere bundle $S^*_FS$ is the space of cotangent vectors having norm 1 with respect to the dual Finsler metric $F^*$ .\",\n \"The canonical Liouville 1-form $p\\\\, dq$ of $T^* S$ restricts to a contact form $\\\\alpha _F$ on $S_F^*S$ .\",\n \"Two different Finsler metrics $F$ and $F^{\\\\prime }$ induce contact forms $\\\\alpha _F$ and $\\\\alpha _{F^{\\\\prime }}$ which correspond to the same contact structure, once $S_F^*S$ and $S_{F^{\\\\prime }}^*S$ are identified by means of the radial projection.\",\n \"The flow of the corresponding Reeb vector field $R_{\\\\alpha _F}$ is precisely the geodesic flow of $F$ , once this is read on the cotangent bundle by the Legendre transform.\",\n \"Therefore, $\\\\ell _{\\\\min }(S,F)=T_{\\\\min }(\\\\alpha _F)$ .\",\n \"Moreover, the volume of $S^*_FS$ with respect to $\\\\alpha _F \\\\wedge d\\\\alpha _F$ is, essentially by definition, $2\\\\pi $ times the Holmes-Thompson area of $(S,F)$ .\",\n \"We conclude that $\\\\rho (S^*_FS,\\\\alpha _F) = \\\\frac{\\\\rho (S,F)}{2\\\\pi },$ and the systolic ratio of a contact form is a genuine generalization of the corresponding notion from Riemannian and Finsler geometry.\",\n \"All of this generalizes to higher dimensions, but here we restrict our attention to three-dimensional contact manifolds.\",\n \"Seeing Riemannian and Finsler geometry in the larger contact setting is often fruitful.\",\n \"On the one hand, results about existence and multiplicity of Riemannian and Finsler closed geodesics have often natural generalizations to Reeb flows, see e.g.\",\n \"[13], and the same holds for some statements about the topological entropy of geodesic flows, see e.g.\",\n \"[14], [9], [1].\",\n \"On the other hand, techniques from contact geometry have been recently found to be useful to address systolic questions in Riemannian and Finlser geometry, such as the local systolic maximality of Zoll metrics, see [4], [2], [3].\",\n \"It is therefore a natural question to ask whether the systolic ratio of contact forms inducing a given contact structure on a closed three-manifold also has uniform upper bounds.\",\n \"The purpose of this paper is to give a negative answer to this question.\",\n \"More precisely, we shall prove the following: Theorem Let $\\\\xi $ be a contact structure on a closed 3-manifold $M$ .\",\n \"For every $c>0$ there exists a contact form $\\\\alpha $ satisfying $\\\\ker \\\\alpha =\\\\xi $ and $\\\\rho (M,\\\\alpha )\\\\ge c.$ When applied to the cotangent sphere bundle of an arbitrary closed surface $S$ , the above theorem has the following consequence: If $T$ is any positive number, then there exists a fiberwise starshaped domain $A\\\\subset T^*S$ (i.e.\",\n \"a connected open neighborhood of the zero-section with smooth boundary $\\\\partial A$ such that for every $q\\\\in S$ the set $\\\\partial A \\\\cap T_q^* S$ is a closed curve in $T^*_q S\\\\setminus \\\\lbrace 0\\\\rbrace $ which is transverse to the radial direction) with volume 1 (with respect to the standard symplectic form $dp\\\\wedge dq$ of $T^*S$ ) and such that every closed orbit of the Reeb vector field on $\\\\partial A$ determined by the restriction of the Liouville form $p\\\\, dq$ has period larger than $T$ .\",\n \"When $A$ is fiberwise convex then $\\\\partial A$ is the unit sphere bundle of some Finsler metric and the results of Àlvarez Paiva, Balacheff and Tzanev mentioned above prevent this phenomenon.\",\n \"This shows that convexity plays an essential role in systolic inequalities.\",\n \"In the special case of the tight three-sphere $(S^3,\\\\xi _{\\\\rm std})$ , that is the standard unit sphere in $\\\\mathbb {R}^4$ endowed with coordinaten $(x_1,y_1,x_2,y_2)$ and with the contact structure $\\\\xi _{\\\\rm std}$ which is given by the kernel of the contact form $\\\\alpha _0 = \\\\frac{1}{2} \\\\sum _{j=1}^2 (x_j \\\\, dy_j - y_j\\\\, dx_j) \\\\Big |_{S^3}$ the unboundedness of the systolic ratio was proved in [3].\",\n \"The argument in [3] starts with the construction of a special symplectomorphism of the disk with a good lower bound on actions of periodic points and with a suitable negative value of the Calabi invariant (see Section and references therein for the definition of these notions).\",\n \"This disk-map is then embedded as a return map to a disk-like global surface of section of some Reeb flow on $(S^3,\\\\xi _{\\\\rm std})$ in a way that there is a precise “dictionary”: Actions of periodic points correspond to periods of closed Reeb orbits and the Calabi invariant corresponds to the contact volume, up to explicit additive constants.\",\n \"The tight three-sphere is very simple from a contact-topological point of view, since it admits the trivial supporting open book decomposition with unknotted binding, disk-like pages and trivial monodromy (see Section below and references therein for the definition of these notions).\",\n \"Moreover, the construction of the disk-map in [3] uses the special symmetries of the disk.\",\n \"In order to deal with a general contact structure $\\\\xi $ on a general orientable closed three-manifold $M$ we argue in the following way.\",\n \"First we use the fact that by a theorem of Giroux [11] $\\\\xi $ is supported by an open book decomposition of $M$ to construct a contact form $\\\\alpha $ which is adapted to the open book decomposition and is such that the supports of the monodromy and return maps are contained in a region of small contact area, as visualized in Figure 1.\",\n \"The construction of this special contact form is explained in Section , where the necessary background about open book decompositions is also recalled.\",\n \"The next step is to construct suitable contact forms on a solid torus which are standard near the boundary, have a small contact volume and no closed Reeb orbits with small period.\",\n \"These are constructed in Section , by building on results from [3].\",\n \"Finally, the latter solid tori are used as plugs to modify the contact form $\\\\alpha $ constructed in Section .\",\n \"Indeed, these plugs can be inserted in the region spanned by the large portions of the pages on which the monodromy and first return map are the identity: Their effect is to eat up volume without creating short periodic orbits.\",\n \"In this way we can reduce the contact volume as much as we wish, while keeping $T_{\\\\min }$ bounded away from zero, and hence making the systolic ratio arbitrarily large.\",\n \"This final step is performed in Section .\"\n ],\n [\n \"The research of A. Abbondandolo and B. Bramham is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the Deutsche Forschungsgemeinschaft.\",\n \"U. Hryniewicz thanks the Floer Center for Geometry (Bochum) for its warm hospitality, and acknowledges the generous support of the Alexander von Humboldt Foundation.\",\n \"U. Hryniewicz is also supported by CNPq grant 309966/2016-7.\",\n \"P. Salomão is supported by FAPESP grant 2016/25053-8 and CNPq grant 306106/2016-7.\",\n \"Figure: On the left a typical page as one would picture it in an open book decomposition.\",\n \"On the right, the same page on an open book supporting a contact structure where the topology, and the supports of the monodromy and return maps are squeezed in a region with small contact area.\",\n \"This leaves lots of space to embed solid tori with high systolic ratio.\"\n ],\n [\n \"A special contact form\",\n \"We recall that a global surface of section for a smooth flow $\\\\phi ^t$ on a 3-dimensional manifold $M$ is a compact surface $S$ smoothly embedded in $M$ whose boundary $\\\\partial S$ is $\\\\phi ^t$ -invariant and such that all orbits which are not contained in $\\\\partial S$ meet the surface transversally infinitely many times in the future and in the past.\",\n \"The corresponding first return time function is the smooth function $\\\\tau : S \\\\setminus \\\\partial S \\\\rightarrow (0,+\\\\infty ), \\\\qquad \\\\tau (p) := \\\\inf \\\\lbrace t>0 \\\\mid \\\\phi ^t(p)\\\\in S\\\\rbrace ,$ and the corresponding first return map is the smooth diffeomorphism $\\\\varphi : S \\\\setminus \\\\partial S \\\\rightarrow S \\\\setminus \\\\partial S, \\\\qquad \\\\varphi (p) := \\\\phi ^{\\\\tau (p)}(p).$ When $\\\\phi ^t$ is the Reeb flow of a contact form $\\\\alpha $ , the transversality of the flow to $S\\\\setminus \\\\partial S$ implies that the restriction of $d\\\\alpha $ to $S\\\\setminus \\\\partial S$ is an area form.\",\n \"The exactness of $d\\\\alpha $ implies that in the Reeb case the boundary of $S$ is necessarily non-empty.\",\n \"Since Reeb vector fields are non-singular, the finitely many circles forming the boundary of $S$ are periodic orbits.\",\n \"Moreover, it is well known that the contact volume of $M$ coincides with the integral of $\\\\tau $ on $S$ , once this surface is equipped with the area form given by the restriction of $\\\\alpha $ : $\\\\mathrm {vol}\\\\,(M,\\\\alpha \\\\wedge d\\\\alpha ) = \\\\int _S \\\\tau \\\\, d\\\\alpha .$ See e.g.\",\n \"[3] for the easy proof.\",\n \"Proposition 1 Let $M$ be a closed connected 3-manifold with a contact structure $\\\\xi $ .\",\n \"Then there is an embedded compact surface $S\\\\subset M$ with the following property: For every $\\\\epsilon >0$ there exists a contact form $\\\\alpha $ on $M$ satisfying $\\\\xi =\\\\ker \\\\alpha $ such that $S$ is a global surface of section for the Reeb flow of $\\\\alpha $ and: If we orient $S$ by $d\\\\alpha $ and give $\\\\partial S$ the boundary orientation, we have that $\\\\int _C \\\\alpha = 1$ for every connected component $C$ of $\\\\partial S$ .\",\n \"In particular, $\\\\int _S d\\\\alpha = \\\\int _{\\\\partial S} \\\\alpha = \\\\ell ,$ where $\\\\ell \\\\ge 1$ denotes the number of circles forming the boundary of $S$ .\",\n \"The first return time function $\\\\tau $ of the Reeb flow of $\\\\alpha $ extends to a smooth function on $S$ and the corresponding first return map $\\\\varphi $ extends to a smooth diffeomorphism of $S$ onto itself.\",\n \"There exists an open tubular neighborhood $U$ of $\\\\partial S$ in $S$ such that the support of $\\\\varphi $ is contained in $S\\\\setminus U$ and $\\\\int _{S\\\\setminus U} d\\\\alpha < \\\\epsilon .$ $\\\\Vert \\\\tau -1\\\\Vert _{\\\\infty } < \\\\epsilon $ and $\\\\tau $ is constantly equal to 1 on the neighborhood $U$ from (iii).\",\n \"In order to prove this proposition, we need to recall some facts about open book decompositions.\",\n \"See [8] for more details.\",\n \"An open book decomposition of an oriented closed 3-manifold $M$ is a pair $(\\\\Pi ,L)$ , where $L\\\\subset M$ is a smooth link and $\\\\Pi :M\\\\setminus L \\\\rightarrow \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}$ is a smooth (locally trivial) fibration which near $L$ is a normal angular coordinate.\",\n \"More precisely, there exists a compact neighborhood $N$ of $L$ and a diffeomorphism $N \\\\simeq L \\\\times such that $ L L {0}$ and $ |N L$ is represented as$$(\\\\lambda ,r e^{ix})\\\\mapsto x,$$where $$ belongs to $ L$ and $ (r,x)(0,1]R/2Z$ are polar coordinates on the punctured closed unit disc $ {0}.\",\n \"The link $L$ is called the binding and the fibers of $\\\\Pi $ are called the pages of the open book decomposition.\",\n \"The closure of each page is a Seifert surface for $L$ , that is, a compact smoothly embedded surface with boundary $L$ .\",\n \"The co-orientation of the pages given by the map $\\\\Pi $ and the ambient orientation induce an orientation of the pages.\",\n \"The link $L$ inherits the boundary orientation.\",\n \"Let $S$ be the closure in $M$ of the page $\\\\Pi ^{-1}(0)$ .\",\n \"The coordinates introduced above induce coordinates $(\\\\lambda ,r) \\\\in \\\\partial S\\\\times [0,1]$ on the compact neighborhood $N\\\\cap S$ of $\\\\partial S$ in $S$ , where $[0,1]$ is seen as the intersection of $ with the positive real half-axis.$ The vector field $\\\\partial _x$ on $N \\\\simeq L \\\\times is smooth, vanishes at $ LL {0}$ and generates the group of rotations\\\\begin{equation}\\\\bigl (t,(\\\\lambda ,\\\\rho e^{ix}) \\\\bigr ) \\\\mapsto (\\\\lambda ,\\\\rho e^{i(x+t)}).\\\\end{equation}Let $ Z$ be a smooth vector field on $ M$ which coincides with $ x$ on $ N$ and is positively transverse to all the pages, meaning that $ dZ >0$ on $ ML$.\",\n \"The set of such vector fields is obviously convex and in particular path connected.$ The surface $S$ is a global surface of section for the flow of $Z$ .\",\n \"The form () of the flow of $Z$ on $N$ implies that the corresponding first return map is the identity on the set $N \\\\cap (S\\\\setminus \\\\partial S)$ , and hence it extends smoothly to the boundary and gives us an orientation preserving diffeomorphism $h: S \\\\rightarrow S,$ which is the identity on $N \\\\cap S$ .\",\n \"The diffeomorphism $h$ is called the monodromy map of the open book decomposition which depends on the choice of the vector field $Z$ .\",\n \"However as the set of such vector fields is path connected the isotopy class of $h$ within the space of diffeomorphisms of $S$ that are the identity in a neighborhood of $\\\\partial S$ is uniquely determined.\",\n \"The mapping torus of $h$ is the smooth 3-manifold with boundary $S(h) := \\\\left([0,2\\\\pi ]\\\\times S\\\\right)/ \\\\sim \\\\qquad (2\\\\pi ,q) \\\\sim (0,h(q)).$ It is equipped with the fibration $x: S(h) \\\\rightarrow \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z},$ given by the projection onto the first component.\",\n \"The piece $([0,2\\\\pi ]\\\\times (S\\\\cap N))/\\\\sim $ is diffeomorphic to $\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times (S\\\\cap N)$ , because $h$ is the identity on $S\\\\cap N$ , and is equipped with coordinates $(x,\\\\lambda ,r) \\\\in \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\partial S \\\\times [0,1].$ In particular, the boundary of $S(h)$ is the product $\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\partial S$ , which is a union of finitely many 2-tori.\",\n \"The flow of $Z$ preserves the family of leaves in the neighborhood $N$ and is transverse to the interiors of the leaves.\",\n \"Thus we may renormalise $Z$ outside of $N$ so that its flow preserves the leaves also in $M\\\\backslash N$ .\",\n \"Then the flow of $Z$ induces a diffeomorphism between the interior of $S(h)$ and $M\\\\setminus L$ , which identifies the fibrations $x$ and $\\\\Pi $ .\",\n \"Moreover, $M$ is obtained from $S(h)$ by collapsing each boundary torus onto a circle.\",\n \"More precisely, we can identify $M$ with the quotient $\\\\bigl (S(h) \\\\sqcup (\\\\partial S\\\\times \\\\bigr )/\\\\sim $ where $(x,\\\\lambda ,r) \\\\sim (\\\\lambda ,r e^{ix})$ .\",\n \"Here, the smooth structure near the link is induced by the identification $\\\\partial S \\\\times N$ .\",\n \"By a theorem of Giroux [11] (see also [8] for a detailed proof), the contact structure $\\\\xi $ is supported by some open book decomposition $(\\\\Pi ,L)$ of $M$ : This means that $\\\\xi =\\\\ker \\\\alpha $ , where $\\\\alpha $ is a contact form such that: the restriction of $d\\\\alpha $ to each page is a positive area form; $\\\\alpha $ is positive on $L$ .\",\n \"As smoothly embedded compact surface $S$ we choose the closure of the page $\\\\Pi ^{-1}(0)$ .\",\n \"We freely draw from the notation about open book decompositions established above; in particular, $N$ is the closed tubular neighborhood of $L=\\\\partial S$ with diffeomorphism $N\\\\simeq L\\\\times and $ h:S S$ is a choice of monodromy map which is supported in $ SN$.$ If $\\\\alpha $ and $\\\\alpha ^{\\\\prime }$ are contact forms satisfying (A1) and (A2), then $\\\\ker \\\\alpha $ and $\\\\ker \\\\alpha ^{\\\\prime }$ are isotopic (see [11] and [8]).\",\n \"In this case, Gray's stability theorem (see [10]) implies the existence of a diffeomorphism bringing $\\\\ker \\\\alpha $ into $\\\\ker \\\\alpha ^{\\\\prime }$ .\",\n \"Since the properties (i), (ii), (iii) and (iv) that we wish our contact form to have are preserved by the action of a diffeomorphism, it is enough to find a contact form $\\\\alpha $ which satisfies them together with the conditions (A1) and (A2) above, i.e.\",\n \"without explicitly assuming that $\\\\ker \\\\alpha $ is the given contact structure.\",\n \"Denote by $C_1,\\\\dots ,C_{\\\\ell }$ the connected components of $\\\\partial S$ .\",\n \"The tubular neighborhood $N\\\\cap S$ of $\\\\partial S$ in $S$ is the union of $\\\\ell $ closed annuli $A_j$ carrying coordinates $(\\\\theta ,r)\\\\in \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times [0,1]$ , where $\\\\theta $ is an orientation preserving parametrization of $C_j$ .\",\n \"Here the boundary component $C_j$ corresponds to $r=0$ .\",\n \"The fact that $C_j$ is given the boundary orientation from $S$ implies that $2r\\\\,d\\\\theta \\\\wedge dr$ is a positive 2-form on each half-open annulus $A_j\\\\setminus \\\\partial S$ .\",\n \"Choose a 2-form $\\\\Omega $ on $S$ such that $\\\\Omega >0$ on $S\\\\setminus L$ , $\\\\int _S\\\\Omega = 2\\\\pi \\\\ell $ and $\\\\Omega = 2r\\\\, d\\\\theta \\\\wedge dr \\\\qquad \\\\mbox{on each closed annulus } A_j^{\\\\prime }:= \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times [0,\\\\rho ]\\\\subset A_j,$ for a suitably small number $\\\\rho \\\\in (0,1]$ .\",\n \"Perhaps after shrinking $\\\\rho $ , we can assume that there exists a primitive $\\\\eta $ of $\\\\Omega $ agreeing with $(1-r^2)d\\\\theta $ on each $A_j^{\\\\prime }$ .\",\n \"To see this, choose any 1-form $\\\\sigma _0$ on $S$ which agrees with $(1-r^2)d\\\\theta $ near the $C_j$ 's.\",\n \"Then $\\\\Omega -d\\\\sigma _0$ is a 2-form with support contained in $S\\\\setminus \\\\partial S$ .\",\n \"Since $\\\\int _S \\\\bigl ( \\\\Omega - d\\\\sigma _0 \\\\bigr ) = 2\\\\pi \\\\ell - \\\\int _{\\\\partial S} \\\\sigma _0 = 2\\\\pi \\\\ell - 2\\\\pi \\\\ell = 0,$ we can invoke de Rham's theorem to find a 1-form $\\\\sigma _1$ supported in $S\\\\setminus \\\\partial S$ and such that $\\\\Omega -d\\\\sigma _0=d\\\\sigma _1$ .\",\n \"Then $\\\\eta = \\\\sigma _0+\\\\sigma _1$ has the desired property.\",\n \"Let $\\\\beta :\\\\mathbb {R}\\\\rightarrow [0,1]$ be a smooth function satisfying $\\\\beta (0)=0$ , $\\\\beta (2\\\\pi )=1$ and $\\\\mathrm {supp\\\\,}(\\\\beta ^{\\\\prime })\\\\subset (0,2\\\\pi )$ .\",\n \"The family of 1-forms $\\\\tilde{\\\\alpha }_s := dx + s \\\\bigl ( (1-\\\\beta (x)) \\\\eta + \\\\beta (x) h^*\\\\eta \\\\bigr ), \\\\qquad s\\\\in (0,+\\\\infty ),$ is well defined on the mapping torus $S(h)$ of $h$ which is defined in (REF ).\",\n \"Here $x$ denotes the fibration (REF ), which corresponds to the fibration $\\\\Pi $ under the identification of the interior of $S(h)$ with $M\\\\setminus L$ .\",\n \"From now on, we shall identify $M$ with the manifold $\\\\left( S(h) \\\\sqcup \\\\bigsqcup _{j=1}^{\\\\ell } \\\\bigl ( \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\rho ) \\\\right) / \\\\sim $ as in (REF ), where a point in $\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times A_j^{\\\\prime }\\\\subset S(h)$ with coordinates $(x,\\\\theta ,r)$ , is identified with the point $(\\\\theta ,r e^{ix})$ in the corresponding copy of $\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\rho .\",\n \"With this identification, the set$$V := \\\\bigsqcup _{j=1}^{\\\\ell } \\\\bigl ( \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\rho )$$is a compact tubular neighborhood of $ L$ in $ M$.$ We will think of $\\\\tilde{\\\\alpha }_s$ as defined in $M\\\\setminus L \\\\simeq S(h)\\\\setminus \\\\partial S(h)$ .\",\n \"The fact that $h$ is the identity on $A_j$ and the form of $\\\\eta $ on $A_j^{\\\\prime }$ imply that $\\\\tilde{\\\\alpha }_s=dx + s (1-r^2)\\\\, d\\\\theta \\\\qquad \\\\mbox{on }V\\\\setminus L,$ with respect to the standard coordinates $(\\\\theta ,r,x)\\\\in \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times (0,\\\\rho ] \\\\times \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}$ on $V \\\\setminus L$ .\",\n \"Fix $\\\\delta >0$ .\",\n \"There exists $s_1>0$ , depending only on $\\\\rho $ and $\\\\delta $ , with the following properties: If $s\\\\in (0,s_1)$ then there are smooth functions $f,g:[0,\\\\rho ] \\\\rightarrow [0,+\\\\infty )$ defining a curve in the complex plane $\\\\gamma : [0,\\\\rho ] \\\\rightarrow \\\\qquad \\\\gamma (r) = f(r) + i g(r),$ satisfying: $\\\\gamma (r) = 1 + is(1-r^2)$ on $[r_1,\\\\rho ]$ , for some $r_1 \\\\in (0,\\\\rho )$ ; $g^{\\\\prime }<0$ on $(0,\\\\rho ]$ .\",\n \"$\\\\gamma ([0,r_0]) \\\\subset \\\\lbrace x+iy \\\\mid x+y=1+\\\\delta \\\\rbrace $ for some $r_0\\\\in (0,r_1)$ such that $g(r_0)-g(\\\\rho )=g(r_0)-s(1-\\\\rho ^2) \\\\le 2\\\\delta .$ Moreover, $\\\\gamma (r) = r^2 + i (1+\\\\delta -r^2)$ when $r$ is close enough to 0.\",\n \"The derivative of the argument of $\\\\gamma $ , that is the function $ \\\\frac{g^{\\\\prime }f-f^{\\\\prime }g}{f^2+g^2} $ is negative on $(0,\\\\rho ]$ .\",\n \"The derivative of the argument of $\\\\gamma ^{\\\\prime }$ , that is the function $ \\\\frac{g^{\\\\prime \\\\prime }f^{\\\\prime }-f^{\\\\prime \\\\prime }g^{\\\\prime }}{(f^{\\\\prime })^2+(g^{\\\\prime })^2} $ is non-positive on $[0,\\\\rho ]$ .\",\n \"Figure: Functions f,gf,g with the above properties exist.Finally, we define a smooth 1-form $\\\\alpha _s$ on $M$ by $\\\\alpha _s = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle {\\\\frac{\\\\tilde{\\\\alpha }_s}{2\\\\pi (1+\\\\delta )}} & \\\\mbox{on } M\\\\setminus V, \\\\\\\\ \\\\\\\\\\\\displaystyle {\\\\frac{f(r)\\\\,dx + g(r)\\\\,d\\\\theta }{2\\\\pi (1+\\\\delta )}} & \\\\mbox{on } V. \\\\end{array} \\\\right.$ The smoothness of $\\\\alpha _{s}$ near the boundary of $V$ follows from (REF ) and (B1).\",\n \"By (B3) we have the formula $\\\\alpha _{s} = \\\\frac{r^2 \\\\, dx + (1+\\\\delta -r^2)\\\\, d\\\\theta }{2\\\\pi (1+\\\\delta )} \\\\qquad \\\\mbox{near $L$},$ which shows that $\\\\alpha _{s}$ is smooth in a neighborhood of $L$ .\",\n \"Therefore, $\\\\alpha _{s}$ is a smooth 1-form on $M$ for every $s\\\\in (0,s_1)$ .\",\n \"We claim that there exists $s_2 \\\\in (0,s_1)$ , depending on $\\\\delta ,\\\\rho ,h,\\\\eta ,\\\\beta $ , such that if $s \\\\in (0,s_2)$ then $\\\\alpha _s$ is a contact form and satisfies conditions (A1) and (A2).\",\n \"Let us prove this claim.\",\n \"By (B3) the 1-form $\\\\alpha _{s}$ has the following expression on $L$ : $\\\\alpha _{s} = \\\\frac{1}{2\\\\pi } \\\\, d\\\\theta \\\\qquad \\\\mbox{on } L \\\\simeq \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\lbrace 0\\\\rbrace \\\\subset \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times \\\\rho $ which shows that condition (A2) holds for every $s\\\\in (0,s_1)$ .\",\n \"On $V$ we have $d\\\\alpha _s = \\\\frac{f^{\\\\prime }dr \\\\wedge dx-g^{\\\\prime }d\\\\theta \\\\wedge dr}{2\\\\pi (1+\\\\delta )}.$ The restriction of $d\\\\alpha _{s}$ on the page $\\\\Pi ^{-1}(x_0) = x^{-1}(x_0)$ intersected with $V$ is $-\\\\frac{g^{\\\\prime }d\\\\theta \\\\wedge dr}{2\\\\pi (1+\\\\delta )},$ which is a positive area form thanks to (B2) and the positivity of $d\\\\theta \\\\wedge dr$ .\",\n \"So the condition (A1) holds on $V$ for every $s\\\\in (0,s_1)$ .\",\n \"Moreover, $\\\\alpha _s \\\\wedge d\\\\alpha _s = \\\\frac{f^{\\\\prime }g-fg^{\\\\prime }}{4\\\\pi ^2 (1+\\\\delta )^2} \\\\ dx\\\\wedge d\\\\theta \\\\wedge dr = \\\\frac{f^{\\\\prime }g-fg^{\\\\prime }}{4\\\\pi ^2 (1+\\\\delta )^2 r} \\\\ d\\\\theta \\\\wedge (r\\\\, dr \\\\wedge dx).$ By the last part of (B3), $\\\\frac{f^{\\\\prime }g-fg^{\\\\prime }}{r} =2(1+\\\\delta )$ when $r$ is close enough to 0.\",\n \"Together with (B4), this implies that the smooth function $(f^{\\\\prime }g-fg^{\\\\prime })/r$ is strictly positive on $V$ .\",\n \"This implies that $\\\\alpha _{s}$ is a contact form on $V$ for every $s\\\\in (0,s_1)$ .\",\n \"Now we analyze $\\\\alpha _s$ on $M\\\\setminus V$ .\",\n \"The formula $\\\\alpha _s=\\\\frac{dx+s \\\\bigl ((1-\\\\beta )\\\\eta +\\\\beta h^*\\\\eta \\\\bigr )}{2\\\\pi (1+\\\\delta )} \\\\qquad \\\\mbox{on }M\\\\setminus V$ gives $d\\\\alpha _s = s \\\\, \\\\frac{\\\\beta ^{\\\\prime }dx\\\\wedge (h^*\\\\eta -\\\\eta ) + \\\\omega }{2\\\\pi (1+\\\\delta )} \\\\qquad \\\\mbox{on }M\\\\setminus V,$ where $\\\\omega $ is the smooth 2-form on $M\\\\setminus L$ with kernel $\\\\partial _x$ and whose restriction to the page $\\\\Pi ^{-1}(x) \\\\simeq S\\\\setminus \\\\partial S$ is the positive area form $\\\\omega _x := (1-\\\\beta (x))d\\\\eta +\\\\beta (x) h^*d\\\\eta .$ The restriction of $d\\\\alpha _{s}$ to the page $\\\\Pi ^{-1}(x_0)=x^{-1}(x_0)\\\\simeq S\\\\setminus S$ intersected with $M\\\\setminus V$ is a positive multiple of the positive area form $\\\\omega _{x_0}$ , so $\\\\alpha _{s}$ satisfies (A1) on $M\\\\setminus V$ for every $s \\\\in (0,s_1)$ .\",\n \"Moreover, $\\\\frac{4\\\\pi ^2 (1+\\\\delta )^2}{s} \\\\, \\\\alpha _s \\\\wedge d\\\\alpha _s = dx\\\\wedge \\\\omega + s \\\\ \\\\beta ^{\\\\prime } \\\\eta \\\\wedge dx \\\\wedge h^*\\\\eta \\\\qquad \\\\mbox{on } M\\\\setminus V.$ Since $dx\\\\wedge \\\\omega $ is a volume form on $M\\\\setminus V$ , the above formula shows that we can choose $s_2\\\\in (0,s_1]$ , depending on the data $h,\\\\eta ,\\\\beta $ , such that $\\\\alpha _s$ is a contact form on $M\\\\setminus V$ whenever $s\\\\in (0,s_2)$ .\",\n \"We conclude that $\\\\alpha _s$ is a contact form satisfying conditions (A1) and (A2), for every $s\\\\in (0,s_2)$ .\",\n \"The next task is to understand the Reeb flow of $\\\\alpha _s$ and to show it fullfills the requirements (i), (ii), (iii) and (iv) with respect to the surface $S$ , when $s$ is small enough.\",\n \"Condition (i) is actually fulfilled for any $s\\\\in (0,s_1)$ , due to (REF ), so we need to focus only on (ii), (iii) and (iv).\",\n \"Denote by $R_{s}$ the Reeb vector field of $\\\\alpha _{s}$ .\",\n \"We start by analysing the flow of $R_{s}$ on $M\\\\setminus V$ .\",\n \"Since $\\\\beta ^{\\\\prime }$ is supported in $(0,2\\\\pi )$ , $h^* \\\\eta -\\\\eta $ is compactly supported in $S\\\\setminus V$ and $\\\\omega $ restricts to an area form on each page, there is a unique smooth vector field $Y$ on $M$ which is supported in $M\\\\setminus V$ , is tangent to the pages and satisfies $\\\\omega (Y,\\\\cdot ) = -\\\\beta ^{\\\\prime }(x) (h^*\\\\eta -\\\\eta ) \\\\qquad \\\\mbox{on } \\\\Pi ^{-1}(x).$ By (REF ), we have the identity $\\\\begin{split}\\\\frac{2\\\\pi (1+\\\\delta )}{s} \\\\ d\\\\alpha _s(\\\\partial _x+Y,\\\\cdot ) &= \\\\beta ^{\\\\prime }(h^*\\\\eta -\\\\eta ) - \\\\beta ^{\\\\prime }(h^*\\\\eta -\\\\eta )(Y)\\\\, dx + \\\\omega (Y,\\\\cdot ) \\\\\\\\ &=\\\\beta ^{\\\\prime }(h^*\\\\eta -\\\\eta ) + \\\\omega (Y,Y)\\\\, dx - \\\\beta ^{\\\\prime }(h^*\\\\eta -\\\\eta ) = 0,\\\\end{split}$ which shows that the non-vanishing vector field $\\\\partial _x+Y$ is in the kernel of $d\\\\alpha _{s}$ .\",\n \"Therefore, the Reeb vector field $R_{s}$ of $\\\\alpha _{s}$ has the form $R_s = \\\\frac{\\\\partial _x+Y}{\\\\alpha _s(\\\\partial _x+Y)} \\\\qquad \\\\mbox{on } M\\\\setminus V,$ for every $s\\\\in (0,s_2)$ .\",\n \"The fact that $Y$ is supported in $M\\\\setminus V$ and the form of $V$ imply that $M\\\\setminus V$ is invariant for the Reeb flow, and hence the same is true for its complement $V$ .\",\n \"Moreover, the above formula implies that $R_{s}$ is transverse to the portion of the pages $x^{-1}(x_0)$ lying in $M\\\\setminus V$ , and in particular to $S\\\\setminus V$ .\",\n \"Next we study the Reeb flow on $V$ .\",\n \"By the form of $\\\\alpha _{s}$ on $V$ , $R_{s}$ has the form $R_{s} = 2\\\\pi (1+\\\\delta ) \\\\frac{f^{\\\\prime } \\\\partial _{\\\\theta } - g^{\\\\prime } \\\\partial _x}{f^{\\\\prime }g-g^{\\\\prime }f} \\\\qquad \\\\mbox{on } V.$ The condition (B2) implies that $R_{s}$ is transverse the the portions of the pages $x^{-1}(x_0)$ which lie in $V$ , and in particular to $(S\\\\setminus \\\\partial S)\\\\cap V$ .\",\n \"The expression for $R_{s}$ near $L$ becomes, thanks to (B3), $R_{s} = 2\\\\pi (\\\\partial _{\\\\theta } + \\\\partial _x) \\\\qquad \\\\mbox{for } (\\\\theta ,r,x)\\\\in \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times [0,r_0] \\\\times \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}.$ In particular, since $\\\\partial _x$ vanishes on $L$ , $R_{s} = 2\\\\pi \\\\,\\\\partial _{\\\\theta }\\\\qquad \\\\mbox{on } L,$ and the link $L$ consists of periodic orbits of $R_{s}$ of period 1.\",\n \"We conclude that the vector field $R_{s}$ is transverse to the pages $x^{-1}(x_0)$ and leaves the binding $L$ invariant.\",\n \"In particular, in view of the form of $R_s$ near $L$ described above, $S$ is a global surface of section for the Reeb flow, and the first return time function and first return map $\\\\tau : S \\\\setminus \\\\partial S \\\\rightarrow (0,+\\\\infty ), \\\\qquad \\\\varphi : S \\\\setminus \\\\partial S \\\\rightarrow S \\\\setminus \\\\partial S$ are well defined.\",\n \"By (REF ), on $(S\\\\setminus \\\\partial S)\\\\cap V$ the first return time function is explicitly given by the formula $\\\\tau = \\\\frac{g^{\\\\prime }f-f^{\\\\prime }g}{(1+\\\\delta )g^{\\\\prime }} \\\\qquad \\\\mbox{on } (S\\\\setminus \\\\partial S)\\\\cap V,$ and the first return time map by $\\\\varphi : (r,\\\\theta ) \\\\mapsto \\\\left( r,\\\\theta -2\\\\pi \\\\frac{f^{\\\\prime }(r)}{g^{\\\\prime }(r)} \\\\right) \\\\qquad \\\\mbox{on } (S\\\\setminus \\\\partial S) \\\\cap V.$ By (B3), we have $\\\\tau (r,\\\\theta ) = 1 \\\\qquad \\\\mbox{and} \\\\qquad \\\\varphi (r,\\\\theta ) = (r,\\\\theta -2\\\\pi ) \\\\qquad \\\\forall (r,\\\\theta )\\\\in (0,r_0]\\\\times \\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}.$ This implies that $\\\\tau $ and $\\\\varphi $ extend smoothly to the boundary of $S$ and proves (ii).\",\n \"By (REF ), the restriction of the first return time function $\\\\tau $ to $S\\\\cap V$ depends only on $r$ and we have $\\\\partial _r \\\\tau = \\\\frac{g(f^{\\\\prime }g^{\\\\prime \\\\prime }-g^{\\\\prime }f^{\\\\prime \\\\prime })}{(1+\\\\delta )(g^{\\\\prime })^2}.$ By (B5), this function is non-positive on $[0,\\\\rho ]$ .\",\n \"Together with (REF ) and $\\\\tau (\\\\rho ,\\\\theta ) = \\\\frac{1}{1+\\\\delta },$ where we have used (REF ) and (B1), we find the bounds $\\\\frac{1}{1+\\\\delta } \\\\le \\\\tau \\\\le 1 \\\\qquad \\\\mbox{on } S\\\\cap V,$ from which $\\\\sup _{S\\\\cap V} |\\\\tau -1| \\\\le 1 - \\\\frac{1}{1+\\\\delta } = \\\\frac{\\\\delta }{1+\\\\delta } < \\\\delta .$ From (REF ) we have $dx(R_{s}) = \\\\frac{1}{\\\\alpha _{s}(\\\\partial _x + Y)} \\\\qquad \\\\mbox{on } M\\\\setminus V.$ Since the function $\\\\alpha _{s}(\\\\partial _x + Y) = \\\\frac{1}{2\\\\pi (1+\\\\delta )} \\\\tilde{\\\\alpha }_{s}(\\\\partial _x + Y) = \\\\frac{1}{2\\\\pi (1+\\\\delta )} \\\\bigl ( 1 + s ((1-\\\\beta ) \\\\eta (Y) + \\\\beta h^* \\\\eta (Y))\\\\bigr )$ converges to $1/(2\\\\pi (1+\\\\delta ))$ for $s\\\\rightarrow 0$ uniformly on $M\\\\setminus V$ , by (REF ) the function $dx(R_{s})$ converges to $2\\\\pi (1+\\\\delta )$ for $s\\\\rightarrow 0$ uniformly on $M\\\\setminus V$ .\",\n \"This implies that $\\\\tau $ converges uniformly to $1/(1+\\\\delta )$ on $S\\\\setminus V$ .\",\n \"Together with (REF ), this implies that there exists $s_3\\\\in (0,s_2]$ such that for every $s\\\\in (0,s_3)$ we have $\\\\Vert \\\\tau -1\\\\Vert _{\\\\infty } < 2\\\\delta .$ Define $U$ to be the open tubular neighborhood of $L=\\\\partial S$ in $S$ consisting of those $(r,\\\\theta )$ in $S\\\\cap V$ with $0\\\\le r < r_0$ .\",\n \"By (REF ), $\\\\tau =1$ and $\\\\varphi =\\\\mathrm {id}$ on $U$ .\",\n \"Together with (REF ), this proves that (iv) holds, by choosing $\\\\delta \\\\le \\\\epsilon /2$ .\",\n \"The identity (REF ) gives us a constant $C$ , depending only on $\\\\beta $ , $\\\\eta $ and $h$ , such that $\\\\int _{S\\\\setminus V} d\\\\alpha _{s} \\\\le C s \\\\qquad \\\\forall s\\\\in (0,s_1).$ We now fix a $s_4\\\\in (0,s_3]$ such that $C s_4<\\\\delta $ .\",\n \"By (REF ), the $d\\\\alpha _{s}$ -area of the annulus in $A_j^{\\\\prime }$ corresponding to the values of $r$ in the interval $[r_0,\\\\rho ]$ is $\\\\frac{1}{2\\\\pi (1+\\\\delta )} \\\\int _{\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times [r_0,\\\\rho ]} (-g^{\\\\prime }) \\\\, d\\\\theta \\\\wedge dr = \\\\frac{1}{1+\\\\delta } \\\\bigl ( g(r_0) - g(\\\\rho ) \\\\bigr ) \\\\le \\\\frac{2\\\\delta }{1+\\\\delta }< 2\\\\delta ,$ where the first upper bound follows from (B3).\",\n \"If $s$ is in the interval $(0,s_4)$ , the above two inequalities imply that $\\\\int _{S\\\\setminus U} d\\\\alpha _{s} = \\\\int _{S\\\\setminus V} d\\\\alpha _{s} + \\\\frac{1}{2\\\\pi (1+\\\\delta )} \\\\int _{\\\\mathbb {R}/2\\\\pi \\\\mathbb {Z}\\\\times [r_0,\\\\rho ]} (-g^{\\\\prime }) \\\\, d\\\\theta \\\\wedge dr < 3\\\\delta ,$ and the conclusion (iii) follows by choosing $\\\\delta =\\\\epsilon /3$ .\"\n ],\n [\n \"Construction of the plug\",\n \"The aim of this section is to show how some results from [3] can be used in order to build a contact form on a solid torus such that the contact volume is small and all closed orbits of the corresponding Reeb flow have large period.\",\n \"In the next section, this solid torus will be used as a plug to modify the special contact form which we constructed in the previous section.\",\n \"Here is the statement which summarizes the properties of the plug.\",\n \"Proposition 2 Fix positive numbers $r$ and $\\\\epsilon $ .\",\n \"Let $\\\\lambda $ be a primitive of the standard area form $dx\\\\wedge dy$ on the closed disc $r\\\\mathbb {D}$ .\",\n \"Then there exists a smooth contact form $\\\\beta $ on the solid torus $r\\\\mathbb {D} \\\\times \\\\mathbb {R}/\\\\mathbb {Z}$ with the following properties: $\\\\beta = \\\\lambda + ds$ in a neighborhood of $\\\\partial (r \\\\times \\\\mathbb {R}/\\\\mathbb {Z}$ , where $s$ denotes the coordinate on $\\\\mathbb {R}/ \\\\mathbb {Z}$ ; in particular, the Reeb vector field $R_{\\\\beta }$ of $\\\\beta $ coincides with $\\\\partial _s$ near the boundary of $r\\\\mathbb {R}/\\\\mathbb {Z}$ , and its flow is globally well-defined; the contact form $\\\\beta $ is smoothly isotopic to the contact form $\\\\lambda + ds$ on $r\\\\mathbb {R}/ \\\\mathbb {Z}$ through a path of contact forms which agree with $\\\\lambda + ds$ in a neighborhood of $\\\\partial (r \\\\times \\\\mathbb {R}/ \\\\mathbb {Z}$ ; all the closed orbits of $R_{\\\\beta }$ have period at least 1; $\\\\mathrm {vol}\\\\,(r\\\\mathbb {R}/ \\\\mathbb {Z}, \\\\beta \\\\wedge d\\\\beta ) < \\\\epsilon $ .\",\n \"Before discussing the proof of the above proposition, we recall the definition of the Calabi invariant for compactly supported area-preserving diffeomorphisms of the plane.\",\n \"Endow the plane $\\\\mathbb {R}^2$ with the standard area form $dx\\\\wedge dy$ and let $\\\\mathrm {Diff}_c(\\\\mathbb {R}^2,dx\\\\wedge dy)$ be the group of compactly supported area-preserving diffeomorphisms of $\\\\mathbb {R}^2$ .\",\n \"Let $\\\\varphi $ be an element of $\\\\mathrm {Diff}_c(\\\\mathbb {R}^2,dx\\\\wedge dy)$ and let $\\\\lambda $ be a primitive of $dx\\\\wedge dy$ .\",\n \"To $\\\\varphi $ and $\\\\lambda $ we can associate the unique compactly supported smooth function $\\\\sigma _{\\\\varphi ,\\\\lambda }: \\\\mathbb {R}^2 \\\\rightarrow \\\\mathbb {R}$ satisfying $\\\\varphi ^* \\\\lambda - \\\\lambda = d\\\\sigma _{\\\\varphi ,\\\\lambda }.$ This function is called the action of $\\\\varphi $ with respect to $\\\\lambda $ .\",\n \"Its value at fixed points of $\\\\varphi $ is independent of the choice of the primitive $\\\\lambda $ , and so is its integral $\\\\mathrm {CAL}(\\\\varphi ) = \\\\int _{\\\\mathbb {R}^2} \\\\sigma _{\\\\varphi ,\\\\lambda } \\\\, dx\\\\wedge dy,$ which is called the Calabi invariant of $\\\\varphi $ .\",\n \"The Calabi invariant defines a homomorphism from $\\\\mathrm {Diff}_c(\\\\mathbb {R}^2,dx\\\\wedge dy)$ onto $\\\\mathbb {R}$ .\",\n \"Let $\\\\lambda _0$ be the following radially symmetric primitive of $dx\\\\wedge dy$ $\\\\lambda _0 := \\\\frac{1}{2} ( x\\\\, dy - y \\\\, dx).$ We shall make use of the following result: Proposition 3 Fix positive numbers $r$ and $L$ .\",\n \"For every $\\\\epsilon >0$ there exists a positive integer $n$ and an area-preserving diffeomorphism $\\\\varphi \\\\in \\\\mathrm {Diff}_c(\\\\mathrm {int}(r\\\\mathbb {D}),dx\\\\wedge dy)$ such that the following properties hold: $\\\\sigma _{\\\\varphi ,\\\\lambda _0} \\\\ge - L + L/n $ ; $\\\\mathrm {CAL}(\\\\varphi )< - L\\\\pi r^2 + \\\\epsilon $ ; all the fixed points of $\\\\varphi $ have non-negative action; all the periodic points of $\\\\varphi $ which are not fixed points have period at least $n$ .\",\n \"This result is proved in [3] for $r=1$ .\",\n \"The general case follows by a simple rescaling argument, using that if $\\\\varphi $ is in $\\\\mathrm {Diff}_c(\\\\mathbb {R}^2,dx\\\\wedge dy)$ then the rescaled map $\\\\varphi _r(z) := r \\\\varphi \\\\left( \\\\frac{z}{r} \\\\right)$ is also in $\\\\mathrm {Diff}_c(\\\\mathbb {R}^2,dx\\\\wedge dy)$ and $\\\\sigma _{\\\\varphi _r,\\\\lambda _0}(z) = r^2 \\\\sigma _{\\\\varphi ,\\\\lambda _0}\\\\left( \\\\frac{z}{r} \\\\right), \\\\qquad \\\\mathrm {CAL}(\\\\varphi _r)= r^4 \\\\, \\\\mathrm {CAL}(\\\\varphi ).$ The last ingredient which we need is the following result from [3][Proposition 3.1], which allows us to lift an area preserving diffeomorphism of a disc to a Reeb flow on a solid torus.\",\n \"Proposition 4 Fix positive numbers $r$ and $L$ .\",\n \"Let $\\\\varphi \\\\in \\\\mathrm {Diff}_c(\\\\mathrm {int}(r\\\\mathbb {D}),dx\\\\wedge dy)$ and assume that the function $\\\\tau := \\\\sigma _{\\\\varphi ,\\\\lambda _0} + L$ is positive on $r.\",\n \"Then there exists a smooth contact form $$ on the solid torus $ rR/L Z$ with the following properties:\\\\begin{enumerate}[({c}1)]\\\\item \\\\beta = \\\\lambda _0 + ds in a neighborhood of \\\\partial (r \\\\times \\\\mathbb {R}/L \\\\mathbb {Z}, where s denotes the coordinate on \\\\mathbb {R}/L \\\\mathbb {Z}; in particular, the Reeb vector field R_{\\\\beta } of \\\\beta coincides with \\\\partial _s near the boundary of r\\\\mathbb {R}/L \\\\mathbb {Z}, and its flow is globally well-defined;\\\\item the surface r\\\\lbrace 0\\\\rbrace is transverse to the flow of R_{\\\\beta }, and the orbit of every point in r\\\\mathbb {R}/L \\\\mathbb {Z} intersects r\\\\lbrace 0\\\\rbrace both in the future and in the past;\\\\item the first return map and the first return time of the flow of R_{\\\\beta } associated to the surface r\\\\lbrace 0\\\\rbrace \\\\cong r are the map \\\\varphi and the function \\\\tau ;\\\\item \\\\mathrm {vol}\\\\,(r\\\\mathbb {R}/L \\\\mathbb {Z}, \\\\beta \\\\wedge d\\\\beta ) = L \\\\,\\\\pi r^2+ \\\\mathrm {CAL}(\\\\varphi );\\\\item the contact form \\\\beta is smoothly isotopic to \\\\lambda _0 + ds on r\\\\mathbb {R}/L \\\\mathbb {Z} through a path of contact forms which agree with \\\\lambda _0 + ds in a neighborhood of \\\\partial (r \\\\times \\\\mathbb {R}/L \\\\mathbb {Z}.\\\\end{enumerate}$ Again, Proposition 3.1 in [3] is stated for $r=1$ , but the general case follows by rescaling.\",\n \"In fact consider $r,L$ and $\\\\varphi $ as in the statement of Proposition REF .\",\n \"Then, as explained above, we have $\\\\sigma _{\\\\varphi _{r^{-1}},\\\\lambda _0}(z) = r^{-2} \\\\sigma _{\\\\varphi ,\\\\lambda _0}\\\\left( rz \\\\right)$ .\",\n \"Applying [3] to the map $\\\\varphi _{r^{-1}}$ with $Lr^{-2}$ in the place of $L$ we get a contact form $\\\\beta _{r^{-1}}$ on $\\\\mathbb {R}/Lr^{-2}\\\\mathbb {Z}$ satisfying the desired conclusions.\",\n \"Consider now the diffeomorphism $\\\\Phi _{r^{-1}}:r\\\\mathbb {R}/L\\\\mathbb {Z}\\\\rightarrow \\\\mathbb {R}/Lr^{-2}\\\\mathbb {Z}$ defined as $(z,s) \\\\mapsto (r^{-1}z,r^{-2}s)$ .\",\n \"Direct calculations reveal that $\\\\beta = r^2\\\\Phi _{r^{-1}}^*\\\\beta _{r^{-1}}$ is the desired contact form.\",\n \"The statement of [3] is actually slightly more general, since it allows $\\\\lambda _0$ to be replaced by a more general primitive of $dx\\\\wedge dy$ , and gives more properties of the contact form $\\\\beta $ .\",\n \"Building on the above two propositions, it is now easy to prove Proposition REF .\",\n \"The first step is to prove the existence of a contact form $\\\\beta _0$ on $r\\\\mathbb {R}/\\\\mathbb {Z}$ which satisfies the required conditions (a1)-(a4), but where in (a1) and (a2) the primitive $\\\\lambda $ of $dx\\\\wedge dy$ is the radially symmetric primitive $\\\\lambda _0$ .\",\n \"Let $n\\\\in \\\\mathbb {N}$ , $\\\\varphi \\\\in \\\\mathrm {Diff}_c(\\\\mathrm {int}(r\\\\mathbb {D}),dx\\\\wedge dy)$ be given by an application of Proposition REF with $L=1$ .\",\n \"By statement (b1) in this proposition, we have the lower bound $1 + \\\\sigma _{\\\\varphi ,\\\\lambda _0} \\\\ge \\\\frac{1}{n}.$ In particular, the function $\\\\tau := 1 + \\\\sigma _{\\\\varphi ,\\\\lambda _0}$ is everywhere positive.\",\n \"Then Proposition REF implies the existence of a contact form $\\\\beta _0$ on the solid torus $r\\\\mathbb {D} \\\\times \\\\mathbb {R}/\\\\mathbb {Z}$ which satisfies the conditions (c1)-(c5) with $L=1$ .\",\n \"We just need to check that $\\\\beta _0$ satisfies (a1)-(a4) with respect to $\\\\lambda _0$ .\",\n \"Conditions (a1) and (a2) are precisely (c1) and (c5).\",\n \"Condition (a4) follows from (c4) and (b2): $\\\\mathrm {vol}\\\\,(r\\\\mathbb {R}/ \\\\mathbb {Z}, \\\\beta _0\\\\wedge d\\\\beta _0) \\\\stackrel{(c4)}{=} \\\\pi r^2 + \\\\mathrm {CAL}(\\\\varphi ) \\\\stackrel{(b2)}{<} \\\\epsilon .$ There remains to prove that all closed orbits of the Reeb flow of $\\\\beta _0$ have period at least 1.\",\n \"By (c2) and (c3), closed orbits of $R_{\\\\beta _0}$ are in one-to-one correspondence with periodic points of $\\\\varphi $ , and if $z\\\\in r\\\\mathbb {D}$ is a periodic point of $\\\\varphi $ with minimal period $k\\\\in \\\\mathbb {N}$ , then the corresponding closed orbit has period $T:= \\\\sum _{j=0}^{k-1} \\\\tau (\\\\varphi ^j(z)).$ When $k=1$ , $z$ is a fixed point of $\\\\varphi $ and by (b3) we have $T=\\\\tau (z) = 1 + \\\\sigma _{\\\\varphi ,\\\\lambda _0}(z) \\\\ge 1.$ By condition (b4), all periodic points of $\\\\varphi $ which are not fixed points have period $k\\\\ge n$ .\",\n \"If $z$ is such a point, condition (b1) gives us $T = \\\\sum _{j=0}^{k-1} \\\\tau (\\\\varphi ^j(z)) = \\\\sum _{j=0}^{k-1} \\\\bigl (1+\\\\sigma _{\\\\varphi ,\\\\lambda _0}(\\\\varphi ^j(z))\\\\bigr ) \\\\ge \\\\sum _{j=0}^{k-1} \\\\frac{1}{n} = \\\\frac{k}{n} \\\\ge 1.$ This proves (a3) and concludes the first step.\",\n \"Now we wish to modify $\\\\beta _0$ in a neighborhood of the boundary in order to obtain (a1) and (a2) with respect to the given primitive $\\\\lambda $ of $dx\\\\wedge dy$ , while keeping conditions (a3) and (a4).\",\n \"The 1-form $\\\\lambda -\\\\lambda _0$ is closed and hence exact on $r.\",\n \"Let $ u$ be a smooth function on $ r such that $\\\\lambda - \\\\lambda _0 = du.$ Let $\\\\lbrace \\\\beta _0^t\\\\rbrace _{t\\\\in [0,1]}$ be a smooth path of contact forms on $r\\\\mathbb {R}/\\\\mathbb {Z}$ going from $\\\\beta _0$ to $\\\\lambda _0+ds$ and agreeing with $\\\\lambda _0+ds$ in a neighborhood $(rr^{\\\\prime }\\\\times \\\\mathbb {R}/\\\\mathbb {Z}$ of the boundary of $r\\\\mathbb {R}/\\\\mathbb {Z}$ , where $r^{\\\\prime }\\\\in (0,r)$ .\",\n \"Let $\\\\chi $ be a smooth function on $r such that $ =0$ on $ r' and $\\\\chi =1$ on a neighborhood of $\\\\partial (r$ .\",\n \"Consider the following smooth contact form on $r\\\\mathbb {R}/\\\\mathbb {Z}$ : $\\\\beta := \\\\beta _0 + d(\\\\chi u).$ This contact form satisfies (a1) with respect to $\\\\lambda $ .\",\n \"The formula $\\\\beta ^t := \\\\beta _0^t + d(\\\\chi u)$ defines a smooth path of contact forms on $r\\\\mathbb {R}/\\\\mathbb {Z}$ going from $\\\\beta $ to the 1-form $\\\\lambda _0 + ds+d(\\\\chi u),$ and agreeing with $\\\\lambda +ds$ in a neighborhood of the boundary of $r\\\\mathbb {R}/\\\\mathbb {Z}$ .\",\n \"The latter contact form can be joined to $\\\\lambda + ds$ by the smooth isotopy $\\\\lambda _0 + t\\\\, du + (1-t) d(\\\\chi u) + ds,$ which consists of contact forms on $r\\\\mathbb {R}/\\\\mathbb {Z}$ agreeing with $\\\\lambda +ds$ in a neighborhood of the boundary of $r\\\\mathbb {R}/\\\\mathbb {Z}$ .\",\n \"This proves that $\\\\beta $ satisfies (a2) with respect to $\\\\lambda $ .\",\n \"The volume form induced by $\\\\beta $ is $\\\\beta \\\\wedge d\\\\beta = \\\\beta _0 \\\\wedge d\\\\beta _0 + d(\\\\chi u) \\\\wedge d\\\\beta _0.$ The 3-form $d(\\\\chi u) \\\\wedge d\\\\beta _0$ vanishes identically, because $d(\\\\chi u)$ is supported in the region in which $d\\\\beta _0=d\\\\lambda _0$ and the contractions of this 1-form and this 2-form along $\\\\partial _s$ are both zero.\",\n \"We deduce that $\\\\beta \\\\wedge d\\\\beta = \\\\beta _0 \\\\wedge d\\\\beta _0$ on the whole $r\\\\mathbb {R}/\\\\mathbb {Z}$ , and the fact that $\\\\beta _0$ satisfies (a4) implies that also $\\\\beta $ does.\",\n \"The fact that $\\\\beta $ differs from $\\\\beta _0$ by an exact 1-form which vanishes along the direction of $R_{\\\\beta _0}$ implies that the Reeb vector field of $\\\\beta $ coincides with the one of $\\\\beta _0$ .\",\n \"Therefore, the fact that $\\\\beta _0$ satisfies (a3) implies that also $\\\\beta $ does.\"\n ],\n [\n \"Proof of the main theorem\",\n \"We are now ready to prove the theorem stated in the introduction.\",\n \"Let $\\\\xi $ be a contact structure on the closed three-manifold $M$ .\",\n \"Let $S$ be the smoothly embedded compact surface $S\\\\subset M$ given by Proposition REF and let $\\\\epsilon <1$ be a fixed positive number.\",\n \"By Proposition REF we can find a contact form $\\\\tilde{\\\\alpha }$ on $M$ such that $\\\\ker \\\\tilde{\\\\alpha } = \\\\xi $ , $S$ is a global surface of section for the Reeb flow of $\\\\tilde{\\\\alpha }$ and statements (i)-(iv) hold, where $\\\\alpha $ is renamed as $\\\\tilde{\\\\alpha }$ .\",\n \"In the following, $U$ , $\\\\varphi $ and $\\\\tau $ are the objects appearing in this proposition.\",\n \"Statements labeled by a roman number refer to the statements of this proposition.\",\n \"Denote by $C_1,\\\\dots ,C_{\\\\ell }$ the circles forming the boundary of $S$ .\",\n \"Each connected component of $U$ is a half-open annulus $A_j$ and $d\\\\tilde{\\\\alpha }$ restricts to an area form on the interior of each $A_j$ .\",\n \"Let $a_j>0$ be the $d\\\\tilde{\\\\alpha }$ -area of each $A_j$ .\",\n \"By (i) and (iii) we have $\\\\ell = \\\\int _S d\\\\tilde{\\\\alpha } = \\\\int _{S\\\\setminus U} d\\\\tilde{\\\\alpha }+ \\\\sum _{j=1}^{\\\\ell } \\\\int _{A_j} d\\\\tilde{\\\\alpha } < \\\\epsilon + \\\\sum _{j=1}^{\\\\ell } a_j.$ For any $j\\\\in \\\\lbrace 1,\\\\dots ,\\\\ell \\\\rbrace $ , let $r_j>0$ be such that $\\\\pi r_j^2 = (1-\\\\epsilon ) a_j,$ and let $\\\\psi _j: r_j \\\\mathbb {D} \\\\hookrightarrow A_j \\\\setminus \\\\partial S$ be a smooth embedding such that $\\\\psi _j^*(d\\\\tilde{\\\\alpha }) = dx\\\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.\",\n \"Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\\\Psi _j: r_j \\\\mathbb {R}/\\\\mathbb {Z}\\\\hookrightarrow M$$such that\\\\begin{equation}\\\\Psi _j(\\\\cdot ,0) = \\\\psi _j \\\\qquad \\\\mbox{and} \\\\qquad \\\\Psi _j^*(R_{\\\\tilde{\\\\alpha }}) = \\\\partial _s,\\\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\\\begin{equation}\\\\Psi _j^*(\\\\tilde{\\\\alpha }) = \\\\lambda _j + ds\\\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.\",\n \"Indeed, set for simplicity $\\\\tilde{\\\\alpha }_j:= \\\\Psi _j^*(\\\\tilde{\\\\alpha })$ .\",\n \"The the second identity in () implies that the Reeb vector field of $\\\\tilde{\\\\alpha }_j$ is $\\\\partial _s$ .\",\n \"Since any contact form is invariant by its Reeb flow, $\\\\tilde{\\\\alpha }_j$ is invariant under the translations $(x,y,s) \\\\mapsto (x,y,s+t)$ and hence has the form $\\\\tilde{\\\\alpha }_j = \\\\lambda _j + u_j(x,y)\\\\, ds,$ where $\\\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.\",\n \"Then the condition $\\\\tilde{\\\\alpha }_j(\\\\partial _s)=1$ implies that $u_j=1$ .\",\n \"Finally, the first identity in () implies that $d\\\\lambda _j = d\\\\tilde{\\\\alpha }_j|_{r_j \\\\lbrace 0\\\\rbrace } = \\\\psi _j^*(d\\\\tilde{\\\\alpha }) = dx\\\\wedge dy,$ which concludes the proof of the claim.\",\n \"Denote by $W_j$ the image of the embedding $\\\\Psi _j$ .\",\n \"Its volume with respect to $\\\\tilde{\\\\alpha } \\\\wedge d\\\\tilde{\\\\alpha }$ is $\\\\begin{split}\\\\mathrm {vol}\\\\,( W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) &=\\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z}, \\\\tilde{\\\\alpha }_j \\\\wedge d\\\\tilde{\\\\alpha }_j ) = \\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z}, d\\\\lambda _j\\\\wedge ds) \\\\\\\\ &= \\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z},dx\\\\wedge dy\\\\wedge ds) = \\\\pi r_j^2 = (1-\\\\epsilon )a_j.\\\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }$ has the upper bound $\\\\mathrm {vol}\\\\,(M,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) = \\\\int _S \\\\tau \\\\, d\\\\tilde{\\\\alpha } \\\\le (1+\\\\epsilon ) \\\\int _S d\\\\tilde{\\\\alpha } = (1+\\\\epsilon ) \\\\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\\\begin{split}\\\\mathrm {vol}\\\\,\\\\Bigl ( M \\\\setminus \\\\bigcup _{j=1}^{\\\\ell } &W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }\\\\Bigr ) = \\\\mathrm {vol}\\\\,(M,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) - \\\\sum _{j=1}^{\\\\ell } \\\\mathrm {vol}\\\\,(W_j,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) \\\\\\\\&\\\\le (1+\\\\epsilon ) \\\\ell - (1-\\\\epsilon ) \\\\sum _{j=1}^{\\\\ell } a_j < (1+\\\\epsilon ) \\\\ell - (1-\\\\epsilon )(\\\\ell - \\\\epsilon ) \\\\\\\\ &= \\\\epsilon (2\\\\ell +1) - \\\\epsilon ^2 < \\\\epsilon (2\\\\ell +1),\\\\end{split}$ where the second inequality follows from (REF ).\",\n \"Thanks to Proposition REF , on every solid torus $r_j \\\\mathbb {R}/\\\\mathbb {Z}$ there is a contact form $\\\\beta _j$ which agrees with $\\\\lambda _j+ds$ near the boundary, has total volume less than $\\\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.\",\n \"Moreover, $\\\\beta _j$ is smoothly isotopic to $\\\\lambda _j+ds$ through contact forms which agree with $\\\\lambda _j+ds$ near the boundary.\",\n \"Let $\\\\alpha $ be the contact form on $M$ which coincides with $\\\\tilde{\\\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\\\beta _j$ by the embedding $\\\\Psi _j$ .\",\n \"This form is indeed smooth due to ().\",\n \"It is smoothly isotopic to the contact form $\\\\tilde{\\\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\\\xi =\\\\ker \\\\tilde{\\\\alpha }$ .\",\n \"By (REF ) and the fact that the volume of each $r_j \\\\mathbb {R}/\\\\mathbb {Z}$ with respect to $\\\\beta _j\\\\wedge d\\\\beta _j$ is smaller than $\\\\epsilon $ , the volume of $M$ with respect to $\\\\alpha \\\\wedge d\\\\alpha $ has the upper bound $\\\\begin{split}\\\\mathrm {vol}\\\\,(M,\\\\alpha \\\\wedge d\\\\alpha ) &= \\\\mathrm {vol}\\\\,\\\\Bigl ( M \\\\setminus \\\\bigcup _{j=1}^{\\\\ell } W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }\\\\Bigr ) + \\\\sum _{j=1}^{\\\\ell } \\\\mathrm {vol}\\\\,(r_j \\\\mathbb {R}/\\\\mathbb {Z},\\\\beta _j\\\\wedge d\\\\beta _j) \\\\\\\\ &< \\\\epsilon (2\\\\ell +1) + \\\\epsilon \\\\ell = \\\\epsilon (3\\\\ell +1).\\\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\\\alpha $ .\",\n \"The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\\\beta _j$ 's.\",\n \"The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .\",\n \"Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\\\alpha $ have period larger than $1-\\\\epsilon $ .\",\n \"We conclude that $T_{\\\\min }(\\\\alpha ) > 1-\\\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\\\alpha $ has the lower bound $\\\\rho _{\\\\mathrm {sys}}(\\\\alpha ) = \\\\frac{T_{\\\\min }(\\\\alpha )^2}{\\\\mathrm {vol}\\\\,(M,\\\\alpha \\\\wedge d\\\\alpha )} > \\\\frac{(1-\\\\epsilon )^2}{\\\\epsilon (3\\\\ell +1)}.$ Since the latter quantity tends to $+\\\\infty $ for $\\\\epsilon \\\\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\\\xi $ on $M$ can be made arbitrarily large.\",\n \"This concludes the proof of the theorem stated in the introduction.\",\n \"We are now ready to prove the theorem stated in the introduction.\",\n \"Let $\\\\xi $ be a contact structure on the closed three-manifold $M$ .\",\n \"Let $S$ be the smoothly embedded compact surface $S\\\\subset M$ given by Proposition REF and let $\\\\epsilon <1$ be a fixed positive number.\",\n \"By Proposition REF we can find a contact form $\\\\tilde{\\\\alpha }$ on $M$ such that $\\\\ker \\\\tilde{\\\\alpha } = \\\\xi $ , $S$ is a global surface of section for the Reeb flow of $\\\\tilde{\\\\alpha }$ and statements (i)-(iv) hold, where $\\\\alpha $ is renamed as $\\\\tilde{\\\\alpha }$ .\",\n \"In the following, $U$ , $\\\\varphi $ and $\\\\tau $ are the objects appearing in this proposition.\",\n \"Statements labeled by a roman number refer to the statements of this proposition.\",\n \"Denote by $C_1,\\\\dots ,C_{\\\\ell }$ the circles forming the boundary of $S$ .\",\n \"Each connected component of $U$ is a half-open annulus $A_j$ and $d\\\\tilde{\\\\alpha }$ restricts to an area form on the interior of each $A_j$ .\",\n \"Let $a_j>0$ be the $d\\\\tilde{\\\\alpha }$ -area of each $A_j$ .\",\n \"By (i) and (iii) we have $\\\\ell = \\\\int _S d\\\\tilde{\\\\alpha } = \\\\int _{S\\\\setminus U} d\\\\tilde{\\\\alpha }+ \\\\sum _{j=1}^{\\\\ell } \\\\int _{A_j} d\\\\tilde{\\\\alpha } < \\\\epsilon + \\\\sum _{j=1}^{\\\\ell } a_j.$ For any $j\\\\in \\\\lbrace 1,\\\\dots ,\\\\ell \\\\rbrace $ , let $r_j>0$ be such that $\\\\pi r_j^2 = (1-\\\\epsilon ) a_j,$ and let $\\\\psi _j: r_j \\\\mathbb {D} \\\\hookrightarrow A_j \\\\setminus \\\\partial S$ be a smooth embedding such that $\\\\psi _j^*(d\\\\tilde{\\\\alpha }) = dx\\\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.\",\n \"Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\\\Psi _j: r_j \\\\mathbb {R}/\\\\mathbb {Z}\\\\hookrightarrow M$$such that\\\\begin{equation}\\\\Psi _j(\\\\cdot ,0) = \\\\psi _j \\\\qquad \\\\mbox{and} \\\\qquad \\\\Psi _j^*(R_{\\\\tilde{\\\\alpha }}) = \\\\partial _s,\\\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\\\begin{equation}\\\\Psi _j^*(\\\\tilde{\\\\alpha }) = \\\\lambda _j + ds\\\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.\",\n \"Indeed, set for simplicity $\\\\tilde{\\\\alpha }_j:= \\\\Psi _j^*(\\\\tilde{\\\\alpha })$ .\",\n \"The the second identity in () implies that the Reeb vector field of $\\\\tilde{\\\\alpha }_j$ is $\\\\partial _s$ .\",\n \"Since any contact form is invariant by its Reeb flow, $\\\\tilde{\\\\alpha }_j$ is invariant under the translations $(x,y,s) \\\\mapsto (x,y,s+t)$ and hence has the form $\\\\tilde{\\\\alpha }_j = \\\\lambda _j + u_j(x,y)\\\\, ds,$ where $\\\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.\",\n \"Then the condition $\\\\tilde{\\\\alpha }_j(\\\\partial _s)=1$ implies that $u_j=1$ .\",\n \"Finally, the first identity in () implies that $d\\\\lambda _j = d\\\\tilde{\\\\alpha }_j|_{r_j \\\\lbrace 0\\\\rbrace } = \\\\psi _j^*(d\\\\tilde{\\\\alpha }) = dx\\\\wedge dy,$ which concludes the proof of the claim.\",\n \"Denote by $W_j$ the image of the embedding $\\\\Psi _j$ .\",\n \"Its volume with respect to $\\\\tilde{\\\\alpha } \\\\wedge d\\\\tilde{\\\\alpha }$ is $\\\\begin{split}\\\\mathrm {vol}\\\\,( W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) &=\\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z}, \\\\tilde{\\\\alpha }_j \\\\wedge d\\\\tilde{\\\\alpha }_j ) = \\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z}, d\\\\lambda _j\\\\wedge ds) \\\\\\\\ &= \\\\mathrm {vol}\\\\,( r_j \\\\mathbb {R}/\\\\mathbb {Z},dx\\\\wedge dy\\\\wedge ds) = \\\\pi r_j^2 = (1-\\\\epsilon )a_j.\\\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }$ has the upper bound $\\\\mathrm {vol}\\\\,(M,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) = \\\\int _S \\\\tau \\\\, d\\\\tilde{\\\\alpha } \\\\le (1+\\\\epsilon ) \\\\int _S d\\\\tilde{\\\\alpha } = (1+\\\\epsilon ) \\\\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\\\begin{split}\\\\mathrm {vol}\\\\,\\\\Bigl ( M \\\\setminus \\\\bigcup _{j=1}^{\\\\ell } &W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }\\\\Bigr ) = \\\\mathrm {vol}\\\\,(M,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) - \\\\sum _{j=1}^{\\\\ell } \\\\mathrm {vol}\\\\,(W_j,\\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }) \\\\\\\\&\\\\le (1+\\\\epsilon ) \\\\ell - (1-\\\\epsilon ) \\\\sum _{j=1}^{\\\\ell } a_j < (1+\\\\epsilon ) \\\\ell - (1-\\\\epsilon )(\\\\ell - \\\\epsilon ) \\\\\\\\ &= \\\\epsilon (2\\\\ell +1) - \\\\epsilon ^2 < \\\\epsilon (2\\\\ell +1),\\\\end{split}$ where the second inequality follows from (REF ).\",\n \"Thanks to Proposition REF , on every solid torus $r_j \\\\mathbb {R}/\\\\mathbb {Z}$ there is a contact form $\\\\beta _j$ which agrees with $\\\\lambda _j+ds$ near the boundary, has total volume less than $\\\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.\",\n \"Moreover, $\\\\beta _j$ is smoothly isotopic to $\\\\lambda _j+ds$ through contact forms which agree with $\\\\lambda _j+ds$ near the boundary.\",\n \"Let $\\\\alpha $ be the contact form on $M$ which coincides with $\\\\tilde{\\\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\\\beta _j$ by the embedding $\\\\Psi _j$ .\",\n \"This form is indeed smooth due to ().\",\n \"It is smoothly isotopic to the contact form $\\\\tilde{\\\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\\\xi =\\\\ker \\\\tilde{\\\\alpha }$ .\",\n \"By (REF ) and the fact that the volume of each $r_j \\\\mathbb {R}/\\\\mathbb {Z}$ with respect to $\\\\beta _j\\\\wedge d\\\\beta _j$ is smaller than $\\\\epsilon $ , the volume of $M$ with respect to $\\\\alpha \\\\wedge d\\\\alpha $ has the upper bound $\\\\begin{split}\\\\mathrm {vol}\\\\,(M,\\\\alpha \\\\wedge d\\\\alpha ) &= \\\\mathrm {vol}\\\\,\\\\Bigl ( M \\\\setminus \\\\bigcup _{j=1}^{\\\\ell } W_j , \\\\tilde{\\\\alpha }\\\\wedge d\\\\tilde{\\\\alpha }\\\\Bigr ) + \\\\sum _{j=1}^{\\\\ell } \\\\mathrm {vol}\\\\,(r_j \\\\mathbb {R}/\\\\mathbb {Z},\\\\beta _j\\\\wedge d\\\\beta _j) \\\\\\\\ &< \\\\epsilon (2\\\\ell +1) + \\\\epsilon \\\\ell = \\\\epsilon (3\\\\ell +1).\\\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\\\alpha $ .\",\n \"The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\\\beta _j$ 's.\",\n \"The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .\",\n \"Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\\\alpha $ have period larger than $1-\\\\epsilon $ .\",\n \"We conclude that $T_{\\\\min }(\\\\alpha ) > 1-\\\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\\\alpha $ has the lower bound $\\\\rho _{\\\\mathrm {sys}}(\\\\alpha ) = \\\\frac{T_{\\\\min }(\\\\alpha )^2}{\\\\mathrm {vol}\\\\,(M,\\\\alpha \\\\wedge d\\\\alpha )} > \\\\frac{(1-\\\\epsilon )^2}{\\\\epsilon (3\\\\ell +1)}.$ Since the latter quantity tends to $+\\\\infty $ for $\\\\epsilon \\\\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\\\xi $ on $M$ can be made arbitrarily large.\",\n \"This concludes the proof of the theorem stated in the introduction.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01621"}}},{"rowIdx":989,"cells":{"text":{"kind":"list like","value":[["On Gauge Invariant Cosmological Perturbations in UV-modified Horava\n Gravity: A Brief Introduction"],["Abstract We revisit gauge invariant cosmological perturbations in UV-modified, z = 3 Horava gravity with one scalar matter field, which has been proposed as a renormalizable gravity theory without the ghost problem in four dimensions.","We confirm that there is no extra graviton modes and general relativity is recovered in IR, which achieves the consistency of the model.","From the UV-modification terms which break the detailed balance condition in UV, we obtain scale-invariant power spectrums for non-inflationary backgrounds, like the power-law expansions, without knowing the details of early expansion history of Universe.","This could provide a new framework for the Big Bang cosmology."],["Introduction","(1).","As a particle physicist's point of view, a surprise of cosmology is as follows.","In particle physics, we need (fundamental) scalar fields (called Higgs fields) for “theoretical\" reasons (i.e., to give masses to fundamental particles with renormalizability), but we have waited for a long time (about 40 years, after 1964 papers) for an experimental confirmation at LHC (4 July, 2012).","This is (thought to be) the first elementary scalar particle discovered in Nature.","However, in cosmology, it has been known for a long time that there is scalar (cosmological fluctuation mode) in the sky after the discovery of CMB (1964, the same year that Higgs field has been proposed !",").","But, the existence of a scalar mode in the sky is a big mystery from the following reasons.","The small fluctuations in CMB are thought to be due to (space-time) fluctuations in the early epoch of our (Big Bang) Universe, which should be described by General Relativity (GR).","However, GR alone can not have scalar (fluctuation) mode but only the (usual) tensor mode (2 polarizations, called gravitational waves) !","The cosmological/primordial tensor modes have not been discovered yet, though astrophysical tensor modes (gravitational waves) from black hole mergers have been detected in LIGO.","So, it would not be quite strange (at least to me) even though we discover the cosmological/primordial tensor modes (called B-mode) in the near future.","Rather, the existence of the scalar mode is much more surprising !","The simplest way to explain the data is introduce a “cosmic scalar\" field and “assume\" some peculiar behaviors in the early epoch (known as the inflationary epoch).","But, there are huge numbers of explicit models and we still do not know what is the right one and its origin–Is Higgs the remnant of the primordial scalar ?","This situation is opposite to that of particle physics: In cosmology, we have discovered scalar mode (CMB) long ago but we do not have its theory yet !","Another way to explain the scalar mode is to modify GR, like $f(R)$ , massive gravity, etc., so that the gravity itself has additional scalar modes.","But, in that case the scalar (gravitation) could be disaster since it could affect the known (or, well-established) GR test in solar system (,i.e., low energy (IR) test of GR), unless we decouple it from low energy GR sector.","And also, this could affect the dark matter and dark energy problems as well.","Sometimes, the scalar could be ghost as well, which should be avoided.","Today, I will consider another modified gravity theory but without the scalar gravitation mode so that the usual scalar matters are needed in explaining the cosmology data.","(2).","One the other hand, our universe is considered to be created from “quantum (vacuum) fluctuations\".","Actually, since the scalar power spectrum in inflationary cosmology contains Planck constant $\\hbar $ as well as Newton's constant $G$ ($H$ is the Hubble parameter), $\\Delta ^2(k)= 8 \\pi G \\hbar ( {2 H^2}/{\\pi ^2}),$ one can consider the power spectrum as a “quantum gravity effect\" and so does the inflationary cosmology as a “quantum cosmology\" !","Moreover, recent LIGO's detections of gravitational waves from merging black holes seems to imply that we need to consider quantum gravity more seriously.","Actually, we open the strong gravity test era of GR, beyond the weak gravity test in solar system.","(3).","Do we have quantum gravity then ?","For quantum theory of particle's interactions, “renormalization\" has been a powerful constraint and Higgs particle is its natural consequence.","Now, what if we require renormalizability in quantum gravity ?","But, it is well-known that the renormalizable quantum gravity can not be realized in Einstein's gravity or its (relativistic) higher-derivative generalizations: There are “ghosts\" which have kinetic terms with a wrong sign, in addition to massless gravitons.","Recently, Hořava (or Hořava-Lifshitz (HL)) gravity has been proposed as a renormalzable gravity [1].","There is no ghosts (in the tensor modes) by abandoning the equal-footing treatment of space and time (i.e., Lorentz symmetry) in the higher energy regime (UV).","It is power-counting renormalizable but no proof of renormalizability yet.","This resembles the situations of Yang-Mills theory when it first appeared.","In cosmology, this theory may provide inflationary effect without “inflationary phase (i.e., early de Sitter or accelerating phase)\" so that scale-invariant (scalar/tensor) spectrum can be also produced.","This has been first argued by Mukohyama [2] but there is no rigorous analysis about this yet.","On the contrary, in the works of Gao et.","al.","[3] and Gong et.","al.","[4], scalar spectrum are not scale-invariant though tensor spectrum does.","This is today's topic and it provides a natural formulation of cosmology, known as “effective field theory (EFT)\", by construction."],["Hořava (-Lifshitz) Gravity: Basic Idea","In 2009, Hořava proposed a renormalizable, higher-derivative gravity theory, without ghost problems, by abandoning Lorentz symmetry in UV but keeping, so called, “foliation preserving\" diffemorphism ($FPDiff$ ) [1].","The basic ingredients of the Hořava are (1) Einstein gravity (with a Lorentz deformation parameter $\\lambda $ ), (2) non-covariant deformations with higher-spatial derivatives (up to six orders), and (3) “detailed balance\" in the coefficients (with five constant parameters, $\\kappa , {\\lambda }, \\nu , \\mu , {\\Lambda }_W$ ), in contrast to the Einstein gravity case with the three physical parameters $c, G, {\\Lambda }$ and ${\\lambda }=1$ .","Here, six orders of spatial derivatives $(z=3)$ came from the power counting renormalizability in $3+1$ dimensions.","(For the limitation of space and time, I can not explain all the details of the construction.","So, please read the original work of Hořava [1] about this.)","From the Hořava's construction, some improved UV behaviors, without ghosts, are expected so that the gravity theory “could\" be renormalizable and one might have predictable quantum gravity theory.","But, in cosmology applications, it seems that the detailed-balance condition is too strong to get the required, scale invariant cosmological perturbations.","For example, tensor spectrum is scale invariant but scalar spectrum is not !","[3], [4].","So, we need to break the detailed balancing in some way but without altering the improved UV behaviors of scale invariant tensor modes in the previous works [3], [4] and we called it as “UV modifications\" [5]."],["Cosmological Perturbations in Hořava gravity","We start by considering the four-dimensional, UV-modified Hořava gravity action with $z=3$ , which is power-counting renormalizable [1], $S_\\mathrm {g} &=& \\int d \\eta d^3x \\sqrt{g} N \\left[ \\frac{2}{\\kappa ^2}\\left(K_{ij}K^{ij} - \\lambda K^2 \\right) - {\\cal V} \\right], \\\\-{\\cal V}&=& \\sigma + \\xi R + \\alpha _1 R^2+ \\alpha _2 R_{ij}R^{ij}+\\alpha _3 \\frac{\\epsilon ^{ijk}}{\\sqrt{g}}R_{il}\\nabla _jR^l{}_k+ \\alpha _4 \\nabla _{i}R_{jk} \\nabla ^{i}{R}^{jk}+{\\alpha }_5 \\nabla _{i}R_{jk}\\nabla ^{j} {R}^{ik}+{\\alpha }_6 \\nabla _{i}R\\nabla ^{i}R , \\numero $ where (the prime $(^{\\prime })$ denotes the derivative with respect to $\\eta $ ) $K_{ij}=\\frac{1}{2N}\\left({g_{ij}}^{\\prime }-\\nabla _i N_j-\\nabla _jN_i\\right)\\, ,~ds^2 =-N^2 c^2 d\\eta ^2 +g_{ij}\\left(dx^i+N^i d\\eta \\right) \\left(dx^j+N^j d\\eta \\right).","\\numero $ With the detailed balance condition, the number of independent coupling constants can be reduced to six, i.e., $\\kappa ,\\lambda ,\\mu ,\\nu ,{\\Lambda }_W,{\\omega }$ for a viable model in IR [8], [9], $\\sigma = {8 (3 {\\lambda }-1)},~\\xi ={8 (3 {\\lambda }-1)},~{\\alpha }_1={32 (3 {\\lambda }-1)}, ~{\\alpha }_2=-{8}, ~{\\alpha }_3 = {2 \\nu ^2},~{\\alpha }_4=-{8 2 \\nu ^4}=-{\\alpha }_5=-8 {\\alpha }_6,$ But, we do not restrict to this case only, at least for the UV couplings ${\\alpha }_4,{\\alpha }_5,{\\alpha }_6$ so that the power-counting renormalizable and scale-invariant cosmological scalar fluctuations can be obtained.","For the power-counting renormalizable matter action, we consider $z=3$ scalar field action [6], [7], $S_\\mathrm {m} = \\int d\\eta d^3x \\sqrt{g} N \\left[ \\frac{1}{2N^2}\\left( \\phi ^{\\prime } - N^i {\\partial }_i \\phi \\right)^2 - V(\\phi )- Z({\\partial }_i\\phi ) \\right] \\, ,$ where $Z({\\partial }_i\\phi ) = \\sum _{n=1}^3 \\xi _n \\partial _i^{(n)}\\phi \\partial ^{i(n)}\\phi \\,$ with the superscript $(n)$ denoting $n$ -th spatial derivatives, and $V(\\phi )$ is the matter's potential without derivatives.","The actions, $S_\\mathrm {g}$ and $S_\\mathrm {m}$ are invariant under the foliation preserving FPDiff [1], $\\delta x^i &=&-\\zeta ^i (\\eta , {\\bf x}), ~\\delta \\eta =-f(\\eta ),~\\delta g_{ij}={\\partial }_i\\zeta ^k g_{jk}+{\\partial }_j \\zeta ^k g_{ik}+\\zeta ^k{\\partial }_k g_{ij}+f g^{\\prime }_{ij},\\\\\\delta N_i &=& {\\partial }_i \\zeta ^j N_j+\\zeta ^j {\\partial }_j N_i+\\zeta ^{^{\\prime }j}g_{ij}+f {N^{\\prime }}_i+f^{\\prime } N_i,~\\delta N= \\zeta ^j {\\partial }_j N+f N^{\\prime }+f^{\\prime } N, ~\\delta \\phi =\\zeta ^j {\\partial }_j \\phi +f \\phi ^{\\prime }.","\\numero $ In order to study the cosmological perturbations around the homogeneous and isotropic backgrounds (as seen in CMB), we expand the metric and the scalar field as, $N = a(\\eta )[1+{\\cal A}(\\eta ,{\\bf x})] \\, ,~N_i = a^2(\\eta ){{\\cal B}(\\eta ,{\\bf x})}_i \\, , ~g_{ij} = a^2(\\eta ) [{\\delta }_{ij}+h_{ij} (\\eta ,{\\bf x})],~\\phi = \\phi _0(\\eta ) + \\delta \\phi (\\eta ,{\\bf x}) \\, ,$ by considering spatially flat ($k=0$ ) backgrounds and the conformal (or comoving) time $\\eta $ , for simplicity.","By substituting the metric and scalar field of (REF ) into the actions one can obtain the linear-order perturbation part of the total action $S=S_g +S_m$ which gives the Friedman's equations for the background, $&&{\\cal H}^2 =-\\frac{\\kappa ^2}{6(1-3\\lambda )} \\left( {2}{\\phi _0^{\\prime }}^2+ a^2 \\left(V_0 -\\sigma \\right) \\right) \\, ,~{\\cal H}^2 +2{\\cal H}^{\\prime } =\\frac{\\kappa ^2}{2(1-3\\lambda )} \\left( {2}{\\phi _0^{\\prime }}^2- a^2 \\left(V_0 -\\sigma \\right) \\right) \\, , \\numero \\\\&&\\phi _0^{\\prime \\prime } + 2{\\cal H}\\phi _0^{\\prime } + a^2V_{\\phi _0} = 0 \\, ,$ with the comoving Hubble parameter ${\\cal H} \\equiv a^{\\prime }/a$ , $V_0 \\equiv V(\\phi _0), V_{\\phi _0}\\equiv ({\\partial }V/{\\partial }\\phi )_{\\phi _0}$ , and $h\\equiv h^i_i$ .","Here, it important to note that there is no higher-derivative corrections to the Friedman's equations for spatially flat case and so the background equations are the same as those of GR [8], [9].","However, even in this case, the higher-derivative effects can reappear in the perturbed parts.","The quadratic part of the total perturbed action is given by ${\\delta }_2 S&=& \\int d \\eta d^3x \\left\\lbrace {{\\kappa }^2} \\left[ (1-3 {\\lambda }) {\\cal H} \\left(3 {\\cal H} {\\cal A}^2 + {\\cal A} (2 {\\partial }{\\cal B}^i-h^{\\prime }) \\right)+ (1-{\\lambda }) ({\\partial }_i {\\cal B}^i)^2+{2} {\\partial }_i {\\cal B}_j {\\partial }^i {\\cal B}^j \\right.","\\right.","\\numero \\\\&&\\left.","\\left.","-{\\partial }_i {\\cal B}_j {h^{ij}}^{\\prime } +{4} {h_{ij}}^{\\prime } {h^{ij}}^{\\prime }+{\\lambda }\\left({\\partial }_i {\\cal B}^i h^{\\prime }-{4}h^{\\prime 2} \\right) \\right]+a^2 \\xi \\left({\\cal A} +{2} h \\right)\\left( {\\partial }_i {\\partial }_j h^{ij} -\\Delta h \\right) \\right.","\\numero \\\\&&\\left.+a^2 \\left[ {2} {\\delta }\\phi ^{\\prime 2}-{\\cal A} \\phi _0^{\\prime } {\\delta }\\phi ^{\\prime } +{2} {\\cal A}^2 \\phi _0^{\\prime 2} +{\\partial }_i {\\cal B}^i \\phi _0^{\\prime } {\\delta }\\phi -{2} V_{\\phi _0 \\phi _0} {\\delta }\\phi ^2 -a^2 V_{\\phi _0} {\\delta }\\phi {\\cal A}-{2} \\phi _0^{\\prime } {\\delta }\\phi h^{\\prime }\\right] \\right.\\numero \\\\&&\\left.","-a^4 \\left({\\cal V}^{(2)}+{\\delta }Z \\right)\\right\\rbrace ,$ where ${\\cal V}^{(2)}$ is the quadratic part of the potential ${\\cal V}$ and $\\Delta \\equiv {\\delta }^{ij} {\\partial }_i {\\partial }_j$ , ${\\delta }Z=\\sum _{n=1}^3 \\xi _n~ \\partial _i^{(n)}{\\delta }\\phi \\partial ^{i(n)}{\\delta }\\phi $ .","Now, in order to separate the scalar, vector, and tensor contributions, we consider the most general (SVT) decompositions $({\\partial }_i S^i = {\\partial }_i F^i = \\tilde{H} = {\\partial }_i \\tilde{H}^i_j = 0 )$ , ${\\cal B}_i = {\\partial }_i {\\cal B} + S_i \\, ,~h_{ij} = 2 {\\cal R} {\\delta }_{ij}+ {\\partial }_i {\\partial }_j {\\cal E} + {\\partial }_{(i} F_{j)} + \\tilde{H}_{ij} \\, .$ Then, the pure tensor, vector, and scalar parts of the total action are given by, respectively, $\\delta _2 S^{(t)} &=& \\int d \\eta d^3x~ a^2\\left[ \\frac{2}{\\kappa ^2} {\\tilde{H}_{ij}}^{\\prime } {\\tilde{H}}^{ij^{\\prime }}+\\xi \\tilde{H}_{ij}\\Delta \\tilde{H}^{ij}+ \\frac{\\alpha _2}{a^2}\\Delta \\tilde{H}_{ij}\\Delta \\tilde{H}^{ij}+ \\frac{\\alpha _3}{a^3} \\epsilon ^{ijk} \\Delta \\tilde{H}_{il} \\Delta {\\partial }_j \\tilde{H}^l_{k} \\right.","\\numero \\\\&&~~~~~~~~~~~~~~~~~~~~\\left.- \\frac{\\alpha _4}{a^4} \\Delta \\tilde{H}_{ij}\\Delta ^2 \\tilde{H}^{ij} \\right],\\\\\\delta _2 S^{(v)} &=& \\frac{1}{\\kappa ^2} \\int d \\eta d^3x~ a^2 {\\partial }_i\\left( S^j-{F^j}^{\\prime } \\right) {\\partial }_i \\left( S_j-F_j^{\\prime } \\right) \\, ,\\\\\\delta _2 S^{(s)}&=& \\int d \\eta d^3x ~a^2 \\left\\lbrace \\frac{2(1-3\\lambda )}{\\kappa ^2}\\left[ 3{{\\cal R}^{\\prime }}^2 - 6{\\cal H}{\\cal A}{\\cal R}^{\\prime } +3{\\cal H}^2 {\\cal A}^2- 2\\left( {\\cal R}^{\\prime } - {\\cal H}{\\cal A}\\right) \\Delta ({\\cal B}-{\\cal E}^{\\prime })\\right] \\right.\\nonumber \\\\&&+ \\frac{2(1-\\lambda )}{\\kappa ^2} \\left[ \\Delta \\left({\\cal B}-{\\cal E}^{\\prime }\\right) \\right]^2- 2\\xi ({\\cal R}+2{\\cal A})\\Delta {\\cal R}+ \\frac{2}{a^2} \\left( 8\\alpha _1+3\\alpha _2 \\right)(\\Delta {\\cal R})^2\\nonumber \\\\&& \\left.-\\frac{2}{a^4} \\left( 3\\alpha _4+2\\alpha _5+8 {\\alpha }_6 \\right)\\Delta {\\cal R}\\Delta ^2 {\\cal R}-a^2V_{\\phi _0} {\\cal A}\\delta \\phi -\\frac{1}{2a^2}V_{\\phi _0\\phi _0}\\delta \\phi ^2 - \\delta {Z} \\right.\\numero \\\\&& \\left.+ \\frac{1}{2}{\\delta \\phi ^{\\prime }}^2 - \\phi _0^{\\prime }\\delta \\phi ^{\\prime } {\\cal A}+{2} {\\phi _0^{\\prime }}^2 {\\cal A}^2+ [\\Delta ({\\cal B}-{\\cal E}^{\\prime })-3 {\\cal R}^{\\prime }]\\phi _0^{\\prime }\\delta \\phi \\right\\rbrace \\, .$ Here, it is important to note that sixth-order-derivative terms in the gravity action contribute to scalar as well as tensor perturbations, through the specific combination of `$3\\alpha _4+2\\alpha _5+8 {\\alpha }_6$ ' for the former but through only `${\\alpha }_4$ ' for the latter.","This is what we need for renormalizability and scale-invariant spectrums (as we can see shortly), for both scalar and tensor.","In the detailed-balanced case, the combination `$3\\alpha _4+2\\alpha _5+8 {\\alpha }_6$ ' vanishes though ${\\alpha }_4$ does not.","So in that case, the theory would not be renormalizable and nor scale invariant for scalar part, which is in contradict to observational data !","Now, in order to exhibit the true dynamical degrees of freedom we consider the Hamiltonian reduction method [10], for the cosmologically perturbed actions [11], [4] and, after some computation, the action reduces to $\\delta _2S_\\star ^{(s)}= \\int d\\eta d^3x \\frac{1}{2}\\left\\lbrace {u^{\\prime }}^2- u \\left[ {2}\\left( {\\cal G}_1^{\\prime }{\\cal G}_1^{-1} \\right)^{\\prime }- {4}\\left({\\cal G}_1^{\\prime }{\\cal G}_1^{-1}\\right)^2- {\\cal G}_1 \\left( {\\cal G}_2 {\\cal G}_1^{-1} \\right)^{\\prime }- {\\cal G}_2^2 + 4{\\cal G}_1{\\cal G}_3 \\right]u \\right\\rbrace \\,$ for a true scalar degree of freedom $u$ .","In UV limit ($3 \\tilde{{\\alpha }_4} \\equiv 3 {\\alpha }_4 +2 {\\alpha }_5+8 {\\alpha }_6, ~z \\equiv a \\phi _0^{\\prime }/{\\cal H}$ ), its equations of motion reduce to $u^{\\prime \\prime } =- {\\omega }_{u(UV)}^2 u \\, , ~~{\\omega }_{u(UV)}^2= {a^2 z^2 } \\left[ 2+{4 (1-3 {\\lambda })}{a^2}\\right] \\Delta ^3.$ Here, it is important to note that there are sixth-spatial derivatives, as required by the scale invariance of the observed power spectrum [2] as well as the (power-counting) renormalizability [1].","This occurs only when there are sixth-derivative terms in the starting scalar action as well as some breaking of the detailed balance condition in sixth-derivative terms for the gravity action (i.e., $\\tilde{{\\alpha }_4} \\ne 0$ ) Regarding the scale invariance of the power spectrums, it has been noted that Hořava gravity could provide an alternative mechanism for the early Universe without introducing the hypothetical inflationary epoch [2]: The basic reason of the alternative mechanism comes from the momentum-dependent speeds of gravitational perturbations which could be much larger than the current, low energy (i.e., IR) speed $c$ so that the exponentially expanding early space-time could be mimicked.","In order to see this explicitly in our case, we consider the power-law expansions [12] ($t$ is the physical time, defined by $dt=a d \\eta $ ), $a =a_0 t^p, ~(1/3 < p <1).$ Then we can produce the scale-invariant power spectrums for the quantum field $\\hat{\\zeta }$ of the $\\zeta $ (scalar) perturbation, as follows, $\\left<0|\\hat{\\zeta }_{\\bf k} (\\eta ) \\hat{\\zeta }_{{\\bf k}^{\\prime }} (\\eta )|0\\right>=(2 \\pi )^3 {\\delta }({\\bf k}+{\\bf k}^{\\prime }) {k^3} \\Delta ^2_{\\zeta } (k),~~\\Delta ^2_{\\zeta }={2 \\pi ^2} | \\zeta _{\\bf k} |^2={8 \\pi ^2} \\sqrt{{6 {\\kappa }^2 \\xi _3 \\tilde{{\\alpha }}_4}}~,$ without knowing the details of the history of the early Universe and the form of the (non-derivative) potential $V (\\phi )$ .","This result shows that one can achieve the \"inflation without inflation\" picture, as argued by Mukohyama [2].","The basic reason of this is that, in Hořava gravity, due to the momentum-dependent, superluminal speeds of fluctuations is possible in the early Universe, which is assumed to be UV region, so that one can mimic \"the inflationary scenario without inflationary epoch\" !","(See Fig.","1 in [5].)"],["Acknowledgments","This was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1A2B401304)."]],"string":"[\n [\n \"On Gauge Invariant Cosmological Perturbations in UV-modified Horava\\n Gravity: A Brief Introduction\"\n ],\n [\n \"Abstract We revisit gauge invariant cosmological perturbations in UV-modified, z = 3 Horava gravity with one scalar matter field, which has been proposed as a renormalizable gravity theory without the ghost problem in four dimensions.\",\n \"We confirm that there is no extra graviton modes and general relativity is recovered in IR, which achieves the consistency of the model.\",\n \"From the UV-modification terms which break the detailed balance condition in UV, we obtain scale-invariant power spectrums for non-inflationary backgrounds, like the power-law expansions, without knowing the details of early expansion history of Universe.\",\n \"This could provide a new framework for the Big Bang cosmology.\"\n ],\n [\n \"Introduction\",\n \"(1).\",\n \"As a particle physicist's point of view, a surprise of cosmology is as follows.\",\n \"In particle physics, we need (fundamental) scalar fields (called Higgs fields) for “theoretical\\\" reasons (i.e., to give masses to fundamental particles with renormalizability), but we have waited for a long time (about 40 years, after 1964 papers) for an experimental confirmation at LHC (4 July, 2012).\",\n \"This is (thought to be) the first elementary scalar particle discovered in Nature.\",\n \"However, in cosmology, it has been known for a long time that there is scalar (cosmological fluctuation mode) in the sky after the discovery of CMB (1964, the same year that Higgs field has been proposed !\",\n \").\",\n \"But, the existence of a scalar mode in the sky is a big mystery from the following reasons.\",\n \"The small fluctuations in CMB are thought to be due to (space-time) fluctuations in the early epoch of our (Big Bang) Universe, which should be described by General Relativity (GR).\",\n \"However, GR alone can not have scalar (fluctuation) mode but only the (usual) tensor mode (2 polarizations, called gravitational waves) !\",\n \"The cosmological/primordial tensor modes have not been discovered yet, though astrophysical tensor modes (gravitational waves) from black hole mergers have been detected in LIGO.\",\n \"So, it would not be quite strange (at least to me) even though we discover the cosmological/primordial tensor modes (called B-mode) in the near future.\",\n \"Rather, the existence of the scalar mode is much more surprising !\",\n \"The simplest way to explain the data is introduce a “cosmic scalar\\\" field and “assume\\\" some peculiar behaviors in the early epoch (known as the inflationary epoch).\",\n \"But, there are huge numbers of explicit models and we still do not know what is the right one and its origin–Is Higgs the remnant of the primordial scalar ?\",\n \"This situation is opposite to that of particle physics: In cosmology, we have discovered scalar mode (CMB) long ago but we do not have its theory yet !\",\n \"Another way to explain the scalar mode is to modify GR, like $f(R)$ , massive gravity, etc., so that the gravity itself has additional scalar modes.\",\n \"But, in that case the scalar (gravitation) could be disaster since it could affect the known (or, well-established) GR test in solar system (,i.e., low energy (IR) test of GR), unless we decouple it from low energy GR sector.\",\n \"And also, this could affect the dark matter and dark energy problems as well.\",\n \"Sometimes, the scalar could be ghost as well, which should be avoided.\",\n \"Today, I will consider another modified gravity theory but without the scalar gravitation mode so that the usual scalar matters are needed in explaining the cosmology data.\",\n \"(2).\",\n \"One the other hand, our universe is considered to be created from “quantum (vacuum) fluctuations\\\".\",\n \"Actually, since the scalar power spectrum in inflationary cosmology contains Planck constant $\\\\hbar $ as well as Newton's constant $G$ ($H$ is the Hubble parameter), $\\\\Delta ^2(k)= 8 \\\\pi G \\\\hbar ( {2 H^2}/{\\\\pi ^2}),$ one can consider the power spectrum as a “quantum gravity effect\\\" and so does the inflationary cosmology as a “quantum cosmology\\\" !\",\n \"Moreover, recent LIGO's detections of gravitational waves from merging black holes seems to imply that we need to consider quantum gravity more seriously.\",\n \"Actually, we open the strong gravity test era of GR, beyond the weak gravity test in solar system.\",\n \"(3).\",\n \"Do we have quantum gravity then ?\",\n \"For quantum theory of particle's interactions, “renormalization\\\" has been a powerful constraint and Higgs particle is its natural consequence.\",\n \"Now, what if we require renormalizability in quantum gravity ?\",\n \"But, it is well-known that the renormalizable quantum gravity can not be realized in Einstein's gravity or its (relativistic) higher-derivative generalizations: There are “ghosts\\\" which have kinetic terms with a wrong sign, in addition to massless gravitons.\",\n \"Recently, Hořava (or Hořava-Lifshitz (HL)) gravity has been proposed as a renormalzable gravity [1].\",\n \"There is no ghosts (in the tensor modes) by abandoning the equal-footing treatment of space and time (i.e., Lorentz symmetry) in the higher energy regime (UV).\",\n \"It is power-counting renormalizable but no proof of renormalizability yet.\",\n \"This resembles the situations of Yang-Mills theory when it first appeared.\",\n \"In cosmology, this theory may provide inflationary effect without “inflationary phase (i.e., early de Sitter or accelerating phase)\\\" so that scale-invariant (scalar/tensor) spectrum can be also produced.\",\n \"This has been first argued by Mukohyama [2] but there is no rigorous analysis about this yet.\",\n \"On the contrary, in the works of Gao et.\",\n \"al.\",\n \"[3] and Gong et.\",\n \"al.\",\n \"[4], scalar spectrum are not scale-invariant though tensor spectrum does.\",\n \"This is today's topic and it provides a natural formulation of cosmology, known as “effective field theory (EFT)\\\", by construction.\"\n ],\n [\n \"Hořava (-Lifshitz) Gravity: Basic Idea\",\n \"In 2009, Hořava proposed a renormalizable, higher-derivative gravity theory, without ghost problems, by abandoning Lorentz symmetry in UV but keeping, so called, “foliation preserving\\\" diffemorphism ($FPDiff$ ) [1].\",\n \"The basic ingredients of the Hořava are (1) Einstein gravity (with a Lorentz deformation parameter $\\\\lambda $ ), (2) non-covariant deformations with higher-spatial derivatives (up to six orders), and (3) “detailed balance\\\" in the coefficients (with five constant parameters, $\\\\kappa , {\\\\lambda }, \\\\nu , \\\\mu , {\\\\Lambda }_W$ ), in contrast to the Einstein gravity case with the three physical parameters $c, G, {\\\\Lambda }$ and ${\\\\lambda }=1$ .\",\n \"Here, six orders of spatial derivatives $(z=3)$ came from the power counting renormalizability in $3+1$ dimensions.\",\n \"(For the limitation of space and time, I can not explain all the details of the construction.\",\n \"So, please read the original work of Hořava [1] about this.)\",\n \"From the Hořava's construction, some improved UV behaviors, without ghosts, are expected so that the gravity theory “could\\\" be renormalizable and one might have predictable quantum gravity theory.\",\n \"But, in cosmology applications, it seems that the detailed-balance condition is too strong to get the required, scale invariant cosmological perturbations.\",\n \"For example, tensor spectrum is scale invariant but scalar spectrum is not !\",\n \"[3], [4].\",\n \"So, we need to break the detailed balancing in some way but without altering the improved UV behaviors of scale invariant tensor modes in the previous works [3], [4] and we called it as “UV modifications\\\" [5].\"\n ],\n [\n \"Cosmological Perturbations in Hořava gravity\",\n \"We start by considering the four-dimensional, UV-modified Hořava gravity action with $z=3$ , which is power-counting renormalizable [1], $S_\\\\mathrm {g} &=& \\\\int d \\\\eta d^3x \\\\sqrt{g} N \\\\left[ \\\\frac{2}{\\\\kappa ^2}\\\\left(K_{ij}K^{ij} - \\\\lambda K^2 \\\\right) - {\\\\cal V} \\\\right], \\\\\\\\-{\\\\cal V}&=& \\\\sigma + \\\\xi R + \\\\alpha _1 R^2+ \\\\alpha _2 R_{ij}R^{ij}+\\\\alpha _3 \\\\frac{\\\\epsilon ^{ijk}}{\\\\sqrt{g}}R_{il}\\\\nabla _jR^l{}_k+ \\\\alpha _4 \\\\nabla _{i}R_{jk} \\\\nabla ^{i}{R}^{jk}+{\\\\alpha }_5 \\\\nabla _{i}R_{jk}\\\\nabla ^{j} {R}^{ik}+{\\\\alpha }_6 \\\\nabla _{i}R\\\\nabla ^{i}R , \\\\numero $ where (the prime $(^{\\\\prime })$ denotes the derivative with respect to $\\\\eta $ ) $K_{ij}=\\\\frac{1}{2N}\\\\left({g_{ij}}^{\\\\prime }-\\\\nabla _i N_j-\\\\nabla _jN_i\\\\right)\\\\, ,~ds^2 =-N^2 c^2 d\\\\eta ^2 +g_{ij}\\\\left(dx^i+N^i d\\\\eta \\\\right) \\\\left(dx^j+N^j d\\\\eta \\\\right).\",\n \"\\\\numero $ With the detailed balance condition, the number of independent coupling constants can be reduced to six, i.e., $\\\\kappa ,\\\\lambda ,\\\\mu ,\\\\nu ,{\\\\Lambda }_W,{\\\\omega }$ for a viable model in IR [8], [9], $\\\\sigma = {8 (3 {\\\\lambda }-1)},~\\\\xi ={8 (3 {\\\\lambda }-1)},~{\\\\alpha }_1={32 (3 {\\\\lambda }-1)}, ~{\\\\alpha }_2=-{8}, ~{\\\\alpha }_3 = {2 \\\\nu ^2},~{\\\\alpha }_4=-{8 2 \\\\nu ^4}=-{\\\\alpha }_5=-8 {\\\\alpha }_6,$ But, we do not restrict to this case only, at least for the UV couplings ${\\\\alpha }_4,{\\\\alpha }_5,{\\\\alpha }_6$ so that the power-counting renormalizable and scale-invariant cosmological scalar fluctuations can be obtained.\",\n \"For the power-counting renormalizable matter action, we consider $z=3$ scalar field action [6], [7], $S_\\\\mathrm {m} = \\\\int d\\\\eta d^3x \\\\sqrt{g} N \\\\left[ \\\\frac{1}{2N^2}\\\\left( \\\\phi ^{\\\\prime } - N^i {\\\\partial }_i \\\\phi \\\\right)^2 - V(\\\\phi )- Z({\\\\partial }_i\\\\phi ) \\\\right] \\\\, ,$ where $Z({\\\\partial }_i\\\\phi ) = \\\\sum _{n=1}^3 \\\\xi _n \\\\partial _i^{(n)}\\\\phi \\\\partial ^{i(n)}\\\\phi \\\\,$ with the superscript $(n)$ denoting $n$ -th spatial derivatives, and $V(\\\\phi )$ is the matter's potential without derivatives.\",\n \"The actions, $S_\\\\mathrm {g}$ and $S_\\\\mathrm {m}$ are invariant under the foliation preserving FPDiff [1], $\\\\delta x^i &=&-\\\\zeta ^i (\\\\eta , {\\\\bf x}), ~\\\\delta \\\\eta =-f(\\\\eta ),~\\\\delta g_{ij}={\\\\partial }_i\\\\zeta ^k g_{jk}+{\\\\partial }_j \\\\zeta ^k g_{ik}+\\\\zeta ^k{\\\\partial }_k g_{ij}+f g^{\\\\prime }_{ij},\\\\\\\\\\\\delta N_i &=& {\\\\partial }_i \\\\zeta ^j N_j+\\\\zeta ^j {\\\\partial }_j N_i+\\\\zeta ^{^{\\\\prime }j}g_{ij}+f {N^{\\\\prime }}_i+f^{\\\\prime } N_i,~\\\\delta N= \\\\zeta ^j {\\\\partial }_j N+f N^{\\\\prime }+f^{\\\\prime } N, ~\\\\delta \\\\phi =\\\\zeta ^j {\\\\partial }_j \\\\phi +f \\\\phi ^{\\\\prime }.\",\n \"\\\\numero $ In order to study the cosmological perturbations around the homogeneous and isotropic backgrounds (as seen in CMB), we expand the metric and the scalar field as, $N = a(\\\\eta )[1+{\\\\cal A}(\\\\eta ,{\\\\bf x})] \\\\, ,~N_i = a^2(\\\\eta ){{\\\\cal B}(\\\\eta ,{\\\\bf x})}_i \\\\, , ~g_{ij} = a^2(\\\\eta ) [{\\\\delta }_{ij}+h_{ij} (\\\\eta ,{\\\\bf x})],~\\\\phi = \\\\phi _0(\\\\eta ) + \\\\delta \\\\phi (\\\\eta ,{\\\\bf x}) \\\\, ,$ by considering spatially flat ($k=0$ ) backgrounds and the conformal (or comoving) time $\\\\eta $ , for simplicity.\",\n \"By substituting the metric and scalar field of (REF ) into the actions one can obtain the linear-order perturbation part of the total action $S=S_g +S_m$ which gives the Friedman's equations for the background, $&&{\\\\cal H}^2 =-\\\\frac{\\\\kappa ^2}{6(1-3\\\\lambda )} \\\\left( {2}{\\\\phi _0^{\\\\prime }}^2+ a^2 \\\\left(V_0 -\\\\sigma \\\\right) \\\\right) \\\\, ,~{\\\\cal H}^2 +2{\\\\cal H}^{\\\\prime } =\\\\frac{\\\\kappa ^2}{2(1-3\\\\lambda )} \\\\left( {2}{\\\\phi _0^{\\\\prime }}^2- a^2 \\\\left(V_0 -\\\\sigma \\\\right) \\\\right) \\\\, , \\\\numero \\\\\\\\&&\\\\phi _0^{\\\\prime \\\\prime } + 2{\\\\cal H}\\\\phi _0^{\\\\prime } + a^2V_{\\\\phi _0} = 0 \\\\, ,$ with the comoving Hubble parameter ${\\\\cal H} \\\\equiv a^{\\\\prime }/a$ , $V_0 \\\\equiv V(\\\\phi _0), V_{\\\\phi _0}\\\\equiv ({\\\\partial }V/{\\\\partial }\\\\phi )_{\\\\phi _0}$ , and $h\\\\equiv h^i_i$ .\",\n \"Here, it important to note that there is no higher-derivative corrections to the Friedman's equations for spatially flat case and so the background equations are the same as those of GR [8], [9].\",\n \"However, even in this case, the higher-derivative effects can reappear in the perturbed parts.\",\n \"The quadratic part of the total perturbed action is given by ${\\\\delta }_2 S&=& \\\\int d \\\\eta d^3x \\\\left\\\\lbrace {{\\\\kappa }^2} \\\\left[ (1-3 {\\\\lambda }) {\\\\cal H} \\\\left(3 {\\\\cal H} {\\\\cal A}^2 + {\\\\cal A} (2 {\\\\partial }{\\\\cal B}^i-h^{\\\\prime }) \\\\right)+ (1-{\\\\lambda }) ({\\\\partial }_i {\\\\cal B}^i)^2+{2} {\\\\partial }_i {\\\\cal B}_j {\\\\partial }^i {\\\\cal B}^j \\\\right.\",\n \"\\\\right.\",\n \"\\\\numero \\\\\\\\&&\\\\left.\",\n \"\\\\left.\",\n \"-{\\\\partial }_i {\\\\cal B}_j {h^{ij}}^{\\\\prime } +{4} {h_{ij}}^{\\\\prime } {h^{ij}}^{\\\\prime }+{\\\\lambda }\\\\left({\\\\partial }_i {\\\\cal B}^i h^{\\\\prime }-{4}h^{\\\\prime 2} \\\\right) \\\\right]+a^2 \\\\xi \\\\left({\\\\cal A} +{2} h \\\\right)\\\\left( {\\\\partial }_i {\\\\partial }_j h^{ij} -\\\\Delta h \\\\right) \\\\right.\",\n \"\\\\numero \\\\\\\\&&\\\\left.+a^2 \\\\left[ {2} {\\\\delta }\\\\phi ^{\\\\prime 2}-{\\\\cal A} \\\\phi _0^{\\\\prime } {\\\\delta }\\\\phi ^{\\\\prime } +{2} {\\\\cal A}^2 \\\\phi _0^{\\\\prime 2} +{\\\\partial }_i {\\\\cal B}^i \\\\phi _0^{\\\\prime } {\\\\delta }\\\\phi -{2} V_{\\\\phi _0 \\\\phi _0} {\\\\delta }\\\\phi ^2 -a^2 V_{\\\\phi _0} {\\\\delta }\\\\phi {\\\\cal A}-{2} \\\\phi _0^{\\\\prime } {\\\\delta }\\\\phi h^{\\\\prime }\\\\right] \\\\right.\\\\numero \\\\\\\\&&\\\\left.\",\n \"-a^4 \\\\left({\\\\cal V}^{(2)}+{\\\\delta }Z \\\\right)\\\\right\\\\rbrace ,$ where ${\\\\cal V}^{(2)}$ is the quadratic part of the potential ${\\\\cal V}$ and $\\\\Delta \\\\equiv {\\\\delta }^{ij} {\\\\partial }_i {\\\\partial }_j$ , ${\\\\delta }Z=\\\\sum _{n=1}^3 \\\\xi _n~ \\\\partial _i^{(n)}{\\\\delta }\\\\phi \\\\partial ^{i(n)}{\\\\delta }\\\\phi $ .\",\n \"Now, in order to separate the scalar, vector, and tensor contributions, we consider the most general (SVT) decompositions $({\\\\partial }_i S^i = {\\\\partial }_i F^i = \\\\tilde{H} = {\\\\partial }_i \\\\tilde{H}^i_j = 0 )$ , ${\\\\cal B}_i = {\\\\partial }_i {\\\\cal B} + S_i \\\\, ,~h_{ij} = 2 {\\\\cal R} {\\\\delta }_{ij}+ {\\\\partial }_i {\\\\partial }_j {\\\\cal E} + {\\\\partial }_{(i} F_{j)} + \\\\tilde{H}_{ij} \\\\, .$ Then, the pure tensor, vector, and scalar parts of the total action are given by, respectively, $\\\\delta _2 S^{(t)} &=& \\\\int d \\\\eta d^3x~ a^2\\\\left[ \\\\frac{2}{\\\\kappa ^2} {\\\\tilde{H}_{ij}}^{\\\\prime } {\\\\tilde{H}}^{ij^{\\\\prime }}+\\\\xi \\\\tilde{H}_{ij}\\\\Delta \\\\tilde{H}^{ij}+ \\\\frac{\\\\alpha _2}{a^2}\\\\Delta \\\\tilde{H}_{ij}\\\\Delta \\\\tilde{H}^{ij}+ \\\\frac{\\\\alpha _3}{a^3} \\\\epsilon ^{ijk} \\\\Delta \\\\tilde{H}_{il} \\\\Delta {\\\\partial }_j \\\\tilde{H}^l_{k} \\\\right.\",\n \"\\\\numero \\\\\\\\&&~~~~~~~~~~~~~~~~~~~~\\\\left.- \\\\frac{\\\\alpha _4}{a^4} \\\\Delta \\\\tilde{H}_{ij}\\\\Delta ^2 \\\\tilde{H}^{ij} \\\\right],\\\\\\\\\\\\delta _2 S^{(v)} &=& \\\\frac{1}{\\\\kappa ^2} \\\\int d \\\\eta d^3x~ a^2 {\\\\partial }_i\\\\left( S^j-{F^j}^{\\\\prime } \\\\right) {\\\\partial }_i \\\\left( S_j-F_j^{\\\\prime } \\\\right) \\\\, ,\\\\\\\\\\\\delta _2 S^{(s)}&=& \\\\int d \\\\eta d^3x ~a^2 \\\\left\\\\lbrace \\\\frac{2(1-3\\\\lambda )}{\\\\kappa ^2}\\\\left[ 3{{\\\\cal R}^{\\\\prime }}^2 - 6{\\\\cal H}{\\\\cal A}{\\\\cal R}^{\\\\prime } +3{\\\\cal H}^2 {\\\\cal A}^2- 2\\\\left( {\\\\cal R}^{\\\\prime } - {\\\\cal H}{\\\\cal A}\\\\right) \\\\Delta ({\\\\cal B}-{\\\\cal E}^{\\\\prime })\\\\right] \\\\right.\\\\nonumber \\\\\\\\&&+ \\\\frac{2(1-\\\\lambda )}{\\\\kappa ^2} \\\\left[ \\\\Delta \\\\left({\\\\cal B}-{\\\\cal E}^{\\\\prime }\\\\right) \\\\right]^2- 2\\\\xi ({\\\\cal R}+2{\\\\cal A})\\\\Delta {\\\\cal R}+ \\\\frac{2}{a^2} \\\\left( 8\\\\alpha _1+3\\\\alpha _2 \\\\right)(\\\\Delta {\\\\cal R})^2\\\\nonumber \\\\\\\\&& \\\\left.-\\\\frac{2}{a^4} \\\\left( 3\\\\alpha _4+2\\\\alpha _5+8 {\\\\alpha }_6 \\\\right)\\\\Delta {\\\\cal R}\\\\Delta ^2 {\\\\cal R}-a^2V_{\\\\phi _0} {\\\\cal A}\\\\delta \\\\phi -\\\\frac{1}{2a^2}V_{\\\\phi _0\\\\phi _0}\\\\delta \\\\phi ^2 - \\\\delta {Z} \\\\right.\\\\numero \\\\\\\\&& \\\\left.+ \\\\frac{1}{2}{\\\\delta \\\\phi ^{\\\\prime }}^2 - \\\\phi _0^{\\\\prime }\\\\delta \\\\phi ^{\\\\prime } {\\\\cal A}+{2} {\\\\phi _0^{\\\\prime }}^2 {\\\\cal A}^2+ [\\\\Delta ({\\\\cal B}-{\\\\cal E}^{\\\\prime })-3 {\\\\cal R}^{\\\\prime }]\\\\phi _0^{\\\\prime }\\\\delta \\\\phi \\\\right\\\\rbrace \\\\, .$ Here, it is important to note that sixth-order-derivative terms in the gravity action contribute to scalar as well as tensor perturbations, through the specific combination of `$3\\\\alpha _4+2\\\\alpha _5+8 {\\\\alpha }_6$ ' for the former but through only `${\\\\alpha }_4$ ' for the latter.\",\n \"This is what we need for renormalizability and scale-invariant spectrums (as we can see shortly), for both scalar and tensor.\",\n \"In the detailed-balanced case, the combination `$3\\\\alpha _4+2\\\\alpha _5+8 {\\\\alpha }_6$ ' vanishes though ${\\\\alpha }_4$ does not.\",\n \"So in that case, the theory would not be renormalizable and nor scale invariant for scalar part, which is in contradict to observational data !\",\n \"Now, in order to exhibit the true dynamical degrees of freedom we consider the Hamiltonian reduction method [10], for the cosmologically perturbed actions [11], [4] and, after some computation, the action reduces to $\\\\delta _2S_\\\\star ^{(s)}= \\\\int d\\\\eta d^3x \\\\frac{1}{2}\\\\left\\\\lbrace {u^{\\\\prime }}^2- u \\\\left[ {2}\\\\left( {\\\\cal G}_1^{\\\\prime }{\\\\cal G}_1^{-1} \\\\right)^{\\\\prime }- {4}\\\\left({\\\\cal G}_1^{\\\\prime }{\\\\cal G}_1^{-1}\\\\right)^2- {\\\\cal G}_1 \\\\left( {\\\\cal G}_2 {\\\\cal G}_1^{-1} \\\\right)^{\\\\prime }- {\\\\cal G}_2^2 + 4{\\\\cal G}_1{\\\\cal G}_3 \\\\right]u \\\\right\\\\rbrace \\\\,$ for a true scalar degree of freedom $u$ .\",\n \"In UV limit ($3 \\\\tilde{{\\\\alpha }_4} \\\\equiv 3 {\\\\alpha }_4 +2 {\\\\alpha }_5+8 {\\\\alpha }_6, ~z \\\\equiv a \\\\phi _0^{\\\\prime }/{\\\\cal H}$ ), its equations of motion reduce to $u^{\\\\prime \\\\prime } =- {\\\\omega }_{u(UV)}^2 u \\\\, , ~~{\\\\omega }_{u(UV)}^2= {a^2 z^2 } \\\\left[ 2+{4 (1-3 {\\\\lambda })}{a^2}\\\\right] \\\\Delta ^3.$ Here, it is important to note that there are sixth-spatial derivatives, as required by the scale invariance of the observed power spectrum [2] as well as the (power-counting) renormalizability [1].\",\n \"This occurs only when there are sixth-derivative terms in the starting scalar action as well as some breaking of the detailed balance condition in sixth-derivative terms for the gravity action (i.e., $\\\\tilde{{\\\\alpha }_4} \\\\ne 0$ ) Regarding the scale invariance of the power spectrums, it has been noted that Hořava gravity could provide an alternative mechanism for the early Universe without introducing the hypothetical inflationary epoch [2]: The basic reason of the alternative mechanism comes from the momentum-dependent speeds of gravitational perturbations which could be much larger than the current, low energy (i.e., IR) speed $c$ so that the exponentially expanding early space-time could be mimicked.\",\n \"In order to see this explicitly in our case, we consider the power-law expansions [12] ($t$ is the physical time, defined by $dt=a d \\\\eta $ ), $a =a_0 t^p, ~(1/3 < p <1).$ Then we can produce the scale-invariant power spectrums for the quantum field $\\\\hat{\\\\zeta }$ of the $\\\\zeta $ (scalar) perturbation, as follows, $\\\\left<0|\\\\hat{\\\\zeta }_{\\\\bf k} (\\\\eta ) \\\\hat{\\\\zeta }_{{\\\\bf k}^{\\\\prime }} (\\\\eta )|0\\\\right>=(2 \\\\pi )^3 {\\\\delta }({\\\\bf k}+{\\\\bf k}^{\\\\prime }) {k^3} \\\\Delta ^2_{\\\\zeta } (k),~~\\\\Delta ^2_{\\\\zeta }={2 \\\\pi ^2} | \\\\zeta _{\\\\bf k} |^2={8 \\\\pi ^2} \\\\sqrt{{6 {\\\\kappa }^2 \\\\xi _3 \\\\tilde{{\\\\alpha }}_4}}~,$ without knowing the details of the history of the early Universe and the form of the (non-derivative) potential $V (\\\\phi )$ .\",\n \"This result shows that one can achieve the \\\"inflation without inflation\\\" picture, as argued by Mukohyama [2].\",\n \"The basic reason of this is that, in Hořava gravity, due to the momentum-dependent, superluminal speeds of fluctuations is possible in the early Universe, which is assumed to be UV region, so that one can mimic \\\"the inflationary scenario without inflationary epoch\\\" !\",\n \"(See Fig.\",\n \"1 in [5].)\"\n ],\n [\n \"Acknowledgments\",\n \"This was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1A2B401304).\"\n ]\n]"},"id":{"kind":"string","value":"1709.01671"}}},{"rowIdx":990,"cells":{"text":{"kind":"list like","value":[["Inner amenability and approximation properties of locally compact\n quantum groups"],["Abstract We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction.","We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\\mathbb{G}$.","Similar homological techniques are used to prove that $\\ell^1(\\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\\mathbb{G}$; a discrete Kac algebra $\\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\\widehat{\\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\\mathbb{G}$ implies $C_u(\\widehat{\\mathbb{G}})=L^1(\\widehat{\\mathbb{G}})\\widehat{\\otimes}_{L^1(\\widehat{\\mathbb{G}})}C_0(\\widehat{\\mathbb{G}})$ completely isometrically.","The latter result allows us to partially answer a conjecture of Voiculescu when $\\mathbb{G}$ has the approximation property."],["Introduction","The class of inner amenable locally compact groups has played an important role in the development of abstract harmonic analysis and its connection to operator algebras [46], [48], [58].","In particular, it provides a sufficiently general class of locally compact groups for which approximation properties of $G$ can be characterized through approximation properties of its reduced $C^*$ -algebra $C^*_\\lambda (G)$ and von Neumann algebra $VN(G)$ .","Indeed, within the class of inner amenable groups, amenability of $G$ is equivalent to nuclearity of $C^*_\\lambda (G)$ as well as injectivity of $VN(G)$ [46]; exactness of $G$ is equivalent to exactness of $C^*_\\lambda (G)$ [1], and the Haagerup property of $G$ is equivalent to the Haagerup property of $VN(G)$ [52].","In [33], a notion of inner amenability was introduced for an arbitrary locally compact quantum group $\\mathbb {G}$ by means of the existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,x\\star f\\rangle =\\langle m,f\\star x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Although such a condition may be interesting in its own right, if $\\mathbb {G}$ is co-amenable, then any cluster point in $L^{\\infty }(\\mathbb {G})^*$ of a bounded approximate identity in $L^1(\\mathbb {G})$ will give rise to such a state.","In particular, any locally compact group is an “inner amenable” locally compact quantum group in the sense of [33].","In this paper we introduce a bona fide generalization of inner amenability to locally compact quantum groups.","We study its basic properties, examples, and related notions of strong and topological inner amenability.","Generalizing a well-known result of Lau–Paterson [46], we show that $\\mathbb {G}$ is amenable if and only if $L^{\\infty }(\\widehat{\\mathbb {G}})$ is injective and $\\mathbb {G}$ is inner amenable.","We also derive sufficient conditions under which the bicrossed product of a matched pair of locally compact groups is inner amenable.","We pursue homological manifestations of this concept in section 4, showing that inner amenability of $\\mathbb {G}$ entails the relative 1-flatness of $L^1(\\widehat{\\mathbb {G}})$ .","As an application of our techniques, we prove that $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.","This resolves a recent conjecture of Ng–Viselter [51], and generalizes a recent result of Ng [50] from locally compact groups to arbitrary Kac algebras, wherein $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.","The method of proof shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.","Indeed, we show that for strongly inner amenable quantum groups, weak amenability and the approximation property follow respectively from the w*CBAP and w*OAP of $L^{\\infty }(\\widehat{\\mathbb {G}})$ .","In particular, for inner amenable locally compact groups $G$ , the w*CBAP of $VN(G)$ implies weak amenability of $G$ and the approximation property is equivalent to the w*OAP of $VN(G)$ .","Similar results are proved in the setting of topological inner amenability and approximation properties of $C_0()$ .","These techniques may be viewed as new tools for the development of harmonic analysis on quantum groups beyond the unimodular discrete setting (for which the above results greatly simplify).","In section 5 we establish the self-duality of (non-relative) 1-biflatness, that is, $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.","This shows, in particular, that $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group $\\mathbb {G}$ .","Section 6 is devoted to the question of operator amenability of $L^1_{cb}(\\mathbb {G})$ , the $cb$ -multiplier closure of $L^1(\\mathbb {G})$ .","For unimodular discrete quantum groups $\\mathbb {G}$ with Kirchberg's factorization property, we show that $\\mathbb {G}$ is weakly amenable if and only if $L^1_{cb}()$ is operator amenable.","This result is new even for the class of weakly amenable residually finite discrete groups such that $C^*(G)$ is not residually finite-dimensional (see [6]).","We finish in section 7 with a strengthening of [14], by showing that $C_u(\\mathbb {G})^*\\cong M_{cb}^l(L^1(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically when $$ is amenable.","As a corollary, when $$ has the approximation property, we prove that $\\mathbb {G}$ is co-amenable if and only if $$ is amenable."],["Operator Modules","Let $\\mathcal {A}$ be a completely contractive Banach algebra.","We say that an operator space $X$ is a right operator $\\mathcal {A}$ -module if it is a right Banach $\\mathcal {A}$ -module such that the module map $m_X:X\\widehat{\\otimes }\\mathcal {A}\\rightarrow X$ is completely contractive, where $\\widehat{\\otimes }$ denotes the operator space projective tensor product.","We say that $X$ is faithful if for every non-zero $x\\in X$ , there is $a\\in \\mathcal {A}$ such that $x\\cdot a\\ne 0$ , and we say that $X$ is essential if $\\langle X\\cdot \\mathcal {A}\\rangle =X$ , where $\\langle \\cdot \\rangle $ denotes the closed linear span.","Note that our definition of faithfulness, the standard notion in operator modules, is opposite in nature to the usual definition in algebra.","We denote by $\\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ the category of right operator $\\mathcal {A}$ -modules with morphisms given by completely bounded module homomorphisms.","Left operator $\\mathcal {A}$ -modules and operator $\\mathcal {A}$ -bimodules are defined similarly, and we denote the respective categories by $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ and $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .","Regarding terminology, in what follows we will often omit the term “operator” when discussing homological properties of operator modules as we will be working exclusively in the operator space category.","Let $\\mathcal {A}$ be a completely contractive Banach algebra, $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ and $Y\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ .","The $\\mathcal {A}$ -module tensor product of $X$ and $Y$ is the quotient space $X\\widehat{\\otimes }_{\\mathcal {A}}Y:=X\\widehat{\\otimes }Y/N$ , where $N=\\langle x\\cdot a\\otimes y-x\\otimes a\\cdot y\\mid x\\in X, \\ y\\in Y, \\ a\\in \\mathcal {A}\\rangle ,$ and, again, $\\langle \\cdot \\rangle $ denotes the closed linear span.","It follows that (see [9]) $\\mathcal {CB}_{\\mathcal {A}}(X,Y^*)\\cong N^{\\perp }\\cong (X\\widehat{\\otimes }_{\\mathcal {A}} Y)^*,$ where $\\mathcal {CB}_{\\mathcal {A}}(X,Y^*)$ is the space of completely bounded right $\\mathcal {A}$ -module maps $\\Phi :X\\rightarrow Y^*$ .","If $Y=\\mathcal {A}$ , then clearly $N\\subseteq \\mathrm {Ker}(m_X)$ where $m_X:X\\widehat{\\otimes }\\mathcal {A}\\rightarrow X$ is the module map.","If the induced mapping $\\widetilde{m}_X:X\\widehat{\\otimes }_{\\mathcal {A}}\\mathcal {A}\\rightarrow X$ is a completely isometric isomorphism we say that $X$ is an induced $\\mathcal {A}$ -module.","A similar definition applies for left modules.","In particular, we say that $\\mathcal {A}$ is self-induced if $\\widetilde{m}_\\mathcal {A}:\\mathcal {A}\\widehat{\\otimes }_{\\mathcal {A}}\\mathcal {A}\\cong \\mathcal {A}$ completely isometrically.","Let $\\mathcal {A}$ be a completely contractive Banach algebra and $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ .","The identification $\\mathcal {A}^+=\\mathcal {A}\\oplus _1 turns the unitization of $ A$ into a unital completely contractive Banach algebra, and it follows that $ X$ becomes a right operator $ A+$-module via the extended action\\begin{equation*}x\\cdot (a+\\lambda e)=x\\cdot a+\\lambda x, \\ \\ \\ a\\in \\mathcal {A}^+, \\ \\lambda \\in \\ x\\in X.\\end{equation*}Let $ C1$.","Then $ X$ is \\emph {relatively $ C$-projective} if there exists a morphism $ +:XXA+$ which is a right inverse to the extended module map $ mX+:XA+X$ such that $ +cbC$.","When $ X$ is essential, then $ X$ is relatively $ C$-projective if and only if there exists a morphism $ :XXA$ satisfying $ cbC$ and $ mX=idX$ by the operator analogue of \\cite [Proposition 1.2]{DP}.","We say that $ X$ is \\emph {$ C$-projective} if for every $ Y,Zmod A$, every complete quotient morphism $ :YZ$, every morphism $ :XZ$, and every $ >0$, there exists a morphism $ :XY$ such that $ cb< Ccb+$ and $ =$, i.e., the following diagram commutes:\\begin{equation*}\\begin{tikzcd}&Y [d, two heads, \"\\Psi \"]\\\\X [ru, dotted, \"\\widetilde{\\Phi }_\\varepsilon \"] [r, \"\\Phi \"] &Z\\end{tikzcd}\\end{equation*}$ Given a completely contractive Banach algebra $\\mathcal {A}$ and $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ , there is a canonical completely contractive morphism $\\Delta ^+:X\\rightarrow \\mathcal {CB}(\\mathcal {A}^+,X)$ given by $\\Delta ^+(x)(a)=x\\cdot a, \\ \\ \\ x\\in X, \\ a\\in \\mathcal {A}^+,$ where the right $\\mathcal {A}$ -module structure on $\\mathcal {CB}(\\mathcal {A}^+,X)$ is defined by $(\\Psi \\cdot a)(b)=\\Psi (ab), \\ \\ \\ a\\in \\mathcal {A}, \\ \\Psi \\in \\mathcal {CB}(\\mathcal {A}^+,X), \\ b\\in \\mathcal {A}^+.$ An analogous construction exists for objects in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ .","Let $C\\ge 1$ .","Then $X$ is relatively $C$ -injective if there exists a morphism $\\Phi ^+:\\mathcal {CB}(\\mathcal {A}^+,X)\\rightarrow X$ such that $\\Phi ^+\\circ \\Delta ^+=\\textnormal {id}_{X}$ and $\\Vert \\Phi ^+\\Vert _{cb}\\le C$ .","When $X$ is faithful, then $X$ is relatively $C$ -injective if and only if there exists a morphism $\\Phi :\\mathcal {CB}(\\mathcal {A},X)\\rightarrow X$ such that $\\Phi \\circ \\Delta =\\textnormal {id}_{X}$ and $\\Vert \\Phi \\Vert _{cb}\\le C$ by the operator analogue of [17], where $\\Delta (x)(a):=\\Delta ^+(x)(a)$ for $x\\in X$ and $a\\in \\mathcal {A}$ .","We say that $X$ is $C$ -injective if for every $Y,Z\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ , every completely isometric morphism $\\Psi :Y\\hookrightarrow Z$ , and every morphism $\\Phi :Y\\rightarrow X$ , there exists a morphism $\\widetilde{\\Phi }:Z\\rightarrow X$ such that $\\Vert \\widetilde{\\Phi }\\Vert _{cb}\\le C\\Vert \\Phi \\Vert _{cb}$ and $\\widetilde{\\Phi }\\circ \\Psi =\\Phi $ , that is, the following diagram commutes: $\\begin{tikzcd}Z [rd, dotted, \"\\widetilde{\\Phi }\"]\\\\Y [u, hook, \"\\Psi \"] [r, \"\\Phi \"] &X\\end{tikzcd}$ There is a natural categorical equivalence between $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ and $\\mathbf {mod\\hspace{2.0pt}\\mathcal {A}^{\\mathrm {op}}\\widehat{\\otimes }\\mathcal {A}}$ given by $axb=x\\cdot (a\\otimes b), \\ \\ \\ a,b\\in \\mathcal {A}, \\ x\\in X, \\ X\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}.$ With this identification, we obtain the following bimodule analogue of [14].","Proposition 2.1 Let $\\mathcal {A}$ be a completely contractive Banach algebra and $X\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .","If $X$ is $C_1$ -injective in $\\mathbf {mod}- and is relatively $ C2$-injective in $ A mod A$, then $ X$ is $ C1C2$-injective in $ A mod A$.$"],["Locally Compact Quantum Groups","A locally compact quantum group is a quadruple $\\mathbb {G}=(L^{\\infty }(\\mathbb {G}),\\Gamma ,\\varphi ,\\psi )$ , where $L^{\\infty }(\\mathbb {G})$ is a Hopf-von Neumann algebra with co-multiplication $\\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , and $\\varphi $ and $\\psi $ are fixed left and right Haar weights on $L^{\\infty }(\\mathbb {G})$ , respectively [43], [44].","For every locally compact quantum group $\\mathbb {G}$ , there exists a left fundamental unitary operator $W$ on $L^2(\\mathbb {G},\\varphi )\\otimes L^2(\\mathbb {G},\\varphi )$ and a right fundamental unitary operator $V$ on $L^2(\\mathbb {G},\\psi )\\otimes L^2(\\mathbb {G},\\psi )$ implementing the co-multiplication $\\Gamma $ via $\\Gamma (x)=W^*(1\\otimes x)W=V(x\\otimes 1)V^*, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Both unitaries satisfy the pentagonal relation; that is, $W_{12}W_{13}W_{23}=W_{23}W_{12}\\hspace{10.0pt}\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}\\hspace{10.0pt}V_{12}V_{13}V_{23}=V_{23}V_{12}.$ By [44], we may identify $L^2(\\mathbb {G},\\varphi )$ and $L^2(\\mathbb {G},\\psi )$ , so we will simply use $L^2(\\mathbb {G})$ for this Hilbert space throughout the paper.","We denote by $R$ the unitary antipode of $\\mathbb {G}$ .","Let $L^1(\\mathbb {G})$ denote the predual of $L^{\\infty }(\\mathbb {G})$ .","Then the pre-adjoint of $\\Gamma $ induces an associative completely contractive multiplication on $L^1(\\mathbb {G})$ , defined by $\\star :L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})\\ni f\\otimes g\\mapsto f\\star g=\\Gamma _*(f\\otimes g)\\in L^1(\\mathbb {G}).$ The multiplication $\\star $ is a complete quotient map from $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ onto $L^1(\\mathbb {G})$ , implying $\\langle L^1(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle =L^1(\\mathbb {G}).$ Moreover, $L^1(\\mathbb {G})$ is always self-induced.","The proof follows from [63] (see [14] for details).","The canonical $L^1(\\mathbb {G})$ -bimodule structure on $L^{\\infty }(\\mathbb {G})$ is given by $f\\star x=(\\textnormal {id}\\otimes f)\\Gamma (x) \\ \\ \\ \\ x\\star f=(f\\otimes \\textnormal {id})\\Gamma (x), \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}), \\ f\\in L^1(\\mathbb {G}).$ A left invariant mean on $L^{\\infty }(\\mathbb {G})$ is a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,x\\star f \\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}), \\ f\\in L^1(\\mathbb {G}).$ Right and two-sided invariant means are defined similarly.","A locally compact quantum group $\\mathbb {G}$ is said to be amenable if there exists a left invariant mean on $L^{\\infty }(\\mathbb {G})$ .","It is known that $\\mathbb {G}$ is amenable if and only if there exists a right (equivalently, two-sided) invariant mean (cf.","[26]).","We say that $\\mathbb {G}$ is co-amenable if $L^1(\\mathbb {G})$ has a bounded left (equivalently, right or two-sided) approximate identity (cf.","[5]).","The left regular representation $\\lambda :L^1(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ of $\\mathbb {G}$ is defined by $\\lambda (f)=(f\\otimes \\textnormal {id})(W), \\ \\ \\ f\\in L^1(\\mathbb {G}),$ and is an injective, completely contractive homomorphism from $L^1(\\mathbb {G})$ into $\\mathcal {B}(L^2(\\mathbb {G}))$ .","Then $L^{\\infty }(\\widehat{\\mathbb {G}}):=\\lbrace \\lambda (f) : f\\in L^1(\\mathbb {G})\\rbrace ^{\\prime \\prime }$ is the von Neumann algebra associated with the dual quantum group $$ .","Analogously, we have the right regular representation $\\rho :L^1(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\rho (f)=(\\textnormal {id}\\otimes f)(V), \\ \\ \\ f\\in L^1(\\mathbb {G}),$ which is also an injective, completely contractive homomorphism from $L^1(\\mathbb {G})$ into $\\mathcal {B}(L^2(\\mathbb {G}))$ .","Then $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime }):=\\lbrace \\rho (f) : f\\in L^1(\\mathbb {G})\\rbrace ^{\\prime \\prime }$ is the von Neumann algebra associated to the quantum group $^{\\prime }$ .","It follows that $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })=L^{\\infty }(\\widehat{\\mathbb {G}})^{\\prime }$ , and the left and right fundamental unitaries satisfy $W\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})$ and $V\\in L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ [44].","Moreover, dual quantum groups always satisfy $L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })=$ [66].","The modular conjugations of the dual Haar weights give rise to conjugate linear isometries $J,\\widehat{J}:L^2(\\mathbb {G})\\rightarrow L^2(\\mathbb {G})$ satisfying $JL^{\\infty }(\\mathbb {G})J=L^{\\infty }(\\mathbb {G})^{\\prime }\\hspace{10.0pt}\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}\\hspace{10.0pt}\\widehat{J}L^{\\infty }(\\widehat{\\mathbb {G}})\\widehat{J}=L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime }).$ Moreover, the unitary $U:=\\widehat{J}J$ intertwines the left and right regular representations via $\\rho (f)=U\\lambda (f)U^*$ , $f\\in L^1(\\mathbb {G})$ .","At the level of the fundamental unitaries, this relation becomes $V=\\sigma (1\\otimes U)W(1\\otimes U^*)\\sigma .$ We also record the adjoint formulas $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*$ and $(J\\otimes \\widehat{J})V(J\\otimes \\widehat{J})=V^*$ .","If $G$ is a locally compact group, we let $\\mathbb {G}_a=(L^{\\infty }(G),\\Gamma _a,\\varphi _a,\\psi _a)$ denote the commutative quantum group associated with the commutative von Neumann algebra $L^{\\infty }(G)$ , where the co-multiplication is given by $\\Gamma _a(f)(s,t)=f(st)$ , and $\\varphi _a$ and $\\psi _a$ are integration with respect to a left and right Haar measure, respectively.","The dual $_a$ of $\\mathbb {G}_a$ is the co-commutative quantum group $\\mathbb {G}_s=(VN(G),\\Gamma _s,\\varphi _s,\\psi _s)$ , where $VN(G)$ is the left group von Neumann algebra with co-multiplication $\\Gamma _s(\\lambda (t))=\\lambda (t)\\otimes \\lambda (t)$ , and $\\varphi _s=\\psi _s$ is Haagerup's Plancherel weight (cf.","[62]).","Then $L^1(\\mathbb {G}_a)$ is the usual group convolution algebra $L^1(G)$ , and $L^1(\\mathbb {G}_s)$ is the Fourier algebra $A(G)$ .","It is known that every commutative locally compact quantum group is of the form $\\mathbb {G}_a$ [60].","Therefore, every commutative locally compact quantum group is co-amenable, and is amenable if and only if the underlying locally compact group is amenable.","By duality, every co-commutative locally compact quantum group is of the form $\\mathbb {G}_s$ , which is always amenable, and is co-amenable if and only if the underlying locally compact group is amenable, by Leptin's theorem [47].","For a locally compact quantum group $\\mathbb {G}$ , we let $C_0(\\mathbb {G}):=\\overline{\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))}^{\\Vert \\cdot \\Vert }$ denote the reduced quantum group $C^*$ -algebra of $\\mathbb {G}$ .","We say that $\\mathbb {G}$ is compact if $C_0(\\mathbb {G})$ is a unital $C^*$ -algebra, in which case we denote $C_0(\\mathbb {G})$ by $C(\\mathbb {G})$ .","We say that $\\mathbb {G}$ is discrete if $L^1(\\mathbb {G})$ is unital, in which case we denote $L^1(\\mathbb {G})$ by $\\ell ^1(\\mathbb {G})$ .","It is well-known that $\\mathbb {G}$ is compact if and only if $$ is discrete, and in that case, $\\ell ^1()\\cong \\bigoplus _{1} \\lbrace T_{n_\\alpha }(\\mid \\alpha \\in \\mathrm {Irr}(\\mathbb {G})\\rbrace $ , where $T_{n_\\alpha }($ is the space of $n_\\alpha \\times n_\\alpha $ trace class operators, and $\\mathrm {Irr}(\\mathbb {G})$ denotes the set of (equivalence classes of) irreducible co-representations of the compact quantum group $\\mathbb {G}$ [68].","For general $\\mathbb {G}$ , the operator dual $M(\\mathbb {G}):=C_0(\\mathbb {G})^*$ is a completely contractive Banach algebra containing $L^1(\\mathbb {G})$ as a norm closed two-sided ideal via the map $L^1(\\mathbb {G})\\ni f\\mapsto f|_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ .","The co-multiplication satisfies $\\Gamma (C_0(\\mathbb {G}))\\subseteq M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))$ , where $M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))$ is the multiplier algebra of the $C^*$ -algebra $C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})$ .","We let $C_u(\\mathbb {G})$ be the universal quantum group $C^*$ -algebra of $\\mathbb {G}$ , and denote the canonical surjective *-homomorphism onto $C_0(\\mathbb {G})$ by $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ [42].","The space $C_u(\\mathbb {G})^*$ then has the structure of a unital completely contractive Banach algebra such that the map $L^1(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})^*$ given by the composition of the inclusion $L^1(\\mathbb {G})\\subseteq M(\\mathbb {G})$ and $\\pi _{\\mathbb {G}}^*:M(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})^*$ is a completely isometric homomorphism, and it follows that $L^1(\\mathbb {G})$ is a norm closed two-sided ideal in $C_u(\\mathbb {G})^*$ [42].","An element $\\hat{b}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ is said to be a completely bounded left multiplier of $L^1(\\mathbb {G})$ if $\\hat{b}\\lambda (f)\\in \\lambda (L^1(\\mathbb {G}))$ for all $f\\in L^1(\\mathbb {G})$ and the induced map $m_{\\hat{b}}^l:L^1(\\mathbb {G})\\ni f\\mapsto \\lambda ^{-1}(\\hat{b}\\lambda (f))\\in L^1(\\mathbb {G})$ is completely bounded on $L^1(\\mathbb {G})$ .","We let $M_{cb}^l(L^1(\\mathbb {G}))$ denote the space of all completely bounded left multipliers of $L^1(\\mathbb {G})$ , which is a completely contractive Banach algebra with respect to the norm $\\Vert [\\hat{b}_{ij}]\\Vert _{M_n(M_{cb}^l(L^1(\\mathbb {G})))}=\\Vert [m^l_{\\hat{b}_{ij}}]\\Vert _{cb}.$ Completely bounded right multipliers are defined analogously and we denote by $M_{cb}^r(L^1(\\mathbb {G}))$ the corresponding completely contractive Banach algebra.","There is a canonical, injective completely contractive homomorphism $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ , extending $\\lambda $ , which maps $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .","In general, given $\\hat{b}\\in M_{cb}^l(L^1(\\mathbb {G}))$ , the adjoint $\\Theta ^l(\\hat{b}):=(m_{\\hat{b}}^l)^*$ defines a normal completely bounded left $L^1(\\mathbb {G})$ -module map on $L^{\\infty }(\\mathbb {G})$ , and by [37] have the completely isometric identification $\\Theta ^l:M_{cb}^l(L^1(\\mathbb {G}))\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G})).$ Moreover, it follows from [37] that $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ .","It is known that $M_{cb}^l(L^1(\\mathbb {G}))$ is a dual operator space [35], with predual $Q_{cb}^l(L^1(\\mathbb {G}))$ .","By the general result [34], it follows from [41] that for arbitrary locally compact quantum groups $Q_{cb}^l(L^1(\\mathbb {G}))=\\lbrace \\Omega _{A,\\rho }\\mid A\\in C_0(\\mathbb {G})\\otimes _{\\min }\\mathcal {K}(\\ell ^2), \\ \\rho \\in L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(\\ell ^2)\\rbrace ,$ where $\\langle \\hat{b},\\Omega _{A,\\rho }\\rangle =\\langle (\\Theta ^l(\\hat{b})\\otimes \\textnormal {id}_{K_{\\infty }})(A),\\rho \\rangle $ , $\\hat{b}\\in M_{cb}^l(L^1(\\mathbb {G}))$ .","We say that $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .","The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .","We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.","Let $\\mathbb {G}$ be a locally compact quantum group.","The right fundamental unitary $V$ of $\\mathbb {G}$ induces a co-associative co-multiplication $\\Gamma ^r:\\mathcal {B}(L^2(\\mathbb {G}))\\ni T\\mapsto V(T\\otimes 1)V^*\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})),$ and the restriction of $\\Gamma ^r$ to $L^{\\infty }(\\mathbb {G})$ yields the original co-multiplication $\\Gamma $ on $L^{\\infty }(\\mathbb {G})$ .","The pre-adjoint of $\\Gamma ^r$ induces an associative completely contractive multiplication on the space of trace class operators $\\mathcal {T}(L^2(\\mathbb {G}))$ , defined by $\\rhd :\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\otimes \\tau \\mapsto \\omega \\rhd \\tau =\\Gamma ^r_*(\\omega \\otimes \\tau )\\in \\mathcal {T}(L^2(\\mathbb {G})).$ Analogously, the left fundamental unitary $W$ of $\\mathbb {G}$ induces a co-associative co-multiplication $\\Gamma ^l:\\mathcal {B}(L^2(\\mathbb {G}))\\ni T\\mapsto W^*(1\\otimes T)W\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})),$ and the restriction of $\\Gamma ^l$ to $L^{\\infty }(\\mathbb {G})$ is also equal to $\\Gamma $ .","The pre-adjoint of $\\Gamma ^l$ induces another associative completely contractive multiplication $\\lhd :\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\otimes \\tau \\mapsto \\omega \\lhd \\tau =\\Gamma ^l_*(\\omega \\otimes \\tau )\\in \\mathcal {T}(L^2(\\mathbb {G})).$ The algebra $\\mathcal {B}(L^2(\\mathbb {G}))$ inherits a canonical $\\mathcal {T}(L^2(\\mathbb {G}))$ -bimodule structure with respect to both the left $\\lhd $ and right $\\rhd $ products.","It was shown in [35] that the pre-annihilator $L^{\\infty }(\\mathbb {G})_{\\perp }$ of $L^{\\infty }(\\mathbb {G})$ in $\\mathcal {T}(L^2(\\mathbb {G}))$ is a norm closed two sided ideal in $(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ and $(\\mathcal {T}(L^2(\\mathbb {G})),\\lhd )$ , respectively, and the complete quotient map $\\pi :\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\mapsto f=\\omega |_{L^{\\infty }(\\mathbb {G})}\\in L^1(\\mathbb {G})$ is an algebra homomorphism from $\\mathcal {T}_{\\rhd }:=(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ , respectively, $\\mathcal {T}_{\\lhd }:=(\\mathcal {T}(L^2(\\mathbb {G})),\\lhd )$ , onto $L^1(\\mathbb {G})$ .","It follows that the right $\\lhd $ -module and left $\\rhd $ -module structures degenerate to an $L^1(\\mathbb {G})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ : $ f\\rhd T = (\\textnormal {id}\\otimes f)\\Gamma ^r(T), \\ \\ \\ T\\lhd f = (f\\otimes \\textnormal {id})\\Gamma ^l(T), \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})).$ By [44] the unitary antipode $R$ satisfies $R(x)=\\widehat{J}x^*\\widehat{J}$ , for $x\\in L^{\\infty }(\\mathbb {G})$ .","It therefore extends to a *-anti-automorphism (still denoted) $R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ , via $R(T)=J̉T^*J̉$ , $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","The extended antipode maps $L^{\\infty }(\\mathbb {G})$ and $L^{\\infty }(\\widehat{\\mathbb {G}})$ onto $L^{\\infty }(\\mathbb {G})$ and $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ , respectively, and satisfies the generalized antipode relations; that is, $( R\\otimes R)\\circ \\Gamma ^r=\\Sigma \\circ \\Gamma ^l\\circ R\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}( R\\otimes R)\\circ \\Gamma ^l=\\Sigma \\circ \\Gamma ^r\\circ R,$ where $\\Sigma $ is the flip map on $\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ .","Let $\\mathbb {G}$ and $\\mathbb {H}$ be two locally compact quantum groups.","Then $\\mathbb {H}$ is said to be a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes if there exists a normal, unital, injective *-homomorphism $\\gamma :L^{\\infty }(\\widehat{\\mathbb {H}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ satisfying $(\\gamma \\otimes \\gamma )\\circ \\Gamma _{}=\\Gamma _{}\\circ \\gamma $ .","This is not the original definition of Vaes [64], but was shown to be equivalent in [24]."],["Inner Amenability","Given a locally compact quantum group $\\mathbb {G}$ the composition $W\\sigma V\\sigma \\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ defines a unitary co-representation of $\\mathbb {G}$ called the conjugation co-representation, where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\sigma V\\sigma \\xi (s,t)=\\xi (s,s^{-1}ts)\\Delta (s)^{1/2}, \\ \\ \\ \\xi \\in L^2(G\\times G), \\ s,t\\in G.$ Thus, $W\\sigma V\\sigma $ is the unitary generator of the conjugation representation $\\beta _2:G\\rightarrow \\mathcal {B}(L^2(G))$ , where $\\beta _2(s)\\xi (t)=\\lambda (s)\\rho (s)\\xi (t)=\\xi (s^{-1}ts)\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\in L^{\\infty }(G)^*$ satisfying $\\langle m,\\beta _{\\infty }(s)f\\rangle =\\langle m,f\\rangle \\ \\ \\ s\\in G, \\ f\\in L^{\\infty }(G),$ where $\\beta _\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\in G$ , $f\\in L^{\\infty }(G)$ , is the conjugation action on $L^{\\infty }(G)$ .","Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\in \\ell ^{\\infty }(G)^*$ such that $m\\ne \\delta _e$ .","In what follows, inner amenability will always refer to the definition (REF ) given above.","In particular, every discrete group is inner amenable.","Definition 3.2 Let $\\mathbb {G}$ be a locally compact quantum group.","We say that $\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\xi _i)$ of unit vectors such that $\\Vert W\\sigma V\\sigma (\\eta \\otimes \\xi _i)-\\eta \\otimes \\xi _i\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ $\\mathbb {G}$ is inner amenable if there exists a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ satisfying $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}}).$ Such a state is said to be inner invariant.","$\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\in C_0()^*$ such that $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in C_0().$ Such a state is said to be topologically inner invariant.","Examples 3.3 The following known examples are worth mentioning.","Any co-commutative quantum group $\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\sigma V_s\\sigma =W_sW_s^*=1$ .","Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\xi :=\\Lambda _\\varphi (1)$ being conjugation invariant, where $\\varphi $ is the Haar trace on the compact dual.","It was shown in [51] that if $\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\mathbb {G}$ is topologically inner amenable.","Further examples, including the bicrossed product construction will be studied below.","The following proposition is standard.","We include the proof for completeness.","Proposition 3.4 Let $\\mathbb {G}$ be a locally compact quantum group.","If $\\mathbb {G}$ is strongly inner amenability then it is inner amenable.","Let $(\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\sigma V\\sigma $ .","Passing to a subnet, we may assume that $(\\omega _{J\\xi _i}|_{L^{\\infty }(\\widehat{\\mathbb {G}})})$ converges weak* to a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ .","For any $f=\\omega _{\\xi ,\\eta }\\in L^1(\\mathbb {G})$ and $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*, \\ \\ \\ (\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\otimes J)=\\sigma V^*\\sigma ,$ we have $\\langle m,\\hat{x}\\lhd f\\rangle &=\\lim _i\\langle \\omega _{J\\xi _i},\\hat{x}\\lhd f\\rangle \\\\&=\\lim _i\\langle W^*(1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),\\eta \\otimes J\\xi _i\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),W(\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})\\sigma V^*\\sigma (\\xi \\otimes J\\xi _i),\\sigma V^*\\sigma (\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\langle m,\\hat{x}\\rangle \\langle f,1\\rangle .$ In the commutative case, the converse of Proposition REF holds.","Proposition 3.5 A commutative quantum group $\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.","If $\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.","Let $m\\in VN(G)^*$ satisfy $\\langle m,x\\rangle =\\langle m,x \\lhd f\\rangle $ for all $x\\in VN(G)$ and all states $f\\in L^1(G)$ .","Then for $t\\in G$ $\\lambda (t)x\\lambda (t)^* \\lhd f=\\int _G \\lambda (s^{-1}t)x\\lambda (t^{-1}s) f(s)ds=\\int _G\\lambda (r)^*x\\lambda (r)f(tr)dr=x \\lhd f_t,$ so that $\\langle m,\\lambda (t)x\\lambda (t)^*\\rangle =\\langle m,\\lambda (t)x\\lambda (t)^* \\lhd f\\rangle =\\langle m,x \\lhd f_t\\rangle =\\langle m,x\\rangle $ for all $x\\in VN(G)$ , $t\\in G$ and states $f\\in L^1(G)$ .","Thus, $G$ is inner amenable by [16].","Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\xi _i)$ in $L^2(G)$ satisfying $\\Vert \\lambda (s)\\xi _i-\\rho (s)^*\\xi _i\\Vert _{L^2(G)}\\rightarrow 0$ uniformly on compact subsets of $G$ .","For any $\\eta \\in C_c(G)$ we then have $\\langle W\\sigma V\\sigma (\\eta \\otimes \\xi _i),\\eta \\otimes \\xi _i\\rangle =\\iint \\xi _i(ts)\\overline{\\xi _i}(st)\\Delta (s)^{1/2}|\\eta (s)|^2 \\ ds \\ dt\\rightarrow 1.$ It follows that $(\\xi _i)$ is strongly inner invariant.","Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].","Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\mathcal {B}(L^2(G))$ , via $\\langle m,f \\rhd T\\rangle =\\langle m, T \\lhd f\\rangle , \\ \\ \\ f\\in L^1(G), \\ T\\in \\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.","For details, see [13].","Proposition 3.6 A commutative quantum group $\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.","The existence of a tracial state on $C_\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.","By [51] $\\mathbb {G}_a$ is topologically inner amenable if $C_\\lambda ^*(G)$ has a tracial state.","Conversely, if $m$ is an inner invariant state on $C_\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.","Viewing $m$ as a positive definite function in $B_\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\in G$ .","A simple integral calculation then shows that $m$ is a tracial state on $C_\\lambda ^*(G)$ .","Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\infty }(\\widehat{\\mathbb {G}})$ to $C_0()$ .","This is not the case however.","In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\bigoplus _{n\\in \\mathbb {N}} \\Gamma _n\\rtimes \\prod _{n\\in \\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\Gamma _n$ for which $C^*_\\lambda (\\Gamma _n\\rtimes F_n)$ admits a unique tracial state.","In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .","Thus, $G$ is inner amenable.","However, in [30], the authors also show that the continuous function $m$ in $B_\\lambda (G)$ associated to a tracial state on $C^*_\\lambda (G)$ must be discontinuous.","Whence, there is no tracial state on $C^*_\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .","Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\lambda (G)$ .","In the other direction, in general it is not possible to lift a tracial state on $C^*_\\lambda (G)$ to an inner invariant mean on $VN(G)$ .","In [30] it is shown that $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\mathbb {R})$ .","Proposition 3.8 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is co-amenable then $\\mathbb {G}$ is strongly inner amenable.","If $\\mathbb {G}$ is amenable then it is inner amenable.","If $$ is co-amenable, let $E\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\omega _{\\xi _i}$ with $J̉\\xi _i=\\xi _i$ (since $L^{\\infty }(\\widehat{\\mathbb {G}})\\curvearrowright L^2(\\mathbb {G})$ is standard).","Then $1=(\\textnormal {id}\\otimes E)(W)=\\lim _i(\\textnormal {id}\\otimes \\omega _{\\xi _i})(W)$ strongly, from which it follows that $\\Vert W(\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Since $\\sigma V\\sigma =(J̉\\otimes J̉)W^*(J̉\\otimes J̉)$ , and $J̉\\xi _i=\\xi _i$ , we also have $\\Vert \\sigma V\\sigma (\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Whence, $\\mathbb {G}$ is strongly inner amenable.","Now, suppose $\\mathbb {G}$ is amenable and let $m\\in L^{\\infty }(\\mathbb {G})^*$ be a two-sided invariant mean.","We will show a stronger statement by providing a state $M\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ such that $\\langle M,\\rho \\rhd T\\rangle =\\langle M,T\\lhd \\rho \\rangle , \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})),$ upon which restriction to $L^{\\infty }(\\widehat{\\mathbb {G}})$ is the desired state.","Letting $m$ also denote its restriction to $\\mathrm {LUC}(\\mathbb {G}):=\\langle L^{\\infty }(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle $ , let $m_0:=\\rho _0\\circ \\Theta ^r(m)\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ , where $\\rho _0\\in \\mathcal {T}(L^2(\\mathbb {G}))$ is a fixed normal state, and $\\Theta ^r:\\mathrm {LUC}(\\mathbb {G})^*\\ni n\\mapsto (T\\mapsto (\\textnormal {id}\\otimes n)V^*(T\\otimes 1)V)\\in \\mathcal {CB}_{\\mathcal {T}_{\\rhd }}(\\mathcal {B}(L^2(\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).","Since $\\mathrm {LUC}(\\mathbb {G})=\\langle \\mathcal {B}(L^2(\\mathbb {G}))\\rhd \\mathcal {T}(L^2(\\mathbb {G}))\\rangle $ [35], one may view the map $\\Theta ^r$ as follows: $\\langle \\Theta ^r(n)(T),\\rho \\rangle =\\langle n,T\\rhd \\rho \\rangle , \\ \\ \\ n\\in \\mathrm {LUC}(\\mathbb {G})^*, \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Consider the state $R^*(m_0)\\square m_0\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\square $ is the left Arens product on $\\mathcal {B}(L^2(\\mathbb {G}))^*$ extending the multiplication in $\\mathcal {T}_\\rhd =(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ .","Fix $\\rho ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","Firstly, since $m\\in \\mathrm {LUC}(\\mathbb {G})^*$ is an invariant mean $m\\square \\pi (\\rho )=\\langle \\pi (\\rho ),1\\rangle m=\\langle \\rho ,1\\rangle m$ .","Thus, $\\langle m_0\\square \\rho ,T\\rangle &=\\langle m_0,\\rho \\rhd T\\rangle =\\langle \\rho _0,\\Theta ^r(m)\\circ \\Theta ^r(\\pi (\\rho ))(T)\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m\\square \\pi (\\rho ))(T)\\rangle =\\langle \\rho ,1\\rangle \\langle \\rho _0,\\Theta ^r(m)(T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle m_0,T\\rangle .$ Hence, $m_0\\square \\rho =\\langle \\rho ,1\\rangle m_0$ .","Secondly, $\\Theta ^r(m)$ is a conditional expectation onto $L^{\\infty }(\\widehat{\\mathbb {G}})$ commuting with both the right $\\mathcal {T}_\\rhd $ - and right $\\mathcal {T}_\\lhd $ -actions [15].","Since $\\hat{x}\\rhd \\omega =\\langle \\omega ,\\hat{x}\\rangle 1$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ and $\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we also have $\\langle m_0\\square (T\\lhd \\rho ),\\omega \\rangle &=\\langle m_0,(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,\\Theta ^r(m)((T\\lhd \\rho )\\rhd \\omega )\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T\\lhd \\rho ),\\omega \\rangle \\\\&=\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T)\\lhd \\rho ,\\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T),\\rho \\lhd \\omega \\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T)\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle \\rho _0,\\Theta ^r(m)(T\\rhd (\\rho \\lhd \\omega ))\\rangle \\\\&=\\langle m_0,T\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle m_0\\square T,\\rho \\lhd \\omega \\rangle \\\\&=\\langle (m_0\\square T)\\lhd \\rho ,\\omega \\rangle .\\\\$ Thus, $m_0\\square (T\\lhd \\rho )=(m_0\\square T)\\lhd \\rho $ .","Putting things together, on the one hand we obtain $\\langle R^*(m_0)\\square m_0,\\rho \\rhd T\\rangle =\\langle R^*(m_0),(m_0\\square \\rho )\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle ,$ and on the other, $\\langle R^*(m_0)\\square m_0,T\\lhd \\rho \\rangle &=\\langle R^*(m_0),m_0\\square (T\\lhd \\rho )\\rangle =\\langle R^*(m_0),(m_0\\square T)\\lhd \\rho \\rangle \\\\&=\\langle m_0,R((m_0\\square T)\\lhd \\rho )\\rangle =\\langle m_0,R_*(\\rho )\\rhd R(m_0\\square T)\\rangle \\\\&=\\langle m_0\\square R_*(\\rho ),R(m_0\\square T)\\rangle =\\langle R_*(\\rho ),1\\rangle \\langle m_0, R(m_0\\square T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0)\\square m_0, T\\rangle .$ Therefore, $M:=R^*(m_0)\\square m_0$ is the required state.","Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].","Corollary 3.9 A locally compact quantum group $\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $2pt\\mathbf {mod}$ .","Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.","Proposition 3.10 Let $\\mathbb {G}$ and $\\mathbb {H}$ be locally compact quantum groups such that $\\mathbb {H}$ is a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes.","If $\\mathbb {G}$ is inner amenable then $\\mathbb {H}$ is inner amenable.","Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\circ \\gamma \\in L^{\\infty }(\\widehat{\\mathbb {H}})^*$ , and let $W_{\\mathbb {G}}$ and $W_{\\mathbb {H}}$ denote the left fundamental unitaries of $\\mathbb {G}$ and $\\mathbb {H}$ , respectively.","We also denote by $\\mathbb {W}_{\\mathbb {G}}\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_u())$ and $\\mathbb {W}_{\\mathbb {H}}\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\pi _{\\mathbb {G}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})=W_{\\mathbb {G}}$ and $(\\pi _{\\mathbb {H}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})=W_{\\mathbb {H}}$ .","Finally, we define $\\W _{\\mathbb {G}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0()), \\ \\ \\ \\W _{\\mathbb {H}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_0()).$ Let $\\pi _{\\mathbb {G},\\mathbb {H}}:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {H})$ be the surjection from [24].","Its dual morphism is a non-degenerate $*$ -homomorphism $\\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}:C_u()\\rightarrow M(C_u())$ satisfying $(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})=(\\textnormal {id}\\otimes \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}}).$ From the relation $\\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}}=\\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}$ [24], we have $(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})&=(\\textnormal {id}\\otimes \\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}})(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})\\\\&=(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}).$ Thus, for any $x̉\\in L^{\\infty }(\\widehat{\\mathbb {H}})$ and $g\\in L^{1}(\\mathbb {H})$ we have $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\gamma ((g\\otimes \\textnormal {id})W_{\\mathbb {H}}^*(1\\otimes x̉)W_{\\mathbb {H}})\\rangle \\\\&=\\langle m̉,\\gamma ((\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\W _{\\mathbb {H}}^*(1\\otimes x̉)\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})^*(1\\otimes \\gamma (x̉))(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}^*(1\\otimes \\gamma (x̉))\\W _{\\mathbb {G}})\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g)))(\\gamma (\\hat{x}))\\rangle ,$ where the last equality follows from Lemma REF .","By invariance of $m̉$ , we may convolve with any state $f\\in L^1(\\mathbb {G})$ to obtain $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(g))(\\gamma (\\hat{x}))\\lhd _{\\mathbb {G}}f\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)(\\gamma (\\hat{x}))\\rangle \\\\&=\\langle m̉,\\gamma (\\hat{x})\\lhd _{\\mathbb {G}}(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)\\rangle \\\\&=\\langle \\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f,1\\rangle \\langle m̉,\\gamma (\\hat{x})\\rangle \\\\&=\\langle g,1\\rangle \\langle n̉,\\hat{x}\\rangle .$"],["Examples arising from the bicrossed product construction","Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\rightarrow G$ and an anti-homomorphism $j:G_2\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.","Suppose further that $G_1\\times G_2\\ni (g,s)\\mapsto i(g)j(s)\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.","Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].","Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\alpha :G_1\\times G_2\\rightarrow G_2$ and $\\beta :G_1\\times G_2\\rightarrow G_1$ satisfying mutual co-cycle relations [65].","It is known that the von Neumann crossed product $G_1\\rtimes _\\alpha L^\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .","The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\infty }(G_1)^\\beta \\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .","Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.","In preparation we collect some useful formulae from [65], to which we refer the reader for details.","To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .","The fundamental unitary $W$ satisfies $W^*\\xi (g,s,h,t)=\\xi (\\beta _{t}(h)^{-1}g,s,h,\\alpha _{\\beta _t(h)^{-1}g}(s)t), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2).$ Letting $\\Delta $ , $\\Delta _1$ , and $\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\xi (g,s)=\\Delta (\\alpha _g(s))^{1/2}\\Delta _1(\\beta _s(g)g^{-1})^{1/2}\\Delta _2(\\alpha _g(s)s^{-1})^{1/2}\\overline{\\xi }(\\beta _s(g),s^{-1}), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2).$ Let $\\Psi :G_2\\times G_1\\rightarrow (0,\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\Psi (s,g):=\\frac{d\\beta _s(g)}{dg}$ .","It follows that $\\Psi (s,g)=\\Delta (\\alpha _g(s))\\Delta _1(\\beta _s(g)g^{-1})\\Delta _2(\\alpha _g(s))$ .","The action $\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )^{1/2}\\xi (\\beta _{s^{-1}}(g)), \\ \\ \\ \\xi \\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )f(\\beta _{s^{-1}}(g)), \\ \\ \\ f\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\beta $ preserves $\\Delta _1$ , there exists an asymptotically $\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\Vert \\cdot \\Vert _1=1}$ with $\\mathrm {supp}(f_i)\\rightarrow \\lbrace e\\rbrace $ satisfying $\\Vert v^1_sf_i-f_i\\Vert _1\\rightarrow 0$ uniformly on compacta.","Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.","Using the co-cycle properties of $\\alpha $ and $\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\otimes J̉)W^*(J̉\\otimes J̉)\\xi (g,s,h,t)\\\\&=\\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2}\\xi (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\alpha _{h^{-1}\\beta _s(g)}(s^{-1})^{-1})$ for all $\\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2)$ .","Let $\\eta _i:=\\sqrt{f_i}$ .","By hypothesis $(iii)$ we have $\\Vert v_s\\eta _i - \\eta _i\\Vert _{L^2(G_1)}^2\\le \\Vert v_s^1f_i-f_i\\Vert _{L^1(G_1)}\\rightarrow 0$ uniformly on compacta.","Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\times G_2\\times G_1\\rightarrow it follows that\\begin{equation} \\int f(g,s,h) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dh \\rightarrow f(g,s,e)\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\xi _j)$ in $C_c(G_2)_{\\Vert \\cdot \\Vert _2=1}$ satisfying $\\Vert \\rho (s)\\xi _j-\\lambda (s)^*\\xi _j\\Vert _{L^2(G_2)}\\rightarrow 0$ uniformly on compacta.","Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\xi (g)=\\xi (g^{-1})\\Delta _1(g^{-1})$ .","Then for any $\\eta \\in C_c(G_1\\times G_2)$ , and any $j$ , we have $&\\langle W\\sigma V\\sigma (\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),\\eta \\otimes U_1\\eta _i\\otimes \\xi _j\\rangle =\\langle (J̉\\otimes J̉)W^*(J̉\\otimes J̉)(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),W^*(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j)\\rangle \\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ U_1\\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{U_1\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ \\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\ \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\ \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ \\overline{\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiint \\bigg (\\int \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ dt\\bigg )\\\\&\\times \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dg \\ ds \\ dh\\\\&\\rightarrow \\iiint \\xi _j(t(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1})\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2} \\ dg \\ ds \\ dt\\\\$ by ().","But $\\alpha _{\\beta _s(g)}(s^{-1})^{-1}=\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2}\\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\Delta _2(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\\\&=\\Delta _2(\\alpha _g(s))^{1/2}, \\ \\ \\ a.e.$ The final integral in the above calculation therefore reduces to $\\iiint \\Delta _2(\\alpha _g(s))^{1/2} \\ \\xi _j(t\\alpha _{g}(s))\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ dg \\ ds \\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\iint |\\eta (g,s)|^2 \\ dg \\ ds = \\Vert \\eta \\Vert _{L^2(G_1\\times G_2)}^2.$ Denoting the index sets of $(\\eta _i)$ and $(\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\mathcal {I}:=J\\times I^{J}$ and for $I=(j,(i_j)_{j\\in J})\\in \\mathcal {I}$ we let $\\xi _I:=U_1\\eta _{i_j}\\otimes \\xi _j\\in L^2(G_1\\times G_2)$ .","By [39] and the above analysis, the resulting net $(\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .","Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.","Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.","Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.","For example, $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\mathbb {R}^2$ and $\\mathbb {F}_6$ are.","Example 3.14 We consider a discretized version of [26].","Let $G=\\bigg \\lbrace \\begin{pmatrix} a & b & x\\\\ c & d & y\\\\ 0 & 0 & 1\\end{pmatrix}\\mid \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\in SL(2,\\mathbb {Q}), \\ x,y\\in \\mathbb {Q}\\bigg \\rbrace ,$ $G_1=(\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\mathbb {Q})$ , viewed as discrete groups.","The embeddings $G_1\\ni (x,y)\\mapsto \\begin{pmatrix} 1 & 0 & -x\\\\ -x & 1 & -y+\\frac{1}{2}x^2\\\\ 0 & 0 & 1\\end{pmatrix}\\in G, \\ \\ G_2\\ni \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\mapsto \\begin{pmatrix}d & -b & 0\\\\ -c & a & 0\\\\ 0 & 0 & 1\\end{pmatrix}\\in G$ determine a matched pair structure for $(G_1,G_2)$ .","The corresponding actions are given by $\\alpha _{(x,y)}\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}&=\\begin{pmatrix}a+bx & b\\\\ c+dx-(a+bx)(ax+b(y+\\frac{1}{2}x^2)) & d-b(ax+b(y+\\frac{1}{2}x^2))\\end{pmatrix} \\\\\\beta _{\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}}(x,y)&=\\bigg (ax+by+\\frac{b}{2}x^2,cx+d\\bigg (y+\\frac{1}{2}x^2\\bigg )-\\frac{1}{2}\\bigg (ax+b\\bigg (y+\\frac{1}{2}x^2\\bigg )\\bigg )^2\\bigg ).$ By Corollary REF , the bicrossed product $\\mathbb {G}:=VN(\\mathbb {Q}^2)^\\beta \\bowtie _\\alpha L^\\infty (SL(2,\\mathbb {Q}))$ is strongly inner amenable.","Since $SL(2,\\mathbb {Q})$ is not amenable, it follows from [26] that $\\mathbb {G}$ is not amenable.","Moreover, since $\\mathbb {Q}^2$ is not compact, $\\mathbb {G}$ is not discrete by [65].","Thus, $\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group."],["Relative Injectivity and the Averaging Technique","In [56] Ruan and Xu showed that the dual $L^{\\infty }(\\widehat{\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\mathbb {G}$ is relatively 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}$ .","We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.","Below we let $\\widetilde{\\mathbb {G}}:=\\mathrm {Gr}()$ denote the intrinsic group of $$ .","Proposition 4.1 Let $\\mathbb {G}$ be a locally compact quantum group.","Consider the following conditions: $$ is inner amenable; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ ; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ ; $\\widetilde{}=\\mathrm {Gr}(\\mathbb {G})$ is inner amenable.","Then $(i)\\Rightarrow (ii)\\Leftrightarrow (iii)\\Rightarrow (iv)$ .","When $\\mathbb {G}$ is co-commutative, the conditions are equivalent.","$(i)\\Rightarrow (ii)$ : Given a state $n\\in L^{\\infty }(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $m:=n\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant.","It suffices to provide a completely contractive morphism which is a left inverse to the map $\\Delta :L^{\\infty }(\\mathbb {G})\\rightarrow \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ given by $\\Delta (x)(f)=x\\rhd f, \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Identifying $\\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))\\cong L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ via $\\langle \\Phi ,f\\otimes g\\rangle =\\langle \\Phi (f),g\\rangle , \\ \\ \\ \\Phi \\in \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G})), \\ f,g\\in L^1(\\mathbb {G}),$ we have $\\Delta =\\Gamma $ , and that the corresponding $L^1(\\mathbb {G})$ -module structure on $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ is defined by $X\\unrhd f=(f\\otimes \\textnormal {id}\\otimes \\textnormal {id})(\\Gamma ^r\\otimes \\textnormal {id})(X)$ for $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $f\\in L^1(\\mathbb {G})$ .","First, consider the map $\\Phi :\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni A\\mapsto (\\textnormal {id}\\otimes m)(V^*AV)\\in \\mathcal {B}(L^2(\\mathbb {G})).$ Clearly, $\\Phi $ is a completely contractive left inverse to $\\Gamma $ .","We show that $\\Phi $ is a right $\\mathcal {T}_\\rhd $ -module map.","This will complete the proof since [14] will entail the invariance $\\Phi (L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ , and the restricted module action $\\mathcal {T}_\\rhd \\curvearrowright L^{\\infty }(\\mathbb {G})$ is the pertinent $L^1(\\mathbb {G})$ -module action.","To this end, fix $A\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $\\rho \\in \\mathcal {T}(L^2(\\mathbb {G}))$ .","Then $\\Phi (A\\unrhd \\rho )&=\\Phi ((\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\prime }=\\sigma V^*\\sigma $ , where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ , for any $\\tau ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we have $&\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*(\\sigma \\otimes 1)V_{23}^*A_{23}V_{23}(\\sigma \\otimes 1)V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\omega \\otimes \\tau \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m)(V^*(1\\otimes (\\tau \\otimes \\textnormal {id})(V^*AV))V),\\omega \\rangle \\\\&=\\langle ( m\\otimes \\textnormal {id})(V̉^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV)\\otimes 1)V̉^{\\prime *}),\\omega \\rangle \\\\&=\\langle m,\\omega \\widehat{\\rhd }^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV))\\rangle \\\\&=\\langle m,(\\tau \\otimes \\textnormal {id})(V^*AV)\\rangle \\langle \\omega ,1\\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m\\otimes \\textnormal {id})(V^*AV\\otimes 1),\\tau \\otimes \\omega \\rangle \\\\&=\\langle \\Phi (A)\\otimes 1,\\tau \\otimes \\omega \\rangle .$ As $\\tau $ and $\\omega $ were arbitrary, we have $\\Phi (A\\unrhd \\rho )&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\Phi (A)\\otimes 1)V^*)\\\\&=\\Phi (A)\\rhd \\rho .$ $(ii)\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\mathbb {G})$ -module left inverse $\\Phi $ to $\\Gamma $ , it follows that $R\\circ \\Phi \\circ \\Sigma \\circ (R\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ .","$(iii)\\Rightarrow (iv)$ : Recall that $\\mathrm {Gr}(\\mathbb {G})$ is a group of unitaries in $L^{\\infty }(\\mathbb {G})$ , so it acts naturally on $L^{\\infty }(\\mathbb {G})$ by conjugation.","The existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ which is $\\mathrm {Gr}(\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\Gamma (L^{\\infty }(\\mathbb {G}))-L^{\\infty }(\\mathbb {G})$ -bimodule property of $\\Phi $ .","Since $\\widetilde{\\widehat{\\mathbb {G}}}$ is a closed quantum subgroup of $\\widehat{\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\gamma :L^{\\infty }(\\widehat{\\widetilde{}})\\rightarrow L^{\\infty }(\\mathbb {G})$ intertwining the co-multiplications.","As $L^{\\infty }(\\widehat{\\widetilde{}})=VN(\\widetilde{\\widehat{\\mathbb {G}}})=VN(\\mathrm {Gr}(\\mathbb {G}))$ , the state $m\\circ \\gamma \\in VN(\\mathrm {Gr}(\\mathbb {G}))^*$ is $\\mathrm {Gr}(\\mathbb {G})$ -invariant, making $\\mathrm {Gr}(\\mathbb {G})$ inner amenable by [16].","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, then $\\mathrm {Gr}(\\mathbb {G}_s)=G$ and the implication $(iv)\\Rightarrow (i)$ follows immediately from [16].","Corollary 4.2 Let $\\mathbb {G}$ be a locally compact quantum group for which $L^{\\infty }(\\widehat{\\mathbb {G}})$ is an injective von Neumann algebra.","Then the following are equivalent: $\\mathbb {G}$ is amenable; $\\mathbb {G}$ is inner amenable; $L^{\\infty }(\\widehat{\\mathbb {G}})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","Propositions REF and REF yield the implications $(i)\\Rightarrow (ii)\\Rightarrow (iii)$ .","Assume $(iii)$ .","Since $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $\\mathbf {mod}\\hspace{2.0pt}it follows from \\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.","Hence, $ G$ is amenable by \\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\mathbb {G}$ and amenability of $$ .","One of their main results is the following.","Theorem 4.3 (Ng–Viselter) Let $\\mathbb {G}$ be a locally compact quantum group.","Consider the following conditions: $\\mathbb {G}$ is co-amenable; $C_0(\\mathbb {G})$ is nuclear and there exists a state $\\rho \\in C_0(\\mathbb {G})^*$ such that $\\langle \\rho ,x\\widehat{\\lhd } \\hat{f}\\rangle =\\langle \\rho ,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ $$ is amenable.","Then $(a)\\Rightarrow (b)\\Rightarrow (c)$ .","Moreover, if $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra), then $(a)\\Rightarrow (b^{\\prime })\\Rightarrow (b)$ , where $C_0(\\mathbb {G})$ is nuclear and has a tracial state.","They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .","We now show that $(b)$ is indeed equivalent to $(a)$ .","Theorem 4.4 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.","Suppose that $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.","Given a state $m\\in C_0(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $n:=m\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant, as in the proof of Proposition REF .","Let $n\\in M(C_0(\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\textnormal {id}\\otimes n):M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ denote the unique extension of the slice map $(\\textnormal {id}\\otimes n):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))$ which is strictly continuous on the unit ball.","By strict density of $C_0(\\mathbb {G})$ in $M(C_0(\\mathbb {G}))$ , it follows that $\\langle n,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle n,x\\rangle \\langle \\hat{f}^{\\prime },1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in M(C_0(\\mathbb {G})).$ Since $V\\in M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))$ , the map $\\Phi :C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\Phi (A)=(\\textnormal {id}\\otimes n)(V^*AV), \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.","Using the extended $\\widehat{\\rhd }^{\\prime }$ -invariance on $M(C_0(\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\Phi (A\\unrhd \\rho )=\\Phi (A)\\rhd \\rho , \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Since $\\mathrm {Ad}(V^*):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})$ and $\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})=(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))^*,$ we have $\\mathrm {Ad}(V^*)^*:\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})\\rightarrow \\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G}).$ Letting $r:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto \\rho |_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\Phi ^*|_{\\mathcal {T}(L^2(\\mathbb {G}))}:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto (r\\otimes \\textnormal {id})(\\mathrm {Ad}(V^*)^*(\\rho \\otimes n))\\in M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ The proof of [14] entails the inclusion $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ .","In fact, since $\\Phi $ is a right $L^1(\\mathbb {G})$ -module map and $C_0(\\mathbb {G})$ is essential, more is true: $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq \\mathrm {LUC}(\\mathbb {G})\\subseteq M(C_0(\\mathbb {G})).$ The unique strict extension $\\widetilde{\\Phi }:M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow M(C_0(\\mathbb {G}))$ , which exists by [45], satisfies $\\widetilde{\\Phi }\\circ \\Gamma |_{C_0(\\mathbb {G})}=\\textnormal {id}_{C_0(\\mathbb {G})}$ .","If $\\rho \\in L^{\\infty }(\\mathbb {G})_{\\perp }$ then the invariance $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ implies $\\Phi ^*(\\rho )=0$ .","Thus, $\\Phi ^*$ induces a completely positive left $L^1(\\mathbb {G})$ -module map $\\Phi ^*:L^1(\\mathbb {G})=(\\mathcal {T}(L^2(\\mathbb {G}))/L^{\\infty }(\\mathbb {G})_{\\perp })\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\mathbb {G})$ , for $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle m_{M(\\mathbb {G})}(\\Phi ^*(f)),x\\rangle =\\langle \\Phi ^*(f),\\Gamma (x)\\rangle =\\langle f,\\widetilde{\\Phi }(\\Gamma (x))\\rangle =\\langle f,x\\rangle .$ Hence, $m_{M(\\mathbb {G})}\\circ \\Phi ^*$ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .","Now, by nuclearity of $C_0(\\mathbb {G})$ , there exists a net $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.","Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*.$ Then $\\psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}),M(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}))$ .","By [20] there exist contractive positive functionals $\\nu _i\\in C_u(\\mathbb {G})^*$ such that $\\psi _i=m^r_{\\nu _i}$ .","Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.","Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .","Since $(b)\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.","Hence, by [14] $M^r_{cb}(L^1(\\mathbb {G}))= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))=C_u(\\mathbb {G})^*$ .","For each $i$ and $k$ the map $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*\\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ , so there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*=m^r_{\\nu _k^i}$ .","Thus, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*(f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi ^*(f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i.$ Hence, $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Passing to a subnet we may assume $\\nu _i\\rightarrow \\nu $ weak* in $M(\\mathbb {G})$ .","But then for any $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle f\\star \\nu , x\\rangle =\\langle \\nu , x\\star f\\rangle =\\lim _i\\langle \\nu _i,x\\star f\\rangle =\\lim _i\\langle f\\star \\nu _i,x\\rangle =\\langle f,x\\rangle .$ It follows that $\\nu $ is a right identity for $M(\\mathbb {G})$ , which implies that $M(\\mathbb {G})$ is unital, whence $\\mathbb {G}$ is co-amenable (cf.","[5]).","Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\lambda (G)$ is nuclear and has a tracial state.","Corollary 4.5 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra).","Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.","The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in L^{\\infty }(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ while $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in C_0(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\mathbb {G})$ is 1-flat [14], while $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective, as we now prove.","Theorem 4.6 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective in $M(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ .","If $\\mathbb {G}$ is co-amenable then $M(\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.","Any unital completely contractive Banach algebra $\\mathcal {A}$ is $\\Vert e_{\\mathcal {A}}\\Vert $ -projective, so the claim follows.","Conversely, if $M(\\mathbb {G})$ is 1-projective, then $C_0(\\mathbb {G})^{**}=M(\\mathbb {G})^*$ is 1-injective in $\\mathbf {mod}-M(\\mathbb {G})$ .","It follows that the inclusion morphism $M(C_0(\\mathbb {G}))\\hookrightarrow C_0(\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\mathbb {G})$ -module map $\\Phi : \\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}$ .","Applying the argument of [51] we see that $(\\textnormal {id}\\otimes \\Phi ):C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}\\overline{\\otimes }C_0(\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\textnormal {id}\\otimes \\Phi )(W̉)=W̉$ .","By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\textnormal {id}\\otimes \\Phi )$ , and hence $(\\textnormal {id}\\otimes \\Phi )(W̉^*XW̉)=W̉^*(\\textnormal {id}\\otimes \\Phi )(X)W̉, \\ \\ \\ X\\in C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})).$ In particular, for every $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ and $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ we have $\\Phi (T\\hat{f})&=\\Phi ((\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes T)W̉))=(\\hat{f}\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\Phi )((W̉^*(1\\otimes T)W̉))\\\\&=(\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes \\Phi (T))W̉)=\\Phi (T)\\hat{f}.$ Thus, $\\Phi $ is a morphism with respect to the canonical $L^1(\\widehat{\\mathbb {G}})$ -module structure on $C_0(\\mathbb {G})^{**}$ .","Moreover, the $M(\\mathbb {G})$ -module property entails $\\pi \\circ \\Phi (L^{\\infty }(\\widehat{\\mathbb {G}}))=$ , where $\\pi :C_0(\\mathbb {G})^{**}\\rightarrow L^{\\infty }(\\mathbb {G})$ is the canonical surjection.","Since $\\pi $ is clearly an $L^1(\\widehat{\\mathbb {G}})$ -module map, it follows that $\\pi \\circ \\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\mathbb {G})$ .","Now, by 1-projectivity of $M(\\mathbb {G})$ , for every $\\varepsilon >0$ there exists a morphism $\\Phi _\\varepsilon :M(\\mathbb {G})\\rightarrow M(\\mathbb {G})^+\\widehat{\\otimes }M(\\mathbb {G})$ such that $m^+\\circ \\Phi _\\varepsilon =\\textnormal {id}_{M(\\mathbb {G})}$ , and $\\Vert \\Phi _\\varepsilon \\Vert _{cb}<1+\\varepsilon $ .","By amenability of $$ , we know $C_u(\\mathbb {G})^*= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\Phi _\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\mu _i)$ in $M(\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\mathbb {G}$ .","The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.","In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to the co-multiplication that is an $L^1(\\mathbb {G})$ -module map.","Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow \\Gamma (L^{\\infty }(\\mathbb {G}))$ onto the image of $\\Gamma $ (see [23] and [41]).","It is the combination of the $L^1(\\mathbb {G})$ -module property of $\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\infty }(\\mathbb {G})$ or $C_0(\\mathbb {G})$ to approximation properties of $$ .","Thus, provided one has a suitably nice $L^1(\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.","This is where inner amenability enters the picture.","Recall that a locally compact quantum group $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .","The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .","We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.","Proposition 4.7 Let $\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.","If $L^{\\infty }(\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .","$L^{\\infty }(\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.","Let $(\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\mathbb {G})$ .","It follows verbatim from [56] that $\\Phi _i:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni X\\mapsto (\\omega _{\\xi _i}\\otimes \\textnormal {id})(W^*(U^*\\otimes 1)X(U\\otimes 1)W)\\in L^{\\infty }(\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\mathbb {G})$ -module maps, which cluster weak* in $\\mathcal {CB}(L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=((L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\widehat{\\otimes }L^1(\\mathbb {G}))^*$ to a module left inverse to $\\Gamma $ .","Passing to a subnet we may assume convergence.","$(i)$ Let $(\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\Vert \\varphi _j\\Vert _{cb}\\le C$ .","Since $\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\in L^1(\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\in L^{\\infty }(\\mathbb {G})$ such that $\\varphi _j=\\sum _{k=1}^{n_j} x^j_k\\otimes f^j_k$ .","Put $\\Phi _{ij}=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G}).$ Then $\\Phi _{ij}$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map with $\\Vert \\Phi _{ij}\\Vert _{cb}\\le C$ .","Also, for each $i,j$ and $1\\le k\\le n_j$ , the map $L^{\\infty }(\\mathbb {G})\\ni x\\mapsto \\Phi _i(x^j_k\\otimes x)\\in L^{\\infty }(\\mathbb {G})$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map.","Since $x^j_k$ is a linear combination of positive elements in $L^{\\infty }(\\mathbb {G})$ , and $\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\mathbb {G})$ -module maps $L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ .","By [20], there exist $\\mu ^{ij}_k\\in C_u(\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\mu ^{ij}_k$ .","Hence, for each $x\\in L^{\\infty }(\\mathbb {G})$ $\\Phi _{ij}(x)&=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma (x)\\\\&=\\sum _{k=1}^{n_j}\\Phi _i(x^j_k \\otimes x\\star f^j_k)\\\\&=\\sum _{k=1}^{n_j}x\\star f^j_k\\star \\mu ^{ij}_k\\\\&=x\\star f_{ij},$ where $f_{ij}=\\sum _{k=1}^{n_i} f^j_k\\star \\mu ^{ij}_k\\in L^1(\\mathbb {G})$ as $L^1(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Thus, $\\Vert f_{ij}\\Vert _{cb}\\le C$ , and it follows that $\\lim _i\\lim _jx\\star f_{ij}=\\lim _i\\Phi _i(\\Gamma (x))=x,$ point-weak* in $L^{\\infty }(\\mathbb {G})$ .","Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .","$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.","[34]).","The converse was shown for Kac algebras in [41].","Their proof extends verbatim to arbitrary locally compact quantum groups.","Corollary 4.8 Let $G$ be an inner amenable locally compact group.","If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.","$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.","Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ that can be approximated by normal $L^1(\\mathbb {G})$ -module maps.","It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .","Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.","That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?","By [34] it is known that in this case $\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\mathcal {CB}^\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\mathcal {CB}^\\sigma (VN(G))$ .","Proposition 4.11 Let $\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.","If $C_0(\\mathbb {G})$ has the CBAP, then there exists a net $(\\nu _i)$ in $M(\\mathbb {G})$ such that $\\Vert \\nu _i\\Vert _{cb}\\le \\Lambda _{cb}(C_0(\\mathbb {G}))$ and $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .","If $C_0(\\mathbb {G})$ has the OAP then $M(\\mathbb {G})$ is $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\mathbb {G}))$ .","Let $\\Phi :L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G})$ be the completely positive left $L^1(\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.","Recall that $m_{M(\\mathbb {G})}\\circ \\Phi $ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .","$(i)$ Let $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\Vert \\varphi _i\\Vert _{cb}\\le C$ .","Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .","For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\mathbb {G})$ , and hence the map $(\\textnormal {id}\\otimes x_k^i)\\Phi \\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\mathbb {G})$ -module maps on $L^1(\\mathbb {G})$ , so by [20] there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi =m^r_{\\nu _k^i}$ .","Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi $ .","Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.","Moreover, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi (f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi (f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i\\\\&=f\\star \\nu _i,$ where $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Then $\\Vert \\nu _i\\Vert _{cb}\\le C$ , and it follows that $\\nu _i\\star x\\rightarrow x$ weak* in $L^{\\infty }(\\mathbb {G})$ for all $x\\in C_0(\\mathbb {G})$ .","Since $(\\nu _i)$ is bounded in $\\mathcal {CB}_{L^1(\\mathbb {G})}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ , we have $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .","$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\mathbb {G}))$ .","Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .","Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .","In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).","We now obtain the same conclusion under a priori weaker hypotheses.","Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.","It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .","Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .","However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.","Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .","Hence, $m$ satisfies (REF ).","A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.","By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .","This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.","In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.","Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.","To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.","Proposition 5.2 Let $G$ be a locally compact group.","Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.","By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.","Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .","Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].","Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.","Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].","We now show that 1-biflatness of quantum convolution algebras is a self dual property.","Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.","Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .","By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .","Also, [15] implies that $E$ is a right $\\lhd $ -module map.","Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .","Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .","Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .","Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.","Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).","Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .","By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .","It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.","Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].","Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.","Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .","Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.","The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].","The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.","In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].","It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].","The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .","We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .","Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.","Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.","$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].","$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.","The bimodule analogue of [14] then yields $(iii)$ .","$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .","As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.","Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.","Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .","Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.","In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.","In [16] the authors gave examples of weakly amenable connected groups (e.g.","$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.","We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].","Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .","As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .","The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .","Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.","There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .","Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .","Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).","Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.","Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .","Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.","Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.","Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.","Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .","It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].","Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .","By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .","Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.","Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .","By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .","Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .","It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .","Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .","By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .","Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .","The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .","By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.","Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .","Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .","Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .","Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .","Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .","It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.","Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.","There are examples of weakly amenable residually finite groups (e.g.","$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].","Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.","When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].","Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .","Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .","If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .","Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.","Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].","We now generalize this implication to arbitrary locally compact quantum groups.","Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.","By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].","Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].","We present the proof for the convenience of the reader.","Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .","Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.","Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .","Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.","Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].","For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .","In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.","We show that it is a complete order bijection.","To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .","On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .","Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .","By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","We now show that $\\tilde{\\lambda }$ is a complete isometry.","To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].","In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.","Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].","By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .","Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .","Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].","The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.","To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.","Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .","Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.","Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.","Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].","Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.","Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.","Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .","Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).","Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .","Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.","Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .","Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.","It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.","Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.","By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .","Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .","Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.","The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.","Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .","Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.","The author was partially supported by the NSERC Discovery Grant 1304873."],["Inner Amenability","Given a locally compact quantum group $\\mathbb {G}$ the composition $W\\sigma V\\sigma \\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ defines a unitary co-representation of $\\mathbb {G}$ called the conjugation co-representation, where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\sigma V\\sigma \\xi (s,t)=\\xi (s,s^{-1}ts)\\Delta (s)^{1/2}, \\ \\ \\ \\xi \\in L^2(G\\times G), \\ s,t\\in G.$ Thus, $W\\sigma V\\sigma $ is the unitary generator of the conjugation representation $\\beta _2:G\\rightarrow \\mathcal {B}(L^2(G))$ , where $\\beta _2(s)\\xi (t)=\\lambda (s)\\rho (s)\\xi (t)=\\xi (s^{-1}ts)\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\in L^{\\infty }(G)^*$ satisfying $\\langle m,\\beta _{\\infty }(s)f\\rangle =\\langle m,f\\rangle \\ \\ \\ s\\in G, \\ f\\in L^{\\infty }(G),$ where $\\beta _\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\in G$ , $f\\in L^{\\infty }(G)$ , is the conjugation action on $L^{\\infty }(G)$ .","Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\in \\ell ^{\\infty }(G)^*$ such that $m\\ne \\delta _e$ .","In what follows, inner amenability will always refer to the definition (REF ) given above.","In particular, every discrete group is inner amenable.","Definition 3.2 Let $\\mathbb {G}$ be a locally compact quantum group.","We say that $\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\xi _i)$ of unit vectors such that $\\Vert W\\sigma V\\sigma (\\eta \\otimes \\xi _i)-\\eta \\otimes \\xi _i\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ $\\mathbb {G}$ is inner amenable if there exists a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ satisfying $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}}).$ Such a state is said to be inner invariant.","$\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\in C_0()^*$ such that $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in C_0().$ Such a state is said to be topologically inner invariant.","Examples 3.3 The following known examples are worth mentioning.","Any co-commutative quantum group $\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\sigma V_s\\sigma =W_sW_s^*=1$ .","Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\xi :=\\Lambda _\\varphi (1)$ being conjugation invariant, where $\\varphi $ is the Haar trace on the compact dual.","It was shown in [51] that if $\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\mathbb {G}$ is topologically inner amenable.","Further examples, including the bicrossed product construction will be studied below.","The following proposition is standard.","We include the proof for completeness.","Proposition 3.4 Let $\\mathbb {G}$ be a locally compact quantum group.","If $\\mathbb {G}$ is strongly inner amenability then it is inner amenable.","Let $(\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\sigma V\\sigma $ .","Passing to a subnet, we may assume that $(\\omega _{J\\xi _i}|_{L^{\\infty }(\\widehat{\\mathbb {G}})})$ converges weak* to a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ .","For any $f=\\omega _{\\xi ,\\eta }\\in L^1(\\mathbb {G})$ and $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*, \\ \\ \\ (\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\otimes J)=\\sigma V^*\\sigma ,$ we have $\\langle m,\\hat{x}\\lhd f\\rangle &=\\lim _i\\langle \\omega _{J\\xi _i},\\hat{x}\\lhd f\\rangle \\\\&=\\lim _i\\langle W^*(1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),\\eta \\otimes J\\xi _i\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),W(\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})\\sigma V^*\\sigma (\\xi \\otimes J\\xi _i),\\sigma V^*\\sigma (\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\langle m,\\hat{x}\\rangle \\langle f,1\\rangle .$ In the commutative case, the converse of Proposition REF holds.","Proposition 3.5 A commutative quantum group $\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.","If $\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.","Let $m\\in VN(G)^*$ satisfy $\\langle m,x\\rangle =\\langle m,x \\lhd f\\rangle $ for all $x\\in VN(G)$ and all states $f\\in L^1(G)$ .","Then for $t\\in G$ $\\lambda (t)x\\lambda (t)^* \\lhd f=\\int _G \\lambda (s^{-1}t)x\\lambda (t^{-1}s) f(s)ds=\\int _G\\lambda (r)^*x\\lambda (r)f(tr)dr=x \\lhd f_t,$ so that $\\langle m,\\lambda (t)x\\lambda (t)^*\\rangle =\\langle m,\\lambda (t)x\\lambda (t)^* \\lhd f\\rangle =\\langle m,x \\lhd f_t\\rangle =\\langle m,x\\rangle $ for all $x\\in VN(G)$ , $t\\in G$ and states $f\\in L^1(G)$ .","Thus, $G$ is inner amenable by [16].","Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\xi _i)$ in $L^2(G)$ satisfying $\\Vert \\lambda (s)\\xi _i-\\rho (s)^*\\xi _i\\Vert _{L^2(G)}\\rightarrow 0$ uniformly on compact subsets of $G$ .","For any $\\eta \\in C_c(G)$ we then have $\\langle W\\sigma V\\sigma (\\eta \\otimes \\xi _i),\\eta \\otimes \\xi _i\\rangle =\\iint \\xi _i(ts)\\overline{\\xi _i}(st)\\Delta (s)^{1/2}|\\eta (s)|^2 \\ ds \\ dt\\rightarrow 1.$ It follows that $(\\xi _i)$ is strongly inner invariant.","Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].","Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\mathcal {B}(L^2(G))$ , via $\\langle m,f \\rhd T\\rangle =\\langle m, T \\lhd f\\rangle , \\ \\ \\ f\\in L^1(G), \\ T\\in \\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.","For details, see [13].","Proposition 3.6 A commutative quantum group $\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.","The existence of a tracial state on $C_\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.","By [51] $\\mathbb {G}_a$ is topologically inner amenable if $C_\\lambda ^*(G)$ has a tracial state.","Conversely, if $m$ is an inner invariant state on $C_\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.","Viewing $m$ as a positive definite function in $B_\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\in G$ .","A simple integral calculation then shows that $m$ is a tracial state on $C_\\lambda ^*(G)$ .","Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\infty }(\\widehat{\\mathbb {G}})$ to $C_0()$ .","This is not the case however.","In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\bigoplus _{n\\in \\mathbb {N}} \\Gamma _n\\rtimes \\prod _{n\\in \\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\Gamma _n$ for which $C^*_\\lambda (\\Gamma _n\\rtimes F_n)$ admits a unique tracial state.","In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .","Thus, $G$ is inner amenable.","However, in [30], the authors also show that the continuous function $m$ in $B_\\lambda (G)$ associated to a tracial state on $C^*_\\lambda (G)$ must be discontinuous.","Whence, there is no tracial state on $C^*_\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .","Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\lambda (G)$ .","In the other direction, in general it is not possible to lift a tracial state on $C^*_\\lambda (G)$ to an inner invariant mean on $VN(G)$ .","In [30] it is shown that $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\mathbb {R})$ .","Proposition 3.8 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is co-amenable then $\\mathbb {G}$ is strongly inner amenable.","If $\\mathbb {G}$ is amenable then it is inner amenable.","If $$ is co-amenable, let $E\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\omega _{\\xi _i}$ with $J̉\\xi _i=\\xi _i$ (since $L^{\\infty }(\\widehat{\\mathbb {G}})\\curvearrowright L^2(\\mathbb {G})$ is standard).","Then $1=(\\textnormal {id}\\otimes E)(W)=\\lim _i(\\textnormal {id}\\otimes \\omega _{\\xi _i})(W)$ strongly, from which it follows that $\\Vert W(\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Since $\\sigma V\\sigma =(J̉\\otimes J̉)W^*(J̉\\otimes J̉)$ , and $J̉\\xi _i=\\xi _i$ , we also have $\\Vert \\sigma V\\sigma (\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Whence, $\\mathbb {G}$ is strongly inner amenable.","Now, suppose $\\mathbb {G}$ is amenable and let $m\\in L^{\\infty }(\\mathbb {G})^*$ be a two-sided invariant mean.","We will show a stronger statement by providing a state $M\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ such that $\\langle M,\\rho \\rhd T\\rangle =\\langle M,T\\lhd \\rho \\rangle , \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})),$ upon which restriction to $L^{\\infty }(\\widehat{\\mathbb {G}})$ is the desired state.","Letting $m$ also denote its restriction to $\\mathrm {LUC}(\\mathbb {G}):=\\langle L^{\\infty }(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle $ , let $m_0:=\\rho _0\\circ \\Theta ^r(m)\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ , where $\\rho _0\\in \\mathcal {T}(L^2(\\mathbb {G}))$ is a fixed normal state, and $\\Theta ^r:\\mathrm {LUC}(\\mathbb {G})^*\\ni n\\mapsto (T\\mapsto (\\textnormal {id}\\otimes n)V^*(T\\otimes 1)V)\\in \\mathcal {CB}_{\\mathcal {T}_{\\rhd }}(\\mathcal {B}(L^2(\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).","Since $\\mathrm {LUC}(\\mathbb {G})=\\langle \\mathcal {B}(L^2(\\mathbb {G}))\\rhd \\mathcal {T}(L^2(\\mathbb {G}))\\rangle $ [35], one may view the map $\\Theta ^r$ as follows: $\\langle \\Theta ^r(n)(T),\\rho \\rangle =\\langle n,T\\rhd \\rho \\rangle , \\ \\ \\ n\\in \\mathrm {LUC}(\\mathbb {G})^*, \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Consider the state $R^*(m_0)\\square m_0\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\square $ is the left Arens product on $\\mathcal {B}(L^2(\\mathbb {G}))^*$ extending the multiplication in $\\mathcal {T}_\\rhd =(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ .","Fix $\\rho ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","Firstly, since $m\\in \\mathrm {LUC}(\\mathbb {G})^*$ is an invariant mean $m\\square \\pi (\\rho )=\\langle \\pi (\\rho ),1\\rangle m=\\langle \\rho ,1\\rangle m$ .","Thus, $\\langle m_0\\square \\rho ,T\\rangle &=\\langle m_0,\\rho \\rhd T\\rangle =\\langle \\rho _0,\\Theta ^r(m)\\circ \\Theta ^r(\\pi (\\rho ))(T)\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m\\square \\pi (\\rho ))(T)\\rangle =\\langle \\rho ,1\\rangle \\langle \\rho _0,\\Theta ^r(m)(T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle m_0,T\\rangle .$ Hence, $m_0\\square \\rho =\\langle \\rho ,1\\rangle m_0$ .","Secondly, $\\Theta ^r(m)$ is a conditional expectation onto $L^{\\infty }(\\widehat{\\mathbb {G}})$ commuting with both the right $\\mathcal {T}_\\rhd $ - and right $\\mathcal {T}_\\lhd $ -actions [15].","Since $\\hat{x}\\rhd \\omega =\\langle \\omega ,\\hat{x}\\rangle 1$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ and $\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we also have $\\langle m_0\\square (T\\lhd \\rho ),\\omega \\rangle &=\\langle m_0,(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,\\Theta ^r(m)((T\\lhd \\rho )\\rhd \\omega )\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T\\lhd \\rho ),\\omega \\rangle \\\\&=\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T)\\lhd \\rho ,\\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T),\\rho \\lhd \\omega \\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T)\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle \\rho _0,\\Theta ^r(m)(T\\rhd (\\rho \\lhd \\omega ))\\rangle \\\\&=\\langle m_0,T\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle m_0\\square T,\\rho \\lhd \\omega \\rangle \\\\&=\\langle (m_0\\square T)\\lhd \\rho ,\\omega \\rangle .\\\\$ Thus, $m_0\\square (T\\lhd \\rho )=(m_0\\square T)\\lhd \\rho $ .","Putting things together, on the one hand we obtain $\\langle R^*(m_0)\\square m_0,\\rho \\rhd T\\rangle =\\langle R^*(m_0),(m_0\\square \\rho )\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle ,$ and on the other, $\\langle R^*(m_0)\\square m_0,T\\lhd \\rho \\rangle &=\\langle R^*(m_0),m_0\\square (T\\lhd \\rho )\\rangle =\\langle R^*(m_0),(m_0\\square T)\\lhd \\rho \\rangle \\\\&=\\langle m_0,R((m_0\\square T)\\lhd \\rho )\\rangle =\\langle m_0,R_*(\\rho )\\rhd R(m_0\\square T)\\rangle \\\\&=\\langle m_0\\square R_*(\\rho ),R(m_0\\square T)\\rangle =\\langle R_*(\\rho ),1\\rangle \\langle m_0, R(m_0\\square T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0)\\square m_0, T\\rangle .$ Therefore, $M:=R^*(m_0)\\square m_0$ is the required state.","Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].","Corollary 3.9 A locally compact quantum group $\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $2pt\\mathbf {mod}$ .","Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.","Proposition 3.10 Let $\\mathbb {G}$ and $\\mathbb {H}$ be locally compact quantum groups such that $\\mathbb {H}$ is a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes.","If $\\mathbb {G}$ is inner amenable then $\\mathbb {H}$ is inner amenable.","Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\circ \\gamma \\in L^{\\infty }(\\widehat{\\mathbb {H}})^*$ , and let $W_{\\mathbb {G}}$ and $W_{\\mathbb {H}}$ denote the left fundamental unitaries of $\\mathbb {G}$ and $\\mathbb {H}$ , respectively.","We also denote by $\\mathbb {W}_{\\mathbb {G}}\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_u())$ and $\\mathbb {W}_{\\mathbb {H}}\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\pi _{\\mathbb {G}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})=W_{\\mathbb {G}}$ and $(\\pi _{\\mathbb {H}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})=W_{\\mathbb {H}}$ .","Finally, we define $\\W _{\\mathbb {G}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0()), \\ \\ \\ \\W _{\\mathbb {H}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_0()).$ Let $\\pi _{\\mathbb {G},\\mathbb {H}}:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {H})$ be the surjection from [24].","Its dual morphism is a non-degenerate $*$ -homomorphism $\\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}:C_u()\\rightarrow M(C_u())$ satisfying $(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})=(\\textnormal {id}\\otimes \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}}).$ From the relation $\\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}}=\\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}$ [24], we have $(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})&=(\\textnormal {id}\\otimes \\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}})(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})\\\\&=(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}).$ Thus, for any $x̉\\in L^{\\infty }(\\widehat{\\mathbb {H}})$ and $g\\in L^{1}(\\mathbb {H})$ we have $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\gamma ((g\\otimes \\textnormal {id})W_{\\mathbb {H}}^*(1\\otimes x̉)W_{\\mathbb {H}})\\rangle \\\\&=\\langle m̉,\\gamma ((\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\W _{\\mathbb {H}}^*(1\\otimes x̉)\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})^*(1\\otimes \\gamma (x̉))(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}^*(1\\otimes \\gamma (x̉))\\W _{\\mathbb {G}})\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g)))(\\gamma (\\hat{x}))\\rangle ,$ where the last equality follows from Lemma REF .","By invariance of $m̉$ , we may convolve with any state $f\\in L^1(\\mathbb {G})$ to obtain $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(g))(\\gamma (\\hat{x}))\\lhd _{\\mathbb {G}}f\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)(\\gamma (\\hat{x}))\\rangle \\\\&=\\langle m̉,\\gamma (\\hat{x})\\lhd _{\\mathbb {G}}(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)\\rangle \\\\&=\\langle \\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f,1\\rangle \\langle m̉,\\gamma (\\hat{x})\\rangle \\\\&=\\langle g,1\\rangle \\langle n̉,\\hat{x}\\rangle .$"],["Examples arising from the bicrossed product construction","Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\rightarrow G$ and an anti-homomorphism $j:G_2\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.","Suppose further that $G_1\\times G_2\\ni (g,s)\\mapsto i(g)j(s)\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.","Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].","Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\alpha :G_1\\times G_2\\rightarrow G_2$ and $\\beta :G_1\\times G_2\\rightarrow G_1$ satisfying mutual co-cycle relations [65].","It is known that the von Neumann crossed product $G_1\\rtimes _\\alpha L^\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .","The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\infty }(G_1)^\\beta \\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .","Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.","In preparation we collect some useful formulae from [65], to which we refer the reader for details.","To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .","The fundamental unitary $W$ satisfies $W^*\\xi (g,s,h,t)=\\xi (\\beta _{t}(h)^{-1}g,s,h,\\alpha _{\\beta _t(h)^{-1}g}(s)t), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2).$ Letting $\\Delta $ , $\\Delta _1$ , and $\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\xi (g,s)=\\Delta (\\alpha _g(s))^{1/2}\\Delta _1(\\beta _s(g)g^{-1})^{1/2}\\Delta _2(\\alpha _g(s)s^{-1})^{1/2}\\overline{\\xi }(\\beta _s(g),s^{-1}), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2).$ Let $\\Psi :G_2\\times G_1\\rightarrow (0,\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\Psi (s,g):=\\frac{d\\beta _s(g)}{dg}$ .","It follows that $\\Psi (s,g)=\\Delta (\\alpha _g(s))\\Delta _1(\\beta _s(g)g^{-1})\\Delta _2(\\alpha _g(s))$ .","The action $\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )^{1/2}\\xi (\\beta _{s^{-1}}(g)), \\ \\ \\ \\xi \\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )f(\\beta _{s^{-1}}(g)), \\ \\ \\ f\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\beta $ preserves $\\Delta _1$ , there exists an asymptotically $\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\Vert \\cdot \\Vert _1=1}$ with $\\mathrm {supp}(f_i)\\rightarrow \\lbrace e\\rbrace $ satisfying $\\Vert v^1_sf_i-f_i\\Vert _1\\rightarrow 0$ uniformly on compacta.","Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.","Using the co-cycle properties of $\\alpha $ and $\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\otimes J̉)W^*(J̉\\otimes J̉)\\xi (g,s,h,t)\\\\&=\\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2}\\xi (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\alpha _{h^{-1}\\beta _s(g)}(s^{-1})^{-1})$ for all $\\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2)$ .","Let $\\eta _i:=\\sqrt{f_i}$ .","By hypothesis $(iii)$ we have $\\Vert v_s\\eta _i - \\eta _i\\Vert _{L^2(G_1)}^2\\le \\Vert v_s^1f_i-f_i\\Vert _{L^1(G_1)}\\rightarrow 0$ uniformly on compacta.","Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\times G_2\\times G_1\\rightarrow it follows that\\begin{equation} \\int f(g,s,h) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dh \\rightarrow f(g,s,e)\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\xi _j)$ in $C_c(G_2)_{\\Vert \\cdot \\Vert _2=1}$ satisfying $\\Vert \\rho (s)\\xi _j-\\lambda (s)^*\\xi _j\\Vert _{L^2(G_2)}\\rightarrow 0$ uniformly on compacta.","Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\xi (g)=\\xi (g^{-1})\\Delta _1(g^{-1})$ .","Then for any $\\eta \\in C_c(G_1\\times G_2)$ , and any $j$ , we have $&\\langle W\\sigma V\\sigma (\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),\\eta \\otimes U_1\\eta _i\\otimes \\xi _j\\rangle =\\langle (J̉\\otimes J̉)W^*(J̉\\otimes J̉)(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),W^*(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j)\\rangle \\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ U_1\\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{U_1\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ \\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\ \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\ \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ \\overline{\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiint \\bigg (\\int \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ dt\\bigg )\\\\&\\times \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dg \\ ds \\ dh\\\\&\\rightarrow \\iiint \\xi _j(t(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1})\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2} \\ dg \\ ds \\ dt\\\\$ by ().","But $\\alpha _{\\beta _s(g)}(s^{-1})^{-1}=\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2}\\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\Delta _2(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\\\&=\\Delta _2(\\alpha _g(s))^{1/2}, \\ \\ \\ a.e.$ The final integral in the above calculation therefore reduces to $\\iiint \\Delta _2(\\alpha _g(s))^{1/2} \\ \\xi _j(t\\alpha _{g}(s))\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ dg \\ ds \\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\iint |\\eta (g,s)|^2 \\ dg \\ ds = \\Vert \\eta \\Vert _{L^2(G_1\\times G_2)}^2.$ Denoting the index sets of $(\\eta _i)$ and $(\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\mathcal {I}:=J\\times I^{J}$ and for $I=(j,(i_j)_{j\\in J})\\in \\mathcal {I}$ we let $\\xi _I:=U_1\\eta _{i_j}\\otimes \\xi _j\\in L^2(G_1\\times G_2)$ .","By [39] and the above analysis, the resulting net $(\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .","Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.","Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.","Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.","For example, $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\mathbb {R}^2$ and $\\mathbb {F}_6$ are.","Example 3.14 We consider a discretized version of [26].","Let $G=\\bigg \\lbrace \\begin{pmatrix} a & b & x\\\\ c & d & y\\\\ 0 & 0 & 1\\end{pmatrix}\\mid \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\in SL(2,\\mathbb {Q}), \\ x,y\\in \\mathbb {Q}\\bigg \\rbrace ,$ $G_1=(\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\mathbb {Q})$ , viewed as discrete groups.","The embeddings $G_1\\ni (x,y)\\mapsto \\begin{pmatrix} 1 & 0 & -x\\\\ -x & 1 & -y+\\frac{1}{2}x^2\\\\ 0 & 0 & 1\\end{pmatrix}\\in G, \\ \\ G_2\\ni \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\mapsto \\begin{pmatrix}d & -b & 0\\\\ -c & a & 0\\\\ 0 & 0 & 1\\end{pmatrix}\\in G$ determine a matched pair structure for $(G_1,G_2)$ .","The corresponding actions are given by $\\alpha _{(x,y)}\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}&=\\begin{pmatrix}a+bx & b\\\\ c+dx-(a+bx)(ax+b(y+\\frac{1}{2}x^2)) & d-b(ax+b(y+\\frac{1}{2}x^2))\\end{pmatrix} \\\\\\beta _{\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}}(x,y)&=\\bigg (ax+by+\\frac{b}{2}x^2,cx+d\\bigg (y+\\frac{1}{2}x^2\\bigg )-\\frac{1}{2}\\bigg (ax+b\\bigg (y+\\frac{1}{2}x^2\\bigg )\\bigg )^2\\bigg ).$ By Corollary REF , the bicrossed product $\\mathbb {G}:=VN(\\mathbb {Q}^2)^\\beta \\bowtie _\\alpha L^\\infty (SL(2,\\mathbb {Q}))$ is strongly inner amenable.","Since $SL(2,\\mathbb {Q})$ is not amenable, it follows from [26] that $\\mathbb {G}$ is not amenable.","Moreover, since $\\mathbb {Q}^2$ is not compact, $\\mathbb {G}$ is not discrete by [65].","Thus, $\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group."],["Relative Injectivity and the Averaging Technique","In [56] Ruan and Xu showed that the dual $L^{\\infty }(\\widehat{\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\mathbb {G}$ is relatively 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}$ .","We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.","Below we let $\\widetilde{\\mathbb {G}}:=\\mathrm {Gr}()$ denote the intrinsic group of $$ .","Proposition 4.1 Let $\\mathbb {G}$ be a locally compact quantum group.","Consider the following conditions: $$ is inner amenable; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ ; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ ; $\\widetilde{}=\\mathrm {Gr}(\\mathbb {G})$ is inner amenable.","Then $(i)\\Rightarrow (ii)\\Leftrightarrow (iii)\\Rightarrow (iv)$ .","When $\\mathbb {G}$ is co-commutative, the conditions are equivalent.","$(i)\\Rightarrow (ii)$ : Given a state $n\\in L^{\\infty }(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $m:=n\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant.","It suffices to provide a completely contractive morphism which is a left inverse to the map $\\Delta :L^{\\infty }(\\mathbb {G})\\rightarrow \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ given by $\\Delta (x)(f)=x\\rhd f, \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Identifying $\\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))\\cong L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ via $\\langle \\Phi ,f\\otimes g\\rangle =\\langle \\Phi (f),g\\rangle , \\ \\ \\ \\Phi \\in \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G})), \\ f,g\\in L^1(\\mathbb {G}),$ we have $\\Delta =\\Gamma $ , and that the corresponding $L^1(\\mathbb {G})$ -module structure on $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ is defined by $X\\unrhd f=(f\\otimes \\textnormal {id}\\otimes \\textnormal {id})(\\Gamma ^r\\otimes \\textnormal {id})(X)$ for $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $f\\in L^1(\\mathbb {G})$ .","First, consider the map $\\Phi :\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni A\\mapsto (\\textnormal {id}\\otimes m)(V^*AV)\\in \\mathcal {B}(L^2(\\mathbb {G})).$ Clearly, $\\Phi $ is a completely contractive left inverse to $\\Gamma $ .","We show that $\\Phi $ is a right $\\mathcal {T}_\\rhd $ -module map.","This will complete the proof since [14] will entail the invariance $\\Phi (L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ , and the restricted module action $\\mathcal {T}_\\rhd \\curvearrowright L^{\\infty }(\\mathbb {G})$ is the pertinent $L^1(\\mathbb {G})$ -module action.","To this end, fix $A\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $\\rho \\in \\mathcal {T}(L^2(\\mathbb {G}))$ .","Then $\\Phi (A\\unrhd \\rho )&=\\Phi ((\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\prime }=\\sigma V^*\\sigma $ , where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ , for any $\\tau ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we have $&\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*(\\sigma \\otimes 1)V_{23}^*A_{23}V_{23}(\\sigma \\otimes 1)V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\omega \\otimes \\tau \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m)(V^*(1\\otimes (\\tau \\otimes \\textnormal {id})(V^*AV))V),\\omega \\rangle \\\\&=\\langle ( m\\otimes \\textnormal {id})(V̉^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV)\\otimes 1)V̉^{\\prime *}),\\omega \\rangle \\\\&=\\langle m,\\omega \\widehat{\\rhd }^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV))\\rangle \\\\&=\\langle m,(\\tau \\otimes \\textnormal {id})(V^*AV)\\rangle \\langle \\omega ,1\\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m\\otimes \\textnormal {id})(V^*AV\\otimes 1),\\tau \\otimes \\omega \\rangle \\\\&=\\langle \\Phi (A)\\otimes 1,\\tau \\otimes \\omega \\rangle .$ As $\\tau $ and $\\omega $ were arbitrary, we have $\\Phi (A\\unrhd \\rho )&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\Phi (A)\\otimes 1)V^*)\\\\&=\\Phi (A)\\rhd \\rho .$ $(ii)\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\mathbb {G})$ -module left inverse $\\Phi $ to $\\Gamma $ , it follows that $R\\circ \\Phi \\circ \\Sigma \\circ (R\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ .","$(iii)\\Rightarrow (iv)$ : Recall that $\\mathrm {Gr}(\\mathbb {G})$ is a group of unitaries in $L^{\\infty }(\\mathbb {G})$ , so it acts naturally on $L^{\\infty }(\\mathbb {G})$ by conjugation.","The existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ which is $\\mathrm {Gr}(\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\Gamma (L^{\\infty }(\\mathbb {G}))-L^{\\infty }(\\mathbb {G})$ -bimodule property of $\\Phi $ .","Since $\\widetilde{\\widehat{\\mathbb {G}}}$ is a closed quantum subgroup of $\\widehat{\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\gamma :L^{\\infty }(\\widehat{\\widetilde{}})\\rightarrow L^{\\infty }(\\mathbb {G})$ intertwining the co-multiplications.","As $L^{\\infty }(\\widehat{\\widetilde{}})=VN(\\widetilde{\\widehat{\\mathbb {G}}})=VN(\\mathrm {Gr}(\\mathbb {G}))$ , the state $m\\circ \\gamma \\in VN(\\mathrm {Gr}(\\mathbb {G}))^*$ is $\\mathrm {Gr}(\\mathbb {G})$ -invariant, making $\\mathrm {Gr}(\\mathbb {G})$ inner amenable by [16].","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, then $\\mathrm {Gr}(\\mathbb {G}_s)=G$ and the implication $(iv)\\Rightarrow (i)$ follows immediately from [16].","Corollary 4.2 Let $\\mathbb {G}$ be a locally compact quantum group for which $L^{\\infty }(\\widehat{\\mathbb {G}})$ is an injective von Neumann algebra.","Then the following are equivalent: $\\mathbb {G}$ is amenable; $\\mathbb {G}$ is inner amenable; $L^{\\infty }(\\widehat{\\mathbb {G}})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","Propositions REF and REF yield the implications $(i)\\Rightarrow (ii)\\Rightarrow (iii)$ .","Assume $(iii)$ .","Since $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $\\mathbf {mod}\\hspace{2.0pt}it follows from \\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.","Hence, $ G$ is amenable by \\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\mathbb {G}$ and amenability of $$ .","One of their main results is the following.","Theorem 4.3 (Ng–Viselter) Let $\\mathbb {G}$ be a locally compact quantum group.","Consider the following conditions: $\\mathbb {G}$ is co-amenable; $C_0(\\mathbb {G})$ is nuclear and there exists a state $\\rho \\in C_0(\\mathbb {G})^*$ such that $\\langle \\rho ,x\\widehat{\\lhd } \\hat{f}\\rangle =\\langle \\rho ,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ $$ is amenable.","Then $(a)\\Rightarrow (b)\\Rightarrow (c)$ .","Moreover, if $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra), then $(a)\\Rightarrow (b^{\\prime })\\Rightarrow (b)$ , where $C_0(\\mathbb {G})$ is nuclear and has a tracial state.","They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .","We now show that $(b)$ is indeed equivalent to $(a)$ .","Theorem 4.4 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.","Suppose that $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.","Given a state $m\\in C_0(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $n:=m\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant, as in the proof of Proposition REF .","Let $n\\in M(C_0(\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\textnormal {id}\\otimes n):M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ denote the unique extension of the slice map $(\\textnormal {id}\\otimes n):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))$ which is strictly continuous on the unit ball.","By strict density of $C_0(\\mathbb {G})$ in $M(C_0(\\mathbb {G}))$ , it follows that $\\langle n,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle n,x\\rangle \\langle \\hat{f}^{\\prime },1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in M(C_0(\\mathbb {G})).$ Since $V\\in M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))$ , the map $\\Phi :C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\Phi (A)=(\\textnormal {id}\\otimes n)(V^*AV), \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.","Using the extended $\\widehat{\\rhd }^{\\prime }$ -invariance on $M(C_0(\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\Phi (A\\unrhd \\rho )=\\Phi (A)\\rhd \\rho , \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Since $\\mathrm {Ad}(V^*):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})$ and $\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})=(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))^*,$ we have $\\mathrm {Ad}(V^*)^*:\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})\\rightarrow \\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G}).$ Letting $r:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto \\rho |_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\Phi ^*|_{\\mathcal {T}(L^2(\\mathbb {G}))}:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto (r\\otimes \\textnormal {id})(\\mathrm {Ad}(V^*)^*(\\rho \\otimes n))\\in M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ The proof of [14] entails the inclusion $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ .","In fact, since $\\Phi $ is a right $L^1(\\mathbb {G})$ -module map and $C_0(\\mathbb {G})$ is essential, more is true: $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq \\mathrm {LUC}(\\mathbb {G})\\subseteq M(C_0(\\mathbb {G})).$ The unique strict extension $\\widetilde{\\Phi }:M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow M(C_0(\\mathbb {G}))$ , which exists by [45], satisfies $\\widetilde{\\Phi }\\circ \\Gamma |_{C_0(\\mathbb {G})}=\\textnormal {id}_{C_0(\\mathbb {G})}$ .","If $\\rho \\in L^{\\infty }(\\mathbb {G})_{\\perp }$ then the invariance $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ implies $\\Phi ^*(\\rho )=0$ .","Thus, $\\Phi ^*$ induces a completely positive left $L^1(\\mathbb {G})$ -module map $\\Phi ^*:L^1(\\mathbb {G})=(\\mathcal {T}(L^2(\\mathbb {G}))/L^{\\infty }(\\mathbb {G})_{\\perp })\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\mathbb {G})$ , for $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle m_{M(\\mathbb {G})}(\\Phi ^*(f)),x\\rangle =\\langle \\Phi ^*(f),\\Gamma (x)\\rangle =\\langle f,\\widetilde{\\Phi }(\\Gamma (x))\\rangle =\\langle f,x\\rangle .$ Hence, $m_{M(\\mathbb {G})}\\circ \\Phi ^*$ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .","Now, by nuclearity of $C_0(\\mathbb {G})$ , there exists a net $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.","Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*.$ Then $\\psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}),M(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}))$ .","By [20] there exist contractive positive functionals $\\nu _i\\in C_u(\\mathbb {G})^*$ such that $\\psi _i=m^r_{\\nu _i}$ .","Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.","Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .","Since $(b)\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.","Hence, by [14] $M^r_{cb}(L^1(\\mathbb {G}))= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))=C_u(\\mathbb {G})^*$ .","For each $i$ and $k$ the map $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*\\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ , so there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*=m^r_{\\nu _k^i}$ .","Thus, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*(f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi ^*(f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i.$ Hence, $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Passing to a subnet we may assume $\\nu _i\\rightarrow \\nu $ weak* in $M(\\mathbb {G})$ .","But then for any $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle f\\star \\nu , x\\rangle =\\langle \\nu , x\\star f\\rangle =\\lim _i\\langle \\nu _i,x\\star f\\rangle =\\lim _i\\langle f\\star \\nu _i,x\\rangle =\\langle f,x\\rangle .$ It follows that $\\nu $ is a right identity for $M(\\mathbb {G})$ , which implies that $M(\\mathbb {G})$ is unital, whence $\\mathbb {G}$ is co-amenable (cf.","[5]).","Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\lambda (G)$ is nuclear and has a tracial state.","Corollary 4.5 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra).","Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.","The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in L^{\\infty }(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ while $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in C_0(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\mathbb {G})$ is 1-flat [14], while $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective, as we now prove.","Theorem 4.6 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective in $M(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ .","If $\\mathbb {G}$ is co-amenable then $M(\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.","Any unital completely contractive Banach algebra $\\mathcal {A}$ is $\\Vert e_{\\mathcal {A}}\\Vert $ -projective, so the claim follows.","Conversely, if $M(\\mathbb {G})$ is 1-projective, then $C_0(\\mathbb {G})^{**}=M(\\mathbb {G})^*$ is 1-injective in $\\mathbf {mod}-M(\\mathbb {G})$ .","It follows that the inclusion morphism $M(C_0(\\mathbb {G}))\\hookrightarrow C_0(\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\mathbb {G})$ -module map $\\Phi : \\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}$ .","Applying the argument of [51] we see that $(\\textnormal {id}\\otimes \\Phi ):C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}\\overline{\\otimes }C_0(\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\textnormal {id}\\otimes \\Phi )(W̉)=W̉$ .","By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\textnormal {id}\\otimes \\Phi )$ , and hence $(\\textnormal {id}\\otimes \\Phi )(W̉^*XW̉)=W̉^*(\\textnormal {id}\\otimes \\Phi )(X)W̉, \\ \\ \\ X\\in C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})).$ In particular, for every $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ and $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ we have $\\Phi (T\\hat{f})&=\\Phi ((\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes T)W̉))=(\\hat{f}\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\Phi )((W̉^*(1\\otimes T)W̉))\\\\&=(\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes \\Phi (T))W̉)=\\Phi (T)\\hat{f}.$ Thus, $\\Phi $ is a morphism with respect to the canonical $L^1(\\widehat{\\mathbb {G}})$ -module structure on $C_0(\\mathbb {G})^{**}$ .","Moreover, the $M(\\mathbb {G})$ -module property entails $\\pi \\circ \\Phi (L^{\\infty }(\\widehat{\\mathbb {G}}))=$ , where $\\pi :C_0(\\mathbb {G})^{**}\\rightarrow L^{\\infty }(\\mathbb {G})$ is the canonical surjection.","Since $\\pi $ is clearly an $L^1(\\widehat{\\mathbb {G}})$ -module map, it follows that $\\pi \\circ \\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\mathbb {G})$ .","Now, by 1-projectivity of $M(\\mathbb {G})$ , for every $\\varepsilon >0$ there exists a morphism $\\Phi _\\varepsilon :M(\\mathbb {G})\\rightarrow M(\\mathbb {G})^+\\widehat{\\otimes }M(\\mathbb {G})$ such that $m^+\\circ \\Phi _\\varepsilon =\\textnormal {id}_{M(\\mathbb {G})}$ , and $\\Vert \\Phi _\\varepsilon \\Vert _{cb}<1+\\varepsilon $ .","By amenability of $$ , we know $C_u(\\mathbb {G})^*= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\Phi _\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\mu _i)$ in $M(\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\mathbb {G}$ .","The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.","In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to the co-multiplication that is an $L^1(\\mathbb {G})$ -module map.","Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow \\Gamma (L^{\\infty }(\\mathbb {G}))$ onto the image of $\\Gamma $ (see [23] and [41]).","It is the combination of the $L^1(\\mathbb {G})$ -module property of $\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\infty }(\\mathbb {G})$ or $C_0(\\mathbb {G})$ to approximation properties of $$ .","Thus, provided one has a suitably nice $L^1(\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.","This is where inner amenability enters the picture.","Recall that a locally compact quantum group $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .","The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .","We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.","Proposition 4.7 Let $\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.","If $L^{\\infty }(\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .","$L^{\\infty }(\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.","Let $(\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\mathbb {G})$ .","It follows verbatim from [56] that $\\Phi _i:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni X\\mapsto (\\omega _{\\xi _i}\\otimes \\textnormal {id})(W^*(U^*\\otimes 1)X(U\\otimes 1)W)\\in L^{\\infty }(\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\mathbb {G})$ -module maps, which cluster weak* in $\\mathcal {CB}(L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=((L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\widehat{\\otimes }L^1(\\mathbb {G}))^*$ to a module left inverse to $\\Gamma $ .","Passing to a subnet we may assume convergence.","$(i)$ Let $(\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\Vert \\varphi _j\\Vert _{cb}\\le C$ .","Since $\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\in L^1(\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\in L^{\\infty }(\\mathbb {G})$ such that $\\varphi _j=\\sum _{k=1}^{n_j} x^j_k\\otimes f^j_k$ .","Put $\\Phi _{ij}=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G}).$ Then $\\Phi _{ij}$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map with $\\Vert \\Phi _{ij}\\Vert _{cb}\\le C$ .","Also, for each $i,j$ and $1\\le k\\le n_j$ , the map $L^{\\infty }(\\mathbb {G})\\ni x\\mapsto \\Phi _i(x^j_k\\otimes x)\\in L^{\\infty }(\\mathbb {G})$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map.","Since $x^j_k$ is a linear combination of positive elements in $L^{\\infty }(\\mathbb {G})$ , and $\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\mathbb {G})$ -module maps $L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ .","By [20], there exist $\\mu ^{ij}_k\\in C_u(\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\mu ^{ij}_k$ .","Hence, for each $x\\in L^{\\infty }(\\mathbb {G})$ $\\Phi _{ij}(x)&=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma (x)\\\\&=\\sum _{k=1}^{n_j}\\Phi _i(x^j_k \\otimes x\\star f^j_k)\\\\&=\\sum _{k=1}^{n_j}x\\star f^j_k\\star \\mu ^{ij}_k\\\\&=x\\star f_{ij},$ where $f_{ij}=\\sum _{k=1}^{n_i} f^j_k\\star \\mu ^{ij}_k\\in L^1(\\mathbb {G})$ as $L^1(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Thus, $\\Vert f_{ij}\\Vert _{cb}\\le C$ , and it follows that $\\lim _i\\lim _jx\\star f_{ij}=\\lim _i\\Phi _i(\\Gamma (x))=x,$ point-weak* in $L^{\\infty }(\\mathbb {G})$ .","Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .","$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.","[34]).","The converse was shown for Kac algebras in [41].","Their proof extends verbatim to arbitrary locally compact quantum groups.","Corollary 4.8 Let $G$ be an inner amenable locally compact group.","If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.","$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.","Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ that can be approximated by normal $L^1(\\mathbb {G})$ -module maps.","It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .","Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.","That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?","By [34] it is known that in this case $\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\mathcal {CB}^\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\mathcal {CB}^\\sigma (VN(G))$ .","Proposition 4.11 Let $\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.","If $C_0(\\mathbb {G})$ has the CBAP, then there exists a net $(\\nu _i)$ in $M(\\mathbb {G})$ such that $\\Vert \\nu _i\\Vert _{cb}\\le \\Lambda _{cb}(C_0(\\mathbb {G}))$ and $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .","If $C_0(\\mathbb {G})$ has the OAP then $M(\\mathbb {G})$ is $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\mathbb {G}))$ .","Let $\\Phi :L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G})$ be the completely positive left $L^1(\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.","Recall that $m_{M(\\mathbb {G})}\\circ \\Phi $ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .","$(i)$ Let $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\Vert \\varphi _i\\Vert _{cb}\\le C$ .","Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .","For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\mathbb {G})$ , and hence the map $(\\textnormal {id}\\otimes x_k^i)\\Phi \\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\mathbb {G})$ -module maps on $L^1(\\mathbb {G})$ , so by [20] there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi =m^r_{\\nu _k^i}$ .","Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi $ .","Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.","Moreover, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi (f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi (f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i\\\\&=f\\star \\nu _i,$ where $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .","Then $\\Vert \\nu _i\\Vert _{cb}\\le C$ , and it follows that $\\nu _i\\star x\\rightarrow x$ weak* in $L^{\\infty }(\\mathbb {G})$ for all $x\\in C_0(\\mathbb {G})$ .","Since $(\\nu _i)$ is bounded in $\\mathcal {CB}_{L^1(\\mathbb {G})}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ , we have $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .","$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\mathbb {G}))$ .","Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .","Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .","In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).","We now obtain the same conclusion under a priori weaker hypotheses.","Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.","It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .","Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .","However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.","Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .","Hence, $m$ satisfies (REF ).","A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.","By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .","This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.","In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.","Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.","To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.","Proposition 5.2 Let $G$ be a locally compact group.","Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.","By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.","Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .","Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].","Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.","Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].","We now show that 1-biflatness of quantum convolution algebras is a self dual property.","Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.","Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .","By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .","Also, [15] implies that $E$ is a right $\\lhd $ -module map.","Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .","Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .","Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .","Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.","Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).","Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .","By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .","It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.","Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].","Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.","Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .","Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.","The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].","The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.","In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].","It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].","The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .","We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .","Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.","Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.","$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].","$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.","The bimodule analogue of [14] then yields $(iii)$ .","$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .","As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.","Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.","Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .","Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.","In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.","In [16] the authors gave examples of weakly amenable connected groups (e.g.","$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.","We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].","Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .","As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .","The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .","Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.","There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .","Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .","Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).","Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.","Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .","Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.","Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.","Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.","Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .","It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].","Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .","By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .","Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.","Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .","By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .","Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .","It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .","Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .","By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .","Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .","The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .","By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.","Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .","Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .","Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .","Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .","Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .","It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.","Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.","There are examples of weakly amenable residually finite groups (e.g.","$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].","Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.","When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].","Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .","Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .","If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .","Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.","Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].","We now generalize this implication to arbitrary locally compact quantum groups.","Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.","By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].","Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].","We present the proof for the convenience of the reader.","Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .","Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.","Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .","Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.","Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].","For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .","In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.","We show that it is a complete order bijection.","To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .","On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .","Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .","By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","We now show that $\\tilde{\\lambda }$ is a complete isometry.","To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].","In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.","Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].","By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .","Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .","Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].","The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.","To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.","Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .","Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.","Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.","Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].","Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.","Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.","Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .","Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).","Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .","Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.","Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .","Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.","It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.","Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.","By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .","Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .","Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.","The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.","Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .","Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.","The author was partially supported by the NSERC Discovery Grant 1304873."],["Self-duality of Biflatness","As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .","Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .","In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).","We now obtain the same conclusion under a priori weaker hypotheses.","Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.","It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .","Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .","However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.","Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .","Hence, $m$ satisfies (REF ).","A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.","By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .","This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.","In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.","Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.","To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.","Proposition 5.2 Let $G$ be a locally compact group.","Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.","By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.","Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .","Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].","Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.","Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].","We now show that 1-biflatness of quantum convolution algebras is a self dual property.","Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.","Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.","Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .","Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .","By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .","Also, [15] implies that $E$ is a right $\\lhd $ -module map.","Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .","Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .","Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .","Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.","Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).","Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .","By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .","It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.","Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].","Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.","Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .","Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .","But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.","The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].","The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.","In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].","It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].","The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .","We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .","Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.","Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.","$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].","$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.","The bimodule analogue of [14] then yields $(iii)$ .","$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .","As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.","Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.","Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .","Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.","In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.","In [16] the authors gave examples of weakly amenable connected groups (e.g.","$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.","We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].","Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .","As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .","The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .","Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.","There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .","Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .","Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).","Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.","Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .","Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.","Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.","Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.","Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .","It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].","Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .","By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .","Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.","Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .","By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .","Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .","It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .","Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .","By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .","Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .","The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .","By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.","Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .","Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .","Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .","Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .","Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .","It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.","Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.","There are examples of weakly amenable residually finite groups (e.g.","$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].","Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.","When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].","Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .","Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .","If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .","Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.","Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].","We now generalize this implication to arbitrary locally compact quantum groups.","Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.","By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].","Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].","We present the proof for the convenience of the reader.","Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .","Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.","Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .","Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.","Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].","For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .","In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.","We show that it is a complete order bijection.","To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .","On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .","Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .","By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","We now show that $\\tilde{\\lambda }$ is a complete isometry.","To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].","In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.","Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].","By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .","Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .","Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].","The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.","To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.","Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .","Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.","Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.","Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].","Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.","Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.","Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .","Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).","Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .","Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.","Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .","Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.","It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.","Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.","By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .","Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .","Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.","The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.","Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .","Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.","The author was partially supported by the NSERC Discovery Grant 1304873."],["Operator Amenability of $L^1_{cb}(\\mathbb {G})$","For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .","Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.","In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.","In [16] the authors gave examples of weakly amenable connected groups (e.g.","$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.","We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].","Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .","As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .","The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .","Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.","There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .","Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .","Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).","Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.","Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .","Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.","Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.","Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.","Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .","It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].","Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .","Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .","By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .","Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.","Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .","By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .","Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .","It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .","Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .","By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .","Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .","Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .","The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .","By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.","Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .","Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .","Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .","Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .","Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .","It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.","Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.","There are examples of weakly amenable residually finite groups (e.g.","$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].","Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.","When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].","Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .","Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .","If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .","Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold."],["Decomposability","For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].","We now generalize this implication to arbitrary locally compact quantum groups.","Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.","By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].","Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].","We present the proof for the convenience of the reader.","Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .","Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.","Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .","Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.","If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.","Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].","For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .","In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.","We show that it is a complete order bijection.","To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .","On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .","Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .","By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .","We now show that $\\tilde{\\lambda }$ is a complete isometry.","To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .","That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].","In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.","Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].","By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .","Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .","Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].","The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.","To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.","Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .","Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.","Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.","Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].","Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.","Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.","Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .","Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).","Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .","Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.","Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .","Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.","It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .","Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.","Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.","By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .","Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .","Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .","Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.","The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.","Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ ."],["Acknowledgements","The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.","The author was partially supported by the NSERC Discovery Grant 1304873."]],"string":"[\n [\n \"Inner amenability and approximation properties of locally compact\\n quantum groups\"\n ],\n [\n \"Abstract We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction.\",\n \"We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\\\\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\\\\mathbb{G}$.\",\n \"Similar homological techniques are used to prove that $\\\\ell^1(\\\\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\\\\mathbb{G}$; a discrete Kac algebra $\\\\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\\\\widehat{\\\\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\\\\mathbb{G}$ implies $C_u(\\\\widehat{\\\\mathbb{G}})=L^1(\\\\widehat{\\\\mathbb{G}})\\\\widehat{\\\\otimes}_{L^1(\\\\widehat{\\\\mathbb{G}})}C_0(\\\\widehat{\\\\mathbb{G}})$ completely isometrically.\",\n \"The latter result allows us to partially answer a conjecture of Voiculescu when $\\\\mathbb{G}$ has the approximation property.\"\n ],\n [\n \"Introduction\",\n \"The class of inner amenable locally compact groups has played an important role in the development of abstract harmonic analysis and its connection to operator algebras [46], [48], [58].\",\n \"In particular, it provides a sufficiently general class of locally compact groups for which approximation properties of $G$ can be characterized through approximation properties of its reduced $C^*$ -algebra $C^*_\\\\lambda (G)$ and von Neumann algebra $VN(G)$ .\",\n \"Indeed, within the class of inner amenable groups, amenability of $G$ is equivalent to nuclearity of $C^*_\\\\lambda (G)$ as well as injectivity of $VN(G)$ [46]; exactness of $G$ is equivalent to exactness of $C^*_\\\\lambda (G)$ [1], and the Haagerup property of $G$ is equivalent to the Haagerup property of $VN(G)$ [52].\",\n \"In [33], a notion of inner amenability was introduced for an arbitrary locally compact quantum group $\\\\mathbb {G}$ by means of the existence of a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,x\\\\star f\\\\rangle =\\\\langle m,f\\\\star x\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Although such a condition may be interesting in its own right, if $\\\\mathbb {G}$ is co-amenable, then any cluster point in $L^{\\\\infty }(\\\\mathbb {G})^*$ of a bounded approximate identity in $L^1(\\\\mathbb {G})$ will give rise to such a state.\",\n \"In particular, any locally compact group is an “inner amenable” locally compact quantum group in the sense of [33].\",\n \"In this paper we introduce a bona fide generalization of inner amenability to locally compact quantum groups.\",\n \"We study its basic properties, examples, and related notions of strong and topological inner amenability.\",\n \"Generalizing a well-known result of Lau–Paterson [46], we show that $\\\\mathbb {G}$ is amenable if and only if $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is injective and $\\\\mathbb {G}$ is inner amenable.\",\n \"We also derive sufficient conditions under which the bicrossed product of a matched pair of locally compact groups is inner amenable.\",\n \"We pursue homological manifestations of this concept in section 4, showing that inner amenability of $\\\\mathbb {G}$ entails the relative 1-flatness of $L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"As an application of our techniques, we prove that $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.\",\n \"This resolves a recent conjecture of Ng–Viselter [51], and generalizes a recent result of Ng [50] from locally compact groups to arbitrary Kac algebras, wherein $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and has a tracial state.\",\n \"The method of proof shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.\",\n \"Indeed, we show that for strongly inner amenable quantum groups, weak amenability and the approximation property follow respectively from the w*CBAP and w*OAP of $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"In particular, for inner amenable locally compact groups $G$ , the w*CBAP of $VN(G)$ implies weak amenability of $G$ and the approximation property is equivalent to the w*OAP of $VN(G)$ .\",\n \"Similar results are proved in the setting of topological inner amenability and approximation properties of $C_0()$ .\",\n \"These techniques may be viewed as new tools for the development of harmonic analysis on quantum groups beyond the unimodular discrete setting (for which the above results greatly simplify).\",\n \"In section 5 we establish the self-duality of (non-relative) 1-biflatness, that is, $L^1(\\\\mathbb {G})$ is 1-biflat if and only if $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat.\",\n \"This shows, in particular, that $\\\\ell ^1(\\\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group $\\\\mathbb {G}$ .\",\n \"Section 6 is devoted to the question of operator amenability of $L^1_{cb}(\\\\mathbb {G})$ , the $cb$ -multiplier closure of $L^1(\\\\mathbb {G})$ .\",\n \"For unimodular discrete quantum groups $\\\\mathbb {G}$ with Kirchberg's factorization property, we show that $\\\\mathbb {G}$ is weakly amenable if and only if $L^1_{cb}()$ is operator amenable.\",\n \"This result is new even for the class of weakly amenable residually finite discrete groups such that $C^*(G)$ is not residually finite-dimensional (see [6]).\",\n \"We finish in section 7 with a strengthening of [14], by showing that $C_u(\\\\mathbb {G})^*\\\\cong M_{cb}^l(L^1(\\\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically when $$ is amenable.\",\n \"As a corollary, when $$ has the approximation property, we prove that $\\\\mathbb {G}$ is co-amenable if and only if $$ is amenable.\"\n ],\n [\n \"Operator Modules\",\n \"Let $\\\\mathcal {A}$ be a completely contractive Banach algebra.\",\n \"We say that an operator space $X$ is a right operator $\\\\mathcal {A}$ -module if it is a right Banach $\\\\mathcal {A}$ -module such that the module map $m_X:X\\\\widehat{\\\\otimes }\\\\mathcal {A}\\\\rightarrow X$ is completely contractive, where $\\\\widehat{\\\\otimes }$ denotes the operator space projective tensor product.\",\n \"We say that $X$ is faithful if for every non-zero $x\\\\in X$ , there is $a\\\\in \\\\mathcal {A}$ such that $x\\\\cdot a\\\\ne 0$ , and we say that $X$ is essential if $\\\\langle X\\\\cdot \\\\mathcal {A}\\\\rangle =X$ , where $\\\\langle \\\\cdot \\\\rangle $ denotes the closed linear span.\",\n \"Note that our definition of faithfulness, the standard notion in operator modules, is opposite in nature to the usual definition in algebra.\",\n \"We denote by $\\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ the category of right operator $\\\\mathcal {A}$ -modules with morphisms given by completely bounded module homomorphisms.\",\n \"Left operator $\\\\mathcal {A}$ -modules and operator $\\\\mathcal {A}$ -bimodules are defined similarly, and we denote the respective categories by $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod}$ and $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"Regarding terminology, in what follows we will often omit the term “operator” when discussing homological properties of operator modules as we will be working exclusively in the operator space category.\",\n \"Let $\\\\mathcal {A}$ be a completely contractive Banach algebra, $X\\\\in \\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ and $Y\\\\in \\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod}$ .\",\n \"The $\\\\mathcal {A}$ -module tensor product of $X$ and $Y$ is the quotient space $X\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y:=X\\\\widehat{\\\\otimes }Y/N$ , where $N=\\\\langle x\\\\cdot a\\\\otimes y-x\\\\otimes a\\\\cdot y\\\\mid x\\\\in X, \\\\ y\\\\in Y, \\\\ a\\\\in \\\\mathcal {A}\\\\rangle ,$ and, again, $\\\\langle \\\\cdot \\\\rangle $ denotes the closed linear span.\",\n \"It follows that (see [9]) $\\\\mathcal {CB}_{\\\\mathcal {A}}(X,Y^*)\\\\cong N^{\\\\perp }\\\\cong (X\\\\widehat{\\\\otimes }_{\\\\mathcal {A}} Y)^*,$ where $\\\\mathcal {CB}_{\\\\mathcal {A}}(X,Y^*)$ is the space of completely bounded right $\\\\mathcal {A}$ -module maps $\\\\Phi :X\\\\rightarrow Y^*$ .\",\n \"If $Y=\\\\mathcal {A}$ , then clearly $N\\\\subseteq \\\\mathrm {Ker}(m_X)$ where $m_X:X\\\\widehat{\\\\otimes }\\\\mathcal {A}\\\\rightarrow X$ is the module map.\",\n \"If the induced mapping $\\\\widetilde{m}_X:X\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}\\\\mathcal {A}\\\\rightarrow X$ is a completely isometric isomorphism we say that $X$ is an induced $\\\\mathcal {A}$ -module.\",\n \"A similar definition applies for left modules.\",\n \"In particular, we say that $\\\\mathcal {A}$ is self-induced if $\\\\widetilde{m}_\\\\mathcal {A}:\\\\mathcal {A}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}\\\\mathcal {A}\\\\cong \\\\mathcal {A}$ completely isometrically.\",\n \"Let $\\\\mathcal {A}$ be a completely contractive Banach algebra and $X\\\\in \\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"The identification $\\\\mathcal {A}^+=\\\\mathcal {A}\\\\oplus _1 turns the unitization of $ A$ into a unital completely contractive Banach algebra, and it follows that $ X$ becomes a right operator $ A+$-module via the extended action\\\\begin{equation*}x\\\\cdot (a+\\\\lambda e)=x\\\\cdot a+\\\\lambda x, \\\\ \\\\ \\\\ a\\\\in \\\\mathcal {A}^+, \\\\ \\\\lambda \\\\in \\\\ x\\\\in X.\\\\end{equation*}Let $ C1$.\",\n \"Then $ X$ is \\\\emph {relatively $ C$-projective} if there exists a morphism $ +:XXA+$ which is a right inverse to the extended module map $ mX+:XA+X$ such that $ +cbC$.\",\n \"When $ X$ is essential, then $ X$ is relatively $ C$-projective if and only if there exists a morphism $ :XXA$ satisfying $ cbC$ and $ mX=idX$ by the operator analogue of \\\\cite [Proposition 1.2]{DP}.\",\n \"We say that $ X$ is \\\\emph {$ C$-projective} if for every $ Y,Zmod A$, every complete quotient morphism $ :YZ$, every morphism $ :XZ$, and every $ >0$, there exists a morphism $ :XY$ such that $ cb< Ccb+$ and $ =$, i.e., the following diagram commutes:\\\\begin{equation*}\\\\begin{tikzcd}&Y [d, two heads, \\\"\\\\Psi \\\"]\\\\\\\\X [ru, dotted, \\\"\\\\widetilde{\\\\Phi }_\\\\varepsilon \\\"] [r, \\\"\\\\Phi \\\"] &Z\\\\end{tikzcd}\\\\end{equation*}$ Given a completely contractive Banach algebra $\\\\mathcal {A}$ and $X\\\\in \\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ , there is a canonical completely contractive morphism $\\\\Delta ^+:X\\\\rightarrow \\\\mathcal {CB}(\\\\mathcal {A}^+,X)$ given by $\\\\Delta ^+(x)(a)=x\\\\cdot a, \\\\ \\\\ \\\\ x\\\\in X, \\\\ a\\\\in \\\\mathcal {A}^+,$ where the right $\\\\mathcal {A}$ -module structure on $\\\\mathcal {CB}(\\\\mathcal {A}^+,X)$ is defined by $(\\\\Psi \\\\cdot a)(b)=\\\\Psi (ab), \\\\ \\\\ \\\\ a\\\\in \\\\mathcal {A}, \\\\ \\\\Psi \\\\in \\\\mathcal {CB}(\\\\mathcal {A}^+,X), \\\\ b\\\\in \\\\mathcal {A}^+.$ An analogous construction exists for objects in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod}$ .\",\n \"Let $C\\\\ge 1$ .\",\n \"Then $X$ is relatively $C$ -injective if there exists a morphism $\\\\Phi ^+:\\\\mathcal {CB}(\\\\mathcal {A}^+,X)\\\\rightarrow X$ such that $\\\\Phi ^+\\\\circ \\\\Delta ^+=\\\\textnormal {id}_{X}$ and $\\\\Vert \\\\Phi ^+\\\\Vert _{cb}\\\\le C$ .\",\n \"When $X$ is faithful, then $X$ is relatively $C$ -injective if and only if there exists a morphism $\\\\Phi :\\\\mathcal {CB}(\\\\mathcal {A},X)\\\\rightarrow X$ such that $\\\\Phi \\\\circ \\\\Delta =\\\\textnormal {id}_{X}$ and $\\\\Vert \\\\Phi \\\\Vert _{cb}\\\\le C$ by the operator analogue of [17], where $\\\\Delta (x)(a):=\\\\Delta ^+(x)(a)$ for $x\\\\in X$ and $a\\\\in \\\\mathcal {A}$ .\",\n \"We say that $X$ is $C$ -injective if for every $Y,Z\\\\in \\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ , every completely isometric morphism $\\\\Psi :Y\\\\hookrightarrow Z$ , and every morphism $\\\\Phi :Y\\\\rightarrow X$ , there exists a morphism $\\\\widetilde{\\\\Phi }:Z\\\\rightarrow X$ such that $\\\\Vert \\\\widetilde{\\\\Phi }\\\\Vert _{cb}\\\\le C\\\\Vert \\\\Phi \\\\Vert _{cb}$ and $\\\\widetilde{\\\\Phi }\\\\circ \\\\Psi =\\\\Phi $ , that is, the following diagram commutes: $\\\\begin{tikzcd}Z [rd, dotted, \\\"\\\\widetilde{\\\\Phi }\\\"]\\\\\\\\Y [u, hook, \\\"\\\\Psi \\\"] [r, \\\"\\\\Phi \\\"] &X\\\\end{tikzcd}$ There is a natural categorical equivalence between $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ and $\\\\mathbf {mod\\\\hspace{2.0pt}\\\\mathcal {A}^{\\\\mathrm {op}}\\\\widehat{\\\\otimes }\\\\mathcal {A}}$ given by $axb=x\\\\cdot (a\\\\otimes b), \\\\ \\\\ \\\\ a,b\\\\in \\\\mathcal {A}, \\\\ x\\\\in X, \\\\ X\\\\in \\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}.$ With this identification, we obtain the following bimodule analogue of [14].\",\n \"Proposition 2.1 Let $\\\\mathcal {A}$ be a completely contractive Banach algebra and $X\\\\in \\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"If $X$ is $C_1$ -injective in $\\\\mathbf {mod}- and is relatively $ C2$-injective in $ A mod A$, then $ X$ is $ C1C2$-injective in $ A mod A$.$\"\n ],\n [\n \"Locally Compact Quantum Groups\",\n \"A locally compact quantum group is a quadruple $\\\\mathbb {G}=(L^{\\\\infty }(\\\\mathbb {G}),\\\\Gamma ,\\\\varphi ,\\\\psi )$ , where $L^{\\\\infty }(\\\\mathbb {G})$ is a Hopf-von Neumann algebra with co-multiplication $\\\\Gamma :L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ , and $\\\\varphi $ and $\\\\psi $ are fixed left and right Haar weights on $L^{\\\\infty }(\\\\mathbb {G})$ , respectively [43], [44].\",\n \"For every locally compact quantum group $\\\\mathbb {G}$ , there exists a left fundamental unitary operator $W$ on $L^2(\\\\mathbb {G},\\\\varphi )\\\\otimes L^2(\\\\mathbb {G},\\\\varphi )$ and a right fundamental unitary operator $V$ on $L^2(\\\\mathbb {G},\\\\psi )\\\\otimes L^2(\\\\mathbb {G},\\\\psi )$ implementing the co-multiplication $\\\\Gamma $ via $\\\\Gamma (x)=W^*(1\\\\otimes x)W=V(x\\\\otimes 1)V^*, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Both unitaries satisfy the pentagonal relation; that is, $W_{12}W_{13}W_{23}=W_{23}W_{12}\\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\mathrm {and}\\\\hspace{10.0pt}\\\\hspace{10.0pt}V_{12}V_{13}V_{23}=V_{23}V_{12}.$ By [44], we may identify $L^2(\\\\mathbb {G},\\\\varphi )$ and $L^2(\\\\mathbb {G},\\\\psi )$ , so we will simply use $L^2(\\\\mathbb {G})$ for this Hilbert space throughout the paper.\",\n \"We denote by $R$ the unitary antipode of $\\\\mathbb {G}$ .\",\n \"Let $L^1(\\\\mathbb {G})$ denote the predual of $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"Then the pre-adjoint of $\\\\Gamma $ induces an associative completely contractive multiplication on $L^1(\\\\mathbb {G})$ , defined by $\\\\star :L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})\\\\ni f\\\\otimes g\\\\mapsto f\\\\star g=\\\\Gamma _*(f\\\\otimes g)\\\\in L^1(\\\\mathbb {G}).$ The multiplication $\\\\star $ is a complete quotient map from $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ onto $L^1(\\\\mathbb {G})$ , implying $\\\\langle L^1(\\\\mathbb {G})\\\\star L^1(\\\\mathbb {G})\\\\rangle =L^1(\\\\mathbb {G}).$ Moreover, $L^1(\\\\mathbb {G})$ is always self-induced.\",\n \"The proof follows from [63] (see [14] for details).\",\n \"The canonical $L^1(\\\\mathbb {G})$ -bimodule structure on $L^{\\\\infty }(\\\\mathbb {G})$ is given by $f\\\\star x=(\\\\textnormal {id}\\\\otimes f)\\\\Gamma (x) \\\\ \\\\ \\\\ \\\\ x\\\\star f=(f\\\\otimes \\\\textnormal {id})\\\\Gamma (x), \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}), \\\\ f\\\\in L^1(\\\\mathbb {G}).$ A left invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ is a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,x\\\\star f \\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}), \\\\ f\\\\in L^1(\\\\mathbb {G}).$ Right and two-sided invariant means are defined similarly.\",\n \"A locally compact quantum group $\\\\mathbb {G}$ is said to be amenable if there exists a left invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"It is known that $\\\\mathbb {G}$ is amenable if and only if there exists a right (equivalently, two-sided) invariant mean (cf.\",\n \"[26]).\",\n \"We say that $\\\\mathbb {G}$ is co-amenable if $L^1(\\\\mathbb {G})$ has a bounded left (equivalently, right or two-sided) approximate identity (cf.\",\n \"[5]).\",\n \"The left regular representation $\\\\lambda :L^1(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ of $\\\\mathbb {G}$ is defined by $\\\\lambda (f)=(f\\\\otimes \\\\textnormal {id})(W), \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}),$ and is an injective, completely contractive homomorphism from $L^1(\\\\mathbb {G})$ into $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Then $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}):=\\\\lbrace \\\\lambda (f) : f\\\\in L^1(\\\\mathbb {G})\\\\rbrace ^{\\\\prime \\\\prime }$ is the von Neumann algebra associated with the dual quantum group $$ .\",\n \"Analogously, we have the right regular representation $\\\\rho :L^1(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ defined by $\\\\rho (f)=(\\\\textnormal {id}\\\\otimes f)(V), \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}),$ which is also an injective, completely contractive homomorphism from $L^1(\\\\mathbb {G})$ into $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Then $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }):=\\\\lbrace \\\\rho (f) : f\\\\in L^1(\\\\mathbb {G})\\\\rbrace ^{\\\\prime \\\\prime }$ is the von Neumann algebra associated to the quantum group $^{\\\\prime }$ .\",\n \"It follows that $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })=L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^{\\\\prime }$ , and the left and right fundamental unitaries satisfy $W\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ and $V\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ [44].\",\n \"Moreover, dual quantum groups always satisfy $L^{\\\\infty }(\\\\mathbb {G})\\\\cap L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})=L^{\\\\infty }(\\\\mathbb {G})\\\\cap L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })=$ [66].\",\n \"The modular conjugations of the dual Haar weights give rise to conjugate linear isometries $J,\\\\widehat{J}:L^2(\\\\mathbb {G})\\\\rightarrow L^2(\\\\mathbb {G})$ satisfying $JL^{\\\\infty }(\\\\mathbb {G})J=L^{\\\\infty }(\\\\mathbb {G})^{\\\\prime }\\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\mathrm {and}\\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\widehat{J}L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\widehat{J}=L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }).$ Moreover, the unitary $U:=\\\\widehat{J}J$ intertwines the left and right regular representations via $\\\\rho (f)=U\\\\lambda (f)U^*$ , $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"At the level of the fundamental unitaries, this relation becomes $V=\\\\sigma (1\\\\otimes U)W(1\\\\otimes U^*)\\\\sigma .$ We also record the adjoint formulas $(\\\\widehat{J}\\\\otimes J)W(\\\\widehat{J}\\\\otimes J)=W^*$ and $(J\\\\otimes \\\\widehat{J})V(J\\\\otimes \\\\widehat{J})=V^*$ .\",\n \"If $G$ is a locally compact group, we let $\\\\mathbb {G}_a=(L^{\\\\infty }(G),\\\\Gamma _a,\\\\varphi _a,\\\\psi _a)$ denote the commutative quantum group associated with the commutative von Neumann algebra $L^{\\\\infty }(G)$ , where the co-multiplication is given by $\\\\Gamma _a(f)(s,t)=f(st)$ , and $\\\\varphi _a$ and $\\\\psi _a$ are integration with respect to a left and right Haar measure, respectively.\",\n \"The dual $_a$ of $\\\\mathbb {G}_a$ is the co-commutative quantum group $\\\\mathbb {G}_s=(VN(G),\\\\Gamma _s,\\\\varphi _s,\\\\psi _s)$ , where $VN(G)$ is the left group von Neumann algebra with co-multiplication $\\\\Gamma _s(\\\\lambda (t))=\\\\lambda (t)\\\\otimes \\\\lambda (t)$ , and $\\\\varphi _s=\\\\psi _s$ is Haagerup's Plancherel weight (cf.\",\n \"[62]).\",\n \"Then $L^1(\\\\mathbb {G}_a)$ is the usual group convolution algebra $L^1(G)$ , and $L^1(\\\\mathbb {G}_s)$ is the Fourier algebra $A(G)$ .\",\n \"It is known that every commutative locally compact quantum group is of the form $\\\\mathbb {G}_a$ [60].\",\n \"Therefore, every commutative locally compact quantum group is co-amenable, and is amenable if and only if the underlying locally compact group is amenable.\",\n \"By duality, every co-commutative locally compact quantum group is of the form $\\\\mathbb {G}_s$ , which is always amenable, and is co-amenable if and only if the underlying locally compact group is amenable, by Leptin's theorem [47].\",\n \"For a locally compact quantum group $\\\\mathbb {G}$ , we let $C_0(\\\\mathbb {G}):=\\\\overline{\\\\hat{\\\\lambda }(L^1(\\\\widehat{\\\\mathbb {G}}))}^{\\\\Vert \\\\cdot \\\\Vert }$ denote the reduced quantum group $C^*$ -algebra of $\\\\mathbb {G}$ .\",\n \"We say that $\\\\mathbb {G}$ is compact if $C_0(\\\\mathbb {G})$ is a unital $C^*$ -algebra, in which case we denote $C_0(\\\\mathbb {G})$ by $C(\\\\mathbb {G})$ .\",\n \"We say that $\\\\mathbb {G}$ is discrete if $L^1(\\\\mathbb {G})$ is unital, in which case we denote $L^1(\\\\mathbb {G})$ by $\\\\ell ^1(\\\\mathbb {G})$ .\",\n \"It is well-known that $\\\\mathbb {G}$ is compact if and only if $$ is discrete, and in that case, $\\\\ell ^1()\\\\cong \\\\bigoplus _{1} \\\\lbrace T_{n_\\\\alpha }(\\\\mid \\\\alpha \\\\in \\\\mathrm {Irr}(\\\\mathbb {G})\\\\rbrace $ , where $T_{n_\\\\alpha }($ is the space of $n_\\\\alpha \\\\times n_\\\\alpha $ trace class operators, and $\\\\mathrm {Irr}(\\\\mathbb {G})$ denotes the set of (equivalence classes of) irreducible co-representations of the compact quantum group $\\\\mathbb {G}$ [68].\",\n \"For general $\\\\mathbb {G}$ , the operator dual $M(\\\\mathbb {G}):=C_0(\\\\mathbb {G})^*$ is a completely contractive Banach algebra containing $L^1(\\\\mathbb {G})$ as a norm closed two-sided ideal via the map $L^1(\\\\mathbb {G})\\\\ni f\\\\mapsto f|_{C_0(\\\\mathbb {G})}\\\\in M(\\\\mathbb {G})$ .\",\n \"The co-multiplication satisfies $\\\\Gamma (C_0(\\\\mathbb {G}))\\\\subseteq M(C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))$ , where $M(C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))$ is the multiplier algebra of the $C^*$ -algebra $C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})$ .\",\n \"We let $C_u(\\\\mathbb {G})$ be the universal quantum group $C^*$ -algebra of $\\\\mathbb {G}$ , and denote the canonical surjective *-homomorphism onto $C_0(\\\\mathbb {G})$ by $\\\\pi _{\\\\mathbb {G}}:C_u(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ [42].\",\n \"The space $C_u(\\\\mathbb {G})^*$ then has the structure of a unital completely contractive Banach algebra such that the map $L^1(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})^*$ given by the composition of the inclusion $L^1(\\\\mathbb {G})\\\\subseteq M(\\\\mathbb {G})$ and $\\\\pi _{\\\\mathbb {G}}^*:M(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})^*$ is a completely isometric homomorphism, and it follows that $L^1(\\\\mathbb {G})$ is a norm closed two-sided ideal in $C_u(\\\\mathbb {G})^*$ [42].\",\n \"An element $\\\\hat{b}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is said to be a completely bounded left multiplier of $L^1(\\\\mathbb {G})$ if $\\\\hat{b}\\\\lambda (f)\\\\in \\\\lambda (L^1(\\\\mathbb {G}))$ for all $f\\\\in L^1(\\\\mathbb {G})$ and the induced map $m_{\\\\hat{b}}^l:L^1(\\\\mathbb {G})\\\\ni f\\\\mapsto \\\\lambda ^{-1}(\\\\hat{b}\\\\lambda (f))\\\\in L^1(\\\\mathbb {G})$ is completely bounded on $L^1(\\\\mathbb {G})$ .\",\n \"We let $M_{cb}^l(L^1(\\\\mathbb {G}))$ denote the space of all completely bounded left multipliers of $L^1(\\\\mathbb {G})$ , which is a completely contractive Banach algebra with respect to the norm $\\\\Vert [\\\\hat{b}_{ij}]\\\\Vert _{M_n(M_{cb}^l(L^1(\\\\mathbb {G})))}=\\\\Vert [m^l_{\\\\hat{b}_{ij}}]\\\\Vert _{cb}.$ Completely bounded right multipliers are defined analogously and we denote by $M_{cb}^r(L^1(\\\\mathbb {G}))$ the corresponding completely contractive Banach algebra.\",\n \"There is a canonical, injective completely contractive homomorphism $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ , extending $\\\\lambda $ , which maps $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ to the operator of left multiplication by $\\\\mu $ on $L^1(\\\\mathbb {G})$ .\",\n \"In general, given $\\\\hat{b}\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))$ , the adjoint $\\\\Theta ^l(\\\\hat{b}):=(m_{\\\\hat{b}}^l)^*$ defines a normal completely bounded left $L^1(\\\\mathbb {G})$ -module map on $L^{\\\\infty }(\\\\mathbb {G})$ , and by [37] have the completely isometric identification $\\\\Theta ^l:M_{cb}^l(L^1(\\\\mathbb {G}))\\\\cong \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G})).$ Moreover, it follows from [37] that $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))=_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"It is known that $M_{cb}^l(L^1(\\\\mathbb {G}))$ is a dual operator space [35], with predual $Q_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"By the general result [34], it follows from [41] that for arbitrary locally compact quantum groups $Q_{cb}^l(L^1(\\\\mathbb {G}))=\\\\lbrace \\\\Omega _{A,\\\\rho }\\\\mid A\\\\in C_0(\\\\mathbb {G})\\\\otimes _{\\\\min }\\\\mathcal {K}(\\\\ell ^2), \\\\ \\\\rho \\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }\\\\mathcal {T}(\\\\ell ^2)\\\\rbrace ,$ where $\\\\langle \\\\hat{b},\\\\Omega _{A,\\\\rho }\\\\rangle =\\\\langle (\\\\Theta ^l(\\\\hat{b})\\\\otimes \\\\textnormal {id}_{K_{\\\\infty }})(A),\\\\rho \\\\rangle $ , $\\\\hat{b}\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"We say that $\\\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\\\widehat{\\\\mathbb {G}}))$ .\",\n \"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\\\mathbb {G}$ , and is denoted $\\\\Lambda _{cb}(\\\\mathbb {G})$ .\",\n \"We say that $\\\\mathbb {G}$ has the approximation property if there exists a net $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ such that $^l(\\\\hat{\\\\lambda }(\\\\hat{f}_i))$ converges to $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ in the stable point-weak* topology.\",\n \"Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"The right fundamental unitary $V$ of $\\\\mathbb {G}$ induces a co-associative co-multiplication $\\\\Gamma ^r:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\ni T\\\\mapsto V(T\\\\otimes 1)V^*\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G})),$ and the restriction of $\\\\Gamma ^r$ to $L^{\\\\infty }(\\\\mathbb {G})$ yields the original co-multiplication $\\\\Gamma $ on $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"The pre-adjoint of $\\\\Gamma ^r$ induces an associative completely contractive multiplication on the space of trace class operators $\\\\mathcal {T}(L^2(\\\\mathbb {G}))$ , defined by $\\\\rhd :\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\omega \\\\otimes \\\\tau \\\\mapsto \\\\omega \\\\rhd \\\\tau =\\\\Gamma ^r_*(\\\\omega \\\\otimes \\\\tau )\\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Analogously, the left fundamental unitary $W$ of $\\\\mathbb {G}$ induces a co-associative co-multiplication $\\\\Gamma ^l:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\ni T\\\\mapsto W^*(1\\\\otimes T)W\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G})),$ and the restriction of $\\\\Gamma ^l$ to $L^{\\\\infty }(\\\\mathbb {G})$ is also equal to $\\\\Gamma $ .\",\n \"The pre-adjoint of $\\\\Gamma ^l$ induces another associative completely contractive multiplication $\\\\lhd :\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\omega \\\\otimes \\\\tau \\\\mapsto \\\\omega \\\\lhd \\\\tau =\\\\Gamma ^l_*(\\\\omega \\\\otimes \\\\tau )\\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ The algebra $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ inherits a canonical $\\\\mathcal {T}(L^2(\\\\mathbb {G}))$ -bimodule structure with respect to both the left $\\\\lhd $ and right $\\\\rhd $ products.\",\n \"It was shown in [35] that the pre-annihilator $L^{\\\\infty }(\\\\mathbb {G})_{\\\\perp }$ of $L^{\\\\infty }(\\\\mathbb {G})$ in $\\\\mathcal {T}(L^2(\\\\mathbb {G}))$ is a norm closed two sided ideal in $(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\rhd )$ and $(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\lhd )$ , respectively, and the complete quotient map $\\\\pi :\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\omega \\\\mapsto f=\\\\omega |_{L^{\\\\infty }(\\\\mathbb {G})}\\\\in L^1(\\\\mathbb {G})$ is an algebra homomorphism from $\\\\mathcal {T}_{\\\\rhd }:=(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\rhd )$ , respectively, $\\\\mathcal {T}_{\\\\lhd }:=(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\lhd )$ , onto $L^1(\\\\mathbb {G})$ .\",\n \"It follows that the right $\\\\lhd $ -module and left $\\\\rhd $ -module structures degenerate to an $L^1(\\\\mathbb {G})$ -bimodule structure on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ : $ f\\\\rhd T = (\\\\textnormal {id}\\\\otimes f)\\\\Gamma ^r(T), \\\\ \\\\ \\\\ T\\\\lhd f = (f\\\\otimes \\\\textnormal {id})\\\\Gamma ^l(T), \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})).$ By [44] the unitary antipode $R$ satisfies $R(x)=\\\\widehat{J}x^*\\\\widehat{J}$ , for $x\\\\in L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"It therefore extends to a *-anti-automorphism (still denoted) $R:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ , via $R(T)=J̉T^*J̉$ , $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"The extended antipode maps $L^{\\\\infty }(\\\\mathbb {G})$ and $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ onto $L^{\\\\infty }(\\\\mathbb {G})$ and $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ , respectively, and satisfies the generalized antipode relations; that is, $( R\\\\otimes R)\\\\circ \\\\Gamma ^r=\\\\Sigma \\\\circ \\\\Gamma ^l\\\\circ R\\\\hspace{10.0pt}\\\\mathrm {and}\\\\hspace{10.0pt}( R\\\\otimes R)\\\\circ \\\\Gamma ^l=\\\\Sigma \\\\circ \\\\Gamma ^r\\\\circ R,$ where $\\\\Sigma $ is the flip map on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Let $\\\\mathbb {G}$ and $\\\\mathbb {H}$ be two locally compact quantum groups.\",\n \"Then $\\\\mathbb {H}$ is said to be a closed quantum subgroup of $\\\\mathbb {G}$ in the sense of Vaes if there exists a normal, unital, injective *-homomorphism $\\\\gamma :L^{\\\\infty }(\\\\widehat{\\\\mathbb {H}})\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ satisfying $(\\\\gamma \\\\otimes \\\\gamma )\\\\circ \\\\Gamma _{}=\\\\Gamma _{}\\\\circ \\\\gamma $ .\",\n \"This is not the original definition of Vaes [64], but was shown to be equivalent in [24].\"\n ],\n [\n \"Inner Amenability\",\n \"Given a locally compact quantum group $\\\\mathbb {G}$ the composition $W\\\\sigma V\\\\sigma \\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ defines a unitary co-representation of $\\\\mathbb {G}$ called the conjugation co-representation, where $\\\\sigma $ is the flip map on $L^2(\\\\mathbb {G})\\\\otimes L^2(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\\\sigma V\\\\sigma \\\\xi (s,t)=\\\\xi (s,s^{-1}ts)\\\\Delta (s)^{1/2}, \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G\\\\times G), \\\\ s,t\\\\in G.$ Thus, $W\\\\sigma V\\\\sigma $ is the unitary generator of the conjugation representation $\\\\beta _2:G\\\\rightarrow \\\\mathcal {B}(L^2(G))$ , where $\\\\beta _2(s)\\\\xi (t)=\\\\lambda (s)\\\\rho (s)\\\\xi (t)=\\\\xi (s^{-1}ts)\\\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\\\in L^{\\\\infty }(G)^*$ satisfying $\\\\langle m,\\\\beta _{\\\\infty }(s)f\\\\rangle =\\\\langle m,f\\\\rangle \\\\ \\\\ \\\\ s\\\\in G, \\\\ f\\\\in L^{\\\\infty }(G),$ where $\\\\beta _\\\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\\\in G$ , $f\\\\in L^{\\\\infty }(G)$ , is the conjugation action on $L^{\\\\infty }(G)$ .\",\n \"Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\\\in \\\\ell ^{\\\\infty }(G)^*$ such that $m\\\\ne \\\\delta _e$ .\",\n \"In what follows, inner amenability will always refer to the definition (REF ) given above.\",\n \"In particular, every discrete group is inner amenable.\",\n \"Definition 3.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"We say that $\\\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\\\xi _i)$ of unit vectors such that $\\\\Vert W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i)-\\\\eta \\\\otimes \\\\xi _i\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ $\\\\mathbb {G}$ is inner amenable if there exists a state $m\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ satisfying $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,\\\\hat{x}\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}).$ Such a state is said to be inner invariant.\",\n \"$\\\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\\\in C_0()^*$ such that $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,\\\\hat{x}\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\hat{x}\\\\in C_0().$ Such a state is said to be topologically inner invariant.\",\n \"Examples 3.3 The following known examples are worth mentioning.\",\n \"Any co-commutative quantum group $\\\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\\\sigma V_s\\\\sigma =W_sW_s^*=1$ .\",\n \"Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\\\xi :=\\\\Lambda _\\\\varphi (1)$ being conjugation invariant, where $\\\\varphi $ is the Haar trace on the compact dual.\",\n \"It was shown in [51] that if $\\\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\\\mathbb {G}$ is topologically inner amenable.\",\n \"Further examples, including the bicrossed product construction will be studied below.\",\n \"The following proposition is standard.\",\n \"We include the proof for completeness.\",\n \"Proposition 3.4 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $\\\\mathbb {G}$ is strongly inner amenability then it is inner amenable.\",\n \"Let $(\\\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\\\sigma V\\\\sigma $ .\",\n \"Passing to a subnet, we may assume that $(\\\\omega _{J\\\\xi _i}|_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})})$ converges weak* to a state $m\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ .\",\n \"For any $f=\\\\omega _{\\\\xi ,\\\\eta }\\\\in L^1(\\\\mathbb {G})$ and $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\\\widehat{J}\\\\otimes J)W(\\\\widehat{J}\\\\otimes J)=W^*, \\\\ \\\\ \\\\ (\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\otimes J)=\\\\sigma V^*\\\\sigma ,$ we have $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle &=\\\\lim _i\\\\langle \\\\omega _{J\\\\xi _i},\\\\hat{x}\\\\lhd f\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle W^*(1\\\\otimes \\\\hat{x})W(\\\\xi \\\\otimes J\\\\xi _i),\\\\eta \\\\otimes J\\\\xi _i\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})W(\\\\xi \\\\otimes J\\\\xi _i),W(\\\\eta \\\\otimes J\\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})(\\\\widehat{J}\\\\otimes J)W^*(\\\\widehat{J}\\\\xi \\\\otimes \\\\xi _i),(\\\\widehat{J}\\\\otimes J)W^*(\\\\widehat{J}\\\\eta \\\\otimes \\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})(\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\xi \\\\otimes \\\\xi _i),(\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\eta \\\\otimes \\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})\\\\sigma V^*\\\\sigma (\\\\xi \\\\otimes J\\\\xi _i),\\\\sigma V^*\\\\sigma (\\\\eta \\\\otimes J\\\\xi _i)\\\\rangle \\\\\\\\&=\\\\langle m,\\\\hat{x}\\\\rangle \\\\langle f,1\\\\rangle .$ In the commutative case, the converse of Proposition REF holds.\",\n \"Proposition 3.5 A commutative quantum group $\\\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.\",\n \"If $\\\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.\",\n \"Let $m\\\\in VN(G)^*$ satisfy $\\\\langle m,x\\\\rangle =\\\\langle m,x \\\\lhd f\\\\rangle $ for all $x\\\\in VN(G)$ and all states $f\\\\in L^1(G)$ .\",\n \"Then for $t\\\\in G$ $\\\\lambda (t)x\\\\lambda (t)^* \\\\lhd f=\\\\int _G \\\\lambda (s^{-1}t)x\\\\lambda (t^{-1}s) f(s)ds=\\\\int _G\\\\lambda (r)^*x\\\\lambda (r)f(tr)dr=x \\\\lhd f_t,$ so that $\\\\langle m,\\\\lambda (t)x\\\\lambda (t)^*\\\\rangle =\\\\langle m,\\\\lambda (t)x\\\\lambda (t)^* \\\\lhd f\\\\rangle =\\\\langle m,x \\\\lhd f_t\\\\rangle =\\\\langle m,x\\\\rangle $ for all $x\\\\in VN(G)$ , $t\\\\in G$ and states $f\\\\in L^1(G)$ .\",\n \"Thus, $G$ is inner amenable by [16].\",\n \"Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\\\xi _i)$ in $L^2(G)$ satisfying $\\\\Vert \\\\lambda (s)\\\\xi _i-\\\\rho (s)^*\\\\xi _i\\\\Vert _{L^2(G)}\\\\rightarrow 0$ uniformly on compact subsets of $G$ .\",\n \"For any $\\\\eta \\\\in C_c(G)$ we then have $\\\\langle W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i),\\\\eta \\\\otimes \\\\xi _i\\\\rangle =\\\\iint \\\\xi _i(ts)\\\\overline{\\\\xi _i}(st)\\\\Delta (s)^{1/2}|\\\\eta (s)|^2 \\\\ ds \\\\ dt\\\\rightarrow 1.$ It follows that $(\\\\xi _i)$ is strongly inner invariant.\",\n \"Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].\",\n \"Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\\\mathcal {B}(L^2(G))$ , via $\\\\langle m,f \\\\rhd T\\\\rangle =\\\\langle m, T \\\\lhd f\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(G), \\\\ T\\\\in \\\\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.\",\n \"For details, see [13].\",\n \"Proposition 3.6 A commutative quantum group $\\\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.\",\n \"The existence of a tracial state on $C_\\\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.\",\n \"By [51] $\\\\mathbb {G}_a$ is topologically inner amenable if $C_\\\\lambda ^*(G)$ has a tracial state.\",\n \"Conversely, if $m$ is an inner invariant state on $C_\\\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.\",\n \"Viewing $m$ as a positive definite function in $B_\\\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\\\in G$ .\",\n \"A simple integral calculation then shows that $m$ is a tracial state on $C_\\\\lambda ^*(G)$ .\",\n \"Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ to $C_0()$ .\",\n \"This is not the case however.\",\n \"In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\\\bigoplus _{n\\\\in \\\\mathbb {N}} \\\\Gamma _n\\\\rtimes \\\\prod _{n\\\\in \\\\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\\\Gamma _n$ for which $C^*_\\\\lambda (\\\\Gamma _n\\\\rtimes F_n)$ admits a unique tracial state.\",\n \"In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .\",\n \"Thus, $G$ is inner amenable.\",\n \"However, in [30], the authors also show that the continuous function $m$ in $B_\\\\lambda (G)$ associated to a tracial state on $C^*_\\\\lambda (G)$ must be discontinuous.\",\n \"Whence, there is no tracial state on $C^*_\\\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .\",\n \"Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\\\lambda (G)$ .\",\n \"In the other direction, in general it is not possible to lift a tracial state on $C^*_\\\\lambda (G)$ to an inner invariant mean on $VN(G)$ .\",\n \"In [30] it is shown that $\\\\mathbb {R}^2\\\\rtimes \\\\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\\\mathbb {R})$ .\",\n \"Proposition 3.8 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is co-amenable then $\\\\mathbb {G}$ is strongly inner amenable.\",\n \"If $\\\\mathbb {G}$ is amenable then it is inner amenable.\",\n \"If $$ is co-amenable, let $E\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\\\omega _{\\\\xi _i}$ with $J̉\\\\xi _i=\\\\xi _i$ (since $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\curvearrowright L^2(\\\\mathbb {G})$ is standard).\",\n \"Then $1=(\\\\textnormal {id}\\\\otimes E)(W)=\\\\lim _i(\\\\textnormal {id}\\\\otimes \\\\omega _{\\\\xi _i})(W)$ strongly, from which it follows that $\\\\Vert W(\\\\eta \\\\otimes \\\\xi _i)-(\\\\eta \\\\otimes \\\\xi _i)\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ Since $\\\\sigma V\\\\sigma =(J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)$ , and $J̉\\\\xi _i=\\\\xi _i$ , we also have $\\\\Vert \\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i)-(\\\\eta \\\\otimes \\\\xi _i)\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ Whence, $\\\\mathbb {G}$ is strongly inner amenable.\",\n \"Now, suppose $\\\\mathbb {G}$ is amenable and let $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ be a two-sided invariant mean.\",\n \"We will show a stronger statement by providing a state $M\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ such that $\\\\langle M,\\\\rho \\\\rhd T\\\\rangle =\\\\langle M,T\\\\lhd \\\\rho \\\\rangle , \\\\ \\\\ \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})),$ upon which restriction to $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is the desired state.\",\n \"Letting $m$ also denote its restriction to $\\\\mathrm {LUC}(\\\\mathbb {G}):=\\\\langle L^{\\\\infty }(\\\\mathbb {G})\\\\star L^1(\\\\mathbb {G})\\\\rangle $ , let $m_0:=\\\\rho _0\\\\circ \\\\Theta ^r(m)\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ , where $\\\\rho _0\\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ is a fixed normal state, and $\\\\Theta ^r:\\\\mathrm {LUC}(\\\\mathbb {G})^*\\\\ni n\\\\mapsto (T\\\\mapsto (\\\\textnormal {id}\\\\otimes n)V^*(T\\\\otimes 1)V)\\\\in \\\\mathcal {CB}_{\\\\mathcal {T}_{\\\\rhd }}(\\\\mathcal {B}(L^2(\\\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).\",\n \"Since $\\\\mathrm {LUC}(\\\\mathbb {G})=\\\\langle \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rhd \\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\rangle $ [35], one may view the map $\\\\Theta ^r$ as follows: $\\\\langle \\\\Theta ^r(n)(T),\\\\rho \\\\rangle =\\\\langle n,T\\\\rhd \\\\rho \\\\rangle , \\\\ \\\\ \\\\ n\\\\in \\\\mathrm {LUC}(\\\\mathbb {G})^*, \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Consider the state $R^*(m_0)\\\\square m_0\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\\\square $ is the left Arens product on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ extending the multiplication in $\\\\mathcal {T}_\\\\rhd =(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\rhd )$ .\",\n \"Fix $\\\\rho ,\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Firstly, since $m\\\\in \\\\mathrm {LUC}(\\\\mathbb {G})^*$ is an invariant mean $m\\\\square \\\\pi (\\\\rho )=\\\\langle \\\\pi (\\\\rho ),1\\\\rangle m=\\\\langle \\\\rho ,1\\\\rangle m$ .\",\n \"Thus, $\\\\langle m_0\\\\square \\\\rho ,T\\\\rangle &=\\\\langle m_0,\\\\rho \\\\rhd T\\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)\\\\circ \\\\Theta ^r(\\\\pi (\\\\rho ))(T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m\\\\square \\\\pi (\\\\rho ))(T)\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle \\\\rho _0,\\\\Theta ^r(m)(T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho ,1\\\\rangle \\\\langle m_0,T\\\\rangle .$ Hence, $m_0\\\\square \\\\rho =\\\\langle \\\\rho ,1\\\\rangle m_0$ .\",\n \"Secondly, $\\\\Theta ^r(m)$ is a conditional expectation onto $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ commuting with both the right $\\\\mathcal {T}_\\\\rhd $ - and right $\\\\mathcal {T}_\\\\lhd $ -actions [15].\",\n \"Since $\\\\hat{x}\\\\rhd \\\\omega =\\\\langle \\\\omega ,\\\\hat{x}\\\\rangle 1$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ and $\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ , we also have $\\\\langle m_0\\\\square (T\\\\lhd \\\\rho ),\\\\omega \\\\rangle &=\\\\langle m_0,(T\\\\lhd \\\\rho )\\\\rhd \\\\omega \\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)((T\\\\lhd \\\\rho )\\\\rhd \\\\omega )\\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T\\\\lhd \\\\rho )\\\\rhd \\\\omega \\\\rangle =\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T\\\\lhd \\\\rho ),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T)\\\\lhd \\\\rho ,\\\\omega \\\\rangle =\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T),\\\\rho \\\\lhd \\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T)\\\\rhd (\\\\rho \\\\lhd \\\\omega )\\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T\\\\rhd (\\\\rho \\\\lhd \\\\omega ))\\\\rangle \\\\\\\\&=\\\\langle m_0,T\\\\rhd (\\\\rho \\\\lhd \\\\omega )\\\\rangle =\\\\langle m_0\\\\square T,\\\\rho \\\\lhd \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (m_0\\\\square T)\\\\lhd \\\\rho ,\\\\omega \\\\rangle .\\\\\\\\$ Thus, $m_0\\\\square (T\\\\lhd \\\\rho )=(m_0\\\\square T)\\\\lhd \\\\rho $ .\",\n \"Putting things together, on the one hand we obtain $\\\\langle R^*(m_0)\\\\square m_0,\\\\rho \\\\rhd T\\\\rangle =\\\\langle R^*(m_0),(m_0\\\\square \\\\rho )\\\\square T\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0),m_0\\\\square T\\\\rangle ,$ and on the other, $\\\\langle R^*(m_0)\\\\square m_0,T\\\\lhd \\\\rho \\\\rangle &=\\\\langle R^*(m_0),m_0\\\\square (T\\\\lhd \\\\rho )\\\\rangle =\\\\langle R^*(m_0),(m_0\\\\square T)\\\\lhd \\\\rho \\\\rangle \\\\\\\\&=\\\\langle m_0,R((m_0\\\\square T)\\\\lhd \\\\rho )\\\\rangle =\\\\langle m_0,R_*(\\\\rho )\\\\rhd R(m_0\\\\square T)\\\\rangle \\\\\\\\&=\\\\langle m_0\\\\square R_*(\\\\rho ),R(m_0\\\\square T)\\\\rangle =\\\\langle R_*(\\\\rho ),1\\\\rangle \\\\langle m_0, R(m_0\\\\square T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0),m_0\\\\square T\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0)\\\\square m_0, T\\\\rangle .$ Therefore, $M:=R^*(m_0)\\\\square m_0$ is the required state.\",\n \"Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].\",\n \"Corollary 3.9 A locally compact quantum group $\\\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $2pt\\\\mathbf {mod}$ .\",\n \"Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.\",\n \"Proposition 3.10 Let $\\\\mathbb {G}$ and $\\\\mathbb {H}$ be locally compact quantum groups such that $\\\\mathbb {H}$ is a closed quantum subgroup of $\\\\mathbb {G}$ in the sense of Vaes.\",\n \"If $\\\\mathbb {G}$ is inner amenable then $\\\\mathbb {H}$ is inner amenable.\",\n \"Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\\\circ \\\\gamma \\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {H}})^*$ , and let $W_{\\\\mathbb {G}}$ and $W_{\\\\mathbb {H}}$ denote the left fundamental unitaries of $\\\\mathbb {G}$ and $\\\\mathbb {H}$ , respectively.\",\n \"We also denote by $\\\\mathbb {W}_{\\\\mathbb {G}}\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u())$ and $\\\\mathbb {W}_{\\\\mathbb {H}}\\\\in M(C_u(\\\\mathbb {H})\\\\otimes _{\\\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\\\pi _{\\\\mathbb {G}}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})=W_{\\\\mathbb {G}}$ and $(\\\\pi _{\\\\mathbb {H}}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {H}})=W_{\\\\mathbb {H}}$ .\",\n \"Finally, we define $\\\\W _{\\\\mathbb {G}}:=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0()), \\\\ \\\\ \\\\ \\\\W _{\\\\mathbb {H}}:=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\in M(C_u(\\\\mathbb {H})\\\\otimes _{\\\\min } C_0()).$ Let $\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {H})$ be the surjection from [24].\",\n \"Its dual morphism is a non-degenerate $*$ -homomorphism $\\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}}:C_u()\\\\rightarrow M(C_u())$ satisfying $(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\mathbb {W}_{\\\\mathbb {G}})=(\\\\textnormal {id}\\\\otimes \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}})(\\\\mathbb {W}_{\\\\mathbb {H}}).$ From the relation $\\\\gamma \\\\circ \\\\pi _{\\\\widehat{\\\\mathbb {H}}}=\\\\pi _{\\\\widehat{\\\\mathbb {G}}}\\\\circ \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}}$ [24], we have $(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}})&=(\\\\textnormal {id}\\\\otimes \\\\gamma \\\\circ \\\\pi _{\\\\widehat{\\\\mathbb {H}}})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\widehat{\\\\mathbb {G}}}\\\\circ \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\widehat{\\\\mathbb {G}}})(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\\\\\&=(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {G}}).$ Thus, for any $x̉\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {H}})$ and $g\\\\in L^{1}(\\\\mathbb {H})$ we have $\\\\langle n̉,x̉\\\\lhd _{\\\\mathbb {H}}g\\\\rangle &=\\\\langle m̉,\\\\gamma ((g\\\\otimes \\\\textnormal {id})W_{\\\\mathbb {H}}^*(1\\\\otimes x̉)W_{\\\\mathbb {H}})\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\gamma ((\\\\pi _{\\\\mathbb {H}}^*(g)\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {H}}^*(1\\\\otimes x̉)\\\\W _{\\\\mathbb {H}}))\\\\rangle \\\\\\\\&=\\\\langle m̉,(\\\\pi _{\\\\mathbb {H}}^*(g)\\\\otimes \\\\textnormal {id})(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}})^*(1\\\\otimes \\\\gamma (x̉))(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}}))\\\\rangle \\\\\\\\&=\\\\langle m̉,(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\gamma (x̉))\\\\W _{\\\\mathbb {G}})\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g)))(\\\\gamma (\\\\hat{x}))\\\\rangle ,$ where the last equality follows from Lemma REF .\",\n \"By invariance of $m̉$ , we may convolve with any state $f\\\\in L^1(\\\\mathbb {G})$ to obtain $\\\\langle n̉,x̉\\\\lhd _{\\\\mathbb {H}}g\\\\rangle &=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(g))(\\\\gamma (\\\\hat{x}))\\\\lhd _{\\\\mathbb {G}}f\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f)(\\\\gamma (\\\\hat{x}))\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\gamma (\\\\hat{x})\\\\lhd _{\\\\mathbb {G}}(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f)\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f,1\\\\rangle \\\\langle m̉,\\\\gamma (\\\\hat{x})\\\\rangle \\\\\\\\&=\\\\langle g,1\\\\rangle \\\\langle n̉,\\\\hat{x}\\\\rangle .$\"\n ],\n [\n \"Examples arising from the bicrossed product construction\",\n \"Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\\\rightarrow G$ and an anti-homomorphism $j:G_2\\\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.\",\n \"Suppose further that $G_1\\\\times G_2\\\\ni (g,s)\\\\mapsto i(g)j(s)\\\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.\",\n \"Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].\",\n \"Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\\\alpha :G_1\\\\times G_2\\\\rightarrow G_2$ and $\\\\beta :G_1\\\\times G_2\\\\rightarrow G_1$ satisfying mutual co-cycle relations [65].\",\n \"It is known that the von Neumann crossed product $G_1\\\\rtimes _\\\\alpha L^\\\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .\",\n \"The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\\\infty }(G_1)^\\\\beta \\\\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ .\",\n \"Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.\",\n \"In preparation we collect some useful formulae from [65], to which we refer the reader for details.\",\n \"To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .\",\n \"The fundamental unitary $W$ satisfies $W^*\\\\xi (g,s,h,t)=\\\\xi (\\\\beta _{t}(h)^{-1}g,s,h,\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1\\\\times G_2\\\\times G_1\\\\times G_2).$ Letting $\\\\Delta $ , $\\\\Delta _1$ , and $\\\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\\\xi (g,s)=\\\\Delta (\\\\alpha _g(s))^{1/2}\\\\Delta _1(\\\\beta _s(g)g^{-1})^{1/2}\\\\Delta _2(\\\\alpha _g(s)s^{-1})^{1/2}\\\\overline{\\\\xi }(\\\\beta _s(g),s^{-1}), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1\\\\times G_2).$ Let $\\\\Psi :G_2\\\\times G_1\\\\rightarrow (0,\\\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\\\Psi (s,g):=\\\\frac{d\\\\beta _s(g)}{dg}$ .\",\n \"It follows that $\\\\Psi (s,g)=\\\\Delta (\\\\alpha _g(s))\\\\Delta _1(\\\\beta _s(g)g^{-1})\\\\Delta _2(\\\\alpha _g(s))$ .\",\n \"The action $\\\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\\\xi (g)=\\\\bigg (\\\\frac{d\\\\beta _{s^{-1}}(g)}{dg}\\\\bigg )^{1/2}\\\\xi (\\\\beta _{s^{-1}}(g)), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\\\xi (g)=\\\\bigg (\\\\frac{d\\\\beta _{s^{-1}}(g)}{dg}\\\\bigg )f(\\\\beta _{s^{-1}}(g)), \\\\ \\\\ \\\\ f\\\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\\\beta $ preserves $\\\\Delta _1$ , there exists an asymptotically $\\\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\\\Vert \\\\cdot \\\\Vert _1=1}$ with $\\\\mathrm {supp}(f_i)\\\\rightarrow \\\\lbrace e\\\\rbrace $ satisfying $\\\\Vert v^1_sf_i-f_i\\\\Vert _1\\\\rightarrow 0$ uniformly on compacta.\",\n \"Then $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ is strongly inner amenable.\",\n \"Using the co-cycle properties of $\\\\alpha $ and $\\\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)\\\\xi (g,s,h,t)\\\\\\\\&=\\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2}\\\\xi (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1})^{-1})$ for all $\\\\xi \\\\in L^2(G_1\\\\times G_2\\\\times G_1\\\\times G_2)$ .\",\n \"Let $\\\\eta _i:=\\\\sqrt{f_i}$ .\",\n \"By hypothesis $(iii)$ we have $\\\\Vert v_s\\\\eta _i - \\\\eta _i\\\\Vert _{L^2(G_1)}^2\\\\le \\\\Vert v_s^1f_i-f_i\\\\Vert _{L^1(G_1)}\\\\rightarrow 0$ uniformly on compacta.\",\n \"Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\\\times G_2\\\\times G_1\\\\rightarrow it follows that\\\\begin{equation} \\\\int f(g,s,h) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\overline{\\\\eta _i}(h) \\\\ dh \\\\rightarrow f(g,s,e)\\\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\\\xi _j)$ in $C_c(G_2)_{\\\\Vert \\\\cdot \\\\Vert _2=1}$ satisfying $\\\\Vert \\\\rho (s)\\\\xi _j-\\\\lambda (s)^*\\\\xi _j\\\\Vert _{L^2(G_2)}\\\\rightarrow 0$ uniformly on compacta.\",\n \"Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\\\xi (g)=\\\\xi (g^{-1})\\\\Delta _1(g^{-1})$ .\",\n \"Then for any $\\\\eta \\\\in C_c(G_1\\\\times G_2)$ , and any $j$ , we have $&\\\\langle W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j),\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j\\\\rangle =\\\\langle (J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)(\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j),W^*(\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j)\\\\rangle \\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ U_1\\\\eta _i(\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{U_1\\\\eta _i}(h) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ \\\\eta _i(\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ \\\\Delta _1(h^{-1}) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h^{-1}) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ \\\\Delta _1(h^{-1}) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h)g,s) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\ \\\\xi _j(t(\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h)^{-1/2} \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h^{-1})^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h^{-1})^{-1}g}(s)t) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiint \\\\bigg (\\\\int \\\\xi _j(t(\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h^{-1})^{-1}g}(s)t) \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h^{-1})^{-1}g,s) \\\\ dt\\\\bigg )\\\\\\\\&\\\\times \\\\Delta (\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h)g,s) \\\\ \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h)^{-1/2} \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\overline{\\\\eta _i}(h) \\\\ dg \\\\ ds \\\\ dh\\\\\\\\&\\\\rightarrow \\\\iiint \\\\xi _j(t(\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1})\\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{g}(s)t) \\\\ |\\\\eta (g,s)|^2 \\\\ \\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),e)^{-1/2} \\\\ dg \\\\ ds \\\\ dt\\\\\\\\$ by ().\",\n \"But $\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}=\\\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2}\\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1/2}\\\\Delta _2(\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1/2}\\\\\\\\&=\\\\Delta _2(\\\\alpha _g(s))^{1/2}, \\\\ \\\\ \\\\ a.e.$ The final integral in the above calculation therefore reduces to $\\\\iiint \\\\Delta _2(\\\\alpha _g(s))^{1/2} \\\\ \\\\xi _j(t\\\\alpha _{g}(s))\\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{g}(s)t) \\\\ |\\\\eta (g,s)|^2 \\\\ dg \\\\ ds \\\\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\\\iint |\\\\eta (g,s)|^2 \\\\ dg \\\\ ds = \\\\Vert \\\\eta \\\\Vert _{L^2(G_1\\\\times G_2)}^2.$ Denoting the index sets of $(\\\\eta _i)$ and $(\\\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\\\mathcal {I}:=J\\\\times I^{J}$ and for $I=(j,(i_j)_{j\\\\in J})\\\\in \\\\mathcal {I}$ we let $\\\\xi _I:=U_1\\\\eta _{i_j}\\\\otimes \\\\xi _j\\\\in L^2(G_1\\\\times G_2)$ .\",\n \"By [39] and the above analysis, the resulting net $(\\\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ .\",\n \"Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.\",\n \"Then $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ is strongly inner amenable.\",\n \"Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.\",\n \"For example, $\\\\mathbb {R}^2\\\\rtimes \\\\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\\\mathbb {R}^2$ and $\\\\mathbb {F}_6$ are.\",\n \"Example 3.14 We consider a discretized version of [26].\",\n \"Let $G=\\\\bigg \\\\lbrace \\\\begin{pmatrix} a & b & x\\\\\\\\ c & d & y\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\mid \\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}\\\\in SL(2,\\\\mathbb {Q}), \\\\ x,y\\\\in \\\\mathbb {Q}\\\\bigg \\\\rbrace ,$ $G_1=(\\\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\\\mathbb {Q})$ , viewed as discrete groups.\",\n \"The embeddings $G_1\\\\ni (x,y)\\\\mapsto \\\\begin{pmatrix} 1 & 0 & -x\\\\\\\\ -x & 1 & -y+\\\\frac{1}{2}x^2\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\in G, \\\\ \\\\ G_2\\\\ni \\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}\\\\mapsto \\\\begin{pmatrix}d & -b & 0\\\\\\\\ -c & a & 0\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\in G$ determine a matched pair structure for $(G_1,G_2)$ .\",\n \"The corresponding actions are given by $\\\\alpha _{(x,y)}\\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}&=\\\\begin{pmatrix}a+bx & b\\\\\\\\ c+dx-(a+bx)(ax+b(y+\\\\frac{1}{2}x^2)) & d-b(ax+b(y+\\\\frac{1}{2}x^2))\\\\end{pmatrix} \\\\\\\\\\\\beta _{\\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}}(x,y)&=\\\\bigg (ax+by+\\\\frac{b}{2}x^2,cx+d\\\\bigg (y+\\\\frac{1}{2}x^2\\\\bigg )-\\\\frac{1}{2}\\\\bigg (ax+b\\\\bigg (y+\\\\frac{1}{2}x^2\\\\bigg )\\\\bigg )^2\\\\bigg ).$ By Corollary REF , the bicrossed product $\\\\mathbb {G}:=VN(\\\\mathbb {Q}^2)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (SL(2,\\\\mathbb {Q}))$ is strongly inner amenable.\",\n \"Since $SL(2,\\\\mathbb {Q})$ is not amenable, it follows from [26] that $\\\\mathbb {G}$ is not amenable.\",\n \"Moreover, since $\\\\mathbb {Q}^2$ is not compact, $\\\\mathbb {G}$ is not discrete by [65].\",\n \"Thus, $\\\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group.\"\n ],\n [\n \"Relative Injectivity and the Averaging Technique\",\n \"In [56] Ruan and Xu showed that the dual $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\\\mathbb {G}$ is relatively 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}$ .\",\n \"We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.\",\n \"Below we let $\\\\widetilde{\\\\mathbb {G}}:=\\\\mathrm {Gr}()$ denote the intrinsic group of $$ .\",\n \"Proposition 4.1 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Consider the following conditions: $$ is inner amenable; $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ ; $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ ; $\\\\widetilde{}=\\\\mathrm {Gr}(\\\\mathbb {G})$ is inner amenable.\",\n \"Then $(i)\\\\Rightarrow (ii)\\\\Leftrightarrow (iii)\\\\Rightarrow (iv)$ .\",\n \"When $\\\\mathbb {G}$ is co-commutative, the conditions are equivalent.\",\n \"$(i)\\\\Rightarrow (ii)$ : Given a state $n\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ which is right $\\\\widehat{\\\\lhd }$ invariant, it follows that $m:=n\\\\circ R$ is left $\\\\widehat{\\\\rhd }^{\\\\prime }$ invariant.\",\n \"It suffices to provide a completely contractive morphism which is a left inverse to the map $\\\\Delta :L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ given by $\\\\Delta (x)(f)=x\\\\rhd f, \\\\ \\\\ \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Identifying $\\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))\\\\cong L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ via $\\\\langle \\\\Phi ,f\\\\otimes g\\\\rangle =\\\\langle \\\\Phi (f),g\\\\rangle , \\\\ \\\\ \\\\ \\\\Phi \\\\in \\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})), \\\\ f,g\\\\in L^1(\\\\mathbb {G}),$ we have $\\\\Delta =\\\\Gamma $ , and that the corresponding $L^1(\\\\mathbb {G})$ -module structure on $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ is defined by $X\\\\unrhd f=(f\\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(\\\\Gamma ^r\\\\otimes \\\\textnormal {id})(X)$ for $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ and $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"First, consider the map $\\\\Phi :\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\ni A\\\\mapsto (\\\\textnormal {id}\\\\otimes m)(V^*AV)\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})).$ Clearly, $\\\\Phi $ is a completely contractive left inverse to $\\\\Gamma $ .\",\n \"We show that $\\\\Phi $ is a right $\\\\mathcal {T}_\\\\rhd $ -module map.\",\n \"This will complete the proof since [14] will entail the invariance $\\\\Phi (L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ , and the restricted module action $\\\\mathcal {T}_\\\\rhd \\\\curvearrowright L^{\\\\infty }(\\\\mathbb {G})$ is the pertinent $L^1(\\\\mathbb {G})$ -module action.\",\n \"To this end, fix $A\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ and $\\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ .\",\n \"Then $\\\\Phi (A\\\\unrhd \\\\rho )&=\\\\Phi ((\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\\\\\&=(\\\\textnormal {id}\\\\otimes m)(\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes m)(\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\\\\\&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\\\prime }=\\\\sigma V^*\\\\sigma $ , where $\\\\sigma $ is the flip map on $L^2(\\\\mathbb {G})\\\\otimes L^2(\\\\mathbb {G})$ , for any $\\\\tau ,\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ , we have $&\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*(\\\\sigma \\\\otimes 1)V_{23}^*A_{23}V_{23}(\\\\sigma \\\\otimes 1)V_{23}),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\\\omega \\\\otimes \\\\tau \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes m)(V^*(1\\\\otimes (\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV))V),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle ( m\\\\otimes \\\\textnormal {id})(V̉^{\\\\prime }((\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV)\\\\otimes 1)V̉^{\\\\prime *}),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle m,\\\\omega \\\\widehat{\\\\rhd }^{\\\\prime }((\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV))\\\\rangle \\\\\\\\&=\\\\langle m,(\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV)\\\\rangle \\\\langle \\\\omega ,1\\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes m\\\\otimes \\\\textnormal {id})(V^*AV\\\\otimes 1),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\Phi (A)\\\\otimes 1,\\\\tau \\\\otimes \\\\omega \\\\rangle .$ As $\\\\tau $ and $\\\\omega $ were arbitrary, we have $\\\\Phi (A\\\\unrhd \\\\rho )&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\\\\\&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\Phi (A)\\\\otimes 1)V^*)\\\\\\\\&=\\\\Phi (A)\\\\rhd \\\\rho .$ $(ii)\\\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\\\mathbb {G})$ -module left inverse $\\\\Phi $ to $\\\\Gamma $ , it follows that $R\\\\circ \\\\Phi \\\\circ \\\\Sigma \\\\circ (R\\\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\\\mathbb {G})$ -module left inverse to $\\\\Gamma $ .\",\n \"$(iii)\\\\Rightarrow (iv)$ : Recall that $\\\\mathrm {Gr}(\\\\mathbb {G})$ is a group of unitaries in $L^{\\\\infty }(\\\\mathbb {G})$ , so it acts naturally on $L^{\\\\infty }(\\\\mathbb {G})$ by conjugation.\",\n \"The existence of a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ which is $\\\\mathrm {Gr}(\\\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\\\Gamma (L^{\\\\infty }(\\\\mathbb {G}))-L^{\\\\infty }(\\\\mathbb {G})$ -bimodule property of $\\\\Phi $ .\",\n \"Since $\\\\widetilde{\\\\widehat{\\\\mathbb {G}}}$ is a closed quantum subgroup of $\\\\widehat{\\\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\\\gamma :L^{\\\\infty }(\\\\widehat{\\\\widetilde{}})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ intertwining the co-multiplications.\",\n \"As $L^{\\\\infty }(\\\\widehat{\\\\widetilde{}})=VN(\\\\widetilde{\\\\widehat{\\\\mathbb {G}}})=VN(\\\\mathrm {Gr}(\\\\mathbb {G}))$ , the state $m\\\\circ \\\\gamma \\\\in VN(\\\\mathrm {Gr}(\\\\mathbb {G}))^*$ is $\\\\mathrm {Gr}(\\\\mathbb {G})$ -invariant, making $\\\\mathrm {Gr}(\\\\mathbb {G})$ inner amenable by [16].\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, then $\\\\mathrm {Gr}(\\\\mathbb {G}_s)=G$ and the implication $(iv)\\\\Rightarrow (i)$ follows immediately from [16].\",\n \"Corollary 4.2 Let $\\\\mathbb {G}$ be a locally compact quantum group for which $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is an injective von Neumann algebra.\",\n \"Then the following are equivalent: $\\\\mathbb {G}$ is amenable; $\\\\mathbb {G}$ is inner amenable; $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is relatively 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Propositions REF and REF yield the implications $(i)\\\\Rightarrow (ii)\\\\Rightarrow (iii)$ .\",\n \"Assume $(iii)$ .\",\n \"Since $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}it follows from \\\\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.\",\n \"Hence, $ G$ is amenable by \\\\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\\\mathbb {G}$ and amenability of $$ .\",\n \"One of their main results is the following.\",\n \"Theorem 4.3 (Ng–Viselter) Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Consider the following conditions: $\\\\mathbb {G}$ is co-amenable; $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $\\\\rho \\\\in C_0(\\\\mathbb {G})^*$ such that $\\\\langle \\\\rho ,x\\\\widehat{\\\\lhd } \\\\hat{f}\\\\rangle =\\\\langle \\\\rho ,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in C_0(\\\\mathbb {G}).$ $$ is amenable.\",\n \"Then $(a)\\\\Rightarrow (b)\\\\Rightarrow (c)$ .\",\n \"Moreover, if $$ has trivial scaling group (for instance, if $\\\\mathbb {G}$ is a Kac algebra), then $(a)\\\\Rightarrow (b^{\\\\prime })\\\\Rightarrow (b)$ , where $C_0(\\\\mathbb {G})$ is nuclear and has a tracial state.\",\n \"They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .\",\n \"We now show that $(b)$ is indeed equivalent to $(a)$ .\",\n \"Theorem 4.4 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.\",\n \"Suppose that $C_0(\\\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.\",\n \"Given a state $m\\\\in C_0(\\\\mathbb {G})^*$ which is right $\\\\widehat{\\\\lhd }$ invariant, it follows that $n:=m\\\\circ R$ is left $\\\\widehat{\\\\rhd }^{\\\\prime }$ invariant, as in the proof of Proposition REF .\",\n \"Let $n\\\\in M(C_0(\\\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\\\textnormal {id}\\\\otimes n):M(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ denote the unique extension of the slice map $(\\\\textnormal {id}\\\\otimes n):\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {K}(L^2(\\\\mathbb {G}))$ which is strictly continuous on the unit ball.\",\n \"By strict density of $C_0(\\\\mathbb {G})$ in $M(C_0(\\\\mathbb {G}))$ , it follows that $\\\\langle n,\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle n,x\\\\rangle \\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in M(C_0(\\\\mathbb {G})).$ Since $V\\\\in M(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))$ , the map $\\\\Phi :C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ defined by $\\\\Phi (A)=(\\\\textnormal {id}\\\\otimes n)(V^*AV), \\\\ \\\\ \\\\ A\\\\in C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.\",\n \"Using the extended $\\\\widehat{\\\\rhd }^{\\\\prime }$ -invariance on $M(C_0(\\\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\\\Phi (A\\\\unrhd \\\\rho )=\\\\Phi (A)\\\\rhd \\\\rho , \\\\ \\\\ \\\\ A\\\\in C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Since $\\\\mathrm {Ad}(V^*):\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})$ and $\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G})=(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))^*,$ we have $\\\\mathrm {Ad}(V^*)^*:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ Letting $r:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\rho \\\\mapsto \\\\rho |_{C_0(\\\\mathbb {G})}\\\\in M(\\\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\\\Phi ^*|_{\\\\mathcal {T}(L^2(\\\\mathbb {G}))}:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\rho \\\\mapsto (r\\\\otimes \\\\textnormal {id})(\\\\mathrm {Ad}(V^*)^*(\\\\rho \\\\otimes n))\\\\in M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ The proof of [14] entails the inclusion $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"In fact, since $\\\\Phi $ is a right $L^1(\\\\mathbb {G})$ -module map and $C_0(\\\\mathbb {G})$ is essential, more is true: $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq \\\\mathrm {LUC}(\\\\mathbb {G})\\\\subseteq M(C_0(\\\\mathbb {G})).$ The unique strict extension $\\\\widetilde{\\\\Phi }:M(C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\rightarrow M(C_0(\\\\mathbb {G}))$ , which exists by [45], satisfies $\\\\widetilde{\\\\Phi }\\\\circ \\\\Gamma |_{C_0(\\\\mathbb {G})}=\\\\textnormal {id}_{C_0(\\\\mathbb {G})}$ .\",\n \"If $\\\\rho \\\\in L^{\\\\infty }(\\\\mathbb {G})_{\\\\perp }$ then the invariance $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ implies $\\\\Phi ^*(\\\\rho )=0$ .\",\n \"Thus, $\\\\Phi ^*$ induces a completely positive left $L^1(\\\\mathbb {G})$ -module map $\\\\Phi ^*:L^1(\\\\mathbb {G})=(\\\\mathcal {T}(L^2(\\\\mathbb {G}))/L^{\\\\infty }(\\\\mathbb {G})_{\\\\perp })\\\\rightarrow M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\\\mathbb {G})$ , for $f\\\\in L^1(\\\\mathbb {G})$ and $x\\\\in C_0(\\\\mathbb {G})$ we have $\\\\langle m_{M(\\\\mathbb {G})}(\\\\Phi ^*(f)),x\\\\rangle =\\\\langle \\\\Phi ^*(f),\\\\Gamma (x)\\\\rangle =\\\\langle f,\\\\widetilde{\\\\Phi }(\\\\Gamma (x))\\\\rangle =\\\\langle f,x\\\\rangle .$ Hence, $m_{M(\\\\mathbb {G})}\\\\circ \\\\Phi ^*$ is the canonical inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ .\",\n \"Now, by nuclearity of $C_0(\\\\mathbb {G})$ , there exists a net $\\\\varphi _i:C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.\",\n \"Define $\\\\psi _i:L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ by $\\\\psi _i=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi ^*.$ Then $\\\\psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}(L^1(\\\\mathbb {G}),M(\\\\mathbb {G}))=_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}(L^1(\\\\mathbb {G}))$ .\",\n \"By [20] there exist contractive positive functionals $\\\\nu _i\\\\in C_u(\\\\mathbb {G})^*$ such that $\\\\psi _i=m^r_{\\\\nu _i}$ .\",\n \"Since $\\\\varphi ^*_i:M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\\\mathbb {G})}$ is separately weak* continuous, it follows that $\\\\psi _i$ converges to the inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ point-weak*.\",\n \"Write $\\\\varphi _i=\\\\sum _{k=1}^{n_i}x_k^i\\\\otimes \\\\mu _k^i$ for some $x_k^i\\\\in C_0(\\\\mathbb {G})$ and $\\\\mu _k^i\\\\in M(\\\\mathbb {G})$ .\",\n \"Since $(b)\\\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.\",\n \"Hence, by [14] $M^r_{cb}(L^1(\\\\mathbb {G}))= _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))=C_u(\\\\mathbb {G})^*$ .\",\n \"For each $i$ and $k$ the map $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*\\\\in _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ , so there exists $\\\\nu _k^i$ such that $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*=m^r_{\\\\nu _k^i}$ .\",\n \"Thus, for $f\\\\in L^1(\\\\mathbb {G})$ we have $\\\\psi _i(f)&=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi ^*(f)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}((\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*(f)\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}(f\\\\star \\\\nu _k^i\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}f\\\\star \\\\nu _k^i\\\\star \\\\mu _k^i.$ Hence, $\\\\nu _i=\\\\sum _{k=1}^{n_i}\\\\nu _k^i\\\\star \\\\mu _k^i\\\\in M(\\\\mathbb {G})$ as $M(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Passing to a subnet we may assume $\\\\nu _i\\\\rightarrow \\\\nu $ weak* in $M(\\\\mathbb {G})$ .\",\n \"But then for any $f\\\\in L^1(\\\\mathbb {G})$ and $x\\\\in C_0(\\\\mathbb {G})$ we have $\\\\langle f\\\\star \\\\nu , x\\\\rangle =\\\\langle \\\\nu , x\\\\star f\\\\rangle =\\\\lim _i\\\\langle \\\\nu _i,x\\\\star f\\\\rangle =\\\\lim _i\\\\langle f\\\\star \\\\nu _i,x\\\\rangle =\\\\langle f,x\\\\rangle .$ It follows that $\\\\nu $ is a right identity for $M(\\\\mathbb {G})$ , which implies that $M(\\\\mathbb {G})$ is unital, whence $\\\\mathbb {G}$ is co-amenable (cf.\",\n \"[5]).\",\n \"Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\\\lambda (G)$ is nuclear and has a tracial state.\",\n \"Corollary 4.5 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\\\mathbb {G}$ is a Kac algebra).\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and has a tracial state.\",\n \"The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ such that $\\\\langle m,x\\\\widehat{\\\\lhd } f̉\\\\rangle =\\\\langle m,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}),$ while $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $m\\\\in C_0(\\\\mathbb {G})^*$ such that $\\\\langle m,x\\\\widehat{\\\\lhd } f̉\\\\rangle =\\\\langle m,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in C_0(\\\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\\\mathbb {G})$ is 1-flat [14], while $\\\\mathbb {G}$ is co-amenable if and only if $M(\\\\mathbb {G})$ is 1-projective, as we now prove.\",\n \"Theorem 4.6 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $M(\\\\mathbb {G})$ is 1-projective in $M(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ .\",\n \"If $\\\\mathbb {G}$ is co-amenable then $M(\\\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.\",\n \"Any unital completely contractive Banach algebra $\\\\mathcal {A}$ is $\\\\Vert e_{\\\\mathcal {A}}\\\\Vert $ -projective, so the claim follows.\",\n \"Conversely, if $M(\\\\mathbb {G})$ is 1-projective, then $C_0(\\\\mathbb {G})^{**}=M(\\\\mathbb {G})^*$ is 1-injective in $\\\\mathbf {mod}-M(\\\\mathbb {G})$ .\",\n \"It follows that the inclusion morphism $M(C_0(\\\\mathbb {G}))\\\\hookrightarrow C_0(\\\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\\\mathbb {G})$ -module map $\\\\Phi : \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow C_0(\\\\mathbb {G})^{**}$ .\",\n \"Applying the argument of [51] we see that $(\\\\textnormal {id}\\\\otimes \\\\Phi ):C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }C_0(\\\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\\\textnormal {id}\\\\otimes \\\\Phi )(W̉)=W̉$ .\",\n \"By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\\\textnormal {id}\\\\otimes \\\\Phi )$ , and hence $(\\\\textnormal {id}\\\\otimes \\\\Phi )(W̉^*XW̉)=W̉^*(\\\\textnormal {id}\\\\otimes \\\\Phi )(X)W̉, \\\\ \\\\ \\\\ X\\\\in C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G})).$ In particular, for every $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ and $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ we have $\\\\Phi (T\\\\hat{f})&=\\\\Phi ((\\\\hat{f}\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes T)W̉))=(\\\\hat{f}\\\\otimes \\\\textnormal {id})(\\\\textnormal {id}\\\\otimes \\\\Phi )((W̉^*(1\\\\otimes T)W̉))\\\\\\\\&=(\\\\hat{f}\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes \\\\Phi (T))W̉)=\\\\Phi (T)\\\\hat{f}.$ Thus, $\\\\Phi $ is a morphism with respect to the canonical $L^1(\\\\widehat{\\\\mathbb {G}})$ -module structure on $C_0(\\\\mathbb {G})^{**}$ .\",\n \"Moreover, the $M(\\\\mathbb {G})$ -module property entails $\\\\pi \\\\circ \\\\Phi (L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}))=$ , where $\\\\pi :C_0(\\\\mathbb {G})^{**}\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ is the canonical surjection.\",\n \"Since $\\\\pi $ is clearly an $L^1(\\\\widehat{\\\\mathbb {G}})$ -module map, it follows that $\\\\pi \\\\circ \\\\Phi |_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\\\mathbb {G})$ .\",\n \"Now, by 1-projectivity of $M(\\\\mathbb {G})$ , for every $\\\\varepsilon >0$ there exists a morphism $\\\\Phi _\\\\varepsilon :M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})^+\\\\widehat{\\\\otimes }M(\\\\mathbb {G})$ such that $m^+\\\\circ \\\\Phi _\\\\varepsilon =\\\\textnormal {id}_{M(\\\\mathbb {G})}$ , and $\\\\Vert \\\\Phi _\\\\varepsilon \\\\Vert _{cb}<1+\\\\varepsilon $ .\",\n \"By amenability of $$ , we know $C_u(\\\\mathbb {G})^*= _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\\\Phi _\\\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\\\mu _i)$ in $M(\\\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\\\mathbb {G}$ .\",\n \"The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.\",\n \"In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ to the co-multiplication that is an $L^1(\\\\mathbb {G})$ -module map.\",\n \"Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow \\\\Gamma (L^{\\\\infty }(\\\\mathbb {G}))$ onto the image of $\\\\Gamma $ (see [23] and [41]).\",\n \"It is the combination of the $L^1(\\\\mathbb {G})$ -module property of $\\\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\\\infty }(\\\\mathbb {G})$ or $C_0(\\\\mathbb {G})$ to approximation properties of $$ .\",\n \"Thus, provided one has a suitably nice $L^1(\\\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.\",\n \"This is where inner amenability enters the picture.\",\n \"Recall that a locally compact quantum group $\\\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\\\widehat{\\\\mathbb {G}}))$ .\",\n \"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\\\mathbb {G}$ , and is denoted $\\\\Lambda _{cb}(\\\\mathbb {G})$ .\",\n \"We say that $\\\\mathbb {G}$ has the approximation property if there exists a net $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ such that $^l(\\\\hat{\\\\lambda }(\\\\hat{f}_i))$ converges to $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ in the stable point-weak* topology.\",\n \"Proposition 4.7 Let $\\\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.\",\n \"If $L^{\\\\infty }(\\\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\\\Lambda _{cb}()\\\\le \\\\Lambda _{cb}(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"$L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.\",\n \"Let $(\\\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\\\mathbb {G})$ .\",\n \"It follows verbatim from [56] that $\\\\Phi _i:L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\ni X\\\\mapsto (\\\\omega _{\\\\xi _i}\\\\otimes \\\\textnormal {id})(W^*(U^*\\\\otimes 1)X(U\\\\otimes 1)W)\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\\\mathbb {G})$ -module maps, which cluster weak* in $\\\\mathcal {CB}(L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=((L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}))^*$ to a module left inverse to $\\\\Gamma $ .\",\n \"Passing to a subnet we may assume convergence.\",\n \"$(i)$ Let $(\\\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\\\Vert \\\\varphi _j\\\\Vert _{cb}\\\\le C$ .\",\n \"Since $\\\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\\\in L^1(\\\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\\\in L^{\\\\infty }(\\\\mathbb {G})$ such that $\\\\varphi _j=\\\\sum _{k=1}^{n_j} x^j_k\\\\otimes f^j_k$ .\",\n \"Put $\\\\Phi _{ij}=\\\\Phi _i\\\\circ (\\\\varphi _j\\\\otimes \\\\textnormal {id})\\\\circ \\\\Gamma :L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G}).$ Then $\\\\Phi _{ij}$ is a normal completely bounded left $L^1(\\\\mathbb {G})$ -module map with $\\\\Vert \\\\Phi _{ij}\\\\Vert _{cb}\\\\le C$ .\",\n \"Also, for each $i,j$ and $1\\\\le k\\\\le n_j$ , the map $L^{\\\\infty }(\\\\mathbb {G})\\\\ni x\\\\mapsto \\\\Phi _i(x^j_k\\\\otimes x)\\\\in L^{\\\\infty }(\\\\mathbb {G})$ is a normal completely bounded left $L^1(\\\\mathbb {G})$ -module map.\",\n \"Since $x^j_k$ is a linear combination of positive elements in $L^{\\\\infty }(\\\\mathbb {G})$ , and $\\\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\\\mathbb {G})$ -module maps $L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"By [20], there exist $\\\\mu ^{ij}_k\\\\in C_u(\\\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\\\mu ^{ij}_k$ .\",\n \"Hence, for each $x\\\\in L^{\\\\infty }(\\\\mathbb {G})$ $\\\\Phi _{ij}(x)&=\\\\Phi _i\\\\circ (\\\\varphi _j\\\\otimes \\\\textnormal {id})\\\\circ \\\\Gamma (x)\\\\\\\\&=\\\\sum _{k=1}^{n_j}\\\\Phi _i(x^j_k \\\\otimes x\\\\star f^j_k)\\\\\\\\&=\\\\sum _{k=1}^{n_j}x\\\\star f^j_k\\\\star \\\\mu ^{ij}_k\\\\\\\\&=x\\\\star f_{ij},$ where $f_{ij}=\\\\sum _{k=1}^{n_i} f^j_k\\\\star \\\\mu ^{ij}_k\\\\in L^1(\\\\mathbb {G})$ as $L^1(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Thus, $\\\\Vert f_{ij}\\\\Vert _{cb}\\\\le C$ , and it follows that $\\\\lim _i\\\\lim _jx\\\\star f_{ij}=\\\\lim _i\\\\Phi _i(\\\\Gamma (x))=x,$ point-weak* in $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\\\Lambda _{cb}()\\\\le \\\\Lambda _{cb}(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.\",\n \"[34]).\",\n \"The converse was shown for Kac algebras in [41].\",\n \"Their proof extends verbatim to arbitrary locally compact quantum groups.\",\n \"Corollary 4.8 Let $G$ be an inner amenable locally compact group.\",\n \"If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.\",\n \"$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.\",\n \"Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\\\mathbb {G})$ -module left inverse to $\\\\Gamma $ that can be approximated by normal $L^1(\\\\mathbb {G})$ -module maps.\",\n \"It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .\",\n \"Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.\",\n \"That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?\",\n \"By [34] it is known that in this case $\\\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\\\mathcal {CB}^\\\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\\\mathcal {CB}^\\\\sigma (VN(G))$ .\",\n \"Proposition 4.11 Let $\\\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.\",\n \"If $C_0(\\\\mathbb {G})$ has the CBAP, then there exists a net $(\\\\nu _i)$ in $M(\\\\mathbb {G})$ such that $\\\\Vert \\\\nu _i\\\\Vert _{cb}\\\\le \\\\Lambda _{cb}(C_0(\\\\mathbb {G}))$ and $\\\\nu _i\\\\rightarrow 1$ $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ .\",\n \"If $C_0(\\\\mathbb {G})$ has the OAP then $M(\\\\mathbb {G})$ is $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"Let $\\\\Phi :L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G})$ be the completely positive left $L^1(\\\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.\",\n \"Recall that $m_{M(\\\\mathbb {G})}\\\\circ \\\\Phi $ is the canonical inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ .\",\n \"$(i)$ Let $\\\\varphi _i:C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\\\Vert \\\\varphi _i\\\\Vert _{cb}\\\\le C$ .\",\n \"Write $\\\\varphi _i=\\\\sum _{k=1}^{n_i}x_k^i\\\\otimes \\\\mu _k^i$ for some $x_k^i\\\\in C_0(\\\\mathbb {G})$ and $\\\\mu _k^i\\\\in M(\\\\mathbb {G})$ .\",\n \"For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\\\mathbb {G})$ , and hence the map $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi \\\\in _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\\\mathbb {G})$ -module maps on $L^1(\\\\mathbb {G})$ , so by [20] there exists $\\\\nu _k^i$ such that $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi =m^r_{\\\\nu _k^i}$ .\",\n \"Define $\\\\psi _i:L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ by $\\\\psi _i=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi $ .\",\n \"Since $\\\\varphi ^*_i:M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\\\mathbb {G})}$ is separately weak* continuous, it follows that $\\\\psi _i$ converges to the inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ point-weak*.\",\n \"Moreover, for $f\\\\in L^1(\\\\mathbb {G})$ we have $\\\\psi _i(f)&=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi (f)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}((\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi (f)\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}(f\\\\star \\\\nu _k^i\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}f\\\\star \\\\nu _k^i\\\\star \\\\mu _k^i\\\\\\\\&=f\\\\star \\\\nu _i,$ where $\\\\nu _i=\\\\sum _{k=1}^{n_i}\\\\nu _k^i\\\\star \\\\mu _k^i\\\\in M(\\\\mathbb {G})$ as $M(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Then $\\\\Vert \\\\nu _i\\\\Vert _{cb}\\\\le C$ , and it follows that $\\\\nu _i\\\\star x\\\\rightarrow x$ weak* in $L^{\\\\infty }(\\\\mathbb {G})$ for all $x\\\\in C_0(\\\\mathbb {G})$ .\",\n \"Since $(\\\\nu _i)$ is bounded in $\\\\mathcal {CB}_{L^1(\\\\mathbb {G})}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ , we have $\\\\nu _i\\\\rightarrow 1$ $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ .\",\n \"$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\\\mathcal {A}$ , we say that an operator $\\\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\\\rightarrow Y\\\\hookrightarrow Z\\\\twoheadrightarrow Z/Y\\\\rightarrow 0$ in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ , the sequence $0\\\\rightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y\\\\hookrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z\\\\twoheadrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z/Y\\\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\\\widehat{\\\\otimes }Y/_{\\\\mathcal {A}}N_{\\\\mathcal {A}}$ , where $_{\\\\mathcal {A}}N_{\\\\mathcal {A}}=\\\\langle a\\\\cdot x\\\\otimes y - x\\\\otimes y\\\\cdot a, \\\\ x^{\\\\prime }\\\\cdot a^{\\\\prime }\\\\otimes y^{\\\\prime } - x^{\\\\prime }\\\\otimes a^{\\\\prime }\\\\cdot y^{\\\\prime }\\\\mid a,a^{\\\\prime }\\\\in \\\\mathcal {A}, \\\\ x,x^{\\\\prime }\\\\in X, \\\\ y,y^{\\\\prime }\\\\in Y\\\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\\\widehat{\\\\mathbb {G}})$ for any Kac algebra $\\\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\\\sigma V \\\\sigma $ and for which $\\\\omega _{\\\\xi _i}|_{L^{\\\\infty }(\\\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\\\mathbb {G})$ .\",\n \"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).\",\n \"We now obtain the same conclusion under a priori weaker hypotheses.\",\n \"Proposition 5.1 Let $\\\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Then $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse holds.\",\n \"It suffices to provide a completely contractive $L^1(\\\\mathbb {G})$ -bimodule map $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ which is a left inverse to $\\\\Gamma $ .\",\n \"Defining $\\\\Phi (X)=(\\\\textnormal {id}\\\\otimes m)(V^*XV)$ , $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\\\Phi $ is a completely contractive right $L^1(\\\\mathbb {G})$ -module map and $\\\\Phi \\\\circ \\\\Gamma =\\\\textnormal {id}_{L^{\\\\infty }(\\\\mathbb {G})}$ .\",\n \"However, since $m$ is also a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ , the module argument from [15] shows that $\\\\Phi $ is also a left $L^1(\\\\mathbb {G})$ -module map.\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.\",\n \"Owing to the fact that $V_s=W_a$ we have $f\\\\widehat{\\\\rhd }_{s}^{\\\\prime }x&=(\\\\textnormal {id}\\\\otimes f)(V̉^{\\\\prime }_s(x\\\\otimes 1)V̉_s^{\\\\prime *})=(f\\\\otimes \\\\textnormal {id})(V_s^*(1\\\\otimes x)V_s)\\\\\\\\&=(f\\\\otimes \\\\textnormal {id})(W_a^*(1\\\\otimes x)W_a)=x\\\\lhd _a f$ for all $f\\\\in L^1(G)$ and $x\\\\in VN(G)$ .\",\n \"Hence, $m$ satisfies (REF ).\",\n \"A completely contractive Banach algebra $\\\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ for some $C\\\\ge 1$ , and has a bounded approximate identity.\",\n \"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ , that is, a bounded net $(A_\\\\alpha )$ in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ satisfying $a\\\\cdot A_\\\\alpha - A_\\\\alpha \\\\cdot a, \\\\ m_{\\\\mathcal {A}}(A_\\\\alpha )\\\\cdot a \\\\rightarrow 0, \\\\ \\\\ \\\\ a\\\\in \\\\mathcal {A}.$ We let $OA(\\\\mathcal {A})$ denote the operator amenability constant of $\\\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ .\",\n \"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.\",\n \"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.\",\n \"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.\",\n \"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.\",\n \"Proposition 5.2 Let $G$ be a locally compact group.\",\n \"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.\",\n \"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.\",\n \"Since $L^{\\\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\\\infty }(G)$ in $L^1(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(G)$ .\",\n \"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ by (the proof of) [54].\",\n \"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.\",\n \"Conversely, if $VN(G)$ is 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}$ , so $G$ is amenable by [14].\",\n \"We now show that 1-biflatness of quantum convolution algebras is a self dual property.\",\n \"Theorem 5.3 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $L^1(\\\\mathbb {G})$ is 1-biflat if and only if $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat.\",\n \"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat, that is, $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Consider the canonical $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule structure on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ given by $\\\\hat{f}\\\\widehat{\\\\rhd }T=(\\\\textnormal {id}\\\\otimes \\\\hat{f})\\\\widehat{V}(T\\\\otimes 1)V̉^* \\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\textnormal {and}\\\\hspace{10.0pt}\\\\hspace{10.0pt}T\\\\widehat{\\\\lhd }\\\\hat{f}=(\\\\hat{f}\\\\otimes \\\\textnormal {id})W̉^*(1\\\\otimes T)W̉,$ for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Then by 1-injectivity, $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ extends to a completely contractive $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule projection $E:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"By the left $\\\\widehat{\\\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\\\infty }(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})\\\\cap L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})=$ .\",\n \"Also, [15] implies that $E$ is a right $\\\\lhd $ -module map.\",\n \"Let $R$ be the extended unitary antipode of $\\\\mathbb {G}$ .\",\n \"Then $(R\\\\otimes R)(V̉^{\\\\prime })=(R\\\\otimes R)(\\\\sigma V^*\\\\sigma )=\\\\Sigma (R\\\\otimes R)(V^*)=\\\\Sigma (J̉\\\\otimes J̉)(V)(J̉\\\\otimes J̉)=\\\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .\",\n \"Let $E_R:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ be the projection of norm one, $E_R=R\\\\circ E\\\\circ R$ .\",\n \"Then for $\\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ , we have $E_R(\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }T)&=R(E(R((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime })V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)(R\\\\otimes R)(V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((R\\\\otimes R)(V̉^{\\\\prime *})(R(T)\\\\otimes 1)(R\\\\otimes R)(V̉^{\\\\prime }))))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((\\\\Sigma W̉^*)(R(T)\\\\otimes 1)(\\\\Sigma W̉))))\\\\\\\\&=R(E((\\\\hat{f}^{\\\\prime }\\\\circ R\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes R(T))W̉)))\\\\\\\\&=R(E(R(T)\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R)))\\\\\\\\&=R(E(R(T))\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R))\\\\\\\\&=\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }E_R(T).$ Thus, $E_R$ is a left $\\\\widehat{\\\\rhd }^{\\\\prime }$ -module map.\",\n \"Since $R(L^{\\\\infty }(\\\\mathbb {G}))=L^{\\\\infty }(\\\\mathbb {G})$ , the restriction $E_R|_{L^{\\\\infty }(\\\\mathbb {G})}$ defines a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,f̉^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ But $E$ was also a right $\\\\lhd $ -module map, which implies that $E_R$ is a left $\\\\rhd $ -module map by the generalized antipode relation (REF ).\",\n \"Thus, we also have $\\\\langle m,f\\\\rhd x\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"By Proposition REF it follows that $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"By 1-injectivity of $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ , there exists a completely contractive morphism $\\\\Phi :L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ which is a left inverse to $$ .\",\n \"It follows that $\\\\Phi |_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\otimes 1}$ defines a state $m̉\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ which is a right $L^1(\\\\widehat{\\\\mathbb {G}})$ -module map, i.e., $$ is amenable.\",\n \"Hence, $L^{\\\\infty }(\\\\mathbb {G})$ is a 1-injective operator space by [5].\",\n \"Proposition REF then implies the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Corollary 5.4 $\\\\ell ^1(\\\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.\",\n \"Since $\\\\ell ^\\\\infty (\\\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\\\mathbb {G}$ , if $\\\\ell ^1(\\\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\\\ell ^\\\\infty (\\\\mathbb {G})$ would be 1-injective in $\\\\ell ^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}\\\\ell ^1(\\\\mathbb {G})$ .\",\n \"Then by Theorem REF $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ would be 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\\\mathbb {G}$ is unimodular.\",\n \"The relative biprojectivity of $L^1(\\\\mathbb {G})$ , that is, relative projectivity of $L^1(\\\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\\\mathbb {G})$ is relatively 1-biprojective if and only if $\\\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].\",\n \"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.\",\n \"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].\",\n \"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\\\Delta $ [16].\",\n \"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .\",\n \"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 5.5 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then the following conditions are equivalent: $\\\\mathbb {G}$ is finite–dimensional.\",\n \"$\\\\mathcal {T}_\\\\rhd $ is relatively 1-biprojective; $L^1(\\\\mathbb {G})$ is 1-biprojective; $(i)\\\\Rightarrow (ii)$ follows from [38].\",\n \"$(ii)\\\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\\\mathbb {G})$ together with the 1-projectivity of $L^1(\\\\mathbb {G})$ as an operator space.\",\n \"The bimodule analogue of [14] then yields $(iii)$ .\",\n \"$(iii)\\\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\\\mathbb {G})$ -bimodule left inverse $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ to $\\\\Gamma $ .\",\n \"As usual, the restriction $\\\\Phi |_{L^{\\\\infty }(\\\\mathbb {G})\\\\otimes 1}:L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.\",\n \"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\\\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.\",\n \"Thus, $ G$ is finite-dimensional by \\\\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\\\mathbb {G})$ For a locally compact quantum group $\\\\mathbb {G}$ , let $L^1_{cb}(\\\\mathbb {G})$ denote the closure of $L^1(\\\\mathbb {G})$ inside $M_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\\\mathbb {G})$ has a bounded approximate identity.\",\n \"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.\",\n \"In [16] the authors gave examples of weakly amenable connected groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.\",\n \"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].\",\n \"Let $\\\\mathbb {G}$ be a compact Kac algebra and let $\\\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\\\mathbb {G})$ , as well as their universal extensions to $C_u(\\\\mathbb {G})$ .\",\n \"As in [7], we define $*$ -homomorphisms $\\\\lambda ,\\\\rho :C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ by $\\\\lambda (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (xy), \\\\ \\\\ \\\\ \\\\rho (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (y R(x)), \\\\ \\\\ \\\\ x,y\\\\in C_u(\\\\mathbb {G}).$ Since $\\\\lambda $ and $\\\\rho $ have commuting ranges, we obtain a canonical representation $\\\\lambda \\\\times \\\\rho :C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .\",\n \"Lemma 6.1 Let $\\\\mathcal {A}$ be a $C^*$ -algebra.\",\n \"There exists a complete isometry $\\\\iota :\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ such that $\\\\iota (\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1})$ is weak* dense in $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ .\",\n \"Let $\\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}$ denote the universal representation of $\\\\mathcal {A}$ .\",\n \"Then the universal cover of the representation $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\\\pi $ of $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{**}$ onto $(\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A})(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }\\\\overline{\\\\otimes }\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^*$ (see [61]).\",\n \"Its pre-adjoint $\\\\iota :=\\\\pi _*:\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ is a complete isometry.\",\n \"Now, Let $F\\\\in (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ , $\\\\Vert F\\\\Vert \\\\le 1$ .\",\n \"Since $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}\\\\subseteq \\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\\\tilde{F}\\\\in (\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**})^*=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^{**}$ satisfying $\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\langle F,A\\\\rangle , \\\\ \\\\ \\\\ A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ such that $f_j\\\\rightarrow \\\\tilde{F}$ weak*.\",\n \"Thus, for all $A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}$ , $\\\\langle F,A\\\\rangle =\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle f_j,\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle \\\\iota (f_j),A\\\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.\",\n \"Then $$ is weakly amenable if and only if $L^1_{cb}(\\\\mathbb {G})$ is operator amenable.\",\n \"Moreover, $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"It is clear that $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].\",\n \"Conversely, by Kirchberg's factorization property the representation $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"Composing with $\\\\omega _{\\\\Lambda _\\\\varphi (1)}$ , we obtain a state $\\\\mu :=\\\\omega _{\\\\Lambda _\\\\varphi (1)}\\\\circ \\\\lambda \\\\times \\\\rho \\\\in (C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ .\",\n \"By $R$ -invariance of $\\\\varphi $ , one can easily verify that $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ .\",\n \"Let $\\\\Gamma _u:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u(\\\\mathbb {G})$ denote the universal co-multiplication.\",\n \"Then, similar to the calculations in [55], for all $u\\\\in \\\\mathrm {Irr}(\\\\mathbb {G})$ , $1\\\\le i,j\\\\le n_u$ , $\\\\langle \\\\Gamma _u^*(\\\\mu )\\\\star f,u_{ij}\\\\rangle &=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\langle \\\\mu ,u_{ik}\\\\otimes u_{kl}\\\\rangle =\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} R(u_{kl}))\\\\\\\\&=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} u_{lk}^{*})=\\\\frac{1}{n_u}\\\\sum _{k=1}^{n_u}\\\\langle f,u_{ij}\\\\rangle \\\\\\\\&=\\\\langle f,u_{ij}\\\\rangle .$ Hence, $\\\\Gamma _u^*(\\\\mu )\\\\star f = f$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"By [42], $R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x^*)(1\\\\otimes y)))=(\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x^*)\\\\Gamma _u(y))$ for all $x,y\\\\in C_u(\\\\mathbb {G})$ , so that $\\\\langle (f\\\\otimes 1)\\\\star \\\\mu ,x\\\\otimes y\\\\rangle &=\\\\varphi ((x\\\\star f) R(y))=(f\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))\\\\\\\\&=f\\\\circ R(R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))))\\\\\\\\&=f\\\\circ R((\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Gamma _u(R(y))))\\\\\\\\&=(f\\\\circ R\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Sigma \\\\circ R\\\\otimes R\\\\circ \\\\Gamma _u(y))\\\\\\\\&=(\\\\varphi \\\\otimes f)(\\\\Gamma _u(y)(R(x)\\\\otimes 1))\\\\\\\\&=\\\\varphi ((f\\\\star y)R(x))\\\\\\\\&=\\\\langle \\\\mu \\\\star (1\\\\otimes f),x\\\\otimes y\\\\rangle .$ Thus, $(f\\\\otimes 1)\\\\star \\\\mu =\\\\mu \\\\star (1\\\\otimes f)$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"Since $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ , we also have $(1\\\\otimes f)\\\\star \\\\mu =\\\\mu \\\\star (f\\\\otimes 1)$ , for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"It follows that $F\\\\star \\\\mu =\\\\mu \\\\star \\\\Sigma F$ for all $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ , where we let $\\\\Sigma $ also denote the flip homomorphism on $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"The proof of Lemma REF implies the existence of a net $(\\\\mu _i)$ of states in $C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*$ such that $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ , and hence in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G}))^*=C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ .\",\n \"Since $m_{C_u(\\\\mathbb {G})^*}=\\\\Gamma _u^*|_{C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*}$ , it follows that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\rightarrow \\\\Gamma _u^*(\\\\mu )$ weak* in $C_u(\\\\mathbb {G})^*$ .\",\n \"By [57] (note that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i),\\\\Gamma _u^*(\\\\mu )$ are states), we have $\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-f\\\\Vert _{L^1(\\\\mathbb {G})}=\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-\\\\Gamma _u^*(\\\\mu )\\\\star f\\\\Vert _{L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}).$ Since $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ , again by [57] we obtain $\\\\Vert F\\\\star \\\\mu _i-\\\\mu _i\\\\star \\\\Sigma F\\\\Vert _{L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}),$ from which we have $&\\\\Vert F\\\\star ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\Vert =\\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-(\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)\\\\Vert +\\\\Vert (\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-\\\\mu _i\\\\star (\\\\Sigma (F\\\\star (1\\\\otimes f)))\\\\Vert +\\\\Vert \\\\mu _i\\\\star \\\\Sigma F-F\\\\star \\\\mu _i\\\\Vert \\\\Vert f\\\\otimes 1\\\\Vert \\\\\\\\&\\\\rightarrow 0$ for every $f\\\\in L^1(\\\\mathbb {G})$ , $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"Similarly, $\\\\Vert ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\star F\\\\Vert \\\\rightarrow 0$ .\",\n \"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\\\mathbb {G})$ in $\\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ such that $\\\\sup _j\\\\Vert f_j\\\\Vert _{cb}<\\\\infty $ .\",\n \"The tensor square of the canonical complete contraction $C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ allows us to view $\\\\mu _i\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))$ with $\\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1$ for all $i$ .\",\n \"By the universal property of the operator space projective tensor product, we may also view each $\\\\mu _i$ as an element of $\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))$ by right multiplication, as well as in $\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))$ by left multiplication.\",\n \"Moreover, $\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))},\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))}\\\\le \\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1.$ Define $\\\\mu _{ij}:=(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)$ .\",\n \"Then $\\\\mu _{ij}\\\\in L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _{ij}\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Vert f_j\\\\Vert _{cb}^2\\\\le \\\\Lambda _{cb}()^2.$ Given $f\\\\in L^1(\\\\mathbb {G})$ , for each $j$ we have $\\\\lim _i f\\\\star \\\\mu _{ij}-\\\\mu _{ij}\\\\star f&=\\\\lim _i(f\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)-(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f)\\\\\\\\&=\\\\lim _i(f\\\\otimes f_j^2)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f_j^2\\\\otimes f)\\\\\\\\&=0$ in $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ and therefore in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ .\",\n \"Furthermore, $\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}(\\\\mu _{ij})\\\\star f&=\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1))\\\\star f\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}(\\\\mu _i\\\\star (f^2_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=\\\\lim _j m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\\\in \\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\\\mu _I)$ in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _I\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\\\mathbb {G})$ in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"It is proved in the exact same way using the fact that $f\\\\star \\\\Gamma _u^*(\\\\mu )=f$ for all $f\\\\in L^1(\\\\mathbb {G})$ , which is easily verified.\",\n \"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.\",\n \"There are examples of weakly amenable residually finite groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {Z}[\\\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].\",\n \"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.\",\n \"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\\\mathbb {G})=L^1_{cb}(\\\\mathbb {G})$ [55].\",\n \"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\\\ne 3$ .\",\n \"Since $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\\\ne 3$ .\",\n \"If $\\\\mathbb {G}_1$ and $\\\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\\\Lambda _{cb}()=\\\\Lambda _{cb}()=1$ , then $\\\\mathbb {G}=\\\\mathbb {G}_1\\\\ast \\\\mathbb {G}_2$ also has the factorization property [11] and $\\\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\\\mathbb {G}))=1$ .\",\n \"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.\",\n \"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].\",\n \"We now generalize this implication to arbitrary locally compact quantum groups.\",\n \"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.\",\n \"By [42], the universal co-representation $\\\\W _{\\\\mathbb {G}}=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0())$ from the proof of Proposition REF satisfies $\\\\hat{\\\\lambda }_u(f̉)=(\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}^*)$ for all $f̉\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ , $(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}$ for all $x\\\\in C_u(\\\\mathbb {G})$ , where $\\\\hat{\\\\lambda }_u$ is the embedding of $L^1_*(\\\\widehat{\\\\mathbb {G}})$ into $C_u(\\\\mathbb {G})$ , and $\\\\pi _{\\\\mathbb {G}}:C_u(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ is the (unique) extension of $\\\\hat{\\\\lambda }:L^1_*(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow C_0(\\\\mathbb {G})$ [42].\",\n \"Moreover, $C_u(\\\\mathbb {G})=\\\\overline{\\\\text{span}\\\\lbrace (\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}) : f̉\\\\in L^1(\\\\widehat{\\\\mathbb {G}})\\\\rbrace }^{\\\\Vert \\\\cdot \\\\Vert _u}.$ We will need the following representation of $\\\\Theta ^l(\\\\mu )$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , which may be found in [19].\",\n \"We present the proof for the convenience of the reader.\",\n \"Lemma 7.1 For $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , $\\\\Theta ^l(\\\\mu )(x)=(\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ First let $x=\\\\hat{\\\\lambda }(\\\\hat{f})\\\\in C_0(\\\\mathbb {G})$ for some $\\\\hat{f}\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Then, for all $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu )(x),f\\\\rangle &=\\\\langle x,m^l_\\\\mu (f)\\\\rangle =\\\\langle \\\\hat{\\\\lambda }(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G}}\\\\circ \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\\\\\&=\\\\langle \\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f})),\\\\mu \\\\otimes \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle (\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})(\\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f}))),\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(\\\\hat{\\\\lambda }_u(\\\\hat{f})))\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle =\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle (\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},f\\\\rangle .$ As $\\\\hat{\\\\lambda }(L^1_*(\\\\widehat{\\\\mathbb {G}}))$ is norm dense in $C_0(\\\\mathbb {G})$ , and since $C_0(\\\\mathbb {G})$ is weak* dense in $L^{\\\\infty }(\\\\mathbb {G})$ , the result follows.\",\n \"Recall that $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is the map taking $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ to the operator of left multiplication by $\\\\mu $ on $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 7.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is amenable then $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.\",\n \"Amenability of $$ entails the surjectivity of $\\\\tilde{\\\\lambda }$ from (the left version of) [14].\",\n \"For simplicity, throughout the proof we denote by $\\\\Theta ^l(\\\\mu )$ the map $\\\\Theta ^l(\\\\tilde{\\\\lambda }(\\\\mu ))$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ .\",\n \"In [19] Daws shows that $\\\\Theta ^l:C_u(\\\\mathbb {G})^*_+\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is an order bijection.\",\n \"We show that it is a complete order bijection.\",\n \"To this end, let $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"By Lemma REF $\\\\Theta ^l(\\\\mu _{ij})(x)=(\\\\textnormal {id}\\\\otimes \\\\mu _{ij})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\\\in L^{\\\\infty }(\\\\mathbb {G})$ we have $((\\\\Theta ^l)^n([\\\\mu _{ij}]))^m([x_k^*x_l])&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}]\\\\\\\\&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})]^m([\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}])\\\\ge 0.$ It follows that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ .\",\n \"On the other hand, suppose $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)$ such that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\\\hat{\\\\tau }$ on $$ , via $\\\\hat{\\\\tau }_t(\\\\hat{x})=P̉^{it}\\\\hat{x}P̉^{-it}$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Using [20], for $\\\\xi _1,...\\\\xi _m\\\\in \\\\mathcal {D}(P̉^{1/2})$ , $\\\\eta _1,...,\\\\eta _m\\\\in \\\\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\\\in {nm}$ , $\\\\langle [\\\\mu _{ij}]^m([\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})])[z_{ik}],[z_{ik}]\\\\rangle &=\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^m\\\\overline{z_{ik}}z_{jl}\\\\langle \\\\mu _{ij},\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\rangle \\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\mu _{ij}^*,\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)\\\\eta _k,\\\\eta _l\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{k,l=1}^m \\\\langle [\\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)]y_k,y_l\\\\rangle }\\\\ge 0,$ where $y_k=[z_{1k}\\\\eta _k \\\\ \\\\cdots \\\\ z_{nk}\\\\eta _k]^T\\\\in L^2(\\\\mathbb {G})^n$ for $1\\\\le k\\\\le m$ .\",\n \"By density of $\\\\lbrace \\\\omega _{\\\\xi ,\\\\eta }\\\\mid \\\\xi \\\\in \\\\mathcal {D}(P̉^{1/2}), \\\\eta \\\\in \\\\mathcal {D}(P̉^{-1/2})\\\\rbrace $ in $L^1_*(\\\\widehat{\\\\mathbb {G}})$ [21], it follows that $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"We now show that $\\\\tilde{\\\\lambda }$ is a complete isometry.\",\n \"To do so we introduce a decomposability norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , given by $\\\\Vert \\\\Phi \\\\Vert _{L^1dec}:=\\\\inf \\\\bigg \\\\lbrace \\\\max \\\\lbrace \\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}\\\\rbrace \\\\mid \\\\begin{bmatrix}\\\\Psi _1 & \\\\Phi \\\\\\\\ \\\\Phi ^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0\\\\bigg \\\\rbrace ,$ where $\\\\Psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"It is evident that $\\\\Vert \\\\cdot \\\\Vert _{L^1dec}$ is a norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"That $\\\\Vert \\\\Phi \\\\Vert _{cb}\\\\le \\\\Vert \\\\Phi \\\\Vert _{L^1dec}$ for all $\\\\Phi \\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ follows verbatim from the first part of [28].\",\n \"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Since $\\\\Theta ^l$ is a completely positive contraction from $(C_u(\\\\mathbb {G})^*)^+$ onto $_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , one easily sees that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}\\\\le \\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}, \\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*),$ where $\\\\Vert \\\\cdot \\\\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.\",\n \"Conversely, if $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{dec}<1$ then there exist $\\\\Psi _1,\\\\Psi _2\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ such that $\\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}<1$ , and $\\\\begin{bmatrix}\\\\Psi _1 & (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\\\\\ (\\\\Theta ^l)^n([\\\\mu _{ij}])^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0.$ Since $(\\\\Theta ^l)^n$ is a complete order bijection there exist $[\\\\nu ^k_{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+=\\\\mathcal {CP}(C_u(\\\\mathbb {G}),M_n)$ such that $\\\\Psi _k=(\\\\Theta ^l)^n([\\\\nu ^k_{ij}])$ , $k=1,2$ , and $\\\\begin{bmatrix}[\\\\nu ^1_{ij}] & [\\\\mu _{ij}]\\\\\\\\ [\\\\mu _{ij}]^* & [\\\\nu ^2_{ij}]\\\\end{bmatrix}\\\\ge _{cp}0.$ It follows that $[\\\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\\\mathbb {G})\\\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\\\widetilde{[\\\\nu ^k_{ij}]}:M(C_u(\\\\mathbb {G}))\\\\rightarrow M_n$ which is strictly continuous on the unit ball [45].\",\n \"By uniqueness, $\\\\widetilde{[\\\\nu ^k_{ij}]}=[\\\\tilde{\\\\nu }^k_{ij}]$ , where $\\\\tilde{\\\\nu }^k_{ij}$ is the unique strict extension of the functional $\\\\nu ^k_{ij}$ .\",\n \"Thus, by completely positivity $\\\\Vert [\\\\nu ^k_{ij}] \\\\Vert _{cb}=\\\\Vert \\\\widetilde{[\\\\nu ^k_{ij}]}(1_{M(C_u(\\\\mathbb {G}))})\\\\Vert =\\\\Vert [\\\\tilde{\\\\nu }^k_{ij}(1_{M(C_u(\\\\mathbb {G}))})]\\\\Vert =\\\\Vert \\\\Psi _k(1)\\\\Vert =\\\\Vert \\\\Psi _k\\\\Vert _{cb}<1,$ so that $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}<1$ .\",\n \"Therefore $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}.$ However, $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}, ,\\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ by the left version of [14].\",\n \"The matricial analogues of the proofs of [14] show that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{cb}=\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}.$ Hence, $\\\\Theta ^l:C_u(\\\\mathbb {G})^*\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is a completely isometric isomorphism.\",\n \"To prove that $\\\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.\",\n \"Let $(\\\\mu _i)$ be a bounded net in $C_u(\\\\mathbb {G})^*$ converging weak* to $\\\\mu $ .\",\n \"Since $C_u(\\\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.\",\n \"Hence, for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu _i\\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\rightarrow \\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle .$ The density of $\\\\hat{\\\\lambda }(L^1(\\\\widehat{\\\\mathbb {G}}))$ in $C_0(\\\\mathbb {G})$ and boundedness of $\\\\Theta ^l(\\\\mu _i)$ in $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=(Q_{cb}^l(\\\\mathbb {G}))^*$ (see [14]) establish the claim.\",\n \"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\\\mathbb {G}$ is co-amenable [35].\",\n \"Corollary 7.4 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $$ is amenable.\",\n \"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\\\mathbb {G})$ .\",\n \"Moreover, as noted in the proof of Proposition REF , $L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).\",\n \"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\\\mathcal {T}(H)\\\\twoheadrightarrow C_0(\\\\mathbb {G})$ .\",\n \"Then $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }\\\\mathcal {T}(H)\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ by projectivity of $\\\\widehat{\\\\otimes }$ , and $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.\",\n \"Hence, $\\\\Theta ^l(f_i)\\\\otimes \\\\textnormal {id}_{M(\\\\mathbb {G})}(X)\\\\rightarrow X,$ weak* for all $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})$ , so that $\\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A)\\\\rightarrow A$ weakly for all $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\\\Vert \\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A) - A\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ .\",\n \"Let $\\\\tilde{m}:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ denote the induced map and $q:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ denote the quotient map.\",\n \"It follows that $q(f\\\\star A)=q(f\\\\otimes m(A))$ for all $f\\\\in L^1(\\\\mathbb {G})$ and $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"Thus, if $m(A)=0$ , then $q(A)=\\\\lim _i q(f_i\\\\star A)=\\\\lim _iq(f_i\\\\otimes m(A))=0,$ so that the induced multiplication $\\\\tilde{m}$ is injective.\",\n \"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\\\mathbb {G})^*\\\\cong \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ completely isometrically.\",\n \"By the left version of [14] $L^1(\\\\mathbb {G})$ is 1-flat in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Thus, the following sequence is 1-exact $0\\\\rightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})\\\\hookrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow 0.$ Since $f\\\\star x=(\\\\textnormal {id}\\\\otimes f\\\\circ \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=(\\\\textnormal {id}\\\\otimes f)\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}=0$ for all $x\\\\in \\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})$ , it follows that $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})=0$ .\",\n \"Hence, $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ .\",\n \"Moreover, as $\\\\Theta ^l(\\\\mu \\\\star f)(x)=\\\\Theta ^l(f)(\\\\Theta ^l(\\\\mu )(x))=(\\\\Theta ^l(\\\\mu )(x))\\\\star f, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\mu \\\\in C_u(\\\\mathbb {G})^*, \\\\ x\\\\in C_0(\\\\mathbb {G}),$ it follows that $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ is an isomorphism of left $L^1(\\\\mathbb {G})$ -modules, i.e., $C_u(\\\\mathbb {G})$ is induced.\",\n \"The commutative diagram $\\\\begin{tikzcd}L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G}) [r, equal][d, equal] &L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})[d, \\\"\\\\tilde{m}\\\"]\\\\\\\\C_u(\\\\mathbb {G}) [r, two heads] &C_0(\\\\mathbb {G})\\\\end{tikzcd}$ then implies that $\\\\tilde{m}$ is a complete quotient map.\",\n \"Thus, $C_0(\\\\mathbb {G})$ is an induced $L^1(\\\\mathbb {G})$ -module and $M(\\\\mathbb {G})\\\\cong (L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G}))^*= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})$ is necessarily a left unit for $M(\\\\mathbb {G})$ , which entails the co-amenability of $\\\\mathbb {G}$ .\",\n \"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.\",\n \"The author was partially supported by the NSERC Discovery Grant 1304873.\"\n ],\n [\n \"Inner Amenability\",\n \"Given a locally compact quantum group $\\\\mathbb {G}$ the composition $W\\\\sigma V\\\\sigma \\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ defines a unitary co-representation of $\\\\mathbb {G}$ called the conjugation co-representation, where $\\\\sigma $ is the flip map on $L^2(\\\\mathbb {G})\\\\otimes L^2(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\\\sigma V\\\\sigma \\\\xi (s,t)=\\\\xi (s,s^{-1}ts)\\\\Delta (s)^{1/2}, \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G\\\\times G), \\\\ s,t\\\\in G.$ Thus, $W\\\\sigma V\\\\sigma $ is the unitary generator of the conjugation representation $\\\\beta _2:G\\\\rightarrow \\\\mathcal {B}(L^2(G))$ , where $\\\\beta _2(s)\\\\xi (t)=\\\\lambda (s)\\\\rho (s)\\\\xi (t)=\\\\xi (s^{-1}ts)\\\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\\\in L^{\\\\infty }(G)^*$ satisfying $\\\\langle m,\\\\beta _{\\\\infty }(s)f\\\\rangle =\\\\langle m,f\\\\rangle \\\\ \\\\ \\\\ s\\\\in G, \\\\ f\\\\in L^{\\\\infty }(G),$ where $\\\\beta _\\\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\\\in G$ , $f\\\\in L^{\\\\infty }(G)$ , is the conjugation action on $L^{\\\\infty }(G)$ .\",\n \"Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\\\in \\\\ell ^{\\\\infty }(G)^*$ such that $m\\\\ne \\\\delta _e$ .\",\n \"In what follows, inner amenability will always refer to the definition (REF ) given above.\",\n \"In particular, every discrete group is inner amenable.\",\n \"Definition 3.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"We say that $\\\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\\\xi _i)$ of unit vectors such that $\\\\Vert W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i)-\\\\eta \\\\otimes \\\\xi _i\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ $\\\\mathbb {G}$ is inner amenable if there exists a state $m\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ satisfying $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,\\\\hat{x}\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}).$ Such a state is said to be inner invariant.\",\n \"$\\\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\\\in C_0()^*$ such that $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,\\\\hat{x}\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\hat{x}\\\\in C_0().$ Such a state is said to be topologically inner invariant.\",\n \"Examples 3.3 The following known examples are worth mentioning.\",\n \"Any co-commutative quantum group $\\\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\\\sigma V_s\\\\sigma =W_sW_s^*=1$ .\",\n \"Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\\\xi :=\\\\Lambda _\\\\varphi (1)$ being conjugation invariant, where $\\\\varphi $ is the Haar trace on the compact dual.\",\n \"It was shown in [51] that if $\\\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\\\mathbb {G}$ is topologically inner amenable.\",\n \"Further examples, including the bicrossed product construction will be studied below.\",\n \"The following proposition is standard.\",\n \"We include the proof for completeness.\",\n \"Proposition 3.4 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $\\\\mathbb {G}$ is strongly inner amenability then it is inner amenable.\",\n \"Let $(\\\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\\\sigma V\\\\sigma $ .\",\n \"Passing to a subnet, we may assume that $(\\\\omega _{J\\\\xi _i}|_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})})$ converges weak* to a state $m\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ .\",\n \"For any $f=\\\\omega _{\\\\xi ,\\\\eta }\\\\in L^1(\\\\mathbb {G})$ and $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\\\widehat{J}\\\\otimes J)W(\\\\widehat{J}\\\\otimes J)=W^*, \\\\ \\\\ \\\\ (\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\otimes J)=\\\\sigma V^*\\\\sigma ,$ we have $\\\\langle m,\\\\hat{x}\\\\lhd f\\\\rangle &=\\\\lim _i\\\\langle \\\\omega _{J\\\\xi _i},\\\\hat{x}\\\\lhd f\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle W^*(1\\\\otimes \\\\hat{x})W(\\\\xi \\\\otimes J\\\\xi _i),\\\\eta \\\\otimes J\\\\xi _i\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})W(\\\\xi \\\\otimes J\\\\xi _i),W(\\\\eta \\\\otimes J\\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})(\\\\widehat{J}\\\\otimes J)W^*(\\\\widehat{J}\\\\xi \\\\otimes \\\\xi _i),(\\\\widehat{J}\\\\otimes J)W^*(\\\\widehat{J}\\\\eta \\\\otimes \\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})(\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\xi \\\\otimes \\\\xi _i),(\\\\widehat{J}\\\\otimes J)\\\\sigma V\\\\sigma (\\\\widehat{J}\\\\eta \\\\otimes \\\\xi _i)\\\\rangle \\\\\\\\&=\\\\lim _i\\\\langle (1\\\\otimes \\\\hat{x})\\\\sigma V^*\\\\sigma (\\\\xi \\\\otimes J\\\\xi _i),\\\\sigma V^*\\\\sigma (\\\\eta \\\\otimes J\\\\xi _i)\\\\rangle \\\\\\\\&=\\\\langle m,\\\\hat{x}\\\\rangle \\\\langle f,1\\\\rangle .$ In the commutative case, the converse of Proposition REF holds.\",\n \"Proposition 3.5 A commutative quantum group $\\\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.\",\n \"If $\\\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.\",\n \"Let $m\\\\in VN(G)^*$ satisfy $\\\\langle m,x\\\\rangle =\\\\langle m,x \\\\lhd f\\\\rangle $ for all $x\\\\in VN(G)$ and all states $f\\\\in L^1(G)$ .\",\n \"Then for $t\\\\in G$ $\\\\lambda (t)x\\\\lambda (t)^* \\\\lhd f=\\\\int _G \\\\lambda (s^{-1}t)x\\\\lambda (t^{-1}s) f(s)ds=\\\\int _G\\\\lambda (r)^*x\\\\lambda (r)f(tr)dr=x \\\\lhd f_t,$ so that $\\\\langle m,\\\\lambda (t)x\\\\lambda (t)^*\\\\rangle =\\\\langle m,\\\\lambda (t)x\\\\lambda (t)^* \\\\lhd f\\\\rangle =\\\\langle m,x \\\\lhd f_t\\\\rangle =\\\\langle m,x\\\\rangle $ for all $x\\\\in VN(G)$ , $t\\\\in G$ and states $f\\\\in L^1(G)$ .\",\n \"Thus, $G$ is inner amenable by [16].\",\n \"Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\\\xi _i)$ in $L^2(G)$ satisfying $\\\\Vert \\\\lambda (s)\\\\xi _i-\\\\rho (s)^*\\\\xi _i\\\\Vert _{L^2(G)}\\\\rightarrow 0$ uniformly on compact subsets of $G$ .\",\n \"For any $\\\\eta \\\\in C_c(G)$ we then have $\\\\langle W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i),\\\\eta \\\\otimes \\\\xi _i\\\\rangle =\\\\iint \\\\xi _i(ts)\\\\overline{\\\\xi _i}(st)\\\\Delta (s)^{1/2}|\\\\eta (s)|^2 \\\\ ds \\\\ dt\\\\rightarrow 1.$ It follows that $(\\\\xi _i)$ is strongly inner invariant.\",\n \"Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].\",\n \"Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\\\mathcal {B}(L^2(G))$ , via $\\\\langle m,f \\\\rhd T\\\\rangle =\\\\langle m, T \\\\lhd f\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(G), \\\\ T\\\\in \\\\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.\",\n \"For details, see [13].\",\n \"Proposition 3.6 A commutative quantum group $\\\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.\",\n \"The existence of a tracial state on $C_\\\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.\",\n \"By [51] $\\\\mathbb {G}_a$ is topologically inner amenable if $C_\\\\lambda ^*(G)$ has a tracial state.\",\n \"Conversely, if $m$ is an inner invariant state on $C_\\\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.\",\n \"Viewing $m$ as a positive definite function in $B_\\\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\\\in G$ .\",\n \"A simple integral calculation then shows that $m$ is a tracial state on $C_\\\\lambda ^*(G)$ .\",\n \"Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ to $C_0()$ .\",\n \"This is not the case however.\",\n \"In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\\\bigoplus _{n\\\\in \\\\mathbb {N}} \\\\Gamma _n\\\\rtimes \\\\prod _{n\\\\in \\\\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\\\Gamma _n$ for which $C^*_\\\\lambda (\\\\Gamma _n\\\\rtimes F_n)$ admits a unique tracial state.\",\n \"In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .\",\n \"Thus, $G$ is inner amenable.\",\n \"However, in [30], the authors also show that the continuous function $m$ in $B_\\\\lambda (G)$ associated to a tracial state on $C^*_\\\\lambda (G)$ must be discontinuous.\",\n \"Whence, there is no tracial state on $C^*_\\\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .\",\n \"Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\\\lambda (G)$ .\",\n \"In the other direction, in general it is not possible to lift a tracial state on $C^*_\\\\lambda (G)$ to an inner invariant mean on $VN(G)$ .\",\n \"In [30] it is shown that $\\\\mathbb {R}^2\\\\rtimes \\\\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\\\mathbb {R})$ .\",\n \"Proposition 3.8 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is co-amenable then $\\\\mathbb {G}$ is strongly inner amenable.\",\n \"If $\\\\mathbb {G}$ is amenable then it is inner amenable.\",\n \"If $$ is co-amenable, let $E\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\\\omega _{\\\\xi _i}$ with $J̉\\\\xi _i=\\\\xi _i$ (since $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\curvearrowright L^2(\\\\mathbb {G})$ is standard).\",\n \"Then $1=(\\\\textnormal {id}\\\\otimes E)(W)=\\\\lim _i(\\\\textnormal {id}\\\\otimes \\\\omega _{\\\\xi _i})(W)$ strongly, from which it follows that $\\\\Vert W(\\\\eta \\\\otimes \\\\xi _i)-(\\\\eta \\\\otimes \\\\xi _i)\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ Since $\\\\sigma V\\\\sigma =(J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)$ , and $J̉\\\\xi _i=\\\\xi _i$ , we also have $\\\\Vert \\\\sigma V\\\\sigma (\\\\eta \\\\otimes \\\\xi _i)-(\\\\eta \\\\otimes \\\\xi _i)\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ \\\\eta \\\\in L^2(\\\\mathbb {G}).$ Whence, $\\\\mathbb {G}$ is strongly inner amenable.\",\n \"Now, suppose $\\\\mathbb {G}$ is amenable and let $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ be a two-sided invariant mean.\",\n \"We will show a stronger statement by providing a state $M\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ such that $\\\\langle M,\\\\rho \\\\rhd T\\\\rangle =\\\\langle M,T\\\\lhd \\\\rho \\\\rangle , \\\\ \\\\ \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})),$ upon which restriction to $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is the desired state.\",\n \"Letting $m$ also denote its restriction to $\\\\mathrm {LUC}(\\\\mathbb {G}):=\\\\langle L^{\\\\infty }(\\\\mathbb {G})\\\\star L^1(\\\\mathbb {G})\\\\rangle $ , let $m_0:=\\\\rho _0\\\\circ \\\\Theta ^r(m)\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ , where $\\\\rho _0\\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ is a fixed normal state, and $\\\\Theta ^r:\\\\mathrm {LUC}(\\\\mathbb {G})^*\\\\ni n\\\\mapsto (T\\\\mapsto (\\\\textnormal {id}\\\\otimes n)V^*(T\\\\otimes 1)V)\\\\in \\\\mathcal {CB}_{\\\\mathcal {T}_{\\\\rhd }}(\\\\mathcal {B}(L^2(\\\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).\",\n \"Since $\\\\mathrm {LUC}(\\\\mathbb {G})=\\\\langle \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rhd \\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\rangle $ [35], one may view the map $\\\\Theta ^r$ as follows: $\\\\langle \\\\Theta ^r(n)(T),\\\\rho \\\\rangle =\\\\langle n,T\\\\rhd \\\\rho \\\\rangle , \\\\ \\\\ \\\\ n\\\\in \\\\mathrm {LUC}(\\\\mathbb {G})^*, \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Consider the state $R^*(m_0)\\\\square m_0\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\\\square $ is the left Arens product on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))^*$ extending the multiplication in $\\\\mathcal {T}_\\\\rhd =(\\\\mathcal {T}(L^2(\\\\mathbb {G})),\\\\rhd )$ .\",\n \"Fix $\\\\rho ,\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Firstly, since $m\\\\in \\\\mathrm {LUC}(\\\\mathbb {G})^*$ is an invariant mean $m\\\\square \\\\pi (\\\\rho )=\\\\langle \\\\pi (\\\\rho ),1\\\\rangle m=\\\\langle \\\\rho ,1\\\\rangle m$ .\",\n \"Thus, $\\\\langle m_0\\\\square \\\\rho ,T\\\\rangle &=\\\\langle m_0,\\\\rho \\\\rhd T\\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)\\\\circ \\\\Theta ^r(\\\\pi (\\\\rho ))(T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m\\\\square \\\\pi (\\\\rho ))(T)\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle \\\\rho _0,\\\\Theta ^r(m)(T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho ,1\\\\rangle \\\\langle m_0,T\\\\rangle .$ Hence, $m_0\\\\square \\\\rho =\\\\langle \\\\rho ,1\\\\rangle m_0$ .\",\n \"Secondly, $\\\\Theta ^r(m)$ is a conditional expectation onto $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ commuting with both the right $\\\\mathcal {T}_\\\\rhd $ - and right $\\\\mathcal {T}_\\\\lhd $ -actions [15].\",\n \"Since $\\\\hat{x}\\\\rhd \\\\omega =\\\\langle \\\\omega ,\\\\hat{x}\\\\rangle 1$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ and $\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ , we also have $\\\\langle m_0\\\\square (T\\\\lhd \\\\rho ),\\\\omega \\\\rangle &=\\\\langle m_0,(T\\\\lhd \\\\rho )\\\\rhd \\\\omega \\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)((T\\\\lhd \\\\rho )\\\\rhd \\\\omega )\\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T\\\\lhd \\\\rho )\\\\rhd \\\\omega \\\\rangle =\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T\\\\lhd \\\\rho ),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T)\\\\lhd \\\\rho ,\\\\omega \\\\rangle =\\\\langle \\\\rho _0,1\\\\rangle \\\\langle \\\\Theta ^r(m)(T),\\\\rho \\\\lhd \\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T)\\\\rhd (\\\\rho \\\\lhd \\\\omega )\\\\rangle =\\\\langle \\\\rho _0,\\\\Theta ^r(m)(T\\\\rhd (\\\\rho \\\\lhd \\\\omega ))\\\\rangle \\\\\\\\&=\\\\langle m_0,T\\\\rhd (\\\\rho \\\\lhd \\\\omega )\\\\rangle =\\\\langle m_0\\\\square T,\\\\rho \\\\lhd \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (m_0\\\\square T)\\\\lhd \\\\rho ,\\\\omega \\\\rangle .\\\\\\\\$ Thus, $m_0\\\\square (T\\\\lhd \\\\rho )=(m_0\\\\square T)\\\\lhd \\\\rho $ .\",\n \"Putting things together, on the one hand we obtain $\\\\langle R^*(m_0)\\\\square m_0,\\\\rho \\\\rhd T\\\\rangle =\\\\langle R^*(m_0),(m_0\\\\square \\\\rho )\\\\square T\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0),m_0\\\\square T\\\\rangle ,$ and on the other, $\\\\langle R^*(m_0)\\\\square m_0,T\\\\lhd \\\\rho \\\\rangle &=\\\\langle R^*(m_0),m_0\\\\square (T\\\\lhd \\\\rho )\\\\rangle =\\\\langle R^*(m_0),(m_0\\\\square T)\\\\lhd \\\\rho \\\\rangle \\\\\\\\&=\\\\langle m_0,R((m_0\\\\square T)\\\\lhd \\\\rho )\\\\rangle =\\\\langle m_0,R_*(\\\\rho )\\\\rhd R(m_0\\\\square T)\\\\rangle \\\\\\\\&=\\\\langle m_0\\\\square R_*(\\\\rho ),R(m_0\\\\square T)\\\\rangle =\\\\langle R_*(\\\\rho ),1\\\\rangle \\\\langle m_0, R(m_0\\\\square T)\\\\rangle \\\\\\\\&=\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0),m_0\\\\square T\\\\rangle =\\\\langle \\\\rho ,1\\\\rangle \\\\langle R^*(m_0)\\\\square m_0, T\\\\rangle .$ Therefore, $M:=R^*(m_0)\\\\square m_0$ is the required state.\",\n \"Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].\",\n \"Corollary 3.9 A locally compact quantum group $\\\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $2pt\\\\mathbf {mod}$ .\",\n \"Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.\",\n \"Proposition 3.10 Let $\\\\mathbb {G}$ and $\\\\mathbb {H}$ be locally compact quantum groups such that $\\\\mathbb {H}$ is a closed quantum subgroup of $\\\\mathbb {G}$ in the sense of Vaes.\",\n \"If $\\\\mathbb {G}$ is inner amenable then $\\\\mathbb {H}$ is inner amenable.\",\n \"Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\\\circ \\\\gamma \\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {H}})^*$ , and let $W_{\\\\mathbb {G}}$ and $W_{\\\\mathbb {H}}$ denote the left fundamental unitaries of $\\\\mathbb {G}$ and $\\\\mathbb {H}$ , respectively.\",\n \"We also denote by $\\\\mathbb {W}_{\\\\mathbb {G}}\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u())$ and $\\\\mathbb {W}_{\\\\mathbb {H}}\\\\in M(C_u(\\\\mathbb {H})\\\\otimes _{\\\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\\\pi _{\\\\mathbb {G}}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})=W_{\\\\mathbb {G}}$ and $(\\\\pi _{\\\\mathbb {H}}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {H}})=W_{\\\\mathbb {H}}$ .\",\n \"Finally, we define $\\\\W _{\\\\mathbb {G}}:=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0()), \\\\ \\\\ \\\\ \\\\W _{\\\\mathbb {H}}:=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\in M(C_u(\\\\mathbb {H})\\\\otimes _{\\\\min } C_0()).$ Let $\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {H})$ be the surjection from [24].\",\n \"Its dual morphism is a non-degenerate $*$ -homomorphism $\\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}}:C_u()\\\\rightarrow M(C_u())$ satisfying $(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\mathbb {W}_{\\\\mathbb {G}})=(\\\\textnormal {id}\\\\otimes \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}})(\\\\mathbb {W}_{\\\\mathbb {H}}).$ From the relation $\\\\gamma \\\\circ \\\\pi _{\\\\widehat{\\\\mathbb {H}}}=\\\\pi _{\\\\widehat{\\\\mathbb {G}}}\\\\circ \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}}$ [24], we have $(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}})&=(\\\\textnormal {id}\\\\otimes \\\\gamma \\\\circ \\\\pi _{\\\\widehat{\\\\mathbb {H}}})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\widehat{\\\\mathbb {G}}}\\\\circ \\\\hat{\\\\pi }_{\\\\mathbb {G},\\\\mathbb {H}})(\\\\mathbb {W}_{\\\\mathbb {H}})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\widehat{\\\\mathbb {G}}})(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\\\\\&=(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {G}}).$ Thus, for any $x̉\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {H}})$ and $g\\\\in L^{1}(\\\\mathbb {H})$ we have $\\\\langle n̉,x̉\\\\lhd _{\\\\mathbb {H}}g\\\\rangle &=\\\\langle m̉,\\\\gamma ((g\\\\otimes \\\\textnormal {id})W_{\\\\mathbb {H}}^*(1\\\\otimes x̉)W_{\\\\mathbb {H}})\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\gamma ((\\\\pi _{\\\\mathbb {H}}^*(g)\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {H}}^*(1\\\\otimes x̉)\\\\W _{\\\\mathbb {H}}))\\\\rangle \\\\\\\\&=\\\\langle m̉,(\\\\pi _{\\\\mathbb {H}}^*(g)\\\\otimes \\\\textnormal {id})(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}})^*(1\\\\otimes \\\\gamma (x̉))(\\\\textnormal {id}\\\\otimes \\\\gamma )(\\\\W _{\\\\mathbb {H}}))\\\\rangle \\\\\\\\&=\\\\langle m̉,(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\otimes \\\\textnormal {id})(\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\gamma (x̉))\\\\W _{\\\\mathbb {G}})\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g)))(\\\\gamma (\\\\hat{x}))\\\\rangle ,$ where the last equality follows from Lemma REF .\",\n \"By invariance of $m̉$ , we may convolve with any state $f\\\\in L^1(\\\\mathbb {G})$ to obtain $\\\\langle n̉,x̉\\\\lhd _{\\\\mathbb {H}}g\\\\rangle &=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(g))(\\\\gamma (\\\\hat{x}))\\\\lhd _{\\\\mathbb {G}}f\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\Theta ^l(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f)(\\\\gamma (\\\\hat{x}))\\\\rangle \\\\\\\\&=\\\\langle m̉,\\\\gamma (\\\\hat{x})\\\\lhd _{\\\\mathbb {G}}(\\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f)\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G},\\\\mathbb {H}}^*(\\\\pi _{\\\\mathbb {H}}^*(g))\\\\star _{\\\\mathbb {G}}f,1\\\\rangle \\\\langle m̉,\\\\gamma (\\\\hat{x})\\\\rangle \\\\\\\\&=\\\\langle g,1\\\\rangle \\\\langle n̉,\\\\hat{x}\\\\rangle .$\"\n ],\n [\n \"Examples arising from the bicrossed product construction\",\n \"Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\\\rightarrow G$ and an anti-homomorphism $j:G_2\\\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.\",\n \"Suppose further that $G_1\\\\times G_2\\\\ni (g,s)\\\\mapsto i(g)j(s)\\\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.\",\n \"Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].\",\n \"Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\\\alpha :G_1\\\\times G_2\\\\rightarrow G_2$ and $\\\\beta :G_1\\\\times G_2\\\\rightarrow G_1$ satisfying mutual co-cycle relations [65].\",\n \"It is known that the von Neumann crossed product $G_1\\\\rtimes _\\\\alpha L^\\\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .\",\n \"The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\\\infty }(G_1)^\\\\beta \\\\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ .\",\n \"Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.\",\n \"In preparation we collect some useful formulae from [65], to which we refer the reader for details.\",\n \"To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .\",\n \"The fundamental unitary $W$ satisfies $W^*\\\\xi (g,s,h,t)=\\\\xi (\\\\beta _{t}(h)^{-1}g,s,h,\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1\\\\times G_2\\\\times G_1\\\\times G_2).$ Letting $\\\\Delta $ , $\\\\Delta _1$ , and $\\\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\\\xi (g,s)=\\\\Delta (\\\\alpha _g(s))^{1/2}\\\\Delta _1(\\\\beta _s(g)g^{-1})^{1/2}\\\\Delta _2(\\\\alpha _g(s)s^{-1})^{1/2}\\\\overline{\\\\xi }(\\\\beta _s(g),s^{-1}), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1\\\\times G_2).$ Let $\\\\Psi :G_2\\\\times G_1\\\\rightarrow (0,\\\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\\\Psi (s,g):=\\\\frac{d\\\\beta _s(g)}{dg}$ .\",\n \"It follows that $\\\\Psi (s,g)=\\\\Delta (\\\\alpha _g(s))\\\\Delta _1(\\\\beta _s(g)g^{-1})\\\\Delta _2(\\\\alpha _g(s))$ .\",\n \"The action $\\\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\\\xi (g)=\\\\bigg (\\\\frac{d\\\\beta _{s^{-1}}(g)}{dg}\\\\bigg )^{1/2}\\\\xi (\\\\beta _{s^{-1}}(g)), \\\\ \\\\ \\\\ \\\\xi \\\\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\\\xi (g)=\\\\bigg (\\\\frac{d\\\\beta _{s^{-1}}(g)}{dg}\\\\bigg )f(\\\\beta _{s^{-1}}(g)), \\\\ \\\\ \\\\ f\\\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\\\beta $ preserves $\\\\Delta _1$ , there exists an asymptotically $\\\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\\\Vert \\\\cdot \\\\Vert _1=1}$ with $\\\\mathrm {supp}(f_i)\\\\rightarrow \\\\lbrace e\\\\rbrace $ satisfying $\\\\Vert v^1_sf_i-f_i\\\\Vert _1\\\\rightarrow 0$ uniformly on compacta.\",\n \"Then $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ is strongly inner amenable.\",\n \"Using the co-cycle properties of $\\\\alpha $ and $\\\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)\\\\xi (g,s,h,t)\\\\\\\\&=\\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2}\\\\xi (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1})^{-1})$ for all $\\\\xi \\\\in L^2(G_1\\\\times G_2\\\\times G_1\\\\times G_2)$ .\",\n \"Let $\\\\eta _i:=\\\\sqrt{f_i}$ .\",\n \"By hypothesis $(iii)$ we have $\\\\Vert v_s\\\\eta _i - \\\\eta _i\\\\Vert _{L^2(G_1)}^2\\\\le \\\\Vert v_s^1f_i-f_i\\\\Vert _{L^1(G_1)}\\\\rightarrow 0$ uniformly on compacta.\",\n \"Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\\\times G_2\\\\times G_1\\\\rightarrow it follows that\\\\begin{equation} \\\\int f(g,s,h) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\overline{\\\\eta _i}(h) \\\\ dh \\\\rightarrow f(g,s,e)\\\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\\\xi _j)$ in $C_c(G_2)_{\\\\Vert \\\\cdot \\\\Vert _2=1}$ satisfying $\\\\Vert \\\\rho (s)\\\\xi _j-\\\\lambda (s)^*\\\\xi _j\\\\Vert _{L^2(G_2)}\\\\rightarrow 0$ uniformly on compacta.\",\n \"Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\\\xi (g)=\\\\xi (g^{-1})\\\\Delta _1(g^{-1})$ .\",\n \"Then for any $\\\\eta \\\\in C_c(G_1\\\\times G_2)$ , and any $j$ , we have $&\\\\langle W\\\\sigma V\\\\sigma (\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j),\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j\\\\rangle =\\\\langle (J̉\\\\otimes J̉)W^*(J̉\\\\otimes J̉)(\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j),W^*(\\\\eta \\\\otimes U_1\\\\eta _i\\\\otimes \\\\xi _j)\\\\rangle \\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ U_1\\\\eta _i(\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{U_1\\\\eta _i}(h) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ \\\\eta _i(\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ \\\\Delta _1(h^{-1}) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h^{-1}) \\\\ \\\\xi _j(t(\\\\alpha _{h^{-1}\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h)^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h)^{-1}g}(s)t) \\\\ \\\\Delta _1(h^{-1}) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiiint \\\\Delta (\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h)g,s) \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\ \\\\xi _j(t(\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{-1})\\\\\\\\&\\\\times \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h)^{-1/2} \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h^{-1})^{-1}g,s) \\\\ \\\\overline{\\\\eta _i}(h) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h^{-1})^{-1}g}(s)t) \\\\ dg \\\\ ds \\\\ dh \\\\ dt\\\\\\\\&=\\\\iiint \\\\bigg (\\\\int \\\\xi _j(t(\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{-1}) \\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{\\\\beta _t(h^{-1})^{-1}g}(s)t) \\\\ \\\\overline{\\\\eta }(\\\\beta _t(h^{-1})^{-1}g,s) \\\\ dt\\\\bigg )\\\\\\\\&\\\\times \\\\Delta (\\\\alpha _{h\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\eta (\\\\beta _{\\\\alpha _{\\\\beta _s(g)}(s^{-1})}(h)g,s) \\\\ \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),h)^{-1/2} \\\\ (v_{\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}}\\\\eta _i)(h) \\\\overline{\\\\eta _i}(h) \\\\ dg \\\\ ds \\\\ dh\\\\\\\\&\\\\rightarrow \\\\iiint \\\\xi _j(t(\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1})\\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{g}(s)t) \\\\ |\\\\eta (g,s)|^2 \\\\ \\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\ \\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),e)^{-1/2} \\\\ dg \\\\ ds \\\\ dt\\\\\\\\$ by ().\",\n \"But $\\\\alpha _{\\\\beta _s(g)}(s^{-1})^{-1}=\\\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2}\\\\Psi (\\\\alpha _{\\\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{1/2} \\\\Delta (\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1/2}\\\\Delta _2(\\\\alpha _{\\\\beta _s(g)}(s^{-1}))^{-1/2}\\\\\\\\&=\\\\Delta _2(\\\\alpha _g(s))^{1/2}, \\\\ \\\\ \\\\ a.e.$ The final integral in the above calculation therefore reduces to $\\\\iiint \\\\Delta _2(\\\\alpha _g(s))^{1/2} \\\\ \\\\xi _j(t\\\\alpha _{g}(s))\\\\ \\\\overline{\\\\xi _j}(\\\\alpha _{g}(s)t) \\\\ |\\\\eta (g,s)|^2 \\\\ dg \\\\ ds \\\\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\\\iint |\\\\eta (g,s)|^2 \\\\ dg \\\\ ds = \\\\Vert \\\\eta \\\\Vert _{L^2(G_1\\\\times G_2)}^2.$ Denoting the index sets of $(\\\\eta _i)$ and $(\\\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\\\mathcal {I}:=J\\\\times I^{J}$ and for $I=(j,(i_j)_{j\\\\in J})\\\\in \\\\mathcal {I}$ we let $\\\\xi _I:=U_1\\\\eta _{i_j}\\\\otimes \\\\xi _j\\\\in L^2(G_1\\\\times G_2)$ .\",\n \"By [39] and the above analysis, the resulting net $(\\\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ .\",\n \"Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.\",\n \"Then $VN(G_1)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (G_2)$ is strongly inner amenable.\",\n \"Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.\",\n \"For example, $\\\\mathbb {R}^2\\\\rtimes \\\\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\\\mathbb {R}^2$ and $\\\\mathbb {F}_6$ are.\",\n \"Example 3.14 We consider a discretized version of [26].\",\n \"Let $G=\\\\bigg \\\\lbrace \\\\begin{pmatrix} a & b & x\\\\\\\\ c & d & y\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\mid \\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}\\\\in SL(2,\\\\mathbb {Q}), \\\\ x,y\\\\in \\\\mathbb {Q}\\\\bigg \\\\rbrace ,$ $G_1=(\\\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\\\mathbb {Q})$ , viewed as discrete groups.\",\n \"The embeddings $G_1\\\\ni (x,y)\\\\mapsto \\\\begin{pmatrix} 1 & 0 & -x\\\\\\\\ -x & 1 & -y+\\\\frac{1}{2}x^2\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\in G, \\\\ \\\\ G_2\\\\ni \\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}\\\\mapsto \\\\begin{pmatrix}d & -b & 0\\\\\\\\ -c & a & 0\\\\\\\\ 0 & 0 & 1\\\\end{pmatrix}\\\\in G$ determine a matched pair structure for $(G_1,G_2)$ .\",\n \"The corresponding actions are given by $\\\\alpha _{(x,y)}\\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}&=\\\\begin{pmatrix}a+bx & b\\\\\\\\ c+dx-(a+bx)(ax+b(y+\\\\frac{1}{2}x^2)) & d-b(ax+b(y+\\\\frac{1}{2}x^2))\\\\end{pmatrix} \\\\\\\\\\\\beta _{\\\\begin{pmatrix}a & b\\\\\\\\ c & d\\\\end{pmatrix}}(x,y)&=\\\\bigg (ax+by+\\\\frac{b}{2}x^2,cx+d\\\\bigg (y+\\\\frac{1}{2}x^2\\\\bigg )-\\\\frac{1}{2}\\\\bigg (ax+b\\\\bigg (y+\\\\frac{1}{2}x^2\\\\bigg )\\\\bigg )^2\\\\bigg ).$ By Corollary REF , the bicrossed product $\\\\mathbb {G}:=VN(\\\\mathbb {Q}^2)^\\\\beta \\\\bowtie _\\\\alpha L^\\\\infty (SL(2,\\\\mathbb {Q}))$ is strongly inner amenable.\",\n \"Since $SL(2,\\\\mathbb {Q})$ is not amenable, it follows from [26] that $\\\\mathbb {G}$ is not amenable.\",\n \"Moreover, since $\\\\mathbb {Q}^2$ is not compact, $\\\\mathbb {G}$ is not discrete by [65].\",\n \"Thus, $\\\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group.\"\n ],\n [\n \"Relative Injectivity and the Averaging Technique\",\n \"In [56] Ruan and Xu showed that the dual $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\\\mathbb {G}$ is relatively 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}$ .\",\n \"We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.\",\n \"Below we let $\\\\widetilde{\\\\mathbb {G}}:=\\\\mathrm {Gr}()$ denote the intrinsic group of $$ .\",\n \"Proposition 4.1 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Consider the following conditions: $$ is inner amenable; $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ ; $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ ; $\\\\widetilde{}=\\\\mathrm {Gr}(\\\\mathbb {G})$ is inner amenable.\",\n \"Then $(i)\\\\Rightarrow (ii)\\\\Leftrightarrow (iii)\\\\Rightarrow (iv)$ .\",\n \"When $\\\\mathbb {G}$ is co-commutative, the conditions are equivalent.\",\n \"$(i)\\\\Rightarrow (ii)$ : Given a state $n\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ which is right $\\\\widehat{\\\\lhd }$ invariant, it follows that $m:=n\\\\circ R$ is left $\\\\widehat{\\\\rhd }^{\\\\prime }$ invariant.\",\n \"It suffices to provide a completely contractive morphism which is a left inverse to the map $\\\\Delta :L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ given by $\\\\Delta (x)(f)=x\\\\rhd f, \\\\ \\\\ \\\\ T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Identifying $\\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))\\\\cong L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ via $\\\\langle \\\\Phi ,f\\\\otimes g\\\\rangle =\\\\langle \\\\Phi (f),g\\\\rangle , \\\\ \\\\ \\\\ \\\\Phi \\\\in \\\\mathcal {CB}(L^1(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})), \\\\ f,g\\\\in L^1(\\\\mathbb {G}),$ we have $\\\\Delta =\\\\Gamma $ , and that the corresponding $L^1(\\\\mathbb {G})$ -module structure on $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ is defined by $X\\\\unrhd f=(f\\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(\\\\Gamma ^r\\\\otimes \\\\textnormal {id})(X)$ for $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ and $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"First, consider the map $\\\\Phi :\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\ni A\\\\mapsto (\\\\textnormal {id}\\\\otimes m)(V^*AV)\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G})).$ Clearly, $\\\\Phi $ is a completely contractive left inverse to $\\\\Gamma $ .\",\n \"We show that $\\\\Phi $ is a right $\\\\mathcal {T}_\\\\rhd $ -module map.\",\n \"This will complete the proof since [14] will entail the invariance $\\\\Phi (L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ , and the restricted module action $\\\\mathcal {T}_\\\\rhd \\\\curvearrowright L^{\\\\infty }(\\\\mathbb {G})$ is the pertinent $L^1(\\\\mathbb {G})$ -module action.\",\n \"To this end, fix $A\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ and $\\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ .\",\n \"Then $\\\\Phi (A\\\\unrhd \\\\rho )&=\\\\Phi ((\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\\\\\&=(\\\\textnormal {id}\\\\otimes m)(\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\\\\\&=(\\\\textnormal {id}\\\\otimes m)(\\\\rho \\\\otimes \\\\textnormal {id}\\\\otimes \\\\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\\\\\&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\\\prime }=\\\\sigma V^*\\\\sigma $ , where $\\\\sigma $ is the flip map on $L^2(\\\\mathbb {G})\\\\otimes L^2(\\\\mathbb {G})$ , for any $\\\\tau ,\\\\omega \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G}))$ , we have $&\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*(\\\\sigma \\\\otimes 1)V_{23}^*A_{23}V_{23}(\\\\sigma \\\\otimes 1)V_{23}),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\\\omega \\\\otimes \\\\tau \\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes m)(V^*(1\\\\otimes (\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV))V),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle ( m\\\\otimes \\\\textnormal {id})(V̉^{\\\\prime }((\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV)\\\\otimes 1)V̉^{\\\\prime *}),\\\\omega \\\\rangle \\\\\\\\&=\\\\langle m,\\\\omega \\\\widehat{\\\\rhd }^{\\\\prime }((\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV))\\\\rangle \\\\\\\\&=\\\\langle m,(\\\\tau \\\\otimes \\\\textnormal {id})(V^*AV)\\\\rangle \\\\langle \\\\omega ,1\\\\rangle \\\\\\\\&=\\\\langle (\\\\textnormal {id}\\\\otimes m\\\\otimes \\\\textnormal {id})(V^*AV\\\\otimes 1),\\\\tau \\\\otimes \\\\omega \\\\rangle \\\\\\\\&=\\\\langle \\\\Phi (A)\\\\otimes 1,\\\\tau \\\\otimes \\\\omega \\\\rangle .$ As $\\\\tau $ and $\\\\omega $ were arbitrary, we have $\\\\Phi (A\\\\unrhd \\\\rho )&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\textnormal {id}\\\\otimes \\\\textnormal {id}\\\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\\\\\&=(\\\\rho \\\\otimes \\\\textnormal {id})(V(\\\\Phi (A)\\\\otimes 1)V^*)\\\\\\\\&=\\\\Phi (A)\\\\rhd \\\\rho .$ $(ii)\\\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\\\mathbb {G})$ -module left inverse $\\\\Phi $ to $\\\\Gamma $ , it follows that $R\\\\circ \\\\Phi \\\\circ \\\\Sigma \\\\circ (R\\\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\\\mathbb {G})$ -module left inverse to $\\\\Gamma $ .\",\n \"$(iii)\\\\Rightarrow (iv)$ : Recall that $\\\\mathrm {Gr}(\\\\mathbb {G})$ is a group of unitaries in $L^{\\\\infty }(\\\\mathbb {G})$ , so it acts naturally on $L^{\\\\infty }(\\\\mathbb {G})$ by conjugation.\",\n \"The existence of a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ which is $\\\\mathrm {Gr}(\\\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\\\Gamma (L^{\\\\infty }(\\\\mathbb {G}))-L^{\\\\infty }(\\\\mathbb {G})$ -bimodule property of $\\\\Phi $ .\",\n \"Since $\\\\widetilde{\\\\widehat{\\\\mathbb {G}}}$ is a closed quantum subgroup of $\\\\widehat{\\\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\\\gamma :L^{\\\\infty }(\\\\widehat{\\\\widetilde{}})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ intertwining the co-multiplications.\",\n \"As $L^{\\\\infty }(\\\\widehat{\\\\widetilde{}})=VN(\\\\widetilde{\\\\widehat{\\\\mathbb {G}}})=VN(\\\\mathrm {Gr}(\\\\mathbb {G}))$ , the state $m\\\\circ \\\\gamma \\\\in VN(\\\\mathrm {Gr}(\\\\mathbb {G}))^*$ is $\\\\mathrm {Gr}(\\\\mathbb {G})$ -invariant, making $\\\\mathrm {Gr}(\\\\mathbb {G})$ inner amenable by [16].\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, then $\\\\mathrm {Gr}(\\\\mathbb {G}_s)=G$ and the implication $(iv)\\\\Rightarrow (i)$ follows immediately from [16].\",\n \"Corollary 4.2 Let $\\\\mathbb {G}$ be a locally compact quantum group for which $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is an injective von Neumann algebra.\",\n \"Then the following are equivalent: $\\\\mathbb {G}$ is amenable; $\\\\mathbb {G}$ is inner amenable; $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is relatively 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Propositions REF and REF yield the implications $(i)\\\\Rightarrow (ii)\\\\Rightarrow (iii)$ .\",\n \"Assume $(iii)$ .\",\n \"Since $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $\\\\mathbf {mod}\\\\hspace{2.0pt}it follows from \\\\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.\",\n \"Hence, $ G$ is amenable by \\\\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\\\mathbb {G}$ and amenability of $$ .\",\n \"One of their main results is the following.\",\n \"Theorem 4.3 (Ng–Viselter) Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Consider the following conditions: $\\\\mathbb {G}$ is co-amenable; $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $\\\\rho \\\\in C_0(\\\\mathbb {G})^*$ such that $\\\\langle \\\\rho ,x\\\\widehat{\\\\lhd } \\\\hat{f}\\\\rangle =\\\\langle \\\\rho ,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in C_0(\\\\mathbb {G}).$ $$ is amenable.\",\n \"Then $(a)\\\\Rightarrow (b)\\\\Rightarrow (c)$ .\",\n \"Moreover, if $$ has trivial scaling group (for instance, if $\\\\mathbb {G}$ is a Kac algebra), then $(a)\\\\Rightarrow (b^{\\\\prime })\\\\Rightarrow (b)$ , where $C_0(\\\\mathbb {G})$ is nuclear and has a tracial state.\",\n \"They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .\",\n \"We now show that $(b)$ is indeed equivalent to $(a)$ .\",\n \"Theorem 4.4 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.\",\n \"Suppose that $C_0(\\\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.\",\n \"Given a state $m\\\\in C_0(\\\\mathbb {G})^*$ which is right $\\\\widehat{\\\\lhd }$ invariant, it follows that $n:=m\\\\circ R$ is left $\\\\widehat{\\\\rhd }^{\\\\prime }$ invariant, as in the proof of Proposition REF .\",\n \"Let $n\\\\in M(C_0(\\\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\\\textnormal {id}\\\\otimes n):M(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ denote the unique extension of the slice map $(\\\\textnormal {id}\\\\otimes n):\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {K}(L^2(\\\\mathbb {G}))$ which is strictly continuous on the unit ball.\",\n \"By strict density of $C_0(\\\\mathbb {G})$ in $M(C_0(\\\\mathbb {G}))$ , it follows that $\\\\langle n,\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle n,x\\\\rangle \\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in M(C_0(\\\\mathbb {G})).$ Since $V\\\\in M(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))$ , the map $\\\\Phi :C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ defined by $\\\\Phi (A)=(\\\\textnormal {id}\\\\otimes n)(V^*AV), \\\\ \\\\ \\\\ A\\\\in C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.\",\n \"Using the extended $\\\\widehat{\\\\rhd }^{\\\\prime }$ -invariance on $M(C_0(\\\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\\\Phi (A\\\\unrhd \\\\rho )=\\\\Phi (A)\\\\rhd \\\\rho , \\\\ \\\\ \\\\ A\\\\in C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}), \\\\ \\\\rho \\\\in \\\\mathcal {T}(L^2(\\\\mathbb {G})).$ Since $\\\\mathrm {Ad}(V^*):\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G})$ and $\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G})=(\\\\mathcal {K}(L^2(\\\\mathbb {G}))\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))^*,$ we have $\\\\mathrm {Ad}(V^*)^*:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ Letting $r:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\rho \\\\mapsto \\\\rho |_{C_0(\\\\mathbb {G})}\\\\in M(\\\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\\\Phi ^*|_{\\\\mathcal {T}(L^2(\\\\mathbb {G}))}:\\\\mathcal {T}(L^2(\\\\mathbb {G}))\\\\ni \\\\rho \\\\mapsto (r\\\\otimes \\\\textnormal {id})(\\\\mathrm {Ad}(V^*)^*(\\\\rho \\\\otimes n))\\\\in M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ The proof of [14] entails the inclusion $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"In fact, since $\\\\Phi $ is a right $L^1(\\\\mathbb {G})$ -module map and $C_0(\\\\mathbb {G})$ is essential, more is true: $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq \\\\mathrm {LUC}(\\\\mathbb {G})\\\\subseteq M(C_0(\\\\mathbb {G})).$ The unique strict extension $\\\\widetilde{\\\\Phi }:M(C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\rightarrow M(C_0(\\\\mathbb {G}))$ , which exists by [45], satisfies $\\\\widetilde{\\\\Phi }\\\\circ \\\\Gamma |_{C_0(\\\\mathbb {G})}=\\\\textnormal {id}_{C_0(\\\\mathbb {G})}$ .\",\n \"If $\\\\rho \\\\in L^{\\\\infty }(\\\\mathbb {G})_{\\\\perp }$ then the invariance $\\\\Phi (C_0(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})$ implies $\\\\Phi ^*(\\\\rho )=0$ .\",\n \"Thus, $\\\\Phi ^*$ induces a completely positive left $L^1(\\\\mathbb {G})$ -module map $\\\\Phi ^*:L^1(\\\\mathbb {G})=(\\\\mathcal {T}(L^2(\\\\mathbb {G}))/L^{\\\\infty }(\\\\mathbb {G})_{\\\\perp })\\\\rightarrow M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\\\mathbb {G})$ , for $f\\\\in L^1(\\\\mathbb {G})$ and $x\\\\in C_0(\\\\mathbb {G})$ we have $\\\\langle m_{M(\\\\mathbb {G})}(\\\\Phi ^*(f)),x\\\\rangle =\\\\langle \\\\Phi ^*(f),\\\\Gamma (x)\\\\rangle =\\\\langle f,\\\\widetilde{\\\\Phi }(\\\\Gamma (x))\\\\rangle =\\\\langle f,x\\\\rangle .$ Hence, $m_{M(\\\\mathbb {G})}\\\\circ \\\\Phi ^*$ is the canonical inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ .\",\n \"Now, by nuclearity of $C_0(\\\\mathbb {G})$ , there exists a net $\\\\varphi _i:C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.\",\n \"Define $\\\\psi _i:L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ by $\\\\psi _i=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi ^*.$ Then $\\\\psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}(L^1(\\\\mathbb {G}),M(\\\\mathbb {G}))=_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}(L^1(\\\\mathbb {G}))$ .\",\n \"By [20] there exist contractive positive functionals $\\\\nu _i\\\\in C_u(\\\\mathbb {G})^*$ such that $\\\\psi _i=m^r_{\\\\nu _i}$ .\",\n \"Since $\\\\varphi ^*_i:M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\\\mathbb {G})}$ is separately weak* continuous, it follows that $\\\\psi _i$ converges to the inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ point-weak*.\",\n \"Write $\\\\varphi _i=\\\\sum _{k=1}^{n_i}x_k^i\\\\otimes \\\\mu _k^i$ for some $x_k^i\\\\in C_0(\\\\mathbb {G})$ and $\\\\mu _k^i\\\\in M(\\\\mathbb {G})$ .\",\n \"Since $(b)\\\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.\",\n \"Hence, by [14] $M^r_{cb}(L^1(\\\\mathbb {G}))= _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))=C_u(\\\\mathbb {G})^*$ .\",\n \"For each $i$ and $k$ the map $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*\\\\in _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ , so there exists $\\\\nu _k^i$ such that $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*=m^r_{\\\\nu _k^i}$ .\",\n \"Thus, for $f\\\\in L^1(\\\\mathbb {G})$ we have $\\\\psi _i(f)&=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi ^*(f)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}((\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi ^*(f)\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}(f\\\\star \\\\nu _k^i\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}f\\\\star \\\\nu _k^i\\\\star \\\\mu _k^i.$ Hence, $\\\\nu _i=\\\\sum _{k=1}^{n_i}\\\\nu _k^i\\\\star \\\\mu _k^i\\\\in M(\\\\mathbb {G})$ as $M(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Passing to a subnet we may assume $\\\\nu _i\\\\rightarrow \\\\nu $ weak* in $M(\\\\mathbb {G})$ .\",\n \"But then for any $f\\\\in L^1(\\\\mathbb {G})$ and $x\\\\in C_0(\\\\mathbb {G})$ we have $\\\\langle f\\\\star \\\\nu , x\\\\rangle =\\\\langle \\\\nu , x\\\\star f\\\\rangle =\\\\lim _i\\\\langle \\\\nu _i,x\\\\star f\\\\rangle =\\\\lim _i\\\\langle f\\\\star \\\\nu _i,x\\\\rangle =\\\\langle f,x\\\\rangle .$ It follows that $\\\\nu $ is a right identity for $M(\\\\mathbb {G})$ , which implies that $M(\\\\mathbb {G})$ is unital, whence $\\\\mathbb {G}$ is co-amenable (cf.\",\n \"[5]).\",\n \"Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\\\lambda (G)$ is nuclear and has a tracial state.\",\n \"Corollary 4.5 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\\\mathbb {G}$ is a Kac algebra).\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and has a tracial state.\",\n \"The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ such that $\\\\langle m,x\\\\widehat{\\\\lhd } f̉\\\\rangle =\\\\langle m,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}),$ while $\\\\mathbb {G}$ is co-amenable if and only if $C_0(\\\\mathbb {G})$ is nuclear and there exists a state $m\\\\in C_0(\\\\mathbb {G})^*$ such that $\\\\langle m,x\\\\widehat{\\\\lhd } f̉\\\\rangle =\\\\langle m,x\\\\rangle \\\\langle \\\\hat{f},1\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}}), \\\\ x\\\\in C_0(\\\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\\\mathbb {G})$ is 1-flat [14], while $\\\\mathbb {G}$ is co-amenable if and only if $M(\\\\mathbb {G})$ is 1-projective, as we now prove.\",\n \"Theorem 4.6 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $M(\\\\mathbb {G})$ is 1-projective in $M(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ .\",\n \"If $\\\\mathbb {G}$ is co-amenable then $M(\\\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.\",\n \"Any unital completely contractive Banach algebra $\\\\mathcal {A}$ is $\\\\Vert e_{\\\\mathcal {A}}\\\\Vert $ -projective, so the claim follows.\",\n \"Conversely, if $M(\\\\mathbb {G})$ is 1-projective, then $C_0(\\\\mathbb {G})^{**}=M(\\\\mathbb {G})^*$ is 1-injective in $\\\\mathbf {mod}-M(\\\\mathbb {G})$ .\",\n \"It follows that the inclusion morphism $M(C_0(\\\\mathbb {G}))\\\\hookrightarrow C_0(\\\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\\\mathbb {G})$ -module map $\\\\Phi : \\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow C_0(\\\\mathbb {G})^{**}$ .\",\n \"Applying the argument of [51] we see that $(\\\\textnormal {id}\\\\otimes \\\\Phi ):C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }C_0(\\\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\\\textnormal {id}\\\\otimes \\\\Phi )(W̉)=W̉$ .\",\n \"By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\\\textnormal {id}\\\\otimes \\\\Phi )$ , and hence $(\\\\textnormal {id}\\\\otimes \\\\Phi )(W̉^*XW̉)=W̉^*(\\\\textnormal {id}\\\\otimes \\\\Phi )(X)W̉, \\\\ \\\\ \\\\ X\\\\in C_0(\\\\mathbb {G})^{**}\\\\overline{\\\\otimes }\\\\mathcal {B}(L^2(\\\\mathbb {G})).$ In particular, for every $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ and $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ we have $\\\\Phi (T\\\\hat{f})&=\\\\Phi ((\\\\hat{f}\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes T)W̉))=(\\\\hat{f}\\\\otimes \\\\textnormal {id})(\\\\textnormal {id}\\\\otimes \\\\Phi )((W̉^*(1\\\\otimes T)W̉))\\\\\\\\&=(\\\\hat{f}\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes \\\\Phi (T))W̉)=\\\\Phi (T)\\\\hat{f}.$ Thus, $\\\\Phi $ is a morphism with respect to the canonical $L^1(\\\\widehat{\\\\mathbb {G}})$ -module structure on $C_0(\\\\mathbb {G})^{**}$ .\",\n \"Moreover, the $M(\\\\mathbb {G})$ -module property entails $\\\\pi \\\\circ \\\\Phi (L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}))=$ , where $\\\\pi :C_0(\\\\mathbb {G})^{**}\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ is the canonical surjection.\",\n \"Since $\\\\pi $ is clearly an $L^1(\\\\widehat{\\\\mathbb {G}})$ -module map, it follows that $\\\\pi \\\\circ \\\\Phi |_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\\\mathbb {G})$ .\",\n \"Now, by 1-projectivity of $M(\\\\mathbb {G})$ , for every $\\\\varepsilon >0$ there exists a morphism $\\\\Phi _\\\\varepsilon :M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})^+\\\\widehat{\\\\otimes }M(\\\\mathbb {G})$ such that $m^+\\\\circ \\\\Phi _\\\\varepsilon =\\\\textnormal {id}_{M(\\\\mathbb {G})}$ , and $\\\\Vert \\\\Phi _\\\\varepsilon \\\\Vert _{cb}<1+\\\\varepsilon $ .\",\n \"By amenability of $$ , we know $C_u(\\\\mathbb {G})^*= _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\\\Phi _\\\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\\\mu _i)$ in $M(\\\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\\\mathbb {G}$ .\",\n \"The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.\",\n \"In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ to the co-multiplication that is an $L^1(\\\\mathbb {G})$ -module map.\",\n \"Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow \\\\Gamma (L^{\\\\infty }(\\\\mathbb {G}))$ onto the image of $\\\\Gamma $ (see [23] and [41]).\",\n \"It is the combination of the $L^1(\\\\mathbb {G})$ -module property of $\\\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\\\infty }(\\\\mathbb {G})$ or $C_0(\\\\mathbb {G})$ to approximation properties of $$ .\",\n \"Thus, provided one has a suitably nice $L^1(\\\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.\",\n \"This is where inner amenability enters the picture.\",\n \"Recall that a locally compact quantum group $\\\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\\\widehat{\\\\mathbb {G}}))$ .\",\n \"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\\\mathbb {G}$ , and is denoted $\\\\Lambda _{cb}(\\\\mathbb {G})$ .\",\n \"We say that $\\\\mathbb {G}$ has the approximation property if there exists a net $(\\\\hat{f}_i)$ in $L^1(\\\\widehat{\\\\mathbb {G}})$ such that $^l(\\\\hat{\\\\lambda }(\\\\hat{f}_i))$ converges to $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ in the stable point-weak* topology.\",\n \"Proposition 4.7 Let $\\\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.\",\n \"If $L^{\\\\infty }(\\\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\\\Lambda _{cb}()\\\\le \\\\Lambda _{cb}(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"$L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.\",\n \"Let $(\\\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\\\mathbb {G})$ .\",\n \"It follows verbatim from [56] that $\\\\Phi _i:L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\ni X\\\\mapsto (\\\\omega _{\\\\xi _i}\\\\otimes \\\\textnormal {id})(W^*(U^*\\\\otimes 1)X(U\\\\otimes 1)W)\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\\\mathbb {G})$ -module maps, which cluster weak* in $\\\\mathcal {CB}(L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=((L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}))^*$ to a module left inverse to $\\\\Gamma $ .\",\n \"Passing to a subnet we may assume convergence.\",\n \"$(i)$ Let $(\\\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\\\Vert \\\\varphi _j\\\\Vert _{cb}\\\\le C$ .\",\n \"Since $\\\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\\\in L^1(\\\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\\\in L^{\\\\infty }(\\\\mathbb {G})$ such that $\\\\varphi _j=\\\\sum _{k=1}^{n_j} x^j_k\\\\otimes f^j_k$ .\",\n \"Put $\\\\Phi _{ij}=\\\\Phi _i\\\\circ (\\\\varphi _j\\\\otimes \\\\textnormal {id})\\\\circ \\\\Gamma :L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G}).$ Then $\\\\Phi _{ij}$ is a normal completely bounded left $L^1(\\\\mathbb {G})$ -module map with $\\\\Vert \\\\Phi _{ij}\\\\Vert _{cb}\\\\le C$ .\",\n \"Also, for each $i,j$ and $1\\\\le k\\\\le n_j$ , the map $L^{\\\\infty }(\\\\mathbb {G})\\\\ni x\\\\mapsto \\\\Phi _i(x^j_k\\\\otimes x)\\\\in L^{\\\\infty }(\\\\mathbb {G})$ is a normal completely bounded left $L^1(\\\\mathbb {G})$ -module map.\",\n \"Since $x^j_k$ is a linear combination of positive elements in $L^{\\\\infty }(\\\\mathbb {G})$ , and $\\\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\\\mathbb {G})$ -module maps $L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"By [20], there exist $\\\\mu ^{ij}_k\\\\in C_u(\\\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\\\mu ^{ij}_k$ .\",\n \"Hence, for each $x\\\\in L^{\\\\infty }(\\\\mathbb {G})$ $\\\\Phi _{ij}(x)&=\\\\Phi _i\\\\circ (\\\\varphi _j\\\\otimes \\\\textnormal {id})\\\\circ \\\\Gamma (x)\\\\\\\\&=\\\\sum _{k=1}^{n_j}\\\\Phi _i(x^j_k \\\\otimes x\\\\star f^j_k)\\\\\\\\&=\\\\sum _{k=1}^{n_j}x\\\\star f^j_k\\\\star \\\\mu ^{ij}_k\\\\\\\\&=x\\\\star f_{ij},$ where $f_{ij}=\\\\sum _{k=1}^{n_i} f^j_k\\\\star \\\\mu ^{ij}_k\\\\in L^1(\\\\mathbb {G})$ as $L^1(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Thus, $\\\\Vert f_{ij}\\\\Vert _{cb}\\\\le C$ , and it follows that $\\\\lim _i\\\\lim _jx\\\\star f_{ij}=\\\\lim _i\\\\Phi _i(\\\\Gamma (x))=x,$ point-weak* in $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\\\Lambda _{cb}()\\\\le \\\\Lambda _{cb}(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.\",\n \"[34]).\",\n \"The converse was shown for Kac algebras in [41].\",\n \"Their proof extends verbatim to arbitrary locally compact quantum groups.\",\n \"Corollary 4.8 Let $G$ be an inner amenable locally compact group.\",\n \"If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.\",\n \"$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.\",\n \"Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\\\mathbb {G})$ -module left inverse to $\\\\Gamma $ that can be approximated by normal $L^1(\\\\mathbb {G})$ -module maps.\",\n \"It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .\",\n \"Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.\",\n \"That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?\",\n \"By [34] it is known that in this case $\\\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\\\mathcal {CB}^\\\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\\\mathcal {CB}^\\\\sigma (VN(G))$ .\",\n \"Proposition 4.11 Let $\\\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.\",\n \"If $C_0(\\\\mathbb {G})$ has the CBAP, then there exists a net $(\\\\nu _i)$ in $M(\\\\mathbb {G})$ such that $\\\\Vert \\\\nu _i\\\\Vert _{cb}\\\\le \\\\Lambda _{cb}(C_0(\\\\mathbb {G}))$ and $\\\\nu _i\\\\rightarrow 1$ $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ .\",\n \"If $C_0(\\\\mathbb {G})$ has the OAP then $M(\\\\mathbb {G})$ is $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"Let $\\\\Phi :L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})\\\\widehat{\\\\otimes }M(\\\\mathbb {G})$ be the completely positive left $L^1(\\\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.\",\n \"Recall that $m_{M(\\\\mathbb {G})}\\\\circ \\\\Phi $ is the canonical inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ .\",\n \"$(i)$ Let $\\\\varphi _i:C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\\\Vert \\\\varphi _i\\\\Vert _{cb}\\\\le C$ .\",\n \"Write $\\\\varphi _i=\\\\sum _{k=1}^{n_i}x_k^i\\\\otimes \\\\mu _k^i$ for some $x_k^i\\\\in C_0(\\\\mathbb {G})$ and $\\\\mu _k^i\\\\in M(\\\\mathbb {G})$ .\",\n \"For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\\\mathbb {G})$ , and hence the map $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi \\\\in _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(L^1(\\\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\\\mathbb {G})$ -module maps on $L^1(\\\\mathbb {G})$ , so by [20] there exists $\\\\nu _k^i$ such that $(\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi =m^r_{\\\\nu _k^i}$ .\",\n \"Define $\\\\psi _i:L^1(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ by $\\\\psi _i=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi $ .\",\n \"Since $\\\\varphi ^*_i:M(\\\\mathbb {G})\\\\rightarrow M(\\\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\\\mathbb {G})}$ is separately weak* continuous, it follows that $\\\\psi _i$ converges to the inclusion $L^1(\\\\mathbb {G})\\\\hookrightarrow M(\\\\mathbb {G})$ point-weak*.\",\n \"Moreover, for $f\\\\in L^1(\\\\mathbb {G})$ we have $\\\\psi _i(f)&=m_{M(\\\\mathbb {G})}\\\\circ (\\\\textnormal {id}\\\\otimes \\\\varphi _i^*)\\\\circ \\\\Phi (f)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}((\\\\textnormal {id}\\\\otimes x_k^i)\\\\Phi (f)\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}m_{M(\\\\mathbb {G})}(f\\\\star \\\\nu _k^i\\\\otimes \\\\mu _k^i)\\\\\\\\&=\\\\sum _{k=1}^{n_i}f\\\\star \\\\nu _k^i\\\\star \\\\mu _k^i\\\\\\\\&=f\\\\star \\\\nu _i,$ where $\\\\nu _i=\\\\sum _{k=1}^{n_i}\\\\nu _k^i\\\\star \\\\mu _k^i\\\\in M(\\\\mathbb {G})$ as $M(\\\\mathbb {G})$ is a closed ideal in $C_u(\\\\mathbb {G})^*$ .\",\n \"Then $\\\\Vert \\\\nu _i\\\\Vert _{cb}\\\\le C$ , and it follows that $\\\\nu _i\\\\star x\\\\rightarrow x$ weak* in $L^{\\\\infty }(\\\\mathbb {G})$ for all $x\\\\in C_0(\\\\mathbb {G})$ .\",\n \"Since $(\\\\nu _i)$ is bounded in $\\\\mathcal {CB}_{L^1(\\\\mathbb {G})}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ , we have $\\\\nu _i\\\\rightarrow 1$ $\\\\sigma (M_{cb}^r(L^1(\\\\mathbb {G})),Q_{cb}^r(L^1(\\\\mathbb {G})))$ .\",\n \"$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\\\mathcal {A}$ , we say that an operator $\\\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\\\rightarrow Y\\\\hookrightarrow Z\\\\twoheadrightarrow Z/Y\\\\rightarrow 0$ in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ , the sequence $0\\\\rightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y\\\\hookrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z\\\\twoheadrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z/Y\\\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\\\widehat{\\\\otimes }Y/_{\\\\mathcal {A}}N_{\\\\mathcal {A}}$ , where $_{\\\\mathcal {A}}N_{\\\\mathcal {A}}=\\\\langle a\\\\cdot x\\\\otimes y - x\\\\otimes y\\\\cdot a, \\\\ x^{\\\\prime }\\\\cdot a^{\\\\prime }\\\\otimes y^{\\\\prime } - x^{\\\\prime }\\\\otimes a^{\\\\prime }\\\\cdot y^{\\\\prime }\\\\mid a,a^{\\\\prime }\\\\in \\\\mathcal {A}, \\\\ x,x^{\\\\prime }\\\\in X, \\\\ y,y^{\\\\prime }\\\\in Y\\\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\\\widehat{\\\\mathbb {G}})$ for any Kac algebra $\\\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\\\sigma V \\\\sigma $ and for which $\\\\omega _{\\\\xi _i}|_{L^{\\\\infty }(\\\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\\\mathbb {G})$ .\",\n \"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).\",\n \"We now obtain the same conclusion under a priori weaker hypotheses.\",\n \"Proposition 5.1 Let $\\\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Then $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse holds.\",\n \"It suffices to provide a completely contractive $L^1(\\\\mathbb {G})$ -bimodule map $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ which is a left inverse to $\\\\Gamma $ .\",\n \"Defining $\\\\Phi (X)=(\\\\textnormal {id}\\\\otimes m)(V^*XV)$ , $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\\\Phi $ is a completely contractive right $L^1(\\\\mathbb {G})$ -module map and $\\\\Phi \\\\circ \\\\Gamma =\\\\textnormal {id}_{L^{\\\\infty }(\\\\mathbb {G})}$ .\",\n \"However, since $m$ is also a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ , the module argument from [15] shows that $\\\\Phi $ is also a left $L^1(\\\\mathbb {G})$ -module map.\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.\",\n \"Owing to the fact that $V_s=W_a$ we have $f\\\\widehat{\\\\rhd }_{s}^{\\\\prime }x&=(\\\\textnormal {id}\\\\otimes f)(V̉^{\\\\prime }_s(x\\\\otimes 1)V̉_s^{\\\\prime *})=(f\\\\otimes \\\\textnormal {id})(V_s^*(1\\\\otimes x)V_s)\\\\\\\\&=(f\\\\otimes \\\\textnormal {id})(W_a^*(1\\\\otimes x)W_a)=x\\\\lhd _a f$ for all $f\\\\in L^1(G)$ and $x\\\\in VN(G)$ .\",\n \"Hence, $m$ satisfies (REF ).\",\n \"A completely contractive Banach algebra $\\\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ for some $C\\\\ge 1$ , and has a bounded approximate identity.\",\n \"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ , that is, a bounded net $(A_\\\\alpha )$ in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ satisfying $a\\\\cdot A_\\\\alpha - A_\\\\alpha \\\\cdot a, \\\\ m_{\\\\mathcal {A}}(A_\\\\alpha )\\\\cdot a \\\\rightarrow 0, \\\\ \\\\ \\\\ a\\\\in \\\\mathcal {A}.$ We let $OA(\\\\mathcal {A})$ denote the operator amenability constant of $\\\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ .\",\n \"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.\",\n \"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.\",\n \"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.\",\n \"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.\",\n \"Proposition 5.2 Let $G$ be a locally compact group.\",\n \"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.\",\n \"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.\",\n \"Since $L^{\\\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\\\infty }(G)$ in $L^1(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(G)$ .\",\n \"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ by (the proof of) [54].\",\n \"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.\",\n \"Conversely, if $VN(G)$ is 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}$ , so $G$ is amenable by [14].\",\n \"We now show that 1-biflatness of quantum convolution algebras is a self dual property.\",\n \"Theorem 5.3 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $L^1(\\\\mathbb {G})$ is 1-biflat if and only if $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat.\",\n \"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat, that is, $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Consider the canonical $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule structure on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ given by $\\\\hat{f}\\\\widehat{\\\\rhd }T=(\\\\textnormal {id}\\\\otimes \\\\hat{f})\\\\widehat{V}(T\\\\otimes 1)V̉^* \\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\textnormal {and}\\\\hspace{10.0pt}\\\\hspace{10.0pt}T\\\\widehat{\\\\lhd }\\\\hat{f}=(\\\\hat{f}\\\\otimes \\\\textnormal {id})W̉^*(1\\\\otimes T)W̉,$ for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Then by 1-injectivity, $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ extends to a completely contractive $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule projection $E:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"By the left $\\\\widehat{\\\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\\\infty }(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})\\\\cap L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})=$ .\",\n \"Also, [15] implies that $E$ is a right $\\\\lhd $ -module map.\",\n \"Let $R$ be the extended unitary antipode of $\\\\mathbb {G}$ .\",\n \"Then $(R\\\\otimes R)(V̉^{\\\\prime })=(R\\\\otimes R)(\\\\sigma V^*\\\\sigma )=\\\\Sigma (R\\\\otimes R)(V^*)=\\\\Sigma (J̉\\\\otimes J̉)(V)(J̉\\\\otimes J̉)=\\\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .\",\n \"Let $E_R:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ be the projection of norm one, $E_R=R\\\\circ E\\\\circ R$ .\",\n \"Then for $\\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ , we have $E_R(\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }T)&=R(E(R((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime })V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)(R\\\\otimes R)(V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((R\\\\otimes R)(V̉^{\\\\prime *})(R(T)\\\\otimes 1)(R\\\\otimes R)(V̉^{\\\\prime }))))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((\\\\Sigma W̉^*)(R(T)\\\\otimes 1)(\\\\Sigma W̉))))\\\\\\\\&=R(E((\\\\hat{f}^{\\\\prime }\\\\circ R\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes R(T))W̉)))\\\\\\\\&=R(E(R(T)\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R)))\\\\\\\\&=R(E(R(T))\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R))\\\\\\\\&=\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }E_R(T).$ Thus, $E_R$ is a left $\\\\widehat{\\\\rhd }^{\\\\prime }$ -module map.\",\n \"Since $R(L^{\\\\infty }(\\\\mathbb {G}))=L^{\\\\infty }(\\\\mathbb {G})$ , the restriction $E_R|_{L^{\\\\infty }(\\\\mathbb {G})}$ defines a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,f̉^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ But $E$ was also a right $\\\\lhd $ -module map, which implies that $E_R$ is a left $\\\\rhd $ -module map by the generalized antipode relation (REF ).\",\n \"Thus, we also have $\\\\langle m,f\\\\rhd x\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"By Proposition REF it follows that $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"By 1-injectivity of $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ , there exists a completely contractive morphism $\\\\Phi :L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ which is a left inverse to $$ .\",\n \"It follows that $\\\\Phi |_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\otimes 1}$ defines a state $m̉\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ which is a right $L^1(\\\\widehat{\\\\mathbb {G}})$ -module map, i.e., $$ is amenable.\",\n \"Hence, $L^{\\\\infty }(\\\\mathbb {G})$ is a 1-injective operator space by [5].\",\n \"Proposition REF then implies the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Corollary 5.4 $\\\\ell ^1(\\\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.\",\n \"Since $\\\\ell ^\\\\infty (\\\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\\\mathbb {G}$ , if $\\\\ell ^1(\\\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\\\ell ^\\\\infty (\\\\mathbb {G})$ would be 1-injective in $\\\\ell ^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}\\\\ell ^1(\\\\mathbb {G})$ .\",\n \"Then by Theorem REF $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ would be 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\\\mathbb {G}$ is unimodular.\",\n \"The relative biprojectivity of $L^1(\\\\mathbb {G})$ , that is, relative projectivity of $L^1(\\\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\\\mathbb {G})$ is relatively 1-biprojective if and only if $\\\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].\",\n \"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.\",\n \"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].\",\n \"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\\\Delta $ [16].\",\n \"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .\",\n \"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 5.5 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then the following conditions are equivalent: $\\\\mathbb {G}$ is finite–dimensional.\",\n \"$\\\\mathcal {T}_\\\\rhd $ is relatively 1-biprojective; $L^1(\\\\mathbb {G})$ is 1-biprojective; $(i)\\\\Rightarrow (ii)$ follows from [38].\",\n \"$(ii)\\\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\\\mathbb {G})$ together with the 1-projectivity of $L^1(\\\\mathbb {G})$ as an operator space.\",\n \"The bimodule analogue of [14] then yields $(iii)$ .\",\n \"$(iii)\\\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\\\mathbb {G})$ -bimodule left inverse $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ to $\\\\Gamma $ .\",\n \"As usual, the restriction $\\\\Phi |_{L^{\\\\infty }(\\\\mathbb {G})\\\\otimes 1}:L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.\",\n \"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\\\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.\",\n \"Thus, $ G$ is finite-dimensional by \\\\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\\\mathbb {G})$ For a locally compact quantum group $\\\\mathbb {G}$ , let $L^1_{cb}(\\\\mathbb {G})$ denote the closure of $L^1(\\\\mathbb {G})$ inside $M_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\\\mathbb {G})$ has a bounded approximate identity.\",\n \"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.\",\n \"In [16] the authors gave examples of weakly amenable connected groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.\",\n \"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].\",\n \"Let $\\\\mathbb {G}$ be a compact Kac algebra and let $\\\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\\\mathbb {G})$ , as well as their universal extensions to $C_u(\\\\mathbb {G})$ .\",\n \"As in [7], we define $*$ -homomorphisms $\\\\lambda ,\\\\rho :C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ by $\\\\lambda (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (xy), \\\\ \\\\ \\\\ \\\\rho (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (y R(x)), \\\\ \\\\ \\\\ x,y\\\\in C_u(\\\\mathbb {G}).$ Since $\\\\lambda $ and $\\\\rho $ have commuting ranges, we obtain a canonical representation $\\\\lambda \\\\times \\\\rho :C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .\",\n \"Lemma 6.1 Let $\\\\mathcal {A}$ be a $C^*$ -algebra.\",\n \"There exists a complete isometry $\\\\iota :\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ such that $\\\\iota (\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1})$ is weak* dense in $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ .\",\n \"Let $\\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}$ denote the universal representation of $\\\\mathcal {A}$ .\",\n \"Then the universal cover of the representation $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\\\pi $ of $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{**}$ onto $(\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A})(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }\\\\overline{\\\\otimes }\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^*$ (see [61]).\",\n \"Its pre-adjoint $\\\\iota :=\\\\pi _*:\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ is a complete isometry.\",\n \"Now, Let $F\\\\in (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ , $\\\\Vert F\\\\Vert \\\\le 1$ .\",\n \"Since $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}\\\\subseteq \\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\\\tilde{F}\\\\in (\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**})^*=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^{**}$ satisfying $\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\langle F,A\\\\rangle , \\\\ \\\\ \\\\ A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ such that $f_j\\\\rightarrow \\\\tilde{F}$ weak*.\",\n \"Thus, for all $A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}$ , $\\\\langle F,A\\\\rangle =\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle f_j,\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle \\\\iota (f_j),A\\\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.\",\n \"Then $$ is weakly amenable if and only if $L^1_{cb}(\\\\mathbb {G})$ is operator amenable.\",\n \"Moreover, $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"It is clear that $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].\",\n \"Conversely, by Kirchberg's factorization property the representation $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"Composing with $\\\\omega _{\\\\Lambda _\\\\varphi (1)}$ , we obtain a state $\\\\mu :=\\\\omega _{\\\\Lambda _\\\\varphi (1)}\\\\circ \\\\lambda \\\\times \\\\rho \\\\in (C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ .\",\n \"By $R$ -invariance of $\\\\varphi $ , one can easily verify that $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ .\",\n \"Let $\\\\Gamma _u:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u(\\\\mathbb {G})$ denote the universal co-multiplication.\",\n \"Then, similar to the calculations in [55], for all $u\\\\in \\\\mathrm {Irr}(\\\\mathbb {G})$ , $1\\\\le i,j\\\\le n_u$ , $\\\\langle \\\\Gamma _u^*(\\\\mu )\\\\star f,u_{ij}\\\\rangle &=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\langle \\\\mu ,u_{ik}\\\\otimes u_{kl}\\\\rangle =\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} R(u_{kl}))\\\\\\\\&=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} u_{lk}^{*})=\\\\frac{1}{n_u}\\\\sum _{k=1}^{n_u}\\\\langle f,u_{ij}\\\\rangle \\\\\\\\&=\\\\langle f,u_{ij}\\\\rangle .$ Hence, $\\\\Gamma _u^*(\\\\mu )\\\\star f = f$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"By [42], $R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x^*)(1\\\\otimes y)))=(\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x^*)\\\\Gamma _u(y))$ for all $x,y\\\\in C_u(\\\\mathbb {G})$ , so that $\\\\langle (f\\\\otimes 1)\\\\star \\\\mu ,x\\\\otimes y\\\\rangle &=\\\\varphi ((x\\\\star f) R(y))=(f\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))\\\\\\\\&=f\\\\circ R(R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))))\\\\\\\\&=f\\\\circ R((\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Gamma _u(R(y))))\\\\\\\\&=(f\\\\circ R\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Sigma \\\\circ R\\\\otimes R\\\\circ \\\\Gamma _u(y))\\\\\\\\&=(\\\\varphi \\\\otimes f)(\\\\Gamma _u(y)(R(x)\\\\otimes 1))\\\\\\\\&=\\\\varphi ((f\\\\star y)R(x))\\\\\\\\&=\\\\langle \\\\mu \\\\star (1\\\\otimes f),x\\\\otimes y\\\\rangle .$ Thus, $(f\\\\otimes 1)\\\\star \\\\mu =\\\\mu \\\\star (1\\\\otimes f)$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"Since $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ , we also have $(1\\\\otimes f)\\\\star \\\\mu =\\\\mu \\\\star (f\\\\otimes 1)$ , for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"It follows that $F\\\\star \\\\mu =\\\\mu \\\\star \\\\Sigma F$ for all $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ , where we let $\\\\Sigma $ also denote the flip homomorphism on $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"The proof of Lemma REF implies the existence of a net $(\\\\mu _i)$ of states in $C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*$ such that $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ , and hence in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G}))^*=C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ .\",\n \"Since $m_{C_u(\\\\mathbb {G})^*}=\\\\Gamma _u^*|_{C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*}$ , it follows that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\rightarrow \\\\Gamma _u^*(\\\\mu )$ weak* in $C_u(\\\\mathbb {G})^*$ .\",\n \"By [57] (note that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i),\\\\Gamma _u^*(\\\\mu )$ are states), we have $\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-f\\\\Vert _{L^1(\\\\mathbb {G})}=\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-\\\\Gamma _u^*(\\\\mu )\\\\star f\\\\Vert _{L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}).$ Since $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ , again by [57] we obtain $\\\\Vert F\\\\star \\\\mu _i-\\\\mu _i\\\\star \\\\Sigma F\\\\Vert _{L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}),$ from which we have $&\\\\Vert F\\\\star ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\Vert =\\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-(\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)\\\\Vert +\\\\Vert (\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-\\\\mu _i\\\\star (\\\\Sigma (F\\\\star (1\\\\otimes f)))\\\\Vert +\\\\Vert \\\\mu _i\\\\star \\\\Sigma F-F\\\\star \\\\mu _i\\\\Vert \\\\Vert f\\\\otimes 1\\\\Vert \\\\\\\\&\\\\rightarrow 0$ for every $f\\\\in L^1(\\\\mathbb {G})$ , $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"Similarly, $\\\\Vert ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\star F\\\\Vert \\\\rightarrow 0$ .\",\n \"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\\\mathbb {G})$ in $\\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ such that $\\\\sup _j\\\\Vert f_j\\\\Vert _{cb}<\\\\infty $ .\",\n \"The tensor square of the canonical complete contraction $C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ allows us to view $\\\\mu _i\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))$ with $\\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1$ for all $i$ .\",\n \"By the universal property of the operator space projective tensor product, we may also view each $\\\\mu _i$ as an element of $\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))$ by right multiplication, as well as in $\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))$ by left multiplication.\",\n \"Moreover, $\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))},\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))}\\\\le \\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1.$ Define $\\\\mu _{ij}:=(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)$ .\",\n \"Then $\\\\mu _{ij}\\\\in L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _{ij}\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Vert f_j\\\\Vert _{cb}^2\\\\le \\\\Lambda _{cb}()^2.$ Given $f\\\\in L^1(\\\\mathbb {G})$ , for each $j$ we have $\\\\lim _i f\\\\star \\\\mu _{ij}-\\\\mu _{ij}\\\\star f&=\\\\lim _i(f\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)-(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f)\\\\\\\\&=\\\\lim _i(f\\\\otimes f_j^2)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f_j^2\\\\otimes f)\\\\\\\\&=0$ in $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ and therefore in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ .\",\n \"Furthermore, $\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}(\\\\mu _{ij})\\\\star f&=\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1))\\\\star f\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}(\\\\mu _i\\\\star (f^2_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=\\\\lim _j m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\\\in \\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\\\mu _I)$ in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _I\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\\\mathbb {G})$ in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"It is proved in the exact same way using the fact that $f\\\\star \\\\Gamma _u^*(\\\\mu )=f$ for all $f\\\\in L^1(\\\\mathbb {G})$ , which is easily verified.\",\n \"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.\",\n \"There are examples of weakly amenable residually finite groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {Z}[\\\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].\",\n \"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.\",\n \"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\\\mathbb {G})=L^1_{cb}(\\\\mathbb {G})$ [55].\",\n \"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\\\ne 3$ .\",\n \"Since $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\\\ne 3$ .\",\n \"If $\\\\mathbb {G}_1$ and $\\\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\\\Lambda _{cb}()=\\\\Lambda _{cb}()=1$ , then $\\\\mathbb {G}=\\\\mathbb {G}_1\\\\ast \\\\mathbb {G}_2$ also has the factorization property [11] and $\\\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\\\mathbb {G}))=1$ .\",\n \"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.\",\n \"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].\",\n \"We now generalize this implication to arbitrary locally compact quantum groups.\",\n \"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.\",\n \"By [42], the universal co-representation $\\\\W _{\\\\mathbb {G}}=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0())$ from the proof of Proposition REF satisfies $\\\\hat{\\\\lambda }_u(f̉)=(\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}^*)$ for all $f̉\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ , $(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}$ for all $x\\\\in C_u(\\\\mathbb {G})$ , where $\\\\hat{\\\\lambda }_u$ is the embedding of $L^1_*(\\\\widehat{\\\\mathbb {G}})$ into $C_u(\\\\mathbb {G})$ , and $\\\\pi _{\\\\mathbb {G}}:C_u(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ is the (unique) extension of $\\\\hat{\\\\lambda }:L^1_*(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow C_0(\\\\mathbb {G})$ [42].\",\n \"Moreover, $C_u(\\\\mathbb {G})=\\\\overline{\\\\text{span}\\\\lbrace (\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}) : f̉\\\\in L^1(\\\\widehat{\\\\mathbb {G}})\\\\rbrace }^{\\\\Vert \\\\cdot \\\\Vert _u}.$ We will need the following representation of $\\\\Theta ^l(\\\\mu )$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , which may be found in [19].\",\n \"We present the proof for the convenience of the reader.\",\n \"Lemma 7.1 For $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , $\\\\Theta ^l(\\\\mu )(x)=(\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ First let $x=\\\\hat{\\\\lambda }(\\\\hat{f})\\\\in C_0(\\\\mathbb {G})$ for some $\\\\hat{f}\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Then, for all $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu )(x),f\\\\rangle &=\\\\langle x,m^l_\\\\mu (f)\\\\rangle =\\\\langle \\\\hat{\\\\lambda }(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G}}\\\\circ \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\\\\\&=\\\\langle \\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f})),\\\\mu \\\\otimes \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle (\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})(\\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f}))),\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(\\\\hat{\\\\lambda }_u(\\\\hat{f})))\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle =\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle (\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},f\\\\rangle .$ As $\\\\hat{\\\\lambda }(L^1_*(\\\\widehat{\\\\mathbb {G}}))$ is norm dense in $C_0(\\\\mathbb {G})$ , and since $C_0(\\\\mathbb {G})$ is weak* dense in $L^{\\\\infty }(\\\\mathbb {G})$ , the result follows.\",\n \"Recall that $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is the map taking $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ to the operator of left multiplication by $\\\\mu $ on $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 7.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is amenable then $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.\",\n \"Amenability of $$ entails the surjectivity of $\\\\tilde{\\\\lambda }$ from (the left version of) [14].\",\n \"For simplicity, throughout the proof we denote by $\\\\Theta ^l(\\\\mu )$ the map $\\\\Theta ^l(\\\\tilde{\\\\lambda }(\\\\mu ))$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ .\",\n \"In [19] Daws shows that $\\\\Theta ^l:C_u(\\\\mathbb {G})^*_+\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is an order bijection.\",\n \"We show that it is a complete order bijection.\",\n \"To this end, let $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"By Lemma REF $\\\\Theta ^l(\\\\mu _{ij})(x)=(\\\\textnormal {id}\\\\otimes \\\\mu _{ij})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\\\in L^{\\\\infty }(\\\\mathbb {G})$ we have $((\\\\Theta ^l)^n([\\\\mu _{ij}]))^m([x_k^*x_l])&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}]\\\\\\\\&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})]^m([\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}])\\\\ge 0.$ It follows that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ .\",\n \"On the other hand, suppose $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)$ such that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\\\hat{\\\\tau }$ on $$ , via $\\\\hat{\\\\tau }_t(\\\\hat{x})=P̉^{it}\\\\hat{x}P̉^{-it}$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Using [20], for $\\\\xi _1,...\\\\xi _m\\\\in \\\\mathcal {D}(P̉^{1/2})$ , $\\\\eta _1,...,\\\\eta _m\\\\in \\\\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\\\in {nm}$ , $\\\\langle [\\\\mu _{ij}]^m([\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})])[z_{ik}],[z_{ik}]\\\\rangle &=\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^m\\\\overline{z_{ik}}z_{jl}\\\\langle \\\\mu _{ij},\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\rangle \\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\mu _{ij}^*,\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)\\\\eta _k,\\\\eta _l\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{k,l=1}^m \\\\langle [\\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)]y_k,y_l\\\\rangle }\\\\ge 0,$ where $y_k=[z_{1k}\\\\eta _k \\\\ \\\\cdots \\\\ z_{nk}\\\\eta _k]^T\\\\in L^2(\\\\mathbb {G})^n$ for $1\\\\le k\\\\le m$ .\",\n \"By density of $\\\\lbrace \\\\omega _{\\\\xi ,\\\\eta }\\\\mid \\\\xi \\\\in \\\\mathcal {D}(P̉^{1/2}), \\\\eta \\\\in \\\\mathcal {D}(P̉^{-1/2})\\\\rbrace $ in $L^1_*(\\\\widehat{\\\\mathbb {G}})$ [21], it follows that $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"We now show that $\\\\tilde{\\\\lambda }$ is a complete isometry.\",\n \"To do so we introduce a decomposability norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , given by $\\\\Vert \\\\Phi \\\\Vert _{L^1dec}:=\\\\inf \\\\bigg \\\\lbrace \\\\max \\\\lbrace \\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}\\\\rbrace \\\\mid \\\\begin{bmatrix}\\\\Psi _1 & \\\\Phi \\\\\\\\ \\\\Phi ^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0\\\\bigg \\\\rbrace ,$ where $\\\\Psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"It is evident that $\\\\Vert \\\\cdot \\\\Vert _{L^1dec}$ is a norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"That $\\\\Vert \\\\Phi \\\\Vert _{cb}\\\\le \\\\Vert \\\\Phi \\\\Vert _{L^1dec}$ for all $\\\\Phi \\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ follows verbatim from the first part of [28].\",\n \"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Since $\\\\Theta ^l$ is a completely positive contraction from $(C_u(\\\\mathbb {G})^*)^+$ onto $_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , one easily sees that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}\\\\le \\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}, \\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*),$ where $\\\\Vert \\\\cdot \\\\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.\",\n \"Conversely, if $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{dec}<1$ then there exist $\\\\Psi _1,\\\\Psi _2\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ such that $\\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}<1$ , and $\\\\begin{bmatrix}\\\\Psi _1 & (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\\\\\ (\\\\Theta ^l)^n([\\\\mu _{ij}])^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0.$ Since $(\\\\Theta ^l)^n$ is a complete order bijection there exist $[\\\\nu ^k_{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+=\\\\mathcal {CP}(C_u(\\\\mathbb {G}),M_n)$ such that $\\\\Psi _k=(\\\\Theta ^l)^n([\\\\nu ^k_{ij}])$ , $k=1,2$ , and $\\\\begin{bmatrix}[\\\\nu ^1_{ij}] & [\\\\mu _{ij}]\\\\\\\\ [\\\\mu _{ij}]^* & [\\\\nu ^2_{ij}]\\\\end{bmatrix}\\\\ge _{cp}0.$ It follows that $[\\\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\\\mathbb {G})\\\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\\\widetilde{[\\\\nu ^k_{ij}]}:M(C_u(\\\\mathbb {G}))\\\\rightarrow M_n$ which is strictly continuous on the unit ball [45].\",\n \"By uniqueness, $\\\\widetilde{[\\\\nu ^k_{ij}]}=[\\\\tilde{\\\\nu }^k_{ij}]$ , where $\\\\tilde{\\\\nu }^k_{ij}$ is the unique strict extension of the functional $\\\\nu ^k_{ij}$ .\",\n \"Thus, by completely positivity $\\\\Vert [\\\\nu ^k_{ij}] \\\\Vert _{cb}=\\\\Vert \\\\widetilde{[\\\\nu ^k_{ij}]}(1_{M(C_u(\\\\mathbb {G}))})\\\\Vert =\\\\Vert [\\\\tilde{\\\\nu }^k_{ij}(1_{M(C_u(\\\\mathbb {G}))})]\\\\Vert =\\\\Vert \\\\Psi _k(1)\\\\Vert =\\\\Vert \\\\Psi _k\\\\Vert _{cb}<1,$ so that $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}<1$ .\",\n \"Therefore $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}.$ However, $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}, ,\\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ by the left version of [14].\",\n \"The matricial analogues of the proofs of [14] show that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{cb}=\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}.$ Hence, $\\\\Theta ^l:C_u(\\\\mathbb {G})^*\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is a completely isometric isomorphism.\",\n \"To prove that $\\\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.\",\n \"Let $(\\\\mu _i)$ be a bounded net in $C_u(\\\\mathbb {G})^*$ converging weak* to $\\\\mu $ .\",\n \"Since $C_u(\\\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.\",\n \"Hence, for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu _i\\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\rightarrow \\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle .$ The density of $\\\\hat{\\\\lambda }(L^1(\\\\widehat{\\\\mathbb {G}}))$ in $C_0(\\\\mathbb {G})$ and boundedness of $\\\\Theta ^l(\\\\mu _i)$ in $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=(Q_{cb}^l(\\\\mathbb {G}))^*$ (see [14]) establish the claim.\",\n \"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\\\mathbb {G}$ is co-amenable [35].\",\n \"Corollary 7.4 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $$ is amenable.\",\n \"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\\\mathbb {G})$ .\",\n \"Moreover, as noted in the proof of Proposition REF , $L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).\",\n \"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\\\mathcal {T}(H)\\\\twoheadrightarrow C_0(\\\\mathbb {G})$ .\",\n \"Then $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }\\\\mathcal {T}(H)\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ by projectivity of $\\\\widehat{\\\\otimes }$ , and $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.\",\n \"Hence, $\\\\Theta ^l(f_i)\\\\otimes \\\\textnormal {id}_{M(\\\\mathbb {G})}(X)\\\\rightarrow X,$ weak* for all $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})$ , so that $\\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A)\\\\rightarrow A$ weakly for all $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\\\Vert \\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A) - A\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ .\",\n \"Let $\\\\tilde{m}:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ denote the induced map and $q:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ denote the quotient map.\",\n \"It follows that $q(f\\\\star A)=q(f\\\\otimes m(A))$ for all $f\\\\in L^1(\\\\mathbb {G})$ and $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"Thus, if $m(A)=0$ , then $q(A)=\\\\lim _i q(f_i\\\\star A)=\\\\lim _iq(f_i\\\\otimes m(A))=0,$ so that the induced multiplication $\\\\tilde{m}$ is injective.\",\n \"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\\\mathbb {G})^*\\\\cong \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ completely isometrically.\",\n \"By the left version of [14] $L^1(\\\\mathbb {G})$ is 1-flat in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Thus, the following sequence is 1-exact $0\\\\rightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})\\\\hookrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow 0.$ Since $f\\\\star x=(\\\\textnormal {id}\\\\otimes f\\\\circ \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=(\\\\textnormal {id}\\\\otimes f)\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}=0$ for all $x\\\\in \\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})$ , it follows that $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})=0$ .\",\n \"Hence, $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ .\",\n \"Moreover, as $\\\\Theta ^l(\\\\mu \\\\star f)(x)=\\\\Theta ^l(f)(\\\\Theta ^l(\\\\mu )(x))=(\\\\Theta ^l(\\\\mu )(x))\\\\star f, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\mu \\\\in C_u(\\\\mathbb {G})^*, \\\\ x\\\\in C_0(\\\\mathbb {G}),$ it follows that $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ is an isomorphism of left $L^1(\\\\mathbb {G})$ -modules, i.e., $C_u(\\\\mathbb {G})$ is induced.\",\n \"The commutative diagram $\\\\begin{tikzcd}L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G}) [r, equal][d, equal] &L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})[d, \\\"\\\\tilde{m}\\\"]\\\\\\\\C_u(\\\\mathbb {G}) [r, two heads] &C_0(\\\\mathbb {G})\\\\end{tikzcd}$ then implies that $\\\\tilde{m}$ is a complete quotient map.\",\n \"Thus, $C_0(\\\\mathbb {G})$ is an induced $L^1(\\\\mathbb {G})$ -module and $M(\\\\mathbb {G})\\\\cong (L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G}))^*= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})$ is necessarily a left unit for $M(\\\\mathbb {G})$ , which entails the co-amenability of $\\\\mathbb {G}$ .\",\n \"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.\",\n \"The author was partially supported by the NSERC Discovery Grant 1304873.\"\n ],\n [\n \"Self-duality of Biflatness\",\n \"As in the one-sided case, for a completely contractive Banach algebra $\\\\mathcal {A}$ , we say that an operator $\\\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ .\",\n \"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\\\rightarrow Y\\\\hookrightarrow Z\\\\twoheadrightarrow Z/Y\\\\rightarrow 0$ in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ , the sequence $0\\\\rightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y\\\\hookrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z\\\\twoheadrightarrow X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Z/Y\\\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\\\mathcal {A}}\\\\widehat{\\\\otimes }_{\\\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\\\widehat{\\\\otimes }Y/_{\\\\mathcal {A}}N_{\\\\mathcal {A}}$ , where $_{\\\\mathcal {A}}N_{\\\\mathcal {A}}=\\\\langle a\\\\cdot x\\\\otimes y - x\\\\otimes y\\\\cdot a, \\\\ x^{\\\\prime }\\\\cdot a^{\\\\prime }\\\\otimes y^{\\\\prime } - x^{\\\\prime }\\\\otimes a^{\\\\prime }\\\\cdot y^{\\\\prime }\\\\mid a,a^{\\\\prime }\\\\in \\\\mathcal {A}, \\\\ x,x^{\\\\prime }\\\\in X, \\\\ y,y^{\\\\prime }\\\\in Y\\\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\\\widehat{\\\\mathbb {G}})$ for any Kac algebra $\\\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\\\sigma V \\\\sigma $ and for which $\\\\omega _{\\\\xi _i}|_{L^{\\\\infty }(\\\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\\\mathbb {G})$ .\",\n \"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).\",\n \"We now obtain the same conclusion under a priori weaker hypotheses.\",\n \"Proposition 5.1 Let $\\\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Then $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse holds.\",\n \"It suffices to provide a completely contractive $L^1(\\\\mathbb {G})$ -bimodule map $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ which is a left inverse to $\\\\Gamma $ .\",\n \"Defining $\\\\Phi (X)=(\\\\textnormal {id}\\\\otimes m)(V^*XV)$ , $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\\\Phi $ is a completely contractive right $L^1(\\\\mathbb {G})$ -module map and $\\\\Phi \\\\circ \\\\Gamma =\\\\textnormal {id}_{L^{\\\\infty }(\\\\mathbb {G})}$ .\",\n \"However, since $m$ is also a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ , the module argument from [15] shows that $\\\\Phi $ is also a left $L^1(\\\\mathbb {G})$ -module map.\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.\",\n \"Owing to the fact that $V_s=W_a$ we have $f\\\\widehat{\\\\rhd }_{s}^{\\\\prime }x&=(\\\\textnormal {id}\\\\otimes f)(V̉^{\\\\prime }_s(x\\\\otimes 1)V̉_s^{\\\\prime *})=(f\\\\otimes \\\\textnormal {id})(V_s^*(1\\\\otimes x)V_s)\\\\\\\\&=(f\\\\otimes \\\\textnormal {id})(W_a^*(1\\\\otimes x)W_a)=x\\\\lhd _a f$ for all $f\\\\in L^1(G)$ and $x\\\\in VN(G)$ .\",\n \"Hence, $m$ satisfies (REF ).\",\n \"A completely contractive Banach algebra $\\\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\\\mathbf {\\\\mathcal {A}\\\\hspace{2.0pt}mod\\\\hspace{2.0pt}\\\\mathcal {A}}$ for some $C\\\\ge 1$ , and has a bounded approximate identity.\",\n \"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ , that is, a bounded net $(A_\\\\alpha )$ in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ satisfying $a\\\\cdot A_\\\\alpha - A_\\\\alpha \\\\cdot a, \\\\ m_{\\\\mathcal {A}}(A_\\\\alpha )\\\\cdot a \\\\rightarrow 0, \\\\ \\\\ \\\\ a\\\\in \\\\mathcal {A}.$ We let $OA(\\\\mathcal {A})$ denote the operator amenability constant of $\\\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\\\mathcal {A}\\\\widehat{\\\\otimes }\\\\mathcal {A}$ .\",\n \"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.\",\n \"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.\",\n \"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.\",\n \"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.\",\n \"Proposition 5.2 Let $G$ be a locally compact group.\",\n \"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.\",\n \"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.\",\n \"Since $L^{\\\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\\\infty }(G)$ in $L^1(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(G)$ .\",\n \"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ by (the proof of) [54].\",\n \"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.\",\n \"Conversely, if $VN(G)$ is 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\\\hspace{2.0pt}\\\\mathbf {mod}$ , so $G$ is amenable by [14].\",\n \"We now show that 1-biflatness of quantum convolution algebras is a self dual property.\",\n \"Theorem 5.3 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then $L^1(\\\\mathbb {G})$ is 1-biflat if and only if $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat.\",\n \"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\\\widehat{\\\\mathbb {G}})$ is 1-biflat, that is, $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ is 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Consider the canonical $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule structure on $\\\\mathcal {B}(L^2(\\\\mathbb {G}))$ given by $\\\\hat{f}\\\\widehat{\\\\rhd }T=(\\\\textnormal {id}\\\\otimes \\\\hat{f})\\\\widehat{V}(T\\\\otimes 1)V̉^* \\\\hspace{10.0pt}\\\\hspace{10.0pt}\\\\textnormal {and}\\\\hspace{10.0pt}\\\\hspace{10.0pt}T\\\\widehat{\\\\lhd }\\\\hat{f}=(\\\\hat{f}\\\\otimes \\\\textnormal {id})W̉^*(1\\\\otimes T)W̉,$ for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"Then by 1-injectivity, $\\\\textnormal {id}_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})}$ extends to a completely contractive $L^1(\\\\widehat{\\\\mathbb {G}})$ -bimodule projection $E:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"By the left $\\\\widehat{\\\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\\\infty }(\\\\mathbb {G}))\\\\subseteq L^{\\\\infty }(\\\\mathbb {G})\\\\cap L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})=$ .\",\n \"Also, [15] implies that $E$ is a right $\\\\lhd $ -module map.\",\n \"Let $R$ be the extended unitary antipode of $\\\\mathbb {G}$ .\",\n \"Then $(R\\\\otimes R)(V̉^{\\\\prime })=(R\\\\otimes R)(\\\\sigma V^*\\\\sigma )=\\\\Sigma (R\\\\otimes R)(V^*)=\\\\Sigma (J̉\\\\otimes J̉)(V)(J̉\\\\otimes J̉)=\\\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .\",\n \"Let $E_R:\\\\mathcal {B}(L^2(\\\\mathbb {G}))\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ be the projection of norm one, $E_R=R\\\\circ E\\\\circ R$ .\",\n \"Then for $\\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime })$ and $T\\\\in \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ , we have $E_R(\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }T)&=R(E(R((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime })V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)(R\\\\otimes R)(V̉^{\\\\prime }(T\\\\otimes 1)V̉^{\\\\prime *})))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((R\\\\otimes R)(V̉^{\\\\prime *})(R(T)\\\\otimes 1)(R\\\\otimes R)(V̉^{\\\\prime }))))\\\\\\\\&=R(E((\\\\textnormal {id}\\\\otimes \\\\hat{f}^{\\\\prime }\\\\circ R)((\\\\Sigma W̉^*)(R(T)\\\\otimes 1)(\\\\Sigma W̉))))\\\\\\\\&=R(E((\\\\hat{f}^{\\\\prime }\\\\circ R\\\\otimes \\\\textnormal {id})(W̉^*(1\\\\otimes R(T))W̉)))\\\\\\\\&=R(E(R(T)\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R)))\\\\\\\\&=R(E(R(T))\\\\widehat{\\\\lhd }(\\\\hat{f}^{\\\\prime }\\\\circ R))\\\\\\\\&=\\\\hat{f}^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }E_R(T).$ Thus, $E_R$ is a left $\\\\widehat{\\\\rhd }^{\\\\prime }$ -module map.\",\n \"Since $R(L^{\\\\infty }(\\\\mathbb {G}))=L^{\\\\infty }(\\\\mathbb {G})$ , the restriction $E_R|_{L^{\\\\infty }(\\\\mathbb {G})}$ defines a state $m\\\\in L^{\\\\infty }(\\\\mathbb {G})^*$ satisfying $\\\\langle m,f̉^{\\\\prime }\\\\widehat{\\\\rhd }^{\\\\prime }x\\\\rangle =\\\\langle \\\\hat{f}^{\\\\prime },1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ \\\\hat{f}^{\\\\prime }\\\\in L^1(\\\\widehat{\\\\mathbb {G}}^{\\\\prime }), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ But $E$ was also a right $\\\\lhd $ -module map, which implies that $E_R$ is a left $\\\\rhd $ -module map by the generalized antipode relation (REF ).\",\n \"Thus, we also have $\\\\langle m,f\\\\rhd x\\\\rangle =\\\\langle f,1\\\\rangle \\\\langle m,x\\\\rangle , \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\\\infty }(\\\\mathbb {G})$ .\",\n \"By Proposition REF it follows that $L^{\\\\infty }(\\\\mathbb {G})$ is relatively 1-injective in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"By 1-injectivity of $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ , there exists a completely contractive morphism $\\\\Phi :L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ which is a left inverse to $$ .\",\n \"It follows that $\\\\Phi |_{L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})\\\\otimes 1}$ defines a state $m̉\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})^*$ which is a right $L^1(\\\\widehat{\\\\mathbb {G}})$ -module map, i.e., $$ is amenable.\",\n \"Hence, $L^{\\\\infty }(\\\\mathbb {G})$ is a 1-injective operator space by [5].\",\n \"Proposition REF then implies the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Corollary 5.4 $\\\\ell ^1(\\\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.\",\n \"Since $\\\\ell ^\\\\infty (\\\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\\\mathbb {G}$ , if $\\\\ell ^1(\\\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\\\ell ^\\\\infty (\\\\mathbb {G})$ would be 1-injective in $\\\\ell ^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}\\\\ell ^1(\\\\mathbb {G})$ .\",\n \"Then by Theorem REF $L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ would be 1-injective in $L^1(\\\\widehat{\\\\mathbb {G}})\\\\hspace{2.0pt}\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\\\mathbb {G}$ is unimodular.\",\n \"The relative biprojectivity of $L^1(\\\\mathbb {G})$ , that is, relative projectivity of $L^1(\\\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\\\mathbb {G})$ is relatively 1-biprojective if and only if $\\\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].\",\n \"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.\",\n \"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].\",\n \"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\\\Delta $ [16].\",\n \"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .\",\n \"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 5.5 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"Then the following conditions are equivalent: $\\\\mathbb {G}$ is finite–dimensional.\",\n \"$\\\\mathcal {T}_\\\\rhd $ is relatively 1-biprojective; $L^1(\\\\mathbb {G})$ is 1-biprojective; $(i)\\\\Rightarrow (ii)$ follows from [38].\",\n \"$(ii)\\\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\\\mathbb {G})$ together with the 1-projectivity of $L^1(\\\\mathbb {G})$ as an operator space.\",\n \"The bimodule analogue of [14] then yields $(iii)$ .\",\n \"$(iii)\\\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\\\mathbb {G})$ -bimodule left inverse $\\\\Phi :L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ to $\\\\Gamma $ .\",\n \"As usual, the restriction $\\\\Phi |_{L^{\\\\infty }(\\\\mathbb {G})\\\\otimes 1}:L^{\\\\infty }(\\\\mathbb {G})\\\\rightarrow L^{\\\\infty }(\\\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.\",\n \"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\\\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.\",\n \"Thus, $ G$ is finite-dimensional by \\\\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\\\mathbb {G})$ For a locally compact quantum group $\\\\mathbb {G}$ , let $L^1_{cb}(\\\\mathbb {G})$ denote the closure of $L^1(\\\\mathbb {G})$ inside $M_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\\\mathbb {G})$ has a bounded approximate identity.\",\n \"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.\",\n \"In [16] the authors gave examples of weakly amenable connected groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.\",\n \"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].\",\n \"Let $\\\\mathbb {G}$ be a compact Kac algebra and let $\\\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\\\mathbb {G})$ , as well as their universal extensions to $C_u(\\\\mathbb {G})$ .\",\n \"As in [7], we define $*$ -homomorphisms $\\\\lambda ,\\\\rho :C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ by $\\\\lambda (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (xy), \\\\ \\\\ \\\\ \\\\rho (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (y R(x)), \\\\ \\\\ \\\\ x,y\\\\in C_u(\\\\mathbb {G}).$ Since $\\\\lambda $ and $\\\\rho $ have commuting ranges, we obtain a canonical representation $\\\\lambda \\\\times \\\\rho :C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .\",\n \"Lemma 6.1 Let $\\\\mathcal {A}$ be a $C^*$ -algebra.\",\n \"There exists a complete isometry $\\\\iota :\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ such that $\\\\iota (\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1})$ is weak* dense in $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ .\",\n \"Let $\\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}$ denote the universal representation of $\\\\mathcal {A}$ .\",\n \"Then the universal cover of the representation $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\\\pi $ of $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{**}$ onto $(\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A})(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }\\\\overline{\\\\otimes }\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^*$ (see [61]).\",\n \"Its pre-adjoint $\\\\iota :=\\\\pi _*:\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ is a complete isometry.\",\n \"Now, Let $F\\\\in (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ , $\\\\Vert F\\\\Vert \\\\le 1$ .\",\n \"Since $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}\\\\subseteq \\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\\\tilde{F}\\\\in (\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**})^*=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^{**}$ satisfying $\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\langle F,A\\\\rangle , \\\\ \\\\ \\\\ A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ such that $f_j\\\\rightarrow \\\\tilde{F}$ weak*.\",\n \"Thus, for all $A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}$ , $\\\\langle F,A\\\\rangle =\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle f_j,\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle \\\\iota (f_j),A\\\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.\",\n \"Then $$ is weakly amenable if and only if $L^1_{cb}(\\\\mathbb {G})$ is operator amenable.\",\n \"Moreover, $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"It is clear that $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].\",\n \"Conversely, by Kirchberg's factorization property the representation $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"Composing with $\\\\omega _{\\\\Lambda _\\\\varphi (1)}$ , we obtain a state $\\\\mu :=\\\\omega _{\\\\Lambda _\\\\varphi (1)}\\\\circ \\\\lambda \\\\times \\\\rho \\\\in (C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ .\",\n \"By $R$ -invariance of $\\\\varphi $ , one can easily verify that $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ .\",\n \"Let $\\\\Gamma _u:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u(\\\\mathbb {G})$ denote the universal co-multiplication.\",\n \"Then, similar to the calculations in [55], for all $u\\\\in \\\\mathrm {Irr}(\\\\mathbb {G})$ , $1\\\\le i,j\\\\le n_u$ , $\\\\langle \\\\Gamma _u^*(\\\\mu )\\\\star f,u_{ij}\\\\rangle &=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\langle \\\\mu ,u_{ik}\\\\otimes u_{kl}\\\\rangle =\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} R(u_{kl}))\\\\\\\\&=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} u_{lk}^{*})=\\\\frac{1}{n_u}\\\\sum _{k=1}^{n_u}\\\\langle f,u_{ij}\\\\rangle \\\\\\\\&=\\\\langle f,u_{ij}\\\\rangle .$ Hence, $\\\\Gamma _u^*(\\\\mu )\\\\star f = f$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"By [42], $R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x^*)(1\\\\otimes y)))=(\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x^*)\\\\Gamma _u(y))$ for all $x,y\\\\in C_u(\\\\mathbb {G})$ , so that $\\\\langle (f\\\\otimes 1)\\\\star \\\\mu ,x\\\\otimes y\\\\rangle &=\\\\varphi ((x\\\\star f) R(y))=(f\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))\\\\\\\\&=f\\\\circ R(R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))))\\\\\\\\&=f\\\\circ R((\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Gamma _u(R(y))))\\\\\\\\&=(f\\\\circ R\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Sigma \\\\circ R\\\\otimes R\\\\circ \\\\Gamma _u(y))\\\\\\\\&=(\\\\varphi \\\\otimes f)(\\\\Gamma _u(y)(R(x)\\\\otimes 1))\\\\\\\\&=\\\\varphi ((f\\\\star y)R(x))\\\\\\\\&=\\\\langle \\\\mu \\\\star (1\\\\otimes f),x\\\\otimes y\\\\rangle .$ Thus, $(f\\\\otimes 1)\\\\star \\\\mu =\\\\mu \\\\star (1\\\\otimes f)$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"Since $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ , we also have $(1\\\\otimes f)\\\\star \\\\mu =\\\\mu \\\\star (f\\\\otimes 1)$ , for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"It follows that $F\\\\star \\\\mu =\\\\mu \\\\star \\\\Sigma F$ for all $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ , where we let $\\\\Sigma $ also denote the flip homomorphism on $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"The proof of Lemma REF implies the existence of a net $(\\\\mu _i)$ of states in $C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*$ such that $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ , and hence in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G}))^*=C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ .\",\n \"Since $m_{C_u(\\\\mathbb {G})^*}=\\\\Gamma _u^*|_{C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*}$ , it follows that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\rightarrow \\\\Gamma _u^*(\\\\mu )$ weak* in $C_u(\\\\mathbb {G})^*$ .\",\n \"By [57] (note that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i),\\\\Gamma _u^*(\\\\mu )$ are states), we have $\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-f\\\\Vert _{L^1(\\\\mathbb {G})}=\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-\\\\Gamma _u^*(\\\\mu )\\\\star f\\\\Vert _{L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}).$ Since $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ , again by [57] we obtain $\\\\Vert F\\\\star \\\\mu _i-\\\\mu _i\\\\star \\\\Sigma F\\\\Vert _{L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}),$ from which we have $&\\\\Vert F\\\\star ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\Vert =\\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-(\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)\\\\Vert +\\\\Vert (\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-\\\\mu _i\\\\star (\\\\Sigma (F\\\\star (1\\\\otimes f)))\\\\Vert +\\\\Vert \\\\mu _i\\\\star \\\\Sigma F-F\\\\star \\\\mu _i\\\\Vert \\\\Vert f\\\\otimes 1\\\\Vert \\\\\\\\&\\\\rightarrow 0$ for every $f\\\\in L^1(\\\\mathbb {G})$ , $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"Similarly, $\\\\Vert ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\star F\\\\Vert \\\\rightarrow 0$ .\",\n \"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\\\mathbb {G})$ in $\\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ such that $\\\\sup _j\\\\Vert f_j\\\\Vert _{cb}<\\\\infty $ .\",\n \"The tensor square of the canonical complete contraction $C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ allows us to view $\\\\mu _i\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))$ with $\\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1$ for all $i$ .\",\n \"By the universal property of the operator space projective tensor product, we may also view each $\\\\mu _i$ as an element of $\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))$ by right multiplication, as well as in $\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))$ by left multiplication.\",\n \"Moreover, $\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))},\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))}\\\\le \\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1.$ Define $\\\\mu _{ij}:=(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)$ .\",\n \"Then $\\\\mu _{ij}\\\\in L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _{ij}\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Vert f_j\\\\Vert _{cb}^2\\\\le \\\\Lambda _{cb}()^2.$ Given $f\\\\in L^1(\\\\mathbb {G})$ , for each $j$ we have $\\\\lim _i f\\\\star \\\\mu _{ij}-\\\\mu _{ij}\\\\star f&=\\\\lim _i(f\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)-(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f)\\\\\\\\&=\\\\lim _i(f\\\\otimes f_j^2)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f_j^2\\\\otimes f)\\\\\\\\&=0$ in $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ and therefore in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ .\",\n \"Furthermore, $\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}(\\\\mu _{ij})\\\\star f&=\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1))\\\\star f\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}(\\\\mu _i\\\\star (f^2_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=\\\\lim _j m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\\\in \\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\\\mu _I)$ in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _I\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\\\mathbb {G})$ in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"It is proved in the exact same way using the fact that $f\\\\star \\\\Gamma _u^*(\\\\mu )=f$ for all $f\\\\in L^1(\\\\mathbb {G})$ , which is easily verified.\",\n \"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.\",\n \"There are examples of weakly amenable residually finite groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {Z}[\\\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].\",\n \"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.\",\n \"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\\\mathbb {G})=L^1_{cb}(\\\\mathbb {G})$ [55].\",\n \"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\\\ne 3$ .\",\n \"Since $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\\\ne 3$ .\",\n \"If $\\\\mathbb {G}_1$ and $\\\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\\\Lambda _{cb}()=\\\\Lambda _{cb}()=1$ , then $\\\\mathbb {G}=\\\\mathbb {G}_1\\\\ast \\\\mathbb {G}_2$ also has the factorization property [11] and $\\\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\\\mathbb {G}))=1$ .\",\n \"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.\",\n \"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].\",\n \"We now generalize this implication to arbitrary locally compact quantum groups.\",\n \"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.\",\n \"By [42], the universal co-representation $\\\\W _{\\\\mathbb {G}}=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0())$ from the proof of Proposition REF satisfies $\\\\hat{\\\\lambda }_u(f̉)=(\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}^*)$ for all $f̉\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ , $(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}$ for all $x\\\\in C_u(\\\\mathbb {G})$ , where $\\\\hat{\\\\lambda }_u$ is the embedding of $L^1_*(\\\\widehat{\\\\mathbb {G}})$ into $C_u(\\\\mathbb {G})$ , and $\\\\pi _{\\\\mathbb {G}}:C_u(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ is the (unique) extension of $\\\\hat{\\\\lambda }:L^1_*(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow C_0(\\\\mathbb {G})$ [42].\",\n \"Moreover, $C_u(\\\\mathbb {G})=\\\\overline{\\\\text{span}\\\\lbrace (\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}) : f̉\\\\in L^1(\\\\widehat{\\\\mathbb {G}})\\\\rbrace }^{\\\\Vert \\\\cdot \\\\Vert _u}.$ We will need the following representation of $\\\\Theta ^l(\\\\mu )$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , which may be found in [19].\",\n \"We present the proof for the convenience of the reader.\",\n \"Lemma 7.1 For $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , $\\\\Theta ^l(\\\\mu )(x)=(\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ First let $x=\\\\hat{\\\\lambda }(\\\\hat{f})\\\\in C_0(\\\\mathbb {G})$ for some $\\\\hat{f}\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Then, for all $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu )(x),f\\\\rangle &=\\\\langle x,m^l_\\\\mu (f)\\\\rangle =\\\\langle \\\\hat{\\\\lambda }(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G}}\\\\circ \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\\\\\&=\\\\langle \\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f})),\\\\mu \\\\otimes \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle (\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})(\\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f}))),\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(\\\\hat{\\\\lambda }_u(\\\\hat{f})))\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle =\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle (\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},f\\\\rangle .$ As $\\\\hat{\\\\lambda }(L^1_*(\\\\widehat{\\\\mathbb {G}}))$ is norm dense in $C_0(\\\\mathbb {G})$ , and since $C_0(\\\\mathbb {G})$ is weak* dense in $L^{\\\\infty }(\\\\mathbb {G})$ , the result follows.\",\n \"Recall that $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is the map taking $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ to the operator of left multiplication by $\\\\mu $ on $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 7.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is amenable then $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.\",\n \"Amenability of $$ entails the surjectivity of $\\\\tilde{\\\\lambda }$ from (the left version of) [14].\",\n \"For simplicity, throughout the proof we denote by $\\\\Theta ^l(\\\\mu )$ the map $\\\\Theta ^l(\\\\tilde{\\\\lambda }(\\\\mu ))$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ .\",\n \"In [19] Daws shows that $\\\\Theta ^l:C_u(\\\\mathbb {G})^*_+\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is an order bijection.\",\n \"We show that it is a complete order bijection.\",\n \"To this end, let $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"By Lemma REF $\\\\Theta ^l(\\\\mu _{ij})(x)=(\\\\textnormal {id}\\\\otimes \\\\mu _{ij})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\\\in L^{\\\\infty }(\\\\mathbb {G})$ we have $((\\\\Theta ^l)^n([\\\\mu _{ij}]))^m([x_k^*x_l])&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}]\\\\\\\\&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})]^m([\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}])\\\\ge 0.$ It follows that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ .\",\n \"On the other hand, suppose $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)$ such that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\\\hat{\\\\tau }$ on $$ , via $\\\\hat{\\\\tau }_t(\\\\hat{x})=P̉^{it}\\\\hat{x}P̉^{-it}$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Using [20], for $\\\\xi _1,...\\\\xi _m\\\\in \\\\mathcal {D}(P̉^{1/2})$ , $\\\\eta _1,...,\\\\eta _m\\\\in \\\\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\\\in {nm}$ , $\\\\langle [\\\\mu _{ij}]^m([\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})])[z_{ik}],[z_{ik}]\\\\rangle &=\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^m\\\\overline{z_{ik}}z_{jl}\\\\langle \\\\mu _{ij},\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\rangle \\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\mu _{ij}^*,\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)\\\\eta _k,\\\\eta _l\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{k,l=1}^m \\\\langle [\\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)]y_k,y_l\\\\rangle }\\\\ge 0,$ where $y_k=[z_{1k}\\\\eta _k \\\\ \\\\cdots \\\\ z_{nk}\\\\eta _k]^T\\\\in L^2(\\\\mathbb {G})^n$ for $1\\\\le k\\\\le m$ .\",\n \"By density of $\\\\lbrace \\\\omega _{\\\\xi ,\\\\eta }\\\\mid \\\\xi \\\\in \\\\mathcal {D}(P̉^{1/2}), \\\\eta \\\\in \\\\mathcal {D}(P̉^{-1/2})\\\\rbrace $ in $L^1_*(\\\\widehat{\\\\mathbb {G}})$ [21], it follows that $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"We now show that $\\\\tilde{\\\\lambda }$ is a complete isometry.\",\n \"To do so we introduce a decomposability norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , given by $\\\\Vert \\\\Phi \\\\Vert _{L^1dec}:=\\\\inf \\\\bigg \\\\lbrace \\\\max \\\\lbrace \\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}\\\\rbrace \\\\mid \\\\begin{bmatrix}\\\\Psi _1 & \\\\Phi \\\\\\\\ \\\\Phi ^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0\\\\bigg \\\\rbrace ,$ where $\\\\Psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"It is evident that $\\\\Vert \\\\cdot \\\\Vert _{L^1dec}$ is a norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"That $\\\\Vert \\\\Phi \\\\Vert _{cb}\\\\le \\\\Vert \\\\Phi \\\\Vert _{L^1dec}$ for all $\\\\Phi \\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ follows verbatim from the first part of [28].\",\n \"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Since $\\\\Theta ^l$ is a completely positive contraction from $(C_u(\\\\mathbb {G})^*)^+$ onto $_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , one easily sees that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}\\\\le \\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}, \\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*),$ where $\\\\Vert \\\\cdot \\\\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.\",\n \"Conversely, if $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{dec}<1$ then there exist $\\\\Psi _1,\\\\Psi _2\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ such that $\\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}<1$ , and $\\\\begin{bmatrix}\\\\Psi _1 & (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\\\\\ (\\\\Theta ^l)^n([\\\\mu _{ij}])^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0.$ Since $(\\\\Theta ^l)^n$ is a complete order bijection there exist $[\\\\nu ^k_{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+=\\\\mathcal {CP}(C_u(\\\\mathbb {G}),M_n)$ such that $\\\\Psi _k=(\\\\Theta ^l)^n([\\\\nu ^k_{ij}])$ , $k=1,2$ , and $\\\\begin{bmatrix}[\\\\nu ^1_{ij}] & [\\\\mu _{ij}]\\\\\\\\ [\\\\mu _{ij}]^* & [\\\\nu ^2_{ij}]\\\\end{bmatrix}\\\\ge _{cp}0.$ It follows that $[\\\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\\\mathbb {G})\\\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\\\widetilde{[\\\\nu ^k_{ij}]}:M(C_u(\\\\mathbb {G}))\\\\rightarrow M_n$ which is strictly continuous on the unit ball [45].\",\n \"By uniqueness, $\\\\widetilde{[\\\\nu ^k_{ij}]}=[\\\\tilde{\\\\nu }^k_{ij}]$ , where $\\\\tilde{\\\\nu }^k_{ij}$ is the unique strict extension of the functional $\\\\nu ^k_{ij}$ .\",\n \"Thus, by completely positivity $\\\\Vert [\\\\nu ^k_{ij}] \\\\Vert _{cb}=\\\\Vert \\\\widetilde{[\\\\nu ^k_{ij}]}(1_{M(C_u(\\\\mathbb {G}))})\\\\Vert =\\\\Vert [\\\\tilde{\\\\nu }^k_{ij}(1_{M(C_u(\\\\mathbb {G}))})]\\\\Vert =\\\\Vert \\\\Psi _k(1)\\\\Vert =\\\\Vert \\\\Psi _k\\\\Vert _{cb}<1,$ so that $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}<1$ .\",\n \"Therefore $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}.$ However, $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}, ,\\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ by the left version of [14].\",\n \"The matricial analogues of the proofs of [14] show that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{cb}=\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}.$ Hence, $\\\\Theta ^l:C_u(\\\\mathbb {G})^*\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is a completely isometric isomorphism.\",\n \"To prove that $\\\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.\",\n \"Let $(\\\\mu _i)$ be a bounded net in $C_u(\\\\mathbb {G})^*$ converging weak* to $\\\\mu $ .\",\n \"Since $C_u(\\\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.\",\n \"Hence, for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu _i\\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\rightarrow \\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle .$ The density of $\\\\hat{\\\\lambda }(L^1(\\\\widehat{\\\\mathbb {G}}))$ in $C_0(\\\\mathbb {G})$ and boundedness of $\\\\Theta ^l(\\\\mu _i)$ in $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=(Q_{cb}^l(\\\\mathbb {G}))^*$ (see [14]) establish the claim.\",\n \"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\\\mathbb {G}$ is co-amenable [35].\",\n \"Corollary 7.4 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $$ is amenable.\",\n \"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\\\mathbb {G})$ .\",\n \"Moreover, as noted in the proof of Proposition REF , $L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).\",\n \"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\\\mathcal {T}(H)\\\\twoheadrightarrow C_0(\\\\mathbb {G})$ .\",\n \"Then $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }\\\\mathcal {T}(H)\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ by projectivity of $\\\\widehat{\\\\otimes }$ , and $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.\",\n \"Hence, $\\\\Theta ^l(f_i)\\\\otimes \\\\textnormal {id}_{M(\\\\mathbb {G})}(X)\\\\rightarrow X,$ weak* for all $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})$ , so that $\\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A)\\\\rightarrow A$ weakly for all $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\\\Vert \\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A) - A\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ .\",\n \"Let $\\\\tilde{m}:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ denote the induced map and $q:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ denote the quotient map.\",\n \"It follows that $q(f\\\\star A)=q(f\\\\otimes m(A))$ for all $f\\\\in L^1(\\\\mathbb {G})$ and $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"Thus, if $m(A)=0$ , then $q(A)=\\\\lim _i q(f_i\\\\star A)=\\\\lim _iq(f_i\\\\otimes m(A))=0,$ so that the induced multiplication $\\\\tilde{m}$ is injective.\",\n \"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\\\mathbb {G})^*\\\\cong \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ completely isometrically.\",\n \"By the left version of [14] $L^1(\\\\mathbb {G})$ is 1-flat in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Thus, the following sequence is 1-exact $0\\\\rightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})\\\\hookrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow 0.$ Since $f\\\\star x=(\\\\textnormal {id}\\\\otimes f\\\\circ \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=(\\\\textnormal {id}\\\\otimes f)\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}=0$ for all $x\\\\in \\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})$ , it follows that $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})=0$ .\",\n \"Hence, $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ .\",\n \"Moreover, as $\\\\Theta ^l(\\\\mu \\\\star f)(x)=\\\\Theta ^l(f)(\\\\Theta ^l(\\\\mu )(x))=(\\\\Theta ^l(\\\\mu )(x))\\\\star f, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\mu \\\\in C_u(\\\\mathbb {G})^*, \\\\ x\\\\in C_0(\\\\mathbb {G}),$ it follows that $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ is an isomorphism of left $L^1(\\\\mathbb {G})$ -modules, i.e., $C_u(\\\\mathbb {G})$ is induced.\",\n \"The commutative diagram $\\\\begin{tikzcd}L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G}) [r, equal][d, equal] &L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})[d, \\\"\\\\tilde{m}\\\"]\\\\\\\\C_u(\\\\mathbb {G}) [r, two heads] &C_0(\\\\mathbb {G})\\\\end{tikzcd}$ then implies that $\\\\tilde{m}$ is a complete quotient map.\",\n \"Thus, $C_0(\\\\mathbb {G})$ is an induced $L^1(\\\\mathbb {G})$ -module and $M(\\\\mathbb {G})\\\\cong (L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G}))^*= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})$ is necessarily a left unit for $M(\\\\mathbb {G})$ , which entails the co-amenability of $\\\\mathbb {G}$ .\",\n \"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.\",\n \"The author was partially supported by the NSERC Discovery Grant 1304873.\"\n ],\n [\n \"Operator Amenability of $L^1_{cb}(\\\\mathbb {G})$\",\n \"For a locally compact quantum group $\\\\mathbb {G}$ , let $L^1_{cb}(\\\\mathbb {G})$ denote the closure of $L^1(\\\\mathbb {G})$ inside $M_{cb}^l(L^1(\\\\mathbb {G}))$ .\",\n \"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\\\mathbb {G})$ has a bounded approximate identity.\",\n \"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.\",\n \"In [16] the authors gave examples of weakly amenable connected groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.\",\n \"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].\",\n \"Let $\\\\mathbb {G}$ be a compact Kac algebra and let $\\\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\\\mathbb {G})$ , as well as their universal extensions to $C_u(\\\\mathbb {G})$ .\",\n \"As in [7], we define $*$ -homomorphisms $\\\\lambda ,\\\\rho :C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ by $\\\\lambda (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (xy), \\\\ \\\\ \\\\ \\\\rho (x)\\\\Lambda _\\\\varphi (y)=\\\\Lambda _\\\\varphi (y R(x)), \\\\ \\\\ \\\\ x,y\\\\in C_u(\\\\mathbb {G}).$ Since $\\\\lambda $ and $\\\\rho $ have commuting ranges, we obtain a canonical representation $\\\\lambda \\\\times \\\\rho :C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G})\\\\rightarrow \\\\mathcal {B}(L^2(\\\\mathbb {G}))$ .\",\n \"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"When $\\\\mathbb {G}=\\\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .\",\n \"Lemma 6.1 Let $\\\\mathcal {A}$ be a $C^*$ -algebra.\",\n \"There exists a complete isometry $\\\\iota :\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ such that $\\\\iota (\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1})$ is weak* dense in $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ .\",\n \"Let $\\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}$ denote the universal representation of $\\\\mathcal {A}$ .\",\n \"Then the universal cover of the representation $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\\\pi $ of $(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{**}$ onto $(\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A})(\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }\\\\overline{\\\\otimes }\\\\pi _\\\\mathcal {A}(\\\\mathcal {A})^{\\\\prime \\\\prime }=\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^*$ (see [61]).\",\n \"Its pre-adjoint $\\\\iota :=\\\\pi _*:\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*\\\\hookrightarrow (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ is a complete isometry.\",\n \"Now, Let $F\\\\in (\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A})^*$ , $\\\\Vert F\\\\Vert \\\\le 1$ .\",\n \"Since $\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}:\\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}\\\\rightarrow \\\\mathcal {A}^{**}\\\\otimes _{\\\\min }\\\\mathcal {A}^{**}\\\\subseteq \\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\\\tilde{F}\\\\in (\\\\mathcal {A}^{**}\\\\overline{\\\\otimes }\\\\mathcal {A}^{**})^*=(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)^{**}$ satisfying $\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\langle F,A\\\\rangle , \\\\ \\\\ \\\\ A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\\\mathcal {A}^*\\\\widehat{\\\\otimes }\\\\mathcal {A}^*)_{\\\\Vert \\\\cdot \\\\Vert \\\\le 1}$ such that $f_j\\\\rightarrow \\\\tilde{F}$ weak*.\",\n \"Thus, for all $A\\\\in \\\\mathcal {A}\\\\otimes _{\\\\min }\\\\mathcal {A}$ , $\\\\langle F,A\\\\rangle =\\\\langle \\\\tilde{F},\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle f_j,\\\\pi _\\\\mathcal {A}\\\\otimes \\\\pi _\\\\mathcal {A}(A)\\\\rangle =\\\\lim _j\\\\langle \\\\iota (f_j),A\\\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.\",\n \"Then $$ is weakly amenable if and only if $L^1_{cb}(\\\\mathbb {G})$ is operator amenable.\",\n \"Moreover, $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"It is clear that $\\\\Lambda _{cb}()\\\\le OA(L^1_{cb}(\\\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].\",\n \"Conversely, by Kirchberg's factorization property the representation $\\\\lambda \\\\times \\\\rho $ factors through $C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G})$ .\",\n \"Composing with $\\\\omega _{\\\\Lambda _\\\\varphi (1)}$ , we obtain a state $\\\\mu :=\\\\omega _{\\\\Lambda _\\\\varphi (1)}\\\\circ \\\\lambda \\\\times \\\\rho \\\\in (C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ .\",\n \"By $R$ -invariance of $\\\\varphi $ , one can easily verify that $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ .\",\n \"Let $\\\\Gamma _u:C_u(\\\\mathbb {G})\\\\rightarrow C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_u(\\\\mathbb {G})$ denote the universal co-multiplication.\",\n \"Then, similar to the calculations in [55], for all $u\\\\in \\\\mathrm {Irr}(\\\\mathbb {G})$ , $1\\\\le i,j\\\\le n_u$ , $\\\\langle \\\\Gamma _u^*(\\\\mu )\\\\star f,u_{ij}\\\\rangle &=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\langle \\\\mu ,u_{ik}\\\\otimes u_{kl}\\\\rangle =\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} R(u_{kl}))\\\\\\\\&=\\\\sum _{k,l=1}^{n_u}\\\\langle f,u_{lj}\\\\rangle \\\\varphi (u_{ik} u_{lk}^{*})=\\\\frac{1}{n_u}\\\\sum _{k=1}^{n_u}\\\\langle f,u_{ij}\\\\rangle \\\\\\\\&=\\\\langle f,u_{ij}\\\\rangle .$ Hence, $\\\\Gamma _u^*(\\\\mu )\\\\star f = f$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"By [42], $R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x^*)(1\\\\otimes y)))=(\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x^*)\\\\Gamma _u(y))$ for all $x,y\\\\in C_u(\\\\mathbb {G})$ , so that $\\\\langle (f\\\\otimes 1)\\\\star \\\\mu ,x\\\\otimes y\\\\rangle &=\\\\varphi ((x\\\\star f) R(y))=(f\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))\\\\\\\\&=f\\\\circ R(R((\\\\textnormal {id}\\\\otimes \\\\varphi )(\\\\Gamma _u(x)(1\\\\otimes R(y)))))\\\\\\\\&=f\\\\circ R((\\\\textnormal {id}\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Gamma _u(R(y))))\\\\\\\\&=(f\\\\circ R\\\\otimes \\\\varphi )((1\\\\otimes x)\\\\Sigma \\\\circ R\\\\otimes R\\\\circ \\\\Gamma _u(y))\\\\\\\\&=(\\\\varphi \\\\otimes f)(\\\\Gamma _u(y)(R(x)\\\\otimes 1))\\\\\\\\&=\\\\varphi ((f\\\\star y)R(x))\\\\\\\\&=\\\\langle \\\\mu \\\\star (1\\\\otimes f),x\\\\otimes y\\\\rangle .$ Thus, $(f\\\\otimes 1)\\\\star \\\\mu =\\\\mu \\\\star (1\\\\otimes f)$ for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"Since $\\\\mu =\\\\mu \\\\circ \\\\Sigma $ , we also have $(1\\\\otimes f)\\\\star \\\\mu =\\\\mu \\\\star (f\\\\otimes 1)$ , for all $f\\\\in L^1(\\\\mathbb {G})$ .\",\n \"It follows that $F\\\\star \\\\mu =\\\\mu \\\\star \\\\Sigma F$ for all $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ , where we let $\\\\Sigma $ also denote the flip homomorphism on $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"The proof of Lemma REF implies the existence of a net $(\\\\mu _i)$ of states in $C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*$ such that $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min }C_u(\\\\mathbb {G}))^*$ , and hence in $(C_u(\\\\mathbb {G})\\\\otimes _{\\\\max }C_u(\\\\mathbb {G}))^*=C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ .\",\n \"Since $m_{C_u(\\\\mathbb {G})^*}=\\\\Gamma _u^*|_{C_u(\\\\mathbb {G})^*\\\\widehat{\\\\otimes }C_u(\\\\mathbb {G})^*}$ , it follows that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\rightarrow \\\\Gamma _u^*(\\\\mu )$ weak* in $C_u(\\\\mathbb {G})^*$ .\",\n \"By [57] (note that $m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i),\\\\Gamma _u^*(\\\\mu )$ are states), we have $\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-f\\\\Vert _{L^1(\\\\mathbb {G})}=\\\\Vert m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)\\\\star f-\\\\Gamma _u^*(\\\\mu )\\\\star f\\\\Vert _{L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}).$ Since $\\\\mu _i\\\\rightarrow \\\\mu $ weak* in $C_u(\\\\mathbb {G}\\\\times \\\\mathbb {G})^*$ , again by [57] we obtain $\\\\Vert F\\\\star \\\\mu _i-\\\\mu _i\\\\star \\\\Sigma F\\\\Vert _{L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})}\\\\rightarrow 0, \\\\ \\\\ \\\\ F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G}),$ from which we have $&\\\\Vert F\\\\star ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\Vert =\\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-(\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)\\\\Vert +\\\\Vert (\\\\mu _i\\\\star \\\\Sigma F)\\\\star (f\\\\otimes 1)-F\\\\star \\\\mu _i\\\\star (f\\\\otimes 1)\\\\Vert \\\\\\\\&\\\\le \\\\Vert (F\\\\star (1\\\\otimes f))\\\\star \\\\mu _i-\\\\mu _i\\\\star (\\\\Sigma (F\\\\star (1\\\\otimes f)))\\\\Vert +\\\\Vert \\\\mu _i\\\\star \\\\Sigma F-F\\\\star \\\\mu _i\\\\Vert \\\\Vert f\\\\otimes 1\\\\Vert \\\\\\\\&\\\\rightarrow 0$ for every $f\\\\in L^1(\\\\mathbb {G})$ , $F\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ .\",\n \"Similarly, $\\\\Vert ((1\\\\otimes f)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f\\\\otimes 1))\\\\star F\\\\Vert \\\\rightarrow 0$ .\",\n \"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\\\mathbb {G})$ in $\\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ such that $\\\\sup _j\\\\Vert f_j\\\\Vert _{cb}<\\\\infty $ .\",\n \"The tensor square of the canonical complete contraction $C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ allows us to view $\\\\mu _i\\\\in M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))$ with $\\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1$ for all $i$ .\",\n \"By the universal property of the operator space projective tensor product, we may also view each $\\\\mu _i$ as an element of $\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))$ by right multiplication, as well as in $\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))$ by left multiplication.\",\n \"Moreover, $\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G}))},\\\\Vert \\\\mu _i\\\\Vert _{\\\\mathcal {CB}(L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G})))}\\\\le \\\\Vert \\\\mu _i\\\\Vert _{ M_{cb}^l(L^1(\\\\mathbb {G}))\\\\widehat{\\\\otimes }M_{cb}^l(L^1(\\\\mathbb {G}))}\\\\le 1.$ Define $\\\\mu _{ij}:=(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)$ .\",\n \"Then $\\\\mu _{ij}\\\\in L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _{ij}\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Vert f_j\\\\Vert _{cb}^2\\\\le \\\\Lambda _{cb}()^2.$ Given $f\\\\in L^1(\\\\mathbb {G})$ , for each $j$ we have $\\\\lim _i f\\\\star \\\\mu _{ij}-\\\\mu _{ij}\\\\star f&=\\\\lim _i(f\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1)-(1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f)\\\\\\\\&=\\\\lim _i(f\\\\otimes f_j^2)\\\\star \\\\mu _i-\\\\mu _i\\\\star (f_j^2\\\\otimes f)\\\\\\\\&=0$ in $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1(\\\\mathbb {G})$ and therefore in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ .\",\n \"Furthermore, $\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}(\\\\mu _{ij})\\\\star f&=\\\\lim _j \\\\lim _i m_{L^1_{cb}(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes 1))\\\\star f\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}((1\\\\otimes f_j)\\\\star \\\\mu _i\\\\star (f_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{L^1(\\\\mathbb {G})}(\\\\mu _i\\\\star (f^2_j\\\\otimes f))\\\\\\\\&=\\\\lim _j \\\\lim _i m_{C_u(\\\\mathbb {G})^*}(\\\\mu _i)m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=\\\\lim _j m_{L^1(\\\\mathbb {G})}(f^2_j\\\\otimes f)\\\\\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\\\in \\\\mathcal {Z}(L^1(\\\\mathbb {G}))$ .\",\n \"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\\\mu _I)$ in $L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})$ with $\\\\Vert \\\\mu _I\\\\Vert _{L^1_{cb}(\\\\mathbb {G})\\\\widehat{\\\\otimes }L^1_{cb}(\\\\mathbb {G})}\\\\le \\\\Lambda _{cb}()^2$ .\",\n \"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\\\mathbb {G})$ in $M_{cb}^r(L^1(\\\\mathbb {G}))$ .\",\n \"It is proved in the exact same way using the fact that $f\\\\star \\\\Gamma _u^*(\\\\mu )=f$ for all $f\\\\in L^1(\\\\mathbb {G})$ , which is easily verified.\",\n \"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.\",\n \"There are examples of weakly amenable residually finite groups (e.g.\",\n \"$G=SL(2,\\\\mathbb {Z}[\\\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].\",\n \"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.\",\n \"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\\\mathbb {G})=L^1_{cb}(\\\\mathbb {G})$ [55].\",\n \"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\\\ne 3$ .\",\n \"Since $\\\\widehat{O_N^+}$ and $\\\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\\\ne 3$ .\",\n \"If $\\\\mathbb {G}_1$ and $\\\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\\\Lambda _{cb}()=\\\\Lambda _{cb}()=1$ , then $\\\\mathbb {G}=\\\\mathbb {G}_1\\\\ast \\\\mathbb {G}_2$ also has the factorization property [11] and $\\\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\\\mathbb {G}))=1$ .\",\n \"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.\"\n ],\n [\n \"Decomposability\",\n \"For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].\",\n \"We now generalize this implication to arbitrary locally compact quantum groups.\",\n \"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.\",\n \"By [42], the universal co-representation $\\\\W _{\\\\mathbb {G}}=(\\\\textnormal {id}\\\\otimes \\\\pi _{})(\\\\mathbb {W}_{\\\\mathbb {G}})\\\\in M(C_u(\\\\mathbb {G})\\\\otimes _{\\\\min } C_0())$ from the proof of Proposition REF satisfies $\\\\hat{\\\\lambda }_u(f̉)=(\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}^*)$ for all $f̉\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ , $(\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}$ for all $x\\\\in C_u(\\\\mathbb {G})$ , where $\\\\hat{\\\\lambda }_u$ is the embedding of $L^1_*(\\\\widehat{\\\\mathbb {G}})$ into $C_u(\\\\mathbb {G})$ , and $\\\\pi _{\\\\mathbb {G}}:C_u(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ is the (unique) extension of $\\\\hat{\\\\lambda }:L^1_*(\\\\widehat{\\\\mathbb {G}})\\\\rightarrow C_0(\\\\mathbb {G})$ [42].\",\n \"Moreover, $C_u(\\\\mathbb {G})=\\\\overline{\\\\text{span}\\\\lbrace (\\\\textnormal {id}\\\\otimes f̉)(\\\\W _{\\\\mathbb {G}}) : f̉\\\\in L^1(\\\\widehat{\\\\mathbb {G}})\\\\rbrace }^{\\\\Vert \\\\cdot \\\\Vert _u}.$ We will need the following representation of $\\\\Theta ^l(\\\\mu )$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , which may be found in [19].\",\n \"We present the proof for the convenience of the reader.\",\n \"Lemma 7.1 For $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ , $\\\\Theta ^l(\\\\mu )(x)=(\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ First let $x=\\\\hat{\\\\lambda }(\\\\hat{f})\\\\in C_0(\\\\mathbb {G})$ for some $\\\\hat{f}\\\\in L^1_*(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Then, for all $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu )(x),f\\\\rangle &=\\\\langle x,m^l_\\\\mu (f)\\\\rangle =\\\\langle \\\\hat{\\\\lambda }(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle \\\\\\\\&=\\\\langle \\\\pi _{\\\\mathbb {G}}\\\\circ \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\\\\\&=\\\\langle \\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f})),\\\\mu \\\\otimes \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle (\\\\textnormal {id}\\\\otimes \\\\pi _{\\\\mathbb {G}})(\\\\Gamma _u(\\\\hat{\\\\lambda }_u(\\\\hat{f}))),\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(\\\\hat{\\\\lambda }_u(\\\\hat{f})))\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle =\\\\langle \\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},\\\\mu \\\\otimes f\\\\rangle \\\\\\\\&=\\\\langle (\\\\mu \\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}},f\\\\rangle .$ As $\\\\hat{\\\\lambda }(L^1_*(\\\\widehat{\\\\mathbb {G}}))$ is norm dense in $C_0(\\\\mathbb {G})$ , and since $C_0(\\\\mathbb {G})$ is weak* dense in $L^{\\\\infty }(\\\\mathbb {G})$ , the result follows.\",\n \"Recall that $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is the map taking $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ to the operator of left multiplication by $\\\\mu $ on $L^1(\\\\mathbb {G})$ .\",\n \"Theorem 7.2 Let $\\\\mathbb {G}$ be a locally compact quantum group.\",\n \"If $$ is amenable then $\\\\tilde{\\\\lambda }:C_u(\\\\mathbb {G})^*\\\\rightarrow M_{cb}^l(L^1(\\\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.\",\n \"Amenability of $$ entails the surjectivity of $\\\\tilde{\\\\lambda }$ from (the left version of) [14].\",\n \"For simplicity, throughout the proof we denote by $\\\\Theta ^l(\\\\mu )$ the map $\\\\Theta ^l(\\\\tilde{\\\\lambda }(\\\\mu ))$ for $\\\\mu \\\\in C_u(\\\\mathbb {G})^*$ .\",\n \"In [19] Daws shows that $\\\\Theta ^l:C_u(\\\\mathbb {G})^*_+\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is an order bijection.\",\n \"We show that it is a complete order bijection.\",\n \"To this end, let $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"By Lemma REF $\\\\Theta ^l(\\\\mu _{ij})(x)=(\\\\textnormal {id}\\\\otimes \\\\mu _{ij})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x)\\\\W _{\\\\mathbb {G}}, \\\\ \\\\ \\\\ x\\\\in L^{\\\\infty }(\\\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\\\in L^{\\\\infty }(\\\\mathbb {G})$ we have $((\\\\Theta ^l)^n([\\\\mu _{ij}]))^m([x_k^*x_l])&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}]\\\\\\\\&=[(\\\\mu _{ij}\\\\otimes \\\\textnormal {id})]^m([\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes x_k^*x_l)\\\\W _{\\\\mathbb {G}}])\\\\ge 0.$ It follows that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ .\",\n \"On the other hand, suppose $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)$ such that $(\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\in \\\\mathcal {CP}(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\\\hat{\\\\tau }$ on $$ , via $\\\\hat{\\\\tau }_t(\\\\hat{x})=P̉^{it}\\\\hat{x}P̉^{-it}$ , $\\\\hat{x}\\\\in L^{\\\\infty }(\\\\widehat{\\\\mathbb {G}})$ .\",\n \"Using [20], for $\\\\xi _1,...\\\\xi _m\\\\in \\\\mathcal {D}(P̉^{1/2})$ , $\\\\eta _1,...,\\\\eta _m\\\\in \\\\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\\\in {nm}$ , $\\\\langle [\\\\mu _{ij}]^m([\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})])[z_{ik}],[z_{ik}]\\\\rangle &=\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^m\\\\overline{z_{ik}}z_{jl}\\\\langle \\\\mu _{ij},\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\rangle \\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\mu _{ij}^*,\\\\lambda _u(\\\\omega _{\\\\xi _l,\\\\eta _l})\\\\lambda _u(\\\\omega _{\\\\xi _k,\\\\eta _k})^*\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{i,j=1}^n\\\\sum _{k,l=1}^mz_{ik}\\\\overline{z_{jl}}\\\\langle \\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)\\\\eta _k,\\\\eta _l\\\\rangle }\\\\\\\\&=\\\\overline{\\\\sum _{k,l=1}^m \\\\langle [\\\\Theta ^l(\\\\mu _{ij})(\\\\xi _l\\\\xi _k^*)]y_k,y_l\\\\rangle }\\\\ge 0,$ where $y_k=[z_{1k}\\\\eta _k \\\\ \\\\cdots \\\\ z_{nk}\\\\eta _k]^T\\\\in L^2(\\\\mathbb {G})^n$ for $1\\\\le k\\\\le m$ .\",\n \"By density of $\\\\lbrace \\\\omega _{\\\\xi ,\\\\eta }\\\\mid \\\\xi \\\\in \\\\mathcal {D}(P̉^{1/2}), \\\\eta \\\\in \\\\mathcal {D}(P̉^{-1/2})\\\\rbrace $ in $L^1_*(\\\\widehat{\\\\mathbb {G}})$ [21], it follows that $[\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+$ .\",\n \"We now show that $\\\\tilde{\\\\lambda }$ is a complete isometry.\",\n \"To do so we introduce a decomposability norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , given by $\\\\Vert \\\\Phi \\\\Vert _{L^1dec}:=\\\\inf \\\\bigg \\\\lbrace \\\\max \\\\lbrace \\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}\\\\rbrace \\\\mid \\\\begin{bmatrix}\\\\Psi _1 & \\\\Phi \\\\\\\\ \\\\Phi ^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0\\\\bigg \\\\rbrace ,$ where $\\\\Psi _i\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"It is evident that $\\\\Vert \\\\cdot \\\\Vert _{L^1dec}$ is a norm on $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ .\",\n \"That $\\\\Vert \\\\Phi \\\\Vert _{cb}\\\\le \\\\Vert \\\\Phi \\\\Vert _{L^1dec}$ for all $\\\\Phi \\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ follows verbatim from the first part of [28].\",\n \"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G}))).$ Since $\\\\Theta ^l$ is a completely positive contraction from $(C_u(\\\\mathbb {G})^*)^+$ onto $_{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}))$ , one easily sees that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}\\\\le \\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}, \\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*),$ where $\\\\Vert \\\\cdot \\\\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.\",\n \"Conversely, if $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{dec}<1$ then there exist $\\\\Psi _1,\\\\Psi _2\\\\in \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CP}^{\\\\sigma }(L^{\\\\infty }(\\\\mathbb {G}),M_n(L^{\\\\infty }(\\\\mathbb {G})))$ such that $\\\\Vert \\\\Psi _1\\\\Vert _{cb},\\\\Vert \\\\Psi _2\\\\Vert _{cb}<1$ , and $\\\\begin{bmatrix}\\\\Psi _1 & (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\\\\\ (\\\\Theta ^l)^n([\\\\mu _{ij}])^* & \\\\Psi _2\\\\end{bmatrix}\\\\ge _{cp}0.$ Since $(\\\\Theta ^l)^n$ is a complete order bijection there exist $[\\\\nu ^k_{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*)^+=\\\\mathcal {CP}(C_u(\\\\mathbb {G}),M_n)$ such that $\\\\Psi _k=(\\\\Theta ^l)^n([\\\\nu ^k_{ij}])$ , $k=1,2$ , and $\\\\begin{bmatrix}[\\\\nu ^1_{ij}] & [\\\\mu _{ij}]\\\\\\\\ [\\\\mu _{ij}]^* & [\\\\nu ^2_{ij}]\\\\end{bmatrix}\\\\ge _{cp}0.$ It follows that $[\\\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\\\mathbb {G})\\\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\\\widetilde{[\\\\nu ^k_{ij}]}:M(C_u(\\\\mathbb {G}))\\\\rightarrow M_n$ which is strictly continuous on the unit ball [45].\",\n \"By uniqueness, $\\\\widetilde{[\\\\nu ^k_{ij}]}=[\\\\tilde{\\\\nu }^k_{ij}]$ , where $\\\\tilde{\\\\nu }^k_{ij}$ is the unique strict extension of the functional $\\\\nu ^k_{ij}$ .\",\n \"Thus, by completely positivity $\\\\Vert [\\\\nu ^k_{ij}] \\\\Vert _{cb}=\\\\Vert \\\\widetilde{[\\\\nu ^k_{ij}]}(1_{M(C_u(\\\\mathbb {G}))})\\\\Vert =\\\\Vert [\\\\tilde{\\\\nu }^k_{ij}(1_{M(C_u(\\\\mathbb {G}))})]\\\\Vert =\\\\Vert \\\\Psi _k(1)\\\\Vert =\\\\Vert \\\\Psi _k\\\\Vert _{cb}<1,$ so that $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}<1$ .\",\n \"Therefore $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}.$ However, $\\\\Vert [\\\\mu _{ij}]\\\\Vert _{dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}, ,\\\\ \\\\ \\\\ [\\\\mu _{ij}]\\\\in M_n(C_u(\\\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\\\infty }(\\\\mathbb {G})$ in $L^1(\\\\mathbb {G})\\\\hspace{2.0pt}\\\\mathbf {mod}$ by the left version of [14].\",\n \"The matricial analogues of the proofs of [14] show that $\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{cb}=\\\\Vert (\\\\Theta ^l)^n([\\\\mu _{ij}])\\\\Vert _{L^1dec}=\\\\Vert [\\\\mu _{ij}]\\\\Vert _{cb}.$ Hence, $\\\\Theta ^l:C_u(\\\\mathbb {G})^*\\\\rightarrow \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}^\\\\sigma (L^{\\\\infty }(\\\\mathbb {G}))$ is a completely isometric isomorphism.\",\n \"To prove that $\\\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.\",\n \"Let $(\\\\mu _i)$ be a bounded net in $C_u(\\\\mathbb {G})^*$ converging weak* to $\\\\mu $ .\",\n \"Since $C_u(\\\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.\",\n \"Hence, for $\\\\hat{f}\\\\in L^1(\\\\widehat{\\\\mathbb {G}})$ and $f\\\\in L^1(\\\\mathbb {G})$ , $\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle =\\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu _i\\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle \\\\rightarrow \\\\langle \\\\hat{\\\\lambda }_u(\\\\hat{f}),\\\\mu \\\\star _u \\\\pi _{\\\\mathbb {G}}^*(f)\\\\rangle =\\\\langle \\\\Theta ^l(\\\\mu _i)(\\\\hat{\\\\lambda }(\\\\hat{f})),f\\\\rangle .$ The density of $\\\\hat{\\\\lambda }(L^1(\\\\widehat{\\\\mathbb {G}}))$ in $C_0(\\\\mathbb {G})$ and boundedness of $\\\\Theta ^l(\\\\mu _i)$ in $_{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))=(Q_{cb}^l(\\\\mathbb {G}))^*$ (see [14]) establish the claim.\",\n \"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\\\mathbb {G}$ is co-amenable [35].\",\n \"Corollary 7.4 Let $\\\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.\",\n \"Then $\\\\mathbb {G}$ is co-amenable if and only if $$ is amenable.\",\n \"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\\\mathbb {G})$ .\",\n \"Moreover, as noted in the proof of Proposition REF , $L^{\\\\infty }(\\\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).\",\n \"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\\\mathcal {T}(H)\\\\twoheadrightarrow C_0(\\\\mathbb {G})$ .\",\n \"Then $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }\\\\mathcal {T}(H)\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ by projectivity of $\\\\widehat{\\\\otimes }$ , and $L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }\\\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.\",\n \"Hence, $\\\\Theta ^l(f_i)\\\\otimes \\\\textnormal {id}_{M(\\\\mathbb {G})}(X)\\\\rightarrow X,$ weak* for all $X\\\\in L^{\\\\infty }(\\\\mathbb {G})\\\\overline{\\\\otimes }M(\\\\mathbb {G})$ , so that $\\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A)\\\\rightarrow A$ weakly for all $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\\\Vert \\\\Theta ^l(f_i)_*\\\\otimes \\\\textnormal {id}_{C_0(\\\\mathbb {G})}(A) - A\\\\Vert \\\\rightarrow 0, \\\\ \\\\ \\\\ A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ .\",\n \"Let $\\\\tilde{m}:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow C_0(\\\\mathbb {G})$ denote the induced map and $q:L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ denote the quotient map.\",\n \"It follows that $q(f\\\\star A)=q(f\\\\otimes m(A))$ for all $f\\\\in L^1(\\\\mathbb {G})$ and $A\\\\in L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }C_0(\\\\mathbb {G})$ .\",\n \"Thus, if $m(A)=0$ , then $q(A)=\\\\lim _i q(f_i\\\\star A)=\\\\lim _iq(f_i\\\\otimes m(A))=0,$ so that the induced multiplication $\\\\tilde{m}$ is injective.\",\n \"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\\\mathbb {G})^*\\\\cong \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ completely isometrically.\",\n \"By the left version of [14] $L^1(\\\\mathbb {G})$ is 1-flat in $\\\\mathbf {mod}\\\\hspace{2.0pt}L^1(\\\\mathbb {G})$ .\",\n \"Thus, the following sequence is 1-exact $0\\\\rightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})\\\\hookrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\twoheadrightarrow L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})\\\\rightarrow 0.$ Since $f\\\\star x=(\\\\textnormal {id}\\\\otimes f\\\\circ \\\\pi _{\\\\mathbb {G}})\\\\circ \\\\Gamma _u(x)=(\\\\textnormal {id}\\\\otimes f)\\\\W _{\\\\mathbb {G}}^*(1\\\\otimes \\\\pi _{\\\\mathbb {G}}(x))\\\\W _{\\\\mathbb {G}}=0$ for all $x\\\\in \\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})$ , it follows that $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}\\\\mathrm {Ker}(\\\\pi _{\\\\mathbb {G}})=0$ .\",\n \"Hence, $L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})$ .\",\n \"Moreover, as $\\\\Theta ^l(\\\\mu \\\\star f)(x)=\\\\Theta ^l(f)(\\\\Theta ^l(\\\\mu )(x))=(\\\\Theta ^l(\\\\mu )(x))\\\\star f, \\\\ \\\\ \\\\ f\\\\in L^1(\\\\mathbb {G}), \\\\ \\\\mu \\\\in C_u(\\\\mathbb {G})^*, \\\\ x\\\\in C_0(\\\\mathbb {G}),$ it follows that $C_u(\\\\mathbb {G})\\\\cong L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})} C_0(\\\\mathbb {G})$ is an isomorphism of left $L^1(\\\\mathbb {G})$ -modules, i.e., $C_u(\\\\mathbb {G})$ is induced.\",\n \"The commutative diagram $\\\\begin{tikzcd}L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_u(\\\\mathbb {G}) [r, equal][d, equal] &L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G})[d, \\\"\\\\tilde{m}\\\"]\\\\\\\\C_u(\\\\mathbb {G}) [r, two heads] &C_0(\\\\mathbb {G})\\\\end{tikzcd}$ then implies that $\\\\tilde{m}$ is a complete quotient map.\",\n \"Thus, $C_0(\\\\mathbb {G})$ is an induced $L^1(\\\\mathbb {G})$ -module and $M(\\\\mathbb {G})\\\\cong (L^1(\\\\mathbb {G})\\\\widehat{\\\\otimes }_{L^1(\\\\mathbb {G})}C_0(\\\\mathbb {G}))^*= \\\\ _{L^1(\\\\mathbb {G})}\\\\mathcal {CB}(C_0(\\\\mathbb {G}),L^{\\\\infty }(\\\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\\\mathbb {G})\\\\hookrightarrow L^{\\\\infty }(\\\\mathbb {G})$ is necessarily a left unit for $M(\\\\mathbb {G})$ , which entails the co-amenability of $\\\\mathbb {G}$ .\"\n ],\n [\n \"Acknowledgements\",\n \"The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.\",\n \"The author was partially supported by the NSERC Discovery Grant 1304873.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01770"}}},{"rowIdx":991,"cells":{"text":{"kind":"list like","value":[["Puiseux monoids and transfer homomorphisms"],["Abstract There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades.","The factorization theory of finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most systematically studied.","Puiseux monoids, which are additive submonoids of $\\mathbb{Q}_{\\ge 0}$ consisting of nonnegative rational numbers, have only been studied recently.","In this paper, we provide evidence that this family comprises plenty of monoids with a basically unexplored atomic structure.","We do this by showing that the arithmetical invariants of the well-studied atomic monoids mentioned earlier cannot be transferred to most Puiseux monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms.","Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux monoid to a finitely generated monoid do not exist.","We also find a large family of Puiseux monoids that fail to be strongly primary.","In addition, we prove that the only nontrivial Puiseux monoid that accepts a transfer homomorphism to a Krull monoid is $\\mathbb{N}_0$.","Finally, we classify the Puiseux monoids that happen to be C-monoids."],["Introduction","The study of the phenomenon of non-unique factorizations in the ring of integers $\\mathcal {O}_K$ of an algebraic number field $K$ was initiated by L. Carlitz in the 1950's, and it was later carried out on more general integral domains.","As a result, many techniques to measure the non-uniqueness of factorizations in several families of integral domains were systematically developed during the second half of the last century (see [2] and references therein).","However, it was not until recently that questions about the non-uniqueness of factorizations were abstractly formulated in the context of commutative cancellative monoids.","This was possible because most of the factorization-related questions inside an integral domain are purely multiplicative in essence.","The fundamental goal of abstract (or modern) factorization theory is to measure how far is a commutative cancellative monoid from being factorial by using different arithmetical invariants.","At this point, the arithmetical invariants of several families of atomic monoids have been intensively studied.","Finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most studied.","These families of monoids not only have very diverse arithmetical properties, but also have proved to be useful in the study of the factorization theory of less-understood atomic monoids via transfer homomorphisms.","A monoid homomorphism is said to be transfer if somehow it allows to shift the atomic structure of its codomain back to its domain (see Definition REF ).","Therefore if one is willing to know the factorization invariants of a given monoid, it suffices to find a transfer homomorphism from such a monoid to a better-understood monoid and carry over the desired factorization properties.","Puiseux monoids were recently introduced as a rational generalization of numerical monoids.","Many families of atomic Puiseux monoids were explored in [16], [17], [19], and their elasticity was studied in [20].","However, it is still unanswered whether the non-unique factorization behavior in Puiseux monoids is somehow similar to that of some of the monoids whose factorization properties are already well-understood.","To give a partial answer to this, we will determine which atomic Puiseux monoids can be the domain of a transfer homomorphism to some of the monoids whose arithmetical invariants have already been studied.","In particular, we consider finitely generated monoids, Krull monoids, and C-monoids as our transfer codomains.","The content of this paper is organized as follows.","In Section , we establish the notation we shall be using later, and we formally present most of the fundamental concepts needed in this paper.","Then, in Section , we show that homomorphisms between Puiseux monoids can only be given by rational multiplication, which will allow us to characterize the transfer homomorphisms between Puiseux monoids.","We also present a family of Puiseux monoids whose members have $\\mathbb {Z}$ as their group of automorphisms.","Section is devoted to characterize the Puiseux monoids admitting a transfer homomorphism to some finitely generated monoid.","Then, in Section we investigate which Puiseux monoids are strongly primary.","Finally, in Section , we prove that the only Puiseux monoid that is transfer Krull is the additive monoid $\\mathbb {N}_0$ .","We use this information to classify the Puiseux monoids which happen to be C-monoids."],["Background","To begin with let us introduce the fundamental concepts related to our exposition as an excuse to establish the notation we need.","The reader can consult Grillet [21] for information on commutative semigroups and Geroldinger and Halter-Koch [10] for extensive background in non-unique factorization theory of atomic monoids.","Throughout this sequel, we let $\\mathbb {N}$ denote the set of positive integers, and we set $\\mathbb {N}_0 := \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .","For $X \\subseteq \\mathbb {R}$ and $r \\in \\mathbb {R}$ , we set $X_{\\le r} := \\lbrace x \\in X \\mid x \\le r\\rbrace $ ; with a similar spirit we use the symbols $X_{\\ge r}$ , $X_{< r}$ , and $X_{> r}$ .","If $q \\in \\mathbb {Q}_{> 0}$ , then we call the unique $a,b \\in \\mathbb {N}$ such that $q = a/b$ and $\\gcd (a,b)=1$ the numerator and denominator of $q$ and denote them by $\\mathsf {n}(q)$ and $\\mathsf {d}(q)$ , respectively.","For each subset $Q$ of $\\mathbb {Q}_{>0}$ , we call the sets $\\mathsf {n}(Q) = \\lbrace \\mathsf {n}(q) \\mid q \\in Q\\rbrace $ and $\\mathsf {d}(Q) = \\lbrace \\mathsf {d}(q) \\mid q \\in Q\\rbrace $ the numerator set and denominator set of $Q$ , respectively.","As usual, a semigroup is a pair $(S, *)$ , where $S$ is a set and $*$ is an associative binary operation in $S$ ; we write $S$ instead of $(S,*)$ provided that $*$ is clear from the context.","However, inside the scope of this paper, a monoid is a commutative cancellative semigroup with identity (cf.","the standard definition of monoid).","To comply with established conventions, we will be using simultaneously additive and multiplicative notations; however, the context will always save us from the risk of ambiguity.","Let $M$ be a monoid written additively.","We set $M^\\bullet := M \\!","\\setminus \\!","\\lbrace 0\\rbrace $ and, as usual, we let $M^\\times $ denote the set of units (i.e., invertible elements) of $M$ .","The monoid $M$ is reduced if $M^\\times = \\lbrace 0\\rbrace $ .","For $a, b \\in M$ , we say that $a$ divides $b$ in $M$ if there exists $c \\in M$ such that $b = a + c$ ; in this case we write $a \\mid _M b$ .","An element $a \\in M \\!","\\setminus \\!","M^\\times $ is an atom if whenever $a = u + v$ for some $u,v \\in M$ , either $u \\in M^\\times $ or $v \\in M^\\times $ .","Atoms are the building blocks in factorization theory; this motivates the especial notation $\\mathcal {A}(M) := \\lbrace a \\in M \\mid a \\text{ is an atom of } M\\rbrace .$ For $S \\subseteq M$ , we let $\\langle S \\rangle $ denote the smallest submonoid of $M$ containing $S$ , and we say that $S$ generates $M$ if $M = \\langle S \\rangle $ .","The monoid $M$ is said to be finitely generated if it can be generated by a finite set.","On the other hand, we say that $M$ is atomic if $M = \\langle \\mathcal {A}(M) \\rangle $ .","A monoid is factorial if every element can be written as a sum of primes.","As every prime is an atom, every factorial monoid is atomic.","Let $\\rho \\subseteq M \\times M$ be an equivalence relation on $M$ , and let $[a]_\\rho $ denote the equivalence class of $a \\in M$ .","We say that $\\rho $ is a congruence if for all $a,b,c \\in M$ such that $(a,b) \\in \\rho $ it follows that $(ca,cb) \\in \\rho $ .","Congruences are precisely the equivalence relations that are compatible with the operation of $M$ , meaning that $M/\\rho := \\lbrace [a]_\\rho \\mid a \\in M\\rbrace $ is a commutative semigroup with identity (no necessarily cancellative).","Two elements $a,b \\in M$ are associates, and we write $a \\simeq b$ , if $a = ub$ for some $u \\in M^\\times $ .","Being associates defines a congruence relation $\\simeq $ on $M$ , and $M_{\\text{red}} := M/\\!\\!\\simeq $ is called the associated reduced semigroup of $M$ .","We say that a multiplicative monoid $F$ is free abelian with basis $P \\subset F$ if every element $a \\in F$ can be written uniquely in the form $a = \\prod _{p \\in P} p^{\\mathsf {v}_p(a)},$ where $\\mathsf {v}_p(a) \\in \\mathbb {N}_0$ and $\\mathsf {v}_p(a) > 0$ only for finitely many elements $p \\in P$ .","The monoid $F$ is determined by $P$ up to canonical isomorphism, so we shall also denote $F$ by $\\mathcal {F}(P)$ .","By the fundamental theorem of arithmetic, the multiplicative monoid $\\mathbb {N}$ is free on the set of prime numbers.","In this case, we can extend $\\mathsf {v}_p$ to $\\mathbb {Q}_{\\ge 0}$ as follows.","For $r \\in \\mathbb {Q}_{> 0}$ let $\\mathsf {v}_p(r) := \\mathsf {v}_p(\\mathsf {n}(r)) - \\mathsf {v}_p(\\mathsf {d}(r))$ and set $\\mathsf {v}_p(0) = \\infty $ .","The free abelian monoid on $\\mathcal {A}(M)$ , denoted by $\\mathsf {Z}(M)$ , is called the factorization monoid of $M$ , and the elements of $\\mathsf {Z}(M)$ are called factorizations.","If $z = a_1 \\dots a_n \\in \\mathsf {Z}(M)$ for some $n \\in \\mathbb {N}_0$ and $a_1, \\dots , a_n \\in \\mathcal {A}(M)$ , then $n$ is the length of the factorization $z$ ; the length of $z$ is denoted by $|z|$ .","The unique homomorphism $\\phi \\colon \\mathsf {Z}(M) \\rightarrow M \\ \\ \\text{satisfying} \\ \\ \\phi (a) = a \\ \\ \\text{for all} \\ \\ a \\in \\mathcal {A}(M)$ is called the factorization homomorphism of $M$ .","Additionally, for $x \\in M^\\bullet $ , $\\mathsf {Z}(x) := \\phi ^{-1}(x) \\subseteq \\mathsf {Z}(M)$ is the set of factorizations of $x$ .","By definition, we set $\\mathsf {Z}(0) = \\lbrace 0\\rbrace $ .","Note that the monoid $M$ is atomic if and only if $\\mathsf {Z}(x)$ is not empty for all $x \\in M$ .","For each $x \\in M$ , the set of lengths of $x$ is defined by $\\mathsf {L}(x) := \\lbrace |z| : z \\in \\mathsf {Z}(x)\\rbrace .$ We say that the monoid $M$ is half-factorial if $|\\mathsf {L}(x)| = 1$ for all $x \\in M$ .","On the other hand, if $\\mathsf {L}(x)$ is a finite set for all $x \\in M$ , then we say that $M$ is a BF-monoid.","The system of sets of lengths of $M$ is defined by $\\mathcal {L}(M) := \\lbrace \\mathsf {L}(x) \\mid x \\in M\\rbrace .$ The system of sets of lengths is an arithmetical invariant of atomic monoids that has received significant attention in recent years (see [1], [4] and the literature cited there).","A very special family of atomic monoids is that one comprising all numerical monoids, cofinite submonoids of the additive monoid $\\mathbb {N}_0$ .","We say that a numerical monoid is proper if it is strictly contained in $\\mathbb {N}_0$ .","Each numerical monoid has a unique minimal set of generators, which is finite.","Moreover, if $\\lbrace a_1, \\dots , a_n\\rbrace $ is the minimal set of generators for a numerical monoid $N$ , then $\\mathcal {A}(N) = \\lbrace a_1, \\dots , a_n\\rbrace $ and $\\gcd (a_1, \\dots , a_n) = 1$ .","As a result, every numerical monoid is atomic and contains only finitely many atoms.","The Frobenius number of $N$ , denoted by $F(N)$ , is the minimum $n \\in \\mathbb {N}$ such that $\\mathbb {Z}_{> n} \\subset N$ .","An introduction to numerical monoids can be found in [7].","An additive submonoid of $\\mathbb {Q}_{\\ge 0}$ is called a Puiseux monoid.","Puiseux monoids are a natural generalization of numerical monoids.","However, the general atomic structure of Puiseux monoids drastically differs from that one of numerical monoids.","Puiseux monoids are not always atomic; for instance, consider $\\langle 1/2^n \\mid n \\in \\mathbb {N}\\rangle $ .","On the other hand, if an atomic Puiseux monoid $M$ is not isomorphic to a numerical monoid, then $\\mathcal {A}(M)$ is infinite.","The atomic structure of Puiseux monoids has been studied in [17] and [19], where several families of atomic Puiseux monoids were described."],["Homomorphisms Between Puiseux Monoids","In this section we present characterizations of homomorphisms and transfer homomorphisms between Puiseux monoids.","Let us start by introducing the concept of a transfer homomorphism, which is going to play a central role in this paper.","Definition 3.1 A monoid homomorphism $\\theta \\colon M \\rightarrow N$ is said to be a transfer homomorphism if the following conditions hold: (T1) $N = \\theta (M) N^\\times $ and $\\theta ^{-1}(N^\\times ) = M^\\times $ ; (T2) if $\\theta (a) = b_1 b_2$ for $a \\in M$ and $b_1,b_2 \\in N$ , then there exist $a_1, a_2 \\in M$ such that $a = a_1 a_2$ and $\\theta (a_i) = b_i$ for $i \\in \\lbrace 1,2\\rbrace $ .","We proceed to characterize the homomorphisms between Puiseux monoids.","This will immediately yield a characterization of those homomorphisms between Puiseux monoids that happen to be transfer homomorphisms.","Proposition 3.2 If $\\phi \\colon M \\rightarrow N$ be a homomorphism between Puiseux monoids, then the following conditions hold.","There exists $q \\in \\mathbb {Q}_{\\ge 0}$ such that $\\phi (x) = qx$ for all $x \\in M$ , i.e., $\\phi $ is given by rational multiplication.","The homomorphism $\\phi $ is a transfer homomorphism if and only if it is surjective.","Let us argue first that $\\phi $ is given by rational multiplication.","It is clear that if a map $P \\rightarrow P^{\\prime }$ between two Puiseux monoids is multiplication by a rational number, then it is a monoid homomorphism.","Thus, it suffices to verify that the only homomorphisms of Puiseux monoids are those given by rational multiplication.","To do this, consider the Puiseux monoid homomorphism $\\varphi \\colon P \\rightarrow P^{\\prime }$ .","Because the trivial homomorphism is multiplication by 0, there is no loss in assuming that $P \\ne \\lbrace 0\\rbrace $ .","Let $\\lbrace n_1, \\dots , n_k\\rbrace $ be a minimal set of generators for the additive monoid $N = P \\cap \\mathbb {N}_0$ .","Notice that $N \\ne \\lbrace 0\\rbrace $ and, therefore, $k \\ge 1$ .","The fact that $\\varphi $ is nontrivial implies that $\\varphi (n_j) \\ne 0$ for some $j \\in \\lbrace 1,\\dots ,k\\rbrace $ .","Set $q = \\varphi (n_j)/n_j$ , and take $r \\in P^\\bullet $ and $c_1, \\dots , c_k \\in \\mathbb {N}_0$ satisfying that $\\mathsf {n}(r) = c_1 n_1 + \\dots + c_k n_k$ .","Since $n_i \\varphi (n_j) = \\varphi (n_i n_j) = n_j \\varphi (n_i)$ for each $i \\in \\lbrace 1,\\dots ,k\\rbrace $ , one obtains $\\varphi (r) = \\frac{1}{\\mathsf {d}(r)} \\varphi (\\mathsf {n}(r)) = \\frac{1}{\\mathsf {d}(r)} \\sum _{i=1}^k c_i \\varphi (n_i) = \\frac{1}{\\mathsf {d}(r)} \\sum _{i=1}^k c_i n_i \\frac{\\varphi (n_j)}{n_j} = rq.$ As a result, the homomorphism $\\varphi $ is just multiplication by $q \\in \\mathbb {Q}_{>0}$ .","It is easy to see that condition (2) is a direct consequence of condition (1), which completes the proof.","Remark 3.3 A Puiseux monoid $M$ is said to be increasing (resp., decreasing) if $M$ can be generated by an increasing (resp., decreasing) sequence of rational numbers.","Also, we say that $M$ is bounded if $M$ can be generated by a bounded sequence of rational numbers, and it is said to be strongly bounded if it can be generated by a sequence of rational numbers whose numerator set is bounded.","Finally, $M$ is called dense if it contains 0 as a limit point.","Although the definitions just given are not algebraic in nature, we should notice that they are all preserved by Puiseux monoid isomorphisms.","This explains why all of them have been useful in the study of the atomic structure of Puiseux monoids (see [17] and [19]).","With notation as in Definition REF , when $M$ and $N$ are reduced, we can restate the first condition above as (T1') $\\theta $ is surjective and $\\theta ^{-1}(1) = 1$ .","We have already mentioned that a transfer homomorphism allows us to shift the atomic structure and the arithmetic of length of factorizations from its codomain to its domain.","This property is formally described in the following proposition.","Proposition 3.4 [9] If $\\theta \\colon M \\rightarrow N$ is a transfer homomorphism of atomic monoids, then the following conditions hold: $a \\in \\mathcal {A}(M)$ if and only if $\\theta (a) \\in \\mathcal {A}(N)$ ; $M$ is atomic if and only if $N$ is atomic; $\\mathsf {L}_M(x) = \\mathsf {L}_N(\\theta (x))$ for all $x \\in M$ ; $\\mathcal {L}(M) = \\mathcal {L}(N)$ , and so $M$ is a BF-monoid if and only if $N$ is a BF-monoid.","For a Puiseux monoid $M$ , let $\\text{Aut}(M)$ denote the group of automorphisms of $M$ .","As we have seen in Proposition REF , the set of homomorphisms between Puiseux monoids is very exclusive.","In particular, we might wonder whether $\\text{Aut}(M)$ is always trivial.","However, it is not hard to verify, for instance, that when $M_1 = \\langle 1/2^n \\mid n \\in \\mathbb {N}\\rangle $ , multiplication by $1/2$ is in $\\text{Aut}(M_1)$ .","This example might not be the most desirable because $M_1$ fails to be atomic; in fact, $M_1$ does not contain any atoms.","The next proposition exhibits a family of atomic monoids whose groups of automorphisms are nontrivial.","First, let us introduce a family of atomic Puiseux monoids whose atomicity is used in the proof.","For $r \\in \\mathbb {Q}_{>0}$ , the monoid $M_r = \\langle r^n \\mid n \\in \\mathbb {N}\\rangle $ is the multiplicatively $r$ -cyclic Puiseux monoid.","If $\\mathsf {n}(r), \\mathsf {d}(r) > 1$ , then $M_r$ is atomic with $\\mathcal {A}(M_r) = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ (see [19]).","Proposition 3.5 Let $r \\in \\mathbb {Q}_{>0}$ such that $\\mathsf {n}(r), \\mathsf {d}(r) > 1$ .","If $M = \\langle r^n \\mid n \\in \\mathbb {Z}\\rangle $ , then $\\emph {Aut}(M) \\cong \\mathbb {Z}$ .","Set $A = \\lbrace r^n \\mid n \\in \\mathbb {Z}\\rbrace $ .","For $n \\in \\mathbb {Z}$ , the fact that $r^n A = A$ implies that multiplication by $r^n$ is an endomorphism of $M$ whose inverse is given by multiplication by $r^{-n}$ .","Thus, multiplication by any integer power of $r$ is an automorphism of $M$ .","To prove that these are the only elements of $\\text{Aut}(M)$ , let us first argue that $M$ is atomic with $\\mathcal {A}(M) = A$ .","Assume first that $r < 1$ .","Fix $k \\in \\mathbb {Z}$ , and let us check that $r^k \\in \\mathcal {A}(M)$ .","To do this notice that the monoid $\\langle r^n \\mid n \\ge k \\rangle $ is the isomorphic image (under multiplication by $r^{k-1}$ ) of the multiplicatively $r$ -cyclic Puiseux monoid $M_r$ , which is atomic with set of atoms $A = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ .","Since $r \\in \\mathcal {A}(M_r)$ , it follows that $r \\notin \\langle r^n \\mid n > 1 \\rangle $ .","Then $r^k \\notin \\langle r^n \\mid n > k \\rangle $ .","As $r < 1$ , no atom in $\\lbrace r^n \\mid n < k\\rbrace $ divides $r^k$ .","Hence $r^k \\notin \\langle A \\setminus \\lbrace r^k\\rbrace \\rangle $ and, therefore, $r^k \\in \\mathcal {A}(M)$ .","As a result, $\\mathcal {A}(M) = A$ .","Now suppose that $r > 1$ .","As before, fix $k \\in \\mathbb {Z}$ .","Because $r > 1$ , proving that $r^k \\in \\mathcal {A}(M)$ amounts to showing that $r^k \\notin \\langle r^n \\mid n < k \\rangle $ .","Let us assume, by way of contradiction, that this is not the case.","Then $r^k = a_1 r^{n_1} + \\dots + a_t r^{n_t}$ for some $a_1, \\dots , a_t \\in \\mathbb {N}$ and $n_1, \\dots , n_t \\in \\mathbb {N}$ with $k > n_1 > \\dots > n_t$ .","As a consequence, $r^{k - n_t + 1} \\in \\langle r^{n_1 - n_t + 1}, r^{n_2 - n_t + 1}, \\dots , r \\rangle $ , which contradicts the fact that $r^{k - n_t + 1} \\in \\mathcal {A}(M_r)$ .","As in the previous case, we conclude that $\\mathcal {A}(M) = A$ .","By Proposition REF , any automorphism of $M$ is given by rational multiplication.","Take $s \\in \\mathbb {Q}_{>0}$ such that $\\phi _s \\in \\text{Aut}(M)$ , where $\\phi _s$ consists in left multiplication by $s$ .","Because $\\phi _s$ must send atoms to atoms, it follows that $sr = \\phi _s(r) \\in A$ .","Therefore $s$ must be an integer power of $r$ .","Hence $\\text{Aut}(M)$ is precisely $A$ when seen as a multiplicative subgroup of $\\mathbb {Q}$ .","As $A$ is the infinite cyclic group, the proof follows."],["Finite Transfer Puiseux Monoids","Now we turn to characterize the transfer homomorphisms from Puiseux monoids to finitely generated monoids.","Definition 4.1 We say that a Puiseux monoid $M$ is transfer finite if there exists a transfer homomorphism from $M$ to a finitely generated monoid.","By the fundamental structure theorem of finitely generated abelian groups, it immediately follows that every finitely generated monoid $F$ is a submonoid of a group $T \\times \\mathbb {Z}^\\beta $ for some finite abelian group $T$ and $\\beta \\in \\mathbb {N}_0$ .","In case of $F$ being reduced, it can be thought of as a submonoid of $T \\times \\mathbb {N}_0^\\beta $ .","Condition (T2) in the definition of a transfer homomorphism $\\theta \\colon M \\rightarrow F$ is crucial to transfer the factorization behavior of $F$ to $M$ .","However, the reader might wonder how much the set $\\text{Hom}(M,F)$ will increase if we drop condition (T2).","Surprisingly, the set of homomorphisms will remain the same as long as we impose $F$ to be reduced.","This fact facilitates to classify the Puiseux monoids that happen to be transfer finite, as we will prove in the next theorem.","First, notice that if $\\phi \\colon M \\rightarrow N$ is a monoid homomorphism, then the map $\\phi _{\\text{red}} \\colon M_{\\text{red}} \\rightarrow N_{\\text{red}}$ defined by $\\phi _{\\text{red}}(aM^\\times ) = \\phi (a)N^\\times $ is also a monoid homomorphism.","Theorem 4.2 Let $M$ be a nontrivial Puiseux monoid, and let $F$ be a finitely generated (additive) monoid.","If $\\theta \\colon M \\rightarrow F$ is a homomorphism satisfying $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ , then $M$ is isomorphic to a numerical monoid.","The Puiseux monoid $M$ is transfer finite if and only if it is isomorphic to a numerical monoid.","We argue first part (1).","It is easy to see that $\\theta _{\\text{red}} \\colon M_{\\text{red}} = M \\rightarrow F_{\\text{red}}$ is also a transfer homomorphism.","So we can assume, without loss of generality, that $F$ is reduced.","Suppose that $F$ is a submonoid of $T \\times \\mathbb {N}_0^\\beta $ , where $T$ is a finite abelian group and $\\beta \\in \\mathbb {N}_0$ .","First, assume, by way of contradiction, that $\\beta = 0$ .","In this case, it is not hard to verify that $\\theta (M)$ must be a subgroup of $T$ .","If $\\alpha = |\\theta (M)|$ and $r \\in M^\\bullet $ , then $\\theta (\\alpha r) = \\alpha \\theta (r) = 0$ .","This contradicts that $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ .","Thus, $\\beta \\ge 1$ .","Define $\\pi \\colon T \\times \\mathbb {N}_0^\\beta \\rightarrow \\mathbb {N}_0^\\beta $ by $\\pi (t,\\mathbf {v}) = \\mathbf {v}$ for all $t \\in T$ and $\\mathbf {v} \\in \\mathbb {N}^\\beta $ .","Let us verify that $\\pi (\\theta (M))$ is finitely generated.","Take $\\textbf {x} = (x_1, \\dots , x_\\beta ) \\in \\pi (\\theta (M))^\\bullet $ , and let $d = \\gcd (x_1, \\dots , x_\\beta )$ .","We shall verify that $\\pi (\\theta (M)) \\subseteq \\langle \\textbf {x}/d \\rangle $ .","To do so, consider $\\textbf {y} = (y_1, \\dots , y_\\beta ) \\in \\pi (\\theta (M))^\\bullet $ .","Now take $r,s \\in M^\\bullet $ such that $\\pi (\\theta (r)) = \\textbf {x}$ and $\\pi (\\theta (s)) = \\textbf {y}$ , and take $m,n \\in \\mathbb {N}$ satisfying that $\\gcd (m,n) = 1$ and $mr = ns$ .","Because $m \\textbf {x} = \\pi (\\theta (mr)) = \\pi (\\theta (ns)) = n \\textbf {y},$ one finds that $mx_i = ny_i$ for $i = 1, \\dots , \\beta $ .","As $\\gcd (m,n) = 1$ , it follows that $n$ divides each $x_i$ , i.e., $d/n \\in \\mathbb {N}$ .","As a result, $\\mathbf {y} = \\frac{m}{n} \\textbf {x} = \\bigg (\\frac{md}{n} \\bigg ) \\frac{\\mathbf {x}}{d} \\in \\bigg \\langle \\frac{\\mathbf {x}}{d} \\bigg \\rangle .$ Hence $\\pi (\\theta (M)) \\subseteq \\langle \\textbf {x}/d \\rangle $ .","Because $\\langle \\mathbf {x}/d \\rangle $ is isomorphic to $\\mathbb {N}_0$ , it follows that $\\pi (\\theta (M))$ is finitely generated.","We show now that $M$ is also finitely generated, which amounts to proving that $\\pi \\circ \\theta \\colon M \\rightarrow \\mathbb {N}_0^\\beta $ is injective.","First, let us verify that $\\pi $ is injective when restricted to $\\theta (M)$ .","As $\\theta (M)$ is a submonoid of the reduced monoid $F$ , it is also reduced.","Suppose that $(t_1, \\mathbf {v}), (t_2, \\mathbf {v}) \\in \\theta (M)$ , and let us check that $t_1 = t_2$ .","If $\\mathbf {v} = \\mathbf {0}$ , then $t_1 = t_2 = 0$ because $\\theta (M)^\\times $ is trivial.","Otherwise, there exist $r,s \\in M^\\bullet $ such that $\\theta (r) = (t_1, \\mathbf {v})$ and $\\theta (s) = (t_2, \\mathbf {v})$ .","Take $m,n \\in \\mathbb {N}$ such that $mr = ns$ .","Since $m (t_1, \\mathbf {v}) = m \\theta (r) = n \\theta (s) = n (t_2, \\mathbf {v})$ and $\\mathbf {v} \\ne \\mathbf {0}$ , one finds that $m = n$ and, therefore, $r = s$ .","This, in turn, implies that $t_1 = t_2$ .","Hence the restriction of $\\pi $ to $\\theta (M)$ is injective.","To conclude the proof of part (1), we show that $\\theta $ is also injective.","Let $r,s \\in M$ such that $\\theta (r) = \\theta (s) \\ne 0$ .","Taking $m,n \\in \\mathbb {N}$ satisfying $mr = ns$ , we have $m \\theta (r) = \\theta (mr) = \\theta (ns) = n \\theta (s).$ Since $\\theta (M)$ is reduced, the element $\\theta (r)$ must be torsion-free in $T \\times \\mathbb {N}_0^\\beta $ .","Thus, $m = n$ , which implies that $r = s$ .","As $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ , it follows that $|\\theta ^{-1}(a)| = 1$ for all $a \\in \\theta (M)$ .","Therefore $\\theta $ is injective, leading us to the injectivity of $\\pi \\circ \\theta $ .","Now that fact that $\\pi (\\theta (M))$ is finitely generated implies that $M$ is also finitely generated.","Hence $M$ must be isomorphic to a numerical monoid.","Finally, let us argue part (2) of the theorem.","For the direct implication, assume that the homomorphism $\\theta \\colon M \\rightarrow F$ is a transfer homomorphism.","As we did in the proof of part (1), we can assume that $F$ is a reduced.","As both $M$ and $F$ are reduced, condition (T1') yields $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ .","Now it follows by part (1) that the Puiseux monoid $M$ is isomorphic to a numerical monoid.","For the reverse implication, just take $\\theta $ to be the identity map.","Imposing the homomorphism $\\theta \\colon M \\rightarrow F$ in Theorem REF to satisfy $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ is not superfluous even if $M$ is atomic.","Then next example sheds some light upon this observation.","Example 4.3 Let $p_1, p_2, \\dots , $ be an enumeration of the odd prime numbers, and let $M = \\langle 1/p_n \\mid n \\in \\mathbb {N}\\rangle $ .","It is not hard to verify that $\\mathcal {A}(M) = \\lbrace 1/p_n \\mid n \\in \\mathbb {N}\\rbrace $ .","This implies that $M$ is atomic.","Now define $\\theta \\colon M \\rightarrow \\mathbb {Z}_2$ by setting $\\theta (0) = 0$ , $\\theta (r) = 0$ if $\\mathsf {n}(r)$ is even, and $\\theta (r) = 1$ if $\\mathsf {n}(r)$ is odd.","It follows immediately that $\\theta $ is a surjective monoid homomorphism.","However, $M$ is not isomorphic to any numerical monoid because it contains infinitely many atoms."],["Strongly Primary Puiseux Monoids","In this section we investigate which Puiseux monoids are strongly primary.","In the case of Puiseux monoids, being strongly primary is equivalent to being finitary.","In general, finitary monoids provide a common algebraic framework to study not only the arithmetic of strongly primary monoids but also that one of $v$ -noetherian $G$ -monoids (see [10]).","The structure of strongly primary monoids was first studied by Satyanarayana in [23] and has received substantial attention in the literature since then (see [15] and references therein).","In particular, the class of strongly primary monoids yields multiplicative models for a large class of one-dimensional local domains (see [10]).","All monoids mentioned in this section are assumed to be reduced.","Definition 5.1 Let $M$ be a monoid.","A submonoid $S$ of $M$ is called divisor-closed provided that for all $x \\in M$ and $s \\in S$ the fact that $x \\mid _M s$ implies that $x \\in S$ .","The monoid $M$ is called primary if it is nontrivial and its only divisor-closed submonoids are $\\lbrace 0\\rbrace $ and $M$ .","The monoid $M$ is called finitary if $M$ is a BF-monoid and there exist $n \\in \\mathbb {N}$ and a finite subset $S \\subseteq M^\\bullet $ such that $n M^\\bullet \\subseteq S + M$ .","The monoid $M$ is called strongly primary provided that $M$ is both primary and finitary.","Let $M$ be a Puiseux monoid, and let $M^{\\prime }$ be a nontrivial proper submonoid of $M$ .","Notice that for all $x \\in M \\setminus M^{\\prime }$ and $y \\in M^{\\prime }$ satisfying that $x \\mid _M y$ , the fact that $x \\mid _M \\mathsf {n}(x) \\mathsf {d}(y) y \\in M^{\\prime }$ immediately implies that $M^{\\prime }$ is not a divisor-closed submonoid of $M$ .","Thus, $M$ is primary.","On the other hand, suppose that $F$ is a finitely generated monoid, say $F = \\langle S \\rangle $ for some finite subset $S$ of $F^\\bullet $ .","Then the fact that $F^\\bullet = S + F$ immediately implies that $F$ is finitary.","In particular, every nontrivial finitely generated Puiseux monoid is finitary and, therefore, strongly primary.","The next proposition summarizes the observations made in this paragraph.","Proposition 5.2 Every nontrivial Puiseux monoid is primary.","Every finitely generated Puiseux monoid is strongly primary.","A Puiseux monoid that is not finitely generated may fail to be strongly primary (see, for example, Proposition REF ).","However, in Proposition REF and Proposition REF we exhibit two infinite families of non-finitely generated Puiseux monoids that are strongly primary.","To argue Proposition REF , we will use the following result, which is a weaker version of [16].","Proposition 5.3 If $M$ is a Puiseux monoid satisfying that 0 is not a limit point of $M^\\bullet $ , then $M$ is a BF-monoid.","The next two propositions introduce two families of (non-finitely generated) strongly primary Puiseux monoids.","Proposition 5.4 Let $p, q \\in \\mathbb {N}$ such that $\\gcd (p,q) = 1$ , and let $\\lbrace S_n\\rbrace $ be an inclusion-decreasing sequence of numerical monoids.","If a function $f \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ satisfies that $f(1) = 1$ and $q^{f(n+1) - f(n)} - p^n > p \\max \\lbrace F(S_n), a \\mid a \\in \\mathcal {A}(S_n) \\rbrace $ for every $n \\in \\mathbb {N}$ , then the Puiseux monoid $\\bigg \\langle \\frac{q^{f(n)}}{p^n} s \\ \\bigg {|} \\ n \\in \\mathbb {N}\\ \\text{ and} \\ s \\in S_n \\bigg \\rangle $ is strongly primary.","Set $M = \\big \\langle q^{f(n)}s/p^n \\mid n \\in \\mathbb {N}\\ \\text{and} \\ s \\in S_n \\big \\rangle $ , and for each $n \\in \\mathbb {N}$ set $A_n = \\mathcal {A}(S_n)$ .","First, we argue that $M$ is a BF-monoid.","To do so, observe that for each $n \\in \\mathbb {N}$ the fact that $q^{f(n+1) - f(n)} > p \\max A_n$ implies that $ \\min \\frac{q^{f(n+1)}}{p^{n+1}} A_{n+1} \\ge \\frac{q^{f(n+1)}}{p^{n+1}} > \\frac{q^{f(n)}}{p^n} \\max A_n.$ Therefore the generating set $\\cup _{n \\in \\mathbb {N}} (q^{f(n)}/p^n) A_n$ of $M$ can be listed as an increasing sequence of rational numbers.","As $M$ is generated by an increasing sequence of rational numbers, 0 cannot be a limit point of $M^\\bullet $ .","Hence $M$ is a BF-monoid by Proposition REF .","We proceed to prove that $M$ is finitary.","Since $M \\cap \\mathbb {N}_0$ is a submonoid of $(\\mathbb {N}_0,+)$ , it is atomic and $\\mathcal {A}(M \\cap \\mathbb {N}_0)$ is finite.","Take $S$ to be the finite set $\\mathcal {A}(M \\cap \\mathbb {N}_0) \\cup q A_1 \\subset M$ .","We are done once we verify the inclusion $p M^\\bullet \\subseteq S + M$ .","Fix $n \\in \\mathbb {N}$ and $a \\in A_{n+1}$ .","Because $q^{f(n)}a \\in q^{f(n)}S_{n+1} \\subseteq q^{f(n)}S_n \\in M \\cap \\mathbb {N}_0$ , the element $q^{f(n)}a$ is divisible in $M$ by an element of $S$ .","On the other hand, $(q^{f(n+1) - f(n)} - p^n)a \\ge q^{f(n+1) - f(n)} - p^n > F(S_n)$ , which implies that $(q^{f(n+1) - f(n)} - p^n)a \\, \\frac{q^{f(n)}}{p^n} \\in M$ .","Thus, $p\\bigg ( \\frac{q^{f(n+1)}}{p^{n+1}} a \\bigg ) = q^{f(n)}a + \\big (q^{f(n+1) - f(n)} - p^n\\big ) \\frac{q^{f(n)}}{p^n} a \\in S + M.$ In addition, the fact that $qa \\in qS_{n+1} \\subseteq qS_1$ guarantees $qa$ is divisible in $M$ by some element of $S$ .","This, in turn, implies that $p \\big (q^{f(1)}/p\\big )a = qa \\in S + M$ .","As a result, for each $x \\in M^\\bullet $ it follows that $px \\in m(S+M) \\subseteq S + M$ for some $m \\in \\mathbb {N}$ .","Hence $pM^\\bullet \\subseteq S + M$ , as desired.","Example 5.5 Consider the Puiseux monoid $M = \\bigg \\langle \\frac{3^{n^2 + 1}}{2^n}, \\frac{5 \\cdot 3^{n^2}}{2^n} \\ \\bigg {|} \\ n \\in \\mathbb {N}\\bigg \\rangle .$ Taking $S_n$ to be the numerical monoid $\\langle 3, 5 \\rangle $ for each $n \\in \\mathbb {N}$ and defining the function $f \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ by $f(n) = n^2$ , we can rewrite the Puiseux monoid $M$ as follows: $M = \\bigg \\langle \\frac{3^{f(n)}}{2^n} s \\ \\bigg {|} \\ n \\in \\mathbb {N}\\ \\text{ and} \\ s \\in S_n \\bigg \\rangle .$ Since $F(S_n) = 7$ for each $n \\in \\mathbb {N}$ , it follows that $3^{f(n+1) - f(n)} - 2^n = 3^{2n+1} - 2^n > 14 = 2 \\max \\lbrace 3,5, F(S_n)\\rbrace .$ As $f(1) = 1$ , Proposition REF guarantees that $M$ is a strongly primary Puiseux monoid.","As in Section , for $r \\in \\mathbb {Q}_{> 0}$ we let $M_r$ denote the multiplicatively $r$ -cyclic Puiseux monoid $\\langle r^n \\mid n \\in \\mathbb {N}\\rangle $ .","Proposition 5.6 For each $r \\in \\mathbb {Q}_{> 1}$ , the Puiseux monoid $M_r$ is strongly primary.","Since $r > 1$ , it follows that 0 is not a limit point of $M_r^\\bullet $ .","Thus, Proposition REF ensures that $M_r$ is a BF-monoid.","On the other hand, it was proved in [19] that $\\mathcal {A}(M_r) = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ .","Now we check that $\\mathsf {n}(r)$ divides $\\mathsf {n}(r)r^j$ in $M_r$ for every $j \\in \\mathbb {N}_0$ .","If $j=0$ , then $\\mathsf {n}(r) \\mid _{M_r} \\mathsf {n}(r)r^j$ follows trivially.","Hence it suffices to assume that $j \\in \\mathbb {N}$ .","In this case, the fact that $\\mathsf {n}(r)(r-1) = r( \\mathsf {n}(r) - \\mathsf {d}(r))$ implies that $\\mathsf {n}(r) r^j - \\mathsf {n}(r) = \\mathsf {n}(r) (r-1) \\sum _{i=0}^{j-1} r^i = r \\big (\\mathsf {n}(r) - \\mathsf {d}(r)\\big ) \\sum _{i=0}^{j-1} r^i = \\big (\\mathsf {n}(r) - \\mathsf {d}(r)\\big ) \\sum _{i=0}^{j-1} r^{i+1} \\in M_r.$ Therefore $\\mathsf {n}(r)$ divides $\\mathsf {n}(r) r^j$ in $M_r$ .","To show now that $M_r$ is finitary, we take $n = \\mathsf {d}(r)$ and $S = \\lbrace \\mathsf {n}(r)\\rbrace $ and verify that $n M_r^\\bullet \\subseteq S + M_r$ .","For $q \\in M_r^\\bullet $ , take $\\alpha _1, \\dots , \\alpha _t \\in \\mathbb {N}_0$ such that $q = \\sum \\alpha _i r^i$ , and fix $k \\in \\lbrace 1, \\dots , t\\rbrace $ such that $\\alpha _k > 0$ .","Because $\\mathsf {n}(r)$ divides $\\mathsf {n}(r) r^{k-1}$ in $M_r$ , there exists $s \\in M_r$ satisfying that $\\mathsf {n}(r) r^{k-1} = \\mathsf {n}(r) + s$ .","Thus, $n q &= \\mathsf {d}(r) \\alpha _k r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\\\&= \\mathsf {n}(r) r^{k-1} + (\\alpha _k - 1) \\mathsf {d}(r) r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\\\&= \\mathsf {n}(r) + \\big ( s + (\\alpha _k - 1) \\mathsf {d}(r) r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\big ),$ which implies that $nq \\in S + M_r$ .","Since $n M_r^\\bullet \\subseteq S + M_r$ , one obtains that $M_r$ is finitary and, therefore, strongly primary.","We conclude this section providing a family of Puiseux monoids that fail to be strongly primary.","Proposition 5.7 Let $\\lbrace a_n\\rbrace $ be a sequence of positive rational numbers satisfying that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ for any $i \\ne j$ .","Then the Puiseux monoid $\\langle a_n \\mid n \\in \\mathbb {N}\\rangle $ is atomic but not strongly primary.","Set $M = \\langle a_n \\mid n \\in \\mathbb {N}\\rangle $ .","It is not difficult to verify that $\\mathcal {A}(M) = \\lbrace a_n \\mid n \\in \\mathbb {N}\\rbrace $ from the fact that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ for any $i \\ne j$ ; we leave the details to the reader.","This implies, in particular, that $M$ is atomic.","Suppose, by way of contradiction, that $M$ is finitary.","Choose $n \\in \\mathbb {N}$ and a finite subset $S$ of $M^\\bullet $ such that $n M^\\bullet \\subseteq S + M$ .","Let $S^{\\prime }$ be a finite subset of $\\mathcal {A}(M)$ such that for each $s \\in S$ there is at least one atom in $S^{\\prime }$ dividing $s$ in $M$ .","After substituting $S$ by $S^{\\prime }$ , we can assume that $S \\subset \\mathcal {A}(M)$ .","Since $n M^\\bullet \\subseteq S + M$ and $\\inf (S+M) \\ge \\min S$ , it follows that 0 cannot be a limit point of $M^\\bullet $ .","Fix $\\epsilon > 0$ such that $\\epsilon < \\inf M^\\bullet $ .","Since $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ , there exists $j \\in \\mathbb {N}$ such that $\\mathsf {d}(a_j) > \\max \\lbrace n, \\max \\mathsf {d}(S)\\rbrace $ .","Because $n M^\\bullet \\subseteq S + M$ , one can write $ n a_j = a_s + \\sum _{i=1}^k \\alpha _i a_i$ for some $k \\in \\mathbb {N}$ , $a_s \\in S$ , and $\\alpha _i \\in \\mathbb {N}_0$ .","Applying the $\\mathsf {d}(a_j)$ -valuation to both sides of (REF ) and using the fact that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ , it is not hard to find that $\\mathsf {d}(a_j)$ divides $n - \\alpha _j$ .","Now $\\mathsf {d}(a_j) > n$ yields $n = \\alpha _j$ .","This, along with (REF ), would force $a_s = 0$ , which contradicts that $S \\subset \\mathcal {A}(M)$ .","Thus, $M$ is not strongly primary.","Remark 5.8 The previous proposition not only shows that Puiseux monoids are not, in general, strongly primary, but also illustrates that a natural bounding, ordering, or topological restriction under which a Puiseux monoid is guaranteed to be strongly primary is rather unlikely.","For example, consider the Puiseux monoids $M_1 = \\bigg \\langle \\frac{1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle , \\ M_2 = \\bigg \\langle \\frac{p-1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle , \\ \\text{and} \\ M_3 = \\bigg \\langle \\frac{p^2+1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle .$ It is not hard to check that the sets of atoms of $M_1$ , $M_2$ , and $M_3$ are precisely the generating sets displayed.","Therefore $M_1$ is strongly bounded, $M_2$ is bounded, and $M_3$ is not bounded.","We can also see that $M_1$ is a decreasing Puiseux monoid, while $M_2$ is increasing.","Furthermore, notice that 0 is a limit point of $M_1^\\bullet $ , but 0 is not a limit point of $M_2^\\bullet $ .","Finally, Proposition REF ensures that $M_1$ , $M_2$ , and $M_3$ are all strongly primary."],["Puiseux Monoids Are Almost Never Transfer Krull","We dedicate this section to show that the atomic structure of Puiseux monoids almost never can be obtained by transferring back that one of Krull monoids; specifically we shall prove that the existence of a transfer homomorphism from a nontrivial Puiseux monoid to a Krull monoid forces the domain to be isomorphic to $(\\mathbb {N}_0,+)$ .","The we use this information to show that only finitely generated Puiseux monoids admit transfer homomorphisms to C-monoids.","Let us start by giving the definition of a Krull monoid.","Definition 6.1 A monoid $K$ is called a Krull monoid if there is a monoid homomorphism $\\varphi \\colon K \\rightarrow D$ , where $D$ is a free abelian monoid and $\\varphi $ satisfies the following two conditions: if $a, b \\in K$ and $\\varphi (a) \\mid _D \\varphi (b)$ , then $a \\mid _K b$ ; for every $d \\in D$ there exist $a_1, \\dots , a_n \\in K$ with $d = \\gcd \\lbrace \\varphi (a_1), \\dots , \\varphi (a_n)\\rbrace $ .","With notation as in Definition REF , it is easy to see that $K$ is a Krull monoid if and only if $K_{\\text{red}}$ is a Krull monoid.","The basis elements of $D$ are called the prime divisors of $K$ .","The abelian group Cl$(K) := D/\\varphi (K)$ is called the class group of $K$ (see [10]).","As Krull monoids are isomorphic to submonoids of free abelian monoids, Krull monoids are atomic.","The factorization theory of Krull monoids has been significantly studied (see [5], [13] and references therein).","The class of Krull monoids contains many well-studied types of monoids, including the multiplicative monoid of the ring of integers of an algebraic number, the Hilbert monoids, and the regular congruence monoids.","These and further examples of Krull monoids are presented in [9] and [10].","From the point of view of factorization theory, perhaps the most important family of Krull monoids is that one consisting of block monoids, which we are about to introduce.","This is because block monoids capture the essence of the arithmetic of lengths of factorizations in Krull monoids.","Let $G$ be an abelian group and $\\mathcal {F}(G)$ the free abelian monoid on $G$ .","An element $X = g_1 \\dots g_l \\in \\mathcal {F}(G)$ is called a sequence over $G$ .","The length of $X$ is defined as $|X| = l =\\sum _{g \\in G} \\mathsf {v}_g(X).$ For every $I \\subseteq [1, l]$ , the sequence $Y = \\prod _{i \\in I} g_i$ is called a subsequence of $X$ .","The subsequences are precisely the divisors of $X$ in the free abelian monoid $\\mathcal {F}(G)$ .","The submonoid $\\mathcal {B}(G) := \\bigg \\lbrace X \\in \\mathcal {F}(G) \\ \\bigg {|} \\ \\sum _{g \\in G} \\mathsf {v}_g(X) g = 0 \\bigg \\rbrace $ of $\\mathcal {F}(G)$ is called the block monoid on $G$ , and its elements are referred to as zero-sum sequences or blocks over $G$ ([10] is a good general reference on block monoids).","Furthermore, if $G_0$ is a subset of $G$ , then the submonoid $\\mathcal {B}(G_0) := \\lbrace X \\in \\mathcal {B}(G) \\mid \\mathsf {v}_g(X) = 0 \\ \\emph { if } \\ g \\notin G_0 \\rbrace $ of $\\mathcal {B}(G)$ is called the restriction of the block monoid $\\mathcal {B}(G)$ to $G_0$ .","For $X \\in \\mathcal {B}(G_0)$ , the support of $X$ in $G_0$ is defined to be $\\text{supp}_{G_0}(X) := \\lbrace g \\in G_0 \\mid \\mathsf {v}_g(X) > 0\\rbrace .$ As mentioned before, the relevance of block monoids in the theory of non-unique factorizations lies in the next result.","Proposition 6.2 [10] Let $K$ be a Krull monoid with class group $G$ and let $G_0$ be the set of classes of $G$ which contain prime divisors.","Then $\\mathcal {L}(K) = \\mathcal {L}(\\mathcal {B}(G_0)).$ As a consequence, understanding the arithmetic of lengths of factorizations in Krull monoids amounts to understanding the same in block monoids.","Definition 6.3 A Puiseux monoid $M$ is transfer Krull if there exist an abelian group $G$ , a subset $G_0$ of $G$ , and a transfer homomorphism $\\theta \\colon M \\rightarrow \\mathcal {B}(G_0)$ .","Remark: Our definition of a transfer Krull monoid coincides with the definition given in [8]; this is because in the present setting the concepts of a transfer homomorphism and the concept of a weak transfer homomorphism coincide by [3].","We denote the field of fractions of an integral domain $R$ by $\\mathsf {q}(R)$ .","For subsets $X,Y$ of $\\mathsf {q}(R)$ we set $(X : Y) := \\lbrace x \\in \\mathsf {q}(R) \\mid xY \\subseteq X\\rbrace $ .","In addition, $R$ is called a Krull domain if $R^\\bullet $ is a Krull monoid.","In this case, the divisor class group of $R$ , denoted by $\\mathcal {C}(R)$ , measures the extent to which factorizations in $R$ fail to be unique (see [10]).","Unlike Krull domains/monoids, which have been central objects in commutative algebra since mid-nineteenth century, transfer Krull monoids (which generalize the concept of Krull monoids) were introduced more recently.","Let us proceed to present a few examples of transfer Krull monoids.","Examples of transfer Krull monoids: Let $H$ be a half-factorial monoid, and let $\\theta \\colon H \\rightarrow \\mathcal {B}(\\lbrace 0\\rbrace )$ be the map defined by $\\theta (h) = 0$ if $h \\in \\mathcal {A}(H)$ and $\\theta (h) = 1$ if $h \\in H^\\times $ .","As the map $\\theta $ is a transfer homomorphism, it follows that $H$ is a transfer Krull monoid.","Let $R$ be a Krull domain, and let $K$ be a subring of $R$ with the same field of fractions.","Suppose, in addition, that the following three conditions hold: $R = KR^\\times $ ; $K \\cap R^\\times = K^\\times $ ; $(K : R)$ is a maximal ideal of $K$ (see [10]).","Then the inclusion map $K^\\bullet \\hookrightarrow R^\\bullet $ is a transfer homomorphism and, therefore, $K^\\bullet $ is a transfer Krull monoid.","Transfer Krull monoids can also be defined in a non-commutative context (see, for instance, [3]).","Let $R$ be a bounded HNP (hereditary Noetherian prime) ring.","If every stably free left $R$ -ideal is free, then $R^\\bullet $ is a transfer Krull monoid (see [24] for details).","There are also many monoids that fail to be transfer Krull.","Examples of non-transfer Krull monoids in a non-commutative setting are provided by [6], [12], and [25].","On the other hand, Theorem REF and the next proposition (which follows from [14]) yield examples of non-transfer Krull monoids in a commutative context.","Proposition 6.4 Every proper numerical monoid fails to be transfer Krull.","The next lemma will be used in the proof of Theorem REF .","Lemma 6.5 If $\\lbrace a_n\\rbrace $ is an infinite sequence of positive integers, then there exists $m \\in \\mathbb {N}$ such that $a_{m+1} \\in \\langle a_1, \\dots , a_m \\rangle $ .","If $\\lbrace a_n\\rbrace $ is bounded there is a term that repeats infinitely many times, making the conclusion of the lemma obvious.","Thus, suppose that $\\lbrace a_n\\rbrace $ is not bounded.","Let $\\lbrace a_{n_j}\\rbrace $ be a subsequence of $\\lbrace a_n\\rbrace $ satisfying that $ a_{n_{j+1}} > \\prod _{i=1}^{j} a_{n_i}$ for every $j \\in \\mathbb {N}$ .","Now, for each natural number $j$ , set $d_j = \\gcd (a_{n_1}, \\dots , a_{n_j})$ , and notice that $d_{j+1} \\mid d_j$ for every $j \\in \\mathbb {N}$ .","Therefore $d_{k+1} = d_k$ must hold for some $k$ .","In particular, $d_k \\mid a_{n_{k+1}}$ .","On the other hand, condition (REF ) ensures that $a_{n_{k+1}}/d_k$ is greater than the Frobenius number of the numerical monoid $\\langle a_{n_1}/d_k, \\dots , a_{n_k}/d_k \\rangle $ .","This implies that $a_{n_{k+1}} \\in \\langle a_{n_1}, \\dots , a_{n_k} \\rangle $ .","The lemma follows by taking $m = n_{k+1}-1$ .","Now we are in a position to prove that atomic Puiseux monoids are almost never transfer Krull.","Theorem 6.6 If a nontrivial Puiseux monoid is transfer Krull, then it must be isomorphic to $(\\mathbb {N}_0,+)$ .","Let $M$ be a nontrivial Puiseux monoid that happens to be transfer Krull.","As Krull monoids are atomic, $M$ is atomic by Proposition REF .","Let $G$ be an abelian group, and let $\\theta \\colon M \\rightarrow \\mathcal {B}(G_0)$ be a transfer homomorphism, where $G_0$ is a subset of $G$ .","Because both $M$ and $\\mathcal {B}(G_0)$ are reduced, $\\theta ^{-1}(\\emptyset ) = \\lbrace 0\\rbrace $ .","Assume, by way of contradiction, that $M$ is not isomorphic to a numerical monoid.","Take $X \\in \\mathcal {B}(G_0)^\\bullet $ and $r,s \\in M^\\bullet $ such that $\\theta (r) = \\theta (s) = X$ .","Taking $m,n \\in \\mathbb {N}$ such that $mr = ns$ , one obtains $ \\prod _{g \\in G_0} g^{m \\mathsf {v}_g(X)} = \\theta (r)^m = \\theta (s)^n = \\prod _{g \\in G_0} g^{n \\mathsf {v}_g(X)}.$ Since $|X| \\ge 1$ and $m \\mathsf {v}_g(X) = n \\mathsf {v}_g(X)$ for every $g \\in G_0$ , it follows that $m=n$ , which yields $r = s$ .","Hence the preimage under $\\theta $ of each element of $\\mathcal {B}(G_0)^\\bullet $ is a singleton.","This, along with the fact that $\\theta ^{-1}(\\emptyset ) = \\lbrace 0\\rbrace $ , implies that $\\theta $ is injective.","In addition, the same equality (REF ) implies that $\\text{supp}_{G_0}(\\theta (a)) = \\text{supp}_{G_0}(\\theta (a^{\\prime }))$ for all $a,a^{\\prime } \\in \\mathcal {A}(M)$ .","As a consequence, any two elements of $\\theta (M^\\bullet )$ have the same support, and we can assume, without loss of generality, that $G_0$ is finite.","Let $G_0 =: \\lbrace g_1, \\dots , g_t\\rbrace $ be the common support.","List the set $\\mathcal {A}(M)$ as a sequence $\\lbrace a_n\\rbrace $ , and let $A_n = \\theta (a_n)$ for each $n \\in \\mathbb {N}$ .","Because $\\theta $ is injective, $A_i \\ne A_j$ when $i \\ne j$ .","Now, for any pair $(i,j) \\in \\mathbb {N}^2$ , there exist $c_i,c_j \\in \\mathbb {N}$ such that $c_i a_i = c_j a_j$ .","For each $n \\in \\lbrace 1, \\dots , t\\rbrace $ , we can apply $\\mathsf {v}_{g_n} \\circ \\theta $ to the equality $c_i a_i = c_j a_j$ to get $c_i \\mathsf {v}_{g_n}(A_i) = c_j \\mathsf {v}_{g_n}(A_j)$ .","After rewriting this equality, one obtains that $ \\frac{\\mathsf {v}_{g_n}(A_i)}{\\mathsf {v}_{g_n}(A_j)} = \\frac{c_j}{c_i} = \\frac{\\mathsf {v}_{g_1}(A_i)}{\\mathsf {v}_{g_1}(A_j)}$ for each $n \\in \\lbrace 1, \\dots , t\\rbrace $ .","On the other hand, notice that Lemma REF guarantees the existence of $m \\in \\mathbb {N}$ and $\\alpha _1, \\dots , \\alpha _m \\in \\mathbb {N}_0$ such that $ \\mathsf {v}_{g_1}(A_{m+1}) = \\sum _{i=1}^m \\alpha _i \\mathsf {v}_{g_1}(A_i).$ By (REF ), it follows that the equality (REF ) holds when we replace $g_1$ by any other element of $G_0$ (exactly with the same $\\alpha _i$ 's).","As a result, we obtain $A_{m+1} = \\prod _{j=1}^{|G_0|} g_j^{\\mathsf {v}_{g_j}(A_{m+1})} = \\prod _{j=1}^{|G_0|} \\prod _{i=1}^m g_j^{\\alpha _i \\mathsf {v}_{g_j}(A_i)} = \\prod _{i=1}^m \\bigg (\\prod _{j=1}^{|G_0|} g_j^{\\mathsf {v}_{g_j}(A_i)}\\bigg )^{\\alpha _i} = \\prod _{i=1}^m A_i^{\\alpha _i}.$ This contradicts the fact that $A_{m+1}$ is an atom of the block monoid $\\mathcal {B}(G_0)$ .","Therefore $M$ must be isomorphic to a numerical monoid.","Now the direct implication of the proof follows by Proposition REF .","For the reverse implication, it suffices to notice that $\\theta \\colon 1 \\mapsto [1_G]$ is an isomorphism from $(\\mathbb {N}_0,+)$ to the block monoid $\\mathcal {B}(G)$ , where $G$ is the trivial group.","Corollary 6.7 A Puiseux monoid is a Krull monoid if and only if it can be generated by one element.","Perhaps the second most-systematically studied family of atomic monoids is that one comprising the C-monoids.","We would like to know under which conditions a Puiseux monoid happens to be a C-monoid.","Any monoid $M$ can be embedded into a quotient group $\\mathsf {g}(M)$ , which is unique up to canonical isomorphism.","Let $D$ be a multiplicative monoid with quotient group $\\mathsf {g}(D)$ , and let $M$ be a submonoid of $D$ .","Two elements $x,y \\in D$ are said to be $M$ -equivalent provided that $x^{-1}M \\cap D = y^{-1}M \\cap D$ .","It can be easily checked that being $M$ -equivalent defines a congruence relation on $D$ .","For each $x \\in D$ , let $[x]^D_M$ denote the congruence class of $x$ .","The set $\\mathcal {C}^*(M,D) := \\big \\lbrace [x]^D_M \\mid x \\in (D \\setminus D^\\times ) \\cup \\lbrace 1\\rbrace \\big \\rbrace $ is a commutative semigroup with identity, which is called the reduced class semigroup of $M$ in $D$ .","Definition 6.8 A monoid $M$ is called a C-monoid if it is a submonoid of a factorial monoid $F$ such that $F^\\times \\cap M = M^\\times $ and $\\mathcal {C}^*(M,F)$ is finite.","With notation as in Definition REF , we say that $M$ is a C-monoid defined in $F$ .","A C-monoid can be defined in more than one factorial monoid $F$ ; however, there is a canonical way of choosing $F$ (see [9]).","Because C-monoids are submonoids of factorial monoids, they are atomic.","The family of C-monoids allows us to study the arithmetic of non-integrally closed Noetherian domains.","Given a multiplicative monoid $M$ with quotient group $\\mathsf {g}(M)$ , we say that $x \\in \\mathsf {g}(M)$ is almost integral over $M$ if there exists $c \\in M$ such that $cx^n \\in M$ for every $n \\in \\mathbb {N}$ .","The subset of $\\mathsf {g}(M)$ consisting of all almost integral elements over $M$ is denoted by $\\widehat{M}$ and called the complete integral closure of $M$ .","Let $R$ be an integral domain with field of fractions $\\mathsf {q}(R)$ .","An ideal $I$ of $R$ is divisorial if $(R : (R : I)) = I$ .","The domain $R$ is called a Mori domain if it satisfies the ascending chain condition on divisorial ideals.","Finally, for the domain $R$ we set $\\widehat{R} = \\widehat{R^\\bullet } \\cup \\lbrace 0\\rbrace $ , where $R^\\bullet $ is the multiplicative monoid of $R$ .","Example 6.9 If $A$ is a Mori domain, then $R = \\widehat{A}$ is a Krull domain.","Moreover, if $\\mathfrak {f} = (A : R)$ is nonzero and both the quotient ring $R/\\mathfrak {f}$ and the class group $\\mathcal {C} (R)$ are finite, then $A^\\bullet $ is a C-monoid (see [10]).","More examples of C-monoids can be found in [11] and [22].","The next theorem is used in the proof of Proposition REF .","Theorem 6.10 [10] The complete integral closure of a C-monoid is a Krull monoid.","Proposition 6.11 A nontrivial Puiseux monoid is a C-monoid if and only if it is isomorphic to a numerical monoid.","Let $M$ be a nontrivial Puiseux monoid that is also a C-monoid.","Let $\\widehat{M}$ be the complete integral closure of $M$ .","Observe first that if $x \\in \\mathsf {g}(M) \\cap \\mathbb {Q}_{<0}$ , then $S_{x,r} := \\lbrace r + nx \\mid n \\in \\mathbb {N}\\rbrace $ contains only finitely many positive rational numbers for all $r \\in M$ .","As a result, $|S_{x,r} \\cap M| < \\infty $ for all $r \\in M$ , which implies that no negative element of $\\mathsf {g}(M)$ is almost integral over $M$ .","Because $\\widehat{M}$ is a monoid and it is contained in $\\mathbb {Q}_{\\ge 0}$ , it must be a Puiseux monoid.","By Theorem REF , the monoid $\\widehat{M}$ is a Krull monoid; in particular, it is transfer Krull.","Now Theorem REF ensures that $\\widehat{M}$ is isomorphic to $(\\mathbb {N}_0,+)$ .","Finally, the fact that $M$ is a submonoid of $\\widehat{M}$ forces $M$ to be isomorphic to a numerical monoid.","For the reverse implication, it suffices to note that for every proper numerical monoid $N$ , any two natural numbers greater than the Frobenius number of $N$ are $N$ -equivalent, which implies that $\\mathcal {C}^*(N, \\mathbb {N}_0)$ is finite.","As $\\mathbb {N}_0$ is also a C-monoid, the proof follows."],["Acknowledgements","While working on this paper, I was supported by the NSF-AGEP fellowship.","I am grateful to Alfred Geroldinger not only for proposing the main questions motivating this paper but also for his enlightening guidance through early drafts.","Also, I would like to thank Salvatore Tringali and the anonymous referee, whose helpful suggestions help me improve the final version of this paper."]],"string":"[\n [\n \"Puiseux monoids and transfer homomorphisms\"\n ],\n [\n \"Abstract There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades.\",\n \"The factorization theory of finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most systematically studied.\",\n \"Puiseux monoids, which are additive submonoids of $\\\\mathbb{Q}_{\\\\ge 0}$ consisting of nonnegative rational numbers, have only been studied recently.\",\n \"In this paper, we provide evidence that this family comprises plenty of monoids with a basically unexplored atomic structure.\",\n \"We do this by showing that the arithmetical invariants of the well-studied atomic monoids mentioned earlier cannot be transferred to most Puiseux monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms.\",\n \"Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux monoid to a finitely generated monoid do not exist.\",\n \"We also find a large family of Puiseux monoids that fail to be strongly primary.\",\n \"In addition, we prove that the only nontrivial Puiseux monoid that accepts a transfer homomorphism to a Krull monoid is $\\\\mathbb{N}_0$.\",\n \"Finally, we classify the Puiseux monoids that happen to be C-monoids.\"\n ],\n [\n \"Introduction\",\n \"The study of the phenomenon of non-unique factorizations in the ring of integers $\\\\mathcal {O}_K$ of an algebraic number field $K$ was initiated by L. Carlitz in the 1950's, and it was later carried out on more general integral domains.\",\n \"As a result, many techniques to measure the non-uniqueness of factorizations in several families of integral domains were systematically developed during the second half of the last century (see [2] and references therein).\",\n \"However, it was not until recently that questions about the non-uniqueness of factorizations were abstractly formulated in the context of commutative cancellative monoids.\",\n \"This was possible because most of the factorization-related questions inside an integral domain are purely multiplicative in essence.\",\n \"The fundamental goal of abstract (or modern) factorization theory is to measure how far is a commutative cancellative monoid from being factorial by using different arithmetical invariants.\",\n \"At this point, the arithmetical invariants of several families of atomic monoids have been intensively studied.\",\n \"Finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most studied.\",\n \"These families of monoids not only have very diverse arithmetical properties, but also have proved to be useful in the study of the factorization theory of less-understood atomic monoids via transfer homomorphisms.\",\n \"A monoid homomorphism is said to be transfer if somehow it allows to shift the atomic structure of its codomain back to its domain (see Definition REF ).\",\n \"Therefore if one is willing to know the factorization invariants of a given monoid, it suffices to find a transfer homomorphism from such a monoid to a better-understood monoid and carry over the desired factorization properties.\",\n \"Puiseux monoids were recently introduced as a rational generalization of numerical monoids.\",\n \"Many families of atomic Puiseux monoids were explored in [16], [17], [19], and their elasticity was studied in [20].\",\n \"However, it is still unanswered whether the non-unique factorization behavior in Puiseux monoids is somehow similar to that of some of the monoids whose factorization properties are already well-understood.\",\n \"To give a partial answer to this, we will determine which atomic Puiseux monoids can be the domain of a transfer homomorphism to some of the monoids whose arithmetical invariants have already been studied.\",\n \"In particular, we consider finitely generated monoids, Krull monoids, and C-monoids as our transfer codomains.\",\n \"The content of this paper is organized as follows.\",\n \"In Section , we establish the notation we shall be using later, and we formally present most of the fundamental concepts needed in this paper.\",\n \"Then, in Section , we show that homomorphisms between Puiseux monoids can only be given by rational multiplication, which will allow us to characterize the transfer homomorphisms between Puiseux monoids.\",\n \"We also present a family of Puiseux monoids whose members have $\\\\mathbb {Z}$ as their group of automorphisms.\",\n \"Section is devoted to characterize the Puiseux monoids admitting a transfer homomorphism to some finitely generated monoid.\",\n \"Then, in Section we investigate which Puiseux monoids are strongly primary.\",\n \"Finally, in Section , we prove that the only Puiseux monoid that is transfer Krull is the additive monoid $\\\\mathbb {N}_0$ .\",\n \"We use this information to classify the Puiseux monoids which happen to be C-monoids.\"\n ],\n [\n \"Background\",\n \"To begin with let us introduce the fundamental concepts related to our exposition as an excuse to establish the notation we need.\",\n \"The reader can consult Grillet [21] for information on commutative semigroups and Geroldinger and Halter-Koch [10] for extensive background in non-unique factorization theory of atomic monoids.\",\n \"Throughout this sequel, we let $\\\\mathbb {N}$ denote the set of positive integers, and we set $\\\\mathbb {N}_0 := \\\\mathbb {N}\\\\cup \\\\lbrace 0\\\\rbrace $ .\",\n \"For $X \\\\subseteq \\\\mathbb {R}$ and $r \\\\in \\\\mathbb {R}$ , we set $X_{\\\\le r} := \\\\lbrace x \\\\in X \\\\mid x \\\\le r\\\\rbrace $ ; with a similar spirit we use the symbols $X_{\\\\ge r}$ , $X_{< r}$ , and $X_{> r}$ .\",\n \"If $q \\\\in \\\\mathbb {Q}_{> 0}$ , then we call the unique $a,b \\\\in \\\\mathbb {N}$ such that $q = a/b$ and $\\\\gcd (a,b)=1$ the numerator and denominator of $q$ and denote them by $\\\\mathsf {n}(q)$ and $\\\\mathsf {d}(q)$ , respectively.\",\n \"For each subset $Q$ of $\\\\mathbb {Q}_{>0}$ , we call the sets $\\\\mathsf {n}(Q) = \\\\lbrace \\\\mathsf {n}(q) \\\\mid q \\\\in Q\\\\rbrace $ and $\\\\mathsf {d}(Q) = \\\\lbrace \\\\mathsf {d}(q) \\\\mid q \\\\in Q\\\\rbrace $ the numerator set and denominator set of $Q$ , respectively.\",\n \"As usual, a semigroup is a pair $(S, *)$ , where $S$ is a set and $*$ is an associative binary operation in $S$ ; we write $S$ instead of $(S,*)$ provided that $*$ is clear from the context.\",\n \"However, inside the scope of this paper, a monoid is a commutative cancellative semigroup with identity (cf.\",\n \"the standard definition of monoid).\",\n \"To comply with established conventions, we will be using simultaneously additive and multiplicative notations; however, the context will always save us from the risk of ambiguity.\",\n \"Let $M$ be a monoid written additively.\",\n \"We set $M^\\\\bullet := M \\\\!\",\n \"\\\\setminus \\\\!\",\n \"\\\\lbrace 0\\\\rbrace $ and, as usual, we let $M^\\\\times $ denote the set of units (i.e., invertible elements) of $M$ .\",\n \"The monoid $M$ is reduced if $M^\\\\times = \\\\lbrace 0\\\\rbrace $ .\",\n \"For $a, b \\\\in M$ , we say that $a$ divides $b$ in $M$ if there exists $c \\\\in M$ such that $b = a + c$ ; in this case we write $a \\\\mid _M b$ .\",\n \"An element $a \\\\in M \\\\!\",\n \"\\\\setminus \\\\!\",\n \"M^\\\\times $ is an atom if whenever $a = u + v$ for some $u,v \\\\in M$ , either $u \\\\in M^\\\\times $ or $v \\\\in M^\\\\times $ .\",\n \"Atoms are the building blocks in factorization theory; this motivates the especial notation $\\\\mathcal {A}(M) := \\\\lbrace a \\\\in M \\\\mid a \\\\text{ is an atom of } M\\\\rbrace .$ For $S \\\\subseteq M$ , we let $\\\\langle S \\\\rangle $ denote the smallest submonoid of $M$ containing $S$ , and we say that $S$ generates $M$ if $M = \\\\langle S \\\\rangle $ .\",\n \"The monoid $M$ is said to be finitely generated if it can be generated by a finite set.\",\n \"On the other hand, we say that $M$ is atomic if $M = \\\\langle \\\\mathcal {A}(M) \\\\rangle $ .\",\n \"A monoid is factorial if every element can be written as a sum of primes.\",\n \"As every prime is an atom, every factorial monoid is atomic.\",\n \"Let $\\\\rho \\\\subseteq M \\\\times M$ be an equivalence relation on $M$ , and let $[a]_\\\\rho $ denote the equivalence class of $a \\\\in M$ .\",\n \"We say that $\\\\rho $ is a congruence if for all $a,b,c \\\\in M$ such that $(a,b) \\\\in \\\\rho $ it follows that $(ca,cb) \\\\in \\\\rho $ .\",\n \"Congruences are precisely the equivalence relations that are compatible with the operation of $M$ , meaning that $M/\\\\rho := \\\\lbrace [a]_\\\\rho \\\\mid a \\\\in M\\\\rbrace $ is a commutative semigroup with identity (no necessarily cancellative).\",\n \"Two elements $a,b \\\\in M$ are associates, and we write $a \\\\simeq b$ , if $a = ub$ for some $u \\\\in M^\\\\times $ .\",\n \"Being associates defines a congruence relation $\\\\simeq $ on $M$ , and $M_{\\\\text{red}} := M/\\\\!\\\\!\\\\simeq $ is called the associated reduced semigroup of $M$ .\",\n \"We say that a multiplicative monoid $F$ is free abelian with basis $P \\\\subset F$ if every element $a \\\\in F$ can be written uniquely in the form $a = \\\\prod _{p \\\\in P} p^{\\\\mathsf {v}_p(a)},$ where $\\\\mathsf {v}_p(a) \\\\in \\\\mathbb {N}_0$ and $\\\\mathsf {v}_p(a) > 0$ only for finitely many elements $p \\\\in P$ .\",\n \"The monoid $F$ is determined by $P$ up to canonical isomorphism, so we shall also denote $F$ by $\\\\mathcal {F}(P)$ .\",\n \"By the fundamental theorem of arithmetic, the multiplicative monoid $\\\\mathbb {N}$ is free on the set of prime numbers.\",\n \"In this case, we can extend $\\\\mathsf {v}_p$ to $\\\\mathbb {Q}_{\\\\ge 0}$ as follows.\",\n \"For $r \\\\in \\\\mathbb {Q}_{> 0}$ let $\\\\mathsf {v}_p(r) := \\\\mathsf {v}_p(\\\\mathsf {n}(r)) - \\\\mathsf {v}_p(\\\\mathsf {d}(r))$ and set $\\\\mathsf {v}_p(0) = \\\\infty $ .\",\n \"The free abelian monoid on $\\\\mathcal {A}(M)$ , denoted by $\\\\mathsf {Z}(M)$ , is called the factorization monoid of $M$ , and the elements of $\\\\mathsf {Z}(M)$ are called factorizations.\",\n \"If $z = a_1 \\\\dots a_n \\\\in \\\\mathsf {Z}(M)$ for some $n \\\\in \\\\mathbb {N}_0$ and $a_1, \\\\dots , a_n \\\\in \\\\mathcal {A}(M)$ , then $n$ is the length of the factorization $z$ ; the length of $z$ is denoted by $|z|$ .\",\n \"The unique homomorphism $\\\\phi \\\\colon \\\\mathsf {Z}(M) \\\\rightarrow M \\\\ \\\\ \\\\text{satisfying} \\\\ \\\\ \\\\phi (a) = a \\\\ \\\\ \\\\text{for all} \\\\ \\\\ a \\\\in \\\\mathcal {A}(M)$ is called the factorization homomorphism of $M$ .\",\n \"Additionally, for $x \\\\in M^\\\\bullet $ , $\\\\mathsf {Z}(x) := \\\\phi ^{-1}(x) \\\\subseteq \\\\mathsf {Z}(M)$ is the set of factorizations of $x$ .\",\n \"By definition, we set $\\\\mathsf {Z}(0) = \\\\lbrace 0\\\\rbrace $ .\",\n \"Note that the monoid $M$ is atomic if and only if $\\\\mathsf {Z}(x)$ is not empty for all $x \\\\in M$ .\",\n \"For each $x \\\\in M$ , the set of lengths of $x$ is defined by $\\\\mathsf {L}(x) := \\\\lbrace |z| : z \\\\in \\\\mathsf {Z}(x)\\\\rbrace .$ We say that the monoid $M$ is half-factorial if $|\\\\mathsf {L}(x)| = 1$ for all $x \\\\in M$ .\",\n \"On the other hand, if $\\\\mathsf {L}(x)$ is a finite set for all $x \\\\in M$ , then we say that $M$ is a BF-monoid.\",\n \"The system of sets of lengths of $M$ is defined by $\\\\mathcal {L}(M) := \\\\lbrace \\\\mathsf {L}(x) \\\\mid x \\\\in M\\\\rbrace .$ The system of sets of lengths is an arithmetical invariant of atomic monoids that has received significant attention in recent years (see [1], [4] and the literature cited there).\",\n \"A very special family of atomic monoids is that one comprising all numerical monoids, cofinite submonoids of the additive monoid $\\\\mathbb {N}_0$ .\",\n \"We say that a numerical monoid is proper if it is strictly contained in $\\\\mathbb {N}_0$ .\",\n \"Each numerical monoid has a unique minimal set of generators, which is finite.\",\n \"Moreover, if $\\\\lbrace a_1, \\\\dots , a_n\\\\rbrace $ is the minimal set of generators for a numerical monoid $N$ , then $\\\\mathcal {A}(N) = \\\\lbrace a_1, \\\\dots , a_n\\\\rbrace $ and $\\\\gcd (a_1, \\\\dots , a_n) = 1$ .\",\n \"As a result, every numerical monoid is atomic and contains only finitely many atoms.\",\n \"The Frobenius number of $N$ , denoted by $F(N)$ , is the minimum $n \\\\in \\\\mathbb {N}$ such that $\\\\mathbb {Z}_{> n} \\\\subset N$ .\",\n \"An introduction to numerical monoids can be found in [7].\",\n \"An additive submonoid of $\\\\mathbb {Q}_{\\\\ge 0}$ is called a Puiseux monoid.\",\n \"Puiseux monoids are a natural generalization of numerical monoids.\",\n \"However, the general atomic structure of Puiseux monoids drastically differs from that one of numerical monoids.\",\n \"Puiseux monoids are not always atomic; for instance, consider $\\\\langle 1/2^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ .\",\n \"On the other hand, if an atomic Puiseux monoid $M$ is not isomorphic to a numerical monoid, then $\\\\mathcal {A}(M)$ is infinite.\",\n \"The atomic structure of Puiseux monoids has been studied in [17] and [19], where several families of atomic Puiseux monoids were described.\"\n ],\n [\n \"Homomorphisms Between Puiseux Monoids\",\n \"In this section we present characterizations of homomorphisms and transfer homomorphisms between Puiseux monoids.\",\n \"Let us start by introducing the concept of a transfer homomorphism, which is going to play a central role in this paper.\",\n \"Definition 3.1 A monoid homomorphism $\\\\theta \\\\colon M \\\\rightarrow N$ is said to be a transfer homomorphism if the following conditions hold: (T1) $N = \\\\theta (M) N^\\\\times $ and $\\\\theta ^{-1}(N^\\\\times ) = M^\\\\times $ ; (T2) if $\\\\theta (a) = b_1 b_2$ for $a \\\\in M$ and $b_1,b_2 \\\\in N$ , then there exist $a_1, a_2 \\\\in M$ such that $a = a_1 a_2$ and $\\\\theta (a_i) = b_i$ for $i \\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"We proceed to characterize the homomorphisms between Puiseux monoids.\",\n \"This will immediately yield a characterization of those homomorphisms between Puiseux monoids that happen to be transfer homomorphisms.\",\n \"Proposition 3.2 If $\\\\phi \\\\colon M \\\\rightarrow N$ be a homomorphism between Puiseux monoids, then the following conditions hold.\",\n \"There exists $q \\\\in \\\\mathbb {Q}_{\\\\ge 0}$ such that $\\\\phi (x) = qx$ for all $x \\\\in M$ , i.e., $\\\\phi $ is given by rational multiplication.\",\n \"The homomorphism $\\\\phi $ is a transfer homomorphism if and only if it is surjective.\",\n \"Let us argue first that $\\\\phi $ is given by rational multiplication.\",\n \"It is clear that if a map $P \\\\rightarrow P^{\\\\prime }$ between two Puiseux monoids is multiplication by a rational number, then it is a monoid homomorphism.\",\n \"Thus, it suffices to verify that the only homomorphisms of Puiseux monoids are those given by rational multiplication.\",\n \"To do this, consider the Puiseux monoid homomorphism $\\\\varphi \\\\colon P \\\\rightarrow P^{\\\\prime }$ .\",\n \"Because the trivial homomorphism is multiplication by 0, there is no loss in assuming that $P \\\\ne \\\\lbrace 0\\\\rbrace $ .\",\n \"Let $\\\\lbrace n_1, \\\\dots , n_k\\\\rbrace $ be a minimal set of generators for the additive monoid $N = P \\\\cap \\\\mathbb {N}_0$ .\",\n \"Notice that $N \\\\ne \\\\lbrace 0\\\\rbrace $ and, therefore, $k \\\\ge 1$ .\",\n \"The fact that $\\\\varphi $ is nontrivial implies that $\\\\varphi (n_j) \\\\ne 0$ for some $j \\\\in \\\\lbrace 1,\\\\dots ,k\\\\rbrace $ .\",\n \"Set $q = \\\\varphi (n_j)/n_j$ , and take $r \\\\in P^\\\\bullet $ and $c_1, \\\\dots , c_k \\\\in \\\\mathbb {N}_0$ satisfying that $\\\\mathsf {n}(r) = c_1 n_1 + \\\\dots + c_k n_k$ .\",\n \"Since $n_i \\\\varphi (n_j) = \\\\varphi (n_i n_j) = n_j \\\\varphi (n_i)$ for each $i \\\\in \\\\lbrace 1,\\\\dots ,k\\\\rbrace $ , one obtains $\\\\varphi (r) = \\\\frac{1}{\\\\mathsf {d}(r)} \\\\varphi (\\\\mathsf {n}(r)) = \\\\frac{1}{\\\\mathsf {d}(r)} \\\\sum _{i=1}^k c_i \\\\varphi (n_i) = \\\\frac{1}{\\\\mathsf {d}(r)} \\\\sum _{i=1}^k c_i n_i \\\\frac{\\\\varphi (n_j)}{n_j} = rq.$ As a result, the homomorphism $\\\\varphi $ is just multiplication by $q \\\\in \\\\mathbb {Q}_{>0}$ .\",\n \"It is easy to see that condition (2) is a direct consequence of condition (1), which completes the proof.\",\n \"Remark 3.3 A Puiseux monoid $M$ is said to be increasing (resp., decreasing) if $M$ can be generated by an increasing (resp., decreasing) sequence of rational numbers.\",\n \"Also, we say that $M$ is bounded if $M$ can be generated by a bounded sequence of rational numbers, and it is said to be strongly bounded if it can be generated by a sequence of rational numbers whose numerator set is bounded.\",\n \"Finally, $M$ is called dense if it contains 0 as a limit point.\",\n \"Although the definitions just given are not algebraic in nature, we should notice that they are all preserved by Puiseux monoid isomorphisms.\",\n \"This explains why all of them have been useful in the study of the atomic structure of Puiseux monoids (see [17] and [19]).\",\n \"With notation as in Definition REF , when $M$ and $N$ are reduced, we can restate the first condition above as (T1') $\\\\theta $ is surjective and $\\\\theta ^{-1}(1) = 1$ .\",\n \"We have already mentioned that a transfer homomorphism allows us to shift the atomic structure and the arithmetic of length of factorizations from its codomain to its domain.\",\n \"This property is formally described in the following proposition.\",\n \"Proposition 3.4 [9] If $\\\\theta \\\\colon M \\\\rightarrow N$ is a transfer homomorphism of atomic monoids, then the following conditions hold: $a \\\\in \\\\mathcal {A}(M)$ if and only if $\\\\theta (a) \\\\in \\\\mathcal {A}(N)$ ; $M$ is atomic if and only if $N$ is atomic; $\\\\mathsf {L}_M(x) = \\\\mathsf {L}_N(\\\\theta (x))$ for all $x \\\\in M$ ; $\\\\mathcal {L}(M) = \\\\mathcal {L}(N)$ , and so $M$ is a BF-monoid if and only if $N$ is a BF-monoid.\",\n \"For a Puiseux monoid $M$ , let $\\\\text{Aut}(M)$ denote the group of automorphisms of $M$ .\",\n \"As we have seen in Proposition REF , the set of homomorphisms between Puiseux monoids is very exclusive.\",\n \"In particular, we might wonder whether $\\\\text{Aut}(M)$ is always trivial.\",\n \"However, it is not hard to verify, for instance, that when $M_1 = \\\\langle 1/2^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ , multiplication by $1/2$ is in $\\\\text{Aut}(M_1)$ .\",\n \"This example might not be the most desirable because $M_1$ fails to be atomic; in fact, $M_1$ does not contain any atoms.\",\n \"The next proposition exhibits a family of atomic monoids whose groups of automorphisms are nontrivial.\",\n \"First, let us introduce a family of atomic Puiseux monoids whose atomicity is used in the proof.\",\n \"For $r \\\\in \\\\mathbb {Q}_{>0}$ , the monoid $M_r = \\\\langle r^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ is the multiplicatively $r$ -cyclic Puiseux monoid.\",\n \"If $\\\\mathsf {n}(r), \\\\mathsf {d}(r) > 1$ , then $M_r$ is atomic with $\\\\mathcal {A}(M_r) = \\\\lbrace r^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ (see [19]).\",\n \"Proposition 3.5 Let $r \\\\in \\\\mathbb {Q}_{>0}$ such that $\\\\mathsf {n}(r), \\\\mathsf {d}(r) > 1$ .\",\n \"If $M = \\\\langle r^n \\\\mid n \\\\in \\\\mathbb {Z}\\\\rangle $ , then $\\\\emph {Aut}(M) \\\\cong \\\\mathbb {Z}$ .\",\n \"Set $A = \\\\lbrace r^n \\\\mid n \\\\in \\\\mathbb {Z}\\\\rbrace $ .\",\n \"For $n \\\\in \\\\mathbb {Z}$ , the fact that $r^n A = A$ implies that multiplication by $r^n$ is an endomorphism of $M$ whose inverse is given by multiplication by $r^{-n}$ .\",\n \"Thus, multiplication by any integer power of $r$ is an automorphism of $M$ .\",\n \"To prove that these are the only elements of $\\\\text{Aut}(M)$ , let us first argue that $M$ is atomic with $\\\\mathcal {A}(M) = A$ .\",\n \"Assume first that $r < 1$ .\",\n \"Fix $k \\\\in \\\\mathbb {Z}$ , and let us check that $r^k \\\\in \\\\mathcal {A}(M)$ .\",\n \"To do this notice that the monoid $\\\\langle r^n \\\\mid n \\\\ge k \\\\rangle $ is the isomorphic image (under multiplication by $r^{k-1}$ ) of the multiplicatively $r$ -cyclic Puiseux monoid $M_r$ , which is atomic with set of atoms $A = \\\\lbrace r^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ .\",\n \"Since $r \\\\in \\\\mathcal {A}(M_r)$ , it follows that $r \\\\notin \\\\langle r^n \\\\mid n > 1 \\\\rangle $ .\",\n \"Then $r^k \\\\notin \\\\langle r^n \\\\mid n > k \\\\rangle $ .\",\n \"As $r < 1$ , no atom in $\\\\lbrace r^n \\\\mid n < k\\\\rbrace $ divides $r^k$ .\",\n \"Hence $r^k \\\\notin \\\\langle A \\\\setminus \\\\lbrace r^k\\\\rbrace \\\\rangle $ and, therefore, $r^k \\\\in \\\\mathcal {A}(M)$ .\",\n \"As a result, $\\\\mathcal {A}(M) = A$ .\",\n \"Now suppose that $r > 1$ .\",\n \"As before, fix $k \\\\in \\\\mathbb {Z}$ .\",\n \"Because $r > 1$ , proving that $r^k \\\\in \\\\mathcal {A}(M)$ amounts to showing that $r^k \\\\notin \\\\langle r^n \\\\mid n < k \\\\rangle $ .\",\n \"Let us assume, by way of contradiction, that this is not the case.\",\n \"Then $r^k = a_1 r^{n_1} + \\\\dots + a_t r^{n_t}$ for some $a_1, \\\\dots , a_t \\\\in \\\\mathbb {N}$ and $n_1, \\\\dots , n_t \\\\in \\\\mathbb {N}$ with $k > n_1 > \\\\dots > n_t$ .\",\n \"As a consequence, $r^{k - n_t + 1} \\\\in \\\\langle r^{n_1 - n_t + 1}, r^{n_2 - n_t + 1}, \\\\dots , r \\\\rangle $ , which contradicts the fact that $r^{k - n_t + 1} \\\\in \\\\mathcal {A}(M_r)$ .\",\n \"As in the previous case, we conclude that $\\\\mathcal {A}(M) = A$ .\",\n \"By Proposition REF , any automorphism of $M$ is given by rational multiplication.\",\n \"Take $s \\\\in \\\\mathbb {Q}_{>0}$ such that $\\\\phi _s \\\\in \\\\text{Aut}(M)$ , where $\\\\phi _s$ consists in left multiplication by $s$ .\",\n \"Because $\\\\phi _s$ must send atoms to atoms, it follows that $sr = \\\\phi _s(r) \\\\in A$ .\",\n \"Therefore $s$ must be an integer power of $r$ .\",\n \"Hence $\\\\text{Aut}(M)$ is precisely $A$ when seen as a multiplicative subgroup of $\\\\mathbb {Q}$ .\",\n \"As $A$ is the infinite cyclic group, the proof follows.\"\n ],\n [\n \"Finite Transfer Puiseux Monoids\",\n \"Now we turn to characterize the transfer homomorphisms from Puiseux monoids to finitely generated monoids.\",\n \"Definition 4.1 We say that a Puiseux monoid $M$ is transfer finite if there exists a transfer homomorphism from $M$ to a finitely generated monoid.\",\n \"By the fundamental structure theorem of finitely generated abelian groups, it immediately follows that every finitely generated monoid $F$ is a submonoid of a group $T \\\\times \\\\mathbb {Z}^\\\\beta $ for some finite abelian group $T$ and $\\\\beta \\\\in \\\\mathbb {N}_0$ .\",\n \"In case of $F$ being reduced, it can be thought of as a submonoid of $T \\\\times \\\\mathbb {N}_0^\\\\beta $ .\",\n \"Condition (T2) in the definition of a transfer homomorphism $\\\\theta \\\\colon M \\\\rightarrow F$ is crucial to transfer the factorization behavior of $F$ to $M$ .\",\n \"However, the reader might wonder how much the set $\\\\text{Hom}(M,F)$ will increase if we drop condition (T2).\",\n \"Surprisingly, the set of homomorphisms will remain the same as long as we impose $F$ to be reduced.\",\n \"This fact facilitates to classify the Puiseux monoids that happen to be transfer finite, as we will prove in the next theorem.\",\n \"First, notice that if $\\\\phi \\\\colon M \\\\rightarrow N$ is a monoid homomorphism, then the map $\\\\phi _{\\\\text{red}} \\\\colon M_{\\\\text{red}} \\\\rightarrow N_{\\\\text{red}}$ defined by $\\\\phi _{\\\\text{red}}(aM^\\\\times ) = \\\\phi (a)N^\\\\times $ is also a monoid homomorphism.\",\n \"Theorem 4.2 Let $M$ be a nontrivial Puiseux monoid, and let $F$ be a finitely generated (additive) monoid.\",\n \"If $\\\\theta \\\\colon M \\\\rightarrow F$ is a homomorphism satisfying $\\\\theta ^{-1}(0) = \\\\lbrace 0\\\\rbrace $ , then $M$ is isomorphic to a numerical monoid.\",\n \"The Puiseux monoid $M$ is transfer finite if and only if it is isomorphic to a numerical monoid.\",\n \"We argue first part (1).\",\n \"It is easy to see that $\\\\theta _{\\\\text{red}} \\\\colon M_{\\\\text{red}} = M \\\\rightarrow F_{\\\\text{red}}$ is also a transfer homomorphism.\",\n \"So we can assume, without loss of generality, that $F$ is reduced.\",\n \"Suppose that $F$ is a submonoid of $T \\\\times \\\\mathbb {N}_0^\\\\beta $ , where $T$ is a finite abelian group and $\\\\beta \\\\in \\\\mathbb {N}_0$ .\",\n \"First, assume, by way of contradiction, that $\\\\beta = 0$ .\",\n \"In this case, it is not hard to verify that $\\\\theta (M)$ must be a subgroup of $T$ .\",\n \"If $\\\\alpha = |\\\\theta (M)|$ and $r \\\\in M^\\\\bullet $ , then $\\\\theta (\\\\alpha r) = \\\\alpha \\\\theta (r) = 0$ .\",\n \"This contradicts that $\\\\theta ^{-1}(0) = \\\\lbrace 0\\\\rbrace $ .\",\n \"Thus, $\\\\beta \\\\ge 1$ .\",\n \"Define $\\\\pi \\\\colon T \\\\times \\\\mathbb {N}_0^\\\\beta \\\\rightarrow \\\\mathbb {N}_0^\\\\beta $ by $\\\\pi (t,\\\\mathbf {v}) = \\\\mathbf {v}$ for all $t \\\\in T$ and $\\\\mathbf {v} \\\\in \\\\mathbb {N}^\\\\beta $ .\",\n \"Let us verify that $\\\\pi (\\\\theta (M))$ is finitely generated.\",\n \"Take $\\\\textbf {x} = (x_1, \\\\dots , x_\\\\beta ) \\\\in \\\\pi (\\\\theta (M))^\\\\bullet $ , and let $d = \\\\gcd (x_1, \\\\dots , x_\\\\beta )$ .\",\n \"We shall verify that $\\\\pi (\\\\theta (M)) \\\\subseteq \\\\langle \\\\textbf {x}/d \\\\rangle $ .\",\n \"To do so, consider $\\\\textbf {y} = (y_1, \\\\dots , y_\\\\beta ) \\\\in \\\\pi (\\\\theta (M))^\\\\bullet $ .\",\n \"Now take $r,s \\\\in M^\\\\bullet $ such that $\\\\pi (\\\\theta (r)) = \\\\textbf {x}$ and $\\\\pi (\\\\theta (s)) = \\\\textbf {y}$ , and take $m,n \\\\in \\\\mathbb {N}$ satisfying that $\\\\gcd (m,n) = 1$ and $mr = ns$ .\",\n \"Because $m \\\\textbf {x} = \\\\pi (\\\\theta (mr)) = \\\\pi (\\\\theta (ns)) = n \\\\textbf {y},$ one finds that $mx_i = ny_i$ for $i = 1, \\\\dots , \\\\beta $ .\",\n \"As $\\\\gcd (m,n) = 1$ , it follows that $n$ divides each $x_i$ , i.e., $d/n \\\\in \\\\mathbb {N}$ .\",\n \"As a result, $\\\\mathbf {y} = \\\\frac{m}{n} \\\\textbf {x} = \\\\bigg (\\\\frac{md}{n} \\\\bigg ) \\\\frac{\\\\mathbf {x}}{d} \\\\in \\\\bigg \\\\langle \\\\frac{\\\\mathbf {x}}{d} \\\\bigg \\\\rangle .$ Hence $\\\\pi (\\\\theta (M)) \\\\subseteq \\\\langle \\\\textbf {x}/d \\\\rangle $ .\",\n \"Because $\\\\langle \\\\mathbf {x}/d \\\\rangle $ is isomorphic to $\\\\mathbb {N}_0$ , it follows that $\\\\pi (\\\\theta (M))$ is finitely generated.\",\n \"We show now that $M$ is also finitely generated, which amounts to proving that $\\\\pi \\\\circ \\\\theta \\\\colon M \\\\rightarrow \\\\mathbb {N}_0^\\\\beta $ is injective.\",\n \"First, let us verify that $\\\\pi $ is injective when restricted to $\\\\theta (M)$ .\",\n \"As $\\\\theta (M)$ is a submonoid of the reduced monoid $F$ , it is also reduced.\",\n \"Suppose that $(t_1, \\\\mathbf {v}), (t_2, \\\\mathbf {v}) \\\\in \\\\theta (M)$ , and let us check that $t_1 = t_2$ .\",\n \"If $\\\\mathbf {v} = \\\\mathbf {0}$ , then $t_1 = t_2 = 0$ because $\\\\theta (M)^\\\\times $ is trivial.\",\n \"Otherwise, there exist $r,s \\\\in M^\\\\bullet $ such that $\\\\theta (r) = (t_1, \\\\mathbf {v})$ and $\\\\theta (s) = (t_2, \\\\mathbf {v})$ .\",\n \"Take $m,n \\\\in \\\\mathbb {N}$ such that $mr = ns$ .\",\n \"Since $m (t_1, \\\\mathbf {v}) = m \\\\theta (r) = n \\\\theta (s) = n (t_2, \\\\mathbf {v})$ and $\\\\mathbf {v} \\\\ne \\\\mathbf {0}$ , one finds that $m = n$ and, therefore, $r = s$ .\",\n \"This, in turn, implies that $t_1 = t_2$ .\",\n \"Hence the restriction of $\\\\pi $ to $\\\\theta (M)$ is injective.\",\n \"To conclude the proof of part (1), we show that $\\\\theta $ is also injective.\",\n \"Let $r,s \\\\in M$ such that $\\\\theta (r) = \\\\theta (s) \\\\ne 0$ .\",\n \"Taking $m,n \\\\in \\\\mathbb {N}$ satisfying $mr = ns$ , we have $m \\\\theta (r) = \\\\theta (mr) = \\\\theta (ns) = n \\\\theta (s).$ Since $\\\\theta (M)$ is reduced, the element $\\\\theta (r)$ must be torsion-free in $T \\\\times \\\\mathbb {N}_0^\\\\beta $ .\",\n \"Thus, $m = n$ , which implies that $r = s$ .\",\n \"As $\\\\theta ^{-1}(0) = \\\\lbrace 0\\\\rbrace $ , it follows that $|\\\\theta ^{-1}(a)| = 1$ for all $a \\\\in \\\\theta (M)$ .\",\n \"Therefore $\\\\theta $ is injective, leading us to the injectivity of $\\\\pi \\\\circ \\\\theta $ .\",\n \"Now that fact that $\\\\pi (\\\\theta (M))$ is finitely generated implies that $M$ is also finitely generated.\",\n \"Hence $M$ must be isomorphic to a numerical monoid.\",\n \"Finally, let us argue part (2) of the theorem.\",\n \"For the direct implication, assume that the homomorphism $\\\\theta \\\\colon M \\\\rightarrow F$ is a transfer homomorphism.\",\n \"As we did in the proof of part (1), we can assume that $F$ is a reduced.\",\n \"As both $M$ and $F$ are reduced, condition (T1') yields $\\\\theta ^{-1}(0) = \\\\lbrace 0\\\\rbrace $ .\",\n \"Now it follows by part (1) that the Puiseux monoid $M$ is isomorphic to a numerical monoid.\",\n \"For the reverse implication, just take $\\\\theta $ to be the identity map.\",\n \"Imposing the homomorphism $\\\\theta \\\\colon M \\\\rightarrow F$ in Theorem REF to satisfy $\\\\theta ^{-1}(0) = \\\\lbrace 0\\\\rbrace $ is not superfluous even if $M$ is atomic.\",\n \"Then next example sheds some light upon this observation.\",\n \"Example 4.3 Let $p_1, p_2, \\\\dots , $ be an enumeration of the odd prime numbers, and let $M = \\\\langle 1/p_n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ .\",\n \"It is not hard to verify that $\\\\mathcal {A}(M) = \\\\lbrace 1/p_n \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ .\",\n \"This implies that $M$ is atomic.\",\n \"Now define $\\\\theta \\\\colon M \\\\rightarrow \\\\mathbb {Z}_2$ by setting $\\\\theta (0) = 0$ , $\\\\theta (r) = 0$ if $\\\\mathsf {n}(r)$ is even, and $\\\\theta (r) = 1$ if $\\\\mathsf {n}(r)$ is odd.\",\n \"It follows immediately that $\\\\theta $ is a surjective monoid homomorphism.\",\n \"However, $M$ is not isomorphic to any numerical monoid because it contains infinitely many atoms.\"\n ],\n [\n \"Strongly Primary Puiseux Monoids\",\n \"In this section we investigate which Puiseux monoids are strongly primary.\",\n \"In the case of Puiseux monoids, being strongly primary is equivalent to being finitary.\",\n \"In general, finitary monoids provide a common algebraic framework to study not only the arithmetic of strongly primary monoids but also that one of $v$ -noetherian $G$ -monoids (see [10]).\",\n \"The structure of strongly primary monoids was first studied by Satyanarayana in [23] and has received substantial attention in the literature since then (see [15] and references therein).\",\n \"In particular, the class of strongly primary monoids yields multiplicative models for a large class of one-dimensional local domains (see [10]).\",\n \"All monoids mentioned in this section are assumed to be reduced.\",\n \"Definition 5.1 Let $M$ be a monoid.\",\n \"A submonoid $S$ of $M$ is called divisor-closed provided that for all $x \\\\in M$ and $s \\\\in S$ the fact that $x \\\\mid _M s$ implies that $x \\\\in S$ .\",\n \"The monoid $M$ is called primary if it is nontrivial and its only divisor-closed submonoids are $\\\\lbrace 0\\\\rbrace $ and $M$ .\",\n \"The monoid $M$ is called finitary if $M$ is a BF-monoid and there exist $n \\\\in \\\\mathbb {N}$ and a finite subset $S \\\\subseteq M^\\\\bullet $ such that $n M^\\\\bullet \\\\subseteq S + M$ .\",\n \"The monoid $M$ is called strongly primary provided that $M$ is both primary and finitary.\",\n \"Let $M$ be a Puiseux monoid, and let $M^{\\\\prime }$ be a nontrivial proper submonoid of $M$ .\",\n \"Notice that for all $x \\\\in M \\\\setminus M^{\\\\prime }$ and $y \\\\in M^{\\\\prime }$ satisfying that $x \\\\mid _M y$ , the fact that $x \\\\mid _M \\\\mathsf {n}(x) \\\\mathsf {d}(y) y \\\\in M^{\\\\prime }$ immediately implies that $M^{\\\\prime }$ is not a divisor-closed submonoid of $M$ .\",\n \"Thus, $M$ is primary.\",\n \"On the other hand, suppose that $F$ is a finitely generated monoid, say $F = \\\\langle S \\\\rangle $ for some finite subset $S$ of $F^\\\\bullet $ .\",\n \"Then the fact that $F^\\\\bullet = S + F$ immediately implies that $F$ is finitary.\",\n \"In particular, every nontrivial finitely generated Puiseux monoid is finitary and, therefore, strongly primary.\",\n \"The next proposition summarizes the observations made in this paragraph.\",\n \"Proposition 5.2 Every nontrivial Puiseux monoid is primary.\",\n \"Every finitely generated Puiseux monoid is strongly primary.\",\n \"A Puiseux monoid that is not finitely generated may fail to be strongly primary (see, for example, Proposition REF ).\",\n \"However, in Proposition REF and Proposition REF we exhibit two infinite families of non-finitely generated Puiseux monoids that are strongly primary.\",\n \"To argue Proposition REF , we will use the following result, which is a weaker version of [16].\",\n \"Proposition 5.3 If $M$ is a Puiseux monoid satisfying that 0 is not a limit point of $M^\\\\bullet $ , then $M$ is a BF-monoid.\",\n \"The next two propositions introduce two families of (non-finitely generated) strongly primary Puiseux monoids.\",\n \"Proposition 5.4 Let $p, q \\\\in \\\\mathbb {N}$ such that $\\\\gcd (p,q) = 1$ , and let $\\\\lbrace S_n\\\\rbrace $ be an inclusion-decreasing sequence of numerical monoids.\",\n \"If a function $f \\\\colon \\\\mathbb {N}\\\\rightarrow \\\\mathbb {N}$ satisfies that $f(1) = 1$ and $q^{f(n+1) - f(n)} - p^n > p \\\\max \\\\lbrace F(S_n), a \\\\mid a \\\\in \\\\mathcal {A}(S_n) \\\\rbrace $ for every $n \\\\in \\\\mathbb {N}$ , then the Puiseux monoid $\\\\bigg \\\\langle \\\\frac{q^{f(n)}}{p^n} s \\\\ \\\\bigg {|} \\\\ n \\\\in \\\\mathbb {N}\\\\ \\\\text{ and} \\\\ s \\\\in S_n \\\\bigg \\\\rangle $ is strongly primary.\",\n \"Set $M = \\\\big \\\\langle q^{f(n)}s/p^n \\\\mid n \\\\in \\\\mathbb {N}\\\\ \\\\text{and} \\\\ s \\\\in S_n \\\\big \\\\rangle $ , and for each $n \\\\in \\\\mathbb {N}$ set $A_n = \\\\mathcal {A}(S_n)$ .\",\n \"First, we argue that $M$ is a BF-monoid.\",\n \"To do so, observe that for each $n \\\\in \\\\mathbb {N}$ the fact that $q^{f(n+1) - f(n)} > p \\\\max A_n$ implies that $ \\\\min \\\\frac{q^{f(n+1)}}{p^{n+1}} A_{n+1} \\\\ge \\\\frac{q^{f(n+1)}}{p^{n+1}} > \\\\frac{q^{f(n)}}{p^n} \\\\max A_n.$ Therefore the generating set $\\\\cup _{n \\\\in \\\\mathbb {N}} (q^{f(n)}/p^n) A_n$ of $M$ can be listed as an increasing sequence of rational numbers.\",\n \"As $M$ is generated by an increasing sequence of rational numbers, 0 cannot be a limit point of $M^\\\\bullet $ .\",\n \"Hence $M$ is a BF-monoid by Proposition REF .\",\n \"We proceed to prove that $M$ is finitary.\",\n \"Since $M \\\\cap \\\\mathbb {N}_0$ is a submonoid of $(\\\\mathbb {N}_0,+)$ , it is atomic and $\\\\mathcal {A}(M \\\\cap \\\\mathbb {N}_0)$ is finite.\",\n \"Take $S$ to be the finite set $\\\\mathcal {A}(M \\\\cap \\\\mathbb {N}_0) \\\\cup q A_1 \\\\subset M$ .\",\n \"We are done once we verify the inclusion $p M^\\\\bullet \\\\subseteq S + M$ .\",\n \"Fix $n \\\\in \\\\mathbb {N}$ and $a \\\\in A_{n+1}$ .\",\n \"Because $q^{f(n)}a \\\\in q^{f(n)}S_{n+1} \\\\subseteq q^{f(n)}S_n \\\\in M \\\\cap \\\\mathbb {N}_0$ , the element $q^{f(n)}a$ is divisible in $M$ by an element of $S$ .\",\n \"On the other hand, $(q^{f(n+1) - f(n)} - p^n)a \\\\ge q^{f(n+1) - f(n)} - p^n > F(S_n)$ , which implies that $(q^{f(n+1) - f(n)} - p^n)a \\\\, \\\\frac{q^{f(n)}}{p^n} \\\\in M$ .\",\n \"Thus, $p\\\\bigg ( \\\\frac{q^{f(n+1)}}{p^{n+1}} a \\\\bigg ) = q^{f(n)}a + \\\\big (q^{f(n+1) - f(n)} - p^n\\\\big ) \\\\frac{q^{f(n)}}{p^n} a \\\\in S + M.$ In addition, the fact that $qa \\\\in qS_{n+1} \\\\subseteq qS_1$ guarantees $qa$ is divisible in $M$ by some element of $S$ .\",\n \"This, in turn, implies that $p \\\\big (q^{f(1)}/p\\\\big )a = qa \\\\in S + M$ .\",\n \"As a result, for each $x \\\\in M^\\\\bullet $ it follows that $px \\\\in m(S+M) \\\\subseteq S + M$ for some $m \\\\in \\\\mathbb {N}$ .\",\n \"Hence $pM^\\\\bullet \\\\subseteq S + M$ , as desired.\",\n \"Example 5.5 Consider the Puiseux monoid $M = \\\\bigg \\\\langle \\\\frac{3^{n^2 + 1}}{2^n}, \\\\frac{5 \\\\cdot 3^{n^2}}{2^n} \\\\ \\\\bigg {|} \\\\ n \\\\in \\\\mathbb {N}\\\\bigg \\\\rangle .$ Taking $S_n$ to be the numerical monoid $\\\\langle 3, 5 \\\\rangle $ for each $n \\\\in \\\\mathbb {N}$ and defining the function $f \\\\colon \\\\mathbb {N}\\\\rightarrow \\\\mathbb {N}$ by $f(n) = n^2$ , we can rewrite the Puiseux monoid $M$ as follows: $M = \\\\bigg \\\\langle \\\\frac{3^{f(n)}}{2^n} s \\\\ \\\\bigg {|} \\\\ n \\\\in \\\\mathbb {N}\\\\ \\\\text{ and} \\\\ s \\\\in S_n \\\\bigg \\\\rangle .$ Since $F(S_n) = 7$ for each $n \\\\in \\\\mathbb {N}$ , it follows that $3^{f(n+1) - f(n)} - 2^n = 3^{2n+1} - 2^n > 14 = 2 \\\\max \\\\lbrace 3,5, F(S_n)\\\\rbrace .$ As $f(1) = 1$ , Proposition REF guarantees that $M$ is a strongly primary Puiseux monoid.\",\n \"As in Section , for $r \\\\in \\\\mathbb {Q}_{> 0}$ we let $M_r$ denote the multiplicatively $r$ -cyclic Puiseux monoid $\\\\langle r^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ .\",\n \"Proposition 5.6 For each $r \\\\in \\\\mathbb {Q}_{> 1}$ , the Puiseux monoid $M_r$ is strongly primary.\",\n \"Since $r > 1$ , it follows that 0 is not a limit point of $M_r^\\\\bullet $ .\",\n \"Thus, Proposition REF ensures that $M_r$ is a BF-monoid.\",\n \"On the other hand, it was proved in [19] that $\\\\mathcal {A}(M_r) = \\\\lbrace r^n \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ .\",\n \"Now we check that $\\\\mathsf {n}(r)$ divides $\\\\mathsf {n}(r)r^j$ in $M_r$ for every $j \\\\in \\\\mathbb {N}_0$ .\",\n \"If $j=0$ , then $\\\\mathsf {n}(r) \\\\mid _{M_r} \\\\mathsf {n}(r)r^j$ follows trivially.\",\n \"Hence it suffices to assume that $j \\\\in \\\\mathbb {N}$ .\",\n \"In this case, the fact that $\\\\mathsf {n}(r)(r-1) = r( \\\\mathsf {n}(r) - \\\\mathsf {d}(r))$ implies that $\\\\mathsf {n}(r) r^j - \\\\mathsf {n}(r) = \\\\mathsf {n}(r) (r-1) \\\\sum _{i=0}^{j-1} r^i = r \\\\big (\\\\mathsf {n}(r) - \\\\mathsf {d}(r)\\\\big ) \\\\sum _{i=0}^{j-1} r^i = \\\\big (\\\\mathsf {n}(r) - \\\\mathsf {d}(r)\\\\big ) \\\\sum _{i=0}^{j-1} r^{i+1} \\\\in M_r.$ Therefore $\\\\mathsf {n}(r)$ divides $\\\\mathsf {n}(r) r^j$ in $M_r$ .\",\n \"To show now that $M_r$ is finitary, we take $n = \\\\mathsf {d}(r)$ and $S = \\\\lbrace \\\\mathsf {n}(r)\\\\rbrace $ and verify that $n M_r^\\\\bullet \\\\subseteq S + M_r$ .\",\n \"For $q \\\\in M_r^\\\\bullet $ , take $\\\\alpha _1, \\\\dots , \\\\alpha _t \\\\in \\\\mathbb {N}_0$ such that $q = \\\\sum \\\\alpha _i r^i$ , and fix $k \\\\in \\\\lbrace 1, \\\\dots , t\\\\rbrace $ such that $\\\\alpha _k > 0$ .\",\n \"Because $\\\\mathsf {n}(r)$ divides $\\\\mathsf {n}(r) r^{k-1}$ in $M_r$ , there exists $s \\\\in M_r$ satisfying that $\\\\mathsf {n}(r) r^{k-1} = \\\\mathsf {n}(r) + s$ .\",\n \"Thus, $n q &= \\\\mathsf {d}(r) \\\\alpha _k r^k + \\\\sum _{i \\\\ne k} \\\\mathsf {d}(r) \\\\alpha _i r^i \\\\\\\\&= \\\\mathsf {n}(r) r^{k-1} + (\\\\alpha _k - 1) \\\\mathsf {d}(r) r^k + \\\\sum _{i \\\\ne k} \\\\mathsf {d}(r) \\\\alpha _i r^i \\\\\\\\&= \\\\mathsf {n}(r) + \\\\big ( s + (\\\\alpha _k - 1) \\\\mathsf {d}(r) r^k + \\\\sum _{i \\\\ne k} \\\\mathsf {d}(r) \\\\alpha _i r^i \\\\big ),$ which implies that $nq \\\\in S + M_r$ .\",\n \"Since $n M_r^\\\\bullet \\\\subseteq S + M_r$ , one obtains that $M_r$ is finitary and, therefore, strongly primary.\",\n \"We conclude this section providing a family of Puiseux monoids that fail to be strongly primary.\",\n \"Proposition 5.7 Let $\\\\lbrace a_n\\\\rbrace $ be a sequence of positive rational numbers satisfying that $\\\\gcd (\\\\mathsf {d}(a_i), \\\\mathsf {d}(a_j)) = 1$ for any $i \\\\ne j$ .\",\n \"Then the Puiseux monoid $\\\\langle a_n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ is atomic but not strongly primary.\",\n \"Set $M = \\\\langle a_n \\\\mid n \\\\in \\\\mathbb {N}\\\\rangle $ .\",\n \"It is not difficult to verify that $\\\\mathcal {A}(M) = \\\\lbrace a_n \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ from the fact that $\\\\gcd (\\\\mathsf {d}(a_i), \\\\mathsf {d}(a_j)) = 1$ for any $i \\\\ne j$ ; we leave the details to the reader.\",\n \"This implies, in particular, that $M$ is atomic.\",\n \"Suppose, by way of contradiction, that $M$ is finitary.\",\n \"Choose $n \\\\in \\\\mathbb {N}$ and a finite subset $S$ of $M^\\\\bullet $ such that $n M^\\\\bullet \\\\subseteq S + M$ .\",\n \"Let $S^{\\\\prime }$ be a finite subset of $\\\\mathcal {A}(M)$ such that for each $s \\\\in S$ there is at least one atom in $S^{\\\\prime }$ dividing $s$ in $M$ .\",\n \"After substituting $S$ by $S^{\\\\prime }$ , we can assume that $S \\\\subset \\\\mathcal {A}(M)$ .\",\n \"Since $n M^\\\\bullet \\\\subseteq S + M$ and $\\\\inf (S+M) \\\\ge \\\\min S$ , it follows that 0 cannot be a limit point of $M^\\\\bullet $ .\",\n \"Fix $\\\\epsilon > 0$ such that $\\\\epsilon < \\\\inf M^\\\\bullet $ .\",\n \"Since $\\\\gcd (\\\\mathsf {d}(a_i), \\\\mathsf {d}(a_j)) = 1$ , there exists $j \\\\in \\\\mathbb {N}$ such that $\\\\mathsf {d}(a_j) > \\\\max \\\\lbrace n, \\\\max \\\\mathsf {d}(S)\\\\rbrace $ .\",\n \"Because $n M^\\\\bullet \\\\subseteq S + M$ , one can write $ n a_j = a_s + \\\\sum _{i=1}^k \\\\alpha _i a_i$ for some $k \\\\in \\\\mathbb {N}$ , $a_s \\\\in S$ , and $\\\\alpha _i \\\\in \\\\mathbb {N}_0$ .\",\n \"Applying the $\\\\mathsf {d}(a_j)$ -valuation to both sides of (REF ) and using the fact that $\\\\gcd (\\\\mathsf {d}(a_i), \\\\mathsf {d}(a_j)) = 1$ , it is not hard to find that $\\\\mathsf {d}(a_j)$ divides $n - \\\\alpha _j$ .\",\n \"Now $\\\\mathsf {d}(a_j) > n$ yields $n = \\\\alpha _j$ .\",\n \"This, along with (REF ), would force $a_s = 0$ , which contradicts that $S \\\\subset \\\\mathcal {A}(M)$ .\",\n \"Thus, $M$ is not strongly primary.\",\n \"Remark 5.8 The previous proposition not only shows that Puiseux monoids are not, in general, strongly primary, but also illustrates that a natural bounding, ordering, or topological restriction under which a Puiseux monoid is guaranteed to be strongly primary is rather unlikely.\",\n \"For example, consider the Puiseux monoids $M_1 = \\\\bigg \\\\langle \\\\frac{1}{p} \\\\ \\\\bigg {|} \\\\ p \\\\ \\\\text{is prime} \\\\bigg \\\\rangle , \\\\ M_2 = \\\\bigg \\\\langle \\\\frac{p-1}{p} \\\\ \\\\bigg {|} \\\\ p \\\\ \\\\text{is prime} \\\\bigg \\\\rangle , \\\\ \\\\text{and} \\\\ M_3 = \\\\bigg \\\\langle \\\\frac{p^2+1}{p} \\\\ \\\\bigg {|} \\\\ p \\\\ \\\\text{is prime} \\\\bigg \\\\rangle .$ It is not hard to check that the sets of atoms of $M_1$ , $M_2$ , and $M_3$ are precisely the generating sets displayed.\",\n \"Therefore $M_1$ is strongly bounded, $M_2$ is bounded, and $M_3$ is not bounded.\",\n \"We can also see that $M_1$ is a decreasing Puiseux monoid, while $M_2$ is increasing.\",\n \"Furthermore, notice that 0 is a limit point of $M_1^\\\\bullet $ , but 0 is not a limit point of $M_2^\\\\bullet $ .\",\n \"Finally, Proposition REF ensures that $M_1$ , $M_2$ , and $M_3$ are all strongly primary.\"\n ],\n [\n \"Puiseux Monoids Are Almost Never Transfer Krull\",\n \"We dedicate this section to show that the atomic structure of Puiseux monoids almost never can be obtained by transferring back that one of Krull monoids; specifically we shall prove that the existence of a transfer homomorphism from a nontrivial Puiseux monoid to a Krull monoid forces the domain to be isomorphic to $(\\\\mathbb {N}_0,+)$ .\",\n \"The we use this information to show that only finitely generated Puiseux monoids admit transfer homomorphisms to C-monoids.\",\n \"Let us start by giving the definition of a Krull monoid.\",\n \"Definition 6.1 A monoid $K$ is called a Krull monoid if there is a monoid homomorphism $\\\\varphi \\\\colon K \\\\rightarrow D$ , where $D$ is a free abelian monoid and $\\\\varphi $ satisfies the following two conditions: if $a, b \\\\in K$ and $\\\\varphi (a) \\\\mid _D \\\\varphi (b)$ , then $a \\\\mid _K b$ ; for every $d \\\\in D$ there exist $a_1, \\\\dots , a_n \\\\in K$ with $d = \\\\gcd \\\\lbrace \\\\varphi (a_1), \\\\dots , \\\\varphi (a_n)\\\\rbrace $ .\",\n \"With notation as in Definition REF , it is easy to see that $K$ is a Krull monoid if and only if $K_{\\\\text{red}}$ is a Krull monoid.\",\n \"The basis elements of $D$ are called the prime divisors of $K$ .\",\n \"The abelian group Cl$(K) := D/\\\\varphi (K)$ is called the class group of $K$ (see [10]).\",\n \"As Krull monoids are isomorphic to submonoids of free abelian monoids, Krull monoids are atomic.\",\n \"The factorization theory of Krull monoids has been significantly studied (see [5], [13] and references therein).\",\n \"The class of Krull monoids contains many well-studied types of monoids, including the multiplicative monoid of the ring of integers of an algebraic number, the Hilbert monoids, and the regular congruence monoids.\",\n \"These and further examples of Krull monoids are presented in [9] and [10].\",\n \"From the point of view of factorization theory, perhaps the most important family of Krull monoids is that one consisting of block monoids, which we are about to introduce.\",\n \"This is because block monoids capture the essence of the arithmetic of lengths of factorizations in Krull monoids.\",\n \"Let $G$ be an abelian group and $\\\\mathcal {F}(G)$ the free abelian monoid on $G$ .\",\n \"An element $X = g_1 \\\\dots g_l \\\\in \\\\mathcal {F}(G)$ is called a sequence over $G$ .\",\n \"The length of $X$ is defined as $|X| = l =\\\\sum _{g \\\\in G} \\\\mathsf {v}_g(X).$ For every $I \\\\subseteq [1, l]$ , the sequence $Y = \\\\prod _{i \\\\in I} g_i$ is called a subsequence of $X$ .\",\n \"The subsequences are precisely the divisors of $X$ in the free abelian monoid $\\\\mathcal {F}(G)$ .\",\n \"The submonoid $\\\\mathcal {B}(G) := \\\\bigg \\\\lbrace X \\\\in \\\\mathcal {F}(G) \\\\ \\\\bigg {|} \\\\ \\\\sum _{g \\\\in G} \\\\mathsf {v}_g(X) g = 0 \\\\bigg \\\\rbrace $ of $\\\\mathcal {F}(G)$ is called the block monoid on $G$ , and its elements are referred to as zero-sum sequences or blocks over $G$ ([10] is a good general reference on block monoids).\",\n \"Furthermore, if $G_0$ is a subset of $G$ , then the submonoid $\\\\mathcal {B}(G_0) := \\\\lbrace X \\\\in \\\\mathcal {B}(G) \\\\mid \\\\mathsf {v}_g(X) = 0 \\\\ \\\\emph { if } \\\\ g \\\\notin G_0 \\\\rbrace $ of $\\\\mathcal {B}(G)$ is called the restriction of the block monoid $\\\\mathcal {B}(G)$ to $G_0$ .\",\n \"For $X \\\\in \\\\mathcal {B}(G_0)$ , the support of $X$ in $G_0$ is defined to be $\\\\text{supp}_{G_0}(X) := \\\\lbrace g \\\\in G_0 \\\\mid \\\\mathsf {v}_g(X) > 0\\\\rbrace .$ As mentioned before, the relevance of block monoids in the theory of non-unique factorizations lies in the next result.\",\n \"Proposition 6.2 [10] Let $K$ be a Krull monoid with class group $G$ and let $G_0$ be the set of classes of $G$ which contain prime divisors.\",\n \"Then $\\\\mathcal {L}(K) = \\\\mathcal {L}(\\\\mathcal {B}(G_0)).$ As a consequence, understanding the arithmetic of lengths of factorizations in Krull monoids amounts to understanding the same in block monoids.\",\n \"Definition 6.3 A Puiseux monoid $M$ is transfer Krull if there exist an abelian group $G$ , a subset $G_0$ of $G$ , and a transfer homomorphism $\\\\theta \\\\colon M \\\\rightarrow \\\\mathcal {B}(G_0)$ .\",\n \"Remark: Our definition of a transfer Krull monoid coincides with the definition given in [8]; this is because in the present setting the concepts of a transfer homomorphism and the concept of a weak transfer homomorphism coincide by [3].\",\n \"We denote the field of fractions of an integral domain $R$ by $\\\\mathsf {q}(R)$ .\",\n \"For subsets $X,Y$ of $\\\\mathsf {q}(R)$ we set $(X : Y) := \\\\lbrace x \\\\in \\\\mathsf {q}(R) \\\\mid xY \\\\subseteq X\\\\rbrace $ .\",\n \"In addition, $R$ is called a Krull domain if $R^\\\\bullet $ is a Krull monoid.\",\n \"In this case, the divisor class group of $R$ , denoted by $\\\\mathcal {C}(R)$ , measures the extent to which factorizations in $R$ fail to be unique (see [10]).\",\n \"Unlike Krull domains/monoids, which have been central objects in commutative algebra since mid-nineteenth century, transfer Krull monoids (which generalize the concept of Krull monoids) were introduced more recently.\",\n \"Let us proceed to present a few examples of transfer Krull monoids.\",\n \"Examples of transfer Krull monoids: Let $H$ be a half-factorial monoid, and let $\\\\theta \\\\colon H \\\\rightarrow \\\\mathcal {B}(\\\\lbrace 0\\\\rbrace )$ be the map defined by $\\\\theta (h) = 0$ if $h \\\\in \\\\mathcal {A}(H)$ and $\\\\theta (h) = 1$ if $h \\\\in H^\\\\times $ .\",\n \"As the map $\\\\theta $ is a transfer homomorphism, it follows that $H$ is a transfer Krull monoid.\",\n \"Let $R$ be a Krull domain, and let $K$ be a subring of $R$ with the same field of fractions.\",\n \"Suppose, in addition, that the following three conditions hold: $R = KR^\\\\times $ ; $K \\\\cap R^\\\\times = K^\\\\times $ ; $(K : R)$ is a maximal ideal of $K$ (see [10]).\",\n \"Then the inclusion map $K^\\\\bullet \\\\hookrightarrow R^\\\\bullet $ is a transfer homomorphism and, therefore, $K^\\\\bullet $ is a transfer Krull monoid.\",\n \"Transfer Krull monoids can also be defined in a non-commutative context (see, for instance, [3]).\",\n \"Let $R$ be a bounded HNP (hereditary Noetherian prime) ring.\",\n \"If every stably free left $R$ -ideal is free, then $R^\\\\bullet $ is a transfer Krull monoid (see [24] for details).\",\n \"There are also many monoids that fail to be transfer Krull.\",\n \"Examples of non-transfer Krull monoids in a non-commutative setting are provided by [6], [12], and [25].\",\n \"On the other hand, Theorem REF and the next proposition (which follows from [14]) yield examples of non-transfer Krull monoids in a commutative context.\",\n \"Proposition 6.4 Every proper numerical monoid fails to be transfer Krull.\",\n \"The next lemma will be used in the proof of Theorem REF .\",\n \"Lemma 6.5 If $\\\\lbrace a_n\\\\rbrace $ is an infinite sequence of positive integers, then there exists $m \\\\in \\\\mathbb {N}$ such that $a_{m+1} \\\\in \\\\langle a_1, \\\\dots , a_m \\\\rangle $ .\",\n \"If $\\\\lbrace a_n\\\\rbrace $ is bounded there is a term that repeats infinitely many times, making the conclusion of the lemma obvious.\",\n \"Thus, suppose that $\\\\lbrace a_n\\\\rbrace $ is not bounded.\",\n \"Let $\\\\lbrace a_{n_j}\\\\rbrace $ be a subsequence of $\\\\lbrace a_n\\\\rbrace $ satisfying that $ a_{n_{j+1}} > \\\\prod _{i=1}^{j} a_{n_i}$ for every $j \\\\in \\\\mathbb {N}$ .\",\n \"Now, for each natural number $j$ , set $d_j = \\\\gcd (a_{n_1}, \\\\dots , a_{n_j})$ , and notice that $d_{j+1} \\\\mid d_j$ for every $j \\\\in \\\\mathbb {N}$ .\",\n \"Therefore $d_{k+1} = d_k$ must hold for some $k$ .\",\n \"In particular, $d_k \\\\mid a_{n_{k+1}}$ .\",\n \"On the other hand, condition (REF ) ensures that $a_{n_{k+1}}/d_k$ is greater than the Frobenius number of the numerical monoid $\\\\langle a_{n_1}/d_k, \\\\dots , a_{n_k}/d_k \\\\rangle $ .\",\n \"This implies that $a_{n_{k+1}} \\\\in \\\\langle a_{n_1}, \\\\dots , a_{n_k} \\\\rangle $ .\",\n \"The lemma follows by taking $m = n_{k+1}-1$ .\",\n \"Now we are in a position to prove that atomic Puiseux monoids are almost never transfer Krull.\",\n \"Theorem 6.6 If a nontrivial Puiseux monoid is transfer Krull, then it must be isomorphic to $(\\\\mathbb {N}_0,+)$ .\",\n \"Let $M$ be a nontrivial Puiseux monoid that happens to be transfer Krull.\",\n \"As Krull monoids are atomic, $M$ is atomic by Proposition REF .\",\n \"Let $G$ be an abelian group, and let $\\\\theta \\\\colon M \\\\rightarrow \\\\mathcal {B}(G_0)$ be a transfer homomorphism, where $G_0$ is a subset of $G$ .\",\n \"Because both $M$ and $\\\\mathcal {B}(G_0)$ are reduced, $\\\\theta ^{-1}(\\\\emptyset ) = \\\\lbrace 0\\\\rbrace $ .\",\n \"Assume, by way of contradiction, that $M$ is not isomorphic to a numerical monoid.\",\n \"Take $X \\\\in \\\\mathcal {B}(G_0)^\\\\bullet $ and $r,s \\\\in M^\\\\bullet $ such that $\\\\theta (r) = \\\\theta (s) = X$ .\",\n \"Taking $m,n \\\\in \\\\mathbb {N}$ such that $mr = ns$ , one obtains $ \\\\prod _{g \\\\in G_0} g^{m \\\\mathsf {v}_g(X)} = \\\\theta (r)^m = \\\\theta (s)^n = \\\\prod _{g \\\\in G_0} g^{n \\\\mathsf {v}_g(X)}.$ Since $|X| \\\\ge 1$ and $m \\\\mathsf {v}_g(X) = n \\\\mathsf {v}_g(X)$ for every $g \\\\in G_0$ , it follows that $m=n$ , which yields $r = s$ .\",\n \"Hence the preimage under $\\\\theta $ of each element of $\\\\mathcal {B}(G_0)^\\\\bullet $ is a singleton.\",\n \"This, along with the fact that $\\\\theta ^{-1}(\\\\emptyset ) = \\\\lbrace 0\\\\rbrace $ , implies that $\\\\theta $ is injective.\",\n \"In addition, the same equality (REF ) implies that $\\\\text{supp}_{G_0}(\\\\theta (a)) = \\\\text{supp}_{G_0}(\\\\theta (a^{\\\\prime }))$ for all $a,a^{\\\\prime } \\\\in \\\\mathcal {A}(M)$ .\",\n \"As a consequence, any two elements of $\\\\theta (M^\\\\bullet )$ have the same support, and we can assume, without loss of generality, that $G_0$ is finite.\",\n \"Let $G_0 =: \\\\lbrace g_1, \\\\dots , g_t\\\\rbrace $ be the common support.\",\n \"List the set $\\\\mathcal {A}(M)$ as a sequence $\\\\lbrace a_n\\\\rbrace $ , and let $A_n = \\\\theta (a_n)$ for each $n \\\\in \\\\mathbb {N}$ .\",\n \"Because $\\\\theta $ is injective, $A_i \\\\ne A_j$ when $i \\\\ne j$ .\",\n \"Now, for any pair $(i,j) \\\\in \\\\mathbb {N}^2$ , there exist $c_i,c_j \\\\in \\\\mathbb {N}$ such that $c_i a_i = c_j a_j$ .\",\n \"For each $n \\\\in \\\\lbrace 1, \\\\dots , t\\\\rbrace $ , we can apply $\\\\mathsf {v}_{g_n} \\\\circ \\\\theta $ to the equality $c_i a_i = c_j a_j$ to get $c_i \\\\mathsf {v}_{g_n}(A_i) = c_j \\\\mathsf {v}_{g_n}(A_j)$ .\",\n \"After rewriting this equality, one obtains that $ \\\\frac{\\\\mathsf {v}_{g_n}(A_i)}{\\\\mathsf {v}_{g_n}(A_j)} = \\\\frac{c_j}{c_i} = \\\\frac{\\\\mathsf {v}_{g_1}(A_i)}{\\\\mathsf {v}_{g_1}(A_j)}$ for each $n \\\\in \\\\lbrace 1, \\\\dots , t\\\\rbrace $ .\",\n \"On the other hand, notice that Lemma REF guarantees the existence of $m \\\\in \\\\mathbb {N}$ and $\\\\alpha _1, \\\\dots , \\\\alpha _m \\\\in \\\\mathbb {N}_0$ such that $ \\\\mathsf {v}_{g_1}(A_{m+1}) = \\\\sum _{i=1}^m \\\\alpha _i \\\\mathsf {v}_{g_1}(A_i).$ By (REF ), it follows that the equality (REF ) holds when we replace $g_1$ by any other element of $G_0$ (exactly with the same $\\\\alpha _i$ 's).\",\n \"As a result, we obtain $A_{m+1} = \\\\prod _{j=1}^{|G_0|} g_j^{\\\\mathsf {v}_{g_j}(A_{m+1})} = \\\\prod _{j=1}^{|G_0|} \\\\prod _{i=1}^m g_j^{\\\\alpha _i \\\\mathsf {v}_{g_j}(A_i)} = \\\\prod _{i=1}^m \\\\bigg (\\\\prod _{j=1}^{|G_0|} g_j^{\\\\mathsf {v}_{g_j}(A_i)}\\\\bigg )^{\\\\alpha _i} = \\\\prod _{i=1}^m A_i^{\\\\alpha _i}.$ This contradicts the fact that $A_{m+1}$ is an atom of the block monoid $\\\\mathcal {B}(G_0)$ .\",\n \"Therefore $M$ must be isomorphic to a numerical monoid.\",\n \"Now the direct implication of the proof follows by Proposition REF .\",\n \"For the reverse implication, it suffices to notice that $\\\\theta \\\\colon 1 \\\\mapsto [1_G]$ is an isomorphism from $(\\\\mathbb {N}_0,+)$ to the block monoid $\\\\mathcal {B}(G)$ , where $G$ is the trivial group.\",\n \"Corollary 6.7 A Puiseux monoid is a Krull monoid if and only if it can be generated by one element.\",\n \"Perhaps the second most-systematically studied family of atomic monoids is that one comprising the C-monoids.\",\n \"We would like to know under which conditions a Puiseux monoid happens to be a C-monoid.\",\n \"Any monoid $M$ can be embedded into a quotient group $\\\\mathsf {g}(M)$ , which is unique up to canonical isomorphism.\",\n \"Let $D$ be a multiplicative monoid with quotient group $\\\\mathsf {g}(D)$ , and let $M$ be a submonoid of $D$ .\",\n \"Two elements $x,y \\\\in D$ are said to be $M$ -equivalent provided that $x^{-1}M \\\\cap D = y^{-1}M \\\\cap D$ .\",\n \"It can be easily checked that being $M$ -equivalent defines a congruence relation on $D$ .\",\n \"For each $x \\\\in D$ , let $[x]^D_M$ denote the congruence class of $x$ .\",\n \"The set $\\\\mathcal {C}^*(M,D) := \\\\big \\\\lbrace [x]^D_M \\\\mid x \\\\in (D \\\\setminus D^\\\\times ) \\\\cup \\\\lbrace 1\\\\rbrace \\\\big \\\\rbrace $ is a commutative semigroup with identity, which is called the reduced class semigroup of $M$ in $D$ .\",\n \"Definition 6.8 A monoid $M$ is called a C-monoid if it is a submonoid of a factorial monoid $F$ such that $F^\\\\times \\\\cap M = M^\\\\times $ and $\\\\mathcal {C}^*(M,F)$ is finite.\",\n \"With notation as in Definition REF , we say that $M$ is a C-monoid defined in $F$ .\",\n \"A C-monoid can be defined in more than one factorial monoid $F$ ; however, there is a canonical way of choosing $F$ (see [9]).\",\n \"Because C-monoids are submonoids of factorial monoids, they are atomic.\",\n \"The family of C-monoids allows us to study the arithmetic of non-integrally closed Noetherian domains.\",\n \"Given a multiplicative monoid $M$ with quotient group $\\\\mathsf {g}(M)$ , we say that $x \\\\in \\\\mathsf {g}(M)$ is almost integral over $M$ if there exists $c \\\\in M$ such that $cx^n \\\\in M$ for every $n \\\\in \\\\mathbb {N}$ .\",\n \"The subset of $\\\\mathsf {g}(M)$ consisting of all almost integral elements over $M$ is denoted by $\\\\widehat{M}$ and called the complete integral closure of $M$ .\",\n \"Let $R$ be an integral domain with field of fractions $\\\\mathsf {q}(R)$ .\",\n \"An ideal $I$ of $R$ is divisorial if $(R : (R : I)) = I$ .\",\n \"The domain $R$ is called a Mori domain if it satisfies the ascending chain condition on divisorial ideals.\",\n \"Finally, for the domain $R$ we set $\\\\widehat{R} = \\\\widehat{R^\\\\bullet } \\\\cup \\\\lbrace 0\\\\rbrace $ , where $R^\\\\bullet $ is the multiplicative monoid of $R$ .\",\n \"Example 6.9 If $A$ is a Mori domain, then $R = \\\\widehat{A}$ is a Krull domain.\",\n \"Moreover, if $\\\\mathfrak {f} = (A : R)$ is nonzero and both the quotient ring $R/\\\\mathfrak {f}$ and the class group $\\\\mathcal {C} (R)$ are finite, then $A^\\\\bullet $ is a C-monoid (see [10]).\",\n \"More examples of C-monoids can be found in [11] and [22].\",\n \"The next theorem is used in the proof of Proposition REF .\",\n \"Theorem 6.10 [10] The complete integral closure of a C-monoid is a Krull monoid.\",\n \"Proposition 6.11 A nontrivial Puiseux monoid is a C-monoid if and only if it is isomorphic to a numerical monoid.\",\n \"Let $M$ be a nontrivial Puiseux monoid that is also a C-monoid.\",\n \"Let $\\\\widehat{M}$ be the complete integral closure of $M$ .\",\n \"Observe first that if $x \\\\in \\\\mathsf {g}(M) \\\\cap \\\\mathbb {Q}_{<0}$ , then $S_{x,r} := \\\\lbrace r + nx \\\\mid n \\\\in \\\\mathbb {N}\\\\rbrace $ contains only finitely many positive rational numbers for all $r \\\\in M$ .\",\n \"As a result, $|S_{x,r} \\\\cap M| < \\\\infty $ for all $r \\\\in M$ , which implies that no negative element of $\\\\mathsf {g}(M)$ is almost integral over $M$ .\",\n \"Because $\\\\widehat{M}$ is a monoid and it is contained in $\\\\mathbb {Q}_{\\\\ge 0}$ , it must be a Puiseux monoid.\",\n \"By Theorem REF , the monoid $\\\\widehat{M}$ is a Krull monoid; in particular, it is transfer Krull.\",\n \"Now Theorem REF ensures that $\\\\widehat{M}$ is isomorphic to $(\\\\mathbb {N}_0,+)$ .\",\n \"Finally, the fact that $M$ is a submonoid of $\\\\widehat{M}$ forces $M$ to be isomorphic to a numerical monoid.\",\n \"For the reverse implication, it suffices to note that for every proper numerical monoid $N$ , any two natural numbers greater than the Frobenius number of $N$ are $N$ -equivalent, which implies that $\\\\mathcal {C}^*(N, \\\\mathbb {N}_0)$ is finite.\",\n \"As $\\\\mathbb {N}_0$ is also a C-monoid, the proof follows.\"\n ],\n [\n \"Acknowledgements\",\n \"While working on this paper, I was supported by the NSF-AGEP fellowship.\",\n \"I am grateful to Alfred Geroldinger not only for proposing the main questions motivating this paper but also for his enlightening guidance through early drafts.\",\n \"Also, I would like to thank Salvatore Tringali and the anonymous referee, whose helpful suggestions help me improve the final version of this paper.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01693"}}},{"rowIdx":992,"cells":{"text":{"kind":"list like","value":[["Nanoscale chemical mapping of laser-solubilized silk"],["Abstract A water soluble amorphous form of silk was made by ultra-short laser pulse irradiation and detected by nanoscale IR mapping.","An optical absorption-induced nanoscale surface expansion was probed to yield the spectral response of silk at IR molecular fingerprinting wavelengths with a high ~20 nm spatial resolution defined by the tip of the probe.","Silk microtomed sections of 1-5 micrometers in thickness were prepared for nanoscale spectroscopy and a laser was used to induce amorphisation.","Comparison of silk absorbance measurements carried out by table-top and synchrotron Fourier transform IR spectroscopy proved that chemical imaging obtained at high spatial resolution and specificity (able to discriminate between amorphous and crystalline silk) is reliably achieved by nanoscale IR.","A nanoscale material characterization using synchrotron IR radiation is discussed."],["Introduction","In analytical material science, absorption of IR light is used for fingerprinting (chemical imaging) of particular molecules, specific compounds, and provides insight into interactions in their immediate vicinity.","However, challenges arise when this information needs to be obtained from sub-wavelength and sub-cellular dimensions, in particular at IR and terahertz spectral bands of absorption [1] or scattering [2].","Optical properties of sub-wavelength structures and patterns have opened an entirely new direction in photonics and design of highly efficient optical elements for control of intensity, phase, polarisation, spin and orbital momenta of light based on flat planar geometries, yet rich in nanoscale features [3], [4].","We aim at reaching that level of control in the domain of chemical spectral imaging.","There, interpretation of data from near-field requires further knowledge of probe interaction with substrate, phase information of the reflected/transmitted light from sub-wavelength structures to reveal complex peculiarities of light-matter interactions at nanoscale and is now advancing with strongly concentrated efforts [5], [6], [7], [8], [9], [10].","Recently, an electron tunneling control by a single-cycle terahertz pulse illuminated onto a tip of a scanning transmission microscope (STM) needle was demonstrated at 10 V/nm fields [11].","STM reaches an atomic precision in surface probing and its spectroscopic characterisation and can be carried out on a water surface [12].","Absorbance spectra quantify and identify the resonant molecules, chemical structures through their individual or collective excitations as detected in transmission (or inferred from reflection).","By sweeping excitation wavelength through the finger printing region of a particular material, the usual optical excitation relaxation pathway ending with a thermal energy deposition into host material can be sensitively measured using an atomic force microscopy (AFM) approach [13], [14] which opened up a rapidly growing AFM-IR field [15] (also known as nano-IR; Fig.","REF (a)).","How reliably one can determine IR properties with an AFM nano-tip based on the thermal expansion is currently still under debate due to a lack of knowledge of the actual anisotropy of thermal and mechanical properties at the nanoscale, 3D molecular conformation, alignment, and the interaction volume.","To establish the correspondence between nano-IR and spectroscopy it is necessary to compare IR spectral imaging in near-field and far-field modes with nano-IR.","Another challenge of the nano-IR technique is that a very thin sub-micrometer film has to be prepared and mounted on a thermally conductive substrate.","Microtomed thin sections usually have thicknesses above the optimum and need to be embedded into an epoxy host which interferes with the nano-IR signal from micro-specimens.","In order to demonstrate that microtomed slices of samples with features at or below micrometers in lateral cross-section can be measured using nano-IR and provide reliable IR spectral information we acquired absorbance spectra from areas down to single pixel hyper-spectral resolution accessible on table-top FT-IR spectrometers, and at high-resolution which was achieved with a solid immersion lens using high-brightness synchrotron FT-IR microscopy.","Such comparison of IR properties read out from nanoscale and far-field was carried out in this study using silk - a bio-polymer complex in its structure comprised of crystalline and amorphous building blocks [16], [17], [18], [19].","Here, micrometer-thick slices of silk were used to measure thermal expansion under a specific wavelength of excitation with $\\sim 20$ -nm-sharp AFM tips and to compare with high-resolution spectra measured with ATR FT-IR method using synchrotron radiation at the Infrared Microspectroscopy (IRM) Beamline (Australian Synchrotron) as well as a table-top FT-IR spectrometer.","Amorphisation of silk induced by single ultra-short laser pulses [19] has been spectroscopically recognised using nano-IR.","Figure: Principle of nano-IR: an AFM tip follows height changesinduced by a spectral sweep of excitation at IR wavelengths (∼6μ\\sim 6~\\mu m).","IR laser beam is p-polarised and incident atΘ≃60 ∘ \\Theta \\simeq 60^\\circ angle to minimise reflective losses (closeto the Brewster angle) and to illuminate a much larger sample areawhile AFM readout occurs from a AFM needle contact (∼20\\sim 20 nm)continuously scanned with 0.1 Hz frequency and digitised to obtainmap with x×y=25×100x\\times y = 25\\times 100 nm 2 ^2 pixels.","AFM and nano-IRmapping was carried out at a selected excitation wavelength.","(b)SEM images of silk fiber melted/amorphised by a single laser pulseas-fabricated and after immersion into water; laser wavelength515 nm, pulse duration 230 fs, focused with objective lens withnumerical aperture NA=0.5NA = 0.5, pulse energy 425 nJ, linearpolarisation was along the fiber (different fibers exposed at thesame conditions were used for the two images).Figure: AFM (a) and chemical map at Amide I 1660 cm -1 ^{-1} band(b) measured with nano-IR; inset in (b) shows the ablation pitregion imaged at crystalline band at 1700 cm -1 ^{-1}.","(c) Pointspectra measured on epoxy, amorphous-rich, and predominantlycrystalline regions of the T-cross-section of a silk fiber; insetshows AFM image of a microtome slice of a silk fiber in the epoxymatrix and laser irradiated crater region."],["Samples and methods","Domestic (Bombyx mori) silk fibers, stripped of their sericin rich cladding [20], were probed during the spectroscopic imaging experiments.","Pulsed laser radiation was used to induce local structural modifications of silk.","The AFM-based nano-IR experiments with $\\sim $ 25-nm-diameter tips were benchmarked against more conventional methods, such as attenuated total reflectance (ATR) at the Australian Synchrotron IRM Beamline ($\\sim 1.9~\\mu $ m resolution), and a table-top FT-IR spectrometer ($\\sim 6~\\mu $ m resolution).","For this set of experiments method-agnostic silk samples, in the form of thin slices, had to be prepared.","For cross-sectional observation, the natural silk fibers were aligned and embedded into an epoxy adhesive (jER 828, Mitsubishi Chemical Co., Ltd.).","Fibers fixed in the epoxy matrix were microtomed into 1-5 $\\mu $ m-thick slices which are mechanically robust enough to be measured using standard FT-IR setups without any supporting substrate.","This was particularly important to increase sensitivity of the far-field absorbance measurements and to diminish reflective losses.","Both longitudinal (L) and transverse (T) slices of the silk fibers were prepared by microtome (RV-240, Yamato Khoki Industrial Co., Ltd).","The slices were thinner than the original silk fibers.","For synchrotron ATR FT-IR, an aluminum disk was used to support the thin silk sections when they were brought into contact with a 100 $\\mu $ m diameter facet tip of the Ge ATR hemisphere (refractive index $n = 4$ ).","Synchrotron ATR-FTIR mapping measurement was performed using a in-house developed ATR-FTIR device at the IR Microspectroscopy Beamline, which has a high-speed, high-resolution surface characterisation capabilities with spatial resolution down to $~1.9~\\mu $ m [21].","A 100 $\\mu $ m tip ATR accessory for a FT-IR spectrometer (Hyperion 3000, Bruker) with Ge contact lens of $NA = n\\sin \\varphi \\simeq 2.4$ of refractive index $n = 4$ and $\\varphi = 36.9^\\circ $ a half-angle of the focusing cone was used.","A deep sub-wavelength resolution $r = 0.61\\lambda _{IR}/NA \\simeq 1.5~\\mu $ m is achievable for the IR wavelengths of interest at the Amide band of $\\lambda _{IR} = 1600-1700$ cm$^{-1}$ or 6.25 - 5.9 $\\mu $ m. The nano-IR experiments, i.e., an AFM readout of the height changes in response to IR sample excitation, were carried out with nano-IR2 (Anasys Instruments, Santa Barbara, CA) tool with a $\\sim $ 20 nm diameter tip.","During continuous scan over the selected region with 0.1 Hz frequency, digitisation was carried out resulting in $25\\times 100$ nm$^2$ pixels in $x\\times y$ .","The oscillation frequency of the AFM needle was 190 kHz.","Table-top FT-IR spectrometer (Spotlight, PerkinElmer) was used with a detector array with $\\sim 6~\\mu $ m pixel resolution.","Localized modification of silk was carried out via exposure to 515 nm wavelength and 230 fs duration pulses (Pharos, Light Conversion Ltd.) in an integrated industrial laser fabrication setup (Workshop of Photonics, Ltd.).","Fibers were imaged and laser radiation was focused using an objective lens of numerical aperture $NA = 0.5$ (Mitutoyo).","Single pulse exposures were carried out with pulse energy, $E_p$ .","Optical and electron scanning microscopy (SEM) were used for structural characterisation of the laser modified regions.","Figure: Point spectra and chemical maps of T cross sections ofsilk (red) in epoxy (blue).","(a) Synchrotron ATR-FTIR spectra andits chemical map based on Amide A band.","(b) Table-top FT-IRspectra and the corresponding Amide A chemical map.","Note that theCCD array has the pixel size of 6.25×6.25μ6.25\\times 6.25~\\mu m 2 ^2comparable with the T-cross-section of the silk fiber.","Chemicalmaps (on the right-side) are presented as measured (withoutsmoothing)."],["Results and Discussion","A highly crystalline natural silk fiber can be thermally amorphised only through rapid $2\\times 10^3$ K/s thermal quenching from melt [22] as demonstrated for a very tiny amounts of silk measured in nanograms (1 ng occupies a sphere of 5.7 $\\mu $ m diameter).","Amorphous silk is water soluble and can be used as 3D printing material for scaffolds, desorbable implants [23], [24], [25], bio-resists [26], and even be utilised as an electron beam resist [27], [28].","Amorphous-to-crystalline transition of silk fibroin is usually achieved via a simple water and alcohol bath processing at moderately elevated temperatures $\\sim 80^\\circ $ [28].","An UV 266 nm wavelength nanosecond pulsed laser irradiation was also used to enrich amorphous silk fibroin with crystalline $\\beta $ -sheets [29].","To realise a fast thermal quenching and to retrieve the amorphous silk phase [19], we used ultrashort 230 fs duration and 515 nm wavelength laser pulses tightly focused into focal spot of $d = 1.22\\lambda /NA \\simeq 1.3~\\mu $ m; the numerical aperture of the objective lens was $NA = 0.5$ and the wavelength was $\\lambda = 515$ nm.","Single pulse irradiation was carried out to ablate nanograms of silk and create a molten phase which is thermally quenched fast enough [22] to prevent crystallisation (Fig.","REF (b)).","Threshold of optically recognisable modification of silk fiber during laser irradiation was at 8 nJ which corresponds to 2.8 TW/cm$^2$ average power (0.6 J/cm$^2$ fluence) per pulse and is typical for polymer glasses [30].","Nano-IR spectrum, the low-frequency band of the AFM detected height changes in response to a spectral sweep of excitation at the IR absorbance bands at 1 kHz frequency was measured on a transverse (T) microtome cross section of silk fiber embedded into a micro-thin epoxy (Fig.","REF (a)).","Single wavelength chemical map measured at the specific Amide I band of 1660 cm$^{-1}$ associated with the amorphous silk components is shown in (b) with clearly discernable amorphous rim of the ablation crater, which, in contrast, is not recognisable at the crystalline $\\beta $ -sheet region of 1700 cm$^{-1}$ (inset of (b)).","Typical single point measurement spectra are shown in Fig.","REF (c) for the Amide I region with clear distinction between epoxy matrix, amorphous, and crystalline counterparts of silk.","Amorphous silk of the molten quenched phase has only 200 nm thickness as observed by SEM and AFM, however, it was distinguished using illumination at the corresponding absorption bands ($\\lambda _{ex} \\simeq 6~\\mu $ m).","Thin microtomed slices placed on a high thermal conductivity substrate have been shown to enhance speed of nano-IR imaging since a thermalisation time scale is $t_{th}\\simeq \\rho c_ph^2/\\eta $ [31], where $h$ is the thickness of the film, $\\eta $ is thermal conductivity, $\\rho $ is the mass density, and $c_p$ is the specific heat capacity."],["Conclusions and outlook","It was shown that by using micro-thin slices which are sub-wavelength at the IR spectral range of interest, it was possible to obtain spectral band readout using nano-IR method - the surface height changes due to thermal expansion following the absorbance spectrum of silk.","Amorphous silk created by ultra-fast thermal quenching at the irradiation location of fs-laser pulse was distinguished on the chemical map and imaged with lateral resolution defined by digitisation $x\\times y \\equiv 25\\times 100$ nm$^2$ and can potentially reach the limit defined by tip [32] which was $20\\times 20$ nm$^2$ in this study.","Chemical mapping result acquired using the nano-IR method is consistent with far-field spectroscopy of silk carried out with table-top and synchrotron FT-IR (see Fig.","REF (b)).","The far-field FT-IR detectors have a typical pixel size of $\\sim 6~\\mu $ m which limits the resolution to the entire T-cross-section of the silk fiber, however, a good correspondence with nanoscale IR spectral mapping is confirmed between different spectroscopic methods.","Laser-induced amorphisation of crystalline silk has been spectroscopically resolved with high spatial resolution.","We can envisage, that a simple principle of the nano-IR technique allows for a coupling with synchrotron light sources and to complement it with a phase and amplitude mapping using scanning near-field microscopy.","The high brightness of synchrotron radiation would also enable a fast mapping required for temporally resolved evolution of photo or thermally excited processes, and even a molecular alignment mapping could be realised by using the four polarisation method [33].","These functionalities will bring new developments into a cutting edge nanoscale molecular characterisation."],["Acknowledgments","J.M.","acknowledges a partial support by a JSPS KAKENHI Grant No.16K06768.","We acknowledge the Swinburne's startup grant for Nanotechnology facility and partial support via ARC Discovery DP130101205 and DP170100131 grants.","Experiments were carried out via beamtime project No.","11119 at the Australian Synchrotron IRM Beamline.","Window-on-Photonics R&D, Ltd. is acknowledged for joint development grant and laser fabrication facility."]],"string":"[\n [\n \"Nanoscale chemical mapping of laser-solubilized silk\"\n ],\n [\n \"Abstract A water soluble amorphous form of silk was made by ultra-short laser pulse irradiation and detected by nanoscale IR mapping.\",\n \"An optical absorption-induced nanoscale surface expansion was probed to yield the spectral response of silk at IR molecular fingerprinting wavelengths with a high ~20 nm spatial resolution defined by the tip of the probe.\",\n \"Silk microtomed sections of 1-5 micrometers in thickness were prepared for nanoscale spectroscopy and a laser was used to induce amorphisation.\",\n \"Comparison of silk absorbance measurements carried out by table-top and synchrotron Fourier transform IR spectroscopy proved that chemical imaging obtained at high spatial resolution and specificity (able to discriminate between amorphous and crystalline silk) is reliably achieved by nanoscale IR.\",\n \"A nanoscale material characterization using synchrotron IR radiation is discussed.\"\n ],\n [\n \"Introduction\",\n \"In analytical material science, absorption of IR light is used for fingerprinting (chemical imaging) of particular molecules, specific compounds, and provides insight into interactions in their immediate vicinity.\",\n \"However, challenges arise when this information needs to be obtained from sub-wavelength and sub-cellular dimensions, in particular at IR and terahertz spectral bands of absorption [1] or scattering [2].\",\n \"Optical properties of sub-wavelength structures and patterns have opened an entirely new direction in photonics and design of highly efficient optical elements for control of intensity, phase, polarisation, spin and orbital momenta of light based on flat planar geometries, yet rich in nanoscale features [3], [4].\",\n \"We aim at reaching that level of control in the domain of chemical spectral imaging.\",\n \"There, interpretation of data from near-field requires further knowledge of probe interaction with substrate, phase information of the reflected/transmitted light from sub-wavelength structures to reveal complex peculiarities of light-matter interactions at nanoscale and is now advancing with strongly concentrated efforts [5], [6], [7], [8], [9], [10].\",\n \"Recently, an electron tunneling control by a single-cycle terahertz pulse illuminated onto a tip of a scanning transmission microscope (STM) needle was demonstrated at 10 V/nm fields [11].\",\n \"STM reaches an atomic precision in surface probing and its spectroscopic characterisation and can be carried out on a water surface [12].\",\n \"Absorbance spectra quantify and identify the resonant molecules, chemical structures through their individual or collective excitations as detected in transmission (or inferred from reflection).\",\n \"By sweeping excitation wavelength through the finger printing region of a particular material, the usual optical excitation relaxation pathway ending with a thermal energy deposition into host material can be sensitively measured using an atomic force microscopy (AFM) approach [13], [14] which opened up a rapidly growing AFM-IR field [15] (also known as nano-IR; Fig.\",\n \"REF (a)).\",\n \"How reliably one can determine IR properties with an AFM nano-tip based on the thermal expansion is currently still under debate due to a lack of knowledge of the actual anisotropy of thermal and mechanical properties at the nanoscale, 3D molecular conformation, alignment, and the interaction volume.\",\n \"To establish the correspondence between nano-IR and spectroscopy it is necessary to compare IR spectral imaging in near-field and far-field modes with nano-IR.\",\n \"Another challenge of the nano-IR technique is that a very thin sub-micrometer film has to be prepared and mounted on a thermally conductive substrate.\",\n \"Microtomed thin sections usually have thicknesses above the optimum and need to be embedded into an epoxy host which interferes with the nano-IR signal from micro-specimens.\",\n \"In order to demonstrate that microtomed slices of samples with features at or below micrometers in lateral cross-section can be measured using nano-IR and provide reliable IR spectral information we acquired absorbance spectra from areas down to single pixel hyper-spectral resolution accessible on table-top FT-IR spectrometers, and at high-resolution which was achieved with a solid immersion lens using high-brightness synchrotron FT-IR microscopy.\",\n \"Such comparison of IR properties read out from nanoscale and far-field was carried out in this study using silk - a bio-polymer complex in its structure comprised of crystalline and amorphous building blocks [16], [17], [18], [19].\",\n \"Here, micrometer-thick slices of silk were used to measure thermal expansion under a specific wavelength of excitation with $\\\\sim 20$ -nm-sharp AFM tips and to compare with high-resolution spectra measured with ATR FT-IR method using synchrotron radiation at the Infrared Microspectroscopy (IRM) Beamline (Australian Synchrotron) as well as a table-top FT-IR spectrometer.\",\n \"Amorphisation of silk induced by single ultra-short laser pulses [19] has been spectroscopically recognised using nano-IR.\",\n \"Figure: Principle of nano-IR: an AFM tip follows height changesinduced by a spectral sweep of excitation at IR wavelengths (∼6μ\\\\sim 6~\\\\mu m).\",\n \"IR laser beam is p-polarised and incident atΘ≃60 ∘ \\\\Theta \\\\simeq 60^\\\\circ angle to minimise reflective losses (closeto the Brewster angle) and to illuminate a much larger sample areawhile AFM readout occurs from a AFM needle contact (∼20\\\\sim 20 nm)continuously scanned with 0.1 Hz frequency and digitised to obtainmap with x×y=25×100x\\\\times y = 25\\\\times 100 nm 2 ^2 pixels.\",\n \"AFM and nano-IRmapping was carried out at a selected excitation wavelength.\",\n \"(b)SEM images of silk fiber melted/amorphised by a single laser pulseas-fabricated and after immersion into water; laser wavelength515 nm, pulse duration 230 fs, focused with objective lens withnumerical aperture NA=0.5NA = 0.5, pulse energy 425 nJ, linearpolarisation was along the fiber (different fibers exposed at thesame conditions were used for the two images).Figure: AFM (a) and chemical map at Amide I 1660 cm -1 ^{-1} band(b) measured with nano-IR; inset in (b) shows the ablation pitregion imaged at crystalline band at 1700 cm -1 ^{-1}.\",\n \"(c) Pointspectra measured on epoxy, amorphous-rich, and predominantlycrystalline regions of the T-cross-section of a silk fiber; insetshows AFM image of a microtome slice of a silk fiber in the epoxymatrix and laser irradiated crater region.\"\n ],\n [\n \"Samples and methods\",\n \"Domestic (Bombyx mori) silk fibers, stripped of their sericin rich cladding [20], were probed during the spectroscopic imaging experiments.\",\n \"Pulsed laser radiation was used to induce local structural modifications of silk.\",\n \"The AFM-based nano-IR experiments with $\\\\sim $ 25-nm-diameter tips were benchmarked against more conventional methods, such as attenuated total reflectance (ATR) at the Australian Synchrotron IRM Beamline ($\\\\sim 1.9~\\\\mu $ m resolution), and a table-top FT-IR spectrometer ($\\\\sim 6~\\\\mu $ m resolution).\",\n \"For this set of experiments method-agnostic silk samples, in the form of thin slices, had to be prepared.\",\n \"For cross-sectional observation, the natural silk fibers were aligned and embedded into an epoxy adhesive (jER 828, Mitsubishi Chemical Co., Ltd.).\",\n \"Fibers fixed in the epoxy matrix were microtomed into 1-5 $\\\\mu $ m-thick slices which are mechanically robust enough to be measured using standard FT-IR setups without any supporting substrate.\",\n \"This was particularly important to increase sensitivity of the far-field absorbance measurements and to diminish reflective losses.\",\n \"Both longitudinal (L) and transverse (T) slices of the silk fibers were prepared by microtome (RV-240, Yamato Khoki Industrial Co., Ltd).\",\n \"The slices were thinner than the original silk fibers.\",\n \"For synchrotron ATR FT-IR, an aluminum disk was used to support the thin silk sections when they were brought into contact with a 100 $\\\\mu $ m diameter facet tip of the Ge ATR hemisphere (refractive index $n = 4$ ).\",\n \"Synchrotron ATR-FTIR mapping measurement was performed using a in-house developed ATR-FTIR device at the IR Microspectroscopy Beamline, which has a high-speed, high-resolution surface characterisation capabilities with spatial resolution down to $~1.9~\\\\mu $ m [21].\",\n \"A 100 $\\\\mu $ m tip ATR accessory for a FT-IR spectrometer (Hyperion 3000, Bruker) with Ge contact lens of $NA = n\\\\sin \\\\varphi \\\\simeq 2.4$ of refractive index $n = 4$ and $\\\\varphi = 36.9^\\\\circ $ a half-angle of the focusing cone was used.\",\n \"A deep sub-wavelength resolution $r = 0.61\\\\lambda _{IR}/NA \\\\simeq 1.5~\\\\mu $ m is achievable for the IR wavelengths of interest at the Amide band of $\\\\lambda _{IR} = 1600-1700$ cm$^{-1}$ or 6.25 - 5.9 $\\\\mu $ m. The nano-IR experiments, i.e., an AFM readout of the height changes in response to IR sample excitation, were carried out with nano-IR2 (Anasys Instruments, Santa Barbara, CA) tool with a $\\\\sim $ 20 nm diameter tip.\",\n \"During continuous scan over the selected region with 0.1 Hz frequency, digitisation was carried out resulting in $25\\\\times 100$ nm$^2$ pixels in $x\\\\times y$ .\",\n \"The oscillation frequency of the AFM needle was 190 kHz.\",\n \"Table-top FT-IR spectrometer (Spotlight, PerkinElmer) was used with a detector array with $\\\\sim 6~\\\\mu $ m pixel resolution.\",\n \"Localized modification of silk was carried out via exposure to 515 nm wavelength and 230 fs duration pulses (Pharos, Light Conversion Ltd.) in an integrated industrial laser fabrication setup (Workshop of Photonics, Ltd.).\",\n \"Fibers were imaged and laser radiation was focused using an objective lens of numerical aperture $NA = 0.5$ (Mitutoyo).\",\n \"Single pulse exposures were carried out with pulse energy, $E_p$ .\",\n \"Optical and electron scanning microscopy (SEM) were used for structural characterisation of the laser modified regions.\",\n \"Figure: Point spectra and chemical maps of T cross sections ofsilk (red) in epoxy (blue).\",\n \"(a) Synchrotron ATR-FTIR spectra andits chemical map based on Amide A band.\",\n \"(b) Table-top FT-IRspectra and the corresponding Amide A chemical map.\",\n \"Note that theCCD array has the pixel size of 6.25×6.25μ6.25\\\\times 6.25~\\\\mu m 2 ^2comparable with the T-cross-section of the silk fiber.\",\n \"Chemicalmaps (on the right-side) are presented as measured (withoutsmoothing).\"\n ],\n [\n \"Results and Discussion\",\n \"A highly crystalline natural silk fiber can be thermally amorphised only through rapid $2\\\\times 10^3$ K/s thermal quenching from melt [22] as demonstrated for a very tiny amounts of silk measured in nanograms (1 ng occupies a sphere of 5.7 $\\\\mu $ m diameter).\",\n \"Amorphous silk is water soluble and can be used as 3D printing material for scaffolds, desorbable implants [23], [24], [25], bio-resists [26], and even be utilised as an electron beam resist [27], [28].\",\n \"Amorphous-to-crystalline transition of silk fibroin is usually achieved via a simple water and alcohol bath processing at moderately elevated temperatures $\\\\sim 80^\\\\circ $ [28].\",\n \"An UV 266 nm wavelength nanosecond pulsed laser irradiation was also used to enrich amorphous silk fibroin with crystalline $\\\\beta $ -sheets [29].\",\n \"To realise a fast thermal quenching and to retrieve the amorphous silk phase [19], we used ultrashort 230 fs duration and 515 nm wavelength laser pulses tightly focused into focal spot of $d = 1.22\\\\lambda /NA \\\\simeq 1.3~\\\\mu $ m; the numerical aperture of the objective lens was $NA = 0.5$ and the wavelength was $\\\\lambda = 515$ nm.\",\n \"Single pulse irradiation was carried out to ablate nanograms of silk and create a molten phase which is thermally quenched fast enough [22] to prevent crystallisation (Fig.\",\n \"REF (b)).\",\n \"Threshold of optically recognisable modification of silk fiber during laser irradiation was at 8 nJ which corresponds to 2.8 TW/cm$^2$ average power (0.6 J/cm$^2$ fluence) per pulse and is typical for polymer glasses [30].\",\n \"Nano-IR spectrum, the low-frequency band of the AFM detected height changes in response to a spectral sweep of excitation at the IR absorbance bands at 1 kHz frequency was measured on a transverse (T) microtome cross section of silk fiber embedded into a micro-thin epoxy (Fig.\",\n \"REF (a)).\",\n \"Single wavelength chemical map measured at the specific Amide I band of 1660 cm$^{-1}$ associated with the amorphous silk components is shown in (b) with clearly discernable amorphous rim of the ablation crater, which, in contrast, is not recognisable at the crystalline $\\\\beta $ -sheet region of 1700 cm$^{-1}$ (inset of (b)).\",\n \"Typical single point measurement spectra are shown in Fig.\",\n \"REF (c) for the Amide I region with clear distinction between epoxy matrix, amorphous, and crystalline counterparts of silk.\",\n \"Amorphous silk of the molten quenched phase has only 200 nm thickness as observed by SEM and AFM, however, it was distinguished using illumination at the corresponding absorption bands ($\\\\lambda _{ex} \\\\simeq 6~\\\\mu $ m).\",\n \"Thin microtomed slices placed on a high thermal conductivity substrate have been shown to enhance speed of nano-IR imaging since a thermalisation time scale is $t_{th}\\\\simeq \\\\rho c_ph^2/\\\\eta $ [31], where $h$ is the thickness of the film, $\\\\eta $ is thermal conductivity, $\\\\rho $ is the mass density, and $c_p$ is the specific heat capacity.\"\n ],\n [\n \"Conclusions and outlook\",\n \"It was shown that by using micro-thin slices which are sub-wavelength at the IR spectral range of interest, it was possible to obtain spectral band readout using nano-IR method - the surface height changes due to thermal expansion following the absorbance spectrum of silk.\",\n \"Amorphous silk created by ultra-fast thermal quenching at the irradiation location of fs-laser pulse was distinguished on the chemical map and imaged with lateral resolution defined by digitisation $x\\\\times y \\\\equiv 25\\\\times 100$ nm$^2$ and can potentially reach the limit defined by tip [32] which was $20\\\\times 20$ nm$^2$ in this study.\",\n \"Chemical mapping result acquired using the nano-IR method is consistent with far-field spectroscopy of silk carried out with table-top and synchrotron FT-IR (see Fig.\",\n \"REF (b)).\",\n \"The far-field FT-IR detectors have a typical pixel size of $\\\\sim 6~\\\\mu $ m which limits the resolution to the entire T-cross-section of the silk fiber, however, a good correspondence with nanoscale IR spectral mapping is confirmed between different spectroscopic methods.\",\n \"Laser-induced amorphisation of crystalline silk has been spectroscopically resolved with high spatial resolution.\",\n \"We can envisage, that a simple principle of the nano-IR technique allows for a coupling with synchrotron light sources and to complement it with a phase and amplitude mapping using scanning near-field microscopy.\",\n \"The high brightness of synchrotron radiation would also enable a fast mapping required for temporally resolved evolution of photo or thermally excited processes, and even a molecular alignment mapping could be realised by using the four polarisation method [33].\",\n \"These functionalities will bring new developments into a cutting edge nanoscale molecular characterisation.\"\n ],\n [\n \"Acknowledgments\",\n \"J.M.\",\n \"acknowledges a partial support by a JSPS KAKENHI Grant No.16K06768.\",\n \"We acknowledge the Swinburne's startup grant for Nanotechnology facility and partial support via ARC Discovery DP130101205 and DP170100131 grants.\",\n \"Experiments were carried out via beamtime project No.\",\n \"11119 at the Australian Synchrotron IRM Beamline.\",\n \"Window-on-Photonics R&D, Ltd. is acknowledged for joint development grant and laser fabrication facility.\"\n ]\n]"},"id":{"kind":"string","value":"1709.01799"}}},{"rowIdx":993,"cells":{"text":{"kind":"list like","value":[["The Cosmological Memory Effect"],["Abstract The \"memory effect\" is the permanent change in the relative separation of test particles resulting from the passage of gravitational radiation.","We investigate the memory effect for a general, spatially flat FLRW cosmology by considering the radiation associated with emission events involving particle-like sources.","We find that if the resulting perturbation is decomposed into scalar, vector, and tensor parts, only the tensor part contributes to memory.","Furthermore, the tensor contribution to memory depends only on the cosmological scale factor at the source and observation events, not on the detailed expansion history of the universe.","In particular, for sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of $(1 + z)$."],["Introduction","The passage of gravitational radiation through a configuration of test particles that make up a gravitational wave detector can induce a permanent change in the relative separation of the test particles, a phenomenon which has come to be known as the gravitational wave memory effect.","The memory effect on a flat background was first recognized in the linear regime for sources in non-relativistic motion by Zel'dovich and Ponarev [1].","Afterwards, Christodoulou [2] discovered that there could be additional contributions to memory arising from the nonlinearity of the Einstein equation and associated with the Bondi flux of the gravitational waves to null infinity.","Shortly thereafter, it was argued that the nonlinear memory of Christodoulou could be interpreted as corresponding to a linear memory caused by the effective stress-energy associated with the primary gravitational radiation [3], [4].","More recently, it has been found to be useful to make a distinction between ordinary and null memory in the linearized gravity context [5],[6]: Ordinary memory is caused by massive matter sources and null memory is caused by null matter sources.","The gravitational radiation emitted from a source event involving both massive and null matter will induce both ordinary and null memory [5],[6],[7].","Neither the ordinary nor null memory should be viewed as a tidal effect with a Newtonian analogue but rather as a byproduct of the gravitational radiation emitted from a burst-type event [7],[8].","The above work has concerned itself with memory on an asymptotically flat spacetime.","If a spacetime is not asymptotically flat, it is not clear what “memory” should mean, even if we treat the gravitational radiation as a perturbation off of a background metric.","Memory has been defined in terms of the net change in the separation of test particles, but this could include motion due to the background curvature rather than the radiation.","For example, in cosmological Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, the proper separation of FLRW observers will change with time.","Furthermore, if there is no notion of null infinity, it is not clear where we should put our detector so that it will be exposed to radiation but isolated from other non-radiative gravitational forces.","Recently, Bieri, Garfinkle and Yau [9], Kehagias and Riotto [10], and Chu [11],[12] have considered the memory effect in FLRW spacetimes.","However, their choice of methods for resolving the above difficulties have so far limited the applicability of their results.","Bieri and Garfinkle consider Weyl tensor perturbations, so their analysis is greatly simplified by working in vacuo, and, consequently, they have considered only memory in a background vacuum de Sitter universe.","Meanwhile, Kehagias and Riotto's use of BMS transformations depends on the existence of a null infinity for the FLRW spacetime, so they restrict consideration to a decelerating universe without a cosmological constant.","Furthermore, both groups consider only null memory.","While Chu does not make either of these specific restrictions, his preferred definition of memory does not distinguish test particle motion due to radiative and non-radiative gravity.","In this paper, we will investigate the total memory effect—both ordinary and null—in a general, spatially flat, FLRW spacetime.","We will simplify the problem not by limiting the cosmological models under consideration, but rather the kinds of radiation sources.","Specifically, we will consider—in the context of linearParticle-like sources in Einsten's equation do not make sense outside of the context of linear perturbation theory [13].","perturbation theory off of an FLRW background—only point-particle sources, i.e., sources whose stress-energy is confined by Dirac delta functions to worldlines that meet at a vertex, which we will call the source event.","We previously considered such sources in Minkowski spacetime [7], [8], and found that the retarded solution gives rise to a contribution to the curvature tensor of the form of the derivative of a delta function in retarded time.","Upon integration of the geodesic deviation equation, this derivative of a delta function in curvature gives rise to a well defined memory effect—i.e., a change in the separation of test particles coincident in time with the passage of the radiation from the source event—that includes both the ordinary and null memory [7], [8].","Thus, for the idealized sources that we consider, the memory effect can be associated with the presence of a derivative of a delta function in the Riemann curvature of the retarded solution arising from the source event.","This enables us to distinguish the memory effect induced by the passage of radiation from non-radiative gravitational effects, and it does not require us to take limits to null infinity.","It thereby yields a well defined notion of the memory effect that is applicable to linearized perturbations about arbitrary background spacetimes.","Another advantage to considering the above idealized particle sources, where the radiation emerges from a single “source event” in the background spacetime, is that—since all spacetimes are “locally flat” on sufficiently small scales—there is a well defined notion of having the “same source” in different spacetimes.","Similarly, there is a well defined notion of having the “same detector”—i.e., geodesic test particles initially at rest and with small separation—in different spacetimes.","Thus, we can compare the memory effect in two different spacetimes provided only that we specify the location and “rest frame” of both the source and the detector in the two spacetimes.","Since any spatially flat FLRW spacetime is conformal to Minkowski spacetime, it is particularly useful to state our results by comparing the memory effect in FLRW spacetime to that in Minkowski spacetime.","Stated in this manner, the main result of our paper is as follows: Consider a spatially flat FLRW solution to Einstein's equation with arbitrary fluid matter, and with a given particle source perturbation, as described above.","Now place the same source and detector in Minkowski spacetime such that the source and detector are at rest with respect to each other and the source is at a distance, $d$ , in Minkowski spacetime equal to the luminosity distance, $d_L$ , in the FLRW spacetime.","Then the memory effect in the FLRW spacetime is enhanced over the Minkowski value by a factor of $(1+z)$ , where $z$ denotes the redshift factor between the source and observer/detector.","This result applies to both ordinary and null memory.","It is in agreement with the results obtained for null memory in the special cases considered in [9] and [10].","Note that in a spatially flat FLRW spacetime, the luminosity distance $d_L$ and the angular diameter distance, $d_A$ , are related by $d_A = d_L/(1+z)^2 $ It follows that for a Minkowski source at $d = d_A$ , the memory effect in the FLRW spacetime will be decreased from the Minkowski value by a factor of $(1+z)^{-1}$ .","Finally, it should be noted that although the memory effect in a spatially flat FLRW spacetime is related to the Minkowski memory effect in this simple way, the waveforms will be different; in particular, there will be “tail effects” in the FLRW spacetime.","Although our analysis is restricted to the context of linear perturbation theory with idealized particle sources, we expect that our main result stated above should be valid completely generally for any sources whose spatial and time variation scales are small compared with the Hubble scale.","Indeed, the main difficulty in generalizing our results to non-particle-like sources and to the nonlinear regime would be to give a precise definition of “memory” outside of the context we consider.","Thus, if one wishes to compute the memory effect resulting from, say, the coalescence of two black holes in a distant galaxy in a spatially flat FLRW spacetime, it should suffice to compute the memory effect arising from a similar coalescence in an asymptotically flat spacetime and then use the above correspondence.","However, we shall not attempt to formulate or prove such a generalization here.","We shall begin in section by describing the particle sources that we shall consider and characterizing the memory effect for perturbations of arbitrary curved spacetimes.","In section , we analyze linearized perturbations off of spatially flat FLRW spacetimes with such particle sources.","In section , we consider the tensor mode contribution to memory.","We show that only the “light cone portion” of the retarded Green's function will contribute to memory, and that its contribution can be related in a simple way to the memory caused by similar sources in a flat spacetime.","In the Appendix, we show that the scalar modes do not contribute to memory.","Latin indices from the early alphabet ($a,b, \\dots $ ) denote abstract spacetime indices.","Greek indices ($\\mu , \\nu , \\dots $ ) denote spacetime components of tensors, whereas Latin indices from the mid-alphabet ($i,j, \\dots $ ) denote spatial components."],["Particle Sources","In asymptotically flat spacetimes, the memory effect can be characterized in a precise manner by considering a detector composed of test particles near null infinity.","Radiation effects fall off as $1/r$ , whereas Newtonian tidal effects fall off as $1/r^3$ , so by considering only the $O(1/r)$ effects on the test particles, we can distinguish between effects produced by gravitational radiation and all other gravitational effects.","However, in a non-asymptotically-flat spacetime, it is not clear how even to define memory, since there is no obvious way to make a clean distinction between effects due to “radiation” as compared with other tidal gravitational effects.","In our previous investigation of the memory effect in linearized gravity [7],[8], we considered the idealized process of the instantaneous decay of a massive particle into two other particles.","We found that the retarded solution to the linearized Einstein equation with such a source has the property that the $O(1/r)$ part of the curvature has the form of a derivative of a delta-function of retarded time at the retarded time of the decay event.","Integration of the geodesic deviation equation then shows that the $O(1/r)$ effect on test particles is to produce a sharp step function in their relative separation.","Thus, for this kind of idealized source, the memory effect can be characterized by the presence of a derivative of a delta-function in the linearized curvature and a corresponding step function behavior in the relative separation of test particlesOf course, if one were to consider a less idealized source with a smoothed out energy-momentum tensor, then the Riemann tensor also will be smoothed out, and the relative separation of the test particles will not undergo a sharp, sudden change in separation; rather, separation of the particles that results in a memory effect would occur continuously on the same timescale as that of the event itself.. For our present purposes, the main advantage of considering sources consisting of particles undergoing instantaneous interactions is that the characterization of the memory effect in terms of derivative of a delta-function behavior of the curvature holds at all distances from the interaction event, i.e., one does not need to go to null infinity to extract this characterization of the memory effect.","This characterization may therefore be imported straightforwardly to other spacetimes.","Thus, in this paper, we shall restrict consideration to linearized gravity off of a smooth background spacetime, with a linearized perturbation sourced by (massive or massless) point particles.","The interactions of the particles will be modeled by having their worldlines intersect (and, possibly, begin or end) at a single event, $q$ , in spacetime as illustrated in figure 1.","Conservation of stress-energy then requires that (i) the particle worldlines are geodesics [14] away from $q$ , and (ii) total 4-momentum is conserved at $q$ .","“Memory” will then be characterized by the presence of a derivative of a delta-function in the curvature of the retarded solution with this source.","This characterization does not require that the detector be placed near “infinity.” Figure: A spacetime diagram of the sort of gravitational wave source we will consider.","Here 5 point particles enter a single “source event” qq, and 3 emerge.","The worldlines of the incoming and outgoing particles must be timelike or null geodesics.To specify more precisely the type of source we consider, we assume that local coordinates $(t,\\mathbf {x})$ have been introduced in a neighborhood of $q$ so that $\\nabla t$ is past-directed timelike and so that the event $q$ is labeled by $t=\\mathbf {x} = 0$ .","The worldline, $\\gamma $ , of each incoming massive particle must be a timelike geodesic [14] with endpoint at $q$ .","We can parametrize $\\gamma $ by $t$ and specify it by giving $\\mathbf {x}(t) = \\mathbf {z}(t)$ , where $\\mathbf {z}(0) = \\mathbf {0}$ .","The stress-energy of each incoming massive particle then takes the form $T^{(M, {\\rm in})}_{ab} = m u_a u_b \\, \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(t)\\right) \\frac{1}{\\sqrt{-g}} \\frac{d\\tau }{dt} \\Theta (-t) \\, .$ Here $u^{a}$ is the unit tangent (4-velocity) to $\\gamma $ , $\\tau $ is the proper time along $\\gamma $ , $\\Theta $ is the Heaviside step function, and $\\delta ^{(3)}$ is the “coordinate delta-function,” i.e., $\\int \\delta ^{(3)}(\\mathbf {x} - \\mathbf {z}(t)) d^3 \\mathbf {x} = 1$ .","Each incoming massless particle moves on a null geodesic, $\\alpha $ , given by $\\mathbf {x}(t) = \\mathbf {y}(t)$ , with $\\mathbf {y}(0) = \\mathbf {0}$ .","The stress-energy of each incoming massless particle takes the form $T^{(N, {\\rm in})}_{ab} = k_a k_b \\, \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {y}(t)\\right) \\frac{1}{\\sqrt{-g}} \\frac{d\\lambda }{dt} \\Theta (-t) \\, .$ Here, $\\lambda $ is an affine parameter of $\\alpha $ and $k^a$ is the corresponding tangent, with the scaling of $\\lambda $ chosen so that (REF ) holds.","The stress-energy of each of the outgoing massive and massless particles takes the form of (REF ) and (REF ) except that $\\Theta (-t)$ is replaced by $\\Theta (t)$ .","The total stress-energy of the particle sources we consider takes the form $T^{(P)}_{ab} = \\sum _{l, {\\rm in}} T^{(M,l)}_{ab} + \\sum _{n, {\\rm in}} T^{(N,n)}_{ab} + \\sum _{l^{\\prime }, {\\rm out}} T^{(M, l^{\\prime })}_{ab} + \\sum _{n^{\\prime }, {\\rm out}} T^{(N, n^{\\prime })}_{ab} \\, ,$ where each $T^{(M,l)}_{ab}$ takes the form of (REF ), each $T^{(N,n)}_{ab}$ takes the form of (REF ), and each $T^{(M, l^{\\prime })}_{ab}$ and $T^{(N,n^{\\prime })}_{ab}$ also take these forms with $\\Theta (t)$ replaced by $\\Theta (-t)$ .","Conservation of stress-energy, $\\nabla ^a T_{ab} = 0$ , holds in the distributional sense away from $q$ by virtue of the fact that each particle moves on a geodesic [14].","Conservation of stress-energy will hold at $q$ if and only if we have at $q$ $\\sum _{l, {\\rm in}} m^{(l)}u^{(l)}_a+ \\sum _{n, {\\rm in}} k^{(n)}_a = \\sum _{l^{\\prime }, {\\rm out}} m^{(l^{\\prime })}u^{(l^{\\prime })}_a + \\sum _{n^{\\prime }, {\\rm out}} k^{(n^{\\prime })}_a \\, ,$ Equation (REF ) with condition (REF ) defines the sources that we consider in this paper.","We are interested in the solution to the linearized Einstein equation “produced by such a source” in an arbitrary background spacetime.","For hyperbolic equations, what we mean by “produced by a source” is the solution obtained by convolving the source with the retarded Green's function.","In general spacetimes, issues of convergence of the retarded solution will arise from the contribution of the sources at arbitrarily early timesEven in Minkowski spacetime, the contribution of null sources at arbitrarily early times does not converge to a distribution [8]..","However, we are not interested in such issues of convergence here, but rather the effects arising from near the source event $q$ .","Thus, we shall simply consider the contributions to the retarded Green's function integral arising from the above particle sources in a small neighborhood of $q$ .","The memory effect will then be identified with the presence of a derivative of a delta-function in the curvature “produced” by these particle sources near event $q$ ."],["Perturbation Theory in Cosmological Spacetimes","We now wish to consider linearized perturbations off of a spatially flat FLRW background, $ds^2= - d \\tau ^2 +a^2(\\tau )(dx^2 + dy^2 + dz^2).$ As usual, it is convenient to introduce conformal time $d\\eta =d\\tau /a$ so that the background FLRW metric takes the manifestly conformally flat form $ds^2 =a^2(\\eta )\\left(- d \\eta ^2 + dx^2 + dy^2 + dz^2\\right).$ Throughout the rest of the paper, “0” and “$i$ ” (i.e., spatial) indices will denote components of tenors with respect to these coordinates, and an overdot will denote a derivative with respect to $\\eta $ .","We will write $\\partial ^i=\\delta ^{ij}\\partial _j$ and $\\nabla ^2 = \\partial ^i\\partial _i = \\delta ^{ij} \\partial _i \\partial _j$ , i.e., $\\nabla ^2$ is the Laplacian with respect to the spatial metric $\\delta _{ij}$ given by $dx^2 + dy^2 +dz^2$ .","We assume that Einstein's equation holds (possibly with a cosmological constant $\\Lambda $ ) and that the matter stress-energy—apart from the particle matter that we will add as a perturbation—is that of a perfect fluid $T^{(F)}_{ab}=(\\rho +p)u_a u_b+pg_{ab} \\, ,$ with 4-velocity $u^a$ , density $\\rho $ and pressure $p$ .","The fluid is assumed to be described by a one-parameter (“barotropic”) equation of state $p=p(\\rho )$ .","The density and pressure are perturbations away from homogeneous background values $\\bar{\\rho }$ and $\\bar{p}$ which satisfy the Friedmann equations $\\left(\\frac{1}{a}\\frac{da}{d\\tau }\\right)^2=\\frac{1}{3}\\left(8\\pi \\bar{\\rho }+\\Lambda \\right) \\, ,\\\\\\frac{1}{a}\\frac{d^2a}{d\\tau ^2}=\\frac{1}{3}\\left(-4\\pi (\\bar{\\rho }+3\\bar{p})+\\Lambda \\right) \\, .","$ We write the perturbed metric as $g_{ab}= \\bar{g}_{ab}+a^2h_{ab}$ where $\\bar{g}_{ab}$ denotes the background FLRW metric.","The perturbed fluid is described by $\\delta u^\\mu $ , $\\delta \\rho $ , and $\\delta p = c_s^2 \\delta \\rho $ , where $c_s^2=\\frac{d p}{d\\rho } \\, .$ We wish to consider the metric perturbation resulting from the presence of a particle stress-energy of the form (REF ).","The particle sources are assumed to have no direct interaction with the fluid present in the FLRW background; the particle stress-energy is separately conserved.","However, since the particles affect the perturbed metric, they automatically affect the fluid (even at the linearized level), so the fluid perturbations cannot be ignored.","Analysis of the perturbations is most easily done using the gauge-invariant methods of Bardeen [15] with modifications by Durrer [16],[17] allowing for additional forms of matter perturbationsBoth Bardeen and Durrer allow for general stress-energies with non-fluid properties like anisotropic pressures.","However, as we have discussed in section , we want our perturbed fluid and particles to interact only gravitationally, which means that the stress-energies of the fluid and the particles must be conserved independently.","Bardeen does not discuss this scenario, but it corresponds to Durrer's notion of cosmological seeds..","These methods rely on decomposing the metric, fluid stress-energy, and particle stress-energy perturbations into scalar, vector, and tensor parts, and working with gauge invariant quantities in each sector.","We can decompose a general symmetric tensor field $X_{ab}$ on spacetime into its scalar, vector, and tensor parts by writing its coordinate components as $X_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi &\\partial _i\\chi \\;\\;+\\;\\;\\xi _i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi \\\\+\\\\ \\xi _i\\end{array}}&{\\begin{array}{c}\\psi \\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega \\\\+\\partial _{(i}\\zeta _{j)}+{X}_{ij}\\end{array}}\\end{array}\\right) \\, ,$ where the scalar parts are given byIf the spatial slices have topology $\\bf {R}^3$ , we need to impose boundary conditions at infinity in order to get a unique solution to the Poisson equations for $\\chi $ and $\\omega $ (and $\\zeta _i$ below), which, in turn, may put restrictions on the asymptotic behavior of $X_{ab}$ .","However, as we are ultimately interested in singular behavior of the perturbations, it does not matter what solutions of the Poisson equations we choose.","For convenience, we shall assume that the spatial slices have the topology of three-tori, with the dimensions of the tori being much larger than the dimensions of the physical problem.","The solutions are then unique up to the addition of constants, which do not affect the decomposition.","=X00 2=iX0i =13ijXij 22=32(ij-13ij2)Xij , the vector parts are given by i=X0i-i 2i=2(jXij-i-232i) , and the tensor part is ${X}_{ij} =X_{ij}-\\psi \\delta _{ij}-\\left(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2\\right)\\omega -\\partial _{(i}\\zeta _{j)} \\, .","$ If the metric perturbation is written in this way, $h_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi ^{(h)}&\\partial _i\\chi ^{(h)}\\;\\;+\\;\\;\\xi ^{(h)}_i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi ^{(h)}\\\\+\\\\ \\xi ^{(h)}_i\\end{array}}&{\\begin{array}{c}\\psi ^{(h)}\\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega ^{(h)}\\\\+\\partial _{(i}\\zeta ^{(h)}_{j)}+{h}_{ij}\\end{array}}\\end{array}\\right),$ then =(h)+2(h)+2aa(h)+(h)+aa(h) =(h)+2aa(h)-132(h)-aa(h) are gauge-invariant scalar quantities, whereas $\\Xi _i=\\xi ^{(h)}_i-\\dot{\\zeta }^{(h)}_i$ is a gauge-invariant vector quantity, and ${h}_{ij}$ is a gauge-invariant tensor quantity.","The above two scalar fields, $\\Phi $ and $\\Psi $ , one transverse three-vector field, $\\Xi _i$ , and one transverse-traceless three-tensor field, ${h}_{ij}$ , contain all of the physical (non-gauge) information concerning the metric perturbation.","The stress-energy tensor of the particles (REF ) can also be decomposed in this way: $T^{(P)}_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi ^{(P)}&\\partial _i\\chi ^{(P)}\\;\\;+\\;\\;\\xi ^{(P)}_i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi ^{(P)}\\\\+\\\\ \\xi ^{(P)}_i\\end{array}}&{\\begin{array}{c}\\psi ^{(P)}\\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega ^{(P)}\\\\+\\partial _{(i}\\zeta ^{(P)}_{j)}+{T}_{ij}\\end{array}}\\end{array}\\right).$ Because there is no “background” particle stress-energy, each of the individual component fields $\\varphi ^{(P)},\\chi ^{(P)},$ etc.","are already gauge invariant to first order.","These quantities are related to $T^{(P)}_{\\mu \\nu }$ by eqs.","()-(REF ).","Since $T^{(P)}_{\\mu \\nu }$ is distributional, these quantities will also be distributional.","We can also find gauge-invariant combinations of the perturbed stress-energy $\\delta T^{(F)}_{\\mu \\nu }$ of the fluid, (REF ).","We define $\\delta _\\rho = \\frac{\\rho -\\bar{\\rho }}{\\bar{\\rho }} \\, .$ We decompose the perturbed 4-velocity as $\\delta u^\\mu =\\frac{1}{a}\\left(\\begin{array}{c}\\delta u^0\\\\\\partial ^i v+v^i\\end{array}\\right)$ with $\\partial _i v^i = 0$ , and we remind the reader that $\\partial ^i v = \\delta ^{ij} \\partial _j v$ .","Note that the quantity $\\delta u^0$ is not independent, since it is fixed by the normalization condition $g_{ab}u^au^b=-1$ .","In terms of these quantities and the perturbed metric, we can obtain the following gauge-invariant fluid variables: $V=v+\\frac{1}{2}\\dot{\\omega }^{(h)}\\\\A=\\delta _\\rho +3\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\frac{1}{2}\\left(\\psi ^{(h)}-\\frac{1}{3}\\nabla ^2\\omega ^{(h)}\\right)-\\frac{\\dot{a}}{a}V-\\Phi \\right)\\\\W_i=\\delta _{ij}v^j+\\frac{1}{2}\\dot{\\xi }^{(h)}_i.$ The fields $V$ , $A$ , and $W_i$ thus provide us, respectively, with gauge-invariant measures of fluid's peculiar velocity with respect to the Hubble flow, its perturbed density, and its vorticity.","The linearized Einstein equation decomposes into decoupled sets of equations involving the scalar, vector, and tensor parts of the perturbations.","These equations can be written entirely in terms of the gauge-invariant quantities introduced above.","The scalar equations are $\\nabla ^2\\Psi -3\\frac{\\dot{a}}{a}\\left(\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi \\right) = -8\\pi \\left(a^2\\bar{\\rho }A-3a\\dot{a}(\\bar{\\rho }+\\bar{p})V+\\varphi ^{(P)}\\right)\\\\\\partial _i\\left(\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi \\right) = -8\\pi \\left(a^2(\\bar{\\rho }+\\bar{p})\\partial _iV-\\partial _i\\chi ^{(P)}\\right)\\\\\\partial _i\\partial _j\\left(\\Psi -\\Phi \\right) = -16\\pi \\partial _i\\partial _j\\omega ^{(P)}$ $\\ddot{\\Psi }+2\\frac{\\dot{a}}{a}\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\dot{\\Phi }+\\left(2\\frac{\\ddot{a}}{a}-\\left(\\frac{\\dot{a}}{a}\\right)^2\\right)\\Phi +\\frac{2}{3}\\nabla ^2\\left(\\Psi -\\Phi \\right)\\\\=-16\\pi \\left(a^2c_s^2(\\bar{\\rho })A-3(\\bar{\\rho }+\\bar{\\phi })\\frac{\\dot{a}}{a}V+\\psi ^{(P)}\\right) \\, .$ The vector equations are $\\nabla ^2\\Xi _i=-8\\pi \\left(2a^2(\\bar{\\rho }+\\bar{p})W_i-\\xi ^{(P)}_i\\right)\\\\\\partial _{(i}\\dot{\\Xi }_{j)}+2\\frac{\\dot{a}}{a}\\partial _{(i}\\Xi _{j)}=-8\\pi \\partial _{(i}\\zeta ^{(P)}_{j)} \\, .$ Finally, the tensor equation is $-\\ddot{{h}}_{ij}-2\\frac{\\dot{a}}{a}\\dot{{h}}_{ij}+\\nabla ^2{h}_{ij} =-16\\pi {T}_{ij} \\, .$ If the various perturbation fields do not grow in an unbounded fashion at large distances, the unique solutions to () and () are $\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi =-8\\pi \\left(a^2(\\bar{\\rho }+\\bar{p})V-\\chi ^{(P)}\\right)\\\\\\Psi -\\Phi =-16\\pi \\omega ^{(P)} \\, .$ Equations (REF ) and (REF ) then simplify to $\\nabla ^2\\Psi =-8\\pi \\left(a^2\\bar{\\rho }A+\\varphi ^{(P)}+3\\chi ^{(P)}\\right)$ $\\ddot{\\Psi }+2\\frac{\\dot{a}}{a}\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\dot{\\Phi }+\\left(2\\frac{\\ddot{a}}{a}-\\left(\\frac{\\dot{a}}{a}\\right)^2\\right)\\Phi \\\\=-16\\pi \\left(a^2c_s^2\\left(\\bar{\\rho }A-3(\\bar{\\rho }+\\bar{p})\\frac{\\dot{a}}{a}V\\right)+\\psi ^{(P)}-\\frac{2}{3}\\nabla ^2\\omega ^{(P)}\\right) \\, .$ Similarly, () implies that $\\dot{\\Xi }_{i}+2\\frac{\\dot{a}}{a}\\Xi _{i}=-8\\pi \\zeta ^{(P)}_{i} \\, .$ The full set of Einstein's equation thus reduces to (REF )-(REF ) together with (REF ) and (REF ).","The perturbed conservation of stress energy for the fluid, $\\delta [\\nabla ^\\mu T^{(F)}_{\\mu \\nu }] = 0$ , yields the scalar equations $\\dot{V}+\\frac{\\dot{a}}{a}V=-\\frac{c_s^2\\bar{\\rho }E}{\\bar{\\rho }+\\bar{p}}-\\frac{1}{2}\\Phi \\\\\\dot{A}-3\\frac{\\bar{p}}{\\bar{\\rho }}\\frac{\\dot{a}}{a}A=-\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\nabla ^2V-3\\chi ^{(P)}\\right)$ as well as the vector equation $\\dot{W}_i-3\\frac{\\dot{a}}{a}c_s^2W_i=0 \\, .$ Similarly, conservation of stress-energy for the particles can also be expressed in terms of the fields (REF ) as $\\dot{\\varphi }^{(P)}+\\frac{\\dot{a}}{a}\\varphi ^{(P)}-\\nabla ^2\\chi ^{(P)}+3\\frac{\\dot{a}}{a}\\psi ^{(P)}=0\\\\\\dot{\\chi }^{(P)}+2\\frac{\\dot{a}}{a}\\chi ^{(P)}-\\psi ^{(P)}-\\frac{2}{3}\\nabla ^2\\omega ^{(P)}=0\\\\\\dot{\\xi }^{(P)}_i+2\\frac{\\dot{a}}{a}\\xi ^{(P)}_i-\\frac{1}{2}\\nabla ^2\\zeta ^{(P)}_i=0.$ A very useful equation for $A$ can be derived as follows [16]: We differentiate () with respect to $\\eta $ , and substitute from (REF ) to eliminate $\\dot{V}$ .","Then we use () to eliminate $\\nabla ^2 V$ , and we use () and (REF ) to eliminate $\\nabla ^2\\Phi $ .","Finally, we use () to eliminate $\\dot{\\chi }^{(P)}$ .","We thereby obtain a wave equation for $A$ with particle sources, $-\\ddot{A}-\\frac{\\dot{a}}{a}\\left(1+3c_s^2-6\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\dot{A}+c_s^2\\nabla ^2 A+3\\left(\\frac{\\ddot{a}}{a}\\frac{\\bar{p}}{\\bar{\\rho }}-3\\left(\\frac{\\dot{a}}{a}\\right)^2\\left(c_s^2-\\frac{\\bar{p}}{\\bar{\\rho }}\\right)+a^2\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\frac{4\\pi }{3}\\bar{\\rho }\\right)A\\\\=-4\\pi a^2\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\varphi ^{(P)}+3\\psi ^{(P)}\\right).$ Physically, this equation describes the propagation of sound waves in the fluid.","Although there is no direct coupling between the particles and the fluid, there are “particle source terms” in (REF ) resulting from the gravitational interactions between the particles and the fluid.","As previously explained, the memory effect will be identified with the presence of a derivative of a delta function in the curvature.","The curvature is given by an expression involving at most 2 derivatives of the metric variables.","In particular, in the Newtonian gauge, the perturbation to the electric components of the Riemann tensor is [18] $\\delta R_{i00}^{\\quad j}=-\\frac{1}{2}\\Bigg (\\left(\\partial _i\\partial _k-\\frac{1}{3}\\delta _{ik}\\nabla ^2\\right)\\Phi +\\left(\\ddot{\\Psi }+\\frac{\\dot{a}}{a}(\\dot{\\Psi }-\\dot{\\Phi })\\right)\\delta _{ik}\\\\-\\partial _{(i}\\left(\\dot{\\Xi }_{k)}+\\frac{\\dot{a}}{a}\\Xi _{k)}\\right)+\\left(\\ddot{{h}}_{ik}+\\frac{\\dot{a}}{a}\\dot{{h}}_{ik}\\right)\\Bigg )\\delta ^{jk}.$ Thus, a derivative of a delta function in the curvature requires a step function (or worse) discontinuity in the gauge invariant metric variables.","We are now in a position to analyze how discontinuities could arise.","First, it is important to note that the equations ()-(REF ) giving the scalar, vector, and tensor parts of a tensor $X_{\\mu \\nu }$ involve solving elliptic and/or algebraic equations, with “source” given by components of $X_{\\mu \\nu }$ and their derivatives.","It follows immediately that the scalar, vector, and tensor parts of $X_{\\mu \\nu }$ are smooth wherever $X_{\\mu \\nu }$ itself is smooth.","In particular, the scalar, vector, and tensor parts of the particle stress-energy (REF ) are smooth away from the worldlines of the particles.","Consider, now, the scalar perturbations.","Eqs.","(REF ) and (REF ) are elliptic in $\\Phi $ and $\\Psi $ , so these quantities—which fully characterize the scalar part of the metric perturbation—can be singular only where the source terms in these equations are singular.","These source terms involve the scalar part of the particle source and the quantity $A$ .","The scalar part of the particle source is smooth away from the particle world lines.","The quantity $A$ satisfies the hyperbolic equation (REF ), which, in turn is sourced by the scalar parts of the particle stress-energy.","We shall analyze the possible singular behavior of $A$ in the Appendix.","We shall show there that, although $A$ can be discontinuous along the sound cone of the source event $q$ , there cannot be any discontinuities in $\\Phi $ or $\\Psi $ .","Thus, no memory effect can occur in the scalar sector.","Consider, now, the vector perturbations.","The quantity $\\Xi _i$ satisfies the elliptic equation (REF ) and thus is smooth wherever the sources are smooth.","However, the particle source term $\\xi _i^{(P)}$ is smooth away from the worldlines of the particles and the fluid source term $W_i$ satisfies the source free evolution equation (REF ) and is thus nonsingular everywhere.","Thus, $\\Xi _i$ is smooth away from the particle worldlines and no memory effect can occur in the vector sector.","In the next section, we calculate the memory effect occurring in the tensor sector."],["The Retarded Gravitational Field and the Memory Effect","The tensor perturbations are described by the quantity ${h}_{ij}$ , which satisfies (see (REF )) $-\\ddot{{h}}_{ij}-2\\frac{\\dot{a}}{a}\\dot{{h}}_{ij}+\\nabla ^2{h}_{ij} =-16\\pi {T}_{ij} \\, , $ where ${T}_{ij}$ is the tensor part of the particle stress energy.","Thus, each component of ${h}_{ij}$ in the coordinates (REF ) satisfies a decoupled scalar wave equation and it suffices to analyze the behavior of solutions to the scalar wave equation.","We are interested in the contribution to the retarded integral ${h}_{ij}(x)=16\\pi \\int \\sqrt{-g(x^{\\prime })}d^4x^{\\prime }G^{\\rm ret}(x,x^{\\prime }){T}_{ij}(x^{\\prime })$ arising from a small neighborhood of the source event $q$ (see section ), where $G^{\\rm ret}(x,x^{\\prime })$ denotes the retarded Green's function for the scalar wave equation $- \\ddot{\\phi } - 2\\frac{\\dot{a}}{a} \\dot{\\phi } + \\nabla ^2 \\phi = -16\\pi T \\, .$ Specifically, we seek to determine whether a discontinuity can arise in ${h}_{ij}$ and, if so, to determine its magnitude.","Such discontinuities will give rise to derivative of delta function contributions to the curvature, which, in turn, will produce a memory effect.","To proceed, we need to know the form of $G^{\\rm ret}(x,x^{\\prime })$ .","Consider an equation of the general form $L[\\phi ]=g^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\phi +b^\\mu \\partial _\\mu \\phi +c\\phi =-16\\pi T \\, ,$ where $g_{\\mu \\nu }$ is a metric of Lorentz signature.","It is well known [19], [20] that, in 4 spacetime dimensions, the retarded Green's function for this equation takes the form $G^{\\rm ret}(x,x^{\\prime })=\\left[U(x,x^{\\prime })\\delta (\\sigma )+V(x,x^{\\prime })\\Theta (-\\sigma )\\right] \\Theta (t-t^{\\prime }) \\, , $ where $\\sigma $ denotes the squared geodesic distance between $x$ and $x^{\\prime }$ in the metric $g_{\\mu \\nu }$ and $t$ is a global time function.","The quantities $U$ (the Van Vleck-Morette determinant) and $V$ are smooth functions; we refer to $U(x,x^{\\prime })\\delta (\\sigma )$ as the “direct part” and $V(x,x^{\\prime })\\Theta (-\\sigma )$ as the “tail part” of $G^{\\rm ret}(x,x^{\\prime })$ .","In general, the form (REF ) for $G^{\\rm ret}(x,x^{\\prime })$ will hold only locally in a convex normal neighborhood, but in the case of (REF ), the spacetime metric corresponding to (REF ) is flat, and this form of $G^{\\rm ret}(x,x^{\\prime })$ holds globally, with $\\sigma (x,x^{\\prime })=-(\\eta -\\eta ^{\\prime })^2+(x-x^{\\prime })^2+(y-y^{\\prime })^2+(z-z^{\\prime })^2 \\, .$ The Van Vleck-Morette $U$ is determined by integrating an ODE along a geodesic connecting $x$ and $x^{\\prime }$ [19], [20].","The quantities appearing in this ODE depend on $g^{\\mu \\nu }$ and $b^\\mu $ but do not depend on $c$ .","We could integrate this ODE directly, but we can greatly simplify the calculation of $U$ by working with the rescaled variable ${\\tilde{\\phi }} = a\\phi $ (where $\\phi $ denotes a component of ${h}_{ij}$ ), which satisfies the equation $- \\ddot{\\tilde{\\phi }} + \\nabla ^2 {\\tilde{\\phi }} + \\frac{\\ddot{a}}{a}{\\tilde{\\phi }} =-16\\pi a T \\, .$ This change of variables eliminates the term involving $b^\\mu $ in (REF ), so $\\tilde{U}$ is determined by exactly the same equation for the wave equation in flat spacetime.","Thus we obtain the unique solution $\\tilde{U}(x,x^{\\prime }) = (4\\pi )^{-1}$ .","However, the retarded Green's function for $\\phi $ is related to the retarded Green's function for $\\tilde{\\phi }$ by [21]-[23] $G^{\\rm ret}(x,x^{\\prime })=\\frac{a(\\eta ^{\\prime })}{a(\\eta )}\\tilde{G}^{\\rm ret}(x,x^{\\prime }) \\, .$ Thus, we obtain $U(x,x^{\\prime })=\\frac{1}{4\\pi }\\frac{a(\\eta ^{\\prime })}{a(\\eta )} \\, .$ This holds for any FLRW universe, i.e., we have not assumed any particular equation of state $\\bar{P}= \\bar{P}(\\bar{\\rho })$ (and, thus, we have not assumed any particular expansion law) in the background spacetime.","By contrast, $V(x,x^{\\prime })$ will depend on the expansion history of the FLRW universe between $\\eta ^{\\prime }$ and $\\eta $ .","Note, however, that by spatial Euclidean invariance, $V$ depends on $(x,x^{\\prime })$ only via $\\eta $ , $\\eta ^{\\prime }$ , and $|\\mathbf {x} - \\mathbf {x}^{\\prime }|$ .","As previously stated, we are interested in the possible discontinuities in ${h}_{ij}$ resulting from the source behavior near $q$ , where we take $q$ to have coordinates $\\mathbf {x} = t =0$ .","To analyze this, let us first introduce a new toy mathematical problem, wherein we consider retarded solutions to the equation $-\\ddot{H}_{ij}-2\\frac{\\dot{a}}{a}\\dot{H}_{ij}+\\nabla ^2H_{ij} =-16\\pi T^{(P)}_{ij} \\, .$ Eq.","(REF ) differs from the equation of interest (REF ) in that we have not taken the tensor part of the particle source and, correspondingly, we do not require $H_{ij}$ to be transverse or traceless.","By (REF ), $T^{(P)}_{ij}$ consists of a sum of terms, each one of which has the form $ \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(t)\\right) \\Theta (\\pm t)$ , where $\\mathbf {z}(t)$ describes a timelike or null geodesic.","The contribution of the tail part, $V(x,x^{\\prime })\\Theta (-\\sigma )$ , of $G^{\\rm ret}(x,x^{\\prime })$ to the retarded integral is thus a sum of terms of the form $H^{\\rm tail}_{ij} (x) =\\int d^4x^{\\prime } f_{ij}(x^{\\prime }) V(x,x^{\\prime })\\Theta \\left(-\\sigma (x,x^{\\prime })\\right) \\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right) \\Theta (\\pm t^{\\prime }) \\, ,$ where $f_{ij}$ is smooth.","It is not difficult to see that $H^{\\rm tail}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ .","On the other hand, if $\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic—i.e., if one of the ingoing or outgoing null particles is “aimed” directly at an observer at $x$ —then the singularities of $\\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right)$ and $\\Theta \\left(-\\sigma (x,x^{\\prime })\\right)$ will coincide and $H^{\\rm tail}_{ij}$ will, in general, be “highly singular” at $x$ in the sense that, in general, it will be defined only distributionally in a neighborhood of $x$ .","We exclude such special points from consideration.","The case of main interest is one where $x$ lies near the future light cone of $q$ but does not lie on the special direction defined by $\\mathbf {z}(t)$ (if null).","Then the $\\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right)$ singularity in the integral will be transverse to the step function singularity of $\\Theta \\left(-\\sigma (x,x^{\\prime })\\right)$ as well as to that of $\\Theta (\\pm t)$ .","One can then integrate over $\\mathbf {x}^{\\prime }$ , leaving one with an integral only over $t^{\\prime }$ .","The integrand will be proportional to $V(x; \\mathbf {z}(t^{\\prime }), t^{\\prime }) \\Theta (u-t^{\\prime }) \\Theta (\\pm t^{\\prime })$ , where $u = t-|\\mathbf {x}|$ denotes the retarded time of $x$ .","The integral over $t^{\\prime }$ thus yields a result of the form $F_{ij}(x) u \\Theta (u)$ , where $F_{ij}$ is smooth.","Thus, we see that $H^{\\rm tail}_{ij}$ is continuous (although not continuously differentiable) at $x$ .","The analysis of the contribution, $H^{\\rm dir}_{ij}$ , of the “direct part,” $U(x,x^{\\prime }) \\delta (\\sigma )$ , of $G^{\\rm ret}(x,x^{\\prime })$ to $H_{ij}$ is similar, with $U(x,x^{\\prime }) \\delta (\\sigma )$ replacing $V(x,x^{\\prime })\\Theta (-\\sigma )$ in (REF ).","Again, $H^{\\rm dir}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ , and is highly singular if $\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic.","If we exclude such special points, then the integral over $\\mathbf {x}^{\\prime }$ can again be done, and we are again left with an integral over $t^{\\prime }$ .","However, now the integrand is proportional to $\\delta (u-t^{\\prime }) \\Theta (\\pm t^{\\prime })$ , where $u$ is the retarded time of $x$ .","Consequently, $H^{\\rm dir}_{ij}$ has the form $\\tilde{F}_{ij}(x) \\Theta (u)$ for some smooth $\\tilde{F}_{ij}$ , and thus it has a discontinuity along the future light cone of $q$ .","The above analysis is for the toy problem (REF ), where we did not take the tensor part, ${T}_{ij}$ , of the source $T^{(P)}_{ij}$ .","It would be cumbersome to compute ${T}_{ij}$ and then perform a similar direct analysis of the behavior of the retarded Green's function integral involving ${T}_{ij}$ .","Fortunately, we can bypass this by noting that the operation of “taking the tensor part\" commutes with the wave operator appearing in (REF ) and (REF ).","It follows that the desired quantity, ${h}_{ij}$ , given by (REF ), is related to $H_{ij}$ by ${h}_{ij}=[H_{ij}]^T \\, ,$ where “$[X_{ij}]^T$ ” denotes the operation of taking the “tensor part” of $X_{ij}$ , as given by (REF ).","Thus, our analysis reduces to extracting information about how the operation of taking the tensor part of a quantity affects the nature of its singularities.","To analyze this, we note first that since “taking the tensor part” consists of algebraic operations involving differentiations and inversions of Laplacians (see (REF )-(REF )), the tensor part, ${X}_{ij}$ , of a distribution $X_{ij}$ must be smooth wherever $X_{ij}$ is smooth.","It then follows that the singular behavior of the tensor part of $X_{ij}$ at $x$ is the same as that of the tensor part of $\\psi X_{ij}$ , where $\\psi $ is any smooth function of compact support with $\\psi = 1$ in a neighborhood of $x$ .","(Proof: $X_{ij} - \\psi X_{ij}$ vanishes in a neighborhood of $x$ and hence is smooth there, so the tensor part of this difference is smooth in a neighborhood of $x$ .)","However, the singular behavior of $\\psi X_{ij}$ is characterized by the decay (or lack thereof) of its Fourier transform at large $k_\\mu $ .","The key point is that in the operation of “taking the tensor part,” there are exactly as many total “inverse derivatives” from Laplacian inversions in (REF ) as there are differentiations.","It follows that the Fourier transform of the tensor part of $\\psi X_{ij}$ is related to the Fourier transform of $\\psi X_{ij}$ by a function that is everywhere bounded in $k_\\mu $ .","In particular, the decay of the Fourier transform of the tensor part of $\\psi X_{ij}$ at large $k_\\mu $ cannot be slower than that of $\\psi X_{ij}$ .","The above argument can be applied to the present case as follows to get the key conclusion that we need.","It is easily seen from the explicit behavior of $H^{\\rm tail}_{ij}$ found above that the Fourier transform of $\\psi H^{\\rm tail}_{ij}$ lies in $L^1$ for any smooth function $\\psi $ of compact support.","Therefore, the Fourier transform of the tensor part of $\\psi H^{\\rm tail}_{ij}$ —which differs from the Fourier transform of $\\psi H^{\\rm tail}_{ij}$ by a bounded function of $k_\\mu $ —also lies in $L^1$ .","But that implies that the tensor part of $\\psi H^{\\rm tail}_{ij}$ is continuous for all $\\psi $ , which implies that the tensor part of $H^{\\rm tail}_{ij}$ is continuous.","Thus, we have shown that the tail contribution to ${h}_{ij}$ is continuous and thus cannot contribute to the memory effect.","The above conclusion is all that is needed to derive our results on the memory effect, because, as we have seen above, the direct contribution to the retarded solution is universal, and does not depend on the expansion history.","Furthermore, Minkowski spacetime lies within the class of $k=0$ FLRW spacetimes to which our analysis applies.","Thus, we can relate the memory effect in an arbitrary FLRW spacetime to that in Minkowski spacetime as follows.","Consider a source event at $q$ in the FLRW spacetime that is observed at event $p$ .","Let $\\eta _s$ and $\\eta _o$ denote the conformal times of the events $q$ and $p$ respectively.","For convenience, rescale the coordinates, if necessary, so that $a(\\eta _s) = 1$ .","This corresponds to choosing the comoving coordinates to corrspond to proper distances at $\\eta = \\eta _s$ .","Now, identify the FLRW spacetime with Minkowski spacetime by identifying the coordinates (REF ) of the FLRW spacetime with global inertial coordinates of Minkowski spacetime.","Place a source and observer at the events ${\\bar{q}}$ and ${\\bar{p}}$ of Minkowski spacetime that are identified in this manner with events $q$ and $p$ in the FLRW spacetime.","Since $a(\\eta _s) =1$ , the Minkowski source will physically correspond to the FLRW source provided that the masses and 4-velocities of each of the particles agree (under this identification) at $q$ .","It follows immediately from (REF ) that the direct part, ${h}_{ij}^{\\rm dir}$ , of ${h}_{ij}(x)$ near $p$ will be a factor of $1/a(\\eta _o)$ times the same function of $x^\\mu $ as it is in the Minkowski case, i.e., near $p$ , ${h}_{ij}^{\\rm dir} (x^\\mu ) = \\frac{1}{a(\\eta _o)} {\\bar{h}}_{ij}^{\\rm dir} (x^\\mu ) \\, .$ It then follows immediately from (REF ) that the direct parts of the linearized Riemann curvature tensor are similarly related, i.e., near $p$ $\\delta {R^{\\rm dir}_{i00}}^{j} (x^\\mu ) = \\frac{1}{a(\\eta _o)} \\delta {\\bar{R}}^{\\rm dir}_{i00}{}^j (x^\\mu ) \\, .$ Suppose, now, that we place a gravitational wave detector at $p$ , composed of two nearby particles initially at rest in the cosmic reference frame.","By the geodesic deviation equation, the deviation vector, $D^i$ , describing the displacement of the particles will satisfy $v^b \\nabla _b (v^c\\nabla _c {D}^a) = R_{def}^{\\quad a}{D}^d v^e v^f \\, ,$ where $v^a$ is the unit tangent to the geodesic.","Since $v^\\mu \\approx 1/a(\\eta _o) (\\partial /\\partial \\eta )^\\mu $ and the Hubble expansion is negligible over the relevant timescale, we can rewrite this equation as $\\frac{d^2}{d \\eta ^2} D^j = R_{i00}{}^j D^i \\, .$ Let $\\Delta D^i$ denote the coordinate components of the “memory displacement,” obtained by integrating (REF ) twice with respect to $\\eta $ .","In view of (REF ) and the fact, proven above, that the “direct part” of the Riemann tensor contains the full memory effect, we see that $\\Delta D^i = \\frac{1}{a(\\eta _o)} \\overline{\\Delta D}^i \\, ,$ where $\\overline{\\Delta D}^i$ denotes the corresponding memory displacement in Minkowski spacetime, assuming that the initial displacement was $\\overline{D}^i = D^i$ .","Thus, the relationship between $\\Delta D^i$ and $D^i$ in an arbitrary FLRW spacetime differs from the corresponding Minkowski result by a factor of $1/a(\\eta _o)$ .","Thus, we have shown that if we identify the FLRW spacetime with Minkowski spacetime via the coordinates (REF ) in such a way that $a(\\eta _s) = 1$ , and we place the same physical source at $q$ and the same physical detector at $p$ in both spacetimes, then the memory effect in the FLRW spacetime will be a factor of $1/a(\\eta _o) = 1/(1 + z)$ smaller than the corresponding memory effect in Minkowski spacetime.","Note that placing the source at the same proper distance at the time of emission corresponds to placing the source at the same angular diameter distance in both spacetimes.","The above result compares the memory effect in FLRW and Minkowski spacetime when the source and detector are at the same proper distance at the source emission time, i.e., when they are at the same location with $a(\\eta _s) = 1$ .","Since the memory effect in Minkowski spacetime falls off as $1/r$ , this result may be reformulated in numerous equivalent ways.","In particular, we have If the source and detector are placed so that they are at the same proper distance at the time of detection (rather than emission), then the memory effect in the FLRW spacetime is identical to the corresponding memory effect in Minkowski spacetime.","If the source and detector are placed so that the source is at the same luminosity distance in both cases, then the memory effect in the FLRW spacetime is larger by a factor of $(1+z)$ as compared with the corresponding memory effect in Minkowski spacetime.","The above result is in agreement with the results of [9], [10], [11], and [12] in the cases where the results of those references apply.","Acknowledgements We wish to thank Lydia Bieri and David Garfinkle for helpful discussions.","This research was supported in part by NSF grants PHY 12-02718 and PHY 15-05124 to the University of Chicago."],["Acoustic Shock Waves and Metric Continuity","In this Appendix, we show that the the scalar sector does not contribute to the memory effect.","The gauge-invariant density perturbation $A$ satisfies (REF ).","Equation (REF ) is a hyperbolic wave equation of the general form (REF ), with the Lorentz metric $g_{\\mu \\nu }$ now being the “acoustic metric,” $ds^2= -d \\eta ^2 + \\frac{1}{c_s^2}[dx^2 + dy^2 + dz^2] \\, ,$ and with the source term $T$ proportional to the scalar particle fields $\\varphi ^{(P)}$ and $\\psi ^{(P)}$ .","Thus, the retarded Green's function for (REF ) takes the general Hadamard form (REF ), with $\\sigma $ replaced by the squared geodesic distance, $\\sigma _s$ , in the acoustic metric (REF ), i.e., we have $G_s(x,x^{\\prime })=\\left[U_s(x,x^{\\prime })\\delta (\\sigma _s)+V_s(x,x^{\\prime })\\Theta (-\\sigma _s) \\right] \\Theta (\\eta -\\eta ^{\\prime }) \\, ,$ where $U_s$ and $V_s$ are again smooth functions in both $x$ and $x^{\\prime }$ Furthermore, it can be seen from (REF ) that both $\\varphi ^{(P)}$ and $\\psi ^{(P)}$ are obtained from $T^{(P)}_{\\mu \\nu }$ by algebraic operations (i.e., no differentiations or Laplace inversions).","It follows immediately that the source term appearing in (REF ) takes the same form (namely, proportional to $\\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(\\eta )\\right) \\Theta (\\pm \\eta )$ ), as considered in Section .","We may therefore repeat the analysis of Section to draw the following conclusion: Suppose that all of the particles in $T^{(P)}_{\\mu \\nu }$ are moving with velocity smaller than the speed of soundIf any of the particles are moving with velocity greater than the speed of sound, there will be additional “Cherenkov radiation” singularities occurring at points $x$ where the past sound cone of $x$ intersects a particle world line orthogonally (in the sound metric).","These additional singularities are not of interest for the memory effect.. Then $A$ is smooth except on the future sound cone of $q$ .","Furthermore, on the sound cone, $A$ will, in general, be discontinuous, but it cannot have “worse” singular behavior.","The metric perturbation variables $\\Phi $ and $\\Psi $ satisfy elliptic equations, with source terms given by $A$ and the scalar parts of the particle sources.","It follows that $\\Phi $ and $\\Psi $ must be smooth everywhere apart from the worldlines of the particles and the points at which $A$ fails to be smooth, i.e., the future sound cone of $q$ .","We are not interested in the singularities at the particle worldlines.","However, on the future sound cone of $q$ , $\\nabla ^2 \\Phi $ and $\\nabla ^2 \\Psi $ are at worst discontinuous, so $\\Phi $ and $\\Psi $ themselves are at least $C^1$ .","Thus, they cannot contribute a derivative of delta-function to the Riemann curvature (REF ), and thus do not contribute to any memory effect."]],"string":"[\n [\n \"The Cosmological Memory Effect\"\n ],\n [\n \"Abstract The \\\"memory effect\\\" is the permanent change in the relative separation of test particles resulting from the passage of gravitational radiation.\",\n \"We investigate the memory effect for a general, spatially flat FLRW cosmology by considering the radiation associated with emission events involving particle-like sources.\",\n \"We find that if the resulting perturbation is decomposed into scalar, vector, and tensor parts, only the tensor part contributes to memory.\",\n \"Furthermore, the tensor contribution to memory depends only on the cosmological scale factor at the source and observation events, not on the detailed expansion history of the universe.\",\n \"In particular, for sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of $(1 + z)$.\"\n ],\n [\n \"Introduction\",\n \"The passage of gravitational radiation through a configuration of test particles that make up a gravitational wave detector can induce a permanent change in the relative separation of the test particles, a phenomenon which has come to be known as the gravitational wave memory effect.\",\n \"The memory effect on a flat background was first recognized in the linear regime for sources in non-relativistic motion by Zel'dovich and Ponarev [1].\",\n \"Afterwards, Christodoulou [2] discovered that there could be additional contributions to memory arising from the nonlinearity of the Einstein equation and associated with the Bondi flux of the gravitational waves to null infinity.\",\n \"Shortly thereafter, it was argued that the nonlinear memory of Christodoulou could be interpreted as corresponding to a linear memory caused by the effective stress-energy associated with the primary gravitational radiation [3], [4].\",\n \"More recently, it has been found to be useful to make a distinction between ordinary and null memory in the linearized gravity context [5],[6]: Ordinary memory is caused by massive matter sources and null memory is caused by null matter sources.\",\n \"The gravitational radiation emitted from a source event involving both massive and null matter will induce both ordinary and null memory [5],[6],[7].\",\n \"Neither the ordinary nor null memory should be viewed as a tidal effect with a Newtonian analogue but rather as a byproduct of the gravitational radiation emitted from a burst-type event [7],[8].\",\n \"The above work has concerned itself with memory on an asymptotically flat spacetime.\",\n \"If a spacetime is not asymptotically flat, it is not clear what “memory” should mean, even if we treat the gravitational radiation as a perturbation off of a background metric.\",\n \"Memory has been defined in terms of the net change in the separation of test particles, but this could include motion due to the background curvature rather than the radiation.\",\n \"For example, in cosmological Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, the proper separation of FLRW observers will change with time.\",\n \"Furthermore, if there is no notion of null infinity, it is not clear where we should put our detector so that it will be exposed to radiation but isolated from other non-radiative gravitational forces.\",\n \"Recently, Bieri, Garfinkle and Yau [9], Kehagias and Riotto [10], and Chu [11],[12] have considered the memory effect in FLRW spacetimes.\",\n \"However, their choice of methods for resolving the above difficulties have so far limited the applicability of their results.\",\n \"Bieri and Garfinkle consider Weyl tensor perturbations, so their analysis is greatly simplified by working in vacuo, and, consequently, they have considered only memory in a background vacuum de Sitter universe.\",\n \"Meanwhile, Kehagias and Riotto's use of BMS transformations depends on the existence of a null infinity for the FLRW spacetime, so they restrict consideration to a decelerating universe without a cosmological constant.\",\n \"Furthermore, both groups consider only null memory.\",\n \"While Chu does not make either of these specific restrictions, his preferred definition of memory does not distinguish test particle motion due to radiative and non-radiative gravity.\",\n \"In this paper, we will investigate the total memory effect—both ordinary and null—in a general, spatially flat, FLRW spacetime.\",\n \"We will simplify the problem not by limiting the cosmological models under consideration, but rather the kinds of radiation sources.\",\n \"Specifically, we will consider—in the context of linearParticle-like sources in Einsten's equation do not make sense outside of the context of linear perturbation theory [13].\",\n \"perturbation theory off of an FLRW background—only point-particle sources, i.e., sources whose stress-energy is confined by Dirac delta functions to worldlines that meet at a vertex, which we will call the source event.\",\n \"We previously considered such sources in Minkowski spacetime [7], [8], and found that the retarded solution gives rise to a contribution to the curvature tensor of the form of the derivative of a delta function in retarded time.\",\n \"Upon integration of the geodesic deviation equation, this derivative of a delta function in curvature gives rise to a well defined memory effect—i.e., a change in the separation of test particles coincident in time with the passage of the radiation from the source event—that includes both the ordinary and null memory [7], [8].\",\n \"Thus, for the idealized sources that we consider, the memory effect can be associated with the presence of a derivative of a delta function in the Riemann curvature of the retarded solution arising from the source event.\",\n \"This enables us to distinguish the memory effect induced by the passage of radiation from non-radiative gravitational effects, and it does not require us to take limits to null infinity.\",\n \"It thereby yields a well defined notion of the memory effect that is applicable to linearized perturbations about arbitrary background spacetimes.\",\n \"Another advantage to considering the above idealized particle sources, where the radiation emerges from a single “source event” in the background spacetime, is that—since all spacetimes are “locally flat” on sufficiently small scales—there is a well defined notion of having the “same source” in different spacetimes.\",\n \"Similarly, there is a well defined notion of having the “same detector”—i.e., geodesic test particles initially at rest and with small separation—in different spacetimes.\",\n \"Thus, we can compare the memory effect in two different spacetimes provided only that we specify the location and “rest frame” of both the source and the detector in the two spacetimes.\",\n \"Since any spatially flat FLRW spacetime is conformal to Minkowski spacetime, it is particularly useful to state our results by comparing the memory effect in FLRW spacetime to that in Minkowski spacetime.\",\n \"Stated in this manner, the main result of our paper is as follows: Consider a spatially flat FLRW solution to Einstein's equation with arbitrary fluid matter, and with a given particle source perturbation, as described above.\",\n \"Now place the same source and detector in Minkowski spacetime such that the source and detector are at rest with respect to each other and the source is at a distance, $d$ , in Minkowski spacetime equal to the luminosity distance, $d_L$ , in the FLRW spacetime.\",\n \"Then the memory effect in the FLRW spacetime is enhanced over the Minkowski value by a factor of $(1+z)$ , where $z$ denotes the redshift factor between the source and observer/detector.\",\n \"This result applies to both ordinary and null memory.\",\n \"It is in agreement with the results obtained for null memory in the special cases considered in [9] and [10].\",\n \"Note that in a spatially flat FLRW spacetime, the luminosity distance $d_L$ and the angular diameter distance, $d_A$ , are related by $d_A = d_L/(1+z)^2 $ It follows that for a Minkowski source at $d = d_A$ , the memory effect in the FLRW spacetime will be decreased from the Minkowski value by a factor of $(1+z)^{-1}$ .\",\n \"Finally, it should be noted that although the memory effect in a spatially flat FLRW spacetime is related to the Minkowski memory effect in this simple way, the waveforms will be different; in particular, there will be “tail effects” in the FLRW spacetime.\",\n \"Although our analysis is restricted to the context of linear perturbation theory with idealized particle sources, we expect that our main result stated above should be valid completely generally for any sources whose spatial and time variation scales are small compared with the Hubble scale.\",\n \"Indeed, the main difficulty in generalizing our results to non-particle-like sources and to the nonlinear regime would be to give a precise definition of “memory” outside of the context we consider.\",\n \"Thus, if one wishes to compute the memory effect resulting from, say, the coalescence of two black holes in a distant galaxy in a spatially flat FLRW spacetime, it should suffice to compute the memory effect arising from a similar coalescence in an asymptotically flat spacetime and then use the above correspondence.\",\n \"However, we shall not attempt to formulate or prove such a generalization here.\",\n \"We shall begin in section by describing the particle sources that we shall consider and characterizing the memory effect for perturbations of arbitrary curved spacetimes.\",\n \"In section , we analyze linearized perturbations off of spatially flat FLRW spacetimes with such particle sources.\",\n \"In section , we consider the tensor mode contribution to memory.\",\n \"We show that only the “light cone portion” of the retarded Green's function will contribute to memory, and that its contribution can be related in a simple way to the memory caused by similar sources in a flat spacetime.\",\n \"In the Appendix, we show that the scalar modes do not contribute to memory.\",\n \"Latin indices from the early alphabet ($a,b, \\\\dots $ ) denote abstract spacetime indices.\",\n \"Greek indices ($\\\\mu , \\\\nu , \\\\dots $ ) denote spacetime components of tensors, whereas Latin indices from the mid-alphabet ($i,j, \\\\dots $ ) denote spatial components.\"\n ],\n [\n \"Particle Sources\",\n \"In asymptotically flat spacetimes, the memory effect can be characterized in a precise manner by considering a detector composed of test particles near null infinity.\",\n \"Radiation effects fall off as $1/r$ , whereas Newtonian tidal effects fall off as $1/r^3$ , so by considering only the $O(1/r)$ effects on the test particles, we can distinguish between effects produced by gravitational radiation and all other gravitational effects.\",\n \"However, in a non-asymptotically-flat spacetime, it is not clear how even to define memory, since there is no obvious way to make a clean distinction between effects due to “radiation” as compared with other tidal gravitational effects.\",\n \"In our previous investigation of the memory effect in linearized gravity [7],[8], we considered the idealized process of the instantaneous decay of a massive particle into two other particles.\",\n \"We found that the retarded solution to the linearized Einstein equation with such a source has the property that the $O(1/r)$ part of the curvature has the form of a derivative of a delta-function of retarded time at the retarded time of the decay event.\",\n \"Integration of the geodesic deviation equation then shows that the $O(1/r)$ effect on test particles is to produce a sharp step function in their relative separation.\",\n \"Thus, for this kind of idealized source, the memory effect can be characterized by the presence of a derivative of a delta-function in the linearized curvature and a corresponding step function behavior in the relative separation of test particlesOf course, if one were to consider a less idealized source with a smoothed out energy-momentum tensor, then the Riemann tensor also will be smoothed out, and the relative separation of the test particles will not undergo a sharp, sudden change in separation; rather, separation of the particles that results in a memory effect would occur continuously on the same timescale as that of the event itself.. For our present purposes, the main advantage of considering sources consisting of particles undergoing instantaneous interactions is that the characterization of the memory effect in terms of derivative of a delta-function behavior of the curvature holds at all distances from the interaction event, i.e., one does not need to go to null infinity to extract this characterization of the memory effect.\",\n \"This characterization may therefore be imported straightforwardly to other spacetimes.\",\n \"Thus, in this paper, we shall restrict consideration to linearized gravity off of a smooth background spacetime, with a linearized perturbation sourced by (massive or massless) point particles.\",\n \"The interactions of the particles will be modeled by having their worldlines intersect (and, possibly, begin or end) at a single event, $q$ , in spacetime as illustrated in figure 1.\",\n \"Conservation of stress-energy then requires that (i) the particle worldlines are geodesics [14] away from $q$ , and (ii) total 4-momentum is conserved at $q$ .\",\n \"“Memory” will then be characterized by the presence of a derivative of a delta-function in the curvature of the retarded solution with this source.\",\n \"This characterization does not require that the detector be placed near “infinity.” Figure: A spacetime diagram of the sort of gravitational wave source we will consider.\",\n \"Here 5 point particles enter a single “source event” qq, and 3 emerge.\",\n \"The worldlines of the incoming and outgoing particles must be timelike or null geodesics.To specify more precisely the type of source we consider, we assume that local coordinates $(t,\\\\mathbf {x})$ have been introduced in a neighborhood of $q$ so that $\\\\nabla t$ is past-directed timelike and so that the event $q$ is labeled by $t=\\\\mathbf {x} = 0$ .\",\n \"The worldline, $\\\\gamma $ , of each incoming massive particle must be a timelike geodesic [14] with endpoint at $q$ .\",\n \"We can parametrize $\\\\gamma $ by $t$ and specify it by giving $\\\\mathbf {x}(t) = \\\\mathbf {z}(t)$ , where $\\\\mathbf {z}(0) = \\\\mathbf {0}$ .\",\n \"The stress-energy of each incoming massive particle then takes the form $T^{(M, {\\\\rm in})}_{ab} = m u_a u_b \\\\, \\\\delta ^{(3)}\\\\left(\\\\mathbf {x} - \\\\mathbf {z}(t)\\\\right) \\\\frac{1}{\\\\sqrt{-g}} \\\\frac{d\\\\tau }{dt} \\\\Theta (-t) \\\\, .$ Here $u^{a}$ is the unit tangent (4-velocity) to $\\\\gamma $ , $\\\\tau $ is the proper time along $\\\\gamma $ , $\\\\Theta $ is the Heaviside step function, and $\\\\delta ^{(3)}$ is the “coordinate delta-function,” i.e., $\\\\int \\\\delta ^{(3)}(\\\\mathbf {x} - \\\\mathbf {z}(t)) d^3 \\\\mathbf {x} = 1$ .\",\n \"Each incoming massless particle moves on a null geodesic, $\\\\alpha $ , given by $\\\\mathbf {x}(t) = \\\\mathbf {y}(t)$ , with $\\\\mathbf {y}(0) = \\\\mathbf {0}$ .\",\n \"The stress-energy of each incoming massless particle takes the form $T^{(N, {\\\\rm in})}_{ab} = k_a k_b \\\\, \\\\delta ^{(3)}\\\\left(\\\\mathbf {x} - \\\\mathbf {y}(t)\\\\right) \\\\frac{1}{\\\\sqrt{-g}} \\\\frac{d\\\\lambda }{dt} \\\\Theta (-t) \\\\, .$ Here, $\\\\lambda $ is an affine parameter of $\\\\alpha $ and $k^a$ is the corresponding tangent, with the scaling of $\\\\lambda $ chosen so that (REF ) holds.\",\n \"The stress-energy of each of the outgoing massive and massless particles takes the form of (REF ) and (REF ) except that $\\\\Theta (-t)$ is replaced by $\\\\Theta (t)$ .\",\n \"The total stress-energy of the particle sources we consider takes the form $T^{(P)}_{ab} = \\\\sum _{l, {\\\\rm in}} T^{(M,l)}_{ab} + \\\\sum _{n, {\\\\rm in}} T^{(N,n)}_{ab} + \\\\sum _{l^{\\\\prime }, {\\\\rm out}} T^{(M, l^{\\\\prime })}_{ab} + \\\\sum _{n^{\\\\prime }, {\\\\rm out}} T^{(N, n^{\\\\prime })}_{ab} \\\\, ,$ where each $T^{(M,l)}_{ab}$ takes the form of (REF ), each $T^{(N,n)}_{ab}$ takes the form of (REF ), and each $T^{(M, l^{\\\\prime })}_{ab}$ and $T^{(N,n^{\\\\prime })}_{ab}$ also take these forms with $\\\\Theta (t)$ replaced by $\\\\Theta (-t)$ .\",\n \"Conservation of stress-energy, $\\\\nabla ^a T_{ab} = 0$ , holds in the distributional sense away from $q$ by virtue of the fact that each particle moves on a geodesic [14].\",\n \"Conservation of stress-energy will hold at $q$ if and only if we have at $q$ $\\\\sum _{l, {\\\\rm in}} m^{(l)}u^{(l)}_a+ \\\\sum _{n, {\\\\rm in}} k^{(n)}_a = \\\\sum _{l^{\\\\prime }, {\\\\rm out}} m^{(l^{\\\\prime })}u^{(l^{\\\\prime })}_a + \\\\sum _{n^{\\\\prime }, {\\\\rm out}} k^{(n^{\\\\prime })}_a \\\\, ,$ Equation (REF ) with condition (REF ) defines the sources that we consider in this paper.\",\n \"We are interested in the solution to the linearized Einstein equation “produced by such a source” in an arbitrary background spacetime.\",\n \"For hyperbolic equations, what we mean by “produced by a source” is the solution obtained by convolving the source with the retarded Green's function.\",\n \"In general spacetimes, issues of convergence of the retarded solution will arise from the contribution of the sources at arbitrarily early timesEven in Minkowski spacetime, the contribution of null sources at arbitrarily early times does not converge to a distribution [8]..\",\n \"However, we are not interested in such issues of convergence here, but rather the effects arising from near the source event $q$ .\",\n \"Thus, we shall simply consider the contributions to the retarded Green's function integral arising from the above particle sources in a small neighborhood of $q$ .\",\n \"The memory effect will then be identified with the presence of a derivative of a delta-function in the curvature “produced” by these particle sources near event $q$ .\"\n ],\n [\n \"Perturbation Theory in Cosmological Spacetimes\",\n \"We now wish to consider linearized perturbations off of a spatially flat FLRW background, $ds^2= - d \\\\tau ^2 +a^2(\\\\tau )(dx^2 + dy^2 + dz^2).$ As usual, it is convenient to introduce conformal time $d\\\\eta =d\\\\tau /a$ so that the background FLRW metric takes the manifestly conformally flat form $ds^2 =a^2(\\\\eta )\\\\left(- d \\\\eta ^2 + dx^2 + dy^2 + dz^2\\\\right).$ Throughout the rest of the paper, “0” and “$i$ ” (i.e., spatial) indices will denote components of tenors with respect to these coordinates, and an overdot will denote a derivative with respect to $\\\\eta $ .\",\n \"We will write $\\\\partial ^i=\\\\delta ^{ij}\\\\partial _j$ and $\\\\nabla ^2 = \\\\partial ^i\\\\partial _i = \\\\delta ^{ij} \\\\partial _i \\\\partial _j$ , i.e., $\\\\nabla ^2$ is the Laplacian with respect to the spatial metric $\\\\delta _{ij}$ given by $dx^2 + dy^2 +dz^2$ .\",\n \"We assume that Einstein's equation holds (possibly with a cosmological constant $\\\\Lambda $ ) and that the matter stress-energy—apart from the particle matter that we will add as a perturbation—is that of a perfect fluid $T^{(F)}_{ab}=(\\\\rho +p)u_a u_b+pg_{ab} \\\\, ,$ with 4-velocity $u^a$ , density $\\\\rho $ and pressure $p$ .\",\n \"The fluid is assumed to be described by a one-parameter (“barotropic”) equation of state $p=p(\\\\rho )$ .\",\n \"The density and pressure are perturbations away from homogeneous background values $\\\\bar{\\\\rho }$ and $\\\\bar{p}$ which satisfy the Friedmann equations $\\\\left(\\\\frac{1}{a}\\\\frac{da}{d\\\\tau }\\\\right)^2=\\\\frac{1}{3}\\\\left(8\\\\pi \\\\bar{\\\\rho }+\\\\Lambda \\\\right) \\\\, ,\\\\\\\\\\\\frac{1}{a}\\\\frac{d^2a}{d\\\\tau ^2}=\\\\frac{1}{3}\\\\left(-4\\\\pi (\\\\bar{\\\\rho }+3\\\\bar{p})+\\\\Lambda \\\\right) \\\\, .\",\n \"$ We write the perturbed metric as $g_{ab}= \\\\bar{g}_{ab}+a^2h_{ab}$ where $\\\\bar{g}_{ab}$ denotes the background FLRW metric.\",\n \"The perturbed fluid is described by $\\\\delta u^\\\\mu $ , $\\\\delta \\\\rho $ , and $\\\\delta p = c_s^2 \\\\delta \\\\rho $ , where $c_s^2=\\\\frac{d p}{d\\\\rho } \\\\, .$ We wish to consider the metric perturbation resulting from the presence of a particle stress-energy of the form (REF ).\",\n \"The particle sources are assumed to have no direct interaction with the fluid present in the FLRW background; the particle stress-energy is separately conserved.\",\n \"However, since the particles affect the perturbed metric, they automatically affect the fluid (even at the linearized level), so the fluid perturbations cannot be ignored.\",\n \"Analysis of the perturbations is most easily done using the gauge-invariant methods of Bardeen [15] with modifications by Durrer [16],[17] allowing for additional forms of matter perturbationsBoth Bardeen and Durrer allow for general stress-energies with non-fluid properties like anisotropic pressures.\",\n \"However, as we have discussed in section , we want our perturbed fluid and particles to interact only gravitationally, which means that the stress-energies of the fluid and the particles must be conserved independently.\",\n \"Bardeen does not discuss this scenario, but it corresponds to Durrer's notion of cosmological seeds..\",\n \"These methods rely on decomposing the metric, fluid stress-energy, and particle stress-energy perturbations into scalar, vector, and tensor parts, and working with gauge invariant quantities in each sector.\",\n \"We can decompose a general symmetric tensor field $X_{ab}$ on spacetime into its scalar, vector, and tensor parts by writing its coordinate components as $X_{\\\\mu \\\\nu }=\\\\left(\\\\begin{array}{c|c}\\\\varphi &\\\\partial _i\\\\chi \\\\;\\\\;+\\\\;\\\\;\\\\xi _i\\\\\\\\\\\\hline {\\\\begin{array}{c}\\\\partial _i\\\\chi \\\\\\\\+\\\\\\\\ \\\\xi _i\\\\end{array}}&{\\\\begin{array}{c}\\\\psi \\\\delta _{ij}+(\\\\partial _i\\\\partial _j-\\\\frac{1}{3}\\\\delta _{ij}\\\\nabla ^2)\\\\omega \\\\\\\\+\\\\partial _{(i}\\\\zeta _{j)}+{X}_{ij}\\\\end{array}}\\\\end{array}\\\\right) \\\\, ,$ where the scalar parts are given byIf the spatial slices have topology $\\\\bf {R}^3$ , we need to impose boundary conditions at infinity in order to get a unique solution to the Poisson equations for $\\\\chi $ and $\\\\omega $ (and $\\\\zeta _i$ below), which, in turn, may put restrictions on the asymptotic behavior of $X_{ab}$ .\",\n \"However, as we are ultimately interested in singular behavior of the perturbations, it does not matter what solutions of the Poisson equations we choose.\",\n \"For convenience, we shall assume that the spatial slices have the topology of three-tori, with the dimensions of the tori being much larger than the dimensions of the physical problem.\",\n \"The solutions are then unique up to the addition of constants, which do not affect the decomposition.\",\n \"=X00 2=iX0i =13ijXij 22=32(ij-13ij2)Xij , the vector parts are given by i=X0i-i 2i=2(jXij-i-232i) , and the tensor part is ${X}_{ij} =X_{ij}-\\\\psi \\\\delta _{ij}-\\\\left(\\\\partial _i\\\\partial _j-\\\\frac{1}{3}\\\\delta _{ij}\\\\nabla ^2\\\\right)\\\\omega -\\\\partial _{(i}\\\\zeta _{j)} \\\\, .\",\n \"$ If the metric perturbation is written in this way, $h_{\\\\mu \\\\nu }=\\\\left(\\\\begin{array}{c|c}\\\\varphi ^{(h)}&\\\\partial _i\\\\chi ^{(h)}\\\\;\\\\;+\\\\;\\\\;\\\\xi ^{(h)}_i\\\\\\\\\\\\hline {\\\\begin{array}{c}\\\\partial _i\\\\chi ^{(h)}\\\\\\\\+\\\\\\\\ \\\\xi ^{(h)}_i\\\\end{array}}&{\\\\begin{array}{c}\\\\psi ^{(h)}\\\\delta _{ij}+(\\\\partial _i\\\\partial _j-\\\\frac{1}{3}\\\\delta _{ij}\\\\nabla ^2)\\\\omega ^{(h)}\\\\\\\\+\\\\partial _{(i}\\\\zeta ^{(h)}_{j)}+{h}_{ij}\\\\end{array}}\\\\end{array}\\\\right),$ then =(h)+2(h)+2aa(h)+(h)+aa(h) =(h)+2aa(h)-132(h)-aa(h) are gauge-invariant scalar quantities, whereas $\\\\Xi _i=\\\\xi ^{(h)}_i-\\\\dot{\\\\zeta }^{(h)}_i$ is a gauge-invariant vector quantity, and ${h}_{ij}$ is a gauge-invariant tensor quantity.\",\n \"The above two scalar fields, $\\\\Phi $ and $\\\\Psi $ , one transverse three-vector field, $\\\\Xi _i$ , and one transverse-traceless three-tensor field, ${h}_{ij}$ , contain all of the physical (non-gauge) information concerning the metric perturbation.\",\n \"The stress-energy tensor of the particles (REF ) can also be decomposed in this way: $T^{(P)}_{\\\\mu \\\\nu }=\\\\left(\\\\begin{array}{c|c}\\\\varphi ^{(P)}&\\\\partial _i\\\\chi ^{(P)}\\\\;\\\\;+\\\\;\\\\;\\\\xi ^{(P)}_i\\\\\\\\\\\\hline {\\\\begin{array}{c}\\\\partial _i\\\\chi ^{(P)}\\\\\\\\+\\\\\\\\ \\\\xi ^{(P)}_i\\\\end{array}}&{\\\\begin{array}{c}\\\\psi ^{(P)}\\\\delta _{ij}+(\\\\partial _i\\\\partial _j-\\\\frac{1}{3}\\\\delta _{ij}\\\\nabla ^2)\\\\omega ^{(P)}\\\\\\\\+\\\\partial _{(i}\\\\zeta ^{(P)}_{j)}+{T}_{ij}\\\\end{array}}\\\\end{array}\\\\right).$ Because there is no “background” particle stress-energy, each of the individual component fields $\\\\varphi ^{(P)},\\\\chi ^{(P)},$ etc.\",\n \"are already gauge invariant to first order.\",\n \"These quantities are related to $T^{(P)}_{\\\\mu \\\\nu }$ by eqs.\",\n \"()-(REF ).\",\n \"Since $T^{(P)}_{\\\\mu \\\\nu }$ is distributional, these quantities will also be distributional.\",\n \"We can also find gauge-invariant combinations of the perturbed stress-energy $\\\\delta T^{(F)}_{\\\\mu \\\\nu }$ of the fluid, (REF ).\",\n \"We define $\\\\delta _\\\\rho = \\\\frac{\\\\rho -\\\\bar{\\\\rho }}{\\\\bar{\\\\rho }} \\\\, .$ We decompose the perturbed 4-velocity as $\\\\delta u^\\\\mu =\\\\frac{1}{a}\\\\left(\\\\begin{array}{c}\\\\delta u^0\\\\\\\\\\\\partial ^i v+v^i\\\\end{array}\\\\right)$ with $\\\\partial _i v^i = 0$ , and we remind the reader that $\\\\partial ^i v = \\\\delta ^{ij} \\\\partial _j v$ .\",\n \"Note that the quantity $\\\\delta u^0$ is not independent, since it is fixed by the normalization condition $g_{ab}u^au^b=-1$ .\",\n \"In terms of these quantities and the perturbed metric, we can obtain the following gauge-invariant fluid variables: $V=v+\\\\frac{1}{2}\\\\dot{\\\\omega }^{(h)}\\\\\\\\A=\\\\delta _\\\\rho +3\\\\left(1+\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)\\\\left(\\\\frac{1}{2}\\\\left(\\\\psi ^{(h)}-\\\\frac{1}{3}\\\\nabla ^2\\\\omega ^{(h)}\\\\right)-\\\\frac{\\\\dot{a}}{a}V-\\\\Phi \\\\right)\\\\\\\\W_i=\\\\delta _{ij}v^j+\\\\frac{1}{2}\\\\dot{\\\\xi }^{(h)}_i.$ The fields $V$ , $A$ , and $W_i$ thus provide us, respectively, with gauge-invariant measures of fluid's peculiar velocity with respect to the Hubble flow, its perturbed density, and its vorticity.\",\n \"The linearized Einstein equation decomposes into decoupled sets of equations involving the scalar, vector, and tensor parts of the perturbations.\",\n \"These equations can be written entirely in terms of the gauge-invariant quantities introduced above.\",\n \"The scalar equations are $\\\\nabla ^2\\\\Psi -3\\\\frac{\\\\dot{a}}{a}\\\\left(\\\\dot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}\\\\Phi \\\\right) = -8\\\\pi \\\\left(a^2\\\\bar{\\\\rho }A-3a\\\\dot{a}(\\\\bar{\\\\rho }+\\\\bar{p})V+\\\\varphi ^{(P)}\\\\right)\\\\\\\\\\\\partial _i\\\\left(\\\\dot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}\\\\Phi \\\\right) = -8\\\\pi \\\\left(a^2(\\\\bar{\\\\rho }+\\\\bar{p})\\\\partial _iV-\\\\partial _i\\\\chi ^{(P)}\\\\right)\\\\\\\\\\\\partial _i\\\\partial _j\\\\left(\\\\Psi -\\\\Phi \\\\right) = -16\\\\pi \\\\partial _i\\\\partial _j\\\\omega ^{(P)}$ $\\\\ddot{\\\\Psi }+2\\\\frac{\\\\dot{a}}{a}\\\\dot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}\\\\dot{\\\\Phi }+\\\\left(2\\\\frac{\\\\ddot{a}}{a}-\\\\left(\\\\frac{\\\\dot{a}}{a}\\\\right)^2\\\\right)\\\\Phi +\\\\frac{2}{3}\\\\nabla ^2\\\\left(\\\\Psi -\\\\Phi \\\\right)\\\\\\\\=-16\\\\pi \\\\left(a^2c_s^2(\\\\bar{\\\\rho })A-3(\\\\bar{\\\\rho }+\\\\bar{\\\\phi })\\\\frac{\\\\dot{a}}{a}V+\\\\psi ^{(P)}\\\\right) \\\\, .$ The vector equations are $\\\\nabla ^2\\\\Xi _i=-8\\\\pi \\\\left(2a^2(\\\\bar{\\\\rho }+\\\\bar{p})W_i-\\\\xi ^{(P)}_i\\\\right)\\\\\\\\\\\\partial _{(i}\\\\dot{\\\\Xi }_{j)}+2\\\\frac{\\\\dot{a}}{a}\\\\partial _{(i}\\\\Xi _{j)}=-8\\\\pi \\\\partial _{(i}\\\\zeta ^{(P)}_{j)} \\\\, .$ Finally, the tensor equation is $-\\\\ddot{{h}}_{ij}-2\\\\frac{\\\\dot{a}}{a}\\\\dot{{h}}_{ij}+\\\\nabla ^2{h}_{ij} =-16\\\\pi {T}_{ij} \\\\, .$ If the various perturbation fields do not grow in an unbounded fashion at large distances, the unique solutions to () and () are $\\\\dot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}\\\\Phi =-8\\\\pi \\\\left(a^2(\\\\bar{\\\\rho }+\\\\bar{p})V-\\\\chi ^{(P)}\\\\right)\\\\\\\\\\\\Psi -\\\\Phi =-16\\\\pi \\\\omega ^{(P)} \\\\, .$ Equations (REF ) and (REF ) then simplify to $\\\\nabla ^2\\\\Psi =-8\\\\pi \\\\left(a^2\\\\bar{\\\\rho }A+\\\\varphi ^{(P)}+3\\\\chi ^{(P)}\\\\right)$ $\\\\ddot{\\\\Psi }+2\\\\frac{\\\\dot{a}}{a}\\\\dot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}\\\\dot{\\\\Phi }+\\\\left(2\\\\frac{\\\\ddot{a}}{a}-\\\\left(\\\\frac{\\\\dot{a}}{a}\\\\right)^2\\\\right)\\\\Phi \\\\\\\\=-16\\\\pi \\\\left(a^2c_s^2\\\\left(\\\\bar{\\\\rho }A-3(\\\\bar{\\\\rho }+\\\\bar{p})\\\\frac{\\\\dot{a}}{a}V\\\\right)+\\\\psi ^{(P)}-\\\\frac{2}{3}\\\\nabla ^2\\\\omega ^{(P)}\\\\right) \\\\, .$ Similarly, () implies that $\\\\dot{\\\\Xi }_{i}+2\\\\frac{\\\\dot{a}}{a}\\\\Xi _{i}=-8\\\\pi \\\\zeta ^{(P)}_{i} \\\\, .$ The full set of Einstein's equation thus reduces to (REF )-(REF ) together with (REF ) and (REF ).\",\n \"The perturbed conservation of stress energy for the fluid, $\\\\delta [\\\\nabla ^\\\\mu T^{(F)}_{\\\\mu \\\\nu }] = 0$ , yields the scalar equations $\\\\dot{V}+\\\\frac{\\\\dot{a}}{a}V=-\\\\frac{c_s^2\\\\bar{\\\\rho }E}{\\\\bar{\\\\rho }+\\\\bar{p}}-\\\\frac{1}{2}\\\\Phi \\\\\\\\\\\\dot{A}-3\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\frac{\\\\dot{a}}{a}A=-\\\\left(1+\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)\\\\left(\\\\nabla ^2V-3\\\\chi ^{(P)}\\\\right)$ as well as the vector equation $\\\\dot{W}_i-3\\\\frac{\\\\dot{a}}{a}c_s^2W_i=0 \\\\, .$ Similarly, conservation of stress-energy for the particles can also be expressed in terms of the fields (REF ) as $\\\\dot{\\\\varphi }^{(P)}+\\\\frac{\\\\dot{a}}{a}\\\\varphi ^{(P)}-\\\\nabla ^2\\\\chi ^{(P)}+3\\\\frac{\\\\dot{a}}{a}\\\\psi ^{(P)}=0\\\\\\\\\\\\dot{\\\\chi }^{(P)}+2\\\\frac{\\\\dot{a}}{a}\\\\chi ^{(P)}-\\\\psi ^{(P)}-\\\\frac{2}{3}\\\\nabla ^2\\\\omega ^{(P)}=0\\\\\\\\\\\\dot{\\\\xi }^{(P)}_i+2\\\\frac{\\\\dot{a}}{a}\\\\xi ^{(P)}_i-\\\\frac{1}{2}\\\\nabla ^2\\\\zeta ^{(P)}_i=0.$ A very useful equation for $A$ can be derived as follows [16]: We differentiate () with respect to $\\\\eta $ , and substitute from (REF ) to eliminate $\\\\dot{V}$ .\",\n \"Then we use () to eliminate $\\\\nabla ^2 V$ , and we use () and (REF ) to eliminate $\\\\nabla ^2\\\\Phi $ .\",\n \"Finally, we use () to eliminate $\\\\dot{\\\\chi }^{(P)}$ .\",\n \"We thereby obtain a wave equation for $A$ with particle sources, $-\\\\ddot{A}-\\\\frac{\\\\dot{a}}{a}\\\\left(1+3c_s^2-6\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)\\\\dot{A}+c_s^2\\\\nabla ^2 A+3\\\\left(\\\\frac{\\\\ddot{a}}{a}\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}-3\\\\left(\\\\frac{\\\\dot{a}}{a}\\\\right)^2\\\\left(c_s^2-\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)+a^2\\\\left(1+\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)\\\\frac{4\\\\pi }{3}\\\\bar{\\\\rho }\\\\right)A\\\\\\\\=-4\\\\pi a^2\\\\left(1+\\\\frac{\\\\bar{p}}{\\\\bar{\\\\rho }}\\\\right)\\\\left(\\\\varphi ^{(P)}+3\\\\psi ^{(P)}\\\\right).$ Physically, this equation describes the propagation of sound waves in the fluid.\",\n \"Although there is no direct coupling between the particles and the fluid, there are “particle source terms” in (REF ) resulting from the gravitational interactions between the particles and the fluid.\",\n \"As previously explained, the memory effect will be identified with the presence of a derivative of a delta function in the curvature.\",\n \"The curvature is given by an expression involving at most 2 derivatives of the metric variables.\",\n \"In particular, in the Newtonian gauge, the perturbation to the electric components of the Riemann tensor is [18] $\\\\delta R_{i00}^{\\\\quad j}=-\\\\frac{1}{2}\\\\Bigg (\\\\left(\\\\partial _i\\\\partial _k-\\\\frac{1}{3}\\\\delta _{ik}\\\\nabla ^2\\\\right)\\\\Phi +\\\\left(\\\\ddot{\\\\Psi }+\\\\frac{\\\\dot{a}}{a}(\\\\dot{\\\\Psi }-\\\\dot{\\\\Phi })\\\\right)\\\\delta _{ik}\\\\\\\\-\\\\partial _{(i}\\\\left(\\\\dot{\\\\Xi }_{k)}+\\\\frac{\\\\dot{a}}{a}\\\\Xi _{k)}\\\\right)+\\\\left(\\\\ddot{{h}}_{ik}+\\\\frac{\\\\dot{a}}{a}\\\\dot{{h}}_{ik}\\\\right)\\\\Bigg )\\\\delta ^{jk}.$ Thus, a derivative of a delta function in the curvature requires a step function (or worse) discontinuity in the gauge invariant metric variables.\",\n \"We are now in a position to analyze how discontinuities could arise.\",\n \"First, it is important to note that the equations ()-(REF ) giving the scalar, vector, and tensor parts of a tensor $X_{\\\\mu \\\\nu }$ involve solving elliptic and/or algebraic equations, with “source” given by components of $X_{\\\\mu \\\\nu }$ and their derivatives.\",\n \"It follows immediately that the scalar, vector, and tensor parts of $X_{\\\\mu \\\\nu }$ are smooth wherever $X_{\\\\mu \\\\nu }$ itself is smooth.\",\n \"In particular, the scalar, vector, and tensor parts of the particle stress-energy (REF ) are smooth away from the worldlines of the particles.\",\n \"Consider, now, the scalar perturbations.\",\n \"Eqs.\",\n \"(REF ) and (REF ) are elliptic in $\\\\Phi $ and $\\\\Psi $ , so these quantities—which fully characterize the scalar part of the metric perturbation—can be singular only where the source terms in these equations are singular.\",\n \"These source terms involve the scalar part of the particle source and the quantity $A$ .\",\n \"The scalar part of the particle source is smooth away from the particle world lines.\",\n \"The quantity $A$ satisfies the hyperbolic equation (REF ), which, in turn is sourced by the scalar parts of the particle stress-energy.\",\n \"We shall analyze the possible singular behavior of $A$ in the Appendix.\",\n \"We shall show there that, although $A$ can be discontinuous along the sound cone of the source event $q$ , there cannot be any discontinuities in $\\\\Phi $ or $\\\\Psi $ .\",\n \"Thus, no memory effect can occur in the scalar sector.\",\n \"Consider, now, the vector perturbations.\",\n \"The quantity $\\\\Xi _i$ satisfies the elliptic equation (REF ) and thus is smooth wherever the sources are smooth.\",\n \"However, the particle source term $\\\\xi _i^{(P)}$ is smooth away from the worldlines of the particles and the fluid source term $W_i$ satisfies the source free evolution equation (REF ) and is thus nonsingular everywhere.\",\n \"Thus, $\\\\Xi _i$ is smooth away from the particle worldlines and no memory effect can occur in the vector sector.\",\n \"In the next section, we calculate the memory effect occurring in the tensor sector.\"\n ],\n [\n \"The Retarded Gravitational Field and the Memory Effect\",\n \"The tensor perturbations are described by the quantity ${h}_{ij}$ , which satisfies (see (REF )) $-\\\\ddot{{h}}_{ij}-2\\\\frac{\\\\dot{a}}{a}\\\\dot{{h}}_{ij}+\\\\nabla ^2{h}_{ij} =-16\\\\pi {T}_{ij} \\\\, , $ where ${T}_{ij}$ is the tensor part of the particle stress energy.\",\n \"Thus, each component of ${h}_{ij}$ in the coordinates (REF ) satisfies a decoupled scalar wave equation and it suffices to analyze the behavior of solutions to the scalar wave equation.\",\n \"We are interested in the contribution to the retarded integral ${h}_{ij}(x)=16\\\\pi \\\\int \\\\sqrt{-g(x^{\\\\prime })}d^4x^{\\\\prime }G^{\\\\rm ret}(x,x^{\\\\prime }){T}_{ij}(x^{\\\\prime })$ arising from a small neighborhood of the source event $q$ (see section ), where $G^{\\\\rm ret}(x,x^{\\\\prime })$ denotes the retarded Green's function for the scalar wave equation $- \\\\ddot{\\\\phi } - 2\\\\frac{\\\\dot{a}}{a} \\\\dot{\\\\phi } + \\\\nabla ^2 \\\\phi = -16\\\\pi T \\\\, .$ Specifically, we seek to determine whether a discontinuity can arise in ${h}_{ij}$ and, if so, to determine its magnitude.\",\n \"Such discontinuities will give rise to derivative of delta function contributions to the curvature, which, in turn, will produce a memory effect.\",\n \"To proceed, we need to know the form of $G^{\\\\rm ret}(x,x^{\\\\prime })$ .\",\n \"Consider an equation of the general form $L[\\\\phi ]=g^{\\\\mu \\\\nu }\\\\partial _\\\\mu \\\\partial _\\\\nu \\\\phi +b^\\\\mu \\\\partial _\\\\mu \\\\phi +c\\\\phi =-16\\\\pi T \\\\, ,$ where $g_{\\\\mu \\\\nu }$ is a metric of Lorentz signature.\",\n \"It is well known [19], [20] that, in 4 spacetime dimensions, the retarded Green's function for this equation takes the form $G^{\\\\rm ret}(x,x^{\\\\prime })=\\\\left[U(x,x^{\\\\prime })\\\\delta (\\\\sigma )+V(x,x^{\\\\prime })\\\\Theta (-\\\\sigma )\\\\right] \\\\Theta (t-t^{\\\\prime }) \\\\, , $ where $\\\\sigma $ denotes the squared geodesic distance between $x$ and $x^{\\\\prime }$ in the metric $g_{\\\\mu \\\\nu }$ and $t$ is a global time function.\",\n \"The quantities $U$ (the Van Vleck-Morette determinant) and $V$ are smooth functions; we refer to $U(x,x^{\\\\prime })\\\\delta (\\\\sigma )$ as the “direct part” and $V(x,x^{\\\\prime })\\\\Theta (-\\\\sigma )$ as the “tail part” of $G^{\\\\rm ret}(x,x^{\\\\prime })$ .\",\n \"In general, the form (REF ) for $G^{\\\\rm ret}(x,x^{\\\\prime })$ will hold only locally in a convex normal neighborhood, but in the case of (REF ), the spacetime metric corresponding to (REF ) is flat, and this form of $G^{\\\\rm ret}(x,x^{\\\\prime })$ holds globally, with $\\\\sigma (x,x^{\\\\prime })=-(\\\\eta -\\\\eta ^{\\\\prime })^2+(x-x^{\\\\prime })^2+(y-y^{\\\\prime })^2+(z-z^{\\\\prime })^2 \\\\, .$ The Van Vleck-Morette $U$ is determined by integrating an ODE along a geodesic connecting $x$ and $x^{\\\\prime }$ [19], [20].\",\n \"The quantities appearing in this ODE depend on $g^{\\\\mu \\\\nu }$ and $b^\\\\mu $ but do not depend on $c$ .\",\n \"We could integrate this ODE directly, but we can greatly simplify the calculation of $U$ by working with the rescaled variable ${\\\\tilde{\\\\phi }} = a\\\\phi $ (where $\\\\phi $ denotes a component of ${h}_{ij}$ ), which satisfies the equation $- \\\\ddot{\\\\tilde{\\\\phi }} + \\\\nabla ^2 {\\\\tilde{\\\\phi }} + \\\\frac{\\\\ddot{a}}{a}{\\\\tilde{\\\\phi }} =-16\\\\pi a T \\\\, .$ This change of variables eliminates the term involving $b^\\\\mu $ in (REF ), so $\\\\tilde{U}$ is determined by exactly the same equation for the wave equation in flat spacetime.\",\n \"Thus we obtain the unique solution $\\\\tilde{U}(x,x^{\\\\prime }) = (4\\\\pi )^{-1}$ .\",\n \"However, the retarded Green's function for $\\\\phi $ is related to the retarded Green's function for $\\\\tilde{\\\\phi }$ by [21]-[23] $G^{\\\\rm ret}(x,x^{\\\\prime })=\\\\frac{a(\\\\eta ^{\\\\prime })}{a(\\\\eta )}\\\\tilde{G}^{\\\\rm ret}(x,x^{\\\\prime }) \\\\, .$ Thus, we obtain $U(x,x^{\\\\prime })=\\\\frac{1}{4\\\\pi }\\\\frac{a(\\\\eta ^{\\\\prime })}{a(\\\\eta )} \\\\, .$ This holds for any FLRW universe, i.e., we have not assumed any particular equation of state $\\\\bar{P}= \\\\bar{P}(\\\\bar{\\\\rho })$ (and, thus, we have not assumed any particular expansion law) in the background spacetime.\",\n \"By contrast, $V(x,x^{\\\\prime })$ will depend on the expansion history of the FLRW universe between $\\\\eta ^{\\\\prime }$ and $\\\\eta $ .\",\n \"Note, however, that by spatial Euclidean invariance, $V$ depends on $(x,x^{\\\\prime })$ only via $\\\\eta $ , $\\\\eta ^{\\\\prime }$ , and $|\\\\mathbf {x} - \\\\mathbf {x}^{\\\\prime }|$ .\",\n \"As previously stated, we are interested in the possible discontinuities in ${h}_{ij}$ resulting from the source behavior near $q$ , where we take $q$ to have coordinates $\\\\mathbf {x} = t =0$ .\",\n \"To analyze this, let us first introduce a new toy mathematical problem, wherein we consider retarded solutions to the equation $-\\\\ddot{H}_{ij}-2\\\\frac{\\\\dot{a}}{a}\\\\dot{H}_{ij}+\\\\nabla ^2H_{ij} =-16\\\\pi T^{(P)}_{ij} \\\\, .$ Eq.\",\n \"(REF ) differs from the equation of interest (REF ) in that we have not taken the tensor part of the particle source and, correspondingly, we do not require $H_{ij}$ to be transverse or traceless.\",\n \"By (REF ), $T^{(P)}_{ij}$ consists of a sum of terms, each one of which has the form $ \\\\delta ^{(3)}\\\\left(\\\\mathbf {x} - \\\\mathbf {z}(t)\\\\right) \\\\Theta (\\\\pm t)$ , where $\\\\mathbf {z}(t)$ describes a timelike or null geodesic.\",\n \"The contribution of the tail part, $V(x,x^{\\\\prime })\\\\Theta (-\\\\sigma )$ , of $G^{\\\\rm ret}(x,x^{\\\\prime })$ to the retarded integral is thus a sum of terms of the form $H^{\\\\rm tail}_{ij} (x) =\\\\int d^4x^{\\\\prime } f_{ij}(x^{\\\\prime }) V(x,x^{\\\\prime })\\\\Theta \\\\left(-\\\\sigma (x,x^{\\\\prime })\\\\right) \\\\delta ^{(3)}\\\\left(\\\\mathbf {x}^{\\\\prime } - \\\\mathbf {z}(t^{\\\\prime })\\\\right) \\\\Theta (\\\\pm t^{\\\\prime }) \\\\, ,$ where $f_{ij}$ is smooth.\",\n \"It is not difficult to see that $H^{\\\\rm tail}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ .\",\n \"On the other hand, if $\\\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic—i.e., if one of the ingoing or outgoing null particles is “aimed” directly at an observer at $x$ —then the singularities of $\\\\delta ^{(3)}\\\\left(\\\\mathbf {x}^{\\\\prime } - \\\\mathbf {z}(t^{\\\\prime })\\\\right)$ and $\\\\Theta \\\\left(-\\\\sigma (x,x^{\\\\prime })\\\\right)$ will coincide and $H^{\\\\rm tail}_{ij}$ will, in general, be “highly singular” at $x$ in the sense that, in general, it will be defined only distributionally in a neighborhood of $x$ .\",\n \"We exclude such special points from consideration.\",\n \"The case of main interest is one where $x$ lies near the future light cone of $q$ but does not lie on the special direction defined by $\\\\mathbf {z}(t)$ (if null).\",\n \"Then the $\\\\delta ^{(3)}\\\\left(\\\\mathbf {x}^{\\\\prime } - \\\\mathbf {z}(t^{\\\\prime })\\\\right)$ singularity in the integral will be transverse to the step function singularity of $\\\\Theta \\\\left(-\\\\sigma (x,x^{\\\\prime })\\\\right)$ as well as to that of $\\\\Theta (\\\\pm t)$ .\",\n \"One can then integrate over $\\\\mathbf {x}^{\\\\prime }$ , leaving one with an integral only over $t^{\\\\prime }$ .\",\n \"The integrand will be proportional to $V(x; \\\\mathbf {z}(t^{\\\\prime }), t^{\\\\prime }) \\\\Theta (u-t^{\\\\prime }) \\\\Theta (\\\\pm t^{\\\\prime })$ , where $u = t-|\\\\mathbf {x}|$ denotes the retarded time of $x$ .\",\n \"The integral over $t^{\\\\prime }$ thus yields a result of the form $F_{ij}(x) u \\\\Theta (u)$ , where $F_{ij}$ is smooth.\",\n \"Thus, we see that $H^{\\\\rm tail}_{ij}$ is continuous (although not continuously differentiable) at $x$ .\",\n \"The analysis of the contribution, $H^{\\\\rm dir}_{ij}$ , of the “direct part,” $U(x,x^{\\\\prime }) \\\\delta (\\\\sigma )$ , of $G^{\\\\rm ret}(x,x^{\\\\prime })$ to $H_{ij}$ is similar, with $U(x,x^{\\\\prime }) \\\\delta (\\\\sigma )$ replacing $V(x,x^{\\\\prime })\\\\Theta (-\\\\sigma )$ in (REF ).\",\n \"Again, $H^{\\\\rm dir}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ , and is highly singular if $\\\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic.\",\n \"If we exclude such special points, then the integral over $\\\\mathbf {x}^{\\\\prime }$ can again be done, and we are again left with an integral over $t^{\\\\prime }$ .\",\n \"However, now the integrand is proportional to $\\\\delta (u-t^{\\\\prime }) \\\\Theta (\\\\pm t^{\\\\prime })$ , where $u$ is the retarded time of $x$ .\",\n \"Consequently, $H^{\\\\rm dir}_{ij}$ has the form $\\\\tilde{F}_{ij}(x) \\\\Theta (u)$ for some smooth $\\\\tilde{F}_{ij}$ , and thus it has a discontinuity along the future light cone of $q$ .\",\n \"The above analysis is for the toy problem (REF ), where we did not take the tensor part, ${T}_{ij}$ , of the source $T^{(P)}_{ij}$ .\",\n \"It would be cumbersome to compute ${T}_{ij}$ and then perform a similar direct analysis of the behavior of the retarded Green's function integral involving ${T}_{ij}$ .\",\n \"Fortunately, we can bypass this by noting that the operation of “taking the tensor part\\\" commutes with the wave operator appearing in (REF ) and (REF ).\",\n \"It follows that the desired quantity, ${h}_{ij}$ , given by (REF ), is related to $H_{ij}$ by ${h}_{ij}=[H_{ij}]^T \\\\, ,$ where “$[X_{ij}]^T$ ” denotes the operation of taking the “tensor part” of $X_{ij}$ , as given by (REF ).\",\n \"Thus, our analysis reduces to extracting information about how the operation of taking the tensor part of a quantity affects the nature of its singularities.\",\n \"To analyze this, we note first that since “taking the tensor part” consists of algebraic operations involving differentiations and inversions of Laplacians (see (REF )-(REF )), the tensor part, ${X}_{ij}$ , of a distribution $X_{ij}$ must be smooth wherever $X_{ij}$ is smooth.\",\n \"It then follows that the singular behavior of the tensor part of $X_{ij}$ at $x$ is the same as that of the tensor part of $\\\\psi X_{ij}$ , where $\\\\psi $ is any smooth function of compact support with $\\\\psi = 1$ in a neighborhood of $x$ .\",\n \"(Proof: $X_{ij} - \\\\psi X_{ij}$ vanishes in a neighborhood of $x$ and hence is smooth there, so the tensor part of this difference is smooth in a neighborhood of $x$ .)\",\n \"However, the singular behavior of $\\\\psi X_{ij}$ is characterized by the decay (or lack thereof) of its Fourier transform at large $k_\\\\mu $ .\",\n \"The key point is that in the operation of “taking the tensor part,” there are exactly as many total “inverse derivatives” from Laplacian inversions in (REF ) as there are differentiations.\",\n \"It follows that the Fourier transform of the tensor part of $\\\\psi X_{ij}$ is related to the Fourier transform of $\\\\psi X_{ij}$ by a function that is everywhere bounded in $k_\\\\mu $ .\",\n \"In particular, the decay of the Fourier transform of the tensor part of $\\\\psi X_{ij}$ at large $k_\\\\mu $ cannot be slower than that of $\\\\psi X_{ij}$ .\",\n \"The above argument can be applied to the present case as follows to get the key conclusion that we need.\",\n \"It is easily seen from the explicit behavior of $H^{\\\\rm tail}_{ij}$ found above that the Fourier transform of $\\\\psi H^{\\\\rm tail}_{ij}$ lies in $L^1$ for any smooth function $\\\\psi $ of compact support.\",\n \"Therefore, the Fourier transform of the tensor part of $\\\\psi H^{\\\\rm tail}_{ij}$ —which differs from the Fourier transform of $\\\\psi H^{\\\\rm tail}_{ij}$ by a bounded function of $k_\\\\mu $ —also lies in $L^1$ .\",\n \"But that implies that the tensor part of $\\\\psi H^{\\\\rm tail}_{ij}$ is continuous for all $\\\\psi $ , which implies that the tensor part of $H^{\\\\rm tail}_{ij}$ is continuous.\",\n \"Thus, we have shown that the tail contribution to ${h}_{ij}$ is continuous and thus cannot contribute to the memory effect.\",\n \"The above conclusion is all that is needed to derive our results on the memory effect, because, as we have seen above, the direct contribution to the retarded solution is universal, and does not depend on the expansion history.\",\n \"Furthermore, Minkowski spacetime lies within the class of $k=0$ FLRW spacetimes to which our analysis applies.\",\n \"Thus, we can relate the memory effect in an arbitrary FLRW spacetime to that in Minkowski spacetime as follows.\",\n \"Consider a source event at $q$ in the FLRW spacetime that is observed at event $p$ .\",\n \"Let $\\\\eta _s$ and $\\\\eta _o$ denote the conformal times of the events $q$ and $p$ respectively.\",\n \"For convenience, rescale the coordinates, if necessary, so that $a(\\\\eta _s) = 1$ .\",\n \"This corresponds to choosing the comoving coordinates to corrspond to proper distances at $\\\\eta = \\\\eta _s$ .\",\n \"Now, identify the FLRW spacetime with Minkowski spacetime by identifying the coordinates (REF ) of the FLRW spacetime with global inertial coordinates of Minkowski spacetime.\",\n \"Place a source and observer at the events ${\\\\bar{q}}$ and ${\\\\bar{p}}$ of Minkowski spacetime that are identified in this manner with events $q$ and $p$ in the FLRW spacetime.\",\n \"Since $a(\\\\eta _s) =1$ , the Minkowski source will physically correspond to the FLRW source provided that the masses and 4-velocities of each of the particles agree (under this identification) at $q$ .\",\n \"It follows immediately from (REF ) that the direct part, ${h}_{ij}^{\\\\rm dir}$ , of ${h}_{ij}(x)$ near $p$ will be a factor of $1/a(\\\\eta _o)$ times the same function of $x^\\\\mu $ as it is in the Minkowski case, i.e., near $p$ , ${h}_{ij}^{\\\\rm dir} (x^\\\\mu ) = \\\\frac{1}{a(\\\\eta _o)} {\\\\bar{h}}_{ij}^{\\\\rm dir} (x^\\\\mu ) \\\\, .$ It then follows immediately from (REF ) that the direct parts of the linearized Riemann curvature tensor are similarly related, i.e., near $p$ $\\\\delta {R^{\\\\rm dir}_{i00}}^{j} (x^\\\\mu ) = \\\\frac{1}{a(\\\\eta _o)} \\\\delta {\\\\bar{R}}^{\\\\rm dir}_{i00}{}^j (x^\\\\mu ) \\\\, .$ Suppose, now, that we place a gravitational wave detector at $p$ , composed of two nearby particles initially at rest in the cosmic reference frame.\",\n \"By the geodesic deviation equation, the deviation vector, $D^i$ , describing the displacement of the particles will satisfy $v^b \\\\nabla _b (v^c\\\\nabla _c {D}^a) = R_{def}^{\\\\quad a}{D}^d v^e v^f \\\\, ,$ where $v^a$ is the unit tangent to the geodesic.\",\n \"Since $v^\\\\mu \\\\approx 1/a(\\\\eta _o) (\\\\partial /\\\\partial \\\\eta )^\\\\mu $ and the Hubble expansion is negligible over the relevant timescale, we can rewrite this equation as $\\\\frac{d^2}{d \\\\eta ^2} D^j = R_{i00}{}^j D^i \\\\, .$ Let $\\\\Delta D^i$ denote the coordinate components of the “memory displacement,” obtained by integrating (REF ) twice with respect to $\\\\eta $ .\",\n \"In view of (REF ) and the fact, proven above, that the “direct part” of the Riemann tensor contains the full memory effect, we see that $\\\\Delta D^i = \\\\frac{1}{a(\\\\eta _o)} \\\\overline{\\\\Delta D}^i \\\\, ,$ where $\\\\overline{\\\\Delta D}^i$ denotes the corresponding memory displacement in Minkowski spacetime, assuming that the initial displacement was $\\\\overline{D}^i = D^i$ .\",\n \"Thus, the relationship between $\\\\Delta D^i$ and $D^i$ in an arbitrary FLRW spacetime differs from the corresponding Minkowski result by a factor of $1/a(\\\\eta _o)$ .\",\n \"Thus, we have shown that if we identify the FLRW spacetime with Minkowski spacetime via the coordinates (REF ) in such a way that $a(\\\\eta _s) = 1$ , and we place the same physical source at $q$ and the same physical detector at $p$ in both spacetimes, then the memory effect in the FLRW spacetime will be a factor of $1/a(\\\\eta _o) = 1/(1 + z)$ smaller than the corresponding memory effect in Minkowski spacetime.\",\n \"Note that placing the source at the same proper distance at the time of emission corresponds to placing the source at the same angular diameter distance in both spacetimes.\",\n \"The above result compares the memory effect in FLRW and Minkowski spacetime when the source and detector are at the same proper distance at the source emission time, i.e., when they are at the same location with $a(\\\\eta _s) = 1$ .\",\n \"Since the memory effect in Minkowski spacetime falls off as $1/r$ , this result may be reformulated in numerous equivalent ways.\",\n \"In particular, we have If the source and detector are placed so that they are at the same proper distance at the time of detection (rather than emission), then the memory effect in the FLRW spacetime is identical to the corresponding memory effect in Minkowski spacetime.\",\n \"If the source and detector are placed so that the source is at the same luminosity distance in both cases, then the memory effect in the FLRW spacetime is larger by a factor of $(1+z)$ as compared with the corresponding memory effect in Minkowski spacetime.\",\n \"The above result is in agreement with the results of [9], [10], [11], and [12] in the cases where the results of those references apply.\",\n \"Acknowledgements We wish to thank Lydia Bieri and David Garfinkle for helpful discussions.\",\n \"This research was supported in part by NSF grants PHY 12-02718 and PHY 15-05124 to the University of Chicago.\"\n ],\n [\n \"Acoustic Shock Waves and Metric Continuity\",\n \"In this Appendix, we show that the the scalar sector does not contribute to the memory effect.\",\n \"The gauge-invariant density perturbation $A$ satisfies (REF ).\",\n \"Equation (REF ) is a hyperbolic wave equation of the general form (REF ), with the Lorentz metric $g_{\\\\mu \\\\nu }$ now being the “acoustic metric,” $ds^2= -d \\\\eta ^2 + \\\\frac{1}{c_s^2}[dx^2 + dy^2 + dz^2] \\\\, ,$ and with the source term $T$ proportional to the scalar particle fields $\\\\varphi ^{(P)}$ and $\\\\psi ^{(P)}$ .\",\n \"Thus, the retarded Green's function for (REF ) takes the general Hadamard form (REF ), with $\\\\sigma $ replaced by the squared geodesic distance, $\\\\sigma _s$ , in the acoustic metric (REF ), i.e., we have $G_s(x,x^{\\\\prime })=\\\\left[U_s(x,x^{\\\\prime })\\\\delta (\\\\sigma _s)+V_s(x,x^{\\\\prime })\\\\Theta (-\\\\sigma _s) \\\\right] \\\\Theta (\\\\eta -\\\\eta ^{\\\\prime }) \\\\, ,$ where $U_s$ and $V_s$ are again smooth functions in both $x$ and $x^{\\\\prime }$ Furthermore, it can be seen from (REF ) that both $\\\\varphi ^{(P)}$ and $\\\\psi ^{(P)}$ are obtained from $T^{(P)}_{\\\\mu \\\\nu }$ by algebraic operations (i.e., no differentiations or Laplace inversions).\",\n \"It follows immediately that the source term appearing in (REF ) takes the same form (namely, proportional to $\\\\delta ^{(3)}\\\\left(\\\\mathbf {x} - \\\\mathbf {z}(\\\\eta )\\\\right) \\\\Theta (\\\\pm \\\\eta )$ ), as considered in Section .\",\n \"We may therefore repeat the analysis of Section to draw the following conclusion: Suppose that all of the particles in $T^{(P)}_{\\\\mu \\\\nu }$ are moving with velocity smaller than the speed of soundIf any of the particles are moving with velocity greater than the speed of sound, there will be additional “Cherenkov radiation” singularities occurring at points $x$ where the past sound cone of $x$ intersects a particle world line orthogonally (in the sound metric).\",\n \"These additional singularities are not of interest for the memory effect.. Then $A$ is smooth except on the future sound cone of $q$ .\",\n \"Furthermore, on the sound cone, $A$ will, in general, be discontinuous, but it cannot have “worse” singular behavior.\",\n \"The metric perturbation variables $\\\\Phi $ and $\\\\Psi $ satisfy elliptic equations, with source terms given by $A$ and the scalar parts of the particle sources.\",\n \"It follows that $\\\\Phi $ and $\\\\Psi $ must be smooth everywhere apart from the worldlines of the particles and the points at which $A$ fails to be smooth, i.e., the future sound cone of $q$ .\",\n \"We are not interested in the singularities at the particle worldlines.\",\n \"However, on the future sound cone of $q$ , $\\\\nabla ^2 \\\\Phi $ and $\\\\nabla ^2 \\\\Psi $ are at worst discontinuous, so $\\\\Phi $ and $\\\\Psi $ themselves are at least $C^1$ .\",\n \"Thus, they cannot contribute a derivative of delta-function to the Riemann curvature (REF ), and thus do not contribute to any memory effect.\"\n ]\n]"},"id":{"kind":"string","value":"1606.04894"}}},{"rowIdx":994,"cells":{"text":{"kind":"list like","value":[["The stochastic heat equation, 2D Toda equations and dynamics for the\n multilayer process"],["Abstract We show that solutions of the stochastic heat equation driven by space-time white noise, although not smooth, meaningfully solve the two-dimensional Toda equations.","Then by extending our arguments we show the time evolution of the multilayer process introduced by O'Connell and Warren is conjugate to a flow induced by the stochastic heat equation.","In particular this establishes a Markov property conjectured by O'Connell and Warren.","It also defines, for the first time, the multilayer process started from a general initial condition."],["Introduction","In [23], O'Connell and Warren introduced the following: for each $n = 1,2,\\ldots $ , $t>0$ and $x$ , $y\\in \\mathbf {R}$ define $Z_n(t,x,y) = p_t(x-y)^n \\bigg (1 + \\sum _{k=1}^\\infty \\int _{\\Delta _k(t)} \\int _{\\mathbf {R}^k} R_k(\\mathbf {s},\\mathbf {y^\\prime }; t,x,y) \\;W^{\\otimes k}(\\mathrm {d}\\mathbf {s},\\mathrm {d}\\mathbf {y}^\\prime ) \\bigg ),$ where $\\Delta _k(t) = \\lbrace 0 < s_1 < s_2 < \\cdots < s_k < t\\rbrace $ .","$\\mathbf {s} = (s_1,\\ldots ,s_k)$ , $\\mathbf {y}^\\prime = (y_1^\\prime ,\\ldots ,y_k^\\prime )$ and $R_k(\\mathbf {s}, \\mathbf {y}^\\prime ; t,x,y)$ is the $k$ -point correlation function for a collection of $n$ non-intersecting Brownian bridges each of which starts at $x$ at time 0 and ends at $y$ at time $t$ .","$p_t(x-y)= (2\\pi t)^{-1/2} e^{-(x-y)^2/2t}$ is the transition density of Brownian motion.","The integral is a multiple stochastic integral with respect to space-time white noise.","It was shown in [23] by considering local times of non-intersecting Brownian bridges that the infinite sum in the definition is convergent in $L^2$ with respect to the white noise.","Observe that $u = Z_1$ is the solution to the (multiplicative) stochastic heat equation (SHE) with delta initial data: ${\\left\\lbrace \\begin{array}{ll}\\partial _t u(t,x,y) = \\Big ( \\frac{1}{2} \\Delta _y + \\dot{W}(t,y) \\Big ) u(t,x,y), \\quad t\\in (0,\\infty ), y\\in \\mathbf {R}, \\\\u(0,x,y) = \\delta (x-y), \\quad x\\in \\mathbf {R}.\\end{array}\\right.","}$ By a solution to the above we mean a random field $u$ which satisfies almost surely the mild form of the equation: $u(t,x,y) = p_t(x-y) + \\int _0^t \\int _\\mathbf {R}p_{t-s}(y-y^\\prime ) u(s,x,y^\\prime ) \\;W(\\mathrm {d}s,\\mathrm {d}y^\\prime ).$ Iterating equation (REF ) multiple times gives the chaos expansion (REF ) for $n=1$ .","One can express $Z_n(t,x,y)$ in a more suggestive notation: $Z_n(t,x,y) = p_t(x-y)^n \\mathbf {E}_{x,y;t}^X \\bigg [ {E}\\mathrm {xp} \\bigg ( \\sum _{i=1}^n \\int _0^t W(s,X_s^i) \\;\\mathrm {d}s \\bigg ) \\bigg ],$ where $(X_s^1,\\ldots ,X_s^n, 0\\le s\\le t)$ denotes the trajectories of the above mentioned collection of $n$ non-intersecting Brownian bridges and $\\mathbf {E}_{x,y;t}^X$ is the corresponding expectation.","${E}\\mathrm {xp}$ is the Wick exponential defined by ${E}\\mathrm {xp}(M_t) := \\exp \\big ( M_t - \\frac{1}{2}\\langle M,M\\rangle _t \\big )$ for a martingale $M$ .","The Feynman–Kac formula (REF ) is not rigorous as it is unclear how one would define the integral of the white noise along a Brownian path and moreover to exponentiate such an expression.","However, one can obtain an rigorous expression by replacing $W$ in (REF ) with a smoothed version of the space-time white noise.","Indeed, Bertini and Cancrini showed in [2] that such expression (in the case $n=1$ ) has a meaningful limit as one takes away the smoothing and that the limit solves the SHE.","Bymeans of the Feynman–Kac formula, one can interpret the solution to the stochastic heat equation as the partition function (up to a multiplication by the heat kernel) of the continuum directed random polymer, and then similarly, $Z_n$ is the partition function of a natural extension of the continuum polymer involving multiple Brownian paths.","The Cole–Hopf solution $h = \\log u$ to the KPZ equation with narrow wedge initial data corresponds via the Feynman–Kac formula to the free energy of the continuum directed random polymer.","With this interpretation $h$ can be regarded as the continuum analogue of the longest increasing subsequence of a random permutation, length of the first row of a random Young diagram, directed last passage percolation and free energy of a discrete/semi-discrete polymer in random media etc., see [3], [4], [5], [16], [17], [24], [18], [10] and the references therein.","In each of these discrete models, there is further structure provided either by multiple non-intersecting up-right paths on lattices, multi-layer growth dynamics or Young diagrams constructed from the RSK correspondence.","The work in the above mentioned references have shown that in some cases, utilisation of this additional structure have lead to derivations of exact formulae for the distribution of quantities of interest.","The above mentioned discrete models provide examples of what is called integrability or exact solvability.","The motivation for introducing the partition functions $Z_n$ , which are the continuum analogue of the structures mentioned above, is that they should provide insight to the integrable structure in the continuum setting.","There has been other recent work on multiple polymer paths and the multilayer process in the stochastic heat equation setting.","In [11] and [12], in a manifestation of the exact solvability, the Bethe Ansatz is used to make exact and asymptotic distributional statements.","In [8], the KPZ line ensemble, which is expected to be given by the logarithm of the multilayer process we are considering here is shown to have a remarkable Gibbs property which generalises that of the Airy line ensemble, [7].","Very recent results in [9] show the multilayer process arises as a scaling limit of discrete polymer models.","It was shown in [23] by considering a smooth space-time potential that $(Z_n, n\\ge 1)$ should satisfy a system of coupled SPDEs, however unfortunately it is not immediately obvious that such SPDEs make sense in the white noise setting.","Nevertheless, it does suggests that the process should have a Markovian evolution.","Indeed, we have the following theorem which is the main result of this paper.","Theorem 1.1 For each $n\\ge 1$ and $x\\in \\mathbf {R}$ , $\\big (Z_1(t,x,\\cdot ),\\ldots ,Z_n(t,x,\\cdot ) ; t\\ge 0\\big ),$ is a Markov process with respect to the filtration generated by the space-time white noise taking values in the space ${\\mathfrak {L}}_n:= C(\\mathbf {R})\\times \\cdots \\times C(\\mathbf {R})$ , where $C(\\mathbf {R}) := C(\\mathbf {R},\\mathbf {R}_+)$ is the space of continuous functions from $\\mathbf {R}$ to $\\mathbf {R}_+ = (0,\\infty )$ .","In the case of the stochastic heat equation ($n=1$ ), the Markov property can be seen from the Feynman–Kac formula since $u(s+t,x,y)$ can be written in the form $\\mathbf {E}_{x,y;t}^X\\Big [ {E}\\mathrm {xp}\\big ( F_X(0,s)\\big ) {E}\\mathrm {xp}\\big ( F_X(s,t)\\big )\\Big ],$ where $F_X(s,t)$ is a function of the Brownian bridge $X$ starting from $(s,X_s)$ and ending at $(t,y)$ and the white noise over the time interval $[s,t]$ , which is independent of the white noise over $[0,s]$ and the bridge from $(0,x)$ to $(s,X_s)$ .","From this one obtains the flow property of $u$ : $u(t,x,y) = \\int _\\mathbf {R}u(s,x,z) u(s,t,z,y) \\;\\mathrm {d}z,$ where $u(s,s+\\cdot ,z,\\cdot )$ is the solution to the SHE driven by the shifted white noise $\\dot{W}(s+\\cdot ,\\cdot )$ .","However, this argument does not apply for $n\\ge 2$ since the definition of $Z_n$ involves non-intersecting Brownian bridges with common starting and ending points but at any intermediate time each of the bridges are at distinct locations.","Nevertheless, Theorem REF is true and we shall prove it by considering a natural extension $M_n$ of $Z_n$ which corresponds to allowing the multiple polymer paths to have differing starting and ending points from one another.","This extended process can easily be seen to have the Markov property, and the key to understanding Theorem REF is that the extension $M_n$ turns out to be able to be recovered from the values of $(Z_1, Z_2, \\ldots , Z_n)$ .","In fact we establish in Theorem REF what amounts to a conjugacy between the random dynamical systems that describe the evolution of $(M_1, M_2, \\ldots , M_n)$ and $(Z_1, Z_2, \\ldots , Z_n)$ as shown in Figure REF .","This develops an idea that was suggested in [23], but only rigorously established when $n=2$ .","Figure: The evolution of the multilayer process is described by a conjugacyLet $W_n = \\lbrace x\\in \\mathbf {R}^n : x_1\\ge \\cdots \\ge x_n\\rbrace $ be the Weyl chamber in $\\mathbf {R}^n$ , then define for $n\\ge 1$ , $(t,\\mathbf {x},\\mathbf {y})\\in (0,\\infty )\\times W_n\\times W_n$ , $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{p_n^*(t,\\mathbf {x},\\mathbf {y})}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})} \\bigg (1 + \\sum _{k=1}^\\infty \\int _{\\Delta _{k}(t)} \\int _{\\mathbf {R}^k} R_k(\\mathbf {s},\\mathbf {y}^\\prime ; t,\\mathbf {x},\\mathbf {y}) \\;W^{\\otimes k}(\\mathrm {d}\\mathbf {s},\\mathrm {d}\\mathbf {y}^\\prime ) \\bigg ),$ where $R_k$ is the $k$ -point correlation function of a collection of $n$ non-intersection Brownian bridges which starts at $\\mathbf {x}$ at time 0 and ends at $\\mathbf {y}$ at time $t$ .","$p_n^*(t,\\mathbf {x},\\mathbf {y}) = \\det [p_t(x_i-x_j)]_{i,j=1}^n$ is by the Karlin–McGregor formula [21] the transition density of Brownian motion killed at the boundary of $W_n$ and $\\Delta _n(\\mathbf {x}) = \\prod _{1\\le i 0 \\text{ for all } t>0 \\text{ and } \\mathbf {x},\\mathbf {y}\\in {\\mathbf {R}}^n] = 1$ .","Moreover, when all the coordinates of $\\mathbf {x}$ and $\\mathbf {y}$ are equal, $M_n$ agrees almost surely up to a strictly positive multiplicative constant with $Z_n$ , that is $M_n(t,a\\mathbf {1}_n,b\\mathbf {1}_n) = c_{n,t} Z_n(t,a,b), \\quad c_{n,t} = c_n t^{-n(n-1)/2}, c_n = \\left(\\prod _{i=1}^{n-1} i!\\right)^{-1},$ where $\\mathbf {1}_n= (1,\\ldots ,1) \\in {\\mathbf {R}}^n$ .","This implies the almost sure continuity and everywhere strict positivity of $Z_n$ for all $n$ .","It was shown in [14] that the difference of two solutions to the KPZ equation (with the same white noise) starting from two different Hölder continuous initial data is in $C^{\\frac{3}{2}-\\varepsilon }$ for every $\\varepsilon >0$ .","Since $M_2(t,\\mathbf {x},\\cdot )$ and $u(t,x,\\cdot )$ are Hölder continuous of order up to 1/2, Theorem REF and equation (REF ) imply that the ratio of $u(t,x_1,\\cdot )$ and $u(t,x_2,\\cdot )$ is in $C^{\\frac{3}{2}-\\varepsilon }$ for every $\\varepsilon >0$ and hence generalises the result of [14] to delta initial data.","In fact Theorem REF implies that there is much more regularity present when considering multiple solutions to the stochastic heat equation driven by the same white noise than might be initially expected.","We have the following determinantal expression for $M_n$ (see [23], and also proved in the Appendix to this paper) $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{\\det [u(t,x_i,y_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {x})}\\Delta _n{(\\mathbf {y})}}, \\quad t>0, \\mathbf {x},\\mathbf {y}\\in {\\mathbf {R}}^n_{\\ne },$ where the entries in the determinant are solutions to (REF ) each driven by the same white noise.","${\\mathbf {R}}^n_{\\ne }$ denotes the subset of points in ${\\mathbf {R}}^n$ whose coordinates are all distinct.","In view of this representation Theorem REF implies we can meaningfully assign values to Wronskians whose entries are formally given as derivatives of solutions of the stochastic heat equation.","Suppose that we replace the space-time white noise with a smooth space-time potential in the definition of $Z_n$ then it can be shown that $Z_n$ is given by the bi-directional Wronskian $Z_n(t,x,y) = c_n t^{n(n-1)/2} \\det [\\partial _x^{i-1} \\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n,$ where $u(t,x,y)$ is the solution to (REF ) driven by the smooth potential.","Now let $\\tau _n(x,y) = \\det [\\partial _x^{i-1} \\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n$ then $\\tau _n$ satisfies the two-dimensional Toda equation (2DTE) $\\partial _{xy} q_n = e^{q_{n+1}-q_n} - e^{q_n-q_{n-1}}, \\quad n\\ge 1,$ where $q_n = \\log (n/{n-1})$ or equivalently, $\\partial _{xy} \\log n = \\frac{{n-1}{n+1}}{n^2},$ with the convention that $0 \\equiv 1$ .","Evaluating the derivative and rearranging we obtain $n \\partial _{xy} n - (\\partial _xn) (\\partial _yn) = {n-1}{n+1}.$ We introduce the following notation.","For an $(n+1)\\times (n+1)$ determinant $D$ , let $D\\begin{bmatrix} i \\\\ j\\end{bmatrix}$ be the $n\\times n$ determinant obtained from $D$ by removing the $i$ th row and the $j$ th column and similarly let $D\\begin{bmatrix} i & j \\\\ k & l\\end{bmatrix}$ be the $(n-1)\\times (n-1)$ determinant obtained from $D$ by removing the $i$ th and $j$ th rows and the $k$ th and $l$ th columns.","Then, by properties of Wronskians (REF ) can be written as ${n+1}\\begin{bmatrix} n+1 \\\\ n+1\\end{bmatrix} {n+1}\\begin{bmatrix} n \\\\ n\\end{bmatrix} - {n+1} \\begin{bmatrix} n \\\\ n+1\\end{bmatrix} {n+1} \\begin{bmatrix} n+1 \\\\ n\\end{bmatrix} = {n+1} \\begin{bmatrix} n & n+1 \\\\ n & n+1\\end{bmatrix} {n+1},$ which is nothing but the Jacobi identity for determinants [15].","In Section we study Wronskians defined as the continuous extensions of determinantal expressions such as (REF ) where $u$ is not necessarily differentiable.","With the help of the Jacobi idenitity we show that such generalised Wronksians satisfy the 2D Toda equations in an integrated form.","Then in Section , using a key lemma from the preceeding section, we construct the conjugacy that describes the evolution of the multilayer process.","A fuller description of this conjugacy is the content of Section which follows now."],["Acknowledgements","The research of C.H.L.","was supported by EPSRC grant number EP/H023364/1 through the MASDOC DTC.","This research of J.W.","was supported in part by the National Science Foundation under Grant No.","NSF PHY11-25915."],["Description of the dynamics of the multilayer process","The multilayer process is, for a fixed $x\\in {\\mathbf {R}}$ , and $n\\ge 1$ the finite sequence of “lines” $\\big (Z_1(t,x,\\cdot ),\\ldots ,Z_n(t,x,\\cdot ) )$ , which we consider to be evolving in the time variable $t>0$ .","Actually it seems more natural to work with a slightly different normalisation than that originally used by [23], and consider instead the process ${\\tau }_{t}(x,\\cdot )=\\big (\\tau _1(t,x,\\cdot ),\\ldots , \\tau _n(t,x,\\cdot ) )$ , where $\\tau _n(t,x,y)=c_{n}^{-1} t ^{-n(n-1)/2} Z_n(t,x,y)= c_n^{-2} M_n( t,x{\\mathbf {1}}_n, y{\\mathbf {1}}_n).$ As discussed in the Introduction, we describe the evolution of the multilayer process with the aid of the extended process $M_n$ .","It was shown in [22] that $M_n$ , defined by the chaos expansion (REF ), satisfies an evolution equation which can be regarded as a type of multi-dimensional SHE.","$M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{p_n^*(t,\\mathbf {x},\\mathbf {y})}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})} + \\frac{1}{(n-1)!}","\\int _0^t \\int _{\\mathbf {R}^n} Q_{t-s}(\\mathbf {y},\\mathbf {y}^\\prime ) M_n(s,\\mathbf {x},\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}_*^\\prime \\;W(\\mathrm {d}s,\\mathrm {d}y_1^\\prime ),$ almost surely for all $(t,x,y)\\in (0,\\infty )\\times \\mathbf {R}^n\\times \\mathbf {R}^n$ where $\\mathrm {d}\\mathbf {y}_*^\\prime = \\mathrm {d}y_2^\\prime \\ldots \\mathrm {d}y_n^\\prime $ .","The function $Q_t(\\mathbf {x},\\mathbf {y}) = \\frac{\\Delta _n(\\mathbf {y})}{\\Delta _n(\\mathbf {x})} p_n^*(t,\\mathbf {x},\\mathbf {y})$ is the transition density of Dyson Brownian motion [13] starting from $\\mathbf {x}\\in W_n$ and ending at $\\mathbf {y}\\in W_n$ .","Note that in (REF ) we have extended $Q_t$ by symmetry to a function on $\\mathbf {R}^n\\times \\mathbf {R}^n$ so the integral over $\\mathbf {R}^n$ is defined.","We can consider this same evolution for initial data $g$ , which is assumed to be a symmetric function on ${\\mathbf {R}}^n$ , $M^g_n(t,\\mathbf {y}) =\\frac{1}{n!}","\\int _{{\\mathbf {R}}^n} g(\\mathbf {y}^\\prime )Q_{t}(\\mathbf {y},\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}^\\prime + \\\\\\frac{1}{(n-1)!}","\\int _0^t \\int _{\\mathbf {R}^n} Q_{t-s}(\\mathbf {y},\\mathbf {y}^\\prime ) M^g_n(s,\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}_*^\\prime \\;W(\\mathrm {d}s,\\mathrm {d}y_1^\\prime ),$ For bounded $g$ , existence and uniqueness for this equation were established in [22].","If $g$ takes the form $g(\\mathbf {x})= \\frac{\\det [f_i(x_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {x})}}, \\quad \\mathbf {x}\\in {\\mathbf {R}}^n_{\\ne },$ then we have the representation $M_n^g(t,\\mathbf {y})=\\frac{\\det [u^{f_i}(t,y_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {y})}}, \\quad t>0, \\mathbf {y}\\in {\\mathbf {R}}^n_{\\ne },$ where $u^{f_i}$ denotes the solution to the solution to the SHE with initial condition at time 0 given by $f_i$ .","This is a consequence of equation (REF ).","From now on we use the notation $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n (\\mathbf {x})= \\det [f_i(x_j)]_{i,j=1}^n$ We let ${\\mathfrak {F}}_n$ be the set of sequences $({\\mathfrak {f}}_1, {\\mathfrak {f}}_2, \\ldots , {\\mathfrak {f}}_n)$ with each ${\\mathfrak {f}}_k:{\\mathbf {R}}^k \\rightarrow {\\mathbf {R}}$ being a continuous, strictly positive symmetric function, and such that there exists a sequence of continuous functions $f_1, f_2, \\ldots , f_k, \\ldots $ each defined on ${\\mathbf {R}}$ so that, for every $k$ , ${\\mathfrak {f}_k}(\\mathbf {x})= \\frac{f_1\\wedge f_2 \\wedge \\ldots \\wedge f_k(\\mathbf {x})}{\\Delta _k(\\mathbf {x})}$ for $\\mathbf {x}\\in {\\mathbf {R}}^k$ with all coordinates distinct.","In view of equation (REF ), for $\\mathfrak {f} \\in {\\mathfrak {F}}_n$ with bounded components, we can define an ${\\mathfrak {F}}_n$ -valued process $(\\mathfrak {f}_t)_{t \\ge 0}$ with initial value $\\mathfrak {f}$ by $\\mathfrak {f}_t= \\bigl ( M_1^{{\\mathfrak {f}}_1}(t,\\cdot ), M_2^{{\\mathfrak {f}}_2}(t,\\cdot ), \\ldots , M_n^{{\\mathfrak {f}}_n}(t,\\cdot ) \\bigr )$ We will denote, for any $t>0$ , the ${\\mathfrak {F}}_n$ -valued random variable $\\mathfrak {f}_t$ by ${\\mathcal {M}}_t \\mathfrak {f}$ .","For $00}=({\\tau }(t,x,\\cdot ))_{t>0}$ is an ${\\mathfrak {L}}_n$ -valued process whose evolution is given by ${\\tau }_{t}={\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1} {\\tau }_{s} \\quad \\text{ for $00$ and $k\\ge 1$ , $M_k(s, x{\\mathbf {1}}_k,\\mathbf {y})$ has uniformly bounded $p$ th moments as $y\\in {\\mathbf {R}}^k$ varies, and consequently we can solve the evolution equation for $M_k$ with shifted white noise and initial data $\\bigl (M_k(s, x{\\mathbf {1}}_k,\\mathbf {y}); \\mathbf {y}\\in {\\mathbf {R}}\\bigr )$ .","Thus, as at equation as a (REF ), it is meaningful to apply ${\\mathcal {M}}_{s,t}$ to $(M_k( s, x{\\mathbf {1}}_k, \\cdot ))_{1\\le k\\le n}$ , giving $(M_k( t, x{\\mathbf {1}}_k, \\cdot ))_{1\\le k\\le n}$ .","It now follows from (REF ), with the time $t$ replaced by the time $s$ , that ${\\tau }_t(x, \\cdot )$ satisfies ${\\tau }_{t}={\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1} {\\tau }_{s}.$ Corollary 6.2 of [23], states that, for given $\\mathbf {x}, \\mathbf {y}\\in {\\mathbf {R}}^n$ and $00$ and $x$ , $y\\\\in \\\\mathbf {R}$ define $Z_n(t,x,y) = p_t(x-y)^n \\\\bigg (1 + \\\\sum _{k=1}^\\\\infty \\\\int _{\\\\Delta _k(t)} \\\\int _{\\\\mathbf {R}^k} R_k(\\\\mathbf {s},\\\\mathbf {y^\\\\prime }; t,x,y) \\\\;W^{\\\\otimes k}(\\\\mathrm {d}\\\\mathbf {s},\\\\mathrm {d}\\\\mathbf {y}^\\\\prime ) \\\\bigg ),$ where $\\\\Delta _k(t) = \\\\lbrace 0 < s_1 < s_2 < \\\\cdots < s_k < t\\\\rbrace $ .\",\n \"$\\\\mathbf {s} = (s_1,\\\\ldots ,s_k)$ , $\\\\mathbf {y}^\\\\prime = (y_1^\\\\prime ,\\\\ldots ,y_k^\\\\prime )$ and $R_k(\\\\mathbf {s}, \\\\mathbf {y}^\\\\prime ; t,x,y)$ is the $k$ -point correlation function for a collection of $n$ non-intersecting Brownian bridges each of which starts at $x$ at time 0 and ends at $y$ at time $t$ .\",\n \"$p_t(x-y)= (2\\\\pi t)^{-1/2} e^{-(x-y)^2/2t}$ is the transition density of Brownian motion.\",\n \"The integral is a multiple stochastic integral with respect to space-time white noise.\",\n \"It was shown in [23] by considering local times of non-intersecting Brownian bridges that the infinite sum in the definition is convergent in $L^2$ with respect to the white noise.\",\n \"Observe that $u = Z_1$ is the solution to the (multiplicative) stochastic heat equation (SHE) with delta initial data: ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\partial _t u(t,x,y) = \\\\Big ( \\\\frac{1}{2} \\\\Delta _y + \\\\dot{W}(t,y) \\\\Big ) u(t,x,y), \\\\quad t\\\\in (0,\\\\infty ), y\\\\in \\\\mathbf {R}, \\\\\\\\u(0,x,y) = \\\\delta (x-y), \\\\quad x\\\\in \\\\mathbf {R}.\\\\end{array}\\\\right.\",\n \"}$ By a solution to the above we mean a random field $u$ which satisfies almost surely the mild form of the equation: $u(t,x,y) = p_t(x-y) + \\\\int _0^t \\\\int _\\\\mathbf {R}p_{t-s}(y-y^\\\\prime ) u(s,x,y^\\\\prime ) \\\\;W(\\\\mathrm {d}s,\\\\mathrm {d}y^\\\\prime ).$ Iterating equation (REF ) multiple times gives the chaos expansion (REF ) for $n=1$ .\",\n \"One can express $Z_n(t,x,y)$ in a more suggestive notation: $Z_n(t,x,y) = p_t(x-y)^n \\\\mathbf {E}_{x,y;t}^X \\\\bigg [ {E}\\\\mathrm {xp} \\\\bigg ( \\\\sum _{i=1}^n \\\\int _0^t W(s,X_s^i) \\\\;\\\\mathrm {d}s \\\\bigg ) \\\\bigg ],$ where $(X_s^1,\\\\ldots ,X_s^n, 0\\\\le s\\\\le t)$ denotes the trajectories of the above mentioned collection of $n$ non-intersecting Brownian bridges and $\\\\mathbf {E}_{x,y;t}^X$ is the corresponding expectation.\",\n \"${E}\\\\mathrm {xp}$ is the Wick exponential defined by ${E}\\\\mathrm {xp}(M_t) := \\\\exp \\\\big ( M_t - \\\\frac{1}{2}\\\\langle M,M\\\\rangle _t \\\\big )$ for a martingale $M$ .\",\n \"The Feynman–Kac formula (REF ) is not rigorous as it is unclear how one would define the integral of the white noise along a Brownian path and moreover to exponentiate such an expression.\",\n \"However, one can obtain an rigorous expression by replacing $W$ in (REF ) with a smoothed version of the space-time white noise.\",\n \"Indeed, Bertini and Cancrini showed in [2] that such expression (in the case $n=1$ ) has a meaningful limit as one takes away the smoothing and that the limit solves the SHE.\",\n \"Bymeans of the Feynman–Kac formula, one can interpret the solution to the stochastic heat equation as the partition function (up to a multiplication by the heat kernel) of the continuum directed random polymer, and then similarly, $Z_n$ is the partition function of a natural extension of the continuum polymer involving multiple Brownian paths.\",\n \"The Cole–Hopf solution $h = \\\\log u$ to the KPZ equation with narrow wedge initial data corresponds via the Feynman–Kac formula to the free energy of the continuum directed random polymer.\",\n \"With this interpretation $h$ can be regarded as the continuum analogue of the longest increasing subsequence of a random permutation, length of the first row of a random Young diagram, directed last passage percolation and free energy of a discrete/semi-discrete polymer in random media etc., see [3], [4], [5], [16], [17], [24], [18], [10] and the references therein.\",\n \"In each of these discrete models, there is further structure provided either by multiple non-intersecting up-right paths on lattices, multi-layer growth dynamics or Young diagrams constructed from the RSK correspondence.\",\n \"The work in the above mentioned references have shown that in some cases, utilisation of this additional structure have lead to derivations of exact formulae for the distribution of quantities of interest.\",\n \"The above mentioned discrete models provide examples of what is called integrability or exact solvability.\",\n \"The motivation for introducing the partition functions $Z_n$ , which are the continuum analogue of the structures mentioned above, is that they should provide insight to the integrable structure in the continuum setting.\",\n \"There has been other recent work on multiple polymer paths and the multilayer process in the stochastic heat equation setting.\",\n \"In [11] and [12], in a manifestation of the exact solvability, the Bethe Ansatz is used to make exact and asymptotic distributional statements.\",\n \"In [8], the KPZ line ensemble, which is expected to be given by the logarithm of the multilayer process we are considering here is shown to have a remarkable Gibbs property which generalises that of the Airy line ensemble, [7].\",\n \"Very recent results in [9] show the multilayer process arises as a scaling limit of discrete polymer models.\",\n \"It was shown in [23] by considering a smooth space-time potential that $(Z_n, n\\\\ge 1)$ should satisfy a system of coupled SPDEs, however unfortunately it is not immediately obvious that such SPDEs make sense in the white noise setting.\",\n \"Nevertheless, it does suggests that the process should have a Markovian evolution.\",\n \"Indeed, we have the following theorem which is the main result of this paper.\",\n \"Theorem 1.1 For each $n\\\\ge 1$ and $x\\\\in \\\\mathbf {R}$ , $\\\\big (Z_1(t,x,\\\\cdot ),\\\\ldots ,Z_n(t,x,\\\\cdot ) ; t\\\\ge 0\\\\big ),$ is a Markov process with respect to the filtration generated by the space-time white noise taking values in the space ${\\\\mathfrak {L}}_n:= C(\\\\mathbf {R})\\\\times \\\\cdots \\\\times C(\\\\mathbf {R})$ , where $C(\\\\mathbf {R}) := C(\\\\mathbf {R},\\\\mathbf {R}_+)$ is the space of continuous functions from $\\\\mathbf {R}$ to $\\\\mathbf {R}_+ = (0,\\\\infty )$ .\",\n \"In the case of the stochastic heat equation ($n=1$ ), the Markov property can be seen from the Feynman–Kac formula since $u(s+t,x,y)$ can be written in the form $\\\\mathbf {E}_{x,y;t}^X\\\\Big [ {E}\\\\mathrm {xp}\\\\big ( F_X(0,s)\\\\big ) {E}\\\\mathrm {xp}\\\\big ( F_X(s,t)\\\\big )\\\\Big ],$ where $F_X(s,t)$ is a function of the Brownian bridge $X$ starting from $(s,X_s)$ and ending at $(t,y)$ and the white noise over the time interval $[s,t]$ , which is independent of the white noise over $[0,s]$ and the bridge from $(0,x)$ to $(s,X_s)$ .\",\n \"From this one obtains the flow property of $u$ : $u(t,x,y) = \\\\int _\\\\mathbf {R}u(s,x,z) u(s,t,z,y) \\\\;\\\\mathrm {d}z,$ where $u(s,s+\\\\cdot ,z,\\\\cdot )$ is the solution to the SHE driven by the shifted white noise $\\\\dot{W}(s+\\\\cdot ,\\\\cdot )$ .\",\n \"However, this argument does not apply for $n\\\\ge 2$ since the definition of $Z_n$ involves non-intersecting Brownian bridges with common starting and ending points but at any intermediate time each of the bridges are at distinct locations.\",\n \"Nevertheless, Theorem REF is true and we shall prove it by considering a natural extension $M_n$ of $Z_n$ which corresponds to allowing the multiple polymer paths to have differing starting and ending points from one another.\",\n \"This extended process can easily be seen to have the Markov property, and the key to understanding Theorem REF is that the extension $M_n$ turns out to be able to be recovered from the values of $(Z_1, Z_2, \\\\ldots , Z_n)$ .\",\n \"In fact we establish in Theorem REF what amounts to a conjugacy between the random dynamical systems that describe the evolution of $(M_1, M_2, \\\\ldots , M_n)$ and $(Z_1, Z_2, \\\\ldots , Z_n)$ as shown in Figure REF .\",\n \"This develops an idea that was suggested in [23], but only rigorously established when $n=2$ .\",\n \"Figure: The evolution of the multilayer process is described by a conjugacyLet $W_n = \\\\lbrace x\\\\in \\\\mathbf {R}^n : x_1\\\\ge \\\\cdots \\\\ge x_n\\\\rbrace $ be the Weyl chamber in $\\\\mathbf {R}^n$ , then define for $n\\\\ge 1$ , $(t,\\\\mathbf {x},\\\\mathbf {y})\\\\in (0,\\\\infty )\\\\times W_n\\\\times W_n$ , $M_n(t,\\\\mathbf {x},\\\\mathbf {y}) = \\\\frac{p_n^*(t,\\\\mathbf {x},\\\\mathbf {y})}{\\\\Delta _n(\\\\mathbf {x})\\\\Delta _n(\\\\mathbf {y})} \\\\bigg (1 + \\\\sum _{k=1}^\\\\infty \\\\int _{\\\\Delta _{k}(t)} \\\\int _{\\\\mathbf {R}^k} R_k(\\\\mathbf {s},\\\\mathbf {y}^\\\\prime ; t,\\\\mathbf {x},\\\\mathbf {y}) \\\\;W^{\\\\otimes k}(\\\\mathrm {d}\\\\mathbf {s},\\\\mathrm {d}\\\\mathbf {y}^\\\\prime ) \\\\bigg ),$ where $R_k$ is the $k$ -point correlation function of a collection of $n$ non-intersection Brownian bridges which starts at $\\\\mathbf {x}$ at time 0 and ends at $\\\\mathbf {y}$ at time $t$ .\",\n \"$p_n^*(t,\\\\mathbf {x},\\\\mathbf {y}) = \\\\det [p_t(x_i-x_j)]_{i,j=1}^n$ is by the Karlin–McGregor formula [21] the transition density of Brownian motion killed at the boundary of $W_n$ and $\\\\Delta _n(\\\\mathbf {x}) = \\\\prod _{1\\\\le i 0 \\\\text{ for all } t>0 \\\\text{ and } \\\\mathbf {x},\\\\mathbf {y}\\\\in {\\\\mathbf {R}}^n] = 1$ .\",\n \"Moreover, when all the coordinates of $\\\\mathbf {x}$ and $\\\\mathbf {y}$ are equal, $M_n$ agrees almost surely up to a strictly positive multiplicative constant with $Z_n$ , that is $M_n(t,a\\\\mathbf {1}_n,b\\\\mathbf {1}_n) = c_{n,t} Z_n(t,a,b), \\\\quad c_{n,t} = c_n t^{-n(n-1)/2}, c_n = \\\\left(\\\\prod _{i=1}^{n-1} i!\\\\right)^{-1},$ where $\\\\mathbf {1}_n= (1,\\\\ldots ,1) \\\\in {\\\\mathbf {R}}^n$ .\",\n \"This implies the almost sure continuity and everywhere strict positivity of $Z_n$ for all $n$ .\",\n \"It was shown in [14] that the difference of two solutions to the KPZ equation (with the same white noise) starting from two different Hölder continuous initial data is in $C^{\\\\frac{3}{2}-\\\\varepsilon }$ for every $\\\\varepsilon >0$ .\",\n \"Since $M_2(t,\\\\mathbf {x},\\\\cdot )$ and $u(t,x,\\\\cdot )$ are Hölder continuous of order up to 1/2, Theorem REF and equation (REF ) imply that the ratio of $u(t,x_1,\\\\cdot )$ and $u(t,x_2,\\\\cdot )$ is in $C^{\\\\frac{3}{2}-\\\\varepsilon }$ for every $\\\\varepsilon >0$ and hence generalises the result of [14] to delta initial data.\",\n \"In fact Theorem REF implies that there is much more regularity present when considering multiple solutions to the stochastic heat equation driven by the same white noise than might be initially expected.\",\n \"We have the following determinantal expression for $M_n$ (see [23], and also proved in the Appendix to this paper) $M_n(t,\\\\mathbf {x},\\\\mathbf {y}) = \\\\frac{\\\\det [u(t,x_i,y_j)]_{i,j=1}^n}{\\\\Delta _n{(\\\\mathbf {x})}\\\\Delta _n{(\\\\mathbf {y})}}, \\\\quad t>0, \\\\mathbf {x},\\\\mathbf {y}\\\\in {\\\\mathbf {R}}^n_{\\\\ne },$ where the entries in the determinant are solutions to (REF ) each driven by the same white noise.\",\n \"${\\\\mathbf {R}}^n_{\\\\ne }$ denotes the subset of points in ${\\\\mathbf {R}}^n$ whose coordinates are all distinct.\",\n \"In view of this representation Theorem REF implies we can meaningfully assign values to Wronskians whose entries are formally given as derivatives of solutions of the stochastic heat equation.\",\n \"Suppose that we replace the space-time white noise with a smooth space-time potential in the definition of $Z_n$ then it can be shown that $Z_n$ is given by the bi-directional Wronskian $Z_n(t,x,y) = c_n t^{n(n-1)/2} \\\\det [\\\\partial _x^{i-1} \\\\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n,$ where $u(t,x,y)$ is the solution to (REF ) driven by the smooth potential.\",\n \"Now let $\\\\tau _n(x,y) = \\\\det [\\\\partial _x^{i-1} \\\\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n$ then $\\\\tau _n$ satisfies the two-dimensional Toda equation (2DTE) $\\\\partial _{xy} q_n = e^{q_{n+1}-q_n} - e^{q_n-q_{n-1}}, \\\\quad n\\\\ge 1,$ where $q_n = \\\\log (n/{n-1})$ or equivalently, $\\\\partial _{xy} \\\\log n = \\\\frac{{n-1}{n+1}}{n^2},$ with the convention that $0 \\\\equiv 1$ .\",\n \"Evaluating the derivative and rearranging we obtain $n \\\\partial _{xy} n - (\\\\partial _xn) (\\\\partial _yn) = {n-1}{n+1}.$ We introduce the following notation.\",\n \"For an $(n+1)\\\\times (n+1)$ determinant $D$ , let $D\\\\begin{bmatrix} i \\\\\\\\ j\\\\end{bmatrix}$ be the $n\\\\times n$ determinant obtained from $D$ by removing the $i$ th row and the $j$ th column and similarly let $D\\\\begin{bmatrix} i & j \\\\\\\\ k & l\\\\end{bmatrix}$ be the $(n-1)\\\\times (n-1)$ determinant obtained from $D$ by removing the $i$ th and $j$ th rows and the $k$ th and $l$ th columns.\",\n \"Then, by properties of Wronskians (REF ) can be written as ${n+1}\\\\begin{bmatrix} n+1 \\\\\\\\ n+1\\\\end{bmatrix} {n+1}\\\\begin{bmatrix} n \\\\\\\\ n\\\\end{bmatrix} - {n+1} \\\\begin{bmatrix} n \\\\\\\\ n+1\\\\end{bmatrix} {n+1} \\\\begin{bmatrix} n+1 \\\\\\\\ n\\\\end{bmatrix} = {n+1} \\\\begin{bmatrix} n & n+1 \\\\\\\\ n & n+1\\\\end{bmatrix} {n+1},$ which is nothing but the Jacobi identity for determinants [15].\",\n \"In Section we study Wronskians defined as the continuous extensions of determinantal expressions such as (REF ) where $u$ is not necessarily differentiable.\",\n \"With the help of the Jacobi idenitity we show that such generalised Wronksians satisfy the 2D Toda equations in an integrated form.\",\n \"Then in Section , using a key lemma from the preceeding section, we construct the conjugacy that describes the evolution of the multilayer process.\",\n \"A fuller description of this conjugacy is the content of Section which follows now.\"\n ],\n [\n \"Acknowledgements\",\n \"The research of C.H.L.\",\n \"was supported by EPSRC grant number EP/H023364/1 through the MASDOC DTC.\",\n \"This research of J.W.\",\n \"was supported in part by the National Science Foundation under Grant No.\",\n \"NSF PHY11-25915.\"\n ],\n [\n \"Description of the dynamics of the multilayer process\",\n \"The multilayer process is, for a fixed $x\\\\in {\\\\mathbf {R}}$ , and $n\\\\ge 1$ the finite sequence of “lines” $\\\\big (Z_1(t,x,\\\\cdot ),\\\\ldots ,Z_n(t,x,\\\\cdot ) )$ , which we consider to be evolving in the time variable $t>0$ .\",\n \"Actually it seems more natural to work with a slightly different normalisation than that originally used by [23], and consider instead the process ${\\\\tau }_{t}(x,\\\\cdot )=\\\\big (\\\\tau _1(t,x,\\\\cdot ),\\\\ldots , \\\\tau _n(t,x,\\\\cdot ) )$ , where $\\\\tau _n(t,x,y)=c_{n}^{-1} t ^{-n(n-1)/2} Z_n(t,x,y)= c_n^{-2} M_n( t,x{\\\\mathbf {1}}_n, y{\\\\mathbf {1}}_n).$ As discussed in the Introduction, we describe the evolution of the multilayer process with the aid of the extended process $M_n$ .\",\n \"It was shown in [22] that $M_n$ , defined by the chaos expansion (REF ), satisfies an evolution equation which can be regarded as a type of multi-dimensional SHE.\",\n \"$M_n(t,\\\\mathbf {x},\\\\mathbf {y}) = \\\\frac{p_n^*(t,\\\\mathbf {x},\\\\mathbf {y})}{\\\\Delta _n(\\\\mathbf {x})\\\\Delta _n(\\\\mathbf {y})} + \\\\frac{1}{(n-1)!}\",\n \"\\\\int _0^t \\\\int _{\\\\mathbf {R}^n} Q_{t-s}(\\\\mathbf {y},\\\\mathbf {y}^\\\\prime ) M_n(s,\\\\mathbf {x},\\\\mathbf {y}^\\\\prime ) \\\\;\\\\mathrm {d}\\\\mathbf {y}_*^\\\\prime \\\\;W(\\\\mathrm {d}s,\\\\mathrm {d}y_1^\\\\prime ),$ almost surely for all $(t,x,y)\\\\in (0,\\\\infty )\\\\times \\\\mathbf {R}^n\\\\times \\\\mathbf {R}^n$ where $\\\\mathrm {d}\\\\mathbf {y}_*^\\\\prime = \\\\mathrm {d}y_2^\\\\prime \\\\ldots \\\\mathrm {d}y_n^\\\\prime $ .\",\n \"The function $Q_t(\\\\mathbf {x},\\\\mathbf {y}) = \\\\frac{\\\\Delta _n(\\\\mathbf {y})}{\\\\Delta _n(\\\\mathbf {x})} p_n^*(t,\\\\mathbf {x},\\\\mathbf {y})$ is the transition density of Dyson Brownian motion [13] starting from $\\\\mathbf {x}\\\\in W_n$ and ending at $\\\\mathbf {y}\\\\in W_n$ .\",\n \"Note that in (REF ) we have extended $Q_t$ by symmetry to a function on $\\\\mathbf {R}^n\\\\times \\\\mathbf {R}^n$ so the integral over $\\\\mathbf {R}^n$ is defined.\",\n \"We can consider this same evolution for initial data $g$ , which is assumed to be a symmetric function on ${\\\\mathbf {R}}^n$ , $M^g_n(t,\\\\mathbf {y}) =\\\\frac{1}{n!}\",\n \"\\\\int _{{\\\\mathbf {R}}^n} g(\\\\mathbf {y}^\\\\prime )Q_{t}(\\\\mathbf {y},\\\\mathbf {y}^\\\\prime ) \\\\;\\\\mathrm {d}\\\\mathbf {y}^\\\\prime + \\\\\\\\\\\\frac{1}{(n-1)!}\",\n \"\\\\int _0^t \\\\int _{\\\\mathbf {R}^n} Q_{t-s}(\\\\mathbf {y},\\\\mathbf {y}^\\\\prime ) M^g_n(s,\\\\mathbf {y}^\\\\prime ) \\\\;\\\\mathrm {d}\\\\mathbf {y}_*^\\\\prime \\\\;W(\\\\mathrm {d}s,\\\\mathrm {d}y_1^\\\\prime ),$ For bounded $g$ , existence and uniqueness for this equation were established in [22].\",\n \"If $g$ takes the form $g(\\\\mathbf {x})= \\\\frac{\\\\det [f_i(x_j)]_{i,j=1}^n}{\\\\Delta _n{(\\\\mathbf {x})}}, \\\\quad \\\\mathbf {x}\\\\in {\\\\mathbf {R}}^n_{\\\\ne },$ then we have the representation $M_n^g(t,\\\\mathbf {y})=\\\\frac{\\\\det [u^{f_i}(t,y_j)]_{i,j=1}^n}{\\\\Delta _n{(\\\\mathbf {y})}}, \\\\quad t>0, \\\\mathbf {y}\\\\in {\\\\mathbf {R}}^n_{\\\\ne },$ where $u^{f_i}$ denotes the solution to the solution to the SHE with initial condition at time 0 given by $f_i$ .\",\n \"This is a consequence of equation (REF ).\",\n \"From now on we use the notation $f_1 \\\\wedge f_2 \\\\wedge \\\\ldots \\\\wedge f_n (\\\\mathbf {x})= \\\\det [f_i(x_j)]_{i,j=1}^n$ We let ${\\\\mathfrak {F}}_n$ be the set of sequences $({\\\\mathfrak {f}}_1, {\\\\mathfrak {f}}_2, \\\\ldots , {\\\\mathfrak {f}}_n)$ with each ${\\\\mathfrak {f}}_k:{\\\\mathbf {R}}^k \\\\rightarrow {\\\\mathbf {R}}$ being a continuous, strictly positive symmetric function, and such that there exists a sequence of continuous functions $f_1, f_2, \\\\ldots , f_k, \\\\ldots $ each defined on ${\\\\mathbf {R}}$ so that, for every $k$ , ${\\\\mathfrak {f}_k}(\\\\mathbf {x})= \\\\frac{f_1\\\\wedge f_2 \\\\wedge \\\\ldots \\\\wedge f_k(\\\\mathbf {x})}{\\\\Delta _k(\\\\mathbf {x})}$ for $\\\\mathbf {x}\\\\in {\\\\mathbf {R}}^k$ with all coordinates distinct.\",\n \"In view of equation (REF ), for $\\\\mathfrak {f} \\\\in {\\\\mathfrak {F}}_n$ with bounded components, we can define an ${\\\\mathfrak {F}}_n$ -valued process $(\\\\mathfrak {f}_t)_{t \\\\ge 0}$ with initial value $\\\\mathfrak {f}$ by $\\\\mathfrak {f}_t= \\\\bigl ( M_1^{{\\\\mathfrak {f}}_1}(t,\\\\cdot ), M_2^{{\\\\mathfrak {f}}_2}(t,\\\\cdot ), \\\\ldots , M_n^{{\\\\mathfrak {f}}_n}(t,\\\\cdot ) \\\\bigr )$ We will denote, for any $t>0$ , the ${\\\\mathfrak {F}}_n$ -valued random variable $\\\\mathfrak {f}_t$ by ${\\\\mathcal {M}}_t \\\\mathfrak {f}$ .\",\n \"For $00}=({\\\\tau }(t,x,\\\\cdot ))_{t>0}$ is an ${\\\\mathfrak {L}}_n$ -valued process whose evolution is given by ${\\\\tau }_{t}={\\\\mathcal {R}} {\\\\mathcal {M}}_{s,t} {\\\\mathcal {R}}^{-1} {\\\\tau }_{s} \\\\quad \\\\text{ for $00$ and $k\\\\ge 1$ , $M_k(s, x{\\\\mathbf {1}}_k,\\\\mathbf {y})$ has uniformly bounded $p$ th moments as $y\\\\in {\\\\mathbf {R}}^k$ varies, and consequently we can solve the evolution equation for $M_k$ with shifted white noise and initial data $\\\\bigl (M_k(s, x{\\\\mathbf {1}}_k,\\\\mathbf {y}); \\\\mathbf {y}\\\\in {\\\\mathbf {R}}\\\\bigr )$ .\",\n \"Thus, as at equation as a (REF ), it is meaningful to apply ${\\\\mathcal {M}}_{s,t}$ to $(M_k( s, x{\\\\mathbf {1}}_k, \\\\cdot ))_{1\\\\le k\\\\le n}$ , giving $(M_k( t, x{\\\\mathbf {1}}_k, \\\\cdot ))_{1\\\\le k\\\\le n}$ .\",\n \"It now follows from (REF ), with the time $t$ replaced by the time $s$ , that ${\\\\tau }_t(x, \\\\cdot )$ satisfies ${\\\\tau }_{t}={\\\\mathcal {R}} {\\\\mathcal {M}}_{s,t} {\\\\mathcal {R}}^{-1} {\\\\tau }_{s}.$ Corollary 6.2 of [23], states that, for given $\\\\mathbf {x}, \\\\mathbf {y}\\\\in {\\\\mathbf {R}}^n$ and $0 R) = M_\\infty = M.$ Assuming a post-TOV correction ${\\cal F} \\equiv 4\\pi |\\mu _1 | (M_\\infty /R)^2 \\ll 1$ we can re-expand our result (REF ), $M = M_\\infty \\left( 1- 2\\pi \\mu _1 \\frac{M^2_\\infty }{R^2} \\right).$ The inverse relation $M_\\infty = M_\\infty (M)$ readsAt first glance, obtaining $M_\\infty (M)$ entails solving a cubic equation.","However, the procedure is greatly simplified if we recall that the post-TOV formalism must reduce to GR for $\\lbrace \\mu _i, \\pi _i\\rbrace \\rightarrow 0$ .","Having that in mind we can treat $\\mu _1$ as a small parameter and solve (REF ) perturbatively.","The only regular solution in the $\\mu _1 \\rightarrow 0$ limit is Eq.","(REF ).","$M_\\infty = M \\left( 1+ 2\\pi \\mu _1 \\frac{M^2}{R^2} \\right).$ The three mass relations (REF ), (REF ) and (REF ) are equivalent in the ${\\cal F} \\ll 1$ limit.","Equations (REF ) and (REF ) are “exact\" post-TOV results and do not require ${\\cal F}$ or $\\mu _1$ to be much smaller than unity, although Eq.","(REF ) does place a lower limit on $\\mu _1$ because the argument of the square root must be nonnegative.","Unfortunately, the use of (REF ) in the calculation of the metric components leads to very cumbersome expressions.","To make progress (while also keeping up with the post-TOV spirit), we hereafter use the ${\\cal F} \\ll 1$ approximations (REF )-(REF ).","This step, however, introduces a certain degree of error.","This is quantified in Fig.","REF (left panel), where we show the relative percent error in calculating $M_{\\infty }$ using the post-TOV expanded Eqs.","(REF ) and (REF ) rather than Eq.","(REF ).","Using EOS Sly4 [13], we considered values of $\\mu _1$ for which Eqs.","(REF ) and (REF ) admit a positive solution for $M_{\\infty }$ .","As test beds, we consider neutron stars with central energy densities which result in a canonical $1.4\\,M_{\\odot }$ and the maximum allowed mass in GR, i.e.","$2.05\\, M_{\\odot }$ .","As evident from Fig.","REF , the error can become significant as we increase the value of $|\\mu _1|$ .","By demanding that the errors remain within $5\\%$ we can narrow down the admissible values of $\\mu _1$ to $[-1.0, 0.1]$ .","We observe that while $M_{\\infty }$ can deviate greatly from the GR value (e.g.","$M_{\\infty }$ reduces by $\\approx 21\\%$ when $\\mu _1 = -1.0$ with respect to a $1.4 \\, M_{\\odot }$ neutron star), ${\\cal F}$ remains below unity (see right panel of Fig.","REF ).","This is because large negative values of the parameter $\\mu _1$ make the star less compact (i.e.","Newtonian), as evidenced in Fig.","1 of [1].","We emphasize that the larger errors for some values of $\\mu _1$ are not an issue with the post-TOV formalism itself, but serve to constrain the values of $\\mu _1$ for which the perturbative expansion is valid.","From a practical point of view, excluding large values of $|\\mu _1|$ is a sensible strategy, since the resulting stellar parameters are so different with respect to their GR values that these cannot be considered as meaningful post-TOV corrections.","Hereafter, whenever we refer to $M_{\\infty }$ we mean the mass calculated using Eq.","(REF ) with $\\mu _1 \\in [-1.0, 0.1]$ .","Figure: Errors in M ∞ M_{\\infty }.We show the percent error [% error ≡100×(x value -x ref )/x ref \\%\\, {\\rm error} \\equiv 100 \\times (x_{\\rm value} - x_{\\rm ref})/x_{\\rm ref}]in calculating M ∞ M_{\\infty } using Eqs.","() and ()with respect to () for various values of μ 1 \\mu _1 using EOS SLy4.The range of μ 1 \\mu _1 is chosen such that using any of Eqs.","(), ()or () one can obtain a real root corresponding to M ∞ M_{\\infty }.The post-TOV models are constructed using a fixed central value of the energy density,which results in either a canonical (1.4M ⊙ 1.4\\, M_{\\odot }) or a maximum-mass(2.05M ⊙ 2.05\\, M_{\\odot }) neutron star in GR.Top panel: errors for a maximum-mass GR star.Bottom panel: errors for a canonical-mass GR star.Right panel: the absolute value of the post-TOV correctionℱ=4πμ 1 (M ∞ /R) 2 {\\cal F} =4 \\pi \\mu _1 (M_{\\infty }/R)^2 as a function of μ 1 \\mu _1.","Thecondition ℱ≪1{\\cal F} \\ll 1 boundsthe range of acceptable values of μ 1 \\mu _1 for which the expansionsleading toEqs.","() and () are valid.Errors are below 5 %\\% when μ 1 ∈[-1.0,0.1]\\mu _1 \\in [-1.0, 0.1].Within this approximation we are free to use the Taylor-expanded form of (REF ), i.e.","$m(r) = M_\\infty \\left( 1 - 2\\pi \\mu _1 M^2_\\infty /r^2 \\right)\\,.$ This expression leads to the exterior $g_{rr}$ metric $g_{rr} (r) = \\left( 1 - \\frac{2 M_\\infty }{r} \\right)^{-1} -4\\pi \\mu _1 \\frac{M^3_\\infty }{r^3}+ {\\cal O} \\left(\\frac{\\mu _1 M_\\infty ^4}{ r^4} \\right).","\\nonumber \\\\$ This expression allows us to identify $M_\\infty $ as the spacetime's gravitating mass (see also the result for $g_{tt}$ below).","The next step is to use our result for $m(r)$ in (REF ) and integrate to obtain $\\nu (r)$ .","After expanding to 2PN post-TOV order we obtain: $\\frac{d\\nu }{dr} = \\frac{2M_\\infty }{r^2} \\left( 1-\\frac{2M_\\infty }{r} \\right)^{-1}+ 2 \\chi \\frac{M^3_\\infty }{r^4}.$ where the parameter $\\chi $ , defined in Eq.","(REF ), quantifies the departure from the Schwarzschild metric.","Integrating, $\\nu (r) = \\log \\left( 1-\\frac{2M_\\infty }{r} \\right) - \\frac{2\\chi }{3}\\frac{M^3_\\infty }{r^3},$ where the integration constant has been eliminated by requiring asymptotic flatness.","The resulting exterior $g_{tt}$ metric component is $g_{tt} (r) = -\\left( 1-\\frac{2M_\\infty }{r} \\right) + \\frac{2\\chi }{3} \\frac{M^3_\\infty }{r^3}+{\\cal O} \\left( \\frac{\\chi M_\\infty ^4}{r^4} \\right)\\,.$ Eqs.","(REF ) and (REF ) represent our final results for the 2PN-accurate exterior post-Schwarzschild metric.","From this construction it follows that post-TOV stars for which $\\mu _1 = \\mu _2 = \\pi _2 = 0$ have the Schwarzschild metric as the exterior spacetime.","The following sections describe how the exterior metric can be used to compute observables of relevance for neutron star astrophysics."],["Surface redshift & stellar compactness","The surface redshift is among the most basic neutron star observables that could be affected by a change in the gravity theory.","The surface redshift is defined in the usual way as $z_s \\equiv \\frac{\\lambda _{\\infty } - \\lambda _{\\rm s}}{\\lambda _{\\rm s}}= \\frac{f_{\\rm s}}{f_{\\infty }} - 1,$ where $\\lambda $ and $f$ are the wavelength and frequency of a photon, respectively.","Here and below, the subscripts $s$ and $\\infty $ will denote the value of various quantities at the stellar surface $r = R$ and at spatial infinity.","The familiar redshift formula $\\frac{f_\\infty }{f_{\\rm s}} = \\left[\\frac{g_{tt} (R)}{g_{tt} (\\infty )}\\right]^{1/2}$ is valid for any static spacetime, regardless of the form of the field equations.","Using the metric (REF ), we easily obtain (at first post-TOV order) $\\frac{f_{\\rm s}}{f_{\\infty }} = \\left( 1 - \\frac{2 M_{\\infty }}{R}\\right)^{-1/2} + \\frac{\\chi }{3} \\frac{M_{\\infty }^3}{R^3}.$ Given that the frequency shift depends only on the ratio $M_\\infty /R$ , it is more convenient to work in terms of the stellar compactness $C = {M_\\infty }/{R}.$ Then from the definition of the surface redshift we obtain $z_{\\rm s} = z_{\\rm GR} + \\frac{\\chi }{3} C^3,$ where $z_{\\rm GR} \\equiv \\left(\\, 1 - 2C \\, \\right)^{-1/2} - 1,$ is the standard redshift formula in GR, while the second term represents the post-TOV correction.","Observe that $z_{\\rm s}$ can be smaller or larger than $z_{\\rm GR}$ depending on the sign of the parameter $\\chi $ .","This is shown in Fig.","REF (left panel), where we plot the percent difference $\\delta z_{\\rm s}/z_{\\rm s}\\equiv 100 \\times (z_s - z_{\\rm GR})/z_{\\rm GR} $ as a function of $C$ for two representative cases ($\\chi = \\pm 0.1$ ).","A characteristic property of the redshift is that it is a function of $C$ , and as such it cannot be used to disentangle mass and radius individually.","A given observed surface redshift $z_{\\rm obs}$ can be experimentally interpreted either as $z_{\\rm obs} = z_{\\rm GR} (C)$ or $z_{\\rm obs} =z_s (C,\\chi )$ , and therefore lead to different estimates for $C$ (for a given $\\chi $ ).","Fig.","REF (right panel), where we plot the percent difference $\\delta C / C \\equiv 100 \\times (C-C_{\\rm GR})/C_{\\rm GR}$ as a function of $z_{\\rm s}$ , shows how much the inferred compactness would differ in the two cases where $\\chi = \\pm 0.1$ .","A positive (negative) $\\chi $ leads to a lower (higher) inferred compactness with respect to GR.","The figure suggests that the “error” in $C$ becomes significant for redshifts $z_{\\rm s} \\gtrsim 1$ .","Figure: Surface redshift and stellar compactness.","Relative percent changeswith respect to GR for both z s z_{\\rm s} and CC for different values of χ\\chi .It is a straightforward exercise to invert the redshift formula and obtain a post-TOV expression $C= C(z_{\\rm s})$ .","We first write $1 + z_{\\rm s} = \\frac{1}{\\sqrt{-g_{tt} (R)}} \\,\\Rightarrow \\,g_{tt} (R) = -\\frac{1}{(1+z_{\\rm s})^2}.$ Upon inserting the post-Schwarzschild metric (REF ) we get a cubic equation for the compactness, $-1 + \\frac{1}{(1+z_{\\rm s})^2} + 2 C + \\frac{2}{3} \\chi C^3 = 0.$ In solving this equation we take into account that the small parameters are $C$ and $z_{\\rm s} \\sim {\\cal O}(C)$ and that the $\\chi \\rightarrow 0$ limit should be smooth.","We find, $C = C_{\\rm GR} \\left(\\, 1 - \\frac{1}{3} \\chi z_{\\rm s}^2 \\, \\right),$ where $C_{\\rm GR} = \\frac{1}{2} \\left[ 1 - (1+z_{\\rm s})^{-2} \\right],$ is the corresponding solution in GR.","As was the case for the post-TOV redshift formula, the compactness of a post-TOV star can be pushed above (below) the GR value by choosing a negative (positive) parameter $\\chi $ .","The two main results of this section, Eqs.","(REF ) and (REF ), are also interesting from a different prespective, namely, their dependence on the single post-TOV parameter $\\chi $ .","This dependence entails a degeneracy with respect to the coefficient triad $\\lbrace \\mu _1, \\mu _2, \\pi _2 \\rbrace $ when (for example) a neutron star redshift observation is used as a gravity theory discriminator.","The redshift/compactness $\\chi $ -degeneracy is another reminder of the intrinsic difficulty in distinguishing non-GR theories of gravity from neutron star physics (see e.g.","discussion around Fig.","1 in [1])."],["Bursting neutron stars","A potential testbed for measuring deviations from GR with a parametrized scheme like our post-TOV formalism is provided by accreting neutron stars exhibiting the so-called type I bursts.","These are X-ray flashes powered by the nuclear detonation of accreted matter on the stellar surface layers [14].","The luminosity associated with these events can reach the Eddington limit and may cause a photospheric radius expansion (see e.g.","[15], [16]), thus offering a number of observational “handles\" to the system (see below for more details).","A paper by Psaltis [7] proposed type I bursting neutron stars as a means to constrain possible deviations from GR.","Psaltis' analysis, based on a static and spherically symmetric model for describing the spacetime outside a non-rotating neutron star, is general enough to allow a direct adaptation to the post-TOV scheme.","For that reason we can omit most of the technical details discussed in [7] and instead focus on the key results derived in that paper.","There is a number of observable quantities associated with type I bursting neutron stars that can be used to set up a test of GR.","The first one is the surface redshift $z_{\\rm s}$ ; in Section we have derived post-TOV formulae for $z_{\\rm s}$ and the stellar compactness $C$ , which are used below in the derivation of a constraint equation between the post-Schwarzschild metric and the various observables.","The luminosity (as measured at infinity) of a source located at a (luminosity) distance $D$ is $L_\\infty = 4\\pi D^2 F_\\infty ,$ where $F_\\infty $ is the (observable) flux.","This luminosity can be written in a black-body form with the help of an apparent surface area $S_{\\rm app}$ and a color temperature (as measured at infinity) $\\bar{T}_{\\infty }$ : $4\\pi D^2 F_\\infty = \\sigma _{\\rm SB} S_{\\rm app} \\bar{T}^4_\\infty ,$ where $\\sigma _{\\rm SB}$ is the Stefan-Boltzmann constant.","We then define the second observable parameter used in this analysis, i.e.","the apparent radius $R_{\\rm app} \\equiv \\left( \\frac{S_{\\rm app}}{4\\pi } \\right)^{1/2} = D \\left( \\frac{F_\\infty }{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4} \\right)^{1/2}.$ As evident from its form, $R_{\\rm app}$ is independent of the underpinning gravitational theory, at least to the extent that the theory does not appreciably modify the (luminosity) distance to the source.","The surface color temperature is related to the intrinsic effective temperature $T_{\\rm eff}$ via the standard color correction factor $f_{\\rm c}$ [17], [18], $\\bar{T}_{\\rm s} = f_{\\rm c} T_{\\rm eff}.$ The observed temperature at infinity picks up a redshift factor with respect to its local surface value, that is, $\\bar{T}_\\infty = f_{\\rm c} \\sqrt{-g_{tt} (R)} T_{\\rm eff}.$ The effective temperature is the one related to the source's intrinsic luminosity, $L_{\\rm s} = 4\\pi R^2 \\sigma _{\\rm SB} T_{\\rm eff}^4.$ As shown in [7], $L_\\infty = - g_{tt} (R) L_{\\rm s} = 4\\pi R^2 \\sigma _{\\rm SB} \\left( \\frac{\\bar{T}_\\infty }{f_{\\rm c}} \\right)^4 [-g_{tt} (R)]^{-1}.$ Combining this with the preceding formulae leads to, $\\frac{R_{\\rm app}}{R} = \\frac{1+z_{\\rm s}}{f_{\\rm c}^2}.$ The third relevant observable is the Eddington luminosity/flux at infinity.","This is given by [7], $L_{\\rm E}^\\infty = 4\\pi D^2 F^\\infty _{\\rm E} = \\frac{4\\pi }{\\kappa } \\frac{R^2}{(1+z_{\\rm s})^2} g_{\\rm eff},$ where $\\kappa $ is the opacity of the matter interacting with the radiation fieldTypically, this interaction manifests itself as Thomson scattering in a hydrogen-helium plasma, in which case the opacity is $\\kappa \\approx 0.2 (1+ X)\\,\\mbox{cm}^2/\\mbox{gr}$ where $X$ is the hydrogen mass fraction [16].","and $g_{\\rm eff}$ is an effective surface gravitational acceleration, defined as: $g_{\\rm eff} =\\frac{1}{2\\sqrt{g_{rr} (R)}} \\frac{g^\\prime _{tt} (R)}{g_{tt} (R)}.$ This parameter is key to the present analysis as it encodes the departure from the general relativistic Schwarzschild metric.","Figure: Bursting neutron star constraints.","The surfaces arecontours of constant1+(2/3)χz s 2 z s (2+z s )/(1+z s ) 4 \\left[\\, 1 + (2/3) \\chi z_{\\rm s}^2 \\, \\right] z_{\\rm s}(2+z_{\\rm s})/(1+z_{\\rm s})^4 in the (z s ,χ)(z_{\\rm s},\\,\\chi ) plane.","This quantity is a combinationof observables – cf.","the right-hand side of Eq.","()– and therefore it is potentially measurable; a measurement willsingle out a specific contour in this plot.","A further measurement of(say) the redshift z s z_{\\rm s} corresponds to the intersectionbetween one such contour and a line with z s = const .z_{\\rm s}={\\rm const.","},so it can lead to a determination of χ\\chi .Having at our disposal the above three observable combinations, the strategy is to combine them and derive a constraint equation between the observables and the spacetime metric.","To this end, we first need to eliminate the not directly observable stellar radius $R$ between (REF ) and (REF ) and subsequently solve with respect to $g_{\\rm eff}$ .","We obtain $g_{\\rm eff} = \\kappa \\sigma _{\\rm SB} \\frac{F_{\\rm E}^\\infty }{F_\\infty } \\left( \\frac{\\bar{T}_\\infty }{f_{\\rm c}} \\right)^4 (1+z_{\\rm s})^4.$ The remaining task is to express $ g_{\\rm eff} $ in terms of $z_{\\rm s}$ .","Using the post-Schwarzschild metric, Eqs.","(REF ) and (REF ), in (REF ) we obtain the post-TOV result $g_{\\rm eff} = \\frac{C}{R} (1+z_{\\rm s} ) \\left(\\, 1 + \\chi C^2 \\, \\right).$ Making use of Eq.","(REF ) for the compactness leads to the desired result [cf.","Eq.","(39) in [7]]: $g_{\\rm eff} = \\frac{z_{\\rm s}}{2R} \\frac{(2+z_{\\rm s})}{(1+z_{\\rm s})} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right),$ where, as evident, the prefactor represents the GR result.","Finally, after eliminating $R$ with the help of (REF ) and (REF ), we obtain the “observable\" effective gravity: $g_{\\rm eff} = \\frac{z_{\\rm s}(2+z_{\\rm s})}{2 D f_{\\rm c}^2} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right)\\left( \\frac{\\sigma _{\\rm SB} \\bar{T}^4_\\infty }{F_\\infty } \\right)^{1/2}.$ This can then be combined with (REF ) to give $\\frac{z_{\\rm s}(2+z_{\\rm s})}{(1+z_{\\rm s})^4} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right)= 2D \\kappa \\frac{F_{\\rm E}^\\infty }{ f_{\\rm c}^2} \\left( \\frac{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4 }{F_\\infty } \\right)^{1/2},$ and consequently $\\chi = \\frac{3}{2 z^2_{\\rm s}} \\left[\\, 2D \\kappa \\frac{ (1+z_{\\rm s})^4}{z_{\\rm s}(2+z_{\\rm s})} \\frac{F_{\\rm E}^\\infty }{ f_{\\rm c}^2}\\left( \\frac{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4 }{F_\\infty } \\right)^{1/2} - 1 \\, \\right].$ This equation is the main result of this section and provides, at least as a proof of principle, a quantitative connection between the post-Schwarzschild correction to the exterior metric [in the form of the $\\chi $ coefficient defined in Eq.","(REF )] and observable quantities in a type I bursting neutron star.","Ref.","[7] arrives at a similar result [their Eq.","(49)] which has the same physical meaning, but is not identical to Eq.","(REF ) due to the different assumed form of the exterior metric.","Our results are illustrated in Fig.","REF , where we show the left-hand side of Eq.","(REF ) as a contour plot in the $(z_{\\rm s},\\,\\chi )$ plane.","Each contour represents a specific measurement of this observable quantity.","An additional surface redshift measurement can lead, at least in principle, to the determination of the post-TOV parameter $\\chi $ , as given by Eq.","(REF )."],["Quasi-periodic oscillations","The post-Schwarzschild metric allows us to compute the geodesic motion of particles in the exterior spacetime of post-TOV neutron stars.","Geodesics in neutron star spacetimes play a key role in the theoretical modelling of the QPOs observed in the X-ray spectra of accreting neutron stars.","The detailed physical mechanism(s) responsible for the QPO-like time variability in the flux of these systems is still a matter of debate, but some of the most popular models are based on the notion of a radiating hot “blob” of matter moving in nearly circular geodesic orbits.","The QPO frequencies are identified either with the orbital frequencies, or with simple combinations of the orbital frequencies.","The most popular models are variants of the relativistic precession [8], [9] and epicyclic resonance [10] models.","In this section we discuss the relevant orbital frequencies within the post-TOV formalism and derive formulae that could easily be used in the aforementioned QPO models.","In principle, matching the orbital frequencies to the QPO data would allow one to extract post-TOV parameters such as $\\chi $ and $\\mu _1$ (see [19], [20], [21] for a similar exercise in the context of GR and scalar-tensor theory).","For nearly circular orbits in a spherically symmetric spacetime, the only perturbations of interest are the radial ones (i.e., there is periastron precession but no Lense-Thirring nodal precession) and therefore we can associate two frequencies to every circular orbit: the orbital azimuthal frequency of the circular orbit $\\Omega _\\varphi $ and the radial epicyclic frequency $\\Omega _r$ .","Geodesics in a static, spherically symmetric spacetime are characterized by the two usual conserved quantities, the energy $E =-g_{tt} \\dot{t}$ and the angular momentum $ L= g_{\\phi \\phi } \\dot{\\phi }$ .","Here both constants are defined per unit particle mass, and the dots stand for differentiation with respect to proper time.","The four-velocity normalization condition $u^au_a=-1$ yields an effective potential equation for the particle's radial motion, $g_{rr} \\dot{r}^2 = -\\frac{E^2}{g_{tt}}-\\frac{L^2}{g_{\\phi \\phi }}-1\\equiv V_{\\rm eff}(r).$ The conditions for circular orbits are $V_{\\rm eff}(r)=V^{\\prime }_{\\rm eff}(r)=0$ , where the prime denotes differentiation with respect to the radial coordinate.","Hereafter $r$ will denote the circular orbit radius.","From these conditions we can determine the orbital frequency $\\Omega _\\varphi \\equiv \\dot{\\phi }/\\dot{t}$ measured by an observer at infinity.Apart from its implications for the QPOs, the post-TOV corrected orbital frequency would imply a shift in the corotation radius $r_{\\rm co}$ in an accreting system.","This radius is defined as $\\Omega _*= \\Omega _\\varphi (r_{\\rm co})$ , where $\\Omega _*$ is the stellar angular frequency, and plays a key role in determining the torque-spin equilibrium in magnetic field-disk coupling models.","Using the above definition we find the following result for the post-TOV corotation radius: $r_{\\rm co}= M_{\\infty }(M_{\\infty } \\Omega _*)^{-2/3} \\left[ \\, 1 + (\\chi /3) (M_{\\infty } \\Omega _*)^{4/3} \\, \\right]$ .","The square of the orbital frequency is then given as $\\Omega _\\varphi ^2 &= -\\frac{g^{\\prime }_{tt}}{g^{\\prime }_{\\phi \\phi }} =\\frac{M_{\\infty }}{r^3}\\left(1+\\chi \\frac{M_{\\infty }^2}{r^2}\\right).$ The Schwarzschild frequency is recovered for $\\chi =0$ .","The radial epicyclic frequency can be calculated from the equation for radially perturbed circular orbits, which follows from Eq.","(REF ): $\\Omega _r^2 &= -\\frac{g^{rr}}{2 \\dot{t}^2} V^{\\prime \\prime }_{\\rm eff}(r)\\approx \\frac{M_{\\infty }}{r^3}\\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{\\chi M_{\\infty }^2}{r^2} + {\\cal O} (r^{-3})\\right] \\\\&= \\Omega _\\varphi ^2 \\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{2 \\chi M_{\\infty }^2}{r^2} + {\\cal O} (r^{-3}) \\right], $ where again the frequency is calculated with respect to observers at infinity.","From the post-TOV expanded result we can see that the first two terms correspond to the Schwarzschild epicyclic frequency.","The additional post-TOV terms in these formulae produce a shift in the frequency and radius of the innermost stable circular orbit (ISCO) with respect to their GR values – the latter quantity is determined by the condition $\\Omega _r^2=0$ , which in GR leads to the well-known result $r_{\\rm isco}=6M_{\\infty }$ .","The corresponding post-TOV ISCO is obtained from (REF ), up to linear order in $\\chi $ , as: $r_{\\rm isco} \\approx 6 M_{\\infty } \\left(1+\\frac{19}{324}\\chi \\right).$ The post-TOV ISCO parameters $r_{\\rm isco}$ and $(\\Omega _\\varphi )_{\\rm isco}$ are plotted in Fig.","REF as functions of the parameter $\\chi $ .","As evident from Eq.","(REF ), a positive (negative) $\\chi $ implies $r_{\\rm isco} > 6M_{\\infty }$ ($r_{\\rm isco} < 6 M_{\\infty }$ ).","If one takes Eq.","(REF ) at face value for the given post-Schwarzschild metric, for negative enough values of $\\chi $ there is no ISCO solution, but this occurs well beyond the point where it is safe to use our perturbative formalism.","The orbital frequency profile remains rather simple, with $(\\Omega _\\varphi )_{\\rm isco}$ exceeding the GR value when $r_{\\rm isco} < 6M_{\\infty }$ (and vice versa).","Figure: ISCO quantities.","ISCO quantities as functions ofχ\\chi .","The solid curve corresponds to the relative difference (inpercent) of r isco r_{\\rm isco} with respect to GR, while the dashedcurve corresponds to the relative difference of the orbitalfrequency at the ISCO, (Ω ϕ ) isco ( \\Omega _\\varphi )_{\\rm isco}.Besides the frequency pair $\\lbrace \\Omega _\\varphi , \\Omega _r \\rbrace $ , a third prominent quantity in the QPO models is the frequency $\\Omega _{\\rm per}=\\Omega _\\varphi -\\Omega _r,$ associated with the orbital periastron precession (for example, in the relativistic precession model [8], [9] this frequency is typically associated with the low-frequency QPO) .","Given our earlier results, it is straightforward to derive a series expansion in powers of $1/r$ for $\\Omega _{\\rm per}$ .","However, it is usually more desirable to produce a series expansion with respect to an observable quantity, such as the circular orbital velocity $U_{\\infty }=(M_{\\infty }\\Omega _\\varphi )^{1/3}$ .","This can be done by first expanding $U_\\infty $ with respect to $1/r$ and then inverting the expansion, thus producing a series in $U_\\infty $ .","The outcome of this recipe is $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 1-\\frac{\\Omega _r}{\\Omega _\\varphi }= 3 U_{\\infty }^2 +\\left(\\frac{9}{2} + \\chi \\right) U_{\\infty }^4 + {\\cal O} (U_{\\infty }^{6}).$ A similar “Keplerian\" version of this expression can be produced if we opt for using the velocity $U_{\\rm K}$ and mass $M_{\\rm K}$ that an observer would infer from the motion of (say) a binary system under the assumption of exactly Keplerian orbits.","These are $U_{\\rm K} =(M_{\\rm K}\\Omega _\\varphi )^{1/3}$ and $M_{\\rm K}=r^3\\Omega ^2_\\varphi $ , so that $M_{\\rm K} =M_{\\infty }\\left(1+\\chi M_{\\infty }^2/r^2\\right)$ .","The resulting series is identical to Eq.","(REF ) when truncated to $U_{\\rm K}^4$ order.","Higher-order terms, however, are different (see the following section).","The above results for the frequencies $\\lbrace \\Omega _\\varphi , \\Omega _r, \\Omega _{\\rm per}\\rbrace $ suggest that a QPO-based test of GR within the post-TOV formalism could in principle allow the extraction of the post-TOV parameter $\\chi $ .","In this sense these frequencies probe the same kind of deviation from GR (and suffer from the same degree of degeneracy) as the observations of bursting neutron stars discussed in Section .","Figure: Orbital frequencies.Plots of Ω r \\Omega _r against Ω ϕ \\Omega _{\\varphi } fordifferent values of χ=±0.1,±0.5\\chi =\\pm 0.1,\\, \\pm 0.5 and ±1\\pm 1.","The black solid curve corresponds to the GR case.The dashed curves correspond to positive values of χ\\chi (and r isco >6M ∞ r_{\\rm isco}>6M_{\\infty })while the dashed-dotted curves correspond to negative values of χ\\chi (andr isco <6M ∞ r_{\\rm isco}<6M_{\\infty }).We conclude this section by sketching how this procedure works in practice in the context of the relativistic precession model.","The twin $\\mbox{kHz}$ QPO frequencies $\\lbrace \\nu _1, \\nu _2 \\rbrace $ seen in the flux of bright low-mass X-ray binaries are identified with the azimuthal and periastron precession orbital frequencies.","More specifically, the high-frequency member of the pair is identified with the azimuthal frequency ($\\nu _2 = \\nu _\\varphi = \\Omega _\\varphi /2\\pi $ ), while the low-frequency member is identified with the periastron precession ($\\nu _1 = \\nu _{\\rm per} = \\Omega _{\\rm per}/2\\pi $ ).","With this interpretation, the QPO separation is equal to the radial epicyclic frequency: $\\Delta \\nu = \\nu _2 - \\nu _1 = \\Omega _r/2\\pi $ .","We use our previous results [Eqs.","(REF ), (), (REF )] to plot these orbital frequencies (clearly, $\\nu _1/\\nu _2 = \\Omega _{\\rm per}/\\Omega _\\varphi $ and $\\Delta \\nu /\\nu _2 = \\Omega _r/\\Omega _\\varphi $ ) as functions of each other and for varying $\\chi $ .","As it turns out, deviations from GR are best illustrated by plotting $\\Omega _r (\\Omega _\\varphi )$ (or equivalenty $\\Delta \\nu (\\nu _2)$ ).","In Fig.","REF we plot the dimensionless combinations $M_\\infty \\Omega _r,~M_\\infty \\Omega _\\varphi $ (in units of kHz for the frequencies and solar masses for $M_{\\infty }$ ).","As we can see, the post-TOV models considered here ($ -1 <\\chi < 1 $ ) are qualitatively similar to the GR result (black solid curve), all cases showing the characteristic hump in $\\Omega _r$ as $\\Omega _\\varphi $ increases (so that the orbital radius decreases).","This feature is evidently associated with the existence of an ISCO (where $\\Omega _r \\rightarrow 0$ ) and is consistent with a similar trend seen in observations [9]."],["Multipolar structure of the spacetime","Expansions like Eq.","(REF ) contain information about the multipolar structure of the background spacetime.","That expansion can be directly compared against a similar expansion derived by Ryan [19] for an axisymmetric, stationary spacetime in GR with an arbitrary set of mass ($M_0=M_{\\infty }, M_2, M_4, ...$ ) and current ($S_1, S_3, S_5, ...$ ) Geroch-Hansen multipole moments [22], [23], [24]:A multipolar expansion in scalar-tensor theory can be found in [25].","Specific calculations were also carried out in other theories: for example, the quadrupole moment was computed in Einstein-dilaton-Gauss-Bonnet gravity [26].","$\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U^2 - 4\\frac{S_1}{M_{\\infty }^2} U^3+ \\left(\\, \\frac{9}{2} - \\frac{3}{2} \\frac{M_2}{M_{\\infty }^3} \\,\\right) U^4 -10 \\frac{S_1}{M_{\\infty }^2} U^5\\nonumber \\\\& + \\left(\\, \\frac{27}{2} -2\\frac{S_1^2}{M_{\\infty }^4} -\\frac{21}{2} \\frac{M_2}{M_{\\infty }^3} \\, \\right) U^6 + {\\cal O} (U^{7} ).$ where $U = (M_{\\infty } \\Omega _\\varphi )^{1/3} $ denotes the orbital velocity.","To understand the PN accuracy of the post-TOV expansion in this context, it is useful to consider the effect of higher PN order terms in the metric.","Imagine that the $g_{tt}$ and $g_{rr}$ metric components [see Eqs.","(REF ), (REF )] included 3PN corrections of the schematic form, $g_{tt}(r)&=g_{tt}^{\\rm 2PN}+\\alpha _{tt} \\frac{M_{\\infty }^3}{r^3},\\\\g_{rr}(r)&=g_{rr}^{\\rm 2PN}+\\alpha _{rr} \\frac{M_{\\infty }^3}{r^3}.$ We can use the coefficients $\\alpha _{tt}$ and $\\alpha _{rr}$ as bookeeping parameters in order to understand how these omitted higher-order contributions affect the results of the previous section.","The recalculation of the various expressions reveals that the orbital frequency remains unchanged to 2PN order; the 3PN term of Eq.","(REF ) contributes at the next order, as expected.","The same applies to the epicyclic frequency, as we can see for example from the modified Eq.","(), where the next-order correction is a mixture of $g_{rr}^{\\rm 2PN}$ and the 3PN term in $g_{tt}$ : $\\Omega _r^2&=\\Omega _\\varphi ^2 \\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{2 \\chi M_{\\infty }^2}{r^2} \\right.\\nonumber \\\\&\\left.+ \\frac{(4 \\pi \\mu _1 -6\\alpha _{tt})M_{\\infty }^3}{r^3} + {\\cal O} (r^{-4}) \\right].$ Proceeding in a similar way we find the next-order correction to the Ryan-like expansion (REF ): $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U_{\\infty }^2 +\\left(\\frac{9}{2} + \\chi \\right) U_{\\infty }^4\\nonumber \\\\&+\\left[\\frac{27}{2} + 2 (\\chi -\\pi \\mu _1 )+3\\alpha _{tt}\\right] U_{\\infty }^6+ {\\cal O} (U_{\\infty }^{8}).$ We can see that the 3PN term “contaminates\" the PN correction that was omitted in Eq.","(REF ).","Repeating the same exercise for the Keplerian version of the multipole expansion (i.e.","where the orbital velocity $U_\\infty $ is replaced by $U_{\\rm K}$ ) we find: $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U^2_{\\rm K} +\\left(\\frac{9}{2} + \\chi \\right) U^4_{\\rm K} \\nonumber \\\\&+\\left(\\frac{27}{2} - 2 \\pi \\mu _1+3\\alpha _{tt}\\right) U^6_{\\rm K} + {\\cal O} (U^{8}_{\\rm K} ).$ At the PN order considered in the previous section the two expressions were identical but, as we can see, they differ at the next order.","We now have Ryan-type multipole expansions of the post-Schwarzschild spacetime up to 3PN in the circular orbital velocity, which we can compare against Eq.","(REF ) to draw (with some caution) analogies and differences between GR and modified theories of gravity.","For instance, odd powers of $U_\\infty $ are missing in Eq.","(REF ) because the nonrotating post-Schwarzschild spacetime has vanishing current multipole moments.","Furthermore, we can see that the quadrupole moment $M_2$ , first appearing in the coefficient of $U^4_\\infty $ in Eq.","(REF ), can be associated with $\\chi $ .","The parameter $\\chi $ is an effective quadrupole moment in the sense that $M_2^{\\rm eff} = -\\frac{2}{3} \\chi M_{\\infty }^3.$ Indeed, this relation implies that a positive (negative) $\\chi $ could be associated with an oblate (prolate) source of the gravitational field.","The identification (REF ) holds at $ {\\cal O} (U_{\\infty }^{4})$ .","The next-order term $U^6_\\infty $ would, in general, lead to a different effective $M_2$ .","Hence, the comparison between the $U_{\\infty }^4$ and $U_{\\infty }^6$ terms could provide a null test for the GR-predicted quadrupole.","However, there is a special case where these two terms could be consistent with the same effective quadrupole (REF ): this occurs when the post-TOV parameters satisfy the condition $5\\chi = -2\\pi \\mu _1+3\\alpha _{tt}$ , in which case the expansion (REF ) behaves as a “GR mimicker\".","A different kind of “multipole” expansion in powers of $1/r$ can be applied to the metric functions $\\nu (r), m(r)$ [see Eqs.","(REF )–(REF )], leading to an alternative calculation of the ADM mass $M_\\infty $ of a post-TOV star.","We consider the expansions $\\nu (r) = \\sum _{n = 0}^{\\infty } \\frac{\\nu _{n}}{r^n}, \\qquad m(r) = \\sum _{n = 0}^{\\infty } \\frac{m_{n}}{r^n},$ where $\\nu _n$ and $m_n$ are constant coefficients.","In addition, we impose that $\\nu _0 = 0$ and $m_0 = M_{\\infty }$ .","We subsequently substitute these expansions into Eqs.","(REF )–(), expand for $r/R \\gg 1$ , and then solve for the coefficients.","The outcome of this exercise in the vacuum exterior spacetime is $m(r) &= M_{\\infty } - 2 \\pi \\mu _1 \\frac{M_{\\infty }^3}{r^2} + {\\cal O}(r^{-4})\\,,\\\\\\nonumber \\\\\\nu (r) &= - \\frac{2 M_{\\infty }}{r} - \\frac{2 M_{\\infty }^2}{r^2}- \\frac{2}{3} \\frac{M_{\\infty }^3}{r^3}\\left( 4 + \\chi \\right) + {\\cal O}(r^{-4})\\,.$ As expected, the top equation is consistent with our earlier result, Eq.","(REF ).","To get an agreement between Eqs.","(REF ) and () we must expand the logarithm appearing in the former equation in powers of $M_{\\infty }/r$ , thus recovering Eq.","()."],["Conclusions","In this paper we have demonstrated the applicability of the post-TOV formalism to a number of facets of neutron star astrophysics.","Let us summarize our main results.","The exterior post-Schwarzschild metric [Eqs.","(REF ) and (REF )] depends only on the ADM mass $M_\\infty $ (given by Eq.","(REF )) and on just two post-TOV parameters $\\mu _1$ and $\\chi $ .","These are subsequently used to produce a post-TOV formula for the surface redshift, Eq.","(REF ), which is a function of the stellar compactness and $\\chi $ .","Next, we have shown how a basic post-TOV model for type I bursting neutron stars can be constructed.","The key equation here is (REF ), which gives $\\chi $ (the only post-TOV parameter appearing in the model) in terms of observable quantities.","We also computed geodesic motion in the post-Schwarzschild exterior of post-TOV neutron star models, finding expressions for the orbital, epicyclic and periastron precession frequencies of nearly circular orbits [Eqs.","(REF ), (), (REF )] and for the ISCO radius [Eq.","(REF )].","These results can be fed into models for QPOs from accreting neutron stars, such as the relativistic precession model.","Finally, on a more theoretical level, we have sketched how the post-TOV parameters enter in the spacetime's multipolar structure [Eq.","(REF )].","The meticulous reader may have noticed that, in spite of the exterior metric being a function $g_{tt} (\\chi )$ and $g_{rr} (\\mu _1)$ , all other post-TOV results feature only $\\chi $ , while $\\mu _1$ is either not present or enters at higher order.","This is not a coincidence: these quantities either depend solely on $g_{tt}$ (e.g.","the redshift) or receive their leading-order contributions from $g_{tt}$ (e.g.","the orbital frequencies).","The post-TOV formalism developed in [1] and in this paper can be viewed as a basic “stage-one” version of a more general framework.","There are several directions one can follow for taking the formalism to a more sophisticated level, and here we discuss just a couple of possibilities.","An obvious improvement is the addition of stellar rotation.","This is necessary because all astrophysical compact stars rotate, some of them quite rapidly, and the influence of rotation is ubiquitous, affecting to some extent all of the effects discussed in this paper.","As a first stab at the problem, it would make sense to work in the Hartle-Thorne slow rotation approximation [27], [28], which should be accurate enough for all but the fastest spinning neutron stars [29].","There are equally important possibilities for improvement on the modified gravity sector of the formalism.","The present post-TOV theory is oblivious to the existence of dimensionful coupling constants, such as the ones appearing in many modified theories of gravity (e.g.","$f(R)$ theories or theories quadratic in the curvature).","These coupling parameters should be added to the existing set of fluid parameters, and participate in the algorithmic generation of “families\" of post-TOV terms (see [1] for details).","The extended set of parameters will most likely lead to a proliferation of post-TOV terms, and result in more complicated stellar structure equations than the ones used so far [i.e.","Eqs. ()].","Among other things, this enhancement may allow one to study in more generality to what extent other theories of gravity are mapped onto the post-TOV formalism.","Another limitation of the formalism is that it is intrinsically perturbative with respect to GR solutions.","It is important to generalize to theories of gravity that present screening mechanisms; the viability of perturbative expansions in these theories is a topic of active research (see e.g.","[30], [31], [32], [33]).","We hope that the astrophysical applications outlined in this work will stimulate more research to address these issues.","K.G.","is supported by the Ramón y Cajal Programme of the Spanish Ministerio de Ciencia e Innovación and by NewCompStar (a COST-funded Research Networking Programme).","H.O.S., G.P.","and E.B.","are supported by NSF CAREER Grant No.","PHY-1055103.","E.B.","is supported by FCT contract IF/00797/2014/CP1214/CT0012 under the IF2014 Programme.","This work was supported by the H2020-MSCA-RISE-2015 Grant No.","StronGrHEP-690904.","H.O.S.","thanks the Instituto Superior Técnico for hospitality, and the Department of Physics and Astronomy of the University of Mississippi for financial support."]],"string":"[\n [\n \"Astrophysical applications of the post-Tolman-Oppenheimer-Volkoff\\n formalism\"\n ],\n [\n \"Abstract The bulk properties of spherically symmetric stars in general relativity can be obtained by integrating the Tolman-Oppenheimer-Volkoff (TOV) equations.\",\n \"In previous work we developed a \\\"post-TOV\\\" formalism - inspired by parametrized post-Newtonian theory - which allows us to classify in a parametrized, phenomenological form all possible perturbative deviations from the structure of compact stars in general relativity that may be induced by modified gravity at second post-Newtonian order.\",\n \"In this paper we extend the formalism to deal with the stellar exterior, and we compute several potential astrophysical observables within the post-TOV formalism: the surface redshift $z_s$, the apparent radius $R_{\\\\rm app}$, the Eddington luminosity at infinity $L_{\\\\rm E}^\\\\infty$ and the orbital frequencies.\",\n \"We show that, at leading order, all of these quantities depend on just two post-TOV parameters $\\\\mu_1$ and $\\\\chi$, and we discuss the possibility to measure (or set upper bounds on) these parameters.\"\n ],\n [\n \"Introduction\",\n \"Compact stars are ideal astrophysical environments to probe the coupling between matter and gravity in a high-density, strong gravity regime not accessible in the laboratory.\",\n \"Cosmological observations and high-energy physics considerations have spurred extensive research on the properties of neutron stars, whether isolated or in binary systems, in modified theories of gravity (see e.g.\",\n \"[2], [3] for reviews).\",\n \"Different extensions of general relativity (GR) affect the bulk properties of the star (such as the mass $M$ and radius $R$ ) in similar ways for given assumptions on the equation of state (EOS) of high-density matter.\",\n \"Therefore it is interesting to understand whether these deviations from the predictions of GR can be understood within a simple parametrized formalism.\",\n \"The development of such a generic framework to understand how neutron star properties are affected in modified gravity is even more pressing now that gravitational-wave observations are finally a reality [4], since the observation of neutron star mergers could allow us to probe the dynamical behavior of these objects in extreme environments.\",\n \"In previous work we developed a post-Tolman-Oppenheimer-Volkoff (henceforth, post-TOV) formalism valid for spherical stars [1].\",\n \"The basic idea is quite simple.\",\n \"The structure of nonrotating, relativistic stars can be determined by integrating two ordinary differential equations: one of these equations gives the “mass function” and the other equation – a generalization of the hydrostatic equilibrium condition in Newtonian gravity – determines the pressure profile and the stellar radius, defined as the point where the pressure vanishes.\",\n \"The post-TOV formalism, reviewed in Section below, adds a relatively small number of parametrized corrections with parameters $\\\\mu _i$ ($i=1, \\\\dots \\\\,, 5$ ) and $\\\\pi _i$ ($i=1, \\\\dots \\\\,, 4$ ) to the mass and pressure equations.\",\n \"These corrections have two properties: (i) they are of second post-Newtonian (PN) order, because first-order deviations are already tightly constrained by observations; and (ii) they are general enough to capture in a phenomenological way all possible deviations from the mass-radius relation in GR.\",\n \"Other parametrizations were explored in [5], [6] by modifying ad hoc the TOV equations.\",\n \"In this paper we turn to the investigation of astrophysical applications of the formalism.\",\n \"Part of our analysis is inspired by previous work by Psaltis [7], who showed that, under the assumption of spherical symmetry, many properties of neutron stars in metric theories of gravity can be calculated using only conservation laws, Killing symmetries, and the Einstein equivalence principle, without requiring the validity of the GR field equations.\",\n \"Psaltis computed the gravitational redshift of a surface atomic line $z_s$ , the Eddington luminosity at infinity $L_{\\\\rm E}^\\\\infty $ (thought to be equal to the touchdown luminosity of a radius-expansion burst), and the apparent surface area of a neutron star (which is potentially measurable during the cooling tails of bursts).\",\n \"We first extend our previous work to study the exterior of neutron stars.\",\n \"Then we compute the surface redshift $z_s$ , the apparent radius $R_{\\\\rm app}$ and the Eddington luminosity at infinity $L_{\\\\rm E}^\\\\infty $ .\",\n \"In addition, we study geodesic motion in the neutron star spacetime within the post-TOV formalism.\",\n \"We focus on the orbital and epicyclic frequencies, that according to some models – such as the relativistic precession model [8], [9] and the epicyclic resonance model [10] – may be related with the quasi-periodic oscillations (QPOs) observed in the X-ray spectra of accreting neutron stars.\",\n \"Our main result is that, at leading order, all of these quantities depend on just two post-TOV parameters: $\\\\mu _1$ and the combination $\\\\chi \\\\equiv \\\\pi _2 - \\\\mu _2 - 2\\\\pi \\\\mu _1\\\\,.$ We also express the leading multipoles in a multipolar expansion of the neutron star spacetime in terms of $\\\\mu _1$ and $\\\\chi $ , and we discuss the possibility to measure (or set upper bounds on) these parameters with astrophysical observations.\",\n \"The plan of the paper is as follows.\",\n \"In Section we present a short review of the post-TOV formalism developed in [1].\",\n \"In Section we extend the formalism to deal with the stellar exterior, computing a “post-Schwarzschild” exterior metric.\",\n \"In Section we compute the surface redshift $z_s$ and relate it to the stellar compactness $M/R$ .\",\n \"In Section , following [7], we study the properties of bursting neutron stars in the post-TOV framework.\",\n \"In Section we calculate the orbital frequencies.\",\n \"In Section we look at the leading-order multipoles of post-TOV stars.\",\n \"Then we present some conclusions and possible directions for future work.\"\n ],\n [\n \"Overview of the post-TOV formalism\",\n \"Let us begin with a review of the post-TOV formalism introduced in [1].\",\n \"The core of this formalism consists of the following set of “post-TOV\\\" structure equations for static spherically symmetric stars (we use geometrical units $G=c=1$ ): $\\\\frac{dp}{dr} &= \\\\left(\\\\frac{dp}{dr} \\\\right)_{\\\\rm GR} -\\\\frac{\\\\rho m}{r^2} \\\\left(\\\\, {\\\\cal P}_1 + {\\\\cal P}_2\\\\, \\\\right),\\\\\\\\\\\\nonumber \\\\\\\\\\\\frac{dm}{dr} & = \\\\left( \\\\frac{dm}{dr} \\\\right)_{\\\\rm GR} + 4\\\\pi r^2\\\\rho \\\\left( {\\\\cal M}_1 + {\\\\cal M}_2\\\\right),$ where ${\\\\cal P}_1 &\\\\equiv \\\\delta _1 \\\\frac{m}{r} + 4\\\\pi \\\\delta _2 \\\\frac{r^3 p}{m},\\\\quad {\\\\cal M}_1 \\\\equiv \\\\delta _3 \\\\frac{m}{r} + \\\\delta _4 \\\\Pi ,\\\\\\\\\\\\nonumber \\\\\\\\{\\\\cal P}_2 &\\\\equiv \\\\pi _1 \\\\frac{m^3}{r^5\\\\rho } + \\\\pi _2 \\\\frac{m^2}{r^2}+ \\\\pi _3 r^2 p + \\\\pi _4 \\\\frac{\\\\Pi p}{\\\\rho },\\\\\\\\\\\\nonumber \\\\\\\\{\\\\cal M}_2 &\\\\equiv \\\\mu _1 \\\\frac{m^3}{r^5\\\\rho } + \\\\mu _2 \\\\frac{m^2}{r^2}+ \\\\mu _3 r^2 p+ \\\\mu _4 \\\\frac{\\\\Pi p}{\\\\rho } + \\\\mu _5 \\\\Pi ^3 \\\\frac{r}{m} .\",\n \"\\\\nonumber \\\\\\\\$ Here $r$ is the circumferential radius, $m$ is the mass function, $p$ is the fluid pressure, $\\\\rho $ is the baryonic rest mass density, $\\\\epsilon $ is the total energy density and $\\\\Pi \\\\equiv (\\\\epsilon -\\\\rho )/\\\\rho $ is the internal energy per unit baryonic mass.\",\n \"A “GR” subscript denotes the standard TOV equations in GR, i.e.\",\n \"$& \\\\left(\\\\frac{dp}{dr} \\\\right)_{\\\\rm GR} = -\\\\frac{(\\\\epsilon + p)}{r^2} \\\\frac{ (m_{\\\\rm T} + 4\\\\pi r^3 p )}{ ( 1-2m_{\\\\rm T} /r )},\\\\\\\\\\\\nonumber \\\\\\\\& \\\\left( \\\\frac{dm}{dr} \\\\right)_{\\\\rm GR} = \\\\frac{d m_{\\\\rm T}}{dr} = 4\\\\pi r^2 \\\\epsilon ,$ where $m_{\\\\rm T}$ is the GR mass function.\",\n \"The dimensionless combinations ${\\\\cal P}_1,{\\\\cal M}_1$ and ${\\\\cal P}_2, {\\\\cal M}_2$ represent a parametrized departure from the GR stellar structure and are linear combinations of 1PN- and 2PN-order terms, respectively.\",\n \"These terms feature the phenomenological post-TOV parameters $\\\\delta _i$ ($i=1,\\\\,\\\\dots \\\\,,4$ ), $\\\\pi _i$ ($i=1,\\\\,\\\\dots \\\\,,4$ ) and $\\\\mu _i$ ($i=1, \\\\dots \\\\,, 5$ ).\",\n \"In particular, the coefficients $\\\\delta _i$ attached to the 1PN terms are simple algebraic combinations of the traditional PPN parameters $\\\\delta _1 \\\\equiv 3 (1+ \\\\gamma ) -6\\\\beta + \\\\zeta _2$ , $\\\\delta _2 \\\\equiv \\\\gamma -1+\\\\zeta _4$ , $\\\\delta _3 \\\\equiv -\\\\frac{1}{2} \\\\left( 11 + \\\\gamma -12\\\\beta + \\\\zeta _2 -2\\\\zeta _4 \\\\right)$ , $\\\\delta _4\\\\equiv \\\\zeta _3.$ As such, they are constrained to be very close to zero by existing Solar System and binary pulsar observationsUsing the latest constraints on the PPN parameters [11] we obtain the following upper limits: $| \\\\delta _1| \\\\lesssim 6\\\\times 10^{-4}, |\\\\delta _2| \\\\lesssim 7\\\\times 10^{-3}, | \\\\delta _3 | \\\\lesssim 7\\\\times 10^{-3}, |\\\\delta _4| \\\\lesssim 10^{-8}$ .\",\n \": $|\\\\delta _i| \\\\ll 1$ .\",\n \"This result translates to negligibly small 1PN terms in Eq.\",\n \"(): ${\\\\cal P}_{1}\\\\ll 1$ , ${\\\\cal M}_1 \\\\ll 1$ .\",\n \"On the other hand, $\\\\pi _i$ and $\\\\mu _i$ are presently unconstrained, and consequently ${\\\\cal P}_2, {\\\\cal M}_2$ should be viewed as describing the dominant (significant) departure from GR.\",\n \"The GR limit of the formalism corresponds to setting all of these parameters to zero, i.e.\",\n \"$\\\\delta _i, \\\\pi _i, \\\\mu _i \\\\rightarrow 0$ .\",\n \"Alternatively, the stellar structure equations () can be formally derived – if we neglect the small terms ${\\\\cal P}_1, {\\\\cal M}_1$ – from a covariantly conserved perfect fluid stress energy tensor [1]: $\\\\nabla _\\\\nu T^{\\\\mu \\\\nu } = 0, \\\\qquad T^{\\\\mu \\\\nu } = (\\\\epsilon _{\\\\rm eff} + p) u^\\\\mu u^\\\\nu + p g^{\\\\mu \\\\nu },$ where the effective, gravity-modified energy density is $\\\\epsilon _{\\\\rm eff} = \\\\epsilon + \\\\rho {\\\\cal M}_2,$ and the covariant derivative is compatible with the effective post-TOV metric $g_{\\\\mu \\\\nu } = \\\\mbox{diag} [\\\\, -e^{\\\\nu (r)}, (1-2m(r)/r)^{-1}, r^2, r^2 \\\\sin ^2\\\\theta \\\\,],$ with $\\\\frac{d\\\\nu }{dr} = \\\\frac{2}{r^2\\\\\\\\} \\\\left[\\\\, (1-{\\\\cal M}_2) \\\\frac{m+4\\\\pi r^3 p}{1-2m/r} + m{\\\\cal P}_2 \\\\, \\\\right].$ This post-TOV metric is valid in the interior of the star.\",\n \"In the following section we discuss how an exterior post-TOV metric can be constructed within our framework.\"\n ],\n [\n \"The exterior “post-Schwarzschild” metric\",\n \"For the applications of the post-TOV formalism considered in this work, we must specify how the $g_{tt}$ and $g_{rr}$ metric elements are calculated in the interior and exterior regions of the fluid distribution.\",\n \"In this section we will construct an exterior spacetime in a “post-Schwarzschild” form.\",\n \"From the effective post-TOV metric, we have that inside the fluid body $g_{tt} = -\\\\exp [\\\\nu (r)]$ , where $\\\\nu (r)$ is determined in terms of the fluid variables and post-TOV parameters from Eq.\",\n \"(REF ).\",\n \"We will assume that outside the fluid distribution the same effective metric expression holdsThis assumption is based on simplicity.\",\n \"While we keep an agnostic view on the validity of Birkhoff's theorem within our formalism (and in modified gravity theories in general), the interior post-TOV metric is arguably the best guide towards the construction of the exterior metric.. Then we get the equations $&& \\\\frac{d\\\\nu }{dr} = \\\\left(\\\\frac{d\\\\nu }{dr} \\\\right)_{\\\\rm GR} + \\\\frac{2}{r^2} \\\\left[\\\\, - \\\\mu _2 \\\\frac{m^2}{r^2} \\\\frac{m}{1-2m/r}+ \\\\pi _2 \\\\frac{m^3}{r^2} \\\\, \\\\right],\\\\nonumber \\\\\\\\\\\\\\\\\\\\nonumber \\\\\\\\&& \\\\frac{dm}{dr} = 4\\\\pi \\\\mu _1 \\\\frac{m^3}{r^3},$ where $\\\\left(\\\\frac{d\\\\nu }{dr} \\\\right)_{\\\\rm GR} \\\\equiv \\\\frac{2}{r^2}\\\\frac{m}{1-2m/r}.$ These equations originate from the general expressions (REF ) and () after setting all fluid parameters to zero, i.e $p=\\\\epsilon =\\\\rho =\\\\Pi =0$ , and keeping the surviving terms in ${\\\\cal M}_2$ and ${\\\\cal P}_2$ .\",\n \"It is not difficult to see that, in the nomenclature of [1], the only 2PN post-TOV terms that can appear in the exterior equations are those of “family F1” and “family F2”.\",\n \"The F1 term (coefficient $\\\\pi _1$ ) should not appear in the ${\\\\cal P}_2$ correction of the interior $d\\\\nu /dr$ equation because it is divergent at the surface.\",\n \"This implies that the F1 term should not appear in the $dp/dr$ equation either.\",\n \"As it stands, Eq.\",\n \"(REF ) contains higher than 2PN order terms.\",\n \"It should therefore be PN-expanded with respect to the post-TOV terms: $\\\\frac{d\\\\nu }{dr} = \\\\left(\\\\frac{d\\\\nu }{dr} \\\\right)_{\\\\rm GR} + 2( \\\\pi _2 - \\\\mu _2) \\\\frac{m^3}{r^4}.$ Thus (REF ) and () are our “final” post-TOV equations for the stellar exterior.\",\n \"The mass equation is decoupled and can be directly integrated.\",\n \"The result is $m(r) = \\\\frac{r}{\\\\sqrt{4\\\\pi \\\\mu _1 + K r^2}},$ where $K$ is an integration constant.\",\n \"The fact that $dm/dr \\\\ne 0$ outside the star implies the presence of an “atmosphere” due to the non-GR degree of freedom.\",\n \"This is reminiscent of the exterior structure of neutron stars in scalar-tensor theories [12].\",\n \"The constant $K$ is fixed by setting $m(r\\\\rightarrow \\\\infty ) $ equal to the system's ADM mass $M_{\\\\infty }$ .\",\n \"Then, $m(r) = M_\\\\infty \\\\left( 1+ 4\\\\pi \\\\mu _1 \\\\frac{M^2_\\\\infty }{r^2} \\\\right)^{-1/2}.$ Thus the ADM mass is related to the Schwarzschild mass $M \\\\equiv m(R)$ by $M = M_\\\\infty \\\\left( 1+ 4\\\\pi \\\\mu _1 \\\\frac{M^2_\\\\infty }{R^2} \\\\right)^{-1/2}.$ As expected, in the GR limit the two masses coincide $m(r> R) = M_\\\\infty = M.$ Assuming a post-TOV correction ${\\\\cal F} \\\\equiv 4\\\\pi |\\\\mu _1 | (M_\\\\infty /R)^2 \\\\ll 1$ we can re-expand our result (REF ), $M = M_\\\\infty \\\\left( 1- 2\\\\pi \\\\mu _1 \\\\frac{M^2_\\\\infty }{R^2} \\\\right).$ The inverse relation $M_\\\\infty = M_\\\\infty (M)$ readsAt first glance, obtaining $M_\\\\infty (M)$ entails solving a cubic equation.\",\n \"However, the procedure is greatly simplified if we recall that the post-TOV formalism must reduce to GR for $\\\\lbrace \\\\mu _i, \\\\pi _i\\\\rbrace \\\\rightarrow 0$ .\",\n \"Having that in mind we can treat $\\\\mu _1$ as a small parameter and solve (REF ) perturbatively.\",\n \"The only regular solution in the $\\\\mu _1 \\\\rightarrow 0$ limit is Eq.\",\n \"(REF ).\",\n \"$M_\\\\infty = M \\\\left( 1+ 2\\\\pi \\\\mu _1 \\\\frac{M^2}{R^2} \\\\right).$ The three mass relations (REF ), (REF ) and (REF ) are equivalent in the ${\\\\cal F} \\\\ll 1$ limit.\",\n \"Equations (REF ) and (REF ) are “exact\\\" post-TOV results and do not require ${\\\\cal F}$ or $\\\\mu _1$ to be much smaller than unity, although Eq.\",\n \"(REF ) does place a lower limit on $\\\\mu _1$ because the argument of the square root must be nonnegative.\",\n \"Unfortunately, the use of (REF ) in the calculation of the metric components leads to very cumbersome expressions.\",\n \"To make progress (while also keeping up with the post-TOV spirit), we hereafter use the ${\\\\cal F} \\\\ll 1$ approximations (REF )-(REF ).\",\n \"This step, however, introduces a certain degree of error.\",\n \"This is quantified in Fig.\",\n \"REF (left panel), where we show the relative percent error in calculating $M_{\\\\infty }$ using the post-TOV expanded Eqs.\",\n \"(REF ) and (REF ) rather than Eq.\",\n \"(REF ).\",\n \"Using EOS Sly4 [13], we considered values of $\\\\mu _1$ for which Eqs.\",\n \"(REF ) and (REF ) admit a positive solution for $M_{\\\\infty }$ .\",\n \"As test beds, we consider neutron stars with central energy densities which result in a canonical $1.4\\\\,M_{\\\\odot }$ and the maximum allowed mass in GR, i.e.\",\n \"$2.05\\\\, M_{\\\\odot }$ .\",\n \"As evident from Fig.\",\n \"REF , the error can become significant as we increase the value of $|\\\\mu _1|$ .\",\n \"By demanding that the errors remain within $5\\\\%$ we can narrow down the admissible values of $\\\\mu _1$ to $[-1.0, 0.1]$ .\",\n \"We observe that while $M_{\\\\infty }$ can deviate greatly from the GR value (e.g.\",\n \"$M_{\\\\infty }$ reduces by $\\\\approx 21\\\\%$ when $\\\\mu _1 = -1.0$ with respect to a $1.4 \\\\, M_{\\\\odot }$ neutron star), ${\\\\cal F}$ remains below unity (see right panel of Fig.\",\n \"REF ).\",\n \"This is because large negative values of the parameter $\\\\mu _1$ make the star less compact (i.e.\",\n \"Newtonian), as evidenced in Fig.\",\n \"1 of [1].\",\n \"We emphasize that the larger errors for some values of $\\\\mu _1$ are not an issue with the post-TOV formalism itself, but serve to constrain the values of $\\\\mu _1$ for which the perturbative expansion is valid.\",\n \"From a practical point of view, excluding large values of $|\\\\mu _1|$ is a sensible strategy, since the resulting stellar parameters are so different with respect to their GR values that these cannot be considered as meaningful post-TOV corrections.\",\n \"Hereafter, whenever we refer to $M_{\\\\infty }$ we mean the mass calculated using Eq.\",\n \"(REF ) with $\\\\mu _1 \\\\in [-1.0, 0.1]$ .\",\n \"Figure: Errors in M ∞ M_{\\\\infty }.We show the percent error [% error ≡100×(x value -x ref )/x ref \\\\%\\\\, {\\\\rm error} \\\\equiv 100 \\\\times (x_{\\\\rm value} - x_{\\\\rm ref})/x_{\\\\rm ref}]in calculating M ∞ M_{\\\\infty } using Eqs.\",\n \"() and ()with respect to () for various values of μ 1 \\\\mu _1 using EOS SLy4.The range of μ 1 \\\\mu _1 is chosen such that using any of Eqs.\",\n \"(), ()or () one can obtain a real root corresponding to M ∞ M_{\\\\infty }.The post-TOV models are constructed using a fixed central value of the energy density,which results in either a canonical (1.4M ⊙ 1.4\\\\, M_{\\\\odot }) or a maximum-mass(2.05M ⊙ 2.05\\\\, M_{\\\\odot }) neutron star in GR.Top panel: errors for a maximum-mass GR star.Bottom panel: errors for a canonical-mass GR star.Right panel: the absolute value of the post-TOV correctionℱ=4πμ 1 (M ∞ /R) 2 {\\\\cal F} =4 \\\\pi \\\\mu _1 (M_{\\\\infty }/R)^2 as a function of μ 1 \\\\mu _1.\",\n \"Thecondition ℱ≪1{\\\\cal F} \\\\ll 1 boundsthe range of acceptable values of μ 1 \\\\mu _1 for which the expansionsleading toEqs.\",\n \"() and () are valid.Errors are below 5 %\\\\% when μ 1 ∈[-1.0,0.1]\\\\mu _1 \\\\in [-1.0, 0.1].Within this approximation we are free to use the Taylor-expanded form of (REF ), i.e.\",\n \"$m(r) = M_\\\\infty \\\\left( 1 - 2\\\\pi \\\\mu _1 M^2_\\\\infty /r^2 \\\\right)\\\\,.$ This expression leads to the exterior $g_{rr}$ metric $g_{rr} (r) = \\\\left( 1 - \\\\frac{2 M_\\\\infty }{r} \\\\right)^{-1} -4\\\\pi \\\\mu _1 \\\\frac{M^3_\\\\infty }{r^3}+ {\\\\cal O} \\\\left(\\\\frac{\\\\mu _1 M_\\\\infty ^4}{ r^4} \\\\right).\",\n \"\\\\nonumber \\\\\\\\$ This expression allows us to identify $M_\\\\infty $ as the spacetime's gravitating mass (see also the result for $g_{tt}$ below).\",\n \"The next step is to use our result for $m(r)$ in (REF ) and integrate to obtain $\\\\nu (r)$ .\",\n \"After expanding to 2PN post-TOV order we obtain: $\\\\frac{d\\\\nu }{dr} = \\\\frac{2M_\\\\infty }{r^2} \\\\left( 1-\\\\frac{2M_\\\\infty }{r} \\\\right)^{-1}+ 2 \\\\chi \\\\frac{M^3_\\\\infty }{r^4}.$ where the parameter $\\\\chi $ , defined in Eq.\",\n \"(REF ), quantifies the departure from the Schwarzschild metric.\",\n \"Integrating, $\\\\nu (r) = \\\\log \\\\left( 1-\\\\frac{2M_\\\\infty }{r} \\\\right) - \\\\frac{2\\\\chi }{3}\\\\frac{M^3_\\\\infty }{r^3},$ where the integration constant has been eliminated by requiring asymptotic flatness.\",\n \"The resulting exterior $g_{tt}$ metric component is $g_{tt} (r) = -\\\\left( 1-\\\\frac{2M_\\\\infty }{r} \\\\right) + \\\\frac{2\\\\chi }{3} \\\\frac{M^3_\\\\infty }{r^3}+{\\\\cal O} \\\\left( \\\\frac{\\\\chi M_\\\\infty ^4}{r^4} \\\\right)\\\\,.$ Eqs.\",\n \"(REF ) and (REF ) represent our final results for the 2PN-accurate exterior post-Schwarzschild metric.\",\n \"From this construction it follows that post-TOV stars for which $\\\\mu _1 = \\\\mu _2 = \\\\pi _2 = 0$ have the Schwarzschild metric as the exterior spacetime.\",\n \"The following sections describe how the exterior metric can be used to compute observables of relevance for neutron star astrophysics.\"\n ],\n [\n \"Surface redshift & stellar compactness\",\n \"The surface redshift is among the most basic neutron star observables that could be affected by a change in the gravity theory.\",\n \"The surface redshift is defined in the usual way as $z_s \\\\equiv \\\\frac{\\\\lambda _{\\\\infty } - \\\\lambda _{\\\\rm s}}{\\\\lambda _{\\\\rm s}}= \\\\frac{f_{\\\\rm s}}{f_{\\\\infty }} - 1,$ where $\\\\lambda $ and $f$ are the wavelength and frequency of a photon, respectively.\",\n \"Here and below, the subscripts $s$ and $\\\\infty $ will denote the value of various quantities at the stellar surface $r = R$ and at spatial infinity.\",\n \"The familiar redshift formula $\\\\frac{f_\\\\infty }{f_{\\\\rm s}} = \\\\left[\\\\frac{g_{tt} (R)}{g_{tt} (\\\\infty )}\\\\right]^{1/2}$ is valid for any static spacetime, regardless of the form of the field equations.\",\n \"Using the metric (REF ), we easily obtain (at first post-TOV order) $\\\\frac{f_{\\\\rm s}}{f_{\\\\infty }} = \\\\left( 1 - \\\\frac{2 M_{\\\\infty }}{R}\\\\right)^{-1/2} + \\\\frac{\\\\chi }{3} \\\\frac{M_{\\\\infty }^3}{R^3}.$ Given that the frequency shift depends only on the ratio $M_\\\\infty /R$ , it is more convenient to work in terms of the stellar compactness $C = {M_\\\\infty }/{R}.$ Then from the definition of the surface redshift we obtain $z_{\\\\rm s} = z_{\\\\rm GR} + \\\\frac{\\\\chi }{3} C^3,$ where $z_{\\\\rm GR} \\\\equiv \\\\left(\\\\, 1 - 2C \\\\, \\\\right)^{-1/2} - 1,$ is the standard redshift formula in GR, while the second term represents the post-TOV correction.\",\n \"Observe that $z_{\\\\rm s}$ can be smaller or larger than $z_{\\\\rm GR}$ depending on the sign of the parameter $\\\\chi $ .\",\n \"This is shown in Fig.\",\n \"REF (left panel), where we plot the percent difference $\\\\delta z_{\\\\rm s}/z_{\\\\rm s}\\\\equiv 100 \\\\times (z_s - z_{\\\\rm GR})/z_{\\\\rm GR} $ as a function of $C$ for two representative cases ($\\\\chi = \\\\pm 0.1$ ).\",\n \"A characteristic property of the redshift is that it is a function of $C$ , and as such it cannot be used to disentangle mass and radius individually.\",\n \"A given observed surface redshift $z_{\\\\rm obs}$ can be experimentally interpreted either as $z_{\\\\rm obs} = z_{\\\\rm GR} (C)$ or $z_{\\\\rm obs} =z_s (C,\\\\chi )$ , and therefore lead to different estimates for $C$ (for a given $\\\\chi $ ).\",\n \"Fig.\",\n \"REF (right panel), where we plot the percent difference $\\\\delta C / C \\\\equiv 100 \\\\times (C-C_{\\\\rm GR})/C_{\\\\rm GR}$ as a function of $z_{\\\\rm s}$ , shows how much the inferred compactness would differ in the two cases where $\\\\chi = \\\\pm 0.1$ .\",\n \"A positive (negative) $\\\\chi $ leads to a lower (higher) inferred compactness with respect to GR.\",\n \"The figure suggests that the “error” in $C$ becomes significant for redshifts $z_{\\\\rm s} \\\\gtrsim 1$ .\",\n \"Figure: Surface redshift and stellar compactness.\",\n \"Relative percent changeswith respect to GR for both z s z_{\\\\rm s} and CC for different values of χ\\\\chi .It is a straightforward exercise to invert the redshift formula and obtain a post-TOV expression $C= C(z_{\\\\rm s})$ .\",\n \"We first write $1 + z_{\\\\rm s} = \\\\frac{1}{\\\\sqrt{-g_{tt} (R)}} \\\\,\\\\Rightarrow \\\\,g_{tt} (R) = -\\\\frac{1}{(1+z_{\\\\rm s})^2}.$ Upon inserting the post-Schwarzschild metric (REF ) we get a cubic equation for the compactness, $-1 + \\\\frac{1}{(1+z_{\\\\rm s})^2} + 2 C + \\\\frac{2}{3} \\\\chi C^3 = 0.$ In solving this equation we take into account that the small parameters are $C$ and $z_{\\\\rm s} \\\\sim {\\\\cal O}(C)$ and that the $\\\\chi \\\\rightarrow 0$ limit should be smooth.\",\n \"We find, $C = C_{\\\\rm GR} \\\\left(\\\\, 1 - \\\\frac{1}{3} \\\\chi z_{\\\\rm s}^2 \\\\, \\\\right),$ where $C_{\\\\rm GR} = \\\\frac{1}{2} \\\\left[ 1 - (1+z_{\\\\rm s})^{-2} \\\\right],$ is the corresponding solution in GR.\",\n \"As was the case for the post-TOV redshift formula, the compactness of a post-TOV star can be pushed above (below) the GR value by choosing a negative (positive) parameter $\\\\chi $ .\",\n \"The two main results of this section, Eqs.\",\n \"(REF ) and (REF ), are also interesting from a different prespective, namely, their dependence on the single post-TOV parameter $\\\\chi $ .\",\n \"This dependence entails a degeneracy with respect to the coefficient triad $\\\\lbrace \\\\mu _1, \\\\mu _2, \\\\pi _2 \\\\rbrace $ when (for example) a neutron star redshift observation is used as a gravity theory discriminator.\",\n \"The redshift/compactness $\\\\chi $ -degeneracy is another reminder of the intrinsic difficulty in distinguishing non-GR theories of gravity from neutron star physics (see e.g.\",\n \"discussion around Fig.\",\n \"1 in [1]).\"\n ],\n [\n \"Bursting neutron stars\",\n \"A potential testbed for measuring deviations from GR with a parametrized scheme like our post-TOV formalism is provided by accreting neutron stars exhibiting the so-called type I bursts.\",\n \"These are X-ray flashes powered by the nuclear detonation of accreted matter on the stellar surface layers [14].\",\n \"The luminosity associated with these events can reach the Eddington limit and may cause a photospheric radius expansion (see e.g.\",\n \"[15], [16]), thus offering a number of observational “handles\\\" to the system (see below for more details).\",\n \"A paper by Psaltis [7] proposed type I bursting neutron stars as a means to constrain possible deviations from GR.\",\n \"Psaltis' analysis, based on a static and spherically symmetric model for describing the spacetime outside a non-rotating neutron star, is general enough to allow a direct adaptation to the post-TOV scheme.\",\n \"For that reason we can omit most of the technical details discussed in [7] and instead focus on the key results derived in that paper.\",\n \"There is a number of observable quantities associated with type I bursting neutron stars that can be used to set up a test of GR.\",\n \"The first one is the surface redshift $z_{\\\\rm s}$ ; in Section we have derived post-TOV formulae for $z_{\\\\rm s}$ and the stellar compactness $C$ , which are used below in the derivation of a constraint equation between the post-Schwarzschild metric and the various observables.\",\n \"The luminosity (as measured at infinity) of a source located at a (luminosity) distance $D$ is $L_\\\\infty = 4\\\\pi D^2 F_\\\\infty ,$ where $F_\\\\infty $ is the (observable) flux.\",\n \"This luminosity can be written in a black-body form with the help of an apparent surface area $S_{\\\\rm app}$ and a color temperature (as measured at infinity) $\\\\bar{T}_{\\\\infty }$ : $4\\\\pi D^2 F_\\\\infty = \\\\sigma _{\\\\rm SB} S_{\\\\rm app} \\\\bar{T}^4_\\\\infty ,$ where $\\\\sigma _{\\\\rm SB}$ is the Stefan-Boltzmann constant.\",\n \"We then define the second observable parameter used in this analysis, i.e.\",\n \"the apparent radius $R_{\\\\rm app} \\\\equiv \\\\left( \\\\frac{S_{\\\\rm app}}{4\\\\pi } \\\\right)^{1/2} = D \\\\left( \\\\frac{F_\\\\infty }{\\\\sigma _{\\\\rm SB} \\\\bar{T}_\\\\infty ^4} \\\\right)^{1/2}.$ As evident from its form, $R_{\\\\rm app}$ is independent of the underpinning gravitational theory, at least to the extent that the theory does not appreciably modify the (luminosity) distance to the source.\",\n \"The surface color temperature is related to the intrinsic effective temperature $T_{\\\\rm eff}$ via the standard color correction factor $f_{\\\\rm c}$ [17], [18], $\\\\bar{T}_{\\\\rm s} = f_{\\\\rm c} T_{\\\\rm eff}.$ The observed temperature at infinity picks up a redshift factor with respect to its local surface value, that is, $\\\\bar{T}_\\\\infty = f_{\\\\rm c} \\\\sqrt{-g_{tt} (R)} T_{\\\\rm eff}.$ The effective temperature is the one related to the source's intrinsic luminosity, $L_{\\\\rm s} = 4\\\\pi R^2 \\\\sigma _{\\\\rm SB} T_{\\\\rm eff}^4.$ As shown in [7], $L_\\\\infty = - g_{tt} (R) L_{\\\\rm s} = 4\\\\pi R^2 \\\\sigma _{\\\\rm SB} \\\\left( \\\\frac{\\\\bar{T}_\\\\infty }{f_{\\\\rm c}} \\\\right)^4 [-g_{tt} (R)]^{-1}.$ Combining this with the preceding formulae leads to, $\\\\frac{R_{\\\\rm app}}{R} = \\\\frac{1+z_{\\\\rm s}}{f_{\\\\rm c}^2}.$ The third relevant observable is the Eddington luminosity/flux at infinity.\",\n \"This is given by [7], $L_{\\\\rm E}^\\\\infty = 4\\\\pi D^2 F^\\\\infty _{\\\\rm E} = \\\\frac{4\\\\pi }{\\\\kappa } \\\\frac{R^2}{(1+z_{\\\\rm s})^2} g_{\\\\rm eff},$ where $\\\\kappa $ is the opacity of the matter interacting with the radiation fieldTypically, this interaction manifests itself as Thomson scattering in a hydrogen-helium plasma, in which case the opacity is $\\\\kappa \\\\approx 0.2 (1+ X)\\\\,\\\\mbox{cm}^2/\\\\mbox{gr}$ where $X$ is the hydrogen mass fraction [16].\",\n \"and $g_{\\\\rm eff}$ is an effective surface gravitational acceleration, defined as: $g_{\\\\rm eff} =\\\\frac{1}{2\\\\sqrt{g_{rr} (R)}} \\\\frac{g^\\\\prime _{tt} (R)}{g_{tt} (R)}.$ This parameter is key to the present analysis as it encodes the departure from the general relativistic Schwarzschild metric.\",\n \"Figure: Bursting neutron star constraints.\",\n \"The surfaces arecontours of constant1+(2/3)χz s 2 z s (2+z s )/(1+z s ) 4 \\\\left[\\\\, 1 + (2/3) \\\\chi z_{\\\\rm s}^2 \\\\, \\\\right] z_{\\\\rm s}(2+z_{\\\\rm s})/(1+z_{\\\\rm s})^4 in the (z s ,χ)(z_{\\\\rm s},\\\\,\\\\chi ) plane.\",\n \"This quantity is a combinationof observables – cf.\",\n \"the right-hand side of Eq.\",\n \"()– and therefore it is potentially measurable; a measurement willsingle out a specific contour in this plot.\",\n \"A further measurement of(say) the redshift z s z_{\\\\rm s} corresponds to the intersectionbetween one such contour and a line with z s = const .z_{\\\\rm s}={\\\\rm const.\",\n \"},so it can lead to a determination of χ\\\\chi .Having at our disposal the above three observable combinations, the strategy is to combine them and derive a constraint equation between the observables and the spacetime metric.\",\n \"To this end, we first need to eliminate the not directly observable stellar radius $R$ between (REF ) and (REF ) and subsequently solve with respect to $g_{\\\\rm eff}$ .\",\n \"We obtain $g_{\\\\rm eff} = \\\\kappa \\\\sigma _{\\\\rm SB} \\\\frac{F_{\\\\rm E}^\\\\infty }{F_\\\\infty } \\\\left( \\\\frac{\\\\bar{T}_\\\\infty }{f_{\\\\rm c}} \\\\right)^4 (1+z_{\\\\rm s})^4.$ The remaining task is to express $ g_{\\\\rm eff} $ in terms of $z_{\\\\rm s}$ .\",\n \"Using the post-Schwarzschild metric, Eqs.\",\n \"(REF ) and (REF ), in (REF ) we obtain the post-TOV result $g_{\\\\rm eff} = \\\\frac{C}{R} (1+z_{\\\\rm s} ) \\\\left(\\\\, 1 + \\\\chi C^2 \\\\, \\\\right).$ Making use of Eq.\",\n \"(REF ) for the compactness leads to the desired result [cf.\",\n \"Eq.\",\n \"(39) in [7]]: $g_{\\\\rm eff} = \\\\frac{z_{\\\\rm s}}{2R} \\\\frac{(2+z_{\\\\rm s})}{(1+z_{\\\\rm s})} \\\\left(\\\\, 1 + \\\\frac{2}{3} \\\\chi z_{\\\\rm s}^2 \\\\, \\\\right),$ where, as evident, the prefactor represents the GR result.\",\n \"Finally, after eliminating $R$ with the help of (REF ) and (REF ), we obtain the “observable\\\" effective gravity: $g_{\\\\rm eff} = \\\\frac{z_{\\\\rm s}(2+z_{\\\\rm s})}{2 D f_{\\\\rm c}^2} \\\\left(\\\\, 1 + \\\\frac{2}{3} \\\\chi z_{\\\\rm s}^2 \\\\, \\\\right)\\\\left( \\\\frac{\\\\sigma _{\\\\rm SB} \\\\bar{T}^4_\\\\infty }{F_\\\\infty } \\\\right)^{1/2}.$ This can then be combined with (REF ) to give $\\\\frac{z_{\\\\rm s}(2+z_{\\\\rm s})}{(1+z_{\\\\rm s})^4} \\\\left(\\\\, 1 + \\\\frac{2}{3} \\\\chi z_{\\\\rm s}^2 \\\\, \\\\right)= 2D \\\\kappa \\\\frac{F_{\\\\rm E}^\\\\infty }{ f_{\\\\rm c}^2} \\\\left( \\\\frac{\\\\sigma _{\\\\rm SB} \\\\bar{T}_\\\\infty ^4 }{F_\\\\infty } \\\\right)^{1/2},$ and consequently $\\\\chi = \\\\frac{3}{2 z^2_{\\\\rm s}} \\\\left[\\\\, 2D \\\\kappa \\\\frac{ (1+z_{\\\\rm s})^4}{z_{\\\\rm s}(2+z_{\\\\rm s})} \\\\frac{F_{\\\\rm E}^\\\\infty }{ f_{\\\\rm c}^2}\\\\left( \\\\frac{\\\\sigma _{\\\\rm SB} \\\\bar{T}_\\\\infty ^4 }{F_\\\\infty } \\\\right)^{1/2} - 1 \\\\, \\\\right].$ This equation is the main result of this section and provides, at least as a proof of principle, a quantitative connection between the post-Schwarzschild correction to the exterior metric [in the form of the $\\\\chi $ coefficient defined in Eq.\",\n \"(REF )] and observable quantities in a type I bursting neutron star.\",\n \"Ref.\",\n \"[7] arrives at a similar result [their Eq.\",\n \"(49)] which has the same physical meaning, but is not identical to Eq.\",\n \"(REF ) due to the different assumed form of the exterior metric.\",\n \"Our results are illustrated in Fig.\",\n \"REF , where we show the left-hand side of Eq.\",\n \"(REF ) as a contour plot in the $(z_{\\\\rm s},\\\\,\\\\chi )$ plane.\",\n \"Each contour represents a specific measurement of this observable quantity.\",\n \"An additional surface redshift measurement can lead, at least in principle, to the determination of the post-TOV parameter $\\\\chi $ , as given by Eq.\",\n \"(REF ).\"\n ],\n [\n \"Quasi-periodic oscillations\",\n \"The post-Schwarzschild metric allows us to compute the geodesic motion of particles in the exterior spacetime of post-TOV neutron stars.\",\n \"Geodesics in neutron star spacetimes play a key role in the theoretical modelling of the QPOs observed in the X-ray spectra of accreting neutron stars.\",\n \"The detailed physical mechanism(s) responsible for the QPO-like time variability in the flux of these systems is still a matter of debate, but some of the most popular models are based on the notion of a radiating hot “blob” of matter moving in nearly circular geodesic orbits.\",\n \"The QPO frequencies are identified either with the orbital frequencies, or with simple combinations of the orbital frequencies.\",\n \"The most popular models are variants of the relativistic precession [8], [9] and epicyclic resonance [10] models.\",\n \"In this section we discuss the relevant orbital frequencies within the post-TOV formalism and derive formulae that could easily be used in the aforementioned QPO models.\",\n \"In principle, matching the orbital frequencies to the QPO data would allow one to extract post-TOV parameters such as $\\\\chi $ and $\\\\mu _1$ (see [19], [20], [21] for a similar exercise in the context of GR and scalar-tensor theory).\",\n \"For nearly circular orbits in a spherically symmetric spacetime, the only perturbations of interest are the radial ones (i.e., there is periastron precession but no Lense-Thirring nodal precession) and therefore we can associate two frequencies to every circular orbit: the orbital azimuthal frequency of the circular orbit $\\\\Omega _\\\\varphi $ and the radial epicyclic frequency $\\\\Omega _r$ .\",\n \"Geodesics in a static, spherically symmetric spacetime are characterized by the two usual conserved quantities, the energy $E =-g_{tt} \\\\dot{t}$ and the angular momentum $ L= g_{\\\\phi \\\\phi } \\\\dot{\\\\phi }$ .\",\n \"Here both constants are defined per unit particle mass, and the dots stand for differentiation with respect to proper time.\",\n \"The four-velocity normalization condition $u^au_a=-1$ yields an effective potential equation for the particle's radial motion, $g_{rr} \\\\dot{r}^2 = -\\\\frac{E^2}{g_{tt}}-\\\\frac{L^2}{g_{\\\\phi \\\\phi }}-1\\\\equiv V_{\\\\rm eff}(r).$ The conditions for circular orbits are $V_{\\\\rm eff}(r)=V^{\\\\prime }_{\\\\rm eff}(r)=0$ , where the prime denotes differentiation with respect to the radial coordinate.\",\n \"Hereafter $r$ will denote the circular orbit radius.\",\n \"From these conditions we can determine the orbital frequency $\\\\Omega _\\\\varphi \\\\equiv \\\\dot{\\\\phi }/\\\\dot{t}$ measured by an observer at infinity.Apart from its implications for the QPOs, the post-TOV corrected orbital frequency would imply a shift in the corotation radius $r_{\\\\rm co}$ in an accreting system.\",\n \"This radius is defined as $\\\\Omega _*= \\\\Omega _\\\\varphi (r_{\\\\rm co})$ , where $\\\\Omega _*$ is the stellar angular frequency, and plays a key role in determining the torque-spin equilibrium in magnetic field-disk coupling models.\",\n \"Using the above definition we find the following result for the post-TOV corotation radius: $r_{\\\\rm co}= M_{\\\\infty }(M_{\\\\infty } \\\\Omega _*)^{-2/3} \\\\left[ \\\\, 1 + (\\\\chi /3) (M_{\\\\infty } \\\\Omega _*)^{4/3} \\\\, \\\\right]$ .\",\n \"The square of the orbital frequency is then given as $\\\\Omega _\\\\varphi ^2 &= -\\\\frac{g^{\\\\prime }_{tt}}{g^{\\\\prime }_{\\\\phi \\\\phi }} =\\\\frac{M_{\\\\infty }}{r^3}\\\\left(1+\\\\chi \\\\frac{M_{\\\\infty }^2}{r^2}\\\\right).$ The Schwarzschild frequency is recovered for $\\\\chi =0$ .\",\n \"The radial epicyclic frequency can be calculated from the equation for radially perturbed circular orbits, which follows from Eq.\",\n \"(REF ): $\\\\Omega _r^2 &= -\\\\frac{g^{rr}}{2 \\\\dot{t}^2} V^{\\\\prime \\\\prime }_{\\\\rm eff}(r)\\\\approx \\\\frac{M_{\\\\infty }}{r^3}\\\\left[1-\\\\frac{6 M_{\\\\infty }}{r} -\\\\frac{\\\\chi M_{\\\\infty }^2}{r^2} + {\\\\cal O} (r^{-3})\\\\right] \\\\\\\\&= \\\\Omega _\\\\varphi ^2 \\\\left[1-\\\\frac{6 M_{\\\\infty }}{r} -\\\\frac{2 \\\\chi M_{\\\\infty }^2}{r^2} + {\\\\cal O} (r^{-3}) \\\\right], $ where again the frequency is calculated with respect to observers at infinity.\",\n \"From the post-TOV expanded result we can see that the first two terms correspond to the Schwarzschild epicyclic frequency.\",\n \"The additional post-TOV terms in these formulae produce a shift in the frequency and radius of the innermost stable circular orbit (ISCO) with respect to their GR values – the latter quantity is determined by the condition $\\\\Omega _r^2=0$ , which in GR leads to the well-known result $r_{\\\\rm isco}=6M_{\\\\infty }$ .\",\n \"The corresponding post-TOV ISCO is obtained from (REF ), up to linear order in $\\\\chi $ , as: $r_{\\\\rm isco} \\\\approx 6 M_{\\\\infty } \\\\left(1+\\\\frac{19}{324}\\\\chi \\\\right).$ The post-TOV ISCO parameters $r_{\\\\rm isco}$ and $(\\\\Omega _\\\\varphi )_{\\\\rm isco}$ are plotted in Fig.\",\n \"REF as functions of the parameter $\\\\chi $ .\",\n \"As evident from Eq.\",\n \"(REF ), a positive (negative) $\\\\chi $ implies $r_{\\\\rm isco} > 6M_{\\\\infty }$ ($r_{\\\\rm isco} < 6 M_{\\\\infty }$ ).\",\n \"If one takes Eq.\",\n \"(REF ) at face value for the given post-Schwarzschild metric, for negative enough values of $\\\\chi $ there is no ISCO solution, but this occurs well beyond the point where it is safe to use our perturbative formalism.\",\n \"The orbital frequency profile remains rather simple, with $(\\\\Omega _\\\\varphi )_{\\\\rm isco}$ exceeding the GR value when $r_{\\\\rm isco} < 6M_{\\\\infty }$ (and vice versa).\",\n \"Figure: ISCO quantities.\",\n \"ISCO quantities as functions ofχ\\\\chi .\",\n \"The solid curve corresponds to the relative difference (inpercent) of r isco r_{\\\\rm isco} with respect to GR, while the dashedcurve corresponds to the relative difference of the orbitalfrequency at the ISCO, (Ω ϕ ) isco ( \\\\Omega _\\\\varphi )_{\\\\rm isco}.Besides the frequency pair $\\\\lbrace \\\\Omega _\\\\varphi , \\\\Omega _r \\\\rbrace $ , a third prominent quantity in the QPO models is the frequency $\\\\Omega _{\\\\rm per}=\\\\Omega _\\\\varphi -\\\\Omega _r,$ associated with the orbital periastron precession (for example, in the relativistic precession model [8], [9] this frequency is typically associated with the low-frequency QPO) .\",\n \"Given our earlier results, it is straightforward to derive a series expansion in powers of $1/r$ for $\\\\Omega _{\\\\rm per}$ .\",\n \"However, it is usually more desirable to produce a series expansion with respect to an observable quantity, such as the circular orbital velocity $U_{\\\\infty }=(M_{\\\\infty }\\\\Omega _\\\\varphi )^{1/3}$ .\",\n \"This can be done by first expanding $U_\\\\infty $ with respect to $1/r$ and then inverting the expansion, thus producing a series in $U_\\\\infty $ .\",\n \"The outcome of this recipe is $\\\\frac{\\\\Omega _{\\\\rm per}}{\\\\Omega _\\\\varphi } &= 1-\\\\frac{\\\\Omega _r}{\\\\Omega _\\\\varphi }= 3 U_{\\\\infty }^2 +\\\\left(\\\\frac{9}{2} + \\\\chi \\\\right) U_{\\\\infty }^4 + {\\\\cal O} (U_{\\\\infty }^{6}).$ A similar “Keplerian\\\" version of this expression can be produced if we opt for using the velocity $U_{\\\\rm K}$ and mass $M_{\\\\rm K}$ that an observer would infer from the motion of (say) a binary system under the assumption of exactly Keplerian orbits.\",\n \"These are $U_{\\\\rm K} =(M_{\\\\rm K}\\\\Omega _\\\\varphi )^{1/3}$ and $M_{\\\\rm K}=r^3\\\\Omega ^2_\\\\varphi $ , so that $M_{\\\\rm K} =M_{\\\\infty }\\\\left(1+\\\\chi M_{\\\\infty }^2/r^2\\\\right)$ .\",\n \"The resulting series is identical to Eq.\",\n \"(REF ) when truncated to $U_{\\\\rm K}^4$ order.\",\n \"Higher-order terms, however, are different (see the following section).\",\n \"The above results for the frequencies $\\\\lbrace \\\\Omega _\\\\varphi , \\\\Omega _r, \\\\Omega _{\\\\rm per}\\\\rbrace $ suggest that a QPO-based test of GR within the post-TOV formalism could in principle allow the extraction of the post-TOV parameter $\\\\chi $ .\",\n \"In this sense these frequencies probe the same kind of deviation from GR (and suffer from the same degree of degeneracy) as the observations of bursting neutron stars discussed in Section .\",\n \"Figure: Orbital frequencies.Plots of Ω r \\\\Omega _r against Ω ϕ \\\\Omega _{\\\\varphi } fordifferent values of χ=±0.1,±0.5\\\\chi =\\\\pm 0.1,\\\\, \\\\pm 0.5 and ±1\\\\pm 1.\",\n \"The black solid curve corresponds to the GR case.The dashed curves correspond to positive values of χ\\\\chi (and r isco >6M ∞ r_{\\\\rm isco}>6M_{\\\\infty })while the dashed-dotted curves correspond to negative values of χ\\\\chi (andr isco <6M ∞ r_{\\\\rm isco}<6M_{\\\\infty }).We conclude this section by sketching how this procedure works in practice in the context of the relativistic precession model.\",\n \"The twin $\\\\mbox{kHz}$ QPO frequencies $\\\\lbrace \\\\nu _1, \\\\nu _2 \\\\rbrace $ seen in the flux of bright low-mass X-ray binaries are identified with the azimuthal and periastron precession orbital frequencies.\",\n \"More specifically, the high-frequency member of the pair is identified with the azimuthal frequency ($\\\\nu _2 = \\\\nu _\\\\varphi = \\\\Omega _\\\\varphi /2\\\\pi $ ), while the low-frequency member is identified with the periastron precession ($\\\\nu _1 = \\\\nu _{\\\\rm per} = \\\\Omega _{\\\\rm per}/2\\\\pi $ ).\",\n \"With this interpretation, the QPO separation is equal to the radial epicyclic frequency: $\\\\Delta \\\\nu = \\\\nu _2 - \\\\nu _1 = \\\\Omega _r/2\\\\pi $ .\",\n \"We use our previous results [Eqs.\",\n \"(REF ), (), (REF )] to plot these orbital frequencies (clearly, $\\\\nu _1/\\\\nu _2 = \\\\Omega _{\\\\rm per}/\\\\Omega _\\\\varphi $ and $\\\\Delta \\\\nu /\\\\nu _2 = \\\\Omega _r/\\\\Omega _\\\\varphi $ ) as functions of each other and for varying $\\\\chi $ .\",\n \"As it turns out, deviations from GR are best illustrated by plotting $\\\\Omega _r (\\\\Omega _\\\\varphi )$ (or equivalenty $\\\\Delta \\\\nu (\\\\nu _2)$ ).\",\n \"In Fig.\",\n \"REF we plot the dimensionless combinations $M_\\\\infty \\\\Omega _r,~M_\\\\infty \\\\Omega _\\\\varphi $ (in units of kHz for the frequencies and solar masses for $M_{\\\\infty }$ ).\",\n \"As we can see, the post-TOV models considered here ($ -1 <\\\\chi < 1 $ ) are qualitatively similar to the GR result (black solid curve), all cases showing the characteristic hump in $\\\\Omega _r$ as $\\\\Omega _\\\\varphi $ increases (so that the orbital radius decreases).\",\n \"This feature is evidently associated with the existence of an ISCO (where $\\\\Omega _r \\\\rightarrow 0$ ) and is consistent with a similar trend seen in observations [9].\"\n ],\n [\n \"Multipolar structure of the spacetime\",\n \"Expansions like Eq.\",\n \"(REF ) contain information about the multipolar structure of the background spacetime.\",\n \"That expansion can be directly compared against a similar expansion derived by Ryan [19] for an axisymmetric, stationary spacetime in GR with an arbitrary set of mass ($M_0=M_{\\\\infty }, M_2, M_4, ...$ ) and current ($S_1, S_3, S_5, ...$ ) Geroch-Hansen multipole moments [22], [23], [24]:A multipolar expansion in scalar-tensor theory can be found in [25].\",\n \"Specific calculations were also carried out in other theories: for example, the quadrupole moment was computed in Einstein-dilaton-Gauss-Bonnet gravity [26].\",\n \"$\\\\frac{\\\\Omega _{\\\\rm per}}{\\\\Omega _\\\\varphi } &= 3 U^2 - 4\\\\frac{S_1}{M_{\\\\infty }^2} U^3+ \\\\left(\\\\, \\\\frac{9}{2} - \\\\frac{3}{2} \\\\frac{M_2}{M_{\\\\infty }^3} \\\\,\\\\right) U^4 -10 \\\\frac{S_1}{M_{\\\\infty }^2} U^5\\\\nonumber \\\\\\\\& + \\\\left(\\\\, \\\\frac{27}{2} -2\\\\frac{S_1^2}{M_{\\\\infty }^4} -\\\\frac{21}{2} \\\\frac{M_2}{M_{\\\\infty }^3} \\\\, \\\\right) U^6 + {\\\\cal O} (U^{7} ).$ where $U = (M_{\\\\infty } \\\\Omega _\\\\varphi )^{1/3} $ denotes the orbital velocity.\",\n \"To understand the PN accuracy of the post-TOV expansion in this context, it is useful to consider the effect of higher PN order terms in the metric.\",\n \"Imagine that the $g_{tt}$ and $g_{rr}$ metric components [see Eqs.\",\n \"(REF ), (REF )] included 3PN corrections of the schematic form, $g_{tt}(r)&=g_{tt}^{\\\\rm 2PN}+\\\\alpha _{tt} \\\\frac{M_{\\\\infty }^3}{r^3},\\\\\\\\g_{rr}(r)&=g_{rr}^{\\\\rm 2PN}+\\\\alpha _{rr} \\\\frac{M_{\\\\infty }^3}{r^3}.$ We can use the coefficients $\\\\alpha _{tt}$ and $\\\\alpha _{rr}$ as bookeeping parameters in order to understand how these omitted higher-order contributions affect the results of the previous section.\",\n \"The recalculation of the various expressions reveals that the orbital frequency remains unchanged to 2PN order; the 3PN term of Eq.\",\n \"(REF ) contributes at the next order, as expected.\",\n \"The same applies to the epicyclic frequency, as we can see for example from the modified Eq.\",\n \"(), where the next-order correction is a mixture of $g_{rr}^{\\\\rm 2PN}$ and the 3PN term in $g_{tt}$ : $\\\\Omega _r^2&=\\\\Omega _\\\\varphi ^2 \\\\left[1-\\\\frac{6 M_{\\\\infty }}{r} -\\\\frac{2 \\\\chi M_{\\\\infty }^2}{r^2} \\\\right.\\\\nonumber \\\\\\\\&\\\\left.+ \\\\frac{(4 \\\\pi \\\\mu _1 -6\\\\alpha _{tt})M_{\\\\infty }^3}{r^3} + {\\\\cal O} (r^{-4}) \\\\right].$ Proceeding in a similar way we find the next-order correction to the Ryan-like expansion (REF ): $\\\\frac{\\\\Omega _{\\\\rm per}}{\\\\Omega _\\\\varphi } &= 3 U_{\\\\infty }^2 +\\\\left(\\\\frac{9}{2} + \\\\chi \\\\right) U_{\\\\infty }^4\\\\nonumber \\\\\\\\&+\\\\left[\\\\frac{27}{2} + 2 (\\\\chi -\\\\pi \\\\mu _1 )+3\\\\alpha _{tt}\\\\right] U_{\\\\infty }^6+ {\\\\cal O} (U_{\\\\infty }^{8}).$ We can see that the 3PN term “contaminates\\\" the PN correction that was omitted in Eq.\",\n \"(REF ).\",\n \"Repeating the same exercise for the Keplerian version of the multipole expansion (i.e.\",\n \"where the orbital velocity $U_\\\\infty $ is replaced by $U_{\\\\rm K}$ ) we find: $\\\\frac{\\\\Omega _{\\\\rm per}}{\\\\Omega _\\\\varphi } &= 3 U^2_{\\\\rm K} +\\\\left(\\\\frac{9}{2} + \\\\chi \\\\right) U^4_{\\\\rm K} \\\\nonumber \\\\\\\\&+\\\\left(\\\\frac{27}{2} - 2 \\\\pi \\\\mu _1+3\\\\alpha _{tt}\\\\right) U^6_{\\\\rm K} + {\\\\cal O} (U^{8}_{\\\\rm K} ).$ At the PN order considered in the previous section the two expressions were identical but, as we can see, they differ at the next order.\",\n \"We now have Ryan-type multipole expansions of the post-Schwarzschild spacetime up to 3PN in the circular orbital velocity, which we can compare against Eq.\",\n \"(REF ) to draw (with some caution) analogies and differences between GR and modified theories of gravity.\",\n \"For instance, odd powers of $U_\\\\infty $ are missing in Eq.\",\n \"(REF ) because the nonrotating post-Schwarzschild spacetime has vanishing current multipole moments.\",\n \"Furthermore, we can see that the quadrupole moment $M_2$ , first appearing in the coefficient of $U^4_\\\\infty $ in Eq.\",\n \"(REF ), can be associated with $\\\\chi $ .\",\n \"The parameter $\\\\chi $ is an effective quadrupole moment in the sense that $M_2^{\\\\rm eff} = -\\\\frac{2}{3} \\\\chi M_{\\\\infty }^3.$ Indeed, this relation implies that a positive (negative) $\\\\chi $ could be associated with an oblate (prolate) source of the gravitational field.\",\n \"The identification (REF ) holds at $ {\\\\cal O} (U_{\\\\infty }^{4})$ .\",\n \"The next-order term $U^6_\\\\infty $ would, in general, lead to a different effective $M_2$ .\",\n \"Hence, the comparison between the $U_{\\\\infty }^4$ and $U_{\\\\infty }^6$ terms could provide a null test for the GR-predicted quadrupole.\",\n \"However, there is a special case where these two terms could be consistent with the same effective quadrupole (REF ): this occurs when the post-TOV parameters satisfy the condition $5\\\\chi = -2\\\\pi \\\\mu _1+3\\\\alpha _{tt}$ , in which case the expansion (REF ) behaves as a “GR mimicker\\\".\",\n \"A different kind of “multipole” expansion in powers of $1/r$ can be applied to the metric functions $\\\\nu (r), m(r)$ [see Eqs.\",\n \"(REF )–(REF )], leading to an alternative calculation of the ADM mass $M_\\\\infty $ of a post-TOV star.\",\n \"We consider the expansions $\\\\nu (r) = \\\\sum _{n = 0}^{\\\\infty } \\\\frac{\\\\nu _{n}}{r^n}, \\\\qquad m(r) = \\\\sum _{n = 0}^{\\\\infty } \\\\frac{m_{n}}{r^n},$ where $\\\\nu _n$ and $m_n$ are constant coefficients.\",\n \"In addition, we impose that $\\\\nu _0 = 0$ and $m_0 = M_{\\\\infty }$ .\",\n \"We subsequently substitute these expansions into Eqs.\",\n \"(REF )–(), expand for $r/R \\\\gg 1$ , and then solve for the coefficients.\",\n \"The outcome of this exercise in the vacuum exterior spacetime is $m(r) &= M_{\\\\infty } - 2 \\\\pi \\\\mu _1 \\\\frac{M_{\\\\infty }^3}{r^2} + {\\\\cal O}(r^{-4})\\\\,,\\\\\\\\\\\\nonumber \\\\\\\\\\\\nu (r) &= - \\\\frac{2 M_{\\\\infty }}{r} - \\\\frac{2 M_{\\\\infty }^2}{r^2}- \\\\frac{2}{3} \\\\frac{M_{\\\\infty }^3}{r^3}\\\\left( 4 + \\\\chi \\\\right) + {\\\\cal O}(r^{-4})\\\\,.$ As expected, the top equation is consistent with our earlier result, Eq.\",\n \"(REF ).\",\n \"To get an agreement between Eqs.\",\n \"(REF ) and () we must expand the logarithm appearing in the former equation in powers of $M_{\\\\infty }/r$ , thus recovering Eq.\",\n \"().\"\n ],\n [\n \"Conclusions\",\n \"In this paper we have demonstrated the applicability of the post-TOV formalism to a number of facets of neutron star astrophysics.\",\n \"Let us summarize our main results.\",\n \"The exterior post-Schwarzschild metric [Eqs.\",\n \"(REF ) and (REF )] depends only on the ADM mass $M_\\\\infty $ (given by Eq.\",\n \"(REF )) and on just two post-TOV parameters $\\\\mu _1$ and $\\\\chi $ .\",\n \"These are subsequently used to produce a post-TOV formula for the surface redshift, Eq.\",\n \"(REF ), which is a function of the stellar compactness and $\\\\chi $ .\",\n \"Next, we have shown how a basic post-TOV model for type I bursting neutron stars can be constructed.\",\n \"The key equation here is (REF ), which gives $\\\\chi $ (the only post-TOV parameter appearing in the model) in terms of observable quantities.\",\n \"We also computed geodesic motion in the post-Schwarzschild exterior of post-TOV neutron star models, finding expressions for the orbital, epicyclic and periastron precession frequencies of nearly circular orbits [Eqs.\",\n \"(REF ), (), (REF )] and for the ISCO radius [Eq.\",\n \"(REF )].\",\n \"These results can be fed into models for QPOs from accreting neutron stars, such as the relativistic precession model.\",\n \"Finally, on a more theoretical level, we have sketched how the post-TOV parameters enter in the spacetime's multipolar structure [Eq.\",\n \"(REF )].\",\n \"The meticulous reader may have noticed that, in spite of the exterior metric being a function $g_{tt} (\\\\chi )$ and $g_{rr} (\\\\mu _1)$ , all other post-TOV results feature only $\\\\chi $ , while $\\\\mu _1$ is either not present or enters at higher order.\",\n \"This is not a coincidence: these quantities either depend solely on $g_{tt}$ (e.g.\",\n \"the redshift) or receive their leading-order contributions from $g_{tt}$ (e.g.\",\n \"the orbital frequencies).\",\n \"The post-TOV formalism developed in [1] and in this paper can be viewed as a basic “stage-one” version of a more general framework.\",\n \"There are several directions one can follow for taking the formalism to a more sophisticated level, and here we discuss just a couple of possibilities.\",\n \"An obvious improvement is the addition of stellar rotation.\",\n \"This is necessary because all astrophysical compact stars rotate, some of them quite rapidly, and the influence of rotation is ubiquitous, affecting to some extent all of the effects discussed in this paper.\",\n \"As a first stab at the problem, it would make sense to work in the Hartle-Thorne slow rotation approximation [27], [28], which should be accurate enough for all but the fastest spinning neutron stars [29].\",\n \"There are equally important possibilities for improvement on the modified gravity sector of the formalism.\",\n \"The present post-TOV theory is oblivious to the existence of dimensionful coupling constants, such as the ones appearing in many modified theories of gravity (e.g.\",\n \"$f(R)$ theories or theories quadratic in the curvature).\",\n \"These coupling parameters should be added to the existing set of fluid parameters, and participate in the algorithmic generation of “families\\\" of post-TOV terms (see [1] for details).\",\n \"The extended set of parameters will most likely lead to a proliferation of post-TOV terms, and result in more complicated stellar structure equations than the ones used so far [i.e.\",\n \"Eqs. ()].\",\n \"Among other things, this enhancement may allow one to study in more generality to what extent other theories of gravity are mapped onto the post-TOV formalism.\",\n \"Another limitation of the formalism is that it is intrinsically perturbative with respect to GR solutions.\",\n \"It is important to generalize to theories of gravity that present screening mechanisms; the viability of perturbative expansions in these theories is a topic of active research (see e.g.\",\n \"[30], [31], [32], [33]).\",\n \"We hope that the astrophysical applications outlined in this work will stimulate more research to address these issues.\",\n \"K.G.\",\n \"is supported by the Ramón y Cajal Programme of the Spanish Ministerio de Ciencia e Innovación and by NewCompStar (a COST-funded Research Networking Programme).\",\n \"H.O.S., G.P.\",\n \"and E.B.\",\n \"are supported by NSF CAREER Grant No.\",\n \"PHY-1055103.\",\n \"E.B.\",\n \"is supported by FCT contract IF/00797/2014/CP1214/CT0012 under the IF2014 Programme.\",\n \"This work was supported by the H2020-MSCA-RISE-2015 Grant No.\",\n \"StronGrHEP-690904.\",\n \"H.O.S.\",\n \"thanks the Instituto Superior Técnico for hospitality, and the Department of Physics and Astronomy of the University of Mississippi for financial support.\"\n ]\n]"},"id":{"kind":"string","value":"1606.05106"}}},{"rowIdx":996,"cells":{"text":{"kind":"list like","value":[["On logarithmic coefficients of some close-to-convex functions"],["Abstract The logarithmic coefficients $\\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\\mathbb{D}=\\{z\\in\\mathbb{C}:|z|<1\\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\\log \\frac{f(z)}{z}= 2\\sum_{n=1}^{\\infty} \\gamma_n z^n$.","Recently, D.K.","Thomas [On the logarithmic coefficients of close to convex functions, {\\it Proc.","Amer.","Math.","Soc.}","{\\bf 144} (2016), 1681--1687] proved that $|\\gamma_3|\\le \\frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and claimed that the estimate is sharp by providing a form of a extremal function.","In the present paper, we pointed out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument $0$).","We also determine a sharp upper bound of $|\\gamma_3|$ for close-to-convex functions (with argument $0$) with respect to the Koebe function."],["Introduction","Let $\\mathcal {A}$ denote the class of analytic functions $f$ in the unit disk $\\mathbb {D}=\\lbrace z\\in \\mathbb {C}:|z|<1\\rbrace $ normalized by $f(0)=0=f^{\\prime }(0)-1$ .","If $f\\in \\mathcal {A}$ then $f(z)$ has the following representation $f(z)= z+\\sum _{n=2}^{\\infty }a_n(f) z^n.$ We will simply write $a_n:=a_n(f)$ when there is no confusion.","Let $\\mathcal {S}$ denote the class of all univalent (i.e.","one-to-one) functions in $\\mathcal {A}$ .","A function $f\\in \\mathcal {A}$ is called starlike (convex respectively) if $f(\\mathbb {D})$ is starlike with respect to the origin (convex respectively).","Let $\\mathcal {S}^*$ and $\\mathcal {C}$ denote the class of starlike and convex functions in $\\mathcal {S}$ respectively.","It is well-known that a function $f\\in \\mathcal {A}$ is in $\\mathcal {S}^*$ if and only if ${\\rm Re\\,}\\left(zf^{\\prime }(z)/f(z)\\right)>0$ for $z\\in \\mathbb {D}$ .","Similarly, a function $f\\in \\mathcal {A}$ is in $\\mathcal {C}$ if and only if ${\\rm Re\\,}\\left(1+(zf^{\\prime \\prime }(z)/f^{\\prime }(z))\\right)>0$ for $z\\in \\mathbb {D}$ .","From the above it is easy to see that $f\\in \\mathcal {C}$ if and only if $zf^{\\prime }\\in \\mathcal {S}^*$ .","Given $\\alpha \\in (-\\pi /2,\\pi /2)$ and $g\\in \\mathcal {S}^*$ , a function $f\\in \\mathcal {A}$ is said to be close-to-convex with argument $\\alpha $ and with respect to $g$ if ${\\rm Re\\,} \\left(e^{i\\alpha }\\frac{zf^{\\prime }(z)}{g(z)}\\right)>0 \\quad z\\in \\mathbb {D}.$ Let $\\mathcal {K}_{\\alpha }(g)$ denote the class of all such functions.","Let $\\mathcal {K}(g):= \\bigcup _{\\alpha \\in (-\\pi /2,\\pi /2)} \\mathcal {K}_{\\alpha }(g) \\quad \\mbox{ and }\\quad \\mathcal {K}_{\\alpha }:= \\bigcup _{g\\in \\mathcal {S}^*} \\mathcal {K}_{\\alpha }(g)$ be the classes of functions called close-to-convex functions with respect to $g$ and close-to-convex functions with argument $\\alpha $ , respectively.","The class $\\mathcal {K}:= \\bigcup _{\\alpha \\in (-\\pi /2,\\pi /2)} \\mathcal {K}_{\\alpha }= \\bigcup _{g\\in \\mathcal {S}^*} \\mathcal {K}(g)$ is the class of all close-to-convex functions.","It is well-known that every close-to-convex function is univalent in $\\mathbb {D}$ (see [2]).","Geometrically, $f\\in \\mathcal {K}$ means that the complement of the image-domain $f(\\mathbb {D})$ is the union of non-intersecting half-lines.","The logarithmic coefficients of $f\\in \\mathcal {S}$ are defined by $\\log \\frac{f(z)}{z}= 2\\sum _{n=1}^{\\infty } \\gamma _n z^n$ where $\\gamma _n$ are known as the logarithmic coefficients.","The logarithmic coefficients $\\gamma _n$ play a central role in the theory of univalent functions.","Very few exact upper bounds for $\\gamma _n$ seem have been established.","The significance of this problem in the context of Bieberbach conjecture was pointed out by Milin in his conjecture.","Milin conjectured that for $f\\in \\mathcal {S}$ and $n\\ge 2$ , $\\sum _{m=1}^{n}\\sum _{k=1}^{m} \\left(k|\\gamma _k|^2-\\frac{1}{k}\\right)\\le 0,$ which led De Branges, by proving this conjecture, to the proof of the Bieberbach conjecture [1].","More attention has been given to the results of an average sense (see [2], [3]) than the exact upper bounds for $|\\gamma _n|$ .","For the Koebe function $k(z)=z/(1-z)^2$ , the logarithmic coefficients are $\\gamma _n=1/n$ .","Since the Koebe function $k(z)$ plays the role of extremal function for most of the extremal problems in the class $\\mathcal {S}$ , it is expected that $|\\gamma _n|\\le \\frac{1}{n}$ holds for functions in $\\mathcal {S}$ .","But this is not true in general, even in order of magnitude [2].","Indeed, there exists a bounded function $f$ in the class $\\mathcal {S}$ with logarithmic coefficients $\\gamma _n\\ne O(n^{-0.83})$ (see [2]).","By differentiating (REF ) and equating coefficients we obtain $\\gamma _1=\\frac{1}{2} a_2$ $\\gamma _2=\\frac{1}{2}(a_3-\\frac{1}{2}a_2^2)$ $\\gamma _3=\\frac{1}{2}(a_4-a_2a_3+\\frac{1}{3}a_2^3).$ If $f\\in \\mathcal {S}$ then $|\\gamma _1|\\le 1$ follows at once from (REF ).","Using Fekete-Szegö inequality [2] in (REF ), we can obtain the sharp estimate $|\\gamma _2|\\le \\frac{1}{2}(1+2e^{-2})=0.635\\ldots .$ For $n\\ge 3$ , the problem seems much harder, and no significant upper bound for $|\\gamma _n|$ when $f\\in \\mathcal {S}$ appear to be known.","If $f\\in \\mathcal {S}^*$ then it is not very difficult to prove that $|\\gamma _n|\\le \\frac{1}{n}$ for $n\\ge 1$ and equality holds for the Koebe function $k(z)=z/(1-z)^2$ .","The inequality $|\\gamma _n|\\le \\frac{1}{n}$ for $n\\ge 2$ extends to the class $\\mathcal {K}$ was claimed in a paper of Elhosh [4].","However, Girela [6] pointed out some error in the proof of Elhosh [4] and, hence, the result is not substantiated.","Indeed, Girela proved that for each $n\\ge 2$ , there exists a function $f\\in \\mathcal {K}$ such that $|\\gamma _n|> \\frac{1}{n}$ .","In the same paper it has been shown that $|\\gamma _n|\\le \\frac{3}{2n}$ holds for $n\\ge 1$ whenever $f$ belongs to the set of extreme points of the closed convex hull of the class $\\mathcal {K}$ .","Recently, Thomas [12] proved that $|\\gamma _3|\\le \\frac{7}{12}$ for functions in $\\mathcal {K}_0$ (close-to-convex functions with argument 0) with the additional assumption that the second coefficient of the corresponding starlike function $g$ is real.","Thomas claimed that this estimate is sharp and has given a form of the extremal function.","But after rigorous reading of the paper [12], we observed that such functions do not belong to the class $\\mathcal {K}_0$ (more details will be given in Section ).","By fixing a starlike function $g$ in the class $\\mathcal {S}^*$ , the inequality (REF ) assertions a specific subclass of close-to-convex functions.","One of such important subclass is the class of close-to-convex functions with respect to the Koebe function $k(z)=z/(1-z)^2$ .","In this case, the inequality (REF ) becomes ${\\rm Re\\,} \\left(e^{i\\alpha }(1-z)^2f^{\\prime }(z)\\right)>0,\\quad z\\in \\mathbb {D}$ and defines the subclass $\\mathcal {K}_{\\alpha }(k)$ .","Several authors have been extensively studied the class of functions $f\\in \\mathcal {S}$ that satisfies the condition (REF ) (see [5], [7], [9], [11]).","Geometrically (REF ) says that the function $h:=e^{i\\delta }f$ has the boundary normalization $\\lim _{t\\rightarrow \\infty } h^{-1}(h(z)+t)=1$ and $h(\\mathbb {D})$ is a domain such that $\\lbrace w + t : t\\ge 0\\rbrace \\subseteq h(\\mathbb {D})$ for every $w\\in h(\\mathbb {D})$ .","Clearly, the image domain $h(\\mathbb {D})$ is convex in the positive direction of the real axis.","Denote by $\\mathcal {CR}^+:=\\mathcal {K}_{0}(k)$ the class of close-to-convex functions with argument 0 and with respect to Koebe function $k(z)$ .","That is $\\mathcal {CR}^+=\\left\\lbrace f\\in \\mathcal {A}: {\\rm Re\\,} (1-z)^2f^{\\prime }(z)>0, ~~ z\\in \\mathbb {D} \\right\\rbrace .$ Then clearly functions in $\\mathcal {CR}^+$ are convex in the positive direction of the real axis.","In the present article, we determine the upper bound of $|\\gamma _3|$ for functions in $\\mathcal {K}_0$ and $\\mathcal {CR}^+$ ."],["Main Results","Let $\\mathcal {P}$ denote the class of analytic functions $P$ with positive real part on $\\mathbb {D}$ which has the form $P(z)= 1+\\sum _{n=1}^{\\infty }c_n z^n.$ Functions in $\\mathcal {P}$ are sometimes called Carathéodory function.","To prove our main results, we need some preliminary lemmas.","The first one is known as Carathéodory's lemma (see [2] for example) and the second one is due to Libera and Złotkiewicz [10].","Lemma 2.1 [2] For a function $P\\in \\mathcal {P}$ of the form (REF ), the sharp inequality $|c_n|\\le 2$ holds for each $n\\ge 1$ .","Equality holds for the function $P(z)=(1+z)/(1-z)$ .","Lemma 2.2 [10] Let $P\\in \\mathcal {P}$ be of the form (REF ).","Then there exist $x, t\\in \\mathbb {C}$ with $|x|\\le 1$ and $|t|\\le 1$ such that $2c_2 = c_1^2 + x(4 - c_1^2)$ and $4c_3= c_1^3+ 2(4-c_1^2)c_1x-c_1(4-c_1^2)x^2+2(4-c_1^2)(1-|x|^2)t.$ In [12], Thomas claimed that his result (i.e.","$|\\gamma _3|\\le 7/12$ ) is sharp for functions in the class $\\mathcal {K}_0$ by ascertaining the equality holds for a function $f$ defined by $zf^{\\prime }(z)=g(z)P(z)$ where $g\\in \\mathcal {S}^*$ with $b_2(g)=b_3(g)=b_4(g)=2$ and $P\\in \\mathcal {P}$ with $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .","But in view of Lemma REF , it is easy to see that there does not exist a function $P\\in \\mathcal {P}$ with the property $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .","Thus we can conclude that the result obtained by Thomas is not sharp.","The main aim of the present paper is to obtain a better upper bound for $|\\gamma _3|$ for functions in the class $\\mathcal {K}_0$ than that of obtained by Thomas [12].","To prove our main results we also need the following Fekete-Szegö inequality for functions in the class $\\mathcal {S}^*$ .","Lemma 2.3 [8] Let $g\\in \\mathcal {S}^*$ be of the form $g(z)=z+\\sum _{n=2}^{\\infty }b_n z^n$ .","Then for any $\\lambda \\in \\mathbb {C}$ , $|b_3-\\lambda b_2^2|\\le \\max \\lbrace 1,|3-4\\lambda |\\rbrace .$ The inequality is sharp for $k(z)=z/(1-z)^2$ if $|3-4\\lambda |\\ge 1$ and for $(k(z^2))^{1/2}$ if $|3-4\\lambda |<1$ .","For $f\\in \\mathcal {K}_0$ (close-to-convex functions with argument 0), we obtained the following improved result for $|\\gamma _3|$ (compare [12]).","Theorem 2.1 If $f\\in \\mathcal {K}_0$ then $|\\gamma _3|\\le \\frac{1}{18} (3+4 \\sqrt{2})=0.4809$ .","Let $f\\in \\mathcal {K}_0$ be of the form (REF ).","Then there exists a starlike function $g(z)=z+\\sum _{n=2}^{\\infty }b_n z^n$ and a Carathéodory function $P\\in \\mathcal {P}$ of the form (REF ) such that $zf^{\\prime }(z)=g(z)P(z).$ A comparison of the coefficients on the both sides of (REF ) yields $a_2&=\\frac{1}{2}(b_2+c_1)\\\\a_3&=\\frac{1}{3}(b_3+b_2c_1+c_2)\\\\a_4&=\\frac{1}{4}(b_4+b_3c_1+b_2c_2+c_3).$ By substituting the above $a_2, a_3$ and $a_4$ in (REF ) and then further simplification gives $2\\gamma _3&= a_4-a_2a_3+\\frac{1}{3}a_2^3\\\\&=\\frac{1}{24}\\left((6b_4-4b_2b_3+b_2^3)+\\frac{c_1}{2}\\left(b_3-\\frac{1}{2}b_2^2\\right)+b_2(2c_2-c_1^2)+c_1^3-4c_1c_2+6c_3\\right)\\nonumber .$ In view of Lemma REF and writing $c_2$ and $c_3$ in terms of $c_1$ we obtain $48\\gamma _3&= (6b_4-4b_2b_3+b_2^3)+ 2c_1\\left(b_3-\\frac{1}{2}b_2^2\\right)+ b_2x(4-c_1^2)\\\\&\\quad +\\frac{1}{2}c_1^3+c_1x(4-c_1^2)-\\frac{3}{2}c_1x^2(4-c_1^2)+3(4-c_1^2)(1-|x|^2)t,\\nonumber $ where $|x|\\le 1$ and $|t|\\le 1$ .","Note that if $\\gamma _3(g)$ denote the third logarithmic coefficient of $g\\in \\mathcal {S}^*$ then $|\\gamma _3(g)|=\\frac{1}{2}|b_4-b_2b_3+\\frac{1}{3}b_2^3|\\le \\frac{1}{3}$ .","Since $g\\in \\mathcal {S}^*$ , in view of Lemma REF we obtain $|6b_4-4b_2b_3+b_2^3|\\le 6|b_4-b_2b_3+\\frac{1}{3}b_2^3|+2|b_2||b_3-\\frac{1}{2}b_2^2|\\le 8.$ Since the class $\\mathcal {K}_0$ is invariant under rotation, without loss of generality we can assume that $c_1=c$ , where $0\\le c\\le 2$ .","Taking modulus on both the sides of (REF ) and then applying triangle inequality and further using the inequality (REF ) and Lemma REF , it follows that $48|\\gamma _3|\\le 8+ 2c+ 2|x|(4-c^2)+\\left|\\frac{1}{2}c^3+cx(4-c^2)-\\frac{3}{2}cx^2(4-c^2)\\right|+3(4-c^2)(1-|x|^2),$ where we have also used the fact $|t|\\le 1$ .","Let $x=re^{i\\theta }$ where $0\\le r\\le 1$ and $0\\le \\theta \\le 2\\pi $ .","For simplicity, by writing $\\cos \\theta =p$ we obtain $48|\\gamma _3|\\le \\psi (c,r)+\\left|\\phi (c,r,p)\\right|=:F(c,r,p)$ where $\\psi (c,r)=8+ 2c+ 2r(4-c^2)+3(4-c^2)(1-r^2)$ and $\\phi (c,r,p)&=\\left(\\frac{1}{4}c^6+c^2r^2(4-c^2)^2+\\frac{9}{4}c^2r^4(4-c^2)^2+c^4(4-c^2)rp\\right.\\\\&\\qquad \\quad \\left.-\\frac{3}{2}c^4r^2(4-c^2)(2p^2-1)-3c^2(4-c^2)r^3p \\right)^{1/2}.$ Thus we need to find the maximum value of $F(c,r,p)$ over the rectangular cube $R:=[0,2]\\times [0,1]\\times [-1,1]$ .","By elementary calculus one can verify the followings: $&\\max _{0\\le r\\le 1} \\psi (0,r)=\\psi \\left(0,\\frac{1}{3}\\right)=\\frac{64}{3},\\quad \\max _{0\\le r\\le 1} \\psi (2,r)=12,\\\\[2mm]&\\max _{0\\le c\\le 2} \\psi (c,0)=\\psi \\left(\\frac{1}{3},0\\right)=\\frac{61}{3},\\quad \\max _{0\\le c\\le 2} \\psi (c,1)=\\psi (0,1)=16 \\quad \\mbox{ and }\\\\[2mm]&\\max _{(c,r)\\in [0,2]\\times [0,1]} \\psi (c,r)=\\psi \\left(\\frac{3}{10},\\frac{1}{3}\\right)=\\frac{649}{30}=21.6333.$ We first find the maximum value of $F(c,r,p)$ on the boundary of $R$ , i.e on the six faces of the rectangular cube $R$ .","On the face $c=0$ , we have $F(0,r,p)=\\psi (0,r)$ , where $(r,p)\\in R_1:=[0,1]\\times [-1,1]$ .","Thus $\\max _{(r,p)\\in R_1} F(0,r,p)= \\max _{0\\le r\\le 1} \\psi (0,r)=\\psi \\left(0,\\frac{1}{3}\\right)=\\frac{64}{3}=21.33.$ On the face $c=2$ , we have $F(2,r,p)= 16$ , where $(r,p)\\in R_1$ .","On the face $r=0$ , we have $F(c,0,p)=8+ 2c+3(4-c^2)+\\frac{1}{2}c^3$ , where $(c,p)\\in R_2:=[0,2]\\times [-1,1]$ .","By using elementary calculus it is easy to see that $\\max _{(c,p)\\in R_2} F(c,0,p)= F\\left(\\frac{2}{3} (3-\\sqrt{6}),0,p\\right)=\\frac{16}{9} \\left(9+\\sqrt{6}\\right)=20.3546.$ On the face $r=1$ , we have $F(c,1,p)=\\psi (c,1)+ |\\phi (c,1,p)|$ , where $(c,p)\\in R_2$ .","We first prove that $\\phi (c,1,p)\\ne 0$ in the interior of $R_2$ .","On the contrary, if $\\phi (c,1,p)=0$ in the interior of $R_2$ then $|\\phi (c,1,p)|^2=\\left|\\frac{1}{2}c^3+ce^{i\\theta }(4-c^2)-\\frac{3}{2}ce^{2i\\theta }(4-c^2)\\right|^2=0$ and hence $\\frac{1}{2}c^3+cp(4-c^2)-\\frac{3}{2}c(4-c^2)(2p^2-1)=0 ~\\mbox{ and } c(4-c^2)\\sin \\theta -\\frac{3}{2}c(4-c^2)\\sin 2\\theta =0.$ Further, (REF ) reduces to $\\frac{1}{2}c^2+p(4-c^2)-\\frac{3}{2}(4-c^2)(2p^2-1)=0 \\quad \\mbox{and}\\quad 1-3p=0,$ which is equivalent to $p=1/3$ and $c^2=6$ .","This contradicts the range of $c\\in (0,2)$ .","Thus $\\phi (c,1,p)\\ne 0$ in the interior of $R_2$ .","Next, we prove that $F(c,1,p)$ has no maximum at any interior point of $R_2$ .","Suppose that $F(c,1,p)$ has the maximum at an interior point of $R_2$ .","Then at such point $\\frac{\\partial F(c,1,p)}{\\partial c}=0$ and $\\frac{\\partial F(c,1,p)}{\\partial p}=0$ .","From $\\frac{\\partial F(c,1,p)}{\\partial p}=0$ , (for points in the interior of $R_2$ ), a straight forward calculation gives $p=\\frac{2 \\left(c^2-3\\right)}{3 c^2}.$ Substituting the value of $p$ as given in (REF ) in the relation $\\frac{\\partial F(c,1,p)}{\\partial c}=0$ and further simplification gives $3c^3-2c+(2c-1) \\sqrt{6(c^2+2)}=0.$ It is easy to show that the function $\\rho (c)=3c^3-2c+(2c-1) \\sqrt{6(c^2+2)}$ is strictly increasing in $(0,2)$ .","Since $\\rho (0)<0$ and $\\rho (2)>0$ , the equation (REF ) has exactly one solution in $(0,2)$ .","By solving the equation (REF ) numerically, we obtain the approximate root in $(0,2)$ as $0.5772$ .","But the corresponding value of $p$ obtained by (REF ) is $-5.3365$ which does not belong to $(-1,1)$ .","Thus $F(c,1,p)$ has no maximum at any interior point of $R_2$ .","Thus we find the maximum value of $F(c,1,p)$ on the boundary of $R_2$ .","Clearly, $F(0,1,p)=F(2,1,p)=16$ , $F(c,1,-1)={\\left\\lbrace \\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(10-3c^2)& \\mbox{for}\\quad 0\\le c\\le \\sqrt{\\frac{10}{3}}\\\\[2mm]8+ 2c+ 2(4-c^2)-c(10-3c^2)& \\mbox{for}\\quad \\sqrt{\\frac{10}{3}}< c\\le 2\\end{array}\\right.","}$ and $F(c,1,1)={\\left\\lbrace \\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(2-c^2)& \\mbox{for}\\quad 0\\le c\\le \\sqrt{2}\\\\[2mm]8+ 2c+ 2(4-c^2)-c(2-c^2)& \\mbox{for}\\quad \\sqrt{2}< c\\le 2.\\end{array}\\right.","}$ By using elementary calculus we find that $\\max _{0\\le c\\le 2}F(c,1,-1)= F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023\\quad \\mbox{ and }$ $\\max _{0\\le c\\le 2}F(c,1,1)= F\\left(\\frac{2}{3},1,1\\right)=\\frac{427}{27} =17.48.$ Hence, $\\max _{(c,p)\\in R_2} F(c,1,p)= F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023.$ On the face $p=-1$ , $F(c,r,-1)={\\left\\lbrace \\begin{array}{ll}\\psi (c,r)+\\eta _1(c,r) & \\mbox{ for }\\quad \\eta _1(c,r)\\ge 0\\\\[2mm]\\psi (c,r)-\\eta _1(c,r)& \\mbox{ for }\\quad \\eta _1(c,r)< 0,\\end{array}\\right.","}$ where $\\eta _1(c,r)=c^3 (3r^2+2r+1)-4cr(3r+2)$ and $(c,r)\\in R_3:=[0,2]\\times [0,1]$ .","Differentiating partially $F(c,r,-1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\max _{(c,r)\\in {\\rm int \\,}R_3\\setminus S_1} F(c,r,-1)= F\\left(2(\\sqrt{2}-1),\\frac{1}{3}(1+\\sqrt{2}),-1\\right)=\\frac{8}{3} (3+4 \\sqrt{2})=23.0849,$ where $S_1=\\lbrace (c,r)\\in R_3: \\eta _1(c,r)=0\\rbrace $ .","Now we find the maximum value of $F(c,r,-1)$ on the boundary of $R_3$ and on the set $S_1$ .","Note that $\\max _{(c,r)\\in S_1} F(c,r,-1)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\frac{649}{30}=21.6333.$ On the other hand by using elementary calculus, as before, we find that $\\max _{(c,r)\\in \\partial R_3} F(c,r,-1)=F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023,$ where $\\partial R_3$ denotes the boundary of $R_3$ .","Hence, by combining the above cases we obtain $\\max _{(c,r)\\in R_3} F(c,r,-1)=F\\left(2(\\sqrt{2}-1),\\frac{1}{3}(1+\\sqrt{2}),-1\\right)=\\frac{8}{3} (3+4 \\sqrt{2})=23.0849.$ On the face $p=1$ , $F(c,r,1)={\\left\\lbrace \\begin{array}{ll}\\psi (c,r)+\\eta _2(c,r) & \\mbox{ for }\\quad \\eta _2(c,r)\\ge 0\\\\[2mm]\\psi (c,r)-\\eta _2(c,r)& \\mbox{ for }\\quad \\eta _2(c,r)< 0,\\end{array}\\right.","}$ where $\\eta _2(c,r)=c^3 (3r^2-2r+1)+4cr(3r-2)$ and $(c,r)\\in R_3$ .","Differentiating partially $F(c,r,1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\max _{(c,r)\\in {\\rm int \\,}R_3\\setminus S_2} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.89,$ where $S_2=\\lbrace (c,r)\\in R_3: \\eta _2(c,r)=0\\rbrace $ .","Now, we find the maximum value of $F(c,r,1)$ on the boundary of $R_3$ and on the set $S_2$ .","By noting that $\\max _{(c,r)\\in S_2} F(c,r,1)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\frac{649}{30}=21.6333$ and proceeding similarly as in the previous case, we find that $\\max _{(c,r)\\in R_3} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.89.$ Let $S^{\\prime }=\\lbrace (c,r,p)\\in R: \\phi (c,r,p)=0\\rbrace $ .","Then $\\max _{(c,r,p)\\in S^{\\prime }} F(c,r,p)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\psi \\left(\\frac{3}{10},\\frac{1}{3}\\right)=\\frac{649}{30}=21.6333.$ We prove that $F(c,r,p)$ has no maximum value at any interior point of $R\\setminus S^{\\prime }$ .","Suppose that $F(c,r,p)$ has a maximum value at an interior point of $R\\setminus S^{\\prime }$ .","Then at such point $\\frac{\\partial F}{\\partial c}=0$ , $\\frac{\\partial F}{\\partial r}=0$ and $\\frac{\\partial F}{\\partial p}=0$ .","Note that $\\frac{\\partial F}{\\partial c}$ , $\\frac{\\partial F}{\\partial r}$ and $\\frac{\\partial F}{\\partial p}$ may not exist at points in $S^{\\prime }$ .","In view of $\\frac{\\partial F}{\\partial p}=0$ (for points in the interior of $R\\setminus S^{\\prime }$ ), a straight forward but laborious calculation gives $p= \\frac{3 c^2 r^2+c^2-12 r^2}{6 c^2 r}.$ Substituting the value of $p$ as given in (REF ) in the relations $\\frac{\\partial F}{\\partial c}=0$ and $\\frac{\\partial F}{\\partial r}=0$ and simplifying (again, a long and laborious calculation), we obtain $\\frac{3 \\sqrt{6} c^3 (1-3 r^2)+12 (c(3r^2-2r-3)+1)\\sqrt{c^2+2} )+4\\sqrt{6}c}{6\\sqrt{c^2+2}}=0$ and $(4-c^2) \\left(( \\sqrt{6(c^2+2)}-6) r+2\\right)=0.$ Since $00$ for $z\\\\in \\\\mathbb {D}$ .\",\n \"Similarly, a function $f\\\\in \\\\mathcal {A}$ is in $\\\\mathcal {C}$ if and only if ${\\\\rm Re\\\\,}\\\\left(1+(zf^{\\\\prime \\\\prime }(z)/f^{\\\\prime }(z))\\\\right)>0$ for $z\\\\in \\\\mathbb {D}$ .\",\n \"From the above it is easy to see that $f\\\\in \\\\mathcal {C}$ if and only if $zf^{\\\\prime }\\\\in \\\\mathcal {S}^*$ .\",\n \"Given $\\\\alpha \\\\in (-\\\\pi /2,\\\\pi /2)$ and $g\\\\in \\\\mathcal {S}^*$ , a function $f\\\\in \\\\mathcal {A}$ is said to be close-to-convex with argument $\\\\alpha $ and with respect to $g$ if ${\\\\rm Re\\\\,} \\\\left(e^{i\\\\alpha }\\\\frac{zf^{\\\\prime }(z)}{g(z)}\\\\right)>0 \\\\quad z\\\\in \\\\mathbb {D}.$ Let $\\\\mathcal {K}_{\\\\alpha }(g)$ denote the class of all such functions.\",\n \"Let $\\\\mathcal {K}(g):= \\\\bigcup _{\\\\alpha \\\\in (-\\\\pi /2,\\\\pi /2)} \\\\mathcal {K}_{\\\\alpha }(g) \\\\quad \\\\mbox{ and }\\\\quad \\\\mathcal {K}_{\\\\alpha }:= \\\\bigcup _{g\\\\in \\\\mathcal {S}^*} \\\\mathcal {K}_{\\\\alpha }(g)$ be the classes of functions called close-to-convex functions with respect to $g$ and close-to-convex functions with argument $\\\\alpha $ , respectively.\",\n \"The class $\\\\mathcal {K}:= \\\\bigcup _{\\\\alpha \\\\in (-\\\\pi /2,\\\\pi /2)} \\\\mathcal {K}_{\\\\alpha }= \\\\bigcup _{g\\\\in \\\\mathcal {S}^*} \\\\mathcal {K}(g)$ is the class of all close-to-convex functions.\",\n \"It is well-known that every close-to-convex function is univalent in $\\\\mathbb {D}$ (see [2]).\",\n \"Geometrically, $f\\\\in \\\\mathcal {K}$ means that the complement of the image-domain $f(\\\\mathbb {D})$ is the union of non-intersecting half-lines.\",\n \"The logarithmic coefficients of $f\\\\in \\\\mathcal {S}$ are defined by $\\\\log \\\\frac{f(z)}{z}= 2\\\\sum _{n=1}^{\\\\infty } \\\\gamma _n z^n$ where $\\\\gamma _n$ are known as the logarithmic coefficients.\",\n \"The logarithmic coefficients $\\\\gamma _n$ play a central role in the theory of univalent functions.\",\n \"Very few exact upper bounds for $\\\\gamma _n$ seem have been established.\",\n \"The significance of this problem in the context of Bieberbach conjecture was pointed out by Milin in his conjecture.\",\n \"Milin conjectured that for $f\\\\in \\\\mathcal {S}$ and $n\\\\ge 2$ , $\\\\sum _{m=1}^{n}\\\\sum _{k=1}^{m} \\\\left(k|\\\\gamma _k|^2-\\\\frac{1}{k}\\\\right)\\\\le 0,$ which led De Branges, by proving this conjecture, to the proof of the Bieberbach conjecture [1].\",\n \"More attention has been given to the results of an average sense (see [2], [3]) than the exact upper bounds for $|\\\\gamma _n|$ .\",\n \"For the Koebe function $k(z)=z/(1-z)^2$ , the logarithmic coefficients are $\\\\gamma _n=1/n$ .\",\n \"Since the Koebe function $k(z)$ plays the role of extremal function for most of the extremal problems in the class $\\\\mathcal {S}$ , it is expected that $|\\\\gamma _n|\\\\le \\\\frac{1}{n}$ holds for functions in $\\\\mathcal {S}$ .\",\n \"But this is not true in general, even in order of magnitude [2].\",\n \"Indeed, there exists a bounded function $f$ in the class $\\\\mathcal {S}$ with logarithmic coefficients $\\\\gamma _n\\\\ne O(n^{-0.83})$ (see [2]).\",\n \"By differentiating (REF ) and equating coefficients we obtain $\\\\gamma _1=\\\\frac{1}{2} a_2$ $\\\\gamma _2=\\\\frac{1}{2}(a_3-\\\\frac{1}{2}a_2^2)$ $\\\\gamma _3=\\\\frac{1}{2}(a_4-a_2a_3+\\\\frac{1}{3}a_2^3).$ If $f\\\\in \\\\mathcal {S}$ then $|\\\\gamma _1|\\\\le 1$ follows at once from (REF ).\",\n \"Using Fekete-Szegö inequality [2] in (REF ), we can obtain the sharp estimate $|\\\\gamma _2|\\\\le \\\\frac{1}{2}(1+2e^{-2})=0.635\\\\ldots .$ For $n\\\\ge 3$ , the problem seems much harder, and no significant upper bound for $|\\\\gamma _n|$ when $f\\\\in \\\\mathcal {S}$ appear to be known.\",\n \"If $f\\\\in \\\\mathcal {S}^*$ then it is not very difficult to prove that $|\\\\gamma _n|\\\\le \\\\frac{1}{n}$ for $n\\\\ge 1$ and equality holds for the Koebe function $k(z)=z/(1-z)^2$ .\",\n \"The inequality $|\\\\gamma _n|\\\\le \\\\frac{1}{n}$ for $n\\\\ge 2$ extends to the class $\\\\mathcal {K}$ was claimed in a paper of Elhosh [4].\",\n \"However, Girela [6] pointed out some error in the proof of Elhosh [4] and, hence, the result is not substantiated.\",\n \"Indeed, Girela proved that for each $n\\\\ge 2$ , there exists a function $f\\\\in \\\\mathcal {K}$ such that $|\\\\gamma _n|> \\\\frac{1}{n}$ .\",\n \"In the same paper it has been shown that $|\\\\gamma _n|\\\\le \\\\frac{3}{2n}$ holds for $n\\\\ge 1$ whenever $f$ belongs to the set of extreme points of the closed convex hull of the class $\\\\mathcal {K}$ .\",\n \"Recently, Thomas [12] proved that $|\\\\gamma _3|\\\\le \\\\frac{7}{12}$ for functions in $\\\\mathcal {K}_0$ (close-to-convex functions with argument 0) with the additional assumption that the second coefficient of the corresponding starlike function $g$ is real.\",\n \"Thomas claimed that this estimate is sharp and has given a form of the extremal function.\",\n \"But after rigorous reading of the paper [12], we observed that such functions do not belong to the class $\\\\mathcal {K}_0$ (more details will be given in Section ).\",\n \"By fixing a starlike function $g$ in the class $\\\\mathcal {S}^*$ , the inequality (REF ) assertions a specific subclass of close-to-convex functions.\",\n \"One of such important subclass is the class of close-to-convex functions with respect to the Koebe function $k(z)=z/(1-z)^2$ .\",\n \"In this case, the inequality (REF ) becomes ${\\\\rm Re\\\\,} \\\\left(e^{i\\\\alpha }(1-z)^2f^{\\\\prime }(z)\\\\right)>0,\\\\quad z\\\\in \\\\mathbb {D}$ and defines the subclass $\\\\mathcal {K}_{\\\\alpha }(k)$ .\",\n \"Several authors have been extensively studied the class of functions $f\\\\in \\\\mathcal {S}$ that satisfies the condition (REF ) (see [5], [7], [9], [11]).\",\n \"Geometrically (REF ) says that the function $h:=e^{i\\\\delta }f$ has the boundary normalization $\\\\lim _{t\\\\rightarrow \\\\infty } h^{-1}(h(z)+t)=1$ and $h(\\\\mathbb {D})$ is a domain such that $\\\\lbrace w + t : t\\\\ge 0\\\\rbrace \\\\subseteq h(\\\\mathbb {D})$ for every $w\\\\in h(\\\\mathbb {D})$ .\",\n \"Clearly, the image domain $h(\\\\mathbb {D})$ is convex in the positive direction of the real axis.\",\n \"Denote by $\\\\mathcal {CR}^+:=\\\\mathcal {K}_{0}(k)$ the class of close-to-convex functions with argument 0 and with respect to Koebe function $k(z)$ .\",\n \"That is $\\\\mathcal {CR}^+=\\\\left\\\\lbrace f\\\\in \\\\mathcal {A}: {\\\\rm Re\\\\,} (1-z)^2f^{\\\\prime }(z)>0, ~~ z\\\\in \\\\mathbb {D} \\\\right\\\\rbrace .$ Then clearly functions in $\\\\mathcal {CR}^+$ are convex in the positive direction of the real axis.\",\n \"In the present article, we determine the upper bound of $|\\\\gamma _3|$ for functions in $\\\\mathcal {K}_0$ and $\\\\mathcal {CR}^+$ .\"\n ],\n [\n \"Main Results\",\n \"Let $\\\\mathcal {P}$ denote the class of analytic functions $P$ with positive real part on $\\\\mathbb {D}$ which has the form $P(z)= 1+\\\\sum _{n=1}^{\\\\infty }c_n z^n.$ Functions in $\\\\mathcal {P}$ are sometimes called Carathéodory function.\",\n \"To prove our main results, we need some preliminary lemmas.\",\n \"The first one is known as Carathéodory's lemma (see [2] for example) and the second one is due to Libera and Złotkiewicz [10].\",\n \"Lemma 2.1 [2] For a function $P\\\\in \\\\mathcal {P}$ of the form (REF ), the sharp inequality $|c_n|\\\\le 2$ holds for each $n\\\\ge 1$ .\",\n \"Equality holds for the function $P(z)=(1+z)/(1-z)$ .\",\n \"Lemma 2.2 [10] Let $P\\\\in \\\\mathcal {P}$ be of the form (REF ).\",\n \"Then there exist $x, t\\\\in \\\\mathbb {C}$ with $|x|\\\\le 1$ and $|t|\\\\le 1$ such that $2c_2 = c_1^2 + x(4 - c_1^2)$ and $4c_3= c_1^3+ 2(4-c_1^2)c_1x-c_1(4-c_1^2)x^2+2(4-c_1^2)(1-|x|^2)t.$ In [12], Thomas claimed that his result (i.e.\",\n \"$|\\\\gamma _3|\\\\le 7/12$ ) is sharp for functions in the class $\\\\mathcal {K}_0$ by ascertaining the equality holds for a function $f$ defined by $zf^{\\\\prime }(z)=g(z)P(z)$ where $g\\\\in \\\\mathcal {S}^*$ with $b_2(g)=b_3(g)=b_4(g)=2$ and $P\\\\in \\\\mathcal {P}$ with $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .\",\n \"But in view of Lemma REF , it is easy to see that there does not exist a function $P\\\\in \\\\mathcal {P}$ with the property $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .\",\n \"Thus we can conclude that the result obtained by Thomas is not sharp.\",\n \"The main aim of the present paper is to obtain a better upper bound for $|\\\\gamma _3|$ for functions in the class $\\\\mathcal {K}_0$ than that of obtained by Thomas [12].\",\n \"To prove our main results we also need the following Fekete-Szegö inequality for functions in the class $\\\\mathcal {S}^*$ .\",\n \"Lemma 2.3 [8] Let $g\\\\in \\\\mathcal {S}^*$ be of the form $g(z)=z+\\\\sum _{n=2}^{\\\\infty }b_n z^n$ .\",\n \"Then for any $\\\\lambda \\\\in \\\\mathbb {C}$ , $|b_3-\\\\lambda b_2^2|\\\\le \\\\max \\\\lbrace 1,|3-4\\\\lambda |\\\\rbrace .$ The inequality is sharp for $k(z)=z/(1-z)^2$ if $|3-4\\\\lambda |\\\\ge 1$ and for $(k(z^2))^{1/2}$ if $|3-4\\\\lambda |<1$ .\",\n \"For $f\\\\in \\\\mathcal {K}_0$ (close-to-convex functions with argument 0), we obtained the following improved result for $|\\\\gamma _3|$ (compare [12]).\",\n \"Theorem 2.1 If $f\\\\in \\\\mathcal {K}_0$ then $|\\\\gamma _3|\\\\le \\\\frac{1}{18} (3+4 \\\\sqrt{2})=0.4809$ .\",\n \"Let $f\\\\in \\\\mathcal {K}_0$ be of the form (REF ).\",\n \"Then there exists a starlike function $g(z)=z+\\\\sum _{n=2}^{\\\\infty }b_n z^n$ and a Carathéodory function $P\\\\in \\\\mathcal {P}$ of the form (REF ) such that $zf^{\\\\prime }(z)=g(z)P(z).$ A comparison of the coefficients on the both sides of (REF ) yields $a_2&=\\\\frac{1}{2}(b_2+c_1)\\\\\\\\a_3&=\\\\frac{1}{3}(b_3+b_2c_1+c_2)\\\\\\\\a_4&=\\\\frac{1}{4}(b_4+b_3c_1+b_2c_2+c_3).$ By substituting the above $a_2, a_3$ and $a_4$ in (REF ) and then further simplification gives $2\\\\gamma _3&= a_4-a_2a_3+\\\\frac{1}{3}a_2^3\\\\\\\\&=\\\\frac{1}{24}\\\\left((6b_4-4b_2b_3+b_2^3)+\\\\frac{c_1}{2}\\\\left(b_3-\\\\frac{1}{2}b_2^2\\\\right)+b_2(2c_2-c_1^2)+c_1^3-4c_1c_2+6c_3\\\\right)\\\\nonumber .$ In view of Lemma REF and writing $c_2$ and $c_3$ in terms of $c_1$ we obtain $48\\\\gamma _3&= (6b_4-4b_2b_3+b_2^3)+ 2c_1\\\\left(b_3-\\\\frac{1}{2}b_2^2\\\\right)+ b_2x(4-c_1^2)\\\\\\\\&\\\\quad +\\\\frac{1}{2}c_1^3+c_1x(4-c_1^2)-\\\\frac{3}{2}c_1x^2(4-c_1^2)+3(4-c_1^2)(1-|x|^2)t,\\\\nonumber $ where $|x|\\\\le 1$ and $|t|\\\\le 1$ .\",\n \"Note that if $\\\\gamma _3(g)$ denote the third logarithmic coefficient of $g\\\\in \\\\mathcal {S}^*$ then $|\\\\gamma _3(g)|=\\\\frac{1}{2}|b_4-b_2b_3+\\\\frac{1}{3}b_2^3|\\\\le \\\\frac{1}{3}$ .\",\n \"Since $g\\\\in \\\\mathcal {S}^*$ , in view of Lemma REF we obtain $|6b_4-4b_2b_3+b_2^3|\\\\le 6|b_4-b_2b_3+\\\\frac{1}{3}b_2^3|+2|b_2||b_3-\\\\frac{1}{2}b_2^2|\\\\le 8.$ Since the class $\\\\mathcal {K}_0$ is invariant under rotation, without loss of generality we can assume that $c_1=c$ , where $0\\\\le c\\\\le 2$ .\",\n \"Taking modulus on both the sides of (REF ) and then applying triangle inequality and further using the inequality (REF ) and Lemma REF , it follows that $48|\\\\gamma _3|\\\\le 8+ 2c+ 2|x|(4-c^2)+\\\\left|\\\\frac{1}{2}c^3+cx(4-c^2)-\\\\frac{3}{2}cx^2(4-c^2)\\\\right|+3(4-c^2)(1-|x|^2),$ where we have also used the fact $|t|\\\\le 1$ .\",\n \"Let $x=re^{i\\\\theta }$ where $0\\\\le r\\\\le 1$ and $0\\\\le \\\\theta \\\\le 2\\\\pi $ .\",\n \"For simplicity, by writing $\\\\cos \\\\theta =p$ we obtain $48|\\\\gamma _3|\\\\le \\\\psi (c,r)+\\\\left|\\\\phi (c,r,p)\\\\right|=:F(c,r,p)$ where $\\\\psi (c,r)=8+ 2c+ 2r(4-c^2)+3(4-c^2)(1-r^2)$ and $\\\\phi (c,r,p)&=\\\\left(\\\\frac{1}{4}c^6+c^2r^2(4-c^2)^2+\\\\frac{9}{4}c^2r^4(4-c^2)^2+c^4(4-c^2)rp\\\\right.\\\\\\\\&\\\\qquad \\\\quad \\\\left.-\\\\frac{3}{2}c^4r^2(4-c^2)(2p^2-1)-3c^2(4-c^2)r^3p \\\\right)^{1/2}.$ Thus we need to find the maximum value of $F(c,r,p)$ over the rectangular cube $R:=[0,2]\\\\times [0,1]\\\\times [-1,1]$ .\",\n \"By elementary calculus one can verify the followings: $&\\\\max _{0\\\\le r\\\\le 1} \\\\psi (0,r)=\\\\psi \\\\left(0,\\\\frac{1}{3}\\\\right)=\\\\frac{64}{3},\\\\quad \\\\max _{0\\\\le r\\\\le 1} \\\\psi (2,r)=12,\\\\\\\\[2mm]&\\\\max _{0\\\\le c\\\\le 2} \\\\psi (c,0)=\\\\psi \\\\left(\\\\frac{1}{3},0\\\\right)=\\\\frac{61}{3},\\\\quad \\\\max _{0\\\\le c\\\\le 2} \\\\psi (c,1)=\\\\psi (0,1)=16 \\\\quad \\\\mbox{ and }\\\\\\\\[2mm]&\\\\max _{(c,r)\\\\in [0,2]\\\\times [0,1]} \\\\psi (c,r)=\\\\psi \\\\left(\\\\frac{3}{10},\\\\frac{1}{3}\\\\right)=\\\\frac{649}{30}=21.6333.$ We first find the maximum value of $F(c,r,p)$ on the boundary of $R$ , i.e on the six faces of the rectangular cube $R$ .\",\n \"On the face $c=0$ , we have $F(0,r,p)=\\\\psi (0,r)$ , where $(r,p)\\\\in R_1:=[0,1]\\\\times [-1,1]$ .\",\n \"Thus $\\\\max _{(r,p)\\\\in R_1} F(0,r,p)= \\\\max _{0\\\\le r\\\\le 1} \\\\psi (0,r)=\\\\psi \\\\left(0,\\\\frac{1}{3}\\\\right)=\\\\frac{64}{3}=21.33.$ On the face $c=2$ , we have $F(2,r,p)= 16$ , where $(r,p)\\\\in R_1$ .\",\n \"On the face $r=0$ , we have $F(c,0,p)=8+ 2c+3(4-c^2)+\\\\frac{1}{2}c^3$ , where $(c,p)\\\\in R_2:=[0,2]\\\\times [-1,1]$ .\",\n \"By using elementary calculus it is easy to see that $\\\\max _{(c,p)\\\\in R_2} F(c,0,p)= F\\\\left(\\\\frac{2}{3} (3-\\\\sqrt{6}),0,p\\\\right)=\\\\frac{16}{9} \\\\left(9+\\\\sqrt{6}\\\\right)=20.3546.$ On the face $r=1$ , we have $F(c,1,p)=\\\\psi (c,1)+ |\\\\phi (c,1,p)|$ , where $(c,p)\\\\in R_2$ .\",\n \"We first prove that $\\\\phi (c,1,p)\\\\ne 0$ in the interior of $R_2$ .\",\n \"On the contrary, if $\\\\phi (c,1,p)=0$ in the interior of $R_2$ then $|\\\\phi (c,1,p)|^2=\\\\left|\\\\frac{1}{2}c^3+ce^{i\\\\theta }(4-c^2)-\\\\frac{3}{2}ce^{2i\\\\theta }(4-c^2)\\\\right|^2=0$ and hence $\\\\frac{1}{2}c^3+cp(4-c^2)-\\\\frac{3}{2}c(4-c^2)(2p^2-1)=0 ~\\\\mbox{ and } c(4-c^2)\\\\sin \\\\theta -\\\\frac{3}{2}c(4-c^2)\\\\sin 2\\\\theta =0.$ Further, (REF ) reduces to $\\\\frac{1}{2}c^2+p(4-c^2)-\\\\frac{3}{2}(4-c^2)(2p^2-1)=0 \\\\quad \\\\mbox{and}\\\\quad 1-3p=0,$ which is equivalent to $p=1/3$ and $c^2=6$ .\",\n \"This contradicts the range of $c\\\\in (0,2)$ .\",\n \"Thus $\\\\phi (c,1,p)\\\\ne 0$ in the interior of $R_2$ .\",\n \"Next, we prove that $F(c,1,p)$ has no maximum at any interior point of $R_2$ .\",\n \"Suppose that $F(c,1,p)$ has the maximum at an interior point of $R_2$ .\",\n \"Then at such point $\\\\frac{\\\\partial F(c,1,p)}{\\\\partial c}=0$ and $\\\\frac{\\\\partial F(c,1,p)}{\\\\partial p}=0$ .\",\n \"From $\\\\frac{\\\\partial F(c,1,p)}{\\\\partial p}=0$ , (for points in the interior of $R_2$ ), a straight forward calculation gives $p=\\\\frac{2 \\\\left(c^2-3\\\\right)}{3 c^2}.$ Substituting the value of $p$ as given in (REF ) in the relation $\\\\frac{\\\\partial F(c,1,p)}{\\\\partial c}=0$ and further simplification gives $3c^3-2c+(2c-1) \\\\sqrt{6(c^2+2)}=0.$ It is easy to show that the function $\\\\rho (c)=3c^3-2c+(2c-1) \\\\sqrt{6(c^2+2)}$ is strictly increasing in $(0,2)$ .\",\n \"Since $\\\\rho (0)<0$ and $\\\\rho (2)>0$ , the equation (REF ) has exactly one solution in $(0,2)$ .\",\n \"By solving the equation (REF ) numerically, we obtain the approximate root in $(0,2)$ as $0.5772$ .\",\n \"But the corresponding value of $p$ obtained by (REF ) is $-5.3365$ which does not belong to $(-1,1)$ .\",\n \"Thus $F(c,1,p)$ has no maximum at any interior point of $R_2$ .\",\n \"Thus we find the maximum value of $F(c,1,p)$ on the boundary of $R_2$ .\",\n \"Clearly, $F(0,1,p)=F(2,1,p)=16$ , $F(c,1,-1)={\\\\left\\\\lbrace \\\\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(10-3c^2)& \\\\mbox{for}\\\\quad 0\\\\le c\\\\le \\\\sqrt{\\\\frac{10}{3}}\\\\\\\\[2mm]8+ 2c+ 2(4-c^2)-c(10-3c^2)& \\\\mbox{for}\\\\quad \\\\sqrt{\\\\frac{10}{3}}< c\\\\le 2\\\\end{array}\\\\right.\",\n \"}$ and $F(c,1,1)={\\\\left\\\\lbrace \\\\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(2-c^2)& \\\\mbox{for}\\\\quad 0\\\\le c\\\\le \\\\sqrt{2}\\\\\\\\[2mm]8+ 2c+ 2(4-c^2)-c(2-c^2)& \\\\mbox{for}\\\\quad \\\\sqrt{2}< c\\\\le 2.\\\\end{array}\\\\right.\",\n \"}$ By using elementary calculus we find that $\\\\max _{0\\\\le c\\\\le 2}F(c,1,-1)= F\\\\left(\\\\frac{2}{9} (2 \\\\sqrt{7}-1),1,-1\\\\right)=\\\\frac{8}{243} \\\\left(403+112 \\\\sqrt{7}\\\\right)=23.023\\\\quad \\\\mbox{ and }$ $\\\\max _{0\\\\le c\\\\le 2}F(c,1,1)= F\\\\left(\\\\frac{2}{3},1,1\\\\right)=\\\\frac{427}{27} =17.48.$ Hence, $\\\\max _{(c,p)\\\\in R_2} F(c,1,p)= F\\\\left(\\\\frac{2}{9} (2 \\\\sqrt{7}-1),1,-1\\\\right)=\\\\frac{8}{243} \\\\left(403+112 \\\\sqrt{7}\\\\right)=23.023.$ On the face $p=-1$ , $F(c,r,-1)={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\psi (c,r)+\\\\eta _1(c,r) & \\\\mbox{ for }\\\\quad \\\\eta _1(c,r)\\\\ge 0\\\\\\\\[2mm]\\\\psi (c,r)-\\\\eta _1(c,r)& \\\\mbox{ for }\\\\quad \\\\eta _1(c,r)< 0,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\eta _1(c,r)=c^3 (3r^2+2r+1)-4cr(3r+2)$ and $(c,r)\\\\in R_3:=[0,2]\\\\times [0,1]$ .\",\n \"Differentiating partially $F(c,r,-1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\\\max _{(c,r)\\\\in {\\\\rm int \\\\,}R_3\\\\setminus S_1} F(c,r,-1)= F\\\\left(2(\\\\sqrt{2}-1),\\\\frac{1}{3}(1+\\\\sqrt{2}),-1\\\\right)=\\\\frac{8}{3} (3+4 \\\\sqrt{2})=23.0849,$ where $S_1=\\\\lbrace (c,r)\\\\in R_3: \\\\eta _1(c,r)=0\\\\rbrace $ .\",\n \"Now we find the maximum value of $F(c,r,-1)$ on the boundary of $R_3$ and on the set $S_1$ .\",\n \"Note that $\\\\max _{(c,r)\\\\in S_1} F(c,r,-1)\\\\le \\\\max _{(c,r)\\\\in R_3} \\\\psi (c,r)=\\\\frac{649}{30}=21.6333.$ On the other hand by using elementary calculus, as before, we find that $\\\\max _{(c,r)\\\\in \\\\partial R_3} F(c,r,-1)=F\\\\left(\\\\frac{2}{9} (2 \\\\sqrt{7}-1),1,-1\\\\right)=\\\\frac{8}{243} \\\\left(403+112 \\\\sqrt{7}\\\\right)=23.023,$ where $\\\\partial R_3$ denotes the boundary of $R_3$ .\",\n \"Hence, by combining the above cases we obtain $\\\\max _{(c,r)\\\\in R_3} F(c,r,-1)=F\\\\left(2(\\\\sqrt{2}-1),\\\\frac{1}{3}(1+\\\\sqrt{2}),-1\\\\right)=\\\\frac{8}{3} (3+4 \\\\sqrt{2})=23.0849.$ On the face $p=1$ , $F(c,r,1)={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\psi (c,r)+\\\\eta _2(c,r) & \\\\mbox{ for }\\\\quad \\\\eta _2(c,r)\\\\ge 0\\\\\\\\[2mm]\\\\psi (c,r)-\\\\eta _2(c,r)& \\\\mbox{ for }\\\\quad \\\\eta _2(c,r)< 0,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\eta _2(c,r)=c^3 (3r^2-2r+1)+4cr(3r-2)$ and $(c,r)\\\\in R_3$ .\",\n \"Differentiating partially $F(c,r,1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\\\max _{(c,r)\\\\in {\\\\rm int \\\\,}R_3\\\\setminus S_2} F(c,r,1)=F\\\\left(\\\\frac{1}{3} (10-2 \\\\sqrt{19}),\\\\frac{1}{3},1\\\\right)=\\\\frac{16}{81} \\\\left(28+19 \\\\sqrt{19}\\\\right)=21.89,$ where $S_2=\\\\lbrace (c,r)\\\\in R_3: \\\\eta _2(c,r)=0\\\\rbrace $ .\",\n \"Now, we find the maximum value of $F(c,r,1)$ on the boundary of $R_3$ and on the set $S_2$ .\",\n \"By noting that $\\\\max _{(c,r)\\\\in S_2} F(c,r,1)\\\\le \\\\max _{(c,r)\\\\in R_3} \\\\psi (c,r)=\\\\frac{649}{30}=21.6333$ and proceeding similarly as in the previous case, we find that $\\\\max _{(c,r)\\\\in R_3} F(c,r,1)=F\\\\left(\\\\frac{1}{3} (10-2 \\\\sqrt{19}),\\\\frac{1}{3},1\\\\right)=\\\\frac{16}{81} \\\\left(28+19 \\\\sqrt{19}\\\\right)=21.89.$ Let $S^{\\\\prime }=\\\\lbrace (c,r,p)\\\\in R: \\\\phi (c,r,p)=0\\\\rbrace $ .\",\n \"Then $\\\\max _{(c,r,p)\\\\in S^{\\\\prime }} F(c,r,p)\\\\le \\\\max _{(c,r)\\\\in R_3} \\\\psi (c,r)=\\\\psi \\\\left(\\\\frac{3}{10},\\\\frac{1}{3}\\\\right)=\\\\frac{649}{30}=21.6333.$ We prove that $F(c,r,p)$ has no maximum value at any interior point of $R\\\\setminus S^{\\\\prime }$ .\",\n \"Suppose that $F(c,r,p)$ has a maximum value at an interior point of $R\\\\setminus S^{\\\\prime }$ .\",\n \"Then at such point $\\\\frac{\\\\partial F}{\\\\partial c}=0$ , $\\\\frac{\\\\partial F}{\\\\partial r}=0$ and $\\\\frac{\\\\partial F}{\\\\partial p}=0$ .\",\n \"Note that $\\\\frac{\\\\partial F}{\\\\partial c}$ , $\\\\frac{\\\\partial F}{\\\\partial r}$ and $\\\\frac{\\\\partial F}{\\\\partial p}$ may not exist at points in $S^{\\\\prime }$ .\",\n \"In view of $\\\\frac{\\\\partial F}{\\\\partial p}=0$ (for points in the interior of $R\\\\setminus S^{\\\\prime }$ ), a straight forward but laborious calculation gives $p= \\\\frac{3 c^2 r^2+c^2-12 r^2}{6 c^2 r}.$ Substituting the value of $p$ as given in (REF ) in the relations $\\\\frac{\\\\partial F}{\\\\partial c}=0$ and $\\\\frac{\\\\partial F}{\\\\partial r}=0$ and simplifying (again, a long and laborious calculation), we obtain $\\\\frac{3 \\\\sqrt{6} c^3 (1-3 r^2)+12 (c(3r^2-2r-3)+1)\\\\sqrt{c^2+2} )+4\\\\sqrt{6}c}{6\\\\sqrt{c^2+2}}=0$ and $(4-c^2) \\\\left(( \\\\sqrt{6(c^2+2)}-6) r+2\\\\right)=0.$ Since $00.$ It is proved in that the Cauchy problem of (1.1) for $n=2$ is globally well-posed for $u_0\\in L^{\\infty }_{\\sigma }$ based on the local solvability result in .","We proved a local solvability on $L^{\\infty }_{\\sigma }$ for exterior domains.","Note that global solutions exist for rotationally symmetric initial data $u_0\\in L^{\\infty }_{\\sigma }$ ; see below (iv).","(iv) (Rotating flows) An example of $u_0\\in L^{\\infty }_{\\sigma }$ which is not asymptotically constant is a vector field rotating at space infinity.","For example, we consider the two-dimensional unit disk $\\Omega ^{c}$ centered at the origin and a rotationally symmetric initial data $u_{0}=u_{0}^{\\theta }(r)e_{\\theta }(\\theta )$ for $e_{\\theta }(\\theta )=(-\\sin {\\theta },\\cos {\\theta })$ .","Observe that $u_0$ is a solenoidal vector field in $\\Omega $ and a direction of $u_0$ varies for $\\theta \\in [0,2\\pi ]$ and $u_{0}^{\\theta }\\in L^{\\infty }(1,\\infty )$ .","Solutions of (1.1) for $u_0$ are rotationally symmetric and given by $u=e^{t\\Delta _{D}}u_0\\qquad p=\\int _{1}^{|x|}\\frac{|u|^{2}}{r}\\textrm {d}r,$ where $\\Delta _{D}$ denotes the Laplace operator subject to the Dirichlet boundary condition.","The solution $u$ is bounded in $\\Omega \\times (0,\\infty )$ and non-decaying as $|x|\\rightarrow \\infty $ .","(v) (Associated pressure) We invoke that the associated pressure of mild solutions on $L^{p}$ ($p\\ge n$ ) is determined by the projection operator $\\mathbb {Q}=I-\\mathbb {P}$ and $\\nabla p=\\mathbb {Q}\\Delta u-\\mathbb {Q}(u\\cdot \\nabla u).$ Since the projection $\\mathbb {Q}$ may not be bounded on $L^{\\infty }$ , this representation is no longer available for mild solutions on $L^{\\infty }_{\\sigma }$ .","When $\\Omega =\\mathbb {R}^{n}$ or $\\mathbb {R}^{n}_{+}$ , the projection $\\mathbb {Q}$ has explicit kernels and we are able to find associated pressure of mild solutions on $L^{\\infty }$ ; see for $\\Omega =\\mathbb {R}^{n}$ and , , for $\\Omega =\\mathbb {R}^{n}_{+}$ .","Although explicit kernels are not available for exterior domains, we are able to find the associated pressure of mild solutions on $L^{\\infty }$ .","We set $\\nabla p=\\mathbb {K}W-\\mathbb {Q}F̥ $ for $W=-(\\nabla u-\\nabla ^{T} u)n_{\\Omega }$ and $F=uu$ , where $n_{\\Omega }$ is the unit outward normal on $\\partial \\Omega $ and $\\mathbb {K}$ is a solution operator of the homogeneous Neumann problem (harmonic-pressure operator) .","Note that $W=-\\textrm {curl}\\ u\\times n_{\\Omega }$ for $n=3$ .","The operators $\\mathbb {K}$ and $\\mathbb {Q} act for bounded functions and the associated pressure on $ L$ is uniquely determined by (1.7) in the sense of distribution; see Remark 3.5 for a detailed discussion.$ For asymptotically constant initial data $u_0$ (i.e., $u_{0}\\rightarrow u_{\\infty }$ as $|x|\\rightarrow \\infty $ ), local solvability of (1.1) for $n=3$ is proved in by means of the Oseen semigroup.","In the paper, the problem (1.1) is reduced to an initial-boundary problem for decaying data by shifting $u$ by a constant $u_{\\infty }$ .","Our analysis is based on the $L^{\\infty }$ -estimates of the Stokes semigroup which yields a local-in-time solvability of (1.1) without conditions for $u_0$ at space infinity.","The $L^{\\infty }$ -theory for the Cauchy problem of the Navier-Stokes equations is developed by Knightly , , Cannon and Knightly , Cannone () and Giga et al.",".","For the whole space, mild solutions on $L^{\\infty }$ are smooth and satisfy (1.1) in a classical sense .","For a half space, mild solutions on $L^{\\infty }$ are constructed in (see also , ).","There are a few results on solvability of the exterior problem for non-decaying data.","In , unique existence of continuous solutions of (1.1) for $n\\ge 3$ is proved for non-decaying and Hölder continuous initial data.","The result is extended in for merely bounded $u_0\\in L^{\\infty }_{\\sigma }$ and $n\\ge 3$ by using $L^{\\infty }$ -estimates of the Stokes semigroup , .","Note that mild solutions on $L^{\\infty }_{\\sigma }$ are not constructed without the composition operator $\\overline{S(t)\\mathbb {P}.","We proved the unique existence of mild solutions on L^{\\infty }_{\\sigma }, which in particular yields a local-in-time solvability for n=2.","The integral form (1.3) is fundamental for studying solutions of (1.1).","We expect that mild solutions on L^{\\infty } are sufficiently smooth and satisfy (1.1) in a classical sense.\\\\}$ The article is organized as follows.","In Section 2, we extend the composition operator $S(t)\\mathbb {P} to a bounded operator from $ W1,0$ to $ L$ by approximation as we did the Stokes semigroup in \\cite {AG2}.","We extend $ S(t)P as a solution operator $F\\longmapsto v(\\cdot ,t)$ for solutions $(v,q)$ of the Stokes equations for $v_0=\\mathbb {P}F$ .","Note that $v_0=\\mathbb {P}F$ for $F\\in W^{1,\\infty }_{0}$ is not an element of $L^{\\infty }$ in general since the projection $\\mathbb {P}$ is not bounded on $L^{\\infty }$ .","We understand $v_0=\\mathbb {P}F$ as distribution by using the fact that $\\nabla \\mathbb {P}\\varphi \\in L^{1}$ for $\\varphi \\in C_{c}^{\\infty }$ (Lemma A.1).","We approximate $F\\in W^{1,\\infty }_{0}$ by a sequence $\\lbrace F_m\\rbrace \\subset C_{c}^{\\infty }$ locally uniformly in $\\overline{\\Omega }$ and obtain a unique extension $\\overline{S(t)\\mathbb {P}: F\\longmapsto v(\\cdot ,t) by a limit v of the sequence v_m=S(t)\\mathbb {P}F_m.","}In Section 3, we prove Theorem 1.1.","We approximate initial data $ u0L$ by a sequence $ {u0,m}Cc,$ satisfying $ u0,mu0$ a.e.","in $$ and $ ||u0,m||C||u0||$.","Since the property of mild solutions (1.4) may not follow from a direct iteration argument on $ L$, we construct mild solutions by approximation.","We apply an existence theorem on $ C0,$ \\cite {A4} and construct a sequence of mild solutions $ umC([0,T]; C0,)$ satisfying (1.4)-(1.6) for $ u0,mC0,$.","We prove that $ um$ subsequently converges to a mild solution $ u$ for $ u0L$ locally uniformly in $(0,T]$.$ In Appendix A, we show that $\\nabla \\mathbb {P}\\varphi \\in L^{1}$ for $\\varphi \\in C_{c}^{\\infty }$ by means of the layer potential."],["An extension of the composition operator","In this section, we prove that the composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\infty }_{0}$ to $L^{\\infty }_{\\sigma }$ .","We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\mathbb {P}\\partial :f\\longmapsto v(\\cdot ,t)$ .","In what follows, $\\partial =\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots ,n$ ."],["The Stokes system","We consider the Stokes equations, $\\begin{aligned}\\partial _{t}v-\\Delta v+\\nabla q&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {on}\\ \\partial \\Omega \\times (0,T), \\\\v&=v_0\\quad \\hspace{-4.0pt} \\textrm {on}\\ \\Omega \\times \\lbrace t=0\\rbrace .\\end{aligned}\\qquad \\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\bigl |v(x,t)\\bigr |+t^{\\frac{1}{2}}\\bigl |\\nabla v(x,t)\\bigr |+t\\bigl |\\nabla ^{2}v(x,t)\\bigr |+t\\bigl |\\partial _{t}v(x,t)\\bigr |+t\\bigl |\\nabla q(x,t)\\bigr |.$ Let $d(x)$ denote the distance from $x\\in \\Omega $ to $\\partial \\Omega $ .","Let $(v, \\nabla q)\\in C^{2+\\mu ,1+\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T])\\times C^{\\mu ,\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T]),\\ \\mu \\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .","We say that $(v,q)$ is a solution of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , if $\\sup _{00$ .","For $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\sup _{00$ , $T>0$ , $R>0$ , there exists a constant $C=C\\bigl (\\mu ,\\delta ,T, R,d\\bigr )$ such that $\\Big [\\nabla ^{2}v\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [v_t\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [\\nabla q\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}\\le CN \\qquad \\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\prime }=B_{x_0}(R)\\times (2\\delta ,T]$ and $x_0\\in \\Omega $ satisfying $\\overline{B_{x_0}(R)}\\subset \\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\partial \\Omega $ .","(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\mu \\in (0,1),\\ \\delta >0$ , $T>0$ and $R\\le R_{0}$ , there exists a constant $C$ depending on $\\mu $ , $\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\partial \\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\prime }=\\Omega _{x_0,R}\\times (2\\delta ,T]$ and $\\Omega _{x_0,R}=B_{x_0}(R)\\cap \\Omega $ , $x_0\\in \\partial \\Omega $ .","We observe the uniqueness of solutions for (2.1).","The uniqueness of the Stokes equations (2.1) for $v_0\\in L^{\\infty }_{\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .","In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , may not be bounded at $t=0$ .","The corresponding uniqueness result for a half space is recently proved in .","We deduce the result for exterior domains by the same blow-up argument as we did in .","Proposition 2.4 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.","Let $(v,\\nabla q)\\in C^{2,1}(\\overline{\\Omega }\\times (0,T])\\times C(\\overline{\\Omega }\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\gamma \\in [0,1/2)$ .","Assume that $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =0,$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ .","Then, $v\\equiv 0$ and $\\nabla q\\equiv 0$ ."],["Approximation","We prove Theorem 2.1.","We show existence of solutions for the Stokes equations (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , by approximation.","We approximate $f\\in W^{1,\\infty }_{0}$ by elements of $C^{\\infty }_{c}$ locally uniformly in $\\overline{\\Omega }$ .","Lemma 2.5 Let $\\Omega $ be an exterior domain with Lipschitz boundary.","There exist constants $C_1$ , $C_2$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C_{c}^{\\infty }(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert f_{m}\\big \\Vert _{\\infty }\\le C_1\\big \\Vert f\\big \\Vert _{\\infty }, \\\\&\\big \\Vert \\nabla f_{m}\\big \\Vert _{\\infty }\\le C_2\\big \\Vert f\\big \\Vert _{1,\\infty }, \\\\&f_m\\rightarrow f\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.","Proposition 2.6 The statement of Lemma 2.5 holds when $\\Omega =\\mathbb {R}^{n}$ with $C_1=1$ .","We cutoff the function $f\\in W^{1,\\infty }(\\mathbb {R}^{n})$ .","Let $\\theta \\in C^{\\infty }_{c}[0,\\infty )$ be a cut-off function satisfying $\\theta \\equiv 1$ in $[0,1]$ , $\\theta \\equiv 0$ in $[2,\\infty )$ and $0\\le \\theta \\le 1$ .","We set $\\theta _{m}(x)=\\theta (|x|/m)$ for $m\\ge 1$ so that $\\theta _{m}\\equiv 1$ for $|x|\\le m$ and $\\theta _{m}\\equiv 0$ for $|x|\\ge 2m$ .","Then, $f_m=f\\theta _{m}$ satisfies (2.7).","Proposition 2.7 Let $\\Omega $ be a bounded domain with Lipschitz boundary.","There exists a constant $C_3$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C^{\\infty }_{c}(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert \\nabla f_m\\big \\Vert _{\\infty }\\le C_3\\big \\Vert \\nabla f\\big \\Vert _{\\infty }\\\\f_m&\\rightarrow f\\quad \\textrm {uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.8)}$ We begin with the case when $\\Omega $ is star-shaped, i.e., $\\lambda \\Omega _{x_0}\\subset \\overline{\\Omega }$ for some $x_0\\in \\Omega $ and all $\\lambda <1$ , where $\\lambda \\Omega _{x_0}=\\lbrace x_0+\\lambda (x-x_0)\\ |\\ x\\in \\Omega \\rbrace $ .","We may assume $x_0=0\\in \\Omega $ and $\\lambda \\Omega \\subset \\overline{\\Omega }$ by translation.","For $f\\in W^{1,\\infty }_{0}(\\Omega )$ , we set $f_{\\lambda }(x)={\\left\\lbrace \\begin{array}{ll}&f\\big (x/\\lambda \\big )\\quad x\\in \\lambda \\Omega ,\\\\&0\\qquad \\hspace{15.0pt} x\\in \\Omega \\backslash \\overline{\\lambda \\Omega } .\\end{array}\\right.","}$ Then, $f_{\\lambda }$ is in $ W^{1,\\infty }(\\Omega )$ since $f$ is vanishing on $\\partial \\Omega $ .","It follows that $\\big \\Vert \\nabla f_{\\lambda }\\big \\Vert _{\\infty }\\le \\frac{1}{\\lambda }\\big \\Vert \\nabla f\\big \\Vert _{\\infty },$ and $f_{\\lambda }\\rightarrow f$ uniformly in $\\overline{\\Omega }$ as $\\lambda \\rightarrow 1$ .","By a mollification of $f_{\\lambda }$ , we obtain a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfying (2.8) with $C_3=2$ .","For general $\\Omega $ , we take an open covering $\\lbrace D_j\\rbrace _{j=1}^{N}$ so that $\\overline{\\Omega }\\subset \\cup _{j=1}^{N}D_j$ and $\\Omega _{j}=\\Omega \\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\in \\Omega _{j}$ .","We take a partition of unity $\\lbrace \\xi _{j}\\rbrace _{j=1}^{N}\\subset C_{c}^{\\infty }(\\mathbb {R}^{n})$ such that $\\sum _{j=1}^{N}\\xi _{j}=1$ , $0\\le \\xi _{j}\\le 1$ , spt $\\xi _{j}\\subset \\overline{D_j}$ and set $f=\\sum _{j=1}^{N}f_j,\\quad f_j=f\\xi _j.$ Since spt $f_j\\subset \\overline{\\Omega }_{j}$ , $\\xi _{j}=0$ on $\\partial D_{j}$ and $f=0$ on $\\partial \\Omega $ , $f_{j}$ is in $W^{1,\\infty }_{0}(\\Omega _{j})$ .","Since $\\Omega _j$ is star-shaped for some $x_j\\in \\Omega _{j}$ , there exists $\\lbrace f_{j,m}\\rbrace \\subset C_{c}^{\\infty }(\\Omega _j)$ satisfying (2.8) in $\\Omega _j$ with $C_3=2$ .","We extend $f_{j,m}\\in C_{c}^{\\infty }(\\Omega _j)$ to $\\Omega \\backslash \\overline{\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\sum _{j=1}^{N}f_{j,m}$ .","Then, $f_m\\in C_{c}^{\\infty }(\\Omega )$ converges to $f$ uniformly in $\\overline{\\Omega }$ .","We estimate $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le \\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j,m}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}\\le 2\\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}.$ Since $\\nabla f_j=\\nabla f\\xi _j+f\\nabla \\xi _j$ and $\\big \\Vert f\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C_{p}\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )}.$ Thus, $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfies (2.8).","The proof is complete.","The assertion follows from Propositions 2.6 and 2.7.","We recall the a priori estimate of $S(t)\\mathbb {P}\\partial $ for $f\\in C_{c}^{\\infty }(\\Omega )$ .","Proposition 2.8 There exists a constant $C$ such that $\\sup _{0\\le t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}S(t)\\mathbb {P}\\partial f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert \\nabla f\\big \\Vert _{\\infty }^{\\alpha } \\qquad \\mathrm {(2.9)}$ for $f\\in C^{\\infty }_{c}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ .","For $f\\in W^{1,\\infty }_{0}$ , we take a sequence $\\lbrace f_m\\rbrace \\subset C^{\\infty }_{c}$ satisfying (2.7).","For $v_{0,m}=\\mathbb {P}\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\int _{0}^{T}\\int _{\\Omega }\\big ( v_{m}\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q_m\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t=\\int _{\\Omega }f_m\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x,$ for $\\varphi \\in C^{\\infty }_{c}\\big (\\Omega \\times [0,T)\\big )$ .","By (2.9) and (2.7), there exists a constant $C$ independent of $m\\ge 1$ such that $\\sup _{0\\le t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v_m,q_m)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d\\nabla q_{m}\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla v_m$ , $\\nabla ^{2}v_m$ , $\\partial _{t}v_m$ and $\\nabla q_{m}$ .","By sending $m\\rightarrow \\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\mathbb {P}\\partial f$ .","By Proposition 2.4, the limit $(v,q)$ is unique.","We proved the unique existence of solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ and $f\\in W^{1,\\infty }_{0}$ satisfying (2.4).","The proof is now complete.","Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\mathbb {P}\\partial $ for $f\\in W^{1,\\infty }_{0}$ .","We take a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\mathbb {P}\\partial f_m$ satisfies $(v_{0,m}, \\varphi )=-(f_m, \\partial \\mathbb {P}\\varphi )\\quad \\textrm {for}\\ \\varphi \\in C^{\\infty }_{c}(\\Omega ).$ Since $\\partial \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ by Lemma A.1, the sequence $\\lbrace v_{0,m}\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\varphi )=-(f,\\partial \\mathbb {P}\\varphi )$ .","Since the limit $v_0$ is unique, the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ .","(ii) We recall that for a sequence $\\lbrace v_{0,m}\\rbrace _{m=1}^{\\infty }\\subset L^{\\infty }_{\\sigma }$ satisfying $||v_{0,m}||_{\\infty }&\\le K_1,\\\\v_{0,m}\\rightarrow v_{0}&\\quad \\textrm {a.e.","}\\ \\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .","From the proof of Theorem 2.1, we observe that for a sequence $\\lbrace f_m\\rbrace \\subset W^{1,\\infty }_{0}$ satisfying $||f_m||_{1,\\infty }&\\le K_2,\\\\f_m\\rightarrow f\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega },$ $\\overline{S(t)\\mathbb {P}\\partial } f_m$ subsequently converges to $\\overline{S(t)\\mathbb {P}\\partial } f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .","(iii) The extension $\\overline{S(t)\\mathbb {P}\\partial }$ satisfies the property $S(t)\\overline{S(s)\\mathbb {P}\\partial } f=\\overline{S(t+s)\\mathbb {P}\\partial } f$ for $t\\ge 0$ , $s>0$ and $f\\in W^{1,\\infty }_{0}$ .","In fact, this property holds for $f_m\\in C_{c}^{\\infty }$ satisfying (2.7).","By choosing a subsequence, $v_m(\\cdot ,t)=S(t)\\mathbb {P}\\partial f_m$ converges to $v(\\cdot ,t)=S(t)\\mathbb {P}\\partial f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ as in the proof of Theorem 2.1.","For fixed $s>0$ , sending $m\\rightarrow \\infty $ implies $S(t)S(s)\\mathbb {P}\\partial f_m=S(t)v_m(s)&\\rightarrow S(t)v(s) \\\\&=S(t)\\overline{S(s)\\mathbb {P}\\partial }f\\quad \\textrm {locally uniformly in \\overline{\\Omega }\\times (0,\\infty )}.$ Thus the property is inherited to $\\overline{S(t)\\mathbb {P}\\partial } f$ ."],["Mild solutions on $L^{\\infty }_{\\sigma }$","We prove Theorem 1.1 by approximation.","We show that a sequence of mild solutions $\\lbrace u_m\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by the $L^{\\infty }$ -estimates (1.5) and (1.6).","Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).","We first recall the existence of mild solutions on $C_{0,\\sigma }$ Proposition 3.1 For $u_0\\in C_{0,\\sigma }$ , there exist $T\\ge \\varepsilon _0/||u_0||_{\\infty }^{2}$ and a unique mild solution $u\\in C([0,T]; C_{0,\\sigma })$ satisfying (1.3)-(1.6).","We approximate $u_0\\in L^{\\infty }_{\\sigma }$ by elements of $C_{c,\\sigma }^{\\infty }\\subset C_{0,\\sigma }$ .","We take a sequence $\\lbrace u_{0,m}\\rbrace _{m=1}^{\\infty }\\subset C_{c,\\sigma }^{\\infty }(\\Omega )$ satisfying $\\begin{aligned}&\\Vert u_{0,m}\\Vert _{\\infty }\\le C\\Vert u_{0}\\Vert _{\\infty }\\\\&u_{0,m}\\rightarrow u_{0}\\quad \\textrm {a.e.","in}\\ \\Omega ,\\end{aligned}\\qquad \\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\ge 1$ .","We apply Proposition 3.1 and observe that there exists $T_m\\ge \\varepsilon _0/||u_{0,m}||_{\\infty }^{2}$ and a unique mild solution $u_m\\in C([0,T_m]; C_{0,\\sigma })$ satisfying $\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P}F_m(s)ds,\\\\F_m&=u_mu_m.","}\\qquad \\mathrm {(3.2)}\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\ge \\varepsilon /||u_0||_{\\infty }^{2} for \\varepsilon =\\varepsilon _0 C^{-2}/2 so that T_m\\ge T and u_m\\in C([0,T]; C_{0,\\sigma }) for m\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla u_m$ .","It follows from (1.5), (1.6) and (3.1) that $\\sup _{0\\le t\\le T}\\Big \\lbrace \\Vert u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1}{2}}\\Vert \\nabla u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1+\\beta }{2}}\\big [\\nabla u_{m}\\big ]_{\\Omega }^{(\\beta )}(t)\\Big \\rbrace &\\le C_1^{\\prime }\\Vert u_0\\Vert _{\\infty }, \\\\\\sup _{x\\in \\Omega }\\Big \\lbrace \\big [u_{m}\\big ]_{[\\delta ,T]}^{(\\gamma )}(x)+\\big [\\nabla u_{m}\\big ]_{[\\delta ,T]}^{(\\frac{\\gamma }{2})}(x)\\Big \\rbrace &\\le C_2^{\\prime }\\Vert u_0\\Vert _{\\infty }, $ for $\\beta ,\\gamma \\in (0,1)$ and $\\delta \\in (0,T]$ with some constants $C_1^{\\prime }$ and $C_2^{\\prime }$ , independent of $m\\ge 1$ .","Since $u_m$ and $\\nabla u_m$ are uniformly bounded and equi-continuous in $\\overline{\\Omega }\\times [\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.","Proposition 3.3 The limit $u\\in C_{w}([0,T]; L^{\\infty })$ is a mild solution for $u_0\\in L^{\\infty }_{\\sigma }$ .","We observe that the limit $u$ satisfies (1.4) by sending $m\\rightarrow \\infty $ .","The estimates (3.3) and (3.4) are inherited to $u$ .","We prove that $u$ satisfies the integral equation (1.3).","By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by Remarks 2.9 (ii).","It follows from (3.3) and Proposition 3.2 that $\\begin{aligned}||F_m||_{\\infty }&\\le K,\\\\||\\nabla F_{m}||_{\\infty }&\\le \\frac{2}{s^{\\frac{1}{2}}}K,\\\\F_m\\rightarrow F\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T]\\ \\textrm {as}\\ m\\rightarrow \\infty ,\\end{aligned}\\qquad \\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\prime }||u_0||_{\\infty }$ .","By choosing a subsequence, we have $\\overline{S(\\eta )\\mathbb {P} F_m\\rightarrow \\overline{S(\\eta )\\mathbb {P} F\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T],}for each s\\in (0,t) as in Remarks 2.9 (ii).","It follows from (3.5) and (2.5) that\\begin{equation*}\\big \\Vert \\overline{S(t-s)\\mathbb {P} F_m\\big \\Vert _{\\infty }\\le \\frac{C}{(t-s)^{\\frac{1-\\alpha }{2}}}\\Bigg (1+\\frac{2}{s^{\\frac{\\alpha }{2}}}\\Bigg )K^{2}}for 00$ .","For $u_{0}^{1}\\in C_{0,\\sigma }$ , there exists $\\lbrace u_{0,m}^{1}\\rbrace \\subset C_{c,\\sigma }^{\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\infty }\\le \\epsilon $ for $m\\ge N^{1}_{\\epsilon }$ .","We apply (3.5) and observe that $t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }&\\le t^{\\frac{1}{2}}C\\big (||S(t)u_{0,m}^{1}||_{\\infty }+||S(t)Au_{0,m}^{1}||_{\\infty }\\big )\\\\&\\le t^{\\frac{1}{2}}C^{\\prime }\\big (||u_{0,m}^{1}||_{\\infty }+||Au_{0,m}^{1}||_{\\infty }\\big )\\rightarrow 0\\quad \\textrm {as}\\ t\\rightarrow 0.$ We estimate $\\overline{\\lim _{t\\rightarrow 0}}t^{\\frac{1}{2}}||\\nabla S(t)u_0^{1}||_{\\infty }&\\le \\overline{\\lim _{t\\rightarrow 0}}\\big (t^{\\frac{1}{2}}||\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\infty }+t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }\\big ) \\\\&\\le C^{\\prime \\prime }\\epsilon .$ We set $u_{0,m}^{2}=\\eta _{\\delta _m}*u_{0}^{2}$ by the mollifier $\\eta _{\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\overline{\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\infty }\\le \\epsilon $ for $m\\ge N_{\\epsilon }^{2}$ .","Since $u_{0,m}^{2}$ is supported away from $\\partial \\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\Delta u_{0,m}^{2}$ (see ).","By a similar way as for $u_{0}^{1}$ , we estimate $\\overline{\\lim }_{t\\rightarrow 0}t^{1/2}||\\nabla S(t)u_0^{2}||_{\\infty }\\le C^{\\prime \\prime }\\epsilon $ .","We proved $\\overline{\\lim }_{t\\rightarrow 0}t^{\\frac{1}{2}}||\\nabla S(t)u_0||_{\\infty }\\le 2C^{\\prime \\prime }\\epsilon .$ Since $\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .","The assertion follows from Propositions 3.1-3.4.","The proof is now complete.","Remark 3.5 We set the associated pressure of mild solutions on $L^{\\infty }$ by (1.7) and the harmonic-pressure operator $\\mathbb {K}: L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )\\longrightarrow L^{\\infty }_{d}(\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\Delta q&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial q}{\\partial n}&={\\partial \\Omega }W\\quad \\textrm {on}\\ \\partial \\Omega .$ Note that $\\Delta u\\cdot n={\\partial \\Omega }W$ by the divergence-free condition of $u$ .","Here, $L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )$ denotes the space of all bounded tangential vector fields on $\\partial \\Omega $ and $L^{\\infty }_{d}(\\Omega )$ is the space of all functions $f\\in L^{1}_{\\textrm {loc}}(\\Omega )$ such that $df$ is bounded in $\\Omega $ for $d(x)=\\inf _{y\\in \\partial \\Omega }|x-y|$ , $x\\in \\Omega $ .","Since $W=-(\\nabla u-\\nabla ^{T}u)n$ is bounded on $\\partial \\Omega $ for mild solutions on $L^{\\infty }$ , $\\nabla q=\\mathbb {K}W$ is defined as an element of $L^{\\infty }_{d}$ .","Moreover, $\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\in W^{1,\\infty }_{0}$ as a distribution by Remarks 2.9 (i).","Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\infty }$ .","acknowledgements The author is grateful to the anonymous referees for their valuable comments.","This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.","appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .","We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.","Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.","Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .","Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .","We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .","It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .","We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .","We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .","We may assume that $0\\in \\Omega ^{c}$ by translation.","We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .","Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.","We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .","In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .","Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .","Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .","In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .","By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .","We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .","We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .","Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .","Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .","The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .","We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .","Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.","By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .","Thus, $g$ is continuous on $\\partial \\Omega $ .","Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .","Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.","The proof is complete.","We estimate $\\Phi _2$ by means of the layer potential.","Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .","The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .","The assertion (i) is based on the Fredholm's theorem.","See .","Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .","Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).","When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .","Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.","Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.","Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.","We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.","The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .","By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .","We shall show that $\\Psi \\equiv 0$ .","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .","Suppose that $x_0\\in \\partial \\Omega $ .","Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .","Thus $x_0\\in \\Omega $ .","We apply the strong maximum principle and conclude that $\\Psi $ is constant.","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .","The proof is complete.","Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .","Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .","Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .","The proof is complete.","By Propositions A.2 and A.6, the assertion follows.","A4article author=Abe, K., title=The Navier-Stokes equations in a space of bounded functions, 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A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.","Math.","Soc.","Transl.","Ser.","2, volume=220, publisher=Amer.","Math.","Soc., Providence, RI, pages=165200,"],["An extension of the composition operator","In this section, we prove that the composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\infty }_{0}$ to $L^{\\infty }_{\\sigma }$ .","We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\mathbb {P}\\partial :f\\longmapsto v(\\cdot ,t)$ .","In what follows, $\\partial =\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots ,n$ ."],["The Stokes system","We consider the Stokes equations, $\\begin{aligned}\\partial _{t}v-\\Delta v+\\nabla q&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {on}\\ \\partial \\Omega \\times (0,T), \\\\v&=v_0\\quad \\hspace{-4.0pt} \\textrm {on}\\ \\Omega \\times \\lbrace t=0\\rbrace .\\end{aligned}\\qquad \\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\bigl |v(x,t)\\bigr |+t^{\\frac{1}{2}}\\bigl |\\nabla v(x,t)\\bigr |+t\\bigl |\\nabla ^{2}v(x,t)\\bigr |+t\\bigl |\\partial _{t}v(x,t)\\bigr |+t\\bigl |\\nabla q(x,t)\\bigr |.$ Let $d(x)$ denote the distance from $x\\in \\Omega $ to $\\partial \\Omega $ .","Let $(v, \\nabla q)\\in C^{2+\\mu ,1+\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T])\\times C^{\\mu ,\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T]),\\ \\mu \\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .","We say that $(v,q)$ is a solution of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , if $\\sup _{00$ .","For $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\sup _{00$ , $T>0$ , $R>0$ , there exists a constant $C=C\\bigl (\\mu ,\\delta ,T, R,d\\bigr )$ such that $\\Big [\\nabla ^{2}v\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [v_t\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [\\nabla q\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}\\le CN \\qquad \\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\prime }=B_{x_0}(R)\\times (2\\delta ,T]$ and $x_0\\in \\Omega $ satisfying $\\overline{B_{x_0}(R)}\\subset \\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\partial \\Omega $ .","(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\mu \\in (0,1),\\ \\delta >0$ , $T>0$ and $R\\le R_{0}$ , there exists a constant $C$ depending on $\\mu $ , $\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\partial \\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\prime }=\\Omega _{x_0,R}\\times (2\\delta ,T]$ and $\\Omega _{x_0,R}=B_{x_0}(R)\\cap \\Omega $ , $x_0\\in \\partial \\Omega $ .","We observe the uniqueness of solutions for (2.1).","The uniqueness of the Stokes equations (2.1) for $v_0\\in L^{\\infty }_{\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .","In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , may not be bounded at $t=0$ .","The corresponding uniqueness result for a half space is recently proved in .","We deduce the result for exterior domains by the same blow-up argument as we did in .","Proposition 2.4 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.","Let $(v,\\nabla q)\\in C^{2,1}(\\overline{\\Omega }\\times (0,T])\\times C(\\overline{\\Omega }\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\gamma \\in [0,1/2)$ .","Assume that $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =0,$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ .","Then, $v\\equiv 0$ and $\\nabla q\\equiv 0$ ."],["Approximation","We prove Theorem 2.1.","We show existence of solutions for the Stokes equations (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , by approximation.","We approximate $f\\in W^{1,\\infty }_{0}$ by elements of $C^{\\infty }_{c}$ locally uniformly in $\\overline{\\Omega }$ .","Lemma 2.5 Let $\\Omega $ be an exterior domain with Lipschitz boundary.","There exist constants $C_1$ , $C_2$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C_{c}^{\\infty }(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert f_{m}\\big \\Vert _{\\infty }\\le C_1\\big \\Vert f\\big \\Vert _{\\infty }, \\\\&\\big \\Vert \\nabla f_{m}\\big \\Vert _{\\infty }\\le C_2\\big \\Vert f\\big \\Vert _{1,\\infty }, \\\\&f_m\\rightarrow f\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.","Proposition 2.6 The statement of Lemma 2.5 holds when $\\Omega =\\mathbb {R}^{n}$ with $C_1=1$ .","We cutoff the function $f\\in W^{1,\\infty }(\\mathbb {R}^{n})$ .","Let $\\theta \\in C^{\\infty }_{c}[0,\\infty )$ be a cut-off function satisfying $\\theta \\equiv 1$ in $[0,1]$ , $\\theta \\equiv 0$ in $[2,\\infty )$ and $0\\le \\theta \\le 1$ .","We set $\\theta _{m}(x)=\\theta (|x|/m)$ for $m\\ge 1$ so that $\\theta _{m}\\equiv 1$ for $|x|\\le m$ and $\\theta _{m}\\equiv 0$ for $|x|\\ge 2m$ .","Then, $f_m=f\\theta _{m}$ satisfies (2.7).","Proposition 2.7 Let $\\Omega $ be a bounded domain with Lipschitz boundary.","There exists a constant $C_3$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C^{\\infty }_{c}(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert \\nabla f_m\\big \\Vert _{\\infty }\\le C_3\\big \\Vert \\nabla f\\big \\Vert _{\\infty }\\\\f_m&\\rightarrow f\\quad \\textrm {uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.8)}$ We begin with the case when $\\Omega $ is star-shaped, i.e., $\\lambda \\Omega _{x_0}\\subset \\overline{\\Omega }$ for some $x_0\\in \\Omega $ and all $\\lambda <1$ , where $\\lambda \\Omega _{x_0}=\\lbrace x_0+\\lambda (x-x_0)\\ |\\ x\\in \\Omega \\rbrace $ .","We may assume $x_0=0\\in \\Omega $ and $\\lambda \\Omega \\subset \\overline{\\Omega }$ by translation.","For $f\\in W^{1,\\infty }_{0}(\\Omega )$ , we set $f_{\\lambda }(x)={\\left\\lbrace \\begin{array}{ll}&f\\big (x/\\lambda \\big )\\quad x\\in \\lambda \\Omega ,\\\\&0\\qquad \\hspace{15.0pt} x\\in \\Omega \\backslash \\overline{\\lambda \\Omega } .\\end{array}\\right.","}$ Then, $f_{\\lambda }$ is in $ W^{1,\\infty }(\\Omega )$ since $f$ is vanishing on $\\partial \\Omega $ .","It follows that $\\big \\Vert \\nabla f_{\\lambda }\\big \\Vert _{\\infty }\\le \\frac{1}{\\lambda }\\big \\Vert \\nabla f\\big \\Vert _{\\infty },$ and $f_{\\lambda }\\rightarrow f$ uniformly in $\\overline{\\Omega }$ as $\\lambda \\rightarrow 1$ .","By a mollification of $f_{\\lambda }$ , we obtain a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfying (2.8) with $C_3=2$ .","For general $\\Omega $ , we take an open covering $\\lbrace D_j\\rbrace _{j=1}^{N}$ so that $\\overline{\\Omega }\\subset \\cup _{j=1}^{N}D_j$ and $\\Omega _{j}=\\Omega \\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\in \\Omega _{j}$ .","We take a partition of unity $\\lbrace \\xi _{j}\\rbrace _{j=1}^{N}\\subset C_{c}^{\\infty }(\\mathbb {R}^{n})$ such that $\\sum _{j=1}^{N}\\xi _{j}=1$ , $0\\le \\xi _{j}\\le 1$ , spt $\\xi _{j}\\subset \\overline{D_j}$ and set $f=\\sum _{j=1}^{N}f_j,\\quad f_j=f\\xi _j.$ Since spt $f_j\\subset \\overline{\\Omega }_{j}$ , $\\xi _{j}=0$ on $\\partial D_{j}$ and $f=0$ on $\\partial \\Omega $ , $f_{j}$ is in $W^{1,\\infty }_{0}(\\Omega _{j})$ .","Since $\\Omega _j$ is star-shaped for some $x_j\\in \\Omega _{j}$ , there exists $\\lbrace f_{j,m}\\rbrace \\subset C_{c}^{\\infty }(\\Omega _j)$ satisfying (2.8) in $\\Omega _j$ with $C_3=2$ .","We extend $f_{j,m}\\in C_{c}^{\\infty }(\\Omega _j)$ to $\\Omega \\backslash \\overline{\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\sum _{j=1}^{N}f_{j,m}$ .","Then, $f_m\\in C_{c}^{\\infty }(\\Omega )$ converges to $f$ uniformly in $\\overline{\\Omega }$ .","We estimate $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le \\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j,m}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}\\le 2\\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}.$ Since $\\nabla f_j=\\nabla f\\xi _j+f\\nabla \\xi _j$ and $\\big \\Vert f\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C_{p}\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )}.$ Thus, $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfies (2.8).","The proof is complete.","The assertion follows from Propositions 2.6 and 2.7.","We recall the a priori estimate of $S(t)\\mathbb {P}\\partial $ for $f\\in C_{c}^{\\infty }(\\Omega )$ .","Proposition 2.8 There exists a constant $C$ such that $\\sup _{0\\le t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}S(t)\\mathbb {P}\\partial f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert \\nabla f\\big \\Vert _{\\infty }^{\\alpha } \\qquad \\mathrm {(2.9)}$ for $f\\in C^{\\infty }_{c}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ .","For $f\\in W^{1,\\infty }_{0}$ , we take a sequence $\\lbrace f_m\\rbrace \\subset C^{\\infty }_{c}$ satisfying (2.7).","For $v_{0,m}=\\mathbb {P}\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\int _{0}^{T}\\int _{\\Omega }\\big ( v_{m}\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q_m\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t=\\int _{\\Omega }f_m\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x,$ for $\\varphi \\in C^{\\infty }_{c}\\big (\\Omega \\times [0,T)\\big )$ .","By (2.9) and (2.7), there exists a constant $C$ independent of $m\\ge 1$ such that $\\sup _{0\\le t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v_m,q_m)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d\\nabla q_{m}\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla v_m$ , $\\nabla ^{2}v_m$ , $\\partial _{t}v_m$ and $\\nabla q_{m}$ .","By sending $m\\rightarrow \\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\mathbb {P}\\partial f$ .","By Proposition 2.4, the limit $(v,q)$ is unique.","We proved the unique existence of solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ and $f\\in W^{1,\\infty }_{0}$ satisfying (2.4).","The proof is now complete.","Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\mathbb {P}\\partial $ for $f\\in W^{1,\\infty }_{0}$ .","We take a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\mathbb {P}\\partial f_m$ satisfies $(v_{0,m}, \\varphi )=-(f_m, \\partial \\mathbb {P}\\varphi )\\quad \\textrm {for}\\ \\varphi \\in C^{\\infty }_{c}(\\Omega ).$ Since $\\partial \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ by Lemma A.1, the sequence $\\lbrace v_{0,m}\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\varphi )=-(f,\\partial \\mathbb {P}\\varphi )$ .","Since the limit $v_0$ is unique, the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ .","(ii) We recall that for a sequence $\\lbrace v_{0,m}\\rbrace _{m=1}^{\\infty }\\subset L^{\\infty }_{\\sigma }$ satisfying $||v_{0,m}||_{\\infty }&\\le K_1,\\\\v_{0,m}\\rightarrow v_{0}&\\quad \\textrm {a.e.","}\\ \\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .","From the proof of Theorem 2.1, we observe that for a sequence $\\lbrace f_m\\rbrace \\subset W^{1,\\infty }_{0}$ satisfying $||f_m||_{1,\\infty }&\\le K_2,\\\\f_m\\rightarrow f\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega },$ $\\overline{S(t)\\mathbb {P}\\partial } f_m$ subsequently converges to $\\overline{S(t)\\mathbb {P}\\partial } f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .","(iii) The extension $\\overline{S(t)\\mathbb {P}\\partial }$ satisfies the property $S(t)\\overline{S(s)\\mathbb {P}\\partial } f=\\overline{S(t+s)\\mathbb {P}\\partial } f$ for $t\\ge 0$ , $s>0$ and $f\\in W^{1,\\infty }_{0}$ .","In fact, this property holds for $f_m\\in C_{c}^{\\infty }$ satisfying (2.7).","By choosing a subsequence, $v_m(\\cdot ,t)=S(t)\\mathbb {P}\\partial f_m$ converges to $v(\\cdot ,t)=S(t)\\mathbb {P}\\partial f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ as in the proof of Theorem 2.1.","For fixed $s>0$ , sending $m\\rightarrow \\infty $ implies $S(t)S(s)\\mathbb {P}\\partial f_m=S(t)v_m(s)&\\rightarrow S(t)v(s) \\\\&=S(t)\\overline{S(s)\\mathbb {P}\\partial }f\\quad \\textrm {locally uniformly in \\overline{\\Omega }\\times (0,\\infty )}.$ Thus the property is inherited to $\\overline{S(t)\\mathbb {P}\\partial } f$ ."],["Mild solutions on $L^{\\infty }_{\\sigma }$","We prove Theorem 1.1 by approximation.","We show that a sequence of mild solutions $\\lbrace u_m\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by the $L^{\\infty }$ -estimates (1.5) and (1.6).","Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).","We first recall the existence of mild solutions on $C_{0,\\sigma }$ Proposition 3.1 For $u_0\\in C_{0,\\sigma }$ , there exist $T\\ge \\varepsilon _0/||u_0||_{\\infty }^{2}$ and a unique mild solution $u\\in C([0,T]; C_{0,\\sigma })$ satisfying (1.3)-(1.6).","We approximate $u_0\\in L^{\\infty }_{\\sigma }$ by elements of $C_{c,\\sigma }^{\\infty }\\subset C_{0,\\sigma }$ .","We take a sequence $\\lbrace u_{0,m}\\rbrace _{m=1}^{\\infty }\\subset C_{c,\\sigma }^{\\infty }(\\Omega )$ satisfying $\\begin{aligned}&\\Vert u_{0,m}\\Vert _{\\infty }\\le C\\Vert u_{0}\\Vert _{\\infty }\\\\&u_{0,m}\\rightarrow u_{0}\\quad \\textrm {a.e.","in}\\ \\Omega ,\\end{aligned}\\qquad \\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\ge 1$ .","We apply Proposition 3.1 and observe that there exists $T_m\\ge \\varepsilon _0/||u_{0,m}||_{\\infty }^{2}$ and a unique mild solution $u_m\\in C([0,T_m]; C_{0,\\sigma })$ satisfying $\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P}F_m(s)ds,\\\\F_m&=u_mu_m.","}\\qquad \\mathrm {(3.2)}\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\ge \\varepsilon /||u_0||_{\\infty }^{2} for \\varepsilon =\\varepsilon _0 C^{-2}/2 so that T_m\\ge T and u_m\\in C([0,T]; C_{0,\\sigma }) for m\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla u_m$ .","It follows from (1.5), (1.6) and (3.1) that $\\sup _{0\\le t\\le T}\\Big \\lbrace \\Vert u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1}{2}}\\Vert \\nabla u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1+\\beta }{2}}\\big [\\nabla u_{m}\\big ]_{\\Omega }^{(\\beta )}(t)\\Big \\rbrace &\\le C_1^{\\prime }\\Vert u_0\\Vert _{\\infty }, \\\\\\sup _{x\\in \\Omega }\\Big \\lbrace \\big [u_{m}\\big ]_{[\\delta ,T]}^{(\\gamma )}(x)+\\big [\\nabla u_{m}\\big ]_{[\\delta ,T]}^{(\\frac{\\gamma }{2})}(x)\\Big \\rbrace &\\le C_2^{\\prime }\\Vert u_0\\Vert _{\\infty }, $ for $\\beta ,\\gamma \\in (0,1)$ and $\\delta \\in (0,T]$ with some constants $C_1^{\\prime }$ and $C_2^{\\prime }$ , independent of $m\\ge 1$ .","Since $u_m$ and $\\nabla u_m$ are uniformly bounded and equi-continuous in $\\overline{\\Omega }\\times [\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.","Proposition 3.3 The limit $u\\in C_{w}([0,T]; L^{\\infty })$ is a mild solution for $u_0\\in L^{\\infty }_{\\sigma }$ .","We observe that the limit $u$ satisfies (1.4) by sending $m\\rightarrow \\infty $ .","The estimates (3.3) and (3.4) are inherited to $u$ .","We prove that $u$ satisfies the integral equation (1.3).","By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by Remarks 2.9 (ii).","It follows from (3.3) and Proposition 3.2 that $\\begin{aligned}||F_m||_{\\infty }&\\le K,\\\\||\\nabla F_{m}||_{\\infty }&\\le \\frac{2}{s^{\\frac{1}{2}}}K,\\\\F_m\\rightarrow F\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T]\\ \\textrm {as}\\ m\\rightarrow \\infty ,\\end{aligned}\\qquad \\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\prime }||u_0||_{\\infty }$ .","By choosing a subsequence, we have $\\overline{S(\\eta )\\mathbb {P} F_m\\rightarrow \\overline{S(\\eta )\\mathbb {P} F\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T],}for each s\\in (0,t) as in Remarks 2.9 (ii).","It follows from (3.5) and (2.5) that\\begin{equation*}\\big \\Vert \\overline{S(t-s)\\mathbb {P} F_m\\big \\Vert _{\\infty }\\le \\frac{C}{(t-s)^{\\frac{1-\\alpha }{2}}}\\Bigg (1+\\frac{2}{s^{\\frac{\\alpha }{2}}}\\Bigg )K^{2}}for 00$ .","For $u_{0}^{1}\\in C_{0,\\sigma }$ , there exists $\\lbrace u_{0,m}^{1}\\rbrace \\subset C_{c,\\sigma }^{\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\infty }\\le \\epsilon $ for $m\\ge N^{1}_{\\epsilon }$ .","We apply (3.5) and observe that $t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }&\\le t^{\\frac{1}{2}}C\\big (||S(t)u_{0,m}^{1}||_{\\infty }+||S(t)Au_{0,m}^{1}||_{\\infty }\\big )\\\\&\\le t^{\\frac{1}{2}}C^{\\prime }\\big (||u_{0,m}^{1}||_{\\infty }+||Au_{0,m}^{1}||_{\\infty }\\big )\\rightarrow 0\\quad \\textrm {as}\\ t\\rightarrow 0.$ We estimate $\\overline{\\lim _{t\\rightarrow 0}}t^{\\frac{1}{2}}||\\nabla S(t)u_0^{1}||_{\\infty }&\\le \\overline{\\lim _{t\\rightarrow 0}}\\big (t^{\\frac{1}{2}}||\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\infty }+t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }\\big ) \\\\&\\le C^{\\prime \\prime }\\epsilon .$ We set $u_{0,m}^{2}=\\eta _{\\delta _m}*u_{0}^{2}$ by the mollifier $\\eta _{\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\overline{\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\infty }\\le \\epsilon $ for $m\\ge N_{\\epsilon }^{2}$ .","Since $u_{0,m}^{2}$ is supported away from $\\partial \\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\Delta u_{0,m}^{2}$ (see ).","By a similar way as for $u_{0}^{1}$ , we estimate $\\overline{\\lim }_{t\\rightarrow 0}t^{1/2}||\\nabla S(t)u_0^{2}||_{\\infty }\\le C^{\\prime \\prime }\\epsilon $ .","We proved $\\overline{\\lim }_{t\\rightarrow 0}t^{\\frac{1}{2}}||\\nabla S(t)u_0||_{\\infty }\\le 2C^{\\prime \\prime }\\epsilon .$ Since $\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .","The assertion follows from Propositions 3.1-3.4.","The proof is now complete.","Remark 3.5 We set the associated pressure of mild solutions on $L^{\\infty }$ by (1.7) and the harmonic-pressure operator $\\mathbb {K}: L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )\\longrightarrow L^{\\infty }_{d}(\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\Delta q&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial q}{\\partial n}&={\\partial \\Omega }W\\quad \\textrm {on}\\ \\partial \\Omega .$ Note that $\\Delta u\\cdot n={\\partial \\Omega }W$ by the divergence-free condition of $u$ .","Here, $L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )$ denotes the space of all bounded tangential vector fields on $\\partial \\Omega $ and $L^{\\infty }_{d}(\\Omega )$ is the space of all functions $f\\in L^{1}_{\\textrm {loc}}(\\Omega )$ such that $df$ is bounded in $\\Omega $ for $d(x)=\\inf _{y\\in \\partial \\Omega }|x-y|$ , $x\\in \\Omega $ .","Since $W=-(\\nabla u-\\nabla ^{T}u)n$ is bounded on $\\partial \\Omega $ for mild solutions on $L^{\\infty }$ , $\\nabla q=\\mathbb {K}W$ is defined as an element of $L^{\\infty }_{d}$ .","Moreover, $\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\in W^{1,\\infty }_{0}$ as a distribution by Remarks 2.9 (i).","Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\infty }$ .","acknowledgements The author is grateful to the anonymous referees for their valuable comments.","This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.","appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .","We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.","Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.","Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .","Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .","We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .","It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .","We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .","We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .","We may assume that $0\\in \\Omega ^{c}$ by translation.","We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .","Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.","We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .","In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .","Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .","Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .","In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .","By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .","We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .","We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .","Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .","Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .","The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .","We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .","Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.","By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .","Thus, $g$ is continuous on $\\partial \\Omega $ .","Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .","Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.","The proof is complete.","We estimate $\\Phi _2$ by means of the layer potential.","Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .","The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .","The assertion (i) is based on the Fredholm's theorem.","See .","Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .","Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).","When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .","Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.","Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.","Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.","We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.","The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .","By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .","We shall show that $\\Psi \\equiv 0$ .","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .","Suppose that $x_0\\in \\partial \\Omega $ .","Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .","Thus $x_0\\in \\Omega $ .","We apply the strong maximum principle and conclude that $\\Psi $ is constant.","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .","The proof is complete.","Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .","Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .","Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .","The proof is complete.","By Propositions A.2 and A.6, the assertion follows.","A4article author=Abe, K., title=The Navier-Stokes equations in a space of bounded functions, date=2015, journal=Commun.","Math.","Phys., volume=338, pages=849865, A3article author=Abe, K., title=On estimates for the Stokes flow in a space of bounded functions, date=2016, journal=J.","Differ.","Equ., volume=261, pages=17561795, 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A., title=Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, date=1976, journal=Zap.","Naučn.","Sem.","Leningrad.","Otdel Mat.","Inst.","Steklov.","(LOMI), volume=59, pages=178254, 257, note=Boundary value problems of mathematical physics and related questions in the theory of functions, 9, Sl03article author=Solonnikov, V. A., title=On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, date=2003, ISSN=1072-3374, journal=J.","Math.","Sci.","(N.","Y.","), volume=114, pages=17261740, url=http://dx.doi.org/10.1023/A:1022317029111, Sl06article author=Solonnikov, V. A., title=Weighted Schauder estimates for evolution Stokes problem, date=2006, ISSN=0430-3202, journal=Ann.","Univ.","Ferrara Sez.","VII Sci.","Mat., volume=52, pages=137172, url=http://dx.doi.org/10.1007/s11565-006-0012-7, Sl07incollection author=Solonnikov, V. A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.","Math.","Soc.","Transl.","Ser.","2, volume=220, publisher=Amer.","Math.","Soc., Providence, RI, pages=165200,"],["acknowledgements","The author is grateful to the anonymous referees for their valuable comments.","This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015."],["$L^{1}$ -estimates for the Neumann problem"," In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .","We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.","Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.","Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .","Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .","We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .","It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .","We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .","We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .","We may assume that $0\\in \\Omega ^{c}$ by translation.","We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .","Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.","We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .","In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .","Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .","Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .","In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .","By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .","We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .","We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .","Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .","Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .","The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .","We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .","Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.","By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .","Thus, $g$ is continuous on $\\partial \\Omega $ .","Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .","Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.","The proof is complete.","We estimate $\\Phi _2$ by means of the layer potential.","Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .","The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .","The assertion (i) is based on the Fredholm's theorem.","See .","Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .","Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .","Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).","When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .","Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.","Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.","Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.","We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .","Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.","The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .","By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .","We shall show that $\\Psi \\equiv 0$ .","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .","Suppose that $x_0\\in \\partial \\Omega $ .","Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .","Thus $x_0\\in \\Omega $ .","We apply the strong maximum principle and conclude that $\\Psi $ is constant.","Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .","The proof is complete.","Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .","Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .","Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .","The proof is complete.","By Propositions A.2 and A.6, the assertion follows.","A4article author=Abe, K., title=The Navier-Stokes equations in a space of bounded functions, date=2015, journal=Commun.","Math.","Phys., volume=338, pages=849865, A3article author=Abe, K., title=On estimates for the Stokes flow in a space of bounded functions, date=2016, journal=J.","Differ.","Equ., volume=261, pages=17561795, AG1article author=Abe, K., author=Giga, Y., title=Analyticity of the Stokes semigroup in spaces of bounded functions, date=2013, journal=Acta Math., volume=211, pages=146, 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A., title=Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, date=1976, journal=Zap.","Naučn.","Sem.","Leningrad.","Otdel Mat.","Inst.","Steklov.","(LOMI), volume=59, pages=178254, 257, note=Boundary value problems of mathematical physics and related questions in the theory of functions, 9, Sl03article author=Solonnikov, V. A., title=On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, date=2003, ISSN=1072-3374, journal=J.","Math.","Sci.","(N.","Y.","), volume=114, pages=17261740, url=http://dx.doi.org/10.1023/A:1022317029111, Sl06article author=Solonnikov, V. A., title=Weighted Schauder estimates for evolution Stokes problem, date=2006, ISSN=0430-3202, journal=Ann.","Univ.","Ferrara Sez.","VII Sci.","Mat., volume=52, pages=137172, url=http://dx.doi.org/10.1007/s11565-006-0012-7, Sl07incollection author=Solonnikov, V. A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.","Math.","Soc.","Transl.","Ser.","2, volume=220, publisher=Amer.","Math.","Soc., Providence, RI, pages=165200,"]],"string":"[\n [\n \"Exterior Navier-Stokes flows for bounded data\"\n ],\n [\n \"Abstract We prove unique existence of mild solutions on $L^{\\\\infty}_{\\\\sigma}$ for the Navier-Stokes equations in an exterior domain in $ \\\\mathbb{R}^{n}$, $n\\\\geq 2$, subject to the non-slip boundary condition.\"\n ],\n [\n \"Introduction\",\n \"We consider the initial-boundary value problem of the Navier-Stokes equations in an exterior domain $\\\\Omega \\\\subset \\\\mathbb {R}^{n}$ , $n\\\\ge 2$ : $\\\\begin{aligned}\\\\partial _t u-\\\\Delta {u}+u\\\\cdot \\\\nabla u+\\\\nabla {p}&= 0 \\\\quad \\\\textrm {in}\\\\quad \\\\Omega \\\\times (0,T), \\\\\\\\u&=0\\\\quad \\\\textrm {in}\\\\quad \\\\Omega \\\\times (0,T), \\\\\\\\u&=0\\\\quad \\\\textrm {on}\\\\quad \\\\partial \\\\Omega \\\\times (0,T), \\\\\\\\u&=u_0\\\\quad \\\\hspace{-4.0pt} \\\\textrm {on}\\\\quad \\\\Omega \\\\times \\\\lbrace t=0\\\\rbrace .\\\\end{aligned}\\\\qquad \\\\mathrm {(1.1)}$ There is a large literature on the solvability of the exterior problem for initial data decaying at space infinity.\",\n \"However, a few results are available for non-decaying data.\",\n \"A typical example of non-decaying flow is a stationary solution of (1.1) having a finite Dirichlet integral, called $D$ -solution .\",\n \"It is known that $D$ -solutions are bounded in $\\\\Omega $ and asymptotically constant as $|x|\\\\rightarrow \\\\infty $ ; see Remarks 1.2 (ii).\",\n \"In this paper, we do not impose on $u_0$ conditions at space infinity.\",\n \"The purpose of this paper is to establish a solvability of (1.1) for merely bounded initial data.\",\n \"We set the solenoidal $L^{\\\\infty }$ -space, $L^{\\\\infty }_{\\\\sigma }(\\\\Omega )=\\\\left\\\\lbrace f\\\\in L^{\\\\infty }(\\\\Omega )\\\\ \\\\Bigg |\\\\ \\\\int _{\\\\Omega }f\\\\cdot \\\\nabla \\\\varphi \\\\textrm {d}x=0 \\\\quad \\\\textrm {for}\\\\ \\\\varphi \\\\in \\\\hat{W}^{1,1}(\\\\Omega ) \\\\right\\\\rbrace ,$ by the homogeneous Sobolev space $\\\\hat{W}^{1,1}(\\\\Omega )=\\\\lbrace \\\\varphi \\\\in L^{1}_{\\\\textrm {loc}}(\\\\Omega )\\\\ |\\\\ \\\\nabla \\\\varphi \\\\in L^{1}(\\\\Omega )\\\\ \\\\rbrace $ .\",\n \"For exterior domains, the space $L^{\\\\infty }_{\\\\sigma }$ agrees with the space of all bounded divergence-free vector fields, whose normal trace is vanishing on $\\\\partial \\\\Omega $ .\",\n \"The $L^{\\\\infty }$ -type solvability for (1.1) is recently established on $C_{0,\\\\sigma }$ in the previous work of the author , where $C_{0,\\\\sigma }$ is the $L^{\\\\infty }$ -closure of $C_{c,\\\\sigma }^{\\\\infty }$ , the space of all smooth solenoidal vector fields with compact support in $\\\\Omega $ .\",\n \"Since the condition $u_0\\\\in C_{0,\\\\sigma }$ imposes the decay $u_0\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ , we develop an existence theorem for non-decaying space $L^{\\\\infty }_{\\\\sigma }$ , which in particular includes asymptotically constant vector fields.\",\n \"Moreover, the space $L^{\\\\infty }_{\\\\sigma }$ includes vector fields rotating at space infinity; see Remarks 1.2 (iv).\",\n \"When $\\\\Omega $ is the whole space or a half space , , the existence of mild solutions of (1.1) on $L^{\\\\infty }_{\\\\sigma }$ is proved by explicit formulas of the Stokes semigroup.\",\n \"In this paper, we prove unique existence of mild solutions on $L^{\\\\infty }_{\\\\sigma }$ for exterior domains based on $L^{\\\\infty }$ -estimates of the Stokes semigroup , .\",\n \"To state a result, let $S(t)$ denote the Stokes semigroup.\",\n \"It is proved in that $S(t)$ is an analytic semigroup on $L^{\\\\infty }_{\\\\sigma }$ for exterior domains of class $C^{3}$ .\",\n \"Let $\\\\mathbb {P}$ denote the Helmholtz projection.\",\n \"We write $F=(\\\\sum _{i=1}^{n}\\\\partial _{i}F_{ij})$ for matrix-valued functions $F=(F_{ij})$ .\",\n \"It is proved in that the composition operator $S(t)\\\\mathbb {P} satisfies an estimate of the form\\\\begin{equation*}\\\\big \\\\Vert S(t)\\\\mathbb {P}F̥\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le \\\\frac{C_{\\\\alpha }}{t^{\\\\frac{1-\\\\alpha }{2}}}\\\\big \\\\Vert F\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}^{1-\\\\alpha }\\\\big \\\\Vert \\\\nabla F\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}^{\\\\alpha }, \\\\end{equation*}for \\\\qquad \\\\mathrm {(1.2)}$ FC10W1,2()$, $ tT0$ and $ (0,1)$.\",\n \"Here, $ W1,2()$ denotes the Sobolev space and $ C01()$ denotes the $ W1,$-closure of $ Cc()$, the space of all smooth functions with compact support in $$.\",\n \"Although the projection $ P$ may not act as a bounded operator on $ L$, the $ L$-estimate (1.2) implies that the composition $ S(t)P is uniquely extendable to a bounded operator from $C_{0}^{1}$ to $C_{0,\\\\sigma }$ .\",\n \"Note that $F\\\\in C^{1}_{0}$ imposes a decay condition at space infinity.\",\n \"Thus the extension to $C_{0}^{1}$ is not sufficient for studying non-decaying solutions.\",\n \"In this paper, we prove that the composition $S(t)\\\\mathbb {P} is uniquely extendable to a bounded operator $S(t)P$ from the non-decaying space $ W1,0$ to $ L$, where $ W1,0$ is the space of all functions in $ W1,$ vanishing on $$.$ By means of the new extension, we study the integral equation on $L^{\\\\infty }_{\\\\sigma }$ of the form $u(t)=S(t)u_{0}-\\\\int _{0}^{t}\\\\overline{S(t-s)\\\\mathbb {P} (uu)(s)ds.\",\n \"\\\\qquad \\\\mathrm {(1.3)}}Here, uu=(u_iu_j) is the tensor product.\",\n \"We call solutions of (1.3) mild solution on L^{\\\\infty }_{\\\\sigma }.\",\n \"Since the projection \\\\mathbb {P} may not be bounded on L^{\\\\infty }, the extension \\\\overline{S(t)\\\\mathbb {P} is not expressed by the individual operators.\",\n \"We thus prove that mild solutions satisfy (1.1) by using a weak form.\",\n \"Let C^{\\\\infty }_{c,\\\\sigma }(\\\\Omega \\\\times [0,T)) denote the space of all smooth solenoidal vector fields with compact support in \\\\Omega \\\\times [0,T).\",\n \"Let C([0,T]; X) (resp.\",\n \"C_{w}([0,T]; X)) denote the space of all (resp.\",\n \"weakly-star) continuous functions from [0,T] to a Banach space X.\",\n \"Let BUC_{\\\\sigma }(\\\\Omega ) denote the space of all solenoidal vector fields in BUC(\\\\Omega ) vanishing on \\\\partial \\\\Omega , where BUC(\\\\Omega ) is the space of all bounded uniformly continuous functions in \\\\overline{\\\\Omega }.\",\n \"Let [\\\\cdot ]_{\\\\Omega }^{(\\\\beta )} denote the \\\\beta -th Hölder semi-norm in \\\\overline{\\\\Omega }.\",\n \"The main result of this paper is the following:}\\\\vspace{10.0pt}$ Theorem 1.1 Let $\\\\Omega $ be an exterior domain with $C^{3}$ -boundary in $\\\\mathbb {R}^{n}$ , $n\\\\ge 2$ .\",\n \"For $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ , there exist $T\\\\ge \\\\varepsilon /||u_0||_{\\\\infty }^{2}$ and a unique mild solution $u\\\\in C_{w}([0,T]; L^{\\\\infty })$ such that $\\\\int _{0}^{T}\\\\int _{\\\\Omega }\\\\big (u\\\\cdot (\\\\partial _{t}\\\\varphi +\\\\Delta \\\\varphi )+u u: \\\\nabla \\\\varphi \\\\big )\\\\textrm {d}x\\\\textrm {d}t=-\\\\int _{\\\\Omega }u_0(x)\\\\cdot \\\\varphi (x,0)\\\\textrm {d}x \\\\qquad \\\\mathrm {(1.4)}$ for all $\\\\varphi \\\\in C^{\\\\infty }_{c,\\\\sigma }(\\\\Omega \\\\times [0,T))$ , with some constant $\\\\varepsilon =\\\\varepsilon _{\\\\Omega }$ .\",\n \"The solution $u$ satisfies $\\\\sup _{0< t\\\\le T}\\\\left\\\\lbrace \\\\big \\\\Vert u\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}(t)+t^{\\\\frac{1}{2}}\\\\big \\\\Vert \\\\nabla u\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}(t)+t^{\\\\frac{1+\\\\beta }{2}}\\\\Big [\\\\nabla u\\\\Big ]^{(\\\\beta )}_{\\\\Omega }(t) \\\\right\\\\rbrace \\\\le C_1\\\\big \\\\Vert u_0\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}, \\\\qquad \\\\mathrm {(1.5)}$ $\\\\sup _{x\\\\in \\\\Omega }\\\\left\\\\lbrace \\\\Big [u\\\\Big ]^{(\\\\gamma )}_{[\\\\delta ,T]}(x)+\\\\Big [\\\\nabla u\\\\Big ]^{(\\\\frac{\\\\gamma }{2})}_{[\\\\delta ,T]}(x) \\\\right\\\\rbrace \\\\le C_2\\\\big \\\\Vert u_0\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}, \\\\qquad \\\\mathrm {(1.6)}$ for $\\\\beta , \\\\gamma \\\\in (0,1)$ and $\\\\delta \\\\in (0,T)$ with the constant $C_1$ , independent of $u_0$ and $T$ .\",\n \"The constant $C_2$ depends on $\\\\gamma $ , $\\\\delta $ and $T$ .\",\n \"If $u_0\\\\in BUC_{\\\\sigma }$ , $u$ , $t^{1/2}\\\\nabla u\\\\in C([0,T]; BUC)$ and $t^{1/2}\\\\nabla u$ vanishes at time zero.\",\n \"Remarks 1.2 (i) (Blow-up rate) By the estimate of the existence time in Theorem 1.1, we obtain a blow-up rate of mild solutions $u\\\\in C_{w}([0,T_*); L^{\\\\infty })$ of the form $\\\\Vert u\\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\ge \\\\frac{\\\\varepsilon ^{\\\\prime }}{\\\\sqrt{T_*-t}}\\\\quad \\\\textrm {for}\\\\ t0.$ It is proved in that the Cauchy problem of (1.1) for $n=2$ is globally well-posed for $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ based on the local solvability result in .\",\n \"We proved a local solvability on $L^{\\\\infty }_{\\\\sigma }$ for exterior domains.\",\n \"Note that global solutions exist for rotationally symmetric initial data $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ ; see below (iv).\",\n \"(iv) (Rotating flows) An example of $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ which is not asymptotically constant is a vector field rotating at space infinity.\",\n \"For example, we consider the two-dimensional unit disk $\\\\Omega ^{c}$ centered at the origin and a rotationally symmetric initial data $u_{0}=u_{0}^{\\\\theta }(r)e_{\\\\theta }(\\\\theta )$ for $e_{\\\\theta }(\\\\theta )=(-\\\\sin {\\\\theta },\\\\cos {\\\\theta })$ .\",\n \"Observe that $u_0$ is a solenoidal vector field in $\\\\Omega $ and a direction of $u_0$ varies for $\\\\theta \\\\in [0,2\\\\pi ]$ and $u_{0}^{\\\\theta }\\\\in L^{\\\\infty }(1,\\\\infty )$ .\",\n \"Solutions of (1.1) for $u_0$ are rotationally symmetric and given by $u=e^{t\\\\Delta _{D}}u_0\\\\qquad p=\\\\int _{1}^{|x|}\\\\frac{|u|^{2}}{r}\\\\textrm {d}r,$ where $\\\\Delta _{D}$ denotes the Laplace operator subject to the Dirichlet boundary condition.\",\n \"The solution $u$ is bounded in $\\\\Omega \\\\times (0,\\\\infty )$ and non-decaying as $|x|\\\\rightarrow \\\\infty $ .\",\n \"(v) (Associated pressure) We invoke that the associated pressure of mild solutions on $L^{p}$ ($p\\\\ge n$ ) is determined by the projection operator $\\\\mathbb {Q}=I-\\\\mathbb {P}$ and $\\\\nabla p=\\\\mathbb {Q}\\\\Delta u-\\\\mathbb {Q}(u\\\\cdot \\\\nabla u).$ Since the projection $\\\\mathbb {Q}$ may not be bounded on $L^{\\\\infty }$ , this representation is no longer available for mild solutions on $L^{\\\\infty }_{\\\\sigma }$ .\",\n \"When $\\\\Omega =\\\\mathbb {R}^{n}$ or $\\\\mathbb {R}^{n}_{+}$ , the projection $\\\\mathbb {Q}$ has explicit kernels and we are able to find associated pressure of mild solutions on $L^{\\\\infty }$ ; see for $\\\\Omega =\\\\mathbb {R}^{n}$ and , , for $\\\\Omega =\\\\mathbb {R}^{n}_{+}$ .\",\n \"Although explicit kernels are not available for exterior domains, we are able to find the associated pressure of mild solutions on $L^{\\\\infty }$ .\",\n \"We set $\\\\nabla p=\\\\mathbb {K}W-\\\\mathbb {Q}F̥ $ for $W=-(\\\\nabla u-\\\\nabla ^{T} u)n_{\\\\Omega }$ and $F=uu$ , where $n_{\\\\Omega }$ is the unit outward normal on $\\\\partial \\\\Omega $ and $\\\\mathbb {K}$ is a solution operator of the homogeneous Neumann problem (harmonic-pressure operator) .\",\n \"Note that $W=-\\\\textrm {curl}\\\\ u\\\\times n_{\\\\Omega }$ for $n=3$ .\",\n \"The operators $\\\\mathbb {K}$ and $\\\\mathbb {Q} act for bounded functions and the associated pressure on $ L$ is uniquely determined by (1.7) in the sense of distribution; see Remark 3.5 for a detailed discussion.$ For asymptotically constant initial data $u_0$ (i.e., $u_{0}\\\\rightarrow u_{\\\\infty }$ as $|x|\\\\rightarrow \\\\infty $ ), local solvability of (1.1) for $n=3$ is proved in by means of the Oseen semigroup.\",\n \"In the paper, the problem (1.1) is reduced to an initial-boundary problem for decaying data by shifting $u$ by a constant $u_{\\\\infty }$ .\",\n \"Our analysis is based on the $L^{\\\\infty }$ -estimates of the Stokes semigroup which yields a local-in-time solvability of (1.1) without conditions for $u_0$ at space infinity.\",\n \"The $L^{\\\\infty }$ -theory for the Cauchy problem of the Navier-Stokes equations is developed by Knightly , , Cannon and Knightly , Cannone () and Giga et al.\",\n \".\",\n \"For the whole space, mild solutions on $L^{\\\\infty }$ are smooth and satisfy (1.1) in a classical sense .\",\n \"For a half space, mild solutions on $L^{\\\\infty }$ are constructed in (see also , ).\",\n \"There are a few results on solvability of the exterior problem for non-decaying data.\",\n \"In , unique existence of continuous solutions of (1.1) for $n\\\\ge 3$ is proved for non-decaying and Hölder continuous initial data.\",\n \"The result is extended in for merely bounded $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ and $n\\\\ge 3$ by using $L^{\\\\infty }$ -estimates of the Stokes semigroup , .\",\n \"Note that mild solutions on $L^{\\\\infty }_{\\\\sigma }$ are not constructed without the composition operator $\\\\overline{S(t)\\\\mathbb {P}.\",\n \"We proved the unique existence of mild solutions on L^{\\\\infty }_{\\\\sigma }, which in particular yields a local-in-time solvability for n=2.\",\n \"The integral form (1.3) is fundamental for studying solutions of (1.1).\",\n \"We expect that mild solutions on L^{\\\\infty } are sufficiently smooth and satisfy (1.1) in a classical sense.\\\\\\\\}$ The article is organized as follows.\",\n \"In Section 2, we extend the composition operator $S(t)\\\\mathbb {P} to a bounded operator from $ W1,0$ to $ L$ by approximation as we did the Stokes semigroup in \\\\cite {AG2}.\",\n \"We extend $ S(t)P as a solution operator $F\\\\longmapsto v(\\\\cdot ,t)$ for solutions $(v,q)$ of the Stokes equations for $v_0=\\\\mathbb {P}F$ .\",\n \"Note that $v_0=\\\\mathbb {P}F$ for $F\\\\in W^{1,\\\\infty }_{0}$ is not an element of $L^{\\\\infty }$ in general since the projection $\\\\mathbb {P}$ is not bounded on $L^{\\\\infty }$ .\",\n \"We understand $v_0=\\\\mathbb {P}F$ as distribution by using the fact that $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}$ for $\\\\varphi \\\\in C_{c}^{\\\\infty }$ (Lemma A.1).\",\n \"We approximate $F\\\\in W^{1,\\\\infty }_{0}$ by a sequence $\\\\lbrace F_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }$ locally uniformly in $\\\\overline{\\\\Omega }$ and obtain a unique extension $\\\\overline{S(t)\\\\mathbb {P}: F\\\\longmapsto v(\\\\cdot ,t) by a limit v of the sequence v_m=S(t)\\\\mathbb {P}F_m.\",\n \"}In Section 3, we prove Theorem 1.1.\",\n \"We approximate initial data $ u0L$ by a sequence $ {u0,m}Cc,$ satisfying $ u0,mu0$ a.e.\",\n \"in $$ and $ ||u0,m||C||u0||$.\",\n \"Since the property of mild solutions (1.4) may not follow from a direct iteration argument on $ L$, we construct mild solutions by approximation.\",\n \"We apply an existence theorem on $ C0,$ \\\\cite {A4} and construct a sequence of mild solutions $ umC([0,T]; C0,)$ satisfying (1.4)-(1.6) for $ u0,mC0,$.\",\n \"We prove that $ um$ subsequently converges to a mild solution $ u$ for $ u0L$ locally uniformly in $(0,T]$.$ In Appendix A, we show that $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}$ for $\\\\varphi \\\\in C_{c}^{\\\\infty }$ by means of the layer potential.\"\n ],\n [\n \"An extension of the composition operator\",\n \"In this section, we prove that the composition operator $S(t)\\\\mathbb {P}\\\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\\\infty }_{0}$ to $L^{\\\\infty }_{\\\\sigma }$ .\",\n \"We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\\\mathbb {P}\\\\partial :f\\\\longmapsto v(\\\\cdot ,t)$ .\",\n \"In what follows, $\\\\partial =\\\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\\\cdots ,n$ .\"\n ],\n [\n \"The Stokes system\",\n \"We consider the Stokes equations, $\\\\begin{aligned}\\\\partial _{t}v-\\\\Delta v+\\\\nabla q&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega \\\\times (0,T), \\\\\\\\v&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega \\\\times (0,T), \\\\\\\\v&=0\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega \\\\times (0,T), \\\\\\\\v&=v_0\\\\quad \\\\hspace{-4.0pt} \\\\textrm {on}\\\\ \\\\Omega \\\\times \\\\lbrace t=0\\\\rbrace .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\\\bigl |v(x,t)\\\\bigr |+t^{\\\\frac{1}{2}}\\\\bigl |\\\\nabla v(x,t)\\\\bigr |+t\\\\bigl |\\\\nabla ^{2}v(x,t)\\\\bigr |+t\\\\bigl |\\\\partial _{t}v(x,t)\\\\bigr |+t\\\\bigl |\\\\nabla q(x,t)\\\\bigr |.$ Let $d(x)$ denote the distance from $x\\\\in \\\\Omega $ to $\\\\partial \\\\Omega $ .\",\n \"Let $(v, \\\\nabla q)\\\\in C^{2+\\\\mu ,1+\\\\frac{\\\\mu }{2}}(\\\\overline{\\\\Omega }\\\\times (0,T])\\\\times C^{\\\\mu ,\\\\frac{\\\\mu }{2}}(\\\\overline{\\\\Omega }\\\\times (0,T]),\\\\ \\\\mu \\\\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .\",\n \"We say that $(v,q)$ is a solution of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , if $\\\\sup _{00$ .\",\n \"For $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\\\sup _{00$ , $T>0$ , $R>0$ , there exists a constant $C=C\\\\bigl (\\\\mu ,\\\\delta ,T, R,d\\\\bigr )$ such that $\\\\Big [\\\\nabla ^{2}v\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}+\\\\Big [v_t\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}+\\\\Big [\\\\nabla q\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}\\\\le CN \\\\qquad \\\\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\\\prime }=B_{x_0}(R)\\\\times (2\\\\delta ,T]$ and $x_0\\\\in \\\\Omega $ satisfying $\\\\overline{B_{x_0}(R)}\\\\subset \\\\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\\\partial \\\\Omega $ .\",\n \"(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\\\mu \\\\in (0,1),\\\\ \\\\delta >0$ , $T>0$ and $R\\\\le R_{0}$ , there exists a constant $C$ depending on $\\\\mu $ , $\\\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\\\partial \\\\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\\\prime }=\\\\Omega _{x_0,R}\\\\times (2\\\\delta ,T]$ and $\\\\Omega _{x_0,R}=B_{x_0}(R)\\\\cap \\\\Omega $ , $x_0\\\\in \\\\partial \\\\Omega $ .\",\n \"We observe the uniqueness of solutions for (2.1).\",\n \"The uniqueness of the Stokes equations (2.1) for $v_0\\\\in L^{\\\\infty }_{\\\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .\",\n \"In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , may not be bounded at $t=0$ .\",\n \"The corresponding uniqueness result for a half space is recently proved in .\",\n \"We deduce the result for exterior domains by the same blow-up argument as we did in .\",\n \"Proposition 2.4 Let $\\\\Omega $ be an exterior domain with $C^{3}$ -boundary.\",\n \"Let $(v,\\\\nabla q)\\\\in C^{2,1}(\\\\overline{\\\\Omega }\\\\times (0,T])\\\\times C(\\\\overline{\\\\Omega }\\\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\\\gamma \\\\in [0,1/2)$ .\",\n \"Assume that $\\\\int _{0}^{T}\\\\int _{\\\\Omega }\\\\big (v\\\\cdot (\\\\partial _{t}\\\\varphi +\\\\Delta \\\\varphi )-\\\\nabla q\\\\cdot \\\\varphi \\\\big )\\\\textrm {d}x\\\\textrm {d}t =0,$ for all $\\\\varphi \\\\in C^{\\\\infty }_{c}(\\\\Omega \\\\times [0,T))$ .\",\n \"Then, $v\\\\equiv 0$ and $\\\\nabla q\\\\equiv 0$ .\"\n ],\n [\n \"Approximation\",\n \"We prove Theorem 2.1.\",\n \"We show existence of solutions for the Stokes equations (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , by approximation.\",\n \"We approximate $f\\\\in W^{1,\\\\infty }_{0}$ by elements of $C^{\\\\infty }_{c}$ locally uniformly in $\\\\overline{\\\\Omega }$ .\",\n \"Lemma 2.5 Let $\\\\Omega $ be an exterior domain with Lipschitz boundary.\",\n \"There exist constants $C_1$ , $C_2$ such that for $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ there exists a sequence of functions $\\\\lbrace f_m\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ such that $\\\\begin{aligned}&\\\\big \\\\Vert f_{m}\\\\big \\\\Vert _{\\\\infty }\\\\le C_1\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }, \\\\\\\\&\\\\big \\\\Vert \\\\nabla f_{m}\\\\big \\\\Vert _{\\\\infty }\\\\le C_2\\\\big \\\\Vert f\\\\big \\\\Vert _{1,\\\\infty }, \\\\\\\\&f_m\\\\rightarrow f\\\\quad \\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\quad \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.\",\n \"Proposition 2.6 The statement of Lemma 2.5 holds when $\\\\Omega =\\\\mathbb {R}^{n}$ with $C_1=1$ .\",\n \"We cutoff the function $f\\\\in W^{1,\\\\infty }(\\\\mathbb {R}^{n})$ .\",\n \"Let $\\\\theta \\\\in C^{\\\\infty }_{c}[0,\\\\infty )$ be a cut-off function satisfying $\\\\theta \\\\equiv 1$ in $[0,1]$ , $\\\\theta \\\\equiv 0$ in $[2,\\\\infty )$ and $0\\\\le \\\\theta \\\\le 1$ .\",\n \"We set $\\\\theta _{m}(x)=\\\\theta (|x|/m)$ for $m\\\\ge 1$ so that $\\\\theta _{m}\\\\equiv 1$ for $|x|\\\\le m$ and $\\\\theta _{m}\\\\equiv 0$ for $|x|\\\\ge 2m$ .\",\n \"Then, $f_m=f\\\\theta _{m}$ satisfies (2.7).\",\n \"Proposition 2.7 Let $\\\\Omega $ be a bounded domain with Lipschitz boundary.\",\n \"There exists a constant $C_3$ such that for $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ there exists a sequence of functions $\\\\lbrace f_m\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C^{\\\\infty }_{c}(\\\\Omega )$ such that $\\\\begin{aligned}&\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{\\\\infty }\\\\le C_3\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty }\\\\\\\\f_m&\\\\rightarrow f\\\\quad \\\\textrm {uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\quad \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.8)}$ We begin with the case when $\\\\Omega $ is star-shaped, i.e., $\\\\lambda \\\\Omega _{x_0}\\\\subset \\\\overline{\\\\Omega }$ for some $x_0\\\\in \\\\Omega $ and all $\\\\lambda <1$ , where $\\\\lambda \\\\Omega _{x_0}=\\\\lbrace x_0+\\\\lambda (x-x_0)\\\\ |\\\\ x\\\\in \\\\Omega \\\\rbrace $ .\",\n \"We may assume $x_0=0\\\\in \\\\Omega $ and $\\\\lambda \\\\Omega \\\\subset \\\\overline{\\\\Omega }$ by translation.\",\n \"For $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , we set $f_{\\\\lambda }(x)={\\\\left\\\\lbrace \\\\begin{array}{ll}&f\\\\big (x/\\\\lambda \\\\big )\\\\quad x\\\\in \\\\lambda \\\\Omega ,\\\\\\\\&0\\\\qquad \\\\hspace{15.0pt} x\\\\in \\\\Omega \\\\backslash \\\\overline{\\\\lambda \\\\Omega } .\\\\end{array}\\\\right.\",\n \"}$ Then, $f_{\\\\lambda }$ is in $ W^{1,\\\\infty }(\\\\Omega )$ since $f$ is vanishing on $\\\\partial \\\\Omega $ .\",\n \"It follows that $\\\\big \\\\Vert \\\\nabla f_{\\\\lambda }\\\\big \\\\Vert _{\\\\infty }\\\\le \\\\frac{1}{\\\\lambda }\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty },$ and $f_{\\\\lambda }\\\\rightarrow f$ uniformly in $\\\\overline{\\\\Omega }$ as $\\\\lambda \\\\rightarrow 1$ .\",\n \"By a mollification of $f_{\\\\lambda }$ , we obtain a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ satisfying (2.8) with $C_3=2$ .\",\n \"For general $\\\\Omega $ , we take an open covering $\\\\lbrace D_j\\\\rbrace _{j=1}^{N}$ so that $\\\\overline{\\\\Omega }\\\\subset \\\\cup _{j=1}^{N}D_j$ and $\\\\Omega _{j}=\\\\Omega \\\\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\\\in \\\\Omega _{j}$ .\",\n \"We take a partition of unity $\\\\lbrace \\\\xi _{j}\\\\rbrace _{j=1}^{N}\\\\subset C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ such that $\\\\sum _{j=1}^{N}\\\\xi _{j}=1$ , $0\\\\le \\\\xi _{j}\\\\le 1$ , spt $\\\\xi _{j}\\\\subset \\\\overline{D_j}$ and set $f=\\\\sum _{j=1}^{N}f_j,\\\\quad f_j=f\\\\xi _j.$ Since spt $f_j\\\\subset \\\\overline{\\\\Omega }_{j}$ , $\\\\xi _{j}=0$ on $\\\\partial D_{j}$ and $f=0$ on $\\\\partial \\\\Omega $ , $f_{j}$ is in $W^{1,\\\\infty }_{0}(\\\\Omega _{j})$ .\",\n \"Since $\\\\Omega _j$ is star-shaped for some $x_j\\\\in \\\\Omega _{j}$ , there exists $\\\\lbrace f_{j,m}\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega _j)$ satisfying (2.8) in $\\\\Omega _j$ with $C_3=2$ .\",\n \"We extend $f_{j,m}\\\\in C_{c}^{\\\\infty }(\\\\Omega _j)$ to $\\\\Omega \\\\backslash \\\\overline{\\\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\\\sum _{j=1}^{N}f_{j,m}$ .\",\n \"Then, $f_m\\\\in C_{c}^{\\\\infty }(\\\\Omega )$ converges to $f$ uniformly in $\\\\overline{\\\\Omega }$ .\",\n \"We estimate $\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le \\\\sum _{j=1}^{N}\\\\big \\\\Vert \\\\nabla f_{j,m}\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega _{j})}\\\\le 2\\\\sum _{j=1}^{N}\\\\big \\\\Vert \\\\nabla f_{j}\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega _{j})}.$ Since $\\\\nabla f_j=\\\\nabla f\\\\xi _j+f\\\\nabla \\\\xi _j$ and $\\\\big \\\\Vert f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le C_{p}\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le C\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}.$ Thus, $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ satisfies (2.8).\",\n \"The proof is complete.\",\n \"The assertion follows from Propositions 2.6 and 2.7.\",\n \"We recall the a priori estimate of $S(t)\\\\mathbb {P}\\\\partial $ for $f\\\\in C_{c}^{\\\\infty }(\\\\Omega )$ .\",\n \"Proposition 2.8 There exists a constant $C$ such that $\\\\sup _{0\\\\le t\\\\le T}t^{\\\\gamma +\\\\frac{|k|}{2}+s}\\\\Big \\\\Vert \\\\partial _{t}^{s}\\\\partial _{x}^{k}S(t)\\\\mathbb {P}\\\\partial f\\\\Big \\\\Vert _{\\\\infty }\\\\le C\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }^{1-\\\\alpha }\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty }^{\\\\alpha } \\\\qquad \\\\mathrm {(2.9)}$ for $f\\\\in C^{\\\\infty }_{c}(\\\\Omega )$ , $0\\\\le 2s+|k|\\\\le 2$ and $\\\\alpha \\\\in (0,1)$ with $\\\\gamma =(1-\\\\alpha )/2$ .\",\n \"For $f\\\\in W^{1,\\\\infty }_{0}$ , we take a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C^{\\\\infty }_{c}$ satisfying (2.7).\",\n \"For $v_{0,m}=\\\\mathbb {P}\\\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\\\int _{0}^{T}\\\\int _{\\\\Omega }\\\\big ( v_{m}\\\\cdot (\\\\partial _{t}\\\\varphi +\\\\Delta \\\\varphi )-\\\\nabla q_m\\\\cdot \\\\varphi \\\\big )\\\\textrm {d}x\\\\textrm {d}t=\\\\int _{\\\\Omega }f_m\\\\cdot \\\\partial \\\\mathbb {P}\\\\varphi _{0}\\\\textrm {d}x,$ for $\\\\varphi \\\\in C^{\\\\infty }_{c}\\\\big (\\\\Omega \\\\times [0,T)\\\\big )$ .\",\n \"By (2.9) and (2.7), there exists a constant $C$ independent of $m\\\\ge 1$ such that $\\\\sup _{0\\\\le t\\\\le T}\\\\Big \\\\lbrace t^{\\\\gamma }\\\\big \\\\Vert N(v_m,q_m)\\\\big \\\\Vert _{\\\\infty }(t)+t^{\\\\gamma +\\\\frac{1}{2}}\\\\big \\\\Vert d\\\\nabla q_{m}\\\\big \\\\Vert _{\\\\infty }(t)\\\\Big \\\\rbrace \\\\le C\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }^{1-\\\\alpha }\\\\big \\\\Vert f\\\\big \\\\Vert _{1,\\\\infty }^{\\\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ together with $\\\\nabla v_m$ , $\\\\nabla ^{2}v_m$ , $\\\\partial _{t}v_m$ and $\\\\nabla q_{m}$ .\",\n \"By sending $m\\\\rightarrow \\\\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ .\",\n \"By Proposition 2.4, the limit $(v,q)$ is unique.\",\n \"We proved the unique existence of solutions of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ and $f\\\\in W^{1,\\\\infty }_{0}$ satisfying (2.4).\",\n \"The proof is now complete.\",\n \"Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\\\mathbb {P}\\\\partial $ for $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"We take a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\\\mathbb {P}\\\\partial f_m$ satisfies $(v_{0,m}, \\\\varphi )=-(f_m, \\\\partial \\\\mathbb {P}\\\\varphi )\\\\quad \\\\textrm {for}\\\\ \\\\varphi \\\\in C^{\\\\infty }_{c}(\\\\Omega ).$ Since $\\\\partial \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ by Lemma A.1, the sequence $\\\\lbrace v_{0,m}\\\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\\\varphi )=-(f,\\\\partial \\\\mathbb {P}\\\\varphi )$ .\",\n \"Since the limit $v_0$ is unique, the operator $\\\\mathbb {P}\\\\partial $ is uniquely extendable for $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"(ii) We recall that for a sequence $\\\\lbrace v_{0,m}\\\\rbrace _{m=1}^{\\\\infty }\\\\subset L^{\\\\infty }_{\\\\sigma }$ satisfying $||v_{0,m}||_{\\\\infty }&\\\\le K_1,\\\\\\\\v_{0,m}\\\\rightarrow v_{0}&\\\\quad \\\\textrm {a.e.\",\n \"}\\\\ \\\\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ .\",\n \"From the proof of Theorem 2.1, we observe that for a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset W^{1,\\\\infty }_{0}$ satisfying $||f_m||_{1,\\\\infty }&\\\\le K_2,\\\\\\\\f_m\\\\rightarrow f\\\\quad &\\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega },$ $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f_m$ subsequently converges to $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ .\",\n \"(iii) The extension $\\\\overline{S(t)\\\\mathbb {P}\\\\partial }$ satisfies the property $S(t)\\\\overline{S(s)\\\\mathbb {P}\\\\partial } f=\\\\overline{S(t+s)\\\\mathbb {P}\\\\partial } f$ for $t\\\\ge 0$ , $s>0$ and $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"In fact, this property holds for $f_m\\\\in C_{c}^{\\\\infty }$ satisfying (2.7).\",\n \"By choosing a subsequence, $v_m(\\\\cdot ,t)=S(t)\\\\mathbb {P}\\\\partial f_m$ converges to $v(\\\\cdot ,t)=S(t)\\\\mathbb {P}\\\\partial f$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ as in the proof of Theorem 2.1.\",\n \"For fixed $s>0$ , sending $m\\\\rightarrow \\\\infty $ implies $S(t)S(s)\\\\mathbb {P}\\\\partial f_m=S(t)v_m(s)&\\\\rightarrow S(t)v(s) \\\\\\\\&=S(t)\\\\overline{S(s)\\\\mathbb {P}\\\\partial }f\\\\quad \\\\textrm {locally uniformly in \\\\overline{\\\\Omega }\\\\times (0,\\\\infty )}.$ Thus the property is inherited to $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f$ .\"\n ],\n [\n \"Mild solutions on $L^{\\\\infty }_{\\\\sigma }$\",\n \"We prove Theorem 1.1 by approximation.\",\n \"We show that a sequence of mild solutions $\\\\lbrace u_m\\\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ by the $L^{\\\\infty }$ -estimates (1.5) and (1.6).\",\n \"Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).\",\n \"We first recall the existence of mild solutions on $C_{0,\\\\sigma }$ Proposition 3.1 For $u_0\\\\in C_{0,\\\\sigma }$ , there exist $T\\\\ge \\\\varepsilon _0/||u_0||_{\\\\infty }^{2}$ and a unique mild solution $u\\\\in C([0,T]; C_{0,\\\\sigma })$ satisfying (1.3)-(1.6).\",\n \"We approximate $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ by elements of $C_{c,\\\\sigma }^{\\\\infty }\\\\subset C_{0,\\\\sigma }$ .\",\n \"We take a sequence $\\\\lbrace u_{0,m}\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C_{c,\\\\sigma }^{\\\\infty }(\\\\Omega )$ satisfying $\\\\begin{aligned}&\\\\Vert u_{0,m}\\\\Vert _{\\\\infty }\\\\le C\\\\Vert u_{0}\\\\Vert _{\\\\infty }\\\\\\\\&u_{0,m}\\\\rightarrow u_{0}\\\\quad \\\\textrm {a.e.\",\n \"in}\\\\ \\\\Omega ,\\\\end{aligned}\\\\qquad \\\\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\\\ge 1$ .\",\n \"We apply Proposition 3.1 and observe that there exists $T_m\\\\ge \\\\varepsilon _0/||u_{0,m}||_{\\\\infty }^{2}$ and a unique mild solution $u_m\\\\in C([0,T_m]; C_{0,\\\\sigma })$ satisfying $\\\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\\\int _{0}^{t}\\\\overline{S(t-s)\\\\mathbb {P}F_m(s)ds,\\\\\\\\F_m&=u_mu_m.\",\n \"}\\\\qquad \\\\mathrm {(3.2)}\\\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\\\ge \\\\varepsilon /||u_0||_{\\\\infty }^{2} for \\\\varepsilon =\\\\varepsilon _0 C^{-2}/2 so that T_m\\\\ge T and u_m\\\\in C([0,T]; C_{0,\\\\sigma }) for m\\\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ together with $\\\\nabla u_m$ .\",\n \"It follows from (1.5), (1.6) and (3.1) that $\\\\sup _{0\\\\le t\\\\le T}\\\\Big \\\\lbrace \\\\Vert u_{m}\\\\Vert _{\\\\infty }(t)+t^{\\\\frac{1}{2}}\\\\Vert \\\\nabla u_{m}\\\\Vert _{\\\\infty }(t)+t^{\\\\frac{1+\\\\beta }{2}}\\\\big [\\\\nabla u_{m}\\\\big ]_{\\\\Omega }^{(\\\\beta )}(t)\\\\Big \\\\rbrace &\\\\le C_1^{\\\\prime }\\\\Vert u_0\\\\Vert _{\\\\infty }, \\\\\\\\\\\\sup _{x\\\\in \\\\Omega }\\\\Big \\\\lbrace \\\\big [u_{m}\\\\big ]_{[\\\\delta ,T]}^{(\\\\gamma )}(x)+\\\\big [\\\\nabla u_{m}\\\\big ]_{[\\\\delta ,T]}^{(\\\\frac{\\\\gamma }{2})}(x)\\\\Big \\\\rbrace &\\\\le C_2^{\\\\prime }\\\\Vert u_0\\\\Vert _{\\\\infty }, $ for $\\\\beta ,\\\\gamma \\\\in (0,1)$ and $\\\\delta \\\\in (0,T]$ with some constants $C_1^{\\\\prime }$ and $C_2^{\\\\prime }$ , independent of $m\\\\ge 1$ .\",\n \"Since $u_m$ and $\\\\nabla u_m$ are uniformly bounded and equi-continuous in $\\\\overline{\\\\Omega }\\\\times [\\\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.\",\n \"Proposition 3.3 The limit $u\\\\in C_{w}([0,T]; L^{\\\\infty })$ is a mild solution for $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ .\",\n \"We observe that the limit $u$ satisfies (1.4) by sending $m\\\\rightarrow \\\\infty $ .\",\n \"The estimates (3.3) and (3.4) are inherited to $u$ .\",\n \"We prove that $u$ satisfies the integral equation (1.3).\",\n \"By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ by Remarks 2.9 (ii).\",\n \"It follows from (3.3) and Proposition 3.2 that $\\\\begin{aligned}||F_m||_{\\\\infty }&\\\\le K,\\\\\\\\||\\\\nabla F_{m}||_{\\\\infty }&\\\\le \\\\frac{2}{s^{\\\\frac{1}{2}}}K,\\\\\\\\F_m\\\\rightarrow F\\\\quad &\\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\times (0,T]\\\\ \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty ,\\\\end{aligned}\\\\qquad \\\\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\\\prime }||u_0||_{\\\\infty }$ .\",\n \"By choosing a subsequence, we have $\\\\overline{S(\\\\eta )\\\\mathbb {P} F_m\\\\rightarrow \\\\overline{S(\\\\eta )\\\\mathbb {P} F\\\\quad \\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\times (0,T],}for each s\\\\in (0,t) as in Remarks 2.9 (ii).\",\n \"It follows from (3.5) and (2.5) that\\\\begin{equation*}\\\\big \\\\Vert \\\\overline{S(t-s)\\\\mathbb {P} F_m\\\\big \\\\Vert _{\\\\infty }\\\\le \\\\frac{C}{(t-s)^{\\\\frac{1-\\\\alpha }{2}}}\\\\Bigg (1+\\\\frac{2}{s^{\\\\frac{\\\\alpha }{2}}}\\\\Bigg )K^{2}}for 00$ .\",\n \"For $u_{0}^{1}\\\\in C_{0,\\\\sigma }$ , there exists $\\\\lbrace u_{0,m}^{1}\\\\rbrace \\\\subset C_{c,\\\\sigma }^{\\\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\\\infty }\\\\le \\\\epsilon $ for $m\\\\ge N^{1}_{\\\\epsilon }$ .\",\n \"We apply (3.5) and observe that $t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_{0,m}^{1}||_{\\\\infty }&\\\\le t^{\\\\frac{1}{2}}C\\\\big (||S(t)u_{0,m}^{1}||_{\\\\infty }+||S(t)Au_{0,m}^{1}||_{\\\\infty }\\\\big )\\\\\\\\&\\\\le t^{\\\\frac{1}{2}}C^{\\\\prime }\\\\big (||u_{0,m}^{1}||_{\\\\infty }+||Au_{0,m}^{1}||_{\\\\infty }\\\\big )\\\\rightarrow 0\\\\quad \\\\textrm {as}\\\\ t\\\\rightarrow 0.$ We estimate $\\\\overline{\\\\lim _{t\\\\rightarrow 0}}t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_0^{1}||_{\\\\infty }&\\\\le \\\\overline{\\\\lim _{t\\\\rightarrow 0}}\\\\big (t^{\\\\frac{1}{2}}||\\\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\\\infty }+t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_{0,m}^{1}||_{\\\\infty }\\\\big ) \\\\\\\\&\\\\le C^{\\\\prime \\\\prime }\\\\epsilon .$ We set $u_{0,m}^{2}=\\\\eta _{\\\\delta _m}*u_{0}^{2}$ by the mollifier $\\\\eta _{\\\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\\\overline{\\\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\\\infty }\\\\le \\\\epsilon $ for $m\\\\ge N_{\\\\epsilon }^{2}$ .\",\n \"Since $u_{0,m}^{2}$ is supported away from $\\\\partial \\\\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\\\Delta u_{0,m}^{2}$ (see ).\",\n \"By a similar way as for $u_{0}^{1}$ , we estimate $\\\\overline{\\\\lim }_{t\\\\rightarrow 0}t^{1/2}||\\\\nabla S(t)u_0^{2}||_{\\\\infty }\\\\le C^{\\\\prime \\\\prime }\\\\epsilon $ .\",\n \"We proved $\\\\overline{\\\\lim }_{t\\\\rightarrow 0}t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_0||_{\\\\infty }\\\\le 2C^{\\\\prime \\\\prime }\\\\epsilon .$ Since $\\\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\\\nabla S(t)u_0||_{\\\\infty }\\\\rightarrow 0$ as $t\\\\rightarrow 0$ .\",\n \"The assertion follows from Propositions 3.1-3.4.\",\n \"The proof is now complete.\",\n \"Remark 3.5 We set the associated pressure of mild solutions on $L^{\\\\infty }$ by (1.7) and the harmonic-pressure operator $\\\\mathbb {K}: L^{\\\\infty }_{\\\\textrm {tan}}(\\\\partial \\\\Omega )\\\\longrightarrow L^{\\\\infty }_{d}(\\\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\\\Delta q&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial q}{\\\\partial n}&={\\\\partial \\\\Omega }W\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .$ Note that $\\\\Delta u\\\\cdot n={\\\\partial \\\\Omega }W$ by the divergence-free condition of $u$ .\",\n \"Here, $L^{\\\\infty }_{\\\\textrm {tan}}(\\\\partial \\\\Omega )$ denotes the space of all bounded tangential vector fields on $\\\\partial \\\\Omega $ and $L^{\\\\infty }_{d}(\\\\Omega )$ is the space of all functions $f\\\\in L^{1}_{\\\\textrm {loc}}(\\\\Omega )$ such that $df$ is bounded in $\\\\Omega $ for $d(x)=\\\\inf _{y\\\\in \\\\partial \\\\Omega }|x-y|$ , $x\\\\in \\\\Omega $ .\",\n \"Since $W=-(\\\\nabla u-\\\\nabla ^{T}u)n$ is bounded on $\\\\partial \\\\Omega $ for mild solutions on $L^{\\\\infty }$ , $\\\\nabla q=\\\\mathbb {K}W$ is defined as an element of $L^{\\\\infty }_{d}$ .\",\n \"Moreover, $\\\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\\\in W^{1,\\\\infty }_{0}$ as a distribution by Remarks 2.9 (i).\",\n \"Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\\\infty }$ .\",\n \"acknowledgements The author is grateful to the anonymous referees for their valuable comments.\",\n \"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.\",\n \"appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ , $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ , for an exterior domain $\\\\Omega $ .\",\n \"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\\\mathbb {R}^{n}$ by using the heat semigroup.\",\n \"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.\",\n \"Lemma 1.1 Let $\\\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\\\mathbb {R}^{n}$ , $n\\\\ge 2$ .\",\n \"Then, $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ for $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ .\",\n \"We set $\\\\nabla \\\\Phi =\\\\varphi $ for $=I-\\\\mathbb {P}$ .\",\n \"It suffices to show that $\\\\nabla ^{2}\\\\Phi $ is integrable in $\\\\Omega $ .\",\n \"We recall that the $\\\\Phi $ solves the Neumann problem $\\\\begin{aligned}\\\\Delta \\\\Phi =\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\varphi }{\\\\partial n}=0\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.1)}$ See .\",\n \"We observe that $\\\\Phi \\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\\\varphi $ is smooth in $\\\\Omega $ and the boundary is $C^{2}$ .\",\n \"We may assume that $0\\\\in \\\\Omega ^{c}$ by translation.\",\n \"We take $R>0$ such that $\\\\Omega ^{c}\\\\subset B_{0}(R)$ .\",\n \"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\\\ge 3$ and $E(x)=-(2\\\\pi )^{-1}\\\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.\",\n \"We first show that the statement of Lemma A.1 is valid for $\\\\Omega =\\\\mathbb {R}^{n}$ .\",\n \"In the sequel, we do not distinguish $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ and its zero extension to $\\\\mathbb {R}^{n}\\\\backslash \\\\Omega $ .\",\n \"Proposition 1.2 Set $h=E*\\\\varphi $ and $\\\\Phi _1=-\\\\textrm {div}\\\\ h$ .\",\n \"Then, $\\\\nabla ^{3}h$ is integrable in $\\\\mathbb {R}^{n}$ .\",\n \"In particular, $\\\\nabla ^{2}\\\\Phi _1\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"By using the heat semigroup, we transform $h$ into $h=\\\\int _{0}^{\\\\infty }e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t.$ We divide $h$ into two terms and observe that $\\\\partial ^{3}_{x}h=\\\\int _{0}^{1}\\\\partial _{x} e^{t\\\\Delta }\\\\partial ^{2}_{x}\\\\varphi \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\partial ^{3}_{x}e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t,$ where $\\\\partial _{x}=\\\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\\\cdots n$ .\",\n \"We estimate $||\\\\partial ^{3}_{x} h||_{L^{1}(\\\\mathbb {R}^{n})}&\\\\lesssim \\\\int _{0}^{1}\\\\frac{1}{t^{1/2}}||\\\\partial ^{2}_{x} \\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\frac{1}{t^{3/2}}||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t\\\\\\\\&\\\\lesssim ||\\\\partial ^{2}_{x}\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}+||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}.$ We proved $\\\\nabla ^{3}h\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"We reduce (A.1) to the homogeneous Neumann problem $\\\\begin{aligned}-\\\\Delta \\\\Phi _2&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\Phi _2}{\\\\partial n}&=g\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.2)}$ We write connected components of $\\\\Omega $ by unbounded $\\\\Omega _0$ and bounded $\\\\Omega _1$ , $\\\\cdots $ , $\\\\Omega _N$ , i.e., $\\\\Omega =\\\\Omega _{0}\\\\cup (\\\\cup _{j=1}^{N}\\\\Omega _{j})$ .\",\n \"Proposition 1.3 Set $\\\\Phi _2=\\\\Phi -\\\\Phi _1$ .\",\n \"Then, $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ solves (A.2) for $g=\\\\textrm {div}_{\\\\partial \\\\Omega }(An)$ and $A=\\\\nabla h-\\\\nabla ^{T} h$ .\",\n \"The function $g\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega _{j}}g \\\\textrm {d}{\\\\mathcal {H}}=0\\\\quad \\\\textrm {for}\\\\ j=0,1,\\\\cdots ,N. $ We observe that $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ satisfies $-\\\\Delta \\\\Phi _2=0$ in $\\\\Omega $ and $\\\\partial \\\\Phi _2/\\\\partial n=\\\\partial (h̥)/\\\\partial n$ on $\\\\partial \\\\Omega $ .\",\n \"We take an arbitrary $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ .\",\n \"Since $An=(\\\\sum _{1\\\\le j\\\\le n}(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j})_{1\\\\le i\\\\le n}$ is a tangential vector field on $\\\\partial \\\\Omega $ (i.e., $An\\\\cdot n=0$ on $\\\\partial \\\\Omega $ ), applying integration by parts yields $\\\\int _{\\\\partial \\\\Omega } g\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }\\\\textrm {div}_{\\\\partial \\\\Omega }(An)\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(An)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}+\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _j \\\\rho \\\\textrm {d}{\\\\mathcal {H}},$ where the symbol of summation is suppressed.\",\n \"By integration by parts, we have $\\\\int _{\\\\partial \\\\Omega }\\\\partial _jh^{i}n^{j}\\\\partial _i\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho +\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\partial _i \\\\rho ) \\\\textrm {d}x\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho -\\\\nabla h̥\\\\cdot \\\\nabla \\\\rho ) \\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\rho n^{i}\\\\textrm {d}{\\\\mathcal {H}}.$ Since $-\\\\Delta h=\\\\varphi $ is supported in $\\\\Omega $ , it follows that $\\\\int _{\\\\partial \\\\Omega }g\\\\rho {\\\\mathcal {H}}&=-\\\\int _{\\\\Omega }(\\\\Delta h-\\\\nabla h)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}x \\\\\\\\&=-\\\\int _{\\\\Omega }(\\\\Delta h\\\\cdot \\\\nabla \\\\rho +\\\\Delta h\\\\rho )\\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}.$ Since $\\\\partial \\\\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\\\partial \\\\Omega $ .\",\n \"Thus, $g$ is continuous on $\\\\partial \\\\Omega $ .\",\n \"Since $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ is arbitrary, we proved $\\\\partial (h)/\\\\partial n=g$ on $\\\\partial \\\\Omega $ .\",\n \"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.\",\n \"The proof is complete.\",\n \"We estimate $\\\\Phi _2$ by means of the layer potential.\",\n \"Proposition 1.4 (i) For $g\\\\in C(\\\\partial \\\\Omega )$ satisfying (A.3), there exists a moment $h\\\\in C(\\\\partial \\\\Omega )$ satisfying $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ and $-g(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla _x E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ (ii) Set the single layer potential $\\\\tilde{\\\\Phi }_{2}(x)=-\\\\int _{\\\\partial \\\\Omega }E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Then, $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative $\\\\partial _n \\\\tilde{\\\\Phi }_{2}$ exits and is continuous on $\\\\partial \\\\Omega $ .\",\n \"The function $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"The assertion (i) is based on the Fredholm's theorem.\",\n \"See .\",\n \"Since $h$ is bounded on $\\\\partial \\\\Omega $ , $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, we have $-\\\\frac{\\\\partial \\\\tilde{\\\\Phi }_{2}}{\\\\partial n}(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ See .\",\n \"Thus $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) by the assertion (i).\",\n \"When $n\\\\ge 3$ , $\\\\tilde{\\\\Phi }_{2}(x)\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ since the fundamental solution decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"Moreover, when $n=2$ , the average of $h$ on $\\\\partial \\\\Omega $ is zero and we have $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\frac{1}{2\\\\pi }\\\\int _{\\\\partial \\\\Omega }\\\\log {\\\\Bigg (\\\\frac{|x-y|}{|x|}\\\\Bigg )}h(y)\\\\textrm {d}\\\\mathcal {H}(y)\\\\rightarrow 0\\\\quad \\\\textrm {as}\\\\ |x|\\\\rightarrow \\\\infty .$ The proof is complete.\",\n \"Proposition 1.5 The function $\\\\tilde{\\\\Phi }_2$ agrees with ${\\\\Phi }_2$ up to constant.\",\n \"Since $\\\\nabla \\\\Phi _2=\\\\nabla \\\\Phi -\\\\nabla \\\\Phi _1$ is $L^{p}$ -integrable in $\\\\Omega $ for all $p\\\\in (1,\\\\infty )$ (e.g., ), we may assume that $\\\\Phi _2\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ by shifting $\\\\Phi _2$ by a constant.\",\n \"We set $\\\\Psi =\\\\Phi _2-\\\\tilde{\\\\Phi }_2 $ and observe that $\\\\Psi $ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative exists and is continuous on $\\\\partial \\\\Omega $ by Proposition A.4.\",\n \"The function $\\\\Psi $ satisfies $-\\\\Delta \\\\Psi =0$ in $\\\\Omega $ , $\\\\partial \\\\Psi /\\\\partial n=0$ on $\\\\partial \\\\Omega $ and $\\\\Psi \\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ .\",\n \"By the elliptic regularity theory , $\\\\Psi $ is smooth in $\\\\Omega $ and continuously differentiable in $\\\\overline{\\\\Omega }$ .\",\n \"We shall show that $\\\\Psi \\\\equiv 0$ .\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , there exits a point $x_0\\\\in \\\\overline{\\\\Omega }$ such that $\\\\sup _{x\\\\in \\\\Omega }\\\\Psi (x)=\\\\Psi (x_0)$ .\",\n \"Suppose that $x_0\\\\in \\\\partial \\\\Omega $ .\",\n \"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\\\partial \\\\Psi (x_0)/\\\\partial n>0$ .\",\n \"Thus $x_0\\\\in \\\\Omega $ .\",\n \"We apply the strong maximum principle and conclude that $\\\\Psi $ is constant.\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , we have $\\\\Psi \\\\equiv 0$ .\",\n \"The proof is complete.\",\n \"Proposition 1.6 $\\\\nabla ^{2}\\\\Phi _2$ is integrable in $\\\\Omega $ .\",\n \"Since $\\\\nabla ^{2}\\\\Phi _2$ is integrable near the boundary $\\\\partial \\\\Omega $ , it suffices to show that $\\\\nabla ^{2}\\\\Phi _2\\\\in L^{1}(\\\\lbrace |x|\\\\ge 2R\\\\rbrace )$ .\",\n \"Since $h\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ , we observe that $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\int _{0}^{1}\\\\textrm {d}t\\\\int _{\\\\partial \\\\Omega }y\\\\cdot (\\\\nabla E)(x-ty)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Since $\\\\Omega ^{c}\\\\subset B_{0}(R)$ , for $|x|\\\\ge 2R$ we observe that $|x-ty|&\\\\ge \\\\big |\\\\ |x|-|ty|\\\\ \\\\big | \\\\\\\\&\\\\ge |x|-R \\\\\\\\&\\\\ge 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H., title=A Cauchy problem for the Navier-Stokes equations in $R^{n}$ , date=1972, ISSN=0036-1410, journal=SIAM J.\",\n \"Math.\",\n \"Anal., volume=3, pages=506511, KOarticle author=Kozono, H., author=Ogawa, T., title=Two-dimensional Navier-Stokes flow in unbounded domains, date=1993, journal=Math.\",\n \"Ann., volume=297, pages=131, url=http://dx.doi.org/10.1007/BF01459486, Lady61book author=Ladyzhenskaya, O.\",\n \"A., title=The mathematical theory of viscous incompressible flow, publisher=Gordon and Breach, Science Publishers, New York-London-Paris, date=1969, Leray1933article author=Leray, J, title=Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique, date=1933, journal=J.\",\n \"Math.\",\n \"Pures Appl., volume=9, pages=182, Leray1934article author=Leray, J., title=Sur le mouvement d'un liquide visqueux emplissant l'espace, date=1934, ISSN=0001-5962, journal=Acta Math., volume=63, pages=193248, url=http://dx.doi.org/10.1007/BF02547354, LMparticle author=Lions, J.-L., author=Magenes, E., title=Problemi ai limiti non omogenei.\",\n \"V, date=1962, journal=Ann.\",\n \"Scuola Norm Sup.\",\n \"Pisa (3), volume=16, pages=144, Mar09article author=Maremonti, P., title=Stokes and Navier-Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity, date=2008, ISSN=0373-2703, journal=Zap.\",\n \"Nauchn.\",\n \"Sem.\",\n \"S.-Peterburg.\",\n \"Otdel.\",\n \"Mat.\",\n \"Inst.\",\n \"Steklov.\",\n \"(POMI), volume=362, pages=176240, url=http://dx.doi.org/10.1007/s10958-009-9458-3, Mar2014article author=Maremonti, P., title=Non-decaying solutions to the Navier Stokes equations in exterior domains: from the weight function method to the well posedness in $L^\\\\infty $ and in Hölder continuous functional spaces, date=2014, journal=Acta Appl.\",\n \"Math., volume=132, pages=411426, url=http://dx.doi.org/10.1007/s10440-014-9914-z, Miyakawa82article author=Miyakawa, T., title=On nonstationary solutions of the Navier-Stokes equations in an exterior domain, date=1982, ISSN=0018-2079, journal=Hiroshima Math.\",\n \"J., volume=12, pages=115140, url=http://projecteuclid.org/euclid.hmj/1206133879, PWbook author=Protter, M. H., author=Weinberger, H. F., title=Maximum principles in differential equations, publisher=Prentice-Hall, Inc., Englewood Cliffs, N.J.,, date=1967, SiSoincollection author=Simader, C. G., author=Sohr, H., title=A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$ -spaces for bounded and exterior domains, date=1992, booktitle=Mathematical problems relating to the Navier-Stokes equation, series=Ser.\",\n \"Adv.\",\n \"Math.\",\n \"Appl.\",\n \"Sci., volume=11, publisher=World Sci.\",\n \"Publ., River Edge, NJ, pages=135, url=http://dx.doi.org/10.1142/97898145035940001, Sl76article author=Solonnikov, V. A., title=Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, date=1976, journal=Zap.\",\n \"Naučn.\",\n \"Sem.\",\n \"Leningrad.\",\n \"Otdel Mat.\",\n \"Inst.\",\n \"Steklov.\",\n \"(LOMI), volume=59, pages=178254, 257, note=Boundary value problems of mathematical physics and related questions in the theory of functions, 9, Sl03article author=Solonnikov, V. A., title=On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, date=2003, ISSN=1072-3374, journal=J.\",\n \"Math.\",\n \"Sci.\",\n \"(N.\",\n \"Y.\",\n \"), volume=114, pages=17261740, url=http://dx.doi.org/10.1023/A:1022317029111, Sl06article author=Solonnikov, V. A., title=Weighted Schauder estimates for evolution Stokes problem, date=2006, ISSN=0430-3202, journal=Ann.\",\n \"Univ.\",\n \"Ferrara Sez.\",\n \"VII Sci.\",\n \"Mat., volume=52, pages=137172, url=http://dx.doi.org/10.1007/s11565-006-0012-7, Sl07incollection author=Solonnikov, V. A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.\",\n \"Math.\",\n \"Soc.\",\n \"Transl.\",\n \"Ser.\",\n \"2, volume=220, publisher=Amer.\",\n \"Math.\",\n \"Soc., Providence, RI, pages=165200,\"\n ],\n [\n \"An extension of the composition operator\",\n \"In this section, we prove that the composition operator $S(t)\\\\mathbb {P}\\\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\\\infty }_{0}$ to $L^{\\\\infty }_{\\\\sigma }$ .\",\n \"We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\\\mathbb {P}\\\\partial :f\\\\longmapsto v(\\\\cdot ,t)$ .\",\n \"In what follows, $\\\\partial =\\\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\\\cdots ,n$ .\"\n ],\n [\n \"The Stokes system\",\n \"We consider the Stokes equations, $\\\\begin{aligned}\\\\partial _{t}v-\\\\Delta v+\\\\nabla q&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega \\\\times (0,T), \\\\\\\\v&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega \\\\times (0,T), \\\\\\\\v&=0\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega \\\\times (0,T), \\\\\\\\v&=v_0\\\\quad \\\\hspace{-4.0pt} \\\\textrm {on}\\\\ \\\\Omega \\\\times \\\\lbrace t=0\\\\rbrace .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\\\bigl |v(x,t)\\\\bigr |+t^{\\\\frac{1}{2}}\\\\bigl |\\\\nabla v(x,t)\\\\bigr |+t\\\\bigl |\\\\nabla ^{2}v(x,t)\\\\bigr |+t\\\\bigl |\\\\partial _{t}v(x,t)\\\\bigr |+t\\\\bigl |\\\\nabla q(x,t)\\\\bigr |.$ Let $d(x)$ denote the distance from $x\\\\in \\\\Omega $ to $\\\\partial \\\\Omega $ .\",\n \"Let $(v, \\\\nabla q)\\\\in C^{2+\\\\mu ,1+\\\\frac{\\\\mu }{2}}(\\\\overline{\\\\Omega }\\\\times (0,T])\\\\times C^{\\\\mu ,\\\\frac{\\\\mu }{2}}(\\\\overline{\\\\Omega }\\\\times (0,T]),\\\\ \\\\mu \\\\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .\",\n \"We say that $(v,q)$ is a solution of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , if $\\\\sup _{00$ .\",\n \"For $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\\\sup _{00$ , $T>0$ , $R>0$ , there exists a constant $C=C\\\\bigl (\\\\mu ,\\\\delta ,T, R,d\\\\bigr )$ such that $\\\\Big [\\\\nabla ^{2}v\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}+\\\\Big [v_t\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}+\\\\Big [\\\\nabla q\\\\Big ]^{(\\\\mu ,\\\\frac{\\\\mu }{2})}_{Q^{\\\\prime }}\\\\le CN \\\\qquad \\\\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\\\prime }=B_{x_0}(R)\\\\times (2\\\\delta ,T]$ and $x_0\\\\in \\\\Omega $ satisfying $\\\\overline{B_{x_0}(R)}\\\\subset \\\\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\\\partial \\\\Omega $ .\",\n \"(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\\\mu \\\\in (0,1),\\\\ \\\\delta >0$ , $T>0$ and $R\\\\le R_{0}$ , there exists a constant $C$ depending on $\\\\mu $ , $\\\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\\\partial \\\\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\\\prime }=\\\\Omega _{x_0,R}\\\\times (2\\\\delta ,T]$ and $\\\\Omega _{x_0,R}=B_{x_0}(R)\\\\cap \\\\Omega $ , $x_0\\\\in \\\\partial \\\\Omega $ .\",\n \"We observe the uniqueness of solutions for (2.1).\",\n \"The uniqueness of the Stokes equations (2.1) for $v_0\\\\in L^{\\\\infty }_{\\\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .\",\n \"In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , may not be bounded at $t=0$ .\",\n \"The corresponding uniqueness result for a half space is recently proved in .\",\n \"We deduce the result for exterior domains by the same blow-up argument as we did in .\",\n \"Proposition 2.4 Let $\\\\Omega $ be an exterior domain with $C^{3}$ -boundary.\",\n \"Let $(v,\\\\nabla q)\\\\in C^{2,1}(\\\\overline{\\\\Omega }\\\\times (0,T])\\\\times C(\\\\overline{\\\\Omega }\\\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\\\gamma \\\\in [0,1/2)$ .\",\n \"Assume that $\\\\int _{0}^{T}\\\\int _{\\\\Omega }\\\\big (v\\\\cdot (\\\\partial _{t}\\\\varphi +\\\\Delta \\\\varphi )-\\\\nabla q\\\\cdot \\\\varphi \\\\big )\\\\textrm {d}x\\\\textrm {d}t =0,$ for all $\\\\varphi \\\\in C^{\\\\infty }_{c}(\\\\Omega \\\\times [0,T))$ .\",\n \"Then, $v\\\\equiv 0$ and $\\\\nabla q\\\\equiv 0$ .\"\n ],\n [\n \"Approximation\",\n \"We prove Theorem 2.1.\",\n \"We show existence of solutions for the Stokes equations (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ , $f\\\\in W^{1,\\\\infty }_{0}$ , by approximation.\",\n \"We approximate $f\\\\in W^{1,\\\\infty }_{0}$ by elements of $C^{\\\\infty }_{c}$ locally uniformly in $\\\\overline{\\\\Omega }$ .\",\n \"Lemma 2.5 Let $\\\\Omega $ be an exterior domain with Lipschitz boundary.\",\n \"There exist constants $C_1$ , $C_2$ such that for $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ there exists a sequence of functions $\\\\lbrace f_m\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ such that $\\\\begin{aligned}&\\\\big \\\\Vert f_{m}\\\\big \\\\Vert _{\\\\infty }\\\\le C_1\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }, \\\\\\\\&\\\\big \\\\Vert \\\\nabla f_{m}\\\\big \\\\Vert _{\\\\infty }\\\\le C_2\\\\big \\\\Vert f\\\\big \\\\Vert _{1,\\\\infty }, \\\\\\\\&f_m\\\\rightarrow f\\\\quad \\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\quad \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.\",\n \"Proposition 2.6 The statement of Lemma 2.5 holds when $\\\\Omega =\\\\mathbb {R}^{n}$ with $C_1=1$ .\",\n \"We cutoff the function $f\\\\in W^{1,\\\\infty }(\\\\mathbb {R}^{n})$ .\",\n \"Let $\\\\theta \\\\in C^{\\\\infty }_{c}[0,\\\\infty )$ be a cut-off function satisfying $\\\\theta \\\\equiv 1$ in $[0,1]$ , $\\\\theta \\\\equiv 0$ in $[2,\\\\infty )$ and $0\\\\le \\\\theta \\\\le 1$ .\",\n \"We set $\\\\theta _{m}(x)=\\\\theta (|x|/m)$ for $m\\\\ge 1$ so that $\\\\theta _{m}\\\\equiv 1$ for $|x|\\\\le m$ and $\\\\theta _{m}\\\\equiv 0$ for $|x|\\\\ge 2m$ .\",\n \"Then, $f_m=f\\\\theta _{m}$ satisfies (2.7).\",\n \"Proposition 2.7 Let $\\\\Omega $ be a bounded domain with Lipschitz boundary.\",\n \"There exists a constant $C_3$ such that for $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ there exists a sequence of functions $\\\\lbrace f_m\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C^{\\\\infty }_{c}(\\\\Omega )$ such that $\\\\begin{aligned}&\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{\\\\infty }\\\\le C_3\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty }\\\\\\\\f_m&\\\\rightarrow f\\\\quad \\\\textrm {uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\quad \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty .\\\\end{aligned}\\\\qquad \\\\mathrm {(2.8)}$ We begin with the case when $\\\\Omega $ is star-shaped, i.e., $\\\\lambda \\\\Omega _{x_0}\\\\subset \\\\overline{\\\\Omega }$ for some $x_0\\\\in \\\\Omega $ and all $\\\\lambda <1$ , where $\\\\lambda \\\\Omega _{x_0}=\\\\lbrace x_0+\\\\lambda (x-x_0)\\\\ |\\\\ x\\\\in \\\\Omega \\\\rbrace $ .\",\n \"We may assume $x_0=0\\\\in \\\\Omega $ and $\\\\lambda \\\\Omega \\\\subset \\\\overline{\\\\Omega }$ by translation.\",\n \"For $f\\\\in W^{1,\\\\infty }_{0}(\\\\Omega )$ , we set $f_{\\\\lambda }(x)={\\\\left\\\\lbrace \\\\begin{array}{ll}&f\\\\big (x/\\\\lambda \\\\big )\\\\quad x\\\\in \\\\lambda \\\\Omega ,\\\\\\\\&0\\\\qquad \\\\hspace{15.0pt} x\\\\in \\\\Omega \\\\backslash \\\\overline{\\\\lambda \\\\Omega } .\\\\end{array}\\\\right.\",\n \"}$ Then, $f_{\\\\lambda }$ is in $ W^{1,\\\\infty }(\\\\Omega )$ since $f$ is vanishing on $\\\\partial \\\\Omega $ .\",\n \"It follows that $\\\\big \\\\Vert \\\\nabla f_{\\\\lambda }\\\\big \\\\Vert _{\\\\infty }\\\\le \\\\frac{1}{\\\\lambda }\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty },$ and $f_{\\\\lambda }\\\\rightarrow f$ uniformly in $\\\\overline{\\\\Omega }$ as $\\\\lambda \\\\rightarrow 1$ .\",\n \"By a mollification of $f_{\\\\lambda }$ , we obtain a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ satisfying (2.8) with $C_3=2$ .\",\n \"For general $\\\\Omega $ , we take an open covering $\\\\lbrace D_j\\\\rbrace _{j=1}^{N}$ so that $\\\\overline{\\\\Omega }\\\\subset \\\\cup _{j=1}^{N}D_j$ and $\\\\Omega _{j}=\\\\Omega \\\\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\\\in \\\\Omega _{j}$ .\",\n \"We take a partition of unity $\\\\lbrace \\\\xi _{j}\\\\rbrace _{j=1}^{N}\\\\subset C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ such that $\\\\sum _{j=1}^{N}\\\\xi _{j}=1$ , $0\\\\le \\\\xi _{j}\\\\le 1$ , spt $\\\\xi _{j}\\\\subset \\\\overline{D_j}$ and set $f=\\\\sum _{j=1}^{N}f_j,\\\\quad f_j=f\\\\xi _j.$ Since spt $f_j\\\\subset \\\\overline{\\\\Omega }_{j}$ , $\\\\xi _{j}=0$ on $\\\\partial D_{j}$ and $f=0$ on $\\\\partial \\\\Omega $ , $f_{j}$ is in $W^{1,\\\\infty }_{0}(\\\\Omega _{j})$ .\",\n \"Since $\\\\Omega _j$ is star-shaped for some $x_j\\\\in \\\\Omega _{j}$ , there exists $\\\\lbrace f_{j,m}\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega _j)$ satisfying (2.8) in $\\\\Omega _j$ with $C_3=2$ .\",\n \"We extend $f_{j,m}\\\\in C_{c}^{\\\\infty }(\\\\Omega _j)$ to $\\\\Omega \\\\backslash \\\\overline{\\\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\\\sum _{j=1}^{N}f_{j,m}$ .\",\n \"Then, $f_m\\\\in C_{c}^{\\\\infty }(\\\\Omega )$ converges to $f$ uniformly in $\\\\overline{\\\\Omega }$ .\",\n \"We estimate $\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le \\\\sum _{j=1}^{N}\\\\big \\\\Vert \\\\nabla f_{j,m}\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega _{j})}\\\\le 2\\\\sum _{j=1}^{N}\\\\big \\\\Vert \\\\nabla f_{j}\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega _{j})}.$ Since $\\\\nabla f_j=\\\\nabla f\\\\xi _j+f\\\\nabla \\\\xi _j$ and $\\\\big \\\\Vert f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le C_{p}\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\\\big \\\\Vert \\\\nabla f_m\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}\\\\le C\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{L^{\\\\infty }(\\\\Omega )}.$ Thus, $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }(\\\\Omega )$ satisfies (2.8).\",\n \"The proof is complete.\",\n \"The assertion follows from Propositions 2.6 and 2.7.\",\n \"We recall the a priori estimate of $S(t)\\\\mathbb {P}\\\\partial $ for $f\\\\in C_{c}^{\\\\infty }(\\\\Omega )$ .\",\n \"Proposition 2.8 There exists a constant $C$ such that $\\\\sup _{0\\\\le t\\\\le T}t^{\\\\gamma +\\\\frac{|k|}{2}+s}\\\\Big \\\\Vert \\\\partial _{t}^{s}\\\\partial _{x}^{k}S(t)\\\\mathbb {P}\\\\partial f\\\\Big \\\\Vert _{\\\\infty }\\\\le C\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }^{1-\\\\alpha }\\\\big \\\\Vert \\\\nabla f\\\\big \\\\Vert _{\\\\infty }^{\\\\alpha } \\\\qquad \\\\mathrm {(2.9)}$ for $f\\\\in C^{\\\\infty }_{c}(\\\\Omega )$ , $0\\\\le 2s+|k|\\\\le 2$ and $\\\\alpha \\\\in (0,1)$ with $\\\\gamma =(1-\\\\alpha )/2$ .\",\n \"For $f\\\\in W^{1,\\\\infty }_{0}$ , we take a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C^{\\\\infty }_{c}$ satisfying (2.7).\",\n \"For $v_{0,m}=\\\\mathbb {P}\\\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\\\int _{0}^{T}\\\\int _{\\\\Omega }\\\\big ( v_{m}\\\\cdot (\\\\partial _{t}\\\\varphi +\\\\Delta \\\\varphi )-\\\\nabla q_m\\\\cdot \\\\varphi \\\\big )\\\\textrm {d}x\\\\textrm {d}t=\\\\int _{\\\\Omega }f_m\\\\cdot \\\\partial \\\\mathbb {P}\\\\varphi _{0}\\\\textrm {d}x,$ for $\\\\varphi \\\\in C^{\\\\infty }_{c}\\\\big (\\\\Omega \\\\times [0,T)\\\\big )$ .\",\n \"By (2.9) and (2.7), there exists a constant $C$ independent of $m\\\\ge 1$ such that $\\\\sup _{0\\\\le t\\\\le T}\\\\Big \\\\lbrace t^{\\\\gamma }\\\\big \\\\Vert N(v_m,q_m)\\\\big \\\\Vert _{\\\\infty }(t)+t^{\\\\gamma +\\\\frac{1}{2}}\\\\big \\\\Vert d\\\\nabla q_{m}\\\\big \\\\Vert _{\\\\infty }(t)\\\\Big \\\\rbrace \\\\le C\\\\big \\\\Vert f\\\\big \\\\Vert _{\\\\infty }^{1-\\\\alpha }\\\\big \\\\Vert f\\\\big \\\\Vert _{1,\\\\infty }^{\\\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ together with $\\\\nabla v_m$ , $\\\\nabla ^{2}v_m$ , $\\\\partial _{t}v_m$ and $\\\\nabla q_{m}$ .\",\n \"By sending $m\\\\rightarrow \\\\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ .\",\n \"By Proposition 2.4, the limit $(v,q)$ is unique.\",\n \"We proved the unique existence of solutions of (2.1) for $v_0=\\\\mathbb {P}\\\\partial f$ and $f\\\\in W^{1,\\\\infty }_{0}$ satisfying (2.4).\",\n \"The proof is now complete.\",\n \"Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\\\mathbb {P}\\\\partial $ for $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"We take a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset C_{c}^{\\\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\\\mathbb {P}\\\\partial f_m$ satisfies $(v_{0,m}, \\\\varphi )=-(f_m, \\\\partial \\\\mathbb {P}\\\\varphi )\\\\quad \\\\textrm {for}\\\\ \\\\varphi \\\\in C^{\\\\infty }_{c}(\\\\Omega ).$ Since $\\\\partial \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ by Lemma A.1, the sequence $\\\\lbrace v_{0,m}\\\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\\\varphi )=-(f,\\\\partial \\\\mathbb {P}\\\\varphi )$ .\",\n \"Since the limit $v_0$ is unique, the operator $\\\\mathbb {P}\\\\partial $ is uniquely extendable for $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"(ii) We recall that for a sequence $\\\\lbrace v_{0,m}\\\\rbrace _{m=1}^{\\\\infty }\\\\subset L^{\\\\infty }_{\\\\sigma }$ satisfying $||v_{0,m}||_{\\\\infty }&\\\\le K_1,\\\\\\\\v_{0,m}\\\\rightarrow v_{0}&\\\\quad \\\\textrm {a.e.\",\n \"}\\\\ \\\\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ .\",\n \"From the proof of Theorem 2.1, we observe that for a sequence $\\\\lbrace f_m\\\\rbrace \\\\subset W^{1,\\\\infty }_{0}$ satisfying $||f_m||_{1,\\\\infty }&\\\\le K_2,\\\\\\\\f_m\\\\rightarrow f\\\\quad &\\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega },$ $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f_m$ subsequently converges to $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ .\",\n \"(iii) The extension $\\\\overline{S(t)\\\\mathbb {P}\\\\partial }$ satisfies the property $S(t)\\\\overline{S(s)\\\\mathbb {P}\\\\partial } f=\\\\overline{S(t+s)\\\\mathbb {P}\\\\partial } f$ for $t\\\\ge 0$ , $s>0$ and $f\\\\in W^{1,\\\\infty }_{0}$ .\",\n \"In fact, this property holds for $f_m\\\\in C_{c}^{\\\\infty }$ satisfying (2.7).\",\n \"By choosing a subsequence, $v_m(\\\\cdot ,t)=S(t)\\\\mathbb {P}\\\\partial f_m$ converges to $v(\\\\cdot ,t)=S(t)\\\\mathbb {P}\\\\partial f$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,\\\\infty )$ as in the proof of Theorem 2.1.\",\n \"For fixed $s>0$ , sending $m\\\\rightarrow \\\\infty $ implies $S(t)S(s)\\\\mathbb {P}\\\\partial f_m=S(t)v_m(s)&\\\\rightarrow S(t)v(s) \\\\\\\\&=S(t)\\\\overline{S(s)\\\\mathbb {P}\\\\partial }f\\\\quad \\\\textrm {locally uniformly in \\\\overline{\\\\Omega }\\\\times (0,\\\\infty )}.$ Thus the property is inherited to $\\\\overline{S(t)\\\\mathbb {P}\\\\partial } f$ .\"\n ],\n [\n \"Mild solutions on $L^{\\\\infty }_{\\\\sigma }$\",\n \"We prove Theorem 1.1 by approximation.\",\n \"We show that a sequence of mild solutions $\\\\lbrace u_m\\\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ by the $L^{\\\\infty }$ -estimates (1.5) and (1.6).\",\n \"Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).\",\n \"We first recall the existence of mild solutions on $C_{0,\\\\sigma }$ Proposition 3.1 For $u_0\\\\in C_{0,\\\\sigma }$ , there exist $T\\\\ge \\\\varepsilon _0/||u_0||_{\\\\infty }^{2}$ and a unique mild solution $u\\\\in C([0,T]; C_{0,\\\\sigma })$ satisfying (1.3)-(1.6).\",\n \"We approximate $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ by elements of $C_{c,\\\\sigma }^{\\\\infty }\\\\subset C_{0,\\\\sigma }$ .\",\n \"We take a sequence $\\\\lbrace u_{0,m}\\\\rbrace _{m=1}^{\\\\infty }\\\\subset C_{c,\\\\sigma }^{\\\\infty }(\\\\Omega )$ satisfying $\\\\begin{aligned}&\\\\Vert u_{0,m}\\\\Vert _{\\\\infty }\\\\le C\\\\Vert u_{0}\\\\Vert _{\\\\infty }\\\\\\\\&u_{0,m}\\\\rightarrow u_{0}\\\\quad \\\\textrm {a.e.\",\n \"in}\\\\ \\\\Omega ,\\\\end{aligned}\\\\qquad \\\\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\\\ge 1$ .\",\n \"We apply Proposition 3.1 and observe that there exists $T_m\\\\ge \\\\varepsilon _0/||u_{0,m}||_{\\\\infty }^{2}$ and a unique mild solution $u_m\\\\in C([0,T_m]; C_{0,\\\\sigma })$ satisfying $\\\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\\\int _{0}^{t}\\\\overline{S(t-s)\\\\mathbb {P}F_m(s)ds,\\\\\\\\F_m&=u_mu_m.\",\n \"}\\\\qquad \\\\mathrm {(3.2)}\\\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\\\ge \\\\varepsilon /||u_0||_{\\\\infty }^{2} for \\\\varepsilon =\\\\varepsilon _0 C^{-2}/2 so that T_m\\\\ge T and u_m\\\\in C([0,T]; C_{0,\\\\sigma }) for m\\\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ together with $\\\\nabla u_m$ .\",\n \"It follows from (1.5), (1.6) and (3.1) that $\\\\sup _{0\\\\le t\\\\le T}\\\\Big \\\\lbrace \\\\Vert u_{m}\\\\Vert _{\\\\infty }(t)+t^{\\\\frac{1}{2}}\\\\Vert \\\\nabla u_{m}\\\\Vert _{\\\\infty }(t)+t^{\\\\frac{1+\\\\beta }{2}}\\\\big [\\\\nabla u_{m}\\\\big ]_{\\\\Omega }^{(\\\\beta )}(t)\\\\Big \\\\rbrace &\\\\le C_1^{\\\\prime }\\\\Vert u_0\\\\Vert _{\\\\infty }, \\\\\\\\\\\\sup _{x\\\\in \\\\Omega }\\\\Big \\\\lbrace \\\\big [u_{m}\\\\big ]_{[\\\\delta ,T]}^{(\\\\gamma )}(x)+\\\\big [\\\\nabla u_{m}\\\\big ]_{[\\\\delta ,T]}^{(\\\\frac{\\\\gamma }{2})}(x)\\\\Big \\\\rbrace &\\\\le C_2^{\\\\prime }\\\\Vert u_0\\\\Vert _{\\\\infty }, $ for $\\\\beta ,\\\\gamma \\\\in (0,1)$ and $\\\\delta \\\\in (0,T]$ with some constants $C_1^{\\\\prime }$ and $C_2^{\\\\prime }$ , independent of $m\\\\ge 1$ .\",\n \"Since $u_m$ and $\\\\nabla u_m$ are uniformly bounded and equi-continuous in $\\\\overline{\\\\Omega }\\\\times [\\\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.\",\n \"Proposition 3.3 The limit $u\\\\in C_{w}([0,T]; L^{\\\\infty })$ is a mild solution for $u_0\\\\in L^{\\\\infty }_{\\\\sigma }$ .\",\n \"We observe that the limit $u$ satisfies (1.4) by sending $m\\\\rightarrow \\\\infty $ .\",\n \"The estimates (3.3) and (3.4) are inherited to $u$ .\",\n \"We prove that $u$ satisfies the integral equation (1.3).\",\n \"By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\\\overline{\\\\Omega }\\\\times (0,T]$ by Remarks 2.9 (ii).\",\n \"It follows from (3.3) and Proposition 3.2 that $\\\\begin{aligned}||F_m||_{\\\\infty }&\\\\le K,\\\\\\\\||\\\\nabla F_{m}||_{\\\\infty }&\\\\le \\\\frac{2}{s^{\\\\frac{1}{2}}}K,\\\\\\\\F_m\\\\rightarrow F\\\\quad &\\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\times (0,T]\\\\ \\\\textrm {as}\\\\ m\\\\rightarrow \\\\infty ,\\\\end{aligned}\\\\qquad \\\\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\\\prime }||u_0||_{\\\\infty }$ .\",\n \"By choosing a subsequence, we have $\\\\overline{S(\\\\eta )\\\\mathbb {P} F_m\\\\rightarrow \\\\overline{S(\\\\eta )\\\\mathbb {P} F\\\\quad \\\\textrm {locally uniformly in}\\\\ \\\\overline{\\\\Omega }\\\\times (0,T],}for each s\\\\in (0,t) as in Remarks 2.9 (ii).\",\n \"It follows from (3.5) and (2.5) that\\\\begin{equation*}\\\\big \\\\Vert \\\\overline{S(t-s)\\\\mathbb {P} F_m\\\\big \\\\Vert _{\\\\infty }\\\\le \\\\frac{C}{(t-s)^{\\\\frac{1-\\\\alpha }{2}}}\\\\Bigg (1+\\\\frac{2}{s^{\\\\frac{\\\\alpha }{2}}}\\\\Bigg )K^{2}}for 00$ .\",\n \"For $u_{0}^{1}\\\\in C_{0,\\\\sigma }$ , there exists $\\\\lbrace u_{0,m}^{1}\\\\rbrace \\\\subset C_{c,\\\\sigma }^{\\\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\\\infty }\\\\le \\\\epsilon $ for $m\\\\ge N^{1}_{\\\\epsilon }$ .\",\n \"We apply (3.5) and observe that $t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_{0,m}^{1}||_{\\\\infty }&\\\\le t^{\\\\frac{1}{2}}C\\\\big (||S(t)u_{0,m}^{1}||_{\\\\infty }+||S(t)Au_{0,m}^{1}||_{\\\\infty }\\\\big )\\\\\\\\&\\\\le t^{\\\\frac{1}{2}}C^{\\\\prime }\\\\big (||u_{0,m}^{1}||_{\\\\infty }+||Au_{0,m}^{1}||_{\\\\infty }\\\\big )\\\\rightarrow 0\\\\quad \\\\textrm {as}\\\\ t\\\\rightarrow 0.$ We estimate $\\\\overline{\\\\lim _{t\\\\rightarrow 0}}t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_0^{1}||_{\\\\infty }&\\\\le \\\\overline{\\\\lim _{t\\\\rightarrow 0}}\\\\big (t^{\\\\frac{1}{2}}||\\\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\\\infty }+t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_{0,m}^{1}||_{\\\\infty }\\\\big ) \\\\\\\\&\\\\le C^{\\\\prime \\\\prime }\\\\epsilon .$ We set $u_{0,m}^{2}=\\\\eta _{\\\\delta _m}*u_{0}^{2}$ by the mollifier $\\\\eta _{\\\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\\\overline{\\\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\\\infty }\\\\le \\\\epsilon $ for $m\\\\ge N_{\\\\epsilon }^{2}$ .\",\n \"Since $u_{0,m}^{2}$ is supported away from $\\\\partial \\\\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\\\Delta u_{0,m}^{2}$ (see ).\",\n \"By a similar way as for $u_{0}^{1}$ , we estimate $\\\\overline{\\\\lim }_{t\\\\rightarrow 0}t^{1/2}||\\\\nabla S(t)u_0^{2}||_{\\\\infty }\\\\le C^{\\\\prime \\\\prime }\\\\epsilon $ .\",\n \"We proved $\\\\overline{\\\\lim }_{t\\\\rightarrow 0}t^{\\\\frac{1}{2}}||\\\\nabla S(t)u_0||_{\\\\infty }\\\\le 2C^{\\\\prime \\\\prime }\\\\epsilon .$ Since $\\\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\\\nabla S(t)u_0||_{\\\\infty }\\\\rightarrow 0$ as $t\\\\rightarrow 0$ .\",\n \"The assertion follows from Propositions 3.1-3.4.\",\n \"The proof is now complete.\",\n \"Remark 3.5 We set the associated pressure of mild solutions on $L^{\\\\infty }$ by (1.7) and the harmonic-pressure operator $\\\\mathbb {K}: L^{\\\\infty }_{\\\\textrm {tan}}(\\\\partial \\\\Omega )\\\\longrightarrow L^{\\\\infty }_{d}(\\\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\\\Delta q&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial q}{\\\\partial n}&={\\\\partial \\\\Omega }W\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .$ Note that $\\\\Delta u\\\\cdot n={\\\\partial \\\\Omega }W$ by the divergence-free condition of $u$ .\",\n \"Here, $L^{\\\\infty }_{\\\\textrm {tan}}(\\\\partial \\\\Omega )$ denotes the space of all bounded tangential vector fields on $\\\\partial \\\\Omega $ and $L^{\\\\infty }_{d}(\\\\Omega )$ is the space of all functions $f\\\\in L^{1}_{\\\\textrm {loc}}(\\\\Omega )$ such that $df$ is bounded in $\\\\Omega $ for $d(x)=\\\\inf _{y\\\\in \\\\partial \\\\Omega }|x-y|$ , $x\\\\in \\\\Omega $ .\",\n \"Since $W=-(\\\\nabla u-\\\\nabla ^{T}u)n$ is bounded on $\\\\partial \\\\Omega $ for mild solutions on $L^{\\\\infty }$ , $\\\\nabla q=\\\\mathbb {K}W$ is defined as an element of $L^{\\\\infty }_{d}$ .\",\n \"Moreover, $\\\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\\\in W^{1,\\\\infty }_{0}$ as a distribution by Remarks 2.9 (i).\",\n \"Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\\\infty }$ .\",\n \"acknowledgements The author is grateful to the anonymous referees for their valuable comments.\",\n \"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.\",\n \"appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ , $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ , for an exterior domain $\\\\Omega $ .\",\n \"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\\\mathbb {R}^{n}$ by using the heat semigroup.\",\n \"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.\",\n \"Lemma 1.1 Let $\\\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\\\mathbb {R}^{n}$ , $n\\\\ge 2$ .\",\n \"Then, $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ for $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ .\",\n \"We set $\\\\nabla \\\\Phi =\\\\varphi $ for $=I-\\\\mathbb {P}$ .\",\n \"It suffices to show that $\\\\nabla ^{2}\\\\Phi $ is integrable in $\\\\Omega $ .\",\n \"We recall that the $\\\\Phi $ solves the Neumann problem $\\\\begin{aligned}\\\\Delta \\\\Phi =\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\varphi }{\\\\partial n}=0\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.1)}$ See .\",\n \"We observe that $\\\\Phi \\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\\\varphi $ is smooth in $\\\\Omega $ and the boundary is $C^{2}$ .\",\n \"We may assume that $0\\\\in \\\\Omega ^{c}$ by translation.\",\n \"We take $R>0$ such that $\\\\Omega ^{c}\\\\subset B_{0}(R)$ .\",\n \"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\\\ge 3$ and $E(x)=-(2\\\\pi )^{-1}\\\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.\",\n \"We first show that the statement of Lemma A.1 is valid for $\\\\Omega =\\\\mathbb {R}^{n}$ .\",\n \"In the sequel, we do not distinguish $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ and its zero extension to $\\\\mathbb {R}^{n}\\\\backslash \\\\Omega $ .\",\n \"Proposition 1.2 Set $h=E*\\\\varphi $ and $\\\\Phi _1=-\\\\textrm {div}\\\\ h$ .\",\n \"Then, $\\\\nabla ^{3}h$ is integrable in $\\\\mathbb {R}^{n}$ .\",\n \"In particular, $\\\\nabla ^{2}\\\\Phi _1\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"By using the heat semigroup, we transform $h$ into $h=\\\\int _{0}^{\\\\infty }e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t.$ We divide $h$ into two terms and observe that $\\\\partial ^{3}_{x}h=\\\\int _{0}^{1}\\\\partial _{x} e^{t\\\\Delta }\\\\partial ^{2}_{x}\\\\varphi \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\partial ^{3}_{x}e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t,$ where $\\\\partial _{x}=\\\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\\\cdots n$ .\",\n \"We estimate $||\\\\partial ^{3}_{x} h||_{L^{1}(\\\\mathbb {R}^{n})}&\\\\lesssim \\\\int _{0}^{1}\\\\frac{1}{t^{1/2}}||\\\\partial ^{2}_{x} \\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\frac{1}{t^{3/2}}||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t\\\\\\\\&\\\\lesssim ||\\\\partial ^{2}_{x}\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}+||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}.$ We proved $\\\\nabla ^{3}h\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"We reduce (A.1) to the homogeneous Neumann problem $\\\\begin{aligned}-\\\\Delta \\\\Phi _2&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\Phi _2}{\\\\partial n}&=g\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.2)}$ We write connected components of $\\\\Omega $ by unbounded $\\\\Omega _0$ and bounded $\\\\Omega _1$ , $\\\\cdots $ , $\\\\Omega _N$ , i.e., $\\\\Omega =\\\\Omega _{0}\\\\cup (\\\\cup _{j=1}^{N}\\\\Omega _{j})$ .\",\n \"Proposition 1.3 Set $\\\\Phi _2=\\\\Phi -\\\\Phi _1$ .\",\n \"Then, $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ solves (A.2) for $g=\\\\textrm {div}_{\\\\partial \\\\Omega }(An)$ and $A=\\\\nabla h-\\\\nabla ^{T} h$ .\",\n \"The function $g\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega _{j}}g \\\\textrm {d}{\\\\mathcal {H}}=0\\\\quad \\\\textrm {for}\\\\ j=0,1,\\\\cdots ,N. $ We observe that $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ satisfies $-\\\\Delta \\\\Phi _2=0$ in $\\\\Omega $ and $\\\\partial \\\\Phi _2/\\\\partial n=\\\\partial (h̥)/\\\\partial n$ on $\\\\partial \\\\Omega $ .\",\n \"We take an arbitrary $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ .\",\n \"Since $An=(\\\\sum _{1\\\\le j\\\\le n}(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j})_{1\\\\le i\\\\le n}$ is a tangential vector field on $\\\\partial \\\\Omega $ (i.e., $An\\\\cdot n=0$ on $\\\\partial \\\\Omega $ ), applying integration by parts yields $\\\\int _{\\\\partial \\\\Omega } g\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }\\\\textrm {div}_{\\\\partial \\\\Omega }(An)\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(An)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}+\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _j \\\\rho \\\\textrm {d}{\\\\mathcal {H}},$ where the symbol of summation is suppressed.\",\n \"By integration by parts, we have $\\\\int _{\\\\partial \\\\Omega }\\\\partial _jh^{i}n^{j}\\\\partial _i\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho +\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\partial _i \\\\rho ) \\\\textrm {d}x\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho -\\\\nabla h̥\\\\cdot \\\\nabla \\\\rho ) \\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\rho n^{i}\\\\textrm {d}{\\\\mathcal {H}}.$ Since $-\\\\Delta h=\\\\varphi $ is supported in $\\\\Omega $ , it follows that $\\\\int _{\\\\partial \\\\Omega }g\\\\rho {\\\\mathcal {H}}&=-\\\\int _{\\\\Omega }(\\\\Delta h-\\\\nabla h)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}x \\\\\\\\&=-\\\\int _{\\\\Omega }(\\\\Delta h\\\\cdot \\\\nabla \\\\rho +\\\\Delta h\\\\rho )\\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}.$ Since $\\\\partial \\\\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\\\partial \\\\Omega $ .\",\n \"Thus, $g$ is continuous on $\\\\partial \\\\Omega $ .\",\n \"Since $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ is arbitrary, we proved $\\\\partial (h)/\\\\partial n=g$ on $\\\\partial \\\\Omega $ .\",\n \"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.\",\n \"The proof is complete.\",\n \"We estimate $\\\\Phi _2$ by means of the layer potential.\",\n \"Proposition 1.4 (i) For $g\\\\in C(\\\\partial \\\\Omega )$ satisfying (A.3), there exists a moment $h\\\\in C(\\\\partial \\\\Omega )$ satisfying $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ and $-g(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla _x E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ (ii) Set the single layer potential $\\\\tilde{\\\\Phi }_{2}(x)=-\\\\int _{\\\\partial \\\\Omega }E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Then, $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative $\\\\partial _n \\\\tilde{\\\\Phi }_{2}$ exits and is continuous on $\\\\partial \\\\Omega $ .\",\n \"The function $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"The assertion (i) is based on the Fredholm's theorem.\",\n \"See .\",\n \"Since $h$ is bounded on $\\\\partial \\\\Omega $ , $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, we have $-\\\\frac{\\\\partial \\\\tilde{\\\\Phi }_{2}}{\\\\partial n}(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ See .\",\n \"Thus $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) by the assertion (i).\",\n \"When $n\\\\ge 3$ , $\\\\tilde{\\\\Phi }_{2}(x)\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ since the fundamental solution decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"Moreover, when $n=2$ , the average of $h$ on $\\\\partial \\\\Omega $ is zero and we have $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\frac{1}{2\\\\pi }\\\\int _{\\\\partial \\\\Omega }\\\\log {\\\\Bigg (\\\\frac{|x-y|}{|x|}\\\\Bigg )}h(y)\\\\textrm {d}\\\\mathcal {H}(y)\\\\rightarrow 0\\\\quad \\\\textrm {as}\\\\ |x|\\\\rightarrow \\\\infty .$ The proof is complete.\",\n \"Proposition 1.5 The function $\\\\tilde{\\\\Phi }_2$ agrees with ${\\\\Phi }_2$ up to constant.\",\n \"Since $\\\\nabla \\\\Phi _2=\\\\nabla \\\\Phi -\\\\nabla \\\\Phi _1$ is $L^{p}$ -integrable in $\\\\Omega $ for all $p\\\\in (1,\\\\infty )$ (e.g., ), we may assume that $\\\\Phi _2\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ by shifting $\\\\Phi _2$ by a constant.\",\n \"We set $\\\\Psi =\\\\Phi _2-\\\\tilde{\\\\Phi }_2 $ and observe that $\\\\Psi $ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative exists and is continuous on $\\\\partial \\\\Omega $ by Proposition A.4.\",\n \"The function $\\\\Psi $ satisfies $-\\\\Delta \\\\Psi =0$ in $\\\\Omega $ , $\\\\partial \\\\Psi /\\\\partial n=0$ on $\\\\partial \\\\Omega $ and $\\\\Psi \\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ .\",\n \"By the elliptic regularity theory , $\\\\Psi $ is smooth in $\\\\Omega $ and continuously differentiable in $\\\\overline{\\\\Omega }$ .\",\n \"We shall show that $\\\\Psi \\\\equiv 0$ .\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , there exits a point $x_0\\\\in \\\\overline{\\\\Omega }$ such that $\\\\sup _{x\\\\in \\\\Omega }\\\\Psi (x)=\\\\Psi (x_0)$ .\",\n \"Suppose that $x_0\\\\in \\\\partial \\\\Omega $ .\",\n \"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\\\partial \\\\Psi (x_0)/\\\\partial n>0$ .\",\n \"Thus $x_0\\\\in \\\\Omega $ .\",\n \"We apply the strong maximum principle and conclude that $\\\\Psi $ is constant.\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , we have $\\\\Psi \\\\equiv 0$ .\",\n \"The proof is complete.\",\n \"Proposition 1.6 $\\\\nabla ^{2}\\\\Phi _2$ is integrable in $\\\\Omega $ .\",\n \"Since $\\\\nabla ^{2}\\\\Phi _2$ is integrable near the boundary $\\\\partial \\\\Omega $ , it suffices to show that $\\\\nabla ^{2}\\\\Phi _2\\\\in L^{1}(\\\\lbrace |x|\\\\ge 2R\\\\rbrace )$ .\",\n \"Since $h\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ , we observe that $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\int _{0}^{1}\\\\textrm {d}t\\\\int _{\\\\partial \\\\Omega }y\\\\cdot (\\\\nabla E)(x-ty)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Since $\\\\Omega ^{c}\\\\subset B_{0}(R)$ , for $|x|\\\\ge 2R$ we observe that $|x-ty|&\\\\ge \\\\big |\\\\ |x|-|ty|\\\\ \\\\big | \\\\\\\\&\\\\ge |x|-R \\\\\\\\&\\\\ge \\\\frac{|x|}{2}.$ Since $\\\\tilde{\\\\Phi }_{2}$ agrees with $\\\\Phi _2$ up to constant, we estimate $|\\\\nabla ^{2}\\\\Phi _2(x)|&\\\\lesssim \\\\int _{0}^{1}\\\\textrm {d}t\\\\int _{\\\\partial \\\\Omega }\\\\frac{|h(y)|}{|x-ty|^{n+1}}\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&\\\\lesssim \\\\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\\\partial \\\\Omega )}.$ Thus, $\\\\nabla ^{2}\\\\Phi _2$ is 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P., author=Maremonti, P., author=Zhou, Y., title=On the Navier-Stokes problem in exterior domains with non decaying initial data, date=2012, ISSN=1422-6928, journal=J.\",\n \"Math.\",\n \"Fluid Mech., volume=14, pages=633652, url=http://dx.doi.org/10.1007/s00021-011-0083-9, GIMarticle author=Giga, Y., author=Inui, K., author=Matsui, S., title=On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, date=1999, journal=Quaderni di Matematica, volume=4, pages=2868, GMSarticle author=Giga, Y., author=Matsui, S., author=Sawada, O., title=Global existence of smooth solutions for two dimensional Navier-Stokes equations with nondecaying initial velocity, date=2001, journal=J.\",\n \"Math.\",\n \"Fluid Mech., volume=3, pages=302315, GWarticle author=Gilbarg, D., author=Weinberger, H. F., title=Asymptotic properties of Leray's solutions of the stationary two-dimensional Navier-Stokes equations, date=1974, journal=Russian Math.\",\n \"Surveys, volume=29, pages=109123, GW2article author=Gilbarg, D., author=Weinberger, H. F., title=Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral, date=1978, journal=Ann.\",\n \"Scuola Norm.\",\n \"Sup.\",\n \"Pisa Cl.\",\n \"Sci.\",\n \"(4), volume=5, pages=381404, url=http://www.numdam.org/item?id=ASNSP19784523810, K1article author=Knightly, G. H., title=On a class of global solutions of the Navier-Stokes equations, date=1966, ISSN=0003-9527, journal=Arch.\",\n \"Rational Mech.\",\n \"Anal., volume=21, pages=211245, K2article author=Knightly, G. H., title=A Cauchy problem for the Navier-Stokes equations in $R^{n}$ , date=1972, ISSN=0036-1410, journal=SIAM J.\",\n \"Math.\",\n \"Anal., volume=3, pages=506511, KOarticle author=Kozono, H., author=Ogawa, T., title=Two-dimensional Navier-Stokes flow in unbounded domains, date=1993, journal=Math.\",\n \"Ann., volume=297, pages=131, url=http://dx.doi.org/10.1007/BF01459486, Lady61book author=Ladyzhenskaya, O.\",\n \"A., title=The mathematical theory of viscous incompressible flow, publisher=Gordon and Breach, Science Publishers, New York-London-Paris, date=1969, Leray1933article author=Leray, J, title=Étude de diverses équations intégrales non linéaires et de quelques problémes que pose l'hydrodynamique, date=1933, journal=J.\",\n \"Math.\",\n \"Pures Appl., volume=9, pages=182, Leray1934article author=Leray, J., title=Sur le mouvement d'un liquide visqueux emplissant l'espace, date=1934, ISSN=0001-5962, journal=Acta Math., volume=63, pages=193248, url=http://dx.doi.org/10.1007/BF02547354, LMparticle author=Lions, J.-L., author=Magenes, E., title=Problemi ai limiti non omogenei.\",\n \"V, date=1962, journal=Ann.\",\n \"Scuola Norm Sup.\",\n \"Pisa (3), volume=16, pages=144, Mar09article author=Maremonti, P., title=Stokes and Navier-Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity, date=2008, ISSN=0373-2703, journal=Zap.\",\n \"Nauchn.\",\n \"Sem.\",\n \"S.-Peterburg.\",\n \"Otdel.\",\n \"Mat.\",\n \"Inst.\",\n \"Steklov.\",\n \"(POMI), volume=362, pages=176240, url=http://dx.doi.org/10.1007/s10958-009-9458-3, Mar2014article author=Maremonti, P., title=Non-decaying solutions to the Navier Stokes equations in exterior domains: from the weight function method to the well posedness in $L^\\\\infty $ and in Hölder continuous functional spaces, date=2014, journal=Acta Appl.\",\n \"Math., volume=132, pages=411426, url=http://dx.doi.org/10.1007/s10440-014-9914-z, Miyakawa82article author=Miyakawa, T., title=On nonstationary solutions of the Navier-Stokes equations in an exterior domain, date=1982, ISSN=0018-2079, journal=Hiroshima Math.\",\n \"J., volume=12, pages=115140, url=http://projecteuclid.org/euclid.hmj/1206133879, PWbook author=Protter, M. H., author=Weinberger, H. F., title=Maximum principles in differential equations, publisher=Prentice-Hall, Inc., Englewood Cliffs, N.J.,, date=1967, SiSoincollection author=Simader, C. G., author=Sohr, H., title=A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$ -spaces for bounded and exterior domains, date=1992, booktitle=Mathematical problems relating to the Navier-Stokes equation, series=Ser.\",\n \"Adv.\",\n \"Math.\",\n \"Appl.\",\n \"Sci., volume=11, publisher=World Sci.\",\n \"Publ., River Edge, NJ, pages=135, url=http://dx.doi.org/10.1142/97898145035940001, Sl76article author=Solonnikov, V. A., title=Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, date=1976, journal=Zap.\",\n \"Naučn.\",\n \"Sem.\",\n \"Leningrad.\",\n \"Otdel Mat.\",\n \"Inst.\",\n \"Steklov.\",\n \"(LOMI), volume=59, pages=178254, 257, note=Boundary value problems of mathematical physics and related questions in the theory of functions, 9, Sl03article author=Solonnikov, V. A., title=On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, date=2003, ISSN=1072-3374, journal=J.\",\n \"Math.\",\n \"Sci.\",\n \"(N.\",\n \"Y.\",\n \"), volume=114, pages=17261740, url=http://dx.doi.org/10.1023/A:1022317029111, Sl06article author=Solonnikov, V. A., title=Weighted Schauder estimates for evolution Stokes problem, date=2006, ISSN=0430-3202, journal=Ann.\",\n \"Univ.\",\n \"Ferrara Sez.\",\n \"VII Sci.\",\n \"Mat., volume=52, pages=137172, url=http://dx.doi.org/10.1007/s11565-006-0012-7, Sl07incollection author=Solonnikov, V. A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.\",\n \"Math.\",\n \"Soc.\",\n \"Transl.\",\n \"Ser.\",\n \"2, volume=220, publisher=Amer.\",\n \"Math.\",\n \"Soc., Providence, RI, pages=165200,\"\n ],\n [\n \"acknowledgements\",\n \"The author is grateful to the anonymous referees for their valuable comments.\",\n \"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.\"\n ],\n [\n \"$L^{1}$ -estimates for the Neumann problem\",\n \" In Appendix A, we prove that $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ , $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ , for an exterior domain $\\\\Omega $ .\",\n \"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\\\mathbb {R}^{n}$ by using the heat semigroup.\",\n \"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.\",\n \"Lemma 1.1 Let $\\\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\\\mathbb {R}^{n}$ , $n\\\\ge 2$ .\",\n \"Then, $\\\\nabla \\\\mathbb {P}\\\\varphi \\\\in L^{1}(\\\\Omega )$ for $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ .\",\n \"We set $\\\\nabla \\\\Phi =\\\\varphi $ for $=I-\\\\mathbb {P}$ .\",\n \"It suffices to show that $\\\\nabla ^{2}\\\\Phi $ is integrable in $\\\\Omega $ .\",\n \"We recall that the $\\\\Phi $ solves the Neumann problem $\\\\begin{aligned}\\\\Delta \\\\Phi =\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\varphi }{\\\\partial n}=0\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.1)}$ See .\",\n \"We observe that $\\\\Phi \\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\\\varphi $ is smooth in $\\\\Omega $ and the boundary is $C^{2}$ .\",\n \"We may assume that $0\\\\in \\\\Omega ^{c}$ by translation.\",\n \"We take $R>0$ such that $\\\\Omega ^{c}\\\\subset B_{0}(R)$ .\",\n \"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\\\ge 3$ and $E(x)=-(2\\\\pi )^{-1}\\\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.\",\n \"We first show that the statement of Lemma A.1 is valid for $\\\\Omega =\\\\mathbb {R}^{n}$ .\",\n \"In the sequel, we do not distinguish $\\\\varphi \\\\in C_{c}^{\\\\infty }(\\\\Omega )$ and its zero extension to $\\\\mathbb {R}^{n}\\\\backslash \\\\Omega $ .\",\n \"Proposition 1.2 Set $h=E*\\\\varphi $ and $\\\\Phi _1=-\\\\textrm {div}\\\\ h$ .\",\n \"Then, $\\\\nabla ^{3}h$ is integrable in $\\\\mathbb {R}^{n}$ .\",\n \"In particular, $\\\\nabla ^{2}\\\\Phi _1\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"By using the heat semigroup, we transform $h$ into $h=\\\\int _{0}^{\\\\infty }e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t.$ We divide $h$ into two terms and observe that $\\\\partial ^{3}_{x}h=\\\\int _{0}^{1}\\\\partial _{x} e^{t\\\\Delta }\\\\partial ^{2}_{x}\\\\varphi \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\partial ^{3}_{x}e^{t\\\\Delta }\\\\varphi \\\\textrm {d}t,$ where $\\\\partial _{x}=\\\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\\\cdots n$ .\",\n \"We estimate $||\\\\partial ^{3}_{x} h||_{L^{1}(\\\\mathbb {R}^{n})}&\\\\lesssim \\\\int _{0}^{1}\\\\frac{1}{t^{1/2}}||\\\\partial ^{2}_{x} \\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t+\\\\int _{1}^{\\\\infty }\\\\frac{1}{t^{3/2}}||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})} \\\\textrm {d}t\\\\\\\\&\\\\lesssim ||\\\\partial ^{2}_{x}\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}+||\\\\varphi ||_{L^{1}(\\\\mathbb {R}^{n})}.$ We proved $\\\\nabla ^{3}h\\\\in L^{1}(\\\\mathbb {R}^{n})$ .\",\n \"We reduce (A.1) to the homogeneous Neumann problem $\\\\begin{aligned}-\\\\Delta \\\\Phi _2&=0\\\\quad \\\\textrm {in}\\\\ \\\\Omega ,\\\\\\\\\\\\frac{\\\\partial \\\\Phi _2}{\\\\partial n}&=g\\\\quad \\\\textrm {on}\\\\ \\\\partial \\\\Omega .\\\\end{aligned}\\\\qquad \\\\mathrm {(A.2)}$ We write connected components of $\\\\Omega $ by unbounded $\\\\Omega _0$ and bounded $\\\\Omega _1$ , $\\\\cdots $ , $\\\\Omega _N$ , i.e., $\\\\Omega =\\\\Omega _{0}\\\\cup (\\\\cup _{j=1}^{N}\\\\Omega _{j})$ .\",\n \"Proposition 1.3 Set $\\\\Phi _2=\\\\Phi -\\\\Phi _1$ .\",\n \"Then, $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ solves (A.2) for $g=\\\\textrm {div}_{\\\\partial \\\\Omega }(An)$ and $A=\\\\nabla h-\\\\nabla ^{T} h$ .\",\n \"The function $g\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega _{j}}g \\\\textrm {d}{\\\\mathcal {H}}=0\\\\quad \\\\textrm {for}\\\\ j=0,1,\\\\cdots ,N. $ We observe that $\\\\Phi _2\\\\in C^{2}(\\\\Omega )\\\\cap C^{1}(\\\\overline{\\\\Omega })$ satisfies $-\\\\Delta \\\\Phi _2=0$ in $\\\\Omega $ and $\\\\partial \\\\Phi _2/\\\\partial n=\\\\partial (h̥)/\\\\partial n$ on $\\\\partial \\\\Omega $ .\",\n \"We take an arbitrary $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ .\",\n \"Since $An=(\\\\sum _{1\\\\le j\\\\le n}(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j})_{1\\\\le i\\\\le n}$ is a tangential vector field on $\\\\partial \\\\Omega $ (i.e., $An\\\\cdot n=0$ on $\\\\partial \\\\Omega $ ), applying integration by parts yields $\\\\int _{\\\\partial \\\\Omega } g\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }\\\\textrm {div}_{\\\\partial \\\\Omega }(An)\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(An)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }(\\\\partial _jh^{i}-\\\\partial _ih^{j})n^{j}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=-\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _i \\\\rho \\\\textrm {d}{\\\\mathcal {H}}+\\\\int _{\\\\partial \\\\Omega }\\\\partial _j h^{i}n^{i}\\\\partial _j \\\\rho \\\\textrm {d}{\\\\mathcal {H}},$ where the symbol of summation is suppressed.\",\n \"By integration by parts, we have $\\\\int _{\\\\partial \\\\Omega }\\\\partial _jh^{i}n^{j}\\\\partial _i\\\\rho \\\\textrm {d}{\\\\mathcal {H}}&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho +\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\partial _i \\\\rho ) \\\\textrm {d}x\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }(\\\\Delta h^{i}\\\\partial _i\\\\rho -\\\\nabla h̥\\\\cdot \\\\nabla \\\\rho ) \\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\nabla h^{i}\\\\cdot \\\\nabla \\\\rho n^{i}\\\\textrm {d}{\\\\mathcal {H}}.$ Since $-\\\\Delta h=\\\\varphi $ is supported in $\\\\Omega $ , it follows that $\\\\int _{\\\\partial \\\\Omega }g\\\\rho {\\\\mathcal {H}}&=-\\\\int _{\\\\Omega }(\\\\Delta h-\\\\nabla h)\\\\cdot \\\\nabla \\\\rho \\\\textrm {d}x \\\\\\\\&=-\\\\int _{\\\\Omega }(\\\\Delta h\\\\cdot \\\\nabla \\\\rho +\\\\Delta h\\\\rho )\\\\textrm {d}x+\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}\\\\\\\\&=\\\\int _{\\\\partial \\\\Omega }\\\\frac{\\\\partial }{\\\\partial n}h\\\\rho \\\\textrm {d}{\\\\mathcal {H}}.$ Since $\\\\partial \\\\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\\\partial \\\\Omega $ .\",\n \"Thus, $g$ is continuous on $\\\\partial \\\\Omega $ .\",\n \"Since $\\\\rho \\\\in C_{c}^{\\\\infty }(\\\\mathbb {R}^{n})$ is arbitrary, we proved $\\\\partial (h)/\\\\partial n=g$ on $\\\\partial \\\\Omega $ .\",\n \"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.\",\n \"The proof is complete.\",\n \"We estimate $\\\\Phi _2$ by means of the layer potential.\",\n \"Proposition 1.4 (i) For $g\\\\in C(\\\\partial \\\\Omega )$ satisfying (A.3), there exists a moment $h\\\\in C(\\\\partial \\\\Omega )$ satisfying $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ and $-g(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla _x E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ (ii) Set the single layer potential $\\\\tilde{\\\\Phi }_{2}(x)=-\\\\int _{\\\\partial \\\\Omega }E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Then, $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative $\\\\partial _n \\\\tilde{\\\\Phi }_{2}$ exits and is continuous on $\\\\partial \\\\Omega $ .\",\n \"The function $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"The assertion (i) is based on the Fredholm's theorem.\",\n \"See .\",\n \"Since $h$ is bounded on $\\\\partial \\\\Omega $ , $\\\\tilde{\\\\Phi }_{2}$ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, we have $-\\\\frac{\\\\partial \\\\tilde{\\\\Phi }_{2}}{\\\\partial n}(x)=\\\\frac{1}{2}h(x)+\\\\int _{\\\\partial \\\\Omega }n(x)\\\\cdot \\\\nabla E(x-y)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\quad x\\\\in \\\\partial \\\\Omega .$ See .\",\n \"Thus $\\\\tilde{\\\\Phi }_{2}$ satisfies (A.2) by the assertion (i).\",\n \"When $n\\\\ge 3$ , $\\\\tilde{\\\\Phi }_{2}(x)\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ since the fundamental solution decays as $|x|\\\\rightarrow \\\\infty $ .\",\n \"Moreover, when $n=2$ , the average of $h$ on $\\\\partial \\\\Omega $ is zero and we have $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\frac{1}{2\\\\pi }\\\\int _{\\\\partial \\\\Omega }\\\\log {\\\\Bigg (\\\\frac{|x-y|}{|x|}\\\\Bigg )}h(y)\\\\textrm {d}\\\\mathcal {H}(y)\\\\rightarrow 0\\\\quad \\\\textrm {as}\\\\ |x|\\\\rightarrow \\\\infty .$ The proof is complete.\",\n \"Proposition 1.5 The function $\\\\tilde{\\\\Phi }_2$ agrees with ${\\\\Phi }_2$ up to constant.\",\n \"Since $\\\\nabla \\\\Phi _2=\\\\nabla \\\\Phi -\\\\nabla \\\\Phi _1$ is $L^{p}$ -integrable in $\\\\Omega $ for all $p\\\\in (1,\\\\infty )$ (e.g., ), we may assume that $\\\\Phi _2\\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ by shifting $\\\\Phi _2$ by a constant.\",\n \"We set $\\\\Psi =\\\\Phi _2-\\\\tilde{\\\\Phi }_2 $ and observe that $\\\\Psi $ is continuous in $\\\\overline{\\\\Omega }$ .\",\n \"Moreover, the normal derivative exists and is continuous on $\\\\partial \\\\Omega $ by Proposition A.4.\",\n \"The function $\\\\Psi $ satisfies $-\\\\Delta \\\\Psi =0$ in $\\\\Omega $ , $\\\\partial \\\\Psi /\\\\partial n=0$ on $\\\\partial \\\\Omega $ and $\\\\Psi \\\\rightarrow 0$ as $|x|\\\\rightarrow \\\\infty $ .\",\n \"By the elliptic regularity theory , $\\\\Psi $ is smooth in $\\\\Omega $ and continuously differentiable in $\\\\overline{\\\\Omega }$ .\",\n \"We shall show that $\\\\Psi \\\\equiv 0$ .\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , there exits a point $x_0\\\\in \\\\overline{\\\\Omega }$ such that $\\\\sup _{x\\\\in \\\\Omega }\\\\Psi (x)=\\\\Psi (x_0)$ .\",\n \"Suppose that $x_0\\\\in \\\\partial \\\\Omega $ .\",\n \"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\\\partial \\\\Psi (x_0)/\\\\partial n>0$ .\",\n \"Thus $x_0\\\\in \\\\Omega $ .\",\n \"We apply the strong maximum principle and conclude that $\\\\Psi $ is constant.\",\n \"Since $\\\\Psi $ decays as $|x|\\\\rightarrow \\\\infty $ , we have $\\\\Psi \\\\equiv 0$ .\",\n \"The proof is complete.\",\n \"Proposition 1.6 $\\\\nabla ^{2}\\\\Phi _2$ is integrable in $\\\\Omega $ .\",\n \"Since $\\\\nabla ^{2}\\\\Phi _2$ is integrable near the boundary $\\\\partial \\\\Omega $ , it suffices to show that $\\\\nabla ^{2}\\\\Phi _2\\\\in L^{1}(\\\\lbrace |x|\\\\ge 2R\\\\rbrace )$ .\",\n \"Since $h\\\\in C(\\\\partial \\\\Omega )$ satisfies $\\\\int _{\\\\partial \\\\Omega }h\\\\textrm {d}{\\\\mathcal {H}}=0$ , we observe that $\\\\tilde{\\\\Phi }_{2}(x)&=-\\\\int _{\\\\partial \\\\Omega }(E(x-y)-E(x))h(y)\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&=\\\\int _{0}^{1}\\\\textrm {d}t\\\\int _{\\\\partial \\\\Omega }y\\\\cdot (\\\\nabla E)(x-ty)h(y)\\\\textrm {d}{\\\\mathcal {H}}(y).$ Since $\\\\Omega ^{c}\\\\subset B_{0}(R)$ , for $|x|\\\\ge 2R$ we observe that $|x-ty|&\\\\ge \\\\big |\\\\ |x|-|ty|\\\\ \\\\big | \\\\\\\\&\\\\ge |x|-R \\\\\\\\&\\\\ge \\\\frac{|x|}{2}.$ Since $\\\\tilde{\\\\Phi }_{2}$ agrees with $\\\\Phi _2$ up to constant, we estimate $|\\\\nabla ^{2}\\\\Phi _2(x)|&\\\\lesssim \\\\int _{0}^{1}\\\\textrm {d}t\\\\int _{\\\\partial \\\\Omega }\\\\frac{|h(y)|}{|x-ty|^{n+1}}\\\\textrm {d}{\\\\mathcal {H}}(y)\\\\\\\\&\\\\lesssim \\\\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\\\partial \\\\Omega )}.$ Thus, $\\\\nabla ^{2}\\\\Phi _2$ is integrable in $\\\\lbrace |x|\\\\ge 2R\\\\rbrace $ .\",\n \"The proof is complete.\",\n \"By Propositions A.2 and A.6, the assertion follows.\",\n \"A4article author=Abe, K., title=The Navier-Stokes equations in a space of bounded functions, date=2015, journal=Commun.\",\n \"Math.\",\n \"Phys., volume=338, pages=849865, A3article author=Abe, K., title=On estimates for the Stokes flow in a space of bounded functions, date=2016, journal=J.\",\n \"Differ.\",\n \"Equ., volume=261, pages=17561795, AG1article author=Abe, K., author=Giga, Y., title=Analyticity of the Stokes semigroup in spaces of bounded functions, date=2013, journal=Acta Math., volume=211, pages=146, url=http://dx.doi.org/10.1007/s11511-013-0098-6, AG2article author=Abe, K., author=Giga, Y., title=The $L^\\\\infty $ -Stokes semigroup in exterior domains, date=2014, journal=J.\",\n \"Evol.\",\n \"Equ., volume=14, pages=128, url=http://dx.doi.org/10.1007/s00028-013-0197-z, AGHarticle author=Abe, K., author=Giga, Y., 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A., title=Schauder estimates for the evolutionary generalized Stokes problem, date=2007, booktitle=Nonlinear equations and spectral theory, series=Amer.\",\n \"Math.\",\n \"Soc.\",\n \"Transl.\",\n \"Ser.\",\n \"2, volume=220, publisher=Amer.\",\n \"Math.\",\n \"Soc., Providence, RI, pages=165200,\"\n ]\n]"},"id":{"kind":"string","value":"1606.05040"}}},{"rowIdx":999,"cells":{"text":{"kind":"list like","value":[["LOLAS-2 : redesign of an optical turbulence profiler"],["Abstract We present the development, tests and first results of the second generation Low Layer Scidar (LOLAS-2).","This instrument constitutes a strongly improved version of the prototype Low Layer Scidar, which is aimed at the measurement of optical turbulence profiles close to the ground, with high altitude-resolution.","The method is based on the Generalised Scidar principle which consists in taking double-star scintillation images on a defocused pupil plane and calculating in real time the autocovariance of the scintillation.","The main components are an open-truss 40-cm Ritchey-Chr\\'etien telescope, a german-type equatorial mount, an Electron Multiplying CCD camera and a dedicated acquisition and real-time data processing software.","The new optical design of LOLAS-2 is significantly simplified compared with the prototype.","The experiments carried out to test the permanence of the image within the useful zone of the detector and the stability of the telescope focus show that LOLAS-2 can function without the use of the autoguiding and autofocus algorithms that were developed for the prototype version.","Optical turbulence profiles obtained with the new Low Layer Scidar have the best altitude-resolution ever achieved with Scidar-like techniques (6.3 m).","The simplification of the optical layout and the improved mechanical properties of the telescope and mount make of LOLAS-2 a more robust instrument."],["Introduction","Turbulent flows in the atmosphere combined with stratified temperatures provoke turbulent fluctuations of the refractive index of air, commonly known as optical turbulence.","The intensity of such fluctuations is determined by the second order structure constant $C_{\\mathrm {N}}^2$ .","The measurement of $C_{\\mathrm {N}}^2(h)$ , $h$ being the altitude, is of major importance for the development of novel adaptive optical (AO) systems that overcome the angular-resolution degradation introduced by optical turbulence.","The statistical study of optical turbulence profiles is also crucial for the characterization of sites where next generation of optical telescopes are to be installed (e.g., [16]).","One important limitation of AO systems that are designed to account for phase fluctuations generated all along the atmosphere is the tiny field of view over which the wavefront is corrected.","One way to improve image quality over a wide field of view is to correct only the wavefront perturbations that come from turbulent layers close to the ground.","This method is known as ground-layer adaptive optics (GLAO) [14], [18].","Indeed, the compensation of lower-altitude turbulent-layers provides wider corrected fields of view [7] and turbulence close to the ground is generally the most intense (e.g., [3]).","To develop a GLAO system for a given site, it is required to have as much knowledge as possible about the vertical distribution of optical turbulence in the ground layer (e.g, [11]).","This requires measurements of $C_{\\mathrm {N}}^2(h)$ with very high altitude-resolution close to the ground.","Scintillation Detection and Ranging (SCIDAR) has extensively been used for $C_{\\mathrm {N}}^2$ profiling since its invention by [15].","The generalised version of the SCIDAR made it possible to detect turbulence near the ground [10], [4].","A more recent implementation of the Generalized SCIDAR on a 40-cm telescope that used a widely-separated double star as a light source and an Electron Multiplying Charge Coupled Device (EMCCD) as the detector, gave place to the Low Layer SCIDAR (LOLAS) [1].","The LOLAS prototype was used to characterize the ground-layer turbulence at Mauna Kea, together with a Slope Detection and Ranging instrument [8].","Although this several-years campaign gave definitive results, the experience showed that a number of aspects on the prototype LOLAS could be improved in order to optimize data acquisition.","In this paper we describe the development and tests of the second generation Low Layer Scidar (LOLAS-2).","A few instruments based on optical methods have been developed to measure $C_{\\mathrm {N}}^2(h)$ profiles with the required vertical resolution close to the ground.","For example, the surface-layer Slope Detection and Ranging (SL-SLODAR) [12] makes use of two Shack-Hartmann wavefront sensors, each looking at one component of a widely-separated double star.","The slope of the wavefronts coming from each star is measured on 5-cm square subapertures whereas LOLAS-2 measures scintillation on square elements of 1-cm side approximately, which makes the SL-SLODAR five times more sensitive than the LOLAS-2 for equal integration times or five times faster for equal number of photons per element.","On the other hand, if the instruments were using the same double star separation, LOLAS-2 provides 5 times better altitude resolution.","A similar concept that also uses a double star but measures the scintillation from each component on two cameras was implemented in the Stereo-Scidar by [17].","The Lunar Scintillometer, which consists of an array of photodiodes that measure scintillation from the moon, has been used to obtain high altitude-resolution turbulence profiles near the ground [19].","[9] use a Generalized SCIDAR to distinguish layers at very similar altitudes but moving at different velocities.","The present paper is organized as follows: § gives an overview of the LOLAS method.","In § REF we briefly describe the prototype version of the instrument and in § REF the second generation of the instrument is presented.","Tests on the instrument performances and some measurements are shown in § and § .","Conclusions are given in § ."],["Method","The Low Layer Scidar is based on the principle of the Generalized Scintillation Detection and Ranging (G-SCIDAR) technique [15], [10], [4].","In this section, we present a brief overview of the LOLAS method, since a complete description can be found in [1].","The G-SCIDAR principle can be summarized as follows: double-star scintillation patterns are recorded on short exposure-time images on a virtual plane located a distance $h_\\mathrm {gs}$ from the telescope pupil.","In G-SCIDAR experiments this virtual plane is located below the telescope pupil, thus $h_\\mathrm {gs}<0$ .","A schematic view of the optical setup is shown in Fig.","REF .","Each image consists of a randomly distributed intensity pattern.","The autocovariance of this stochastic illumination is obtained by computing the spatial normalized autocorrelation of each image and averaging those autocorrelations over thousands of statistically independent image samples.","Each layer of optical turbulence contributes to the resulting scintillation autocovariance with three covariance peaks: one centred at the autocovariance origin and two others, identical to each other, separated from the origin by $\\mathbf {d}_\\mathrm {l}=-\\rho \\vert h-h_\\mathrm {gs}\\vert $ and $\\mathbf {d}_\\mathrm {r}=\\rho \\vert h-h_\\mathrm {gs} \\vert $ , respectively, where $\\rho $ denotes the angular separation of the double star and $h$ the layer altitude above the ground.","Knowing $\\rho $ and the conjugation altitude $h_\\mathrm {gs}$ , the experimental determination of $\\mathbf {d}_\\mathrm {l}$ (or $\\mathbf {d}_\\mathrm {r}$ ) leads to an estimate of the layer altitude $h$ .","The measured autocovariance peaks $C_\\mathrm {gs}(\\mathrm {\\mathbf {r}})$ are proportional to the optical turbulence strength at altitude $h$ , $C_\\mathrm {N}^2(h)$ , and to the scintillation autocovariance function $K(\\mathbf {r}, \\vert h-h_\\mathrm {gs} \\vert )$ .","In the realistic case of multiple layers, the response of each layer adds up to the measured autocovariance, resulting in Eq.","1 of [1].","This equation consists of an integral with respect the altitude $h$ that is similar to a convolution, except that the kernel $K(\\mathbf {r}, \\vert h-h_\\mathrm {gs} \\vert )$ depends on the integration variable.","One needs to invert this integral equation to retrieve $C_\\mathrm {N}^2(h)$ .","Due to its similarity with a convolution integral, we developed an algorithm based on the CLEAN method, but in which the kernel is recalculated for each different value of $h$ .","Our modified CLEAN algorithm is based on that reported by [13].","As explained by [1], the achievable altitude resolution using the modified CLEAN method, when the target is at the zenith is $\\Delta h=0.52\\sqrt{\\lambda (\\vert h-h_\\mathrm {gs} \\vert )}/\\rho $ , where $\\lambda $ is the wavelength and $\\rho =\\vert \\rho \\vert $ .","When the star is located at an elevation angle $\\theta $ , $\\Delta h$ is decreased by a factor $\\cos \\theta $ .","We take $\\lambda =0.5 \\mu \\mathrm {m}$ , which corresponds to the maximum sensitivity of our detector.","For a telescope having an aperture $D$ , the maximum altitude for which the $C_{\\mathrm {N}}^2$ value can be measured is given by $h_\\mathrm {max}=D/\\rho $ [1].","The Low Layer Scidar concept consists of putting into practice a G-SCIDAR on a dedicated portable telescope, using widely separated double stars as light sources.","As can be seen from the above expressions for $\\Delta h$ and $h_\\mathrm {max}$ , the wider the separation, the better the altitude resolution but the shorter the maximum altitude.","LOLAS was designed to use a 40-cm telescope, an EMCCD to improve sensitivity and a real-time computation of the scintillation autocovariance.","Sections REF and REF describe the prototype LOLAS version and the second generation instrument, respectively."],["Prototype version","The instrumental setup of LOLAS has been widely explained elsewhere [1], [8], [6].","We present a summary of the most important instrumental characteristics.","Figure REF shows a schematic view of the prototype LOLAS.","The scintillation images are obtained with a Schmidt-Cassegrain telescope of focal ratio f/10 and diameter $D = 40.64$ cm and installed on an equatorial mount, manufactured by Meade.","The optics consists of two achromatic lenses of 50 mm focal-length.","With this optical arrangement, the virtual analysis plane is located 1.94 km below the pupil.","The diameter of the pupil image on the detector is $D^\\prime = 24.5$ mm.","The scintillation images are captured by an EMCCD camera (Andor iXon) with $512\\times 512$ square pixels of $16~\\mu \\mathrm {m}$ .","The frames are binned 2 $\\times $ 2, and the active zone is limited to an array of $256 \\times 80$ binned pixels.","The exposure time of each frame ranges from 3 to 10 ms, depending on the wind conditions.","The typical number of images to obtain one autocovariance is set to 30000.","The EMCCD camera is mounted on a base that is attached to the rear of the telescope (see Fig.","REF ).","The same equipment, but with different optics, is used to form the SLODAR instrument.","To switch between each instrument with the required positioning accuracy, a manual exchange mechanism was installed in front of the camera."],["Second generation","As seen in Fig.","REF , LOLAS-2 uses a Ritchey-Chrétien open telescope of focal ratio f/9 and diameter $D = 40.64$ cm, manufactured by RC Optical Systems, a German equatorial mount (1200GTO) manufactured by Astro-Physics, an EMCCD camera (Andor iXon) to acquire scintillation images and a Sbig St-402ME camera for the finder telescope.","One advantage of using a Ritchey-Chrétien telescope is that when the position of the focal plane is changed, the effective focal length of the telescope remains unchanged.","Our RC Optical Systems telescope is equipped with a system that maintains the focus by monitoring the secondary mirror position in closed loop at a frequency of 6 kHz, reaching an accuracy in the secondary mirror and the focus positions of 0.6 and 25.4 $\\mu \\mathrm {m}$ , respectively.","This control system is part of the telescope.","In addition, the fact that the telescope is open prevents air at different temperatures to get trapped inside the tube in a turbulent convective flow, which would add an instrumental bias to the $C_{\\mathrm {N}}^2$ measurements at ground level.","This was the case with the prototype LOLAS.","Removal of the spurious turbulence from the measurements was performed in a post-processing procedure using the method described by [5], as reported by [8].","In LOLAS-2, this post-processing step is avoided, making the data reduction faster and simpler.","Concerning the mount, LOLAS-2 incorporates a German-type mount that reduces considerably the lever arm between the equatorial and declination axis.","Moreover, the Astro-Physics mount has a higher stiffness and the worm gear accuracy is significantly better, compared to those of the Meade mount.","The prototype version uses the optics of the telescope and achromatic doublets to define the spatial sampling and conjugation distance $h_{gs}$ below the pupil.","The second generation instrument was developed so as to dispense the use of the achromatic doublets and the exchange mechanism.","It only uses the telescope optics, making it a simpler and more robust instrument.","The beam is no longer collimated, like in the prototype version.","The EMCDD is placed directly a distance $L$ before the telescope focal plane.","Distance $L$ is chosen such that the detector plane is made the conjugate of the a virtual plane located a distance $h_\\mathrm {gs}$ below the telescope pupil (see Fig.","REFa) and the spatial sampling on this plane is well-suited to sample the scintillation speckles (see Fig.","REFb).","Using the thin lens equation, it can easily be shown that $h_\\mathrm {gs}$ is related to $L$ by the following expression: $h_\\mathrm {gs} = -\\frac{{F_\\mathrm {tel}}^2-F_\\mathrm {tel}L}{L},$ where $F_\\mathrm {tel}$ is the focal length of the telescope.","For the spatial sampling, Fig.","REFb illustrates the demagnification relation: $\\frac{\\mathcal {L}_{D}}{L}= \\frac{\\mathcal {L}_\\mathrm {min}}{F_\\mathrm {tel}}$ where $\\mathcal {L}_\\mathrm {min}$ represents the typical size of the smallest scintillation speckles on the pupil and $\\mathcal {L}_{D}$ is its corresponding size on the detector plane.","[13] showed that the full width at half maximum of the autocovariance of the scintillation produced at altitude $h$ is given by $\\mathcal {L}(h)=0.78\\sqrt{\\lambda \\vert h-h_\\mathrm {gs} \\vert )}.$ This is, the typical size of the smallest speckle is $\\mathcal {L}_\\mathrm {min}\\equiv \\mathcal {L}(0)=0.78\\sqrt{\\lambda \\vert h_\\mathrm {gs}\\vert }$ .","Solving Eq.","REF for $\\mathcal {L}_{D}$ and replacing the above expression for $\\mathcal {L}_\\mathrm {min}$ gives: $\\mathcal {L}_{D}= L\\frac{0.78\\sqrt{\\lambda \\vert h_\\mathrm {gs}\\vert }}{F_\\mathrm {tel}}.$ For ground-level turbulence to be detectable, $h_\\mathrm {gs}$ must be smaller than $-1000$ m. A good spatial sampling of the scintillation speckles is obtained when $\\mathcal {L}_{D}\\simeq 2p$ , where $p$ is the size of the elementary sampling element.","We chose to acquire images with the camera pixels binned $2\\times 2$ .","In that case, $p=2d_\\mathrm {pix}$ .","The camera pixel size is $d_\\mathrm {pix}=16\\;\\mu \\mathrm {m}$ .","Table gives values of different parameters of interest for different values of $L$ .","It can be seen that a good compromise is obtained when the detector is located a distance $L=11$ mm before the telescope focal plane, as the conjugation distance is large enough ($h_\\mathrm {gs} = -1212$ m) and the number of spatial samples per smallest speckle width is 1.8, which is very close to 2, the Nyquist criterion.","Even though the undersampling is small, it is taken into account in the kernel of the inversion process that calculates $C_{\\mathrm {N}}^2$ profiles from the measured autocovariances.","As a consequence of this analysis, the value for $L$ is set to 11 mm in LOLAS-2.","For this value of $L$ , the distance separating two contiguous binned pixels corresponds to an angular separation of $1.8^{\\prime \\prime }$ ."],["Instrument performance","In this section we present results concerning the focus stability and guiding performance obtained the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM), Baja California, Mexico.","The data was obtained on the nights of 2013 June 15, 16 and 17.","Table summarises the pertinent characteristics of the double-star targets used."],["Focus stability","Focus stability is extremely important to maintain the spatial sampling and conjugation altitude along data acquisition.","A variation of the telescope focus position translates into a variation of the pupil image diameter $d$ on the detector.","This diameter is continuously monitored during the standard data acquisition of the instrument.","Images are sent by the EMCCD to the computer in packets of 200 consecutive 256x80 pixels frames.","In each frame, the image is centred (as explained in §REF ) and then co-added to form a mean image made of 200 frames.","The pupil diameters are calculated from this mean image as follows (see Fig.","REF ): the mean image is integrated along columns to form a row of the accumulated values.","The width at half maximum of the left and right pupils on the accumulated-values row determines their diameter in number of pixels $N_{d,l}$ and $N_{d,r}$ , respectively.","To estimate the focus stability, the diameter of each pupil was monitored during several hours on three nights, using the same double star as source.","The total number of pupil size measurements was 4566.","The mean and standard deviation of the measured diameters are $\\left=38.13$ and $\\sigma _{N_{d}}=0.40$ pixels.","The average of the temperature and its variation for each night was $13.5\\pm 0.22$ , $14.6\\pm 0.23$ and $14.1\\pm 0.07$ Celcius degrees, according to the weather station at SPM.","The obtained standard deviations can be due to uncertainties in the estimation procedure and/or to actual physical variation in the focus or camera positions.","If we consider the latter to be the cause of the measured diameter fluctuations, the consequence of the deviation of 0.40 pixels in pupil diameter implies a variation of 13 m in altitude conjugation and 0.32 $\\mu $ m in the sampling element size on the detector, approximately, which are tolerable values.","This result suggests that the use of an auto-focus algorithm, which was developed for the prototype LOLAS, could be avoided in the second generation instrument, although more tests under varying temperature conditions should be performed."],["Guiding test","Maintaining a constant position of the pupil images on the EMCCD is important to correctly compute the mean image, the autocorrelation of which is used to normalize the mean autocorrelation of scintillation images, so obtaining the scintillation autocovariance.","On wind conditions commonly encountered in astronomical observatories, telescope shake may cause pupil images to move on the detector.","This eventual image wander is corrected for by centering the image in every single frame captured by the detector.","The guiding performance of the telescope mount is important to keep pupil images within the useful window of the EMCCD during data acquisition that may last hours.","The image position on the active area of the EMCCD is determined as follows: we construct an artificial reference image $I_\\mathrm {r}(\\mathbf {r})$ formed by two disks of 38 pixels in diameter each and separated from each other by the same distance as the observed double star.","We compute the cross-correlation $C_\\mathrm {c}(\\mathbf {r})$ of the current image $I_i(\\mathbf {r})$ with $I_\\mathrm {r}(\\mathbf {r})$ .","The position of the pupil images $(X,Y)$ in $I_i(\\mathbf {r})$ is set as the position of the maximum value of $C_\\mathrm {c}(\\mathbf {r})$ with respect the frame centre.","In standard operation of LOLAS-2, image positions are not saved on disk.","To test the guiding performance and image stability we recorded image positions $(X,Y)$ during several observations.","The $(X,Y)$ coordinates on the EMCCD were rotated according to the double-star position angle to obtain image positions in right ascension ($\\alpha $ ) and declination ($\\delta $ ) coordinates system.","This allows us to investigate separately the guiding and stability behaviour on the two rotation axis of the mount.","Figures REF and REF show two examples.","For the results obtained on 2013 June 17 UT (Fig.","REF ) wind was blowing from the South-West with mean speed of 15.5 km h$^{-1}$ and a maximum value of 18.0 km h$^{-1}$ .","The image positions remained very stable apart from a slow continuous drift on $\\delta $ and a slow oscillation in $\\alpha $ , presumably due to an inexact alignment of the mount and a periodic error on the right-ascension gear-mechanism, respectively.","On 2013 June 15 UT wind was less benevolent, blowing from the West at a mean speed of 26.6 km h$^{-1}$ with gusts reaching 40 km h$^{-1}$ .","Figure REF shows an example of that night.","Even though images were moving significantly, the mean position remained constant and pupil images remained within the EMCCD working window, which enabled turbulence profiles to be measured in these conditions.","To investigate further the dependence of image jitter on prevailing wind speed, in Fig.","REF it is shown a plot of the standard deviation of image position as a function of wind speed.","For each of the available 4566 frame-packets, the standard deviation $\\sigma _\\eta $ of the image position was calculated.","The frame rate within each packet is 13-ms per frame.","The wind conditions that prevailed at the time of each measurement were obtained from the database of the OAN-SPM weather stationwww.astrossp.unam.mx/weather15/.","Bar extremities on Fig.","REF indicate the 25 and 75 percentiles of the $\\sigma _\\eta $ values for a given wind speed.","As expected, strong winds produce large image jitter, but surprisingly, $\\sigma _\\eta $ values remain lower than $5^{\\prime \\prime }$ for wind speeds lower than 30 $\\mathrm {ms^{-1}}$ .","Even though the force exerted by the wind on the telescope is proportional to the wind speed, the mount and telescope structures move significantly only if the wind speed exceeds a certain value that seems to lie between 30 and 35 $\\mathrm {ms^{-1}}$ .","The telescope and mount stiffness, together with the guiding performance have proven to suffice for conducting observations in moderate to strong wind conditions, without the need of an auto guiding procedure like in the prototype LOLAS."],[" $C_{\\mathrm {N}}^2$ Measurement examples","In this section we present a few examples of measured scintillation autocorrelations and corresponding turbulence profiles.","The first results of LOLAS-2 were obtained in 2013 November at the OAN-SPM.","The instrument was installed on a concrete pillar (see Fig.","REF ) to reduce vibrations.","Figure REF shows examples of autocovariance maps obtained from 30 000 scintillation images, during the nights 2013 November 16 and 17.","In the central peaks of Figs.","REFa and REFc , the contribution of all turbulent layers in the atmosphere are added up.","The correlated speckles produced by turbulent layers above $h_{\\mathrm {max}}$ are separated by a distance longer than the pupil diameter, thereby not giving rise to lateral peaks inside the autocovariance-map boundaries and avoiding the corresponding ${C_{\\mathrm {N}}^2}$ estimate.","Only the layers below $h_{\\mathrm {max}}$ form visible lateral peaks from which ${C_{\\mathrm {N}}^2}$ values are retrieved.","A white rectangle frames the right-hand side lateral peaks in Figs.","REFa and REFc.","Enlarged views are shown in Fig.","REFb and REFd.","The most intense central peak inside each of those frames corresponds to the turbulence at ground level.","Weaker peaks can clearly be seen in Fig.","REFa, which correspond to turbulence above the ground.","Fig.","REFd only exhibits ground-level turbulence.","The ${C_{\\mathrm {N}}^2(h)}$ profiles are obtained using a modified CLEAN algorithm [1] based on the one presented by [13].","In Fig.","REF we present a few profiles obtained on 2013 November 16 and 17 using autocovariance maps whose examples are shown in Fig REF .","The purpose of Fig.","REF is only to display the ability of the instrument in measuring ${C_{\\mathrm {N}}^2}$ profiles and their characteristics.","It is not intended for studying the optical turbulence at the site.","Altitude resolution, maximum sensed altitude and noise level values are summarised in Table .","Note that the altitude resolution obtained when using 12 Cam as a target is the best ever achieved with Scidar-like techniques.","The noise levels indicated in Table are the $C_{\\mathrm {N}}^2$ values that correspond to the $3\\sigma $ , i.e.","three times the standard deviation on the autocovariance-map background [1].","The profiles corresponding to November 16 show three turbulent layers: one at ground level, the second at 81 m and the weakest at 390 m. On November 17, almost all turbulence below 400 m was concentrated at ground level.","Only a weak turbulent layer is detected at 170 m above the ground at 11:27 UT that night.","It is worth recalling that $C_{\\mathrm {N}}^2$ values obtained with Scidar techniques in which the pupil images are not superimposed on the detector, like LOLAS, do not need to be corrected for the normalization error pointed out by [2]."],["Conclusions","The second generation of the Low Layer Scidar incorporates a simplified optical layout, a stiff and precise telescope mount and an open Ritchey-Chrétien telescope with excellent focus stability.","The analysis of the image positions showed that the guiding quality of the mount permits to avoid the autoguiding algorithm that was used in the prototype LOLAS.","Similarly, the excellent focus stability of the telescope allows for observations without the autofocus algorithm employed in the former version of the instrument.","Examples of measurements obtained with LOLAS-2 illustrate the capability of the instrument for very high altitude-resolution profiling of the optical turbulence.","A statistical analysis of $C_{\\mathrm {N}}^2(h)$ profiles obtained so far at the OAN-SPM will be presented in a forthcoming paper.","We are deeply grateful to the staff of the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM) for their kind help in all the logistics for the observations.","Wind data used in the work was provided by the OAN-SPM wheather station (http://tango.astrosen.unam.mx).","Financial support was provided by DGAPA–UNAM through grants IN103913 and IN115013.","1 lcccc Instrumental parameters for different values of $L$ 0pt Parameter $L_1$ =15 mm $\\mathbf {L_2=}$11 mm $L_3$ =7.5 mm $L_4$ =5 mm $h_\\mathrm {gs}$ [m] -886 -1212 -1777 -2668 $d$ [mm]a 1.66 1.22 0.83 0.55 $N_{d}$ [pixels]b 52 38 26 17 $\\mathcal {L}_{D}/p$ c 2.58 1.80 1.29 0.86 aDiameter of the pupil image on the detector plane.","bNumber of binned pixels along a pupil image diameter (Fig.","REF ).","cNumber of binned pixels along a speckle (for $h=0$ ).","lccccc Targets 0pt Name ${\\alpha _{2000}}$ a ${\\delta _{2000}}$ a ${m_{1}}$ b ${m_{2}}$ b $\\rho $ c [$^{\\prime \\prime }$ ] 15 Tri 2$^{\\mathrm {h}}$ :35 34$^{\\circ }$ :41 5.5 6.7 138.9 12 Cam 5$^{\\mathrm {h}}$ :06 58$^{\\circ }$ :58 5.2 6.2 181.3 a Right ascension ($\\alpha _{2000}$ ) and Declination ($\\delta _{2000}$ ).","b Visible magnitude of each star.","c Angular separation.","lccccc Observational parameters 0pt Date Target $\\Delta h$ a ${h_{\\mathrm {max}}}$ b $\\tau $ c N. L.d 2013/11/16 15 Tri 11.7 603 3 9.5$\\times 10^{-16}$ 2013/11/17 12 Cam 6.3 462 2 1.3$\\times 10^{-15}$ a Altitude resolution [m].","b Maximum altitude [m].","c Exposure time [ms].","d Noise level [$\\mathrm {m}^{-2/3}$ ]."]],"string":"[\n [\n \"LOLAS-2 : redesign of an optical turbulence profiler\"\n ],\n [\n \"Abstract We present the development, tests and first results of the second generation Low Layer Scidar (LOLAS-2).\",\n \"This instrument constitutes a strongly improved version of the prototype Low Layer Scidar, which is aimed at the measurement of optical turbulence profiles close to the ground, with high altitude-resolution.\",\n \"The method is based on the Generalised Scidar principle which consists in taking double-star scintillation images on a defocused pupil plane and calculating in real time the autocovariance of the scintillation.\",\n \"The main components are an open-truss 40-cm Ritchey-Chr\\\\'etien telescope, a german-type equatorial mount, an Electron Multiplying CCD camera and a dedicated acquisition and real-time data processing software.\",\n \"The new optical design of LOLAS-2 is significantly simplified compared with the prototype.\",\n \"The experiments carried out to test the permanence of the image within the useful zone of the detector and the stability of the telescope focus show that LOLAS-2 can function without the use of the autoguiding and autofocus algorithms that were developed for the prototype version.\",\n \"Optical turbulence profiles obtained with the new Low Layer Scidar have the best altitude-resolution ever achieved with Scidar-like techniques (6.3 m).\",\n \"The simplification of the optical layout and the improved mechanical properties of the telescope and mount make of LOLAS-2 a more robust instrument.\"\n ],\n [\n \"Introduction\",\n \"Turbulent flows in the atmosphere combined with stratified temperatures provoke turbulent fluctuations of the refractive index of air, commonly known as optical turbulence.\",\n \"The intensity of such fluctuations is determined by the second order structure constant $C_{\\\\mathrm {N}}^2$ .\",\n \"The measurement of $C_{\\\\mathrm {N}}^2(h)$ , $h$ being the altitude, is of major importance for the development of novel adaptive optical (AO) systems that overcome the angular-resolution degradation introduced by optical turbulence.\",\n \"The statistical study of optical turbulence profiles is also crucial for the characterization of sites where next generation of optical telescopes are to be installed (e.g., [16]).\",\n \"One important limitation of AO systems that are designed to account for phase fluctuations generated all along the atmosphere is the tiny field of view over which the wavefront is corrected.\",\n \"One way to improve image quality over a wide field of view is to correct only the wavefront perturbations that come from turbulent layers close to the ground.\",\n \"This method is known as ground-layer adaptive optics (GLAO) [14], [18].\",\n \"Indeed, the compensation of lower-altitude turbulent-layers provides wider corrected fields of view [7] and turbulence close to the ground is generally the most intense (e.g., [3]).\",\n \"To develop a GLAO system for a given site, it is required to have as much knowledge as possible about the vertical distribution of optical turbulence in the ground layer (e.g, [11]).\",\n \"This requires measurements of $C_{\\\\mathrm {N}}^2(h)$ with very high altitude-resolution close to the ground.\",\n \"Scintillation Detection and Ranging (SCIDAR) has extensively been used for $C_{\\\\mathrm {N}}^2$ profiling since its invention by [15].\",\n \"The generalised version of the SCIDAR made it possible to detect turbulence near the ground [10], [4].\",\n \"A more recent implementation of the Generalized SCIDAR on a 40-cm telescope that used a widely-separated double star as a light source and an Electron Multiplying Charge Coupled Device (EMCCD) as the detector, gave place to the Low Layer SCIDAR (LOLAS) [1].\",\n \"The LOLAS prototype was used to characterize the ground-layer turbulence at Mauna Kea, together with a Slope Detection and Ranging instrument [8].\",\n \"Although this several-years campaign gave definitive results, the experience showed that a number of aspects on the prototype LOLAS could be improved in order to optimize data acquisition.\",\n \"In this paper we describe the development and tests of the second generation Low Layer Scidar (LOLAS-2).\",\n \"A few instruments based on optical methods have been developed to measure $C_{\\\\mathrm {N}}^2(h)$ profiles with the required vertical resolution close to the ground.\",\n \"For example, the surface-layer Slope Detection and Ranging (SL-SLODAR) [12] makes use of two Shack-Hartmann wavefront sensors, each looking at one component of a widely-separated double star.\",\n \"The slope of the wavefronts coming from each star is measured on 5-cm square subapertures whereas LOLAS-2 measures scintillation on square elements of 1-cm side approximately, which makes the SL-SLODAR five times more sensitive than the LOLAS-2 for equal integration times or five times faster for equal number of photons per element.\",\n \"On the other hand, if the instruments were using the same double star separation, LOLAS-2 provides 5 times better altitude resolution.\",\n \"A similar concept that also uses a double star but measures the scintillation from each component on two cameras was implemented in the Stereo-Scidar by [17].\",\n \"The Lunar Scintillometer, which consists of an array of photodiodes that measure scintillation from the moon, has been used to obtain high altitude-resolution turbulence profiles near the ground [19].\",\n \"[9] use a Generalized SCIDAR to distinguish layers at very similar altitudes but moving at different velocities.\",\n \"The present paper is organized as follows: § gives an overview of the LOLAS method.\",\n \"In § REF we briefly describe the prototype version of the instrument and in § REF the second generation of the instrument is presented.\",\n \"Tests on the instrument performances and some measurements are shown in § and § .\",\n \"Conclusions are given in § .\"\n ],\n [\n \"Method\",\n \"The Low Layer Scidar is based on the principle of the Generalized Scintillation Detection and Ranging (G-SCIDAR) technique [15], [10], [4].\",\n \"In this section, we present a brief overview of the LOLAS method, since a complete description can be found in [1].\",\n \"The G-SCIDAR principle can be summarized as follows: double-star scintillation patterns are recorded on short exposure-time images on a virtual plane located a distance $h_\\\\mathrm {gs}$ from the telescope pupil.\",\n \"In G-SCIDAR experiments this virtual plane is located below the telescope pupil, thus $h_\\\\mathrm {gs}<0$ .\",\n \"A schematic view of the optical setup is shown in Fig.\",\n \"REF .\",\n \"Each image consists of a randomly distributed intensity pattern.\",\n \"The autocovariance of this stochastic illumination is obtained by computing the spatial normalized autocorrelation of each image and averaging those autocorrelations over thousands of statistically independent image samples.\",\n \"Each layer of optical turbulence contributes to the resulting scintillation autocovariance with three covariance peaks: one centred at the autocovariance origin and two others, identical to each other, separated from the origin by $\\\\mathbf {d}_\\\\mathrm {l}=-\\\\rho \\\\vert h-h_\\\\mathrm {gs}\\\\vert $ and $\\\\mathbf {d}_\\\\mathrm {r}=\\\\rho \\\\vert h-h_\\\\mathrm {gs} \\\\vert $ , respectively, where $\\\\rho $ denotes the angular separation of the double star and $h$ the layer altitude above the ground.\",\n \"Knowing $\\\\rho $ and the conjugation altitude $h_\\\\mathrm {gs}$ , the experimental determination of $\\\\mathbf {d}_\\\\mathrm {l}$ (or $\\\\mathbf {d}_\\\\mathrm {r}$ ) leads to an estimate of the layer altitude $h$ .\",\n \"The measured autocovariance peaks $C_\\\\mathrm {gs}(\\\\mathrm {\\\\mathbf {r}})$ are proportional to the optical turbulence strength at altitude $h$ , $C_\\\\mathrm {N}^2(h)$ , and to the scintillation autocovariance function $K(\\\\mathbf {r}, \\\\vert h-h_\\\\mathrm {gs} \\\\vert )$ .\",\n \"In the realistic case of multiple layers, the response of each layer adds up to the measured autocovariance, resulting in Eq.\",\n \"1 of [1].\",\n \"This equation consists of an integral with respect the altitude $h$ that is similar to a convolution, except that the kernel $K(\\\\mathbf {r}, \\\\vert h-h_\\\\mathrm {gs} \\\\vert )$ depends on the integration variable.\",\n \"One needs to invert this integral equation to retrieve $C_\\\\mathrm {N}^2(h)$ .\",\n \"Due to its similarity with a convolution integral, we developed an algorithm based on the CLEAN method, but in which the kernel is recalculated for each different value of $h$ .\",\n \"Our modified CLEAN algorithm is based on that reported by [13].\",\n \"As explained by [1], the achievable altitude resolution using the modified CLEAN method, when the target is at the zenith is $\\\\Delta h=0.52\\\\sqrt{\\\\lambda (\\\\vert h-h_\\\\mathrm {gs} \\\\vert )}/\\\\rho $ , where $\\\\lambda $ is the wavelength and $\\\\rho =\\\\vert \\\\rho \\\\vert $ .\",\n \"When the star is located at an elevation angle $\\\\theta $ , $\\\\Delta h$ is decreased by a factor $\\\\cos \\\\theta $ .\",\n \"We take $\\\\lambda =0.5 \\\\mu \\\\mathrm {m}$ , which corresponds to the maximum sensitivity of our detector.\",\n \"For a telescope having an aperture $D$ , the maximum altitude for which the $C_{\\\\mathrm {N}}^2$ value can be measured is given by $h_\\\\mathrm {max}=D/\\\\rho $ [1].\",\n \"The Low Layer Scidar concept consists of putting into practice a G-SCIDAR on a dedicated portable telescope, using widely separated double stars as light sources.\",\n \"As can be seen from the above expressions for $\\\\Delta h$ and $h_\\\\mathrm {max}$ , the wider the separation, the better the altitude resolution but the shorter the maximum altitude.\",\n \"LOLAS was designed to use a 40-cm telescope, an EMCCD to improve sensitivity and a real-time computation of the scintillation autocovariance.\",\n \"Sections REF and REF describe the prototype LOLAS version and the second generation instrument, respectively.\"\n ],\n [\n \"Prototype version\",\n \"The instrumental setup of LOLAS has been widely explained elsewhere [1], [8], [6].\",\n \"We present a summary of the most important instrumental characteristics.\",\n \"Figure REF shows a schematic view of the prototype LOLAS.\",\n \"The scintillation images are obtained with a Schmidt-Cassegrain telescope of focal ratio f/10 and diameter $D = 40.64$ cm and installed on an equatorial mount, manufactured by Meade.\",\n \"The optics consists of two achromatic lenses of 50 mm focal-length.\",\n \"With this optical arrangement, the virtual analysis plane is located 1.94 km below the pupil.\",\n \"The diameter of the pupil image on the detector is $D^\\\\prime = 24.5$ mm.\",\n \"The scintillation images are captured by an EMCCD camera (Andor iXon) with $512\\\\times 512$ square pixels of $16~\\\\mu \\\\mathrm {m}$ .\",\n \"The frames are binned 2 $\\\\times $ 2, and the active zone is limited to an array of $256 \\\\times 80$ binned pixels.\",\n \"The exposure time of each frame ranges from 3 to 10 ms, depending on the wind conditions.\",\n \"The typical number of images to obtain one autocovariance is set to 30000.\",\n \"The EMCCD camera is mounted on a base that is attached to the rear of the telescope (see Fig.\",\n \"REF ).\",\n \"The same equipment, but with different optics, is used to form the SLODAR instrument.\",\n \"To switch between each instrument with the required positioning accuracy, a manual exchange mechanism was installed in front of the camera.\"\n ],\n [\n \"Second generation\",\n \"As seen in Fig.\",\n \"REF , LOLAS-2 uses a Ritchey-Chrétien open telescope of focal ratio f/9 and diameter $D = 40.64$ cm, manufactured by RC Optical Systems, a German equatorial mount (1200GTO) manufactured by Astro-Physics, an EMCCD camera (Andor iXon) to acquire scintillation images and a Sbig St-402ME camera for the finder telescope.\",\n \"One advantage of using a Ritchey-Chrétien telescope is that when the position of the focal plane is changed, the effective focal length of the telescope remains unchanged.\",\n \"Our RC Optical Systems telescope is equipped with a system that maintains the focus by monitoring the secondary mirror position in closed loop at a frequency of 6 kHz, reaching an accuracy in the secondary mirror and the focus positions of 0.6 and 25.4 $\\\\mu \\\\mathrm {m}$ , respectively.\",\n \"This control system is part of the telescope.\",\n \"In addition, the fact that the telescope is open prevents air at different temperatures to get trapped inside the tube in a turbulent convective flow, which would add an instrumental bias to the $C_{\\\\mathrm {N}}^2$ measurements at ground level.\",\n \"This was the case with the prototype LOLAS.\",\n \"Removal of the spurious turbulence from the measurements was performed in a post-processing procedure using the method described by [5], as reported by [8].\",\n \"In LOLAS-2, this post-processing step is avoided, making the data reduction faster and simpler.\",\n \"Concerning the mount, LOLAS-2 incorporates a German-type mount that reduces considerably the lever arm between the equatorial and declination axis.\",\n \"Moreover, the Astro-Physics mount has a higher stiffness and the worm gear accuracy is significantly better, compared to those of the Meade mount.\",\n \"The prototype version uses the optics of the telescope and achromatic doublets to define the spatial sampling and conjugation distance $h_{gs}$ below the pupil.\",\n \"The second generation instrument was developed so as to dispense the use of the achromatic doublets and the exchange mechanism.\",\n \"It only uses the telescope optics, making it a simpler and more robust instrument.\",\n \"The beam is no longer collimated, like in the prototype version.\",\n \"The EMCDD is placed directly a distance $L$ before the telescope focal plane.\",\n \"Distance $L$ is chosen such that the detector plane is made the conjugate of the a virtual plane located a distance $h_\\\\mathrm {gs}$ below the telescope pupil (see Fig.\",\n \"REFa) and the spatial sampling on this plane is well-suited to sample the scintillation speckles (see Fig.\",\n \"REFb).\",\n \"Using the thin lens equation, it can easily be shown that $h_\\\\mathrm {gs}$ is related to $L$ by the following expression: $h_\\\\mathrm {gs} = -\\\\frac{{F_\\\\mathrm {tel}}^2-F_\\\\mathrm {tel}L}{L},$ where $F_\\\\mathrm {tel}$ is the focal length of the telescope.\",\n \"For the spatial sampling, Fig.\",\n \"REFb illustrates the demagnification relation: $\\\\frac{\\\\mathcal {L}_{D}}{L}= \\\\frac{\\\\mathcal {L}_\\\\mathrm {min}}{F_\\\\mathrm {tel}}$ where $\\\\mathcal {L}_\\\\mathrm {min}$ represents the typical size of the smallest scintillation speckles on the pupil and $\\\\mathcal {L}_{D}$ is its corresponding size on the detector plane.\",\n \"[13] showed that the full width at half maximum of the autocovariance of the scintillation produced at altitude $h$ is given by $\\\\mathcal {L}(h)=0.78\\\\sqrt{\\\\lambda \\\\vert h-h_\\\\mathrm {gs} \\\\vert )}.$ This is, the typical size of the smallest speckle is $\\\\mathcal {L}_\\\\mathrm {min}\\\\equiv \\\\mathcal {L}(0)=0.78\\\\sqrt{\\\\lambda \\\\vert h_\\\\mathrm {gs}\\\\vert }$ .\",\n \"Solving Eq.\",\n \"REF for $\\\\mathcal {L}_{D}$ and replacing the above expression for $\\\\mathcal {L}_\\\\mathrm {min}$ gives: $\\\\mathcal {L}_{D}= L\\\\frac{0.78\\\\sqrt{\\\\lambda \\\\vert h_\\\\mathrm {gs}\\\\vert }}{F_\\\\mathrm {tel}}.$ For ground-level turbulence to be detectable, $h_\\\\mathrm {gs}$ must be smaller than $-1000$ m. A good spatial sampling of the scintillation speckles is obtained when $\\\\mathcal {L}_{D}\\\\simeq 2p$ , where $p$ is the size of the elementary sampling element.\",\n \"We chose to acquire images with the camera pixels binned $2\\\\times 2$ .\",\n \"In that case, $p=2d_\\\\mathrm {pix}$ .\",\n \"The camera pixel size is $d_\\\\mathrm {pix}=16\\\\;\\\\mu \\\\mathrm {m}$ .\",\n \"Table gives values of different parameters of interest for different values of $L$ .\",\n \"It can be seen that a good compromise is obtained when the detector is located a distance $L=11$ mm before the telescope focal plane, as the conjugation distance is large enough ($h_\\\\mathrm {gs} = -1212$ m) and the number of spatial samples per smallest speckle width is 1.8, which is very close to 2, the Nyquist criterion.\",\n \"Even though the undersampling is small, it is taken into account in the kernel of the inversion process that calculates $C_{\\\\mathrm {N}}^2$ profiles from the measured autocovariances.\",\n \"As a consequence of this analysis, the value for $L$ is set to 11 mm in LOLAS-2.\",\n \"For this value of $L$ , the distance separating two contiguous binned pixels corresponds to an angular separation of $1.8^{\\\\prime \\\\prime }$ .\"\n ],\n [\n \"Instrument performance\",\n \"In this section we present results concerning the focus stability and guiding performance obtained the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM), Baja California, Mexico.\",\n \"The data was obtained on the nights of 2013 June 15, 16 and 17.\",\n \"Table summarises the pertinent characteristics of the double-star targets used.\"\n ],\n [\n \"Focus stability\",\n \"Focus stability is extremely important to maintain the spatial sampling and conjugation altitude along data acquisition.\",\n \"A variation of the telescope focus position translates into a variation of the pupil image diameter $d$ on the detector.\",\n \"This diameter is continuously monitored during the standard data acquisition of the instrument.\",\n \"Images are sent by the EMCCD to the computer in packets of 200 consecutive 256x80 pixels frames.\",\n \"In each frame, the image is centred (as explained in §REF ) and then co-added to form a mean image made of 200 frames.\",\n \"The pupil diameters are calculated from this mean image as follows (see Fig.\",\n \"REF ): the mean image is integrated along columns to form a row of the accumulated values.\",\n \"The width at half maximum of the left and right pupils on the accumulated-values row determines their diameter in number of pixels $N_{d,l}$ and $N_{d,r}$ , respectively.\",\n \"To estimate the focus stability, the diameter of each pupil was monitored during several hours on three nights, using the same double star as source.\",\n \"The total number of pupil size measurements was 4566.\",\n \"The mean and standard deviation of the measured diameters are $\\\\left=38.13$ and $\\\\sigma _{N_{d}}=0.40$ pixels.\",\n \"The average of the temperature and its variation for each night was $13.5\\\\pm 0.22$ , $14.6\\\\pm 0.23$ and $14.1\\\\pm 0.07$ Celcius degrees, according to the weather station at SPM.\",\n \"The obtained standard deviations can be due to uncertainties in the estimation procedure and/or to actual physical variation in the focus or camera positions.\",\n \"If we consider the latter to be the cause of the measured diameter fluctuations, the consequence of the deviation of 0.40 pixels in pupil diameter implies a variation of 13 m in altitude conjugation and 0.32 $\\\\mu $ m in the sampling element size on the detector, approximately, which are tolerable values.\",\n \"This result suggests that the use of an auto-focus algorithm, which was developed for the prototype LOLAS, could be avoided in the second generation instrument, although more tests under varying temperature conditions should be performed.\"\n ],\n [\n \"Guiding test\",\n \"Maintaining a constant position of the pupil images on the EMCCD is important to correctly compute the mean image, the autocorrelation of which is used to normalize the mean autocorrelation of scintillation images, so obtaining the scintillation autocovariance.\",\n \"On wind conditions commonly encountered in astronomical observatories, telescope shake may cause pupil images to move on the detector.\",\n \"This eventual image wander is corrected for by centering the image in every single frame captured by the detector.\",\n \"The guiding performance of the telescope mount is important to keep pupil images within the useful window of the EMCCD during data acquisition that may last hours.\",\n \"The image position on the active area of the EMCCD is determined as follows: we construct an artificial reference image $I_\\\\mathrm {r}(\\\\mathbf {r})$ formed by two disks of 38 pixels in diameter each and separated from each other by the same distance as the observed double star.\",\n \"We compute the cross-correlation $C_\\\\mathrm {c}(\\\\mathbf {r})$ of the current image $I_i(\\\\mathbf {r})$ with $I_\\\\mathrm {r}(\\\\mathbf {r})$ .\",\n \"The position of the pupil images $(X,Y)$ in $I_i(\\\\mathbf {r})$ is set as the position of the maximum value of $C_\\\\mathrm {c}(\\\\mathbf {r})$ with respect the frame centre.\",\n \"In standard operation of LOLAS-2, image positions are not saved on disk.\",\n \"To test the guiding performance and image stability we recorded image positions $(X,Y)$ during several observations.\",\n \"The $(X,Y)$ coordinates on the EMCCD were rotated according to the double-star position angle to obtain image positions in right ascension ($\\\\alpha $ ) and declination ($\\\\delta $ ) coordinates system.\",\n \"This allows us to investigate separately the guiding and stability behaviour on the two rotation axis of the mount.\",\n \"Figures REF and REF show two examples.\",\n \"For the results obtained on 2013 June 17 UT (Fig.\",\n \"REF ) wind was blowing from the South-West with mean speed of 15.5 km h$^{-1}$ and a maximum value of 18.0 km h$^{-1}$ .\",\n \"The image positions remained very stable apart from a slow continuous drift on $\\\\delta $ and a slow oscillation in $\\\\alpha $ , presumably due to an inexact alignment of the mount and a periodic error on the right-ascension gear-mechanism, respectively.\",\n \"On 2013 June 15 UT wind was less benevolent, blowing from the West at a mean speed of 26.6 km h$^{-1}$ with gusts reaching 40 km h$^{-1}$ .\",\n \"Figure REF shows an example of that night.\",\n \"Even though images were moving significantly, the mean position remained constant and pupil images remained within the EMCCD working window, which enabled turbulence profiles to be measured in these conditions.\",\n \"To investigate further the dependence of image jitter on prevailing wind speed, in Fig.\",\n \"REF it is shown a plot of the standard deviation of image position as a function of wind speed.\",\n \"For each of the available 4566 frame-packets, the standard deviation $\\\\sigma _\\\\eta $ of the image position was calculated.\",\n \"The frame rate within each packet is 13-ms per frame.\",\n \"The wind conditions that prevailed at the time of each measurement were obtained from the database of the OAN-SPM weather stationwww.astrossp.unam.mx/weather15/.\",\n \"Bar extremities on Fig.\",\n \"REF indicate the 25 and 75 percentiles of the $\\\\sigma _\\\\eta $ values for a given wind speed.\",\n \"As expected, strong winds produce large image jitter, but surprisingly, $\\\\sigma _\\\\eta $ values remain lower than $5^{\\\\prime \\\\prime }$ for wind speeds lower than 30 $\\\\mathrm {ms^{-1}}$ .\",\n \"Even though the force exerted by the wind on the telescope is proportional to the wind speed, the mount and telescope structures move significantly only if the wind speed exceeds a certain value that seems to lie between 30 and 35 $\\\\mathrm {ms^{-1}}$ .\",\n \"The telescope and mount stiffness, together with the guiding performance have proven to suffice for conducting observations in moderate to strong wind conditions, without the need of an auto guiding procedure like in the prototype LOLAS.\"\n ],\n [\n \" $C_{\\\\mathrm {N}}^2$ Measurement examples\",\n \"In this section we present a few examples of measured scintillation autocorrelations and corresponding turbulence profiles.\",\n \"The first results of LOLAS-2 were obtained in 2013 November at the OAN-SPM.\",\n \"The instrument was installed on a concrete pillar (see Fig.\",\n \"REF ) to reduce vibrations.\",\n \"Figure REF shows examples of autocovariance maps obtained from 30 000 scintillation images, during the nights 2013 November 16 and 17.\",\n \"In the central peaks of Figs.\",\n \"REFa and REFc , the contribution of all turbulent layers in the atmosphere are added up.\",\n \"The correlated speckles produced by turbulent layers above $h_{\\\\mathrm {max}}$ are separated by a distance longer than the pupil diameter, thereby not giving rise to lateral peaks inside the autocovariance-map boundaries and avoiding the corresponding ${C_{\\\\mathrm {N}}^2}$ estimate.\",\n \"Only the layers below $h_{\\\\mathrm {max}}$ form visible lateral peaks from which ${C_{\\\\mathrm {N}}^2}$ values are retrieved.\",\n \"A white rectangle frames the right-hand side lateral peaks in Figs.\",\n \"REFa and REFc.\",\n \"Enlarged views are shown in Fig.\",\n \"REFb and REFd.\",\n \"The most intense central peak inside each of those frames corresponds to the turbulence at ground level.\",\n \"Weaker peaks can clearly be seen in Fig.\",\n \"REFa, which correspond to turbulence above the ground.\",\n \"Fig.\",\n \"REFd only exhibits ground-level turbulence.\",\n \"The ${C_{\\\\mathrm {N}}^2(h)}$ profiles are obtained using a modified CLEAN algorithm [1] based on the one presented by [13].\",\n \"In Fig.\",\n \"REF we present a few profiles obtained on 2013 November 16 and 17 using autocovariance maps whose examples are shown in Fig REF .\",\n \"The purpose of Fig.\",\n \"REF is only to display the ability of the instrument in measuring ${C_{\\\\mathrm {N}}^2}$ profiles and their characteristics.\",\n \"It is not intended for studying the optical turbulence at the site.\",\n \"Altitude resolution, maximum sensed altitude and noise level values are summarised in Table .\",\n \"Note that the altitude resolution obtained when using 12 Cam as a target is the best ever achieved with Scidar-like techniques.\",\n \"The noise levels indicated in Table are the $C_{\\\\mathrm {N}}^2$ values that correspond to the $3\\\\sigma $ , i.e.\",\n \"three times the standard deviation on the autocovariance-map background [1].\",\n \"The profiles corresponding to November 16 show three turbulent layers: one at ground level, the second at 81 m and the weakest at 390 m. On November 17, almost all turbulence below 400 m was concentrated at ground level.\",\n \"Only a weak turbulent layer is detected at 170 m above the ground at 11:27 UT that night.\",\n \"It is worth recalling that $C_{\\\\mathrm {N}}^2$ values obtained with Scidar techniques in which the pupil images are not superimposed on the detector, like LOLAS, do not need to be corrected for the normalization error pointed out by [2].\"\n ],\n [\n \"Conclusions\",\n \"The second generation of the Low Layer Scidar incorporates a simplified optical layout, a stiff and precise telescope mount and an open Ritchey-Chrétien telescope with excellent focus stability.\",\n \"The analysis of the image positions showed that the guiding quality of the mount permits to avoid the autoguiding algorithm that was used in the prototype LOLAS.\",\n \"Similarly, the excellent focus stability of the telescope allows for observations without the autofocus algorithm employed in the former version of the instrument.\",\n \"Examples of measurements obtained with LOLAS-2 illustrate the capability of the instrument for very high altitude-resolution profiling of the optical turbulence.\",\n \"A statistical analysis of $C_{\\\\mathrm {N}}^2(h)$ profiles obtained so far at the OAN-SPM will be presented in a forthcoming paper.\",\n \"We are deeply grateful to the staff of the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM) for their kind help in all the logistics for the observations.\",\n \"Wind data used in the work was provided by the OAN-SPM wheather station (http://tango.astrosen.unam.mx).\",\n \"Financial support was provided by DGAPA–UNAM through grants IN103913 and IN115013.\",\n \"1 lcccc Instrumental parameters for different values of $L$ 0pt Parameter $L_1$ =15 mm $\\\\mathbf {L_2=}$11 mm $L_3$ =7.5 mm $L_4$ =5 mm $h_\\\\mathrm {gs}$ [m] -886 -1212 -1777 -2668 $d$ [mm]a 1.66 1.22 0.83 0.55 $N_{d}$ [pixels]b 52 38 26 17 $\\\\mathcal {L}_{D}/p$ c 2.58 1.80 1.29 0.86 aDiameter of the pupil image on the detector plane.\",\n \"bNumber of binned pixels along a pupil image diameter (Fig.\",\n \"REF ).\",\n \"cNumber of binned pixels along a speckle (for $h=0$ ).\",\n \"lccccc Targets 0pt Name ${\\\\alpha _{2000}}$ a ${\\\\delta _{2000}}$ a ${m_{1}}$ b ${m_{2}}$ b $\\\\rho $ c [$^{\\\\prime \\\\prime }$ ] 15 Tri 2$^{\\\\mathrm {h}}$ :35 34$^{\\\\circ }$ :41 5.5 6.7 138.9 12 Cam 5$^{\\\\mathrm {h}}$ :06 58$^{\\\\circ }$ :58 5.2 6.2 181.3 a Right ascension ($\\\\alpha _{2000}$ ) and Declination ($\\\\delta _{2000}$ ).\",\n \"b Visible magnitude of each star.\",\n \"c Angular separation.\",\n \"lccccc Observational parameters 0pt Date Target $\\\\Delta h$ a ${h_{\\\\mathrm {max}}}$ b $\\\\tau $ c N. L.d 2013/11/16 15 Tri 11.7 603 3 9.5$\\\\times 10^{-16}$ 2013/11/17 12 Cam 6.3 462 2 1.3$\\\\times 10^{-15}$ a Altitude resolution [m].\",\n \"b Maximum altitude [m].\",\n \"c Exposure time [ms].\",\n \"d Noise level [$\\\\mathrm {m}^{-2/3}$ ].\"\n ]\n]"},"id":{"kind":"string","value":"1606.05217"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":9,"numItemsPerPage":100,"numTotalItems":1867602,"offset":900,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MDU2NDgxNywic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUwNTY4NDE3LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.rBWxiE2JAY9_l9OtRZK6YVqQTAhGkszK90-tNIB1raKFxFb4P7OyKb5x4LDe_zBO1fi-hwzLDxgbG-CFAkx0Cg","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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"Localisation, Communication and Networking with VLC: Challenges and\n Opportunities"
],
[
"Abstract The forthcoming Fifth Generation (5G) era raises the expectation for ubiquitous wireless connectivity to enhance human experiences in information and knowledge sharing as well as in entertainment and social interactions.",
"The promising Visible Light Communications (VLC) lies in the intersection field of optical and wireless communications, where substantial amount of new knowledge has been generated by multi-faceted investigations ranging from the understanding of optical communications and signal processing techniques to the development of disruptive networking solutions and to the exploitation of joint localisation and communications.",
"Building on these new understandings and exciting developments, this paper provides an overview on the three inter-linked research strands of VLC, namely localisation, communications and networking.",
"Advanced recent research activities are comprehensively reviewed and intriguing future research directions are actively discussed, along with the identifications of a range of challenges, both for enhancing the established applications and for stimulating the emerging applications."
],
[
"Introduction",
"We are at the dawn of an era in information and communications technology with unprecedented demand for digitalised everything, for connected everything and for automated everything [1].",
"The next decade will witness a range of disruptive changes in wireless technologies, embracing all aspects of cross-disciplinary innovations [2].",
"Fundamental challenges arise when we have reached the limit of the conventional Radio Frequency (RF) based wireless technologies, which are increasingly less capable of meeting the escalating traffic-demands and of satisfying the emerging use-cases.",
"Especially, there have been substantial research efforts dedicated to the high carrier frequencies, including the millimetre wave [3] and the visible light spectrum [4] in the forthcoming Fifth Generation (5G) wireless networks landscape.",
"In this paper, we provide an overview on the fast growing technology of Visible Light Communications (VLC), which lies at the cross-section of optical and wireless communications, and focuses on the human perceivable part of the electromagnetic spectrum, corresponding to wavelengths from 380nm to 780nm.",
"The use of the visible light spectrum for wireless communications has gained great interests.",
"This is because the visible light spectrum is licence-free, has a vast bandwidth and does not interfere with the RF band.",
"Historically, in the late 19th-century, Alexander Graham Bell invented the photo-phone by transmitting voice signals over modulated sunlight [5].",
"Almost a century later, artificial light generated by fluorescent lamps was also successfully demonstrated for supporting low data-rate communications [6].",
"It is these exciting experiments that inspired the modern VLC using Light Emitting Diodes (LEDs).",
"In addition to facilitating communications, being a modern illumination technology as their main function, LEDs have been increasingly dominating over the traditional incandescent lamps and fluorescent lamps, owing to their higher energy-efficiency, colour-rendering capability and longevity [7].",
"Hence, the potential for VLC is further supported by the anticipated presence of a ubiquitous and efficient LED lighting infrastructure.",
"The pioneering implementation of VLC using LEDs for the dual purpose of indoor illumination and communications was carried out by the Nakagawa laboratory in the early 2000s [8].",
"Subsequently, tremendous research efforts have been invested in improving the link-level performance between a single LEDs array and a single receiver, focusing on both the LED components and on the VLC transceivers, where ambitious multi Gbps targets have been achieved.",
"These exciting link-level achievements set the basis for broadening the scope of VLC research beyond point-to-point applications [9], with the focus on VLC aided networking, where various promising protocols, architectures and cross-layer solutions have been proposed.",
"Furthermore, LEDs based localisation has also attracted dedicated research interests, where recent advances have demonstrated sub-centimetre accuracy and 3D positioning capability [10].",
"In addition to the main thrust research in localisation, communications and networking, VLC can also provide innovative solutions for a number of use-cases [11], including vehicular, Device-to-Device (D2D) and underwater applications, just to name a few.",
"Along with the above technical advances, there have been significantly increased activities in the VLC domain.",
"Large scale research programmes were launched, bringing together research-led universities and industries, as exemplified by the Visible Light Communications Consortium (VLCC), the EU-FP7 project OMEGA, the consortium on ultra parallel VLC.",
"Most recently, VLC was also included in the scope for networking research beyond 5G under the framework of EU H2020.",
"Dedicated research centres have been also established, such as the Smart Lighting Engineering Research Center in US, the Li-Fi Research and Development Centre in UK and the Optical Wireless Communication and Network Centre in China, etc.",
"Furthermore, in recent years, ComSoc has published three special issues on VLC in the Communications Magazine [12], [13] and in the Wireless Communications Magazine [14].",
"Moreover, three consecutive international workshops on VLC have also been sponsored by ComSoc along with ICC'15, ICC'16 and ICC'17.",
"Meanwhile, GlobeCom has continuously offered annual workshops on optical wireless since 2010.",
"These investments and efforts have all contributed to the growing success of the subject.",
"Figure: Biannual statistics on the number of papers published in IEEE and OSA in the topic of visible light communications and positioning.",
"These data are gathered by searching for the keywords (Visible Light Communications, VLC, Visible light Positioning, VLP).Hence, we see a strong momentum on the research of VLC, as evidenced by the publication statistics in Fig REF , that motivates this special issue and our paper is organised as follows.",
"We will first introduce some basics in Section REF .",
"We will then review the achievements of three main research strands of VLC, namely localisation in Section REF , communications in Section REF and networking in Section REF .",
"After that, we will discuss about the challenges of VLC in Section REF and paint our vision on the future of VLC in Section REF .",
"Finally, we will conclude our discourse in Section .",
"The modern VLC relies on LEDs for the dual purpose of illumination and communications.",
"LEDs are solid-state semiconductor devices [7], [15], which are capable of converting electrical energy to incoherent light through the process of electro-luminescence.",
"When using LEDs for VLC, data can be readily modulated by electronically flickering the intensity of light at a rate fast enough to be imperceivable by human eyes.",
"At the receiver, data can be directly detected using inexpensive photo-detectors or imaging sensors.",
"This transceiver structure may be referred to as the Intensity Modulation (IM) and Direct Detection (DD), which is much simpler than the conventional coherent RF receivers relying on complicated and energy-consuming heterodyne structure.",
"Although there exist implementations of Laser Diodes (LDs) for VLC [16], [17] that can provide an even higher data-rate, our following discussions will be based on the more popular LEDs.",
"The primary function of LEDs is for illumination, hence the performance of VLC is naturally dependent on the specific characteristics of LEDs [18].",
"White light is the most widely used color for many applications, where there are three different methods to produce while light LEDs.",
"The most common method is the use of blue light LEDs with yellow phosphor coating layer [19].",
"This is a low-cost method, but the coating layer limits the modulation bandwidth to only tens of MHz.",
"Early experiments have demonstrated tens of Mbps data-rate [20], while hundreds of Mbps data-rate may also be achieved with advanced processing [21] and adaptive receiver [22].",
"The second method to generate white light is to carefully mix the color of Red, Green and Blue (RGB) emitted from the RGB LEDs [23].",
"This method is more expensive, but it offers much wider modulation bandwidth upto hundreds of MHz, supporting multi Gbps data-rate [24].",
"The third method is resulted from the recent advances on micro-LED array [25], which produces white light through wavelength conversion.",
"This method is capable of facilitating parallel communications with even higher data-rate [26].",
"When the receiving side is considered, there are generally two types of detectors that are often employed, namely the photo-detectors and the imaging sensors [27].",
"In particular, the Positive-Intrinsic-Negative (PIN) photo-detectors are most widely considered in current studies, owing to its low cost to implement and high sensitivity to strong incident light.",
"By contrast, in the scenario of weak incident light, avalanche photo-detectors are often more preferred.",
"On the other hand, imaging sensors are constituted by an array of photo-detectors, which inherently support the use of pixelated reception owing to its desirable capability in better separating the optical channels [28].",
"Another advantage of imaging sensors is that they can be readily integrated with camera lens in smart hand-held devices [29].",
"Since the frame-rate of the commonly used camera lens is low, this type of receiver is particularly suitable for applications requiring low date-rate [30], such as the project of CeilingCast [31].",
"Sometimes, focusing lens may also be used on top of photo-detectors [32] or imaging sensors [33] to enhance the reception by collecting more light energy from diffuse links with increased Field of View (FoV).",
"In addition to the above receivers, other interesting developments include prism-array receiver [34], solar-cell receiver [35], rotational receiver [36] and aperture-based receiver [37], etc.",
"Since the surface diameter of typical photo-detectors is several thousand times of the light wavelength, the fast fading effects are averaged out at the receiver.",
"Hence, when the indoor propagation is considered, the Line of Sight (LoS) path-loss effect is typically observed, plus the reflections from diffuse links [38].",
"In most of the studies, (dispersive) channel modelling is carried out by assuming Lambertian type emitters and for simplicity, we adopt the well-established Infra-Red (IR) channel modelling [39], with the aid of the convolution-based simulation method to properly characterize the Power Delay Profile (PDP) of the propagation channels [40].",
"This method comes at the expense of complexity, hence more efficient computational methods were proposed [41].",
"Furthermore, there are three typical sources of noise, namely the ambient noise, short noise and thermal noise.",
"In fact, they are mutually dependent, hence it is important to carry out system design by taking into account this property [42]."
],
[
"Localisation",
"The widely used Global Navigation Satellite System (GNSS) technology provides a coarse localisation capability that works just fine for the majority of outdoor applications [43].",
"However, its use for indoor localisation fails, since the satellite signal has a poor indoor penetration and the multi-path reflections are highly complex in indoor environment.",
"Various alternatives were proposed for indoor localisation services [44], mainly based on RF techniques such as Wireless Fidelity (Wi-Fi) or Ultra-Wide Band (UWB).",
"However, the associated cost is often too high to reach the desired accuracy.",
"Hence, LEDs based localisation becomes an interesting option to meet the above demand [45].",
"This is because the LEDs can be readily integrated in the existed lightening infrastructure and several nearby LEDs may provide joint localisation to achieve a very high level of accuracy.",
"Furthermore, the visible light spectrum can be used in RF sensitive environment and the imaging sensors built-in the smart hand-held devices constitute a convenient receiver for LEDs based localisation, or inexpensive PDs may be purposely installed.",
"In the future 5G systems, both manufacturing and service robots will become dominant Machine Type Communication (MTC) devices.",
"In homes, factories and warehouses, light is useful for accurate position control of robots [46], again thanks to its linear propagation property and short wavelength.",
"Explicitly, there exist several approaches for LEDs based localisation, including the proximity based approach, the triangulation based approach and the fingerprinting based approach, where each of them will be elaborated in detail as follows.",
"The simplest way to realise LEDs based localisation is the proximity based approach, where the position of the object is roughly classified to its nearest LED transmitter.",
"This procedure can be conveniently integrated in the cell search stage, where no complicated algorithms are imposed [47].",
"Hence, the achievable localisation accuracy is very coarse, in terms of decimetres.",
"When further exploiting the geometric properties of multiple LEDs, the triangulation based approach is capable of determining the absolute position of the object and reaching a localisation accuracy in terms of centimetres.",
"This approach exploits the measured distance (angles) between the multiple LEDs and the localisation object, where the measurement could be Time of Arrival (ToA) [48], Time Difference of Arrival (TDoA) [49], Received Signal Strength (RSS) [50] and Angle of Arrival (AoA) [51].",
"There are several challenges associated with these measurements that may deteriorate the localisation accuracy.",
"For example, both ToA and TDoA require strict timing synchronisation, RSS requires detailed knowledge of radiation pattern and AoA requires careful calibration of light directionality.",
"Hence, localisation based on a combination of different measurements is a good practice [52], [53], particularly for 3D localisation [54].",
"Finally, the fingerprinting based approach determines the position of the object by comparing on-line measured data with pre-stored off-line measured data [55].",
"Depending on the richness of the pre-stored data set, this approach may provide a higher level of localisation accuracy, but at the cost of an increased complexity.",
"Nevertheless, this approach has a poor scalability to use owing to its scene-dependence.",
"Unique approach exists for LEDs based localisation using camera sensors available in smart hand-held devices [56].",
"Thanks to the high density of pixels, camera sensors can extract detailed spatial information in order to determine the position of the object with high localisation accuracy by using vision analysis and image processing.",
"In addition to only use camera sensors, accelerometer build-in the smart hand-held devices can also be exploited together to achieve 3D localisation [57], [58].",
"To conclude, despite the existence of various LEDs based localisation approaches and their achievements, LoS blockage constitutes the single dominant problem, which is lacking of considerations in the current research.",
"Hence, the recent study on the effect of multi-path reflections on positioning accuracy becomes highly practical [59].",
"More importantly, when localisation is considered, it should be included in the entire design chain for achieving a better integrity [60].",
"We believe that with further scientific advances, powerful and robust solutions for LEDs based localisation will be developed to meet diverse requirements."
],
[
"Communications",
"As discussed above, VLC is capable of realising the dual purpose of illumination and communications.",
"Amongst the cascaded physical layer components of VLC, we elaborate on those that require contrived design and dedicated discussions, namely optical domain modulation and Multiple Input Multiple Output (MIMO).",
"Generally, there are two types of modulation techniques that are commonly employed for VLC, namely the single-carrier modulation and multi-carrier modulation.",
"To elaborate on the former, On-Off Keying (OOK) constitutes the simplest technique for VLC modulation.",
"Hence, it is the most commonly studied and experimented technique together with various different types of LEDs.",
"Despite its simplicity, multi Gbps data-rate has been reported in recent experiment [61].",
"Developed from the plain OOK, there is a family of pulse-based modulations, such as Pulse Width Modulation (PWM) [62] and Pulse Position Modulation (PPM) [63].",
"These modulations belong to the M-ary orthogonal signalling, which is particularly suitable for IM.",
"Unique to VLC, Color Shift Keying (CSK) has been introduced in the VLC standardisation 802.15.7 for simultaneously achieving a higher data-rate and a better dimming support [64], [65].",
"Specifically, CSK modulates the signal based on the intensity of the RGB color, relying on the employment of RGB LEDs.",
"Other interesting schemes exploiting the color property of VLC may be found in [66], [67].",
"It is worth noting that as a modulation technique, CSK is different from the concept of Wavelength Division Multiplexing (WDM) [68].",
"This is because in WDM system, additional modulation may be designed together with each of the multiplexing layers.",
"When considering the multi-carrier modulation family, the celebrated Optical Orthogonal Frequency Division Multiplexing (OOFDM) schemes are often employed.",
"This is because OOFDM scheme allows parallel data transmissions, and it is also capable of combating the detrimental channel dispersion without complex time domain equalisations.",
"Different from the conventional RF based OFDM schemes, transmitted signals of OOFDM need to be real-valued and positive in order to facilitate IM.",
"There are a family of OOFDM schemes, typically including the Asymmetrically Clipped OOFDM (ACO-OFDM) scheme and the DC-biased OOFDM (DCO-OFDM) scheme [69].",
"In both schemes, the property of Hermitian symmetry is exploited for rendering the complex-valued signal to be real-valued, at the cost of halving the bandwidth efficiency.",
"In order to maintain positivity of the transmitted signal, ACO-OFDM scheme resorts to use only the odd sub-carriers, while DCO-OFDM scheme resorts to apply sufficient DC bias.",
"Hence, the former approach is more power efficient, while the latter approach is more bandwidth efficient [70].",
"Moreover, there exist many other interesting realisations of OOFDM, such as the flip OOFDM [71], unipolar OOFDM [72], multi-layer OOFDM [73], hybrid OOFDM [74], and DC-informative OOFDM [75].",
"In general, OOFDM schemes suffer from the classic problem of high Peak to Average Power Ratio (PAPR) [76] and their performance is also limited by the clipping distortion [77] and the LEDs' non-linearity [78], where many countermeasures have thus been proposed [79], [80], [81].",
"Despite all these challenges, OOFDM schemes have attracted great attentions owing to their high data-rate potential, robustness to channel dispersion and flexibility in resource allocation [82].",
"It is important that the above modulation techniques should be co-designed with LEDs' illumination requirements to avoid flickering and support dimming [83].",
"Flickering refers to the undesirable effect of human perceivable brightness fluctuation, or in other words light intensity changes.",
"This is relatively easy to cope with by using Run Length Limited (RLL) coding in order to balance the resultant zeros and ones [84], [85].",
"On the other hand, modern LEDs have now been capable of supporting arbitrary levels of dimming for energy saving and environmental-friendly illumination.",
"Hence, a more intriguing aspect is to jointly design modulation and dimming [86].",
"In general, there are two approaches to support dimming with modulation, namely to control either the light intensity or the signal duty cycle, where the former is easier to implement and the latter achieves higher precision [87].",
"As an example, in OOK, its on or off levels can be defined to support dimming or one may introduce compensation period along with the signalling period without resorting to intensity modification.",
"Amongst others, pulse-based modulations show great flexibility in support dimming, such as the variable PPM scheme proposed in the 802.15.7 standard [88].",
"On the other hand, dimming support in multi-carrier modulation requires further investigations, where recent research has been dedicated to this direction [89], [90].",
"In addition to modulation, channel coding schemes can also be co-designed with dimming support in mind, as demonstrated in various research efforts, including turbo codes [91], Reed-Muller codes [92], adaptive codes [93] and concatenated codes [94].",
"Similar to the conventional RF based wireless communications systems, MIMO in VLC is also capable of providing promising throughput enhancements [95].",
"It is known that the full potential of MIMO can only be achieved in a fully scattered environment.",
"However, in VLC, the particular challenge is that the optical MIMO channels are often highly correlated and the resultant MIMO channel matrix appears to be rank deficient [96].",
"To create full rank MIMO channel matrix, it is crucial to maintain clear separation of the transmitters at the (non)-imaging receiver by careful calibration.",
"In addition to support full rank MIMO channel matrix, robustness to receiver tilts and blockage is also desirable, where it has been shown that specifically designed receivers to harness angle diversity constitutes a good design practice [97], [98].",
"Most recently, the research on MIMO in VLC under diffuse channel is also emerging, leading to substantial practical insights [99].",
"As far as the MIMO functions are considered, MIMO in VLC can achieve diversity gain by using space time coding [100], multiplexing gain by using parallel transmission [101] and beamforming gain by using electronic beam steering [102].",
"Multiple transmit luminaries can also be used to improve the security of VLC transmission by making the VLC signal difficult to intercept by an eavesdropper.",
"Several linear beam-forming schemes for active and passive eavesdroppers have recently been presented in [103], [104], [105].",
"Being an important generalisation, Multiple Input Single Output (MISO) transmission and in particular, Multi-User MISO (MU-MISO) transmission have attracted substantial research interests [106].",
"This is because the MU-MISO scheme provides beneficial multi-user diversity gain without incurring rank deficient MIMO channel matrix.",
"However, in this multi-user scenario, challenges arise when performing inter-user interference cancellation, where (non)-linear transmit pre-coding scheme is required."
],
[
"Networking",
"The above mentioned advances in physical layer research of VLC have lead to the development of VLC aided networking [107].",
"Being next to the physical layer, there are several different candidates for the Medium Access Control (MAC) layer, including both the multiple access and random access schemes.",
"Let us now elaborate on the multiple access scheme first, the most straightforward arrangement is the Time Division Multiple Access (TDMA) scheme [108], [109], where users simply access the network in different time slots.",
"When multi-carrier modulation is employed, the Orthogonal Frequency Division Multiple Access (OFDMA) scheme allows users to be allocated different Time and Frequency (TF) resource blocks [110].",
"When compared to TDMA scheme, OFDMA scheme provides a higher flexibility in terms of resource allocation and user scheduling, at a modestly increased complexity.",
"In addition to the above orthogonal multiple access schemes, in Non-orthogonal Multiple Access (NOMA) scheme [111], [112], two or more users may be multiplexed in power domain in addition to the conventional orthogonal TF domain.",
"At the receiver side, onion-stripping type interference cancellation is required to separate the users from power-domain non-orthogonal multiplexing [113].",
"Other than relying on the power-domain, spatial domain could also be exploited at the transmitter to realise NOMA [114].",
"Differently, (multi-carrier) Optical Code Division Multiple Access (OCDMA) scheme relies on assigning each user a unique and specific optical code [115], [116].",
"Finally, when random access is considered, the classic Carrier Sense Multiple Access (CSMA) scheme remains highly attractive.",
"Importantly, early implementations have already shown its successful usage [117] and in slotted random access, both contention access periods and free periods are included, where the latter ensures guaranteed time slots for resource limited applications.",
"The most straightforward way of constructing an indoor VLC cell is to simply consider each Access Point (AP) function as an individual cell and to adopt the unity frequency reuse across all cells.",
"This construction would result in the highest spatial reuse but it tends to suffer from the typical problem of Inter-Cell Interference (ICI) amongst neighbouring cells.",
"Following the traditional cellular design principle [118], different (fractional) frequency reuse patterns could be used to mitigate ICI, at the cost of a reduced bandwidth efficiency.",
"An effective method of improving the efficiency, whilst mitigating the detrimental ICI is to employ cell merging [119], where a group of neighbouring APs jointly form an enlarged VLC cluster.",
"In this way, the previously ICI-contaminated area becomes the cluster-centre of the newly formed cluster.",
"Multiple users can be served simultaneously by using sophisticated Vectored Transmission (VT) techniques.",
"The underlying principle is to totally eliminate the ICI at the multiple transmitters side by using transmit pre-coding, so that the multiple users receive mutually interference-free signals [120].",
"However, this technique requires that both Channel State Information (CSI) and the users' data have to be shared amongst multiple APs.",
"All of the above cell formations follow the conventional cell-centric design approach, which is based on defining a cell constituted by a fixed set of one or more APs and then associating the users with it.",
"By contrast, the newly proposed user-centric design approach relies on the dynamic association between APs and users [121].",
"More explicitly, by taking into account the users' geo-locations, the new user-centric design flow is based on grouping the users together and then associating the APs with them, leading to amorphous cells [122], which is capable of supporting video service on the move [123].",
"Finally, an intriguing piece of research emerges to consider the optimal placement of LEDs [124], resulting into rich implications on throughput, delay and mobility.",
"Holistically, owing to the existence of lightening infrastructure, Power-Line Communications (PLC) constitute a convenient back-haul for VLC as indoor access technology [125], [126].",
"PLC can reach luminaries that serve as VLC transmitters to supply the data streams as well as to coordinate transmission between multiple VLC transmitters to support multi-user broadcasting [127].",
"The VLC transceivers can be considered as relays that can operate in a full-duplex model and different relaying paradigms such as amplify-and-forward and decode-and-forward are possible [125].",
"From a networking perspective, VLC can be considered as a new member in the small-cell family of the Heterogeneous Networks (HetNet) landscape for complementing the over-loaded Radio Access Technology (RAT).",
"Indeed, the interplay between VLC and RAT system has been an active area of research [128], [129], where there are two different interplay scenarios that may be envisioned, namely single-homing and multi-homing.",
"In the single-homing scenario, only one access system is allowed to maintain its association at any instant.",
"In this scenario, dynamic Load Balancing (LB) will prevent traffic congestion caused by blockage or mobility through diverting the traffic flow appropriately [130].",
"To better exploit the access system's diversity potentials, in the multi-homing scenario, each user maintains multiple associations at the same time by using the Multipath Transport Control Protocol (TCP) to connect multiple interfaces [131].",
"In either scenario, robust vertical handover has to be properly designed to mitigate any ping-pong effect, where the load-ware mobility management appears to be a promising solution [132].",
"Different from using higher layer converging approaches, seamless rate adaptation between VLC and RAT systems may also be achieved by network coding and rate-less coding [133].",
"To sum up, in the light of the information and communications technology convergence, the above mentioned network layer functions should be soft-ware defined to maximise its full potential [134], [135]."
],
[
"Channel Modelling",
"Most of the current channel modelling in VLC was directly adapted from the IR communications.",
"However, it would be ideal to develop specific VLC channel modelling, corresponding to different types of LEDs for the ease of cross-calibration.",
"In particular, with regards to shadowing, there is a lack of both empirical and statistical modelling.",
"In most of the studies, the shadowing effect was often assumed to follow the over-simplified Bernoulli distribution.",
"However, given the sensitivity of VLC over shadowing in all aspects of localisation, communications and networking, a well-calibrated model is indeed of critical importance.",
"Also importantly, VLC channel modelling for vehicular applications is still in its infancy, which requires dedicated efforts."
],
[
"Interference Mitigation and Management",
"VLC relies on visible light spectrum which overlaps with solar radiation and indoor/outdoor lighting and display.",
"A VLC system is inevitably interfered by those sources, from day and night time.",
"Therefore, effective techniques are needed to mitigate not only inter-system interference from neighbouring LEDs, but also external light interference.",
"Considering those interference sources typically emit a large dynamic range of light intensities, a robust and sensitive detector needs to respond reliably to transmitted signals while suppressing interference to certain extent.",
"It is necessary but very challenging to develop advanced methods and algorithms to recover useful signals, weak or strong, from noisy signals overwhelmed by interference, weak or strong.",
"For example, normal operation of a vehicular receiver or an indoor receiver under direct sunlight through a window at noon is extremely difficult."
],
[
"Channel Feedback",
"Several contributions make a massive use of channel state information.",
"While this aspect is not critical in approaches that move all the processing toward the receiver since estimation techniques are not so difficult to implement both with training-based or (semi)-blind, when the communications requires a feedback link, as for pre-equalization, bit loading, retransmission schemes, the feedback channel performance becomes an issue.",
"The problem of how a feedback channel is used is a very challenging task either if it is an optical one and if it is based on RF.",
"In fact, important aspects for example quantisation and error control must be properly taken into account, in addition to delay and jitter."
],
[
"Access Techniques",
"A high percentage of access techniques proposed in the literature and described above start from the tentative of adapting access mechanisms and procedures already used in the RF for implementing them into the VLC context.",
"However, mechanisms that take into account the fact that VLC receivers usually have one or more photo-detectors that are not omnidirectional as well as for emerging VLC applications such as vehicular and camera communications, should be investigated.",
"Access techniques become more complicated, when a user is covered by some LEDs in indoor environment, due to the tilt of the device, the signal can be received from another set of LEDs.",
"Hence, access techniques are strictly related also to the localisation techniques."
],
[
"Mobility Management",
"In many VLC applications, VLC terminals or transceivers are mobile, such as hand-held devices, vehicles, and robots.",
"Maintaining a quality communications link in a point-to-point case and network connectivity in a multiple-user case are important to avoid communications losses.",
"Protocols for mobile optical wireless communication networks are in urgent need, for dynamic system resource adaptation to communications environments, easy node access and drop-off, smooth handover from one AP to another and from one network to another.",
"Meanwhile, mobility prediction and network topology modelling are under-explored, and these topics open new room for further investigation."
],
[
"Integrated Smart Lighting",
"Since VLC is not a paradigm separated from illumination, the ultimate future challenge can be a full integration that takes into account not only communications performance related to the single link under illumination constraints but also a holistic optimisation regarding both the networking (system performance and seamless handover) and the perceived illumination (a good level of light all over the room).",
"This is an issue especially in large rooms such as museums, where multimedia content may be based on the position of the user in the information centric system.",
"Hence, the aim is to integrate lighting, communications, networking and positioning in a single homogeneous framework."
],
[
"5G-Home",
"The concept of 5G-Home is based on the unified heterogeneous access-home network with wired and (optical) wireless connectivity, which is capable of creating tens of millions of additional 5G-Home sites.",
"Technically, to succeed in a timely and affordable manner, both the existing copper or cable based fixed access and in-home wiring technique need to be exploited in addition to fibre [136], [137].",
"Deep in the home, the number of wireless APs and terminals will be going up corresponding to the increasing bandwidth, delay and coverage requirements.",
"At least one AP per room may become the norm given the higher carrier frequencies that will be used for 5G, including Wi-Fi standards 802.11ac/ad and VLC aided networking [138], [139].",
"Hence, we envision beneficial convergence between fixed access network infrastructure and in-home (optical) wireless networks, in order to eliminate the boundary between these two domains [140]."
],
[
"5G-Vehicle",
"Future 5G systems will embody a number of new applications which span across vast areas beyond enhanced mobile broadband services, such as media distribution, Smart Cities, and Internet of Things (IoT).",
"In Smart Cities, light-enabled vehicular communications networks utilize a large number of densely distributed street lamp poles as APs, while vehicular lights, roadside street lights, pedestrian signage and traffic lights can all act as ubiquitous transmitters to meet the need of massive connectivity and high throughput [141], [142].",
"Equipped with image sensor or other types of receivers, these nodes in the immediate vicinity of vehicles provide ultra-reliable and low latency (mission critical) communications and control links [143], [144], which will serve well intelligent transportation, collision avoidance, autonomous driving and telematics.",
"However, challenges will arise when experiencing strong ambient noise, while uneven speed and diverting route may be also difficult to handle [145], [146].",
"Nevertheless, vehicles become mobile offices and entertaining homes, enjoying amount of exterior resources."
],
[
"Underwater Communications",
"In addition to ground-based indoor and outdoor applications, VLC also paves the way for outreach applications deep into water.",
"Currently, underwater acoustics is the major wireless technology for underwater object detection and communications.",
"Due to the very low data rate allowed by acoustic communications in underwater environment and also the poor performance from the propagation point of view in the RF bands, light sources offer a unique opportunity for short-range high speed communications potentially at low cost and low power [147], [148], [149], [150].",
"Data rate of hundreds to thousands of Mbps is possible, and the communications distance is extendible to hundreds of meters in clear water.",
"The VLC technology will undoubtedly play a critical role in marine resource exploration, water sensing, monitoring and tracking of marine organism, and saving endangered species."
],
[
"Emerging Applications",
"LEDs are natural cheap communications transmitters, and CMOS image sensors are equipped in pervasive consumer electronics as detectors.",
"New applications emerge with these tiny optical sensors.",
"Image Sensor Communications (ISC) and positioning is easily realized based on CMOS sensors built upon smart phones.",
"Millions of pixels can be maximally explored for communications [151] and accurate positioning [152].",
"The LED screen-to-camera communications constitute another interesting application, which may facilitate unimaginable potentials in entertainment, education, broadcasting and near-field communications, etc.",
"Several early experiments have been already carried out with exciting findings, including projects of SoftLight [153], Uber-in-light [154] and SBVLC [155] etc.",
"In addition to traditional LEDs, emerging Organic LEDs (OLEDs) are attractive in flexibility, easy integration and fabrication, convenient color selectivity, and wide viewing angle.",
"Thus they serve as wearable and portable VLC transmitters as well as fixed-location large screen communication transmitters [156], [157], [158].",
"Application domains encompass those of LEDs as well as new body area networks and sensor networks.",
"Last but not least, LED based lighting infrastructures are becoming ubiquitous and have already been dubbed as the `eyes and ears of the IoT' in the context of smart lighting systems.",
"Hence, VLC is a promising solution for indoor communications to a broad class of smart objects in particular in scenarios with high density of IoT devices.",
"Steps along this direction have been presented in [159], [160], [161]."
],
[
"Commercialisation",
"To create a larger economic and societal impact, the above mentioned academic research has to be in collaboration with industries.",
"More importantly, the future success of VLC urges the joint efforts from both information industry and illumination industry, driven by demands from verticals, such as health, automotive, manufacturing, entertainment etc.",
"The inclusive of mobile phones within the ecosystem would be highly desirable to the wide public acceptance.",
"One promising area for real-world VLC to grow is the future pervasive IoT in consumer electronics, retailers, warehouses, offices and hospitals, etc.",
"For example, Philips has recently commercialised VLC aided indoor localisation for the hypermarket in France http://www.lighting.philips.co.uk/systems/themes/led-based-indoor-positioning.html.",
"They have also teamed up with Cisco on an exciting IoT project called `Digital Ceiling' http://www.cisco.com/c/en/us/solutions/digital-ceiling/partner-ecosystem.html, which connects all of a building's services in a single and converged network, where empowered LEDs can be used to collect, send and analyse data."
],
[
"Conclusion",
"This paper provides an overview on the localization, communications and networking aspects of VLC, along with the discussions on various challenges and opportunities.",
"It is envisioned that the future research of VLC will open up new scientific areas in the wider academic context, which will be extremely beneficial to the whole community.",
"VLC will create a unique opportunity to innovate all areas related to the future evolution, deployment and operation of ultra-dense small-cell networks, where potentially hundreds of people and thousands of appliances will be connected.",
"More broadly, VLC will stimulate commercial solutions for supporting new killer applications, facilitating innovations for entertainment, collaborative design and vehicular networking, etc.",
"By simultaneously exploiting illumination and communications, VLC will directly contribute towards the all-important `green' agenda, which has been one of the salient topics in the 21st-century."
]
] | 1709.01899 |
[
[
"Quasi-normal modes of black holes in scalar-tensor theories with\n non-minimal derivative couplings"
],
[
"Abstract We study the quasi-normal modes of asymptotically anti-de Sitter black holes in a class of shift-symmetric Horndeski theories where a gravitational scalar is derivatively coupled to the Einstein tensor.",
"The space-time differs from exact Schwarzschild-anti-de Sitter, resulting in a different effective potential for the quasi-normal modes and a different spectrum.",
"We numerically compute this spectrum for a massless test scalar coupled both minimally to the metric, and non-minimally to the gravitational scalar.",
"We find interesting differences from the Schwarzschild-anti-de Sitter black hole found in general relativity."
],
[
"Introduction",
"The mysterious nature of dark energy [1] has galvanized a recent theoretical study of alternative gravity theories as one potential driving mechanism for the acceleration of the cosmic expansion.",
"The search for new and phenomenologically interesting theories has led to a proliferation of scalar-tensor extensions of general relativity (GR) [2], [3], [4], [5], [6].",
"Many of these differ from the classical theories of modified gravity (such as Brans-Dickie) in that they include higher-derivative interactions, yet they are free of any Ostrogradski ghost instabilities because the equations of motion are second-order.",
"These theories have received particular attention because they can self-accelerate cosmologically whilst simultaneously satisfying solar system tests of gravity by utilising the Vainshtein screening mechanism [7], [8], [9], [10], [11], [12], which uses non-linearities in the field equations to suppress deviations from GR.",
"Any scalar-tensor theory that has second-order equations of motion falls into the class of theories first derived by Horndeski [13] and independently re-derived by [14], [15], [16].",
"This class is defined by four free functions of a the scalar $\\varphi $ and its kinetic energy $X=-g^{\\mu \\nu }\\partial _\\mu \\varphi \\partial _\\nu \\varphi /2$ and a set of essential building blocks.",
"Such an expansive theory has found use in a variety of cosmological and astrophysical scenarios from inflation [17], [18] to dark energy [19], [20], [21], [22], [23] to neutron stars [24], [25], [26], [27] and other astrophysical objects [28], [29], [30], [31], [32], [33], [34], [35].",
"The enormous freedom in constructing models has enabled several examples of black holes (BHs) with scalar hair to be found.",
"These circumvent the no-hair theorem [36], [37], [38] because it was derived assuming only first-order derivatives of the scalar and non-derivative couplings to curvature tensors (a no-hair theorem has been proved for asymptotic BHs in shift-symmetric Horndeski theories [39] with one loophole [40], [41], [42]).",
"A comprehensive and systematic review of hairy solutions in Horndeski theories as well as how to construct them can be found in reference [43].",
"In this work, we are concerned with the specific theory with a non-minimal derivative coupling of the scalar to the graviton $S=&\\int \\mathrm {d}^4 x\\sqrt{-g}\\left[\\frac{m_p^2}{2}\\left(R-2\\Lambda \\right)\\right.\\nonumber \\\\&\\left.-\\frac{1}{2}\\left(g^{\\mu \\nu }-\\frac{z}{m_p^2}G^{\\mu \\nu }\\right)\\partial _\\mu \\varphi \\partial _\\nu \\varphi \\right],$ which has been well-studied in the literatureThis specific theory does not include a screening mechanism but passes solar system tests nonetheless since derivatively coupled scalars only source scalar field gradients through their (weak) cosmological dynamics [44], [45], [46], [47]..",
"Here $m_p$ is the reduced Planck mass, $R$ is Ricci scalar and $G^{\\mu \\nu }$ is the Einstein tensor.",
"In particular, in the absence of the canonical kinetic term, this theory is a specific example of the John class of fab-four theories, which can self-tune away a large cosmological constant [48].",
"The parameter $z$ is a free coupling constant and $\\Lambda $ is a bare cosmological constant.",
"Many cosmological and astrophysical scenarios have been studied in this theory, including inflation [49], dark matter [50], neutron stars [51], etc.",
"Besides, it has been shown that this theory admits hairy BHs that are asymptotically anti-de Sitter (AdS) [52], [53], [54], [55].",
"In the original construction [52], the bare cosmological constant was absent and the scalar derivative $\\varphi ^{\\prime 2}<0$ outside the horizon.",
"Here prime is the derivative with the radial coordinate in the Schwarzschild system.",
"This is problematic since it violates the null energy condition and $\\varphi $ is ultimately coupled to matter.",
"Later, [53], [54], [55] showed that this pathology could be ameliorated by including a bare cosmological constant.",
"Whilst not particularly relevant for cosmology, the study of AdS BHs is especially important for the AdS/CFT correspondence [56], [57], [58].",
"Large AdS BHs describe (approximate) thermal states of the boundary CFT and it may be the case that AdS BHs in these theories are dual to an interesting strongly coupled three-dimensional gauge theory.",
"Similarly, the decay of a scalar outside the BH—quasi-normal modes (QNMs)—corresponds to perturbations of these states, and contain information about the time-scale for the system to reach equilibrium [59], [60], [61], [62], [63], [64].",
"In particular, the QNMs of AdS BHs correspond to poles of the retarded Green's function for the boundary CFT; we refer the reader to [63] and references therein for the applications of this to hydrodynamic systems.",
"Motivated by this, Minamitsuji has numerically calculated the fundamental QNM for a massless test scalar outside an AdS BH for this theory [65].",
"The purpose of this work is two-fold.",
"First, we extend this calculation to the higher overtones and non-radial modes.",
"Second, we calculate the QNMs for the case where the scalar is non-minimally coupled to the gravitational scalar $\\varphi $ ; we investigate the lowest-order coupling that preserves the symmetries of $\\varphi $ and test scalar.",
"In the former case, we find qualitatively similar behaviour as the fundamental QNMs calculated by reference [65].",
"In the latter case, we find that there is a critical value of the non-minimal coupling below which the effective potential has a different behaviour at asymptotic infinity so that the QNMs are not well-defined.",
"We numerically calculate the QNMs for parameter choices where this is not the case and find that stronger non-minimal couplings increase the oscillation period and decay rate of the QNMs (at fixed BH horizon and derivative coupling constant).",
"This paper is organized as follows: in section we introduce the specific BH studied in this work.",
"The QNMs are calculated and discussed in section (for both the minimal and non-minimal coupling) before concluding in section ."
],
[
"AdS black holes in derivatively-coupled theories",
"The theory defined by the action (REF ) admits AdS BH solutions of the form [52], [53], [54], [55] $ds^2=-F(r)dt^2+\\frac{h^2(r)}{F(r)}dr^2+r^2(d\\theta ^2+\\sin ^2\\theta d\\phi ^2),$ with $F(r)&=1-\\frac{2M}{r}+\\frac{r^2}{l^2}\\nonumber +\\frac{(3z-m_p^2l^2)^2}{12zm_p^2l^2}\\frac{\\arctan (m_pr/\\sqrt{z})}{m_pr/\\sqrt{z}},\\nonumber \\\\h(r)&=\\frac{z(6m_p^2r^2+m_p^2l^2+3z)}{\\sqrt{12z}m_pl(m_p^2r^2+z)},\\nonumber \\\\\\varphi ^{\\prime 2}(r)&=\\frac{4m_p^6r^2(3z-m_p^2l^2)}{z(m_p^2r^2+z)(3z+m_p^2l^2)}\\frac{h^2(r)}{F(r)},$ where the AdS length $l$ is related to the coupling constant $z$ and the cosmological constant $\\Lambda $ viaNote that it is not possible to choose a value of $\\Lambda $ such that the solution is an asymptotically de Sitter (dS) BH.",
"Such a choice cannot lead to the formation of a cosmological horizon.",
"One could choose $z<0$ , but this results in a naked curvature singularity that is not hidden behind a horizon [65].",
"$l^2=\\frac{3z(3m_p^2+z\\Lambda )}{m_p^2(m_p^2-z\\Lambda )}.$ The BH horizon radius $r_h$ is the only real solution of $F(r_h)=0$ , and the Hawking temperature of this horizon is $T=\\frac{F^{\\prime }(r_h)}{4\\pi h(r_h)}=\\frac{6m_p^2r_h^2+m_p^2l^2+3z}{8\\sqrt{3}\\pi z m_p l r_h}.$ In order for the solution for $\\varphi $ to be real, we need to impose $z\\ge m_p^2l^2/3.$ When this lower bound is saturated, one has $\\varphi ^{\\prime }(r)=0$ , $h(r)=1$ , and the $\\arctan $ term in $F(r)$ vanishes.",
"Therefore, we have an exact Schwarzschild-anti-de Sitter (SAdS) BHIt is important to note that the cosmological constant for this black hole differs from $\\Lambda $ (see Eqn.",
"(REF )) so that this still represents a non-GR solution.",
"In particular, one would expect metric perturbations to differ from their GR counterparts.",
"In the limit $z\\rightarrow 0$ the theory reduces to GR, in which case the vacuum solution is an SAdS black hole with AdS length set by $\\Lambda $ .",
"Since SAdS BHs are solutions of both theories, they are observationally indeistinguishable if one only considers their static, stationary properties, but their different dynamics, such as metric perturbations and interaction with matter, can be used to distingish between the two theories..",
"In the following, we will first study the QNMs for this case and compare them with known results in the literature [62], [65] as a test of our numerical procedure.",
"We will then consider more general values of $z$ where $\\varphi ^{\\prime }(r)$ is nonzero and the BH deviates from exact SAdS, as well as non-minimal couplings of $\\varphi $ to the test scalar.",
"In what follows, we will work in units where $m_p=1$ .",
"Furthermore, we will rescale our distances so that $l=1$ i.e.",
"$r$ and $M$ both have units of $l$ ."
],
[
"Minimal Coupling",
"We first consider a test scalar field $\\Phi $ , minimally coupled to the metric but not to $\\varphi $ .",
"This is the simplest situation one can envision and we will henceforth refer to it as the minimally coupled case.",
"This is in contrast to the case where one has direct couplings between $\\varphi $ and $\\Phi $ , which we will refer to as the non-minimally coupled case.",
"The minimally-coupled Lagrangian is $L_{\\rm MC}=-\\frac{1}{2}\\sqrt{-g}\\partial _{\\mu }\\Phi \\partial ^{\\mu }\\Phi ,$ and, ignoring the back-reaction of $\\Phi $ on the spacetime, the metric is given by Eq.",
"(REF ) so that the equation of motion of $\\Phi $ is $\\Box \\Phi =0.$ For our static and spherically symmetric background, one can separate the dependence on coordinates as $\\Phi =\\frac{1}{r}\\psi (r)Y_j^m(\\theta ,\\phi )e^{-i\\omega t}$ where $Y_j^m(\\theta ,\\phi )$ are the usual spherical harmonics with degree $j$ and order $m$ .",
"Defining $f(r)=F(r)/h(r),$ and introducing the tortoise coordinate $r^*$ given by $dr^*=dr/f(r)$ , Eq.",
"(REF ) can be written in a similar form to the Schrödinger equation: $\\frac{d^2\\psi }{d{r^*}^2}+(\\omega ^2-V(r))\\psi =0,$ where the effective potential is $V(r)=\\frac{f(r)f^{\\prime }(r)}{r}+j(j+1)\\frac{h(r)f(r)}{r^2}.$ Figure: Effective potential for a scalar perturbation given in Eq.",
"().",
"The tortoise coordinate r * r^* takes values from -∞\\infty to π/2\\pi /2.",
"z=1/3z=1/3 and j=0j=0.We plot $V(r)$ for large and small BHs for the SAdS case $z=1/3$ in Fig.",
"REF .",
"Consider perturbations outside the BH, i.e.",
"$r_h<r<\\infty $ .",
"From Eqs.",
"(REF ) and (REF ), we see that $f\\approx 4\\pi T(r-r_h)$ as $r\\rightarrow r_h$ , and $f\\approx C_1 r^2$ as $r\\rightarrow \\infty $ .",
"$C_1$ is a positive constant.",
"From the definition of the tortoise coordinate, we find $r^*&\\approx &\\frac{1}{4\\pi T}\\ln (r-r_h),\\quad r\\rightarrow r_h;\\nonumber \\\\r^*&\\approx &C_2-\\frac{1}{C_1 r},\\quad r\\rightarrow \\infty .$ Here $C_2$ is an integration constant which can be freely chosen.",
"We set it to $\\pi /2$ .",
"Clearly, $r^*$ tends to $-\\infty $ as $r$ approaches $r_h$ so this coordinate takes values in the range $-\\infty <r^*<\\pi /2$ .",
"It is evident that, for any value of $z$ , $V(r)$ vanishes as $r^*$ goes to $-\\infty $ (as $r$ approaches $r_h$ ), and $V(r)$ diverges as $r^*$ goes to its upper bound $\\frac{\\pi }{2}$ (as $r$ goes to $\\infty $ ), as shown in Fig.",
"REF .",
"The QNMs are then naturally defined as the complex values of $\\omega =\\omega _{ST}$ , so that the solution of Eq.",
"(REF ) has the following asymptotic form, $\\psi &\\sim & e^{-i\\omega r^*},\\, r\\rightarrow r_h;\\nonumber \\\\\\psi &\\rightarrow &0,\\, r\\rightarrow \\infty .$ We apply the numerical approach proposed in [62] to solve for the QNMs; the details of this method are outlined in appendix .",
"Note that for SAdS BHs in GR, the imaginary parts of the QNM are always negative ($\\Im (\\omega _{GR})<0$ ) [62].",
"This implies that the modes always decay.",
"The same is true for all asymptotically AdS BHs in the theory we consider here, i.e.",
"$\\Im (\\omega _{ST})<0$ ; we refer the reader to reference [62] for a formal proof.",
"As discussed above, when $z=1/3$ we have $h(r)=1$ and $f(r)=F(r)=1-\\frac{2M}{r}+r^2$ so that the metric has precisely SAdS form (note that $\\varphi ^{\\prime }(r)=0$ ).",
"The QNMs will then be those of the SAdS BH, even if there is a finite coupling between $\\Phi $ and $\\partial \\varphi $ .",
"We begin by studying the QNMs for the radial perturbations, i.e.",
"$j=0$ modes, for $r_h$ between 10 and $1/4$ , for the principal QNM and the first two overtones.",
"As $r_h$ becomes smaller, a larger order of expansion is needed to get precise solutions.",
"For example, 50 orders are sufficient for $r_h=10$ , while 450 orders are considered for $r_h=1/4$ in order to be precise to 3 decimal places.",
"Our results for typical values of $r_h$ are shown in Tab.",
"(REF ) in Appendix B and are plotted in Fig.",
"REF .",
"Figure: Real (a) and imaginary (b) parts of the QNMs, (ω ST \\omega _{ST}), as a function of the BH horizon radius r h r_h.",
"Here z=1/3z=1/3.",
"The data points are the principal QNM (black circles), and the first (red squares) and second (blue diamonds) overtones.",
"The solid continuous curves are (7.75-11.16i)T(7.75-11.16i)T (principal), (13.24-20.59i)T(13.24-20.59i)T (first overtone) and (18.70-30.02i)T(18.70-30.02i)T (second overtone), where TT is the Hawking temperature of the BH horizon.",
"The dashed continuous curves in (b) are 2.66r h 2.66r_h (principal), 4.98r h 4.98r_h (first overtone) and 7.18r h 7.18r_h (second overtone).",
"The units are chosen by setting m p =1m_p=1 and l=1l=1.For large BHs, the relations $\\omega _{ST}^{(0)}=(7.75-11.16i)T$ , $\\omega _{ST}^{(1)}=(13.24-20.59i)T$ , and $\\omega _{ST}^{(2)}=(18.70-30.02i)T$ , as found for SAdS BHs in [62], hold.",
"Here $\\omega _{ST}^{(0)},~\\omega _{ST}^{(1)},~\\omega _{ST}^{(2)}$ are the principal QNM and the first and second overtones respectively, and $T$ is the Hawking temperature of the BH horizon given in Eq.",
"(REF ).",
"As $r_h$ decreases, the above linear relations no longer hold.",
"For intermediate-size BHs, a linear relation between $\\Im (\\omega _{ST})$ and $r_h$ holds, i.e.",
"$\\Im (\\omega _{ST}^{(0)})=-2.66r_h$ , $\\Im (\\omega _{ST}^{(1)})=-4.98r_h$ and $\\Im (\\omega _{ST}^{(2)})=-7.18r_h$ .",
"This relation breaks down for smaller BHs.",
"As found by [66], the QNMs of SAdS BHs approach those of a pure AdS space as the hole becomes very small.",
"Next, we consider non-radial perturbations, i.e.",
"$j>0$ .",
"It is necessary to go to larger orders in the expansion in order to get convergent results for larger values of $j$ .",
"We were able to calculate the principal QNM for $j$ up to 30, for $r_h$ down to 4.",
"Typical results are listed in Tab.",
"(REF ) in Appendix B, and these are plotted in Fig.",
"REF .",
"As seen, both of the real and imaginary parts of QNMs increase with $j$ , with the change becoming less significant for larger BHs.",
"Our results are consistent with those of [62], who have studied SAdS BHs previously.",
"Figure: Contour plot of the principal QNMs for different values of jj and r h r_h.",
"Different jj's are denoted by different marker shapes.",
"Different r h r_h's are represented by different marker colors.",
"Here z=1/3z=1/3."
],
[
"Quasi-normal Modes for $z>1/3$",
"When $z>1/3$ , two factors contribute to the change in QNMs: the spacetime (REF ) deviates from SAdS, and there is a potential coupling between the test field and $\\varphi $ .",
"In this section, we consider the former, the latter is the topic of the next section.",
"As before, we begin by considering radial perturbations, $j=0$ .",
"Examining Eq.",
"(REF ) (using $V(r)$ as given in Eq.",
"(REF )), the differences from SAdS are due to the different form of $f(r)$ in Eq.",
"(REF ).",
"For BHs with $r_h>>\\sqrt{z}$ we have, $\\arctan (r/\\sqrt{z})&\\approx \\pi /2,\\quad \\textrm {and}\\\\h(r)&\\approx \\sqrt{3z},$ and therefore $f(r)$ differs from the SAdS form by a constant factor $\\sqrt{\\frac{1}{3z}}$ .",
"The potential $V(r)$ then differs from the SAdS potential by $\\frac{1}{3z}$ .",
"As a result, $\\omega _{ST}\\approx \\sqrt{\\frac{1}{3z}}\\omega _{GR},$ where $\\omega _{GR}$ is the corresponding QNM for an SAdS BH (solution in GR) with the same horizon $r_h$ .",
"We plot the above relation as dashed lines in Fig.",
"REF for the principle QNM as well as the first and second overtones, and compare them with our numerically computed QNMs (only the principal mode is shown for $r_h=0.6$ since it is sufficient to illustrate that the relation breaks down for small BHs).",
"As expected, our data points follow these lines very well for large BHs but the deviation is significant for small BHs, i.e.",
"$r_h<\\sqrt{z}$ .",
"This can be seen in the figure as small deviations at large $z$ for the case $r_h=5$ .",
"In general, $\\Re (\\omega _{ST})$ and $-\\Im (\\omega _{ST})$ decrease with increasing $z$ .",
"Physically, this means the dominant perturbation oscillates with a longer period and decays more slowly as $z$ increases.",
"From an AdS/CFT correspondence point of view, this means it takes a longer time to reach equilibrium.",
"For large BHs, this change follows the $z^{-1/2}$ trend as discussed above.",
"For small BHs, $\\Re (\\omega _{ST})$ changes more slowly while $-\\Im (\\omega _{ST})$ changes more rapidly with $z$ .",
"Figure: The real (a) and imaginary (b) parts of the QNMs as a function of the constant zz.",
"The dashed lines are given by Eq.",
"() with the corresponding BH horizon radius and QNM order.",
"Different r h r_h's are represented by different colors.",
"The integers between data points are the order of QNMs: \"0\" for principal mode, \"1\" for the first overtone and \"2\" for the second overtones.We also plot the imaginary part of the QNMs as a function of $r_h$ for $z=2$ in Fig.",
"REF .",
"As discussed in the previous subsection, $\\Im (\\omega _{ST})$ is proportional to $T$ for large BHs, and to $r_h$ for intermediate-size BHs.",
"As seen here, the linear relation with $T$ still holds for large BHs for $z>1/3$ , with different proportionality.",
"And the linear relation with $r_h$ also holds for intermediate-size BHs, with the proportionality reduced by a factor of $(3z)^{1/2}$ .",
"Note that these linear relations break down at larger values of $r_h$ than the SAdS BH relations.",
"As discussed by [62], $\\Re (\\omega _{ST})$ never scales as $r_h$ no matter the value of $z$ .",
"Figure: Imaginary part of QNMs as a function of the BH horizon radius, for z=1/3z=1/3 (blue) and z=2z=2 (Red).",
"The filled circles are data from our calculation, and the solid/dashed curved are the analytic functions shown in the legend.Next, we compute the non-radial QNMs for different values of $z$ and plot them in Fig.",
"REF .",
"We choose $r_h=10$ .The QNMs for different values of $r_h$ (provided $r_h>1$ and $r_h>\\sqrt{z}$ ) have similar dependencies on $j$ and $z$ .",
"This constitutes one of the new results of this work.",
"For each value of $z$ , higher-$j$ order QNMs have both larger real and imaginary parts.",
"And the change of QNMs with $j$ becomes more significant for larger values of $z$ .",
"Comparing Fig.",
"REF with Fig.",
"REF shows that increasing $z$ has a similar effect to decreasing $r_h$ .",
"The reason for this can be seen from the the metric.",
"For $r_h\\gg 1$ and $r_h\\gg \\sqrt{z}$ , the metric functions (Eq.",
"(REF )) show that the metric is approximately SAdS with $r_h^3\\approx 2M-\\frac{\\pi (3z-1)^2}{24\\sqrt{z}}.$ This clearly shows that, for large BHs and not too large $z$ , increasing $z$ would reduce $r_h$ while keeping the approximate SAdS form and AdS radius $l$ of the BH fixed.Of course, as discussed in Eq.",
"(REF ), increasing $z$ causes a decrease of $\\Re (\\omega _{ST})$ and $-\\Im (\\omega _{ST})$ , due to the first term in $V(r)$ (Eq.",
"(REF )).",
"The second term in $V(r)$ is exactly the same as the SAdS counterpart for large BHs.",
"As $j$ becomes larger, this term is more important and increasing $z$ has an effect more similar to that of decreasing $r_h$ .",
"Figure: Contour plot of the principal QNMs for different jj's and zz's for r h =10r_h=10.",
"Different jj's are denoted by different marker shapes shown in the figure.",
"For each jj, zz takes values uniformly from 1/3 to 20/9, from top to bottom.In this subsection we consider a coupling between the test scalar and the gravitational scalar $\\varphi $ .",
"The simplest form of coupling preserving the shift symmetry of the field $\\varphi $ , $\\varphi \\rightarrow \\varphi +c$ with $c$ constant, and the reflection symmetry of $\\Phi $ ($\\Phi \\rightarrow -\\Phi $ ) and $\\varphi $ ($\\varphi \\rightarrow -\\varphi $ ) is $L_{\\rm NMC}=\\sqrt{-g}\\left(-\\frac{1}{2}\\partial _{\\mu }\\Phi \\partial ^{\\mu }\\Phi -\\frac{\\xi }{2 m_p^2}\\Phi ^2\\partial _{\\mu }\\varphi \\partial ^{\\mu }\\varphi \\right),$ where $\\xi $ is a dimensionless coupling constant.",
"The equation of motion for $\\Phi $ is now modified to $\\Box \\Phi -\\frac{\\xi }{m_p^2}(\\partial _{\\mu }\\varphi \\partial ^{\\mu }\\varphi )\\Phi =0,$ which still reduces to the form of a Schrödinger-like equation for $\\psi $ when written in terms of the tortoise coordinate, but with the effective potential $V(r)=\\frac{ff^{\\prime }}{r}+j(j+1)\\frac{hf}{r^2}+\\frac{\\xi }{m_p^2}f^2\\varphi ^{\\prime 2}(r),$ where $\\varphi ^{\\prime }(r)$ is given by Eq.",
"(REF ).",
"The first and third terms both vary as $r^2$ when $r\\rightarrow \\infty $ , and therefore there is a critical value of $\\xi $ above which $V(r)\\rightarrow +\\infty $ as $r\\rightarrow \\infty $ , but below which $V(r)\\rightarrow -\\infty $ as $r\\rightarrow \\infty $ .",
"This critical value is $\\xi _c=-\\frac{3z+1}{6(3z-1)}.$ At $\\xi =\\xi _c$ , $V(r)$ is dominated by the second term and becomes constant as $r\\rightarrow \\infty $ .",
"For $j=0$ , $V(r)\\sim 1/r\\rightarrow 0$ as $r\\rightarrow \\infty $ .",
"For $\\xi \\le \\xi _c$ , our boundary conditions (Eq.",
"(REF )) in the definition of QNMs no longer apply.",
"For this reason, we will only consider values of $\\xi $ above $\\xi _c$ .",
"The effective potential is plotted in Fig.",
"REF for this case for different values of $\\xi $ .",
"Figure: Real part (upper half plane) and imaginary part (lower half plane) of the principal QNMs as a function of zz for different values of the coupling constant ξ\\xi (defined in Eq.",
"()) given in the figure.",
"Here r h =10r_h=10 and j=0j=0.",
"Neighboring data points are joined by line segments for better illustration.The principal QNMs are plotted in Fig.",
"REF , for $j=0$ and $r_h=10$ .",
"As seen, both $\\Re (\\omega _{ST})$ and $-\\Im (\\omega _{ST})$ increase as $\\xi $ increases.",
"For low values of $\\xi $ , $\\Re (\\omega _{ST})$ and $-\\Im (\\omega _{ST})$ decrease as $z$ increases but for large $\\xi $ , $\\Re (\\omega _{ST})$ and $-\\Im (\\omega _{ST})$ increase with $z$ for small $z$ , and then decrease for large $z$ .",
"Recall that when $z=1/3$ the solution is exactly SAdS with $\\varphi ^{\\prime }=0$ so that the non-minimal coupling is not relevant and the QNMs converge to the same value whatever the value of $\\xi $ .",
"At $z=1/3$ , $\\xi _c$ becomes $-\\infty $ (Eq.",
"(REF )).",
"Therefore, for $\\xi $ very close to $\\xi _c$ , the QNMs have a very steep change with $z$ close to $z=1/3$ .",
"This is clearly seen in Fig.",
"REF .",
"For a fixed value of $z$ , the real part of QNM increases while the imaginary part becomes more negative with increasing $\\xi $ .",
"Physically this means the dominant perturbation will have a shorter oscillation period and will decay more rapidly.",
"In terms of AdS/CFT, if such BHs have a CFT dual, this means the equilibrium state is reached faster."
],
[
"Conclusions",
"In this work we have studied the quasi-normal modes of asymptotically Anti-de Sitter black holes that are analytic solutions of a class of shift-symmetric Horndeski theories where a gravitational scalar $\\varphi $ is derivatively coupled to the Einstein tensor.",
"We have calculated the QNMs numerically for a massless test scalar both minimally coupled to the metric, and non-minimally coupled to $\\varphi $ .",
"In the case of minimal coupling, we have calculated the principal radial as well as the first two overtones for parameter choices that give exact SAdS solutions.",
"A linear relation between the (complex) frequency $\\omega _{ST}$ and the Hawking temperature was observed in all cases for large black holes, confirming known analytic expectations.",
"We also calculated the principal mode for non-radial perturbations and found that increasing the horizon radius increases the real part of $\\omega _{ST}$ and makes the imaginary part more negative.",
"Moving away from exact Schwarzschild-Anti-de Sitter black holes, we calculated the principal radial mode and first two overtones and found that increasing the coupling $z$ of $\\varphi $ to gravity decreases the real part and makes the imaginary part of $\\omega _{ST}$ less negative at fixed black hole radius.",
"We predict and numerically confirm the relation $\\omega _{ST}\\propto z^{-1/2}$ for large black holes.",
"In the context of the AdS/CFT correspondence, the dual theory exhibits perturbations from the thermal state that decay more slowly.",
"We also calculated the principal non-radial mode and found that, for a black hole with a fixed radius, stronger couplings to gravity have a similar effect to decreasing the horizon radius in the case of Schwarzschild-Anti-de Sitter black holes.",
"Finally, we considered, for the first time, a non-minimal coupling between $\\varphi $ and the test scalar $\\Phi $ ; we chose the lowest-order operator that respects the symmetries of both fields.",
"We found that there is a critical value of the dimensionless coupling constant $\\xi $ below which the structure of the effective potential for perturbations changes so that $\\lim _{r\\rightarrow \\infty }V(r)=-\\infty $ and the QNMs satisfying the usual boundary conditions (Eq.",
"()) are not well-defined.",
"We numerically calculated the principal radial mode for values of $\\xi $ larger than this critical value and found that stronger non-minimal couplings increase the real parts and decrease (make more negative) the imaginary parts of the frequency i.e.",
"they give rise to QNMs that oscillate with a shorter period and decay faster than the equivalent (same $z$ ) minimally coupled models.",
"We are grateful to Eugeny Babichev and Onkar Parrikar for useful discussions.",
"D.S.",
"and R.D.",
"were partially supported by the US National Science Foundation, under Grant No.",
"PHY-1417317.",
"JS is supported by funds provided to the Center for Particle Cosmology by the University of Pennsylvania."
],
[
"Numerical Procedure",
"In what follows, we review the numerical procedure of Horrowitz and Hubeny [62].",
"Defining $\\Psi =\\psi e^{i\\omega r^*}$ , we can write Eq.",
"(REF ) in terms of $\\Psi $ using the $r$ coordinate so that $f\\frac{d^2\\Psi }{dr^2}+[f^{\\prime }-2i\\omega ]\\frac{d\\Psi }{dr}-U(r)\\Psi =0,$ where $U(r)=V(r)/f(r)$ .",
"Introducing the new variable $x=1/r$ , Eq.",
"(REF ) can be written as $s(x)\\frac{d^2\\Psi }{dx^2}+\\frac{t(x)}{x-x_h}\\frac{d\\Psi }{dx}+\\frac{u(x)}{(x-x_h)^2}\\Psi =0.$ Here $x_h=1/r_h$ , $s(x)=\\frac{x^4f(r)}{x-x_h}$ , $t(x)=2x^3f(r)-x^2f^{\\prime }(r)+2i\\omega x^2$ and $u(x)=-(x-x_h)U(r)$ .",
"From our boundary conditions (Eq.",
"(REF )), $\\Psi $ should be finite as $x\\rightarrow x_h$ , and vanish as $x\\rightarrow 0$ .",
"Expanding $\\Psi $ as $\\Psi =\\sum ^\\infty _{n=0}a_n(x-x_h)^{n+\\alpha },$ where $\\alpha $ is a constant.",
"We can solve perturbatively by matching every order of the series in $(x-x_h)$ .",
"The lowest-order term is $s_0\\alpha (\\alpha -1)+t_0\\alpha =0,$ where $s_0=-x_h^2f^{\\prime }(r_h)$ and $t_0=-x_h^2f^{\\prime }(r_h)+2i\\omega x_h^2$ are the zeroth-order term in the expansion of $s(x)$ and $t(x)$ respectively.",
"There are two solutions given by $\\alpha =0, \\frac{2i\\omega }{f^{\\prime }(r_h)};$ the former corresponds to an incoming wave at the BH horizon, and the latter an outgoing wave there.",
"Physically, as measured by an observer at the horizon, a wave can only travel into the BH, and not out.",
"Therefore, we choose the former solution, i.e.",
"$\\alpha =0$ .",
"Next, we need to satisfy the other boundary condition, i.e.",
"$\\Psi (x=0)=0$ .",
"This is achieved by solving the series in $a_n$ to order N, and setting $\\Psi (x=0)=\\sum ^N_{n=0}a_n(\\omega )(0-x_h)^n=0.$ We solve the resulting polynomial equation in $\\omega $ numerically; the precision of the solution can be checked by varying N and checking the convergence of the results."
],
[
"Data tables",
"In this appendix, we tabulate the QNMs for typical values of the BH horizon radii $r_h$ , for $z=1/3$ (SAdS BH).",
"Table REF shows the first three QNMs for radial perturbations ($j=0$ ).",
"Table REF shows the principal QNMs for $j$ up to 30.",
"Table: The principal QNM (ω ST (0) \\omega _{ST}^{(0)}) and the first two overtones (ω ST (1,2) \\omega _{ST}^{(1,2)}), for different BH horizon radii.",
"Here z=1/3z=1/3 and j=0j=0.Table: The principal QNM for the z=1/3z=1/3 case (SAdS BH), for different jj-degrees and BH horizon radii r h r_h."
]
] | 1709.01641 |
[
[
"Kaon femtoscopy in Pb-Pb collisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV"
],
[
"Abstract We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV.",
"Femtoscopy is used to measure the space-time characteristics of particle production from the effects of quantum statistics and final-state interactions in two-particle correlations.",
"Kaon femtoscopy is an important supplement to that of pions because it allows one to distinguish between different model scenarios working equally well for pions.",
"In particular, we compare the measured 3D kaon radii with a purely hydrodynamical calculation and a model where the hydrodynamic phase is followed by a hadronic rescattering stage.",
"The former predicts an approximate transverse mass ($m_{\\mathrm{T}}$) scaling of source radii obtained from pion and kaon correlations.",
"This $m_{\\mathrm{T}}$ scaling appears to be broken in our data, which indicates the importance of the hadronic rescattering phase at LHC energies.",
"A $k_{\\mathrm{T}}$ scaling of pion and kaon source radii is observed instead.",
"The time of maximal emission of the system is estimated using the three-dimensional femtoscopic analysis for kaons.",
"The measured emission time is larger than that of pions.",
"Our observation is well supported by the hydrokinetic model predictions."
],
[
"Introduction",
"Extremely high energy densities achieved in heavy-ion collisions at the Large Hadron Collider (LHC) are expected to lead to the formation of the quark-gluon plasma (QGP), a state characterized by partonic degrees of freedom [1],[2].",
"The systematic study of many observables (transverse momentum spectra, elliptic flow, jets, femtoscopy correlations) measured at the Relativistic Heavy Ion Collider (RHIC) and the LHC confirmed the presence of strong collective motion and the hydrodynamic behavior of the system (see e.g.",
"[3],[4],[3],[5] and [6],[7],[8],[9]).",
"Whereas since quite a long time hydrodynamics describes momentum based observables, it could not describe spatial distributions at decoupling.",
"Correlation femtoscopy (commonly referred to as femtoscopy or HBT, Hanbury Brown and Twiss interferometry), measures the space-time characteristics of particle production using particle correlations due to the effects of quantum statistics and strong and Coulomb final-state interactions [10],[11],[12],[13],[14].",
"The problem to describe the spatio-temporal scales derived from femtoscopy in heavy-ion collisions at RHIC was solved only a few years ago, strongly constraining the hydrodynamical models [15],[16],[17].",
"The following factors were understood to be important: existence of prethermal transverse flow, a crossover transition between quark-gluon and hadron matter, non-hydrodynamic behavior of the hadron gas at the latest stage (hadronic cascade phase), and correct matching between hydrodynamic and non-hydrodynamics phases (see e.g.",
"[15]).",
"New challenges for hydrodynamics appeared when data were obtained at the LHC: the large statistics now allows one to investigate not only pion femtoscopy, which is the most common femtoscopic analysis, but also femtoscopy of heavier particles in differential analyses with high precision.",
"The main objective of ALICE [18] at the LHC is to study the QGP.",
"ALICE has excellent capabilities to study femtoscopy observables due to good track-by-track particle identification (PID), particle acceptance down to low transverse momenta $p_{\\rm T}$ , and good resolution of secondary vertices.",
"We already studied pion correlation radii in Pb–Pb collisions at 2.76 TeV [19],[20].",
"Pion femtoscopy showed genuine effects originating from collective flow in heavy-ion collisions, manifesting as a decrease of the source radii with increasing pair transverse mass $m_{\\rm T} = \\sqrt{k_{\\rm T}^2+m^2}$ [14],[21], where $k_{\\rm T}=|\\mathbf {p}_{\\mathrm {T,1}}+\\mathbf {p}_{\\mathrm {T,2}}|/2$ is the average transverse momentum of the corresponding pair and $m$ is the particles mass.",
"The next most numerous particle species after pions are kaons.",
"The kaon analyses are expected to offer a cleaner signal compared to pions, as they are less affected by resonance decays.",
"Studying charged and neutral kaon correlations together provides a convenient experimental consistency check, since they require different detection techniques.",
"The theoretical models which describe pion femtoscopy well, should describe kaon results with equal precision.",
"Of particular interest is the study of the $m_{\\rm T}$ -dependence of pion and kaon source radii.",
"It was shown that the hydrodynamic picture of nuclear collisions for the particular case of small transverse flow leads to the same $m_{\\rm T}$ behavior of the longitudinal radii ($R_{\\rm long}$ ) for pions and kaons [22].",
"This common $m_{\\rm T}$ -scaling for $\\pi $ and $\\rm K$ is an indication that the thermal freeze-out occurs simultaneously for $\\pi $ and $\\rm K$ and that these two particle species are subject to the same velocity boost from collective flow.",
"Previous kaon femtoscopy studies carried out in Pb–Pb collisions at the SPS by the NA44 and NA49 Collaborations [23],[24] reported the decrease of $R_{\\rm long}$ with $m_{\\rm T}$ as $\\sim m_{\\rm T}^{-0.5}$ as a consequence of the boost-invariant longitudinal flow.",
"Subsequent studies carried out in Au–Au collisions at RHIC [25],[26],[27],[28] have shown the same power in the $m_{\\rm T}$ -dependencies for $\\pi $ and $\\rm K$ radii, consistent with a common freeze-out hypersurface.",
"Like in the SPS data, no exact universal $m_{\\rm T}$ -scaling for the 3D radii was observed at RHIC, but still these experiments observed an approximate $m_{\\rm T}$ -scaling for pions and kaons.",
"The recent study of the $m_{\\rm T}$ -dependence of kaon three-dimensional radii performed by the PHENIX collaboration [29] demonstrated breaking of this scaling especially for the “long” direction.",
"PHENIX reported that the Hydro-Kinetic Model (HKM) describes well the overall trend of femtoscopic radii for pions and kaons [30],[31].",
"We have published previously the study of one-dimensional correlation radii of different particle species: $\\pi ^{\\pm }\\pi ^{\\pm }$ , $\\rm K^{\\pm }\\rm K^{\\pm }$ , $\\rm K^{ 0}_S\\rm K^{ 0}_S$ , $\\rm p\\rm p$ , and $\\rm \\overline{p}\\rm \\overline{p}$ correlations in Pb–Pb collisions at $\\sqrt{s_{\\mathrm {NN}}}=2.76$ TeV for several intervals of centrality and transverse mass [32].",
"The decrease of the source radii with increasing transverse mass was observed for all types of particles, manifesting a fingerprint of collective flow in heavy-ion collisions.",
"The one-dimensional femtoscopic radii demonstrated the approximate $m_{\\rm T}$ -scaling as it was expected by hydrodynamic model considerations [14].",
"Recent calculations made within a 3+1D hydrodynamical model coupled with a statistical hadronization code taking into account the resonance contribution, THERMINATOR-2, showed the approximate scaling of the three dimensional radii with transverse mass for pions, kaons and protons [33].",
"An alternative calculation, that is the Hydro-Kinetic Model, including a hydrodynamic phase as well as a hadronic rescattering stage, predicts violation of such a scaling between pions and kaons at LHC energies [34].",
"Both models observe approximate scaling if there is no rescattering phase.",
"It is suggested in [34] that rescattering has significantly different influence on pions and kaons and is responsible for the violation of $m_{T}$ -scaling at the LHC energies.",
"Moreover, the analysis of the emission times of pions and kaons obtained within HKM in [35] showed that kaons are emitted later than pions due to rescattering through the rather long-lived ${\\rm K}^{*}(892)$ resonance.",
"This effect can explain the $m_{\\rm T}$ -scaling violation predicted in [34].",
"In [34] it was found that immediately decaying the ${\\rm K}^{*}(892)$ and $\\phi $ (1020) resonances at the chemical freeze-out hypersurface has only a negligible influence on the kaon radii.",
"In this scenario, resonances were allowed to be regenerated in the hadronic phase.",
"Further analysis in [35] showed that it is indeed the regeneration of the ${\\rm K}^{*}(892)$ resonance through hadronic reactions which is responsible for the $m_{\\rm T}$ -scaling violation predicted in [34].",
"This mechanism clearly manifests itself in the prolonged emission time of kaons caused by the rather long lifetime of the ${\\rm K}^{*}(892)$ resonance [34].",
"The approximate scaling of pion and kaon radii was predicted by investigating 3+1D hydrodynamical model + THERMINATOR-2 in [33] to hold for each of the three-dimensional radii separately.",
"The scaling of one-dimensional pion and kaon radii was also studied in [33].",
"It was shown that after averaging the three-dimensional radii and taking into account a mass-dependent Lorentz-boost factor, a deviation between one-dimensional pion and kaon radii appeared.",
"These circumstances made it impossible to discriminate between THERMINATOR-2 [33] and the HKM calculation [34] in the earlier published one-dimensional analysis of pion and kaon radii by ALICE [32].",
"The three-dimensional study presented here is not impeded by these effects and allows one to discriminate between the hypothesis of approximate scaling of three-dimensional radii predicted in [33] and strong scaling violation proposed in [34].",
"Thus the study of the $m_{\\rm T}$ -dependence of three-dimensional pion and kaon radii can unambiguously distinguish between the different freeze-out scenarios and clarify the existence of a significant hadronic phase.",
"One more interesting feature of femtoscopy studies of heavy-ion collisions concerns the ratio of radius components in the transverse plane.",
"The strong hydrodynamic flow produces significant positive space-time correlations during the evolution of the freeze-out hypersurface.",
"This influences the extracted radius parameters of the system in the plane perpendicular to the beam axis.",
"The radius along the pair transverse momentum is reduced by the correlation with respect to the perpendicular one in the transverse plane.",
"This effect appears to be stronger at LHC than at RHIC energies [36],[37].",
"It was studied by the ALICE collaboration for pions in Pb–Pb collisions at 2.76 TeV [20] at different centralities.",
"This work extends this study to kaons and compares the obtained transverse radii with those found in the analysis for pions and to the model calculations discussed above.",
"The paper is organized as follows.",
"Section 2 explains the data selection and describes the identification of charged and neutral kaons.",
"In Section 3 the details of the analysis of the correlation functions are discussed together with the investigation of the systematic uncertainties.",
"Section 4 presents the measured source radii as well as the extracted emission times and compares them to model predictions.",
"Finally, Section 5 summarizes the obtained results and discusses them within the hydrokinetic approach."
],
[
"Data selection",
"Large sets of data were recorded by the ALICE collaboration at $\\sqrt{s_{\\rm NN}}=2.76$ TeV in Pb–Pb collisions.",
"The about 8 million events from 2010 (used only in the $\\rm K^{ 0}_S\\rm K^{ 0}_S$ analysis) and about 40 million events from 2011 made it possible to perform the three-dimensional analyses of neutral and charged kaon correlations differentially in centrality and pair transverse momentum $k_{\\rm T}$ .",
"Three trigger types were used: minimum bias, semi-central (10-50% collision centrality), and central (0-10% collision centrality) [38].",
"The analyses were performed in the centrality ranges: (0–5%), (0–10%), (10–30%), and (30–50%).",
"The centrality was determined using the measured amplitudes in the V0 detector [38].",
"The following transverse momentum $k_{\\mathrm {T}}$ bins were considered: (0.2–0.4), (0.4–0.6), and (0.6–0.8) GeV/$c$ for charged kaons and (0.2–0.6), (0.6–0.8), (0.8–1.0), and (1.0–1.5) GeV/$c$ for neutral kaons.",
"Charged particle tracking is generally performed using the Time Projection Chamber (TPC) [39] and the Inner Tracking System (ITS) [18].",
"The ITS also provides high spatial resolution in determining the primary collision vertex.",
"Particle identification (PID) for reconstructed tracks was carried out using both the TPC and the Time-of-Flight (TOF) detector [40].",
"For TPC PID, a parametrization of the Bethe-Bloch formula was employed to calculate the specific energy loss (d$E$ /d$x$ ) in the detector expected for a particle with a given mass and momentum.",
"For PID with TOF, the particle mass hypothesis was used to calculate the expected time-of-flight as a function of track length and momentum.",
"For each PID method, a value $N_{\\sigma }$ was assigned to each track denoting the number of standard deviations between the measured track d$E$ /d$x$ or time-of-flight and the calculated one as described above.",
"Different cut values of $N_{\\sigma }$ were chosen based on detector performance for various particle types and track momenta (see Table REF for specific values used in both analyses).",
"More details on PID can be found in Secs.",
"7.2–7.5 of [41].",
"The analysis details for charged and neutral kaons are discussed separately below.",
"All major selection criteria are also listed in Table REF ."
],
[
"Charged kaon selection",
"Track reconstruction for the charged kaon analysis was performed using the tracks' signal in the TPC.",
"The TPC is divided by the central electrode into two halves, each of them is composed of 18 sectors (covering the full azimuthal angle) with 159 padrows placed radially in each sector.",
"A track signal in the TPC consists of space points (clusters), each of which is reconstructed in one of the padrows.",
"A track was required to be composed out of at least 70 such clusters.",
"The parameters of the track are determined by performing a Kalman fit to a set of clusters with an additional constraint that the track passes through the primary vertex.",
"The quality of the fit is requested to have $\\chi ^2/{\\mathrm {NDF}} $ better than 2.",
"The transverse momentum of each track was determined from its curvature in the uniform magnetic field.",
"The momentum from this fit in the TPC was used in the analysis.",
"Tracks were selected based on their distance of closest approach (DCA) to the primary vertex, which was required to be less than 2.4 cm in the transverse and less than 3.0 cm in the longitudinal direction.",
"$\\rm K^{\\pm }$ identification was performed using the TPC (for all momenta) and the TOF detector (for $p > 0.5$ GeV/$c$ ).",
"The use of different values for $N_{\\sigma ,{\\mathrm {TPC}}}$ and $N_{\\sigma ,{\\mathrm {TOF}}}$ was the result of studies to obtain the best kaon purity, defined as the fraction of accepted kaon tracks that correspond to true kaon particles, while retaining a decent efficiency.",
"The estimation of purity for $p < 0.5$ GeV/c was performed by parametrizing the TPC d$E$ /d$x$ distribution in momentum slices for the contributing species [41].",
"The dominant contamination for charged kaons comes from $\\rm e^{\\pm }$ in the momentum range $0.4<p<0.5$ GeV/$c$ .",
"The purity for $p>0.5$ GeV/$c$ , where the TOF information was employed, was studied with HIJING [42] simulations using GEANT [43] to model particle transport through the detector; the charged kaon purity was estimated to be greater than 99%.",
"The momentum dependence of the single kaon purity is shown in Fig.",
"REF (a).",
"The pair purity is calculated as the product of two single-particle purities, where the momenta are taken from the experimentally determined distribution.",
"The $\\rm K^{\\pm }$ pair purity as a function of $k_{\\rm T}$ at three different centralities is shown in Fig.REF (b).",
"Kaon pair transverse momentum is an averaged $p_{\\rm T}$ of single kaons taken from the whole $p_{\\rm T}$ range, which is the reason why the pair purities are larger than single particle ones.",
"Figure: Single K ± \\rm K^{\\pm } purity (a) and pair purity for small relative momenta (b) for different centralities.",
"In (b) the k T k_{\\rm T} values for different centrality intervals are slightly offset for clarity.Two kinds of two-track effects have been investigated: splitting, where a signal produced by one particle is incorrectly reconstructed as two tracks, and merging, where two particles are reconstructed as only one track.",
"These detector inefficiencies can be suppressed by employing specific pair selection criteria.",
"We used the same procedure as in [20] which works here as well with slightly modified cut values.",
"Charged kaon pairs were required to have a separation of $|\\Delta \\varphi ^{*}|>0.04$ and $|\\Delta \\eta |>0.02$.",
"Here, $\\varphi ^{*}$ is the azimuthal position of the track in the TPC at $R$ = 1.2 m, taking into account track curvature in the magnetic field, and $\\eta $ is the pseudorapidity.",
"Also, all track pairs sharing more than $5 \\%$ of TPC clusters were rejected.",
"Table: Single particle selection criteria."
],
[
"Neutral kaon selection",
"The decay channel $\\rm K^{ 0}_S\\rightarrow \\pi ^+\\pi ^-$ was used for the identification of neutral kaons.",
"The secondary pion tracks were reconstructed using TPC and ITS information.",
"The single-particle cuts for parents ($\\rm K^{ 0}_S$ ) and daughters ($\\pi ^{\\pm }$ ) used in the decay-vertex reconstruction are shown in Table REF .",
"The daughter-daughter DCA, that is the distance of closest approach of the two daughter pions from a candidate $\\rm K^{ 0}_S$ decay, proved useful in rejecting background topologies.",
"PID for the pion daughters was performed using both TPC (for all momenta) and TOF (for $p>0.8$ GeV/$c$ ).",
"The very good detector performance is reflected in the FWHM of the $\\rm K^{ 0}_S$ peak of only 8 MeV/$c^2$ .",
"The selection criteria used in this analysis were chosen as a compromise to maximize statistics while keeping a high signal purity.",
"The neutral kaon purity (defined as Sig./[Sig.+Bkg.]",
"for $0.480 < m_{\\pi ^+ \\pi ^-} < 0.515$ GeV/$c^2$ ) was larger than 0.95.",
"Two main two-particle cuts were used in the neutral kaon analysis.",
"To resolve two-track inefficiencies associated with the daughter tracks, such as the splitting or merging of tracks discussed above, a separation cut was employed in the following way.",
"For each kaon pair, the spatial separation between the same-sign pion daughters was calculated at several points throughout the TPC (every 20 cm radially from 85 cm to 245 cm) and averaged.",
"If the average separation of either pair of tracks was below 5 cm, the kaon pair was not used.",
"Another cut was used to prevent two reconstructed kaons from using the same daughter track.",
"If two kaons shared a daughter track, one of them was excluded using a procedure which compared the two $\\rm K^{ 0}_S$ candidates and kept the candidate whose reconstructed parameters best matched those expected for a true $\\rm K^{ 0}_S$ particle in two of three categories (smaller $\\rm K^{ 0}_S$ DCA to primary vertex, smaller daughter-daughter DCA, and $\\rm K^{ 0}_S$ mass closer to the PDG value [44]).",
"This procedure was shown, using HIJING+GEANT simulations, to have a success rate of about 95% in selecting a true $\\rm K^{ 0}_S$ particle over a fake one.",
"More details about the $\\rm K^{ 0}_S\\rm K^{ 0}_S$ analysis can be found in Refs. [45],[46].",
"$\\rm K^{ 0}_S$ candidate selection criteria developed in other works [32] were used here as well; they are included in Table REF ."
],
[
"Correlation functions",
"The femtoscopic correlation function $C$ is constructed experimentally as the ratio $C({\\bf q}) = A({\\bf q})/B({\\bf q})$ , where $A({\\bf q})$ is the measured distribution of the difference ${\\bf q} = {\\bf p}_{2}-{\\bf p}_{1}$ between the three-momenta of the two particles ${\\bf p_1}$ and ${\\bf p_2}$ taken from the same event, $B({\\bf q})$ is a reference distribution of pairs of particles taken from different events (mixed).",
"For a detailed description of the formalism, see e.g.",
"[13].",
"The pairs in the denominator distribution $B(\\textbf {q})$ are constructed by taking a particle from one event and pairing it with a particle from another event with a similar centrality and primary vertex position along the beam direction.",
"Each event is mixed with five (ten) others for the $\\rm K^{ 0}_S$ ($\\rm K^{\\pm }$ ) analysis.",
"The numerator and denominator are normalized in the full $q = \\sqrt{|{\\bf q}|^{2} - q_{0}^{2}}$ range used (0–0.3 GeV/$c$ ) such that $C(q) \\rightarrow 1$ means no correlation.",
"Pair cuts have been applied in exactly the same way for the same-event (signal) and mixed-event (background) pairs.",
"The momentum difference is calculated in the longitudinally co-moving system (LCMS), where the longitudinal pair momentum vanishes, and is decomposed into ($q_{\\rm out}$ , $q_{\\rm side}$ , $q_{\\rm long}$ ), with the “long” axis going along the beam, “out” along the pair transverse momentum, and “side” perpendicular to the latter in the transverse plane (Bertsch-Pratt convention).",
"The correlation functions have been corrected for momentum resolution effects, by using the HIJING event generator and assigning a quantum-statistical weight to each particle pair.",
"Further, these modified events were propagated through the full simulation of the ALICE detectors [18].",
"The ratios of the correlation functions obtained before and after this full event simulation have been taken as the correction factors.",
"The correlation function from the data has been divided by this $q$ -dependent factor.",
"The correction increases the obtained radii by 3–5%."
],
[
"Charged kaon",
"The three-dimensional correlation functions were fitted by the Bowler-Sinyukov formula [47],[48]: $C({\\bf q}) = N \\left(1 -\\lambda \\right) + N\\lambda K(q)\\left[ 1+\\exp \\left(-R_{\\rm out}^{2} q_{\\rm out}^{2}-R_{\\rm side}^{2} q_{\\rm side}^{2}-R_{\\rm long}^{2} q_{\\rm long}^{2}\\right)\\right],$ where $R_{\\rm out}$ , $R_{\\rm side}$ , and $R_{\\rm long}$ are the Gaussian femtoscopic radii in the LCMS frame, $N$ is the normalization factor, and $q$ is the momentum difference in the pair rest frame (PRF) Average $q$ in PRF for the given “out-side-long” bin is determined during the $C({\\bf q})$ construction and used as an argument of the $K$ -function.",
".",
"The $\\lambda $ parameter, which characterizes the correlation strength, can be affected by long-lived resonances, coherent sources [49],[50],[51], and non-Gaussian features of the particle-emission distribution.",
"We account for Coulomb effects through $K(q)$ , calculated according to Ref.",
"[48],[50] as $K(q) = C({\\rm QS+ Coulomb}) / C(\\rm QS).",
"$ Figure: A sample projected K ± K ± \\rm K^{\\pm }\\rm K^{\\pm } correlation function with fit.",
"The error bars are statistical only.",
"Systematic uncertainties on the points are equal to or less than the statistical error bars shown.Here, the theoretical correlation function $C(\\rm QS)$ takes into account quantum statistics only and $C(\\rm QS+\\rm Coulomb)$ considers quantum statistics and the Coulomb final-state interaction (FSI) contribution to the wave function [13].",
"The experimental correlation functions have been corrected for purity according to: $C_{\\mathrm {corrected}} = (C_{\\mathrm {raw}}-1+\\zeta )/\\zeta ,$ where $\\zeta $ is the pair purity taken from Fig.",
"REF .",
"Figure REF shows a sample projected $\\rm K^{\\pm }\\rm K^{\\pm }$ correlation function with a fit performed according to Eq.",
"(REF ).",
"When the 3D correlation function is projected on one axis, the momentum differences in the two other directions are required to be within (-0.04,0.04) GeV/$c$ ."
],
[
"Neutral kaon",
"$\\rm K^{ 0}_S\\rm K^{ 0}_S$ correlation functions were fitted using a parametrization which includes Bose-Einstein statistics as well as strong final-state interactions [52],[27].",
"Strong final-state interactions have an important effect on $\\rm K^{ 0}_S\\rm K^{ 0}_S$ correlations.",
"Particularly, the $\\rm K^0\\rm \\overline{K}{}^0$ channel is affected by the near-threshold resonances $\\mathrm {f}_0(980)$ and $\\mathrm {a}_0(980)$ .",
"Using the equal emission time approximation in the pair rest frame (PRF) [52], the elastic $\\rm K^0\\rm \\overline{K}{}^0$ transition is written as a stationary solution $\\Psi _{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ of the scattering problem in the PRF, where $\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*$ and $\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*$ represent the momentum of a particle and the emission separation of the pair in the PRF (the $-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*$ subscript refers to a reversal of time from the emission process), which at large distances has the asymptotic form of a superposition of a plane wave and an outgoing spherical wave, $\\Psi _{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*) = e^{-i\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*\\cdot \\vec{r}\\hspace{1.111pt}\\vphantom{r}^*} + g(k^*) \\dfrac{e^{ik^*r^*}}{r^*} \\;,$ where $g(k^*)$ is the s-wave scattering amplitude for a given system.",
"For $\\rm K^0\\rm \\overline{K}{}^0$ , $g(k^*)$ is dominated by the $\\mathrm {f}_0$ and $\\mathrm {a}_0$ resonances and written in terms of the resonance masses and decay couplings [27]: g(k*) = 12 [ g0(k*) + g1(k*)] , gI(k*) = rmr2-s-irk*-ir'kr' .",
"Here, $s = 4(m^2_K+k^{*2})$ ; $\\gamma _r (\\gamma _r^{\\prime })$ refers to the couplings of the resonances to the $\\mathrm {f}_0 \\rightarrow \\rm K^0\\rm \\overline{K}{}^0(\\mathrm {f}_0 \\rightarrow \\pi \\pi )$ and $\\mathrm {a}_0 \\rightarrow \\rm K^0\\rm \\overline{K}{}^0(\\mathrm {a}_0 \\rightarrow \\pi \\eta )$ channels; $m_r$ is the resonance mass; and $k_r^{\\prime }$ refers to the momentum in the PRF of the second decay channel ($\\mathrm {f}_0 \\rightarrow \\pi \\pi $ or $\\mathrm {a}_0 \\rightarrow \\pi \\eta $ ) with the corresponding partial width $\\Gamma _r^{\\prime } = \\gamma _r^{\\prime } k_r^{\\prime }/m_r$ .",
"The amplitudes $g_I$ of isospin $I=0$ and $I=1$ refer to the $\\mathrm {f}_0$ and $\\mathrm {a}_0$ , respectively.",
"The parameters associated with the resonances and their decays are taken from several experiments [53],[54],[55],[56], and the values are listed in Table REF .",
"Table: The f 0 \\mathrm {f}_0 and a 0 \\mathrm {a}_0 masses and coupling parameters, all in GeV.The correlation function is then calculated by integrating $\\Psi _{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ in the Koonin-Pratt equation [57],[58] $C(\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*,\\vec{K}) = \\int d^3 \\, \\vec{r}\\hspace{1.111pt}\\vphantom{r}^*\\, S_{\\vec{K}}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*) \\vert \\Psi ^S_{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*) \\vert ^2 \\, ,$ where $S_{\\vec{K}}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ is the Gaussian source distribution in terms of $R_{\\rm out}$ , $R_{\\rm side}$ , and $R_{\\rm long}$ , $\\vec{K}$ is the average pair momentum, and $\\Psi ^S_{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ is the symmetrized version of $\\Psi _{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ for bosons.",
"Although Eq.",
"(REF ) can be integrated analytically for $\\rm K^0_S\\rm K^0_S$ correlations with FSI for the one-dimensional case [27], for the three-dimensional case this integration cannot be performed analytically.",
"In order to form the 3D correlation function, we combine a Monte Carlo emission simulation with a calculation of the two-particle wavefunction, thus performing a numerical integration of Eq.",
"(REF ).",
"The Monte Carlo emission simulation consists of generating the pair positions sampled from a three-dimensional Gaussian in the PRF, with three input radii as the width parameters, and generating the particle momenta sampled from a distribution taken from data.",
"Using the MC-sampled positions and momenta, we calculate $\\Psi ^S_{-\\vec{k}\\hspace{1.111pt}\\vphantom{k}^*}(\\vec{r}\\hspace{1.111pt}\\vphantom{r}^*)$ .",
"We then build a correlation function using the wavefunction weights to form the signal distribution, and an unweighted distribution acts as a background.",
"This theoretical correlation function is then used to fit the data.",
"Finally, we make a Lorentz boost, $\\gamma $ , of $R_{\\rm out}$ from the PRF to the LCMS frame ($R_{\\rm side}$ and $R_{\\rm long}$ are not affected by the boost).",
"More details on the 3D fitting procedure can be found in Ref. [45].",
"Figure REF shows a sample projected $\\rm K^{ 0}_S\\rm K^{ 0}_S$ correlation function with fit.",
"Also shown is the contribution to the fit from the quantum statistics part only.",
"As seen, the FSI part produces a significant depletion of the correlation function in the $q$ range 0–0.1 GeV/$c$ in each case.",
"Figure: A sample projected K S 0 K S 0 \\rm K^{ 0}_S\\rm K^{ 0}_S correlation function with fit.",
"Also shown is the contribution to the fit from the quantum statistics part only.",
"The error bars are statistical only.",
"Systematic uncertainties on the points are equal to or less than the statistical error bars shown."
],
[
"Systematic uncertainties",
"The effects of various sources of systematic uncertainty on the extracted fit parameters were studied as functions of centrality and $k_{\\rm T}$ .",
"For each source, we take the maximal deviation and apply it symmetrically as the uncertainty.",
"Table REF shows minimum and maximum uncertainty values for various sources of systematic uncertainty for charged and neutral kaons.",
"The systematic errors are summed up quadratically.",
"The values of the total uncertainty are not necessarily equal to the sum of the individual uncertainties, as the latter can come from different centrality or $k_{\\rm T}$ bins.",
"Both analyses studied the effects of changing the selection criteria used for the events, particles, and pairs (variation of cut values up to $\\pm $ 25$\\%$ ) and varying the range of $q$ values over which the fit is performed (variation of $q$ limits up to $\\pm $ 25$\\%$ ).",
"Uncertainties associated with momentum resolution corrections are included into the $\\rm K^{\\pm }$ analysis; for the $\\rm K^{ 0}_S$ analysis, these uncertainties are found to be small compared to other contributions.",
"Both analyses were performed separately for the two different polarities of the ALICE solenoid magnetic field, the difference was found to be negligible.",
"For the $\\rm K^{ 0}_S$ fitting procedure, the mean $\\gamma $ value is calculated for each centrality and $k_{\\rm T}$ selection and used to scale $R_{\\rm out}$ .",
"However, each bin has a spread of $\\gamma $ values associated with it.",
"The standard deviation of the mean $\\gamma $ value for each $k_{\\rm T}$ bin was used as an additional source of systematic error for $R_{\\rm out}$ .",
"For $\\rm K^{ 0}_S$ , an uncertainty on the strong FSI comes from the fact that several sets of $\\rm \\mathrm {f}_0(980)$ and $\\rm \\mathrm {a}_0(980)$ parameters are available [56],[53],[54],[55]; each set is used to fit the data, the results are averaged, and the maximal difference was taken as a systematic error.",
"The $\\rm K^{\\pm }$ analysis has uncertainties associated with the choice of the radius for the Coulomb function.",
"For each correlation function it is set to the value from the one-dimensional analysis [32].",
"Its variation by $\\pm 1$ fm is a source of systematic uncertainty.",
"Another source of systematic uncertainty is misidentification of particles and the associated purity correction.",
"A 10% variation of the parameters in the purity correction was performed.",
"We also incorporated sets with a reduced electron contamination by I) tightening the PID criteria, in particular extending the momentum range where the TOF signal was used and requiring the energy-loss measurement to be consistent with the kaon hypothesis within one sigma, and II) completely excluding the momentum range 0.4–0.5 GeV/$c$ .",
"Table: Minimum and maximum uncertainty values for various sources of systematic uncertainty for charged and neutral kaons (in percent).Note that each value is the maximum uncertainty from a specific source,but can pertain to a different centrality or k T k_{\\rm T} bin.Thus, the maximum total uncertainties are smaller than (or equal to) the quadratic sum of the maximum individual uncertainties."
],
[
"Results and discussion",
"Figure REF shows the $m_{\\mathrm {T}}$ -dependence of the extracted femtoscopic radii $R_{\\rm out}$ , $R_{\\rm side}$ , and $R_{\\rm long}$ in three centrality selections for pions [20] and charged and neutral kaons.",
"The obtained radii are smaller for more peripheral collisions than for central ones.",
"The radii decrease with increasing $m_{\\mathrm {T}}$ and each particle species roughly follows an $m_{\\mathrm {T}}^{-1/2}$ dependence.",
"The radii in “out” and “long” directions exhibit larger values for kaons than for pions at the same transverse mass demonstrating that the $m_{\\mathrm {T}}$ -scaling is broken.",
"This difference increases with centrality and is maximal for the most central collisions.",
"Also presented in Fig.",
"REF are the predictions of the (3+1)D hydrodynamical model coupled with the statistical hadronization code THERMINATOR-2 [33].",
"The model describes well the $m_{\\mathrm {T}}$ -dependence of pion radii, but underestimates kaon radii.",
"Consistent with the data, the (3+1)D Hydro+THERMINATOR-2 model shows mild breaking in the “long” direction for central collisions, but it underestimates the breaking in the “out” direction.",
"The significance of this breaking of the scaling is discussed further in this section.",
"In addition to the aforementioned three-dimensional radii, here for the 0–5% most central events, Fig.",
"REF also shows the $m_{\\mathrm {T}}$ dependence of the ratio $R_{\\rm out}/R_{\\rm side}$ for charged and neutral kaons in comparison with HKM predictions [34] with and without the hadronic rescattering phase.",
"The HKM calculations without rescattering exhibit an approximate $m_{\\mathrm {T}}$ -scaling but do not describe the data, while the data are well reproduced by the full hydro-kinetic model calculations thereby showing the importance of the rescattering phase at LHC energies.",
"The $R_{\\rm out}$ and $R_{\\rm side}$ radii are both influenced by flow and rescatterings, so their ratio is rather robust against these effects.",
"The fact that $R_{\\rm out}/R_{\\rm side}$ ratio of pions and kaons coincide in the HKM simulations (Fig.",
"REF ) is related to some underestimation of $R_{\\rm side}$ radii for pions while pion $R_{\\rm out}$ radii are slightly overestimated in the model.",
"Figure: The 3D LCMS radii vs. m T m_{\\mathrm {T}} for charged (light green crosses) and neutral (dark green squares) kaons and pions (blue circles) in comparison withthe theoretical predictions of the (3+1)D Hydro + THERMINATOR-2 model for pions (blue solid lines) and kaons (red solid lines).Figure: The 3D LCMS radii vs. m T m_{\\mathrm {T}}for 0–5% most central collisionsin comparison with the theoretical predictions of HKM for pions (blue lines) and kaons (red lines).Figure: The 3D LCMS radii vs. k T k_{\\mathrm {T}} for charged (light green crosses) and neutral (dark green squares) kaons and pions (blue circles).It was predicted in [34] that the radii scale better with $k_{\\mathrm {T}}$ at LHC energies as a result of the interplay of different factors in the model, including the particular initial conditions.",
"Figure REF illustrates the $k_{\\mathrm {T}}$ -dependence of the femtoscopic radii $R_{\\rm out}$ , $R_{\\rm side}$ , and $R_{\\rm long}$ .",
"Unlike the $m_{\\mathrm {T}}$ -dependence, the radii seem to scale better with $k_{\\mathrm {T}}$ in accordance with this prediction.",
"The ratio $R_{\\rm out}/R_{\\rm side}$ appears to be sensitive to the space-time correlations present at the freeze-out hypersurface [36],[37],[20].",
"As it was observed in [20], the ratio for pions is consistent with unity, slowly decreasing for more peripheral collisions and higher $k_{\\mathrm {T}}$ .",
"In Fig.",
"REF , the ratio $R_{\\rm out}/R_{\\rm side}$ is shown for pions and kaons at different centralities.",
"The systematic uncertainties partially cancel in the ratio.",
"Systematic uncertainties are correlated in $m_{\\rm T}$ for each type of particle pair; no correlation between the systematic uncertainties of the charged and neutral species exists.",
"The measured $R_{\\rm out}/R_{\\rm side}$ ratios are slightly larger for kaons than for pions.",
"This is an indication of different space-time correlations for pions and kaons, and a more prolonged emission duration for kaons.",
"Figure: R out /R side R_{\\rm out}/R_{\\rm side} vs. m T m_{\\mathrm {T}} for pions and kaonsfor different centrality intervals.Figure: R long 2 R^{2}_{\\rm long} vs. m T m_{\\mathrm {T}} for kaons and pions.The solid lines show the fit using Eq.",
"()for pions and kaons to extract the emission times (τ\\tau ); the dashed and dotted linesshow the fit using Eq.",
"() with T kin T_{\\rm kin} = 0.144 GeV andT kin T_{\\rm kin} = 0.120 GeV, respectively.",
"For pions at small m T m_{\\mathrm {T}}, the dashed and dotted line coincide.In our previous pion femtoscopy analysis [19] the information about the emission time (decoupling time) at kinetic freeze-out $\\tau \\sim 10$ fm/$c$ was extracted by fitting the $m_{\\mathrm {T}}$ -dependence of $R_{\\rm long}^{2}$ using the blast-wave expression [59]: $R_{\\rm long}^{2} = \\tau ^{2} \\frac{T_{\\rm kin}}{m_{\\mathrm {T}}}\\frac{K_{2}(m_{\\mathrm {T}})}{K_{1}(m_{\\mathrm {T}})},$ where $T_{\\rm kin}$ is the temperature at kinetic freeze-out, and $K_n$ are the integer-order modified Bessel functions.",
"We tried to use Eq.",
"(REF ) to fit the $R_{\\rm long}^{2}$ $m_{\\mathrm {T}}$ -dependence (Fig.",
"REF ) for pions and kaons taking the thermal freeze-out temperature $T_{\\rm kin}$ = 0.120 GeV as in [19] (dotted lines) and $T_{\\rm kin}$ = 0.144 GeV (dashed lines).",
"The emission times extracted from the fit are presented in Table REF .",
"However, though this formula works well for pions, it fails to describe kaon longitudinal radii.",
"Large transverse flow may be partially responsible for this failure [35].",
"The following analytical formula for the time of maximal emission, $\\tau _{max}$ , is proposed in [35]: $R_{\\rm long}^{2} = \\tau _{\\rm max}^{2} \\frac{T_{\\rm max}}{m_{\\mathrm {T}} \\cosh {y_{\\mathrm {T}}}}(1+\\frac{3 T_{\\rm max}}{2 m_{\\mathrm {T}} \\cosh {y_{\\mathrm {T}}}} ),$ where $\\cosh {y_{\\mathrm {T}}} = (1-v_{\\mathrm {T}}^2)^{-1/2}$ , $v_{\\mathrm {T}} = \\frac{\\beta p_{\\rm T}}{\\beta m_{\\mathrm {T}}+\\alpha }$ , $T_{\\rm max}$ is the temperature at the hypersurface of maximal emission, $\\beta = 1/T_{\\rm max}$ , and $\\alpha $ is a free parameter determining the intensity of flow The authors of [35] use full evolutionary model (HKM) that has no sharp/sudden kinetic freeze-out.",
"For such type of models a continuous hadron emission takes place instead.",
"Then for each particle species, considered within certain transverse momentum bin, there is a 4D layer, adjacent to the space-like hypersurface of maximal emission, where most of selected particles are emitted from.",
"This non-enclosed hypersurface is characterized by the (average) proper time $\\tau _{\\rm max}$ – time of maximal emission, and the effective temperature $T_{\\rm max}$ .",
"The proposed phenomenological expression for $R_{\\rm long}$ is associated just with this hypersurface and is based on the model that is different from the blast-wave parameterization for sudden freeze-out.",
"So the blast-wave temperature $T_{\\rm kin}$ can differ from the temperature parameter $T_{\\rm max}$ ..",
"The advantage of the formula given in Eq.",
"(REF ) is that it is derived for a scenario with transverse flow of any intensity, which is especially important for LHC energies.",
"The analytical formula Eq.",
"(REF ) was used to fit the $m_{\\mathrm {T}}$ -dependence of $R_{\\rm long}^{2}$ (Fig.",
"REF ).",
"The fit was performed using the following parameters determined in [35] by fitting light flavor particle spectra [60]: $T_{\\rm max}$ = 0.144 GeV, and $\\alpha _{\\pi }$ = 5.0 and $ \\alpha _{K}$ = 2.2.",
"The extracted times of maximal emission are presented in Table REF .",
"Table: Emission times for pions and kaons extracted using the Blast-wave formulaEq.",
"() and the analytical formula Eq.",
"().In order to estimate the systematic errors of the extracted times of maximal emission we also have performed fitting with $T_{\\rm max}$ , $\\alpha _{\\pi }$ and $\\alpha _{K}$ varied within the range of their uncertainty [35]: $\\pm $ 0.03 GeV, $\\pm $ 3.5 and $\\pm $ 0.7, respectively.",
"The maximum deviations from the central values appeared to be (+1.8, -0.5) fm/$c$ for pions and (+0.5, -0.1) fm/$c$ for kaons.",
"These systematic errors are fully correlated.",
"Regardless of the specific parameter choice, we consistently observe the time of maximal emission for kaons to be larger than the one for pions.",
"The extracted times of maximal emission are rather close to those obtained within the HKM model [35]: $\\tau _{\\pi } = 9.44 \\pm 0.02$ fm/$c$ , $\\tau _{K} = 12.40 \\pm 0.04$ fm/$c$ These results were obtained in [35] using the small interval $q$ =0-0.04 GeV/$c$ in order to minimize influence of the non-Gaussian tails.",
"It is found in [35] that if even strong non-Gaussian behavior is observed for the kaon correlation function in wide $q$ -interval, one can nevertheless utilize the same formula Eq.",
"(REF ), but making free the parameter $\\alpha $ for kaons.",
"Then one gets practically the same effective time for kaon emission, as it is obtained from the fit of the correlation function in the small interval $q$ =0-0.04 GeV/$c$ ; for pions there is no such problem..",
"There is evidence that the time of maximal emission for pions is smaller than the one for kaons.",
"This observation can explain the observed breaking of $m_{\\mathrm {T}}$ -scaling between pions and kaons.",
"It is interesting to note that in [35] this difference in the emission times is explained by the different influence of resonances on pions and kaons during the rescattering phase due to kaon rescattering through the $\\rm {K}^{*}$ (892) resonance (with lifetime of 4–5 fm/$c$ ).",
"It was shown in [35] that a significant regeneration of the $\\rm {K}^{*}(892)$ takes place in full HKM simulations with rescatterings (UrQMD cascade), whereas this process is not present in a scenario where only resonance decays are taken into account.",
"Similar findings were reported in [61], where the production yield of $\\rm {K}^*$ (892) in heavy-ion collisions at the LHC was studied.",
"Also there, the inclusion of a hadronic phase in the theoretical modeling of the production process proved to be essential in order to reproduce the experimentally found suppression pattern of $\\rm {K}^{*}(892)$ production when compared to pp collisions [62]."
],
[
"Summary",
"We presented the first results of three-dimensional femtoscopic analyses for charged and neutral kaons in Pb–Pb collisions at $\\sqrt{s_{\\rm {NN}}}$ = 2.76 TeV.",
"A decrease of source radii with increasing transverse mass and decreasing event multiplicity was observed.",
"The $m_{\\mathrm {T}}$ scaling expected by pure hydro-dynamical models appears to be broken in our data.",
"A scaling of pion and kaon radii with $k_{\\rm T}$ was observed instead.",
"The measured ratio of transverse radii $R_{\\rm out}/R_{\\rm side}$ is larger for kaons than for pions, indicating different space-time correlations.",
"A new approach [35] for extracting the emission times for pions and especially for kaons was applied.",
"It was shown that the measured time of maximal emission for kaons is larger than that of pions.",
"The comparison of measured three-dimensional radii with a model, wherein the hydrodynamic phase is followed by the hadronic rescattering phase [34], and pure hydrodynamical calculations [33],[34] has shown that pion femtoscopic radii are well reproduced by both approaches while the behavior of the three-dimensional kaon radii can be described only if the hadronic rescattering phase is present in the model."
],
[
"Acknowledgements",
"The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex.",
"The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration.",
"The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Universidade Federal do Rio Grande do Sul (UFRGS), Financiadora de Estudos e Projetos (Finep) and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Brazil; Ministry of Science & Technology of China (MSTC), National Natural Science Foundation of China (NSFC) and Ministry of Education of China (MOEC) , China; Ministry of Science, Education and Sport and Croatian Science Foundation, Croatia; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the Carlsberg Foundation and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l'Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE) and Council of Scientific and Industrial Research (CSIR), New Delhi, India; Indonesian Institute of Science, Indonesia; Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi and Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japan Society for the Promotion of Science (JSPS) KAKENHI and Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education and National Science Centre, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics and Romanian National Agency for Science, Technology and Innovation, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation and National Research Centre Kurchatov Institute, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba, Ministerio de Ciencia e Innovacion and Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Spain; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; National Science and Technology Development Agency (NSDTA), Suranaree University of Technology (SUT) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Atomic Energy Agency (TAEK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America.",
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"Rodríguez Cahuantziorg2K. Røedorg21E. Rogochayaorg78D. Rohrorg42,org35D. Röhrichorg22P.S. Rokitaorg140F. Ronchettiorg51E.D. Rosasorg73P. Rosnetorg82A. Rossiorg29,org57A. Rotondiorg136F. Roukoutakisorg87A. Royorg49C. Royorg135P. Royorg112A.J.",
"Rubio Monteroorg10O.V. Ruedaorg73R. Ruiorg25B. Rumyantsevorg78A. Rustamovorg91E. Ryabinkinorg92Y. Ryabovorg98A. Rybickiorg121S. Saarinenorg46S. Sadhuorg139S. Sadovskyorg115K. Šafaříkorg35S.K. Sahaorg139B. Sahlmullerorg71B. Sahooorg48P. Sahooorg49R. Sahooorg49S. Sahooorg68P.K. Sahuorg68J. Sainiorg139S. Sakaiorg133M.A. Salehorg141J. Salzwedelorg18S. Sambyalorg103V. Samsonovorg85,org98A. Sandovalorg75D. Sarkarorg139N. Sarkarorg139P. Sarmaorg44M.H.P. Sasorg64E. Scapparoneorg54F. Scarlassaraorg29R.P. Scharenbergorg108H.S. Scheidorg71C. Schiauaorg89R. Schickerorg106C. Schmidtorg109H.R. Schmidtorg105M.O. Schmidtorg106M. Schmidtorg105N.V. Schmidtorg71,org97S. Schuchmannorg106J. Schukraftorg35Y. Schutzorg135,org117,org35K. Schwarzorg109K. Schwedaorg109G. Scioliorg27E. Scomparinorg59R. Scottorg130M. Šefčíkorg40J.E. Segerorg99Y. Sekiguchiorg132D. Sekihataorg47I. Selyuzhenkovorg85,org109K. Senosiorg77S. Senyukovorg35,org3,org135E. Serradillaorg10,org75P. Settorg48A. Sevcencoorg69A. Shabanovorg63A. Shabetaiorg117R. Shahoyanorg35W. Shaikhorg112A. Shangaraevorg115A. Sharmaorg101A. Sharmaorg103M. Sharmaorg103M. Sharmaorg103N. Sharmaorg101,org130A.I. Sheikhorg139K. Shigakiorg47Q. Shouorg7K. Shtejerorg9,org26Y. Sibiriakorg92S. Siddhantaorg55K.M. Sielewiczorg35T. Siemiarczukorg88D. Silvermyrorg34C. Silvestreorg83G. Simatovicorg100G. Simonettiorg35R. Singarajuorg139R. Singhorg90V. Singhalorg139T. Sinhaorg112B. Sitarorg38M. Sittaorg32T.B. Skaaliorg21M. Slupeckiorg128N. Smirnovorg143R.J.M. Snellingsorg64T.W. Snellmanorg128J. Songorg19M. Songorg144F. Soramelorg29S. Sorensenorg130F. Sozziorg109E. Spiritiorg51I. Sputowskaorg121B.K. Srivastavaorg108J. Stachelorg106I. Stanorg69P. Stankusorg97E. Stenlundorg34D. Stoccoorg117M.M. Storetvedtorg37P. Strmenorg38A.A.P. Suaideorg124T. Sugitateorg47C. Suireorg62M. Suleymanovorg15M. Suljicorg25R. Sultanovorg65M. Šumberaorg96S. Sumowidagdoorg50K. Suzukiorg116S. Swainorg68A. Szaboorg38I. Szarkaorg38U. Tabassamorg15J. Takahashiorg125G.J. Tambaveorg22N. Tanakaorg133M. Tarhiniorg62M. Tariqorg17M.G. Tarzilaorg89A. Tauroorg35G.",
"Tejeda Muñozorg2A. Telescaorg35K. Terasakiorg132C. Terrevoliorg29B. Teyssierorg134D. Thakurorg49S. Thakurorg139D. Thomasorg122F. Thoresenorg93R. Tieulentorg134A. Tikhonovorg63A.R. Timminsorg127A. Toiaorg71S.R. Torresorg123S. Tripathyorg49S. Trogoloorg26G. Trombettaorg33L. Tropporg40V. Trubnikovorg3W.H. Trzaskaorg128B.A. Trzeciakorg64T. Tsujiorg132A. Tumkinorg111R. Turrisiorg57T.S. Tveterorg21K. Ullalandorg22E.N. Umakaorg127A. Urasorg134G.L. Usaiorg24A. Utrobicicorg100M. Valaorg119,org66J.",
"Van Der Maarelorg64J.W.",
"Van Hoorneorg35M.",
"van Leeuwenorg64T. Vanatorg96P.",
"Vande Vyvreorg35D. Vargaorg142A. Vargasorg2M. Vargyasorg128R. Varmaorg48M. Vasileiouorg87A. Vasilievorg92A. Vauthierorg83O.",
"Vázquez Doceorg107,org36V. Vecherninorg138A.M. Veenorg64A. Velureorg22E. Vercellinorg26S.",
"Vergara Limónorg2R. Vernetorg8R. Vértesiorg142L. Vickovicorg120S. Vigoloorg64J. Viinikainenorg128Z. Vilakaziorg131O.",
"Villalobos Baillieorg113A.",
"Villatoro Telloorg2A. Vinogradovorg92L. Vinogradovorg138T. Virgiliorg30V. Vislaviciusorg34A. Vodopyanovorg78M.A. Völklorg106,org105K. Voloshinorg65S.A. Voloshinorg141G. Volpeorg33B.",
"von Hallerorg35I. Vorobyevorg107,org36D. Voscekorg119D. Vranicorg35,org109J. Vrlákováorg40B. Wagnerorg22H. Wangorg64M. Wangorg7D. Watanabeorg133Y. Watanabeorg132,org133M. Weberorg116S.G. Weberorg109D.F. Weiserorg106S.C. Wenzelorg35J.P. Wesselsorg72U. Westerhofforg72A.M. Whiteheadorg102J. Wiechulaorg71J. Wikneorg21G. Wilkorg88J. Wilkinsonorg106,org54G.A. Willemsorg72M.C.S. Williamsorg54E. Willsherorg113B. Windelbandorg106W.E. Wittorg130S. Yalcinorg81K. Yamakawaorg47P. Yangorg7S. Yanoorg47Z. Yinorg7H. Yokoyamaorg133,org83I.-K. Yooorg35,org19J.H. Yoonorg61V. Yurchenkoorg3V. Zaccoloorg59,org93A. Zamanorg15C. Zampolliorg35H.J.C. Zanoliorg124N. Zardoshtiorg113A. Zarochentsevorg138P. Závadaorg67N. Zaviyalovorg111H. Zbroszczykorg140M. Zhalovorg98H. Zhangorg22,org7X. Zhangorg7Y. Zhangorg7C. Zhangorg64Z. Zhangorg82,org7C. Zhaoorg21N. Zhigarevaorg65D. Zhouorg7Y. Zhouorg93Z. Zhouorg22H. Zhuorg22J. Zhuorg7A. Zichichiorg12,org27A. Zimmermannorg106M.B. Zimmermannorg35G. Zinovjevorg3J. Zmeskalorg116S.",
"Zouorg7 Affiliation notes org*Deceased orgIDipartimento DET del Politecnico di Torino, Turin, Italy orgIIGeorgia State University, Atlanta, Georgia, United States orgIIIM.V.",
"Lomonosov Moscow State University, D.V.",
"Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia orgIVDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India orgVInstitute of Theoretical Physics, University of Wroclaw, Poland Collaboration Institutes org1A.I.",
"Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine org4Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India org5Budker Institute for Nuclear Physics, Novosibirsk, Russia org6California Polytechnic State University, San Luis Obispo, California, United States org7Central China Normal University, Wuhan, China org8Centre de Calcul de l'IN2P3, Villeurbanne, Lyon, France org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi', Rome, Italy org13Chicago State University, Chicago, Illinois, United States org14China Institute of Atomic Energy, Beijing, China org15COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan org16Departamento de Física de Partículas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain org17Department of Physics, Aligarh Muslim University, Aligarh, India org18Department of Physics, Ohio State University, Columbus, Ohio, United States org19Department of Physics, Pusan National University, Pusan, Republic of Korea org20Department of Physics, Sejong University, Seoul, Republic of Korea org21Department of Physics, University of Oslo, Oslo, Norway org22Department of Physics and Technology, University of Bergen, Bergen, Norway org23Dipartimento di Fisica dell'Università 'La Sapienza' and Sezione INFN, Rome, Italy org24Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy org25Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy org26Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy org27Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy org28Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy org29Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy org30Dipartimento di Fisica `E.R.",
"Caianiello' dell'Università and Gruppo Collegato INFN, Salerno, Italy org31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy org32Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy org33Dipartimento Interateneo di Fisica `M. Merlin' and Sezione INFN, Bari, Italy org34Division of Experimental High Energy Physics, University of Lund, Lund, Sweden org35European Organization for Nuclear Research (CERN), Geneva, Switzerland org36Excellence Cluster Universe, Technische Universität München, Munich, Germany org37Faculty of Engineering, Bergen University College, Bergen, Norway org38Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia org39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic org40Faculty of Science, P.J.",
"Šafárik University, Košice, Slovakia org41Faculty of Technology, Buskerud and Vestfold University College, Tonsberg, Norway org42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org43Gangneung-Wonju National University, Gangneung, Republic of Korea org44Gauhati University, Department of Physics, Guwahati, India org45Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany org46Helsinki Institute of Physics (HIP), Helsinki, Finland org47Hiroshima University, Hiroshima, Japan org48Indian Institute of Technology Bombay (IIT), Mumbai, India org49Indian Institute of Technology Indore, Indore, India org50Indonesian Institute of Sciences, Jakarta, Indonesia org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy org52INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy org53INFN, Sezione di Bari, Bari, Italy org54INFN, Sezione di Bologna, Bologna, Italy org55INFN, Sezione di Cagliari, Cagliari, Italy org56INFN, Sezione di Catania, Catania, Italy org57INFN, Sezione di Padova, Padova, Italy org58INFN, Sezione di Roma, Rome, Italy org59INFN, Sezione di Torino, Turin, Italy org60INFN, Sezione di Trieste, Trieste, Italy org61Inha University, Incheon, Republic of Korea org62Institut de Physique Nucléaire d'Orsay (IPNO), Université Paris-Sud, CNRS-IN2P3, Orsay, France org63Institute for Nuclear Research, Academy of Sciences, Moscow, Russia org64Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands org65Institute for Theoretical and Experimental Physics, Moscow, Russia org66Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia org67Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic org68Institute of Physics, Bhubaneswar, India org69Institute of Space Science (ISS), Bucharest, Romania org70Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org71Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org72Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany org73Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico org74Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil org75Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico org76IRFU, CEA, Université Paris-Saclay, Saclay, France org77iThemba LABS, National Research Foundation, Somerset West, South Africa org78Joint Institute for Nuclear Research (JINR), Dubna, Russia org79Konkuk University, Seoul, Republic of Korea org80Korea Institute of Science and Technology Information, Daejeon, Republic of Korea org81KTO Karatay University, Konya, Turkey org82Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France org83Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France org84Lawrence Berkeley National Laboratory, Berkeley, California, United States org85Moscow Engineering Physics Institute, Moscow, Russia org86Nagasaki Institute of Applied Science, Nagasaki, Japan org87National and Kapodistrian University of Athens, Physics Department, Athens, Greece org88National Centre for Nuclear Studies, Warsaw, Poland org89National Institute for Physics and Nuclear Engineering, Bucharest, Romania org90National Institute of Science Education and Research, HBNI, Jatni, India org91National Nuclear Research Center, Baku, Azerbaijan org92National Research Centre Kurchatov Institute, Moscow, Russia org93Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark org94Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam, Netherlands org95Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom org96Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u Prahy, Czech Republic org97Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States org98Petersburg Nuclear Physics Institute, Gatchina, Russia org99Physics Department, Creighton University, Omaha, Nebraska, United States org100Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia org101Physics Department, Panjab University, Chandigarh, India org102Physics Department, University of Cape Town, Cape Town, South Africa org103Physics Department, University of Jammu, Jammu, India org104Physics Department, University of Rajasthan, Jaipur, India org105Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany org106Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany org107Physik Department, Technische Universität München, Munich, Germany org108Purdue University, West Lafayette, Indiana, United States org109Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany org110Rudjer Bošković Institute, Zagreb, Croatia org111Russian Federal Nuclear Center (VNIIEF), Sarov, Russia org112Saha Institute of Nuclear Physics, Kolkata, India org113School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom org114Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru org115SSC IHEP of NRC Kurchatov institute, Protvino, Russia org116Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria org117SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France org118Suranaree University of Technology, Nakhon Ratchasima, Thailand org119Technical University of Košice, Košice, Slovakia org120Technical University of Split FESB, Split, Croatia org121The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland org122The University of Texas at Austin, Physics Department, Austin, Texas, United States org123Universidad Autónoma de Sinaloa, Culiacán, Mexico org124Universidade de São Paulo (USP), São Paulo, Brazil org125Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil org126Universidade Federal do ABC, Santo Andre, Brazil org127University of Houston, Houston, Texas, United States org128University of Jyväskylä, Jyväskylä, Finland org129University of Liverpool, Liverpool, United Kingdom org130University of Tennessee, Knoxville, Tennessee, United States org131University of the Witwatersrand, Johannesburg, South Africa org132University of Tokyo, Tokyo, Japan org133University of Tsukuba, Tsukuba, Japan org134Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France org135Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France org136Università degli Studi di Pavia, Pavia, Italy org137Università di Brescia, Brescia, Italy org138V.",
"Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia org139Variable Energy Cyclotron Centre, Kolkata, India org140Warsaw University of Technology, Warsaw, Poland org141Wayne State University, Detroit, Michigan, United States org142Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary org143Yale University, New Haven, Connecticut, United States org144Yonsei University, Seoul, Republic of Korea org145Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany"
],
[
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"Alfaro Molinaorg75A. Aliciorg54,org27,org12A. Alkinorg3J. Almeorg22T. Altorg71L. Altenkamperorg22I. Altsybeevorg138C.",
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"Bello Martinezorg2R. Bellwiedorg127L.G.E. Beltranorg123V. Belyaevorg85G. Bencediorg142S. Beoleorg26A. Bercuciorg89Y. Berdnikovorg98D. Berenyiorg142R.A. Bertensorg130D. Berzanoorg35L. Betevorg35A. Bhasinorg103I.R. Bhatorg103A.K. Bhatiorg101B. Bhattacharjeeorg44J. Bhomorg121L. Bianchiorg127N. Bianchiorg51C. Bianchinorg141J. Bielčíkorg39J. Bielčíkováorg96A. Bilandzicorg107,org36G. Biroorg142R. Biswasorg4S. Biswasorg4J.T. Blairorg122D. Blauorg92C. Blumeorg71G. Bocaorg136F. Bockorg84,org35,org106A. Bogdanovorg85L. Boldizsárorg142M. Bombaraorg40G. Bonomiorg137M. Bonoraorg35J. Bookorg71H. Borelorg76A. Borissovorg19M. Borriorg129E. Bottaorg26C. Bourjauorg93L. Bratrudorg71P. Braun-Munzingerorg109M. Bregantorg124T.A. Brokerorg71M. Brozorg39E.J. Bruckenorg46E. Brunaorg59G.E. Brunoorg33D. Budnikovorg111H. Bueschingorg71S. Bufalinoorg31P. Buhlerorg116P. Buncicorg35O. Buschorg133Z. Butheleziorg77J.B. Buttorg15J.T. Buxtonorg18J. Cabalaorg119D. Caffarriorg35,org94H. Cainesorg143A. Calivaorg64E.",
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"Castillo Castellanosorg76A.J. Castroorg130E.A.R. Casulaorg55C.",
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"Conesa del Valleorg62M.E. Connorsorg143orgIIJ.G. Contrerasorg39T.M. Cormierorg97Y.",
"Corrales Moralesorg59I.",
"Cortés Maldonadoorg2P. Corteseorg32M.R. Cosentinoorg126F. Costaorg35S. Costanzaorg136J. Crkovskáorg62P. Crochetorg82E. Cuautleorg73L. Cunqueiroorg72T. Dahmsorg36,org107A. Daineseorg57M.C. Danischorg106A. Danuorg69D. Dasorg112I. Dasorg112S. Dasorg4A. Dashorg90S. Dashorg48S. Deorg124,org49A.",
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"de Cataldoorg53C.",
"de Contiorg124J.",
"de Cuvelandorg42A.",
"De Falcoorg24D.",
"De Gruttolaorg30,org12N.",
"De Marcoorg59S.",
"De Pasqualeorg30R.D.",
"De Souzaorg125H.F. Degenhardtorg124A. Deistingorg109,org106A. Delofforg88C. Deplanoorg94P. Dhankherorg48D.",
"Di Bariorg33A.",
"Di Mauroorg35P.",
"Di Nezzaorg51B.",
"Di Ruzzaorg57M.A.",
"Diaz Corcheroorg10T. Dietelorg102P. Dillensegerorg71R. Diviàorg35Ø. Djuvslandorg22A. Dobrinorg35D.",
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"Natal da Luzorg124C. Nattrassorg130S.R. Navarroorg2K. Nayakorg90R. Nayakorg48T.K. Nayakorg139S. Nazarenkoorg111A. Nedosekinorg65R.A.",
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"Oliveira Da Silvaorg124M.H. Oliverorg143J. Onderwaaterorg109C. Oppedisanoorg59R. Oravaorg46M. Oravecorg119A.",
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"Perez Lezamaorg71V. Peskovorg71Y. Pestovorg5V. Petráčekorg39V. Petrovorg115M. Petroviciorg89C. Pettaorg28R.P. Pezziorg74S. Pianoorg60M. Piknaorg38P. Pillotorg117L.O.D.L. Pimentelorg93O. Pinazzaorg35,org54L. Pinskyorg127D.B. Piyarathnaorg127M. Płoskońorg84M. Planinicorg100F. Pliquettorg71J. Plutaorg140S. Pochybovaorg142P.L.M. Podesta-Lermaorg123M.G. Poghosyanorg97B. Polichtchoukorg115N. Poljakorg100W. Poonsawatorg118A. Poporg89H. Poppenborgorg72S. Porteboeuf-Houssaisorg82V. Pozdniakovorg78S.K. Prasadorg4R. Preghenellaorg54F. Prinoorg59C.A. Pruneauorg141I. Pshenichnovorg63M. Puccioorg26G. Pudduorg24P. Pujahariorg141V. Puninorg111J. Putschkeorg141A. Rachevskiorg60S. Rahaorg4S. Rajputorg103J. Rakorg128A. Rakotozafindrabeorg76L. Ramelloorg32F. Ramiorg135D.B. Ranaorg127R. Raniwalaorg104S. Raniwalaorg104S.S. Räsänenorg46B.T. Rascanuorg71D. Ratheeorg101V. Ratzaorg45I. Ravasengaorg31K.F. Readorg97,org130K. Redlichorg88orgVA. Rehmanorg22P. Reicheltorg71F. Reidtorg35X. Renorg7R. Renfordtorg71A.R. Reolonorg51A. Reshetinorg63K. Reygersorg106V. Riabovorg98R.A. Ricciorg52T. Richertorg64M. Richterorg21P. Riedlerorg35W. Rieglerorg35F. Riggiorg28C. Risteaorg69M.",
"Rodríguez Cahuantziorg2K. Røedorg21E. Rogochayaorg78D. Rohrorg42,org35D. Röhrichorg22P.S. Rokitaorg140F. Ronchettiorg51E.D. Rosasorg73P. Rosnetorg82A. Rossiorg29,org57A. Rotondiorg136F. Roukoutakisorg87A. Royorg49C. Royorg135P. Royorg112A.J.",
"Rubio Monteroorg10O.V. Ruedaorg73R. Ruiorg25B. Rumyantsevorg78A. Rustamovorg91E. Ryabinkinorg92Y. Ryabovorg98A. Rybickiorg121S. Saarinenorg46S. Sadhuorg139S. Sadovskyorg115K. Šafaříkorg35S.K. Sahaorg139B. Sahlmullerorg71B. Sahooorg48P. Sahooorg49R. Sahooorg49S. Sahooorg68P.K. Sahuorg68J. Sainiorg139S. Sakaiorg133M.A. Salehorg141J. Salzwedelorg18S. Sambyalorg103V. Samsonovorg85,org98A. Sandovalorg75D. Sarkarorg139N. Sarkarorg139P. Sarmaorg44M.H.P. Sasorg64E. Scapparoneorg54F. Scarlassaraorg29R.P. Scharenbergorg108H.S. Scheidorg71C. Schiauaorg89R. Schickerorg106C. Schmidtorg109H.R. Schmidtorg105M.O. Schmidtorg106M. Schmidtorg105N.V. Schmidtorg71,org97S. Schuchmannorg106J. Schukraftorg35Y. Schutzorg135,org117,org35K. Schwarzorg109K. Schwedaorg109G. Scioliorg27E. Scomparinorg59R. Scottorg130M. Šefčíkorg40J.E. Segerorg99Y. Sekiguchiorg132D. Sekihataorg47I. Selyuzhenkovorg85,org109K. Senosiorg77S. Senyukovorg35,org3,org135E. Serradillaorg10,org75P. Settorg48A. Sevcencoorg69A. Shabanovorg63A. Shabetaiorg117R. Shahoyanorg35W. Shaikhorg112A. Shangaraevorg115A. Sharmaorg101A. Sharmaorg103M. Sharmaorg103M. Sharmaorg103N. Sharmaorg101,org130A.I. Sheikhorg139K. Shigakiorg47Q. Shouorg7K. Shtejerorg9,org26Y. Sibiriakorg92S. Siddhantaorg55K.M. Sielewiczorg35T. Siemiarczukorg88D. Silvermyrorg34C. Silvestreorg83G. Simatovicorg100G. Simonettiorg35R. Singarajuorg139R. Singhorg90V. Singhalorg139T. Sinhaorg112B. Sitarorg38M. Sittaorg32T.B. Skaaliorg21M. Slupeckiorg128N. Smirnovorg143R.J.M. Snellingsorg64T.W. Snellmanorg128J. Songorg19M. Songorg144F. Soramelorg29S. Sorensenorg130F. Sozziorg109E. Spiritiorg51I. Sputowskaorg121B.K. Srivastavaorg108J. Stachelorg106I. Stanorg69P. Stankusorg97E. Stenlundorg34D. Stoccoorg117M.M. Storetvedtorg37P. Strmenorg38A.A.P. Suaideorg124T. Sugitateorg47C. Suireorg62M. Suleymanovorg15M. Suljicorg25R. Sultanovorg65M. Šumberaorg96S. Sumowidagdoorg50K. Suzukiorg116S. Swainorg68A. Szaboorg38I. Szarkaorg38U. Tabassamorg15J. Takahashiorg125G.J. Tambaveorg22N. Tanakaorg133M. Tarhiniorg62M. Tariqorg17M.G. Tarzilaorg89A. Tauroorg35G.",
"Tejeda Muñozorg2A. Telescaorg35K. Terasakiorg132C. Terrevoliorg29B. Teyssierorg134D. Thakurorg49S. Thakurorg139D. Thomasorg122F. Thoresenorg93R. Tieulentorg134A. Tikhonovorg63A.R. Timminsorg127A. Toiaorg71S.R. Torresorg123S. Tripathyorg49S. Trogoloorg26G. Trombettaorg33L. Tropporg40V. Trubnikovorg3W.H. Trzaskaorg128B.A. Trzeciakorg64T. Tsujiorg132A. Tumkinorg111R. Turrisiorg57T.S. Tveterorg21K. Ullalandorg22E.N. Umakaorg127A. Urasorg134G.L. Usaiorg24A. Utrobicicorg100M. Valaorg119,org66J.",
"Van Der Maarelorg64J.W.",
"Van Hoorneorg35M.",
"van Leeuwenorg64T. Vanatorg96P.",
"Vande Vyvreorg35D. Vargaorg142A. Vargasorg2M. Vargyasorg128R. Varmaorg48M. Vasileiouorg87A. Vasilievorg92A. Vauthierorg83O.",
"Vázquez Doceorg107,org36V. Vecherninorg138A.M. Veenorg64A. Velureorg22E. Vercellinorg26S.",
"Vergara Limónorg2R. Vernetorg8R. Vértesiorg142L. Vickovicorg120S. Vigoloorg64J. Viinikainenorg128Z. Vilakaziorg131O.",
"Villalobos Baillieorg113A.",
"Villatoro Telloorg2A. Vinogradovorg92L. Vinogradovorg138T. Virgiliorg30V. Vislaviciusorg34A. Vodopyanovorg78M.A. Völklorg106,org105K. Voloshinorg65S.A. Voloshinorg141G. Volpeorg33B.",
"von Hallerorg35I. Vorobyevorg107,org36D. Voscekorg119D. Vranicorg35,org109J. Vrlákováorg40B. Wagnerorg22H. Wangorg64M. Wangorg7D. Watanabeorg133Y. Watanabeorg132,org133M. Weberorg116S.G. Weberorg109D.F. Weiserorg106S.C. Wenzelorg35J.P. Wesselsorg72U. Westerhofforg72A.M. Whiteheadorg102J. Wiechulaorg71J. Wikneorg21G. Wilkorg88J. Wilkinsonorg106,org54G.A. Willemsorg72M.C.S. Williamsorg54E. Willsherorg113B. Windelbandorg106W.E. Wittorg130S. Yalcinorg81K. Yamakawaorg47P. Yangorg7S. Yanoorg47Z. Yinorg7H. Yokoyamaorg133,org83I.-K. Yooorg35,org19J.H. Yoonorg61V. Yurchenkoorg3V. Zaccoloorg59,org93A. Zamanorg15C. Zampolliorg35H.J.C. Zanoliorg124N. Zardoshtiorg113A. Zarochentsevorg138P. Závadaorg67N. Zaviyalovorg111H. Zbroszczykorg140M. Zhalovorg98H. Zhangorg22,org7X. Zhangorg7Y. Zhangorg7C. Zhangorg64Z. Zhangorg82,org7C. Zhaoorg21N. Zhigarevaorg65D. Zhouorg7Y. Zhouorg93Z. Zhouorg22H. Zhuorg22J. Zhuorg7A. Zichichiorg12,org27A. Zimmermannorg106M.B. Zimmermannorg35G. Zinovjevorg3J. Zmeskalorg116S.",
"Zouorg7"
],
[
"Affiliation notes",
"org*Deceased orgIDipartimento DET del Politecnico di Torino, Turin, Italy orgIIGeorgia State University, Atlanta, Georgia, United States orgIIIM.V.",
"Lomonosov Moscow State University, D.V.",
"Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia orgIVDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India orgVInstitute of Theoretical Physics, University of Wroclaw, Poland Collaboration Institutes org1A.I.",
"Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine org4Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India org5Budker Institute for Nuclear Physics, Novosibirsk, Russia org6California Polytechnic State University, San Luis Obispo, California, United States org7Central China Normal University, Wuhan, China org8Centre de Calcul de l'IN2P3, Villeurbanne, Lyon, France org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi', Rome, Italy org13Chicago State University, Chicago, Illinois, United States org14China Institute of Atomic Energy, Beijing, China org15COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan org16Departamento de Física de Partículas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain org17Department of Physics, Aligarh Muslim University, Aligarh, India org18Department of Physics, Ohio State University, Columbus, Ohio, United States org19Department of Physics, Pusan National University, Pusan, Republic of Korea org20Department of Physics, Sejong University, Seoul, Republic of Korea org21Department of Physics, University of Oslo, Oslo, Norway org22Department of Physics and Technology, University of Bergen, Bergen, Norway org23Dipartimento di Fisica dell'Università 'La Sapienza' and Sezione INFN, Rome, Italy org24Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy org25Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy org26Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy org27Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy org28Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy org29Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy org30Dipartimento di Fisica `E.R.",
"Caianiello' dell'Università and Gruppo Collegato INFN, Salerno, Italy org31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy org32Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy org33Dipartimento Interateneo di Fisica `M. Merlin' and Sezione INFN, Bari, Italy org34Division of Experimental High Energy Physics, University of Lund, Lund, Sweden org35European Organization for Nuclear Research (CERN), Geneva, Switzerland org36Excellence Cluster Universe, Technische Universität München, Munich, Germany org37Faculty of Engineering, Bergen University College, Bergen, Norway org38Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia org39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic org40Faculty of Science, P.J.",
"Šafárik University, Košice, Slovakia org41Faculty of Technology, Buskerud and Vestfold University College, Tonsberg, Norway org42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org43Gangneung-Wonju National University, Gangneung, Republic of Korea org44Gauhati University, Department of Physics, Guwahati, India org45Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany org46Helsinki Institute of Physics (HIP), Helsinki, Finland org47Hiroshima University, Hiroshima, Japan org48Indian Institute of Technology Bombay (IIT), Mumbai, India org49Indian Institute of Technology Indore, Indore, India org50Indonesian Institute of Sciences, Jakarta, Indonesia org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy org52INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy org53INFN, Sezione di Bari, Bari, Italy org54INFN, Sezione di Bologna, Bologna, Italy org55INFN, Sezione di Cagliari, Cagliari, Italy org56INFN, Sezione di Catania, Catania, Italy org57INFN, Sezione di Padova, Padova, Italy org58INFN, Sezione di Roma, Rome, Italy org59INFN, Sezione di Torino, Turin, Italy org60INFN, Sezione di Trieste, Trieste, Italy org61Inha University, Incheon, Republic of Korea org62Institut de Physique Nucléaire d'Orsay (IPNO), Université Paris-Sud, CNRS-IN2P3, Orsay, France org63Institute for Nuclear Research, Academy of Sciences, Moscow, Russia org64Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands org65Institute for Theoretical and Experimental Physics, Moscow, Russia org66Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia org67Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic org68Institute of Physics, Bhubaneswar, India org69Institute of Space Science (ISS), Bucharest, Romania org70Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org71Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org72Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany org73Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico org74Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil org75Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico org76IRFU, CEA, Université Paris-Saclay, Saclay, France org77iThemba LABS, National Research Foundation, Somerset West, South Africa org78Joint Institute for Nuclear Research (JINR), Dubna, Russia org79Konkuk University, Seoul, Republic of Korea org80Korea Institute of Science and Technology Information, Daejeon, Republic of Korea org81KTO Karatay University, Konya, Turkey org82Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France org83Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France org84Lawrence Berkeley National Laboratory, Berkeley, California, United States org85Moscow Engineering Physics Institute, Moscow, Russia org86Nagasaki Institute of Applied Science, Nagasaki, Japan org87National and Kapodistrian University of Athens, Physics Department, Athens, Greece org88National Centre for Nuclear Studies, Warsaw, Poland org89National Institute for Physics and Nuclear Engineering, Bucharest, Romania org90National Institute of Science Education and Research, HBNI, Jatni, India org91National Nuclear Research Center, Baku, Azerbaijan org92National Research Centre Kurchatov Institute, Moscow, Russia org93Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark org94Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam, Netherlands org95Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom org96Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u Prahy, Czech Republic org97Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States org98Petersburg Nuclear Physics Institute, Gatchina, Russia org99Physics Department, Creighton University, Omaha, Nebraska, United States org100Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia org101Physics Department, Panjab University, Chandigarh, India org102Physics Department, University of Cape Town, Cape Town, South Africa org103Physics Department, University of Jammu, Jammu, India org104Physics Department, University of Rajasthan, Jaipur, India org105Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany org106Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany org107Physik Department, Technische Universität München, Munich, Germany org108Purdue University, West Lafayette, Indiana, United States org109Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany org110Rudjer Bošković Institute, Zagreb, Croatia org111Russian Federal Nuclear Center (VNIIEF), Sarov, Russia org112Saha Institute of Nuclear Physics, Kolkata, India org113School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom org114Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru org115SSC IHEP of NRC Kurchatov institute, Protvino, Russia org116Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria org117SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France org118Suranaree University of Technology, Nakhon Ratchasima, Thailand org119Technical University of Košice, Košice, Slovakia org120Technical University of Split FESB, Split, Croatia org121The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland org122The University of Texas at Austin, Physics Department, Austin, Texas, United States org123Universidad Autónoma de Sinaloa, Culiacán, Mexico org124Universidade de São Paulo (USP), São Paulo, Brazil org125Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil org126Universidade Federal do ABC, Santo Andre, Brazil org127University of Houston, Houston, Texas, United States org128University of Jyväskylä, Jyväskylä, Finland org129University of Liverpool, Liverpool, United Kingdom org130University of Tennessee, Knoxville, Tennessee, United States org131University of the Witwatersrand, Johannesburg, South Africa org132University of Tokyo, Tokyo, Japan org133University of Tsukuba, Tsukuba, Japan org134Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France org135Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France org136Università degli Studi di Pavia, Pavia, Italy org137Università di Brescia, Brescia, Italy org138V.",
"Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia org139Variable Energy Cyclotron Centre, Kolkata, India org140Warsaw University of Technology, Warsaw, Poland org141Wayne State University, Detroit, Michigan, United States org142Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary org143Yale University, New Haven, Connecticut, United States org144Yonsei University, Seoul, Republic of Korea org145Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany org1A.I.",
"Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine org4Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India org5Budker Institute for Nuclear Physics, Novosibirsk, Russia org6California Polytechnic State University, San Luis Obispo, California, United States org7Central China Normal University, Wuhan, China org8Centre de Calcul de l'IN2P3, Villeurbanne, Lyon, France org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi', Rome, Italy org13Chicago State University, Chicago, Illinois, United States org14China Institute of Atomic Energy, Beijing, China org15COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan org16Departamento de Física de Partículas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain org17Department of Physics, Aligarh Muslim University, Aligarh, India org18Department of Physics, Ohio State University, Columbus, Ohio, United States org19Department of Physics, Pusan National University, Pusan, Republic of Korea org20Department of Physics, Sejong University, Seoul, Republic of Korea org21Department of Physics, University of Oslo, Oslo, Norway org22Department of Physics and Technology, University of Bergen, Bergen, Norway org23Dipartimento di Fisica dell'Università 'La Sapienza' and Sezione INFN, Rome, Italy org24Dipartimento di Fisica dell'Università and Sezione INFN, Cagliari, Italy org25Dipartimento di Fisica dell'Università and Sezione INFN, Trieste, Italy org26Dipartimento di Fisica dell'Università and Sezione INFN, Turin, Italy org27Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Bologna, Italy org28Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Catania, Italy org29Dipartimento di Fisica e Astronomia dell'Università and Sezione INFN, Padova, Italy org30Dipartimento di Fisica `E.R.",
"Caianiello' dell'Università and Gruppo Collegato INFN, Salerno, Italy org31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy org32Dipartimento di Scienze e Innovazione Tecnologica dell'Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy org33Dipartimento Interateneo di Fisica `M. Merlin' and Sezione INFN, Bari, Italy org34Division of Experimental High Energy Physics, University of Lund, Lund, Sweden org35European Organization for Nuclear Research (CERN), Geneva, Switzerland org36Excellence Cluster Universe, Technische Universität München, Munich, Germany org37Faculty of Engineering, Bergen University College, Bergen, Norway org38Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia org39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic org40Faculty of Science, P.J.",
"Šafárik University, Košice, Slovakia org41Faculty of Technology, Buskerud and Vestfold University College, Tonsberg, Norway org42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org43Gangneung-Wonju National University, Gangneung, Republic of Korea org44Gauhati University, Department of Physics, Guwahati, India org45Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany org46Helsinki Institute of Physics (HIP), Helsinki, Finland org47Hiroshima University, Hiroshima, Japan org48Indian Institute of Technology Bombay (IIT), Mumbai, India org49Indian Institute of Technology Indore, Indore, India org50Indonesian Institute of Sciences, Jakarta, Indonesia org51INFN, Laboratori Nazionali di Frascati, Frascati, Italy org52INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy org53INFN, Sezione di Bari, Bari, Italy org54INFN, Sezione di Bologna, Bologna, Italy org55INFN, Sezione di Cagliari, Cagliari, Italy org56INFN, Sezione di Catania, Catania, Italy org57INFN, Sezione di Padova, Padova, Italy org58INFN, Sezione di Roma, Rome, Italy org59INFN, Sezione di Torino, Turin, Italy org60INFN, Sezione di Trieste, Trieste, Italy org61Inha University, Incheon, Republic of Korea org62Institut de Physique Nucléaire d'Orsay (IPNO), Université Paris-Sud, CNRS-IN2P3, Orsay, France org63Institute for Nuclear Research, Academy of Sciences, Moscow, Russia org64Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands org65Institute for Theoretical and Experimental Physics, Moscow, Russia org66Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia org67Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic org68Institute of Physics, Bhubaneswar, India org69Institute of Space Science (ISS), Bucharest, Romania org70Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org71Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org72Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany org73Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico org74Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil org75Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico org76IRFU, CEA, Université Paris-Saclay, Saclay, France org77iThemba LABS, National Research Foundation, Somerset West, South Africa org78Joint Institute for Nuclear Research (JINR), Dubna, Russia org79Konkuk University, Seoul, Republic of Korea org80Korea Institute of Science and Technology Information, Daejeon, Republic of Korea org81KTO Karatay University, Konya, Turkey org82Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France org83Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France org84Lawrence Berkeley National Laboratory, Berkeley, California, United States org85Moscow Engineering Physics Institute, Moscow, Russia org86Nagasaki Institute of Applied Science, Nagasaki, Japan org87National and Kapodistrian University of Athens, Physics Department, Athens, Greece org88National Centre for Nuclear Studies, Warsaw, Poland org89National Institute for Physics and Nuclear Engineering, Bucharest, Romania org90National Institute of Science Education and Research, HBNI, Jatni, India org91National Nuclear Research Center, Baku, Azerbaijan org92National Research Centre Kurchatov Institute, Moscow, Russia org93Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark org94Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam, Netherlands org95Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom org96Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u Prahy, Czech Republic org97Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States org98Petersburg Nuclear Physics Institute, Gatchina, Russia org99Physics Department, Creighton University, Omaha, Nebraska, United States org100Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia org101Physics Department, Panjab University, Chandigarh, India org102Physics Department, University of Cape Town, Cape Town, South Africa org103Physics Department, University of Jammu, Jammu, India org104Physics Department, University of Rajasthan, Jaipur, India org105Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen, Germany org106Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany org107Physik Department, Technische Universität München, Munich, Germany org108Purdue University, West Lafayette, Indiana, United States org109Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany org110Rudjer Bošković Institute, Zagreb, Croatia org111Russian Federal Nuclear Center (VNIIEF), Sarov, Russia org112Saha Institute of Nuclear Physics, Kolkata, India org113School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom org114Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru org115SSC IHEP of NRC Kurchatov institute, Protvino, Russia org116Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria org117SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France org118Suranaree University of Technology, Nakhon Ratchasima, Thailand org119Technical University of Košice, Košice, Slovakia org120Technical University of Split FESB, Split, Croatia org121The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland org122The University of Texas at Austin, Physics Department, Austin, Texas, United States org123Universidad Autónoma de Sinaloa, Culiacán, Mexico org124Universidade de São Paulo (USP), São Paulo, Brazil org125Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil org126Universidade Federal do ABC, Santo Andre, Brazil org127University of Houston, Houston, Texas, United States org128University of Jyväskylä, Jyväskylä, Finland org129University of Liverpool, Liverpool, United Kingdom org130University of Tennessee, Knoxville, Tennessee, United States org131University of the Witwatersrand, Johannesburg, South Africa org132University of Tokyo, Tokyo, Japan org133University of Tsukuba, Tsukuba, Japan org134Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France org135Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France org136Università degli Studi di Pavia, Pavia, Italy org137Università di Brescia, Brescia, Italy org138V.",
"Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia org139Variable Energy Cyclotron Centre, Kolkata, India org140Warsaw University of Technology, Warsaw, Poland org141Wayne State University, Detroit, Michigan, United States org142Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary org143Yale University, New Haven, Connecticut, United States org144Yonsei University, Seoul, Republic of Korea org145Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany"
]
] | 1709.01731 |
[
[
"Stocator: A High Performance Object Store Connector for Spark"
],
[
"Abstract We present Stocator, a high performance object store connector for Apache Spark, that takes advantage of object store semantics.",
"Previous connectors have assumed file system semantics, in particular, achieving fault tolerance and allowing speculative execution by creating temporary files to avoid interference between worker threads executing the same task and then renaming these files.",
"Rename is not a native object store operation; not only is it not atomic, but it is implemented using a costly copy operation and a delete.",
"Instead our connector leverages the inherent atomicity of object creation, and by avoiding the rename paradigm it greatly decreases the number of operations on the object store as well as enabling a much simpler approach to dealing with the eventually consistent semantics typical of object stores.",
"We have implemented Stocator and shared it in open source.",
"Performance testing shows that it is as much as 18 times faster for write intensive workloads and performs as much as 30 times fewer operations on the object store than the legacy Hadoop connectors, reducing costs both for the client and the object storage service provider."
],
[
"Introduction",
"Data is the natural resource of the 21st century.",
"It is being produced at dizzying rates, e.g., for genomics by sequencers, for healthcare through a variety of imaging modalities, and for Internet of Things (IoT) by multitudes of sensors.",
"This data increasingly resides in cloud object stores, such as AWS S3[4], Azure Blob storage[15], and IBM Cloud Object Storage[20], which are highly scalable distributed cloud storage systems that offer high capacity, cost effective storage.",
"But it is not enough just to store data; we also need to derive value from it, in particular, through analytics engines such as Apache Hadoop[8] and Apache Spark[14].",
"However, these highly distributed analytics engines were originally designed work on data stored in HDFS (Hadoop Distributed File System) where the storage and processing are co-located in the same server cluster.",
"Moving data from object storage to HDFS in order to process it and then moving the results back to object storage for long term storage is inefficient.",
"In this paper we present Stocator[22], a high performance storage connector, that enables Hadoop-based analytics engines to work directly on data stored in object storage systems.",
"Here we focus on Spark; our work can be extended to work with the other parts of the Hadoop ecosystem.",
"Current connectors to object stores for Spark, e.g., S3a[7] and the Hadoop Swift Connector[38] are notorious for their poor performance[40] for write workloads and sometimes leaving behind temporary objects that do not get deleted.",
"The poor performance of these connectors follows from their assumption of file system semantics, a natural assumption given that their model of operation is based on the way that Hadoop interacts with its original storage system, HDFS[9].",
"In particular, Spark and Hadoop achieve fault tolerance and enable speculative execution by creating temporary files and then renaming these files.",
"This paradigm avoids interference between threads doing the same work and thus writing output with the same name.",
"Notice, however, that rename is not a native object store operation; not only is it not atomic, but it must be implemented using a costly copy operation, followed by a delete.",
"Current connectors can also lead to failures and incorrect executions because the list operation on containers/buckets is eventually consistent.",
"EMRFS[3] from Amazon and S3mper[26] from Netflix overcome eventual consistency by storing file metadata in DynamoDB[1], an additional strongly consistent storage system separate from the object store.",
"A similar feature called S3Guard[19] that also requires an additional strongly consistent storage system is being developed by the Hadoop open source community for the S3a connector.",
"Solutions like these, which require multiple storage systems, are complex and can introduce issues of consistency between the stores.",
"They also add cost since users must pay for the additional strongly consistent storage.",
"Others have tried to improve the performance of object store connectors, e.g., the DirectOutputCommitter[29] for S3a introduced by Databricks, but have failed to preserve the fault tolerance and speculation properties of the temporary file/rename paradigm.",
"There are also recommendations in the Hadoop open source community to abandon speculation and employ an optimization[10] that renames files to their final names when tasks complete (commit) instead of waiting for the completion of the entire job.",
"However, incorrect executions, though rare, can still occur even with speculation turned off due to the eventually consistent list operations employed at task commit to determine which objects to rename.",
"In this paper we present a high performance object store connector for Apache Spark that takes full advantage of object store semantics, enables speculative execution and also deals correctly with eventual consistency.",
"Our connector eliminates the rename paradigm by writing each output object to its final name.",
"The name includes both the part number and the attempt number, so that multiple attempts to write the same part due to speculation or fault tolerance use different object names.",
"Avoiding rename also removes the necessity to execute list operations to determine which objects to rename at task and job commit, so that a Spark job writes all of the parts constituting its output dataset correctly despite eventual consistency.",
"This reduces the issue of eventual consistency to ensuring that a subsequent job correctly determines the constituent parts when it reads the output of previous jobs.",
"Accordingly we extend an already existing success indicator object written at the end of a Spark job to include a manifest to indicate the part names that actually compose the final output.",
"A subsequent job reads the indicator object to determine which objects are part of the dataset.",
"Overall, our approach increases performance by greatly decreasing the number of operations on the object store and ensures correctness despite eventual consistency by greatly decreasing complexity.",
"Our connector also takes advantage of HTTP Chunked Transfer Encoding to stream the data being written to the object store as it is produced, thereby avoiding the need to write objects to local storage prior to being written to the object store.",
"We have implemented our connector for the OpenStack Swift API[28] and shared it in open source[23].",
"We have compared its performance with the S3a and Hadoop Swift connectors over a range of workloads and found that it executes far less operations on the object store, in some cases as little as one thirtieth of the operations.",
"Since the price for an object store service typically includes charges based on the number of operations executed, this reduction in the number of operations lowers the costs for clients in addition to reducing the load on client software.",
"It also reduces costs and load for the object store provider since it can serve more clients with the same amount of processing power.",
"Stocator also substantially increases performance for Spark workloads running over object storage, especially for write intensive workloads, where it is as much as 18 times faster.",
"In summary our contributions include: The design of a novel storage connector for Spark that leverages object storage semantics, avoiding costly copy operations and providing correct execution in the face of faults and speculation.",
"A solution that works correctly despite the eventually consistent semantics of object storage, yet without requiring additional strongly consistent storage.",
"An implementation that has been contributed to open source.",
"Stocator is in production in IBM Analytics for Apache Spark, a Bluemix service, and has enabled the SETI project to perform computationally intensive Spark workloads on multi-terabyte binary signal files[33].",
"The remainder of this paper is structured as follows.",
"In sec:background we present background on object storage and Apache Spark as well as the motivation for our work.",
"In sec:algorithm we describe how Stocator works.",
"In sec:methodology we present the methodology for our performance evaluation, including our experimental set up and a description of our workloads.",
"In sec:evaluation we present a detailed evaluation of Stocator, comparing its performance with existing Hadoop object storage connectors, from the point of view of runtime, number of operations and resource utilization.",
"sec:related discusses related work and finally in sec:conclusion we conclude."
],
[
"Background",
"We provide background material necessary for understanding the remainder of the paper.",
"First, we describe object storage and then the background on Spark[14] and its implementation that have implications on the way that it uses object storage.",
"Finally, we motivate the need for Stocator."
],
[
"Cloud Object Storage",
"An object encapsulates data and metadata describing the object and its data.",
"An entire object is created at once and cannot be updated in place, although the entire value of an object can be replaced.",
"Object storage is typically accessed through RESTful HTTP, which is a good fit for cloud applications.",
"This simple object semantics enables the implementation of highly scalable, distributed and durable object storage that can provide very large storage capacities at low cost.",
"Object storage is ideal for storing unstructured data, e.g., video, images, backups and documents such as web pages and blogs.",
"Examples of object storage systems include AWS S3[4], Azure Blob storage[15], OpenStack Swift[27] and IBM Cloud Object Storage[20].",
"Object storage has a shallow hierarchy.",
"A storage account may contain one or more buckets or containers (hereafter we use the term container), where each container may contain many objects.",
"Typically there is no hierarchy in a container, e.g., no containers within a container, although there is support for hierarchical naming.",
"In particular, when listing the contents of a container a separator character, e.g., “/” or “*”, between levels of naming can be specified as well as a prefix string, so that only the names for objects in the container starting with the prefix will be included.",
"This is different than file systems where there is both hierarchy in the implementation as well as in naming, i.e., a directory is a special file that can contain other files and directories.",
"Common operations on object storage include: PUT Object, which creates an object, with the name, data and metadata provided with the operation, GET object, which returns the data and metadata of the object, HEAD Object, which returns just the metadata of the object, GET Container, which lists the objects in a container, HEAD Container, which returns the metadata of a container, and DELETE Object, which deletes an object.",
"Object creation is atomic, so that two simultaneous PUTs on the same name will create an object with the data of one PUT , but not some combination of the two.",
"In order to enable a highly distributed implementation the consistency semantics for object storage often include some degree of eventual consistency[47].",
"Eventual consistency guarantees that if no new updates are made to a given data item, then eventually all accesses to that item will return the same value.",
"There are various aspects of eventual consistency.",
"For example, AWS[5] guarantees read after write consistency for its S3 object storage system, i.e., that a newly created object will be instantly visible.",
"Note that this does not necessarily include read after update, i.e., that a new value for an existing object name will be instantly visible, or read after delete, i.e., that a delete will make an object instantly invisible.",
"Another aspect of eventual consistency concerns the listing of the objects in a container; the creation and deletion of an object may be eventually consistent with respect to the listing of its container.",
"In particular, a container listing may not include a recently created object and may not exclude a recently deleted object."
],
[
"Spark",
"We describe Spark's execution model, how Spark interacts with storage, pointing out some of the problems that arise when Spark works on data in object storage."
],
[
"Spark execution model",
"The execution of a Spark application is orchestrated by the driver.",
"The driver divides the application into jobs and jobs into stages.",
"One stage does not begin execution until the previous stage has completed.",
"Stages consists of tasks, where each task is totally independent of the other tasks in that stage, so that the tasks can be executed in parallel.",
"The output of one stage is typically passed as the input to the next stage, so that a task reads its input from the output of the previous stage and/or from storage.",
"Similarly, a task writes its output to the next stage and/or to storage.",
"The driver creates worker processes called executors to which it assigns the execution of the tasks.",
"The execution of a task may fail.",
"In that case the driver may start a new execution of the same task.",
"The execution of a task may also be slow and in some cases the driver cannot tell whether the execution has failed or is just slow.",
"Spark has an important feature to deal with these cases called speculation, where it speculatively executes multiple executions of the same task in parallel.",
"Speculation can cut down on the total elapsed time for a Spark application/job.",
"Thus, a task may be executed multiple times and each such attempt to execute a task is assigned a unique identifier, containing a job identifier, a task identifier and an execution attempt number."
],
[
"Spark and its underlying storage",
"Spark interacts with its storage system through Hadoop[8], primarily through a component called the Hadoop Map Reduce Client Core (HMRCC) as shown in the diagram on the left side in fig:HadoopStorageConnectors.",
"Figure: Hadoop Storage ConnectorsHMRCC interacts with its underlying storage through the Hadoop File System Interface.",
"A connector that implements the interface must be implemented for each underlying storage system.",
"For example, the Hadoop distribution includes a connector for HDFS, as well as an S3a connector for the S3 object store API and a Swift connector for the OpenStack Swift object store API.",
"A task writes output to storage through the Hadoop FileOutputCommitter.",
"Since each task execution attempt needs to write an output file of the same name, Hadoop employs a rename strategy, where each execution attempt writes its own task temporary file.",
"At task commit, the output committer renames the task temporary file to a job temporary file.",
"Task commit is done by the executors, so it occurs in parallel.",
"And then when all of the tasks of a job complete, the driver calls the output committer to do job commit, which renames the job temporary files to their final names.",
"Job commit occurs in the driver after all of the tasks have committed and does not benefit from parallelism.",
"Figure: Sequence of names for part 0 of output from task temporary name to job temporary name to final name.fig:SequenceOfNames shows the names for a task's output.",
"This two stage strategy of task commit and then job commit was chosen to avoid the case where incomplete results might be interpreted as complete results.",
"However, Hadoop also writes a zero length object with the name _SUCCESS when a job completes successfully, so the case of incomplete results can easily by identified by the absence of a _SUCCESS object.",
"Accordingly, there is now a new version of the file output committer algorithm (version 2), where the task temporary files are renamed to their final names at task commit and job commit is largely reduced to the writing of the _SUCCESS object.",
"However, as of Hadoop 2.7.3, this algorithm is not yet the default output committer.",
"Hadoop is highly distributed and thus it keeps its state in its storage system, e.g., HDFS or object storage.",
"In particular, the output committer determines what temporary objects need to be renamed through directory listings, i.e., it lists the directory of the output dataset to find the directories and files holding task temporary and job temporary output.",
"In object stores this is done through container listing operations.",
"However, due to eventual consistency a container listing may not contain an object that was just successfully created, or it may still contain an object that was just successfully deleted.",
"This can lead to situations where some of the legitimate output objects do not get renamed by the output committer, so that the output of the Spark/Hadoop job will be incomplete.",
"This danger is compounded when speculation is enabled, and thus, despite the benefits of speculation, Spark users are encouraged to run with it disabled.",
"Furthermore, in order to avoid the dangers of eventual consistency entirely, Spark users are often encouraged to copy their input data to HDFS, run their Spark job over the data in HDFS, and then when it is complete, copy the output from HDFS back to object storage.",
"Note, however, that this adds considerable overhead.",
"Existing solutions to this problem require a consistent storage system in addition to object storage[26], [3], [12]."
],
[
"Motivation",
"To motivate the need for Stocator we show the sequence of interactions between Spark and its storage system for a program that executes a single task that produces a single output object as shown in fig:ProduceOneObject.",
"Figure: A Spark program that executes a single task that produces a single output object.Spark and Hadoop were originally designed to work with a file system.",
"Accordingly, tab:FileSystemOperations shows the series of file system operations that Spark carries out for the sample program.",
"Table: The file system operations executed on behalf of a Spark program that executes a single task to produces a single output object.",
"The Spark driver and executor recursively create the directories for the task temporary, job temporary and final output (steps 1–2).",
"The task outputs the task temporary file (step 3).",
"At task commit the executor lists the task temporary directory, and renames the file it finds to its job temporary name (steps 4-5).",
"At job commit the driver recursively lists the job temporary directories and renames the file it finds to its final names (steps 6-7).",
"The driver writes the _SUCCESS object.",
"When this same Spark program runs with the Hadoop Swift or S3a connectors, these file operations are translated to equivalent operations on objects in the object store.",
"These connectors use PUT to create zero byte objects representing the directories, after first using HEAD to check if objects for the directories already exist.",
"When listing the contents of a directory, these connectors descend the “directory tree” listing each directory.",
"To rename objects these connectors use PUT or COPY to copy the object to its new name and then use DELETE on the object at the old name.",
"All of the zero byte directory objects also need to be deleted.",
"Overall the Hadoop Swift connector executes 48 REST operations and the S3a connector executes 117 operations.",
"tab:RESTOperations shows the breakdown according to operation type.",
"Table: Breakdown of REST operations by type for the Spark program that creates an output consisting of a single object.In the next section we describe Stocator, which leverages object storage semantics to replace the temporary file/rename paradigm and takes advantage of hierarchal naming to avoid the creation of directory objects.",
"For the Spark program in fig:ProduceOneObject Stocator executes just 8 REST operations: 3 PUT object, 4 HEAD object and 1 GET container."
],
[
"Stocator algorithm",
"The right side of fig:HadoopStorageConnectors shows how Stocator fits underneath HMRCC; it implements the Hadoop Filesystem Interface just like the other storage connectors.",
"Below we describe the basic Stocator protocol; and then how it streams data, deals with eventual consistency, and reduces operations on the read path.",
"Finally we provide several examples of the protocol in action."
],
[
"Basic Stocator protocol",
"The overall strategy used by Stocator to avoid rename is to write output objects directly to their final name and then to determine which objects actually belong to the output at the time that the output is read by its consumer, e.g., the next Spark job in a sequence of jobs.",
"Stocator does this in a way that preserves the fault tolerance model of Spark/Hadoop and enables speculation.",
"Below we describe the components of this strategy.",
"As described in sec:background the driver orchestrates the execution of a Spark application.",
"In particular, the driver is responsible for creating a “directory” to hold an application's output dataset.",
"Stocator uses this “directory” as a marker to indicate that it wrote the output.",
"In particular, Stocator writes a zero byte object with the name of the dataset and object metadata that indicates that the object was written by Stocator.",
"All of the dataset's parts are stored hierarchically under this name.",
"Then when a Spark task asks to create a temporary object for its part through HMRCC, Stocator recognizes the pattern of the name and writes the object directly to its final name so it will not need to be renamed.",
"If Spark executes a task multiple times due to failures, slow execution or speculative execution, each execution attempt is assigned a number.",
"The Stocator object naming scheme includes this attempt number so that individual attempts can be distinguished.",
"In particular, HMRCC asks to write a temporary file/object in a temporary directory of the form <output-dataset-name>/_temporary/0/_temporary/attempt_<job-timestamp>_0000_m_000000_<attempt-number>/part-<part-number>, where <job-timestamp> is the timestamp of the Spark job, <attempt-number> is the number of attempt, and <part-number> is the number of the part.",
"Stocator notices this pattern and in place of the temporary object in the temporary directory, it writes an object with the name <output-dataset-name>/part-<part-number>_attempt_<job-timestamp>_0000_m_000000_<attempt-number>.",
"Finally, when all tasks have completed successfully, Spark writes a _SUCCESS object through HMRCC.",
"Notice that by avoiding rename, Stocator also avoids the need for list operations during task and job commit that may lead to incorrect results due to eventual consistency; thus, the presence of a _SUCCESS object means that there was a correct execution for each task and that there is an object for each part in the output."
],
[
"Alternatives for reading an input dataset",
"Stocator delays the determination of which parts belong to an output dataset until it reads the dataset as input.",
"We consider two options.",
"The first option is simpler to implement since it can be done entirely in the implementation of Stocator.",
"It depends on the assumption that Spark exhibits fail-stop behavior, i.e., that a Spark server executes correctly until it halts.",
"After determining that the dataset was produced by Stocator through reading the metadata from the object written with the dataset's name, and checking that the _SUCCESS object exists, Stocator lists the object parts belonging to the dataset through a GET container operation.",
"If there are objects in the list representing multiple execution attempts for same task, Stocator will choose the one that has the most data.",
"Given the fail-stop assumption, the fact that all successful execution attempts write the same output, and that it is certain that at least one attempt succeeded (otherwise there would not be a _SUCCESS object), this is the correct choice.",
"The second option is more complex to implement.",
"Here at the time the _SUCCESS object is written, Stocator includes in it a list of all the successful execution attempts completed by the Spark job.",
"Now after determining that the dataset was produced by Stocator through reading the metadata from the object written with the dataset's name, and checking that the _SUCCESS object exists, Stocator reads the manifest of successful task execution attempts from the _SUCCESS object.",
"Stocator uses the manifest to reconstruct the list of constituent object parts of the dataset.",
"In particular, the construction of the object part names follows the same pattern outlined above that was used when the parts were written.",
"The benefit of the second option is that it solves the remaining eventual consistency issue by constructing the object names from the manifest rather than issuing a REST command to list the object parts, which may not return a correct result in the presence of eventual consistency.",
"The second option also does not need the fail-stop assumption.",
"However, due to its simplicity we have implemented the first option in our Stocator prototype."
],
[
"Streaming of output",
"When Stocator outputs data it streams the data to the object store as the data is produced using chunked transfer encoding.",
"Normally the total length of the object is one of the parameters of a PUT operation and thus needs to be known before starting the operation.",
"Since Spark produces the data for an object on the fly and the final length of the data is not known until all of its data is produced, this would mean that Spark would need to store the entire object data prior to starting the PUT.",
"To avoid running out of memory, a storage connector for Spark can store the object in the Spark server's local file system as the connector produces the object's content, and then read the object back from the file to do the PUT operation on the object store.",
"Indeed this is what the default Hadoop Swift and S3a connectors do.",
"Instead Stocator leverages HTTP chunked transfer encoding, which is supported by the Swift API.",
"In chunked transfer encoding the object data is sent in chunks, the sender needs to know the length of each chunk, but it does not need to know the final length of the object content before starting the PUT operation.",
"S3a has an optional feature, not activated by default, called fast upload, where it leverages the multi-part upload feature of the S3 API.",
"This achieves a similar effect to chunked transfer encoding except that it uses more memory since the minimum part size for multi-part upload is 5 MB.",
"Table: Possible operations performed by the Spark application showed in fig:ProduceThreeObject"
],
[
"Optimizing the read path",
"We describe several optimizations that Stocator uses to reduce the number of operations on the read path.",
"The first optimization can remove a HEAD operation that occurs just before a GET operation for the same object.",
"In particular, the storage connector often reads the metadata of an object just before its data.",
"Typically this is to check that the object exists and to obtain the size of the object.",
"In file systems this is performed by two different operations.",
"Accordingly a naive implementation for object storage would read object metadata through a HEAD operation, and then read the data of the object itself through a GET operation.",
"However, object store GET operations also return the metadata of an object together with its data.",
"In many of these cases Stocator is able to remove the HEAD operation, which can greatly reduce the overall number of operations invoked on the underlying object storage system.",
"A second optimization is caching the results of HEAD operations.",
"A basic assumption of Spark is that the input is immutable.",
"Thus, if a HEAD is called on the same object multiple times, it should return the same result.",
"Stocator uses a small cache to reduce these calls."
],
[
"Examples",
"We show here some examples of Stocator at work.",
"For simplicity we focus on Stocator's interaction with HMRCC to eliminate the rename paradigm and so we do not show all of the requests that HMRCC makes on Stocator, e.g., to create/delete “directories” and check their status.",
"fig:ProduceThreeObject shows a simple Spark program that will be executed by three tasks, each task writing its part to the output dataset called $data.txt$ in a container called $res$ .",
"The swift2d: prefix in the URI for the output dataset indicates that Stocator is to be used as the storage connector.",
"tab:possibleOperations shows the operations that can be executed by our example in different situations.",
"Lines 1-3 and 8-9 are executed when each task runs exactly once and the program completes successfully.",
"We show the requests that HMRCC generates; for each task it issues one request to create a temporary object and two requests to “rename” it (copy to a new name and delete the object at the former name).",
"We see that Stocator intercepts the pattern for the temporary name that it receives from HMRCC, and creates the final names for the objects directly.",
"At the end of the run Spark creates the _SUCCESS object.",
"Lines 1-5, instead, shows an execution where Spark decides to execute Task 2 three times, i.e., three attempts.",
"This could be because the first and second attempts failed or due to speculation because they were slow.",
"Notice that Stocator includes the attempt number as part of the name of the objects that it creates.",
"By adding lines 6-9 to the previous, we show what happens when Spark is able to clean up the results from the duplicate attempts to execute Task 2.",
"In particular, Spark aborts attempts 0 and 2, and commits attempt 1.",
"When Spark aborts attempts 0 and 2, HMRCC deletes their corresponding temporary objects.",
"Stocator recognizes the pattern for the temporary objects and deletes the corresponding objects that it created.",
"If Spark is not able to clean up the results from the duplicate attempts to execute Task 2, we have lines 1-5 and 8-9.",
"In particular, we see that Stocator created five object parts, one each for Tasks 0 and 1, and three for Task 2 due to its extra attempts.",
"We assume as in the previous situation that it is attempt 1 for Task 2 that succeeded.",
"Stocator recognizes this through the manifest stored in the _SUCCESS object."
],
[
"Methodology",
"We describe the experimental platform, deployment scenarios, workloads and performance metrics that we use to evaluate Stocator."
],
[
"Experimental Platform",
"Our experimental infrastructure includes a Spark cluster, an IBM Cloud Object Storage (formerly Cleversafe) cluster, Keystone, and Graphite/Grafana.",
"The Spark cluster consists of three bare metal servers.",
"Each server has a dual Intel Xeon E52690 processor with 12 hyper-threaded 2.60 GHz cores (so 24 hyper-threaded cores per server), 256 GB memory, a 10 Gbps NIC and a 1 TB SATA disk.",
"That means that the total parallelism of the Spark cluster is 144.",
"We run 12 executors on each server; each executor gets 4 cores and 16 GB of memory.",
"We use Spark submit to run the workloads and the driver runs on one of the Spark servers (always the same server).",
"We use the standalone Spark cluster manager.",
"Our IBM Cloud Object Storage (COS) [42] cluster also runs on bare metal.",
"It consists of two Accessers, front end servers that receive the REST commands and then orchestrate their execution across twelve Slicestors, which hold the storage.",
"Each Accesser has two 10 Gbps NICs bonded to yield 20 Gbps.",
"Each Slicestor has twelve 1 TB SATA disks for data.",
"The Information Dispersal Algorithm (IDA) or erasure code is (12, 8, 10), which means that the erasure code splits the data into 12 parts, 8 parts are needed to read the data, and at least 10 parts need to be written for a write to complete.",
"IBM COS exposes multiple object APIs; we use the Swift and S3 APIs.",
"We employ HAProxy for load balancing.",
"It is installed on each of the Spark servers and configured with round-robin so that connections opened by a Spark server with the object storage alternate between Accessers.",
"Given that each of the three Spark servers has a 10 Gbps NIC, the maximum network bandwidth between the Spark cluster and the COS cluster is 30 Gbps.",
"Keystone and Graphite/Grafana run on virtual machines.",
"Keystone provides authentication/authorization for the Swift API.",
"We collect monitoring data on Graphite and view it through Grafana to check that there are no unexpected bottlenecks during the performance runs.",
"In particular we use the Spark monitoring interface and the collectd daemon to collect monitoring data from the Spark servers, and we use the Device API of IBM COS to collect monitoring data from the Accessers and the Slicestors."
],
[
"Deployment scenarios",
"In our experiments, we compare Stocator with the Hadoop Swift and S3a connectors.",
"By using different configurations of these two connectors, we define six scenarios: (i) Hadoop-Swift Base (H-S Base), (ii) S3a Base (S3a Base), (iii) Stocator Base (Stocator), (iv) Hadoop-Swift Commit V2 (H-S Cv2), (v) S3a Commit V2 (S3a Cv2) and (vi) S3a Commit V2 + Fast Upload (S3a Cv2+FU).",
"These scenarios are split into 3 groups according to the optional optimization features that are active.",
"The first group, with the suffix Base, uses connectors out of the box, meaning that no optional features are active.",
"The second group, with the suffix Commit V2, uses the version 2 of Hadoop FileOutputCommitter that reduces the number of copy operations towards the object storage (as described in Section ).",
"The last group, with the suffix Commit V2 + Fast Upload, uses both version 2 of Hadoop FileOutputCommitter and an optimization feature of S3a called S3AFastOutputStream that streams data to the object storage as it is produced (as described in Section ).",
"All experiments run on Spark 2.0.1 with a patched [11] version of Hadoop 2.7.3 infrastructure.",
"This patch allows us to use, for the S3a scenarios, Amazon SDK version 1.11.53 instead of version 1.7.4.",
"The Hadoop-Swift scenarios run with the default Hadoop-Swift connector that comes with Hadoop 2.7.3.",
"Finally, the Stocator scenario runs with stocator 1.0.8."
],
[
"Benchmark and Workloads",
"To study the performance of our solution we use several workloads (described in tab:workloads-details), that are currently used in popular benchmark suites and cover different kinds of applications.",
"The workloads span from simple applications that target a single and specific feature of the connectors (micro benchmarks), to real complex applications composed by several jobs (macro benchmarks).",
"The micro benchmarks use three different applications: (i) Read-only, (ii) Write-only and (iii) Copy.",
"The Read-only application reads two different text datasets, one whose size is 46.5 GB and the second 465.6 GB, and counts the number of lines in them.",
"For the Write-only application we use the popular Teragen application, available in the Spark example suite, that only performs write operations creating a dataset of 46.5 GB.",
"The last application that we use for our micro benchmark set is what we call the Copy application; it copies the small dataset used by the Read-only application.",
"We also use three macro benchmarks.",
"The first, Wordcount from Intel Hi-Bench [24], [35] test suite, is the “Hello World” application for parallel computing.",
"It is a read-intensive workload, that reads an input text file, computes the number of times each word occurs in the file and then writes a much smaller output file containing the word counts.",
"The second macro benchmark, Terasort, is a popular application used to understand the performance of large scale computing frameworks like Spark and Hadoop.",
"Its input dataset is the output of the Teragen application used in the micro benchmarks.",
"The third macro benchmark,TPC-DS, is the Transaction Processing Performance Council's decision-support benchmark test [30], [36] implemented with DataBricks' Spark-Sql-Perf library [17].",
"It executes several complex queries on files stored in Parquet format [13]; the input dataset size is 50 GB, which is compressed to 13.8 GB when converted to Parquet.",
"The query set that we use to perform our experiments is composed of the following 8 TPC-DS queries: q34, q43, q46, q59, q68, q73, q79 and ss_max.",
"These are the queries from the Impala subset that work with the Hadoop-Swift connector.",
"Stocator and S3a support all of the queries in the Impala subset.",
"The inputs for the Read-only, Copy, Wordcount and Terasort benchmarks are divided into 128 MB objects.",
"The outputs of the Copy, Teragen and Terasort benchmarks are also divided into 128 MB objects.",
"We also run Spark with a partition size of 128 MB."
],
[
"Performance metrics",
"We evaluate the different connectors and scenarios by using metrics that target the various optimization features.",
"As a general metric we use the total runtime of the application; this provides a quick overview of the performance of a specific scenario.",
"To delve into the reason behind the performance we use two additional metrics.",
"The first is the number of REST calls – and their type; with this metric we are able to understand the load on the object storage imposed by the connector.",
"The second metric is the number of bytes read from, written to and copied in the object storage; this also help us to understand the load on the object storage imposed by the connectors."
],
[
"Experimental Evaluation",
"We now present a comparative analysis between the different scenarios that we defined in subsec:scenarios.",
"We first show the benefit of Stocator through the average run time of the different workloads.",
"Then we compare the number of REST operations issued by the Compute Layer toward the Object Storage and the relative cost for these operations charged by cloud object store services.",
"Finally we compare the number of bytes transferred between the Compute Layer and the Object Storage."
],
[
"Reduction in run time",
"For each workload we ran each scenario ten times.",
"We report the average and standard deviation in tab:runtime-comparison.",
"The results shows that, when using a connector out of the box and under workloads that perform write operations, Stocator performs much better than Hadoop-Swift and S3a.",
"Only by activating and configuring optimization features provided by the Hadoop ecosystem, Hadoop-Swift and S3a manage to close the gap with Stocator, but they still fall behind.",
"tab:speedups shows the speedups that we obtain when using Stocator with respect to the other connectors.",
"We see a relationship between Stocator performance and the workload; the more write operations performed, the greater the benefit obtained.",
"On the one hand the write-only workloads, like Teragen, run 18 time faster with Stocator compared to the other out of the box connectors, 4 time faster when we enable FileOutputCommitter Version 2, and 1.5 times faster when we also add the S3AFastOutputStream feature.",
"On the other hand, workloads more skewed toward read operations, like Wordcount, have lower speedups.",
"These results are possible thanks to the algorithm implemented in Stocator.",
"Unlike the alternatives, Stocator removes the rename – and thus copy – operations completely.",
"In contrast, the other connectors, even with FileOutputCommitter Version 2, must still rename each output object once, although the overhead of the remaining renames is partially masked since they are carried out by the executors in parallel.",
"Stocator performs slightly worse than S3a on two of the workloads that contain only read operations (no writes), Read-only 50 GB and TPC-DS, and virtually the same for the larger 500 GB Read-only workload.",
"We have identified a small start-up cost that we have not yet removed from Stocator that can explain the difference between the results for the 50 GB and 500 GB Read-only workload.",
"As expected the results for the read-only workloads for S3a and Hadoop-Swift connectors are virtually the same with and without the FileOutputCommitter Version 2 and S3AFastOutputStream features; these features optimize the write path and do not affect the read path."
],
[
"Reduction in the number of REST calls",
"Next we look at the number of REST operations executed by Spark in order to understand the load generated on the object storage infrastructure.",
"fig:micro-rest-comparison,fig:macro-rest-comparison show that, in all the workloads, the scenario that uses Stocator achieves the lowest number of REST calls and thus the lowest load on the object storage.",
"When looking at Read-only with both 50 and 500 GB dataset, the scenario with Hadoop-Swift has the highest number of REST calls and more than double compared to the scenario with Stocator.",
"The Hadoop-Swift connector does many more GET calls on containers to list their contents.",
"Compared to S3a, Stocator is optimized to reduce the number of HEAD calls on the objects.",
"We see this consistently for all of the workloads.",
"In write-intensive workloads, Teragen and Copy, we see that the scenarios that use S3a as the connector have the highest number of REST calls while Stocator still has the lowest.",
"Compared to Hadoop-Swift and Stocator, S3a performs many more HEAD calls for the objects and GET for the containers.",
"Stocator also does not need to create temporary directories objects, thus uses far fewer HEAD requests, and does not need to DELETE objects; this is possible because our algorithm is conceived to avoid renaming objects after a task or job completes.",
"tab:extra-rest-calls shows the number of REST calls that is possible to save by using Stocator.",
"We observe that, for write-intensive workloads, Stocator issues 6 to 11 times less REST calls compared to Hadoop-Swift and 15 to 33 times less compared to S3a, depending on the optimization features active.",
"Having a low load on the Object Storage has advantages both for the data scientist and the storage providers.",
"On the one hand, cloud providers will be able to serve a bigger pool of consumers and give them a better experience.",
"On the other hand, since most public providers charge fees based on the number of operations performed on the storage tier, reducing the operations results in a lower cost for the data scientists.",
"tab:relative-costs shows the relative costs for the REST operations.",
"For the workloads with write (Teragen, Copy, Terasort and Wordcount) Stocator is 16 to 18 times less expensive than S3a run with FileOutputCommitter version 2, and 5 to 6 times less expensive than Hadoop-Swift.",
"To calculate the cost ratio we used the pricing models of IBM [21], AWS [6], Google [18] and Azure [16]; given that the models are very similar we report the average price.",
"Figure: Object Storage bytes read/written comparisonAs an additional way of measuring the load on the object storage and confirming the fact that Stocator does not perform COPY (or DELETE) operations we present the number of bytes read and written to the object storage.",
"From fig:bytes-comparison we see that Stocator does not write more data than needed on the storage.",
"In contrast we confirm that Hadoop-Swift and S3a base write each object three times – one from the PUT and two from the COPY – while Stocator only does it once.",
"Only by enabling FileOutputCommitter Version 2 in Hadoop, it is possible to reduce the COPY operations to one, but this is still one more object copy compared to Stocator.",
"We show only the workloads that have write operations since during a read-only workload, the number of bytes read from the object storage are identical for all of the connectors and scenarios (as we see from the Wordcount workload in fig:bytes-comparison where the number of bytes written is very small).",
"As expected the S3a scenario that uses the S3AFastOutputStream optimization gains no benefit with respect to the number of bytes written to the object storage."
],
[
"Related Work",
"In the past several years there has been a variety of work both from academia and industry [39], [31], [37], [49], [34], [41], [48], [43], [46], [44], [45], [40], [47], [26], [25], [3], [2], [12], [19] that target the performance of analytics frameworks with different configurations of Compute and Storage layers.",
"We can divide this work into two major categories that tackle performance analysis and eventual consistency.",
"Performance Analysis.",
"Work from [39], [31], [37], [49], [34], [41], [48], [43], [46], [44] analyze the performance of analytics frameworks with different configuration of the Compute and Storage layer.",
"All this work, albeit valid, base their conclusion on limited information, workloads and configurations that may not highlight some problems that exist when analytic applications connect to a specific Data or Storage layer solution.",
"In particular Ousterhout et al.",
"[39] use an ideal configuration (Compute and Data layer on the same Virtual Machine), with limited knowledge of the underlying storage system.",
"With the help of an analysis performed on network, disk block time and percentages of resource utilization, such work states that the runtime of analytics applications is generally CPU-bound rather than I/O intensive.",
"A recent work [45] shows that this is not always true; moving from a 1Gbps to a 10Gbps network can have a huge impact on the application runtime.",
"Another work [40] shows that is possible to further improve the run times by eliminating impedance mismatch between the layers, which can highly affect the run times of such applications; one in particular when using an Object Storage solution (e.g.",
"; Openstack Swift [27], [32]) as the Storage layer.",
"Concurrently there has also been some work from industry and open source to improve this impedance mismatch.",
"Databricks introduced something called the DirectOutputCommitter [29] for S3, but it failed to preserve the fault tolerance and speculation properties of the temporary file / rename paradigm.",
"At the same time Hadoop developed version 2 of the FileOutputCommitter [10], which renames files when tasks complete instead of waiting for the completion (commit) of the entire job.",
"However, this solution does not solve the entire problem.",
"Eventual Consistency.",
"Vogels [47] addresses the relationship between high-availability, replication and eventual consistency.",
"Eventual consistency guarantees that if no new updates are made to a given data item, then eventually all accesses to that item will return the same value.",
"In particular, when there is eventual consistency on the list operations over containers/buckets, current connectors from the Hadoop community for Swift API [38] and the S3 API [7], can also lead to failures and incorrect executions.",
"EMRFS [3], [2] from Amazon and S3mper [26], [25] from Netflix overcome eventual consistency by storing file metadata in DynamoDB [1], an additional storage system separate from the object store that is strongly consistent.",
"A similar feature called S3Guard [12], [19] that also requires an additional strongly consistent storage system is being developed by the Hadoop open source community for the S3a connector.",
"Solutions such as these that require multiple storage systems are complex and can introduce issues of consistency between the stores.",
"They also add cost since users must pay for the additional strongly consistent storage.",
"Our solution does not require any extra storage system."
],
[
"Conclusion and Future Work",
"We have presented a high performance object storage connector for Apache Spark called Stocator, which has been made available to the open source community [23].",
"Stocator overcomes the impedance mismatch of previous open source connectors with their storage, by leveraging object storage semantics rather than trying to treat object storage as a file system.",
"In particular Stocator eliminates the rename paradigm without sacrificing fault tolerance or speculative execution.",
"It also deals correctly with the eventually consistent semantics of object stores without the need to use an additional consistent storage system.",
"Finally, Stocator leverages HTTP chunked transfer encoding to stream data as it is produced to object storage, thereby avoiding the need to first write output to local storage.",
"We have compared Stocator's performance with the Hadoop Swift and S3a connectors over a range of workloads and found that it executes far less operations on object storage, in some cases as little as one thirtieth.",
"This reduces the load both for client software and the object storage service, as well as reducing costs for the client.",
"Stocator also substantially increases the performance of Spark workloads, especially write intensive workloads, where it is as much as 18 times faster than alternatives.",
"In the future we plan to continue improving the read performance of Stocator and extending it to support additional elements of the Hadoop ecosystem such as MapReduce (which should primarily require testing) and Hive."
]
] | 1709.01812 |
[
[
"Group actions on Smale space C*-algebras"
],
[
"Abstract Group actions on a Smale space and the actions induced on the C*-algebras associated to such a dynamical system are studied.",
"We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra.",
"We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C*-algebras.",
"In each of these cases, we discuss the preservation of properties---such as finite nuclear dimension, Z-stability, and classification by Elliott invariants---in the resulting crossed products."
],
[
"Introduction",
"Topological dynamical systems have long been a source for constructing interesting examples of $\\mathrm {C}^*$ -algebras.",
"Via the Gelfand transform, compact Hausdorff spaces and continuous maps are in one-to-one contravariant correspondence to unital commutative $\\mathrm {C}^*$ -algebras and unital $^*$ -homomorphisms.",
"To study topological dynamical systems using $\\mathrm {C}^*$ -algebraic techniques, loosely speaking, one encodes dynamical aspects of the system in a noncommutative $\\mathrm {C}^*$ -algebra.",
"For example, to encode the orbit equivalence classes of a topological dynamical system, one uses the transformation group construction to produce a so-called crossed product $\\mathrm {C}^*$ -algebra.",
"These have been widely studied for minimal dynamical systems because in such a case, the associated $\\mathrm {C}^*$ -algebra is simple.",
"When the space is a Cantor set with a single automorphism, it was shown by Putnam that the $\\mathrm {C}^*$ -algebras are classifiable by $K$ -theoretic data [50].",
"Moreover, he proved that they are all approximately circle algebras.",
"Subsequently, Giordano, Putnam and Skau showed that there is an isomorphism of the corresponding $\\mathrm {C}^*$ -algebras if and only if the systems are strong orbit equivalent [16].",
"The question of classification of such $\\mathbb {Z}$ -actions proved much more difficult for higher dimensional spaces, but was eventually settled by Lin in [37] following many partial results (see for example [38], [68], [72], [62]): the $\\mathrm {C}^*$ -algebras associated to minimal dynamical systems on compact metric spaces with finite covering dimension are classifiable by the Elliott invariant for simple separable unital nuclear $\\mathrm {C}^*$ -algebras.",
"For a given $\\mathrm {C}^*$ -algebra, the Elliott invariant consists of its ordered $K$ -theory, tracial state space, and a pairing map between these two objects.",
"Unlike in the zero-dimensional case, however, how this classification result relates to the dynamical systems involved is unclear.",
"In the present paper, we are interested in nonminimal systems called Smale spaces.",
"These were defined by Ruelle [56] based on the behaviour of the basic sets associated to Smale's Axiom A diffeomorphisms [60].",
"In the present paper, a dynamical system is a compact metric space with a self-homeomorphism.",
"A Smale space is a dynamical system $(X, \\varphi )$ which has a local hyperbolic structure: at every point $x \\in X$ there is a small neighbourhood which decomposes into the product of a stable and unstable set.",
"For a Smale space, the dynamical behaviour which we seek to study is the asymptotic behaviour of points.",
"The appropriate $\\mathrm {C}^*$ -algebras, defined by Putnam [51] following earlier work by Ruelle [55] encode this behaviour via the groupoid $\\mathrm {C}^*$ -algebras, $ \\mathrm {C}^*(S)$ , $ \\mathrm {C}^*(U)$ and $ {\\mathrm {C}}^*(H)$ , associated to stable, unstable, and homoclinic equivalence relations, respectively.",
"When a Smale space is mixing, these three $\\mathrm {C}^*$ -algebras are each simple, separable and nuclear.",
"Furthermore, the homoclinic $\\mathrm {C}^*$ -algebra is unital.",
"From the point of view of the structure and classification of $\\mathrm {C}^*$ -algebras, the path of investigation has shared many similarities to the case of minimal systems: the first, and most successful, results come from the zero dimensional case.",
"Here, a Smale space is a shift of finite type and the associated $\\mathrm {C}^*$ -algebras are all approximately finite (AF) algebras, which can be classified by $K$ -theory.",
"Moreover, two shifts of finite type are eventually conjugate if and only if the associated $\\mathrm {C}^*$ -algebras are equivariantly isomorphic with respect to the shift maps [39].",
"As in the case of minimal systems, classification for homoclinic $\\mathrm {C}^*$ -algebras of mixing Smale spaces in higher-dimensional settings proved to be more involved.",
"However, the authors of the present paper showed in [12] that these $\\mathrm {C}^*$ -algebras are always classifiable using advanced techniques from the classification programme, rather than tools from dynamics.",
"Thus it is once again difficult to glean information about the underlying Smale spaces.",
"In addition, [12] showed that both the stable and unstable $\\mathrm {C}^*$ -algebra have finite nuclear dimension and are $\\mathcal {Z}$ -stable.",
"However since these $\\mathrm {C}^*$ -algebras are nonunital, classification by the Elliott invariant is still out of reach, see [12].",
"Here we turn our attention to group actions on Smale spaces and the corresponding actions induced on the associated $\\mathrm {C}^*$ -algebras.",
"Such group actions have been studied at the Smale space level in various places.",
"A number of explicit examples are discussed in Section REF .",
"The specific case of shifts of finite type is already well-studied; see [1], [7], [6], [43] among many others.",
"Boyle's survey article [6] and the references within are a good place to get an introduction to this vast field, in particular see open problems 12 and 13 in [6].",
"The goal of the present paper is to study how an action of a group on a Smale space relates to the induced action on its $\\mathrm {C}^*$ -algebras as well as to the associated crossed product.",
"Sufficient conditions for a group action on a general $\\mathrm {C}^*$ -algebra which ensure structural properties of the crossed product (simplicity, for example, or finite nuclear dimension, or $\\mathcal {Z}$ -stability) have been studied.",
"These might be thought of as the noncommutative interpretation of a free action.",
"Of particular interest for this paper are when an action has finite Rokhlin dimension or is strongly outer.",
"We establish sufficient conditions for when a group acting on a mixing Smale space induces an action on each of its associated $\\mathrm {C}^*$ -algebras with one of these properties.",
"From there, we may deduce results about the crossed products.",
"The Rokhlin property takes its motivation from the Rokhlin Lemma of ergodic theory.",
"The Rokhlin Lemma says that a measure-preserving, aperiodic integer action can be approximated by cyclic shifts.",
"Connes successfully adapted this to the noncommutative setting of von Neumann algebras to classify automorphisms of the hyperfinite II$_1$ factor up to outer conjugacy [9], [10].",
"Many generalisations followed in the von Neumann setting.",
"Taking motivation from von Neumann algebras, the corresponding $\\mathrm {C}^*$ -algebraic theory gradually emerged, first for the restricted case of UHF algebras (the most straightforward $\\mathrm {C}^*$ -algebraic interpretation of a II$_1$ factor) [19], [20], and more generally in the work of Kishimoto (see for example [32], [33], [34]) and Izumi [26], [27].",
"The presence of the Rokhlin property for an action of a group on a $\\mathrm {C}^*$ -algebra allows that certain properties from the $\\mathrm {C}^*$ -algbera to the crossed product.",
"However, since arbitrary $\\mathrm {C}^*$ -algebras need not have many projections, the Rokhlin property itself is often too strict.",
"For this reason, weaker properties, via higher dimensional versions or tracial versions, emerged, see for examples [23], [63], [22], [15], [44], [48], [3].",
"These are much more general but still allow for good preservation of properties in the crossed product.",
"For further details on the Rokhlin property, Rokhlin dimension, and related $\\mathrm {C}^*$ -algebraic properties; see for example [23].",
"To the authors' knowledge, this is the first time general actions on Smale space $\\mathrm {C}^*$ -algebras have been studied.",
"However, a number of special cases have been considered.",
"In particular, the most obvious action on the stable, unstable, and homoclinic algebras is the one induced from $\\varphi $ .",
"In [52], Putnam and Spielberg showed that if $(X, \\varphi )$ is mixing, then $ \\mathrm {C}^*(S)\\rtimes _{\\varphi } {\\mathbb {Z}}$ and $ \\mathrm {C}^*(U)\\rtimes _{\\varphi } {\\mathbb {Z}}$ are purely infinite.",
"On the other hand, $ {\\mathrm {C}}^*(H)\\rtimes _{\\varphi } {\\mathbb {Z}}$ is stably finite and in the special case of a shift of finite type, Holton [24] showed that the shift map (that is, $\\varphi $ in the case of a shift of finite type) leads to an action on $ {\\mathrm {C}}^*(H)$ with the Rokhlin property.",
"Automorphisms of subshifts and their associated $\\mathrm {C}^*$ -algebras have been studied by a number of authors, see for example [40] and references therein.",
"Finally, in [61], Starling studies certain finite group actions on tiling spaces and a $\\mathrm {C}^*$ -algebra associated to them which is related to the $\\mathrm {C}^*$ -algebras in the present paper through work of Anderson and Putnam [2].",
"This is discussed briefly in Example REF .",
"It is worth noting that an action at the Smale space level also induces an action on each of $ \\mathrm {C}^*(S)\\rtimes _{\\varphi } {\\mathbb {Z}}$ , $ \\mathrm {C}^*(U)\\rtimes _{\\varphi } {\\mathbb {Z}}$ and $ {\\mathrm {C}}^*(H)\\rtimes _{\\varphi } {\\mathbb {Z}}$ .",
"Properties of the resulting crossed product algebras can be inferred from results in this paper by considering the action of the group generated by the original group and $\\varphi $ on $ \\mathrm {C}^*(S)$ , $ \\mathrm {C}^*(U)$ and $ {\\mathrm {C}}^*(H)$ .",
"For example, if $\\alpha $ is an automorphism of the given Smale space, then often (if $\\alpha $ has finite order or is a power of $\\varphi $ some care would be required) the study of induced actions on $ \\mathrm {C}^*(S)\\rtimes _{\\varphi } {\\mathbb {Z}}$ , $ \\mathrm {C}^*(U)\\rtimes _{\\varphi } {\\mathbb {Z}}$ and $ {\\mathrm {C}}^*(H)\\rtimes _{\\varphi } {\\mathbb {Z}}$ amounts to studying the associated ${\\mathbb {Z}}^2$ -actions on $ \\mathrm {C}^*(S)$ , $ \\mathrm {C}^*(U)$ and $ {\\mathrm {C}}^*(H)$ .",
"The paper is structured as follows.",
"In Section 1 we introduce Smale spaces.",
"Section 2 provides examples of group actions on Smale spaces as well as a couple of short proofs about free actions and effective actions.",
"We move on to $\\mathrm {C}^*$ -algebras in Section 3.",
"In Section 4 we consider the induced action on the homoclinic $\\mathrm {C}^*$ -algebra, $ {\\mathrm {C}}^*(H)$ .",
"Two of our main results are included in this section: that actions of finite groups on mixing $(X, \\varphi )$ which act freely on $X$ induce actions on $ {\\mathrm {C}}^*(H)$ with finite Rokhlin dimension and actions of discrete groups acting effectively on $X$ induce strongly outer actions on $ {\\mathrm {C}}^*(H)$ .",
"We also consider $\\mathbb {Z}$ -actions in the case that $(X, \\varphi )$ is irreducible, but not necessarily mixing.",
"In Section 5 we look at the induced actions on the stable and unstable $\\mathrm {C}^*$ -algebras of a Smale space, $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"We show that we can use what we know about the induced action on $ {\\mathrm {C}}^*(H)$ to determine the behaviour on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ and consider the resulting properties of the crossed products.",
"Finally, in Section 6 we collect some results about $\\mathcal {Z}$ -stability, nuclear dimension, and classification of the crossed products.",
"In summary, the main results of the present paper are the following: Theorem: Let $G$ be a discrete group acting effectively on a mixing Smale space $(X, \\varphi )$ .",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ is strongly outer.",
"When $G$ is a countable amenable group, then the fact that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable follows from [58].",
"However, since $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are nonunital, we cannot apply that result directly.",
"Nevertheless, we show they are indeed $\\mathcal {Z}$ -stable.",
"Theorem: Suppose $G$ is a countable amenable group acting on a mixing Smale space, $(X, \\varphi )$ .",
"Then, for the induced actions $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}(\\mathrm {C}^*(S))$ and $\\beta ^{(U)} : G \\rightarrow \\operatorname{Aut}(\\mathrm {C}^*(U))$ , the crossed products $\\mathrm {C}^*(S) \\rtimes _{\\beta ^{(U)}} G$ and $\\mathrm {C}^*(U) \\rtimes _{\\beta ^{(U)}} G$ are both $\\mathcal {Z}$ -stable.",
"Theorem: Let $G$ be a residually finite group acting on an irreducible Smale space $(X, \\varphi )$ .",
"Suppose the induced action of G on $ {\\mathrm {C}}^*(H)$ has Rokhlin dimension at most $d$ .",
"Then the induced actions of $G$ on $\\mathrm {C}^*(S)$ and $\\mathrm {C}^*(U)$ both have Rokhlin dimension at most $d$ .",
"The same statement holds for Rokhlin dimension with commuting towers."
],
[
"Preliminaries",
"Smale spaces were defined by Ruelle [56].",
"1.1 1.1 Definition: [56] Let $(X, d)$ be a compact metric space and let $\\varphi : X \\rightarrow X$ be a homeomorphism.",
"The dynamical system $(X, \\varphi )$ is called a Smale space if there are two constants $\\epsilon _X > 0$ and $0<\\lambda _X<1$ and a map, called the bracket map, $ [ \\cdot , \\cdot ] : X \\times X \\rightarrow X $ which is defined for $x, y \\in X$ such that $d(x,y)< \\epsilon _X$ .",
"The bracket map is required to satisify the following axioms: B1.",
"$\\left[ x, x \\right] = x$ , B2.",
"$\\left[ x, [ y, z] \\right] = [ x, z]$ , B3.",
"$\\left[ [ x, y], z \\right] = [ x,z ]$ , B4.",
"$\\varphi [x, y] = [ \\varphi (x), \\varphi (y)]$ ; for $x, y, z \\in X$ whenever both sides in the above equations are defined.",
"The system also satisfies C1.",
"For $x,y \\in X$ such that $[x,y]=y$ , we have $d(\\varphi (x),\\varphi (y)) \\le \\lambda _X d(x,y)$ and C2.",
"For $x,y \\in X$ such that $[x,y]=x$ , we have $d(\\varphi ^{-1}(x),\\varphi ^{-1}(y)) \\le \\lambda _X d(x,y)$ .",
"If the bracket map exists, it is unique (see for example [51]).",
"Also, in order to avoid certain trivial cases, we will always assume that $X$ is infinite.",
"1.2 1.2 Definition: Suppose $(X, \\varphi )$ is a Smale space, $x \\in X$ , and $0<\\epsilon \\le \\epsilon _X$ and $Y, Z \\subset X$ a subset of points.",
"Then we define the following sets (i) $X^S(x, \\varepsilon ) := \\left\\lbrace y \\in X \\mid d(x,y) < \\varepsilon , [y,x]=x \\right\\rbrace ,$ $X^U(x, \\varepsilon ) := \\left\\lbrace y \\in X \\mid d(x,y) < \\varepsilon , [x,y]=x \\right\\rbrace , $ $X^S(x) := \\left\\lbrace y\\in X \\mid \\lim _{n \\rightarrow + \\infty } d(\\varphi ^n(x), \\varphi ^n(y)) =0 \\right\\rbrace , $ $X^U(x) := \\left\\lbrace y\\in X \\mid \\lim _{n \\rightarrow - \\infty } d(\\varphi ^n(x), \\varphi ^n(y)) =0 \\right\\rbrace ,$ $X^S(Z) := \\cup _{x\\in Z} X^S(x)$ and $X^U(Z):=\\cup _{x\\in Z} X^U(x)$ , and $X^H(Y, Z) := X^S(Y) \\cap X^U(Z).$ We say $x$ and $y$ are stably equivalent and write $x \\sim _s y$ if $y \\in X^S(x)$ .",
"Similarly, we say $x$ and $y$ are unstably equivalent, written $x \\sim _u y$ , if $y \\in X^U(x)$ .",
"Points $x,y \\in X$ are homoclinic if $y \\in X^S(x) \\cap X^U(x)$ , meaning both $\\lim _{n \\rightarrow \\infty } d(\\varphi ^n(x), \\varphi ^n(y)) =0$ and $\\lim _{n \\rightarrow -\\infty } d(\\varphi ^n(x), \\varphi ^n(y)) =0$ .",
"1.3 1.3 A topological dynamical system $(X, \\varphi )$ (not necessarily a Smale space) is called mixing if, for every ordered pair of nonempty open sets $U, V \\subset X$ , there exists $N \\in \\mathbb {Z}_{>0}$ such that $\\varphi ^n(U) \\cap V$ is nonempty for every $n \\ge N$ .",
"A Smale space $(X, \\varphi )$ is irreducible if for every ordered pair of nonempty open sets $U, V \\subset X$ there exists $N \\in \\mathbb {Z}_{>0}$ such that $\\varphi ^N(U) \\cap V$ is nonempty.",
"In this case, Smale's decomposition theorem states that $X$ can be written as a disjoint union of finitely many clopen subspaces $X_1, \\dots , X_N$ which are cyclically permuted by $\\varphi $ and each system $(X_i, \\varphi ^N|_{X_i})$ is mixing.",
"1.4 1.4 For $k \\in \\mathbb {N} \\setminus \\lbrace 0\\rbrace $ , let $ {\\rm Per}_k(X, \\varphi ) = \\lbrace x \\in X \\mid \\varphi ^k(x) = x \\rbrace $ denote the set of points of period $k$ , and let $ {\\rm Per}(X, \\varphi ) = \\cup _{k \\in \\mathbb {N} \\setminus \\lbrace 0\\rbrace } {\\rm Per}_k(X, \\varphi ) $ denote the set of all periodic points.",
"If $(X, \\varphi )$ is irreducible, then the set ${\\rm Per}(X, \\varphi )$ is dense in $X$ and for each $k$ , the set ${\\rm Per}_k(X, \\varphi )$ is finite [56].",
"1.5 1.5 Theorem: [57] Given a mixing Smale space $(X, \\varphi )$ there exists a unique $\\varphi $ -invariant, entropy-maximizing probability measure $\\mu _X$ on $X$ which, for every $x \\in X$ and $0< \\epsilon < \\epsilon _X$ , can be written locally as a product measure supported on $X^{u}(x, \\epsilon ) \\times X^{s}(x, \\epsilon )$ .",
"The measure described in Theorem is called the Bowen measure, which in the case of a shift of finite type is the Parry measure, see for example [39].",
"Group actions on Smale spaces Let $G$ be a topological group.",
"By an action of $G$ on a Smale space $(X, \\varphi )$ we mean a continuous group homomorphism $G \\rightarrow \\operatorname{Homeo}(X)$ such that $ g \\varphi (x) = \\varphi (gx) \\text{ for every } x \\in X \\text{ and } g \\in G.$ For a given Smale space $(X, \\varphi )$ , its automorphism group is defined to be ${\\rm Aut}(X, \\varphi ) := \\lbrace \\beta : X \\rightarrow X \\: | \\: \\beta \\hbox{ is a homeomorphism and }\\beta \\circ \\varphi = \\varphi \\circ \\beta \\rbrace .$ Let $X$ be a compact metric space and $G$ a topological group.",
"An action of group $G \\rightarrow \\operatorname{Homeo}(X)$ is free if, for every $x \\in X$ , we have that $g(x) = x$ if and only if $g = \\operatorname{\\textup {id}}$ .",
"The action of $G$ is effective (or faithful), if for every $g \\in G\\setminus \\lbrace e\\rbrace $ , there is an $x \\in X$ such that $gx \\ne x$ .",
"If $(X, \\varphi )$ is a Smale space then we say the action is free, respectively effective, if the action on $X$ is free, respectively effective.",
"Examples of automorphisms on Smale spaces Group actions on Smale spaces are quite ubiquitous.",
"Here we discuss four familiar classes of Smale spaces, three of which are treated in Putnam's work on Smale spaces and $\\mathrm {C}^*$ -algebras [51], and give examples of their automorphisms.",
"In each case, these actions will be free or effective and thus are covered by the results in Sections and .",
"2.1 2.1 Example: [Smale space automorphism] Let $(X, \\varphi )$ be a Smale space.",
"Our first and most obvious example is the action induced by the homeomorphism $\\varphi $ of $X$ .",
"Clearly $\\varphi $ commutes with itself and hence defines an action of $\\mathbb {Z}$ on $(X, \\varphi )$ .",
"The action induced on the associated $\\mathrm {C}^*$ -algebras will prove to be a useful tool in the sequel.",
"Of course, powers of $\\varphi $ are also automorphisms of $(X, \\varphi )$ .",
"2.2 2.2 Example: [Shifts of finite type] In the irreducible case, shifts of finite type are exactly the zero dimensional Smale spaces, though of course they were well known before Ruelle's work.",
"The automorphism group and group actions on shifts of finite type have been studied by many authors, see for example [1], [7], [6], [43] along with references therein.",
"The full two-shift is the Smale space $(X, \\varphi )$ where $X = \\Sigma _{[2]}=\\lbrace 0, 1\\rbrace ^{{\\mathbb {Z}}}$ with $\\varphi = \\sigma $ defined to be the left shift.",
"The automorphism group of the full two-shift is large: it contains, for example, every finite group and the free group on two generators [7].",
"Here we highlight two automorphisms of finite order.",
"The latter comes from [39].",
"Define $\\beta _1 : \\Sigma _{[2]} \\rightarrow \\Sigma _{[2]}$ via $(a_n)_{n\\in {\\mathbb {Z}}} \\mapsto (a_n+ 1$ mod$(2))_{n \\in {\\mathbb {Z}}}$ and $\\beta _2 : \\Sigma _{[2]} \\rightarrow \\Sigma _{[2]}$ via $(a_n)_{n\\in {\\mathbb {Z}}} \\mapsto (b_n)_{n\\in {\\mathbb {Z}}}$ where $b_n := a_n + a_{n-1}( a_{n+1}+1)a_{n+2} \\hbox{ mod}(2) .$ One can compute that $\\beta _1^2$ and $\\beta _2^2$ are both the identity.",
"Hence they are order two automorphisms of the full two shift.",
"The $\\frac{{\\mathbb {Z}}}{2{\\mathbb {Z}}}$ -action induced from $\\beta _1$ is free while that of $\\beta _2$ is effective but not free.",
"2.3 2.3 Example: [Solenoids] A solenoid is a Smale space obtained from a stationary inverse limit of a metric space equipped with a surjective continuous map that is subject to certain conditions such as those in [71] or in [70].",
"Some prototypical examples are obtained as follows.",
"Let $S^1 \\subseteq (as the unit circle), $ k2$ be an integer, and define $ g: S1 S1$ by $ g(z)=zk$.",
"Then $ (X, )$ is the Smale space given by $ X= (S1, g)$ and the map $ : X X$ defined as$$(z_n)_{n \\in {\\mathbb {N}}} \\mapsto (g(z_0), g(z_1), g(z_2), \\ldots ) = (g(z_0) , z_0, z_1, \\ldots ).$$$ Let $\\beta _1: X \\rightarrow X$ be the map $(z_n)_{n\\in {\\mathbb {N}}} \\mapsto (\\bar{z}_n)_{n\\in {\\mathbb {N}}}$ , where $\\bar{z}$ denotes complex conjugate of $z$ .",
"For any $k$ , $\\beta _1$ defines an order two automorphism.",
"There are other automorphisms.",
"For example, if $g(z) = z^6$ , then we have automorphisms defined by $\\beta _2( (z_n)_{n\\in {\\mathbb {N}}}) = (z^2_n)_{n\\in {\\mathbb {N}}} \\quad \\hbox{ and } \\quad \\beta _3( (z_n)_{n\\in {\\mathbb {N}}}) = (z^3_n)_{n\\in {\\mathbb {N}}}.$ The ${{\\mathbb {Z}}}/{2{\\mathbb {Z}}}$ -action induced from $\\beta _1$ is effective but not free.",
"While the ${\\mathbb {Z}}$ -actions induced by $\\beta _2$ and $\\beta _3$ are each effective but not free.",
"The automorphism group of these examples are known (see [65] for details).",
"Further results concerning the automorphism group of similar examples can also be found in [65].",
"2.4 2.4 Example: [Hyperbolic toral automorphisms] Let $d\\ge 2$ be an integer.",
"A hyperbolic toral automorphism is a Smale space $(X, \\varphi )$ were $X={\\mathbb {R}}^d/ {\\mathbb {Z}}^d$ and $\\varphi $ is induced by a $d \\times d$ integer matrix $A$ with the following properties: (i) $|\\det (A)| =1$ ; no eigenvalue of $A$ has modulus one.",
"A specific example when $d=2$ is $A = \\left( \\begin{array}{cc} 2 & 1 \\\\ 1 & 1 \\end{array} \\right).$ To obtain an automorphism of $(X, \\varphi )$ one can take $B\\in M_d({\\mathbb {Z}})$ with det$(B)=\\pm 1$ such that $AB=BA$ .",
"For example, again when $d=2$ , $B=\\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & -1 \\end{array} \\right)$ defines an order two automorphism of any hyperbolic toral automorphism; the induced $\\frac{{\\mathbb {Z}}}{2{\\mathbb {Z}}}$ -action is effective.",
"For larger $d$ there are more interesting automorphisms.",
"An explicit example taken from [49] is $A = \\left( \\begin{array}{ccc} 1 & -1 & 0 \\\\ -1 & 2 & -1 \\\\ 0 & -1 & 2 \\end{array} \\right) \\hbox{ and } B= \\left( \\begin{array}{ccc} 2 & 0 & -1 \\\\ 0 & 1 & 1 \\\\ -1 & 1 & 2 \\end{array} \\right)$ In this example, both $A$ and $B$ are hyperbolic.",
"More on automorphisms of hyperbolic toral automorphisms can be found in [4] and [49] along with the references therein.",
"2.5 2.5 Example: [Substitution tilings] Many people have studied the dynamics of substitution tiling systems.",
"In [11], Connes associates an AF algebra to the Penrose tiling by kites and darts; other $\\mathrm {C}^*$ -algebraic constructions were considered by Kellendonk (see for example [30]).",
"It was Putnam and Anderson who showed that, under the assumptions that the substitution map $\\omega $ is one-to-one, primitive and the tiling space $\\Omega $ is of finite type (that is, has finite local complexity), that $(\\Omega , \\omega )$ is in fact a Smale space [2].",
"Moreover, the unstable $\\mathrm {C}^*$ -algebra associated to $(\\Omega , \\omega )$ is Morita equivalent to the tiling $\\mathrm {C}^*$ -algebra studied by Kellendonk [2].",
"Many aperiodic tilings have interesting rotational symmetries.",
"For example, the dihedral group, $D_5$ acts on the Penrose tiling [59].",
"In [61], Starling studied free actions of finite subgroups of the symmetry group of these substitution tilings.",
"The results of the present paper, in the special case of tilings, can be related to those in [61] by showing that the Morita equivalence considered by Anderson and Putnam can be made equivariant with respect to the given group action.",
"Returning to general actions on Smale spaces, we begin by showing free actions are quite rare.",
"However, as we shall see in Section REF , an effective action is sufficiently strong to guarantee good properties of the induced action on the associated $\\mathrm {C}^*$ -algebras.",
"The next two propositions are likely not new, but we could not find a reference for them, except for the case of a shift of finite type [1].",
"2.6 2.6 Proposition: Suppose $G$ has an element of infinite order and acts on a Smale space $(X, \\varphi )$ .",
"Then $G$ does not act freely.",
"Proof.",
"Let $g\\in G$ be an element of infinite order.",
"The set periodic points of $(X, \\varphi )$ is non-empty because the non-wandering set is non-empty by [56] and the periodic points are dense in the non-wandering set by [56].",
"Let $x\\in X$ be a periodic point with minimal period $k$ .",
"Then, for any $n \\in {\\mathbb {Z}}$ , $ \\varphi ^k (g^n x) = g^n \\varphi ^k (x) = g^n x$ Hence, for any $n \\in Z$ , $g^n x \\in {\\rm Per}_k(X, \\varphi )$ .",
"Since ${\\rm Per}_k (X, \\varphi )$ is finite, there are $n_1 \\ne n_2 \\in {\\mathbb {Z}}$ such that $g^{n_1} x = g^{n_2}x$ It follows that the action is not free.",
"2.7 2.7 Proposition: Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"For each $g\\in G\\setminus \\lbrace e\\rbrace $ , the set $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is dense and open.",
"Moreover, the Bowen measure of this set is one.",
"Proof.",
"The set $\\lbrace x \\in X \\: | \\: gx =x \\rbrace $ is closed since the map $G \\times X \\rightarrow X$ is continuous.",
"Thus $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is open.",
"Since $(X, \\varphi )$ is mixing, there is a point $x_0 \\in X$ with dense orbit.",
"We show $x_0 \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Suppose not.",
"Then for each $n\\in {\\mathbb {Z}}$ , $g(\\varphi ^n(x_0))= \\varphi ^n( g x_0) = \\varphi ^n(x_0).$ Thus, because the orbit of $x_0$ is dense, $g$ is the identity on $X$ .",
"This contradicts the assumption that $G$ acts effectively, so $x_0 \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Moreover, for each $n\\in {\\mathbb {Z}}$ , $\\varphi ^n(x_0) \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Since the orbit of $x_0$ is dense, $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is also dense.",
"Finally, the set of points with dense orbit has Bowen measure one, which follows from Bowen's theorem and the fact that this is true for a shift of finite type [39].",
"The set $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ contains the set of points with dense orbit and hence also has Bowen measure one.",
"From Smale spaces to $\\mathrm {C}^*$ -algebras Ruelle was the first person to associate operator algebras to Smale spaces in [55].",
"We follow the approach introduced by Putnam and Spielberg [51], [52]: three $\\mathrm {C}^*$ -algebras are constructed via the groupoid $\\mathrm {C}^*$ -algebra construction for étale equivalence relations which capture the contracting, expanding, and asymptotic behaviour of the system given in Definition .",
"Fix an irreducible Smale space $(X, \\varphi )$ .",
"To define topologies on each of our equivalence relations, we first note the following.",
"Using the notation from Definition , it is not difficult to show that for any $0 < \\epsilon \\le \\epsilon _X$ we have $ X^U(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon )),$ and similarly that $ X^S(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{-n}(X^S(\\varphi ^n(x), \\epsilon )).$ Each $\\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon ))$ , $n \\in \\mathbb {N}$ is given the relative topology from $X$ while $X^U(x)$ and $X^S(x)$ are given the topology coming from these inductive unions, see [51] for the precise details.",
"We could proceed to construct $\\mathrm {C}^*$ -algebras directly from the equivalence relations in Definition following the construction of Putnam in [51].",
"However neither the stable nor the unstable groupoids would have a natural étale topology.",
"Instead, following [52], we restrict our relation to those points equivalent to periodic points.",
"3.1 3.1 Definition: Let $P$ and $Q$ be finite $\\varphi $ -invariant sets of periodic points of $(X, \\varphi )$ .",
"Define the stable and unstable groupoids of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_S(P) := \\lbrace (x,y) \\in X^U(P) \\times X^U(P) \\mid x \\sim _s y \\rbrace ,$ and $ \\operatorname{\\mathcal {G}}_U(Q) := \\lbrace (x,y) \\in X^S(Q) \\times X^S(Q) \\mid x \\sim _u y \\rbrace .$ Up to Morita equivalence of groupoids these constructions do not depend on the choice of periodic points.",
"3.2 3.2 Definition: Define the homoclinic groupoid of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_H := \\lbrace (x, y) \\in X \\times X \\mid x \\sim _h y\\rbrace .$ Now, if $(v, w) \\in X^S(P)$ , then $v \\sim _s w$ so there is some sufficiently large $N \\in \\mathbb {N}$ such that $d(\\varphi ^N(v), \\varphi ^N(w)) < \\epsilon _X/2$ .",
"By continuity of $\\varphi $ , we may choose $\\delta > 0$ small enough so that $\\varphi ^N(X^U(w , \\delta )) \\subset X^U(\\varphi ^N(w), \\epsilon _X/2)$ and also $\\varphi ^N(X^U(v , \\delta )) \\subset X^U(\\varphi ^N(v), \\epsilon _X/2)$ .",
"Then define $ h^s := h^s(v, w, N, \\delta ) : X^U(w, \\delta ) \\rightarrow X^U(v, \\epsilon _X), \\quad x \\mapsto \\varphi ^{-N}([\\varphi ^N(x), \\varphi ^N(v)]).",
"$ By [56] this is a local homeomorphism.",
"For any such $v, w, \\delta , h , N$ , we then define an open set by $ V(v, w, \\delta , h^s, N) := \\lbrace (h^s(x), x) \\mid x \\in X^U(w, \\delta ) \\rbrace \\subset \\operatorname{\\mathcal {G}}_S(P).$ These sets generate an étale topology for $\\operatorname{\\mathcal {G}}_S(P)$ [52].",
"The construction for the topologies of $\\operatorname{\\mathcal {G}}_U(Q)$ and $\\operatorname{\\mathcal {G}}_H$ are similar; we refer the reader to [52] for details.",
"$\\mathrm {C}^*$ -algebras Fix finite $\\varphi $ -invariant sets $P$ and $Q$ and let $\\mathcal {H} = \\ell ^2(X^H(P,Q))$ where $X^H(P,Q)$ is the set of points in $X$ which are both stably equivalent to a point in $P$ and unstably equivalent to a point in $Q$ .",
"It is shown in [56] that $X^H(P, Q)$ is countable.",
"If $\\operatorname{\\mathcal {G}}$ is one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ , let $C_c(\\operatorname{\\mathcal {G}})$ denote the a compactly supported functions on $\\operatorname{\\mathcal {G}}$ with convolution product, $ (f \\ast g) (x, y) = \\sum _{z \\sim x, z \\sim y} f(x,z) g(z, y), \\quad (x,y) \\in \\operatorname{\\mathcal {G}}, $ and $f^*(x, y) = \\overline{f(y,x)}, \\quad (x,y) \\in \\operatorname{\\mathcal {G}}.$ We can represent each of $C_c(\\operatorname{\\mathcal {G}}_H)$ , $C_c(\\operatorname{\\mathcal {G}}_S(P))$ and $C_c(\\operatorname{\\mathcal {G}}_U(Q))$ on the Hilbert space $\\mathcal {H}$ , and define the homoclinic algebra by $ {\\mathrm {C}}^*(H):= \\overline{C_c(\\operatorname{\\mathcal {G}}_H)}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ the stable algebra $ \\mathrm {C}^*(S):= \\overline{C_c(\\operatorname{\\mathcal {G}}_S(P))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ and the unstable algebra $ \\mathrm {C}^*(U):= \\overline{C_c(\\operatorname{\\mathcal {G}}_U(Q))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}}.$ 3.3 3.3 Remarks: 1.",
"We suppress the reference to $P$ and $Q$ in the notation of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"For any choice of such $P$ and $Q$ , the resulting groupoids are Morita equivalent, and hence so are their $\\mathrm {C}^*$ -algebras.",
"Thus from the perspective of most $\\mathrm {C}^*$ -algebraic properties, we don't need to keep track of the original choice.",
"2.",
"In the usual groupoid $\\mathrm {C}^*$ -algebra construction for a groupoid $\\mathcal {G}$ , the algebra of compactly supported functions $C_c(\\mathcal {G})$ is represented on the Hilbert space $\\ell ^2(\\mathcal {G})$ and the completion is the reduced groupoid $\\mathrm {C}^*$ -algebra $\\mathrm {C}_r(\\mathcal {G})$ .",
"However, when the groupoid is amenable, the completion of any faithful representation will result in the same $\\mathrm {C}^*$ -algebra.",
"Here $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each amenable [52].",
"It is convenient, however, to represent them all on the same Hilbert space (namely $\\ell ^2(X^H(P,Q))$ ) because we can consider the interactions between operators coming from the different algebras.",
"This will be particularly useful when showing how finite Rokhlin dimension passes from $ {\\mathrm {C}}^*(H)$ to $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ in Section .",
"3.4 3.4 Let $(X, \\varphi )$ be a mixing Smale space.",
"Then $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are simple by [51] and [52], separable (since $X$ is a metric space) and nuclear by [52] and [51].",
"The homoclinic $\\mathrm {C}^*$ -algebra $ {\\mathrm {C}}^*(H)$ is unital since the diagonal $X \\times X$ is open in $\\operatorname{\\mathcal {G}}_H$ and $X$ is compact.",
"Each of $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ admit a trace [51], hence are stably finite.",
"The traces on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are not bounded while $ {\\mathrm {C}}^*(H)$ admits a tracial state.",
"Moreover, when $(X, \\varphi )$ is mixing, this trace is unique [25].",
"In [12], the authors showed that for a mixing Smale space, $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ have finite nuclear dimension and hence are $\\mathcal {Z}$ -stable.",
"(There it was not noted that $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are $\\mathcal {Z}$ -stable; however this follows from [66].",
"In fact, it can be proved directly in a similar manner to Theorem REF below, once we know that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.)",
"It then follows that $ {\\mathrm {C}}^*(H)\\otimes \\operatorname{\\mathcal {U}}$ is tracially approximately finite (TAF) in the sense of [36] for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"In particular, the $\\mathrm {C}^*$ -algebras coming from the homoclinic relation on a mixing Smale space is classified by the Elliott invariant, see [12].",
"Suppose $G$ is a discrete group acting on a mixing Smale space.",
"For the action to induce a well-defined action on $\\operatorname{\\mathcal {G}}_U$ and $\\operatorname{\\mathcal {G}}_S$ , the choice of finite sets of $\\varphi $ -invariant periodic points must be $G$ -invariant.",
"Fortunately, this can always be arranged.",
"3.5 3.5 Lemma: Let $G$ be a discrete group acting effectively on a mixing Smale space $(X , \\varphi )$ .",
"Then there exists a finite set of $\\varphi $ -invariant periodic points $P$ such that $gp \\in P$ for every $g \\in G$ and every $p \\in P$ .",
"Proof.",
"Let $P^{\\prime }$ be any finite $\\varphi $ -invariant set of periodic points.",
"Then $P^{\\prime } \\subseteq {\\rm Per}_n(X, \\varphi )$ for some $n\\in {\\mathbb {N}}$ .",
"We know that ${\\rm Per}_n(X, \\varphi )$ is finite.",
"Also, for any $g\\in G$ and $x\\in {\\rm Per}_n(X, \\varphi )$ it was shown in the proof of Proposition REF that $gx \\in {\\rm Per}_n(X, \\varphi )$ .",
"It follows that the set $P=\\lbrace p \\in X \\mid p=gx \\hbox{ for some }g\\in G, x\\in P^{\\prime }\\rbrace $ is contained in ${\\rm Per}_n(X, \\varphi )$ and hence is finite.",
"It is $G$ -invariant by construction and it is $\\varphi $ -invariant because $g \\varphi (x) = \\varphi ( gx)$ for any $g\\in G$ and $x\\in X$ .",
"For the remainder of the paper, we will assume that $P$ and $Q$ are $G$ -invariant.",
"Let $\\operatorname{\\mathcal {G}}$ be one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ and let $G$ be a group acting on $(X, \\varphi )$ .",
"Since $P$ and $Q$ are assumed to be $G$ -invariant, $g(x,y) \\mapsto (gx, gy)$ defines an induced action of $G$ on $\\operatorname{\\mathcal {G}}$ .",
"The action of $G$ on $\\operatorname{\\mathcal {G}}$ in turn induces an action on $\\mathrm {C}^*(\\operatorname{\\mathcal {G}})$ .",
"3.6 3.6 Example: Following Example REF , the Smale space homeomorphism $\\varphi $ induces a $\\mathbb {Z}$ -action on each of $ {\\mathrm {C}}^*(H), \\mathrm {C}^*(S), \\mathrm {C}^*(U)$ .",
"These actions are denoted by $\\alpha $ , $\\alpha _S$ and $\\alpha _U$ , respectively.",
"Properties of $\\alpha $ (in particular, [51]) will prove indispensable for the results in the next section and also in Section , where we seek to pass from known properties about $ {\\mathrm {C}}^*(H)$ to the nonunital $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"3.7 3.7 Example: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type.",
"Then the $C^*$ -algebras associated to $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each AF.",
"Moreover if $G$ is a finite group acting effectively on $(\\Sigma , \\sigma )$ , then using [1] one can show that the induced action of $G$ on each of these AF $\\mathrm {C}^*$ -algebras is locally representable in the sense of [18].",
"This is the case for the automorphisms $\\beta _1$ and $\\beta _2$ on the full two-shift given in Example REF .",
"It follows from results in [18] that the crossed products associated to the action of $G$ are each also AF.",
"3.8 3.8 Example: Suppose $(X, \\varphi )$ is the Smale space obtained via the solenoid construction in Example REF with $Y=S^1$ and $g(z)=z^n$ ($n\\ge 2$ ).",
"Then the stable and unstable algebras are the stabilisation of a Bunce–Deddens algebra.",
"For details in the case $n=2$ , see page 28 of [51].",
"Automorphisms of Bunce–Deddens algebras are considered in [45], for example.",
"3.9 3.9 Example: If $(X, \\varphi )$ is a hyperbolic toral automorphism, as in Example REF , then the stable and unstable algebras are the stabilization of irrational rotation algebras as is shown on pages 27-28 of [51].",
"Automorphisms of irrational rotation algebras are well-studied, see [14] and reference therein.",
"The induced action on $ {\\mathrm {C}}^*(H)$ In the sequel, the aim is to provide conditions of a group action on a mixing Smale space which will allow us to determine structural properties of the crossed products of the associated $\\mathrm {C}^*$ -algebras by the induced group action.",
"The idea is to determine what properties are preserved when passing from the $\\mathrm {C}^*$ -algebra to its crossed product.",
"If we interpret a $\\mathrm {C}^*$ -crossed product as a “noncommutative orbit space” then what we are asking for is some sort of “freeness” condition.",
"In the $\\mathrm {C}^*$ -algebraic context this might take a number of different forms.",
"Here, we focus on the Rokhlin dimension of an action (which is akin to a “coloured” version of noncommutative freeness), initially proposed for finite group and $\\mathbb {Z}$ -actions by Hirshberg, Winter and Zacharias [23] and subsequently generalised to other groups [63], [15], [64], [22], [8], as well as the notion of a “strongly outer action” (which can be thought of as a noncommutative approximation of freeness in trace) (see for example, [41]).",
"Recent results of Sato [58] also play a key role, although for finite group actions and ${\\mathbb {Z}}^d$ -actions these new results are not required.",
"Finite group actions on $ {\\mathrm {C}}^*(H)$ For this subsection, we fix a mixing Smale space $(X, \\varphi )$ .",
"Let $ {\\mathrm {C}}^*(H)$ denotes its homoclinic algebra.",
"Here we study actions induced on $ {\\mathrm {C}}^*(H)$ by free actions of $G$ on $(X, \\varphi )$ .",
"Given an action of a group $G$ on a $C^*$ -algebra, $A$ , $\\beta : G \\rightarrow {\\rm Aut}(A)$ we will write $\\beta _h$ for $\\beta (h)$ .",
"Based on Proposition REF which shows that freeness is unlikely to hold for infinite groups, we restrict to the case that $G$ is finite.",
"4.1 4.1 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra and let $G$ be a finite group.",
"An action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and each finite subset $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A $ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ $g \\ne h$ in $G$ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ ; $\\Vert [ f^{( l )}_g, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ .",
"When $d = 0$ the action is said to have the Rokhlin property.",
"In this case the contractions $(f_g)_{g \\in G}$ can in fact be taken to be projections.",
"The definition of the Rokhlin property for $\\mathrm {C}^*$ -algebras was introduced by Izumi [26], [27].",
"4.2 4.2 Lemma: Let $G$ be a finite group acting on $(X, \\varphi )$ and denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Let $d$ be a nonnegative integer.",
"Suppose that for any $\\epsilon >0$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for $g \\ne h \\in G$ , $l \\in \\lbrace 0, \\dots , d\\rbrace $ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ .",
"Proof.",
"Let $\\epsilon >0$ and $F \\subset {\\mathrm {C}}^*(H)$ be a finite set.",
"Take positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ with the properties assumed in the statement of the theorem.",
"By [51], there is $n \\in {\\mathbb {N}}$ such that $\\Vert [ \\alpha ^n( f^{( l )}_g) , a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ , where $\\alpha $ is the automorphism from Example REF .",
"Since $\\alpha $ is an automorphism, $\\left( \\alpha ^n( f^{(l)}_g) \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ satisfies the other requirements of Definition REF .",
"4.3 4.3 Corollary: Suppose $A$ is a $G$ -invariant unital $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ .",
"If $G$ acting on $A$ has Rokhlin dimension at most $d$ then the action of $G$ on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"In particular, if $G$ acting on $C(X)$ has Rokhlin dimension at most $d$ , then $G$ acting on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"Proof.",
"By assumption, given $\\epsilon >0$ , there exists positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A \\subset {\\mathrm {C}}^*(H)$ such that the hypotheses of Lemma REF hold; this then implies the result.",
"When considering the statement concerning $C(X)$ one need only note that $C(X)$ is a $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ and that it is invariant under the action of $G$ .",
"4.4 4.4 Corollary: Suppose $(X, \\varphi )$ is a mixing Smale space and a finite group $G$ acts on $(X, \\varphi )$ freely.",
"Then $G$ acting on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"For finite group actions on a compact space, freeness implies finite Rokhlin dimension [21].",
"The result then follows from Corollary REF .",
"4.5 4.5 Corollary: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type and a finite group $G$ acts on $(\\Sigma , \\sigma )$ freely.",
"Then the action of $G$ on $ {\\mathrm {C}}^*(H)$ has the Rokhlin property.",
"Proof.",
"Since $\\Sigma $ is the Cantor set and $G$ acts freely, the action of $G$ on $C(\\Sigma )$ has the Rokhlin property.",
"Corollary REF then implies the result.",
"Strongly outer actions Using [41], we prove the first of the three results listed at the end of the introduction; it appears as Theorem REF .",
"To begin, we recall the Vitali covering theorem and the definitions needed for its statement.",
"4.6 4.6 Definition: A finite measure $\\mu $ on a metric space $(X, d)$ is said to be doubling if there exists a constant $M > 0$ such that $ \\mu (B(x, 2\\epsilon )) \\le M \\mu (B(x, \\epsilon )) $ for any $x \\in X$ and any $\\epsilon >0$ .",
"4.7 4.7 Definition: Suppose $(Y,d)$ is a metric space and $A \\subseteq Y$ .",
"Then a Vitali cover of $A$ is a collection of closed balls $\\mathcal {B}$ such that inf$\\lbrace r>0 \\mid B(x,r) \\in \\mathcal {B} \\rbrace =0$ for all $x\\in A$ .",
"4.8 4.8 Theorem: [Vitali Covering Theorem] Suppose $(Y,d)$ is a compact metric space, $\\mu $ is a doubling measure, $A \\subseteq Y$ and $\\mathcal {F}$ is Vitali cover of $A$ .",
"Then, for any $\\epsilon >0$ , there exists finite disjoint family $\\lbrace F_1, F_2, \\ldots F_n \\rbrace \\subseteq \\mathcal {F}$ such that $\\mu ( A - \\cup ^n_{i=1} F_i) < \\epsilon $ .",
"4.9 4.9 Lemma: Suppose $(X, \\varphi )$ is a mixing Smale space with Bowen measure $\\mu $ and $G$ is a discrete group acting effectively on $(X, \\varphi )$ .",
"Let $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ denote the induced action.",
"Then, for any $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ , there exists positive contractions $(f_i)_{i=1}^k \\subset {\\mathrm {C}}^*(H)$ such that (i) $f_i \\cdot f_j=0$ for all $i\\ne j$ ; $f_i \\cdot \\beta _g(f_i) =0$ ; $\\tau (\\sum _{i=1}^k f_i)> 1- \\epsilon $ where $\\tau $ denotes the (unique) trace on $ {\\mathrm {C}}^*(H)$ obtained from $\\mu $ .",
"Proof.",
"Fix $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ .",
"Let $D=\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Then $D$ is open and has full measure by Proposition REF .",
"Moreover, for each $x\\in D$ there exists $\\delta _x>0$ such for any $0<\\delta \\le \\delta _x$ we have $\\overline{B_{\\delta }(x)} \\cap g(\\overline{ B_{\\delta }(x)}) = \\emptyset $ and $\\overline{B_{\\delta }(x)} \\subseteq D$ .",
"Let $\\mathcal {F}= \\left\\lbrace F_x \\: | \\: F_x = \\overline{ B_{\\delta }(x)} \\hbox{ for some }x\\in D \\hbox{ and some }0<\\delta \\le \\frac{\\delta _x}{2} \\right\\rbrace .$ By construction, the collection $\\mathcal {F}$ is a Vitali covering of $D$ .",
"The Bowen measure is doubling (see for example [46]) so we may apply the Vitali Covering Theorem to obtain a finite subcollection of $\\mathcal {F}$ , $\\lbrace F_{x_1}, \\ldots , F_{x_k} \\rbrace $ , with the following properties (i) $F_{x_i} \\cap F_{x_j}=\\emptyset $ ; $\\mu (\\cup _{i=1}^k F_{x_i}) > 1- \\epsilon $ .",
"Recall that $\\mu (D)=1$ .",
"Since each $F_{x_i}$ is compact (they are closed in a compact space) and pairwise disjoint, we can use Urysohn's lemma and the Tietze extension theorem to obtain pairwise disjoint open sets $(U_i)_{i=1}^k$ such that for each $i$ , $F_{x_i}\\subseteq U_{x_i}\\subseteq B_{\\delta _{x_i}}(x_i)$ and functions $(f_i)_{i=1}^k \\subseteq C(X) \\subseteq {\\mathrm {C}}^*(H)$ with the following properties: (i) $0\\le f_i \\le 1$ , $ {\\rm supp}(f_i) \\subseteq U_i$ , $F_{x_i} \\subseteq \\lbrace x \\: |\\: f_i(x)=1 \\rbrace $ ; for every $i \\in \\lbrace 1, \\dots , k\\rbrace $ .",
"Finally, we show that $(f_i)_{i=1}^k$ has the required properties.",
"They are by definition positive contractions.",
"Moreover, (i) $f_i \\cdot f_j =0$ for every $i \\ne j \\in \\lbrace 1, \\dots , k\\rbrace $ , since ${\\rm supp}(f_i) \\subseteq U_i$ and $U_{i} \\cap U_{j}=\\emptyset $ ; $f_i \\cdot \\beta _g(f_i)=0$ for every $i \\in \\lbrace 0, \\dots , k\\rbrace $ since $U_i \\subseteq B_{\\delta _{x_i}}(x_i)$ implies that $U_i \\cap g( U_i) = \\emptyset $ .",
"Finally, $\\tau \\left( \\sum _{i=1}^k f_i \\right) & \\ge & \\sum _{i=1}^k \\mu ( \\lbrace x | f_i(x) =1 \\rbrace ) \\\\& \\ge & \\sum _{i=1}^k \\mu (F_i) \\\\& = & \\mu ( \\cup _{i=1}^k F_i ) \\\\& > & 1 - \\epsilon ,$ showing (iii) holds.",
"4.10 4.10 Let $A$ be a $\\mathrm {C}^*$ -algebra and $\\tau $ a state on $A$ .",
"We denote by $\\pi _{\\tau }$ the representation of $A$ corresponding to the GNS construction with respect to $\\tau $ .",
"In this case, $\\pi _{\\tau }(A)^{\\prime \\prime }$ is the enveloping von Neumann algebra of $\\pi _{\\tau }(A)$ .",
"Definition: [41] Let $A$ be a unital simple $\\mathrm {C}^*$ -algebra with nonempty tracial state space $T(A)$ .",
"An automorphism $\\beta $ of $A$ is not weakly inner if, for every $\\tau \\in T(A)$ such that $\\tau \\circ \\beta = \\tau $ , the weak extension of $\\beta $ to $\\pi _{\\tau }(A)^{\\prime \\prime }$ is outer.",
"If $G$ is a discrete group, then an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is strongly outer if, for every $g \\in G \\setminus \\lbrace e\\rbrace $ , the automorphism $\\beta _g$ is not weakly inner.",
"The next lemma and theorem are based on arguments due to Kishimoto in [33].",
"4.11 4.11 Lemma: Let $A$ be a unital $\\mathrm {C}^*$ -algebra and $\\tau \\in T(A)$ .",
"Then for every $\\epsilon > 0$ there is $\\delta > 0$ such that for any positive contraction $f \\in A$ such that $\\tau (f) > 1 - \\delta $ we have $\\tau (a) \\le \\tau (fa) + \\epsilon $ for every $a \\in A$ .",
"Proof.",
"It is enough to show this holds when $a \\in A_+$ .",
"Given $\\epsilon > 0$ let $\\delta = \\epsilon ^2$ .",
"Then since $a = af + a(1-f)$ we have $\\tau (a) - \\tau (af) &=& \\tau (a(1-f)) \\\\&\\le & \\tau (a^2)^{1/2}\\tau ((1-f)^2)^{1/2} \\\\&\\le & \\tau ((1-f)^2)^{1/2}\\\\&\\le & \\tau (1-f)^{1/2} \\\\&\\le & \\epsilon .$ 4.12 4.12 Theorem: Let $(X, \\varphi )$ be a mixing Smale space with an effective action of a discrete group $G$ .",
"Then the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"Proof.",
"Let $\\tau $ denote the tracial state on $ {\\mathrm {C}}^*(H)$ corresponding to the Bowen measure on $(X, \\varphi )$ .",
"Fix $g \\in G\\setminus \\lbrace e\\rbrace $ .",
"Since $\\tau $ is the unique tracial state, we have $\\tau = \\tau \\circ \\beta _g$ .",
"Let $\\epsilon > 0$ and let $F \\subset A$ be a finite subset.",
"Let $\\delta $ be the $\\delta $ of Lemma REF with respect to $\\epsilon /2$ .",
"By Lemma REF , we can find positive contractions $f_1, \\dots , f_k \\in {\\mathrm {C}}^*(H)$ such that (i) $f_if_j = 0$ for every $0 \\le i \\ne j \\le k$ , $f_i \\beta _g(f_i) = 0$ , $\\tau (f_1 + \\dots + f_k) > 1 - \\delta $ .",
"There is a sufficiently large $N \\in \\mathbb {N}$ such that, for any $i =1, \\dots , k$ , we have $ \\Vert \\alpha ^N(f_i) a - a \\alpha ^N(f_i) \\Vert < \\epsilon /2 \\text{, for any } a \\in F.$ Since $\\alpha ^N : {\\mathrm {C}}^*(H)\\rightarrow {\\mathrm {C}}^*(H)$ is an automorphism, we have that (i) $\\alpha ^N(f_i) \\alpha ^N(f_j) = 0$ for every $1 \\le i \\ne j \\le k$ and $\\tau (\\alpha ^N(f_1) + \\dots + \\alpha ^N(f_k)) > 1 - \\delta .$ Moreover, since the action of $G$ on $X$ commutes with $\\varphi $ , we have, for any $f\\in {\\mathrm {C}}^*(H)$ , that $\\alpha ^N(\\beta _g(f)) = \\beta _g(\\alpha ^N(f))$ .",
"Hence (i) $\\alpha ^N(f_i) \\beta _g(\\alpha ^N(f_i)) = 0.$ Thus we have $ \\Vert \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) \\Vert < \\epsilon /2$ for any $a \\in F$ .",
"Consider the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z}$ .",
"It is generated by $ {\\mathrm {C}}^*(H)$ and $u_g$ where $u_g$ is unitary satisfying $u_g a u_g^* = \\beta _g(a)$ for every $a \\in {\\mathrm {C}}^*(H)$ .",
"Let $\\sigma \\in T( {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z})$ and note that $\\sigma |_{ {\\mathrm {C}}^*(H)} = \\tau $ .",
"Put $f = \\sum _{i=1}^k \\alpha ^N(f_i)$ .",
"By the above and Lemma REF we have $\\sigma (a u_g) &\\le & \\sigma (f a u_g f) + \\epsilon \\\\&=& \\sum _{i=1}^k \\sigma ( \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) u_g ) + \\epsilon \\\\&<& 2\\epsilon .$ Since $\\epsilon $ and $F$ were arbitrary, it follows that $\\sigma (a u_g) = 0$ for all $a \\in A$ .",
"That $\\beta _g$ is not weakly inner now follows from the proof of [33].",
"Since this holds for every $g \\in G\\setminus \\lbrace e\\rbrace $ , it follows that the action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"4.13 4.13 Proposition: Suppose $(X, \\varphi )$ is a mixing Smale space with an effective action of ${\\mathbb {Z}}^m$ .",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"In fact, the Rokhlin dimension of the action is bounded by $4^m -1$ .",
"If $m = 1$ then the action has Rokhlin dimension no more than one.",
"Proof.",
"This result follows directly from [35].",
"$\\mathbb {Z}$ -actions on irreducible Smale spaces Throughout this subsection, we fix an irreducible Smale space $(X, \\varphi )$ with decomposition into mixing components given by $X = X_1 \\sqcup \\dots \\sqcup X_N$ , as in .",
"To begin, we recall the definition of Rokhlin dimension for integer actions as given in [23].",
"4.14 4.14 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra.",
"An action of the integers $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least natural number such that the following holds: for any finite subset $\\mathcal {F} \\subset A$ , and $p \\in \\mathbb {N}$ and any $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ in $A$ satisfying (i) for any $l \\in \\lbrace 0, \\dots , d\\rbrace $ we have $\\Vert f^{(l)}_{r,i} f^{(l)}_{s, j} \\Vert < \\epsilon $ whenever $(r,i) \\ne (s, j)$ , $\\Vert \\sum _{l=0}^d \\sum _{r =0}^1 \\sum _{j = 0}^{p-1+r} f^{(l)}_{r,j} -1\\Vert < \\epsilon $ , $\\Vert \\beta _1(f^{(l)}_{r, j}) - f^{(l)}_{r, j+1} \\Vert < \\epsilon $ for every $r \\in \\lbrace 0,1\\rbrace $ , $j \\in \\lbrace 0, \\dots , p-2+r\\rbrace $ and $l \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert \\beta _1(f^{(l)}_{0,p-1} + f^{(l)}_{1, p}) - (f^{(l)}_{0,0} + f^{(l)}_{1,0}) \\Vert < \\epsilon $ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ $\\Vert [ f^{(l)}_{r,j}, a ] \\Vert < \\epsilon $ for every $r, j, l$ and $a \\in \\mathcal {F}$ .",
"As in the case of a finite group, in the special case of the homoclinic algebra, we don't need to worry about satisfying the last condition.",
"The proof is similar to Lemma REF and is omitted.",
"4.15 4.15 Lemma: Let $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ be an action of the integers.",
"Suppose that for any $p \\in \\mathbb {N}$ and $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ satisfying (i)–(iv) of Definition REF .",
"Then $\\beta $ has Rokhlin at most dimension $d$ .",
"4.16 4.16 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there is a $\\sigma \\in S_N$ such that $ \\beta |_{X_i} : X_i \\rightarrow X_{\\sigma (i)} $ for each $i = 1,\\dots , N$ .",
"Proof.",
"Since $(X_i, \\varphi ^N|_{X_i})$ is mixing, there is a point, call it $x_i$ , with dense $\\varphi ^N$ -orbit in $X_i$ .",
"Since $X_1, \\dots , X_N$ are disjoint, there is $k(i) \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta (x_i) \\in X_{k(i)}$ and $\\beta (x_i) \\notin X_k$ for $k \\ne k(i)$ .",
"Now $\\varphi ^N \\circ \\beta (x_i) \\in X_{k(i)}$ , so we have $\\varphi ^N \\circ \\beta (x_i) = \\beta \\circ \\varphi ^N(x_i) \\in X_{k(i)}$ .",
"The fact that $x_i$ has dense $\\varphi ^N$ -orbit then implies $\\beta |_{X_i} : X_i \\rightarrow X_{k(i)}$ .",
"Suppose that $\\beta |_{X_j}(X_j) \\subset X_{k(i)}$ .",
"If $j \\ne i$ , there exists some $l \\in \\lbrace 1, \\dots , N\\rbrace $ such that $X_l$ is not in the image of any $\\beta (X_i)$ , $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"Choose $x \\in X_l$ .",
"Then $\\beta (x )\\in X_{k(l)}$ and there is some $m$ , $0<m<N$ satisfying $\\varphi ^m(X_{k(l)}) = X_{l}$ .",
"But this implies $\\beta \\circ \\varphi ^m(x) = \\varphi ^m \\circ \\beta (x) \\in X_l$ .",
"So we have $j= i$ , which proves the theorem.",
"4.17 4.17 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there are $L, S \\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ such that $N = LS$ and partition of $\\lbrace 1, \\dots , N\\rbrace $ into $S$ subsets of $L$ elements $\\lbrace r_{s,1}, \\dots , r_{s,L}\\rbrace $ , $1 \\le s \\le S$ satisfying, for each $s$ , $ \\beta (X_{r_{s, l}}) = X_{s, r_{l+1 \\text{mod} L}}.$ Proof.",
"Let $r_{1,1} = 1$ .",
"By the previous lemma there is $r_{1,2} \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta : X_{r_{1,1}} \\rightarrow X_{r_{1,2}}$ .",
"If $r_{1, 2} \\ne 1$ then there is $r_{1, 3} \\ne r_{1,2}$ such that $\\beta : X_{r_{1,2}} \\rightarrow X_{r_{1,3}}$ .",
"By induction and the pigeonhole principal, there is some $0 < L \\le N$ such that $r_{1, l}$ are all distinct, $\\beta : X_{r_{1,l}} \\rightarrow X_{r_{1,l+1}}$ , for $1 \\le l \\le L-1$ and $\\beta : X_{r_{1,L}} \\rightarrow X_{r_{1,1}}$ .",
"If $L= N$ , we are done.",
"Otherwise, there is some $X_{r_{2,1}} \\ne X_{r_{2, l}}$ for every $1 \\le l \\le L$ .",
"Arguing as above and using the previous lemma, there is some $L^{\\prime } \\in \\lbrace 1, \\dots , N-L\\rbrace $ such that $r_{2, l}$ are all distinct, $r_{2,l^{\\prime }} \\ne r_{1, l}$ for any $1 \\le l^{\\prime } \\le L^{\\prime }$ and $1 \\le l \\le L$ , $\\beta : X_{r_{2,l}} \\rightarrow X_{r_{2,l+1}}$ , for $1 \\le l \\le L^{\\prime }-1$ and $\\beta : X_{r_{2,L^{\\prime }}} \\rightarrow X_{r_{2,1}}$ .",
"Without loss of generality, we may assume that $L \\le L^{\\prime }$ .",
"Let $1 \\le c \\le N$ satisfy $\\varphi ^c : X_{r_{1,i}} \\rightarrow X_{r_{2,2}}$ .",
"Then we have a diagram ${ X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & \\dots [r]^{\\beta } & X_{r_{1,L}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,1 \\,}} [d]^{\\varphi ^c}\\\\X_{r_{2,2}} [r]^{\\beta } & X_{r_{2,2}} [r]^{\\beta } & \\dots [r]^{\\beta }& X_{r_{2,L}} [r]^{\\beta } & X_{r_{2,L+1}} \\\\}$ which commutes.",
"If follows that $X_{r_{2, L+1}} = Y_{r_{2, 1}}$ , that is, $L = L^{\\prime }$ .",
"The proof now follows from induction.",
"4.18 4.18 Theorem: Suppose $(X, \\varphi )$ is an irreducible Smale space with an effective $\\mathbb {Z}$ -action.",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"Let $\\beta : X \\rightarrow X$ be the homeomorphism generating the $\\mathbb {Z}$ action and let $X = X_1 \\sqcup \\cdots \\sqcup X_N$ be the Smale decomposition into mixing components $(X_i, \\varphi ^N|_{X_i})$ .",
"Denote by $\\mathrm {C^*}(H_i)$ the homoclinic algebra for the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"By Lemmas REF and REF , there exists $L$ such that $ \\beta ^L : X_i \\rightarrow X_i $ for every $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"For each $i$ , we consider the action $\\beta ^L|_{X_i}$ .",
"Since $\\beta $ induces an effective ${\\mathbb {Z}}$ -action on $(X, \\varphi )$ (in particular $\\beta ^L \\circ \\varphi ^N= \\varphi ^N\\circ \\beta ^L$ ) it follows that $\\beta ^L|_{X_i}$ induces an effective ${\\mathbb {Z}}$ -action on the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"Hence, we can apply Theorem REF to $\\beta ^L|_{X_i}$ acting on $(X_i, \\varphi ^N|_{X_i})$ to conclude that the action induced by $\\beta ^L|_{X_i}$ on $\\mathrm {C}^*(H_i)$ has Rokhlin dimension $d_i$ for some $d_i < \\infty $ .",
"Let $\\epsilon > 0$ , $p \\in \\mathbb {N} \\setminus \\lbrace 0\\rbrace $ .",
"Let $f^{(k)}_{i, j}$ , $i\\in \\lbrace 0,1\\rbrace $ , $0 \\le j \\le p-1+i$ , $0 \\le k \\le d_1$ be two Rokhlin towers for $\\beta ^L|_{X_1}$ with respect to $\\epsilon $ and $p$ and any finite subset $\\mathcal {F}_1 \\subset \\mathrm {C}^*(H_1)$ .",
"We claim that the elements $\\beta ^l(f^{(k)}_{i, j})$ , $1 \\le l \\le L-1$ satisfy (i) – (v) of Lemma REF with respect to $ {\\mathrm {C}}^*(H), \\epsilon $ and $p$ .",
"Indeed, (i), (iii) and (iv) are clear by construction.",
"To see (ii), we note that $\\Vert \\sum _{i,j,k} f^{(k)}_{i, j} - 1_{\\mathrm {C}^*(H_1)}\\Vert < \\epsilon $ and since $\\beta $ is a homeomorphism, it follows that $\\Vert \\sum _{i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{\\mathrm {C}^*(H_l)})\\Vert < \\epsilon $ for every $l$ , and hence $\\Vert \\sum _{l,i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{ {\\mathrm {C}}^*(H)}\\Vert < \\epsilon .$ 4.19 4.19 Remark: Let $Y$ be a compact Hausdorff space.",
"It is not difficult to check that if a $\\mathbb {Z}$ -action on $C(Y)$ has finite Rokhlin dimension, then the action on $Y$ must be free.",
"Thus when ${\\mathbb {Z}}$ acts on an irreducible Smale space $(X, \\varphi )$ , the action of ${\\mathbb {Z}}$ on $C(X)$ cannot have finite Rokhlin dimension because the action of ${\\mathbb {Z}}$ on $X$ is not free by Proposition REF .",
"Nevertheless, Proposition REF implies that a ${\\mathbb {Z}}$ -action on $ {\\mathrm {C}}^*(H)$ induced from an effective action on $(X, \\varphi )$ has Rokhlin dimension at most one.",
"The induced action on the stable and unstable algebras In this section we use what we have already proved about actions on the homoclinic algebra to deduce results for actions on the stable and unstable algebras.",
"To do so, we take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"It is easy to check that if $(X, \\varphi )$ is a Smale space, then $(X, \\varphi ^{-1})$ is also a Smale space with the bracket reversed.",
"The unstable relation of $(X, \\varphi )$ is then the stable relation of $(X, \\varphi ^{-1})$ .",
"Thus, it is enough to show something holds for $ \\mathrm {C}^*(S)$ of an arbitrary (irreducible, mixing) Smale space to imply the same for $ \\mathrm {C}^*(U)$ of an arbitrary (irreducible, mixing) Smale space.",
"Throughout this section, we again assume that $(X, \\varphi )$ is irreducible, but is not necessarily mixing unless explicitly stated.",
"5.1 5.1 Let $(X, \\varphi )$ be an irreducible Smale space.",
"Fix a finite set $P$ of $\\varphi $ -invariant periodic points.",
"Let $ {\\mathrm {C}}^*(H)$ denote the associated homoclinic and $ \\mathrm {C}^*(S)$ the stable algebra.",
"As in [51], we define, for each $a \\in C_c(\\mathcal {G}_H)$ , an element $(\\rho (a), \\rho (a))$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ by $ (\\rho (a) b)(x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} a(x,z)b(z,y) $ and $ (b \\rho (a)) (x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} b(x,z) a(z,y)$ where $b\\in C_c(S)$ .",
"This extends to a map $\\rho : {\\mathrm {C}}^*(H)\\operatorname{\\hookrightarrow }\\mathcal {M}( \\mathrm {C}^*(S)).$ We note that $\\rho $ and the representations of these algebras on the Hilbert space $l^2(X^H(P,Q))$ (see Remark REF ) are compatible.",
"5.2 5.2 Lemma: Let $F \\subset {C}_c(S)$ be a finite subset and let $r_1, \\dots , r_N \\in {C}_c(\\mathcal {G}_H)$ .",
"Then, for every $\\epsilon > 0$ there exists a $k\\in \\mathbb {N}$ such that, viewing $a$ as an element of the multiplier algebra $\\mathcal {M}( \\mathrm {C}^*(S))$ , we have $ \\Vert \\rho (\\alpha ^k(r_i) ) a - a \\rho (\\alpha ^k(r_i )) \\Vert < \\epsilon $ for every $i = 1, \\dots , N$ and every $a \\in F$ .",
"Proof.",
"Follows from [51] or [31].",
"5.3 5.3 Definition: Suppose $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is an action of a countable discrete group.",
"Let $I$ be a separable $G$ -invariant ideal in $A$ and $B$ be a $\\sigma $ -unital $G$ -$\\mathrm {C}^*$ -subalgebra of $A$ .",
"Then there exists a countable $G$ -quasi-invariant quasicentral approximate unit $(w_n)_{n\\in N}$ of $I$ in $B$ .",
"That is, there exists $(w_n)_{n\\in {\\mathbb {N}}}$ an approximate identity for $I$ such that (i) for any $a \\in B$ , $\\Vert aw_n - w_n a\\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ ; for each $g\\in G$ , $\\Vert \\beta _g(w_n) -w_n \\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"The existence of a $G$ -quasi-invariant quasicentral approximate unit is shown in [29], but also see [17] and [13].",
"Rokhlin dimension for actions on the stable and unstable algebra To take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , we will find it convenient to introduce the definition of multiplier Rokhlin dimension with repect to a finite index subgroup.",
"It is inspired by [47], [23] and [64].",
"For a given $\\mathrm {C}^*$ -algebra $A$ and group $G$ with action $\\beta : G \\rightarrow {\\rm Aut}(A)$ we denote by $\\mathcal {M}(\\beta )$ the induced action of $G$ on the multiplier algebra $\\mathcal {M}(A)$ .",
"5.4 5.4 Definition: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"We say the action has multilplier Rokhlin dimension $d$ with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K)$ , if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and finite subsets $M \\subset G$ , $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_{\\overline{g}} \\right)_{l=0, \\ldots d; \\overline{g}\\in G/K} \\subset \\mathcal {M}(A) $ such that (i) for any $l$ , $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert < \\epsilon $ , for $\\overline{g} \\ne \\overline{h}$ in $G/K$ ; $\\Vert \\sum _{l=0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert < \\epsilon $ ; $\\Vert \\mathcal {M}(\\beta _h)(f^{(l)}_{\\overline{g}}) - f^{(l)}_{\\overline{hg}} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $h \\in M$ ; $\\Vert [ f^{( l )}_{\\overline{g}}, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $a \\in F$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) = \\infty $ .",
"We say the action has multilplier Rokhlin dimension $d$ with commuting towers with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K)$ , if, in addition, (i) $\\Vert f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}}- f^{(k)}_{\\overline{h}}f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for every $k,l = 0, \\dots , d$ , $\\overline{g}, \\overline{h} \\in G/K$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) = \\infty $ .",
"The multiplier Rokhlin dimension with respect to a finite subgroup is used to define the Rokhlin dimension for an action of a countable residually finite group.",
"5.5 5.5 Definition: [cf.",
"[64]] Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable, residually finite group, and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ .",
"The Rokhlin dimension of $\\beta $ is defined by $ \\mathrm {dim}_{\\mathrm {Rok}}(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ The Rokhlin dimension of $\\beta $ with commuting towers is given by $ \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ It will be easier to show finite multiplier Rokhlin dimension relative to a finite subgroup, which we show implies the definition of Rokhlin dimension relative to a finite index subgroup for $\\mathrm {C}^*$ -algebras given in [64], recalled in Definition REF below.",
"Some further notation is required to do so.",
"First, we need to define central sequence algebras.",
"Loosely speaking, working in a central sequence algebra allows one to turn statements such as approximate commutativity in the original algebra into honest commutativity in the central sequence algebra.",
"As such, central sequence arguments often allow one to streamline proofs.",
"In the case of discrete groups, an action on a $\\mathrm {C}^*$ -algebra induces an action on its central sequence algebra, and it is possible to reformulate definitions for both Rokhlin dimension of finite group actions and integer actions on separable unital $\\mathrm {C}^*$ -algebras in terms of induced actions on central sequence algebras [64].",
"5.6 5.6 Definition: Let $A$ be a separable $\\mathrm {C}^*$ -algebra.",
"We denote the sequence algebra of $A$ by $ A_{\\infty } := \\prod _{n \\in \\mathbb {N}} A / \\bigoplus _{n \\in \\mathbb {N}} A.",
"$ We view $A$ as a subalgebra of $A_{\\infty }$ by mapping an element $a \\in A$ to the constant sequence consisting of $a$ in every entry.",
"The central sequence algebra is then defined to be $ A^{\\infty } := A_{\\infty } \\cap A^{\\prime } = \\lbrace x \\in A_{\\infty } \\mid ax = xa \\text{ for every } a \\in A\\rbrace , $ the relative commutant of $A$ in $A_{\\infty }$ .",
"Let $ \\mathrm {Ann}(A, A_{\\infty }) := \\lbrace x \\in A_{\\infty } \\mid ax = xa = 0 \\text{ for every } a \\in A\\rbrace ,$ which is evidently an ideal in $A^{\\infty }$ .",
"Finally, we define $ F(A) := A^{\\infty }/ \\mathrm {Ann}(A, A_{\\infty }).",
"$ When $A$ is not separable, one can define the above with respect to a given separable subalgebra $D$ , as is done in [64].",
"However, since all our $\\mathrm {C}^*$ -algebras will be separable, we will not require this.",
"A completely positive contractive (c.p.c.)",
"map $\\varphi : A \\rightarrow B$ between $\\mathrm {C}^*$ -algebras $A$ and $B$ is said to be order zero if it is orthogonality preserving, that is, for every $a, b \\in A_+$ with $ab = ba =0$ we have $\\varphi (a) \\varphi (b) = 0$ .",
"Any $^*$ -homomorphism is of course order zero, but a c.p.c.",
"order zero map is not in general a $^*$ -homomorphism.",
"For more about c.p.c.",
"order zero maps, see [73].",
"5.7 5.7 Definition: [64] Let $A$ be a separable $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"Let $\\tilde{a}_{\\infty }$ denote the induced action on $F_{\\infty }(A)$ .",
"We say the action has Rokhlin dimension $d$ with respect to $K$ if $d$ is the least integer such that there exists equivariant c.p.c.",
"order zero maps $ \\varphi _l : (C(G/K), G\\text{-shift}) \\rightarrow (F_{\\infty }(A), \\tilde{a}_{\\infty }), \\quad l = 0, \\dots , d$ with $ \\varphi _0(1) + \\cdots + \\varphi _d(1) = 1.",
"$ If moreover $\\varphi _0 ,\\dots , \\varphi _d$ can be chosen to have commuting ranges, then we say the action has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"The next result is an obvious generalisation of the equivalence of (1) and (3) of [64] to the case of commuting towers.",
"5.8 5.8 Lemma: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group and $K$ a subgroup of finite index.",
"The following are equivalent for an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ .",
"(i) The action $\\beta $ has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"For every finite subset $M \\subset G$ , finite subset $F \\subset A$ and $\\epsilon >0$ there are positive contractions $(f^{(l)}_{\\overline{g}})_{\\overline{g} \\subset H}^{l= 0, \\dots , d}$ in $A$ such that (i) $\\Vert (\\sum _{l =0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}}) \\cdot a - a \\Vert < \\epsilon $ for all $a \\in F$ ; $\\Vert f_{\\overline{g}}^{(l)}f_{\\overline{h}}^{(l)}a \\Vert \\le \\epsilon $ for all $a\\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\ne \\overline{h} \\in G/K$ ; $\\Vert (\\beta _g(f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}})a \\Vert < \\epsilon $ for all $a \\in F$ , $l \\in 0, \\dots , d$ and $g \\in M$ and $\\overline{h} \\in G/K$ ; $\\Vert f^{(l)}_{\\overline{g}}a - a f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for all $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ ; $\\Vert (f^{(k)}_{\\overline{g}} f^{(l)}_{\\overline{h}} - f^{(l)}_{\\overline{h}} f^{(k)}_{\\overline{g}})a \\Vert < \\epsilon $ for all $a \\in F$ , $k,l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g}, \\overline{h} \\in G/K$ .",
"Proof.",
"The only thing that one needs to check is that asking for the $f^{(l)}_{\\overline{g}}$ to approximately commute is equivalent to having the images of the order zero maps of [64] commute, but this is obvious.",
"5.9 5.9 Theorem: Let $G$ be a countable discrete group, $K$ a subgroup of $G$ with finite index, $A$ a separable $\\mathrm {C}^*$ -algebra and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action with multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ respect to $K$ .",
"If $\\beta $ has multiplier Rokhlin dimension at most $d$ with commuting towers with respect to $K$ , then $\\beta $ has Rokhlin dimension at most $d$ with commuting towers respect to $K$ .",
"Proof.",
"We show the action satisfies the criteria of [64] (and in the commuting tower case Lemma REF (ii)).",
"Note that [64] is exactly (a)-(d) in Lemma REF (ii).",
"Let $M \\subset G$ and $F \\subset A$ be finite subsets and let $\\epsilon > 0$ .",
"Without loss of generality we may assume that every $a \\in F$ is a positive contraction.",
"Since $\\beta $ has multiplier Rokhlin dimension less than or equal to $d$ with respect to $K$ we can find positive contractions $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ satisfying Definition REF with respect to $M$ , $F$ and $\\epsilon /2$ .",
"Let $(w_n)_{n \\in \\mathbb {N}}$ be an $G$ -quasi-invariant quasicentral approximate unit for $A$ in $\\mathcal {M}(A)$ .",
"Since $F \\subset A$ and $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ are in $\\mathcal {M}(A)$ , there exists $N \\in \\mathbb {N}$ sufficiently large so that $ \\Vert w_N a - a \\Vert < \\epsilon /4 \\text{ for every } a \\in F,$ and $ \\Vert [f_{\\overline{g}}^{(l)}, w_N] \\Vert < \\epsilon /8.$ and $ \\Vert \\beta _g(f_{\\overline{h}}^{(l)}) - f_{\\overline{gh}}^{(l)} \\Vert < \\epsilon /8.$ By increasing $N$ if necessary, we may also assume, since $M$ is finite, that $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ .",
"Then let $ r_{\\overline{g}}^{(l)} := w_N f_{\\overline{g}}^{(l)} w_N,$ for $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $g \\in G$ .",
"Then each $r_{\\overline{g}}^{(l)}$ is a positive contraction in $A$ and $\\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}}) a - a\\Vert &=& \\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N) a - a\\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N - 1 \\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert \\\\&<& \\epsilon ,$ showing that (a) of Lemma REF (ii) holds.",
"Next, $\\Vert r_{\\overline{g}}^{(l)} r_{\\overline{h}}^{(l)} a \\Vert &=& \\Vert w_N f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(l)} w_N a \\Vert \\\\&=& \\epsilon /2 + \\Vert w_N^2 f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(l)} w_N^2 a \\Vert \\\\&<& \\epsilon $ for every $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , showing that (b) of Lemma REF holds.",
"Using $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ , we obtain $\\Vert (\\beta _g(r_{\\overline{h}}^{(l)}) - r_{\\overline{gh}}^{(l)})a\\Vert &=& \\epsilon /4 + \\Vert w_N \\mathcal {M}(\\beta _g)(f^{(l)}_{\\overline{h}}) w_N - w_N f^{(l)}_{\\overline{gh}} w_N \\Vert \\\\&<& \\epsilon ,$ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ , every $\\overline{h} \\in G/K$ , every $g \\in M$ and every $a \\in F$ , showing (c) of Lemma REF .",
"For (d) of Lemma REF we have $\\Vert r_{\\overline{g}}^{(l)} a - a r_{\\overline{g}}^{(l)} \\Vert = \\epsilon / 2 + \\Vert f_{\\overline{g}}^{(l)} a - a f_{\\overline{g}}^{(l)} \\Vert < \\epsilon ,$ for every $a \\in F$ , every $l \\in \\lbrace 0, \\dots , d\\rbrace $ and every $\\overline{g} \\in G/K$ .",
"Finally, in the commuting tower case, we must show (e): $\\Vert (r_{\\overline{g}}^{(k)} r_{\\overline{h}}^{(l)} - r_{\\overline{h}}^{(l)} r_{\\overline{g}}^{(k))} a\\Vert &\\le & \\Vert f_{\\overline{g}}^{(k)} w_N^2 f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(k)}\\Vert \\\\&\\le & \\epsilon /4 + \\Vert f_{\\overline{g}}^{(k)} f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(k)}\\Vert \\\\&<& \\epsilon ,$ for every $\\overline{g}, \\overline{h} \\in G/K$ and $k, l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"The proof of the next lemma is obvious and hence omitted.",
"It will, however, prove useful in what follows.",
"5.10 5.10 Lemma: Suppose that the action of $G$ on $ {\\mathrm {C}}^*(H)$ has Rokhlin dimension at most $d$ .",
"Then we may choose the Rokhlin elements to satisfy $f^{(l)}_g \\in C_c(\\mathcal {G}_H)$ for $l = 0, \\dots , d$ and $ g \\in G$ .",
"5.11 5.11 Proposition: Let $G$ be a countable group acting on an irreducible Smale space $(X, \\varphi )$ and let $K \\subset G$ be a subset of finite index.",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ with respect to $K$ so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"If $\\beta $ has Rokhlin dimension at most $d$ with commuting towers with respect to $K$ so does the action $\\beta ^{(S)}$ .",
"Proof.",
"We will show that $\\beta ^{(S)}$ has multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"The result then follows from Theorem REF .",
"Let $M$ be a finite subset of $G$ , $F$ a finite subset of $ \\mathrm {C}^*(S)$ and $\\epsilon >0$ .",
"Without loss of generality, we may assume that $F \\subset C_c(S)$ .",
"Since $\\beta $ has Rokhlin dimension at most $d$ with respect to $K$ there are contractions $r^{(l)}_g \\in {\\mathrm {C}}^*(H)$ such that (i) $\\Vert r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}} \\Vert < \\epsilon /2$ for $l = 0, \\dots , d$ and any $g, h \\in G$ with $g \\ne h$ , $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1\\Vert < \\epsilon /2,$ $\\Vert \\beta _h(r^{(l)}_{\\overline{g}}) - r^{(l)}_{\\overline{hg}}\\Vert < \\epsilon /2$ for $l =0, \\dots , d$ , every $h \\in M$ and $\\overline{g} \\in G/K$ , and $\\Vert r^{(k)}_{\\overline{g}}r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}}r^{(k)}_{\\overline{g}}\\Vert < \\epsilon /2$ for every $\\overline{g}$ , $\\overline{h} \\in G/K$ and $k,l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"By the previous lemma, we may moreover assume that each $r^{(l)}_{\\overline{g}} \\in C_c(\\mathcal {G}_H)$ .",
"Now we can find a natural number $k$ such that $ \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert < \\epsilon /2,$ for every $a \\in F$ .",
"For $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , let $ f^{(l)}_{\\overline{g}} := \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}} )).$ Note that each $f^{(l)}_g$ is a positive contraction in $\\mathcal {M}( \\mathrm {C}^*(S))$ .",
"We will show that the $f^{(l)}_{\\overline{g}}$ satisfy (i) – (iv) of Definition REF .",
"For (i) we have $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} )) \\Vert \\\\&=& \\Vert \\alpha ^k(r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) \\Vert \\\\&<& \\epsilon ,$ for any $\\overline{g} \\ne \\overline{h} \\in G/K$ , any $l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"For (ii) $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f_{\\overline{g}}^{(l)} - 1_{\\mathcal {M}( \\mathrm {C}^*(S))} \\Vert &=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g}\\in G} \\rho (\\alpha ^k( r^{(l)}_{\\overline{g}} )) - \\rho (1_{ {\\mathrm {C}}^*(H)})\\Vert \\\\&=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1_{ {\\mathrm {C}}^*(H)} \\Vert \\\\&<& \\epsilon ,$ for any $a \\in F$ .",
"For (iii) we have $\\Vert f^{(l)}_{\\overline{g}} a - a f^{(l)}_{\\overline{g}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert \\\\&<& \\epsilon /2,$ for all $l \\in \\lbrace 0, \\dots d\\rbrace , \\overline{g} \\in G/K$ and $a \\in F$ .",
"For (iv), let $g \\in G$ with $\\overline{h} \\in G/K $ and let $a \\in F$ .",
"Then, $\\Vert \\beta ^{(S)}_g (f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}} \\Vert &=& \\Vert \\alpha ^k(\\beta _g(r^{(l)}_{\\overline{h}}) - r^{(l)}_{\\overline{gh}}) \\Vert \\\\&<& \\epsilon .$ Finally, in the commuting towers case, for (v), let $\\overline{g}, \\overline{h} \\in G/K$ and $l,k \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert f^{(k)}_{\\overline{h}} f^{(l)}_{\\overline{g}} - f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) - \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}) \\Vert \\\\&=& \\Vert r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}\\Vert \\\\&<& \\epsilon .$ The result now follows.",
"This gives us the next corollary: 5.12 5.12 Corollary: Let $G$ be a countable residually finite group acting on an irreducible Smale space $(X, \\varphi )$ .",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ (with commuting towers) so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"By Proposition REF and Theorem REF respectively, we get the next two corollaries.",
"5.13 5.13 Corollary: Let $d\\in {\\mathbb {N}}$ and $G={\\mathbb {Z}}^d$ .",
"Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"5.14 5.14 Corollary: Let $\\mathbb {Z}$ be an effective action on an irreducible Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"For more general group actions the situation is less clear.",
"In particular, to the authors' knowledge, it is not known if strong outerness implies finite Rokhlin dimensionAt the workshop “Future Targets in the Classification Program for Amenable $\\mathrm {C}^*$ -Algebras” held at BIRS, Eusebio Gardella presented results in this direction, but they have yet to appear..",
"In fact, for general discrete groups, there are obstructions to a (strongly outer) action having finite Rokhlin dimension with commuting towers, see [21].",
"Such examples can occur in the context considered in the present paper.",
"An explicit example is the following, let $(\\Sigma _{[3]}, \\sigma )$ be the full three shift (so $\\Sigma _{[3]}= \\lbrace 0, 1, 2\\rbrace ^{{\\mathbb {Z}}}$ and $\\sigma $ is the left sided shift).",
"Then $ {\\mathrm {C}}^*(H)$ is the UHF-algebra with supernatural number $3^{\\infty }$ .",
"The action induced from the permutation $0 \\mapsto 1$ , $1 \\mapsto 0$ , and $2 \\mapsto 2$ is an effective order two automorphism.",
"Thus the action induced on $ {\\mathrm {C}}^*(H)$ is strongly outer, but it follows from [21] that it does not have finite Rokhlin dimension with commuting towers.",
"$\\mathcal {Z}$ -stability, nuclear dimension and classification We begin this section with two theorems that follow quickly from the work done above.",
"As well as being interesting observations on their own, they will allow us to say something about the $\\mathcal {Z}$ -stability of the crossed products of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ below.",
"6.1 6.1 Theorem: Let $G$ be a countable discrete amenable group.",
"Suppose $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action of $G$ on $ {\\mathrm {C}}^*(H)$ .",
"Then $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable.",
"Proof.",
"Since $(X, \\varphi )$ is mixing, $ {\\mathrm {C}}^*(H)$ is simple and so the classification results of [12] imply that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace, it must be fixed by the action.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable by [58].",
"6.2 6.2 Theorem: Let $G$ be a countable amenable group acting effectively on a mixing Smale space $(X, \\varphi )$ and $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Then the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is a simple unital nuclear $\\mathcal {Z}$ -stable $\\mathrm {C}^*$ -algebra with unique tracial state and nuclear dimension (in fact, decomposition rank) at most one.",
"In particular, it belongs to the class of $\\mathrm {C}^*$ -algebras that are classified by the Elliott invariant and $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF, for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"Proof.",
"Since $ {\\mathrm {C}}^*(H)$ is amenable, it follows from [53] that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G \\cong \\mathrm {C}^*(\\mathcal {G}_H \\rtimes G)$ is also amenable and hence by [69] it satisfies the UCT.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is quasidiagonal [67] and by [67] together with Theorem classified by the Elliott invariant.",
"The nuclear dimension and decomposition rank bounds are given by [5].",
"Finally, by [42] $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF.",
"$\\mathcal {Z}$ -stability of crossed products $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ Let $A$ be separable $\\mathrm {C}^*$ -algebra and $G$ a discrete group.",
"For an action $\\beta : G\\rightarrow \\operatorname{Aut}(A)$ the fixed point algebra is given by $ A^{\\beta } := \\lbrace a \\in A \\mid \\beta _g(a) = a \\text{ for all } g \\in G\\rbrace .",
"$ Let $ A^{\\infty } := \\ell ^{\\infty }(\\mathbb {N}, A) / c_0(A) .$ The central sequence algebra of $A$ is defined by $ A_{\\infty } := A^{\\infty } \\cap A^{\\prime },$ where $A$ is considered as the subalgebra of $A^{\\infty }$ by viewing an element as a constant sequence.",
"We will denote by $\\overline{\\beta }$ the induced action on $A_{\\infty }$ .",
"Let $p$ and $q$ be positive integers.",
"The dimension drop algebra $I(p,q)$ is given by $I(p, q) := \\lbrace f \\in C([0,1], M_p( \\otimes M_q( \\mid f(0) \\in M_p( \\otimes { and }f(1)\\in M_q(\\rbrace .$ The Jiang–Su algebra, $\\mathcal {Z}$ , is an inductive limit of such algebras [28].",
"6.3 6.3 Theorem: Suppose $G$ is a countable discrete group acting on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"If, for each $k\\in {\\mathbb {N}}$ , there exists a unital equivariant embedding $I(k, k+1) \\rightarrow {\\mathrm {C}}^*(H)$ , then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"As usual, it suffices to prove the result for $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ .",
"To do so, we will show that the hypotheses of Lemma 2.6 in [21] hold.",
"We will show that, for every $k \\in \\mathbb {N}$ , there is a completely positive contractive map $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty } $ satisfying (i) $ (\\overline{\\beta }^S( \\gamma (x)) - \\gamma (x))a = $ for every $x \\in I(k, k+1)$ , $g \\in G$ and $a \\in \\mathrm {C}^*(S)$ , $a \\gamma (1) = a$ for every $a \\in \\mathrm {C}^*(S)$ , and $a(\\gamma (xy) - \\gamma (x)\\gamma (y)) = 0$ for every $x, y \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Suppose $F \\subset I(k, j+1)$ a finite subset and $\\epsilon > 0$ are given.",
"Let $w_n$ be a $G$ -invariant quasicentral approximate unit for $ \\mathrm {C}^*(S)$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ and $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ be a unital embedding (which exists by assumption).",
"Define $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)^{\\infty } $ via $ \\gamma (d) = (w_n \\rho (\\alpha ^n(d_n)) w_n)_{n \\in \\mathbb {N}}, $ where $(d_n)_{n \\in \\mathbb {N}}$ is a representative sequence for $\\tilde{\\gamma }(d)$ .",
"Then $\\gamma $ gives a c.p.c.",
"map.",
"Moreover, if $a \\in \\mathrm {C}^*(S)$ we have $ \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\alpha ^n(d_n) )w_n a - a w_n \\rho (\\alpha ^n(d_n)) w_n \\Vert = 0,$ so in fact $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty }.",
"$ Let us check that $\\gamma $ satisfies (i), (ii) and (iii).",
"Let $a \\in A$ and for any $d \\in I(k, k+1)$ , let $(d_n)_{n \\in \\mathbb {N}}$ be a representative of $\\tilde{\\gamma }(d)$ in $( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }$ .",
"$\\Vert \\overline{\\beta }^S(\\gamma (d) - \\gamma (d)) a \\Vert &=& \\lim _{n \\rightarrow \\infty } \\Vert \\beta ^{(S)}(w_n \\rho (\\alpha ^n(d_n)) w_n) - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&=& \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\beta (\\alpha ^n(d_n)) w_n - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&\\le & \\lim _{n \\rightarrow \\infty } \\Vert \\beta (\\alpha ^n(d_n) ) - \\alpha ^n(d_n) \\Vert \\\\&=& 0,$ showing $(i)$ .",
"To show (ii), we have $a \\gamma (1) &=& (a w_n \\rho (\\tilde{\\gamma }(1)) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a \\rho (1) w_n^2)_{n \\in \\mathbb {N}} \\\\&=& a,$ for every $a \\in \\mathrm {C}^*(S)$ Finally, for (iii), let $d, d^{\\prime } \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Then $a(\\gamma (d d^{\\prime }) - \\gamma (d)\\gamma (d^{\\prime })) &=& (a w_n \\rho (d_n d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}} - (a w_n \\rho (d_n) w_n^2 \\rho (d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a w_n^2 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a w_n^4 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} \\\\&=& (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}}\\\\&=& 0.$ Thus $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ is $\\mathcal {Z}$ -stable.",
"6.4 6.4 Corollary: Suppose $G$ is a discrete amenable group and $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"Then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"By Theorem REF (v), $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable and hence has strict comparison [54].",
"This in turn implies $ {\\mathrm {C}}^*(H)$ has property (TI) of Sato [58].",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace $\\tau $ which is therefore fixed by $\\beta $ , the hypotheses of [58] are satisfied.",
"Thus from the proof of [58] we get a unital embedding $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ The result then follows from the previous theorem.",
"Acknowledgments.",
"The authors thank Ian Putnam for many useful discussions concerning the content of this paper, Smale spaces, group actions and dynamics in general.",
"We also thank Magnus Goffeng for a number of useful comments.",
"The authors thank the referee for reading the paper carefully and making a number of useful comments.",
"The authors wish to thank Matrix at the University of Melbourne for hosting them during the programme Refining $\\mathrm {C}^*$ -algebraic Invariants for Dynamics using KK-theory in July 2016, the Banach Centre at the Institute of Mathematics of the Polish Academy of Sciences for hosting the first listed author during the conference Index Theory in October 2016, the University of Hawaii, Manoa for hosting the second listed author during the workshop Computability of K-theory in November 2016 and the Centre Rercerca Matemàtica, Barcelona, for their stay during the Intensive Research on Operator Algebras: Dynamics and Interactions in July 2017.",
"The above research visits were partially supported through NSF grants DMS 1564281 and DMS 1665118."
],
[
"Group actions on Smale spaces",
"Let $G$ be a topological group.",
"By an action of $G$ on a Smale space $(X, \\varphi )$ we mean a continuous group homomorphism $G \\rightarrow \\operatorname{Homeo}(X)$ such that $ g \\varphi (x) = \\varphi (gx) \\text{ for every } x \\in X \\text{ and } g \\in G.$ For a given Smale space $(X, \\varphi )$ , its automorphism group is defined to be ${\\rm Aut}(X, \\varphi ) := \\lbrace \\beta : X \\rightarrow X \\: | \\: \\beta \\hbox{ is a homeomorphism and }\\beta \\circ \\varphi = \\varphi \\circ \\beta \\rbrace .$ Let $X$ be a compact metric space and $G$ a topological group.",
"An action of group $G \\rightarrow \\operatorname{Homeo}(X)$ is free if, for every $x \\in X$ , we have that $g(x) = x$ if and only if $g = \\operatorname{\\textup {id}}$ .",
"The action of $G$ is effective (or faithful), if for every $g \\in G\\setminus \\lbrace e\\rbrace $ , there is an $x \\in X$ such that $gx \\ne x$ .",
"If $(X, \\varphi )$ is a Smale space then we say the action is free, respectively effective, if the action on $X$ is free, respectively effective."
],
[
"Examples of automorphisms on Smale spaces",
"Group actions on Smale spaces are quite ubiquitous.",
"Here we discuss four familiar classes of Smale spaces, three of which are treated in Putnam's work on Smale spaces and $\\mathrm {C}^*$ -algebras [51], and give examples of their automorphisms.",
"In each case, these actions will be free or effective and thus are covered by the results in Sections and .",
"2.1 2.1 Example: [Smale space automorphism] Let $(X, \\varphi )$ be a Smale space.",
"Our first and most obvious example is the action induced by the homeomorphism $\\varphi $ of $X$ .",
"Clearly $\\varphi $ commutes with itself and hence defines an action of $\\mathbb {Z}$ on $(X, \\varphi )$ .",
"The action induced on the associated $\\mathrm {C}^*$ -algebras will prove to be a useful tool in the sequel.",
"Of course, powers of $\\varphi $ are also automorphisms of $(X, \\varphi )$ .",
"2.2 2.2 Example: [Shifts of finite type] In the irreducible case, shifts of finite type are exactly the zero dimensional Smale spaces, though of course they were well known before Ruelle's work.",
"The automorphism group and group actions on shifts of finite type have been studied by many authors, see for example [1], [7], [6], [43] along with references therein.",
"The full two-shift is the Smale space $(X, \\varphi )$ where $X = \\Sigma _{[2]}=\\lbrace 0, 1\\rbrace ^{{\\mathbb {Z}}}$ with $\\varphi = \\sigma $ defined to be the left shift.",
"The automorphism group of the full two-shift is large: it contains, for example, every finite group and the free group on two generators [7].",
"Here we highlight two automorphisms of finite order.",
"The latter comes from [39].",
"Define $\\beta _1 : \\Sigma _{[2]} \\rightarrow \\Sigma _{[2]}$ via $(a_n)_{n\\in {\\mathbb {Z}}} \\mapsto (a_n+ 1$ mod$(2))_{n \\in {\\mathbb {Z}}}$ and $\\beta _2 : \\Sigma _{[2]} \\rightarrow \\Sigma _{[2]}$ via $(a_n)_{n\\in {\\mathbb {Z}}} \\mapsto (b_n)_{n\\in {\\mathbb {Z}}}$ where $b_n := a_n + a_{n-1}( a_{n+1}+1)a_{n+2} \\hbox{ mod}(2) .$ One can compute that $\\beta _1^2$ and $\\beta _2^2$ are both the identity.",
"Hence they are order two automorphisms of the full two shift.",
"The $\\frac{{\\mathbb {Z}}}{2{\\mathbb {Z}}}$ -action induced from $\\beta _1$ is free while that of $\\beta _2$ is effective but not free.",
"2.3 2.3 Example: [Solenoids] A solenoid is a Smale space obtained from a stationary inverse limit of a metric space equipped with a surjective continuous map that is subject to certain conditions such as those in [71] or in [70].",
"Some prototypical examples are obtained as follows.",
"Let $S^1 \\subseteq (as the unit circle), $ k2$ be an integer, and define $ g: S1 S1$ by $ g(z)=zk$.",
"Then $ (X, )$ is the Smale space given by $ X= (S1, g)$ and the map $ : X X$ defined as$$(z_n)_{n \\in {\\mathbb {N}}} \\mapsto (g(z_0), g(z_1), g(z_2), \\ldots ) = (g(z_0) , z_0, z_1, \\ldots ).$$$ Let $\\beta _1: X \\rightarrow X$ be the map $(z_n)_{n\\in {\\mathbb {N}}} \\mapsto (\\bar{z}_n)_{n\\in {\\mathbb {N}}}$ , where $\\bar{z}$ denotes complex conjugate of $z$ .",
"For any $k$ , $\\beta _1$ defines an order two automorphism.",
"There are other automorphisms.",
"For example, if $g(z) = z^6$ , then we have automorphisms defined by $\\beta _2( (z_n)_{n\\in {\\mathbb {N}}}) = (z^2_n)_{n\\in {\\mathbb {N}}} \\quad \\hbox{ and } \\quad \\beta _3( (z_n)_{n\\in {\\mathbb {N}}}) = (z^3_n)_{n\\in {\\mathbb {N}}}.$ The ${{\\mathbb {Z}}}/{2{\\mathbb {Z}}}$ -action induced from $\\beta _1$ is effective but not free.",
"While the ${\\mathbb {Z}}$ -actions induced by $\\beta _2$ and $\\beta _3$ are each effective but not free.",
"The automorphism group of these examples are known (see [65] for details).",
"Further results concerning the automorphism group of similar examples can also be found in [65].",
"2.4 2.4 Example: [Hyperbolic toral automorphisms] Let $d\\ge 2$ be an integer.",
"A hyperbolic toral automorphism is a Smale space $(X, \\varphi )$ were $X={\\mathbb {R}}^d/ {\\mathbb {Z}}^d$ and $\\varphi $ is induced by a $d \\times d$ integer matrix $A$ with the following properties: (i) $|\\det (A)| =1$ ; no eigenvalue of $A$ has modulus one.",
"A specific example when $d=2$ is $A = \\left( \\begin{array}{cc} 2 & 1 \\\\ 1 & 1 \\end{array} \\right).$ To obtain an automorphism of $(X, \\varphi )$ one can take $B\\in M_d({\\mathbb {Z}})$ with det$(B)=\\pm 1$ such that $AB=BA$ .",
"For example, again when $d=2$ , $B=\\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & -1 \\end{array} \\right)$ defines an order two automorphism of any hyperbolic toral automorphism; the induced $\\frac{{\\mathbb {Z}}}{2{\\mathbb {Z}}}$ -action is effective.",
"For larger $d$ there are more interesting automorphisms.",
"An explicit example taken from [49] is $A = \\left( \\begin{array}{ccc} 1 & -1 & 0 \\\\ -1 & 2 & -1 \\\\ 0 & -1 & 2 \\end{array} \\right) \\hbox{ and } B= \\left( \\begin{array}{ccc} 2 & 0 & -1 \\\\ 0 & 1 & 1 \\\\ -1 & 1 & 2 \\end{array} \\right)$ In this example, both $A$ and $B$ are hyperbolic.",
"More on automorphisms of hyperbolic toral automorphisms can be found in [4] and [49] along with the references therein.",
"2.5 2.5 Example: [Substitution tilings] Many people have studied the dynamics of substitution tiling systems.",
"In [11], Connes associates an AF algebra to the Penrose tiling by kites and darts; other $\\mathrm {C}^*$ -algebraic constructions were considered by Kellendonk (see for example [30]).",
"It was Putnam and Anderson who showed that, under the assumptions that the substitution map $\\omega $ is one-to-one, primitive and the tiling space $\\Omega $ is of finite type (that is, has finite local complexity), that $(\\Omega , \\omega )$ is in fact a Smale space [2].",
"Moreover, the unstable $\\mathrm {C}^*$ -algebra associated to $(\\Omega , \\omega )$ is Morita equivalent to the tiling $\\mathrm {C}^*$ -algebra studied by Kellendonk [2].",
"Many aperiodic tilings have interesting rotational symmetries.",
"For example, the dihedral group, $D_5$ acts on the Penrose tiling [59].",
"In [61], Starling studied free actions of finite subgroups of the symmetry group of these substitution tilings.",
"The results of the present paper, in the special case of tilings, can be related to those in [61] by showing that the Morita equivalence considered by Anderson and Putnam can be made equivariant with respect to the given group action.",
"Returning to general actions on Smale spaces, we begin by showing free actions are quite rare.",
"However, as we shall see in Section REF , an effective action is sufficiently strong to guarantee good properties of the induced action on the associated $\\mathrm {C}^*$ -algebras.",
"The next two propositions are likely not new, but we could not find a reference for them, except for the case of a shift of finite type [1].",
"2.6 2.6 Proposition: Suppose $G$ has an element of infinite order and acts on a Smale space $(X, \\varphi )$ .",
"Then $G$ does not act freely.",
"Proof.",
"Let $g\\in G$ be an element of infinite order.",
"The set periodic points of $(X, \\varphi )$ is non-empty because the non-wandering set is non-empty by [56] and the periodic points are dense in the non-wandering set by [56].",
"Let $x\\in X$ be a periodic point with minimal period $k$ .",
"Then, for any $n \\in {\\mathbb {Z}}$ , $ \\varphi ^k (g^n x) = g^n \\varphi ^k (x) = g^n x$ Hence, for any $n \\in Z$ , $g^n x \\in {\\rm Per}_k(X, \\varphi )$ .",
"Since ${\\rm Per}_k (X, \\varphi )$ is finite, there are $n_1 \\ne n_2 \\in {\\mathbb {Z}}$ such that $g^{n_1} x = g^{n_2}x$ It follows that the action is not free.",
"2.7 2.7 Proposition: Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"For each $g\\in G\\setminus \\lbrace e\\rbrace $ , the set $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is dense and open.",
"Moreover, the Bowen measure of this set is one.",
"Proof.",
"The set $\\lbrace x \\in X \\: | \\: gx =x \\rbrace $ is closed since the map $G \\times X \\rightarrow X$ is continuous.",
"Thus $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is open.",
"Since $(X, \\varphi )$ is mixing, there is a point $x_0 \\in X$ with dense orbit.",
"We show $x_0 \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Suppose not.",
"Then for each $n\\in {\\mathbb {Z}}$ , $g(\\varphi ^n(x_0))= \\varphi ^n( g x_0) = \\varphi ^n(x_0).$ Thus, because the orbit of $x_0$ is dense, $g$ is the identity on $X$ .",
"This contradicts the assumption that $G$ acts effectively, so $x_0 \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Moreover, for each $n\\in {\\mathbb {Z}}$ , $\\varphi ^n(x_0) \\in \\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Since the orbit of $x_0$ is dense, $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ is also dense.",
"Finally, the set of points with dense orbit has Bowen measure one, which follows from Bowen's theorem and the fact that this is true for a shift of finite type [39].",
"The set $\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ contains the set of points with dense orbit and hence also has Bowen measure one.",
"From Smale spaces to $\\mathrm {C}^*$ -algebras Ruelle was the first person to associate operator algebras to Smale spaces in [55].",
"We follow the approach introduced by Putnam and Spielberg [51], [52]: three $\\mathrm {C}^*$ -algebras are constructed via the groupoid $\\mathrm {C}^*$ -algebra construction for étale equivalence relations which capture the contracting, expanding, and asymptotic behaviour of the system given in Definition .",
"Fix an irreducible Smale space $(X, \\varphi )$ .",
"To define topologies on each of our equivalence relations, we first note the following.",
"Using the notation from Definition , it is not difficult to show that for any $0 < \\epsilon \\le \\epsilon _X$ we have $ X^U(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon )),$ and similarly that $ X^S(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{-n}(X^S(\\varphi ^n(x), \\epsilon )).$ Each $\\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon ))$ , $n \\in \\mathbb {N}$ is given the relative topology from $X$ while $X^U(x)$ and $X^S(x)$ are given the topology coming from these inductive unions, see [51] for the precise details.",
"We could proceed to construct $\\mathrm {C}^*$ -algebras directly from the equivalence relations in Definition following the construction of Putnam in [51].",
"However neither the stable nor the unstable groupoids would have a natural étale topology.",
"Instead, following [52], we restrict our relation to those points equivalent to periodic points.",
"3.1 3.1 Definition: Let $P$ and $Q$ be finite $\\varphi $ -invariant sets of periodic points of $(X, \\varphi )$ .",
"Define the stable and unstable groupoids of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_S(P) := \\lbrace (x,y) \\in X^U(P) \\times X^U(P) \\mid x \\sim _s y \\rbrace ,$ and $ \\operatorname{\\mathcal {G}}_U(Q) := \\lbrace (x,y) \\in X^S(Q) \\times X^S(Q) \\mid x \\sim _u y \\rbrace .$ Up to Morita equivalence of groupoids these constructions do not depend on the choice of periodic points.",
"3.2 3.2 Definition: Define the homoclinic groupoid of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_H := \\lbrace (x, y) \\in X \\times X \\mid x \\sim _h y\\rbrace .$ Now, if $(v, w) \\in X^S(P)$ , then $v \\sim _s w$ so there is some sufficiently large $N \\in \\mathbb {N}$ such that $d(\\varphi ^N(v), \\varphi ^N(w)) < \\epsilon _X/2$ .",
"By continuity of $\\varphi $ , we may choose $\\delta > 0$ small enough so that $\\varphi ^N(X^U(w , \\delta )) \\subset X^U(\\varphi ^N(w), \\epsilon _X/2)$ and also $\\varphi ^N(X^U(v , \\delta )) \\subset X^U(\\varphi ^N(v), \\epsilon _X/2)$ .",
"Then define $ h^s := h^s(v, w, N, \\delta ) : X^U(w, \\delta ) \\rightarrow X^U(v, \\epsilon _X), \\quad x \\mapsto \\varphi ^{-N}([\\varphi ^N(x), \\varphi ^N(v)]).",
"$ By [56] this is a local homeomorphism.",
"For any such $v, w, \\delta , h , N$ , we then define an open set by $ V(v, w, \\delta , h^s, N) := \\lbrace (h^s(x), x) \\mid x \\in X^U(w, \\delta ) \\rbrace \\subset \\operatorname{\\mathcal {G}}_S(P).$ These sets generate an étale topology for $\\operatorname{\\mathcal {G}}_S(P)$ [52].",
"The construction for the topologies of $\\operatorname{\\mathcal {G}}_U(Q)$ and $\\operatorname{\\mathcal {G}}_H$ are similar; we refer the reader to [52] for details.",
"$\\mathrm {C}^*$ -algebras Fix finite $\\varphi $ -invariant sets $P$ and $Q$ and let $\\mathcal {H} = \\ell ^2(X^H(P,Q))$ where $X^H(P,Q)$ is the set of points in $X$ which are both stably equivalent to a point in $P$ and unstably equivalent to a point in $Q$ .",
"It is shown in [56] that $X^H(P, Q)$ is countable.",
"If $\\operatorname{\\mathcal {G}}$ is one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ , let $C_c(\\operatorname{\\mathcal {G}})$ denote the a compactly supported functions on $\\operatorname{\\mathcal {G}}$ with convolution product, $ (f \\ast g) (x, y) = \\sum _{z \\sim x, z \\sim y} f(x,z) g(z, y), \\quad (x,y) \\in \\operatorname{\\mathcal {G}}, $ and $f^*(x, y) = \\overline{f(y,x)}, \\quad (x,y) \\in \\operatorname{\\mathcal {G}}.$ We can represent each of $C_c(\\operatorname{\\mathcal {G}}_H)$ , $C_c(\\operatorname{\\mathcal {G}}_S(P))$ and $C_c(\\operatorname{\\mathcal {G}}_U(Q))$ on the Hilbert space $\\mathcal {H}$ , and define the homoclinic algebra by $ {\\mathrm {C}}^*(H):= \\overline{C_c(\\operatorname{\\mathcal {G}}_H)}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ the stable algebra $ \\mathrm {C}^*(S):= \\overline{C_c(\\operatorname{\\mathcal {G}}_S(P))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ and the unstable algebra $ \\mathrm {C}^*(U):= \\overline{C_c(\\operatorname{\\mathcal {G}}_U(Q))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}}.$ 3.3 3.3 Remarks: 1.",
"We suppress the reference to $P$ and $Q$ in the notation of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"For any choice of such $P$ and $Q$ , the resulting groupoids are Morita equivalent, and hence so are their $\\mathrm {C}^*$ -algebras.",
"Thus from the perspective of most $\\mathrm {C}^*$ -algebraic properties, we don't need to keep track of the original choice.",
"2.",
"In the usual groupoid $\\mathrm {C}^*$ -algebra construction for a groupoid $\\mathcal {G}$ , the algebra of compactly supported functions $C_c(\\mathcal {G})$ is represented on the Hilbert space $\\ell ^2(\\mathcal {G})$ and the completion is the reduced groupoid $\\mathrm {C}^*$ -algebra $\\mathrm {C}_r(\\mathcal {G})$ .",
"However, when the groupoid is amenable, the completion of any faithful representation will result in the same $\\mathrm {C}^*$ -algebra.",
"Here $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each amenable [52].",
"It is convenient, however, to represent them all on the same Hilbert space (namely $\\ell ^2(X^H(P,Q))$ ) because we can consider the interactions between operators coming from the different algebras.",
"This will be particularly useful when showing how finite Rokhlin dimension passes from $ {\\mathrm {C}}^*(H)$ to $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ in Section .",
"3.4 3.4 Let $(X, \\varphi )$ be a mixing Smale space.",
"Then $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are simple by [51] and [52], separable (since $X$ is a metric space) and nuclear by [52] and [51].",
"The homoclinic $\\mathrm {C}^*$ -algebra $ {\\mathrm {C}}^*(H)$ is unital since the diagonal $X \\times X$ is open in $\\operatorname{\\mathcal {G}}_H$ and $X$ is compact.",
"Each of $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ admit a trace [51], hence are stably finite.",
"The traces on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are not bounded while $ {\\mathrm {C}}^*(H)$ admits a tracial state.",
"Moreover, when $(X, \\varphi )$ is mixing, this trace is unique [25].",
"In [12], the authors showed that for a mixing Smale space, $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ have finite nuclear dimension and hence are $\\mathcal {Z}$ -stable.",
"(There it was not noted that $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are $\\mathcal {Z}$ -stable; however this follows from [66].",
"In fact, it can be proved directly in a similar manner to Theorem REF below, once we know that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.)",
"It then follows that $ {\\mathrm {C}}^*(H)\\otimes \\operatorname{\\mathcal {U}}$ is tracially approximately finite (TAF) in the sense of [36] for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"In particular, the $\\mathrm {C}^*$ -algebras coming from the homoclinic relation on a mixing Smale space is classified by the Elliott invariant, see [12].",
"Suppose $G$ is a discrete group acting on a mixing Smale space.",
"For the action to induce a well-defined action on $\\operatorname{\\mathcal {G}}_U$ and $\\operatorname{\\mathcal {G}}_S$ , the choice of finite sets of $\\varphi $ -invariant periodic points must be $G$ -invariant.",
"Fortunately, this can always be arranged.",
"3.5 3.5 Lemma: Let $G$ be a discrete group acting effectively on a mixing Smale space $(X , \\varphi )$ .",
"Then there exists a finite set of $\\varphi $ -invariant periodic points $P$ such that $gp \\in P$ for every $g \\in G$ and every $p \\in P$ .",
"Proof.",
"Let $P^{\\prime }$ be any finite $\\varphi $ -invariant set of periodic points.",
"Then $P^{\\prime } \\subseteq {\\rm Per}_n(X, \\varphi )$ for some $n\\in {\\mathbb {N}}$ .",
"We know that ${\\rm Per}_n(X, \\varphi )$ is finite.",
"Also, for any $g\\in G$ and $x\\in {\\rm Per}_n(X, \\varphi )$ it was shown in the proof of Proposition REF that $gx \\in {\\rm Per}_n(X, \\varphi )$ .",
"It follows that the set $P=\\lbrace p \\in X \\mid p=gx \\hbox{ for some }g\\in G, x\\in P^{\\prime }\\rbrace $ is contained in ${\\rm Per}_n(X, \\varphi )$ and hence is finite.",
"It is $G$ -invariant by construction and it is $\\varphi $ -invariant because $g \\varphi (x) = \\varphi ( gx)$ for any $g\\in G$ and $x\\in X$ .",
"For the remainder of the paper, we will assume that $P$ and $Q$ are $G$ -invariant.",
"Let $\\operatorname{\\mathcal {G}}$ be one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ and let $G$ be a group acting on $(X, \\varphi )$ .",
"Since $P$ and $Q$ are assumed to be $G$ -invariant, $g(x,y) \\mapsto (gx, gy)$ defines an induced action of $G$ on $\\operatorname{\\mathcal {G}}$ .",
"The action of $G$ on $\\operatorname{\\mathcal {G}}$ in turn induces an action on $\\mathrm {C}^*(\\operatorname{\\mathcal {G}})$ .",
"3.6 3.6 Example: Following Example REF , the Smale space homeomorphism $\\varphi $ induces a $\\mathbb {Z}$ -action on each of $ {\\mathrm {C}}^*(H), \\mathrm {C}^*(S), \\mathrm {C}^*(U)$ .",
"These actions are denoted by $\\alpha $ , $\\alpha _S$ and $\\alpha _U$ , respectively.",
"Properties of $\\alpha $ (in particular, [51]) will prove indispensable for the results in the next section and also in Section , where we seek to pass from known properties about $ {\\mathrm {C}}^*(H)$ to the nonunital $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"3.7 3.7 Example: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type.",
"Then the $C^*$ -algebras associated to $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each AF.",
"Moreover if $G$ is a finite group acting effectively on $(\\Sigma , \\sigma )$ , then using [1] one can show that the induced action of $G$ on each of these AF $\\mathrm {C}^*$ -algebras is locally representable in the sense of [18].",
"This is the case for the automorphisms $\\beta _1$ and $\\beta _2$ on the full two-shift given in Example REF .",
"It follows from results in [18] that the crossed products associated to the action of $G$ are each also AF.",
"3.8 3.8 Example: Suppose $(X, \\varphi )$ is the Smale space obtained via the solenoid construction in Example REF with $Y=S^1$ and $g(z)=z^n$ ($n\\ge 2$ ).",
"Then the stable and unstable algebras are the stabilisation of a Bunce–Deddens algebra.",
"For details in the case $n=2$ , see page 28 of [51].",
"Automorphisms of Bunce–Deddens algebras are considered in [45], for example.",
"3.9 3.9 Example: If $(X, \\varphi )$ is a hyperbolic toral automorphism, as in Example REF , then the stable and unstable algebras are the stabilization of irrational rotation algebras as is shown on pages 27-28 of [51].",
"Automorphisms of irrational rotation algebras are well-studied, see [14] and reference therein.",
"The induced action on $ {\\mathrm {C}}^*(H)$ In the sequel, the aim is to provide conditions of a group action on a mixing Smale space which will allow us to determine structural properties of the crossed products of the associated $\\mathrm {C}^*$ -algebras by the induced group action.",
"The idea is to determine what properties are preserved when passing from the $\\mathrm {C}^*$ -algebra to its crossed product.",
"If we interpret a $\\mathrm {C}^*$ -crossed product as a “noncommutative orbit space” then what we are asking for is some sort of “freeness” condition.",
"In the $\\mathrm {C}^*$ -algebraic context this might take a number of different forms.",
"Here, we focus on the Rokhlin dimension of an action (which is akin to a “coloured” version of noncommutative freeness), initially proposed for finite group and $\\mathbb {Z}$ -actions by Hirshberg, Winter and Zacharias [23] and subsequently generalised to other groups [63], [15], [64], [22], [8], as well as the notion of a “strongly outer action” (which can be thought of as a noncommutative approximation of freeness in trace) (see for example, [41]).",
"Recent results of Sato [58] also play a key role, although for finite group actions and ${\\mathbb {Z}}^d$ -actions these new results are not required.",
"Finite group actions on $ {\\mathrm {C}}^*(H)$ For this subsection, we fix a mixing Smale space $(X, \\varphi )$ .",
"Let $ {\\mathrm {C}}^*(H)$ denotes its homoclinic algebra.",
"Here we study actions induced on $ {\\mathrm {C}}^*(H)$ by free actions of $G$ on $(X, \\varphi )$ .",
"Given an action of a group $G$ on a $C^*$ -algebra, $A$ , $\\beta : G \\rightarrow {\\rm Aut}(A)$ we will write $\\beta _h$ for $\\beta (h)$ .",
"Based on Proposition REF which shows that freeness is unlikely to hold for infinite groups, we restrict to the case that $G$ is finite.",
"4.1 4.1 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra and let $G$ be a finite group.",
"An action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and each finite subset $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A $ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ $g \\ne h$ in $G$ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ ; $\\Vert [ f^{( l )}_g, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ .",
"When $d = 0$ the action is said to have the Rokhlin property.",
"In this case the contractions $(f_g)_{g \\in G}$ can in fact be taken to be projections.",
"The definition of the Rokhlin property for $\\mathrm {C}^*$ -algebras was introduced by Izumi [26], [27].",
"4.2 4.2 Lemma: Let $G$ be a finite group acting on $(X, \\varphi )$ and denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Let $d$ be a nonnegative integer.",
"Suppose that for any $\\epsilon >0$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for $g \\ne h \\in G$ , $l \\in \\lbrace 0, \\dots , d\\rbrace $ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ .",
"Proof.",
"Let $\\epsilon >0$ and $F \\subset {\\mathrm {C}}^*(H)$ be a finite set.",
"Take positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ with the properties assumed in the statement of the theorem.",
"By [51], there is $n \\in {\\mathbb {N}}$ such that $\\Vert [ \\alpha ^n( f^{( l )}_g) , a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ , where $\\alpha $ is the automorphism from Example REF .",
"Since $\\alpha $ is an automorphism, $\\left( \\alpha ^n( f^{(l)}_g) \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ satisfies the other requirements of Definition REF .",
"4.3 4.3 Corollary: Suppose $A$ is a $G$ -invariant unital $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ .",
"If $G$ acting on $A$ has Rokhlin dimension at most $d$ then the action of $G$ on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"In particular, if $G$ acting on $C(X)$ has Rokhlin dimension at most $d$ , then $G$ acting on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"Proof.",
"By assumption, given $\\epsilon >0$ , there exists positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A \\subset {\\mathrm {C}}^*(H)$ such that the hypotheses of Lemma REF hold; this then implies the result.",
"When considering the statement concerning $C(X)$ one need only note that $C(X)$ is a $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ and that it is invariant under the action of $G$ .",
"4.4 4.4 Corollary: Suppose $(X, \\varphi )$ is a mixing Smale space and a finite group $G$ acts on $(X, \\varphi )$ freely.",
"Then $G$ acting on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"For finite group actions on a compact space, freeness implies finite Rokhlin dimension [21].",
"The result then follows from Corollary REF .",
"4.5 4.5 Corollary: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type and a finite group $G$ acts on $(\\Sigma , \\sigma )$ freely.",
"Then the action of $G$ on $ {\\mathrm {C}}^*(H)$ has the Rokhlin property.",
"Proof.",
"Since $\\Sigma $ is the Cantor set and $G$ acts freely, the action of $G$ on $C(\\Sigma )$ has the Rokhlin property.",
"Corollary REF then implies the result.",
"Strongly outer actions Using [41], we prove the first of the three results listed at the end of the introduction; it appears as Theorem REF .",
"To begin, we recall the Vitali covering theorem and the definitions needed for its statement.",
"4.6 4.6 Definition: A finite measure $\\mu $ on a metric space $(X, d)$ is said to be doubling if there exists a constant $M > 0$ such that $ \\mu (B(x, 2\\epsilon )) \\le M \\mu (B(x, \\epsilon )) $ for any $x \\in X$ and any $\\epsilon >0$ .",
"4.7 4.7 Definition: Suppose $(Y,d)$ is a metric space and $A \\subseteq Y$ .",
"Then a Vitali cover of $A$ is a collection of closed balls $\\mathcal {B}$ such that inf$\\lbrace r>0 \\mid B(x,r) \\in \\mathcal {B} \\rbrace =0$ for all $x\\in A$ .",
"4.8 4.8 Theorem: [Vitali Covering Theorem] Suppose $(Y,d)$ is a compact metric space, $\\mu $ is a doubling measure, $A \\subseteq Y$ and $\\mathcal {F}$ is Vitali cover of $A$ .",
"Then, for any $\\epsilon >0$ , there exists finite disjoint family $\\lbrace F_1, F_2, \\ldots F_n \\rbrace \\subseteq \\mathcal {F}$ such that $\\mu ( A - \\cup ^n_{i=1} F_i) < \\epsilon $ .",
"4.9 4.9 Lemma: Suppose $(X, \\varphi )$ is a mixing Smale space with Bowen measure $\\mu $ and $G$ is a discrete group acting effectively on $(X, \\varphi )$ .",
"Let $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ denote the induced action.",
"Then, for any $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ , there exists positive contractions $(f_i)_{i=1}^k \\subset {\\mathrm {C}}^*(H)$ such that (i) $f_i \\cdot f_j=0$ for all $i\\ne j$ ; $f_i \\cdot \\beta _g(f_i) =0$ ; $\\tau (\\sum _{i=1}^k f_i)> 1- \\epsilon $ where $\\tau $ denotes the (unique) trace on $ {\\mathrm {C}}^*(H)$ obtained from $\\mu $ .",
"Proof.",
"Fix $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ .",
"Let $D=\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Then $D$ is open and has full measure by Proposition REF .",
"Moreover, for each $x\\in D$ there exists $\\delta _x>0$ such for any $0<\\delta \\le \\delta _x$ we have $\\overline{B_{\\delta }(x)} \\cap g(\\overline{ B_{\\delta }(x)}) = \\emptyset $ and $\\overline{B_{\\delta }(x)} \\subseteq D$ .",
"Let $\\mathcal {F}= \\left\\lbrace F_x \\: | \\: F_x = \\overline{ B_{\\delta }(x)} \\hbox{ for some }x\\in D \\hbox{ and some }0<\\delta \\le \\frac{\\delta _x}{2} \\right\\rbrace .$ By construction, the collection $\\mathcal {F}$ is a Vitali covering of $D$ .",
"The Bowen measure is doubling (see for example [46]) so we may apply the Vitali Covering Theorem to obtain a finite subcollection of $\\mathcal {F}$ , $\\lbrace F_{x_1}, \\ldots , F_{x_k} \\rbrace $ , with the following properties (i) $F_{x_i} \\cap F_{x_j}=\\emptyset $ ; $\\mu (\\cup _{i=1}^k F_{x_i}) > 1- \\epsilon $ .",
"Recall that $\\mu (D)=1$ .",
"Since each $F_{x_i}$ is compact (they are closed in a compact space) and pairwise disjoint, we can use Urysohn's lemma and the Tietze extension theorem to obtain pairwise disjoint open sets $(U_i)_{i=1}^k$ such that for each $i$ , $F_{x_i}\\subseteq U_{x_i}\\subseteq B_{\\delta _{x_i}}(x_i)$ and functions $(f_i)_{i=1}^k \\subseteq C(X) \\subseteq {\\mathrm {C}}^*(H)$ with the following properties: (i) $0\\le f_i \\le 1$ , $ {\\rm supp}(f_i) \\subseteq U_i$ , $F_{x_i} \\subseteq \\lbrace x \\: |\\: f_i(x)=1 \\rbrace $ ; for every $i \\in \\lbrace 1, \\dots , k\\rbrace $ .",
"Finally, we show that $(f_i)_{i=1}^k$ has the required properties.",
"They are by definition positive contractions.",
"Moreover, (i) $f_i \\cdot f_j =0$ for every $i \\ne j \\in \\lbrace 1, \\dots , k\\rbrace $ , since ${\\rm supp}(f_i) \\subseteq U_i$ and $U_{i} \\cap U_{j}=\\emptyset $ ; $f_i \\cdot \\beta _g(f_i)=0$ for every $i \\in \\lbrace 0, \\dots , k\\rbrace $ since $U_i \\subseteq B_{\\delta _{x_i}}(x_i)$ implies that $U_i \\cap g( U_i) = \\emptyset $ .",
"Finally, $\\tau \\left( \\sum _{i=1}^k f_i \\right) & \\ge & \\sum _{i=1}^k \\mu ( \\lbrace x | f_i(x) =1 \\rbrace ) \\\\& \\ge & \\sum _{i=1}^k \\mu (F_i) \\\\& = & \\mu ( \\cup _{i=1}^k F_i ) \\\\& > & 1 - \\epsilon ,$ showing (iii) holds.",
"4.10 4.10 Let $A$ be a $\\mathrm {C}^*$ -algebra and $\\tau $ a state on $A$ .",
"We denote by $\\pi _{\\tau }$ the representation of $A$ corresponding to the GNS construction with respect to $\\tau $ .",
"In this case, $\\pi _{\\tau }(A)^{\\prime \\prime }$ is the enveloping von Neumann algebra of $\\pi _{\\tau }(A)$ .",
"Definition: [41] Let $A$ be a unital simple $\\mathrm {C}^*$ -algebra with nonempty tracial state space $T(A)$ .",
"An automorphism $\\beta $ of $A$ is not weakly inner if, for every $\\tau \\in T(A)$ such that $\\tau \\circ \\beta = \\tau $ , the weak extension of $\\beta $ to $\\pi _{\\tau }(A)^{\\prime \\prime }$ is outer.",
"If $G$ is a discrete group, then an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is strongly outer if, for every $g \\in G \\setminus \\lbrace e\\rbrace $ , the automorphism $\\beta _g$ is not weakly inner.",
"The next lemma and theorem are based on arguments due to Kishimoto in [33].",
"4.11 4.11 Lemma: Let $A$ be a unital $\\mathrm {C}^*$ -algebra and $\\tau \\in T(A)$ .",
"Then for every $\\epsilon > 0$ there is $\\delta > 0$ such that for any positive contraction $f \\in A$ such that $\\tau (f) > 1 - \\delta $ we have $\\tau (a) \\le \\tau (fa) + \\epsilon $ for every $a \\in A$ .",
"Proof.",
"It is enough to show this holds when $a \\in A_+$ .",
"Given $\\epsilon > 0$ let $\\delta = \\epsilon ^2$ .",
"Then since $a = af + a(1-f)$ we have $\\tau (a) - \\tau (af) &=& \\tau (a(1-f)) \\\\&\\le & \\tau (a^2)^{1/2}\\tau ((1-f)^2)^{1/2} \\\\&\\le & \\tau ((1-f)^2)^{1/2}\\\\&\\le & \\tau (1-f)^{1/2} \\\\&\\le & \\epsilon .$ 4.12 4.12 Theorem: Let $(X, \\varphi )$ be a mixing Smale space with an effective action of a discrete group $G$ .",
"Then the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"Proof.",
"Let $\\tau $ denote the tracial state on $ {\\mathrm {C}}^*(H)$ corresponding to the Bowen measure on $(X, \\varphi )$ .",
"Fix $g \\in G\\setminus \\lbrace e\\rbrace $ .",
"Since $\\tau $ is the unique tracial state, we have $\\tau = \\tau \\circ \\beta _g$ .",
"Let $\\epsilon > 0$ and let $F \\subset A$ be a finite subset.",
"Let $\\delta $ be the $\\delta $ of Lemma REF with respect to $\\epsilon /2$ .",
"By Lemma REF , we can find positive contractions $f_1, \\dots , f_k \\in {\\mathrm {C}}^*(H)$ such that (i) $f_if_j = 0$ for every $0 \\le i \\ne j \\le k$ , $f_i \\beta _g(f_i) = 0$ , $\\tau (f_1 + \\dots + f_k) > 1 - \\delta $ .",
"There is a sufficiently large $N \\in \\mathbb {N}$ such that, for any $i =1, \\dots , k$ , we have $ \\Vert \\alpha ^N(f_i) a - a \\alpha ^N(f_i) \\Vert < \\epsilon /2 \\text{, for any } a \\in F.$ Since $\\alpha ^N : {\\mathrm {C}}^*(H)\\rightarrow {\\mathrm {C}}^*(H)$ is an automorphism, we have that (i) $\\alpha ^N(f_i) \\alpha ^N(f_j) = 0$ for every $1 \\le i \\ne j \\le k$ and $\\tau (\\alpha ^N(f_1) + \\dots + \\alpha ^N(f_k)) > 1 - \\delta .$ Moreover, since the action of $G$ on $X$ commutes with $\\varphi $ , we have, for any $f\\in {\\mathrm {C}}^*(H)$ , that $\\alpha ^N(\\beta _g(f)) = \\beta _g(\\alpha ^N(f))$ .",
"Hence (i) $\\alpha ^N(f_i) \\beta _g(\\alpha ^N(f_i)) = 0.$ Thus we have $ \\Vert \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) \\Vert < \\epsilon /2$ for any $a \\in F$ .",
"Consider the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z}$ .",
"It is generated by $ {\\mathrm {C}}^*(H)$ and $u_g$ where $u_g$ is unitary satisfying $u_g a u_g^* = \\beta _g(a)$ for every $a \\in {\\mathrm {C}}^*(H)$ .",
"Let $\\sigma \\in T( {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z})$ and note that $\\sigma |_{ {\\mathrm {C}}^*(H)} = \\tau $ .",
"Put $f = \\sum _{i=1}^k \\alpha ^N(f_i)$ .",
"By the above and Lemma REF we have $\\sigma (a u_g) &\\le & \\sigma (f a u_g f) + \\epsilon \\\\&=& \\sum _{i=1}^k \\sigma ( \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) u_g ) + \\epsilon \\\\&<& 2\\epsilon .$ Since $\\epsilon $ and $F$ were arbitrary, it follows that $\\sigma (a u_g) = 0$ for all $a \\in A$ .",
"That $\\beta _g$ is not weakly inner now follows from the proof of [33].",
"Since this holds for every $g \\in G\\setminus \\lbrace e\\rbrace $ , it follows that the action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"4.13 4.13 Proposition: Suppose $(X, \\varphi )$ is a mixing Smale space with an effective action of ${\\mathbb {Z}}^m$ .",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"In fact, the Rokhlin dimension of the action is bounded by $4^m -1$ .",
"If $m = 1$ then the action has Rokhlin dimension no more than one.",
"Proof.",
"This result follows directly from [35].",
"$\\mathbb {Z}$ -actions on irreducible Smale spaces Throughout this subsection, we fix an irreducible Smale space $(X, \\varphi )$ with decomposition into mixing components given by $X = X_1 \\sqcup \\dots \\sqcup X_N$ , as in .",
"To begin, we recall the definition of Rokhlin dimension for integer actions as given in [23].",
"4.14 4.14 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra.",
"An action of the integers $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least natural number such that the following holds: for any finite subset $\\mathcal {F} \\subset A$ , and $p \\in \\mathbb {N}$ and any $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ in $A$ satisfying (i) for any $l \\in \\lbrace 0, \\dots , d\\rbrace $ we have $\\Vert f^{(l)}_{r,i} f^{(l)}_{s, j} \\Vert < \\epsilon $ whenever $(r,i) \\ne (s, j)$ , $\\Vert \\sum _{l=0}^d \\sum _{r =0}^1 \\sum _{j = 0}^{p-1+r} f^{(l)}_{r,j} -1\\Vert < \\epsilon $ , $\\Vert \\beta _1(f^{(l)}_{r, j}) - f^{(l)}_{r, j+1} \\Vert < \\epsilon $ for every $r \\in \\lbrace 0,1\\rbrace $ , $j \\in \\lbrace 0, \\dots , p-2+r\\rbrace $ and $l \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert \\beta _1(f^{(l)}_{0,p-1} + f^{(l)}_{1, p}) - (f^{(l)}_{0,0} + f^{(l)}_{1,0}) \\Vert < \\epsilon $ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ $\\Vert [ f^{(l)}_{r,j}, a ] \\Vert < \\epsilon $ for every $r, j, l$ and $a \\in \\mathcal {F}$ .",
"As in the case of a finite group, in the special case of the homoclinic algebra, we don't need to worry about satisfying the last condition.",
"The proof is similar to Lemma REF and is omitted.",
"4.15 4.15 Lemma: Let $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ be an action of the integers.",
"Suppose that for any $p \\in \\mathbb {N}$ and $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ satisfying (i)–(iv) of Definition REF .",
"Then $\\beta $ has Rokhlin at most dimension $d$ .",
"4.16 4.16 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there is a $\\sigma \\in S_N$ such that $ \\beta |_{X_i} : X_i \\rightarrow X_{\\sigma (i)} $ for each $i = 1,\\dots , N$ .",
"Proof.",
"Since $(X_i, \\varphi ^N|_{X_i})$ is mixing, there is a point, call it $x_i$ , with dense $\\varphi ^N$ -orbit in $X_i$ .",
"Since $X_1, \\dots , X_N$ are disjoint, there is $k(i) \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta (x_i) \\in X_{k(i)}$ and $\\beta (x_i) \\notin X_k$ for $k \\ne k(i)$ .",
"Now $\\varphi ^N \\circ \\beta (x_i) \\in X_{k(i)}$ , so we have $\\varphi ^N \\circ \\beta (x_i) = \\beta \\circ \\varphi ^N(x_i) \\in X_{k(i)}$ .",
"The fact that $x_i$ has dense $\\varphi ^N$ -orbit then implies $\\beta |_{X_i} : X_i \\rightarrow X_{k(i)}$ .",
"Suppose that $\\beta |_{X_j}(X_j) \\subset X_{k(i)}$ .",
"If $j \\ne i$ , there exists some $l \\in \\lbrace 1, \\dots , N\\rbrace $ such that $X_l$ is not in the image of any $\\beta (X_i)$ , $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"Choose $x \\in X_l$ .",
"Then $\\beta (x )\\in X_{k(l)}$ and there is some $m$ , $0<m<N$ satisfying $\\varphi ^m(X_{k(l)}) = X_{l}$ .",
"But this implies $\\beta \\circ \\varphi ^m(x) = \\varphi ^m \\circ \\beta (x) \\in X_l$ .",
"So we have $j= i$ , which proves the theorem.",
"4.17 4.17 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there are $L, S \\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ such that $N = LS$ and partition of $\\lbrace 1, \\dots , N\\rbrace $ into $S$ subsets of $L$ elements $\\lbrace r_{s,1}, \\dots , r_{s,L}\\rbrace $ , $1 \\le s \\le S$ satisfying, for each $s$ , $ \\beta (X_{r_{s, l}}) = X_{s, r_{l+1 \\text{mod} L}}.$ Proof.",
"Let $r_{1,1} = 1$ .",
"By the previous lemma there is $r_{1,2} \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta : X_{r_{1,1}} \\rightarrow X_{r_{1,2}}$ .",
"If $r_{1, 2} \\ne 1$ then there is $r_{1, 3} \\ne r_{1,2}$ such that $\\beta : X_{r_{1,2}} \\rightarrow X_{r_{1,3}}$ .",
"By induction and the pigeonhole principal, there is some $0 < L \\le N$ such that $r_{1, l}$ are all distinct, $\\beta : X_{r_{1,l}} \\rightarrow X_{r_{1,l+1}}$ , for $1 \\le l \\le L-1$ and $\\beta : X_{r_{1,L}} \\rightarrow X_{r_{1,1}}$ .",
"If $L= N$ , we are done.",
"Otherwise, there is some $X_{r_{2,1}} \\ne X_{r_{2, l}}$ for every $1 \\le l \\le L$ .",
"Arguing as above and using the previous lemma, there is some $L^{\\prime } \\in \\lbrace 1, \\dots , N-L\\rbrace $ such that $r_{2, l}$ are all distinct, $r_{2,l^{\\prime }} \\ne r_{1, l}$ for any $1 \\le l^{\\prime } \\le L^{\\prime }$ and $1 \\le l \\le L$ , $\\beta : X_{r_{2,l}} \\rightarrow X_{r_{2,l+1}}$ , for $1 \\le l \\le L^{\\prime }-1$ and $\\beta : X_{r_{2,L^{\\prime }}} \\rightarrow X_{r_{2,1}}$ .",
"Without loss of generality, we may assume that $L \\le L^{\\prime }$ .",
"Let $1 \\le c \\le N$ satisfy $\\varphi ^c : X_{r_{1,i}} \\rightarrow X_{r_{2,2}}$ .",
"Then we have a diagram ${ X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & \\dots [r]^{\\beta } & X_{r_{1,L}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,1 \\,}} [d]^{\\varphi ^c}\\\\X_{r_{2,2}} [r]^{\\beta } & X_{r_{2,2}} [r]^{\\beta } & \\dots [r]^{\\beta }& X_{r_{2,L}} [r]^{\\beta } & X_{r_{2,L+1}} \\\\}$ which commutes.",
"If follows that $X_{r_{2, L+1}} = Y_{r_{2, 1}}$ , that is, $L = L^{\\prime }$ .",
"The proof now follows from induction.",
"4.18 4.18 Theorem: Suppose $(X, \\varphi )$ is an irreducible Smale space with an effective $\\mathbb {Z}$ -action.",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"Let $\\beta : X \\rightarrow X$ be the homeomorphism generating the $\\mathbb {Z}$ action and let $X = X_1 \\sqcup \\cdots \\sqcup X_N$ be the Smale decomposition into mixing components $(X_i, \\varphi ^N|_{X_i})$ .",
"Denote by $\\mathrm {C^*}(H_i)$ the homoclinic algebra for the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"By Lemmas REF and REF , there exists $L$ such that $ \\beta ^L : X_i \\rightarrow X_i $ for every $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"For each $i$ , we consider the action $\\beta ^L|_{X_i}$ .",
"Since $\\beta $ induces an effective ${\\mathbb {Z}}$ -action on $(X, \\varphi )$ (in particular $\\beta ^L \\circ \\varphi ^N= \\varphi ^N\\circ \\beta ^L$ ) it follows that $\\beta ^L|_{X_i}$ induces an effective ${\\mathbb {Z}}$ -action on the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"Hence, we can apply Theorem REF to $\\beta ^L|_{X_i}$ acting on $(X_i, \\varphi ^N|_{X_i})$ to conclude that the action induced by $\\beta ^L|_{X_i}$ on $\\mathrm {C}^*(H_i)$ has Rokhlin dimension $d_i$ for some $d_i < \\infty $ .",
"Let $\\epsilon > 0$ , $p \\in \\mathbb {N} \\setminus \\lbrace 0\\rbrace $ .",
"Let $f^{(k)}_{i, j}$ , $i\\in \\lbrace 0,1\\rbrace $ , $0 \\le j \\le p-1+i$ , $0 \\le k \\le d_1$ be two Rokhlin towers for $\\beta ^L|_{X_1}$ with respect to $\\epsilon $ and $p$ and any finite subset $\\mathcal {F}_1 \\subset \\mathrm {C}^*(H_1)$ .",
"We claim that the elements $\\beta ^l(f^{(k)}_{i, j})$ , $1 \\le l \\le L-1$ satisfy (i) – (v) of Lemma REF with respect to $ {\\mathrm {C}}^*(H), \\epsilon $ and $p$ .",
"Indeed, (i), (iii) and (iv) are clear by construction.",
"To see (ii), we note that $\\Vert \\sum _{i,j,k} f^{(k)}_{i, j} - 1_{\\mathrm {C}^*(H_1)}\\Vert < \\epsilon $ and since $\\beta $ is a homeomorphism, it follows that $\\Vert \\sum _{i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{\\mathrm {C}^*(H_l)})\\Vert < \\epsilon $ for every $l$ , and hence $\\Vert \\sum _{l,i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{ {\\mathrm {C}}^*(H)}\\Vert < \\epsilon .$ 4.19 4.19 Remark: Let $Y$ be a compact Hausdorff space.",
"It is not difficult to check that if a $\\mathbb {Z}$ -action on $C(Y)$ has finite Rokhlin dimension, then the action on $Y$ must be free.",
"Thus when ${\\mathbb {Z}}$ acts on an irreducible Smale space $(X, \\varphi )$ , the action of ${\\mathbb {Z}}$ on $C(X)$ cannot have finite Rokhlin dimension because the action of ${\\mathbb {Z}}$ on $X$ is not free by Proposition REF .",
"Nevertheless, Proposition REF implies that a ${\\mathbb {Z}}$ -action on $ {\\mathrm {C}}^*(H)$ induced from an effective action on $(X, \\varphi )$ has Rokhlin dimension at most one.",
"The induced action on the stable and unstable algebras In this section we use what we have already proved about actions on the homoclinic algebra to deduce results for actions on the stable and unstable algebras.",
"To do so, we take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"It is easy to check that if $(X, \\varphi )$ is a Smale space, then $(X, \\varphi ^{-1})$ is also a Smale space with the bracket reversed.",
"The unstable relation of $(X, \\varphi )$ is then the stable relation of $(X, \\varphi ^{-1})$ .",
"Thus, it is enough to show something holds for $ \\mathrm {C}^*(S)$ of an arbitrary (irreducible, mixing) Smale space to imply the same for $ \\mathrm {C}^*(U)$ of an arbitrary (irreducible, mixing) Smale space.",
"Throughout this section, we again assume that $(X, \\varphi )$ is irreducible, but is not necessarily mixing unless explicitly stated.",
"5.1 5.1 Let $(X, \\varphi )$ be an irreducible Smale space.",
"Fix a finite set $P$ of $\\varphi $ -invariant periodic points.",
"Let $ {\\mathrm {C}}^*(H)$ denote the associated homoclinic and $ \\mathrm {C}^*(S)$ the stable algebra.",
"As in [51], we define, for each $a \\in C_c(\\mathcal {G}_H)$ , an element $(\\rho (a), \\rho (a))$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ by $ (\\rho (a) b)(x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} a(x,z)b(z,y) $ and $ (b \\rho (a)) (x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} b(x,z) a(z,y)$ where $b\\in C_c(S)$ .",
"This extends to a map $\\rho : {\\mathrm {C}}^*(H)\\operatorname{\\hookrightarrow }\\mathcal {M}( \\mathrm {C}^*(S)).$ We note that $\\rho $ and the representations of these algebras on the Hilbert space $l^2(X^H(P,Q))$ (see Remark REF ) are compatible.",
"5.2 5.2 Lemma: Let $F \\subset {C}_c(S)$ be a finite subset and let $r_1, \\dots , r_N \\in {C}_c(\\mathcal {G}_H)$ .",
"Then, for every $\\epsilon > 0$ there exists a $k\\in \\mathbb {N}$ such that, viewing $a$ as an element of the multiplier algebra $\\mathcal {M}( \\mathrm {C}^*(S))$ , we have $ \\Vert \\rho (\\alpha ^k(r_i) ) a - a \\rho (\\alpha ^k(r_i )) \\Vert < \\epsilon $ for every $i = 1, \\dots , N$ and every $a \\in F$ .",
"Proof.",
"Follows from [51] or [31].",
"5.3 5.3 Definition: Suppose $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is an action of a countable discrete group.",
"Let $I$ be a separable $G$ -invariant ideal in $A$ and $B$ be a $\\sigma $ -unital $G$ -$\\mathrm {C}^*$ -subalgebra of $A$ .",
"Then there exists a countable $G$ -quasi-invariant quasicentral approximate unit $(w_n)_{n\\in N}$ of $I$ in $B$ .",
"That is, there exists $(w_n)_{n\\in {\\mathbb {N}}}$ an approximate identity for $I$ such that (i) for any $a \\in B$ , $\\Vert aw_n - w_n a\\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ ; for each $g\\in G$ , $\\Vert \\beta _g(w_n) -w_n \\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"The existence of a $G$ -quasi-invariant quasicentral approximate unit is shown in [29], but also see [17] and [13].",
"Rokhlin dimension for actions on the stable and unstable algebra To take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , we will find it convenient to introduce the definition of multiplier Rokhlin dimension with repect to a finite index subgroup.",
"It is inspired by [47], [23] and [64].",
"For a given $\\mathrm {C}^*$ -algebra $A$ and group $G$ with action $\\beta : G \\rightarrow {\\rm Aut}(A)$ we denote by $\\mathcal {M}(\\beta )$ the induced action of $G$ on the multiplier algebra $\\mathcal {M}(A)$ .",
"5.4 5.4 Definition: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"We say the action has multilplier Rokhlin dimension $d$ with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K)$ , if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and finite subsets $M \\subset G$ , $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_{\\overline{g}} \\right)_{l=0, \\ldots d; \\overline{g}\\in G/K} \\subset \\mathcal {M}(A) $ such that (i) for any $l$ , $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert < \\epsilon $ , for $\\overline{g} \\ne \\overline{h}$ in $G/K$ ; $\\Vert \\sum _{l=0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert < \\epsilon $ ; $\\Vert \\mathcal {M}(\\beta _h)(f^{(l)}_{\\overline{g}}) - f^{(l)}_{\\overline{hg}} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $h \\in M$ ; $\\Vert [ f^{( l )}_{\\overline{g}}, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $a \\in F$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) = \\infty $ .",
"We say the action has multilplier Rokhlin dimension $d$ with commuting towers with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K)$ , if, in addition, (i) $\\Vert f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}}- f^{(k)}_{\\overline{h}}f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for every $k,l = 0, \\dots , d$ , $\\overline{g}, \\overline{h} \\in G/K$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) = \\infty $ .",
"The multiplier Rokhlin dimension with respect to a finite subgroup is used to define the Rokhlin dimension for an action of a countable residually finite group.",
"5.5 5.5 Definition: [cf.",
"[64]] Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable, residually finite group, and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ .",
"The Rokhlin dimension of $\\beta $ is defined by $ \\mathrm {dim}_{\\mathrm {Rok}}(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ The Rokhlin dimension of $\\beta $ with commuting towers is given by $ \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ It will be easier to show finite multiplier Rokhlin dimension relative to a finite subgroup, which we show implies the definition of Rokhlin dimension relative to a finite index subgroup for $\\mathrm {C}^*$ -algebras given in [64], recalled in Definition REF below.",
"Some further notation is required to do so.",
"First, we need to define central sequence algebras.",
"Loosely speaking, working in a central sequence algebra allows one to turn statements such as approximate commutativity in the original algebra into honest commutativity in the central sequence algebra.",
"As such, central sequence arguments often allow one to streamline proofs.",
"In the case of discrete groups, an action on a $\\mathrm {C}^*$ -algebra induces an action on its central sequence algebra, and it is possible to reformulate definitions for both Rokhlin dimension of finite group actions and integer actions on separable unital $\\mathrm {C}^*$ -algebras in terms of induced actions on central sequence algebras [64].",
"5.6 5.6 Definition: Let $A$ be a separable $\\mathrm {C}^*$ -algebra.",
"We denote the sequence algebra of $A$ by $ A_{\\infty } := \\prod _{n \\in \\mathbb {N}} A / \\bigoplus _{n \\in \\mathbb {N}} A.",
"$ We view $A$ as a subalgebra of $A_{\\infty }$ by mapping an element $a \\in A$ to the constant sequence consisting of $a$ in every entry.",
"The central sequence algebra is then defined to be $ A^{\\infty } := A_{\\infty } \\cap A^{\\prime } = \\lbrace x \\in A_{\\infty } \\mid ax = xa \\text{ for every } a \\in A\\rbrace , $ the relative commutant of $A$ in $A_{\\infty }$ .",
"Let $ \\mathrm {Ann}(A, A_{\\infty }) := \\lbrace x \\in A_{\\infty } \\mid ax = xa = 0 \\text{ for every } a \\in A\\rbrace ,$ which is evidently an ideal in $A^{\\infty }$ .",
"Finally, we define $ F(A) := A^{\\infty }/ \\mathrm {Ann}(A, A_{\\infty }).",
"$ When $A$ is not separable, one can define the above with respect to a given separable subalgebra $D$ , as is done in [64].",
"However, since all our $\\mathrm {C}^*$ -algebras will be separable, we will not require this.",
"A completely positive contractive (c.p.c.)",
"map $\\varphi : A \\rightarrow B$ between $\\mathrm {C}^*$ -algebras $A$ and $B$ is said to be order zero if it is orthogonality preserving, that is, for every $a, b \\in A_+$ with $ab = ba =0$ we have $\\varphi (a) \\varphi (b) = 0$ .",
"Any $^*$ -homomorphism is of course order zero, but a c.p.c.",
"order zero map is not in general a $^*$ -homomorphism.",
"For more about c.p.c.",
"order zero maps, see [73].",
"5.7 5.7 Definition: [64] Let $A$ be a separable $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"Let $\\tilde{a}_{\\infty }$ denote the induced action on $F_{\\infty }(A)$ .",
"We say the action has Rokhlin dimension $d$ with respect to $K$ if $d$ is the least integer such that there exists equivariant c.p.c.",
"order zero maps $ \\varphi _l : (C(G/K), G\\text{-shift}) \\rightarrow (F_{\\infty }(A), \\tilde{a}_{\\infty }), \\quad l = 0, \\dots , d$ with $ \\varphi _0(1) + \\cdots + \\varphi _d(1) = 1.",
"$ If moreover $\\varphi _0 ,\\dots , \\varphi _d$ can be chosen to have commuting ranges, then we say the action has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"The next result is an obvious generalisation of the equivalence of (1) and (3) of [64] to the case of commuting towers.",
"5.8 5.8 Lemma: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group and $K$ a subgroup of finite index.",
"The following are equivalent for an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ .",
"(i) The action $\\beta $ has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"For every finite subset $M \\subset G$ , finite subset $F \\subset A$ and $\\epsilon >0$ there are positive contractions $(f^{(l)}_{\\overline{g}})_{\\overline{g} \\subset H}^{l= 0, \\dots , d}$ in $A$ such that (i) $\\Vert (\\sum _{l =0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}}) \\cdot a - a \\Vert < \\epsilon $ for all $a \\in F$ ; $\\Vert f_{\\overline{g}}^{(l)}f_{\\overline{h}}^{(l)}a \\Vert \\le \\epsilon $ for all $a\\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\ne \\overline{h} \\in G/K$ ; $\\Vert (\\beta _g(f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}})a \\Vert < \\epsilon $ for all $a \\in F$ , $l \\in 0, \\dots , d$ and $g \\in M$ and $\\overline{h} \\in G/K$ ; $\\Vert f^{(l)}_{\\overline{g}}a - a f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for all $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ ; $\\Vert (f^{(k)}_{\\overline{g}} f^{(l)}_{\\overline{h}} - f^{(l)}_{\\overline{h}} f^{(k)}_{\\overline{g}})a \\Vert < \\epsilon $ for all $a \\in F$ , $k,l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g}, \\overline{h} \\in G/K$ .",
"Proof.",
"The only thing that one needs to check is that asking for the $f^{(l)}_{\\overline{g}}$ to approximately commute is equivalent to having the images of the order zero maps of [64] commute, but this is obvious.",
"5.9 5.9 Theorem: Let $G$ be a countable discrete group, $K$ a subgroup of $G$ with finite index, $A$ a separable $\\mathrm {C}^*$ -algebra and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action with multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ respect to $K$ .",
"If $\\beta $ has multiplier Rokhlin dimension at most $d$ with commuting towers with respect to $K$ , then $\\beta $ has Rokhlin dimension at most $d$ with commuting towers respect to $K$ .",
"Proof.",
"We show the action satisfies the criteria of [64] (and in the commuting tower case Lemma REF (ii)).",
"Note that [64] is exactly (a)-(d) in Lemma REF (ii).",
"Let $M \\subset G$ and $F \\subset A$ be finite subsets and let $\\epsilon > 0$ .",
"Without loss of generality we may assume that every $a \\in F$ is a positive contraction.",
"Since $\\beta $ has multiplier Rokhlin dimension less than or equal to $d$ with respect to $K$ we can find positive contractions $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ satisfying Definition REF with respect to $M$ , $F$ and $\\epsilon /2$ .",
"Let $(w_n)_{n \\in \\mathbb {N}}$ be an $G$ -quasi-invariant quasicentral approximate unit for $A$ in $\\mathcal {M}(A)$ .",
"Since $F \\subset A$ and $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ are in $\\mathcal {M}(A)$ , there exists $N \\in \\mathbb {N}$ sufficiently large so that $ \\Vert w_N a - a \\Vert < \\epsilon /4 \\text{ for every } a \\in F,$ and $ \\Vert [f_{\\overline{g}}^{(l)}, w_N] \\Vert < \\epsilon /8.$ and $ \\Vert \\beta _g(f_{\\overline{h}}^{(l)}) - f_{\\overline{gh}}^{(l)} \\Vert < \\epsilon /8.$ By increasing $N$ if necessary, we may also assume, since $M$ is finite, that $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ .",
"Then let $ r_{\\overline{g}}^{(l)} := w_N f_{\\overline{g}}^{(l)} w_N,$ for $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $g \\in G$ .",
"Then each $r_{\\overline{g}}^{(l)}$ is a positive contraction in $A$ and $\\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}}) a - a\\Vert &=& \\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N) a - a\\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N - 1 \\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert \\\\&<& \\epsilon ,$ showing that (a) of Lemma REF (ii) holds.",
"Next, $\\Vert r_{\\overline{g}}^{(l)} r_{\\overline{h}}^{(l)} a \\Vert &=& \\Vert w_N f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(l)} w_N a \\Vert \\\\&=& \\epsilon /2 + \\Vert w_N^2 f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(l)} w_N^2 a \\Vert \\\\&<& \\epsilon $ for every $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , showing that (b) of Lemma REF holds.",
"Using $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ , we obtain $\\Vert (\\beta _g(r_{\\overline{h}}^{(l)}) - r_{\\overline{gh}}^{(l)})a\\Vert &=& \\epsilon /4 + \\Vert w_N \\mathcal {M}(\\beta _g)(f^{(l)}_{\\overline{h}}) w_N - w_N f^{(l)}_{\\overline{gh}} w_N \\Vert \\\\&<& \\epsilon ,$ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ , every $\\overline{h} \\in G/K$ , every $g \\in M$ and every $a \\in F$ , showing (c) of Lemma REF .",
"For (d) of Lemma REF we have $\\Vert r_{\\overline{g}}^{(l)} a - a r_{\\overline{g}}^{(l)} \\Vert = \\epsilon / 2 + \\Vert f_{\\overline{g}}^{(l)} a - a f_{\\overline{g}}^{(l)} \\Vert < \\epsilon ,$ for every $a \\in F$ , every $l \\in \\lbrace 0, \\dots , d\\rbrace $ and every $\\overline{g} \\in G/K$ .",
"Finally, in the commuting tower case, we must show (e): $\\Vert (r_{\\overline{g}}^{(k)} r_{\\overline{h}}^{(l)} - r_{\\overline{h}}^{(l)} r_{\\overline{g}}^{(k))} a\\Vert &\\le & \\Vert f_{\\overline{g}}^{(k)} w_N^2 f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(k)}\\Vert \\\\&\\le & \\epsilon /4 + \\Vert f_{\\overline{g}}^{(k)} f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(k)}\\Vert \\\\&<& \\epsilon ,$ for every $\\overline{g}, \\overline{h} \\in G/K$ and $k, l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"The proof of the next lemma is obvious and hence omitted.",
"It will, however, prove useful in what follows.",
"5.10 5.10 Lemma: Suppose that the action of $G$ on $ {\\mathrm {C}}^*(H)$ has Rokhlin dimension at most $d$ .",
"Then we may choose the Rokhlin elements to satisfy $f^{(l)}_g \\in C_c(\\mathcal {G}_H)$ for $l = 0, \\dots , d$ and $ g \\in G$ .",
"5.11 5.11 Proposition: Let $G$ be a countable group acting on an irreducible Smale space $(X, \\varphi )$ and let $K \\subset G$ be a subset of finite index.",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ with respect to $K$ so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"If $\\beta $ has Rokhlin dimension at most $d$ with commuting towers with respect to $K$ so does the action $\\beta ^{(S)}$ .",
"Proof.",
"We will show that $\\beta ^{(S)}$ has multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"The result then follows from Theorem REF .",
"Let $M$ be a finite subset of $G$ , $F$ a finite subset of $ \\mathrm {C}^*(S)$ and $\\epsilon >0$ .",
"Without loss of generality, we may assume that $F \\subset C_c(S)$ .",
"Since $\\beta $ has Rokhlin dimension at most $d$ with respect to $K$ there are contractions $r^{(l)}_g \\in {\\mathrm {C}}^*(H)$ such that (i) $\\Vert r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}} \\Vert < \\epsilon /2$ for $l = 0, \\dots , d$ and any $g, h \\in G$ with $g \\ne h$ , $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1\\Vert < \\epsilon /2,$ $\\Vert \\beta _h(r^{(l)}_{\\overline{g}}) - r^{(l)}_{\\overline{hg}}\\Vert < \\epsilon /2$ for $l =0, \\dots , d$ , every $h \\in M$ and $\\overline{g} \\in G/K$ , and $\\Vert r^{(k)}_{\\overline{g}}r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}}r^{(k)}_{\\overline{g}}\\Vert < \\epsilon /2$ for every $\\overline{g}$ , $\\overline{h} \\in G/K$ and $k,l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"By the previous lemma, we may moreover assume that each $r^{(l)}_{\\overline{g}} \\in C_c(\\mathcal {G}_H)$ .",
"Now we can find a natural number $k$ such that $ \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert < \\epsilon /2,$ for every $a \\in F$ .",
"For $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , let $ f^{(l)}_{\\overline{g}} := \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}} )).$ Note that each $f^{(l)}_g$ is a positive contraction in $\\mathcal {M}( \\mathrm {C}^*(S))$ .",
"We will show that the $f^{(l)}_{\\overline{g}}$ satisfy (i) – (iv) of Definition REF .",
"For (i) we have $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} )) \\Vert \\\\&=& \\Vert \\alpha ^k(r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) \\Vert \\\\&<& \\epsilon ,$ for any $\\overline{g} \\ne \\overline{h} \\in G/K$ , any $l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"For (ii) $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f_{\\overline{g}}^{(l)} - 1_{\\mathcal {M}( \\mathrm {C}^*(S))} \\Vert &=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g}\\in G} \\rho (\\alpha ^k( r^{(l)}_{\\overline{g}} )) - \\rho (1_{ {\\mathrm {C}}^*(H)})\\Vert \\\\&=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1_{ {\\mathrm {C}}^*(H)} \\Vert \\\\&<& \\epsilon ,$ for any $a \\in F$ .",
"For (iii) we have $\\Vert f^{(l)}_{\\overline{g}} a - a f^{(l)}_{\\overline{g}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert \\\\&<& \\epsilon /2,$ for all $l \\in \\lbrace 0, \\dots d\\rbrace , \\overline{g} \\in G/K$ and $a \\in F$ .",
"For (iv), let $g \\in G$ with $\\overline{h} \\in G/K $ and let $a \\in F$ .",
"Then, $\\Vert \\beta ^{(S)}_g (f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}} \\Vert &=& \\Vert \\alpha ^k(\\beta _g(r^{(l)}_{\\overline{h}}) - r^{(l)}_{\\overline{gh}}) \\Vert \\\\&<& \\epsilon .$ Finally, in the commuting towers case, for (v), let $\\overline{g}, \\overline{h} \\in G/K$ and $l,k \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert f^{(k)}_{\\overline{h}} f^{(l)}_{\\overline{g}} - f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) - \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}) \\Vert \\\\&=& \\Vert r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}\\Vert \\\\&<& \\epsilon .$ The result now follows.",
"This gives us the next corollary: 5.12 5.12 Corollary: Let $G$ be a countable residually finite group acting on an irreducible Smale space $(X, \\varphi )$ .",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ (with commuting towers) so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"By Proposition REF and Theorem REF respectively, we get the next two corollaries.",
"5.13 5.13 Corollary: Let $d\\in {\\mathbb {N}}$ and $G={\\mathbb {Z}}^d$ .",
"Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"5.14 5.14 Corollary: Let $\\mathbb {Z}$ be an effective action on an irreducible Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"For more general group actions the situation is less clear.",
"In particular, to the authors' knowledge, it is not known if strong outerness implies finite Rokhlin dimensionAt the workshop “Future Targets in the Classification Program for Amenable $\\mathrm {C}^*$ -Algebras” held at BIRS, Eusebio Gardella presented results in this direction, but they have yet to appear..",
"In fact, for general discrete groups, there are obstructions to a (strongly outer) action having finite Rokhlin dimension with commuting towers, see [21].",
"Such examples can occur in the context considered in the present paper.",
"An explicit example is the following, let $(\\Sigma _{[3]}, \\sigma )$ be the full three shift (so $\\Sigma _{[3]}= \\lbrace 0, 1, 2\\rbrace ^{{\\mathbb {Z}}}$ and $\\sigma $ is the left sided shift).",
"Then $ {\\mathrm {C}}^*(H)$ is the UHF-algebra with supernatural number $3^{\\infty }$ .",
"The action induced from the permutation $0 \\mapsto 1$ , $1 \\mapsto 0$ , and $2 \\mapsto 2$ is an effective order two automorphism.",
"Thus the action induced on $ {\\mathrm {C}}^*(H)$ is strongly outer, but it follows from [21] that it does not have finite Rokhlin dimension with commuting towers.",
"$\\mathcal {Z}$ -stability, nuclear dimension and classification We begin this section with two theorems that follow quickly from the work done above.",
"As well as being interesting observations on their own, they will allow us to say something about the $\\mathcal {Z}$ -stability of the crossed products of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ below.",
"6.1 6.1 Theorem: Let $G$ be a countable discrete amenable group.",
"Suppose $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action of $G$ on $ {\\mathrm {C}}^*(H)$ .",
"Then $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable.",
"Proof.",
"Since $(X, \\varphi )$ is mixing, $ {\\mathrm {C}}^*(H)$ is simple and so the classification results of [12] imply that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace, it must be fixed by the action.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable by [58].",
"6.2 6.2 Theorem: Let $G$ be a countable amenable group acting effectively on a mixing Smale space $(X, \\varphi )$ and $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Then the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is a simple unital nuclear $\\mathcal {Z}$ -stable $\\mathrm {C}^*$ -algebra with unique tracial state and nuclear dimension (in fact, decomposition rank) at most one.",
"In particular, it belongs to the class of $\\mathrm {C}^*$ -algebras that are classified by the Elliott invariant and $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF, for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"Proof.",
"Since $ {\\mathrm {C}}^*(H)$ is amenable, it follows from [53] that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G \\cong \\mathrm {C}^*(\\mathcal {G}_H \\rtimes G)$ is also amenable and hence by [69] it satisfies the UCT.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is quasidiagonal [67] and by [67] together with Theorem classified by the Elliott invariant.",
"The nuclear dimension and decomposition rank bounds are given by [5].",
"Finally, by [42] $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF.",
"$\\mathcal {Z}$ -stability of crossed products $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ Let $A$ be separable $\\mathrm {C}^*$ -algebra and $G$ a discrete group.",
"For an action $\\beta : G\\rightarrow \\operatorname{Aut}(A)$ the fixed point algebra is given by $ A^{\\beta } := \\lbrace a \\in A \\mid \\beta _g(a) = a \\text{ for all } g \\in G\\rbrace .",
"$ Let $ A^{\\infty } := \\ell ^{\\infty }(\\mathbb {N}, A) / c_0(A) .$ The central sequence algebra of $A$ is defined by $ A_{\\infty } := A^{\\infty } \\cap A^{\\prime },$ where $A$ is considered as the subalgebra of $A^{\\infty }$ by viewing an element as a constant sequence.",
"We will denote by $\\overline{\\beta }$ the induced action on $A_{\\infty }$ .",
"Let $p$ and $q$ be positive integers.",
"The dimension drop algebra $I(p,q)$ is given by $I(p, q) := \\lbrace f \\in C([0,1], M_p( \\otimes M_q( \\mid f(0) \\in M_p( \\otimes { and }f(1)\\in M_q(\\rbrace .$ The Jiang–Su algebra, $\\mathcal {Z}$ , is an inductive limit of such algebras [28].",
"6.3 6.3 Theorem: Suppose $G$ is a countable discrete group acting on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"If, for each $k\\in {\\mathbb {N}}$ , there exists a unital equivariant embedding $I(k, k+1) \\rightarrow {\\mathrm {C}}^*(H)$ , then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"As usual, it suffices to prove the result for $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ .",
"To do so, we will show that the hypotheses of Lemma 2.6 in [21] hold.",
"We will show that, for every $k \\in \\mathbb {N}$ , there is a completely positive contractive map $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty } $ satisfying (i) $ (\\overline{\\beta }^S( \\gamma (x)) - \\gamma (x))a = $ for every $x \\in I(k, k+1)$ , $g \\in G$ and $a \\in \\mathrm {C}^*(S)$ , $a \\gamma (1) = a$ for every $a \\in \\mathrm {C}^*(S)$ , and $a(\\gamma (xy) - \\gamma (x)\\gamma (y)) = 0$ for every $x, y \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Suppose $F \\subset I(k, j+1)$ a finite subset and $\\epsilon > 0$ are given.",
"Let $w_n$ be a $G$ -invariant quasicentral approximate unit for $ \\mathrm {C}^*(S)$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ and $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ be a unital embedding (which exists by assumption).",
"Define $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)^{\\infty } $ via $ \\gamma (d) = (w_n \\rho (\\alpha ^n(d_n)) w_n)_{n \\in \\mathbb {N}}, $ where $(d_n)_{n \\in \\mathbb {N}}$ is a representative sequence for $\\tilde{\\gamma }(d)$ .",
"Then $\\gamma $ gives a c.p.c.",
"map.",
"Moreover, if $a \\in \\mathrm {C}^*(S)$ we have $ \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\alpha ^n(d_n) )w_n a - a w_n \\rho (\\alpha ^n(d_n)) w_n \\Vert = 0,$ so in fact $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty }.",
"$ Let us check that $\\gamma $ satisfies (i), (ii) and (iii).",
"Let $a \\in A$ and for any $d \\in I(k, k+1)$ , let $(d_n)_{n \\in \\mathbb {N}}$ be a representative of $\\tilde{\\gamma }(d)$ in $( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }$ .",
"$\\Vert \\overline{\\beta }^S(\\gamma (d) - \\gamma (d)) a \\Vert &=& \\lim _{n \\rightarrow \\infty } \\Vert \\beta ^{(S)}(w_n \\rho (\\alpha ^n(d_n)) w_n) - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&=& \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\beta (\\alpha ^n(d_n)) w_n - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&\\le & \\lim _{n \\rightarrow \\infty } \\Vert \\beta (\\alpha ^n(d_n) ) - \\alpha ^n(d_n) \\Vert \\\\&=& 0,$ showing $(i)$ .",
"To show (ii), we have $a \\gamma (1) &=& (a w_n \\rho (\\tilde{\\gamma }(1)) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a \\rho (1) w_n^2)_{n \\in \\mathbb {N}} \\\\&=& a,$ for every $a \\in \\mathrm {C}^*(S)$ Finally, for (iii), let $d, d^{\\prime } \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Then $a(\\gamma (d d^{\\prime }) - \\gamma (d)\\gamma (d^{\\prime })) &=& (a w_n \\rho (d_n d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}} - (a w_n \\rho (d_n) w_n^2 \\rho (d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a w_n^2 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a w_n^4 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} \\\\&=& (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}}\\\\&=& 0.$ Thus $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ is $\\mathcal {Z}$ -stable.",
"6.4 6.4 Corollary: Suppose $G$ is a discrete amenable group and $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"Then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"By Theorem REF (v), $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable and hence has strict comparison [54].",
"This in turn implies $ {\\mathrm {C}}^*(H)$ has property (TI) of Sato [58].",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace $\\tau $ which is therefore fixed by $\\beta $ , the hypotheses of [58] are satisfied.",
"Thus from the proof of [58] we get a unital embedding $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ The result then follows from the previous theorem.",
"Acknowledgments.",
"The authors thank Ian Putnam for many useful discussions concerning the content of this paper, Smale spaces, group actions and dynamics in general.",
"We also thank Magnus Goffeng for a number of useful comments.",
"The authors thank the referee for reading the paper carefully and making a number of useful comments.",
"The authors wish to thank Matrix at the University of Melbourne for hosting them during the programme Refining $\\mathrm {C}^*$ -algebraic Invariants for Dynamics using KK-theory in July 2016, the Banach Centre at the Institute of Mathematics of the Polish Academy of Sciences for hosting the first listed author during the conference Index Theory in October 2016, the University of Hawaii, Manoa for hosting the second listed author during the workshop Computability of K-theory in November 2016 and the Centre Rercerca Matemàtica, Barcelona, for their stay during the Intensive Research on Operator Algebras: Dynamics and Interactions in July 2017.",
"The above research visits were partially supported through NSF grants DMS 1564281 and DMS 1665118."
],
[
"From Smale spaces to $\\mathrm {C}^*$ -algebras",
"Ruelle was the first person to associate operator algebras to Smale spaces in [55].",
"We follow the approach introduced by Putnam and Spielberg [51], [52]: three $\\mathrm {C}^*$ -algebras are constructed via the groupoid $\\mathrm {C}^*$ -algebra construction for étale equivalence relations which capture the contracting, expanding, and asymptotic behaviour of the system given in Definition .",
"Fix an irreducible Smale space $(X, \\varphi )$ .",
"To define topologies on each of our equivalence relations, we first note the following.",
"Using the notation from Definition , it is not difficult to show that for any $0 < \\epsilon \\le \\epsilon _X$ we have $ X^U(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon )),$ and similarly that $ X^S(x) = \\cup _{n \\in \\mathbb {N}} \\varphi ^{-n}(X^S(\\varphi ^n(x), \\epsilon )).$ Each $\\varphi ^{n} (X^U(\\varphi ^{-n}(x), \\epsilon ))$ , $n \\in \\mathbb {N}$ is given the relative topology from $X$ while $X^U(x)$ and $X^S(x)$ are given the topology coming from these inductive unions, see [51] for the precise details.",
"We could proceed to construct $\\mathrm {C}^*$ -algebras directly from the equivalence relations in Definition following the construction of Putnam in [51].",
"However neither the stable nor the unstable groupoids would have a natural étale topology.",
"Instead, following [52], we restrict our relation to those points equivalent to periodic points.",
"3.1 3.1 Definition: Let $P$ and $Q$ be finite $\\varphi $ -invariant sets of periodic points of $(X, \\varphi )$ .",
"Define the stable and unstable groupoids of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_S(P) := \\lbrace (x,y) \\in X^U(P) \\times X^U(P) \\mid x \\sim _s y \\rbrace ,$ and $ \\operatorname{\\mathcal {G}}_U(Q) := \\lbrace (x,y) \\in X^S(Q) \\times X^S(Q) \\mid x \\sim _u y \\rbrace .$ Up to Morita equivalence of groupoids these constructions do not depend on the choice of periodic points.",
"3.2 3.2 Definition: Define the homoclinic groupoid of $(X, \\varphi )$ by $ \\operatorname{\\mathcal {G}}_H := \\lbrace (x, y) \\in X \\times X \\mid x \\sim _h y\\rbrace .$ Now, if $(v, w) \\in X^S(P)$ , then $v \\sim _s w$ so there is some sufficiently large $N \\in \\mathbb {N}$ such that $d(\\varphi ^N(v), \\varphi ^N(w)) < \\epsilon _X/2$ .",
"By continuity of $\\varphi $ , we may choose $\\delta > 0$ small enough so that $\\varphi ^N(X^U(w , \\delta )) \\subset X^U(\\varphi ^N(w), \\epsilon _X/2)$ and also $\\varphi ^N(X^U(v , \\delta )) \\subset X^U(\\varphi ^N(v), \\epsilon _X/2)$ .",
"Then define $ h^s := h^s(v, w, N, \\delta ) : X^U(w, \\delta ) \\rightarrow X^U(v, \\epsilon _X), \\quad x \\mapsto \\varphi ^{-N}([\\varphi ^N(x), \\varphi ^N(v)]).",
"$ By [56] this is a local homeomorphism.",
"For any such $v, w, \\delta , h , N$ , we then define an open set by $ V(v, w, \\delta , h^s, N) := \\lbrace (h^s(x), x) \\mid x \\in X^U(w, \\delta ) \\rbrace \\subset \\operatorname{\\mathcal {G}}_S(P).$ These sets generate an étale topology for $\\operatorname{\\mathcal {G}}_S(P)$ [52].",
"The construction for the topologies of $\\operatorname{\\mathcal {G}}_U(Q)$ and $\\operatorname{\\mathcal {G}}_H$ are similar; we refer the reader to [52] for details."
],
[
"$\\mathrm {C}^*$ -algebras",
"Fix finite $\\varphi $ -invariant sets $P$ and $Q$ and let $\\mathcal {H} = \\ell ^2(X^H(P,Q))$ where $X^H(P,Q)$ is the set of points in $X$ which are both stably equivalent to a point in $P$ and unstably equivalent to a point in $Q$ .",
"It is shown in [56] that $X^H(P, Q)$ is countable.",
"If $\\operatorname{\\mathcal {G}}$ is one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ , let $C_c(\\operatorname{\\mathcal {G}})$ denote the a compactly supported functions on $\\operatorname{\\mathcal {G}}$ with convolution product, $ (f \\ast g) (x, y) = \\sum _{z \\sim x, z \\sim y} f(x,z) g(z, y), \\quad (x,y) \\in \\operatorname{\\mathcal {G}}, $ and $f^*(x, y) = \\overline{f(y,x)}, \\quad (x,y) \\in \\operatorname{\\mathcal {G}}.$ We can represent each of $C_c(\\operatorname{\\mathcal {G}}_H)$ , $C_c(\\operatorname{\\mathcal {G}}_S(P))$ and $C_c(\\operatorname{\\mathcal {G}}_U(Q))$ on the Hilbert space $\\mathcal {H}$ , and define the homoclinic algebra by $ {\\mathrm {C}}^*(H):= \\overline{C_c(\\operatorname{\\mathcal {G}}_H)}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ the stable algebra $ \\mathrm {C}^*(S):= \\overline{C_c(\\operatorname{\\mathcal {G}}_S(P))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}},$ and the unstable algebra $ \\mathrm {C}^*(U):= \\overline{C_c(\\operatorname{\\mathcal {G}}_U(Q))}^{\\Vert \\cdot \\Vert _{\\mathcal {H}}}.$ 3.3 3.3 Remarks: 1.",
"We suppress the reference to $P$ and $Q$ in the notation of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"For any choice of such $P$ and $Q$ , the resulting groupoids are Morita equivalent, and hence so are their $\\mathrm {C}^*$ -algebras.",
"Thus from the perspective of most $\\mathrm {C}^*$ -algebraic properties, we don't need to keep track of the original choice.",
"2.",
"In the usual groupoid $\\mathrm {C}^*$ -algebra construction for a groupoid $\\mathcal {G}$ , the algebra of compactly supported functions $C_c(\\mathcal {G})$ is represented on the Hilbert space $\\ell ^2(\\mathcal {G})$ and the completion is the reduced groupoid $\\mathrm {C}^*$ -algebra $\\mathrm {C}_r(\\mathcal {G})$ .",
"However, when the groupoid is amenable, the completion of any faithful representation will result in the same $\\mathrm {C}^*$ -algebra.",
"Here $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each amenable [52].",
"It is convenient, however, to represent them all on the same Hilbert space (namely $\\ell ^2(X^H(P,Q))$ ) because we can consider the interactions between operators coming from the different algebras.",
"This will be particularly useful when showing how finite Rokhlin dimension passes from $ {\\mathrm {C}}^*(H)$ to $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ in Section .",
"3.4 3.4 Let $(X, \\varphi )$ be a mixing Smale space.",
"Then $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are simple by [51] and [52], separable (since $X$ is a metric space) and nuclear by [52] and [51].",
"The homoclinic $\\mathrm {C}^*$ -algebra $ {\\mathrm {C}}^*(H)$ is unital since the diagonal $X \\times X$ is open in $\\operatorname{\\mathcal {G}}_H$ and $X$ is compact.",
"Each of $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ admit a trace [51], hence are stably finite.",
"The traces on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are not bounded while $ {\\mathrm {C}}^*(H)$ admits a tracial state.",
"Moreover, when $(X, \\varphi )$ is mixing, this trace is unique [25].",
"In [12], the authors showed that for a mixing Smale space, $ {\\mathrm {C}}^*(H)$ , $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ have finite nuclear dimension and hence are $\\mathcal {Z}$ -stable.",
"(There it was not noted that $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ are $\\mathcal {Z}$ -stable; however this follows from [66].",
"In fact, it can be proved directly in a similar manner to Theorem REF below, once we know that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.)",
"It then follows that $ {\\mathrm {C}}^*(H)\\otimes \\operatorname{\\mathcal {U}}$ is tracially approximately finite (TAF) in the sense of [36] for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"In particular, the $\\mathrm {C}^*$ -algebras coming from the homoclinic relation on a mixing Smale space is classified by the Elliott invariant, see [12].",
"Suppose $G$ is a discrete group acting on a mixing Smale space.",
"For the action to induce a well-defined action on $\\operatorname{\\mathcal {G}}_U$ and $\\operatorname{\\mathcal {G}}_S$ , the choice of finite sets of $\\varphi $ -invariant periodic points must be $G$ -invariant.",
"Fortunately, this can always be arranged.",
"3.5 3.5 Lemma: Let $G$ be a discrete group acting effectively on a mixing Smale space $(X , \\varphi )$ .",
"Then there exists a finite set of $\\varphi $ -invariant periodic points $P$ such that $gp \\in P$ for every $g \\in G$ and every $p \\in P$ .",
"Proof.",
"Let $P^{\\prime }$ be any finite $\\varphi $ -invariant set of periodic points.",
"Then $P^{\\prime } \\subseteq {\\rm Per}_n(X, \\varphi )$ for some $n\\in {\\mathbb {N}}$ .",
"We know that ${\\rm Per}_n(X, \\varphi )$ is finite.",
"Also, for any $g\\in G$ and $x\\in {\\rm Per}_n(X, \\varphi )$ it was shown in the proof of Proposition REF that $gx \\in {\\rm Per}_n(X, \\varphi )$ .",
"It follows that the set $P=\\lbrace p \\in X \\mid p=gx \\hbox{ for some }g\\in G, x\\in P^{\\prime }\\rbrace $ is contained in ${\\rm Per}_n(X, \\varphi )$ and hence is finite.",
"It is $G$ -invariant by construction and it is $\\varphi $ -invariant because $g \\varphi (x) = \\varphi ( gx)$ for any $g\\in G$ and $x\\in X$ .",
"For the remainder of the paper, we will assume that $P$ and $Q$ are $G$ -invariant.",
"Let $\\operatorname{\\mathcal {G}}$ be one of $\\operatorname{\\mathcal {G}}_H, \\operatorname{\\mathcal {G}}_S(P), \\operatorname{\\mathcal {G}}_U(Q)$ and let $G$ be a group acting on $(X, \\varphi )$ .",
"Since $P$ and $Q$ are assumed to be $G$ -invariant, $g(x,y) \\mapsto (gx, gy)$ defines an induced action of $G$ on $\\operatorname{\\mathcal {G}}$ .",
"The action of $G$ on $\\operatorname{\\mathcal {G}}$ in turn induces an action on $\\mathrm {C}^*(\\operatorname{\\mathcal {G}})$ .",
"3.6 3.6 Example: Following Example REF , the Smale space homeomorphism $\\varphi $ induces a $\\mathbb {Z}$ -action on each of $ {\\mathrm {C}}^*(H), \\mathrm {C}^*(S), \\mathrm {C}^*(U)$ .",
"These actions are denoted by $\\alpha $ , $\\alpha _S$ and $\\alpha _U$ , respectively.",
"Properties of $\\alpha $ (in particular, [51]) will prove indispensable for the results in the next section and also in Section , where we seek to pass from known properties about $ {\\mathrm {C}}^*(H)$ to the nonunital $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"3.7 3.7 Example: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type.",
"Then the $C^*$ -algebras associated to $\\operatorname{\\mathcal {G}}_H$ , $\\operatorname{\\mathcal {G}}_S(P)$ and $\\operatorname{\\mathcal {G}}_U(Q)$ are each AF.",
"Moreover if $G$ is a finite group acting effectively on $(\\Sigma , \\sigma )$ , then using [1] one can show that the induced action of $G$ on each of these AF $\\mathrm {C}^*$ -algebras is locally representable in the sense of [18].",
"This is the case for the automorphisms $\\beta _1$ and $\\beta _2$ on the full two-shift given in Example REF .",
"It follows from results in [18] that the crossed products associated to the action of $G$ are each also AF.",
"3.8 3.8 Example: Suppose $(X, \\varphi )$ is the Smale space obtained via the solenoid construction in Example REF with $Y=S^1$ and $g(z)=z^n$ ($n\\ge 2$ ).",
"Then the stable and unstable algebras are the stabilisation of a Bunce–Deddens algebra.",
"For details in the case $n=2$ , see page 28 of [51].",
"Automorphisms of Bunce–Deddens algebras are considered in [45], for example.",
"3.9 3.9 Example: If $(X, \\varphi )$ is a hyperbolic toral automorphism, as in Example REF , then the stable and unstable algebras are the stabilization of irrational rotation algebras as is shown on pages 27-28 of [51].",
"Automorphisms of irrational rotation algebras are well-studied, see [14] and reference therein."
],
[
"The induced action on $ {\\mathrm {C}}^*(H)$",
"In the sequel, the aim is to provide conditions of a group action on a mixing Smale space which will allow us to determine structural properties of the crossed products of the associated $\\mathrm {C}^*$ -algebras by the induced group action.",
"The idea is to determine what properties are preserved when passing from the $\\mathrm {C}^*$ -algebra to its crossed product.",
"If we interpret a $\\mathrm {C}^*$ -crossed product as a “noncommutative orbit space” then what we are asking for is some sort of “freeness” condition.",
"In the $\\mathrm {C}^*$ -algebraic context this might take a number of different forms.",
"Here, we focus on the Rokhlin dimension of an action (which is akin to a “coloured” version of noncommutative freeness), initially proposed for finite group and $\\mathbb {Z}$ -actions by Hirshberg, Winter and Zacharias [23] and subsequently generalised to other groups [63], [15], [64], [22], [8], as well as the notion of a “strongly outer action” (which can be thought of as a noncommutative approximation of freeness in trace) (see for example, [41]).",
"Recent results of Sato [58] also play a key role, although for finite group actions and ${\\mathbb {Z}}^d$ -actions these new results are not required."
],
[
"Finite group actions on $ {\\mathrm {C}}^*(H)$",
"For this subsection, we fix a mixing Smale space $(X, \\varphi )$ .",
"Let $ {\\mathrm {C}}^*(H)$ denotes its homoclinic algebra.",
"Here we study actions induced on $ {\\mathrm {C}}^*(H)$ by free actions of $G$ on $(X, \\varphi )$ .",
"Given an action of a group $G$ on a $C^*$ -algebra, $A$ , $\\beta : G \\rightarrow {\\rm Aut}(A)$ we will write $\\beta _h$ for $\\beta (h)$ .",
"Based on Proposition REF which shows that freeness is unlikely to hold for infinite groups, we restrict to the case that $G$ is finite.",
"4.1 4.1 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra and let $G$ be a finite group.",
"An action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and each finite subset $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A $ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ $g \\ne h$ in $G$ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ ; $\\Vert [ f^{( l )}_g, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ .",
"When $d = 0$ the action is said to have the Rokhlin property.",
"In this case the contractions $(f_g)_{g \\in G}$ can in fact be taken to be projections.",
"The definition of the Rokhlin property for $\\mathrm {C}^*$ -algebras was introduced by Izumi [26], [27].",
"4.2 4.2 Lemma: Let $G$ be a finite group acting on $(X, \\varphi )$ and denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Let $d$ be a nonnegative integer.",
"Suppose that for any $\\epsilon >0$ there are positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ such that (i) $\\Vert f^{(l)}_g f^{(l)}_h \\Vert < \\epsilon $ , for $g \\ne h \\in G$ , $l \\in \\lbrace 0, \\dots , d\\rbrace $ ; $\\Vert \\sum _{l=0}^d \\sum _{g \\in G} f^{(l)}_g - 1 \\Vert < \\epsilon $ ; $\\Vert \\beta _h(f^{(l)}_g) - f^{(l)}_{hg} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g, h \\in G$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ .",
"Proof.",
"Let $\\epsilon >0$ and $F \\subset {\\mathrm {C}}^*(H)$ be a finite set.",
"Take positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ with the properties assumed in the statement of the theorem.",
"By [51], there is $n \\in {\\mathbb {N}}$ such that $\\Vert [ \\alpha ^n( f^{( l )}_g) , a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $g \\in G$ and $a \\in F$ , where $\\alpha $ is the automorphism from Example REF .",
"Since $\\alpha $ is an automorphism, $\\left( \\alpha ^n( f^{(l)}_g) \\right)_{l=0, \\ldots d; g\\in G} \\subset {\\mathrm {C}}^*(H)$ satisfies the other requirements of Definition REF .",
"4.3 4.3 Corollary: Suppose $A$ is a $G$ -invariant unital $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ .",
"If $G$ acting on $A$ has Rokhlin dimension at most $d$ then the action of $G$ on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"In particular, if $G$ acting on $C(X)$ has Rokhlin dimension at most $d$ , then $G$ acting on $ {\\mathrm {C}}^*(H)$ also has Rokhlin dimension at most $d$ .",
"Proof.",
"By assumption, given $\\epsilon >0$ , there exists positive contractions $ \\left( f^{(l)}_g \\right)_{l=0, \\ldots d; g\\in G} \\subset A \\subset {\\mathrm {C}}^*(H)$ such that the hypotheses of Lemma REF hold; this then implies the result.",
"When considering the statement concerning $C(X)$ one need only note that $C(X)$ is a $\\mathrm {C}^*$ -subalgebra of $ {\\mathrm {C}}^*(H)$ and that it is invariant under the action of $G$ .",
"4.4 4.4 Corollary: Suppose $(X, \\varphi )$ is a mixing Smale space and a finite group $G$ acts on $(X, \\varphi )$ freely.",
"Then $G$ acting on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"For finite group actions on a compact space, freeness implies finite Rokhlin dimension [21].",
"The result then follows from Corollary REF .",
"4.5 4.5 Corollary: Suppose $(\\Sigma , \\sigma )$ is a mixing shift of finite type and a finite group $G$ acts on $(\\Sigma , \\sigma )$ freely.",
"Then the action of $G$ on $ {\\mathrm {C}}^*(H)$ has the Rokhlin property.",
"Proof.",
"Since $\\Sigma $ is the Cantor set and $G$ acts freely, the action of $G$ on $C(\\Sigma )$ has the Rokhlin property.",
"Corollary REF then implies the result.",
"Strongly outer actions Using [41], we prove the first of the three results listed at the end of the introduction; it appears as Theorem REF .",
"To begin, we recall the Vitali covering theorem and the definitions needed for its statement.",
"4.6 4.6 Definition: A finite measure $\\mu $ on a metric space $(X, d)$ is said to be doubling if there exists a constant $M > 0$ such that $ \\mu (B(x, 2\\epsilon )) \\le M \\mu (B(x, \\epsilon )) $ for any $x \\in X$ and any $\\epsilon >0$ .",
"4.7 4.7 Definition: Suppose $(Y,d)$ is a metric space and $A \\subseteq Y$ .",
"Then a Vitali cover of $A$ is a collection of closed balls $\\mathcal {B}$ such that inf$\\lbrace r>0 \\mid B(x,r) \\in \\mathcal {B} \\rbrace =0$ for all $x\\in A$ .",
"4.8 4.8 Theorem: [Vitali Covering Theorem] Suppose $(Y,d)$ is a compact metric space, $\\mu $ is a doubling measure, $A \\subseteq Y$ and $\\mathcal {F}$ is Vitali cover of $A$ .",
"Then, for any $\\epsilon >0$ , there exists finite disjoint family $\\lbrace F_1, F_2, \\ldots F_n \\rbrace \\subseteq \\mathcal {F}$ such that $\\mu ( A - \\cup ^n_{i=1} F_i) < \\epsilon $ .",
"4.9 4.9 Lemma: Suppose $(X, \\varphi )$ is a mixing Smale space with Bowen measure $\\mu $ and $G$ is a discrete group acting effectively on $(X, \\varphi )$ .",
"Let $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ denote the induced action.",
"Then, for any $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ , there exists positive contractions $(f_i)_{i=1}^k \\subset {\\mathrm {C}}^*(H)$ such that (i) $f_i \\cdot f_j=0$ for all $i\\ne j$ ; $f_i \\cdot \\beta _g(f_i) =0$ ; $\\tau (\\sum _{i=1}^k f_i)> 1- \\epsilon $ where $\\tau $ denotes the (unique) trace on $ {\\mathrm {C}}^*(H)$ obtained from $\\mu $ .",
"Proof.",
"Fix $g\\in G \\setminus \\lbrace e\\rbrace $ and $\\epsilon >0$ .",
"Let $D=\\lbrace x \\in X \\: | \\: gx \\ne x \\rbrace $ .",
"Then $D$ is open and has full measure by Proposition REF .",
"Moreover, for each $x\\in D$ there exists $\\delta _x>0$ such for any $0<\\delta \\le \\delta _x$ we have $\\overline{B_{\\delta }(x)} \\cap g(\\overline{ B_{\\delta }(x)}) = \\emptyset $ and $\\overline{B_{\\delta }(x)} \\subseteq D$ .",
"Let $\\mathcal {F}= \\left\\lbrace F_x \\: | \\: F_x = \\overline{ B_{\\delta }(x)} \\hbox{ for some }x\\in D \\hbox{ and some }0<\\delta \\le \\frac{\\delta _x}{2} \\right\\rbrace .$ By construction, the collection $\\mathcal {F}$ is a Vitali covering of $D$ .",
"The Bowen measure is doubling (see for example [46]) so we may apply the Vitali Covering Theorem to obtain a finite subcollection of $\\mathcal {F}$ , $\\lbrace F_{x_1}, \\ldots , F_{x_k} \\rbrace $ , with the following properties (i) $F_{x_i} \\cap F_{x_j}=\\emptyset $ ; $\\mu (\\cup _{i=1}^k F_{x_i}) > 1- \\epsilon $ .",
"Recall that $\\mu (D)=1$ .",
"Since each $F_{x_i}$ is compact (they are closed in a compact space) and pairwise disjoint, we can use Urysohn's lemma and the Tietze extension theorem to obtain pairwise disjoint open sets $(U_i)_{i=1}^k$ such that for each $i$ , $F_{x_i}\\subseteq U_{x_i}\\subseteq B_{\\delta _{x_i}}(x_i)$ and functions $(f_i)_{i=1}^k \\subseteq C(X) \\subseteq {\\mathrm {C}}^*(H)$ with the following properties: (i) $0\\le f_i \\le 1$ , $ {\\rm supp}(f_i) \\subseteq U_i$ , $F_{x_i} \\subseteq \\lbrace x \\: |\\: f_i(x)=1 \\rbrace $ ; for every $i \\in \\lbrace 1, \\dots , k\\rbrace $ .",
"Finally, we show that $(f_i)_{i=1}^k$ has the required properties.",
"They are by definition positive contractions.",
"Moreover, (i) $f_i \\cdot f_j =0$ for every $i \\ne j \\in \\lbrace 1, \\dots , k\\rbrace $ , since ${\\rm supp}(f_i) \\subseteq U_i$ and $U_{i} \\cap U_{j}=\\emptyset $ ; $f_i \\cdot \\beta _g(f_i)=0$ for every $i \\in \\lbrace 0, \\dots , k\\rbrace $ since $U_i \\subseteq B_{\\delta _{x_i}}(x_i)$ implies that $U_i \\cap g( U_i) = \\emptyset $ .",
"Finally, $\\tau \\left( \\sum _{i=1}^k f_i \\right) & \\ge & \\sum _{i=1}^k \\mu ( \\lbrace x | f_i(x) =1 \\rbrace ) \\\\& \\ge & \\sum _{i=1}^k \\mu (F_i) \\\\& = & \\mu ( \\cup _{i=1}^k F_i ) \\\\& > & 1 - \\epsilon ,$ showing (iii) holds.",
"4.10 4.10 Let $A$ be a $\\mathrm {C}^*$ -algebra and $\\tau $ a state on $A$ .",
"We denote by $\\pi _{\\tau }$ the representation of $A$ corresponding to the GNS construction with respect to $\\tau $ .",
"In this case, $\\pi _{\\tau }(A)^{\\prime \\prime }$ is the enveloping von Neumann algebra of $\\pi _{\\tau }(A)$ .",
"Definition: [41] Let $A$ be a unital simple $\\mathrm {C}^*$ -algebra with nonempty tracial state space $T(A)$ .",
"An automorphism $\\beta $ of $A$ is not weakly inner if, for every $\\tau \\in T(A)$ such that $\\tau \\circ \\beta = \\tau $ , the weak extension of $\\beta $ to $\\pi _{\\tau }(A)^{\\prime \\prime }$ is outer.",
"If $G$ is a discrete group, then an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is strongly outer if, for every $g \\in G \\setminus \\lbrace e\\rbrace $ , the automorphism $\\beta _g$ is not weakly inner.",
"The next lemma and theorem are based on arguments due to Kishimoto in [33].",
"4.11 4.11 Lemma: Let $A$ be a unital $\\mathrm {C}^*$ -algebra and $\\tau \\in T(A)$ .",
"Then for every $\\epsilon > 0$ there is $\\delta > 0$ such that for any positive contraction $f \\in A$ such that $\\tau (f) > 1 - \\delta $ we have $\\tau (a) \\le \\tau (fa) + \\epsilon $ for every $a \\in A$ .",
"Proof.",
"It is enough to show this holds when $a \\in A_+$ .",
"Given $\\epsilon > 0$ let $\\delta = \\epsilon ^2$ .",
"Then since $a = af + a(1-f)$ we have $\\tau (a) - \\tau (af) &=& \\tau (a(1-f)) \\\\&\\le & \\tau (a^2)^{1/2}\\tau ((1-f)^2)^{1/2} \\\\&\\le & \\tau ((1-f)^2)^{1/2}\\\\&\\le & \\tau (1-f)^{1/2} \\\\&\\le & \\epsilon .$ 4.12 4.12 Theorem: Let $(X, \\varphi )$ be a mixing Smale space with an effective action of a discrete group $G$ .",
"Then the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"Proof.",
"Let $\\tau $ denote the tracial state on $ {\\mathrm {C}}^*(H)$ corresponding to the Bowen measure on $(X, \\varphi )$ .",
"Fix $g \\in G\\setminus \\lbrace e\\rbrace $ .",
"Since $\\tau $ is the unique tracial state, we have $\\tau = \\tau \\circ \\beta _g$ .",
"Let $\\epsilon > 0$ and let $F \\subset A$ be a finite subset.",
"Let $\\delta $ be the $\\delta $ of Lemma REF with respect to $\\epsilon /2$ .",
"By Lemma REF , we can find positive contractions $f_1, \\dots , f_k \\in {\\mathrm {C}}^*(H)$ such that (i) $f_if_j = 0$ for every $0 \\le i \\ne j \\le k$ , $f_i \\beta _g(f_i) = 0$ , $\\tau (f_1 + \\dots + f_k) > 1 - \\delta $ .",
"There is a sufficiently large $N \\in \\mathbb {N}$ such that, for any $i =1, \\dots , k$ , we have $ \\Vert \\alpha ^N(f_i) a - a \\alpha ^N(f_i) \\Vert < \\epsilon /2 \\text{, for any } a \\in F.$ Since $\\alpha ^N : {\\mathrm {C}}^*(H)\\rightarrow {\\mathrm {C}}^*(H)$ is an automorphism, we have that (i) $\\alpha ^N(f_i) \\alpha ^N(f_j) = 0$ for every $1 \\le i \\ne j \\le k$ and $\\tau (\\alpha ^N(f_1) + \\dots + \\alpha ^N(f_k)) > 1 - \\delta .$ Moreover, since the action of $G$ on $X$ commutes with $\\varphi $ , we have, for any $f\\in {\\mathrm {C}}^*(H)$ , that $\\alpha ^N(\\beta _g(f)) = \\beta _g(\\alpha ^N(f))$ .",
"Hence (i) $\\alpha ^N(f_i) \\beta _g(\\alpha ^N(f_i)) = 0.$ Thus we have $ \\Vert \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) \\Vert < \\epsilon /2$ for any $a \\in F$ .",
"Consider the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z}$ .",
"It is generated by $ {\\mathrm {C}}^*(H)$ and $u_g$ where $u_g$ is unitary satisfying $u_g a u_g^* = \\beta _g(a)$ for every $a \\in {\\mathrm {C}}^*(H)$ .",
"Let $\\sigma \\in T( {\\mathrm {C}}^*(H)\\rtimes _{\\beta _g} \\mathbb {Z})$ and note that $\\sigma |_{ {\\mathrm {C}}^*(H)} = \\tau $ .",
"Put $f = \\sum _{i=1}^k \\alpha ^N(f_i)$ .",
"By the above and Lemma REF we have $\\sigma (a u_g) &\\le & \\sigma (f a u_g f) + \\epsilon \\\\&=& \\sum _{i=1}^k \\sigma ( \\alpha ^N(f_i) a \\beta _g(\\alpha ^N(f_i)) u_g ) + \\epsilon \\\\&<& 2\\epsilon .$ Since $\\epsilon $ and $F$ were arbitrary, it follows that $\\sigma (a u_g) = 0$ for all $a \\in A$ .",
"That $\\beta _g$ is not weakly inner now follows from the proof of [33].",
"Since this holds for every $g \\in G\\setminus \\lbrace e\\rbrace $ , it follows that the action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ is strongly outer.",
"4.13 4.13 Proposition: Suppose $(X, \\varphi )$ is a mixing Smale space with an effective action of ${\\mathbb {Z}}^m$ .",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"In fact, the Rokhlin dimension of the action is bounded by $4^m -1$ .",
"If $m = 1$ then the action has Rokhlin dimension no more than one.",
"Proof.",
"This result follows directly from [35].",
"$\\mathbb {Z}$ -actions on irreducible Smale spaces Throughout this subsection, we fix an irreducible Smale space $(X, \\varphi )$ with decomposition into mixing components given by $X = X_1 \\sqcup \\dots \\sqcup X_N$ , as in .",
"To begin, we recall the definition of Rokhlin dimension for integer actions as given in [23].",
"4.14 4.14 Definition: [23] Let $A$ be a unital $\\mathrm {C}^*$ -algebra.",
"An action of the integers $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}(A)$ has Rokhlin dimension $d$ if $d$ is the least natural number such that the following holds: for any finite subset $\\mathcal {F} \\subset A$ , and $p \\in \\mathbb {N}$ and any $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ in $A$ satisfying (i) for any $l \\in \\lbrace 0, \\dots , d\\rbrace $ we have $\\Vert f^{(l)}_{r,i} f^{(l)}_{s, j} \\Vert < \\epsilon $ whenever $(r,i) \\ne (s, j)$ , $\\Vert \\sum _{l=0}^d \\sum _{r =0}^1 \\sum _{j = 0}^{p-1+r} f^{(l)}_{r,j} -1\\Vert < \\epsilon $ , $\\Vert \\beta _1(f^{(l)}_{r, j}) - f^{(l)}_{r, j+1} \\Vert < \\epsilon $ for every $r \\in \\lbrace 0,1\\rbrace $ , $j \\in \\lbrace 0, \\dots , p-2+r\\rbrace $ and $l \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert \\beta _1(f^{(l)}_{0,p-1} + f^{(l)}_{1, p}) - (f^{(l)}_{0,0} + f^{(l)}_{1,0}) \\Vert < \\epsilon $ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ $\\Vert [ f^{(l)}_{r,j}, a ] \\Vert < \\epsilon $ for every $r, j, l$ and $a \\in \\mathcal {F}$ .",
"As in the case of a finite group, in the special case of the homoclinic algebra, we don't need to worry about satisfying the last condition.",
"The proof is similar to Lemma REF and is omitted.",
"4.15 4.15 Lemma: Let $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ be an action of the integers.",
"Suppose that for any $p \\in \\mathbb {N}$ and $\\epsilon > 0$ there are positive contractions $ f^{(l)}_{0,0}, \\dots , f^{(l)}_{0, p-1}, f^{(l)}_{1, 0}, \\dots , f^{(l)}_{1, p}, \\qquad l \\in \\lbrace 0, \\dots , d\\rbrace ,$ satisfying (i)–(iv) of Definition REF .",
"Then $\\beta $ has Rokhlin at most dimension $d$ .",
"4.16 4.16 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there is a $\\sigma \\in S_N$ such that $ \\beta |_{X_i} : X_i \\rightarrow X_{\\sigma (i)} $ for each $i = 1,\\dots , N$ .",
"Proof.",
"Since $(X_i, \\varphi ^N|_{X_i})$ is mixing, there is a point, call it $x_i$ , with dense $\\varphi ^N$ -orbit in $X_i$ .",
"Since $X_1, \\dots , X_N$ are disjoint, there is $k(i) \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta (x_i) \\in X_{k(i)}$ and $\\beta (x_i) \\notin X_k$ for $k \\ne k(i)$ .",
"Now $\\varphi ^N \\circ \\beta (x_i) \\in X_{k(i)}$ , so we have $\\varphi ^N \\circ \\beta (x_i) = \\beta \\circ \\varphi ^N(x_i) \\in X_{k(i)}$ .",
"The fact that $x_i$ has dense $\\varphi ^N$ -orbit then implies $\\beta |_{X_i} : X_i \\rightarrow X_{k(i)}$ .",
"Suppose that $\\beta |_{X_j}(X_j) \\subset X_{k(i)}$ .",
"If $j \\ne i$ , there exists some $l \\in \\lbrace 1, \\dots , N\\rbrace $ such that $X_l$ is not in the image of any $\\beta (X_i)$ , $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"Choose $x \\in X_l$ .",
"Then $\\beta (x )\\in X_{k(l)}$ and there is some $m$ , $0<m<N$ satisfying $\\varphi ^m(X_{k(l)}) = X_{l}$ .",
"But this implies $\\beta \\circ \\varphi ^m(x) = \\varphi ^m \\circ \\beta (x) \\in X_l$ .",
"So we have $j= i$ , which proves the theorem.",
"4.17 4.17 Lemma: Let $\\beta \\in \\operatorname{Aut}(X, \\varphi )$ .",
"Then there are $L, S \\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ such that $N = LS$ and partition of $\\lbrace 1, \\dots , N\\rbrace $ into $S$ subsets of $L$ elements $\\lbrace r_{s,1}, \\dots , r_{s,L}\\rbrace $ , $1 \\le s \\le S$ satisfying, for each $s$ , $ \\beta (X_{r_{s, l}}) = X_{s, r_{l+1 \\text{mod} L}}.$ Proof.",
"Let $r_{1,1} = 1$ .",
"By the previous lemma there is $r_{1,2} \\in \\lbrace 1, \\dots , N\\rbrace $ such that $\\beta : X_{r_{1,1}} \\rightarrow X_{r_{1,2}}$ .",
"If $r_{1, 2} \\ne 1$ then there is $r_{1, 3} \\ne r_{1,2}$ such that $\\beta : X_{r_{1,2}} \\rightarrow X_{r_{1,3}}$ .",
"By induction and the pigeonhole principal, there is some $0 < L \\le N$ such that $r_{1, l}$ are all distinct, $\\beta : X_{r_{1,l}} \\rightarrow X_{r_{1,l+1}}$ , for $1 \\le l \\le L-1$ and $\\beta : X_{r_{1,L}} \\rightarrow X_{r_{1,1}}$ .",
"If $L= N$ , we are done.",
"Otherwise, there is some $X_{r_{2,1}} \\ne X_{r_{2, l}}$ for every $1 \\le l \\le L$ .",
"Arguing as above and using the previous lemma, there is some $L^{\\prime } \\in \\lbrace 1, \\dots , N-L\\rbrace $ such that $r_{2, l}$ are all distinct, $r_{2,l^{\\prime }} \\ne r_{1, l}$ for any $1 \\le l^{\\prime } \\le L^{\\prime }$ and $1 \\le l \\le L$ , $\\beta : X_{r_{2,l}} \\rightarrow X_{r_{2,l+1}}$ , for $1 \\le l \\le L^{\\prime }-1$ and $\\beta : X_{r_{2,L^{\\prime }}} \\rightarrow X_{r_{2,1}}$ .",
"Without loss of generality, we may assume that $L \\le L^{\\prime }$ .",
"Let $1 \\le c \\le N$ satisfy $\\varphi ^c : X_{r_{1,i}} \\rightarrow X_{r_{2,2}}$ .",
"Then we have a diagram ${ X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,2}} [r]^{\\beta } [d]^{\\varphi ^c} & \\dots [r]^{\\beta } & X_{r_{1,L}} [r]^{\\beta } [d]^{\\varphi ^c} & X_{r_{1,1 \\,}} [d]^{\\varphi ^c}\\\\X_{r_{2,2}} [r]^{\\beta } & X_{r_{2,2}} [r]^{\\beta } & \\dots [r]^{\\beta }& X_{r_{2,L}} [r]^{\\beta } & X_{r_{2,L+1}} \\\\}$ which commutes.",
"If follows that $X_{r_{2, L+1}} = Y_{r_{2, 1}}$ , that is, $L = L^{\\prime }$ .",
"The proof now follows from induction.",
"4.18 4.18 Theorem: Suppose $(X, \\varphi )$ is an irreducible Smale space with an effective $\\mathbb {Z}$ -action.",
"Then the induced action on $ {\\mathrm {C}}^*(H)$ has finite Rokhlin dimension.",
"Proof.",
"Let $\\beta : X \\rightarrow X$ be the homeomorphism generating the $\\mathbb {Z}$ action and let $X = X_1 \\sqcup \\cdots \\sqcup X_N$ be the Smale decomposition into mixing components $(X_i, \\varphi ^N|_{X_i})$ .",
"Denote by $\\mathrm {C^*}(H_i)$ the homoclinic algebra for the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"By Lemmas REF and REF , there exists $L$ such that $ \\beta ^L : X_i \\rightarrow X_i $ for every $i \\in \\lbrace 1, \\dots , N\\rbrace $ .",
"For each $i$ , we consider the action $\\beta ^L|_{X_i}$ .",
"Since $\\beta $ induces an effective ${\\mathbb {Z}}$ -action on $(X, \\varphi )$ (in particular $\\beta ^L \\circ \\varphi ^N= \\varphi ^N\\circ \\beta ^L$ ) it follows that $\\beta ^L|_{X_i}$ induces an effective ${\\mathbb {Z}}$ -action on the mixing Smale space $(X_i, \\varphi ^N|_{X_i})$ .",
"Hence, we can apply Theorem REF to $\\beta ^L|_{X_i}$ acting on $(X_i, \\varphi ^N|_{X_i})$ to conclude that the action induced by $\\beta ^L|_{X_i}$ on $\\mathrm {C}^*(H_i)$ has Rokhlin dimension $d_i$ for some $d_i < \\infty $ .",
"Let $\\epsilon > 0$ , $p \\in \\mathbb {N} \\setminus \\lbrace 0\\rbrace $ .",
"Let $f^{(k)}_{i, j}$ , $i\\in \\lbrace 0,1\\rbrace $ , $0 \\le j \\le p-1+i$ , $0 \\le k \\le d_1$ be two Rokhlin towers for $\\beta ^L|_{X_1}$ with respect to $\\epsilon $ and $p$ and any finite subset $\\mathcal {F}_1 \\subset \\mathrm {C}^*(H_1)$ .",
"We claim that the elements $\\beta ^l(f^{(k)}_{i, j})$ , $1 \\le l \\le L-1$ satisfy (i) – (v) of Lemma REF with respect to $ {\\mathrm {C}}^*(H), \\epsilon $ and $p$ .",
"Indeed, (i), (iii) and (iv) are clear by construction.",
"To see (ii), we note that $\\Vert \\sum _{i,j,k} f^{(k)}_{i, j} - 1_{\\mathrm {C}^*(H_1)}\\Vert < \\epsilon $ and since $\\beta $ is a homeomorphism, it follows that $\\Vert \\sum _{i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{\\mathrm {C}^*(H_l)})\\Vert < \\epsilon $ for every $l$ , and hence $\\Vert \\sum _{l,i,j,k} \\beta ^l(f^{(k)}_{i, j}) - 1_{ {\\mathrm {C}}^*(H)}\\Vert < \\epsilon .$ 4.19 4.19 Remark: Let $Y$ be a compact Hausdorff space.",
"It is not difficult to check that if a $\\mathbb {Z}$ -action on $C(Y)$ has finite Rokhlin dimension, then the action on $Y$ must be free.",
"Thus when ${\\mathbb {Z}}$ acts on an irreducible Smale space $(X, \\varphi )$ , the action of ${\\mathbb {Z}}$ on $C(X)$ cannot have finite Rokhlin dimension because the action of ${\\mathbb {Z}}$ on $X$ is not free by Proposition REF .",
"Nevertheless, Proposition REF implies that a ${\\mathbb {Z}}$ -action on $ {\\mathrm {C}}^*(H)$ induced from an effective action on $(X, \\varphi )$ has Rokhlin dimension at most one.",
"The induced action on the stable and unstable algebras In this section we use what we have already proved about actions on the homoclinic algebra to deduce results for actions on the stable and unstable algebras.",
"To do so, we take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"It is easy to check that if $(X, \\varphi )$ is a Smale space, then $(X, \\varphi ^{-1})$ is also a Smale space with the bracket reversed.",
"The unstable relation of $(X, \\varphi )$ is then the stable relation of $(X, \\varphi ^{-1})$ .",
"Thus, it is enough to show something holds for $ \\mathrm {C}^*(S)$ of an arbitrary (irreducible, mixing) Smale space to imply the same for $ \\mathrm {C}^*(U)$ of an arbitrary (irreducible, mixing) Smale space.",
"Throughout this section, we again assume that $(X, \\varphi )$ is irreducible, but is not necessarily mixing unless explicitly stated.",
"5.1 5.1 Let $(X, \\varphi )$ be an irreducible Smale space.",
"Fix a finite set $P$ of $\\varphi $ -invariant periodic points.",
"Let $ {\\mathrm {C}}^*(H)$ denote the associated homoclinic and $ \\mathrm {C}^*(S)$ the stable algebra.",
"As in [51], we define, for each $a \\in C_c(\\mathcal {G}_H)$ , an element $(\\rho (a), \\rho (a))$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ by $ (\\rho (a) b)(x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} a(x,z)b(z,y) $ and $ (b \\rho (a)) (x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} b(x,z) a(z,y)$ where $b\\in C_c(S)$ .",
"This extends to a map $\\rho : {\\mathrm {C}}^*(H)\\operatorname{\\hookrightarrow }\\mathcal {M}( \\mathrm {C}^*(S)).$ We note that $\\rho $ and the representations of these algebras on the Hilbert space $l^2(X^H(P,Q))$ (see Remark REF ) are compatible.",
"5.2 5.2 Lemma: Let $F \\subset {C}_c(S)$ be a finite subset and let $r_1, \\dots , r_N \\in {C}_c(\\mathcal {G}_H)$ .",
"Then, for every $\\epsilon > 0$ there exists a $k\\in \\mathbb {N}$ such that, viewing $a$ as an element of the multiplier algebra $\\mathcal {M}( \\mathrm {C}^*(S))$ , we have $ \\Vert \\rho (\\alpha ^k(r_i) ) a - a \\rho (\\alpha ^k(r_i )) \\Vert < \\epsilon $ for every $i = 1, \\dots , N$ and every $a \\in F$ .",
"Proof.",
"Follows from [51] or [31].",
"5.3 5.3 Definition: Suppose $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is an action of a countable discrete group.",
"Let $I$ be a separable $G$ -invariant ideal in $A$ and $B$ be a $\\sigma $ -unital $G$ -$\\mathrm {C}^*$ -subalgebra of $A$ .",
"Then there exists a countable $G$ -quasi-invariant quasicentral approximate unit $(w_n)_{n\\in N}$ of $I$ in $B$ .",
"That is, there exists $(w_n)_{n\\in {\\mathbb {N}}}$ an approximate identity for $I$ such that (i) for any $a \\in B$ , $\\Vert aw_n - w_n a\\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ ; for each $g\\in G$ , $\\Vert \\beta _g(w_n) -w_n \\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"The existence of a $G$ -quasi-invariant quasicentral approximate unit is shown in [29], but also see [17] and [13].",
"Rokhlin dimension for actions on the stable and unstable algebra To take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , we will find it convenient to introduce the definition of multiplier Rokhlin dimension with repect to a finite index subgroup.",
"It is inspired by [47], [23] and [64].",
"For a given $\\mathrm {C}^*$ -algebra $A$ and group $G$ with action $\\beta : G \\rightarrow {\\rm Aut}(A)$ we denote by $\\mathcal {M}(\\beta )$ the induced action of $G$ on the multiplier algebra $\\mathcal {M}(A)$ .",
"5.4 5.4 Definition: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"We say the action has multilplier Rokhlin dimension $d$ with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K)$ , if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and finite subsets $M \\subset G$ , $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_{\\overline{g}} \\right)_{l=0, \\ldots d; \\overline{g}\\in G/K} \\subset \\mathcal {M}(A) $ such that (i) for any $l$ , $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert < \\epsilon $ , for $\\overline{g} \\ne \\overline{h}$ in $G/K$ ; $\\Vert \\sum _{l=0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert < \\epsilon $ ; $\\Vert \\mathcal {M}(\\beta _h)(f^{(l)}_{\\overline{g}}) - f^{(l)}_{\\overline{hg}} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $h \\in M$ ; $\\Vert [ f^{( l )}_{\\overline{g}}, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $a \\in F$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) = \\infty $ .",
"We say the action has multilplier Rokhlin dimension $d$ with commuting towers with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K)$ , if, in addition, (i) $\\Vert f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}}- f^{(k)}_{\\overline{h}}f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for every $k,l = 0, \\dots , d$ , $\\overline{g}, \\overline{h} \\in G/K$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) = \\infty $ .",
"The multiplier Rokhlin dimension with respect to a finite subgroup is used to define the Rokhlin dimension for an action of a countable residually finite group.",
"5.5 5.5 Definition: [cf.",
"[64]] Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable, residually finite group, and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ .",
"The Rokhlin dimension of $\\beta $ is defined by $ \\mathrm {dim}_{\\mathrm {Rok}}(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ The Rokhlin dimension of $\\beta $ with commuting towers is given by $ \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ It will be easier to show finite multiplier Rokhlin dimension relative to a finite subgroup, which we show implies the definition of Rokhlin dimension relative to a finite index subgroup for $\\mathrm {C}^*$ -algebras given in [64], recalled in Definition REF below.",
"Some further notation is required to do so.",
"First, we need to define central sequence algebras.",
"Loosely speaking, working in a central sequence algebra allows one to turn statements such as approximate commutativity in the original algebra into honest commutativity in the central sequence algebra.",
"As such, central sequence arguments often allow one to streamline proofs.",
"In the case of discrete groups, an action on a $\\mathrm {C}^*$ -algebra induces an action on its central sequence algebra, and it is possible to reformulate definitions for both Rokhlin dimension of finite group actions and integer actions on separable unital $\\mathrm {C}^*$ -algebras in terms of induced actions on central sequence algebras [64].",
"5.6 5.6 Definition: Let $A$ be a separable $\\mathrm {C}^*$ -algebra.",
"We denote the sequence algebra of $A$ by $ A_{\\infty } := \\prod _{n \\in \\mathbb {N}} A / \\bigoplus _{n \\in \\mathbb {N}} A.",
"$ We view $A$ as a subalgebra of $A_{\\infty }$ by mapping an element $a \\in A$ to the constant sequence consisting of $a$ in every entry.",
"The central sequence algebra is then defined to be $ A^{\\infty } := A_{\\infty } \\cap A^{\\prime } = \\lbrace x \\in A_{\\infty } \\mid ax = xa \\text{ for every } a \\in A\\rbrace , $ the relative commutant of $A$ in $A_{\\infty }$ .",
"Let $ \\mathrm {Ann}(A, A_{\\infty }) := \\lbrace x \\in A_{\\infty } \\mid ax = xa = 0 \\text{ for every } a \\in A\\rbrace ,$ which is evidently an ideal in $A^{\\infty }$ .",
"Finally, we define $ F(A) := A^{\\infty }/ \\mathrm {Ann}(A, A_{\\infty }).",
"$ When $A$ is not separable, one can define the above with respect to a given separable subalgebra $D$ , as is done in [64].",
"However, since all our $\\mathrm {C}^*$ -algebras will be separable, we will not require this.",
"A completely positive contractive (c.p.c.)",
"map $\\varphi : A \\rightarrow B$ between $\\mathrm {C}^*$ -algebras $A$ and $B$ is said to be order zero if it is orthogonality preserving, that is, for every $a, b \\in A_+$ with $ab = ba =0$ we have $\\varphi (a) \\varphi (b) = 0$ .",
"Any $^*$ -homomorphism is of course order zero, but a c.p.c.",
"order zero map is not in general a $^*$ -homomorphism.",
"For more about c.p.c.",
"order zero maps, see [73].",
"5.7 5.7 Definition: [64] Let $A$ be a separable $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"Let $\\tilde{a}_{\\infty }$ denote the induced action on $F_{\\infty }(A)$ .",
"We say the action has Rokhlin dimension $d$ with respect to $K$ if $d$ is the least integer such that there exists equivariant c.p.c.",
"order zero maps $ \\varphi _l : (C(G/K), G\\text{-shift}) \\rightarrow (F_{\\infty }(A), \\tilde{a}_{\\infty }), \\quad l = 0, \\dots , d$ with $ \\varphi _0(1) + \\cdots + \\varphi _d(1) = 1.",
"$ If moreover $\\varphi _0 ,\\dots , \\varphi _d$ can be chosen to have commuting ranges, then we say the action has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"The next result is an obvious generalisation of the equivalence of (1) and (3) of [64] to the case of commuting towers.",
"5.8 5.8 Lemma: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group and $K$ a subgroup of finite index.",
"The following are equivalent for an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ .",
"(i) The action $\\beta $ has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"For every finite subset $M \\subset G$ , finite subset $F \\subset A$ and $\\epsilon >0$ there are positive contractions $(f^{(l)}_{\\overline{g}})_{\\overline{g} \\subset H}^{l= 0, \\dots , d}$ in $A$ such that (i) $\\Vert (\\sum _{l =0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}}) \\cdot a - a \\Vert < \\epsilon $ for all $a \\in F$ ; $\\Vert f_{\\overline{g}}^{(l)}f_{\\overline{h}}^{(l)}a \\Vert \\le \\epsilon $ for all $a\\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\ne \\overline{h} \\in G/K$ ; $\\Vert (\\beta _g(f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}})a \\Vert < \\epsilon $ for all $a \\in F$ , $l \\in 0, \\dots , d$ and $g \\in M$ and $\\overline{h} \\in G/K$ ; $\\Vert f^{(l)}_{\\overline{g}}a - a f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for all $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ ; $\\Vert (f^{(k)}_{\\overline{g}} f^{(l)}_{\\overline{h}} - f^{(l)}_{\\overline{h}} f^{(k)}_{\\overline{g}})a \\Vert < \\epsilon $ for all $a \\in F$ , $k,l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g}, \\overline{h} \\in G/K$ .",
"Proof.",
"The only thing that one needs to check is that asking for the $f^{(l)}_{\\overline{g}}$ to approximately commute is equivalent to having the images of the order zero maps of [64] commute, but this is obvious.",
"5.9 5.9 Theorem: Let $G$ be a countable discrete group, $K$ a subgroup of $G$ with finite index, $A$ a separable $\\mathrm {C}^*$ -algebra and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action with multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ respect to $K$ .",
"If $\\beta $ has multiplier Rokhlin dimension at most $d$ with commuting towers with respect to $K$ , then $\\beta $ has Rokhlin dimension at most $d$ with commuting towers respect to $K$ .",
"Proof.",
"We show the action satisfies the criteria of [64] (and in the commuting tower case Lemma REF (ii)).",
"Note that [64] is exactly (a)-(d) in Lemma REF (ii).",
"Let $M \\subset G$ and $F \\subset A$ be finite subsets and let $\\epsilon > 0$ .",
"Without loss of generality we may assume that every $a \\in F$ is a positive contraction.",
"Since $\\beta $ has multiplier Rokhlin dimension less than or equal to $d$ with respect to $K$ we can find positive contractions $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ satisfying Definition REF with respect to $M$ , $F$ and $\\epsilon /2$ .",
"Let $(w_n)_{n \\in \\mathbb {N}}$ be an $G$ -quasi-invariant quasicentral approximate unit for $A$ in $\\mathcal {M}(A)$ .",
"Since $F \\subset A$ and $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ are in $\\mathcal {M}(A)$ , there exists $N \\in \\mathbb {N}$ sufficiently large so that $ \\Vert w_N a - a \\Vert < \\epsilon /4 \\text{ for every } a \\in F,$ and $ \\Vert [f_{\\overline{g}}^{(l)}, w_N] \\Vert < \\epsilon /8.$ and $ \\Vert \\beta _g(f_{\\overline{h}}^{(l)}) - f_{\\overline{gh}}^{(l)} \\Vert < \\epsilon /8.$ By increasing $N$ if necessary, we may also assume, since $M$ is finite, that $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ .",
"Then let $ r_{\\overline{g}}^{(l)} := w_N f_{\\overline{g}}^{(l)} w_N,$ for $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $g \\in G$ .",
"Then each $r_{\\overline{g}}^{(l)}$ is a positive contraction in $A$ and $\\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}}) a - a\\Vert &=& \\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N) a - a\\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N - 1 \\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert \\\\&<& \\epsilon ,$ showing that (a) of Lemma REF (ii) holds.",
"Next, $\\Vert r_{\\overline{g}}^{(l)} r_{\\overline{h}}^{(l)} a \\Vert &=& \\Vert w_N f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(l)} w_N a \\Vert \\\\&=& \\epsilon /2 + \\Vert w_N^2 f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(l)} w_N^2 a \\Vert \\\\&<& \\epsilon $ for every $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , showing that (b) of Lemma REF holds.",
"Using $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ , we obtain $\\Vert (\\beta _g(r_{\\overline{h}}^{(l)}) - r_{\\overline{gh}}^{(l)})a\\Vert &=& \\epsilon /4 + \\Vert w_N \\mathcal {M}(\\beta _g)(f^{(l)}_{\\overline{h}}) w_N - w_N f^{(l)}_{\\overline{gh}} w_N \\Vert \\\\&<& \\epsilon ,$ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ , every $\\overline{h} \\in G/K$ , every $g \\in M$ and every $a \\in F$ , showing (c) of Lemma REF .",
"For (d) of Lemma REF we have $\\Vert r_{\\overline{g}}^{(l)} a - a r_{\\overline{g}}^{(l)} \\Vert = \\epsilon / 2 + \\Vert f_{\\overline{g}}^{(l)} a - a f_{\\overline{g}}^{(l)} \\Vert < \\epsilon ,$ for every $a \\in F$ , every $l \\in \\lbrace 0, \\dots , d\\rbrace $ and every $\\overline{g} \\in G/K$ .",
"Finally, in the commuting tower case, we must show (e): $\\Vert (r_{\\overline{g}}^{(k)} r_{\\overline{h}}^{(l)} - r_{\\overline{h}}^{(l)} r_{\\overline{g}}^{(k))} a\\Vert &\\le & \\Vert f_{\\overline{g}}^{(k)} w_N^2 f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(k)}\\Vert \\\\&\\le & \\epsilon /4 + \\Vert f_{\\overline{g}}^{(k)} f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(k)}\\Vert \\\\&<& \\epsilon ,$ for every $\\overline{g}, \\overline{h} \\in G/K$ and $k, l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"The proof of the next lemma is obvious and hence omitted.",
"It will, however, prove useful in what follows.",
"5.10 5.10 Lemma: Suppose that the action of $G$ on $ {\\mathrm {C}}^*(H)$ has Rokhlin dimension at most $d$ .",
"Then we may choose the Rokhlin elements to satisfy $f^{(l)}_g \\in C_c(\\mathcal {G}_H)$ for $l = 0, \\dots , d$ and $ g \\in G$ .",
"5.11 5.11 Proposition: Let $G$ be a countable group acting on an irreducible Smale space $(X, \\varphi )$ and let $K \\subset G$ be a subset of finite index.",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ with respect to $K$ so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"If $\\beta $ has Rokhlin dimension at most $d$ with commuting towers with respect to $K$ so does the action $\\beta ^{(S)}$ .",
"Proof.",
"We will show that $\\beta ^{(S)}$ has multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"The result then follows from Theorem REF .",
"Let $M$ be a finite subset of $G$ , $F$ a finite subset of $ \\mathrm {C}^*(S)$ and $\\epsilon >0$ .",
"Without loss of generality, we may assume that $F \\subset C_c(S)$ .",
"Since $\\beta $ has Rokhlin dimension at most $d$ with respect to $K$ there are contractions $r^{(l)}_g \\in {\\mathrm {C}}^*(H)$ such that (i) $\\Vert r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}} \\Vert < \\epsilon /2$ for $l = 0, \\dots , d$ and any $g, h \\in G$ with $g \\ne h$ , $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1\\Vert < \\epsilon /2,$ $\\Vert \\beta _h(r^{(l)}_{\\overline{g}}) - r^{(l)}_{\\overline{hg}}\\Vert < \\epsilon /2$ for $l =0, \\dots , d$ , every $h \\in M$ and $\\overline{g} \\in G/K$ , and $\\Vert r^{(k)}_{\\overline{g}}r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}}r^{(k)}_{\\overline{g}}\\Vert < \\epsilon /2$ for every $\\overline{g}$ , $\\overline{h} \\in G/K$ and $k,l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"By the previous lemma, we may moreover assume that each $r^{(l)}_{\\overline{g}} \\in C_c(\\mathcal {G}_H)$ .",
"Now we can find a natural number $k$ such that $ \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert < \\epsilon /2,$ for every $a \\in F$ .",
"For $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , let $ f^{(l)}_{\\overline{g}} := \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}} )).$ Note that each $f^{(l)}_g$ is a positive contraction in $\\mathcal {M}( \\mathrm {C}^*(S))$ .",
"We will show that the $f^{(l)}_{\\overline{g}}$ satisfy (i) – (iv) of Definition REF .",
"For (i) we have $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} )) \\Vert \\\\&=& \\Vert \\alpha ^k(r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) \\Vert \\\\&<& \\epsilon ,$ for any $\\overline{g} \\ne \\overline{h} \\in G/K$ , any $l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"For (ii) $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f_{\\overline{g}}^{(l)} - 1_{\\mathcal {M}( \\mathrm {C}^*(S))} \\Vert &=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g}\\in G} \\rho (\\alpha ^k( r^{(l)}_{\\overline{g}} )) - \\rho (1_{ {\\mathrm {C}}^*(H)})\\Vert \\\\&=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1_{ {\\mathrm {C}}^*(H)} \\Vert \\\\&<& \\epsilon ,$ for any $a \\in F$ .",
"For (iii) we have $\\Vert f^{(l)}_{\\overline{g}} a - a f^{(l)}_{\\overline{g}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert \\\\&<& \\epsilon /2,$ for all $l \\in \\lbrace 0, \\dots d\\rbrace , \\overline{g} \\in G/K$ and $a \\in F$ .",
"For (iv), let $g \\in G$ with $\\overline{h} \\in G/K $ and let $a \\in F$ .",
"Then, $\\Vert \\beta ^{(S)}_g (f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}} \\Vert &=& \\Vert \\alpha ^k(\\beta _g(r^{(l)}_{\\overline{h}}) - r^{(l)}_{\\overline{gh}}) \\Vert \\\\&<& \\epsilon .$ Finally, in the commuting towers case, for (v), let $\\overline{g}, \\overline{h} \\in G/K$ and $l,k \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert f^{(k)}_{\\overline{h}} f^{(l)}_{\\overline{g}} - f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) - \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}) \\Vert \\\\&=& \\Vert r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}\\Vert \\\\&<& \\epsilon .$ The result now follows.",
"This gives us the next corollary: 5.12 5.12 Corollary: Let $G$ be a countable residually finite group acting on an irreducible Smale space $(X, \\varphi )$ .",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ (with commuting towers) so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"By Proposition REF and Theorem REF respectively, we get the next two corollaries.",
"5.13 5.13 Corollary: Let $d\\in {\\mathbb {N}}$ and $G={\\mathbb {Z}}^d$ .",
"Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"5.14 5.14 Corollary: Let $\\mathbb {Z}$ be an effective action on an irreducible Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"For more general group actions the situation is less clear.",
"In particular, to the authors' knowledge, it is not known if strong outerness implies finite Rokhlin dimensionAt the workshop “Future Targets in the Classification Program for Amenable $\\mathrm {C}^*$ -Algebras” held at BIRS, Eusebio Gardella presented results in this direction, but they have yet to appear..",
"In fact, for general discrete groups, there are obstructions to a (strongly outer) action having finite Rokhlin dimension with commuting towers, see [21].",
"Such examples can occur in the context considered in the present paper.",
"An explicit example is the following, let $(\\Sigma _{[3]}, \\sigma )$ be the full three shift (so $\\Sigma _{[3]}= \\lbrace 0, 1, 2\\rbrace ^{{\\mathbb {Z}}}$ and $\\sigma $ is the left sided shift).",
"Then $ {\\mathrm {C}}^*(H)$ is the UHF-algebra with supernatural number $3^{\\infty }$ .",
"The action induced from the permutation $0 \\mapsto 1$ , $1 \\mapsto 0$ , and $2 \\mapsto 2$ is an effective order two automorphism.",
"Thus the action induced on $ {\\mathrm {C}}^*(H)$ is strongly outer, but it follows from [21] that it does not have finite Rokhlin dimension with commuting towers.",
"$\\mathcal {Z}$ -stability, nuclear dimension and classification We begin this section with two theorems that follow quickly from the work done above.",
"As well as being interesting observations on their own, they will allow us to say something about the $\\mathcal {Z}$ -stability of the crossed products of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ below.",
"6.1 6.1 Theorem: Let $G$ be a countable discrete amenable group.",
"Suppose $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action of $G$ on $ {\\mathrm {C}}^*(H)$ .",
"Then $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable.",
"Proof.",
"Since $(X, \\varphi )$ is mixing, $ {\\mathrm {C}}^*(H)$ is simple and so the classification results of [12] imply that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace, it must be fixed by the action.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable by [58].",
"6.2 6.2 Theorem: Let $G$ be a countable amenable group acting effectively on a mixing Smale space $(X, \\varphi )$ and $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Then the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is a simple unital nuclear $\\mathcal {Z}$ -stable $\\mathrm {C}^*$ -algebra with unique tracial state and nuclear dimension (in fact, decomposition rank) at most one.",
"In particular, it belongs to the class of $\\mathrm {C}^*$ -algebras that are classified by the Elliott invariant and $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF, for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"Proof.",
"Since $ {\\mathrm {C}}^*(H)$ is amenable, it follows from [53] that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G \\cong \\mathrm {C}^*(\\mathcal {G}_H \\rtimes G)$ is also amenable and hence by [69] it satisfies the UCT.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is quasidiagonal [67] and by [67] together with Theorem classified by the Elliott invariant.",
"The nuclear dimension and decomposition rank bounds are given by [5].",
"Finally, by [42] $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF.",
"$\\mathcal {Z}$ -stability of crossed products $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ Let $A$ be separable $\\mathrm {C}^*$ -algebra and $G$ a discrete group.",
"For an action $\\beta : G\\rightarrow \\operatorname{Aut}(A)$ the fixed point algebra is given by $ A^{\\beta } := \\lbrace a \\in A \\mid \\beta _g(a) = a \\text{ for all } g \\in G\\rbrace .",
"$ Let $ A^{\\infty } := \\ell ^{\\infty }(\\mathbb {N}, A) / c_0(A) .$ The central sequence algebra of $A$ is defined by $ A_{\\infty } := A^{\\infty } \\cap A^{\\prime },$ where $A$ is considered as the subalgebra of $A^{\\infty }$ by viewing an element as a constant sequence.",
"We will denote by $\\overline{\\beta }$ the induced action on $A_{\\infty }$ .",
"Let $p$ and $q$ be positive integers.",
"The dimension drop algebra $I(p,q)$ is given by $I(p, q) := \\lbrace f \\in C([0,1], M_p( \\otimes M_q( \\mid f(0) \\in M_p( \\otimes { and }f(1)\\in M_q(\\rbrace .$ The Jiang–Su algebra, $\\mathcal {Z}$ , is an inductive limit of such algebras [28].",
"6.3 6.3 Theorem: Suppose $G$ is a countable discrete group acting on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"If, for each $k\\in {\\mathbb {N}}$ , there exists a unital equivariant embedding $I(k, k+1) \\rightarrow {\\mathrm {C}}^*(H)$ , then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"As usual, it suffices to prove the result for $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ .",
"To do so, we will show that the hypotheses of Lemma 2.6 in [21] hold.",
"We will show that, for every $k \\in \\mathbb {N}$ , there is a completely positive contractive map $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty } $ satisfying (i) $ (\\overline{\\beta }^S( \\gamma (x)) - \\gamma (x))a = $ for every $x \\in I(k, k+1)$ , $g \\in G$ and $a \\in \\mathrm {C}^*(S)$ , $a \\gamma (1) = a$ for every $a \\in \\mathrm {C}^*(S)$ , and $a(\\gamma (xy) - \\gamma (x)\\gamma (y)) = 0$ for every $x, y \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Suppose $F \\subset I(k, j+1)$ a finite subset and $\\epsilon > 0$ are given.",
"Let $w_n$ be a $G$ -invariant quasicentral approximate unit for $ \\mathrm {C}^*(S)$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ and $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ be a unital embedding (which exists by assumption).",
"Define $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)^{\\infty } $ via $ \\gamma (d) = (w_n \\rho (\\alpha ^n(d_n)) w_n)_{n \\in \\mathbb {N}}, $ where $(d_n)_{n \\in \\mathbb {N}}$ is a representative sequence for $\\tilde{\\gamma }(d)$ .",
"Then $\\gamma $ gives a c.p.c.",
"map.",
"Moreover, if $a \\in \\mathrm {C}^*(S)$ we have $ \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\alpha ^n(d_n) )w_n a - a w_n \\rho (\\alpha ^n(d_n)) w_n \\Vert = 0,$ so in fact $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty }.",
"$ Let us check that $\\gamma $ satisfies (i), (ii) and (iii).",
"Let $a \\in A$ and for any $d \\in I(k, k+1)$ , let $(d_n)_{n \\in \\mathbb {N}}$ be a representative of $\\tilde{\\gamma }(d)$ in $( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }$ .",
"$\\Vert \\overline{\\beta }^S(\\gamma (d) - \\gamma (d)) a \\Vert &=& \\lim _{n \\rightarrow \\infty } \\Vert \\beta ^{(S)}(w_n \\rho (\\alpha ^n(d_n)) w_n) - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&=& \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\beta (\\alpha ^n(d_n)) w_n - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&\\le & \\lim _{n \\rightarrow \\infty } \\Vert \\beta (\\alpha ^n(d_n) ) - \\alpha ^n(d_n) \\Vert \\\\&=& 0,$ showing $(i)$ .",
"To show (ii), we have $a \\gamma (1) &=& (a w_n \\rho (\\tilde{\\gamma }(1)) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a \\rho (1) w_n^2)_{n \\in \\mathbb {N}} \\\\&=& a,$ for every $a \\in \\mathrm {C}^*(S)$ Finally, for (iii), let $d, d^{\\prime } \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Then $a(\\gamma (d d^{\\prime }) - \\gamma (d)\\gamma (d^{\\prime })) &=& (a w_n \\rho (d_n d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}} - (a w_n \\rho (d_n) w_n^2 \\rho (d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a w_n^2 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a w_n^4 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} \\\\&=& (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}}\\\\&=& 0.$ Thus $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ is $\\mathcal {Z}$ -stable.",
"6.4 6.4 Corollary: Suppose $G$ is a discrete amenable group and $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"Then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"By Theorem REF (v), $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable and hence has strict comparison [54].",
"This in turn implies $ {\\mathrm {C}}^*(H)$ has property (TI) of Sato [58].",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace $\\tau $ which is therefore fixed by $\\beta $ , the hypotheses of [58] are satisfied.",
"Thus from the proof of [58] we get a unital embedding $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ The result then follows from the previous theorem.",
"Acknowledgments.",
"The authors thank Ian Putnam for many useful discussions concerning the content of this paper, Smale spaces, group actions and dynamics in general.",
"We also thank Magnus Goffeng for a number of useful comments.",
"The authors thank the referee for reading the paper carefully and making a number of useful comments.",
"The authors wish to thank Matrix at the University of Melbourne for hosting them during the programme Refining $\\mathrm {C}^*$ -algebraic Invariants for Dynamics using KK-theory in July 2016, the Banach Centre at the Institute of Mathematics of the Polish Academy of Sciences for hosting the first listed author during the conference Index Theory in October 2016, the University of Hawaii, Manoa for hosting the second listed author during the workshop Computability of K-theory in November 2016 and the Centre Rercerca Matemàtica, Barcelona, for their stay during the Intensive Research on Operator Algebras: Dynamics and Interactions in July 2017.",
"The above research visits were partially supported through NSF grants DMS 1564281 and DMS 1665118."
],
[
"The induced action on the stable and unstable algebras",
"In this section we use what we have already proved about actions on the homoclinic algebra to deduce results for actions on the stable and unstable algebras.",
"To do so, we take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ .",
"It is easy to check that if $(X, \\varphi )$ is a Smale space, then $(X, \\varphi ^{-1})$ is also a Smale space with the bracket reversed.",
"The unstable relation of $(X, \\varphi )$ is then the stable relation of $(X, \\varphi ^{-1})$ .",
"Thus, it is enough to show something holds for $ \\mathrm {C}^*(S)$ of an arbitrary (irreducible, mixing) Smale space to imply the same for $ \\mathrm {C}^*(U)$ of an arbitrary (irreducible, mixing) Smale space.",
"Throughout this section, we again assume that $(X, \\varphi )$ is irreducible, but is not necessarily mixing unless explicitly stated.",
"5.1 5.1 Let $(X, \\varphi )$ be an irreducible Smale space.",
"Fix a finite set $P$ of $\\varphi $ -invariant periodic points.",
"Let $ {\\mathrm {C}}^*(H)$ denote the associated homoclinic and $ \\mathrm {C}^*(S)$ the stable algebra.",
"As in [51], we define, for each $a \\in C_c(\\mathcal {G}_H)$ , an element $(\\rho (a), \\rho (a))$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ by $ (\\rho (a) b)(x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} a(x,z)b(z,y) $ and $ (b \\rho (a)) (x,y) = \\sum _{z \\in X^U(P), \\ z \\sim _s x} b(x,z) a(z,y)$ where $b\\in C_c(S)$ .",
"This extends to a map $\\rho : {\\mathrm {C}}^*(H)\\operatorname{\\hookrightarrow }\\mathcal {M}( \\mathrm {C}^*(S)).$ We note that $\\rho $ and the representations of these algebras on the Hilbert space $l^2(X^H(P,Q))$ (see Remark REF ) are compatible.",
"5.2 5.2 Lemma: Let $F \\subset {C}_c(S)$ be a finite subset and let $r_1, \\dots , r_N \\in {C}_c(\\mathcal {G}_H)$ .",
"Then, for every $\\epsilon > 0$ there exists a $k\\in \\mathbb {N}$ such that, viewing $a$ as an element of the multiplier algebra $\\mathcal {M}( \\mathrm {C}^*(S))$ , we have $ \\Vert \\rho (\\alpha ^k(r_i) ) a - a \\rho (\\alpha ^k(r_i )) \\Vert < \\epsilon $ for every $i = 1, \\dots , N$ and every $a \\in F$ .",
"Proof.",
"Follows from [51] or [31].",
"5.3 5.3 Definition: Suppose $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ is an action of a countable discrete group.",
"Let $I$ be a separable $G$ -invariant ideal in $A$ and $B$ be a $\\sigma $ -unital $G$ -$\\mathrm {C}^*$ -subalgebra of $A$ .",
"Then there exists a countable $G$ -quasi-invariant quasicentral approximate unit $(w_n)_{n\\in N}$ of $I$ in $B$ .",
"That is, there exists $(w_n)_{n\\in {\\mathbb {N}}}$ an approximate identity for $I$ such that (i) for any $a \\in B$ , $\\Vert aw_n - w_n a\\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ ; for each $g\\in G$ , $\\Vert \\beta _g(w_n) -w_n \\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"The existence of a $G$ -quasi-invariant quasicentral approximate unit is shown in [29], but also see [17] and [13].",
"Rokhlin dimension for actions on the stable and unstable algebra To take advantage of the embedding of the homoclinic algebra into the multiplier algebras of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , we will find it convenient to introduce the definition of multiplier Rokhlin dimension with repect to a finite index subgroup.",
"It is inspired by [47], [23] and [64].",
"For a given $\\mathrm {C}^*$ -algebra $A$ and group $G$ with action $\\beta : G \\rightarrow {\\rm Aut}(A)$ we denote by $\\mathcal {M}(\\beta )$ the induced action of $G$ on the multiplier algebra $\\mathcal {M}(A)$ .",
"5.4 5.4 Definition: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"We say the action has multilplier Rokhlin dimension $d$ with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K)$ , if $d$ is the least integer such that the following holds: for each $\\epsilon >0$ and finite subsets $M \\subset G$ , $F \\subset A$ there are positive contractions $ \\left( f^{(l)}_{\\overline{g}} \\right)_{l=0, \\ldots d; \\overline{g}\\in G/K} \\subset \\mathcal {M}(A) $ such that (i) for any $l$ , $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert < \\epsilon $ , for $\\overline{g} \\ne \\overline{h}$ in $G/K$ ; $\\Vert \\sum _{l=0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert < \\epsilon $ ; $\\Vert \\mathcal {M}(\\beta _h)(f^{(l)}_{\\overline{g}}) - f^{(l)}_{\\overline{hg}} \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $h \\in M$ ; $\\Vert [ f^{( l )}_{\\overline{g}}, a] \\Vert < \\epsilon $ for all $l \\in \\lbrace 0, \\ldots , d\\rbrace $ , $\\overline{g} \\in G/K$ and $a \\in F$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) = \\infty $ .",
"We say the action has multilplier Rokhlin dimension $d$ with commuting towers with respect to $K$ , denoted $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K)$ , if, in addition, (i) $\\Vert f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}}- f^{(k)}_{\\overline{h}}f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for every $k,l = 0, \\dots , d$ , $\\overline{g}, \\overline{h} \\in G/K$ .",
"If no such $d$ exists, then we write $\\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) = \\infty $ .",
"The multiplier Rokhlin dimension with respect to a finite subgroup is used to define the Rokhlin dimension for an action of a countable residually finite group.",
"5.5 5.5 Definition: [cf.",
"[64]] Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable, residually finite group, and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ .",
"The Rokhlin dimension of $\\beta $ is defined by $ \\mathrm {dim}_{\\mathrm {Rok}}(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ The Rokhlin dimension of $\\beta $ with commuting towers is given by $ \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta ) := \\sup \\lbrace \\mathrm {dim}_{\\mathrm {Rok}}^c(\\beta , K) \\mid K \\le G, [G:H] < \\infty \\rbrace .$ It will be easier to show finite multiplier Rokhlin dimension relative to a finite subgroup, which we show implies the definition of Rokhlin dimension relative to a finite index subgroup for $\\mathrm {C}^*$ -algebras given in [64], recalled in Definition REF below.",
"Some further notation is required to do so.",
"First, we need to define central sequence algebras.",
"Loosely speaking, working in a central sequence algebra allows one to turn statements such as approximate commutativity in the original algebra into honest commutativity in the central sequence algebra.",
"As such, central sequence arguments often allow one to streamline proofs.",
"In the case of discrete groups, an action on a $\\mathrm {C}^*$ -algebra induces an action on its central sequence algebra, and it is possible to reformulate definitions for both Rokhlin dimension of finite group actions and integer actions on separable unital $\\mathrm {C}^*$ -algebras in terms of induced actions on central sequence algebras [64].",
"5.6 5.6 Definition: Let $A$ be a separable $\\mathrm {C}^*$ -algebra.",
"We denote the sequence algebra of $A$ by $ A_{\\infty } := \\prod _{n \\in \\mathbb {N}} A / \\bigoplus _{n \\in \\mathbb {N}} A.",
"$ We view $A$ as a subalgebra of $A_{\\infty }$ by mapping an element $a \\in A$ to the constant sequence consisting of $a$ in every entry.",
"The central sequence algebra is then defined to be $ A^{\\infty } := A_{\\infty } \\cap A^{\\prime } = \\lbrace x \\in A_{\\infty } \\mid ax = xa \\text{ for every } a \\in A\\rbrace , $ the relative commutant of $A$ in $A_{\\infty }$ .",
"Let $ \\mathrm {Ann}(A, A_{\\infty }) := \\lbrace x \\in A_{\\infty } \\mid ax = xa = 0 \\text{ for every } a \\in A\\rbrace ,$ which is evidently an ideal in $A^{\\infty }$ .",
"Finally, we define $ F(A) := A^{\\infty }/ \\mathrm {Ann}(A, A_{\\infty }).",
"$ When $A$ is not separable, one can define the above with respect to a given separable subalgebra $D$ , as is done in [64].",
"However, since all our $\\mathrm {C}^*$ -algebras will be separable, we will not require this.",
"A completely positive contractive (c.p.c.)",
"map $\\varphi : A \\rightarrow B$ between $\\mathrm {C}^*$ -algebras $A$ and $B$ is said to be order zero if it is orthogonality preserving, that is, for every $a, b \\in A_+$ with $ab = ba =0$ we have $\\varphi (a) \\varphi (b) = 0$ .",
"Any $^*$ -homomorphism is of course order zero, but a c.p.c.",
"order zero map is not in general a $^*$ -homomorphism.",
"For more about c.p.c.",
"order zero maps, see [73].",
"5.7 5.7 Definition: [64] Let $A$ be a separable $\\mathrm {C}^*$ -algebra, $G$ a countable group with finite index subgroup $K$ , $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action of $G$ on $A$ and $d \\in \\mathbb {N}$ .",
"Let $\\tilde{a}_{\\infty }$ denote the induced action on $F_{\\infty }(A)$ .",
"We say the action has Rokhlin dimension $d$ with respect to $K$ if $d$ is the least integer such that there exists equivariant c.p.c.",
"order zero maps $ \\varphi _l : (C(G/K), G\\text{-shift}) \\rightarrow (F_{\\infty }(A), \\tilde{a}_{\\infty }), \\quad l = 0, \\dots , d$ with $ \\varphi _0(1) + \\cdots + \\varphi _d(1) = 1.",
"$ If moreover $\\varphi _0 ,\\dots , \\varphi _d$ can be chosen to have commuting ranges, then we say the action has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"The next result is an obvious generalisation of the equivalence of (1) and (3) of [64] to the case of commuting towers.",
"5.8 5.8 Lemma: Let $A$ be a $\\mathrm {C}^*$ -algebra, $G$ a countable group and $K$ a subgroup of finite index.",
"The following are equivalent for an action $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ .",
"(i) The action $\\beta $ has Rokhlin dimension $d$ with commuting towers with respect to $K$ .",
"For every finite subset $M \\subset G$ , finite subset $F \\subset A$ and $\\epsilon >0$ there are positive contractions $(f^{(l)}_{\\overline{g}})_{\\overline{g} \\subset H}^{l= 0, \\dots , d}$ in $A$ such that (i) $\\Vert (\\sum _{l =0}^d \\sum _{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}}) \\cdot a - a \\Vert < \\epsilon $ for all $a \\in F$ ; $\\Vert f_{\\overline{g}}^{(l)}f_{\\overline{h}}^{(l)}a \\Vert \\le \\epsilon $ for all $a\\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\ne \\overline{h} \\in G/K$ ; $\\Vert (\\beta _g(f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}})a \\Vert < \\epsilon $ for all $a \\in F$ , $l \\in 0, \\dots , d$ and $g \\in M$ and $\\overline{h} \\in G/K$ ; $\\Vert f^{(l)}_{\\overline{g}}a - a f^{(l)}_{\\overline{g}} \\Vert < \\epsilon $ for all $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ ; $\\Vert (f^{(k)}_{\\overline{g}} f^{(l)}_{\\overline{h}} - f^{(l)}_{\\overline{h}} f^{(k)}_{\\overline{g}})a \\Vert < \\epsilon $ for all $a \\in F$ , $k,l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g}, \\overline{h} \\in G/K$ .",
"Proof.",
"The only thing that one needs to check is that asking for the $f^{(l)}_{\\overline{g}}$ to approximately commute is equivalent to having the images of the order zero maps of [64] commute, but this is obvious.",
"5.9 5.9 Theorem: Let $G$ be a countable discrete group, $K$ a subgroup of $G$ with finite index, $A$ a separable $\\mathrm {C}^*$ -algebra and $\\beta : G \\rightarrow \\operatorname{Aut}(A)$ an action with multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"Then $\\beta $ has Rokhlin dimension at most $d$ respect to $K$ .",
"If $\\beta $ has multiplier Rokhlin dimension at most $d$ with commuting towers with respect to $K$ , then $\\beta $ has Rokhlin dimension at most $d$ with commuting towers respect to $K$ .",
"Proof.",
"We show the action satisfies the criteria of [64] (and in the commuting tower case Lemma REF (ii)).",
"Note that [64] is exactly (a)-(d) in Lemma REF (ii).",
"Let $M \\subset G$ and $F \\subset A$ be finite subsets and let $\\epsilon > 0$ .",
"Without loss of generality we may assume that every $a \\in F$ is a positive contraction.",
"Since $\\beta $ has multiplier Rokhlin dimension less than or equal to $d$ with respect to $K$ we can find positive contractions $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ satisfying Definition REF with respect to $M$ , $F$ and $\\epsilon /2$ .",
"Let $(w_n)_{n \\in \\mathbb {N}}$ be an $G$ -quasi-invariant quasicentral approximate unit for $A$ in $\\mathcal {M}(A)$ .",
"Since $F \\subset A$ and $(f_g^{(l)})_{\\overline{g} \\in G/K}$ , $l = 0 , \\dots d$ are in $\\mathcal {M}(A)$ , there exists $N \\in \\mathbb {N}$ sufficiently large so that $ \\Vert w_N a - a \\Vert < \\epsilon /4 \\text{ for every } a \\in F,$ and $ \\Vert [f_{\\overline{g}}^{(l)}, w_N] \\Vert < \\epsilon /8.$ and $ \\Vert \\beta _g(f_{\\overline{h}}^{(l)}) - f_{\\overline{gh}}^{(l)} \\Vert < \\epsilon /8.$ By increasing $N$ if necessary, we may also assume, since $M$ is finite, that $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ .",
"Then let $ r_{\\overline{g}}^{(l)} := w_N f_{\\overline{g}}^{(l)} w_N,$ for $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $g \\in G$ .",
"Then each $r_{\\overline{g}}^{(l)}$ is a positive contraction in $A$ and $\\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}}) a - a\\Vert &=& \\Vert (\\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N) a - a\\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} w_N f^{(l)}_{\\overline{g}} w_N - 1 \\Vert \\\\&\\le & \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d\\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f^{(l)}_{\\overline{g}} - 1 \\Vert \\\\&<& \\epsilon ,$ showing that (a) of Lemma REF (ii) holds.",
"Next, $\\Vert r_{\\overline{g}}^{(l)} r_{\\overline{h}}^{(l)} a \\Vert &=& \\Vert w_N f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(l)} w_N a \\Vert \\\\&=& \\epsilon /2 + \\Vert w_N^2 f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(l)} w_N^2 a \\Vert \\\\&<& \\epsilon $ for every $a \\in F$ , $l\\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , showing that (b) of Lemma REF holds.",
"Using $\\Vert \\beta _g(w_N) - w_N \\Vert < \\epsilon /8$ for each $g\\in M$ , we obtain $\\Vert (\\beta _g(r_{\\overline{h}}^{(l)}) - r_{\\overline{gh}}^{(l)})a\\Vert &=& \\epsilon /4 + \\Vert w_N \\mathcal {M}(\\beta _g)(f^{(l)}_{\\overline{h}}) w_N - w_N f^{(l)}_{\\overline{gh}} w_N \\Vert \\\\&<& \\epsilon ,$ for every $l \\in \\lbrace 0, \\dots , d\\rbrace $ , every $\\overline{h} \\in G/K$ , every $g \\in M$ and every $a \\in F$ , showing (c) of Lemma REF .",
"For (d) of Lemma REF we have $\\Vert r_{\\overline{g}}^{(l)} a - a r_{\\overline{g}}^{(l)} \\Vert = \\epsilon / 2 + \\Vert f_{\\overline{g}}^{(l)} a - a f_{\\overline{g}}^{(l)} \\Vert < \\epsilon ,$ for every $a \\in F$ , every $l \\in \\lbrace 0, \\dots , d\\rbrace $ and every $\\overline{g} \\in G/K$ .",
"Finally, in the commuting tower case, we must show (e): $\\Vert (r_{\\overline{g}}^{(k)} r_{\\overline{h}}^{(l)} - r_{\\overline{h}}^{(l)} r_{\\overline{g}}^{(k))} a\\Vert &\\le & \\Vert f_{\\overline{g}}^{(k)} w_N^2 f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} w_N^2 f_{\\overline{g}}^{(k)}\\Vert \\\\&\\le & \\epsilon /4 + \\Vert f_{\\overline{g}}^{(k)} f_{\\overline{h}}^{(l)} - f_{\\overline{h}}^{(l)} f_{\\overline{g}}^{(k)}\\Vert \\\\&<& \\epsilon ,$ for every $\\overline{g}, \\overline{h} \\in G/K$ and $k, l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"The proof of the next lemma is obvious and hence omitted.",
"It will, however, prove useful in what follows.",
"5.10 5.10 Lemma: Suppose that the action of $G$ on $ {\\mathrm {C}}^*(H)$ has Rokhlin dimension at most $d$ .",
"Then we may choose the Rokhlin elements to satisfy $f^{(l)}_g \\in C_c(\\mathcal {G}_H)$ for $l = 0, \\dots , d$ and $ g \\in G$ .",
"5.11 5.11 Proposition: Let $G$ be a countable group acting on an irreducible Smale space $(X, \\varphi )$ and let $K \\subset G$ be a subset of finite index.",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ with respect to $K$ so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"If $\\beta $ has Rokhlin dimension at most $d$ with commuting towers with respect to $K$ so does the action $\\beta ^{(S)}$ .",
"Proof.",
"We will show that $\\beta ^{(S)}$ has multiplier Rokhlin dimension at most $d$ with respect to $K$ .",
"The result then follows from Theorem REF .",
"Let $M$ be a finite subset of $G$ , $F$ a finite subset of $ \\mathrm {C}^*(S)$ and $\\epsilon >0$ .",
"Without loss of generality, we may assume that $F \\subset C_c(S)$ .",
"Since $\\beta $ has Rokhlin dimension at most $d$ with respect to $K$ there are contractions $r^{(l)}_g \\in {\\mathrm {C}}^*(H)$ such that (i) $\\Vert r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}} \\Vert < \\epsilon /2$ for $l = 0, \\dots , d$ and any $g, h \\in G$ with $g \\ne h$ , $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1\\Vert < \\epsilon /2,$ $\\Vert \\beta _h(r^{(l)}_{\\overline{g}}) - r^{(l)}_{\\overline{hg}}\\Vert < \\epsilon /2$ for $l =0, \\dots , d$ , every $h \\in M$ and $\\overline{g} \\in G/K$ , and $\\Vert r^{(k)}_{\\overline{g}}r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}}r^{(k)}_{\\overline{g}}\\Vert < \\epsilon /2$ for every $\\overline{g}$ , $\\overline{h} \\in G/K$ and $k,l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"By the previous lemma, we may moreover assume that each $r^{(l)}_{\\overline{g}} \\in C_c(\\mathcal {G}_H)$ .",
"Now we can find a natural number $k$ such that $ \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert < \\epsilon /2,$ for every $a \\in F$ .",
"For $l \\in \\lbrace 0, \\dots , d\\rbrace $ and $\\overline{g} \\in G/K$ , let $ f^{(l)}_{\\overline{g}} := \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}} )).$ Note that each $f^{(l)}_g$ is a positive contraction in $\\mathcal {M}( \\mathrm {C}^*(S))$ .",
"We will show that the $f^{(l)}_{\\overline{g}}$ satisfy (i) – (iv) of Definition REF .",
"For (i) we have $\\Vert f^{(l)}_{\\overline{g}} f^{(l)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} )) \\Vert \\\\&=& \\Vert \\alpha ^k(r^{(l)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) \\Vert \\\\&<& \\epsilon ,$ for any $\\overline{g} \\ne \\overline{h} \\in G/K$ , any $l \\in \\lbrace 0, \\dots , d\\rbrace $ .",
"For (ii) $\\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} f_{\\overline{g}}^{(l)} - 1_{\\mathcal {M}( \\mathrm {C}^*(S))} \\Vert &=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g}\\in G} \\rho (\\alpha ^k( r^{(l)}_{\\overline{g}} )) - \\rho (1_{ {\\mathrm {C}}^*(H)})\\Vert \\\\&=& \\Vert \\operatorname{\\textstyle {\\sum }}_{l=0}^d \\operatorname{\\textstyle {\\sum }}_{\\overline{g} \\in G/K} r^{(l)}_{\\overline{g}} - 1_{ {\\mathrm {C}}^*(H)} \\Vert \\\\&<& \\epsilon ,$ for any $a \\in F$ .",
"For (iii) we have $\\Vert f^{(l)}_{\\overline{g}} a - a f^{(l)}_{\\overline{g}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) a - a \\rho (\\alpha ^k(r^{(l)}_{\\overline{g}})) \\Vert \\\\&<& \\epsilon /2,$ for all $l \\in \\lbrace 0, \\dots d\\rbrace , \\overline{g} \\in G/K$ and $a \\in F$ .",
"For (iv), let $g \\in G$ with $\\overline{h} \\in G/K $ and let $a \\in F$ .",
"Then, $\\Vert \\beta ^{(S)}_g (f^{(l)}_{\\overline{h}}) - f^{(l)}_{\\overline{gh}} \\Vert &=& \\Vert \\alpha ^k(\\beta _g(r^{(l)}_{\\overline{h}}) - r^{(l)}_{\\overline{gh}}) \\Vert \\\\&<& \\epsilon .$ Finally, in the commuting towers case, for (v), let $\\overline{g}, \\overline{h} \\in G/K$ and $l,k \\in \\lbrace 0, \\dots , d\\rbrace $ , $\\Vert f^{(k)}_{\\overline{h}} f^{(l)}_{\\overline{g}} - f^{(l)}_{\\overline{g}}f^{(k)}_{\\overline{h}} \\Vert &=& \\Vert \\rho (\\alpha ^k(r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}}) - \\rho (\\alpha ^k(r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}) \\Vert \\\\&=& \\Vert r^{(k)}_{\\overline{g}} r^{(l)}_{\\overline{h}} - r^{(l)}_{\\overline{h}} r^{(k)}_{\\overline{g}}\\Vert \\\\&<& \\epsilon .$ The result now follows.",
"This gives us the next corollary: 5.12 5.12 Corollary: Let $G$ be a countable residually finite group acting on an irreducible Smale space $(X, \\varphi )$ .",
"Then if the induced action $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ has Rokhlin dimension at most $d$ (with commuting towers) so does the action $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ .",
"By Proposition REF and Theorem REF respectively, we get the next two corollaries.",
"5.13 5.13 Corollary: Let $d\\in {\\mathbb {N}}$ and $G={\\mathbb {Z}}^d$ .",
"Suppose $G$ acts effectively on a mixing Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : G \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"5.14 5.14 Corollary: Let $\\mathbb {Z}$ be an effective action on an irreducible Smale space $(X, \\varphi )$ .",
"Then the induced actions $\\beta : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ , $\\beta ^{(S)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(S))$ and $\\beta ^{(U)} : \\mathbb {Z} \\rightarrow \\operatorname{Aut}( \\mathrm {C}^*(U))$ each have finite Rokhlin dimension.",
"For more general group actions the situation is less clear.",
"In particular, to the authors' knowledge, it is not known if strong outerness implies finite Rokhlin dimensionAt the workshop “Future Targets in the Classification Program for Amenable $\\mathrm {C}^*$ -Algebras” held at BIRS, Eusebio Gardella presented results in this direction, but they have yet to appear..",
"In fact, for general discrete groups, there are obstructions to a (strongly outer) action having finite Rokhlin dimension with commuting towers, see [21].",
"Such examples can occur in the context considered in the present paper.",
"An explicit example is the following, let $(\\Sigma _{[3]}, \\sigma )$ be the full three shift (so $\\Sigma _{[3]}= \\lbrace 0, 1, 2\\rbrace ^{{\\mathbb {Z}}}$ and $\\sigma $ is the left sided shift).",
"Then $ {\\mathrm {C}}^*(H)$ is the UHF-algebra with supernatural number $3^{\\infty }$ .",
"The action induced from the permutation $0 \\mapsto 1$ , $1 \\mapsto 0$ , and $2 \\mapsto 2$ is an effective order two automorphism.",
"Thus the action induced on $ {\\mathrm {C}}^*(H)$ is strongly outer, but it follows from [21] that it does not have finite Rokhlin dimension with commuting towers.",
"$\\mathcal {Z}$ -stability, nuclear dimension and classification We begin this section with two theorems that follow quickly from the work done above.",
"As well as being interesting observations on their own, they will allow us to say something about the $\\mathcal {Z}$ -stability of the crossed products of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ below.",
"6.1 6.1 Theorem: Let $G$ be a countable discrete amenable group.",
"Suppose $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action of $G$ on $ {\\mathrm {C}}^*(H)$ .",
"Then $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable.",
"Proof.",
"Since $(X, \\varphi )$ is mixing, $ {\\mathrm {C}}^*(H)$ is simple and so the classification results of [12] imply that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace, it must be fixed by the action.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable by [58].",
"6.2 6.2 Theorem: Let $G$ be a countable amenable group acting effectively on a mixing Smale space $(X, \\varphi )$ and $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Then the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is a simple unital nuclear $\\mathcal {Z}$ -stable $\\mathrm {C}^*$ -algebra with unique tracial state and nuclear dimension (in fact, decomposition rank) at most one.",
"In particular, it belongs to the class of $\\mathrm {C}^*$ -algebras that are classified by the Elliott invariant and $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF, for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"Proof.",
"Since $ {\\mathrm {C}}^*(H)$ is amenable, it follows from [53] that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G \\cong \\mathrm {C}^*(\\mathcal {G}_H \\rtimes G)$ is also amenable and hence by [69] it satisfies the UCT.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is quasidiagonal [67] and by [67] together with Theorem classified by the Elliott invariant.",
"The nuclear dimension and decomposition rank bounds are given by [5].",
"Finally, by [42] $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF.",
"$\\mathcal {Z}$ -stability of crossed products $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ Let $A$ be separable $\\mathrm {C}^*$ -algebra and $G$ a discrete group.",
"For an action $\\beta : G\\rightarrow \\operatorname{Aut}(A)$ the fixed point algebra is given by $ A^{\\beta } := \\lbrace a \\in A \\mid \\beta _g(a) = a \\text{ for all } g \\in G\\rbrace .",
"$ Let $ A^{\\infty } := \\ell ^{\\infty }(\\mathbb {N}, A) / c_0(A) .$ The central sequence algebra of $A$ is defined by $ A_{\\infty } := A^{\\infty } \\cap A^{\\prime },$ where $A$ is considered as the subalgebra of $A^{\\infty }$ by viewing an element as a constant sequence.",
"We will denote by $\\overline{\\beta }$ the induced action on $A_{\\infty }$ .",
"Let $p$ and $q$ be positive integers.",
"The dimension drop algebra $I(p,q)$ is given by $I(p, q) := \\lbrace f \\in C([0,1], M_p( \\otimes M_q( \\mid f(0) \\in M_p( \\otimes { and }f(1)\\in M_q(\\rbrace .$ The Jiang–Su algebra, $\\mathcal {Z}$ , is an inductive limit of such algebras [28].",
"6.3 6.3 Theorem: Suppose $G$ is a countable discrete group acting on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"If, for each $k\\in {\\mathbb {N}}$ , there exists a unital equivariant embedding $I(k, k+1) \\rightarrow {\\mathrm {C}}^*(H)$ , then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"As usual, it suffices to prove the result for $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ .",
"To do so, we will show that the hypotheses of Lemma 2.6 in [21] hold.",
"We will show that, for every $k \\in \\mathbb {N}$ , there is a completely positive contractive map $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty } $ satisfying (i) $ (\\overline{\\beta }^S( \\gamma (x)) - \\gamma (x))a = $ for every $x \\in I(k, k+1)$ , $g \\in G$ and $a \\in \\mathrm {C}^*(S)$ , $a \\gamma (1) = a$ for every $a \\in \\mathrm {C}^*(S)$ , and $a(\\gamma (xy) - \\gamma (x)\\gamma (y)) = 0$ for every $x, y \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Suppose $F \\subset I(k, j+1)$ a finite subset and $\\epsilon > 0$ are given.",
"Let $w_n$ be a $G$ -invariant quasicentral approximate unit for $ \\mathrm {C}^*(S)$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ and $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ be a unital embedding (which exists by assumption).",
"Define $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)^{\\infty } $ via $ \\gamma (d) = (w_n \\rho (\\alpha ^n(d_n)) w_n)_{n \\in \\mathbb {N}}, $ where $(d_n)_{n \\in \\mathbb {N}}$ is a representative sequence for $\\tilde{\\gamma }(d)$ .",
"Then $\\gamma $ gives a c.p.c.",
"map.",
"Moreover, if $a \\in \\mathrm {C}^*(S)$ we have $ \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\alpha ^n(d_n) )w_n a - a w_n \\rho (\\alpha ^n(d_n)) w_n \\Vert = 0,$ so in fact $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty }.",
"$ Let us check that $\\gamma $ satisfies (i), (ii) and (iii).",
"Let $a \\in A$ and for any $d \\in I(k, k+1)$ , let $(d_n)_{n \\in \\mathbb {N}}$ be a representative of $\\tilde{\\gamma }(d)$ in $( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }$ .",
"$\\Vert \\overline{\\beta }^S(\\gamma (d) - \\gamma (d)) a \\Vert &=& \\lim _{n \\rightarrow \\infty } \\Vert \\beta ^{(S)}(w_n \\rho (\\alpha ^n(d_n)) w_n) - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&=& \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\beta (\\alpha ^n(d_n)) w_n - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&\\le & \\lim _{n \\rightarrow \\infty } \\Vert \\beta (\\alpha ^n(d_n) ) - \\alpha ^n(d_n) \\Vert \\\\&=& 0,$ showing $(i)$ .",
"To show (ii), we have $a \\gamma (1) &=& (a w_n \\rho (\\tilde{\\gamma }(1)) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a \\rho (1) w_n^2)_{n \\in \\mathbb {N}} \\\\&=& a,$ for every $a \\in \\mathrm {C}^*(S)$ Finally, for (iii), let $d, d^{\\prime } \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Then $a(\\gamma (d d^{\\prime }) - \\gamma (d)\\gamma (d^{\\prime })) &=& (a w_n \\rho (d_n d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}} - (a w_n \\rho (d_n) w_n^2 \\rho (d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a w_n^2 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a w_n^4 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} \\\\&=& (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}}\\\\&=& 0.$ Thus $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ is $\\mathcal {Z}$ -stable.",
"6.4 6.4 Corollary: Suppose $G$ is a discrete amenable group and $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"Then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"By Theorem REF (v), $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable and hence has strict comparison [54].",
"This in turn implies $ {\\mathrm {C}}^*(H)$ has property (TI) of Sato [58].",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace $\\tau $ which is therefore fixed by $\\beta $ , the hypotheses of [58] are satisfied.",
"Thus from the proof of [58] we get a unital embedding $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ The result then follows from the previous theorem.",
"Acknowledgments.",
"The authors thank Ian Putnam for many useful discussions concerning the content of this paper, Smale spaces, group actions and dynamics in general.",
"We also thank Magnus Goffeng for a number of useful comments.",
"The authors thank the referee for reading the paper carefully and making a number of useful comments.",
"The authors wish to thank Matrix at the University of Melbourne for hosting them during the programme Refining $\\mathrm {C}^*$ -algebraic Invariants for Dynamics using KK-theory in July 2016, the Banach Centre at the Institute of Mathematics of the Polish Academy of Sciences for hosting the first listed author during the conference Index Theory in October 2016, the University of Hawaii, Manoa for hosting the second listed author during the workshop Computability of K-theory in November 2016 and the Centre Rercerca Matemàtica, Barcelona, for their stay during the Intensive Research on Operator Algebras: Dynamics and Interactions in July 2017.",
"The above research visits were partially supported through NSF grants DMS 1564281 and DMS 1665118."
],
[
"$\\mathcal {Z}$ -stability, nuclear dimension and classification",
"We begin this section with two theorems that follow quickly from the work done above.",
"As well as being interesting observations on their own, they will allow us to say something about the $\\mathcal {Z}$ -stability of the crossed products of $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ below.",
"6.1 6.1 Theorem: Let $G$ be a countable discrete amenable group.",
"Suppose $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Denote by $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action of $G$ on $ {\\mathrm {C}}^*(H)$ .",
"Then $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable.",
"Proof.",
"Since $(X, \\varphi )$ is mixing, $ {\\mathrm {C}}^*(H)$ is simple and so the classification results of [12] imply that $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable.",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace, it must be fixed by the action.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is $\\mathcal {Z}$ -stable by [58].",
"6.2 6.2 Theorem: Let $G$ be a countable amenable group acting effectively on a mixing Smale space $(X, \\varphi )$ and $\\beta : G \\rightarrow \\operatorname{Aut}( {\\mathrm {C}}^*(H))$ the induced action.",
"Then the crossed product $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is a simple unital nuclear $\\mathcal {Z}$ -stable $\\mathrm {C}^*$ -algebra with unique tracial state and nuclear dimension (in fact, decomposition rank) at most one.",
"In particular, it belongs to the class of $\\mathrm {C}^*$ -algebras that are classified by the Elliott invariant and $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF, for any UHF algebra of infinite type $\\operatorname{\\mathcal {U}}$ .",
"Proof.",
"Since $ {\\mathrm {C}}^*(H)$ is amenable, it follows from [53] that $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G \\cong \\mathrm {C}^*(\\mathcal {G}_H \\rtimes G)$ is also amenable and hence by [69] it satisfies the UCT.",
"Thus $ {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G$ is quasidiagonal [67] and by [67] together with Theorem classified by the Elliott invariant.",
"The nuclear dimension and decomposition rank bounds are given by [5].",
"Finally, by [42] $( {\\mathrm {C}}^*(H)\\rtimes _{\\beta } G) \\otimes \\operatorname{\\mathcal {U}}$ is TAF."
],
[
"$\\mathcal {Z}$ -stability of crossed products {{formula:ce845875-fc73-47e5-9f4d-83d1d2402608}} and {{formula:f7d68908-2e6e-4024-b18f-eb9d6710dd2a}}",
"Let $A$ be separable $\\mathrm {C}^*$ -algebra and $G$ a discrete group.",
"For an action $\\beta : G\\rightarrow \\operatorname{Aut}(A)$ the fixed point algebra is given by $ A^{\\beta } := \\lbrace a \\in A \\mid \\beta _g(a) = a \\text{ for all } g \\in G\\rbrace .",
"$ Let $ A^{\\infty } := \\ell ^{\\infty }(\\mathbb {N}, A) / c_0(A) .$ The central sequence algebra of $A$ is defined by $ A_{\\infty } := A^{\\infty } \\cap A^{\\prime },$ where $A$ is considered as the subalgebra of $A^{\\infty }$ by viewing an element as a constant sequence.",
"We will denote by $\\overline{\\beta }$ the induced action on $A_{\\infty }$ .",
"Let $p$ and $q$ be positive integers.",
"The dimension drop algebra $I(p,q)$ is given by $I(p, q) := \\lbrace f \\in C([0,1], M_p( \\otimes M_q( \\mid f(0) \\in M_p( \\otimes { and }f(1)\\in M_q(\\rbrace .$ The Jiang–Su algebra, $\\mathcal {Z}$ , is an inductive limit of such algebras [28].",
"6.3 6.3 Theorem: Suppose $G$ is a countable discrete group acting on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"If, for each $k\\in {\\mathbb {N}}$ , there exists a unital equivariant embedding $I(k, k+1) \\rightarrow {\\mathrm {C}}^*(H)$ , then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"As usual, it suffices to prove the result for $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ .",
"To do so, we will show that the hypotheses of Lemma 2.6 in [21] hold.",
"We will show that, for every $k \\in \\mathbb {N}$ , there is a completely positive contractive map $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty } $ satisfying (i) $ (\\overline{\\beta }^S( \\gamma (x)) - \\gamma (x))a = $ for every $x \\in I(k, k+1)$ , $g \\in G$ and $a \\in \\mathrm {C}^*(S)$ , $a \\gamma (1) = a$ for every $a \\in \\mathrm {C}^*(S)$ , and $a(\\gamma (xy) - \\gamma (x)\\gamma (y)) = 0$ for every $x, y \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Suppose $F \\subset I(k, j+1)$ a finite subset and $\\epsilon > 0$ are given.",
"Let $w_n$ be a $G$ -invariant quasicentral approximate unit for $ \\mathrm {C}^*(S)$ in $\\mathcal {M}( \\mathrm {C}^*(S))$ and $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ be a unital embedding (which exists by assumption).",
"Define $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)^{\\infty } $ via $ \\gamma (d) = (w_n \\rho (\\alpha ^n(d_n)) w_n)_{n \\in \\mathbb {N}}, $ where $(d_n)_{n \\in \\mathbb {N}}$ is a representative sequence for $\\tilde{\\gamma }(d)$ .",
"Then $\\gamma $ gives a c.p.c.",
"map.",
"Moreover, if $a \\in \\mathrm {C}^*(S)$ we have $ \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\alpha ^n(d_n) )w_n a - a w_n \\rho (\\alpha ^n(d_n)) w_n \\Vert = 0,$ so in fact $ \\gamma : I(k, k+1) \\rightarrow \\mathrm {C}^*(S)_{\\infty }.",
"$ Let us check that $\\gamma $ satisfies (i), (ii) and (iii).",
"Let $a \\in A$ and for any $d \\in I(k, k+1)$ , let $(d_n)_{n \\in \\mathbb {N}}$ be a representative of $\\tilde{\\gamma }(d)$ in $( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }$ .",
"$\\Vert \\overline{\\beta }^S(\\gamma (d) - \\gamma (d)) a \\Vert &=& \\lim _{n \\rightarrow \\infty } \\Vert \\beta ^{(S)}(w_n \\rho (\\alpha ^n(d_n)) w_n) - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&=& \\lim _{n \\rightarrow \\infty } \\Vert w_n \\rho (\\beta (\\alpha ^n(d_n)) w_n - w_n \\rho (\\alpha ^n(d_n)) w_n\\Vert \\\\&\\le & \\lim _{n \\rightarrow \\infty } \\Vert \\beta (\\alpha ^n(d_n) ) - \\alpha ^n(d_n) \\Vert \\\\&=& 0,$ showing $(i)$ .",
"To show (ii), we have $a \\gamma (1) &=& (a w_n \\rho (\\tilde{\\gamma }(1)) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a \\rho (1) w_n^2)_{n \\in \\mathbb {N}} \\\\&=& a,$ for every $a \\in \\mathrm {C}^*(S)$ Finally, for (iii), let $d, d^{\\prime } \\in I(k, k+1)$ and $a \\in \\mathrm {C}^*(S)$ .",
"Then $a(\\gamma (d d^{\\prime }) - \\gamma (d)\\gamma (d^{\\prime })) &=& (a w_n \\rho (d_n d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}} - (a w_n \\rho (d_n) w_n^2 \\rho (d^{\\prime }_n) w_n)_{n \\in \\mathbb {N}}\\\\&=& (a w_n^2 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a w_n^4 \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} \\\\&=& (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}} - (a \\rho (d_n d_n^{\\prime }))_{n \\in \\mathbb {N}}\\\\&=& 0.$ Thus $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ is $\\mathcal {Z}$ -stable.",
"6.4 6.4 Corollary: Suppose $G$ is a discrete amenable group and $G$ acts on a mixing Smale space $(X, \\varphi )$ .",
"Let $\\beta ^{(S)}$ and $\\beta ^{(U)}$ denote the induced actions on $ \\mathrm {C}^*(S)$ and $ \\mathrm {C}^*(U)$ , respectively.",
"Then $ \\mathrm {C}^*(S)\\rtimes _{\\beta ^{(S)}} G$ and $ \\mathrm {C}^*(U)\\rtimes _{\\beta ^{(U)}} G$ are $\\mathcal {Z}$ -stable.",
"Proof.",
"By Theorem REF (v), $ {\\mathrm {C}}^*(H)$ is $\\mathcal {Z}$ -stable and hence has strict comparison [54].",
"This in turn implies $ {\\mathrm {C}}^*(H)$ has property (TI) of Sato [58].",
"Since $ {\\mathrm {C}}^*(H)$ has unique trace $\\tau $ which is therefore fixed by $\\beta $ , the hypotheses of [58] are satisfied.",
"Thus from the proof of [58] we get a unital embedding $ \\tilde{\\gamma } : I(k, k+1) \\rightarrow ( {\\mathrm {C}}^*(H)_{\\infty })^{\\beta }.",
"$ The result then follows from the previous theorem.",
"Acknowledgments.",
"The authors thank Ian Putnam for many useful discussions concerning the content of this paper, Smale spaces, group actions and dynamics in general.",
"We also thank Magnus Goffeng for a number of useful comments.",
"The authors thank the referee for reading the paper carefully and making a number of useful comments.",
"The authors wish to thank Matrix at the University of Melbourne for hosting them during the programme Refining $\\mathrm {C}^*$ -algebraic Invariants for Dynamics using KK-theory in July 2016, the Banach Centre at the Institute of Mathematics of the Polish Academy of Sciences for hosting the first listed author during the conference Index Theory in October 2016, the University of Hawaii, Manoa for hosting the second listed author during the workshop Computability of K-theory in November 2016 and the Centre Rercerca Matemàtica, Barcelona, for their stay during the Intensive Research on Operator Algebras: Dynamics and Interactions in July 2017.",
"The above research visits were partially supported through NSF grants DMS 1564281 and DMS 1665118."
]
] | 1709.01897 |
[
[
"High energy solution of the Choquard equation"
],
[
"Abstract The present paper is concerned with the existence of positive high energy solution of the Choquard equation.",
"Under certain assumptions, the ground state of Choquard equation can not be achieved.",
"However, by global compactness analysis, we prove that there exists a positive high energy solution."
],
[
"Introduction",
"In this paper, we study the following Choquard equation ${\\left\\lbrace \\begin{array}{ll}-\\Delta u+u=Q(x)\\left(I_\\alpha *\\vert u\\vert ^p\\right)\\vert u\\vert ^{p-2}u \\ \\ \\ in\\ \\ \\mathbb {R}^N,\\\\u\\in H^1(\\mathbb {R}^N),\\end{array}\\right.",
"}$ where $I_\\alpha (x)$ is the Riesz potential of order $\\alpha \\in (0,N)$ on the Euclidean space $\\mathbb {R}^N$ , defined for each point $x \\in \\mathbb {R}^N\\backslash \\lbrace 0\\rbrace $ by $\\nonumber I_\\alpha (x)=\\frac{A_\\alpha }{\\vert x\\vert ^{N-\\alpha }},\\ \\ where\\ A_\\alpha =\\frac{\\Gamma (\\frac{N-\\alpha }{2})}{\\Gamma (\\frac{\\alpha }{2})\\pi ^{\\frac{N}{2}}2^\\alpha }$ and $Q(x)$ is a positive bounded continuous function on $\\mathbb {R}^N$ .",
"We consider the existence of high energy solutions under the assumptions that $\\alpha =2, p=2, N=3,4,5$ or $\\alpha =2,\\, N=3,\\ 2<p<\\frac{7}{3}$ .",
"When $\\alpha =2, p=2,\\, N=3$ , (REF ) is usually called the Choquard-Pekar equation which can be traced back to the 1954's work by Pekar on quantum theory of a Polaron [12] and to 1976's model of Choquard of an electron trapped in its own hole, in an approximation to Hartree-Fock theory of one-component plasma [5].",
"What's more, some Schr$\\ddot{o}$ dinger-Newton equations were regarded as the Choquard type equation.",
"When $Q(x)$ is a positive constant and $1+\\frac{\\alpha }{N}\\le p\\le \\frac{N+\\alpha }{N-2},\\, N\\ge 3$ , the existence of positive ground state solutions of (REF ) has been studied in many papers,see [5], [7], [8], [9], [10], for instance.",
"In addition, uniqueness of positive solutions of the Choquard equations has also been widely discussed in recent years by a lot of papers [6], [5], [13], [14], [15].",
"In [14], T.Wang and Taishan Yi proved that the positive solution of (REF ) is uniquely determined, up to translation provided $\\alpha =2, p=2,N=3,4,5$ .",
"The assumption on $p=2$ can be extended to $p>2$ and close to 2 when $N=3,\\ \\alpha =2$ , and under these assumptions C.L.",
"Xiang proved that the positive solution of (REF ) is unique in [15].",
"What's more, in [10],V. Moroz and J.Van Schaftingen gave some results on the decay of ground state solutions of the Choquard equation which will be used in the proof of our results.",
"(Another result on decay of ground states was shown in [15]) Motivated by D. Cao's work , [1], we prove in this paper that there exists a positive high energy solution of the Choquard equation under the following condition on $Q(x)$ : (C): $\\lim _{\\vert x\\vert \\rightarrow +\\infty } Q(x)=\\bar{Q}>0,\\,\\,\\, Q(x)\\ge \\frac{\\sqrt{2}}{2}\\bar{Q},\\,x\\in \\mathbb {R}^N $ .",
"One can also find some other assumptions on $Q(x)$ under which similar results can be obtained.",
"In particular we would like to mention the results in in which A. Bahri and Y.Y.",
"Li showed that there exists a positive solution of certain semilinear elliptic equations in $\\mathbb {R}^N$ even if the ground state can not be achieved.",
"The limiting problem of (REF ) is as following ${\\left\\lbrace \\begin{array}{ll}-\\Delta u+u=\\bar{Q}\\left(I_\\alpha *\\vert u\\vert ^p\\right)\\vert u\\vert ^{p-2}u \\ \\ \\ in\\ \\ \\mathbb {R}^N,\\\\u\\in H^1(\\mathbb {R}^N),\\end{array}\\right.",
"}$ where $\\bar{Q}$ is the positive constant given in condition (C).",
"When $\\alpha =2, p=2, N=3$ , Schr$\\ddot{o}$ dinger-Newton equation can be regarded as the Choquard type equation.",
"In [2], Giusi Vaira proved existence of positive bound solutions of a particular Schr$\\ddot{o}$ dinger-Newton type systems.",
"However the structure of equation in [2] is different from ours and we extend the assumption on $N$ and $p$ as well.",
"One of the difficulties to prove our results is that the Brezis-Lieb lemma can not be applied directly to our proof.",
"In order to overcome the difficulty we improve the results of lemma 2.2 in [3] for $N=3,4,5$ or $N=3,\\ 2<p<\\frac{7}{3}$ .",
"What's more, the main method of our proof depends on global compactness analysis and min-max method.",
"Our main results are as following Theorem 1.1 Assume that condition (C) holds, $\\alpha =2, p=2, N=3,4,5$ , then when the ground state level can not be achieved (REF ) has a positive high energy solution.",
"Remark 1.2 Suppose that $\\bar{Q}-Q(x)\\ge 0$ holds in $\\mathbb {R}^N$ and $Q(x)$ is not a constant, then it is not difficult to see that the ground state does not exist.",
"Remark 1.3 When $N=3, \\alpha =2$ and $2<p<\\frac{7}{3}$ the uniqueness result in [15] implies that the positive solution of (REF ) is unique up to a translation.",
"Moreover if we replace condition (C) by the following condition: ($\\textbf {C}^{\\,*}$ ): $\\lim _{\\vert x\\vert \\rightarrow +\\infty } Q(x)=\\bar{Q}>0,\\ Q(x)\\ge 2^{1-p}\\bar{Q},\\,\\,\\,x\\in \\mathbb {R}^N$ then using the uniqueness result and condition ($\\textbf {C}^{\\,*}$ ) we can see that the result of Theorem REF is also true by a similar discussion without bringing about new difficulties.",
"Our paper is organized as follows.",
"In section 2 we first give some notations and preliminary results for our proof of Theorem REF .",
"In section 3, we give the proof of Theorem REF ."
],
[
"Some Notations and Preliminary Results",
"In this section we give some preliminary results which will be used in our discussion in next section.",
"To start with, let us first give some definition.",
"Define $I(u)&=&\\frac{1}{2}\\int _{\\mathbb {R}^N}\\vert \\nabla u\\vert ^2+u^2-\\frac{1}{4}\\int _{\\mathbb {R}^N}Q(x)I_\\alpha *\\vert u\\vert ^2\\vert u\\vert ^2,\\\\I^*(u)&=&\\frac{1}{2}\\int _{\\mathbb {R}^N}\\vert \\nabla u\\vert ^2+u^2-\\frac{1}{4}\\int _{\\mathbb {R}^N}\\bar{Q}I_\\alpha *\\vert u\\vert ^2\\vert u\\vert ^2,\\\\J(u)&=&\\int _{\\mathbb {R}^N}\\vert \\nabla u\\vert ^2+u^2,\\\\=&\\lbrace u\\,\\,|\\,\\,u \\in H^1(\\mathbb {R}^N),u\\ge 0, \\int _{\\mathbb {R}^N}Q(x)I_\\alpha *\\vert u\\vert ^2\\vert u\\vert ^2=1\\rbrace ,\\\\*&=&\\lbrace u\\,\\,|\\,\\,u \\in H^1(\\mathbb {R}^N),u\\ge 0, \\int _{\\mathbb {R}^N}\\bar{Q}I_\\alpha *\\vert u\\vert ^2\\vert u\\vert ^2=1\\rbrace .$ Let $M$ and $M^*$ be defined respectively by $\\nonumber M=inf\\lbrace J(u)\\,\\,|\\,\\,u \\in \\ \\ \\ \\ and\\ \\ \\ \\ \\ M^*=inf\\lbrace J(u)\\,\\,|\\,\\,u \\in *\\rbrace .$ $D^{1,2}(\\mathbb {R}^N)$ is the completion of $C^\\infty _0(\\mathbb {R}^3)$ with respect to the norm $\\Vert u\\Vert ^2_{D^{1,2}}=\\int _{\\mathbb {R}^3}\\vert \\nabla u\\vert ^2dx.$ For later discussion, we introduce an inequality given in [4].",
"Proposition 2.1 (Hardy-Littlewood-Sobolev inequality [4]) Let $q \\in (1,+\\infty )$ and $\\alpha < \\frac{N}{q}$ , then for every $f\\in L^q(\\mathbb {R}^N)$ , $I_\\alpha *f \\in L^\\frac{Nq}{N-\\alpha q}(\\mathbb {R}^N)$ and $(\\int _{\\mathbb {R}^N}\\vert I_\\alpha *f\\vert ^\\frac{Nq}{N-\\alpha q})^\\frac{N-\\alpha q}{Nq}\\le C_{N,\\alpha ,q}(\\int _{\\mathbb {R}^N}\\vert f\\vert ^q)^\\frac{1}{q}.$ By Proposition REF , we have $\\int _{\\mathbb {R}^N}(I_\\alpha *\\vert u\\vert ^p)\\vert u\\vert ^p&\\le & C_{N,\\alpha }(\\int _{\\mathbb {R}^N}\\vert u\\vert ^\\frac{2Np}{N+\\alpha })^{1+\\frac{\\alpha }{N}}\\nonumber \\\\&\\le & C(\\int _{\\mathbb {R}^N} \\vert \\nabla u\\vert ^2+\\vert u\\vert ^2)^p.$ As a consequence of (REF ) we can easily get Proposition 2.2 Suppose $1+\\frac{\\alpha }{N}\\le p< \\frac{N+\\alpha }{N-2}$ and $\\alpha \\in (0,N)$ .",
"If $u_m\\rightharpoonup 0$ in $H^1(\\mathbb {R}^N)$ , then for any bounded domain $\\Omega $ in $\\mathbb {R}^N$ , $\\int _\\Omega (I_\\alpha *\\vert u_m\\vert ^p)\\vert u_m\\vert ^p\\rightarrow 0.$ Let $u$ be a positive ground state solution of (REF ), following [10], [15] the decay of $u$ is as follows Proposition 2.3 Assume that $\\alpha =2, p=2,N=3,4,5$ or $\\alpha =2,\\ N=3,\\ 2<p<\\frac{7}{3}$ , then $u=O(e^{-\\sigma \\vert x\\vert })$ for $\\vert x\\vert $ large enough, where $\\sigma $ is a positive constant.",
"Next we shall prove a proposition on the weak convergence of a nonlinear operator.",
"Denote $T(u,v,w,z)=\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)w(y)z(y)}{\\vert x-y\\vert ^{N-2}}dxdy.$ Proposition 2.4 Assume that $3\\le N\\le 6$ and that there are three weekly convergent sequences in $H^1(\\mathbb {R}^N)$ such that $u_m\\rightharpoonup u,\\ v_m\\rightharpoonup v,\\ w_m\\rightharpoonup w$ and $z \\in \\ H^1(\\mathbb {R}^N)$ , then as $m\\rightarrow +\\infty $ $T(u_m,v_m,w_m,z)\\rightarrow T(u,v,w,z).$ Firstly assume that $u_m\\equiv u$ for all $m$ , we claim that $T(u,v_m,w_m,z)\\rightarrow T(u,v,w,z)$ .",
"$T(u,v_m,w_m,z)=T(u,v_m-v,w_m,z)+T(u,v,w_m,z).$ Since $w_m\\rightharpoonup w$ in $H^1(\\mathbb {R}^N)$ , then $w_m\\rightharpoonup w$ in both $L^2(\\mathbb {R}^N)$ and $L^\\frac{2N}{N-2}(\\mathbb {R}^N)$ .",
"When $N=3,4$ , since $&&\\int _{\\mathbb {R}^N}(\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dxz(y))^2dy\\\\&\\le &(\\int _{\\mathbb {R}^N}(\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dx)^\\frac{2N}{N-2})^{\\frac{N-2}{N}}(\\int _{\\mathbb {R}^N}\\vert z\\vert ^N)^{\\frac{2}{N}},$ which implies that $\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dxz(y)\\in \\ L^2(\\mathbb {R}^N)$ .",
"Therefore it is easy to prove that $T(u,v,w_m,z)\\rightarrow T(u,v,w,z)$ .",
"For $5\\le N\\le 6$ , similarly we have $&&\\int _{\\mathbb {R}^N}(\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dxz(y))^{\\frac{2N}{N+2}}dy\\\\&\\le &(\\int _{\\mathbb {R}^N}(\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dx)^{\\frac{N}{2}})^{\\frac{4}{N+2}}(\\int _{\\mathbb {R}^N}\\vert z\\vert ^\\frac{2N}{N-2})^{\\frac{N-2}{N+2}}.$ As a consequence, $\\int _{\\mathbb {R}^N}\\frac{u(x)v(x)}{\\vert x-y\\vert ^{N-2}}dxz(y)\\in \\ L^{\\frac{2N}{N+2}}$ from which we get $T(u,v,w_m,z)\\rightarrow T(u,v,w,z)$ .",
"In addition, using Holder inequality we have $&&T(u,v_m-v,w_m,z)^2\\\\&\\le &\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N}\\frac{(v_m-v)^2(x)z^2(y)}{\\vert x-y\\vert ^{N-2}}dxdy\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N}\\frac{u^2(x)w_m^2(y)}{\\vert x-y\\vert ^{N-2}}dxdy\\\\&=&T(v_m-v,v_m-v,z,z)T(u,u,w_m,w_m).$ It is easy to see that $T(u,u,w_m,w_m)$ is bounded.",
"Set $\\phi _{u^2}(y)=\\int _{\\mathbb {R}^N}\\frac{u^2(x)}{\\vert x-y\\vert ^{N-2}}dx$ , then $\\phi _{u^2} \\in D^{1,2}(\\mathbb {R}^N)$ is a solution of $-\\Delta \\phi =u^2 \\ \\ \\ in\\ \\mathbb {R}^N$ and we have, as $m\\rightarrow +\\infty $ , $T(v_m-v,v_m-v,z,z)=\\int _{\\mathbb {R}^N}\\phi _{z^2}(v_m-v)^2dx\\rightarrow 0.$ Thus we complete the claim.",
"Now consider that $T(u_m,v_m,w_m,z)=T(u,v_m,w_m,z)+T(u_m-u,v_m,w_m,z)$ $T(u,v_m,w_m,z)\\rightarrow T(u,v,w,z)$ and with respect to the above discussion we get that $T(u_m-u,v_m,w_m,z)\\rightarrow 0$ as $m\\rightarrow +\\infty $ .",
"Lemma 2.5 Assume that $3\\le N\\le 6,\\ \\alpha =2$ and that $\\lbrace u_m\\rbrace $ is bounded in $H^1(\\mathbb {R}^N)$ .",
"If $u_m\\rightarrow u$ almost everywhere on $\\mathbb {R}^N$ as $m\\rightarrow +\\infty $ , then $T(u_m,u_m,u_m,u_m)-T(u,u,u,u)=T(u_m-u,u_m-u,u_m-u,u_m-u)+o(1).$ $&&T(u_m,u_m,u_m,u_m)\\\\&=&T(u_m,u_m,u_m,u_m-u)+T(u_m,u_m,u_m,u),\\\\&=&T(u_m,u_m,u_m,u_m-u)+T(u,u,u,u)+o(1),\\\\&=&T(u_m-u,u_m-u,u_m-u,u_m-u)+T(u,u,u,u)+o(1).$ Remark 2.6 For $N=3, \\alpha =2$ and $2<p<\\frac{7}{3}$ the results of Proposition REF and Lemma REF are also true by a similar calculation.",
"Next, we establish a global compactness lemma.",
"Lemma 2.7 Let $\\lbrace u_m\\rbrace \\subset H^1(\\mathbb {R}^N)$ be a sequence such that as $m\\rightarrow +\\infty $ $I(u_m)\\rightarrow C$ , $I^{^{\\prime }}(u_m)\\rightarrow 0$ in $H^{-1}(\\mathbb {R}^N)$ .",
"Then, there exists a number $k \\in \\mathbb {N}$ , $k$ sequences of points $\\lbrace y^j_m\\rbrace $ such that $\\vert y^j_m\\vert \\rightarrow +\\infty $ as $m\\rightarrow +\\infty $ , $1\\le j\\le k$ , $k+1$ sequence of functions $\\lbrace u^j_m\\rbrace \\subset H^1(\\mathbb {R}^N)$ , $0\\le j\\le k$ , such that for some subsequences ${\\left\\lbrace \\begin{array}{ll}u^0_m\\equiv u_m\\rightharpoonup u^0,\\\\u^j_m=(u^{j-1}_m-u^{j-1})(x-y^j_m)\\rightharpoonup u^j,\\\\1\\le j\\le k.\\end{array}\\right.",
"}$ where $u^0$ is a solution of (REF ) and $u^j, 1\\le j\\le k$ are nontrivial positive solutions of (REF ).",
"Moreover as $m\\rightarrow +\\infty $ $J(u_m)&\\rightarrow & \\sum ^k_{j=0} J(u^j),\\\\I(u_m)&\\rightarrow & I(u^0)+\\sum ^k_{j=1}I^*(u^j).$ Our proof is similar to the those in and [16].",
"Since $\\lbrace u_m\\rbrace $ is $(PS)_C$ sequence of $I(u)$ , it is easy to prove that $u_m$ is bounded in $H^1(\\mathbb {R}^N)$ .",
"Then we can assume that $u_m\\rightharpoonup u^0$ in $H^1(\\mathbb {R}^N)$ .",
"Set $v_m=u_m-u^0$ , then $v_m\\rightharpoonup 0$ in $H^1(\\mathbb {R}^N)$ .",
"If $v_m\\rightarrow 0$ in $H^1(\\mathbb {R}^N)$ , we are done.",
"Now suppose that $v_m\\nrightarrow 0$ in $H^1(\\mathbb {R}^N)$ .",
"By Proposition REF and Lemma REF , we get $I(v_m)&=&I^*(v_m)+o(1),\\\\I^{\\prime }(v_m)&=&(I^*)^{\\prime }(v_m)+o(1)=o(1).$ Moreover there exists $\\lambda \\in (0,+\\infty )$ such that $I^*(v_m)\\ge \\lambda >0$ for $m$ large enough.",
"In fact, otherwise $I^*(v_m)=o(1), \\ \\ (I^*)^{\\prime }(v_m)=o(1)$ would imply $\\Vert v_m\\Vert _{H^1}\\rightarrow 0$ , which is a contradiction to $v_m\\nrightarrow 0$ in $H^1(\\mathbb {R}^N)$ .",
"Let us decompose $\\mathbb {R}^N$ into N-dim hypercubes $\\Omega _i$ and define $d_m=\\sup _{\\Omega _i}(\\int _{\\Omega _i}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2)^\\frac{1}{4}.$ Claim $d_m\\ge \\gamma >0$ .",
"Since $(I^*)^{\\prime }(v_m)=o(1)$ as $m\\rightarrow +\\infty $ , then $\\Vert v_m\\Vert _{H^1}&=&\\int _{\\mathbb {R}^N}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2+o(1),\\\\I^*(v_m)&=&\\frac{1}{4}\\int _{\\mathbb {R}^N}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2+o(1).$ Thus, we have $4I^*(v_m)+o(1)&=&\\int _{\\mathbb {R}^N}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2\\\\&=&\\sum _i\\int _{\\Omega _i}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2\\\\&\\le & d^2_m\\sum _i(\\int _{\\Omega _i}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2)^\\frac{1}{2}\\\\&\\le & C_Nd^2_m\\sum _i\\Vert v_m\\Vert ^2_{H^1(\\Omega _i)}\\ \\ (by\\ (\\ref {2-1}))\\\\&=&C_Nd^2_m\\Vert v_m\\Vert ^2_{H^1(\\mathbb {R}^N)},$ where $C_N$ is a positive constant.",
"Since $I^*(v_m)\\ge \\lambda >0$ then $d_m\\ge \\gamma >0$ .",
"Now, let us call $y_m$ the center of $\\Omega _m$ such that $(\\int _{\\Omega _m}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2)^\\frac{1}{4}\\ge d_m-\\frac{1}{m}$ and put $\\widetilde{v_m}=v_m(x+y_m)$ .",
"It is easy to prove that $\\widetilde{v_m}\\rightharpoonup v_0\\lnot \\equiv 0$ .",
"In fact, letting $\\Omega $ be the hypercube centered at the origin,then we have $(\\int _{\\Omega }\\bar{Q}I_\\alpha *\\vert \\widetilde{v_m}\\vert ^2\\vert \\widetilde{v_m}\\vert ^2)^\\frac{1}{4}=(\\int _{\\Omega _m}\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2)^\\frac{1}{4}\\ge d_m-\\frac{1}{m}\\ge \\gamma +o(1)$ If $\\widetilde{v_m}\\rightharpoonup 0$ , then $\\int _{\\Omega }\\bar{Q}I_\\alpha *\\vert \\widetilde{v_m}\\vert ^2\\vert \\widetilde{v_m}\\vert ^2\\rightarrow 0$ as $m\\rightarrow +\\infty $ , we get a contradiction.",
"Iterating the above procedure, if $\\widetilde{v_m}\\rightarrow v_0$ we are done, otherwise setting $w_m=\\widetilde{v_m}-v_0\\rightharpoonup 0$ and $w_m\\nrightarrow 0$ , continue the above procedure.",
"Since $\\lbrace u_m\\rbrace $ is bounded away from zero, by Brezis-Lieb lemma and Lemma REF we know that the iteration must terminate at some index $k>0$ and $I(u_m)&=&I(u^0)+\\sum ^k_{j=1}I^*(u^j)+o(1),\\\\J(u_m)&=&\\sum ^k_{j=0}J(u^j)+o(1).$ Moreover we claim as $m\\rightarrow +\\infty $ , $\\vert y_m\\vert \\rightarrow +\\infty $ , otherwise $\\vert y_m\\vert $ is bounded, we can choose a bounded domain $\\Sigma $ such that $\\bigcup \\Omega _m\\subset \\Sigma $ .",
"As a consequence, by $v_m\\rightharpoonup 0$ , we get $\\int _{\\Sigma }\\bar{Q}I_\\alpha *\\vert v_m\\vert ^2\\vert v_m\\vert ^2\\rightarrow 0$ , which is a contradiction to (REF ).",
"Remark 2.8 For $1\\le j\\le k$ , $J(u^j)\\ge {M^*}^2$ and $I^*(u^j)\\ge \\frac{1}{4}{M^*}^2$ .",
"If $c \\in (0,\\frac{1}{4}{M^*}^2)$ , then we can see that $k=0$ and therefore $u_m\\rightarrow u^0\\lnot \\equiv 0.$ If $c \\in [\\frac{1}{4}{M^*}^2,\\frac{1}{2}{M^*}^2)$ , then either $k=0$ or $k=1$ .",
"${\\left\\lbrace \\begin{array}{ll}u_m(x)\\rightarrow u^0(x)\\ \\ \\ \\ k=0,\\\\u_m(x)=u^0(x)+u(x+y_m)+w_m(x)\\ \\ \\ k=1,\\end{array}\\right.",
"}$ where $w_n(x)\\rightarrow 0$ in $H^1(\\mathbb {R}^N)$ .",
"Lemma 2.9 Assume that $\\lbrace u_m\\rbrace $ is a $(PS)_c$ sequence of $I(u)$ and $M=M^*$ .",
"If $0<c<\\frac{1}{4}{M^*}^2$ or $\\frac{1}{4}{M^*}^2<c<\\frac{1}{2}{M^*}^2$ , then $\\lbrace u_m\\rbrace $ contains a strongly convergent subsequence.",
"If $0<c<\\frac{1}{4}{M^*}^2$ we are done.",
"If $\\frac{1}{4}{M^*}^2<c<\\frac{1}{2}{M^*}^2$ , since $M=M^*$ , we get $J(u^0)\\ge \\frac{1}{4}{M^*}^2$ , then $u^0\\equiv 0$ or $u\\equiv 0$ .",
"If $u\\equiv 0$ , we are done.",
"Otherwise, $u^0\\equiv 0$ and $u\\lnot \\equiv 0$ is a positive solution of (REF ).",
"By the uniqueness of positive solutions of (REF ), $I^*(u)=\\frac{1}{4}{M^*}^2$ which contradicts to the value of $c$ .",
"Another form of Lemma REF is as following Lemma 2.10 Assume that $\\lbrace u_m\\rbrace \\subset such that\\begin{equation}{\\left\\lbrace \\begin{array}{ll}(i) \\ \\ J(u_m)\\rightarrow c \\ \\ \\ \\in (0,M^*) \\ or \\ c \\ \\in (M^*,\\sqrt{2}M^*),\\\\(ii)\\ \\ dJ|_{(u_m)\\rightarrow 0.",
"}\\end{array}\\right.then, J|_{ has a critical point v_0 such that J(v_0)=c.",
"}}\\end{equation}$"
],
[
"Proof of Theorem ",
"It is easy to see that $0<M\\le M^*$ , from the fact that $Q(x)\\rightarrow \\bar{Q}$ as $\\vert x\\vert \\rightarrow +\\infty $ .",
"Under the assumptions in Remark REF , when the ground state is achieved it is easy to see that $ M^*< M$ which is a contradiction.",
"If $M<M^*$ , there must exist a sequence $\\lbrace u_m\\rbrace \\subset such that as $ m+$\\begin{equation}J(u_m)\\rightarrow M\\ \\ \\ \\ dJ|_{(u_m)\\rightarrow 0.",
"}Consequently, by Lemma \\ref {2-7}, J|_{ has a critical point v_0 \\in such that J(v_0)=M,\\ \\ \\ dJ|_{(v_0)=0.Taking u^0=M^\\frac{1}{2}v_0, it is easy to see that u^0 is a positive solution of (\\ref {1-0}) and I(u^0)=\\frac{1}{4}M^2.If M=M^* and M can be achieved in , there also a positive solution of (\\ref {1-0}).",
"Next we always assume M=M^* and M can not be achieved.Defined \\beta (u):\\ H^1(\\mathbb {R}^N)\\rightarrow \\mathbb {R}^N as following\\begin{equation}\\beta (u)=\\int _{\\mathbb {R}^N}u^2\\chi (\\vert x\\vert )\\cdot x,\\end{equation}where\\begin{eqnarray}\\chi (t)={\\left\\lbrace \\begin{array}{ll}1\\ \\ \\ \\ \\ \\ \\ 0\\le t\\le 1\\\\\\frac{1}{t}\\ \\ \\ \\ \\ \\ t>1.\\end{array}\\right.",
"}\\end{eqnarray}Let \\overline{ be defined as \\overline{=\\lbrace u|u\\in \\beta (u)=0\\rbrace and \\bar{u} be a positive solution of (\\ref {1-1}) achieving its maximum at the origin.\\begin{Lem}Let \\bar{M}=\\inf \\lbrace J(u)\\,|\\,u\\in \\overline{\\rbrace .",
"If M=M^* can not be achieved in , then M<\\bar{M} and there exists R>0 such that\\begin{eqnarray*}{\\left\\lbrace \\begin{array}{ll}(i)\\ \\ \\ J(h(y))\\ \\in \\ \\ (M,\\frac{M+\\bar{M}}{2})\\ \\ \\ if \\ \\vert y\\vert \\ge R,\\\\(ii)\\ \\ \\ (\\beta \\cdot h(y),y)>0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ if\\ \\ \\vert y\\vert =R,\\end{array}\\right.",
"}\\end{eqnarray*}where h(y)=\\bar{u}(x-y)/(\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2)^\\frac{1}{4}.",
"}\\begin{proof}It is obvious that \\bar{M}\\ge M=M^*.",
"To prove \\bar{M}>M, we shall argue by contradiction.",
"Suppose \\bar{M}=M, then there exists a sequence \\lbrace u_m\\rbrace \\subset \\overline{ such that J(u_m)\\rightarrow M and dJ|_{V}(u_m)\\rightarrow 0 as m\\rightarrow +\\infty .",
"There exists u^0 such that u_m\\rightharpoonup u^0 in H^1(\\mathbb {R}^N).Let v_m=M^\\frac{1}{2}u_m, we deduced that as m\\rightarrow +\\infty \\begin{eqnarray}I(v_m)&\\rightarrow &\\frac{1}{4}M^2,\\\\dI(v_m)&\\rightarrow & 0.\\end{eqnarray}Denote v^0=M^\\frac{1}{2}u^0 from Remark \\ref {2-5}, we have\\begin{equation}v_m(x)=v^0+u(x-y_m)+w_m(x),\\end{equation}where u is either 0 or positive solution of (\\ref {1-1}) and w_m\\rightarrow 0.If u\\equiv 0, we get v^0\\lnot \\equiv 0,\\ v_m\\rightarrow v^0.",
"Thus u_m\\rightarrow u^0 and u^0 \\in \\ J(u^0)=M which is a contradiction.So u\\lnot \\equiv 0, hence v^0\\equiv 0.",
"}Let us set (\\mathbb {R}^N)^{+}_m=\\lbrace x\\in \\mathbb {R}^N : (x,y_m)>0\\rbrace and (\\mathbb {R}^N)^{-}_m=\\mathbb {R}^N\\backslash (\\mathbb {R}^N)^{+}_m.",
"Choosing m large enough, since \\vert y_m\\vert \\rightarrow +\\infty , we can assert that there is a ball B_r(y_m)=\\lbrace x \\in \\mathbb {R}^N : \\vert x-y_m\\vert <r\\rbrace \\subset (\\mathbb {R}^N)^{+}_m such that \\forall \\ x\\in B_r(y_m),\\ \\ u(x-y_m)\\ge \\frac{1}{2}u(0)>0.By Proposition \\ref {1-2}\\begin{eqnarray}&&(\\beta (u(x-y_m)),y_m)\\nonumber \\\\&=&\\int _{(\\mathbb {R}^N)^{+}_m}u(x-y_m)\\chi (\\vert x\\vert )(x,y_m)+\\int _{(\\mathbb {R}^N)^{-}_m}u(x-y_m)\\chi (\\vert x\\vert )(x,y_m)\\nonumber \\\\&\\ge &\\int _{B_r(y_m)}\\frac{1}{2}u(0)\\chi (\\vert x\\vert )(x,y_m)-\\int _{(\\mathbb {R}^N)^{-}_m}\\frac{kR\\vert y_m\\vert }{e^{\\sigma \\vert x-y_m\\vert }}\\nonumber \\\\&\\ge & C-o(\\frac{1}{\\vert y_m\\vert }).\\end{eqnarray}where C is a positive constant.Thus \\beta (v_m)\\ne 0 for m large enough.",
"So \\beta (u_m)\\ne 0 for large m which is a contradiction.\\end{proof}By Q(x)\\rightarrow \\bar{ Q} as \\vert x\\vert \\rightarrow +\\infty , it is easy to check that h(y) is continuous on y and J(h(y))\\rightarrow M^*=M, then (i) is satisfied by choosing R>0 large enough.",
"(ii) is analogous to the calculation of (\\ref {3-1}), (\\beta (h(y)),y)>0 if \\vert y\\vert =R.\\end{Lem}}For fixed R define\\begin{eqnarray}F&=&\\lbrace f\\in C(\\overline{B_R},:f|_{\\partial B_R}=h|_{\\partial B_R}\\rbrace ,\\\\c&=&\\inf _{f\\in F}\\max _{y\\in \\overline{B_R}}J(f(y)).\\end{eqnarray}}Since Lemma \\ref {3-0} (ii), by Brouwer degree, for any f\\in F there exists a point y\\in B_R such that \\beta (f(y))=0 and consequently f(y)\\in \\overline{.",
"So c\\ge \\bar{M}>M=M^*.",
"Condition (\\textbf {C}) deduces for y\\in \\mathbb {R}^N\\begin{equation*}\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2>\\frac{\\sqrt{2}}{2}\\int _{\\mathbb {R}^N}\\bar{Q}(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2\\end{equation*}As a consequence, we get\\begin{eqnarray}\\max _{y\\in \\overline{B_R}}J(h(y))&=&\\max _{y\\in \\overline{B_R}}\\frac{\\int _{\\mathbb {R}^N}\\vert \\nabla \\bar{u}\\vert ^2+\\vert \\bar{u}\\vert ^2}{(\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2)^\\frac{1}{2}}\\nonumber \\\\&<&\\sqrt{2}\\frac{\\int _{\\mathbb {R}^N}\\vert \\nabla \\bar{u}\\vert ^2+\\vert \\bar{u}\\vert ^2}{(\\int _{\\mathbb {R}^N}\\bar{Q}(I_2*\\vert \\bar{u}(x)\\vert ^2)\\vert \\bar{u}(x)\\vert ^2)^\\frac{1}{2}}\\nonumber \\\\&=&\\sqrt{2}M^*.\\end{eqnarray}M^*<\\bar{M}\\le c<\\sqrt{2}M^*.",
"By Lemma \\ref {3-0} (i)\\begin{equation}\\max _{y\\in \\partial B_R}J(h(y))<\\frac{M+\\bar{M}}{2}<\\bar{M}<c.\\end{equation}Thus by Lemma \\ref {2-7}, we conclude that J|_{ has a critical point v_0 such that J(v_0)=c, dJ|_{(v_0)=0.Let u^0=c^\\frac{1}{2}v_0, then it is easy to see that u^0 is a positive high energy solution of (\\ref {1-0}) and I(u^0)=\\frac{1}{4}c^2<\\frac{1}{2}{M^*}^2.Thus we complete the proof of Theorem \\ref {1-3}.",
"}\\vspace{14.22636pt}{\\bf Acknowledgments:}This work was partially supported by NSFC grants (No.11771469 and No.11688101).",
"Cao was also supported by the Key Laboratory of Random Complex Structures and Data Science, AMSS, Chinese Academy of Sciences (2008DP173182).",
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"}\\bibitem {B.V} V. Benci, G. Cerami, \\textit {Positive solutions of some nonlinear elliptic problems in exterior domains}, Arch.",
"Rational Mech.",
"Analysis 99, 283-300, (1987).\\end{equation}\\bibitem {D.C} Daomin Cao, \\textit {Positive solution and bifurcation from the essential spactrum of a semilinear elliptic equation on $ RN$}, Nonlinear Analysis 15(11), 1045-1052, (1990).$ Daomin Cao, Positive solution of a semilinear elliptic equation on $\\mathbb {R}^N$, J.Partial Differential Equations 8(3), 261-272, (1995).",
"Giusi Vaira, Existence of Bound States for Schrodinger-Newton Type Systems, Advanced Nonlinear Studies 13, 495-516, (2013).",
"Giusi Vaira, Ground states for Schrodinger-Poisson type systems, Ricerche mat 60, 263-297, (2011) G.H.",
"Hardy, J.E.",
"Littlewood, G. Polya, Inequalities, 2nd ed.",
"Cambridge University Press, (1952).",
"E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard¡¯s nonlinear equation, Stud.",
"Appl.",
"Math.",
"57(2), 93-105, (1976).",
"E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal.",
"PDE 2(1), 1-27, (2009).",
"P.L.",
"Lions, The concentration-compactness principle in the calculus of variations.",
"The locally compact case.",
"I., Ann.",
"Inst.",
"H. Poincar¡äe Anal.",
"Non Lineaire 1(2), 109-145, (1984).",
"Li Ma, Lin Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch.",
"Rational Mech.",
"Anal.",
"195(2), 455-467, (2010).",
"G.P.",
"Menzala, On regular solutions of a nonlinear equation of Choquard¡¯s type, Proc.",
"Royal Soc.",
"Edinburg Sect.",
"A 86(3-4), 291-301, (1980).",
"V. Moroz, J.Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct.",
"Anal.",
"265, 153-184, (2013).",
"V. Moroz, J.Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Comm.",
"Cont.",
"Math., 17(5), (2015).",
"S. Pekar, Untersuchung $\\ddot{u}$ ber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954).",
"Tod, K.P., Moroz, I.M., An analytical approach to the Schr¡§odinger-Newton equations, Nonlinearity 12(2), 201-216, (1999).",
"Tao Wang, Yia, T., Uniqueness of positive solutions of the Choquard type equations, Appl.",
"Anal.",
"96, 409-417, (2017).",
"Changlin Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc.",
"Var.& PDE.",
"55, 134-159, (2016).",
"Xiping Zhu, Daomin Cao, The concentration-compact principle in nonlinear equations, Acta Math.",
"Sci.",
"9(3),307-328, (1989).",
"It is easy to see that $0<M\\le M^*$ , from the fact that $Q(x)\\rightarrow \\bar{Q}$ as $\\vert x\\vert \\rightarrow +\\infty $ .",
"Under the assumptions in Remark REF , when the ground state is achieved it is easy to see that $ M^*< M$ which is a contradiction.",
"If $M<M^*$ , there must exist a sequence $\\lbrace u_m\\rbrace \\subset such that as $ m+$\\begin{equation}J(u_m)\\rightarrow M\\ \\ \\ \\ dJ|_{(u_m)\\rightarrow 0.",
"}Consequently, by Lemma \\ref {2-7}, J|_{ has a critical point v_0 \\in such that J(v_0)=M,\\ \\ \\ dJ|_{(v_0)=0.Taking u^0=M^\\frac{1}{2}v_0, it is easy to see that u^0 is a positive solution of (\\ref {1-0}) and I(u^0)=\\frac{1}{4}M^2.If M=M^* and M can be achieved in , there also a positive solution of (\\ref {1-0}).",
"Next we always assume M=M^* and M can not be achieved.Defined \\beta (u):\\ H^1(\\mathbb {R}^N)\\rightarrow \\mathbb {R}^N as following\\begin{equation}\\beta (u)=\\int _{\\mathbb {R}^N}u^2\\chi (\\vert x\\vert )\\cdot x,\\end{equation}where\\begin{eqnarray}\\chi (t)={\\left\\lbrace \\begin{array}{ll}1\\ \\ \\ \\ \\ \\ \\ 0\\le t\\le 1\\\\\\frac{1}{t}\\ \\ \\ \\ \\ \\ t>1.\\end{array}\\right.",
"}\\end{eqnarray}Let \\overline{ be defined as \\overline{=\\lbrace u|u\\in \\beta (u)=0\\rbrace and \\bar{u} be a positive solution of (\\ref {1-1}) achieving its maximum at the origin.\\begin{Lem}Let \\bar{M}=\\inf \\lbrace J(u)\\,|\\,u\\in \\overline{\\rbrace .",
"If M=M^* can not be achieved in , then M<\\bar{M} and there exists R>0 such that\\begin{eqnarray*}{\\left\\lbrace \\begin{array}{ll}(i)\\ \\ \\ J(h(y))\\ \\in \\ \\ (M,\\frac{M+\\bar{M}}{2})\\ \\ \\ if \\ \\vert y\\vert \\ge R,\\\\(ii)\\ \\ \\ (\\beta \\cdot h(y),y)>0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ if\\ \\ \\vert y\\vert =R,\\end{array}\\right.",
"}\\end{eqnarray*}where h(y)=\\bar{u}(x-y)/(\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2)^\\frac{1}{4}.",
"}\\begin{proof}It is obvious that \\bar{M}\\ge M=M^*.",
"To prove \\bar{M}>M, we shall argue by contradiction.",
"Suppose \\bar{M}=M, then there exists a sequence \\lbrace u_m\\rbrace \\subset \\overline{ such that J(u_m)\\rightarrow M and dJ|_{V}(u_m)\\rightarrow 0 as m\\rightarrow +\\infty .",
"There exists u^0 such that u_m\\rightharpoonup u^0 in H^1(\\mathbb {R}^N).Let v_m=M^\\frac{1}{2}u_m, we deduced that as m\\rightarrow +\\infty \\begin{eqnarray}I(v_m)&\\rightarrow &\\frac{1}{4}M^2,\\\\dI(v_m)&\\rightarrow & 0.\\end{eqnarray}Denote v^0=M^\\frac{1}{2}u^0 from Remark \\ref {2-5}, we have\\begin{equation}v_m(x)=v^0+u(x-y_m)+w_m(x),\\end{equation}where u is either 0 or positive solution of (\\ref {1-1}) and w_m\\rightarrow 0.If u\\equiv 0, we get v^0\\lnot \\equiv 0,\\ v_m\\rightarrow v^0.",
"Thus u_m\\rightarrow u^0 and u^0 \\in \\ J(u^0)=M which is a contradiction.So u\\lnot \\equiv 0, hence v^0\\equiv 0.",
"}Let us set (\\mathbb {R}^N)^{+}_m=\\lbrace x\\in \\mathbb {R}^N : (x,y_m)>0\\rbrace and (\\mathbb {R}^N)^{-}_m=\\mathbb {R}^N\\backslash (\\mathbb {R}^N)^{+}_m.",
"Choosing m large enough, since \\vert y_m\\vert \\rightarrow +\\infty , we can assert that there is a ball B_r(y_m)=\\lbrace x \\in \\mathbb {R}^N : \\vert x-y_m\\vert <r\\rbrace \\subset (\\mathbb {R}^N)^{+}_m such that \\forall \\ x\\in B_r(y_m),\\ \\ u(x-y_m)\\ge \\frac{1}{2}u(0)>0.By Proposition \\ref {1-2}\\begin{eqnarray}&&(\\beta (u(x-y_m)),y_m)\\nonumber \\\\&=&\\int _{(\\mathbb {R}^N)^{+}_m}u(x-y_m)\\chi (\\vert x\\vert )(x,y_m)+\\int _{(\\mathbb {R}^N)^{-}_m}u(x-y_m)\\chi (\\vert x\\vert )(x,y_m)\\nonumber \\\\&\\ge &\\int _{B_r(y_m)}\\frac{1}{2}u(0)\\chi (\\vert x\\vert )(x,y_m)-\\int _{(\\mathbb {R}^N)^{-}_m}\\frac{kR\\vert y_m\\vert }{e^{\\sigma \\vert x-y_m\\vert }}\\nonumber \\\\&\\ge & C-o(\\frac{1}{\\vert y_m\\vert }).\\end{eqnarray}where C is a positive constant.Thus \\beta (v_m)\\ne 0 for m large enough.",
"So \\beta (u_m)\\ne 0 for large m which is a contradiction.\\end{proof}By Q(x)\\rightarrow \\bar{ Q} as \\vert x\\vert \\rightarrow +\\infty , it is easy to check that h(y) is continuous on y and J(h(y))\\rightarrow M^*=M, then (i) is satisfied by choosing R>0 large enough.",
"(ii) is analogous to the calculation of (\\ref {3-1}), (\\beta (h(y)),y)>0 if \\vert y\\vert =R.\\end{Lem}}For fixed R define\\begin{eqnarray}F&=&\\lbrace f\\in C(\\overline{B_R},:f|_{\\partial B_R}=h|_{\\partial B_R}\\rbrace ,\\\\c&=&\\inf _{f\\in F}\\max _{y\\in \\overline{B_R}}J(f(y)).\\end{eqnarray}}Since Lemma \\ref {3-0} (ii), by Brouwer degree, for any f\\in F there exists a point y\\in B_R such that \\beta (f(y))=0 and consequently f(y)\\in \\overline{.",
"So c\\ge \\bar{M}>M=M^*.",
"Condition (\\textbf {C}) deduces for y\\in \\mathbb {R}^N\\begin{equation*}\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2>\\frac{\\sqrt{2}}{2}\\int _{\\mathbb {R}^N}\\bar{Q}(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2\\end{equation*}As a consequence, we get\\begin{eqnarray}\\max _{y\\in \\overline{B_R}}J(h(y))&=&\\max _{y\\in \\overline{B_R}}\\frac{\\int _{\\mathbb {R}^N}\\vert \\nabla \\bar{u}\\vert ^2+\\vert \\bar{u}\\vert ^2}{(\\int _{\\mathbb {R}^N}Q(x)(I_2*\\vert \\bar{u}(x-y)\\vert ^2)\\vert \\bar{u}(x-y)\\vert ^2)^\\frac{1}{2}}\\nonumber \\\\&<&\\sqrt{2}\\frac{\\int _{\\mathbb {R}^N}\\vert \\nabla \\bar{u}\\vert ^2+\\vert \\bar{u}\\vert ^2}{(\\int _{\\mathbb {R}^N}\\bar{Q}(I_2*\\vert \\bar{u}(x)\\vert ^2)\\vert \\bar{u}(x)\\vert ^2)^\\frac{1}{2}}\\nonumber \\\\&=&\\sqrt{2}M^*.\\end{eqnarray}M^*<\\bar{M}\\le c<\\sqrt{2}M^*.",
"By Lemma \\ref {3-0} (i)\\begin{equation}\\max _{y\\in \\partial B_R}J(h(y))<\\frac{M+\\bar{M}}{2}<\\bar{M}<c.\\end{equation}Thus by Lemma \\ref {2-7}, we conclude that J|_{ has a critical point v_0 such that J(v_0)=c, dJ|_{(v_0)=0.Let u^0=c^\\frac{1}{2}v_0, then it is easy to see that u^0 is a positive high energy solution of (\\ref {1-0}) and I(u^0)=\\frac{1}{4}c^2<\\frac{1}{2}{M^*}^2.Thus we complete the proof of Theorem \\ref {1-3}.",
"}\\vspace{14.22636pt}{\\bf Acknowledgments:}This work was partially supported by NSFC grants (No.11771469 and No.11688101).",
"Cao was also supported by the Key Laboratory of Random Complex Structures and Data Science, AMSS, Chinese Academy of Sciences (2008DP173182).",
"}}\\newpage {\\bf References}}\\begin{thebibliography}{1}\\end{thebibliography}\\bibitem {A.B} Abbas Bahri, Yanyan Li, \\textit {On a Min-Max Procedure for the Existence of a Positive Solution for Certain Scalar Field Equations in \\mathbb {R}^N}, Rev.",
"Mat.",
"Iberoamericana 6, 1-15, (1990).",
"}\\bibitem {B.V} V. Benci, G. Cerami, \\textit {Positive solutions of some nonlinear elliptic problems in exterior domains}, Arch.",
"Rational Mech.",
"Analysis 99, 283-300, (1987).\\end{equation}\\bibitem {D.C} Daomin Cao, \\textit {Positive solution and bifurcation from the essential spactrum of a semilinear elliptic equation on $ RN$}, Nonlinear Analysis 15(11), 1045-1052, (1990).$ Daomin Cao, Positive solution of a semilinear elliptic equation on $\\mathbb {R}^N$, J.Partial Differential Equations 8(3), 261-272, (1995).",
"Giusi Vaira, Existence of Bound States for Schrodinger-Newton Type Systems, Advanced Nonlinear Studies 13, 495-516, (2013).",
"Giusi Vaira, Ground states for Schrodinger-Poisson type systems, Ricerche mat 60, 263-297, (2011) G.H.",
"Hardy, J.E.",
"Littlewood, G. Polya, Inequalities, 2nd ed.",
"Cambridge University Press, (1952).",
"E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard¡¯s nonlinear equation, Stud.",
"Appl.",
"Math.",
"57(2), 93-105, (1976).",
"E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal.",
"PDE 2(1), 1-27, (2009).",
"P.L.",
"Lions, The concentration-compactness principle in the calculus of variations.",
"The locally compact case.",
"I., Ann.",
"Inst.",
"H. Poincar¡äe Anal.",
"Non Lineaire 1(2), 109-145, (1984).",
"Li Ma, Lin Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch.",
"Rational Mech.",
"Anal.",
"195(2), 455-467, (2010).",
"G.P.",
"Menzala, On regular solutions of a nonlinear equation of Choquard¡¯s type, Proc.",
"Royal Soc.",
"Edinburg Sect.",
"A 86(3-4), 291-301, (1980).",
"V. Moroz, J.Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct.",
"Anal.",
"265, 153-184, (2013).",
"V. Moroz, J.Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Comm.",
"Cont.",
"Math., 17(5), (2015).",
"S. Pekar, Untersuchung $\\ddot{u}$ ber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954).",
"Tod, K.P., Moroz, I.M., An analytical approach to the Schr¡§odinger-Newton equations, Nonlinearity 12(2), 201-216, (1999).",
"Tao Wang, Yia, T., Uniqueness of positive solutions of the Choquard type equations, Appl.",
"Anal.",
"96, 409-417, (2017).",
"Changlin Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc.",
"Var.& PDE.",
"55, 134-159, (2016).",
"Xiping Zhu, Daomin Cao, The concentration-compact principle in nonlinear equations, Acta Math.",
"Sci.",
"9(3),307-328, (1989)."
]
] | 1709.01817 |
[
[
"Tunable quantum spin liquidity in the 1/6th-filled breathing kagome\n lattice"
],
[
"Abstract We present measurements on a series of materials, Li$_2$In$_{1-x}$Sc$_x$Mo$_3$O$_8$, that can be described as a 1/6th-filled breathing kagome lattice.",
"Substituting Sc for In generates chemical pressure which alters the breathing parameter non-monotonically.",
"$\\mu$SR experiments show that this chemical pressure tunes the system from antiferromagnetic long range order to a quantum spin liquid phase.",
"A strong correlation with the breathing parameter implies that it is the dominant parameter controlling the level of magnetic frustration, with increased kagome symmetry generating the quantum spin liquid phase.",
"Magnetic susceptibility measurements suggest that this is related to distinct types of charge order induced by changes in lattice symmetry, in line with the theory of Chen et al.",
"[Phys.",
"Rev.",
"B 93, 245134 (2016)].",
"The specific heat for samples at intermediate Sc concentration and with minimal breathing parameter, show consistency with the predicted $U(1)$ quantum spin liquid."
],
[
"Tunable quantum spin liquidity in the 1/6th-filled breathing kagome lattice A. Akbari-Sharbaf Institut Quantique and Département de Physique, Université de Sherbrooke, 2500 boul.",
"de l'Université, Sherbrooke (Québec) J1K 2R1 Canada R. Sinclair Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee, 37996-1200, USA A. Verrier D. Ziat Institut Quantique and Département de Physique, Université de Sherbrooke, 2500 boul.",
"de l'Université, Sherbrooke (Québec) J1K 2R1 Canada H. D. Zhou Key laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai JiaoTong University, Shanghai, 200240, China Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee, 37996-1200, USA X. F. Sun Department of Physics, Hefei National Laboratory for Physical Sciences at Microscale, and Key Laboratory of Strongly-Coupled Quantum Matter Physics (CAS), University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China Institute of Physical Science and Information Technology, Anhui University, Hefei, Anhui 230601, People's Republic of China Collaborative Innovation Center of Advanced Microstructures, Nanjing, Jiangsu 210093, People's Republic of China J. A.",
"Quilliam [email protected] Institut Quantique and Département de Physique, Université de Sherbrooke, 2500 boul.",
"de l'Université, Sherbrooke (Québec) J1K 2R1 Canada We present measurements on a series of materials, Li$_2$ In$_{1-x}$ Sc$_x$ Mo$_3$ O$_8$ , that can be described as a 1/6th-filled breathing kagome lattice.",
"Substituting Sc for In generates chemical pressure which alters the breathing parameter non-monotonically.",
"$\\mu $ SR experiments show that this chemical pressure tunes the system from antiferromagnetic long range order to a quantum spin liquid phase.",
"A strong correlation with the breathing parameter implies that it is the dominant parameter controlling the level of magnetic frustration, with increased kagome symmetry generating the quantum spin liquid phase.",
"Magnetic susceptibility measurements suggest that this is related to distinct types of charge order induced by changes in lattice symmetry, in line with the theory of Chen et al.",
"[Phys.",
"Rev.",
"B 93, 245134 (2016)].",
"The specific heat for samples at intermediate Sc concentration and with minimal breathing parameter, show consistency with the predicted $U(1)$ quantum spin liquid.",
"75.50.Lk, 75.50.Ee, 75.40.Cx One of the most sought after magnetic phases is the quantum spin liquid (QSL), wherein spins form a highly-entangled quantum ground state that supports fractional spin excitations [1].",
"Two main approaches to the discovery of QSL materials have been especially fruitful in recent years: spin-1/2 kagome antiferromagnets [2], [3], [4] and triangular-lattice antiferromagnets near a Mott transition [5], [6], [7], [8], [9], [10].",
"However, much remains to be understood about these experimental QSL candidates and some properties remain difficult to reconcile with theory [11], [12], [8].",
"Hence the search for new QSL candidates based on different mechanisms, for example [13], [14], remains a valuable pursuit.",
"In particular, systems in which Hamiltonian parameters can be continuously tuned may provide a prime opportunity to link theoretical models to experimental phenomena.",
"In this Letter, we demonstrate that a high degree of tunability can be achieved with the materials, Li$_2$ In$_{1-x}$ Sc$_x$ Mo$_3$ O$_8$ , that incorporate both spin and charge degrees of freedom.",
"This family of materials consists of a “breathing” kagome lattice (BKL) of Mo ions wherein the triangles that point upward are slightly smaller than those that point downward [15], [16], [13], with a “breathing ratio” $\\lambda = d_\\nabla / d_\\Delta $ .",
"In these particular materials the lattice is 1/6th filled, with one unpaired electron for every 3 Mo sites, and its insulating character is ensured by strong next-nearest-neighbor interactions ($V_1$ on up-triangles and $V_2$ on down-triangles) [17].",
"Figure: Illustrations of (a) the type-I cluster Mott insulator, where electrons are localized on Mo 3 _3 units, leading to 120 ∘ ^{\\circ } antiferromagnetic order and (b) the PCO state.",
"Resonating hexagons are depicted by dashed circles, and the two spatial configurations of the collective tunnelling electrons are depicted by the open and full circles.",
"(c) The ratio of lattice parameters, a/ca/c and (d) breathing parameter λ\\lambda as a function of xx.",
"The shaded region is a guide to the eye.",
"Error bars from the refinement are smaller than the data points.As proposed by Sheckelton et al.",
"[17] for LiZn$_2$ Mo$_3$ O$_8$ (LZMO), a similar QSL candidate material [18], a plausible charge configuration consists of each electron delocalized over one “up-triangle”, ultimately leading to a triangular lattice of spin-1/2 moments on Mo$_3$ O$_{13}$ clusters, as depicted in Fig.",
"REF (a).",
"However, it has been pointed out [19] that, due to the large spatial extent of the 4$d$ electrons, the single unpaired electrons may have a non-zero probability of tunnelling between adjacent clusters.",
"When $\\lambda $ is large, the electrons are expected to localize on the smaller triangles, recovering the Type-I cluster Mott insulator (CMI) proposed by Sheckelton et al. [17].",
"When $V_2$ becomes comparable to $V_1$ it is energetically favorable for electrons to collectively tunnel between the small triangles, giving rise to a long range plaquette charge order (PCO), or Type-II CMI, as depicted in Fig.",
"REF (b).",
"We will show that $x$ in Li$_2$ In$_{1-x}$ Sc$_x$ Mo$_3$ O$_8$ , tunes the system from a long range ordered (LRO) magnetic phase to a QSL phase and propose that these distinct magnetic phases are a result of the distinct charge configurations.",
"Although the end points of this family (at $x=0$ and $x=1$ ) have been studied previously [20], [21], [22], we show that intermediate stoichiometries are essential to generating a homogeneous QSL.",
"Our experimental results agree well with the theoretical framework developed by Chen et al.",
"[19] and highlight a valuable new system for the study of QSL physics.",
"Polycrystalline samples of Li$_2$ In$_{1-x}$ Sc$_x$ Mo$_3$ O$_8$ were synthesized by solid-state reaction.",
"A stoichiometric mixture of Li$_2$ MoO$_4$ , Sc$_2$ O$_3$ , In$_2$ O$_3$ , MoO$_3$ , and Mo were ground together and pressed into 6 mm diameter, 60 mm long rods under 400 atm hydrostatic pressure which were placed in alumina crucibles and sealed in silica tubes at a pressure of $10^{-4}$ mbar.",
"Finally, the samples were annealed for 48 hours at 850 C. Powder X-ray diffraction (XRD) patterns were recorded at room temperature with a HUBER Imaging Plate Guinier Camera 670 with Ge monochromatized Cu K $\\alpha $ 1 radiation (1.54059 Å).",
"Mo-Mo bond lengths were refined by the Rietveld method [23] with $\\chi ^2$ in the range 1-2 for all samples.",
"Susceptibility measurements were performed at 2 T and specific heat measurements were carried out in zero field, with Quantum Design MPMS and PPMS systems.",
"$\\mu $ SR measurements were carried out at TRIUMF in zero field (ZF) and longitudinal field (LF).",
"Measurements in the range from 25 mK up to 3 K were performed with the samples affixed to an Ag cold finger of a dilution refrigerator.",
"Higher temperature measurements were carried out in veto mode to eliminate the background asymmetry and were used to correct for the background present at low temperatures.",
"XRD spectra [24] reveal that as the In ions are replaced by smaller Sc ions, the lattice parameters decrease and, as seen in Fig.",
"REF (c), the ratio $a/c$ varies monotonically with a total change of about 1.4%.",
"It is important to investigate the evolution of the breathing parameter with $x$ and the XRD measurements reveal a non-monotonic behavior of $\\lambda (x)$ , as can be seen in Fig.",
"REF (d).",
"The parent compound ($x = 0$ ) has the highest average degree of asymmetry, whereas at a concentration of 60% In and 40% Sc ($x = 0.6$ ) the lowest degree of asymmetry is attained.",
"Meanwhile, the reported structure of LZMO [17] corresponds to a breathing parameter of $\\lambda \\simeq 1.23$ , making it closer to an ideal kagome lattice than the most symmetric sample in the series studied here.",
"Figure: (a) Zero-field muon spin polarization P(t)P(t) for Li 2 _2InMo 3 _3O 8 _8 (x=0x = 0).",
"(b) Zero-field P(t)P(t) measured at 25 mK for LiIn 1-x _{1-x}Sc x _xMo 3 _3O 8 _8 for different values of xx.",
"Polarization in various longitudinal fields for (c) x=0.6x=0.6 and (d) x=0.2x=0.2.",
"The black lines are fits as described in the text.In general, the $\\mu $ SR polarization measured for our samples shows that the muon spins are influenced by a mix of fluctuating and static electron spins and the data are fitted with a two-component polarization function, $P_{\\mathrm {tot}} = fP_S(t) + (1-f)P_D(t)$ , where $P_S(t)$ is the polarization for the fraction $f$ of muons stopping in a static fraction (ordered or frozen) and $P_D(t)$ is the contribution from regions with dynamic electron spins, either QSL or paramagnetic phases.",
"For the dynamic fraction, $P_D(t) = P_N(t)e^{-t/T_1}$ where $P_N(t)$ is a nuclear Gaussian Kubo-Toyabe function and $1/T_1$ is the spin-lattice relaxation rate.",
"The ZF $\\mu $ SR asymmetry measured at 1.9 K for Li$_2$ InMo$_3$ O$_8$ ($x = 0$ ) shown in Fig.",
"REF (a) features a slowly decaying oscillation, demonstrating LRO with well defined internal fields consistent with NMR measurements of the same stoichiometry [20].",
"$P_S(t)$ for this sample has thus been fitted to the static Lorentzian Koptev-Tarasov polarization function.",
"Four distinct frequencies (1.1, 1.4, 2.0 and 3.3 MHz) are extracted, which correspond well to the four inequivalent oxygen sites.",
"Select polarization curves at different temperatures in Fig.",
"REF (a) show the reduction of the oscillation frequencies (and order parameter) and the appearance of a dynamic fraction of the sample as the temperature is raised.",
"The smallness of the observed frequencies is consistent with each spin-1/2 moment being highly distributed over a Mo$_3$ O$_{13}$ cluster, similar to observations in systems of mixed-valence Ru dimers [25].",
"For $x=0.2$ , $x=0.4$ and $x=1$ , we find an inhomogeneous mix of disordered static magnetism (giving a quickly relaxing signal) and a weakly relaxing dynamic fraction as shown in Fig.",
"REF (b).",
"The frozen fraction represents 49%, 25% and 43% of these samples, respectively.",
"On the other hand, $P(t)$ for $x = 0.6$ shows no indication of static fields originating from electron spins to as low as 25 mK, which suggests that the entire sample is in a homogeneous QSL phase.",
"In fact, the $\\mu $ SR asymmetry profile for $x =0.6$ is very similar to that of LZMO [26].",
"To fit the inhomogeneous samples, a Lorentzian Kubo-Toyabe function was used for $P_S(t)$ .",
"This fitting has been performed in zero- and longitudinal-field, $B_L$ , as shown in Fig.",
"REF (d) and in supplemental material [24].",
"This analysis conclusively demonstrates that we have correctly identified the frozen and dynamic fractions of the sample since the muon spins are much more quickly decoupled from static than dynamic magnetism.",
"For the homogeneous QSL sample, small $B_L$ quickly decouples the muon spins from the nuclear moments, but at higher field relaxation persists, indicating relaxation that is purely of dynamic origin, as seen in Fig.",
"REF (c).",
"As shown in Fig.",
"REF (a), $1/T_1(B_L)$ for the QSL fractions is fairly well fit with Redfield theory [27] using a sum of two characteristic fluctuation frequencies.",
"Meanwhile $1/T_1$ of the liquid fractions shows relaxation plateaus in temperature below $\\sim 1$ K, a common but still poorly understood feature of QSL candidates [28], [29], [30], [3].",
"Figure: (a) Spin-lattice relaxation rate vs. longitudinal field at base temperature for the liquid phase of several samples, with fits given by Redfield theory with two different fluctuation frequencies.",
"(b) Relaxation rate as a function of temperature in longitudinal field of 55 G, showing relaxation plateaus typical of QSL materials.",
"Curves are guides to the eye.Evidently the concentration of Sc does not monotonically change the ratio of static and QSL fraction, but rather there is an optimal concentration of $x = 0.6$ where a homogeneous QSL is stabilized.",
"The phase diagram as a function of $x$ , presented in Fig.",
"REF (c), is highly correlated with the behavior of the breathing parameter, $\\lambda (x)$ , as shown in Fig.",
"REF (d).",
"This suggests that the magnetic phenomenology of this material is intimately connected to the symmetry of the BKL and that past a critical value of $\\lambda $ the system passes from antiferromagnetic to QSL.",
"At critical values of $\\lambda $ , such as for $x=0.2$ and $x=1$ , inhomogeneous phases result.",
"Figure: (a) Temperature dependent inverse magnetic susceptibility χ -1 \\chi ^{-1} for select samples.",
"For x=0x = 0 a sharp feature at the onset of AFM order is indicated by an arrow at 11 K. χ -1 (T)\\chi ^{-1}(T) for the homogeneous QSL sample, x=0.6x = 0.6, shows two apparent Curie-Weiss regimes.",
"The fit is described in the text.",
"(b) The magnetic specific heat C M C_M of select samples.",
"The fit to the x=0x=0 data is a T 3 T^3 power law plus a C N ∝T -2 C_N \\propto T^{-2} nuclear contribution.",
"The specific heat of x=0.6x=0.6 is compared with a T 2/3 T^{2/3} power law plus nuclear contribution, as well as a TT-linear dependence.",
"(c) Magnetic phase diagram for Li 2 _2In 1-x _{1-x}Sc x _xMo 3 _3O 8 _8.",
"Red squares show the onset of freezing determined by specific heat (for x=x=0 and 0.2) μ\\mu SR (for the remaining samples).",
"The dark red region shows antiferromagnetic (AFM) ordering, whereas pink regions show spin freezing, either spin glass (SG) or disordered antiferromagnetism.",
"The blue region shows the approximate temperature onset of the relaxation plateau in μ\\mu SR.The way in which $\\lambda $ influences the charge degrees of freedom, and consequently the spins, may be better understood with the magnetic susceptibility, $\\chi $ , measurements in Fig.",
"REF (a).",
"Our measurements of the end points of the series ($x=0$ and $x=1$ ) are consistent with previous work [20].",
"For intermediate concentrations $\\chi (T)$ is very different.",
"For the homogeneous QSL sample, $x=0.6$ , $\\chi ^{-1}(T)$ displays two apparent linear Curie-Weiss regimes distinguished by different Curie constants and a smooth crossover between the two regimes.",
"The $x=0.4$ and $x=0.8$ samples show similar behavior [24].",
"This strong, qualitative change in $\\chi (T)$ , even at high temperatures, implies that the effect of $x$ on the magnetic ground state is not simply an effect of disorder.",
"The temperature dependence of the susceptibility has been a central focus of the discussion surrounding the Mo$_3$ O$_{13}$ cluster magnet family.",
"Sheckelton et al.",
"first reported two Curie-Weiss regimes for the compound LZMO, where the Curie constant reduces to 1/3 of the high temperature value below a crossover at 96 K [17].",
"They attributed this to the condensation of two-thirds of the spins into singlets [17], [26].",
"Chen et al.",
"[19] proposed an alternative theory for the “1/3-anomaly” in $\\chi ^{-1}(T)$ whereby the low temperature regime corresponds to plaquette charge order (PCO).",
"The PCO reconstructs the spinon bands with the lowest band splitting into 3 sub-bands.",
"The lowest sub-band is completely filled with 2/3 of the spinons, becoming magnetically inert.",
"The upper sub-band is partially filled with the remaining 1/3 spinons and these spinons contribute to $\\chi $ .",
"Chen et al.",
"[31], [19] argue that at the crossover temperature, PCO is destroyed and the full spin degrees of freedom are recovered.",
"However, a transition between these two phases involves a spontaneous breaking of symmetry and should normally give rise to sharp thermodynamic features, the absence of which has been attributed to disorder [19].",
"We propose an alternative mechanism for 1/3-anomaly.",
"If the compounds $x=0.6$ and LiZn$_2$ Mo$_3$ O$_8$ are in the strong PCO regime, the energy scale required to break the PCO ($E_\\mathrm {PCO} \\sim t_1^3/V_2^2$ ) ought to be significantly larger than the energy gap, $\\Delta E$ , between filled and partially filled spinon sub-bands (which is governed by the next-nearest-neighbor interaction strength, $J^{\\prime }$ ), allowing for thermal excitation of spinons across the spinon gap while preserving PCO [32].",
"From a local perspective, each resonating hexagon in the PCO phase is composed of three coupled spins with a $S_\\mathrm {tot} = 1/2$ ground state manifold and a $S_\\mathrm {tot} = 3/2$ excited state.",
"The magnetic susceptibility for non-interacting hexagons can be written as $\\chi _0 = \\frac{\\mu _0N_Ag^2\\mu _{\\beta }^2}{4k_B T} \\frac{1+5e^{-\\Delta E/k_B T}}{1+e^{-\\Delta E/k_B T}} = \\beta (T)\\frac{C_0}{T}.$ The $S_\\mathrm {tot} = 1/2$ ground state is doubly degenerate due to a pseudospin that represents the spatial configuration of entanglement in the resonating hexagon [19].",
"In a mean-field approximation, the interacting susceptibility then gives $\\chi = \\beta (T) C_0/[T-\\ \\beta (T)\\theta _W]$ , naturally leading to two Curie-Weiss regimes with a ratio of 1/3 between the effective Curie constants $C_\\mathrm {eff}= \\beta (T) C_0$ .",
"Eq.",
"REF gives an excellent fit of $\\chi ^{-1}(T)$ measured for sample $x=0.6$ , shown in Fig.",
"REF (a), where the parameters extracted from the fit are $C_0 = 0.264 \\pm 0.001$ emu K Oe$^{-1}$ mol$^{-1}$ , $\\Delta E/k_B = 109 \\pm 1$ K, and $ \\theta _W = -46.3 \\pm 0.5$ K. A fit of Eq.6 to the susceptibility data reported for LiZn$_2$ Mo$_3$ O$_8$ [17] is also successful (see supplemental material [24]), with fitting parameters $C_0 = 0.277 \\pm 0.002$ emu K Oe$^{-1}$ mol$^{-1}$ , $\\Delta E/k_B = 300 \\pm 20$ K, and $\\theta _W = -20 \\pm 10$ K. The same analysis can also be applied to other samples that are primarily spin liquids ($x=0.4$ and $x=0.8$ ) giving slightly different energy gaps.",
"The magnetic specific heat, after lattice subtraction, for select samples is displayed in Fig.",
"REF (b).",
"As expected for LRO, the $x=0$ sample displays a peak at $T_N \\simeq 12$ K and the appropriate power law, $C_M \\propto T^3$ , for gapless magnons.",
"Below 1 K, the specific heat turns upward with a $T^{-2}$ power law which we attribute to the upper limit of a nuclear Schottky anomaly, $C_N$ , likely originating from the $^{95}$ Mo and $^{97}$ Mo hyperfine couplings since the quadrupolar energy of $^{115}$ In is not large enough [22].",
"For samples that are primarily or entirely QSL ($x=$ 0.4, 0.6 and 0.8), there is no sharp peak and the $C_M(T)$ is much shallower.",
"Between 1 and 10 K, $C_M\\propto T$ , but below 1 K $C_M$ becomes even shallower than linear.",
"This shallow temperature dependence of the specific heat in the order-free phase of this series of materials lends further evidence for a $U(1)$ QSL as predicted [19], [5], [33].",
"It can be seen in Fig.",
"REF (b) that if we apply the same nuclear contribution to the specific heat for the $x=0.6$ sample as was determined for the $x=0$ sample, a $T^{2/3}$ power law provides a reasonable fit to the data below $\\sim 2$ K. Hence it is tempting to propose that this intermediate concentration has a $U(1)$ spin liquid state, similar to what has been proposed for the triangular organic QSLs [5], [6], [7], [8], [9], [10], although there $C_M\\propto T$ is observed [7], [9].",
"For $x=1$ a somewhat steeper, $C_M\\sim T^{1.4}$ , is observed similar to the $T^{1.5}$ power law obtained in Ref. [22].",
"The mixture of QSL and magnetic freezing may lead to an intermediate temperature dependence.",
"In conclusion, we have demonstrated a high degree of tunability of the series Li$_2$ In$_{1-x}$ Sc$_x$ Mo$_3$ O$_8$ , through isovalant substitution of In with Sc.",
"The magnetic phase diagram, Fig.",
"REF (c), shows a strong correlation with the breathing parameter, with a homogeneous QSL phase in the most symmetric sample at $x=0.6$ , suggesting that $\\lambda $ is the principal controlling parameter.",
"The nature of $\\chi (T)$ also varies substantially with $x$ .",
"Notably, in the range of $0.4 < x <0.8$ , $\\chi ^{-1}(T)$ is very similar to that of the QSL LZMO, with two apparent Curie-Weiss regimes.",
"This observation fits well with the theory of Chen et al.",
"[19] predicting the 1/3-anomaly in the PCO phase, which should be stabilized by small $\\lambda $ .",
"We propose that the 1/3-anomaly originates from thermal excitations of the resonating hexagons from the $S_\\mathrm {tot} = 1/2$ ground state to a $S_\\mathrm {tot} = 3/2$ excited state.",
"Since smaller $\\lambda $ and the 1/3-anomaly seem to be associated with a QSL ground state, the spins in the PCO phase appear to be more frustrated than in the Type-I CMI.",
"Indeed the specific heat of the homogeneous QSL at $x=0.6$ has a particularly shallow temperature dependence, possibly consistent with a $U(1)$ QSL [5], [33].",
"This work has therefore provided a likely resolution to the debate surrounding LZMO [17].",
"An alternative scenario to explain the 1/3-anomaly in LZMO has been put forward by Flint and Lee [34], wherein the electrons are localized on the up-triangles but two thirds of the clusters rotate, generating an emergent honeycomb lattice, thereby leaving 1/3 of the spins as weakly connected “orphan” spins.",
"However, we find no natural reason that changes in $\\lambda $ would encourage rotation of Mo$_3$ O$_{13}$ clusters and the 1/3 of the spins that remain active at low temperature exhibit a strongly negative Curie-Weiss constant, $\\Theta _W \\simeq -46$ K, meaning they cannot be described as orphan spins.",
"Valuable future work on this series could include direct measurements of charge order with resonant X-ray spectroscopy, although the changes in local charge density will be rather small, as well as a search for thermodynamic indications of charge-ordering at higher temperatures.",
"Furthermore, it would be interesting to study the parent compounds under applied pressure instead of chemical pressure, potentially tuning the system into a QSL phase without introducing structural disorder.",
"Indeed the role of disorder in either destabilizing or even generating QSL-like phases remains a contentious issue in the field [35].",
"Furthermore, although the model proposed by Chen et al.",
"[19] is consistent with our observations, many assumptions have been made regarding the appropriate Hamiltonian for these materials which should be validated with detailed electronic structure calculations.",
"We are grateful to the staff of the Centre for Molecular and Materials Science at TRIUMF for extensive technical support, in particular G. Morris, B. Hitti, D. Arseneau, and I. MacKenzie.",
"We also acknowledge helpful conversations with Y.",
"B. Kim, G. Chen, H.-Y.",
"Kee, M. Gingras, A.-M. Tremblay, F. Bert and P. Mendels.",
"A. A.-S. and J. Q. acknowledge funding through NSERC, FRQNT, CFI and CFREF grants.",
"H. D. Z. acknowledges support from the Ministry of Science and Technology of China with grant number 2016YFA0300500.",
"R. S. and H. D. Z. acknowledge support from NSF-DMR with grant number NSF-DMR-1350002.",
"X. F. S. acknowledges support from the National Natural Science Foundation of China (Grant Nos.",
"11374277, U1532147) and the National Basic Research Program of China (Grant Nos.",
"2015CB921201, 2016YFA0300103)."
]
] | 1709.01904 |
[
[
"Dynamic Relaxations for Online Bipartite Matching"
],
[
"Abstract Online bipartite matching (OBM) is a fundamental model underpinning many important applications, including search engine advertisement, website banner and pop-up ads, and ride-hailing.",
"We study the i.i.d.",
"OBM problem, where one side of the bipartition is fixed and known in advance, while nodes from the other side appear sequentially as i.i.d.",
"realizations of an underlying distribution, and must immediately be matched or discarded.",
"We introduce dynamic relaxations of the set of achievable matching probabilities, show how they theoretically dominate lower-dimensional, static relaxations from previous work, and perform a polyhedral study to theoretically examine the new relaxations' strength.",
"We also discuss how to derive heuristic policies from the relaxations' dual prices, in a similar fashion to dynamic resource prices used in network revenue management.",
"We finally present a computational study to demonstrate the empirical quality of the new relaxations and policies."
],
[
"Introduction",
"Many important and emerging applications in e-commerce and in the internet more generally can be modeled as online two-sided markets, with buyers and sellers dynamically appearing and conducting transactions.",
"When a platform or other entity controls or manages one side of this market (usually the supply) and can choose what product to offer to dynamically appearing buyers, the system in question can be modeled as an online bipartite matching (OBM) problem.",
"As more services move to online platforms in the coming years, the ubiquity and importance of OBM models will only increase.",
"An important application of OBM and its generalizations is in the rapidly growing sector of digital advertisement; in their US Ad Spending Estimates and Forecast for 2017, eMarketer reports that digital ad spending reached $83 billion last year, an almost 16% year-over-year increase.",
"Within digital marketing, search engine advertisement yields one application of OBM and similar models [18], [19]: Users input search terms, and the engine displays ad links in addition to the actual search results.",
"The engine chooses the ad(s) to display (i.e.",
"matches an ad to a user) based on the search term, user information, and advertisers' preferences and budget, with one typical objective being to maximize the expected revenue collected from advertisers.",
"Similarly, OBM models can be applied to website banner and pop-up ads; here, each time a user loads a website, the site manager can choose ad(s) to display based on the user's information and browsing history as well as the advertisers' budget and target market.",
"OBM also finds applications in ride-hailing [21], another rapidly growing sector – one study by Goldman Sachs in 2017 projected that global revenues in the industry could reach $285 billion by 2030C.",
"Huston.",
"“Ride-hailing industry expected to grow eightfold to $285 billion by 2030.” MarketWatch, published May 27, 2017..",
"Within these systems, when a user requests a ride, the ride-hailing platform must match them to an available driver, with the overall goal of maximizing some measure of customer satisfaction or utility (for example, by minimizing users' average waiting time before a pickup).",
"As in classical deterministic bipartite matching, OBM involves matching nodes on opposite sides of a bipartite graph, with the objective of maximizing the cardinality of the matching or a more general weight function.",
"In online versions of the problem, nodes on one or both sides of the bipartition may appear and/or disappear dynamically, matches are often irrevocable, and decisions must usually be made with only partial information about the underlying graph.",
"In the version of OBM we study here, one side of the bipartition is fixed and known, representing the goods or resources the platform can offer; the nodes from the other side, representing customers, arrive sequentially, one at a time.",
"Upon each arrival, the platform must immediately and irrevocably match the arriving node to a remaining compatible node from the other side, or discard it.",
"We assume arriving nodes are i.i.d.",
"draws from an underlying uniform distribution over possible node “types”, representing customer classes that may or may not be compatible with different resources or goods.",
"For example, in search engine advertisement, advertisers indicate which search terms they wish their ads displayed with, and the search engine can only match ads with terms in these classes.",
"The i.i.d.",
"assumption implies this model is applicable in situations where the platform can forecast customer behavior, e.g.",
"based on past arrival data, and where this behavior is relatively stable over time.",
"The model may not be as applicable in data-poor situations where the platform cannot confidently forecast customer behavior; the literature includes more conservative models for such cases [18], culminating with the adversarial model studied in [13].",
"Conversely, if customer behavior can be forecast but is not necessarily stable over time, the assumption of identical distributions may be problematic.",
"While we are not aware of OBM models for this case, the revenue management literature includes many works in this vein, particularly in network revenue management, e.g.",
"[22].",
"We briefly survey related work below.",
"Perhaps because of its applicability in search engine advertisement, the algorithms community has extensively studied i.i.d.",
"OBM and related models for the past decade, starting with [8].",
"For the i.i.d.",
"variant with cardinality objective, this work typically focuses on developing heuristic matching policies with multiplicative worst-case performance guarantees.",
"It is straightforward to see that a simple policy based on solving a max-flow relaxation achieves at least $ 1 - 1/e \\approx 0.63 $ of the optimal policy, and this ratio in fact matches the best possible competitive ratio in the adversarial case [13]; [8] established that in the less conservative i.i.d.",
"model a better ratio is indeed possible.",
"Currently, the best guarantee of this type is roughly $ 0.71 $ [11] (and it is slightly better under additional assumptions).",
"The analysis of these heuristic policies and their worst-case guarantees relies on simple linear programming (LP) relaxations, often with network flow structure.",
"However, there is comparably far less work focusing directly on the derivation of strong relaxations for i.i.d.",
"OBM, even though these relaxations provide dual upper bounds useful for benchmarking any new heuristic policies and often can be employed in policy design as well.",
"The only work along these lines we are aware are of is [24], which builds on the network flow relaxations used in [8], and subsequent papers, by adding more sophisticated valid inequalities derived from probabilistic arguments.",
"Though somewhat unusual in the algorithms or optimization literature, this approach can be interpreted as a version of achievable region techniques from queueing theory and applied probability; see e.g.",
"[3], [6]."
],
[
"Contribution",
"Starting with [8], to the best of our knowledge all known relaxations for i.i.d.",
"OBM are “static”: Although the process occurs dynamically over a horizon with sequential decision epochs, the relaxations use as their primary variables the probability that an arriving customer node of some type is ever matched to a fixed resource node.",
"Though this reduces the number of variables to consider, it also means the corresponding relaxations are coarser and looser, as they cannot easily capture the model's dynamics.",
"Furthermore, with few exceptions, the policies derived from such relaxations are also mostly static in nature; that is, though a decision may depend on the arriving node type and the remaining available resource nodes, it usually does not depend on the decision epoch itself and how far or close it might be from the end of the horizon.",
"Our main contribution is to explicitly account for the problem's sequential nature and consider dynamic relaxations.",
"Specifically, these relaxations use as decision variables the probability that a particular match occurs in a particular stage.",
"Using these time-indexed probabilities affords several modeling advantages, such as allowing us to include edge weights that vary by time and thus simplify the analysis by capturing all compatibility information in the objective.",
"Of course, the primary appeal of dynamic relaxations is the possibility of providing tighter dual bounds for the model.",
"As one of our main results, we establish that our simplest dynamic relaxation is provably at least as tight as the best-performing relaxation from [24]; furthermore, the latter relaxation includes exponentially many inequalities and relies on a separation algorithm, whereas our new relaxation has polynomially many variables and constraints.",
"To further understand our new relaxation, we also perform a polyhedral study, demonstrating that all of its inequalities are facet-defining for the underlying polytope of achievable probabilities.",
"We then extend this polyhedral study and introduce more complex inequalities, all facet-defining as well.",
"Our empirical study verifies the strength of the new relaxation; it improves the previous best gaps by $4\\%$ to $5\\%$ in absolute terms on average.",
"As a secondary contribution, we also show how our new relaxation can be leveraged to construct a dynamic heuristic policy.",
"Although this kind of policy is new in OBM to our knowledge, our policy can be viewed as the OBM analogue to dynamic bid price policies, introduced in [1] for network revenue management.",
"To design the policy, we establish a connection between our relaxation and a value function approximation of the model's dynamic programming (DP) formulation.",
"Our empirical results also verify the new policy's quality in comparison to the best empirically performing policies from the literature.",
"The remainder of the paper is organized in the following way: After a brief literature survey at the end of this section, Section formulates the problem, and summarizes pertinent previous results.",
"Section introduces our relaxations and gives our theoretical results, while Section outlines our computational study.",
"Section concludes and discusses possible future work.",
"An Appendix has mathematical proofs not included in the body of the paper."
],
[
"Literature Review",
"The OBM model was introduced by [13], who studied the adversarial version in which node arrivals are not governed by a distribution, but rather by an adversary whose objective is to maximize the difference between the cardinality of the decision maker's matching and the offline optimum.",
"The authors show that a randomized ranking algorithm that chooses a random permutation of the resource nodes and matches the highest-ranked available and compatible node according to the permutation yields an optimal competitive ratio of $ 1 - 1/e $ ; see e.g.",
"[4] for a simplified and corrected proof.",
"Most work on OBM since has focused on less conservative variants.",
"The i.i.d.",
"version with cardinality objective was first studied in [8], who showed that in this version the performance guarantee could be strictly better than $ 1 - 1/e $ ; they also used a network flow relaxation to design their two suggested matchings policy.",
"Subsequent work has focused on improving the performance guarantee and/or generalizing the objective, and typically relies on LP relaxations with network flow structure, e.g.",
"[2], [5], [10], [17].",
"For the cardinality case, [11] have the current state of the art, a policy based on a max-flow relaxation with a guarantee of approximately $ 0.71 $ .",
"The i.i.d.",
"version of OBM (sometimes also called the known i.i.d.",
"model) is in some sense the least conservative OBM variant, compared to the most conservative adversarial version.",
"Some authors have studied models that compromise between the two.",
"For example, in the random permutation model an adversary chooses the graph, but the arriving nodes are revealed in a random order not controlled by the adversary.",
"With this slight relaxation of the adversarial framework, [9] show that for the cardinality objective a simple greedy algorithm, which matches an arriving node to any remaining compatible node, achieves a competitive ratio of $1-1/e$ ; later improvements came in [12], [16].",
"Other models, variants and extensions have appeared in the algorithms literature; we refer the reader to the survey [18].",
"While the notion of dynamic relaxations appears to be new in the OBM context, there is a stream of related literature in network revenue management, beginning with [1], who introduced dynamic bid relaxations and their corresponding policies.",
"In this literature, the goal is often to show that a particular relaxation can be computed efficiently, e.g.",
"[15], [23], [25], as a naive formulation involves a separation problem solved via an integer program.",
"These dynamic relaxations have also been extended to customer choice models, e.g.",
"[25], [26]."
],
[
"Model Description and Preliminaries",
"The OBM model is formulated using two finite disjoint sets $N$ and $V$ , with the process occurring dynamically in the following way.",
"The right-hand node set $V$ , with $ \\vert V \\vert =m$ , is known and given ahead of time.",
"The left-hand set $N$ with $ \\vert N \\vert = n $ represents different node types that may appear, but we do not know which ones will appear and how often.",
"We know only that $ T $ left-hand nodes in total will appear sequentially, each one drawn independently from the uniform distribution over node types $ N $ .",
"That is, at each epoch a node from one of the types in $N$ appears with probability $1/n$ and must be immediately (and irrevocably) matched to a remaining available node in $V$ or discarded; two or more nodes from the same type may appear throughout the process, each treated as a separate copy.",
"Matching $ i \\in N $ to $ j \\in V $ in stage $ t $ yields a (known) reward or weight $ w_{ij}^t $ , and the objective is to maximize the expected weight of the matching.",
"Following convention from previous literature and the motivating application of search engine advertisement, we call $i\\in N$ an impression, and each $j\\in V$ an ad.",
"By considering time-indexed weights $w_{ij}^t$ , we generalize much of the existing literature and can avoid dealing with specific graph structure.",
"In particular, we may assume that the process occurs in a complete bipartite graph, i.e.",
"every node type in $N$ is connected or compatible with every node in $V$ ; non-existent edges simply get weight zero.",
"Moreover, we can assume $m=n=T$ without loss of generality.",
"Indeed, if $m<T$ we add dummy nodes to $V$ and assign zero weight to all corresponding edges.",
"Similarly, if $m>T$ we increase the number of stages and give zero weight to all edges in the new stages.",
"If $n>m = T$ , we again add dummy nodes and stages.",
"Finally, if $n< m = T$ we make $\\kappa $ copies of every node type in $N$ (and the corresponding edges) for the smallest $ \\kappa $ with $ \\kappa n \\ge m$ , then proceed as before.",
"To ease notation, in the remainder of the paper we write $n$ for $m$ and $T$ , but we use the indices $i$ for impressions, $j$ for ads, and $t$ for stages.",
"We use the shorthand $ [n] := \\lbrace 1, \\cdots , n\\rbrace $ , and identify singleton sets with their unique element."
],
[
"DP and LP Formulations",
"Let $\\eta $ be the random variable with uniform distribution over $N$ .",
"We count stages down from $n$ , meaning stage $t$ occurs when $t$ decision epochs (including the current one) remain in the process.",
"We can now give a DP formulation for this OBM model.",
"Let $v^*_t(i,S)$ denote the optimal expected value given that $i\\in N$ appears in stage $t$ when the set of ads $S\\subseteq V$ is available.",
"Then, for all $t=1,\\cdots ,n $ , $ i\\in N $ and $ S\\subseteq V$ , $v^*_t(i,S) & = \\max {\\left\\lbrace \\begin{array}{ll}\\max _{j\\in S}\\lbrace w^t_{ij}+{\\mathbb {E}}_{\\eta }[v^*_{t-1}(\\eta ,S\\backslash j)]\\rbrace \\\\{\\mathbb {E}}_{\\eta }[v^*_{t-1}(\\eta ,S)],\\end{array}\\right.}",
"$ where $v_0^*(\\cdot ,\\cdot )$ is identically zero, and the optimal expected value of the model is given by ${\\mathbb {E}}_{\\eta }[v^*_{n}(\\eta ,V)] = 1/n \\sum _{i \\in N} v_n^*(i, V)$ .",
"The first term in this recursion corresponds to matching $i$ with one of the remaining ads $j\\in S$ ; the second corresponds to discarding $i$ .",
"As with any DP, the optimal value function $v^*$ induces an optimal policy: At any state $(t,i,S)$ , we choose an action that attains the maximum in (REF ).",
"Using a standard reformulation (see e.g.",
"[20]), we can capture the recursion (REF ) with the linear program $\\min _{ v \\ge 0 } ~ & {\\mathbb {E}}_{\\eta }[v_n(\\eta ,V)] \\\\\\text{s.t.\\ } & v_{t}(i,S\\cup j)-{\\mathbb {E}}_{\\eta }[v_{t-1}(\\eta ,S)]\\ge w^t_{ij}, & & t\\in [n], \\ i\\in N, \\ j\\in V, \\ S\\subseteq V\\backslash j \\\\& v_{t}(i,S)-{\\mathbb {E}}_{\\eta }[v_{t-1}(\\eta ,S)]\\ge 0, & & t\\in [n], \\ i\\in N, \\ S\\subseteq V .",
"$ The value function $v^*$ defined in (REF ) is optimal for (REF ).",
"Moreover, this LP is a strong dual for OBM; any feasible $v$ has an objective greater than or equal to ${\\mathbb {E}}_{\\eta }[v^*_{n}(\\eta ,V)]$ .",
"The dual of (REF ) is a primal formulation where any feasible solution encodes a feasible policy and its probability of choosing any action from any state in the DP.",
"That formulation is the LP $\\max _{x,y \\ge 0} & \\quad \\sum _{i\\in N} \\sum _{j\\in V} \\sum _{t\\in [n]} \\sum _{S\\subseteq V\\backslash j} w^t_{ij}x_{i,j}^{t,S}\\\\\\text{s.t.\\ } & \\sum _{j\\in V}x_{i,j}^{n,V\\backslash j}+y_i^{n,V}\\le \\frac{1}{n}, \\quad i\\in N, \\\\\\begin{split}& \\sum _{j\\in S}x_{i,j}^{t,S\\backslash j} +y_i^{t,S}\\cdot 1_{\\lbrace t\\ne 1\\rbrace }-\\frac{1}{n}\\cdot 1_{\\lbrace |S|> t\\rbrace }\\cdot \\sum _{k\\in N}y_{k}^{t+1,S} \\\\& \\qquad -\\frac{1}{n}\\sum _{k\\in N}\\sum _{j\\in V\\backslash S}x_{k,j}^{t+1,S}\\le 0, \\qquad t\\in [n-1], \\ i\\in N, \\ \\varnothing \\ne S\\subset V, \\ |S|\\ge t,\\end{split} \\\\& \\sum _{j\\in V} x_{i,j}^{t,V\\backslash j} +y_i^{t,V}\\cdot 1_{\\lbrace t\\ne 1\\rbrace } -\\frac{1}{n}\\sum _{k\\in N}y_k^{t+1,V}\\le 0, \\qquad i\\in N, \\ t\\in [n-1].",
"$ We denote by $1_{\\mathcal {A}}$ the indicator function for a condition $\\mathcal {A}$ , which takes value one if condition $\\mathcal {A}$ is satisfied, zero otherwise.",
"Decision variable $ x_{i,j}^{t,S} $ represents the probability that the policy chooses to match impression $ i $ to ad $ j $ in state $ ( t, i, S \\cup j ) $ , and $ y_i^{t,S} $ similarly represents a discarding action.",
"As with its dual, (REF ) has exponentially many variables and constraints, and is therefore difficult to analyze directly.",
"However, we can equivalently consider the probability that a feasible policy makes a particular match between $ i $ and $ j $ in stage $ t $ without tracking the other remaining ads $ S \\subseteq V \\setminus j $ ; this corresponds to optimizing over a projection of the feasible region of (REF ), $\\max \\Biggl \\lbrace \\sum _{ i \\in N } \\sum _{ j \\in V}\\sum _{t\\in [n]} w^t_{ij}z^t_{ij} : \\exists ~ (x, y) \\ge 0 \\text{ satisfying } (\\ref {eq:obm_lp_pol1})\\text{--}(\\ref {eq:obm_lp_pol2}) \\text{ with } z^t_{ij} =\\sum _{S\\subseteq V\\backslash j}x_{i,j}^{t,S} \\Biggr \\rbrace ,$ where $ z^t_{ij} $ is the probability that impression $ i $ is matched to ad $ j $ in stage $ t $ .",
"Any such $ z $ is a vector of matching probabilities that is achievable by at least one feasible policy.",
"Let $ Q $ denote this projected polyhedron in the space of $ z_{ij}^t $ variables, and note that $ Q $ is full-dimensional in $ {\\mathbb {R}}^{n^3}$ .",
"Optimizing over $Q$ is as difficult as solving the original DP formulation (REF ), but optimizing over any relaxation of $Q$ yields a valid upper bound; this is our main goal."
],
[
"Relevant Previous Work",
"Most previous results concerning relaxations for OBM use a lower-dimensional projection of the feasible region of (REF ).",
"Specifically, assuming edge weights are static across stages, $ w_{ij}^t = w_{ij} $ for $ t \\in [n] $ , consider $\\max \\Biggl \\lbrace \\sum _{ i \\in N } \\sum _{ j \\in V} w_{ij} z_{ij} : \\exists ~ (x, y) \\ge 0 \\text{ satisfying } (\\ref {eq:obm_lp_pol1})\\text{--}(\\ref {eq:obm_lp_pol2}) \\text{ with } z_{ij} =\\sum _{t\\in [n]}\\sum _{S\\subseteq V\\backslash j}x_{i,j}^{t,S} \\Biggr \\rbrace ,$ where $ z_{ij} $ is the probability that impression $ i $ is ever matched to ad $ j $ .",
"Let $ Q^{\\prime } $ denote this projected polyhedron in the space of $z_{ij}$ variables, and observe that $ Q^{\\prime } $ is also a projection of $Q$ via $z_{ij}=\\sum _{t\\in [n]}z_{ij}^t$ .",
"The following max flow (or deterministic bipartite matching) LP is known to be a relaxation of $Q^{\\prime }$ and has been used to study $Q^{\\prime }$ in several works starting with [8]: $\\max _{z \\ge 0} ~ & \\sum _{i \\in N} \\sum _{j \\in V} w_{ij} z_{ij} \\\\\\text{s.t.\\ } & \\sum _{j\\in V}z_{ij}\\le T/n=1, \\qquad i\\in N \\\\& \\sum _{i\\in N}z_{ij}\\le 1, \\qquad j\\in V .",
"$ In this relaxation, constraints (REF ) limit the expected number of times an impression type can be matched to $ T/n = 1 $ , the expected number of times it will appear, while () state that each ad is matched at most once.",
"To our knowledge, the only past work that specifically focuses on polyhedral relaxations of $Q^{\\prime }$ is [24], which presents several classes of valid inequalities, including the right-star inequalities, $\\sum _{i\\in I}z_{ij}\\le 1- (1- \\vert I \\vert /n )^n, \\qquad j\\in V, \\ I\\subseteq N,$ which yield the best empirical bounds when added to (REF ).",
"Although exponential in number, these inequalities can be separated over in polynomial time by a simple greedy algorithm.",
"We use the bound given by (REF ) with (REF ) as a theoretical and empirical benchmark to test our new relaxations."
],
[
"Dynamic Relaxations",
"We introduce various classes of valid inequalities for $Q$ and study their facial dimension.",
"These inequalities always include variables corresponding to complete bipartite subgraphs; therefore, to ease notation we define $Z_{I,J}^{t} := \\sum _{i\\in I}\\sum _{j\\in J}z_{ij}^{t}, \\qquad I \\subseteq N, J \\subseteq V .$ We begin by presenting a simple inequality class to motivate our approach.",
"For an impression $i\\in N$ , the probability of matching $i$ in each stage $t\\in [n]$ is at most $1/n$ ; this corresponds to $Z_{i,V}^t\\le 1/n, \\qquad \\ i\\in N, \\ t\\in [n].$ Note that by summing these constraints over all $ t $ for a fixed $ i $ , we obtain (REF ).",
"Proposition 1 Constraints (REF ) are facet-defining for the polyhedron of achievable probabilities $Q$ .",
"Fix $i\\in N$ and $t\\in [n]$ .",
"We use $e_{k,j}^\\tau \\in {\\mathbb {R}}^{n^3}$ to denote the canonical vector, i.e., a vector with a one in the coordinate $(k,j,\\tau )$ and zero elsewhere, indicating that we match impression $k$ with ad $j$ in stage $\\tau $ .",
"We construct the following $n^3$ affinely independent points corresponding to policies that satisfy (REF ) with equality: Policy for $(i,j,t)$ with $j\\in V$ : If $i$ appears in stage $t$ , which happens with probability $1/n$ , we match it with $j$ .",
"This corresponds to the point $\\frac{1}{n}e_{i,j}^t$ .",
"Policy for $(k,j,\\tau )$ with $j\\in V$ , $\\tau \\ne t$ , and $k\\in N$ : If $k$ appears in stage $\\tau $ (with probability $1/n$ ), we match it to $ j $ .",
"Then, if $i$ appears in stage $t$ with probability $1/n$ , we match it to some $\\ell \\in V$ , $\\ell \\ne j$ , so we have the point $\\frac{1}{n}e_{k,j}^\\tau +\\frac{1}{n}e_{i,\\ell }^t$ .",
"Policy for $(k,j,t)$ with $j\\in V$ , and $k\\ne i$ : If $k$ appears in stage $t$ (with probability $1/n$ ), we match it to $j$ .",
"On the other hand, if $i$ appears in stage $t$ with probability $1/n$ , we match it to some $\\ell \\in V$ , $\\ell \\ne j$ , so we have the point $\\frac{1}{n}e_{k,j}^t+\\frac{1}{n}e_{i,\\ell }^t$ .",
"These points are linearly independent, which implies they are affinely independent.",
"We now introduce our general inequality family.",
"Fix a set of ads $J\\subseteq V$ and a family of impression sets $ I_t \\subseteq N $ , $ t \\in [n] $ .",
"For any vector $\\alpha \\in {\\mathbb {R}}^n_+$ , we have a valid inequality for $Q$ of the form $\\sum _{t=1}^n \\alpha _t Z_{I_t,J}^t \\le R(\\alpha , (I_t), J ), $ where $R(\\alpha , (I_t), J )$ defines the maximum of the left-hand side over $Q$ .",
"As one example, inequalities (REF ) are a special case of (REF ) where $ J = V $ , $ I_t = \\lbrace i\\rbrace $ , $ \\alpha _t = 1 $ , and $ I_\\tau = \\varnothing $ , $ \\alpha _\\tau = 0 $ for $ \\tau \\ne t $ .",
"Proposition 2 For $\\alpha \\in {\\mathbb {R}}^n_+$ , set family $I_t\\subseteq N$ , $ t \\in [n] $ , and $J\\subseteq V$ , constraints (REF ) are valid for the polyhedron of achievable probabilities $Q$ , and $R(\\alpha , (I_t), J )$ can be computed in polynomial time via a DP.",
"Define variables $p_t\\in \\lbrace 0,1\\rbrace $ to indicate whether a node from $I_t$ appears in stage $t$ or not, and denote by $d\\in \\lbrace 0,\\ldots ,|J|\\rbrace $ the number of remaining nodes from $J$ .",
"Given this, we can state a DP recursion using the value function $R(t,d,p_t)$ , the expected value in stage $t$ when $d$ nodes from $J$ are available and $ p_t $ has occurred.",
"For example, if only one stage remains, $d$ nodes from $J$ are available, and no element of $I_1$ appears, $R(1,d,0)=0$ since we cannot match any node in $ I_1 $ .",
"Conversely, $R(1,d,1)= \\alpha _1 \\min \\lbrace 1, d \\rbrace $ , since we can match a node and obtain value $ \\alpha _1 $ as long as at least one element of $ J $ remains.",
"In general, if $d$ nodes are available in stage $t$ and no node from $I_t$ appears ($p_t=0$ ), then the expected value $R(t,d,0)$ can be computed recursively by conditioning on terms from stage $ t - 1 $ : $R(t,d,0) = \\frac{ n - \\vert I_t \\vert }{n} R( t-1, d, 0 ) + \\frac{\\vert I_t \\vert }{n} R( t-1, d, 1 ).$ On the other hand, to compute $R(t,d,1)$ we choose the maximum between discarding or matching, with value $R(t,d,1) = \\max \\lbrace R(t-1,d,0), \\alpha _t + R(t-1,d-1,0) \\rbrace $ Finally, the value of the right-hand side is $R(\\alpha , (I_t),J )= \\frac{ n - \\vert I_n \\vert }{n} R(n,|J|,0) + \\frac{\\vert I_n \\vert }{n} R(n,|J|,1).$ The number of states is $ n \\times \\vert J \\vert \\times 2 = O(n^2) $ and the number of operations to calculate a state's value is constant, so the entire recursion takes $ O(n^2) $ time.",
"In the remainder of this section, we study particular cases of inequalities (REF ).",
"We construct them intuitively using probabilistic arguments, but their right-hand sides can also be calculated directly using the DP from Proposition REF .",
"As a first example, let $i\\in N$ , $j\\in V$ and $t\\in [n-1]$ .",
"Matching $i$ to $j$ in stage $t$ implies the intersection of two independent events.",
"First, $j$ is not matched in any previous stage $[t+1,n]$ , and second, $i$ appears in stage $t$ .",
"In terms of probability this means ${\\mathbb {P}}(\\text{match $i$ with $j$ in $t$})\\le \\frac{1}{n}(1-{\\mathbb {P}}(\\text{match $j$ in $[t+1,n]$})),$ which is equivalent to ${\\mathbb {P}}(\\text{match $j$ in stages $[t+1,n]$})+n{\\mathbb {P}}(\\text{match $i$ with $j$ in $t$})\\le 1.$ The previous expression is equivalent to $\\sum _{\\tau =t+1}^n Z_{N,j}^{\\tau }+nz_{i,j}^{t}\\le 1 \\qquad \\forall \\ i\\in N, \\ j\\in V, \\ t\\in [n].$ Inequality family (REF ) corresponds to a particular case of (REF ), with $\\vert J \\vert = 1$ , $I_\\tau =N$ for $\\tau \\in [t+1,n]$ , $\\vert I_t \\vert = 1$ , $I_\\tau =\\varnothing $ for $\\tau \\le t-1$ , $\\alpha _\\tau =1$ for $\\tau \\in [t+1,n]$ , $\\alpha _{t}=n$ , and $\\alpha _\\tau =0$ for $\\tau \\le t-1$ .",
"Furthermore, for a fixed $ j \\in V $ and $ t = 1 $ , by summing the inequalities over all $ i \\in N $ we obtain ().",
"Proposition 3 Constraints (REF ) are facet-defining for the polyhedron of achievable probabilities $Q$ when $ t \\le n - 1 $ .",
"Fix $i\\in N, \\ j\\in V, \\ t\\in [n-1]$ .",
"We construct the following $n^3$ affinely independent points corresponding to policies that satisfy (REF ) with equality: 1.",
"Policy for $(i,j,t)$ : if $i$ appears in stage $t$ , then match it to $j$ with probability $1/n$ .",
"This corresponds to the point $\\frac{1}{n}e^t_{i,j}.$ 2.",
"Policy for $(k,j,\\tau )$ with any $k\\in N$ , and any $\\tau \\in [t+1,n]$ : If $k$ appears in stage $\\tau $ , match it to $j$ with probability $1/n$ , but if $k$ does not appear and $i$ appears in stage $t$ , we match $i$ to $j$ with probability $\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)$ .",
"This corresponds to the point $\\frac{1}{n}e_{k,j}^\\tau + \\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e^t_{i,j}.$ As we chose any $k$ and any $\\tau $ , we have $n(n-t)$ points.",
"So far, we only have $n(n-t)+1$ points.",
"For the remaining points, we can use modifications of policy 1 above.",
"Policy for $(k,j,\\tau )$ with any $k\\in N$ and $\\tau \\le t-1$ : If $i$ appears in stage $t$ with probability $1/n$ , then match it with $j$ ; if $i$ does not appear (with probability $1-1/n$ ), and if $k$ appears in stage $\\tau $ (with probability $1/n$ ), then match it with $j$ .",
"This corresponds to $\\frac{1}{n}e^t_{i,j}+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e^\\tau _{k,j}$ .",
"As we chose any $k$ , and any $\\tau \\le t-1$ , we have $n(t-1)$ points.",
"Policy for $(k,\\ell ,\\tau )$ with any $k\\in V$ , $\\ell \\in V$ such that $\\ell \\ne j$ , and $\\tau \\in [n]$ : if $i$ appears in stage $t$ with probability $1/n$ , then match it with $j$ ; if $k$ appears in stage $\\tau $ (with probability $1/n$ ), then match it with $\\ell $ .",
"This corresponds to $\\frac{1}{n}e^t_{i,j}+\\frac{1}{n}e^\\tau _{k,\\ell }$ .",
"In total, this yields $n(n-1)n$ points.",
"Policy for $(k,j,t)$ with $k\\in V$ such that $k\\ne i$ : if $i$ appears in stage $t$ with probability $1/n$ , then match it with $j$ ; if $k$ appears in stage $t$ (with probability $1/n$ ), then match it with $j$ .",
"This corresponds to $\\frac{1}{n}e^t_{i,j}+\\frac{1}{n}e^t_{k,j}$ .",
"In this family, we have $n-1$ points.",
"If we order these points in a suitable way, they form the columns of a block matrix $A=\\begin{pmatrix} A_1 & A_2\\\\ 0 & A_3\\end{pmatrix},$ where $A_1$ is upper triangular and $A_3$ is a diagonal matrix.",
"$A_1$ is formed by the first $n(n-t)+1$ points from policy 1 and the policies of item 2, while $ A_2 $ and $ A_3 $ are given by the remaining points.",
"All diagonal entries of $A_1$ and $A_3$ are positive, implying that $A$ has positive determinant.",
"This shows that the points previously described are linearly independent, completing the proof.",
"We next compare the inequalities we have introduced so far to the known results for the lower-dimensional polyhedron $Q^{\\prime }$ of achievable probabilities that are not time-indexed, which we detail in Section .",
"Recall that $Q^{\\prime }$ is a projection of $Q$ obtained by aggregating variables $z_{ij}^t$ over all stages, $z_{ij}=\\sum _{t\\in [n]}z_{ij}^t$ .",
"We already indicated how inequality families (REF ) and () are respectively implied by (REF ) and (REF ).",
"We next discuss the right-star inequalities (REF ).",
"Theorem 4 Inequalities (REF ) imply the right-star inequalities (REF ).",
"Fix $ j \\in V $ and $ I \\subseteq N $ .",
"First, for $ t = n $ (REF ) is simply $n z_{ij}^n \\le 1$ (it is also a weakened version of (REF )), and summing over $I$ we get $n\\sum _{i\\in I}z_{ij}^n\\le |I|$ .",
"For $ t \\le n - 1 $ , if we sum over $i\\in I$ in (REF ) we get $|I|\\sum _{\\tau \\in [t+1,n]}\\sum _{k\\in N}z_{k,j}^{\\tau }+n\\sum _{i\\in I}z_{i,j}^{t}\\le |I|, \\qquad \\forall \\ t\\in [n-1],$ and since $\\sum _{i\\in I}z_{i,j}^{\\tau }\\le \\sum _{k\\in N}z_{k,j}^{\\tau }$ , we have $|I|\\sum _{\\tau \\in [t+1,n]}\\sum _{i\\in I}z_{i,j}^{\\tau }+n\\sum _{i\\in I}z_{i,j}^{t}\\le |I|, \\qquad \\forall \\ t\\in [n-1].$ Then, multiply each inequality for $t\\in [n-1]$ by $\\frac{1}{n}\\left(1-\\frac{|I|}{n}\\right)^{t-1}$ , and add all of them (including the one for $t = n$ ); the resulting coefficient for each $ z_{ij}^t $ is $\\left(1-\\frac{|I|}{n}\\right)^{t-1} + \\sum _{ \\tau \\le t - 1 } \\frac{\\vert I \\vert }{n}\\left(1-\\frac{|I|}{n}\\right)^{\\tau -1} = 1.$ We thus obtain $\\sum _{t\\in [n]}\\sum _{i\\in I}z_{ij}^t=\\sum _{i\\in I}z_{ij}$ in the left-hand side.",
"In the right-hand side, we get $\\frac{|I|}{n}\\sum _{t=1}^n\\left(1-\\frac{|I|}{n}\\right)^{t-1}=1-\\left(1-\\frac{|I|}{n}\\right)^n.",
"$ This result shows that inequalities (REF ) and (REF ) yield an upper bound that theoretically dominates the bound given by LP (REF ) with additional inequalities (REF ), the best empirical bound previously known for OBM [24].",
"In terms of dimension, the LP given by (REF ) and (REF ) with non-negativity constraints has $ O(n^3) $ inequalities in $ {\\mathbb {R}}^{n^3} $ , while (REF ) with (REF ) has exponentially many inequalities in $ {\\mathbb {R}}^{n^2} $ ."
],
[
"Policy Design",
"Theorem REF establishes that an LP in the space of $ z_{ij}^t $ variables with inequalities (REF ) and (REF ), $\\max _{z \\ge 0} \\biggl \\lbrace \\sum _{i \\in N} \\sum _{j \\in V} \\sum _{t \\in [n]} w_{ij}^t z_{ij}^t : (\\ref {eq:prob_bound}), (\\ref {eq:ineq_Jsize_1}) \\biggr \\rbrace ,$ is guaranteed to provide a bound at least as good as the state of the art.",
"We can also devise a policy from the LP (REF ), in a similar fashion to dynamic bid policies from network revenue management [1].",
"Denote by $\\lambda _i^t\\ge 0$ and $\\mu _{ij}^t\\ge 0$ the dual multipliers corresponding to constraints (REF ) and (REF ) respectively.",
"Along the lines of [1], [24] and other approximate DP approaches, we construct an approximation of the true value function (REF ): Interpret each $ \\lambda _i^t $ as the value of having an impression of type $ i $ appear in period $ t $ , and similarly interpret each $ \\mu _{ij}^t $ as the value of having impression $ i $ appear in period $ t $ when ad $ j $ is available to match.",
"For state $ (t, i, S) $ this yields the value function approximation $v_t(i,S)\\approx \\lambda _i^t+\\sum _{\\tau \\in [t-1]}{\\mathbb {E}}_\\eta [\\lambda _{\\eta }^\\tau ]+\\sum _{j\\in S} \\biggl ( \\mu _{ij}^t+ \\sum _{\\tau \\in [t-1]} {\\mathbb {E}}_\\eta [\\mu _{\\eta j}^\\tau ] \\biggr ).$ By imposing the constraints from (REF ) on this approximation of the value function, we obtain the dual of (REF ): $\\min _{ v \\ge 0 } ~ & {\\mathbb {E}}_{\\eta }[v_n(\\eta ,V)] & \\min _{ \\lambda , \\mu \\ge 0 } ~ & \\sum _{ t \\in [n] } \\biggl ( {\\mathbb {E}}_\\eta [ \\lambda _\\eta ^t ] + \\sum _{ j \\in V } {\\mathbb {E}}_\\eta [ \\mu _{\\eta j}^t ] \\biggr ) \\\\\\text{s.t.\\ } & v_{t}(i,S\\cup j)-{\\mathbb {E}}_{\\eta }[v_{t-1}(\\eta ,S)]\\ge w^t_{ij}, \\quad \\xrightarrow{} & \\text{s.t.\\ } & \\lambda _i^t + \\mu _{ij}^t + \\sum _{\\tau \\in [t-1]} {\\mathbb {E}}_\\eta [ \\mu _{\\eta j}^\\tau ] \\ge w_{ij}^t .",
"\\\\ & v_{t}(i,S)-{\\mathbb {E}}_{\\eta }[v_{t-1}(\\eta ,S)]\\ge 0, $ Furthermore, by replacing (REF ) in the DP recursion (REF ) for a state $ (t, i, S) $ , we get the heuristic policy $\\operatornamewithlimits{arg \\ max}\\Bigl \\lbrace \\max _{ j \\in S } \\lbrace w_{ij}^t + {\\mathbb {E}}_\\eta [ v_{t-1}( \\eta , S \\setminus j ) ] \\rbrace , {\\mathbb {E}}_\\eta [ v_{t-1}( \\eta , S ) ] \\Bigr \\rbrace \\\\\\overset{(\\ref {eq:approx_function})}{\\approx } \\operatornamewithlimits{arg \\ max}\\Biggl \\lbrace \\max _{ j \\in S } \\biggl \\lbrace w_{ij}^t + \\sum _{\\tau \\in [t-1]} \\biggl ( {\\mathbb {E}}_\\eta [\\lambda _{\\eta }^\\tau ] + \\sum _{\\ell \\in S \\setminus j} {\\mathbb {E}}_\\eta [\\mu _{\\eta \\ell }^\\tau ] \\biggr ) \\biggr \\rbrace , \\\\\\sum _{\\tau \\in [t-1]}{\\mathbb {E}}_\\eta [\\lambda _{\\eta }^\\tau ] + \\sum _{\\ell \\in S}\\sum _{\\tau \\in [t-1]} {\\mathbb {E}}_\\eta [\\mu _{\\eta \\ell }^\\tau ] \\Biggr \\rbrace \\\\= \\operatornamewithlimits{arg \\ max}\\Biggl \\lbrace \\max _{j\\in S}\\biggl \\lbrace w^t_{ij} - \\sum _{ \\tau \\in [t-1] } {\\mathbb {E}}_\\eta [ \\mu _{\\eta j}^\\tau ] \\biggr \\rbrace ,0 \\Biggr \\rbrace .",
"$ Intuitively, this policy evaluates the net benefit of a potential match of impression $ i $ to ad $ j $ in period $ t $ as the match's weight minus the value we give up by losing ad $ j $ in the subsequent remaining periods.",
"The policy chooses the match with the largest such benefit (if positive), and otherwise discards the impression."
],
[
"Polyhedral Study",
"Inequalities (REF ) correspond to a particular case of (REF ), when the fixed set of ads $J$ has one element.",
"We can apply a similar idea to a subset of any size; take the next simplest case of (REF ), a set of size two, say $J=\\lbrace j_1,j_2\\rbrace $ .",
"Consider also two impressions $i_1, i_2\\in N$ , where we may have $ i_1 = i_2 $ .",
"In terms of probability, the event of matching $i_2$ with $j_1$ or $j_2$ in stage $t$ implies $i_2$ must appear in stage $t$ with probability $1/n$ and either of two events happens: First, neither $j_1$ nor $j_2$ are matched in stages $[t+2,n]$ , and then $i_1$ appears in stage $t+1$ with probability $1/n$ (and can be matched to one of the ads or not); and second, $j_1$ or $j_2$ (but not both) are matched in stages $[t+2,n]$ , and $i_1$ is not matched to $j_1$ nor $j_2$ in stage $t+1$ (this includes the case of another impression being matched to one of them).",
"Since matching $j_1$ or $j_2$ in $ t $ are mutually exclusive events, we have the inequality ${\\mathbb {P}}(\\text{match $i_2$ with $j_1$ or $j_2$ in $t$})\\le \\\\ \\frac{1}{n}\\bigg [\\frac{1}{n}\\left(1-{\\mathbb {P}}(\\text{match $j_1$ or $j_2$ in $[t+2,n]$})\\right)+\\left(1-{\\mathbb {P}}(\\text{match $i_1$ with $j_1$ or $j_2$ in $t+1$})\\right)\\bigg ] .$ In terms of variables $z$ , this is equivalent to $\\begin{split}& \\sum _{\\tau \\in [t+2,n]} Z_{N,J}^{\\tau }+n Z_{i_{t+1},J}^{t+1}+ n^2 Z_{i_t,J}^{t}\\le 1+n \\\\& \\qquad \\qquad \\qquad \\forall \\ i_t,i_{t+1}\\in N, \\ J\\subseteq V, \\ |J|=2, \\ t\\in [n-2].\\end{split}$ This probabilistic argument can be generalized for any set $J\\subseteq V$ with $|J| = h \\in [n-1]$ and any $ t \\le n-h $ .",
"Let $(i_1,\\ldots ,i_h)$ be a sequence of nodes in $N$ allowing repeats; the general constraint corresponds to $&{\\mathbb {P}}(\\text{match $i_{h} $ to some $ j \\in J $ in $t$})\\\\&\\le \\frac{1}{n}\\left[\\frac{1}{n^{h-1}}\\left(1-{\\mathbb {P}}(\\text{match $j_1$ or $j_2$ or $\\ldots $ or $j_h$ in $[t+h,n]$})\\right)\\right.\\\\&+\\frac{1}{n^{h-2}}\\left(1-{\\mathbb {P}}(\\text{match $i_{1} $ to some $ j \\in J $ in $t+h-1$})\\right)\\\\&+\\frac{1}{n^{h-3}}\\left(1-{\\mathbb {P}}(\\text{match $i_{2} $ to some $ j \\in J $ in $t+h-2$})\\right)\\\\&+\\cdots +\\left(1-{\\mathbb {P}}(\\text{match $i_{h-1} $ to some $ j \\in J $ in $t+1$})\\right)\\bigg ].$ Therefore, we can give a general expression for this particular subclass of inequalities (REF ): $\\begin{split}\\sum _{\\tau =t+h}^{n} Z_{N,J}^{\\tau }&+\\sum _{\\tau =t}^{t+h-1}n^{t+h-\\tau } Z_{i_{\\tau },J}^{\\tau }\\le 1+\\sum _{\\tau =1}^{h-1}n^{\\tau }, \\\\& \\forall \\ J\\subseteq V, \\ \\vert J \\vert = h\\in [n-1], \\ t\\in [n-h], \\ i_t,\\cdots ,i_{t+h-1}\\in N.\\end{split}$ Theorem 5 Constraints (REF ) are facet-defining for $Q$ .",
"The proof of this theorem can be found in the Appendix.",
"So far we have only considered either $I_\\tau =N$ or $ \\vert I_\\tau \\vert = 1$ within inequalities (REF ).",
"We next propose a generalization for other sets $I$ .",
"Consider the case $J = \\lbrace j\\rbrace $ , and any subset $I\\subseteq N$ ; suppose we naively apply the same argument behind inequality (REF ).",
"Matching an element of $I$ with $j$ in stage $t$ implies the intersection of two independent events: First, $j$ is not matched in stages $[t+1,n]$ , and second, some element in $I$ appears in stage $t$ .",
"In probabilistic terms, ${\\mathbb {P}}(\\text{match any element in $I$ with $j$ in $t$})\\le \\frac{|I|}{n}(1-{\\mathbb {P}}(\\text{match $j$ in $[t+1,n]$})),$ which is equivalent to $|I|\\sum _{\\tau =t+1}^n Z_{N,j}^{\\tau }+n Z_{I,j}^{t}\\le |I|.$ However, this inequality is made redundant by (REF ), because we can sum over $i\\in I$ for the same fixed $ t $ to get it.",
"Consider instead $J=\\lbrace j_1,j_2\\rbrace $ , any $I_1 \\subseteq N$ with $ \\vert I_1 \\vert \\ge 2 $ , and another impression $i_2\\in N$ ; we apply the same argument used for (REF ), but substituting $ I_1 $ for the single impression $ i_1 $ .",
"Matching $i_2$ with $j_1$ or $j_2$ in stage $t$ implies $i_2$ appears in stage $t$ with probability $1/n$ , and either of two previous events happens: First, neither $j_1$ nor $j_2$ are matched in stages $[t+2,n]$ , and then any element in $I_1$ appears in stage $t+1$ with probability $|I_1|/n$ (and is matched to one of the ads or not); and second, one of $j_1$ or $j_2$ is matched in stages $[t+2,n]$ , and no element from $I_1$ is matched to $j_1$ nor $j_2$ in $t+1$ (this includes the case of another impression being matched to one of them).",
"Since matching $j_1$ or $j_2$ in $ t $ are mutually exclusive, we have ${\\mathbb {P}}(\\text{match $i_2$ with $j_1$ or $j_2$ in $t$})&\\le \\frac{1}{n}\\bigg [\\frac{|I_1|}{n}\\left(1-{\\mathbb {P}}(\\text{match $j_1$ or $j_2$ in $[t+2,n]$})\\right)\\\\&+\\left(1-{\\mathbb {P}}(\\text{match some $i \\in I_1$ with $j_1$ or $j_2$ in $t+1$})\\right)\\bigg ],$ which is equivalent to $\\vert I_1 \\vert \\sum _{\\tau =t+2}^n Z_{N,J}^{\\tau }+n Z_{I_1,J}^{t+1}+n^2 Z_{i_2,J}^{t}\\le \\vert I_1 \\vert +n .$ As with inequalities (REF ), if we attempt to naively extend this argument by considering a larger set $ I_2 $ instead of the single impression $ i_2 $ , we simply get redundant inequalities.",
"However, we can generalize (REF ) using the same argument for (REF ): For any $ I \\subseteq N $ with $ \\vert I \\vert = r \\le n-1 $ and any $ J \\subseteq V $ with $ \\vert J \\vert = h \\le n -1 $ , we obtain the inequalities $\\begin{split}& r \\sum _{\\tau =t+h}^{n} Z_{N,J}^{\\tau }+n Z_{I,J}^{t+h-1}+\\sum _{\\tau =t}^{t+h-2}n^{t+h-\\tau } Z_{i_{\\tau },J}^{\\tau }\\le r+\\sum _{\\tau =1}^{h-1}n^{\\tau }, \\\\& \\forall \\ J\\subseteq V, \\vert J \\vert = h\\in [n-1], \\ I\\subseteq N, \\vert I \\vert = r\\in [n-1], t\\in [n-h], \\ i_t,\\cdots ,i_{t+h-2}\\in N.\\end{split}$ Theorem 6 Constraints (REF ) are facet-defining for $Q$ .",
"For a proof of this theorem, see the Appendix.",
"In inequalities (REF ), we do not consider $ I = N $ .",
"Suppose we apply the same argument for (REF ) in this case; we then obtain $n\\sum _{\\tau =t+h}^{n} Z_{N,J}^{\\tau }+n Z_{N,J}^{t+h-1}+\\sum _{\\tau =t}^{t+h-2}n^{t+h-\\tau } Z_{i_{\\tau },J}^{\\tau }\\le n+\\sum _{\\tau =1}^{h-1}n^{\\tau }.$ Dividing by $n$ , we get $\\sum _{\\tau =t+h}^{n} Z_{N,J}^{\\tau }+ Z_{N,J}^{t+h-1}+\\sum _{\\tau =t}^{t+h-2}n^{t+h-\\tau -1} Z_{i_{\\tau },J}^{\\tau }\\le 1+\\sum _{\\tau =1}^{h-1}n^{\\tau -1},$ which is equivalent to $\\sum _{\\tau =t+h}^{n} Z_{N,J}^{\\tau }+ Z_{N,J}^{t+h-1}+n Z_{i_{t+h-2},J}^{t+h-2}+\\sum _{\\tau =t}^{t+h-3}n^{t+h-\\tau -1} Z_{i_{\\tau },J}^{\\tau }\\le 2+\\sum _{\\tau =1}^{h-2}n^{\\tau }.$ This idea also generates valid inequalities for $Q$ , but we can generalize it even more.",
"Thus far, we consider an arbitrary subset $I$ in stage $t+h-1$ , but in this last inequality the arbitrary subset can “shift” to stage $t+h-2$ , so we can actually state a more general valid inequality $r\\sum _{\\tau =t+h-1}^{n}& Z_{N,J}^{\\tau }+n Z_{I,J}^{t+h-2}+\\sum _{\\tau =t}^{t+h-2}n^{t+h-\\tau -1} Z_{i_{\\tau },J}^{\\tau }\\le 2r+\\sum _{\\tau =1}^{h-2}n^{\\tau }, \\\\& \\forall \\ t\\in [n-h], \\ J\\subseteq V, \\ l\\in [n-1], \\ I\\subseteq N, \\ r\\in [n-1], \\ i_t,\\cdots ,i_{t+h-2}\\in N.$ In this last inequality we only consider $r\\in [n-1]$ , but as before, we can actually again take $I=N$ in stage $t+h-2$ .",
"After dividing by $n$ , we get $\\sum _{\\tau =t+h-1}^{n} Z_{N,J}^{\\tau }+ Z_{N,J}^{t+h-2}+\\sum _{\\tau =t}^{t+h-2}n^{t+h-\\tau -2} Z_{i_{\\tau },J}^{\\tau }\\le 2+\\sum _{\\tau =1}^{h-2}n^{\\tau -1},$ equivalent to $\\sum _{\\tau =t+h-2}^{n} Z_{N,J}^{\\tau }+n Z_{i_{t+h-3},J}^{t+h-3}+\\sum _{\\tau =t}^{t+h-4}n^{t+h-\\tau -2} Z_{i_{\\tau },J}^{\\tau }\\le 3+\\sum _{\\tau =1}^{h-3}n^{\\tau } .$ This is also a valid inequality for $Q$ , and we can continue doing this process as many as $h-2$ times until we get $r\\sum _{\\tau =t+2}^{n} Z_{N,J}^{\\tau }+n Z^{t+1}_{i_{t+1},J}+n^2 Z_{i_{t},J}^{t}\\le r(h-1)+n,$ which is a generalization of (REF ).",
"Finally, if we do this process one more time we get $\\sum _{\\tau =t+1}^{n} Z_{N,J}^{\\tau }+n Z_{i_{t},J}^{t}\\le h,$ which is clearly implied by summing over $j \\in J$ in (REF ).",
"Denote by $q\\in [0,h-2]$ the number of times we apply this procedure.",
"We now state the most general family of valid inequalities we have obtained as a specific subclass of (REF ).",
"We subdivide this class using a 4-tuple $(h,r,t,q)$ , which respectively identifies the size of $J$ , the size of $I$ , the stage, and the number of times we apply the previous procedure.",
"So, for any $J\\subseteq V$ with $\\vert J \\vert = h\\in [2,n-1]$ , any $I\\subseteq N$ with $ \\vert I \\vert = r\\in [n-1]$ , any $t\\in [n-h]$ , and any $q\\in [0,h-2]$ , we have the following valid inequality $r \\sum _{\\tau = t+h-q}^n Z_{N,J}^{\\tau } +n Z^{t+h-q-1}_{I,J}&+\\sum _{\\tau =t}^{t+h-q-2}n^{t+h-q-\\tau } Z^{\\tau }_{i_{\\tau },J} \\le r(q+1)+\\sum _{\\tau =1}^{h-q-1}n^{h-q-\\tau } .$ We have already proved that the inequalities given by $(h,r,t,0)$ are facet-defining; here we give the general result.",
"Theorem 7 Inequalities (REF ) identified by $(h,r,t,q)$ are facet-defining for $Q$ when $ h \\in [2, n-1] $ , $ r \\in [n-1] $ , $ t \\in [n - h] $ and $ q \\in [0, h-2] $ .",
"Finally, we show the following complexity result for this general family of facet-defining inequalities.",
"Proposition 8 It is NP-hard to separate inequalities (REF ), and thus also (REF ).",
"Fix $ h = r $ and $ t $ .",
"Suppose we have a solution $ z $ that is zero (or constant) in all values except for stage $ t + h - 1 $ .",
"In this case, the separation problem for this $ h $ , $ r $ and $ t $ is equivalent to $\\max \\lbrace Z_{I,J}^{t+h-1} : I \\subseteq N, J \\subseteq V, \\vert I \\vert = \\vert J \\vert = h = r \\rbrace .$ This is a weighted version of the maximum balanced biclique problem, which is NP-hard [7].",
"For $ h \\ne r $ , the problem can be transformed to make the two cardinalities equal."
],
[
"Description of Experiments",
"Our main experimental goal is testing the effectiveness of our new dynamic relaxations and comparing the new bounds given by these relaxations to several benchmarks.",
"As a secondary goal, we also study the heuristic policy (REF ) implied by our relaxation and compare it with the best empirically performing policy from the literature.",
"The best empirical bound previously known for OBM is the LP (REF ) with additional inequalities (REF ) [24].",
"Our results in the previous section establish that (REF ) is guaranteed to be no worse.",
"So we compare these two bounds to determine how much of an improvement the latter LP (REF ) offers over the former.",
"In addition, we would like to examine if some of the other inequalities we introduce can further improve the bound.",
"However, testing these additional inequality classes involves computational challenges.",
"In particular, the LP's dimension grows as $ n^3 $ , implying a relatively large number of variables even for moderately sized instances.",
"This practically limits both the number of inequalities we consider, and the actual number we can dynamically add to the LP.",
"To this end, we test adding inequalities (REF ) to (REF ); these inequalities are still polynomially many, $ \\Theta (n^5) $ , and relatively efficient to separate over.",
"We also considered including inequalities (REF ), that is, the special case of (REF ) with $ h = n - 1 $ and $ t = 1 $ , as they are also simple to separate over despite numbering $ \\Theta (n^n) $ .",
"However, our preliminary experiments revealed numerical difficulties with these inequalities; the smallest non-zero coefficient is 1, while the largest is $ n^{n-1} $ , and although these numbers (and all of the coefficients and right-hand sides of our inequalities) require $ O(n \\log n) $ space in binary representation and are thus of polynomial size, in practical terms these differences in scale make it difficult to even determine whether a particular inequality is violated, and thus to separate over the entire family.",
"We therefore did not include these inequalities in our experiments.",
"As for lower bounds given by heuristic policies, [24] introduce a time-dependent ranking policy derived from (REF ) with additional inequalities (REF ), and results in this paper establish it as the best performing policy among several from the literature.",
"We use it as a benchmark to test policy (REF ).",
"Finally, we include as additional benchmarks the optimal value given by the DP recursion (REF ) (for small instances where it can be computed), as well as the max-weight expected off-line matching, the expected value of the matching we would choose if we could observe the entire sequence of realized impressions before making a decision.",
"This latter benchmark is also an upper bound on the optimal value, as it relaxes non-anticipativity."
],
[
"Instance Design and Implementation",
"All of the instances we tested have $ n = m = T $ , with binary edge weights constant over time, $ w_{ij}^t = w_{ij} \\in \\lbrace 0, 1\\rbrace $ .",
"In other words, all the instances are max-cardinality OBM problems with static edges; the static weights are required because the benchmarks we use to compare against do not accommodate weights that vary over time.",
"We generate instances with the following rubrics: 20 small instances with $n=10$ , each one randomly generated by having a possible edge in $ N \\times V $ be present independently with a probability of $ 25\\% $ , so the expected average degree is 2.5.",
"20 large, dense instances with $n=100$ , each one randomly generated by having a possible edge in $N\\times V$ be present independently with a probability of $ 10\\% $ , so the expected average degree is 10.",
"20 large, sparse instances with $n=100$ , each one randomly generated by having a possible edge in $N\\times V$ be present independently with probability of 2.5%, so the expected average degree is 2.5.",
"A set of large, $k$ -regular graphs with $ n = 100 $ and $ k \\in \\lbrace 3, 4, 5, 6 \\rbrace $ , constructed in the following way: Indexing both impressions and ads from 0 to $n-1$ , each impression $ i $ is adjacent to ads $ \\lbrace i,i+1,\\ldots ,i+k-1\\rbrace \\mod {k} $ .",
"The motivation for this last set of experiments is that the relaxations and policies may behave differently on instances with a high degree of symmetry, as opposed to randomly generated instances.",
"For any experiment requiring simulation, including computing the expected value of the heuristic policies and the max-weight off-line matching, we used $ 20,000 $ simulations and report the sample mean and sample standard deviation.",
"For small instances, all the bound experiments took a few seconds on average.",
"For the larger instances, we solved the benchmark LP's following the approach from [24].",
"For the new bounds, we formulated (REF ) but eliminated all variables corresponding to missing edges; this results in models with an average of 25,000, 100,000 and $ 10,000 \\times k $ variables for sparse, dense and regular instances respectively.",
"The solution times for these LP's were roughly one hour for dense instances, and under a minute for sparse instances, with regular instances varying as $k$ grows.",
"After solving this LP, we switched to constraint generation for inequalities (REF ); however, after preliminary experiments we did this only for small and large sparse instances, because of protracted solve times with minimal bound improvement in the other cases."
],
[
"Summary of Results",
"Table REF summarizes the experiment results for all instances except regular ones, which are detailed individually below.",
"For each instance class, in each row we present the geometric mean of each bound or policy's ratio to a fixed benchmark – the DP value for small instances and the max-weight expected off-line matching for large ones.",
"We also report the sample standard deviation of these ratios in parenthesis.",
"Table: Summary of experiment results.We know from our results in the previous section that the bound given by (REF ) is guaranteed to outperform the bound given by (REF ) with (REF ).",
"However, our results show that the improvement is significant, with the new bound cutting the gap by about $ 4\\% $ on average for small instances and approximately $3\\%$ to $ 5\\% $ for large ones.",
"Furthermore, the improvement is consistent across all the tested instances; in particular, the two bounds never match.",
"The results for large, dense instances are particularly noteworthy; the new bound from (REF ) also beats the max-weight expected off-line matching, not only on average but in every instance.",
"Our intuition for this result is the following.",
"In dense instances, there is likely a perfect or near-perfect matching in every realization, and thus the off-line matching will be very close to $ n $ in expectation.",
"Of course, even in a dense instance it may be that no online policy can guarantee a perfect or near-perfect matching, and explicitly accounting for temporal aspects of the problem, particularly as inequalities (REF ) do, captures this phenomenon and tightens the bound, unlike the off-line matching or the more static approach of the benchmark LP.",
"Interestingly, our results also reveal that the bound from (REF ) is not improved much with the addition of inequalities (REF ), especially considering the significant additional computing time.",
"In light of these results, we also performed experiments to test the bound given by (REF ) and (REF ) only (without inequalities (REF )).",
"However, the resulting bounds were much looser, confirming that inequalities (REF ) are crucial to providing a tight bound.",
"In terms of policies, our new heuristic (REF ) is consistently better than the time-dependent ranking policy, the best performing policy from the literature.",
"This improvement occurs in almost every tested instance, though the magnitude of the improvement varies.",
"The new policy is near-optimal for small instances, and cuts the gap for large instances, by about $0.7\\%$ to $1\\%$ on average in absolute terms.",
"This improvement in policy quality mirrors results in other areas, such as revenue management, where heuristic policies derived from time-indexed relaxations also outperform policies stemming from “static” LP's; see e.g.",
"[1], [26].",
"The results for regular graphs are in Table REF , shown here in absolute terms since we are not averaging multiple experiments.",
"We observe similar improvements in terms of upper bounds, where our new bound significantly cuts the gap, by around $7\\%$ .",
"On the other hand, we observe no improvement on the policy side.",
"Intuitively, this last result is unsurprising, since both heuristic policies depend on dual multipliers of LP's that are symmetric for regular instances, in the sense that they both have dual optimal solutions in which every value at some stage is equal.",
"Both policies are thus choosing a match uniformly at random.",
"Table: Experiment results for regular graphs."
],
[
"Conclusions",
"This work proposes dynamic relaxations for the i.i.d.",
"OBM problem and studies them from a polyhedral point of view.",
"While several past results have used different LP relaxations, ours is the first to explicitly consider the time dimension.",
"Among various benefits of the approach, this allows for the model to accommodate time-varying edge weights, and also allows us to elide the instance's structure in the analysis, by capturing all of this information in the problem's objective.",
"Our study centers on the polyhedron of time-indexed achievable probabilities $Q$ , and includes a large class of facet-defining inequalities for this polytope based on choosing complete bipartite subgraphs.",
"Furthermore, our experiments confirm that the time-indexed approach offers significant benefits; the bound given by the simplest members of our proposed inequality family already significantly outperforms the best empirical bounds given by static LP's, and a heuristic policy derived from this new bound also significantly outperforms the best policy based on a static relaxation.",
"Our results motivate a variety of questions for future work.",
"For example, we would like to understand the structure of valid inequalities that are not based on complete bipartite subgraphs, to potentially further improve the dual bound.",
"Using Fourier-Motzkin elimination and the software PORTA, we have derived the full description of $Q$ for small cases, such as $ n = m = T = 3 $ .",
"We observed many different inequalities, including some that are somewhat similar to our general family (REF ), so there may be a more general class to propose that still lends itself to analysis similar to ours.",
"Much of the literature on OBM studies the worst-case performance of heuristic policies based on relaxations.",
"Although that was not our goal in this work, the positive empirical results we observed when implementing our new heuristic policy suggest a similar analysis for that policy, especially since it appears to differ in structural terms from many OBM heuristics.",
"More generally, an interesting question is whether a polyhedral analysis similar to ours can be applied to derive new bounds and policies in related online matching and resource allocation contexts."
],
[
"Acknowledgments",
"The authors' work was partially supported by the National Science Foundation under grant CMMI 1552479."
],
[
"Remaining Proofs",
"[Proof of Theorem REF .]",
"The case $h=1$ is already covered by the proof of Proposition REF .",
"Consider the case $h=n-1$ and $t=1$ ; the other cases follow a similar construction of linearly independent points.",
"Let $J=\\lbrace 0,\\ldots , n-2\\rbrace $ , and assume without loss of generality that $i_\\tau =i$ for all $\\tau \\in [n-1]$ .",
"The specific inequality is $Z_{N,J}^n+\\sum _{\\tau =1}^{n-1} n^{n-\\tau } Z_{i,J}^\\tau \\le 1+\\sum _{\\tau =1}^{n-2} n^{\\tau }.$ We know that $z\\in [0,1]^{n^3}$ , but for the description of the points (and the proof) we will just consider the coordinates involved in the inequality, i.e., $z\\in [0,1]^{p}$ , where $p:=(2n-1)(n-1)$ .",
"For the rest of the points, the construction is similar to the one in the proof of Proposition REF .",
"Recall that $e_{k,j}^{\\tau }$ denotes the canonical vector in $[0,1]^{p}$ , i.e.",
"a vector with a 1 in coordinate $ (k, j, \\tau ) $ and zero elsewhere, indicating a match of impression $k$ with ad $j$ in stage $\\tau $ .",
"Consider the elements of $J$ as an $(n-1)$ -tuple, i.e., $(0,\\ldots ,n-2)$ .",
"For $j\\in J$ , we define $j+(0,\\ldots ,n-2):= (j,\\ldots ,j+n-2) \\mod {(}n-1).$ Any addition or substraction with $j\\in J$ is modulo $(n-1)$ for the remainder of the proof.",
"We denote the circulation of $J$ as the following set of $(n-1)$ -tuples: $\\operatornamewithlimits{circ}(J)&:=\\lbrace j+(0,\\ldots ,n-2)\\rbrace _{j\\in J}\\\\&=\\lbrace (0,\\ldots ,n-2),(1,\\ldots ,n-2,0),\\ldots ,(n-2,0,\\ldots ,n-3)\\rbrace .$ Note that $\\operatornamewithlimits{circ}(J)$ can be viewed as a matrix.",
"Each element of $\\operatornamewithlimits{circ}(J)$ corresponds to a sequence of ads in the process from stage $n$ to stage 1.",
"Since we have $n$ stages and any of those sequences has size $n-1$ , then clearly there is no matching in some stage or an element repeats.",
"We now describe the family of linearly independent points.",
"Fix $k\\in N$ and $j\\in J$ .",
"In stage $n$ , if node $k$ appears, then match it to node $j$ , with probability $1/n$ .",
"For the remaining stages match according to $(j,j+1,\\ldots ,j+n-2) \\in \\operatornamewithlimits{circ}(J)$ .",
"In terms of probability, if $i$ appears in stage $n-1$ , then it is matched to $j$ with probability $(1-1/n)\\cdot 1/n$ .",
"For the rest, the probability is $1/n$ .",
"So, we have the point $\\frac{1}{n}e_{k,j}^n+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e_{i,j}^{n-1}+\\frac{1}{n}\\sum _{\\tau =1}^{n-2}e_{i,j+\\tau }^{n-1-\\tau }.$ By a simple calculation, it is easy to see that each of these points achieves the righ-hand side of (REF ).",
"Since we chose an arbitrary $k\\in N$ and $j\\in J$ , we have $n(n-1)$ points in this family.",
"Fix $j\\in J$ .",
"In this family we repeat $ j $ in stages $n-1$ and $n-2$ .",
"If $i$ appears in stage $n-1$ , match it to node $j$ with probability $1/n$ .",
"If $i$ appears in stage $n-2$ and it did not appear in $n-1$ , match it to $j$ with probability $(1-1/n)\\cdot 1/n$ .",
"For the remaining stages match according to $(j+n-2,j,j+1,\\ldots ,j+n-3) \\in \\operatornamewithlimits{circ}(J)$ ; in particular, in stage $n$ match any $k\\in N$ that appears with node $j+n-2$ , in stage $ n-3 $ match $i$ to $j+1$ if it appears, and so forth.",
"So, we have the point $\\frac{1}{n}\\sum _{k\\in N}e_{k,j+n-2}^n+\\frac{1}{n}e^{n-1}_{i,j}+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e_{i,j}^{n-2}+\\frac{1}{n}\\sum _{\\tau =1}^{n-3}e_{i,j+\\tau }^{n-\\tau -2}$ By a simple calculation, we get the right-hand side of (REF ).",
"Since we chose an arbitrary $j\\in J$ , we have $n-1$ points in this family.",
"Fix $j\\in J$ ; in this family we have two different options in stage $n-3$ .",
"If $i$ appears in stage $n-1$ , match it to $j$ with probability $1/n$ .",
"If $i$ appears in stage $n-2$ , match it to $j+1$ , also with probability $1/n$ .",
"If $i$ appears in stage $n-3$ , match it to $ j+1 $ with probability $(1-1/n)\\cdot 1/n$ , or if $ j+1 $ was matched in stage $n-2$ , then to node $j$ with probability $(1-1/n)\\cdot 1/n^2$ .",
"For the remaining stages match according to $(j+n-2,j,j+1,\\ldots ,j+n-3)$ ; in stage $n$ match any $k\\in N$ that appears to $j+n-2$ , in stage $ n - 4 $ match $i$ to $ j+2 $ if it appears, and so forth.",
"So, we have the point $\\frac{1}{n}\\sum _{k\\in N}e_{k,j+n-2}^n&+\\frac{1}{n}e^{n-1}_{i,j}+\\frac{1}{n}e_{i,j+1}^{n-2} \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\frac{1}{n^2}e_{i,j}^{n-3}+\\frac{1}{n}e_{i,j+1}^{n-3}\\right]+\\frac{1}{n}\\sum _{\\tau =1}^{n-4}e_{i,j+\\tau +1}^{n-\\tau -3}$ By a simple calculation, we get the right-hand side of (REF ).",
"Since we chose an arbitrary $j\\in J$ , we have $n-1$ points in this family.",
"Fix $j\\in J$ and stage $s\\in [n-4]$ ; the previous family can be generalized for stage $s$ , but increasing the number of options, i.e., in stage $s$ we have $n-s-1$ options from the previous stages.",
"If $i$ appears in stage $n-1$ , match it to $j$ with probability $1/n$ , if $i$ appears in stage $n-2$ , match it to $j+1$ with probability $1/n$ , and continue in this way until stage $s+1$ , where if $i$ appears, match it to node $j+n-s-2$ with probability $1/n$ .",
"If $i$ appears in stage $s$ , we consider ads $ ( j+n-s-2, \\cdots , j+1, j ) $ in this order of priority, so that $i$ is matched to $j+n-s-2$ with probability $(1-1/n)\\cdot 1/n$ ; each subsequent ad's probability of being matched to $i$ decreases exponentially until $j$ , which has probability $(1-1/n)\\cdot 1/n^{n-s-1}$ .",
"For the remaining stages (including stage $n$ ) match according to $(j+n-2,j,j+1,\\ldots ,j+n-3)$ ; in stage $n$ match any $k\\in N$ that appears with $j+n-2$ , in $s-1$ match $i$ to $ j+ n - s - 1 $ if it appears, etc.",
"So, we have the point $\\frac{1}{n}\\sum _{k\\in N}&e_{k,j+n-2}^n+\\frac{1}{n}\\sum _{\\tau =0}^{n-s-2}e^{n-\\tau -1}_{i,j+\\tau }\\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,j+\\tau }\\right]+\\frac{1}{n}\\sum _{\\tau =1}^{s-1}e_{i,j+n-s-2+\\tau }^{s-\\tau }$ The left-hand side of (REF ) evaluated at this point is $1+\\sum _{\\tau =s+1}^{n-1} \\frac{n^{n-\\tau }}{n}+\\left(1-\\frac{1}{n}\\right)\\sum _{\\tau =0}^{n-s-2}\\frac{n^{n-s}}{n^{n-\\tau -s-1}}+\\sum _{\\tau =1}^{s-1}\\frac{n^{n-\\tau }}{n} =1+\\sum _{\\tau =1}^{n-2} n^{\\tau }.$ Finally, since we chose an arbitrary $j\\in J$ and $s\\in [n-4]$ , we have $(n-1)(n-4)$ points in this family.",
"Fix $j\\in J$ .",
"For this family we do not match in stage $n$ , and in the remaining stages we match according to $(j,j+1,\\ldots ,j+n-2) \\in \\operatornamewithlimits{circ}(J)$ .",
"If $i$ appears in stage $n-1$ match it to $j$ with probability $1/n$ , if $i$ appears in stage $n-2$ , match it to $j+1$ , and so on.",
"So we have the point $\\frac{1}{n}\\sum _{\\tau =0}^{n-2}e_{i,j+\\tau }^{n-\\tau -1}$ By a simple calculation, we get the right-hand side of (REF ).",
"Finally, since we chose an arbitrary $j\\in J$ , then we have $n-1$ points in this family.",
"With these families, we have $p$ points in total.",
"Denote by $(k,j,\\tau )$ the index of a vector $z\\in [0,1]^p$ , which indicates that $k\\in N$ is matched to $j\\in J$ in stage $\\tau $ .",
"In any of these points consider the following order of components (starting from the first one): $(1,0,n)$ , $(1,1,n)$ , $\\ldots $ , $(1,n-2,n)$ , $\\ldots $ , $(n,0,n)$ , $\\ldots $ , $(n,n-2,n)$ , $(i,0,n-1)$ , $\\ldots $ , $(i,n-2,n-1)$ , $\\ldots $ , $(i,0,1)$ , $\\ldots $ , $(i,n-2,1)$ .",
"The rest of the proof consists of showing that these families define a set of linearly independent points, and we prove this using Gaussian elimination.",
"Arrange these points as column vectors in a matrix $A$ , $A=[\\text{I} , \\text{II}, \\text{III}, \\text{IV}, \\text{V}]=\\begin{pmatrix}B_1 & B_2 \\\\ B_3 & B_4\\end{pmatrix},$ where $B_1$ is a $n(n-1)\\times n(n-1)$ diagonal matrix with entries $1/n$ .",
"These columns can be used to make $B_2$ a zero matrix, yielding $\\bar{A}=\\begin{pmatrix}B_1 & 0 \\\\ B_3 & C\\end{pmatrix}.$ Consider how the columns from families II, III, and IV look like after this elimination procedure (family V is not affected).",
"Fix $g\\in J$ and sum every point (REF ) over $k\\in N$ ; this yields $\\frac{1}{n}\\sum _{k\\in N}e_{k,g}^n+\\left(1-\\frac{1}{n}\\right)e_{i,g}^{n-1}+\\sum _{\\tau =1}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } .$ Pick the point (REF ) associated with $g+1\\in J$ , $\\frac{1}{n}\\sum _{k\\in N}e_{k,g}^n+\\frac{1}{n}e^{n-1}_{i,g+1}+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e_{i,g+1}^{n-2}+\\frac{1}{n}\\sum _{\\tau =1}^{n-3}e_{i,g+1+\\tau }^{n-\\tau -2} .$ Subtract (REF ) from (REF ) to get $\\frac{1}{n}e^{n-1}_{i,g+1}+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e_{i,g+1}^{n-2}+\\frac{1}{n}\\sum _{\\tau =1}^{n-3}e_{i,g+1+\\tau }^{n-\\tau -2}-\\left(1-\\frac{1}{n}\\right)e_{i,g}^{n-1}-\\sum _{\\tau =1}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } ,$ which is equivalent to $\\frac{1}{n} e^{n-1}_{i,g+1} + \\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}+\\left(\\frac{1}{n}-\\frac{1}{n^2}-1\\right)e_{i,g+1}^{n-2}+\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =2}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } .$ Pick the point (REF ) associated with $g+1\\in J$ , $\\frac{1}{n}\\sum _{k\\in N}e_{k,g}^n&+\\frac{1}{n}e^{n-1}_{i,g+1}+\\frac{1}{n}e_{i,g+2}^{n-2} \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\frac{1}{n^2}e_{i,g+1}^{n-3}+\\frac{1}{n}e_{i,g+2}^{n-3}\\right]+\\frac{1}{n}\\sum _{\\tau =1}^{n-4}e_{i,g+\\tau +2}^{n-\\tau -3} .$ Subtract (REF ) from (REF ) to get $& \\begin{split}&\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1} +\\frac{1}{n}e^{n-1}_{i,g+1}+\\frac{1}{n}e_{i,g+2}^{n-2}-e_{i,g+1}^{n-2} \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\frac{1}{n^2}e_{i,g+1}^{n-3} +\\frac{1}{n}e_{i,g+2}^{n-3}\\right] -e_{i,g+2}^{n-3}+\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =3}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } .\\end{split}$ Pick the point (REF ) associated with $g+1\\in J$ and any $s\\in [n-4]$ , $\\begin{split}& \\frac{1}{n}\\sum _{k\\in N}e_{k,g}^n +\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n}e^{n-\\tau -1}_{i,g+\\tau +1} \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right]+\\frac{1}{n}\\sum _{\\tau =1}^{s-1}e_{i,g+n-1-s+\\tau }^{s-\\tau } .\\end{split}$ Subtract (REF ) from (REF ) to get $\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n}e^{n-\\tau -1}_{i,g+\\tau +1}&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] \\\\&+\\frac{1}{n}\\sum _{\\tau =1}^{s-1}e_{i,g+n-1-s+\\tau }^{s-\\tau }-\\left(1-\\frac{1}{n}\\right)e_{i,g}^{n-1}-\\sum _{\\tau =1}^{n-2}e_{i,g+\\tau }^{n-1-\\tau },$ which is equivalent to $\\begin{split}& \\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1} +\\frac{1}{n}e^{n-1}_{i,g+1}+\\sum _{\\tau =1}^{n-s-2}\\left[\\frac{1}{n}e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -e_{i,g+n-s-1}^s +\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =n-s}^{n-2}e_{i,g+\\tau }^{n-1-\\tau }.\\end{split}$ Since $B_1$ is a diagonal matrix, for the rest of the proof we focus on the matrix $C$ , formed by points in families II$^a$ , III$^a$ , IV$^a$ and V. Next, we apply Gaussian elimination on $C$ .",
"Subtract (REF ) from (REF ), $\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}&+\\frac{1}{n}e^{n-1}_{i,g+1}+\\frac{1}{n}e_{i,g+2}^{n-2}-e_{i,g+1}^{n-2} +\\left(1-\\frac{1}{n}\\right)\\left[\\frac{1}{n^2}e_{i,g+1}^{n-3}\\right.",
"\\\\&\\left.+\\frac{1}{n}e_{i,g+2}^{n-3}\\right] -e_{i,g+2}^{n-3}+\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =3}^{n-2}e_{i,g+\\tau }^{n-1-\\tau }\\\\&-\\frac{1}{n}e^{n-1}_{i,g+1}-\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}-\\left(\\frac{1}{n}-\\frac{1}{n^2}-1\\right)e_{i,g+1}^{n-2}\\\\&-\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =2}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } ,$ which is equivalent to $\\left(-\\frac{1}{n}+\\frac{1}{n^2}\\right)e_{i,g+1}^{n-2}+\\frac{1}{n}e_{i,g+2}^{n-2}+\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}-\\frac{1}{n^2}e_{i,g+2}^{n-3} .$ For every $s\\in [n-4]$ , subtract (REF ) from (REF ), $\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}&+\\frac{1}{n}e^{n-1}_{i,g+1}+\\sum _{\\tau =1}^{n-s-2}\\left[\\frac{1}{n}e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-2}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -e_{i,g+n-s-1}^s\\\\&+\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =n-s}^{n-2}e_{i,g+\\tau }^{n-1-\\tau }\\\\&-\\frac{1}{n}e^{n-1}_{i,g+1}-\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}-\\left(\\frac{1}{n}-\\frac{1}{n^2}-1\\right)e_{i,g+1}^{n-2}\\\\&-\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =2}^{n-2}e_{i,g+\\tau }^{n-1-\\tau } ,$ which is equivalent to $\\left(-\\frac{1}{n}+\\frac{1}{n^2}\\right)&e_{i,g+1}^{n-2}+\\frac{1}{n}e_{i,g+2}^{n-2}+\\frac{1}{n}\\sum _{\\tau =2}^{n-s-2}\\left[e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-3}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -\\frac{1}{n^2}e_{i,g+n-s-1}^s .$ For every $s\\in [n-4]$ , subtract (REF ) from (REF ), $\\left(-\\frac{1}{n}+\\frac{1}{n^2}\\right)e_{i,g+1}^{n-2}&+\\frac{1}{n}e_{i,g+2}^{n-2}+\\frac{1}{n}\\sum _{\\tau =2}^{n-s-2}\\left[e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-3}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -\\frac{1}{n^2}e_{i,g+n-s-1}^s\\\\&-\\left(-\\frac{1}{n}+\\frac{1}{n^2}\\right)e_{i,g+1}^{n-2}-\\frac{1}{n}e_{i,g+2}^{n-2}-\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}+\\frac{1}{n^2}e_{i,g+2}^{n-3} ,$ which is equivalent to $\\frac{1}{n}e^{n-3}_{i,g+3}&-\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}+\\left(\\frac{1}{n^2}-\\frac{1}{n}\\right)e_{i,g+2}^{n-3}+\\frac{1}{n}\\sum _{\\tau =3}^{n-s-2}\\left[e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-3}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -\\frac{1}{n^2}e_{i,g+n-s-1}^s .$ For every $s\\in [n-5]$ , subtract (REF ) corresponding to $s+1$ from (REF ) corresponding to $s$ , $\\frac{1}{n}e^{n-3}_{i,g+3}&-\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}+\\left(\\frac{1}{n^2}-\\frac{1}{n}\\right)e_{i,g+2}^{n-3}+\\frac{1}{n}\\sum _{\\tau =3}^{n-s-2}\\left[e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&+\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-3}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] -\\frac{1}{n^2}e_{i,g+n-s-1}^s\\\\&-\\frac{1}{n}e^{n-3}_{i,g+3}+\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}-\\left(\\frac{1}{n^2}-\\frac{1}{n}\\right)e_{i,g+2}^{n-3}\\\\ &-\\frac{1}{n}\\sum _{\\tau =3}^{n-s-3}\\left[e^{n-\\tau -1}_{i,g+\\tau +1}-e_{i,g+\\tau }^{n-\\tau -1}\\right] \\\\&-\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-4}\\frac{1}{n^{n-\\tau -s-2}}e^{s+1}_{i,g+\\tau +1}\\right] +\\frac{1}{n^2}e_{i,g+n-s-2}^{s+1} ,$ which is equivalent to $\\frac{1}{n}&e^{s+1}_{i,g+n-s-1}+\\left(\\frac{1}{n^2}-\\frac{1}{n}\\right)e_{i,g+n-s-2}^{s+1}+\\left(-1+\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-4}\\frac{1}{n^{n-\\tau -s-2}}e^{s+1}_{i,g+\\tau +1}\\right] \\\\& \\qquad -\\frac{1}{n^2}e_{i,g+n-s-1}^s+\\left(\\frac{1}{n^{2}}-\\frac{1}{n^{3}}\\right)e^{s}_{i,g+n-s-2} \\\\& \\qquad +\\left(1-\\frac{1}{n}\\right)\\left[\\sum _{\\tau =0}^{n-s-4}\\frac{1}{n^{n-\\tau -s-1}}e^{s}_{i,g+\\tau +1}\\right] .$ For $s=n-4$ , we do not need this step, since from (REF ) we have $\\frac{1}{n}e^{n-3}_{i,g+3}&+\\left(\\frac{1}{n^2}-\\frac{1}{n}\\right)e_{i,g+2}^{n-3}+\\left(-\\frac{1}{n^2}+\\frac{1}{n^3}\\right)e_{i,g+1}^{n-3}\\\\&-\\frac{1}{n^2}e_{i,g+3}^{n-4}+\\left(\\frac{1}{n^2}-\\frac{1}{n^3}\\right)e^{n-4}_{i,g+2}+\\left(\\frac{1}{n^3}-\\frac{1}{n^4}\\right)e^{n-4}_{i,g+1} .$ Observe that for any $s\\in [n-4]$ and $g\\in J$ , we can multiply row $(i,g,s+1)$ by $-1/n$ and we get the entry in row $(i,g,s)$ .",
"Finally, pick a point (REF ) in family V for $g\\in J$ , $\\frac{1}{n}e_{i,g}^{n-1}+\\frac{1}{n}e_{i,g+1}^{n-2}+\\cdots +\\frac{1}{n}e_{i,g+n-2}^1 .$ Now multiply (REF ) by $(1-n)$ and subtract it from (REF ) for $g\\in J$ , yielding $\\frac{1}{n}e^{n-1}_{i,g+1}+\\left(-1+\\frac{1}{n}\\right)e_{i,g}^{n-1}&+\\left(\\frac{1}{n}-\\frac{1}{n^2}-1\\right)e_{i,g+1}^{n-2} \\\\&+\\left(-1+\\frac{1}{n}\\right)\\sum _{\\tau =2}^{n-2}e_{i,g+\\tau }^{n-1-\\tau }-\\frac{1-n}{n}\\sum _{\\tau =0}^{n-2}e_{i,g+\\tau }^{n-\\tau -1} ,$ which is equivalent to $\\frac{1}{n}e^{n-1}_{i,g+1}-\\frac{1}{n^2}e_{i,g+1}^{n-2}.$ Since we have $g+1$ in those two stages, we have a general expression for any $g\\in J$ , $\\frac{1}{n}e^{n-1}_{i,g}-\\frac{1}{n^2}e_{i,g}^{n-2}.$ As before, we can multiply row $(i,g,n-1)$ by $-1/n$ to get the entry in row $(i,g,n-2)$ .",
"Now, we can organize the points in $C$ as $C=[\\text{V}, \\text{II}^b, \\text{III}^b, \\text{IV}^d_{n-4},\\ldots , \\text{IV}^d_s, \\ldots , \\text{IV}^d_1],$ where $\\text{IV}^d_s$ corresponds to the block of points ($g\\in J$ ) with $s\\in [n-4]$ .",
"$C$ has the form $C=\\begin{pmatrix}C_{n-1} & D_{n-2} & 0 & 0& 0& \\ldots &0&0 \\\\C_{n-2} & -\\frac{1}{n}D_{n-2} & D_{n-3} & 0 &0&\\ldots &0&0\\\\C_{n-3} & 0 & -\\frac{1}{n}D_{n-3} & D_{n-4} &0&\\ldots &0&0\\\\\\vdots & & &\\ddots &&& \\vdots \\\\C_{2} & 0 & 0 & 0 &0&\\ldots &-\\frac{1}{n}D_{2}&D_{1}\\\\C_{1} & 0 & 0 & 0 &0&\\ldots &0&-\\frac{1}{n}D_{1}\\\\\\end{pmatrix},$ where every $C_i$ and $D_i$ are circulant matrices [14] of size $(n-1)\\times (n-1)$ .",
"Since the determinant is invariant under elementary row and column operations, we can perform Gaussian elimination (of rows) from bottom to top, and we get $\\bar{C}=\\begin{pmatrix}\\bar{C}_{n-1} & 0 & 0 & 0& 0& \\ldots &0&0 \\\\\\bar{C}_{n-2} & -\\frac{1}{n}D_{n-2} & 0 & 0 &0&\\ldots &0&0\\\\\\bar{C}_{n-3} & 0 & -\\frac{1}{n}D_{n-3} & 0 &0&\\ldots &0&0\\\\\\vdots & & &\\ddots &&& \\vdots \\\\\\bar{C}_{2} & 0 & 0 & 0 &0&\\ldots &-\\frac{1}{n}D_{2}&0\\\\C_{1} & 0 & 0 & 0 &0&\\ldots &0&-\\frac{1}{n}D_{1}\\\\\\end{pmatrix},$ where $\\bar{C}_1&=C_1+n(C_2-\\cdots n(C_{n-2}+nC_{n-1})\\cdots )\\\\&=\\operatornamewithlimits{circ}(1/n,1,n,n^2,\\ldots ,n^{n-3}),\\\\D_{n-2}&=\\operatornamewithlimits{circ}(1/n,0,\\ldots 0),\\\\D_{n-3}&=\\operatornamewithlimits{circ}\\left(\\frac{1}{n},-\\frac{1}{n}+\\frac{1}{n^2},0,\\ldots 0\\right)\\\\&\\vdots \\\\D_s&=\\operatornamewithlimits{circ}\\left(\\frac{1}{n},-\\frac{1}{n}+\\frac{1}{n^2},-\\frac{1}{n^2}+\\frac{1}{n^3},\\ldots ,-\\frac{1}{n^{n-s-2}}+\\frac{1}{n^{n-s-1}},0,\\ldots 0\\right)\\\\&\\vdots \\\\D_1&=\\operatornamewithlimits{circ}\\left(\\frac{1}{n},-\\frac{1}{n}+\\frac{1}{n^2},-\\frac{1}{n^2}+\\frac{1}{n^3},\\ldots ,-\\frac{1}{n^{n-3}}+\\frac{1}{n^{n-2}}\\right)$ For $\\bar{C}_1$ , the last entry, $n^{n-3}$ , is greater than the sum of the remaining entries.",
"The same applies for $D_s$ , with entry $1/n$ .",
"Due to Proposition 18 in [14] all these matrices are nonsingular, so $\\bar{C}$ is nonsingular, and therefore $C$ is nonsingular.",
"This implies that $A$ is nonsingular, showing that these points are linearly independent, and proceeding in the same way as we did in the proof of Proposition REF for the remaining components, we can show (REF ) is facet-defining.",
"[Proof of Theorem REF .]",
"The proof is similar to Theorem REF .",
"Again, assume $h=n-1$ and $t=1$ ; the remaining cases follow a similar construction of linearly independent points.",
"Without loss of generality we can assume that $i_\\tau =i$ for all $\\tau \\in [n-2]$ .",
"So, we have an inequality of the form $r Z_{N,J}^n+n Z_{I,J}^{n-1}+\\sum _{\\tau =1}^{n-2} n^{n-\\tau } Z_{i,J}^{\\tau }\\le r+\\sum _{\\tau =1}^{n-2} n^{\\tau }.$ We construct the following linearly independent points.",
"Fix $k\\in N$ and $j\\in J$ .",
"In stage $n$ , if node $k$ appears, match it to node $j$ , with probability $1/n$ .",
"For the remaining stages match according to $(j,j+1,\\ldots ,j+n-2) \\in \\operatornamewithlimits{circ}(J)$ .",
"In terms of probability, if any $k^{\\prime }\\in I$ appears in stage $n-1$ , then it is matched to $j$ with probability $(1-1/n)\\cdot 1/n$ , so the probability of matching $j$ in stage $n-1$ is $(1-1/n)\\cdot r/n$ .",
"For the rest, the probability is $1/n$ .",
"So, we have the point $\\frac{1}{n}e_{k,j}^n+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)\\sum _{k^{\\prime }\\in I}e_{k^{\\prime },j}^{n-1}+\\frac{1}{n}\\sum _{\\tau =1}^{n-2}e_{i,j+\\tau }^{n-1-\\tau }.$ By a simple calculation, it is easy to see that each of these points achieves the right-hand side of (REF ).",
"Since we chose any arbitrary $k\\in N$ and $j\\in J$ , we have $n(n-1)$ points in this family.",
"Fix $j\\in J$ and $k\\in I$ .",
"In this family we repeat the same ad to match in stages $n-1$ and $n-2$ .",
"If $k$ appears in stage $n-1$ , match it to $j$ with probability $1/n$ .",
"Then, if $i$ appears in stage $n-2$ and $k$ did not appear in $ n-1 $ , match it to $j$ with probability $(1-1/n)\\cdot 1/n$ .",
"For the remaining stages match according to $(j+n-2,j,j+1,\\ldots ,j+n-3) \\in \\operatornamewithlimits{circ}(J)$ ; in stage $n$ match any $k^{\\prime }\\in N$ that appears with $j+n-2$ , in stage $n-3$ match $i$ to $ j + 1 $ if it appears, and so on.",
"So, we have the point $\\frac{1}{n}\\sum _{k^{\\prime }\\in N}e_{k^{\\prime },j+n-2}^n+\\frac{1}{n}e^{n-1}_{k,j}+\\frac{1}{n}\\left(1-\\frac{1}{n}\\right)e_{i,j}^{n-2}+\\frac{1}{n}\\sum _{\\tau =1}^{n-3}e_{i,j+\\tau }^{n-\\tau -2} .$ By a simple calculation, we get the right-hand side of (REF ).",
"Finally, since we chose an arbitrary $j\\in J$ and $k^{\\prime }\\in I$ , we have $r(n-1)$ points in this family.",
"Fix $j\\in J$ .",
"In this family we do not match in stage $n$ , and in the remaining stages we match according to a vector in $\\operatornamewithlimits{circ}(J)$ .",
"If any $k\\in I$ appears in stage $n-1$ , match it to $j$ with probability $1/n$ , if $i$ appears in stage $n-2$ , match it to node $j+1$ , and so forth.",
"So we have the point $\\frac{1}{n}\\sum _{k\\in I}e_{k,j}^{n-1}+\\frac{1}{n}\\sum _{\\tau =1}^{n-2}e_{i,j+\\tau }^{n-\\tau -1} .$ By a simple calculation, we get the right-hand side of (REF ).",
"Since we chose an arbitrary $j\\in J$ , we have $n-1$ points in this family.",
"Families III, and IV remain the same as in the proof of Theorem REF , so in total we have $(n-1)(r+2n-2)$ points.",
"The rest of the proof follows the same argument as Theorem REF .",
"[Proof of Theorem REF .]",
"The proof follows the same argument as the previous two theorems, the only difference being that the collection of points given by policies corresponding to $I$ is bigger; however, they still form a diagonal block, so we can apply the same procedure."
]
] | 1709.01557 |
[
[
"Machine Learning and Social Robotics for Detecting Early Signs of\n Dementia"
],
[
"Abstract This paper presents the EACare project, an ambitious multi-disciplinary collaboration with the aim to develop an embodied system, capable of carrying out neuropsychological tests to detect early signs of dementia, e.g., due to Alzheimer's disease.",
"The system will use methods from Machine Learning and Social Robotics, and be trained with examples of recorded clinician-patient interactions.",
"The interaction will be developed using a participatory design approach.",
"We describe the scope and method of the project, and report on a first Wizard of Oz prototype."
],
[
"Introduction",
"With an increasing number of elderly in the population, dementia is expected to increase.",
"Dementia disorders and related illnesses such as depression in the aging population are devastating for the quality of life of the afflicted individuals, their families, and caregivers.",
"Moreover, the increase in such disorders imply enormous costs for the health sectors of all developed countries.",
"There is still no cure available.",
"However, studies, e.g.",
"[18], show that it is possible to reduce the risk for dementia in later life through a number of preventative measures.",
"Early diagnoses and prevention measures are thus key to counteract dementia.",
"Diagnoses of dementia are now made by teams of expert clinicians with standardized neuropsychological tests (Section ) as well as EEG and MRI examinations.",
"A number of additional factors in the behavior are taken into regard apart from the actual test scores and measurements (Section ).",
"Using these methods, dementia in early stages are often missed, since the time intervals between the sessions usually are quite large, and symptoms often come and go over time.",
"Once a diagnosis has been made, the caring medical expert typically prescribes a specific therapeutic intervention of the type of memory training.",
"Training sessions are often scarce and require the presence of a medically trained person.",
"The main goal of the multi-disciplinary research described in this position paper, is to develop an embodied agent (Section ) capable of interacting with elderly people in their home, assessing their mental abilities using both standardized test and analysis of their non-verbal behavior, in order to identify subjects at the first stages of dementia.",
"The contributions of such a system to the diagnostics and treatment of dementia would be twofold: Firstly, since the proposed embodied system would enable tests to be carried out much more often, at the test subject's home, subtle and temporarily appearing symptoms might be detected that are missed in regular screenings at the clinic.",
"Moreover, the computerized test procedure makes it possible to collect quantitative measurements of the subject's cognitive development over time, enabling a clinician overseeing the test process to base their diagnosis on rich statistical data instead of few, more qualitative, test results.",
"All this will make it possible to introduce preventive therapy much earlier.",
"Secondly, after a mild cognitive impairment or early dementia diagnosis, the proposed system may also be used for patients to train regularly at home as a complement to sessions with an expert, greatly enhancing the efficiency of the therapy.",
"The research efforts needed to achieve these goals span over such diverse research areas of Social Robotics, Geriatrics, and Machine Learning, and include: a user- and caregiver-friendly embodied agent that carries out established neuropsychological tests.",
"This includes the development of an adaptive and sustainable dialogue system, with elements of gamification that encourages the subject to carry out the tests as effectively as possible (Section ).",
"pioneering Machine Learning solutions to achieve diagnostics from the user's performance in the neuropsychological tests, as well as their non-verbal behavior during the test (Section ).",
"systematic evaluation of the developed systems in both large and focused social groups, using feedback from each evaluation to guide further scientific research and design.",
"Research efforts in this direction have been made before, e.g. [23].",
"The novelty of the present project with respect to this work is that we propose a more advanced interaction and a more sophisticated embodiment that supports, among other things, mutual gaze and shared attention."
],
[
"Survey of related work",
"Several efforts have been made to investigate the acceptance of social robots by the elderly [13], some of them with a particular focus on home environments [9], [27].",
"For example, through a series of questionnaires and interviews to understand the attitudes and preferences of older adults for robot assistance with everyday tasks, Smarr et al.",
"[27] found that this user group is quite open for robots to perform a wide range of tasks in their homes.",
"Surprisingly, for tasks such as chores or information management (e.g., reminders and monitoring their daily activity), older adults reported to prefer robot assistance over human assistance.",
"One of the main motivations of developing assistive robots for the elderly is the fact that this technology can promote longer independent living [9].",
"However, in-home experiments are still scarce because of privacy concerns and limited robot autonomy.",
"One of the few exceptions is the work by de Graaf et al.",
"[11], who investigated reasons for technology abandonment in a study where 70 autonomous robots were deployed in people’s homes for about six months.",
"Regardless of age (their participant pool ranged from 8 to 77 years), they found that the main reasons why people stopped using the robot were lack of enjoyment and perceived usefulness.",
"These findings indicate that involving the target users from the early stages of development can be crucial for the success and acceptance of social robots.",
"While most of the human-robot interaction studies with older adults have been conducted with neurotypical participants [31], [3], a few authors have included patients with dementia in their research.",
"Sabanovic et al.",
"[24], for instance, evaluated the effects of PARO, a seal-like robot, in group sensory therapy for older adults with different levels of dementia.",
"The results of a seven-week study indicate that the robot's presence contributed to participants’ higher levels of engagement not only with the robot but also with the other people in the environment.",
"Similar findings were obtained by Iacono and Marti [14], who compared the effects of the presence of a PARO robot and a stuffed toy during a group storytelling task.",
"Furthermore, the authors found that while in the presence of the robot, participants' stories were much more articulated in terms of number of words, characters and narrative details.",
"Perhaps most similar to our work both in terms of the interaction modality (language-based) and condition of the user group (older adults diagnosed with dementia), Rudzicz et al.",
"[23] conducted a laboratory Wizard of Oz experiment to evaluate the challenges in speech recognition and dialogue between participants and a personal assistant robot while performing daily household tasks such as making tea.",
"Their results suggest that autonomous language-based interactions in this setting can be challenging not only because of speech recognition errors but also because robots will often need to proactively employ conversational repair strategies during moments of confusion."
],
[
"The Furhat dialogue system",
"The project will make use of the robot platform Furhat [1] (Figure REF ).",
"Furhat is a unique social robotic head based on back-projected animation that has proven capable of exhibiting verbal and non-verbal cues in a manner that is in many ways superior to on-screen avatars.",
"Furhat also offers advantages over mechatronic robot heads in terms of high facial expressivity, very low noise level and fast animation allowing e.g.",
"for highly accurate lip-synch.",
"Since the face is projected on a mask, the robot’s visual appearance can be easily modified e.g.",
"to be more or less photo realistic or cartoonish, by changing the projection and/or the mask.",
"This feature will be exploited in the participatory design study (Section REF ).",
"The Furhat robot includes a multi-modal dialog system, implemented using the IrisTK framework [26].",
"It uses statecharts as a powerful formalism for complex, reactive, event-driven spoken interaction management.",
"It also includes a 3D situation model that makes it possible to handle situated interaction, including multiple users talking to the system and objects being discussed.",
"A customizable Wizard of Oz mode as well as API:s for external control makes it straightforward to set up supervised experiments and data collection studies when a fully autonomous system can not be implemented."
],
[
"Clinical assessment of cognitive function",
"As described in the introduction, dementia is detected through clinical cognitive assessments.",
"They are carried out under supervision of a clinician and evaluate attention, memory, language, visuospatial/perceptual functions, psycho- motor speed, and executive functions through a wide range of tests.",
"Common and valid tests include learning and remembering a wordlist, e.g.",
"Rey Auditory Verbal Learning Test (RAVLT) [32], visuo-spatial and memory abilities, e.g.",
"Rey-Osterrieth Complex Figure Test (ROCF) [19], and test of mental and motor speed, e.g.",
"Wechsler Adult Intelligence Scale (WAIS) subtest Coding [15].",
"Speech and language assessment includes, e.g., tests of picture naming [17], word fluency [29], and aspects of high-level language comprehension [2]."
],
[
"Montreal Cognitive Assessment test (MoCA)",
"The Montreal Cognitive Assessment (MoCA) [21] is a brief test that evaluates several of the cognitive domains mentioned above in a time-effective manner.",
"Visuospatial and executive functions are here assessed using trail making, figure copying and clock drawing tasks.",
"Language is assessed using object naming, sentence repetition, phonemic fluency and abstraction tasks.",
"Attention is assessed using digit repetition, target detection and serial subtraction tasks.",
"Memory is assessed using a delayed verbal recall task after initial learning trials.",
"Temporal and spatial orientation is also assessed.",
"MoCA has proven robust in identifying subjects with mild cognitive impairment (MCI) and early Alzheimer’s disease (AD) and in distinguishing them from healthy controls, thereby becoming an important screening tool for clinicians all over the world [16].",
"Due to an increasing interest in using MoCA as a monitoring tool and the need of minimizing practice effects associated with repeated assessment, alternate forms of the test have also been developed [6]."
],
[
"Automating assessment of cognitive function",
"We will in this project implement clinical neuropsychological assessments with the Furhat system (Section ).",
"The Furhat agent will take the clinician's role, supervising and driving the test procedure.",
"The assessments with Furhat will take place in the user's home and complement the regular assessments at the memory clinic."
],
[
"Automating the Montreal Cognitive Assessment test (MoCA)",
"We employ the Furhat system to implement the Montreal Cognitive Assessment test (MoCA).",
"The intent is replicate the typical interaction between patient and clinician during the administration of the test, with the robot acting as the clinician.",
"In order to rapidly get feedback from participants, a Wizard of Oz [7] prototype has been developed where participants can interact with the embodied agent.",
"The prototype consists of a Furhat robot and a touch-enabled tabletop computer (Figure REF ) connected to a remote wizarding interface.",
"The system is set up to run through MoCA as described above.",
"The prototype follows a scripted interaction, but also gives the wizard a large set of additional utterances in order to handle small deviations and make the interaction more life-like.",
"These include typical conversational devices, such as affirmations, discourse markers, and the ability to add continuity or “flow\" to the interaction, such as “ok, let's move on to the next section\".",
"The interaction is primarily centered around speech, which is realized by converting pre-set text prompts into speech output via a built in text-to-speech (TTS) system.",
"However, some tasks require the participant to use the touch table in order to, for example, draw an image or recall the name of certain objects displayed on the screen.",
"Besides providing the wizard with a set of utterances, the prototype also provides the wizard with the ability to perform certain facial expressions, head nods and sharing attention towards a given point.",
"Using all of these modalities simultaneously allows the wizard to convey a highly engaging and believable embodiment of a clinician who can understand the user's actions and responses, and respond appropriately, just as a real clinician would.",
"The wizard interface consists of “buttons\" on a standard computer screen (Figure REF ) that can be triggered from keys on its keyboard.",
"These include the aforementioned pre-set text prompts which are realized as a TTS voice, as well as dedicated keys for facial expressions, head nods, and head movement to look at the touch screen.",
"The wizard can hear participants via a microphone, and see them through a glass partition or via a webcam.",
"Figure: Wizard of Oz interface to the first Furhat system prototype."
],
[
"Future Work: Participatory Design of the dialogue system",
"A series of workshops, each lasting 1-2 hours, will be conducted with around 10 participants.",
"These workshops will be based on the concept of Participatory Design, wherein the various stakeholders in the project will participate in the design of both the physical robot and the content of the proposed interactions.",
"Stakeholders include potential users from the target demographic (volunteers and patients), clinicians with relevant expertise (KI), and the robot/interaction researchers (KTH).",
"Specific activities during the workshops will include: introductory interaction with the robot system, discussion/feedback of various aspects of system, e.g.",
"look, sound, content, discussion of volunteers' and clinicians' preferences/expectations/concerns regarding system.",
"The process will be iterative: after each workshop session, the researchers will integrate ideas generated during the workshop into a new version of the system, which they will demonstrate at the subsequent session, which the stakeholders can then re-evaluate.",
"Interactions with the prototype will be recorded in order to improve the current system, but also in order to collect data used for building the automated system in the future (Section )."
],
[
"Additional factors used in assessment of cognitive function",
"During the clinical assessment, the clinician also make use of additional information in the diagnostic process.",
"Different neurodegenerative diseases often differ in socioemotional presentation; one example is mutual gaze [28].",
"Studies have also shown changes in eye movements due to Alzheimer’s disease, with e.g.",
"alterations in gaze behavior [20].",
"Spoken interaction offers clues to cognitive functioning that are not usually measured or rated in clinical assessment.",
"The temporal organization of speech, such as the incidence and duration of pauses, as well as the overall speaking rate, may signal word-finding problems and difficulties with discourse planning [10].",
"The prosodic organization, including pitch and loudness, may be abnormal in right-hemisphere brain lesions and in striatal loop dysfunction as in Parkinson’s disease and related cognitive disorders [22]."
],
[
"Future Work: Structured description of the diagnostic process",
"To be able to incorporate such reasoning in the diagnostic process of the automatic dialogue system, we need to develop simplified and structured descriptions of what clinicians actually do during different kinds of assessment; pseudo-code-like scripts suitable for computer implementation.",
"These descriptions include both the verbal and non-verbal communication of the clinician during the cognitive assessment, but also the aspects of the patient's verbal and non-verbal communication relevant to assessment of cognitive function."
],
[
"Automating inference from additional factors",
"During the last decades, there has been a rapid development of methods for machine analysis of human spoken language.",
"However, the information communicated between interacting humans is only to a small part verbal; humans transfer huge amounts of information through non-verbal cues such as face and body motion, gaze, and tone of voice, and these signals can be analyzed automatically to a certain degree [4].",
"As described in Section , dementia diagnostics relies heavily on such cues, and we aim to equip the system with the capability to take non-verbal cues into account in the diagnostics.",
"We will model the cognitive processes of non-verbal communication in the human brain (Figure REF ), on such a level that they explain the correlation between what the human perceives from the clinician's communication, and what the human in turn communicates.",
"The underlying condition of an observed human can then be inferred from the recorded interaction with the clinician.",
"Data will be collected during user studies (Section REF ).",
"Non-verbal signals of both clinician and evaluated human will be recorded using sensors such as Kinect human tracker, Tobii gaze detector, but also state-of-the-art techniques for extracting social signals from RGB-D video, e.g.",
"facial action units [30], and speech sound, e.g.",
"prosody, laughter and pauses [12]."
],
[
"Future Work: Development of Machine Learning methodology",
"The processes depicted in Figure REF represent incredibly complex, non-smooth, and non-linear mappings and representations, which indicates that it will be suitable to use a deep neural network [25] approach.",
"Our initial studies [5] show that it is beneficial to use generative, probabilistic deep models, in order to be able to incorporate prior information in a principled manner.",
"Such information includes clinical expert knowledge and logic reasoning.",
"Moreover, deep probabilistic approaches, such as [8], provide both more interpretability and lowers the needed amount of training data – important aspects for a diagnostics method."
],
[
"Conclusions",
"This position paper presented the EACare project, where we aim to develop a system with an embodied agent, that can carry out neuropsychological clinical tests and detect early signs of dementia from both the test results and from the user's non-verbal behavior.",
"This is intended as a stand-alone system, which the user brings home, and which interacts with the user in between regular screenings at the clinic.",
"Two different “spin-off products\" could be a system without embodiment or dialogue generation, which only serves as decision support to a clinician during a neuropsychological screening at the clinic, a system which passively monitors the user's communication behavior during daily activities, e.g.",
"as a back-end to Skype."
],
[
"Acknowledgements",
" The project described in this paper is funded by the Swedish Foundation for Strategic Research (SSF)."
]
] | 1709.01613 |
[
[
"PageNet: Page Boundary Extraction in Historical Handwritten Documents"
],
[
"Abstract When digitizing a document into an image, it is common to include a surrounding border region to visually indicate that the entire document is present in the image.",
"However, this border should be removed prior to automated processing.",
"In this work, we present a deep learning based system, PageNet, which identifies the main page region in an image in order to segment content from both textual and non-textual border noise.",
"In PageNet, a Fully Convolutional Network obtains a pixel-wise segmentation which is post-processed into the output quadrilateral region.",
"We evaluate PageNet on 4 collections of historical handwritten documents and obtain over 94% mean intersection over union on all datasets and approach human performance on 2 of these collections.",
"Additionally, we show that PageNet can segment documents that are overlayed on top of other documents."
],
[
"Introduction",
"When digitizing a document into an image, it is common to include a surrounding border region to visually indicate that the entire document is present in the image.",
"The border noise resulting from this process can interfere with document analysis algorithms and cause undesirable results.",
"For example, text may be detected and transcribed outside of the main page region when textual noise is present due to a partially visible neighboring page.",
"Thus, it is beneficial to preprocess the image to remove the border region.",
"Some examples of border noise in historical documents include background, book edges, overlayed documents, and parts of neighboring pages (see Figure REF ).",
"While removing background is feasible using simple segmentation techniques, other types of border noise are more challenging.",
"A number of approaches have been developed to remove border noise (e.g., [7], [19], [21], [3], [5]).",
"However, as noted in [4], many prior methods make assumptions that do not necessarily hold for historical handwritten documents.",
"These assumptions about the input image include consistent text size, absolute location of border noise, straight text lines, and distances between page text and border [4].",
"In this work, we propose PageNet, a deep learning based system that, given an input image, predicts a bounding quadrilateral for the main page region (see Figure REF ).",
"PageNet is composed of a Fully Convolutional Network (FCN) and post-processing.",
"The FCN component outputs a pixel-wise segmentation which is post-processed to extract a quadrilateral shaped region.",
"Detecting the main page region is similar to finding the page frame [19], but the former task detects the entire page while the latter crops the detected region to the page text.",
"PageNet is able to robustly handle a variety of border noise because, as a learning based system, it does not make any explicit assumptions about the border noise, layout, or content of the input image.",
"Though learning methods can potentially overfit the training collection, we demonstrate that PageNet generalizes to other collections of documents.",
"We experimentally evaluated PageNet on 5 collections of historical documents that we manually annotated with quadrilateral regions.",
"To estimate human agreement on this task, we made a second set of annotations for a subset of images.",
"On our primary test-set, PageNet achieves 97.4% mean Intersection over Union (mIoU) while the human agreement is 98.3% mIoU.",
"On all collections tested, we achieve a mIoU of $>$ 94%.",
"Additionally, we show that PageNet is capable of segmenting documents that are overlayed on top of other documents.",
"In order to support the reproducibility of our work, we are releasing our code, models, and dataset annotations, available at www.example.com."
],
[
"Related Work",
"We take a segmentation approach to removing border noise, so we review the literature on traditional border noise removal techniques and on segmentation.",
"Fan et al.",
"remove non-textual marginal scanning noise (e.g., large black regions) from printed documents by detecting noise with a resolution reduction approach and removing noise through region growing or local thresholding [7].",
"Shafait et al.",
"handle both textual and non-textual marginal noise by finding the page frame of an image that maximizes a quality function w.r.t.",
"an input layout analysis composed of sets of connected components, text lines, and zones [20].",
"The method of Shafait and Bruel examines the local densities of black and white pixels in fixed image regions to identify noise and also removes connected components near the image border [19].",
"Stamatopoulos et al.",
"proposed a system based on projection profiles to find the two individual page frames in images of books where two pages are shown.",
"They report an average F-measure of 99.2% on 15 historical books [21].",
"Bukhari et al.",
"find the page frame for camera captured documents by detecting text lines, aligning the text line end points, and estimating a straight line from the endpoints using RANdom SAmple Consensus (RANSAC) linear regression [3].",
"For further reading, we refer the reader to a recent survey on border noise removal [4].",
"Another formulation of the border noise removal problem is to find the four corners of the bounding quadrilateral of the page, which is a sub-task shared with perspective dewarping techniques.",
"Jagannathan et al.",
"find page corners in camera captured documents by identifying two sets of parallel lines and two sets of perpendicular lines in the perspective transformed image [12].",
"Yang et al.",
"use a Hough line transform to detect boundaries in binarized images of license plates [24].",
"Figure: Post processing to extract quadrilateral from FCN predictions.Intelligent scissors segments an object from background by finding a least cost path through a weighted graph defined over pixels, subject to input constraints [17].",
"Active Contours or Snakes formulate segmentation as a continuous optimization problem and finds object boundaries by minimizing a boundary energy cost and the cost of deformation from some prior shape [14].",
"Graph Cut methods (e.g.",
"[1]) formulate image segmentation as finding the minimum cut over a graph constructed from the image.",
"Weights in the graph are determined by pixel colors and by per-pixel apriori costs of being assigned to the foreground and background segments.",
"GrabCut iteratively performs graph cut segmentations, starting from an initial rough bounding box.",
"The result of each iteration is used to refine a color model which is used to construct the edge weights for the next iteration [18].",
"Several neural network approaches have been proposed for image segmentation.",
"The Fully Convolution Network (FCN) learns an end-to-end classification function for each pixel in the image [16].",
"However, the output is poorly localized due to downsampling in the FCN architecture used.",
"Zheng et al.",
"integrated a Conditional Random Field (CRF) graphical model into the FCN to improve segmentation localization [25].",
"In contrast, our approach maintains the input resolution and therefore does not suffer from poor localization.",
"The Spatial Transformer Network [11] learns a latent affine transformation in conjunction with a task specific objective, effectively learning cropping, rotation, and skew correct in an end-to-end fashion.",
"In our case, we are interested in directly learning a pre-processing transformation from ground truth.",
"Chen and Seuret used a convolutional network to classify super pixels as background, text, decoration, and comments [6].",
"Super pixel based features would not work in our case as neighboring pages are identical to the main page region in local appearance."
],
[
"Method",
"In this section, we describe PageNet, which takes in a document image and outputs the coordinates of four corners of the quadrilateral that encloses the main page region.",
"PageNet has two parts: A Fully Convolutional Neural Network (FCN) to classify pixels as page or background Post processing to extract a quadrilateral region"
],
[
"Pixel Classification",
"FCNs are learning models that alternate convolutions with element-wise non-linear functions [16].",
"They differ from traditional CNNs (e.g., AlexNet) that have fully connected layers which constrain the input image size.",
"In particular, the FCN used in PageNet maps an input RGB image $x \\in \\mathbb {R}^{3 \\times H \\times W} \\rightarrow y \\in \\mathbb {R}^{H \\times W}$ , where $y_{ij} \\in [0,1]$ is the probability that pixel $x_{ij}$ is part of the main page region.",
"Each layer of a basic FCN performs the operation $ x_{\\ell } = g(W_{\\ell } \\ast x_{\\ell - 1} + b_{\\ell })$ where $\\ell $ is a layer index, $\\ast $ indicates multi-channel 2D convolution, $W_{\\ell }$ is a set of learnable convolution filters, $b_{\\ell }$ is a learnable bias term, and $g$ is a non-linear function.",
"In our case, we use $g(z) = \\operatorname{ReLU}(z) = \\max (0, z)$ as element-wise non-linear rectification.",
"In some FCN architectures, spatial resolution is decreased at certain layers through pooling or strided convolution and increased through bilinear interpolation or backwards convolution [16].",
"In the last layer of the network, $\\operatorname{ReLU}$ is replaced with a channel-wise softmax operation (over 2 channels in our case) to obtain output probabilities for each pixel.",
"PageNet uses a successful multi-scale FCN architecture originally designed for binarizing handwritten text [22].",
"This FCN operates on 4 image scales: $1, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{8}$ .",
"The full resolution image scale uses 7 sequential layers (Eq.",
"REF ), and each smaller layer uses one layer less than the previous (e.g.",
"$\\frac{1}{8}$ scale uses 4 layers).",
"The input feature maps of the 3 smallest scales are obtained by $2\\times 2$ average-pooling over the output of the first layer of the next highest image scale.",
"The FCN concatenates the output feature maps of each scale, upsampling them to the input image size using bilinear interpolation.",
"Thus the architecture both preserves the original input resolution and benefits from a larger context window obtained through downsampling.",
"This is followed by 2 more convolution layers and the softmax operation.",
"For full details on the FCN architecture, we refer the reader to [22].",
"We performed initial experiments (not shown) with a single scale FCN but it performed worse, likely due to smaller surrounding context for each pixel."
],
[
"Quadrilaterals",
"Thresholding output of the FCN yields a binary image (Figure REF b), which is converted to the coordinates of a quadrilateral around the main page region in the image.",
"While the pixel representation may be already useful for some applications (e.g., masking text detection regions), it can lack global and local spatial consistency due to the FCN predicting pixels independently based on local context.",
"Representing the detected page region as a quadrilateral fixes some errors in the FCN output and makes it easier for potential downstream processing to incorporate this information.",
"Figure REF demonstrates the following post-processing steps that converts the binary per-pixel output of the FCN (after thresholding at 0.5 probability) to a quadrilateral region: Remove all but the largest foreground components.",
"Fill in any holes in the remaining foreground component.",
"Find the minimum area oriented bounding rectangle using Rotating Calipers method [23].",
"Iteratively perturb corners in greedy fashion to maximize IoU between the quadrilateral and the predicted pixels.",
"Step 1 helps remove any extraneous components (false positives) that were predicted as page regions.",
"Some of these errors occur because the FCN makes local classification decisions.",
"Similarly, Step 2 removes false negatives.",
"In Step 3, we find a (rotated) bounding box for the main page region (OpenCV implementation [2]), but the bounding box encloses all predicted foreground pixels and is therefore sensitive to any false positive pixels that are outside the true page boundary.",
"This bounding box is used to initialize the corners for iterative refinement in Step 4.",
"At each refinement iteration, we measure the IoU of 16 perturbations of the corners (4 corners moved 1 pixel in 4 directions) and greedily update with the perturbation that has the highest IoU w.r.t.",
"the FCN output.",
"We stop the process when no perturbation improves IoU.",
"Post-processing is done on 256x256 images, so there may be quantization artifacts after the quadrilaterals are upsampled to the original size."
],
[
"PageNet Implementation Details",
"We implemented the FCN part of PageNet using the popular deep learning library Caffe [13].",
"The dataset used for training is detailed in Section .",
"For preprocessing, color images are first resized to 256x256 pixels and pixel intensities are shifted and scaled to the range $[-0.5,0.5]$ .",
"For ground truth, we label each pixel inside the image's annotated quadrilateral as foreground and all other pixels as background.",
"The ground truth images are also resized to 256x256.",
"While all input images yield a probability map of the same size, we used 256x256 images in training and evaluation.",
"While a larger input sizes could lead to slightly higher segmentation accuracy, we achieve good results with the computationally faster 256x256 size.",
"Initial experiments with 128x128 inputs were less accurate.",
"To train the FCN, we used Stochastic Gradient Descent for 15000 weight updates with a mini-batch size of 2 images.",
"We used an initial learning rate of 0.001, which was reduced to 0.0001 after 10000 weight updates.",
"We used a momentum of 0.9, L2 regularization of 0.0005, and clipped gradients to have an L2 norm of 10.",
"We trained 10 networks and used the validation set to select the best network for the results reported in Section ."
],
[
"Dataset",
"Our main dataset is the ICDAR 2017 Competition on Baseline Detection (CBAD) dataset [10], which are handwritten documents with varying levels of layout complexity (see Figure REF a,b,c).",
"We combined both tracks of the competition data and separated the images into training, validation and test sets with 1635, 200, and 200 images respectively.",
"We train PageNet on the training split of CBAD and evaluate on the validation and test splits of the same dataset.",
"We also evaluated on a subset of the CODH PMJT datasethttp://codh.rois.ac.jp/char-shape/, and on all images from the Saint Gall and Parzival datasets.",
"The PMJT (Pre-Modern Japanese Text) dataset is taken from the Center for Open Data in the Humanities (CODH) [15] and consists of handwritten literature (see Figure REF d) and some graphics.",
"We randomly sampled 10 pages from each the collections, excluding ID 20003967, to create an evaluation set of 140 images.",
"The Saint Gall dataset [8] is a collection of 9th century manuscripts put together by the FKI: Research Group on Computer Vision and Artificial Intelligence.",
"The Parzival dataset [9] is also put together by FKI and consists of 13th century Medieval German texts.",
"We also trained and evaluated PageNet on a private collection of Ohio death recordsData provided by FamilySearch, having a training set of 800 images, and validation and testing sets of 100 images each.",
"This dataset, while relatively uniform in the types of documents, presents a unique challenge of overlay documents (see Figure REF e).",
"While much of the document underneath an overlay is visible, much is still occluded, thus it is most desirable to localize just the overlay, which is a task that has not received much attention in the literature.",
"These overlays represent approximately 12% of the images in the dataset."
],
[
"Ground Truth Annotation",
"Our ground truth annotations consist of quadrilaterals encompassing the pages fully present in an image, not including any partial pages or page edges (if possible).",
"We chose to use quadrilaterals rather than pixel level annotations as most pages are quadrilateral-shaped, and it is much faster to annotate polygon regions than pixel regions.",
"Regions were manually annotated using an interface where the annotator clicks on each of the four corner vertices.",
"In the case of multiple full pages, the quadrilateral encloses all pages present regardless of their orientation to each other.",
"This typically occurs when two pages of a book are captured in a single image.",
"In order to give an upper bound on expected automated performance, we measured human agreement on the quadrilateral regions of our datasets.",
"A second annotator provided region annotations for validation and test sets.",
"To measure human agreement, this second set of regions was treated the same as the output of an automated system and scored w.r.t.",
"the first set of annotations."
],
[
"Results",
"In this section, we quantitatively and qualitatively compare PageNet with baseline systems and with human annotators.",
"To evaluate system performance, we use Intersection over Union (IoU) averaged over all images in a dataset (mean IoU).",
"We chose mIoU as our metric because it is commonly used to evaluate segmentation task."
],
[
"Baselines systems",
"We compare PageNet with three baseline systems: full image, mean quadrilateral, and GrabCut [18].",
"For the full image baseline, the entire image is predicted as the main page region.",
"The mean quadrilateral can be computed as $\\bar{x}_n = \\frac{1}{N} \\sum _{i=1}^N \\frac{x_{in}}{w_i} \\;\\;\\; \\bar{y}_n = \\frac{1}{N} \\sum _{i=1}^N \\frac{y_{in}}{h_i}$ where the mean quadrilateral is $(\\bar{x}_1, \\bar{y}_1, \\dots , \\bar{x}_4, \\bar{y}_4)$ , the $i$ th annotated quadrilateral is $(x_{i1}, y_{i1}, \\dots , x_{i4}, y_{i4})$ , $N$ is the number of training images, $w_i$ and $h_i$ are respectively the width and height of the $i$ th image.",
"The predicted quadrilateral for this baseline only relies on the height and width on the test image.",
"For the $j$ th image, the prediction is $(w_j \\bar{x}_1,$ $h_j \\bar{y}_1, \\dots , w_j \\bar{x}_4, h_j \\bar{y}_4)$ .",
"Our mean quadrilateral was computed from the CBAD training split.",
"For the GrabCut baseline, we used the implementation in the OpenCV library [2].",
"For an initial object bounding box, we include the whole image except for a 5 pixel wide border around the image edges.",
"Unlike other baselines and the full PageNet results, GrabCut outputs a pixel mask (i.e., not a quadrilateral).",
"Therefore, it is directly comparable to PageNet before we extract a quadrilateral."
],
[
"Overall Results",
"We trained PageNet on CBAD-train and tested it and the baseline methods on 4 datasets of historical handwritten documents.",
"Numerical results are shown in Table REF , and some example images are shown in Figure REF .",
"For all datasets, except CBAD-train, we manually annotated each image twice in order to estimate human agreement on this task.",
"On all datasets, the full PageNet system performed the best of all automated systems and strongly outperformed the baseline methods.",
"Notably, outputting quadrilaterals improves the pixel segmentation produced by the FCN, which shows that outputting a simpler region description does not decrease segmentation quality.",
"There is little difference in the results for the different splits of CBAD, which indicates that PageNet does not overfit the training images.",
"Performance is highest on Saint Gall because the border noise is largely limited to a black background.",
"On PMJT, PageNet performed worst and most errors can be attributed to incorrectly identifying page boundaries between pages, perhaps because the Japanese text is vertically aligned.",
"The full image baseline performs worst as it simply measures the average normalized area of the main page region.",
"With the exception of Saint Gall, GrabCut only marginally outperforms the mean quadrilateral.",
"As GrabCut is based on colors and edges, it often fails to exclude partial page regions (e.g., Figure REF ) and sometimes labels dark text regions the same as the dark background.",
"A few images in CBAD are well cropped and contain only the main page region, which is problematic for GrabCut because it will always attempt to find two distinct regions.",
"In contrast, once trained, PageNet can classify an image as entirely the main page region if it does not contain border noise."
],
[
"Comparison to Human Agreement",
"The last column of Table REF shows the performance of a second annotator scored w.r.t.",
"the original annotations.",
"This performance captures the degree of error, or ambiguity, inherent with human annotations on this task, a level of performance which would be difficult for any automated system to surpass.",
"For CBAD, PageNet is roughly 1% below human agreement, which indicates the network's proficiency at the task.",
"On the PMJT dataset there is a larger gap between automated and human performance and indicates that there is still room for improvement.",
"The human agreement results in Table REF show the agreement between two annotators.",
"We also measured the agreement of the same annotator labeling the same images on a different day.",
"The same-annotator mIOU are 99.0% and 98.4% for CBAD-test and CBAD-val respectively.",
"These are slightly higher than the mIOU of 98.3% and 97.8% obtained by a different annotator.",
"This highlights the inherit ambiguity in labeling the corners of the main page region."
],
[
"Overlay Performance",
"We also trained PageNet on a private dataset of Ohio death records.",
"This dataset has several images where one document is overlayed on top of another document (e.g., Figure REF e), which creates particularly challenging textual noise.",
"Table REF shows results on this dataset.",
"In Figure REF , we show predicted segmentation masks for two images, which together show that PageNet correctly segments the overlayed image when present.",
"Figure REF a contains a document overlayed on top of the document shown in Figure REF b.",
"With the overlayed document, PageNet segments only the overlayed document, but when the overlayed document is removed, it segments the document underneath from the background."
],
[
"Conclusion",
"We have presented a deep learning system, PageNet, which removes border noise by segmenting the main page region from the rest of the image.",
"An FCN first predicts a class for each input pixel, and then a quadrilateral region is extracted from the output of the FCN.",
"We demonstrated near human performance on images similar to the training set and showed good performance on images from other collections.",
"On an additional collection, we showed that PageNet can correctly segment overlayed documents."
]
] | 1709.01618 |
[
[
"A Note on Iterated Consistency and Infinite Proofs"
],
[
"Abstract Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $\\Pi^0_1$-ordinal, characterizing the $\\Pi^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the $\\Pi^0_1$-ordinals for a range of theories.",
"The present note demonstrates that these achievements are independent: We read off $\\Pi^0_1$-ordinals from a Sch\\\"utte-style ordinal analysis via infinite proofs, in a direct and transparent way."
],
[
"Iterated Consistency",
"On an intuitive level, we define iterations of consistency over primitive recursive arithmetic by the recursion $\\mathbf {T}_\\alpha =\\mathbf {PRA}+\\forall _{\\gamma <\\alpha }\\operatorname{Con}(\\mathbf {T}_\\gamma ).$ We point out that this deviates from the definition used by Beklemishev, who takes $\\mathbf {T}^{\\prime }_\\alpha =\\mathbf {PRA}+\\lbrace \\operatorname{Con}(\\mathbf {T}^{\\prime }_\\gamma )\\,|\\,\\gamma <\\alpha \\rbrace .$ The difference is just an index shift at limit stages: As $\\mathbf {T}^{\\prime }_{\\alpha +1}$ entails $\\operatorname{Con}(\\mathbf {T}^{\\prime }_\\alpha )$ it proves that all $\\Pi _1$ -consequences of $\\mathbf {T}^{\\prime }_\\alpha $ are true, which gives $\\mathbf {T}^{\\prime }_{\\alpha +1}\\vdash \\forall _{\\gamma <\\alpha }\\operatorname{Con}(\\mathbf {T}^{\\prime }_\\gamma )$ .",
"The stronger limit step will make for more appealing bounds in our context.",
"More formally, the definition of the theories $\\mathbf {T}_\\alpha $ depends on an arithmetization of the relevant ordinals.",
"In this note we work with the usual notation system for $\\varepsilon _0$ .",
"Following [1], it will be most convenient to formalize the relation “$\\varphi $ is provable in $\\mathbf {T}_\\alpha $ ”: Using an arithmetization $\\operatorname{Pr_{\\mathbf {PRA}}}(\\cdot )$ of provability in $\\mathbf {PRA}$ the fixed point lemma yields a formula $\\Box _\\alpha (\\varphi )$ with $\\mathbf {PRA}\\vdash \\Box _\\alpha (\\varphi )\\leftrightarrow \\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\lnot \\Box _{\\gamma }(0=1)\\rightarrow \\varphi ).$ This very equivalence shows that $\\Box _\\alpha (\\varphi )$ is $\\Sigma _1$ in $\\mathbf {PRA}$ .",
"Thus $\\operatorname{Con}_\\alpha (\\mathbf {PRA}):\\equiv \\lnot \\Box _\\alpha (0=1)$ is a $\\Pi _1$ -formula.",
"Negating both sides of (REF ) we see $\\mathbf {PRA}\\vdash \\operatorname{Con}_\\alpha (\\mathbf {PRA})\\leftrightarrow \\operatorname{Con}(\\mathbf {PRA}+\\forall _{\\gamma <\\dot{\\alpha }}\\operatorname{Con}_\\gamma (\\mathbf {PRA})).$ This suggests that $\\mathbf {T}_\\alpha :=\\mathbf {PRA}+\\forall _{\\gamma <\\alpha }\\operatorname{Con}_\\gamma (\\mathbf {PRA})$ captures the above intuition.",
"Further justification comes from the result that equivalences such as (REF ) determine the relation $\\Box _\\alpha (\\varphi )$ completely (see [1]).",
"It may be astonishing that this holds over $\\mathbf {PRA}$ , where we have limited access to transfinite induction.",
"Instead, the result relies on a principle of “reflexive induction”, due to Schmerl and Girard (see [4]).",
"Beklemishev states a slightly strengthened version, which will be important later: Lemma 1 For any formula $\\varphi (\\alpha )$ in the language of $\\mathbf {PRA}$ , if we have $\\mathbf {PRA}\\vdash \\forall _\\alpha (\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\varphi (\\gamma ))\\rightarrow \\varphi (\\alpha ))$ — we say that $\\varphi $ is reflexively progressive — then we can conclude $\\mathbf {PRA}\\vdash \\forall _\\alpha \\varphi (\\alpha ).$ We cite the short argument from [1]: To conclude by Löb's theorem it suffices to establish $\\mathbf {PRA}\\vdash \\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _\\alpha \\varphi (\\alpha ))\\rightarrow \\forall _\\alpha \\varphi (\\alpha ).$ Working in $\\mathbf {PRA}$ , assume $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _\\alpha \\varphi (\\alpha ))$ .",
"In particular $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\varphi (\\gamma ))$ holds for any $\\alpha $ .",
"By assumption this implies $\\varphi (\\alpha )$ .",
"Since $\\alpha $ was arbitrary we have $\\forall _\\alpha \\varphi (\\alpha )$ , as required.",
"We will also need the following easy observation: Lemma 2 If the formula $\\varphi (x_1,\\dots ,x_n)$ is $\\Pi _1$ in $\\mathbf {PRA}$ then we have $\\mathbf {PRA}\\vdash \\forall _\\alpha (\\operatorname{Con}_\\alpha (\\mathbf {PRA})\\wedge \\Box _\\alpha (\\varphi (\\dot{x}_1,\\dots ,\\dot{x}_n))\\rightarrow \\varphi (x_1,\\dots ,x_n)).$ Arguing in $\\mathbf {PRA}$ , consider an arbitrary ordinal $\\alpha $ .",
"To establish the contrapositive of the claim, assume that we have $\\lnot \\varphi (x_1,\\dots ,x_n)$ .",
"By $\\Sigma _1$ -completeness we get $\\operatorname{Pr_{\\mathbf {PRA}}}(\\varphi (\\dot{x}_1,\\dots ,\\dot{x}_n)\\rightarrow 0=1)$ .",
"Also assume $\\Box _\\alpha (\\varphi (\\dot{x}_1,\\dots ,\\dot{x}_n))$ , and observe that this means $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\lnot \\Box _\\gamma (0=1)\\rightarrow \\varphi (\\dot{x}_1,\\dots ,\\dot{x}_n))$ .",
"Combining the two implications we obtain $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\lnot \\Box _\\gamma (0=1)\\rightarrow 0=1)$ .",
"This amounts to $\\Box _\\alpha (0=1)$ , which is equivalent to $\\lnot \\operatorname{Con}_\\alpha (\\mathbf {PRA})$ , completing the proof of the contrapositive.",
"Let us briefly review the ordinal analysis of Peano arithmetic via infinite proofs, and its formalization in $\\mathbf {PRA}$ .",
"We will follow the approach of Buchholz [2], where the reader can find all missing details.",
"The infinitary proof system is based on (Tait-style) sequent calculus: Rather than single formulas one derives finite sets (“sequents”), which are to be read disjunctively.",
"Thus, the sequent $\\Gamma \\equiv \\lbrace \\varphi _1,\\dots ,\\varphi _n\\rbrace $ is interpreted as the disjunction $\\bigvee \\Gamma \\equiv \\varphi _1\\vee \\dots \\vee \\varphi _n$ .",
"As usual we write $\\Gamma ,\\varphi $ rather than $\\Gamma \\cup \\lbrace \\varphi \\rbrace $ .",
"The characteristic feature of our infinite calculus is the $\\omega $ -rule ${\\leavevmode \\hbox{{}{$\\Gamma ,\\varphi (0)$}{$\\Gamma ,\\varphi (1)$}{$\\cdots $}{,}{$\\Gamma ,\\forall _n\\varphi (n)$}}}$ with a premise for each numeral.",
"In other words, if we know that the disjunction $\\bigvee \\Gamma \\vee \\varphi (n)$ holds for each number $n$ then we can infer $\\bigvee \\Gamma \\vee \\forall _n\\varphi (n)$ .",
"Similarly, there are rules to introduce propositional connectives and existential quantifiers, in these cases with finitely many premises.",
"As axioms we take true prime formulas.",
"To derive a formula we combine these rules into a proof tree.",
"In the presence of the infinitary $\\omega $ -rule these trees will generally have infinite height.",
"The reader may observe that any true arithmetical formula can be derived with finite height — but this is not much use if we work in $\\mathbf {PRA}$ or another weak theory, which recognizes few formulas as true.",
"What $\\mathbf {PRA}$ does know (see below) is that proofs in Peano arithmetic can be translated into infinite proofs.",
"The resulting proof trees have height below $\\omega \\cdot 2$ .",
"At the same time, this embedding introduces cut rules ${\\leavevmode \\hbox{{}{$\\Gamma ,\\varphi $}{$\\Gamma ,\\lnot \\varphi $}{.",
"}{$\\Gamma $}}}$ Note that the cut formula $\\varphi $ may be very complex, even if we know that the end-sequent $\\Gamma $ is very simple.",
"This can be problematic when we check properties by induction over proofs.",
"The famous method of cut elimination removes that obstacle: It allows us to reduce the complexity of cut formulas, increasing the height of the proof by a power of $\\omega $ in each step.",
"In the end we obtain a cut-free proof with height below $\\varepsilon _0$ .",
"From this proof we can read off the desired bounds.",
"There are several ways to formalize infinite proofs in $\\mathbf {PRA}$ .",
"One option would be to work with primitive recursive proof trees, represented by numerical codes.",
"We choose the particularly elegant approach of Buchholz [2]: The idea is to name proofs by the role they play in the cut elimination process.",
"For each (finite) proof $d$ in Peano arithmetic one introduces a constant symbol $[d]$ , which names the embedding of $d$ into the infinite system.",
"Furthermore, one adds a unary function symbol $E$ : If $h$ is a name for an infinite proof tree then the term $E h$ names the result of cut elimination (with cut rank reduced by one).",
"Auxiliary function symbols $I_{k,\\varphi }$ and $R_\\psi $ refer to inversion and reduction, the usual ingredients of cut elimination.",
"The resulting set of terms is denoted by $\\mathbf {Z^*}$ .",
"Crucially, there is a primitive recursive function $\\mathfrak {s}:\\mathbb {N}\\times \\mathbf {Z^*}\\rightarrow \\mathbf {Z^*}$ which computes codes for the immediate subtrees of a proof tree.",
"For example, if the last rule of $h$ (also read off by a primitive recursive function) is not a cut then we have $\\mathfrak {s}_n(E h)=E\\mathfrak {s}_n(h)$ .",
"Intuitively this means that we apply cut elimination to the $n$ -th subtree of $h$ in order to get the $n$ -th subtree of $Eh$ .",
"Officially, the equation $\\mathfrak {s}_n(E h)=E\\mathfrak {s}_n(h)$ is part of the definition of $\\mathfrak {s}$ , which goes by recursion over the terms in $\\mathbf {Z^*}$ .",
"Similarly, we have functions to read off the end sequent, the cut rank, and the ordinal height of (the proof represented by) a given term.",
"We should mention that it is crucial to extend the infinitary system by a “repetition rule”, which simply repeats its premise: It permits us to “call” a proof even if we cannot immediately determine its last rule.",
"The point is that Buchholz' approach allows a very smooth formalization in $\\mathbf {PRA}$ : We simply work with a system of terms, without official reference to their interpretation as infinite trees.",
"In $\\mathbf {PRA}$ we can show that the proof terms are “locally correct”.",
"Amongst other things, this means that the ordinal height of $\\mathfrak {s}_n(h)$ is smaller than the ordinal height of $h$ .",
"Having explained this approach we can revert to more traditional notation: By $h\\vdash ^\\alpha _0\\Gamma $ we express that $h\\in \\mathbf {Z^*}$ codes a cut-free proof of (a sub-sequent of) $\\Gamma $ with ordinal height $\\alpha $ .",
"We write $\\vdash ^\\alpha _0\\Gamma $ to indicate that this holds for some term $h\\in \\mathbf {Z^*}$ .",
"When we try to track usual (i.e.",
"finite) provability through an infinite proof, the $\\omega $ -rule poses an obstruction: From $\\mathbf {PRA}\\vdash \\varphi (n)$ for all $n\\in \\mathbb {N}$ we cannot infer $\\mathbf {PRA}\\vdash \\forall _n\\varphi (n)$ .",
"On the other hand, $\\mathbf {PRA}\\vdash \\forall _n\\operatorname{Pr_{\\mathbf {PRA}}}(\\varphi (\\dot{n}))$ does imply $\\mathbf {PRA}+\\operatorname{Con}(\\mathbf {PRA})\\vdash \\forall _n\\varphi (n)$ , provided that $\\varphi $ is a $\\Pi _1$ -formula.",
"This explains why iterated consistency is needed.",
"Iterations of consistency are important for a second reason as well: They will make it possible to use the reflexive induction principle.",
"These ideas lead to the main result of the present note: Theorem 3 Provably in $\\mathbf {PRA}$ , we have $\\vdash ^\\alpha _0\\Gamma \\quad \\Rightarrow \\quad \\Box _\\alpha (\\bigvee \\Gamma )$ for any sequent $\\Gamma $ that consists of $\\Pi _1$ -formulas only.",
"We argue by reflexive induction on $\\alpha $ , as established in Lemma REF .",
"Working in $\\mathbf {PRA}$ , we may thus assume $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\forall _{h\\in \\mathbf {Z^*}}(h\\vdash ^\\gamma _0\\Gamma \\wedge \\text{``$\\Gamma $ consists of $\\Pi _1$-formulas''}\\rightarrow \\Box _\\gamma (\\bigvee \\Gamma )))$ in order to deduce the claim for $\\alpha $ .",
"The antecendent of that claim provides $h\\vdash ^\\alpha _0\\Gamma $ for some $h\\in \\mathbf {Z^*}$ , where $\\Gamma $ is a sequent of $\\Pi _1$ -formulas.",
"We must deduce $\\Box _\\alpha (\\bigvee \\Gamma )$ .",
"Let us assume that $h$ ends in an $\\omega $ -rule that introduces the formula $\\forall _n\\varphi (n)\\in \\Gamma $ — in all other cases the argument is similar and easier.",
"By $\\Sigma _1$ -completeness we can recover the assumptions inside $\\mathbf {PRA}$ , i.e.",
"we get $\\operatorname{Pr_{\\mathbf {PRA}}}(\\dot{h}\\vdash ^{\\dot{\\alpha }}_0\\dot{\\Gamma }\\wedge \\text{``$\\dot{h}$ ends in an $\\omega $-rule that introduces $\\forall _n\\varphi (n)$''}).$ Recall that $\\mathfrak {s}_n(h)\\in \\mathbf {Z^*}$ denotes the $n$ -th immediate subtree of the infinite proof denoted by $h$ .",
"By the local correctness of $\\mathbf {Z^*}$ , this subtree deduces the premise $\\Gamma ,\\varphi (n)$ of the $\\omega $ -rule, with ordinal height $\\alpha _n<\\alpha $ .",
"Applying this argument in the meta theory $\\mathbf {PRA}$ would only give the subtrees $\\mathfrak {s}_n(h)$ for “standard” numbers $n$ .",
"This is not enough for our purpose, but the same argument in the object theory does yield $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _n\\exists _{\\alpha _n<\\dot{\\alpha }}\\,\\mathfrak {s}_n(\\dot{h})\\vdash ^{\\alpha _n}_0\\dot{\\Gamma },\\dot{\\varphi }(n)).$ Note that $\\varphi (n)$ is a $\\Pi _1$ -formula (indeed a $\\Delta _0$ -formula), since $\\forall _n\\varphi (n)$ occurs in $\\Gamma $ .",
"Thus the reflexive induction hypothesis implies $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _n\\exists _{\\alpha _n<\\dot{\\alpha }}\\,\\Box _{\\alpha _n}(\\bigvee \\Gamma \\vee \\varphi (\\dot{n}))).$ Formalizing Lemma REF in $\\mathbf {PRA}$ we get $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\operatorname{Con}_\\gamma (\\mathbf {PRA})\\rightarrow \\forall _n(\\bigvee \\Gamma \\vee \\varphi (n))).$ In view of $\\forall _n\\varphi (n)\\in \\Gamma $ one obtains $\\operatorname{Pr_{\\mathbf {PRA}}}(\\forall _{\\gamma <\\dot{\\alpha }}\\operatorname{Con}_\\gamma (\\mathbf {PRA})\\rightarrow \\bigvee \\Gamma ).$ By equivalence (REF ) this amounts to $\\Box _\\alpha (\\bigvee \\Gamma )$ , as required.",
"The result cited in the introduction is an easy consequence: Corollary 4 (Schmerl) We have $\\mathbf {PA}\\equiv _{\\Pi _1}\\mathbf {PRA}+\\lbrace \\operatorname{Con}_\\alpha (\\mathbf {PRA})\\,|\\,\\alpha <\\varepsilon _0\\rbrace .$ First, assume that $\\varphi $ is a $\\Pi _1$ -theorem of Peano arithmetic.",
"By embedding and cut elimination (see [2]) we get $\\vdash ^\\alpha _0 \\varphi $ for some ordinal $\\alpha <\\varepsilon _0$ , provably in $\\mathbf {PRA}$ .",
"In this situation, the previous theorem yields $\\mathbf {PRA}\\vdash \\Box _\\alpha (\\varphi )$ .",
"Using Lemma REF we can infer $\\mathbf {PRA}+\\operatorname{Con}_\\alpha (\\mathbf {PRA})\\vdash \\varphi $, as required.",
"In fact, equivalence (REF ) and the soundness of $\\mathbf {PRA}$ give the sharper bound $\\mathbf {PRA}+\\forall _{\\gamma <\\alpha }\\operatorname{Con}_\\gamma (\\mathbf {PRA})\\vdash \\varphi ,$ as promised in the introduction.",
"Conversely, Peano arithmetic proves $\\operatorname{Con}_\\alpha (\\mathbf {PA})$ by transfinite induction up to any fixed $\\alpha <\\varepsilon _0$ : Assume $\\forall _{\\beta <\\gamma }\\operatorname{Con}_\\beta (\\mathbf {PRA})$ by induction hypothesis.",
"The contrapositive of $\\Sigma _1$ -reflection over $\\mathbf {PRA}$ , which is provable in $\\mathbf {PA}$ , yields $\\operatorname{Con}(\\mathbf {PRA}+\\forall _{\\beta <\\dot{\\gamma }}\\operatorname{Con}_\\beta (\\mathbf {PRA}))$ , or equivalently $\\operatorname{Con}_\\gamma (\\mathbf {PRA})$ ."
]
] | 1709.01540 |
[
[
"On non-supersymmetric conformal manifolds: field theory and holography"
],
[
"Abstract We discuss the constraints that a conformal field theory should enjoy to admit exactly marginal deformations, i.e.",
"to be part of a conformal manifold.",
"In particular, using tools from conformal perturbation theory, we derive a sum rule from which one can extract restrictions on the spectrum of low spin operators and on the behavior of OPE coefficients involving nearly marginal operators.",
"We then consider conformal field theories admitting a gravity dual description, and as such a large-$N$ expansion.",
"We discuss the relation between conformal perturbation theory and loop expansion in the bulk, and show how such connection could help in the search for conformal manifolds beyond the planar limit.",
"Our results do not rely on supersymmetry, and therefore apply also outside the realm of superconformal field theories."
],
[
"Introduction",
"It has been known since many years that there exist families of superconformal field theories (SCFTs) connected by exactly marginal deformations [1] (see, e.g., [2], [3], [4], [5] for generalizations).",
"The corresponding exactly marginal couplings parametrize what is known as the conformal manifold.",
"An obvious question is whether conformal manifolds can exist even in absence of supersymmetry.",
"Unless there exist some other underlying extended symmetries, general arguments suggest this to be hardly possible.",
"Upon deforming a conformal field theory (CFT) as $S_{CFT} \\rightarrow S_{CFT} + g \\int d^dx \\,{\\cal O}~,$ where ${\\cal O}$ is a scalar primary of the CFT with scaling dimension $\\Delta _{\\cal O}=d$ , a $\\beta $ function for the coupling $g$ is induced, at the quantum level.",
"The existence of a conformal manifold requires $\\beta (g)=0$ and it is hard to believe this to be possible without supersymmetry, which, in some circumstances [1], can in fact protect ${\\cal O}$ from acquiring an anomalous dimension.",
"Moreover, the deformation triggered by the coupling $g$ could also generate new couplings at the quantum level, and the corresponding $\\beta $ functions should also be set to zero, if we were to preserve conformal invariance.",
"Therefore, constraints look rather tight.",
"One could wonder whether there exist some consistency constraints that forbid a non-supersymmetric conformal manifold to exist, to start with.",
"While we are not aware of any no-go theorem, the following simple argument shows that non-supersymmetric conformal manifolds can be consistent at least with unitarity and crossing symmetry.",
"As we will review later, the requirement of vanishing $\\beta $ functions imposes stringent constraints on the CFT data, but only regarding operators with integer spins.",
"One can then take any of the known SCFTs belonging to a conformal manifold and truncate the spectrum of operators, excluding all operators with half-integer spins, while leaving CFT data of integer-spin operators unmodified.",
"This is consistent, because half-integer spin operators cannot appear in the OPE of two integer spin operators.",
"The operator algebra one ends up with is crossing-symmetric because initially it was, and also the truncated Hilbert space does not contain any negative-norm states, because the original one did not, consistently with unitarity.",
"CFT data still obey the $\\beta $ -function constraints, because the original theory had a conformal manifold by assumption.",
"And, finally, the resulting operator algebra does not form a representation of the supersymmetry algebra, because it contains only integer spin operators.",
"This might suggest it to be simple, eventually, to construct non-supersymmetric CFTs living on a conformal manifold.",
"In fact, unitarity and crossing symmetry are necessary but not sufficient conditions to get a consistent theory.We thank Alexander Zhiboedov for a discussion on this point.",
"For instance, there are further conditions coming from modular invariance in two dimensions or, more generally, by requiring the consistency of the CFT at finite temperature in any number of dimensions, see, e.g., [6].",
"This is why the truncation described above does not allow for getting non-supersymmetric conformal manifolds for free.",
"The truncated operator algebra might not form a consistent CFT, eventually.",
"It will be interesting to investigate this issue further.",
"In this work, we will just assume that non-supersymmetric conformal manifolds can exist, and elaborate upon the corresponding constraints.",
"The very possibility for a conformal manifold to exist requires the presence of one (or more) marginal scalar operator in the undeformed CFT, an operator ${\\cal O}$ with scaling dimension $\\Delta _{\\cal O} = d$ .",
"This implies that $\\beta (g)$ vanishes, at tree level in $g$ .",
"We want to investigate which further conditions the requirement of vanishing $\\beta $ function at the quantum level imposes on the CFT.",
"To put things the simplest, we will focus on one-dimensional conformal manifolds, described by deformations like (REF ).",
"In section 2, using conformal perturbation theory, we start by reviewing the conditions that the vanishing of the $\\beta $ function up to two-loop order imposes on the OPE of the operator ${\\cal O}$ .",
"Then, using also recent numerical bootstrap results, we show what other information on the spectrum of low dimension operators other than ${\\cal O}$ , can be extracted.",
"This includes, in particular, the dependence on scaling dimension of OPE coefficients involving nearly marginal operators $\\Delta \\sim d$ , as well as a prediction on the content of low spin operators in the spectrum of the CFT.",
"In section 3, we focus on CFTs admitting a gravity dual description.",
"First, we discuss the relation between conformal perturbation theory and the $1/N$ expansion, and the role that Witten diagrams play in this matter.",
"Then, focusing on a toy-model, we investigate under which conditions a conformal manifold existing at leading order in $1/N$ , can survive at non-planar level, and show that, even in absence of supersymmetry, this is a non-empty set.",
"On the way, we also provide a nice AdS/CFT consistency check regarding non-supersymmetric AdS (in)stability and CFTs RG flows.",
"Section 4, which is our last section, contains a discussion on models with richer dynamics, and an outlook on what one can do next using our results."
],
[
"Constraints from conformal perturbation theory",
"Given a CFT and a deformation as that in eq.",
"(REF ), one expects that a $\\beta $ function for the coupling $g$ is generated and that conformal invariance is lost.",
"The $\\beta $ function reads $\\beta (g) = \\beta _1\\, g^2 + \\beta _2 \\, g^3 + \\dots ~.$ Loop coefficients are expected to depend on the data of undeformed CFT.",
"In order to find such dependence a perturbative analysis can be conveniently done in the context of conformal perturbation theory (CPT) [7].",
"One can extract the $\\beta $ function by considering cleverly chosen physical observables and demand them to be UV-cutoff independent.",
"Following [8] (see also [9]), we consider the overlap $\\langle {\\cal O}(\\infty )|0\\rangle _{g,V}$ where ${\\cal O}(\\infty ) = \\lim _{x\\rightarrow \\infty } x^{2d} {\\cal O}(x)$ , while $|0\\rangle _{g,V} = e^{ g \\int _V d^dx \\,{\\cal O}(x)} |0\\rangle $ is the state obtained by deforming the theory by (REF ) in a finite region around the origin.",
"The choice of a finite volume $V$ allows one to get rid of IR divergences, while not affecting the UV behavior we are interested in.",
"Expanding (REF ) in $g$ one gets a perturbative expansion in terms of integrals of $n$ -point functions of ${\\cal O}$ .",
"These are generically plagued by logarithmic divergences, which can be absorbed by demanding that the coupling $g$ runs with scale $\\mu $ in a way that the final result is $\\mu $ -independent.",
"This, in turns, lets one extract the $\\beta $ function.",
"Proceeding this way one gets for the $\\beta $ function at two loops (which to this order is universal, hence independent of the renormalization scheme) the following expressions $\\beta _1 &=& - \\frac{1}{2} \\, S_{d-1} C_{{\\cal O}{\\cal O}{\\cal O}} \\\\\\beta _2 &=&-\\frac{1}{6} \\, S_{d-1} \\int d^dx \\Big [ \\langle {\\cal O}(0) {\\cal O}(x) {\\cal O}(e) {\\cal O}(\\infty )\\rangle _c- \\sum _{\\Phi }\\frac{1}{2} C_{{\\cal O O} \\Phi }^2\\left( \\frac{1}{x^d (x -e)^d} + \\frac{1}{x^d} + \\frac{1}{(x -e)^d} \\right) \\nonumber \\\\&& - \\sum _{\\Psi } C_{{\\cal O O} \\Psi }^2\\left( \\frac{1}{x^{2d-\\Delta _{\\Psi }}} + \\frac{1}{(x-e)^{2d-\\Delta _{\\Psi }}} + x^{-\\Delta _{\\Psi }} \\right)\\Big ]~,$ where $S_{d-1}$ is the volume of the ($d-1$ )-dimensional unit sphere, $e$ is a unit vector in some fixed direction and the subscript $c$ in the four-point function refers to the connected contribution.",
"Sums are over marginal operators $\\Phi $ and relevant operators $\\Psi $ appearing in the $\\cal O O$ OPE.",
"In principle, one can go to higher orders in $g$ .",
"In particular, marginality of ${\\cal O}$ at order $O(g^{n-1})$ would require the vanishing of logarithmic divergences of an integral in $d^dx_1 \\cdots d^dx_{n-3}$ of the $n$ -point function $\\langle {\\cal O} \\dots {\\cal O}\\rangle $ .",
"The deformation (REF ) does not cause the running of $g$ , only.",
"In general, any coupling $g_\\Phi $ dual to a marginal operator $\\Phi $ appearing in the OPE of ${\\cal O}(x){\\cal O}(0)$ will start running, due to quantum effects.Runnings are also induced for relevant operators appearing in the OPE.",
"However, these effects are associated to power-law divergences and can be reabsorbed by local counter-terms.",
"This is equivalent to be at a fixed point, to ${\\cal O}(g^2)$ order, of the corresponding $\\beta $ functions $\\beta (g_\\Psi )$ [7].",
"Following the same procedure described above, one gets the following contribution at order $g^2$ to $\\beta (g_)$ $\\beta (g_) \\supset - \\frac{1}{2} \\, S_{d-1} C_{{\\cal O}{\\cal O}} \\, g^2 ~.$ Therefore, at one loop in CPT, the persistence of a conformal manifold under the deformation (REF ) implies the following constraints on the OPE coefficients of the CFT $C_{{\\cal O}{\\cal O}} = 0 ~~,~~\\forall \\,~~\\mbox{such~that}~~\\Delta _= d~.$ Taking into account the above constraint, eq.",
"() simplifies and we get the following condition at two-loops, eventually $\\int d^dx \\left[\\, \\langle {\\cal O}(0) {\\cal O}(x) {\\cal O}(e) {\\cal O}(\\infty )\\rangle _c- \\sum _{\\Psi } C_{{\\cal O O} \\Psi }^2\\left( \\frac{1}{x^{2d-\\Delta _{\\Psi }}} + \\frac{1}{(x-e)^{2d-\\Delta _{\\Psi }}} + x^{-\\Delta _{\\Psi }} \\right)\\right] = 0~.$ Eqs.",
"(REF ) and (REF ) are the two constraints the existence of a conformal manifold under the deformation (REF ) imposes on the CFT at two-loop order in CPT.One can obtain similar expressions for two-loop $\\beta $ function of other marginal operators, if there are any, and get additional constraints."
],
[
"Two-loop constraint and integrated conformal blocks",
"One can try to translate the constraint (REF ) into a sum rule in terms of conformal blocks, which can provide, in turn, constraints on the CFT data.",
"Let us first rewrite (REF ) as an integral of the full four-point function, that is $\\int d^dx \\, &\\Big (\\langle {\\cal O}(0) {\\cal O}(x) {\\cal O}(e) {\\cal O}(\\infty )\\rangle - \\frac{1}{x^{2d}} - \\frac{1}{(x-e)^{2d}} - 1 -\\nonumber \\\\&\\sum _{\\Psi } C_{{\\cal O O} \\Psi }^2\\left( \\frac{1}{x^{2d-\\Delta _{\\Psi }}} + \\frac{1}{(x-e)^{2d-\\Delta _{\\Psi }}} + x^{-\\Delta _{\\Psi }} \\right)\\Big )= 0~.$ The integrand above is axial-symmetric, hence the integration can be seen as an integration over a two-plane $(z , \\bar{z})$ containing the unit vector $e$ , followed by integration over a $(d-2)$ -dimensional sphere, whose coordinates the integrand does not depend on.",
"So, for the integration measure, we get $d^dx\\rightarrow \\frac{\\pi ^{\\frac{d-1}{2}}}{2\\Gamma \\left( \\frac{d-1}{2}\\right)}d^2z\\, \\left( \\frac{z-\\bar{z}}{2i}\\right)^{d-2}~.$ Notice that the integrand together with the measure is inversion-invariant.",
"Therefore, instead of integrating over the whole $R^d$ , one can integrate over a unit disk, $B_{r=1}(0)=\\lbrace z\\in C \\,,\\, |z|\\le 1\\rbrace $ , where the coordinate $z$ is chosen such that $x=e$ corresponds to $z=1$ .",
"The integrand in eq.",
"(REF ) is expected to be a singularity-free function, but among the terms coming with a minus sign, there are some which have manifest singularities.",
"Hence, they must be compensated by the corresponding singularities of the four-point function.",
"Due to divergences both at $z=0$ and $z=1$ , one cannot use just one OPE channel.",
"However, it turns out that one can reduce the integration domain to a fundamental one [10], for which a single channel suffices.",
"The integral (REF ) is invariant under transformations generated by $z\\rightarrow 1/z$ and $z\\rightarrow 1-z$ and complex conjugation.",
"Hence, choosing one of the following domains $&&D_1=\\lbrace z\\in C|~|1-z|^2<1, ~ {\\rm Re\\,}(z)<1/2, ~ {\\rm Im\\,}(z)>0\\rbrace \\nonumber \\\\&&D_2=\\lbrace z\\in C|~|1-z|^2<1, ~ {\\rm Re\\,}(z)<1/2, ~ {\\rm Im\\,}(z)<0\\rbrace \\nonumber \\\\&&D_3=\\lbrace z\\in C|~|1-z|^2>1, ~ |z|^2<1, ~ {\\rm Im\\,}(z)>0\\rbrace \\nonumber \\\\&&D_4=\\lbrace z\\in C|~|1-z|^2>1, ~ |z|^2<1, ~ {\\rm Im\\,}(z)<0\\rbrace ~,$ one can use $s$ -channel OPE only.",
"Figure: Integration in the (z,z ¯)(z, \\bar{z}) plane.",
"The fundamental domain D 1 D_1 is the violet region.",
"The regions D 2 ,D 3 D_2, D_3 and D 4 D_4 are defined in () and are easily recognizable in the figure.For the sake of computational convenience we will not do the minimal choice, but use the union of all four domains, $D=D_1\\cup D_2\\cup D_3\\cup D_4$ .",
"Using $s$ -channel OPE, we get $\\langle \\mathcal {O}(0)\\mathcal {O}(x)\\mathcal {O}(e)\\mathcal {O}(\\infty )\\rangle =\\frac{\\sum _{\\mathcal {O}^{\\prime }}C_{\\mathcal {O}\\mathcal {O}\\mathcal {O}^{\\prime }}^2g_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}}{x^{2d}}~,$ where $g_{\\Delta _{\\mathcal {O}^{\\prime }},\\,l_{\\mathcal {O}^{\\prime }}}$ are conformal blocks corresponding to the exchange of an operator $\\mathcal {O}^{\\prime }$ with dimension $\\Delta _{\\mathcal {O}^{\\prime }}$ and spin $l_{\\mathcal {O}^{\\prime }}$ (with $l_{\\mathcal {O}^{\\prime }}$ even, as in the OPE of two identical scalars only operators with even spin appear).",
"The identity operator contribution cancels the $1/x^{2d}$ divergent contribution in eq.",
"(REF ).",
"Let us now define the following quantities $G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}&=&\\frac{\\pi ^{\\frac{d-1}{2}}}{2\\Gamma \\left( \\frac{d-1}{2}\\right)} \\int _{D}d^2z\\, \\left( \\frac{z-\\bar{z}}{2i}\\right)^{d-2}\\frac{g_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}(z,\\bar{z})}{|z|^{2d}}, ~ \\Delta >d~, \\\\G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,0}&=&\\frac{\\pi ^{\\frac{d-1}{2}}}{2\\Gamma \\left( \\frac{d-1}{2}\\right)} \\int _{D}d^2z\\, \\left( \\frac{z-\\bar{z}}{2i}\\right)^{d-2}\\Big (\\frac{g_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,0}(z,\\bar{z})}{|z|^{2d}} - \\frac{1}{|z|^{2d-\\Delta }} - \\frac{1}{|1-z|^{2d-\\Delta }} - |z|^{-\\Delta }\\Big ), ~\\Delta <d~, \\nonumber \\\\ \\\\A&=&\\frac{\\pi ^{\\frac{d-1}{2}}}{2\\Gamma \\left( \\frac{d-1}{2}\\right)}~\\int _{D}d^2z\\, \\left( \\frac{z-\\bar{z}}{2i}\\right)^{d-2}~\\left( \\frac{1}{|1-z|^{2d}}+1\\right)~,$ where $G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}$ are integrated conformal blocks (note that, for $\\Delta < d$ , that is eq.",
"(), only scalar operators are above the unitarity bound) and $A$ is a positive, dimension-dependent number, which in, e.g., $d=4$ dimensions reads $A=\\frac{\\pi }{24}\\left( 9\\sqrt{3}+16\\pi \\right)~.$ Using all above definitions, eq.",
"(REF ) can be rewritten as the following sum rule $\\sum _{\\mathcal {O}^{\\prime }} C_{\\mathcal {O}\\mathcal {O}\\mathcal {O}^{\\prime }}^2G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}=A~.$ Note that now the contribution of the identity operator is excluded from the sum.",
"Equation (REF ) is valid in $d$ dimensions, and can be evaluated using known expressions for conformal blocks.",
"Focusing, again, on $d=4$ , they read $g_{\\Delta ,\\, l}(z,\\bar{z})=\\frac{z\\bar{z}}{z-\\bar{z}}\\left( K_{\\Delta +l}(z)K_{\\Delta -l-2}(\\bar{z})-K_{\\Delta +l}(\\bar{z})K_{\\Delta -l-2}(z) \\right),$ where $K_{\\beta }$ is given in terms of hypergeometric functions, $K_{\\beta }(x)=x^{\\beta /2}{}_2F_1\\left( \\tfrac{\\beta }{2},\\tfrac{\\beta }{2},\\beta ;x\\right)$ .",
"From these, one can then compute integrated conformal blocks $G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}$ defined in eqs.",
"(REF ) and ().",
"In figure REF , integrated conformal blocks as functions of dimensions $\\Delta $ and spin $l$ are provided.",
"Relevant scalar operators have negative integrated conformal blocks and therefore give a negative contribution to the sum rule (REF ).",
"The opposite holds for irrelevant scalar operators which give instead a positive contribution.",
"All other operators display an alternating behavior: contributions are positive for $l=4,\\, 8, \\,...$ and negative for $l=2, \\,6, \\,...$ (our numerics suggests this behavior to hold for arbitrary values of $l$ ).",
"One can repeat the above analysis in spacetime dimensions other than four, and it turns out that exactly the same pattern holds.",
"Figure: l=8l=8A point worth stressing is that the sum rule (REF ) is not unique.",
"For one thing, it depends upon the choice of the integration domain $D$ .",
"More generally, this ambiguity comes from crossing symmetry.",
"Indeed, the crossing symmetry equation for a marginal operator is given by $\\sum _{\\mathcal {O}^{\\prime }}C_{\\cal O O O^{\\prime }}^2\\left(v^d ~g_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}(u, v)-u^d ~g_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}(v,u)\\right)=0~,$ where $u$ and $v$ are conformal cross-ratios which, in our case, are $u= z \\bar{z}$ and $v=(1-z)(1- \\bar{z})$ .",
"For any point $z, \\bar{z}$ this gives a sum of the same form as eq.",
"(REF ) but with a zero on the r.h.s.",
".",
"Any such sum, or linear combinations thereof, can be added to eq.",
"(REF ), modifying the coefficients in front of $C_{\\cal O O O^{\\prime }}$ 's without changing the r.h.s., hence giving, eventually, a different sum rule.",
"It would be interesting to see whether there exists a choice which makes all terms in the l.h.s.",
"of (REF ) being positive definite.",
"From such a sum rule it would be possible to get very stringent constraints on CFT data as, e.g., a lower bound on the central charge of the theory.",
"We were not able to find such linear combination for arbitrary $d$ , if it exists at all.",
"For the sake of what we will do in later sections, let us finally notice that if there are no relevant scalar operators in the ${\\cal O O}$ OPE, eq.",
"(REF ) simplifies to $\\int d^dx \\langle {\\cal O}(0) {\\cal O}(x) {\\cal O}(e) {\\cal O}(\\infty )\\rangle _c = 0~,$ and integrated conformal blocks in eq.",
"(), hence contributions as in figure REF , would not contribute to (REF ).",
"Still, this would not change the alternate sign behavior of the sum rule (REF ), since also operators with $l=2 ~\\mbox{mod}~4$ contribute with a negative sign."
],
[
"Constraints and bounds on CFT data",
"The alternating sign behavior in the sum (REF ) makes it impossible to get straight bounds on $C_{\\cal O O O^{\\prime }}$ coefficients, as one might have hoped.",
"Nevertheless, one can still extract useful information out of (REF ) , as we are going to discuss below."
],
[
"Nearly marginal operators.",
"Along a conformal manifold the dimension of a generic (that is, non-protected) operator changes continuously as a function of the couplings $g$ parametrizing the conformal manifold.",
"In particular, it can happen that an operator ${\\cal K}$ is relevant for $g < g_*$ , irrelevant for $g>g_*$ , and becomes marginal at $g=g_*$ .",
"From (REF ) it follows that, at $g=g_*$ , $C_{\\cal O O K}=0$ .",
"On the other hand, figures REF and REF show that integrated conformal blocks of scalar operators blow up when $\\Delta \\rightarrow d$ .",
"More precisely, one can see that $G_{\\Delta , 0}\\propto \\frac{1}{\\Delta -d} ~~\\mbox{when}~~ \\Delta \\rightarrow d~.$ In order to keep the two-loop beta function coefficient finite, it should be thatThis holds unless for (the very fine-tuned) situations in which there exists a second marginal operator at $g=g_*$ that changes its dimension from being irrelevant to be relevant, in such a way that the two singularities compensate each other.",
"$\\lim _{\\Delta \\rightarrow d}C^2_{\\cal O O {\\cal K}} \\, G_{\\Delta , 0} =\\;\\mbox{finite}~,$ implying that as $\\Delta \\rightarrow d$ , $C_{\\cal O O {\\cal K}}$ must approach zero at least as fast as $(\\Delta -d)^{1/2}$ .",
"This gives a prediction on how the OPE coefficient approaches zero as a function of $g - g_*$ (in fact, a lower bound on such a dependence).",
"The simplest testing ground one can think of to put this prediction at work is ${\\cal N}=4$ SYM, which admits an exactly marginal deformation associated to the gauge coupling itself.",
"Indeed, the free theory is part of the conformal manifold and one can work at arbitrary small coupling, where computations can be reliably done.",
"As an example, one can consider the $D$ -component (in ${\\cal N}=1$ language) of the Konishi multiplet, $\\mathcal {K}=\\mbox{Tr}\\,X^i X_i$ , which is marginal at $g_{YM}=0$ and becomes marginally irrelevant in the interacting theory.",
"Its anomalous dimensions is known [11], [12] and one could then give a prediction, via (REF ), on the behaviour of $C_{\\mathcal {O}\\mathcal {O}\\mathcal {K}}$ , where $\\mathcal {O}$ is the marginal operator dual to the (complexified) gauge coupling.",
"However, due to a $U(1)$ bonus symmetry enjoyed by ${\\cal N}=4$ SYM and a corresponding selection rule [13], such OPE coefficient is predicted to vanish.",
"Hence, in this specific case, the constraint (REF ) does not provide any new information.",
"In fact, ${\\cal N}=4$ SYM admits a larger conformal manifold, along which the predictions coming from eq.",
"(REF ) become relevant.",
"Using again an ${\\cal N}=1$ notation, ${\\cal N}=4$ has three chiral superfields $\\Phi _i$ that transform in the fundamental representation of the $SU(3)$ flavor symmetry.",
"These chiral superfields can be used to construct an exactly marginal $SU(3)$ invariant superpotential (the ${\\cal N}=4$ cubic superpotential) and ten classically marginal superpotential terms that transform as a 10 of $SU(3)$ .",
"Two out of the ten marginal superpotentials are exactly marginal [1].",
"Deforming the ${\\cal N}=4$ theory by these exactly marginal operators explicitly breaks the $SU(3)$ symmetry and lifts the dimension of other classically marginal operators along with the $SU(3)$ broken currents.",
"These operators acquire anomalous dimension at the quadratic order in the deformation and were explicitly obtained in [14].",
"The constraint (REF ) then predicts that the OPE coefficient $C_{\\cal O O K}$ , ($\\cal O$ being the $SU(3)$ breaking exactly marginal operator and ${\\cal K}$ any of the marginally irrelevant operators) scales at least linearly in the exactly marginal coupling.",
"Note that these statements are independent from $g_{YM}$ so they hold also at strong coupling.",
"Very similar behavior occurs for a large class of ${\\cal N}=1$ superconformal quiver gauge theories obtained by considering D-branes at Calabi-Yau singularities [2], [3], [4], [5].",
"There again, non-trivial conformal manifolds exist, along which operators which are marginal in the undeformed theory acquire an anomalous dimensions, which can be computed using similar techniques as for ${\\cal N}=4$ SYM (see [14] for details).",
"What is interesting in these models is that, unlike ${\\cal N}=4$ SYM, there is no point whatsoever on the conformal manifold in which the theory is weakly coupled.",
"So these results are intrinsically at strong coupling.",
"The fact that $A$ in eq.",
"(REF ) is a positive number implies that the $\\cal O O$ OPE must contain at least one operator with positive integrated conformal block.",
"From the results reported in figure REF it follows that at least an irrelevant scalar operator or else a spinning operator with $l=4$ mod 4 must be present.",
"In principle, this can be interesting since to date numerical bootstrap results are less powerful as far as OPE of operators of dimension $\\Delta \\gtrsim d$ are concerned.",
"When a marginal operator $\\mathcal {O}$ exists, instead, one gets constraints also about the spectrum of other such operators.",
"This can be seen as follows.",
"Let us consider a given value $\\Delta =\\Delta _*$ and divide the sum (REF ) as $\\sum _{\\mathcal {O}^{\\prime }: \\Delta <\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} ~G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}} + \\sum _{\\mathcal {O}^{\\prime }: \\Delta >\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} ~G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}} = A~.$ Since the series is expected to converge, there should exist (large enough) values of $\\Delta _*$ for which $\\sum _{\\mathcal {O}^{\\prime }: \\Delta >\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} ~G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}} < A~.$ This means that $\\sum _{\\mathcal {O}^{\\prime }: \\Delta <\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} ~G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}} > 0~,$ which implies, in turn, that among the operators with dimension $\\Delta <\\Delta _*$ , at least one operator with positive integrated conformal block should exist.",
"If $\\Delta _*$ is parametrically large this is something not very informative.",
"If $\\Delta _*$ is not too large, instead, one can get interesting constraints on the spectrum of low dimension operators.",
"One can try to give an estimate of the values of $\\Delta = \\Delta _*$ for which (REF ) is satisfied, e.g., using the approach of [15], [16], where the question of convergence of OPE expansion was addressed, and an estimate of the tail was given.",
"For example, for $d=4$ this takes the form $\\sum _{\\mathcal {O}^{\\prime }: \\Delta >\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} g_{\\Delta _{\\mathcal {O}^{\\prime }},l_{\\mathcal {O}^{\\prime }}}(z, \\bar{z})\\lesssim \\frac{2^{16}\\Delta _*^{16}}{\\Gamma (17)}\\left|\\frac{z}{(1+\\sqrt{1-z})^2}\\right|^{\\Delta _*}.$ One can then define $\\Sigma (\\Delta _*)\\equiv \\pi \\int _{D} d^2z ~\\left( \\frac{z-\\bar{z}}{2i}\\right)^2\\frac{2^{16}\\Delta _*^{16}}{\\Gamma (17)|z|^8}\\left|\\frac{z}{(1+\\sqrt{1-z})^2}\\right|^{\\Delta _*}~,$ which means that $\\sum _{\\mathcal {O}^{\\prime }: \\Delta >\\Delta _*} ~C^2_{\\cal O O O^{\\prime }} ~G_{\\Delta _{\\mathcal {O}^{\\prime }}, \\,l_{\\mathcal {O}^{\\prime }}}\\lesssim \\Sigma (\\Delta _*)~.$ The function $\\Sigma (\\Delta _*)$ is shown in figure REF .",
"Figure: The estimate Σ(Δ * )\\Sigma (\\Delta _*) as a function of Δ * \\Delta _*.In principle, the estimate (REF ) is valid only asymptotically, namely in the limit $\\Delta _* \\rightarrow \\infty $ .",
"Moreover, the actual value above which the error one is making can be neglected is theory-dependent.",
"Therefore, one should be careful using (REF ) for too low values of $\\Delta _*$ and/or to make generic predictions.",
"In fact, numerical bootstrap results suggest that a value of, say, ${\\cal O}(10)$ , can already be in a safe region for a large class of CFTs (see [17] for a discussion on this point).",
"Looking at (REF ), it is clear that the lower $\\Delta _*$ the more stringent the constraints on low dimension operators.",
"Requiring the l.h.s.",
"of eq.",
"(REF ) to saturate the inequality, which is the best one can do, and evaluate it using (REF ), we get that $\\Sigma (\\Delta _*)=A$ for $\\Delta _*=16.3$ .",
"This is already a large enough value for which the estimate (REF ) can be trusted, for a large class of CFTs [17].",
"Looking at figure REF we then conclude that in the OPE of an exactly marginal scalar operator there must be either an irrelevant scalar operator and/or some spin $l=4,8,12$ operators with dimensions $\\Delta \\lesssim 16$ (recall that the unitarity bound is $\\Delta = d -2 +l $ ).",
"In all above discussion we have been focusing, for definiteness, on $d=4$ dimensions, but similar conclusions can be drawn in any dimensions $d$ .",
"Let us finally note, in passing, that the same approach used here could more generally be used to constrain the spectra of a CFT whenever the two loop $\\beta $ function coefficient is known."
],
[
"Conformal manifolds and holography",
"In this section we want to focus our attention on CFTs admitting a gravity dual description.",
"These can be characterized as CFTs which admit a large-$N$ expansion and whose single-trace operators with spin greater than two have a parametrically large dimension [18].",
"More precisely, in the large-$N$ limit the CFT reduces to a subset of operators having small dimension (i.e., a dimension $\\Delta $ that does not scale with $N$ ), and whose connected $n$ -point functions are suppressed by powers of $1/N$ .",
"This implies, in particular, that for $N \\rightarrow \\infty $ the four-point function factorizes and hence the connected four-point function vanishes, like for free operators.",
"However, unlike the latter, these operators, also known as generalized free fields, do not saturate the unitarity bound (see [6] for a nice review).",
"Scalar operators are dual to scalar fields in the bulk.",
"From the mass/dimension relation, which (for scalars and in units of the AdS radius) reads $m^2 = \\Delta (\\Delta - d)~,$ it follows that in order for the dual operator ${\\cal O}$ to be marginal, one needs to consider a massless scalar in the bulk.",
"Its non-normalizable mode acts as a source for ${\\cal O}$ , and thus corresponds to a deformation in the dual field theory described by eq.",
"(REF ) (in other words, the non-normalizable mode is dual to the coupling $g$ ).",
"The conformal manifold ${\\cal M}_c$ is hence mapped into the moduli space ${\\cal M}$ of AdS vacua of the dual gravitational theory, i.e., AdS solutions of bulk equations of motion parametrized by massless, constant scalar fields [19].",
"The duality between ${\\cal M}_c$ and ${\\cal M}$ makes it manifest the difficulty to have conformal manifolds in absence of supersymmetry.",
"A non-supersymmetric CFT is dual to a non-supersymmetric gravitational theory.",
"Differently from supersymmetric moduli spaces, non-supersymmetric moduli spaces are expected to be lifted at the quantum level.",
"Quantum corrections in the bulk are weighted by powers of $1/N$ .",
"Hence, one would expect that a moduli space of AdS vacua existing at the classical level, would be lifted at finite $N$ .",
"For theories with a gravity dual description, this is the simplest argument one can use to argue that conformal manifolds without supersymmetry are something difficult to achieve.",
"In this respect, it is already interesting to find non-supersymmetric conformal manifolds persisting at first non-planar level.",
"One of our aims, in what follows, is to show that this is not an empty set.",
"We will consider the simplest model one can think of, namely a massless scalar field $\\phi $ minimally coupled to gravity.",
"This corresponds to CFTs which, as far as single-trace operators are concerned, in the large-$N$ limit reduce to a single low-dimension scalar operator ${\\cal O}$ , dual to $\\phi $ .A CFT must include the energy-momentum tensor.",
"Our toy-model could be thought of as a sector of an AdS compactification in which there is a self-interacting scalar in the approximation that gravity decouples, as in e.g.",
"[18].",
"Most of what we will do, does not depend on this approximation."
],
[
"Conformal perturbation theory and the 1/N expansion",
"Our first goal is to discuss how the two perturbative expansions we have to deal with in the CFT, that is, conformal perturbation theory, which is an expansion in $g$ , and the $1/N$ expansion, are related to one another from a holographic dual perspective.",
"Let us consider a bulk massless scalar $\\phi $ having polynomial interactions of the form $\\sum _{n} \\lambda _n \\left[ \\phi ^n\\right] ~,$ where $n \\ge 3$ and $\\left[ \\phi ^n\\right] $ stands for Lorentz invariant operators made of $n$ fields $\\phi $ 's.",
"For the time being, we do not need to specify their explicit form, which can also include derivative couplings.",
"Let us consider the one-loop coefficient $\\beta _1$ , eq.",
"(REF ).",
"In order to compute it holographically, one needs to evaluate Witten diagrams [20] with three external lines.",
"Witten diagrams are weighted with different powers of $1/N$ , corresponding to tree-level and loop contributions in the bulk.",
"As shown in figure REF , at tree level only the cubic vertex can contribute to the three-point function.",
"At higher loops, instead, also couplings with $n>3$ may contribute to $\\beta _1$ .",
"Figure: Witten diagrams contributing to C 𝒪𝒪𝒪 C_{\\cal OOO}.",
"Violet lines correspond to propagation of φ\\phi fields and may have spacetime derivatives acting on them, depending on the specific structure of the operators ().",
"At tree-level, only cubic couplings can contribute to the three-point function.",
"At loop level, also couplings with n>3n>3 can contribute, e.g., the quintic coupling shown in the figure.A similar story holds for the two-loop coefficient $\\beta _2$ (note that in our one-field model eq.",
"(REF ) simplifies just to the integral of the four-point function, eq.",
"(REF )).",
"To leading order, there are two contributions.",
"The contact quartic interaction and the cubic scalar exchange, as shown in figure REF .",
"Again, at higher-loops in the bulk coupling, one can get contributions also from operators with $n >4$ .",
"The analysis applies unchanged to the three-loop coefficient $\\beta _3$ and higher.",
"In particular, only operators $\\left[ \\phi ^n\\right] $ with $n \\le m$ can contribute to the $m$ -point function of ${\\cal O}$ at tree level.",
"Conversely, at loop level, also operators with $n>m$ may contribute.",
"Figure: Structure of Witten diagrams contributing to the two-loop coefficient of β(g)\\beta (g), after integration in d d xd^dx.",
"Conventions are as in figure .What we would like to emphasize with this discussion is that by doing tree-level computations in the bulk, one can extract the leading, planar contribution to $\\beta (g)$ at all loops in $g$ .",
"In other words, classical gravity provides an exact answer, in conformal perturbation theory, to the existence of a conformal manifold, at leading order $1/N$ .",
"To get this, rather than computing Witten diagrams, it is clearly much simpler to solve bulk equations of motion and see which constraints on the structure of the operators (REF ) does the existence of AdS solutions with constant $\\phi $ impose.",
"This is what we will do, first.",
"Then, we will compute explicitly tree-level Witten diagrams contributing to $\\beta _1$ and $\\beta _2$ , and check that the constraints one gets by requiring them to vanish, are in agreement with those coming from equations of motion analysis.",
"A non-trivial question one can ask is whether the vanishing of $\\beta (g)$ at two-loops leaves some freedom in the scalar couplings compared to the equation of motion analysis.",
"And, if this is the case, at which loop order in CPT one should go, to fix such freedom.",
"The answer turns out to be rather simple: admissible operators of the form $[\\phi ^n]$ will be fully determined by imposing the vanishing of the $\\beta $ -function coefficient $\\beta _{n-2}$ , and no higher orders will be needed.",
"The toy model we are going to discuss has operators with $n=3,4$ only, and, consistently, we will see that the constraints coming just from the vanishing of $\\beta _1$ and $\\beta _2$ , will provide the full gravity answer.",
"Another interesting question is which further constraints the vanishing of the one and two-loop coefficients of $\\beta (g)$ put on the CFT taking into account $1/N$ corrections, that is, going beyond planar level.",
"As already emphasized, one does not expect exact conformal manifolds to survive at finite $N$ , without supersymmetry.",
"However, one can ask whether non-trivial CFTs with non-supersymmetric conformal manifolds persisting at first non-planar level could exist.",
"That this can be, it is not obvious, and this is what we will address next."
],
[
"Scalar fields in AdS",
"We want to compare the holographic analysis with CPT at two-loops, which, as such, involves at most four-point functions, eqs.",
"(REF ) and (REF ).",
"Therefore, for simplicity, we will focus on models with cubic and quartic couplings, only.",
"The bulk action reads $S = \\frac{1}{2 \\kappa ^2_{d+1}} \\int d^{d+1}x \\sqrt{-g} \\left(R - \\frac{1}{2} g^{\\mu \\nu } \\partial _\\mu \\phi \\,\\partial _\\nu \\phi - 2 \\Lambda + \\left[ \\phi ^3\\right] + \\left[ \\phi ^4\\right] \\right)~,$ where $\\Lambda $ is the (negative) cosmological constant and the last two terms represent cubic and quartic interactions.",
"The absence of a mass term for $\\phi $ guarantees that the dual operator ${\\cal O}$ is marginal, i.e.",
"$\\Delta _{\\mathcal {O}}=d$ .",
"We would like to constrain the explicit form of cubic and quartic couplings by requiring the existence of a conformal manifold under a deformation parametrized by $\\phi $ itself.",
"We take $\\kappa _{d+1}\\sim N^{-1}$ to match holographic correlators with CFT correlation functions in the large-$N$ limit.",
"In the above normalization, the two point function $\\langle {\\cal O}{\\cal O}\\rangle $ scales as $N^2$ .",
"Such unusual normalization has the advantage to treat democratically all Witten diagrams (as well as the dual $n$ -point functions, and so the $\\beta $ -function coefficients $\\beta _n$ ), in the sense that, regardless the number of external legs, they all scale the same with $N$ , at any fixed order in the bulk loop expansion.The interested reader can explicitly check this statement, after having properly chosen the normalization of the bulk-to-boundary propagator.",
"This is the most natural choice that avoids mixing-up the expansion in $1/N$ with that in $g$ .",
"From the action (REF ) one can derive the equations of motion, which read $R_{\\mu \\nu } - \\frac{1}{2} g_{\\mu \\nu } R &=& \\frac{1}{2} \\partial _\\mu \\phi \\partial _\\nu \\phi - \\frac{1}{2} g_{\\mu \\nu } \\,( \\frac{1}{2} \\partial _\\rho \\phi \\,\\partial ^\\rho \\phi + 2 \\Lambda ) -\\frac{1}{\\sqrt{-g}} \\frac{\\delta }{\\delta g_{\\mu \\nu }} \\sqrt{-g} \\left( \\left[ \\phi ^3\\right] + \\left[ \\phi ^4\\right] \\right)\\ \\nonumber \\\\\\\\\\Box _g \\, \\phi &=& - \\frac{\\delta }{\\delta \\phi } \\left(\\left[ \\phi ^3\\right] + \\left[ \\phi ^4\\right] \\right)~,$ where $\\Box _g = g^{\\mu \\nu } \\nabla _\\mu \\nabla _\\nu = g^{\\mu \\nu } (\\partial _\\mu \\partial _\\nu - \\Gamma ^\\rho _{\\mu \\nu } \\partial _\\rho )$ .",
"We need to look for pure AdS solutions with constant scalar profile.",
"In absence of interactions, that is in the strict generalized free-field limit, the large-$N$ CFT reduces to a massless free scalar $\\phi $ propagating in a rigid AdS background.",
"The equations of motion admit a solution with AdS metric and constant scalar field $\\phi = \\phi _0$ which, in Poincaré coordinates, reads $ds^2 &=& \\frac{L^2}{z^2} \\left(dz^2 + dx_i dx^i \\right) \\\\\\phi &=& \\phi _0$ with $L=\\sqrt{d (1-d) \\Lambda }$ being the AdS radius and the AdS boundary sitting at $z=0$ .",
"The modulus $\\phi _0$ parametrizes the dual conformal manifold, described by the deformation $g \\int d^dx \\,{\\cal O}$ .",
"Eqs.",
"(REF ) and (REF ) are trivially satisfied: since $\\phi $ is a free field, Witten diagrams vanish identically (in particular, in eq.",
"(REF ) the integrand itself vanishes).",
"Let us now consider possible cubic and quartic interactions.",
"From eqs.",
"(REF )-() it follows that couplings compatible with solutions with AdS metric and a constant scalar profile are couplings where spacetime derivatives appear (note that, due to Lorentz invariance, only even numbers of derivatives are allowed).",
"Schematically, acceptable operators look like $\\nabla \\nabla \\dots \\phi \\; \\nabla \\nabla \\dots \\phi \\; \\nabla \\nabla \\dots \\phi \\nabla \\nabla \\dots \\phi \\dots ~,$ where full contraction on Lorentz indexes is understood and some (but not all) naked $\\phi $ 's, that is $\\phi $ 's without derivatives acting on them, can appear.",
"Therefore, at the classical level, i.e.",
"to leading order in $1/N$ , the requirement of existence of a conformal manifold under the deformation (REF ) rules out the non-derivative couplings $\\phi ^3$ and $\\phi ^4$ , only.One can consider the more general structure (REF ) and the same conclusion holds.",
"Any coupling $[\\phi ^n]$ with (an even number of) derivatives is allowed, classically.",
"As anticipated, we want to compare the above analysis with a direct computation of three and (integrated) four-point functions, which are related to the one and two-loop coefficients of $\\beta (g)$ via eqs.",
"(REF )-(), by means of tree-level Witten diagrams.",
"This could be seen as a simple AdS/CFT self-consistency check, but one can in fact learn from it some interesting lessons, which could be useful when considering more involved models, as well as when taking into account loop corrections in the bulk."
],
[
"Tree-level Witten diagrams",
"Let us consider the one-loop coefficient $\\beta _1$ , which is proportional to $C_{\\cal OOO}$ .",
"To leading order at large $N$ , this corresponds to the Witten diagram shown in figure REF , to which only cubic couplings $[\\phi ^3]$ can contribute.",
"Figure: Witten diagram contributing to β 1 \\beta _1 at leading order in 1/N1/N.The pure non-derivative coupling $\\phi ^3$ provides a non-vanishing contribution to $C_{\\cal OOO}$ .",
"Therefore, it is excluded.",
"The first non-trivial couplings are then two-derivative interactions.",
"In principle, the following interaction terms are allowed $\\phi \\nabla _\\mu \\phi \\nabla ^\\mu \\phi ~~,~\\phi ^2 \\, \\nabla _\\mu \\nabla ^\\mu \\phi ~.$ Upon using integration by parts and the equation of motion which, at lowest order in the couplings, is just $\\nabla _\\mu \\nabla ^\\mu \\phi = 0$ , these interactions are either total derivatives or vanish on-shell.",
"Therefore, they do not contribute to $C_{\\cal OOO}$ (this is to be contrasted with the case of a massive scalar, where these interactions are proportional to $\\phi ^3$ ).",
"Next, one can consider interactions with four spacetime derivatives, that is $\\phi \\nabla _\\mu \\nabla _\\nu \\phi \\nabla ^\\mu \\nabla ^\\nu \\phi ~~,~~\\nabla _\\mu \\phi \\nabla _\\nu \\phi \\nabla ^\\mu \\nabla ^\\nu \\phi ~~,~~\\phi ^2 \\nabla _\\mu \\nabla _\\nu \\nabla ^\\mu \\nabla ^\\nu \\phi ~.$ These terms are also either vanishing on-shell or total derivatives, and do not provide any contribution to the three-point function $\\langle \\cal OOO\\rangle $ , at leading order.",
"Let us briefly see this.",
"Using integration by parts, the second term in (REF ) can be written as $\\int ~\\nabla _\\mu \\phi \\nabla _\\nu \\phi \\nabla ^\\mu \\nabla ^\\nu \\phi = -\\frac{1}{2} \\int ~ \\nabla _\\nu \\nabla ^\\nu \\phi ~ \\nabla _\\mu \\phi \\nabla ^\\mu \\phi ~, \\\\$ which vanishes upon using the equation of motion.",
"As for the other two terms in (REF ), using the identity $[\\Box , \\nabla _\\mu ]\\phi =-d~\\nabla _\\mu \\phi $ , they can be re-written, respectively, as $\\int ~ \\phi \\nabla _\\mu \\nabla _\\nu \\phi \\nabla ^\\mu \\nabla ^\\nu \\phi &= \\int \\left( \\frac{1}{2} ~ \\nabla _\\nu \\nabla ^\\nu \\phi ~ \\nabla _\\mu \\phi \\nabla ^\\mu \\phi -\\frac{d}{2}~ \\phi ^2 \\nabla _\\nu \\nabla ^\\nu \\phi \\right)~,\\\\\\phi ^2 \\nabla _\\mu \\nabla _\\nu \\nabla ^\\mu \\nabla ^\\nu \\phi &= \\phi ^2 \\nabla _\\mu \\nabla ^\\mu \\nabla _\\nu \\nabla ^\\nu \\phi - d\\phi ^2 \\nabla _\\mu \\nabla ^\\mu \\phi ~.$ Again, both terms vanish upon using the equation of motion, and hence provide no contribution to $C_{\\cal OOO}$ .",
"One can proceed further, and consider couplings with an increasing number of derivatives, with structures that generalize (REF ).",
"Using previous results and proceeding by induction, one can prove that contributions vanish for any number of derivatives.",
"The upshot is that all operators with two or more derivatives either vanish or can be turned into total spacetime derivatives, and hence give a vanishing contribution to the Witten diagram in figure REF and, in turn, to $C_{\\cal OOO}$ .",
"Although derivative couplings provide a vanishing contribution to cubic Witten diagrams, they can provide non-vanishing contribution to the four-point function by exchange Witten diagrams like the one depicted in figure REF (which include, in the dual CFT, the exchange of double-trace operators).",
"Therefore, these interactions can potentially contribute to the two-loop coefficient $\\beta _2$ .",
"Figure: Exchange Witten diagram contributing to β 2 \\beta _2, after integration in ∫d d x\\int d^dx.The pure non-derivative coupling $\\phi ^3$ is already excluded by previous analysis (and it would also contribute to the Witten diagram in figure REF , in fact).",
"Let us then start considering contributions from operators having one field $\\phi $ not being acted by derivatives, i.e.",
"the first ones in (REF ) and (REF ) and generalizations thereof, that is operators of the form $\\phi \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi ~.$ There are two possible types of exchange Witten diagrams: (a) diagrams where all external lines are acted by derivatives, (b) diagrams where at least one external line is free of derivatives.",
"Focusing, for definiteness, on two-derivative couplings, contributions of type (a) and (b) correspond to the following integrals, respectively $\\int d^d x_1 \\int d^d w_1 dz_1 \\int d^d w_2 dz_2 &~\\nabla _\\mu K(z_1,w_1-x_1)\\nabla ^\\mu K(z_1,w_1-x_2)G(z_1-z_2, w_1- ,w_2)\\nonumber \\\\& \\nabla _\\nu K(z_2,w_2-x_3)\\nabla ^\\nu K(z_2,w_2-x_4)\\\\\\int d^d x_1 \\int d^d w_1 dz_1 \\int d^d w_2 dz_2 &~K(z_1,w_1-x_1)\\nabla ^\\mu K(z_1,w_1-x_2)\\nabla _\\mu ^{(1)}\\nabla _\\nu ^{(2)} G(z_1-z_2, w_1- ,w_2)\\nonumber \\\\& K(z_2,w_2-x_3)\\nabla ^\\nu K(z_2,w_2-x_4)~.$ $K(z,w-x_i)$ is the bulk-to-boundary propagator which, for massless scalars, reads $K(z,w-x)=\\left( \\frac{z}{z^2+(w-x)^2}\\right)^d~,$ and satisfies the equation $\\nabla _{\\mu }\\nabla ^{\\mu } K(z,w-x) = \\Box _g K(z,w-x) = 0$ .",
"$G(z_1-z_2 , w_1-w_2)$ is instead the bulk-to-bulk propagator which, for massless scalars, reads $G(z_1 - z_2,w_1-w_2)=\\frac{2^{-d}C_d}{d}\\xi ^d F\\left( \\frac{d}{2},\\frac{d}{2}+\\frac{1}{2};\\frac{d}{2}+1;\\xi ^2\\right)~,~~~~C_d=\\frac{\\Gamma (d)}{\\pi ^{d/2}\\Gamma (d/2)}~,$ where $\\xi $ is the geodesic distance between the two points in the bulk where interactions occur, $(z_1 , w_1)$ and $(z_2 , w_2)$ , $\\xi =\\frac{2 z_1z_2}{z_1^2+z_2^2+(w_1-w_2)^2}~.$ The bulk-to-bulk propagator satisfies the equation $\\Box _g G(z_1,w_1;z_2,w_2)=\\frac{1}{\\sqrt{g}}\\delta \\left( z_1-z_2,w_1-w_2\\right)$ .",
"Diagrams of type (a) vanish because the integrated bulk-to-boundary propagator $K(z,w-x)$ is independent of $z$ and $w$ , namely $\\int d^dx ~K(z,w-x)= \\frac{\\pi ^{d/2}\\Gamma (d/2)}{\\Gamma (d)}~,$ and, plugging (REF ) into (REF ), one gets $\\int d^d x_1 \\nabla _\\mu K(z_1,w_1;x_1)=0~.$ Diagrams of type (b), after $x$ -integration, also vanish.",
"Indeed, the integral () becomes $\\frac{\\pi ^{d/2}\\Gamma (d/2)}{\\Gamma (d)}\\int d^d w_1 dz_1 \\int d^d,w_2 dz_2 &~\\nabla ^\\mu K(z_1,w_1-x_2)\\nabla _\\mu ^{(1)}\\nabla _\\nu ^{(2)} G(z_1-z_2 , w_1-w_2)\\nonumber \\\\& K(z_2,w_2-x_3)\\nabla ^\\nu K(z_2,w_2-x_4)~,$ and, integrating by parts, one can transfer the covariant derivative $\\nabla _\\mu ^{(1)}$ acting on the bulk-to-bulk propagator onto $K(z_1,w_1-x_2)$ , getting $-\\frac{\\pi ^{d/2}\\Gamma (d/2)}{\\Gamma (d)}\\int d^d w_1 dz_1 \\int d^d w_2 dz_2 &~\\Box _g K(z_1,w_1-x_2)\\nabla _\\nu ^{(2)} G(z_1 - z_2 ,w_1-w_2)\\nonumber \\\\& K(z_2,w_2-x_3)\\nabla ^\\nu K(z_2,w_2-x_4)~,$ which vanishes because $\\Box _g K=0$ .",
"This computation can be repeated for terms with four or more derivatives, just replacing single derivatives acting on the propagators in (REF ) and () with multiple derivatives.",
"The end result can again be shown to be zero.",
"The second possible cubic vertexes which could contribute to the exchange Witten diagram are those with derivatives acting on one field only, schematically $\\phi ^2 \\, \\nabla \\nabla \\nabla \\dots \\phi ~.$ Using properties of Ricci and Riemann tensors in AdS, one can show that these couplings can be re-written as sums of terms of the form $\\phi ^2 \\,\\Box ^p \\phi $ , with $p$ an integer.",
"Due to the property $\\Box _g K=0$ , if derivatives are acting on at least one external line, the result is zero.",
"If not, namely if derivatives act only on the bulk-to-bulk propagator, then the corresponding diagram is a special instance of a (b)-type diagram previously discussed and, following similar steps as in eqs.",
"(REF )-(REF ), one gets again a vanishing result.",
"Finally, let us consider shift-symmetric couplings, that is couplings without naked $\\phi $ 's.",
"This kind of couplings give rise to diagrams of type (a), very much like (REF ), where all external lines (in fact any line) contain derivatives.",
"Therefore, they do not contribute to exchange Witten diagrams, either.",
"This ends our analysis of cubic operators, which fully agrees with equations of motion analysis.",
"Let us emphasize that while all cubic couplings but $\\phi ^3$ do not contribute at the level of three-point functions, they do, in general, as far as exchange Witten diagrams are concerned.",
"There, it matters that, in computing the two-loop coefficient $\\beta _2$ , integration in $d^dx$ is required, and this plays a crucial role in providing a vanishing result, in the end.",
"Let us now consider quartic couplings.",
"At tree level they do not contribute to $\\beta _1$ , but they can contribute to $\\beta _2$ , instead, via contact-terms, as the one depicted in figure REF .",
"Figure: Contact Witten diagram contributing to β 2 \\beta _2, after integration in d d xd^dx.The operators we should consider are just obtained by adding an extra field $\\phi $ to all cubic vertexes previously considered.",
"Again, the pure non-derivative coupling $\\phi ^4$ is excluded from the outset, since it clearly gives a non-vanishing contribution.",
"The other operators have the following structures $\\phi \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi ~~,~~\\phi ^2 \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi ~~,~~\\phi ^3 \\, \\nabla \\nabla \\dots \\phi ~,$ as well as the shift-symmetric one $\\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi \\, \\nabla \\nabla \\dots \\phi ~.$ Given our previous analysis it is not difficult to compute the contribution of these diagrams to the integrated four-point function and hence to the $\\beta $ function two-loop coefficient $\\beta _2$ .",
"Upon integration, the diagram in figure REF either gives zero, when the $x$ -dependence is on a line where bulk derivatives act, see eq.",
"(REF ), or, after $x$ -integration, it reduces to the effective vertex of one of the cubic vertices discussed previously, which vanish.",
"We thus see that all operators (REF ) and (REF ) do not give any contribution to $\\beta _2$ .",
"Note, again, that $x$ -integration plays a crucial role.",
"To summarize, the constraints on cubic and quartic couplings coming from CPT at two-loops, already capture the (full) gravity answer, as anticipated.",
"From the analysis in section , it is not difficult to get convinced that operators with $n$ fields $\\phi $ will be univocally fixed by computing tree-level Witten diagrams with $n$ external legs, which contribute to the $\\beta $ function at $n-2$ loop order.",
"As already emphasized, a CFT must include the energy-momentum tensor in the spectrum of primary operators, which amounts to include dynamical gravity in the bulk.",
"At tree-level, this would contribute to the exchange Witten diagram in figure REF , since now also graviton exchange should be considered in the bulk-to-bulk propagator.",
"For a minimally coupled scalar, which is the case here, the only such contribution would arise from a vertex of the following kind $h^{\\mu \\nu } \\partial _\\mu \\phi \\, \\partial _\\nu \\phi ~,$ where $h_{\\mu \\nu }$ denotes the fluctuations of the $AdS_{d+1}$ metric.",
"It is not difficult to see that, because the scalar field $\\phi $ enters under derivatives, the integrated four-point function is of (a)-type, following our previous terminology, and it vanishes, because of eqs.",
"(REF ) and (REF ).",
"So, our conclusions are unchanged also once gravity is taken into account.For non-minimally coupled scalars one could have other operators contributing to the exchange Witten diagram.",
"Couplings of the type, e.g., $ R^{\\mu \\nu } \\partial _\\mu \\phi \\, \\partial _\\nu \\phi ~$ would again be allowed since the resulting integrated four-point function would also be of the (a)-type.",
"Conversely, couplings like $R \\,\\phi ^2$ (and, more generally, any non-derivative coupling) would not be permitted because they would instead contribute to the integrated four-point function via exchange Witten diagrams.",
"Before closing this section, let us note the following interesting fact.",
"Suppose we add a quartic, non-derivative coupling $\\lambda \\phi ^4$ to the free scalar theory.",
"This lifts the flat direction associated to $\\phi $ .",
"In the dual CFT, a non-vanishing $\\beta $ function for the dual coupling $g$ is generated at two-loops, at leading order in $1/N$ (recall that a one-loop coefficient $\\beta _1$ cannot be generated by a quartic interaction at tree level in the bulk).",
"In the bulk, the sign of $\\lambda $ matters.",
"In particular, the quartic interaction destabilizes the AdS background for $\\lambda <0$ , while it leaves AdS as a stationary point for $\\lambda >0$ .",
"One can then try to understand what this instability corresponds to, in the dual CFT.",
"The two-loop coefficient of the $\\beta $ function in CPT is proportional to the (integrated) contact Witten diagram of figure REF , which in this case is non-vanishing, i.e.",
"$\\beta _2= a \\lambda $ , with $a$ a positive $d$ -dependent number, $a= \\pi ^d\\Gamma (d/2)^4/2 \\Gamma (d)^3$ .",
"Therefore, $\\beta _2$ has the same sign as $\\lambda $ .",
"This means that for $\\lambda >0$ the operator ${\\cal O}$ becomes marginally irrelevant, while for $\\lambda <0$ it becomes marginally relevant.",
"Hence, in the latter case, a deformation triggered by ${\\cal O}$ induces an RG-flow which brings the theory away from the fixed point.",
"On the contrary, for $\\lambda >0$ the deformation is marginally irrelevant and the undeformed CFT remains, consistently, a stable point.",
"Note how different this is from the case of SCFTs.",
"There, marginal operators may either remain marginal or become marginally irrelevant, but never marginally relevant [4], which agrees with the fact that AdS backgrounds are stable in supersymmetric setups."
],
[
"Loops in AdS",
"An obvious question is whether one can push the above analysis to higher orders in $1/N$ .",
"This corresponds to take into account loop corrections in the bulk.",
"Already at one-loop, this is something very hard to do (see, e.g., [21], [22], [23], and, more recently, [24], [25], where interesting progress have been obtained from complementary perspectives).",
"The main issue in this matter is not really to compute loop amplitudes per sé, but to make their relation to tree-level amplitudes precise, and this is something non-trivial to do in AdS.",
"In fact, the question we are mostly interested in, here, is slightly different.",
"Starting from the effective action (REF ), which is valid up to some energy cut-off $E$ , in computing quantum corrections we are not much interested on how the couplings run with the scale but else on which (new) operators would be generated at energies lower than $E$ .In doing so, we can use the intuition from flat-space physics, since we are dealing with local effects in the bulk.",
"More precisely, what we have to do is to pinpoint, between the operators having passed our tree-level bulk analysis, i.e., operators of the form (REF ), those which could induce, at loop level, effective couplings which have instead been excluded at tree-level, that is the pure non-derivative couplings $\\phi ^3$ and $\\phi ^4$ , as well as a mass term, which was set to zero from the outset.",
"Such operators would spoil the vanishing of the $\\beta $ function, see figure REF (the generation of a $\\phi ^2$ -term would modify the scaling dimension of ${\\cal O}$ , which should instead remain a marginal operator).",
"Figure: One-loop Witten diagrams contributing to Δ 𝒪 \\Delta _{\\cal O}, β 1 \\beta _1 and β 2 \\beta _2.",
"Cubic and quartic Witten diagrams should include also those with loop corrections to propagators, but we have not drawn them explicitly.So, the basic question we have to answer is whether (one and higher) loop analysis still leaves some of the operators (REF ) being compatible with the vanishing of the $\\beta $ function (REF ) and with $\\Delta _{\\mathcal {O}}=d$ .",
"That this is not an empty set can be easily seen as follows.",
"Out of the full set (REF ), let us consider shift-symmetric operators, only, namely operators which are invariant under the shift symmetry $\\phi \\rightarrow \\phi +a~.$ In perturbation theory, such operators cannot generate effective operators not respecting (REF ), hence in particular $\\phi ^n$ terms.",
"Therefore, at least perturbatively, a conformal manifold does persist, if only shift-symmetric couplings are allowed in the action (REF ).",
"Let us now consider all other couplings, those with at least one naked $\\phi $ , which do not respect the shift-symmetry (REF ).",
"Generically, these operators would generate any effective operator of the form $\\phi ^n$ , quantum mechanically.",
"In particular, regardless of spacetime dimension, $\\phi ^2$ and $\\phi ^3$ will be generated at one-loop by any (non shift-symmetric) operator of the form (REF ).",
"Operators $\\phi ^n$ with $n \\ge 4$ , instead, will be generated at one-loop or higher, depending on spacetime dimension and the specific operator (REF ) one is considering.",
"In any event, the upshot is that, unless one invokes some unnatural tuning between the a priori independent couplings $\\lambda _n$ , any operator with at least one naked $\\phi $ should be excluded, eventually, by requiring a conformal manifold to persist at finite $N$ .",
"This leaves only shift-symmetric couplings in business, meaning that the shift symmetry (REF ) should be imposed on the bulk action (REF ) altogether.Following the discussion in the previous section, one can easily get convinced that the inclusion of a dynamical graviton, hence of the energy-momentum tensor in the low-dimension CFT operators, would not affect this result.",
"As already noticed, shift-symmetric couplings would not contribute to (integrated) Witten diagrams not just at one loop but at any loop order in the bulk.",
"Therefore, the final answer we got may be extended as a statement on the existence of a conformal manifold generated by ${\\cal O}$ at all orders in the $1/N$ perturbative expansion.",
"This apparently strong statement is just due to the axion-like behavior of an operator subject to eq.",
"(REF ), which, as such, is expected to be lifted by non-perturbative effects only.",
"The latter are suppressed as, say, $e^{-N}$ .",
"Richer holographic models would behave differently, and not share such perturbative non-renormalization property.",
"Our analysis just aims at showing that, in principle, non-supersymmetric conformal manifolds can exist also beyond planar limit.",
"It would be very interesting to consider models with richer structure.",
"We will offer a few more comments on this issue in the next, concluding section."
],
[
"Discussion",
"In general, it is hard to find non-supersymmetric interacting CFTs in $d>2$ , notable exceptions being, e.g., the 3d Ising model, the critical $O(N)$ model and Banks-Zaks fixed point.In the context of boundary conformal field theories (bCFT) there also exist examples.",
"One such example, the mixed dimensional QED discussed in [26], [27], is even believed to admit a (perturbative-stable) conformal manifold.",
"We thank Chris Herzog for making us aware of this possibility.",
"Since its early days, the AdS/CFT correspondence has been a natural framework where to look for novel examples.",
"Besides the limiting case of generalized free fields, most attempts have encountered obstructions.",
"Starting from the original ${\\cal N}=4$ SYM/$\\mbox{AdS}_5\\times S^5$ duality, a very natural possibility is to consider non-supersymmetric orbifold thereof.",
"It was shown in [28] that (unlike in the parent supersymmetric theory) conformal invariance is broken already at leading order in $1/N$ , by the logarithmic running of double-trace operators.",
"This looks like a generic phenomenon which has been proposed in [29], [30] to be related to the presence of tachyonic instabilities in the gravity dual [31].Eventually, these problems might also be connected with recent claims about the non-perturbative instability of non-supersymmetric AdS vacua [32].",
"In this context, the only model we are aware of which evades this problem, is a non-tachyonic orientifold of Type 0B string theory, discussed in [33].",
"However, it turns out that the absence of tachyons is not dual to the existence of fixed points in the dangerous double-trace operator running, but rather to the absence of such operators, at least at leading order in $1/N$ [34].",
"Hence, conformal invariance is preserved (and a fixed line exists in the space of couplings) but in a rather trivial sense, because of the exact equivalence of this theory with a subsector of the original ${\\cal N}=4$ SYM, at large $N$ .",
"More recently, another class of non-supersymmetric models obtained as a suitable double scaling limit of $\\gamma $ -deformed ${\\cal N}=4$ SYM has been proposed [35] (see also [36]).",
"For example, there exists a four-dimensional two (complex) scalar theory which looks particularly simple at face value.",
"These models, although not being unitary, are interesting in many respect, but they also share the presence of double-trace operators in the effective action which spoil conformal invariance at leading order in $1/N$ .",
"In a more recent work [37], it was suggested that a suitable refinement of these models (that is, introducing an extra flavor structure for the component fields) could project out the double-trace operators, at least at leading order in $1/N$ , similarly to [34].",
"And that also three and six-dimensional versions of the same model are not plagued by double-trace operator running, at large $N$ .",
"It would be interesting to see whether conformal invariance is preserved beyond leading order and, if this is the case, if a conformal manifold exists.",
"These models look tractable enough, with respect to full-fledged top-down models, to make one hope that some concrete progress could be possible.",
"More difficult, here, is to have some intuition about what the gravity dual description could be.",
"Within less ambitious, bottom-up models one can try to consider simple improvements of our one-field model.",
"The basic reason why supersymmetric theories can admit conformal manifolds is due to the knowledge of the (perturbatively exact) $\\beta $ function for elementary fields and the possibility that some linear combinations have vanishing anomalous dimension $\\gamma $ , that is $\\gamma _j(g_1,\\dots , g_n)=0~~,~~j=1,2,\\dots ,m~,$ where $g_i$ are the couplings associated to classically marginal operators ${\\cal O}_i$ .",
"If $n>m$ , the above equations describe a $n-m$ dimensional manifold of exactly marginal deformations, the conformal manifold.",
"In a non-supersymmetric, bottom-up context, one can imagine to deform a CFT by (say) two scalar marginal operators ${\\cal O}_i$ as $\\delta S = g_1 \\int d^d x \\,{\\cal O}_1 + g_2 \\int d^d x \\, {\\cal O}_2 ~,$ with the couplings subject to the constrain $F(g_1,g_2)=0~.$ This equation defines a line in the space of couplings.",
"One can demand that on (REF ), at one and two loops in CPT, $\\beta $ functions vanish and, generalizing the analysis of section , read-off the corresponding constraints that the existence of such one-dimensional conformal manifold imposes on the original CFT.",
"Two-field models like the one above can be cooked-up holographically, and one might hope to get some richer answers with respect to the one-field model we have considered here."
],
[
"Acknowledgements",
"We would like to thank Riccardo Argurio, Agnese Bissi, Lorenzo Di Pietro, Denis Karateev, Per Kraus, Petr Kravchuk, Zohar Komargodski, Ioannis Papadimitriou, Rodolfo Russo, Slava Rychkov, Marco Serone, River Snively, Massimo Taronna, Giovanni Villadoro and Alexander Zhiboedov for helpful discussions and comments at different stages of this work.",
"We are indebted with Riccardo Argurio, Lorenzo Di Pietro and Rodolfo Russo for very useful feedbacks on a preliminary draft version.",
"H. R. is grateful to the Department of Physics and Astronomy of UCLA for kind hospitality during the completion of this work.",
"We acknowledge partial support from the MIUR-PRIN Project “Non-perturbative Aspects Of Gauge Theories And Strings” (Contract 2015MP2CX4)."
]
] | 1709.01749 |
[
[
"On a characterization of path connected topological fields"
],
[
"Abstract The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space $X$ and the space of maximal ideals of the ring of real valued continuous functions $C(X,\\mathbb{R})$.",
"More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces?",
"It turns out that such a perfect correspondence exists if and only if $F$ is a path connected topological field."
],
[
"§11em all The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space $X$ and the space of maximal ideals of the ring of real valued continuous functions $C(X, \\mathbb {R})$ .",
"More explicitly, our motivation is the following question: What is the essential property of the topological field $F= \\mathbb {R}$ that makes such a correspondence valid for all compact Hausdorff spaces?",
"It turns out that such a perfect correspondence exists if and only if $F$ is a path connected topological field."
],
[
"Introduction",
"Let us recall briefly what happens with the real field.",
"Let $X\\ne \\emptyset $ be a compact Hausdorff topological space, and let $C(X, \\mathbb {R})$ be the set of continuous functions from $X$ to $ \\mathbb {R}$ .",
"Since evaluating at a point $x_{0}\\in X$ is a surjective ring homomorphism $\\mathrm {ev}_{x_{0}}:C(X, \\mathbb {R})\\rightarrow \\mathbb {R};\\ f\\mapsto f(x_{0})$ , the kernel of such morphism, $I_{ \\mathbb {R}}(x_{0}):= \\text{Ker}(\\mathrm {ev}_{x_{0}})$ , is a maximal ideal of $C(X, \\mathbb {R})$ .",
"By endowing the set of maximal ideals $\\mathrm {Max}(C(X, \\mathbb {R}))$ with the Zariski topology, Gelfand's theorem tells us that this correspondence is actually a homeomorphism, explicitly: Theorem Let $X$ be a non-empty compact Hausdorff topological space.",
"Then the map $\\begin{array}{cccccc}I_{ \\mathbb {R}}: & X & \\rightarrow & \\mathrm {Max}(C(X, \\mathbb {R})) & & \\\\& x_{0} & \\mapsto & I_{ \\mathbb {R}}(x_{0}) & &\\end{array}$ is a homeomorphism.",
"In other words the algebraic structure of the ring $C(X, \\mathbb {R})$ completely characterizes the space $X$ .",
"What is so special about $ \\mathbb {R}$ that makes such a powerful characterization possible?",
"Could we ensure the same kind of result if we replace $ \\mathbb {R}$ by another topological field $F$ ?",
"Let $F$ be a topological field.",
"Let $X\\ne \\emptyset $ be a compact Hausdorff topological space, and let $C(X,F)$ be the set of continuous functions from $X$ to $F$ .",
"As in the real case given $x_{0}\\in X$ if $I_{F}(x_{0})$ is the kernel of the evaluation morphism at the point $x_{0}$ then $I_{F}(x_{0})\\in \\mathrm {Max}(C(X,F))$ .",
"The question we address is the following: What conditions on a topological field $F$ are sufficient and necessary such that for every compact Hausdorff topological space $X$ the function $\\begin{array}{cccccc}I_{F}: & X & \\rightarrow & \\mathrm {Max}(C(X,F)) & & \\\\& x_{0} & \\mapsto & I_{F}(x_{0}) & &\\end{array}$ is a homeomorphism?",
"The answer is given by the main result of the paper: Theorem (cf.",
"Theorem REF ) Let $F$ be topological field.",
"Then the Gelfand map $I_{F}$ is a homeomorphism for every compact Hausdorff topological space $X$ if and only if $F$ is path connected.",
"The continuity of the Gelfand map holds more generally for any topological field.",
"Injectivity and bicontinuity will be shown to be equivalent to path connectedness.",
"And, remarkably, surjectivity will be seen to hold for any topological field.",
"Our proof of the later fact relays on the famous classification of non-discrete locally compact fields ([5], [10], or [2]), and the observation (Proposition REF ) that for a non algebraically closed field $F$ there is a polynomial function $\\psi :F^{2}\\rightarrow F$ such that $\\psi ^{-1}(\\lbrace 0_{F}\\rbrace )=(0_{F},0_{F}).$ This polynomial must have the form $\\psi (x,y)=x\\phi _{1}(x,y)+y\\phi _{2}(x,y)$ , and it generalizes the function $\\psi (x,y)=x^{2}+y^{2}$ for $\\mathbb {R}$ .",
"In the algebraically closed case such polynomial can not exist, but surjectivity of $I_{\\mathbb {C}}$ may be shown utilizing the function $\\psi (x,y)=x\\overline{x}+y\\overline{y}$ which has the displayed property.",
"By analyzing the cases we have in hand, we suspect that there is for any path connected topological field $F$ a similar function $\\psi (x,y)=x\\phi _{1}(x,y)+y\\phi _{2}(x,y)$ with $\\phi _{1},$ $\\phi _{2}:F^{2}\\rightarrow F$ continuous, but have not been able to prove this.",
"The subject of rings of continuos functions with values in a field is classic and has been studied by several authors, see for instance [1], [3], [6].",
"A survey article on the subject containing a great deal of results and many references is [9].",
"Although our characterization of path connected fields given by Theorem REF is quite natural, it seems to have been missed in previous characterizations, see [6]."
],
[
"Path connected fields",
"In this paper a topological field is a field $F$ which is also a topological space in which the operations are continuous and the points are closed.",
"Since the topology must be regular by general properties of topological groups, $F$ must be Hausdorff.",
"We denote by $0_{F}$ and $1_{F}$ the zero the and identity of $F$ respectively.",
"Recall that a topological space $X$ is path connected if for every two points $x,y\\in X$ there is a continuous path $\\gamma :[0,1]\\rightarrow X$ joining $x$ and $y,$ and it is arcwise connected if $\\gamma $ may be always chosen as a homeomorphism between $[0,1]$ and its image.",
"The following observation shows that the various forms of path connectivity are equivalent in fields.",
"Lemma 2.1 Let $F$ be a topological field.",
"The following are equivalent: (i) There is a continuous path $\\gamma :[0,1]\\rightarrow F$ joining $0_{F}$ and $1_{F}$ .",
"(ii) $\\ F$ is contractible.",
"(iii) $F$ is path connected.",
"(iv) $F$ is arcwise connected.",
"Path and arcwise connectendness are known to be equivalent in Hausdorff spaces ([11], Corollary 31.6).",
"It is enough then to show the implications (i) $\\Longrightarrow $ (ii) and (iii) $\\Longrightarrow $ (i).",
"Given a path $\\gamma $ joining $0_{F}$ and $1_{F}$ , the function $H:[0,1]\\times F\\rightarrow F;$ $(t,\\lambda )\\mapsto (1-\\gamma (t))\\lambda $ gives the result.",
"For the later case, given a path $\\gamma _{1} $ joining two different points $a,b\\in F$ the function $\\displaystyle \\gamma (t)=\\frac{\\gamma _{1}(t)-a}{b-a}$ gives a path between $0_{F}$ and $1_{F}$ ."
],
[
"A first characterization",
"We prove in this section a first approximation to our main result (Theorem REF below) by following essentially the lines of the proof that the map $I_{ \\mathbb {R}}$ is injective and bicontinuous for any compact Hausdorff $X $ .",
"Continuity of $I_{ \\mathbb {R}}$ follows from definition of the Zariski topology, and from the fact that 0 is closed in $ \\mathbb {R}$ .",
"This holds for every topological field $F$ and every space $X$ (not necessarily compact or Hausdorff).",
"Lemma 3.1 For any space $X$ and any topological field the Gelfand map $I_{F}:X\\rightarrow \\mathrm {Max}(C(X,F))$ is continuous.",
"By definition of the Zariski topology a closed subset of $\\mathrm {Max}(C(X,F))$ is of the form $C_{S}:=\\lbrace M\\in \\mathrm {Max}(C(X,F))\\mid S\\subseteq M\\rbrace $ for some $S\\subseteq C(X,F).$ Continuity of $I_{F}$ follows since $\\lbrace 0_{F}\\rbrace $ is closed and $I_{F}^{-1}(C_{S})=\\lbrace x:S\\subseteq I_{F}(x)\\rbrace =\\bigcap _{f\\in S}f^{-1}(0_{F}).$ Injectivity of $I_{ \\mathbb {R}}$ follows from the fact that $ \\mathbb {R}$ is path connected and compact Hausdorff spaces are completely regular.",
"These ideas can be generalized as follows: Lemma 3.2 Let $F$ be a topological field.",
"Then $F$ is path connected if and only if for any compact Hausdorff space $X$ the Gelfand map $I_{F}:X\\rightarrow \\mathrm {Max}(C(X,F))$ is injective.",
"Assume $I_{F}$ is injective for the space $X=[0,1]$ then $I_{F}(0)\\ne I_{F}(1)$ , pick $f\\in I_{F}(0)\\setminus I_{F}(1)$ then $f(0)=0_{F}$ and $f(1)\\ne 0_{F}.$ Hence, $\\gamma =$ $f(1)^{-1}f$ defines a path connecting $0_{F}$ and $1_{F}$ .",
"Reciprocally, assume there is a continuous path $\\gamma $ as described.",
"Let $x\\ne y\\in X$ .",
"Since $X$ is completely regular it follows that there is a continuous function $f:X\\rightarrow [0,1]$ such that $f(x)=0$ and $f(y)=1$ .",
"Notice that $\\gamma \\circ f(x)=0_{F}$ and $\\gamma \\circ f(y)=1_{F}$ .",
"In particular, $\\gamma \\circ f\\in I_{F}(x)\\setminus I_{F}(y)$ hence $I_{F}$ is injective.",
"Path connectedness of $F$ implies also that the image of the embedding is Hausdorff for any completely regular space.",
"Recall that an open basis for $\\mathrm {Max}(C(X,F))$ is given by the sets $D(f)=\\lbrace M:f\\notin M\\rbrace .$ Lemma 3.3 Let $F$ be a path connected topological field.",
"Then the image of $I_{F}$ is Hausdorff for every compact Hausdorff space $X$ .",
"To separate $I_{F}(x_{0})\\ne I_{F}(x_{1})$ in $\\mathrm {Max}(C(X,F))$ it is enough to show that there are $f,g\\in C(X,F)$ such that $f(x_{0})g(x_{1})\\ne 0_{F}$ (thus, $I_{F}(x_{0})\\in D(f),$ $I_{F}(x_{1})\\in D(g))$ and $fg\\equiv 0_{F}$ (thus $D(f)\\cap D(g)=\\emptyset )$ Since $x_{0}\\ne x_{1}$ and $X$ is Hausdorff there are disjoint open sets $U_{i},$ $i=0,1$ , such that $x_{i}\\in U_{i}$ .",
"Letting $C_{i}:=X\\setminus U_{i}$ we see that the $C_{i}$ are closed subsets of $X$ .",
"Therefore, by complete regularity of $X$ there are $\\widetilde{f},\\widetilde{g}\\in C(X,[0,1])$ such that $\\widetilde{g}(x_{1})=1=\\widetilde{f}(x_{0})\\ \\mathrm {and}\\ \\widetilde{g}(C_{0})=0=\\widetilde{f}(C_{1}).$ Fix a continuous path $\\gamma :[0,1]\\rightarrow F$ joining $0_{F}$ and $1_{F} $ , and let $f:=\\gamma \\circ \\widetilde{f}$ and $g:=\\gamma \\circ \\widetilde{g} $ .",
"By the above $f(x_{0})=1_{F}$ and $g(x_{1})=1_{F}$ .",
"Let $x\\in X$ , then $x\\in C_{0}$ or $x\\in C_{1}$ , which by construction implies that $f(x)g(x)=0_{F}.$ By standard topology a continuous injection from a compact space in a Hausdorff space is bicontinuous.",
"Hence, combining lemmas 2.1, 2,1 and 2.3, we obtain a first characterization.",
"Theorem 3.4 Let $F$ be a topological field.",
"Then $F$ is path connected if and only if for every compact Hausdorff space $X$ the Gelfand map $I_{F}$ induces a homeomorphism between $X$ and its image."
],
[
"Surjectivity",
"We will show in this section that the Gelfand map is surjective for any compact Hausdorff space $X$ and any topological field $F.$ In this case the techniques for $I_{ \\mathbb {R}}$ do not generalize fully.",
"Surjectivity of $I_{ \\mathbb {R}}$ follows basically from the fact that for every positive $n$ the vanishing locus of the polynomial $f_{n}:=x_{1}^{2}+...+x_{n}^{2}$ in $ \\mathbb {R}^{n}$ is the single point $(0_{F},...,0_{F})$ .",
"Such a polynomial function can not exist in algebraically closed fields, but we notice that it exists for any non algebraically closed field $F,$ which will lead us to the proof of surjectivity in this case.",
"Proposition 4.1 Let $F$ be a non algebraically closed field.",
"Then for each positive integer $n$ there is a polynomial map $f_{n}:F^{n}\\rightarrow F $ such that $f_{n}^{-1}(0_{F})=\\left\\lbrace (0_{F},...,0_{F})\\right\\rbrace $ .",
"Since $F$ is not algebraically closed there is a monic polynomial $f(x)\\in F[x]$ , of positive degree $m$ , which has no zeros in $F$ .",
"Let $f_{2}(x,y):=y^{m}f(\\frac{x}{y})\\in F[x,y]$ be the homogenization of $f $ .",
"Since $f(x)$ has no roots in $F$ then $f_{2}^{-1}(0_{F})=\\left\\lbrace (0_{F},0_{F})\\right\\rbrace $ .",
"Now proceed by induction.",
"Let $f_{1}:F\\rightarrow F$ be the identity function, and suppose that for $n\\ge 1$ we have defined $f_{n}$ .",
"Then $\\displaystyle f_{n+1}(x_{1},...,x_{n+1}):=f_{2}(f_{n}(x_{1},...,x_{n}),x_{n+1})$ is also polynomial and $f_{n+1}^{-1}(0_{F})=(f_{n} \\times f_{1})^{-1}(\\left\\lbrace (0_{F},0_{F})\\right\\rbrace )=\\left\\lbrace (0_{F},...,0_{F})\\right\\rbrace \\times \\lbrace 0_{F}\\rbrace =\\left\\lbrace (0_{F},...,0_{F})\\right\\rbrace .$ For the complex numbers surjectivity of $I_{ \\mathbb {C}}$ may be shown as for $I_{ \\mathbb {R}}$ by taking the continuous function $f_{n}:=x_{1}\\overline{x_{1}}+...+x_{n}\\overline{x_{n}}$ .",
"It is possible that for any field there is continuous function vanishing only at $(0_{F},...,0_{F})$ of the form $\\displaystyle f=x_{1}\\phi _{1}+...+x_{n}\\phi _{n},$ where $x_{i}:F^{n}\\rightarrow F$ denotes the $i$ -th coordinate function and $\\phi _{i}:F^{n}\\rightarrow F$ is continuous$.$ However, we have not been able to prove this.",
"Fortunately, a weaker version of this idea gives an alternative argument that works for all fields.",
"Theorem 4.2 Let $F$ be a topological field.",
"Then for any compact Hausdorff space $X$ the Gelfand map $I_{F}:X\\rightarrow \\mathrm {Max}(C(X,F))$ is surjective.",
"Let $M\\in \\mathrm {Max}(C(X,F))$ and for $\\psi \\in M$ let $D(\\psi ):=\\psi ^{-1}(F\\setminus \\lbrace 0_{F}\\rbrace )$ .",
"Notice that $I_{F}^{-1}(M)=\\lbrace x:I_{F}(x)\\supseteq M\\rbrace =\\bigcap _{\\psi \\in M}\\psi ^{-1}(0_{F}).$ Hence, $I_{F}^{-1}(M)=X\\setminus \\bigcup _{\\psi \\in M}D(\\psi ).$ Suppose that $M$ is not in the image of $I_{F}$ .",
"It follows from the compactness of $X$ that there are finitely many $\\psi _{1},...,\\psi _{n}\\in M $ such that $X=D(\\psi _{1})\\cup ...\\cup D(\\psi _{n})$ .",
"Hence, any element is not zero by some $\\psi _{i}.$ We will show that there is a continuous function $f_{n}: F^{n} \\rightarrow F$ such that $f_{n}(\\psi _{1},...,\\psi _{n})\\in M$ and $f_{n}(\\psi _{1}(x),...,\\psi _{n}(x))$ is never 0 in $X$ .",
"This contradicts the fact that $M$ is a proper ideal.",
"To achieve this we consider several cases.",
"Case I.",
"$F$ is non-discrete locally compact then $F$ must be $\\mathbb {C}$ , $\\mathbb {R}$ , a finite extension of $\\mathbb {Q}_{p}$ or a finite extension of $\\mathbb {F}_{p}((t)).$ For $\\mathbb {C}$ take $f_{n}:=x_{1}\\overline{x_{1}}+...+x_{n}\\overline{x_{n}}$ .",
"In the remaining cases $F$ is not algebraically closed.",
"Therefore, the polynomial $f_{n}$ provided by Proposition REF will do: $f_{n}(\\psi _{1}(x),...,\\psi _{n}(x))$ never vanishes because $(\\psi _{1}(x),...,\\psi _{n}(x))$ is never $(0_{F},...,0_{F})$ and $f_{n}(\\psi _{1},..,\\psi _{n})\\in M$ since $f_{n}$ does not have constant term.",
"Case II.",
"$F$ is discrete$.$ Consider the continuous map $x\\mapsto (\\psi _{1}(x),...,\\psi _{n}(x))\\ $ from $X$ into $F^{n}.$ Its image $J=\\lbrace (\\psi _{1}(x),...,\\psi _{n}(x)):x\\in X\\rbrace $ is compact and thus necessarily finite since $F^{n}$ is discrete, moreover, it does not include $(0_{F},...,0_{F})$ by hypothesis.",
"Find a polynomial $f_{n}(x_{1},...,x_{n})$ identically $1_{F}$ in $J$ and $0_{F}$ in $(0_{F},...,0_{F})$ (Lagrange interpolation).",
"Then $f_{n}(\\psi _{1}(x),...,\\psi _{n}(x))=1_{F}$ for any $x\\in X$ and belongs to $M$ because $f_{n}$ does not have constant term.",
"Case III.",
"$F$ is not locally compact.",
"Consider first $D(\\psi _{1})\\cup D(\\psi _{2})$ restricted to the compact space $Y=X\\setminus D(\\psi _{3})\\cup ...\\cup D(\\psi _{n}),$ and consider the map $x\\mapsto [\\psi _{1}(x),\\psi _{2}(x)]\\ $ from $Y$ into $P_{1}(F),$ the projective $F$ -line with the quotient topology, and let $I$ be the image of this map.",
"Recall that $P_{1}(F)=F\\cup \\lbrace [1,0]\\rbrace $ where $F$ is topologically embedded in $P_{1}(F)$ via the identification $F\\sim \\lbrace [a,1]:a\\in F\\rbrace $ .",
"As $I $ is compact, the set $P_{1}(F)\\setminus I$ must be infinite, as otherwise $P_{1}(F)$ would be compact and thus $F$ would be locally compact.",
"Then we may find $[a,1]\\in F$ such that $[\\psi _{1}(x),\\psi _{2}(x)]\\ne [a,1]\\ $ for each $x\\in Y.$ If $\\psi _{2}(x)=0$ then $\\psi _{1}(x)\\ne 0$ and thus $\\psi _{1}(x)-a\\psi _{2}(x)\\ne 0$ .",
"If $\\psi _{2}(x)\\ne 0$ then $\\psi _{1}(x)/\\psi _{2}(x)\\ne a$ and also $\\psi _{1}(x)-a\\psi _{2}(x)\\ne 0.$ Therefore $\\psi _{1}-a\\psi _{2}$ does not vanish in $Y$ and we have $X=D(\\psi _{1}-a\\psi _{2})\\cup D(\\psi _{3})\\cup ...\\cup D(\\psi _{n})$ Repeating the procedure with the pair $[\\psi _{1}-a\\psi _{2},\\psi _{3}],$ and continuing inductively, we obtain finally $X=D(a_{1}\\psi _{1}+a_{2}\\psi _{2}+...+a_{n}\\psi _{n})$ where evidently $a_{1}\\psi _{1}+a_{2}\\psi _{2}+...+a_{n}\\psi _{n}\\in M.$"
],
[
"Conclusion",
"Thanks to theorems REF and REF , we have our main result: Theorem 5.1 A topological field $F$ is path connected if and only if for every compact Hausdorff topological space $X$ the Gelfand map $I_{F}:X\\rightarrow \\mathrm {Max}(C(X,F))$ is a homeomorphism.",
"An important question to ask at this point is: are there other examples besides $ \\mathbb {R}$ and $ \\mathbb {C}$ that make Gelfand correspondence work?",
"As it turns out there are many other path connected topological fields of 0 characteristic not isomorphic to either $ \\mathbb {R}$ or $ \\mathbb {C}$ .",
"There are even examples in positive characteristic; see for instance [8], where it is shown that any discrete field may be embedded in a path connected field.",
"However, by Pontryagin's classification theorem [7] the only locally compact ones are $ \\mathbb {R}$ and $ \\mathbb {C}$ .",
"An inspection of the proof of Theorem REF shows that for any topological field $F,$ given functions $\\psi _{1},...,\\psi _{n}\\in C(X,F)$ such that $X=D(\\psi _{1})\\cup ...\\cup D(\\psi _{n})$ there exist continuous functions $\\phi _{1},...,\\phi _{m}:F^{n}\\rightarrow F$ such that $\\displaystyle \\Sigma _{i=1}^{n}\\psi _{i}\\phi _{1}(\\psi _{1},..,\\psi _{n})$ does not vanishes in $X.$ But the $\\phi _{i}$ depend on the $\\psi _{i}$ .",
"One may wonder if the $\\phi _{i}$ may be chosen independently; that is, whether there exists $f=x_{1}\\phi _{1}+...+x_{n}\\phi _{n}$ such that $f^{-1}(0_{F})=(0_{F},...,0_{F}).$ We call this polynomially generated functions.",
"We are able to show: Proposition 5.2 Let $F$ be a path connected metrizable topological field.",
"Then for every positive integer $n$ there is a continuous function $f_{n}:F^{n}\\rightarrow F$ such that $f_{n}^{-1}(0_{F})=\\left\\lbrace (0_{F},...,0_{F})\\right\\rbrace .$ Let $d$ be a metric on $F^{n}$ and let $\\bar{d}:=\\min \\lbrace 1, d\\rbrace $ be the standard bounded metric on $F^{n}$ .",
"Define $\\phi :F^{n}\\rightarrow [0,1]$ as the bounded distance to $(0_{F},...,0_{F}).$ As $F$ is arcwise connected (Lemma REF ), there is an embbeding $[0,1]\\overset{\\gamma }{\\rightarrow }F$ sending 0 to $0_{F}.$ Take $f_{n}=\\gamma \\circ \\phi .$ We finish with the following question, that appeared naturally during the progress of the paper, for which we don't have an answer.",
"Question 5.3 Let $F$ be a path connected topological field.",
"Is there is a polynomially generated function $f_{2} :F^{2}\\rightarrow F$ such that $\\displaystyle f_{2}^{-1}(\\lbrace 0_{F}\\rbrace )=(0_{F},0_{F})$ ?",
"Xavier Caicedo, Department of Mathematics, Universidad de los Andes, Bogotá, Colombia ([email protected]) Guillermo Mantilla-Soler, Department of Mathematics, Universidad Konrad Lorenz, Bogotá, Colombia ([email protected])"
]
] | 1709.01538 |
[
[
"Learning to Compose Domain-Specific Transformations for Data\n Augmentation"
],
[
"Abstract Data augmentation is a ubiquitous technique for increasing the size of labeled training sets by leveraging task-specific data transformations that preserve class labels.",
"While it is often easy for domain experts to specify individual transformations, constructing and tuning the more sophisticated compositions typically needed to achieve state-of-the-art results is a time-consuming manual task in practice.",
"We propose a method for automating this process by learning a generative sequence model over user-specified transformation functions using a generative adversarial approach.",
"Our method can make use of arbitrary, non-deterministic transformation functions, is robust to misspecified user input, and is trained on unlabeled data.",
"The learned transformation model can then be used to perform data augmentation for any end discriminative model.",
"In our experiments, we show the efficacy of our approach on both image and text datasets, achieving improvements of 4.0 accuracy points on CIFAR-10, 1.4 F1 points on the ACE relation extraction task, and 3.4 accuracy points when using domain-specific transformation operations on a medical imaging dataset as compared to standard heuristic augmentation approaches."
],
[
"Introduction",
"Modern machine learning models, such as deep neural networks, may have billions of free parameters and accordingly require massive labeled data sets for training.",
"In most settings, labeled data is not available in sufficient quantities to avoid overfitting to the training set.",
"The technique of artificially expanding labeled training sets by transforming data points in ways which preserve class labels – known as data augmentation – has quickly become a critical and effective tool for combatting this labeled data scarcity problem.",
"Data augmentation can be seen as a form of weak supervision, providing a way for practitioners to leverage their knowledge of invariances in a task or domain.",
"And indeed, data augmentation is cited as essential to nearly every state-of-the-art result in image classification , , , (see Appendix REF ), and is becoming increasingly common in other modalities as well .",
"Even on well studied benchmark tasks, however, the choice of data augmentation strategy is known to cause large variances in end performance and be difficult to select , , with papers often reporting their heuristically found parameter ranges .",
"In practice, it is often simple to formulate a large set of primitive transformation operations, but time-consuming and difficult to find the parameterizations and compositions of them needed for state-of-the-art results.",
"In particular, many transformation operations will have vastly different effects based on parameterization, the set of other transformations they are applied with, and even their particular order of composition.",
"For example, brightness and saturation enhancements might be destructive when applied together, but produce realistic images when paired with geometric transformations.",
"Given the difficulty of searching over this configuration space, the de facto norm in practice consists of applying one or more transformations in random order and with random parameterizations selected from hand-tuned ranges.",
"Recent lines of work attempt to automate data augmentation entirely, but either rely on large quantities of labeled data , , restricted sets of simple transformations , , or consider only local perturbations that are not informed by domain knowledge , (see Section ).",
"In contrast, our aim is to directly and flexibly leverage domain experts' knowledge of invariances as a valuable form of weak supervision in real-world settings where labeled training data is limited.",
"In this paper, we present a new method for data augmentation that directly leverages user domain knowledge in the form of transformation operations, and automates the difficult process of composing and parameterizing them.",
"We formulate the problem as one of learning a generative sequence model over black-box transformation functions (TFs): user-specified operators representing incremental transformations to data points that need not be differentiable nor deterministic.",
"For example, TFs could rotate an image by a small degree, swap a word in a sentence, or translate a segmented structure in an image (Fig.",
"REF ).",
"We then design a generative adversarial objective which allows us to train the sequence model to produce transformed data points which are still within the data distribution of interest, using unlabeled data.",
"Because the TFs can be stochastic or non-differentiable, we present a reinforcement learning-based training strategy for this model.",
"The learned model can then be used to perform data augmentation on labeled training data for any end discriminative model.",
"Given the flexibility of our representation of the data augmentation process, we can apply our approach in many different domains, and on different modalities including both text and images.",
"On a real-world mammography image task, we achieve a 3.4 accuracy point boost above randomly composed augmentation by learning to appropriately combine standard image TFs with domain-specific TFs derived in collaboration with radiology experts.",
"Using novel language model-based TFs, we see a 1.4 F1 boost over heuristic augmentation on a text relation extraction task from the ACE corpus.",
"And on a 10%-subsample of the CIFAR-10 dataset, we achieve a 4.0 accuracy point gain over a standard heuristic augmentation approach and are competitive with comparable semi-supervised approaches.",
"Additionally, we show empirical results suggesting that the proposed approach is robust to misspecified TFs.",
"Our hope is that the proposed method will be of practical value to practitioners and of interest to researchers, so we have open-sourced the code at https://github.com/HazyResearch/tanda."
],
[
"Modeling Setup and Motivation",
"In the standard data augmentation setting, our aim is to expand a labeled training set by leveraging knowledge of class-preserving transformations.",
"For a practitioner with domain expertise, providing individual transformations is straightforward.",
"However, high performance augmentation techniques use compositions of finely tuned transformations to achieve state-of-the-art results , , , and heuristically searching over this space of all possible compositions and parameterizations for a new task is often infeasible.",
"Our goal is to automate this task by learning to compose and parameterize a set of user-specified transformation operators in ways that are diverse but still preserve class labels.",
"In our method, transformations are modeled as sequences of incremental user-specified operations, called transformation functions (TFs) (Fig.",
"REF ).",
"Rather than making the strong assumption that all the provided TFs preserve class labels, as existing approaches do, we assume a weaker form of class invariance which enables us to use unlabeled data to learn a generative model over transformation sequences.",
"We then propose two representative model classes to handle modeling both commutative and non-commutative transformations."
],
[
"Augmentation as Sequence Modeling",
"In our approach, we represent transformations as sequences of incremental operations.",
"In this setting, the user provides a set of $K$ TFs, $h_i : \\mathcal {X} \\mapsto \\mathcal {X}$ , $i\\in [1,K]$ .",
"Each TF performs an incremental transformation: for example, $h_i$ could rotate an image by five degrees, swap a word in a sentence, or move a segmented tumor mass around a background mammography image (see Fig.",
"REF ).",
"In order to accommodate a wide range of such user-defined TFs, we treat them as black-box functions which need not be deterministic nor differentiable.",
"This formulation gives us a tractable way to tune both the parameterization and composition of the TFs in a discretized but fine-grained manner.",
"Our representation can be thought of as an implicit binning strategy for tuning parameterizations – e.g.",
"a 15 degree rotation might be represented as three applications of a five-degree rotation TF.",
"It also provides a direct way to represent compositions of multiple transformation operations.",
"This is critical as a multitude of state-of-the-art results in the literature show the importance of using compositions of more than one transformations per image , , , which we also confirm experimentally in Section ."
],
[
"Weakening the Class-Invariance Assumption",
"Any data augmentation technique fundamentally relies on some assumption about the transformation operations' relation to the class labels.",
"Previous approaches make the unrealistic assumption that all provided transformation operations preserve class labels for all data points.",
"That is, $y(h_{\\tau _L} \\circ \\hdots \\circ h_{\\tau _1}(x)) = y(x) $ for label mapping function $y$ , any sequence of TF indices $\\tau _1,...,\\tau _L$ , and all data points $x$ .",
"This assumption puts a large burden of precise specification on the user, and based on our observations, is violated by many real-world data augmentation strategies.",
"Instead, we consider a weaker modeling assumption.",
"We assume that transformation operations will not map between classes, but might destructively map data points out of the distribution of interest entirely: $y(h_{\\tau _L} \\circ \\hdots \\circ h_{\\tau _1}(x)) \\in \\lbrace y(x), y_\\emptyset \\rbrace $ where $y_\\emptyset $ represents an out-of-distribution null class.",
"Intuitively, this weaker assumption is motivated by the categorical image classification setting, where we observe that transformation operations provided by the user will almost never turn, for example, a plane into a car, but may often turn a plane into an indistinguishable “garbage” image (Fig.",
"REF ).",
"We are the first to consider this weaker invariance assumption, which we believe more closely matches various practical data augmentation settings of interest.",
"In Section , we also provide empirical evidence that this weaker assumption is useful in binary classification settings and over modalities other than image data.",
"Critically, it also enables us to learn a model of TF sequences using unlabeled data alone."
],
[
"Minimizing Null Class Mappings Using Unlabeled Data",
"Given assumption (REF ), our objective is to learn a model $G_\\theta $ which generates sequences of TF indices $\\tau \\in \\lbrace 1,K\\rbrace ^L$ with fixed length $L$ , such that the resulting TF sequences $h_{\\tau _1},...,h_{\\tau _L}$ are not likely to map data points into $y_\\emptyset $ .",
"Crucially, this does not involve using the class labels of any data points, and so we can use unlabeled data.",
"Our goal is then to minimize the the probability of a generated sequence mapping unlabeled data points into the null class, with respect to $\\theta $ : $J_{\\emptyset } &= \\mathbb {E}_{\\tau \\sim G_\\theta }\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[P(y(h_{\\tau _L}\\circ \\hdots \\circ h_{\\tau _1}(x)) = y_\\emptyset )\\right]$ where $\\mathcal {U}$ is some distribution of unlabeled data."
],
[
"Generative Adversarial Objective",
"In order to approximate $P(y(h_{\\tau _1}\\circ \\hdots \\circ h_{\\tau _L}(x)) = y_\\emptyset )$ , we jointly train the generator $G_{\\theta }$ and a discriminative model $D_\\phi ^\\emptyset $ using a generative adversarial network (GAN) objective , now minimizing with respect to $\\theta $ and maximizing with respect to $\\phi $ : $\\tilde{J}_{\\emptyset } &= \\mathbb {E}_{\\tau \\sim G_{\\theta }} \\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\log (1 - D_\\phi ^\\emptyset (h_{\\tau _L}\\circ \\hdots \\circ h_{\\tau _1}(x)))\\right]+ \\mathbb {E}_{x^{\\prime }\\sim \\mathcal {U}}\\left[ \\log (D_\\phi ^\\emptyset (x^{\\prime })) \\right] $ As in the standard GAN setup, the training procedure can be viewed as a minimax game in which the discriminator’s goal is to assign low values to transformed, out-of-distribution data points and high values to real in-distribution data points, while simultaneously, the generator's goal is to generate transformation sequences which produce data points that are indistinguishable from real data points according to the discriminator.",
"For $D^\\emptyset _\\phi $ , we use an all-convolution CNN as in .",
"For further details, see Appendix REF .",
"An additional concern is that the model will learn a variety of null transformation sequences (e.g.",
"rotating first left than right repeatedly).",
"Given the potentially large state-space of actions, and the black-box nature of the user-specified TFs, it seems infeasible to hard-code sets of inverse operations to avoid.",
"To mitigate this, we instead consider a second objective term: $J_{d} &= \\mathbb {E}_{\\tau \\sim G_\\theta }\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[d(h_{\\tau _L}\\circ \\hdots \\circ h_{\\tau _1}(x), x)\\right] $ where $d:\\mathcal {X}\\times \\mathcal {X}\\rightarrow \\mathbb {R}$ is some distance function.",
"For $d$ , we evaluated using both distance in the raw input space, and in the feature space learned by the final pre-softmax layer of the discriminator $D_\\phi ^\\emptyset $ .",
"Combining eqns.",
"REF and REF , our final objective is then $J = \\tilde{J}_{\\emptyset } + \\alpha J_{d}^{-1}$ where $\\alpha > 0$ is a hyperparameter.",
"We minimize $J$ with respect to $\\theta $ and maximize with respect to $\\phi $ ."
],
[
"Modeling Transformation Sequences",
"We now consider two model classes for $G_\\theta $ :"
],
[
"Independent Model",
"We first consider a mean field model in which each sequential TF is chosen independently.",
"This reduces our task to one of learning $K$ parameters, which we can think of as representing the task-specific “accuracies” or “frequencies” of each TF.",
"For example, we might want to learn that elastic deformations or swirls should only rarely be applied to images in CIFAR-10, but that small rotations can be applied frequently.",
"In particular, a mean field model also provides a simple way of effectively learning stochastic, discretized parameterizations of the TFs.",
"For example, if we have a TF representing five-degree rotations, Rotate5Deg, a marginal value of $P_{G_\\theta }(\\texttt {Rotate5Deg}) = 0.1$ could be thought of as roughly equivalent to learning to rotate $0.5L$ degrees on average.",
"There are important cases, however, where the independent representation learned by the mean field model could be overly limited.",
"In many settings, certain TFs may have very different effects depending on which other TFs are applied with them.",
"As an example, certain similar pairs of image transformations might be overly lossy when applied together, such as a blur and a zoom operation, or a brighten and a saturate operation.",
"A mean field model could not represent such disjunctions as these.",
"Another scenario where an independent model fails is where the TFs are non-commutative, such as with lossy operators (e.g.",
"image transformations which use aliasing).",
"In both of these cases, modeling the sequences of transformations could be important.",
"Therefore we consider a long short-term memory (LSTM) network as as a representative sequence model.",
"The output from each cell of the network is a distribution over the TFs.",
"The next TF in the sequence is then sampled from this distribution, and is fed as a one-hot vector to the next cell in the network."
],
[
"Learning a Transformation Sequence Model",
"The core challenge that we now face in learning $G_\\theta $ is that it generates sequences over TFs which are not necessarily differentiable or deterministic.",
"This constraint is a critical facet of our approach from the usability perspective, as it allows users to easily write TFs as black-box scripts in the language of their choosing, leveraging arbitrary subfunctions, libraries, and methods.",
"In order to work around this constraint, we now describe our model in the syntax of reinforcement learning (RL), which provides a convenient framework and set of approaches for handling computation graphs with non-differentiable or stochastic nodes ."
],
[
"Reinforcement Learning Formulation",
"Let $\\tau _i$ be the index of the $i$ th TF applied, and $\\tilde{x}_i$ be the resulting incrementally transformed data point.",
"Then we consider $s_t = \\left({x, \\tilde{x}_1, \\tilde{x}_2,..., \\tilde{x}_t, \\tau _1, ...., \\tau _t}\\right)$ as the state after having applied $t$ of the incremental TFs.",
"Note that we include the incrementally transformed data points $\\tilde{x}_1, ..., \\tilde{x}_t$ in $s_t$ since the TFs may be stochastic.",
"Each of the model classes considered for $G_\\theta $ then uses a different state representation $\\hat{s}$ .",
"For the mean field model, the state representation used is $\\hat{s}_t^{\\text{MF}} = \\emptyset $ .",
"For the LSTM model, we use $\\hat{s}_t^{\\text{LSTM}} = \\textsf {LSTM}(\\tau _t, s_{t-1})$ , the state update operation performed by a standard LSTM cell parameterized by $\\theta $ .",
"Let $\\ell _t(x,\\tau ) = \\log (1 -D_\\phi ^\\emptyset (\\tilde{x}_t) )$ be the cumulative loss for a data point $x$ at step $t$ , with $\\ell _0(x) = \\ell _0(x,\\tau ) \\equiv \\log (1-D_\\phi ^\\emptyset (x))$ .",
"Let $R(s_t) = \\ell _t(x,\\tau ) - \\ell _{t-1}(x,\\tau )$ be the incremental reward, representing the difference in discriminator loss at incremental transformation step $t$ .",
"We can now recast the first term of our objective $\\tilde{J}_\\emptyset $ as an expected sum of incremental rewards: $U(\\theta ) &\\equiv \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\log (1 - D_\\phi ^\\emptyset (h_{\\tau _1}\\circ \\hdots \\circ h_{\\tau _L}(x)))\\right]= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\ell _0(x) + \\sum _{t=1}^L R(s_t)\\right]$ We omit $\\ell _0$ in practice, equivalent to using the loss of $x$ as a baseline term.",
"Next, let $\\pi _\\theta $ be the stochastic transition policy implictly defined by $G_\\theta $ .",
"We compute the recurrent policy gradient of the objective $U(\\theta )$ as: $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L R(s_t) \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1})\\right]$ Following standard practice, we approximate this quantity by sampling batches of $n$ data points and $m$ sampled action sequences per data point.",
"We also use standard techniques of discounting with factor $\\gamma \\in [0,1]$ and considering only future rewards .",
"See Appendix for details."
],
[
"Related Work",
"We now review related work, both to motivate comparisons in the experiments section and to present complementary lines of work."
],
[
"Heuristic Data Augmentation",
"Most state-of-the-art image classification pipelines use some limited form of data augmentation , .",
"This generally consists of applying crops, flips, or small affine transformations, in fixed order or at random, and with parameters drawn randomly from hand-tuned ranges.",
"In addition, various studies have applied heuristic data augmentation techniques to modalities such as audio and text .",
"As reported in the literature, the selection of these augmentation strategies can have large performance impacts, and thus can require extensive selection and tuning by hand , (see Appendix REF as well).",
"Some techniques have explored generating augmented training sets by interpolating between labeled data points.",
"For example, the well-known SMOTE algorithm applies this basic technique for oversampling in class-imbalanced settings , and recent work explores using a similar interpolation approach in a learned feature space .",
"proposes learning a class-conditional model of diffeomorphisms interpolating between nearest-neighbor labeled data points as a way to perform augmentation.",
"We view these approaches as complementary but orthogonal, as our goal is to directly exploit user domain knowledge of class-invariant transformation operations.",
"Several lines of recent work have explored techniques which can be viewed as forms of data augmentation that are adversarial with respect to the end classification model.",
"In one set of approaches, transformation operations are selected adaptively from a given set in order to maximize the loss of the end classification model being trained , .",
"These procedures make the strong assumption that all of the provided transformations will preserve class labels, or use bespoke models over restricted sets of operations .",
"Another line of recent work has showed that augmentation via small adversarial linear perturbations can act as a regularizer , .",
"While complimentary, this work does not consider taking advantage of non-local transformations derived from user knowledge of task or domain invariances.",
"Finally, generative adversarial networks (GANs) have recently made great progress in learning complete data generation models from unlabeled data.",
"These can be used to augment labeled training sets as well.",
"Class-conditional GANs , generate artificial data points but require large sets of labeled training data to learn from.",
"Standard unsupervised GANs can be used to generate additional out-of-class data points that can then augment labeled training sets , .",
"We compare our proposed approach with these methods empirically in Section ."
],
[
"Experiments",
"We experimentally validate the proposed framework by learning augmentation models for several benchmark and real-world data sets, exploring both image recognition and natural language understanding tasks.",
"Our focus is on the performance of end classification models trained on labeled datasets augmented with our approach and others used in practice.",
"We also examine robustness to user misspecification of TFs, and sensitivity to core hyperparameters."
],
[
"Benchmark Image Datasets",
"We ran experiments on the MNIST and CIFAR-10 datasets, using only a subset of the class labels to train the end classification models and treating the rest as unlabeled data.",
"We used a generic set of TFs for both MNIST and CIFAR-10: small rotations, shears, central swirls, and elastic deformations.",
"We also used morphologic operations for MNIST, and adjustments to hue, saturation, contrast, and brightness for CIFAR-10.",
"We applied our approach to the Employment relation extraction subtask from the NIST Automatic Content Extraction (ACE) corpus , where the goal is to identify mentions of employer-employee relations in news articles.",
"Given the standard class imbalance in information extraction tasks like this, we used data augmentation to oversample the minority positive class.",
"The flexibility of our TF representation allowed us to take a straightforward but novel approach to data augmentation in this setting.",
"We constructed a trigram language model using the ACE corpus and Reuters Corpus Volume I from which we can sample a word conditioned on the preceding words.",
"We then used this model as the basis for a set of TFs that select words to swap based on the part-of-speech tag and location relative to entities of interest (see Appendix REF for details).",
"To demonstrate the effectiveness of our approach on real-world applications, we also considered the task of classifying benign versus malignant tumors from images in the Digital Database for Screening Mammography (DDSM) dataset , , , which is a class-balanced dataset consisting of 1506 labeled mammograms.",
"In collaboration with domain experts in radiology, we constructed two basic TF sets.",
"The first set consisted of standard image transformation operations subselected so as not to break class-invariance in the mammography setting.",
"For example, brightness operations were excluded for this reason.",
"The second set consisted of both the first set as well as several novel segmentation-based transplantation TFs.",
"Each of these TFs utilized the output of an unsupervised segmentation algorithm to isolate the tumor mass, perform a transformation operation such as rotation or shifting, and then stitch it into a randomly-sampled benign tissue image.",
"See Fig.",
"REF (right panel) for an illustrative example, and Appendix REF for further details."
],
[
"End Classifier Performance",
"We evaluated our approach by using it to augment labeled training sets for the tasks mentioned above, and show that we achieve strong gains over heuristic baselines.",
"In particular, for a given set of TFs, we evaluate the performance of mean field (MF) and LSTM generators trained using our approach against two standard data augmentation techniques used in practice.",
"The first (Basic) consists of applying random crops to images, or performing simple minority class duplication for the ACE relation extraction task.",
"The second (Heur.)",
"is the standard heuristic approach of applying random compositions of the given set of transformation operations, the most common technique used in practice , , .",
"For both our approaches (MF and LSTM) and Heur., we additionally use the same random cropping technique as in the Basic approach.",
"We present these results in Table REF , where we report test set accuracy (or F1 score for ACE), and use a random subsample of the available labeled training data.",
"Additionally, we include an extra row for the DDSM task highlighting the impact of adding domain-specific (DS) TFs – the segmentation-based operations described above – on performance.",
"In Table REF we additionally compare to two related generative-adversarial methods, the Categorical GAN (CatGAN) , and the semi-supervised GAN (SS-GAN) from .",
"Both of these methods use GAN-based architectures trained on unlabeled data to generate new out-of-class data points with which to augment a labeled training set.",
"Following their protocol for CIFAR-10, we train our generator on the full set of unlabeled data, and our end discriminator on ten disjoint random folds of the labeled training set not including the validation set (i.e.",
"$n=4000$ each), averaging the results.",
"Table: Reported end model accuracies, averaged across 10% subsample folds, on CIFAR-10 for comparable GAN methods.In all settings, we train our TF sequence generator on the full set of unlabeled data.",
"We select a fixed sequence length for each task via an initial calibration experiment (Fig.",
"REF ).",
"We use $L=5$ for ACE, $L=7$ for DDSM + DS, and $L=10$ for all other tasks.",
"We note that our findings here mirrored those in the literature, namely that compositions of multiple TFs lead to higher end model accuracies.",
"We selected hyperparameters of the generator via performance on a validation set.",
"We then used the trained generator to transform the entire training set at each epoch of end classification model training.",
"For MNIST and DDSM we use a four-layer all-convolutional CNN, for CIFAR10 we use a 56-layer ResNet , and for ACE we use a bi-directional LSTM.",
"Additionally, we incorporate a basic transformation regularization term as in (see Appendix REF ), and train for the last ten epochs without applying any transformations as in .",
"In all cases, we use hyperparameters as reported in the literature.",
"For further details of generator and end model training see the Appendix REF .",
"We see that across the applications studied, our approach outperforms the heuristic data augmentation approach most commonly used in practice.",
"Furthermore, the LSTM generator outperforms the simple mean field one in most settings, indicating the value of modeling sequential structure in data augmentation.",
"In particular, we realize significant gains over standard heuristic data augmentation on CIFAR-10, where we are competitive with comparable semi-supervised GAN approaches, but with significantly smaller variance.",
"We also train the same CIFAR-10 end model using the full labeled training dataset, and again see strong relative gains (2.1 pts.",
"in accuracy over heuristic), coming within 2.1 points of the current state-of-the-art using our much simpler end model.",
"On the ACE and DDSM tasks, we also achieve strong performance gains, showing the ability of our method to productively incorporate more complex transformation operations from domain expert users.",
"In particular, in DDSM we observe that the addition of the segmentation-based TFs causes the heuristic augmentation approach to perform significantly worse, due to a large number of new failure modes resulting from combinations of the segmentation-based TFs – which use gradient-based blending – and the standard TFs such as zoom and rotate.",
"In contrast, our LSTM model learns to avoid these destructive subsequences and achieves the highest score, resulting in a 9.0 point boost over the comparable heuristic approach."
],
[
"Robustness to TF Misspecification",
"One of the high-level goals of our approach is to enable an easier interface for users by not requiring that the TFs they specify be completely class-preserving.",
"The lack of any assumption of well-specified transformation operations in our approach, and the strong empirical performance realized, is evidence of this robustness.",
"To additionally illustrate the robustness of our approach to misspecified TFs, we train a mean field generator on MNIST using the standard TF set, but with two TFs (shear operations) parameterized so as to map almost all images to the null class.",
"We see in Fig.",
"REF that the generator learns to avoid applying the misspecified TFs (red lines) almost entirely."
],
[
"Conclusion and Future Work",
"We presented a method for learning how to parameterize and compose user-provided black-box transformation operations used for data augmentation.",
"Our approach is able to model arbitrary TFs, allowing practitioners to leverage domain knowledge in a flexible and simple manner.",
"By training a generative sequence model over the specified transformation functions using reinforcement learning in a GAN-like framework, we are able to generate realistic transformed data points which are useful for data augmentation.",
"We demonstrated that our method yields strong gains over standard heuristic approaches to data augmentation for a range of applications, modalities, and complex domain-specific transformation functions.",
"There are many possible future directions of research for learning data augmentation strategies in the proposed model, such as conditioning the generator's stochastic policy on a featurized version of the data point being transformed, and generating TF sequences of dynamic length.",
"More broadly, we are excited about further formalizing data augmentation as a novel form of weak supervision, allowing users to directly encode domain knowledge about invariants into machine learning models."
],
[
"Acknowledgements",
"We would like to thank Daniel Selsam, Ioannis Mitliagkas, Christopher De Sa, William Hamilton, and Daniel Rubin for valuable feedback and conversations.",
"We gratefully acknowledge the support of the Defense Advanced Research Projects Agency (DARPA) SIMPLEX program under No.",
"N66001-15-C-4043, the DARPA D3M program under No.",
"FA8750-17-2-0095, DARPA programs No.",
"FA8750-12-2-0335 and FA8750-13-2-0039, DOE 108845, National Institute of Health (NIH) U54EB020405, the Office of Naval Research (ONR) under awards No.",
"N000141210041 and No.",
"N000141310129, the Moore Foundation, the Okawa Research Grant, American Family Insurance, Accenture, Toshiba, and Intel.",
"This research was also supported in part by affiliate members and other supporters of the Stanford DAWN project: Intel, Microsoft, Teradata, and VMware.",
"This material is based on research sponsored by DARPA under agreement number FA8750-17-2-0095.",
"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.",
"Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, policies, or endorsements, either expressed or implied, of DARPA, AFRL, NSF, NIH, ONR, or the U.S. Government."
],
[
"Review of Data Augmentation Use in State-of-the-Art",
"To underscore both the omnipresence and diversity of heuristic data augmentation in practice, we compiled a list of the top ten models for the well documented CIFAR-10 and CIFAR-100 tasks (Table REF ).",
"We see that in 10 out of 10 of the top CIFAR-10 results and 9 out of 10 of the top CIFAR-100 results use data augmentation, for average boosts (when reported) of 3.71 and 13.39 points in accuracy, respectively.",
"Moreover, we see that while some sets of papers inherit a simple data augmentation strategy from prior work (in particular, all the recent ResNet variants), there are still a large variety of approaches.",
"And in general, the particular choice of data augmentation strategy is widely reported to have large effects on performance.",
"Note that the below table is compiled from a well-known online compendium and from the latest CVPR best paper (indicated by a *) which achieves new state-of-the-art results.",
"We compile it for illustrative purposes and it is not necessarily comprehensive.",
"Also note that we selected CIFAR-10/100 both as a representative and well-studied task, but also due to the availability of published results.",
"For competitions such as ImageNet, although data augmentation is widely reported to be critical, many top results are reported very opaquely, with little described about implementation details such as data augmentation.",
"Table: Current state-of-the-art image classification models as ranked by reported performance on the CIFAR-10 and CIFAR-100 tasks, and their error with (Err.",
"w/ DA) and without (Err.",
"w/o DA) data augmentation.",
"We include both scores and particular data augmentation techniques when reported, although the latter is rarely reported with great precision."
],
[
"Simple Synthetic Setup",
"As both a diagnostic tool and as a simple way to probe the properties of our approach, we construct a simple synthetic dataset consisting of points in two dimensions, uniformly selected in a ball of radius $r=1$ around the origin.",
"We then consider various displacement vectors as our TFs.",
"We consider the same two generative models as in our main experiments – mean field and LSTM – and use either a basic fully-connected two-layer neural network or an oracle discriminator $f(x) = 1\\lbrace ||x|| < 1\\rbrace $ .",
"Figure: Original data points (blue) are transformed using sequences of vector displacement TFs (L=10L=10) drawn from G θ G_\\theta , producing augmented data points (red).",
"G θ G_\\theta is either a mean field model or an LSTM, trained with an orcale discriminator D ∅ D^\\emptyset for 15 epochs.In this setting, we define $y_\\emptyset (x) = 1\\lbrace ||x|| \\ge 1 \\rbrace $ , and consider two different TF sets: Good vs. Bad TFs: In a first toy scenario we consider TFs which are vector displacements of random direction, with magnitude drawn from one of two distributions, $\\mathcal {N}(\\mu _1, \\sigma _1)$ or $\\mathcal {N}(\\mu _2, \\sigma _2)$ , where $\\mu _1 > 1 > \\mu _2$ .",
"In other words, the model should learn not to select certain individual TFs.",
"Lossy TFs: We consider a second toy setting where random-direction displacement TFs have magnitude drawn uniformly from $\\mathcal {N}(\\mu , \\sigma )$ , $\\mu < 1$ ; however the magnitude of each TF decays exponentially with the distance a point is outside of the unit ball.",
"This simulates the setting where TFs are irrecoverably lossy when applied in certain sequences.",
"As expected, we see that while the mean field model is able to model the first setting (Figure REF ), it fails to adequately represent the second one (Figure REF ), whereas the RNN model is able to (Figure REF ).",
"Robustness to Transformed Test Data We run a simple experiment to test the robustness of the trained end classification models to the individual TFs in the TF sets used.",
"Specifically, on CIFAR-10 we create ten transformed copies a 10% subsample of the test data by transforming with a single TF, and then test the end model on this set.",
"We compare our approach with heuristic random augmentation and no data augmentation of the model during training, and consider rotations, zooms, shears, and hue shifts.",
"Results are presented in Figure REF .",
"Figure: Accuracy scores on random 10% subsamples of test data (dotted lines) and on versions augmented with a single transformation (vertical bars) with parameters drawn uniformly at random.We can consider evaluating the results in terms of absolute robustness – i.e.",
"model accuracy – and relative robustness, i.e.",
"the change in model score when applied to the transformed test set.",
"Roughly we see that our approach is most absolutely robust.",
"Random appears to be most relatively robust, in particular on larger transformations, which we hypothesize our approach mostly learned to avoid applying during training.",
"Reinforcement Learning Formulation Details Variance reduction methods In Section , the vanilla policy gradient of our objective was given as $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L R(s_t) \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1})\\right]$ Noting that actions $\\tau _t$ only impact future outcomes, following standard practice, we apply only future rewards in order to reduce variance: $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1}) \\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(s_{t^{\\prime }})\\right]$ where $\\gamma \\in [0,1]$ is a discounting factor.",
"Additionally, we use a baseline term $b_t$ to reduce variance when estimating $\\nabla _\\theta U(\\theta )$ : $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1}) \\left(\\left(\\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(\\hat{s}_{t^{\\prime }})\\right) - b_t\\right)\\right]$ Policy gradient estimation Using a batch of $n$ data points and $m$ sampled action sequences per data point – given by the state representations $\\lbrace \\hat{s}^{(i,j)}\\rbrace $ and action sequences $\\lbrace \\tau ^{(i,j)}\\rbrace $ – the gradient estimate is computed as: $\\nabla _\\theta \\hat{U}(\\theta ) &= \\frac{1}{nm}\\sum _{i=1}^n\\sum _{j=1}^m \\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau ^{(i,j)}_t\\ |\\ \\hat{s}^{(i,j)}_{t-1}) \\left(\\left(\\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(\\hat{s}^{(i,j)}_{t^{\\prime }})\\right) - \\hat{b}_t\\right)\\right]$ where the baseline term $\\hat{b}_t$ is also computed using the batch: $\\hat{b}_t = \\frac{1}{nm}\\sum _{i=1}^n\\sum _{j=1}^m \\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R\\left(\\hat{s}^{(i,j)}_{t^{\\prime }}\\right)$ In our experiments, we fixed $n=32$ and $m=5$ .",
"Experimental Details Benchmark Image Datasets We use the MNIST dataset with 5000 training data points used as a validation set.",
"We use the following TFs: Rotation (2.5, -2.5, 5, -5, 10, and -10 degrees) Zoom (0.9x, 1.1x) Shear (0.1, -0.1, 0.2, -0.2, 0.4, and -0.4 degrees) Swirl (0.1, -0.1, 0.2, -0.2, 0.4, and -0.4 degrees) Random elastic deformations ($\\alpha $ = 1.0, 1.25, and 1.5) Erosion Dilation For the CIFAR-10 dataset, we use the following TFs: Rotation (2.5, -2.5, 5, -5 degrees) Zoom (0.9x, 1.1x, 0.75x, 1.25x) Shear (0.1, -0.1, 0.25, and -0.25 degrees) Swirl (0.1, -0.1, 0.25, -0.25 degrees) Hue Shift (by 0.1, -0.1, 0.25, and -0.25) Enhance contrast (by 0.75, 1.25, 0.5, and 1.5) Enhance brightness (by 0.75, 1.25, 0.5, and 1.5) Enhance color (by 0.75, 1.25, 0.5, and 1.5) Horizontal flip For both datasets, we also applied random padding (by 4 pixels on each side) followed by random crops back to the original dimensions during training.",
"We note that the choice of certain TFs to use in certain datasets was deliberate – for example, we would not expect horizontal flips or hue shifts to be appropriate in MNIST, or erosion and dilation to be useful in CIFAR-10.",
"However, the particular choice of parameterizations was mainly due to disjoint implementations of the two experiments.",
"For further details of the TF implementations used, see our code, which will be open-sourced after the review process.",
"Benchmark Text Dataset The ACE corpus consists of news articles and broadcast transcripts, all of which are pretagged with entity mentions.",
"The objective of the Employment relation subtask is to extract Person-Organization entity pairs which are implied to have an affiliation in the text.",
"We pose this as a binary classification problem by first identifying relation candidates: any pair of Person-Organization entities which occur in the same sentence.",
"As noted in Section , there are far more true negative candidates than true positive candidates.",
"The end model is trained to classify relation candidates as either true or false relations based on the raw text of the sentence in which they occur.",
"The language model described in Section was constructed by recording counts of unigrams following each unique trigram, bigram, and unigram in the corpus.",
"Laplace smoothing was applied to the counts, and basic filtering was applied to the $n$ -grams.",
"The sampler falls back to using bigrams then unigrams if the trigram preceding the word we want to swap was filtered out of the corpus.",
"We used the following TFs in all experiments: Replace a noun to the left of both entities Replace a noun between the two entities Replace a noun to the right of both entities Replace a verb to the left of both entities Replace a verb between the two entities Replace a verb to the right of both entities Replace an adjective to the left of both entities Replace an adjective between the two entities Replace an adjective to the right of both entities DDSM Mammography Task We use the following transformation functions: Rotate Image: Rotate the entire mammogram by a deterministic angle $\\theta $ .",
"Tumor geometry is fundamentally invariant to 2-D orientation.",
"TF run with $\\theta \\in [-5^o,-2.5^o,2.5^o,5^o]$ .",
"Zoom Image: Zoom in on the entire mammogram by a deterministic factor $\\gamma $ .",
"Tumor classification is insensitive to $\\gamma $ for $\\gamma $ close to one.",
"TF run with $\\gamma \\in \\lbrace 0.98,1.02\\rbrace $ Enhance Image Contrast: Enhance contrast values of grayscale image by a deterministic factor $\\gamma $ .",
"Tumor classification is insensitive to $\\gamma $ for $\\gamma $ close to one.",
"TF run with $\\gamma \\in \\lbrace 0.95,1.05\\rbrace $ Translate and Transplant Image: Extract pixels within mass segmentation.",
"Perform translation of a bounding box of side length $N$ pixels about mass center by a deterministic vector $\\hat{g}$ .",
"Transplant the translated bounding box containing the mass onto a randomly sampled normal tissue image using Poisson blending.",
"Retains information about the mass itself within the context of a different set of normal tissue background.",
"The bounding box also retains information about the tissue in the tumor near field.",
"TF run with $N = 10$ , $\\hat{g}\\in \\lbrace (-3,0),(3,0),(0,-3),(0,3),(0,0)\\rbrace $ .",
"Rotate and Transplant Image: Extract pixels within mass segmentation.",
"Perform rotation of a bounding box of side length $N$ pixels about mass center by a deterministic angle $\\theta $ .",
"Transplant the rotated bounding box containing the mass onto a randomly sampled normal tissue image using Poisson blending .",
"Retains information about the mass itself within the context of a different set of normal tissue background.",
"The bounding box also retains information about the tissue in the tumor near field.",
"TF run with $N = 10$ , $\\theta \\in \\lbrace -5^o,-2.5^o,2.5^o,5^o\\rbrace $ .",
"Note that for the Poisson blending TFs, it is important that the translation and rotation domains be specified such that excessive proximity to the boundary of the destination image does not introduce spurious gradient information into the blended image.",
"Details of Generative Adversarial Network Models All models, both for the generator training as described in this section, and the end classification model training described next, were implemented in Tensorflow https://www.tensorflow.org.",
"Discriminator For image tasks, the discriminator used in the training of the generator in our approach was the same model as in , an all-convolutional CNN with four convolutional layers and leaky ReLU activations.",
"For the text task, we used a unidirectional RNN with basic LSTM cells.",
"Mean Field Model The mean field model is represented simply as a length $K$ vector of unbounded variables, where $K$ is the number of TFs.",
"Applying the softmax function to this vector yields the TF sampling distribution.",
"LSTM In the LSTM generative model, we create a length-$L$ RNN with basic LSTM cells.",
"The input and output size for each cell is $K$ .",
"We feed an indicator vector of the last TF used as the input to each cell, except for the first cell, which recieves a randomly initialized variable vector as its input.",
"The output of each cell is shifted and scaled to range from $-r$ to $r$ , where $r$ is a hyperparameter.",
"Applying the softmax function to the shifted and scaled output yields the stochastic policy: a sampling distribution over the $K$ TFs.",
"In our experiments, we fix $r=2$ to avoid overfitting.",
"Training and Model Selection Procedure We trained the TF sequence generators jointly with the discriminator using SGD with momentum (fixed at 0.9), in an adversarial manner as described in .",
"We performed an initial search over the TF sequence length $L$ as described in Section , and then held it fixed at $L=10$ for all subsequent experiments.",
"We searched over a range of values for learning rates for the generator and discriminator, as well as for hyperparameters specific to our formulation, such as the diversity objective term coefficient $\\alpha $ , the diversity objective term distance metric $d$ (choosing between distance in the raw input space or in the feature space learned by the final pre-softmax layer of the discriminator), and whether or not to split the data used for the discriminator and generator training steps.",
"We selected final generators to use for test set evaluation by using them to augment training data for end classification models then evaluated on the validation set.",
"In addition, we filtered some generators out based on their loss (according to the discriminator $D_\\phi ^\\emptyset $ ) as compared to that of random TF sequences.",
"Diversity Objective For the diversity objective term, we tried both distance in the raw pixel-level input space and distance in the feature space learned by the final pre-softmax layer of the discriminator as choices for distance metric $d$ .",
"During training of the generators, we measured the average pairwise generalized Jaccard distance.",
"For CIFAR-10, as an example, the final batches had an average distance of 0.52 compared to 0.86 for randomly generated sequences, which implied diversity in the learned sequences.",
"We also computed the ratio of unique TF n-grams to total possible n-grams, and measured 0.37 compared to 0.98 for random sequences as expected.",
"End Classification Models MNIST and DDSM For MNIST and DDSM we use a similar architecture to the discriminator in the previous section, adapted for the multinomial classification setting: a four-layer all-convolution CNN with leaky ReLUs and batch norm.",
"CIFAR-10 Given the flexibility of end classifier choice with our approach, for CIFAR-10 we used a more computationally expensive but standard model: a 56-layer ResNet as described in .",
"We used batch norm, regularization, learning rate schedule, and all other hyperparameters as reported in .",
"ACE The end model used for the ACE task was a bidirectional recurrent neural network using LSTM cells with attention mechanisms.",
"The maximum sentence length and attention window length were both 50.",
"Word embeddings were initialized from pretrained vectors via , and updated during training.",
"Hyperparameters were selected via a cursory grid search, and fixed for experiments.",
"End Model Training Basic Training Procedure with Data Augmentation We trained all end models using minibatch stochastic gradient descent with momentum (fixed at 0.9), using a fixed learning rate schedule set once for each model and then fixed for all experiments.",
"To perform data augmentation, during end classifier training we transformed some portion of each minibatch, $p_{\\text{transform}}$ .",
"For all experiments we used $p_{\\text{transform}}=1.0$ .",
"Additionally, for the last ten epochs of training, we switched to $p_{\\text{transform}}=0.0$ following reported practice in the literature .",
"For all other hyperparameters we used default values as reported in the respective literature, held fixed at these values for all experiments.",
"Transformation Regularization Term We additionally apply a transformation regularization (TR) term to the transformed data points for all image experiments by adding a term to the loss function which is the distance between the pre-softmax layer logits for each data point and its transformed copy, similar to the term in .",
"Given the fact that we are producing these transformed data points anyway, incorporating this term introduces little additional overhead.",
"Table: A simple study of the effect of adding a transformation regularization (TR) term to the objective function, evaluated on a labeled validation set.",
"We see that adding the term improves performance for both heuristic (random) TF sequences and for TF sequences generated by the trained LSTM model, and that there is a larger positive effect for the latter.In an early calibration experiment (Table REF ), we found that introducing this regularization term (using a coefficient of $0.1$ and unlabeled data batch size of $20\\%$ that of the labeled data batch size) yielded improvements in performance to the end model with both learned transformation sequences and random sequences.",
"However, we see that the positive effect is much larger for the trained LSTM sequences (1.2 points versus 0.1 points in accuracy).",
"We chose to subsequently keep this term fixed, viewing further calibration and exploration of this term as largely orthogonal to our central experimental questions.",
"However, we believe that this is an extremely interesting and empirically proimising area for future study, especially given the indication that this term may be more effective when used in conjunction with a trained augmentation model such as ours."
],
[
"Variance reduction methods",
"In Section , the vanilla policy gradient of our objective was given as $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L R(s_t) \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1})\\right]$ Noting that actions $\\tau _t$ only impact future outcomes, following standard practice, we apply only future rewards in order to reduce variance: $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1}) \\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(s_{t^{\\prime }})\\right]$ where $\\gamma \\in [0,1]$ is a discounting factor.",
"Additionally, we use a baseline term $b_t$ to reduce variance when estimating $\\nabla _\\theta U(\\theta )$ : $\\nabla _\\theta U(\\theta ) &= \\mathbb {E}_{\\tau \\sim G_{\\theta }}\\mathbb {E}_{x\\sim \\mathcal {U}}\\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau _t\\ |\\ \\hat{s}_{t-1}) \\left(\\left(\\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(\\hat{s}_{t^{\\prime }})\\right) - b_t\\right)\\right]$"
],
[
"Policy gradient estimation",
"Using a batch of $n$ data points and $m$ sampled action sequences per data point – given by the state representations $\\lbrace \\hat{s}^{(i,j)}\\rbrace $ and action sequences $\\lbrace \\tau ^{(i,j)}\\rbrace $ – the gradient estimate is computed as: $\\nabla _\\theta \\hat{U}(\\theta ) &= \\frac{1}{nm}\\sum _{i=1}^n\\sum _{j=1}^m \\left[\\sum _{t=1}^L \\nabla _\\theta \\log \\pi _\\theta (\\tau ^{(i,j)}_t\\ |\\ \\hat{s}^{(i,j)}_{t-1}) \\left(\\left(\\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R(\\hat{s}^{(i,j)}_{t^{\\prime }})\\right) - \\hat{b}_t\\right)\\right]$ where the baseline term $\\hat{b}_t$ is also computed using the batch: $\\hat{b}_t = \\frac{1}{nm}\\sum _{i=1}^n\\sum _{j=1}^m \\sum _{t^{\\prime }=t}^L \\gamma ^{t^{\\prime }-t} R\\left(\\hat{s}^{(i,j)}_{t^{\\prime }}\\right)$ In our experiments, we fixed $n=32$ and $m=5$ ."
],
[
"Benchmark Image Datasets",
"We use the MNIST dataset with 5000 training data points used as a validation set.",
"We use the following TFs: Rotation (2.5, -2.5, 5, -5, 10, and -10 degrees) Zoom (0.9x, 1.1x) Shear (0.1, -0.1, 0.2, -0.2, 0.4, and -0.4 degrees) Swirl (0.1, -0.1, 0.2, -0.2, 0.4, and -0.4 degrees) Random elastic deformations ($\\alpha $ = 1.0, 1.25, and 1.5) Erosion Dilation For the CIFAR-10 dataset, we use the following TFs: Rotation (2.5, -2.5, 5, -5 degrees) Zoom (0.9x, 1.1x, 0.75x, 1.25x) Shear (0.1, -0.1, 0.25, and -0.25 degrees) Swirl (0.1, -0.1, 0.25, -0.25 degrees) Hue Shift (by 0.1, -0.1, 0.25, and -0.25) Enhance contrast (by 0.75, 1.25, 0.5, and 1.5) Enhance brightness (by 0.75, 1.25, 0.5, and 1.5) Enhance color (by 0.75, 1.25, 0.5, and 1.5) Horizontal flip For both datasets, we also applied random padding (by 4 pixels on each side) followed by random crops back to the original dimensions during training.",
"We note that the choice of certain TFs to use in certain datasets was deliberate – for example, we would not expect horizontal flips or hue shifts to be appropriate in MNIST, or erosion and dilation to be useful in CIFAR-10.",
"However, the particular choice of parameterizations was mainly due to disjoint implementations of the two experiments.",
"For further details of the TF implementations used, see our code, which will be open-sourced after the review process."
],
[
"Benchmark Text Dataset",
"The ACE corpus consists of news articles and broadcast transcripts, all of which are pretagged with entity mentions.",
"The objective of the Employment relation subtask is to extract Person-Organization entity pairs which are implied to have an affiliation in the text.",
"We pose this as a binary classification problem by first identifying relation candidates: any pair of Person-Organization entities which occur in the same sentence.",
"As noted in Section , there are far more true negative candidates than true positive candidates.",
"The end model is trained to classify relation candidates as either true or false relations based on the raw text of the sentence in which they occur.",
"The language model described in Section was constructed by recording counts of unigrams following each unique trigram, bigram, and unigram in the corpus.",
"Laplace smoothing was applied to the counts, and basic filtering was applied to the $n$ -grams.",
"The sampler falls back to using bigrams then unigrams if the trigram preceding the word we want to swap was filtered out of the corpus.",
"We used the following TFs in all experiments: Replace a noun to the left of both entities Replace a noun between the two entities Replace a noun to the right of both entities Replace a verb to the left of both entities Replace a verb between the two entities Replace a verb to the right of both entities Replace an adjective to the left of both entities Replace an adjective between the two entities Replace an adjective to the right of both entities"
],
[
"DDSM Mammography Task",
"We use the following transformation functions: Rotate Image: Rotate the entire mammogram by a deterministic angle $\\theta $ .",
"Tumor geometry is fundamentally invariant to 2-D orientation.",
"TF run with $\\theta \\in [-5^o,-2.5^o,2.5^o,5^o]$ .",
"Zoom Image: Zoom in on the entire mammogram by a deterministic factor $\\gamma $ .",
"Tumor classification is insensitive to $\\gamma $ for $\\gamma $ close to one.",
"TF run with $\\gamma \\in \\lbrace 0.98,1.02\\rbrace $ Enhance Image Contrast: Enhance contrast values of grayscale image by a deterministic factor $\\gamma $ .",
"Tumor classification is insensitive to $\\gamma $ for $\\gamma $ close to one.",
"TF run with $\\gamma \\in \\lbrace 0.95,1.05\\rbrace $ Translate and Transplant Image: Extract pixels within mass segmentation.",
"Perform translation of a bounding box of side length $N$ pixels about mass center by a deterministic vector $\\hat{g}$ .",
"Transplant the translated bounding box containing the mass onto a randomly sampled normal tissue image using Poisson blending.",
"Retains information about the mass itself within the context of a different set of normal tissue background.",
"The bounding box also retains information about the tissue in the tumor near field.",
"TF run with $N = 10$ , $\\hat{g}\\in \\lbrace (-3,0),(3,0),(0,-3),(0,3),(0,0)\\rbrace $ .",
"Rotate and Transplant Image: Extract pixels within mass segmentation.",
"Perform rotation of a bounding box of side length $N$ pixels about mass center by a deterministic angle $\\theta $ .",
"Transplant the rotated bounding box containing the mass onto a randomly sampled normal tissue image using Poisson blending .",
"Retains information about the mass itself within the context of a different set of normal tissue background.",
"The bounding box also retains information about the tissue in the tumor near field.",
"TF run with $N = 10$ , $\\theta \\in \\lbrace -5^o,-2.5^o,2.5^o,5^o\\rbrace $ .",
"Note that for the Poisson blending TFs, it is important that the translation and rotation domains be specified such that excessive proximity to the boundary of the destination image does not introduce spurious gradient information into the blended image."
],
[
"Details of Generative Adversarial Network Models",
"All models, both for the generator training as described in this section, and the end classification model training described next, were implemented in Tensorflow https://www.tensorflow.org."
],
[
"Discriminator",
"For image tasks, the discriminator used in the training of the generator in our approach was the same model as in , an all-convolutional CNN with four convolutional layers and leaky ReLU activations.",
"For the text task, we used a unidirectional RNN with basic LSTM cells.",
"The mean field model is represented simply as a length $K$ vector of unbounded variables, where $K$ is the number of TFs.",
"Applying the softmax function to this vector yields the TF sampling distribution.",
"In the LSTM generative model, we create a length-$L$ RNN with basic LSTM cells.",
"The input and output size for each cell is $K$ .",
"We feed an indicator vector of the last TF used as the input to each cell, except for the first cell, which recieves a randomly initialized variable vector as its input.",
"The output of each cell is shifted and scaled to range from $-r$ to $r$ , where $r$ is a hyperparameter.",
"Applying the softmax function to the shifted and scaled output yields the stochastic policy: a sampling distribution over the $K$ TFs.",
"In our experiments, we fix $r=2$ to avoid overfitting.",
"We trained the TF sequence generators jointly with the discriminator using SGD with momentum (fixed at 0.9), in an adversarial manner as described in .",
"We performed an initial search over the TF sequence length $L$ as described in Section , and then held it fixed at $L=10$ for all subsequent experiments.",
"We searched over a range of values for learning rates for the generator and discriminator, as well as for hyperparameters specific to our formulation, such as the diversity objective term coefficient $\\alpha $ , the diversity objective term distance metric $d$ (choosing between distance in the raw input space or in the feature space learned by the final pre-softmax layer of the discriminator), and whether or not to split the data used for the discriminator and generator training steps.",
"We selected final generators to use for test set evaluation by using them to augment training data for end classification models then evaluated on the validation set.",
"In addition, we filtered some generators out based on their loss (according to the discriminator $D_\\phi ^\\emptyset $ ) as compared to that of random TF sequences.",
"For the diversity objective term, we tried both distance in the raw pixel-level input space and distance in the feature space learned by the final pre-softmax layer of the discriminator as choices for distance metric $d$ .",
"During training of the generators, we measured the average pairwise generalized Jaccard distance.",
"For CIFAR-10, as an example, the final batches had an average distance of 0.52 compared to 0.86 for randomly generated sequences, which implied diversity in the learned sequences.",
"We also computed the ratio of unique TF n-grams to total possible n-grams, and measured 0.37 compared to 0.98 for random sequences as expected.",
"For MNIST and DDSM we use a similar architecture to the discriminator in the previous section, adapted for the multinomial classification setting: a four-layer all-convolution CNN with leaky ReLUs and batch norm.",
"Given the flexibility of end classifier choice with our approach, for CIFAR-10 we used a more computationally expensive but standard model: a 56-layer ResNet as described in .",
"We used batch norm, regularization, learning rate schedule, and all other hyperparameters as reported in .",
"The end model used for the ACE task was a bidirectional recurrent neural network using LSTM cells with attention mechanisms.",
"The maximum sentence length and attention window length were both 50.",
"Word embeddings were initialized from pretrained vectors via , and updated during training.",
"Hyperparameters were selected via a cursory grid search, and fixed for experiments.",
"We trained all end models using minibatch stochastic gradient descent with momentum (fixed at 0.9), using a fixed learning rate schedule set once for each model and then fixed for all experiments.",
"To perform data augmentation, during end classifier training we transformed some portion of each minibatch, $p_{\\text{transform}}$ .",
"For all experiments we used $p_{\\text{transform}}=1.0$ .",
"Additionally, for the last ten epochs of training, we switched to $p_{\\text{transform}}=0.0$ following reported practice in the literature .",
"For all other hyperparameters we used default values as reported in the respective literature, held fixed at these values for all experiments.",
"We additionally apply a transformation regularization (TR) term to the transformed data points for all image experiments by adding a term to the loss function which is the distance between the pre-softmax layer logits for each data point and its transformed copy, similar to the term in .",
"Given the fact that we are producing these transformed data points anyway, incorporating this term introduces little additional overhead.",
"Table: A simple study of the effect of adding a transformation regularization (TR) term to the objective function, evaluated on a labeled validation set.",
"We see that adding the term improves performance for both heuristic (random) TF sequences and for TF sequences generated by the trained LSTM model, and that there is a larger positive effect for the latter.In an early calibration experiment (Table REF ), we found that introducing this regularization term (using a coefficient of $0.1$ and unlabeled data batch size of $20\\%$ that of the labeled data batch size) yielded improvements in performance to the end model with both learned transformation sequences and random sequences.",
"However, we see that the positive effect is much larger for the trained LSTM sequences (1.2 points versus 0.1 points in accuracy).",
"We chose to subsequently keep this term fixed, viewing further calibration and exploration of this term as largely orthogonal to our central experimental questions.",
"However, we believe that this is an extremely interesting and empirically proimising area for future study, especially given the indication that this term may be more effective when used in conjunction with a trained augmentation model such as ours."
]
] | 1709.01643 |
[
[
"Model Checking for Fragments of Halpern and Shoham's Interval Temporal\n Logic Based on Track Representatives"
],
[
"Abstract Model checking allows one to automatically verify a specification of the expected properties of a system against a formal model of its behaviour (generally, a Kripke structure).",
"Point-based temporal logics, such as LTL, CTL, and CTL*, that describe how the system evolves state-by-state, are commonly used as specification languages.",
"They proved themselves quite successful in a variety of application domains.",
"However, properties constraining the temporal ordering of temporally extended events as well as properties involving temporal aggregations, which are inherently interval-based, can not be properly dealt with by them.",
"Interval temporal logics (ITLs), that take intervals as their primitive temporal entities, turn out to be well-suited for the specification and verification of interval properties of computations (we interpret all the tracks of a Kripke structure as computation intervals).",
"We study the model checking problem for some fragments of Halpern and Shoham's modal logic of time intervals (HS).",
"HS features one modality for each possible ordering relation between pairs of intervals (the so-called Allen's relations).",
"First, we describe an EXPSPACE model checking algorithm for the HS fragment of Allen's relations meets, met-by, starts, started-by, and finishes, which exploits the possibility of finding, for each track (of unbounded length), an equivalent bounded-length track representative.",
"While checking a property, it only needs to consider tracks whose length does not exceed the given bound.",
"We prove the model checking problem for such a fragment to be PSPACE-hard.",
"Finally, we identify other well-behaved HS fragments which are expressive enough to capture meaningful interval properties of systems, such as mutual exclusion, state reachability, and non-starvation, and whose complexity is less than or equal to that of LTL."
],
[
"Introduction",
"One of the most notable techniques for system verification is model checking, which allows one to verify the desired properties of a system against a model of its behaviour [9].",
"Properties are usually formalized by means of temporal logics, such as LTL and CTL, and systems are represented as labelled state-transition graphs (Kripke structures).",
"Model checking algorithms perform, in a fully automatic way, an (implicit or explicit) exhaustive enumeration of all the states reachable by the system, and either terminate positively, proving that all properties are met, or produce a counterexample, witnessing that some behavior falsifies a property.",
"The model checking problem has systematically been investigated in the context of classical, point-based temporal logics, like LTL, CTL, and CTL$^*$ , which predicate over single computation points/states, while it is still largely unexplored in the interval logic setting.",
"Interval temporal logics (ITLs) have been proposed as a formalism for temporal representation and reasoning more expressive than standard point-based ones [13], [34], [35].",
"They take intervals, instead of points, as their primitive temporal entities.",
"Such a choice gives them the ability to cope with advanced temporal properties, such as actions with duration, accomplishments, and temporal aggregations, which can not be properly dealt with by standard, point-based temporal logics.",
"Expressiveness of ITLs makes them well suited for many applications in a variety of computer science fields, including artificial intelligence (reasoning about action and change, qualitative reasoning, planning, configuration and multi-agent systems, and computational linguistics), theoretical computer science (formal verification, synthesis), and databases (temporal and spatio-temporal databases) [2], [10], [18], [30], [8], [27], [26], [19], [11].",
"However, this great expressiveness is a double-edged sword: in most cases the satisfiability problem for ITLs turns out to be undecidable, and, in the few cases of decidable ITLs, the standard proof machinery, like Rabin's theorem, is usually not applicable.",
"The most prominent ITL is Halpern and Shoham's modal logic of time intervals (HS, for short) [13].",
"HS features one modality for each of the 13 possible ordering relations between pairs of intervals (the so-called Allen's relations [1]), apart from the equality relation.",
"In [13], it has been shown that the satisfiability problem for HS interpreted over all relevant (classes of) linear orders is undecidable.",
"Since then, a lot of work has been done on the satisfiability problem for HS fragments, which has shown that undecidability prevails over them (see [4] for an up-to-date account of undecidable fragments).",
"However, meaningful exceptions exist, including the interval logic of temporal neighbourhood $\\mathsf {A\\overline{A}}$ and the interval logic of sub-intervals $~\\cite {BGMS10,BGMS09,BMSS11,MPS10}.$ In this paper, we focus our attention on the model checking problem for HS, for which, as we said, little work has been done [24], [20], [15], [16], [17] (it is worth pointing out that, in contrast to the case of point-based, linear temporal logics, there is not an easy reduction from the model checking problem to validity/satisfiability for ITL).",
"In the classical formulation of the model checking problem [9], point-based temporal logics are used to analyze, for each path in a Kripke structure, how proposition letters labelling the states change from one state to the next one along the path.",
"In interval-based model checking, in order to check interval properties of computations, one needs to collect information about states into computation stretches.",
"This amounts to interpreting each finite path of a Kripke structure (a track) as an interval, and to suitably defining its labelling on the basis of the proposition letters that hold on the states composing it.",
"In [24], Montanari et al.",
"give a first characterization of the model checking problem for full HS, interpreted over finite Kripke structures (under the homogeneity assumption [31], according to which a proposition letter holds on an interval if and only if it holds on all its sub-intervals).",
"In that paper, the authors introduce the basic elements of the general picture, namely, the interpretation of HS formulas over (abstract) interval models, the mapping of finite Kripke structures into (abstract) interval models, the notion of track descriptor, and a small model theorem proving (with a non-elementary procedure) the decidability of the model checking problem for full HS against finite Kripke structures.",
"Many of these notions will be recalled in the following section.",
"In [20], Molinari et al.",
"work out the model checking problem for full HS in all its details, and prove that it is EXPSPACE-hard, if a succinct encoding of formulas is allowed, and PSPACE-hard otherwise.",
"In [15], [16], [17], Lomuscio and Michaliszyn address the model checking problem for some fragments of HS extended with epistemic modalities.",
"Their semantic assumptions differ from those made in [24], making it difficult to compare the outcomes of the two research directions.",
"In both cases, formulas of interval temporal logic are evaluated over finite paths/tracks obtained from the unravelling of a finite Kripke structure.",
"However, while in [24] a proposition letter holds over an interval (track) if and only if it holds over all its states (homogeneity assumption), in [15], [16] truth of proposition letters on a track/interval depends only on their values at its endpoints.",
"In [15], the authors focus their attention on the HS fragment $\\mathsf {BED}$ of Allen's relations started-by, finished-by, and contains (since modality $\\operatorname{\\langle D\\rangle }$ is definable in terms of modalities $\\operatorname{\\langle B\\rangle }$ and $\\operatorname{\\langle E\\rangle }$ , $\\mathsf {BED}$ is actually as expressive as $\\mathsf {BE}$ ), extended with epistemic modalities.",
"They consider a restricted form of model checking, which verifies the given specification against a single (finite) initial computation interval.",
"Their goal is indeed to reason about a given computation of a multi-agent system, rather than on all its admissible computations.",
"They prove that the considered model checking problem is PSPACE-complete; moreover, they show that the same problem restricted to the pure temporal fragment $\\mathsf {BED}$ , that is, the one obtained by removing epistemic modalities, is in PTIME.",
"These results do not come as a surprise as they trade expressiveness for efficiency: modalities $\\operatorname{\\langle B\\rangle }$ and $\\operatorname{\\langle E\\rangle }$ allow one to access only sub-intervals of the initial one, whose number is quadratic in the length (number of states) of the initial interval.",
"In [16], they show that the picture drastically changes with other fragments of HS, that allow one to access infinitely many tracks/intervals.",
"In particular, they prove that the model checking problem for the HS fragment $\\mathsf {A\\overline{B}L}$ of Allen's relations meets, starts, and before (since modality $\\operatorname{\\langle L\\rangle }$ is definable in terms of modality $\\operatorname{\\langle A\\rangle }$ , $\\mathsf {A\\overline{B}L}$ is actually as expressive as $\\mathsf {A\\overline{B}}$ ), extended with epistemic modalities, is decidable with a non-elementary upper bound.",
"Note that, thanks to modalities $\\operatorname{\\langle A\\rangle }$ and $\\operatorname{\\langle \\overline{B}\\rangle }$ , formulas of $\\mathsf {A\\overline{B}L}$ can possibly refer to infinitely many (future) tracks/intervals.",
"Finally, in [17], Lomuscio and Michaliszyn show how to use regular expressions in order to specify the way in which tracks/intervals of a Kripke structure get labelled.",
"Such an extension leads to a significant increase in expressiveness, as the labelling of an interval is no more determined by that of its endpoints, but it depends on the ordered sequence of states the interval consists of.",
"They also prove that there is not a corresponding increase in computational complexity, as the complexity bounds given in [15], [16] still hold with the new semantics: the model checking problem for $\\mathsf {BED}$ is still in PSPACE, and it is non-elementarily decidable for $\\mathsf {A\\overline{B}L}$ ."
],
[
"Main contributions",
"In this paper, we elaborate on the approach to ITL model checking outlined in [24] and we propose an original solution to the problem for some relevant HS fragments based on the notion of track representative.",
"We first prove that the model checking problem for two large HS fragments, namely, the fragment $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ (resp., $\\mathsf {A\\overline{A}E\\overline{B}\\overline{E}}$ ) of Allen's relations meets, met-by, started-by (resp., finished-by), starts and finishes, is in EXPSPACE.",
"Moreover, we show that it is PSPACE-hard (NEXP-hard, if a succinct encoding of formulas is used).",
"Then, we identify some well-behaved HS fragments, which are still expressive enough to capture meaningful interval properties of state-transition systems, such as mutual exclusion, state reachability, and non-starvation, whose model checking problem exhibits a considerably lower computational complexity, notably, $(i)$ the fragment $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ , whose model checking problem is PSPACE-complete, and $(ii)$ the fragment $\\mathsf {\\forall A\\overline{A}BE}$ , including formulas of $\\mathsf {A\\overline{A}BE}$ where only universal modalities are allowed and negation can be applied to propositional formulas only, whose model checking problem is coNP-complete.",
"Figure: Complexity of model checking for HS fragments.In Figure REF , we summarize known (white boxes) and new (grey boxes) results about complexity of model checking for HS fragments.",
"The main technical contributions of the paper can be summarized as follows.",
"Track descriptors.",
"We start with some background knowledge about HS and Kripke structures, and then we show how the latter can be mapped into interval-based structures, called abstract interval models, over which HS formulas are evaluated.",
"Each track in a Kripke structure is interpreted as an interval, which becomes an (atomic) object of the domain of an abstract interval model.",
"The labeling of an interval is defined on the basis of the states that compose it, according to the homogeneity assumption [31].",
"Then, we introduce track descriptors [24].",
"A track descriptor is a tree-like structure providing information about a possibly infinite set of tracks (the number of admissible track descriptors for a given Kripke structure is finite).",
"Being associated with the same descriptor is indeed a sufficient condition for two tracks to be indistinguishable with respect to satisfiability of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas, provided that the nesting depth of $\\operatorname{\\langle B\\rangle }$ modality is less than or equal to the depth of the descriptor itself.",
"Finally, we introduce the key notions of descriptor sequence for a track and cluster, and the relation of descriptor element indistinguishability, which allow us to determine when two prefixes of some track are associated with the same descriptor, avoiding the expensive operation of explicitly constructing track descriptors.",
"A small model theorem.",
"The main result of the paper is a small model theorem, showing that we can restrict the verification of an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula to a finite number of bounded-length track representatives.",
"A track representative is a track that can be analyzed in place of all—possibly infinitely many—tracks associated with its descriptor.",
"We use track representatives to devise an EXPSPACE model checking algorithm for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ .",
"Descriptor element indistinguishability plays a fundamental role in the proof of the bound to the maximum length of representatives, and it allows us to show the completeness of the algorithm, which considers all the possible representatives.",
"In addition, we prove that the model checking problem for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ is PSPACE-hard, NEXP-hard if a succinct encoding of formulas is used (it is worth noticing that the proposed algorithm requires exponential working space also in the latter case).",
"Well-behaved HS fragments.",
"We first show that the proposed model checking algorithm can verify formulas with a constant nesting depth of $\\operatorname{\\langle B\\rangle }$ modality by using polynomial working space.",
"This allows us to conclude that the model checking problem for $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ formulas (which lack modality $\\operatorname{\\langle B\\rangle }$ ) is in PSPACE.",
"Then, we prove that the model checking problem for $\\mathsf {A\\overline{B}}$ is PSPACE-hard.",
"PSPACE-completeness of $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ (and $\\mathsf {A\\overline{B}}$ ) immediately follows.",
"Next, we deal with the fragment $\\mathsf {\\forall A\\overline{A}BE}$ .",
"We first provide a coNP model checking algorithm for $\\mathsf {\\forall A\\overline{A}BE}$ , and then we show that model checking for the pure propositional fragment $\\mathsf {Prop}$ is coNP-hard.",
"The two results together allow us to conclude that the model checking problem for both $\\mathsf {Prop}$ and $\\mathsf {\\forall A\\overline{A}BE}$ is coNP-complete.",
"In addition, upper and lower bounds to the complexity of the problem for $\\mathsf {A\\overline{A}}$ (the logic of temporal neighbourhood) directly follow: since $\\mathsf {A\\overline{A}}$ is a fragment of $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ and $\\mathsf {Prop}$ is a fragment of $\\mathsf {A\\overline{A}}$ , complexity of model checking for $\\mathsf {A\\overline{A}}$ is in between coNP and PSPACE."
],
[
"Organization of the paper",
"In Section , we provide some background knowledge.",
"Then, in Section , we introduce track descriptors [24] and, in Section , we formally define the key relation of indistinguishability over descriptor elements.",
"In Section , we describe an EXPSPACE model checking algorithm for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ based on track representatives.",
"We also show how to obtain a PSPACE model checking algorithm for $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ by suitably tailoring the one for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ .",
"In Section , we prove that model checking for $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ is PSPACE-hard; PSPACE-completeness immediately follows.",
"Moreover, we get for free a lower bound to the complexity of the model checking problem for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ , which turns out to be PSPACE-hard (in the appendix, we show that the problem is NEXP-hard if a succinct encoding of formulas is used).",
"Finally, in Section we provide a coNP model checking algorithm for $\\mathsf {\\forall A\\overline{A}BE}$ and then we show that the problem is actually coNP-complete.",
"Conclusions give a short assessment of the work done and describe future research directions."
],
[
"The interval temporal logic HS",
"Interval-based approaches to temporal representation and reasoning have been successfully pursued in computer science and artificial intelligence.",
"An interval algebra to reason about intervals and their relative order was first proposed by Allen [1].",
"Then, a systematic logical study of ITLs was done by Halpern and Shoham, who introduced the logic HS featuring one modality for each Allen interval relation [13], except for equality.",
"Table: Allen's interval relations and corresponding HS modalities.Table REF depicts 6 of the 13 Allen's relations together with the corresponding HS (existential) modalities.",
"The other 7 are equality and the 6 inverse relations (given a binary relation $\\mathpzc {R}$ , the inverse relation $\\overline{\\mathpzc {R}}$ is such that $b \\overline{\\mathpzc {R}} a$ if and only if $a \\mathpzc {R} b$ ).",
"The language of HS features a set of proposition letters $\\mathpzc {AP}$ , the Boolean connectives $\\lnot $ and $\\wedge $ , and a temporal modality for each of the (non trivial) Allen's relations, namely, $\\operatorname{\\langle A\\rangle }$ , $\\operatorname{\\langle L\\rangle }$ , $\\operatorname{\\langle B\\rangle }$ , $\\operatorname{\\langle E\\rangle }$ , $\\operatorname{\\langle D\\rangle }$ , $\\operatorname{\\langle O\\rangle }$ , $\\operatorname{\\langle \\overline{A}\\rangle }$ , $\\operatorname{\\langle \\overline{L}\\rangle }$ , $\\operatorname{\\langle \\overline{B}\\rangle }$ , $\\operatorname{\\langle \\overline{E}\\rangle }$ , $\\operatorname{\\langle \\overline{D}\\rangle }$ and $\\operatorname{\\langle \\overline{O}\\rangle }$ .",
"HS formulas are defined by the following grammar: $\\psi ::= p \\;\\vert \\; \\lnot \\psi \\;\\vert \\; \\psi \\wedge \\psi \\;\\vert \\; \\langle X\\rangle \\psi \\;\\vert \\; \\langle \\overline{X}\\rangle \\psi , \\ \\ \\mbox{ with } p\\in \\mathpzc {AP},\\; X\\in \\lbrace A,L,B,E,D,O\\rbrace .$ We will make use of the standard abbreviations of propositional logic, e.g., we will write $\\psi \\vee \\phi $ for $\\lnot (\\lnot \\psi \\wedge \\lnot \\phi )$ , $\\psi \\rightarrow \\phi $ for $\\lnot \\psi \\vee \\phi $ , and $\\psi \\leftrightarrow \\phi $ for $\\left(\\psi \\rightarrow \\phi \\right)\\wedge \\left(\\phi \\rightarrow \\psi \\right)$ .",
"Moreover, for all $X$ , dual universal modalities $[X]\\psi $ and $[\\overline{X}]\\psi $ are defined as $\\lnot \\langle X\\rangle \\lnot \\psi $ and $\\lnot \\langle \\overline{X} \\rangle \\lnot \\psi $ , respectively.",
"We will assume the strict semantics of HS: only intervals consisting of at least two points are allowed.",
"Under that assumption, HS modalities are mutually exclusive and jointly exhaustive, that is, exactly one of them holds between any two intervals.",
"However, the strict semantics can easily be “relaxed” to include point intervals, and all results we are going to prove hold for the non-strict HS semantics as well.",
"All HS modalities can be expressed in terms of $\\operatorname{\\langle A\\rangle }$ , $\\operatorname{\\langle B\\rangle }$ , and $\\operatorname{\\langle E\\rangle }$ , and the inverse modalities $\\operatorname{\\langle \\overline{A}\\rangle }, \\operatorname{\\langle \\overline{B}\\rangle }$ , and $\\operatorname{\\langle \\overline{E}\\rangle }$ , as follows: $\\begin{array}{cc}\\operatorname{\\langle L\\rangle }\\psi \\equiv \\operatorname{\\langle A\\rangle }\\operatorname{\\langle A\\rangle }\\psi & \\qquad \\operatorname{\\langle \\overline{L}\\rangle }\\psi \\equiv \\operatorname{\\langle \\overline{A}\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }\\psi \\\\\\operatorname{\\langle D\\rangle }\\psi \\equiv \\operatorname{\\langle B\\rangle }\\operatorname{\\langle E\\rangle }\\psi \\equiv \\operatorname{\\langle E\\rangle }\\operatorname{\\langle B\\rangle }\\psi & \\qquad \\operatorname{\\langle \\overline{D}\\rangle }\\psi \\equiv \\operatorname{\\langle \\overline{B}\\rangle }\\operatorname{\\langle \\overline{E}\\rangle }\\psi \\equiv \\operatorname{\\langle \\overline{E}\\rangle }\\operatorname{\\langle \\overline{B}\\rangle }\\psi \\\\\\operatorname{\\langle O\\rangle }\\psi \\equiv \\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{B}\\rangle }\\psi & \\qquad \\operatorname{\\langle \\overline{O}\\rangle }\\psi \\equiv \\operatorname{\\langle B\\rangle }\\operatorname{\\langle \\overline{E}\\rangle }\\psi \\end{array}.$ We denote by $\\mathsf {X_1\\cdots X_n}$ the fragment of HS that features modalities $\\langle X_1\\rangle ,\\cdots , \\langle X_n\\rangle $ only.",
"HS can be viewed as a multi-modal logic with the 6 primitive modalities $\\operatorname{\\langle A\\rangle }$ , $\\operatorname{\\langle B\\rangle }$ , $\\operatorname{\\langle E\\rangle }$ , $\\operatorname{\\langle \\overline{A}\\rangle }$ , $\\operatorname{\\langle \\overline{B}\\rangle }$ , and $\\operatorname{\\langle \\overline{E}\\rangle }$ .",
"Accordingly, HS semantics can be defined over a multi-modal Kripke structure, called here an abstract interval model, in which (strict) intervals are treated as atomic objects and Allen's relations as simple binary relations between pairs of them.",
"Definition 1 ([20]) An abstract interval model is a tuple $\\mathpzc {A}=(\\mathpzc {AP},\\mathbb {I},A_\\mathbb {I},B_\\mathbb {I},E_\\mathbb {I},\\sigma )$ , where $\\mathpzc {AP}$ is a finite set of proposition letters, $\\mathbb {I}$ is a possibly infinite set of atomic objects (worlds), $A_\\mathbb {I}$ , $B_\\mathbb {I}$ , and $E_\\mathbb {I}$ are three binary relations over $\\mathbb {I}$ , and $\\sigma :\\mathbb {I}\\mapsto 2^{\\mathpzc {AP}}$ is a (total) labeling function which assigns a set of proposition letters to each world.",
"Intuitively, in the interval setting, $\\mathbb {I}$ is a set of intervals, $A_\\mathbb {I}$ , $B_\\mathbb {I}$ , and $E_\\mathbb {I}$ are interpreted as Allen's interval relations $A$ (meets), $B$ (started-by), and $E$ (finished-by), respectively, and $\\sigma $ assigns to each interval the set of proposition letters that hold over it.",
"Given an abstract interval model $\\mathpzc {A}=(\\mathpzc {AP},\\mathbb {I},A_\\mathbb {I},B_\\mathbb {I},E_\\mathbb {I}, \\sigma )$ and an interval $I\\in \\mathbb {I}$ , truth of an HS formula over $I$ is defined by structural induction on the formula as follows: $\\mathpzc {A},I\\models p$ if and only if $p\\in \\sigma (I)$ , for any proposition letter $p\\in \\mathpzc {AP}$ ; $\\mathpzc {A},I\\models \\lnot \\psi $ if and only if it is not true that $\\mathpzc {A},I\\models \\psi $ (also denoted as $\\mathpzc {A},I\\lnot \\models \\psi $ ); $\\mathpzc {A},I\\models \\psi \\wedge \\phi $ if and only if $\\mathpzc {A},I\\models \\psi $ and $\\mathpzc {A},I\\models \\phi $ ; $\\mathpzc {A},I\\models \\langle X\\rangle \\psi $ , for $X \\in \\lbrace A,B,E\\rbrace $ , if and only if there exists $J\\in \\mathbb {I}$ such that $I\\, X_\\mathbb {I}\\, J$ and $\\mathpzc {A},J\\models \\psi $ ; $\\mathpzc {A},I\\models \\langle \\overline{X}\\rangle \\psi $ , for $\\overline{X} \\in \\lbrace \\overline{A},\\overline{B},\\overline{E}\\rbrace $ , if and only if there exists $J\\in \\mathbb {I}$ such that $J\\, X_\\mathbb {I}\\, I$ and $\\mathpzc {A},J\\models \\psi $ ."
],
[
"Kripke structures and abstract interval models",
"In this section, we define a mapping from Kripke structures to abstract interval models that makes it possible to specify properties of systems by means of HS formulas.",
"Definition 2 A finite Kripke structure $\\mathpzc {K}$ is a tuple $(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ , where $\\mathpzc {AP}$ is a set of proposition letters, $W$ is a finite set of states, $\\delta \\subseteq W\\times W$ is a left-total relation between pairs of states, $\\mu :W\\mapsto 2^\\mathpzc {AP}$ is a total labelling function, and $w_0\\in W$ is the initial state.",
"For all $w\\in W$ , $\\mu (w)$ is the set of proposition letters which hold at that state, while $\\delta $ is the transition relation which constrains the evolution of the system over time.",
"Figure: The Kripke structure K 2 \\mathpzc {K}_{2}.Figure REF depicts a Kripke structure, $\\mathpzc {K}_{2}$ , with two states (the initial state is identified by a double circle).",
"Formally, $\\mathpzc {K}_{2}$ is defined by the following quintuple: $(\\lbrace p,q\\rbrace ,\\lbrace v_0,v_1\\rbrace ,\\lbrace (v_0,v_0),(v_0,v_1),(v_1,v_0),(v_1,v_1)\\rbrace ,\\mu ,v_0),$ where $\\mu (v_0)=\\lbrace p\\rbrace $ and $\\mu (v_1)=\\lbrace q\\rbrace $ .",
"Definition 3 A track $\\rho $ over a finite Kripke structure $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ is a finite sequence of states $v_0\\cdots v_n$ , with $n\\ge 1$ , such that for all $i\\in \\lbrace 0,\\cdots ,n-1\\rbrace $ , $(v_i,v_{i+1})\\in \\delta $ .",
"Let $\\operatorname{Trk}_\\mathpzc {K}$ be the (infinite) set of all tracks over a finite Kripke structure $\\mathpzc {K}$ .",
"For any track $\\rho =v_0\\cdots v_n \\in \\operatorname{Trk}_\\mathpzc {K}$ , we define: $|\\rho |=n+1$ ; $\\rho (i)=v_i$ , for $0\\le i\\le |\\rho |-1$ ; $\\operatorname{states}(\\rho )=\\lbrace v_0,\\cdots ,v_n\\rbrace \\subseteq W$ ; $\\operatorname{intstates}(\\rho )=\\lbrace v_1,\\cdots ,v_{n-1}\\rbrace \\subseteq W$ ; $\\operatorname{fst}(\\rho )=v_0$ and $\\operatorname{lst}(\\rho )=v_n$ ; $\\rho (i,j)=v_i\\cdots v_j$ is a subtrack of $\\rho $ , for $0\\le i < j\\le |\\rho |-1$ ; $\\operatorname{Pref}(\\rho )=\\lbrace \\rho (0,i) \\mid 1\\le i\\le |\\rho |-2\\rbrace $ is the set of all proper prefixes of $\\rho $ .",
"Note that $\\operatorname{Pref}(\\rho )=\\emptyset $ if $|\\rho |=2$ ; $\\operatorname{Suff}(\\rho )=\\lbrace \\rho (i,|\\rho |-1) \\mid 1\\le i\\le |\\rho |-2\\rbrace $ is the set of all proper suffixes of $\\rho $ .",
"Note that $\\operatorname{Suff}(\\rho )=\\emptyset $ if $|\\rho |=2$ .",
"It is worth pointing out that the length of tracks, prefixes, and suffixes is greater than 1, as they will be mapped into strict intervals.",
"If $\\operatorname{fst}(\\rho )=w_0$ (the initial state of $\\mathpzc {K}$ ), $\\rho $ is said to be an initial track.",
"In the following, we will denote by $\\rho \\cdot \\rho ^{\\prime }$ the concatenation of the tracks $\\rho $ and $\\rho ^{\\prime }$ , assuming that $(\\operatorname{lst}(\\rho ),\\operatorname{fst}(\\rho ^{\\prime }))\\in \\delta $ hence $\\rho \\cdot \\rho ^{\\prime } \\in \\operatorname{Trk}_\\mathpzc {K}$ ; moreover, by $\\rho ^n$ we will denote the track obtained by concatenating $n$ copies of $\\rho $ .",
"An abstract interval model (over $\\operatorname{Trk}_\\mathpzc {K}$ ) can be naturally associated with a finite Kripke structure by interpreting every track as an interval bounded by its first and last states.",
"Definition 4 ([20]) The abstract interval model induced by a finite Kripke structure $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ is the abstract interval model $\\mathpzc {A}_\\mathpzc {K}=(\\mathpzc {AP},\\mathbb {I},A_\\mathbb {I},B_\\mathbb {I},E_\\mathbb {I},\\sigma )$ , where: $\\mathbb {I}=\\operatorname{Trk}_\\mathpzc {K}$ , $A_\\mathbb {I}=\\left\\lbrace (\\rho ,\\rho ^{\\prime })\\in \\mathbb {I}\\times \\mathbb {I}\\mid \\operatorname{lst}(\\rho )=\\operatorname{fst}(\\rho ^{\\prime })\\right\\rbrace $ , $B_\\mathbb {I}=\\left\\lbrace (\\rho ,\\rho ^{\\prime })\\in \\mathbb {I}\\times \\mathbb {I}\\mid \\rho ^{\\prime }\\in \\operatorname{Pref}(\\rho )\\right\\rbrace $ , $E_\\mathbb {I}=\\left\\lbrace (\\rho ,\\rho ^{\\prime })\\in \\mathbb {I}\\times \\mathbb {I}\\mid \\rho ^{\\prime }\\in \\operatorname{Suff}(\\rho )\\right\\rbrace $ , and $\\sigma :\\mathbb {I}\\mapsto 2^\\mathpzc {AP}$ where $\\sigma (\\rho )=\\bigcap _{w\\in \\operatorname{states}(\\rho )}\\mu (w)$ , for all $\\rho \\in \\mathbb {I}$ .",
"In Definition REF , relations $A_\\mathbb {I},B_\\mathbb {I}$ , and $E_\\mathbb {I}$ are interpreted as Allen's interval relations meets, started-by, and finished-by, respectively.",
"Moreover, according to the definition of $\\sigma $ , a proposition letter $p\\in \\mathpzc {AP}$ holds over $\\rho =v_0\\cdots v_n$ if and only if it holds over all the states $v_0, \\ldots , v_n$ of $\\rho $ .",
"This conforms to the homogeneity principle, according to which a proposition letter holds over an interval if and only if it holds over all of its subintervals.",
"Satisfiability of an HS formula over a finite Kripke structure can be given in terms of induced abstract interval models.",
"Definition 5 Let $\\mathpzc {K}$ be a finite Kripke structure, $\\rho $ be a track in $\\operatorname{Trk}_\\mathpzc {K}$ , and $\\psi $ be an HS formula.",
"We say that the pair $(\\mathpzc {K},\\rho )$ satisfies $\\psi $ , denoted by $\\mathpzc {K},\\rho \\models \\psi $ , if and only if it holds that $\\mathpzc {A}_\\mathpzc {K},\\rho \\models \\psi $ .",
"Definition 6 Let $\\mathpzc {K}$ be a finite Kripke structure and $\\psi $ be an HS formula.",
"We say that $\\mathpzc {K}$ models $\\psi $ , denoted by $\\mathpzc {K}\\models \\psi $ , if and only if for all initial tracks $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ , it holds that $\\mathpzc {K},\\rho \\models \\psi .$ The model checking problem for HS over finite Kripke structures is the problem of deciding whether $\\mathpzc {K}\\models \\psi $ .",
"Since Kripke structures feature an infinite number of tracks, the problem is not trivially decidable.",
"We end the section by providing some meaningful examples of properties of tracks and/or transition systems that can be expressed in HS.",
"Example 1 The formula $[B]\\bot $ can be used to select all and only the tracks of length 2.",
"Given any $\\rho $ , with $|\\rho |=2$ , independently of $\\mathpzc {K}$ , it indeed holds that $\\mathpzc {K},\\rho \\models [B]\\bot $ , because $\\rho $ has no (strict) prefixes.",
"On the other hand, it holds that $\\mathpzc {K},\\rho \\models \\operatorname{\\langle B\\rangle }\\top $ if (and only if) $|\\rho |>2$ .",
"Finally, let $\\ell (k)$ be a shorthand for $[B]^{k-1}\\bot \\wedge \\operatorname{\\langle B\\rangle }^{k-2}\\top $ .",
"It holds that $\\mathpzc {K},\\rho \\models \\ell (k)$ if and only if $|\\rho |=k$ .",
"Example 2 Let us consider the finite Kripke structure $\\mathpzc {K}_{2}$ depicted in Figure REF .",
"The truth of the following statements can be easily checked: $\\mathpzc {K}_{2},(v_0v_1)^2\\models \\operatorname{\\langle A\\rangle }q$ ; $\\mathpzc {K}_{2},v_0v_1v_0\\lnot \\models \\operatorname{\\langle A\\rangle }q$ ; $\\mathpzc {K}_{2},(v_0v_1)^2\\models \\operatorname{\\langle \\overline{A}\\rangle }p$ ; $\\mathpzc {K}_{2},v_1v_0v_1\\lnot \\models \\operatorname{\\langle \\overline{A}\\rangle }p$ .",
"The above statements show that modalities $\\operatorname{\\langle A\\rangle }$ and $\\operatorname{\\langle \\overline{A}\\rangle }$ can be used to distinguish between tracks that start or end at different states.",
"In particular, note that $\\operatorname{\\langle A\\rangle }$ (resp., $\\operatorname{\\langle \\overline{A}\\rangle }$ ) allows one to “move” to any track branching on the right (resp., left) of the considered one, e.g., if $\\rho =v_0v_1v_0$ , then $\\rho \\, A_\\mathbb {I}\\, v_0v_0$ , $\\rho \\, A_\\mathbb {I}\\, v_0v_1$ , $\\rho \\, A_\\mathbb {I}\\, v_0v_0v_0$ , $\\rho \\, A_\\mathbb {I}\\, v_0v_0v_1$ , $\\rho \\, A_\\mathbb {I}\\, v_0v_1v_0v_1$ , and so on.",
"Modalities $\\operatorname{\\langle B\\rangle }$ and $\\operatorname{\\langle E\\rangle }$ can be used to distinguish between tracks encompassing a different number of iterations of a given loop.",
"This is the case, for instance, with the following statements: $\\mathpzc {K}_{2},(v_1v_0)^3 v_1\\models \\operatorname{\\langle B\\rangle }\\big (\\operatorname{\\langle A\\rangle }p \\wedge \\operatorname{\\langle B\\rangle }\\left(\\operatorname{\\langle A\\rangle }p \\wedge \\operatorname{\\langle B\\rangle }\\operatorname{\\langle A\\rangle }p\\right)\\big )$ ; $\\mathpzc {K}_{2},(v_1v_0)^2 v_1\\lnot \\models \\operatorname{\\langle B\\rangle }\\big (\\operatorname{\\langle A\\rangle }p \\wedge \\operatorname{\\langle B\\rangle }\\left(\\operatorname{\\langle A\\rangle }p \\wedge \\operatorname{\\langle B\\rangle }\\operatorname{\\langle A\\rangle }p\\right)\\big )$ .",
"Finally, HS makes it possible to distinguish between $\\rho _1=v_0^3v_1v_0$ and $\\rho _2=v_0v_1v_0^3$ , which feature the same number of iterations of the same loops, but differ in the order of loop occurrences: $\\mathpzc {K}_{2},\\rho _1\\models \\operatorname{\\langle B\\rangle }\\left(\\operatorname{\\langle A\\rangle }q \\wedge \\operatorname{\\langle B\\rangle }\\operatorname{\\langle A\\rangle }p\\right)$ but $\\mathpzc {K}_{2},\\rho _2\\lnot \\models \\operatorname{\\langle B\\rangle }\\left(\\operatorname{\\langle A\\rangle }q \\wedge \\operatorname{\\langle B\\rangle }\\operatorname{\\langle A\\rangle }p\\right)$ .",
"Example 3 In Figure REF , we give an example of a finite Kripke structure $\\mathpzc {K}_{Sched}$ that models the behaviour of a scheduler serving three processes which are continuously requesting the use of a common resource.",
"The initial state is $v_0$ : no process is served in that state.",
"In any other state $v_i$ and $\\overline{v}_i$ , with $i \\in \\lbrace 1,2,3\\rbrace $ , the $i$ -th process is served (this is denoted by the fact that $p_i$ holds in those states).",
"For the sake of readability, edges are marked either by $r_i$ , for $request(i)$ , or by $u_i$ , for $unlock(i)$ .",
"However, edge labels do not have a semantic value, i.e., they are neither part of the structure definition, nor proposition letters; they are simply used to ease reference to edges.",
"Process $i$ is served in state $v_i$ , then, after “some time”, a transition $u_i$ from $v_i$ to $\\overline{v}_i$ is taken; subsequently, process $i$ cannot be served again immediately, as $v_i$ is not directly reachable from $\\overline{v}_i$ (the scheduler cannot serve the same process twice in two successive rounds).",
"A transition $r_j$ , with $j\\ne i$ , from $\\overline{v}_i$ to $v_j$ is then taken and process $j$ is served.",
"This structure can be easily generalised to a higher number of processes.",
"Figure: The Kripke structure K Sched \\mathpzc {K}_{Sched}.We show how some meaningful properties to check against $\\mathpzc {K}_{Sched}$ can be expressed in HS, and, in particular, by means of formulas of the fragment $\\mathsf {\\overline{A}E}$ —a subfragment of the fragment $\\mathsf {A\\overline{A}E\\overline{B}\\overline{E}}$ , on which we will focus in the following.",
"In all formulas, we force the validity of the considered property over all legal computation sub-intervals by using modality $[E]$ (all computation sub-intervals are suffixes of at least one initial track).",
"Truth of the following statements can be easily checked: $\\mathpzc {K}_{Sched}\\models [E]\\big (\\operatorname{\\langle E\\rangle }^4\\top \\rightarrow (\\chi (p_1,p_2) \\vee \\chi (p_1,p_3) \\vee \\chi (p_2,p_3))\\big )$ , with $\\chi (p,q):=\\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }p \\wedge \\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }q$ ; $\\mathpzc {K}_{Sched}\\lnot \\models [E](\\operatorname{\\langle E\\rangle }^{10}\\top \\rightarrow \\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }p_3)$ ; $\\mathpzc {K}_{Sched}\\lnot \\models [E](\\operatorname{\\langle E\\rangle }^6 \\rightarrow (\\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }p_1 \\wedge \\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }p_2 \\wedge \\operatorname{\\langle E\\rangle }\\operatorname{\\langle \\overline{A}\\rangle }p_3))$ .",
"The first formula requires that in any suffix of length at least 6 of an initial track, at least 2 proposition letters are witnessed.",
"$\\mathpzc {K}_{Sched}$ satisfies the formula since a process cannot be executed twice consecutively.",
"The second formula requires that in any suffix of length at least 12 of an initial track, process 3 is executed at least once in some internal states.",
"$\\mathpzc {K}_{Sched}$ does not satisfy the formula since the scheduler, being unfair, can avoid executing a process ad libitum.",
"The third formula requires that in any suffix of length at least 8 of an initial track, $p_1$ , $p_2$ , and $p_3$ are all witnessed.",
"The only way to satisfy this property would be to constrain the scheduler to execute the three processes in a strictly periodic manner, which is not the case."
],
[
"The notion of $B_k$ -descriptor",
"For any finite Kripke structure $\\mathpzc {K}$ , one can find a corresponding induced abstract interval model $\\mathpzc {A}_\\mathpzc {K}$ , featuring one interval for each track of $\\mathpzc {K}$ .",
"As we already pointed out, since $\\mathpzc {K}$ has loops (each state must have at least one successor, as the transition relation $\\delta $ is left-total), the number of its tracks, and thus the number of intervals of $\\mathpzc {A}_\\mathpzc {K}$ , is infinite.",
"In [20], Molinari et al.",
"showed that, given a bound $k$ on the structural complexity of HS formulas (that is, on the nesting depth of $\\operatorname{\\langle B\\rangle }$ and $\\operatorname{\\langle E\\rangle }$ modalities), it is possible to obtain a finite representation for $\\mathpzc {A}_\\mathpzc {K}$ , which is equivalent to $\\mathpzc {A}_\\mathpzc {K}$ with respect to satisfiability of HS formulas with structural complexity less than or equal to $k$ .",
"By making use of such a representation, they prove that the model checking problem for (full) HS is decidable (with a non-elementary upper bound).",
"In this paper, we first restrict our attention to $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ and provide a model checking algorithm of lower complexity.",
"All the results we are going to prove hold also for the fragment $\\mathsf {A\\overline{A}E\\overline{B}\\overline{E}}$ by symmetry.",
"We start with the definition of some basic notions.",
"Definition 7 Let $\\psi $ be an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula.",
"The B-nesting depth of $\\psi $ , denoted by $\\operatorname{Nest_B}(\\psi )$ , is defined by induction on the complexity of the formula as follows: $\\operatorname{Nest_B}(p)=0$ , for any proposition letter $p\\in \\mathpzc {AP}$ ; $\\operatorname{Nest_B}(\\lnot \\psi )=\\operatorname{Nest_B}(\\psi )$ ; $\\operatorname{Nest_B}(\\psi \\wedge \\phi )=\\max \\lbrace \\operatorname{Nest_B}(\\psi ),\\operatorname{Nest_B}(\\phi )\\rbrace $ ; $\\operatorname{Nest_B}(\\operatorname{\\langle B\\rangle }\\psi )=1+\\operatorname{Nest_B}(\\psi )$ ; $\\operatorname{Nest_B}(\\operatorname{\\langle X\\rangle }\\psi )=\\operatorname{Nest_B}(\\psi )$ , for $X\\in \\lbrace A, \\overline{A}, \\overline{B}, \\overline{E}\\rbrace $ .",
"Making use of Definition REF , we can introduce the relation(s) of $k$ -equivalence over tracks.",
"Definition 8 Let $\\mathpzc {K}$ be a finite Kripke structure, $\\rho $ and $\\rho ^{\\prime }$ be two tracks in $\\operatorname{Trk}_\\mathpzc {K}$ , and $k\\in \\mathbb {N}$ .",
"We say that $\\rho $ and $\\rho ^{\\prime }$ are $k$ -equivalent if and only if, for every $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula $\\psi $ with $\\operatorname{Nest_B}(\\psi )=k$ , $\\mathpzc {K},\\rho \\models \\psi $ if and only if $\\mathpzc {K},\\rho ^{\\prime }\\models \\psi $ .",
"It can be easily proved that $k$ -equivalence propagates downwards.",
"Proposition 9 Let $\\mathpzc {K}$ be a finite Kripke structure, $\\rho $ and $\\rho ^{\\prime }$ be two tracks in $\\operatorname{Trk}_\\mathpzc {K}$ , and $k\\in \\mathbb {N}$ .",
"If $\\rho $ and $\\rho ^{\\prime }$ are $k$ -equivalent, then they are $h$ -equivalent, for all $0\\le h\\le k$ .",
"Let us assume that $\\mathpzc {K},\\rho \\models \\psi $ , with $0\\le \\operatorname{Nest_B}(\\psi )=h\\le k$ .",
"Consider the formula $\\operatorname{\\langle B\\rangle }^k\\top $ , whose B-nesting depth is equal to $k$ .",
"It holds that either $\\mathpzc {K},\\rho \\models \\operatorname{\\langle B\\rangle }^k\\top $ or $\\mathpzc {K},\\rho \\models \\lnot \\operatorname{\\langle B\\rangle }^k\\top $ .",
"In the first case, we have that $\\mathpzc {K},\\rho \\models \\operatorname{\\langle B\\rangle }^k\\top \\wedge \\psi $ .",
"Since $\\operatorname{Nest_B}(\\operatorname{\\langle B\\rangle }^k\\top \\wedge \\psi )=k$ , from the hypothesis, it immediately follows that $\\mathpzc {K},\\rho ^{\\prime }\\models \\operatorname{\\langle B\\rangle }^k\\top \\wedge \\psi $ , and thus $\\mathpzc {K},\\rho ^{\\prime }\\models \\psi $ .",
"The other case can be dealt with in a symmetric way.",
"We are now ready to define the key notion of descriptor for a track of a Kripke structure.",
"Definition 10 ([20]) Let $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,v_0)$ be a finite Kripke structure, $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ , and $k\\in \\mathbb {N}$ .",
"The $B_k$ -descriptor for $\\rho $ is a labelled tree $\\mathpzc {D}=(V,E,\\lambda )$ of depth $k$ , where $V$ is a finite set of vertices, $E\\subseteq V\\times V$ is a set of edges, and $\\lambda :V\\mapsto W\\times 2^W\\times W$ is a node labelling function, inductively defined as follows: for $k=0$ , the $B_k$ -descriptor for $\\rho $ is the tree $\\mathpzc {D} = (\\lbrace \\operatorname{root}(\\mathpzc {D})\\rbrace ,\\emptyset ,\\lambda )$ , where $\\lambda (\\operatorname{root}(\\mathpzc {D}))=(\\operatorname{fst}(\\rho ),\\operatorname{intstates}(\\rho ), \\operatorname{lst}(\\rho ));$ for $k>0$ , the $B_k$ -descriptor for $\\rho $ is the tree $\\mathpzc {D} = (V,E,\\lambda )$ , where $\\lambda (\\operatorname{root}(\\mathpzc {D}))=(\\operatorname{fst}(\\rho ),\\operatorname{intstates}(\\rho ),\\operatorname{lst}(\\rho )),$ which satisfies the following conditions: for each prefix $\\rho ^{\\prime }$ of $\\rho $ , there exists $v\\in V$ such that $(\\operatorname{root}(\\mathpzc {D}),v)\\in E$ and the subtree rooted in $v$ is the $B_{k-1}$ -descriptor for $\\rho ^{\\prime }$ ; for each vertex $v\\in V$ such that $(\\operatorname{root}(\\mathpzc {D}),v)\\in E$ , there exists a prefix $\\rho ^{\\prime }$ of $\\rho $ such that the subtree rooted in $v$ is the $B_{k-1}$ -descriptor for $\\rho ^{\\prime }$ ; for all pairs of edges $(\\operatorname{root}(\\mathpzc {D}),v^{\\prime }), (\\operatorname{root}(\\mathpzc {D}),v^{\\prime \\prime })\\in E$ , if the subtree rooted in $v^{\\prime }$ is isomorphic to the subtree rooted in $v^{\\prime \\prime }$ , then $v^{\\prime }=v^{\\prime \\prime }$ Here and in the following, we write subtree for maximal subtree.",
"Moreover, isomorphism between descriptors accounts for node labels, as well (not only for the structure of descriptors)..",
"Condition REF of Definition REF simply states that no two subtrees whose roots are siblings can be isomorphic.",
"A $B_0$ -descriptor $\\mathpzc {D}$ for a track consists of its root only, which is denoted by $\\operatorname{root}(\\mathpzc {D})$ .",
"A label of a node will be referred to as a descriptor element: the notion of descriptor element bears analogies with an abstraction technique for discrete time Duration Calculus proposed by Hansen et al.",
"in [14], which, on its turn, is connected to Parikh images [29] (a descriptor element can be seen as a qualitative analogue of this).",
"Basically, for any $k \\ge 0$ , the label of the root of the $B_k$ -descriptor $\\mathpzc {D}$ for $\\rho $ is the triple $(\\operatorname{fst}(\\rho ),\\operatorname{intstates}(\\rho ),\\operatorname{lst}(\\rho ))$ .",
"Each prefix $\\rho ^{\\prime }$ of $\\rho $ is associated with some subtree whose root is labelled with $(\\operatorname{fst}(\\rho ^{\\prime }),\\operatorname{intstates}(\\rho ^{\\prime }),\\operatorname{lst}(\\rho ^{\\prime }))$ and is a child of the root of $\\mathpzc {D}$ .",
"Such a construction is then iteratively applied to the children of the root until either depth $k$ is reached or a track of length 2 is being considered on a node.",
"Hereafter equality between descriptors is considered up to isomorphism.",
"As an example, in Figure REF we show the $B_2$ -descriptor for the track $\\rho = v_0v_1v_0v_0v_0v_0v_1$ of $\\mathpzc {K}_{2}$ (Figure REF ).",
"It is worth noting that there exist two distinct prefixes of $\\rho $ , that is, the tracks $\\rho ^{\\prime }=v_0v_1v_0v_0v_0v_0$ and $\\rho ^{\\prime \\prime }=v_0v_1v_0v_0v_0$ , which have the same $B_1$ -descriptor.",
"Since, according to Definition REF , no tree can occur more than once as a subtree of the same node (in this example, the root), in the $B_2$ -descriptor for $\\rho $ , prefixes $\\rho ^{\\prime }$ and $\\rho ^{\\prime \\prime }$ are represented by the same tree (the first subtree of the root on the left).",
"This shows that, in general, the root of a descriptor for a track with $h$ proper prefixes does not necessarily have $h$ children.",
"Figure: The B 2 B_2-descriptor for the track v 0 v 1 v 0 v 0 v 0 v 0 v 1 v_0v_1v_0v_0v_0v_0v_1 of K 2 \\mathpzc {K}_{2}.$B$ -descriptors do not convey, in general, enough information to determine which track they were built from; however, they can be used to determine which $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas are satisfied by the track from which they were built.",
"In [20], the authors prove that, for a finite Kripke structure $\\mathpzc {K}$ , there exists a finite number (non-elementary w.r.t.",
"$|W|$ and $k$ ) of possible $B_k$ -descriptors.",
"Moreover, the number of nodes of a descriptor has a non-elementary upper bound as well.",
"Since the number of tracks of $\\mathpzc {K}$ is infinite, and for any $k\\in \\mathbb {N}$ the set of $B_k$ -descriptors for its tracks is finite, at least one $B_k$ -descriptor must be the $B_k$ -descriptor of infinitely many tracks.",
"Thus, $B_k$ -descriptors naturally induce an equivalence relation of finite index over the set of tracks of a finite Kripke structure ($k$ -descriptor equivalence relation).",
"Definition 11 Let $\\mathpzc {K}$ be a finite Kripke structure, $\\rho ,\\rho ^{\\prime } \\in \\operatorname{Trk}_\\mathpzc {K}$ , and $k \\in \\mathbb {N}$ .",
"We say that $\\rho $ and $\\rho ^{\\prime }$ are $k$ -descriptor equivalent (denoted as $\\rho \\sim _k\\rho ^{\\prime }$ ) if and only if the $B_k$ -descriptors for $\\rho $ and $\\rho ^{\\prime }$ coincide.",
"Lemma 12 Let $k\\in \\mathbb {N}$ , $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,v_0)$ be a finite Kripke structure and $\\rho _1$ , $\\rho _1^{\\prime }$ , $\\rho _2$ , $\\rho _2^{\\prime }$ be tracks in $\\operatorname{Trk}_\\mathpzc {K}$ such that $\\left(\\operatorname{lst}(\\rho _1),\\operatorname{fst}(\\rho _1^{\\prime })\\right)\\in \\delta $ , $\\left(\\operatorname{lst}(\\rho _2),\\operatorname{fst}(\\rho _2^{\\prime })\\right)\\in \\delta $ , $\\rho _1\\sim _k\\rho _2$ and $\\rho _1^{\\prime }\\sim _k\\rho _2^{\\prime }$ .",
"Then $\\rho _1\\cdot \\rho _1^{\\prime }\\sim _k\\rho _2\\cdot \\rho _2^{\\prime }$ .",
"The proof is reported in REF .",
"The next proposition immediately follows from Lemma REF .",
"Proposition 13 (Left and right extensions) Let $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,v_0)$ be a finite Kripke structure, $\\rho , \\rho ^{\\prime }$ be two tracks in $\\operatorname{Trk}_\\mathpzc {K}$ such that $\\rho \\sim _k\\rho ^{\\prime }$ , and $\\overline{\\rho }\\in \\operatorname{Trk}_\\mathpzc {K}$ .",
"If $\\left(\\operatorname{lst}(\\rho ),\\operatorname{fst}(\\overline{\\rho })\\right)\\in \\delta $ , then $\\rho \\cdot \\overline{\\rho }\\sim _k\\rho ^{\\prime }\\cdot \\overline{\\rho }$ , and if $\\left(\\operatorname{lst}(\\overline{\\rho }),\\operatorname{fst}(\\rho )\\right)\\in \\delta $ , then $\\overline{\\rho }\\cdot \\rho \\sim _k\\overline{\\rho }\\cdot \\rho ^{\\prime }$ .",
"The next theorem proves that, for any pair of tracks $\\rho ,\\rho ^{\\prime }\\in \\operatorname{Trk}_\\mathpzc {K}$ , if $\\rho \\sim _k\\rho ^{\\prime }$ , then $\\rho $ and $\\rho ^{\\prime }$ are $k$ -equivalent (see Definition REF ).",
"Theorem 14 ([20]) Let $\\mathpzc {K}$ be a finite Kripke structure, $\\rho $ and $\\rho ^{\\prime }$ be two tracks in $\\operatorname{Trk}_\\mathpzc {K}$ , and $\\psi $ be a formula of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ with $\\operatorname{Nest_B}(\\psi )=k$ .",
"If $\\rho \\sim _k\\rho ^{\\prime }$ , then $\\mathpzc {K},\\rho \\models \\psi \\iff \\mathpzc {K},\\rho ^{\\prime }\\models \\psi $ .",
"Since the set of $B_k$ -descriptors for the tracks of a finite Kripke structure $\\mathpzc {K}$ is finite, i.e., the equivalence relation $\\sim _k$ has a finite index, there always exists a finite number of $B_k$ -descriptors that “satisfy” an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula $\\psi $ with $\\operatorname{Nest_B}(\\psi )= k$ (this can be formally proved by a quotient construction [20])."
],
[
"Clusters and descriptor element indistinguishability",
"A $B_k$ -descriptor provides a finite encoding for a possibly infinite set of tracks (the tracks associated with that descriptor).",
"Unfortunately, the representation of $B_k$ -descriptors as trees labelled over descriptor elements is highly redundant.",
"For instance, given any pair of subtrees rooted in some children of the root of a descriptor, it is always the case that one of them is a subtree of the other: the two subtrees are associated with two (different) prefixes of a track and one of them is necessarily a prefix of the other.",
"In practice, the size of the tree representation of $B_k$ -descriptors prevents their direct use in model checking algorithms, and makes it difficult to determine the intrinsic complexity of $B_k$ -descriptors.",
"In this section, we devise a more compact representation of $B_k$ -descriptors.",
"Each class of the $k$ -descriptor equivalence relation is a set of $k$ -equivalent tracks.",
"For any such class, we select (at least) one track representative whose length is (exponentially) bounded in both the size of $W$ (the set of states of the Kripke structure) and $k$ .",
"In order to determine such a bound, we consider suitable ordered sequences (possibly with repetitions) of descriptor elements of a $B_k$ -descriptor.",
"Let the descriptor sequence for a track be the ordered sequence of descriptor elements associated with its prefixes.",
"It can be easily checked that in a descriptor sequence descriptor elements can be repeated.",
"We devise a criterion to avoid such repetitions whenever they cannot be distinguished by an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula of $B$ -nesting depth up to $k$ .",
"Definition 15 Let $\\rho =v_0v_1 \\cdots v_n$ be a track of a finite Kripke structure.",
"The descriptor sequence $\\rho _{ds}$ for $\\rho $ is $d_0 \\cdots d_{n-1}$ , where $d_i = \\rho _{ds}(i)=(v_0, \\operatorname{intstates}(v_0\\cdots v_{i+1}),v_{i+1})$ , for $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace $ .",
"We denote by $DElm(\\rho _{ds})$ the set of descriptor elements occurring in $\\rho _{ds}$ .",
"Figure: An example of finite Kripke structure.As an example, let us consider the finite Kripke structure of Figure REF and the track $\\rho = v_0v_0v_0v_1v_2v_1v_2v_3v_3v_2v_2$ .",
"The descriptor sequence for $\\rho $ is: $\\rho _{ds}=(v_0,\\emptyset , v_0)\\boxed{(v_0,\\lbrace v_0\\rbrace ,v_0)}(v_0,\\lbrace v_0\\rbrace ,v_1)(v_0,\\lbrace v_0,v_1\\rbrace ,v_2)\\boxed{(v_0,\\Gamma ,v_1)(v_0,\\Gamma ,v_2)}\\\\ (v_0,\\Gamma ,v_3) \\boxed{(v_0,\\Delta ,v_3)(v_0,\\Delta ,v_2)(v_0,\\Delta ,v_2)},\\qquad \\mathrm {(*)}$ where $\\Gamma \\!=\\!\\lbrace v_0,v_1,v_2\\rbrace $ , $\\Delta \\!=\\!\\lbrace v_0,v_1,v_2,v_3\\rbrace $ , and $DElm(\\rho _{ds})$ is the set $\\lbrace (v_0,\\emptyset , v_0),(v_0,\\lbrace v_0\\rbrace ,v_0),(v_0,\\lbrace v_0\\rbrace ,v_1),\\, (v_0,\\lbrace v_0,v_1\\rbrace ,v_2),\\,(v_0,\\Gamma ,v_1),\\, (v_0,\\Gamma ,v_2),\\, (v_0,\\Gamma ,v_3),\\, (v_0,\\Delta ,v_2),\\, (v_0,\\Delta ,v_3)\\rbrace $ .",
"The meaning of boxes in (REF ) will be clear later.",
"To express the relationships between descriptor elements occurring in a descriptor sequence, we introduce a binary relation $\\operatorname{R_t}$ .",
"Intuitively, given two descriptor elements $d^{\\prime }$ and $d^{\\prime \\prime }$ of a descriptor sequence, the relation $d^{\\prime }\\operatorname{R_t}d^{\\prime \\prime }$ holds if $d^{\\prime }$ and $d^{\\prime \\prime }$ are the descriptor elements of two tracks $\\rho ^{\\prime }$ and $\\rho ^{\\prime \\prime }$ , respectively, and $\\rho ^{\\prime }$ is a prefix of $\\rho ^{\\prime \\prime }$ .",
"Definition 16 Let $\\rho _{ds}$ be the descriptor sequence for a track $\\rho $ and let $d^{\\prime }=(v_{in},S^{\\prime },v_{fin}^{\\prime })$ and $d^{\\prime \\prime }=(v_{in},S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ be two descriptor elements in $\\rho _{ds}$ .",
"It holds that $d^{\\prime }\\operatorname{R_t}d^{\\prime \\prime }$ if and only if $S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace \\subseteq S^{\\prime \\prime }$ .",
"Note that the relation $\\operatorname{R_t}$ is transitive.",
"In fact for all descriptor elements $d^{\\prime }\\!=\\!",
"(v_{in},S^{\\prime },v_{fin}^{\\prime })$ , $d^{\\prime \\prime }=(v_{in},S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ and $d^{\\prime \\prime \\prime }=(v_{in},S^{\\prime \\prime \\prime },v_{fin}^{\\prime \\prime \\prime })$ , if $d^{\\prime }\\operatorname{R_t}d^{\\prime \\prime }$ and $d^{\\prime \\prime }\\operatorname{R_t}d^{\\prime \\prime \\prime }$ , then $S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace \\subseteq S^{\\prime \\prime }$ and $S^{\\prime \\prime }\\cup \\lbrace v_{fin}^{\\prime \\prime }\\rbrace \\subseteq S^{\\prime \\prime \\prime }$ ; it follows that $S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace \\subseteq S^{\\prime \\prime \\prime }$ , and thus $d^{\\prime }\\operatorname{R_t}d^{\\prime \\prime \\prime }$ .",
"The relation $\\operatorname{R_t}$ is neither an equivalence relation nor a quasiorder, since $\\operatorname{R_t}$ is neither reflexive (e.g., $(v_0,\\lbrace v_0\\rbrace ,v_1)\\operatorname{{R}_t}(v_0,\\lbrace v_0\\rbrace ,v_1)$ ), nor symmetric (e.g., $(v_0,\\lbrace v_0\\rbrace ,v_1)\\operatorname{R_t}(v_0,\\lbrace v_0,v_1\\rbrace ,v_1)$ and $(v_0, \\lbrace v_0,v_1\\rbrace ,v_1)\\operatorname{{R}_t}(v_0,\\lbrace v_0\\rbrace ,v_1)$ ), nor antisymmetric (e.g., $(v_0,\\lbrace v_1,v_2\\rbrace ,v_1)\\operatorname{R_t}(v_0,\\lbrace v_1,v_2\\rbrace ,v_2)$ and $(v_0,\\lbrace v_1,v_2\\rbrace ,v_2)\\operatorname{R_t}(v_0,\\lbrace v_1,v_2\\rbrace ,v_1)$ , but the two elements are distinct).",
"It can be easily shown that $\\operatorname{R_t}$ pairs descriptor elements of increasing prefixes of a track.",
"Proposition 17 Let $\\rho _{ds}$ be the descriptor sequence for the track $\\rho =v_0v_1\\cdots v_n$ .",
"Then, $\\rho _{ds}(i)\\operatorname{R_t}\\rho _{ds}(j)$ , for all $0\\le i<j<n$ .",
"We now partition descriptor elements into two different types.",
"Definition 18 A descriptor element $(v_{in},S,v_{fin})$ is a Type-1 descriptor element if $v_{fin}\\notin S$ , while it is a Type-2 descriptor element if $v_{fin}\\in S$ .",
"It can be easily checked that a descriptor element $d=(v_{in},S,v_{fin}$ ) is Type-1 if and only if $\\operatorname{R_t}$ is not reflexive for $d$ .",
"In fact, if $d\\operatorname{{R}_t}d$ , then $S\\cup \\lbrace v_{fin}\\rbrace \\lnot \\subseteq S$ , and thus $v_{fin}\\notin S$ .",
"Conversely, if $v_{fin}\\notin S$ , then $d\\operatorname{{R}_t}d$ .",
"It follows that a Type-1 descriptor element cannot occur more than once in a descriptor sequence.",
"On the other hand, Type-2 descriptor elements may occur multiple times, and if a descriptor element occurs more than once in a descriptor sequence, then it is necessarily of Type-2.",
"Proposition 19 If both $d^{\\prime }\\operatorname{R_t}d^{\\prime \\prime }$ and $d^{\\prime \\prime }\\operatorname{R_t}d^{\\prime }$ , for $d^{\\prime } = (v_{in},S^{\\prime },v^{\\prime }_{fin})$ and $d^{\\prime \\prime } = (v_{in},S^{\\prime \\prime },v^{\\prime \\prime }_{fin})$ , then $v_{fin}^{\\prime }\\in S^{\\prime }$ , $v_{fin}^{\\prime \\prime }\\in S^{\\prime \\prime }$ , and $S^{\\prime }=S^{\\prime \\prime }$ ; thus, both $d^{\\prime }$ and $d^{\\prime \\prime }$ are Type-2 descriptor elements.",
"We are now ready to give a general characterization of the descriptor sequence $\\rho _{ds}$ for a track $\\rho $ : $\\rho _{ds}$ is composed of some (maximal) subsequences, consisting of occurrences of Type-2 descriptor elements on which $\\operatorname{R_t}$ is symmetric, separated by occurrences of Type-1 descriptor elements.",
"This can be formalized by means of the following notion of cluster.",
"Definition 20 A cluster $\\mathpzc {C}$ of (Type-2) descriptor elements is a maximal set of descriptor elements $\\lbrace d_1,\\ldots , d_s\\rbrace \\subseteq DElm(\\rho _{ds})$ such that $d_i\\operatorname{R_t}d_j$ and $d_j\\operatorname{R_t}d_i$ for all $i,j\\in \\lbrace 1,\\ldots , s\\rbrace $ .",
"Thanks to maximality, clusters are pairwise disjoint: if $\\mathpzc {C}$ and $\\mathpzc {C}^{\\prime }$ are distinct clusters, $d\\in \\mathpzc {C}$ and $d^{\\prime }\\in \\mathpzc {C}^{\\prime }$ , either $d\\operatorname{R_t}d^{\\prime }$ and $d^{\\prime }\\operatorname{{R}_t}d$ , or $d^{\\prime }\\operatorname{R_t}d$ and $d\\operatorname{{R}_t}d^{\\prime }$ .",
"It can be easily checked that the descriptor elements of a cluster $\\mathpzc {C}$ are contiguous in $\\rho _{ds}$ (in other words, they form a subsequence of $\\rho _{ds}$ ), that is, occurrences of descriptor elements of $\\mathpzc {C}$ are never shuffled with occurrences of descriptor elements not belonging to $\\mathpzc {C}$ .",
"Definition 21 Let $\\rho _{ds}$ be a descriptor sequence and $\\mathpzc {C}$ be one of its clusters.",
"The subsequence of $\\rho _{ds}$ associated with $\\mathpzc {C}$ is the subsequence $\\rho _{ds}(i,j)$ , with $i\\le j < |\\rho _{ds}|$ , including all and only the occurrences of the descriptor elements in $\\mathpzc {C}$ .",
"Note that two subsequences associated with two distinct clusters $\\mathpzc {C}$ and $\\mathpzc {C}^{\\prime }$ in a descriptor sequence must be separated by at least one occurrence of a Type-1 descriptor element.",
"For instance, with reference to the descriptor sequence (REF ) for the track $\\rho =v_0v_0v_0v_1v_2v_1v_2v_3v_3v_2v_2$ of the Kripke structure in Figure REF , the subsequences associated with clusters are enclosed in boxes.",
"While $\\operatorname{R_t}$ allows us to order any pair of Type-1 descriptor elements, as well as any Type-1 descriptor element with respect to a Type-2 one, it does not give us any means to order Type-2 descriptor elements belonging to the same cluster.",
"This, together with the fact that Type-2 elements may have multiple occurrences in a descriptor sequence, implies that we need to somehow limit the number of occurrences of Type-2 elements in order to determine a bound on the length of track representatives of $B_{k}$ -descriptors.",
"To this end, we introduce an equivalence relation that allows us to put together indistinguishable occurrences of the same descriptor element in a descriptor sequence, that is, to detect those occurrences which are associated with prefixes of the track with the same $B_{k}$ -descriptor.",
"The idea is that a track representative for a $B_{k}$ -descriptor should not feature indistinguishable occurrences of the same descriptor element.",
"Definition 22 Let $\\rho _{ds}$ be a descriptor sequence and $k \\ge 1$ .",
"We say that $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $0 \\le i<j<|\\rho _{ds}|$ , are $k$ -indistinguishable if (and only if) they are occurrences of the same descriptor element $d$ and: (for $k = 1$ ) $DElm(\\rho _{ds}(0, i-1))=DElm(\\rho _{ds}(0, j-1))$ ; (for $k \\ge 2$ ) for all $i \\le \\ell \\le j-1$ , there exists $0\\le \\ell ^{\\prime }\\le i-1$ such that $\\rho _{ds}(\\ell )$ and $\\rho _{ds}(\\ell ^{\\prime })$ are $(k-1)$ -indistinguishable.",
"From Definition REF , it follows that two indistinguishable occurrences $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ of the same descriptor element necessarily belong to the same subsequence of $\\rho _{ds}$ associated with a cluster.",
"In general, it is always the case that $DElm(\\rho _{ds}(0, i-1))\\subseteq DElm(\\rho _{ds}(0, j-1))$ for $i<j$ .",
"Moreover, note that the two first occurrences of a descriptor element, say $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i<j$ , are never 1-indistinguishable as a consequence of the fact that 1-indistinguishability requires that $DElm(\\rho _{ds}(0, i-1))= DElm(\\rho _{ds}(0, j-1))$ .",
"Proposition REF and REF state some basic properties of the $k$ -indistinguishability relation.",
"Proposition 23 Let $k\\ge 2$ and $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $0 \\le i<j<|\\rho _{ds}|$ , be two $k$ -indistinguishable occurrences of the same descriptor element in a descriptor sequence $\\rho _{ds}$ .",
"Then, $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ are also $(k-1)$ -indistinguishable.",
"The proof is by induction on $k \\ge 2$ .",
"Base case ($k= 2$ ).",
"Let $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ be two 2-indistinguishable occurrences of a descriptor element $d$ .",
"By definition, for any $\\rho _{ds}(i^{\\prime })$ , with $i\\le i^{\\prime }<j$ , an occurrence of the descriptor element $d^{\\prime }=\\rho _{ds}(i^{\\prime })$ must exist before position $i$ , and thus $DElm(\\rho _{ds}(0,i-1))=DElm(\\rho _{ds}(0,j-1))$ .",
"It immediately follows that $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ are 1-indistinguishable.",
"Inductive step ($k\\ge 3$ ).",
"By definition, for all $i\\le \\ell \\le j-1$ , there exists $0\\le \\ell ^{\\prime } \\le i-1$ such that $\\rho _{ds}(\\ell )$ and $\\rho _{ds}(\\ell ^{\\prime })$ are $(k-1)$ -indistinguishable.",
"By the inductive hypothesis, $\\rho _{ds}(\\ell )$ and $\\rho _{ds}(\\ell ^{\\prime })$ are $(k-2)$ -indistinguishable, which implies that $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ are $(k-1)$ -indistinguishable.",
"Proposition 24 Let $k\\ge 1$ and $\\rho _{ds}(i)$ and $\\rho _{ds}(m)$ , with $0\\!",
"\\le \\!",
"i\\!<\\!m\\!<\\!|\\rho _{ds}|$ , be two $k$ -indistinguishable occurrences of the same descriptor element in a descriptor sequence $\\rho _{ds}$ .",
"If $\\rho _{ds}(j)=\\rho _{ds}(m)$ , for some $i<j<m$ , then $\\rho _{ds}(j)$ and $\\rho _{ds}(m)$ are also $k$ -indistinguishable.",
"For $k=1$ , we have $DElm(\\rho _{ds}(0,i-1))=DElm(\\rho _{ds}(0,m-1))$ ; moreover, $DElm(\\rho _{ds}(0,i-1))\\subseteq DElm(\\rho _{ds}(0,j-1))\\subseteq DElm(\\rho _{ds}(0,m-1))$ .",
"Thus $DElm(\\rho _{ds}(0,i-1))=DElm(\\rho _{ds}(0, m-1))= DElm(\\rho _{ds}(0,j-1))$ , proving the property.",
"If $k\\ge 2$ , all occurrences $\\rho _{ds}(i^{\\prime })$ , with $i\\le i^{\\prime }<m$ , are $(k-1)$ -indistinguishable from some occurrence of the same descriptor element before $i$ , by hypothesis.",
"In particular, this is true for all occurrences $\\rho _{ds}(j^{\\prime })$ , with $j\\le j^{\\prime }<m$ .",
"The thesis trivially follows.",
"Example 4 In Figure REF , we give some examples of $k$ -indistinguishability relations, for $k \\in \\lbrace 1,2,3\\rbrace $ , considering the track $\\rho =v_0v_1v_2v_3 v_3v_2v_3v_3 v_2v_3v_2v_3 v_3v_2v_3v_2 v_1v_3v_2v_3 v_2v_1v_2v_1 v_3v_2v_2v_3v_2$ of the finite Kripke structure depicted in Figure REF .",
"The track $\\rho $ generates the descriptor sequence $\\rho _{ds}=(v_0,\\emptyset ,v_1) (v_0,\\lbrace v_1\\rbrace ,v_2) (v_0,\\lbrace v_1,v_2\\rbrace ,v_3)abaababaababcababcbcabbab$ , where $a$ , $b$ , and $c$ stand for $(v_0,\\lbrace v_1,v_2,v_3\\rbrace ,v_3)$ , $(v_0,\\lbrace v_1,v_2,v_3\\rbrace ,v_2)$ , and $(v_0,\\lbrace v_1,v_2,v_3\\rbrace ,v_1)$ , respectively.",
"The figure shows the subsequence $\\rho _{ds}(3, |\\rho _{ds}|-1)$ associated with the cluster $\\mathpzc {C}=\\lbrace a,b,c\\rbrace $ .",
"Pairs of $k$ -indistinguishable consecutive occurrences of descriptor elements are connected by a rounded edge labelled by $k$ .",
"Edges labelled by $\\times $ link occurrences which are not 1-indistinguishable.",
"The values of all missing edges can easily be derived using the property stated by Corollary REF below.",
"The meaning of numerical strings at the bottom of the figure will be clear later.",
"Figure: Examples of kk-indistinguishability relations.The next theorem establishes a fundamental connection between $k$ -indistinguishability of descriptor elements and $k$ -descriptor equivalence of tracks.",
"Theorem 25 Let $\\rho _{ds}$ be the descriptor sequence for a track $\\rho $ .",
"Two occurrences $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $0\\le i <j<|\\rho _{ds}|$ , of the same descriptor element are $k$ -indistinguishable if and only if $\\rho (0, i+1)\\sim _k\\rho (0,j+1)$ .",
"Let us assume that $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i<j$ , are $k$ -indistinguishable.",
"We prove by induction on $k \\ge 1$ that $\\rho (0, i+1)$ and $\\rho (0,j+1)$ are associated with the same $B_k$ -descriptor.",
"Base case ($k=1$ ).",
"Since $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ are occurrences of the same descriptor element, the roots of the $B_1$ -descriptors for $\\rho (0,i+1)$ and for $\\rho (0,j+1)$ are labelled by the same descriptor element.",
"Moreover, for each leaf of the $B_1$ -descriptor for $\\rho (0, i+1)$ there is a leaf of the $B_1$ -descriptor for $\\rho (0, j+1)$ with the same label, and vice versa, as by 1-indistinguishability $DElm(\\rho _{ds}(0,i-1))=DElm(\\rho _{ds}(0,j-1))$ .",
"Inductive step ($k\\ge 2$ ).",
"Since all the prefixes of $\\rho (0,i+1)$ are also prefixes of $\\rho (0,j+1)$ , we just need to focus on the prefixes $\\rho (0,t)$ , with $i+1 \\le t \\le j$ , in order to show that $\\rho (0, i+1)$ and $\\rho (0,j+1)$ have the same $B_k$ -descriptor.",
"By definition, any occurrence $\\rho _{ds}(i^{\\prime })$ with $i\\le i^{\\prime }<j$ , is $(k-1)$ -indistinguishable from another occurrence $\\rho _{ds}(i^{\\prime \\prime })$ , with $i^{\\prime \\prime }<i$ , of the same descriptor element.",
"By the inductive hypothesis, $\\rho (0, i^{\\prime }+1)$ and $\\rho (0, i^{\\prime \\prime }+1)$ are associated with the same $B_{k-1}$ -descriptor.",
"It follows that, for any proper prefix of $\\rho (0, j+1)$ (of length at least 2), there exists a proper prefix of $\\rho (0, i+1)$ with the same $B_{k-1}$ -descriptor, which implies that the tracks $\\rho (0,i+1)$ and $\\rho (0,j+1)$ are associated with the same $B_{k}$ -descriptor.",
"Conversely, we prove by induction on $k\\ge 1$ that if $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i<j$ , are not $k$ -indistinguishable, then the $B_k$ -descriptors for $\\rho (0, i+1)$ and $\\rho (0, j+1)$ are different.",
"We assume $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ to be occurrences of the same descriptor element (if this was not the case, the thesis would trivially follow, since the roots of the $B_k$ -descriptors for $\\rho (0,i+1)$ and $\\rho (0,j+1)$ would be labelled by different descriptor elements).",
"Base case ($k=1$ ).",
"If $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i < j$ , are not 1-indistinguishable, $DElm(\\rho _{ds}(0,i-1))\\subset DElm(\\rho _{ds}(0,j-1))$ .",
"Hence, there is $d\\in DElm(\\rho _{ds}(0,j-1))$ such that $d\\notin DElm(\\rho _{ds}(0,i-1))$ , and thus the $B_1$ -descriptor for $\\rho (0,j+1)$ has a leaf labelled by $d$ which is not present in the $B_1$ -descriptor for $\\rho (0,i+1)$ .",
"Inductive step ($k\\ge 2$ ).",
"If $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i<j$ , are not $k$ -indistinguishable, then there exists (at least) one occurrence $\\rho _{ds}(i^{\\prime })$ , with $i\\le i^{\\prime }<j$ , of a descriptor element $d$ which is not $(k-1)$ -indistinguishable from any occurrence of $d$ before position $i$ .",
"By the inductive hypothesis, $\\rho (0,i^{\\prime }+1)$ is associated to a $B_{k-1}$ -descriptor which is not equal to any $B_{k-1}$ -descriptors associated with proper prefixes of $\\rho (0, i+1)$ .",
"Thus, in the $B_k$ -descriptor for $\\rho (0,j+1)$ there exists a subtree of depth $k-1$ such that there is no isomorphic subtree of depth $k-1$ in the $B_k$ -descriptor for $\\rho (0,i+1)$ .",
"Note that $k$ -indistinguishability between occurrences of descriptor elements is defined only for pairs of prefixes of the same track, while the relation of $k$ -descriptor equivalence can be applied to pairs of any tracks of a Kripke structure.",
"The next corollary easily follows from Theorem REF .",
"Corollary 26 Let $\\rho _{ds}(i)$ , $\\rho _{ds}(j)$ , and $\\rho _{ds}(m)$ , with $0 \\le i<j<m<|\\rho _{ds}|$ , be three occurrences of the same descriptor element in a descriptor sequence $\\rho _{ds}$ .",
"If both the pair $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ and the pair $\\rho _{ds}(j)$ and $\\rho _{ds}(m)$ are $k$ -indistinguishable, for some $k\\ge 1$ , then $\\rho _{ds}(i)$ and $\\rho _{ds}(m)$ are also $k$ -indistinguishable."
],
[
"A model checking procedure for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ based on track representatives",
"In this section, we will exploit the $k$ -indistinguishability relation(s) between descriptor elements in a descriptor sequence $\\rho _{ds}$ for a track $\\rho $ to possibly replace $\\rho $ by a $k$ -descriptor equivalent, shorter track $\\rho ^{\\prime }$ of bounded length.",
"This allows us to find, for each $B_k$ -descriptor $\\mathpzc {D}_{B_k}$ (witnessed by a track of a finite Kripke structure $\\mathpzc {K}$ ), a track representative $\\tilde{\\rho }$ in $\\mathpzc {K}$ such that $(i)$ $\\mathpzc {D}_{B_k}$ is the $B_k$ -descriptor for $\\tilde{\\rho }$ and $(ii)$ the length of $\\tilde{\\rho }$ is bounded.",
"Thanks to property $(ii)$ , we can check all the track representatives of a finite Kripke structure by simply visiting its unravelling up to a bounded depth.",
"The notion of track representative can be explained as follows.",
"Let $\\rho _{ds}$ be the descriptor sequence for a track $\\rho $ .",
"If there are two occurrences of the same descriptor element $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $i<j$ , which are $k$ -indistinguishable—let $\\rho = \\rho (0,j+1)\\cdot \\overline{\\rho }$ , with $\\overline{\\rho }=\\rho (j+2,|\\rho |-1)$ —then we can replace $\\rho $ by the $k$ -descriptor equivalent, shorter track $\\rho (0, i+1)\\cdot \\overline{\\rho }$ .",
"By Theorem REF , $\\rho (0, i+1)$ and $\\rho (0, j+1)$ have the same $B_k$ -descriptor and thus, by Proposition REF , $\\rho =\\rho (0, j+1)\\cdot \\overline{\\rho }$ and $\\rho (0, i+1)\\cdot \\overline{\\rho }$ have the same $B_k$ -descriptor.",
"Moreover, since $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ are occurrences of the same descriptor element, $\\rho (i+1)=\\rho (j+1)$ and thus the track $\\rho (0,i+1)\\cdot \\overline{\\rho }$ is witnessed in the finite Kripke structure.",
"By iteratively applying such a contraction method, we can find a track $\\rho ^{\\prime }$ which is $k$ -descriptor equivalent to $\\rho $ , whose descriptor sequence is devoid of $k$ -indistinguishable occurrences of descriptor elements.",
"A track representative is a track that fulfils this property.",
"We now show how to calculate a bound to the length of track representatives.",
"We start by stating some technical properties.",
"The next proposition provides a bound to the distance within which we necessarily observe a repeated occurrence of some descriptor element in the descriptor sequence for a track.",
"We preliminarily observe that, for any track $\\rho $ , $|DElm(\\rho _{ds})|\\le 1+|W|^2$ , where $W$ is the set of states of the finite Kripke structure.",
"Indeed, in the descriptor sequence, the sets of internal states of prefixes of $\\rho $ increase monotonically with respect to the “$\\subseteq $ ” relation.",
"As a consequence, at most $|W|$ distinct sets may occur—excluding $\\emptyset $ which can occur only in the first descriptor element.",
"Moreover, these sets can be paired with all possible final states, which are at most $|W|$ .",
"Proposition 27 For each track $\\rho $ of $\\mathpzc {K}$ , associated with a descriptor element $d$ , there exists a track $\\rho ^{\\prime }$ of $\\mathpzc {K}$ , associated with the same descriptor element $d$ , such that $|\\rho ^{\\prime }|\\le 2+|W|^2$ .",
"By induction on the length $\\ell \\ge 2$ of $\\rho $ .",
"Base case ($\\ell =2$ ).",
"The track $\\rho $ satisfies the condition $\\ell \\le 2+|W|^2$ .",
"Inductive step ($\\ell >2$ ).",
"We distinguish two cases.",
"If $\\rho _{ds}$ has no duplicated occurrences of the same descriptor element, then $|\\rho _{ds}|\\le 1+|W|^2$ , since $|DElm(\\rho _{ds})|\\le 1+|W|^2$ , and thus $\\ell \\le 2+|W|^2$ (the length of $\\rho $ is equal to the length of $\\rho _{ds}$ plus 1).",
"On the other hand, if $\\rho _{ds}(i)=\\rho _{ds}(j)$ , for some $0\\le i<j<|\\rho _{ds}|$ , $\\rho (0,i+1)$ and $\\rho (0,j+1)$ are associated with the same descriptor element.",
"Now, $\\rho ^{\\prime }=\\rho (0,i+1)\\cdot \\rho (j+2, |\\rho |-1)$ is a track of $\\mathpzc {K}$ since $\\rho (i+1)=\\rho (j+1)$ , and, by Proposition REF , $\\rho =\\rho (0, j+1)\\cdot \\rho (j+2, |\\rho |-1)$ and $\\rho ^{\\prime }$ are associated with the same descriptor element.",
"By the inductive hypothesis, there exists a track $\\rho ^{\\prime \\prime }$ of $\\mathpzc {K}$ , associated with the same descriptor element of $\\rho ^{\\prime }$ (and of $\\rho $ ), with $|\\rho ^{\\prime \\prime }|\\le 2+|W|^2$ .",
"Proposition REF will be used in the following unravelling Algorithm as a termination criterion (referred to as 0-termination criterion) for unravelling a finite Kripke structure when it is not necessary to observe multiple occurrences of the same descriptor element: to get a track representative for every descriptor element with initial state $v$ , witnessed in a finite Kripke structure with set of states $W$ , we can avoid considering tracks longer than $2+|W|^2$ while exploring the unravelling of the Kripke structure from $v$ .",
"Let us now consider the (more difficult) problem of establishing a bound for tracks devoid of pairs of $k$ -indistinguishable occurrences of descriptor elements.",
"We first note that, in a descriptor sequence $\\rho _{ds}$ for a track $\\rho $ , there are at most $|W|$ occurrences of Type-1 descriptor elements.",
"On the other hand, Type-2 descriptor elements can occur multiple times and thus, to bound the length of $\\rho _{ds}$ , one has to constrain the number and the length of the subsequences of $\\rho _{ds}$ associated with clusters.",
"As for their number, it suffices to observe that they are separated by Type-1 descriptor elements, and hence at most $|W|$ of them, related to distinct clusters, can occur in a descriptor sequence.",
"As for their length, we can proceed as follows.",
"First, for any cluster $\\mathpzc {C}$ , it holds that $|\\mathpzc {C}|\\le |W|$ , as all (Type-2) descriptor elements of $\\mathpzc {C}$ share the same set $S$ of internal states and their final states $v_{fin}$ must belong to $S$ .",
"In the following, we consider the (maximal) subsequence $\\rho _{ds}(u,v)$ of $\\rho _{ds}$ associated with a specific cluster $\\mathpzc {C}$ , for some $0\\le u\\le v\\le |\\rho _{ds}|-1$ , and when we mention an index $i$ , we implicitly assume that $u\\le i\\le v$ , that is, $i$ refers to a position in the subsequence.",
"We sequentially scan such a subsequence suitably recording the multiplicity of occurrences of descriptor elements into an auxiliary structure.",
"To detect indistinguishable occurrences of descriptor elements up to indistinguishability $s \\ge 1$ , we use $s + 3$ arrays $Q_{-2}()$ , $Q_{-1}()$ , $Q_0()$ , $Q_1()$ , $\\ldots $ , $Q_s()$ .",
"Array elements are sets of descriptor elements of $\\mathpzc {C}$ : given an index $i$ , the sets at position $i$ , $Q_{-2}(i)$ , $Q_{-1}(i)$ , $Q_0(i)$ , $Q_1(i)$ , $\\ldots $ , $Q_s(i)$ , store information about indistinguishability for multiple occurrences of descriptor elements in the subsequence up to position $i>u$ .",
"To exemplify, if we find an occurrence of the descriptor element $d \\in \\mathpzc {C}$ at position $i$ , that is, $\\rho _{ds}(i)=d$ , we have that: $Q_{-2}(i)$ contains all descriptor elements of $\\mathpzc {C}$ which have never occurred in $\\rho _{ds}(u,i)$ ; $d\\in Q_{-1}(i)$ if $d$ has never occurred in $\\rho _{ds}(u,i-1)$ and $\\rho _{ds}(i)=d$ , that is, $\\rho _{ds}(i)$ is the first occurrence of $d$ in $\\rho _{ds}(u,i)$ ; $d \\in Q_{0}(i)$ if $d$ occurs at least twice in $\\rho _{ds}(u,i)$ and the occurrence $\\rho _{ds}(i)$ of $d$ is not 1-indistinguishable from the last occurrence of $d$ in $\\rho _{ds}(u,i-1)$ ; $d \\in Q_{t}(i)$ (for some $t\\ge 1$ ) if the occurrence $\\rho _{ds}(i)$ of $d$ is $t$ -indistinguishable, but not also $(t+1)$ -indistinguishable, from the last occurrence of $d$ in $\\rho _{ds}(u,i-1)$ .",
"In particular, at position $u$ (the first of the subsequence), $Q_{-1}(u)$ contains only the descriptor element $d =\\rho _{ds}(u)$ , $Q_{-2}(u)$ is the set $\\mathpzc {C}\\setminus \\lbrace d\\rbrace $ , and $Q_0(u)$ , $Q_1(u)$ , $\\dots $ are empty sets.",
"Arrays $Q_{-2}()$ , $Q_{-1}()$ , $Q_0()$ , $Q_1()$ , $\\ldots $ , $Q_s()$ satisfy the following constraints: for all positions $i$ , $\\bigcup ^s_{m=-2} Q_m(i)=\\mathpzc {C}$ and, for all $i$ and all $m\\ne m^{\\prime }$ , $Q_m(i)\\cap Q_{m^{\\prime }}(i)=\\emptyset $ .",
"Intuitively, at every position $i$ , $Q_{-2}(i)$ , $Q_{-1}(i), Q_0(i), Q_1(i)$ , $\\ldots $ , $Q_s(i)$ describe a state of the scanning process of the subsequence.",
"The change of state produced by the transition from position $i-1$ to position $i$ while scanning the subsequence is formally defined by the function $f$ , reported in Figure REF , which maps the descriptor sequence $\\rho _{ds}$ and a position $i$ to the tuple of sets $\\big (Q_{-2}(i),Q_{-1}(i),Q_0(i), Q_1(i), \\ldots , Q_s(i)\\big )$ .",
"Figure: Definition of the scan function ff.Note that, whenever a descriptor element $\\rho _{ds}(i)=d$ is such that $d\\in Q_z(i-1)$ and $d\\in Q_{z^{\\prime }}(i)$ , with $z<z^{\\prime }$ (cases (a), (b), and (d) of the definition of $f$ ), all $Q_{z^{\\prime \\prime }}(i)$ , with $z^{\\prime \\prime }>z^{\\prime }$ , are empty sets and, for all $z^{\\prime \\prime }\\ge z^{\\prime }$ , all elements in $Q_{z^{\\prime \\prime }}(i-1)$ belong to $Q_{z^{\\prime }}(i)$ .",
"As an intuitive explanation, consider, for instance, the following scenario: in a subsequence of $\\rho _{ds}$ , associated with some cluster $\\mathpzc {C}$ , $\\rho _{ds}(h)=\\rho _{ds}(i)=d\\in \\mathpzc {C}$ and $\\rho _{ds}(h^{\\prime })=\\rho _{ds}(i^{\\prime })=d^{\\prime }\\in \\mathpzc {C}$ , for some $h<h^{\\prime }<i<i^{\\prime }$ and $d \\ne d^{\\prime }$ , and there are not other occurrences of $d$ and $d^{\\prime }$ in $\\rho _{ds}(h,i^{\\prime })$ .",
"If $\\rho _{ds}(h)$ and $\\rho _{ds}(i)$ are exactly $z^{\\prime }$ -indistinguishable, by definition of the indistinguishability relation, $\\rho _{ds}(h^{\\prime })$ and $\\rho _{ds}(i^{\\prime })$ can be no more than $(z^{\\prime }+1)$ -indistinguishable.",
"Thus, if $d^{\\prime }$ is in $Q_{z^{\\prime \\prime }}(i-1)$ , for some $z^{\\prime \\prime }>z^{\\prime }$ , we can safely “downgrade” it to $Q_{z^{\\prime }}(i)$ , because we know that, when we meet the next occurrence of $d^{\\prime }$ ($\\rho _{ds}(i^{\\prime })$ ), $\\rho _{ds}(h^{\\prime })$ and $\\rho _{ds}(i^{\\prime })$ will be no more than $(z^{\\prime }+1)$ -indistinguishable.",
"In the following, we will make use of an abstract characterization of the state of arrays at a given position $i$ , as determined by the scan function $f$ , called configuration, that only accounts for the cardinality of sets in arrays.",
"Theorem REF states that, when a descriptor subsequence is scanned, configurations never repeat, since the sequence of configurations is strictly decreasing according to the lexicographical order $>_{lex}$ .",
"This property will allow us to establish the desired bound on the length of track representatives.",
"Definition 28 Let $\\rho _{ds}$ be the descriptor sequence for a track $\\rho $ and $i$ be a position in the subsequence of $\\rho _{ds}$ associated with a given cluster.",
"The configuration at position $i$ , denoted as $c(i)$ , is the tuple $c(i)=(|Q_{-2}(i)|,|Q_{-1}(i)|,|Q_{0}(i)|,|Q_{1}(i)|,\\cdots ,|Q_{s}(i)|),$ where $f(\\rho _{ds},i) = (Q_{-2}(i),Q_{-1}(i),Q_{0}(i),Q_{1}(i),\\cdots ,Q_{s}(i))$ .",
"An example of a sequence of configurations is given in Figure REF of Example REF , where, for each position in the subsequence $\\rho _{ds}(3, |\\rho _{ds}|-1)$ , we give the associated configuration: $c(3)=(2,1,0,0,0,0)$ , $c(4)=(1,2,0,0,0,0)$ , and so forth.",
"Theorem 29 Let $\\rho _{ds}$ be the descriptor sequence for a track $\\rho $ and $\\rho _{ds}(u,v)$ , for some $u <v$ , be the subsequence associated with a cluster $\\mathpzc {C}$ .",
"For all $u<i\\le v$ , if $\\rho _{ds}(i)=d$ , then it holds that $d\\in Q_t(i-1)$ , $d\\in Q_{t+1}(i)$ , for some $t\\in \\lbrace -2,-1\\rbrace \\cup \\mathbb {N}$ , and $c(i-1)>_{lex}c(i)$ .",
"The proof is given in REF .",
"We show now how to select all and only those tracks which do not feature any pair of $k$ -indistinguishable occurrences of descriptor elements.",
"To this end, we make use of a scan function $f$ which uses $k + 3$ arrays (the value $k+3$ accounts for the parameter $k$ of descriptor element indistinguishability, plus the three arrays $Q_{-2}()$ , $Q_{-1}()$ , $Q_0()$ ).",
"Theorem REF guarantees that, while scanning a subsequence, configurations never repeat.",
"This allows us to set an upper bound to the length of a track such that, whenever exceeded, the descriptor sequence for the track features at least a pair of $k$ -indistinguishable occurrences of some descriptor element.",
"The bound is essentially given by the number of possible configurations for $k + 3$ arrays.",
"By an easy combinatorial argument, we can prove the following proposition.",
"Proposition 30 For all $n,t\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ , the number of distinct $t$ -tuples of natural numbers whose sum equals $n$ is $\\varepsilon (n,t)=\\binom{n+t-1}{n}=\\binom{n+t-1}{t-1}$ .",
"The following figure suggests an alternative representation of a tuple, in the form of a configuration of separators/bullets: Table: NO_CAPTION $\\leftrightsquigarrow $ $(5,3,1,0,1)$ It can be easily checked that such a representation is unambiguous, i.e., there exists a bijection between configurations of separators/bullets and tuples.",
"The sum of the natural numbers of the tuple equals the number of bullets, and the size of the tuple is the number of separators plus 1.",
"Since there are $\\varepsilon (n,t)=\\binom{n+t-1}{t-1}$ distinct ways of choosing $t-1$ separators among $n+t-1$ different places—and places which are not chosen must contain bullets—there are exactly $\\varepsilon (n,t)$ distinct $t$ -tuples of natural numbers whose sum equals $n$ .",
"Proposition REF provides two upper bounds for $\\varepsilon (n,t)$ : $\\varepsilon (n,t)\\le (n+1)^{t-1}$ and $\\varepsilon (n,t)\\le t^n$ .",
"Since a configuration $c(i)$ of a cluster $\\mathpzc {C}$ is a $(k+3)$ -tuple whose elements add up to $|\\mathpzc {C}|$ , by Proposition REF we conclude that there are at most $\\varepsilon (|\\mathpzc {C}|,k+3)=\\binom{|\\mathpzc {C}|+k+2}{k+2}$ distinct configurations of size $(k+3)$ , whose natural numbers add up to $|\\mathpzc {C}|$ .",
"Moreover, since configurations never repeat while scanning a subsequence associated with a cluster $\\mathpzc {C}$ , $\\varepsilon (|\\mathpzc {C}|,k+3)$ is an upper bound to the length of such a subsequence.",
"Now, for any track $\\rho $ , $\\rho _{ds}$ features at most $|W|$ subsequences associated with distinct clusters $\\mathpzc {C}_1,\\mathpzc {C}_2,\\dots $ , and thus, if the following upper bound to the length of $\\rho $ is exceeded, then there is at least one pair of $k$ -indistinguishable occurrences of some descriptor element in $\\rho _{ds}$ : $|\\rho |\\le 1+(|\\mathpzc {C}_1|+1)^{k+2}+(|\\mathpzc {C}_2|+1)^{k+2}+\\cdots + (|\\mathpzc {C}_s|+1)^{k+2}+|W|$ , where $s\\le |W|$ , and the last addend is to count occurrences of Type-1 descriptor elements.",
"Since clusters are disjoint, their union is a subset of $DElm(\\rho _{ds})$ , and $|DElm(\\rho _{ds})|\\le 1+|W|^2$ , we get: $|\\rho |\\le 1+(|\\mathpzc {C}_1| + |\\mathpzc {C}_2| +\\cdots + |\\mathpzc {C}_s| + |W|)^{k+2}+|W|\\le 1+(|DElm(\\rho _{ds})| + |W|)^{k+2}+|W|\\\\ \\le 1+(1+|W|^2 + |W|)^{k+2}+|W|\\le 1+(1+|W|)^{2k+4}+|W|.$ Analogously, by using the alternative bound to $\\varepsilon (|\\mathpzc {C}|,k+3)$ , we have that $|\\rho |\\le 1+(k+3)^{|\\mathpzc {C_1}|}+(k+3)^{|\\mathpzc {C_2}|}+ \\cdots + (k+3)^{|\\mathpzc {C_s}|} +|W|\\le 1+(k+3)^{|\\mathpzc {C_1}|+|\\mathpzc {C_2}|+\\cdots + |\\mathpzc {C_s}|}+|W|\\\\ \\le 1+(k+3)^{|DElm(\\rho _{ds})|}+|W|\\le 1+(k+3)^{|W|^2+1}+|W|.$ The upper bound for $|\\rho |$ is then the least of the two given upper bounds: $\\tau (|W|,k)= \\min \\big \\lbrace 1+(1+|W|)^{2k+4}+|W|, 1+(k+3)^{|W|^2+1}+|W|\\big \\rbrace .$ Theorem 31 Let $\\mathpzc {K}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ be a finite Kripke structure and $\\rho $ be a track in $\\operatorname{Trk}_\\mathpzc {K}$ .",
"If $|\\rho |>\\tau (|W|,k)$ , then there exists another track in $\\operatorname{Trk}_\\mathpzc {K}$ , whose length is less than or equal to $\\tau (|W|,k)$ , associated with the same $B_k$ -descriptor as $\\rho $ .",
"[Proof (sketch)] If $|\\rho |>\\tau (|W|,k)$ , then there exists (at least) a subsequence of $\\rho _{ds}$ , associated with some cluster $\\mathpzc {C}$ , which contains (at least) a pair of $k$ -indistinguishable occurrences of some descriptor element $d\\in \\mathpzc {C}$ , say $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ , with $j<i$ .",
"By Theorem REF , the two tracks $\\tilde{\\rho }_1=\\rho (0, j+1)$ and $\\tilde{\\rho }_2=\\rho (0, i+1)$ have the same $B_k$ -descriptor.",
"Now, let us rewrite the track $\\rho $ as the concatenation $\\tilde{\\rho }_2\\cdot \\overline{\\rho }$ for some $\\overline{\\rho }$ .",
"By Proposition REF , the tracks $\\rho =\\tilde{\\rho }_2\\cdot \\overline{\\rho }$ and $\\rho ^{\\prime }=\\tilde{\\rho }_1\\cdot \\overline{\\rho }$ are associated with the same $B_k$ -descriptor.",
"Since $\\operatorname{lst}(\\tilde{\\rho }_1)=\\operatorname{lst}(\\tilde{\\rho }_2)$ ($\\rho _{ds}(j)$ and $\\rho _{ds}(i)$ are occurrences of the same descriptor element $d$ ), $\\rho ^{\\prime }=\\tilde{\\rho }_1\\cdot \\overline{\\rho }$ is a track of $\\mathpzc {K}$ shorter than $\\rho $ .",
"If $|\\rho ^{\\prime }|\\le \\tau (|W|,k)$ , we have proved the thesis; otherwise, we can iterate the process by applying the above contraction to $\\rho ^{\\prime }$ .",
"Theorem REF allows us to define a termination criterion to bound the depth of the unravelling of a finite Kripke structure ($(k\\ge 1)$ -termination criterion), while searching for track representatives for witnessed $B_k$ -descriptors: for any $k\\ge 1$ , to get a track representative for every $B_k$ -descriptor, with initial state $v$ , and witnessed in a finite Kripke structure with set of states $W$ , we can avoid taking into consideration tracks longer than $\\tau (|W|,k)$ while exploring the unravelling of the structure from $v$ .",
"Thanks to the above results, we are now ready to define a model checking algorithm for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas.",
"First, we introduce the unravelling Algorithm , which explores the unravelling of the input Kripke structure $\\mathpzc {K}$ to find track representatives for all witnessed $B_k$ -descriptors.",
"It features two modalities, forward mode (which is active when its fourth parameter, direction, is forw) and backward mode (active when the parameter direction is backw), in which the unravelling of $\\mathpzc {K}$ is visited following the direction of edges and against their direction (that is equivalent to visiting the transposed graph $\\overline{\\mathpzc {K}}$ of $\\mathpzc {K}$ ), respectively.",
"In both cases, if there exist $k$ -indistinguishable occurrences of a descriptor element in $\\rho _{ds}$ , the track $\\rho $ is never returned.",
"[tb] [1] direction = forw Unravel $\\mathpzc {K}$ starting from $v$ according to $\\ll $ “$\\ll $ ” is an arbitrary order of the nodes of $\\mathpzc {K}$ For every new node of the unravelling met during the visit, return the track $\\rho $ from $v$ to the current node only if: $k=0$ Apply the 0-termination criterion The last descriptor element $d$ of (the descriptor sequence of) the current track $\\rho $ is $k$ -indistinguishable from a previous occurrence of $d$ skip $\\rho $ and backtrack to $\\rho (0, |\\rho |-2)\\cdot \\overline{v}$ , where $\\overline{v}$ is the minimum state (w.r.t.",
"$\\ll $ ), greater than $\\rho (|\\rho |-1)$ , such that $(\\rho (|\\rho |-2),\\overline{v})$ is an edge of $\\mathpzc {K}$ .",
"direction = backw Unravel $\\overline{\\mathpzc {K}}$ starting from $v$ according to $\\ll $$\\overline{\\mathpzc {K}}$ is $\\mathpzc {K}$ with transposed edges For every new node of the unravelling met during the visit, consider the track $\\rho $ from the current node to $v$, and recalculate descriptor element indistinguishability from scratch (left to right); return the track only if: $k=0$ Apply the 0-termination criterion There exist two $k$ -indistinguishable occurrences of a descriptor element $d$ in (the descriptor sequence of) the current track $\\rho $ skip $\\rho $ Do not visit tracks of length greater than $\\tau (|W|,k)$ Unrav$(\\mathpzc {K},v,k,\\text{direction})$ In the forward mode (which will be used to deal with $\\operatorname{\\langle A\\rangle }$ and $\\operatorname{\\langle \\overline{B}\\rangle }$ modalities), the direction of track exploration and that of indistinguishability checking are the same, so we can stop extending a track as soon as the first pair of $k$ -indistinguishable occurrences of a descriptor element is found in the descriptor sequence, suggesting an easy termination criterion for stopping the unravelling of tracks.",
"In the backward mode (used in the case of $\\operatorname{\\langle \\overline{A}\\rangle }$ and $\\operatorname{\\langle \\overline{E}\\rangle }$ modalities), such a straightforward criterion cannot be adopted, because tracks are explored right to left (the opposite direction with respect to edges of the Kripke structure), while the indistinguishability relation over descriptor elements is computed left to right.",
"In general, changing the prefix of a considered track requires recomputing from scratch the descriptor sequence and the indistinguishability relation over descriptor elements.",
"In particular, $k$ -indistinguishable occurrences of descriptor elements can be detected in the middle of a subsequence, and not necessarily at the end.",
"In this latter case, however, the upper bound $\\tau (|W|,k)$ on the maximum depth of the unravelling ensures the termination of the algorithm (line 17).",
"The next theorem proves soundness and completeness of Algorithm for the forward mode.",
"The proof for the backward one is quite similar, and thus omitted.",
"Theorem 32 Let $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ be a finite Kripke structure, $v\\in W$ , and $k\\in \\mathbb {N}$ .",
"For every track $\\rho $ of $\\mathpzc {K}$ , with $\\operatorname{fst}(\\rho )=v$ and $|\\rho |\\ge 2$ , the unravelling Algorithm returns a track $\\rho ^{\\prime }$ of $\\mathpzc {K}$ , with $\\operatorname{fst}(\\rho ^{\\prime })=v$ , such that $\\rho $ and $\\rho ^{\\prime }$ are associated with the same $B_k$ -descriptor and $|\\rho ^{\\prime }|\\le \\tau (|W|,k)$ .",
"If $k=0$ the thesis follows immediately by the 0-termination criterion.",
"So let us assume $k\\ge 1$ .",
"The proof is by induction on $\\ell =|\\rho |$ .",
"(Case $\\ell =2$ ) In this case, $\\rho _{ds}=(\\operatorname{fst}(\\rho ),\\emptyset ,\\operatorname{lst}(\\rho ))$ , and the only descriptor element of the sequence is Type-1.",
"Thus, $\\rho $ itself is returned by the algorithm.",
"(Case $\\ell >2$ ) If in $\\rho _{ds}$ there are no pairs of $k$ -indistinguishable occurrences of some descriptor element, the termination criterion of Algorithm can never be applied.",
"Thus, $\\rho $ itself is returned (as soon as it is visited) and its length is at most $\\tau (|W|,k)$ .",
"Otherwise, the descriptor sequence of any track $\\rho $ can be split into 3 parts: $\\rho _{ds}=\\rho _{ds1}\\cdot \\rho _{ds2}\\cdot \\rho _{ds3}$ , where $\\rho _{ds1}$ ends with a Type-1 descriptor element and it does not contain pairs of $k$ -indistinguishable occurrences of any descriptor element; $\\rho _{ds2}$ is a subsequence associated with a cluster $\\mathpzc {C}$ of (Type-2) descriptor elements with at least a pair of $k$ -indistinguishable occurrences of descriptor elements; $\\rho _{ds3}$ (if it is not the empty sequence) begins with a Type-1 descriptor element.",
"This amounts to say that $\\rho _{ds2}$ is the “leftmost” subsequence of $\\rho _{ds}$ consisting of elements of a cluster $\\mathpzc {C}$ , with at least a pair of $k$ -indistinguishable occurrences of some descriptor element.",
"Therefore, there are two indexes $i,j$ , with $j<i$ , such that $\\rho _{ds2}(j)$ and $\\rho _{ds2}(i)$ are two $k$ -indistinguishable occurrences of some $d\\in \\mathpzc {C}$ in $\\rho _{ds}$ .",
"By Proposition REF , there exists a pair of indexes $i^{\\prime },j^{\\prime }$ , with $j^{\\prime }<i^{\\prime }$ , such that $\\rho _{ds2}(j^{\\prime })$ and $\\rho _{ds2}(i^{\\prime })$ are two consecutive $k$ -indistinguishable occurrences of $d$ (by consecutive we mean that, for all $t\\in [j^{\\prime }+1,i^{\\prime }-1]$ , $\\rho _{ds2}(t)\\ne d$ ).",
"If there are many such pairs (even for different elements in $\\mathpzc {C}$ ), let us consider the one with the lower index $i^{\\prime }$ (namely, precisely the pair which is found earlier by the unravelling algorithm).",
"By Theorem REF , the two tracks associated with $\\rho _{ds1}\\cdot \\rho _{ds2}(0, j^{\\prime })$ and $\\rho _{ds1}\\cdot \\rho _{ds2}(0, i^{\\prime })$ , say $\\tilde{\\rho }_1$ and $\\tilde{\\rho }_2$ respectively, have the same $B_k$ -descriptor.",
"Then, by Proposition REF , the tracks $\\rho =\\tilde{\\rho }_2\\cdot \\overline{\\rho }$ (for some $\\overline{\\rho }$ ) and $\\rho ^{\\prime }=\\tilde{\\rho }_1\\cdot \\overline{\\rho }$ have the same $B_k$ -descriptor.",
"Algorithm does not return $\\tilde{\\rho }_2$ and, due to the backtrack step, neither $\\rho =\\tilde{\\rho }_2\\cdot \\overline{\\rho }$ is returned.",
"But since $\\operatorname{lst}(\\tilde{\\rho }_1)=\\operatorname{lst}(\\tilde{\\rho }_2)$ ($\\rho _{ds2}(j^{\\prime })$ and $\\rho _{ds2}(i^{\\prime })$ are occurrences of the same descriptor element), the unravelling of $\\mathpzc {K}$ features $\\rho ^{\\prime }=\\tilde{\\rho }_1\\cdot \\overline{\\rho }$ , as well.",
"Now, by induction hypothesis, a track $\\rho ^{\\prime \\prime }$ of $\\mathpzc {K}$ is returned, such that $\\rho ^{\\prime }$ and $\\rho ^{\\prime \\prime }$ have the same $B_k$ -descriptor, and $|\\rho ^{\\prime \\prime }|\\le \\tau (|W|,k)$ .",
"$\\rho $ has in turn the same $B_k$ -descriptor as $\\rho ^{\\prime \\prime }$ .",
"The above proof shows how a “contracted variant” of a track $\\rho $ is (indirectly) computed by Algorithm .",
"As an example, $\\rho ^{\\prime }=v_0v_1v_2v_3v_3v_2v_3v_3v_2v_3v_2v_3v_2v_1v_3v_2v_3v_2v_1v_2v_1v_3v_2$ is returned by Algorithm in place of the track $\\rho $ of Example REF , and it can be checked that $\\rho ^{\\prime }_{ds}$ does not contain any pair of 3-indistinguishable occurrences of a descriptor element and that $\\rho $ and $\\rho ^{\\prime }$ have the same $B_3$ -descriptor.",
"[p] [1] $k\\leftarrow \\operatorname{Nest_B}(\\psi )$ $u\\leftarrow New\\left(\\texttt {Unrav}(\\mathpzc {K},w_0,k,\\textsc {forw})\\right)$$w_0$ is the initial state of $\\mathpzc {K}$ $u.\\texttt {hasMoreTracks()}$ $\\tilde{\\rho }\\leftarrow u.\\texttt {getNextTrack()}$ $\\texttt {Check}(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=0$ 0: “$\\mathpzc {K},\\tilde{\\rho }\\lnot \\models \\psi $ ” 1: “$\\mathpzc {K}\\models \\psi $ ” ModCheck$(\\mathpzc {K},\\psi )$ [p] 2 [1] $\\psi =\\top $ 1 $\\psi =\\bot $ 0 $\\psi =p\\in \\mathpzc {AP}$ $p\\in \\bigcap _{s\\in \\operatorname{states}(\\tilde{\\rho })}\\mu (s)$ return 1 else return 0 $\\psi =\\lnot \\varphi $ 1 $-$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho })$ $\\psi =\\varphi _1\\wedge \\varphi _2$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi _1,\\tilde{\\rho })=0$ 0 $\\texttt {Check}(\\mathpzc {K},k,\\varphi _2,\\tilde{\\rho })$ $\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ $u\\leftarrow New\\left(\\texttt {Unrav}(\\mathpzc {K},\\operatorname{lst}(\\tilde{\\rho }),k,\\textsc {forw})\\right)$ $u.\\texttt {hasMoreTracks()}$ $\\rho \\leftarrow u.\\texttt {getNextTrack()}$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\rho )=1$ 1 0 $\\psi =\\operatorname{\\langle \\overline{A}\\rangle }\\varphi $ $u\\leftarrow New\\left(\\texttt {Unrav}(\\mathpzc {K},\\operatorname{fst}(\\tilde{\\rho }),k,\\textsc {backw})\\right)$ $u.\\texttt {hasMoreTracks()}$ $\\rho \\leftarrow u.\\texttt {getNextTrack()}$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\rho )=1$ 1 0$\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ each $\\overline{\\rho }$ prefix of $\\tilde{\\rho }$ $\\texttt {Check}(\\mathpzc {K},k-1,\\varphi ,\\overline{\\rho })=1$ 1 0 $\\psi =\\operatorname{\\langle \\overline{B}\\rangle }\\varphi $ each $v\\in W$ s.t.",
"$(\\operatorname{lst}(\\tilde{\\rho }),v)\\in \\delta $ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho }\\cdot v)=1$ 1 $u\\leftarrow New\\left(\\texttt {Unrav}(\\mathpzc {K},v,k,\\textsc {forw})\\right)$ $u.\\texttt {hasMoreTracks()}$ $\\rho \\leftarrow u.\\texttt {getNextTrack()}$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho }\\cdot \\rho )=1$ 1 0 $\\psi =\\operatorname{\\langle \\overline{E}\\rangle }\\varphi $ each $v\\in W$ s.t.",
"$(v,\\operatorname{fst}(\\tilde{\\rho }))\\in \\delta $ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,v \\cdot \\tilde{\\rho })=1$ 1 $u\\leftarrow New\\left(\\texttt {Unrav}(\\mathpzc {K},v,k,\\textsc {backw})\\right)$ $u.\\texttt {hasMoreTracks()}$ $\\rho \\leftarrow u.\\texttt {getNextTrack()}$ $\\texttt {Check}(\\mathpzc {K},k,\\varphi ,\\rho \\cdot \\tilde{\\rho })=1$ 1 0 Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })$ Algorithm can be used to define the model checking procedure ModCheck$(\\mathpzc {K},\\psi )$ (Algorithm ).",
"ModCheck$(\\mathpzc {K},\\psi )$ exploits the procedure $\\texttt {Check}(\\mathpzc {K},k, \\psi ,\\tilde{\\rho })$ (Algorithm ), which checks a formula $\\psi $ of B-nesting depth $k$ against a track $\\tilde{\\rho }$ of the Kripke structure $\\mathpzc {K}$ .",
"$\\texttt {Check}(\\mathpzc {K},k, \\psi ,\\tilde{\\rho })$ basically calls itself recursively on the subformulas of $\\psi $ , and uses the unravelling Algorithm to deal with $\\operatorname{\\langle A\\rangle }$ , $\\operatorname{\\langle \\overline{A}\\rangle }$ , $\\operatorname{\\langle \\overline{B}\\rangle }$ , and $\\operatorname{\\langle \\overline{E}\\rangle }$ modalities.",
"Soundness and completeness of these two procedures are stated by Lemma REF and Theorem REF below, whose proofs can be found in REF and REF , respectively.",
"Lemma 33 Let $\\psi $ be an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula with $\\operatorname{Nest_B}(\\psi )=k$ , $\\mathpzc {K}$ be a finite Kripke structure, and $\\tilde{\\rho }$ be a track in $\\operatorname{Trk}_\\mathpzc {K}$ .",
"It holds that $\\texttt {Check}(\\mathpzc {K},k, \\psi ,\\tilde{\\rho })=1$ if and only if $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ .",
"Theorem 34 Let $\\psi $ be an $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula and $\\mathpzc {K}$ be a finite Kripke structure.",
"It holds that ModCheck$(\\mathpzc {K},\\psi )=1$ if and only if $\\mathpzc {K}\\models \\psi $ .",
"The model checking algorithm ModCheck requires exponential working space, as it uses an instance of the unravelling algorithm and some additional space for a track $\\tilde{\\rho }$ .",
"Analogously, every recursive call to Check (possibly) needs an instance of the unravelling algorithm and space for a track.",
"There are at most $|\\psi |$ jointly active calls to Check (plus one to ModCheck), thus the maximum space needed by the considered algorithms is $\\left(|\\psi |+1\\right)\\cdot O(|W|+\\operatorname{Nest_B}(\\psi ))\\cdot \\tau (|W|,\\operatorname{Nest_B}(\\psi ))$ bits overall, where $\\tau (|W|,\\operatorname{Nest_B}(\\psi ))$ is the maximum length of track representatives, and $O(|W|+\\operatorname{Nest_B}(\\psi ))$ bits are needed to represent a state of $\\mathpzc {K}$ , a descriptor element, and a counter for $k$ -indistinguishability.",
"In conclusion, we have proved that the model checking problem for formulas of the HS fragment $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ over finite Kripke structures is in EXPSPACE.",
"As a particular case, formulas $\\psi $ of the fragment $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ can be checked in polynomial working space by ModCheck, as its formulas do not feature $\\operatorname{\\langle B\\rangle }$ modality (hence $\\operatorname{Nest_B}(\\psi )=0$ ).",
"Thus, the model checking problem for $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ is in PSPACE.",
"In the next section, we prove that it is actually PSPACE-complete.",
"As a direct consequence, $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ turns out to be PSPACE-hard.",
"The next theorem proves that the model checking problem for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ is NEXP-hard if a succinct encoding of formulas is adopted (the proof is given in REF ).",
"Theorem REF The model checking problem for succinctly encoded formulas of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ over finite Kripke structures is NEXP-hard (under polynomial-time reductions)."
],
[
"The fragment $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$",
"In this section, we prove that the model checking algorithm described in the previous section, applied to $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ formulas, is optimal by showing that model checking for $\\mathsf {A\\overline{B}}$ is a PSPACE-hard problem (Theorem REF ).",
"PSPACE-completeness of $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ (and $\\mathsf {A\\overline{B}}$ ) immediately follows.",
"As a by-product, model checking for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ is PSPACE-hard as well.",
"Before proving Theorem REF , we give an example showing that the three HS fragments $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ , $\\mathsf {\\forall A\\overline{A}BE}$ , and $\\mathsf {A\\overline{A}}$ , on which we focus in this (and the next) section, are expressive enough to capture meaningful properties of state-transition systems.",
"Example 5 Figure: A simple state-transition system.Let $\\mathpzc {K}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ , with $\\mathpzc {AP}=\\lbrace r_0, r_1,e_0,e_1,x_0\\rbrace $ , be the Kripke structure of Figure REF , that models the interactions between a scheduler $\\mathpzc {S}$ and two processes, $\\mathcal {P}_0$ and $\\mathcal {P}_1$ , which possibly ask for a shared resource.",
"At the initial state $w_0$ , $\\mathpzc {S}$ has not received any request from the processes yet, while in $w_1$ (resp., $w_2$ ) only $\\mathcal {P}_0$ (resp., $\\mathcal {P}_1$ ) has sent a request, and thus $r_0$ (resp., $r_1$ ) holds.",
"As long as at most one process has issued a request, $\\mathpzc {S}$ is not forced to allocate the resource ($w_1$ and $w_2$ have self loops).",
"At state $w_3$ , both $\\mathcal {P}_0$ and $\\mathcal {P}_1$ are waiting for the shared resource (both $r_0$ and $r_1$ hold).",
"State $w_3$ has transitions only towards $w_4$ , $w_6$ , and $w_8$ .",
"At state $w_4$ (resp., $w_6$ ) $\\mathcal {P}_1$ (resp., $\\mathcal {P}_0$ ) can access the resource and $e_1$ (resp., $e_0$ ) holds in the interval $w_4w_5$ (resp., $w_6w_7$ ).",
"In addition, a faulty transition may be taken from $w_3$ leading to states $w_8$ and $w_9$ where both $\\mathcal {P}_0$ and $\\mathcal {P}_1$ use the resource (both $e_0$ and $e_1$ hold in the interval $w_8w_9$ ).",
"Finally, from $w_5$ , $w_7$ , and $w_9$ the system can only move to $w_0$ , where $\\mathpzc {S}$ waits for new requests from $\\mathcal {P}_0$ and $\\mathcal {P}_1$ .",
"Let $\\mathpzc {P}$ be the set $\\lbrace r_0,r_1,e_0,e_1\\rbrace $ and let $x_0$ be an auxiliary proposition letter labelling the states $w_0$ , $w_1$ , $w_6$ , and $w_7$ , where $\\mathpzc {S}$ and $\\mathcal {P}_0$ , but not $\\mathcal {P}_1$ , are active.",
"We now give some examples of formulas in the fragments $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ , $\\mathsf {\\forall A\\overline{A}BE}$ , and $\\mathsf {A\\overline{A}}$ that encode requirements for $\\mathpzc {K}$ .",
"As in Example REF , we force the validity of the considered property over all legal computation sub-intervals by using the modality $[E]$ , or alternatively the modality $[A]$ (any computation sub-interval occurs after at least one initial track).",
"It can be checked that $\\mathpzc {K}\\lnot \\models [E]\\lnot (e_0\\wedge e_1)$ (the formula is in $\\mathsf {\\forall A\\overline{A}BE}$ ), i.e., mutual exclusion is not guaranteed, as the faulty transition leading to $w_8$ may be taken at $w_3$ , and then both $\\mathcal {P}_0$ and $\\mathcal {P}_1$ access the resource in the interval $w_8w_9$ where $e_0\\wedge e_1$ holds.",
"On the contrary, it holds that $\\mathpzc {K}\\models [A]\\big (r_0\\rightarrow \\operatorname{\\langle A\\rangle }e_0 \\vee \\operatorname{\\langle A\\rangle }\\operatorname{\\langle A\\rangle }e_0\\big )$ (the formula is in $\\mathsf {A\\overline{A}}$ and $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ ).",
"Such a formula expresses the following reachability property: if $r_0$ holds over some interval, then it is always possible to reach an interval where $e_0$ holds.",
"Obviously, this does not mean that all possible computations will necessarily lead to such an interval, but that the system is never trapped in a state from which it is no more possible to satisfy requests from $\\mathcal {P}_0$ .",
"It also holds that $\\mathpzc {K}\\models [A]\\big (r_0\\wedge r_1\\rightarrow [A](e_0\\vee e_1\\vee \\bigwedge _{p\\in \\mathpzc {P}}\\lnot p)\\big )$ (in $\\mathsf {A\\overline{A}}$ and $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ ).",
"Indeed, if both processes send a request (state $w_3$ ), then $\\mathpzc {S}$ immediately allocates the resource.",
"In detail, if $r_0\\wedge r_1$ holds over some tracks (the only possible intervals are $w_3w_4$ , $w_3w_6$ , and $w_3w_8$ ), then in any possible subsequent interval of length 2 $e_0\\vee e_1$ holds, that is, $\\mathcal {P}_0$ or $\\mathcal {P}_1$ are executed, or, considering tracks longer than 2, $\\bigwedge _{p\\in \\mathpzc {P}}\\lnot p$ holds.",
"On the contrary, if only one process asks for the resource, then $\\mathpzc {S}$ can arbitrarily delay the allocation, and therefore $\\mathpzc {K}\\lnot \\models [A]\\big (r_0\\rightarrow [A](e_0\\vee \\bigwedge _{p\\in \\mathpzc {P}}\\lnot p)\\big )$ .",
"Finally, it holds that $\\mathpzc {K}\\models x_0\\rightarrow \\operatorname{\\langle \\overline{B}\\rangle }x_0$ (in $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ ), that is, any initial track satisfying $x_0$ (any such track involves states $w_0$ , $w_1$ , $w_6$ , and $w_7$ only) can be extended to the right in such a way that the resulting track still satisfies $x_0$ .",
"This amounts to say that there exists a computation in which $\\mathcal {P}_1$ starves.",
"Note that $\\mathpzc {S}$ and $\\mathcal {P}_0$ can continuously interact without waiting for $\\mathcal {P}_1$ .",
"This is the case, for instance, when $\\mathcal {P}_1$ is not asking for the shared resource at all.",
"Now, in order to prove Theorem REF , we provide a reduction from the QBF problem (i.e., the problem of determining the truth of a fully-quantified Boolean formula in prenex normal form)—which is known to be PSPACE-complete (see, for example, [33])—to the model checking problem for $\\mathsf {A\\overline{B}}$ formulas over finite Kripke structures.",
"We consider a quantified Boolean formula $\\psi \\!",
"= \\!",
"Q_n x_n Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\!\\cdots \\!",
",x_1)$ where $Q_i\\in \\lbrace \\exists , \\forall \\rbrace $ for all $i=1,\\cdots ,n$ , and $\\phi (x_n,x_{n-1},\\cdots ,x_1)$ is a quantifier-free Boolean formula.",
"Let $Var = \\lbrace x_n,\\ldots ,x_1\\rbrace $ be the set of variables of $\\psi $ .",
"We define the Kripke structure $\\mathpzc {K}_{QBF}^{Var}$ , whose initial tracks represent all the possible assignments to the variables of $Var$ .",
"For each $x \\in Var$ , $\\mathpzc {K}_{QBF}^{Var}$ features four states, $w_x^{\\top 1}$ , $w_x^{\\top 2}$ , $w_x^{\\bot 1}$ , and $w_x^{\\bot 2}$ : the first two represent a $\\top $ truth assignment to $x$ and the last two a $\\bot $ one.",
"$\\mathpzc {K}_{QBF}^{Var}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ is formally defined as follows: $\\mathpzc {AP}= Var \\cup \\lbrace start\\rbrace \\cup \\lbrace x_{i\\,aux} \\mid 1\\le i\\le n\\rbrace $ ; $W= \\lbrace w_{x_i}^\\ell \\mid 1\\le i\\le n, \\ell \\in \\lbrace \\bot _1,\\bot _2,\\top _1,\\top _2\\rbrace \\rbrace \\cup \\lbrace w_0,w_1,sink\\rbrace $ ; if $n=0$ , $\\delta =\\lbrace (w_0,w_1),(w_1,sink),(sink,sink)\\rbrace $ ; if $n>0$ , $\\delta = \\lbrace (w_0,w_1),(w_1,w_{x_n}^{\\top _1}),(w_1,w_{x_n}^{\\bot _1})\\rbrace \\cup \\lbrace (w_{x_i}^{\\top _1},w_{x_i}^{\\top _2}), (w_{x_i}^{\\bot _1},w_{x_i}^{\\bot _2}) \\mid 1 \\le i \\le n\\rbrace \\cup \\lbrace (w_{x_i}^\\ell ,w_{x_{i-1}}^m) \\mid \\ell \\in \\lbrace \\bot _2,\\top _2\\rbrace , m \\in \\lbrace \\bot _1,\\top _1\\rbrace , 2 \\le i \\le n\\rbrace \\cup \\lbrace (w_{x_1}^{\\top _2},sink),(w_{x_1}^{\\bot _2},sink)\\rbrace \\cup \\lbrace (sink,sink) \\rbrace $ .",
"$\\mu (w_0) = \\mu (w_1) = Var \\cup \\lbrace start\\rbrace $ ; $\\mu (w_{x_i}^\\ell ) = Var \\cup \\lbrace x_{i\\, aux}\\rbrace $ , for $1\\le i\\le n$ and $\\ell \\in \\lbrace \\top _1,\\top _2\\rbrace $ ; $\\mu (w_{x_i}^\\ell ) = (Var \\setminus \\lbrace x_i\\rbrace ) \\cup \\lbrace x_{i\\, aux}\\rbrace $ , for $1\\le i\\le n$ and $\\ell \\in \\lbrace \\bot _1,\\bot _2\\rbrace $ ; $\\mu (sink) = Var$ .",
"Figure: Kripke structure K QBF x,y,z \\mathpzc {K}_{QBF}^{x,y,z} associated with a quantified Boolean formula with variables xx, yy, zz.An example of such a Kripke structure, for $Var=\\lbrace x,y,z\\rbrace $ , is given in Figure REF .",
"From $\\psi $ , we obtain the $\\mathsf {A\\overline{B}}$ formula $\\xi =start\\rightarrow \\xi _n$ , where $\\xi _i={\\left\\lbrace \\begin{array}{ll}\\phi (x_n,x_{n-1},\\cdots ,x_1) & i=0\\\\\\operatorname{\\langle \\overline{B}\\rangle }\\big ((\\operatorname{\\langle A\\rangle }x_{i\\, aux}) \\wedge \\xi _{i-1}\\big ) & i>0 \\wedge Q_i=\\exists \\\\[\\overline{B}]\\big ((\\operatorname{\\langle A\\rangle }x_{i\\, aux}) \\rightarrow \\xi _{i-1}\\big ) & i>0 \\wedge Q_i=\\forall \\end{array}\\right.",
"}$ Both $\\mathpzc {K}_{QBF}^{Var}$ and $\\xi $ can be built by using logarithmic working space.",
"We will show (proof of Theorem REF ) that $\\psi $ is true if and only if $\\mathpzc {K}_{QBF}^{Var}\\models \\xi $ .",
"As a preliminary step, we introduce some technical definitions.",
"Given a Kripke structure $\\mathpzc {K}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ and an $\\mathsf {A\\overline{B}}$ formula $\\psi $ , we denote by $p\\ell (\\psi )$ the set of proposition letters occurring in $\\psi $ and by $\\mathpzc {K}_{\\,|p\\ell (\\psi )}$ the structure obtained from $\\mathpzc {K}$ by restricting the labelling of each state to $p\\ell (\\psi )$ , namely, the Kripke structure $(\\overline{\\mathpzc {AP}},W, \\delta ,\\overline{\\mu },w_0)$ , where $\\overline{\\mathpzc {AP}}=\\mathpzc {AP}\\cap p\\ell (\\psi )$ and $\\overline{\\mu }(w)=\\mu (w)\\cap p\\ell (\\psi )$ , for all $w\\in W$ .",
"Moreover, for $v\\in W$ , we denote by $reach(\\mathpzc {K},v)$ the subgraph of $\\mathpzc {K}$ , with $v$ as its initial state, consisting of all and only the states which are reachable from $v$ , namely, the Kripke structure $(\\mathpzc {AP},W^{\\prime },\\delta ^{\\prime },\\mu ^{\\prime },v)$ , where $W^{\\prime }=\\lbrace w\\in W \\mid \\text{ there exists } \\rho \\in \\operatorname{Trk}_\\mathpzc {K} \\text{ with } \\operatorname{fst}(\\rho )=v \\text{ and } \\operatorname{lst}(\\rho )=w\\rbrace $ , $\\delta ^{\\prime }=\\delta \\cap (W^{\\prime }\\times W^{\\prime })$ , and $\\mu ^{\\prime }(w)=\\mu (w)$ , for all $w\\in W^{\\prime }$ .",
"As usual, two Kripke structures $\\mathpzc {K}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ and $\\mathpzc {K}^{\\prime }=(\\mathpzc {AP}^{\\prime },W^{\\prime }, \\delta ^{\\prime },\\mu ^{\\prime },w_0^{\\prime })$ are said to be isomorphic ($\\mathpzc {K}\\sim \\mathpzc {K}^{\\prime }$ for short) if and only if there is a bijection $f:W\\mapsto W^{\\prime }$ such that $(i)$ $f(w_0)=w_0^{\\prime }$ ; $(ii)$ for all $u,v\\in W$ , $(u,v)\\in \\delta $ if and only if $(f(u),f(v))\\in \\delta ^{\\prime }$ ; $(iii)$ for all $v\\in W$ , $\\mu (v)=\\mu ^{\\prime }(f(v))$ .",
"Finally, if $\\mathpzc {A}_\\mathpzc {K}=(\\mathpzc {AP},\\mathbb {I},A_\\mathbb {I},B_\\mathbb {I},E_\\mathbb {I},\\sigma )$ is the abstract interval model induced by a Kripke structure $\\mathpzc {K}$ and $\\rho \\in \\operatorname{Trk}_{\\mathpzc {K}}$ , we denote $\\sigma (\\rho )$ by $\\mathpzc {L}(\\mathpzc {K},\\rho )$ .",
"Let $\\mathpzc {K}$ and $\\mathpzc {K}^{\\prime }$ be two Kripke structures.",
"The following lemma states that, for any $\\mathsf {A\\overline{B}}$ formula $\\psi $ , if the same set of proposition letters, restricted to $p\\ell (\\psi )$ , holds over two tracks $\\rho \\in \\operatorname{Trk}_{\\mathpzc {K}}$ and $\\rho ^{\\prime } \\in \\operatorname{Trk}_{\\mathpzc {K}^{\\prime }}$ , and the subgraphs consisting of the states reachable from, respectively, $\\operatorname{lst}(\\rho )$ and $\\operatorname{lst}(\\rho ^{\\prime })$ are isomorphic, then $\\rho $ and $\\rho ^{\\prime }$ are equivalent with respect to $\\psi $ .",
"Lemma 35 Given an $\\mathsf {A\\overline{B}}$ formula $\\psi $ , two Kripke structures $\\mathpzc {K}=(\\mathpzc {AP},W, \\delta ,\\mu ,w_0)$ and $\\mathpzc {K}^{\\prime }=(\\mathpzc {AP}^{\\prime },W^{\\prime }, \\delta ^{\\prime },\\mu ^{\\prime },w_0^{\\prime })$ , and two tracks $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ and $\\rho ^{\\prime }\\in \\operatorname{Trk}_{\\mathpzc {K}^{\\prime }}$ such that $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\rho )=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\rho ^{\\prime })\\quad \\text{and}\\quad reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ^{\\prime })),$ it holds that $\\mathpzc {K},\\rho \\models \\psi \\iff \\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models \\psi $ .",
"The proof of this lemma can be found in REF .",
"Theorem 36 The model checking problem for $\\mathsf {A\\overline{B}}$ formulas over finite Kripke structures is PSPACE-hard (under LOGSPACE reductions).",
"We prove that the quantified Boolean formula $\\psi =Q_n x_n Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)$ is true if and only if $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}\\models \\xi $ by induction on the number of variables $n\\ge 0$ of $\\psi $ .",
"In the following, $\\phi (x_n,x_{n-1},\\cdots ,x_1)\\lbrace x_i/\\upsilon \\rbrace $ , with $\\upsilon \\in \\lbrace \\top ,\\bot \\rbrace $ , denotes the formula obtained from $\\phi (x_n,x_{n-1},\\cdots ,x_1)$ by replacing all occurrences of $x_i$ by $\\upsilon $ .",
"It is worth noticing that $\\mathpzc {K}_{QBF}^{x_n,x_{n-1},\\cdots , x_1}$ and $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}$ are isomorphic when they are restricted to the states $w_{x_{n-1}}^{\\top 1}$ , $w_{x_{n-1}}^{\\top 2}$ , $w_{x_{n-1}}^{\\bot 1}$ , $w_{x_{n-1}}^{\\bot 2}$ , $\\cdots $ , $w_{x_1}^{\\top 1}$ , $w_{x_1}^{\\top 2}$ , $w_{x_1}^{\\bot 1}$ , $w_{x_1}^{\\bot 2}$ , $sink$ (i.e., the leftmost part of both Kripke structures is omitted), and the labelling of states is suitably restricted accordingly.",
"Note that only the track $w_0w_1$ satisfies $start$ and, for $i=n,\\cdots , 1$ , the proposition letter $x_{i\\, aux}$ is satisfied by the two tracks $w_{x_i}^{\\top 1}w_{x_i}^{\\top 2}$ and $w_{x_i}^{\\bot 1}w_{x_i}^{\\bot 2}$ only.",
"(Case $n=0$ ) $\\psi $ equals $\\phi $ and it has no variables.",
"The states of $\\mathpzc {K}_{QBF}^\\emptyset $ are $W=\\lbrace w_0,w_1,sink\\rbrace $ and $\\xi =start\\rightarrow \\phi $ .",
"Let us assume $\\phi $ to be true.",
"All initial tracks of length greater than 2 trivially satisfy $\\xi $ , as $start$ does not hold on them.",
"As for $w_0w_1$ , it is true that $\\mathpzc {K}_{QBF}^\\emptyset ,w_0w_1\\models \\phi $ , since $\\phi $ is true (its truth does not depend on the proposition letters that hold on $w_0w_1$ , because it has no variables).",
"Thus $\\mathpzc {K}_{QBF}^\\emptyset \\models \\xi $ .",
"Vice versa, if $\\mathpzc {K}_{QBF}^\\emptyset \\models \\xi $ , then in particular $\\mathpzc {K}_{QBF}^\\emptyset ,w_0w_1\\models \\phi $ .",
"But $\\phi $ has no variables, hence it is true.",
"(Case $n\\ge 1$ ) Let us consider the formula $\\psi =Q_n x_n Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)$ .",
"We distinguish two cases, depending on whether $Q_n=\\exists $ or $Q_n=\\forall $ , and for both we prove the two implications.",
"$\\circ $ Case $Q_n=\\exists $ : $(\\Rightarrow )$ If the formula $\\psi $ is true, then, by definition, there exists $\\upsilon \\in \\lbrace \\top , \\bot \\rbrace $ such that if we replace all occurrences of $x_n$ in $\\phi (x_n,x_{n-1},\\cdots ,x_1)$ by $\\upsilon $ , we get the formula $\\phi ^{\\prime }(x_{n-1},\\cdots ,x_1)=\\phi (x_n,x_{n-1},\\cdots ,x_1)\\lbrace x_n/\\upsilon \\rbrace $ such that $\\psi ^{\\prime }=Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi ^{\\prime }(x_{n-1},\\cdots ,x_1)$ is a true quantified Boolean formula.",
"By the inductive hypothesis $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}\\models \\xi ^{\\prime }$ , where $\\xi ^{\\prime }=start\\rightarrow \\xi _{n-1}^{\\prime }$ is obtained from $\\psi ^{\\prime }$ and $\\xi _{n-1}^{\\prime }=\\xi _{n-1}\\lbrace x_n/\\upsilon \\rbrace $ .",
"It follows that $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1},w_0^{\\prime }w_1^{\\prime }\\models \\xi ^{\\prime }_{n-1}$ , where $w_0^{\\prime }$ and $w_1^{\\prime }$ are the two “leftmost” states of $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}$ (corresponding to $w_0$ and $w_1$ of $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}$ ).",
"We now prove that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}\\models \\xi $ .",
"Let us consider a generic initial track $\\rho $ in $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}$ .",
"If it does not satisfy $start$ , then it trivially holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},\\rho \\models \\xi $ .",
"Otherwise $\\rho =w_0w_1$ , and we have to show that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1\\models \\operatorname{\\langle \\overline{B}\\rangle }((\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\wedge \\xi _{n-1})$ ($=\\xi _n$ ).",
"If $\\upsilon =\\top $ , we consider $w_0w_1w_{x_n}^{\\top 1}$ ; otherwise, we consider $w_0w_1w_{x_n}^{\\bot 1}$ .",
"In the first case (the other is symmetric), we must prove that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models (\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\wedge \\xi _{n-1}$ .",
"It trivially holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\operatorname{\\langle A\\rangle }x_{n\\, aux}$ .",
"Hence, we only need to prove that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\xi _{n-1}$ .",
"As we have shown, by the inductive hypothesis, it holds that $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1},w_0^{\\prime }w_1^{\\prime }\\models \\xi ^{\\prime }_{n-1}$ ($=\\xi _{n-1}\\lbrace x_n/\\top \\rbrace $ ).",
"Now, since $p\\ell (\\xi _{n-1}\\lbrace x_n/\\top \\rbrace )=\\lbrace x_1,\\cdots ,x_{n-1},x_{1\\, aux},\\cdots ,x_{n-1\\, aux}\\rbrace $ , $\\mathpzc {L}({\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}}_{|p\\ell (\\xi _{n-1}\\lbrace x_n/\\top \\rbrace )},w_0^{\\prime }w_1^{\\prime })= \\lbrace x_{n-1},\\cdots ,x_1\\rbrace $ , $\\mathpzc {L}({\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}}_{|p\\ell (\\xi _{n-1}\\lbrace x_n/\\top \\rbrace )},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2})=\\lbrace x_{n-1}, \\cdots ,x_1\\rbrace $ , and $reach({\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}}_{|p\\ell (\\xi _{n-1}\\lbrace x_n/\\top \\rbrace )},w_{x_n}^{\\top 2})\\sim reach({\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}}_{|p\\ell (\\xi _{n-1}\\lbrace x_n/\\top \\rbrace )},w_1^{\\prime })$ , by Lemma REF , $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models \\xi ^{\\prime }_{n-1}$ .",
"Hence, $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models \\xi _{n-1}$ as $x_n$ is in the labelling of the track $w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}$ and of any $\\overline{\\rho }$ such that $w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\in \\operatorname{Pref}(\\overline{\\rho })$ .",
"Now, if $n=1$ , then $\\xi _{n-1}=\\phi (x_n)$ and it holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\xi _{n-1}$ .",
"If $n>1$ , either $\\xi _{n-1}=\\operatorname{\\langle \\overline{B}\\rangle }((\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\wedge \\xi _{n-2})$ or $\\xi _{n-1}=[\\overline{B}] ((\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\rightarrow \\xi _{n-2})$ .",
"In the first case, since $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models \\operatorname{\\langle \\overline{B}\\rangle }((\\operatorname{\\langle A\\rangle }x_{n-1\\, aux})\\wedge \\xi _{n-2})$ , there are only two possibilities: $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}w_{x_{n-1}}^{\\top 1}\\models (\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\wedge \\xi _{n-2}$ or $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}w_{x_{n-1}}^{\\bot 1}\\models (\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\wedge \\xi _{n-2}$ .",
"In both cases, $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\operatorname{\\langle \\overline{B}\\rangle }((\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\wedge \\xi _{n-2})$ .",
"Otherwise, it holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models [\\overline{B}] ((\\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\rightarrow \\xi _{n-2})$ .",
"It follows that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}w_{x_{n-1}}^{\\top 1}\\models \\xi _{n-2}$ and $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}w_{x_{n-1}}^{\\bot 1}\\models \\xi _{n-2}$ .",
"As a consequence, $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models [\\overline{B}] ((\\lnot \\operatorname{\\langle A\\rangle }x_{n-1\\, aux}) \\vee \\xi _{n-2})$ $(=\\xi _{n-1})$ (recall that the only successor of $w_{x_n}^{\\top 1}$ in $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}$ is $w_{x_n}^{\\top 2}$ and, in particular, $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models \\lnot \\operatorname{\\langle A\\rangle }x_{n-1\\, aux}$ ).",
"$(\\Leftarrow )$ If $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}\\models \\xi $ , it holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1\\models \\operatorname{\\langle \\overline{B}\\rangle }(\\operatorname{\\langle A\\rangle }x_{n\\, aux} \\wedge \\xi _{n-1})$ .",
"Hence, either $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models (\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\wedge \\xi _{n-1}$ or $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\bot 1}\\models (\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\wedge \\xi _{n-1}$ .",
"Let us consider the first case (the other is symmetric).",
"It holds that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\xi _{n-1}\\lbrace x_n/\\top \\rbrace $ and $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}w_{x_n}^{\\top 2}\\models \\xi _{n-1}\\lbrace x_n/\\top \\rbrace $ (as before).",
"By Lemma REF we get that $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1},w_0^{\\prime }w_1^{\\prime }\\models \\xi _{n-1}\\lbrace x_n/\\top \\rbrace (=\\xi _{n-1}^{\\prime })$ and thus $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}\\models start \\rightarrow \\xi _{n-1}^{\\prime }$ , namely, $\\mathpzc {K}_{QBF}^{x_{n-1},\\cdots , x_1}\\models \\xi ^{\\prime }$ .",
"By the inductive hypothesis, $\\psi ^{\\prime }=Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)\\lbrace x_n/\\top \\rbrace $ is true.",
"Hence, $\\psi =\\exists x_n Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)$ is true.",
"$\\circ $ Case $Q_n=\\forall $ : $(\\Rightarrow )$ Assume that both $\\psi ^{\\prime }=Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots , x_1) \\lbrace x_n/\\top \\rbrace $ and $\\psi ^{\\prime \\prime }=\\linebreak Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1}, \\cdots , x_1)\\lbrace x_n/\\bot \\rbrace $ are true quantified Boolean formulas.",
"We show that $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1\\models [\\overline{B}]((\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\rightarrow \\xi _{n-1})$ .",
"To this end, we prove that both $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\xi _{n-1}$ and $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\bot 1}\\models \\xi _{n-1}$ .",
"This can be shown exactly as in the $\\exists $ case.",
"$(\\Leftarrow )$ If $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1}\\models \\xi $ , then $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1\\models [\\overline{B}]((\\operatorname{\\langle A\\rangle }x_{n\\, aux}) \\rightarrow \\xi _{n-1})$ .",
"Hence, both $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\top 1}\\models \\xi _{n-1}$ and $\\mathpzc {K}_{QBF}^{x_n,\\cdots , x_1},w_0w_1w_{x_n}^{\\bot 1}\\models \\xi _{n-1}$ .",
"Reasoning as in the $\\exists $ case and by applying the inductive hypothesis twice, we get that the quantified Boolean formulas $Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)\\lbrace x_n/\\top \\rbrace $ and $Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)\\lbrace x_n/\\bot \\rbrace $ are true; thus $\\forall x_n Q_{n-1} x_{n-1} \\cdots Q_1 x_1 \\phi (x_n,x_{n-1},\\cdots ,x_1)$ is true."
],
[
"The fragment $\\mathsf {\\forall A\\overline{A}BE}$",
"In this section, we introduce and study the complexity of the model checking problem for the universal fragment of $\\mathsf {A\\overline{A}BE}$ , denoted by $\\mathsf {\\forall A\\overline{A}BE}$ .",
"Its formulas are defined as follows: $\\psi ::= \\beta \\;\\vert \\; \\psi \\wedge \\psi \\;\\vert \\; [A]\\psi \\;\\vert \\; [B]\\psi \\;\\vert \\; [E]\\psi \\;\\vert \\; [\\overline{A}]\\psi ,$ where $\\beta $ is a pure propositional formula, $\\beta ::= p \\;\\vert \\; \\beta \\vee \\beta \\;\\vert \\; \\beta \\wedge \\beta \\;\\vert \\; \\lnot \\beta \\;\\vert \\; \\bot \\;\\vert \\; \\top \\mbox{ with } p \\in \\mathpzc {AP}.$ Formulas of $\\mathsf {\\forall A\\overline{A}BE}$ can thus be constructed starting from pure propositional formulas (a fragment of HS that we denote by $\\mathsf {Prop}$ ); subsequently, formulas with universal modalities $[A]$ , $[B]$ , $[E]$ , and $[\\overline{A}]$ can be combined only by conjunctions, but not by negations or disjunctions (which may occur in pure propositional formulas only).",
"We will prove that the model checking problem for $\\mathsf {\\forall A\\overline{A}BE}$ formulas (as well as for $\\mathsf {Prop}$ ) over finite Kripke structures is coNP-complete.",
"To start with, we need to introduce the (auxiliary) fragment $\\mathsf {\\exists A\\overline{A}BE}$ , which can be regarded as the “dual” of $\\mathsf {\\forall A\\overline{A}BE}$ .",
"Its formulas are defined as: $\\psi ::= \\beta \\;\\vert \\; \\psi \\vee \\psi \\;\\vert \\; \\langle A\\rangle \\psi \\;\\vert \\; \\langle B\\rangle \\psi \\;\\vert \\; \\langle E\\rangle \\psi \\;\\vert \\; \\langle \\overline{A}\\rangle \\psi .$ $\\mathsf {\\exists A\\overline{A}BE}$ formulas feature $\\operatorname{\\langle A\\rangle }$ , $\\operatorname{\\langle B\\rangle }$ , $\\operatorname{\\langle E\\rangle }$ , and $\\operatorname{\\langle \\overline{A}\\rangle }$ existential modalities; negation and conjunction symbols may occur only in pure propositional formulas.",
"The intersection of $\\mathsf {\\forall A\\overline{A}BE}$ and $\\mathsf {\\exists A\\overline{A}BE}$ is precisely $\\mathsf {Prop}$ .",
"The negation of any $\\mathsf {\\forall A\\overline{A}BE}$ formula can be transformed into an equivalent $\\mathsf {\\exists A\\overline{A}BE}$ formula (of at most double length), and vice versa, by using De Morgan's laws and the equivalences $[X]\\psi \\equiv \\lnot \\langle X\\rangle \\lnot \\psi $ and $\\lnot \\lnot \\psi \\equiv \\psi $ .",
"In the following, we outline a non-deterministic algorithm to decide the model checking problem for a $\\mathsf {\\forall A\\overline{A}BE}$ formula $\\psi $ (Algorithm ).",
"As usual, the algorithm searches for a counterexample to $\\psi $ , that is, an initial track satisfying $\\lnot \\psi $ .",
"Since, as we already pointed out, $\\lnot \\psi $ is equivalent to a suitable formula $\\psi ^{\\prime }$ of the dual fragment $\\mathsf {\\exists A\\overline{A}BE}$ , the algorithm looks for an initial track satisfying $\\psi ^{\\prime }$ .",
"Algorithm makes use of descriptor elements: we remind that they are the labels of the nodes of $B_k$ -descriptors.",
"By Proposition REF , if a descriptor element $d$ is witnessed in $\\mathpzc {K}$ , i.e., there exists some $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ associated with $d$ , then there exists a track of length at most $2+|W|^2$ associated with $d$ .",
"Thus, to generate a (all) witnessed descriptor element(s) with initial state $v$ , we just need to non-deterministically visit the unravelling of $\\mathpzc {K}$ from $v$ up to depth $2+|W|^2$ .",
"This property is fundamental for the completeness of the algorithm, and also for bounding the length of tracks we need to consider.",
"Before presenting Algorithm , we need to describe the non-deterministic auxiliary procedure Check$\\exists $ (see Algorithm ), which takes as input a Kripke structure $\\mathpzc {K}$ , a formula $\\psi $ of $\\mathsf {\\exists A\\overline{A}BE}$ , and a witnessed descriptor element $d=(v_{in},S,v_{fin})$ and it returns Yes if and only if there exists a track $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\psi $ .",
"[tp] Check$\\exists $$(\\mathpzc {K},\\psi ,(v_{in},S,v_{fin}))$ [1] $\\psi =\\beta $$\\beta $ is a pure propositional formula $VAL(\\beta ,(v_{in},S,v_{fin}))=\\top $ Yes else No $\\psi =\\varphi _1\\vee \\varphi _2$ Check$\\exists $$(\\mathpzc {K},\\varphi _1,(v_{in},S,v_{fin}))$ Check$\\exists $$(\\mathpzc {K},\\varphi _2,(v_{in},S,v_{fin}))$ $\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ $(v_{fin},S^{\\prime },v_{fin}^{\\prime })\\leftarrow \\texttt {aDescrEl}(\\mathpzc {K},v_{fin},\\textsc {forw})$ Check$\\exists $$(\\mathpzc {K},\\varphi ,(v_{fin},S^{\\prime },v_{fin}^{\\prime }))$ $\\psi =\\operatorname{\\langle \\overline{A}\\rangle }\\varphi $ $(v_{in}^{\\prime },S^{\\prime },v_{in})\\leftarrow \\texttt {aDescrEl}(\\mathpzc {K},v_{in},\\textsc {backw})$ Check$\\exists $$(\\mathpzc {K},\\varphi ,(v_{in}^{\\prime },S^{\\prime },v_{in}))$ $\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ $(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime })\\leftarrow \\texttt {aDescrEl}(\\mathpzc {K},v_{in},\\textsc {forw})$$v_{in}^{\\prime }=v_{in}$ $(v_{in}^{\\prime },S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace ,v_{fin})=(v_{in},S,v_{fin})$ and $(v_{fin}^{\\prime },v_{fin})$ is an edge of $\\mathpzc {K}$ Check$\\exists $$(\\mathpzc {K},\\varphi ,(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime }))$ No $(v_{in}^{\\prime \\prime },S^{\\prime \\prime },v_{fin}^{\\prime \\prime })\\leftarrow \\texttt {aDescrEl}(\\mathpzc {K},v_{in}^{\\prime \\prime },\\textsc {forw})$ , where $(v_{fin}^{\\prime },v_{in}^{\\prime \\prime })$ is an edge of $\\mathpzc {K}$ chosen non-deterministically $\\texttt {concat}\\left((v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime }),(v_{in}^{\\prime \\prime },S^{\\prime \\prime },v_{fin}^{\\prime \\prime })\\right)=(v_{in},S,v_{fin})$ Check$\\exists $$(\\mathpzc {K},\\varphi ,(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime }))$ No $\\psi =\\operatorname{\\langle E\\rangle }\\varphi $ Symmetric to $\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ The procedure is recursively defined as follows.",
"When it is called on a pure propositional formula $\\beta $ (base of the recursion), $VAL(\\beta ,d)$ evaluates $\\beta $ over $d$ in the standard way.",
"The evaluation can be performed in deterministic polynomial time, and if $VAL(\\beta ,d)$ returns $\\top $ , then there exists a track associated with $d$ (of length at most quadratic in $|W|$ ) that satisfies $\\beta $ .",
"If $\\psi =\\psi ^{\\prime }\\vee \\psi ^{\\prime \\prime }$ , where $\\psi ^{\\prime }$ or $\\psi ^{\\prime \\prime }$ feature some temporal modality, the procedure non-deterministically calls itself on $\\psi ^{\\prime }$ or $\\psi ^{\\prime \\prime }$ (the construct Either $c_1$ Or $c_2$ EndOr denotes a non-deterministic choice between commands $c_1$ and $c_2$ ).",
"If $\\psi =\\operatorname{\\langle A\\rangle }\\psi ^{\\prime }$ (respectively, $\\operatorname{\\langle \\overline{A}\\rangle }\\psi ^{\\prime }$ ), the procedure looks for a new descriptor element for a track starting from the final state (respectively, leading to the initial state) of the current descriptor element $d$ .",
"To this aim, we use the procedure aDescrEl$(\\mathpzc {K},v, \\textsc {forw})$ (resp., aDescrEl$(\\mathpzc {K},v,\\textsc {backw})$ ) which non-deterministically returns a descriptor element $(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime })$ , with $v_{in}^{\\prime }=v$ (resp., $v_{fin}^{\\prime }=v$ ), witnessed in $\\mathpzc {K}$ by exploring forward (resp., backward) the unravelling of $\\mathpzc {K}$ from $v_{in}^{\\prime }$ (resp., from $v_{fin}^{\\prime }$ ).",
"Its complexity is polynomial in $|W|$ , since it needs to examine the unravelling of $\\mathpzc {K}$ from $v$ up to depth $2+|W|^2$ .",
"If $\\psi =\\operatorname{\\langle B\\rangle }\\psi ^{\\prime }$ , the procedure looks for a new descriptor element $d_1$ and eventually calls itself on $\\psi ^{\\prime }$ and $d_1$ only if the current descriptor element $d$ results from the “concatenation” of $d_1$ with a suitable descriptor element $d_2$ : if $d_1=(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime })$ and $d_2=(v_{in}^{\\prime \\prime },S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ , then concat$(d_1,d_2)$ returns $(v_{in}^{\\prime },S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime },v_{in}^{\\prime \\prime }\\rbrace \\cup S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ .",
"Notice that if $\\rho _1$ and $\\rho _2$ are tracks associated with $d_1$ and $d_2$ , respectively, then $\\rho _1\\cdot \\rho _2$ is associated with concat$(d_1,d_2)$ .",
"The following theorem proves soundness and completeness of the Check$\\exists $ procedure.",
"Theorem 37 For any formula $\\psi $ of the fragment $\\mathsf {\\exists A\\overline{A}BE}$ and any witnessed descriptor element $d=(v_{in},S,v_{fin})$ , the procedure Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation if and only if there exists a track $\\rho $ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\psi $ .",
"(Soundness) The proof is by induction on the structure of the formula $\\psi $ .",
"$\\psi $ is a pure propositional formula $\\beta $ : let $\\rho $ be a witness track for $d$ ; if Check$\\exists $$(\\mathpzc {K},\\beta ,d)$ has a successful computation, then $VAL(\\beta ,d)$ is true and so $\\mathpzc {K},\\rho \\models \\psi $ .",
"$\\psi =\\varphi _1\\vee \\varphi _2$ : if Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation, then, for some $i \\in \\lbrace 1,2\\rbrace $ , Check$\\exists $$(\\mathpzc {K},\\varphi _i,d)$ has a successful computation.",
"By the inductive hypothesis, there exists $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ associated with $d$ such that $\\mathpzc {K},\\rho \\models \\varphi _i$ , and thus $\\mathpzc {K},\\rho \\models \\varphi _1\\vee \\varphi _2$ .",
"$\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ : if Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation, then there exists a witnessed $d^{\\prime }=(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime })$ , with $v_{in}^{\\prime }=v_{fin}$ , such that Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"By the inductive hypothesis, there exists a track $\\rho ^{\\prime }$ , associated with $d^{\\prime }$ , such that $\\mathpzc {K},\\rho ^{\\prime }\\models \\varphi $ .",
"If $\\rho $ is a track associated with $d$ (which is witnessed by hypothesis), we have that $\\operatorname{lst}(\\rho )=\\operatorname{fst}(\\rho ^{\\prime })=v_{fin}$ and, by definition, $\\mathpzc {K},\\rho \\models \\psi $ .",
"$\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ : if Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation, then we must distinguish two possible cases.",
"$(i)$ There exists $d^{\\prime }=(v_{in},S^{\\prime },v_{fin}^{\\prime })$ , witnessed by a track with $(v_{fin}^{\\prime },v_{fin})\\in \\delta $ , such that $(v_{in},S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace ,v_{fin})=d$ , and Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"By the inductive hypothesis, there exists a track $\\rho ^{\\prime }$ , associated with $d^{\\prime }$ , such that $\\mathpzc {K},\\rho ^{\\prime }\\models \\varphi $ .",
"Hence $\\mathpzc {K},\\rho ^{\\prime }\\cdot v_{fin}\\models \\psi $ and $\\rho ^{\\prime }\\cdot v_{fin}$ is associated with $d$ .",
"$(ii)$ There exist $d^{\\prime }=(v_{in},S^{\\prime },v_{fin}^{\\prime })$ , witnessed by a track, and $d^{\\prime \\prime }=(v_{in}^{\\prime \\prime },S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ , witnessed by a track as well, such that $(v_{fin}^{\\prime },v_{in}^{\\prime \\prime })\\in \\delta $ , concat$(d^{\\prime },d^{\\prime \\prime })=d$ , and Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"By the inductive hypothesis, there exists a track $\\rho ^{\\prime }$ , associated with $d^{\\prime }$ , such that $\\mathpzc {K},\\rho ^{\\prime }\\models \\varphi $ .",
"Hence $\\mathpzc {K},\\rho ^{\\prime }\\cdot \\rho ^{\\prime \\prime }\\models \\psi $ , where $\\rho ^{\\prime \\prime }$ is any track associated with $d^{\\prime \\prime }$ and $\\rho ^{\\prime }\\cdot \\rho ^{\\prime \\prime }$ is associated with $d$ .",
"The case $\\psi =\\operatorname{\\langle \\overline{A}\\rangle }\\varphi $ (respectively, $\\psi =\\operatorname{\\langle E\\rangle }\\varphi $ ) can be dealt with as $\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ (respectively, $\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ ).",
"(Completeness) The proof is by induction on the structure of the formula $\\psi $ .",
"$\\psi $ is a pure propositional formula $\\beta $ : if $\\rho $ is associated with $d$ and $\\mathpzc {K},\\rho \\models \\beta $ , then $VAL(\\beta ,d)=\\top $ , and thus Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation.",
"$\\psi = \\varphi _1\\vee \\varphi _2$ : if there exists a track $\\rho $ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\varphi _1\\vee \\varphi _2$ , then $\\mathpzc {K},\\rho \\models \\varphi _i$ , for some $i \\in \\lbrace 1,2\\rbrace $ .",
"By the inductive hypothesis, Check$\\exists $$(\\mathpzc {K},\\varphi _i,d)$ has a successful computation, and hence Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation.",
"$\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ : if there exists a track $\\rho $ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\operatorname{\\langle A\\rangle }\\varphi $ , then, by definition, there exists a track $\\overline{\\rho }$ , with $\\operatorname{fst}(\\overline{\\rho })=\\operatorname{lst}(\\rho )=v_{fin}$ , such that $\\mathpzc {K},\\overline{\\rho }\\models \\varphi $ .",
"If $d^{\\prime }=(v_{fin},S^{\\prime },v_{fin}^{\\prime })$ is the descriptor element for $\\overline{\\rho }$ , then, by the inductive hypothesis, Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"Since there exists a computation where the non-deterministic call to $\\texttt {aDescrEl}(\\mathpzc {K},v_{fin}, \\textsc {forw})$ returns the descriptor element $d^{\\prime }$ for $\\overline{\\rho }$ , it follows that Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation.",
"$\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ : if there exists a track $\\rho $ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\operatorname{\\langle B\\rangle }\\varphi $ , there are two possible cases.",
"$(i)$ $\\mathpzc {K},\\overline{\\rho }\\models \\varphi $ , with $\\rho =\\overline{\\rho }\\cdot v_{fin}$ for some $\\overline{\\rho } \\in \\operatorname{Trk}_\\mathpzc {K}$ .",
"If $d^{\\prime }=(v_{in},S^{\\prime },v_{fin}^{\\prime })$ is the descriptor element for $\\overline{\\rho }$ , by the inductive hypothesis Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"Since there is a computation where $\\texttt {aDescrEl}(\\mathpzc {K},v_{in},\\textsc {forw})$ returns $d^{\\prime }$ and both $(v_{fin}^{\\prime },v_{fin}) \\in \\delta $ and $(v_{in},S^{\\prime }\\cup \\lbrace v_{fin}^{\\prime }\\rbrace ,v_{fin})=d$ , it follows that Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation.",
"$(ii)$ $\\mathpzc {K},\\overline{\\rho }\\models \\varphi $ with $\\rho =\\overline{\\rho }\\cdot \\tilde{\\rho }$ for some $\\overline{\\rho },\\tilde{\\rho }\\in \\operatorname{Trk}_\\mathpzc {K}$ .",
"Let $d^{\\prime }=(v_{in},S^{\\prime },v_{fin}^{\\prime })$ and $d^{\\prime \\prime }=(v_{in}^{\\prime \\prime },S^{\\prime \\prime },v_{fin}^{\\prime \\prime })$ be the descriptor elements for $\\overline{\\rho }$ and $\\tilde{\\rho }$ , respectively.",
"Obviously, it holds that $\\texttt {concat}(d^{\\prime },d^{\\prime \\prime })=d$ .",
"By the inductive hypothesis, Check$\\exists $$(\\mathpzc {K},\\varphi ,d^{\\prime })$ has a successful computation.",
"Since both $\\overline{\\rho }$ and $\\tilde{\\rho }$ are witnessed, there is a computation where the calls to $\\texttt {aDescrEl}(\\mathpzc {K},v_{in},\\textsc {forw})$ and $\\texttt {aDescrEl}(\\mathpzc {K},v_{in}^{\\prime \\prime },\\textsc {forw})$ non-deterministically return $d^{\\prime }$ and $d^{\\prime \\prime }$ , respectively, and $(v_{fin}^{\\prime },v_{in}^{\\prime \\prime })\\in \\delta $ is non-deterministically chosen.",
"Hence, Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ has a successful computation.",
"The case $\\psi =\\operatorname{\\langle \\overline{A}\\rangle }\\varphi $ (respectively, $\\psi =\\operatorname{\\langle E\\rangle }\\varphi $ ) can be dealt with as $\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ (respectively, $\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ ).",
"It is worth pointing out that Check$\\exists $$(\\mathpzc {K},\\psi ,d)$ cannot deal with $\\operatorname{\\langle \\overline{B}\\rangle }$ and $\\operatorname{\\langle \\overline{E}\\rangle }$ modalities.",
"To cope with them, descriptor elements are not enough: the whole descriptors must be considered.",
"[tbp] ProvideCounterex$(\\mathpzc {K},\\psi )$ [1] $(v_{in},S,v_{fin})\\leftarrow \\texttt {aDescrEl}(\\mathpzc {K},w_0,\\textsc {forw})$$v_{in}=w_0$ is the initial state of $\\mathpzc {K}$ Check$\\exists $$(\\mathpzc {K},\\texttt {to}\\mathsf {\\exists A\\overline{A}BE}(\\lnot \\psi ),(v_{in},S,v_{fin}))$ We can finally introduce the procedure ProvideCounterex$(\\mathpzc {K},\\psi )$ (Algorithm ), which searches for counterexamples to the input $\\mathsf {\\forall A\\overline{A}BE}$ formula $\\psi $ ; indeed, it is possible to prove that it has a successful computation if and only if $\\mathpzc {K}\\lnot \\models \\psi $ .",
"In the pseudocode of procedure ProvideCounterex, $\\texttt {to}\\mathsf {\\exists A\\overline{A}BE}(\\lnot \\psi )$ denotes the $\\mathsf {\\exists A\\overline{A}BE}$ formula equivalent to $\\lnot \\psi $ .",
"On the one hand, if ProvideCounterex$(\\mathpzc {K},\\psi )$ has a successful computation, then there exists a witnessed descriptor element $d=(v_{in},S,v_{fin})$ , where $v_{in}$ is $w_0$ (the initial state of $\\mathpzc {K}$ ), such that Check$\\exists $$(\\mathpzc {K},\\texttt {to}\\mathsf {\\exists A\\overline{A}BE}(\\lnot \\psi ),d)$ has a successful computation.",
"This means that there exists a track $\\rho $ , associated with $d$ , such that $\\mathpzc {K},\\rho \\models \\lnot \\psi $ , and thus $\\mathpzc {K}\\lnot \\models \\psi $ .",
"On the other hand, if $\\mathpzc {K}\\lnot \\models \\psi $ , then there exists an initial track $\\rho $ such that $\\mathpzc {K},\\rho \\models \\lnot \\psi $ .",
"Let $d$ be the descriptor element for $\\rho $ : Check$\\exists $$(\\mathpzc {K},\\texttt {to}\\mathsf {\\exists A\\overline{A}BE}(\\lnot \\psi ),d)$ has a successful computation.",
"Since $d$ is witnessed by an initial track, some non-deterministic instance of $\\texttt {aDescrEl}(\\mathpzc {K},w_0,\\textsc {forw})$ returns $d$ .",
"Hence ProvideCounterex$(\\mathpzc {K},\\psi )$ has a successful computation.",
"As for the complexity, ProvideCounterex$(\\mathpzc {K},\\psi )$ runs in non-deterministic polynomial time (it is in NP), since the number of recursive invocations of the procedure Check$\\exists $ is $O(|\\psi |)$ , and each invocation requires time polynomial in $|W|$ while generating descriptor elements.",
"Therefore, the model checking problem for $\\mathsf {\\forall A\\overline{A}BE}$ belongs to coNP.",
"We conclude the section by proving that the model checking problem for $\\mathsf {\\forall A\\overline{A}BE}$ is coNP-complete.",
"Such a result is an easy corollary of the following theorem.",
"Theorem 38 Let $\\mathpzc {K}$ be a finite Kripke structure and $\\beta \\in \\mathsf {Prop}$ be a pure propositional formula.",
"The problem of deciding whether $\\mathpzc {K}\\lnot \\models \\beta $ is NP-hard (under LOGSPACE reductions).",
"We provide a reduction from the NP-complete SAT problem to the considered problem.",
"Let $\\beta $ be a Boolean formula over a set of variables $Var=\\lbrace x_1,\\ldots , x_n\\rbrace $ .",
"We build a Kripke structure, $\\mathpzc {K}_{SAT}^{Var}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ , with: $\\mathpzc {AP}=Var$ ; $W= \\lbrace w_0\\rbrace \\cup \\lbrace w_i^\\ell \\mid \\ell \\in \\lbrace \\top ,\\bot \\rbrace ,\\; 1\\le i \\le n\\rbrace $ ; $\\delta = \\lbrace (w_0,w_1^\\top ), (w_0,w_1^\\bot )\\rbrace \\cup \\lbrace (w_i^\\ell ,w_{i+1}^m) \\mid \\ell ,m \\in \\lbrace \\top ,\\bot \\rbrace , 1 \\le i \\le n-1\\rbrace \\cup \\lbrace (w_n^\\top ,w_n^\\top )\\rbrace \\cup \\lbrace (w_n^\\bot ,w_n^\\bot )\\rbrace $ ; $\\mu (w_0)= \\mathpzc {AP}$ ; for $1\\le i \\le n$ , $\\mu (w_i^{\\top })= \\mathpzc {AP}$ and $\\mu (w_i^{\\bot })= \\mathpzc {AP} \\setminus \\lbrace x_i\\rbrace $ .",
"See Figure REF for an example of $\\mathpzc {K}_{SAT}^{Var}$ , with $Var=\\lbrace x_1, \\ldots ,x_4\\rbrace $ .",
"Figure: Kripke structure K SAT Var \\mathpzc {K}_{SAT}^{Var} associated with a SAT formula with variables Var={x 1 ,x 2 ,x 3 ,x 4 }Var=\\lbrace x_1,x_2,x_3,x_4\\rbrace .It is immediate to see that any initial track $\\rho $ of any length induces a truth assignment to the variables of $Var$ : for any $x_i \\in Var$ , $x_i$ evaluates to $\\top $ if and only if $x_i \\in \\bigcap _{w\\in \\operatorname{states}(\\rho )}\\mu (w)$ .",
"Conversely, for any possible truth assignment to the variables in $Var$ , there exists an initial track $\\rho $ that induces such an assignment: we include in the track the state $w_i^{\\top }$ if $x_i$ is assigned to $\\top $ , $w_i^{\\bot }$ otherwise.",
"Let $\\gamma =\\lnot \\beta $ .",
"It holds that $\\beta $ is satisfiable if and only if there exists an initial track $\\rho \\in \\operatorname{Trk}_{\\mathpzc {K}_{SAT}^{Var}}$ such that $\\mathpzc {K}_{SAT}^{Var},\\rho \\models \\beta $ , that is, if and only if $\\mathpzc {K}_{SAT}^{Var}\\lnot \\models \\gamma $ .",
"To conclude, we observe that $\\mathpzc {K}_{SAT}^{Var}$ can be built with logarithmic working space.",
"It immediately follows that checking whether $\\mathpzc {K}\\lnot \\models \\beta $ for $\\beta \\in \\mathsf {Prop}$ is NP-complete, thus model checking for formulas of $\\mathsf {Prop}$ is coNP-complete.",
"Moreover, since a pure propositional formula in $\\mathsf {Prop}$ is also a $\\mathsf {\\forall A\\overline{A}BE}$ formula, ProvideCounterex$(\\mathpzc {K},\\psi )$ is at least as hard as checking whether $\\mathpzc {K}\\lnot \\models \\beta $ for $\\beta \\in \\mathsf {Prop}$ .",
"Thus, ProvideCounterex$(\\mathpzc {K},\\psi )$ is NP-complete, hence the model checking problem for $\\mathsf {\\forall A\\overline{A}BE}$ is coNP-complete.",
"We conclude the section spending a few words about the complexity of the model checking problem for the fragment $\\mathsf {A\\overline{A}}$ , also known as the logic of temporal neighborhood.",
"As a consequence of the lower bound for $\\mathsf {Prop}$ , model checking for $\\mathsf {A\\overline{A}}$ turns out to be coNP-hard as well.",
"Moreover, the problem is in PSPACE, as $\\mathsf {A\\overline{A}}$ is a subfragment of $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ .",
"Actually, in [23], the authors proved that $\\mathsf {A\\overline{A}}$ belongs to $\\mbox{P}^{\\mbox{\\scriptsize NP}[O(\\log ^2 n)]}$ and is $\\mbox{P}^{\\mbox{\\scriptsize NP}[O(\\log n)]}$ -hard: the complexity class $\\mbox{P}^{\\mbox{\\scriptsize NP}[O(\\log n)]}$ (respectively, $\\mbox{P}^{\\mbox{\\scriptsize NP}[O(\\log ^2 n)]}$ ) contains the problems decided by a deterministic polynomial time algorithm which requires only $O(\\log n)$ (respectively, $O(\\log ^2 n)$ ) queries to an NP oracle, being $n$ the input size [12], [32].",
"Hence, such classes are higher than both NP and coNP in the polynomial time hierarchy."
],
[
"Conclusions and future work",
"In this paper, we have studied the model checking problem for some fragments of Halpern and Shoham's modal logic of time intervals.",
"First, we have considered the large fragment $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ , and devised an EXPSPACE model checking algorithm for it, which rests on a contraction method that allows us to restrict the verification of the input formula to a finite subset of tracks of bounded size, called track representatives.",
"We have also proved that the problem is PSPACE-hard, NEXP-hard if a suitable succinct encoding of formulas is allowed.",
"As a matter of fact, in the latter case, the problem can also be proved coNEXP-hard, and thus we conjecture that a tighter lower bound can be established (for instance, EXPSPACE-hardness).",
"Then, we identified some other HS fragments, namely, $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ , $\\mathsf {\\forall A\\overline{A}BE}$ , and $\\mathsf {A\\overline{A}}$ , whose model checking problem turns out to be (computationally) much simpler than that of full HS and of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ , and comparable to that of point-based temporal logics (as an example, the model checking problem for $\\mathsf {A\\overline{A}\\overline{B}\\overline{E}}$ is PSPACE-complete, and has thus the same complexity as LTL).",
"Luckily, these fragments are expressive enough to capture meaningful properties of state-transition systems, such as, for instance, mutual exclusion, state reachability, and non-starvation.",
"One may wonder whether, given the homogeneity assumption, there is the possibility to reduce the model checking problem for HS fragments over finite Kripke structures to a point-based setting.",
"Such an issue has been systematically dealt with in [3].",
"Together with Laura Bozzelli and Pietro Sala, we consider three semantic variants of HS: the one we introduced in [24] and we used in the subsequent papers, including the present one, called state-based semantics, which allows branching in the past and in the future, the computation-tree-based semantics, allowing branching only in the future, and the linear semantics, disallowing branching.",
"These variants are compared, as for their expressiveness, among themselves and to standard temporal logics, getting a complete picture.",
"In particular, we show that (i) HS with computation-tree-based semantics is equivalent to finitary CTL* and strictly included in HS with state-based semantics, and (ii) HS with linear semantics is equivalent to LTL and incomparable to HS with state-based semantics.",
"As for future work, we are currently exploring two main research directions.",
"On the one hand, we are looking for other well-behaved fragments of HS; on the other hand, we are thinking of possible ways of relaxing the homogeneity assumption.",
"As for the latter, a promising direction has been recently outlined by Lomuscio and Michaliszyn, who proposed to use regular expressions to define the behavior of proposition letters over intervals in terms of the component states [17].",
"Our ultimate goal is to be able to deal with interval properties that can only be predicated over time intervals considered as a whole.",
"This is the case, for instance, of temporal aggregations (think of a constraint on the average speed of a moving device during a given time period).",
"In this respect, the existing work on Duration Calculus (DC) model checking seems to be relevant.",
"DC extends interval temporal logic with an explicit notion of state: states are denoted by state expressions and characterized by a duration (the time period during which the system remains in a given state).",
"Recent results on DC model checking and an account of related work can be found in [14]."
],
[
"Acknowledgements",
"The work by Adriano Peron has been supported by the SHERPA collaborative project, which has received funding from the European Community 7-th Framework Programme (FP7/2007-2013) under grant agreements ICT-600958.",
"He is solely responsible for its content.",
"The paper does not represent the opinion of the European Community and the Community is not responsible for any use that might be made of the information contained therein.",
"The work by Alberto Molinari and Angelo Montanari has been supported by the GNCS project Logic, Automata, and Games for Auto-Adaptive Systems."
],
[
"Proof of Lemma ",
"In the proof, we will exploit the fact that if two tracks in $\\operatorname{Trk}_\\mathpzc {K}$ have the same $B_{k+1}$ -descriptor, then they also have the same $B_k$ -descriptor.",
"The latter can indeed be obtained from the former by removing the nodes at depth $k+1$ (leaves) and then deleting isomorphic subtrees possibly originated by the removal.",
"By induction on $k\\ge 0$ .",
"Base case ($k=0$ ): let us assume $\\rho _1$ and $\\rho _2$ are associated with the descriptor element $(v_{in},S,v_{fin})$ and $\\rho _1^{\\prime }$ and $\\rho _2^{\\prime }$ with $(v_{in}^{\\prime },S^{\\prime },v_{fin}^{\\prime })$ .",
"Thus $\\rho _1\\cdot \\rho _1^{\\prime }$ and $\\rho _2\\cdot \\rho _2^{\\prime }$ are both described by the descriptor element $(v_{in},S\\cup \\lbrace v_{fin},v_{in}^{\\prime }\\rbrace \\cup S^{\\prime },v_{fin}^{\\prime })$ .",
"Inductive step ($k>0$ ): let $\\mathpzc {D}_{B_k}$ be the $B_k$ -descriptor for $\\rho _1\\cdot \\rho _1^{\\prime }$ and $\\mathpzc {D}_{B_k}^{\\prime }$ be the one for $\\rho _2\\cdot \\rho _2^{\\prime }$ : their roots are the same, as for $k=0$ ; let us now consider a prefix $\\rho $ of $\\rho _1\\cdot \\rho _1^{\\prime }$ : if $\\rho $ is a proper prefix of $\\rho _1$ , since $\\rho _1$ and $\\rho _2$ have the same $B_k$ -descriptor, there exists a prefix $\\overline{\\rho }$ of $\\rho _2$ associated with the same subtree as $\\rho $ of depth $k-1$ in the descriptor for $\\rho _1$ (and $\\rho _2$ ); for $\\rho =\\rho _1$ , it holds that $\\rho _1$ and $\\rho _2$ have the same $B_{k-1}$ -descriptor because they have the same $B_k$ -descriptor; if $\\rho $ is a proper prefix of $\\rho _1\\cdot \\rho _1^{\\prime }$ such that $\\rho =\\rho _1\\cdot \\tilde{\\rho }_1$ for some prefix $\\tilde{\\rho }_1$ of $\\rho _1^{\\prime }$ , then two cases have to be taken into account: if $|\\tilde{\\rho }_1|=1$ , then $\\tilde{\\rho }_1=v_{in}^{\\prime }$ ; but also $\\operatorname{fst}(\\rho _2^{\\prime })=v_{in}^{\\prime }$ .",
"Let us now consider the $B_{k-1}$ -descriptors for $\\rho _1\\cdot v_{in}^{\\prime }$ and $\\rho _2\\cdot v_{in}^{\\prime }$ : the labels of the roots are the same, namely $(v_{in},S\\cup \\lbrace v_{fin}\\rbrace ,v_{in}^{\\prime })$ , then the subtrees of depth $k-2$ are exactly the same as in $\\rho _1$ and $\\rho _2$ 's $B_{k-1}$ -descriptor, (possibly) with the addition of the $B_{k-2}$ -descriptor for $\\rho _1$ (which is equal to that for $\\rho _2$ ).",
"Thus $\\rho _1\\cdot v_{in}^{\\prime }$ and $\\rho _2\\cdot v_{in}^{\\prime }$ have the same $B_{k-1}$ -descriptor; otherwise, since $\\tilde{\\rho }_1$ is a prefix of $\\rho _1^{\\prime }$ of length at least 2, and $\\rho _1^{\\prime }$ and $\\rho _2^{\\prime }$ have the same $B_k$ -descriptor, there exists a prefix $\\tilde{\\rho }_2$ of $\\rho _2^{\\prime }$ associated with the same subtree of depth $k-1$ as $\\tilde{\\rho }_1$ (in the $B_k$ -descriptor for $\\rho _1^{\\prime }$ ).",
"Hence, by inductive hypothesis, $\\rho _1\\cdot \\tilde{\\rho }_1$ and $\\rho _2\\cdot \\tilde{\\rho }_2$ have the same $B_{k-1}$ -descriptor.",
"Therefore we have shown that for any proper prefix of $\\rho _1\\cdot \\rho _1^{\\prime }$ there exists a proper prefix of $\\rho _2\\cdot \\rho _2^{\\prime }$ having the same $B_{k-1}$ -descriptor.",
"The inverse can be shown by symmetry.",
"Thus $\\mathpzc {D}_{B_k}$ is equal to $\\mathpzc {D}_{B_k}^{\\prime }$ ."
],
[
"Proof of Theorem ",
"The proof is by induction on $i \\ge u+1$ .",
"(Case $i=u+1$ ) We consider two cases: if $\\rho _{ds}(u)=\\rho _{ds}(u+1)=d\\in \\mathpzc {C}$ , then we have $Q_{-2}(u)=\\mathpzc {C}\\setminus \\lbrace d\\rbrace $ , and $Q_{-1}(u)=\\lbrace d\\rbrace $ , $Q_0(u)=Q_1(u)=\\cdots =Q_s(u)=\\emptyset $ .",
"Moreover, it holds that $Q_{-2}(u+1)=\\mathpzc {C}\\setminus \\lbrace d\\rbrace $ , $Q_{-1}(u)=\\emptyset $ , $Q_0(u)=\\lbrace d\\rbrace $ , and $Q_1(u)=Q_2(u)=\\cdots = Q_s(u)=\\emptyset $ .",
"$c(u)>_{lex}c(u+1)$ and the thesis follows.",
"if $d,d^{\\prime }\\in \\mathpzc {C}$ , with $d\\ne d^{\\prime }$ , $\\rho _{ds}(u)=d$ , and $\\rho _{ds}(u+1)=d^{\\prime }$ , then we have $Q_{-2}(u)=\\mathpzc {C}\\setminus \\lbrace d\\rbrace $ , $Q_{-1}(u)=\\lbrace d\\rbrace $ , and $Q_0(u)=Q_1(u)=\\cdots =Q_s(u)=\\emptyset $ .",
"Moreover, it holds that $Q_{-2}(u+1)=\\mathpzc {C}\\setminus \\lbrace d,d^{\\prime }\\rbrace $ , $Q_{-1}(u)=\\lbrace d,d^{\\prime }\\rbrace $ , $Q_0(u)=Q_1(u)=\\cdots =Q_s(u)=\\emptyset $ , and $c(u)>_{lex}c(u+1)$ , implying the thesis.",
"(Case $i>u+1$ ) In the following, we say that $\\rho _{ds}(\\ell )$ and $\\rho _{ds}(m)$ ($\\ell <m$ ) are consecutive occurrences of a descriptor element $d$ if there are no other occurrences of $d$ in $\\rho _{ds}(\\ell +1, m-1)$ .",
"We consider the following cases: If $\\rho _{ds}(i)$ is the first occurrence of $d\\in \\mathpzc {C}$ , then $d\\in Q_{-2}(i-1)$ , $d\\in Q_{-1}(i)$ , and it holds that $c(i-1)>_{lex}c(i)$ .",
"If $\\rho _{ds}(i)$ is the second occurrence of $d\\in \\mathpzc {C}$ , according to the definition, $\\rho _{ds}(i)$ can not be 1-indistinguishable from the previous occurrence of $d$ , and thus $d\\in Q_{-1}(i-1)$ ($\\rho _{ds}(u,i-1)$ contains the first occurrence of $d$ ) and $d\\in Q_0(i)$ , proving that $c(i-1)>_{lex}c(i)$ .",
"If $\\rho _{ds}(i)$ is at least the third occurrence of $d\\in \\mathpzc {C}$ , but $\\rho _{ds}(i)$ is not 1-indistinguishable from the immediately preceding occurrence of $d$ , $\\rho _{ds}(i^{\\prime })$ , with $i^{\\prime }<i$ , then $DElm(\\rho _{ds}(u,i^{\\prime }-1))\\subset DElm(\\rho _{ds}(u,i-1))$ .",
"Hence, there exists a first occurrence of some $d^{\\prime }\\in \\mathpzc {C}$ in $\\rho _{ds}(i^{\\prime }+1, i-1)$ , say $\\rho _{ds}(j)=d^{\\prime }$ , for $i^{\\prime }+1\\le j\\le i-1$ .",
"Thus, $d\\in Q_{-1}(j)$ , $\\cdots $ , $d\\in Q_{-1}(i-1)$ , and $d\\in Q_0(i)$ , proving that $c(i-1)>_{lex}c(i)$ .",
"In the remaining cases, we assume that $\\rho _{ds}(i)$ is at least the third occurrence of $d\\in \\mathpzc {C}$ .",
"If $\\rho _{ds}(i-1)$ and $\\rho _{ds}(i)$ are both occurrences of $d\\in \\mathpzc {C}$ and $\\rho _{ds}(i-1)$ is $t$ -indistinguishable, for some $t>0$ , and not $(t+1)$ -indistinguishable, from the immediately preceding occurrence of $d$ , then $\\rho _{ds}(i-1)$ and $\\rho _{ds}(i)$ are exactly $(t+1)$ -indistinguishable.",
"Thus, $d\\in Q_{t}(i-1)$ and $d\\in Q_{t+1}(i)$ , implying that $c(i-1)>_{lex}c(i)$ (as a particular case, if $\\rho _{ds}(i-1)$ and the immediately preceding occurrence are not 1-indistinguishable, then $\\rho _{ds}(i-1)$ and $\\rho _{ds}(i)$ are at most 1-indistinguishable).",
"If $\\rho _{ds}(i)$ is exactly 1-indistinguishable from the immediately preceding occurrence of $d$ , $\\rho _{ds}(j)$ , with $j<i-1$ , then $DElm(\\rho _{ds}(u,j-1))= DElm(\\rho _{ds}(u,i-1))$ , and there are no first occurrences of any $d^{\\prime }\\in \\mathpzc {C}$ in $\\rho _{ds}(j,i-1)$ .",
"If $\\rho _{ds}(j)$ is not 1-indistinguishable from its previous occurrence of $d$ , it immediately follows that $d\\in Q_0(j)$ , $\\cdots $ , $d\\in Q_0(i-1)$ and $d\\in Q_1(i)$ , implying that $c(i-1)>_{lex}c(i)$ .",
"Otherwise, there exists $j< i^{\\prime }<i$ such that $\\rho _{ds}(i^{\\prime })=d^{\\prime \\prime }\\in \\mathpzc {C}$ is not 1-indistinguishable from any occurrence of $d^{\\prime \\prime }$ before $j$ (as a matter of fact, if this was not the case, $\\rho _{ds}(i)$ and $\\rho _{ds}(j)$ would be 2-indistinguishable); in particular, $\\rho _{ds}(i^{\\prime })$ is not 1-indistinguishable from the last occurrence of $d^{\\prime \\prime }$ before $j$ , say $\\rho _{ds}(j^{\\prime })$ , for some $j^{\\prime }<j$ (such a $j^{\\prime }$ exists since there are no first occurrences in $\\rho _{ds}(j+1,i-1)$ ).",
"Now, if by contradiction every pair of consecutive occurrences of $d^{\\prime \\prime }$ in $\\rho _{ds}(j^{\\prime },i^{\\prime })$ were 1-indistinguishable, then by Corollary REF $\\rho _{ds}(j^{\\prime })$ and $\\rho _{ds}(i^{\\prime })$ would be 1-indistinguishable.",
"Thus, a pair of consecutive occurrences of $d^{\\prime \\prime }$ exists, where the second element in the pair is $\\rho _{ds}(\\ell )=d^{\\prime \\prime }$ , with $j<\\ell <i$ , such that they are not 1-indistinguishable.",
"By inductive hypothesis, $d^{\\prime \\prime }\\in Q_{-1}(\\ell -1)$ and $d^{\\prime \\prime }\\in Q_0(\\ell )$ .",
"Therefore, $d\\in Q_0(\\ell )$ , $\\cdots $ , $d\\in Q_0(i-1)$ (recall that there are no first occurrences between $j$ and $i$ ) and $d\\in Q_1(i)$ , proving that $c(i-1)>_{lex}c(i)$ .",
"If $\\rho _{ds}(j)=d\\in \\mathpzc {C}$ is at most $t$ -indistinguishable (for some $t\\ge 1$ ) from a preceding occurrence of $d$ and $\\rho _{ds}(j)$ and $\\rho _{ds}(i)=d$ , with $j<i-1$ , are $(t+1)$ -indistinguishable consecutive occurrences of $d$ (by definition of indistinguishability, $\\rho _{ds}(j)$ and $\\rho _{ds}(i)$ can not be more than $(t+1)$ -indistinguishable), any occurrence of $d^{\\prime }\\in \\mathpzc {C}$ in $\\rho _{ds}(j+1,i-1)$ is (at least) $t$ -indistinguishable from another occurrence of $d^{\\prime }$ before $j$ .",
"By Proposition REF , all pairs of consecutive occurrences of $d^{\\prime }$ in $\\rho _{ds}(j+1,i-1)$ are (at least) $t$ -indistinguishable, hence $d\\in Q_t(j)$ , $\\cdots $ , $d\\in Q_t(i-1)$ and finally $d\\in Q_{t+1}(i)$ , proving that $c(i-1)>_{lex}c(i)$ .",
"If $\\rho _{ds}(j)=d\\in \\mathpzc {C}$ is at most $t$ -indistinguishable (for some $t\\ge 1$ ) from a preceding occurrence of $d$ , and $\\rho _{ds}(j)$ and $\\rho _{ds}(i)=d$ , with $j<i-1$ , are consecutive occurrences of $d$ which are at most $\\overline{t}$ -indistinguishable, for some $1\\le \\overline{t}\\le t$ , we preliminarily observe that $DElm(\\rho _{ds}(u,j-1))= DElm(\\rho _{ds}(u,i-1))$ .",
"Then, if some $d^{\\prime \\prime }\\in \\mathpzc {C}$ , with $d^{\\prime \\prime } \\ne d$ , occurs in $\\rho _{ds}(j+1,i-1)$ and it is not 1-indistinguishable from any occurrence of $d^{\\prime \\prime }$ before $j$ , then $\\overline{t}=1$ and we are again in case REF .",
"Otherwise, all the occurrences of descriptor elements in $\\rho _{ds}(j+1,i-1)$ are (at least) 1-indistinguishable from other occurrences before $j$ .",
"Moreover, there exists $j<i^{\\prime }<i$ such that $\\rho _{ds}(i^{\\prime })=d^{\\prime }\\in \\mathpzc {C},d\\ne d^{\\prime }$ , and it is at most $(\\overline{t}-1)$ -indistinguishable from another occurrence of $d^{\\prime }$ before $j$ .",
"Analogously to case REF , by Proposition REF , $\\rho _{ds}(i^{\\prime })$ must be $(\\overline{t}-1)$ -indistinguishable from the last occurrence of $d^{\\prime }$ before $j$ , say $\\rho _{ds}(j^{\\prime })$ , with $j^{\\prime }<j$ .",
"But two consecutive occurrences of $d^{\\prime }$ in $\\rho _{ds}(j^{\\prime },i^{\\prime })$ must then be at most $(\\overline{t}-1)$ -indistinguishable (if all pairs of occurrences of $d^{\\prime }$ in $\\rho _{ds}(j^{\\prime }, i^{\\prime })$ were $\\overline{t}$ -indistinguishable, $\\rho _{ds}(i^{\\prime })$ and $\\rho _{ds}(j^{\\prime })$ would be $\\overline{t}$ -indistinguishable as well), where the second occurrence is $\\rho _{ds}(\\ell )=d^{\\prime }$ for some $j<\\ell \\le i^{\\prime }$ .",
"By applying the inductive hypothesis, we have $d^{\\prime }\\in Q_{\\overline{t}-2}(\\ell -1)$ and $d^{\\prime }\\in Q_{\\overline{t}-1}(\\ell )$ .",
"As a consequence, we have $d\\in Q_{\\overline{t}-1}(\\ell )$ , $\\cdots $ , $d\\in Q_{\\overline{t}-1}(i-1)$ (all descriptor elements in $\\rho _{ds}(j, i)$ are at least $(\\overline{t}-1)$ -indistinguishable from other occurrences before $j$ ) and finally $d\\in Q_{\\overline{t}}(i)$ , implying that $c(i-1)>_{lex}c(i)$ .",
"It is worth pointing out that, from the proof of the theorem, it follows that the definition of $f$ is in fact redundant: cases (c) and (e) never occur."
],
[
"Proof of Lemma ",
"The proof is by induction on the structure of $\\psi $ .",
"The cases in which $\\psi =\\top $ , $\\psi =\\bot $ , $\\psi =p\\in \\mathpzc {AP}$ are trivial.",
"The cases in which $\\psi =\\lnot \\varphi $ , $\\psi =\\varphi _1\\wedge \\varphi _2$ are also trivial and omitted.",
"We focus on the remaining cases.",
"$\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ .",
"If $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ , then there exists $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ such that $\\operatorname{lst}(\\tilde{\\rho })=\\operatorname{fst}(\\rho )$ and $\\mathpzc {K},\\rho \\models \\varphi $ .",
"By Theorem REF the unravelling procedure returns $\\overline{\\rho }\\in \\operatorname{Trk}_\\mathpzc {K}$ such that $\\operatorname{fst}(\\overline{\\rho })=\\operatorname{fst}(\\rho )$ and $\\overline{\\rho }$ and $\\rho $ have the same $B_k$ -descriptor, thus $\\mathpzc {K},\\overline{\\rho }\\models \\varphi $ .",
"By the inductive hypothesis, Check$(\\mathpzc {K},k,\\varphi ,\\overline{\\rho })=1$ , hence Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ .",
"Vice versa, if Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ , there exists $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ such that $\\operatorname{lst}(\\tilde{\\rho })=\\operatorname{fst}(\\rho )$ and Check$(\\mathpzc {K},k,\\varphi ,\\rho )=1$ .",
"By the inductive hypothesis, $\\mathpzc {K},\\rho \\models \\varphi $ , hence $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ .",
"$\\psi =\\operatorname{\\langle \\overline{A}\\rangle }\\varphi $ .",
"The proof is symmetric to the case $\\psi =\\operatorname{\\langle A\\rangle }\\varphi $ .",
"$\\psi =\\operatorname{\\langle B\\rangle }\\varphi $ .",
"If $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ , there exists $\\rho \\in \\operatorname{Pref}(\\tilde{\\rho })$ such that $\\mathpzc {K},\\rho \\models \\varphi $ .",
"By the inductive hypothesis, Check$(\\mathpzc {K},k-1,\\varphi ,\\rho )=1$ .",
"Since all prefixes of $\\tilde{\\rho }$ are checked, Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ .",
"Note that, by definition of descriptor, if $\\tilde{\\rho }$ is a track representative of a $B_k$ -descriptor $\\mathpzc {D}_{B_k}$ , a prefix of $\\tilde{\\rho }$ is a representative of a $B_{k-1}$ -descriptor, whose root is a child of the root of $\\mathpzc {D}_{B_k}$ .",
"Vice versa, if Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ , then for some track $\\rho \\in \\operatorname{Pref}(\\tilde{\\rho })$ , we have Check$(\\mathpzc {K},k-1,\\varphi ,\\rho )=1$ .",
"By the inductive hypothesis $\\mathpzc {K},\\rho \\models \\varphi $ , hence $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ .",
"$\\psi =\\operatorname{\\langle \\overline{B}\\rangle }\\varphi $ .",
"If $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ , then there exists $\\rho $ such that $\\tilde{\\rho }\\cdot \\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ for which $\\mathpzc {K},\\tilde{\\rho }\\cdot \\rho \\models \\varphi $ .",
"If $|\\rho |=1$ , since by the inductive hypothesis Check$(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho }\\cdot \\rho )=1$ , then Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ .",
"Otherwise, the unravelling algorithm returns a track $\\overline{\\rho }$ with the same $B_k$ -descriptor as $\\rho $ .",
"Thus, by the extension Proposition REF , $\\tilde{\\rho }\\cdot \\rho $ and $\\tilde{\\rho }\\cdot \\overline{\\rho }$ have the same $B_k$ -descriptor.",
"Thus $\\mathpzc {K},\\tilde{\\rho }\\cdot \\overline{\\rho }\\models \\varphi $ .",
"So (by inductive hypothesis) Check$(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho }\\cdot \\overline{\\rho })=1$ implying that Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ .",
"Note that, given two tracks $\\rho ,\\rho ^{\\prime }$ of $\\mathpzc {K}$ , if we are considering $\\overline{\\rho }$ as the track representative of the $B_k$ -descriptor of $\\rho $ , and the unravelling algorithm returns $\\overline{\\rho }^{\\prime }$ as the representative of the $B_k$ -descriptor of $\\rho ^{\\prime }$ , since by Lemma REF $\\rho \\cdot \\rho ^{\\prime }$ and $\\overline{\\rho }\\cdot \\overline{\\rho }^{\\prime }$ have the same $B_k$ -descriptor, we have that $\\overline{\\rho }\\cdot \\overline{\\rho }^{\\prime }$ is the representative of the $B_k$ -descriptor of $\\rho \\cdot \\rho ^{\\prime }$ .",
"Vice versa, if Check$(\\mathpzc {K},k,\\psi ,\\tilde{\\rho })=1$ , there exists $\\rho $ such that $\\tilde{\\rho }\\cdot \\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ and Check$(\\mathpzc {K},k,\\varphi ,\\tilde{\\rho }\\cdot \\rho )=1$ .",
"By the inductive hypothesis, $\\mathpzc {K},\\tilde{\\rho }\\cdot \\rho \\models \\varphi $ , hence $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ .",
"$\\psi =\\operatorname{\\langle \\overline{E}\\rangle }\\varphi $ .",
"The proof is symmetric to the case $\\psi =\\operatorname{\\langle \\overline{B}\\rangle }\\varphi $ ."
],
[
"Proof of Theorem ",
"If $\\mathpzc {K}\\models \\psi $ , then for all $\\rho \\in \\operatorname{Trk}_\\mathpzc {K}$ such that $\\operatorname{fst}(\\rho )=w_0$ is the initial state of $\\mathpzc {K}$ , we have $\\mathpzc {K},\\rho \\models \\psi $ .",
"By Lemma REF , it follows that $\\texttt {Check}(\\mathpzc {K},\\operatorname{Nest_B}(\\psi ),\\psi ,\\rho )=1$ .",
"Now, the unravelling procedure returns a subset of the initial tracks.",
"This implies that ModCheck$(\\mathpzc {K},\\psi )=1$ .",
"On the other hand, if ModCheck$(\\mathpzc {K},\\psi )=1$ , then for any track $\\rho $ with $\\operatorname{fst}(\\rho )=w_0$ returned by the unravelling algorithm, $\\texttt {Check}(\\mathpzc {K},\\operatorname{Nest_B}(\\psi ),\\psi ,\\rho )=1$ and, by Lemma REF , $\\mathpzc {K},\\rho \\models \\psi $ .",
"Assume now that a track $\\tilde{\\rho }$ , with $\\operatorname{fst}(\\tilde{\\rho })=w_0$ , is not returned by the unravelling algorithm.",
"By Theorem REF , there exists a track $\\overline{\\rho }$ , with $\\operatorname{fst}(\\overline{\\rho })=w_0$ , which is returned in place of $\\tilde{\\rho }$ and $\\overline{\\rho }$ has the same $B_k$ -descriptor as $\\tilde{\\rho }$ (with $k=\\operatorname{Nest_B}(\\psi )$ ).",
"Since $\\mathpzc {K},\\tilde{\\rho }\\models \\psi \\iff \\mathpzc {K},\\overline{\\rho }\\models \\psi $ (by Theorem REF ) and $\\mathpzc {K},\\overline{\\rho }\\models \\psi $ , we get that $\\mathpzc {K},\\tilde{\\rho }\\models \\psi $ .",
"So all tracks starting from state $w_0$ model $\\psi $ , implying that $\\mathpzc {K}\\models \\psi $ ."
],
[
"NEXP-hardness of succinct $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$",
"In Section , we proved that the model checking problem for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas is in EXPSPACE, and, in Section , that it is PSPACE-hard.",
"Here we prove that the model checking problem for $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ is in between EXPSPACE and NEXP when a suitable encoding of formulas is exploited.",
"Such an encoding is succinct, in the sense that the following binary-encoded shorthands are used: $\\operatorname{\\langle B\\rangle }^k\\psi $ stands for $k$ repetitions of $\\operatorname{\\langle B\\rangle }$ before $\\psi $ , where $k$ is represented in binary (the same for all the other HS modalities); moreover, $\\bigwedge _{i=l,\\cdots ,r} \\psi (i)$ denotes a conjunction of formulas which contain some occurrences of the index $i$ as exponents ($l$ and $r$ are binary encoded naturals), e.g., $\\bigwedge _{i=1,\\cdots ,5}\\operatorname{\\langle B\\rangle }^i \\top $ .",
"Finally, we denote by $\\operatorname{expand}(\\psi )$ the expanded form of $\\psi $ , where all exponents $k$ are removed from $\\psi $ , by explicitly repeating $k$ times each HS modality with such an exponent, and big conjunctions are replaced by conjunctions of formulas without indexes.",
"It is not difficult to show that there exists a constant $c>0$ such that, for all succinct $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas $\\psi $ , $|\\operatorname{expand}(\\psi )|\\le 2^{|\\psi |^c}$ .",
"Therefore the model checking algorithm ModCheck of Section still runs in exponential working space with respect to the succinct input formula $\\psi $ —by preliminarily expanding $\\psi $ to $\\operatorname{expand}(\\psi )$ —as $\\tau (|W|,\\operatorname{Nest_B}(\\operatorname{expand}(\\psi )))$ is exponential in $|W|$ and $|\\psi |$ .",
"Moreover, the following result holds: Theorem 39 The model checking problem for succinctly encoded formulas of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ over finite Kripke structures is NEXP-hard (under polynomial-time reductions).",
"The theorem is proved by means of a reduction from the acceptance problem for a (generic) language $L$ decided by a non-deterministic one-tape Turing machine $M$ (w.l.o.g.)",
"that halts in $O(2^{n^{k}})$ computation steps on any input of size $n$ , where $k>0$ is a constant.",
"We suitably define a Kripke structure $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ and a succinct $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formula $\\psi $ such that $\\mathpzc {K}\\models \\psi $ if and only if $M$ accepts its input string $c_{0}c_{1}\\cdots c_{n-1}$ .",
"This allows us to conclude that the model checking problem for succinct $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ formulas over finite Kripke structures is between NEXP and EXPSPACE.",
"We end this section by proving Theorem REF .",
"Let us consider a language $L$ decided by a non-deterministic one-tape Turing machine $M$ (w.l.o.g.)",
"that halts after no more than $2^{n^{k}}-3$ computation steps on an input of size $n$ (assuming a sufficiently high constant $k\\in \\mathbb {N}$ ).",
"Hence, $L$ belongs to NEXP.",
"Let $\\Sigma $ and $Q$ be the alphabet and the set of states of $M$ , respectively, and let $\\#$ be a special symbol not in $\\Sigma $ used as separator for configurations (in the following we let $\\Sigma ^{\\prime }=\\Sigma \\cup \\left\\lbrace \\#\\right\\rbrace $ ).",
"The alphabet $\\Sigma $ is assumed to contain the blank symbol $\\sqcup $ .",
"As usual, a computation of $M$ is a sequence of configurations of $M$ , where each configuration fixes the content of the tape, the position of the head on the tape and the internal state of $M$ .",
"We use a standard encoding for computations called computation table (or tableau) (see [28], [33] for further details).",
"Each configuration of $M$ is a sequence over the alphabet $\\Gamma =\\Sigma ^{\\prime }\\cup (Q\\times \\Sigma )$ ; a symbol in $(q,c) \\in Q\\times \\Sigma $ occurring in the $i$ -th position encodes the fact that the machine has internal state $q$ and its head is currently on the $i$ -th position of the tape (obviously exactly one occurrence of a symbol in $Q\\times \\Sigma $ occurs in each configuration).",
"Since $M$ halts after no more than $2^{n^{k}}-3$ computation steps, $M$ uses at most $2^{n^{k}}-3$ cells on its tape, so the size of a configuration is $2^{n^{k}}$ (we need 3 occurrences of the auxiliary symbol $\\#$ , two for delimiting the beginning of the configuration, and one for the end; additionally $M$ never overwrites delimiters $\\#$ ).",
"If a configuration is actually shorter than $2^{n^k}$ , it is padded with $\\sqcup $ symbols in order to reach length $2^{n^k}$ (which is a fixed number, once the input length is known).",
"Moreover, since $M$ halts after no more than $2^{n^{k}}-3$ computation steps, the number of configurations is $2^{n^{k}}-3$ .",
"The computation table is basically a matrix of $2^{n^{k}}-3$ rows and $2^{n^{k}}$ columns, where the $i$ -th row records the configuration of $M$ at the $i$ -th computation step.",
"Figure: An example of computation table (tableau).As an example, a possible table is depicted in Figure REF .",
"In the first configuration (row) the head is in the leftmost position (on the right of delimiters $\\#$ ) and $M$ is in state $q_0$ .",
"In addition, we have the string symbols $c_0c_1\\cdots c_{n-1}$ padded with occurrences of $\\sqcup $ to reach length $2^{n^k}$ .",
"In the second configuration, the head has moved one position to the right, $c_0$ has been overwritten by $c_0^{\\prime }$ , and $M$ is in state $q_1$ .",
"From the first two rows, we can deduce that the tuple $(q_{0},c_{0},q_{1},c_{0}^{\\prime },\\rightarrow )$ belongs to the transition relation $\\delta _M$ of $M$ (we assume that $\\delta _M \\subseteq Q \\times \\Sigma \\times Q \\times \\Sigma \\times \\lbrace \\rightarrow ,\\leftarrow ,\\bullet \\rbrace $ with the obvious standard meaning).",
"Following [28], [33], we now introduce the notion of (legal) window.",
"A window is a $2\\times 3$ matrix, in which the first row represents three consecutive symbols of a possible configuration.",
"The second row represents the three symbols which are placed exactly in the same position in the next configuration.",
"A window is legal when the changes from the first to the second row are coherent with $\\delta _M$ in the obvious sense.",
"Actually, the set of legal windows, which we denote by $Wnd\\subseteq \\left(\\Gamma ^{3}\\right)^{2}$ , is a tabular representation of the transition relation $\\delta _M$ .",
"For example, two legal windows associated with the table of the previous example are: $\\begin{array}{|c|c|c|}\\hline \\# & (q_{0},c_{0}) & c_{1} \\\\\\hline \\# & c_{0}^{\\prime } & (q_{1},c_{1}) \\\\\\hline \\end{array}\\hspace{28.45274pt}\\begin{array}{|c|c|c|}\\hline (q_{0},c_{0}) & c_{1} & c_2\\\\\\hline c_{0}^{\\prime } & (q_{1},c_{1}) & c_2 \\\\\\hline \\end{array}$ Formally, a $((x,y,z),(x^{\\prime },y^{\\prime },z^{\\prime }))\\in Wnd$ can be represented as $\\begin{array}{|c|c|c|}\\hline x & y & z \\\\\\hline x^{\\prime } & y^{\\prime } & z^{\\prime } \\\\\\hline \\end{array}\\qquad \\text{with } x,x^{\\prime },y,y^{\\prime },z,z^{\\prime }\\in \\Gamma ,$ where the following constraints must hold: if all $x,y,z\\in \\Sigma ^{\\prime }$ ($x$ , $y$ , $z$ are not state-symbol pairs), then $y=y^{\\prime }$ ; if one of $x$ , $y$ and $z$ belongs to $Q\\times \\Sigma $ , then $x^{\\prime }$ , $y^{\\prime }$ and $z^{\\prime }$ are coherent with $\\delta _M$ , and $(x=\\#\\Rightarrow x^{\\prime }=\\#)\\wedge (y=\\#\\Rightarrow y^{\\prime }=\\#)\\wedge (z=\\#\\Rightarrow z^{\\prime }=\\#)$ .",
"As we said, $M$ never overwrites a $\\#$ and we can assume that the head never visits a $\\#$ , as well (some more windows can be possibly added if necessary, see [28]).",
"In the following we define a Kripke structure $\\mathpzc {K}=(\\mathpzc {AP},W,\\delta ,\\mu ,w_0)$ and a (succinct) formula $\\psi $ of $\\mathsf {A\\overline{A}B\\overline{B}\\overline{E}}$ such that $\\mathpzc {K}\\models \\psi $ if and only if $M$ accepts its input string $c_{0}c_{1}\\cdots c_{n-1}$ .",
"The set of propositional letters is $\\mathpzc {AP}=\\Gamma \\cup \\Gamma ^{3}\\cup \\left\\lbrace start\\right\\rbrace $ .",
"The Kripke structure $\\mathpzc {K}$ is obtained by suitably composing a basic pattern called gadget.",
"An instance of the gadget is associated with a triple of symbols $(a,b,c)\\in \\Gamma ^{3}$ (i.e., a sequence of three adjacent symbols in a configuration) and consists of 3 states: $q_{(a,b,c)}^{0}$ , $q_{(a,b,c)}^{1}$ , $q_{(a,b,c)}^{2}$ such that $\\mu \\left(q_{(a,b,c)}^{0}\\right)=\\mu \\left(q_{(a,b,c)}^{1}\\right)=\\left\\lbrace (a,b,c),c\\right\\rbrace \\text{ and }\\mu \\left(q_{(a,b,c)}^{2}\\right)=\\emptyset .$ Moreover, $\\delta \\left(q_{(a,b,c)}^{0}\\right)=\\left\\lbrace q_{(a,b,c)}^{1}\\right\\rbrace \\text{ and } \\delta \\left(q_{(a,b,c)}^{1}\\right)=\\left\\lbrace q_{(a,b,c)}^{2}\\right\\rbrace .$ (See Figure REF .)",
"The underlying idea is that a gadget associated with $(x,y,z)\\in \\Gamma ^{3}$ “records” the current proposition letter $z$ , as well as two more “past” letters ($x$ and $y$ ).",
"Figure: An instance of the described gadget for (a,b,c)∈Γ 3 (a,b,c)\\in \\Gamma ^{3}.The Kripke structure $\\mathpzc {K}$ has (an instance of) a gadget for every $(x,y,z)\\in \\Gamma ^{3}$ and for all $(x,y,z)$ and $(x^{\\prime },y^{\\prime },z^{\\prime })$ in $\\Gamma ^{3}$ , we have $q_{(x^{\\prime },y^{\\prime },z^{\\prime })}^{0}\\in \\delta \\left(q_{(x,y,z)}^{2}\\right)$ if and only if $x^{\\prime }=y$ and $y^{\\prime }=z$ .",
"Moreover, $\\mathpzc {K}$ has some additional (auxiliary) states $w_0,\\cdots , w_6$ described in Figure REF and $\\delta (w_{6})=\\left\\lbrace q_{(\\#,\\#,x)}^{0}\\mid x\\in \\Gamma \\right\\rbrace $ .",
"Note that the overall size of $\\mathpzc {K}$ only depends on $|\\Gamma |$ and it is constant w.r. to the input string $c_{0}c_{1}\\cdots c_{n-1}$ of $M$ .",
"Figure: Initial part of K\\mathpzc {K}.Now we want to decide whether an input string belongs to the language $L$ by solving the model checking problem $\\mathpzc {K}\\models start\\rightarrow \\operatorname{\\langle A\\rangle }\\xi $ where $\\xi $ is satisfied only by tracks which represent a successful computation of $M$ .",
"Since the only (initial) track which satisfies $start$ is $w_{0}w_{1}$ , we are actually verifying the existence of a track which begins with $w_{1}$ and satisfies $\\xi $ .",
"As for $\\xi $ , it requires that a track $\\rho $ , for which $\\mathpzc {K},\\rho \\models \\xi $ (with $\\operatorname{fst}(\\rho )=w_1$ ), mimics a successful computation of $M$ in this way: every interval $\\rho (i,i+1)$ , for $i\\mod {3}=0$ , satisfies the proposition letter $p\\in \\mathpzc {AP}$ if and only if the $\\frac{i}{3}$ -th character of the computation represented by $\\rho $ is $p$ (note that as a consequence of the gadget structure, only $\\rho $ 's subtracks $\\overline{\\rho }=\\rho (i, i+1)$ for $i\\mod {3}=0$ can satisfy some proposition letters).",
"A symbol of a configuration is mapped to an occurrence of an instance of a gadget in $\\rho $ ; $\\rho $ , in turn, encodes a computation of $M$ through the concatenation of the first, second, third...rows of the computation table (two consecutive configurations are separated by 3 occurrences of $\\#$ , which require 9 states overall).",
"Let us now define the HS formula $\\xi =\\psi _{accept}\\wedge \\psi _{input}\\wedge \\psi _{window}$ , where $\\psi _{accept}=\\operatorname{\\langle B\\rangle }\\operatorname{\\langle A\\rangle }\\bigvee _{a\\in \\Sigma }(q_{yes},a)$ requires a track to contain an occurrence of the accepting state of $M$ , $q_{yes}$ ; $\\psi _{input}$ is a bit more involved and demands that the subtrack corresponding to the first configuration of $M$ actually “spells” the input $c_0c_1\\cdots c_{n-1}$ , suitably padded with occurrences of $\\sqcup $ and terminated by a $\\#$ (in the following, $\\ell (k)$ , introduced in Example REF , is satisfied only by those tracks whose length equals $k$ ($k\\ge 2$ ) and it has a binary encoding of $O(\\log k)$ bits): $\\psi _{input}=[B]\\Big (\\ell (7)\\rightarrow \\operatorname{\\langle A\\rangle }(q_{0},c_{0})\\Big )\\wedge [B]\\Big (\\ell (10)\\rightarrow \\operatorname{\\langle A\\rangle }c_1\\Big )\\wedge [B]\\Big (\\ell (13)\\rightarrow \\operatorname{\\langle A\\rangle }c_2\\Big )\\wedge \\\\\\vdots \\\\[B]\\Big (\\ell (7+3(n-1))\\rightarrow \\operatorname{\\langle A\\rangle }c_{n-1}\\Big )\\wedge \\\\[B]\\Bigg (\\operatorname{\\langle B\\rangle }^{5+3n}\\top \\wedge [B]^{3\\cdot 2^{n^{k}}-6}\\bot \\rightarrow \\operatorname{\\langle A\\rangle }\\bigg (\\Big (\\ell (2)\\wedge \\bigwedge _{a\\in \\Gamma }\\lnot a \\Big )\\vee \\sqcup \\bigg )\\Bigg )\\wedge \\\\[B]\\Big (\\ell \\big (3\\cdot 2^{n^{k}}-2\\big )\\rightarrow \\operatorname{\\langle A\\rangle }\\#\\Big ).$ Finally $\\psi _{window}$ enforces the window constraint: if the proposition $(d,e,f)\\in \\Gamma ^3$ is witnessed in a subinterval (of length 2) in the subtrack of $\\rho $ corresponding to the $j$ -th configuration of $M$ , then in the same position of (the subtrack of $\\rho $ associated with) configuration $j-1$ , some $(a,b,c)\\in \\Gamma ^3$ must be there, such that $((a,b,c),(d,e,f))\\in Wnd$ .",
"$\\psi _{window}= [B]\\Bigg (\\bigwedge _{i=2,\\cdots ,t}\\bigwedge _{(d,e,f)\\in \\Gamma ^3}\\Big (\\ell (3\\cdot 2^{n^k}+3i+1)\\wedge \\operatorname{\\langle A\\rangle }(d,e,f)\\\\\\rightarrow [B] \\big (\\ell (3i+1)\\rightarrow \\bigvee _{((a,b,c),(d,e,f))\\in Wnd}\\operatorname{\\langle A\\rangle }(a,b,c)\\big )\\Big )\\Bigg ).$ where $t=2^{n^k}\\cdot (2^{n^k}-4)-1$ is encoded in binary.",
"All the integers which must be stored in the formula are less than $(2^{n^k})^2$ , thus they need $O(n^{k})$ bits to be encoded; in this way the formula can be generated in polynomial time."
],
[
"Proof of Lemma ",
"The proof is by induction on the complexity of $\\psi $ .",
"$\\psi =p$ , with $p\\in \\mathpzc {AP}$ ($p\\ell (p)=\\lbrace p\\rbrace $ ).",
"If $\\mathpzc {K},\\rho \\models p$ , then $p\\in \\mathpzc {L}(\\mathpzc {K},\\rho )$ and hence $p\\in \\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\rho )$ .",
"By hypothesis, it immediately follows that $p\\in \\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\rho ^{\\prime })$ , and thus $p\\in \\mathpzc {L}(\\mathpzc {K}^{\\prime },\\rho ^{\\prime })$ and $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models p$ .",
"$\\psi =\\lnot \\phi $ ($p\\ell (\\phi )=p\\ell (\\psi )$ ).",
"If $\\mathpzc {K},\\rho \\models \\lnot \\phi $ , then $\\mathpzc {K},\\rho \\lnot \\models \\phi $ .",
"By the inductive hypothesis, $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\lnot \\models \\phi $ and thus $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models \\lnot \\phi $ .",
"$\\psi =\\phi _1\\wedge \\phi _2$ .",
"If $\\mathpzc {K},\\rho \\models \\phi _1\\wedge \\phi _2$ , then in particular $\\mathpzc {K},\\rho \\models \\phi _1$ .",
"Since, by hypothesis, $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\rho )=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\rho ^{\\prime })$ and $reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ^{\\prime }))$ , it holds that $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi _1)},\\rho )=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi _1)},\\rho ^{\\prime })$ and $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi _1)},\\operatorname{lst}(\\rho ))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi _1)},\\operatorname{lst}(\\rho ^{\\prime }))$ , as $p\\ell (\\phi _1)\\subseteq p\\ell (\\psi )$ .",
"By the inductive hypothesis, $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models \\phi _1$ .",
"The same argument works for $\\phi _2$ .",
"The thesis follows.",
"$\\psi =\\operatorname{\\langle A\\rangle }\\phi $ .",
"If $\\mathpzc {K},\\rho \\models \\operatorname{\\langle A\\rangle }\\phi $ , there exists a track $\\overline{\\rho }\\in \\operatorname{Trk}_\\mathpzc {K}$ such that $\\operatorname{fst}(\\overline{\\rho })=\\operatorname{lst}(\\rho )$ and $\\mathpzc {K},\\overline{\\rho }\\models \\phi $ , with $p\\ell (\\phi ) = p\\ell (\\psi )$ .",
"By hypothesis, it holds that $reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ^{\\prime }))$ .",
"Hence, there exists a track $\\overline{\\rho }^{\\prime }\\in \\operatorname{Trk}_{\\mathpzc {K}^{\\prime }}$ , with $\\operatorname{fst}(\\overline{\\rho }^{\\prime })=\\operatorname{lst}(\\rho ^{\\prime })$ , such that $|\\overline{\\rho }|=|\\overline{\\rho }^{\\prime }|$ and for all $0\\le i \\le |\\overline{\\rho }|-1$ , $f(\\overline{\\rho }(i)) = \\overline{\\rho }^{\\prime }(i)$ , where $f$ is the (an) isomorphism between $reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))$ and $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ^{\\prime }))$ .",
"It immediately follows that $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\overline{\\rho })=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\overline{\\rho }^{\\prime })$ .",
"We now prove that $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))\\sim reach(\\mathpzc {K}^{\\prime }_{\\, |p\\ell (\\phi )}, \\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ .",
"To this end, it suffices to prove that the restriction of the isomorphism $f$ to the states of $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ , say $f^{\\prime }$ , is an isomorphism between $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ and $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ (note that $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ is a subgraph of $reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))$ ).",
"First, it holds that $f(\\operatorname{lst}(\\overline{\\rho }))=f^{\\prime }(\\operatorname{lst}(\\overline{\\rho }))=\\operatorname{lst}(\\overline{\\rho }^{\\prime })$ .",
"Next, if $w$ is any state of $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ , then $f(w)=f^{\\prime }(w)=w^{\\prime }$ is a state of $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ , as from the existence of a track from $\\operatorname{lst}(\\overline{\\rho })$ to $w$ , it follows that there is an isomorphic track (w.r.t.",
"$f$ ) from $\\operatorname{lst}(\\overline{\\rho }^{\\prime })$ to $w^{\\prime }$ .",
"Moreover, if $(w,\\overline{w})\\in \\delta $ , then $\\overline{w}$ belongs to $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ , and thus $(w^{\\prime },f(\\overline{w}))\\in \\delta ^{\\prime }$ and $f(\\overline{w})=f^{\\prime }(\\overline{w})$ belongs to $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ .",
"We can conclude that, for any two states $v, v^{\\prime }$ of $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))$ , it holds that $(v,v^{\\prime })$ is an edge if and only if $(f^{\\prime }(v),f^{\\prime }(v^{\\prime }))$ is an edge of $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ .",
"By the inductive hypothesis, $\\mathpzc {K}^{\\prime },\\overline{\\rho }^{\\prime }\\models \\phi $ and hence $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models \\operatorname{\\langle A\\rangle }\\phi $ .",
"$\\psi =\\operatorname{\\langle \\overline{B}\\rangle }\\phi $ .",
"If $\\mathpzc {K},\\rho \\models \\operatorname{\\langle \\overline{B}\\rangle }\\phi $ , then $\\mathpzc {K},\\rho \\cdot \\overline{\\rho }\\models \\phi $ , with $p\\ell (\\psi )=p\\ell (\\phi )$ , where $\\rho \\cdot \\overline{\\rho }\\in \\operatorname{Trk}_\\mathpzc {K}$ and $\\overline{\\rho }$ is either a single state or a proper track.",
"In analogy to the previous case, let $\\overline{\\rho }^{\\prime }\\in \\operatorname{Trk}_{\\mathpzc {K}^{\\prime }}$ such that $|\\overline{\\rho }|=|\\overline{\\rho }^{\\prime }|$ and, for all $0\\le i <|\\overline{\\rho }|$ , $f(\\overline{\\rho }(i))=\\overline{\\rho }^{\\prime }(i)$ , where $f$ is the isomorphism between $reach(\\mathpzc {K}_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ))$ and $reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\psi )},\\operatorname{lst}(\\rho ^{\\prime }))$ .",
"Since $f(\\operatorname{lst}(\\rho ))=\\operatorname{lst}(\\rho ^{\\prime })$ , by definition of isomorphism, $(\\operatorname{lst}(\\rho ),\\operatorname{fst}(\\overline{\\rho }))\\in \\delta $ implies $(\\operatorname{lst}(\\rho ^{\\prime }),\\operatorname{fst}(\\overline{\\rho }^{\\prime }))\\in \\delta ^{\\prime }$ .",
"Therefore $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\overline{\\rho })=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\overline{\\rho }^{\\prime })$ and $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\overline{\\rho }^{\\prime }))$ .",
"Finally, $\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\rho \\cdot \\overline{\\rho })=\\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\rho )\\cap \\mathpzc {L}(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\overline{\\rho })= \\\\\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\rho ^{\\prime })\\cap \\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\overline{\\rho }^{\\prime })=\\mathpzc {L}(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\rho ^{\\prime }\\cdot \\overline{\\rho }^{\\prime })$ and $reach(\\mathpzc {K}_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\rho \\cdot \\overline{\\rho }))\\sim reach(\\mathpzc {K}^{\\prime }_{\\,|p\\ell (\\phi )},\\operatorname{lst}(\\rho ^{\\prime }\\cdot \\overline{\\rho }^{\\prime }))$ .",
"By the inductive hypothesis, $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\cdot \\overline{\\rho }^{\\prime }\\models \\phi $ and thus $\\mathpzc {K}^{\\prime },\\rho ^{\\prime }\\models \\operatorname{\\langle \\overline{B}\\rangle }\\phi $ ."
]
] | 1709.01849 |
[
[
"Edge states in a ferromagnetic honeycomb lattice with armchair\n boundaries"
],
[
"Abstract We investigate the properties of magnon edge states in a ferromagnetic honeycomb lattice with armchair boundaries.",
"In contrast with fermionic graphene, we find novel edge states due to the missing bonds along the boundary sites.",
"After introducing an external on-site potential at the outermost sites we find that the energy spectra of the edge states are tunable.",
"Additionally, when a non-trivial gap is induced, we find that some of the edge states are topologically protected and also tunable.",
"Our results may explain the origin of the novel edge states recently observed in photonic lattices.",
"We also discuss the behavior of these edge states for further experimental confirmations"
],
[
"colorlinks = true, citecolor = blue Edge states in a ferromagnetic honeycomb lattice with armchair boundaries Pierre A. Pantaleón [email protected] Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom Y. Xian [email protected] Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom We investigate the properties of magnon edge states in a ferromagnetic honeycomb lattice with armchair boundaries.",
"In contrast with fermionic graphene, we find novel edge states due to the missing bonds along the boundary sites.",
"After introducing an external on-site potential at the outermost sites we find that the energy spectra of the edge states are tunable.",
"Additionally, when a non-trivial gap is induced, we find that some of the edge states are topologically protected and also tunable.",
"Our results may explain the origin of the novel edge states recently observed in photonic lattices.",
"We also discuss the behavior of these edge states for further experimental confirmations.",
"Introduction.— One intriguing aspect of electrons moving in finite-sized honeycomb lattices is the presence of edge states, which have strong implications in the electronic properties and play an essential role in the electronic transport [1], [2], [3].",
"It is well known that natural graphene exhibits edge states under some particular boundaries [4], [5].",
"For example, there are flat edge states connecting the two Dirac points in a lattice with zig-zag [1] or bearded edges [6].",
"On the contrary, there are no edge states in a lattice with armchair boundary [7], unless a boundary potential is applied [8].",
"The edge states have also been studied in magnetic insulators [9], [10], [11], where the spin moments are carried by magnons.",
"Recently, it has been shown that the magnonic equivalence for the Kane-Mele-Haldane model is a ferromagnetic Heisenberg Hamiltonian with the Dzialozinskii-Moriya interaction [12], [13].",
"Firstly, while the energy band structure of the magnons of ferromagnets on the honeycomb lattice closely resembles that of the fermionic graphene [14], [15], it is not clear whether or not they show similar edge states, particularly in view of the interaction terms in the bosonic models which are usually ignored in graphene [16].",
"Secondly, most recent experiments in photonic lattices have observed novel edge states in honeycomb lattices with bearded [17] and armchair [18] boundaries, which are not present in fermionic graphene.",
"The main purpose of this paper is to address these two issues.",
"By considering a ferromagnetic honeycomb lattice with armchair boundaries, we find that the bosonic nature of the Hamiltonian reveals novel edge states which are not present in their fermionic counterpart.",
"After introducing an external on-site potential at the outermost sites, we find that the edge states are tunable.",
"Interestingly, we find that the nature of such edge states is Tamm-like [19], in contrast with the equivalent model for the armchair graphene [8] but, as mentioned earlier, in agreement with the experiments in photonic lattices [17], [18].",
"Furthermore, after introducing a Dzialozinskii-Moriya interaction (DMI), we find that the topologically protected edge states are sensitive to the presence of the Tamm-like states and they also become tunable.",
"Model.— We consider the following Hamiltonian for a ferromagnetic honeycomb lattice, $H=-J\\sum _{\\left\\langle i,j\\right\\rangle }\\mathbf {S}_{i}\\cdot \\mathbf {S}_{j}+\\sum _{\\left\\langle \\left\\langle i,j\\right\\rangle \\right\\rangle }D_{ij}\\cdot \\left(\\mathbf {S}_{i}\\times \\mathbf {S}_{j}\\right),$ where the first summation runs over the nearest-neighbors (NN) and the second over the next-nearest-neighbors (NNN), $J\\left(>0\\right)$ is the isotropic ferromagnetic coupling, $S_{i}$ is the spin moment at site $i$ and $D_{ij}$ is the DMI vector between NNN sites [20].",
"If we assume that the lattice is at the $x$ -$y$ plane, according to Moriya's rules [20], the DMI vector vanishes for the NN but has non-zero component along the $z$ direction for the NNN.",
"Hence, we can assume $D_{ij}=D\\nu _{ij}\\hat{z}$ , where $\\nu _{ij}=\\pm 1$ is an orientation dependent coefficient in analogy with the Kane-Mele model [21].",
"For the infinite system in the linear spin-wave approximation (LSWA), the Hamiltonian in Eq.",
"(REF ) can be reduced to a bosonic equivalent of the Kane-Mele-Haldane model [12], [13], [14].",
"To investigate the edge states we consider an armchair boundary along the $x$ direction, with a large $N$ sites in the $y$ direction, as shown in Fig.",
"(REF ).",
"A partial Fourier transform is made and the Hamiltonian given in Eq.",
"(REF ) in LSWA can be written in the form, $H=-t\\sum _{k}\\Psi _{k}^{\\dagger }M\\Psi _{k},$ where $\\Psi _{k}^{\\dagger }=\\left[\\Psi _{k,A}^{\\dagger },\\,\\Psi _{k,B}^{\\dagger }\\right]$ is a $2N\\times 2N$ , 2-component spinor, $k$ is the Bloch wave number in the $x$ direction and $t=JS$ .",
"The matrix elements of $M$ are $N\\times N$ matrices given by: $M_{11}=\\left(1-\\delta _{1}\\right)T^{\\dagger }T+\\left(1-\\delta _{N}\\right)TT^{\\dagger }+\\left(1+\\delta _{1}+\\delta _{N}\\right)I+M_{D}$ , $M_{12}=-J_{1}I-J_{2}\\left(T+T^{\\dagger }\\right)$ , $M_{21}=M_{12}^{\\dagger }$ , $M_{22}=M_{11}-2M_{D}$ and $M_{D}=J_{3}\\left(TT-T^{\\dagger }T^{\\dagger }\\right)+J_{4}\\left(T+T^{\\dagger }\\right)$ the DMI contribution.",
"Here, $T$ is a displacement matrix as defined in Ref.",
"[22] and $I$ a $N\\times N$ identity matrix.",
"We have also introduced two on-site energies $\\delta _{1}$ and $\\delta _{N}$ at the outermost sites of each boundary, respectively.",
"The coupling terms are: $J_{1}=e^{-ik}$ , $J_{2}=e^{\\frac{1}{2}ik}$ , $J_{3}=i\\,D^{\\prime }$ , $J_{4}=2i\\,D^{\\prime }\\,\\cos \\left(3k/2\\right)$ and $D^{\\prime }=D/J$ .",
"The numerical diagonalization of the matrix given by Eq.",
"(REF ) reveals that the bulk spectra is gapless only if $N=3m+1$ , with $m$ a positive integer [23].",
"However, to avoid size-dependent bulk gaps or hybridization between edge states of opposite edges [8], we consider a large $N$ where the edge states are independent of the size [24], [25].",
"Figure: Squematics of the upper armchair edge of a honeycomb lattice.The external on-site potential δ 1 \\delta _{1} is applied at the outermostsites.",
"Here, nn is a real-space row index in yy direction perpendicularto the edge.",
"For a large NN, we consider the opposite edge withthe same structure and with an on-site potential δ N \\delta _{N}.",
"Boundary conditions.— From the explicit form of the matrix elements in Eq.",
"(REF ), the coupled Harper equations can be obtained [26].",
"If we assume that the edge states are exponentially decaying from the armchair boundary, we can consider the following anzats [27], [28] for the eigenstates of $M$ in Eq.",
"(REF ), $\\Psi _{k}\\left(n\\right)=\\left[\\begin{array}{c}\\psi _{k,A}\\left(n\\right)\\\\\\psi _{k,B}\\left(n\\right)\\end{array}\\right]=z^{n}\\left[\\begin{array}{c}\\phi _{k,A}\\\\\\phi _{k,B}\\end{array}\\right],$ where $\\left[\\phi _{k,A},\\,\\phi _{k,B}\\right]^{t}$ is an eigenvector of $M$ , $z$ is a complex number and $n\\,\\left\\lbrace =1,\\,2,\\,3\\,...\\right\\rbrace $ is a real space lattice index in the $y$ direction, as shown in Fig.",
"(REF ).",
"Upon substitution of the anzats in the coupled Harper equations, the complex number $z$ obey the following polynomial equation, $\\sum _{\\mu =0}^{4}a_{\\mu }(z+z^{-1})^{\\mu }=0,$ with coefficients: $a_{0}=1-\\left(3-\\varepsilon \\right)^{2}-4J_{4}^{2}$ , $a_{1}=8J_{3}J_{4}+J_{1}^{\\ast }J_{2}+J_{2}^{\\ast }J_{1}$ , $a_{2}=-4J_{3}^{2}+J_{4}^{2}+1$ , $a_{3}=-2J_{3}J_{4}$ and $a_{4}=J_{3}^{2}$ .",
"For a given $k$ and energy $\\varepsilon $ , such a polynomial always yields four solutions for $\\left(z+z^{-1}\\right)$ .",
"Since we require a decaying wave from the boundary, only the solutions with $\\left|z\\right|<1$ are relevant for the description of the edge states at the upper edge and $\\left|z\\right|>1$ for the lower (opposite) edge.",
"The eigenfunction of Eq.",
"(REF ) satisfying $\\underset{n\\rightarrow \\infty }{lim}\\,\\Psi _{k}\\left(n\\right)=0$ may now in general be written as, $\\psi _{k,l}\\left(n\\right)=\\sum _{\\upsilon =1}^{4}c_{v}z_{v}^{n}\\phi _{l,v},$ where the coefficients $c_{\\upsilon }$ are determined by the boundary conditions and $\\phi _{l,v}$ is the two-component eigenvector $\\left(l=A,\\,B\\right)$ of $M$ .",
"From the Harper equations provided by the Eq.",
"(REF ) and Eq.",
"(REF ), the boundary conditions are satisfied by, $\\left(1-\\delta _{1}\\right)\\psi _{k,A}\\left(1\\right)-J_{2}\\psi _{k,B}\\left(0\\right) & =0,\\\\\\left(1-\\delta _{1}\\right)\\psi _{k,B}\\left(1\\right)-J_{2}^{\\ast }\\psi _{k,A}\\left(0\\right) & =0,\\\\J_{4}\\psi _{k,A}\\left(0\\right)-J_{3}\\psi _{k,A}\\left(-1\\right) & =0,\\\\J_{4}\\psi _{k,B}\\left(0\\right)-J_{3}\\psi _{k,B}\\left(-1\\right) & =0.$ By Eq.",
"(REF ), the above relations can be written as a set of equations for the unknown coefficients $c_{v}$ .",
"The non-trivial solution and the polynomial given by Eq.",
"(REF ), provide us a complete set of equations for the edge state energy dispersion and they can be solved numerically.",
"The same procedure can be followed to obtain the solutions for the opposite edge.",
"Figure: (Color online) a) Edge state energy dispersion for D=0D=0and δ 1 =δ N =0\\delta _{1}=\\delta _{N}=0.",
"The blue regions are the bulk energyspectra.",
"The green (dotted) and red (continuous) lines are the edgestate energy bands.",
"In b) their corresponding penetration lengthsare shown.",
"The magnon density profile for the edge magnon is shownin c) for ε=2±2t\\varepsilon =\\left(2\\pm \\sqrt{2}\\right)t at k=±π/3k=\\pm \\pi /3,and in d) for ε=0.298t\\varepsilon =0.298\\,t at k=±0.65k=\\pm 0.65.",
"The radiusof each circle is proportional to the magnon density.Zero DMI.— For the system without DMI, the coupling terms invoving $J_{3}$ and $J_{4}$ vanish, and the boundary conditions are reduced to the Eqs.",
"(REF ) and () with a quadratic polynomial in $(z+z^{-1})$ of Eq.",
"(REF ).",
"In particular, for the (uniform) case with $\\delta _{1}=\\delta _{N}=1$ , the edge and the bulk sites have the same on-site potential and the boundary conditions provide us with two bulk solutions with $z^{2}=1$ .",
"Therefore, in analogy with graphene with armchair edges, there are not edge states [7].",
"However, as shown in Fig.",
"(REF a), in the absence of external on-site potential $\\left(\\delta _{1}=\\delta _{N}=0\\right)$ , two new dispersive localized modes are obtained.",
"Located between (red, continuous line) and below (green, dotted) the bulk bands, such edge states are well defined along the Brillouin zone and their energy bands are doubly degenerated due to the fact that there are two edges in the ribbon.",
"These edge states have not been previously predicted or observed in magnetic insulators.",
"However, we believe that they are analogous to the novel edge states recently observed in a photonic honeycomb lattice with armchair edges [18].",
"Although in Ref.",
"[18] these edge states may be attributed to the dangling bonds along the boundary sites (the details have been given for zig-zag and bearded but not for armchair edges), and since these dangling bonds can be viewed as effective defects along the edges, similar physics is contained in our model where the effective defects are described by the different on-site potential at the boundaries.",
"We believe that our approach has the advantage of simple implementation for various boundary conditions.",
"In particular, we have obtained analytical expressions for the wavefunctions and their confinement along the boundary.",
"The latter is given by the penetration length (or width) of the edge state [29] defined as, $\\xi _{i}\\left(k\\right)\\equiv \\frac{\\sqrt{3}}{2}\\left[\\ln \\left|\\frac{1}{z_{i}\\left(k\\right)}\\right|\\right]^{-1},$ indicating a decay of the form $\\sim e^{-y/\\xi _{i}\\left(k\\right)}$ .",
"In the above equation, $z_{i}$ is the $i$ -th decaying factor in the linear combination, Eq.",
"(REF ).",
"Since we require two decaying factors to construct the edge state, we have two penetration lengths as mentioned in Ref.",
"[18].",
"The penetration lengths for the edge states with $\\delta _{1}=\\delta _{N}=0$ are shown in the Fig.",
"(REF b).",
"The dotted (green) and continuous (red) lines are the corresponding penetration lengths for the edge states in the Fig.",
"(REF a).",
"The edge state between the bulk bands (red, continuous) is composed by two penetration lengths but they are indistinguishable to each other.",
"This indicates that the decaying factors are complex conjugates to each other, $z_{1}=z_{2}^{\\ast }$ [29], [14].",
"The edge state below the lower bulk band (green, dotted) depends on two penetration lengths in the region around $k=\\pm \\pi /3$ .",
"However, outside such region, one penetration length diverges, $\\left|z\\right|\\rightarrow 1$ , while the other one decreases to a minimum value.",
"This means that the edge state tends to merge with the bulk and is almost indistinguishable at $k=0,\\,\\pm 2\\pi /3$ .",
"Furthermore, as we can see in the Fig.",
"(REF b), at $k=\\pm \\pi /3$ , the penetration length of both edge states is identical hence they have their maximum confinement along the boundary at the same Bloch wave-vector.",
"This is shown in the Fig.",
"(REF c) where we plot the magnon density, $\\left|\\psi _{k,l}\\left(n\\right)/\\psi _{k,l}\\left(1\\right)\\right|^{2}$ , for both edge states at $k=\\pm \\pi /3$ with their corresponding energies, $\\varepsilon =\\left(2\\pm \\sqrt{2}\\right)t$ .",
"In addition, in the Fig.",
"(REF d) the magnon density for the edge state below the lower bulk band with energy $\\varepsilon =0.298\\,t$ at $k=\\pm 0.65$ is shown, where as we mentioned before, the edge state tends to spread to the inner sites.",
"Figure: (Color online) (Color online) a) Edge state energy dispersionfor D=0D=0 with δ 1 =2.5\\delta _{1}=2.5 and δ N =0\\delta _{N}=0.",
"The blue regionsare the bulk energy spectra.",
"The dotted (green) and continuous (red)lines are the edge state energy bands at the lower edge.",
"The dot-dashed(purple) line, continuous (black) line and the circle (orange) arethe edge states at the upper edge.",
"In b) the penetration lengths ofthe corresponding edge states at the upper edge is shown.",
"The edge states discussed above have been obtained with no gap in the bulk and their dispersion relations are between the Dirac points.",
"Their existence without external on-site potentials indicates that the edge states are “Tamm-like” [19], [17].",
"Such type of states are usually associated with surface perturbations or defects.",
"However, in our system no defects are present.",
"The nature of such edge states can be explained in terms of the intrinsic on-site potential, where each site along the armchair boundary has two nearest-neighbors and the intrinsic on-site potential is lower than in the bulk.",
"Such a difference plays the same role as an effective defect which allows the existence of a Tamm-like state.",
"Figure: (Color online) Edge state energy dispersion for D=0.1JD=0.1\\,J.The blue regions are the bulk energy spectra.",
"For δ N =1\\delta _{N}=1(lower edge) there is an edge state crossing the gap (green, continuous).The red (dotted) lines are the edge states for a) δ 1 =0\\delta _{1}=0,b) δ 1 =1\\delta _{1}=1, c) δ 1 =2\\delta _{1}=2, d) δ 1 =3\\delta _{1}=3 and e)δ 1 =4\\delta _{1}=4.",
"The circles in d) and e) are edge states within thebulk energy bands.",
"We next discuss the effects of edge potentials.",
"It has been shown that edge states can be induced by edge potentials in the armchair graphene [8], however, the edge states that we found are a consequence of the bosonic nature of the lattice as discussed above.",
"Since they exist without opening a gap and they are dispersive, it is not clear if they can be predicted by a topological approach.",
"Our approach reveals that the edge states in a bosonic lattice are strongly dependent on the on-site potential along the boundary.",
"For example, if $\\delta _{1}=2.5$ (with $\\delta _{N}=0$ ) some new features are obtained.",
"As shown in Fig.",
"(REF a), the presence of the strong external on-site potential reveals three edge states at the upper boundary: a high energy edge state over the bulk bands (black, continuous line), an edge state between the bulk bands (purple, dot-dashed line), and interestingly, an edge state within the bulk bands (orange circle).",
"Such edge state is strongly localized and is highly dispersive.",
"It merges into the bulk with an small change in their Bloch wave-number and may be difficult to detect in a magnetic insulator.",
"It is therefore very encouraging that similar edge state was observed in a photonic lattice [18].",
"The penetration lengths of the corresponding edge states are shown in the Fig.",
"(REF b), firstly, the edge state over the bulk bands (black, continuous line) depends on two decaying factors and the oscillating behavior of their corresponding penetration lengths reveals that is strongly localized along the boundary sites and it never merges into the bulk.",
"Secondly, the edge state within the bulk bands (orange, dashed line) depends on a single decaying factor and is highly dispersive.",
"Thirdly, the edge state between the bulk bands (purple, dot-dashed) depends on a single penetration length since the two decaying factors are conjugate to each other [29].",
"Non-zero DMI.— It is well known that a non-zero DMI in a bosonic honeycomb lattice makes the band structure topologically non-trivial and reveals metallic edge states which transverse the gap [12].",
"However, the edge states that appear, under, within and over the bulk bands in Fig.",
"(REF a) and Fig.",
"(REF a) are distinct to the edge states predicted by topological arguments.",
"In the Fig.",
"(REF ) we show the energy bands for a DMI strength of $D=0.1\\,J$ , where we keep a fixed $\\delta _{N}=1$ and we modified $\\delta _{1}$ .",
"The continuous (green) line that cross the gap from the lower to the upper bulk bands is the edge state at the lower edge.",
"The dotted (red) lines correspond to the edge states at the upper edge.",
"If we follow the edge state energy spectra at the upper boundary from the Fig.",
"(REF a) to the Fig.",
"(REF c), we observe that the edge state within the bulk gap change its concavity.",
"The Tamm-like state below the bulk bands, Fig.",
"(REF a), merge with the bulk and a new Tamm-like state appears at the top of the upper bulk band, as shown in Fig.",
"(REF c).",
"If we keep increasing the value of the external on-site potential the Tamm-like state over the bulk band in Fig.",
"(REF c) moves away from the upper bulk band, Fig.",
"(REF d).",
"Furthermore, a second Tamm-like state appears with components within the bulk, as shown in Fig.",
"(REF d) and Fig.",
"(REF e).",
"The boundary conditions suggest that the existence of these two tunable edge states is due to the two sites in the unit cell of the armchair boundary and, by symmetry, the same behavior is expected at the opposite edge.",
"These edge states can be made to locate below, within and over the bulk bands.",
"If a non-trivial gap is induced the topologically protected edge states are also tunable.",
"Finally, a similar phenomena is expected for a lattice with zig-zag or bearded boundaries.",
"In both cases there is a single outermost site and Tamm-like edge states may appear due to the missing bond and/or by the external on-site potential.",
"Since the outermost site at the lattice with a bearded boundary has three missing bonds, the effective defect should be stronger than the corresponding to a zig-zag boundary.",
"This may be related with the existence of the “unconventional” edge states found in optical lattices [17].",
"A more extensive investigation of Tamm-like edge states along different boundaries will be reported elsewhere.",
"Conclusions.— We have analyzed the edge states in a ferromagnetic honeycomb lattice with armchair edges and an external on-site potential at the outermost sites.",
"In contrast with graphene, our system without external on-site potential reveals two edge states.",
"It is clear that the open boundary in a bosonic lattice creates an effective defect by a difference in the on-site potential between the bulk and boundary sites.",
"This effective defect is responsible for the existence of the novel edge states.",
"By introducing an external on-site potential at the outermost sites we found that the nature of this edge states is Tamm-like.",
"We also found that these edge states are tunable in their shapes and positions depending on the external on-site potential strength.",
"Such tunability can be used to modify the topologically protected edge states when a non-trivial gap is induced.",
"Finally, we found that the number of these tunable edge states is related to the number of sites in a unit cell along the boundary.",
"We believe that our results may explain the edge states recently found in optical lattices [17], [18] and motivate new experiments in both magnonic and photonic lattices.",
"Pierre.",
"A. Pantaleón is sponsored by Mexico's National Council of Science and Technology (CONACYT) under the scholarship 381939."
]
] | 1709.01855 |
[
[
"Evaluating Partisan Gerrymandering in Wisconsin"
],
[
"Abstract We examine the extent of gerrymandering for the 2010 General Assembly district map of Wisconsin.",
"We find that there is substantial variability in the election outcome depending on what maps are used.",
"We also found robust evidence that the district maps are highly gerrymandered and that this gerrymandering likely altered the partisan make up of the Wisconsin General Assembly in some elections.",
"Compared to the distribution of possible redistricting plans for the General Assembly, Wisconsin's chosen plan is an outlier in that it yields results that are highly skewed to the Republicans when the statewide proportion of Democratic votes comprises more than 50-52% of the overall vote (with the precise threshold depending on the election considered).",
"Wisconsin's plan acts to preserve the Republican majority by providing extra Republican seats even when the Democratic vote increases into the range when the balance of power would shift for the vast majority of redistricting plans."
],
[
"The Inherent Variability of Election Results",
"For each redistricting plan in the ensemble, the outcome of the election is computed using votes from either the Wisconsin General Assembly elections from 2012 (denoted WSA12), from 2014 (denoted WSA16) and from 2016 (denoted WSA16).",
"In all cases, the actual votes were used at the ward level.",
"However, the existence of unopposed races necessitated interpolating the data using votes from other elections in a number of wards: 27% in WSA12, 46% in WSA14, and 49% in WSA16.",
"The details of this interpolation are given in Section , but the vote counts are based on actual Wisconsin election data in the years given.",
"Figure: Distribution of election outcomes in the ensemble of 19,184 redistricting plans,interpolated for the WSA12,WSA14, and WSA16 election data.",
"With a fixed number and location of votes, the outcome of the election varies based on the choice of redistricting plan chosen.Figure REF shows the frequency of different election outcomes in our ensemble using the votes from WSA12, WSA14, and WSA16.",
"Across the redistricting plans for the 99-seat Wisconsin General Assembly, the expected number of seats won by Republicans was typically concentrated within a range of 3-5 seats.",
"However, a small proportion of redistricting plans are outliers, which extend the range to as much as 10 seats.",
"This wide range of possible outcomes shows that the state's choice of redistricting plan can have an effect on the same order as the typical changes in the popular vote across elections (e.g.",
"a swing of 60 to 64 elected Republicans from 2012 to 2016 in the Wisconsin General Assembly).",
"The fact that the different redistricting plans in the ensemble give such different results speaks to the need for a concept of acceptable redistricting, lest the state's redistricting exercise become as, or more important than, the democratic expression of voters.",
"While the precise definition of a typical result may be debatable, it is clear that some extreme ranges clearly represent anomalous behavior: the results should be labeled as outliers.",
"The view that some points would clearly be labeled as outliers is the starting point for our analysis."
],
[
"Situating the Wisconsin Act 43 Redistricting in the Ensemble",
"We now turn to situating the actual redistricting plan established by Act 43 of the 2011 Wisconsin General Assembly within our ensemble of $19,184$ redistricting plans.",
"This was the redistricting plan actually used in the WSA12, WSA14, and WSA16 elections.",
"The annotation “WI” on each plot in Figure REF indicates the number of seats produced by this redistricting.",
"We note that the use of our modified election data in 2012 and 2014, which interpolates the missing data caused by unopposed races, does not change the balance of power.",
"However, in 2016 the results of the actual election differed from those our interpolated vote data produces.",
"The actual results had three fewer Republican seats than the interpolated results would have had, due to unopposed races in which Democrats ran unopposed in districts that tended to vote Republican.",
"The number of unopposed races was least in 2012 with 27%, growing to 46% in 2014, and then to 49% in 2016.",
"Figure: Distribution of election outcomes in the ensemble of 19,184 redistricting plans,interpolated for the WSA12,WSA14, and WSA16 election data.",
"The outcome using the Wisconsin Act 43 redistrictingis marked with “WI”.",
"For the WSA16 outcome, there were three unopposed Democrats that ran in districts that voted more Republican across the interpolated data; thus we have marked the actual result (64), along with the interpolated result (67).Any reasonable sense of outlier would label the Wisconsin result in 2012 as anomalous.",
"Yet, the actual result produced by the same map is well within the distribution for 2014 and 2016, in which the Republican share of the vote was considerably higher.",
"As we see below, this behavior turns out to reflects an unusual property of Wisconsin's redistricting plan: it gives an anomalously high number of seats to Republicans in elections in which Democrats perform well, but a typical number of seats in elections in which Republicans perform well.",
"To better understand this situation, we consider the outcome that would occur if votes from a number of other elections were used as if they had been cast for the Wisconsin General Assembly.",
"Specifically we compare the effect of using results from U.S. House, U.S. Senate, and Presidential elections from 2012, 2014, and 2016 in addition to our interpolated results for WSA12, WSA14, and WSA16.",
"Figure: A number of election seat result histograms situated on alarger plot by the number Republican seats and the overallfraction of Republican vote.",
"The circles mark the outcome usingthe Wisconsin Act 43 redistricting.",
"Election plotted: Wisconsin StateAssembly 2012 (WSA12), Wisconsin State Assembly 2016 (WSA16),Presidential 2012 (PRE12),Presidential 2016 (PRE16), US House 2012 (USH12),US House 2014 (USH14), US Senate 2012 (USS12),US Senate 2016 (USS16), Wisconsin Secretary of State 2014 (SOS14)Figure REF presents an interesting trend: whenever the election would have typically produced around 55 or fewer Republican seats, the Wisconsin plan behaves very anomalously in the sense that it is far to the right in the histogram.",
"In fact, even though the expected number of Republican seats falls below 50 in one election and the statewide percentage of Republican votes falls well below 50% in three elections, the number of seats elected stays pinned in the high 50s; it is almost constant despite the fact that the Republican vote continues to fall as one moves down the plot.",
"The plot shows that to determine whether the outcome of a given map will be anomalous for a particular election, it is not enough to consider only the total vote count or expected number of seats.",
"The USH12 and WSA12 are similar by those two metrics, yet the outcome in the first is typical but the second is anomalous.",
"This detail shows the importance of the geopolitical structure of the votes in determining the outcome and the pitfalls of coarse, global measures.",
"Based on these insights, we propose a method to evaluate the extent to which a state redistricting plan is an outlier with respect to its ability to protect a party from losing seats: we examine the impact of shifting the proportion of votes up or down within each election examined.",
"We shift the proportions in WSA14 and WSA16 uniformly up and down in all districts so that the statewide Republican vote fraction varies from 45% to 55%.",
"We then plot the histograms and the election results for each shifted vote count in Figure REF .",
"Figure: The number of seats elected when the percentage in eachdistrict is shifted so that the global fraction of the vote for theRepublicans rangesbetween 0.45 and 0.55.",
"Results of WSA14(left) and WSA16(right) areshown.",
"Horizontal lines mark the level of theoriginal vote.",
"Vertical lines mark the number of seats require fora majority and a super-majority.Unlike the plots in Figure REF , the geopolitical structure of all of these shifted votes is identical.",
"Both plots in Figure REF exhibit the trend we already observed in Figure REF .",
"As the percentage of Republican votes decreases, the election results for both WSA14 and WSA16 (shown with red dots in Figure REF ) move from being representative (located in the center of the histograms) to being outliers (located in the extrema of the histograms).",
"The Wisconsin redistricting seems to create a firewall which resists Republicans falling below 50 seats.",
"The effect is striking around the mark of 60 seats where the number of Republican seats remains constant, despite the fraction of votes dropping from 51% to 48%.",
"Figure: Partisan composition of Wisconsin General Assemble as afunction of global Republican vote usingshifted WSA12(top), WSA14(middle), WSA16(bottom) votes.Vertical line indicates the actual votes in unshifted data.",
"Horizontal lines mark seatsneeded for majority and super-majority.",
"Thin line shows seats in Wisconsin redistricting.Figure REF gives a more stylized version of Figure REF .",
"Rather than the entire histogram, we plot the mean, variance, a region containing 90% of all sampled redistricting plans, and the extrema for a larger number of finer shifts that swing the election 10 percentage points in both directions of the observed result.",
"The fine black line gives the number of seats produced by the Wisconsin Act 43 redistricting.",
"Though the local geography of the votes in the three elections is different, each produces a clear deviation from the typical results starting around 50%.",
"This deviation continues as the fraction of Republican votes decreases.",
"All three elections (especially WSA14 and WSA16) show a significant range of Republican votes where the partisan outcome of the election (expressed in the number of Republican seats) does not change even though the percentage of the Republican votes decreases substantially.",
"Figure: We plot the distribution of HH indices, defined inSection , for each set of voting data.",
"In all cases, we find that the Wisconsin Act 43 redistricting plan is an extreme outlier when compared with the ensemble.Finally, we seek to summarize the extent to which Wisconsin's redistricting plan is an outlier (compared to the ensemble of redistricting plans).",
"Toward this end, we defined a statistic as follows: for each of various shifts around an equally split election, we (i) calculate the extent to which a state's redistricting plan produces results different from the results produced by the distribution of possible redistricting plans, and (ii) take the average across the shifts.",
"For any election, we then measure the extent to which a state's plan is outlier with respect to it statistic.",
"(See Section for more details.)",
"As show in Figure REF , we find that the Wisconsin plan is an extreme outlier.",
"In each of the three elections (2012, 2014, and 2016), it is more extreme than 99% of all possible redistricting plans in our ensemble (See the values of $H$ in Table REF in Section ).",
"This statistic is essentially the log-likelihood of seeing the election outcome produced in the Wisconsin plan averaged across the shifted elections.",
"The above statistic is symmetric in that it measures if anomalous results favors one political party over the other, not which party.",
"Using a second set of statistics, we measure if one party is favored over the other by the Wisconsin plan.",
"For each party, we measure (i) what fraction of redistricting plans from our ensemble produce fewer legislative seats than the state's plan, and (ii) take the minimum of these fractions across all the shifts considered above.",
"As before, for any election, we then measure the extent to which a state's plan is an outlier with respect to these statistics measuring party bias.",
"In each of the three elections (2012, 2014, and 2016), the Wisconsin plan is more favorable to the Republicans with respect to this statistic than 99% of all possible redistricting plans in our ensemble.",
"In contrast, the percentage of plans that favor the Democrats less at some shift is much smaller (only 3%, 73%, and 8% in the 2012, 2014, and 2016 elections respectively).",
"See the values of $L_{\\mathrm {rep}}$ and $L_\\mathrm {dem}$ in Table REF in Section ).",
"We will further see that, in the contexts in which the Act 43 map aids Democrats, its effect is to make a large GOP majority somewhat smaller.",
"In Section , we also consider a complementary approach in which we assess shifts of up to $\\pm 7.5\\%$ centered around the outcomes of each election.",
"Wisconsin's plan is again seen to be an extreme outlier.",
"In the next section, we explain why the Wisconsin plan is such an outlier by exploring the structure of the vote in more detail.",
"The graphical understanding of the structure of the vote developed in the next section, and Figure REF in particular, is encapsulated in the Gerrymandering Index defined in [3].",
"In Section , we explore the Gerrymandering Index of the Wisconsin plan over a number of historical Wisconsin elections (WSA12, WSA14, WSA16, Governor 2012, US House 2012 and 2014, US Senate 2012 and 2016, Wisconsin Secretary of State 2014, and Presidential 2012 and 2016).",
"Again situating the result in our ensemble, we fine that at worse 98% of our ensemble had a better Gerrymandering Index.",
"For the majority of the elections considered, none of the redistrictings in our ensemble had a worst Gerrymandering score (See Table REF in Section REF )."
],
[
"Exposing the Geopolitical Structure of Wisconsin",
"To understand the structure which leads to the results of the previous section, we repeat the marginal analysis developed in [3], [2].",
"Fixing a set of votes, for each redistricting we calculate the percentage of Republican votes and then place this vector of 99 numbers in increasing order.",
"To gain insight into the distribution of this 99-dimensional vector when varied over our ensemble, we plot a box-plot for each marginal direction.",
"As standard in box-plots, the box contains 50% of the values, the outer whiskers bracket whichever is smaller – 1.5 times the interquartile range from each quartile or the furthest outlier – and the central line through the box marks the median value.",
"The resulting 99 box-plots arranged on one graph for WSA12, WSA14, and WSA16 give insight into the inherent geopolitical geometry of Wisconsin due to the interaction of the state's geometry with population density and partisan distribution (Figure REF ).",
"We see that typically there are at least 25 districts with less then 40% Democratic vote.",
"Figure: Box-plot summary of districts ordered from mostRepublican to most Democratic, for the voting data from WSA12(top), WSA14 (middle), WSA16 (bottom).",
"We compare our statistical resultswith Wisconsin redistricting in each case.These plots give provide a method to determine the typical partisan makeup of each district.",
"This inherently reflects the geopolitical structure of the state.",
"For a given redistricting plan, if a given district's percentage falls below the horizontal 50% line then the district elects a Republican.",
"If it falls above the line, it elects a Democrat.",
"This plot has proven useful in detecting redistricting plans with packed or fractured districts.",
"See [14], [3].",
"In some sense, they give quantitative definitions to these concepts.",
"If a given district's Democratic vote percentage is at the bottom or below the box plot, the district has fewer Democrats than expected.",
"If the percentage is above or at the top of the box plot, the district has more Democrats than expected.",
"It is clear that the Wisconsin Act 43 redistricting plan produces election results with Democratic votes depleted from the center of the plot and places those votes in the districts which already have a large number of Democratic voters.",
"We also understand why the actual results of the WSA12 elections were not representative while the actual results of the WSA14 and WSA16 elections were representative.",
"It is simply a result of where the 50% line hits the box plot graph in Figure REF .",
"If the 50% line crosses the graph in the region in which the location of the current Wisconsin redistricting plan (the red dots) falls outside of the boxplot (which encodes typical behavior) then the results will be anomalous.",
"This is the case in WSA12 but not in WSA 14 and WSA16.",
"On the other hand, in all three years the district corresponding to the 50th seat, which dictates who is in the majority, is always an outlier, requiring less Republican votes than expected."
],
[
"Inherent Lack of Proportionality",
"Notice that there is a structural tilt to the Republicans – in all of our analyses, a 50% vote fraction for both parties leads to a majority of Republicans.",
"We see that one only needs the Republican vote to be around 47% to 49% to obtain 50 seats with the structure of the WSA12 votes over the majority of redistricting plans.",
"Similarly, WSA14 and WSA16 require between 46% and 47% Republican vote fractions and between 45% and 46% Republican vote fractions, respectively, to obtain 50 seats.",
"This shows clearly that it is not reasonable to expect that 50% of the vote leads to 50% of the seats.",
"This does not explain all of the shift in the Republican favor produced by the Wisconsin Act 43 redistricting plan.",
"Our analysis allows us to separate out the effect of the geopolitical landscape, and to show that the Act 43 map generates extreme partisan asymmetry above and beyond this effect."
],
[
"Exploring Parity",
"To further explore the impact of the structure in Figures REF and REF , we explore two ideas around parity.",
"We begin by shifting the votes in WSA12, WSA14, and WSA16 so that there are an equal number of redistricting plans in the ensemble in which the Republicans and Democrats are in the majority.",
"When shifting the votes in this way, the Wisconsin Act 43 redistricting produces significantly more Republican seats – 56 with WSA12, 57 with WSA14, and 54 with WSA16.",
"In the first two cases, this is a result seen in very few redistricting plans of the ensemble redistricting plans while in WSA16 it has a very low probability (that is to say a relatively small number redistricting plans compared to the just under 20,000 plans considered in the ensemble).",
"Of course, one could perform a similar analysis around another point than the 50% mark.",
"One can see whether if the Wisconsin redistricting would still be an outlier when the votes are centered around a different line by drawing a vertical line in Figure REF at a different value and noting where the thin black line corresponding to the Wisconsin Act 43 redistricting crosses this vertical line.",
"For instance for WSA12, any vertical line up to at least 52% and above 41% results in a result with Wisconsin Act 43 which is well outside the results of 90% of the ensemble – very few redistricting plans which give this result exist in the ensemble.",
"Similar in WSA14 from about 43% to 50.5% the results produced by Wisconsin Act 43 are outliers as they lie outside the region containing 5% to 95% of the ensemble.",
"Lastly for WSA16, from 42% to 50% the Wisconsin redistricting produces results which are outside the region containing 5% to 95% of the ensemble.",
"In all cases the results are skewed to the Republicans precisely in the region where the Democrats threaten to move into the majority.",
"Figure: Histogram of Republican seats won when WSA12 (top), WSA14(middle), WSA16 (botom) shifted so that half of the redistricting plans lead to a majority for either party.A complimentary perspective is instead to ask to what percentage of Republican vote does one have to shift the election to produce a 50/50 split of the seats with a given redistricting.",
"A histogram of the quantity tabulated over the ensemble is shown in Figure REF along with the percentage needed for the Wisconsin Act 43 redistricting plan.",
"Again we see there is a systematic tilt towards the Republicans built into the geopolitical structure of the state.",
"However, in all cases the percentage needed to produce parity in the Wisconsin Act 43 plan is abnormally low.",
"Figure: Votes fraction needed so both parties have an equal chance at majority"
],
[
"Summary Statistics",
"We now develop a number of summary statistics that highlight and make quantitative the graphical analysis developed in the last section."
],
[
"Gerrymandering Index",
"We begin by calculating the Gerrymandering Index developing in [3].",
"It measures the extent to which a particular redistricting has districts whose vote margins for each election deviate from what is expected in Figure REF .",
"For a given election, the square of the Gerrymandering index is the sum of the square deviations of each of the sorted Democratic percentages from the means of the marginals in the ninety-nine box-plots in Figure REF .",
"To contextualize the Gerrymandering Index, we situate a given score within the distribution of scores from our ensemble of redistricting plans.",
"Redistricting plans which have unusually large Gerrymandering index should be view as Gerrymandered.",
"The percentage of the ensemble with Gerrymandering Index worst than the Wisconsin Act 43 redistricting is shown in Table REF for a number of different sets of votes from different years.",
"In all cases, the Wisconsin Act 43 redistricting is seen to have an unusually high level of the Gerrymandering Index.",
"Table: We show the percentage of redistricting plans within the ensemblethat are (i) more gerrymandered and (ii) less representative than WisconsinAct 43 redistricting; we also display the republican vote fraction.",
"According to all vote counts, the current Wisconsin plan is highly gerrymandered.",
"There is a strong correlation between Republican vote fraction and Representativeness.",
"(PRE =President, WAG=Attorney General, GOV= Governor, USS= US Senate,USH=US House)The Gerrymandering Index directly measures how anomalous the partisan composition of a redistricting is.",
"It is possible for a redistricting to be gerrymandered, yet still be representative of the vote count, as we have seen in Figure REF for WSA14 and WSA16.",
"Therefore, we also measure how representative a redistricting is in the context of different vote counts.",
"In [3], we also define a Representative Index which quantifies how representative the result obtained by using a particular redistricting and vote combination is.",
"It is essentially the distance from the mean value in the histograms in Figure REF when one extends the number of seats won by a given party to a continuous variable in a natural way.",
"See [3] for the details.",
"As with the Gerrymandering Index, in Table REF we postion the Representative Index inside the ensemble of redistricting plans by reporting the percentage of the ensemble with a larger index.",
"It is worth noting that Table REF shows the same dependence of representativeness reflected in the histograms in Figures REF and REF and the plots in Figure REF .",
"As the global percentage of Republicans decreases towards 50% the representative score begins to drop.",
"The effect is not strictly monotone as the geopolitical structure of each vote also plays a role.",
"From the preceding section, it is clear that the overall percentage of the vote as well as its geopolitical structure can have a large effect on the perceived representativeness of a redistricting, even when the Gerrymandering Index reports a high level of gerrymandering.",
"To control for this, we consider shifts of a given collection of votes much in the spirit of Figure REF .",
"Rather than use the Representative Index from [3], we consider an alternative formulation which measures the negative log probability of the observed elevation outcome using the probabilities from our ensemble.",
"We then sum these values over a number of shifts of the original election.",
"The logarithmic measure more naturally lives on the same scale across different elections and hence seems more appropriate for this context.",
"This measure, which we will denote by $H$ , is essentially an average log-likelihood across the different shifts We compliment this nonpartisan statistic with one designed to measure deviations in the Republicans's advantage, denoted by $L_\\mathrm {rep}$ , and one to measure deviations in the Democrats's advantage, denoted by $L_\\mathrm {dem}$ .",
"In Table REF , we see based on the $H$ statistics that the Wisconsin Act 43 districts are outliers being much less representative than most of the redistricting plans in the ensemble.",
"The $L$ statistic shows that the Wisconsin redistricting is tilted to favor the Republicans.",
"In one year the $L_\\mathrm {dem}$ raises to the almost 75%; however the box-plots in Figure REF show that the benefit to Democrats comes in elections where the Republicans hold a strong majority in any event, so that the benefit does not affect majority control.",
"To capture the representativeness over a range of election outcomes, we consider shifted election votes over a range of outcomes.",
"We consider a measure which registers both the worst-case deviation from the typical and one which measures the average deviation.",
"Fixing a set of votes to evaluate election outcomes, we define the index $\\ell _{\\mathrm {rep}}$ to be the minimum, overall shifts of the percentage Republican vote between 45% and 55%, of the probability that the number of Republican seats for a given map is greater than one drawn from our distribution.",
"We estimate this probability using the ensemble we generated.",
"We then define $L_{\\mathrm {rep}}$ to be the fraction of maps in our ensemble for which $\\ell _{\\mathrm {rep}}$ is greater than it is for the map in question.",
"We define $\\ell _{\\mathrm {dem}}$ and $L_{\\mathrm {dem}}$ in the same fashion but with Republicans replaced by Democrats.",
"The $L$ statistics described above compare the worst case between two redistricting plans and are inherently one-sided, hence the Democratic and Republican versions.",
"It is also useful to consider a statistic which is an average over a range of shifts.",
"Again fixing a set of votes and a redistricting to be investigated, we define $h$ to be the sum over a set of shifts of the logarithm of the probability that two of the redistricting plans in question produce the same number of Republicans as a random redistricting drawn from our distribution.",
"As with the preceding statistic, we determine a sense of scale for $h$ by defining $H$ to be the probability that the $h$ of a given redistricting is greater than a randomly drawn redistricting from our ensemble.",
"Table: Summary statistics measuring representativeness for Wisconsin Act 43redistricting.",
"These numbers give the percent of redistricting plans the Wisconsin Act 43 redistricting is worse than in terms of average (HH), Republican favoritism (L rep )L_{rep}), and Democratic favoritism (L dem L_{dem}).The results in Table REF show that the results are clearly anomalous.",
"The values of $H$ for the Wisconsin plan are extreme outliers within our ensemble.",
"By detecting unrepresentativeness over a range of shifts, the $H$ statistic assess the level of gerrymandering in the range of total vote fractions where elections typically occur.",
"We now clarify these definitions by restating them in more mathematical notation.",
"We begin by fixing some notation.",
"For any redistricting $\\pi $ and $s \\in \\mathbf {R}$ , we let $\\pi +s$ to be the vote obtained by shifting the partisan vote $s\\%$ to the Republicans.",
"Let $\\mathrm {rep}(\\pi )$ and $\\mathrm {dem}(\\pi )$ denote respectively the total percent Republican or Democratic vote in the election $\\pi $ .",
"Now for any redistricting $\\xi $ , we let $\\mathrm {Rep}(\\xi ,\\pi )$ and $\\mathrm {Dem}(\\xi ,\\pi )$ be the total number of seats won by respectively the Republicans and Democrats with vote $\\pi $ and redistricting $\\xi $ .",
"Now we define $\\ell _{\\mathrm {rep}}(\\xi ,\\pi ) &= \\min _{s \\in [45,55]-r(\\pi )} \\mathbf {P}\\Big ( \\mathrm {Rep}\\big (\\Xi ,\\pi +s\\big )\\ge \\mathrm {Rep}(\\xi ,\\pi +s\\big ) \\Big )\\\\\\ell _{\\mathrm {dem}}(\\xi ,\\pi ) &= \\min _{s \\in [45,55]-r(\\pi )} \\mathbf {P}\\Big ( \\mathrm {Dem}\\big (\\Xi ,\\pi +s\\big )\\ge \\mathrm {Dem}\\big (\\xi ,\\pi +s\\big ) \\Big )$ where $\\Xi $ is a redistricting chosen uniformly from our ensemble and $[45,55]-r(\\pi )$ is compact notation for the set of shifts $[45 -r(\\pi ),55 -r(\\pi )]$ .",
"We then situate these probabilities in the ensemble by defining $L_{\\mathrm {rep}}(\\xi ,\\pi )&= \\mathbf {P}\\big ( \\ell _{\\mathrm {rep}}(\\Xi ,\\pi ) \\le \\ell _{\\mathrm {rep}}(\\xi ,\\pi )\\big ) \\\\L_{\\mathrm {dem}}(\\xi ,\\pi )&= \\mathbf {P}\\big ( \\ell _{\\mathrm {dem}}(\\Xi ,\\pi ) \\le \\ell _{\\mathrm {dem}}(\\xi ,\\pi )\\big )$ where $\\Xi $ is again a randomly chosen redistricting from the ensemble.",
"To define the averaged representative index, we define the average log-likelihood $h(\\xi ,\\pi ) =- \\frac{1}{|I|}\\sum _{s \\in I-r(\\pi )} \\log \\, \\mathbf {P}\\Big ( \\mathrm {Rep}\\big (\\Xi ,\\pi +s\\big ) = \\mathrm {Rep}\\big (\\xi ,\\pi +s\\big ) \\Big )$ where $\\Xi $ is chosen according to our distribution on redistricting plans and $I=\\lbrace 45,45.5,\\dots ,54.5,55\\rbrace $ , $I-x$ is the shifted set defined as before by $I-x=\\lbrace y-x: y \\in I\\rbrace $ , and $|I|$ is the number of points in $I$ .",
"We then situate these in the ensemble by defining $H(\\xi ,\\pi ) &= \\mathbf {P}\\big ( h(\\xi ,\\pi ) \\ge h(\\Xi ,\\pi )\\big )\\,.$ In calculating $H$ , we extrapolate the observed histogram using a Gaussian tail approximation whenever a values is needed outside the range observed in the histogram.",
"We report the summary statistics in Table REF .",
"We find that the Wisconsin Act 43 redistricting is an extreme outlier in terms of how probable it is to observe its value of $H$ .",
"We also find that in the worst case, it can benefit the Republicans by more than 99% of all redistricting plans in our ensemble.",
"Conversely, when shifting between 45%-55% of the vote fraction, the Democrats are significantly impeded in WSA12 and WSA16, and are aided to a much lesser degree in other elections and vote shifts.",
"We remark that when we re-examine Figure REF , the Democrats are only `aided,' once the Republicans have obtained a super majority, as can be seen by the thin continuous line falling below the 90% region."
],
[
"Generating the Ensemble of Redistricting Plans",
"Our method begins by first placing a probability distribution on all the reasonable redistricting plans.",
"The probability distribution will be concentrated on redistricting plans which better satisfy the design certain specified in the laws and legal precedents covering redistricting plans in Wisconsin.",
"We then draw an ensemble of redistricting using the classical Markov Chain Monte Carlo algorithm.",
"The frequency of redistricting plans with different qualities will depend on how well those districts satisfy the design criteria."
],
[
"The Distribution on Redistricting Plans",
"Following the prescription from [3] (see also [1], [14]), we consider distributions with a density proportional to $e^{-\\beta J(\\xi )}$ where $\\xi $ is the function which assigns to each ward a district which we label with the numbers 1 to 99 for convenience.",
"The score function $J$ will be the sum of a number of different score functions $J(\\xi ) = &w_\\mathrm {comp}J_\\mathrm {comp}(\\xi ) +w_\\mathrm {pop} J_\\mathrm {pop}(\\xi )\\\\&+ w_{county}J_\\mathrm {county}(\\xi )+w_\\mathrm {vra}J_{\\mathrm {vra}}(\\xi )$ where $J_\\mathrm {comp}(\\xi )$ measures compactness, $J_\\mathrm {pop}(\\xi )$ measure population deviation from the ideal, $J_\\mathrm {county}(\\xi )$ the number of counties split across counties, and $J_\\mathrm {vra}(\\xi )$ measures the compliance with the Voting Rights Act (VRA); the $w_i$ 's are positive weights.",
"In all cases, low scores will correspond to better compliance with the associated design criteria.",
"We will use the population and compactness score functions from [3].",
"The county and VRA score functions will versions of those from [3] with modifications to adapt to the details of the Wisconsin redistricting context.",
"The Wisconsin State Assembly (WSA) districting required that many counties and towns be split into more than two districts (in contrast to the work in [3]).",
"Hence minor alterations were required to our previous score functions.",
"We determine the weight parameters with a nearly identical process to that described in [3].",
"We have determined $w_{pop}=2200$ , $w_{comp}=0.8$ , $w_{county} = 0.6$ , $w_{VRA}=100$ ."
],
[
"Markov Chain Monte Carlo Sampling",
"We sample redistricting plans according to the algorithm presented [3].",
"For simulated annealing parameters, we take 20,000 accepted steps at $\\beta =0$ , 80,000 accepted steps as $\\beta $ linearly increasing to one, and $20,000$ steps for $\\beta =1$ ( see [3] for more details about the meaning of these parameters).",
"In our reported ensemble we take 19,184 samples.",
"In Section , we show evidence that this is sufficient to correctly sample the distribution on redistricting plans."
],
[
"Redistricting Plans in the Ensemble",
"We ensure that the districts are contiguous, all redistricting plans are more compact than the Wisconsin Act 43 plan.",
"We only kept samples such that the maximum population deviation is below 5%.",
"To account for the VRA, we only retain redistricting plans containing six districts that have at least 40% African Americans and one district that has at least a 40% Hispanic population.",
"All sampled redistricting plans are described at the ward level, and no ward is split.",
"With the above criteria, we have account for all districting criteria present in the Wisconsin constitution, with the exception of splitting townships.",
"We also gather a smaller number of samples (2043) from a distribution that concentrates on redistricting plans that also preserve townships.",
"The township consideration requires an additional term in the score function, and is similar in form to the county splitting energy.",
"We compare the effect of preserving townships below in Section ."
],
[
"Interpolating Election Data",
"We now explain how we interpolate the election data which is missing due to unopposed races.",
"Let $V_{\\mathrm {tot}}(i)$ , $V_{\\mathrm {dem}}(i)$ , and $V_{\\mathrm {rep}}(i)$ be respectively the total vote, the Democratic vote, and the Republican vote in ward $i$ .",
"We split the ward indices into the good $\\mathcal {G}$ and $\\mathcal {B}$ wards.",
"Typically the wards in $\\mathcal {B}$ are the wards where the race is unopposed.",
"We also make use of a second set of voting data $(U_{\\mathrm {tot}}(i), U_{\\mathrm {dem}}(i), U_{\\mathrm {rep}}(i))$ for which no data is missing.",
"We begin by considering the pairs ${(U_{\\mathrm {tot}}(i),V_{\\mathrm {tot}}(i)) : i \\in \\lbrace 1,\\dots 99}\\rbrace $ which we assume to be sorted by the first value.",
"To interpolate $V_{\\mathrm {tot}}(i)$ for some $i \\in \\mathcal {B}$ , we select the pairs $\\big \\lbrace (U_{\\mathrm {tot}}(i_{-2}), V_{\\mathrm {tot}}(i_{-2})), (U_{\\mathrm {tot}}(i_{-1}),V_{\\mathrm {tot}}(i_{-1})), \\\\(U_{\\mathrm {tot}}(i_{1}), V_{\\mathrm {tot}}(i_{1})), (U_{\\mathrm {tot}}(i_{2}), V_{\\mathrm {tot}}(i_{2}))\\big \\rbrace $ where $i_1$ and $i_2$ are the next two elements in the increasing ordered sequence of $U_{\\mathrm {tot}}$ values after $U_{\\mathrm {tot}}(i)$ so that $i_1, i_2 \\in \\mathcal {G}$ .",
"Similarly $U_{\\mathrm {tot}}(i_{-2})$ and $U_{\\mathrm {tot}}(i_{-1})$ are the previous two elements in the ordered sequence again so that both indices are in $\\mathcal {G}$ .",
"If no such point exists, we proceed with the points we have.",
"We then perform a linear least-squares fit to this collection of points.",
"Observe that there are always at least two points in the collection.",
"We then evaluate this linear fit at the point $U_{\\mathrm {tot}}(i)$ to obtain our estimate of $V_{\\mathrm {tot}}(i)$ which we will denote by $\\widehat{V}_{\\mathrm {tot}}(i)$ .",
"Then, in the same fashion, we estimate $\\rho _{\\mathrm {rep}}(i) =U_{\\mathrm {rep}}(i)/U_{\\mathrm {tot}}(i)$ and $\\rho _{\\mathrm {dem}}(i) =U_{\\mathrm {dem}}(i)/U_{\\mathrm {tot}}(i)$ with $r_{ \\mathrm {rep}}(i) =V_{\\mathrm {rep}}(i)/V_{\\mathrm {tot}}(i)$ and $r_{\\mathrm {dem}}(i) =V_{\\mathrm {dem}}(i)/V_{\\mathrm {tot}}(i)$ to obtain $\\widehat{\\rho }_{\\mathrm {rep}}(i)$ and $\\widehat{\\rho }_{\\mathrm {dem}}(i)$ .",
"We then set $\\widehat{V}_{\\mathrm {rep}}(i)$ to be the average of $\\mathrm {floor}(\\widehat{\\rho }_{\\mathrm {rep}}(i)\\widehat{V}_{\\mathrm {tot}}(i))$ and $\\widehat{V}_{\\mathrm {tot}}(i)) - \\widehat{\\rho }_{\\mathrm {dem}}(i)\\widehat{V}_{\\mathrm {tot}}(i))$ and similarly $\\widehat{V}_{\\mathrm {dem}}(i)$ to be the average of $\\widehat{\\rho }_{\\mathrm {dem}}(i)\\widehat{V}_{\\mathrm {tot}}(i)$ and $\\widehat{V}_{\\mathrm {tot}}(i)) - \\widehat{\\rho }_{\\mathrm {rep}}(i)\\widehat{V}_{\\mathrm {tot}}(i)$ .",
"For each choice of reference vote $(U_{\\mathrm {tot}}(i), U_{\\mathrm {dem}}(i), U_{\\mathrm {rep}}(i))$ , we obtain such an estimate.",
"In some cases, we obtain multiple such estimates associated with different reference votes.",
"We then average all of the estimates to obtain a finial estimate which we then round to the nearest integer.",
"To decide which of the many possible combinations of reference votes $(U_{\\mathrm {tot}}(i), U_{\\mathrm {dem}}(i), U_{\\mathrm {rep}}(i))$ produce the best results, we also predict the values for the $i \\in \\mathcal {G}$ and select the collection of reference votes which produces the smallest total squared error.",
"This leads to the choices of reference votes presented in Table REF in the following elections with unopposed elections.",
"Table: Data used to interpolate Wisconsin State Assembly date.",
"SeeTable for abbreviations.In 2012 and 2014, the interpolated votes yield the same number of seats with the Wisconsin Act 43 maps as the original vote counts with the unopposed races not interpolated.",
"In 2016, the number of seats changed from 64 to 67.",
"To understand why this occurred, observe that districts 54, 73 and 74 were uncontested by Republicans and thus went to the Democratic candidate; however the votes in these districts leaned Republican for the President and the Senate, explaining why the interpolated result disagrees with the actual result."
],
[
"Robustness of Results",
"To check the robustness of our results, we (i) take longer runs, and (ii) generate a second ensemble in which we additionally account for town splitting.",
"In considering more runs, we extend our sampling algorithms to examine 84500 redistricting plans.",
"This extended sampling tests whether we have appropriately sampled the space of redistricting plans.",
"The box plots are the most detailed of our results and all other results may be derived from the data contained within them; to show even more detail we present marginal histogram plots.",
"We plot the marginal histograms of the extended samples compared with the reported samples in Figure REF .",
"We find the histogram structures are visually identical for WSA12, WSA14, and WSA16 voting data which provides evidence that we have appropriately sampled the space of redistricting plans.",
"Figure: Testing the effect of using an ensemble with more samples.To consider the effect of keeping townships contiguous, we add a fifth term to the score function that is similar to the county splitting score reported in [3].",
"We consider townships to be all wards with the same name within the shapefile provided by the Legislative Technology Services Bureau [15].",
"For example, using this criteria the city of Wausau is comprised of 41 wards.",
"The new score function is weighted with a value of 0.005, which we have found only marginally affects the overall districting compactness and keeps townships together in a similar way to that of the current plan in Wisconsin.",
"We sample 2043 redistricting plans that preserve townships.",
"We compare the marginal histogram plots when considering township splitting and for the ensemble we have reported above in Figure REF .",
"We find the histogram structures are visually identical for WSA12, WSA14, and WSA16 voting data.",
"Because this new ensemble predicts identical district level results, we have evidence that (1) the ensemble used throughout the paper is robust and (2) reflects all of Wisconsin's stated redistricting criteria according to the state constitution.",
"Figure: Testing the effect of favoring townships not being split bydistrict boundaries on the districting results.Lastly, we considered alternative definitions of the summary statistics $H$ , $L_\\mathrm {rep}$ , and $L_\\mathrm {dem}$ .",
"Instead of shifting the election data so the resulting global elections margins varied between 45% and 55% on might want to take a symmetric interval around the actual global elections margins.",
"Taking a range of $\\pm 7.5\\%$ for the shift, we produced a second set statistics: $\\widetilde{H}$ , $\\widetilde{L}_\\mathrm {rep}$ , and $\\widetilde{L}_\\mathrm {dem}$ .",
"Table: NO_CAPTIONAgain we see that the Wisconsin's plan is still an extreme outlier.",
"The only change is that the $\\widetilde{L}_\\mathrm {rep}$ statistic is much higher.",
"As we discuss bellow, this is because the range now includes a range of percentages where the Wisconsin plan causes the Democrats to perform better than expected in the typical plan.",
"However the results in this range have little effect on the balance of power as the Republicans are already solidly in the majority in those elections.",
"We prefer $H$ , $L_\\mathrm {rep}$ , and $L_\\mathrm {dem}$ to $\\widetilde{H}$ , $\\widetilde{L}_\\mathrm {rep}$ , and $\\widetilde{L}_\\mathrm {dem}$ because the range is limited to 45% to 55%.",
"While the others are more symmetric, they often pull information from the low 60% or high 30% in global vote.",
"These ranges seem less relevant.",
"The effect of this difference is seen in the values of $\\widetilde{L}_\\mathrm {dem}$ which is much higher than $L_\\mathrm {dem}$ because in includes elections with a large global percentage of Republican votes.",
"From Figure REF , we see that the Democratic votes depleted from districts with partisen make up around 50% often is packed into districts with more that 60%.",
"This causes a tilt in favore of the Democrats from what is expected should the global vote get that high.",
"Of course if the vote is above 60% Republican, a few seats shifted to the Democrats will have little effect operationally."
],
[
"Adjustments to Wisconsin General Assembly Redistricting",
"Data provided in [15] is incomplete in terms of the current redistricting plan for Wisconsin.",
"We provide the script that we used to assign districts to unreported wards in our repository.",
"The number of wards affected is relatively small."
],
[
"Supplementary Materials",
"Database with redistricting plans and other data: [email protected]:gjh/WIRedistrictingData.git"
],
[
"Acknowledgements",
"This work uses a code base initiated by Han Sung Kang and Justin Luo as part of a Data+ project under the supervision of the authors at Duke University.",
"We thank the Information Initiative at Duke and the Mathematics Department for their support.",
"We would also like to thank Moon Duchin, Assaf Bar-Natan, and Mira Bernstein for their guidance on districting criteria in Wisconsin and assistance with gathering and extracting data.",
"We are also indebted to Eric Lander for useful discussions and debates around the meaning and presentation of these results as well as Jordan Ellenberg's insightful comments on a previous draft.",
"We are also indebted to Venessa Barnett-Loro for helping to polish this report."
]
] | 1709.01596 |
[
[
"Active Sampling for Large-scale Information Retrieval Evaluation"
],
[
"Abstract Evaluation is crucial in Information Retrieval.",
"The development of models, tools and methods has significantly benefited from the availability of reusable test collections formed through a standardized and thoroughly tested methodology, known as the Cranfield paradigm.",
"Constructing these collections requires obtaining relevance judgments for a pool of documents, retrieved by systems participating in an evaluation task; thus involves immense human labor.",
"To alleviate this effort different methods for constructing collections have been proposed in the literature, falling under two broad categories: (a) sampling, and (b) active selection of documents.",
"The former devises a smart sampling strategy by choosing only a subset of documents to be assessed and inferring evaluation measure on the basis of the obtained sample; the sampling distribution is being fixed at the beginning of the process.",
"The latter recognizes that systems contributing documents to be judged vary in quality, and actively selects documents from good systems.",
"The quality of systems is measured every time a new document is being judged.",
"In this paper we seek to solve the problem of large-scale retrieval evaluation combining the two approaches.",
"We devise an active sampling method that avoids the bias of the active selection methods towards good systems, and at the same time reduces the variance of the current sampling approaches by placing a distribution over systems, which varies as judgments become available.",
"We validate the proposed method using TREC data and demonstrate the advantages of this new method compared to past approaches."
],
[
"Introduction",
"Evaluation is crucial in Information Retrieval (IR).",
"The development of models, tools and methods has significantly benefited from the availability of reusable test collections formed through a standardized and thoroughly tested methodology, known as the Cranfield paradigm [8].",
"Under the Cranfield paradigm the evaluation of retrieval systems typically involves assembling a document collection, creating a set of information needs (topics), and identifying a set of documents relevant to the topics.",
"One of the simplifying assumptions made by the Cranfield paradigm is that the relevance judgments are complete, i.e.",
"for each topic all relevant documents in the collection have been identified.",
"When the document collection is large identifying all relevant documents is difficult due to the immense human labor required.",
"In order to avoid judging the entire document collection depth-k pooling [24] is being used: a set of retrieval systems (also called runs) ranks the document collection against each topic, and only the union of the top-$k$ retrieved documents is being assessed by human assessors.",
"Documents outside the depth-k pool are considered irrelevant.",
"Pooling aims at being fair to all runs and hopes for a diverse set of submitted runs that can provide a good coverage of all relevant documents.",
"Nevertheless, the underestimation of recall [29] and the pooling bias generated when re-using these pooled collections to evaluate novel systems that retrieve relevant but unjudged documents [29], [5], [26], [17] are well-known problems.",
"The literature suggests a number of approaches to cope with missing judgments (an overview can be found in [22] and [12]): (1) Defining IR measures that are robust to missing judgments, like bpref [6].",
"The developed measures however may not precisely capture the notion of retrieval effectiveness one requires, while some have been shown to remain biased [27].",
"(2) Running a meta-experiment where runs are “left out” from contributing to the pool and measuring the bias experienced by these left-out runs compared to the original pool, which is then used to correct measurements over new retrieval systems [26], [17], [14], [15].",
"(3) Leaving the design of the evaluation measure unrestricted, but instead introducing a document selection methodology that carefully chooses which documents to be judged.",
"Methods proposed under this approach belong to two categories: (a) sample-based methods [3], [27], [19], [28], [23], and (b) active selection methods [9], [1], [18], [16].",
"Sample-based methods devise a sampling strategy that randomly selects a subset of documents to be assessed; evaluation measures are then inferred on the basis of the obtained sample.",
"Different methods employ different sampling distributions.",
"[3] and [27] use a uniform distribution over the ranked document collection, while [19] and [28] recognize that relevant documents typically reside at the top of the ranked lists returned by participating runs and use stratified sampling to draw larger sample from the top ranks.",
"[23] also use a weighted-importance sampling method on documents with the sampling distribution optimized for a comparative evaluation between runs.",
"In all aforementioned work, an experiment that dictates the probability distribution under which documents are being sampled is being designed in such a way that evaluation measures can be defined as the expected outcome of this experiment.",
"Evaluation measures can then be estimated by the judged documents sampled.",
"In all cases the sampling distribution is being defined at the beginning of the sampling process and remains fixed throughout the experiment.",
"Sample-based methods have the following desirable properties: (1) on average, estimates have no systematic error, (2) past data can be re-used by new, novel runs without introducing bias, and (3) sampling distributions can be designed to optimize the number of judgments needed to confidently and accurately estimate a measure.",
"On the other hand, active-selection methods recognize that systems contributing documents to the pool vary in quality.",
"Based on this observation they bias the selection of documents towards those retrieved by good retrieval systems.",
"The selection process is deterministic and depends on how accurately the methods can estimate the quality of each retrieval system.",
"Judging is performed in multiple rounds: at each round the best system is identified, and the next unjudged document in the ranked list of this system is selected to be judged.",
"The quality of systems is calculated at the end of each round, as soon as a new judgment becomes available.",
"Active-selection methods include Move-to-Front [9], Fixed-Budget Pooling [16], and Multi-Armed Bandits [18].",
"[18] considers the problem as an exploration-exploitation dilemma, balancing between selecting documents from the best-quality run, and exploring the possibility that the quality of some runs might be underestimated at different rounds of the experiment.",
"The advantage of active-selection methods compared to sample-based methods is that they are designed to identify as many relevant document as possible, by selecting documents with the highest relevance probability.",
"The disadvantage is that the judging process is not fair to all runs, with the selection of documents being biased towards good-performing runs.",
"In this paper, we follow a sample-based approach for an efficient large-scale evaluation.",
"Different from past sample-based approaches we account for the fact that some systems are of higher quality than others, and we design our sampling distribution to over-sample documents from these systems.",
"At the same time, given that our approach is a sample-based approach the estimated evaluation measures are, by construction, unbiased on average, and judgments can be used to evaluate new, novel systems without introducing any systematic error.",
"The method we propose therefore is an active sampling method with the probability distribution over documents changing at every round of judgments through the re-estimation of the quality of the participating runs.",
"Accordingly, our solution consists of a sampling step and an estimation step.",
"In the sampling step, we construct a distribution over runs and a distribution over documents in a ranked list and calculate a joint distribution over documents to sample from.",
"In the estimation step, we use the Horvitz-Thompson estimator to correct for the bias in the sampling distribution and estimate evaluation measure values for all the runs.",
"The estimated measures then dictate the new sampling distribution over systems, and hence a new joint distribution over the ranked collection of documents.",
"Therefore, the contribution of this paper is a new sampling methodology for large-scale retrieval evaluation that combines the advantages of the sample-based and the active-selection approaches.",
"We demonstrate that the proposed method outperforms state-of-the-art methods in terms of effectiveness, efficiency, and reusability."
],
[
"Active sampling",
"In this section we introduce our new sampling method.",
"Table: Notation used throughout this paper"
],
[
"Active sampling algorithm",
"The key idea underlying our sampling strategy is to place a probability distribution over runs and a probability distribution over documents in the ranked lists of the runs, and iteratively sample documents from the joint distribution.",
"At each round, we sample a set of documents from the joint probability distribution (batch sampling) and request relevance judgments by human assessors.",
"The judged documents are then used to update the probability distribution over runs.",
"The process is repeated until we reach a fixed budget of human assessments (Figure REF ).",
"Figure: Active sampling and retrieval performance estimationActive sampling [1] Prior distribution over runs $\\big \\lbrace p_1(k) \\big \\rbrace _{k=1}^{K}$ , prior distributions over document ranks $\\big \\lbrace p_{1}(k, r(i)) \\big \\rbrace _{i=1}^{N}$ , document collection $\\mathcal {C}$ , batch size $N_b$ Sampled documents $S$ , associated with relevance judgment and selection probability: $ \\big \\lbrace (d_{t,j}, y_{t,j}, p_t(j)) \\big \\rbrace _{j=1}^{N_bT}$ $t = 1, 2, ..., T $ Calculate the joint document sampling distribution $p_t(i) = \\sum _{k=1}^{K}{ p_{t}(k) p_{t}(k, r(i)) , i=1, ..., N}$ Sample $N_t$ documents with replacement (so that it contains $N_b$ unique documents) from $p_t(i)$ Let the sampled document be $d_{t, j}$ ; judge relevance of the sampled documents $y_{t, j}, j=1,...,N_t$ Augment data $S_{t+1} = S_{t} \\bigcup \\big \\lbrace (d_{t,j}, y_{t,j}) \\big \\rbrace _{j=1}^{N_{t}} $ Update distribution over runs $p_{t+1}(k), k=1,...,K $ The process is illustrated in Algorithm REF , while Table 1 shows the notation used throughout the paper.",
"Initially, we provide a prior distribution over runs $\\big \\lbrace p_1(k)\\big \\rbrace _{k=1}^{K}$ , a prior distribution over the ranks of the documents $\\big \\lbrace p_1(k, r(i)) \\big \\rbrace _{i=1}^{N}$ for each run $k$ , and the document collection $\\mathcal {C}$ .",
"Given that we have no prior knowledge of the system quality it is reasonable to use a uniform probability distribution over runs, i.e.",
"$p_1 (1) = p_1 (2) = ... = p_1 (K)$ .",
"At each round $t$ , we calculate the selection probabilities of the documents (that is the probability that a document is selected at each sampling time) $p_t(i)$ for each document $i$ , and then sample a document on the basis of this distribution.",
"We use sampling with replacement with varying probabilities to sample documents, which is closely related to how we calculate the unbiased estimators and it is describe in Section 3.",
"The sampled documents $d_{t, j}$ are then judged by human assessors, with the relevance of these documents denoted as $y_{t, j}$ , and the new data are added to $S_{t}$ which is used to update the ${(t+1)}^{th}$ posterior distribution over runs."
],
[
"Distribution over runs",
"The distribution over runs determines the probability of sampling documents from each run.",
"Similar to active-selection methods, we make the assumption that good systems retrieve more relevant documents at the top of their rankings compared to bad systems.",
"Based on this assumption we wish to over-sample from rankings of good systems.",
"Any distribution that places a higher probability to better performing systems could be used here.",
"In this work we consider the estimated performance of the retrieval systems on the basis of the relevance judgments accumulated at each round of assessments as system weights and normalize these weights to obtain a probability distribution over runs.",
"Different evaluation measures can be used to estimate the performance of each run after every sampling round.",
"Here we define a probability distribution proportional to estimated average precision $\\widehat{AP}$ introduced in Section REF .",
"$p_t(k) = \\frac{\\widehat{AP_t}(k)}{\\sum _{k=1}^{K}{\\widehat{AP_t}(k)}}, k=1,...,K; t=1,...,T$ Figure REF demonstrates the accuracy of the estimated (normalized) average precision at the end of four sampling rounds compared to the (normalized) average precision when the entire document collection (or to be more accurate the depth-100 pool for topic 251 in TREC 5) is used.",
"At every round the estimates (denoted with circular markers of different sizes for different rounds) better approximates the target values (denoted with a line).",
"The details of the measure approximations are provided at Section .",
"Figure: Probability distribution over runs on topic 251 in TREC 5.",
"The black curve is the probability induced by the actual average precisions based on depth-100 pooling, while circular markers of different sizes denote the approximate probabilities for different runs.",
"Runs have been sorted according to their actual average precision values."
],
[
"Distribution over document ranks",
"The distribution over document ranks for a system $k$ determines the probability of sampling a document at a certain rank of the ordered list returned by run $k$ .",
"The underlying assumption that defines this probability distribution is that runs satisfy the Probability Ranking Principle (PRP) [21] which dictates that the probability of relevance monotonically decreases with the rank of the document.",
"Hence, if we let $p$ denote the probability of sampling a document at rank $r$ , then it is natural to assume $p$ is a function of $r$ and $p(r)$ monotonically decreases with $r$ .",
"Once again, any distribution that agrees with PRP can be used; researcher have used a number of such distributions (e.g.",
"see [2], [19], [10]).",
"In this work we consider an AP-prior distribution proposed by [2] and [19] which aims to define the probability at each rank on the basis of the contribution of this rank in the calculation of average precision.",
"The intuition is that when rewriting $AP=\\frac{1}{N}\\sum _{1\\leqslant j \\leqslant i \\leqslant N} \\frac{1}{i} y_i y_j$ , the implicit weight associated with rank $r$ can be obtained by summing weights associated with all pairs involving $r$ , i.e.",
"$\\frac{1}{N}(1+\\frac{1}{r}+\\frac{1}{r+1}+...+\\frac{1}{N})$ .",
"Then the AP-prior distribution is defined as follows: $&w(r)=\\frac{1}{N}(1+\\frac{1}{r}+\\frac{1}{r+1}+...+\\frac{1}{N}) \\approx \\frac{1}{N}\\log {\\frac{N}{r}} \\\\&p(r)=\\frac{w(r)}{\\sum _{r=1}^{N}{w(r)}}$ where $r$ is the rank of a document and $N$ the total number of documents in the collection.",
"Similar to [2], [19] and all other sample-based methods, this distribution is defined at the beginning of the sampling process and remains fixed throughout the experiment.",
"In this section, we discuss the estimation of evaluation measures on the basis of the sampling procedure described in Algorithm REF .",
"We first calculate the inclusion probabilities of each document in the collection, and then demonstrate how these probabilities can be used by a Horvitz-Thompson estimator to produce unbiased estimators of the population mean, and subsequently of some popular evaluation measures.",
"The Horvitz-Thompson estimator, together with the calculated inclusion probabilities can be used to calculate the majority of the evaluation measures used in IR; in this paper we focus on three of them, Precision, Recall, and Average Precision.",
"Other measures can be derived in similar ways (e.g.",
"see Table 1 in [23])."
],
[
"Sampling with replacement with varying probabilities",
"Sampling procedure.",
"At each round of our iterative sampling process described in Algorithm REF , $n$ documents are sampled from a collection of size $N$ .",
"At each round, the unconditional probability of sampling a document $d_i$ (selection probability) is $p_t(i)$ , as defined in Step 2 of Algorithm REF , with $\\sum _{i=1}^{N}{p_t(i)} = 1 & \\textrm {and} & p_t(i) \\ge 0 \\\\& \\textrm {for} & i = 1,2,...,N\\\\& & t = 1, 2, ..., T.$ Let $i$ denote the index of the $n$ documents composing the sample set.",
"The probability of a document $d_i$ being sampled (first-order inclusion probability) at the end of the sampling process is given by $\\pi _i = 1 - \\prod _{t=1}^{T}\\prod _{z=1}^{N_t}{(1- p_t(i))}$ which accounts for varying probabilities across different rounds, while the probability of any two different document $d_i$ and $d_j$ being sampled (second-order inclusion probability) is given by $\\pi _{ij} = \\pi _i + \\pi _j - [ 1 - \\prod _{t=1}^{T}\\prod _{z=1}^{N_t}{ (1- p_t(i) - p_t(j)) } ]$ For the details of the derivation of the inclusion probabilities the reader can refer to [25].",
"Using these inclusion probabilities together with the Horvitz-Thompson estimator allows us to construct unbiased estimators for different evaluation measures in IR.",
"Horvitz-Thompson estimator of population total.",
"[11] propose a general sampling theory for constructing unbiased estimators of population totals.",
"With any sampling design, with or without replacement, the unbiased Horvitz-Thompson estimator of the population total is $\\widehat{\\tau } = \\sum _{i \\in S^{\\prime }}{\\frac{y_i}{\\pi _i}}$ where $S^{\\prime }$ is the subset of $S$ , only containing unique documents.",
"An unbiased estimator of the variance of the population total estimator is given by: $\\widehat{var}(\\mu ) = \\sum _{i \\in S^{\\prime }}{(\\frac{1}{{\\pi _i}^2} - \\frac{1}{\\pi _i}){y_i}^2 } + 2 \\sum _{i > j \\in S^{\\prime }}{(\\frac{1}{{\\pi _i \\pi _j}} - \\frac{1}{\\pi _{ij}})y_i y_j }$ For the details of these derivations the reader can refer to [25]."
],
[
"Evaluation metrics",
"In this work we consider three of the most popular evaluation measures in IR, precision at a certain cut-off, PC(r), average precision, AP, and R-precision, RP.",
"We first clarify the exact expressions of the evaluation metrics with regard to the population, then introduce the estimators of these evaluation metrics on the sample set.",
"Let $C = \\lbrace d_i\\rbrace _{i=1}^{N}$ denote a population of documents and let $y_i$ be an indicator variable of $d_i$ , with $y_i = 1$ if the document $d_i$ is relevant, and $y_i = 0$ otherwise.",
"The population total is the summation of all $y_i$ , i.e.",
"the total number of relevant documents in the collection, while the population mean is the population total divided by the population size.",
"If the population of documents considered is the documents ranked in the top-$r$ for some run $k$ then the population mean is the precision at cut-off $r$ .",
"Based on the definition, precision at cutoff r, average precision, and precision at rank R are defined as: $PC(r) = \\frac{ \\sum _{ d_i \\in C, r(i) \\le r }{y_i} }{r} \\\\AP = \\frac{ \\sum _{d_i \\in C }{PC(r(i))y_i} }{R} \\\\RP = \\frac{ \\sum _{ d_i \\in C, r(i) \\le R }{y_i} }{R} \\\\$ Suppose that we have sampled $n$ documents $S=\\lbrace d_i\\rbrace _{i=1}^{n}$ , with associated relevance labels $\\lbrace y_i\\rbrace _{i=1}^{n}$ .",
"We wish to estimate the total number of relevant documents in the collection, R, PC(r), AP and RP.",
"Note that AP and RP, as many other evaluation measures that are normalized are ratios.",
"For these measures, similar to previous work [19] we can estimate the numerator and denominator separately, and while this ratio estimator is not guaranteed to be unbiased, the bias tends to be small and decrease with an increasing sample size [25], [20].",
"The unbiased estimators for the four aforementioned measures based on Horvitz-Thompson can be calculated by: $&\\widehat{R} = \\sum _{d_i \\in S^{\\prime }}{\\frac{y_i}{\\pi _i}} \\\\&\\widehat{PC}(r) =\\frac{\\sum _{d_i \\in S^{\\prime }, r(i) \\le r }{\\frac{y_i}{\\pi _i}} }{r} \\\\&\\widehat{AP} = \\frac{\\sum _{d_i \\in S^{\\prime }}{\\frac{\\widehat{PC}(r(i))y_i}{\\pi _i}}}{\\widehat{R}} \\\\&\\widehat{RP} = \\frac{\\sum _{d_i \\in S^{\\prime }, r(i) \\le \\widehat{R} }{\\frac{y_i}{\\pi _i}}}{\\widehat{R}} \\\\$"
],
[
"Experiment setup",
"In this section we introduce our research questions, the statistics we use to evaluate the performance of the proposed estimators, and the data sets and baselines used in our experiments The implementation of the algorithm and the experiments run can be found at https://github.com/dli1/activesampling.",
"The batch size $N_b$ for all the experiments has been set to 3."
],
[
"Research questions",
"In the remainder of the paper we aim to answer the following research questions: RQ1 How does active sampling perform compared to other sample-based and active-selection methods regarding bias and variance in the calculated effectiveness measures?",
"RQ2 How fast active sampling estimators approximate the actual evaluation measures compared to other sample-based and active-selection methods?",
"RQ3 Is the test collection generated by active sampling reusable for new runs that do not contribute in the construction of the collection?",
"The aforementioned questions allow us to have a thorough examination of the effectiveness as well as the robustness of the proposed method."
],
[
"Statistics",
"To answer the research questions put forward in the previous section, we need to quantify the performance of different document selection methods.",
"Our first goal is to measure how close the estimation of an evaluation measure is to its actual value when the full judgment set is being used.",
"Assume that a document selection algorithm chooses a set of documents $S$ to calculate an evaluation measure.",
"Let's denote the estimated measure with $f(k|S)$ , for some run $k$ .",
"Let's also assume that the actual value of that evaluation measure, when the full judgment set is used, is $h(k)$ .",
"The root mean squared error (rms) of the estimator over a sample set measures how close on average the estimated and the actual values are.",
"We follow the definition in [19] : $rms=\\mathbb {E}_{S}\\sqrt{\\mathbb {E}_{k}{\\big ( f(k|S) - h(k) \\big )}^2} .$ To further decompose the estimation errors made by different methods we also calculate the bias, and the variance decomposed from the mean square error (mse) between the estimator and the corresponding real value.",
"Bias expresses the extent to which the average estimator over all sample sets differs from the actual value of a measure, while variance expresses the extent to which the estimator is sensitive to the particular choice of a sample set (see [4]).",
"The mse, $\\mathbb {E}_{S}\\mathbb {E}_{k}{\\big ( f(k|S) - h(k) \\big )}^2$ , can be rewritten as $\\mathbb {E}_{k}\\mathbb {E}_{S}{\\big ( f(k|S) - h(k) \\big )}^2$ , which can further be rewritten as ${ { {\\Big ( \\mathbb {E}_{S}{ \\big ( f(k|S) - h(k) \\big )}\\Big ) }^2 } } + { { \\mathbb {VAR}_{S} f(k|S) } }$ .",
"The first term denotes the bias and second the variance of the estimator.",
"Taking all runs into account, we have $&bias = \\mathbb {E}_{k} \\mathbb {E}_{S}{ \\big ( f(k|S) - h(k) \\big )}, \\\\&variance = \\mathbb {E}_{k} \\mathbb {VAR}_{S} f(k|S).",
"\\\\$ A second measurement we are interested in is how far the inferred ranking of systems when estimating an evaluation measure is to the actual ranking of systems when the entire judged collection is being used.",
"Following previous work [1], [3], [19], [27], [28] we also report the Kendall's $\\tau $ between estimated and actual rankings.",
"Even though the Kendall's $\\tau $ is an important measure when it comes to comparative evaluation, rms error remains our focus, since test collections have found use not only in the evaluation of retrieval systems but also in learning retrieval functions [13].",
"In the latter case, for some algorithms, the accuracy of the estimated values is more important than just the correct ordering of systems."
],
[
"Test collections",
"We conduct our experiments on TREC 5–8 AdHoc and TREC 9–11 Web tracks.",
"The details of the data sets can be found in Table REF .",
"In our experiments we did not exclude any participating run, and we considered the relevance judgments released by NIST (qrels) as the complete set of judgment over which the actual values of measures are being computed.",
"Table: Test collections"
],
[
"Baselines",
"We use two active-selection and one sample-based methods as baselines: Move-to-Front (MTF) [9].",
"MTF is a deterministic, iterative selection method.",
"At the first round, all runs are given equal priorities.",
"At each round, the method selects the run with the highest priority and obtains the judgment of the first unjudged document in the ranked list of the given run.",
"If the document is relevant the method selects the next unjudged document until a non-relevant document is judged.",
"If that happens the priority of the current run is being reduced and the run with the highest priority is selected next.",
"Multi-armed Bandits (MAB) [18].",
"Similar to MTF, MAB aims to find as many relevant documents as possible.",
"MAB casts document selection as a multi-armed bandit problem, and different to MTF it randomly decides whether to select documents from the best run on the current stage, or sample a document across the entire collection.",
"For the MAB baseline we used the best method MM-NS with its default setting reported in [18] http://tec.citius.usc.es/ir/code/poolingbandits.html.",
"Stratified sampling [19].",
"Stratified sampling is a stochastic method based on importance sampling.",
"The probability distribution over documents used is the AP-prior distribution, which remains unchanged throughout the sampling process.",
"Similar to our approach, the Horvitz-Thompson estimator is used to estimate the evaluation metrics.",
"The stratified sampling approach proposed by [19] has been used in the construction of the TREC Million Query track collection [7], it outperforms methods using uniform random sampling [3], [27] and demonstrate similar performance to [28]."
],
[
"Bias and Variance",
"This first experiment is designed to answer RQ1 and is conducted on TREC 5.",
"We reduce the retrieved document lists of all runs to the top-100 ranks (so that all documents in the ranked lists are judged) and consider this the ground truth rankings, based on which the actual values of MAP, RP and P@30 are calculated.",
"The judgment effort is set to 10% of the depth-100 pool for each query, and different methods are used to obtain the corresponding subset and calculate the estimated MAP, RP and P@30 for each run.",
"For any stochastic method (i.e.",
"the sampling methods and MAB) the experiment is repeated 30 times.",
"Based on the estimated and actual values we calculate $mse$ , and its decomposition to $bias$ and $variance$ for each estimator.",
"Figure REF shows a number of scatter plots for MTF, MAB, Stratif (stratified sampling), and our method denoted as Active (active sampling).",
"Each point in the plots corresponds to a given run.",
"To declutter the figure, the shown points for the sample-based methods are computed over a single sample.",
"An unbiased estimator should lead to points that lie on the x=y line.",
"As it can be observed the active sampling estimated values are the ones that are closer to the diagonal.",
"As expected, and by construction, precision is unbiased, while the bias introduced in the ratio estimators of AP and RP is smaller that all active-selection methods, and comparable to the stratified sampling method.",
"A decomposition of the mse into bias and variance can be found in Figure REF .",
"As expected the variance of active-selection method is zero (or close to zero) since MTF is a deterministic method, while the randomness of MAB is only in the decision between exploration and exploitation.",
"Active sampling has a much lower variance than stratified sampling, which demonstrates one of the main contribution of the our sampling method: biasing the sampling distribution towards good performing runs improves the estimation of the evaluation measures.",
"The bias of the sample-based methods, as expected, is near-zero, while it is smaller than zero for the active-selection methods, since they do not correct for their preference to select documents from good performing runs.",
"For example, the bias on P@30 of active-selection methods are much smaller than zero, because the greedy strategies only count the number of relevant documents and thus underestimate P@30; while the sampling methods can avoid the problem by using unbiased estimators.",
"This demonstrates the second main contribution of our approach: using sampling avoids any systematic error in the calculation of measures.",
"Therefore, the proposed sampling method indeed combines the advantages of both sample-based and active-selection methods that have been proposed in the literature."
],
[
"Effectiveness",
"This second experiment is designed to answer RQ2 and is conducted on TREC 5-11.",
"In this experiment we vary the judgment effort from 1% to 20% of the depth-100 pool.",
"At each sampling percentage, when sample-based methods are used, we first calculate the $rms$ error and Kendall's $\\tau $ values for a given sample and then average these values over 30 sample sets.",
"Figure REF shows the average $rms$ and $\\tau $ value at different sample sizes.",
"For all TREC tracks active sampling demonstrates a lower rms error than stratified sampling, MTF, and MAB for sampling rates greater than 3-5%.",
"At lower sampling rates active-selection methods show an advantage compared to sample-based methods that suffer from high variance.",
"Regarding Kendall's $\\tau $ active sampling outperforms all methods for TREC 5–8, for sampling rates greater than 5%, while for TREC 10 and 11 it picks up at sampling rates greater than 10%.",
"TREC 10 and 11 are the two collection with the smallest number of relevant documents per query, hence finding these document using active-selection methods leads to a better ordering of systems when the percentage of judged documents is very small.",
"For those small percentages the sample-based methods demonstrate high variance, and it really depends on how lucky one is when drawing the sample of documents.",
"The variance of $rms$ error and Kendall's $\\tau $ across the 30 different samples drawn in this experiment for the estimation of MAP on TREC 11 can be seen in Figure REF .",
"Figure: Variance of rms error (solid lines) and Kendall's τ\\tau values (dashed lines) when estimating MAP, over 30 sample for different sample sizes TREC 11.Overall, when comparing active sampling with MTF and MAB, we find that our method outperforms them regarding $rms$ .",
"This indicates once again that the calculated inclusion probabilities and the Horvitz-Thompson estimator allows active sampling to produce an unbiased estimation of the actual value of the evaluation measures.",
"When comparing active sampling with stratified sampling, both of which use the Horvitz-Thompson estimator, we can find that our method outperforms stratified sampling regarding Kendall's $\\tau $ .",
"This indicates that the dynamic strategy we employ is beneficial compared to a static sampling distribution.",
"Therefore, active sampling indeed combines the advantage of both methods."
],
[
"Reusability",
"Constructing a test collection is a laborious task, hence it is very important that the proposed document selection methods construct test collections that can be used to evaluate new, novel algorithms without introducing any systematic errors.",
"This experiment is designed to answer RQ3 and is conducted on TREC 5-11.",
"In this experiment we split the runs into contributing runs and left-out runs.",
"Using the contributing runs we construct a test collection for each different document selection method.",
"We then calculate the estimated measures for all runs including those that were left out from the collection construction experiment.",
"In our experiment, we use a one-group-out split of the runs.",
"Runs that contributed in the sampling procedure come from different participating groups.",
"Groups often submit different versions of the same retrieval algorithm, hence, typically, all the runs submitted by the same participating group differ very little in the ranking of the documents.",
"To ensure that left-out runs are as novel as possible we leave out all runs for a given group.",
"Regarding the calculation of rms error and Kendall's $\\tau $ we compute rms error and Kendall's $\\tau $ considering both participating and left-out runs.",
"Figure REF shows the average $rms$ error and Kendall's $\\tau $ values at different sample sizes using the latter afore-described option to isolate the effect of the different document selection methods on new, novel systems.",
"In general, the trends observed in Figure REF can also be observed in Figure REF , with active sampling outperforming all other methods regarding rms error and Kendall's $\\tau $ for sampling rates greater than 5%.",
"For sampling rates lower than 5% in collections with very few relevant documents per topic (such as TREC 10 and 11) the active-selection methods perform better than the sample-based methods, however we can also conclude that at these low sampling rates none of the methods lead to reliably reusable collections."
],
[
"Conclusion",
"In this paper we consider the problem of large-scale retrieval evaluation.",
"We tackle the problem of devising a sample-based approach - active sampling.",
"Our method consists of a sampling step and an unbiased estimation step.",
"In the sampling step, we construct two distributions, one over retrieval systems that is updated at every round of relevance judgments giving larger probabilities to better quality runs, and one over document ranks that is defined at the beginning of the sampling process and remains static throughout the experiment.",
"Document samples are drawn from the joint probability distribution, and inclusion probabilities are computed at the end of the entire sampling process accounting for varying probabilities across sampling rounds.",
"In the estimation step, we use the well-known Horvitz-Thompson estimator to estimate evaluation metrics for all system runs.",
"The proposed method is designed to combine the advantages of two different families of methods that have appeared in the literature: sample-based and active-selection approaches.",
"Similar to the former, our method leads to unbiased, by construction, estimators of evaluation measures, and can safely be used to evaluate new, novel runs that have not contributed to the generation of the test collection.",
"Similar to the latter, the attention of our method is put on good quality runs with the hope of identifying more relevant documents and reduce the variability naturally introduced in the estimation of a measure due to sampling.",
"To examine the performance of the proposed method, we tested against state-of-the-art sample-based and active-selection methods over seven TREC AdHoc and Web collections, TREC 5–11.",
"Compared to sample-based approaches, such as stratified sampling, out method indeed demonstrated lower variance, while compared against active-selection approaches, such as Move-to-Front, and Multi-Armed Bandits, our method, as expected, has lower, near-zero bias.",
"For sampling rates as low as 5% of the entire depth-100 pool, the proposed method outperforms all other methods regarding effectiveness and efficiency and leads to reusable test collections.",
"This research was supported by the Google Faculty Research Award program.",
"All content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and/or sponsors."
]
] | 1709.01709 |
[
[
"On the Oberlin affine curvature condition"
],
[
"Abstract In this paper we generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\\mathbb R^n$ , $1 \\leq d \\leq n-1$.",
"We show that a canonical affine invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of D. Oberlin with an exponent which is best possible.",
"The proof combines aspects of Geometric Invariant Theory, convex geometry, and frame theory.",
"A significant new element of the proof is a generalization to higher dimensions of an earlier result of the author concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line."
],
[
"Introduction",
"Many of the deep questions in harmonic analysis, including Fourier restriction, decoupling theory, or $L^p$ -improving estimates for geometric averages, deal with certain operators associated to submanifolds of Euclidean space.",
"In most cases, the “nicest possible” submanifolds are, informally, as far as possible from lying in any affine hyperplane.",
"Many of these problems also exhibit natural affine invariance, meaning that when the underlying Euclidean space is transformed by a measure-preserving affine linear mapping, the the relevant quantities (norms, etc.)",
"are unchanged.",
"This simple observation leads naturally to the question of how in general to properly quantify this sort of well-curvedness in a way that respects affine invariance.",
"Of the many approaches to this question, one particularly successful strategy has been the use of the so-called affine arclength measure for curves and the analogous notion of affine hypersurface measures (sometimes called the equiaffine measure).",
"In the former case, affine arclength is defined on a curve parametrized by $\\gamma : I \\rightarrow {\\mathbb {R}}^n$ by $ \\int f d \\mu _{\\!_{\\mathcal {A}}}:= \\int _{I} f(\\gamma (t)) \\left| \\det (\\gamma ^{\\prime }(t),\\ldots ,\\gamma ^{(n)}(t))\\right|^{\\frac{2}{n(n+1)}} dt, $ while equiaffine measure on the graph $(x,\\varphi (x))$ over $U \\subset {\\mathbb {R}}^{n-1}$ is given by $ \\int f d \\mu _{\\!_{\\mathcal {A}}}:= \\int _U f(x,\\varphi (x)) |\\det \\nabla ^2 \\varphi (x)|^{\\frac{1}{n+1}} dx, $ where $\\nabla ^2 \\varphi $ is the Hessian matrix of second derivatives of $\\varphi $ .",
"Though these measures were well-known outside harmonic analysis for quite some time (see, for example, guggenheimer1963,lutwak1991), their first appearances within the field are somewhat more recent, in work of Sölin (in two dimensions, generalized later by Drury and Marshall ) and Carbery and Ziesler , respectively.",
"Both measures have the property that they are independent of the parametrization and that they are unchanged when the curve or surface is transformed by a measure-preserving affine mapping.",
"These measures and certain “variable coefficient” generalizations to families of curves and hypersurfaces have played a central role in the Fourier restriction problem as well as the problem of characterizing the $L^p$ –$L^q$ mapping properties of geometrically-constructed convolution operators, two problems which have been of sustained interest for many years drury1990,choi1999,oberlin1999II,dw2010,stovall2016,dlw2009,stovall2014,gressman2013,oberlin2012.",
"The deep connections between analysis and geometry enjoyed by affine arclength and hypersurface measures naturally lead to the problem of generalizing these objects to manifolds of arbitrary dimension or even to abstract measure-theoretic settings.",
"One particularly interesting approach is due to D. Oberlin (which generalizes an earlier observation of Graham, Hare, and Ritter in one dimension), who introduced the following condition on nonnegative measures $\\mu $ associated to submanifolds: a measure $\\mu $ on a $d$ -dimensional immersed submanifold of ${\\mathbb {R}}^n$ will be said to satisfy the Oberlin condition with exponent $\\alpha > 0$ when there exists a finite positive constant $C$ such that for every $K$ in the set $\\mathcal {K}_n$ of compact convex subsets of ${\\mathbb {R}}^n$ , $\\mu (K) \\le C |K|^\\alpha , $ where $|K|$ represents the usual Lebesgue measure of $K$ in ${\\mathbb {R}}^n$ .",
"When restricted to the class of balls with respect to the standard metric on ${\\mathbb {R}}^n$ , the condition (REF ) becomes a familiar inequality from geometric measure theory.",
"Unlike in that setting, here the exponent $\\alpha $ measures not just dimension of the measure, but also a certain kind of curvature (for the simple reason that (REF ) cannot hold for any $\\alpha > 0$ when $\\mu $ is supported on a hyperplane, which can be seen by taking $K$ to be increasingly thin in the direction transverse to such a hyperplane).",
"Oberlin observed that this condition is necessary for Fourier restriction or $L^p$ -improving estimates to hold; in particular, $ \\ \\left( \\int |\\hat{f}|^q d \\mu \\right)^\\frac{1}{q} \\lesssim ||f||_{L^p({\\mathbb {R}}^n)} \\ \\forall f \\in L^p({\\mathbb {R}}^n) \\Rightarrow \\mu (K) \\lesssim |K|^{\\frac{q}{p^{\\prime }}} \\ \\forall K \\in {\\mathcal {K}}_n$ and $ || f * \\mu ||_{L^q({\\mathbb {R}}^n)} \\lesssim ||f||_{L^p({\\mathbb {R}}^n)} \\ \\forall f \\in L^p({\\mathbb {R}}^n) \\Rightarrow \\mu (K) \\lesssim |K|^{\\frac{1}{p} - \\frac{1}{q}} \\ \\forall K \\in {\\mathcal {K}}_n, $ where $\\hat{\\cdot }$ is the Fourier transform, $*$ is convolution.",
"Here and throughout the paper, the notation $A \\lesssim B$ means that there is a finite positive constant $C$ such that $A \\le C B$ and this constant $C$ is independent of the relevant variables (functions and sets in this case) appearing in the expressions or quantities $A$ and $B$ .",
"By virtue of known results for these two problems, the affine arclength and hypersurface measures must satisfy (REF ) for appropriate exponents $\\alpha $ when suitable regularity hypotheses on the submanifolds are imposed.",
"The significance of the Oberlin condition (REF ) for curves and hypersurfaces in ${\\mathbb {R}}^n$ is that, up to a constant factor, the affine arclength and affine hypersurface measures on an immersed submanifold are the unique largest measures on the manifolds satisfying (REF ) when $\\alpha = 2 / (n^2+n)$ and $\\alpha = (n-1)/(n+1)$ , respectively.",
"More precisely, in the case of hypersurfaces (first established by Oberlin ), if $\\mu $ is any nonnegative measure on an immersed hypersurface $\\mathcal {M} \\subset {\\mathbb {R}}^n$ , if $\\mu $ satisfies (REF ) with $\\alpha = (n+1)/(n-1)$ , then $\\mu \\lesssim \\mu _{\\!_{\\mathcal {A}}}$ for affine hypersurface measure $\\mu _{\\!_{\\mathcal {A}}}$ (where $\\lesssim $ here means $\\mu (E) \\lesssim \\mu _{\\!_{\\mathcal {A}}}(E)$ uniformly for all Borel sets $E$ ).",
"Moreover, subject to certain algebraic limits on the complexity of the immersion, $\\mu _{\\!_{\\mathcal {A}}}$ itself satisfies (REF ) for this same exponent.",
"The condition (REF ) also turns out to be equivalent to the boundedness of certain geometrically-constructed multilinear determinant functionals and leads to a natural affine generalization of the classical Hausdorff measure .",
"This paper examines the Oberlin condition for arbitrary $d$ -dimensional submanifolds of ${\\mathbb {R}}^n$ (where $1 \\le d \\le n$ ) and characterizes it in the case of maximal nondegeneracy.",
"Specifically, the analogous results to those just mentioned above are established in all dimensions and codimensions: an affine invariant measure is constructed which is essentially the largest-possible measure satisfying the Oberlin condition for the largest nontrivial choice of $\\alpha $ .",
"To say that $\\alpha $ is nontrivial means simply that there is a nonzero measure, satisfying (REF ) for this $\\alpha $ , on some immersed submanifold of the given dimension and codimension.",
"As in the case of curves and hypersurfaces, the largest nontrivial $\\alpha $ can be understood as a ratio of the intrinsic dimension of the submanifold and its “homogeneous dimension,” which captures information about scaling and curvature-like properties to be measured.",
"The correct value of homogeneous dimension is defined as follows: when $d$ and $n$ are fixed, let the homogeneous dimension $Q$ be defined to be the smallest positive integer which equals the sum of the degrees of some collection of $n$ distinct, nonconstant monomials in $d$ variables (see Figure REF ).",
"The main result of this paper is Theorem REF : Theorem 1 Suppose $\\mathcal {M}$ is an immersed $d$ -dimensional submanifold of ${\\mathbb {R}}^n$ equipped with a nonnegative measure $\\mu $ .",
"Then the following are true: If (REF ) holds for $\\mu $ with exponent $\\alpha > d/Q$ , then $\\mu $ is the zero measure.",
"There is a nonnegative measure $\\mu _{\\!_{\\mathcal {A}}}$ on $\\mathcal {M}$ such that if (REF ) holds for $\\mu $ when $\\alpha = d/Q$ , then $\\mu \\lesssim \\mu _{\\!_{\\mathcal {A}}}$ .",
"Under certain algebraic constraints on the immersion of $\\mathcal {M}$ in ${\\mathbb {R}}^d$ (which are satisfied globally for polynomial embeddings and locally for real analytic embeddings), $\\mu _{\\!_{\\mathcal {A}}}$ satisfies the Oberlin condition (REF ) with $\\alpha = d/Q$ .",
"There is a $d$ -dimensional submanifold ${\\mathcal {M}}$ of ${\\mathbb {R}}^n$ for which $\\mu _{\\!_{\\mathcal {A}}}$ has everywhere strictly positive density with respect to Lebesgue measure on ${\\mathcal {M}}$ and satisfies (REF ) with $\\alpha = d/Q$ .",
"Figure: This plot shows the homogeneous dimension QQ as a function of nn for dd fixed.",
"The graph is piecewise linear with slope k+1k+1 from the point (d+k d-1,kd d+1d+k d)(\\binom{d+k}{d}-1,\\frac{k d}{d+1} \\binom{d+k}{d}) to the point (d+k+1 d-1,(k+1)d d+1d+k+1 d)(\\binom{d+k+1}{d}-1,\\frac{(k+1) d}{d+1} \\binom{d+k+1}{d }) for each k≥1k \\ge 1.Condition REF shows that the measure $\\mu _{\\!_{\\mathcal {A}}}$ to be constructed (which is intrinsic, invariant under measure-preserving affine linear transformations of ${\\mathbb {R}}^n$ , and agrees up to normalization with affine arclength and equiaffine measure when $d=1,n-1$ , respectively) is, up to a constant factor, the unique maximal measure on ${\\mathcal {M}}$ which satisfies (REF ) for $\\alpha = d/Q$ .",
"This extends the result of Oberlin for equiaffine measure to submanifolds of any dimension.",
"The structure of the rest of this paper is as follows.",
"In Section , the measure $\\mu _{\\!_{\\mathcal {A}}}$ is constructed by combining ideas of Kempf and Ness from Geometric Invariant Theory together with a simple but far-reaching observation that any covariant tensor field on a manifold can be used to construct an associated measure on that manifold in a way that generalizes the relationship between the Riemanninan metric tensor and the Riemannian volume.",
"In particular, the measure $\\mu _{\\!_{\\mathcal {A}}}$ will be the measure associated to an “affine curvature tensor” on the manifold $\\mathcal {M}$ immersed in ${\\mathbb {R}}^n$ .",
"After these constructions are complete, Parts REF and REF of Theorem REF follow in a rather immediate way.",
"Section is devoted to the proof of Parts REF and REF of Theorem 1.",
"Part REF is proved by first generalizing Theorem 1 of to higher dimensions.",
"The result, Lemma REF , is interesting in its own right and will have important implications for the theory of $L^p$ -improving estimates for averages over submanifolds in much the same way that Theorem 1 of formed the basis for a new proof of a restricted version Tao and Wright's result for averages over curves.",
"The final part of Section shows that the measure $\\mu _{\\!_{\\mathcal {A}}}$ is not trivial by constructing submanifolds on which it is possible to say with certainty that the density of $\\mu _{\\!_{\\mathcal {A}}}$ with respect to Lebesgue measure is never zero.",
"Finally, Section establishes uniform estimates for the number of nondegenerate solutions of certain systems of equations.",
"These estimates are important for Part REF of Theorem REF ."
],
[
"Geometric Invariant Theory",
"The main ideas and results from Geometric Invariant Theory that will be used in this paper come from the seminal work of Kempf and Ness and its subsequent extension to real reductive algebraic groups by Richardson and Slodowsky .",
"The idea of interest is that, for suitable representations of such groups, one can study group orbits by understanding the infimum over the orbit of a certain vector space norm.",
"For the purposes of this paper, it suffices to consider only representations of ${\\mathrm {SL}}(d,{\\mathbb {R}})$ or ${\\mathrm {SL}}(m,{\\mathbb {R}}) \\times {\\mathrm {SL}}(d,{\\mathbb {R}})$ on vector spaces of tensors.",
"In this context, the associated minimum vectors can be understood as normal forms of tensors and the actual numerical value of the infimum carries meaningful and important quantitative information about these tensors (in contrast to the usual situation in GIT in which one cares only about whether the infimum is zero or nonzero and whether or not it is attained).",
"To begin the construction, suppose that $\\mathcal {A}$ is any $k$ -linear functional on a real vector space $V$ of dimension $d$ .",
"Appropriating the Kempf-Ness minimum vector calculations of GIT, it becomes possible to canonically associate a density functional $\\mu _{\\!_{\\mathcal {A}}}: V^d \\rightarrow {\\mathbb {R}}_{\\ge 0}$ to any such $\\mathcal {A}$ .",
"Specifically, for any such $\\mathcal {A}$ and any vectors $v_1,\\ldots ,v_d$ , let $\\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d)$ be the quantity given by $\\begin{split}\\mu _{\\!_{\\mathcal {A}}}(v_1,& \\ldots ,v_d) := \\\\& \\left[ \\inf _{M \\in \\mathrm {SL}(d,{\\mathbb {R}})} \\sum _{j_1,\\ldots ,j_k=1}^d \\left| \\sum _{i_1,\\ldots ,i_k = 1}^d {\\mathcal {A}} (M_{j_1 i_1} v_{i_1},\\ldots , M_{j_k i_k} v_{i_k}) \\right|^{2} \\right]^{\\frac{d}{2k}}.\\end{split} $ Before showing that the quantity (REF ) is a density functional, it is worthwhile to acknowledge the algebraic structure that lies behind it.",
"When the $d$ -tuple of vectors $v := (v_1,\\ldots ,v_d) \\in V^{d}$ are linearly independent, one may define a representation $\\rho ^v_{\\cdot } : {\\mathrm {SL}}(d,{\\mathbb {R}}) \\times V \\rightarrow V$ by setting $ \\rho ^v_M(v_j) := \\sum _{i=1}^d M_{ij} v_i $ for each $j=1,\\ldots ,d$ and then extending to all of $V$ by linearity.",
"This representation extends to act on $k$ -linear functionals by duality, i.e., $ \\left( \\rho _{M}^v \\mathcal {A} \\right) (v_{j_1},\\ldots ,v_{j_k}) := \\mathcal {A} (\\rho ^v_{M^T} v_{j_1},\\ldots ,\\rho ^v_{M^T} v_{j_k}), $ where $M^T$ is the transpose of $M$ .",
"If we further define a norm on the space of $k$ -linear functionals by means of the formula $ || \\mathcal {A} ||_v^2 := \\sum _{j_1=1}^d \\cdots \\sum _{j_k = 1}^d |\\mathcal {A}(v_{j_1},\\ldots ,v_{j_k})|^2 $ then the formula (REF ) becomes $ \\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d) = \\left( \\inf _{M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})} || \\rho _M^v \\mathcal {A}||_v\\right)^{\\frac{d}{k}}.",
"$ To see that $\\mu _{\\!_{\\mathcal {A}}}$ is a density as promised, the first step is to demonstrate that $\\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d) = 0$ when $v_1,\\ldots ,v_d$ are linearly dependent.",
"In this case, there must exist an invertible matrix $M$ such that $\\sum _{i=1}^d M_{1 i} v_i = 0$ , and without loss of generality, one may assume that this matrix $M$ has been normalized so as to belong to $\\mathrm {SL}(d,{\\mathbb {R}})$ .",
"Now for each $t > 0$ , let $M^{(t)}$ be the transpose of the matrix obtained by scalar multiplying the first row of $M$ by $t^{d-1}$ and all remaining rows by $t^{-1}$ .",
"These matrices $M^{(t)}$ belong to $\\mathrm {SL}(d,{\\mathbb {R}})$ for all $t > 0$ , and $ \\rho ^v_{M^{(t)}} {\\mathcal {A}} = t^{-k} \\rho ^{v}_{M} {\\mathcal {A}} $ by multilinearlity of $\\mathcal {A}$ because $ {\\mathcal {A}}(M_{j_1 i_1}^{(t)} v_{i_1}, \\ldots , M_{j_k i_k}^{(t)} v_{i_k}) = t^{-k} {\\mathcal {A}} (M_{j_1 i_1} v_{i_1},\\ldots , M_{j_k i_k} v_{i_k}) $ by homogeneity if each $j_1,\\ldots ,j_k$ is not equal to one, and if any index $j_\\ell $ does equal one, then both sides vanish, making the equality (REF ) true trivially.",
"Taking $t \\rightarrow \\infty $ shows that the infimum in (REF ) over all $\\mathrm {SL}(d,{\\mathbb {R}})$ must vanish when $v_1,\\ldots ,v_d$ are linearly dependent.",
"Now let $T$ be any linear transformation of $V$ .",
"When $v_1,\\ldots ,v_d$ are linearly dependent or when $T$ is not invertible, $T v_1,\\ldots ,T v_d$ will be linearly dependent, so it must hold that $ \\mu _{\\!_{\\mathcal {A}}}(T v_1,\\ldots , T v_d) = |\\det T| \\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d) = 0.",
"$ Otherwise, when $v_1,\\ldots ,v_d$ are linearly independent and $T$ is invertible, there is a matrix $P \\in \\mathrm {GL}(d,{\\mathbb {R}})$ with $\\det P = \\det T$ such that $T v_j = \\sum _{i=1}^d P_{ji} v_i$ for each $j=1,\\ldots ,d$ .",
"Factor $P$ as $\\pm |\\det P|^{\\frac{1}{d}} P^{\\prime }$ for some $P^{\\prime }$ with $|\\det P^{\\prime }| = 1$ in general and $\\det P^{\\prime } =1$ when $d$ is odd.",
"Once again, by multilinearity of $\\mathcal {A}$ , $& \\sum _{j_1,\\ldots ,j_k=1}^d \\left| \\sum _{i_1,\\ldots ,i_k = 1}^d {\\mathcal {A}} (M_{j_1 i_1} T v_{i_1},\\ldots , M_{j_k i_k} T v_{i_k}) \\right|^2 \\nonumber \\\\ & \\ \\ = | \\det T|^{\\frac{2k}{d}} \\sum _{j_1,\\ldots ,j_k=1}^d \\left| \\sum _{i_1,\\ldots ,i_k,\\ell _1,\\ldots ,\\ell _k = 1}^d {\\mathcal {A}} (M_{j_1 i_1} P_{i_1 \\ell _1} ^{\\prime } v_{\\ell _1},\\ldots ,M_{j_k i_k} P_{i_k \\ell _k}^{\\prime } v_{\\ell _k}) \\right|^2 \\nonumber \\\\& \\ \\ = |\\det T|^{\\frac{2k}{d}} \\sum _{j_1,\\ldots ,j_k=1}^d \\left| \\sum _{i_1,\\ldots ,i_k = 1}^d {\\mathcal {A}} ((MP^{\\prime })_{j_1 i_1} v_{i_1},\\ldots , (MP^{\\prime })_{j_k i_k} v_{i_k}) \\right|^2.",
"$ Since $\\mathrm {SL}(d,{\\mathbb {R}})$ is a group, the set of matrices of the form $MP^{\\prime }$ when $M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})$ is itself exactly ${\\mathrm {SL}}(d,{\\mathbb {R}})$ assuming that $\\det P^{\\prime } = 1$ .",
"If $\\det P^{\\prime } = -1$ , then the matrices $M P^{\\prime }$ for $M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})$ are exactly those matrices $N$ which belong to ${\\mathrm {SL}}(d,{\\mathbb {R}})$ after the first two rows of $N$ are interchanged.",
"Since (REF ) is invariant under permutations of the rows of $MP^{\\prime }$ , it follows in both cases ($\\det P^{\\prime } = \\pm 1$ ) that $\\inf _{M \\in {\\mathrm {SL}}(d,R)} \\sum _{j_1,\\ldots ,j_k=1}^d & \\left| \\sum _{i_1,\\ldots ,i_k = 1}^d {\\mathcal {A}} ((MP^{\\prime })_{j_1 i_1} v_{i_1},\\ldots , (MP^{\\prime })_{j_k i_k} v_{i_k}) \\right|^2 \\\\ & = \\left[ \\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d) \\right]^{\\frac{2k}{d}},$ which gives the desired identity $ \\mu _{\\mathcal {A}}( T v_1,\\ldots ,T v_d ) = |\\det T| ~ \\mu _{\\mathcal {A}}(v_1,\\ldots ,v_d) $ for any $v_1,\\ldots ,v_d$ and any linear transformation $T$ , as asserted.",
"Example It is illuminating to compute $\\mu _{\\!_{\\mathcal {A}}}$ in the specal case when $\\mathcal {A}$ is a symmetric bilinear form.",
"Fix linearly independent vectors $v_1,\\ldots ,v_d$ and define the matrix $A$ by $A_{i j} := {\\mathcal {A}}(v_i,v_j)$ .",
"It follows that $ (\\rho ^v_M \\mathcal {A})(v_{j_1},v_{j_2}) = (M A M^T)_{j_1 j_2} \\mbox{ and } || \\rho ^v_M \\mathcal {A}||_v^2 = \\mathrm {tr} ( M A M^T M A M^T).",
"$ Now $M A M^T M A M^T$ is symmetric and positive semidefinite, so its eigenvalues are all nonnegative.",
"Thus the AM-GM inequality implies that $ d (\\det ( M A M^T M A M^T))^\\frac{1}{d} \\le \\mathrm {tr} (M A M^T M A M^T) $ with equality when all eigenvalues are equal (which, when $A$ is invertible, can be attained for some $M \\in \\mathrm {SL}(d,{\\mathbb {R}})$ by building $M$ from a basis of unit-length eigenvectors of $A$ with respect to some inner product and then rescaling the eigenvectors appropriately).",
"Because $\\det M = 1$ , $\\det (M A M^T M A M^T) = (\\det A)^2$ and therefore $ \\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d) = d^\\frac{d}{4} |\\det A|^{\\frac{1}{2}}.",
"$ In particular, on a Riemannian manifold, setting $\\mathcal {A}$ equal to the metric tensor $g$ yields a tensor density $\\mu _{\\!_{\\mathcal {A}}}$ which is exactly equal to a dimensional constant times the corresponding Riemannian volume density.",
"At this point, the reader may be somewhat understandably disappointed by the abstract nature of the infimum appearing in (REF ) since it is not immediately apparent how to compute the infimum in finitely many operations.",
"However, the abstract nature of the definition (REF ) turns out to be a blessing rather than a curse, because it effectively allows the analysis to entirely sidestep the very deep and rich algebraic question of what (REF ) computes.",
"It turns out that (REF ) is deeply connected, both algebraically and analytically, to the problem of finding polynomials in the entries of the tensor $A_{j_1,\\ldots ,j_k} = \\mathcal {A}(v_{j_1},\\ldots ,v_{j_k})$ which are invariant under the action of the representation $\\rho ^v_\\cdot $ .",
"Hilbert showed that, when the group ${\\mathrm {SL}}(d,{\\mathbb {R}})$ is replaced by ${\\mathrm {SL}}(d,$ , there are finitely many polynomials invariant under the action of ${\\mathrm {SL}}(d,$ which generate the algebra of all such invariant polynomials.",
"From this fact it is easy to see that the same result must be true for ${\\mathrm {SL}}(d,{\\mathbb {R}})$ itself.",
"A vast body of literature shows (via Weyl's unitarian trick) shows that the same result holds for representations of any group $G$ which is a real reductive algebraic group (which, as far as the present paper is concerned, is a class which includes ${\\mathrm {SL}}(d,{\\mathbb {R}})$ and is closed under Cartesian products).",
"It is possible in principle to compute these polynomials explicitly in finite time (see Sturmfels ), but in general going about the calculation in this way is somewhat unwieldy and akin to the computation of the determinant via its permutation expansion rather than by more efficient, symmetry-exploiting techniques.",
"In any case, the density (REF ) simultaneously captures the behavior of all invariant polynomials at once, as demonstrated by the following lemma: Lemma 1 Suppose that $G$ is a real reductive algebraic group and that $\\rho $ is a $G$ -representation on some finite-dimensional real vector space $V$ equipped with a norm $|| \\cdot ||$ .",
"Let $p_1,\\ldots ,p_N$ be any collection of homogeneous polynomials of positive degree in $V$ which generate the algebra of all $G$ -invariant polynomials.",
"Then there exist constants $0 < C_1 \\le C_2 < \\infty $ such that $ C_1 \\max _{j=1,\\ldots ,N} |p_j(\\mathcal {A})|^{\\frac{1}{d_j}} \\le \\inf _{M \\in G} ||\\rho _M \\mathcal {A}|| \\le C_2 \\max _{j=1,\\ldots ,N} |p_j(\\mathcal {A})|^{\\frac{1}{d_j}} \\mbox{ for all } \\mathcal {A} \\in V. $ To prove the first inequality, observe by scaling that $ ||p_j||^{-\\frac{1}{d_j}}_\\infty |p_j(\\mathcal {A})|^{\\frac{1}{d_j}} \\le ||\\mathcal {A}|| $ for all $j$ and all $\\mathcal {A} \\in V$ , where $||p_j||_\\infty $ is the supremum of $p_j$ on the unit sphere of $||\\cdot ||$ .",
"Moreover, because each $p_j$ is invariant under $\\rho $ , $ ||p_j||^{-\\frac{1}{d_j}}_\\infty |p_j(\\mathcal {A})|^{\\frac{1}{d_j}} = ||p_j||^{-\\frac{1}{d_j}}_\\infty |p_j(\\rho _M \\mathcal {A})|^{\\frac{1}{d_j}} \\le ||\\rho _M \\mathcal {A}||, $ so taking an infimum in $M$ and a supremum in $j$ gives $ \\left[ \\min _{j=1,\\ldots ,N} ||p_j||_\\infty ^{-\\frac{1}{d_j}} \\right] \\max _{j = 1,\\ldots ,N} |p_j(\\mathcal {A})|^{\\frac{1}{d_j}} \\le \\inf _{M \\in G} ||\\rho _M \\mathcal {A}||.",
"$ To prove the reverse inequality, suppose for the sake of contradiction that it does not hold for any finite $C$ .",
"Because the inequality is homogeneous in the norm $|| \\cdot ||$ , its failure would imply that one could find a sequence $\\mathcal {A}_k$ with $\\inf _M || \\rho _M \\mathcal {A}_k|| = 1$ for all $k$ such that $ \\max _{j} |p_j (\\mathcal {A}_k)|^{\\frac{1}{d_j}} \\le k^{-1} \\inf _{M \\in G} || \\rho _M \\mathcal {A}_k|| = k^{-1}.",
"$ Moreover, by replacing $\\mathcal {A}_k$ by $\\rho _{M_k} \\mathcal {A}_k$ for suitable $M_k$ and taking a subsequence by compactness, it may be assumed that $\\mathcal {A}_k$ converges to some $\\mathcal {A}$ in the unit sphere as $k \\rightarrow \\infty $ .",
"By continuity of the polynomials $p_j$ , $p_j(\\mathcal {A}) = 0$ for all $j$ .",
"Therefore $\\mathcal {A}$ belongs to the so-called nullcone of the representation and by the real Hilbert-Mumford criterion, first proved by Birkes , there must exist a one-parameter subgroup $\\rho _{\\exp (t X)}$ of $G$ such that $\\rho _{\\exp (t X)}\\mathcal {A} \\rightarrow 0$ as $t \\rightarrow \\infty $ .",
"This, of course, implies that $\\inf _M ||\\rho _M \\mathcal {A}|| = 0$ .",
"However $\\inf _{M} ||\\rho _M \\mathcal {A}_k|| = 1$ for all $k$ implies that $||\\rho _M \\mathcal {A}_k|| \\ge 1$ for all $M \\in G$ and all $k$ , which means by continuity that $||\\rho _M \\mathcal {A}|| \\ge 1$ for all $M$ , so $\\inf _M ||\\rho _M \\mathcal {A}|| = 0$ must be contradicted.",
"The inequality (REF ) shows that the numerical value of $\\mu _{\\!_{\\mathcal {A}}}(v_1,\\ldots ,v_d)$ is, in rough analogy with the symmetric bilinear form example just examined, comparable to the maximum of appropriate powers of the invariant polynomials applied to $A_{j_1,\\ldots ,j_k} = \\mathcal {A}(v_{j_1},\\ldots ,v_{j_k})$ .",
"It is also worth observing that when many invariant polynomials exist (which, unlike the symmetric bilinear form case, is generally the more common situation), the nullcone of tensors $\\mathcal {A}$ for which $\\mu _{\\!_{\\mathcal {A}}}= 0$ will have codimension greater than one.",
"In terms of affine curvature, this will mean that for general submanifolds of dimension $d$ in ${\\mathbb {R}}^n$ , it is typically “easier” to have nonvanishing affine curvature than it is in the case of hypersurfaces because the space of “flat” Taylor polynomial jets which must be avoided is often of codimension greater than one."
],
[
"Construction of the affine curvature tensor and associated measure",
"unit=15pt Figure: In the diagram above, squares represent points in ℤ×ℤ{\\mathbb {Z}}\\times {\\mathbb {Z}}.",
"The index set Λ d,n \\Lambda _{d,n} is simply the union of the first nn columns, and the homogeneous dimension QQ is simply the cardinality of Λ d,n \\Lambda _{d,n}.",
"In terms of the tensor 𝒜 p \\mathcal {A}_p, the number of boxes in each column indicates how many derivatives are applied to ff in the corresponding factor of the wedge product (or equivalently, the corresponding column of the matrix).We move now to the construction of a covariant tensor which captures the affine curvature we are interested in.",
"This tensor will be given an associated density using the formula (REF ) which can be integrated to give a canonical measure on immersed submanifolds $\\mathcal {M} \\subset {\\mathbb {R}}^n$ .",
"Suppose that $\\mathcal {M}$ is a manifold of dimension $d$ which is equipped with a smooth immersion $f : \\mathcal {M} \\rightarrow {\\mathbb {R}}^n$ .",
"For convenience, let the values of $f$ be regarded as column vectors.",
"For any positive integer $j$ , let $\\kappa _j$ be the smallest integer such that the dimension of the space $P_d^{\\kappa _j}$ of real polynomials of degree $\\kappa _j$ in $d$ variables has dimension at least $j+1$ , and let $\\Lambda _{d,n}$ be the index set $ \\Lambda _{d,n} := \\left\\lbrace (j,k) \\in {\\mathbb {Z}}\\times {\\mathbb {Z}} \\ \\left| \\ 1 \\le j \\le n \\mbox{ and } 1 \\le k \\le \\kappa _j \\right.",
"\\right\\rbrace .",
"$ The index set $\\Lambda _{d,n}$ is represented pictorially in Figure REF as the first $n$ columns of boxes.",
"The cardinality of $\\Lambda _{d,n}$ is exactly the homogeneous dimension $Q$ defined in the introduction.",
"We are going to define a $Q$ -linear covariant tensor $\\mathcal {A}_p$ at each point $p \\in \\mathcal {M}$ which captures the affine geometry of the immersion $f$ .",
"We will denote the action of $\\mathcal {A}_p$ on $Q$ -tuples of vectors by either $ {\\mathcal {A}}_p(X_1,\\ldots ,X_Q) \\mbox{ or } {\\mathcal {A}}_p ((X_\\lambda )_{\\lambda \\in \\Lambda _{d,n}}) $ depending on which approach is most convenient at the moment (where we lexicographically order the elements of $\\Lambda _{d,n}$ when such an order is not otherwise specified).",
"Now for any finite sequence of vector fields $X_\\lambda $ indexed by $\\lambda \\in \\Lambda _{d,n}$ , let $\\begin{split}{\\mathcal {A}}_p( (X_\\lambda )_{\\lambda \\in \\Lambda _{d,n}}) := & \\det ( X_{(1,1)} f(p) \\wedge \\cdots \\wedge \\\\& X_{(j,1)} \\cdots X_{(j,\\kappa _j)} f(p) \\wedge \\cdots \\wedge X_{(n,1)} \\cdots X_{(n,\\kappa _n)} f(p)).\\end{split} $ Here the determinant of an $n$ -fold wedge of vectors in ${\\mathbb {R}}^n$ is understood to equal the determinant of the $n \\times n$ matrix whose columns are the factors of the wedge (technically these factors are not unique, but the antisymmetry of the determinant and of wedge products guarantees the same determinant for any factorization).",
"In other words, (REF ) equals the determinant of an $n \\times n$ matrix whose $j$ -th column is the column vector $X_{(j,1)} \\cdots X_{(j,\\kappa _j)} f(p)$ .",
"(Note also that the lexicographic order on $\\Lambda _{d,n}$ corresponds exactly to the order that each $\\lambda \\in \\Lambda _{d,n}$ appears in the above formula when moving from left to right; with respect to Figure REF , the order is left-to-right followed by bottom-to-top.)",
"This object $\\mathcal {A}_p$ will be called the affine curvature tensor at $p$ .",
"First observe that it is certainly linear in $X_\\lambda $ for each $\\lambda \\in \\Lambda _{d,n}$ .",
"To see that $\\mathcal {A}_p$ depends only on the pointwise values of the $X_\\lambda $ at $p$ and not any derivatives of these vector fields, it suffices to show that any single one of the vector fields $X_\\lambda $ may be replaced by any other vector field $X_\\lambda ^{\\prime }$ agreeing with $X_\\lambda $ at $p$ without changing the value of $\\mathcal {A}_p$ .",
"For any indices $\\lambda = (j,k)$ such that $\\kappa _j = 1$ , this invariance under replacement follows immediately from the fact that these vector fields appear alone in their own column (i.e., the formula (REF ) contains no derivatives of $X_\\lambda $ to begin with).",
"For any $\\lambda = (j,k)$ with $\\kappa _j > 1$ , the identity $X_{(j,1)} & \\cdots X_{(j,k)} \\cdots X_{(j,\\kappa _j)} f(p) - X_{(j,1)} \\cdots X^{\\prime }_{(j,k)} \\cdots X_{(j,\\kappa _j)} f(p) \\\\& = X_{(j,1)} \\cdots X_{(j,k-1)} [ X_{(j,k-1)}, X_{(j,k)} - X^{\\prime }_{(j,k)}] X_{(j,k+1)} \\cdots X_{(j,\\kappa _j)} f(p) \\\\& + \\cdots + [X_{(j,1)}, X_{(j,k)} - X_{(j,k)}^{\\prime }] X_{(j,2)} \\cdots \\widehat{X_{(j,k)}} \\cdots X_{(j,\\kappa _j)} f(p)$ (where $\\widehat{\\cdot }$ indicates omission of $X_{(j,k)}$ in its usual place) shows that $\\mathcal {A}_p$ vanishes when $X_{\\lambda }$ is replaced by $X_{\\lambda } - X^{\\prime }_{\\lambda }$ : the number of columns of the matrix in (REF ) for which $f$ is differentiated to some order between 1 and $\\kappa _j - 1$ is strictly greater than the dimension of the vector space generated by such operators, so there must be linearly dependent columns in the matrix, which forces $\\mathcal {A}_p$ to vanish.",
"The measure $\\mu _{\\!_{\\mathcal {A}}}$ on the submanifold which will be shown under suitable additional hypotheses to satisfy (REF ) is exactly that measure whose density is given from the tensor $\\mathcal {A}_p$ by the formula (REF )."
],
[
"Proof of Parts ",
"We begin with the following elementary lemma which gives an estimate for the volume of the convex hull of certain sets $S \\subset {\\mathbb {R}}^n$ : Lemma 2 Suppose $S \\subset {\\mathbb {R}}^n$ is a compact set containing the origin, and let $K$ be its convex hull.",
"There exist $v_1,\\ldots ,v_n \\in S$ such that the sets $ K_1 := \\left\\lbrace v \\in {\\mathbb {R}}^n \\ \\left| \\ v = \\sum _{i=1}^n c_i v_i \\mbox{ for coefficients } c_i \\mbox{ such that } \\sum _{i=1}^n |c_i| \\le 1 \\right.",
"\\right\\rbrace $ and $ K_\\infty := \\left\\lbrace v \\in {\\mathbb {R}}^n \\ \\left| \\ v = \\sum _{i=1}^n c_i v_i \\mbox{ for coefficients } c_i \\in [-1,1], \\ i = 1,\\ldots ,n \\right.",
"\\right\\rbrace $ satisfy $ K_1 \\subset K \\subset K_\\infty .",
"$ In particular, $ \\frac{2^n}{n!}",
"|\\det (v_1 \\wedge \\cdots \\wedge v_n)| \\le |K| \\le 2^n |\\det (v_1 \\wedge \\cdots \\wedge v_n)|.",
"$ Let $V$ be the unique vector subspace of ${\\mathbb {R}}^n$ of smallest dimension which contains $S$ (where uniqueness holds because the intersection of two subspaces containing $S$ would be a subspace of smaller dimension also containing $S$ ).",
"Let $m$ denote the dimension of $V$ , and let $\\det _V$ be any nontrivial alternating $m$ -linear form on $V$ .",
"Let $(v_1,\\ldots ,v_m) \\in S^m$ be any $m$ -tuple at which the maximum of the function $ (v_1,\\ldots ,v_m) \\rightarrow |\\det _V (v_1 \\wedge \\cdots \\wedge v_m)| $ is attained.",
"Since $S$ is not contained in any subspace of smaller dimension, $|\\det _V (v_1 \\wedge \\cdots \\wedge v_m)| > 0$ unless $m=0$ (in which case $S = \\lbrace 0\\rbrace $ and the lemma is trivial).",
"Now by Cramer's rule, for any $v \\in V$ , $ v = \\sum _{i=1}^m (-1)^{i-1}\\frac{\\det _V ( v \\wedge v_1 \\wedge \\cdots \\wedge \\widehat{v_i} \\wedge \\cdots \\wedge v_m)}{\\det _V (v_1 \\wedge \\cdots \\wedge v_m)} v_i, $ where, in this case, the circumflex $\\widehat{\\cdot }$ indicates that a vector is to be omitted from the determinant.",
"In the particular case when $v \\in S$ , the $m$ -tuple $(v,v_1,\\ldots ,\\widehat{v_i},\\ldots ,v_m)$ belongs to the set $S^m$ over which the supremum of $|\\det _V|$ was taken; therefore each numerator has magnitude less than or equal to the denominator.",
"Thus $S$ belongs to the parallelepiped $ P := \\left\\lbrace v \\in {\\mathbb {R}}^n \\ \\left| \\ v = \\sum _{i=1}^m c_i v_i \\mbox{ for some } c_1,\\ldots ,c_m \\in [-1,1] \\right.",
"\\right\\rbrace .",
"$ Since $P$ is convex and contains $S$ , it must contain $K$ as well.",
"To establish the lemma, we extend the sequence $v_1,\\ldots ,v_m$ to a sequence of length $n$ by fixing $v_j = 0$ for $j > m$ .",
"Trivially $P = K_\\infty $ for this choice, so the containment $K \\subset K_\\infty $ must hold.",
"For the remaining containment, observe that $v_1,\\ldots ,v_n$ must belong to $K$ since they belong to $S$ .",
"Therefore, by convexity of $K$ , the set $K_1$ must be contained in $K$ .",
"The volume inequality (REF ) follows from the elementary calculation of the volumes of $K_1$ and $K_\\infty $ .",
"With Lemma REF in place, we turn now to the proof of Parts REF and REF of Theorem REF .",
"Pick any point $p \\in {\\mathcal {M}}$ and fix any smooth coordinate system $(t_1,\\ldots ,t_d)$ near $p$ so that the immersion $f : \\mathcal {M} \\rightarrow {\\mathbb {R}}^d$ may be regarded in these coordinates as a function from a $3 \\delta $ neighborhood of the origin (chosen so that $t=0$ are the coordinates of $p$ ) into ${\\mathbb {R}}^n$ .",
"By Taylor's formula, for all $t_0,t \\in {\\mathbb {R}}^d$ with $|t_0| \\le \\delta , |t| \\le 2\\delta $ , $ f(t)-f(t_0) = \\sum _{0< |\\alpha | \\le \\ell } \\frac{(t-t_0)^\\alpha }{\\alpha !}",
"\\partial _t^\\alpha f(t_0) + \\sum _{|\\beta |=\\ell +1} \\frac{(t-t_0)^\\beta }{\\beta !}",
"R^\\beta _{t_0}(t), $ for any finite $\\ell $ , where each remainder term $R^\\beta _{t_0}(t)$ is continuous on $|t| \\le 2 \\delta $ and equals $\\partial _t^\\beta f(t_0)$ when $t = t_0$ .",
"(For most of what follows, $t_0$ will be regarded as a fixed but otherwise arbitrary point with $|t_0| \\le \\delta $ .)",
"For definiteness, let $\\ell := \\kappa _n$ , i.e., $\\ell $ equals the highest order of differentiation that appears in a column of the matrix whose determinant forms $\\mathcal {A}$ (or equivalently, $\\ell $ is the number of boxes in column $n$ of the diagram given in Figure REF ).",
"This choice of $\\ell $ implies that the dimension of the space of polynomials of degree $\\ell $ with no constant term is at least equal to $n$ .",
"For any $r \\in (0,\\delta ]$ , let $S_{t_0,r}$ be the compact subset of ${\\mathbb {R}}^n$ given by $ S_{t_0,r} := \\lbrace 0 \\rbrace \\cup \\bigcup _{|\\alpha | \\le \\ell } \\left\\lbrace \\frac{r^{|\\alpha |}}{\\alpha !}",
"\\partial _t^\\alpha f(t_0) \\right\\rbrace \\cup \\bigcup _{|\\beta | = \\ell +1, |t| \\le 2\\delta } \\left\\lbrace \\frac{r^{\\ell +1}}{\\beta !}",
"R^\\beta _{t_0} (t) \\right\\rbrace $ and let $K_{t_0,r}$ be the convex hull of $S_{t_0,r} \\cup (-S_{t_0,r})$ .",
"Now each term in either sum on the right-hand side of (REF ) belongs to $K_{t_0,r}$ whenever $|t| \\le 2 \\delta $ and $|t-t_0| \\le r$ .",
"Because the total number of summands on the right-hand side is at most some constant $C$ depending only on $d$ and $n$ , the difference vector $f(t) - f(t_0)$ must belong to the dilated set $C K_{t_0,r}$ whenever $|t| \\le 2 \\delta $ and $|t-t_0| \\le r$ .",
"In particular, this implies that the translated set $C K_{t_0,r} + f(t_0)$ must contain the vector $f(t)$ whenever $|t| \\le 2\\delta $ and $|t-t_0| \\le r \\le \\delta $ .",
"By virtue of (REF ), the Lebesgue measure of the set $C K_{t_0,r} + f(t_0)$ is $O(r^Q)$ as $r \\rightarrow 0^+$ since it is dominated by a constant depending on $d$ and $n$ times a determinant $|\\det (v_1 \\wedge \\cdots \\wedge v_n)|$ for some $v_1,\\ldots ,v_n \\in S_{t_0,r} \\cup (-S_{t_0,r})$ and since $Q$ is by definition the smallest integer which it is possible to express as a sum of degrees of distinct, nonconstant monomials in $d$ variables (thus $Q$ corresponds to the smallest possible factor of $r$ which will appear via scaling in such determinants).",
"In fact, a slightly stronger result is also true: namely, that it is possible to quantify the implied constant in this $O(r^Q)$ estimate in terms of the affine curvature tensor $\\mathcal {A}$ at $t_0$ .",
"For any collection $\\alpha _1,\\ldots ,\\alpha _n$ of monomials such that $|\\alpha _1| + \\cdots + |\\alpha _n| = Q$ , it is possible to find indices $i_\\lambda $ for each $\\lambda \\in \\Lambda _{d,n}$ (these indices being obtained by “expanding” each $\\alpha _i$ as a composition of first-order coordinate derivatives) so that $ |\\det (\\partial ^{\\alpha _1}_t f(t_0) \\wedge \\cdots \\wedge \\partial ^{\\alpha _n}_t f(t_0))| = | {\\mathcal {A}}_{t_0} ((\\partial _{t_{i_\\lambda }})_{\\lambda \\in \\Lambda _{d,n}})|.",
"$ Therefore it follows from (REF ) that when $r \\le \\delta $ , the image $f(B_r(t_0))$ is contained in $C K_{t_0,r} + f(t_0)$ , which is a compact convex set with volume no greater than $ C^{\\prime } r^Q \\left[ \\sum _{j_1,\\ldots ,j_Q=1}^d \\left| {\\mathcal {A}}_{t_0} (\\partial _{t_{j_1}},\\ldots ,\\partial _{t_{j_Q}}) \\right|^2 \\right]^{\\frac{1}{2}} + O(r^{Q+1}) $ as $r \\rightarrow 0^+$ , where $C^{\\prime }$ is some new constant depending only on $d$ and $n$ .",
"Consequently, if $\\mu $ is any measure on $\\mathcal {M}$ satisfying the restricted Oberlin condition (REF ) with exponent $\\alpha $ and constant $C_\\mu $ , then $ \\limsup _{r \\rightarrow 0^+} r^{-\\alpha Q} \\mu (B_r(t_0)) \\lesssim C_\\mu \\left| \\left[ \\sum _{j_1,\\ldots ,j_Q=1}^d \\left| {\\mathcal {A}}_{t_0} (\\partial _{t_{j_1}},\\ldots ,\\partial _{t_{j_Q}}) \\right|^2 \\right]^{\\frac{1}{2}} \\right|^{\\alpha } $ for any $t_0$ with $|t_0| \\le \\delta $ with an implied constant depending only on $d$ and $n$ .",
"If $\\alpha \\ge d/Q$ , this implies that $\\mu $ must be absolutely continuous with respect to Lebesgue measure on $\\mathcal {M}$ on a $\\delta $ -neighborhood of the chosen origin point $p$ , and if $\\alpha > d/Q$ , it further implies that $\\mu $ must be the zero measure on that neighborhood (since the Radon-Nykodym derivative of $\\mu $ with respect to Lebesgue measure must vanish at every Lebesgue point, which is almost every point in the neighborhood), thus establishing Part REF of Theorem REF .",
"When $\\alpha = d/Q$ , because $\\mu $ must be absolutely continuous with respect to Lebesgue measure, it must follow that $ \\limsup _{r \\rightarrow 0^+} r^{-d} \\mu (B_r(t_0)) = c_d \\frac{d \\mu }{dt} (t_0) $ for almost every $t_0$ with $|t_0| \\le \\delta $ , where $d \\mu / dt$ is the Radon-Nykodym derivative of $\\mu $ with respect to Lebesgue measure $dt$ in the chosen coordinate system.",
"By (REF ), then, $ \\left.",
"\\frac{d \\mu }{dt} \\right|_q \\lesssim C_\\mu \\left[ \\sum _{j_1,\\ldots ,j_Q=1}^d \\left| {\\mathcal {A}}_{q} (\\partial _{t_{j_1}},\\ldots ,\\partial _{t_{j_Q}}) \\right|^2 \\right]^{\\frac{d}{2Q}} $ for almost every point $q$ in some neighborhood of the original point $p$ (where, once again, the implied constant depends only on $d$ and $n$ ).",
"Now, by transforming the coordinates $(t_1,\\ldots ,t_d)$ by matrices $M \\in \\mathrm {SL}(d,{\\mathbb {R}})$ to produce new coordinate systems, it follows by the same reasoning as above that $\\left.",
"\\frac{d \\mu }{dt} \\right|_q \\lesssim C_\\mu \\left[ \\sum _{j_1,\\ldots ,j_Q=1}^d \\left| \\sum _{i_1,\\ldots ,i_Q=1}^d {\\mathcal {A}}_{q} (M_{j_1 i_1} \\partial _{t_{i_1}},\\ldots ,M_{j_Q i_Q} \\partial _{t_{i_Q}}) \\right|^2 \\right]^{\\frac{d}{2Q}}$ for every $M \\in \\mathrm {SL}(d,{\\mathbb {R}})$ and almost every $q$ in a neighborhood of $p$ (by continuity of the right-hand side as a function of $M$ , it suffices to consider only some countable dense subset of $\\mathrm {SL}(d,{\\mathbb {R}})$ so that the set on which the inequality fails is clearly null).",
"Taking an infimum over $M$ gives that $ \\frac{d \\mu }{dt} \\lesssim C_\\mu \\frac{d \\mu _{\\!_{\\mathcal {A}}}}{dt} $ almost everywhere on the coordinate patch.",
"Because the coordinates and patch were arbitrary, it follows that Part REF of Theorem REF must hold with an implicit constant which equals a dimensional quantity (depending only on $d$ and $n$ ) times the Oberlin constant $C_\\mu $ from (REF ) for the measure $\\mu $ itself."
],
[
"On the geometry of functions on measurable sets",
"This section begins with a construction generalizing the results of Theorem 1 of .",
"Roughly stated, that theorem indicated that for single-variable real polynomials of a given degree, every measurable subset of the real line has a “core” which contains a nontrivial fraction of the set such that the supremum of any such polynomial (or appropriately weighted derivatives) on the core is bounded above by the average of the polynomial on the entire set.",
"The proof involved careful analysis of Vandermonde determinants and has no immediate generalization to other dimensions or families of functions.",
"In the arguments below, an entirely different approach will be used which is based on convex geometry and admits extensions to a variety of new contexts.",
"In particular, the setting of polynomials is no simpler to study than any other finite-dimensional family of real analytic functions, which will be the preferred formulation of the result.",
"To formulate the result, it is convenient to make the following definition.",
"Let ${\\mathcal {M}}$ be any real analytic manifold of dimension $d$ and let ${\\mathcal {F}}$ be a finite-dimensional vector space of real analytic functions on ${\\mathcal {M}}$ whose differentials span the cotangent space at every point of ${\\mathcal {M}}$ .",
"Any such pair $({\\mathcal {M}},{\\mathcal {F}})$ will be called a geometric function system.",
"Such a system will be called compact when either ${\\mathcal {M}}$ is compact or has a compact closure in some larger real analytic manifold ${\\mathcal {M}}^+$ such that the functions of ${\\mathcal {F}}$ extend to functions ${\\mathcal {F}}^+$ on ${\\mathcal {M}}^+$ in such a way that $({\\mathcal {M}}^+,{\\mathcal {F}}^+)$ is also a geometric function system.",
"The first result is a “zeroth order” version of the results of : Lemma 3 Suppose $({\\mathcal {M}},{\\mathcal {F}})$ is a compact geometric function system.",
"Then for any finite positive measure $\\mu $ on ${\\mathcal {M}}$ absolutely continuous with respect to Lebesgue measure and any measurable set $E \\subset {\\mathcal {M}}$ of positive measure, there is a measurable subset $E^{\\prime } \\subset E$ such that $\\mu (E^{\\prime }) \\gtrsim \\mu (E)$ and $ \\sup _{p \\in E^{\\prime }} |f(p)| \\lesssim \\frac{1}{\\mu (E)} \\int _E |f| ~ d \\mu \\mbox{ for all } f \\in {\\mathcal {F}}.",
"$ The implicit constants in both inequalities depend only on the pair $({\\mathcal {M}},{\\mathcal {F}})$ .",
"For each $f \\in {\\mathcal {F}}$ , consider the following norm: $ ||f|| := \\frac{1}{\\mu (E)} \\int _E |f| d \\mu .",
"$ Compactness of the geometric function system implies that $||f||$ is finite for every $f \\in {\\mathcal {F}}$ .",
"Because each function $f \\in {\\mathcal {F}}$ is real analytic and the measure of $E$ is strictly positive, no $f \\in {\\mathcal {F}}$ aside from the zero function can have $||f|| = 0$ , which is what guarantees that $|| \\cdot ||$ is a norm rather than merely a seminorm.",
"Assuming that the dimension of $\\mathcal {F}$ is $k$ , applying Lemma REF to the set $S$ which is the unit sphere of $|| \\cdot ||$ and using homogeneity of the norm, there must be functions $f_1,\\ldots ,f_k$ with $||f_i||=1$ for all $i$ (none of the functions $f_i$ will be identically zero because the unit sphere does not lie in any nontrivial subspace of $\\mathcal {F}$ ) such that every $f \\in {\\mathcal {F}}$ has the property that $ f = \\sum _{i=1}^k c_i f_i $ with $|c_i| \\le ||f||$ for each $i$ .",
"In particular, this implies that $ |f(p)| = \\left| \\sum _{i=1}^k c_i f_i(p) \\right| \\le ||f|| \\sum _{i=1}^k |f_i(p)| $ for each $f \\in {\\mathcal {F}}$ .",
"Let $E^{\\prime }$ be the subset of $E$ on which $\\sum _{i=1}^k |f_i(p)| \\le 2k$ ; by Tchebyshev's inequality, $\\mu (E^{\\prime }) \\ge \\mu (E) - \\frac{1}{2k} \\int _E \\sum _{i=1}^k |f_i(p)| d \\mu \\ge \\frac{1}{2} \\mu (E)$ and $ \\sup _{p \\in E^{\\prime }} |f(p)| \\le \\sup _{p \\in E^{\\prime }} \\left[ ||f|| \\sum _{i=1}^k |f_i(p)| \\right] \\le \\frac{2k}{\\mu (E)} \\int _E |f| d \\mu $ for all $f \\in {\\mathcal {F}}$ .",
"The extension of the results of to derivative estimates in higher dimensions is necessarily much more subtle than the one-dimensional case because of inherent issues of anisotropy of differentiation in differing directions.",
"Any proper formulation will necessarily be phrased in terms of vector fields which capture (either implicitly or explicitly) this anisotropy.",
"The formulation to be used here is as follows: Lemma 4 Suppose $({\\mathcal {M}},{\\mathcal {F}})$ is a compact geometric function system and let $N$ be any positive integer.",
"Then for any finite positive measure $\\mu $ on ${\\mathcal {M}}$ absolutely continuous with respect to Lebesgue measure and any measurable set $E \\subset {\\mathcal {M}}$ of positive measure, there is an open set $U \\subset {\\mathcal {M}}$ , a family of smooth vector fields $\\lbrace X_{j,i}\\rbrace _{j,i}$ with $j \\in \\lbrace 1,\\ldots ,N\\rbrace $ and $i \\in \\lbrace 1,\\ldots ,d\\rbrace $ and a measurable set $E^{\\prime } \\subset E \\cap U$ such that the following are true with implicit constants depending only on the pair $({\\mathcal {M}},{\\mathcal {F}})$ and the integer $N$ : The subset $E^{\\prime } \\subset E \\cap U$ satisfies $\\mu (E^{\\prime }) \\gtrsim \\mu (E)$ .",
"The vector fields $X_{j,i}$ satisfy $\\inf _{p \\in E^{\\prime }} \\left.",
"\\mu ( X_{j,1} \\wedge \\cdots \\wedge X_{j,d}) \\right|_p \\gtrsim \\mu (E)$ and $ X_{j,i} = \\sum _{i^{\\prime } = 1}^d c_{j,i,i^{\\prime }} X_{j-1,i^{\\prime }} $ with $|c_{j,i,i^{\\prime }}| \\lesssim 1$ for each $j \\in \\lbrace 2,\\ldots ,N\\rbrace $ and each $i,i^{\\prime } \\in \\lbrace 1,\\ldots ,d\\rbrace $ .",
"Here $\\mu (X_1 \\wedge \\cdots \\wedge X_d)$ equals the volume of the parallelepiped generated by $X_1,\\ldots ,X_d$ as measured by $\\mu $ , which more formally is defined to equal Radon-Nykodym derivative $d \\mu / dt$ with respect to some coordinate system $(t_1,\\ldots ,t_d)$ times the absolute value of the determinant of the matrix ${\\mathbf {X}}$ with entries ${\\mathbf {X}}_{ij} = dt_i(X_j)$ .",
"For any indices $i_1,\\ldots ,i_N \\in \\lbrace 1,\\ldots ,d\\rbrace $ , $ \\sup _{ p \\in E^{\\prime }} \\left| X_{N,i_N} \\cdots X_{1,i_1} f(p) \\right| \\lesssim \\frac{1}{\\mu (E)} \\int _E |f| d \\mu $ uniformly for all $f \\in {\\mathcal {F}}$ .",
"By induction (the base case of which is taken to be Lemma REF ), for a given measurable set $E \\subset {\\mathcal {M}}$ of positive measure, we may assume that there exist a nested family of open sets ${\\mathcal {M}}=: U_0 \\supset U_1 \\supset U_2 \\supset \\cdots \\supset U_{N-1}$ and vector fields $\\lbrace X_{j,i}\\rbrace $ , $i=1,\\ldots ,d$ , defined on $U_j$ for each $j \\in \\lbrace 1,\\ldots ,N-1\\rbrace $ satisfying all the stated properties.",
"Next, let ${\\mathcal {F}}_0 := {\\mathcal {F}}$ and then take ${\\mathcal {F}}_j$ to be the vector space of real analytic functions on $U_j$ spanned by ${\\mathcal {F}}$ and all functions of the form $X_{j,i} f$ for $i = 1,\\ldots ,d$ and $f \\in {\\mathcal {F}}$ .",
"By construction, $\\mu (E \\cap U_{N-1}) \\gtrsim \\mu (E) > 0$ , so in particular, $ ||f||_{N-1} := \\frac{1}{\\mu (E \\cap U_{N-1})} \\int _{E \\cap U_{N-1}} |f| d \\mu $ will be a norm on ${\\mathcal {F}}_{N-1}$ .",
"Let $f_1,\\ldots ,f_k$ be a basis of ${\\mathcal {F}}_{N-1}$ given by applying Lemma REF to the unit sphere of $||\\cdot ||_{N-1}$ , let $\\mathcal {I}$ be the set of $d$ -tuples $\\beta := (\\beta _1,\\ldots ,\\beta _d)$ of indices satisfying $1 \\le \\beta _1 < \\beta _2 < \\cdots < \\beta _d \\le k$ , and let $V_\\beta $ be the open set $ \\left\\lbrace p \\in U_{N-1} \\ \\left| \\ \\left| \\left.",
"d f_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _d} \\right|_p \\right| > \\frac{1}{2} \\left| \\left.",
"d f_{\\beta _1^{\\prime }} \\wedge \\cdots \\wedge df_{\\beta _d}^{\\prime } \\right|_p \\right| \\ \\forall \\beta ^{\\prime } \\in {\\mathcal {I}} \\setminus \\lbrace \\beta \\rbrace \\right.",
"\\right\\rbrace .",
"$ Because the cardinality of $\\mathcal {I}$ is bounded by a constant depending only on $N$ and the dimensions of ${\\mathcal {F}}$ and ${\\mathcal {M}}$ , there is at least one $\\beta $ such that $\\mu (E \\cap V_\\beta ) \\gtrsim \\mu (E \\cap U_{N-1})$ , where the implicit constant may simply be taken to be $(\\# \\Lambda )^{-1}$ .",
"If we now define the vector field $X_{N,i}$ on the set $U_N := V_\\beta $ to equal $ X_{N,i} f := \\frac{ d f_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _{i-1}} \\wedge df \\wedge df_{\\beta _{i+1}} \\wedge \\cdots \\wedge df_{\\beta _d}}{df_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _d}}, $ then it must be the case that $ X_{N,i} f(p) = \\sum _{i=1}^k c_i \\frac{ \\left.",
"d f_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _{i-1}} \\wedge df_i \\wedge df_{\\beta _{i+1}} \\wedge \\cdots \\wedge df_{\\beta _d} \\right|_p }{ \\left.",
"df_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _d} \\right|_p } $ for constants $c_i$ satisfying $|c_i| \\le ||f||_{N-1}$ .",
"Since the ratio is bounded above by 2 on $U_N$ , it follows that $ \\sup _{p \\in U_N} |X_{N,i} f(p)| \\le \\frac{2k}{\\mu (E \\cap U_{N-1})} \\int _{E \\cap U_{N-1}} |f| d \\mu $ for all $f \\in {\\mathcal {F}}_{N-1}$ .",
"Since $\\mu (E \\cap U_{N-1}) \\gtrsim \\mu (E)$ , it follows by induction that $ \\sup _{p \\in U_N} |X_{N,i_N} \\cdots X_{1,i_1} f(p)| \\lesssim \\frac{1}{\\mu (E)} \\int _{E} |f| d \\mu $ for all $f \\in {\\mathcal {F}}$ .",
"This establishes (REF ) for any set $E^{\\prime } \\subset E \\cap U_N$ .",
"Next observe that each vector field $X_{N,i}$ is locally a coordinate vector field relative to the coordinate functions $(f_{\\beta _1},\\ldots ,f_{\\beta _d}) \\in {\\mathcal {F}}_{N-1}^d$ such that the average value of $|f_{\\beta _i}|$ on $E \\cap U_{N-1}$ is 1.",
"By induction, we may assume the corresponding fact is true for the vector fields $X_{N-1,i}$ .",
"In particular, if $(g_{\\alpha _1},\\ldots ,g_{\\alpha _n})$ are the coordinate functions for the vector fields $X_{N-1,i}$ , then it follows that $ X_{N,i} = \\sum _{i^{\\prime }=1}^d (X_{N,i} g_{\\alpha _{i^{\\prime }}}) X_{N-1,i^{\\prime }}.",
"$ By the derivative estimate (REF ), assuming $N \\ge 2$ , $\\sup _{p \\in U_N} |X_{N,i} g_{\\alpha _{i^{\\prime }}}(p)| & \\le \\frac{2 \\dim {\\mathcal {F}}_{N-1}}{\\mu (E \\cap U_{N-1})} \\int _{U_{N-1} \\cap E} |g_{\\alpha _{i^{\\prime }}}| d \\mu \\\\& \\lesssim \\frac{1}{\\mu (E \\cap U_{N-2})} \\int _{E \\cap U_{N-2}} |g_{\\alpha _{i^{\\prime }}}| d \\mu \\lesssim 1,$ which is exactly the bound on the coefficients $c_{j,i,i^{\\prime }}$ claimed for (REF ).",
"Lastly, the quantity $\\mu (X_{j,1} \\wedge \\cdots \\wedge X_{j,d})$ must be estimated.",
"Observe that $\\mu (X_{j,1} \\wedge \\cdots \\wedge X_{j,d})$ is exactly the Radon-Nykodym derivative of $\\mu $ with respect to Lebesgue measure in coordinates given by $f_{\\beta _1},\\ldots ,f_{\\beta _d}$ .",
"This implies that meaning that $ \\int _{E^{\\prime }} \\left[ \\mu ( X_{N,1} \\wedge \\cdots \\wedge X_{N,d}) \\right]^{-1} d \\mu = \\int _{E^{\\prime }} | d f_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _d}|.",
"$ Because the functions are real analytic, we know that there is a finite number $M$ independent of the choice of the functions $f_{\\beta _i}$ such that the system $f_{\\beta _i}(p) = c_i$ has at most $M$ nondegenerate solutions (at which the Jacobian is nonzero).",
"For polynomial functions, this is a simple consequence of Bézout's Theorem.",
"In our case, however, even if the original function system ${\\mathcal {F}}_0$ consists only of polynomial functions, the definition of the $X_{j,i}$ lead naturally to the inclusion of certain rational functions in ${\\mathcal {F}}_i$ , at which point there is not much additional difficulty in going to the more general context of real analytic functions.",
"The algebraic argument is given in Section and for now may be safely postponed.",
"Assuming the existence of such an $M$ depending only on the geometric function system, by the change of variables formula, $ \\int _{E^{\\prime }} |d f_{\\beta _1} \\wedge \\cdots \\wedge df_{\\beta _d}| \\le M \\prod _{j=1}^d | f_{\\beta _i}(E^{\\prime })|, $ where $|f_{\\beta _i}(E^{\\prime })|$ refers to the one-dimensional Lebesgue measure of the image of $E^{\\prime }$ via $f_{\\beta _i}$ .",
"Because the average value of $f_{\\beta _i}$ on $E \\cap U_{N-1}$ is 1, by Lemma REF , there is a subset $E^{\\prime } \\subset E \\cap U_{N}$ with $\\mu (E^{\\prime }) \\gtrsim \\mu (E \\cap U_{N}) \\gtrsim \\mu (E)$ such that $ \\sup _{p \\in E^{\\prime }} |f_{\\beta _i}(p)| \\lesssim \\frac{1}{\\mu (E \\cap U_N)} \\int _{E \\cap U_N} |f_{\\beta _i}| d \\mu \\lesssim \\frac{1}{\\mu (E \\cap U_{N-1})} \\int _{E \\cap U_{N-1}} |f_{\\beta _i}| d \\mu \\lesssim 1 $ for each $i$ , which implies that $|f_{\\beta _i}(E^{\\prime })| \\lesssim 1$ for each $i$ as well.",
"For this set $E^{\\prime }$ , it follows that $ \\int _{E^{\\prime }} \\left[ \\mu ( X_{N,1} \\wedge \\cdots \\wedge X_{N,d}) \\right]^{-1} d \\mu \\lesssim 1.",
"$ Further restricting $E^{\\prime }$ using Tchebyshev's inequality, we may assume that $ \\inf _{E^{\\prime }} \\mu (X_{N,1} \\wedge \\cdots \\wedge X_{N,n}) \\gtrsim \\mu (E).",
"$ This completes the proof."
],
[
"Proof of Part ",
"We now return to the proof of Part REF of Theorem REF .",
"The proof combines Lemma REF with the geometric framework introduced in Section REF .",
"Suppose that $\\mathcal {M}$ is a real analytic manifold of dimension $d$ and that $f$ is a real analytic immersion of ${\\mathcal {M}}$ into ${\\mathbb {R}}^n$ in such a way that the component functions $f_1,\\ldots ,f_n$ of the immersion together with the constant function belong to some compact geometric function system $({\\mathcal {M}},{\\mathcal {F}})$ .",
"Fix any compact convex set $K \\in \\mathcal {K}_n$ , let $E := f^{-1}(K)$ , and let $p_0 \\in E$ .",
"Now the integral $ I(E) := \\frac{1}{\\mu _{\\!_{\\mathcal {A}}}(E)^n} \\int _{E^n} \\!",
"\\!",
"\\!",
"|\\det ( f(p_1) - f(p_0),\\ldots ,f(p_n) - f(p_0))| d \\mu _{\\!_{\\mathcal {A}}}(p_1) \\cdots d \\mu _{\\!_{\\mathcal {A}}}(p_n) $ must be bounded above by $n!",
"|K|$ since the integrand equals $n!$ times the volume of the simplex generated by $f(p_0),\\ldots ,f(p_n)$ , which has volume bounded by $|K|$ since each point belongs to $K$ and $K$ is convex.",
"In this case Lemma REF can be applied to each integral iteratively to prove a lower bound for the functional.",
"Specifically, the lemma is applied to the innermost integral, which is then replaced by a supremum over some set $E^{\\prime }$ of some derivative in the parameter $p_1$ .",
"As a result, the lemma establishes that $I(E) \\gtrsim \\sup _{(p_1,\\ldots ,p_n) \\in (E^{\\prime })^n} | \\det (& X_{1,i_{(1,1)}} f(p_1) \\wedge \\cdots \\wedge \\\\& X_{\\kappa _j,i_{(j,1)}} \\cdots X_{1,i_{(j,\\kappa _j)}} f(p_j) \\wedge \\cdots \\wedge \\\\& X_{\\kappa _n,i_{(n,1)}} \\cdots X_{1,i_{(n,\\kappa _n)}} f(p_n)) |$ for any choice of indices $i_{\\lambda }$ for $\\lambda \\in \\Lambda _{d,n}$ .",
"Next replace the supremum over $(p_1,\\ldots ,p_n) \\in E^{\\prime n}$ by a supremum over $p \\in E^{\\prime }$ assuming $p_1 = \\cdots = p_n = p$ .",
"It is also advantageous to use only vector fields $X_{\\kappa _n,i^{\\prime }}$ rather than using any $X_{j,i}$ for $j < \\kappa _n$ .",
"Thanks to (REF ) it must be the case that $\\det & \\left(X_{\\kappa _n,i_{(1,1)}^{\\prime }} f(p) \\wedge \\cdots \\wedge X_{\\kappa _n,i_{(n,1)}} \\cdots X_{\\kappa _n,i_{(n,\\kappa _n)}^{\\prime }} f(p) \\right) \\\\& = \\sum _{i} c_{ii^{\\prime }} \\det \\left(X_{1,i_{(1,1)}} f(p) \\wedge \\cdots \\wedge X_{\\kappa _n,i_{(n,1)}} \\cdots X_{1,i_{(n,\\kappa _n)}} f(p) \\right)$ with coefficients $|c_{ii^{\\prime }}| \\lesssim 1$ where the sum is over all possible choices of the indices $i_\\lambda $ .",
"This identity holds because the change of basis formula may be simply substituted term-by-term in the left-hand side of the equation; any terms in which the coefficients of the change of basis happened to be differentiated by some subsequent vector field would ultimately have determinant zero since (assuming the column in which the derivative appears is column $j$ ), the number of derivatives acting directly on $f$ would be strictly less than $\\kappa _j$ , which means that column $j$ and all preceding columns would be linearly dependent.",
"Therefore by the triangle inequality, it must be the case that $I(E) \\gtrsim \\sup _{p \\in E^{\\prime }} | \\det (& X_{\\kappa _n,i_{(1,1)}^{\\prime }} f(p) \\wedge \\cdots \\wedge X_{\\kappa _n,i_{(n,1)}^{\\prime }} \\cdots X_{\\kappa _n,i_{(n,\\kappa _n)}} f(p)) |$ uniformly for any choice of $i^{\\prime }_\\lambda $ .",
"Taking an $\\ell ^2$ norm over all such choices and invoking the definition (REF ) of the density $\\mu _{\\!_{\\mathcal {A}}}$ $I(E) \\gtrsim \\sup _{p \\in E^{\\prime }} \\left[ \\left.",
"\\mu _{\\!_{\\mathcal {A}}}(X_{\\kappa _n,1},\\ldots ,X_{\\kappa _n,d}) \\right|_p \\right]^\\frac{Q}{d}.$ To conclude, observe that for the measure $\\mu _{\\!_{\\mathcal {A}}}$ , the quantity $\\mu _{\\!_{\\mathcal {A}}}(X_{\\kappa _n,1},\\ldots ,X_{\\kappa _n,d})$ exactly equals the geometric density $\\mu _{\\!_{\\mathcal {A}}}(X_{\\kappa _n,n} \\wedge \\cdots \\wedge X_{\\kappa _n,n})$ bounded below by Lemma REF .",
"Therefore $ |K| \\gtrsim I(E)| \\gtrsim \\left[ \\left| \\mu _{\\!_{\\mathcal {A}}}(X_{\\kappa _n,1},\\ldots ,X_{\\kappa _n,n} ) \\right|_p \\right]^\\frac{Q}{d} \\gtrsim (\\mu _{\\!_{\\mathcal {A}}}(E))^{\\frac{Q}{d}} $ uniformly in $K$ and $E$ .",
"This is exactly Part REF of Theorem REF ."
],
[
"Proof of Part ",
"The final piece of Theorem REF is to show that $\\alpha = d/Q$ is a nontrivial exponent in the sense that there is always some submanifold $\\mathcal {M}$ of dimension $d$ in ${\\mathbb {R}}^n$ for which the Oberlin condition (REF ) is satisfied with exponent $\\alpha $ for some nonzero measure on $\\mathcal {M}$ .",
"In fact, it suffices to consider the case when $\\mathcal {M}$ is essentially ${\\mathbb {R}}^d$ and the immersion $f$ is a polynomial embedding.",
"In principle, one needs only to show that $\\mu _{\\!_{\\mathcal {A}}}$ is nonzero in some such case (since the arguments of the previous section apply to show that (REF ) holds locally on $\\mathcal {M}$ , and then a scaling argument establishes the same result globally).",
"In light of the estimate (REF ) for the value of $\\mu _{\\!_{\\mathcal {A}}}$ , the existence of submanifolds with nonzero affine measure is exactly equivalent to the existence of submanifolds for which the affine curvature tensor does not belong to the nullcone of the space of $Q$ -linear covariant tensors.",
"The nullcone is difficult if not impossible to describe explicitly, and determining whether a tensor of the very special form (REF ) belongs to it or not turns out to be a significant challenge.",
"The key observation is that it so happens that critical points (as a function of $M$ ) in the infimum definition (REF ) of $\\mu _{\\!_{\\mathcal {A}}}$ must be points at which the infimum is attained.",
"This will be the main observation to be exploited; a secondary observation, encapsulated in the following lemma, allows one to simplify the structure of the affine curvature tensor $\\mathcal {A}_p$ at the expense of infimizing over a larger group: Lemma 5 Let $A$ be a real $n \\times m$ matrix where $m \\ge n$ , and let $[A]_{i_1 \\cdots i_n}$ be the $n \\times n$ matrix formed by combining columns $i_1,\\ldots ,i_n$ into a square matrix, i.e., the $(j,k)$ entry of this matrix is $A_{ji_k}$ .",
"Then $ \\sum _{i_1,\\ldots ,i_n=1}^m |\\det [A]_{i_1 \\cdots i_n}|^2 = \\frac{n!",
"}{n^n} \\left[ \\inf _{M \\in {\\mathrm {SL}}(n,{\\mathbb {R}})} \\sum _{j=1}^n \\sum _{i=1}^m \\left| \\sum _{k=1}^n M_{jk} A_{ki} \\right|^2 \\right]^n.",
"$ First observe that both $ A \\mapsto \\sum _{i_1,\\ldots ,i_n = 1}^m |\\det [A]_{i_1 \\cdots i_n} |^2 \\mbox{ and } A \\mapsto \\sum _{j=1}^n \\sum _{i=1}^m |A_{ji}|^2 $ are invariant under the action of $O(n,{\\mathbb {R}})$ on the columns of $A$ as well as the action of $O(m,{\\mathbb {R}})$ on the rows of $A$ (the former assertion is relatively simple; the latter case rests on the observation that tensor products of elements of an orthonormal basis generate an orthonormal basis on the space of tensors in a natural way).",
"In particular, this means that we may, by the singular value decomposition, assume without loss of generality that $ A_{ji} = \\sigma _j \\delta _{ji} $ where $\\sigma _j$ is the $j$ -th singular value of $A$ .",
"Thus $ \\sum _{i_1,\\ldots ,i_n=1}^m |\\det [A]_{i_1 \\cdots i_n}|^2 = n!",
"\\sigma _1^2 \\cdots \\sigma _n^2 \\mbox{ and } \\sum _{j=1}^n \\sum _{i=1}^m |A_{ji}|^2 = \\sigma _1^2 + \\cdots + \\sigma _n^2.",
"$ By the AM-GM inequality, $ \\frac{1}{n!}",
"\\sum _{i_1,\\ldots ,i_n=1}^m |\\det [A]_{i_1 \\cdots i_n}|^2 \\le \\left[ \\frac{1}{n} \\sum _{j=1}^n \\sum _{i=1}^m |A_{ji}|^2 \\right]^n $ with equality if and only if the singular values of $A$ are all equal.",
"Now multiplication of $A$ on the left by a matrix $M \\in {\\mathrm {SL}}(n,{\\mathbb {R}})$ preserves the left-hand side but not necessarily the right-hand side; taking an infimum of the right-hand side over all $M$ gives that $ \\sum _{i_1=1,\\ldots ,i_n=1}^m |\\det [A]_{i_1 \\cdots i_n}|^2 \\le \\frac{n!",
"}{n^n} \\left[ \\inf _{M \\in {\\mathrm {SL}}(n,{\\mathbb {R}})} \\sum _{j=1}^n \\sum _{i=1}^m \\left| \\sum _{k=1}^n M_{jk} A_{ki} \\right|^2 \\right]^n.",
"$ To show equality, assume without loss of generality that $A$ is diagonal in the standard basis of ${\\mathbb {R}}^{n \\times m}$ and let $M$ be the diagonal matrix such that $M_{ii} := \\sigma _i^{-1} (\\sigma _1 \\cdots \\sigma _n)^{1/n}$ assuming none of the singular values are zero.",
"In this case, $MA$ has all diagonal entries equal, and consequently (REF ) holds with equality when $A$ is replaced by $MA$ , giving equality in (REF ) as well.",
"If, on the other hand, some singular value $\\sigma _{i^{\\prime }}$ of $A$ is zero, let $M^{(t)}$ be another diagonal matrix such that $M_{ii}^{(t)} = t$ for all entries $i \\ne i^{\\prime }$ and let $M_{i^{\\prime } i^{\\prime }}^{(t)} = t^{-n+1}$ .",
"Then for $t > 0$ , $M^{(t)} \\in SL(n,{\\mathbb {R}})$ and $ \\lim _{t \\rightarrow 0^+} \\left[ \\sum _{j=1}^n \\sum _{i=1}^m \\left| \\sum _{k=1}^n M_{jk}^{(t)} A_{ki} \\right|^2 \\right]^m = \\lim _{t \\rightarrow 0^+} \\left[ \\sum _{i \\ne i^{\\prime }} t^2 \\sigma _i^2 \\right]^n = 0 $ so (REF ) holds with equality again in this case as well.",
"In showing that for any pair $(d,n)$ with $1 \\le d < n$ , there is a $d$ -dimensional submanifold of ${\\mathbb {R}}^n$ for which the corresponding measure $\\mu _{\\!_{\\mathcal {A}}}$ is not trivial, it is clear from the definition of $\\mathcal {A}$ and the pigeonhole principle that nontriviality of $\\mu _{\\!_{\\mathcal {A}}}$ requires that the vectors $\\lbrace X^\\alpha f(p)\\rbrace _{1 \\le |\\alpha | < \\kappa _n }$ be linearly independent for any system of coordinate vectors $X_1,\\ldots ,X_n$ .",
"Knowing a priori that this must be the case, it is possible to essentially factor $\\mathcal {A}$ in such a way that only the highest-order behavior of $f$ at $p$ is relevant for purposes of calculation.",
"To that end, fix any $p$ and let $V_p$ be the $n_V$ -dimensional subspace of ${\\mathbb {R}}^n$ spanned by the vectors $X^{\\alpha } f(p)$ as $\\alpha $ ranges over all multiindices $\\alpha $ with $1 \\le |\\alpha | < \\kappa _{n}$ (where $\\kappa _n$ is the highest order of differentiation that one finds in any column of $\\mathcal {A}$ ), and let $W_p$ be any $n_W$ -dimensional subspace of ${\\mathbb {R}}^n$ chosen so that $ V_p \\cap W_p = \\lbrace 0\\rbrace \\mbox{ and } V_p + W_p = {\\mathbb {R}}^n.",
"$ It is then possible to uniquely and smoothly write $f$ as a sum $f = f_V + f_W$ such that $f_V$ takes values in $V_p$ and $f_W$ takes values in $W_p$ .",
"By definition of $V_p$ , it must also be the case that, modulo a constant vector, $f_W$ vanishes to order $\\kappa _n$ at the point $p \\in {\\mathcal {M}}$ .",
"Next fix determinant functionals on $V_p$ and $W_p$ compatible with the determinant on ${\\mathbb {R}}^n$ , meaning that $ \\det (v_1 \\wedge \\cdots \\wedge v_{n_V} \\wedge w_1 \\wedge \\cdots \\wedge w_{n_W}) = \\det _V(v_1 \\wedge \\cdots \\wedge v_{n_V}) \\det _W(w_1 \\wedge \\cdots \\wedge w_{n_W}) $ when $\\lbrace v_1,\\ldots ,v_{n_V}\\rbrace $ and $\\lbrace w_1,\\ldots ,w_{n_W}\\rbrace $ are bases of $V$ and $W$ , respectively.",
"By the multilinearity of the determinant on ${\\mathbb {R}}^n$ , the tensor $\\mathcal {A}_p$ factors at $p$ into pieces that depend on $f_V$ and $f_W$ separately, namely $ {\\mathcal {A}}_p & ( (X_\\lambda )_{\\lambda \\in \\Lambda _{d,n}} ) = \\det _V \\left(X_{(1,1)} f_V(p) \\wedge \\cdots \\wedge X_{(j_*,1)} \\cdots X_{(j_*,\\kappa _{n}-1)} f_V(p) \\right) \\\\ & \\times \\det _W\\left(X_{(j_*+1,1)} \\cdots X_{(j_*+1,\\kappa _{n})} f_W(p) \\wedge \\cdots \\wedge X_{(n,1)} \\cdots X_{(n-1,\\kappa _{n})} f_W(p)\\right).$ where $j_*$ is the largest index for which $\\kappa _{j_*} < \\kappa _{n} $ .",
"Splitting the index set $\\Lambda _{d,n}$ into subsets $\\Lambda _V$ and $\\Lambda _W$ for those indices which appear in the first and second terms of this factorization, respectively, it follows that the terms in the factorization are themselves tensors (which up to a normalization constant, are defined intrinsically and smoothly in a neighborhood of the chosen point $p$ ) which will be called ${\\mathcal {A}}_{V_p} ((X_\\lambda )_{\\lambda \\in \\Lambda _V})$ and ${\\mathcal {A}}_{W_p} ((X_\\lambda )_{\\lambda \\in \\Lambda _W})$ , respectively, so that $ {\\mathcal {A}}_p( (X_\\lambda )_{\\lambda \\in \\Lambda } ) = {\\mathcal {A}}_{V_p} ((X_\\lambda )_{\\lambda \\in \\Lambda _V}) {\\mathcal {A}}_{W_p} ((X_\\lambda )_{\\lambda \\in \\Lambda _W}).",
"$ The fundamental consequence of the factorization (REF ) is that it allows one to fully separate the contributions of the “lower order” parts $\\mathcal {A}_{V_p}$ and the “higher order” parts $\\mathcal {A}_{W_p}$ .",
"In the former case, it turns out that $\\mathcal {A}_{V_p}$ expressed in coordinates with respect to the basis $X_1,\\ldots ,X_d$ is actually invariant under the representation $\\rho _\\cdot ^X$ defined by (REF ).",
"This is because the vector space of differential operators generated by $X^{\\alpha }$ for $1 \\le |\\alpha | < \\kappa _n$ is invariant under the action of $\\rho _M^X$ , so there must be a matrix $[\\rho _M^X]$ which acts on the column space spanned by $X_{(1,1)} f(p),\\ldots ,X_{(j_*,1)} \\cdots X_{(j_*,\\kappa _{j_*})} f(p) $ which is equal to the action of $\\rho _M^X$ on this space.",
"For every $M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})$ , the matrix $[\\rho _M^X]$ must have determinant of magnitude 1.",
"This follows by symmetry when $M$ is a diagonal matrix (since $\\mathcal {A}_p((X_\\lambda )_{\\lambda \\in \\Lambda _V}$ will necessarily vanish by the pigeonhole principle unless each vector field $X_j$ occurs an equal number of times, i.e., unless the number of $\\lambda \\in \\Lambda _V$ for which $X_\\lambda = X_j$ is a constant function of $j$ ).",
"Likewise $|\\det [\\rho ^X_M] | = 1$ when $M$ is an orthogonal matrix since continuity of the map $M \\mapsto |\\det [\\rho ^X_M] |$ together with the identity $|\\det [\\rho _{MN}^X]| = |\\det [\\rho _{M}^X]| \\cdot |\\det [\\rho _N^X]|$ shows that if the maximum or minimum values of $|\\det [\\rho _M^X]|$ as a function on the orthogonal group were different from 1, they could not be attained (since one could always use the group law to find a new othogonal matrix with strictly greater or smaller absolute determinant).",
"However, any $M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})$ can always be factored as a product of a diagonal and orthogonal matrix, so $|\\det [\\rho _M^X]| = 1$ must hold in all cases.",
"Since $\\mathcal {A}_{V_p}$ is invariant under $\\rho _M^X$ , one needs merely to show that there is some lower-order part $f_V$ for which it is not identically zero.",
"In this case, $ f_V(t_1,\\ldots ,t_d) := (t^\\alpha )_{1 \\le |\\alpha | < \\kappa _n} $ (where we interpret the coordinates on the right-hand side as being relative to some choice of basis of $V$ ) suffices, since with respect to the standard coordinate vectors $\\partial _{t_i}$ the matrix of ${\\mathcal {A}}_{V_p}$ is seen to be lower triangular with nonzero diagonal entries.",
"Thus the problem is now fully reduced to the study of $\\mathcal {A}_{W_p}$ .",
"In this case, we will set $ f_W(t_1,\\ldots ,t_d) := (p_1(t),\\ldots ,p_m(t)) $ for some polynomials $p_1,\\ldots ,p_m$ which are homogeneous of degree $\\kappa _n$ , where $m$ is less than or equal to the dimension of the vector space of all such homogeneous polynomials.",
"By work of Richardson and Slodowy (which is the real analogue of ideas introduced by Kempf and Ness ) it suffices to show that there is a choice of $p_1,\\ldots ,p_m$ such that the map $ M \\mapsto || \\rho _M \\mathcal {A}_{W_p} ||^2 $ has a critical point (where from here forward, $\\rho $ and the norm $||\\cdot ||$ will be taken with respect to the standard coordinates $\\partial _{t_1},\\ldots ,\\partial _{t_d}$ ) since they showed that all critical points are points where the infimum over all $M \\in {\\mathrm {SL}}(d,{\\mathbb {R}})$ is actually attained.",
"Moreover, it suffices to show that such a critical point exists when $M$ is the identity.",
"By (REF ), the problem can be further reduced to showing that the function $ (N,M) \\mapsto \\sum _{k=1}^m \\sum _{j_1,\\ldots ,j_{\\kappa _n} = 1}^d \\left| \\sum _{\\ell =1}^m \\sum _{i_1,\\ldots ,i_{\\kappa _n}=1}^d N_{\\ell k} M_{i_1 j_1} \\cdots M_{i_{\\kappa _n} j_{\\kappa _n}} \\partial _{t_{i_1}} \\cdots \\partial _{t_{i_{\\kappa _n}}} p_\\ell (t) \\right|^2 $ has a critical point at the identity as a function of $(N,M) \\in {\\mathrm {SL}}(m,{\\mathbb {R}}) \\times {\\mathrm {SL}}(d,{\\mathbb {R}})$ for appropriate choice of $p_1,\\ldots ,p_m$ .",
"Differentiating in $N$ at the identity along some $E \\in \\mathfrak {sl}(m,{\\mathbb {R}})$ gives that $ 2 \\sum _{k=1}^m \\sum _{\\ell =1}^m E_{\\ell k} \\sum _{j_1,\\ldots ,j_{\\kappa _n}=1}^d \\partial _{t_{j_1}} \\cdots \\partial _{t_{j_{\\kappa _n}}} p_\\ell (t) \\partial _{t_{j_1}} \\cdots \\partial _{t_{j_{\\kappa _n}}} p_{k}(t) = 0$ for all traceless $m \\times m$ matrices $E$ .",
"A similar calculation differentiating $M$ ultimately gives that critical points are those which satisfy the system $\\sum _{i_1,\\ldots ,i_{\\kappa _n}} \\partial _{t_{i_1}} \\cdots \\partial _{t_{i_{\\kappa _n}}} p_\\ell (t) \\partial _{t_{i_1}} \\cdots \\partial _{t_{i_{\\kappa _n}}} p_{\\ell ^{\\prime }}(t) & = \\lambda _1 \\delta _{\\ell ,\\ell ^{\\prime }}, \\\\\\sum _{\\ell , i_2,\\ldots ,i_{\\kappa _n}} \\partial _{t_{j}} \\cdots \\partial _{t_{i_{\\kappa _n}}} p_\\ell (t) \\partial _{t_{j^{\\prime }}} \\cdots \\partial _{t_{i_{\\kappa _n}}} p_\\ell (t) & = \\lambda _2 \\delta _{j,j^{\\prime }} $ for some real numbers $\\lambda _1,\\lambda _2$ and all indices $j,j^{\\prime },\\ell ,\\ell ^{\\prime }$ .",
"At any such critical point, ${\\mathcal {A}}_{W_p}$ will be nonzero exactly when the constants $\\lambda _1$ and $\\lambda _2$ are nonzero.",
"To simplify matters somewhat, observe that for any real homogeneous polynomial $p(t) = \\sum _{|\\alpha | = k} c_\\alpha t^\\alpha $ of degree $k$ , $ ||p||_{k}^2 := \\sum _{i_1,\\ldots ,i_k} |\\partial _{t_{i_1}} \\cdots \\partial _{t_{i_k}} p(t)|^2 = \\sum _{|\\alpha | = k} k!",
"\\alpha !",
"|c_\\alpha |^2 $ since $\\partial ^\\beta _t t^\\alpha = \\alpha !",
"\\delta _{\\alpha ,\\beta }$ and for any multiindex $\\beta $ , there are $k!",
"/ \\beta !$ ways to write $\\partial ^\\beta _t$ as an iterated derivative $\\partial _{t_{i_1}} \\cdots \\partial _{t_{i_k}}$ .",
"By polarization, the norm $||\\cdot ||_k$ has an immediate corresponding inner product.",
"In this notation, (REF ) and () become $ \\sum _{i=1}^d \\left< \\partial _{t_i} p_\\ell , \\partial _{t_i} p_{\\ell ^{\\prime }} \\right>_{\\kappa _{n}-1} = \\lambda _1 \\delta _{\\ell ,\\ell ^{\\prime }} \\mbox{ and } \\sum _{\\ell =1}^m \\left< \\partial _{t_i} p_\\ell , \\partial _{t_{i^{\\prime }}} p_\\ell \\right>_{\\kappa _{n}-1} = \\lambda _2 \\delta _{i,i^{\\prime }}.",
"$ It is similarly elementary to compute inner products of monomials: $ \\left< \\partial _{t_i} t^\\alpha , \\partial _{t_{i^{\\prime }}} t^\\beta \\right>_{\\kappa _n - 1} = \\alpha _i \\beta _{i^{\\prime }} (\\kappa _n-1)!",
"(\\alpha -e_i)!",
"\\delta _{\\alpha -e_i,\\beta -e_{i^{\\prime }}} $ where $e_i$ is the multiindex which is zero except in position $i$ , where it equals 1 (and note that the right-hand side of (REF ) is to be interpreted as zero if $\\alpha _i = 0$ or $\\beta _{i^{\\prime }} = 0$ ).",
"For simplicity, fix $\\kappa := \\kappa _n$ .",
"To build a nontrivial $f_W$ , we will chose each polynomial $p_1,\\ldots ,p_m$ to have one of two types.",
"The first type is of the form $ p_\\ell (t) := \\frac{t^{\\alpha }}{\\sqrt{\\alpha !}}",
"$ for some multiindex with $|\\alpha | = \\kappa $ which is not a pure $\\kappa $ power (i.e., $t^\\alpha \\ne t^\\kappa _i$ for any $i$ ).",
"We impose a compatibility condition that if $p_j(t) = t^{\\alpha } / \\sqrt{\\alpha !",
"}$ for some $j$ , then for every cyclic permutation $\\alpha ^{\\prime }$ of $\\alpha $ , there is another index $j^{\\prime }$ such that $p_{j^{\\prime }}(t) = t^{\\alpha ^{\\prime }}/\\sqrt{\\alpha ^{\\prime }!",
"}$ .",
"Assuming that most $p_j$ are of this form, we additionally allow for up to $d$ more polynomials which depend only on the pure $\\kappa $ power monomials $t_i^\\kappa $ as follows.",
"Suppose that $\\lbrace \\varphi _j\\rbrace _{j=1,\\ldots ,d}$ is a uniform, normalized tight frame (UNTF) on ${\\mathbb {R}}^{d_0}$ for some $d_0 \\le d$ , which means that $ \\sum _{j=1}^{d} |\\left< v,\\varphi _j \\right>|^2 = ||v||^2 \\mbox{ for all } v \\in {\\mathbb {R}}^{d_0} \\mbox{ and } ||\\varphi _j||^2 = \\frac{d_0}{d}, \\ j=1,\\ldots ,d. $ Such collections of vectors are guaranteed to exist for any $d_0 \\le d$ (see for existence; a general algorithm based on Theorem 7 of which can convert a NTF to a UNTF is also known ).",
"With such a UNTF, one may optionally chose to add exactly $d_0$ polynomials to the collection constituting $f_W$ provided these new polynomials have the form $ \\sum _{j=1}^d \\frac{t_j^\\kappa }{\\sqrt{\\kappa !}}",
"\\varphi _{j,k} \\mbox{ for } k=1,\\ldots ,d_0, $ where $\\varphi _{j,k}$ is the $k$ -th coordinate of $\\varphi _j$ in the standard basis.",
"(Note that these optional UNTF-generated polynomials can be added for at most a single choice of UNTF.)",
"To verify the first condition of (REF ), notice that when $\\ell \\ne \\ell ^{\\prime }$ and one of $\\ell $ or $\\ell ^{\\prime }$ correspond to indices of a monomial-type polynomial, every inner product in the sum must be zero because $\\partial _{t_i} p_\\ell $ and $\\partial _{t_{i}} p_{\\ell ^{\\prime }}$ have no monomials in common and are consequently orthogonal.",
"If $\\ell = \\ell ^{\\prime }$ and the polynomial $p_\\ell $ is monomial type, then $ \\sum _{i=1}^d \\left< \\partial _{t_i} \\frac{t^\\alpha }{\\sqrt{\\alpha !",
"}}, \\partial _{t_{i}} \\frac{t^\\alpha }{\\sqrt{\\alpha !}}",
"\\right>_{\\kappa -1} = \\sum _{i=1}^d \\frac{(\\kappa -1)!",
"\\alpha _i^2 (\\alpha -e_i)!",
"}{\\alpha !}",
"\\delta _{\\alpha _i > 0} = \\kappa !.$ If, in the final case, both $l$ and $l^{\\prime }$ arise from UNTF terms, the left-hand side of the first equality of (REF ) must equal $ \\sum _{i=1}^d \\frac{1}{\\kappa !}",
"\\varphi _{i,\\ell } \\varphi _{i,\\ell ^{\\prime }} \\left< \\partial _{t_i} t_i^\\kappa , \\partial _{t_i} t_i^\\kappa \\right>_{\\kappa -1} = \\sum _{i=1}^d \\frac{1}{\\kappa !}",
"\\varphi _{i,\\ell } \\varphi _{i,\\ell ^{\\prime }} (\\kappa !",
")^2 = \\kappa !",
"\\delta _{\\ell ,\\ell ^{\\prime }} $ since the $\\varphi _j$ are a normalized tight frame.",
"As for the second condition of (REF ), by (REF ), the polynomials $p_\\ell $ of monomial type have norms that equal $ \\left< \\partial _{t_i} \\frac{t^\\alpha }{\\sqrt{\\alpha !",
"}}, \\partial _{t_{i^{\\prime }}} \\frac{t^\\alpha }{\\sqrt{\\alpha !}}",
"\\right>_{\\kappa -1} = (\\kappa -1)!",
"\\alpha !",
"\\delta _{i,i^{\\prime }} \\delta _{\\alpha _i > 0}.",
"$ Summing over all monomial-type polynomials gives a matrix (as a function of $i$ and $i^{\\prime }$ ) which is a multiple of the identity: simply by symmetry, any monomial appearing in the sum also appears with all its cyclic permutations, so all diagonal entries must be equal.",
"As for the terms of the sum which arise from UNTF polynomials, $ \\left< \\partial _{t_i} \\sum _{j=1}^d \\frac{t_j^\\kappa }{\\sqrt{\\kappa !}}",
"\\varphi _{j,k}, \\partial _{t_{i^{\\prime }}} \\sum _{j=1}^d \\frac{t_j^\\kappa }{\\sqrt{\\kappa !}}",
"\\varphi _{j,k} \\right>_{\\kappa -1} = \\frac{|\\varphi _{i,k}|^2}{\\kappa !}",
"\\delta _{i,i^{\\prime }} ||t_i^\\kappa ||_\\kappa ^2 = \\kappa !",
"\\delta _{i,i^{\\prime }} |\\varphi _{i,k}|^2, $ which again sums to a multiple of the identity since, after the sum, the $i$ -th diagonal entry equals $||\\varphi _i||^2$ .",
"By the results of Richardson and Slodowy , it is possible to find a nondegenerate highest-order part $f_W$ of the embedding $f$ provided that the dimension $m$ of this highest order part corresponds to the cardinality of a collection of polynomials of the type considered above: monomial-type polynomials for a set of monomials excluding pure powers and invariant under cyclic permutations together with $d_0$ UNTF-type polynomials for any $d_0 \\in \\lbrace 0,\\ldots ,d\\rbrace $ .",
"To see that any integer $m$ between 1 and the total number of monomials of degree $\\kappa $ (inclusive) can admit such a collection, observe that the possible cardinalities of just the collection of monomial-type polynomials range—with gaps, of course—from 0 up to the total number of monomials minus $d$ .",
"The size of any gap (i.e., consecutive values of $m$ which are not cardinalities of an admissible set of monomials) must be strictly less than $d$ for the simple reason that no equivalence class of monomials modulo cyclic permutation has cardinality greater than $d$ .",
"In other words, if any non-pure power polynomials happen not already to belong to the collection, including any such monomial together with its cyclic permutations (which is a total of $d$ or fewer new monomials) will again make a larger admissible set.",
"Since the gaps are size strictly less than $d$ and since $d_0$ can be chosen as desired in $\\lbrace d,\\ldots ,d\\rbrace $ combining both types of polynomials leads to a nondegenerate measure $\\mu _{\\!_{\\mathcal {A}}}$ for any possible value of $m$ given the dimension $d$ ."
],
[
"Appendix: Uniform bounds on the number of solutions of real analytic systems of equations",
"We finish with a brief discussion of the problem of uniformly bounding the number of nondegenerate solutions to any system of equations that arises in a geometric function system.",
"The precise statement that is needed is that for arbitrary positive integers $d$ and $n$ (no longer retaining their previous definitions) when $f_1,\\ldots ,f_d$ are real analytic functions on a neighborhood of the unit cube $[0,1]^n$ , then any system of equations $(\\Phi _1(x),\\ldots ,\\Phi _n(x)) = (y_1,\\ldots ,y_n)$ must have bounded nondegenerate multiplicity when the functions $\\Phi _i$ are rational functions of the $f_i$ and finitely many derivatives of each $f_i$ .",
"In other words, the number of solutions in $[0,1]^n$ at which the Jacobian is nonzero is bounded above by a constant that depends only on the functions $f_i$ and the complexity of the system, in this case meaning the degrees of the numerators and denominators and the order of the highest derivative of an $f_i$ .",
"To see this, let $S$ be the Cartesian product of $\\lbrace 1,\\ldots ,d\\rbrace $ with the set of multiindices $\\alpha := (\\alpha _1,\\ldots ,\\alpha _n)$ such that $|\\alpha | := \\alpha _1 + \\cdots + \\alpha _n \\le N$ .",
"For any $\\beta $ which is a multiindex on $S$ (i.e., a map from $S$ into nonnegative integers), we define $s^\\beta :=\\prod _{(j,\\alpha ) \\in S} (s_{j,\\alpha })^{\\beta _{j,\\alpha }}$ for every $s \\in {\\mathbb {R}}^S$ in analogy with the usual notation.",
"Lastly, define $P$ be the Cartesian product of $\\lbrace 1,\\ldots ,n\\rbrace $ and multiindices $\\beta $ of size at most $N$ on the set $S$ .",
"We can then define a mapping $F$ from $[0,1]^n \\times {\\mathbb {R}}^n \\times {\\mathbb {R}}^{P} \\times {\\mathbb {R}}^{P} \\times {\\mathbb {R}}^{S}$ into ${\\mathbb {R}}^n \\times {\\mathbb {R}}^n \\times {\\mathbb {R}}^P \\times {\\mathbb {R}}^P \\times {\\mathbb {R}}^S$ by means of the formula $F(&x,y,p,q,s) := \\\\ & \\left(\\left( \\sum _{(1,\\beta ) \\in P} (p_{1,\\beta } - y_1 q_{1,\\beta }) s^\\beta , \\ldots , \\sum _{(n,\\beta ) \\in P} (p_{n,\\beta } - y_n q_{n,\\beta }) s^\\beta \\right) \\right.,y,p,q,\\\\ & \\qquad \\left.",
"\\vphantom{ \\sum _{(1,\\beta ) \\in P}} \\lbrace s_{j,\\alpha } - \\partial ^\\alpha f_j(x)\\rbrace _{(j,\\alpha ) \\in S} \\right).$ For a given triple $(y_0,p_0,q_0) \\in {\\mathbb {R}}^n \\times {\\mathbb {R}}^P \\times {\\mathbb {R}}^P$ and any positive scalar $C$ , nondegenerate solutions of the system $ \\frac{\\sum _{|\\beta | \\le N} (p_0)_{j,\\beta } \\prod _{(j^{\\prime },\\alpha ) \\in S} (\\partial ^\\alpha f_{j^{\\prime }}(x))^{\\beta _{j^{\\prime },\\alpha }}}{\\sum _{|\\beta | \\le N} (q_0)_{j,\\beta } \\prod _{(j^{\\prime },\\alpha ) \\in S} (\\partial ^\\alpha f_{j^{\\prime }}(x))^{\\beta _{j^{\\prime },\\alpha }}} = (y_0)_j, \\qquad j = 1,\\ldots ,n, $ will also be nondegenerate solutions of the system $ F(x,y,p,q,s) = \\left( 0, \\frac{1}{C} y_0, \\frac{1}{C^2} p_0, \\frac{1}{C} q_0, 0 \\right).",
"$ Choosing $C$ so that the right-hand always belongs to a fixed neighborhood of the origin with compact closure, we may use the fact that $F$ is itself real analytic in all parameters and so the number of connected components of the fiber $F^{-1}(0,y_0/C,p_0/C^2,q_0/C,0)$ is bounded uniformly in $y_0, p_0,$ and $q_0$ (which holds, in fact, for any analytic-geometric category in the sense of van den Dries and Miller ), which gives exactly the desired property that there is also a uniform bound on the number of isolated solutions of (REF ).",
"If the functions $f_j$ are all polynomial, Bézout's Theorem gives a similar global bound on the number of nondegenerate solutions, i.e., for all nondegenerate solutions $x \\in {\\mathbb {R}}^n$ rather than simply $[0,1]^n$ .",
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[
[
"Gaseous viscous peeling of linearly elastic substrates"
],
[
"Abstract We study pressure-driven propagation of gas into a micron-scale gap between two linearly elastic substrates.",
"Applying the lubrication approximation, the flow-field is governed by the interaction between elasticity and viscosity, as well as weak rarefaction and low-Mach-compressibility, characteristic to gaseous microflows.",
"Several physical limits allow simplification of the governing evolution equation and enable solution by self-similarity.",
"These limits correspond to different time-scales and physical regimes which include compressiblity-elasticity-viscosity, compressiblity-viscosity and elasticity-viscosity dominant balances.",
"For a prewetting layer thickness which is similar to the elastic deformation generated by the background pressure, a symmetry between compressibility and elasticity allows to obtain a self-similar solution which includes weak rarefaction effects.",
"The results are validated by numerical solutions of the evolution equation"
],
[
"Introduction",
"In this work we analyze the propagation of a Newtonian ideal gas into a thin gas-filled gap, with thickness of the order of microns, bounded by linearly elastic substrates.",
"At standard atmospheric conditions, pressure-driven gaseous flows within micron-sized configurations involve significant viscous resistance, yielding 'low-Mach-compressibility' with negligible inertial effects [27], [2].",
"In addition, weak rarefaction effects emanating from Knudsen numbers at the range of $\\mbox{\\textit {Kn}}\\approx 0.01$ to $\\approx 0.1$ yield velocity- and temperature-slip at the solid boundaries [5].",
"Thus, gaseous viscous peeling is governed by interaction of elasticity of the boundaries, gas viscosity, low-Mach-compressibility and weak-rarefaction.",
"The limit of large deformations compared with the initial gap corresponds to viscous peeling dynamics which are characterized by a distinct peeling front, similarly to the fronts in free-surface flows [25] and gravity currents [18].",
"[24] were the first to examine viscous peeling, and studied the removal of an adhesive strip from a rigid surface.",
"While previous studies modelled the adhesive as a Hookean elastic material, [24] examined the opposite limit of a Newtonian viscous fluid, which enabled calculation of the peeling speed as a function of the applied tension.",
"Other works involving viscous peeling dynamics include [17], who examined the peeling and levitation of a elastic sheet over a thin viscous film and [22] who studied axisymmetric viscous peeling of an elastic sheet from a flat rigid surface by injection of fluid between the surface and the sheet [16], [14], [28], [7], [29].",
"Effects of weak rarefaction and 'low-Mach-compressibility' on pressure driven flows were extensively studied in the context of gaseous micro-fluidics [8], [15].",
"The first experimental works were conducted by [26] and [23] and presented non-constant pressure gradient in uniform micro-channels associated with 'low-Mach-compressibility' effects.",
"[2] and [31] analytically and experimentally studied gas flow through a uniform long micro-channel with both compressibiliy and velocity-slip effects [3], [19].",
"Gaseous flows through shallow non-uniform micro-channels involving bends, constrictions and cavities were studied experimentally by [30], [20], [21] and treated analytically by [9], [10], [12], [11].",
"The aim of the current work is to study gaseous viscous peeling dynamics involving low-Mach-compressibility and weak-rarefaction.",
"The structure of this work is as follows: In §2 we define the problem and develop the evolution equation.",
"In §3.1 we present an implicit steady-state solution.",
"In §3.2 we present self-similar solutions of the evolution equation for various limits and map the transitions between the different regimes.",
"In §3.3 we develop a self-similar solution which includes weak-rarefaction effects for configurations which involve symmetry between elasticity and compressibility.",
"Concluding remarks are presented in §4."
],
[
"Problem formulation and derivation of the evolution equation",
"We examine pressure-driven gaseous viscous peeling of a two dimensional gap bounded by linear elastic substrates.",
"The configuration (similar to [13]) is illustrated in figure REF .",
"The $x-y$ coordinate system is located at the center of the gap at rest, where $x$ is parallel to the gap streamwise direction, time is $t$ , and temperature is $\\theta $ .",
"At rest, the constant gap between the lower and upper substrates is denoted by $h_0$ , and contains gas at the background pressure $p_a$ .",
"Film height is denoted by $h=h_0+d$ , where $d=d_u+d_l$ is the combined pressure induced vertical deformation of the upper and lower surfaces.",
"The stiffness coefficients of the upper and lower distributed spring substrates are $k_u$ and $k_l$ , respectively, where we define total channel stiffness by $k=(k_u^{-1}+k_l^{-1})^{-1}$ .",
"Gas velocity is $(u,v)$ , absolute pressure is $p$ , gas viscosity is $\\mu $ , gas density is $\\rho $ , the gas constant is $r_g$ and the gas mean-free-path is $\\lambda $ .",
"We define $l$ as the axial length-scale of the configuration and $p^*$ as characteristic gauge pressure (representing the characteristic value of $p-p_a$ ).",
"Thus the characteristic elastic displacement is given by $p^*/k$ .",
"We define the dimensionless ratios $Kn=\\frac{\\lambda }{h_0+d}=Kn_a \\left(\\frac{p_a}{p}\\right)\\left(\\frac{h_0}{h_0+d}\\right),\\quad \\Pi _H=\\frac{h_0 k}{p^*},\\quad \\Pi _P=\\frac{p_a}{p^*}, $ where $Kn$ is the Knudsen number representing the validity of the continuum assumption; $\\mbox{\\textit {Kn}}_a$ corresponds to the Knudsen number at the background pressure and the initial gap $h_0$ ; $\\Pi _H$ is the ratio of initial gap to the elastic displacements and $\\Pi _P$ is the ratio of the external background pressure to $p^*$ .",
"The limit $\\Pi _P\\rightarrow \\infty $ corresponds to negligible low-Mach-compressibility and the limit $\\Pi _H\\rightarrow \\infty $ corresponds to negligible elastic deformations.",
"Hereafter we denote normalized variables by Capital letters.",
"Scaling according to the lubrication approximation, the corresponding normalized parameters and variables are the coordinates $(X,Y)=(x/l,yk/p^*)$ , time $T=t (p^*)^2/k^2l^2\\mu $ , total elastic vertical displacement $D=d k/p^*$ , film height $H=h k/p^*=\\Pi _H+D$ , fluid velocity $(U,V)=(u k^2\\mu l/(p^*)^3,v k^3\\mu l^2/(p^*)^4)$ , pressure $P=p/p^*$ and density $\\Lambda =\\rho /(p^*/r_g\\theta )$ .",
"Figure: A schematic description of the configuration: A gas-filled gap separates two parallel distributed linear spring arrays bounded by rigid surfaces.",
"The initial prewetting gas layer is denoted by h 0 h_0 and upper and lower vertical deformations are denoted by d u d_u and d d d_d, respectively.We assume isothermal flow and negligible body forces, which is common practice for flows through micron-sized configurations [1], .",
"Applying the above scaling, the requirements for the validity of the lubrication approximation are given by the following relations, $\\varepsilon =\\frac{p^*}{k l}\\ll 1,\\quad \\alpha ^2=\\varepsilon Re=\\frac{(p^*)^6}{r_g\\theta k^4\\mu ^2 l^2}\\ll 1$ where $\\varepsilon $ represents the slenderness of the configuration, $\\alpha ^2$ is the Womersley number and $\\varepsilon Re$ is the reduced Reynolds number.",
"Applying (REF ) allows to utilize the standard lubrication form of the momentum equations, ${\\partial P}/{\\partial X} \\sim {\\partial ^2U}/{\\partial {Y^2}}$ , ${\\partial P}/{\\partial Y }\\sim 0$ , compressible conservation of mass equation, ${{\\partial \\Lambda }}/{{\\partial T}} + {{\\partial \\left( {\\Lambda U} \\right)}}/{{\\partial X}} + {{\\partial \\left( {\\Lambda V} \\right)}}/{{\\partial Y}} = 0$ and isothermal equation of state $\\Lambda =P$ .",
"The validity of the continuum assumption requires a sufficiently small Knudsen number $Kn=\\lambda /h_0$ , defined here as the ratio between the molecular mean-free-path $\\lambda $ and the prewetting layer thickness $h_0$ .",
"While for $Kn<10^{-3}$ use of the no-slip boundary condition is appropriate, for gas flows through micron-sized configurations at standard atmospheric conditions $\\mbox{\\textit {Kn}}\\approx 10^{-1}-10^{-2}$ .",
"This Knudsen regime requires the incorporation of velocity-slip, and thus the boundary conditions are given by the Navier-slip condition, $ \\left[U(Y = D_u),U(Y = D_l)\\right] = \\sigma \\frac{{K{n_a}{\\Pi _H}{\\Pi _P}}}{P}\\left[-\\frac{{\\partial U(Y= D_u)}}{{\\partial Y}},\\frac{{\\partial U(Y=D_l)}}{{\\partial Y}}\\right]$ as well as the kinematic boundary condition at the gas-substrate interface $\\left[V(Y = D_u),V(Y = D_l)\\right] = \\frac{1}{2}\\left[ {\\frac{{\\partial D_u}}{{\\partial T}} + U(Y = D_u)\\frac{{\\partial D_u}}{{\\partial X}}},{\\frac{{\\partial D_l}}{{\\partial T}} + U(Y = D_l)\\frac{{\\partial D_l}}{{\\partial X}}} \\right],$ where the coefficient $\\sigma $ represents the interaction between the gas molecules and the solid wall [6].",
"Applying (REF ) to the $X$ -momentum equation yields the velocity profile, $U=\\frac{1}{2}\\frac{{\\partial P}}{{\\partial X}}\\left( {{Y^2} - \\frac{{{(D+\\Pi _H)^2}}}{4} - \\frac{{\\sigma K{n_a}{\\Pi _H}{\\Pi _P}}}{2}\\frac{(D+\\Pi _H)}{P}} \\right).$ Integration of mass conservation equation in conjunction with (REF )- (REF ), and applying the normalized linear elastic relation $D=P-\\Pi _P$ , yields the evolution equation $ \\frac{{\\partial }}{{\\partial \\mathfrak {T}}}\\left[(D+\\Pi _H)(D + \\Pi _P)\\right] = \\\\ \\frac{\\partial }{{\\partial X}}\\left[ {\\left( {(D + {\\Pi _P}){{(D + {\\Pi _H})}^3} + 6\\sigma K{n_a}{\\Pi _H}{\\Pi _P}{{(D + {\\Pi _H})}^2}} \\right)\\frac{{\\partial D}}{{\\partial X}}} \\right],$ where hereafter $\\mathfrak {T}=T/12$ for convenience."
],
[
"Steady-state",
"An implicit solution of (REF ) may be obtained for steady-state flow in a finite configuration with prescribed pressures at the inlet and outlet sections ($D(0)=P(0)-\\Pi _P$ and $D(1)=P(1)-\\Pi _P$ , respectively), $X(D) = \\frac{{F(D) - F(D(0))}}{{F(D(1)) - F(D(0))}},$ where $F(D) = 40\\sigma {K{n_a}}{\\Pi _H}{\\Pi _P} {(D+\\Pi _H)^3} + 5(D-\\Pi _P){(D+\\Pi _H )^4} - {(D+\\Pi _H )^5}.$ Solution (REF ) is depicted in figure REF (a) for the case of $\\Pi _P=1$ , $P(0)/P(1)=2$ , $Kn_a=0$ (smooth lines) and $Kn_a=0.1$ (dashed lines) for various values of $\\Pi _H$ .",
"Small values of $\\Pi _H$ represent large ratios of elastic deformation to initial gap $h_0$ , which reduce the pressure gradient near the inlet while increasing it towards the outlet.",
"For constant prewetting layer thickness and background pressure (i.e.",
"constant $Kn_a$ ), decreasing $\\Pi _H$ decreases the local Knudsen number (as seen in panel (b)) and thus decrease the effect of weak rarefaction on the pressure distribution.",
"Panel (c) presents the effect of weak rarefaction on the mass-flow-rate vs. $\\Pi _H$ .",
"Weak rarefaction effects on mass-flow-rate tend to a constant finite value for $\\Pi _H\\rightarrow \\infty $ , and decreases with $\\Pi _H$ .",
"Panel (d) presents the mass-flow-rate vs. the pressure difference for various values of $\\Pi _H$ .",
"While the gradient of the different lines vary significantly with $\\Pi _H$ for the limit of small pressures at the inlet, as the inlet pressure increases the viscous resistance no longer depends on $\\Pi _H$ (or $h_0$ ) and the gradients converge.",
"Figure: Steady-state solution () of gaseous viscous flow in a 2D gap bounded by linearly elastic substrates.",
"Gas pressure (a) and local Knudsen (b) profiles vs. XX for varying Π H \\Pi _H for P(0)/P(1)=2P(0)/P(1)=2.",
"(c) Mass flow rate vs. Π H \\Pi _H for Kn a =0Kn_a=0 and Kn a =0.1Kn_a=0.1.",
"(d) Mass flow rate vs. pressure ratio P(X=0)/P(X=1)P(X=0)/P(X=1) for various values of Π H \\Pi _H.",
"For all panels Kn a =0.1Kn_a=0.1, σ=1\\sigma =1 and Π P =1\\Pi _P=1."
],
[
"Self-similar Barenblatt solutions for negligible rarefaction effects",
"While exact solutions of (REF ) are not available, several limits involving negligible rarefaction effects yield known self-similar solutions.",
"Furthermore, the flow-field may be described by different approximate solutions during different time-scales of observation of the peeling process.",
"Physical insight may thus be gained by mapping the different approximate solutions and the corresponding time-scales and transitions.",
"We focus on fundamental solutions for the case of impulse driven peeling - an abrupt release of a finite mass at $T=0$ into the inlet at $X=0$ .",
"This process is characterized by a compactly supported region of displacement and a distinct front, denoted by $X_F$ .",
"The relevant integral form of mass conservation is $\\int _0^{{X_F}} {\\left[(D+\\Pi _H)(D+\\Pi _P)-\\Pi _H\\Pi _P\\right]dX = M}\\,,$ where $M$ is a constant representing the mass injected into the interface at $T=0$ .",
"The conditions near the contact line $X\\rightarrow X_F$ are $D\\rightarrow 0$ and $P\\rightarrow \\Pi _P$ .",
"Due to the sudden injection of mass at the inlet, for all values of $\\Pi _P,\\Pi _H$ , we obtain that for sufficiently early times $D\\gg \\Pi _P,\\Pi _H$ , yielding an early regime in which both elasticity and compressibility contribute to the peeling process.",
"However, for injection of mass at small but finite time-scales there is an initial value of the characteristic gauge pressure (denoted hereafter by $p_0^*$ ), scaling by which yields $D\\sim O(1)$ for early times.",
"This sets the requirement $\\Pi _P,\\Pi _H\\ll 1$ as a condition for the appearance of the early time regime (i.e.",
"large gauge pressure compared with background pressure and large displacement to prewetting thickness ratio).",
"In this early time regime the leading order of equation (REF ) is a porous-medium-equation of order $2.5$ for the variable $D^2$ .",
"Applying ZKB's solution [4] yields the time propagation rate $X_F=O(\\mathfrak {T}^{2/7})$ and the peeling dynamics are given by, $D(X,\\mathfrak {T}) = {\\left( {\\frac{1}{5}} \\right)^{ - 1/7}}{\\mathfrak {T}^{ - 1/7}}\\left[ {{C_1} - \\frac{3}{{35}}{{\\left( {\\frac{1}{5}} \\right)}^{ - 4/7}}{X^2}{\\mathfrak {T}^{ - 4/7}}} \\right]_ + ^{1/3}{\\hspace{1.0pt}} ,$ ${C_1} = {\\left( {\\frac{{2M}}{{{S_1}}}} \\right)^{6/7}},\\,\\,\\,\\,{S_1} = \\sqrt{\\frac{{35\\pi }}{3}} \\frac{{B(1/2,5/3)}}{{\\Gamma (1/2)}}\\,,$ where $B$ is the beta function, $\\Gamma $ the gamma function and $(s)_{_+}=\\max (s,0)$ .",
"As the added mass propagates and expands into the substrate, the gas pressure decreases and thus $D$ decreases, eventually invalidating the requirement of $D\\gg \\Pi _P,\\Pi _H$ .",
"This sets a validity time range of $\\mathfrak {T}\\ll 5 C_1^{7/3}/ \\text{max}(\\Pi _P^7,\\Pi _H^7)$ for (REF ) based on $D(X=0)$ .",
"For the case of $\\Pi _H/\\Pi _P=h_0/(p_a/k)\\gg 1$ (corresponding to negligible effects of elasticity and dominant effects of low-Mach-number gas compressibility) an intermediate regime exists where $\\Pi _P\\ll D \\ll \\Pi _H$ .",
"In this regime, the leading order of equation (REF ) is a porous-medium-equation of order 2 for $D$ .",
"Thus the solution will transition to a propagation rate of $X_F=O(\\mathfrak {T}^{1/3})$ and the resulting profile, $D(X,\\mathfrak {T}) = {\\left( {\\frac{{\\Pi _H^2}}{{2}}} \\right)^{ - 1/3}}{\\mathfrak {T}^{ - 1/3}}{\\left[ {C_2 - \\frac{1}{{12}}{{\\left( {\\frac{{\\Pi _H^2}}{{2}}} \\right)}^{ - 2/3}}{X^2}{\\mathfrak {T}^{ - 2/3}}} \\right]_ + }\\,,$ ${C_2} = {\\left( {\\frac{{2M}}{{{\\Pi _H}{S_2}}}} \\right)^{2/3}},\\,\\,\\,\\,{S_2} = \\sqrt{12\\pi } \\frac{{B(1/2,2)}}{{\\Gamma (1/2)}}\\,,$ will emerge in intermediate times with a validity range of $ 2C_2^{3} / \\Pi _H^5 \\ll \\mathfrak {T} \\ll 2 C_2^{3} / \\Pi _H^3 \\Pi _P^2$ .",
"Solution (REF ) represents the limit of dominant gas compressibility, and is identical to the evolution of compressible low-Reynolds-number gas flow in rigid configurations.",
"The relevant time-scale of this limit is $t^*=\\mu /p^* \\varepsilon ^2$ .",
"Alternatively, for $\\Pi _P / \\Pi _H=p_a/kh_0 \\gg 1$ , a different intermediate region exists for which $\\Pi _H\\ll D\\ll \\Pi _P$ and the leading order evolution equation is a porous-medium-equation of order 4 for $D$ , yielding $D(X,\\mathfrak {T}) = {\\left( {\\frac{1}{{4}}} \\right)^{ - 1/5}}{\\mathfrak {T}^{ - 1/5}}\\left[ {C_3 - \\frac{3}{{40}}{{\\left( {\\frac{1}{{4}}} \\right)}^{ - 2/5}}{X^2}{\\mathfrak {T}^{ - 2/5}}} \\right]_ + ^{1/3}\\,,$ ${C_3} = {\\left( {\\frac{{2M}}{{{\\Pi _P}{S_3}}}} \\right)^{6/5}},\\,\\,\\,\\,{S_3} = \\sqrt{\\frac{{40\\pi }}{3}} \\frac{{B(1/2,4/3)}}{{\\Gamma (1/2)}}\\,,$ with an $X_F=O(\\mathfrak {T}^{1/5})$ spread-rate, typical of the early time propagation of the incompressible peeling problem [7], and a validity range of $ 4C_3^{5/3} / \\Pi _P^5 \\ll \\mathfrak {T} \\ll 4 C_3^{5/3} / \\Pi _H^5$ .",
"By setting larger values for $\\Pi _H$ or $\\Pi _P$ , propagation dynamics may skip or move across a certain stage in the sequence.",
"All solutions will ultimately settle on $X_F=O(\\mathfrak {T}^{1/2})$ propagation as $\\mathfrak {T}\\rightarrow \\infty $ whether be it the prewetting thickness ratio $\\Pi _H$ or background to gauge pressure ratio $\\Pi _P$ the final regularization mechanism which linearizes (REF ).",
"The evolution of the solution through the various regimes and corresponding propagation rates is validated numerically in figure 3(a).",
"A flow-chart illustrating the transitions and presenting the requirements for the different limits, as well as the time-ranges in which the limits are valid, is presented in figure 3(b).",
"The validity range for each limit is calculated by requiring the appropriate order of magnitude of $D$ for the examined limit from the solutions (REF ), (REF ) and (REF ) at $X=0$ .",
"The conditions $(1)-(6)$ presented in figure 3(b) may be represented in dimensionless form as $(1)$ $\\Pi _P \\gg 1$ and $\\Pi _H\\ll 1$ ; $(2)$ $\\Pi _P \\ll 1$ and $\\Pi _H\\ll 1$ ; $(3)$ $\\Pi _P \\ll 1$ and $\\Pi _H\\gg 1$ ; $(4)$ $\\Pi _H\\ll \\Pi _P$ ; $(5)$ $\\Pi _H\\sim \\Pi _P$ ; and $(6)$ $\\Pi _H\\gg \\Pi _P$ .",
"An additional solution with velocity-slip (dashed line) was also considered along with its no-slip counterpart marked by $(2)+(5)$ .",
"For intermediate times Knudsen-diffusion is shown to mildly alter the spread rate, but this effect is reduced at the early time limit (REF ) as well as in the linearized regime.",
"Figure 3(a) is supplemented by figure 4 which presents the numerical deformation profiles for various limits and the convergence of the numerical profile to the theoretical results presented in (REF ), (REF ) and (REF ), corresponding to panels (a), (b) and (c), respectively.",
"Figure: Propagation regimes of gaseous viscous peeling in a 2D gap bounded by linearly elastic substrates for different time-scales and physical parameters.",
"(a) Numerical computation of the propagation front velocity (logarithmic derivative) from () vs. the theoretical limits given by ()-().",
"Line (1)(1) corresponds to (M,Π H ,Π P )=(3.8×10 -3 ,2×10 -3 ,2)(M,\\Pi _H,\\Pi _P)=(3.8 \\times 10^{-3},2\\times 10^{-3},2); line (2)+(4)(2)+(4) corresponds to (M,Π H ,Π P )=(2.86×10 -4 ,0,1×10-2)(M,\\Pi _H,\\Pi _P)=(2.86\\times 10^{-4},0,1\\times 10{-2}); line (2)+(5)(2)+(5) corresponds to (M,Π H ,Π P )=(2.95×10 -4 ,5×10 -3 ,1×10 -2 )(M,\\Pi _H,\\Pi _P)=(2.95 \\times 10^{-4},5\\times 10^{-3},1\\times 10^{-2}); line (2)+(6)(2)+(6) corresponds to (M,Π H ,Π P )=(2.77×10 -4 ,5×10 -3 ,0)(M,\\Pi _H,\\Pi _P)=(2.77 \\times 10^{-4},5\\times 10^{-3},0); line (3)(3) corresponds to (M,Π H ,Π P )=(5.35×10 -4 ,0.15,10 -4 )(M,\\Pi _H,\\Pi _P)=(5.35 \\times 10^{-4},0.15,10^{-4}).",
"Time is rescaled according to the average time presented in the figure, 𝔗 ˜=10 7 𝔗\\tilde{\\mathfrak {T}}=10^7\\mathfrak {T}.",
"(b) A flow-chart diagram describing the different dominant balances, transitions between regimes and propagation time-scales as a function of initial gauge pressure p 0 * p_0^*, background pressure p a p_a, channel stiffness kk and prewetting layer thickness h 0 h_0.Figure: Convergence of the numerical deformation profiles to the theoretical results (), () and () corresponding to the respective values of (M,Π H ,Π P M,\\Pi _H,\\Pi _P) presented in figure .",
"Panels (a), (b) and (c) present convergence along lines (2)+(6) (stage (2)), (2)+(4) (stage (4)) and (2)+(6) (stage (6)), respectively."
],
[
"Self-similar solution with weak rarefaction effects for $\\Pi _P\\rightarrow \\Pi _H$",
"For the limit of $\\Pi _H\\rightarrow \\Pi _P$ (or $k\\rightarrow p_a/h_0$ ), applying $f=D+\\Pi _H=D+\\Pi _P$ allows to obtain an additional self-similar solution for the case of suddenly applied fixed inlet pressure.",
"In this limit both the viscous and Knudsen diffusion terms of equation (REF ) will enforce a $O(\\mathfrak {T}^{1/2})$ spread-rate and an exact self-similar solution with velocity-slip may be attained for $f(\\eta )$ , where $\\eta =X\\mathfrak {T}^{-1/2}$ .",
"Substitution of $f(\\eta )$ into (REF ) yields the self-similar boundary value problem, ${f^5}^{\\prime \\prime } + 10 \\sigma Kn_a \\Pi _H^2 {f^3}^{\\prime \\prime } + \\frac{{5\\eta }}{2}{f^2}^{\\prime } = 0\\,,$ supplemented by $f(0) = 1 + {\\Pi _H},\\,\\,\\,\\,\\,f(\\infty ) = {\\Pi _H}\\,,$ and $\\int _0^\\infty {{H^2}(X,\\mathfrak {T})dX = M} {\\mathfrak {T}^{1/2}},\\,\\,\\,M = \\int _0^\\infty {{f^2}(\\eta )d\\eta }\\,.$ Self-similar profiles for various values of $\\Pi _H$ are presented in figure REF (a) for $ Kn_a=0$ (smooth lines) and $\\sigma =1,\\, Kn_a=0.1$ (dashed lines).",
"The effect of weak rarefaction is shown to increase the speed of gas propagation, and reduce the gradients of the deformation.",
"This effect decreases as $\\Pi _H$ and $\\Pi _P$ decrease, since $Kn_a$ is defined ahead of the front, while the local Knudsen decreases as the gap and pressure increase (see (REF )).",
"For small $\\Pi _P$ and $\\Pi _H$ , the pressure and gap in the peeled region are greater compared with the background pressure $p_a$ and prewetting layer thickness $h_0$ .",
"This yields a significantly smaller effective Knudsen number in the peeled region, as illustrated in figure REF (b).",
"Figure: Self-similar solutions with weak-rarefaction effects for Π P =Π H \\Pi _P=\\Pi _H and σ=1\\sigma =1.",
"(a) Self-similar deformation profile as a function of the prewetting thickness ratio Π H \\Pi _H for Kn a =0Kn_a=0 (smooth lines) and Kn a =0Kn_a=0 (dashed lines).",
"(b) Local Knudsen as a function of η\\eta for various values of Π H \\Pi _H for Kn a =0.1Kn_a=0.1.",
"Solution based on numerical integration of ()."
],
[
"Concluding remarks",
"The propogation of a gas into micron-sized configurations with linearly elastic boundaries is governed by interaction between effects of low-Mach-compressibility, weak rarefaction, elasticity and viscosity.",
"While exact solutions of the governing nonlinear evolution equation are not available, several limiting cases allow solution by self-similarity.",
"These limits correspond to different physical regimes, including: (i) dominant balance between compressiblity and viscosity, (PME of order 2) characterizing compressible flow in rigid micro-channels, (ii) dominant balance between elasticity and viscosity, (PME of order 4) characterizing incompressible flow in elastic micro-channels, and (iii) dominant balance involving viscosity, elasticity and compressiblity (PME of order $2.5$ ).",
"During gas film propagation, the flow-field transitions between the aforementioned regimes and corresponding exact solutions.",
"A map of these transitions was presented as a function of the prewetting layer thickness, the background pressure and stiffness of the spring array.",
"The case where $k h_0\\approx p_a$ represents symmetry between compressibility and elasticity, and allowed to obtain an additional self-similar solution accounting for weak rarefaction effects.",
"While the steady-state solution presented in §3.1 is implicit, explicit solutions may be obtained for the same physical limits examined in the transient dynamics, as presented in §3.2.",
"For negligible slip $Kn_0\\rightarrow 0$ , steady-state solutions corresponding to the Barrenblatt self-similar limits (REF )-(REF ) can be presented by the relation $D(X)=\\left[\\left( (D(1)+C)^{N}-(D(0)+C)^{N} \\right)X+(D(0)+C)^{N}\\right]^{\\frac{1}{N}}-C,$ where $(N,C)=(5,\\Pi _P)$ for the limit of $\\Pi _P,\\Pi _H\\ll D$ (viscous-elastic-compressibility regime) or the symmetric case of $\\Pi _P=\\Pi _H$ ; $(N,C)=(2,\\Pi _P)$ for the limit of $\\Pi _H\\gg \\Pi _P,D$ (viscous-compressibility regime); $(N,C)=(4,\\Pi _H)$ for the limit of $\\Pi _P\\gg \\Pi _H,D$ (viscous-elastic regime) and finally, $(N,C)=(1,0)$ for the linear limit of $\\Pi _P,\\Pi _H\\gg D$ (viscous regime) ."
]
] | 1709.01694 |
[
[
"A near-infrared interferometric survey of debris-disk stars. VI.\n Extending the exozodiacal light survey with CHARA/JouFLU"
],
[
"Abstract We report the results of high-angular-resolution observations that search for exozodiacal light in a sample of main sequence stars and sub-giants.",
"Using the \"jouvence\" of the fiber linked unit for optical recombination (JouFLU) at the center for high angular resolution astronomy (CHARA) telescope array, we have observed a total of 44 stars.",
"Out of the 44 stars, 33 are new stars added to the initial, previously published survey of 42 stars performed at CHARA with the fiber linked unit for optical recombiation (FLUOR).",
"Since the start of the survey extension, we have detected a K-band circumstellar excess for six new stars at the ~ 1\\% level or higher, four of which are known or candidate binaries, and two for which the excess could be attributed to exozodiacal dust.",
"We have also performed follow-up observations of 11 of the stars observed in the previously published survey and found generally consistent results.",
"We do however detect a significantly larger excess on three of these follow-up targets: Altair, $\\upsilon$ And and $\\kappa$ CrB.",
"Interestingly, the last two are known exoplanet host stars.",
"We perform a statistical analysis of the JouFLU and FLUOR samples combined, which yields an overall exozodi detection rate of $21.7^{+5.7}_{-4.1}\\%$.",
"We also find that the K-band excess in FGK-type stars correlates with the existence of an outer reservoir of cold ($\\lesssim 100\\,$K) dust at the $99\\%$ confidence level, while the same cannot be said for A-type stars."
],
[
"Introduction",
"Observations have recently confirmed that extrasolar planets are ubiquitous, and an important objective of several future space-based missionsSee for example [57], which discusses the use of the upcoming Wide Field Infrared Survey Telescope (WFIRST) for high-contrast imaging.",
"is to directly image earth-like companions in the habitable zone of main-sequence stars.",
"Several planets, lying relatively far from the habitable zone, have been directly imaged using high-contrast techniques, and a common feature has been the prior detection of a debris disk at far-infrared (FIR) wavelengths.",
"Planets have created gaps or warps in these cold ($\\sim 10\\,\\mathrm {K}$ ) debris disks (e.g., [27]), located as far as $\\sim 100\\,\\mathrm {AU}$ from the star, and debris disks are now considered a sign-post for the existence of planets.",
"Debris disks lying close to the star can also hinder the detection of exoplanets, so a primary motivation for this work is to better understand the environment closer to the habitable zone of main sequence stars, and to quantify the level of exozodiacal light originating from this region (Fig.",
"REF ), where the debris disk is thought to have been mostly cleared out.",
"Figure: K-band excesses detected with interferometry possibly originate from hot (∼1000\\sim 1000K) dust within an AU from the star.",
"The left sketch shows the typical field of view attainable with a single telescope, e.g., Spitzer and Hershel , which can resolve the outer cold-debris disk, while the magnified sketch shows the interferometric FOV, which can resolve the inner hot component of the circumstellar disk.The small mass, and presumably low optical depth of these debris disks, makes them difficult targets for indirect detection techniques such as radial velocity (RV) and transits [51].",
"Spectroscopic studies, although complementary, do not easily allow the disentanglement of the light from the star from that of its immediate environment.",
"Long-baseline infrared interferometers permit resolving the inner-most region close to the star, and the use of high-precision beam-combiners has allowed interferometrists to achieve dynamic ranges of the visibility between $\\sim 100$ and $\\sim 1000$ at near- and mid-infrared wavelengths, for example [2], [41].",
"Our own solar system contains dust, thermally emitting zodiacal light at mid-infrared (MIR) wavelengths, which motivated ground-based surveys to search for exozodiacal light using long-baseline interferometers.",
"Interestingly, the vast majority of close-in exozodiacal light detections have been in the near-infrared (NIR) [13], [1], [3], [16], [2], [20], which are likely due to hot ($\\sim 1000\\,\\mathrm {K}$ ) dust lying within an AU from the star.",
"The presence of hot-circumstellar dust, as bright as a few percent relative to the star, and thousands of times brighter than our own zodiacal light, is still poorly understood.",
"This phenomenon has been met with some skepticism in the scientific community for two main reasons: i) NIR exozodiacal light detections still lie very close to the limit of detectability of long-baseline interferometers, and detections are generally claimed in the few sigma range.",
"ii) hot and small dust particles, lying close to the star, should have a short lifetime because Poynting-Robertson drag and radiation pressure should efficiently remove dust within time-scales of a few years.",
"Also, collision rates of larger bodies are too high for them to survive over the age of the stars around which dust is observed [60].",
"Therefore, if hot dust is the origin of the NIR exozodi detections, there needs to be an efficient dust-replenishment or dust-trapping mechanism.",
"In this work we will mainly discuss the recent survey results of K-band observations performed with the “jouvence” of the fiber linked unit for optical recombination (JouFLU) [50] at the Center for High Angular Resolution Astronomy (CHARA) Array [55], and combine our results with the survey performed by [2].",
"To establish confidence in the results, and to deal with item (i), we first developed a data analysis strategy for minimizing the bias and uncertainty of exozodiacal light observations with two-beam Michelson interferometers, which is described in detail in [46].",
"Using this analysis strategy we also performed a study of calibrator stars to verify that the measured circumstellar excesses are not a result of systematic errors.",
"Here we present the full survey results, explore possible correlations with basic stellar parameters, and discuss the plausibility of different physical mechanisms that may be responsible.",
"As mentioned above in item (ii), the main difficulty is to find a physical mechanism that can be responsible for the hot dust phenomenon.",
"Some explanations involve comet in-fall or out-gassing [60], [54], [33], [7].",
"Another explanation proposed by [54], and further developed by [47], is that magnetic trapping of small enough dust grains (nano-grains), close to the star's sublimation radius, is possible with typical magnetic fields of main sequence stars, and predicts that detection rates should increase with stellar rotational velocities.",
"We search for the predicted correlations in our sample.",
"The outline of this paper is the following: in Sections and we discuss the JouFLU stellar sample and observing strategy.",
"In Sections and we discuss the interferometric visibility and circumstellar emission level estimation procedures.",
"In Sections and , we report the measured circumstellar emission levels for the JouFLU survey extension and follow-up observations of excess stars from the original [2] survey.",
"In Section , we perform a statistical analysis of the measured excess levels, and explore correlations with basic stellar parameters."
],
[
"The JouFLU stellar sample",
"Since 2013, we have extended the initial K-band survey of 42 stars performed by [2].",
"The main selection criteria were that targets should have not departed significantly from the main sequence and that they should not have known multiplicity.",
"In addition, we aimed to have a target list that covers many spectral types, and as many stars with known cold dust reservoirs (MIR-FIR excess) as without.",
"However, the main restriction is that targets be bright enough (K$\\lesssim 4.2$ ) to achieve sub-percent precision on the squared visibility.",
"There are approximately $\\sim 214$ main sequence stars with no known multiplicity and brighter than $K=4.2$ observable at CHARA.",
"We have added 33 new stars to the survey as shown in Table REF , distributed across many spectral types as shown in Fig.",
"REF .",
"In comparison with the initial FLUOR sample of [2], which had a median K magnitude of 2.9, the JouFLU stellar sample is somewhat fainter, as shown in Fig.",
"REF , with a median K magnitude of 3.5.",
"The FLUOR sample contains 19 stars with either a MIR or FIR excess, which is attributed to a cold-dust ($\\lesssim 100\\,$ K) reservoir.",
"The JouFLU stellar sample only contains five stars with a such a suspected cold-dust reservoir, mainly because these stars are less common and quickly exceed the current limiting magnitude attainable by JouFLU (K$\\sim 4.5$ ).",
"The JouFLU sample contains eight targetsCommon targets with the VLTI's survey with the Precision Integrated Optics Near Infrared ExpeRiment (PIONIER) are: HD23249, HD28355, HD33111, HD164259, HD165777, HD182572, HD210418, HD215648.",
"in common with another H-band exozodi survey conducted in the southern hemisphere with the Very Large Telescope Interferometer (VLTI) [20].",
"We also note that the JouFLU sample also has eight common targetsCommon targets with the LBTI survey are: HD33111, HD26965, HD19373, HD182572, HD182640, HD185144, HD210418, HD219134.",
"with the Large Binocular Telescope Interferometer survey [59], which will enable a comparison with high resolution and high-contrast mid-infrared observations.",
"Figure: Distribution of spectral types for the stars in the survey (green), the stars in the JouFLU survey (blue) in this work, and the FLUOR follow-up stars (red).Figure: Histogram of K-band magnitude for the stars in the FLUOR survey (green), the JouFLU survey (blue, this work), and the FLUOR follow-up stars (red).Table: New stars observed between 2013 and 2015 for the JouFLU exozodi survey extension.",
"Shown here are the HD number, common name, K-band magnitude, Age, baseline used, and date of observation.In addition to the JouFLU survey, we performed follow-ups on 11 stars of the [2] FLUOR sample in order to confirm some previous detections (see Table REF ), and search for possibly time variability of the exozodiacal light level.",
"Table: Stars observed between 2013 and 2015 for the exozodi follow-up program.",
"Shown here are the HD number, common name, K-band magnitude, Age, baseline used, and date of observation.",
"HD185144 is a common target with the LBTI survey."
],
[
"Observing strategy",
"We used the JouFLU beam combiner at the CHARA array to perform observations.",
"JouFLU, an upgraded version of FLUOR [14], [15], is a two-telescope beam combiner which uses single-mode fibers and photometric channels to enable precise visibility measurements at the $\\sim 1\\%$ level on bright (K$\\lesssim 3$ ) stars [50], [14], [15].",
"Interference fringes are scanned at high frequency (100 Hz) in order to minimize the effects due to atmospheric piston, and we nominally collect 150-200 fringe scans for each target or calibrator star.",
"To minimize calibration biases, we generally use three different calibrator stars (cals) for each science target, hereafter “obj”.",
"Cal$_1$ and Cal$_2$ have respectively a lower and higher right-ascension than the target, and Cal$_3$ is located as close as possible to the target.",
"An observing block is generally of the form (cal$_1$ , obj, cal$_{2}$ , obj, cal$_3$ , obj, cal$_1$ ).",
"We generally obtain approximately six calibrated points (two observing blocks) for each science target.",
"We have mainly used the smallest baselines of the CHARA array S1-S2 ($33\\,\\mathrm {m}$ ) and E1-E2 ($60\\,\\mathrm {m}$ ), for which stars remain mostly unresolved, and with intrinsic visibilities that hence have a very weak dependence on the assumed stellar model diameter.",
"For example, the largest and brightest (K=2) calibrator used in the survey has a uniform angular diameter estimated at $1.71 \\pm 0.024\\,\\mathrm {mas}$ [42].",
"This corresponds to an interferometric visibility of $0.979\\pm 0.0006$ at the $33\\,\\mathrm {m}$ baseline.",
"Even assuming a more conservative diameter uncertainty of 5% on this worst case calibrator, the resulting visibility uncertainty is 0.7%.",
"In fact, V-K surface brightness angular diameters are sufficient to achieve the sub-percent precision level required for the survey.",
"In [46] we also show that the difference in brightness between the target and the calibrators does not bias the visibility measurements to levels higher than $\\sim 0.1\\%$ , particularly when we use the visibility estimation and the calibration methods described in the above reference, which is used for all the data reported here."
],
[
"Visibility estimation procedure",
"We start by obtaining raw (uncalibrated) visibilities: from the ensemble of fringe scans obtained for the science target and calibrators, we computed the median visibility and bootstrapped uncertainty.",
"We then calibrated the raw visibilities by estimating the transfer function at the time of the science target observation using the “hybrid interpolation” as described in [46].",
"The final uncertainty in the calibrated visibility takes into account the statistical error from the median bootstrapping, and the systematic uncertainty from changes in the transfer function.",
"This procedure is discussed in detail in [46]."
],
[
"Exozodiacal light-level estimation procedure",
"The main idea behind the interferometric detection of exozodiacal light is that the star remains mostly unresolved at short baselines, while exozodiacal light emission spreads over a much bigger region and is partially or fully resolved by the interferometer.",
"Faint off-axis emission is therefore detected as a small decrease of the fringe visibility (contrast) at short baselines as shown in Fig.",
"REF .",
"If exozodiacal light uniformly fills the entire FOV, then the circumstellar emission level $f_{\\mathrm {cse}}$ relative to the stellar emission level, is approximately related to the squared fringe visibility as [17] $V^2 = V_{\\star }^2(1-2f_{\\mathrm {cse}}), $ Figure: Filled red curve represents the predicted squared visibility for a uniform-disk type star, with a diameter of (1±0.05) mas (1\\pm 0.05)\\,\\mathrm {mas}.",
"The blue line represents the squared visibility of the star with a circumstellar disk of 1%1\\% brightness relative to the star, that uniformily fills the entire field of view.",
"We note that at short baselines, the uncertainty of the stellar model, i.e., the width of the red curve, is much smaller than the visibility deficit that results from fully resolving the faint circumstellar disk.",
"The squared visibility is equal to one at zero baseline for both models.",
"Also shown are the 33\\,m and 65\\,m baselines used for this study.where $V_{\\star }$ is the expected visibility of the stellar photosphere.",
"This is clearly a simple model, but it allows estimating the circumstellar excess within the FOV when no information is available about the spatial distribution of the excess [1].",
"To estimate the expected photosphere visibility, we used interferometric stellar radii and linear limb-darkenned coefficients when available.",
"When stellar radii were not directly available, we used surface brightness relations to estimate the limb darkened angular diameter as described in [24].",
"Next, we performed a fit of $f_{\\rm {cse}}$ in Eq.",
"REF , which is the only free parameter of the model.",
"To estimate the uncertainty $\\sigma _{\\rm {cse}}$ on the circumstellar emission level relative to the star, we identified the values of $f_{\\rm {cse}}$ that increase the best fit $\\chi ^2$ by 1.",
"An excess detection is claimed when $f_{cse}/\\sigma _{cse}\\ge 3$ .",
"A major strength of this detection strategy is that it is largely insensitive to the uncertainty of the stellar model when observations are performed at short baselines as shown in Fig REF , and discussed in Section 4 of [46].",
"Also, while it is possible that calibrator stars display a circumstellar excess, this would tend to underestimate any excess found around the science target, and tend to decrease the number of detections.",
"Since we used several (approximately three) calibrator stars for each science target, we further verified that the measured excesses were not an effect of systematic errors by checking that the calibrators can be used to calibrate each other (see Section REF )."
],
[
"Results of circumstellar excess levels",
"For each target we tested two main models: a stellar photosphere model without circumstellar disk, and a stellar model with a circumstellar disk as described in Section .",
"In Table REF we present the best-fit circumstellar emission levels for the JouFLU exozodi survey, and the reduced $\\tilde{\\chi ^2}$ for both models.",
"Table: Circumstellar emission level for the exozodi survey.",
"For each target, the table shows the identifiers, K-band magnitude, interferometric baselines used (S1-S2∼33m\\sim 33\\,\\mathrm {m}, E1-E2∼60m\\sim 60\\,\\mathrm {m}), circumstellar emission level, reduced χ ˜ cse 2 \\tilde{\\chi }_{\\mathrm {cse}}^2 for the star + pole-on-disk model (Eq.",
"), and reduced χ ˜ ☆ 2 \\tilde{\\chi }_{\\star }^2 for the photosphere model.",
"The highlighted boxes correspond to statistically significant excesses: red boxes indicate that there is no evidence of binarity, while yellow boxes indicate that there is separate evidence of binarity.The survey results for the new JouFLU targets, summarized in Table REF , show six new circumstellar excesses, including two possibly associated with circumstellar dust around HD210418 and HD222368.",
"The other four cases (HD162003, HD182640, HD33111, and HD5448) are likely binaries, and in the cases of HD182640 and HD5448, we checked for compatibility of the data with binary orbit solutions obtained by other authors.",
"We highlight the first direct detection of a previously unseen companion for HD162003A as discussed in Section REF .",
"Below we present reduced data for some representative stars, which fall into three main categories: non-excess detections, excess detections attributed to binarity, and excess detections attributed to dust."
],
[
"Examples of survey stars with no excess detected: HD34411, HD190360",
"HD34411 (lam Aur) is a solar-type star with a known far-infrared ($70\\mu \\mathrm {m}$ ) excess detected with the Spitzer space telescope [58], which implies the existence of a cold debris disk.",
"The 2013 K-band observations with the $65\\,\\mathrm {m}$ baseline are consistent with a non-excess as shown in Fig.",
"REF , and are compatible with the stellar photosphere model that uses the limb darkened photospheric model derived by [10].",
"HD190360 is a solar-type exoplanet host star [45], with two detected Jovian planets, the farthest one being $\\sim 3.9\\,\\mathrm {AU}$ ($\\sim 4.2\\,\\mathrm {mas}$ ) from the star.",
"The K-band interferometric data, obtained with the $65\\,\\mathrm {m}$ baseline, are compatible with the stellar photosphere parameters derived by [34], and therefore consistent with a non-excess as shown in Fig.",
"REF .",
"Figure: Squared visibility as a function of hour angle for the E1-E2 (65m65\\,\\mathrm {m}) baseline.",
"The dashed curve represents the stellar photosphere model, with a model uncertainty of <0.1%<0.1\\% (not shown).",
"The solid black curve represents the best fit squared visibility model using excess emission that uniformly fills the field of view.",
"Both objects are consistent with a non-excess and compatible with the stellar photosphere model as indicated by the low reduced χ ˜ 2 \\tilde{\\chi }^2 values of the star only model."
],
[
"Survey excess stars with no evidence for binarity: HD210418, HD222368",
"HD222368 is an F-type main sequence star with a cold debris disk as implied by a detected far-infrared excess at $70\\mu \\mathrm {m}$ with the Spitzer space telescope [58].",
"The angular diameter and linear limb-darkening coefficient were inferred from long baseline observations performed by [10].",
"This stellar photosphere model was used to generate the dashed line in Fig.",
"REF , which strongly disagrees with the $60\\,\\mathrm {m}$ baseline data ($\\tilde{\\chi }^2=4.88$ ).",
"We report a K-band circumstellar emission of $(1.58\\pm 0.36)\\%$ relative to the star (4.4$\\sigma $ detection), with a reduced $\\tilde{\\chi }^2=1.41$ .",
"The data are consistent with a uniform emission over the covered azimuthal angle range ($31^\\circ -45^\\circ $ ).",
"This star was also observed with the Keck Interferometric Nuller at $\\sim 10\\mu \\mathrm {m}$ , and no excess was detected since the measured null-depth between 8-13$\\,\\mu $ m was $-(0.24\\pm 1.4)\\times 10^{-3}$ [41].",
"HD210418 is an A-type main sequence star that also seems to display a circumstellar excess of $(1.69\\pm 0.54)\\%$ ($\\tilde{\\chi }^2=1.67$ ) relative to the star.",
"For this target we also used a stellar photosphere model derived from [10].",
"The detection has a lower significance ($3.1\\sigma $ ) and we note that only one data excess point is highly significant (see Fig.",
"REF ), so most of the excess comes from a single data point, and more data are required to confirm this detection.",
"This target was observed in the H band with VLTI/PIONIER, and no excess was detected ($-0.43\\%\\pm 0.29\\%$ ).",
"The JouFLU excess, along with a non-detection using VLTI/PIONIER, allow us to make a rough estimate of the maximum dust temperature.",
"To estimate this temperature, we assume that the excess is due to dust emitting as a graybody, and we further assume that dust is not confined to a certain distance to the star.",
"We estimate that $\\sim 1000\\,\\mathrm {K}$ dust, potentially responsible for the $\\sim 1.7\\%$ excess at $2.1\\,\\mu \\mathrm {m}$ , would likely be undetected by VLTI/PIONIER, since it would correspond to a $\\sim 0.3\\%$ excess at $1.6\\,\\mu \\mathrm {m}$ .",
"Figure: Squared visibility as a function of hour angle for the E1-E2 (65m65\\,\\mathrm {m}) baseline.",
"For these excess star candidates, there is a clear discrepancy between the data and the photosphere model.",
"For both stars the detection significance is above 3σ\\sigma with an acceptable reduced χ ˜ 2 \\tilde{\\chi }^2.",
"However, most of the detection significance of HD210418 is due to a single data point, so more data are needed to confirm this excess."
],
[
"Survey excess stars with evidence for binarity: HD33111, HD182640, HD162003, HD5448",
"HD33111 is an A-type sub-giant displaying a highly significant excess of $(3.21\\pm 0.37)\\%$ .",
"However, as shown in Fig.",
"REF , the high reduced $\\tilde{\\chi }^2$ of 6.34 indicates a strong disagreement with the pole-on dust model adopted here (Eq.",
"REF ), that is, emission is not uniform in the azimuthal angle range of $\\sim 82^{\\circ }-102^{\\circ }$ .",
"In view of the large visibility fluctuations over small ($\\sim 10^{\\circ }$ ) changes in azimuthal angles, we do not attribute the excess to uniform circumstellar emission, but possibly to binarity.",
"Instead of reporting an excess level for suspected binaries, it is more relevant to report a visibility deficit, which depends on the separation vector at the time of observation, but is more directly related to the brightness ratio between binary components.",
"The visibility deficit for this object is $1-V/V_{\\star }=(3.26\\pm 0.38)\\%$ .",
"However, we note that this target was observed with the VLTI/PIONIER in the H band by [20], who did not report an excess for this target.",
"It is possible that the faint companion was outside the PIONIER FoV during the observations (2012-12-17), or that it was too close to the main star to be resolved by their observations.",
"Figure: Squared visibility as a function of hour angle.",
"The S1-S2 (33m33\\,\\mathrm {m}) baseline was used for observing HD33111, HD182640, and HD5448, and the E1-E2 baseline (65m\\,\\mathrm {m}) was used for HD1620003.",
"All stars shown here display a highly significant K-band excesses that we attribute to binarity.",
"See Section .HD182640 is an F-type sub-giant of mass $1.65M_{\\odot }$ for which we detect a highly significant visibility deficit of $1-V/V_{\\star }=(7.5\\pm 0.4)\\%$ .",
"This star is part of a spectroscopic binary system with a semi-major axis of $\\sim 50\\,\\mathrm {mas}$ , well within the FOV of JouFLU.",
"Since this is a known binary, we mainly use these data to check that our measured visibilities are compatible with a binary model.",
"The stars of this system have masses of $0.67M_{\\odot }$ and $1.65M_{\\odot }$ [23].",
"Assuming temperatures of $7200\\,$ K, typical for an F0IV, and $4300\\,$ K for the faint companion, we estimate the K-band flux ratio to be $\\sim 6.6\\%$ .",
"Using this estimate of the contrast ratio, we make a detailed analysis of the compatibility of the data with a binary model, by using the orbital solution obtained with HIPPARCOS [22], as well as the ASPRO software [9], we simulate the squared visibilities at the time of our observations, shown in Fig.",
"REF .",
"The binary orbital parameters and the data are in reasonable agreement in view of the reduced $\\tilde{\\chi }^2=1.96$ .",
"We also report a significant K-band visibility deficit for HD5448 (Fig.",
"REF , bottom-left panel), namely $1-V/V_{\\star }=(3.0\\pm 0.5)\\%$ , using the S1-S2 ($\\sim 33\\,\\mathrm {m}$ ) baseline.",
"In view of the recent results obtained by [6] and [48], using the MIRC beam-combiner at the CHARA array, this visibility deficit is not likely related to hot dust, but rather due to a faint companion.",
"[48] report an H-band flux ratio of $(1.25\\pm 0.3)\\%$ using closure phase measurements, and their orbital parameters correspond to a highly eccentric orbit with a $\\sim 47\\,\\mathrm {mas}$ semimajor axis approximately oriented in the north-south direction, that is, closely parallel to the S1-S2 baseline used in the JouFLU observations reported here.",
"The mass of HD5448A is $\\sim 2M_{\\odot }$ , and the orbital parameters measured by [48] allow estimating the mass of the faint companion to $\\sim 0.81M_{\\odot }$ .",
"With these mass estimates, we assume temperatures of $8200\\,$ K and $5300\\,$ K, which allow us to estimate the K-band flux ratio of $\\sim (1.4\\pm 0.5)\\%$ , which is in turn $\\sim 2.3\\sigma $ lower than our measurement.",
"If we use the orbital parameters found by [48], and the ASPRO software to simulate the squared visibilities at the time of observation, the best fit binary model predicts an essentially constant squared visibility of $\\sim 0.97$ at the time of observations, which is in acceptable agreement with the data as evidenced by a reduced $\\tilde{\\chi }^2=1.9$ .",
"One of the highest detected K-band excess of all the targets is for HD162003A as shown in Fig.",
"REF (botton-right panel).",
"We report a visibility deficit of $1-V/V_{\\star }=(7.7\\pm 0.75)\\%$ at the E1-E2 ($\\sim 65\\,\\mathrm {m}$ ) baseline, where the photosphere model was derived from long-baseline measurements obtained by [10].",
"This object is part of a visual binary star with an angular separation of $\\sim 30.1^{\\prime \\prime }$ , which is well beyond the FOV of JouFLU ($\\sim 0.5^{\\prime \\prime }$ in radius), so we are not detecting HD162003B.",
"HD162003A, with a mass of $1.43M_{\\odot }$ , does have a long period radial velocity trend as reported by [56], corresponding to a low mass M-dwarf of $(0.526\\pm 0.005)M_{\\odot }$ [19], but a complete binary model cannot yet be determined from observations.",
"We can use the stellar masses to assume the temperatures of the binary components to be $6500\\,$ K and $3700\\,$ K, to estimate the K-band flux ratio of $\\sim 6.2\\%$ .",
"This estimated K-band ratio is compatible with the measured visibility deficit, so we are possibly directly detecting this unseen companion for the first time."
],
[
"Results of the follow-up observations of the initial FLUOR survey",
"All of the stars that were part of the follow-up program had been reported as hot dust candidates, with the exception of HD9826 and HD142860.",
"The follow-up measured excess levels are shown in Table REF .",
"Most excess levels are compatible with those reported by [2], as shown in the ninth column of Table REF , but four stars differ significantly: namely HD9826, HD142091, 187642, and HD173667, which is a newly suspected binary as we discuss below.",
"We also note that the typical uncertainties for the follow-up program, shown in Table REF , are larger than those published by [2], by about a factor of between approximately two and three, for reasons that are currently under investigationSome polarization mismatches were detected in the JouFLU beam combiner back in 2013 [49].",
"They caused a significant (x2.5) decrease in the instrumental point source visibility, and were also suspected to amplify the instrument sensitivity to varying observing conditions (e.g temperature or zenith angle).",
"In June of 2016, we installed Lithium Niobate plates that compensate for differential birefringence, as described by [32], which resulted in a notable increase in the raw visibility values, from $\\sim 0.3$ in 2013, to $\\sim 0.6$ in 2016..",
"Therefore, of the nine stars that were reported as displaying a circumstellar excess by [2], we only re-detect four as having a significant circumstellar excess attributed to dust.",
"We discuss some results for individual targets below.",
"The object HD9826 ($\\upsilon $ And) is known to host four Jovian (RV) planets as close as $\\sim 0.06\\,$ AU ($\\sim 4.4\\,$ mas).",
"$\\upsilon $ And does have known stellar companions [35], but these are well outside of the FOV at 55, 114, and 287”.",
"[2] reported a circumstellar excess for this star of $(0.53\\pm 0.17)\\%$ (S1-S2, 33m baseline), which has a significance of $3.1\\sigma $ , but did not report it as a hot dust candidate because the excess level significance remained marginal.",
"We report a much higher K-band circumstellar excess of $(3.62\\pm 0.61)\\%$ when using the same $\\sim 33\\,\\mathrm {m}$ baseline as shown in Fig.",
".",
"The data are consistent with no evidence for a significant azimuthal variation in the range $\\sim 75^{\\circ } - 95^{\\circ }$ , for both the JouFLU and the previous FLUOR observations.",
"Since the position angles of the FLUOR and JouFLU observations cover essentially the same azimuthal range, it is much less likely that the excess varibility is due to the visibility signature of an inclined disk.",
"If this excess is indeed due to exozodiacal dust emission, we are therefore detecting variabilityVariability on the exozodi light level has also been detected on at least one other object (HD 7788) by [21].",
"in the exozodi light level of this object.",
"This star has been observed extensively by RV instruments (e.g., [12]), and a bright stellar companion contributing several percent of the infrared flux within the 0.5” JouFLU FOV would have been likely detected, unless its orbit is seen very close to pole-on and in a very different plane than the planets, which is unlikely.",
"We also exclude the possiblity of binarity based on other interferometric campaigns: an extensive search for close companions on HD9826 was made with the CLASSIC beam combiner [5] in the K band, and with the MIRC beam combiner (H-K bands) [61] at the CHARA array, and none found any stellar companions within 1”.",
"MIRC in particular would have detected a close companion contributing $\\sim 1\\%$ of the flux, and located anywhere between a few milliarcseconds to 1” away from the star.",
"Table: Circumstellar emission for the follow-up targets of .",
"The highlighted boxes in the sixth column correspond to statistically significant excesses: red boxes indicate that the excess is statistically significant, and also consistent with the circumstellar dust model of Eq.",
".",
"We highlight in yellow the excess of HD 173667 because it is likely due to binarity.",
"We also report the FLUOR excess levels reported by (Col. 9), and the change in circumstellar emission Δf cse =f cse, JouFLU -f cse, FLUOR \\Delta f_{cse} = f_{cse,\\mathrm {JouFLU}}-f_{cse,\\mathrm {FLUOR}} (Col. 10), the difference between the results obtained here, and those published by .",
"We find discrepant results for four stars, which are highlighted in the last column.",
"See text for details.In addition, speckle observations on HD9826 were recently made with NESSI (NN-EXPLORE Exoplanet & Stellar Speckle Imager) at the WIYN observatory in October of 2016.",
"This instrument is a dual-channel (562 and 832 nm) speckle camera that uses two Andor EMCCD cameras based off of the previous DSSI (Differential Speckle Survey Instrument) for diffraction-limited imaging (see [25]).",
"We use these data to constrain the possibility that the excess flux thought to be due to exozodiacal dust is the result of a faint companion.",
"The wider FOV of 2.3” from the speckle observations rules out any such companion at the edge of the JouFLU FOV.",
"An upper limit of the flux ratio at 832$\\,\\mathrm {nm}$ of a companion located 0.1” away was found to be 1.6%, which allows us to exclude any hypothetical companion of earlier spectral type than M8V/M9V (T $\\ge $ 2500$\\,$ K).",
"This contrast limit increases with greater separation of the hypothetical companion, reaching 0.4% (T=2100$\\,$ K) at the extent of the JouFLU FOV and 0.06% (T=1700$\\,$ K) at the extent of the speckle FOV.",
"Even in the minimal separation case the speckle observations exclude companions of earlier spectral type than M8V/M9V (T $\\ge $ 2500 K), and provides supporting evidence that the observed JouFLU excess is not due to binarity.",
"Figure: Measured K-band excess for HD9826 (left panel) with the S1-S2 (33m) baseline.",
"Also shown is the binary behavior of HD173667 (right panel).In the case of HD142091 ($\\kappa $ CrB), we detect an excess of $(3.4\\pm 0.52)\\%$ , with no significant azimuthal variation.",
"This detection is also particularly interesting since this star is known to host two exoplanets [26].",
"Moreover, resolved images of a debris disk extending as far as $\\sim 300\\,\\mathrm {AU}$ from the star, were obtained by [8] using Hershel in the far-infrared.",
"[8] also performed high contrast observations in the H band with NIRC2 and the Keck adaptive optics system, and upper limits allow discarding a $\\sim 0.06\\%$ companion located $\\sim 0.2^{\\prime \\prime }$ away from the star, with their contrast limit increasing with greater separation of a hypothetical companion.",
"The JouFLU observations were performed at two different baselines, and we measured a $(3.8\\pm 0.8)\\%$ excess with the E1-E2 (63$\\,$ m) baseline, and $(3.7\\pm 1.4)\\%$ with the S1-S2 (33$\\,$ m) baseline, both measurements consistent with each other, although with some very low visibility outliers that can be seen in Fig.",
"REF .",
"The radial velocity and high contrast observations allow ruling out any stellar companion closer than 0.2”, so the K-band circumstellar emission detected by JouFLU, and previously by [2], most likely comes from hot dust within $\\sim 12\\,$ AU of the star, where at least one of the planets is thought to lie.",
"This star has started to depart from its main sequence, but K-band excess is not likely due to partially resolved atmospheric structures, because we note that in order to explain the K-band excess, the apparent angular diameter would need to be $\\sim 20\\%$ greater than the adopted value of $(1.54\\pm 0.009)\\,\\mathrm {mas}$ found by [4].",
"We see a statistically significant increase of $\\sim 2\\%$ in the K-band excess level compared to results published by [2].",
"However, it is not straightforward to claim a variability in the circumstellar emission level at this point, since the significant JouFLU excess was measured using the E1-E2 ($63\\,$ m) baseline, whereas the excess reported by [2] relied on S1-S2 (33$\\,$ m) baseline data.",
"Figure: Left Panel: Measured a significant K-band excess for the E1-E2 (63\\,m) baseline.",
"Right panel: measurements using the S1-S2 (33\\,m) baseline.HD187642 (Altair) is displaying significant variability in its circumstellar emission.",
"This star displayed the highest K-band excess reported in the [2] sample, namely $(3.07\\pm 0.24)\\%$ , using the S1-S2 ($33\\,$ m) baseline.",
"Using the same baseline, we report a very high excess of $(6.11\\pm 0.74)\\%$ as shown in Fig.",
"REF , as infered by the visibility deficit relative to the stellar model derived by [43].",
"We can exclude a faint companion based on extensive observations with CHARA/MIRC in the H band, which resulted in detailed images of the surface of this star [43].",
"The closure-phase signals of the MIRC observations have been shown to be capable of detecting faint ($\\sim 1\\%$ ) [48] and close ($<0.5^{\\prime \\prime }$ ) stellar companions, and would have easily detected a companion responsible for the K-band excess.",
"Figure: K-band excess of 6.1%6.1\\% was detected for Altair, using the S1-S2 (3333\\,m) baseline.Another case where we see a discrepancy is HD173667 (110 Her), which was previously reported to have an excess of $(0.94\\pm 0.25)\\%$ by [2].",
"We detect a higher excess of $(2.34\\pm 0.37)\\%$ using the same baseline (S1-S2, 33$\\,$ m).",
"The data display large ($\\sim 20\\%$ ) fluctuations in the squared visibility as shown in Fig.",
", which result in a very poor fit to both the photosphere model, derived from parameters obtained by [10], and the star and dust model.",
"The large modulation of the squared visibility measurements obtained versus hour-angle suggests that the detected excess is likely due to a stellar companion.",
"However, a nearly edge on disk with very bright structures can not be entirely discarded given our limited observations.",
"Interestingly, we can discard the possibility that any of these changes are due to the new data reduction software (DRS): we used the new DRS to re-analyze some of the data previously obtained with FLUOR, and found excess levels and uncertainties compatible with those published by [2]."
],
[
"Overall statistical analysis",
"For the analysis presented below we did not include JouFLU survey targets which show strong evidence of binarity (HD33111, HD162003, HD182640, HD5448).",
"This conservative approach leads to two new JouFLU excess detections imputable to dust among the 29 - presumably- single stars observed with the JouFLU survey.",
"Similarly, for the FLUOR sample [2], we did not include the confirmed binary HD211336 [39], resulting in 12/41 FLUOR detections."
],
[
"Significance distribution of the JouFLU survey",
"First we examine the circumstellar excess levels, and their uncertainties, for the new stars added through the CHARA/JouFLU 2013-2016 survey.",
"In the left panel of Fig.",
"REF we compare the measured circumstellar emission levels measured in the [2] FLUOR survey, with the emission levels measured by the JouFLU survey (this work).",
"The weighted mean of the measured circumstellar emission ($f_{\\mathrm {cse}}$ ) of the FLUOR targets is $(0.06\\pm 0.05)\\%$ and $(0.36\\pm 0.12)\\%$ for the JouFLU targets.",
"We note that there is a significant positive bias on the measured excess levels for the JouFLU survey with a significance of $\\sim 0.36/0.12=3.0$ , which may be evidence of an underlying population of undetected excess stars.",
"As shown in the right panel of Fig.",
"REF , there is a notable increase in the median uncertainty: from $0.27\\%$ for FLUOR to $0.78\\%$ for JouFLU.",
"To some extent, this higher uncertainty is expected, given that stars in the JouFLU sample are typically fainter than in the FLUOR survey by a magnitude of approximately one.",
"Figure: Measured circumstellar emission (left panel) and its 1σ1\\sigma uncertainty (right panel) for the new JouFLU targets (blue), and the FLUOR circumstellar emission values reported by (green)Now we compare the significance distributions of the JouFLU and FLUOR samples separately, where by significance we mean the ratio of the measured excess level to its measured uncertainty ($f_{\\mathrm {cse}}/\\sigma _{\\mathrm {cse}}$ ).",
"In Fig.",
"REF , we show the significance histogram reported by [2] and compare it with the signficance histogram derived for the JouFLU targets.",
"Both significance distributions are biased toward positive excess.",
"In particular, the distribution of the JouFLU survey has a median of 0.73.",
"As stated above, this bias may be evidence of a population of undetected excess stars that we can further quantify below.",
"By assuming that the negative significance bins are due to Gaussian noise, we can symmetrize the distribution around zero significance to estimate the instrumental noise distribution.",
"We then fit a Gaussian to this JouFLU instrumental noise distribution, taking into account the finite bin-width, and we find a best-fit standard deviation of 0.84.",
"This is fairly close to unity, although a bit smaller, which may be expected from a limited number (12) of independent realizations of $f_{\\mathrm {cse}}/\\sigma _{\\mathrm {cse}}$ .",
"This may also suggest that our error bars are slightly overestimated.",
"Figure REF shows the instrumental noise distribution and the excess significance measured for the JouFLU targets.",
"As already discussed in Section REF , two of the presumably single stars retained for this analysis do show a bona fide K-band excess, meaning that they are detected at the $3\\,\\sigma $ level or higher.",
"But there are also many more targets with excess significance levels between 1 and $3\\,\\sigma $ than predicted by instrumental noise alone.",
"In addition to the two stars with significant excesses, this suggests that approximately ten more targets have excesses close to the JouFLU detection limit.",
"We just do not know which ones.",
"Regarding our finding of a positive mean value of the circumstellar emission: it is possible that calibration biases could result in a decrease of the transfer function, either because the calibrator itself exhibits circumstellar emission, or because of a decrease in atmospheric coherence time [21].",
"However, this would result in a negative bias, rather than a the positive which we find.",
"Interestingly, the approximately ten undetected excesses that we estimate above would yield a detection rate that is compabible with that reported by [2].",
"Also, since the estimate of the instrumental noise from the negative significance values is consistent with a normal distribution, it is most likely that the origin of the positive mean circumstellar emission is indeed astrophysical.",
"Figure: Significance histogram for the JouFLU survey shown in blue, and for the FLUOR survey shown in green.",
"The red curve is an estimate of the instrumental noise of JouFLU, which is computed by fitting a Gaussian to a symmetrized JouFLU distribution around zero.",
"The difference of between the instrumental noise and the JouFLU significance distribution allows to estimate the number of undetected excesses, which is around ∼9\\sim 9."
],
[
"Significance distribution of the JouFLU extension and follow-up observations ",
"Next, we examine all the targets observed with JouFLU, including the follow-up targets of the [2] survey presented in Table REF , yielding a total of 41 stars after removing all the excesses due to binarity.",
"This is clearly a biased sample since most of the follow-up targets were previously reported as excess stars.",
"For this biased sample, we report 7/41 excesses attributed to uniform circumstellar emission.",
"The significance distribution of all of these JouFLU observations is shown in Fig.",
"REF .",
"It has a median of 0.86, naturally more biased than the distribution of the JouFLU survey alone (Fig.",
"REF ).",
"We again estimate the JouFLU instrumental noise by using the negative significance values (Fig.",
"REF ), find the best fit standard deviation of the noise is 1.02.",
"This is even closer to unity than with the JouFLU survey targets alone, again supporting our errorbar calculations.",
"Figure: Significance histogram that includes all the targets observed with JouFLU, including the follow-up targets of presented in Table"
],
[
"Significance distribution of calibrators",
"To further confirm the validity of our results, and in particular, to confirm that the positive mean of the significance distributions of the science targets is indeed of astrophysical origin, we computed the circumstellar excess levels of the calibrators.",
"Since we nominally use three different calibrators for each science target, we can treat each calibrator as if it were a science target by calibrating it with the remaining two calibrators.",
"So, for a set of three calibrators $cal_1$ , $cal_2$ , $cal_3$ , we can compute the circumstellar excess of $cal_1$ , by using $cal_2$ and $cal_3$ to calibrate $cal_1$ , and similarly measure the excess of either $cal_2$ or $cal_3$ .",
"In Fig.",
"REF (left panel) we show the resulting circumstellar excess distribution of calibrators, and we note that this distribution is indeed unbiased, with a mean circumstellar excess of $(0.24\\pm 0.37)\\%$ .",
"For the particular case of the FLUOR follow-up targets, we find an unbiased mean circumstellar excess level of $(-0.21 \\pm 0.73) \\%$ .",
"From Fig.",
"REF (right panel), which shows the uncertainty distribution of the circumstellar excess levels, we note that the median uncertainty is $2\\%$ , somewhat noisier than for the science targets because each calibrator was observed fewer times ($\\sim 1/3$ ) than its corresponding science target.",
"Figure: Circumstellar excess level of the calibrator stars used in this study (left panel) and its 1σ1\\sigma uncertainty (right panel).",
"For each calibrator triplet, we used two calibrators to calibrate the remaining calibrator.",
"We note that this distribution is unbiased, with a mean of 0.24±0.37%0.24\\pm 0.37\\%.Figure: Circumstellar excess significance level of the calibrator stars used in this study.",
"We note that this distribution is unbiased, with a mean of 0.12, and a standard deviation of 1.03.In Fig.",
"REF we show the significance distribution of the calibrator excess level, with a mean significance of $0.12$ and a standard deviation of $1.03$ .",
"We also find no calibrators with excesses above $3\\sigma $ or below $-3\\sigma $ .",
"In contrast, we do find several target stars above the $3\\sigma $ detection threshold.",
"The fact that we find no significant bias in this calibator analysis strongly supports the validity of our results."
],
[
"Significance distribution of the combined JouFLU and FLUOR samples",
"Now we consider a couple of ways of combining the JouFLU and FLUOR samples, which differ in the way we include the follow-up targets.",
"First, we use the 40 FLUOR measurements and add the 29 single new star measurements from JouFLU.",
"This is shown in Fig.",
"REF , which has a corresponding detection rate of 15/69, or $21.7^{+5.7}_{-4.1}\\%$ .",
"Second, we use the same sample of stars but now use the JouFLU measurements for the 11 FLUOR stars that JouFLU followed up, which results in a detection rate of 11/69, or $15.9^{+5.3}_{-3.8}\\%$ , as shown in Fig.",
"REF .",
"The detection rates for these two ways of combining the JouFLU and FLUOR samples are compatible with each other.",
"That is, by computing the Kolmogorov-Smirnov statistic, we note that these two distributions are likely derived from the same distribution, since we obtain a p-value of 0.94.",
"Also, according to the results of the Shapiro-Wilk test, the probabilities that these two distributions are derived from a normal distribution are $5.6\\times 10^{-7}$ and $0.0120$ respectively.",
"We also estimate the significance distributions of the noise, again by using the negative significance bins, and we find that the standard deviations using the old and new follow-up data are 1.07 and 0.92.",
"Both are close to unity, as expected.",
"Figure: JouFLU and FLUOR samples combined.",
"The blue histogram includes the JouFLU survey and the revisited (follow-up) targets with JouFLU.",
"The blue histogram shows uses the original significance values for the follow-up targets obtained by .",
"The dotted lines are the estimated instrumental noise significance distributions for the two different ways of combining the JouFLU and FLUOR observations.Since the FLUOR targets have detections with a higher signal-to-noise ratio, in the subsequent analyses of the joint FLUOR+JouFLU sample, we used the older FLUOR results along with results obtained for new targets by JouFLU, shown in Fig.",
"12."
],
[
"Correlations with spectral type and detected cold dust",
"Now we look at how the detection rate of 15/69 is distributed for different spectral types and the presence of an outer cold-dust reservoir as shown in Table REF and Fig.",
"REF .",
"In Fig.",
"REF , we can see that the K-band excess occurrence rates for all spectral types are compatible with each other, showing a small decrease in excess rate for later-type stars, but not statistically significant.",
"This tentative trend was also seen in the H-band interferometry data by [20].",
"It is reminiscent of the spectral dependency seen in far infrared excess rates, which are tracing the presence of colder outer debris disks [53], [11].",
"The initial K-band FLUOR data had a significantly higher detection rate for A-type stars [2], but this is no longer the case when the FLUOR and JouFLU samples are combined.",
"This could be an effect of the increased error bars of the JouFLU survey.",
"Table: K-band excess detection rate for stars of different spectral types, with and without an outer cold dust reservoir (as inferred from a MIR or FIR excess)Figure: Excess rates of Table .",
"Left panel: excess rate for different spectral types.",
"Right panel: excess rate with or without a detected cold-dust reservoir.",
"The error-bars are asymmetric, are and computed by numerically integrating the binomial distribution.From Fig.",
"REF we also note that the excess rate for stars with a detected outer reservoir is higher, but once again, this is not statistically significant.",
"However, if we split the cold-dust and no-cold-dust detections into their respective spectral types, as shown in Fig.",
"REF , we note that FGK-type stars tend to display excesses more frequently when they have a corresponding cold dust reservoir, while the same cannot be said about A-type stars.",
"To quantify the significance of this trend, we assume binomial statistics for the excess rates shown in Table REF , and we find that there is a 99% probability that FGK stars display an excess more frequently when they have a known cold reservoir, than when they do not have a known cold reservoir.",
"This trend was also present in the initial FLUOR survey performed by [2].",
"The above analysis leads us to suspect that the mechanism responsible for the circumstellar excess is different for A-type stars than for FGK stars; or that there are two mechanisms behind A-type excesses, and one for FGK excesses.",
"Figure: Excess rates for different spectral types.",
"The blue points represent excesses that have a corresponding detected cold reservoir.",
"The green points represent excesses that do not have a detected cold reservoir.",
"The red points represent the combined excesses with and without detected cold dust.",
"Note that FGK stars have higher excess rate when they are known to have a bright reservoir of cold dust."
],
[
"Correlations with rotational velocity",
"Next we investigated the excess rate as a function of the rotational velocity, or rather its apparent rotational velocity $v\\sin i$ , given the system's inclination angle $i$ .",
"From Fig.",
"REF , it is not obvious that detections favor either higher or lower values of $v\\sin i$ .",
"However, we note a small accumulation of non-detections at lower values of $v\\sin i$ as can be seen in the figure.",
"If we take the median values of $v\\sin i$ for the 14 detections and the 54 non-detections we find Figure: Rotational velocity (vsiniv sini) as function of detection significance (f cse /σ cse f_{\\mathrm {cse}}/\\sigma _{\\mathrm {cse}}).",
"Red points correspond to A-type stars, and blue points correspond to FGK-type stars.$\\mathrm {Median}&(v\\sin i)_{\\mathrm {non-det}}= 6.6\\pm ^{4.4}_{2.1}\\mathrm {km/s}\\\\ \\nonumber \\mathrm {Median}& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"(v\\sin i)_{\\mathrm {det}}= 17 \\pm ^{10.3}_{5.0}\\mathrm {km/s},$ where the uncertainties for the median where estimated by generating many bootstrapped $v\\sin i$ samples, and finding the $68\\%$ confidence interval of the distribution of bootstrapped medians.",
"The median value of $v\\sin i$ for the detections is higher than for the non-detections, and we can estimate the significance of this result using bootstrapping: we generated many bootstrap samples of the detections and non-detections and count the number of times that the median $v\\sin i$ of the detections is higher than for the non-detections.",
"We find a probability of $\\sim 88\\%$ that the median $v\\sin i $ for detections is larger than the median $v\\sin i$ of non-detections.",
"While this is not highly significant, it is an interesting trend that we can further explore by adding the 92 stars observed by [20].",
"In this case we find $\\mathrm {median}(v\\sin i)_{\\mathrm {non-det}}= 9.1\\pm ^{2.4}_{1.9}\\mathrm {km/s}$ and $\\mathrm {median}(v\\sin i)_{\\mathrm {det}}= 16 \\pm ^{6.0}_{3.0}\\mathrm {km/s}$ , and the probability that excess stars have a higher median $v\\sin i$ now increases to $91\\%$ .",
"This statistical trend is particularly interesting in view of the recent theoretical study performed by [47], which attributes the hot dust phenomenon to magnetic trapping of dust nano-grains close to the star, without the need for very high magnetic fields, and predict that there should be a correlation with high rotational velocities.",
"We also note that there are a wide variety of stellar spectral types and hence rotational velocity ranges in the detection and non-detection samples.",
"So to verify that this trend is not an effect of spectral type rather than rotational velocity, we compute the median spectral type of detections and non-detections, and find F6 for F6.5 respectively, which are very close to each other."
],
[
"Other possible correlations",
"We have also investigated possible correlations of the excess levels with the existence of detected exoplanets, but have not found such a correlation.",
"There are 8 known exoplanet host stars in the FLUOR and JouFLU combined samples: HD142091, HD219134, HD217014, HD190360, HD117176, HD69830, HD20630, and HD9826.",
"We detect a K-band circumstellar excess for two such planet host stars: HD9826 ($\\upsilon $ And) and HD142091 ($\\kappa $ CrB).",
"The other stars are consistent with a non-excess, and with good fits to stellar photosphere models.",
"Last, we investigated possible correlations with stellar age.",
"Similar to [2] and [20], we do not see any significant correlation with stellar age."
],
[
"Discussions and conclusions",
"The CHARA/JouFLU beam combiner was used to extend the initial CHARA/FLUOR survey and search for exozodiacal light originating from within a few AU from main sequence and sub-giant stars.",
"In addition, JouFLU performed follow-up observations on most of the FLUOR excess stars to search for possible variability in the circumstellar emission level.",
"By extending the survey, JouFLU observed targets that were fainter in the K band by a magnitude of approximately one.",
"We report 6/33 new circumstellar excesses at the $\\sim 1\\%$ level or higher, relative to the star, out of which two detections are attributed to uniform circumstellar emission, and the other four being known or suspected binaries.",
"Our detection rate of 2/29, after removing binaries from the sample, is clearly incompatible with the rate of 12/41 reported earlier by [2], but this is most likely an effect of our degraded precision by a factor of between approximately two and three.",
"Indeed, we note that if we artificially increase the original FLUOR error bars by a factor of two, we obtain a detection rate of 3/41, which is compatible with the JouFLU detection rate presented here.",
"The fact that we are still detecting K-band excesses for some of the targets originally observed by [2], with a modified instrument, and new data reduction software, is supporting evidence that these K-band excesses are real, and not an instrumental or data reduction artifact.",
"Also, the fact that our data for binaries with known orbital parameters are compatible with expected values strengthens the reliability of our results.",
"After combining the FLUOR and JouFLU samples as described in Section REF , we have a detection rate of 15/69 or $21.7^{+5.7}_{-4.1}\\%$ .",
"Since there is some disagreement between the JouFLU and FLUOR results, it is worth discussing some potential causes: Underestimation of the circumstellar excess uncertainty.",
"As discussed in Section , we estimated the instrumental noise significance distribution and verified that it is indeed consistent with a normal distribution.",
"An underestimation of the circumstellar excess uncertainty would result in a standard deviation greater than 1 in the instrumental noise significance distribution, which is not the case.",
"A systematic bias in the measured excess level.",
"As discussed in Section REF , we measured the circumstellar excess level of the calibrators, and showed that there is no statistically significant bias in their excess levels.",
"For the follow-up targets, and in particular for the targets whose excess level changed between the JouFLU survey and the FLUOR survey, we verified that calibrators could be calibrated against each other.",
"Intrinsic variability in the near-infrared excess.",
"Since we found no evidence for potential causes 1 and 2, we attribute the discrepancies between the JouFLU and FLUOR results to intrinsic variability which we further discuss below.",
"Among the JouFLU circumstellar excess detections which seem to display variability are two exoplanet host stars: HD9826 ($\\upsilon $ And) with a measured circumstellar excess of $(3.62\\pm 0.61)\\%$ , and HD142091 ($\\kappa $ CrB) with an excess of $(3.4\\pm 0.52)\\%$ .",
"Both of these systems had been previously detected by [2] with FLUOR, although with a much lower circumstellar emission level, namely $(0.53\\pm 0.17)\\%$ for HD9826, and $(1.18\\pm 0.19)\\%$ for HD142091.",
"These systems have been observed extensively with other techniques, that allow excluding the possibility that a faint stellar companion could account for the measured JouFLU and FLUOR excesses.",
"Therefore, these excesses are most likely due to extended emission, which we attribute to hot dust, and point to large dynamical activity in these systems.",
"We note that the typical orbital period of dust at the sublimation radius is of the order of a few days, so there is really no reason to expect a constant excess measurement between the FLUOR and JouFLU observations, which were made a few years apart.",
"These two stars should be considered as high priority targets for follow-up observations using high contrast high resolution instruments in the near- to mid-infrared.",
"The origin of these excesses remains difficult to explain.",
"A possible explanation, alternative to hot dust, is that these excess stars have undetected faint companions.",
"A-type stars are more likely to have undetected faint compaions, as they show fewer spectral lines than cooler stars and also tend to be more active, making spectroscopic searches for companions more difficult.",
"An interferometric search for faint companions on the [20] sample was carried out by [36] using precise closure phase measurements.",
"There are 92 stars in the sample of [20], among which 30 are A-type stars, and the five new binaries found in the survey were all discovered around A-type stars.",
"This binary detection rate of 5/30 is actually compatible with the (FLUOR+JouFLU) detection rate of 6/20 for A-type stars (within $\\sim 1\\,\\sigma $ ).",
"Additionally, in the particular case of A stars, we found that there was no correlation with a detected cold-dust reservoir (Sec.",
"REF ).",
"So it is possible that the excesses detected around A-type stars are simply due to binarity.",
"However, even if we assume that all of the JouFLU+FLUOR A-type excesses (6/20) are due to binarity, which is very unlikely for the well studied ones such as Vega and Altair, we are still left with 8/48 detections for all the other spectral types, still a significant detection rate.",
"If we assume that the circumstellar excesses are not due to binarity, then they are most likely due to extended emission, which we attribute to hot dust.",
"Assuming a dust sublimation temperature of $\\sim 1500\\,\\mathrm {K}$ , we estimate the dust sublimation radius to be between $\\sim 2-30\\,\\mathrm {mas}$ for the JouFLU+FLUOR stellar sample, so the CHARA baselines should allow resolving dust for all targets.",
"The dust properties have been somewhat constrained by other observations at different wavelengths and spatial resolutions.",
"In Figure REF we show the expected K-band excess for thermally emitting dust with a K-band flux ratio of $1\\%$ in the K band, relative to a $6000\\,$ K star, and assuming that dust is not confined to be at a certain distance from the star.",
"According to the Figure, dust with gray blackbody emission in the near to MIR, should be more easily detected at longer wavelengths, for example $\\sim 10\\mu \\mathrm {m}$ , the central wavelength of the Keck Interferomer Nuller (KIN).",
"[41] used the KIN to search for exozodiacal light in the $8\\mu \\mathrm {m}-10 \\mu \\mathrm {m}$ band, and their sample included the FLUOR K-band excess stars detected by [2].",
"Even though the KIN excess levels predicted at 10$\\,\\mu $ m for gray emission dust are higher than in the near-infrared, none of the FLUOR excess stars showed a significant excess at 10$\\,\\mu $ m. The spatial resolution of the KIN ($\\sim $ 10$\\,$ mas) should also have allowed to resolve such dust in most cases.",
"This implies a very high dust temperature ($>1000\\,$ K), with grains located close to the dust sublimation radius ($\\sim 0.1\\,$ AU), and with typical grain sizes smaller than the blowout-size ($\\sim 1\\mu \\mathrm {m}$ ) to escape detection in the mid-infrared; or the dust emission is due to scattering, in which case it is more difficult to constrain its distance to the star.",
"However, scattering seems less likely in view of the lower excess detection rate in the H band of $10.6^{+4.3}_{-2.5}\\%$ [20], and in view of polarimetric observations performed by [37], which found no evidence of scattered light emitted from the NIR excess stars.",
"[28] modeled many hot dust systems, including KIN upper limits, and found that a population of reasonably hot, small grains is well consistent with the KIN results.",
"They also found that a fraction of the light can indeed be scattered light for their realistic grains which would still be consistent with a low polarization fraction from [37].",
"In the case of Vega, the dust location has been constrained using the Palomar Fiber Nuller, a near-infrared interferometer with a $3.4\\,$ m baseline, to be either within $0.2\\,$ AU, or beyond $2\\,$ AU [40].",
"Figure: Assuming thermally emitting dust, with a K-band flux ratio of 1%1\\% relative to a star of 60006000\\,K, the curves show the expected contrast as a function of wavelength, and each curve corresponds to a different dust temperature.",
"The shaded regions correspond to the accessible spectral bands for different interferometers, including second-generation instruments such as GRAVITY and MATISSE .Close-in sub-micron dust should have short lifetimes of the order of years [60], so if dust is really responsible for the K-band excesses, then there must be an efficient replenishing or dust trapping mechanism.",
"Several models exist to explain the persistence of dust, the most prominent are: comet in-fall and out-gassing [60], [54], [33], or magnetic trapping of small grains [47].",
"The JouFLU+FLUOR data allowed to test for the magnetic trapping model of [47], which predicts a correlation with high rotational velocities, and the data show a tentative correlation with angular velocity, but more data are needed to confirm this statistical trend.",
"In order to test for episodic events, we are currently performing an exozodi monitoring program with JouFLU/CHARA, led by Nicholas Scott, where we are revisiting excess targets every few months to search for variability in the circumstellar emission level.",
"It has been known for many years that a density enhancement can occur near the dust sublimation radius; as particles sublimate, the effect of stellar radiation pressure grows, slowing and eventually reversing their migration [44].",
"The effect of sublimation on the particle orbits has been explored in detail with numerical and analytic models [30], [31], [29].",
"The optical depth of the disk increases by factors of $\\sim 3-10$ near the sublimation radius, depending on the grain composition and stellar spectral type.",
"This increase in dust is promising, but is still not enough to explain the observations.",
"A possibility that is currently being investigated is the effect of the gas that is produced by the sublimation.",
"Gas can further exaggerate the inner accumulation of dust (and thereby help explain the observations) in several ways.",
"Firstly, the gas will circularize the eccentric orbits produced as the particles sublimate.",
"This circularization acts to increase the outward migration already found in the Kobayashi models, helping the dust to pile up somewhat farther outward and to sublimate more slowly.",
"Second, there will be direct gas drag on the dust particles.",
"While gas drag is often assumed to be inward (e.g., for dust in thick protostellar disks), in the gas and dust environment considered here, drag forces actually push the dust outward (with radiation pressure offsetting the central gravity, small particles orbit slower, not faster, than the gas).",
"Lastly, the gas acts to regularize the dust orbits, thereby lowering both their collision rate and their relative velocities.",
"As such, the combined effect of these gas forces serves to not only enhance the amount of hot dust, but also to decrease its removal rate, allowing for a smaller supply of mass to produce strong emission over the lifetime of a star."
],
[
"Acknowledgments:",
"This research was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory administered by Universities Space Research Association under contract with NASA.",
"PN and BM are grateful for support from the NASA Exoplanet Research Program element, though grant number NNN13D460T.",
"This work is based upon observations obtained with the Georgia State University Center for High Angular Resolution Astronomy Array at Mount Wilson Observatory.",
"The CHARA Array is supported by the National Science Foundation under Grant No.",
"AST-1211929.",
"Institutional support has been provided from the GSU College of Arts and Sciences and the GSU Office of the Vice President for Research and Economic Development.",
"This research has made use of the Simbad database, operated at the Centre de Données Astronomiques de Strasbourg (CDS), and of NASA's Astrophysics Data System Bibliographic Services (ADS).",
"Thanks to those developping the Aspro2 software and those maintaining the Jean Marie Mariotti Center (JMMC).",
"We would also like to thank Elliot Horch for providing a reduction of complementary speckle imaging data for HD9826 obtained from NESSI (NN-EXPLORE Exoplanet & Stellar Speckle Imager)."
]
] | 1709.01655 |
[
[
"The Voynich Manuscript is Written in Natural Language: The Pahlavi\n Hypothesis"
],
[
"Abstract The late medieval Voynich Manuscript (VM) has resisted decryption and was considered a meaningless hoax or an unsolvable cipher.",
"Here, we provide evidence that the VM is written in natural language by establishing a relation of the Voynich alphabet and the Iranian Pahlavi script.",
"Many of the Voynich characters are upside-down versions of their Pahlavi counterparts, which may be an effect of different writing directions.",
"Other Voynich letters can be explained as ligatures or departures from Pahlavi with the intent to cope with known problems due to the stupendous ambiguity of Pahlavi text.",
"While a translation of the VM text is not attempted here, we can confirm the Voynich-Pahlavi relation at the character level by the transcription of many words from the VM illustrations and from parts of the main text.",
"Many of the transcribed words can be identified as terms from Zoroastrian cosmology which is in line with the use of Pahlavi script in Zoroastrian communities from medieval times."
],
[
"Why is Voynichese difficult? ",
"All writing systems in the world [8], [5] require some effort in acquisition and use.",
"While for some groups of languages, difficulty and differences may be comparatively small [17], in others the complexity of the script can appear forbidding for all but a minority of scribes.",
"Religious observance, for example, may require the adherents to continue using a script or language that no longer adapts to its language environment and that may thus tend to become ambiguous or incomprehensible.",
"In order to retain a unique pronunciation and, supported by extensive commentaries, continuing understandability, glyphs (diacritics) from were added to letters to distinguish them, or additional letters (matres lectionis) were inserted to represent sounds (such as vowels in the consonant-based (abjad) scripts.",
"However, such additional efforts may not be considered necessary, if the oral tradition in the community is sufficiently strong, such that the texts do not have to be extracted from the writing itself, but are rather remembered while being read.",
"If the Voynich Manuscript (VM, MS 408 in the Beinecke Rare Book & Manuscript Library at Yale University) derives from such a tradition, the difficulty in reading it may be understandable.",
"The Voynich Manuscript (VM, MS 408 in the Beinecke Rare Book & Manuscript Library at Yale University) which is written on more than 200 vellum pages has been dated between 1404 and 1438 (University of Arizona, 2011), but its history is largely unknown until the discovery by the bookseller Voynich in 1912.",
"Apart from a few cautious attempts, such as Ref.",
"[3], so far little progress has been achieved in deciphering the VM nor even a decision was reached whether the VM has any meaningful content at all [19].",
"Our hypothesis that the VM is written in natural language, is to be evidenced by showing that the script used in the VM is directly related to Pahlavi, a writing system that was in use for several Iranian languages from before the current era at least until 900 [9].",
"Pahlavi is a particular case of a language that is notoriously difficult to read.",
"It was used in medieval scriptures, commentaries, and a few other texts [2] related to Zoroastrianism, the pre-Islamic religion of Persia.",
"Over the few centuries of the language evolution, many Pahlavi letters have coalesced, e.g.",
"for the phonemes d, g, j, and y, only a single letters is retained in Pahlavi.",
"Moreover, letters are usually joined in Pahlavi script and can appear thus similar to other letters: E.g., in addition to its proper meaning, a letter can be indistinguishable from as much a sixteen different phoneme or letter combinations [11].",
"In some words, corrupted forms of letters have become a standard that is accepted to various degrees by the scribes.",
"In addition to Persian words, Pahlavi contains also a large number of heterograms, i.e.",
"around a thousand, partially very common words of Aramaic origin that are meant to be read in Persian (like the Latin abbreviation i.e.",
"is read in English as that is).",
"Finally, as for many other ancient texts, material decay, language drift, scribe errors, unfamiliarity with the original cultural context, and, possibly, the need of the writers to hide the content from contemporary hostility, also contribute to the difficulty of reading the text.",
"Concerning recent work on the VM, statistical approaches [1], [10], [14] that search for non-random features in data may be bound to fail if the target is quite random to begin with.",
"The standard Voynich character set (EVA) [7] is not too helpful either, because it is unrelated to the phonemics, it breaks some of the letters into smaller parts, and fails to identify ligatures, all of which may further reduce the strength of the statistical analysis, cf.",
"[20], [21], [19], [10].",
"In addition, the extensive 19th century literature dedicated to religious writing, see e.g.",
"[15] was difficult to access until scanned copies became available online recently, and, finally, it may be construed that our academic habits thwart the systematic study of matters as obscure as the VM.",
"The Pahlavi hypothesis was proposed informally already in 2005 [18]I was not aware of this news-group post until I found a Twitter comment on the first version of the current paper where Ref.",
"[18] was mentioned..",
"The hypothesis is based there on the similarity of the numbers of letters (“14 - 17”) in the Voynich and Pahlavi alphabets and on a general perception of a topical relation to the Bundahesh and the Denkard.",
"Also a small sample of words lengths from a Pahlavi text was included, but was not compared to a transliteration of the Voynich text.",
"The present paper aims at providing evidence for the hypothesis that the VM is a readable text with an interest in itself.",
"Our approach consists in establishing a relation between the Voynich and Pahlavi scripts (see Section 2).",
"It will also become clear that only within a cooperation among experts in Pahlavi philology, Zoroastrianism, history of medicine, botany, astronomy and palaeography, the content of the VM can be revealed.",
"We will provide evidence for the proposed relation between the two alphabets by a number of examples from VM illustrations as well as from its running text (Section 3).",
"Finally, we will draw (in Section 4) some rather speculative conclusions on the context in which the content of the VM may have originated."
],
[
"Letters are reverted Pahlavi characters",
"Comparing the Voynichese and Pahlavi scripts, we find that many of the characters are upside-down versions of each other, see Table REF .",
"This may be due to the different writing direction of the two scripts.",
"A similar effect that was observed also in the earlier sinistrodextral Brahmi script [4], in which also some of the letters appear as upside-down adoptions from its likely predecessor Aramaic (right to left).",
"Pahlavi, that ultimately derives also from the Aramaic alphabet, has retained the dextrosinistral direction, while the VM is written in the opposite direction.",
"In this way, six of the about 20 Voynich letters can be explained directly (a, h, s, S, r, in our notation, see Table REF , and K, see Table REF ).",
"Two more letters (d, c) differ from s and S, respectively, only by an inverted breve diacritic.",
"In addition, there are three more letters that obtain by rotation about a different angle (t, y) or by mirroring (z).",
"The similarity of eleven out of the comparatively small number of letters of the two alphabets can be considered as a clear indication of a relation between Voynich (V) and Pahlavi (P).",
"Below we will see that the relation extends also to the phonemic level.",
"Two letters o and n that occur frequently in the VM, differ from their counterpart in the P alphabet.",
"It is tempting to relate V o to P pe, but we suggest rather an association to waw.",
"This also supported by the frequent use of o as a word separator in the VM.",
"In Pahlavi a vertical bar is used for this purpose, which is of similar shape as P v, while in the VM apparently the more distinctive letter o has been preferred.",
"Further analysis of the V text will shown whether o, y or a also have a grammatical function.",
"Based on the phonetic content (Sect.",
"3) of the letters, we assume that, in contrast to Pahlavi, the nasal alveolar is not part of the spectrum of V o, but is represented by V n. The remaining V letters are the “capitals” B, K, M, P, or occur only very rarely (see f57v for a number of other characters, some of which, however, occur nowhere else in the VM).",
"The shape of the V “capitals” may have arisen from a fusion of the respective Pahlavi characters with a vertical stroke (P word separator).",
"We do not consider the capitals to be functional ligatures, though, as they are used also within words or after the word separator (o).",
"Table: Voynich characters and initials together with variants of the correspondingPahlavi letters.",
"The last column shows the notation used here.",
"SeeBox 1 for comments.Box 1: Comments on Table REF .",
"The letters are given in the order to the Aramaic alphabet with resh taking the place of phonetically similar lamed, and jod is placed near daleth and gimel with which it is interchangeable in Pahlavi.",
"Frequently occurring corruptions are given in [brackets] [13].",
"Strokes from neighbouring characters are removed from the Voynich letters.",
"[a] Appears in B-Pahlavi as a raised character.",
"$\\aleph $ represents a glottal stop.",
"[B] We could not find enough evidence for systematic use of two variants (B and H) of this character.",
"[g] Occurs usually in final position, elsewhere y is used instead.",
"[o] V v resembles Syriac vav (Figure: NO_CAPTIONvav is identical to resh [y] the letter represented here is daleth.",
"The actual P-Pahlavi letter yod (Figure: NO_CAPTIONd and V c. Many words have an otiose y ending.",
"[c] This character occurs rarely in the VM, the mere fact that we did not identify a distinctive character for P č does not justify the transliteration of c by č.",
"[z] is often (or easily) confused with r. [P] occurs often at the beginning of paragraphs.",
"It may be an abbreviation of pad for to, at, in or on.",
"Table: Main ligatures and letter combinations from the VM.",
"Only part of theimplied phonemes are given in the third column.",
"The last column refersto the transliteration in Table .",
"Thetwo or three strokes of n or m have a similar functionsas h in the final or penultimate position.",
"Ligatures involvingthe letter V S (“table”) represent the succession of twoconsonants usually in the beginning of a word.",
"In some cases it iss rather than S that is represented.",
"While sP and sTare obvious from the vocabulary, the remaining combinations will haveto be reconsidered.",
"First part of ko occurs rarely if everalone.",
"This ligature can represent m, q, h, r,mn, mv, mr, mℵ\\aleph , etc.",
"The combinationscy and co appear to represent single phonemes in somecases, see Appendix .",
"All ligatures arecopied from f37r, the componentsin the second column are from Table .Strokes belonging to neighbouring characters were removed."
],
[
"Vocabulary relates to Zoroastrian religion",
"Voynichese and Pahlavi are not identical.",
"By the introduction of a number of additional characters, such as to distinguish d, g, n, reading a Voynichese text may have been easier than a Pahlavi text.",
"It is not clear why the history of the deciphering of the VM, does not support this claim.",
"Analysing plant and star names, Bax [3] has suggested a similar reading for some but not all of the letters.",
"We base our transcription on a larger number of samples from the manuscript and compare the results with names from the Zoroastrian cosmological scripture Bundahesh [24], [11], which was composed in the 11th century, and with general vocabulary [13], such that we arrive at a more complete and more reliable transcription that is based not only on the similarity of the letter shapes.",
"The translations given below should not be expected to do justice to the VM text.",
"They are solely included to provide evidence for the proposed transcription."
],
[
"Zodiac symbols",
"In the appendix, we show two sets of words from the manuscript.",
"The first (App. )",
"gives the names of the zodiac symbols and the corresponding month names both of which were passed down in the Bundahesh [24], [11] in paragraphs II, 2 and XXV, 20, respectively.",
"Based on the well known symbols shown in the centres of f70v1 – f73v, the identification with the Pahlavi names is straight-forward, expect for the two pages f71r and f72r1 which show the same symbols (Aries and Taurus) as f70v1 and f71v, respectively.",
"We cannot answer the question whether the two repeated signs do in fact represent the missing Capricorn and Aquarius.",
"Because two words (on f72r2 and f72v2) are unreadable due to creases, we are left with 18 words that are identifiable to a reasonable degree of certainty."
],
[
"Plant names ",
"Ancient plant names are occur in manifold variants and are often ambiguous.",
"The same seems true for the plant drawings in the VM, where, in some case, it seems even plausible that the artist followed merely a verbal description rather than an own view or any original drawings.",
"We can thus expect only a few characteristics to be identifiable.",
"In addition, only a few plant names are included in the standard dictionaries (e.g.",
"[13]), such that most of the VM plant depictions will require more research.",
"We will first consider two plants (henbane and cannabis) whose names are easily identifiable and where a visual comparison, see Fig.",
"REF can be considered as additional evidence for the text-based identification.",
"After this will report on some preliminary attempts, i.e.",
"we are not attempting a botanical identification of the plants [23] and should take into account that, even in comparison to other medieval depictions, the drawings are far from perfect."
],
[
"Henbane",
"The first word of f31r is BccNcy which can be transcribed as bang, see Tab.",
"REF , which uniquely translates [13] as henbane (hyoscyamus niger), a poisonous plant of the nightshades family.",
"The similarity of the drawing on f31r and the plant henbane is illustrated by Fig.",
"REF and can be considered as additional evidence for the translation.",
"Figure: Henbane from VM f31r (left) and from Martin Cilenšek's Našeškodljive rastline, 1892 (wikimedia file: Nsr-slika-088), (right),see Sect. .",
"While the leaf shape and the unilateralposition of the flowers roughly coincide, the shape of the flowersis dissimilar.",
"The VM may represent the ripened fruits of the plant."
],
[
"Cannabis",
"The first word of f16r is šcoN which can be transcribed as šān, see Tab.",
"REF , which uniquely translates [13] as hemp (cannabis).",
"The similarity of the drawing on f16r and to the cannabis plant is obvious: Leaves are neither clearly opposite nor alternating, they are shown to consist of seven to nine finger-like leaflets.",
"The spike-shaped flowers are probably female and are riddled with elongated leaflets, see Fig.",
"REF .",
"Figure: Cannabis.",
"from VM f16r (left) and from FranzEugen Köhler's Medizinal-Pflantzen, 1887.",
"(wikimedia id: 1739269),see Sect.",
"."
],
[
"Further observations on the plant pages",
"Among the first words on the plant pages, we find often šPīg (sprout), dān (grain), or dār (tree) which may be a general term or a component of a plant name that consists of more than one word.",
"E.g.",
"on f17r we find don (dān) which here, however, may refer to buckwheat.",
"Folio 21r shows a plant similar to box (buxus) or P šimšār.",
"The first word of the text is Sor (šār).",
"Near the end of the 7th line we find šomšor (with the middle m and š as an odd ligature).",
"The first word of Folio 24r can be read as alālag which would mean anemone (anemone blanda (?",
")), but the picture does hardly match, although the anemone family has a wide variety of leave shapes and numbers of petals.",
"The must be said for f45v which starts with the same word.",
"A problem with this reading is also that we otherwise ignored the waw after the initial “paragraph marker” P while it would be part of the plant name here.",
"Chick-pea in P is naxōd which can be found in the beginning of 5th line of 26v.",
"While the leaves are may be plausible, the drawn flowers are less typical, perhaps chick-peas are mentioned only for comparison here and, therefore, not in the beginning of the text.",
"The drawing on f41v is identified as coriander (Coriandrum sativum) [3], or gišnīz in P. Ref.",
"[13] gives also the variant kišnīz.",
"While the single word that makes up the first line is unrelated to this P word, the second line appears to give several variants, e.g.",
"the 2nd word contains Kš, the third word reads Kšnd.",
"Date palm in P us mu$\\gamma $ [13] which can be found as the third word of the first line in f56r.",
"The drawing shows a plant with at least the base of the stem and the lower pair of leaves reminiscent of a palm tree."
],
[
"Plant parts",
"Folio 100r gives an overview of shape types, most likely of leaves.",
"It contains six descriptive words for five pictures, see Fig.",
"REF .",
"We therefore, consider the first word in the upper row (dōspīg, i.e.",
"double spouted) is seen as the last word of the previous text.",
"The second word in the upper row Mht could be mih (false, opposite) or mahist (greatest).",
"The four words in the second row are less ambiguous.",
"We have Bīor (bahr, part) for a leaf consisting of three parts, rot (rōd, river) for a set of leaf stems that branch off like a river delta, tšhg, which may relate to (tašt, bowl), and Botr (BATR, a heterogram for pas [13], behind) for one leave behind the other.",
"This again is to be seen as evidence for our main hypothesis rather than as an exercise in Pahlavi transcription.",
"Figure: Shapes of plant parts (f100r)"
],
[
"Lunar mansions",
"In a similar way, it is possible to transcribe from the illustration on f69v most of the 28 lunar mansions that are also listed in Bundahesh II, 2 [11].",
"Because of the short and repeated Pahlavi names of the mansions, a unambiguous correspondence was possible for only 20 of the mansions, such that we did not include it here.",
"Interestingly, the 1247 stone representation of the Suzhou star chart (1193) that shows the related 28 Chinese constellations has a “cartouche” title beginning with the ideogram for sky that can also be seen in a corrupted and reverted form on f1r of the VM.",
"This is not implausible considering the continuous exchange between Persia and China in historic times."
],
[
"Zoroastrian material",
"The four words in the center of f67v2 are (with transcription) zoahd (zohr), oBarao (bahr), zary (zōr) and natag (nihadag).",
"The translation yields the words sacrifice, lot, power, foundation [13] that appear, given a Zoroastrian parentage, semantically related.",
"The words are grouped around a small square-shaped picture of a swirl-radiating star which could represent a sacrificial fire.",
"App.",
"includes a transcription of words from the beginning of the third paragraph of f1r.",
"This sample is included not only to show that the Pahlavi transcription applies to the main text, too, but also to demonstrate the difficulty of a translation of the text, which has, however, been noted by all translators of Pahlavi documents.",
"In the illustration on f77v, we find the words oBam yHat otBaNat orShNat oMot dhNy oMotor which can be transcribed as bīm duxt wad-baxt rēšinad mīh dēn wizār and is translated word-by-word as fear daughter unfortunately wounded: false (alternative?)",
"religion explanation [13].",
"This sample, nevertheless, suggests that the “nudes” pages (f75r – f84v) represent medieval medical content.",
"While the representation of nude bodies is rare in such contexts, similar scenes appear in contemporary miniatures from Mughal India, where, however, an erotic perspective is taken, which is not obvious in the VM."
],
[
"The colophon (f116v)",
"Further evidence for the proposed transcription can be obtained from the “colophon” (f116v).",
"The last line of the short text contains the words arar dccy that are, in contrast to the seemingly Latin script on this page, clearly readable.",
"We propose the transcription xwar day, which would refer to the 11th day of the 10th month of the Zoroastrian calendar [13].",
"The question whether the $\\cap \\!\\cap $ character before the lacuna at the end of this line was originally the initial character of a year, cannot be answered without further analysis of the velum.",
"Based on the Pahlavi hypothesis presented here, it seems possible to extract more information from the colophon.",
"In the App.",
"C, we present an attempt to read the colophon, which, however, is largely speculative, even if we assume that the Pahlavi hypotheses is true."
],
[
"Discussion ",
"We have not been presented more than a few words, which is mainly due to the inherent difficulties in reading Pahlavi.",
"Therefore, at this point it is not clear, whether the VM contains also words of a different idiom, such as the northwest Indian language Gujarati, whose Parsi dialect contains many Iranian words due to the Zoroastrian influence, although Gujarati does not itself identify as an Iranian language.",
"It is striking that the manuscript does not contain any obvious religious symbolism (apart from the crucifix on f79v, which may well be a later insertion) nor any other culturally identifiable elements.",
"However, the astronomical charts of the VM are related to the world of the Zoroastrian culture in the middle East or South Asia.",
"They do not show any awareness of (earlier?)",
"Arabic astronomy, but seem to follow the cosmological view in the Bundahesh.",
"Finally, we want to emphasise that we have no evidence that the VM was produced in Persia (or perhaps even western India).",
"It is also possible that it originates from the regions near the Black Sea where an exchange between Persia and the Italian cities of Genoa and Venice took place around the presupposed time of the production of the VM.",
"Our opinion that the content of the VM is meaningful does not exclude the possibility that it is still a “hoax”, in the sense that it was copied to be sold rather than read.",
"In this process or by later action, foliae with critical content may have been removed to further obscure the origin of the manuscript.",
"Although the proposed transcription is obviously tentative, it is now possible to find many of the VM words in a Pahlavi dictionary [13], [16], [11] using Table REF , which will give at least partial insight into the content of the VM.",
"We are also unable to provide a more precise phonemic account at this stage, although some of the differences (e.g.",
"between V d and P t) may allow for such discussions.",
"It will require a substantial effort to provide a complete translation of the VM, as it seems unlikely that large parts of the text have been passed down also from other sources, i.e.",
"the VM does not appear to be identical to any of the better known Zoroastrism-related scriptures or commentaries, so its content may as well have an interest on its own."
],
[
"Acknowledgement ",
"The author acknowledges the use of the high-resolution scans made available by Jason Davies' Voynich Voyager [6].",
"We are also grateful to two anonymous reviewers for their comments on an earlier version of this manuscript.",
"This earlier version is available as a working paper from the PURE repository (University of Edinburgh) since 1st of August, 2017.",
"This is the second version of the paper.",
"It differs from the first version by the deletion of some but not all superfluous text and by a reference [18] to an earlier mentioning of the Pahlavi hypothesis."
],
[
"Zodiac pages (f70v1-f73v)",
"All descriptions were found within the V script around the margin (for f70v2, within the margin) of the central image that shows a depiction of the zodiacal sign."
],
[
"$\\star $ Notes",
"10 The centre image shows Aries in repetition of f70v1, but also the text does not show much evidence for the interpretation as Capricorn.",
"The first letters of the V constellation are ignored, so the transcription is questionable.",
"11 The centre image shows Taurus in repetition of f71v, but also the text does not show much evidence for the interpretation as Aquarius.",
"The first letters of the V month and V constellation are ignored, so the transcription is questionable.",
"Alternative spelling: Ašwahišt Figure: NO_CAPTION 8 Alternative spelling: Figure: NO_CAPTION 9 Alternative spelling: Figure: NO_CAPTIONFigure: NO_CAPTION$\\bar{\\mbox{a}}$ sp.",
"Figure: NO_CAPTION"
],
[
"First folio text (f1r)",
"Passage from the beginning of the third paragraph of f1r.",
"Not all translations from [13] are shown.",
"Our transliteration shows several inconsistencies, which may be due to the complexity and development of the Pahlavi language and will require further analysis.",
"E.g.",
"V otr retains P t, while V dody uses d for P t in accordance with the transliteration [13].",
"Final o, as in P for dody, is often ignored as an otiose stroke [13], see also the final character of Spandarmad.",
"In VM, more often leading o are otiose, e.g.",
"in Spandarmad after the line break, while in V otr the leading o is considered as part of the word.",
"Figure: NO_CAPTION"
],
[
"Notes",
"Curly brackets enclose {ligatures}.",
"Square brackets indicate [inserted characters].",
"Round brackets indicate (ignored characters).",
"A hyphen stands for a line break.",
"Small strokes appearing in V either as c or ı are transcribed here as h, i.e.",
"are considered to indicate a lengthening of the nearest vowel.",
"To explain the ignored V a in Spandarmad, the P d could be considered as a contraction of V a and V t. The chapter on The Nature Of Plants (Bundahesh, Ch.",
"XXVII) mentions Spandarmad [24]"
],
[
"Introduction",
"The last page (f116v) of the Voynich Manuscript (VM) can be assumed to show a colophon, i.e.",
"an addendum that occurs frequently usually on the final page of medieval manuscripts and early modern prints, which usually contains information on the author, production, provenance etc.",
"Following the hypothesis of the main text that the VM is written in a Pahlavi-like script, we present a translation of most of the colophon text.",
"We can identify a place of origin and a date, but not the year in which the manuscript was written.",
"Also we believe to be able to identify the scribe's name which may refer to a female writer from a medieval Zoroastrian community possible in the city of Trebisonta, the a gateway between Persia, Byzantium and early renaissance Italy.",
"Figure: Full view of the relevant part of the colophon page (f116v).The interpretation is in large part speculative and in need of further research, but is included here in order to stimulate further study as well as to provide in turn additional evidence for the Pahlavi hypothesis.",
"Below we will consider the last page of the VM line-by-line in some detail based on the Pahlavi hypothesis.",
"This will enable us to draw some conclusions on the context in which the content of the VM may have originated."
],
[
"Pahlavi hypothesis",
"The claim that the VM is written in natural language rests on the observations that most of the Voynich (V) characters have counterparts in the middle-Iranian Pahlavi script that was used in medieval Zoroastrian scriptures, commentaries, and a few other texts [2].",
"Some of the V characters characters are upside-down versions of their Pahlavi counterparts, which may be an effect of different writing directions.",
"Other V letters are related in another obvious way or can be explained as ligatures.",
"Finally, two letters are added, but are easily identifiable from the vocabulary.",
"In principle, it is thus possible to translate Voynich words using a Pahlavi dictionary such as Ref. [13].",
"Although this process is in many cases successful, is is not always straightforward.",
"The colophon of the Voynich manuscript contains only a few Voynichese letters.",
"Most of the characters resemble Latin letters, but the awkwardness of their shapes contrasts strikingly with the fluency of the proper Voynichese letters.",
"This indicates the possibility that the colophon text was written by a scribe not well acquainted with in Latin letters.",
"Also the impossibility of identifying any other of the language that typically uses Latin script, justifies the attempt to identify the Latin letters as transcription from Voynichese."
],
[
"Colophon ",
"A colophon in medieval European manuscripts usually starts with the explicit that contains the Latin phrase explicit liber (the book is “spread out”, i.e.",
"finished), although since early modern times the words colophon and explicit are used interchangeably.",
"After the explicit, the colophon may give information about content, author, place, date, producer, commissioner, the publication process etc.",
"Figure: The “title” of the colophon page.Figure: The colophon."
],
[
"Explicit",
"The beginning of colophons often reads explicit liber.",
"We will try to identify a similar Pahlavi expression in the beginning of the first line.",
"With this bias, we propose to read the first word as mādayān (book).",
"The letters a in the manuscript are seen to indicate lengthening of the vowel that is not written in abjad alphabets.",
"The first a touches the following letter and the second one is corrected ā by an overlapping o.",
"The i is meant to represent Pahlavi y.",
"The word “be finished” in Pahlavi is frazaftan (alternatively hanaftan), which we can read in the second word of the line.",
"We need to assume the leading o is not a word separator but stands as a waw for the f sound (usually f is expressed by p in Pahlavi).",
"Reading the second letter as r is consistent with all occurrences of this character on this page.",
"Another critical assumption is the interpretation of the sixth letter as a ligature of f and t or by an omission of one of the two letter.",
"Also the other letters are unambiguous including the final figure-8 character that always represents an $n$ .",
"We should note again, that an interpretation of the first two words without being biased by the expectation of the explicit, would be very difficult.",
"The last word in Fig.",
"REF is read as Pahlavi mārdan (spelt with t [13]) for perceive, notice or feel.",
"Also here a correction of a (the penultimate) letter is seen.",
"It may emphasise the fact stated by the first to words or could, in the sense of done at relate to the place name that is seen to follow in this line, see Fig.",
"REF .",
"Figure: First part of the first line.",
"This figures as well as the followingones have the same scale."
],
[
"Trabzon",
"Today's Trabzon, was the antique town Trapezos ($T\\!\\rho \\alpha \\pi \\varepsilon \\zeta o\\tilde{\\upsilon }\\varsigma $ ) on the south-eastern shore of the Black Sea.",
"It had an important role as a trade gateway to Persia and was regularly called at by Venetian trading ships during 13th and 14th centuries.",
"As the capital of the Empire of Trebisonta is was a melting pot of religions.",
"In this way it would be a plausible location for a Zoroastrian book to be transferred from Persia to Europe.",
"In Fig.",
"REF , we note that first two characters (with a $+$ sign between them) appear as unsuccessful attempts to construct a ligature that does not exist in Pahlavi.",
"The combination šr does not occur in initial position [13], where ligatures are mostly used in other parts of the VM.",
"Only at the third attempt, the c-shape is correctly placed between the legs of the $\\pi $ -shape that is usually expressing the sound š, but appears here to represent t. Although there is evidence elsewhere for this corruption, it is clearly a weak point of the interpretation.",
"Also the split of the word into the parts treb and isonta casts doubts on the identification.",
"The final n (figure 8-shape) is less critical as it can be seen as a locative ending.",
"We, nevertheless, propose Trebisonta as the putative location of the production of the VM.",
"For the last letter of the first line, M, refer to Sect.",
"REF Figure: Second part of the first line."
],
[
"Second line",
"If we take the beginning of the second line (Figs.",
"REF and REF ) as a direct continuation of the first one (see, however, Sect.",
"REF ) and identify the first letter as an r (compare Sect.",
"REF ), which is also used to denote the number 20 [13].",
"It may not seem straightforward to explain why the Pahlavi numeral is followed by a Roman IX (Pahlavi for number 9 would be 333), but it is not fully unexpected considering the organisation of the Pahlavi tens in steps of 20.",
"In combination with the M in the preceding line leads to an year 1029 which can refer (based on the date in Sect.",
"REF ) to the Christian (11th of November 1029), the Muslim (9th of September 1620) or the Zoroastrian era (19th of July, 18th of August, or 26th of December in 1660, resp., for the Kadmi, Shenshay or Fasli calendars [12]).",
"However, from the dating of the velum to 14th century, neither of these dates appears likely.",
"One possibility is to use the velum date to justify a lost Roman CD after the M at the end of the first line, such that a date of 1429 is implied, which is, however, highly speculative and contradicts the use of the Zoroastrian calendar for the month (Sect.",
"REF ).",
"Whether or not the space after the M contains indeed the minuscule letters cd cannot be decided from the available scans of the VM.",
"Figure: First part of the second line.In the remainder of the line, we can identify in this line three attempts to write the word māh meaning moon or month.",
"It consists of the letter m followed by a for the lengthening of the vowel and three strokes representing h. In the second occurrence of the word instead of a the letter $o$ is written.",
"The first two occurrences precede what appears to be the Roman numeral X.",
"The third occurrence of māh follows a word with the possible spelling abha (Fig.",
"REF ).",
"The reading is not clear, but the word may be Latin for the Zoroastrian day name xwar that occurs also in the third line, see Sect.",
"REF .",
"Figure: Second part of the second line."
],
[
"Date",
"The last line of the short text starts with the words aror dccy, see Fig.",
"REF which are, in contrast to most of the awkward Latin script on this page, unambiguously identifiable: The letters a, o and r can express the same Latin phonemes.",
"The combination cc is a single letter which can refer to s or ī and the last character is the ambiguous d-g-y letter mentioned in the introduction.",
"The remaining letter $\\pi $ functions as a d. The Pahlavi correspondence is nevertheless more complicated, but suggests unambiguously the irregular transcription xwar day, which refers to the 11th day of the 10th month of the Zoroastrian calendar [13].",
"After the mediocre attempts to give the date in Latin script in the previous line, it seemed necessary to return to the more familiar and less ambiguous Voynichese expression.",
"Figure: First part of the third line."
],
[
"Name",
"Observing the descenders in the two initials in Fig.",
"REF , we read two first letters of the words as g, as in the “title” of the folio f116v.",
"The forth letter of the first word is, as in Figs.",
"REF and REF , an n, and the following letter an r, see Fig.",
"REF .",
"Considering the remaining characters as Latin letters, we can identify the string Galnr Gbrey.",
"The Persian name Gōlnar refers to the flower of a pomegranate tree, and is used since medieval times, as obvious from its prevalence among the Parsis in India.",
"Since Pahlavi uses essentially an abjad alphabet, it seems natural that the vowel between the Voynichese letters is omitted.",
"Likewise, the second letter (a) does not stand for the vowel o, but expresses the lengthening of the sound.",
"Gbrey is a Pahlavi from of Gabr or Gabrī, a term that was used for non-Muslim people in Iran.",
"It seems to have been applied mainly to members of the Zoroastrian faith [22].",
"Considering that, when the VM was written, it would not have had the later pejorative meaning, it could be have well been used as a byname, and in fact has survived in several variants as a surname.",
"Gōlnar Gabrī is likely to a female name, although also unisex names with the compound Gōl (rose) exist.",
"If the scribe was indeed one of the rare female authors or writers from that time, then the preservation of the VM is indeed very interesting, even if we do not yet have much insight into the actual content of the manuscript.",
"Figure: Second part of the third line.",
"Phrase interpreted as the name of theauthor: Gōlnar Gabrī."
],
[
"Finis",
"As the least phrase of the colophon is particularly obscure, it is hard to resist reading the last words of the colophon, Fig.",
"REF , as the German phrase So nimm gar mich. (Thus take even me), which appears to be out off context.",
"However, the identification of the first letter as an descending s and of the similar letter in the middle as an r is not entirely inconsistent within the Pahlavi hypothesis, although we would expect the writer to return rather to the Voynich r in case of ambiguity.",
"A slightly more likely alternative reading can be found by considering the phrase in Fig.",
"REF again as Pahlavi written in Latin letters.",
"In this case we would ignore the first part If we read the first letter as b, then we can identify the first word as band which refers to bastan and is occasionally written as bn [13], noting that the letter waw (o) and nun (n) are of identical shape in Pahlavi.",
"The word bastan means tie or bind.",
"Other options of starting with b would require the presence of a third letter.",
"We prefer to read the first letter as r (as in the third word in Fig.",
"REF ), for which MacKenzie [13] suggests the transliteration raw which then refers to raftan meaning go or move.",
"The following words can be transcribed [13] as nim[ay] (nimudan) meaning guard, gōr for nature (or jewel) and mizd which can mean reward.",
"Although the rough translation of the phrase as “Go, guard nature's reward!” appears utterly anachronistic, it would be within context and could be considered as a final message of the author to posterity.",
"Nevertheless, as many of our conclusions are rather speculative, this translation is even more so.",
"Figure: Final phrase of the colophon."
],
[
"Drawings",
"The velum of the last page has tear that apparently has been mended before the use of the page.",
"The scribe used the large part right of the tear for text of the colophon, and decorated the margin left of the tear by a few small drawings, see Fig.",
"REF .",
"The top picture (Fig.",
"REF left) has a conspicuous likeness to a chicken corpse.",
"The middle picture represents a billy goat or a similar animal.",
"The bottom pictures is a female nude in the style of the figures in the “nudes” pages (f75r – f84v).",
"It would be strange to assume that it represents the author.",
"Although we are unable to give a interpretation of the pictures here, we note the letters in the top figure (Fig.",
"REF left).",
"Although the first letter as a similarity with F, we should stay with the earlier reading of the character as r, which is followed by a (or o) and n (figure-8 shape).",
"The word rān means fight [13], but our confidence in this reading is low.",
"Figure: Drawings on left margin of f116v.",
"The thee images (left to right)are positioned on top of each other and separated from the main textby a repaired tear in the velum."
],
[
"Illegible characters on right margin",
"One of the most important information to obtain from a colophon would be the year of the production of the document.",
"We have touched upon this question above, but are unable to give a definite answer.",
"Fig.",
"REF shows the ending of the first line of the colophon and the right margin of the page next to the colophon.",
"It is possible that characters after the letter M are lost, although they may become visible in a multi-spectral analysis of the velum.",
"In the left middle and lower part of Fig.",
"REF a few blurred characters (such as a question mark) can be seen, but due to the difference in the strokes and unrelatedness to the main part of the colophon, they should be considered as later additions.",
"Figure: Ending of the first line of the colophon and lacuna due to abrasionat the right margin of f116v."
],
[
"Conclusion",
"All of the presented results from our reading of the colophon are to a larger or lesser extent speculative, but at least we should admit that the combined evidence provided here shows that the text on f116v indeed represents a colophon.",
"Further analysis of the text as well as of the material document may lead to more reliable information about the manuscript.",
"Colophons occur already on Ancient Near East clay tablets, but in the case of the VM, the addition of a colophon appears as an ineffectual attempt to adopt a Western custom, which may have be seen as suboptimal when taken, but we can now appreciate it as potentially very useful."
]
] | 1709.01634 |
[
[
"$A_4$ realization of Linear Seesaw and Neutrino Phenomenology"
],
[
"Abstract Motivated by the crucial role played by the discrete flavour symmetry groups in explaining the observed neutrino oscillation data, we consider the $A_4$ realization of linear seesaw by extending the standard model (SM) particle content with two types of right-handed neutrinos along with the flavon fields, and the SM symmetry with $A_4\\times Z_4\\times Z_2$ and a global symmetry $U(1)_X$ which is broken explicitly by the Higgs potential.",
"We scrutinize whether this model can explain the recent results from neutrino oscillation experiments by searching for parameter space that can accommodate the observables such as the reactor mixing angle $\\theta_{13}$, the CP violating phase $\\delta_{CP}$, sum of active neutrino masses $\\Sigma_{i} m_i$, solar and atmospheric mass squared differences, and the lepton number violating parameter called as effective Majorana mass parameter, in line with recent experimental results.",
"We also discuss the scope of this model to explain the baryon asymmetry of the universe through Leptogenesis.",
"We also investigate the possibility of probing the non-unitarity effect in this scenario, but it is found to be rather small."
],
[
"Introduction",
"The Standard Model (SM) of particle physics predicts massless neutrinos contradicting the experimental results on neutrino oscillation, according to which the three neutrino flavors mix with each other and at least two of the neutrinos have non-vanishing mass.",
"Due to the absence of right-handed (RH) neutrinos in the SM, neutrinos do not have Dirac mass like other charged fermions and their mass generation in the SM is generally expected to arise from a dimension-five operator [1], which violates lepton number.",
"However, very little is known about the origin of this operator and the underlying mechanism or its flavour structure.",
"Hence, to generate non-zero neutrino mass, one resorts to some beyond the standard model scenarios.",
"There are many such models where the SM is extended by including the right-handed neutrinos to its particle content.",
"The inclusion of right-handed neutrinos $N_{R_i}$ not only generates the Dirac mass term but also leads to Majorana mass for the right handed neutrinos, which is of the form $\\overline{N_{R_i}}N_{R_i}^c$ and violates $B-L$ symmetry.",
"The smallness of active neutrino mass is ensured by the high value of Majorana mass of the right handed neutrino [2], [3].",
"In these class of models, if Dirac mass of neutrinos are of the order of lightest charged lepton mass i.e., electron mass, the Majorana mass has to be in TeV range to get the observed value of active neutrino mass [4].",
"But if such models have to be embedded in Grand Unified Theories (GUTs) where both Quarks and Leptons are treated on the same footing, the Dirac mass of neutrinos will be of the order of that of up-type quark [5] and the observed value of active neutrino mass requires Majorana mass to be of the order of $10^{15}$ GeV, which is beyond the access of present and future experiments.",
"Many possibilities were proposed to have not so heavy Majorana mass and the existence of other type of neutrinos called sterile neutrinos ($S$ ) is one among them [6].",
"Now the neutrino mass can be expressed in the form of a $3\\times 3$ matrix with each element represents a matrix.",
"Depending on the position of the zero elements in the mass matrix in the basis $(\\nu , N_R, S)$ , active neutrinos receive masses through two different mechanisms called inverse seesaw [6], [7] and linear seesaw [8].",
"If 11 and 13 elements are zero then it is called inverse seesaw and if 11 and 33 elements are zero with non zero off-diagonal elements then we have the linear seesaw.",
"In all those cases the smallness of neutrino mass is independent of the ratio of Dirac mass to heavy neutrino mass, hence, allows to have heavy neutrinos in TeV range and bound on the ratio comes from non-unitarity effect and neutrinoless double beta decay experiments.",
"All those seesaw mechanisms require some of the elements of mass matrix to be zero or very small but none of them are prevented by SM symmetry.",
"All those terms except 33 element in the matrix will be prohibited if the SM symmetry is extended to $SU(2)_L\\times SU(2)_R\\times SU(3)_C$ or $B-L$ , since in those symmetry groups right handed neutrinos are no longer singlets.",
"But linear seesaw requires 33 element of the mass matrix to be zero or very small which is difficult to obtain with gauge symmetry as sterile neutrinos are singlets in all gauge groups.",
"But such terms will be absent if there is flavour symmetry under which sterile neutrinos have non trivial representation.",
"The $A_4$ discrete symmetry group is the group of even permutations of four elements has attracted a lot of attention since it is the smallest one which admits one three-dimensional representation and three inequivalent one-dimensional representations.",
"Then, the choice of the $A_4$ symmetry is natural since there are three families of fermions, i.e, the left-handed leptons can be unified in triplet representation of $A_4$ while the right-handed leptons can be assigned to $A_4$ singlets.",
"This set-up was first proposed in Ref.",
"[9] to study the lepton masses and mixing obtaining nearly degenerate neutrino masses and allowing realistic charged leptons masses after the $A_4$ symmetry is spontaneously broken.",
"Latter $A_4$ symmetry was proved to be very successful in generating Tribimaximal mixing pattern for Lepton mixing [10], which well supported the trends of oscillation data at that time.",
"The Tribimaximal mixing pattern predicts solar mixing and atmospheric mixing angles consistent with the experimental data but yields a vanishing reactor mixing angle [11], [12], [13] contradicting the recent experimental results from the Daya Bay [14], [15], T2K[16], [17], MINOS [18], Double CHOOZ [19] and RENO [20] experiments.",
"In view of this the Tribimaximal mixing pattern has to be modified.",
"Here we consider the realization of linear seesaw with $A_4$ symmetry.",
"We extend SM symmetry with $A_4\\times Z_4\\times Z_2$ along with an extra global symmetry $U(1)_X$ , as discussed in Ref.",
"[21].",
"The SM particle content has been extended by introducing three RH neutrinos, $N_{R_i}$ and three singlet fermions, $S_{R_i}$ along with the flavon fields ($\\phi _S$ , $\\phi _T$ , $\\xi $ , $\\xi ^{\\prime }$ , $\\rho $ , $\\rho ^{\\prime }$ ), to understand the flavour structure of the lepton mixing.",
"The proposed model gives almost similar result as in [21] in the context of neutrino oscillation, but has a different physics aspect in the case of heavy neutrinos.",
"In [21], the active neutrinos get their mass through inverse seesaw with the prediction of six nearly degenerate heavy neutrinos but in our case there are three very different mass state with each state is nearly doubly degenerated.",
"Also, our proposed scenario is very much suitable for Leptogenesis as discussed in [22], [23], where the analytic expression for CP asymmetry and corresponding baryon asymmetry for the case of three pairs of nearly degenerate heavy neutrinos can be found.",
"In Ref.",
"[23], the contributions of the absorptive part of Higgs self-energy to CP violation in heavy particle decays termed as $\\epsilon $ -type CP violation, has been discussed elaborately.",
"Such contributions are neglected in many cases as they are small compared to $\\epsilon ^{\\prime }$ -type, the CP violation in heavy neutrino decays due to the overlapping of tree-level with one-loop vertex diagram.",
"They have provided the formalism to deal with mixing of states during the decay of the particles and have shown that there is resonant enhancement of $\\epsilon $ -type CP violation, if mixing states are nearly degenerate.",
"The CP asymmetries due to both types of CP violations for a model with a pair of nearly degenerate heavy neutrinos are also calculated and it was shown that the CP asymmetry due to $\\epsilon $ -type CP violation is 100 times more than that of due to $\\epsilon ^{\\prime }$ -type, which in turn predicts the correct baryon asymmetry of the Universe.",
"The outline of the paper is as follows.",
"In section II, we present the model framework for linear seesaw.",
"The $A_4$ realization of linear seesaw and its implication to neutrino oscillation parameters is discussed in Section III.",
"Section IV contains the discussion on Leptogenesis and summary and Conclusions are presented in Section V."
],
[
"The model framework for linear seesaw",
"We consider the minimal extension of Standard Model $\\mathcal {G}_{\\rm SM} \\equiv SU(2)_L \\times U(1)_Y$ , omitting the $SU(3)_C$ structure for simplicity, with two types of singlet neutrinos which are complete singlet under $\\mathcal {G}_{\\rm SM}$ for implementation of linear seesaw.",
"We denote these neutral fermion singlets as right-handed sterile neutrinos $N_{R_i}$ and $S_{R_i}$ .",
"Both these neutral fermion species have Yukawa coupling with the lepton doublet $L$ .",
"In addition, one can write down a mixing term connecting these two species of neutrinos.",
"The bare Majorana mass terms for $N_{R_i}$ and $S_{R_i}$ are either assumed to be zero or forbidden by some symmetry arguments.",
"The leptonic Lagrangian for linear seesaw mechanism is given by $-\\mathcal {L} &=&y \\overline{L} \\tilde{H} N_\\text{R} +h \\overline{L} \\tilde{H} S_\\text{R} +\\overline{N_R} m_{RS} S^c_\\text{R} +\\text{~h.c.}",
"\\nonumber \\\\&=&\\overline{\\nu }_\\text{L} m_\\text{D} N_\\text{R} +\\overline{\\nu }_\\text{L} m_\\text{LS} S_\\text{R} +\\overline{N}_\\text{R} m_{RS} S^c_\\text{R} +\\text{~h.c.",
"}\\;.$ The full mass matrix for neutral leptons in the basis $N = (\\nu _\\text{L}, ~N^c_\\text{R}, ~S^c_\\text{R})^\\text{T}$ is given as $\\mathbb {M} =\\begin{pmatrix}0 & m_\\text{D} & m_\\text{LS} \\\\m_\\text{D}^\\text{T} & 0 & m_{RS} \\\\m_\\text{LS}^\\text{T} & m^\\text{T}_{RS} & 0\\end{pmatrix}.",
"$ The resulting mass formula for light neutrinos is governed by linear seesaw mechanism, $m_\\nu &= m_\\text{D} m_{RS}^{-1} m_\\text{LS}^\\text{T} \\mbox{+ transpose}\\;.$"
],
[
"An $A_4$ realization of linear seesaw",
"In this section, we wish to present an $A_4$ realization of linear seesaw which has been discussed in the previous section.",
"The particle content of the model and their representations under flavour symmetries are presented in Table REFThe implication of linear seesaw can be found in [22].. We introduce an extra global symmetry $U(1)_X$ which is broken explicitly but softly by the term $\\mu ^2_{\\rho \\xi }\\rho \\xi +{\\rm h.c.}$ , in the Higgs potential to prevent Goldstone boson [24].",
"This term not only breaks $U(1)_X$ symmetry but also gives non-zero vacuum expectation value to $\\rho $ , $\\langle \\rho \\rangle =\\frac{\\mu ^2_{\\rho \\xi }\\langle \\xi \\rangle }{m_{\\rho }^2}\\ll \\langle \\xi \\rangle $ as $\\mu ^2_{\\rho \\xi }$ is very small compared to $m_{\\rho }$ , the mass of $\\rho $ .",
"Table: The particle content and their charge assignments for an A 4 A_4 realization oflinear seesaw mechanism.The Yukawa Lagrangian for the charged lepton sector is given as $\\nonumber &&\\mathcal {L}_l =-\\left\\lbrace \\left[ \\frac{\\lambda _e}{\\Lambda } \\left( \\bar{L}\\phi _T\\right)H e_{R}\\right]+\\left[ \\frac{\\lambda _{\\mu }}{\\Lambda }\\left(\\bar{L}\\phi _T\\right)^{\\prime }H\\mu _{R}\\right]+\\left[ \\frac{\\lambda _{\\tau }}{\\Lambda }\\left(\\bar{L}\\phi _T\\right)^{\\prime \\prime }H\\tau _{R}\\right]\\right\\rbrace \\,.\\nonumber $ After giving non-zero VEVs to SM Higgs as well as flavon fields and breaking all symmetries, the mass matrix for charged leptons is found to be $M_l=v\\frac{v_T}{\\Lambda }{\\rm diag}\\left(\\lambda _e,\\lambda _{\\mu },\\lambda _{\\tau }\\right),$ where the vacuum expectation values (vevs) of the scalar fields are given as $v=\\langle H \\rangle ,~~v_T=\\langle \\phi _T\\rangle \\;.$ For linear seesaw mechanism, the Lagrangian involved in the generation of the mass matrices for an $A_4$ flavor symmetric framework can be written as, $-\\mathcal {L_{\\nu }}=&\\mathcal {L}_{\\nu N}+\\mathcal {L}_{N S}+\\mathcal {L}_{\\nu S}\\;,$ where $\\mathcal {L}_{\\nu N} & = y_1 \\overline{L}\\widetilde{H} N_R \\frac{\\rho ^{\\prime }}{\\Lambda }\\;, \\\\\\mathcal {L}_{\\nu S} & = y_2 \\overline{L} \\widetilde{H} S_R\\frac{\\rho }{\\Lambda }\\;, \\\\\\mathcal {L}_{N S} & = \\left(\\lambda _{NS}^{\\phi }\\phi _s+\\lambda _{NS}^{\\xi }\\xi +\\lambda _{NS}^{\\xi ^{\\prime }}\\xi ^{\\prime }\\right) \\overline{N_R} S^c_R\\;.$ It should be noted that the terms $\\mathcal {L}_{\\nu N}$ , $\\mathcal {L}_{\\nu S}$ and $\\mathcal {L}_{N S}$ represent the contributions for Dirac neutrino mass connecting $\\nu _L-N_R$ , $\\nu _L-S_R$ mixing and $N_R-S^c_R$ mixing terms.",
"If one looks at the mass formula for light neutrinos governed by linear seesaw mechanism given in Eq.",
"(REF ), one can use the mass hierarchy as $m_{RS} \\gg m_D, m_{LS}$ .",
"That is the reason why we forbid $\\overline{\\nu } N$ and $\\overline{\\nu } S$ terms at tree level and generate them by dimension five operator while the heavy mixing term $N-S$ is generated at tree level.",
"Using the following vevs for the scalar and flavon fields $\\langle \\phi _S \\rangle = v_S(1,1,1),~~ \\langle \\xi \\rangle =v_{\\xi },~ \\langle \\xi ^{\\prime }\\rangle =v_{\\xi ^{\\prime }},~~\\langle \\rho \\rangle =v_{\\rho },~~ \\langle \\rho ^{\\prime }\\rangle =v_{\\rho ^{\\prime }}\\;, \\nonumber $ the various mass matrices are found to be $&&m_\\text{D}=y_1v\\frac{v_{\\rho ^{\\prime }}}{\\Lambda }\\left(\\begin{array}{ccc}1 & 0 & 0\\\\0 & 0 &1 \\\\0 & 1 &0\\end{array}\\right),~~~~m_\\text{LS}=y_2v\\frac{v_{\\rho }}{\\Lambda }\\left(\\begin{array}{ccc}1 & 0 & 0\\\\0 & 0 &1 \\\\0 & 1 &0\\end{array}\\right),\\\\ &&m_{RS}=\\frac{a}{3}\\left(\\begin{array}{ccc}2 & -1 & -1\\\\-1 & 2 &-1 \\\\-1 & -1 &2\\end{array}\\right)+b\\left(\\begin{array}{ccc}1 & 0& 0\\\\0 & 0 &1 \\\\0& 1 &0\\end{array}\\right)+d\\left(\\begin{array}{ccc}0 & 0& 1\\\\0 & 1 &0 \\\\1& 0 &0\\end{array}\\right),$ where $a=\\lambda _{NS}^{\\phi }v_S$ , $b=\\lambda _{NS}^{\\xi }v_{\\xi }$ and $d=\\lambda _{NS}^{\\xi ^{\\prime }}v_{\\xi ^{\\prime }}$ .",
"The first term in Eqn.",
"(REF ) comes from $\\lambda _1\\phi _s\\left(\\overline{N_R}S_R^c\\right)_{3s}$ , where $\\left(\\overline{N_R}S_R^c\\right)_{3s}$ is a triplet which is symmetric under exchange of $N_R$ and $S_R$ .",
"The product of two triplets can also form a triplet which is antisymmetric under the exchange of the particles.",
"In linear seesaw, the mass of the light neutrino is represented as $m_{\\nu }=m_\\text{D}m_\\text{RS}^{-1}m_\\text{LS}^{T}$ +transpose, and as seen from Eqn (REF ) the mass matrices $m_\\text{D}$ and $m_\\text{LS}$ are symmetric and are related as $m_\\text{D} \\propto m_\\text{LS}$ .",
"Hence, in $m_{\\nu }=m_\\text{D}^{T}(m_\\text{RS}^{-1}+{m_\\text{RS}^{-1}}^T)m_\\text{LS}$ , the antisymmetric part cancels out and only symmetric part survives."
],
[
"Neutrino Masses and Mixing",
"For calculational convenience one can rewrite the $m_\\text{RS}$ mass matrix (REF ) as $m_{RS}=\\left(\\begin{array}{ccc}2a/3+b & -a/3& -a/3\\\\-a/3 & 2a/3 &-a/3+b \\\\-a/3& -a/3+b &2a/3\\end{array}\\right)+\\left(\\begin{array}{ccc}0 & 0& d\\\\0 & d &0 \\\\d& 0 &0\\end{array}\\right).$ Thus, with Eqns.",
"(REF ), (REF ) and (REF ), one can obtain the the light neutrino mass $m_{\\nu }&=&m_\\text{D} m_{RS}^{-1}m_\\text{LS}^T+\\text{transpose} \\nonumber \\\\&=&k_1k_2\\left(\\begin{array}{ccc}1 & 0 & 0\\\\0 & 0 &1 \\\\0 & 1 &0\\end{array}\\right) m_{RS}^{-1}\\left(\\begin{array}{ccc}1 & 0 & 0\\\\0 & 0 &1 \\\\0 & 1 &0\\end{array}\\right),$ where the parameters $k_1$ and $k_2$ are related to the vevs through $k_1=\\sqrt{2} y_1v\\frac{v_{\\rho ^{\\prime }}}{\\Lambda }\\;, ~~~~k_2=\\sqrt{2}y_2v\\frac{v_{\\rho }}{\\Lambda }.\\nonumber $ Hence, the inverse of light neutrino mass matrix is given by $m_{\\nu }^{-1}&=&\\frac{1}{k_1k_2}\\left(\\begin{array}{ccc}2a/3+b & -a/3& -a/3\\\\-a/3 & 2a/3 &-a/3+b \\\\-a/3& -a/3+b &2a/3\\end{array}\\right)+\\frac{1}{k_1k_2}\\left(\\begin{array}{ccc}0 & d& 0\\\\d & 0 &0 \\\\0& 0 &d\\end{array}\\right),$ which in TBM basis will have the form, i.e., $m_{\\nu }^{-1^{\\prime }}=U_\\text{TBM}^T m_{\\nu }^{-1}U_\\text{TBM}$ , $m_{\\nu }^{-1^{\\prime }}=\\left(\\begin{array}{ccc}a+b-d/2 & 0 & -\\frac{\\sqrt{3}}{2}d\\\\0 &~~ b+d~~ &0 \\\\-\\frac{\\sqrt{3}}{2}d & 1 &a-b+d/2\\end{array}\\right)\\;.$ The inverse mass matrix $m_{\\nu }^{-1^{\\prime }}$ can be diagonalized by $U_{13}^*$ .",
"Hence, the matrix $m_\\nu ^{-1}$ can be diagonalized by $U_{TBM}\\cdot U_{13}^*$ , and thus, the matrix $m_{\\nu }$ can be diagonalized by $U_{TBM}\\cdot U_{13}$ , while $m_{RS}$ by $U_{TBM}\\cdot U_{13}^T$ .",
"The complex unitary matrix $U_{13}$ has the form $U_{13}=\\left(\\begin{array}{ccc}\\cos \\theta & 0 & \\sin \\theta e^{-i\\psi }\\\\0 & 1 & 0 \\\\-\\sin \\theta e^{i\\psi }& 0 & \\cos \\theta \\end{array}\\right),$ where the parameters $\\theta $ and $\\psi $ are expressed in terms of the mass matrix parameters $d/b=\\lambda _1e^{\\phi _{db}}$ , $a/b=\\lambda _2e^{\\phi _{ab}}$ as $\\tan 2\\theta =-\\frac{\\sqrt{3}\\lambda _1\\cos \\phi _{db}}{(\\lambda _1 \\cos \\phi _{db}-2)\\cos \\psi +(2\\lambda _2\\sin \\phi _{ab})\\sin \\psi }\\;,$ and $\\tan \\psi =\\frac{\\sin \\phi _{db}}{\\lambda _2\\cos (\\phi _{ab}-\\phi _{db})}\\;.$ The eigenvalues of $m_{\\nu }$ and $m_{RS}$ are related to each other as $\\tilde{m}_i=\\frac{k_1k_2}{\\tilde{M}_i}\\;.$ where $\\tilde{m}_i$ and $\\tilde{M}_i$ are $i^\\text{th}$ eigenvalues of $m_{\\nu }$ and $m_{RS}$ respectively.",
"The eigenvalues of $m_{RS}$ can be expressed as $\\tilde{M}_1&=&b\\left[\\lambda _2 e^{i\\phi _{ab}}-\\sqrt{1+\\lambda _1^2e^{2i\\phi _{db}}-\\lambda _1 e^{i\\phi _{db}}}\\right], \\nonumber \\\\\\tilde{M}_2&=&b\\left[1+\\lambda _1 e^{i\\phi _{db}}\\right], \\nonumber \\\\\\tilde{ M}_3&=&b\\left[\\lambda _2 e^{i\\phi _{ab}}+\\sqrt{1+\\lambda _1^2e^{2i\\phi _{db}}-\\lambda _1 e^{i\\phi _{db}}}\\right],$ which give the mass of the heavy neutrinos as $M_i=|\\tilde{M_i}|$ .",
"Explicitly, one can write the heavy neutrino masses as $M_1&=& |b|M_1^{\\prime }=|b|\\left[(\\lambda _2\\cos \\phi _{ab}-C)^2+(\\lambda _2\\sin \\phi _{ab}-D)^2\\right]^{1/2} \\nonumber \\\\M_2&=&|b|M_2^{\\prime }=|b|\\left[1+\\lambda _1^2+2\\lambda _1\\cos \\phi _{db}\\right]^{1/2} \\nonumber \\\\M_3&=&|b|M_3^{\\prime }=|b|\\left[(\\lambda _2\\cos \\phi _{ab}+C)^2+(\\lambda _2\\sin \\phi _{ab}+D)^2\\right]^{1/2}\\;,$ where $C&=&\\pm \\left[ \\frac{A+\\sqrt{A^2+B^2}}{2}\\right]^{1/2}\\;,~~~~~~~D=\\pm \\left[ \\frac{-A+\\sqrt{A^2+B^2}}{2}\\right]^{1/2}\\;,\\nonumber \\\\A&=&1+\\lambda _1^2\\cos 2\\phi _{db}-\\lambda _1\\cos \\phi _{db}\\;,~~~~B=\\lambda _1^2\\sin 2\\phi _{db}-\\lambda \\sin \\phi _{db}\\;.$ and the phases $\\phi _i$ s as $\\phi _1&=&\\tan ^{-1}\\left[\\frac{\\lambda _2\\sin \\phi _{ab}-D}{\\lambda _2\\sin \\phi _{ab}-C}\\right]\\;,\\nonumber \\\\\\phi _2&=&\\tan ^{-1}\\left[\\frac{\\lambda _1\\sin \\phi _{db}}{1+\\lambda _1\\cos \\phi _{db}}\\right]\\;, \\nonumber \\\\\\phi _3&=&\\tan ^{-1}\\left[\\frac{\\lambda _2\\sin \\phi _{ab}+D}{\\lambda _2\\sin \\phi _{ab}+C}\\right].$ Thus, the active neutrino masses $m_i=|\\tilde{m}_i|$ and the matrix which diagonalize active neutrino mass matrix, $U_{\\nu }$ are given by $m_i&=&\\frac{|k_1k_2|}{M_i}\\;,\\nonumber \\\\U_{\\nu }&=&U_{TBM}\\cdot U_{13}\\cdot P\\;,$ with $P=\\text{diag}(e^{-i\\phi _1/2},e^{-i\\phi _2/2},e^{-i\\phi _3/2})$ .",
"The lepton mixing matrix, known as PMNS matrix is given by [25], [26] $U_{PMNS}=U_l^{\\dagger }\\cdot U_{\\nu }\\;,$ where $U_l$ and $U_{\\nu }$ are the matrices which diagonalize charged lepton and neutrino mass matrices.",
"Here $U_l=I$ and $U_{\\nu }=U_{TBM}\\cdot U_{13}\\cdot P$ , hence, $U_{PMNS}=U_{TBM}\\cdot U_{13}\\cdot P, $ which is proved to be in good agreement with the experimental observations [27], [28].",
"The PMNS matrix can be parametrized in terms of three mixing angles ($\\theta _{13}$ , $\\theta _{23}$ and $\\theta _{12}$ ) and three phases (one Dirac phase $\\delta _{CP}$ , and two Majorana phases $\\rho $ and $\\sigma $ ) as $U_\\text{PMNS}=\\left( \\begin{array}{ccc} c^{}_{12} c^{}_{13} & s^{}_{12}c^{}_{13} & s^{}_{13} e^{-i\\delta _{CP}} \\\\ -s^{}_{12} c^{}_{23} -c^{}_{12} s^{}_{13} s^{}_{23} e^{i\\delta _{CP}} & c^{}_{12} c^{}_{23} -s^{}_{12} s^{}_{13} s^{}_{23} e^{i\\delta _{CP}} & c^{}_{13} s^{}_{23} \\\\s^{}_{12} s^{}_{23} - c^{}_{12} s^{}_{13} c^{}_{23} e^{i\\delta _{CP}} &-c^{}_{12} s^{}_{23} - s^{}_{12} s^{}_{13} c^{}_{23} e^{i\\delta _{CP}} &c^{}_{13} c^{}_{23} \\end{array} \\right) P^{}_\\nu \\;,$ where $c_{ij}=\\cos \\theta _{ij}$ and $s_{ij}=\\sin \\theta _{ij}$ and $P_{\\nu }=\\text{diag}(e^{i\\rho },e^{i\\sigma },1)$ .",
"From Eqns.",
"(REF ) and (REF ), one can find $&&\\sin \\theta =\\sqrt{\\frac{3}{2}}\\sin \\theta _{13} \\nonumber \\\\&&\\sin \\delta _\\text{CP}=-\\frac{\\sin \\psi }{\\displaystyle {\\sqrt{1-\\frac{3(2-3\\sin ^2\\theta _{13})}{(1-\\sin ^2\\theta _{13})^2}\\sin ^2\\theta _{13}\\cos ^2\\psi }}}\\approx -\\sin \\psi \\;.$ The above expressions relate the parameters of the model, i.e., $\\theta $ and $\\psi $ to the mixing observables $\\sin ^2\\theta _{13}$ and $\\delta _{CP}$ respectively.",
"Since $\\sin ^2\\theta _{13}$ is known more precisely than $\\delta _\\text{CP}$ , in our calculation we fix $\\theta $ by fixing $\\sin ^2\\theta _{13}$ at its best fit value while considering all possible value of $\\psi $ for which $\\delta _\\text{CP}$ falls within its $3\\sigma $ experimental range."
],
[
"Numerical results",
"Using Eqns.",
"(REF ) and (REF ), the light neutrino masses are found to be $m_i=\\frac{|k_1k_2|}{M_i}=\\frac{|k_1k_2|}{|b|}\\frac{1}{M_i^{\\prime }}\\;.$ Since only the mass squared differences, $\\Delta m^2_{21}$ (solar mass squared difference) and $|\\Delta m^2_{32}|$ (atmospheric mass squared difference) are measured in neutrino oscillation experiments, we calculate the mass squared differences from Eqn.",
"(REF ) as $\\Delta m^2_{21}&=&\\left|\\frac{k_1k_2}{b}\\right|^2\\left(\\frac{1}{{M_2^{\\prime }}^2}-\\frac{1}{{M_1^{\\prime }}^2}\\right)\\;, \\nonumber \\\\\\left|\\Delta m^2_{31}\\right|&=&\\left|\\frac{k_1k_2}{b}\\right|^2\\left|\\left(\\frac{1}{{M_3^{\\prime }}^2}-\\frac{1}{{M_1^{\\prime }}^2}\\right)\\right|\\;.",
"$ Substituting the set of Eqns.",
"(REF ) in the above equations, we find the ratio of the two mass squared differences as $r&=&\\frac{\\Delta m_{21}^2}{|\\Delta m_{31}^2|}=\\left[\\frac{(\\lambda _2\\cos \\phi _{ab}+C)^2+(\\lambda _2\\sin \\phi _{ab}+D)^2}{1+\\lambda _1^2+2\\lambda _1\\cos \\phi _{db}}\\right] \\nonumber \\\\&\\times &\\left[\\frac{(\\lambda _2\\cos \\phi _{ab}-C)^2+(\\lambda _2\\sin \\phi _{ab}-D)^2-\\left(1+\\lambda _1^2+2\\lambda _1\\cos \\phi _{db}\\right)}{4\\lambda _2|C \\cos \\phi _{ab}+D\\sin \\phi _{ab}|}\\right]\\;.$ Now using equations (REF ), (REF ), (REF ), (REF ) and REF , and by fixing the parameters $\\phi _{db}$ , $\\psi $ and $\\theta $ , one can find numerical values of $M_i^{\\prime }$ 's.",
"Once $M_i^{\\prime }$ 's are known $\\left|\\displaystyle {\\frac{k_1k_2}{b}}\\right|$ can be calculated from (REF ) as $\\left|\\frac{k_1k_2}{b}\\right|=\\sqrt{\\frac{\\Delta m^2_{21}}{\\displaystyle { \\left(\\frac{1}{M_2^{\\prime 2}}-\\frac{1}{M_1^{\\prime 2}}\\right)}}}=\\sqrt{\\left|\\frac{\\Delta m^2_{31}}{\\left(\\frac{1}{M_3^{\\prime 2}}-\\frac{1}{M_1^{\\prime 2}}\\right)}\\right|}\\;,$ which will also give the absolute value of light neutrino masses as all the quantities on the right hand side of (REF ) are now known.",
"We now rewrite the expression $\\tan \\psi $ (REF ) in terms of $\\phi _{db}$ as $\\phi _{db}=0,\\pi ,~~~\\text{for}~ \\tan \\psi =0\\;,$ and $\\phi _{ab}=\\phi _{db}+ \\cos ^{-1}\\left(\\frac{\\sin \\phi _{db}}{\\lambda _2\\tan \\psi }\\right),~~~\\text{for}~~ \\tan \\psi \\ne 0,$ and consider the following cases to see the implications.",
"Figure: Variation of λ 2 \\lambda _2, the lightest neutrino mass (m l m_l) and Σ i m i \\Sigma _im_i with φ ab \\phi _{ab}, red lines are for inverted hierarchy and green lines are for normal hierarchy."
],
[
"Correlation between model parameters with $\\tan \\psi =0$",
"In this case $\\phi _{db}$ will be either 0 or $\\pi $ , and for $\\phi _{db}=0$ one can obtain from Eq.",
"(REF ) $\\lambda _1=\\frac{2\\tan 2\\theta }{\\sqrt{3}+\\tan 2\\theta }\\;,$ and the ratio of the mass square differences $r$ satisfies the relation $r=0.03=\\left[\\frac{\\lambda _2^2+2\\lambda _2C\\cos \\phi _{ab}+C^2}{(1+\\lambda _1)^2}\\right]\\left[\\frac{\\lambda _2^2-2\\lambda _2C\\cos \\phi _{ab}+C^2-(1+\\lambda _1)^2}{4\\lambda _2|C\\cos \\phi _{ab}|}\\right], $ where $C=\\sqrt{\\frac{1-\\lambda _1+\\lambda _1^2}{2}}$ .",
"The eigenvalues of $m_{RS}$ in this case become $M_1&=&|b|\\sqrt{\\lambda _2^2-2\\lambda _2C\\cos \\phi _{ab}+C^2}\\;, \\nonumber \\\\M_2&=&|b|(1+\\lambda _1)\\;,\\nonumber \\\\M_3&=&|b|\\sqrt{\\lambda _2^2+2\\lambda _2C\\cos \\phi _{ab}+C^2}\\;.$ Now from Eq.",
"(REF ), using the measured values of $r~(0.03)$ , variation of the parameter $\\lambda _2$ , the lightest neutrino mass ($m_l$ ) and the sum of active neutrino masses $\\sum m_i$ with $\\phi _{ab}$ are shown in Fig.REF .",
"Figure: Variation of λ 2 \\lambda _2, m l m_l and Σ i m i \\Sigma _im_i with φ ab \\phi _{ab}, red points are for inverted hierarchy and green points are for normal hierarchy.While for $\\phi _{db}=\\pi $ $\\lambda _1=\\frac{2\\tan 2\\theta }{\\sqrt{3}-\\tan 2\\theta }\\;,$ and the ratio of the mass square differences $r$ satisfies the relation $r=0.03=\\left[\\frac{\\lambda _2^2+2\\lambda _2C\\cos \\phi _{ab}+C^2}{(1-\\lambda _1)^2}\\right]\\times \\left[\\frac{\\lambda _2^2-2\\lambda _2C\\cos \\phi _{ab}+C^2-(1-\\lambda _1)^2}{4\\lambda _2|C\\cos \\phi _{ab}|}\\right], $ with $C=\\sqrt{\\frac{1+\\lambda _1+\\lambda _1^2}{2}}$ , and the eigenvalues of $m_{RS}$ are given as $M_1&=&|b|\\sqrt{\\lambda _2^2-2\\lambda _2C\\cos \\phi _{ab}+C^2}\\;, \\nonumber \\\\M_2&=&|b|(1-\\lambda _1)\\;,\\nonumber \\\\M_3&=&|b|\\sqrt{\\lambda _2^2+2\\lambda _2C\\cos \\phi _{ab}+C^2}\\;.$ Analogous to Fig.1, the variation of various parameters with $\\phi _{ab}$ is shown in Fig.REF .",
"From the plots it can be seen that for normal ordering, the allowed parameter space is severely constrained.",
"Figure: Correlation plots between λ 1 \\lambda _1 and λ 2 \\lambda _2 for normal hierarchy (top left panel), for inverted hierarchy (top right panel) andbetween Σ i m i \\Sigma _im_i, m i m_i and δ CP \\delta _{CP} in the bottom left (right) panel for normal (inverted) hierarchy.",
"The vertical and horizontal bandsrepresents the values of δ CP \\delta _{CP} beyond its 3σ3\\sigma range and Σ i m i >0.23\\Sigma _i m_i>0.23 eV, the upper bound on sum of active neutrino massesgiven by Planck data, respectively.Figure: Correlation plots between φ db \\phi _{db}, φ ab \\phi _{ab} and δ CP \\delta _{CP} for normal (left panel) and inverted (right panel) hierarchy.",
"The vertical band represents the values of δ CP \\delta _{CP} beyond its 3σ3\\sigma range."
],
[
"Correlation between model parameters with $\\tan \\psi \\ne 0$ .",
"With $\\tan \\psi \\ne 0$ , the analytic expression for $\\lambda _1$ is given by $\\lambda _1=\\frac{2\\lambda _2\\tan 2\\theta \\cos \\phi _{ab}\\sin \\psi }{\\sin \\phi _{ab}\\left[\\sqrt{3}+\\tan 2\\theta \\cos \\psi \\right]}\\;.$ We obtain the correlation plots between various parameters as given in Figs.",
"REF and Fig.",
"REF , by varying $\\phi _{db}$ between 0 to $2\\pi $ and $\\delta _{CP}$ in its $3\\sigma $ range $(0-0.17\\pi \\oplus 0.76\\pi -2\\pi )$ while fixing $\\sin ^2\\theta _{13}$ at its best fit value [29].",
"Comment on Neutrinoless double beta decay: The experimental observation of neutrinoless double beta decay would not only ascertain the Lepton Number Violation (LNV) in nature but it can also give absolute scale of lightest active neutrino mass.",
"The experimental non-observation of such a event puts a bound on half-life of this process on various isotopes which can be translated as a bound on particle physics parameter called as Effective Majorana Mass.",
"In the linear seesaw model, the light Majorana neutrinos contribute to neutrinoless double beta decay while the heavy pseudo-Dirac neutrinos give suppressed contribution.",
"The measure of LNV can be understood with the key parameter called Effective Majorana Mass which is defined as $\\left| M_{ee} \\right| \\equiv \\left| m^\\nu _{ee} \\right|&=&\\bigg | U^2_{e1}\\, m_1 + U^2_{e2}\\, m_2 e^{i \\rho } + U^2_{e3}\\, m_3 e^{i \\sigma } \\bigg |.$ The light neutrino mass eigenvalues $m_1, m_2, m_3$ depend on input model parameters.",
"These input model parameters are constrained to their allowed range in order to satisfy the oscillation data giving correct values of mass-squared differences and mixings.",
"The Majorana phases $\\rho $ and $\\sigma $ are related to $\\phi _{ab}$ and $\\phi _{db}$ in some way and thus, they are constrained to take limited value.",
"The element of PMNS mixing matrix derived from the knowledge of tribimaximal mixing multiplied by rotation matrix in 13 plane.",
"The estimation of Effective Majorana mass parameter using these already constrained input model parameters with the variation of lightest neutrino mass in displayed in Fig.REF where left-panel is for NH and right-panel is for IH pattern of light neutrino masses.",
"The current limit on half-life (or translated bound on Effective Majorana Mass parameter $m^\\nu _{ee}$ ) from GERDA Phase-I [30] is $T_{1/2}^{0\\nu }(^{76}\\text{Ge}) > 2.3\\times 10^{25}$ yr implies $|m_{ee}| \\le \\mbox{0.21\\,eV}$ and from KamLAND-Zen [31] as $T_{1/2}^{0\\nu }(^{136}\\text{Xe}) > 1.07\\times 10^{26}$ yr implies $|m_{ee}| \\le \\mbox{0.15\\,eV}$ .",
"There is also bound from CUORE experiment on effective Majorana mass parameter as $|m_{ee}| \\le \\mbox{0.073\\,eV}$ [32].",
"The expected reach of the future planned $0\\nu \\beta \\beta $ experiments including nEXO experiment gives $T_{1/2}^{0\\nu }(^{136}\\text{Xe}) \\approx 6.6\\times 10^{27}$ yr [33].",
"The variation of Effective mass parameter in green points with lightest neutrino mass is shown in Fig.REF for $\\tan \\psi =0$ and the same is plotted in Fig.",
"REF for $\\tan \\psi \\ne 0$ .",
"The left-panel is for NH pattern and right-panel is for IH patten of light neutrino masses.",
"The horizontal lines represent the bounds on Effective Majorana mass from various neutrinoless double beta decay experiments while the vertical shaded region are disfavored from Planck data.",
"The present bound is $m_\\Sigma < 0.23$ eV from Planck+WP+highL+BAO data (Planck1) at 95% C.L.",
"and $m_\\Sigma < 1.08$ eV from Planck+WP+highL (Planck2) at 95% C.L.",
"[34], [35].",
"Figure: Variation of effective Majorana parameter M ee M_{ee} which is a measure of lepton number violation with lightest neutrino mass for the case of normal (left panel)and inverted hierarchy (right panel) for tanψ≠0\\tan \\psi \\ne 0.This plot shows that quasi-degenerate pattern of light neutrinos are disfavoured if we consider the bound on sum of light neutrino masses from cosmology.",
"The current bound on Effective mass parameters from GERDA Phase-I and KamLAND-Zen proves that NH and IH pattern of light neutrinos are not sensitive.",
"However, the future planned nEXO experiment is sensitive to both pattern of light neutrinos."
],
[
"Leptogenesis",
"It is well known that leptogenesis is one of the most elegant frameworks for dynamically generating the observed baryon asymmetry of the Universe.",
"In the resonance leptogenesis scenarios, since the mass difference between two or more heavy neutrinos is much smaller than their masses and comparable to their widths, the CP asymmetry in their decays occurs primarily through self-energy effects ($\\epsilon $ -type) rather than vertex effect ($\\epsilon ^{\\prime }$ -type) and gets resonantly enhanced.",
"In the present $A_4$ realization, since the mass splitting between the two heavy neutrinos is rather tiny, it provides the opportunity for resonant leptogenesis, which will be discussed in this section.",
"During the calculation of light neutrino masses and mixing, we have neglected the higher order terms in the Lagrangian ${\\cal L}_{\\nu }$ as displayed in Eq.",
"(REF ), which are given with extra dimension six operators as follows $-\\left\\lbrace \\left[\\lambda _{N\\phi }\\phi _S+\\lambda _{N\\xi }\\xi +\\lambda _{N\\xi ^{\\prime }}\\xi ^{\\prime }\\right]\\frac{\\rho \\rho ^{\\prime }}{\\Lambda ^2}\\overline{N}_RN_R^c+\\left[\\lambda _{S\\phi }\\phi _S^{\\dagger }+\\lambda _{S\\xi }\\psi ^{\\dagger }+\\lambda _{S\\xi ^{\\prime }}{\\xi ^{\\prime }}^{\\dagger }\\right]\\frac{\\rho {\\rho ^{\\prime }}^{\\dagger }}{\\Lambda ^2}\\overline{S}_R S_R^c\\right\\rbrace ,$ as these extra terms do not make much difference in those calculations, but they make tiny mass splitting in doubly degenerate mass states of heavy neutrinos.",
"Including these additional terms, the Majorana mass matrix $\\mathbb {M}_2$ becomes $\\mathbb {M}_2=\\left(\\begin{array}{cc}m_R &m_{RS} \\\\m_{RS}^T & m_S\\end{array}\\right),$ where $\\nonumber m_R&=&\\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\left(\\begin{array}{c c c}\\frac{2}{3}\\lambda _{N\\phi }v_S+\\lambda _{N\\xi }v_{\\xi }&-\\frac{1}{3}\\lambda _{N\\phi }v_S&-\\frac{1}{3}\\lambda _{N\\phi }v_S\\\\-\\frac{1}{3}\\lambda _{N\\phi }v_S &\\frac{2}{3}\\lambda _{N\\phi }v_S &-\\frac{1}{3}\\lambda _{N\\phi }v_S+\\lambda _{N\\xi }v_{\\xi } \\\\-\\frac{1}{3}\\lambda _{N\\phi }v_S &-\\frac{1}{3}\\lambda _{N\\phi }v_S+\\lambda _{N\\xi }v_{\\xi } &\\frac{2}{3}\\lambda _{N\\phi }v_S\\end{array}\\right)\\\\ \\nonumber &+&{\\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}}\\left(\\begin{array}{ccc}0 & 0 &\\lambda _{N\\xi ^{\\prime }}v_{\\xi ^{\\prime }} \\\\0&\\lambda _{N\\xi ^{\\prime }}v_{\\xi ^{\\prime }}&0\\\\\\lambda _{N\\xi ^{\\prime }}v_{\\xi ^{\\prime }}&0 &0\\end{array}\\right)\\;,$ and $\\nonumber m_S &=&\\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\left(\\begin{array}{c c c}\\frac{2}{3}\\lambda _{S\\phi }v_S+\\lambda _{S\\xi }v_{\\xi }&-\\frac{1}{3}\\lambda _{S\\phi }v_S&-\\frac{1}{3}\\lambda _{S\\phi }v_S\\\\-\\frac{1}{3}\\lambda _{S\\phi }v_S &\\frac{2}{3}\\lambda _{S\\phi }v_S &-\\frac{1}{3}\\lambda _{S\\phi }v_S+\\lambda _{S\\xi }v_{\\xi } \\\\-\\frac{1}{3}\\lambda _{S\\phi }v_S &-\\frac{1}{3}\\lambda _{S\\phi }v_S+\\lambda _{S\\xi }v_{\\xi } &\\frac{2}{3}\\lambda _{S\\phi }v_S\\end{array}\\right)\\\\&+&{\\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}}\\left(\\begin{array}{ccc}0 & 0 &\\lambda _{S\\xi ^{\\prime }}v_{\\xi ^{\\prime }} \\\\0&\\lambda _{S\\xi ^{\\prime }}v_{\\xi ^{\\prime }}&0\\\\\\lambda _{S\\xi ^{\\prime }}v_{\\xi ^{\\prime }}&0 &0\\end{array}\\right)\\;.$ The mass matrix $\\mathbb {M}_2$ can be approximately block diagonalized by the unitary matrix $\\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cc}I &-I \\\\I &I\\end{array}\\right)$ and becomes $\\mathbb {M}_2^{\\prime }=\\left(\\begin{array}{cc}m_{RS}+\\displaystyle {\\frac{m_R+m_S}{2}} &m_S-m_R \\\\m_S-m_R &-m_{RS}+\\displaystyle {\\frac{m_R+m_S}{2}}\\end{array}\\right)\\approx \\left(\\begin{array}{cc}m_{RS}+\\displaystyle {\\frac{m_R+m_S}{2}} &0 \\\\0 &-m_{RS}+\\displaystyle {\\frac{m_R+m_S}{2}}\\end{array}\\right),\\hspace{14.22636pt}$ with eigenvalues $\\nonumber {M_1^{\\prime }}^{\\pm }\\approx M_1\\left(1\\pm \\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\frac{m_1^{\\prime }}{M_1}\\right)\\;, \\nonumber \\\\{M_2^{\\prime }}^{\\pm }\\approx M_2\\left(1\\pm \\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\frac{m_2^{\\prime }}{M_2}\\right)\\;, \\nonumber \\\\{M_3^{\\prime }}^{\\pm }\\approx M_3\\left(1\\pm \\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\frac{m_3^{\\prime }}{M_3}\\right)\\;,$ where $m_1^{\\prime }&=& 2\\text{Re}\\left\\lbrace \\left[a^{\\prime }-\\left(\\frac{bb^{\\prime }-\\frac{1}{2}\\left(bd^{\\prime }+b^{\\prime }d\\right)+dd^{\\prime }}{\\sqrt{b^2-bd+d^2}}\\right)\\right]e^{-i\\phi _1}\\right\\rbrace \\;, \\nonumber \\\\m_2^{\\prime }&=&2\\text{Re}\\left[\\left(b^{\\prime }+d^{\\prime } \\right)e^{-i\\phi _2}\\right]\\;,\\nonumber \\\\m_3^{\\prime }&=&2\\text{Re}\\left\\lbrace \\left[a^{\\prime }+\\left(\\frac{bb^{\\prime }-\\frac{1}{2}\\left(bd^{\\prime }+b^{\\prime }d\\right)+dd^{\\prime }}{\\sqrt{b^2-bd+d^2}}\\right)\\right]e^{-i\\phi _3}\\right\\rbrace \\;, \\nonumber \\\\a^{\\prime }&=&\\frac{1}{2}\\left(\\lambda _{N\\phi }+\\lambda _{S\\phi }\\right)v_S ,~~b^{\\prime }=\\frac{1}{2}\\left(\\lambda _{N\\xi }+\\lambda _{S\\xi }\\right)v_{\\xi }\\;,\\nonumber \\\\d^{\\prime }&=&\\frac{1}{2}\\left(\\lambda _{N\\xi ^{\\prime }}+\\lambda _{S\\xi ^{\\prime }}\\right)v_{\\xi ^{\\prime }}\\;.$ and $\\phi _i$ is the phase associated with $\\tilde{M}_i$ .",
"The above set of equations show that $m_i^{\\prime }$ can be of the order of $M_i$ since $a$ , $a^{\\prime }$ are of the order of $v_S$ , $b$ , $b^{\\prime }$ are of the order of $v_{\\xi }$ and $d$ , $d^{\\prime }$ are of the order of $v_{\\xi ^{\\prime }}$ .",
"The decay of nearly degenerate heavy neutrinos creates lepton asymmetry, and is given as [22] $\\epsilon _{N_i^{\\pm }}=&-&\\frac{1}{4\\pi A_{N_i^{\\pm }}} \\left[\\displaystyle {\\left(\\frac{\\tilde{m}_{D}}{v}\\right)^{\\dagger }\\left(\\frac{\\tilde{m}_{D}}{v}\\right)-\\left(\\frac{\\tilde{m}_{LS}}{v}\\right)^{\\dagger }\\left(\\frac{\\tilde{m}_{LS}}{v}\\right)}\\right]_{ii} \\displaystyle { \\text{Im}\\left[\\frac{\\tilde{m}_{D}^{\\dagger }\\tilde{m}_{LS}}{v^2}\\right]}_{ii} \\nonumber \\\\&\\times & \\frac{r_{N_i}}{\\displaystyle {{r_{N_i}}^2+\\frac{1}{64\\pi ^2}{A_{N_i^{\\pm }}}^2}}\\;,$ where $A_{N_i^{\\pm }}&=&\\frac{1}{2}\\left[\\left(\\frac{{\\tilde{m}_D}^{\\dagger }}{v}\\pm \\frac{{\\tilde{m}_{LS}}^{\\dagger }}{v}\\right)\\left(\\frac{{\\tilde{m}_D}}{v}\\pm \\frac{{\\tilde{m}_{LS}}}{v}\\right)\\right]_{ii}\\\\ \\nonumber r_{N_i}&=&\\frac{{{M_i^{\\prime }}^+}^2-{{M_i^{\\prime }}^-}^2}{{{M_i^{\\prime }}^+}{{M_i^{\\prime }}^-}}\\approx 4\\left(\\frac{v_{\\rho }v_{\\rho ^{\\prime }}}{\\Lambda ^2}\\frac{m_i^{\\prime }}{M_i}\\right)\\;,\\nonumber \\\\\\nonumber \\\\\\tilde{m}_D&=&m_D U_\\text{TBM} U_{13}^T,~~~~\\tilde{m}_{LS}=m_{LS}U_\\text{TBM}U_{13}^T\\;.$ Since $r_{N_i}\\ll A_{N_i^{\\pm }}$ , ${r_{N_i}}^2+\\frac{1}{64\\pi ^2}A_{N_i^{\\pm }}^2\\approx \\frac{1}{64\\pi ^2}A_{N_i^{\\pm }}^2$ , for $\\tilde{m}_{LS}\\ll \\tilde{m}_D$ $\\epsilon _{N_i^{\\pm }}\\approx -128\\pi \\text{Im}\\left[\\tilde{m}_{LS}^{\\dagger }\\tilde{m}_D\\right]_{ii}\\frac{r_{N_i}v^2}{\\left(\\tilde{m}_D^{\\dagger }\\tilde{m}_D\\right)^2}\\;.$ Substituting $\\tilde{m}_D^{\\dagger }\\tilde{m}_D=|y_1|^2\\left(v v_{\\rho ^{\\prime }}/\\Lambda \\right)^2$ , $\\tilde{m}_{D}^{\\dagger }\\tilde{m}_{LS}=y_1^*y_2 v^2 \\left(v_{\\rho } v_{\\rho ^{\\prime }}/{\\Lambda }^2 \\right)$ and $r_{N_i}\\approx 4\\left(v_{\\rho }v_{\\rho ^{\\prime }}/{\\Lambda ^2}\\right) \\left(m_i^{\\prime }/{M_i}\\right)$ in the above equation, we obtain $\\epsilon _{N_i^{\\pm }}\\approx -512\\pi \\left(\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}\\right)^2 \\frac{\\text{Im}\\left[y_1^*y_2 \\right]}{|y_1|^4}\\frac{m_i^{\\prime }}{M_i}\\;.$ Writing $y_1^*y_2=|y_1y_2|e^{i\\theta _{\\epsilon }}$ , one can have $\\epsilon _{N_i^{\\pm }}\\approx -512\\pi \\left(\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}\\right)^2 \\frac{|y_2|}{|y_1|^3}\\frac{m_i^{\\prime }}{M_i}\\sin \\theta _{\\epsilon }\\;.$ Here we calculate the baryon asymmetry for the case $M_3 \\ll M_2< M_1$ , i.e., normal hierarchy in active neutrino sector.",
"It is mainly the decay of $M_3^{\\pm }$ that contributes to the final baryon asymmetry.",
"Since the decay is in strong wash out region, final baryon asymmetry is given by [22], $\\eta _B=-\\frac{28}{79}\\left(\\frac{0.3\\epsilon _{N_3^{\\pm }}}{g_*K_{N_3^{\\pm }}\\left(\\ln K_{N_3^{\\pm }}\\right)^{0.6}}\\right),$ where $K_{N_i^{\\pm }}=\\displaystyle {\\frac{1}{8\\pi } \\left(\\frac{8\\pi ^3 g_*}{90}\\right)^{-1/2} }\\left( \\frac{M_{Pl}}{ M_{N_i^{\\pm }}}\\right)A_{N_i^{\\pm }}$ , $g_*\\approx 106.75$ and $M_\\text{Pl}=2.435\\times 10^{18}~\\text{GeV}$ are relativistic degrees of freedom of SM particles and Planck mass respectively.",
"Here $K_{N_3^{\\pm }}=K_{N_3}\\approx 0.234\\left[\\frac{m_3~(\\text{(eV)}}{10^{-2}}\\right]\\frac{v_{\\rho ^{\\prime }}}{v_{\\rho }} \\gg 1 \\;,$ as $m_3$ is of the order of $10^{-2} ~\\text{eV}$ and $\\frac{v_{\\rho ^{\\prime }}}{v_{\\rho }}\\gg 1$ .",
"Substituting $K_{N_3^{\\pm }}$ and $\\epsilon _{N_3^{\\pm }}$ in Eqn.",
"(REF ) gives $\\eta _B\\approx 0.174\\left(\\frac{\\left(\\frac{m_3(\\text{eV})}{10^{-2}}\\right)^2}{K_{N_3}^3(\\ln K_{N_3})^{0.6}}\\right)\\frac{|y_2|m_3^{\\prime }}{|y_1|^3M_3}\\sin \\theta _{\\epsilon }\\;.$ For $y_1\\approx y_2$ and $\\frac{m_3^{\\prime }}{M_3}\\approx 1$ the above equation gives $\\eta _B\\le 0.174\\left(\\frac{\\left(\\frac{m_3(\\text{eV})}{10^{-2}}\\right)^2}{|y_1|^2K_{N_3}^3(\\ln K_{N_3})^{0.6}}\\right).$ For $m_1<0.005~{\\rm eV}$ , $m_3\\approx 0.05~\\text{eV}$ , with this value of $m_3$ ,$|y_1|^2=10^{-3}$ and $\\eta _B=6.9\\times 10^{-10}$ from REF and REF we found the minimum value of $\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}$ requires to generate observed baryon asymmetry as $\\left.\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}\\right|_{min}=5.07\\times 10^{-5}\\;.$ Comment on Non-unitarity in leptonic sector: In usual case, the light active Majorana neutrino mass matrix is diagonalized by the PMNS mixing matrix $U_{\\rm PMNS}$ as $U_{\\rm PMNS}^{\\dagger }\\, m_{\\nu }\\, U^*_{\\rm PMNS} = \\text{diag}\\left(m_{1},m_{2},m_{3}\\right)$ where $m_1,m_2,m_3$ are mass eigenvalues for light neutrinos.",
"However, the diagonalizing mixing matrix in case of linear seesaw mechanism–where the neutral lepton sector is comprising of light active Majorana neutrinos plus additional two types of right-handed sterile neutrinos–is given by ${\\cal N}\\simeq (1-\\eta )U_{\\rm PMNS}\\, ,$ where the non-unitarity effect is parametrized as [36], $\\eta =\\frac{1}{2}m_D^*{m_{RS}^{\\dagger ~ -1}} m_{RS}^{-1}m_D^T\\;.$ In the linear seesaw framework under consideration, the $N-S$ mixing matrix $m_{RS}$ is symmetric and with $y_1\\approx y_2$ , the $\\nu -S$ mass term can be expressed as $m_{LS}^{\\dagger }m_{LS}=\\frac{1}{2}m_lM_0 (v_{\\rho }/v_{\\rho ^{\\prime }})$ where $m_0$ and $M_0$ are the masses of heaviest active and lightest heavy neutrinos respectively.",
"Thus, the above relation for $\\eta $ can be written in terms of light neutrino mass matrix and other input model parameters as $\\eta =\\frac{m_{\\nu }^*m_{\\nu }^T}{4m_0M_0\\frac{v_{\\rho }}{v_{\\rho {\\prime }}}}\\;.$ The maximum value of $\\eta $ for inverted mass hierarchy with lightest neutrino mass $m_l \\simeq 0.005~{\\rm eV}$ while considering the constrained value of the ratio of VEV $\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}=5.07\\times 10^{-5}$ as derived from the discussion of leptogenesis and using $M_0=5~\\text{TeV}$ can be obtained as follows $|\\eta |\\approx \\frac{1}{2}\\left[\\begin{array}{c c c}4\\times 10^{-12} & 10^{-11} & 10^{-11}\\\\10^{-11} & 5\\times 10^{-11} &5\\times 10^{-11}\\\\10^{-11} & 5\\times 10^{-11} &5\\times 10^{-11}\\end{array}\\right].$ Using the representative set of model parameters $m_0$ and $M_0$ , the mass matrices $m_D$ and $m_{LS}$ are expressed as follows $m_D=\\sqrt{\\frac{M_0m_0}{v_{\\rho }/v_{\\rho ^{\\prime }}}},~~~ m_{LS}=\\sqrt{\\frac{v_{\\rho }}{v_{\\rho ^{\\prime }}}M_0m_0}\\;.$ Using the constrained value of these model parameters $m_0$ and $M_0$ , the Dirac neutrino mass connecting $\\nu -N$ is found to be $m_D\\approx 70 $ MeV and the other mass term connecting $\\nu -S$ is $m_{LS}\\approx 3.5$ keV."
],
[
"Conclusion",
"In this paper we have considered the realization of linear seesaw by extending SM symmetry with $A_4\\times Z_4\\times Z_2$ along with a global symmetry $U(1)_X$ which is broken explicitly in Higgs potential.",
"In addition to SM fermions, the model has six heavy fermions, three right-handed neutrinos $(N_i)$ and three sterile neutrinos $(S_i)$ .",
"We found that each mass state of heavy neutrino is nearly doubly degenerate with a small mass splitting, which can be neglected for the calculation of active neutrino mass and mixing parameters.",
"The mass of active neutrinos are found to be inversely proportional to that of heavy neutrinos.",
"The model predicts lepton mixing matrix i.e., the PMNS as $U_{TBM}\\cdot U_{13}\\cdot P$ , where $U_{13}$ is the rotation in 13 plane and hence, explains well the results on mixing angles and $\\delta _{CP}$ from oscillation experiments.",
"We obtained the parametric space and correlation plots between various observables by fixing $\\theta _{13}$ at its best-fit value and the ratio mass squared differences, $\\Delta m^2_{21}/\\left|\\Delta m^2_{13}\\right|$ at 0.03 and varying $\\delta _{CP}$ in its $3\\sigma $ range.",
"We have demonstrated that pairs of nearly degenerate Majorana neutrinos in the model opens up the scope to resonant leptogenesis to account for the baryon asymmetry of the universe.",
"We calculated the minimum value of $v_{\\rho }/v_{\\rho ^{\\prime }}$ to generate observed baryon asymmetry by fixing the mass of lightest heavy neutrino in TeV for the case where heavy neutrino mass are highly hierarchical so that the only contribution to baryon asymmetry is from the decay of two lightest heavy neutrinos, the parameter space which satisfies this condition predicts normal hierarchy in active neutrino sector with lightest on less than $0.005$ eV.",
"In this case the maximum non-unitarity value, the model can accommodate in leptonic sector is very small and is of the order of $10^{-11}$ and the mass parameters are found to be $m_D\\approx 70 $ MeV and $m_{LS}\\approx $ 3.5 keV.",
"SM would like to thank University Grants Commission for financial support.",
"RM acknowledges the support from the Science and Engineering Research Board (SERB), Government of India through grant No.",
"SB/S2/HEP-017/2013."
]
] | 1709.01737 |
[
[
"Antenna Selection in MIMO Cognitive Radio-Inspired NOMA Systems"
],
[
"Abstract This letter investigates a joint antenna selection (AS) problem for a MIMO cognitive radio-inspired non-orthogonal multiple access (CR-NOMA) network.",
"In particular, a new computationally efficient joint AS algorithm, namely subset-based joint AS (SJ-AS), is proposed to maximize the signal-to-noise ratio of the secondary user under the condition that the quality of service (QoS) of the primary user is satisfied.",
"The asymptotic closed-form expression of the outage performance for SJ-AS is derived, and the minimal outage probability achieved by SJ-AS among all possible joint AS schemes is proved.",
"The provided numerical results demonstrate the superior performance of the proposed scheme."
],
[
"Introduction",
"Non-orthogonal multiple access (NOMA) and cognitive radio (CR) have emerged as efficient techniques to improve the spectral efficiency [1], [2].",
"By naturally combining the concepts of both NOMA and CR, a cognitive radio-inspired NOMA (CR-NOMA) scheme was proposed and studied in [3].",
"In CR-NOMA, the unlicensed secondary users (SU) is opportunistically served under the condition that the quality of service (QoS) of the licensed primary users (PU) is satisfied.",
"As a result, the transmit power allocated to the SU is constrained by the instantaneous signal-to-interference-plus-noise ratio (SINR) of the PU.",
"Compared to the conventional CR systems, higher spectral efficiency can be achieved by CR-NOMA because both the PU and SU can be served simultaneously using the same spectrum.",
"Recently, multiple-input multiple-output (MIMO) techniques have been considered in CR-NOMA systems to exploit the spatial degrees of freedom [4].",
"To avoid high hardware costs and computational burden while preserving the diversity and throughput benefits from MIMO, the antenna selection (AS) problem for MIMO CR-NOMA systems has been investigated in [5], wherein the SU was assumed to be rate adaptive and the design criterion was to maximize the SU's rate subject to the QoS of PU.",
"On the other hand, the outage probability has also been commonly used to quantify the performance of AS for an alternative scenario, wherein users have fixed transmission rates [6].",
"To the best of the authors' knowledge, the outage-oriented AS schemes for CR-NOMA systems have not been studied in open literature.",
"Motivated by this, the design and analysis of the outage-oriented joint AS algorithm for MIMO CR-NOMA networks is studied in this letter, which is fundamentally different from that for orthogonal multiple access (OMA) networks.",
"This is because there is severe inter-user interference in NOMA scenarios, wherein the signals are transmitted in an interference-free manner in OMA scenarios.",
"Moreover, the transmit power allocated to the SU in CR-NOMA scenarios is constrained by the instantaneous SINR of the PU, which is affected by the antenna selection result.",
"In this case, the joint antenna selection for NOMA networks is coupled with the power allocation design at the BS, which makes the design and analysis of the joint AS problem for CR-NOMA networks more challenging.",
"In this letter, we propose a new low-complexity joint AS scheme, namely subset-based joint AS (SJ-AS), to maximize the signal-to-noise ratio (SNR) of the SU under the condition that the QoS of the PU is satisfied.",
"The asymptotic closed-form expression of the outage performance for SJ-AS is derived, and the minimal outage probability achieved by SJ-AS among all possible joint AS schemes is proved.",
"Numerical results demonstrate the superior performance of the proposed scheme."
],
[
"System Model and Proposed Joint AS Scheme",
"Consider a MIMO CR-NOMA downlink scenario as [4], wherein two users including one PU and one SU are paired in one group to perform NOMA.",
"We consider that BS, PU and SU are equipped with $N$ , $M$ and $K$ antennas, respectively.",
"We assume that the channels between the BS and users undergo spatially independent flat Rayleigh fading, then the entries of the channel matrix, e.g., ${\\tilde{h}_{nm}}$ (${\\tilde{g}_{nk}}$ ), can be modelled as independent and identically distributed complex Gaussian random variables, where $\\tilde{h}_{nm}$ ($\\tilde{g}_{nk}$ ) represents the channel coefficient between the $n$ th antenna of the BS and the $m$ th ($k$ th) antenna of the PU (SU).",
"For notation simplicity, we define ${h}_{nm}=|\\tilde{h}_{nm}|^2$ and ${g}_{nk}=|\\tilde{g}_{nk}|^2$ .",
"As in [7], we consider that the BS selects one (e.g., $n$ th) out of its $N$ antennas to transmit information, while the users select one (e.g., $m$ th and $k$ th) out of $M$ and $K$ available antennas respectively to receive massages.",
"In this sense, only one RF chain is needed at each node to reduce the hardware cost, power consumption and complexity, and only the partial channel state information, i.e., the channel amplitudes, is needed at the BS, which is assumed perfectly known at the BS through the control signalling.",
"According to the principle of NOMA, the BS broadcasts the superimposed signals $\\sqrt{a}s_p + \\sqrt{b}s_s$, where $s_p$ ($s_s$ ) denotes the signal to the PU (SU) with $\\mathbb {E}[{\\left| {{s_p}} \\right|^2}]=\\mathbb {E}[{\\left| {{s_s}} \\right|^2}]=1$, and $a$ and $b$ are the power allocation coefficients satisfying $a+b=1$ .",
"Then the received signals at the PU and SU are given by ${y_p}&=&\\sqrt{P}\\tilde{h}_{nm}\\left(\\sqrt{a}s_p + \\sqrt{b}s_s\\right)+n_p,\\\\{y_s}&=&\\sqrt{P}\\tilde{g}_{nk}\\left(\\sqrt{a}s_p + \\sqrt{b}s_s\\right)+n_s,$ where $P$ is the transmit power at the BS, and $n_p$ ($n_s$ ) is the complex additive white Gaussian noise with variance $\\sigma _p^2$ ($\\sigma _s^2$).",
"For simplicity, we assume $\\sigma _p^2 = \\sigma _s^2=\\sigma ^2$.",
"Following the principle of CR-NOMA, $s_p$ is decoded by treating $s_s$ as noise at both users, and $s_s$ may be recovered at the SU when $s_p$ has been successfully subtracted in the SIC procedure.",
"By denoting the transmit SNR as $\\rho =P/\\sigma ^2$ , the received SINR of decoding $s_p$ at the PU is given by $\\gamma _{p}\\!\\!&=&\\!\\!",
"{ah_{nm}}/\\left(bh_{nm}+1/\\rho \\right).$ Similarly, the received SINR to detect $s_p$ at the SU is given by $\\gamma _{s\\rightarrow p}\\!\\!&=&\\!\\!",
"{ag_{nk}}/\\left(bg_{nk}+1/\\rho \\right).$ When $s_p$ is successfully removed, the SNR to detect $s_s$ at the SU is given by $\\gamma _s&=&bg_{nk}\\rho .$ Let ${{\\gamma }}^{th}_p$ (${\\gamma }^{th}_s$) denotes the predetermined detecting threshold of $s_p$ ($s_s$ ).",
"As the SU is served on the condition that ${\\gamma }^{th}_p$ is met, mathematically, $\\gamma _{p}$ and $\\gamma _{s\\rightarrow p}$ should satisfy the following constraint simultaneously: $\\min \\left(\\gamma _{p},\\gamma _{s\\rightarrow p}\\right)\\geqslant \\gamma ^{th}_p$ ."
],
[
"The Formulation of Joint AS Optimization Problem",
"In order to maximize the received SNR of the SU, we would like to solve the following optimization problem: $\\mathbf {P}:~\\lbrace b^*, n^*,m^*,k^*\\rbrace =&\\mathop {\\arg \\max }\\limits _{b,n \\in {\\mathcal {N}},m \\in {\\mathcal {M}},k \\in {\\mathcal {K}}} {\\gamma _s}\\left(b,{g}_{nk}\\right),\\\\&\\mathrm {s.t.",
"}~\\min \\left(\\gamma _{p},\\gamma _{s\\rightarrow p}\\right)\\geqslant \\gamma ^{th}_p,\\\\&~~~~~0\\le b<1.$ where $\\mathcal {N}\\!\\!=\\!\\!\\lbrace 1,\\cdots ,N\\rbrace , \\mathcal {M}\\!\\!=\\!\\!\\lbrace 1,\\cdots ,M\\rbrace , \\small \\mathrm {and}~\\mathcal {K}\\!\\!=\\!\\!\\lbrace 1,\\cdots ,K\\rbrace $, and $\\mathbf {P}$ is the joint optimization problem of antenna selection and power allocation.",
"Specifically, similar to [8], given the antenna indexes $n$ , $m$ and $k$ , the optimal power allocation strategy $b$ can be obtained based on Lemma 1.",
"Lemma 1 Given the antenna indexes $n$ , $m$ and $k$ , the optimal power allocation strategy $b$ is given by $b=\\max \\left(\\left(\\beta \\rho -{\\gamma }^{th}_p\\right)/\\left(\\left({\\gamma }^{th}_p+1\\right)\\beta \\rho \\right),0\\right),$ where $\\beta =\\min \\left(h_{nm},~g_{nk}\\right)$ .",
"Given antenna indexes $n$ , $m$ and $k$ , by substituting (REF )-(REF ) into (REF ), the power coefficient $b$ should satisfy the condition: $b\\!\\leqslant \\!\\frac{\\beta \\rho -{\\gamma }^{th}_p}{\\left({\\gamma }^{th}_p+1\\right)\\beta \\rho }$.",
"Meanwhile, $\\gamma _s$ is an increasing function of $b$ as shown in (REF ).",
"In this case, in order to maximize $\\gamma _s$ , $b$ should take the maximum value in its range.",
"By noting that $0\\leqslant b<1$ , we then can express the optimal power allocation coefficient $b$ as in (REF ).",
"The proof is completed.",
"By substituting (REF ) into (REF ) and when $b>0$ , we have $\\gamma _s\\left(h_{nm},g_{nk}\\right)=\\frac{\\min \\left(h_{nm},g_{nk}\\right)\\rho -{\\gamma }^{th}_p}{\\left({\\gamma }^{th}_p+1\\right)\\min \\left(h_{nm},g_{nk}\\right)} g_{nk},$ otherwise $\\gamma _s\\!=\\!0$ .",
"At this point, the joint optimization problem $\\mathbf {P}$ is simplified into the joint antenna selection problem.",
"It is straightforward to see that finding the optimal antenna indexes $\\lbrace n^*,m^*,k^*\\rbrace $ may require an exhaustive search (ES) over all possible antenna combinations with the complexity of $\\mathcal {O}$ is usually used in the efficiency analysis of algorithms and $q(x)=\\mathcal {O}\\left(p(x)\\right)$ when $\\lim \\limits _{x\\rightarrow \\infty }|\\frac{q(x)}{p(x)}|=c, 0<c<\\infty $ .$\\mathcal {O}\\left(NMK\\right)$ .",
"When $N$ , $M$ and $K$ become large, the computational burden of ES may become unaffordable.",
"Motivated by this, an computationally efficient joint AS algorithm for MIMO CR-NOMA systems will be developed in the next subsection."
],
[
"Proposed Subset-based Joint AS (SJ-AS) Scheme",
"The aim of SJ-AS algorithm is to decrease the computational complexity by greatly reducing the searching set, while ensuring the QoS of the PU and maximizing the achievable SNR of the SU.",
"Specifically, SJ-AS mainly consists of the following three stages.",
"Stage 1.",
"Build the subset $\\mathcal {S}_1=\\big \\lbrace \\left(h^{(n)},g^{(n)}\\right),n\\in \\mathcal {N}\\big \\rbrace $ to reduce the search space, where $h^{(n)}$ and $g^{(n)}$ are the maximum-value elements in the $n$ th row of $\\bf {H}$ and $\\bf {G}$ , respectively.",
"Mathematically, we have $h^{(n)}&=&\\max \\left(h_{n1},\\cdots ,h_{nM}\\right),\\\\g^{(n)}&=&\\max \\left(g_{n1},\\cdots ,g_{nK}\\right).$ Stage 2.",
"Build the subset $\\mathcal {S}_2$ by selecting the pairs from $\\mathcal {S}_1$ , in which each pair ensures the target SINR of the PU can be supported and $s_p$ can be subtracted successfully at the SU.",
"That is, $\\mathcal {S}_2=\\left\\lbrace \\min \\left(\\gamma _{p}^{(n)},\\gamma _{s\\rightarrow p}^{(n)}\\right)\\geqslant \\gamma ^{th}_p,~n\\in \\mathcal {S}_1\\right\\rbrace ,$ where $\\gamma _{p}^{(n)}$ and $\\gamma _{s\\rightarrow p}^{(n)}$ can be obtained by substituting $h^{(n)}$ and $g^{(n)}$ into (REF ) and (REF ), respectively.",
"Specifically, $b^{(n)}$ is given in (REF ) with $\\beta ^{(n)}=\\min (h^{(n)},g^{(n)})$ .",
"Stage 3.",
"When $|\\mathcal {S}_2|>0$ , select the antenna triple which can maximize the SNR for the SU, i.e., $\\lbrace n^*, m^*, k^*\\rbrace =\\arg \\max \\left\\lbrace \\gamma _s\\left(h^{(n)},g^{(n)}\\right), n\\in \\mathcal {S}_2\\right\\rbrace .$ Let $m^*$ and $k^*$ denote the original column indexes of $h^{(n^*)}$ and $g^{(n^*)}$ , respectively.",
"That is, the $n^*$ th antenna at the BS, and the $m^*$ th and $k^*$ th antennas at the PU and SU are jointly selected.",
"In contrast, when $|\\mathcal {S}_2|=0$ , the system suffers from an outage.",
"As mentioned before, the complexity of the ES-based scheme is as high as $\\mathcal {O}\\left(NMK\\right)$ .",
"In contrast, the complexity of the proposed SJ-AS scheme is upper bounded by $\\mathcal {O}\\left(N\\left(M+K+2\\right)\\right)$ .",
"For the case $N\\!=\\!M\\!=\\!K$ , we can find that the complexity of SJ-AS is approximately $\\mathcal {O}\\left(N^2\\right)$ , which is an order of magnitude lower than $\\mathcal {O}\\left(N^3\\right)$ of the optimal ES-based scheme."
],
[
"Performance Evaluation",
"In this section, we will analyse the system outage performance achieved by SJ-AS.",
"By using the assumption that channel coefficients are Rayleigh distributed, the cumulative density functions (CDF) and the probability density functions (PDF) of $h^{(n)}$ and $g^{(n)}$ in $\\mathcal {S}_1$ can be expressed as in [5], $F_{h^{(n)}}(x)\\!\\!\\!\\!&=&\\!\\!\\!\\!\\left(1-e^{-\\Omega _hx}\\right)^M, ~F_{g^{(n)}}(x)\\!=\\!\\left(1-e^{-\\Omega _gx}\\right)^K,~\\\\f_{h^{(n)}}(x)\\!\\!\\!\\!&=&\\!\\!-\\sum \\nolimits _{m=0}^M(-1)^m\\binom{M}{m}m\\Omega _he^{-m\\Omega _hx},\\\\f_{g^{(n)}}(x)\\!\\!\\!\\!&=&\\!\\!\\!\\!-\\sum \\nolimits _{k=0}^K(-1)^k\\binom{K}{k}k\\Omega _ge^{-k\\Omega _gx},$ where $\\Omega _h=1/\\mathbb {E}\\left[h_{im}\\right]$, $\\Omega _g=1/\\mathbb {E}\\left[g_{nk}\\right]$, and $f_{h^{(n)}}(x)$ and $f_{g^{(n)}}(x)$ are expanded based on the binomial theorem.",
"Let $\\mathcal {O}_1$ denote the event $|\\mathcal {S}_2|=0$ , and $\\mathcal {O}_2$ denote the event $\\gamma _s^{(n^*)}<{\\gamma }^{th}_s$ while $|\\mathcal {S}_2|>0$ .",
"As in [9], the overall system outage is defined as the event that any user in the system cannot achieve reliable detection, i.e., $\\mathrm {Pr}\\left(\\mathcal {O}\\right)={\\mathrm {Pr}\\left(\\mathcal {O}_1\\right)}+{\\mathrm {Pr}\\left(\\mathcal {O}_2\\right)}.$ In this case, the asymptotic system outage probability can be obtained according to the following lemma.",
"Lemma 2 When the transmit SNR $\\rho \\rightarrow \\infty $ , the system outage probability achieved by SJ-AS can be approximated as $\\mathrm {P}(\\mathcal {O})\\!\\!&\\approx &\\!\\!\\sum _{\\ell =0}^N\\binom{N}{\\ell }\\left(\\sum _{m=1}^M\\sum _{k=1}^Kc_{m,k}\\left(e^{-\\varphi _{m,k}{c_1}}-e^{-\\phi _{m,k}}\\right.\\right.\\nonumber \\\\\\!\\!\\!\\!&+&\\!\\!\\!\\!\\left.\\left.\\frac{k\\Omega _ge^{-\\varphi _{m,k}c_2}\\left(1-e^{-\\varphi _{m,k}c_1}\\right)}{\\varphi _{m,k}}\\right)\\right)^{\\ell }F_{\\beta ^{(n)}}\\!\\!\\left(c_1\\right)^{N-\\ell }.~~~$ where $c_{m,k}=(-1)^{m+k}\\binom{M}{m}\\binom{K}{k}$, $\\varphi _{m,k}=m\\Omega _h+k\\Omega _g$, $c_1=\\frac{{\\gamma }^{th}_p}{\\rho }$, $c_2={{\\gamma }^{th}_s\\left({\\gamma }^{th}_p+1\\right)}/{\\rho }$, $\\phi _{m,k}=m\\Omega _hc_1+k\\Omega _gc_2$, and $ F_{\\beta ^{(n)}}(x)=1-\\left(F_{h^{(n)}}(x)-1\\right)\\left(F_{g^{(n)}}(x)-1\\right)$.",
"we can first calculate the term $\\mathrm {Pr}\\left(\\mathcal {O}_1\\right)$ as $\\mathrm {Pr}\\left(\\mathcal {O}_1\\right)\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!\\mathrm {Pr}\\left(|\\mathcal {S}_2|=0\\right)\\!=\\!\\prod _{n=1}^{N}\\mathrm {Pr}\\left(\\min \\left(\\gamma _{p}^{(n)},\\gamma _{s\\rightarrow p}^{(n)}\\right)<{\\gamma }^{th}_p\\right)\\nonumber \\\\\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!\\prod _{n=1}^{N}\\!\\!\\mathrm {Pr}\\!\\left(b^{(n)}\\le 0\\right)\\!=\\!\\prod _{n=1}^{N}\\!\\!\\mathrm {Pr}\\!\\left(\\beta ^{(n)}\\le \\frac{{\\gamma }^{th}_p}{\\rho }\\right)\\nonumber \\\\\\!\\!\\!\\!\\!&=&\\!\\!\\!\\!\\!\\left(F_{\\beta ^{(n)}}\\left(c_1\\right)\\right)^{N}.~~~$ where $c_1={{\\gamma }^{th}_p}/{\\rho }$ and the CDF of $\\beta ^{(n)}$ is given by $F_{\\beta ^{(n)}}(x)\\!\\!\\!&=&\\!\\!\\!1-\\mathrm {Pr}\\left(\\beta ^{(n)}>x\\right)\\nonumber \\\\\\!\\!\\!&=&\\!\\!\\!1-{\\mathrm {Pr}\\left(h^{(n)}>g^{(n)}> x\\right)}-{\\mathrm {Pr}\\left(g^{(n)}>h^{(n)}> x\\right)}\\nonumber \\\\\\!\\!\\!&=&\\!\\!\\!1-\\left(F_{h^{(n)}}(x)-1\\right)\\left(F_{g^{(n)}}(x)-1\\right).$ By substituting (REF ) into (REF ), $\\mathrm {Pr}\\left(\\mathcal {O}_1\\right)$ is obtained.",
"We then turn to the calculation of $\\mathrm {Pr}\\left(\\mathcal {O}_2\\right)$ , $\\mathrm {Pr}\\left(\\mathcal {O}_2\\right)&=&\\mathrm {Pr}\\left(\\rho g^{(n^*)}b^{(n^*)}<{\\gamma }^{th}_s,~|\\mathcal {S}_2|>0\\right)\\nonumber \\\\&=&\\mathrm {Pr}\\left(g^{(n^*)}b^{(n^*)}<\\frac{{\\gamma }^{th}_s}{\\rho },~|\\mathcal {S}_2|>0\\right).$ Let $\\alpha ^{(n)}=g^{(n)}b^{(n)}$ for $\\forall n\\in \\mathcal {S}_2$ .",
"Since $b^{(n)}>0$ for $\\forall n\\in \\mathcal {S}_2$ , the product $\\alpha ^{(n^*)}=g^{(n^*)}b^{(n^*)}$ in (REF ) can be presented as $\\alpha ^{(n^*)}=\\max \\left(\\alpha ^{(n)}\\right),~~\\mathrm {for}~~\\forall n\\in \\mathcal {S}_2.$ Then $\\mathrm {Pr}\\left(\\mathcal {O}_2\\right)$ can be further expressed as $\\mathrm {Pr}\\left(\\mathcal {O}_2\\right)\\!\\!\\!&=&\\!\\!\\!\\mathrm {Pr}\\left(\\alpha ^{(n^*)}<\\frac{{\\gamma }^{th}_s}{\\rho },~|\\mathcal {S}_2|>0\\right)\\nonumber \\\\\\!\\!\\!&=&\\!\\!\\!\\sum _{\\ell =1}^{N}\\mathrm {Pr}\\left(\\alpha ^{(n^*)}<\\frac{{\\gamma }^{th}_s}{\\rho }\\mid |\\mathcal {S}_2|=\\ell \\right)\\mathrm {Pr}\\left(|\\mathcal {S}_2|=\\ell \\right)\\nonumber \\\\\\!\\!\\!&=&\\!\\!\\!\\sum _{\\ell =1}^{N}\\left(\\mathrm {Pr}\\left(\\alpha ^{(n)}<\\frac{{\\gamma }^{th}_s}{\\rho }\\mid |\\mathcal {S}_2|=\\ell \\right)\\right)^{\\ell }\\mathrm {Pr}\\left(|\\mathcal {S}_2|=\\ell \\right)\\nonumber \\\\\\!\\!\\!&=&\\!\\!\\!\\sum _{\\ell =1}^{N}\\left(F_{\\alpha ^{(n)}}\\left(\\frac{{\\gamma }^{th}_s}{\\rho }\\right)\\right)^{\\ell }\\mathrm {Pr}\\left(|\\mathcal {S}_2|=\\ell \\right),$ in which, $F_{\\alpha ^{(n)}}(\\frac{{\\gamma }^{th}_s}{\\rho })\\!\\!&=&\\!\\!\\mathrm {Pr}\\left(\\frac{g^{(n)}\\left(\\beta ^{(n)}-\\frac{{\\gamma }^{th}_p}{\\rho }\\right)}{({\\gamma }^{th}_p+1)\\beta ^{(n)}}<\\frac{{\\gamma }^{th}_s}{\\rho }\\mid n\\in \\mathcal {S}_2, |\\mathcal {S}_2|>0\\right)\\nonumber \\\\\\!\\!&=&\\!\\!\\frac{{\\mathrm {Pr}\\left(\\frac{g^{(n)}\\rho -{{\\gamma }^{th}_p}}{{\\gamma }^{th}_p+1}<{\\gamma }^{th}_s,h^{(n)}\\geqslant g^{(n)}>\\frac{{\\gamma }^{th}_p}{\\rho }\\right)}}{\\mathrm {Pr}\\left(\\beta ^{(n)}>\\frac{{\\gamma }^{th}_p}{\\rho }\\right)}\\nonumber \\\\\\!\\!&+&\\!\\!\\frac{{\\mathrm {Pr}\\left(\\frac{g^{(n)}\\left(h^{(n)}\\rho -{{\\gamma }^{th}_p}\\right)}{({\\gamma }^{th}_p+1)h^{(n)}}<{\\gamma }^{th}_s,\\frac{{\\gamma }^{th}_p}{\\rho }<h^{(n)}<g^{(n)}\\right)}}{\\mathrm {Pr}\\left(\\beta ^{(n)}>\\frac{{\\gamma }^{th}_p}{\\rho }\\right)}\\nonumber \\\\\\!\\!&=&\\!\\!\\left(Q_1+Q_2\\right)/\\left(1-F_{\\beta ^{(n)}}(c_1)\\right),$ Where $Q_1\\!\\!&=&\\!\\!\\mathrm {Pr}\\left(\\frac{{\\gamma }^{th}_p}{\\rho }<g^{(n)}<\\frac{{\\gamma }^{th}_s\\left({\\gamma }^{th}_p+1\\right)+{\\gamma }^{th}_p}{\\rho }, h^{(n)}\\geqslant g^{(n)}\\right),~~~\\\\Q_2\\!\\!&=&\\!\\!\\mathrm {Pr}\\left(\\frac{{\\gamma }^{th}_p}{\\rho }<h^{(n)}<g^{(n)}<\\frac{{\\gamma }^{th}_s({\\gamma }^{th}_p+1)h^{(n)}}{h^{(n)}\\rho -{\\gamma }^{th}_p}\\right).$ Let $c_2={{\\gamma }^{th}_s\\left({\\gamma }^{th}_p+1\\right)}/{\\rho }$, $c_{m,k}=(-1)^{m+k}\\binom{M}{m}\\binom{K}{k}$, and $\\varphi _{m,k}=m\\Omega _h+k\\Omega _g$, then $Q_1$ can be obtained as follows, $Q_1\\!\\!&=&\\!\\!\\mathrm {Pr}\\left(c_1<g^{(n)}<c_1+c_2, h^{(n)}\\geqslant g^{(n)}\\right)\\nonumber \\\\\\!\\!&=&\\!\\!\\sum _{m=1}^M\\sum _{k=1}^Kc_{m,k}\\frac{k\\Omega _ge^{-\\varphi _{m,k}{c_1}}\\left(1-e^{-\\varphi _{m,k}{c_2}}\\right)}{\\varphi _{m,k}}.$ Similarly, when $\\rho \\rightarrow \\infty $ , $Q_2$ can be approximated as $Q_2\\!\\!\\!\\!&\\approx &\\!\\!\\!\\!\\mathrm {Pr}\\left(c_1<h^{(n)}<g^{(n)}<c_2\\right)\\nonumber \\\\\\!\\!\\!\\!&=&\\!\\!\\!\\!\\sum _{m=1}^M\\!\\sum _{k=1}^K\\!c_{m,k}\\!\\left(\\!\\frac{m\\Omega _he^{-\\varphi _{m,k}{c_1}}\\!+\\!k\\Omega _ge^{-\\varphi _{m,k}c_2}}{\\varphi _{m,k}}\\!-\\!e^{-\\phi _{m,k}}\\!\\right),~~~~$ where $\\phi _{m,k}=m\\Omega _hc_1+k\\Omega _gc_2$ .",
"On the other hand, the probability that $|\\mathcal {S}_2|=\\ell $ can be calculated as below, $\\mathrm {Pr}\\left(\\mid \\mathcal {S}_2\\mid =\\ell \\right)\\!\\!\\!\\!&=&\\!\\!\\!\\!\\binom{N}{\\ell }\\left(F_{\\beta ^{(n)}}\\!\\left(c_1\\right)\\right)^{N-\\ell }\\left(1\\!\\!-\\!\\!F_{\\beta ^{(n)}}\\left(c_1\\right)\\right)\\!^{\\ell }.~~~~$ By combining (REF )-(REF ) and applying some algebraic manipulations, $\\mathrm {P}(\\mathcal {O}_2)$ can be expressed as $\\mathrm {P}(\\mathcal {O}_2)\\!\\!&\\approx &\\!\\!\\sum _{\\ell =1}^N\\binom{N}{\\ell }\\left(\\sum _{m=1}^M\\sum _{k=1}^Kc_{m,k}\\left(e^{-\\varphi _{m,k}{c_1}}-e^{-\\phi _{m,k}}\\right.\\right.\\nonumber \\\\\\!\\!\\!\\!&+&\\!\\!\\!\\!\\left.\\left.\\frac{k\\Omega _ge^{-\\varphi _{m,k}c_2}\\left(1-e^{-\\varphi _{m,k}c_1}\\right)}{\\varphi _{m,k}}\\right)\\right)^{\\ell }F_{\\beta ^{(n)}}\\!\\!\\left(c_1\\right)^{N-\\ell }.~~~$ By summing (REF ) and (REF ), the proof of (REF ) is completed.",
"Remark 1: When $\\rho $ approaches infinity, $c_1$ , $c_2$ and $\\phi _{m,k}$ approach zero.",
"By using the binomial theorem and the approximation $1\\!-\\!e^{-x}\\!\\overset{x\\rightarrow 0}{\\approx }\\!x$, the system outage probability can be further approximated as follows: $\\mathrm {P}(\\mathcal {O})\\!\\!&\\approx &\\!\\!\\left(1-\\left(1-F_{h^{(n)}}(c_1)\\right)\\left(1-F_{g^{(n)}}(c_1)\\right)\\right)^N\\nonumber \\\\&\\approx &{\\zeta ^N}/{\\rho ^{N\\min (M,K)}},$ where $\\zeta \\!\\!=\\!\\!\\frac{{\\tilde{c}_1}^M}{\\rho ^{M-\\min (M,K)}}\\!\\!+\\!\\!\\frac{{\\tilde{c}_2}^K}{\\rho ^{K-\\min (M,K)}}\\!\\!-\\!\\!\\frac{{\\tilde{c}_1}^M{\\tilde{c}_2}^K}{\\rho ^{M+K-\\min (M,K)}}$ , $\\tilde{c}_1=\\Omega _h{\\gamma }^{th}_p$, and $\\tilde{c}_2=\\Omega _g{\\gamma }^{th}_p$.",
"From (REF ), we can see that the SJ-AS scheme can realize a diversity of $N\\min \\left(M,K\\right)$ .",
"Remark 2: The optimality of the proposed SJ-AS is illustrated in the following lemma.",
"Lemma 3 The proposed SJ-AS scheme minimizes the system outage probability of the considered MIMO CR-NOMA system.",
"This lemma can be proved by contradiction.",
"Suppose there exists another joint AS strategy achieving a lower system outage probability than SJ-AS.",
"Let $\\left(\\hat{n}^*, \\hat{m}^*,\\hat{k}^*\\right)\\ne \\left({n}^*, {m}^*,{k}^*\\right)$ denote the antenna triple selected by the new strategy.",
"According to the assumption, it is possible that there is no outage with $\\left(\\hat{n}^*, \\hat{m}^*,\\hat{k}^*\\right)$ antennas while an outage occurs with $\\left({n}^*, {m}^*,{k}^*\\right)$ antennas.",
"In this case, the pair $\\left(h_{\\hat{n}^*\\hat{m}^*},g_{\\hat{n}^*\\hat{k}^*}\\right)$ must be in $|\\mathcal {S}_2|$ to satisfy the target SINR of the PU, i.e., $|\\mathcal {S}_2|>0$.",
"Recall that the pair $\\left(h_{{n}^*{m}^*},g_{{n}^*{k}^*}\\right)\\in |\\mathcal {S}_2|$ is selected according to (REF ) to maximize ${\\gamma }_s$.",
"In this case, if the maximized $\\gamma _s$ cannot meet $\\gamma _s^{th}$, the antennas selected by other scheme which provides smaller ${\\gamma }_s$ cannot meet $\\gamma _s^{th}$, either.",
"Therefore, one can conclude that there is NO other joint AS strategies can achieve a lower outage probability than SJ-AS, which is contradicted to the assumption made earlier.",
"The lemma is proved."
],
[
"Numerical Studies",
"In this section, the performance of the proposed SJ-AS algorithm for MIMO CR-NOMA networks is evaluated by Monte Carlo simulations.",
"Let $\\Omega _h=d_p^\\varepsilon $ ($\\Omega _g=d_s^\\varepsilon $ ), where $d_p$ ($d_s$ ) is the distance between the BS and PU (SU), and the path-loss exponent is set as $\\varepsilon =3$ .",
"Fig.",
"(a) and (b) compare the received SNR of the SU and the system outage performance between SJ-AS and other AS strategies.",
"As illustrated in both figures, over the entire SNR region, SJ-AS outperforms the conventional max-min scheme, in which the antenna selection is executed under the max-min criteria, i.e., $\\max (\\min (h_{nm}, g_{nk}))$ for all $n\\in \\mathcal {N}, m\\in \\mathcal {M}$ and $k\\in \\mathcal {K}$.",
"Furthermore, the performance of both SJ-AS and the max-min scheme are much better than that of random AS, since both SJ-AS and the max-min scheme utilize the spatial degrees of freedom brought by the multiple antennas at each node.",
"We also see that the analytical results match the simulation results for SJ-AS, which validates our theoretical analysis in Sec.",
"III.",
"Moreover, compared to the optimal ES scheme, SJ-AS can achieve the optimal outage performance as discussed in Remark 2, but with significantly reduced computational complexity.",
"In particular, the corresponding average power allocation coefficient $b$ for each scheme is illustrate in Table.",
"I.",
"Again we can find that the SJ-AS can achieve the same power allocation of the optimal ES scheme.",
"Figure: Comparison of AS schemes, N=M=K=2N\\!=\\!M\\!=\\!K\\!=\\!2, d p =350d_p\\!=\\!350m, d s =250d_s\\!=\\!250m, σ=-70\\sigma \\!=\\!-70dBm, γ p =2 0.5 -1\\gamma _p\\!=\\!2^{0.5}-1 and γ s =2 2.5 -1\\gamma _s\\!=\\!2^{2.5}-1.",
"Table: Average power allocation coefficient bb."
],
[
"Conclusion",
"In this letter, we studied the joint AS and power allocation problem for a MIMO CR-NOMA system.",
"A computationally efficient SJ-AS scheme was proposed, and the asymptotic closed-form expression for the system outage performance and the diversity order for SJ-AS were both obtained.",
"Numerical results demonstrated that SJ-AS can outperform both the conventional max-min approach and the random selection scheme, and can achieve the optimal performance of the ES algorithm."
]
] | 1709.01629 |
[
[
"Evolution of the magnetic and structural properties of\n Fe$_{1-x}$Co$_x$V$_2$O$_4$"
],
[
"Abstract The magnetic and structural properties of single crystal Fe$_{1-x}$Co$_x$V$_2$O$_{4}$ samples have been investigated by performing specific heat, susceptibility, neutron diffraction, and X-ray diffraction measurements.",
"As the orbital-active Fe$^{2+}$ ions with larger ionic size are gradually substituted by the orbital-inactive Co$^{2+}$ ions with smaller ionic size, the system approaches the itinerant electron limit with decreasing V-V distance.",
"Then, various factors such as the Jahn-Teller distortion and the spin-orbital coupling of the Fe$^{2+}$ ions on the A sites and the orbital ordering and electronic itinerancy of the V$^{3+}$ ions on the B sites compete with each other to produce a complex magnetic and structural phase diagram.",
"This phase diagram is compared to those of Fe$_{1-x}$Mn$_x$V$_2$O$_{4}$ and Mn$_{1-x}$Co$_x$V$_2$O$_{4}$ to emphasize several distinct features."
],
[
"=1 Evolution of the magnetic and structural properties of Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ R. Sinclair Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA J. Ma Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA H.B.",
"Cao Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37381, USA T. Hong Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37381, USA M. Matsuda Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37381, USA Z. L. Dun Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA H. D. Zhou Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA The magnetic and structural properties of single crystal Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ samples have been investigated by performing specific heat, susceptibility, neutron diffraction, and X-ray diffraction measurements.",
"As the orbital-active Fe$^{2+}$ ions with larger ionic size are gradually substituted by the orbital-inactive Co$^{2+}$ ions with smaller ionic size, the system approaches the itinerant electron limit with decreasing V-V distance.",
"Then, various factors such as the Jahn-Teller distortion and the spin-orbital coupling of the Fe$^{2+}$ ions on the A sites and the orbital ordering and electronic itinerancy of the V$^{3+}$ ions on the B sites compete with each other to produce a complex magnetic and structural phase diagram.",
"This phase diagram is compared to those of Fe$_{1-x}$ Mn$_x$ V$_2$ O$_{4}$ and Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ to emphasize several distinct features.",
"72.80.Ga, 75.25.Dk, 75.50.Dd, 61.05.cp Normal spinels[1] AV$_2$ O$_4$ (A = Cd, Mn, Fe, Mg, Zn, and Co) have received considerable attention due to their physical properties resulting from the interplay among the spin-lattice coupling from the localized 3$d$ electrons, the orbital degrees of freedom, and the geometrically frustrated structure.",
"Furthermore, the AV$_2$ O$_4$ normal spinels can be divided into two groups based on the A ions.",
"One group includes AV$_2$ O$_4$ (A = Cd[2], [3], [4], Mg[5], [6], [7], [8], [9], Zn[10], [11]) with non-magnetic A ions.",
"In these three materials, the orbital ordering (OO) transition drives a cubic to tetragonal structural phase transition at low temperatures which relieves the geometrical frustration of the V-pyrochlore sublattice and leads to an antiferromagnetic transition of the V$^{3+}$ ions.",
"The other group includes AV$_2$ O$_4$ (A = Mn, Fe, Co) with magnetic A ions in which the additional A-B magnetic interactions or the Jahn-Teller (JT) active Fe$^{2+}$ ions lead to more complex physical properties.",
"For example, (i) MnV$_2$ O$_4$[12], [13], [14], [15], [16], [17], [18] exhibits a magnetic phase transition at 56 K with a collinear ferrimagnetic (CF) structure where the Mn$^{2+}$ moments are antiparallel to the V$^{3+}$ moments.",
"Then, an antiferro-OO transition in the $t_{2g}$ orbitals of the V$^{3+}$ ions occurs at 53 K, where the $d_{xy}$ orbital is occupied by one electron and the other electron occupies the $d_{yz}$ and $d_{zx}$ orbitals alternately along the $c$ -axis.",
"The characteristic feature of this OO transition is the accompanied cubic to tetragonal structural transition involving a compressed tetragonal distortion ($c$ $<$ $a$ ).",
"This OO transition also results in a non-collinear ferrimagnetic (NCF) ordering below 53 K where the V$^{3+}$ moments are canted from the $[$ 111$]$ direction; (ii) CoV$_2$ O$_4$[19], [20], [21] exhibits two magnetic transitions at 150 K and 75 K which are CF and NCF transitions, respectively.",
"This sample also shows no OO transition due to the fact that it is approaching the itinerant electron behavior with the small V-V distance.",
"In the AV$_2$ O$_4$ system, this increased electronic itinerancy due to the decreased V-V distance has been theoretically predicted[22], [23] and experimentally confirmed[24], [25], [26]; (iii) FeV$_2$ O$_4$[27], [28], [29], [30], [31], [32] exhibits at least three transitions.",
"It is unique since the Fe$^{2+}$ (3$d^6$ ) ions have orbital degrees of freedom in the doubly degenerate $e_g$ states.",
"First, a structural transition from a cubic to a tetragonal phase ($c$ $<$ $a$ ) occurs at 140 K which mainly involves the OO transition of Fe$^{2+}$ ions.",
"Then, a second structural transition from a tetragonal to an orthorhombic phase occurs at 110 K which is accompanied by a CF transition.",
"Finally, a third structural transition from an orthorhombic to another tetragonal phase ($c$ $>$ $a$ ) occurs at 60 K which is accompanied with a NCF transition.",
"In this low temperature tetragonal phase with $c$ $>$ $a$ , a ferro-OO transition containing a complex orbital of the V$^{3+}$ ions has been proposed[31] which is in contrast to the OO of the real V-orbitals observed in the tetragonal phase with $c$ $<$ $a$ for MnV$_2$ O$_4$ .",
"To better understand the distinct physical properties among AV$_2$ O$_4$ (A = Mn, Fe, Co), several studies on these solid solutions have been conducted.",
"For example, the resistivity and X-ray diffraction (XRD) studies on Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ show that with increasing Co-doping, the system approaches the itinerant electron limit with decreasing resistivity[33].",
"Around $x$ = 0.8, the system shows no structural phase transition down to 10 K[33], [34].",
"Recently, the neutron scattering experiments and first principle calculations have revealed that the strong competition between the orbital ordering and itinerancy in Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ is the key factor for its complex magnetic and structural phase diagram.",
"Interestingly, both the orbital ordering in the low Co-doping samples and the magnetic isotropy in the high Co-doping samples lead to the NCF states[35].",
"Modern studies on Fe$_{1-x}$ Mn$_x$ V$_2$ O$_{4}$[36], [37] also reveal a complex phase diagram in which the ferro-OO is gradually suppressed with increasing $x$ and changes to the antiferro-OO for $x$ $>$ 0.6.",
"Around $x$ = 0.6, the long range orbital ordering of the Fe $^{2+}$ ions also disappears.",
"This indicates that the ferro-OO is possibly stabilized by the orbital degrees of freedom of the Fe$^{2+}$ ions located at the A site.",
"In this paper, we aim to study the magnetic and structural properties of another solid solution of V-spinels: Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ .",
"The detailed specific heat, susceptibility, neutron diffraction, and XRD measurements performed on single crystals of Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ reveal a complex magnetic and structural phase diagram.",
"Single crystals of Fe$_{1-x}$ Co$_x$ V$_{2}$ O$_{4}$ were grown by the traveling-solvent floating-zone (TSFZ) technique.",
"The feed and seed rods for the crystal growth were prepared by solid state reactions.",
"Appropriate mixtures of FeO, CoO and V$_{2}$ O$_{3}$ were ground together and pressed into 6-mm-diameter 60-mm rods under 400 atm hydrostatic pressure and then calcined in vacuum in a sealed quartz tube at 950 $^{\\circ }$ C for 12 hours.",
"The crystal growth was carried out in argon in an IR-heated image furnace (NEC) equipped with two halogen lamps and double ellipsoidal mirrors with feed and seed rods rotating in opposite directions at 25 rpm during crystal growth at a rate of 15 mm/h.",
"Small pieces of single crystals were ground into fine, flat-plate powder samples for XRD, and the diffraction patterns were recorded with a HUBER Imaging Plate Guinier Camera 670 with Ge monochromatized Cu $K_{\\alpha 1}$ radiation (1.54059 Å).",
"Data was collected at temperatures down to 10 K with a cryogenic helium compressor unit.",
"The lattice parameters were refined from the XRD patterns by using the program $FullProf$ with typical refinements for all samples having $\\chi ^2$ $\\approx $ 0.3.",
"The refinements also corrected for the absorbed radiation.",
"X-ray Laue diffraction was used to align the crystals.",
"The dc magnetic-susceptibility measurements were performed using a Quantum Design superconducting interference device (SQUID) magnetometer using a magnetic field of 0.01 T. The specific heat measurements were performed on a Quantum Design Physical Property Measurement System (PPMS).",
"The crystal samples used for the magnetic-susceptibility and the specific heat measurements were not aligned.",
"Neutron-diffraction experiments were performed at the four-circle diffractometer (HB-3A) and the cold neutron triple-axis spectrometer (CG-4C, CTAX) configured to measure along the (H,H,K) planes at the High Flux Isotope Reactor (HFIR) of the Oak Ridge National Laboratory (ORNL).",
"Based on the following structural and magnetic data, a phase diagram of Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ is constructed, as shown in Fig.",
"1.",
"Due to the large number of transitions, the phase diagram is presented first to introduce the general trends observed in the data.",
"In total, our samples exhibited six different transitions which were corroborated through several experimental techniques.",
"The description of each transition is detailed in the caption of Figure 1.",
"Figure 2 shows the temperature dependence of the dc magnetic susceptibility and specific heat for Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ .",
"For $x$ = 0.05, the specific heat shows three transitions at $T_1$ = 129 K, $T_2$ = 108 K, and $T_3$ = 57 K. At $T_2$ , the susceptibility shows a sharp increase.",
"At $T_3$ , the zero field cooling susceptibility (ZFC) shows a sharp drop.",
"By comparing these transitions to FeV$_2$ O$_4$ ($T_1$ = 139 K, $T_2$ = 109 K, and $T_3$ = 60 K), it is obvious that $T_1$ represents the cubic to high temperature (HT) tetragonal ($c$ $<$ $a$ ) phase transition, $T_2$ represents the CF transition with the tetragonal to orthorhombic phase transition, and $T_3$ represents the NCF transition with the orthorhombic to low temperature (LT) tetragonal ($c$ $>$ $a$ ) phase transition.",
"It is also obvious that with 5% Co doping, both $T_1$ and $T_3$ decrease but $T_2$ increases.",
"For $x$ = 0.1, the specific heat still shows three transitions.",
"The susceptibility also still shows a sharp increase at $T_2$ , but the ZFC susceptibility does not show a sharp decrease at $T_3$ any more.",
"For $x$ = 0.2, the specific heat just shows two peaks at $T_2$ and $T_3$ .",
"Since at the first peak temperature the susceptibility shows a sharp increase, we assigned this as $T_2$ .",
"For 0.3 $\\le $ $x$ $\\le $ 0.8, the specific heat shows only one peak at $T_2$ , where again the related susceptibility shows a sharp increase.",
"For $x$ = 0.9, the susceptibility shows a sharp increase at $T_2$ and another clear cusp around $T_4$ = 75 K. By comparing these transitions to CoV$_2$ O$_4$ , one would expect that $T_2$ and $T_4$ correspond to the CF and NCF ordering temperatures, respectively.",
"Figure: The derivative of the ZFC susceptibility for Fe 1-x _{1-x}Co x _xV 2 _2O 4 _4.In order to probe the magnetic phase transition of Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ in more detail, the derivative of the ZFC susceptibility is shown in Fig.",
"3.",
"For $x$ = 0.05, the derivative shows two sharp peaks at $T_2$ and $T_3$ .",
"For 0.1 $\\le $ $x$ $\\le $ 0.7, every sample's derivative shows a broad peak around 60 K as well as the sharp peak at $T_2$ .",
"It is noteworthy that this 60 K (we assigned this temperature as $T_5$ ) feature is not exactly at the $T_3$ temperatures for $x$ = 0.1 ($T_3$ = 51 K) and 0.2 ($T_3$ = 42 K) samples observed from the specific heat.",
"For $x$ = 0.8, below the broad peak at 60 K, there is another sharp peak around 40 K. For $x$ = 0.9, the derivative shows a sharp peak at $T_2$ and a jump at $T_4$ .",
"The specific heat and susceptibility show complex magnetic and structural evolution for Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ .",
"Several general trends are that with increasing Co-doping ($x$ ), (i) $T_1$ decreases and disappears with $x$ $\\ge $ 0.2; (ii) $T_2$ increases; (iii) $T_3$ decreases and disappears with $x$ $\\ge $ 0.3; (iv) $T_5$ ($\\sim $ 60 K) seems to be Co-doping independent for 0.1 $\\le $ $x$ $\\le $ 0.7.",
"To further clarify the magnetic phase transitions in Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ , single crystal neutron diffraction measurements have been performed on selective samples.",
"Figure 4 shows the temperature dependence of the intensity of several Bragg peaks ((002), (220), (111)) of these samples.",
"With increasing Co-doping, both the magnetic moments and the V-canting angles decrease compared with FeV$_2$ O$_4$ ; however, the structural transition that the $x$ = 0.2 and $x$ = 0.5 samples undergo make it difficult to determine the exact values of the moments and canting angles from single crystal neutron diffraction.",
"For the $x$ = 0.8 sample at 5 K, the total moment of the A site ions is 3.2(1) $\\mu _{B}$ and the total moment of the B site (V$^{3+}$ ) ions is 0.8(2) $\\mu _{B}$ , and the V-canting angle decreases from 55(4)$$ for FeV$_2$ O$_{4}$ to 38(3)$$[30].",
"For $x$ = 0.2, a ferrimagnetic (FIM) signal develops below 109 K ($T_2$ ) at the symmetry-allowed Bragg positions (220) and (111) which confirms the paramagnetic to CF transition.",
"While the (002) peak is forbidden by the symmetry, the observed scattering intensity below 60 K signals the formation of an antiferromagnetic (AFM) spin structure in the $ab$ plane.",
"Therefore, the onset of the (002) magnetic reflection marks the CF-NCF transition at $T_5$ .",
"For the $x$ = 0.5 and $x$ = 0.8 samples, the onset of (002) peak occurs around 60 K ($T_5$ ) as well.",
"Similar behaviors of (220) and (111) peaks of the $x$ = 0.5, 0.8, and 0.9 samples confirm the CF transition at $T_2$ .",
"For $x$ = 0.9, the (002) peak behavior also confirms its CF-NCF transition at 75 K ($T_4$ ).",
"Also note that for the $x$ = 0.1 and 0.2 samples, $T_3$ no longer represents the CF-NCF transition since at higher temperatures, $T_5$ , the NCF ordering already occurs.",
"To better understand the structural phase transition in Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ , XRD measurements down to 10 K were performed.",
"Figure 5 shows the measured patterns and related refinements for $x$ = 0.1 at 280 K, 90 K, and 40 K, respectively.",
"At high temperature (280 K), the sample has a cubic phase.",
"At 90 K $<$ $T_1$ = 111 K, the best refinement of the pattern leads to a tetragonal structure ($I4$$_1$$/amd$ ) with $c$ $>$ $a$ .",
"Then at 40 K $ < $ $T_3$ = 51 K, the refinement shows that it keeps the same tetragonal structure.",
"Here, we tested the XRD pattern at 40 K with all the three possible tetragonal phases reported for FeV$_2$ O$_4$ (the HT tetragonal phase($I4$$_1$$/amd$ ) with $c$ $<$ $a$ and the LT tetragonal phase ($I4$$_1$$/amd$ ) with $c$ $>$ $a$ ) and MnV$_2$ O$_4$ (the tetragonal phase ($I4$$_1$$/a$ ) with $c$ $<$ $a$ ).",
"The major difference among these three phases are the atomic positions for Fe(Mn) and V ions[30], [31].",
"The refinements using the three phases lead to consistent results with $c$ $>$ $a$ , and the tetragonal phase ($I4$$_1$$/amd$ ) with $c$ $>$ $a$ gives the best fitting results.",
"The temperature dependence of the lattice parameters for $x$ = 0.1 obtained from the detailed XRD measurements is shown in Fig.",
"6(a).",
"Around $T_1$ = 111 K, the cubic phase changes to the tetragonal phase with $c$ $>$ $a$ .",
"Then below $T_3$ = 51 K, the lattice parameter $c$ slightly decreases and $a$ slightly increases which leads to a decrease of the $c/a$ ratio .",
"It is obvious that the structural transitions for $x$ = 0.1 are different from those of FeV$_2$ O$_4$ .",
"To further demonstrate this difference, the temperature dependence of the (400) peak for both samples are shown in Fig.",
"7.",
"For FeV$_2$ O$_4$ , the single (400) peak splits to two peaks below $T_1$ = 139 K (cubic to HT tetragonal phase), then splits to three peaks below $T_2$ = 107 K (HT tetragonal to orthorhombic phase), then merges to two peaks again (orthorhombic to LT tetragonal phase) below $T_3$ = 60 K. But for $x$ = 0.1, the single (400) peak splits to two peaks ((400) and (004)) just below $T_1$ = 111 K. With decreasing temperature these two peaks move away from each other or the splitting 2$\\theta $ within these two peaks increases, which means the $c/a$ ratio increases.",
"Then below $T_3$ = 51 K, these two peaks begin to move towards each other, which means the $c/a$ ratio decreases.",
"As shown in Fig.",
"7, the splitting for $x$ = 0.1 sample is 0.756 degree at 60 K but 0.674 degree at 10 K. This subtle structural distortion at $T_3$ occurs below its CF-NCF magnetic transition at $T_5$ = 60 K. While both the HT tetragonal and orthorhombic phases still manifest in the $x$ = 0.05 sample, in the $x$ = 0.1 sample the HT tetragonal ($c$ $<$ $a$ ) and orthorhombic phases do not exist.",
"Its structure changes from the cubic to the LT tetragonal ($c$ $>$ $a$ ) phase directly.",
"The refinements of the XRD data for the $x$ = 0.2 sample show similar results as those of the $x$ = 0.1 sample (not shown here).",
"Figure: (color online) The temperature dependence of the (400) peak for the xx = 0.5 sample.",
"The green dashed line represents the Lorentzian fit.",
"The red solid line represents the total fitting.The temperature dependence of the lattice parameters and $c/a$ ratio for $x$ = 0.5 (Fig.",
"6(b)) show that there is a cubic to tetragonal phase (($I4$$_1$$/amd$ ) with $c > a$ ) transition at 84 K. This temperature is below $T_2$ ( the CF ordering temperature) and above $T_5$ = 60 K (the NCF ordering temperature).",
"We assigned this temperature as $T_6$ .",
"As shown in Fig.",
"8, the single (400) peak splits around 84 K, which confirms the structural phase transition at $T_6$ .",
"The XRD refinements for samples with 0.3 $\\le $ $x$ $\\le $ 0.6 show similar structural phase transition at 106 K for $x$ = 0.3, 96 K for $x$ = 0.4 and 78 K for $x$ = 0.6, respectively.",
"The general trend is that with increasing $x$ , $T_6$ decreases.",
"There is no further structural phase transition or distortion below $T_6$ for 0.3 $\\le $ $x$ $\\le $ 0.6.",
"For the $x$ $\\ge $ 0.7 samples, there is no structural phase transition down to 10 K according to the XRD data (not shown here).",
"Figure: (color online) The Co-doping dependence of (a) the lattice parameter aa and d V-V d_{V-V} at room temperature; and (b) the c/ac/a ratio at 10 K.Another general rule obtained from the XRD refinements is that at room temperature the lattice parameter $a$ and the distance between the nearest V ions ($d_{V-V}$ ) decreases with increasing Co-doping, as shown in Fig.",
"9(a).",
"At 10 K, the $c/a$ ratio increases (the distortion decreases) with increasing Co-doping.",
"The structural parameters for the $x$ = 0.1 and 0.5 samples at room temperature and 10 K are listed in Table I.",
"Table: Structural parameters for the xx = 0.1 and 0.5 samples at 280 K (space group Fd-3mFd-3m) and 10 K (space group I4I4 1 _1/amd/amd).",
"The B-values for the Oxygen atoms presented below were optimized to find the best fit, and the values are have larger uncertainty due to the relatively low energy and flux of the laboratory X-ray diffractometer used.Based on the magnetic and structural data, several observations can be made.",
"First, the transition from the cubic to the HT tetragonal ($c$ $<$ $a$ ) phase at $T_1$ and the transition from the HT tetragonal to the orthorhombic phase at $T_2$ appear for the $x$ = 0 and 0.05 samples but disappear for the $x$ $\\ge $ 0.1 samples.",
"This suggests that slight disorder or Co-doping on the Fe sites is sufficient to suppress both transitions, behavior which confirms that both structural phase transitions are dominated by the A site Fe$^{2+}$ ions.",
"The transition at $T_1$ is due to the JT type compression of the FeO$_4$ tetrahedron, and the transition at $T_2$ is due to the spin-orbital interaction of the Fe$^{2+}$ ions in the magnetic ordered phase[30], [31].",
"Second, for the $x$ $\\ge $ 0.1 samples, the paramagnetic to CF transition temperature ($T_2$ ) increases with increasing Co-doping.",
"As shown in Fig.",
"9(a), the V-V distance decreases with increasing Co-doping.",
"This is similar to the chemical pressure effects on Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ .",
"The resistivity studies on Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$[33] have shown that with decreasing V-V distance the system approaches the itinerant electron behavior.",
"The DFT calculation on CoV$_2$ O$_4$[35] then shows that this increasing electronic itinerancy can lessen the magnetic anisotropies and enhance the A-B site's magnetic exchange interactions to increase the CF transition temperature.",
"This increase of $T_2$ with increasing Co-doping in Mn$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ has been experimentally confirmed, and we believe a similar situation occurs with increasing Co-doping in Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ .",
"Third, for the $x$ = 0.1 and 0.2 samples, a cubic to LT tetragonal ($c$ $>$ $a$ ) phase transition occurs around the paramagnetic to CF transition at $T_2$ , but for the 0.3 $\\le $ $x$ $\\le $ 0.6 samples, this structural phase transition occurs at $T_6$ which is below the CF transition temperature $T_2$ .",
"For the $x$ $\\ge $ 0.7 samples, no structural phase transition is observed down to 10 K. The direct change from the cubic to tetragonal ( $c$ $>$ $a$ ) phase shows that for the 0.1 $\\le $ $x$ $\\le $ 0.6 samples with larger doping on the Fe$^{2+}$ sites, this transition is controlled by the ferroic-orbital ordering of the V$^{3+}$ ions.",
"The decoupling of the magnetic phase transition at $T_2$ and structural phase transition at $T_6$ for the 0.3 $\\le $ $x$ $\\le $ 0.6 samples show the competition between the orbital ordering and itinerancy of V$^{3+}$ electrons.",
"With increasing Co-doping, the increasing electronic itinerancy leads to enhanced magnetic ordering that contrasts with the decreasing orbital ordering which is completely suppressed for the $x$ $\\ge $ 0.7 samples.",
"This is also revealed by the decreasing $c/a$ ratio (decreasing distortion, without distortion $c/a$ = 1.0 for the $x$ $\\ge $ 0.7 samples) with increasing Co-doping.",
"Some other details of the phase diagram are: (i) the CF-NCF transition temperature $T_5$ ($\\sim $ 60 K) for 0 $\\le $ x $\\le $ 0.8 is doping-independent.",
"It jumps to 75 K ($T_4$ ) for $x$ = 0.9 and 1.0 samples.",
"For the $x$ = 0 and 0.05 samples, the orthorhombic to tetragonal ($c$ $>$ $a$ ) structural phase transition ($T_3$ ) occurs simultaneously at $T_5$ .",
"However, for the 0.1 $\\le $ x $\\le $ 0.7 samples, there is no structural phase transition at $T_5$ .",
"This indicates that in this regime, the $T_5$ (NCF magnetic ordering) is controlled only by the V$^{3+}$ ions.",
"Then for the $x$ = 0.9 and 1.0 samples, the enhanced magnetic exchange isotropy due to the stronger electronic itinerancy stabilizes the CF-NCF transition at 75 K[35] which has been demonstrated by the DFT calculations on CoV$_2$ O$_4$ .",
"The derivative of the susceptibility of the $x$ = 0.8 sample shows two features for the NCF ordering: a broad peak at $T_5$ = 60 K similar to that of the 0.1 $\\le $ x $\\le $ 0.7 samples and a sharp peak at 40 K similar to that of the $x$ $\\le $ 0.9 samples.",
"This suggests that the $x$ = 0.8 sample is on the boundary for the competitions between the orbital ordering of the localized V$^{3+}$ spins and the enhanced exchange isotropy due to the itinerancy.",
"The former stabilizes the NCF ordering at $T_5$ while the latter stabilizes the NCF ordering at 40 K and then improves it to $T_4$ =75 K for the $x$ $\\ge $ 0.9 samples; (ii) for the $x$ = 0.1 and 0.2 samples, there is no structural phase transition at $T_3$ .",
"They instead exhibit a subtle structural distortion with decreased $c/a$ ratio.",
"Moreover, this particular $T_3$ is below the CF-NCF transition temperature, $T_5$ .",
"In the Fe$_{1-x}$ Mn$_x$ V$_2$ O$_{4}$ system, a similar decreased $c/a$ ratio has also been observed at the CF-NCF transition temperature which indicates this subtle structural distortion is due to the spin-lattice coupling of the V spin-canting process.",
"Despite the decoupling of $T_3$ and $T_5$ here, a similar situation may occur around $T_3$ for the $x$ = 0.1 and 0.2 samples.",
"We compare the phase diagram between Fe$_{1-x}$ Co$_x$ V$_2$ O$_4$ and Fe$_{1-x}$ Mn$_x$ V$_2$ O$_4$ .",
"The similarity is that in both systems, the HT tetragonal and orthorhombic phases disappear quickly with small doping.",
"This again confirms both phases are due to the presence of the Fe$^{2+}$ ions on the A sites.",
"The main difference is that in Fe$_{1-x}$ Mn$_x$ V$_2$ O$_4$ , the paramagnetic to CF transition is always accompanied with the cubic to tetragonal phase transition for the $x$ $\\le $ 0.6 samples, and the CF to NCF transition is always accompanied with another type of cubic to tetragonal phase transition for the $x$ $\\ge $ 0.7 samples.",
"In other words, the spin ordering and structural phase transition are always strongly coupled for Fe$_{1-x}$ Mn$_x$ V$_2$ O$_4$ .",
"However, in Fe$_{1-x}$ Co$_x$ V$_2$ O$_4$ these two transitions are decoupled with the structural phase transition occurring below the paramagnetic to CF transition.",
"Meanwhile, in the Mn$_{1-x}$ Co$_x$ V$_2$ O$_4$ phase diagram, the CF-NCF transition is decoupled from the cubic-tetragonal structural phase transition.",
"For MnV$_2$ O$_4$ , both transitions occur at the same temperature, but with increasing Co-doping in Mn$_{1-x}$ Co$_x$ V$_2$ O$_4$ the CF-NCF transition occurs at higher temperatures and is followed by the structural phase transition at lower temperatures.",
"This is similar to the separation between $T_3$ and $T_5$ for the $x$ = 0.1 and 0.2 samples in Fe$_{1-x}$ Co$_x$ V$_2$ O$_4$ .",
"Therefore, one general behavior for Co-doping systems seems to be the separation of the magnetic and structural phase transitions.",
"This separation should be due to the induced competition between the orbital ordering and electronic itinerancy.",
"With increasing Co-doping, the increased electronic itinerancy tends to enhance the A-B magnetic interaction and magnetic exchange isotropy ( to increase the CF and NCF transition temperatures) and suppress the orbital ordering (the structural phase transition temperature).",
"In summary, the single crystals of Fe$_{1-x}$ Co$_x$ V$_2$ O$_{4}$ were studied by specific heat, susceptibility, elastic neutron scattering, and XRD measurements.",
"The main findings are with increasing Co-doping: (i) the HT tetragonal and orthorhombic phases disappear quickly due to the small disorder on the Fe$^{2+}$ sites.",
"This confirms these two phases are due to the JT type distortion and spin-orbital coupling of the Fe$^{2+}$ ions; (ii) the increased electronic intinerancy results in enhanced magnetic ordering but suppressed orbital ordering.",
"The consequence is a complex magnetic and structural phase diagram with decoupled magnetic and structural phase transition boundaries.",
"R. S., Z.L.D., and H.D.Z.",
"thank the support from NSF-DMR through Award DMR-1350002.",
"The research at HFIR/ORNL, were sponsored by the Scientific User Facilities Division (J.M., H.B.C., T.H., M.M.,), Office of Basic Energy Sciences, US Department of Energy."
]
] | 1709.01842 |
[
[
"Control of light polarization by voltage tunable excitonic metasurfaces"
],
[
"Abstract We propose a light emitting device with voltage controlled degree of linear polarization of emission.",
"The device combines the ability of metasurfaces to control light with an energy-tunable light source based on indirect excitons in coupled quantum well heterostructures."
],
[
"Control of light polarization by voltage tunable excitonic metasurfaces S.V.",
"Lobanov School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom Skolkovo Institute of Science and Technology, Moscow 143026, Russia N.A.",
"Gippius Skolkovo Institute of Science and Technology, Moscow 143026, Russia S.G. Tikhodeev A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova Street 38, Moscow 119991, Russia M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia L.V.",
"Butov Department of Physics, University of California at San Diego, La Jolla, California 92093-0319, USA We propose a light emitting device with voltage controlled degree of linear polarization of emission.",
"The device combines the ability of metasurfaces to control light with an energy-tunable light source based on indirect excitons in coupled quantum well heterostructures.",
"Plasmonic resonances in metal gratings on a dielectric substrate, or plasmonic metasurfaces, are now widely used for various purposes such as to enhance the efficiency of light emitting devices (LED) [1], to magnify the light absorption in the photodetectors [2] and thin film solar cells [3], [4], to control the intensity and directivity of light emission (as nanoantennas) [5], to enhance the magnetooptical effects [6], [7], [8], the sensitivity of optical sensors for gas detecting, chemical and biosensing [9], [10], [11], to magnify nonlinear optical effects such as 2nd and 3rd harmonic generation [12], in photochemistry [13].",
"Figure: (a) Schematic diagram of the device.",
"Coupled quantum well, CQW, (black dashed lines) is positioned within an undoped 1 μ\\mu m thick Al 0.33 _{0.33}Ga 0.67 _{0.67}Aslayer (yellow) between a conducting n + n^+-GaAs layer serving as a homogeneous bottom electrode (green) and a striped topelectrode (silvery).",
"(b) Energy band diagram of the CQW; e, electron; h, hole.",
"The cyan dashed ellipse indicates an indirect exciton (IX).",
"(c) Energy of IX as a function of applied voltage.",
"The data is taken from Ref.High2008.",
"(d,e) Schematic of SPP propagation along a metal-dielectric boundary (d) and a thin metal film in asymmetric dielectric surrounding (e).",
"(f) Calculated dispersions of Ag/AlGaAs SPP (blue dashed line) and air/Ag/AlGaAs thin-layer SPP for silver film with thickness t=10t=10 nm(green solid line), 15 nm (black dash-dotted line), 20 nm (cyan dash-double-dotted line), and 30 nm (magenta dash-triple-dotted line).Red dotted line shows the light cone in Al 0.33 _{0.33}Ga 0.67 _{0.67}As.In this letter we propose to combine the ability of plasmonic metasurfaces to control light with an energy-tunable light source based on indirect excitons (IXs) for creating integrated optoelectronic devices where the polarization state of emitted light is controlled by voltage.",
"IXs are composed of electrons and holes in spatially separated quantum well layers (Fig. 1b).",
"Due to the IX built-in electric dipole moment $ed$ , the energy of light emitted by IXs is effectively controlled by voltage $\\delta E = - edF_z$ , where $d$ is the separation between the electron and hole layers (for coupled quantum wells, CQWs, $d$ is close to the distance between the QW centers) and $F_z \\propto V$ is an electric field perpendicular to the QW plane.",
"The IX energy control by voltage allowed creating a variety of excitonic devices, including traps and lattices for studying basic phenomena in IXs, as well as excitonic transistors, routers, and photon storage devices, which form the potential for creating excitonic signal processing devices and excitonic circuits [14], [15], [16].",
"Specifically, in this letter, we present a device demonstrating proof-of-principle for tunable excitonic metasurfaces, namely, a light emitting device with voltage controlled degree of linear polarization (DLP) of emission made of plasmonic metal grating deposited on top of an AlGaAs/GaAs CQW structure.",
"The metal grating is dual-purpose.",
"On the one hand, it serves as the upper electrode to control the IX emission frequency.",
"On the other hand, it provides plasmonic resonances, to control the DLP of IX photoluminescence.",
"The schematic diagram of the proposed device is shown in Fig. 1a.",
"CQW (black dashed lines) is positioned within an undoped 1 $\\mu $ m thick Al$_{0.33}$ Ga$_{0.67}$ As layer between a conducting $n^+$ -GaAs layer serving as a homogeneous bottom electrode and a top electrode made of Ag nanowires grating.",
"The nanowires have thickness $t$ , width $w$ and the grating period is $p$ .",
"Figure: (a) Calculated dependence of xx-linear polarized component of the IX device emission intensity I x I_x along zz-axis on the nanowires width wwand period pp for the photon energy ℏω=1.55\\hbar \\omega =1.55 eV and thickness t=10t=10 nm.",
"(b) Respective degree of linear polarization ρ\\rho .",
"(c,d) Calculated xx- (blue solid line) and yy- (green dashed line) linear polarized components of the emission spectra and respective degreeof linear polarization ρ\\rho (red dotted line) for the structures with the stripe thickness tt, width ww, and period pp, where(c) (t,w,p)=(10,40,100)(t,w,p)=(10,40,100) nm and (d) (10,150,210)(10,150,210) nm.The IX emission energy is controlled by voltage.",
"(e,f) Calculated electric field distribution of the light propagating along zz-axis with IX emission energy of (e) ℏω=\\hbar \\omega =1531 meV and (f) 1563 meV for the structures with(e) (t,w,p)=(10,40,100)(t,w,p)=(10,40,100) nm and (f) (10,150,210)(10,150,210) nm.Background color shows the absolute value of electric field.The CQW band diagram is shown in Fig.",
"1b; a typical dependence of the IX energy $E(V)$ on the applied voltage is shown in Fig. 1c[14].",
"Note however that $E(V)$ can be optimized for a tunable metasurface device by adjusting the CQW parameters including the QW widths, width of the barrier between the QWs, and the well and barrier materials.",
"Figure 1c shows that changing the applied voltage allows to scan the energy of IX photoluminescence in the range of 1520–1580 meV.",
"If the metal grating is engineered in such a way that one of its plasmon resonances is in the same energy range, it becomes possible to change the DLP of the device emission, due to pronounced changes of $x$ -polarized light transmission of the grating in the vicinity of the plasmonic resonances.",
"The characteristic energy dispersions of plasmons in metal graings can be simply understood as follows.",
"We start from a continuous metal film.",
"Figure 1d shows schematically the electric field of a surface plasmon (SP) polariton (SPP), propagating along a boundary between a dielectric and semi-infinite metal.",
"The calculated SPP energy dispersion as a function of wavenumber $k_\\mathrm {sp}=2\\pi /\\lambda _\\mathrm {sp}$ (where $\\lambda _\\mathrm {sp}$ is the SPP wavelength) is shown for Ag/Al$_{0.33}$ Ga$_{0.67}$ As boundary by blue dashed line in Fig. 1f.",
"It is located below the Al$_{0.33}$ Ga$_{0.67}$ As light cone (red dotted line in Fig. 1f).",
"The SPP dispersion curve for air/Ag boundary (not shown) lies below the air light cone (also not shown), both are significantly blue shifted due to a large refraction index of semicondicting AlGaAs.",
"If a semi-infinite metal is replaced by a metal film of finite thickness in asymmetric dielectric surrounding (air and Al$_{0.33}$ Ga$_{0.67}$ As in our case), the plasmons propagating on the opposite sides of the metal film interact with each other.",
"The lower-energy SPP (localized predominantly on the AlGaAs side of the film) repulses from the upper-energy SPP (localized predominantly on the air side).",
"Resultantly, that yields a red energy shift of the lower-energy SPP with the decrease of film thickness.",
"The calculated dispersions of the lower-energy SPP for silver film with thickness $t=10$ nm (green solid line), 15 nm (black dash-dotted line), 20 nm (cyan dash-double-dotted line), and 30 nm (magenta dash-triple-dotted line) are shown in Fig. 1f.",
"The dispersions are calculated neglecting absorption loses in silver and Al$_{0.33}$ Ga$_{0.67}$ As.",
"Note that the polarization of all such SPPs is always magnetic field $\\mathbf {H}\\Vert y$ (see in Fig.",
"1a), i. e., along the boundaries and perpendicular to the SPP propagation direction $x$ .",
"As to the SSP electric field (shown schematically in Fig.",
"1d-e), there are both transverse and longitudinal components, $E_z$ and $E_x$ , respectively.",
"If metal film is truncated from both sides into a single metal nanowire of width $w$ , the localized Mie plasmons arise due to SPP reflections back and forth, forming standing waves.",
"Formation of such localized plasmon requires the width of the wire to be roughly an integer number of thin-layer plasmon half wavelength, i.e.",
"$w_m=m\\lambda _\\mathrm {lp}/2,$ where $\\lambda _\\mathrm {lp}$ is wavelength of thin-layer plasmon and $m$ is a positive integer.",
"One can see from Fig.",
"1f, that, e.g., the 10-nm-thick-layer plasmon wavelength is $\\lambda _\\mathrm {lp}\\approx 120$ nm for the IX emission energy $\\hbar \\omega =1.55$ eV.",
"Calculated dependence of $x$ -linear polarized component of the IX device emission intensity $I_x$ along $z$ -axis on the nanowires width $w$ and period $p$ for the photon energy $\\hbar \\omega =1.55$ eV and thickness $t=10$ nm is shown in Fig. 2a.",
"In this and other figures, the emission intensity is normalized to the maximum emission intensity of an equivalent IX in a homogeneous semiconductor Al$_{0.33}$ Ga$_{0.67}$ As.",
"In order to calculate the emission intensity, we employ the optical scattering matrix method [17], [18] and the electrodynamical reciprocity principle as described in Refs.Maksimov2014,Lobanov2015,Lobanov2015a, and use the dielectric susceptibility of Ag from Ref.Johnson1972 and Al$_{0.33}$ Ga$_{0.67}$ As from Ref.Sopra1998.",
"Three resonance branches are clearly seen in Fig.",
"2a that correspond to excitation of localized SPPs.",
"These SPPs weakly depend on the period $p$ .",
"In a good qualitative agreement with Eq.",
"(1) for odd $m$ , these SPPs occur at nanowire widths $w_1\\approx 40\\,\\mathrm {nm}\\approx \\lambda _\\mathrm {lp}/2$ , $w_3\\approx 150\\,\\mathrm {nm}\\approx 3\\lambda _\\mathrm {lp}/2$ , and $w_5\\approx 270\\,\\mathrm {nm}\\approx 5\\lambda _\\mathrm {lp}/2$ .",
"The respective DLP $\\rho =(I_x-I_y)/(I_x+I_y)$ is shown in Fig. 2b.",
"Here, $I_y$ is $y$ -linear polarized component of the IX device emission intensity.",
"One can see that DLP is varied in the range from about 0% (unpolarized emission) to about -99% for the first plasmon branch ($w\\approx 40\\,\\mathrm {nm}$ ) and to about -85% for the second branch ($w\\approx 150\\,\\mathrm {nm}$ ).",
"Calculated $x$ - (blue solid line) and $y$ - (green dashed line) linear polarized components of the IX device emission spectra and respective DLP $\\rho $ (red dotted line) for the structures with the nanowires thickness $t$ , width $w$ , and period $p$ , where $(t,w,p)=(10,40,100)$ nm and $(10,150,210)$ nm, are shown in Fig.",
"2c and 2d, respectively.",
"In order to understand the effectiveness of the excitation of localized SPPs, we show in Figs.",
"2e and 2f the calculated electric field distribution of the light propagating along $z$ -axis with photon energies 1531 meV and 1563 meV, for structures in Figs.",
"2c and 2d, respectively.",
"The distributions have one and three anti-nodes, respectively, in agreement with the above discussion.",
"Note that SPPs with even $m$ have an even distribution of the electric field with respect to reflection in the vertical symmetry plane of the structure and, consequently, can not emit light along the $z$ -axis for the structure of interest with spatially uniform excitation of CQW.",
"Figure: (a) As Fig.",
"2c, but for (t,w,p)=(45,190,230)(t,w,p)=(45,190,230) nm.",
"(b) As Fig.",
"2e, but for ℏω=1557\\hbar \\omega =1557 meV and (t,w,p)=(45,190,230)(t,w,p)=(45,190,230) nm.",
"(c)-(d) As Figs.",
"2a and 2b, but for t=45t=45 nm.In the described above cases of thin metal wires the transmissivity of the grating in both linear polarisations along and perpendicular to the wires is relatively high away from the SPP resonances.",
"The emission of our device becomes strongly linearly polarized along the wires ($\\Vert y$ , $\\rho \\sim -1$ ) at the SPP resonance frequency, because the grating becomes nearly opaque for $x$ -polarized light.",
"Using thick metal gratings, and the famous effect of the extraordinary optical transmission (EOT) through arrays of subwavelength holes [24], [25], [26], [27], it is also possible to receive nearly completely $x$ -polarized light emission of IXs near the SPP resonance frequency.",
"If thickness of a homogeneous metal film is larger than the skin depth of metal, which is about 20 nm for silver at $\\hbar \\omega =1.55$ eV, interaction of surface plasmons propagating along opposite sides of the film is weak.",
"Therefore, one can consider them independently and use surface-plasmon dispersion curve (blue dashed line in Fig.",
"1f) for analytical estimations.",
"One can see from Fig.",
"1f that the wavelength of surface plasmon for photon energy $\\hbar \\omega =1.55$ eV is $\\lambda _\\mathrm {sp}=184$ nm.",
"If one makes narrow slits in a thick silver film, light can transmit through them.",
"The narrower are slits the smaller is transmission.",
"However, if the slits are ordered in a periodic array with period $p$ , both light propagating along $z$ -axis in Al$_{0.33}$ Ga$_{0.67}$ As and surface plasmon propagating along $x$ -axis along Ag–Al$_{0.33}$ Ga$_{0.67}$ As boundary undergo Bragg diffraction and interact with electromagnetic waves, in-plane wavenumber of which differ by $2\\pi m/p$ , where $m$ is integer.",
"Taking into account that light propagating along $z$ -axis has zero in-plane wavenumber, one conclude that this light can excite surface plasmon with wavelength $\\lambda _\\mathrm {sp}$ if period of the grating is $p=m\\lambda _\\mathrm {sp}$ , where $m$ is a positive integer.",
"The surface plasmon in turn enhances electric field in the hole region, thus yielding to EOT through the metal film with slits.",
"Calculated $x$ - (blue solid line) and $y$ - (green dashed line) linear polarized components of the IX device emission spectra and respective DLP $\\rho $ (red dotted line) for the structure with $(t,w,p)=(45,190,230)$ nm is shown in Fig. 3a.",
"One can see a narrow resonance with extraordinary emission of $x$ -polarized light here.",
"Note, since the wavelengths of surface plasmon and light in Al$_{0.33}$ Ga$_{0.67}$ As are about the same for the photon energy of 1.55 eV (compare blue dashed and red dotted lines in Fig.",
"1f), $x$ -linear polarized component of the IX's emission decreases rapidly from the right side of the resonance (blue line in Fig.",
"3a) due to opening new diffraction channels in AlGaAs (a Wood-Rayleigh anomaly).",
"The calculated electric field distribution of light propagating along $z$ -axis with photon energy equal to the resonance energy (1557 meV) is shown in Fig. 3b.",
"One can clearly see the excitation of SPP propagating along Ag/Al$_{0.33}$ Ga$_{0.67}$ As boundary and enhancement of electric field in the hole region as it was discussed above.",
"Note also, surface plasmon form a standing wave or, in other words, propagates in both direction due to mirror symmetry of the considered structure and normal light propagation.",
"Calculated dependence of $x$ -linear polarized component of the IX's emission intensity $I_x$ along $z$ -axis on the nanowires width $w$ and period $p$ for the photon energy $\\hbar \\omega =1.55$ eV and stripe thickness $t=45$ nm is shown in Fig. 3c.",
"A narrow resonance with extraordinary emission of $x$ -polarized light occurs around the nanowires width $w=190$ nm and period $p=230$ nm.",
"These parameters correspond to the case of small slit between neighbouring nanowires $g = 40$ nm.",
"Respective DLP $\\rho $ is shown in Fig. 3d.",
"One can see that DLP is varied in the range from about -100% to about +85%.",
"We also compared the device performance for various electrode materials.",
"The estimates in the Supplementary Materials show linear polarized components of the IX device emission spectra and respective DLP for voltage tunable excitonic metasurfaces with gold gratings (Figs.",
"S1-S3) operating in the same modes (see Figs.",
"2c, 2d, 3a) as the light emission device with silver gratings.",
"The DLP for these structures is varied in smaller range than for the silver ones because of larger absorptive losses of gold in the investigated frequency range.",
"Smaller DLP variations are estimated for metal nitride plasmonic materials [28] with larger losses such as titanium nitride, see in Fig.",
"S4.",
"To conclude, we propose voltage tunable excitonic metasurfaces — integrated optoelectronic devices with voltage control of the polarization state of light, based on tunable emission of indirect excitons in coupled well semiconductor heterostructure and plasmonic metasurface made of metal grating.",
"Several modifications of such excitonic metasurfaces are considered, which allow to tune the degree of linear polarization of emission in a broad range from linearly polarized along grating to perpendicular to grating, depending on the parameters of the gratings.",
"This work was supported by Russian Science Foundation (Grant No.",
"16-12-10538), NSF Grant No.",
"1640173 and NERC, a subsidiary of SRC, through the SRC-NRI Center for Excitonic Devices.",
"The authors are thankful to T. Weiss for fruitful discussions.",
"Supplementary Materials: Control of light polarization by voltage tunable excitonic metasurfaces The Supplementary Materials presents calculated characteristics (linear polarized components of the emission spectra and respective DLP) of the proposed light emitting device with gold and titanium nitride gratings.",
"Figures S1 and S2 correspond to parameters of gold gratings where localized SPPs with $m=1$ and 3, respectively, are excited.",
"Figures S3 and S4 correspond to parameters of EOT through arrays of subwavelength slits in gold and titanium nitride layers, respectively.",
"Figure: As Fig.",
"2c, but for gold gratings and (t,w,p)=(10,23,50)(t,w,p)=(10,23,50) nm.Figure: As Fig.",
"2c, but for gold gratings and (t,w,p)=(10,120,160)(t,w,p)=(10,120,160) nm.Figure: As Fig.",
"2c, but for gold gratings and (t,w,p)=(45,160,230)(t,w,p)=(45,160,230) nm.Figure: As Fig.",
"2c, but for titanium nitride gratings and (t,w,p)=(100,150,230)(t,w,p)=(100,150,230) nm."
]
] | 1709.01771 |
[
[
"Dynamical instability of the electric transport in strongly fluctuating\n superconductors"
],
[
"Abstract Theory of the influence of the thermal fluctuations on the electric transport beyond linear response in superconductors is developed within the framework of the time dependent Ginzburg - Landau approach.",
"The I - V curve is calculated using the dynamical self - consistent gaussian approximation.",
"Under certain conditions it exhibits a reentrant behaviour acquiring an S - shape form.",
"The unstable region below a critical temperature $T^{\\ast }$ is determined for arbitrary dimensionality ($D=1,2,3$) of the thermal fluctuations.",
"The results are applied to analyse the transport data on nanowires and several classes of 2D superconductors: metallic thin films, layered and atomically thick novel materials."
],
[
"Introduction",
"In most electrical transport phenomena in condensed matter the current in a conductor is a monotonic function of the applied voltage.",
"Moreover at small current densities the I-V curve is nearly linear (see the dark green line in Fig.1 representing a normal metal), so that only the linear response theory[1] is generally needed to describe the transport via conductivity $\\sigma _{n}$ .",
"However in certain types of materials the linearity does not extend to higher current densities.",
"In superconductors close to the normal state (when temperature for example is just below critical, see the purple curve in Fig.1), at small currents the I-V curve slope $\\sigma $ is very large $\\sigma >>\\sigma _{n}$ , however at higher currents it diminishes and then smoothly approaches the normal line.",
"It turns out that under certain conditions (for example at yet lower temperatures, the solid cyan curve in Fig.1), the initial slope is even steeper and moreover at certain current density the differential resistivity becomes negative signalling a dynamical instability.",
"This possibility was envisioned theoretically by Gorkov[2] and Masker, Marcelja and Parks[3], before strongly fluctuating superconductors like the high $T_{c}$ cuprates were discovered.",
"The arguments required strong fluctuations that enhance conductivity of a metal, beyond the parameter range in which the coherent condensate is not formed.",
"The theory in the one - dimensional geometry was discussed in a comprehensive paper by Tucker and Halperin[4].",
"Different versions of the dynamical Hartree - Fock approximation were critically compared.",
"The focus on wires (one dimensions, 1D) was justified, since low $T_{c}$ superconductors have very small Ginzburg number $Gi$ and the fluctuations are detectable only when the dimensionality is reduced (or strong magnetic fields applied).",
"The Tucker and Halperin conclusion was that the approximation is probably inapplicable for currents for which differential resistivity is negative, but qualitatively the phenomenon should be observable in 1D.",
"Later several experiments indeed appeared both in 1D (thin metallic nanowires)[5] and in 2D both in thin metallic films[6] and layered high $T_{c}$ materials[7], [8], that have a much larger Ginzburg number, so that thermal fluctuations in them are much easier to observe.",
"Moreover recently purely 2D superconductors (with thickness of just one or very few unit cells) appeared[9], [10], [11] and similar phenomenon was observed.",
"It is important to note that, due to experimental reasons, only in the first two experiments[5], [6] the voltage drive was used, so that the full I-V curve including the “unstable\" parts was observed.",
"In rest of experiments the current drive was employed, so one observed that at certain current the voltage “jumped\" over the unstable state.",
"Many more experiments observing instability (with jumps due to the current drive) were performed in superconducting films[7], [12], [13] and wires under strong magnetic field.",
"In the presence of magnetic field in type II superconductors the dynamical problem becomes more complex due to effects of the vortex pinning and theoretical explanations invoke thermal transport (hot spots[14]).",
"The experiment on 1D nanowires was qualitatively explained[15], using dynamics of the condensate, rather then utilizing Tucker-Halperin theory.",
"As was noted early on[2], [6], [14], the dynamical instability, is firmly established, can leads to dynamical phase separation patterns and other phenomena and applications.",
"In this paper we revisit and expand the self - consistent theory of the nonlinear response in superconductors and demonstrate that the old and the new experiments on the dynamical instability can in fact be explained by it, not just qualitatively, but quantitatively.",
"The conditions for the instability are derived in D=1,2 and even D=3 (in which case these are almost impossible to observe even for the most “fluctuating\" materials).",
"It seems that a covariant version of the dynamical gaussian approximation[16] in the framework of the Ginzburg - Landau phenomenological approach[17] is precise and universal enough to quantitatively describe the phenomenon including the unstable regions.",
"A qualitative argument ensuring the emergence of the dynamical instability for superconductors that posses large enough thermal fluctuations is as follows.",
"The superconducting fluctuations contribution[18] to the voltage has the following form, see dashed lines in Fig.1.",
"It rises very fast at small currents and gradually decreases to zero when the virtual Cooper pairs are broken by the electric field.",
"The negative slope at some point becomes equal or larger that the normal electrons conductivity that is roughly independent of the transport current (dark green line in Fig.1).",
"The appearance of the S-shaped I-V curves at certain value of temperature (the blue curve) is the crossover temperature that will be determined in the paper.",
"Figure: Schematic I-V curves at different temperatures.",
"The dashed linesare superconducting fluctuation contributions at different temperatures andthe solid lines are total currents contributed both from the normal part(straight line)and superconducting part.",
"Below transition temperature T * T^{\\ast }, non-monotonic S shaped I-V curves appear.The rest of the paper is organized as follows.",
"In Section II the time dependent Ginzburg - Landau model incorporating the effects of thermal fluctuations is specified, while in Section III the I-V curves are derived in $D=1,2,3$ within the gaussian approximation.",
"The instability is analyzed in Section IV by considering a quantum wire experiment and several 2D materials ranging from thin films to layered superconductors and few atomic thick new materials.",
"Section V contains conclusions."
],
[
"Thermal fluctuations and electric field in the time-dependent GL\nmodel",
"Unlike in many other second order transitions in condensed matter, some superconductor - normal transitions exhibit a wide thermal fluctuation region.",
"Since the discovery of high $T_{c}$ superconductors, the superconducting fluctuations have been demonstrated to be the prime cause of many interesting phenomena.",
"For example, fluctuations broaden the critical region of resistivity in the vicinity of the transition temperature[19], lead to large diamagnetism[20] and Nernst effect [21] far above $T_{c}$ etc.",
"The influence is especially enhanced under strong magnetic fields."
],
[
"The model",
"While it is impossible at the present level of our understanding of superconductivity in these materials to describe the effect of thermal fluctuations on transport within a microscopic model, the Ginzburg-Landau (GL) phenomenological description in terms of the order parameter field $\\Psi $ is a method of choice[18], [17] for that purpose.",
"To describe the thermal fluctuations of the order parameter in $D$ $\\,$ - dimensional superconductors a starting point is the GL free energy as a functional of the order parameter field $\\Psi $ : $F_{GL}=A\\int d^{D}\\mathbf {r}\\frac{{\\hbar }^{2}}{2m^{\\ast }}|\\mathbf {\\nabla }\\Psi |^{2}+\\alpha (T-T_{\\Lambda })|\\Psi |^{2}+\\frac{b}{2}|\\Psi |^{4}\\text{.",
"}$ For low dimensional superconductors the cross - section “area\" is indeed area, $A=L_{y}L_{z}$ , for $D=1$ , while in $D=2$ it is the sample effective thickness $A=L_{z}$ .",
"In the GL potential term, $T_{\\Lambda }$ is the mean-field critical temperature, that can be significantly larger than the measured critical temperature $T_{c}$ due to strong thermal fluctuations on the mesoscopic scale[22] and $m^{\\ast }$ is effective Cooper pair mass.",
"For strong fluctuation superconductors away from both the critical range and the gaussian fluctuations regime at very low temperatures, one have to take the quartic term in GL free energy into account.",
"The other two parameters $\\alpha $ and $b$ determine the two characteristic length scales, the coherence length $\\xi ^{2}=\\hbar ^{2}/\\left( 2m^{\\ast }\\alpha T_{\\Lambda }\\right) $ and the penetration depth $\\lambda ^{2}=bc^{2}m^{\\ast }/\\left(16\\pi e^{2}\\alpha T_{\\Lambda }\\right) $ .",
"The relaxational dynamics and thermal fluctuations of the superconducting order parameter in the presence of electric field $E$ are conveniently described by the gauge-invariant time-dependent GL (TDGL) equation[23] with the Langevin white noise: $\\Gamma _{0}^{-1}\\left( \\frac{\\partial }{\\partial \\tau }-i\\frac{e^{\\ast }\\varphi }{\\hbar }\\right) \\Psi =-\\frac{1}{A}\\frac{\\delta F_{GL}}{\\delta \\Psi ^{\\ast }}+\\zeta \\left( \\mathbf {r},\\tau \\right) \\text{.}",
"$ Here the order parameter relaxation time is given by $\\Gamma _{0}^{-1}={\\hbar }^{2}\\gamma /\\left( 2m^{\\ast }\\right) $ , where the inverse diffusion constant $\\gamma /2$ , controlling the time scale of dynamical processes via dissipation, is assumed to be real[24].",
"$e^{\\ast }=2\\left|e\\right|$ .",
"The scalar potential for constant homogeneous electric field (assume to be applied along the $x$ axis) is $\\varphi =-Ex$ .",
"The white-noise forces, which induce the thermodynamical fluctuations, satisfy the fluctuation-dissipation theorem $\\left\\langle \\zeta ^{\\ast }(\\mathbf {r},\\tau )\\zeta (\\mathbf {r}^{\\prime },\\tau ^{\\prime })\\right\\rangle =\\frac{2T}{A\\Gamma _{0}}\\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })\\delta (\\tau -\\tau ^{\\prime })\\text{.}",
"$ The electric current density includes two components, $\\mathbf {J=J}_{n}+\\mathbf {J}_{s},$ where $\\mathbf {J}_{n}=\\sigma _{n}\\mathbf {E}$ is the current density contributed by the Ohmic normal part, and $\\mathbf {J}_{s}$ is fluctuation supercurrent density given by $\\mathbf {J}_{s}=\\frac{ie^{\\ast }{\\hbar }}{2m^{\\ast }}\\left( \\Psi ^{\\ast }\\mathbf {D}\\Psi -\\Psi \\mathbf {D}\\Psi ^{\\ast }\\right) \\text{.",
"}$"
],
[
"Characteristic scales and dimensionless variables",
"In order to facilitate the following discussion and fitting of experimental I-V curves, let us use characteristic units of length, the coherence length $\\xi $ , time, the Ginzburg-Landau “relaxation” time[17] $\\tau _{GL}=\\gamma \\xi ^{2}/2$ .",
"The order parameter is normalized by $\\Psi ^{2}=\\left( 2\\alpha T_{\\Lambda }/b\\right) \\psi ^{2}$ and electric field by $E=E_{GL}\\mathcal {E}$ , where $E_{GL}=2\\hbar /\\gamma e^{\\ast }\\xi ^{3}\\text{.}",
"$ The fluctuation strength is conveniently characterised by the parameter $\\omega $ , $\\omega =\\sqrt{2Gi}\\pi \\text{,} $ related to the $D$ - dimensional Ginzburg number (consistent with the original definitions in $D=2$ ) by $Gi_{D}=2\\left( \\frac{T_{\\Lambda }e^{\\ast 2}\\lambda ^{2}}{Ac^{2}\\hbar ^{2}\\xi ^{D-2}}\\right) ^{2}\\text{.}",
"$ The TDGL Eq.",
"(REF ), written in dimensionless units reads, $\\left( D_{\\tau }-\\frac{1}{2}\\mathbf {\\nabla }^{2}\\right) \\psi +\\frac{t-1}{2}\\psi +\\left|\\psi \\right|^{2}\\psi =\\bar{\\zeta }\\text{,}$ where $t\\equiv T/T_{\\Lambda }$ , $D_{\\tau }=\\frac{\\partial }{\\partial \\tau }+i\\mathcal {E}y$ and $\\zeta =\\bar{\\zeta }\\left( 2\\alpha T_{\\Lambda }\\right)^{3/2}/b^{1/2}$ , the white noise correlation takes a dimensionless form: $\\left\\langle \\bar{\\zeta }^{\\ast }(\\mathbf {r},\\tau )\\bar{\\zeta }(\\mathbf {r}^{\\prime },\\tau ^{\\prime })\\right\\rangle =2\\omega t\\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })\\delta (\\tau -\\tau ^{\\prime })\\text{.",
"}$ Finally, the dimensionless current density $\\mathbf {j}_{s}=\\mathbf {J}_{s}/J_{GL}$ , with $J_{GL}=cH_{c2}\\xi /2\\pi \\lambda ^{2}$ as the unit of the current density, is $\\mathbf {j}_{s}=\\frac{i}{2}\\left( \\psi ^{\\ast }\\mathbf {D}\\psi -\\psi \\mathbf {D}\\psi ^{\\ast }\\right) \\text{.}",
"$ The problem is clearly nonperturbative, so that one should rely on methods of a variational nature that are outlined next.",
"The relevant unit of conductivity is therefore $\\sigma _{GL}\\equiv J_{GL}/E_{GL}=c^{2}\\gamma \\xi ^{2}/4\\pi \\lambda ^{2}$ ."
],
[
"The self - consistent approximation calculation of the I-V curve",
"A sufficiently simple nonperturbative method is the Hartree - Fock type self-consistent Gaussian approximation (SCGA)[23], [25], [22].",
"It has already been applied to other fluctuations phenomena like magnetization[26], Nernst effect[22] and conductivity above $T_{c}$[27]."
],
[
"Dynamical gaussian approximation",
"The TDGL in the presence of the Langevin white noise, Eq.",
"(REF ), is nonlinear, so cannot generally be solved.",
"Since we will need only the thermal averages of quadratic in $\\psi $ quantities, like the superfluid density and the electric current, a sufficiently simple and accurate approximation (similar in nature to the Hartree-Fock approximation in the fermionic models) is the gaussian approximation [25], [22], [26].",
"The nonlinear $\\left|\\psi \\right|^{2}\\psi $ term in the TDGL Eq.",
"(REF ) is approximated by a linear one $2\\left\\langle \\left|\\psi \\right|^{2}\\right\\rangle \\psi $ (there are two possible contractions between $\\psi ^{\\ast }$ , $\\psi $ in $\\left|\\psi \\right|^{2}\\psi $ , see discussion of this point in [16]): $\\left( D_{\\tau }-\\frac{1}{2}\\mathbf {\\nabla }^{2}+\\frac{t-1}{2}+2\\left\\langle \\left|\\psi \\right|^{2}\\right\\rangle \\right) \\psi (\\mathbf {r},\\tau )=\\bar{\\zeta }(\\mathbf {r},\\tau )\\text{.}",
"$ For stationary homogeneous processes considered here, the superfluid density $\\left\\langle \\left|\\psi \\right|^{2}\\right\\rangle $ is just a constant.",
"Now it takes a form, $\\left[ D_{\\tau }-\\frac{1}{2}\\mathbf {\\nabla }^{2}+\\varepsilon \\right] \\psi (\\mathbf {r},\\tau )=\\bar{\\zeta }(\\mathbf {r},\\tau )\\text{,} $ where the excitations energy gap[17] is, $\\varepsilon =-\\frac{1-t}{2}+2\\left\\langle \\left|\\psi \\right|^{2}\\right\\rangle \\text{.}",
"$ The solution therefore can be written via the Green's function, $\\psi (\\mathbf {r}_{1},\\tau _{1})=\\int d\\mathbf {r}_{2}\\int d\\tau _{2}G\\left(\\mathbf {r}_{1},\\tau _{1};\\mathbf {r}_{2},\\tau _{2}\\right) \\bar{\\zeta }\\left(\\mathbf {r}_{2}\\mathbf {,}\\tau _{2}\\right) \\text{.}",
"$ Then the superfluid density, using the noise correlator, Eq.",
"(REF ), can be expressed via the Green's function as, $\\left\\langle \\left|\\psi \\left( \\mathbf {r}_{1},\\tau _{1}\\right) \\right|^{2}\\right\\rangle =2\\omega t\\int d\\mathbf {r}_{2}\\int d\\tau _{2}G^{\\ast }\\left( \\mathbf {r}_{1},\\tau _{1};\\mathbf {r}_{2},\\tau _{2}\\right)G\\left( \\mathbf {r}_{1},\\tau _{1};\\mathbf {r}_{2},\\tau _{2}\\right) \\text{,}$ and is a function of the parameter $\\varepsilon $ which is determined self consistently by Eq.",
"(REF )."
],
[
"Green's function for a homogeneous constant electric field",
"To calculate the response of the system, one needs the well known Green's function in the presence of electric field: $G\\left( \\mathbf {r}_{1},\\mathbf {r}_{2},\\tau \\right) =\\theta \\left( \\tau \\right) \\frac{1}{\\left( 2\\pi \\tau \\right) ^{D/2}}\\exp \\left[ -\\varepsilon \\tau -\\mathcal {E}^{2}\\frac{\\tau ^{3}}{24}-\\frac{i\\mathcal {E}}{2}\\tau \\left(x_{1}+x_{2}\\right) -\\frac{\\left( \\mathbf {r}_{1}\\mathbf {-r}_{2}\\right) ^{2}}{2\\tau }\\right] \\text{.}",
"$ The invariance with respect to the time translations is already taken into account by setting $\\tau =\\tau _{1}-\\tau _{2}$ .",
"Using these expressions, the superfluid density of Eq.",
"(REF ) takes a form, $\\left\\langle \\left|\\psi \\left( \\mathbf {r},\\tau \\right) \\right|^{2}\\right\\rangle =\\frac{\\omega t}{2^{D-1}\\pi ^{D/2}}\\int _{0}^{\\infty }\\frac{d\\tau }{\\tau ^{D/2}}\\exp \\left[ -2\\varepsilon \\tau -\\mathcal {E}^{2}\\frac{\\tau ^{3}}{12}\\right] \\text{.}",
"$ The integrand in Eq.",
"(REF ) is divergent as $1/\\tau $ when $\\tau \\rightarrow 0$ when $D>1$ .",
"The cutoff $\\tau _{cut}$ is thus required to account for the inherent UV divergence of the Ginzburg-Landau theory and it will be addressed below.",
"Finally the gap equation takes a form $\\varepsilon =-\\frac{1-t}{2}+\\frac{\\omega t}{2^{D-2}\\pi ^{D/2}}\\int _{\\tau _{cut}}^{\\infty }\\frac{d\\tau }{\\tau ^{D/2}}\\exp \\left[ -2\\varepsilon \\tau -\\mathcal {E}^{2}\\frac{\\tau ^{3}}{12}\\right] \\text{.}",
"$ After (numerical) solution for the energy gap $\\varepsilon $ , we turn to calculation of the supercurrent.",
"While the upper limit of the integration in Eq.",
"(REF ) is safe (both terms in exponent are positive), the lower limit (UV) depends on dimensionality.",
"In Ref.",
"Bu2, it was shown that $\\tau _{cut}$ in time dependent Ginzburg Landau and the energy cutoff $\\Lambda $ in static Ginzburg Landau theory are related by $\\tau _{cut}=\\frac{\\hbar ^{2}}{2m^{\\ast }\\xi ^{2}\\Lambda e^{\\gamma _{E}}}$ where $\\gamma _{E}$ is Euler constant and $\\Lambda $ is the energy cutoff[22], [26].",
"Our calculation show that taking value $\\tau _{cut}$ from $0.1$ to 10, the physical quantities is essentially unchanged, and is taken as $\\tau _{cut}=1$ in what follows."
],
[
"The electric current density",
"The dimensionless supercurrent density along the electric field direction $x$ , defined by Eq.",
"(REF ), expressed via the Green's functions is $\\left\\langle j_{x}^{s}\\right\\rangle =i\\omega t\\int d\\mathbf {r}_{2}d\\tau ^{\\prime }G^{\\ast }\\left( \\mathbf {r}_{1},\\mathbf {r}_{2},\\tau -\\tau ^{\\prime }\\right) \\frac{\\partial }{\\partial x}G\\left( \\mathbf {r}_{1},\\mathbf {r}_{2},\\tau -\\tau ^{\\prime }\\right) +c.c $ Performing the integrals, one obtains, $\\left\\langle j_{x}^{s}\\right\\rangle =\\frac{\\omega t\\mathcal {E}}{2^{D}\\pi ^{D/2}}\\int \\frac{d\\tau }{\\tau ^{D/2-1}}\\exp \\left[ -2\\varepsilon \\tau -\\mathcal {E}^{2}\\frac{\\tau ^{3}}{12}\\right] \\text{.}",
"$ Returning to the physical units, the total electric current density reads $J_{x}=E\\left\\lbrace \\sigma _{n}+\\frac{\\omega T\\sigma _{GL}}{2^{D}\\pi ^{D/2}T_{\\Lambda }}\\int \\frac{d\\tau }{\\tau ^{D/2-1}}\\exp \\left[-2\\varepsilon \\tau -\\left( \\frac{E}{E_{GL}}\\right) ^{2}\\frac{\\tau ^{3}}{12}\\right] \\right\\rbrace \\text{,} $ where $E_{GL}$ was defined in Eq.",
"(REF ) and the dimensionless fluctuation stress parameter $\\omega $ in Eq.",
"(REF ).",
"The gap equation determining the dimensionless energy gap $\\varepsilon $ in this units is $\\varepsilon =-\\frac{1-T/T_{\\Lambda }}{2}+\\frac{\\omega T}{2^{D-2}\\pi ^{D/2}T_{\\Lambda }}\\int \\frac{d\\tau }{\\tau ^{D/2}}\\exp \\left[ -2\\varepsilon \\tau -\\left( \\frac{E}{E_{GL}}\\right) ^{2}\\frac{\\tau ^{3}}{12}\\right] \\text{.",
"}$ In general there is a factor $k$ relating the two conductivities: $k=\\sigma _{n}/\\sigma _{GL}$ .",
"The (obtained numerically) value of the energy gap $\\varepsilon $ should be used.",
"Illustrative results are presented and compared with experiments in the next section and discussed in the following one."
],
[
"The dynamical instability point.",
"The dynamical instability transition temperature on the phase diagram, $T^{\\ast }$ , see Fig.1, defined as a maximal temperature at which the instability appears.",
"Mathematically is determined by vanishing of the first two derivatives, $\\frac{dJ_{x}}{dE}=0\\ $ and $\\frac{d^{2}J_{x}}{dE^{2}}=0$ .",
"Differentiating the current, Eq.",
"(REF ) (via chain rule of the gap equation), results in: $&&\\frac{\\sigma _{n}T_{\\Lambda }}{\\sigma _{GL}T^{\\ast }}+\\frac{\\omega }{2^{D}\\pi ^{D/2}}\\int \\frac{d\\tau }{\\tau ^{D/2-1}}\\exp \\left[ -2\\varepsilon \\tau -\\left( \\frac{E}{E_{GL}}\\right) ^{2}\\frac{\\tau ^{3}}{12}\\right] \\\\&=&E\\frac{\\omega }{2^{D}\\pi ^{D/2}}\\int \\frac{d\\tau }{\\tau ^{D/2-2}}\\left( 2\\frac{\\partial \\varepsilon }{\\partial E}+\\frac{\\tau ^{2}E}{6E_{GL}^{2}}\\right) \\exp \\left[ -2\\varepsilon \\tau -\\left( \\frac{E}{E_{GL}}\\right) ^{2}\\frac{\\tau ^{3}}{12}\\right] ; $ $\\int \\frac{d\\tau }{\\tau ^{D/2-2}}\\left\\lbrace \\begin{array}{c}-\\frac{E\\tau ^{2}}{2E_{GL}}+\\frac{E^{3}\\tau ^{5}}{36E_{GL}^{3}}+E_{GL}\\frac{d\\varepsilon }{dE}\\left( \\frac{2E^{2}\\tau ^{3}}{3E_{GL}^{2}}-4\\right) \\\\+4\\frac{E\\tau }{E_{GL}}\\left( E_{GL}\\frac{d\\varepsilon }{dE}\\right)^{2}-2EE_{GL}\\frac{d^{2}\\varepsilon }{dE^{2}}\\end{array}\\right\\rbrace \\exp \\left[ -2\\varepsilon \\tau -\\left( \\frac{E}{E_{GL}}\\right) ^{2}\\frac{\\tau ^{3}}{12}\\right] =0 $ Together the gap equation (REF ), the dynamical instability transition temperature $T^{\\ast }$ is determined numerically.",
"The results are first applied to a one dimensional superconductors - metallic wires, and then for several qualitatively different types of 2D superconductors (as explained above, it is very difficult to observe the instability phenomenon in purely 3D materials, although in layered high $T_{c} $ cuprates close to $T_{c}$ the fluctuations become nearly 3D and the phenomenon was observed in magnetic field[13])."
],
[
"I-V curves of 1D Sn nanowires",
"We start with 1D nanowires.",
"Granular superconducting $Pb$ and $Sn$ nanowires of quite regular cross - section and length have been produced by electro - deposition in nanoporous membranes[5].",
"It is important to note that the series of experiments of Ref.",
"1Dexperiment on $Pb$ and $Sn$ nanowires is the only one (known to us) in which both the current and the voltage drives were employed.",
"This allows a qualitative understanding of the important difference between the dynamical behaviour two.",
"We focus on the voltage drive I-V curves of $Sn$ .",
"Figure: The I-V curves of 1D1D Sn nanowires with different temperature.",
"Thepoints are the experimental data and the solid lines are the theoreticalresults.The I-V curves, measured using the voltage drive at three temperatures, are shown in Fig.",
"2 (points).",
"The voltage drive employed clearly demonstrates the non - monotonic character below the onset $T_{c}\\approx T_{\\Lambda }=3.8K $ slightly above the bulk temperature of $Sn$ ($3.72K$ ).",
"The current drive experiment on the same sample (see Fig.",
"3b in Ref.",
"1Dexperiment) demonstrates the voltage jumps over unstable domains of the dynamical phase diagrams.",
"The jumps are more pronounced in $Pb$ , see Fig.",
"3a of Ref.",
"1Dexperiment.",
"This is consistent with the existence of the dynamical instability and was observed in numerous experiments (see 2D examples below).",
"The experimental data are fitted by Eqs.",
"(REF ,REF ) for $D=1$ , see solid curves.",
"The normal-state conductivity is given, $\\sigma _{n}=3.6\\cdot 10^{4}$ $\\left( \\Omega \\ast m\\right) ^{-1}$ , nanowires are $50\\mu m$ long with $55nm$ in diameter.",
"Measured material parameters are: coherence length[28] $\\xi =210nm$ , penetration depth $\\lambda =420nm$ and the normal conductivity was obtained from the red doted line in Fig.2.",
"The value of fitting parameters are: the fluctuation strength parameter $\\omega =0.0043,$ corresponding to the Ginzburg number $Gi=9.4\\cdot 10^{-7}$ , consistent with one dimension Ginzburg number formula, $Gi_{D=1}=2\\left( T_{\\Lambda }e^{\\ast 2}\\lambda ^{2}\\xi /Ac^{2}\\hbar ^{2}\\right) ^{2}\\approx 2.9\\ast 10^{-7}$ and the conductivity ratio $k=\\sigma _{n}/\\sigma _{GL}=0.08$ .",
"This experiment was already discussed in the framework of TDGL equations neglecting thermal fluctuations in Ref.",
"Vodolazov assuming the current drive.",
"The dynamical equations were solved numerically and the focus was on the jumps.",
"It seems to us that the origin of instability cannot ignore the thermal fluctuations, as explained above.",
"Many more experiments were performed in 2D."
],
[
"Instability in 2D",
"Several 2D superconductors exhibit the dynamical instability.",
"We start with metallic thin films, then proceed to the customary layered materials in which the coupling between layers is sufficiently small to ensure that thermal fluctuations dimensionality is 2.",
"Novel purely 2D materials are then mentioned.",
"Of course in a 2D superconductor one should measure close to $T_{c}$ to be able to detect the thermal fluctuations effects like the dynamical instability.",
"The only possible exceptions are high $T_{c}$ cuprates and novel 2D atomically thick superconductors."
],
[
"Thin metallic films near $T_{c}$",
"In experiments on $In$ thin films[6] the voltage drive was applied in a narrow temperature range very close to $T_{c}$ .",
"For the low critical temperature superconductor, $T_{\\Lambda }$ is very near the bulk critical temperature.",
"The temperature range (only superconducting states for $0=1-T/T_{\\Lambda }$ $<2.1\\%$ are replotted in Fig.3 as dots) nevertheless is wide enough to exhibit the dynamical instability, for $T/T_{\\Lambda }=0.9821$ and $T/T_{\\Lambda }=0.9793$ .",
"The coherence length is approximately[29] $\\xi =300nm$ , while the thickness $d$ of $In$ thin films is ranged[6] from $10nm$ to $300nm$ (less than $\\xi $ ), therefore in this temperature range coherence length $\\xi _{z}>d$ and the thermal fluctuations are 2D.",
"The normal-state conductivity is $\\sigma _{n}=9\\ast 10^{4}\\left( \\Omega \\ast cm\\right) ^{-1}$ .",
"The cross section area (perpendicular to the current direction) is approximately equal to $3.62\\times 10^{-7}cm^{2}$ in the fitting, which is consistent with the data provided in Ref.",
"Ivanchenko.",
"Figure: The I-V curves of InIn thin films andtheoretical fittings at different temperatures.The calculated I-V curves according Eqs.",
"(REF ,REF ) for $D=2$ , for different temperature are shown in Fig.",
"3 as solid curves.",
"The experimental data are fitted best for the following values of parameters: $k=\\sigma _{n}/\\sigma _{GL}=0.075$ and the fluctuation strength parameter $\\omega =\\sqrt{2Gi}\\pi =0.001$ corresponding to the Ginzburg number $Gi=5.1\\cdot 10^{-8}$ , consistent with $Gi_{D=2}=2\\left( T_{\\Lambda }e^{\\ast 2}\\lambda ^{2}/L_{z}c^{2}\\hbar ^{2}\\right) ^{2}\\approx 5\\cdot 10^{-8}$ with[29] $\\lambda =296nm$ and[6] $L_{z}=24.1nm$ .",
"The fit is generally good except very low currents.",
"The reason is obvious: critical current due to disorder on the mesoscopic scale is not present in the model.",
"Sometimes the state close to “criticality\" of the Berezinskii - Kosterlitz - Thouless variety is theoretically considered as a collection of the bound vortex - antivortex pairs[30].",
"The critical current clearly seen in Fig.3 as associated with the pairs “pinning\".",
"In fact in this 2D system strictly speaking critical current is zero (also seen in data), but it vanishes exponentially fast as $I\\rightarrow 0.$ Much more common superconductors with 2D fluctuations are layered materials (will be discussed below)."
],
[
"Layered materials",
"Instability in the form of the voltage jumps was observed recently in $FeSeTe $ thin film on $Pb(MgNb)TiO$ substrate[9].",
"Only the current drive was used, so that the S-shaped I-V curved cannot be determined.",
"Only the voltage jumps were observed close to $T_{c}$ .",
"The thickness of $FeSeTe$ thin films is $200nm$ .",
"The layer distance $L_{z}=0.55nm$ [31].",
"Here, the normal-state conductivity is taken to be $\\sigma _{n}=1.3\\cdot 10^{4}\\left( \\Omega \\ast cm\\right) ^{-1}$ .",
"Figure: The I-V curves of FeSeTeFeSeTe thin film at different temperatures.",
"Thepoints are the experimental data and the solid lines are the theoreticalfitting results.The calculated I-V curves of the 2D $FeSeTe$ thin film with different temperature are shown in Fig.",
"4 as solid curves.",
"The experimental data of $FeSeTe$ in a current driving setup from Ref.",
"Lin15 are fitted best for the following values of parameters: $T_{\\Lambda }=8K$ , $k=\\sigma _{n}/\\sigma _{GL}=0.07$ and the fluctuation strength parameter $\\omega =\\sqrt{2Gi}\\pi =0.018$ corresponding to the Ginzburg number $Gi=1.6\\cdot 10^{-5}$ .",
"According to $Gi_{D=2}=2\\left( T_{\\Lambda }e^{\\ast 2}\\lambda ^{2}/L_{z}c^{2}\\hbar ^{2}\\right) ^{2}$ , we deduce the sample's effective penetration depth $\\lambda =123.8nm$ (we are not aware of an experimental determination of the penetration depth from a magnetic measurement).",
"The I-V curves clearly exhibit a re-entrant behavior for $T<T^{\\ast }\\approx 6K$ .",
"This is hard to observe directly in the current driving experimental setup.",
"Experiments show that the current driving lead to the “jump\" I-V Curve and the voltage driving lead to the re-entrant S-Shaped I-V Curve in the superconducting nanowires at low temperature [15]."
],
[
"Other layered materials",
"The instability in the ultra-thin granular $YBa_{2}Cu_{3}O_{7-\\delta }$ nanobridges was clearly observed in a series of works in Ref.",
"Yeshurun.",
"Unfortunately a 2D or a 3D model cannot quantitatively describe these I-V curves since the fluctuations in this layered material and the temperature range can be described by a more complicated Lawrence - Doniach model.",
"The generalization is possible but was not attempted in the present work.",
"Also, the “jump\" I-V curves in a current driving setup was also reported in BSCCO[7] that is clearly 2D.",
"Unfortunately I-V curve at zero magnetic field (actually field perpendicular to the layers) at one value of temperature $\\left( 76K\\text{ for }T_{c}=85.2K\\right) $ was measured.",
"As noted above, the instability has been observed in numerous layered superconductors under strong magnetic field, but a quantitative interpretation requires additional parameters describing the magnetic vortex pinning."
],
[
"Conclusions",
"In this paper, I - V curve of a $D$ dimensional superconductor including the thermal fluctuations effects is calculated in arbitrary dimension using the dynamical self consistent gaussian approximation method.",
"An unstable region is found when currents flow through superconductor with temperature below a critical value $T^{\\ast }$ at which the I-V curve become S-shaped.",
"It is shown how the thermal fluctuations generate the instability.",
"The results are applied to analyse the transport data on various materials that possess sufficiently strong fluctuations in 1D or 2D.",
"While it is found that the unstable region can exist also in 3D, the S-Shaped I-V curve in realistic materials show only in 1D superconductors.",
"Let us stress that the majority of recent experiments on the resistive state are performed in the constant current (current driving) regime and at temperatures close to $T_{c}$ .",
"It would be very interesting to observe the whole S-shaped I-V curve using the voltage drive in novel atomically thick 2D materials as in extensively studied layered ones like $BSCCO$ .",
"Acknowledgments.",
"We are grateful to Professor Jian Wang, Professor Guang-Ming Zhang, and Dr. Ying Xing for valuable discussions.",
"B.R.",
"was supported by NSC of R.O.C.",
"Grants No.",
"103-2112-M-009-014-MY3 and is grateful to School of Physics of Peking University and Bar Ilan Center for Superconductivity for hospitality.",
"The work of D.L.",
"is supported by National Natural Science Foundation of China (No.",
"11674007) and is grateful to NCTS of Taiwan for hospitality."
]
] | 1709.01750 |
[
[
"On the domain of Dirac and Laplace type operators on stratified spaces"
],
[
"Abstract We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric.",
"Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces.",
"This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces.",
"Our argument does not rely on microlocal techniques and is very explicit.",
"The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales.",
"Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition."
],
[
"Introduction and statement of the main results",
"Singular spaces arise naturally in various parts of mathematics.",
"Important examples of singular spaces include algebraic varieties and various moduli spaces; singular spaces also appear naturally as compactifications of smooth spaces or as limits of families of smooth spaces under controlled degenerations.",
"The development of analytic techniques to study partial differential equations in the singular setting is a central issue in modern geometry.",
"Figure: Simple Edge as a Cone bundle over BB.Cheeger [14] was the first to initiate an influential program on spectral analysis on smoothly stratified spaces with singular Riemannian metrics.",
"Analysis of the associated geometric operators on spaces with conical singularities was the focal point of the research by Brüning and Seeley [8], [9], [11], Lesch [25], Melrose [30], Schulze [36], [37], Schrohe and Schulze [39], [40], Gil, Krainer and Mendoza [16], [17] to name just a few.",
"Extensions to spaces with simple edge singularities were developed by Mazzeo [29], as well as Schulze [35], [38] and his collaborators, see also Gil, Krainer and Mendoza [18].",
"Various questions in spectral geometry and index theory on spaces with simple edge singularities have been addressed by Brüning and Seeley [11], Mazzeo and Vertman [27], [28], Krainer and Mendoza [23], [22], Albin and Gell-Redman [3], Piazza and Vertman [34].",
"There have also been recent advances to lift the analysis to a very general setting of stratified spaces with iterated cone-edge singularities.",
"Index theoretic questions for geometric Dirac operators on a general class of compact stratified Witt spaces with iterated cone-edge metrics have been studied by Albin, Leichtnam, Piazza and Mazzeo in [4], [5], [6].",
"The Yamabe problem on stratified spaces has been solved by Akutagawa, Carron and Mazzeo in [1].",
"If we wish to go a step further and do spectral geometry on stratified spaces, the crucial difficulty appears already in the setting of a stratified space of depth two, illustrated as in Figure REF below with fibers $F_y$ , at each $y \\in B=Y_2$ , being simple edge spaces.",
"Consider the family of Gauss–Bonnet operators on the fibers $F_y, y \\in B$ .",
"Even if we impose a spectral Witt condition so that the Gauss–Bonnet operators on the fibers are essentially self-adjoint, their domains may still vary with the base point across $B$ .",
"In case of variable domains however, smoothness of the operator family becomes a much more complicated issue, which needs to be resolved before any meaningful spectral geometric questions may be addressed.",
"Our main result is formulated using the concept of a spectral Witt condition and the weighted edge Sobolev space $^{1,1}_e(M)$ on a stratified Witt space $M$ with an iterated cone-edge metric, which will be made explicit below.",
"Elements of the edge Sobolev spaces take values in a Hermitian vector bundle $E$ , which is suppressed from the notation.",
"For the moment, the spectral Witt condition is a spectral gap condition on certain operators on fibers $F$ , see Witt-condition and Definition , and in case of the Gauss–Bonnet operator on a stratified Witt space it can always be achieved by scaling the iterated cone-edge metric appropriately.",
"The weighted edge Sobolev space $^{s,\\delta }_e(M)= \\rho ^\\delta ^s_e(M)$ is the Sobolev space $^s_e(M)$ of all square integrable sections of the Hermitian vector bundle $E$ that remain square integrable under weak application of $s\\in $ edge vector fields, weighted with a $\\delta $ -th power of a smooth function $\\rho $ that vanishes at the singular strata to first order.",
"Our main theorem is now as follows.",
"Let $M$ be a compact stratified space with an iterated cone-edge metric.",
"Let $D$ denote either the Gauss–Bonnet or the spin Dirac operator, and assume the spectral Witt condition holds,Definition .",
"Then both $D$ and $D^2$ are essentially self-adjoint with domains $\\begin{split}&_{\\max }(D) = _{\\min }(D) = ^{1,1}_e(M), \\\\&_{\\max }(D^2) = _{\\min }(D^2) = ^{2,2}_e(M).\\end{split}$ In case of the Gauss–Bonnet operator, sections take values in the exterior algebra $^* ({}^{ie}T^*M)$ of the incomplete edge cotangent space $^* ({}^{ie}T^*M)$ .",
"In case of the spin Dirac operator, sections take values in the spinor bundle $S$ .",
"Let us comment on related work in connection to Theorem .",
"Gil, Krainer and Mendoza [18] prove that for an elliptic differential wedge operator $A$ of order $m$ on a simple edge space, under an assumption on indicial roots, $_{\\min }(A)= ^{m,m}_e(M)$ .",
"Our theorem here extends this statement to compact stratified spaces in the special case of the Gauss–Bonnet and Spin Dirac operators.",
"Moreover, Albin, Leichtnam, Piazza and Mazzeo in [4] prove that under the spectral Witt condition the minimal domain $_{\\min }(D)$ of the Gauss–Bonnet operator is included in the intersection of $^{1,\\delta }_e(M)$ for all $\\delta <1$ .",
"Our theorem here sharpens this statement into an equality instead of an inclusion.",
"In addition we emphasize that we employ different methods which are more elementary and do have a functional analytic flavor.",
"Furthermore we also do not need singular pseudo-differential calculi."
],
[
"Smoothly stratified iterated edge spaces",
"In this section we recall basic aspects of the definition of a compact smoothly stratified space of depth $k\\in _0$ , referring the reader for a complete discussion to a very thorough analysis in [4], [5], [2]."
],
[
"Smoothly stratified iterated edge spaces of depth zero and one",
"A compact stratified space of depth $k=0$ is simply a compact Riemannian manifold.",
"A compact stratified space of depth $k=1$ is a compact simple edge space $\\overline{M}$ with smooth open interior $M$ , as discussed in in [29], [28].",
"More precisely, $\\overline{M}$ admits a single stratum $B\\subset \\overline{M}$ which is a smooth compact manifold.",
"The edge $B$ comes with an open tubular neighborhood $\\subset \\overline{M}$ , a radial function $x$ defined on $$ , and a smooth fibration $\\phi :\\rightarrow B$ with preimages $\\phi ^{-1}(q)\\setminus \\lbrace q\\rbrace $ , $q\\in B$ , being all diffeomorphic to open cones $C(F)=(0,1)\\times F$ over a smooth compact manifold $F$ .",
"The restriction $x$ to each fiber $\\phi ^{-1}(q)$ is a radial function on that cone.",
"We also write $\\phi : \\rightarrow B$ for the fibration of the $\\lbrace x=1\\rbrace $ level set over $B$ .",
"The tubular neighborhood $\\subset \\overline{M}$ is illustrated in the Figure REF .",
"The resolution $\\widetilde{M}$ is defined by replacing the cones in the tubular neighborhood $$ by finite cylinders $[0,1) \\times F$ .",
"This defines a compact manifold with smooth boundary $\\partial \\widetilde{M}$ given by the total space of the fibration $\\phi $ .",
"The resolution $\\widetilde{}$ of the singular neighborhood $$ is defined analogously.",
"We equip the simple edge space with an edge metric $g$ , which is smooth on $\\overline{M}\\setminus $ and which over $\\backslash B$ takes the following form $g|_{} = dx^2 + \\phi ^{\\ast }g_B + x^2 g_F + h =: g_0 + h$ where $g_B$ is a Riemannian metric on $B$ , $g_F$ is a smooth family of bilinear forms on the tangent bundle of the total space of the fibration $\\phi :\\rightarrow B$ , restricting to a Riemannian metric on fibers $F$ , $h$ is smooth on $\\widetilde{}$ and $|h|_{g_0} = O(x)$ , when $x\\rightarrow 0$ .",
"We also require that $\\phi : (, g_F +\\phi ^{\\ast }g_B)\\rightarrow (B,g_B)$ is a Riemannian submersion.",
"Consider local coordinates $(x,y,\\theta )$ on $\\backslash B \\subset M$ near the edge, where $x$ is as before the radial coordinate, $y$ is the lift of a local coordinate system on $B$ and $\\theta $ restricts to local coordinates on each fiber $F$ .",
"Then, in terms of symmetric 2-tensors $\\textup {Sym}^2\\lbrace dx, x d\\theta , dy\\rbrace $ , generated by the 1-tensors $\\lbrace dx, x d\\theta , dy\\rbrace $ , the higher order term $h$ satisfies over $\\widetilde{}$ $h \\in x \\cdot C^\\infty (\\widetilde{}, \\textup {Sym}^2\\lbrace dx, x d\\theta , dy\\rbrace ).$ We finish with the standard definition of edge vector fields.",
"The edge vector fields $\\mathcal {V}_{e,1}$ are defined to be smooth on $\\widetilde{M}$ and tangent to the fibers $F$ at $\\partial \\widetilde{M}$ .",
"We also write $\\mathcal {V}_{ie,1} := x^{-1} \\mathcal {V}_{e,1}$ , which we call the incomplete edge vector fields.",
"In the chosen local coordinate system $(x,y,\\theta )$ we have explicitly $\\begin{split}&\\mathcal {V}_{e,1}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace x\\partial _x, x\\partial _{y_1}, ..., x\\partial _{y_{\\dim B}}, \\partial _{\\theta _1},...,\\partial _{\\theta _{\\dim F}}\\rbrace , \\\\&\\mathcal {V}_{ie,1}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\partial _x, \\partial _{y_1}, ..., \\partial _{y_{\\dim B}}, x^{-1}\\partial _{\\theta _1},...,x^{-1}\\partial _{\\theta _{\\dim F}}\\rbrace .\\end{split}$"
],
[
"Smoothly stratified iterated edge spaces of depth two",
"A stratified space of depth 2 is modelled as above but allowing the links $F$ to be stratified spaces of depth 1, with smooth links.",
"This is illustrated in Figure REF , and we proceed with studying this case in detail to provide a basis for a definition of smoothly stratified iterated edge spaces of arbitrary depth.",
"Figure: Tubular neighborhood ⊂M ¯\\subset \\overline{M},M ¯\\overline{M} of depth 2.The fibration of cones with singular links defines an open edge space itself with an open edge singularity in $Y_1$ , which fibers over $Y_2$ and contains $Y_2$ in its closure.",
"We now have two strata $\\lbrace Y_1,Y_2\\rbrace $ satisfying the following fundamental properties.",
"$Y_2 \\subset \\overline{Y}_1$ , and $Y_2$ is compact and smooth.",
"Any point $q\\in Y_1 = \\overline{Y}_1\\setminus Y_2$ has a tubular neighborhood of cones with smooth links.",
"We say that $Y_1$ is a stratum of depth 1.",
"Any point $q\\in Y_2$ has a tubular neighborhood of cones $[0,1)\\times F/_{(0, \\theta _1) \\sim (0, \\theta _2)}$ with links $F$ being stratified spaces of depth 1.",
"We say that $Y_2$ is a stratum of depth 2.",
"We have the following sequence of inclusions $\\overline{M} \\supset \\overline{Y}_1 \\supseteq Y_2 \\supseteq \\varnothing .$ Then $\\overline{M} \\setminus \\overline{Y}_1$ is an open Riemannian manifold dense in $\\overline{M}$ , and the strata of $\\overline{M}$ are $Y_2,\\;\\; Y_1 = \\overline{Y}_1\\setminus Y_2,\\;\\; \\overline{M}\\setminus \\overline{Y}_1.$ The resolution $\\widetilde{M}$ is defined as in the depth one case by replacing the cones in the fibration $\\phi : \\rightarrow Y_2$ by finite cylinders $[0,1)\\times F$ , and subsequently replacing the simple edge space $F$ with its resolution as well.",
"This defines a compact manifold with corners.",
"The resolution $\\widetilde{}$ of $$ is defined analogously.",
"We denote the radial function on each cone in the fibration $\\phi $ by $x$ , and write $x^{\\prime }$ for the radial function of the simple edge space $F$ .",
"We can now define an iterated cone-edge metric $g$ as before by specifying $g|_{} = dx^2 + \\phi ^{\\ast } g_B + x^2 g_F + h=:g_0+h,$ where $B=Y_2$ , $g_B$ is a smooth Riemannian metric, $g_F$ restricting on the links $F$ to iterated cone-edge metrics of depth 1 (simple edge space).",
"As before, these metrics $g_B$ and $g_F$ do not depend on the radial function $x$ , and the higher order terms of the metric are included in the tensor $h$ , which is smooth on $\\widetilde{}$ with $|h|_{g_0}= O(x)$ as $x\\rightarrow 0$ .",
"We require that $\\phi \\upharpoonright : (, g_F + \\phi ^{\\ast }g_B)\\rightarrow (B,g_B)$ is a Riemannian submersion and put the same condition on the fibers $(F,g_F)$ .",
"The edge vector fields $\\mathcal {V}_{e, 2}$ , as well as the incomplete edge vector fields $\\mathcal {V}_{ie, 2}$ , are defined similarly to $\\mathcal {V}_{e, 1}$ and $\\mathcal {V}_{ie, 1}$ .",
"$\\begin{split}&\\mathcal {V}_{e,2}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace (xx^{\\prime })\\partial _x, (xx^{\\prime })\\partial _{y_1}, ..., (xx^{\\prime })\\partial _{y_{\\dim B}}, \\mathcal {V}_{e, 1}(F)\\rbrace , \\\\&\\mathcal {V}_{ie,2}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\partial _x, \\partial _{y_1}, ..., \\partial _{y_{\\dim B}}, (xx^{\\prime })^{-1}\\mathcal {V}_{e, 1}(F)\\rbrace .\\end{split}$ where $\\mathcal {V}_{e, 1}(F)$ refers to the edge vector fields on the simple edge space $F$ ."
],
[
"Smoothly stratified iterated edge spaces of arbitrary depth",
"At an informal level we can now say that $\\overline{M}$ is a compact smoothly stratified iterated edge space of arbitrary depth $k\\ge 2$ with strata $\\lbrace Y_{\\alpha }\\rbrace _{\\alpha \\in A}$ if $\\overline{M}$ is compact and the following, inductively defined, properties are satisfied.",
"If $Y_\\alpha \\cap \\overline{Y}_\\beta \\ne \\varnothing $ then $Y_{\\alpha }\\subset \\overline{Y}_\\beta $ (each stratum is identified with its open interior).",
"The depth of a stratum $Y$ is the largest $(j-1) \\in _0$ such that there exists a chain of pairwise distinct strata $Y=Y_j,\\; Y_{j-1},\\ldots , Y_1$ with $Y_i \\subset \\overline{Y}_{i-1}$ for all $2 \\le i \\le j$ .",
"The stratum of maximal depth is smooth and compact.",
"The maximal depth of any stratum of $\\overline{M}$ is called the depth of $\\overline{M}$ .",
"Any point of $Y_\\alpha $ , a stratum of depth $j$ , has a tubular neighborhood of cones with links being stratified spaces of depth $j-1$ , for all $1\\le j \\le k$ .",
"Setting $\\overline{M} = X_n \\supset X_{n-1}=X_{n-2} \\supseteq X_{n-3}\\supseteq \\cdots \\supseteq X_1 \\supseteq X_0,$ where $X_j$ is the union of all strata of dimension less or equal than $j$ , $X_n\\setminus X_{n-2}$ is an open Riemannian manifold, dense in $\\overline{M}$ .",
"We call the union $X_{n-2}$ of all $Y_{\\alpha }$ , for $\\alpha \\in A$ the singular part of $\\overline{M}$ , and its complement in $\\overline{M}$ the regular part, denoted by $M$ .",
"The precise definition of smoothly stratified spaces contains some other technical conditions, Thom–Mather-spaces [2].",
"The resolution $\\widetilde{M}$ is a manifold with corners defined iteratively by resolving in each step the highest codimension singular strata as before.",
"Each tubular neighborhood $_\\alpha $ of any point in $Y_\\alpha $ admits a resolution $\\widetilde{}_\\alpha $ in an analogous way.",
"We define an iterated cone-edge metric $g$ on $M$ by asking $g$ to be an arbitrary smooth Riemannian metric away from singular strata, and requiring in each tubular neighborhood $_\\alpha $ of any point in $Y_\\alpha $ to have the following form $g|_{_\\alpha } = dx^2 + \\phi ^{\\ast }_\\alpha g_{Y_\\alpha } + x^2 g_{F_\\alpha } +h=:g_0 + h,$ where $\\phi _{\\alpha }:_\\alpha \\rightarrow \\phi _{\\alpha } (_\\alpha ) \\subseteq Y_\\alpha $ is the obvious fibration, $\\phi _{\\alpha } (_\\alpha )$ is open in $Y_\\alpha $ , the restriction $g_{Y_\\alpha } \\upharpoonright \\phi _{\\alpha } (_\\alpha )$ is a smooth Riemannian metric, $g_F$ is a symmetric two tensor on the level set $\\lbrace x=1\\rbrace $ , whose restriction to the links $F_\\alpha $ (smoothly stratified iterated edge spaces of depth at most $(k-1)$ ) is an iterated cone-edge metric.",
"The higher order term $h$ is smooth on $\\widetilde{}_\\alpha $ and satisfies $|h|_{g_0}=O(x)$ , when $x\\rightarrow 0$ .",
"We also assume that $\\phi _\\alpha \\upharpoonright _\\alpha : (_\\alpha , g_{F_\\alpha } + \\phi ^{\\ast }_\\alpha g_{Y_\\alpha })\\rightarrow (\\phi _{\\alpha } (_\\alpha ), g_{Y_\\alpha })$ is a Riemannian submersion and put the same condition in the lower depth.",
"Existence of such smooth iterated cone-edge metrics is discussed in [4].",
"The definition of edge vector fields $\\mathcal {V}_{e, k}$ and incomplete edge vector fields $\\mathcal {V}_{ie, k}$ , extends to the smoothly stratified space $M$ by an inductive procedure as in case of $k=2$ , cf.",
"(REF ).",
"To be precise, denote by $\\rho $ a smooth function on the resolution $\\widetilde{M}$ , nowhere vanishing in its open interior, and vanishing to first order at each boundary face.",
"Then $\\mathcal {V}_{e,k} = \\rho \\mathcal {V}_{ie,k}$ and $\\begin{split}&\\mathcal {V}_{e,k}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\rho \\partial _x, \\rho \\partial _{s_1}, ...,\\rho \\partial _{s_{\\dim Y_\\alpha }}, \\mathcal {V}_{e, k-1}(F_\\alpha )\\rbrace , \\\\&\\mathcal {V}_{ie,k}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\partial _x, \\partial _{s_1}, ..., \\partial _{s_{\\dim Y_\\alpha }}, \\rho ^{-1}\\mathcal {V}_{e,k-1}(F_\\alpha )\\rbrace .\\end{split}$"
],
[
"Sobolev spaces on smoothly stratified iterated edge spaces",
"We may now define the edge Sobolev spaces in the setup of a compact stratified space $M$ of depth $k$ with an iterated cone-edge metric.",
"Let ${}^{ie}TM$ denote the canonical vector bundle defined by the condition that the incomplete edge vector fields $\\mathcal {V}_{ie, k}$ form locally a spanning set of sections $\\mathcal {V}_{ie, k} = C^\\infty (M, {}^{ie}TM)$ .",
"We denote by ${}^{ie}T^*M$ the dual of ${}^{ie}TM$ , also referred to as the incomplete edge cotangent bundle.",
"We write $E=\\Lambda ^* ({}^{ie}T^*M)$ , when discussing the Gauss–Bonnet operator, and we set $E$ to be the spinor bundle, when discussing the spin Dirac operator.",
"In either of these cases we define the edge Sobolev spaces with values in $E$ as follows.",
"Let $M$ be a compact smoothly stratified iterated edge space of arbitrary depth $k\\in $ with an iterated cone-edge metric $g$ .",
"We denote by $L^2(M, E)$ the $L^2$ completion of smooth compactly supported differential forms $C^\\infty _0(M, E)$ .",
"Denote by $\\rho $ a smooth function on the resolution $\\widetilde{M}$ , nowhere vanishing in its open interior, and vanishing to first order at each boundary face.",
"Then, for any $s\\in $ and $\\delta \\in $ we define the weighted edge Sobolev spaces by $\\begin{split}&_e^s(M):= \\lbrace \\in L^2(M) \\mid V_1 \\circ \\cdots \\circ V_s \\in L^2(M,E), \\ \\textup {for} \\ V_j \\in \\mathcal {V}_{e, k}\\rbrace , \\\\&_e^{s, \\delta }(M):= \\lbrace = \\rho ^\\delta u \\mid u \\in _e^s(M)\\rbrace ,\\end{split}$ where $V_1 \\circ \\cdots \\circ V_s \\in L^2(M,E)$ is understood in the distributional senseThis is not the ordinary Sobolev space $H^s(_+)$ if $M=_+$ .."
],
[
"Preliminaries",
"Let $H_1, H_2$ be Hilbert spaces which are assumed to be embedded into a barrelled locally convex topological vector space, such that it makes sense to talk about $H_1+H_2$ (non-direct sum space) and $H_1\\cap H_2$ .",
"Let $[H_1,H_2]_{\\theta }$ , $0\\le \\theta \\le 1$ , be their complex interpolation space.",
"For Calderón's complex interpolation theory [12] we refer to [41].",
"The space of bounded linear operators between $H_1, H_2$ is denoted by $(H_1,H_2)$ , resp.",
"if $H_1=H_2=H$ we just write $(H)$ .",
"If $H_2 \\hookrightarrow H_1$ is densely embedded such that the norm of $H_2$ is the graph norm of the nonnegative self-adjoint operator $$ in $H_1$ , then by [41] $[H_1,H_2]_{\\theta } = (^{\\theta }).$ In fact, there is a converse to this statement.",
"Let $T: H_1 \\rightarrow H_2$ be a bounded operator between Hilbert spaces $H_1$ and $H_2$ .",
"Then we have the equality of ranges $T = \\sqrt{T\\ T^*}$ .",
"Let $T^* = U|T^*| = U\\sqrt{T\\ T^\\ast }$ be the polar decomposition of $T^*$ ; $U$ is a partial isometry with $U = {T^*} = (\\ker T)^\\perp $ and $\\ker U = \\ker T^* = (T)^\\perp $ .",
"Then, taking adjoints $T = \\sqrt{T\\ T^*} \\ U^*$ , and hence $\\sqrt{T \\ T^\\ast }\\supset T$ .",
"Since $U^* = (\\ker U)^\\perp = (\\ker T^*)^\\perp = (\\ker \\sqrt{T \\ T^*})^\\perp $ , the equality follows.",
"[[26]] Let $H$ be a Hilbert space with a dense subspace $\\subset H$ .",
"Assume that $$ carries a Hilbert space structure such that the inclusion map $i:\\hookrightarrow H$ is continuous.",
"Then $= \\sqrt{i \\ i^*}$ and $\\sqrt{i \\ i^*}: H \\rightarrow $ is a unitary isomorphism.",
"$:=(\\sqrt{i \\ i^*})^{-1}$ is a self-adjoint operator with domain $$ , hence $[H,]_\\theta = (^\\theta ),\\;\\theta \\in [0,1].$ By (REF ), see [41], the last claim follows once the claims about the operator $$ are established.",
"From Lemma REF we know that $= \\sqrt{i \\ i^*}$ .",
"Note that $\\ker i = \\lbrace 0\\rbrace $ , ${i} = H$ and hence $i^*$ and $\\sqrt{i \\ i^*}$ are injective with dense range.",
"Consequently, $= (\\sqrt{i \\ i^*})^{-1}$ is self-adjoint with domain $$ .",
"For $y\\in $ we find $\\begin{aligned}\\Vert \\sqrt{i\\ i^*} \\ y\\Vert _{}^2 &=\\langle \\sqrt{i\\ i^*} \\ y, \\sqrt{i\\ i^*} \\ y\\rangle _= \\langle i^* \\ y, \\sqrt{i\\ i^*} \\ y\\rangle _\\\\&=\\langle \\ y, i\\ \\sqrt{i\\ i^*} \\ y\\rangle _H=\\langle y, \\ \\sqrt{i\\ i^*} \\ y\\rangle _H = \\Vert y\\Vert ^2_H.\\end{aligned}$ Since $$ is dense the claim follows."
],
[
"Scales of Hilbert Spaces",
"From Brüning and Lesch [7] we recall the useful concept of a scale of Hilbert spaces, which has been used in various forms by several authors, see Connes and Moscovici [15], Higson [20], Otgonbayar [32] and Paycha [33].",
"Let $H$ be a Hilbert space and $A$ a self-adjoint operator in $H$ .",
"Then $H^{\\infty }:= \\bigcap _{n=0}^\\infty (|A|^n) = \\bigcap _{n=0}^\\infty (A^n)$ is dense in $H$ .",
"For $s\\in $ , let $H^s(A)$ be the completion of $H^\\infty $ with respect to the scalar product $\\langle x , y \\rangle _s:=\\langle (I+ A^2)^{\\frac{s}{2}} \\ x ,(I+A^2)^{\\frac{s}{2}}\\ y \\rangle .$ Then $H^n(A) = (A^n) = (|A|^n)$ for $n\\in _+$ , respectively, $H^s(A) =(|A|^s)$ , for any $s\\ge 0$ .",
"The properties of the family $\\lbrace H^s\\rbrace _{s\\in }$ are reminiscent of properties of Sobolev spaces and they are summarized in the following proposition.",
"The family $(H^s(A))_{s\\ge 0}$ satisfies: $H^s$ is a Hilbert space, for all $s\\ge 0$ .",
"For $s^{\\prime }\\ge s$ we have a continuous embedding $H^{s^{\\prime }}\\hookrightarrow H^s$ .",
"$[H_s,H_t]_\\theta = H_{\\theta t + (1-\\theta ) s }$ , for $0\\le \\theta \\le 1$ , in the sense of complex interpolation.",
"$H^\\infty = \\bigcap \\limits _{s\\ge 0} H^s$ is dense in $H^t$ for each $t$ .",
"An abstract family $(H^s)_{s\\ge 0}$ of Hilbert spaces satisfying (1)–(4) is called an (interpolation) scale of Hilbert spaces.",
"If there exists a self-adjoint operator $A$ such that $H^s=H^s(A)$ for $s\\ge 0$ then we call $A$ a generator of the scale.",
"The item (4) implies that $H^{s^{\\prime }}$ is dense in $H^s$ for $s^{\\prime }\\ge s$ .",
"Proposition REF implies that for $N>0$ there exists a self-adjoint operator $\\ge 0$ with $(^N) = H^N$ and hence $H^s = (^s) =: H^s()$ for $0\\le s\\le N$ .",
"Given a scale $(H^s)_{s\\ge 0}$ of Hilbert spaces as in Proposition REF , one may ask whether there exists a generator $$ , such that $H^s = (^s)$ for all $s\\ge 0$ , and not only for $0\\le s\\le N$ .",
"We believe that in general the answer is no.",
"E.g., the scale of Sobolev spaces $H^s([0,\\infty ))$ does not have a natural generator, although we cannot prove that there does not exist one.",
"We leave this open question to the reader.",
"This does not affect the discussion below.",
"Nonetheless, in the sequel we will for convenience assume that the scales do have a global generator $$ .",
"As the arguments will always only concern a compact set of $s$ -values, in light of the discussion above, this is not really a loss of generality.",
"Thus for all practical purposes we may think of a Hilbert space scale being the scale of a positive operator $$ .",
"We note that if two positive self-adjoint operators $_1$ , $_2$ have the same domain $(_1) = (_2)$ then $H^1(_1)=H^1(_2)$ , and by complex interpolation $(_1^s) = [H,H^1(_1)= H^2(_2)]_s = (_2^s),\\;{\\text{for}}\\;0\\le s \\le 1.$ In general, however, we will have $(_1^s)\\ne (_2^s),\\;\\text{for}\\; s>1.$ To illustrate this by example consider $_1 := \\left(\\begin{array}{cc}0 & \\partial _x\\\\-\\partial _x & 0\\end{array}\\right), \\quad _2 := \\left(\\begin{array}{cc}0 & \\partial _x+a\\\\-\\partial _x+a & 0\\end{array}\\right),$ acting in the Hilbert space $L^2(_+,2)=L^2(_+)\\otimes 2$ with domain $(_1) = (_2)= {f={f_1\\atopwithdelims ()f_2}\\in H^1(_+)\\otimes 2}{f_1(0)=0}.$ It is straightforward to see that $_j, j=1,2$ are self-adjoint.",
"However, the domains of the squares are given by $\\begin{split}(_1^2) &= {f\\in H^2(_+)\\otimes 2}{ f_1(0) =0,\\ f_2^{\\prime }(0) = 0 }, \\\\(_2^2) &= {f\\in H^2(_+)\\otimes 2 }{ f_1(0) =0,\\ f_2^{\\prime }(0)+a\\cdot f_2(0) = 0 },\\end{split}$ thus $H^s(_1) \\ne H^s(_2)$ for $1<s\\le 2$ .",
"In view of Example REF , we may now ask for criteria such that two self-adjoint operators generate the same interpolation scale.",
"Let $$ be a self-adjoint operator in the Hilbert space $H$ with interpolation scale $H^s()_{s\\ge 0}$ .",
"A linear operator $P:H^\\infty () \\rightarrow H^\\infty ()$ is said to be of order $\\mu $ if $P$ admits a formal adjointThis means that there is $P^t:H^\\infty \\rightarrow H^\\infty $ such that for all $x,y\\in H^\\infty $ , ${Px,y} = {x,P^ty}$ .",
"with respect to the scalar product of $H$ , and for any $s\\in $ , $P$ and $P^t$ extend by continuity $H^s() \\rightarrow H^{s-\\mu }()$ .",
"I. e. there are constants $C_s(P), C_s(P^t)$ such that for $x\\in H^{\\infty }$ we have $\\Vert Px\\Vert _{s-\\mu } \\le C_s(P)\\cdot \\Vert x\\Vert _s$ and $\\Vert P^tx\\Vert _{s-\\mu } \\le C_s(P)\\cdot \\Vert x\\Vert _s$ .",
"By $^\\mu ( )$ we denote the operators of order $\\mu $ .",
"Clearly, $^\\bullet () = \\bigcup _\\mu ^\\mu ()$ is a filtered algebra of operators acting on $H^\\infty ()$ .",
"Ref to Higson, Connes maybe To show that an operator $P$ is of order $\\mu $ it suffices to check the estimates in the definition on a sequence $(t_j)_j$ of $t$ -values with $\\lim t_j = \\infty $ .",
"This follows again from complex interpolation.",
"The continuity condition can equivalently be formulated in terms of the resolvent of $$ : [email protected]{H^t() [rr]^{P} [dd]_{(I+||)^t} & &H^{t-\\mu }()[dd]^{(I+||)^{t-\\mu }}\\\\& \\circlearrowleft & \\\\H=H^0()[rr] & & H^0()=H.",
"}$ Here, the lower arrow is given by the operator $(I+||)^{t-\\mu } \\circ P \\circ (I+||)^{-t},$ which is required to be bounded on $H$ for all $t\\in $ .",
"If $P=_2$ is a selfadjoint operator of order 1 on the Sobolev-scale $H^\\bullet (_1)$ , then we have an equality of interpolation scales $H^\\bullet (_1) = H^\\bullet (_2)$ , and hence we conclude using the interpolation property with the following observation.",
"Assume that for any $n\\in \\setminus \\lbrace 0\\rbrace $ $ (I+|_1|)^{n-1} \\circ _2 \\circ (I+|_1|)^{-n}$ is bounded on $H$ .",
"Then $_1$ and $_2$ generate the same interpolation scales.",
"If (REF ) is bounded only for $n=1$ then we can only infer that $H^s(_1)= H^s(_2)$ for $0 \\le s \\le 1$ .",
"Tensor products of interpolation scales In this section we follow in part [7].",
"We fix two interpolation scales $\\lbrace H_j^s\\rbrace _{s\\ge 0}$ , $j=1,2$ with generators $_1$ , $_2$ .",
"Without loss of generality we may choose $_1$ , $_2$ such that they are greater or equal to $I$ and hence we may define the scalar product on $H_j^s$ by $\\langle x,y\\rangle _{H^s_j}:=\\langle _j^s x, _j^s y \\rangle _{H_j}.$ For tensor products of (unbounded) operators we refer to the Appendix , in particular Proposition REF .",
"$H_1H_2$ resp.",
"$AB$ denotes the completed Hilbert space tensor product resp.",
"the tensor product of (unbounded) operators $A,B$ .",
"$\\lbrace H_1^s H^s_2\\rbrace _{s\\ge 0}$ is an interpolation scale with generator $_1 \\hat{\\otimes }_2$ .",
"By Proposition REF , we have $_1_2 \\ge I$ , hence the graph norm of $(_1_2)^s$ is equivalent to $\\Vert (_1_2)^s x\\Vert $ .",
"Note furthermore, that $_1I$ and $I_2$ are commuting self-adjoint operators greater or equal to $I$ , thus $(_1_2)^s= (_1I \\cdot I_2)^s= ^s_1I \\cdot I_2^s=^s_1_2^s.$ Furthermore, for $x_j\\in H_1^\\infty $ , $y_j\\in H_2^\\infty , j=1,\\ldots ,r$ we have with each summation index running from $j=1,\\ldots ,r$ : $\\begin{aligned}\\Bigl \\Vert \\sum _j x_j\\otimes y_j\\Bigr \\Vert ^2_{H_1^sH_2^s}&= \\sum _{k,l}\\langle x_k \\otimes y_k, x_l\\otimes y_l \\rangle _{H_1^s H_2^s}\\\\&= \\sum _{k,l}\\langle x_k, x_l\\rangle _{H_1^s} \\langle y_k, y_l \\rangle _{H_2^s}\\\\&= \\sum _{k,l}\\langle _1^s x_k, _1^s x_l\\rangle _{H_1} \\langle _2^sy_k, _2^s y_l \\rangle _{H_2}\\\\&= \\Bigl \\langle _1^s _2^s \\sum _j x_jy_j,_1^s _2^s \\sum _j x_jy_j\\Bigr \\rangle _{H_1\\otimes H_2}.\\end{aligned}$ This shows that the tensor product norm on $H_1^sH_2^s$ is equivalent to the graph norm of $_1^s_2^s$ which proves the claim.",
"As a consequence we get for $s,t\\ge 0$ $\\bigl [H_1^sH_2^s, H_1^t H_2^t\\bigr ]_\\theta =H_1^{\\theta t + (1-\\theta s )} H_2^{\\theta t + (1-\\theta )s}=[H_1^s,H_1^t]_\\theta [H_2^s,H_2^t]_\\theta .$ Since every interpolation pair of Hilbert spaces may be embedded into an interpolation scale (Proposition REF ) we obtain If $E^{\\prime }\\subset E$ , $F^{\\prime }\\subset E$ are interpolation pairs of Hilbert spaces then, for $0\\le \\theta \\le 1$ , $[EF,E^{\\prime }F^{\\prime }]_{\\theta } =[E,E^{\\prime }]_\\theta [F,F^{\\prime }]_\\theta .$ The tensor product of Lemma REF should not be confused with the Sobolev spaces on product spaces.",
"Note that on $^n$ we have $H^s(^n) = H^s(\\Delta _{^n})$ , where $\\Delta _{^n} = - \\sum _{j=1}^n \\partial _{x_j}^2$ is the Laplace operator.",
"Now it is not true that $H^s(^n \\times ^m) = H^s(^n)H^s(^m).$ Rather we have the following equalities $\\begin{aligned}H^s(^n \\times ^m)&= H^s(\\Delta _{^n\\times ^m} = \\Delta _{^n}I + I \\Delta _{^m})\\\\&\\stackrel{!",
"}{=} H^s(\\Delta _{^n})L^2(^m) \\ \\cap \\ L^2(^n)H^s(^m), \\;\\text{ for } s\\ge 0.\\end{aligned}$ This is due to the equality of domains $((\\Delta _{^n}I + I\\Delta _{^m})^s)=(\\Delta ^s_{^n}I)\\cap (I \\Delta ^s_{^m}),$ as we will see below.",
"Given two interpolation scales $\\lbrace H^s_j\\rbrace _{s\\ge 0}, j=1,2$ , we put for $s\\ge 0$ $^s:= ^s(\\lbrace H_1^{\\bullet } \\rbrace ,\\lbrace H_2^{\\bullet } \\rbrace ):=H_1^s H_2^0 \\cap H_1^0 H_2^s.$ This is a Hilbert space with scalar product being the sum of the scalar products of $H_1^sH_2^0$ and $H_1^0 H_2^s$ .",
"Let $_1,_2 \\ge I$ be generators of $\\lbrace H_1^\\bullet \\rbrace $ , $\\lbrace H_2^\\bullet \\rbrace $ , respectively.",
"Then $\\lbrace ^s\\rbrace _{s\\ge 0}$ is an interpolation scale with generator $_1I + I _2$ and $^s = \\bigcap _{0\\le t \\le s} H_1^t H_2^{s-t}= \\left(H_1^s H_2^{0}\\right) \\cap \\left(H_1^0 H_2^{s}\\right).$ Recall that $_1I$ , $I_2$ are commuting self-adjoint operators greater or equal to $I$ .",
"Now from $\\frac{1}{2} (b^s+c^s) \\le (b+c)^s\\le 2^s (b^s+c^s),$ for $b,c,s\\ge 0$ and the Spectral Theorem we infer $^s = (_1^s I) \\cap (I _2^s) = ((_1I + I _2)^s),$ hence the first part of the statement follows.",
"For the second part, we first note that the concavity of the $\\log $ –function implies the inequality $a^{\\theta }\\cdot b^{1-\\theta } \\le \\theta \\cdot a + (1-\\theta )\\cdot b,$ for $a,b\\ge 0$ and $0 \\le \\theta \\le 1$ .",
"For the commuting operators $_1I$ , $I_2$ and $0\\le \\theta \\le 1$ the inequality implies $\\begin{aligned}H_1^0H_2^s \\cap H_1^sH_2^0&= (I_2^s) \\cap (_1^sI)\\\\&\\subset ((_1^sI)^\\theta \\cdot (I_2^s)^{1-\\theta })\\\\&=(_1^{\\theta s} _2^{s-\\theta s})=H_1^{\\theta s} H_2^{s-\\theta s}.\\end{aligned}$ Consequently, $^s \\subset \\bigcap _{0\\le t \\le s} H_1^t H_2^{s-t}$ .",
"Dirac operators on an abstract edge Generalized Dirac operators on an abstract edge Let $S$ be a smooth family of self-adjoint operators in a Hilbert space $H$ with parameter $y\\in ^b$ and a fixed domain $_S$ .",
"We assume that each $S(y)$ is discrete.",
"A generalized Dirac Operator $D$ acting on $C_0^\\infty (_+\\times ^b,H^\\infty )$ is defined by the following (differential) expression $D:=\\Gamma (\\partial _x + X^{-1} S)+ T,$ where $x\\in _+$ , $X$ denotes the multiplication operator by $X$ , $\\Gamma $ is skew-adjoint and a unitary operator on the Hilbert space $L^2(_+\\times ^b,H)$ , and $T$ is a symmetric generalized Dirac Operator on $^b$ , given in terms of coordinates $(y_1,\\ldots ,y_b)\\in ^b$ and smooth families $(c_1(y), c_b(y))$ of bounded linear operators on $H$ , which satisfy Clifford relations for each fixed $y\\in ^b$ , by $T= \\sum _{j=1}^{b}c_j(y) \\frac{\\partial }{\\partial y_j}.$ Here, we have hid the vector bundle value action of the Dirac Operator $T$ into the Hilbert space $H$ .",
"We assume that the following standard commutator relations hold $\\begin{aligned}\\Gamma \\ S + S \\ \\Gamma &= 0,\\\\\\Gamma \\ T + T \\ \\Gamma &= 0,\\\\T \\ S - S \\ T &=0.\\end{aligned}$ In §REF we show that the Gauss–Bonnet operator on a simple edge satisfies these relations, cf.",
"(REF ).",
"The same relations hold for the spin Dirac operator, as shown in [3].",
"We shall also consider $D$ with coefficients frozen at some $y_0 \\in ^b$ $D_{y_0}:=\\Gamma (\\partial _x + X^{-1} S(y_0))+ T_{y_0}, \\quad \\textup {where} \\ T_{y_0}= \\sum _{j=1}^{b}c_j(y_0) \\frac{\\partial }{\\partial y_j}.$ Consider the Fourier transform $\\mathfrak {F}_{y\\rightarrow \\xi }$ on the $L^2(^b)$ -component of $L^2(_+\\times ^b,H)$ .",
"We use Hörmander's normalization and write $\\left(\\mathfrak {F}_{y\\rightarrow \\xi } f\\right)(\\xi ) =\\int _{^b} e^{-i \\langle y, \\xi \\rangle } f(y) dy,\\quad \\left(\\mathfrak {F}^{-1}_{y\\rightarrow \\xi } g\\right)(y) =\\int _{^b} e^{i \\langle y, \\xi \\rangle } g(\\xi ) \\frac{d\\xi }{(2\\pi )^b}.$ We compute $\\mathfrak {F}_{y\\rightarrow \\xi } \\circ D_{y_0} \\circ \\mathfrak {F}_{y\\rightarrow \\xi }^{-1} = \\Gamma (\\partial _x + X^{-1}S(y_0)) + ic(\\xi ; y_0) =: L(y_0,\\xi ),$ where $c(\\xi ; y_0) := \\sum _{j=1}^b c_j(y_0) \\xi _j.$ The usual strategy is now to study invertibility of $L(y_0,\\xi )$ on appropriate spaces, which is then used to construct the parametrix for $D$ and analysis of its domain.",
"The spectral Witt condition We also impose a spectral Witt condition, which asserts that $\\forall \\, y \\in ^b: \\ S(y) \\cap \\left[-1, 1\\right] = \\varnothing ,$ We should point out that Albin and Gell-Redman [3] require a smaller spectral gap $S(y) \\cap -1/2, 1/2= \\varnothing $ .",
"However, when proving an analogue of the crucial [3] by explicit computations, it seems that a smaller spectral gap may not be sufficient for our purposes.",
"In any case, if $D$ is the Gauss–Bonnet operator on a stratified Witt space, one can always achieve $S(y) \\cap -R, R= \\varnothing $ for any $R>0$ by a simple rescaling of the metric.",
"Squares of generalized Dirac operators In view of the commutator relations (REF ), the generalized Laplace operators $D^2$ and $D_{y_0}^2$ , acting both on $C_0^\\infty (_+\\times ^b,H^\\infty )$ , are of the following form $\\begin{split}&D^2 = -x^2 + X^{-2}\\ S \\ (S+1) + T^2, \\\\&D^2_{y_0} = -x^2 + X^{-2} \\ S(y_0) \\ (S(y_0) + 1) + T^2_{y_0}.\\end{split}$ We set $A := \\left|S\\right| +\\frac{1}{2}$ .",
"Assuming $S \\cap \\left[-1, 1\\right] = \\varnothing $ , we find $S(S+1) = A^2 -1/4$ and rewrite the generalized Laplacians $D^2$ and $D_{y_0}^2$ as follows $\\begin{split}&D^2 = -x^2 + X^{-2} A^2 - \\frac{1}{4} + T^2, \\\\&D^2_{y_0} = -x^2 + X^{-2}A^2(y_0) - \\frac{1}{4} + T^2_{y_0}.\\end{split}$ As before, we may apply the Fourier transform $\\mathfrak {F}_{y\\rightarrow \\xi }$ and compute $\\begin{split}\\mathfrak {F}_{y\\rightarrow \\xi } \\circ D^2_{y_0} \\circ \\mathfrak {F}_{y\\rightarrow \\xi }^{-1} &= -x^2 + X^{-2}\\left(A^2(y_0) - \\frac{1}{4}\\right) + c(\\xi ,y_0)^2 \\\\~&=: L^2(y_0,\\xi ), \\ \\textup {where} \\ c(\\xi ; y_0)^2 =- \\sum _{j,k=1}^b c_j(y_0) c_k(y_0)\\xi _j \\xi _k.\\end{split}$ Sobolev-spaces of an abstract edge Recall the definition of interpolation scales of Hilbert spaces in §.",
"This defines for each $y_0\\in ^b$ an interpolation scale $H^s(S(y_0))$ , $s\\in $ .",
"We can now define the Sobolev-scales on the model cone and the model edge in our abstract setting.",
"Consider for this the Sobolev-scale $H^\\bullet _e(_+)$ generated byThe edge Sobolev scale $H^\\bullet _e(_+)$ prescribes regularity under differentiation by $x\\partial _x$ .",
"However, $x\\partial _x$ is not a symmetric operator and hence we take its symmetrization $(i x\\partial _x + i/ 2)$ as the generator of the Sobolev scale.",
"Alternatively we can replace the definition of Sobolev scales to allow for closed not necessarily symmetric operators.",
"$(i x\\partial _x + i/ 2)$ ; and the Sobolev-scale There is an arrow in the note that I do not understand.",
"$H_e^\\bullet (_+\\times ^b)$ generated by $= (i x\\partial _x + i/ 2)I+Ix T_{y_0}$ .",
"The lower index $e$ indicates that these interpolation scales coincide with the edge Sobolev spaces for integer orders.",
"Let $y_0 \\in ^b$ be fixed.",
"The Sobolev-scale $W^\\bullet (_+,H)$ of an abstract model cone is defined as an interpolation scale with generator $(i x\\partial _x + i/ 2)I+IS(y_0)$ .",
"By Proposition REF $W^s(_+,H):=(H^s_e(^+)H) \\cap (L^2(_+)H^s(S(y_0))).$ The Sobolev-scale $W^\\bullet (_+\\times ^b, H)$ of an abstract model edge is defined as an interpolation scale with generator $I + I S(y_0)$ , where $$ is the generator of the Sobolev-scale $H^s_e(_+\\times ^b)$ .",
"By Proposition REF $W^s(_+\\times ^b,H):=(H^s_e(_+\\times ^b)H) \\cap (L^2(_+\\times ^b)H^s(S(y_0))).$ In view of Proposition REF , for $y,y_0\\in ^b$ , the interpolation scales of $S(y)$ and $S(y_0)$ need not coincide.",
"However, since for any $y\\in ^b$ , the domain of $S(y)$ is fixed and given by $_S$ , we have $H^s(S(y))= H^s(S(y_0))$ for $0\\le s\\le 1$ .",
"In particular the Sobolev scales $W^s(_+, H)$ and $W^s(_+\\times ^b, H)$ do not depend on $y_0 \\in ^b$ for $0\\le s\\le 1$ .",
"In fact, in our arguments below we will require independence of the Sobolev spaces for $0 \\le s \\le 2$ .",
"We conclude with a definition of weighted Sobolev-spaces, where we denote by $X$ the multiplication operator by $x~\\in _+$ .",
"The weighted Sobolev-scales are defined by $W^{s,\\delta ,l}:= X^\\delta (1+X)^{-l} W^s(_+,H), \\quad W^{s,\\delta }:= W^{s,\\delta ,0}.$ Examples of generalized Dirac operators on an abstract edge The spin Dirac operator on a model edge space is indeed a generalized Dirac operator in the sense that it is given by the differential expression (REF ) and satisfies the commutator relations (REF ).",
"This has been established by Albin and Gell-Redman [3].",
"In this subsection we prove that the Gauss–Bonnet operator on a model edge space is a generalized Dirac operator in the sense above as well.",
"Let $M^m$ and $N^n$ be Riemannian manifolds.",
"Given forms $\\omega _p \\in \\Omega ^p(M)$ and $\\eta _q \\in \\Omega ^q(N)$ , we will write $\\omega _p \\wedge \\eta _q$ for the form $\\pi _M^* (\\omega _p) \\wedge \\pi _N^* (\\eta _q) \\in \\Omega ^{p+q}(M\\times N)$ , where $\\pi _M:M\\times N \\rightarrow M$ and $\\pi _N:M\\times N \\rightarrow N$ are projections onto the first and second factors respectively.",
"It is well known that the exterior derivative $d:\\Omega ^{\\ast }(M\\times N) \\rightarrow \\Omega ^{\\ast }(M\\times N)$ satisfies the Leibniz rule, if $\\omega _p\\in \\Omega ^{p}(M)$ and $\\eta _q\\in \\Omega ^q(N)$ then $d(\\omega _p\\wedge \\eta _q) \\ = \\ (d^M\\omega _p)\\wedge \\eta _q +(-1)^{p} \\ \\omega _p\\wedge (d^N\\eta _q).$ The same Leibniz rule holds for the adjoint of the exterior derivative $d^t$ in $\\Omega ^{\\ast }(M\\times N)$ .",
"Note that $\\Omega ^{\\ast }(M\\times N)$ can be decomposed into a direct sum of subspaces of the form $\\Omega ^{\\ast }(M) \\wedge \\Omega ^{\\ast }(N)$ .",
"Hence it suffices to study the action of $d^t$ on differential forms in $\\Omega ^{p+1}(M) \\wedge \\Omega ^q(N)$ , where we have $d^t: \\Omega ^{p+1}(M) \\wedge \\Omega ^q(N) \\rightarrow (\\Omega ^{p}(M) \\wedge \\Omega ^q(N)) \\oplus (\\Omega ^{p}(M) \\wedge \\Omega ^{q-1}(N)).$ Consider $\\tilde{\\omega }_{p}\\in \\Omega ^{p}(M)$ , $\\omega _{p+1},\\tilde{\\omega }_{p+1}\\in \\Omega ^{p+1}(M)$ , $\\tilde{\\eta }_q, \\eta _q\\in \\Omega ^{q}(N)$ and $\\tilde{\\eta }_{q-1}\\in \\Omega ^{q-1}(N)$ , then we have for the first component of $d^t$ $\\begin{aligned}\\langle d_{p+q}^t (\\omega _{p+1}\\wedge &\\eta _q),\\tilde{\\omega }_{p}\\wedge \\tilde{\\eta }_q\\rangle \\\\&= \\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p})\\wedge \\tilde{\\eta }_q + (-1)^p\\tilde{\\omega }_{p}\\wedge (d^N\\tilde{\\eta }_q)\\rangle \\\\&=\\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p})\\wedge \\tilde{\\eta }_q \\rangle \\\\&=\\langle \\omega _{p+1}, (d^M\\tilde{\\omega }_{p})\\rangle _M\\langle \\eta _{q}, \\tilde{\\eta }_q\\rangle _N\\\\&=\\langle (d^{M,t}\\omega _{p+1}), \\tilde{\\omega }_{p}\\rangle _M\\langle \\eta _{q}, \\tilde{\\eta }_q\\rangle _N\\\\&=\\langle (d^{M,t}\\omega _{p+1})\\wedge \\eta _{q},\\tilde{\\omega }_{p}\\wedge \\tilde{\\eta }_{q}\\rangle .\\end{aligned}$ For the second component of $d^t$ , we obtain $\\begin{aligned}\\langle d_{p+q}^t &(\\omega _{p+1}\\wedge \\eta _q),\\tilde{\\omega }_{p+1}\\wedge \\tilde{\\eta }_{q-1}\\rangle \\\\&= \\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p+1})\\wedge \\tilde{\\eta }_{q-1} + (-1)^{p+1}\\tilde{\\omega }_{p+1}\\wedge (d^N\\tilde{\\eta }_{q-1})\\rangle \\\\&=(-1)^{p+1}\\langle \\omega _{p+1}\\wedge \\eta _q,\\tilde{\\omega }_{p+1}\\wedge (d^N\\tilde{\\eta }_{q-1}) \\rangle \\\\&=(-1)^{p+1}\\langle \\omega _{p+1}, \\tilde{\\omega }_{p+1}\\rangle _M\\langle \\eta _{q}, (d^N\\tilde{\\eta }_{q-1})\\rangle _N\\\\&=(-1)^{p+1}\\langle \\omega _{p+1}, \\tilde{\\omega }_{p+1}\\rangle _M\\langle (d^{N,t}\\eta _{q}), \\tilde{\\eta }_{q-1}\\rangle _N\\\\&=(-1)^{p+1}\\langle \\omega _{p+1}\\wedge (d^{N,t}\\eta _{q}),\\tilde{\\omega }_{p+1}\\wedge \\tilde{\\eta }_{q-1}\\rangle .\\end{aligned}$ Altogether, we arrive at the result $d_{p+q}^t (\\omega _{p+1}\\wedge \\eta _q) = (d_p^{M,t}\\omega _{p+1})\\wedge \\eta _{q} + (-1)^{p+1} \\omega _{p+1}\\wedge (d^{N,t}_{q} \\eta _q).$ We now apply Lemma REF to the case of a model edge $C(F) \\times Y$ of cones $C(F) = _+\\times F$ fibered over an edge manifold $Y$ .",
"Recall that on a cone $C(F) = _+\\times F$ we have as in [10] the following isometric identifications $\\Omega ^{\\rm ev}(C(F)) \\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F)), \\qquad \\Omega ^{\\rm odd}(C(F)) \\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F)).$ Under these identifications the Gauss–Bonnet operator $D=d+d^t$ acting now from $\\Omega ^{\\rm ev}(C(F))\\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F))$ to $\\Omega ^{\\rm odd}(C(F))\\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F))$ , takes the form cf.",
"[10] $D = \\frac{d}{dx}+X^{-1} A.$ Respectively, the full operator $D$ acts on $C^{\\infty }(_+,\\Omega ^{\\ast }(C(F))\\oplus \\Omega ^{\\ast }(C(F)))$ as $\\left(\\begin{array}{cc}0 & -\\frac{d}{dx}+X^{-1} A\\\\\\frac{d}{dx}+X^{-1} A & 0\\end{array}\\right) =\\left(\\begin{array}{cc}0 & -1\\\\1 & 0\\end{array}\\right)\\left(\\frac{d}{dx} + X^{-1}\\left(\\begin{array}{cc}A & 0\\\\0 & -A\\end{array}\\right)\\right).$ Note that the grading operator on $\\Omega ^{\\ast }(C(F))\\oplus \\Omega ^{\\ast }(C(F))$ is $\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right)$ .",
"Taking now the cartesian product by a manifold $Y$ (the edge), we have $\\begin{aligned}\\Omega ^{\\rm ev}(C(F)\\times Y) &= \\Omega ^{\\rm ev}(C(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus \\Omega ^{\\rm odd}(C(F))\\otimes \\Omega ^{\\rm odd}(Y)\\\\&\\cong C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm odd}(Y),\\end{aligned}$ where we used the identifications (REF ) in the second equality.",
"In exactly the same manner we find for differential forms of odd degree $\\begin{aligned}\\Omega ^{\\rm odd}(C(F)\\times Y) &= \\Omega ^{\\rm odd}(C(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus \\Omega ^{\\rm ev}(C(F))\\otimes \\Omega ^{\\rm odd}(Y)\\\\&\\cong C^{\\infty }(_+, \\Omega ^{\\ast }(L))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm odd}(Y).\\end{aligned}$ So again we have an identification of the space $\\Omega ^{\\rm ev}(C(F)\\times Y)$ with the space $\\Omega ^{\\rm odd}(C(F)\\times Y)$ .",
"For $\\omega _1 \\in \\Omega ^{\\rm ev}(C(F))$ , $\\omega _2 \\in \\Omega ^{\\rm odd}(C(F))$ , $\\eta _1 \\in \\Omega ^{\\rm ev}(Y)$ and $\\eta _2 \\in \\Omega ^{\\rm odd}(Y)$ , we have $\\omega _1\\otimes \\eta _1 \\oplus \\omega _2\\otimes \\eta _2\\in \\Omega ^{\\rm ev}(C(F)\\times Y)$ .",
"Using Lemma REF we now find for $D= d + d^t$ , $\\begin{aligned}D(\\omega _1\\otimes \\eta _1 \\oplus \\omega _2\\otimes \\eta _2) &=D^{C(F)}\\omega _1\\otimes \\eta _1 + \\omega _1 \\otimes D^Y\\eta _1\\\\&+ D^{C(F)}\\omega _2\\otimes \\eta _2 - \\omega _2\\otimes D^Y\\eta _2\\\\&=\\left(\\begin{array}{cc}\\partial _x + X^{-1}A & - D^Y\\\\D^Y & -\\partial _x + X^{-1}A\\end{array}\\right)\\left(\\begin{array}{c}\\omega _1\\otimes \\eta _1\\\\\\omega _2\\otimes \\eta _2\\end{array}\\right).\\end{aligned}$ Note that by construction $A$ and $D^Y$ commute.",
"By abuse of notation $A$ acts as $A\\otimes I$ and $D^Y$ acts as $I\\otimes D^Y$ on the tensors.",
"The full Gauss–Bonnet then becomes $D= \\left(\\begin{array}{cccc}0 & 0 & -\\partial _x + X^{-1}A & D^Y\\\\0 & 0 & -D^Y & \\partial _x + X^{-1}A\\\\\\partial _x + X^{-1}A & -D^Y & 0 & 0\\\\D^Y & -\\partial _x + X^{-1}A & 0 & 0\\end{array}\\right).$ This expression can rewritten as follows.",
"$\\begin{aligned}D = \\left(\\begin{array}{cccc}0 & 0 & -1 & 0\\\\0 & 0 & 0 & 1\\\\1 & 0 & 0 & 0\\\\0 &-1 & 0 & 0\\end{array}\\right) &\\left(\\partial _x + X^{-1} \\left(\\begin{array}{cccc}1 & 0 & 0 & 0\\\\0 & -1 &0 & 0\\\\0 & 0 & -1&0\\\\0 & 0 & 0 & 1\\end{array}\\right)\\cdot A\\right)\\\\&+\\left(\\begin{array}{cccc}0 & 0 & 0 & 1\\\\0 & 0 & -1 & 0\\\\0 & -1 & 0 & 0\\\\1 & 0 & 0 & 0\\end{array}\\right)\\cdot D^Y,\\end{aligned}$ with grading operator $\\left(\\begin{array}{cc}I_2 & 0\\\\0 & - I_2\\end{array}\\right)$ where $I_2$ is the identity in $M_2()$ .",
"Define the following matrices $\\Gamma = \\left(\\begin{array}{cccc}0 & 0 & -1 & 0\\\\0 & 0 & 0 & 1\\\\1 & 0 & 0 & 0\\\\0 &-1 & 0 & 0\\end{array}\\right)\\!\\!,S =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0\\\\0 & -1 &0 & 0\\\\0 & 0 & -1&0\\\\0 & 0 & 0 & 1\\end{array}\\right) A,T = \\left(\\begin{array}{cccc}0 & 0 & 0 & 1\\\\0 & 0 & -1 & 0\\\\0 & -1 & 0 & 0\\\\1 & 0 & 0 & 0\\end{array}\\right) D^Y\\!.$ We introduce the usual Clifford matrices $\\sigma _1 = \\left(\\begin{array}{cc}0 & -1\\\\1 & 0\\end{array}\\right),\\ \\sigma _2=\\left(\\begin{array}{cc}0 & i \\\\i & 0\\end{array}\\right),\\ \\omega =\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right) = i \\cdot \\sigma _1 \\cdot \\sigma _2.$ We have, $\\begin{aligned}\\Gamma &= \\left(\\begin{array}{cc}0 & -\\omega \\\\\\omega & 0\\end{array}\\right) = \\sigma _1 \\otimes \\omega ,\\\\S &= \\left(\\begin{array}{cc}\\omega & 0\\\\0 & -\\omega \\end{array}\\right)\\otimes A = \\omega \\otimes \\omega \\otimes A,\\\\T &= \\left(\\begin{array}{cc}0 & -\\sigma _1\\\\\\sigma _1 & 0\\end{array}\\right) \\otimes D^Y = \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y.\\end{aligned}$ We can now easily compute the commutator relations $\\begin{aligned}\\Gamma \\ S + S \\ \\Gamma &= \\sigma _1 \\otimes \\omega \\cdot \\omega \\otimes \\omega \\otimes A + \\omega \\otimes \\omega \\otimes A \\cdot \\sigma _1 \\otimes \\omega \\\\&= (\\sigma _1 \\omega + \\omega \\sigma _1)\\otimes \\omega ^2 \\otimes A = 0.\\\\\\Gamma \\ T + T \\ \\Gamma &= \\sigma _1 \\otimes \\omega \\cdot \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y + \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y \\cdot \\sigma _1\\otimes \\omega \\\\&= \\sigma _1 \\otimes (\\omega \\sigma _1 + \\sigma _1 \\omega )\\otimes D^Y = 0.\\\\T \\ S - S \\ T &= \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y \\cdot \\omega \\otimes \\omega \\otimes A - \\omega \\otimes \\omega \\otimes A \\cdot \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y\\\\&= (\\sigma _1\\cdot \\omega \\otimes \\sigma _1\\cdot \\omega - \\omega \\cdot \\sigma _1 \\otimes \\omega \\cdot \\sigma _1)\\otimes D^Y \\cdot A\\\\&=(\\omega \\cdot \\sigma _1 \\otimes \\sigma _1 \\cdot \\omega + \\sigma _1\\cdot \\omega \\otimes \\sigma _1 \\cdot \\omega )\\otimes D^Y \\cdot A\\\\&=(\\omega \\cdot \\sigma _1 + \\sigma _1\\cdot \\omega )\\otimes \\sigma _1 \\cdot \\omega \\otimes D^Y \\cdot A = 0.\\end{aligned}$ Some integral operators and auxiliary estimates In this section we study boundedness properties of certain integral operators that appear below when inverting the model Bessel operator $L^2(y_0,\\xi )$ and its square $L^2(y_0,\\xi )^2$ .",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and consider the integral operator $K$ acting on $C^\\infty _0(_+)$ with integral kernel given by $k(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{2\\nu } \\frac{y}{x}^{\\nu } (xy)^{\\frac{1}{2}}, & y\\le x,\\\\\\frac{1}{2\\nu } \\frac{y}{x}^{-\\nu } (xy)^{\\frac{1}{2}}, & x\\le y.\\end{array} \\right.$ Then $X^{-2} \\circ K$ defines a bounded operator on $L^2(0,\\infty )$ and there exists a constant $C>0$ depending only on $\\delta >0$ such that $\\begin{split}&\\Vert X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le \\nu ^{2}-\\frac{9}{4}^{-1}, \\\\&\\Vert (X \\partial _x) \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le \\nu -\\frac{3}{2}^{-1}, \\\\&\\Vert (X \\partial _x)^2 \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C.\\end{split}$ We apply Schur's test, cf.",
"Halmos and Sunder [19].",
"We have $\\begin{aligned}\\int _{0}^{x}x^{-2}k(x,y) \\ dy &+ \\int _{x}^{\\infty } x^{-2} k(x,y) \\ dy \\\\~&=\\frac{1}{2\\nu } x^{-\\nu - \\frac{3}{2}}\\int _{0}^x y^{\\nu +\\frac{1}{2}} \\ dy + x^{\\nu - \\frac{3}{2}} \\int _{x}^{\\infty } y^{-\\nu + \\frac{1}{2}}dy\\\\&= \\frac{1}{2\\nu } \\frac{1}{\\nu + \\frac{3}{2}} + \\frac{1}{\\nu -\\frac{3}{2}} = \\nu ^{2} - \\frac{9}{4} ^{-1}.\\end{aligned}$ Similarly, we integrate in the $x$ variable and find $\\begin{aligned}\\int _{0}^{y}x^{-2}k(x,y) \\ dx &+ \\int _{y}^{\\infty } x^{-2} k(x,y) \\ dx \\\\&=\\frac{1}{2\\nu } y^{-\\nu + \\frac{1}{2}}\\int _{0}^y x^{\\nu -\\frac{3}{2}} \\ dx + y^{\\nu + \\frac{1}{2}} \\int _{y}^{\\infty } x^{-\\nu - \\frac{3}{2}}dx\\\\&= \\frac{1}{2\\nu } \\frac{1}{\\nu - \\frac{1}{2}} + \\frac{1}{\\nu +\\frac{1}{2}} = \\nu ^{2} - \\frac{1}{4} ^{-1}\\end{aligned}$ From there one concludes that $\\Vert X^{-2} \\circ K\\Vert _{L^2\\rightarrow L^2}\\le \\left( \\nu ^2 - \\frac{9}{4}\\nu ^{2}-\\frac{1}{4}\\right)^{-\\frac{1}{2}} \\le \\nu ^2-\\frac{9}{4}^{-1}.$ This proves the first estimate.",
"The second and third estimates are established ad verbatim.",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and let $\\beta >0$ be positive real number.",
"Consider integral operator $K$ acting on $C^\\infty _0(_+)$ with integral kernel given in terms of modified Bessel functions by $k(x,y) = \\left\\lbrace \\begin{array}{cc}(xy)^{\\frac{1}{2}} I_\\nu (\\beta \\ y) K_{\\nu }(\\beta \\ x), & y\\le x,\\\\(xy)^{\\frac{1}{2}} I_\\nu (\\beta \\ x) K_{\\nu }(\\beta \\ y), & x\\le y.\\end{array}\\right.$ Then $X^{-2} \\circ K$ defines a bounded operator on $L^2(0,\\infty )$ and there exists a constant $C>0$ depending only on $\\delta >0$ such that $\\begin{split}&\\Vert X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C \\nu ^{2}-\\frac{9}{4}^{-1}, \\\\&\\Vert (X \\partial _x) \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C \\nu -\\frac{3}{2}^{-1}, \\\\&\\Vert (X \\partial _x)^2 \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C.\\end{split}$ Following Olver [31], we note the asymptotic expansions for Bessel functions as $\\nu \\rightarrow \\infty $ $I_\\nu (\\nu x) \\sim \\frac{1}{\\sqrt{2\\pi \\nu }}\\cdot \\frac{e^{\\nu \\cdot \\eta (x)}}{(1+x^2)^{1/4}},\\;\\;K_\\nu (\\nu x) \\sim \\sqrt{\\frac{2\\pi }{\\nu }}\\cdot \\frac{e^{-\\nu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}$ where $\\eta (x) = \\sqrt{1+x^2} +\\ln \\frac{x}{1+\\sqrt{1+x^2}}$ .",
"By (REF ), these expansions are uniform in $x\\in (0,\\infty )$ .",
"We define an auxiliary function $E(x, \\nu ):=\\frac{e^{\\nu (\\eta (x)-\\ln x)}}{(1+x^2)^{\\frac{1}{4}}}.",
"$ Note that $\\eta (x) - \\ln x$ is increasing as $x\\rightarrow \\infty $ , since $\\begin{aligned}(\\eta (x)-\\ln x)^{\\prime } &= (\\sqrt{1+x^2} - \\ln (1+\\sqrt{1+x^2})^{\\prime }\\\\&=2x\\frac{1}{2\\sqrt{1+x^2}} - \\frac{1}{2\\sqrt{1+x^2}}\\cdot \\frac{1}{1+\\sqrt{1+x^2}} \\\\&=\\frac{x}{\\sqrt{1+x^2}}\\cdot \\frac{\\sqrt{1+x^2}}{1+\\sqrt{1+x^2}}>0.\\end{aligned}$ Consequently $E(x,-\\nu )$ is decreasing and for $y \\le x$ $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| &\\le C\\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\frac{y}{\\nu }^\\nu \\frac{x}{\\nu +\\alpha }^{-(\\nu +\\alpha )} \\\\~&\\times E\\left(\\frac{y}{\\nu },\\nu \\right)\\cdot E\\left(\\frac{y}{\\nu +\\alpha },-(\\nu +\\alpha )\\right),\\end{aligned}$ for some uniform constant $C>0$ and $\\alpha \\in \\lbrace 0,1\\rbrace $ .",
"In fact, below we will always denote uniform positive constants by $C$ .",
"We proceed with a technical calculation $\\begin{aligned}&(\\nu +\\alpha )\\eta \\frac{y}{\\nu +\\alpha }- \\ln \\frac{y}{\\nu +\\alpha } -\\nu \\eta \\frac{y}{\\nu }-\\ln \\frac{y}{\\nu } \\\\&=\\sqrt{(\\nu +\\alpha )^2 +y^2} - \\sqrt{\\nu ^2 +y^2} +\\nu \\ln \\frac{\\nu +\\alpha }{\\nu } \\\\&- \\nu \\ln \\frac{\\nu +\\alpha +\\sqrt{(\\nu +\\alpha )^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}-\\alpha \\ln 1+\\sqrt{1+\\frac{y}{\\nu +\\alpha }^2} \\\\&=\\sqrt{\\nu ^2 +y^2}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1+ \\nu \\ln 1+\\frac{\\alpha }{\\nu }\\\\&-\\nu \\ln 1+ \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1\\\\&-\\alpha \\ln 1 + \\sqrt{1+\\frac{y}{\\nu +\\alpha }^2}.\\end{aligned}$ In order to continue with our estimates we write $O(f)$ for any function whose absolute value is bounded by $f$ , with a uniform constant that is independent of $\\nu $ and $y$ , and note $1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}$ is always positive, $\\ln \\frac{\\nu +\\alpha }{\\nu } = \\frac{\\alpha }{\\nu } + O\\frac{1}{\\nu ^2}$ where the $O$ -constant depends on $\\delta >0$ , $\\ln 1+ \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1\\\\~= \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}} + O\\frac{1}{\\nu ^2},$ where the $O$ -constant may be chosen independently of $y\\in (0,\\infty )$ , but depends on $\\delta >0$ .",
"Plugging in these observations, we arrive at the following estimate, $\\begin{aligned}&(\\nu +\\alpha )\\eta \\frac{y}{\\nu +\\alpha }- \\ln \\frac{y}{\\nu +\\alpha } -\\nu \\eta \\frac{y}{\\nu }-\\ln \\frac{y}{\\nu } \\\\&=-\\alpha \\ln 1+\\sqrt{1+\\frac{y}{\\nu }^2} +O1.\\end{aligned}$ Plugging this into the estimate (REF ) we obtain: $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| &\\le C \\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{\\nu +\\alpha }{\\nu } ^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha } \\\\~&\\le C \\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha }, \\\\& \\textup {where} \\ F\\frac{y}{\\nu }:= 1+\\sqrt{1+\\frac{y}{\\nu }^2}^{\\alpha } / \\sqrt{1+\\frac{y}{\\nu }^2},\\end{aligned}$ for some uniform constants $C>0$ , depending only on $\\delta $ , where in the second inequality we noted that $\\lim \\limits _{\\nu \\rightarrow \\infty } \\frac{\\nu +\\alpha }{\\nu } ^\\nu =e^\\alpha $ and hence $\\frac{\\nu +\\alpha }{\\nu } ^\\nu $ is bounded uniformly for large $\\nu $ .",
"We also note that $(\\nu (\\nu +\\alpha ))^{-1} \\le C \\nu ^{-2}$ , as long as $\\nu $ and $(\\nu +\\alpha )$ are positive bounded away from zero.",
"Hence we arrive at the following estimate $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| \\le C \\cdot \\frac{1}{\\nu } \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha }.",
"\\end{aligned}$ Note that for $\\alpha = 1$ , $F(y/\\nu )$ is uniformly bounded and for $\\alpha =0$ , $F(y/\\nu )\\le C(y/\\nu )^{-1}$ .",
"Hence we conclude for $x \\ge y$ and some uniform constant $C>0$ $\\begin{split}&\\left| \\, K_{\\nu }(\\beta \\ x) \\cdot I_\\nu (\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{1}{\\nu } \\cdot \\frac{y}{x}^{\\nu }.",
"\\\\&\\left| \\, x K_{\\nu +1}(\\beta \\ x) \\cdot I_\\nu (\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{y}{x}^{\\nu }.",
"\\\\&\\left| \\, y K_{\\nu }(\\beta \\ x) \\cdot I_{\\nu - 1}(\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{y}{x}^{\\nu }.",
"\\end{split}$ By the formulae for the derivatives of modified Bessel functions $\\begin{split}& (x\\partial _x) I_\\nu (x) = xI_{\\nu -1}(x) - \\nu I_\\nu (x), \\\\& (x\\partial _x) K_\\nu (x) = \\nu K_\\nu (x) - xK_{\\nu +1}(x),\\end{split}$ the derivatives $(x\\partial _x) k(x,y)$ and $(x\\partial _x)^2 k(x,y)$ can be written as combinations of the products in (REF ).",
"In view of Proposition , we obtain the result.",
"The statement of Proposition corresponds to Brüning-Seeley [9].",
"However, the latter reference does not provide an exact lower bound on $\\nu $ , which is crucial in order to establish the optimal spectral gap in the spectral Witt condition.",
"We close the section with a crucial observation.",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and let $K$ denote either the integral operator in Proposition or in Proposition .",
"Then for any $u \\in L^2(_+)$ with compact support in $[0,1]$ , $Ku$ admits the following estimates $Ku$ is continuously differentiable on $(0,\\infty )$ .",
"$| Ku(x) | \\le \\frac{C}{\\nu } \\Vert u\\Vert _{L^2} \\ x^{-1-\\delta }, \\quad | (x\\partial _x) Ku(x) |\\le C \\Vert u\\Vert _{L^2} \\ x^{-1-\\delta },$ for a constant $C>0$ independent of $u$ and $\\nu $ .",
"It suffices to prove the statement for $K$ in Proposition , since by (REF ) the integral kernels in Proposition , and their derivatives, can be estimated against those in Proposition .",
"Consider $u \\in L^2(_+)$ such that $u \\subset [0,1]$ .",
"Then for $x>1$ we find using $\\nu \\ge \\frac{3}{2} + \\delta $ $| Ku(x) | &\\le \\frac{x^{-\\nu +\\frac{1}{2}}}{2\\nu } \\int _0^1 y^{\\nu +\\frac{1}{2}} | u(y) | dy\\le \\frac{C}{\\nu } \\Vert u\\Vert _{L^2}\\ x^{-1-\\delta }, \\\\| x\\partial _x Ku(x) | &\\le \\frac{(-\\nu +\\frac{1}{2})}{2\\nu } x^{-\\nu +\\frac{1}{2}} \\int _0^1 y^{\\nu +\\frac{1}{2}} |u(y)| dy\\le C \\Vert u\\Vert _{L^2}\\ x^{-1-\\delta }, \\\\$ for a constant $C>0$ independent of $u$ and $\\nu $ .",
"Invertibility of the model Bessel operators In this section we prove invertibility of $L(y_0,\\xi ) = \\Gamma (\\partial _x + X^{-1}S(y_0)) + ic(\\xi ; y_0),$ cf.",
"(REF ), and its square $L(y_0,\\xi )^2$ .",
"We will work with the Sobolev scale $W^s(_+, H)$ , defined in terms of the interpolation scale $H^s\\equiv H^s(S(y_0))$ .",
"As noted in Remark REF , the interpolation scales $H^s(S(y_0))$ in general depend on the base point $y_0\\in ^b$ .",
"This does not play a role here, since in the present section $y_0$ is fixed.",
"Assuming the spectral Witt condition (REF ), the mapping $L(y,0)^2:W^{2,2}(_+, H) \\rightarrow W^{0,0}(_+, H),$ is bijective with bounded inverse.",
"Consider the following commutative diagram: [email protected]{W^{2,2} @<1ex>[rrrr]^{L^2(y,0)}@<1ex>@{-->}[dddd]^{X^{-2}} & & & &W^{0,0}@{=}[dddd] @{-->}[llll]^{K(y,0)} \\\\& & & & \\\\& & \\circlearrowright & & \\\\& & & & \\\\W^{2,0}[rrrr]^{\\tilde{L}^2} [uuuu]^{X^2}& & & & W^{0,0},@<1ex>@{-->}[llll]^{\\tilde{K} }}$ where $\\tilde{L}^2(y,0) = L^2(y,0)\\circ X^2$ and the inverse maps $K$ and $\\tilde{K}$ are constructed as follows.",
"Let $\\lbrace \\phi _{j}\\rbrace _{j\\in }$ be an orthonormal base of $H$ consisting of eigenvectors of $A^2(y)$ such that $A^2(y) \\ \\phi _{j} = \\nu _j^2 \\ \\phi _{j}$ , where by convention we assume $\\nu _j>0$ .",
"The geometric Witt condition (REF ) implies $\\nu _j > \\frac{3}{2}$ and by discreteness we conclude $\\exists \\, \\delta > 0 \\ \\forall \\, j \\in : \\ \\nu _j \\ge \\frac{3}{2}+ \\delta .$ For any $j\\in $ we define $E_j := \\langle \\phi _{j}\\rangle $ .",
"For any $g \\in L^2(_+)$ the equation $L^2(y,0) \\ f \\cdot \\phi _{j} = g \\cdot \\phi _j \\in L^2(_+, E_j)$ reduces to a scalar equation $- x^2 + \\frac{1}{x^2} \\nu _j^2-\\frac{1}{4}f = g. $ The fundamental solutions for that equation are $\\psi ^+_{\\nu _j} (x) = x^{\\nu _j + \\frac{1}{2}}, \\ \\text{and} \\ \\psi ^-_{\\nu _j} (x) = x^{-\\nu _j +\\frac{1}{2}}.$ In view of (REF ), neither of them lies in $W^{2,2}(_+)$ and hence $L^2(y,0)$ is injective on $W^{2,2}(_+,H)$ .",
"It remains to prove surjectivity.",
"The fundamental solutions $\\psi ^\\pm _{\\mu _j}$ yield a solution of the equation eq1 with $f(x) = \\int _{0}^{\\infty } k_j(x,y) g(y) dy =: K_j g,$ where $K_j$ is an integral operator with the integral kernel $k_j(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{2\\nu _j} \\frac{y}{x}^{\\nu _j} (xy)^{\\frac{1}{2}}, & y\\le x,\\\\\\frac{1}{2\\nu _j} \\frac{y}{x}^{-\\nu _j} (xy)^{\\frac{1}{2}}, & x\\le y.\\end{array} \\right.$ Accordingly, a solution of the scalar equation for any $\\tilde{g} \\in L^2(_+)$ $(L^2(y,0) \\circ X^2) \\ \\tilde{f} \\cdot \\phi _{j}= \\tilde{g} \\cdot \\phi _{j} \\in L^2(_+, E_j),$ is given in terms of $\\tilde{K}_j = X^{-2} \\circ K_j$ by $\\tilde{f} = \\tilde{K}_j \\tilde{g}$ .",
"The integral operators $\\tilde{K}_j$ have been studied in Proposition , which proves in view of (REF ) that for each $E_j$ the restriction $\\tilde{L}(y,0)|_{E_j}$ admits an inverse $\\tilde{K}_j:W^{0,0}(_+,E_j) \\rightarrow W^{2,0}(_+,E_j)$ with norm bounded uniformly in $j\\in $ .",
"Equivalently, the restriction $L(y,0)|_{E_j}$ admits an inverse $K_j : W^{0,0}(_+,E_j) \\rightarrow W^{2,2}(_+,E_j)$ with norm bounded uniformly in $j\\in $ .",
"By (REF ), the operator norms of $\\nu _j \\cdot (X\\partial _x) \\circ K_j$ and $\\nu _j^2 \\cdot K_j$ are bounded uniformly in $j$ as well.",
"Hence there exists a bounded inverse $(L^2(y,0))^{-1} : W^{0,0}(_+,H) \\rightarrow W^{2,2}(_+,H).$ This proves the statement.",
"Assume the spectral Witt condition (REF ).",
"Then for fixed parameters $(y,\\xi ) \\in ^b\\times ^b$ , the operator $L^2(y,\\xi ):W^{2,2}(_+, H)\\rightarrow W^{0,0,-2}(_+,H)$ is injective with a right-inverse $L^2(y,\\xi )^{-1}: W^{0,0}(_+, H)\\rightarrow W^{2,2}(_+,H)$ , bounded uniformly in the parameters $(y,\\xi )$ .",
"The case $\\xi = 0$ has been established in Proposition .",
"We proceed with the case $\\xi \\ne 0$ .",
"The commutator relations Eq1 imply that $A^2(y)$ and $c(\\xi ,y)^2$ may be simultaneously diagonalized and hence an orthonormal base $\\lbrace \\phi _{j}\\rbrace _{j \\in }$ of $H$ can be chosen such that $\\begin{aligned}&A^2(y) \\ \\phi _{j} = \\nu _j^2 \\phi _{j},\\; \\text{where wefix}\\;\\nu _j>0, \\\\&c(\\hat{\\xi }, y)^2 \\phi _{j} =\\phi _{j},\\;\\text{where}\\; \\hat{\\xi } =\\frac{\\xi }{\\Vert \\xi \\Vert }.\\end{aligned}$ We write $E_j = \\langle \\phi _{j}\\rangle $ .",
"Then, similar to Proposition , $L^2(y,\\xi )$ reduces over each $E_j$ to the scalar operator $L^2(y,\\xi )|_{E_j} = -x^2 + X^{-2} \\nu _j^2 - \\frac{1}{4}+\\Vert \\xi \\Vert ^2.$ The solutions to $L^2(y,\\xi )|_{E_j} \\phi = 0$ are given by linear combination of modified Bessel-functions $\\sqrt{x} I_{\\nu _j}(\\Vert \\xi \\Vert x)$ and $\\sqrt{x} K_{\\nu _j}(\\Vert \\xi \\Vert x)$ , which are not elements of $W^{2,2}(_+)$ for any $j\\in $ and any $\\xi \\ne 0$ .",
"This proves injectivity of $L^2(y,\\xi )$ on $W^{2,2}(_+,H)$ .",
"For the right-inverse we note the following commutative diagram [email protected]{W^{2,2} @<1ex>[rrrr]^{L^2(y,\\xi )}@<1ex>@{-->}[dddd]^{X^{-2}} & & & &W^{0,0,-2}@{=}[dddd] @{-->}[llll]^{K_1(y,\\xi )}\\\\& & & &\\\\& & \\circlearrowright & &\\\\& & & &\\\\W^{2,0}[rrrr]^{\\tilde{L}^2} [uuuu]^{X^2}& & & & W^{0,0,-2}@<1ex>@{-->}[llll]^{\\tilde{K}_1 }}.$ The equation $[ L^2(y,\\xi )\\circ X^2|_{E_j}]f \\cdot \\phi _j= g \\cdot \\phi _j\\in L^2(_+,E_j)$ admits a solution $\\begin{aligned}X^{-2}\\circ K_{j}(y,\\xi ) g &:= \\int _{_+}x^{-2}\\ k_{j}(x,\\tilde{x})\\ g(\\tilde{x})d\\tilde{x},\\end{aligned}$ where the kernel $k_{j}(x,\\tilde{x})$ is $k_{j}(x,\\tilde{x}) = \\left\\lbrace \\begin{array}{cc}(x\\tilde{x})^{\\frac{1}{2}} I_{\\nu _j} (\\Vert \\xi \\Vert \\ \\tilde{x})K_{\\nu _j}(\\Vert \\xi \\Vert \\ x),& \\tilde{x}\\le x,\\\\(x\\tilde{x})^{\\frac{1}{2}} I_{\\nu _j} (\\Vert \\xi \\Vert \\ x) K_{\\nu _j}(\\Vert \\xi \\Vert \\ \\tilde{x}),& x\\le \\tilde{x}.\\end{array}\\right.$ Therefore, by Proposition , $X^{-2}\\circ K_{j}$ is bounded, uniformly in $j\\in $ and $\\xi > 0$ .",
"Then, in view of the uniform bounds (REF ), $L^2(y,\\xi ) \\circ X^2$ admits a right-inverse $X^{-2} \\circ K(y,\\xi ): W^{0,0}(_+,H) \\rightarrow W^{2,0}(_+,H)$ .",
"Equivalently, $L^2(y,\\xi )$ admits a right-inverse $K(y,\\xi ): W^{0,0}(_+,H) \\rightarrow W^{2,2}(_+,H)$ , which proves the statement in view of continuity at $\\xi = 0$ .",
"Assume the spectral Witt condition (REF ).",
"Then $L(y,\\xi ):W^{1,1}(_+,H)\\rightarrow W^{0,0,-1}(_+,H),$ is injective with right-inverse $L(y,\\xi )^{-1}: W^{0,0}(_+, H)\\rightarrow W^{1,1}(_+,H)$ , bounded uniformly in $(y,\\xi )$ .",
"The commutator relations Eq1 imply that $S,\\Gamma $ and $ic(\\xi )$ may be simultaneously diagonalized and hence an orthonormal base $\\lbrace \\phi _{j,\\pm }\\rbrace $ of $H$ can be chosen such that $\\begin{aligned}S\\phi _{j,\\pm } &= \\pm \\mu _j \\phi _{j,\\pm },\\; \\text{where we fix}\\;\\mu _j>0, \\\\ic(\\hat{\\xi }, y) \\phi _{j,\\pm } &= \\pm \\phi _{j,\\pm },\\;\\text{where}\\; \\hat{\\xi } =\\frac{\\xi }{\\Vert \\xi \\Vert }\\\\\\Gamma \\phi _{j,\\pm } &= \\pm \\phi _{j,\\mp }.\\end{aligned}$ We define $E_j = \\langle \\phi _{j,+};\\phi _{j,-}\\rangle $ .",
"Then $L(y,\\xi )$ reduces over each $E_j$ to $L(y,\\xi )|_{E_j} = \\left(\\begin{array}{cc}0 & -I\\\\I & 0\\end{array}\\right)\\left[\\left(\\begin{array}{cc}\\partial _x & 0 \\\\0 & \\partial _x\\end{array}\\right)+ X^{-1} \\left(\\begin{array}{cc}\\mu _j & 0\\\\0 & -\\mu _j\\end{array}\\right)\\right] + \\left(\\begin{array}{cc}\\Vert \\xi \\Vert & 0 \\\\0 & -\\Vert \\xi \\Vert \\end{array}\\right).$ Like in [3], solutions to $L(y,\\xi )|_{E_j} \\phi = 0$ are given by linear combination of modified Bessel-functions, which are not elements of $W^{1,1}$ for any $j\\in $ and any $\\xi \\ne 0$ .",
"Same can be checked explicitly for $\\xi = 0$ .",
"This proves injectivity of $L(y,\\xi )$ .",
"The right-inverse is obtained by $L(y,\\xi )^{-1} := L(y,\\xi ) \\circ \\left(L^2(y,\\xi )\\right)^{-1}: L^2(_+,H) \\rightarrow W^{1,1} (_+,H),$ where the composition is well-defined for $\\xi = 0$ by Proposition , and for $\\xi \\ne 0$ by the fact that $\\left(L^2(y,\\xi )\\right)^{-1}$ maps $L^2(_+, H)$ to $W^{2,2} \\cap L^2(_+,H)$ , since $L(y,0) \\circ \\left(L^2(y,\\xi )\\right)^{-1} = \\textup {Id} - \\Vert \\xi \\Vert ^2 \\cdot \\left(L^2(y,\\xi )\\right)^{-1}.$ In the Corollary there is a certain overlap with the work of Albin and Gell-Redman [3], where in [3] they assert invertibility of $L(y,\\xi )$ for $\\xi \\ne 0$ , and do not prove uniform bounds for the inverse.",
"Here, we invert $L(y,\\xi )$ for all $\\xi \\in ^b$ and establish uniform bounds for the inverse.",
"Parametrices for generalized Dirac and Laplace operators We define subspaces of functions with compact support in $[0,1]$ $\\begin{split}&W^\\bullet _{\\rm comp}(_+, H):= \\lbrace \\phi u \\mid u \\in W^\\bullet (_+, H), \\phi \\in C^\\infty _0[0,\\infty ), \\phi \\subset [0,1]\\rbrace , \\\\&W^\\bullet _{\\rm comp}(_+\\times ^b, H):= \\lbrace \\phi u \\mid u \\in W^\\bullet (_+\\times ^b, H), \\phi \\in C^\\infty _0([0,\\infty ) \\times ^b), \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\phi \\subset [0,1] \\times ^b\\rbrace .\\end{split}$ Subspaces of weighted Sobolev scales, consisting of functions with compact support in $[0,1]$ and $[0,1] \\times ^b$ as above, are denoted analogously.",
"The Sobolev scales are defined in terms of the interpolation scales $H^s\\equiv H^s(S(y_0))$ , which a priori depend on the base point $y_0 \\in ^b$ .",
"This does not play a role here, since in the present section $y_0$ is fixed.",
"Assume the spectral Witt condition (REF ).",
"Then there exists $\\delta >0$ such that for any $u \\in W^0_{\\rm comp}(_+, H)$ and any $\\xi \\in ^b$ $\\Vert L(y_0,\\xi )^{-1} u(x) \\Vert _H = O(x^{-1-\\delta }), \\quad \\Vert L^2(y_0,\\xi )^{-1} u(x) \\Vert _H = O(x^{-1-\\delta }), \\quad x\\rightarrow \\infty .$ In particular, $L(y_0,\\xi )^{-1} u$ and $L^2(y_0,\\xi )^{-1} u$ are both in $L^2(_+,H)$ .",
"Here, $\\Vert \\cdot \\Vert _H$ denotes the norm of the Hilbert space $H$ .",
"Consider $u \\in W^0_{\\rm comp}(_+, H)$ .",
"Note first that $L(y_0,\\xi )^{-1} u \\in W^{1,1} (_+, H)$ and $L^2(y_0,\\xi )^{-1} u \\in W^{2,2} (_+, H)$ by Proposition and Corollary .",
"By the characterization (REF ) of Sobolev scales and the Sobolev embedding $H^1_e(_+) \\subset C(0,\\infty )$ into continuous functions, $L(y_0,\\xi )^{-1} u$ and $L^2(y_0,\\xi )^{-1} u$ are continuous on $(0,\\infty )$ with values in $H$ , and in that sense their pointwise evaluations are well-defined.",
"Recall $L^2(y_0,\\xi ) = -x^2 + X^{-2}\\left(A^2(y_0) - \\frac{1}{4}\\right) + c(\\xi ,y_0)^2.$ By the spectral Witt condition, $\\textup {Spec} \\, A(y_0) \\cap [0,\\frac{3}{2}] = \\varnothing $ and by discreteness of the spectrum there exists $\\delta >0$ such that $\\textup {Spec} \\, A(y_0) \\cap [0,\\frac{3}{2}+\\delta ) = \\varnothing .$ The integral kernel of $L^2(y_0,\\xi )^{-1}$ is given in terms of (REF ) for $\\xi \\ne 0$ and (REF ) for $\\xi = 0$ .",
"In view of (REF ), in both cases, the asymptotics $\\Vert L^2(y_0,\\xi )^{-1} u(x) \\Vert _H= O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ follows from Corollary .",
"The asymptotics of $\\Vert L(y_0,\\xi )^{-1} u(x)\\Vert _H $ now follows also by Corollary , once we observe that $L(y_0,\\xi )^{-1} u = L(y_0,\\xi ) (L^2(y_0,\\xi )^{-1} u) \\in W^{1,1} (_+, H)$ .",
"Consider $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ and denote its Fourier transform on $^b$ by $\\hat{u} (\\xi )$ .",
"Fix $y_0 \\in ^b$ and consider a generalized Dirac operator $D_{y_0}$ satisfying the spectral Witt condition (REF ).",
"We define $Qu(y):= \\int _{^b} e^{i \\langle y, \\xi \\rangle } L(y_0,\\xi )^{-1} \\hat{u} (\\xi )đ\\xi , \\quad đ\\xi := \\frac{d \\xi }{(2\\pi )^b}.$ Then $Q$ is a right-inverse to $D_{y_0}$ and defines a bounded operator $Q:W^{0}_{\\rm comp}(_+\\times ^b,H) \\subset W^{0} \\rightarrow X\\cdot W^{1}(_+\\times ^b,H) = W^{1,1}.$ By the Plancherel theorem we find for any $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ $\\Vert X^{-1} Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\Vert X^{-1} L(y_0,\\cdot )^{-1} \\hat{u} \\Vert ^2_{L^2(_+\\times ^b_\\xi , H)}\\\\~&= \\int _{^b} \\Vert X^{-1} L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi .$ By Corollary , the operator $X^{-1} L(y_0,\\xi )^{-1}$ defines a bounded map from $L^2(_+, H)$ to itself, with the operator norm bounded uniformly in $\\xi \\in ^b$ .",
"Denote its uniform bound by $C>0$ and compute again by Plancherel theorem $\\Vert X^{-1} Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert X^{-1} L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ & \\le C \\int _{^b} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi =C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Consequently, $Q: L^2(_+\\times ^b,H) \\rightarrow X\\cdot L^2(_+\\times ^b,H) = W^{0,1}$ is bounded.",
"Furthermore, by Corollary the operators $(X\\partial _x) \\circ X^{-1} L(y_0,\\xi )^{-1}$ and $S \\circ X^{-1} L(y_0,\\xi )^{-1}$ are bounded on $L^2(_+, H)$ and by the same argument as before $(X\\partial _x) \\circ Q$ and $S \\circ Q$ define bounded operators from $L^2$ to $W^{0,1}$ .",
"In order to prove the statement, it remains to establish boundedness of $(X \\partial _y) \\circ Q: L^2 \\rightarrow W^{0,1}$ .",
"For $u \\in L^2_{\\rm comp}(_+\\times ^b,H)$ with compact support in $[0,1] \\times ^b$ , its Fourier transform $\\hat{u} (\\xi )$ in the $^b$ component, is still an element of $L^2_{\\rm comp}(_+,H)$ with compact support in $[0,1]$ .",
"By Corollary there exists a preimage $v = L(y_0,\\xi )^{-1} \\hat{u} (\\xi )$ , and by Proposition its norm in $H$ is $O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"In particular, $v \\in L^2(_+,H)$ .",
"We compute using commutator relations Eq1, $\\begin{aligned}\\langle \\ L(y_0,\\xi ) v, &L(y_0,\\xi ) v \\ \\rangle _{L^2} =\\langle L(y_0,\\xi )^2 v ,v \\rangle _{L^2}\\\\&=\\langle (-\\partial _x^2 + X^{-2}(S(y_0)^2+S(y_0))) v , v \\rangle _{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&=\\Vert (\\partial _x+X^{-1}S(y_0)) v \\Vert ^2_{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&\\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2},\\end{aligned}$ where boundary terms at $x=0$ do not arise due to the weight $x$ in $W^{1,1} = X W^{1,0}$ .",
"Boundary terms at $x=\\infty $ do not arise since $\\Vert ~v(x) \\Vert _H =O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"We arrive at the following estimate $\\frac{\\Vert L(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2}}{\\Vert \\hat{u} (\\xi ) \\Vert _{L^2}} =\\frac{\\Vert L(y_0,\\xi )^{-1}L(y_0,\\xi ) v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}} =\\frac{\\Vert v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}} \\le \\Vert \\xi \\Vert ^{-1}.$ By continuity at $\\xi = 0$ we conclude for some constant $C>0$ $\\Vert L(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2} \\le C \\cdot (1+\\Vert \\xi \\Vert ) ^{-1} \\Vert \\hat{u} (\\xi )\\Vert _{L^2}.$ We may now estimate for any $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ $\\Vert X^{-1} (X \\partial _{y_i}) Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert \\xi _i \\cdot L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ &\\le C \\int _{^b} \\frac{\\Vert \\xi \\Vert ^2}{\\Vert 1+\\xi \\Vert ^2} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\~&\\le C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ This finishes the proof.",
"We point out that it is precisely the fact that we have established invertibility of $L(y_0,\\xi )$ for any $\\xi \\in ^b$ instead of $\\xi \\ne 0$ , which allows us to write down the parametrix $Q$ explicitly using Fourier transform and establish its mapping properties as a simple consequence of the Plancherel theorem.",
"In case of $L(y_0,\\xi )$ being invertible only for $\\xi \\ne 0$ the parametrix construction needs to take care of a singularity at $\\xi =0$ via cutoff functions, in which case one cannot deduce its mapping properties by a simple application of the Plancherel theorem and is forced to employ an operator valued version of the theorem by Calderon and Vaillancourt [13].",
"We conclude with construction of a parametrix for $D_{y_0}^2$ , cf.",
"Theorem .",
"Assume the spectral Witt condition (REF ).",
"Consider $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ and denote its Fourier transform on $^b$ by $\\hat{u} (\\xi )$ .",
"Fix $y_0 \\in ^b$ and consider the square $D^2_{y_0}$ of a generalized Dirac operator.",
"We define $Q^2u(y):= \\int _{^b} e^{i \\langle y, \\xi \\rangle } (L^2(y_0,\\xi ))^{-1} \\hat{u} (\\xi )đ\\xi , \\quad đ\\xi := \\frac{d \\xi }{(2\\pi )^b}.$ Then $Q^2$ is a right-inverse to $D^2_{y_0}$ and defines a bounded operator $Q^2:W^{0}_{\\rm comp}(_+\\times ^b,H) \\subset W^{0} \\rightarrow X^2\\cdot W^{2}(_+\\times ^b,H) = W^{2,2}.$ By the Plancherel theorem we find for any $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ $\\Vert X^{-2} \\circ Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\Vert X^{-2} \\circ (L^2(y_0,\\cdot ))^{-1} \\hat{u} \\Vert ^2_{L^2(_+\\times ^b_\\xi , H)}\\\\~&= \\int _{^b} \\Vert X^{-2} \\circ (L^2(y_0,\\xi ))^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi .$ By Proposition , the operator $X^{-2} (L^2(y_0,\\xi ))^{-1}$ defines a bounded map from $L^2(_+, H)$ to itself, with the operator norm bounded uniformly in $\\xi \\in ^b$ .",
"Denote its uniform bound by $C>0$ and compute again by Plancherel theorem $\\Vert X^{-2} \\circ Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert X^{-2} \\circ (L^2(y_0,\\xi ))^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ & \\le C \\int _{^b} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi =C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Consequently, $Q^2: L^2(_+\\times ^b,H) \\rightarrow X^2\\cdot L^2(_+\\times ^b,H) = W^{0,2}$ is bounded.",
"Furthermore, by Proposition we find for any $ V_1, V_2 \\in \\mathcal {K} := \\lbrace (X\\partial _x), S\\rbrace ,$ that the operators $V_1 \\circ X^{-2} \\circ (L^2(y_0,\\xi ))^{-1}$ and $V_1 \\circ V_2 \\circ X^{-2} \\circ (L^2(y_0,\\xi ))^{-1}$ are bounded on $L^2(_+, H)$ .",
"By the same argument as before, $V_1 \\circ Q^2$ and $V_2 \\circ V_2 \\circ Q^2$ define bounded operators from $L^2(_+\\times ^b,H)$ to $W^{0,2}(_+\\times ^b,H)$ .",
"In order to prove the statement, it remains to establish boundedness of $(X \\partial _y) \\circ V_1 \\circ Q^2$ and $(X \\partial _y)^2 \\circ Q^2$ as maps from $L^2(_+\\times ^b,H)$ to $W^{0,2}(_+\\times ^b,H)$ .",
"For $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ with compact support in $[0,1] \\times ^b$ , its Fourier transform $\\hat{u} (\\xi )$ in the $^b$ component, is still an element of $W^{0}_{\\rm comp}(_+,H)$ with compact support in $[0,1]$ .",
"By Proposition , $v = L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi ) \\in W^{2,2}(_+,H)$ .",
"By Proposition , $\\Vert v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"In particular, $v \\in L^2(_+,H)$ .",
"We compute using commutator relations Eq1, $\\begin{aligned}\\langle \\ L(y_0,\\xi ) v, \\, &L(y_0,\\xi ) v \\ \\rangle _{L^2} =\\langle L(y_0,\\xi )^2 v ,v \\rangle _{L^2}\\\\&=\\langle (-\\partial _x^2 + X^{-2}(S(y_0)^2+S(y_0))) v , v \\rangle _{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&=\\Vert (\\partial _x+X^{-1}S(y_0)) v \\Vert ^2_{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&\\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2},\\end{aligned}$ where there are no boundary terms after integration by parts.",
"More precisely, boundary terms at $x=0$ do not arise due to the weight $x^2$ in $W^{2,2} = X^2 W^{2,0}$ .",
"Boundary terms at $x=\\infty $ do not arise since $\\Vert v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ .",
"By Proposition , $L(y_0,\\xi ) v \\in W^{1,1}(_+,H)$ with the asymptotic expansion $\\Vert L(y_0,\\xi ) v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ as well.",
"Hence, in the estimates above, we can replace $v$ with $w = L(y_0,\\xi ) v$ and still conclude $\\langle \\ L(y_0,\\xi ) w, L(y_0,\\xi ) w \\ \\rangle _{L^2} \\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert w \\Vert ^2_{L^2}.$ We arrive at the following estimate $\\frac{\\Vert L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2}}{\\Vert \\hat{u} (\\xi ) \\Vert _{L^2}} =\\frac{\\Vert v \\Vert _{L^2}}{\\Vert L^2(y_0,\\xi ) v \\Vert _{L^2}} \\le \\Vert \\xi \\Vert ^{-1} \\frac{\\Vert v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}}\\le \\Vert \\xi \\Vert ^{-2}.$ By continuity at $\\xi = 0$ we conclude for some constant $C>0$ $\\Vert L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2} \\le C \\cdot (1+\\Vert \\xi \\Vert ) ^{-2} \\Vert \\hat{u} (\\xi )\\Vert _{L^2}.$ We may now estimate for any $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ $\\Vert X^{-2} (X \\partial _{y_i}) (X \\partial _{y_j}) Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert \\xi _i \\xi _j \\cdot L^2(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ &\\le C \\int _{^b} \\frac{\\Vert \\xi \\Vert ^2}{(1+\\Vert \\xi \\Vert )^2} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\~&\\le C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Similar estimate holds for $(X \\partial _y) V \\, Q^2 u$ with $V \\in \\mathcal {K}$ .",
"This finishes the proof.",
"Minimal domain of a Dirac Operator on an abstract edge We now employ the previous parametrix construction in order to deduce statements on the minimal and maximal domains of $D_{y_0}$ and consequently for $D$ .",
"Recall $H=H(S(y_0))$ and the basic definitions of minimal and maximal domains.",
"As noted in Remark REF , the interpolation scales $H^s(S(y))$ and $H^s(S(y_0))$ coincide for $0\\le s \\le 1$ .",
"The maximal and minimal domain of $D$ are defined as follows: $\\begin{aligned}(D_{\\max })&:=\\lbrace u\\in L^2(_+\\times ^b,H)\\ | \\ Du\\in L^2(_+\\times ^b,H) \\rbrace \\\\(D_{\\min })&:=\\lbrace u\\in (D_{\\max }) \\ | \\ \\exists (u_n)\\subset C_0^\\infty (_+\\times ^b,H^\\infty ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\text{with}\\; u_n \\stackrel{L^2}{\\rightarrow } u,\\;Du_n \\stackrel{L^2}{\\rightarrow } Du \\rbrace .\\end{aligned}$ Using smooth cutoff functions we define localized versions of domains: $\\begin{aligned}_{\\rm comp}(D_{\\max })&:=\\lbrace \\varphi u\\ | u\\in (D_{\\max }),\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b) \\rbrace ,\\\\_{\\rm comp}(D_{\\min })&:=\\lbrace \\varphi u\\ | u\\in (D_{\\min }),\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b)\\rbrace ,\\\\W^{s,\\delta }_{\\rm }(_+\\times ^b,H)&:=\\lbrace \\varphi u\\ | u\\in X^\\delta W^s,\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b)\\rbrace ,\\end{aligned}$ where in each case we additionally requireRestriction of the support to be in $[0,1] \\times ^b$ is necessary to achieve uniformity of the estimates in Corollary and for the consequence in Proposition to hold.",
"$\\varphi \\subset [0,1] \\times ^b$ .",
"One checks directly from the definitions $_{\\rm comp}(D_{\\max / \\min }) \\subseteq (D_{\\max /\\min }).$ The maximal and minimal domains $(D_{y_0, \\, \\max }), (D_{y_0, \\, \\min })$ and their respective localized versions $_{\\rm comp}(D_{y_0, \\, \\max }), _{\\rm comp}(D_{y_0, \\, \\min })$ are defined analogously.",
"$_{\\rm comp}(D_{y_0, \\, \\max / \\min }) \\subseteq W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ .",
"Since $_{\\rm comp}(D_{y_0, \\, \\min }) \\subseteq _{\\rm comp}(D_{y_0, \\, \\max })$ , it suffices to show $_{\\rm comp}(D_{y_0, \\, \\max }) \\subseteq W^{1,1}_{\\rm comp}(_+\\times ^b,H).$ Note that the differential expression $D_{y_0}$ induces two mappings $&D_{y_0} : (D_{y_0, \\, \\max }) \\rightarrow L^2(_+\\times ^b,H), \\\\&D_{y_0} : W^{1,1}(_+\\times ^b,H) \\rightarrow L^2(_+\\times ^b,H),$ where the former is an unbounded self-adjoint operator in the Hilbert space $L^2(_+\\times ^b,H)$ , and the latter is a bounded operator between Sobolev spacesNote that $W^{1,1}(_+\\times ^b,H) \\nsubseteq L^2(_+\\times ^b,H)$ .. Theorem provides the right inverse $Q:L^2_{\\rm comp}(_+\\times ^b,H)\\rightarrow W^{1,1}(_+\\times ^b,H)$ to the latter mapping, but not to the former.",
"More precisely, we only have $\\forall \\, u \\in L^2_{\\rm comp}(_+\\times ^b,H): \\quad D_{y_0} (Q u) = u.$ The same holds for the formal adjoints $D^t_{y_0}$ and $Q^t$ in $L^2(_+\\times ^b,H)$ $&D^t_{y_0} : W^{1,1} \\rightarrow L^2, \\quad Q^t:L^2_{\\rm comp}\\rightarrow W^{1,1}\\\\&\\forall \\, u \\in L^2_{\\rm comp}(_+\\times ^b,H): \\quad D^t_{y_0} (Q^t u) = u.$ Consider $u \\in _{\\rm comp}(D_{y_0, \\, \\max })$ and a test function $\\phi \\in C_0^\\infty (_+\\times ^b,H^\\infty )$ .",
"We fix a smooth cutoff function $\\psi \\in C_0^\\infty (_+\\times ^b,H^\\infty )$ , such that $\\psi \\equiv 1$ on $u \\cup \\phi $ .",
"We compute with $L^2=L^2(_+\\times ^b,H)$ $\\langle u, \\phi \\rangle _{L^2} &= \\langle u, \\psi D^t_{y_0} (Q^t \\phi ) \\rangle _{L^2}\\\\ &= \\langle u, [\\psi , D^t_{y_0}] (Q^t \\phi ) \\rangle _{L^2} + \\langle u, D^t_{y_0} \\psi (Q^t \\phi ) \\rangle _{L^2}$ Note that $\\, [\\psi , D^t_{y_0}]$ is by construction disjoint from $u$ and consequently the first summand above is zero.",
"Using $u \\in _{\\rm comp}(D_{y_0, \\, \\max })$ we can integrate by parts and conclude $\\langle u, \\phi \\rangle _{L^2} &= \\langle u, D^t_{y_0} \\psi (Q^t \\phi ) \\rangle _{L^2}\\\\ &= \\langle D_{y_0} u, \\psi (Q^t \\phi ) \\rangle _{L^2} = \\langle D_{y_0} u, Q^t \\phi \\rangle _{L^2}\\\\ &= \\langle Q D_{y_0} u, \\phi \\rangle _{L^2}.$ We conclude that $u = Q(D_{y_0}u)$ as distributions.",
"By Theorem $u = Q(D_{y_0}u)\\in W^{1,1}_{\\rm comp}(_+\\times ^b,H).$ $_{\\rm comp}(D_{y_0, \\, \\min }) = _{\\rm comp}(D_{y_0, \\, \\max }) = W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ .",
"By Lemma it suffices to show that $W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ is included in $_{\\rm comp}(D_{y_0, \\, \\min })$ .",
"Note that $D_{y_0}: W^{1,1}_{\\rm comp}(_+\\times ^b,H)\\rightarrow L^2(_+\\times ^b,H)$ is continuous, and $C_0^\\infty (_+\\times ^b,H^\\infty )\\subset W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ is dense.",
"Consider $u\\in W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ and some $(u_n)\\subset C^\\infty _0(_+\\times ^b,H^\\infty )$ such that $u_n \\stackrel{W^{1,1}}{\\rightarrow } u$ .",
"By continuity, $D_{y_0}u_n \\rightarrow D_{y_0} u$ in $L^2$ .",
"Hence by definition, $u\\in _{\\rm comp}(D_{y_0, \\, \\min })$ .",
"Now we want to extend this statement to a perturbation of $D_{y_0}$ $P = \\Gamma (\\partial _x + X^{-1}S(y)) + T + V:=D_{y_0} + D_{1,y_0}$ where $V:W^{1,1}(_+ \\times ^b, H) \\rightarrow W^{0,1}(_+ \\times ^b, H)$ is a bounded linear operator, preserving compact supports and usually referred to as a higher order term.",
"Assume in addition to the spectral Witt condition (REF ) that $\\partial _y S(y)(|S(y_0)|+1)^{-1}$ are bounded operators on $H$ for any $y, y_0\\in ^b$ .",
"Then $_{\\rm comp}(P_{\\min }) = _{\\rm comp}(P_{\\max }) =W^{1,1}_{\\rm comp}(_+ \\times ^b, H).$ Step 1: Consider $u\\in C_0^\\infty ((0,\\infty )\\times ^b, H^\\infty )$ and smooth cutoff functions $\\phi , \\psi \\in C_0^\\infty ([0,\\infty )\\times ^b)$ taking values in $[0,1]$ , such that $\\phi \\subset [0,\\epsilon )\\times B_\\epsilon (y_0)$ and $\\psi \\upharpoonright u \\equiv 1$ .",
"We compute using (REF ) $\\begin{aligned}\\Vert \\phi D_{1,y_0} u\\Vert _{L^2} &= \\Vert \\phi D_{1,y_0} Q D_{y_0} u\\Vert _{L^2}= \\Vert \\phi D_{1,y_0} Q \\psi D_{y_0} u\\Vert _{L^2}\\\\&\\le \\Vert \\phi D_{1,y_0} Q \\psi \\Vert _{L^2\\rightarrow L^2} \\cdot \\Vert D_{y_0}u\\Vert _{L^2}.\\end{aligned}$ In order to estimate the norm of $\\phi D_{1,y_0} Q \\psi $ , note that $\\begin{aligned}D_{1,y_0} Q & = \\Gamma \\, X^{-1} \\left(S(y)-S(y_0)\\right) Q + \\left(T-T_{y_0}\\right) Q + V Q\\\\ & = \\Gamma \\, X^{-1} (y-y_0) \\int _0^1 \\frac{\\partial S}{\\partial t}\\left(y_0 + t(y-y_0)\\right) dt\\ Q\\\\ & + (y-y_0) \\int _0^1 \\frac{\\partial T}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt \\ Q + VQ.\\end{aligned}$ In view of the assumption (REF ) and boundedness of the higher order term $V:W^{1,1} \\rightarrow W^{0,1}$ we conclude from Theorem that $X^{-1} \\frac{\\partial S}{\\partial t}\\left(y_0 + t(y-y_0)\\right) Q \\psi ,\\quad \\frac{\\partial T}{\\partial t}\\left(y_0 + t(y-y_0)\\right) \\circ Q \\psi ,\\quad X^{-1} V \\circ Q \\psi $ are bounded operators on $L^2(_+ \\times ^b, H)$ with bound uniform in $t\\in [0,1]$ and $\\psi $ .",
"Hence we conclude for some unform constant $C>0$ $\\begin{aligned}\\Vert \\phi D_{1,y_0} Q \\psi \\Vert _{L^2\\rightarrow L^2} \\le C \\left( \\sup _{q\\in \\phi } x(q)+ \\sup _{q\\in \\phi } \\Vert y(q) -y_0\\Vert \\right) \\le 2 \\epsilon C.\\end{aligned}$ Thus we may choose $\\epsilon >0$ sufficiently small such that $\\Vert \\phi D_{1,y_0} u\\Vert _{L^2} \\le q \\cdot \\Vert D_{y_0} u\\Vert _{L^2}, \\;\\;\\text{for}\\;\\; q<1.$ Then the following inequalities hold for $u\\in C^\\infty _0 (_+\\times ^b, H^\\infty )$ , $\\begin{aligned}\\Vert (D_{y_0} + \\phi D_{1,y_0})u\\Vert _{L^2} &\\le \\Vert D_{y_0} u\\Vert _{L^2} + q \\Vert D_{y_0}u\\Vert _{L^2}\\\\ &\\le (1+q)\\cdot \\Vert D_{y_0} u\\Vert _{L^2}.\\end{aligned}$ On the other hand $\\begin{aligned}\\Vert D_{y_0} u\\Vert _{L^2} &\\le \\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2} + \\Vert \\phi D_{1,y_0}u\\Vert _{L^2}\\\\&\\le \\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2} + q\\Vert D_{y_0} u\\Vert _{L^2}.",
"\\\\\\Rightarrow \\ \\Vert D_{y_0} u\\Vert _{L^2} &\\le (1-q) ^{-1}\\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2}.\\end{aligned}$ Thus the graph-norms of $D_{y_0}$ and $(D_{y_0}+\\phi D_{1,y_0})$ are equivalent and hence their minimal domains coincide.",
"Same statement holds for the maximal as well as the localized domains.",
"Thus we have the following equalities.",
"$\\begin{split}&_{\\rm comp}(D_{y_0, \\, \\min }) = _{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min }), \\\\&_{\\rm comp}(D_{y_0, \\, \\max }) = _{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\max }).\\end{split}$ The equalities continue to hold for a cutoff function $\\phi \\in C_0^\\infty ((0,\\infty )\\times ^b)$ such that for some $x_0 > \\epsilon $ , $\\phi \\subset (x_0 - \\epsilon , x_0 + \\epsilon ) \\times B_\\epsilon (y_0)$ by a similar argument.",
"Step 2: We now prove the following inclusion $_{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min })\\subseteq _{\\rm comp}((D_{y_0} + D_{1,y_0})_{\\min }).$ Indeed, for any $u\\in _{\\min }(D_{y_0} + \\phi D_{1,y_0})$ there exists $(u_n)\\subset C^\\infty _0((0,\\infty )\\times ^b, H^\\infty )$ converging to $u$ in the graph norm of $(D_{y_0} + \\phi D_{1,y_0})$ .",
"By (REF ) and Corollary , $(u_n)$ converges to $u$ in $W^{1,1}$ .",
"Hence, using continuity of $D_{1,y_0}: W^{1,1}\\rightarrow L^2$ we conclude $\\begin{aligned}(D_{y_0}+D_{1,y_0})u_n &= (D_{y_0} + \\phi D_{1,y_0})u_n + (1-\\phi )D_{1,y_0} u_n\\\\&\\stackrel{L^2}{\\rightarrow } (D_{y_0}+\\omega D_{1,y_0})u + (1-\\phi )D_{1,y_0} u = (D_{y_0} +D_{1,y_0})u.\\end{aligned}$ Hence $u\\in _{\\rm comp}((D_{y_0} + D_{1,y_0})_{\\min })$ and (REF ) follows.",
"Step 3: Consider now $u \\in _{\\rm comp}(P_{\\max / \\min })$ .",
"Due to compact support there exist finitely many points $\\lbrace (x_1,y_1), \\ldots , (x_N,y_N)\\rbrace \\subset _+ \\times ^b$ and smooth cutoff functions $\\lbrace \\psi _1, \\ldots , \\psi _N\\rbrace \\subset C_0^\\infty ([0,\\infty )\\times ^b)$ such that $u = \\sum _{j=1}^N \\psi _j u, \\quad (\\psi _j u) \\subset \\left(\\left(x_j - \\frac{\\epsilon }{2}, x_j + \\frac{\\epsilon }{2}\\right)\\cap [0, \\epsilon ) \\right)\\times B_{\\frac{\\epsilon }{2}}(y_j).$ The maximal and minimal domains are stable under multiplication with cutoff functions and hence each $\\psi _j u \\in _{\\rm comp}(P_{\\max / \\min })$ .",
"Consider for each $j=1, \\ldots , N$ a cutoff function $\\phi _j \\in C_0^\\infty ([0,\\infty )\\times ^b)$ such that $\\phi _j \\subset ((x_j - \\epsilon , x_j + \\epsilon )\\cap [0,\\epsilon )) \\times B_\\epsilon (y_j)$ and $\\phi _j \\upharpoonright (\\psi _j u) \\equiv 1$ .",
"Then as distributions $P (\\psi _j u) = (D_{y_j} + D_{1,y_j}) \\psi _j u = (D_{y_j} + \\phi _j D_{1,y_j}) \\psi _j u.$ We conclude $\\psi _j u \\in _{\\rm comp}((D_{y_j} + \\phi _j D_{1,y_j})_{\\max / \\min })$ .",
"In view of (REF ) and Corollary we find $_{\\rm comp}(P_{\\min }) \\subseteq _{\\rm comp}(P_{\\max }) \\subseteq W^{1,1}_{\\rm comp}(_+ \\times ^b, H).$ Step 4: The statement now follows from a sequence of inclusions $\\begin{aligned}W^{1,1}_{\\rm comp} &\\ = _{\\rm comp}(D_{y_0, \\, \\min }) \\\\~&\\stackrel{(\\ref {a})}{=}_{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min })\\stackrel{(\\ref {c})}{\\subseteq } _{\\rm comp}(P_{\\min })\\\\&\\ \\subseteq _{\\rm comp}(P_{\\max }) \\stackrel{(\\ref {d})}{=} W^{1,1}_{\\rm comp}.\\end{aligned}$ The first equality is due to Corollary .",
"Hence all inclusions are in fact equalities and the statement follows.",
"Minimal domain of a Laplace Operator on an abstract edge Definition extends to define the notion of minimal and maximal domain for the squares $D_{y_0}^2$ and $D^2$ of the generalized Dirac operators.",
"Their localized versions are defined as in (REF ).",
"In this section, we discuss the minimal and maximal domains of $D_{y_0}^2$ and $D^2$ by repeating the arguments of § with appropriate changes.",
"We also note as in Remark REF , the interpolation scales $H^s(S(y))$ and $H^s(S(y_0))$ coincide for $0\\le s \\le 1$ , but a priori may differ for $s>1$ .",
"While this was sufficient for the discussion of the domain of $D$ in the previous section, it is insufficient for the discussion of the domain of $D^2$ .",
"Hence, within the scope of this section we pose the following The interpolation scales $H^s(S(y))$ are independent of $y\\in ^b$ for $0 \\le s \\le 2$ , in which case we write $H^s\\equiv H^s(S(y))$ .",
"The following result follows by repeating the arguments of Lemma and Corollary ad verbatim, where $D_{y_0}$ is replaced by $D_{y_0}^2$ , $W^{1,1}$ by $W^{2,2}$ and $Q$ by $Q^2$ .",
"These changes do not affect the overall argument.",
"$_{\\rm comp}(D^2_{y_0, \\, \\min }) = _{\\rm comp}(D^2_{y_0, \\, \\max }) = W^{2,2}_{\\rm comp}(_+\\times ^b,H)$ .",
"Now we want to extend this statement to a perturbation of $D^2_{y_0}$ $G = -x^2 + X^{-2}\\ S(y) \\ (S(y)+1) + T^2 + W:=D^2_{y_0} + R_{y_0}$ where $W:W^{2,2}(_+ \\times ^b, H) \\rightarrow W^{0,1}(_+ \\times ^b, H)$ is a bounded linear operator, preserving compact supports, and is referred to as a higher order term.",
"Assume in addition to the spectral Witt condition (REF ) that $\\partial _y S(y) \\circ (|S(y_0)|+1)^{-1}, \\quad (|S(y_0)|+1) \\circ \\partial _y S(y) \\circ (|S(y_0)|+1)^{-2}$ are bounded operators on $H$ for any $y, y_0\\in ^b$ .",
"Then $_{\\rm comp}(G_{\\min }) = _{\\rm comp}(G_{\\max }) =W^{2,2}_{\\rm comp}(_+ \\times ^b, H).$ The assumption (REF ) translates into the condition that for $A=|S|+\\frac{1}{2}$ $\\partial _y A^2(y) \\circ (|S(y_0)|+1)^{-2}$ is bounded.",
"From there we proceed exactly as in Theorem .",
"Consider $u\\in C_0^\\infty ((0,\\infty )\\times ^b, H^\\infty )$ and smooth cutoff functions $\\phi , \\psi \\in C_0^\\infty ([0,\\infty )\\times ^b)$ taking values in $[0,1]$ , such that $\\phi \\subset [0,\\epsilon )\\times B_\\epsilon (y_0)$ and $\\psi \\upharpoonright u \\equiv 1$ .",
"We compute using the analogue of (REF ) for $D^2_{y_0}$ $\\begin{aligned}\\Vert \\phi R_{y_0} u\\Vert _{L^2} &= \\Vert \\phi R_{y_0} Q^2 D^2_{y_0} u\\Vert _{L^2}= \\Vert \\phi R_{y_0} Q^2 \\psi D^2_{y_0} u\\Vert _{L^2}\\\\&\\le \\Vert \\phi R_{y_0} Q^2 \\psi \\Vert _{L^2\\rightarrow L^2} \\cdot \\Vert D^2_{y_0}u\\Vert _{L^2}.\\end{aligned}$ In order to estimate the norm of $\\phi R_{y_0} Q^2 \\psi $ , note that $\\begin{aligned}R_{y_0} Q ^2 & = X^{-2} \\left(A^2(y)-A^2(y_0)\\right) Q^2+ \\left(T^2-T^2_{y_0}\\right) Q^2 + W Q^2\\\\ & = X^{-2} (y-y_0) \\int _0^1 \\frac{\\partial A^2}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt\\ Q^2\\\\ & + (y-y_0) \\int _0^1 \\frac{\\partial T^2}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt\\ Q^2 + WQ^2.\\end{aligned}$ In view of (REF ) and boundedness of the higher order term $W:W^{2,2} \\rightarrow W^{0,1}$ we conclude from Theorem that $X^{-2} \\frac{\\partial A^2}{\\partial t}\\left(y_0 + t(y-y_0)\\right) Q^2 \\psi ,\\quad \\frac{\\partial T^2}{\\partial t}\\left(y_0 + t(y-y_0)\\right) \\, Q^2 \\psi ,\\quad X^{-1} W \\, Q^2 \\psi $ are bounded operators on $L^2(_+ \\times ^b, H)$ with bound uniform in $t\\in [0,1]$ and $\\psi $ .",
"Hence we conclude for some unform constant $C>0$ $\\begin{aligned}\\Vert \\phi R_{y_0} Q^2 \\psi \\Vert _{L^2\\rightarrow L^2} \\le C \\left( \\sup _{q\\in \\phi } x(q)+ \\sup _{q\\in \\phi } \\Vert y(q) -y_0\\Vert \\right) \\le 2 \\epsilon C.\\end{aligned}$ Thus we may choose $\\epsilon >0$ sufficiently small such that $\\Vert \\phi R_{y_0} u\\Vert _{L^2} \\le q \\cdot \\Vert D^2_{y_0} u\\Vert _{L^2}, \\;\\;\\text{for}\\;\\; q<1.$ Then the following inequalities hold for $u\\in C^\\infty _0 (_+\\times ^b, H^\\infty )$ , $\\begin{aligned}\\Vert (D^2_{y_0} + \\phi R_{y_0})u\\Vert _{L^2} &\\le \\Vert D^2_{y_0} u\\Vert _{L^2} + q \\Vert D^2_{y_0}u\\Vert _{L^2}\\\\ &\\le (1+q)\\cdot \\Vert D^2_{y_0} u\\Vert _{L^2}.\\end{aligned}$ On the other hand $\\begin{aligned}\\Vert D^2_{y_0} u\\Vert _{L^2} &\\le \\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2} + \\Vert \\phi R_{y_0}u\\Vert _{L^2}\\\\&\\le \\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2} + q\\Vert D^2_{y_0} u\\Vert _{L^2}.",
"\\\\\\Rightarrow \\ \\Vert D^2_{y_0} u\\Vert _{L^2} &\\le (1-q) ^{-1}\\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2}.\\end{aligned}$ Thus the graph-norms of $D^2_{y_0}$ and $(D^2_{y_0}+\\phi R_{y_0})$ are equivalent and hence their minimal domains coincide.",
"Same statement holds for the maximal as well as the localized domains.",
"Thus we have the following equalities.",
"$\\begin{split}&_{\\rm comp}(D^2_{y_0, \\, \\min }) = _{\\rm comp}((D^2_{y_0} + \\phi R_{y_0})_{\\min }), \\\\&_{\\rm comp}(D^2_{y_0, \\, \\max }) = _{\\rm comp}((D^2_{y_0} + \\phi R_{y_0})_{\\max }).\\end{split}$ The equalities continue to hold for a cutoff function $\\phi \\in C_0^\\infty ((0,\\infty )\\times ^b)$ such that for some $x_0 > \\epsilon $ , $\\phi \\subset (x_0 - \\epsilon , x_0 + \\epsilon ) \\times B_\\epsilon (y_0)$ by a similar argument.",
"From there we may repeat the arguments of the proof of Theorem ad verbatim, replacing $D_{y_0}$ by $D^2_{y_0}$ , $D_{1,y_0}$ by $R_{y_0}$ , $P$ by $G$ , $W^{1,1}$ by $W^{2,2}$ .",
"These replacements do not affect the overall argument.",
"Domains of Dirac and Laplace Operators on a Stratified Space Consider a compact stratified space $M_k$ of depth $k\\in $ with an iterated cone-edge metric $g_k$ .",
"Each singular stratum $B$ of $M_k$ admits an open neighbourhood $\\subset M_k$ with local coordinates $y$ and a defining function $x_k$ such that $g|_{} = dx_k^2 + x_k^2 \\ g_{k-1}(x_k, y) + g_B(y) + h =: \\overline{g} + h,$ where $g_{k-1}(x_k,y)$ is a smooth family of iterated cone-edge metrics on a compact stratified space $M_{k-1}$ of lower depth and $h$ is a higher order symmetric 2-tensor, smooth on the resolution $\\widetilde{}$ with $|h|_{\\overline{g}} = O(x_k)$ as $x_k \\rightarrow 0$ .",
"The associated Sobolev-spaces are defined in Definition REF .",
"Recall, their elements take values in the vector bundle $E$ , which denotes the exterior algebra of the incomplete edge cotangent bundle $\\Lambda ^* {}^{ie}T^*$ in case of the Gauss–Bonnet operator, and the spinor bundle in case of the spin Dirac operator.",
"We usually omit $E$ from the notation.",
"We introduce here the localized versions of the Sobolev spaces ($s\\in $ ) $\\begin{aligned}^{s,\\delta }_{e,{\\rm comp}} :=\\lbrace \\phi \\cdot u \\ | \\ \\phi \\in C^\\infty _0(\\widetilde{}),u\\in ^{s,\\delta }_{e} \\rbrace .\\end{aligned}$ Consider the unitary transformation $\\Phi $ in (REF ), [10], which maps $L^2(,E,\\overline{g})$ to $L^2(, E, \\overline{g}_{\\textup {prod}})$ , where we recall $\\overline{g}$ from (REF ) and set $\\overline{g}_{\\textup {prod}}:= dx_k^2 + g_{k-1}(x_k, y) + g_B(y)$ .",
"The spaces $^{*,*}_{e,{\\rm comp}}$ with compact support in $$ may be defined with respect to $\\overline{g}$ and $\\overline{g}_{\\textup {prod}}$ .",
"We indicate the choice of the metric when necessary, $^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}}), L^2_{{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}})$ , and do not specify the metric when the statement holds for both choices.",
"Note $^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}}) =\\Phi ^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}).$ Whenever we use the Sobolev spaces $^{*,*}_{e}(M_k)$ or $L^2(M_k)$ without compact support in the open interior of $M_k$ , we use the iterated cone-edge metric $g_k$ in the definition of the $L^2$ -structure.",
"We write $L^2_{\\rm comp}:=^{0,0}_{e,{\\rm comp}}$ and denote by $\\rho _k$ a smooth function on the resolution $\\widetilde{M}_k$ , nowhere vanishing in its open interior, and vanishing to first order at each boundary face of $\\widetilde{M}_k$ .",
"Iteratively, $\\rho _k = x_k \\rho _{k-1}$ .",
"Then $\\begin{split}^{1,1}_{e,{\\rm comp}} &= \\rho _k \\ \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\rho _k \\partial _x u, \\rho _k \\partial _y u, \\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} \\ \\rbrace \\\\&= \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\frac{u}{\\rho _k}, \\partial _x u, \\partial _y u, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} \\ \\rbrace .\\end{split}$ Here, the first equality in (REF ) follows by Definition REF , once we recall from (REF ) the following iterative structure of edge vector fields $\\mathcal {V}_{e,k}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\rho _k \\partial _x, \\rho _k\\partial _y,\\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace .$ The second equality in (REF ) is now straightforward.",
"Similarly $\\begin{split}^{2,2}_{e,{\\rm comp}} &= \\rho ^2_k \\ \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\lbrace \\rho _k \\partial _x, \\rho _k \\partial _y, \\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace ^j \\, u \\in L^2_{\\rm comp}, \\ j=1,2 \\rbrace \\\\&= \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\lbrace \\rho _k^{-1}, \\partial _x, \\partial _y, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace ^j \\, u \\in L^2_{\\rm comp}, \\ j=1,2 \\rbrace .\\end{split}$ The spin Dirac and the Gauss–Bonnet operators $D_k$ on $(M_k,g_k)$ admit under a rescaling $\\Phi $ as in (REF ) the following form over the singular neighbourhood $\\subset M_k$ $\\Phi \\circ D_k \\circ \\Phi ^{-1}= \\Gamma (\\partial _{x_k} + X_k^{-1}S_{k-1}(y))+T + V,$ which satisfies the following iterative properties $S_{k-1}(y) = D_{k-1}(y)+R_{k-1}(y)$ , where $D_{k-1}(y)$ is a smooth family of differential operators (spin Dirac or the Gauss–Bonnet operators) on $(M_{k-1}, g_{k-1}(0,y))$ .",
"The operators $S_{k-1}(y),D_{k-1}(y)$ extend continuously to bounded maps $^{1,1}_e(M_{k-1}) \\rightarrow L^2(M_{k-1})$ .",
"Moreover, $R_{k-1}(y)$ extends continuously to a bounded operator on $L^2(M_{k-1})$ ; $x_k^{-1}V$ extends continuously to a map from $^{1,1}_{e, {\\rm comp}}$ to $L^2_{\\rm comp}$ ; $T$ is a Dirac Operator on $B$ .",
"Since at this stage essential self-adjointness of each $S_{k-1}(y)$ and discreteness of its self-adjoint extension is yet to be established, we reformulate the spectral Witt condition (REF ) in terms of quadratic forms.",
"Here, we employ the notions introduced in Kato [21].",
"We define for any smooth compactly supported $u \\in C^\\infty _0(M_{k-1})$ using the inner product of $L^2(M_{k-1}, g_{k-1}(0,y))$ $t(S_{k-1}(y))[u] := \\Vert S_{k-1}(y) u\\Vert ^2_{L^2}.$ This is the quadratic form associated to the symmetric differential operator $S_{k-1}(y)^2$ , densely defined with domain $C^\\infty _0(M_{k-1})$ in the Hilbert space $L^2(M_{k-1}, g_{k-1}(0,y))$ .",
"The numerical range of $t(S_{k-1}(y))$ is defined by $\\Theta (S_{k-1}(y)) := \\left\\lbrace t(S_{k-1}(y))[u] \\in \\mid u \\in C^\\infty _0(M_{k-1}),\\Vert u\\Vert ^2_{L^2} = 1\\right\\rbrace .$ We can now reformulate the spectral Witt condition, cf.",
"(REF ), as follows.",
"The operator $D_k$ on the stratified space $M_k$ satisfies the spectral Witt condition, if there exists $\\delta > 0$ such that in all depths $j \\le k$ the numerical ranges $\\Theta (S_{k-1}(y))$ are subsets of $[1+\\delta , \\infty )$ for any $y\\in B$ .",
"Assume that $S_{k-1}(y)$ with domain $C^\\infty _0(M_{k-1})$ in the Hilbert space $L^2(M_{k-1}, g_{k-1}(0,y))$ is essentially self-adjoint and its self adjoint realization is discrete.",
"Then $\\Theta (S_{k-1}(y)) \\subset [1 + \\delta , \\infty )$ for some $\\delta > 0$ if and only if $\\textup {Spec} S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ .",
"By Kato [21], $\\Theta (S_{k-1}(y))$ is a dense subset of $\\textup {Spec} \\, S_{k-1}(y)^2$ .",
"If the spectral Witt condition in the sense of Definition holds, this implies that $\\textup {Spec} \\, S_{k-1}(y)^2 \\subset [1 + \\delta , \\infty )$ for some $\\delta >0$ .",
"By discreteness this is equivalent to $\\textup {Spec} S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ .",
"Conversely, if $\\textup {Spec} \\, S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ , then by discreteness of the spectrum, $S_{k-1}(y)^2 > 4+\\delta $ for some $\\delta > 0$ .",
"The spectral Witt condition in the sense of Definition now follows, since by Kato [21], $\\Theta (S_{k-1}(y))$ is a dense subset of $\\textup {Spec} \\, S_{k-1}(y)^2$ .",
"We can now prove our main result.",
"Let $M_k$ be a compact stratified Witt space.",
"Let $D_k$ denote either the Gauss–Bonnet or the spin Dirac operator.",
"Assume that $D_k$ satisfies the spectral Witt conditionIn case of the Gauss–Bonnet operator on a stratified Witt space this can always be achieved by scaling the iterated cone-edge metric on fibers accordingly.. Then $_{\\max }(D_k) = _{\\min }(D_k) = ^{1,1}_e(M_k)$ .",
"We prove the result by induction on the following statement.",
"On any compact stratified space $M_j$ the operator $D_j$ satisfies the following conditions near each stratum $B$ : For $y\\in B$ , $S_{j-1}(y)$ admits a unique self-adjoint extension in $L^2(M_{j-1})$ with discrete spectrum and $S_{j-1}\\cap [-1, 1] = \\varnothing $ .",
"The unique self-adjoint domain of $S_{j-1}(y)$ is given by $^{1,1}_e(M_{j-1})$ .",
"The compositions $S_{j-1}(y) (|S_{j-1}(y_0)|+1)^{-1}$ and $\\partial _y S_{j-1}(y) (|S_{j-1}(y_0)|+1)^{-1}$ are bounded on $L^2(M_{j-1})$ for $y,y_0\\in B$ .",
"These assumptions are trivially satisfied if $j=1$ .",
"Assume that Assumption is satisfied for $j\\le k$ .",
"We need to prove that Assumption is then satisfied for $j \\le k+1$ .",
"Let $_{\\rm comp}(D_k)$ denote elements in the maximal domain of $D_k$ with compact support in $\\widetilde{}$ .",
"Then by Theorem , we conclude $\\begin{aligned}\\Phi _{\\rm comp}(D_k) &\\equiv _{\\rm comp}(\\Phi \\circ D_k \\circ \\Phi ^{-1})\\\\~&= W^{1,1}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1}))\\\\&= ^{1,1}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\cap ^{0,1}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1})\\\\&\\subseteq \\lbrace u \\in L^2_{\\rm comp} (M_k,\\overline{g}_{\\textup {prod}})\\ | \\ \\frac{u}{\\rho _k}, \\partial _x u, \\partial _y u, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1}) u\\\\ &\\in L^2_{\\rm comp} (M_k,\\overline{g}_{\\textup {prod}}) \\rbrace =^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}_{\\textup {prod}}) \\equiv \\Phi ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}),\\end{aligned}$ where we used (REF ) in the last line.",
"On the other hand it is straightforward to check that $\\begin{aligned}&\\Phi ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}) \\equiv ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}_{\\textup {prod}}) \\\\~&= \\rho _k \\ \\lbrace u \\in L^2_{\\rm comp}(M_k, \\overline{g}_{\\textup {prod}}) \\ | \\ \\rho _k \\partial _x u, \\rho _k \\partial _y u, \\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} (M_k, \\overline{g}_{\\textup {prod}}) \\rbrace \\\\&\\subseteq ^{1,1}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\cap ^{0,1}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1}) \\\\& = W^{1,1}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1})) = _{\\rm comp}(\\Phi \\circ D_k \\circ \\Phi ^{-1})\\equiv \\Phi _{\\rm comp}(D_k).\\end{aligned}$ We conclude $_{\\rm comp}(D_k) = ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g})$ and hence $(D_k) = ^{1,1}_{e}(M_k)$ .",
"Essential self-adjointness of $D_k$ implies essential self-adjointness of $S_k$ with the domain of both given by $^{1,1}_{e}(M_k)$ independently of parameters.",
"The domain $^{1,1}_{e}(M_k)$ embeds compactly into $L^2(M_k)$ and hence both $D_k$ and $S_k$ are discrete.",
"Since $S_k$ is discrete, the spectral Witt condition of Definition implies $S_{k} \\cap \\left[-1, 1\\right] = \\varnothing .$ The mapping properties of $(|S_{k}|+1)^{-1}$ are derived from the mapping properties of the model parametrix in Theorem in the usual way and hence $(|S_{k}|+1)^{-1}:L^2(M_k) \\rightarrow ^{1,1}_{e}(M_k)$ is bounded.",
"Since $S_k, \\partial _y S_k$ are bounded maps from $^{1,1}_{e}(M_k)$ to $L^2(M_k)$ by the iterative properties of the individual operators in (REF ), we conclude that Assumption is satisfied for $j \\le k+1$ and hence holds for all $j \\in $ .",
"Similar arguments apply for the Laplace operators.",
"Let $M_k$ be a compact stratified Witt space.",
"Let $D_k$ denote either the Gauss–Bonnet or the spin Dirac operator.",
"Assume that $D_k$ satisfies the spectral Witt condition.",
"Then $_{\\max }(D^2_k) = _{\\min }(D^2_k) = ^{2,2}_e(M_k)$ .",
"We prove the result by induction.",
"The statement is trivially satisfied if $k=0$ .",
"Assume that the statement holds for $(k-1)\\in _0$ .",
"In particular, by induction hypothesis and by Theorem $\\begin{split}&H^1(S_{k-1})\\equiv (S_{k-1}) = ^{1,1}_e(M_{k-1}), \\\\&H^2(S_{k-1}) \\equiv (S^2_{k-1}) = ^{2,2}_e(M_{k-1}).\\end{split}$ Since the domains $(S^2_{k-1}(y))$ are independent of $y$ by the induction hypothesis, their interpolation scales $H^s(S_{k-1}(y))$ coincide for $0\\le s \\le 2$ and the Assumption is satisfied.",
"The spectral Witt condition is satisfied in each depth by Theorem .",
"We need to prove the statement for $k$ .",
"Let $_{\\rm comp}(D^2_k)$ denote elements in the maximal domain of $D^2_k$ with compact support in $\\widetilde{}$ .",
"Then by Theorem and (REF ) we conclude $\\begin{aligned}\\Phi _{\\rm comp}(D^2_k) &\\equiv _{\\rm comp}(\\Phi \\circ D^2_k \\circ \\Phi ^{-1})\\\\&= W^{2,2}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1}))\\\\&= ^{2,2}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\\\~&\\cap ^{0,2}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1}) \\\\~&\\cap ^{0,2}_{e,{\\rm comp}}(_+\\times ^b)^{2,2}_e(M_{k-1}) \\\\ &= ^{2,2}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}})\\equiv \\Phi ^{2,2}_{e,{\\rm comp}}(M_k,\\overline{g}).\\end{aligned}$ where we used (REF ) in the last equality.",
"The statement follows.",
"We conclude the section with pointing out that while we cannot geometrically control the spectral Witt condition in case of the spin Dirac operator, for the Gauss–Bonnet operator on a stratified Witt space, we find $0 \\notin S_{k}$ in each iteration step, and can scale the spectral gap up by a simple rescaling of the metric to achieve the spectral Witt condition.",
"appendix Notation In this section matrices $(a_{ij})_{1\\le i,j\\le n}$ will often be abbreviated $(a_{ij})_{ij}$ as long as the size $n$ is clear from the context.",
"Summations $\\sum _{i,j,k,\\ldots }$ will always denote a finite sum where all summation indices run independently from 1 to $n$ .",
"Positivity of Matrices of Operators on Hilbert spaces The following result is based on Lance [24].",
"Let $a=(a_{ij})_{1\\le i, j \\le n}$ , $b=(b_{ij})_{1\\le i,j \\le n}$ be matrices of operators on Hilbert spaces $H_1$ , $H_2$ , respectively.",
"I.e., $a_{ij}\\in (H_1)$ , $b_{ij}\\in (H_2)$ .",
"We may view $a$ as an element of $_n((H_1))$ or of $(H_1^n)$ .",
"Assume that $a\\ge 0$ and $b\\ge 0$ .",
"Then the following holds.",
"$(a_{ij}\\otimes b_{ij})_{ij}\\ge 0$ in $((H_1H_2)^n) = _n((H_1H_2))$ ; $\\sum _{i,j} a_{ij} \\otimes b_{ij} \\ge 0$ in $(H_1H_2)$ .",
"If $a\\le c=(c_{ij})_{ij} \\in (H_1^n)$ , $b\\le d=(d_{ij})_{ij}\\in (H_2^n)$ then $(a_{ij} \\otimes b_{ij})_{ij} \\le (c_{ij}\\otimes d_{ij})_{ij}.$ Note that for $H_1 = H_2 = this is an elementary statement about positivesemi-definite matrices.$ (1) Write $a=s^{\\ast } s$ , $s=(s_{ij})$ , $b=t^{\\ast } t$ , $t=(t_{ij})$ .",
"Thus $a_{ij} = \\sum _k s^{\\ast }_{ki}s_{kj}$ , $b_{ij} = \\sum _k t^{\\ast }_{ki} t_{kj}$ , and $a_{ij} \\otimes b_{ij}= \\sum _{k,l} s^{\\ast }_{ki}s_{kj} \\otimes t_{li}^{\\ast } t_{lj}= \\sum _{k,l} (s_{ki}\\otimes t_{li})^{\\ast }(s_{kj}\\otimes t_{lj}).$ So it suffices to prove that the matrices $\\bigl \\lbrace (s_{ki}\\otimes t_{li})^{\\ast } (s_{kj}\\otimes t_{lj})\\bigr \\rbrace _{ij} \\ge 0.$ For fixed $k,l$ let $T_{i}:=s_{ki}\\otimes t_{li}$ .",
"Then for $\\xi =(\\xi _i)_{1\\le i \\le n} \\in (H_1H_2)^n$ we have $\\begin{split}\\langle (T_i^\\ast T_j)_{ij} \\xi , \\xi \\rangle &= \\Big \\langle \\sum _k T_i^\\ast T_k \\xi _k_i, \\xi \\Bigr \\rangle = \\sum _{i,k} \\langle T_i^\\ast T_k \\xi _k , \\xi _i \\rangle \\\\&= \\sum _{i,k} \\langle T_k \\xi _k , T_i \\xi _i \\rangle = \\Vert (T_i \\xi _i)_i \\Vert ^2 \\ge 0.\\end{split}$ So indeed the matrix $(T_i^\\ast T_j)_{ij}$ is $\\ge 0$ .",
"(2) It suffices to show that if $(f_{ij})_{ij} := (a_{ij}\\otimes b_{ij})_{ij} \\ge 0$ then $\\sum _{i,j} f_{ij} \\ge 0$ .",
"Given $x\\in H$ put $y_i = x$ , $y = (y_i)_{1\\le i \\le n}\\in H^n$ .",
"Then $\\begin{split}0\\le \\langle (a_{ij})\\cdot (y_i), (y_i) \\rangle &=\\sum _i\\Bigl \\langle \\sum _j a_{ij}y_j,y_i\\Bigr \\rangle \\\\&=\\Bigl \\langle \\sum _{i,j} a_{ij} x, x\\Bigr \\rangle = \\Bigl \\langle \\sum _{i,j} a_{ij}x,x\\Bigr \\rangle .\\end{split}$ (3) From $c-a\\ge 0$ and $d-b\\ge 0$ and (1) we infer that the matrices $(c_{ij}-a_{ij})\\otimes b_{ij} $ and $c_{ij}\\otimes ( d_{ij}-b_{ij}) $ are $\\ge 0$ and hence $0 \\le (c_{ij}-a_{ij})\\otimes b_{ij} +c_{ij}\\otimes ( d_{ij}-b_{ij}) = ( c_{ij}\\otimes d_{ij} ) - ( a_{ij} \\otimes b_{ij}).$ Let $A,B$ be self-adjoint operators in Hilbert spaces $H_1$ , $H_2$ , respectively, and let $AB:= \\text{ closure of } A\\otimes _B \\text{ on }^{\\infty }(A)\\otimes _^{\\infty } (B),$ where $^{\\infty }(A):= \\bigcap _{s\\ge 0} (|A|^{s})$ .",
"Then $AB$ is self-adjoint and $^{\\infty }(A)\\otimes _^{\\infty } (B)= ^{\\infty }(A) \\otimes _H_2 \\cap H_1 \\otimes _^{\\infty }(B)$ .",
"It is straightforward to see that $A\\otimes B$ is symmetric on $^{\\infty }(A)\\otimes _^{\\infty } (B)$ and hence $AB$ is a symmetric closed operator.",
"It remains to show self-adjointness which is equivalent to the denseness of the ranges $(AB\\pm i I)$ .",
"First we prove the statement for $B = I$ being the identity on $H_2$ .",
"Then the graph norm of $A\\otimes I$ on $^{\\infty }(A)\\otimes H_2$ is the Hilbert space tensor norm for $(A)H_2$ .",
"Hence $(AI) = (A)H_2$ .",
"The resolvent of $AI$ is obviously $(AI - \\lambda I I)^{-1} = (A-\\lambda I )^{-1}I$ .",
"Thus the denseness of $(AI \\pm II)$ follows from the denseness of $(A\\pm I)$ .",
"Hence $AI$ is self-adjoint.",
"For general $B$ we now know that $AI$ and $IB$ are commuting self-adjoint operators.",
"Hence $(AI)\\cdot (I B)$ is essentially self-adjoint on $^{\\infty }(A) \\otimes _H_2 \\bigcap H_1 \\otimes _^{\\infty }(B).$ It remains to see that the latter equals $^{\\infty }(A) \\otimes _^{\\infty }(B)$ .",
"Because then $(A\\otimes I)\\cdot (I\\otimes B) = A\\otimes _B$ and we conclude the essential self-adjointness of $A\\otimes _B$ .",
"To this end consider $\\xi \\in ^{\\infty }(A) \\otimes _H_2 \\bigcap H_1 \\otimes _^{\\infty }(B)$ .",
"Then there exist $x_i\\in ^{\\infty }(A)$ , $y_i\\in H_2$ , $\\tilde{x}_i\\in H_1$ , $\\tilde{y}_i\\in ^{\\infty }(B)$ , $i=1,\\ldots n$ such that $\\sum _i x_i \\otimes y_i = \\xi = \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i,$ where without loss of generality we may assume that $\\tilde{y}_i$ is orthonormal in $^{\\infty }(B)$ .",
"There is an obvious pairing $H_1 \\otimes _H_2 \\times H_2 \\rightarrow H_1,$ induced by the $H_2$ scalar product.",
"Pick an index $j$ .",
"Then on the one hand $\\Bigl \\langle \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i, (I+B^2)\\tilde{y}_j\\Bigr \\rangle = \\sum _i \\tilde{x}_i \\langle \\tilde{y}_i, \\tilde{y}_j\\rangle _B = \\tilde{x}_j,$ and on the other hand $\\begin{split}\\Bigl \\langle \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i, (I+B^2)\\tilde{y}_j \\Bigr \\rangle &= \\Bigl \\langle \\sum _i x_i \\otimes y_i , (I+B^2) \\tilde{y}_j\\Bigr \\rangle \\\\&= \\sum _i x_i\\, \\langle y_i, (I+B^2) \\tilde{y}_j\\rangle \\in ^\\infty (A).\\end{split}$ This proves $\\tilde{x}_j\\in ^{\\infty }(A)$ for any $j=1,\\ldots ,n$ and the statement follows.",
"Let $A,C\\ge 0$ be self-adjoint operators in $H_1$ ; $B,D\\ge 0$ self-adjoint operators in $H_2$ .",
"If $A\\le C$ , $(C)\\subset (A)$ and $B\\le D$ , $(D)\\subset (B)$ then $AB \\le CD,\\quad (CD) \\subset (AB).$ The domain inclusion is clear from Proposition REF .",
"To prove the inequality, let $\\sum _{i=1}^n x_i \\otimes y_i\\in ^{\\infty }(C)\\otimes _^{\\infty }(D)$ be given.",
"Consider the matrices $(\\langle A x_i, x_j\\rangle )_{ij}$ , $(\\langle C x_i, x_j\\rangle )_{ij}$ , $(\\langle B y_i, y_j\\rangle )_{ij}$ , and $(\\langle D y_i, y_j\\rangle )_{ij}$ .",
"For complex numbers $\\lambda _i$ we have $\\begin{split}\\sum \\overline{\\lambda }_i \\langle A x_i, x_j\\rangle \\lambda _j&= \\langle A \\sum \\lambda x_i , \\lambda _i x_i\\rangle \\ge 0\\\\\\langle A \\sum \\lambda x_i , \\lambda _i x_i\\rangle &\\le \\langle C \\sum \\lambda x_i , \\lambda _i x_i\\rangle =\\sum \\overline{\\lambda }_i \\langle C x_i, x_j\\rangle \\lambda _j.\\end{split}$ Thus we have the matrix inequalities $0 \\le \\left(\\langle A x_i, x_j\\rangle \\right)_{ij} \\le \\left(\\langle C x_i, x_j\\rangle \\right)_{ij}$ and analogously $0 \\le \\left(\\langle B y_i, y_j\\rangle \\right)_{ij} \\le \\left(\\langle D y_i, y_j\\rangle \\right)_{ij}.$ Proposition REF implies $\\begin{split}0 &\\le \\sum _{i,j} \\langle (C-A) x_i, x_j\\rangle \\langle D y_i,y_j\\rangle + \\sum _{i,j} \\langle A x_i, x_j\\rangle \\langle (D-B) y_i,y_j\\rangle \\\\&= \\sum _{i,j} \\langle C x_i, x_j\\rangle \\langle D y_i, y_j\\rangle -\\langle A x_i, x_j\\rangle \\langle B y_i, y_j\\rangle \\\\&= \\Bigl \\langle (C\\otimes D) \\sum x_i \\otimes y_i, \\sum x_i \\otimes y_i\\Bigr \\rangle - \\Bigl \\langle (A\\otimes B) \\sum x_i \\otimes y_i,\\sum x_i \\otimes y_i\\Bigr \\rangle ,\\end{split}$ and hence $AB \\le CD$ .",
"Uniform asymptotic expansions of modified Bessel functions According to Olver [31], we may write for any $\\mu > 0$ and $x>0$ $\\begin{split}I_\\mu (\\mu x) &= \\frac{1}{\\sqrt{2\\pi \\mu }}\\cdot \\frac{e^{\\mu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}\\left(\\sum _{j=0}^{n-1} \\frac{U_j(p(x))}{\\mu ^j} + \\eta _{n,1}(\\mu , x)\\right)\\frac{1}{1+\\eta _{n,1}(\\mu , \\infty )}, \\\\K_\\mu (\\mu x) &= \\sqrt{\\frac{2\\pi }{\\mu }}\\cdot \\frac{e^{-\\mu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}\\left(\\sum _{j=0}^{n-1} (-1)^j \\frac{U_j(p(x))}{\\mu ^j} + \\eta _{n,2}(\\mu ,x)\\right)\\end{split}$ where $p(x)= \\sqrt{1+x^2}$ , $\\eta (x) = p(x) +\\ln \\frac{x}{1+p(x)}$ and $U_j(p)$ are iteratively defined polynomials in $p$ with $U_0 \\equiv 1$ .",
"By Olver [31], the error terms $\\eta _{n,1}$ and $\\eta _{n,2}$ admit the following bounds $\\begin{split}|\\eta _{n,1}(\\mu , x)| &\\le 2 \\exp \\left(\\frac{2{V}_{(1,p(x))}(U_1)}{\\mu }\\right)\\frac{{V}_{(1,p(x))}(U_n)}{\\mu ^n}, \\\\~|\\eta _{n,2}(\\mu , x)| &\\le 2 \\exp \\left(\\frac{2{V}_{(0,p(x))}(U_1)}{\\mu }\\right)\\frac{{V}_{(0,p(x))}(U_n)}{\\mu ^n}\\end{split}$ where ${V}_{(a,b)}(f)$ denotes the total variation of a differentiable function $f$ along an interval $(a,b)$ .",
"In case of complex-valued arguments $x$ , one takes here the variation along $\\eta (x)$ -progressive paths.",
"However, here $x, p(x), \\eta (x)$ are all real-valued, and $\\eta (x)$ is monotonously increasing as $x\\rightarrow \\infty $ by (REF ).",
"Since $p((0,\\infty )) = (0,1)$ , we may take in (REF ) variation over $(0,1)$ for both error terms.",
"Since for any $j\\in $ the total variations ${V}_{(0,1)}(U_j)$ are taken along finite paths and since $U_j$ are polynomials, we conclude that for any $n\\in _0$ $\\eta _{n,1}(\\mu , x) = O(\\mu ^{-n}), \\quad \\eta _{n,2}(\\mu , x) = O(\\mu ^{-n}),\\ \\textup {as} \\ \\mu \\rightarrow \\infty .$ uniformly in $x\\in (0,\\infty )$ .",
"Hence the expansions (REF ) are uniform in $x\\in (0,\\infty )$ as well."
],
[
"Generalized Dirac operators on an abstract edge",
"Let $S$ be a smooth family of self-adjoint operators in a Hilbert space $H$ with parameter $y\\in ^b$ and a fixed domain $_S$ .",
"We assume that each $S(y)$ is discrete.",
"A generalized Dirac Operator $D$ acting on $C_0^\\infty (_+\\times ^b,H^\\infty )$ is defined by the following (differential) expression $D:=\\Gamma (\\partial _x + X^{-1} S)+ T,$ where $x\\in _+$ , $X$ denotes the multiplication operator by $X$ , $\\Gamma $ is skew-adjoint and a unitary operator on the Hilbert space $L^2(_+\\times ^b,H)$ , and $T$ is a symmetric generalized Dirac Operator on $^b$ , given in terms of coordinates $(y_1,\\ldots ,y_b)\\in ^b$ and smooth families $(c_1(y), c_b(y))$ of bounded linear operators on $H$ , which satisfy Clifford relations for each fixed $y\\in ^b$ , by $T= \\sum _{j=1}^{b}c_j(y) \\frac{\\partial }{\\partial y_j}.$ Here, we have hid the vector bundle value action of the Dirac Operator $T$ into the Hilbert space $H$ .",
"We assume that the following standard commutator relations hold $\\begin{aligned}\\Gamma \\ S + S \\ \\Gamma &= 0,\\\\\\Gamma \\ T + T \\ \\Gamma &= 0,\\\\T \\ S - S \\ T &=0.\\end{aligned}$ In §REF we show that the Gauss–Bonnet operator on a simple edge satisfies these relations, cf.",
"(REF ).",
"The same relations hold for the spin Dirac operator, as shown in [3].",
"We shall also consider $D$ with coefficients frozen at some $y_0 \\in ^b$ $D_{y_0}:=\\Gamma (\\partial _x + X^{-1} S(y_0))+ T_{y_0}, \\quad \\textup {where} \\ T_{y_0}= \\sum _{j=1}^{b}c_j(y_0) \\frac{\\partial }{\\partial y_j}.$ Consider the Fourier transform $\\mathfrak {F}_{y\\rightarrow \\xi }$ on the $L^2(^b)$ -component of $L^2(_+\\times ^b,H)$ .",
"We use Hörmander's normalization and write $\\left(\\mathfrak {F}_{y\\rightarrow \\xi } f\\right)(\\xi ) =\\int _{^b} e^{-i \\langle y, \\xi \\rangle } f(y) dy,\\quad \\left(\\mathfrak {F}^{-1}_{y\\rightarrow \\xi } g\\right)(y) =\\int _{^b} e^{i \\langle y, \\xi \\rangle } g(\\xi ) \\frac{d\\xi }{(2\\pi )^b}.$ We compute $\\mathfrak {F}_{y\\rightarrow \\xi } \\circ D_{y_0} \\circ \\mathfrak {F}_{y\\rightarrow \\xi }^{-1} = \\Gamma (\\partial _x + X^{-1}S(y_0)) + ic(\\xi ; y_0) =: L(y_0,\\xi ),$ where $c(\\xi ; y_0) := \\sum _{j=1}^b c_j(y_0) \\xi _j.$ The usual strategy is now to study invertibility of $L(y_0,\\xi )$ on appropriate spaces, which is then used to construct the parametrix for $D$ and analysis of its domain."
],
[
"The spectral Witt condition",
"We also impose a spectral Witt condition, which asserts that $\\forall \\, y \\in ^b: \\ S(y) \\cap \\left[-1, 1\\right] = \\varnothing ,$ We should point out that Albin and Gell-Redman [3] require a smaller spectral gap $S(y) \\cap -1/2, 1/2= \\varnothing $ .",
"However, when proving an analogue of the crucial [3] by explicit computations, it seems that a smaller spectral gap may not be sufficient for our purposes.",
"In any case, if $D$ is the Gauss–Bonnet operator on a stratified Witt space, one can always achieve $S(y) \\cap -R, R= \\varnothing $ for any $R>0$ by a simple rescaling of the metric."
],
[
"Squares of generalized Dirac operators",
"In view of the commutator relations (REF ), the generalized Laplace operators $D^2$ and $D_{y_0}^2$ , acting both on $C_0^\\infty (_+\\times ^b,H^\\infty )$ , are of the following form $\\begin{split}&D^2 = -x^2 + X^{-2}\\ S \\ (S+1) + T^2, \\\\&D^2_{y_0} = -x^2 + X^{-2} \\ S(y_0) \\ (S(y_0) + 1) + T^2_{y_0}.\\end{split}$ We set $A := \\left|S\\right| +\\frac{1}{2}$ .",
"Assuming $S \\cap \\left[-1, 1\\right] = \\varnothing $ , we find $S(S+1) = A^2 -1/4$ and rewrite the generalized Laplacians $D^2$ and $D_{y_0}^2$ as follows $\\begin{split}&D^2 = -x^2 + X^{-2} A^2 - \\frac{1}{4} + T^2, \\\\&D^2_{y_0} = -x^2 + X^{-2}A^2(y_0) - \\frac{1}{4} + T^2_{y_0}.\\end{split}$ As before, we may apply the Fourier transform $\\mathfrak {F}_{y\\rightarrow \\xi }$ and compute $\\begin{split}\\mathfrak {F}_{y\\rightarrow \\xi } \\circ D^2_{y_0} \\circ \\mathfrak {F}_{y\\rightarrow \\xi }^{-1} &= -x^2 + X^{-2}\\left(A^2(y_0) - \\frac{1}{4}\\right) + c(\\xi ,y_0)^2 \\\\~&=: L^2(y_0,\\xi ), \\ \\textup {where} \\ c(\\xi ; y_0)^2 =- \\sum _{j,k=1}^b c_j(y_0) c_k(y_0)\\xi _j \\xi _k.\\end{split}$"
],
[
"Sobolev-spaces of an abstract edge",
"Recall the definition of interpolation scales of Hilbert spaces in §.",
"This defines for each $y_0\\in ^b$ an interpolation scale $H^s(S(y_0))$ , $s\\in $ .",
"We can now define the Sobolev-scales on the model cone and the model edge in our abstract setting.",
"Consider for this the Sobolev-scale $H^\\bullet _e(_+)$ generated byThe edge Sobolev scale $H^\\bullet _e(_+)$ prescribes regularity under differentiation by $x\\partial _x$ .",
"However, $x\\partial _x$ is not a symmetric operator and hence we take its symmetrization $(i x\\partial _x + i/ 2)$ as the generator of the Sobolev scale.",
"Alternatively we can replace the definition of Sobolev scales to allow for closed not necessarily symmetric operators.",
"$(i x\\partial _x + i/ 2)$ ; and the Sobolev-scale There is an arrow in the note that I do not understand.",
"$H_e^\\bullet (_+\\times ^b)$ generated by $= (i x\\partial _x + i/ 2)I+Ix T_{y_0}$ .",
"The lower index $e$ indicates that these interpolation scales coincide with the edge Sobolev spaces for integer orders.",
"Let $y_0 \\in ^b$ be fixed.",
"The Sobolev-scale $W^\\bullet (_+,H)$ of an abstract model cone is defined as an interpolation scale with generator $(i x\\partial _x + i/ 2)I+IS(y_0)$ .",
"By Proposition REF $W^s(_+,H):=(H^s_e(^+)H) \\cap (L^2(_+)H^s(S(y_0))).$ The Sobolev-scale $W^\\bullet (_+\\times ^b, H)$ of an abstract model edge is defined as an interpolation scale with generator $I + I S(y_0)$ , where $$ is the generator of the Sobolev-scale $H^s_e(_+\\times ^b)$ .",
"By Proposition REF $W^s(_+\\times ^b,H):=(H^s_e(_+\\times ^b)H) \\cap (L^2(_+\\times ^b)H^s(S(y_0))).$ In view of Proposition REF , for $y,y_0\\in ^b$ , the interpolation scales of $S(y)$ and $S(y_0)$ need not coincide.",
"However, since for any $y\\in ^b$ , the domain of $S(y)$ is fixed and given by $_S$ , we have $H^s(S(y))= H^s(S(y_0))$ for $0\\le s\\le 1$ .",
"In particular the Sobolev scales $W^s(_+, H)$ and $W^s(_+\\times ^b, H)$ do not depend on $y_0 \\in ^b$ for $0\\le s\\le 1$ .",
"In fact, in our arguments below we will require independence of the Sobolev spaces for $0 \\le s \\le 2$ .",
"We conclude with a definition of weighted Sobolev-spaces, where we denote by $X$ the multiplication operator by $x~\\in _+$ .",
"The weighted Sobolev-scales are defined by $W^{s,\\delta ,l}:= X^\\delta (1+X)^{-l} W^s(_+,H), \\quad W^{s,\\delta }:= W^{s,\\delta ,0}.$"
],
[
"Examples of generalized Dirac operators on an abstract edge",
"The spin Dirac operator on a model edge space is indeed a generalized Dirac operator in the sense that it is given by the differential expression (REF ) and satisfies the commutator relations (REF ).",
"This has been established by Albin and Gell-Redman [3].",
"In this subsection we prove that the Gauss–Bonnet operator on a model edge space is a generalized Dirac operator in the sense above as well.",
"Let $M^m$ and $N^n$ be Riemannian manifolds.",
"Given forms $\\omega _p \\in \\Omega ^p(M)$ and $\\eta _q \\in \\Omega ^q(N)$ , we will write $\\omega _p \\wedge \\eta _q$ for the form $\\pi _M^* (\\omega _p) \\wedge \\pi _N^* (\\eta _q) \\in \\Omega ^{p+q}(M\\times N)$ , where $\\pi _M:M\\times N \\rightarrow M$ and $\\pi _N:M\\times N \\rightarrow N$ are projections onto the first and second factors respectively.",
"It is well known that the exterior derivative $d:\\Omega ^{\\ast }(M\\times N) \\rightarrow \\Omega ^{\\ast }(M\\times N)$ satisfies the Leibniz rule, if $\\omega _p\\in \\Omega ^{p}(M)$ and $\\eta _q\\in \\Omega ^q(N)$ then $d(\\omega _p\\wedge \\eta _q) \\ = \\ (d^M\\omega _p)\\wedge \\eta _q +(-1)^{p} \\ \\omega _p\\wedge (d^N\\eta _q).$ The same Leibniz rule holds for the adjoint of the exterior derivative $d^t$ in $\\Omega ^{\\ast }(M\\times N)$ .",
"Note that $\\Omega ^{\\ast }(M\\times N)$ can be decomposed into a direct sum of subspaces of the form $\\Omega ^{\\ast }(M) \\wedge \\Omega ^{\\ast }(N)$ .",
"Hence it suffices to study the action of $d^t$ on differential forms in $\\Omega ^{p+1}(M) \\wedge \\Omega ^q(N)$ , where we have $d^t: \\Omega ^{p+1}(M) \\wedge \\Omega ^q(N) \\rightarrow (\\Omega ^{p}(M) \\wedge \\Omega ^q(N)) \\oplus (\\Omega ^{p}(M) \\wedge \\Omega ^{q-1}(N)).$ Consider $\\tilde{\\omega }_{p}\\in \\Omega ^{p}(M)$ , $\\omega _{p+1},\\tilde{\\omega }_{p+1}\\in \\Omega ^{p+1}(M)$ , $\\tilde{\\eta }_q, \\eta _q\\in \\Omega ^{q}(N)$ and $\\tilde{\\eta }_{q-1}\\in \\Omega ^{q-1}(N)$ , then we have for the first component of $d^t$ $\\begin{aligned}\\langle d_{p+q}^t (\\omega _{p+1}\\wedge &\\eta _q),\\tilde{\\omega }_{p}\\wedge \\tilde{\\eta }_q\\rangle \\\\&= \\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p})\\wedge \\tilde{\\eta }_q + (-1)^p\\tilde{\\omega }_{p}\\wedge (d^N\\tilde{\\eta }_q)\\rangle \\\\&=\\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p})\\wedge \\tilde{\\eta }_q \\rangle \\\\&=\\langle \\omega _{p+1}, (d^M\\tilde{\\omega }_{p})\\rangle _M\\langle \\eta _{q}, \\tilde{\\eta }_q\\rangle _N\\\\&=\\langle (d^{M,t}\\omega _{p+1}), \\tilde{\\omega }_{p}\\rangle _M\\langle \\eta _{q}, \\tilde{\\eta }_q\\rangle _N\\\\&=\\langle (d^{M,t}\\omega _{p+1})\\wedge \\eta _{q},\\tilde{\\omega }_{p}\\wedge \\tilde{\\eta }_{q}\\rangle .\\end{aligned}$ For the second component of $d^t$ , we obtain $\\begin{aligned}\\langle d_{p+q}^t &(\\omega _{p+1}\\wedge \\eta _q),\\tilde{\\omega }_{p+1}\\wedge \\tilde{\\eta }_{q-1}\\rangle \\\\&= \\langle \\omega _{p+1}\\wedge \\eta _q,(d^M\\tilde{\\omega }_{p+1})\\wedge \\tilde{\\eta }_{q-1} + (-1)^{p+1}\\tilde{\\omega }_{p+1}\\wedge (d^N\\tilde{\\eta }_{q-1})\\rangle \\\\&=(-1)^{p+1}\\langle \\omega _{p+1}\\wedge \\eta _q,\\tilde{\\omega }_{p+1}\\wedge (d^N\\tilde{\\eta }_{q-1}) \\rangle \\\\&=(-1)^{p+1}\\langle \\omega _{p+1}, \\tilde{\\omega }_{p+1}\\rangle _M\\langle \\eta _{q}, (d^N\\tilde{\\eta }_{q-1})\\rangle _N\\\\&=(-1)^{p+1}\\langle \\omega _{p+1}, \\tilde{\\omega }_{p+1}\\rangle _M\\langle (d^{N,t}\\eta _{q}), \\tilde{\\eta }_{q-1}\\rangle _N\\\\&=(-1)^{p+1}\\langle \\omega _{p+1}\\wedge (d^{N,t}\\eta _{q}),\\tilde{\\omega }_{p+1}\\wedge \\tilde{\\eta }_{q-1}\\rangle .\\end{aligned}$ Altogether, we arrive at the result $d_{p+q}^t (\\omega _{p+1}\\wedge \\eta _q) = (d_p^{M,t}\\omega _{p+1})\\wedge \\eta _{q} + (-1)^{p+1} \\omega _{p+1}\\wedge (d^{N,t}_{q} \\eta _q).$ We now apply Lemma REF to the case of a model edge $C(F) \\times Y$ of cones $C(F) = _+\\times F$ fibered over an edge manifold $Y$ .",
"Recall that on a cone $C(F) = _+\\times F$ we have as in [10] the following isometric identifications $\\Omega ^{\\rm ev}(C(F)) \\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F)), \\qquad \\Omega ^{\\rm odd}(C(F)) \\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F)).$ Under these identifications the Gauss–Bonnet operator $D=d+d^t$ acting now from $\\Omega ^{\\rm ev}(C(F))\\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F))$ to $\\Omega ^{\\rm odd}(C(F))\\cong C^{\\infty }(_+,\\Omega ^{\\ast }(F))$ , takes the form cf.",
"[10] $D = \\frac{d}{dx}+X^{-1} A.$ Respectively, the full operator $D$ acts on $C^{\\infty }(_+,\\Omega ^{\\ast }(C(F))\\oplus \\Omega ^{\\ast }(C(F)))$ as $\\left(\\begin{array}{cc}0 & -\\frac{d}{dx}+X^{-1} A\\\\\\frac{d}{dx}+X^{-1} A & 0\\end{array}\\right) =\\left(\\begin{array}{cc}0 & -1\\\\1 & 0\\end{array}\\right)\\left(\\frac{d}{dx} + X^{-1}\\left(\\begin{array}{cc}A & 0\\\\0 & -A\\end{array}\\right)\\right).$ Note that the grading operator on $\\Omega ^{\\ast }(C(F))\\oplus \\Omega ^{\\ast }(C(F))$ is $\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right)$ .",
"Taking now the cartesian product by a manifold $Y$ (the edge), we have $\\begin{aligned}\\Omega ^{\\rm ev}(C(F)\\times Y) &= \\Omega ^{\\rm ev}(C(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus \\Omega ^{\\rm odd}(C(F))\\otimes \\Omega ^{\\rm odd}(Y)\\\\&\\cong C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm odd}(Y),\\end{aligned}$ where we used the identifications (REF ) in the second equality.",
"In exactly the same manner we find for differential forms of odd degree $\\begin{aligned}\\Omega ^{\\rm odd}(C(F)\\times Y) &= \\Omega ^{\\rm odd}(C(F))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus \\Omega ^{\\rm ev}(C(F))\\otimes \\Omega ^{\\rm odd}(Y)\\\\&\\cong C^{\\infty }(_+, \\Omega ^{\\ast }(L))\\otimes \\Omega ^{\\rm ev}(Y) \\oplus C^{\\infty }(_+, \\Omega ^{\\ast }(F))\\otimes \\Omega ^{\\rm odd}(Y).\\end{aligned}$ So again we have an identification of the space $\\Omega ^{\\rm ev}(C(F)\\times Y)$ with the space $\\Omega ^{\\rm odd}(C(F)\\times Y)$ .",
"For $\\omega _1 \\in \\Omega ^{\\rm ev}(C(F))$ , $\\omega _2 \\in \\Omega ^{\\rm odd}(C(F))$ , $\\eta _1 \\in \\Omega ^{\\rm ev}(Y)$ and $\\eta _2 \\in \\Omega ^{\\rm odd}(Y)$ , we have $\\omega _1\\otimes \\eta _1 \\oplus \\omega _2\\otimes \\eta _2\\in \\Omega ^{\\rm ev}(C(F)\\times Y)$ .",
"Using Lemma REF we now find for $D= d + d^t$ , $\\begin{aligned}D(\\omega _1\\otimes \\eta _1 \\oplus \\omega _2\\otimes \\eta _2) &=D^{C(F)}\\omega _1\\otimes \\eta _1 + \\omega _1 \\otimes D^Y\\eta _1\\\\&+ D^{C(F)}\\omega _2\\otimes \\eta _2 - \\omega _2\\otimes D^Y\\eta _2\\\\&=\\left(\\begin{array}{cc}\\partial _x + X^{-1}A & - D^Y\\\\D^Y & -\\partial _x + X^{-1}A\\end{array}\\right)\\left(\\begin{array}{c}\\omega _1\\otimes \\eta _1\\\\\\omega _2\\otimes \\eta _2\\end{array}\\right).\\end{aligned}$ Note that by construction $A$ and $D^Y$ commute.",
"By abuse of notation $A$ acts as $A\\otimes I$ and $D^Y$ acts as $I\\otimes D^Y$ on the tensors.",
"The full Gauss–Bonnet then becomes $D= \\left(\\begin{array}{cccc}0 & 0 & -\\partial _x + X^{-1}A & D^Y\\\\0 & 0 & -D^Y & \\partial _x + X^{-1}A\\\\\\partial _x + X^{-1}A & -D^Y & 0 & 0\\\\D^Y & -\\partial _x + X^{-1}A & 0 & 0\\end{array}\\right).$ This expression can rewritten as follows.",
"$\\begin{aligned}D = \\left(\\begin{array}{cccc}0 & 0 & -1 & 0\\\\0 & 0 & 0 & 1\\\\1 & 0 & 0 & 0\\\\0 &-1 & 0 & 0\\end{array}\\right) &\\left(\\partial _x + X^{-1} \\left(\\begin{array}{cccc}1 & 0 & 0 & 0\\\\0 & -1 &0 & 0\\\\0 & 0 & -1&0\\\\0 & 0 & 0 & 1\\end{array}\\right)\\cdot A\\right)\\\\&+\\left(\\begin{array}{cccc}0 & 0 & 0 & 1\\\\0 & 0 & -1 & 0\\\\0 & -1 & 0 & 0\\\\1 & 0 & 0 & 0\\end{array}\\right)\\cdot D^Y,\\end{aligned}$ with grading operator $\\left(\\begin{array}{cc}I_2 & 0\\\\0 & - I_2\\end{array}\\right)$ where $I_2$ is the identity in $M_2()$ .",
"Define the following matrices $\\Gamma = \\left(\\begin{array}{cccc}0 & 0 & -1 & 0\\\\0 & 0 & 0 & 1\\\\1 & 0 & 0 & 0\\\\0 &-1 & 0 & 0\\end{array}\\right)\\!\\!,S =\\left(\\begin{array}{cccc}1 & 0 & 0 & 0\\\\0 & -1 &0 & 0\\\\0 & 0 & -1&0\\\\0 & 0 & 0 & 1\\end{array}\\right) A,T = \\left(\\begin{array}{cccc}0 & 0 & 0 & 1\\\\0 & 0 & -1 & 0\\\\0 & -1 & 0 & 0\\\\1 & 0 & 0 & 0\\end{array}\\right) D^Y\\!.$ We introduce the usual Clifford matrices $\\sigma _1 = \\left(\\begin{array}{cc}0 & -1\\\\1 & 0\\end{array}\\right),\\ \\sigma _2=\\left(\\begin{array}{cc}0 & i \\\\i & 0\\end{array}\\right),\\ \\omega =\\left(\\begin{array}{cc}1 & 0\\\\0 & -1\\end{array}\\right) = i \\cdot \\sigma _1 \\cdot \\sigma _2.$ We have, $\\begin{aligned}\\Gamma &= \\left(\\begin{array}{cc}0 & -\\omega \\\\\\omega & 0\\end{array}\\right) = \\sigma _1 \\otimes \\omega ,\\\\S &= \\left(\\begin{array}{cc}\\omega & 0\\\\0 & -\\omega \\end{array}\\right)\\otimes A = \\omega \\otimes \\omega \\otimes A,\\\\T &= \\left(\\begin{array}{cc}0 & -\\sigma _1\\\\\\sigma _1 & 0\\end{array}\\right) \\otimes D^Y = \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y.\\end{aligned}$ We can now easily compute the commutator relations $\\begin{aligned}\\Gamma \\ S + S \\ \\Gamma &= \\sigma _1 \\otimes \\omega \\cdot \\omega \\otimes \\omega \\otimes A + \\omega \\otimes \\omega \\otimes A \\cdot \\sigma _1 \\otimes \\omega \\\\&= (\\sigma _1 \\omega + \\omega \\sigma _1)\\otimes \\omega ^2 \\otimes A = 0.\\\\\\Gamma \\ T + T \\ \\Gamma &= \\sigma _1 \\otimes \\omega \\cdot \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y + \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y \\cdot \\sigma _1\\otimes \\omega \\\\&= \\sigma _1 \\otimes (\\omega \\sigma _1 + \\sigma _1 \\omega )\\otimes D^Y = 0.\\\\T \\ S - S \\ T &= \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y \\cdot \\omega \\otimes \\omega \\otimes A - \\omega \\otimes \\omega \\otimes A \\cdot \\sigma _1 \\otimes \\sigma _1 \\otimes D^Y\\\\&= (\\sigma _1\\cdot \\omega \\otimes \\sigma _1\\cdot \\omega - \\omega \\cdot \\sigma _1 \\otimes \\omega \\cdot \\sigma _1)\\otimes D^Y \\cdot A\\\\&=(\\omega \\cdot \\sigma _1 \\otimes \\sigma _1 \\cdot \\omega + \\sigma _1\\cdot \\omega \\otimes \\sigma _1 \\cdot \\omega )\\otimes D^Y \\cdot A\\\\&=(\\omega \\cdot \\sigma _1 + \\sigma _1\\cdot \\omega )\\otimes \\sigma _1 \\cdot \\omega \\otimes D^Y \\cdot A = 0.\\end{aligned}$"
],
[
"Some integral operators and auxiliary estimates",
"In this section we study boundedness properties of certain integral operators that appear below when inverting the model Bessel operator $L^2(y_0,\\xi )$ and its square $L^2(y_0,\\xi )^2$ .",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and consider the integral operator $K$ acting on $C^\\infty _0(_+)$ with integral kernel given by $k(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{2\\nu } \\frac{y}{x}^{\\nu } (xy)^{\\frac{1}{2}}, & y\\le x,\\\\\\frac{1}{2\\nu } \\frac{y}{x}^{-\\nu } (xy)^{\\frac{1}{2}}, & x\\le y.\\end{array} \\right.$ Then $X^{-2} \\circ K$ defines a bounded operator on $L^2(0,\\infty )$ and there exists a constant $C>0$ depending only on $\\delta >0$ such that $\\begin{split}&\\Vert X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le \\nu ^{2}-\\frac{9}{4}^{-1}, \\\\&\\Vert (X \\partial _x) \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le \\nu -\\frac{3}{2}^{-1}, \\\\&\\Vert (X \\partial _x)^2 \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C.\\end{split}$ We apply Schur's test, cf.",
"Halmos and Sunder [19].",
"We have $\\begin{aligned}\\int _{0}^{x}x^{-2}k(x,y) \\ dy &+ \\int _{x}^{\\infty } x^{-2} k(x,y) \\ dy \\\\~&=\\frac{1}{2\\nu } x^{-\\nu - \\frac{3}{2}}\\int _{0}^x y^{\\nu +\\frac{1}{2}} \\ dy + x^{\\nu - \\frac{3}{2}} \\int _{x}^{\\infty } y^{-\\nu + \\frac{1}{2}}dy\\\\&= \\frac{1}{2\\nu } \\frac{1}{\\nu + \\frac{3}{2}} + \\frac{1}{\\nu -\\frac{3}{2}} = \\nu ^{2} - \\frac{9}{4} ^{-1}.\\end{aligned}$ Similarly, we integrate in the $x$ variable and find $\\begin{aligned}\\int _{0}^{y}x^{-2}k(x,y) \\ dx &+ \\int _{y}^{\\infty } x^{-2} k(x,y) \\ dx \\\\&=\\frac{1}{2\\nu } y^{-\\nu + \\frac{1}{2}}\\int _{0}^y x^{\\nu -\\frac{3}{2}} \\ dx + y^{\\nu + \\frac{1}{2}} \\int _{y}^{\\infty } x^{-\\nu - \\frac{3}{2}}dx\\\\&= \\frac{1}{2\\nu } \\frac{1}{\\nu - \\frac{1}{2}} + \\frac{1}{\\nu +\\frac{1}{2}} = \\nu ^{2} - \\frac{1}{4} ^{-1}\\end{aligned}$ From there one concludes that $\\Vert X^{-2} \\circ K\\Vert _{L^2\\rightarrow L^2}\\le \\left( \\nu ^2 - \\frac{9}{4}\\nu ^{2}-\\frac{1}{4}\\right)^{-\\frac{1}{2}} \\le \\nu ^2-\\frac{9}{4}^{-1}.$ This proves the first estimate.",
"The second and third estimates are established ad verbatim.",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and let $\\beta >0$ be positive real number.",
"Consider integral operator $K$ acting on $C^\\infty _0(_+)$ with integral kernel given in terms of modified Bessel functions by $k(x,y) = \\left\\lbrace \\begin{array}{cc}(xy)^{\\frac{1}{2}} I_\\nu (\\beta \\ y) K_{\\nu }(\\beta \\ x), & y\\le x,\\\\(xy)^{\\frac{1}{2}} I_\\nu (\\beta \\ x) K_{\\nu }(\\beta \\ y), & x\\le y.\\end{array}\\right.$ Then $X^{-2} \\circ K$ defines a bounded operator on $L^2(0,\\infty )$ and there exists a constant $C>0$ depending only on $\\delta >0$ such that $\\begin{split}&\\Vert X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C \\nu ^{2}-\\frac{9}{4}^{-1}, \\\\&\\Vert (X \\partial _x) \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C \\nu -\\frac{3}{2}^{-1}, \\\\&\\Vert (X \\partial _x)^2 \\circ X^{-2} \\circ K \\Vert _{L^2\\rightarrow L^2}\\le C.\\end{split}$ Following Olver [31], we note the asymptotic expansions for Bessel functions as $\\nu \\rightarrow \\infty $ $I_\\nu (\\nu x) \\sim \\frac{1}{\\sqrt{2\\pi \\nu }}\\cdot \\frac{e^{\\nu \\cdot \\eta (x)}}{(1+x^2)^{1/4}},\\;\\;K_\\nu (\\nu x) \\sim \\sqrt{\\frac{2\\pi }{\\nu }}\\cdot \\frac{e^{-\\nu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}$ where $\\eta (x) = \\sqrt{1+x^2} +\\ln \\frac{x}{1+\\sqrt{1+x^2}}$ .",
"By (REF ), these expansions are uniform in $x\\in (0,\\infty )$ .",
"We define an auxiliary function $E(x, \\nu ):=\\frac{e^{\\nu (\\eta (x)-\\ln x)}}{(1+x^2)^{\\frac{1}{4}}}.",
"$ Note that $\\eta (x) - \\ln x$ is increasing as $x\\rightarrow \\infty $ , since $\\begin{aligned}(\\eta (x)-\\ln x)^{\\prime } &= (\\sqrt{1+x^2} - \\ln (1+\\sqrt{1+x^2})^{\\prime }\\\\&=2x\\frac{1}{2\\sqrt{1+x^2}} - \\frac{1}{2\\sqrt{1+x^2}}\\cdot \\frac{1}{1+\\sqrt{1+x^2}} \\\\&=\\frac{x}{\\sqrt{1+x^2}}\\cdot \\frac{\\sqrt{1+x^2}}{1+\\sqrt{1+x^2}}>0.\\end{aligned}$ Consequently $E(x,-\\nu )$ is decreasing and for $y \\le x$ $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| &\\le C\\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\frac{y}{\\nu }^\\nu \\frac{x}{\\nu +\\alpha }^{-(\\nu +\\alpha )} \\\\~&\\times E\\left(\\frac{y}{\\nu },\\nu \\right)\\cdot E\\left(\\frac{y}{\\nu +\\alpha },-(\\nu +\\alpha )\\right),\\end{aligned}$ for some uniform constant $C>0$ and $\\alpha \\in \\lbrace 0,1\\rbrace $ .",
"In fact, below we will always denote uniform positive constants by $C$ .",
"We proceed with a technical calculation $\\begin{aligned}&(\\nu +\\alpha )\\eta \\frac{y}{\\nu +\\alpha }- \\ln \\frac{y}{\\nu +\\alpha } -\\nu \\eta \\frac{y}{\\nu }-\\ln \\frac{y}{\\nu } \\\\&=\\sqrt{(\\nu +\\alpha )^2 +y^2} - \\sqrt{\\nu ^2 +y^2} +\\nu \\ln \\frac{\\nu +\\alpha }{\\nu } \\\\&- \\nu \\ln \\frac{\\nu +\\alpha +\\sqrt{(\\nu +\\alpha )^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}-\\alpha \\ln 1+\\sqrt{1+\\frac{y}{\\nu +\\alpha }^2} \\\\&=\\sqrt{\\nu ^2 +y^2}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1+ \\nu \\ln 1+\\frac{\\alpha }{\\nu }\\\\&-\\nu \\ln 1+ \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1\\\\&-\\alpha \\ln 1 + \\sqrt{1+\\frac{y}{\\nu +\\alpha }^2}.\\end{aligned}$ In order to continue with our estimates we write $O(f)$ for any function whose absolute value is bounded by $f$ , with a uniform constant that is independent of $\\nu $ and $y$ , and note $1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}$ is always positive, $\\ln \\frac{\\nu +\\alpha }{\\nu } = \\frac{\\alpha }{\\nu } + O\\frac{1}{\\nu ^2}$ where the $O$ -constant depends on $\\delta >0$ , $\\ln 1+ \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}}\\sqrt{1+\\frac{2\\alpha \\nu +\\alpha ^2}{\\nu ^2+y^2}}-1\\\\~= \\frac{\\alpha }{\\nu +\\sqrt{\\nu ^2+y^2}}+\\frac{\\sqrt{\\nu ^2+y^2}}{\\nu +\\sqrt{\\nu ^2+y^2}} + O\\frac{1}{\\nu ^2},$ where the $O$ -constant may be chosen independently of $y\\in (0,\\infty )$ , but depends on $\\delta >0$ .",
"Plugging in these observations, we arrive at the following estimate, $\\begin{aligned}&(\\nu +\\alpha )\\eta \\frac{y}{\\nu +\\alpha }- \\ln \\frac{y}{\\nu +\\alpha } -\\nu \\eta \\frac{y}{\\nu }-\\ln \\frac{y}{\\nu } \\\\&=-\\alpha \\ln 1+\\sqrt{1+\\frac{y}{\\nu }^2} +O1.\\end{aligned}$ Plugging this into the estimate (REF ) we obtain: $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| &\\le C \\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{\\nu +\\alpha }{\\nu } ^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha } \\\\~&\\le C \\cdot \\frac{1}{\\sqrt{\\nu (\\nu +\\alpha ))}} \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha }, \\\\& \\textup {where} \\ F\\frac{y}{\\nu }:= 1+\\sqrt{1+\\frac{y}{\\nu }^2}^{\\alpha } / \\sqrt{1+\\frac{y}{\\nu }^2},\\end{aligned}$ for some uniform constants $C>0$ , depending only on $\\delta $ , where in the second inequality we noted that $\\lim \\limits _{\\nu \\rightarrow \\infty } \\frac{\\nu +\\alpha }{\\nu } ^\\nu =e^\\alpha $ and hence $\\frac{\\nu +\\alpha }{\\nu } ^\\nu $ is bounded uniformly for large $\\nu $ .",
"We also note that $(\\nu (\\nu +\\alpha ))^{-1} \\le C \\nu ^{-2}$ , as long as $\\nu $ and $(\\nu +\\alpha )$ are positive bounded away from zero.",
"Hence we arrive at the following estimate $\\begin{aligned}\\left|K_{\\nu +\\alpha }(x) \\cdot I_\\nu (y)\\right| \\le C \\cdot \\frac{1}{\\nu } \\cdot F\\frac{y}{\\nu }\\cdot \\frac{y}{x}^\\nu \\cdot \\frac{x}{\\nu +\\alpha } ^{-\\alpha }.",
"\\end{aligned}$ Note that for $\\alpha = 1$ , $F(y/\\nu )$ is uniformly bounded and for $\\alpha =0$ , $F(y/\\nu )\\le C(y/\\nu )^{-1}$ .",
"Hence we conclude for $x \\ge y$ and some uniform constant $C>0$ $\\begin{split}&\\left| \\, K_{\\nu }(\\beta \\ x) \\cdot I_\\nu (\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{1}{\\nu } \\cdot \\frac{y}{x}^{\\nu }.",
"\\\\&\\left| \\, x K_{\\nu +1}(\\beta \\ x) \\cdot I_\\nu (\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{y}{x}^{\\nu }.",
"\\\\&\\left| \\, y K_{\\nu }(\\beta \\ x) \\cdot I_{\\nu - 1}(\\beta \\ y) \\, \\right|\\le C \\cdot \\frac{y}{x}^{\\nu }.",
"\\end{split}$ By the formulae for the derivatives of modified Bessel functions $\\begin{split}& (x\\partial _x) I_\\nu (x) = xI_{\\nu -1}(x) - \\nu I_\\nu (x), \\\\& (x\\partial _x) K_\\nu (x) = \\nu K_\\nu (x) - xK_{\\nu +1}(x),\\end{split}$ the derivatives $(x\\partial _x) k(x,y)$ and $(x\\partial _x)^2 k(x,y)$ can be written as combinations of the products in (REF ).",
"In view of Proposition , we obtain the result.",
"The statement of Proposition corresponds to Brüning-Seeley [9].",
"However, the latter reference does not provide an exact lower bound on $\\nu $ , which is crucial in order to establish the optimal spectral gap in the spectral Witt condition.",
"We close the section with a crucial observation.",
"Let $\\nu \\ge \\frac{3}{2} + \\delta $ for some $\\delta >0$ and let $K$ denote either the integral operator in Proposition or in Proposition .",
"Then for any $u \\in L^2(_+)$ with compact support in $[0,1]$ , $Ku$ admits the following estimates $Ku$ is continuously differentiable on $(0,\\infty )$ .",
"$| Ku(x) | \\le \\frac{C}{\\nu } \\Vert u\\Vert _{L^2} \\ x^{-1-\\delta }, \\quad | (x\\partial _x) Ku(x) |\\le C \\Vert u\\Vert _{L^2} \\ x^{-1-\\delta },$ for a constant $C>0$ independent of $u$ and $\\nu $ .",
"It suffices to prove the statement for $K$ in Proposition , since by (REF ) the integral kernels in Proposition , and their derivatives, can be estimated against those in Proposition .",
"Consider $u \\in L^2(_+)$ such that $u \\subset [0,1]$ .",
"Then for $x>1$ we find using $\\nu \\ge \\frac{3}{2} + \\delta $ $| Ku(x) | &\\le \\frac{x^{-\\nu +\\frac{1}{2}}}{2\\nu } \\int _0^1 y^{\\nu +\\frac{1}{2}} | u(y) | dy\\le \\frac{C}{\\nu } \\Vert u\\Vert _{L^2}\\ x^{-1-\\delta }, \\\\| x\\partial _x Ku(x) | &\\le \\frac{(-\\nu +\\frac{1}{2})}{2\\nu } x^{-\\nu +\\frac{1}{2}} \\int _0^1 y^{\\nu +\\frac{1}{2}} |u(y)| dy\\le C \\Vert u\\Vert _{L^2}\\ x^{-1-\\delta }, \\\\$ for a constant $C>0$ independent of $u$ and $\\nu $ ."
],
[
"Invertibility of the model Bessel operators",
"In this section we prove invertibility of $L(y_0,\\xi ) = \\Gamma (\\partial _x + X^{-1}S(y_0)) + ic(\\xi ; y_0),$ cf.",
"(REF ), and its square $L(y_0,\\xi )^2$ .",
"We will work with the Sobolev scale $W^s(_+, H)$ , defined in terms of the interpolation scale $H^s\\equiv H^s(S(y_0))$ .",
"As noted in Remark REF , the interpolation scales $H^s(S(y_0))$ in general depend on the base point $y_0\\in ^b$ .",
"This does not play a role here, since in the present section $y_0$ is fixed.",
"Assuming the spectral Witt condition (REF ), the mapping $L(y,0)^2:W^{2,2}(_+, H) \\rightarrow W^{0,0}(_+, H),$ is bijective with bounded inverse.",
"Consider the following commutative diagram: [email protected]{W^{2,2} @<1ex>[rrrr]^{L^2(y,0)}@<1ex>@{-->}[dddd]^{X^{-2}} & & & &W^{0,0}@{=}[dddd] @{-->}[llll]^{K(y,0)} \\\\& & & & \\\\& & \\circlearrowright & & \\\\& & & & \\\\W^{2,0}[rrrr]^{\\tilde{L}^2} [uuuu]^{X^2}& & & & W^{0,0},@<1ex>@{-->}[llll]^{\\tilde{K} }}$ where $\\tilde{L}^2(y,0) = L^2(y,0)\\circ X^2$ and the inverse maps $K$ and $\\tilde{K}$ are constructed as follows.",
"Let $\\lbrace \\phi _{j}\\rbrace _{j\\in }$ be an orthonormal base of $H$ consisting of eigenvectors of $A^2(y)$ such that $A^2(y) \\ \\phi _{j} = \\nu _j^2 \\ \\phi _{j}$ , where by convention we assume $\\nu _j>0$ .",
"The geometric Witt condition (REF ) implies $\\nu _j > \\frac{3}{2}$ and by discreteness we conclude $\\exists \\, \\delta > 0 \\ \\forall \\, j \\in : \\ \\nu _j \\ge \\frac{3}{2}+ \\delta .$ For any $j\\in $ we define $E_j := \\langle \\phi _{j}\\rangle $ .",
"For any $g \\in L^2(_+)$ the equation $L^2(y,0) \\ f \\cdot \\phi _{j} = g \\cdot \\phi _j \\in L^2(_+, E_j)$ reduces to a scalar equation $- x^2 + \\frac{1}{x^2} \\nu _j^2-\\frac{1}{4}f = g. $ The fundamental solutions for that equation are $\\psi ^+_{\\nu _j} (x) = x^{\\nu _j + \\frac{1}{2}}, \\ \\text{and} \\ \\psi ^-_{\\nu _j} (x) = x^{-\\nu _j +\\frac{1}{2}}.$ In view of (REF ), neither of them lies in $W^{2,2}(_+)$ and hence $L^2(y,0)$ is injective on $W^{2,2}(_+,H)$ .",
"It remains to prove surjectivity.",
"The fundamental solutions $\\psi ^\\pm _{\\mu _j}$ yield a solution of the equation eq1 with $f(x) = \\int _{0}^{\\infty } k_j(x,y) g(y) dy =: K_j g,$ where $K_j$ is an integral operator with the integral kernel $k_j(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{2\\nu _j} \\frac{y}{x}^{\\nu _j} (xy)^{\\frac{1}{2}}, & y\\le x,\\\\\\frac{1}{2\\nu _j} \\frac{y}{x}^{-\\nu _j} (xy)^{\\frac{1}{2}}, & x\\le y.\\end{array} \\right.$ Accordingly, a solution of the scalar equation for any $\\tilde{g} \\in L^2(_+)$ $(L^2(y,0) \\circ X^2) \\ \\tilde{f} \\cdot \\phi _{j}= \\tilde{g} \\cdot \\phi _{j} \\in L^2(_+, E_j),$ is given in terms of $\\tilde{K}_j = X^{-2} \\circ K_j$ by $\\tilde{f} = \\tilde{K}_j \\tilde{g}$ .",
"The integral operators $\\tilde{K}_j$ have been studied in Proposition , which proves in view of (REF ) that for each $E_j$ the restriction $\\tilde{L}(y,0)|_{E_j}$ admits an inverse $\\tilde{K}_j:W^{0,0}(_+,E_j) \\rightarrow W^{2,0}(_+,E_j)$ with norm bounded uniformly in $j\\in $ .",
"Equivalently, the restriction $L(y,0)|_{E_j}$ admits an inverse $K_j : W^{0,0}(_+,E_j) \\rightarrow W^{2,2}(_+,E_j)$ with norm bounded uniformly in $j\\in $ .",
"By (REF ), the operator norms of $\\nu _j \\cdot (X\\partial _x) \\circ K_j$ and $\\nu _j^2 \\cdot K_j$ are bounded uniformly in $j$ as well.",
"Hence there exists a bounded inverse $(L^2(y,0))^{-1} : W^{0,0}(_+,H) \\rightarrow W^{2,2}(_+,H).$ This proves the statement.",
"Assume the spectral Witt condition (REF ).",
"Then for fixed parameters $(y,\\xi ) \\in ^b\\times ^b$ , the operator $L^2(y,\\xi ):W^{2,2}(_+, H)\\rightarrow W^{0,0,-2}(_+,H)$ is injective with a right-inverse $L^2(y,\\xi )^{-1}: W^{0,0}(_+, H)\\rightarrow W^{2,2}(_+,H)$ , bounded uniformly in the parameters $(y,\\xi )$ .",
"The case $\\xi = 0$ has been established in Proposition .",
"We proceed with the case $\\xi \\ne 0$ .",
"The commutator relations Eq1 imply that $A^2(y)$ and $c(\\xi ,y)^2$ may be simultaneously diagonalized and hence an orthonormal base $\\lbrace \\phi _{j}\\rbrace _{j \\in }$ of $H$ can be chosen such that $\\begin{aligned}&A^2(y) \\ \\phi _{j} = \\nu _j^2 \\phi _{j},\\; \\text{where wefix}\\;\\nu _j>0, \\\\&c(\\hat{\\xi }, y)^2 \\phi _{j} =\\phi _{j},\\;\\text{where}\\; \\hat{\\xi } =\\frac{\\xi }{\\Vert \\xi \\Vert }.\\end{aligned}$ We write $E_j = \\langle \\phi _{j}\\rangle $ .",
"Then, similar to Proposition , $L^2(y,\\xi )$ reduces over each $E_j$ to the scalar operator $L^2(y,\\xi )|_{E_j} = -x^2 + X^{-2} \\nu _j^2 - \\frac{1}{4}+\\Vert \\xi \\Vert ^2.$ The solutions to $L^2(y,\\xi )|_{E_j} \\phi = 0$ are given by linear combination of modified Bessel-functions $\\sqrt{x} I_{\\nu _j}(\\Vert \\xi \\Vert x)$ and $\\sqrt{x} K_{\\nu _j}(\\Vert \\xi \\Vert x)$ , which are not elements of $W^{2,2}(_+)$ for any $j\\in $ and any $\\xi \\ne 0$ .",
"This proves injectivity of $L^2(y,\\xi )$ on $W^{2,2}(_+,H)$ .",
"For the right-inverse we note the following commutative diagram [email protected]{W^{2,2} @<1ex>[rrrr]^{L^2(y,\\xi )}@<1ex>@{-->}[dddd]^{X^{-2}} & & & &W^{0,0,-2}@{=}[dddd] @{-->}[llll]^{K_1(y,\\xi )}\\\\& & & &\\\\& & \\circlearrowright & &\\\\& & & &\\\\W^{2,0}[rrrr]^{\\tilde{L}^2} [uuuu]^{X^2}& & & & W^{0,0,-2}@<1ex>@{-->}[llll]^{\\tilde{K}_1 }}.$ The equation $[ L^2(y,\\xi )\\circ X^2|_{E_j}]f \\cdot \\phi _j= g \\cdot \\phi _j\\in L^2(_+,E_j)$ admits a solution $\\begin{aligned}X^{-2}\\circ K_{j}(y,\\xi ) g &:= \\int _{_+}x^{-2}\\ k_{j}(x,\\tilde{x})\\ g(\\tilde{x})d\\tilde{x},\\end{aligned}$ where the kernel $k_{j}(x,\\tilde{x})$ is $k_{j}(x,\\tilde{x}) = \\left\\lbrace \\begin{array}{cc}(x\\tilde{x})^{\\frac{1}{2}} I_{\\nu _j} (\\Vert \\xi \\Vert \\ \\tilde{x})K_{\\nu _j}(\\Vert \\xi \\Vert \\ x),& \\tilde{x}\\le x,\\\\(x\\tilde{x})^{\\frac{1}{2}} I_{\\nu _j} (\\Vert \\xi \\Vert \\ x) K_{\\nu _j}(\\Vert \\xi \\Vert \\ \\tilde{x}),& x\\le \\tilde{x}.\\end{array}\\right.$ Therefore, by Proposition , $X^{-2}\\circ K_{j}$ is bounded, uniformly in $j\\in $ and $\\xi > 0$ .",
"Then, in view of the uniform bounds (REF ), $L^2(y,\\xi ) \\circ X^2$ admits a right-inverse $X^{-2} \\circ K(y,\\xi ): W^{0,0}(_+,H) \\rightarrow W^{2,0}(_+,H)$ .",
"Equivalently, $L^2(y,\\xi )$ admits a right-inverse $K(y,\\xi ): W^{0,0}(_+,H) \\rightarrow W^{2,2}(_+,H)$ , which proves the statement in view of continuity at $\\xi = 0$ .",
"Assume the spectral Witt condition (REF ).",
"Then $L(y,\\xi ):W^{1,1}(_+,H)\\rightarrow W^{0,0,-1}(_+,H),$ is injective with right-inverse $L(y,\\xi )^{-1}: W^{0,0}(_+, H)\\rightarrow W^{1,1}(_+,H)$ , bounded uniformly in $(y,\\xi )$ .",
"The commutator relations Eq1 imply that $S,\\Gamma $ and $ic(\\xi )$ may be simultaneously diagonalized and hence an orthonormal base $\\lbrace \\phi _{j,\\pm }\\rbrace $ of $H$ can be chosen such that $\\begin{aligned}S\\phi _{j,\\pm } &= \\pm \\mu _j \\phi _{j,\\pm },\\; \\text{where we fix}\\;\\mu _j>0, \\\\ic(\\hat{\\xi }, y) \\phi _{j,\\pm } &= \\pm \\phi _{j,\\pm },\\;\\text{where}\\; \\hat{\\xi } =\\frac{\\xi }{\\Vert \\xi \\Vert }\\\\\\Gamma \\phi _{j,\\pm } &= \\pm \\phi _{j,\\mp }.\\end{aligned}$ We define $E_j = \\langle \\phi _{j,+};\\phi _{j,-}\\rangle $ .",
"Then $L(y,\\xi )$ reduces over each $E_j$ to $L(y,\\xi )|_{E_j} = \\left(\\begin{array}{cc}0 & -I\\\\I & 0\\end{array}\\right)\\left[\\left(\\begin{array}{cc}\\partial _x & 0 \\\\0 & \\partial _x\\end{array}\\right)+ X^{-1} \\left(\\begin{array}{cc}\\mu _j & 0\\\\0 & -\\mu _j\\end{array}\\right)\\right] + \\left(\\begin{array}{cc}\\Vert \\xi \\Vert & 0 \\\\0 & -\\Vert \\xi \\Vert \\end{array}\\right).$ Like in [3], solutions to $L(y,\\xi )|_{E_j} \\phi = 0$ are given by linear combination of modified Bessel-functions, which are not elements of $W^{1,1}$ for any $j\\in $ and any $\\xi \\ne 0$ .",
"Same can be checked explicitly for $\\xi = 0$ .",
"This proves injectivity of $L(y,\\xi )$ .",
"The right-inverse is obtained by $L(y,\\xi )^{-1} := L(y,\\xi ) \\circ \\left(L^2(y,\\xi )\\right)^{-1}: L^2(_+,H) \\rightarrow W^{1,1} (_+,H),$ where the composition is well-defined for $\\xi = 0$ by Proposition , and for $\\xi \\ne 0$ by the fact that $\\left(L^2(y,\\xi )\\right)^{-1}$ maps $L^2(_+, H)$ to $W^{2,2} \\cap L^2(_+,H)$ , since $L(y,0) \\circ \\left(L^2(y,\\xi )\\right)^{-1} = \\textup {Id} - \\Vert \\xi \\Vert ^2 \\cdot \\left(L^2(y,\\xi )\\right)^{-1}.$ In the Corollary there is a certain overlap with the work of Albin and Gell-Redman [3], where in [3] they assert invertibility of $L(y,\\xi )$ for $\\xi \\ne 0$ , and do not prove uniform bounds for the inverse.",
"Here, we invert $L(y,\\xi )$ for all $\\xi \\in ^b$ and establish uniform bounds for the inverse."
],
[
"Parametrices for generalized Dirac and Laplace operators",
"We define subspaces of functions with compact support in $[0,1]$ $\\begin{split}&W^\\bullet _{\\rm comp}(_+, H):= \\lbrace \\phi u \\mid u \\in W^\\bullet (_+, H), \\phi \\in C^\\infty _0[0,\\infty ), \\phi \\subset [0,1]\\rbrace , \\\\&W^\\bullet _{\\rm comp}(_+\\times ^b, H):= \\lbrace \\phi u \\mid u \\in W^\\bullet (_+\\times ^b, H), \\phi \\in C^\\infty _0([0,\\infty ) \\times ^b), \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\phi \\subset [0,1] \\times ^b\\rbrace .\\end{split}$ Subspaces of weighted Sobolev scales, consisting of functions with compact support in $[0,1]$ and $[0,1] \\times ^b$ as above, are denoted analogously.",
"The Sobolev scales are defined in terms of the interpolation scales $H^s\\equiv H^s(S(y_0))$ , which a priori depend on the base point $y_0 \\in ^b$ .",
"This does not play a role here, since in the present section $y_0$ is fixed.",
"Assume the spectral Witt condition (REF ).",
"Then there exists $\\delta >0$ such that for any $u \\in W^0_{\\rm comp}(_+, H)$ and any $\\xi \\in ^b$ $\\Vert L(y_0,\\xi )^{-1} u(x) \\Vert _H = O(x^{-1-\\delta }), \\quad \\Vert L^2(y_0,\\xi )^{-1} u(x) \\Vert _H = O(x^{-1-\\delta }), \\quad x\\rightarrow \\infty .$ In particular, $L(y_0,\\xi )^{-1} u$ and $L^2(y_0,\\xi )^{-1} u$ are both in $L^2(_+,H)$ .",
"Here, $\\Vert \\cdot \\Vert _H$ denotes the norm of the Hilbert space $H$ .",
"Consider $u \\in W^0_{\\rm comp}(_+, H)$ .",
"Note first that $L(y_0,\\xi )^{-1} u \\in W^{1,1} (_+, H)$ and $L^2(y_0,\\xi )^{-1} u \\in W^{2,2} (_+, H)$ by Proposition and Corollary .",
"By the characterization (REF ) of Sobolev scales and the Sobolev embedding $H^1_e(_+) \\subset C(0,\\infty )$ into continuous functions, $L(y_0,\\xi )^{-1} u$ and $L^2(y_0,\\xi )^{-1} u$ are continuous on $(0,\\infty )$ with values in $H$ , and in that sense their pointwise evaluations are well-defined.",
"Recall $L^2(y_0,\\xi ) = -x^2 + X^{-2}\\left(A^2(y_0) - \\frac{1}{4}\\right) + c(\\xi ,y_0)^2.$ By the spectral Witt condition, $\\textup {Spec} \\, A(y_0) \\cap [0,\\frac{3}{2}] = \\varnothing $ and by discreteness of the spectrum there exists $\\delta >0$ such that $\\textup {Spec} \\, A(y_0) \\cap [0,\\frac{3}{2}+\\delta ) = \\varnothing .$ The integral kernel of $L^2(y_0,\\xi )^{-1}$ is given in terms of (REF ) for $\\xi \\ne 0$ and (REF ) for $\\xi = 0$ .",
"In view of (REF ), in both cases, the asymptotics $\\Vert L^2(y_0,\\xi )^{-1} u(x) \\Vert _H= O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ follows from Corollary .",
"The asymptotics of $\\Vert L(y_0,\\xi )^{-1} u(x)\\Vert _H $ now follows also by Corollary , once we observe that $L(y_0,\\xi )^{-1} u = L(y_0,\\xi ) (L^2(y_0,\\xi )^{-1} u) \\in W^{1,1} (_+, H)$ .",
"Consider $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ and denote its Fourier transform on $^b$ by $\\hat{u} (\\xi )$ .",
"Fix $y_0 \\in ^b$ and consider a generalized Dirac operator $D_{y_0}$ satisfying the spectral Witt condition (REF ).",
"We define $Qu(y):= \\int _{^b} e^{i \\langle y, \\xi \\rangle } L(y_0,\\xi )^{-1} \\hat{u} (\\xi )đ\\xi , \\quad đ\\xi := \\frac{d \\xi }{(2\\pi )^b}.$ Then $Q$ is a right-inverse to $D_{y_0}$ and defines a bounded operator $Q:W^{0}_{\\rm comp}(_+\\times ^b,H) \\subset W^{0} \\rightarrow X\\cdot W^{1}(_+\\times ^b,H) = W^{1,1}.$ By the Plancherel theorem we find for any $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ $\\Vert X^{-1} Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\Vert X^{-1} L(y_0,\\cdot )^{-1} \\hat{u} \\Vert ^2_{L^2(_+\\times ^b_\\xi , H)}\\\\~&= \\int _{^b} \\Vert X^{-1} L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi .$ By Corollary , the operator $X^{-1} L(y_0,\\xi )^{-1}$ defines a bounded map from $L^2(_+, H)$ to itself, with the operator norm bounded uniformly in $\\xi \\in ^b$ .",
"Denote its uniform bound by $C>0$ and compute again by Plancherel theorem $\\Vert X^{-1} Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert X^{-1} L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ & \\le C \\int _{^b} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi =C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Consequently, $Q: L^2(_+\\times ^b,H) \\rightarrow X\\cdot L^2(_+\\times ^b,H) = W^{0,1}$ is bounded.",
"Furthermore, by Corollary the operators $(X\\partial _x) \\circ X^{-1} L(y_0,\\xi )^{-1}$ and $S \\circ X^{-1} L(y_0,\\xi )^{-1}$ are bounded on $L^2(_+, H)$ and by the same argument as before $(X\\partial _x) \\circ Q$ and $S \\circ Q$ define bounded operators from $L^2$ to $W^{0,1}$ .",
"In order to prove the statement, it remains to establish boundedness of $(X \\partial _y) \\circ Q: L^2 \\rightarrow W^{0,1}$ .",
"For $u \\in L^2_{\\rm comp}(_+\\times ^b,H)$ with compact support in $[0,1] \\times ^b$ , its Fourier transform $\\hat{u} (\\xi )$ in the $^b$ component, is still an element of $L^2_{\\rm comp}(_+,H)$ with compact support in $[0,1]$ .",
"By Corollary there exists a preimage $v = L(y_0,\\xi )^{-1} \\hat{u} (\\xi )$ , and by Proposition its norm in $H$ is $O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"In particular, $v \\in L^2(_+,H)$ .",
"We compute using commutator relations Eq1, $\\begin{aligned}\\langle \\ L(y_0,\\xi ) v, &L(y_0,\\xi ) v \\ \\rangle _{L^2} =\\langle L(y_0,\\xi )^2 v ,v \\rangle _{L^2}\\\\&=\\langle (-\\partial _x^2 + X^{-2}(S(y_0)^2+S(y_0))) v , v \\rangle _{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&=\\Vert (\\partial _x+X^{-1}S(y_0)) v \\Vert ^2_{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&\\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2},\\end{aligned}$ where boundary terms at $x=0$ do not arise due to the weight $x$ in $W^{1,1} = X W^{1,0}$ .",
"Boundary terms at $x=\\infty $ do not arise since $\\Vert ~v(x) \\Vert _H =O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"We arrive at the following estimate $\\frac{\\Vert L(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2}}{\\Vert \\hat{u} (\\xi ) \\Vert _{L^2}} =\\frac{\\Vert L(y_0,\\xi )^{-1}L(y_0,\\xi ) v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}} =\\frac{\\Vert v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}} \\le \\Vert \\xi \\Vert ^{-1}.$ By continuity at $\\xi = 0$ we conclude for some constant $C>0$ $\\Vert L(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2} \\le C \\cdot (1+\\Vert \\xi \\Vert ) ^{-1} \\Vert \\hat{u} (\\xi )\\Vert _{L^2}.$ We may now estimate for any $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ $\\Vert X^{-1} (X \\partial _{y_i}) Q u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert \\xi _i \\cdot L(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ &\\le C \\int _{^b} \\frac{\\Vert \\xi \\Vert ^2}{\\Vert 1+\\xi \\Vert ^2} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\~&\\le C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ This finishes the proof.",
"We point out that it is precisely the fact that we have established invertibility of $L(y_0,\\xi )$ for any $\\xi \\in ^b$ instead of $\\xi \\ne 0$ , which allows us to write down the parametrix $Q$ explicitly using Fourier transform and establish its mapping properties as a simple consequence of the Plancherel theorem.",
"In case of $L(y_0,\\xi )$ being invertible only for $\\xi \\ne 0$ the parametrix construction needs to take care of a singularity at $\\xi =0$ via cutoff functions, in which case one cannot deduce its mapping properties by a simple application of the Plancherel theorem and is forced to employ an operator valued version of the theorem by Calderon and Vaillancourt [13].",
"We conclude with construction of a parametrix for $D_{y_0}^2$ , cf.",
"Theorem .",
"Assume the spectral Witt condition (REF ).",
"Consider $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ and denote its Fourier transform on $^b$ by $\\hat{u} (\\xi )$ .",
"Fix $y_0 \\in ^b$ and consider the square $D^2_{y_0}$ of a generalized Dirac operator.",
"We define $Q^2u(y):= \\int _{^b} e^{i \\langle y, \\xi \\rangle } (L^2(y_0,\\xi ))^{-1} \\hat{u} (\\xi )đ\\xi , \\quad đ\\xi := \\frac{d \\xi }{(2\\pi )^b}.$ Then $Q^2$ is a right-inverse to $D^2_{y_0}$ and defines a bounded operator $Q^2:W^{0}_{\\rm comp}(_+\\times ^b,H) \\subset W^{0} \\rightarrow X^2\\cdot W^{2}(_+\\times ^b,H) = W^{2,2}.$ By the Plancherel theorem we find for any $u\\in C_0^\\infty (_+\\times ^b,H^\\infty )$ $\\Vert X^{-2} \\circ Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\Vert X^{-2} \\circ (L^2(y_0,\\cdot ))^{-1} \\hat{u} \\Vert ^2_{L^2(_+\\times ^b_\\xi , H)}\\\\~&= \\int _{^b} \\Vert X^{-2} \\circ (L^2(y_0,\\xi ))^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi .$ By Proposition , the operator $X^{-2} (L^2(y_0,\\xi ))^{-1}$ defines a bounded map from $L^2(_+, H)$ to itself, with the operator norm bounded uniformly in $\\xi \\in ^b$ .",
"Denote its uniform bound by $C>0$ and compute again by Plancherel theorem $\\Vert X^{-2} \\circ Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert X^{-2} \\circ (L^2(y_0,\\xi ))^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ & \\le C \\int _{^b} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi =C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Consequently, $Q^2: L^2(_+\\times ^b,H) \\rightarrow X^2\\cdot L^2(_+\\times ^b,H) = W^{0,2}$ is bounded.",
"Furthermore, by Proposition we find for any $ V_1, V_2 \\in \\mathcal {K} := \\lbrace (X\\partial _x), S\\rbrace ,$ that the operators $V_1 \\circ X^{-2} \\circ (L^2(y_0,\\xi ))^{-1}$ and $V_1 \\circ V_2 \\circ X^{-2} \\circ (L^2(y_0,\\xi ))^{-1}$ are bounded on $L^2(_+, H)$ .",
"By the same argument as before, $V_1 \\circ Q^2$ and $V_2 \\circ V_2 \\circ Q^2$ define bounded operators from $L^2(_+\\times ^b,H)$ to $W^{0,2}(_+\\times ^b,H)$ .",
"In order to prove the statement, it remains to establish boundedness of $(X \\partial _y) \\circ V_1 \\circ Q^2$ and $(X \\partial _y)^2 \\circ Q^2$ as maps from $L^2(_+\\times ^b,H)$ to $W^{0,2}(_+\\times ^b,H)$ .",
"For $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ with compact support in $[0,1] \\times ^b$ , its Fourier transform $\\hat{u} (\\xi )$ in the $^b$ component, is still an element of $W^{0}_{\\rm comp}(_+,H)$ with compact support in $[0,1]$ .",
"By Proposition , $v = L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi ) \\in W^{2,2}(_+,H)$ .",
"By Proposition , $\\Vert v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ for some $\\delta >0$ .",
"In particular, $v \\in L^2(_+,H)$ .",
"We compute using commutator relations Eq1, $\\begin{aligned}\\langle \\ L(y_0,\\xi ) v, \\, &L(y_0,\\xi ) v \\ \\rangle _{L^2} =\\langle L(y_0,\\xi )^2 v ,v \\rangle _{L^2}\\\\&=\\langle (-\\partial _x^2 + X^{-2}(S(y_0)^2+S(y_0))) v , v \\rangle _{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&=\\Vert (\\partial _x+X^{-1}S(y_0)) v \\Vert ^2_{L^2} + \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2}\\\\&\\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert v \\Vert ^2_{L^2},\\end{aligned}$ where there are no boundary terms after integration by parts.",
"More precisely, boundary terms at $x=0$ do not arise due to the weight $x^2$ in $W^{2,2} = X^2 W^{2,0}$ .",
"Boundary terms at $x=\\infty $ do not arise since $\\Vert v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ .",
"By Proposition , $L(y_0,\\xi ) v \\in W^{1,1}(_+,H)$ with the asymptotic expansion $\\Vert L(y_0,\\xi ) v(x) \\Vert _H = O(x^{-1-\\delta })$ as $x\\rightarrow \\infty $ as well.",
"Hence, in the estimates above, we can replace $v$ with $w = L(y_0,\\xi ) v$ and still conclude $\\langle \\ L(y_0,\\xi ) w, L(y_0,\\xi ) w \\ \\rangle _{L^2} \\ge \\Vert \\xi \\Vert ^2\\cdot \\Vert w \\Vert ^2_{L^2}.$ We arrive at the following estimate $\\frac{\\Vert L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2}}{\\Vert \\hat{u} (\\xi ) \\Vert _{L^2}} =\\frac{\\Vert v \\Vert _{L^2}}{\\Vert L^2(y_0,\\xi ) v \\Vert _{L^2}} \\le \\Vert \\xi \\Vert ^{-1} \\frac{\\Vert v \\Vert _{L^2}}{\\Vert L(y_0,\\xi ) v \\Vert _{L^2}}\\le \\Vert \\xi \\Vert ^{-2}.$ By continuity at $\\xi = 0$ we conclude for some constant $C>0$ $\\Vert L^2(y_0,\\xi )^{-1} \\hat{u} (\\xi )\\Vert _{L^2} \\le C \\cdot (1+\\Vert \\xi \\Vert ) ^{-2} \\Vert \\hat{u} (\\xi )\\Vert _{L^2}.$ We may now estimate for any $u \\in W^{0}_{\\rm comp}(_+\\times ^b,H)$ $\\Vert X^{-2} (X \\partial _{y_i}) (X \\partial _{y_j}) Q^2 u \\Vert ^2_{L^2(_+\\times ^b_y, H)}&= \\int _{^b} \\Vert \\xi _i \\xi _j \\cdot L^2(y_0,\\xi )^{-1} \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\ &\\le C \\int _{^b} \\frac{\\Vert \\xi \\Vert ^2}{(1+\\Vert \\xi \\Vert )^2} \\Vert \\hat{u}(\\xi ) \\Vert ^2_{L^2(_+, H)} đ\\xi \\\\~&\\le C \\Vert u \\Vert ^2_{L^2(_+\\times ^b_y, H)}.$ Similar estimate holds for $(X \\partial _y) V \\, Q^2 u$ with $V \\in \\mathcal {K}$ .",
"This finishes the proof."
],
[
"Minimal domain of a Dirac Operator on an abstract edge",
"We now employ the previous parametrix construction in order to deduce statements on the minimal and maximal domains of $D_{y_0}$ and consequently for $D$ .",
"Recall $H=H(S(y_0))$ and the basic definitions of minimal and maximal domains.",
"As noted in Remark REF , the interpolation scales $H^s(S(y))$ and $H^s(S(y_0))$ coincide for $0\\le s \\le 1$ .",
"The maximal and minimal domain of $D$ are defined as follows: $\\begin{aligned}(D_{\\max })&:=\\lbrace u\\in L^2(_+\\times ^b,H)\\ | \\ Du\\in L^2(_+\\times ^b,H) \\rbrace \\\\(D_{\\min })&:=\\lbrace u\\in (D_{\\max }) \\ | \\ \\exists (u_n)\\subset C_0^\\infty (_+\\times ^b,H^\\infty ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\text{with}\\; u_n \\stackrel{L^2}{\\rightarrow } u,\\;Du_n \\stackrel{L^2}{\\rightarrow } Du \\rbrace .\\end{aligned}$ Using smooth cutoff functions we define localized versions of domains: $\\begin{aligned}_{\\rm comp}(D_{\\max })&:=\\lbrace \\varphi u\\ | u\\in (D_{\\max }),\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b) \\rbrace ,\\\\_{\\rm comp}(D_{\\min })&:=\\lbrace \\varphi u\\ | u\\in (D_{\\min }),\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b)\\rbrace ,\\\\W^{s,\\delta }_{\\rm }(_+\\times ^b,H)&:=\\lbrace \\varphi u\\ | u\\in X^\\delta W^s,\\varphi \\in C^\\infty _0([0,\\infty ) \\times ^b)\\rbrace ,\\end{aligned}$ where in each case we additionally requireRestriction of the support to be in $[0,1] \\times ^b$ is necessary to achieve uniformity of the estimates in Corollary and for the consequence in Proposition to hold.",
"$\\varphi \\subset [0,1] \\times ^b$ .",
"One checks directly from the definitions $_{\\rm comp}(D_{\\max / \\min }) \\subseteq (D_{\\max /\\min }).$ The maximal and minimal domains $(D_{y_0, \\, \\max }), (D_{y_0, \\, \\min })$ and their respective localized versions $_{\\rm comp}(D_{y_0, \\, \\max }), _{\\rm comp}(D_{y_0, \\, \\min })$ are defined analogously.",
"$_{\\rm comp}(D_{y_0, \\, \\max / \\min }) \\subseteq W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ .",
"Since $_{\\rm comp}(D_{y_0, \\, \\min }) \\subseteq _{\\rm comp}(D_{y_0, \\, \\max })$ , it suffices to show $_{\\rm comp}(D_{y_0, \\, \\max }) \\subseteq W^{1,1}_{\\rm comp}(_+\\times ^b,H).$ Note that the differential expression $D_{y_0}$ induces two mappings $&D_{y_0} : (D_{y_0, \\, \\max }) \\rightarrow L^2(_+\\times ^b,H), \\\\&D_{y_0} : W^{1,1}(_+\\times ^b,H) \\rightarrow L^2(_+\\times ^b,H),$ where the former is an unbounded self-adjoint operator in the Hilbert space $L^2(_+\\times ^b,H)$ , and the latter is a bounded operator between Sobolev spacesNote that $W^{1,1}(_+\\times ^b,H) \\nsubseteq L^2(_+\\times ^b,H)$ .. Theorem provides the right inverse $Q:L^2_{\\rm comp}(_+\\times ^b,H)\\rightarrow W^{1,1}(_+\\times ^b,H)$ to the latter mapping, but not to the former.",
"More precisely, we only have $\\forall \\, u \\in L^2_{\\rm comp}(_+\\times ^b,H): \\quad D_{y_0} (Q u) = u.$ The same holds for the formal adjoints $D^t_{y_0}$ and $Q^t$ in $L^2(_+\\times ^b,H)$ $&D^t_{y_0} : W^{1,1} \\rightarrow L^2, \\quad Q^t:L^2_{\\rm comp}\\rightarrow W^{1,1}\\\\&\\forall \\, u \\in L^2_{\\rm comp}(_+\\times ^b,H): \\quad D^t_{y_0} (Q^t u) = u.$ Consider $u \\in _{\\rm comp}(D_{y_0, \\, \\max })$ and a test function $\\phi \\in C_0^\\infty (_+\\times ^b,H^\\infty )$ .",
"We fix a smooth cutoff function $\\psi \\in C_0^\\infty (_+\\times ^b,H^\\infty )$ , such that $\\psi \\equiv 1$ on $u \\cup \\phi $ .",
"We compute with $L^2=L^2(_+\\times ^b,H)$ $\\langle u, \\phi \\rangle _{L^2} &= \\langle u, \\psi D^t_{y_0} (Q^t \\phi ) \\rangle _{L^2}\\\\ &= \\langle u, [\\psi , D^t_{y_0}] (Q^t \\phi ) \\rangle _{L^2} + \\langle u, D^t_{y_0} \\psi (Q^t \\phi ) \\rangle _{L^2}$ Note that $\\, [\\psi , D^t_{y_0}]$ is by construction disjoint from $u$ and consequently the first summand above is zero.",
"Using $u \\in _{\\rm comp}(D_{y_0, \\, \\max })$ we can integrate by parts and conclude $\\langle u, \\phi \\rangle _{L^2} &= \\langle u, D^t_{y_0} \\psi (Q^t \\phi ) \\rangle _{L^2}\\\\ &= \\langle D_{y_0} u, \\psi (Q^t \\phi ) \\rangle _{L^2} = \\langle D_{y_0} u, Q^t \\phi \\rangle _{L^2}\\\\ &= \\langle Q D_{y_0} u, \\phi \\rangle _{L^2}.$ We conclude that $u = Q(D_{y_0}u)$ as distributions.",
"By Theorem $u = Q(D_{y_0}u)\\in W^{1,1}_{\\rm comp}(_+\\times ^b,H).$ $_{\\rm comp}(D_{y_0, \\, \\min }) = _{\\rm comp}(D_{y_0, \\, \\max }) = W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ .",
"By Lemma it suffices to show that $W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ is included in $_{\\rm comp}(D_{y_0, \\, \\min })$ .",
"Note that $D_{y_0}: W^{1,1}_{\\rm comp}(_+\\times ^b,H)\\rightarrow L^2(_+\\times ^b,H)$ is continuous, and $C_0^\\infty (_+\\times ^b,H^\\infty )\\subset W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ is dense.",
"Consider $u\\in W^{1,1}_{\\rm comp}(_+\\times ^b,H)$ and some $(u_n)\\subset C^\\infty _0(_+\\times ^b,H^\\infty )$ such that $u_n \\stackrel{W^{1,1}}{\\rightarrow } u$ .",
"By continuity, $D_{y_0}u_n \\rightarrow D_{y_0} u$ in $L^2$ .",
"Hence by definition, $u\\in _{\\rm comp}(D_{y_0, \\, \\min })$ .",
"Now we want to extend this statement to a perturbation of $D_{y_0}$ $P = \\Gamma (\\partial _x + X^{-1}S(y)) + T + V:=D_{y_0} + D_{1,y_0}$ where $V:W^{1,1}(_+ \\times ^b, H) \\rightarrow W^{0,1}(_+ \\times ^b, H)$ is a bounded linear operator, preserving compact supports and usually referred to as a higher order term.",
"Assume in addition to the spectral Witt condition (REF ) that $\\partial _y S(y)(|S(y_0)|+1)^{-1}$ are bounded operators on $H$ for any $y, y_0\\in ^b$ .",
"Then $_{\\rm comp}(P_{\\min }) = _{\\rm comp}(P_{\\max }) =W^{1,1}_{\\rm comp}(_+ \\times ^b, H).$ Step 1: Consider $u\\in C_0^\\infty ((0,\\infty )\\times ^b, H^\\infty )$ and smooth cutoff functions $\\phi , \\psi \\in C_0^\\infty ([0,\\infty )\\times ^b)$ taking values in $[0,1]$ , such that $\\phi \\subset [0,\\epsilon )\\times B_\\epsilon (y_0)$ and $\\psi \\upharpoonright u \\equiv 1$ .",
"We compute using (REF ) $\\begin{aligned}\\Vert \\phi D_{1,y_0} u\\Vert _{L^2} &= \\Vert \\phi D_{1,y_0} Q D_{y_0} u\\Vert _{L^2}= \\Vert \\phi D_{1,y_0} Q \\psi D_{y_0} u\\Vert _{L^2}\\\\&\\le \\Vert \\phi D_{1,y_0} Q \\psi \\Vert _{L^2\\rightarrow L^2} \\cdot \\Vert D_{y_0}u\\Vert _{L^2}.\\end{aligned}$ In order to estimate the norm of $\\phi D_{1,y_0} Q \\psi $ , note that $\\begin{aligned}D_{1,y_0} Q & = \\Gamma \\, X^{-1} \\left(S(y)-S(y_0)\\right) Q + \\left(T-T_{y_0}\\right) Q + V Q\\\\ & = \\Gamma \\, X^{-1} (y-y_0) \\int _0^1 \\frac{\\partial S}{\\partial t}\\left(y_0 + t(y-y_0)\\right) dt\\ Q\\\\ & + (y-y_0) \\int _0^1 \\frac{\\partial T}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt \\ Q + VQ.\\end{aligned}$ In view of the assumption (REF ) and boundedness of the higher order term $V:W^{1,1} \\rightarrow W^{0,1}$ we conclude from Theorem that $X^{-1} \\frac{\\partial S}{\\partial t}\\left(y_0 + t(y-y_0)\\right) Q \\psi ,\\quad \\frac{\\partial T}{\\partial t}\\left(y_0 + t(y-y_0)\\right) \\circ Q \\psi ,\\quad X^{-1} V \\circ Q \\psi $ are bounded operators on $L^2(_+ \\times ^b, H)$ with bound uniform in $t\\in [0,1]$ and $\\psi $ .",
"Hence we conclude for some unform constant $C>0$ $\\begin{aligned}\\Vert \\phi D_{1,y_0} Q \\psi \\Vert _{L^2\\rightarrow L^2} \\le C \\left( \\sup _{q\\in \\phi } x(q)+ \\sup _{q\\in \\phi } \\Vert y(q) -y_0\\Vert \\right) \\le 2 \\epsilon C.\\end{aligned}$ Thus we may choose $\\epsilon >0$ sufficiently small such that $\\Vert \\phi D_{1,y_0} u\\Vert _{L^2} \\le q \\cdot \\Vert D_{y_0} u\\Vert _{L^2}, \\;\\;\\text{for}\\;\\; q<1.$ Then the following inequalities hold for $u\\in C^\\infty _0 (_+\\times ^b, H^\\infty )$ , $\\begin{aligned}\\Vert (D_{y_0} + \\phi D_{1,y_0})u\\Vert _{L^2} &\\le \\Vert D_{y_0} u\\Vert _{L^2} + q \\Vert D_{y_0}u\\Vert _{L^2}\\\\ &\\le (1+q)\\cdot \\Vert D_{y_0} u\\Vert _{L^2}.\\end{aligned}$ On the other hand $\\begin{aligned}\\Vert D_{y_0} u\\Vert _{L^2} &\\le \\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2} + \\Vert \\phi D_{1,y_0}u\\Vert _{L^2}\\\\&\\le \\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2} + q\\Vert D_{y_0} u\\Vert _{L^2}.",
"\\\\\\Rightarrow \\ \\Vert D_{y_0} u\\Vert _{L^2} &\\le (1-q) ^{-1}\\Vert (D_{y_0}+\\phi D_{1,y_0})u\\Vert _{L^2}.\\end{aligned}$ Thus the graph-norms of $D_{y_0}$ and $(D_{y_0}+\\phi D_{1,y_0})$ are equivalent and hence their minimal domains coincide.",
"Same statement holds for the maximal as well as the localized domains.",
"Thus we have the following equalities.",
"$\\begin{split}&_{\\rm comp}(D_{y_0, \\, \\min }) = _{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min }), \\\\&_{\\rm comp}(D_{y_0, \\, \\max }) = _{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\max }).\\end{split}$ The equalities continue to hold for a cutoff function $\\phi \\in C_0^\\infty ((0,\\infty )\\times ^b)$ such that for some $x_0 > \\epsilon $ , $\\phi \\subset (x_0 - \\epsilon , x_0 + \\epsilon ) \\times B_\\epsilon (y_0)$ by a similar argument.",
"Step 2: We now prove the following inclusion $_{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min })\\subseteq _{\\rm comp}((D_{y_0} + D_{1,y_0})_{\\min }).$ Indeed, for any $u\\in _{\\min }(D_{y_0} + \\phi D_{1,y_0})$ there exists $(u_n)\\subset C^\\infty _0((0,\\infty )\\times ^b, H^\\infty )$ converging to $u$ in the graph norm of $(D_{y_0} + \\phi D_{1,y_0})$ .",
"By (REF ) and Corollary , $(u_n)$ converges to $u$ in $W^{1,1}$ .",
"Hence, using continuity of $D_{1,y_0}: W^{1,1}\\rightarrow L^2$ we conclude $\\begin{aligned}(D_{y_0}+D_{1,y_0})u_n &= (D_{y_0} + \\phi D_{1,y_0})u_n + (1-\\phi )D_{1,y_0} u_n\\\\&\\stackrel{L^2}{\\rightarrow } (D_{y_0}+\\omega D_{1,y_0})u + (1-\\phi )D_{1,y_0} u = (D_{y_0} +D_{1,y_0})u.\\end{aligned}$ Hence $u\\in _{\\rm comp}((D_{y_0} + D_{1,y_0})_{\\min })$ and (REF ) follows.",
"Step 3: Consider now $u \\in _{\\rm comp}(P_{\\max / \\min })$ .",
"Due to compact support there exist finitely many points $\\lbrace (x_1,y_1), \\ldots , (x_N,y_N)\\rbrace \\subset _+ \\times ^b$ and smooth cutoff functions $\\lbrace \\psi _1, \\ldots , \\psi _N\\rbrace \\subset C_0^\\infty ([0,\\infty )\\times ^b)$ such that $u = \\sum _{j=1}^N \\psi _j u, \\quad (\\psi _j u) \\subset \\left(\\left(x_j - \\frac{\\epsilon }{2}, x_j + \\frac{\\epsilon }{2}\\right)\\cap [0, \\epsilon ) \\right)\\times B_{\\frac{\\epsilon }{2}}(y_j).$ The maximal and minimal domains are stable under multiplication with cutoff functions and hence each $\\psi _j u \\in _{\\rm comp}(P_{\\max / \\min })$ .",
"Consider for each $j=1, \\ldots , N$ a cutoff function $\\phi _j \\in C_0^\\infty ([0,\\infty )\\times ^b)$ such that $\\phi _j \\subset ((x_j - \\epsilon , x_j + \\epsilon )\\cap [0,\\epsilon )) \\times B_\\epsilon (y_j)$ and $\\phi _j \\upharpoonright (\\psi _j u) \\equiv 1$ .",
"Then as distributions $P (\\psi _j u) = (D_{y_j} + D_{1,y_j}) \\psi _j u = (D_{y_j} + \\phi _j D_{1,y_j}) \\psi _j u.$ We conclude $\\psi _j u \\in _{\\rm comp}((D_{y_j} + \\phi _j D_{1,y_j})_{\\max / \\min })$ .",
"In view of (REF ) and Corollary we find $_{\\rm comp}(P_{\\min }) \\subseteq _{\\rm comp}(P_{\\max }) \\subseteq W^{1,1}_{\\rm comp}(_+ \\times ^b, H).$ Step 4: The statement now follows from a sequence of inclusions $\\begin{aligned}W^{1,1}_{\\rm comp} &\\ = _{\\rm comp}(D_{y_0, \\, \\min }) \\\\~&\\stackrel{(\\ref {a})}{=}_{\\rm comp}((D_{y_0} + \\phi D_{1,y_0})_{\\min })\\stackrel{(\\ref {c})}{\\subseteq } _{\\rm comp}(P_{\\min })\\\\&\\ \\subseteq _{\\rm comp}(P_{\\max }) \\stackrel{(\\ref {d})}{=} W^{1,1}_{\\rm comp}.\\end{aligned}$ The first equality is due to Corollary .",
"Hence all inclusions are in fact equalities and the statement follows."
],
[
"Minimal domain of a Laplace Operator on an abstract edge",
"Definition extends to define the notion of minimal and maximal domain for the squares $D_{y_0}^2$ and $D^2$ of the generalized Dirac operators.",
"Their localized versions are defined as in (REF ).",
"In this section, we discuss the minimal and maximal domains of $D_{y_0}^2$ and $D^2$ by repeating the arguments of § with appropriate changes.",
"We also note as in Remark REF , the interpolation scales $H^s(S(y))$ and $H^s(S(y_0))$ coincide for $0\\le s \\le 1$ , but a priori may differ for $s>1$ .",
"While this was sufficient for the discussion of the domain of $D$ in the previous section, it is insufficient for the discussion of the domain of $D^2$ .",
"Hence, within the scope of this section we pose the following The interpolation scales $H^s(S(y))$ are independent of $y\\in ^b$ for $0 \\le s \\le 2$ , in which case we write $H^s\\equiv H^s(S(y))$ .",
"The following result follows by repeating the arguments of Lemma and Corollary ad verbatim, where $D_{y_0}$ is replaced by $D_{y_0}^2$ , $W^{1,1}$ by $W^{2,2}$ and $Q$ by $Q^2$ .",
"These changes do not affect the overall argument.",
"$_{\\rm comp}(D^2_{y_0, \\, \\min }) = _{\\rm comp}(D^2_{y_0, \\, \\max }) = W^{2,2}_{\\rm comp}(_+\\times ^b,H)$ .",
"Now we want to extend this statement to a perturbation of $D^2_{y_0}$ $G = -x^2 + X^{-2}\\ S(y) \\ (S(y)+1) + T^2 + W:=D^2_{y_0} + R_{y_0}$ where $W:W^{2,2}(_+ \\times ^b, H) \\rightarrow W^{0,1}(_+ \\times ^b, H)$ is a bounded linear operator, preserving compact supports, and is referred to as a higher order term.",
"Assume in addition to the spectral Witt condition (REF ) that $\\partial _y S(y) \\circ (|S(y_0)|+1)^{-1}, \\quad (|S(y_0)|+1) \\circ \\partial _y S(y) \\circ (|S(y_0)|+1)^{-2}$ are bounded operators on $H$ for any $y, y_0\\in ^b$ .",
"Then $_{\\rm comp}(G_{\\min }) = _{\\rm comp}(G_{\\max }) =W^{2,2}_{\\rm comp}(_+ \\times ^b, H).$ The assumption (REF ) translates into the condition that for $A=|S|+\\frac{1}{2}$ $\\partial _y A^2(y) \\circ (|S(y_0)|+1)^{-2}$ is bounded.",
"From there we proceed exactly as in Theorem .",
"Consider $u\\in C_0^\\infty ((0,\\infty )\\times ^b, H^\\infty )$ and smooth cutoff functions $\\phi , \\psi \\in C_0^\\infty ([0,\\infty )\\times ^b)$ taking values in $[0,1]$ , such that $\\phi \\subset [0,\\epsilon )\\times B_\\epsilon (y_0)$ and $\\psi \\upharpoonright u \\equiv 1$ .",
"We compute using the analogue of (REF ) for $D^2_{y_0}$ $\\begin{aligned}\\Vert \\phi R_{y_0} u\\Vert _{L^2} &= \\Vert \\phi R_{y_0} Q^2 D^2_{y_0} u\\Vert _{L^2}= \\Vert \\phi R_{y_0} Q^2 \\psi D^2_{y_0} u\\Vert _{L^2}\\\\&\\le \\Vert \\phi R_{y_0} Q^2 \\psi \\Vert _{L^2\\rightarrow L^2} \\cdot \\Vert D^2_{y_0}u\\Vert _{L^2}.\\end{aligned}$ In order to estimate the norm of $\\phi R_{y_0} Q^2 \\psi $ , note that $\\begin{aligned}R_{y_0} Q ^2 & = X^{-2} \\left(A^2(y)-A^2(y_0)\\right) Q^2+ \\left(T^2-T^2_{y_0}\\right) Q^2 + W Q^2\\\\ & = X^{-2} (y-y_0) \\int _0^1 \\frac{\\partial A^2}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt\\ Q^2\\\\ & + (y-y_0) \\int _0^1 \\frac{\\partial T^2}{\\partial t}\\left(y_0 +t(y-y_0)\\right)dt\\ Q^2 + WQ^2.\\end{aligned}$ In view of (REF ) and boundedness of the higher order term $W:W^{2,2} \\rightarrow W^{0,1}$ we conclude from Theorem that $X^{-2} \\frac{\\partial A^2}{\\partial t}\\left(y_0 + t(y-y_0)\\right) Q^2 \\psi ,\\quad \\frac{\\partial T^2}{\\partial t}\\left(y_0 + t(y-y_0)\\right) \\, Q^2 \\psi ,\\quad X^{-1} W \\, Q^2 \\psi $ are bounded operators on $L^2(_+ \\times ^b, H)$ with bound uniform in $t\\in [0,1]$ and $\\psi $ .",
"Hence we conclude for some unform constant $C>0$ $\\begin{aligned}\\Vert \\phi R_{y_0} Q^2 \\psi \\Vert _{L^2\\rightarrow L^2} \\le C \\left( \\sup _{q\\in \\phi } x(q)+ \\sup _{q\\in \\phi } \\Vert y(q) -y_0\\Vert \\right) \\le 2 \\epsilon C.\\end{aligned}$ Thus we may choose $\\epsilon >0$ sufficiently small such that $\\Vert \\phi R_{y_0} u\\Vert _{L^2} \\le q \\cdot \\Vert D^2_{y_0} u\\Vert _{L^2}, \\;\\;\\text{for}\\;\\; q<1.$ Then the following inequalities hold for $u\\in C^\\infty _0 (_+\\times ^b, H^\\infty )$ , $\\begin{aligned}\\Vert (D^2_{y_0} + \\phi R_{y_0})u\\Vert _{L^2} &\\le \\Vert D^2_{y_0} u\\Vert _{L^2} + q \\Vert D^2_{y_0}u\\Vert _{L^2}\\\\ &\\le (1+q)\\cdot \\Vert D^2_{y_0} u\\Vert _{L^2}.\\end{aligned}$ On the other hand $\\begin{aligned}\\Vert D^2_{y_0} u\\Vert _{L^2} &\\le \\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2} + \\Vert \\phi R_{y_0}u\\Vert _{L^2}\\\\&\\le \\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2} + q\\Vert D^2_{y_0} u\\Vert _{L^2}.",
"\\\\\\Rightarrow \\ \\Vert D^2_{y_0} u\\Vert _{L^2} &\\le (1-q) ^{-1}\\Vert (D^2_{y_0}+\\phi R_{y_0})u\\Vert _{L^2}.\\end{aligned}$ Thus the graph-norms of $D^2_{y_0}$ and $(D^2_{y_0}+\\phi R_{y_0})$ are equivalent and hence their minimal domains coincide.",
"Same statement holds for the maximal as well as the localized domains.",
"Thus we have the following equalities.",
"$\\begin{split}&_{\\rm comp}(D^2_{y_0, \\, \\min }) = _{\\rm comp}((D^2_{y_0} + \\phi R_{y_0})_{\\min }), \\\\&_{\\rm comp}(D^2_{y_0, \\, \\max }) = _{\\rm comp}((D^2_{y_0} + \\phi R_{y_0})_{\\max }).\\end{split}$ The equalities continue to hold for a cutoff function $\\phi \\in C_0^\\infty ((0,\\infty )\\times ^b)$ such that for some $x_0 > \\epsilon $ , $\\phi \\subset (x_0 - \\epsilon , x_0 + \\epsilon ) \\times B_\\epsilon (y_0)$ by a similar argument.",
"From there we may repeat the arguments of the proof of Theorem ad verbatim, replacing $D_{y_0}$ by $D^2_{y_0}$ , $D_{1,y_0}$ by $R_{y_0}$ , $P$ by $G$ , $W^{1,1}$ by $W^{2,2}$ .",
"These replacements do not affect the overall argument."
],
[
"Domains of Dirac and Laplace Operators on a Stratified Space",
"Consider a compact stratified space $M_k$ of depth $k\\in $ with an iterated cone-edge metric $g_k$ .",
"Each singular stratum $B$ of $M_k$ admits an open neighbourhood $\\subset M_k$ with local coordinates $y$ and a defining function $x_k$ such that $g|_{} = dx_k^2 + x_k^2 \\ g_{k-1}(x_k, y) + g_B(y) + h =: \\overline{g} + h,$ where $g_{k-1}(x_k,y)$ is a smooth family of iterated cone-edge metrics on a compact stratified space $M_{k-1}$ of lower depth and $h$ is a higher order symmetric 2-tensor, smooth on the resolution $\\widetilde{}$ with $|h|_{\\overline{g}} = O(x_k)$ as $x_k \\rightarrow 0$ .",
"The associated Sobolev-spaces are defined in Definition REF .",
"Recall, their elements take values in the vector bundle $E$ , which denotes the exterior algebra of the incomplete edge cotangent bundle $\\Lambda ^* {}^{ie}T^*$ in case of the Gauss–Bonnet operator, and the spinor bundle in case of the spin Dirac operator.",
"We usually omit $E$ from the notation.",
"We introduce here the localized versions of the Sobolev spaces ($s\\in $ ) $\\begin{aligned}^{s,\\delta }_{e,{\\rm comp}} :=\\lbrace \\phi \\cdot u \\ | \\ \\phi \\in C^\\infty _0(\\widetilde{}),u\\in ^{s,\\delta }_{e} \\rbrace .\\end{aligned}$ Consider the unitary transformation $\\Phi $ in (REF ), [10], which maps $L^2(,E,\\overline{g})$ to $L^2(, E, \\overline{g}_{\\textup {prod}})$ , where we recall $\\overline{g}$ from (REF ) and set $\\overline{g}_{\\textup {prod}}:= dx_k^2 + g_{k-1}(x_k, y) + g_B(y)$ .",
"The spaces $^{*,*}_{e,{\\rm comp}}$ with compact support in $$ may be defined with respect to $\\overline{g}$ and $\\overline{g}_{\\textup {prod}}$ .",
"We indicate the choice of the metric when necessary, $^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}}), L^2_{{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}})$ , and do not specify the metric when the statement holds for both choices.",
"Note $^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}}) =\\Phi ^{*,*}_{e,{\\rm comp}}(M_k,\\overline{g}).$ Whenever we use the Sobolev spaces $^{*,*}_{e}(M_k)$ or $L^2(M_k)$ without compact support in the open interior of $M_k$ , we use the iterated cone-edge metric $g_k$ in the definition of the $L^2$ -structure.",
"We write $L^2_{\\rm comp}:=^{0,0}_{e,{\\rm comp}}$ and denote by $\\rho _k$ a smooth function on the resolution $\\widetilde{M}_k$ , nowhere vanishing in its open interior, and vanishing to first order at each boundary face of $\\widetilde{M}_k$ .",
"Iteratively, $\\rho _k = x_k \\rho _{k-1}$ .",
"Then $\\begin{split}^{1,1}_{e,{\\rm comp}} &= \\rho _k \\ \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\rho _k \\partial _x u, \\rho _k \\partial _y u, \\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} \\ \\rbrace \\\\&= \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\frac{u}{\\rho _k}, \\partial _x u, \\partial _y u, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} \\ \\rbrace .\\end{split}$ Here, the first equality in (REF ) follows by Definition REF , once we recall from (REF ) the following iterative structure of edge vector fields $\\mathcal {V}_{e,k}\\upharpoonright \\widetilde{} = C^\\infty (\\widetilde{})\\textup {- span}\\,\\lbrace \\rho _k \\partial _x, \\rho _k\\partial _y,\\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace .$ The second equality in (REF ) is now straightforward.",
"Similarly $\\begin{split}^{2,2}_{e,{\\rm comp}} &= \\rho ^2_k \\ \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\lbrace \\rho _k \\partial _x, \\rho _k \\partial _y, \\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace ^j \\, u \\in L^2_{\\rm comp}, \\ j=1,2 \\rbrace \\\\&= \\lbrace u \\in L^2_{\\rm comp} \\ | \\ \\lbrace \\rho _k^{-1}, \\partial _x, \\partial _y, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1})\\rbrace ^j \\, u \\in L^2_{\\rm comp}, \\ j=1,2 \\rbrace .\\end{split}$ The spin Dirac and the Gauss–Bonnet operators $D_k$ on $(M_k,g_k)$ admit under a rescaling $\\Phi $ as in (REF ) the following form over the singular neighbourhood $\\subset M_k$ $\\Phi \\circ D_k \\circ \\Phi ^{-1}= \\Gamma (\\partial _{x_k} + X_k^{-1}S_{k-1}(y))+T + V,$ which satisfies the following iterative properties $S_{k-1}(y) = D_{k-1}(y)+R_{k-1}(y)$ , where $D_{k-1}(y)$ is a smooth family of differential operators (spin Dirac or the Gauss–Bonnet operators) on $(M_{k-1}, g_{k-1}(0,y))$ .",
"The operators $S_{k-1}(y),D_{k-1}(y)$ extend continuously to bounded maps $^{1,1}_e(M_{k-1}) \\rightarrow L^2(M_{k-1})$ .",
"Moreover, $R_{k-1}(y)$ extends continuously to a bounded operator on $L^2(M_{k-1})$ ; $x_k^{-1}V$ extends continuously to a map from $^{1,1}_{e, {\\rm comp}}$ to $L^2_{\\rm comp}$ ; $T$ is a Dirac Operator on $B$ .",
"Since at this stage essential self-adjointness of each $S_{k-1}(y)$ and discreteness of its self-adjoint extension is yet to be established, we reformulate the spectral Witt condition (REF ) in terms of quadratic forms.",
"Here, we employ the notions introduced in Kato [21].",
"We define for any smooth compactly supported $u \\in C^\\infty _0(M_{k-1})$ using the inner product of $L^2(M_{k-1}, g_{k-1}(0,y))$ $t(S_{k-1}(y))[u] := \\Vert S_{k-1}(y) u\\Vert ^2_{L^2}.$ This is the quadratic form associated to the symmetric differential operator $S_{k-1}(y)^2$ , densely defined with domain $C^\\infty _0(M_{k-1})$ in the Hilbert space $L^2(M_{k-1}, g_{k-1}(0,y))$ .",
"The numerical range of $t(S_{k-1}(y))$ is defined by $\\Theta (S_{k-1}(y)) := \\left\\lbrace t(S_{k-1}(y))[u] \\in \\mid u \\in C^\\infty _0(M_{k-1}),\\Vert u\\Vert ^2_{L^2} = 1\\right\\rbrace .$ We can now reformulate the spectral Witt condition, cf.",
"(REF ), as follows.",
"The operator $D_k$ on the stratified space $M_k$ satisfies the spectral Witt condition, if there exists $\\delta > 0$ such that in all depths $j \\le k$ the numerical ranges $\\Theta (S_{k-1}(y))$ are subsets of $[1+\\delta , \\infty )$ for any $y\\in B$ .",
"Assume that $S_{k-1}(y)$ with domain $C^\\infty _0(M_{k-1})$ in the Hilbert space $L^2(M_{k-1}, g_{k-1}(0,y))$ is essentially self-adjoint and its self adjoint realization is discrete.",
"Then $\\Theta (S_{k-1}(y)) \\subset [1 + \\delta , \\infty )$ for some $\\delta > 0$ if and only if $\\textup {Spec} S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ .",
"By Kato [21], $\\Theta (S_{k-1}(y))$ is a dense subset of $\\textup {Spec} \\, S_{k-1}(y)^2$ .",
"If the spectral Witt condition in the sense of Definition holds, this implies that $\\textup {Spec} \\, S_{k-1}(y)^2 \\subset [1 + \\delta , \\infty )$ for some $\\delta >0$ .",
"By discreteness this is equivalent to $\\textup {Spec} S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ .",
"Conversely, if $\\textup {Spec} \\, S_{k-1}(y) \\cap [-1, 1] = \\varnothing $ , then by discreteness of the spectrum, $S_{k-1}(y)^2 > 4+\\delta $ for some $\\delta > 0$ .",
"The spectral Witt condition in the sense of Definition now follows, since by Kato [21], $\\Theta (S_{k-1}(y))$ is a dense subset of $\\textup {Spec} \\, S_{k-1}(y)^2$ .",
"We can now prove our main result.",
"Let $M_k$ be a compact stratified Witt space.",
"Let $D_k$ denote either the Gauss–Bonnet or the spin Dirac operator.",
"Assume that $D_k$ satisfies the spectral Witt conditionIn case of the Gauss–Bonnet operator on a stratified Witt space this can always be achieved by scaling the iterated cone-edge metric on fibers accordingly.. Then $_{\\max }(D_k) = _{\\min }(D_k) = ^{1,1}_e(M_k)$ .",
"We prove the result by induction on the following statement.",
"On any compact stratified space $M_j$ the operator $D_j$ satisfies the following conditions near each stratum $B$ : For $y\\in B$ , $S_{j-1}(y)$ admits a unique self-adjoint extension in $L^2(M_{j-1})$ with discrete spectrum and $S_{j-1}\\cap [-1, 1] = \\varnothing $ .",
"The unique self-adjoint domain of $S_{j-1}(y)$ is given by $^{1,1}_e(M_{j-1})$ .",
"The compositions $S_{j-1}(y) (|S_{j-1}(y_0)|+1)^{-1}$ and $\\partial _y S_{j-1}(y) (|S_{j-1}(y_0)|+1)^{-1}$ are bounded on $L^2(M_{j-1})$ for $y,y_0\\in B$ .",
"These assumptions are trivially satisfied if $j=1$ .",
"Assume that Assumption is satisfied for $j\\le k$ .",
"We need to prove that Assumption is then satisfied for $j \\le k+1$ .",
"Let $_{\\rm comp}(D_k)$ denote elements in the maximal domain of $D_k$ with compact support in $\\widetilde{}$ .",
"Then by Theorem , we conclude $\\begin{aligned}\\Phi _{\\rm comp}(D_k) &\\equiv _{\\rm comp}(\\Phi \\circ D_k \\circ \\Phi ^{-1})\\\\~&= W^{1,1}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1}))\\\\&= ^{1,1}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\cap ^{0,1}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1})\\\\&\\subseteq \\lbrace u \\in L^2_{\\rm comp} (M_k,\\overline{g}_{\\textup {prod}})\\ | \\ \\frac{u}{\\rho _k}, \\partial _x u, \\partial _y u, \\rho _k^{-1}\\mathcal {V}_{e, k-1}(M_{k-1}) u\\\\ &\\in L^2_{\\rm comp} (M_k,\\overline{g}_{\\textup {prod}}) \\rbrace =^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}_{\\textup {prod}}) \\equiv \\Phi ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}),\\end{aligned}$ where we used (REF ) in the last line.",
"On the other hand it is straightforward to check that $\\begin{aligned}&\\Phi ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}) \\equiv ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g}_{\\textup {prod}}) \\\\~&= \\rho _k \\ \\lbrace u \\in L^2_{\\rm comp}(M_k, \\overline{g}_{\\textup {prod}}) \\ | \\ \\rho _k \\partial _x u, \\rho _k \\partial _y u, \\mathcal {V}_{e, k-1}(M_{k-1}) u \\in L^2_{\\rm comp} (M_k, \\overline{g}_{\\textup {prod}}) \\rbrace \\\\&\\subseteq ^{1,1}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\cap ^{0,1}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1}) \\\\& = W^{1,1}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1})) = _{\\rm comp}(\\Phi \\circ D_k \\circ \\Phi ^{-1})\\equiv \\Phi _{\\rm comp}(D_k).\\end{aligned}$ We conclude $_{\\rm comp}(D_k) = ^{1,1}_{e,{\\rm comp}}(M_k, \\overline{g})$ and hence $(D_k) = ^{1,1}_{e}(M_k)$ .",
"Essential self-adjointness of $D_k$ implies essential self-adjointness of $S_k$ with the domain of both given by $^{1,1}_{e}(M_k)$ independently of parameters.",
"The domain $^{1,1}_{e}(M_k)$ embeds compactly into $L^2(M_k)$ and hence both $D_k$ and $S_k$ are discrete.",
"Since $S_k$ is discrete, the spectral Witt condition of Definition implies $S_{k} \\cap \\left[-1, 1\\right] = \\varnothing .$ The mapping properties of $(|S_{k}|+1)^{-1}$ are derived from the mapping properties of the model parametrix in Theorem in the usual way and hence $(|S_{k}|+1)^{-1}:L^2(M_k) \\rightarrow ^{1,1}_{e}(M_k)$ is bounded.",
"Since $S_k, \\partial _y S_k$ are bounded maps from $^{1,1}_{e}(M_k)$ to $L^2(M_k)$ by the iterative properties of the individual operators in (REF ), we conclude that Assumption is satisfied for $j \\le k+1$ and hence holds for all $j \\in $ .",
"Similar arguments apply for the Laplace operators.",
"Let $M_k$ be a compact stratified Witt space.",
"Let $D_k$ denote either the Gauss–Bonnet or the spin Dirac operator.",
"Assume that $D_k$ satisfies the spectral Witt condition.",
"Then $_{\\max }(D^2_k) = _{\\min }(D^2_k) = ^{2,2}_e(M_k)$ .",
"We prove the result by induction.",
"The statement is trivially satisfied if $k=0$ .",
"Assume that the statement holds for $(k-1)\\in _0$ .",
"In particular, by induction hypothesis and by Theorem $\\begin{split}&H^1(S_{k-1})\\equiv (S_{k-1}) = ^{1,1}_e(M_{k-1}), \\\\&H^2(S_{k-1}) \\equiv (S^2_{k-1}) = ^{2,2}_e(M_{k-1}).\\end{split}$ Since the domains $(S^2_{k-1}(y))$ are independent of $y$ by the induction hypothesis, their interpolation scales $H^s(S_{k-1}(y))$ coincide for $0\\le s \\le 2$ and the Assumption is satisfied.",
"The spectral Witt condition is satisfied in each depth by Theorem .",
"We need to prove the statement for $k$ .",
"Let $_{\\rm comp}(D^2_k)$ denote elements in the maximal domain of $D^2_k$ with compact support in $\\widetilde{}$ .",
"Then by Theorem and (REF ) we conclude $\\begin{aligned}\\Phi _{\\rm comp}(D^2_k) &\\equiv _{\\rm comp}(\\Phi \\circ D^2_k \\circ \\Phi ^{-1})\\\\&= W^{2,2}_{\\rm comp} (_+\\times ^b,H^{\\bullet }(S_{k-1}))\\\\&= ^{2,2}_{e,{\\rm comp}}(_+\\times ^b)L^2(M_{k-1})\\\\~&\\cap ^{0,2}_{e,{\\rm comp}}(_+\\times ^b)^{1,1}_e(M_{k-1}) \\\\~&\\cap ^{0,2}_{e,{\\rm comp}}(_+\\times ^b)^{2,2}_e(M_{k-1}) \\\\ &= ^{2,2}_{e,{\\rm comp}}(M_k,\\overline{g}_{\\textup {prod}})\\equiv \\Phi ^{2,2}_{e,{\\rm comp}}(M_k,\\overline{g}).\\end{aligned}$ where we used (REF ) in the last equality.",
"The statement follows.",
"We conclude the section with pointing out that while we cannot geometrically control the spectral Witt condition in case of the spin Dirac operator, for the Gauss–Bonnet operator on a stratified Witt space, we find $0 \\notin S_{k}$ in each iteration step, and can scale the spectral gap up by a simple rescaling of the metric to achieve the spectral Witt condition."
],
[
"Notation",
"In this section matrices $(a_{ij})_{1\\le i,j\\le n}$ will often be abbreviated $(a_{ij})_{ij}$ as long as the size $n$ is clear from the context.",
"Summations $\\sum _{i,j,k,\\ldots }$ will always denote a finite sum where all summation indices run independently from 1 to $n$ ."
],
[
"Positivity of Matrices of Operators on Hilbert spaces",
" The following result is based on Lance [24].",
"Let $a=(a_{ij})_{1\\le i, j \\le n}$ , $b=(b_{ij})_{1\\le i,j \\le n}$ be matrices of operators on Hilbert spaces $H_1$ , $H_2$ , respectively.",
"I.e., $a_{ij}\\in (H_1)$ , $b_{ij}\\in (H_2)$ .",
"We may view $a$ as an element of $_n((H_1))$ or of $(H_1^n)$ .",
"Assume that $a\\ge 0$ and $b\\ge 0$ .",
"Then the following holds.",
"$(a_{ij}\\otimes b_{ij})_{ij}\\ge 0$ in $((H_1H_2)^n) = _n((H_1H_2))$ ; $\\sum _{i,j} a_{ij} \\otimes b_{ij} \\ge 0$ in $(H_1H_2)$ .",
"If $a\\le c=(c_{ij})_{ij} \\in (H_1^n)$ , $b\\le d=(d_{ij})_{ij}\\in (H_2^n)$ then $(a_{ij} \\otimes b_{ij})_{ij} \\le (c_{ij}\\otimes d_{ij})_{ij}.$ Note that for $H_1 = H_2 = this is an elementary statement about positivesemi-definite matrices.$ (1) Write $a=s^{\\ast } s$ , $s=(s_{ij})$ , $b=t^{\\ast } t$ , $t=(t_{ij})$ .",
"Thus $a_{ij} = \\sum _k s^{\\ast }_{ki}s_{kj}$ , $b_{ij} = \\sum _k t^{\\ast }_{ki} t_{kj}$ , and $a_{ij} \\otimes b_{ij}= \\sum _{k,l} s^{\\ast }_{ki}s_{kj} \\otimes t_{li}^{\\ast } t_{lj}= \\sum _{k,l} (s_{ki}\\otimes t_{li})^{\\ast }(s_{kj}\\otimes t_{lj}).$ So it suffices to prove that the matrices $\\bigl \\lbrace (s_{ki}\\otimes t_{li})^{\\ast } (s_{kj}\\otimes t_{lj})\\bigr \\rbrace _{ij} \\ge 0.$ For fixed $k,l$ let $T_{i}:=s_{ki}\\otimes t_{li}$ .",
"Then for $\\xi =(\\xi _i)_{1\\le i \\le n} \\in (H_1H_2)^n$ we have $\\begin{split}\\langle (T_i^\\ast T_j)_{ij} \\xi , \\xi \\rangle &= \\Big \\langle \\sum _k T_i^\\ast T_k \\xi _k_i, \\xi \\Bigr \\rangle = \\sum _{i,k} \\langle T_i^\\ast T_k \\xi _k , \\xi _i \\rangle \\\\&= \\sum _{i,k} \\langle T_k \\xi _k , T_i \\xi _i \\rangle = \\Vert (T_i \\xi _i)_i \\Vert ^2 \\ge 0.\\end{split}$ So indeed the matrix $(T_i^\\ast T_j)_{ij}$ is $\\ge 0$ .",
"(2) It suffices to show that if $(f_{ij})_{ij} := (a_{ij}\\otimes b_{ij})_{ij} \\ge 0$ then $\\sum _{i,j} f_{ij} \\ge 0$ .",
"Given $x\\in H$ put $y_i = x$ , $y = (y_i)_{1\\le i \\le n}\\in H^n$ .",
"Then $\\begin{split}0\\le \\langle (a_{ij})\\cdot (y_i), (y_i) \\rangle &=\\sum _i\\Bigl \\langle \\sum _j a_{ij}y_j,y_i\\Bigr \\rangle \\\\&=\\Bigl \\langle \\sum _{i,j} a_{ij} x, x\\Bigr \\rangle = \\Bigl \\langle \\sum _{i,j} a_{ij}x,x\\Bigr \\rangle .\\end{split}$ (3) From $c-a\\ge 0$ and $d-b\\ge 0$ and (1) we infer that the matrices $(c_{ij}-a_{ij})\\otimes b_{ij} $ and $c_{ij}\\otimes ( d_{ij}-b_{ij}) $ are $\\ge 0$ and hence $0 \\le (c_{ij}-a_{ij})\\otimes b_{ij} +c_{ij}\\otimes ( d_{ij}-b_{ij}) = ( c_{ij}\\otimes d_{ij} ) - ( a_{ij} \\otimes b_{ij}).$ Let $A,B$ be self-adjoint operators in Hilbert spaces $H_1$ , $H_2$ , respectively, and let $AB:= \\text{ closure of } A\\otimes _B \\text{ on }^{\\infty }(A)\\otimes _^{\\infty } (B),$ where $^{\\infty }(A):= \\bigcap _{s\\ge 0} (|A|^{s})$ .",
"Then $AB$ is self-adjoint and $^{\\infty }(A)\\otimes _^{\\infty } (B)= ^{\\infty }(A) \\otimes _H_2 \\cap H_1 \\otimes _^{\\infty }(B)$ .",
"It is straightforward to see that $A\\otimes B$ is symmetric on $^{\\infty }(A)\\otimes _^{\\infty } (B)$ and hence $AB$ is a symmetric closed operator.",
"It remains to show self-adjointness which is equivalent to the denseness of the ranges $(AB\\pm i I)$ .",
"First we prove the statement for $B = I$ being the identity on $H_2$ .",
"Then the graph norm of $A\\otimes I$ on $^{\\infty }(A)\\otimes H_2$ is the Hilbert space tensor norm for $(A)H_2$ .",
"Hence $(AI) = (A)H_2$ .",
"The resolvent of $AI$ is obviously $(AI - \\lambda I I)^{-1} = (A-\\lambda I )^{-1}I$ .",
"Thus the denseness of $(AI \\pm II)$ follows from the denseness of $(A\\pm I)$ .",
"Hence $AI$ is self-adjoint.",
"For general $B$ we now know that $AI$ and $IB$ are commuting self-adjoint operators.",
"Hence $(AI)\\cdot (I B)$ is essentially self-adjoint on $^{\\infty }(A) \\otimes _H_2 \\bigcap H_1 \\otimes _^{\\infty }(B).$ It remains to see that the latter equals $^{\\infty }(A) \\otimes _^{\\infty }(B)$ .",
"Because then $(A\\otimes I)\\cdot (I\\otimes B) = A\\otimes _B$ and we conclude the essential self-adjointness of $A\\otimes _B$ .",
"To this end consider $\\xi \\in ^{\\infty }(A) \\otimes _H_2 \\bigcap H_1 \\otimes _^{\\infty }(B)$ .",
"Then there exist $x_i\\in ^{\\infty }(A)$ , $y_i\\in H_2$ , $\\tilde{x}_i\\in H_1$ , $\\tilde{y}_i\\in ^{\\infty }(B)$ , $i=1,\\ldots n$ such that $\\sum _i x_i \\otimes y_i = \\xi = \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i,$ where without loss of generality we may assume that $\\tilde{y}_i$ is orthonormal in $^{\\infty }(B)$ .",
"There is an obvious pairing $H_1 \\otimes _H_2 \\times H_2 \\rightarrow H_1,$ induced by the $H_2$ scalar product.",
"Pick an index $j$ .",
"Then on the one hand $\\Bigl \\langle \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i, (I+B^2)\\tilde{y}_j\\Bigr \\rangle = \\sum _i \\tilde{x}_i \\langle \\tilde{y}_i, \\tilde{y}_j\\rangle _B = \\tilde{x}_j,$ and on the other hand $\\begin{split}\\Bigl \\langle \\sum _i \\tilde{x}_i \\otimes \\tilde{y}_i, (I+B^2)\\tilde{y}_j \\Bigr \\rangle &= \\Bigl \\langle \\sum _i x_i \\otimes y_i , (I+B^2) \\tilde{y}_j\\Bigr \\rangle \\\\&= \\sum _i x_i\\, \\langle y_i, (I+B^2) \\tilde{y}_j\\rangle \\in ^\\infty (A).\\end{split}$ This proves $\\tilde{x}_j\\in ^{\\infty }(A)$ for any $j=1,\\ldots ,n$ and the statement follows.",
"Let $A,C\\ge 0$ be self-adjoint operators in $H_1$ ; $B,D\\ge 0$ self-adjoint operators in $H_2$ .",
"If $A\\le C$ , $(C)\\subset (A)$ and $B\\le D$ , $(D)\\subset (B)$ then $AB \\le CD,\\quad (CD) \\subset (AB).$ The domain inclusion is clear from Proposition REF .",
"To prove the inequality, let $\\sum _{i=1}^n x_i \\otimes y_i\\in ^{\\infty }(C)\\otimes _^{\\infty }(D)$ be given.",
"Consider the matrices $(\\langle A x_i, x_j\\rangle )_{ij}$ , $(\\langle C x_i, x_j\\rangle )_{ij}$ , $(\\langle B y_i, y_j\\rangle )_{ij}$ , and $(\\langle D y_i, y_j\\rangle )_{ij}$ .",
"For complex numbers $\\lambda _i$ we have $\\begin{split}\\sum \\overline{\\lambda }_i \\langle A x_i, x_j\\rangle \\lambda _j&= \\langle A \\sum \\lambda x_i , \\lambda _i x_i\\rangle \\ge 0\\\\\\langle A \\sum \\lambda x_i , \\lambda _i x_i\\rangle &\\le \\langle C \\sum \\lambda x_i , \\lambda _i x_i\\rangle =\\sum \\overline{\\lambda }_i \\langle C x_i, x_j\\rangle \\lambda _j.\\end{split}$ Thus we have the matrix inequalities $0 \\le \\left(\\langle A x_i, x_j\\rangle \\right)_{ij} \\le \\left(\\langle C x_i, x_j\\rangle \\right)_{ij}$ and analogously $0 \\le \\left(\\langle B y_i, y_j\\rangle \\right)_{ij} \\le \\left(\\langle D y_i, y_j\\rangle \\right)_{ij}.$ Proposition REF implies $\\begin{split}0 &\\le \\sum _{i,j} \\langle (C-A) x_i, x_j\\rangle \\langle D y_i,y_j\\rangle + \\sum _{i,j} \\langle A x_i, x_j\\rangle \\langle (D-B) y_i,y_j\\rangle \\\\&= \\sum _{i,j} \\langle C x_i, x_j\\rangle \\langle D y_i, y_j\\rangle -\\langle A x_i, x_j\\rangle \\langle B y_i, y_j\\rangle \\\\&= \\Bigl \\langle (C\\otimes D) \\sum x_i \\otimes y_i, \\sum x_i \\otimes y_i\\Bigr \\rangle - \\Bigl \\langle (A\\otimes B) \\sum x_i \\otimes y_i,\\sum x_i \\otimes y_i\\Bigr \\rangle ,\\end{split}$ and hence $AB \\le CD$ .",
"Uniform asymptotic expansions of modified Bessel functions According to Olver [31], we may write for any $\\mu > 0$ and $x>0$ $\\begin{split}I_\\mu (\\mu x) &= \\frac{1}{\\sqrt{2\\pi \\mu }}\\cdot \\frac{e^{\\mu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}\\left(\\sum _{j=0}^{n-1} \\frac{U_j(p(x))}{\\mu ^j} + \\eta _{n,1}(\\mu , x)\\right)\\frac{1}{1+\\eta _{n,1}(\\mu , \\infty )}, \\\\K_\\mu (\\mu x) &= \\sqrt{\\frac{2\\pi }{\\mu }}\\cdot \\frac{e^{-\\mu \\cdot \\eta (x)}}{(1+x^2)^{1/4}}\\left(\\sum _{j=0}^{n-1} (-1)^j \\frac{U_j(p(x))}{\\mu ^j} + \\eta _{n,2}(\\mu ,x)\\right)\\end{split}$ where $p(x)= \\sqrt{1+x^2}$ , $\\eta (x) = p(x) +\\ln \\frac{x}{1+p(x)}$ and $U_j(p)$ are iteratively defined polynomials in $p$ with $U_0 \\equiv 1$ .",
"By Olver [31], the error terms $\\eta _{n,1}$ and $\\eta _{n,2}$ admit the following bounds $\\begin{split}|\\eta _{n,1}(\\mu , x)| &\\le 2 \\exp \\left(\\frac{2{V}_{(1,p(x))}(U_1)}{\\mu }\\right)\\frac{{V}_{(1,p(x))}(U_n)}{\\mu ^n}, \\\\~|\\eta _{n,2}(\\mu , x)| &\\le 2 \\exp \\left(\\frac{2{V}_{(0,p(x))}(U_1)}{\\mu }\\right)\\frac{{V}_{(0,p(x))}(U_n)}{\\mu ^n}\\end{split}$ where ${V}_{(a,b)}(f)$ denotes the total variation of a differentiable function $f$ along an interval $(a,b)$ .",
"In case of complex-valued arguments $x$ , one takes here the variation along $\\eta (x)$ -progressive paths.",
"However, here $x, p(x), \\eta (x)$ are all real-valued, and $\\eta (x)$ is monotonously increasing as $x\\rightarrow \\infty $ by (REF ).",
"Since $p((0,\\infty )) = (0,1)$ , we may take in (REF ) variation over $(0,1)$ for both error terms.",
"Since for any $j\\in $ the total variations ${V}_{(0,1)}(U_j)$ are taken along finite paths and since $U_j$ are polynomials, we conclude that for any $n\\in _0$ $\\eta _{n,1}(\\mu , x) = O(\\mu ^{-n}), \\quad \\eta _{n,2}(\\mu , x) = O(\\mu ^{-n}),\\ \\textup {as} \\ \\mu \\rightarrow \\infty .$ uniformly in $x\\in (0,\\infty )$ .",
"Hence the expansions (REF ) are uniform in $x\\in (0,\\infty )$ as well."
]
] | 1709.01636 |
[
[
"Characterizing a benchmark scenario for heavy Higgs boson searches in\n the Georgi-Machacek model"
],
[
"Abstract The Georgi-Machacek model is used to motivate and interpret LHC searches for doubly- and singly-charged Higgs bosons decaying into vector boson pairs.",
"In this paper we study the constraints on and phenomenology of the \"H5plane\" benchmark scenario in the Georgi-Machacek model, which has been proposed for use in these searches.",
"We show that the entire H5plane benchmark is compatible with the LHC measurements of the 125 GeV Higgs boson couplings.",
"We also point out that, over much of the H5plane benchmark, the lineshapes of the two CP-even neutral heavy Higgs bosons $H$ and $H_5^0$ will overlap and interfere when produced in vector boson fusion with decays to $W^+W^-$ or $ZZ$.",
"Finally we compute the decay branching ratios of the additional heavy Higgs bosons within the H5plane benchmark to facilitate the development of search strategies for these additional particles."
],
[
"Introduction",
"Since the discovery of a Standard Model (SM)-like Higgs boson at the CERN Large Hadron Collider (LHC) in 2012 [1], much experimental and theoretical attention has been devoted to testing the possibility that the Higgs sector contains additional scalars beyond the single SM isospin doublet.",
"An interesting possibility among these extensions is that part of electroweak symmetry breaking—and hence part of the masses of the $W$ and $Z$ bosons—could be generated by scalars in isospin representations larger than the doublet.",
"A prototype model in this class is the Georgi-Machacek (GM) model [2], [3], which contains a real and a complex isospin-triplet scalar in addition to the usual SM Higgs doublet.",
"A key feature of the GM model is the presence of doubly- and singly-charged Higgs bosons, $H_5^{\\pm \\pm }$ and $H_5^{\\pm }$ , that couple to SM vector boson pairs with an interaction strength proportional to the vacuum expectation value (vev) of the triplets.",
"Constraining this coupling therefore directly constrains the allowed contribution of the triplets to the masses of the $W$ and $Z$ bosons.",
"LHC searches for these scalars have been performed with production via vector boson fusion and decays to a pair of vector bosons [4], [5], [6]; the LHC measurement of the like-sign $W$ boson cross section in vector boson fusion [7] also provides sensitivity to the doubly-charged scalar [8].",
"When the branching ratios of $H_5^{\\pm \\pm }$ and $H_5^{\\pm }$ to vector boson pairs are essentially 100%, these searches directly constrain the triplet vev $v_{\\chi }$ as a function of the common mass $m_5$ of these scalars.",
"To aid the interpretation of these and future similar searches, the LHC Higgs Cross Section Working Group recently developed the “H5plane” benchmark scenario for the GM model [9].",
"The H5plane benchmark depends on two free input parameters, $m_5$ and $s_H \\equiv \\sqrt{8} v_{\\chi }/v$ (where $v = (\\sqrt{2} G_F)^{-1/2}$ is the SM Higgs vev), and the production cross sections for $H_5^{\\pm \\pm }$ and $H_5^{\\pm }$ in vector boson fusion are proportional to $s_H^2$ .",
"The other parameters of the model are fixed in the benchmark so that BR($H_5 \\rightarrow VV) = 1$ to a very good approximation.",
"Predictions for the production cross sections (at next-to-next-to-leading order in QCD) and decay widths of these scalars have been provided in the context of the H5plane benchmark for LHC collisions at 8 [10] and 13 TeV [9].",
"In this paper we perform the first comprehensive survey of the phenomenology of the H5plane benchmark in the GM model.",
"We show that the entire H5plane benchmark is compatible with the LHC measurements of the 125 GeV Higgs boson couplings from 7 and 8 TeV data [11].",
"We point out that, over much of the H5plane benchmark, the lineshapes of the two CP-even neutral heavy Higgs bosons $H$ and $H_5^0$ will overlap and interfere when these scalars are produced in vector boson fusion with decays to $W^+W^-$ or $ZZ$ .",
"We also display the decay branching ratios of the additional heavy Higgs bosons within the H5plane benchmark to facilitate the development of search strategies for these additional particles.",
"Our numerical work is done using the public code GMCALC 1.2.1 [12].",
"This paper is organized as follows.",
"In the next section we review the GM model and the specification of the H5plane benchmark.",
"Section contains the bulk of our results.",
"We conclude in Sec.",
"."
],
[
"Georgi-Machacek model",
"The scalar sector of the GM model [2], [3] consists of the usual complex doublet $(\\phi ^+,\\phi ^0)^T$ with hyperchargeWe use $Q = T^3 + Y/2$ .",
"$Y = 1$ , a real triplet $(\\xi ^+,\\xi ^0, -\\xi ^{+*})^T$ with $Y = 0$ , and a complex triplet $(\\chi ^{++},\\chi ^+,\\chi ^0)^T$ with $Y=2$ .",
"The doublet is responsible for the fermion masses as in the SM.",
"Custodial symmetry, required in order to avoid stringent constraints from the $\\rho $ parameter, is preserved at tree level by imposing a global SU(2)$_L \\times $ SU(2)$_R$ symmetry on the scalar potential.",
"To make this symmetry explicit, we write the doublet in the form of a bidoublet $\\Phi $ and combine the triplets into a bitriplet $X$ : $\\Phi = \\left( \\begin{array}{cc}\\phi ^{0*} &\\phi ^+ \\\\-\\phi ^{+*} & \\phi ^0 \\end{array} \\right), \\qquad X =\\left(\\begin{array}{ccc}\\chi ^{0*} & \\xi ^+ & \\chi ^{++} \\\\-\\chi ^{+*} & \\xi ^{0} & \\chi ^+ \\\\\\chi ^{++*} & -\\xi ^{+*} & \\chi ^0\\end{array}\\right).$ The vevs are given by $\\langle \\Phi \\rangle = \\frac{ v_{\\phi }}{\\sqrt{2}} I_{2\\times 2}$ and $\\langle X \\rangle = v_{\\chi } I_{3 \\times 3}$ , where $I_{n \\times n}$ is the $n\\times n$ unit matrix and the $W$ and $Z$ boson masses constrain $v_{\\phi }^2 + 8 v_{\\chi }^2 \\equiv v^2 = \\frac{1}{\\sqrt{2} G_F} \\approx (246~{\\rm GeV})^2.$ The most general gauge-invariant scalar potential involving these fields that conserves custodial SU(2) is given, in the conventions of Ref.",
"[13], byA translation table to other parameterizations in the literature has been given in the appendix of Ref. [13].",
"$V(\\Phi ,X) &= & \\frac{\\mu _2^2}{2} \\text{Tr}(\\Phi ^\\dagger \\Phi )+ \\frac{\\mu _3^2}{2} \\text{Tr}(X^\\dagger X)+ \\lambda _1 [\\text{Tr}(\\Phi ^\\dagger \\Phi )]^2+ \\lambda _2 \\text{Tr}(\\Phi ^\\dagger \\Phi ) \\text{Tr}(X^\\dagger X) \\nonumber \\\\& & + \\lambda _3 \\text{Tr}(X^\\dagger X X^\\dagger X)+ \\lambda _4 [\\text{Tr}(X^\\dagger X)]^2- \\lambda _5 \\text{Tr}( \\Phi ^\\dagger \\tau ^a \\Phi \\tau ^b) \\text{Tr}( X^\\dagger t^a X t^b)\\nonumber \\\\& & - M_1 \\text{Tr}(\\Phi ^\\dagger \\tau ^a \\Phi \\tau ^b)(U X U^\\dagger )_{ab}- M_2 \\text{Tr}(X^\\dagger t^a X t^b)(U X U^\\dagger )_{ab}.$ Here the SU(2) generators for the doublet representation are $\\tau ^a = \\sigma ^a/2$ with $\\sigma ^a$ being the Pauli matrices, the generators for the triplet representation are $t^1= \\frac{1}{\\sqrt{2}} \\left( \\begin{array}{ccc}0 & 1 & 0 \\\\1 & 0 & 1 \\\\0 & 1 & 0 \\end{array} \\right), \\qquad t^2= \\frac{1}{\\sqrt{2}} \\left( \\begin{array}{ccc}0 & -i & 0 \\\\i & 0 & -i \\\\0 & i & 0 \\end{array} \\right), \\qquad t^3= \\left( \\begin{array}{ccc}1 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & -1 \\end{array} \\right),$ and the matrix $U$ , which rotates $X$ into the Cartesian basis, is given by [14] $U = \\left( \\begin{array}{ccc}- \\frac{1}{\\sqrt{2}} & 0 & \\frac{1}{\\sqrt{2}} \\\\- \\frac{i}{\\sqrt{2}} & 0 & - \\frac{i}{\\sqrt{2}} \\\\0 & 1 & 0 \\end{array} \\right).$ The physical fields can be organized by their transformation properties under the custodial SU(2) symmetry into a fiveplet, a triplet, and two singlets.",
"The fiveplet and triplet states are given by $&&H_5^{++} = \\chi ^{++}, \\qquad H_5^+ = \\frac{\\left(\\chi ^+ - \\xi ^+\\right)}{\\sqrt{2}}, \\qquad H_5^0 = \\sqrt{\\frac{2}{3}} \\xi ^{0,r} - \\sqrt{\\frac{1}{3}} \\chi ^{0,r}, \\nonumber \\\\&&H_3^+ = - s_H \\phi ^+ + c_H \\frac{\\left(\\chi ^++\\xi ^+\\right)}{\\sqrt{2}}, \\qquad H_3^0 = - s_H \\phi ^{0,i} + c_H \\chi ^{0,i},$ where the vevs are parameterized by $c_H \\equiv \\cos \\theta _H = \\frac{v_{\\phi }}{v}, \\qquad s_H \\equiv \\sin \\theta _H = \\frac{2\\sqrt{2}\\,v_\\chi }{v},$ and we have decomposed the neutral fields into real and imaginary parts according to $\\phi ^0 \\rightarrow \\frac{v_{\\phi }}{\\sqrt{2}} + \\frac{\\phi ^{0,r} + i \\phi ^{0,i}}{\\sqrt{2}},\\qquad \\chi ^0 \\rightarrow v_{\\chi } + \\frac{\\chi ^{0,r} + i \\chi ^{0,i}}{\\sqrt{2}},\\qquad \\xi ^0 \\rightarrow v_{\\chi } + \\xi ^{0,r}.$ The masses within each custodial multiplet are degenerate at tree level and can be written (after eliminating $\\mu _2^2$ and $\\mu _3^2$ in favor of the vevs) asNote that the ratio $M_1/v_{\\chi }$ can be written using the minimization condition $\\partial V/ \\partial v_{\\chi } = 0$ as $\\frac{M_1}{v_{\\chi }} = \\frac{4}{v_{\\phi }^2}\\left[ \\mu _3^2 + (2 \\lambda _2 - \\lambda _5) v_{\\phi }^2+ 4(\\lambda _3 + 3 \\lambda _4) v_{\\chi }^2 - 6 M_2 v_{\\chi } \\right],$ which is finite in the limit $v_{\\chi } \\rightarrow 0$ .",
"$m_5^2 &=& \\frac{M_1}{4 v_{\\chi }} v_\\phi ^2 + 12 M_2 v_{\\chi }+ \\frac{3}{2} \\lambda _5 v_{\\phi }^2 + 8 \\lambda _3 v_{\\chi }^2, \\nonumber \\\\m_3^2 &=& \\frac{M_1}{4 v_{\\chi }} (v_\\phi ^2 + 8 v_{\\chi }^2)+ \\frac{\\lambda _5}{2} (v_{\\phi }^2 + 8 v_{\\chi }^2)= \\left( \\frac{M_1}{4 v_{\\chi }} + \\frac{\\lambda _5}{2} \\right) v^2.$ The two custodial-singlet mass eigenstates are given by $h = \\cos \\alpha \\, \\phi ^{0,r} - \\sin \\alpha \\, H_1^{0\\prime }, \\qquad H = \\sin \\alpha \\, \\phi ^{0,r} + \\cos \\alpha \\, H_1^{0\\prime },$ where $H_1^{0 \\prime } = \\sqrt{\\frac{1}{3}} \\xi ^{0,r} + \\sqrt{\\frac{2}{3}} \\chi ^{0,r},$ and we will use the shorthand $c_{\\alpha } \\equiv \\cos \\alpha $ , $s_{\\alpha } \\equiv \\sin \\alpha $ .",
"The mixing angle $\\alpha $ and masses are given by $&&\\sin 2 \\alpha = \\frac{2 \\mathcal {M}^2_{12}}{m_H^2 - m_h^2}, \\qquad \\cos 2 \\alpha = \\frac{ \\mathcal {M}^2_{22} - \\mathcal {M}^2_{11} }{m_H^2 - m_h^2},\\nonumber \\\\&&m^2_{h,H} = \\frac{1}{2} \\left[ \\mathcal {M}_{11}^2 + \\mathcal {M}_{22}^2\\mp \\sqrt{\\left( \\mathcal {M}_{11}^2 - \\mathcal {M}_{22}^2 \\right)^2+ 4 \\left( \\mathcal {M}_{12}^2 \\right)^2} \\right],$ where we choose $m_h < m_H$ , and $\\mathcal {M}_{11}^2 &=& 8 \\lambda _1 v_{\\phi }^2, \\nonumber \\\\\\mathcal {M}_{12}^2 &=& \\frac{\\sqrt{3}}{2} v_{\\phi }\\left[ - M_1 + 4 \\left(2 \\lambda _2 - \\lambda _5 \\right) v_{\\chi } \\right], \\nonumber \\\\\\mathcal {M}_{22}^2 &=& \\frac{M_1 v_{\\phi }^2}{4 v_{\\chi }} - 6 M_2 v_{\\chi }+ 8 \\left( \\lambda _3 + 3 \\lambda _4 \\right) v_{\\chi }^2.$"
],
[
"H5plane benchmark",
"The H5plane benchmark scenario for the GM model was introduced in Ref. [9].",
"It is designed to facilitate LHC searches for $H_5^{\\pm \\pm }$ and $H_5^{\\pm }$ in vector boson fusion with decays to $W^{\\pm }W^{\\pm }$ and $W^{\\pm }Z$ , respectively.",
"It is specified as in Table REF , in a form that is easily implemented in the model calculator GMCALC [12].",
"After imposing the existing direct search constraints on $H_5^{\\pm \\pm }$ , the benchmark has the following features: It comes close to fully populating the theoretically-allowed region of the $m_5$ –$s_H$ plane for $m_5 \\in [200,3000]~{\\rm GeV}$ , as shown in Fig.",
"REF (see below).",
"It has $m_3 > m_5$ over the whole benchmark plane, so that the Higgs-to-Higgs decays $H_5 \\rightarrow H_3 H_3$ and $H_5 \\rightarrow H_3 V$ are kinematically forbidden, leaving only the decays $H_5 \\rightarrow VV$ at tree level; i.e., ${\\rm BR}(H_5 \\rightarrow VV) = 1$ .",
"The entire benchmark satisfies indirect constraints from $B$ physics, the most stringent of which is $b \\rightarrow s \\gamma $ [15].",
"The region still allowed by direct searches is currently unconstrained by LHC measurements of the couplings of the 125 GeV Higgs boson, as we will show in this paper.",
"Table: Specification of the H5plane benchmark scenario for the Georgi-Machacek model.",
"These input parameters correspond to INPUTSET = 4 in GMCALC .In INPUTSET = 4 of GMCALC, the nine parameters of the scalar potential in Eq.",
"(REF ) are fixed in terms of the nine input parameters $m_h$ , $m_5$ , $s_H$ , $\\lambda _2$ , $\\lambda _3$ , $\\lambda _4$ , $M_1$ , $M_2$ , and $v = (\\sqrt{2} G_F)^{-1/2}$ .",
"The quartic coupling $\\lambda _5$ is computed from these using $\\lambda _5 = \\frac{2 m_5^2}{3 c_H^2 v^2} - \\frac{\\sqrt{2} M_1}{3 s_H v}- \\frac{2\\sqrt{2} M_2 \\, s_H}{c_H^2 v} - \\frac{2 \\lambda _3 \\, s_H^2}{3 c_H^2}.$ The quartic coupling $\\lambda _1$ (which depends on $\\lambda _5$ ) is computed using $\\lambda _1 = \\frac{1}{8 c_H^2 v^2} \\left\\lbrace m_h^2 + \\frac{3 c_H^2 v^2 \\left[ -M_1 + \\sqrt{2}(2\\lambda _2-\\lambda _5)s_H v \\right]^2}{2\\sqrt{2}M_1\\frac{c_H^2}{s_H}v - 6\\sqrt{2} M_2 s_H v + 4(\\lambda _3+3\\lambda _4)s_H^2 v^2 - 4m_h^2} \\right\\rbrace .$ The mass-squared parameter $\\mu _2^2$ (which depends on $\\lambda _1$ and $\\lambda _5$ ) is computed using $\\mu _2^2 = -4\\lambda _1 c_H^2 v^2 - \\frac{3}{8}(2\\lambda _2-\\lambda _5)s_H^2 v^2+ \\frac{3\\sqrt{2}}{8}M_1 s_H v,$ and $\\mu _3^2$ is computed using $\\mu _3^2 = \\frac{2}{3} m_5^2 + \\frac{\\sqrt{2}M_1 c_H^2 v}{6s_H} - 2\\lambda _2 c_H^2 v^2- \\frac{1}{6}(7\\lambda _3 + 9\\lambda _4)s_H^2 v^2 - \\frac{\\sqrt{2}}{2}M_2 s_H v.$ In Fig.",
"REF we show the allowed region in the $m_5$ –$s_H$ plane for the full GM model (red points) and the allowed region for the H5plane benchmark scenario (entire region below both the black and blue curves), as generated using GMCALC 1.2.1 with $m_h = 125$ GeV.",
"In both cases we impose the theoretical constraints from perturbative unitarity of the scalar quartic couplings, bounded-from-belowness of the scalar potential, and the absence of deeper alternative minima, as described in Ref.",
"[13], as well as the indirect constraints from $b \\rightarrow s \\gamma $ and the $S$ parameter following Ref.",
"[15] (we use the “loose” constraint on $b \\rightarrow s \\gamma $ as described in Ref.",
"[15]); all of these constraints are implemented in GMCALC.",
"We also impose the direct experimental constraint from a CMS search for $H_5^{\\pm \\pm }$ [4] (described in more detail below), which excludes the area above the blue curve in the context of the H5plane benchmark.",
"The red points represent a scan over the full GM model parameter space.",
"The entire area below the black curve (obtained by scanning $m_5$ and $s_H$ in the H5plane benchmark) represents the theoretically-allowed region in the H5plane benchmark: as advertised, it nearly, but not quite entirely, populates the entire range of $s_H$ that is accessible in the full GM model for any given value of $m_5$ between 200 and 3000 GeV.",
"This makes the H5plane scenario a good benchmark for the interpretation of searches for $H_5^{\\pm }$ and $H_5^{\\pm \\pm }$ in vector boson fusion, for which the signal rate and kinematics depend only on $m_5$ , $s_H$ , and the $H_5$ branching ratios into vector boson pairs.",
"We note however that the accessible ranges of other observables are not necessarily fully populated by the H5plane benchmark; this will be particularly dramatic for the mass splittings among the heavy Higgs bosons.",
"Figure: Theoretically and experimentally allowed parameter region in the m 5 m_5–s H s_H plane in the H5plane benchmark (entire region below both the black and blue curves) and the full GM model (red points).",
"The black curve delimits the region allowed by theoretical constraints in the H5plane benchmark and the blue curve represents the upper bound on s H s_H from a direct search for H 5 ±± H_5^{\\pm \\pm } from Ref. .",
"See text for details.The CMS search in Ref.",
"[4] currently provides the most stringent direct experimental constraint on the GM model for $m_5$ above 200 GeV.For comparison, the 95% confidence level constraint obtained in Ref.",
"[8] from an ATLAS measurement of the cross section for like-sign $W$ boson pairs in vector boson fusion [7] excludes $s_H$ values above 0.39 for $m_5 = 200$ GeV, rising to 0.74 for $m_5 = 600$ GeV.",
"LHC searches for $H_5^{\\pm }$ in the $WZ$ final state [5], [6] are currently slightly less constraining than the search for $H_5^{\\pm \\pm }$ .",
"This search looked for a doubly-charged scalar produced in vector boson fusion (VBF) and decaying to two like-sign $W$ bosons which in turn decay leptonically, using 19.4 fb$^{-1}$ of proton-proton collision data at a centre-of-mass energy of 8 TeV.",
"This search set a 95% confidence level upper bound on the cross section times branching ratio, $\\sigma ({\\rm VBF} \\rightarrow H^{\\pm \\pm }) \\times {\\rm BR}(H^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm })$ , as a function of the doubly-charged Higgs boson mass.",
"The H5plane benchmark is designed so that ${\\rm BR}(H_5^{\\pm \\pm } \\rightarrow W^{\\pm } W^{\\pm }) = 1$ , so that the CMS constraint becomes an upper bound on the cross section $\\sigma ({\\rm VBF} \\rightarrow H_5^{\\pm \\pm })$ , which is proportional to $s_H^2$ .",
"We translated this into an upper bound on $s_H$ in the H5plane benchmark using the ${\\rm VBF} \\rightarrow H_5^{\\pm \\pm }$ cross sections calculated for the 8 TeV LHC at next-to-next-to-leading order in QCD in Ref.",
"[10] (we did not take into account the theoretical uncertainties in these predictions in computing the limit).",
"This constraint in the H5plane benchmark is shown as the blue curve in Fig.",
"REF ; when combined with the theoretical constraints, it limits $s_H < 0.55$ in the H5plane benchmark.",
"In a full scan of the GM model, some allowed points appear that have ${\\rm BR}(H_5^{\\pm \\pm } \\rightarrow W^{\\pm } W^{\\pm }) < 1$ , because decays into $H_3^{\\pm } W^{\\pm }$ are kinematically allowed.",
"Since the CMS constraint applies to the product $\\sigma ({\\rm VBF} \\rightarrow H_5^{\\pm \\pm }) \\times {\\rm BR}(H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm })$ , this results in a few of the allowed red points in Fig.",
"REF falling above the blue curve.",
"The number of such points is quite small, though, because most points in the full GM model scan that have ${\\rm BR}(H_5^{\\pm \\pm } \\rightarrow W^{\\pm } W^{\\pm }) < 1$ also have small $s_H$ , putting them below the blue curve anyway.",
"The H5plane benchmark was designed so that $m_3 > m_5$ over the entire benchmark plane, so that the decay $H_5^{\\pm \\pm } \\rightarrow W^{\\pm } W^{\\pm }$ is the only kinematically-allowed decay for $H_5^{\\pm \\pm }$ .",
"This makes direct searches for $H_5^{\\pm \\pm }$ in the $WW$ final state particularly easy to interpret.",
"Decays of $H_5^{\\pm }$ to $W^{\\pm }Z$ are then also the only kinematically-accessible tree-level decay of $H_5^{\\pm }$ (the loop-induced decay $H_5^{\\pm } \\rightarrow W^{\\pm }\\gamma $ is allowed, but has a very small branching ratio for $m_5 \\ge 200$ GeV), so that direct searches for the singly-charged state in this final state are also easy to interpret.",
"This was used in the GM model interpretation of the ATLAS and CMS searches for $H_5^{\\pm }$ in Refs.",
"[5], [6] (these searches are less constraining on the GM model parameter space than that of Ref. [4]).",
"In the left panel of Fig.",
"REF we show the total width of $H_5^{\\pm \\pm }$ normalized to its mass.",
"This width-to-mass ratio reaches a maximum of 8% for the largest theoretically-allowed values of $s_H$ when $m_5 > 800$ GeV.",
"The right panel of Fig.",
"REF shows the deviation from unity of the ratio of partial widths of $H_5^{\\pm }$ and $H_5^0$ divided by that of $H_5^{\\pm \\pm }$ as a function of $m_5$ .",
"These ratios are independent of $s_H$ .",
"The widths of $H_5^+$ and $H_5^0$ are about 10% smaller than that of $H_5^{++}$ for $m_5 \\sim 200$ GeV, with the difference decreasing to less than 1% for $m_5 \\gtrsim 1000$ GeV.",
"In the H5plane benchmark, this width difference is solely due to the kinematic effect of the different masses of the $WW$ , $WZ$ , and $ZZ$ final states.",
"Figure: Left: Contours of Γ tot /m 5 \\Gamma _\\text{tot} / m_5 for H 5 ++ H_5^{++} in the GM model H5plane benchmark.",
"The value of Γ tot /m 5 \\Gamma _\\text{tot} / m_5 reaches a maximum of 0.08 along the upper boundary of the allowed region for m 5 ≳800m_5 \\gtrsim 800 GeV, and goes to zero at s H =0s_H = 0.Right: Deviation from unity of the ratio of total widths of scalar s=H 5 + s = H_5^+ and H 5 0 H_5^0 to that of H 5 ++ H_5^{++} as a function of m 5 m_5 in the H5plane benchmark.",
"Direct constraints from a CMS search for H 5 ±± →W ± W ± H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm } in vector boson fusion have been applied."
],
[
"$H_3$ –{{formula:ba60ea6c-a918-4e7b-b0ab-e1c5b2d343ff}} mass splitting",
"In the left panel of Fig.",
"REF we show the mass splitting $m_3 - m_5$ in the H5plane benchmark.",
"This splitting depends mainly on $m_5$ , and varies from 84 GeV at $m_5 = 200$ GeV to about 7 GeV at $m_5 = 3000$ GeV.",
"In the right panel of Fig.",
"REF we plot $m_3-m_5$ as a function of $m_5$ scanning over all the other free parameters in the H5plane benchmark (black points) and the full GM model (red points), where we have imposed the indirect constraints from $b \\rightarrow s \\gamma $ and the $S$ parameter [15] and direct constraints from the CMS search for $H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm }$ in vector boson fusion [4].",
"It is clear that the variation in the mass difference $m_3 - m_5$ is much greater in the full model scan than it is in the H5plane benchmark.",
"We can understand this as follows.",
"Figure: Left: Contours of m 3 -m 5 m_3-m_5 in the H5plane benchmark.",
"The value of m 3 -m 5 m_3 - m_5 ranges from 6.7 GeV to 84 GeV.Right: Mass difference m 3 -m 5 m_3 - m_5 as a function of m 5 m_5 in the H5plane benchmark (black points) and in a full scan of the GM model parameter space (red points).",
"Indirect constraints from b→sγb \\rightarrow s \\gamma and the SS parameter and direct constraints from a CMS search for H 5 ±± →W ± W ± H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm } in vector boson fusion have been applied.The difference between $m_3^2$ and $m_5^2$ can be written in the full GM model as $m_3^2 - m_5^2 = (M_1 - 6M_2) \\frac{s_H v}{\\sqrt{2}}+ \\left[ \\lambda _5\\left(\\frac{1}{2}s_H^2-c_H^2\\right) - \\lambda _3 s_H^2\\right] v^2.$ In the H5plane benchmark, the parameter relations simplify this down to $m_3^2-m_5^2 = (m_3 - m_5)(m_3 + m_5) = \\left( \\frac{2}{3} - \\frac{0.3 s_H^2}{c_H^2} \\right) v^2.$ The variation of this expression with $s_H$ is fairly minimal: $m_3^2-m_5^2$ changes by less than 10% between $s_H=0$ and $s_H=0.4$ .",
"This leads to the very narrow range of $m_3-m_5$ covered by the H5plane benchmark scan (black points) in the right panel of Fig.",
"REF .",
"Solving Eq.",
"(REF ) for $m_3-m_5$ , the dependence on $m_5$ is due only to a factor of $1/(m_3+m_5) \\simeq 1/(2 m_5)$ .",
"In contrast, in the full GM model scan (red points in the right panel of Fig.",
"REF ), $m_3-m_5$ varies by hundreds of GeV.",
"This is mostly due to the term proportional to $(M_1 - 6 M_2)$ in Eq.",
"(REF ), which is zero in the H5plane benchmark due to the choice $M_2 = M_1/6$ , and the term $-\\lambda _5 c_H^2 v^2$ , which is not suppressed at small $s_H$ .",
"In the full GM model, $\\lambda _5$ can vary between $-8 \\pi /3$ and $+8\\pi /3$ [13], while in the H5plane benchmark Eq.",
"(REF ) reduces to $\\lambda _5 = - \\frac{2}{3 c_H^2} (1 - 0.1 s_H^2),$ so that the term $-\\lambda _5 c_H^2 v^2$ varies from $2v^2/3$ by less than 2% for $s_H$ between zero and 0.4 in the H5plane benchmark.",
"The preference for positive values of $m_3-m_5$ in the full GM model scan is due to the interplay of the theoretical constraints on the model parameters and is apparent already in Fig.",
"3 of Ref. [15].",
"Viable mass spectra in the full GM model, and their implications for cascade decays of the heavier Higgs bosons, have previously been studied in Ref.",
"[16]."
],
[
"Couplings and decays of $h$",
"The tree-level couplings of the 125 GeV Higgs boson $h$ in the GM model are given in terms of the underlying parameters by $\\kappa ^h_f = \\frac{c_{\\alpha }}{c_H}, \\qquad \\qquad \\kappa ^h_V = c_{\\alpha } c_H - \\sqrt{\\frac{8}{3}} s_{\\alpha } s_H,$ where $\\kappa $ is defined in the usual way as the ratio of the coupling in the GM model to the corresponding coupling of the SM Higgs boson [17].",
"We first illustrate the variation of the custodial-singlet scalar mixing angle $\\sin \\alpha $ over the H5plane benchmark in the left panel of Fig.",
"REF .",
"$\\sin \\alpha $ varies between zero and $-0.64$ in the H5plane benchmark.",
"It is strongly correlated with $s_H$ , as shown in the right panel of Fig.",
"REF .",
"This correlation also appears in a full scan of the GM model (red points in the right panel of Fig.",
"REF ), but is stronger in the H5plane benchmark (black points).",
"Figure: Left: Contours of sinα\\sin \\alpha in the H5plane benchmark.",
"The value of sinα\\sin \\alpha varies between -0.64-0.64 and 0.Right: Correlation between sinα\\sin \\alpha and s H s_H in the H5plane benchmark (black points) and in a general GM model scan with m 5 ≥200m_5 \\ge 200 GeV (red points).",
"Indirect constraints from b→sγb \\rightarrow s \\gamma and the SS parameter and direct constraints from a CMS search for H 5 ±± →W ± W ± H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm } in vector boson fusion have been applied.In Fig.",
"REF we plot $\\kappa _f^h$ (left panel) and $\\kappa _V^h$ (right panel) in the H5plane benchmark.",
"These couplings remain reasonably close to their SM value of 1 everywhere in the benchmark plane.",
"The coupling of $h$ to fermions $\\kappa _f^h$ varies between 0.902 and 1.014, reaching its smallest values when $s_H$ is large, and the coupling of $h$ to vector bosons $\\kappa _V^h$ varies between 1 and 1.21, reaching its largest values when $s_H$ is large.",
"Figure: Left: Contours of κ f h \\kappa _f^h in the H5plane benchmark.The value of κ f h \\kappa _f^h ranges from 0.9020.902 to 1.0141.014.Right: Contours of κ V h \\kappa _V^h in the H5plane benchmark.The value of κ V h \\kappa _V^h ranges from 1.001.00 to 1.211.21.The coupling of $h$ to photon pairs is affected by the modifications of these tree-level couplings, as well as by contributions from loop diagrams involving $H_3^{\\pm }$ , $H_5^{\\pm }$ , and $H_5^{\\pm \\pm }$ .",
"Defining $\\kappa ^h_{\\gamma }$ in the usual way as [17]In GMCALC 1.2.1 the computation of the fermion loop contribution to Higgs decays to two photons includes only the top quark loop.",
"$\\kappa ^h_{\\gamma } = \\sqrt{\\frac{\\Gamma (h \\rightarrow \\gamma \\gamma )}{\\Gamma (h_{\\rm SM} \\rightarrow \\gamma \\gamma )}},$ we plot this coupling in the H5plane benchmark in the left panel of Fig.",
"REF .",
"The coupling of $h$ to photons $\\kappa ^h_{\\gamma }$ varies between 0.99 and 1.24, reaching its largest values when $s_H$ is large.",
"To isolate the effect of the loop diagrams involving $H_3^{\\pm }$ , $H_5^{\\pm }$ , and $H_5^{\\pm \\pm }$ , in the right panel of Fig.",
"REF we plot $\\Delta \\kappa ^h_{\\gamma }$ , which is defined as the contribution to $\\kappa ^h_{\\gamma }$ made by the scalar loops, i.e., $\\Delta \\kappa ^h_{\\gamma } = \\kappa ^h_{\\gamma }({\\rm full}) - \\kappa ^h_{\\gamma }(t~{\\rm and}~W~{\\rm loops~only}).$ $\\Delta \\kappa _{\\gamma }^h$ varies between $\\pm 0.05$ in the H5plane benchmark.",
"It is positive only for $m_5$ below 300 GeV, where it contributes to a slight enhancement of $\\kappa _{\\gamma }^h$ to values up to 1.05.",
"It reaches its most negative value at large $s_H$ , where it limits the enhancement of $\\kappa _{\\gamma }^h$ through destructive interference with the dominant $W$ loop contribution.",
"Figure: Left: Contours of κ γ h \\kappa _\\gamma ^h in the H5plane benchmark.The value of κ γ h \\kappa _\\gamma ^h ranges from 0.9870.987 to 1.241.24.Right: Contours of Δκ γ h \\Delta \\kappa _\\gamma ^h in the H5plane benchmark.The value of Δκ γ h \\Delta \\kappa _\\gamma ^h ranges from -0.054-0.054 to 0.0520.052.We also examine the total width of $h$ in the H5plane benchmark.",
"We define the scaling factor $\\kappa _h$ as [17] $\\kappa _h = \\sqrt{\\frac{\\Gamma _{\\rm tot}(h)}{\\Gamma _{\\rm tot}(h_{\\rm SM})}},$ and calculate it using the formula $\\kappa _h^2 = \\frac{(\\kappa ^{h}_f)^2 \\left(B^{SM}_{h \\rightarrow b \\bar{b}} + B^{SM}_{h \\rightarrow \\tau ^+ \\tau ^-}+ B^{SM}_{h \\rightarrow c \\bar{c}} + B^{SM}_{h \\rightarrow g g}\\right)+ (\\kappa ^{h}_V)^2 \\left(B^{SM}_{h \\rightarrow W^+ W^-} + B^{SM}_{h \\rightarrow ZZ}\\right)+ (\\kappa ^{h}_{\\gamma })^2 B^{SM}_{h \\rightarrow \\gamma \\gamma }+ (\\kappa ^{h}_{Z\\gamma })^2 B^{SM}_{h \\rightarrow \\gamma Z}}{B^{SM}_{h \\rightarrow b \\bar{b}} + B^{SM}_{h \\rightarrow \\tau ^+ \\tau ^-}+ B^{SM}_{h \\rightarrow c \\bar{c}} + B^{SM}_{h \\rightarrow g g}+ B^{SM}_{h \\rightarrow W^+ W^-} + B^{SM}_{h \\rightarrow ZZ}+ B^{SM}_{h \\rightarrow \\gamma \\gamma } + B^{SM}_{h \\rightarrow \\gamma Z}} .$ The values for the SM Higgs branching ratios $B^{SM}_{h \\rightarrow X}$ were taken from Tables 174–178 of Ref.",
"[9] for a SM Higgs mass of 125.09 GeV and are reproduced in Table REF .",
"We use this more precise value of the SM Higgs boson mass in this calculation because the LHC Higgs coupling measurements in Ref.",
"[11] have been extracted for this mass value.",
"Table: Branching ratios of the SM Higgs boson with mass 125.09 GeV, from Ref.",
", used in the calculation of κ h \\kappa _h.We plot $\\kappa _h$ in the H5plane benchmark in the left panel of Fig.",
"REF .",
"$\\kappa _h$ remains very close to one over the entire benchmark, varying between $0.985$ and $1.017$ , which is surprising considering that the tree-level couplings of $h$ to vector bosons are modified by as much as 21% and those of $h$ to fermions by as much as 10% compared to the SM Higgs couplings.",
"The very SM-like values of the $h$ total width are due to an accidental cancellation between an enhancement of the $h$ partial width to vector bosons and a suppression of its partial width to fermions.",
"This cancellation also occurs, though less severely, in a full scan of the GM model, as shown by the red points in the right panel of Fig.",
"REF .",
"$\\kappa _h$ is slightly greater than one in most of the H5plane benchmark, falling below one in a small sliver at high $s_H$ and $m_5$ between 700 and 1800 GeV, and in a thin band for $s_H < 0.04$ .",
"Figure: Left: Contours of κ h \\kappa _h in the H5plane benchmark.",
"The value of κ h \\kappa _h ranges from 0.9850.985 to 1.0171.017.Right: κ h \\kappa _h as a function of m 5 m_5 in the H5plane benchmark (black points) and in a full scan of the GM model parameter space (red points).",
"Indirect constraints from b→sγb \\rightarrow s \\gamma and the SS parameter and direct constraints from a CMS search for H 5 ±± →W ± W ± H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm } in vector boson fusion have been applied.In order to evaluate the consistency of the H5plane benchmark with LHC measurements of the couplings of the 125 GeV Higgs boson, we compute a $\\chi ^2$ using the combined ATLAS and CMS Higgs production and decay measurements in Ref.",
"[11] from data collected at LHC centre-of-mass energies of 7 and 8 TeV.",
"We use the observables and the corresponding correlation matrix $\\rho $ summarized in Table 9 and Fig.",
"28, respectively, of Ref. [11].",
"The $\\chi ^2$ is defined according to $\\chi ^2 = (\\vec{x}-\\vec{\\mu })^T V^{-1} (\\vec{x}-\\vec{\\mu }), \\quad V_{ij} = \\rho _{ij} \\sigma _i \\sigma _j,$ where $\\vec{x}$ is the vector of observed values, $\\vec{\\mu }$ is the vector of theoretical values at a particular point in the H5plane benchmark, and $\\vec{\\sigma }$ is the vector of the combined theoretical and experimental uncertainties.",
"Where the experimental uncertainties in Table 9 of Ref.",
"[11] are asymmetric, we symmetrize them by averaging the upper and lower uncertainty.",
"We then combine the (symmetrized) experimental uncertainties with the theoretical uncertainties quoted in Table 9 of Ref.",
"[11] in quadrature.",
"The results are shown in Fig.",
"REF .",
"The $\\chi ^2$ in the H5plane benchmark of the GM model ranges from a maximum of 29.9 for $s_H$ near zero to a minimum of 16.2 for $s_H$ around 0.5 and $m_5$ around 800–1000 GeV.",
"For comparison, the $\\chi ^2$ for the SM Higgs, computed in the same way, is 29.4.",
"The lower $\\chi ^2$ values in the GM model reflect a pull in the data towards slightly lower $\\kappa _f^h$ and higher $\\kappa _V^h$ values.",
"In particular, we observe that the entire H5plane benchmark is currently consistent with LHC Higgs coupling data.",
"Figure: Contours of the χ 2 \\chi ^2 value for a fit of the hh cross sections and branching ratios in the H5plane benchmark to LHC Higgs boson measurements from Ref. .",
"The χ 2 \\chi ^2 ranges from 16.216.2 to 29.929.9.",
"Compare the χ 2 \\chi ^2 of 29.4 for the SM Higgs boson."
],
[
"Couplings and decays of $H$",
"We now examine the couplings and decays of the heavier custodial-singlet Higgs boson $H$ .",
"The tree-level couplings of $H$ in the GM model are given in terms of the underlying parameters by $\\kappa _f^H = \\frac{s_{\\alpha }}{c_H}, \\qquad \\qquad \\kappa _V^H = s_{\\alpha } c_H + \\sqrt{\\frac{8}{3}} c_{\\alpha } s_H,$ where the $\\kappa $ factors are again defined as the ratio of the $H$ coupling in the GM model to the corresponding coupling of the SM Higgs boson.",
"In Fig.",
"REF we plot $\\kappa _f^H$ (left panel) and $\\kappa _V^H$ (right panel) in the H5plane benchmark.",
"These couplings are interesting mainly because they control the production of $H$ via gluon fusion and vector boson fusion, respectively.",
"The coupling of $H$ to fermions is largest in magnitude at large $s_H$ , reaching $-0.76$ times the corresponding SM Higgs coupling.",
"The coupling of $H$ to vector boson pairs is largest at low $m_5 \\sim 200$ –300 GeV and large $s_H$ , reaching 0.22 times the corresponding SM Higgs coupling strength.",
"Squaring these, the cross sections for $H$ production by gluon fusion and vector boson fusion reach at most 0.58 and 0.048 times the corresponding SM Higgs cross sections for a Higgs boson of the same mass as $H$ , respectively.",
"Figure: Contours of κ f H \\kappa _f^H (left) and κ V H \\kappa _V^H (right) in the H5plane benchmark.κ f H \\kappa _f^H ranges from zero to -0.76-0.76 andκ V H \\kappa _V^H ranges from zero to 0.220.22.In Figs.",
"REF and REF we plot the branching ratios of $H$ to $W^+W^-$ , $ZZ$ , $hh$ , and $t \\bar{t}$ .",
"These are the dominant decays of $H$ over the entire H5plane benchmark.",
"The branching ratios of $H$ to $W^+W^-$ and $ZZ$ dominate for $m_5$ below 600 GeV, with branching ratios above 40% and 20%, respectively.",
"These decays reach maximum branching ratios of 65% and 30%, respectively, for low $m_5 \\sim 200$ –300 GeV.",
"The branching ratio of $H$ to $W^+W^-$ ($ZZ$ ) remains above 20% (10%) over most of the benchmark plane, out to the highest $m_5$ values.",
"The branching ratio of $H$ to $hh$ dominates at high masses, reaching 50% for $m_5 \\sim 1000$ GeV and a maximum of 71% for the highest $s_H$ values at large $m_5 > 1500$ GeV.",
"The branching ratio of $H$ to $t \\bar{t}$ reaches a maximum of 37% for $m_5 \\sim 500$ –600 GeV and high $s_H$ , but falls below 10% for $m_5 \\gtrsim 1400$ GeV.",
"Note that, because $m_H > m_5$ in the H5plane benchmark, the kinematic threshold for $H \\rightarrow t \\bar{t}$ at $m_H = 2m_t$ occurs when $m_5 \\simeq 250$ GeV.",
"Figure: Contours of BR(H→W + W - H \\rightarrow W^+ W^-) (left) and BR(H→ZZH \\rightarrow ZZ) (right) in the H5plane benchmark.BR(H→W + W - H \\rightarrow W^+ W^-) ranges from 0.050.05 to 0.650.65 andBR(H→ZZH \\rightarrow ZZ) ranges from 0.020.02 to 0.300.30.Figure: Contours of BR(H→hhH \\rightarrow hh) (left) and BR(H→tt ¯H \\rightarrow t\\bar{t}) (right) in the H5plane benchmark.BR(H→hhH \\rightarrow hh) ranges from zero to 0.710.71 andBR(H→tt ¯H \\rightarrow t\\bar{t}) ranges from zero to 0.370.37.BR(H→tt ¯H \\rightarrow t\\bar{t}) drops abruptly to zero when m H <2m t m_H < 2m_t because off-shell decays to tt ¯t \\bar{t} are not calculated in GMCALC 1.2.1."
],
[
"$H$ –{{formula:76c62c13-30db-42f6-ade8-a74f85430b21}} mass splitting",
"Decays of $H_5^+$ to $H W^+$ and of $H_5^0$ to $HZ$ or $HH$ are forbidden by custodial symmetry.",
"Therefore our interest in the mass splitting between $H$ and $H_5^0$ is due to the fact that both of these states can be produced in vector boson fusion with decays to $W^+W^-$ and $ZZ$ , which opens the possibility of interference between their lineshapes if the resonances are close enough together.",
"In the left panel of Fig.",
"REF we show the mass splitting $m_H - m_5$ in the H5plane benchmark.",
"The splitting varies from 120 GeV at $m_5 = 200$ GeV to about 9 GeV at $m_5 = 3000$ GeV.",
"In the right panel of Fig.",
"REF we plot $m_H - m_5$ as a function of $m_5$ scanning over all the other free parameters in the H5plane benchmark (black points) and the full GM model (red points), where we have imposed the indirect constraints from $b \\rightarrow s \\gamma $ and the $S$ parameter [15] and direct constraints from the CMS search for $H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm }$ in vector boson fusion [4].",
"Similarly to the case of $m_3-m_5$ , we see that the variation in the mass difference $m_H - m_5$ is much greater in the full model scan than it is in the H5plane benchmark.",
"Figure: Left: Contours of m H -m 5 m_H-m_5 in the H5plane benchmark.m H -m 5 m_H-m_5 ranges from 8.98.9 GeV to 120 GeV.Right: Mass difference m H -m 5 m_H - m_5 as a function of m 5 m_5 in the H5plane benchmark (black points) and in a full scan of the GM model parameter space (red points).",
"Indirect constraints from b→sγb \\rightarrow s \\gamma and the SS parameter and direct constraints from a CMS search for H 5 ±± →W ± W ± H_5^{\\pm \\pm } \\rightarrow W^{\\pm }W^{\\pm } in vector boson fusion have been applied.To understand the experimental implications of this mass splitting, we compare it to the intrinsic widths of $H$ and $H_5^0$ .",
"In Fig.",
"REF we first plot the total width of $H$ (top left panel) and the ratio $\\Gamma _{\\rm tot}(H)/\\Gamma _{\\rm tot}(H_5^0)$ (top right panel) in the H5plane benchmark.",
"The total widths of $H$ and $H_5^0$ are very similar for $m_5 \\gtrsim 500$ GeV.",
"For lower masses, the fact that $H$ is significantly heavier than $H_5^0$ allows its width to become more than twice as large as that of $H_5^0$ for $m_5 < 450$ GeV.",
"Over the entire H5plane benchmark, the width of $H$ is never less than 89% of the width of $H_5^0$ .",
"Therefore we can quantify the $H$ –$H_5^0$ mass splitting by comparing it to the total width of $H$ .",
"We do this in the bottom panel of Fig.",
"REF , in which we plot $(m_H-m_5)/\\Gamma _{\\rm tot}(H)$ over the H5plane benchmark.",
"This ratio varies widely over the benchmark.",
"For low $m_5$ and low $s_H$ , $(m_H-m_5)/\\Gamma _{\\rm tot}(H)$ is large, which means that the $H$ and $H_5^0$ resonances are well separated compared to their intrinsic widths.",
"However, there is a sizable region of parameter space in which $(m_H-m_5)/\\Gamma _{\\rm tot}(H) < 1$ , which means that the mass splitting is less than the intrinsic width of $H$ .",
"In this region of the H5plane benchmark, the total width of $H_5^0$ is within 10% of that of $H$ .",
"In this case the two resonances overlap significantly and interfere, so that experimental searches for these two states in vector boson fusion with decays to $W^+ W^-$ or $ZZ$ must be performed taking into account both resonances and their interference.",
"Interference can be avoided by searching for $H$ produced in gluon fusion, or decaying to $hh$ or $t \\bar{t}$ .",
"Figure: Top left: Contours of the total width of HH, Γ tot (H)\\Gamma _{\\rm tot}(H), in the H5plane benchmark.Γ tot (H)\\Gamma _{\\rm tot}(H) ranges from 0.00130.0013 GeV to 170 GeV.Top right: Contours of the ratio Γ tot (H)/Γ tot (H 5 0 )\\Gamma _{\\rm tot}(H)/\\Gamma _{\\rm tot}(H_5^0) in the H5plane benchmark.Γ tot (H)/Γ tot (H 5 0 )\\Gamma _{\\rm tot}(H)/\\Gamma _{\\rm tot}(H_5^0) ranges from 0.890.89 to 16.Bottom: Contours of (m H -m 5 )/Γ tot (H)(m_H - m_5) / \\Gamma _{\\rm tot}(H) in the H5plane benchmark.",
"(m H -m 5 )/Γ tot (H)(m_H - m_5) / \\Gamma _{\\rm tot}(H) ranges from 0.0540.054 to 89000."
],
[
"Decays of $H_3$",
"The dominant decays of $H_3^0$ in the H5plane benchmark are to $t \\bar{t}$ , $hZ$ , $H_5^0 Z$ , and $H_5^{\\pm } W^{\\mp }$ .",
"($H_3^0$ can also decay to two photons; however, BR($H_3^0 \\rightarrow \\gamma \\gamma $ ) stays below $1.8 \\times 10^{-4}$ over the entire H5plane benchmark.)",
"We plot the branching ratios for these modes in Figs.",
"REF and REF .",
"The kinematic threshold for $H_3^0 \\rightarrow t \\bar{t}$ at $m_3 = 2 m_t$ occurs at $m_5$ just below 300 GeV.",
"Once above this threshold, BR($H_3^0 \\rightarrow t \\bar{t}$ ) quickly rises to a maximum of 79% for $m_5 \\sim 300$ –400 GeV, and then falls with increasing $m_5$ .",
"The next-largest fermionic decay branching ratio of $H_3^0$ is to $b \\bar{b}$ , which is below 1% over almost all of the H5plane benchmark.",
"The branching ratio of $H_3^0$ to $h Z$ exhibits complementary behaviour, growing with $m_5$ to become the dominant decay mode ($>50\\%$ ) for $m_5 \\gtrsim 500$ GeV and surpassing 90% branching ratio for $m_5 \\gtrsim 1200$ GeV.",
"Figure: Left: Contours of BR(H 3 0 →tt ¯H_3^0 \\rightarrow t\\bar{t}) in the H5plane benchmark.BR(H 3 0 →tt ¯H_3^0 \\rightarrow t\\bar{t}) ranges from zero to 0.790.79.Below the kinematic threshold at m 3 =2m t m_3 = 2m_t, the branching ratio drops to zero because off-shell decays to tt ¯t \\bar{t} are not calculated in GMCALC 1.2.1.BR(H 3 0 →tt ¯H_3^0 \\rightarrow t \\bar{t}) reaches a maximum of 0.79 and falls to 0.013 at m 5 =3000m_5 = 3000 GeV.Right: Contours of BR(H 3 0 →hZH_3^0 \\rightarrow h Z) in the H5plane benchmark.BR(H 3 0 →hZH_3^0 \\rightarrow h Z) ranges from 2×10 -4 2 \\times 10^{-4} to 0.9870.987.In the band m 5 ∈(200GeV,300GeV)m_5 \\in (200~\\text{GeV}, 300~\\text{GeV}), the branching ratio increases rapidly,up to nearly 0.90.9 for m 5 =280GeVm_5=280~\\text{GeV}, before collapsing down to about 0.20.2; BR(H 3 0 →hZH_3^0 \\rightarrow h Z) then rises with increasing m 5 m_5.",
"The sudden drop in BR(H 3 0 →hZH_3^0 \\rightarrow h Z) is due to crossing the kinematic threshold for H 3 0 →tt ¯H_3^0 \\rightarrow t\\bar{t}.The branching ratios of $H_3^0$ to $H_5^0 Z$ and $H_5^{\\pm } W^{\\mp }$ (we plot the sum of the branching ratios to $H_5^+W^-$ and $H_5^- W^+$ ) are significant only for very low $m_5$ , below the kinematic threshold for the $t \\bar{t}$ decay.",
"For these low masses, the branching ratios of these modes can be quite large, reaching respective values of 85% and 82% in our calculation, in slightly different areas of parameter space.",
"However, these numbers should be treated with caution because the implementation in GMCALC 1.2.1 of scalar decays to scalar plus vector at and below the kinematic threshold is still rather primitive.",
"At $m_5 = 200$ GeV, the mass splitting between $H_3$ and $H_5$ in the H5plane benchmark is 84 GeV, so that the on-shell decay $H_3^0 \\rightarrow H_5^{\\pm } W^{\\mp }$ is barely kinematically allowed, while $H_3^0 \\rightarrow H_5^0 Z$ is off shell.",
"As $m_5$ increases, the mass splitting decreases, and $H_3^0 \\rightarrow H_5^{\\pm } W^{\\mp }$ goes off shell at $m_5 \\simeq 210$ GeV.",
"Above threshold, GMCALC 1.2.1 computes these decay widths using the two-body on-shell decay formula, while below threshold the computation takes into account the offshellness of the vector boson only.",
"This is a reasonable approximation at $m_5 \\sim 200$ GeV where the $H_5$ scalars are very narrow; however, the transition from the on-shell to off-shell decay widths is not smooth.",
"The handling of this transition, along with off-shell decays of $H_3^0 \\rightarrow t \\bar{t}$ , should be improved if detailed predictions for the $H_3^0$ branching ratios for $m_5 \\lesssim 280$ GeV are needed.",
"The branching ratios for $H_3^0$ to $H_5^0 Z$ and $H_5^{\\pm } W^{\\mp }$ fall below 1% for $m_5 \\gtrsim 500$ GeV.",
"Figure: Contours of BR(H 3 0 →H 5 0 ZH_3^0 \\rightarrow H_5^0 Z) (left) and BR(H 3 0 →H 5 + W - +H 5 - W + H_3^0 \\rightarrow H_5^+ W^- + H_5^- W^+) (right) in the H5plane benchmark.See text for further discussion.The dominant decays of $H_3^+$ in the H5plane benchmark are to $t \\bar{b}$ , $hW^+$ , $H_5^0 W^+$ , $H_5^+Z$ , and $H_5^{++}W^-$ .",
"We plot the branching ratios for these modes in Figs.",
"REF and REF .",
"The decay to $t \\bar{b}$ dominates at low $m_5$ , reaching a maximum of more than 95% for $m_5 \\sim 250$ GeV.",
"This branching ratio falls with increasing $m_5$ and is supplanted by the decay to $h W^+$ .",
"The branching ratio for $H_3^+ \\rightarrow h W^+$ becomes dominant ($> 50\\%$ ) for $m_5 \\gtrsim 500$ GeV and surpasses 90% when $m_5 \\gtrsim 1200$ GeV.",
"Figure: Contours of BR(H 3 + →tb ¯H_3^+ \\rightarrow t \\bar{b}) (left) and BR(H 3 + →hW + H_3^+ \\rightarrow h W^+) (right) in the H5plane benchmark.BR(H 3 + →tb ¯H_3^+ \\rightarrow t \\bar{b}) ranges from 0.0130.013 to 0.9640.964 andBR(H 3 + →hW + H_3^+ \\rightarrow h W^+) ranges from 3×10 -4 3 \\times 10^{-4} to 0.9870.987.The branching ratios of $H_3^+$ to $H_5^0 W^+$ , $H_5^+ Z$ , and $H_5^{++} W^-$ are significant only for very low values of both $m_5$ and $s_H$ within the H5plane benchmark.",
"In this corner of parameter space, the branching ratios of these modes can be significant, reaching maxima of 25%, 79%, and 49%, respectively, in slightly different regions of parameter space.",
"Again, though, these numbers should be treated with caution because the decays of $H_3^+$ to $H_5V$ face the same issues with the transition from on shell to off shell as the decays of $H_3^0$ to $H_5V$ .",
"All three of these branching ratios quickly fall below the 1% level for $m_5 \\gtrsim 500$ GeV.",
"These decay modes also decline quickly with increasing $s_H$ , due to an increase in the partial width for $H_3^+ \\rightarrow t \\bar{b}$ with increasing $s_H$ .",
"Figure: Contours of BR(H 3 + →H 5 0 W + H_3^+ \\rightarrow H_5^0 W^+) (top left), BR(H 3 + →H 5 + ZH_3^+ \\rightarrow H_5^+ Z) (top right), and BR(H 3 + →H 5 ++ W - H_3^+ \\rightarrow H_5^{++} W^-) (bottom) in the H5plane benchmark.See text for further discussion.Finally, we plot the total widths of $H_3^0$ and $H_3^+$ in Fig.",
"REF .",
"They both remain quite small over the entire allowable region: although they do increase with increasing $s_H$ and $m_5$ , the width-to-mass ratio $\\Gamma _\\text{tot}(H_3)/m_3$ never rises above 8% for either $H_3^0$ or $H_3^+$ .",
"Figure: Contours of Γ tot (H 3 0 )/m 3 \\Gamma _\\text{tot}(H^0_3)/m_3 (left) and Γ tot (H 3 + )/m 3 \\Gamma _\\text{tot}(H^+_3)/m_3 (right) in the H5plane benchmark.Γ tot (H 3 0 )/m 3 \\Gamma _\\text{tot}(H^0_3)/m_3 ranges from 6.6×10 -6 6.6 \\times 10^{-6} to 0.0770.077 andΓ tot (H 3 + )/m 3 \\Gamma _\\text{tot}(H^+_3)/m_3 ranges from 6.2×10 -6 6.2 \\times 10^{-6} to 0.0770.077."
],
[
"Conclusions",
"In this paper we studied the constraints on and phenomenology of the H5plane benchmark scenario in the Georgi-Machacek model.",
"The H5plane benchmark has two free parameters, $m_5$ and $s_H$ , where $s_H^2$ is equal to the fraction of $M_W^2$ and $M_Z^2$ that is generated by the vev of the isospin triplets.",
"The H5plane benchmark is defined for $m_5 \\in [200,3000]$ GeV.",
"Existing theoretical and experimental constraints limit $s_H$ to be below 0.55 in the H5plane benchmark, so that at most 30% of the $W$ and $Z$ boson squared-masses can be generated by the triplets.",
"A full parameter scan of the GM model yields an allowed region in the $m_5$ –$s_H$ plane only slightly larger than in the H5plane benchmark for $m_5 \\in [200,3000]$ GeV.",
"Our numerical work has been done using the public code GMCALC 1.2.1.",
"We showed that the couplings of the 125 GeV Higgs boson $h$ in the H5plane benchmark are sufficiently SM-like that the benchmark is not further constrained by the ATLAS and CMS measurements of Higgs production and decay at LHC center-of-mass energies of 7 and 8 TeV—in fact, over most of the H5plane benchmark, the fit to LHC data is slightly better than in the SM.",
"Over the H5plane benchmark, compared to their SM values, the $h$ coupling to fermions can be suppressed by up to 10% or enhanced by up to 1.4%, its coupling to vector boson pairs can be enhanced by up to 21%, and its loop-induced coupling to photon pairs can be suppressed by up to 1.3% or enhanced by up to 24% (loops involving the charged scalars in the GM model contribute non-negligibly to this).",
"The total width of $h$ can be suppressed by up to 2.9% or enhanced by up to 3.5% compared to that of the SM Higgs boson; the smallness of this range is due to an accidental cancellation among the fermionic and bosonic contributions.",
"By design, the mass-degenerate $H_5^{\\pm \\pm }$ , $H_5^{\\pm }$ , and $H_5^0$ scalars are the lightest new scalars in the H5plane benchmark, and hence decay only to vector boson pairs at tree level.",
"Due to the parameter specifications in the benchmark, the mass splittings $m_3-m_5$ and $m_H-m_5$ are almost constant with $s_H$ , depending primarily on $m_5$ .",
"They fall from maxima of 84 and 120 GeV, respectively, at $m_5 = 200$ GeV to minima of 7 and 9 GeV, respectively, at $m_5 = 3000$ GeV.",
"(These mass splittings vary much more freely in the full GM model.)",
"While the mass-to-width ratios of all the new scalars in the GM model remain below 8% in the H5plane benchmark, the fairly small mass splitting between $H_5^0$ and $H$ means that these two resonances can overlap and interfere when produced in vector boson fusion and decaying to $W^+W^-$ or $ZZ$ .",
"Their mass splitting becomes smaller than their intrinsic widths when $m_5 \\gtrsim 700$ GeV, unless $s_H$ is small.",
"Finally we studied the production and decays of the new heavy Higgs bosons in the GM model in the H5plane benchmark.",
"We found that, due to coupling suppressions, the production cross section of $H$ in gluon fusion (vector boson fusion) can be at most 58% (4.8%) as large as that of a SM Higgs boson of the same mass.",
"$H$ decays mainly to $W^+W^-$ and $ZZ$ for $m_5$ below 600–1000 GeV (depending on $s_H$ ), and mainly to $hh$ for $m_5$ above 700–1300 GeV.",
"Its branching ratio to $t \\bar{t}$ can top 30% for $m_5$ between 400 and 700 GeV.",
"$H_3^0$ decays predominantly to $t \\bar{t}$ from the kinematic threshold at $m_5 = 280$ GeV up to $m_5 \\simeq 500$ GeV, where $hZ$ takes over as the dominant decay mode.",
"Below the $t \\bar{t}$ threshold, decays to $H_5^0 Z$ and $H_5^{\\pm } W^{\\mp }$ can be significant, but improvements to the handling of near-threshold decays in GMCALC are needed to fully explore the branching ratios in this region.",
"$H_3^+$ decays predominantly to $t \\bar{b}$ for $m_5$ values up to about 500 GeV, where $h W^+$ takes over as the dominant decay mode.",
"We thank Dag Gillberg for helpful conversations.",
"This work was supported by the Natural Sciences and Engineering Research Council of Canada.",
"H.E.L.",
"was also partially supported through the grant H2020-MSCA-RISE-2014 no.",
"645722 (NonMinimalHiggs)."
]
] | 1709.01883 |
[
[
"Cross-Domain Image Retrieval with Attention Modeling"
],
[
"Abstract With the proliferation of e-commerce websites and the ubiquitousness of smart phones, cross-domain image retrieval using images taken by smart phones as queries to search products on e-commerce websites is emerging as a popular application.",
"One challenge of this task is to locate the attention of both the query and database images.",
"In particular, database images, e.g.",
"of fashion products, on e-commerce websites are typically displayed with other accessories, and the images taken by users contain noisy background and large variations in orientation and lighting.",
"Consequently, their attention is difficult to locate.",
"In this paper, we exploit the rich tag information available on the e-commerce websites to locate the attention of database images.",
"For query images, we use each candidate image in the database as the context to locate the query attention.",
"Novel deep convolutional neural network architectures, namely TagYNet and CtxYNet, are proposed to learn the attention weights and then extract effective representations of the images.",
"Experimental results on public datasets confirm that our approaches have significant improvement over the existing methods in terms of the retrieval accuracy and efficiency."
],
[
"Introduction",
"With the ubiquitousness of smart phones, it is convenient for people to take photos anytime and anywhere.",
"For example, we usually take photos of beautiful sceneries in our outings, candid photos of funny moments, and well-presented dishes in restaurants.",
"These photos can be used as queries to search visually similar images on the Internet.",
"With the wide acceptance of e-commerce, product image search becomes part-and-parcel of online shopping, where users submit a photo taken via their cell phones to look for visually similar products [11].",
"Product image search is a challenging problem as it involves images from two heterogeneous domains, namely the user domain and the shop domain.",
"The user domain consists of the query images taken by users, and the shop domain consists of the database images taken by professional photographers for the e-commerce websites.",
"Images from the two domains exhibit different characteristics.",
"For instance, images from user domain (see Figure REF (a), REF (b)) typically have large variations in orientation and lighting, whereas images from shop domain are taken under good condition(see Figure REF (c)).",
"How to model the domain-specific and domain-invariant features is a challenge.",
"Traditional image retrieval approaches designed for a single domain cannot model the domain-specific features of different domains.",
"DARN[11] and MSAE [28], [29] create different deep learning branches for each domain separately to model domain-specific features.",
"However, they cannot capture the common features shared by both domains.",
"Figure: Example images of the same product.Another challenge comes from the difficulty of attention modeling for the user domain and the shop domain.",
"Attention modeling is to find the focus of images to extract effective image representations from the salient areas, i.e.",
"spatial locations.",
"For images from user domain, however, they exhibit large variations and have noisy background as shown by the example in Figure REF (a) and REF (b).",
"For shop images, it is common that the whole-body clothes along with some accessories are displayed in an image as shown in Figure REF (c), whereas the target is the upper clothes.",
"Without external information, it is difficult to find the attention of the query image (from the user domain) and the database image (from the shop domain).",
"Existing approaches, including DARN [11] and FashionNet [16], reply on human annotated bounding boxes and landmarks for attention modeling.",
"However, it is costly to do such annotations for large image databases from e-commerce websites.",
"In this paper, we propose a neural network architecture to extract effective image representations for cross-domain product image search.",
"Inspired by [35], which shows that the bottom layers of a convolutional neural network (CNN) learn domain-invariant features and the top layers learn domain-specific features, we propose a `Y-shape' network architecture that shares a set of convolution layers at the bottom for both domains, and has separate branches on the top for each domain.",
"Each branch on the top contains an attention modeling layer to learn the spatial attention weights of the feature maps from the convolution layer.",
"The attention modeling layer of the shop branch exploits the tag information of shop images, which are widely available on e-commerce websites.",
"Tags like product category and attributes can help us to locate the attention easily.",
"For instance, in Figure REF (c), from the tag `Flare Sleeves', we know that the focus of this image is sleeves rather than trousers.",
"DARN and FashionNet also use tags to train their models.",
"Different from them, we use tags as inputs instead of prediction outputs.",
"For the attention modeling layer of the user branch, since user-provided tags are not widely available and usually contain much noise, we use the candidate shop image as the context to help locate the attention.",
"For example, in Figure REF (b), the focus of the user image is not clear (T-shirt or dress) until we compare it with the shop image.",
"The final image representation is generated by aggregating the features at each spatial location according to the attention weight.",
"We adapt the triplet loss [20] as the training objective function.",
"A training triple consists of an anchor image from the user domain, a positive image from the shop domain that has the same product as the anchor image, and a negative image from the shop domain that does not have the same product as the anchor image.",
"We first forward them through the shared layers to get a set of feature maps for each image.",
"Second, we forward the feature maps of the positive (resp.",
"negative) image and its tags into the shop branch.",
"Third, the generated representation of the positive (resp.",
"negative) image and the feature maps of the anchor image are fed together into the user branch to learn the representation of the anchor image w.r.t the positive (resp.",
"negative) image.",
"We extend the triplet loss to accept four inputs, namely the representations for the positive image, the negative image, and the anchor image w.r.t the positive and negative image.",
"The training procedure updates the network parameters to move the similar pairs closer than dissimilar pairs.",
"During querying, our approach needs to extract the query representation w.r.t each database image, which is expensive.",
"To improve the search efficiency, we follow [4] to do an initial search with the query representation generated by aggregating its feature maps equally across all spatial locations (i.e.",
"all locations have the same attention weight [1]).",
"The returned top-K candidate images are re-ranked using the query representation extracted by attention modeling.",
"Our contributions include: A novel neural network architecture to extract effective features for cross-domain product image retrieval.",
"Two attention modeling approaches for the user and shop domain respectively, i.e., exploiting tag information to help locate the attention of the database images (denoted as TagYNet) and exploiting the candidate database images to locate the attention of the query images (denoted as CtxYNet).",
"Comprehensive experimental study, which confirms that our approaches outperform existing solutions[11], [16] significantly."
],
[
"Related Work",
"Our cross-domain product image retrieval is one type of Content Based Image Retrieval (CBIR).",
"CBIR has been studied for decades [7] to search for visually similar images, and it consists of two major steps.",
"First, a feature vector is extracted to represent each image, including the query image and images in the database.",
"Second, the similarity (e.g.",
"cosine similarity) of the query image and the database images are computed based on the feature vectors.",
"Top-K similar images are usually returned to the user.",
"Many research works have been proposed to improve the search performance from the feature extraction and similarity measurement perspectives.",
"For instance, local descriptors such as SIFT [17], and feature aggregation algorithm such as VLAD [12], have been proposed for improving the feature representation.",
"Metric learning [3] is a research area which focuses on learning mapping functions to project features into an embedding space for better similarity measurement.",
"With the resurgence of deep learning [13], [30], many approaches based on deep learning models have been proposed towards learning semantic-rich features and similarity metric.",
"We review some related works as follows."
],
[
"Feature Learning",
"Feature learning, in contrast to feature engineering, learns representation of data (i.e.",
"images) directly.",
"It becomes popular due to its superior performance over hand-crafted features, such as SIFT [17], for many computer vision tasks such as image classification [13] and retrieval[26].",
"Deep convolutional neural networks (DCNN) [13], [10] consists of multiple convolutions, pooling, and other layers.",
"These layers are tuned over a large training dataset to extract semantic-rich features for the tasks of interest, e.g.",
"image classification.",
"Rather than training a DCNN from scratch, recent works [24] have shown that transfer learning, by introducing knowledge from large-scale well-labeled dataset such as ImageNet [8], can achieve competitive performance with less training time.",
"Apart from transferring knowledge from domain to domain [33], [34], Ji et al.",
"[26] shows that introducing knowledge from one task to another, i.e., from image classification to image retrieval, also works well.",
"For example, DARN [11] and FashionNet [16] fine-tune the DCNNs to extract features for attribute prediction and cross-domain image retrieval.",
"These work incorporate the attribute information into the model to capture more semantics in feature representation.",
"In addition, FashionNet trains a subnetwork to predict the landmarks (a.k.a.",
"attention areas) by extracting features from local areas.",
"Different from these methods, we exploit the attributes of database images, which are easier to collect than landmarks, to locate the spatial attention areas of images directly and then extract features from these areas to construct our image feature for retrieval.",
"Moreover, for the query image, we extract context-dependent attention when calculating its similarity with database images (see Section REF )."
],
[
"Attention Modeling",
"Attention modeling differentiates inputs with different weights.",
"It has been exploited in computer vision to extract features from salient areas of an image [32].",
"Recently, it is applied in machine translation models [2] to locate the attention of source words for generating the next target word, and in trajectory prediction to track the attention of dynamic objects like pedestrians and bikers [25].",
"Typically, external information is required to infer the attention weights.",
"For example, the image caption generation model [32] uses the previous word to infer the attention weights for features from different locations of the image, and then generates the next word using the feature from weighted aggregation.",
"There are different approaches to compute the attention weights by aligning the external information with the visual feature of each location, e.g., via a multi-layer perceptron (MLP) [2], inner-product [23], outer-product [9], etc.",
"We explicitly exploit the image attributes as the external information to locate the attention of images in the database, and exploit the database image as the context to infer the attention of the query image.",
"Attention modeling ignores noisy background (occlusion), and thus extracts discriminative features for retrieval.",
"Deep metric learning trains the neural networks to project the images into metric space for similarity measurement.",
"Contrastive loss based on Siamese network [6] is the most widely used pairwise loss for metric learning.",
"Wang et al.",
"[31] optimizes the contrastive loss by adding a constraint on the penalty to maintain robustness on noise positive pairs.",
"Unlike pairwise loss that considers the absolute distance of pairs, triplet loss [20] computes the relative distance between a positive pair and a negative pair of the same anchor.",
"Yuan et al.",
"[36] changes the original triplet loss into a cascaded way in order to mine hard examples.",
"Song et al.",
"[18] designs a mini-batch triplet loss that considered all possible triplet correlations inside a mini-batch.",
"Liu et al.",
"[15] proposes a cluster-level triplet loss that considered the correlation of cluster center, positive samples and the nearest negative sample.",
"Query adaptive matching proposed in [4] computes the similarity based on the aggregated local features of each candidate image.",
"The aggregation weights are determined by solving an optimization problem using the query as the context.",
"However, it incurs extra computation time.",
"In this paper, we generate different representations for the query image to compute its similarity with different database images.",
"In other words, the query representation is adaptive to the compared database image."
],
[
"Problem Statement",
"In this paper, we study the problem of cross-domain product image retrieval where the query image comes from user domain $I_u$ (i.e.",
"taken by users) and the database images are from shop domain $I_s$ (i.e.",
"from the e-commerce websites).",
"A database image is matched with the query image if they contain the same product.",
"Our task is to train a model to extract effective image representations where matched pairs have smaller distance.",
"We adopt CNN in our model for image feature extraction as CNN has shown outstanding performance in learning features for variant computer vision tasks [5].",
"Denote the feature vector at location $l$ of the convolution layer as $\\mathbf {x}_l\\in R^C$ , where $C$ is the number of channels, i.e.",
"the number of feature maps.",
"To aggregate the features across all locations effectively, we train an attention modeling subnetwork for each domain to learn the spatial weights (i.e.",
"attention).",
"Once we get the weight of each location, denoted as $a_l\\in R$ , the final image feature is aggregated as $\\sum _{l}a_l * \\mathbf {x}_l$ .",
"Our training dataset consists of user images and shop images, where each image has a product ID.",
"The product ID is used for constructing triples for metric learning, where each triple consists of an anchor image $o\\in I_u$ , a positive image $p\\in I_s$ which contains the same product as the anchor image, and a negative image $q\\in I_s$ which does not contain the same product as the anchor image.",
"In addition, each shop image $x$ has a set of tags (e.g.",
"category or attributes) represented using a binary vector $\\mathbf {x}^t\\in \\lbrace 0, 1\\rbrace ^T$with the value 1 indicting the presence of the tag and 0 otherwise, where $T$ is the total number of tags.",
"These tags are exploited for attention modeling (see Section REF ).",
"The notations used in this paper is summarized in Table REF .",
"Scalar values are in normal fonts, whereas bold fonts are for vectors and matrices.",
"We refer to $o, p, q, x$ as images and $\\mathbf {o,p,q,x}$ as image feature maps (i.e.",
"output of convolution layers) or vectors.",
"We reuse the notation $\\mathbf {o,p,q,x}$ as the input and output of a subnetwork.",
"Table: Notations."
],
[
"Our Approach",
"In this section, we introduce our approach for inferring the spatial attention of both the query and database images to extract effective features for product image search.",
"The network architecture, attention models, training and retrieval algorithms shall be discussed."
],
[
"Overview",
"Our network architecture is illustrated in Figure REF , which is like the character 'Y' in landscape.",
"It consists of convolution layers (subnetwork 1, 2 and 3) for image feature extraction, attention layers for spatial attention modeling (subnetwork 4 and 5), and a triplet ranking loss layer for metric learning.",
"During training, each training triple $<o, p, q>$ is forwarded through a shared subnetwork 1 and then passed to two separate subnetwork 2 and 3.",
"This design is to learn domain-invariant features by 1 and learn domain-specific features by subnetwork 2 and 3.",
"The shared subnetwork 1 also saves memory compared with the dual network architecture in [11] which needs to store the parameters for both branches.",
"We denote the subnetwork 1 2 3 together as YNet.",
"Subnetwork 4 infers the spatial attention weights by exploiting the tags (e.g.",
"attributes) of shop products.",
"With the attention weights, it aggregates the features over all locations to get the feature vector of $p$ (resp.",
"$q$ ).",
"The subnetwork 1 2 3 4 is denoted as TagYNet.",
"By using $p$ (resp.",
"$q$ ) as the context, subnetwork 5 computes the attention of the anchor image $o$ w.r.t the $p$ (resp.",
"$q$ ).",
"The whole network is denoted as CtxYNet.",
"The image representations are fed into an adapted triplet-loss function.",
"After training, we extract the representation of each database image via subnetwork 1, 2 and 4.",
"When a query arrives, we extract a simple representation of the query by forwarding the image through subnetwork 1 and 3, and then averaging features at all spatial locations.",
"This simple representation is used to conduct an initial retrieval.",
"For each candidate image in the top-K (K=256 in our experiments) list, we use it to infer the query's attention via subnetwork 5 and then compute the similarity.",
"These images are finally re-ranked based on the new similarity score.",
"Specifications of the CNN layers in subnetwork 1, 2 and 3 will be given in Section .",
"We next introduce the subnetwork 4, 5 for attention modeling and the adapted triplet loss function."
],
[
"Attention Modeling for Shop Images",
"Shop images are usually taken in good condition by professional photographers.",
"Hence, they have better quality than user images.",
"However, some images, especially for fashion products, are typically presented with accessories or other items as shown in Figure REF (c).",
"To get the attention, we exploit the tags (or attributes) associated with the shop image, which can be collected easily from the shopping websites.",
"Take the shop image in Figure REF (c) as an example, the tags `Crew neck', `Short sleeves' and `Rectangle-shaped' are useful for locating the attention of the image.",
"$g_s(\\mathbf {x}_l, \\mathbf {x}^t) &=& \\mathbf {x}_l \\cdot (\\mathbf {W}^t \\cdot \\mathbf {x}^t) \\\\a_l &=& \\frac{e^{g_s(\\mathbf {x}_l, \\mathbf {x}^t)}}{\\sum _k^L e^{g_s(\\mathbf {x}_k, \\mathbf {x}^t)}} \\\\f_{{4}}(\\mathbf {x}, \\mathbf {a})&=&\\sum _l^L a_l*\\mathbf {x}_l $ Given a shop image associated with the tag vector $\\mathbf {x}^t\\in \\lbrace 0, 1\\rbrace ^T$ , we forward the image's raw feature (i.e.",
"the RGB feature) through subnetwork 1 and 2 to get the feature maps $\\mathbf {x}$ .",
"Subnetwork 4 computes the attention of each spatial location through Equation REF -.",
"The intuition is to find a projection matrix that embeds (via $\\mathbf {W}^t \\in R^{T*C}$ ) the tags into a common latent space as the image features.",
"If the image feature at one position matches (i.e.",
"aligned well with) the embedded tag feature, we assign a larger attention weight for that position to let it contribute more in the final image representation.",
"We adopt inner-product (Equation REF ) as the alignment function and the Softmax function (Equation ) to generate the weights.",
"The output of the subnetwork is a $C$ -dimensional feature vector aggregated across all spatial locations according to Equation ."
],
[
"Attention Modeling for User Images",
"For user images, i.e.",
"the query images, they are usually taken occasionally using smart phones.",
"Consequently, the focus of the image may not be clear, e.g.",
"when there are multiple (background) objects in the image.",
"In addition, for some kinds of products like clothes, the images are subject to deformations.",
"It is thus important to locate the attention of the image for extracting effective features.",
"However, the attention is not easy to get without any context information (the tags of the query images are usually not available).",
"Given a query image and a shop image, we infer the spatial attention of the query image using the shop image as the context.",
"Denote $\\mathbf {x}\\in R^C$ as the output of 4, which could be either a positive image $p$ or a negative image.",
"Following Equation -, we infer the attention of the anchor image $o$ in subnetwork 5.",
"$g_u(\\mathbf {o}_l, \\mathbf {x}) &=& \\mathbf {v}\\cdot \\mathbf {o}_l + \\mathbf {U}_{l\\cdot }\\cdot \\mathbf {x} \\\\a_l &=& \\frac{e^{g_u(\\mathbf {o}_l, \\mathbf {x})}}{\\sum _{k}^L e^{g_u(\\mathbf {o}_k, \\mathbf {x)}}} \\\\f_{{5}}(\\mathbf {o}, \\mathbf {a})&=&\\sum _l^L a_l*\\mathbf {o}_l $ where Equation REF is a linear alignment function, which has better performance than the inner-product alignment function for our experiment.",
"$\\mathbf {v} \\in R^C$ , $\\mathbf {U} \\in R^{L*C}$ are weights to learn."
],
[
"Loss Function",
"We adapt the triplet loss function as the training objective, which is shown in Equation REF .",
"$L(o, p, q) = \\max (0, d(\\mathbf {o}^p,\\mathbf {p}) - d(\\mathbf {o}^q, \\mathbf {q}) + \\alpha ) $ where $d(\\cdot ,\\cdot )$ measures the distance of two images based on their final representation.",
"Euclidean distance is used for $d(\\cdot ,\\cdot )$ .",
"Following [20], in order to make the training converge stable, we normalize the output of subnetwork 4 and 5 via L2 norm before feeding them into the loss function.",
"The margin value $\\alpha $ is tuned on a validation dataset (0.5 for our experiments).",
"Different to the existing approaches [16], [11] that use the same representation for $o$ in Equation REF , i.e.",
"$d(\\mathbf {o,p})$ and $d(\\mathbf {o,q})$ .",
"We have different representations for $o$ , i.e.",
"$\\mathbf {o}^p$ and $\\mathbf {o}^q$ for $d(\\mathbf {o,p})$ and $d(\\mathbf {o,q})$ respectively.",
"This is because two sets of attention weights are generated against $p$ and $q$ respectively.",
"The loss function penalizes the triples if $d(\\mathbf {o}^p,\\mathbf {p}) +\\alpha > d(\\mathbf {o}^q,\\mathbf {q})$ by updating the model parameters to make matched images close and unmatched images far away in the embedded Euclidean space."
],
[
"Experiments",
"In this section, we conduct experimental study by comparing our proposed approaches in Section with baseline methods in terms of search effectiveness and efficiency on two real-life datasets."
],
[
"Dataset",
"In our experimental study, we use the DARN dataset [11] and DeepFashion dataset [16].",
"The DARN dataset is collected for street-to-shop retrieval, i.e.",
"matching street images taken by users with professional photos provided by online shopping sites.",
"After removing corrupted images, we get a subset of 62,812 street images and 238,499 shop images of 13598 distinct products distributed over 20 categories.",
"Each street image has a matched shop image.",
"This dataset also provides semantic attributes for the products.",
"Detailed information of the attributes is available in [11].",
"We use 7 types of attributes except the color attribute since we observe that same product may be displayed with different colors in this dataset.",
"We partition the dataset into three subsets for training, validation and test, with no overlap of products (see Table REF ).",
"The DeepFashion dataset includes over 800,000 images with various labeled information in terms of categories, clothes attributes, landmarks, and image correspondences for cross-domain/in-shop scenario.",
"We do not explore the landmark information since it is beyond the scope of this paper.",
"Additionally, we only use a subset of images from street2shop set, i.e.",
"19,387 distinct upper clothes of over 130,000 images.",
"We select 11 types of tags that are related to the upper clothes of the product images.",
"Similar partition method is applied for this dataset (see Table REF ).",
"Different from DARN dataset, DeepFashion dataset has more street images (130,000+) than shop images (21,377).",
"Table: Dataset Partition DARN[11]We use the notation DARN for both the dataset and the method.",
"has two branches of NIN (Network in Network) networks, one for the street domain and one for the shop domain.",
"It is different to YNet which shares the same bottom layers for both domains.",
"On top of the NIN networks, there are several fully connected layers for category and attribute prediction.",
"The training loss is a weighted combination of the prediction losses and the triplet loss.",
"The triplet loss is calculated using a long feature vector by concatenating the features from convolution layers and fully connected layers.",
"We adjust the shapes of lower convolution layers to be the same as the original NIN model [14] in order to use the pretrained parameters of NIN over ImageNet.",
"FashionNet[16] FashionNet shares the convolution layers for both domains.",
"It has a landmark prediction subnetwork whose results are used to subsample the feature maps of the last convolution layer.",
"The top branches are for different tasks (including tag prediction and landmark prediction) instead of for different domains as in YNet.",
"In other words, all images are passed through the same set of layers in FasionNet, whereas street and shop images are passed through different top layers in YNet.",
"Due to the memory limit, we replace the VGG-16 model [22] used in the DeepFashion paper with VGG-CNN-S [5] https://gist.github.com/ksimonyan/fd8800eeb36e276cd6f9.",
"We also remove the landmark prediction subnetwork since exploring the effect of landmark is out of the scope of this paper.",
"TripletNIN and TripletVGG By removing the category and attribute prediction subnetworks of DARN and FashionNet, we get two networks whose loss function only includes triplet loss.",
"NIN and VGG-CNN-S We use the NIN and VGG-CNN-S trained on ImageNet to extract the feature for both user images and shop images."
],
[
"Network Configuration of Our Approaches",
" TagYNIN and CtxYNIN based on NIN.",
"TagYNIN uses the first 4 convolution layer blocks of NIN as subnetwork 1.",
"The 5-th convolution layer block is used for subnetwork 2 and 3.",
"Subnetwork 4 uses inner-product as the attention alignment function.",
"The output of subnetwork 2 and 4 are fed into the triplet loss.",
"CtxYNIN adds the subnetwork 5 on top of TagYNIN.",
"TagYVGG and CtxYVGG based on VGG-CNN-S. TagYVGG is similar to TagYNIN except the NIN layers are replaced with layers from VGG-CNN-S. CtxYVGG is similar to CtxYNIN except the NIN layers are replaced with layers from VGG-CNN-S."
],
[
"Implementation Details",
"We use backpropagation for parameter gradient calculation and mini-batch Stochastic Gradient Descent (SGD) for parameter value updating.",
"The batch size is 32 and momentum is 0.9.",
"Learning rate is set to 0.01 at the initial state and decays by 0.1 for every 30 epochs.",
"Margin in the loss function is set to 0.5 for the baseline experiments.",
"We set the margin to 0.3 for YNet and TagYNet, and 0.5 for CtxYNet.",
"We train our networks in the order of YNet, TagYNet, and CtxYNet by using the parameters trained from the previous network to initialize the subsequent network.",
"The pretrained parameters over ImageNet (e.g.",
"using NIN) is used to initialize the corresponding YNet (e.g.",
"YNIN).",
"For YNet and TagYNet, we use the standard triplet loss function; For CtxYNet, we use the adapted triplet loss function (Equation REF ).",
"When training TagYNet and CtxYNet, we freeze the parameters in subnetwork 1.",
"Considering that bounding boxes and landmarks are costly to annotate for large image databases in real applications, we do not use these information for training.",
"In contrast, the attribute and category information is typically available on e-commerce websites.",
"Therefore, we use them in training DARN, FashionNet and our approaches.",
"Following [11], [16], we evaluate the retrieval performance by top-K precision, which is defined as follows: $P@K = \\frac{\\sum _{q\\in Q}hit(q, K)}{|Q|}$ where $Q$ is the total number of queries; $hit(q, K)=1$ if at least one image of the same product as the query image $q$ appears in the returned top-K ranking list; otherwise $hit(q, K)=0$ .",
"For most queries, there is only one matched database image in both the DARN and DeepFashion datasets."
],
[
"Comparison on DARN Dataset",
"Figure REF shows the top-K precision of the baseline approaches and our approaches on DARN dataset.",
"We can see that the pretrained NIN performs worst since the task and dataset are different to our product image retrieval application.",
"TripletNIN is better than NIN as it is fine-tuned with metric learning over the DARN dataset to distinguish the inner-category images (e.g.",
"different T-shirts).",
"However, it is not as good as DARN which considers the tag information during training through attribute and category prediction.",
"Meanwhile, we can observe that our TagNIN and CtxNIN outperform a lot than YNIN which does not incorporate tag information.",
"These prove the effectiveness of exploring tag information for image retrieval.",
"We further compare DARN, FashionNet and our attention modeling based approaches.",
"From Figure REF , we observe that both of our attention modeling approaches significantly outperform DARN (over 10% improvement on top-20 accuracy) and FashionNet (1.4% and 3.4% improvement on top-20 accuracy).",
"DARN and FashionNet only use tags during training, which serves as a regularization on the extracted feature (require the feature to capture tag information).",
"Our TagYNet uses tags in both training and querying phases for the shop images, which helps to locate the attention of the shop images especially when noisy background or multiple products occur in one image.",
"Consequently, it captures more discriminative features.",
"In addition, we do not need to tune the weights of prediction loss and triplet loss as in DARN and FashionNet.",
"In terms of network architecture, our YNet architecture involves fewer parameters compared with the dual network architecture of DARN, and thus is more robust to overfitting.",
"CtxYNet performs slightly better than TagYNet by reranking the top-256 list retrieved from TagYNet.It indicates that attention modeling of the query images indeed helps to obtain better feature representations."
],
[
"Comparison on DeepFashion Dataset",
"Figure REF shows the detailed top-k precision of baseline approaches and our approaches on DeepFashion dataset.",
"We notice that the retrieval performance of all approaches on this dataset is much better than those on DARN's dataset, mainly due to the smaller size of the database.",
"We observe that our attention modeling approaches again perform best over the other baselines approaches.",
"Compared with DARN, FashionNet achieves significantly better performance than DARN, which is consistent with results shown in the paper [16].",
"However, our proposed network performs even better than FashionNet.",
"Our tag-based attention mechanism (TagYNet) and context-based attention mechanism (CtxYnet) gain 2% and 4% improvement on top-20 precision over FashionNet.",
"There are still some noisy tags (not visually relevant to the image content, e.g.",
"'thickness of clothes') in DeepFashion dataset, therefore, we expect to see better performance if these tags/attributes are filtered out."
],
[
"Query Efficiency",
"Our retrieval system runs on a server with Intel i7-4930K CPU and three GTX Titan X GPU cards.",
"We develop the system using SINGA [19], [27] — a deep learning library.",
"We measure the query efficiency on the two datasets as shown in Figure REF .",
"The feature extracted by our approaches is aggregated from the features across spatial locations of the convolution layer.",
"Therefore, the feature dimension is the number of the feature maps in the last convolution layer, which is 1000 and 512 for NIN-based and VGG-based networks respectively.",
"DARN and FashionNet[11], [15] concatenate the local features from convolution layers and global features from fully connected layers.",
"Consequently, their feature dimension is larger than ours.",
"Even after applying PCA, the feature dimension is still up to 8196-D, which is 8x and 16x larger than our NIN-based and VGG-based features, respectively.",
"As a result, our approaches achieve better efficiency as shown in Figure REF ."
],
[
"Attention Visualization",
"For better understanding of the attention modeling mechanism, we visualize the attention maps of two sample images from DARN in Figure REF .",
"The `max attention' (resp.",
"min attention) map is generated following [21]: first, setting the gradients of the final convolution layer as 0 except those for the location (denoted as $l$ ) with the maximum (resp.",
"minimum) attention weight computed from TagNIN, whose gradients are set to 1; second, back-propagating the gradients to get the gradients of the input data, which reflect the activations of location $l$ and are used to plot the attention map.",
"From Figure REF , we can see that the activations of the maximum weighted position matches the attention of the input image better than that of the minimum weighted position.",
"In addition, the noisy background is filtered in the attention maps, where the activations mainly cover the clothes.",
"In other words, the attention weights have semantic explanations."
],
[
"Example Queries and Results",
"We analyze some sample queries and corresponding results for better understanding the task and feature extraction models.",
"The DARN dataset and TagYNet are used for this experiment.",
"Four queries with matched results in the top-5 list are shown In FigureREF .",
"First, we can see that the result images (i.e.",
"shop images in column 2-6) have much better quality than the query images (i.e.",
"user image in the first column).",
"Therefore, we need separate branches for extracting the features from the two domains.",
"Second, some shop images, e.g.",
"the matched result of the third query, also have noisy background.",
"It is necessary to incorporate extra information like tags to locate the image attention.",
"Third, the query image and the matched result may look quite different from the global view, as shown in the second row.",
"In spite of such difference, our model is able to focus on local pattens (e.g.",
"the collar and pocket) to find matched results.",
"We also sample some queries for which our model fails to find matched images in the top-5 list, as shown in Figure REF .",
"First, we can see that these query images are either cropped (the first query) or taken under insufficient lighting (the second query).",
"Second, the evaluation criteria is very strict.",
"As shown in the third row, some images are not considered as matched results although they are visually very similar to the query images.",
"In fact, only images of the exact same product as the query are considered matched results."
],
[
"Conclusion",
"To tackle the problem of cross-domain product image search, we have presented a novel neural network architecture which shares bottom convolutional layers to learn domain-invariant features and separates top convolutional layers to learn domain-specific features.",
"Different from other approaches, we introduce a tag-based attention modeling mechanism denoted as TagYNet by exploiting product tag information, and a context-based attention modeling mechanism denoted as CtxYNet, using the candidate image as the context.",
"Experiments on two public datasets confirm the efficiency and effectiveness of the features extracted using our attention based approaches."
],
[
"Acknowledgment",
"This work is supported by the National Research Foundation, Prime Minister's Office, Singapore, under its Competitive Research Programme (CRP Award No.",
"NRF-CRP8-2011-08), and FY2017 SUG Grant, National University of Singapore."
]
] | 1709.01784 |
[
[
"Truncation of the Accretion Disk at One Third of the Eddington Limit in\n the Neutron Star Low-Mass X-ray Binary Aquila X-1"
],
[
"Abstract We perform a reflection study on a new observation of the neutron star low-mass X-ray binary Aquila X-1 taken with NuSTAR during the August 2016 outburst and compare with the July 2014 outburst.",
"The source was captured at $\\sim32\\%\\ L_{\\mathrm{Edd}}$, which is over four times more luminous than the previous observation during the 2014 outburst.",
"Both observations exhibit a broadened Fe line profile.",
"Through reflection modeling, we determine that the inner disk is truncated $R_{in,\\ 2016}=11_{-1}^{+2}\\ R_{g}$ (where $R_{g}=GM/c^{2}$) and $R_{in,\\ 2014}=14\\pm2\\ R_{g}$ (errors quoted at the 90% confidence level).",
"Fiducial neutron star parameters (M$_{NS}=1.4$ M$_{\\odot}$, $R_{NS}=10$ km) give a stellar radius of $R_{NS}=4.85\\ R_{g}$; our measurements rule out a disk extending to that radius at more than the $6\\sigma$ level of confidence.",
"We are able to place an upper limit on the magnetic field strength of $B\\leq3.0-4.5\\times10^{9}$ G at the magnetic poles, assuming that the disk is truncated at the magnetospheric radius in each case.",
"This is consistent with previous estimates of the magnetic field strength for Aquila X-1.",
"However, if the magnetosphere is not responsible for truncating the disk prior to the neutron star surface, we estimate a boundary layer with a maximum extent of $R_{BL,\\ 2016}\\sim10\\ R_{g}$ and $R_{BL,\\ 2014}\\sim6\\ R_{g}$.",
"Additionally, we compare the magnetic field strength inferred from the Fe line profile of Aquila X-1 and other neutron star low-mass X-ray binaries to known accreting millisecond X-ray pulsars."
],
[
"Introduction",
"Aquila X-1 is a neutron star (NS) residing in a low-mass X-ray binary (LMXB) that has exhibited X-ray pulsations, if intermittently.",
"A LMXB consists of an accreting compact object with a companion star of approximately solar mass.",
"The companion star in Aquila X-1 is categorized as a K0 V spectral type ([52]; [36]).",
"Coherent millisecond X-ray pulsations were detected for 150 s during persistent emission imply a spin frequency of 550 Hz [12].",
"Type-I X-ray bursts place an upper limit on the distance to Aquila X-1 of 5.9 kpc away, assuming the bursts are Eddington limited [27].",
"The inclination of the system is constrained to be $<31^{\\circ }$ by infrared photometry measurements performed by [24].",
"Intermittent dipping episodes may indicate an inclination as high as $72-79^{\\circ }$ [23].",
"However, intermittent dipping may not be indicative of a high inclination.",
"Another low inclination system, 4U 1543-47, exhibited intermittent dipping that was suggestive of an accretion instability [45].",
"Additionally, recent near-infrared spectroscopy rules out a high inclination and implies an inclination $23^{\\circ }<i<53^{\\circ }$ when considering conservative constraints [36].",
"The magnetic field strength is estimated to be $0.4-31\\times 10^{8}$ G. This is inferred from pulsations signifying magnetically channeled accretion in $\\emph {Rossi X-ray Timing Explorer}$ ($\\emph {RXTE}$ ) observations [41].",
"Additionally, the “propeller\" phase, where material is thrown off from the disk at low luminosity and can no longer accrete onto the NS, implies a similar magnetic field strength ([9]; [2]).",
"Broadened and skewed Fe line profiles have been detected from accretion disks in NS LMXBs for the last decade (e.g.",
"[4]; [43]; [6], [8]; [19]; [21]; [39]).",
"These profiles are shaped from Doppler and relativistic effects [22] and, as a consequence, the red wing can be used to determine the location of the inner edge of the disk.",
"The accretion disk must extend down to or truncate prior to the surface of the NS.",
"Disk truncation can occur above $\\sim 1\\%$ L$_{\\mathrm {Edd}}$ in one of two ways: either pressure balance between the accreting material and magnetosphere or a boundary layer of material extending from the surface.",
"Below $\\sim 1\\%$ L$_{\\mathrm {Edd}}$ , accretion in LMXBs can become inefficient and disk truncation can occur through other mechanisms, such as disk evaporation ([42]; [53]; [17]).",
"By studying sources with truncated accretion disks at sufficiently high L$_{\\mathrm {Edd}}$ , we can obtain estimates of magnetic field strengths ([26]; [7]; [44]; [38]; [18], [16]; [28]; [33]) and/or extent of potential boundary layers ([46]; [28]; [33], [14]).",
"It remains unclear whether the magnetic field is dynamically important in Aquila X-1 and other non-pulsating NS LMXBs.",
"Aquila X-1 is frequently active with outbursts occurring about once a year ([10]; [54]) making it a key target.",
"[28] obtained observations of Aquila X-1 in the soft state with $\\emph {NuSTAR}$ and $\\emph {Swift}$ during the July 2014 outburst.",
"They found that the disk was truncated at $15\\pm 3 \\ R_{g}$ (where $R_{g}=GM/c^{2}$ ) at $\\sim 7\\%$ of the empirical Eddington luminosity ($L_{\\mathrm {Edd}}=3.8\\times 10^{38}$ ergs s$^{-1}$ ; [29]).",
"This placed a limit on the strength of the equatorial magnetic field of $B<7\\times 10^{8}$ G that is consistent with previous estimates.",
"The $\\emph {Swift}$ /BAT detected renewed activity on 2016 July 29 [48] that was confirmed to be a new outburst with a 500 s follow up $\\emph {Swift}$ /XRT observation [49].",
"Observations were taken with $\\emph {NuSTAR}$ [25] on 2016 August 7 when Aql X-1 was in the soft state at $\\sim 0.32\\ L_{\\mathrm {Edd}}$ during the outburst.",
"We perform a reflection study on the prominent Fe K$_{\\alpha }$ feature for this observation and compare with the 2014 outburst."
],
[
"Observations and Data Reduction",
"$\\emph {NuSTAR}$ observations were taken of Aquila X-1 on 2014 July 17 and 18 (Obsids 80001034002 and 80001034003) and 2016 August 7 (Obsid 90202033002).",
"Figure 1 shows the $\\emph {Swift}$ /BAT and MAXI daily monitoring lightcurves with vertical dashed lines to indicate when the $\\emph {NuSTAR}$ observations were taken.",
"Using the nuproducts tool from nustardas v1.5.1 with caldb 20170503, we created lightcurves and spectra for the 2016 observations.",
"We used a circular extraction region with a radius of 100$^{\\prime \\prime }$ centered around the source and another region away from the source for the purpose of background subtraction.",
"No Type-I X-ray bursts occurred during the 2016 observation.",
"Initial modeling of the spectra with a constant fixed to 1 for the FPMA, found the floating constant for the FPMB to be within 0.95-1.05.",
"We combine the two source spectra, background spectra, ancillary response matrices and redistribution matrix files via addascaspec and addrmf.",
"Each of these have been weighted by exposure time.",
"The 2014 observations were reduced using the most recent caldb, 20170503, which has been updated since the reduction and analysis reported in [28].",
"The combined spectra were grouped to have a minimum of 25 counts per bin [13] using grppha.",
"The net count rate for the combined spectra were 126.8 counts s$^{-1}$ in 2014 and 424.3 counts s$^{-1}$ in 2016.",
"Figure: Swift/BAT 15-5015-50 keV and MAXI 2-202-20 keV daily monitoring lightcurves.",
"The dashed lines represent the NuSTAR\\emph {NuSTAR} observations taken in July 2014 and August 2016.We do not utilize the 2014 $\\emph {Swift}$ observations as per [28] due to major flux differences between the $\\emph {NuSTAR}$ and $\\emph {Swift}$ spectra.",
"The $\\emph {Swift}$ spectrum required a multiplicative constant of 3.75 to match the $\\emph {NuSTAR}$ flux.",
"This flux difference is likely due to the need to exclude the PSF core to avoid pile-up in the $\\emph {Swift}$ data.",
"Additionally, excluding the core of the PSF further limits the sensitivity of the $\\emph {Swift}$ spectrum and, as a result, the reflection spectrum cannot be detected in the data.",
"Furthermore, $\\emph {Swift}$ only performed a short exposure observation (under 200 s) on the same day as the $\\emph {NuSTAR}$ observation in 2016 that do not provide constraints.",
"As a consequence, we opted to focus on the comparison of $\\emph {NuSTAR}$ observations only in this study."
],
[
"Spectral Analysis and Results",
"We utilize XSPEC version 12.9.1 [1] in this work with fits performed over the 3.0-30.0 keV energy range (the spectrum is dominated by background above 30 keV).",
"All errors were calculated using a Monte Carlo Markov Chain (MCMC) of length 100,000 and are quoted at 90$\\%$ confidence level.",
"We use tbnewerWilms, Juett, Schulz, Nowak, in prep, http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/index.html to account for the absorption along the line of sight.",
"Since $\\emph {NuSTAR}$ has a limited lower energy bandpass it is unable to constrain the equivalent neutral hydrogen column density its own.",
"We therefore set the equivalent neutral hydrogen column density to the [20] value of $4.0\\times 10^{21}$ cm$^{-2}$ .",
"Moreover, this value is very close to column densities found with low energy spectral fitting to $\\emph {XMM-Newton}$ and $\\emph {Chandra}$ data [11].",
"[28] modeled the 2014 data using a Comptonized thermal continuum with a relativistically blurred emergent reflection emission.",
"We chose to forego this combination of models in an effort to provide a self-consistent approach between components.",
"The reflection model in [28] assumes that a blackbody continuum is illuminating the disk, though the continuum is modeled with Comptonization.",
"Further, the assumed blackbody in the reflection model that is providing the emergent reflection spectrum does not peak at the same energy as the Comptonized continuum.",
"This means that the component assumed to illuminate the accretion disk is not consistent with the emergent reflection spectrum.",
"We chose to adopt a continuum model akin to [30] for NS transients in the soft state.",
"The continuum is described by two thermal components: a single temperature blackbody component (bbodyrad) and a multi-temperature blackbody (diskbb).",
"The single temperature blackbody component is used to model the emission from the corona or boundary layer.",
"The multi-temperature blackbody is used to account for the thermal emission from different radii in the accretion disk.",
"The addition of a power-law component may be needed in some cases and is suggestive of weak Comptonization.",
"Initial fits were performed with two thermal components, which gave a poor fit in each case ($\\chi ^{2}_{2014}/d.o.f.=4088.70/591$ and $\\chi ^{2}_{2016}/d.o.f.=3946.47/585$ ), partly due to the presence of strong reflection within the spectrum.",
"We added a power-law component with the photon index bound at a hard limit of 4.0.",
"Steep indices of this nature have been observed in [51] and [45] for black hole X-ray binaries.",
"The additional power-law component improved the the overall fit at more than the $9\\sigma $ level of confidence, as determined via F-test, in each case.",
"However, the reflection is still unaccounted for by this model.",
"The broadened Fe K emission line can be seen in Figure 2 for each outburst.",
"Figure: Comparison of Fe line profiles for Aql X-1 during the 2014 and 2016 outbursts created by taking the ratio of the data to the continuum model.",
"The continuum model was fit over the energies of 3.0-5.03.0-5.0 keV and 8.0-10.08.0-10.0 keV.",
"The iron line region was ignored (5.0-8.05.0-8.0 keV) to prevent the feature from skewing the fit.",
"Ignoring above 10.0 keV gives an unhindered view of the Fe K α _{\\alpha } line, though it models both the continuum and some reflection continuum.We account for the emergent reflection from an ionized disk by convolving reflionxhttp://www-xray.ast.cam.ac.uk/$\\sim $ mlparker/reflionx$\\_$ models/reflionx$\\_$ bb.mod [47] with the relativistic blurring kernal relconv [15].",
"The reflionx model has been modified to assume the disk is illuminated by a blackbody.",
"We tie the blackbody temperature of the reflection and continuum emission.",
"We use a constant emissivity index, $q$ , fixed at 3 as would be expected for an accretion disk illuminated by a point source in an assumed geometry of flat, Euclidean space ([55]).",
"Different geometries, such as a boundary layer surrounding the NS or hot spots on the surface illuminating the disk, replicate the same $r^{-3}$ emissivity profile (D. Wilkins, priv.",
"comm.).",
"The iron abundance, $A_{Fe}$ , is fixed at half solar abundance in agreement with the previous analysis on Aql X-1 [28].",
"We fix the dimensionless spin parameter, $a_{*}$ (where $a_{*}=cJ/GM^{2}$ ), to 0.259 which is implied from the pulsation spin frequency of 550 Hz ([5]; [12]; [28]).",
"This assumes a NS mass of 1.4 M$_{\\odot }$ , radius of 10 km, and a moderately soft equation of state [5].",
"The inner disk radius, $R_{in}$ , is given in units of innermost stable circular orbit (ISCO).",
"We convert this value to $R_{g}$ given that 1 ISCO $= 5.2\\ R_{g}$ for $a_{*}=0.259$ [3].",
"The xspec model we used for each spectrum was tbnewer*(diskbb+bbodyrad+pow+relconv*reflionx).",
"This provided an improvement in the overall fit at more than the 25$\\sigma $ level of confidence ($\\chi ^{2}_{2014}/d.o.f.=620.29/583$ and $\\chi ^{2}_{2016}/d.o.f.=603.08/579$ ) over the prior model that did not account for reflection within the spectra.",
"Figure 3 shows the best fit spectra and model components.",
"Model parameters and values are listed in Table 1.",
"The exact nature of the power-law component is unknown as it may or may not be physical, but it is statistically needed at more than the $15\\sigma $ level of confidence for each case.",
"Table: Aql X-1 Reflionx ModelingFigure: Aql X-1 spectrum fit from 3.0-30.0 keV with a diskbb (red dash line), blackbody (purple dot dot dot dash line), power-law (orange dot line), and reflionx (blue dot dash line).",
"The ratio of the data to the model is shown in the lower panel.",
"The data were rebinned for clarity.",
"Table 1 lists parameter values for each model.For the data taken during the 2014 outburst, the diskbb component has a temperature of $kT=1.64\\pm 0.02$ keV and norm$=12.0_{-0.5}^{+0.3}$ km$^{2}$ /100 kpc$^{2}$ cos($i$ ).",
"The bbodyrad component has a temperature of $kT=2.27\\pm 0.02$ keV and normalization of $1.2\\pm 0.1$ km$^{2}$ /100 kpc$^{2}$ .",
"The power-law has a steep photon index of $\\Gamma =3.7\\pm 0.1$ with a normalization of $1.2\\pm 0.1$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 keV.",
"The inner disk radius is truncated at $R_{in}=2.7\\pm 0.4$ ISCO ($14\\pm 2\\ R_{g}$ ).",
"The inclination was found to be $26_{-3}^{+2}\\ ^{\\circ }$ .",
"For the data taken during the 2016 outburst, the diskbb component has a temperature of $kT=1.69_{-0.02}^{+0.01}$ keV and norm$=62\\pm 2$ km$^{2}$ /100 kpc$^{2}$ cos($i$ ).",
"The bbodyrad component has a temperature of $kT=2.33_{-0.02}^{+0.01}$ keV and normalization of $4.1_{-0.2}^{+0.4}$ km$^{2}$ /100 kpc$^{2}$ .",
"Again, the photon index is steep at $\\Gamma =3.96_{-0.21}^{+0.03}$ with a normalization of $4.8_{-0.9}^{+0.2}$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 keV.",
"The inner disk radius is truncated at $R_{in}=2.1_{-0.2}^{+0.3}$ ISCO ($11_{-1}^{+2}\\ R_{g}$ ).",
"The inclination is $26\\pm 2\\ ^{\\circ }$ , which also agrees with the previous observation.",
"The blackbody and disk blackbody normalizations in both fits are implausibly small when used to infer a radial extent of the emitting region.",
"This systematic underestimation was proposed by [31] to be the result of spectral hardening as photons travel through an atmosphere above pure blackbody emission and is supported through numerical simulations ([50]; [37]).",
"The consistency in model parameter values with only the normalization changing between the two soft state observations likely indicates similar accretion geometries.",
"We allow the emissivity parameter to be free to check if our results are dependent on the emissivity index being fixed at 3.",
"The emissivity index tends towards a slightly higher value of $q=3.1$ for the 2014 observation and $q=2.5$ , which is consistent with the disk extending down to a smaller radii in the most recent observation.",
"All model parameters are consistent within the $3\\sigma $ level of confidence with those reported in Table 1.",
"Figure 4 shows how the goodness-of-fit changes with inner disk radius for each observation.",
"We use the XSPEC “steppar\" command to determine how the goodness-of-fit changed as a function of inner disk radius.",
"At each evenly placed step, $R_{in}$ was fixed while the other parameters were free to adjust to find the best fit.",
"The ISCO is ruled out at more than the $6\\sigma $ level of confidence in each case.",
"Figure: Change in goodness-of-fit with inner disk radius for the 2014 (top) and 2016 (bottom) outbursts taken over evenly spaced steps generated with XSPEC “steppar\".",
"The inner disk radius was held constant while the other parameters were free to adjust to find the minimum χ 2 \\chi ^2 value at each step.",
"The dashed lines represent the 1σ1\\sigma , 2σ2\\sigma , and 3σ3\\sigma confidence intervals."
],
[
"Discussion",
"We present a new observation of Aquila X-1 taken with $\\emph {NuSTAR}$ during its August 2016 outburst and compare it to the July 2014 outburst.",
"We perform reflection fits that indicate the disk is truncated prior to the surface of the neutron star.",
"The location of the inner disk radius during the 2014 observation is $14\\pm 2\\ R_{g}$ .",
"This is consistent with the previous results found in [28], although we modeled the continuum in a different way.",
"The location of the inner disk radius remains truncated ($11_{-1}^{+2}\\ R_{g}$ ) during the 2016 observation even though the flux is over four times larger.",
"Additionally, both spectra imply an inclination of $26\\pm 2^{\\circ }$ which is consistent with infrared photometric and spectroscopic measurements ([24]; [36]).",
"By assuming that the ram pressure in the disk is balanced by the outward pressure of the magnetic field, we can place an upper limit on the magnetic field strength using the maximum extent the inner disk of $R_{in}=13\\ R_{g}$ from the 2016 spectrum.",
"Assuming a mass of 1.4 M$_{\\odot }$ , taking the maximum distance to be 5.9 kpc, and using the maximum unabsorbed flux from $0.5-50.0$ keV of $33\\times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$ as the bolometric flux, the magnetic dipole moment, $\\mu $ , can be estimated from Equation (1): $\\small \\begin{aligned}\\mu = 3.5 \\times 10^{23} \\ k_{A}^{-7/4} \\ x^{7/4} \\left(\\frac{M}{1.4\\ M_{\\odot }}\\right)^{2} \\\\ \\times \\left(\\frac{f_{ang}}{\\eta } \\frac{F_{bol}}{10^{-9}\\ \\mathrm {erg \\ cm^{-2} \\ s^{-1}}}\\right)^{1/2} \\frac{D}{3.5\\ \\mathrm {kpc}} \\ \\mathrm {G\\ cm}^{3}\\end{aligned}$ with $x$ being the number of gravitational radii ([26]; [7]).",
"If we assume an accretion efficiency of $\\eta =0.2$ and unity for the angular anisotropy, $f_{ang}$ , and conversion factor, $k_{A}$ , then $\\mu \\sim 6.7\\times 10^{26}$ G cm$^{3}$ .",
"For a NS of 10 km, this implies a magnetic field strength at the poles of $B\\le 1.3\\times 10^{9}$ G. Alternatively, if we assume a different conversion factor between disk and spherical accretion of $k_{A}=0.5$ as proposed in [32], the strength of the magnetic field increases to $B\\le 4.5\\times 10^{9}$ G. For the 2014 outburst, we use the upper limit of $R_{in}=16\\ R_{g}$ and the maximum unabsorbed flux from $0.5-50.0$ keV of $7\\times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$ to place a limit on the magnetic field strength to be $B\\le 0.9\\times 10^{9}$ G for $k_{A}=1.0$ and $B\\le 3.0\\times 10^{9}$ for $k_{A}=0.5$ .",
"Note that the magnetic field strength at the equator is half as strong as at the pole.",
"[28] found a similar value for the maximum strength of the magnetic field for Aquila X-1 of $B\\simeq 1.4\\times 10^{9}$ G at the magnetic poles.",
"We report the upper limit on the magnetic field strength using the conversion factor of $k_A=0.5$ hereafter since it encompasses the value for $k_A=1.0$ .",
"If, however, the magnetosphere was not responsible for truncating the disk, a boundary layer extending from the surface of the NS could plausibly halt the accretion flow.",
"Equation 2, taken from [46], provides a way to estimate the maximum radial extent of this region from the mass accretion rate.",
"$\\small \\begin{aligned}\\log \\left(R_{max}-R_{NS}\\right) \\simeq 5.02+0.245\\left| \\log \\left(\\frac{\\dot{M}}{10^{-9.85}\\ \\mathrm {M}_{\\odot }\\ \\mathrm {yr}^{-1}}\\right) \\right| ^{2.19}\\end{aligned}$ We determine the mass accretion rate using the unabsorbed luminosity from $0.5-50.0$ keV and an accretion efficiency of $\\eta =0.2$ to be $1.1_{-0.3}^{+0.1}\\times 10^{-8}\\ \\mathrm {M}_{\\odot }\\ \\mathrm {yr}^{-1}$ during the 2016 observation and $2.2\\pm 0.4\\times 10^{-9}\\ \\mathrm {M}_{\\odot }\\ \\mathrm {yr}^{-1}$ during the 2014 observation.",
"This gives a maximum radial extent of $\\sim 10\\ R_{g}$ for the boundary layer during 2016 and $\\sim 6\\ R_{g}$ during 2014 (assuming canonical values of $\\mathrm {M}_{NS}=1.4\\ \\mathrm {M}_{\\odot }$ and $R_{NS}=10$ km).",
"This is consistent with the location of the inner disk radius during the 2016 outburst, but falls short of the inner disk radius in our 2014 fits.",
"[28] found a similar radial extent of the boundary layer of $\\sim 7.8\\ R_{g}$ , but this can be increased by rotation of the NS or a change in viscosity to be consistent with the truncation radius.",
"It is more likely that the magnetic field is responsible for disk truncation in this source.",
"The equatorial magnetic field strength inferred from the Fe line profile ($B\\le 15.0-22.5\\times 10^{8}$ G) is consistent with other estimates of the magnetic field strength ($0.4-31\\times 10^{8}$ G: [9]; [2]; [41]) and are well within the range to truncate an accretion disk ([41]).",
"Following Equation (1) and rearranging for inner disk radius in terms of flux, the inner disk radius should scale like $F_{bol}^{-2/7}$ .",
"Thus for magnetic truncation the inner disk radius should decrease as the flux increases, which is what we see for the different observations.",
"Conversely, if the boundary layer were responsible for disk truncation in each case, we should see the inner disk radius increase.",
"Additionally, the maximum extent of the boundary layer during the 2014 observation does not agree with the location of the inner disk radius, pointing to the magnetic field being a more probable explanation for disk truncation.",
"Moreover, although the extent of the boundary layer is consistent with the inner disk radius in the 2016 fits, the behavior of decreasing inner disk radius with increasing flux is indicative of magnetic truncation."
],
[
"Comparison of Magnetic Field Strengths",
"$\\emph {NuSTAR}$ has observed a number of NS LMXBs with Fe lines that imply truncated disks.",
"This has provided a means of placing an upper limit on the strength of their magnetic fields, assuming the disk is truncated at the Alfvén radius (where the ram pressure of the accreting material is balanced by the magnetic pressure outwards).",
"The implied magnetic field strengths reside between $10^{8}-10^{9}$ G and are similar to accreting millisecond X-ray pulsars (AMXPs).",
"[41] systematically estimated the upper and lower limits to the equatorial magnetic field strengths of 14 known AMXPs using $\\emph {Rossi X-ray Timing Explorer}$ ($\\emph {RXTE}$ ).",
"They used the highest flux that the source exhibited pulsations and the radius of the NS to determine $B_{min}$ and the lowest flux that exhibited pulsations and corotation radius with the disk to determine $B_{max}$ in each case.",
"Figure: Comparison of equatorial magnetic field strengths of NSs in LMXBs (red) inferred from Fe line profiles to known AMXPs (black) reported in versus Eddington fraction.",
"The stars represent estimates for Aquila X-1.",
"See Table 2 for magnetic field strengths and Eddington fraction values.Table: Magnetic Field Strengths Versus Eddington FractionFigure 5 presents a comparison of magnetic field strengths of known AMXPs to NS LMXBs observed with $\\emph {NuSTAR}$ versus Eddington fraction, $F_{\\mathrm {Edd}}$ .",
"As can be seen, the NS LMXBs populate higher values of Eddington fraction.",
"Each point from [41] represents a range in magnetic field strength and $F_{\\mathrm {Edd}}$ that the AMXP lies and does not embody an actual measurement.",
"Values can be found in Table 2.",
"The advantage of magnetic field strengths inferred from the Fe line profiles using $\\emph {NuSTAR}$ is that they do not suffer from pile-up or instrumental effects until a source reaches $\\sim 10^{5}$ counts s$^{-1}$ .",
"We use the maximum Eddington luminosity of $3.8\\times 10^{38}$ ergs s$^{-1}$ from [29] when calculating the Eddington fraction for each source.",
"If the Eddington luminosity is smaller, all points would be shifted to higher values of Eddington fraction.",
"Therefore, these are all lower limits.",
"Another caveat of this comparison is that pulsations have not been detected yet for the sources observed with $\\emph {NuSTAR}$ .",
"For Aquila X-1 in particular the 2014 observation is within the same $F_{\\mathrm {Edd}}$ range as the observation taken by $\\emph {RXTE}$ when pulsations were detected.",
"Additionally, our upper limit on the strength of the magnetic field agrees with the estimate when pulsations were detected.",
"It is clear that the strengths implied from Fe line profiles are valuable and consistent with those seen for AMXPs.",
"Therefore, Fe lines can be used to estimate magnetic field strengths to first order."
],
[
"Summary",
"We present a reflection study of Aquila X-1 observed with $\\emph {NuSTAR}$ during the July 2014 and August 2016 outbursts.",
"We find the disk to be truncated prior to the surface of the NS at $14\\pm 2\\ R_{g}$ during 2014 observation when the source was at 7% of Eddington and $11_{-1}^{+2}\\ R_{g}$ during the 2016 observation when the source was at 32% of Eddington.",
"This implies an upper limit on the strength of the magnetic field at the poles of $3.0-4.5\\times 10^{9}$ G, if the magnetosphere is responsible for truncating the disk in each case.",
"If a boundary layer is responsible for halting the accretion flow instead, we estimate the maximal radial extent to be $\\sim 6\\ R_{g}$ for the 2014 observation and $\\sim 10\\ R_{g}$ during 2016.",
"These values can be increased through viscous and spin effects, but the behavior of decreasing inner disk radius with increasing flux favors magnetic truncation.",
"Finally, when comparing the strength of magnetic fields in NS LMXBs to those of known AMXPs we find that they are consistent while probing a higher value of Eddington fraction.",
"We thank the referee for their prompt and thoughtful comments that have improved the quality of this work.",
"This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (Caltech, USA).",
"ND is supported by a Vidi grant from the Netherlands Organization for Scientific Research (NWO).",
"EMC gratefully acknowledges support from the National Science Foundation through CAREER award number AST-1351222.",
"DA acknowledges support from the Royal Society."
]
] | 1709.01559 |
[
[
"Model-Based Control Using Koopman Operators"
],
[
"Abstract This paper explores the application of Koopman operator theory to the control of robotic systems.",
"The operator is introduced as a method to generate data-driven models that have utility for model-based control methods.",
"We then motivate the use of the Koopman operator towards augmenting model-based control.",
"Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis.",
"Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems.",
"Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems.",
"Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand."
],
[
"Introduction",
"Modeling for complex dynamical systems has typically been the first step when designing, control, planning, or state-estimation algorithms.",
"System design and specifications have been dependent on the use of high-fidelity models.",
"However, any derivation of a dynamical model from first principles is typically a demanding task when the complexity of state interactions is high.",
"Moreover, analytical models do not capture external disturbances.",
"As a result, derived models, for use in model-based control settings, often have limited use or poor prediction over longer time spans.",
"Nevertheless, a representation of the behavior of a dynamical system is central to most model-based engineering and scientific application.",
"Within the field of systems and control theory, model uncertainty has typically been mitigated with the use of robust and adaptive control architectures.",
"Typically, adaptive controllers are self tuning and reactive to incoming state information while robust controllers are designed to be invariant to model uncertainty [1], [2], [3], [4].",
"Motion planning for uncertain dynamical systems have also been extensively investigated.",
"Generally, in this approach, uncertainty is explicitly modeled and incorporated into the decision making process [5], [6], [7].",
"However, like robust and adaptive control approaches, the need for an explicit uncertainty model often limits its utility in general settings.",
"Machine learning, offers a much more general approach [8], [9], [10].",
"In particular, recent advances have utilized large sets of data to perform model-based control of various dynamical systems [11].",
"Nonetheless, several questions about the training data, stability, convergence properties, computational complexity, and mechanical property conservation of the models are still open questions that need to be addressed.",
"Recently, the use of data-driven techniques to mitigate the effects of model uncertainty have sparked interest in the Koopman operator [12].",
"The Koopman operator is a infinite-dimensional linear operator that is able to exactly capture the behavior of nonlinear dynamical systems.",
"In application, the Koopman operator is approximated with a finite-dimensional linear operator [13].",
"This approximation can be computed in a solely data-driven manner without any prior information of the dynamical system.",
"Complex fluid flow systems have accurately been modeled using this approach [14].",
"Furthermore, it has been shown that the spectral properties of the approximate Koopman operator can be examined to investigate system-level behavior like ergodicity and stability [12], [15], [16].",
"In addition, recent work has shown its utility in human-machine systems [17].",
"In this paper, we investigate the utility of Koopman operator theory for control in robotic systems.",
"The work is motivated by the desire to generate or augment dynamical models of robotic systems through data collection.",
"In particular, it is of interest to synthesize model-based controllers using these data-driven models.",
"Thus, the main contribution of this paper is the application of Koopman operator theory to the control of robotic systems.",
"The Koopman operator is shown to have a linearizable data-driven model of the dynamical system that is amenable to model-based control methods.",
"Closed-loop and open-loop controllers are then formulated using the proposed data-driven model.",
"Furthermore, we explore the consequences of the specific choice of basis function as well as complexity order for swing up control of a simulated cart- and vertical take-off and landing (VTOL)-pendulum systems.",
"Last, experiments using the Koopman operator using a Sphero SPRK robot are shown.",
"We conclude the paper with recommendations for future work.",
"The organization of this paper is as follows.",
"Section II gives an overview of the Koopman operator theory and its application to data-driven approximations of dynamical systems.",
"In addition, Sections III and IV explore the implementation of Koopman operator theory in simulation and experimentation, respectively.",
"Conclusions are in Section V."
],
[
"Koopman Operator",
"An overview of Koopman operator theory is given in this section.",
"For the purposes of this paper, we focus more on the practical implementation of the theory and omit much of the theoretical presentation.",
"However, the interested reader can find a complete treatment of the Koopman operator in [13].",
"To begin, consider a discrete-time dynamical system evolving as $ x_{k+1} = F(x_k),$ where $x_k\\in M$ is the, possibly unobserved, state of the system and $y_k \\in \\mathbb {C}$ .",
"Furthermore, define an observation function $ y_k = g(x_k),$ where $g\\in \\mathbb {G}: M\\rightarrow \\mathbb {C}$ and $\\mathbb {G}$ is a function space.",
"For the purposes on this paper, we assume that $\\mathbb {G}$ is the $L^2$ space.",
"The Koopman operator, $\\mathcal {K}: \\mathbb {G}\\rightarrow \\mathbb {G}$ , is defined as $ [\\mathcal {K}g](x) = g(F(x)).$ Note that the Koopman operator maps elements in $\\mathbb {G}$ to elements in $\\mathbb {G}$ .",
"Therefore, it does not, as done by $F$ , map system states to system states.",
"Furthermore, note that (REF ) can be written as $[\\mathcal {K}g](x_k) = g(F(x_k)) = g(x_{k+1}).$ Therefore, the Koopman operator propagates the output of the system forward.",
"Finally, the observable equation can be easily extended to the case where multiple observations are available, $g: M\\rightarrow \\mathbb {C}^K$ .",
"The Koopman operator defined in (REF ) is linear when $\\mathbb {G}$ is a vector space.",
"This property holds even if the considered discrete-time dynamical system is nonlinear.",
"However, since the Koopman operator maps $\\mathbb {G}$ to elements in $\\mathbb {G}$ it is infinite dimensional.",
"Therefore, a nonlinear dynamical system given by (REF ) can be equivalently described by a linear infinite dimensional operator.",
"From a practical standpoint, there is not much benefit from this infinite dimensional representation even if the operator could be defined for a specific system of interest.",
"However, the Koopman operator can be approximated with a linear finite dimensional operator using data-driven approaches."
],
[
"Approximating a Koopman Operator",
"In order to define an approximate Koopman operator the observation function (REF ) is redefined as $y_k = g(x_k) = \\Psi (x_k),$ where $\\Psi (x)$ is a user-defined vector-valued function $\\Psi (x) = [\\psi _1(x), \\psi _2(x), \\dots , \\psi _{N}(x)].$ Next, the relation described by (REF ) is now given as $ \\Psi (x_{k+1}) = \\Psi (x_k)K + r(x_k).$ where $K\\in \\mathbb {C}^{N \\times N}$ and $r(x_k)$ is a residual (approximation error).",
"Note that the matrix $K$ advances $\\Psi $ forward one time step.",
"Next, it is assumed that the trajectory of the system has been collected such that $ X &= [x_1, \\dots , x_P]$ where $P$ is the number of recorded data points.",
"The matrix $K$ can be computed in a number of ways.",
"In this paper, we adopt the least-squares approach, described in [18], where $K$ is determined by minimizing $J &= \\frac{1}{2}\\sum ^{P-1}_{p=1}|r(x_p)|^2, \\\\&=\\frac{1}{2}\\sum ^{P-1}_{p=1}|\\Psi (x_{p+1}) - \\Psi (x_p)K|^2.$ Solving the least-squares problem yields $K = G^\\dagger A,$ where $\\dagger $ denotes the Moore–Penrose pseudoinverse and $G &= \\frac{1}{P}\\sum ^{P-1}_{p=1}\\Psi (x_{p})^\\textrm {T}\\Psi (x_{p}), \\\\A &=\\frac{1}{P}\\sum ^{P-1}_{p=1}\\Psi (x_{p})^\\textrm {T}\\Psi (x_{p+1}).",
"$ Note that the computational burden of this approach grows as the dimension of $\\Psi $ increases.",
"The approach generally yields a better approximation as the dimension of $\\Psi $ increases.",
"Furthermore, the number of data points and their distribution across the state space will have a large effect on the computed $K$ matrix.",
"The definitions of (REF -REF ) can be generalized.",
"The recorded data points need not come from a single trajectory nor be sequential [18].",
"Multiple trajectories and trajectories with missing data points can be used.",
"The only requirement is the sum of residuals given in (REF ) be defined by consecutive states $(x_k,x_{k+1})$ spaced equally in time.",
"Even this could be avoided by choosing another optimization to solve for $K$ ."
],
[
"Approximating Dynamical Systems",
"For predicting dynamical systems, the approximation to the Koopman operator can be used to generate a data-driven model of a system by defining $\\Psi $ as $\\Psi (x) = [x^\\textrm {T}, \\psi _{n+1}(x), \\dots , \\psi _{N}(x)].$ Note that the state of the system, $x\\in \\mathbb {R}^n$ , is now included in $\\Psi (x)$ .",
"Thus we can write the approximate dynamical equations of the considered system as $ x_{k+1} \\approx \\hat{K}^\\textrm {T}\\Psi (x_k)^\\textrm {T},$ where $\\hat{K}^T \\in \\mathbb {R}^{n \\times N}$ is the first $n$ columns of $K$ .",
"Note that equation (REF ) simply propagates forward the quantities of interest (e.g.",
"system states).",
"Furthermore, in this work, $x_{k+1}$ is described as a linear combination of the system state, $x_k$ , and the functions $\\psi _i(x_k)$ ."
],
[
"Control Synthesis: Open- and Closed-loop Controllers",
"In this section we formulate open- and closed-loop model-based controllers using the Koopman operator.",
"It is first shown that for a differentiable choice of basis function $\\Psi $ , the Koopman operator has a linearization that can be computed for model-based control methods.",
"Given the linearizable Koopman operator, a model-based optimal control problem is formulated for open- and closed-loop controllers."
],
[
"Koopman Operator Linearization",
"By choosing a $\\Psi $ that is differentiable, the Koopman operator approximation to the dynamical system can be linearized: $x_{k+1} &\\approx \\hat{K}^\\textrm {T}\\frac{\\partial \\Psi }{\\partial x}x_k \\\\&\\approx A(x_k)x_k.$ Control inputs are readily incorporated to the definition of $\\Psi $ as an augmented state, $\\Psi (x,u) = [x^\\textrm {T}, u^\\textrm {T},\\psi _1(x,u), \\psi _2(x,u), \\dots , \\psi _{N}(x,u)].$ This yields the approximate dynamical equations, $x_{k+1} &\\approx \\hat{K}^\\textrm {T}\\Psi (x_k,u_k)^\\textrm {T}$ and the linearization of the approximate dynamical equations, $x_{k+1} &\\approx \\hat{K}^\\textrm {T}\\frac{\\partial \\Psi }{\\partial x}x_k + \\hat{K}^\\textrm {T}\\frac{\\partial \\Psi }{\\partial u}u_k \\\\&\\approx A(x_k,u_k)x_k + B(x_k,u_k)u_k.$ Note that linearizable equations of motion of a dynamical system can be computed solely from data."
],
[
"Optimal Control Problem",
"Control synthesis for trajectory optimization is generated for mobile robot dynamics of the form $ x_{k+1} = f(x_k, u_k),$ where $x \\in \\mathbb {R}^n$ is the state and $u \\in \\mathbb {R}^m$ is the control input.",
"For a discrete system, we can solve for a trajectory that minimizes the objective defined as $ J = \\sum _{k=0}^{N} \\frac{1}{2}(x_k - \\tilde{x}_k)^T{P} (x_k - \\tilde{x}_k) + \\frac{1}{2}u_k^T {R} u_k,$ where ${P} \\in \\mathbb {R}^{n \\times n}$ and ${R} \\in \\mathbb {R}^{m \\times m}$ are positive definite weight matrices on state and control and $\\tilde{x}_k$ is the reference trajectory at time $k$ .",
"Note that the accuracy of the system model (REF ) will largely determine the effectiveness of the synthesized optimal control."
],
[
"Open-Loop",
"Open-loop trajectory optimization precomputes the set of trajectory and control actions that minimize the objective function (REF ) subject to the modeled dynamical constraints in (REF ).",
"Projection-based optimization [19] is used in discrete time to generate the set of trajectory and control actions given an initial trajectory $x_k$ and control $u_k$ for $k \\in \\left[ 0, N \\right] $ .",
"In the experiment, the projection-based optimization algorithm first generates the control actions based on the dynamical model and then at a fixed rate the command signals are sent via Bluetooth communication to the robot.",
"Odometry data is collected only for post-processing and is not used to update the command signals.",
"In the simulated and the experimental work, a discrete-time version of Sequential Action Control (SAC) [20] is used with the Koopman operator to generate closed-loop optimal control calculations.",
"However, any MPC technique can be used with the Koopman operator.",
"Here, SAC operates by first forward simulating an open-loop trajectory for some horizon $N$ for a control-affine dynamical system given by $x_{k+1} = f(x_k, u_k) = g(x_k) + h(x_k)u_k .$ The sensitivity to a control injection for any given discrete time of the objective function is given as $\\frac{dJ}{d\\lambda _k} = \\rho _k (f_2(k) - f_1(k))$ where $f_1(k) &=& f(x_k, u_{0,k}), \\\\f_2(k) &=& f(x_k, u_{k}^\\star )$ are the dynamics subject to the default control $u_{0,k}$ and derived control $u_k^\\star $ .",
"The co-state variable $\\rho _k \\in \\mathbb {R}^n$ is computed by backwards simulating the following discrete equation $\\rho _{k-1} = \\frac{\\partial l_{k}}{\\partial x} + \\frac{\\partial f_{k}}{\\partial x}^T \\rho _{k},$ where $l_k = \\frac{1}{2}(x_k - \\tilde{x}_k)^T {P} (x_k - \\tilde{x}_k) + \\frac{1}{2}u_k^T {R} u_k$ and $f_k = f(x_k, u_{0,k})$ for some default $u_{0,k}$ subject to $\\rho _N = \\vec{0}$ .",
"The optimal control $u^*_k$ is computed by first defining a secondary objective function as $ J_u = \\sum _{k=0}^{N} \\frac{1}{2} (\\frac{dJ}{d\\lambda _k} - \\alpha _d)^2 + \\frac{1}{2} \\Vert u_k^\\star - u_{0,k} \\Vert _{R}^2.$ The objective (REF ) is now convex in $u_k^*$ and has a minimizer when $u^*_k = (\\Lambda + R^T)^{-1} h(x_k)^T \\rho _k \\alpha _d + u_{0,k},$ where $\\Lambda = h(x_k)^T \\rho _k \\rho _k^T h(x_k)$ .",
"Given the sequence of actions $u^\\star _k$ , it is then possible to calculate the time of control application $t_k^\\star $ as $t_k^\\star = \\operatornamewithlimits{argmin}\\frac{dJ}{d\\lambda _k}.$ The control duration in discrete time is found using an outward line search [21] for a sufficient descent on the cost."
],
[
"Experiments Using Sphero SPRK",
"In this section, we describe the experimental set-up for use of the Sphero SPRK robot with model-based control algorithms that utilize a state-space model generated via the Koopman operator.",
"In particular, we define data-driven closed- and open-loop model predictive controllers as well as motivate and explore the utility of Koopman operator for control of a robotic system.",
"In the experiments with the SPRK, trajectory optimization is run both in open-loop form and closed-loop feedback form.",
"Here, the tracked states of the robot are position $x,y$ and velocity $\\dot{x},\\dot{y}$ and inputs to the robot are desired velocities $u_1, u_2$ .",
"The objective function parameters are defined as ${P} = \\text{diag}([60,60,0.1,0.1])$ and ${R} = \\text{diag}([20,20])$ and are maintained constant through both open-loop and closed-loop experiments.",
"An additional set of experiments are done to show the use of the Koopman operator for control in a sand environment."
],
[
"SPRK",
"The SPRK is a differential drive mobile robot enclosed in a spherical case.",
"The dynamics of the SPRK are driven by the nonlinear coupling between the internal mechanism and the outer spherical encasing.",
"In addition, proprietary underlying controllers govern how the command velocities are interpreted to low-level motors.",
"The proprietary embedded software uses the on-board gyro-accelerometers to balance the robot upright while rolling.",
"The caster wheels on top of the internal mechanism ensures constant contact of the lower wheels that are driven via two motors.",
"The embedded software interfaces with heading and velocity (or $x-y$ velocity) command inputs sent via Bluetooth communication.",
"A high fidelity model of the robot would include several internal states characterize the internal mechanism and controller.",
"However, rather than seeking to approximated a high dimensional model, a reduced state model was sought.",
"Figure REF shows a closer look at the SPRK robot.",
"Odometry is collected using a Xbox Kinect with OpenCV [23] image processing.",
"More details about odometry and motion capture are stated in the caption of Fig.",
"REF ."
],
[
"SPRK Koopman Operator",
"The representation of the system consists of the position of the robot $(x,y)$ , its velocity $(\\dot{x}, \\dot{y})$ , and the commanded velocity $(u_x, u_y)$ .",
"Odometry data from the Kinect paired with recorded velocity commands are used to generate the approximate Koopman operator.",
"The vector-valued functions used in this experiment are polynomial basis functions given as $\\Psi (x) &= [x,y,\\dot{x}, \\dot{y}, u_x, u_y, 1, \\psi _1,\\psi _2,\\dots ,\\psi _M] \\\\\\psi _i(x) &= \\dot{x}^{\\alpha _{i}}\\dot{y}^{\\beta _{i}}$ where $\\alpha _i$ , and $\\beta _i$ are nonnegative integers, index $i$ tabulates all the combinations such that $\\alpha _i + \\beta _i\\le Q$ and $Q>1$ defines the largest allowed polynomial degree.",
"We ignore higher order position dependence in the operator in order to prevent any possible overfitting of position-based external disturbances.",
"The approximated Koopman operator was computed using data captured when the robot was operating at velocity under $1 \\ m/s$ for the open-loop trails."
],
[
"Simulation: Mechanical Energy",
"In this section, the equations of motion of a double pendulum are approximated with the method described in Section II.",
"The mass of both pendulums are 1 kilogram and the lengths of both are 1 meter.",
"The mass of the pendulums are assumed to be concentrated at their ends.",
"The system is conservative and subject to a gravitational field ($\\textrm {9.81 m/s}^2$ ).",
"The state of the system, $x$ , is described by the relative angles of the pendulums with respect to the vertical ($\\theta _1$ and $\\theta _2$ ) and their time derivatives ($\\dot{\\theta }_1$ and $\\dot{\\theta }_2$ ).",
"Data was collected by simulating the system multiple times with random initial conditions given by $x_0 = [\\mathcal {U}(-1,1)l_{\\theta _1},\\mathcal {U}(-1,1)l_{\\theta _2}, \\mathcal {U}(-1,1)l_{\\dot{\\theta }_1}, \\mathcal {U}(-1,1)l_{\\dot{\\theta }_2}] \\nonumber $ where $\\mathcal {U}(-1,1)$ is an uniformly distributed random variable with range $-1$ to 1.",
"Furthermore, $l_{\\theta _1}=l_{\\theta _2}=\\frac{\\pi }{3}$ and $l_{\\dot{\\theta }_1}=l_{\\dot{\\theta }_2}=0.5$ .",
"Therefore, the initial condition is uniformly distributed around the origin (and the stable equilibrium) and its range is defined by $L = [l_{\\theta _1},l_{\\theta _2}, l_{\\dot{\\theta }_1}, l_{\\dot{\\theta }_2}]$ .",
"Any data point that fall outside of the range defined by $L$ was not used to approximate the Koopman operator.",
"Data collection occurred at 100 Hz and was stopped when $2,000$ data points were collected.",
"The vector-valued functions used in this numerical experiment are polynomial basis functions give as $\\Psi (x) &= [\\theta _1,\\theta _2,\\dot{\\theta }_1,\\dot{\\theta }_2, 1, \\psi _1,\\psi _2,\\dots ,\\psi _M] \\\\\\psi _i(x) &= (\\theta _1/l_{\\theta _1})^{\\alpha _{i}}(\\theta _2/l_{\\theta _2})^{\\beta _{i}}(\\dot{\\theta }_1/l_{\\dot{\\theta }_1})^{\\gamma _{i}}(\\dot{\\theta }_2/l_{\\dot{\\theta }_2})^{\\delta _{i}}$ where $\\alpha _i, \\beta _i,\\gamma _i$ , and $\\delta _i$ are nonnegative integers, index $i$ tabulates all the combinations such that $\\alpha _i + \\beta _i + \\gamma _i + \\delta _i \\le Q$ and $Q>1$ defines the largest allowed polynomial degree.",
"Note that $-1\\le \\psi _i\\le 1$ when the state of the system is within the defined range.",
"The polynomial basis functions were scaled by the maximum expected value of the state to prevent numerical instability when higher order polynomials were utilized.",
"Figure REF shows a simulated trajectory and the corresponding predicted trajectories when approximated Koopman operators were used to propagate the system's configuration.",
"As expected, the accuracy of the predicted trajectories are improved when Q is increased.",
"Figure REF also shows how the accuracy of the predicted trajectories are dependent on the initial conditions.",
"The prediction error of a trajectory is computed as $\\frac{1}{N}\\sum _i^N (x_{\\textrm {sim},i} - x_{\\textrm {K},i})^2$ where $x_\\textrm {sim}$ is the simulated trajectory, $x_\\textrm {K}$ is the system's trajectory predicted by the approximated Koopman operator, and $N$ is the total run-time of the simulation.",
"The prediction error tended to increase with total mechanical energy.",
"Recall that the dynamics of a double pendulum are described by transcendental functions.",
"Therefore, any approximation by polynomials of these dynamics will deteriorate as the relative angle increases in magnitude.",
"However, when the relative angles are small (total mechanical energy is small) a polynomial approximation is accurate.",
"As expected, selection of $\\Psi $ plays a critical role in determining the quality of the computed Koopman operator."
],
[
"Simulation: Inversion and Stabilization of Pendulum Systems",
"In this section, we describe the results of utilizing the Koopman operator for inverting a cart-pendulum system and a VTOL-pendulum system.",
"In particular, this section overviews the effect that the choice of basis functions has on systems that have components in $SO(n)$ for $n > 1$ .",
"For the cart-pendulum system, the Koopman states are given as $\\Psi (x) = [\\theta ,x,\\dot{\\theta },\\dot{x}, u, 1, \\psi _1,\\psi _2,\\dots ,\\psi _M]$ where we use $\\psi _i(x) = \\theta ^{\\alpha _{i}}x^{\\beta _{i}}\\dot{\\theta }^{\\gamma _{i}}\\dot{x}^{\\delta _{i}} u$ as the polynomial basis function set and compare with a Fourier basis function, $\\psi _i(x) = \\prod _{[x]_i} \\prod _{\\kappa _j}\\cos ([x]_i \\kappa _j) \\sin ([x]_i \\kappa _j) u,$ where $[x]_i$ is the $i^{th}$ state of the system and $\\kappa _j$ is the $j^{th}$ basis order such that $\\sum _j \\kappa _j \\le Q$ .",
"In this simulation, a nominal model given by $x_{k+1} = x_k +\\begin{bmatrix}\\dot{\\theta }_k \\\\\\dot{x}_k \\\\u\\\\u\\end{bmatrix} \\delta t$ is utilized as an initial guess for the controller in order to boot-strap the data-driven process.",
"Figure REF presents the use of increasing complexity orders of a polynomial and Fourier basis function for the cart-pendulum system.",
"Both test cases begin with the same initial condition and the same nominal model.",
"At intervals of $20s$ , a Koopman operator is computed with either the polynomial or Fourier basis functions using the initial $20s$ of data collected.",
"Due to the existence of the pendulum on $SO(1)$ , the Fourier basis function immediately generates a Koopman operator model that allows the controller to balance and stabilize the pendulum.",
"Moreover, the use of the Fourier basis illustrates the concept that increasing complexity on the operator basis set is not always guaranteed to return an improved data-driven model.",
"In particular, when $Q=2$ , the Koopman operator matches the system model identically.",
"As a result, any further additions in complexity using the Fourier basis for this system is not beneficial (this is not always the case if the system has higher order dependencies).",
"In contrast, the polynomial basis function does show improvement as complexity is increased.",
"Although it would require an infinite set of polynomials to approximate a cosine or sine function, the controller using this operator model provides the desired energy pumping cart motion that is commonly witnessed in inverting a pendulum.",
"Figure: The progressive improvement in control as the Koopman operator increases the basis order of complexity QQ is shown.",
"Each pendulum configuration is taken as a snapshot in time.",
"Koopman operators with complexity QQ are trained on the initial first 20 seconds with the nominal model.",
"Note that because of the SO(1)SO(1) configuration of the pendulum, a Fourier basis of complexity Q=1Q=1 is sufficient to invert at stabilize the cart-pendulum.",
"Adding a higher complexity Q=2Q=2 does not provide a different Koopman matrix (this does not necessarily hold true for non-simulated systems).",
"It is interesting to note that as the complexity of the polynomial basis increases, so do the number of attempts at swinging up the cart-pendulum.",
"Link to multimedia provided: https://vimeo.com/219458009 .Simulated examples are further investigated with the use of a vertical take-off and landing (VTOL) pendulum system [24].",
"For this example, the problem of inverting the pendulum attached to a VTOL is slightly modified.",
"Specifically, it is assumed that a well known model of the VTOL exists, but the interaction between the VTOL and the pendulum remains unknown.",
"Thus, the goal of this simulated example is to generate a Koopman operator that describes the interaction of the VTOL on the pendulum.",
"In this example, the Koopman operator is redefined as an augmentation to a dynamical system $ x_{k+1} = f(x_k, u_k) + \\tilde{K}^{T} \\Psi (x_k, u_k)^T.$ By subtracting the current nominal model of the system $f(x_k, u_k)$ from both side in equation (REF ) and treating $x_{k+1}$ as the measurement of state, we can define the following as a nonlinear process that can be used to generate a Koopman operator: $\\Psi (x_{k+1}) = x_{k+1} - f(x_k, u_k) = \\tilde{K}^T \\Psi (x_k, u_k)^T.$ Given the previous cart-pendulum result, we see that the interaction between the VTOL and the pendulum can be captured solely via a vast set of basis functions across the state of the VTOL-pendulum system.",
"In Fig.",
"REF , the VTOL is shown attempting to invert and balance the pendulum attached with the use of the Koopman operator.",
"Each sequential Koopman operator with increasing complexity is generated from the first 20 seconds worth of data.",
"Originating from the nominal model, it can be seen that the swinging behavior captures a portion of the energy pumping maneuvers required to invert the pendulum.",
"As the Koopman basis order increases, so does the refinement in control authority.",
"When $Q=2$ for the polynomial basis, it can be seen that swing up attempts are more successful.",
"Once the Koopman operator generated from the Fourier basis functions is used, the controller generates the appropriate control strategy to swing up and invert the pendulum.",
"Figure: Each Koopman operator is trained on the residual modeling error of 20 seconds attempted pendulum inversion using the nominal model.",
"As the order of the polynomial basis increases from 1→21 \\rightarrow 2, the number of swing up attempts also increases.",
"Notably, a first order Fourier basis captures the necessary features that allow the controller to invert and stabilize the pendulum.",
"Link to multimedia provided: https://vimeo.com/219458009 .In the following section, our discussion on the use of the Koopman operator is extended to control of a Sphero SPRK robot in a reduced state setting."
],
[
"Open-Loop Trajectory Optimization",
"Figure REF shows trajectories generated using the open-loop controller with varying $Q$ .",
"The reference trajectory is given as $ \\begin{bmatrix}\\tilde{x}\\\\\\tilde{y}\\\\\\dot{\\tilde{x}}\\\\\\dot{\\tilde{y}}\\\\\\end{bmatrix}=\\begin{bmatrix}r \\cos (v t) \\\\r \\sin (2v t) \\\\-r v \\sin (v t ) \\\\2rv\\cos (2v t)\\end{bmatrix}.$ where $r=0.5$ and $v = 1.3$ .",
"The reference trajectory was made sufficiently aggressive to excite the system's internal nonlinearities.",
"As expected, the system improves in performance when tracking the reference trajectory with increasing $Q$ .",
"In particular, as $Q$ goes from 1 to 2, less drift in the resulting open-loop trajectory is visually noted at the end of the path.",
"As $Q$ is further increased, more complexity is added to the description of the SPRK via the Koopman operator which in turn reduces drift and improves the tracking performance.",
"Furthermore, the standard deviation of tracking error across trials is shown to reduce as $Q$ is increased.",
"This implies both consistency in the behavior of the robot subject to the controller.",
"Therefore, it can be concluded that the approximated Koopman operator is better able to represent the dynamics of the system by increasing the complexity of $\\Psi $ ."
],
[
"Closed-Loop Trajectory Tracking",
"Figure REF shows the experimental results for trajectory tracking on a tarp and sand terrain using closed-loop model-based controllers with the Koopman operator.",
"The optimal control signal was updated at 20 Hz and the reference trajectory was given by equation (REF ) where $r$ is split into two components, $r_x=0.7$ and $r_y=0.4$ , with $v=0.9$ .",
"The nominal linear model is given by $x_{k+1} = A x_k + B u_k,$ where $A$ and $B$ are defined as a fully controllable double integrator system.",
"The effectiveness of the closed-loop controller is benchmarked by comparing the model generated from the Koopman operator to that of a simulated example of the controller knowing the true system model (Fig.",
"REF ).",
"Using only the first 20 seconds worth of data from the nominal model controller, we can see in Fig.",
"REF A) that as the operator increases in complexity, so shows the performance of the controller relative to the benchmark test.",
"Specifically, Fig.",
"REF B) shows the tracking error for experimental trials with increasing complexity of the Koopman operator.",
"Notably, when $Q=3$ in sand, the Koopman operator did not have a sparse enough data set that spans the higher order terms in the operator.",
"This can be fixed by collecting more data that spans the robot's operating region.",
"Here, the nonlinear dynamics driven by the internal mechanism become more apparent as the order of the operator is increased.",
"In particular, equation (REF ) provides some insight into the output of the data-driven model of the Koopman operator for the update equation of the SPRK's velocity subject to control inputs.",
"Because the effect of the internal mechanism's configuration (typically described on $SO(3)$ ) cannot be linearly approximated, the Koopman operator begins to approximate a Taylor expansion (REF ).",
"Therefore, the Koopman operator captures the inherent nonlinearities that are utilized by the model-based controller with respect to the terrain.",
"However, achieving a representation that performs consistently across all operating terrains seems infeasible with such limited information, without extra structure on the Koopman operator, such as global Lie group structure or mechanical properties (e.g.",
"symmetries).",
"Figure: NO_CAPTION"
],
[
"Conclusion",
"We present Koopman operator theory and focus on the practical implementation of the theory for model-based control.",
"We derive a linearizable data-driven model using the Koopman operator.",
"Closed-loop and open-loop controllers were formulated using the proposed data-driven model.",
"The open-loop experiments reveal the Koopman operator improves performance as the complexity of the basis increases.",
"Closed-loop experiments reveal the Koopman operator is able to capture the nonlinear dynamics of simulated examples with the cart- and VTOL-pendulum and the SPRK robot.",
"Future research directions include an in-depth analysis of the choice of basis for dynamical system with distinct structure (e.g.",
"conservative systems, mechanical systems, etc.).",
"The relationship between available states and the accuracy of the approximate Koopman operator needs rigorous stability analysis.",
"Moreover, numerical stability analysis and algorithmic optimization is another possible research avenue."
]
] | 1709.01568 |
[
[
"Radial Velocity Measurements of an Orbiting Star Around Sgr A*"
],
[
"Abstract During the next closest approach of the orbiting star S2/S0-2 to the Galactic supermassive black hole (SMBH), it is estimated that RV uncertainties of ~ 10 km/s allow us to detect post-Newtonian effects throughout 2018.",
"To evaluate an achievable uncertainty in RV and its stability, we have carried out near-infrared, high resolution (R ~ 20,000) spectroscopic monitoring observations of S2 using the Subaru telescope and the near-infrared spectrograph IRCS from 2014 to 2016.",
"The Br-gamma absorption lines are used to determine the RVs of S2.",
"The RVs we obtained are 497 km/s, 877 km/s, and 1108 km/s in 2014, 2015, and 2016, respectively.",
"The statistical uncertainties are derived using the jackknife analysis.",
"The wavelength calibrations in our three-year monitoring are stable: short-term (hours to days) uncertainties in RVs are < 0.5 km/s, and a long-term (three years) uncertainty is 1.2 km/s.",
"The uncertainties from different smoothing parameter, and from the partial exclusion of the spectra, are found to be a few km/s.",
"The final results using the Br-gamma line are 497 +- 17 (stat.)",
"+- 3 (sys.)",
"km/s in 2014, 877 +- 15 (stat.)",
"+- 4 (sys.)",
"km/s in 2015, and 1108 +- 12 (stat.)",
"+- 4 (sys.)",
"km/s in 2016.",
"When we use two He I lines at 2.113\\mum in addition to Br-gamma, the mean RVs are 513 km/s and 1114 km/s for 2014 and 2016, respectively.",
"The standard errors of the mean are 16.2 km/s (2014) and 5.4 km/s (2016), confirming the reliability of our measurements.",
"The difference between the RVs estimated by Newtonian mechanics and general relativity will reach about 200 km/s near the next pericenter passage in 2018.",
"Therefore our RV uncertainties of 13 - 17 km/s with Subaru enable us to detect the general relativistic effects in the RV measurements with more than 10 sigma in 2018."
],
[
"Introduction",
"At the center of our Galaxy, a dark mass of $\\sim 4 \\times 10^6 M_{\\odot }$ is likely to be associated with the compact radio source Sgr A*.",
"In the immediate vicinity of Sgr A*, a number of rapidly orbiting stars (called S-stars) has been detected (e.g., [4], [6], [7]), and their orbits have been determined (e.g., [29], [8]).",
"The motions of the stars around Sgr A* have given a lower limit of the mass density inside their pericenter, which provides one of the most compelling cases so far for the existence of supermassive black holes (SMBHs; [3], [12], [24], for most recent works).",
"The stellar system around Sgr A* provides an unique test bed for probing the strong gravitational field around a SMBH.",
"The orbiting S-stars can be regarded as test particles moving in the gravitational field generated by the Galactic SMBH, and particularly important is the star S2 (in the VLT nomenclature) or S0-2 (in the Keck nomenclature).",
"S2/S0-2 is orbiting Sgr A* in $\\approx 16$ yr, which is one of the shortest periods among the orbiting stars, and has a large orbital eccentricity of $e \\approx 0.88 - 0.89$ [3], [12], [24].",
"With a magnitude of $K_{S} \\sim 14$ , S2 is the brightest of the short-period stars.",
"These mean that S2 is an ideal target to observe the strong gravitational field around the SMBH, and we expect to detect the post-Newtonian (PN) effects (i.e., deviation from the Newtonian gravity) with current telescopes.",
"The closest approach of S2 to Sgr A* is expected to be at $2018.29 - 2018.59$ [3], [12], [24] .",
"General relativistic (GR) effects are strongest near the pericenter of the orbit, where the pericenter distance is only about 120 AU, and the speed of S2 reaches a few % of the speed of light.",
"The observations of the orbital motion and light trajectories of S2 in 2018 therefore provide an opportunity to test unobserved predictions of GR around the SMBH, and to understand the nature of gravity.",
"The directly observable quantities of S2 dynamics are its positions in the sky ($\\alpha $ and $\\delta $ ) in astrometric measurements, and radial velocities (RVs) in spectroscopic measurements.",
"Given the current measurement precision, establishing an accurate astrometric reference frame is the greatest challenge, and identifying sources of systematic uncertainties is the ongoing study (see, e.g., [11], [31], [26]).",
"On the other hand, when we obtain new spectroscopic data, RV simply refers to the Local Standard of Rest (LSR).",
"More direct comparison of observed RVs with theoretical prediction is possible with respect to the astrometric observations.",
"We have been focusing on the spectroscopic measurements of S2.",
"Note that, in the context of both of general and special relativities, the redshift of observed photons, $z$ , and the RV of S2, $v_{\\mathrm {S2}}$ , are in a complicated non-linear relation.",
"Our observable quantity is not exactly $v_{\\mathrm {S2}}$ , but $z = \\frac{\\lambda _{\\mathrm {obs}}}{\\lambda _{\\mathrm {S2}}} -1 \\ne \\frac{v_{\\mathrm {S2}}}{c}$ where $\\lambda _{\\mathrm {S2}}$ and $\\lambda _{\\mathrm {obs}}$ are wavelengths emitted from S2 and measured in an observation, respectively, and $c$ is the speed of light.",
"However, following the traditional nomenclature, we show our observed value $cz$ in the unit of km/s and call this value not the redshift but the “radial velocity (RV)\".",
"The RV measurements of S2 near its pericenter passage allow us to detect the PN effects [35], [19], [1], [2], [34], [32], [33], [13], [18].",
"In the complicated relation between $cz$ and $v_{\\mathrm {S2}}$ , the kinematic Doppler-shift and the gravitational redshift appear as the strongest effects and they show comparable amplitudes.",
"Those two effects are estimated to be as large as about 200 km/s at the pericenter passage of S2, and current instruments are capable of detecting at least these RV shifts in the spectroscopic measurements [35].",
"However, we have significant constraints for observations of the S-stars [28].",
"The interstellar extinction toward the Galactic center is extreme, more than $A_V = 30$ mag.",
"It means that we cannot observe stars near Sgr A* in the optical wavelength.",
"In addition, stellar number density is so high that without an adaptive optics (AO) system, our observations are limited to relatively bright magnitudes due to the crowding.",
"Therefore NIR instruments with AO are crucial for observational studies to resolve the S-stars.",
"In the past, most of the spectroscopic observations of S2 were carried out with medium spectral resolution instruments.",
"The instruments used are NACO/VLT (long slit spectroscopy with $R \\sim 1,400$ ; [5]), SPPIFI/VLT (IFU with $R \\sim 3,500$ ; [5]), SINFONI/VLT (IFU with $R \\sim 1,500 - 4,000$ ; [11], [12]), NIRSPEC/Keck (long slit spectroscopy with $R \\sim 2,600$ ; [10]), NIRC2/Keck (long slit spectroscopy with $R \\sim 4,000$ ; [10]), OSIRIS/Keck (IFU $R \\sim 3,600$ ; [10], [23], [3]).",
"The mean of the RV uncertainties since 2010 is $\\sim 34$ km/s, with the best of 15 km/s.",
"Hence, the two leading PN effects (the transverse Doppler and gravitational redshift) can be detected with uncertainties on the order of $5 \\sigma $ through the observations in 2018, if the astrometrically measured positions were known with infinite precision.",
"However, it is still difficult to monitor the time evolution of the PN effects with an uncertainty of more than $\\sim 30\\,$ km/s.",
"We focus on the spectroscopic observations of S2 to determine the time variation of its RVs, as precise as possible.",
"We have carried out near-infrared (NIR), high spectral resolution spectroscopy using the Subaru telescope.",
"The aim of this paper is to evaluate the possibilities of more accurate and stable RV measurements of S2 than the past ones.",
"Below, we show results of our spectroscopic monitoring observations of S2 from 2014 to 2016.",
"We also discuss shortly the expectation for the detection of the PN effects in the spectroscopic measurements during 2018."
],
[
"Data and Observation",
"We have conducted out NIR spectroscopic observations using the Subaru telescope [20] and IRCS [21].",
"The observations were carried out during the nights of 18 May 2014, 20 Aug 2015, and $17 - 18$ May 2016 (Table REF ).",
"The IRCS echelle mode provides a spectral resolution of $\\lambda / \\Delta \\lambda \\approx 20,000$ in the $K$ band.",
"The slit length and width are 517 and 014, respectively.",
"We took spectra in the $K+$ setting, to include Br-$\\gamma $ absorption line at 2.16612 $\\mu $ m, and He I absorption lines at 2.11260 $\\mu $ m and 2.11378 $\\mu $ mVacuum wavelengths from http://www.pa.uky.edu/~peter/newpage/ (Table REF ).",
"Table: Summary of observations.Table: Wavelength coverage of the IRCS echelle K+K+ mode andthe number of OH emission lines used for wavelength calibration.On the nights in 2014, 2015, and the first night in 2016, the position angle of the slit was set to be $\\approx 8$ The position angle of the slit is defined as the angular offset in degrees relative to the north celestial pole.",
"The angle is measured from North to East, counterclockwise direction..",
"In this setting, a bright star IRS 16NW (Ofpe/WN9, $K=10.1$ ; [25]) is observable as well as S2.",
"On the second night in 2016, the position angle was set to be 128, and a bright star IRS 29N (WC9, $K=10.0$ ; [25]) is on the slit simultaneously.",
"The bright stars are used to trace the positions of the S2's spectra on the array in the data reduction procedure.",
"The adaptive optics (AO) system on Subaru, AO188 [16], [17], was used in our observations.",
"The laser guide star (LGS) was propagated at the center of our field in the 2014 and 2016 runs.",
"Since the LGS system did not work well in 2015, instead, an $R = 13.8$ mag star USNO 0600-28577051 was used as a natural guide star.",
"USNO 0600-28577051 was used as a tip-tilt guide star in 2014 and 2016.",
"Thirty-two exposures were obtained on 18 May 2014.",
"Due to thin clouds, we cannot find spectra of S2 for two exposures, and we thus use 30 exposures in the following analysis.",
"In 2015 and 2016, 24 and 48 exposures were obtained, respectively.",
"In 2016, S2 spectra cannot be clearly seen in four exposures.",
"The AO guide star was lost in the exposures due to thin clouds coming.",
"The exposure time is always 300 sec from 2014 to 2016.",
"We have observed standard stars during our runs.",
"HD 152521 (A0-1V), HD 171296 (A0V), or HD 183997 (A0IV-V), was observed once or twice per night.",
"Since the region around S2 is very crowded, we have observed a dark cloud located at a few arcmin northwest from S2, to obtain sky measurements.",
"During the sky observations, we confirmed that no object is included in the slit position."
],
[
"Data Reduction and Analysis",
"The reduction procedure includes dark subtraction, flat-fielding, bad pixel correction, cosmic-ray removal, sky subtraction, spectrum extraction, wavelength calibration, telluric correction, and spectrum continuum fitting.",
"Flat field images were obtained through observations of a continuum source.",
"The sky field, a few arcmin northwest from S2, was observed once or twice per night, and time differences between the sky and objects are as large as 2 hrs.",
"This is longer than the typical variability of the sky in the NIR wavelength.",
"We thus scaled the sky images to subtract atmospheric OH emission lines as cleanly as possible.",
"We have extracted S2 spectra using the IRAF apall task in the echelle package.",
"The S2 spectra are faint, and thus bright spectra of IRS 16NW or IRS 29N are used to trace the positions of S2 spectra, in the assumption that the spectra of S2 and that of the bright stars run parallel to each other.",
"Background subtraction is carried out in the apall task.",
"For each exposure, we selected some blank regions around S2 by eye using “aperture editor\".",
"The median value of the selected regions are used for the background subtraction."
],
[
"Wavelength calibration",
"The wavelength calibration is carried out for the extracted spectra.",
"The wavelength solutions are obtained by identifying the atmospheric OH emission lines, or atmospheric absorption lines.",
"At the end of our observations, we took arc lamp exposures for the wavelength calibration.",
"However, the calibration with the arc lamp resulted in large uncertainties, probably because they were taken at different time, and the number of lines is small, only 12 lines in the echelle orders from 25 to 29.",
"We also tried to obtain the wavelength solutions by using the OH lines in the sky frames.",
"However, we have observed the sky fields only once or twice per night to maximize the exposure time of S2, leading to a time difference of $\\lesssim 2$ hrs between the sky and S2 observations.",
"This is likely to be a main reason of the low accuracy of the wavelength calibration using the OH lines in the sky frames.",
"When we use the OH lines as the wavelength calibration, telluric-corrected spectra show many residuals at the position of the atmospheric absorption lines.",
"We therefore decided to determine the wavelength calibration directly from the science data in the following way: We have taken S2 spectra in a traditional A position $-$ B position manner.",
"For the wavelength calibration of S2 spectra in an A position exposure, we use OH lines in spectra at the same coordinates on the array in the B-position exposure.",
"Similarly, for the calibration of B-position spectra, OH lines in the A-position exposure is used.",
"As a result, it becomes possible to use the OH emission lines at the same coordinates on the array, taken at almost the same time as the spectra of S2, for the wavelength calibration.",
"We have used as many OH lines as possible for the wavelength calibration.",
"The number of the OH lines detected in our spectra are 41 in total, in the echelle orders from 25 to 29 (Table REF ).",
"In some exposures, the S/N ratio of some OH lines was too low to be useful for the calibration because the OH lines were contaminated by weak and noisy stellar continuum spectra, and thus the smaller number of the OH lines was used."
],
[
"Telluric Correction using Standard Stars and Continuum Fitting",
"We observed standard star(s) once or twice per night.",
"Since isolated standard stars are selected, we carried out a traditional A$-$ B and B$-$ A reduction.",
"The standard stars are bright enough to obtain continuum spectra with a high S/N ratio, and the atmospheric absorption lines are used for the wavelength calibration.",
"The number of the absorption lines used for the calibration is 269 in the echelle orders from 25 to 29.",
"Prior to division of S2 spectra by the standard star spectra, the Br-$\\gamma $ absorption line was removed from the standard star spectra by fitting the absorption profile with a Moffat function using the IRAF splot task.",
"The S2 spectra were then divided by the standard star spectra using the telluric task.",
"We have carried out continuum fit with the IRAF continuum task for the telluric-corrected spectrum of each exposure.",
"After the fitting, the spectra are median combined with the scombine task.",
"The combined spectra around Br-$\\gamma $ emission are shown in Fig.",
"REF .",
"Figure: Spectra including Br-γ\\gamma absorption and emission lines of S2in May 2014 (top), Aug 2015 (middle), and May 2016 (bottom).The positions of the Br-γ\\gamma absorption features are shown by white arrows.In all cases the smoothing parameters was chosen as s=9s = 9.The Br-γ\\gamma emission lines from ambient gas are seenat 2.166μ2.166 \\mu m -2.167μ- 2.167 \\mu m.The emission line was not removed to improve the line detection in low S/N spectra.The spectra are not corrected to have a LSR wavelength."
],
[
"Correction to LSR",
"To obtain RVs in the LSR reference frame, we need to take into account the following motions: the rotation of the Earth; the orbital motion around the Sun; and the Sun's peculiar motion with respect to the LSR.",
"The amount of the RV correction, $\\Delta \\mathrm {RV}_{\\mathrm {LSR}}$ , was calculated using the IRAF rvcorrect task.",
"The calculated LSR velocities are shown in Table REF .",
"We have used the mean of the values, the start and the end of the observations, for the correction of the motions: $+24.6$ km/s for 2014; $-15.7$ km/s for 2015; and $+24.5$ km/s for 2016.",
"Table: RV correction with respect to LSR."
],
[
"Measured Radial Velocities and their Uncertainties",
"RVs of S2 are determined for each spectrum on the basis of the location of the Br-$\\gamma $ absorption line.",
"The combined spectra around the Br-$\\gamma $ absorption line are given in Fig.",
"REF , which shows how the Br-$\\gamma $ line has shifted from 2014 to 2016.",
"To determine the central wavelength of the lines, a combination of Gaussian plus Lorentzian functions (i.e., Moffat function) is used to fit the Br-$\\gamma $ line profiles.",
"The peaks of both functions are set to have the same value.",
"Probably due to a noisy continuum and wide profile, the 2014 spectrum cannot be fit with a Moffat function.",
"Hence it is fit with a Gaussian function.",
"The wavelength of the best fit peak is compared to the rest wavelength, $2.166120\\,\\mu $ m, to determine RV$_{\\mathrm {LSR}}$ , with the correction of $\\Delta \\mathrm {RV}_{\\mathrm {LSR}}$ .",
"Figure: Combined spectra around the Br-γ\\gamma absorption linefor 2014 (top), 2015 (middle, blue dots), and 2016 (bottom).The Br-γ\\gamma profiles are fit with a Moffat functionfor 2015 and 2016 (red lines).The 2014 spectrum is fit with a Gaussian function.The smoothing parameter for the shown spectra is s=11s = 11."
],
[
"Uncertainty from Spectrum Smoothing",
"Due to the faintness of S2 and confusion of unresolved sources around it, the S/N ratios of the full resolution S2 spectra are only about 15.",
"We have thus used “smoothed\" spectra to determine the peak wavelengths of the lines.",
"The smoothing parameter, $s$ , represents the box size in pixels used for the spectrum smoothing.",
"The mean flux of the consecutive $s$ pixels is adopted as the flux at the mean wavelength $\\lambda $ of the $s$ pixels.",
"Fig.",
"REF represents spectra in 2016 around Br-$\\gamma $ for different smoothing parameters.",
"As one can see, spectra with small $s$ are noisy, but the peak wavelengths obtained by fitting the profiles with different $s$ parameters are almost the same.",
"Table REF gives the peak wavelengths of the Br-$\\gamma $ absorption line and resultant RV$_{\\mathrm {LSR}}$ for different smoothing parameters.",
"In 2014, the standard deviation of the peak wavelengths for the different smoothing parameters $(s = 3 - 23)$ is $0.00011 \\mu $ m, which corresponds to a RV of $1.5$ km/s.",
"Similarly, the standard deviations of the RV$_{\\mathrm {LSR}}$ are 2.7 km/s and 1.9 km/s for 2015 and 2016, respectively.",
"These are significantly smaller than the statistical uncertainties derived below.",
"Figure: Combined 2016 spectra around the Br-γ\\gamma absorption linewith different smoothing parameters.From top to bottom, s=3s = 3, 7, 11, 15, 19, and 23.The Br-γ\\gamma profiles are fit with a Moffat function (red lines).Table: Wavelength of Br-γ\\gamma and RV LSR _{\\mathrm {LSR}} for combined spectra,with different smoothing parameters."
],
[
"Jackknife Analysis of RV Uncertainty",
"We have used a jackknife analysis to estimate RV statistical uncertainties.",
"In the case of the 2016 data sets, 44 exposures were obtained.",
"The $j$ th partial data set consists of 43 exposures without the $j$ th exposure.",
"These 43 spectra were median combined, leading to the $j$ th spectrum of S2.",
"In total, we obtain 44 partial-data spectra of S2.",
"The central wavelengths of the Br-$\\gamma $ lines were obtained by fitting the profiles with a combined function of Gaussian and Lorentzian.",
"The jackknife uncertainty, $\\sigma _{\\mathrm {JK}}$ , is given by $\\sigma _{\\mathrm {JK}} = \\sqrt{ \\frac{1}{N(N-1)} \\sum _{j=1}^{N} (\\alpha ^*_j - \\alpha ^*)^2},$ where $\\alpha ^*_j = N \\alpha - (N-1) \\alpha _j,$ and $\\alpha ^* = \\frac{1}{N} \\sum _{j=1}^{N} \\alpha ^*_j.$ Here $N$ is the number of exposures, and $\\alpha $ and $\\alpha _j$ are the central wavelengths of the Br-$\\gamma $ absorption line in the full combined spectrum and in the $j$ th partial spectra, respectively [30].",
"Fig.",
"REF shows four partial-data spectra (from 1st to 4th) for 2016.",
"We obtained $\\sigma _{\\mathrm {JK}} = 11.8$ km/s for $s = 11$ .",
"Figure: Jackknife partial spectra around the Br-γ\\gamma absorption line for 2016.From top to bottom, they are spectra for the 1st, 2nd, 3rd, and 4th partial data sets.The Br-γ\\gamma profiles are fit with a Moffat function (red lines).The smoothing parameter for the shown spectra is s=11s = 11.We have carried out the same analysis for different smoothing parameters.",
"The results are shown in Fig.",
"REF .",
"For the 2016 data sets (triangles), when the smoothing parameter increases, $\\sigma _{\\mathrm {JK}}$ slightly increases but decreases at $s = 11$ , and increases again for large $s$ .",
"The minimum $\\sigma _{\\mathrm {JK}}$ appears at $s = 11 - 13$ .",
"For 2015 data sets (squares in Fig.",
"REF ), the variation in $\\sigma _{\\mathrm {JK}}$ is larger than 2016 and 2014, and there is no clear local minimum.",
"For 2014 (circles in Fig.",
"REF ), we have found a local minimum at around $s = 9 - 15$ .",
"As shown in Fig.",
"REF and Table REF , the standard deviations in RV$_{\\mathrm {LSR}}$ for different smoothing parameter with $s \\le 23$ are only a few km/s.",
"So in the following analysis, we will use results with $s = 11$ .",
"Figure: Plot of σ JK \\sigma _{\\mathrm {JK}} as a function of smoothing parameter ss,for 2014 (circle), 2015 (squares), and 2016 (triangles).The resultant RV$_{\\mathrm {LSR}}$ and $\\sigma _{\\mathrm {JK}}$ for $s=11$ are listed in Table REF .",
"The shown jackknife RV$_{\\mathrm {LSR}}$ are the mean of all the partial-data sets.",
"As shown in Table REF , RV$_{\\mathrm {LSR}}$ from the fitting of the all combined spectra with $s = 11$ are 497.3 km/s, 876.5 km/s, and 1108.9 km/s for 2014, 2015, and 2016, respectively.",
"The differences in RV$_{\\mathrm {LSR}}$ from the jackknife analysis are very small.",
"Table: RV LSR _{\\mathrm {LSR}} and σ JK \\sigma _{\\mathrm {JK}} from the jackknife analysis for s=11s = 11."
],
[
"Short Term Stability of Wavelength Calibration",
"Our monitoring observations of S2 were carried out for $\\sim 4$ hrs in 2014, $\\sim 2$ hrs in 2015, and 2 nights in 2016.",
"Any uncertainty from unstable wavelength calibrations on scales of hours or days are included in the uncertainties derived using the jackknife analysis.",
"However, it is important to constrain the absolute uncertainty of the wavelength calibration.",
"In our S2 spectra, Br-$\\gamma $ emission from the local, interstellar gas around S2 is seen.",
"The interstellar gas is ionized by UV radiation from high mass stars around Sgr A*.",
"Assuming that the RV of the local gas is constant, the Br-$\\gamma $ emission lines can be used to examine how accurate our wavelength calibrations are.",
"To this aim, we fitted the Br-$\\gamma $ emission profile with a Gaussian function for each exposure, to understand the stability of the wavelength calibration.",
"We have found that RV$_{\\mathrm {LSR}}$ of the Br-$\\gamma $ emission line is stable.",
"Fig.",
"REF shows the RV$_{\\mathrm {LSR}}$ for each exposure with $s = 1$ as a function of the order of the observations.",
"The standard deviations of RV$_{\\mathrm {LSR}}$ in 2014 (black circles in Fig.",
"REF ) is 0.24 km/s.",
"That for 2015 is slightly larger, 0.53 km/s, but still much smaller than $\\sigma _{\\mathrm {JK}}$ .",
"The RV$_{\\mathrm {LSR}}$ in 2016 are most stable, and the standard deviation is only 0.16 km/s.",
"Therefore we conclude that our wavelength calibration is stable during one or two nights, at least in the echelle order 26 where Br-$\\gamma $ emission and absorption lines are found.",
"The short-term uncertainty in the wavelength calibration is negligible compared to the jackknife uncertainty.",
"Figure: RV LSR _{\\mathrm {LSR}} of the Br-γ\\gamma emission linefor each exposure as a function of the order of observations.RV LSR _{\\mathrm {LSR}} for 2014, 2015, and 2016 are represented byblack circles, blue squares, and red triangles, respectively."
],
[
"Long Term Stability of Wavelength Calibration",
"The short term (from a few hours to 2 days) stability of the wavelength calibration was examined in the previous section.",
"However, since our observations were carried out for three years, an examination of a long-term (months or years) stability is necessary.",
"We have derived RV$_{\\mathrm {LSR}}$ of the Br-$\\gamma $ emission line for each year, by fitting the combined spectra ($s=1$ ) with a Gaussian function (Fig.",
"REF ).",
"The resultant peak wavelengths and RV$_{\\mathrm {LSR}}$ are shown in Table REF .",
"The standard deviation of RV$_{\\mathrm {LSR}}$ from 2014 to 2016 is $1.2\\,$ km/s.",
"If we simply use all the data points shown in Fig.",
"REF , the standard deviation is 0.93 km/s.",
"These results suggest that the long-term, systematic uncertainty in our monitoring observations of S2 is small compared to the statistical uncertainty derived by the jackknife analysis.",
"Table: Observed peak wavelength and RV LSR _{\\mathrm {LSR}} of the Br-γ\\gamma emission lines.Figure: Combined, full resolution (s=1s = 1) spectra around the Br-γ\\gamma emission linesfor 2014 (top), 2015 (middle), and 2016 (bottom).The emission profiles are fit with a Gaussian function.The difference of the wavelengths among the three spectraare mainly due to the motions of the Earth and the Solar system."
],
[
"Partly Excluded Spectrum",
"One of the difficulties in data reduction of NIR high-resolution spectroscopy is the telluric correction.",
"Although we have tried to find better wavelength solutions, we see uncorrected atmospheric lines, especially in the orders 28 and 29.",
"The number of the telluric lines is much smaller around the Br-$\\gamma $ wavelength, but weak, uncorrected lines could be included in the spectra we used to determine RV.",
"We have examined how the central wavelength of the Br-$\\gamma $ absorption line changes when a part of the spectra is excluded.",
"In the case of 2016, we have made 20 spectra without several data points in the fitting range of the Br-$\\gamma $ line.",
"The width of the excluded wavelength range is $0.0003\\,\\mu $ m $= 3\\,$ Å, in which $6 -7$ data points are included.",
"Five of the partly excluded spectra are shown in Fig.",
"REF .",
"The top one shows the combined 2016 spectra of $s = 11$ , without data points in the range of $2.1710\\,\\mu $ m $< \\lambda < 2.1713\\,\\mu $ m. Table REF shows the results of the fitting of the Br-$\\gamma $ line for the partly excluded spectra in 2016.",
"When a part of the wing region is excluded, the shift of the central wavelength is very small.",
"On the other hand, when a part of the core region is excluded, the peak wavelength shifts slightly larger.",
"We can see substructures in the core region of the Br-$\\gamma $ absorption line, even with $s = 11$ .",
"There is likely to be some peaks in the profile, although whether the substructures are real or not is still not clear due to the low S/N ratio of our spectra.",
"When a substructure is in the excluded region, the shape of the fitting curve changes, and the resultant central wavelength shifts.",
"However, as shown in Table REF , the standard deviation of the RV$_{\\mathrm {LSR}}$ is small, 3.6 km/s.",
"Even if we focus on the spectra with the excluded region near the spectral core, $2.173\\,\\mu $ m $\\lesssim \\lambda \\lesssim 2.175\\,\\mu $ m, the standard deviation appears to be $\\sim 6$ km/s.",
"Figure: 2016 combined spectra with s=11s = 11, but a part of the data points is excluded.From top to bottom, the wavelength ranges excluded are2.1710-2.1713μ2.1710 - 2.1713\\,\\mu m,2.1731-2.1734μ2.1731 - 2.1734\\,\\mu m,2.1740-2.1743μ2.1740 - 2.1743\\,\\mu m,2.1743-2.1746μ2.1743 - 2.1746\\,\\mu m, and2.1749-2.1752μ2.1749 - 2.1752\\,\\mu m.The Br-γ\\gamma features are fit with a Moffat function (red lines).Table: Wavelength of Br-γ\\gamma and RV LSR _{\\mathrm {LSR}} for partially excluded spectra for s=11s = 11.A similar analysis was done for 2014 and 2015.",
"The fitting wavelength range for the 2014 spectrum is smaller than 2016.",
"We have made 13 spectra excluding $6 - 7$ consecutive data points ($= 3$ Å) in the range of $2.1680\\,\\mu $ m $< \\lambda < 2.1720\\,\\mu $ m for the 2014 data sets.",
"The result of the fitting of the 13 spectra is shown in Fig.",
"REF , left panel.",
"The standard deviation of RV$_{\\mathrm {LSR}}$ is 1.9 km/s.",
"We have made 20 spectra for the 2015 data sets, and carried out the same analysis.",
"The result is shown in the middle panel in Fig.",
"REF , and the standard deviation of RV$_{\\mathrm {LSR}}$ is 2.0 km/s.",
"Figure: Distributions of RV LSR _{\\mathrm {LSR}} for partly excluded spectrafor 2014 (left), 2015 (middle), and 2016 (right).The standard deviations of RV LSR _{\\mathrm {LSR}} are1.9, 2.0, and 3.6 km/s for 2014, 2015, and 2016, respectively.As shown above, even if a part of the spectra is excluded, the resultant RV$_{\\mathrm {LSR}}$ changes only a few km/s.",
"Hence we conclude that the RV$_{\\mathrm {LSR}}$ of S2 derived from the Br-$\\gamma $ absorption is not strongly affected by the incomplete telluric correction."
],
[
"Summary of RV$_{\\mathrm {LSR}}$ and Uncertainties",
"The obtained RV$_{\\mathrm {LSR}}$ and uncertainties discussed above are summarized in Table REF .",
"We use the jackknife $\\sigma _{\\mathrm {JK}}$ as statistical uncertainties.",
"As total systematic uncertainties, we quadratically added the uncertainties from the spectrum smoothing, the long-term stability of the wavelength calibration, and the partly excluded spectrum analysis.",
"The uncertainty in the short-term stability of the wavelength calibration is included in the uncertainties from the jackknife analysis, and thus we do not add them separately.",
"Table: RV LSR _{\\mathrm {LSR}} Error Budget in the unit of km/s."
],
[
"He I absorption lines at 2.1126 $\\mu $ m",
"Although the S/N ratio is low, He I absorption lines at 2.112597 $\\mu $ m and 2.113780 $\\mu $ m are detected in our 2014 and 2016 spectra (Fig.",
"REF ).",
"We use the He I lines to check the reliability of RV$_{\\mathrm {LSR}}$ and their uncertainties from the Br-$\\gamma $ line.",
"We fitted the two He I lines with a double Gaussian function (red curves in Fig.",
"REF ).",
"The two He I lines are so close that we cannot resolve them with the smoothing parameters of $s \\ge 11$ .",
"The wavelength difference between the Gaussian peaks are fixed to be $2.1137800 - 2.1125965 = 0.00011835\\,\\mu $ m, and the peak wavelength of the He I 2.112597 $\\mu $ m line is set to be a free parameter in the fitting procedure.",
"The smoothing parameters for the spectra in Fig.",
"REF are $s = 7$ and 9 for 2014 and 2016, respectively.",
"We cannot find a clear He I feature for the 2015 combined spectrum.",
"We have derived RV$_{\\mathrm {LSR}}$ from the measured He I 2.112597 $\\mu $ m line peak wavelengths.",
"The peak wavelengths and corresponding RV$_{\\mathrm {LSR}}$ are shown in Table REF .",
"The obtained He I RV$_{\\mathrm {LSR}}$ for 2014, 529.6 km/s, is larger than that for the Br-$\\gamma $ line ($\\approx 497\\,$ km/s), although the difference is less than $2\\,\\sigma $ .",
"The He I RV$_{\\mathrm {LSR}}$ for 2016 is in good agreement with that from the Br-$\\gamma $ line ($\\approx 1108\\,$ km/s), and the difference is smaller than $1\\,\\sigma $ uncertainty of 12.5 km/s.",
"We thus conclude that the RV$_{\\mathrm {LSR}}$ values from the Br-$\\gamma $ and He I lines are consistent with each other.",
"If we simply calculate means and standard deviations of the mean from the Br-$\\gamma $ and He I lines, the derived RV$_{\\mathrm {LSR}}$ are $513.3 \\pm 16.2$ km/s and $1113.6 \\pm 5.4$ km/s for 2014 and 2016, respectively.",
"Figure: 2014 (top) and 2016 (bottom) combined spectra around He I lines.The smoothing parameters for the shown spectra ares=7s = 7 and s=9s = 9 for 2014 and 2016, respectively.The positions of the 2.112597 μ\\mu m and 2.113780 μ\\mu m linesare indicated by arrows.The profiles are fit with a double Gaussian function to determine the peak wavelengths (red curves).Table: Observed He I 2.112597 μ\\mu m wavelengths and RV LSR _{\\mathrm {LSR}}."
],
[
"Wider Br-$\\gamma $ Absorption Profile in 2014",
"As shown in Fig.",
"REF , the Br-$\\gamma $ profile in the 2014 spectrum is wider than those for 2015 and 2016.",
"One possible reason for the wider profile is an imperfect correction of the telluric absorption.",
"There is a strong atmospheric absorption line at $\\lambda \\approx 2.1687\\,\\mu $ m, and this absorption profile is within the 2014 Br-$\\gamma $ profile.",
"The telluric profile is well corrected in 2016, but there seems to be residual in 2015.",
"Hence the 2014 Br-$\\gamma $ profile could be extended to the bluer wavelength due to the residual of the telluric line at $\\approx 2.1687\\,\\mu $ m, and this could lead to the difference in RV$_{\\mathrm {LSR}}$ between the Br-$\\gamma $ and He I lines.",
"To check instrumental effects for the width of the Br-$\\gamma $ profile, we have compared the widths of the Br-$\\gamma $ emission line at $\\sim 2.1665\\,\\mu $ m (see §REF ).",
"As shown in Fig.",
"REF , the Br-$\\gamma $ emissions were fit with a Gaussian function.",
"The obtained Gaussian sigmas are 2.4 Å, 3.3 Å, and $2.8\\,\\mathrm {Å}$ in 2014, 2015, and 2016, respectively.",
"The difference in sigma could be explained by the difference of the observation modes; In 2014 and 2016, we could use the LGS system, which makes an AO guide star at a closer position to S2 than natural guide stars.",
"In 2015, we could not use the LGS system, and thus the spatial resolution was worse than other 2 epochs.",
"It could lead to observations of ionized gas at larger region.",
"However, the Gaussian sigma for 2014 is smaller than 2015 and 2016, and it cannot explain the wider Br-$\\gamma $ profile in the 2014 spectrum.",
"The wider profile might be explained by intrinsic properties of S2.",
"Possible origin is a change of the direction of the S2's rotation axis, or binarity of S2.",
"However, no flux variation due to a binary eclipse has been detected [27], [10], [11].",
"We continue to investigate the intrinsic properties of S2 in upcoming monitoring observations."
],
[
"S2 RV Curve since 2000",
"In Fig.",
"REF , we show the plots of RV$_{\\mathrm {LSR}}$ and uncertainties in RV$_{\\mathrm {LSR}}$ as a function of time, combined with the past observations using the Keck telescope [3] and VLT [12].",
"In the past measurements, the uncertainties mainly range from $\\sim 20$ km/s to $\\sim 60$ km/s.",
"The mean RV$_{\\mathrm {LSR}}$ uncertainty is 34 km/s since 2010, and the best measurement was in 2013 with an uncertainty of 16 km/s using Keck/OSIRIS with a total exposure time of $\\approx 7.8$ hrs.",
"As shown in Fig.",
"REF , our RV$_{\\mathrm {LSR}}$ uncertainties are stable, and the mean of them are smaller than that of the past measurements.",
"One of the reasons why the uncertainties of our measurements are smaller than the past ones is probably an accuracy of the wavelength calibration.",
"We have used the atmospheric OH emission lines for the calibration.",
"They have narrow features even in our full resolution spectra, and it means that the peak wavelengths of the OH lines can be determined with a better accuracy if we use higher-spectral resolution spectrograph.",
"As shown in §, the uncertainties of the wavelength calibration on the scales of hours or days are less than $\\sim 0.5$ km/s in 2014, 2015, and 2016.",
"The long-term uncertainty in the calibration is also small with a standard deviation of 1.2 km/s.",
"In the past observations, the uncertainties in the wavelength calibration were $\\sim 9$ km/s [9], [10] for S2/S0-2, although it was estimated to be in the order of $2-3$ km/s using VLT [11].",
"Another reason is that our spectral resolution is high enough to separate the Br-$\\gamma $ line profile from nearby He I absorption lines.",
"As shown in [22] and [15], in the medium-resolution spectroscopy, the He I lines at $\\sim 2.162\\,\\mu $ m are in the wing of the Br-$\\gamma $ profile.",
"However, in our spectra for 2015 and 2016, the He I lines are clearly separated from the Br-$\\gamma $ line, and are at the edge or out of our fitting range.",
"Hence the He I lines do not affect the peak wavelength measurements of the Br-$\\gamma $ line.",
"Figure: Measured RV LSR _{\\mathrm {LSR}} (top) and uncertainties in RV LSR _{\\mathrm {LSR}} (bottom)as a function of time.Our results using the Subaru telescope are shown by green circles.The red data, using the Keck telescopes, are from ,and the blue data, using VLT, are from ."
],
[
"RV Measurements of S2 in 2018",
"To show the importance of RV measurements of S2 in 2018, in Fig.",
"REF , we compare the expected RV curves using orbital parameters derived by the most recent works: [3]; [12]; and [24].",
"Here the curves in Fig.",
"REF show the expected RV curves from pure Keplerian motions, where no relativistic effect is included.",
"A number of astrometric observations with the Keck telescope and NTT/VLT, and careful data analysis have provided us with strong constraints on the mass $M_{\\mathrm {Sgr\\,A*}}$ and the distance to the Galactic SMBH.",
"The amount of mass concentrated around Sgr A* has been estimated with an uncertainty of $3 - 4$ % [3], [12].",
"However, the predicted next pericenter passages are $2018.29 \\pm 0.04$ [3], $2018.35 \\pm 0.02$ [12], and $2018.59 \\pm 0.21$ [24]; the difference is as large as 0.3 yr $\\approx 110$ days.",
"Although this is only a few % uncertainty of the S2's orbital period, this is still large for a detailed, appropriate planning of observations in 2018.",
"As one can see, the differences among the expected RV curves could be more than 1,000 km/s in 2018.",
"Frequent spectroscopic measurements of S2 in 2018, especially during the steep decline phase of RV, will allow us to reduce the uncertainty in the orbital period of S2.",
"Figure: Expected S2 RV curves in 2018, using the parametersin (black curve), (red curve), and (blue curve).Next, let us discuss the expectation for the detection of the PN effects in RV measurements.",
"As noted in §, the relation between the redshift of photons coming from S2, $z$ , and the radial velocity of S2, $v_{\\mathrm {S2}}$ , is complicated and our observable quantity is not exactly $v_{\\mathrm {S2}}$ but $z$ (see equation (REF )).",
"Thus, we define the GR effect measured in spectroscopic observations as $c \\Delta z = cz_{\\mathrm {Einstein}} - cz_{\\mathrm {Newton}},$ where $cz_{\\mathrm {Einstein}}$ is the redshift estimated by GR (S2 motion and photon propagation in the rotating BH spacetime), and $cz_{\\mathrm {Newton}}$ is the redshift by the Newtonian mechanics (S2 motion in the point mass Newtonian gravitational potential).",
"Our estimation of $cz_{\\mathrm {Einstein}}$ and $c \\Delta z$ are shown in Fig.",
"REF .",
"In our theoretical calculation, the mass of Sgr A*, $M_{\\mathrm {Sgr\\,A*}}$ , and the orbital elements of S2, which provide us with the initial condition for the S2 motion, are taken from [12].",
"Fig.",
"REF shows that $c \\Delta z$ will become a few 10 km/s in the latter half of 2017, reach about 200 km/s near the next pericenter passage in 2018, and fall to a few 10 km/s at the end of 2018.",
"Our observational uncertainties in $cz $ of $12 - 17\\,$ km/s, enable us to detect the GR effects in the spectroscopic measurements within the next one and a half years.",
"Figure: Top: Time evolution of cz Einstein cz_{\\mathrm {Einstein}},where the BH mass and S2's orbital parameters arethe best-fit values given by .We assumed that the spin direction is pointing the Galactic southand the spin magnitude is 0.98 M Sgr A* M_{\\mathrm {Sgr\\,A*}}(98% of the theoretically allowed maximum value).Bottom: Time evolution of the general relativistic effect cΔzc \\Delta z.The timing of the pericenter passage is shown by black dot.Note that the so-called PN expansion of $cz_{\\mathrm {Einstein}}$ can be expressed as a series expansion (polynomial in $1/r$ ), $cz_{\\mathrm {Einstein}} = cz_{\\mathrm {Newton}} + \\mathrm {1st~PN~term} + \\mathrm {higher~order~PN~terms} ,$ where the order of the small parameter of this expansion is estimated by $G M_{\\mathrm {Sgr\\,A*}}/(c^2 r_{\\mathrm {peri}}) \\sim 10^{-3}$ , where $r_{\\mathrm {peri}} \\simeq 121$ AU is the pericenter distance of S2, and $G$ is the gravitational constant.",
"The 1st PN term for the S2 dynamics consists mainly of the kinematic Doppler effect and the gravitational redshift.",
"Each of these two effects is estimated to be about 100 km/s ($\\sim c \\times 10^{-3}$ ) and the total of the 1st PN terms becomes 200 km/s near the pericenter passage (Fig.",
"REF , bottom panel).",
"This 1st PN effect includes the effect of the BH mass but not the effect of the BH spin.",
"The 2nd order PN effect in Equ.",
"(REF ), which includes the effect of the BH spin, is typically about $200 \\times 10^{-3} \\sim 0.2$ km/s.",
"Therefore the GR effects we can detect with Subaru/IRCS is the 1st PN order effects.",
"Fig.",
"REF shows the time evolution of $cz_{\\mathrm {Einstein}}$ near its peak, assuming three different BH masses: the best-fit value by [12] (red curve); and 1% larger (green dashed curve) and smaller (blue) than the best value.",
"The BH mass difference of 1% makes $\\sim 17$ km/s shifts in the RV peaks.",
"The difference is almost the same with the parameters derived by [3].",
"These are almost the same amplitudes as the RV uncertainties we obtained with Subaru/IRCS.",
"Note that the measurements of $cz$ is almost independent of the distance from us to the Galactic center, $R_{\\mathrm {GC}}$ , because spectroscopic measurements do not strongly depend on the measurements of visible angle between the S2 and Sgr A*, and we do not need the value of $R_{\\mathrm {GC}}$ to determine the value of $cz$ .",
"In the astrometric measurements, the degeneracy between $R_{\\mathrm {GC}}$ and the mass of Sgr A* is a source of uncertainty (e.g., Equ.",
"(9) in [12]).",
"Hence, a combination of astrometric measurements and accurate spectroscopic measurements, which is almost insensitive to $R_{\\mathrm {GC}}$ , is expected to decrease the uncertainty of the mass of Sgr A*.",
"Figure: Enlarged graph of cz Einstein cz_{\\mathrm {Einstein}} near the maximum value at 2018.2498.Red curve is cz Einstein cz_{\\mathrm {Einstein}} estimated with the best-fit value ofM Sgr A* M_{\\mathrm {Sgr\\,A*}} given by .Green and blue dashed curves are estimatedwith 1% larger and smaller SMBH masses, respectively,than the best-fit value."
],
[
"Conclusion",
"We have carried out near-infrared, high resolution spectroscopic observations of S2 using Subaru/IRCS from 2014 to 2016.",
"The radial velocities of S2 were determined using the Br-$\\gamma $ absorption line.",
"The total uncertainties in the radial velocity measurements are 17.3 km/s, 15.8 km/s, and 12.5 km/s for 2014, 2015, and 2016, respectively.",
"We have confirmed the long-term stability of our radial velocity monitoring observations.",
"The uncertainties are smaller than those in the past, medium resolution spectroscopies, and small enough to detect post-Newtonian effects in 2018."
],
[
"Acknowledgement",
"We thank the Subaru Telescope staff for the support for our observations.",
"This work was supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists (A) 25707012, Grant-in-Aid for Challenging Exploratory Research 15K13463, H. S. was supproted by KAKENHI Grant-in-Aid for Challenging Exploratory Research 26610050.",
"Y. T. was supproted by KAKENHI Grant-in-Aid for Young Scientists (B) 26800150.",
"M.T.",
"was supported by KAKENHI Grant Number 17K05439 and DAIKO FOUNDATION."
]
] | 1709.01598 |
[
[
"A Neural Language Model for Dynamically Representing the Meanings of\n Unknown Words and Entities in a Discourse"
],
[
"Abstract This study addresses the problem of identifying the meaning of unknown words or entities in a discourse with respect to the word embedding approaches used in neural language models.",
"We proposed a method for on-the-fly construction and exploitation of word embeddings in both the input and output layers of a neural model by tracking contexts.",
"This extends the dynamic entity representation used in Kobayashi et al.",
"(2016) and incorporates a copy mechanism proposed independently by Gu et al.",
"(2016) and Gulcehre et al.",
"(2016).",
"In addition, we construct a new task and dataset called Anonymized Language Modeling for evaluating the ability to capture word meanings while reading.",
"Experiments conducted using our novel dataset show that the proposed variant of RNN language model outperformed the baseline model.",
"Furthermore, the experiments also demonstrate that dynamic updates of an output layer help a model predict reappearing entities, whereas those of an input layer are effective to predict words following reappearing entities."
],
[
"Introduction",
"Language models that use probability distributions over sequences of words are found in many natural language processing applications, including speech recognition, machine translation, text summarization, and dialogue utterance generation.",
"Recent studies have demonstrated that language models trained using neural network [3], [29] such as recurrent neural network (RNN) [20] and convolutional neural network [9] achieve the best performance across a range of corpora [29], [4], [28], [12].",
"Figure: Dynamic Neural Text Modeling: the embeddings of unknown words, denoted by coreference indexes “[ k ]” are dynamically computed and used in both the input and output layers (x [k] x_{[\\textrm {k}]} and y [k] y_{[\\textrm {k}]})of a RNN language model.These are constructed from contextual information (d [k],i d_{[\\textrm {k}],\\textrm {i}}) preceding the current (i+1)(i+1)-th sentence.However, current neural language models have a major drawback: the language model works only when applied to a closed vocabulary of fixed size (usually comprising high-frequency words from the given training corpus).",
"All occurrences of out-of-vocabulary words are replaced with a single dummy token “<unk>”, showing that the word is unknown.",
"For example, the word sequence, Pikotaro sings PPAP on YouTube is treated as <unk> sings <unk> on <unk> assuming that the words Pikotaro, PPAP, and YouTube are out of the vocabulary.",
"The model therefore assumes that these words have the same meaning, which is clearly incorrect.",
"The derivation of meanings of unknown words remains a persistent and nontrivial challenge when using word embeddings.",
"Figure: Dynamic Neural Text Modeling: the meaning representation of each unknown word, denoted by a coreference index “[ k ]”, is inferred from the local contexts in which it occurs.In addition, existing language models further assume that the meaning of a word is the same and universal across different documents.",
"Neural language models also make this assumption and represent all occurrences of a word with a single word vector across all documents.",
"However, the assumption of a universal meaning is also unlikely correct.",
"For example, the name John is likely to refer to different individuals in different documents.",
"In one story, John may be a pianist while another John denoted in a second story may be an infant.",
"A model that represents all occurrences of John with the same vector fails to capture the very different behavior expected from John as a pianist and John as an infant.",
"In this study, we address these issues and propose a novel neural language model that can build and dynamically change distributed representations of words based on the multi-sentential discourse.",
"The idea of incorporating dynamic meaning representations into neural networks is not new.",
"In the context of reading comprehension, [23] proposed a model that dynamically computes the representation of a named entity mention from the local context given by its prior occurrences in the text.",
"In neural machine translation, the copy mechanism was proposed as a way of improving the handling of out-of-vocabulary words (e.g., named entities) in a source sentence [13], [14].",
"We use a variant of recurrent neural language model (RNLM), that combines dynamic representation and the copy mechanism.",
"The resulting novel model, Dynamic Neural Text Model, uses the dynamic word embeddings that are constructed from the context in the output and input layers of an RNLM, as shown in Figures REF and REF .",
"The contributions of this paper are three-fold.",
"First, we propose a novel neural language model, which we named the Dynamic Neural Text Model.",
"Second, we introduce a new evaluation task and dataset called Anonymized Language Modeling.",
"This dataset can be used to evaluate the ability of a language model to capture word meanings from contextual information (Figure REF ).",
"This task involves a kind of one-shot learning tasks, in which the meanings of entities are inferred from their limited prior occurrences.",
"Third, our experimental results indicate that the proposed model outperforms baseline models that use only global and static word embeddings in the input and/or output layers of an RNLM.",
"Dynamic updates of the output layer helps the RNLM predict reappearing entities, whereas those of the input layer are effective to predict words following reappearing entities.",
"A more detailed analysis showed that the method was able to successfully capture the meanings of words across large contexts, and to accumulate multiple context information."
],
[
"RNN Language Model",
"Given a sequence of $N$ tokens of a document $D=(w_1, w_2, ..., w_{N})$ , an RNN language model computes the probability $p(D)=\\prod _{t=1}^{N} p(w_t|w_1, ..., w_{t-1})$ .",
"The computation of each factorized probability $p(w_t|w_1, ..., w_{t-1})$ can also be viewed as the task of predicting a following word $w_t$ from the preceding words $(w_1, ..., w_{t-1})$ .",
"Typically, RNNs recurrently compute the probability of the following word $w_{t}$ by using a hidden state $\\mathbf {h}_{t-1}$ at time step $t-1$ , $p(w_t|w_1, ..., w_{t-1}) =\\frac{ \\exp ({\\vec{\\mathbf {h}}_{t-1}}^{\\intercal }\\mathbf {y}_{w_t} + b_{w_t}) }{\\sum _{w \\in V} \\exp ({\\vec{\\mathbf {h}}_{t-1}}^{\\intercal }\\mathbf {y}_{w} + b_{w})} , \\\\\\vec{\\mathbf {h}}_{t} = \\operatornamewithlimits{\\overrightarrow{\\mathrm {RNN}}}(\\mathbf {x}_{w_{t}}, \\vec{\\mathbf {h}}_{t-1}) .$ Here, $\\mathbf {x}_{w_{t}}$ and $\\mathbf {y}_{w_{t}}$ denote the input and output word embeddings of $w_{t}$ respectively, $V$ represents the set of words in the vocabulary, and $b_{w}$ is a bias value applied when predicting the word $w$ .",
"The function $\\operatornamewithlimits{\\overrightarrow{\\mathrm {RNN}}}$ is often replaced with LSTM [18] or GRU [5] to improve performance."
],
[
"Dynamic Entity Representation",
"RNN-based models have been reported to achieve better results on the CNN QA reading comprehension dataset [17], [23].",
"In the CNN QA dataset, every named entity in each document is anonymized.",
"This is done to allow the ability to comprehend a document using neither prior nor external knowledge to be evaluated.",
"To capture the meanings of such anonymized entities, [23] proposed a new model that they named dynamic entity representation.",
"This encodes the local contexts of an entity and uses the resulting context vector as the word embedding of a subsequent occurrence of that entity in the input layer of the RNN.",
"This model: (1) constructs context vectors $\\mathbf {d}_{e,i}^{\\prime }$ from the local contexts of an entity $e$ at the $i$ -th sentence; (2) merges multiple contexts of the entity $e$ through max pooling and produces the dynamic representation $\\mathbf {d}_{e,i}$ ; and (3) replaces the embedding of the entity $e$ in the ($i+1$ )-th sentence with the dynamic embedding $\\mathbf {x}_{e,i+1}$ produced from $\\mathbf {d}_{e,i}$ .",
"More formally, $\\mathbf {x}_{e,i+1} & = W_{dc} \\mathbf {d}_{e,i} + \\mathbf {b}_{e} , \\\\\\mathbf {d}_{e,i} & = \\mathrm {maxpooling}(\\mathbf {d}_{e,i}^{\\prime }, \\mathbf {d}_{e,i-1}) , \\\\\\mathbf {d}_{e,i}^{\\prime } & = \\mathrm {ContextEncoder}(e, i) .",
"$ Here, $\\mathbf {b}_{e}$ denotes a bias vector, $\\mathrm {maxpooling}$ is a function that yields the largest value from the elementwise inputs, and $\\mathrm {ContextEncoder}$ is an encoding function.",
"Figure REF gives an example of the process of encoding and merging contexts from sentences.",
"An arbitrary encoder can be used for $\\mathrm {ContextEncoder}$ ; [23] used bidirectional RNNs, encoding the words surrounding the entity $e$ of a sentence in both directions.",
"If the entity $e$ fails to appear in the $i$ -th sentence, the embedding is not updated, i.e., $\\mathbf {d}_{e,i} = \\mathbf {d}_{e,i-1}$ ."
],
[
"Proposed Method: Dynamic Neural Text Modeling",
"In this section, we introduce the extension of dynamic entity representation to language modeling.",
"From Equations REF and , RNLM uses a set of word embeddings in the input layer to encode the preceding contextual words, and another set of word embeddings in the output layer to predict a word from the encoded context.",
"Therefore, we consider incorporating the idea of dynamic representation into the word embeddings in the output layer ($\\mathbf {y}_w$ in Equation REF ) as well as in the input layer ($\\mathbf {x}_w$ in Equation ; refer to Figure REF ).",
"The novel extension of dynamic representation to the output layer affects predictions made for entities that appear repeatedly, whereas that in the input layer is expected to affect the prediction of words that follow the entities.",
"Figure: An example document for Anonymized Language Modeling.Token “[ k ]” is an anonymized token that appears k-th in the entities in a document.Language models predict the next word from the preceding words, and calculate probabilities for whole word sequences.The procedure for constructing dynamic representations of $e$ , $\\mathbf {d}_{e,i}$ is the same as that introduced in Section REF .",
"Before reading the ($i+1$ )-th sentence, the model constructs the context vectors $[\\mathbf {d}_{e,1}^{\\prime }, ..., \\mathbf {d}_{e,i}^{\\prime }]$ from the local contexts of $e$ in every preceding sentence.",
"Here, $\\mathbf {d}_{e,j}^{\\prime }$ denotes the context vector of $e$ in the $j$ -th sentence.",
"$\\mathrm {ContextEncoder}$ in the model produces a context vector $\\mathbf {d}_{e}^{\\prime }$ for $e$ at the $t$ -th position in a sentence, using a bidirectional RNNEquations and are identical but do not share internal parameters.",
"as follows: $\\mathbf {d}_{e}^{\\prime } & = \\mathrm {ReLU}( W_{hd} [\\vec{\\mathbf {h}}_{t-1},\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\mathbf {h}}}}_{t+1} ] \\!+\\!",
"\\mathbf {b}_{d} ) , \\\\\\vec{\\mathbf {h}}_{t} & = \\operatornamewithlimits{\\overrightarrow{\\mathrm {RNN}}}(\\mathbf {x}_{w_{t}}, \\vec{\\mathbf {h}}_{t-1}) ,\\\\\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\mathbf {h}}}}_{t} & = \\operatornamewithlimits{\\overleftarrow{\\mathrm {RNN}}}(\\mathbf {x}_{w_{t}}, \\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\mathbf {h}}}}_{t+1}) .$ Here, $\\mathrm {ReLU}$ denotes the ReLU activation function [31], while $W_{dc}$ and $W_{hd}$ correspond to learnable matrices; $b_{d}$ is a bias vector.",
"As in the RNN language model, $\\vec{\\mathbf {h}}_{t-1}$ and $\\scalebox {-1}[1]{\\vec{\\scalebox {-1}[1]{\\mathbf {h}}}}_{t+1}$ as well as their composition $\\mathbf {d}_{e}^{\\prime }$ can capture information necessary to predict the features of the target $e$ at the $t$ -th word.",
"Following context encoding, the model merges the multiple context vectors, $[\\mathbf {d}_{e,1}^{\\prime }, ..., \\mathbf {d}_{e,i}^{\\prime }]$ , into the dynamic representation $\\mathbf {d}_{e,i}$ using a merging function.",
"A range of functions are abailable for merging multiple vectors, while [23] used only max pooling (Equation ).",
"In this study, we explored three further functions: GRU, GRU followed by ReLU ($\\mathbf {d}_{e,i} = \\mathrm {ReLU}(\\mathrm {GRU}(\\mathbf {d}_{e,i}^{\\prime }, \\mathbf {d}_{e,i-1}))$ ) and a function that selects only the latest context, i.e., $\\mathbf {d}_{e,i} = \\mathbf {d}_{e,i}^{\\prime }$ .",
"This comparison clarifies the effect of the accumulation of contexts as the experiments proceededNote that merging functions are not restricted to considering two arguments (a new context and a merged past context) recurrently but can consider all vectors over the whole history $[\\mathbf {d}_{e,1}^{\\prime }, ..., \\mathbf {d}_{e,i}^{\\prime }]$ (e.g., by using attention mechanism [2]).",
"However, for simplicity, this research focuses only on the case of a function with two arguments..",
"The merging function produces the dynamic representation $\\mathbf {d}_{e,i}$ of $e$ .",
"In language modeling, to read the $(i+1)$ -th sentence, the model uses two dynamic word embeddings of $e$ in the input and output layers.",
"The input embedding $\\mathbf {x}_{e}$ , used to encode contexts (Equation ), and the output embedding $\\mathbf {y}_{e}$ , used to predict the occurrence of $e$ (Equation REF ), are replaced with dynamic versions: $\\mathbf {x}_{e} & = W_{dx}\\mathbf {d}_{e,i} + \\mathbf {b}_{e}^x ,\\\\\\mathbf {y}_{e} & = W_{dy}\\mathbf {d}_{e,i} + \\mathbf {b}_{e}^y ,$ where $W_{dx}$ and $W_{dy}$ denote learnable matrices, and $\\mathbf {b}_{e}^x$ and $\\mathbf {b}_{e}^y$ denote learnable vectors tied to $e$ .",
"We can observe that a conventional RNN language model is a variant that removes the dynamic terms ($W_{dx}\\mathbf {d}_{e,i}$ and $W_{dy}\\mathbf {d}_{e,i}$ ) using only the static terms ($\\mathbf {b}_{e}^x$ and $\\mathbf {b}_{e}^y$ ) to represent $e$ .",
"The initial dynamic representation $\\mathbf {d}_{e,0}$ is defined as a zero vector, so that the initial word embeddings ($\\mathbf {x}_{e}$ and $\\mathbf {y}_{e}$ ) are identical to the static terms ($\\mathbf {b}_{e}^x$ and $\\mathbf {b}_{e}^y$ ) until the point at which the first context of the target word $e$ is observed.",
"All parameters in the end-to-end model are learned entirely by backpropagation, maximizing the log-likelihood in the same way as a conventional RNN language model.",
"We can view the approach in [23] as a variant on the proposed method, but using the dynamic terms only in the input layer (for $\\mathbf {x}_{e}$ ).",
"We can also view the copy mechanism [13], [14] as a variant on the proposed method, in which specific embeddings in the output layer are replaced with special dynamic vectors."
],
[
"Anonymized Language Modeling",
"This study explores methods for on-the-fly capture and exploitation of the meanings of unknown words or entities in a discourse.",
"To do this, we introduce a novel evaluation task and dataset that we called Anonymized Language Modeling.",
"Figure REF gives an example from the dataset.",
"Briefly, the dataset anonymizes certain noun phrases, treating them as unknown words and retaining their coreference relations.",
"This allows a language model to track the context of every noun phrase in the discourse.",
"Other words are left unchanged, allowing the language model to preserve the context of the anonymized (unknown) words, and to infer their meanings from the known words.",
"The process was inspired by [17], whose approach has been explored by the research on reading comprehension.",
"More precisely, we used the OntoNotes [33] corpus, which includes documents with coreferences and named entity tags manually annotated.",
"We assigned an anonymous identifier to every coreference chain in the corpusWe used documents with no more than 50 clusters, which covered more than 97% of the corpus.",
"in order of first appearanceFollowing the study of [26], we assigned “|<|unk1|>|”, “|<|unk2|>|”, ... to coreference clusters in order of first appearance., and replaced mentions of a coreference chain with its identifier.",
"In our experiments, each coreference chain was given a dynamic representation.",
"Following [29], we limited the vocabulary to 10,000 words appearing frequently in the corpus.",
"Finally, we inserted “<bos>” and “<eos>” tokens to mark the beginning and end of each sentence.",
"An important difference between this dataset and the one presented in [17] is in the way that coreferences are treated.",
"[17] used automatic resolusion of coreferences, whereas our study made use of the manual annotations in the OntoNotes.",
"Thus, the process of [17] introduced (intentional and unintentional) errors into the dataset.",
"Additionally, the dataset did not assign an entity identifier to a pronoun.",
"In contrast, as our dataset has access to the manual annotations of coreferences, we are able to investigate the ability of the language model to capture meanings from contexts.",
"Dynamic updating could be applied to words in all lexical categories, including verbs, adjectives, and nouns without requiring additional extensions.",
"However, verbs and adjectives were excluded from targets of dynamic updates in the experiments, for two reasons.",
"First, proper nouns and nouns accounted for the majority (70%) of the low-frequency (unknown) words, followed by verbs (10%) and adjectives (9%).",
"Second, we assumed that the meaning of a verb or adjective would shift less over the course of a discourse than that of a noun.",
"When semantic information of unknown verbs and adjectives is required, their embeddings may be extracted from ad-hoc training on a different larger corpus.",
"This, however, was beyond the scope of this study.",
"Table: Statistics of Anonymized Language Modeling dataset."
],
[
"Setting",
"An experiment was conducted to investigate the effect of Dynamic Neural Text Model on the Anonymized Language Modeling dataset.",
"The split of dataset followed that of the original corpus [33].",
"Table REF summarizes the statistics of the dataset.",
"The baseline model was a typical LSTM RNN language model with 512 units.",
"We compared three variants of the proposed model, using different applications of dynamic embedding: in the input layer only (as in [23]), in the output layer only, and in both the input and output layers.",
"The context encoders were bidirectional LSTMs with 512 units, the parameters of which were not the same as those in the LSTM RNN language models.",
"All models were trained by maximizing the likelihood of correct tokens, to achieve best perplexity on the validation datasetWe performed a validation at the end of every half epoch out of five epochs..",
"Most hyper-parameters were tuned and fixed by the baseline model on the validation datasetBatchsize was 8.",
"Adam [22] with learning rate $10^{-3}$ .",
"Gradients were normalized so that their norm was smaller than 1.",
"Truncation of backpropagation and updating was performed after every 20 sentences and at the end of document..",
"It is difficult to adequately train the all parts of a model using only the small dataset of Anonymized Language Modeling.",
"We therefore pretrained word embeddings and $\\mathrm {ContextEncoder}$ (the bi-directional RNNs and matrices in Equations REF –) on a sentence completion task in which clozes were predicted from the surrounding words in a large corpus [27]We pretrained a model on the Gigaword Corpus, excluding sentences with more than 32 tokens.",
"We performed training for 50000 iterations with a batch size of 128 and five negative samples.",
"Only words that occurred no fewer than 500 times are used; other words were treated as unknown tokens.",
"[27] used three different sets of word embeddings for the two inputs with respect to the encoders ($\\operatornamewithlimits{\\overrightarrow{\\mathrm {RNN}}}$ and $\\operatornamewithlimits{\\overleftarrow{\\mathrm {RNN}}}$ ) and the output (target).",
"However, we forced the sets of word embeddings to share a single set of word embeddings in pretraining.",
"We initialized the word embeddings in both the input layer ($\\mathbf {x}_w$ ) and the output layer ($\\mathbf {y}_w$ ) of the novel models, including the baseline model, with this single set.",
"The word embeddings of all anonymized tokens were initialized as unknown words with the word embedding of “|<|unk|>|”.. We used the objective function with negative sampling [30]: $\\sum _e (\\log \\sigma ({\\hat{\\mathbf {x}}_{e}}^{\\intercal } {\\mathbf {x}_{e}}) +\\sum _{v\\in Neg}(\\log \\sigma ( -{\\hat{\\mathbf {x}}_{e}}^{\\intercal } {\\mathbf {x}_{v}} )))$ .",
"Here, $\\hat{\\mathbf {x}}_e$ is a context vector predicted by $\\mathrm {ContextEncoder}$ , $\\mathbf {x}_e$ denotes the word embedding of a target word $e$ appearing in the corpus, and $Neg$ represents randomly sampled words.",
"These pretrained parameters of $\\mathrm {ContextEncoder}$ were fixed when the whole language model was trained on the Anonymized Language Modeling dataset.",
"We implemented models in Python using the Chainer neural network library [38].",
"The code and the constructed dataset are publicly availablehttps://github.com/soskek/dynamic_neural_text_model."
],
[
"Perplexity",
"Table REF shows performance of the baseline model and the three variants of the proposed method in terms of perplexity.",
"The table reports the mean and standard error of three perplexity values after training using three different randomly chosen initializations (we used the same convention throughout this paper).",
"Here, we discuss the proposed method using GRU followed by ReLU as the merging function, as this achieved the best perplexity (see Section REF for a comparison of functions).",
"We also show perplexitiy values when evaluating words of specific categories: (1) all words; (2) reappearing entity words; (3) words following entities; and (4) non-entity words.",
"All variants of the proposed method outperformed the baseline model.",
"Focusing on the categories (2) and (3) highlights the roles of dynamic updates of the input and output layers.",
"Dynamic updates of the input layer (B) had a larger improvement for predicting words following entities (3) than those of the output layer (C).",
"In contrast, dynamic updates of the output layer (C) were quite effective for predicting reappearing entities (2) whereas those of the input layer (B) were not.",
"These facts confirm that: dynamic updates of the input layer help a model predict words following entities by supplying on-the-fly context information; and those of the output layer are effective to predict entity words appearing multiple times.",
"In addition, dynamic updates of both the input and output layers (D) further improved the performance from those of either the output (C) or input (B) layer.",
"Thus, the proposed dynamic output was shown to be compatible with dynamic input, and vice versa.",
"These results demonstrated the positive effect of capturing and exploiting the context-sensitive meanings of entities.",
"In order to examine whether dynamic updates of the input and output embeddings capture context-sensitive meanings of entities, we present Figures REF , REF and REF .",
"Figure REF depicts the perplexity of words with different positions in a documentIt is more difficult to predict tokens appearing latter in a document because the number of new (unknown) tokens increases as a model reads the document..",
"The figure confirms that the advantage of the proposed method over the baseline is more evident especially in the latter part of documents, where repeated words are more likely to occur.",
"Figure REF shows the perplexity with respect to the frequency of words $t$ within documents.",
"Note that the word embedding at the first occurrence of an entity is static.",
"This figure indicates that entities appearing many times enjoy the benefit of the dynamic language model.",
"Figure REF visualizes the perplexity of entities with respect to the numbers of their antecedent candidates.",
"It is clear from this figure that the proposed method is better at memorizing the semantic information of entities appearing repeatedly in documents than the baseline.",
"These results also demonstrated the contribution of dynamic updates of word embeddings.",
"Figure: Perplexity of all tokens relative to the time at which they appear in the document.Figure: Perplexity of tokens following the entities relative to the time at which the entity occurs.Figure: Perplexity of entities relative to the number of antecedent entities.Table: Results for models with different merging functions on the test set of the Anonymized Language Modeling dataset, as same as in Table 2."
],
[
"Comparison of Merging functions",
"Table REF compares models with different merging functions; GRU-ReLU, GRU, max pooling, and the use of the latest context.",
"The use of the latest context had the worst performance for all variants of the proposed method.",
"Thus, a proper accumulation of multiple contexts is indispensable for dynamic updates of word embeddings.",
"Although [23] used only max pooling as the merging function, GRU and GRU-ReLU were shown to be comparable in performance and superior to max pooling when predicting tokens related to entities (2) and (3)."
],
[
"Predicting Entities by Likelihood of a Sentence",
"In order to examine contribution of the dynamic language models on a downstream task, we conducted cloze tests for comprehension of a sentence with reappearing entities in a discourse.",
"Given multiple preceding entities $E = \\lbrace e^{+}, e^{1}, e^{2}, ...\\rbrace $ followed by a cloze sentence, the models were required to predict the true antecedent $e^{+}$ which allowed the cloze to be correctly filled, among the other alternatives $E^{-} = \\lbrace e^{1}, e^{2}, ...\\rbrace $ .",
"Language models solve this task by comparing the likelihoods of sentences filled with antecedent candidates in $E$ and returning the entity with the highest likelihood of the sentence.",
"In this experiment, the performance of a model was represented by the Mean Quantile (MQ) [15].",
"The MQ computes the mean ratio at which the model predicts a correct antecedent $e^{+}$ more likely than negative antecedents in $E^{-}$ , $\\mathrm {MQ} & = \\frac{|\\lbrace e^{-} \\in E^{-}: p(e^{-}) < p(e^{+})\\rbrace |}{|E^{-}|} .",
"$ Here, $p(e)$ denotes the likelihood of a sentence whose cloze is filled with $e$ .",
"If the correct antecedent $e^{+}$ yields highest likelihood, MQ gets 1.",
"Table REF reports MQs for the three variants and merging functions.",
"Dynamic updates of the input layer greatly boosted the performance by approximately 10%, while using both dynamic input and output improved it further.",
"In this experiment, the merging functions with GRUs outperform the others.",
"These results demonstrated that Dynamic Neural Text Models can accumulate a new information in word embeddings and contribute to modeling the semantic changes of entities in a discourse.",
"Table: Mean Quantile of a true coreferent entity among antecedent entities."
],
[
"Related Work",
"An approach to addressing the unknown word problem used in recent studies [21], [35], [25], [34] comprises the embeddings of unknown words from character embeddings or subword embeddings.",
"[24] applied word disambiguation and use a sense embedding to the target word.",
"[6] captured the context-sensitive meanings of common words using word embeddings, applied through a gating function controlled by history words, in the context of machine translation.",
"In future work, we will explore a wider range of models, to integrate our dynamic text modeling with methods that estimate the meaning of unknown words or entities from their constituents.",
"When addressing well-known entities such as Obama and Trump, it makes sense to learn their embeddings from external resources, as well as dynamically from the preceding context in a given discourse (as in our Dynamic Neural Text Model).",
"The integration of these two sources of information is an intriguing challenge in language modeling.",
"A key aspect of our model is its incorporation of the copy mechanism [13], [14], using dynamic word embeddings in the output layer.",
"Independently of this study, several research groups have explored the use of variants of the copy mechanisms in language modeling [28], [12], [32].",
"These studies, however, did not incorporate dynamic representations in the input layer.",
"In contrast, our proposal incorporates the copy mechanism through the use of dynamic representations in the output layer, integrating them with dynamic mechanisms in both the input and output layers by applying dynamic entity-wise representation.",
"Our experiments have demonstrated the benefits of such integration.",
"Another related trend in recent studies is the use of neural network to capture the information flow of a discourse.",
"One approach has been to link RNNs across sentences [41], [36], while a second approach has expolited a type of memory space to store contextual information [37], [39], [28].",
"Research on reading comprehension [23], [16] and coreference resolution [43], [8], [7] has shown the salience of entity-wise context information.",
"Our model could be located within such approaches, but is distinct in being the first model to make use of entity-wise context information in both the input and output layers for sentence generation.",
"We summarize and compare works for entity-centric neural networks that read a document.",
"[23] pioneered entity-centric neural models tracking states in a discourse.",
"They proposed Dynamic Entity Representation, which encodes contexts of entities and updates the states using entity-wise memories.",
"[43] also proposed a method for managing similar entity-wise features on neural networks and improved a coreference resolution model.",
"[8], [7] incorporated such entity-wise representations in mention-ranking coreference models.",
"Our paper follows [23] and exploits dynamic entity reprensetions in a neural language model, where dynamic reporesentations are used not only in the neural encoder but also in the decoder, applicable to various sequence generation tasks, e.g., machine translation and dialog response generation.",
"Simultaneously with our paper, [19] use dynamic entity representation in a neural language model for reranking outputs of a coreference resolution system.",
"[44] experiment language modeling with referring to internal contexts or external data.",
"[16] focus on neural networks tracking contexts of entities, achieving the state-of-the-art result in bAbI [42], a reading comprehension task.",
"They encode the contexts of each entity by an attention-like gated RNN instead of using coreference links directly.",
"[10] also try to improve a reading comprehension model using coreference links.",
"Similarly to our dynamic entity representation, [1] construct on-the-fly word embeddings of rare words from dictionary definitions.",
"The first key component of dynamic entity representation is a function to merge more than one contexts about an entity into a consistent representation of the entity.",
"Various choices for the function exist, e.g., max or average-pooling [23], [8], RNN (GRU, LSTM [43], [44] or other gated RNNs [16], [19]), or using the latest context only (without any merging) [44].",
"This paper is the first work comparing the effects of those choices (see Section REF ).",
"The second component is a function to encode local contexts from a given text, e.g., bidirectional RNN encoding [23], unidirectional RNN used in a language model [19], [44], feedforward neural network with a sentence vector and an entity's word vector [16] or hand-crafted features with word embeddings [43], [8].",
"This study employs bi-RNN analogously to [23], which can access full context with powerful learnable units.",
"In the task setting proposed in this study, a model must capture the meaning of a given specific word from a small number of its contexts in a given discourse.",
"The task could also be seen as novel one-shot learning [11] of word meanings.",
"One-shot learning for NLP like this has been little studied, with the exception of the study by [40], which used a task in which the context of a target word is matched with a different context of the same word."
],
[
"Conclusion",
"This study addressed the problem of identifying the meaning of unknown words or entities in a discourse with respect to the word embedding approaches used in neural language models.",
"We proposed a method for on-the-fly construction and exploitation of word embeddings in both the input layer and output layer of a neural model by tracking contexts.",
"This extended the dynamic entity representation presented in [23], and incorporated a copy mechanism proposed independently by [13] and [14].",
"In the course of the study, we also constructed a new task and dataset, called Anonymized Language Modeling, for evaluating the ability of a model to capture word meanings while reading.",
"Experiments conducted using our novel dataset demonstrated that the RNN language model variants proposed in this study outperformed the baseline model.",
"More detailed analysis indicated that the proposed method was particularly successful in capturing the meaning of an unknown words from texts containing few instances."
],
[
"Acknowledgments",
"This work was supported by JSPS KAKENHI Grant Number 15H01702 and JSPS KAKENHI Grant Number 15H05318.",
"We thank members of Preferred Networks, Inc., Makoto Miwa and Daichi Mochihashi for suggestive discussions."
]
] | 1709.01679 |
[
[
"A pointing solution for the medium size telescopes for the Cherenkov\n Telescope Array"
],
[
"Abstract The pointing capability of a telescope in the Cherenkov Telescope Array (CTA) is a crucial aspect in the calibration of the instrument.",
"It describes how a position in the sky is transformed to the focal plane of the telescope and allows precise directional reconstructions of atmospheric particle showers.",
"The favoured approach for pointing calibrations of the Medium Size Telescopes (MST) is the utilisation of an CCD-camera installed in the centre of the dish, which images the night sky and the focal plane simultaneously.",
"The technical implementation of this solution and test results taken over a period of one year at the MST prototype in Berlin/Adlershof are presented.",
"Investigations of pointing calibration precision with simulated data and real data taken during test runs of the prototype telescope will also be shown."
],
[
"Introduction",
"CTA, the next generation ground based gamma-ray telescope, is currently in a pre-construction phase.",
"It will consist of three different sizes of Imaging Atmospheric Cherenkov Telescopes (IACT) to detect Cherenkov light from gamma-ray induced air-showers in a wide energy range.",
"This light is collected by mirror dishes and projected onto specialized Cherenkov cameras.",
"For the reconstruction of the arrival direction of the original gamma-ray photon it is important to know the exact orientation of the telescope with respect to the sky and the alignment of the Cherenkov camera with respect to the telescope's optical axis at the time of its detection.",
"This geometric configuration is also called the pointing of a telescope.",
"The favoured approach for the calibration of the pointing of the MST uses an optical CCD-camera (the SingleCCD) that is mounted in the centre of the mirror dish of the telescope and images the Cherenkov camera and the night sky around it simultaneously.",
"The Single-CCD concept has also been successfully tested by the H.E.S.S.",
"collaboration [1]."
],
[
"Single-CCD Concept",
"The pointing calibration is performed in two steps.",
"In the first step, the telescope is pointed at different positions in the night sky during special pointing runs.",
"In these runs, a screen in front of the Cherenkov camera is used to see light spots from bright stars.",
"These spots are captured by the SingleCCD, and their positions on the screen for a given telescope alignment are used for the derivation of a pointing model for the telescope.",
"This technique has been used successfully e.g.",
"by the H.E.S.S.",
"collaboration [2].",
"The second step is performed during data taking, while the Cherenkov camera is operative.",
"The SingleCCD now images the spots of LEDs installed on the Cherenkov camera housing surrounding the centre of the focal plane and the stars behind the Cherenkov camera.",
"The positions of the LEDs can be used to determine the position of the Cherenkov camera with respect to the optical axis while the positions of the stars can be used to deduce the orientation of the telescope on the sky.",
"With this information the previously derived pointing model can be refined and short term deviations can be corrected.",
"An alternative approach utilizes two separate CCD-cameras: A LidCCD, which images reflected star spots and positioning LEDs on the Cherenkov camera, and a SkyCCD which tracks star positions for pointing corrections during data taking.",
"The downsides of this method are an increased complexity and uncertainties in the stability of the relative orientation between the two cameras.",
"The advantage is a possible increase of precision in the determination of the pointing due to a smaller field of view of the SkyCCD compared to the SingleCCD."
],
[
"Camera Hardware",
"For the SingleCCD camera an Apogee ASPEN CG8050 with electronic shutter, chip dimensions 3296 $\\times $ 2472 pixels, and pixel size 5.5 µm is used.",
"With a 50 mm f/1.8 Nikkor lens, images with a field-of-view of 20.5 $\\times $ 15.5 deg$^2$ can be taken.",
"In these images, a pixel diameter corresponds to about 22” in the sky.",
"The CCD chip of the camera can be cooled to a constant temperature via an integrated Peltier element to avoid thermal expansion and to preserve a constant imaging geometry.",
"The camera data is read out via a standard Ethernet interface.",
"Figure: Schematic view of the SingleCCD camera in its casing.",
"The rigid support structure (focus fixing) and heat conducting elements (contact disk, contact shoes, cooling fins, heat sink, and hollow cylinder) are highlighted.",
"The heating element for the heat sink is also shown.The camera is mounted inside a custom-made casing that shields the electronics and optics from environmental impacts (see Figure REF ).",
"The housing is fully IP67 compliant and connects the SingleCCD rigidly with the telescope dish.",
"It comprises several heat conducting structures that grant a good heat dissipation from the camera to the environment.",
"A heating element is installed to the front of the casing to keep the front window free of ice at low temperatures.",
"Another heating element at the backside of the housing assures that the heat sink temperature stays above the desired CCD temperature as the camera internal electronics do not support chip heating.",
"This is the second version of the casing prototype.",
"It is currently in construction and will replace the currently used version, which has been introduced in [3], at the prototype telescope."
],
[
"Precision Studies",
"The images taken by the SingleCCD are analysed with the software astrometry.net [4].",
"This calibration software calculates a world coordinate system (WCS) for a given image of the night sky.",
"This information is then used to derive the absolute pointing position of the telescope.",
"One challenge of the SingleCCD approach is that a large area in the pointing images is occupied by the Cherenkov telescope camera and its support structure, in particular the central part of the images, where the pointing direction of the telescope is positioned (see Figure REF left).",
"The fact that no stars can be detected in large areas of the images separates this application of astrometry.net from its standard use case.",
"In order to determine the impact of this shadowing effect on the precision of the pointing measurement, a simulation software for SingleCCD images has been developed.",
"Figure: (Left:) SingleCCD image taken at the MST prototype in Berlin Adlershof.",
"Colors are inverted.",
"Red contour marks the area which is obstructed by the telescope structure.",
"(Right:) Distributions of mean WCS residuals for images taken at the MST prototype (black histogram) and simulated images (red histogram).",
"Blue plus signs indicate the corresponding pixel scales.",
"See text for explanations.In [3] it has been shown that the pointing of simulated images shadowed by a square-shaped Cherenkov camera can be reconstructed to a precision of $\\sim 1$ arcsecond.",
"The simulation is now used to reproduce images that have been taken in a test run during one night at the prototype telescope with the full camera support structure.",
"Both real images and simulated images are analysed with astrometry.net.",
"For each identified star in an image, the residual angular difference between expected position and calculated WCS position is determined.",
"In Figure REF right, the distribution of the mean residual is shown both for simulations and for real images.",
"The very good correspondence between simulation and real data indicates that simulation and imaging hardware work as expected and that effects like distortions which are not implemented in the simulation do not affect the calibration of real images.",
"It can be seen that there are two separate classes of solutions.",
"For about 25% of the image calibrations, the residuals become quite large (> 0.5').",
"These are suspected to originate from local minima that are reached by the WCS fit routine.",
"The larger class of calibrations show mean residuals in the range of a few arcseconds (< 20”).",
"The two classes can also be discriminated via the derived pixel scale of the WCS.",
"This scale specifies the angular correspondence of a pixel diameter in an image.",
"While for the higher precision calibrations, this quantity is close to the correct value, it fluctuates strongly for the lower precision fits.",
"These two classes, which also show up for simulated images, indicate that there are still some possibilities to optimize the image calibration procedure.",
"It is currently under investigation how the WCS fit can be improved further.",
"Moreover, the order of magnitude of the observed mean WCS residuals is equal to that of the residuals of first derived pointing models (cf. Sect. )",
"which can be taken as a hint that both originate from the same imprecision in the fit."
],
[
"Pointing Models",
"For MSTs the pointing precision needs to be such that any position in the focal plane can be mapped to a sky position with an accuracy of better than $7^{^{\\prime \\prime }}$ (space angle).",
"To stay within this error budget the pointing accuracy of the SingleCCD alone must be better than $7^{^{\\prime \\prime }}$ .",
"Using data recorded at the MST prototype in Berlin, pointing models for the SingleCCD have been derived in order to study the resulting precision and the time evolution of the model parameters.",
"The input data for the calculation of pointing models were obtained in clear nights where the MST tracked about 100 sky positions at elevations greater than $45^{\\circ }$ .",
"The observations were conducted in a robotic fashion and took several hours since 2.5 min were needed to slew to a new position and to track it while the SingleCCD was read out.",
"The exposure of the SingleCCD was set to 20 s; the resulting images were stored in FITS format along with the time evolution of the azimuth and elevation targeted by the drive system (az$_D$ , el$_D$ ).",
"All measurements were conducted in direction of either continuously increasing or decreasing elevation to test for hysteresis effects.",
"The images were solved using the astrometry.net software to find the equatorial coordinate of the center of the SingleCCD FoV; the 90% of the images for which a good-quality solution (cf. Sect. )",
"could be found were then used to derive a pointing model.",
"To this end, the time $t_0$ corresponding to the centre of the exposure interval was estimated and used to derive the coordinate targeted by the drive system (az$_D(t_0)$ ,el$_D(t_0)$ ).",
"In a similar way, the found equatorial coordinate was converted to obtain the azimuth and elevation (az$_C(t_0)$ , el$_C(t_0)$ ) that corresponded to the center of the FoV of the SingleCCD at time $t_0$ .",
"A pointing model is defined by describing the difference in elevation and azimuth, $\\mbox{el}_D - \\mbox{el}_C = f_1(\\mbox{el}_C,\\mbox{az}_C,\\vec{q}) \\mbox{\\ \\ and\\ \\ }\\mbox{az}_D - \\mbox{az}_C = f_2(\\mbox{el}_C,\\mbox{az}_C,\\vec{q}) \\mbox{,}$ by suitable functions $f_1$ and $f_2$ that depend on a vector of adjustable parameters ($\\vec{q}$ ).",
"The best parameter values were found by minimizing the angular distance between the predicted and the measured az$_D$ and el$_D$ positions over all images recorded in one night.",
"Figure: Results of pointing studies for a SingleCCD mounted on the MST prototypein Berlin Adlershof.",
"(Left:) Time evolution of the 11 parameters of thefitted pointing models.",
"The parameters values (in degrees) are shown vsthe parameter number (0,...,100,\\ldots ,10).",
"The colors denote four different parametersets derived for nights in a period of 2.5 months.",
"Note that thevalues have been slightly shifted horizontally for better visibility.",
"(Middle, Right:) Distributions of the residuals between measured and predicted elevation andazimuth positions.",
"See text for explanations.The employed pointing model had 11 parameters, the most basic ones being constant offsets in azimuth and elevation along with tilts of the azimuth axis in the East-West ($\\Theta _{EW}$ ) and the North-South ($\\Theta _{NS}$ ) direction, respectively.",
"Figure REF (left) shows the time evolution of the 11 parameters over a period of 2.5 months.",
"It is evident that the parameter values are fairly stable; it was also found that the SingleCCD shows no sagging with elevation and that the values of the tilt parameters ($\\Theta _{EW}$ and $\\Theta _{NS}$ ) agree with values derived for a SkyCCD that exclusively observes stars in the sky.",
"Figure REF also presents the residuals between model and prediction in elevation (middle) and azimuth (right).",
"Both distribution are centred at zero, demonstrating that model describes the data on average.",
"The width of the two distributions is, however, at the level of $20^{^{\\prime \\prime }}$ .",
"This implies that the current per-axis error exceeds the MST pointing precision that is aimed for.",
"It is currently under study in how far the broadening of the residuals is due to environmental effects (e.g.",
"the enhanced night-sky background levels in Berlin Adlershof), the telescope structure and drives (e.g.",
"vibrations and tracking deviations), the SingleCCD hardware, or the analysis procedure (e.g.",
"the determination of equatorial coordinates by the astrometry.net software)."
],
[
"Conclusion",
"Results from one year of testing at the MST prototype show that the SingleCCD pointing solution can be used for calibrations and to derive pointing models.",
"The residuals of the calculated pointing models imply that the desired precision is not yet fully reached.",
"Detailed studies give a hint that the algorithm of the used astrometry.net WCS fit routine might be one reason for this imprecision but environmental influences e.g.",
"from the high night sky background in Berlin can not be ruled out yet.",
"Further tests are still needed to improve the method and to show that it can be applied in the final state for the MSTs in CTA."
],
[
"Acknowledgments",
"This work was conducted in the context of the CTA MST Structure Working Group.",
"We gratefully acknowledge financial support from the agencies and organizations listed here: http://www.cta-observatory.org/consortium_acknowledgments.",
"This work was supported by the German Ministry of Education and Research under grant identifier 05A14WE2."
]
] | 1709.01811 |
[
[
"Modeling Quantum Behavior in the Framework of Permutation Groups"
],
[
"Abstract Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group.",
"Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation.",
"Thus, quantum-mechanical problems can be expressed in terms of permutation groups.",
"This approach allows us to clarify the meaning of a number of physical concepts.",
"Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups."
],
[
"Introduction",
"Since the time of Newton, differential calculus demonstrates high efficiency in describing physical phenomena.",
"However, infinitesimal analysis introduces infinities in physical theories.",
"This is often considered as a serious conceptual flaw: recall, for example, Dirac's frequently quoted claim that the most important challenge in physics is “to get rid of infinity”.",
"Moreover, differential calculus, being, in fact, a kind of approximation, may lead to descriptive losses in some problems — an illustrative example is given below in Sect.",
"REF .",
"In the paper, we describe a constructive version of quantum formalism that does not involve any concepts associated with actual infinities.",
"The main part of the paper starts with Sect.",
", which contains a summary of the basic concepts of the standard quantum mechanics with emphasis on the aspects important for our purposes.",
"Sect.",
"describes a constructive modification of the quantum formalism.",
"We start with replacing a continuous group of symmetries of quantum states by a finite group.",
"The natural consequence of this replacement is unitarity, since any linear representation of a finite group is unitary.",
"Further, any finite group is naturally associated with some cyclotomic field.",
"Generally, a cyclotomic field is a dense subfield of the field of complex numbers.",
"This can be regarded as an explanation of the presence of complex numbers in the quantum formalism.",
"Any linear representation of a finite group over the associated cyclotomic field can be obtained from a permutation action of the group on vectors with natural components by projecting into suitable invariant subspace.",
"All this allows us to reproduce all the elements of quantum formalism in invariant subspaces of permutation representations.",
"In Sect.",
"we consider a model of quantum evolution inspired by the quantum Zeno effect — the most convincing manifestation of the role of observation in the dynamics of quantum systems.",
"The model represents the quantum evolution as a sequence of observations with unitary transitions between them.",
"Standard quantum mechanics assumes a single deterministic unitary transition between observations.",
"In our model we generalize this assumption.",
"We treat a unitary transition as a kind of gauge connection — a way of identifying indistinguishable entities at different times.",
"A priori, any unitary transformation can be used as a data identification rule.",
"So, we assume that all unitary transformations participate in transitions between observations with appropriate weights.",
"We call a unitary evolution dominant if it provides the maximum transition probability.In fact, the principle of least action in physical theories implies the selection of dominant evolutions among all possible (“virtual”) evolutions.",
"The apparent determinism of these evolutions can be explained by the sharpness of their dominance.",
"The Monte Carlo simulation shows a sharp dominance of such evolutions over other evolutions.",
"To compare with a continuous description, we present also the Lagrangian of the continuum approximation of the model."
],
[
"Formalism of quantum mechanics",
"Here is a brief outline of the basic concepts of quantum mechanics.",
"We divide these concepts into three categories: states, observations and measurements, and time evolution."
],
[
"States",
"$~~~~$ A pure quantum state is a ray in a Hilbert space $\\color {black}{}\\mathcal {H}$ over the complex field $\\color {black}{}\\mathbb {C}$ , i.e.",
"an equivalence class of vectors $\\color {black}{}\\left|\\psi \\right\\rangle \\in \\mathcal {H}$ with respect to the equivalence relation $\\color {black}{}\\left|\\psi \\right\\rangle \\sim {}a\\left|\\psi \\right\\rangle $ , where $\\color {black}{}a\\in \\mathbb {C},~a\\ne 0$ .",
"We can reduce the equivalence classes by normalization: $\\color {black}{}\\left|\\psi \\right\\rangle \\sim {}\\operatorname{e}^{\\mathrm {i}\\alpha }\\left|\\psi \\right\\rangle , \\left\\Vert \\psi \\right\\Vert =1, \\alpha \\in \\mathbb {R}$ .",
"Finally, we can eliminate the phase “degree of freedom” $\\color {black}{}\\alpha $ by transition to the rank one projector $\\color {black}{}\\Pi _{\\psi }=\\left|\\psi \\right\\rangle \\!\\left\\langle \\psi \\right|$ , which is a special case of a density matrix.",
"$~~~~$ A mixed quantum state is described by a general density matrix $\\color {black}{}{\\rho }$ characterized by the properties: (a) $\\color {black}{}{\\rho }={\\rho }^\\dagger $ , (b) $\\color {black}{}\\left\\langle \\psi \\left|{\\rho }\\right|\\psi \\right\\rangle \\ge 0$ for any $\\color {black}{}\\left|\\psi \\right\\rangle \\in \\mathcal {H}$ , (c) $\\color {black}{}\\operatorname{tr}{\\rho }=1$ .",
"In fact, any mixed state is a weighted mixture of pure states, i.e.",
"its density matrix can be represented as a weighted sum of the rank one projectors.",
"We will denote the set of all density matrices by $\\color {black}{}\\mathcal {D}\\!\\left(\\mathcal {H}\\right)$ .",
"$~~~~$ The Hilbert space of a composite system, $\\color {black}{}XY=X\\times {}Y$ , is the tensor product of the Hilbert spaces for the constituents: $\\color {black}{}\\mathcal {H}_{{XY}}=\\mathcal {H}_{{X}}{\\operatornamewithlimits{\\text{\\raisebox {0.3ex}{${\\bigotimes }$}}}}\\mathcal {H}_{{Y}}$ .",
"The states of composite system, $\\color {black}{}\\mathcal {D}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ , are classified into two types: separable and entangled states.",
"The set of separable states, $\\color {black}{}\\mathcal {D}_\\mathrm {S}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ , consists of the states $\\color {black}{}{\\rho }_{XY}\\in \\mathcal {D}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ that can be represented as weighted sums of the tensor products of states of the constituents: $\\color {black}{}{\\rho }_{{XY}}=\\sum _{k}w_k{\\rho }_{{X}}^k{\\operatornamewithlimits{\\text{\\raisebox {0.3ex}{${\\bigotimes }$}}}}{\\rho }_{{Y}}^k,~~w_k\\ge 0,~~\\sum _{k}w_k=1$ .",
"The set of entangled states, $\\color {black}{}\\mathcal {D}_\\mathrm {E}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ , is by definition the complement of $\\color {black}{}\\mathcal {D}_\\mathrm {S}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ in the set of all states: $\\color {black}{}\\mathcal {D}_\\mathrm {E}\\!\\left(\\mathcal {H}_{{XY}}\\right)=\\mathcal {D}\\!\\left(\\mathcal {H}_{{XY}}\\right)\\setminus \\mathcal {D}_\\mathrm {S}\\!\\left(\\mathcal {H}_{{XY}}\\right)$ ."
],
[
"Observations and measurements",
"The terms `observation' and `measurement' are often used as synonyms.",
"However, it makes sense to separate these concepts: we treat observation as a more general concept which does not imply, in contrast to measurement, obtaining numerical information.",
"$~~~~$Observation is the detection (“click of detector”) of a system, that is in the state $\\color {black}{}{\\rho }$ , in the subspace $\\color {black}{}\\mathcal {S}\\le \\mathcal {H}$ .",
"The mathematical abstraction of the “detector in the subspace” $\\color {black}{}\\mathcal {S}$ of a Hilbert space is the operator of projection, $\\color {black}{}\\Pi _{\\mathcal {S}}$ , into this subspace.",
"The result of quantum observation is random and its statistics is described by a probability measure defined on subspaces of the Hilbert space.",
"Any such measure $\\color {black}{}\\mu \\left(\\cdot \\right)$ must be additive on any set of mutually orthogonal subspaces of a Hilbert space: if, e.g., $\\color {black}{}\\mathcal {A}$ and $\\color {black}{}\\mathcal {B}$ are mutually orthogonal subspaces, then $\\color {black}{}\\mu \\left(\\operatorname{span}\\left(\\mathcal {A}, \\mathcal {B}\\right)\\right)=\\mu \\left(\\mathcal {A}\\right)+\\mu \\left(\\mathcal {B}\\right)$ .",
"Gleason proved [1] that, excepting the case $\\color {black}{}\\dim \\mathcal {H}=2$ , the only such measures have the form $\\color {black}{}\\mu _{\\rho }\\left(\\mathcal {S}\\right)=\\operatorname{tr}\\left({\\rho }\\Pi _{\\mathcal {S}}\\right)$ , where $\\color {black}{}{\\rho }$ is an arbitrary density matrix.",
"If, in particular, $\\color {black}{}{\\rho }$ describes a pure state, $\\color {black}{}{\\rho }=\\left|\\psi \\right\\rangle \\!\\left\\langle \\psi \\right|$ , and $\\color {black}{}\\mathcal {S}$ is one-dimensional, $\\color {black}{}\\mathcal {S}=\\operatorname{span}\\left(\\left|\\varphi \\right\\rangle \\right)$ , we come to the familiar Born rule: $\\color {black}{}\\operatorname{tr}\\left({\\rho }\\Pi _{\\mathcal {S}}\\right)=\\mathrm {\\mathbf {P}}_{\\text{Born}}=\\left|\\left\\langle \\varphi \\mid \\psi \\right\\rangle \\right|^2$ .",
"$~~~~$Measurement is a special case of observation, when the partition of a Hilbert space into mutually orthogonal subspaces is provided by a Hermitian operator $\\color {black}{}A$ .",
"Any such operator can be written as $\\color {black}{}A=\\sum _ka_k\\Pi _{e_k}$ , where $\\color {black}{}a_1, a_2,\\ldots \\in \\mathbb {R}$ is the spectrum of $\\color {black}{}A$ , and $\\color {black}{}e_1, e_2,\\ldots $ is an orthonormal basis of eigenvectors of $\\color {black}{}A$ .",
"“Click of the detector” $\\color {black}{}\\Pi _{e_k}$ is interpreted as that the eigenvalue $\\color {black}{}a_k$ is the result of the measurement.",
"The mean for multiple measurements tends to the expectation value of $\\color {black}{}A$ in the state $\\color {black}{}{\\rho }$ : $\\color {black}{}\\left\\langle {A}\\right\\rangle _{\\!",
"{\\rho }}=\\operatorname{tr}\\left({\\rho }{A}\\right)$ ."
],
[
"Time evolution",
"$~~~~$ The time evolution of a quantum system is a unitary transformation of data between observations.",
"For a density matrix, unitary evolution takes the form $\\color {black}{{\\rho }_{t^{\\prime }}=U_{t^{\\prime }t}{\\rho }_tU_{t^{\\prime }t}^{\\dagger },}$ where $\\color {black}{}{\\rho }_{t}$ is the state after observation at the time $\\color {black}{}t$ , $\\color {black}{}{\\rho }_{t^{\\prime }}$ is the state before observation at the time $\\color {black}{}t^{\\prime }$ , and $\\color {black}{}U_{t^{\\prime }t}$ is the unitary transition between the observation times $\\color {black}{}t$ and $\\color {black}{}t^{\\prime }$ .",
"In standard quantum formalism, time is considered as a continuous parameter, and relation (REF ) becomes the von Neumann equation in the infinitesimal limit.",
"The evolution of a pure state can be written as $\\color {black}{}\\left|\\psi _{t^{\\prime }}\\right\\rangle =U_{t^{\\prime }t}\\left|\\psi _t\\right\\rangle $ , and the corresponding infinitesimal limit is the Schrödinger equation.",
"To emphasize the role of observation in quantum physics, we note that unitary evolution is simply a change of coordinates in Hilbert space and is not sufficient to describe observable physical phenomena."
],
[
"Emergence of geometry within large Hilbert space via entanglement",
"Quantum-mechanical theory does not need a geometric space as a fundamental concept — everything can be formulated using only the Hilbert space formalism.",
"In this view, the observed geometry must emerge as an approximation.",
"The currently popular idea [2], [3], [4] of the emergence of geometry within a Hilbert space is based on the notion of entanglement.",
"Briefly, the scheme of extracting geometric manifold from the entanglement structure of a quantum state $\\color {black}{}{\\rho }$ in a Hilbert space $\\color {black}{}\\mathcal {H}$ is as follows: The Hilbert space decomposes into a large number of tensor factors: $\\color {black}{}\\mathcal {H}=\\operatornamewithlimits{\\text{\\raisebox {0.3ex}{${\\bigotimes }$}}}_x\\mathcal {H}_x,~ x\\in {X}$ .",
"Each factor is treated as a point (or bulk) of geometric space to be built.",
"A graph $\\color {black}{}G$ — called tensor network — with vertices $\\color {black}{}x\\in {X}$ and edges $\\color {black}{}\\left\\lbrace x,y\\right\\rbrace \\in {}X\\times {}X$ is introduced.",
"The edges of $\\color {black}{}G$ are assigned weights based on a measure of entanglement, a function that vanishes on separable states and is positive on entangled states.",
"A typical such measure is the mutual information: $\\color {black}{}{I}\\!\\left({\\rho }_{xy}\\right)=S\\!\\left({\\rho }_{x}\\right)+S\\!\\left({\\rho }_{y}\\right)-S\\!\\left({\\rho }_{xy}\\right)$ , where $\\color {black}{}{\\rho }_{x}$ denotes the result of taking traces of $\\color {black}{}{\\rho }$ over all tensor factors excepting the $\\color {black}{}x$ -th (and similarly for $\\color {black}{}{\\rho }_{x},$ $\\color {black}{}{\\rho }_{xy}$ ); $\\color {black}{}S\\!\\left(a\\right)=-\\operatorname{tr}\\left(a\\log {a}\\right)$ is the von Neumann entropy.",
"The graph $\\color {black}{}G$ is supplied with a metric derived from the weights of the edges.",
"Finally, the graph $\\color {black}{}G$ is approximately isometrically embedded in a smooth metric manifold of as small as possible dimension using algorithms like multidimensional scaling (MDS)."
],
[
"Constructive modification of quantum formalism",
"David Hilbert, a prominent advocate of the free use of the concept of infinity in mathematics, wrote the following about the relation of the infinite to the reality: “Our principal result is that the infinite is nowhere to be found in reality.",
"It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought.” Adopting this view, we reformulate the quantum formalism in constructive finite terms without distorting its empirical content [5], [6], [7]."
],
[
"Losses due to continuum and differential calculus",
"Differential calculus (including differential equations, differential geometry, etc.)",
"forms the basis of mathematical methods in physics.",
"The applicability of differential calculus is based on the assumption that any relevant function can be approximated by linear relations at small scales.",
"This assumption simplifies many problems in physics and mathematics, but at the cost of loss of completeness.",
"As an example, consider the problem of classifying simple groups.",
"The concept of a group is an abstraction of the properties of permutations (also called one-to-one mappings or bijections) of a set.",
"Namely, an abstract group is a set with an associative operation, an identity element, and an invertibility for each element.",
"There are two most common additional assumptions that make the notion of a group more meaningful: (a) the group is a differentiable manifold — such a group is called Lie group; (b) the group is finite.",
"It is clear that empirical physics is insensitive to assumption (b) — ultimately, any empirical description is reduced to a finite set of data.",
"On the contrary, assumption (a) implies severe constraints on possible physical models.",
"The problem of classification of simple groupsSimple groups, i.e.",
"groups that do not contain nontrivial normal subgroups, are “building blocks” for all other groups.",
"under assumption (a) turned out to be rather easy and was solved by two people (Killing and Cartan) in a few years.",
"The result is four infinite series: $\\color {black}{}A_n$ , $\\color {black}{}B_n$ , $\\color {black}{}C_n$ , $\\color {black}{}D_n$ ; and five exceptional groups: $\\color {black}{}E_6$ , $\\color {black}{}E_7$ , $\\color {black}{}E_8$ , $\\color {black}{}F_4$ , $\\color {black}{}G_2$ .",
"The solution of the classification problem under assumption (b) required the efforts of about a hundred people for over a hundred years [8].",
"But the result — “the enormous theorem” — turned out to be much richer.",
"The list of finite simple groups contains $\\color {black}{}16+1+1$ infinite series: groups of Lie type: $\\color {black}{}A_n(q)$ , $\\color {black}{}B_n(q)$ , $\\color {black}{}C_n(q)$ , $\\color {black}{}D_n(q)$ , $\\color {black}{}E_6(q)$ , $\\color {black}{}E_7(q)$ , $\\color {black}{}E_8(q)$ , $\\color {black}{}F_4(q)$ , $\\color {black}{}G_2(q)$ , $\\color {black}{}^2A_n\\left(q^2\\right)$ , $\\color {black}{}^2B_n\\left(2^{2n+1}\\right)$ , $\\color {black}{}^2D_n\\left(q^2\\right)$ , $\\color {black}{}^3D_4\\left(q^3\\right)$ , $\\color {black}{}^2E_6\\left(q^2\\right)$ , $\\color {black}{}^2F_4\\left(2^{2n+1}\\right)$ , $\\color {black}{}^2G_2\\left(3^{2n+1}\\right)$ ; cyclic groups of prime order, $\\color {black}{}\\mathbb {Z}_p$ ; alternating groups, $\\color {black}{}A_n,~n\\ge 5$ ; and $\\color {black}{}26$ sporadic groups: $\\color {black}{}M_{11}$ , $\\color {black}{}M_{12}$ , $\\color {black}{}M_{22}$ , $\\color {black}{}M_{23}$ , $\\color {black}{}M_{24}$ , $\\color {black}{}J_1$ , $\\color {black}{}J_2$ , $\\color {black}{}J_3$ , $\\color {black}{}J_4$ , $\\color {black}{}Co_1$ , $\\color {black}{}Co_2$ , $\\color {black}{}Co_3$ , and $\\color {black}{}26$ sporadic groups :$\\color {black}{}Fi_{22}$ , $\\color {black}{}Fi_{23}$ , $\\color {black}{}Fi_{24}$ , $\\color {black}{}HS$ , $\\color {black}{}McL$ , $\\color {black}{}He$ , $\\color {black}{}Ru$ , $\\color {black}{}Suz$ , $\\color {black}{}O^{\\prime }N$ , $\\color {black}{}HN$ , $\\color {black}{}Ly$ , $\\color {black}{}Th$ , $\\color {black}{}B$ , $\\color {black}{}M$ .",
"Note that finite groups have an advantage over Lie groups in the sense that in empirical applications any Lie group can be modeled by some finite group, but not vice versa."
],
[
"Replacing unitary group by finite group",
"The main non-constructive element of the standard quantum formalism is the unitary group $\\color {black}{}\\mathsf {U}\\!\\left(n\\right)$ , a set of cardinality of the continuum.",
"Formally, the group $\\color {black}{}\\mathsf {U}\\!\\left(n\\right)$ can be replaced by some finite group which is empirically equivalent to $\\color {black}{}\\mathsf {U}\\!\\left(n\\right)$ as follows.",
"From the theory of quantum computing it is known that $\\color {black}{}\\mathsf {U}\\!\\left(n\\right)$ contains a dense finitely generated — and, hence, countable — matrix subgroup $\\color {black}{}\\mathsf {U}_*\\!\\left(n\\right)$ .",
"The group $\\color {black}{}\\mathsf {U}_*\\!\\left(n\\right)$ is residually finite, i.e.",
"it has a reach set of non-trivial homomorphisms to finite groups.",
"In essence, it is more natural to assume that at the fundamental level there are finite symmetry groups, and $\\color {black}{}\\mathsf {U}\\!\\left(n\\right)$ 's are just continuum approximations of their unitary representations.",
"The following properties of finite groups are important for our purposes: any finite group is a subgroup of a symmetric group, any linear representation of a finite group is unitary, any linear representation is subrepresentation of some permutation representation."
],
[
"“Physical” numbers",
"The basic number system in quantum formalism is the complex field $\\color {black}{}\\mathbb {C}$ .",
"This non-constructive field can be obtained as a metric completion of many algebraic extensions of rational numbers.",
"We consider here constructive numbers that are closely related to finite groups and are based on two primitives with a clear intuitive meaning: natural numbers (“counters”): $\\color {black}{}\\mathbb {N}=\\left\\lbrace 0,1,\\ldots \\right\\rbrace $ ; $\\color {black}{}k$ th roots of unityThere are $\\color {black}{}k$ different $\\color {black}{}k$ th roots of unity.",
"A $\\color {black}{}k$ th root of unity is called primitive if $\\color {black}{}\\mathsf {r}_{k}^m\\ne 1$ for any $\\color {black}{}m:~0<m<k$ .",
"(“algebraic form of the idea of $\\color {black}{}k$ -periodicity”): $\\color {black}{}\\mathsf {r}_{k}\\mid \\mathsf {r}_{k}^k=1$ .",
"These basic concepts are sufficient to represent all physically meaningful numbers.",
"We start by introducing $\\color {black}{}\\mathbb {N}\\!\\left[\\mathsf {r}_{k}\\right]$ , the extension of the semiring $\\color {black}{}\\mathbb {N}$ by primitive $\\color {black}{}k$ th root of unity.",
"$\\color {black}{}\\mathbb {N}\\!\\left[\\mathsf {r}_{k}\\right]$ is a ring if $\\color {black}{}k\\ge 2$ .",
"This construction allows, in particular, to add negative numbers to the naturals: $\\color {black}{}\\mathbb {Z}=\\mathbb {N}\\!\\left[\\mathsf {r}_{2}\\right]$ is the extension of $\\color {black}{}\\mathbb {N}$ by the primitive square root of unity.",
"Further, by a standard mathematical procedure, we obtain the $\\color {black}{}k$ th cyclotomic field $\\color {black}{}\\mathbb {Q}\\!\\left(\\mathsf {r}_{k}\\right)$ as the fraction field of the ring $\\color {black}{}\\mathbb {N}\\!\\left[\\mathsf {r}_{k}\\right]$ .",
"If $\\color {black}{}k\\ge 3$ , then the field $\\color {black}{}\\mathbb {Q}\\!\\left(\\mathsf {r}_{k}\\right)$ is a dense subfield of $\\color {black}{}\\mathbb {C}$ , i.e.",
"(constructive) cyclotomic fields are empirically indistinguishable from the (non-constructive) complex field.",
"Note that $\\color {black}{}\\mathbb {Q}\\cong \\mathbb {Q}\\!\\left(\\mathsf {r}_{2}\\right)$ .",
"The importance of cyclotomic numbers for constructive quantum mechanics is explained by the following.",
"Let us recall some terms.",
"The exponent of a group $\\color {black}{}\\mathsf {G}$ is the least common multiple of the orders of its elements.",
"A splitting field for a group $\\color {black}{}\\mathsf {G}$ is a field that allows to split completely any linear representation of $\\color {black}{}\\mathsf {G}$ into irreducible components.",
"A minimal splitting field is a splitting field that does not contain proper splitting subfields.",
"Although minimal splitting field for a given group $\\color {black}{}\\mathsf {G}$ may be non-unique, any minimal splitting field is a subfield of some cyclotomic field $\\color {black}{}\\mathbb {Q}\\!\\left(\\mathsf {r}_{k}\\right)$ , where $\\color {black}{}k$ is a divisor of the exponent of $\\color {black}{}\\mathsf {G}$ .",
"Thus, to work with any unitary representation of $\\color {black}{}\\mathsf {G}$ it is sufficient to use the $\\color {black}{}k$ th cyclotomic field, where $\\color {black}{}k$ is related to the structure of $\\color {black}{}\\mathsf {G}$ ."
],
[
"Constructive representations of a finite group",
"Let a group $\\color {black}{}\\mathsf {G}$ act by permutations on a set $\\color {black}{}\\Omega ,~ \\left|\\Omega \\right|=\\mathsf {N}$ .",
"If we assume that the elements of $\\color {black}{}\\Omega $ are “types” of some discrete entities (“ontological entities”, “elements of reality”), then the collections of these entities can be described as elements of the module $\\color {black}{}H=\\mathbb {N}^\\mathsf {N}$ over the semiring $\\color {black}{}\\mathbb {N}$ with the basis $\\color {black}{}\\Omega $ .",
"The decomposition of the action of $\\color {black}{}\\mathsf {G}$ in the module $\\color {black}{}H$ into irreducible components reflects the structure of the invariants of the action.",
"In order for the decomposition to be complete, it is necessary to extend the semiring $\\color {black}{}\\mathbb {N}$ to a splitting field, e.g., to a cyclotomic field $\\color {black}{}\\mathbb {Q}\\left(\\mathsf {r}_{k}\\right)$ , where $\\color {black}{}k$ is a suitable divisor of the exponent of $\\color {black}{}\\mathsf {G}$ .",
"With such an extension of the scalars, the module $\\color {black}{}H$ is transformed into the Hilbert space $\\color {black}{}\\mathcal {H}$ over $\\color {black}{}\\mathbb {Q}\\left(\\mathsf {r}_{k}\\right)$ .",
"This construction, with a suitable choice of the permutation domain $\\color {black}{}\\Omega $ , allows us to obtain any representation of the group $\\color {black}{}\\mathsf {G}$ in some invariant subspace of the Hilbert space $\\color {black}{}\\mathcal {H}$ .",
"We obtain “quantum mechanics” within an invariant subspace if, in addition to unitary evolutions, projective measurements are also restricted by this subspace.",
"The above is illustrated in Figure REF by the example of the natural action of the symmetric group $\\color {black}{}\\mathsf {S}_{{\\mathsf {N}}}$ on the set $\\color {black}{}\\Omega =\\left\\lbrace e_1,\\ldots ,e_{\\mathsf {N}}\\right\\rbrace $ .",
"Note, that any symmetric group is a rational-representation group, i.e.",
"the field of rational numbers $\\color {black}{}\\mathbb {Q}$ is a splitting field for $\\color {black}{}\\mathsf {S}_{{\\mathsf {N}}}$ .",
"Figure: Natural representation of black𝖲 𝖭 \\color {black}{}\\mathsf {S}_{{\\mathsf {N}}} decomposes into two irreducibles: black1\\color {black}{}1D trivial and black𝖭-1\\color {black}{}\\left(\\mathsf {N}-1\\right)D standardrepresentations.Canonical bases: in trivial subspace blacke 1 +e 2 +⋯+e 𝖭 \\color {black}{}~~~e_1+e_2+\\cdots +e_{\\mathsf {N}} in standard subspace blacke 1 -e 2 \\color {black}{}~~~e_1-e_2 blacke 2 -e 3 \\color {black}{}~~~e_2-e_3 black⋮\\color {black}{}~~~~~~~~\\vdots blacke 𝖭-1 -e 𝖭 \\color {black}{}~~~e_{\\mathsf {N}-1}-e_{\\mathsf {N}}The fundamental discrete time $\\color {black}{}\\mathcal {T}$ is represented by an ordered sequence of integers: $\\color {black}{}\\mathcal {T}=\\mathbb {N}$ or $\\color {black}{}\\mathcal {T}=\\mathbb {Z}$ .",
"We define a finite sequence of “instants of observations” as a subsequence of $\\color {black}{}\\mathcal {T}$ : $\\color {black}{\\left[t_{0},t_{1},\\ldots ,t_{k-1},t_{k},\\ldots ,t_n\\right].",
"}$ The data of the model of quantum evolution include the sequence of the length $\\color {black}{}n+1$ for states $\\color {black}{\\left[{\\rho }_{0},{\\rho }_{1},\\ldots ,{\\rho }_{k-1},{\\rho }_{k},\\ldots ,{\\rho }_n\\right]}$ and the sequence of the length $\\color {black}{}n$ for unitary transitions between observations $\\color {black}{\\left[U_{1},\\ldots ,U_{k},\\ldots ,U_n\\right].",
"}$ Standard quantum mechanics presupposes a single unitary evolution, $\\color {black}{}U_k$ , between observations at times $\\color {black}{}t_{k-1}$ and $\\color {black}{}t_k$ .",
"The single-step transition probability takes the form $\\color {black}{\\mathrm {\\mathbf {P}}_k=\\operatorname{tr}\\left(U_k{\\rho }_{k-1}U_k^\\dagger {\\rho }_k\\right).",
"}$ The evolution can be expressed via the Hamiltonian: $\\color {black}{}U_{k}=\\operatorname{e}^{-\\mathrm {i}{H}\\left(t_k-t_{k-1}\\right)}$ .",
"In physical theories, Hamiltonians are usually derived from the principle of least action, which, like any extremal principle, implies the selection of a small subset of dominant elements in a large set of candidates.",
"Thus it is natural to assume that, in fact, all unitary evolutions take part in the transition between observations with their weights, but only the dominant evolutions are manifested in observations.",
"Therefore, in our model, we use the following modification of the single-step transition probability $\\color {black}{\\mathrm {\\mathbf {P}}_k=\\sum _{m=1}^{\\mathsf {M}}w_{km}\\operatorname{tr}\\left(U_{k,m}{\\rho }_{k-1}U_{k,m}^\\dagger {\\rho }_k\\right),}$ where $\\color {black}{}U_{k,m}=\\mathsf {U}\\left(g_m\\right)$ , $\\color {black}{}g_m\\in \\mathsf {G}$ ; $\\color {black}{}\\mathsf {G}=\\left\\lbrace \\mathsf {g}_1,\\ldots ,\\mathsf {g}_\\mathsf {M}\\right\\rbrace $ is a finite group; $\\color {black}{}\\mathsf {U}$ is a unitary representation of $\\color {black}{}\\mathsf {G}$ ; $\\color {black}{}w_{km}$ is the weight of $\\color {black}{}m$ th group element at $\\color {black}{}k$ th transition.",
"The operators $\\color {black}{}U_{k,m}$ , that maximize $\\color {black}{}\\operatorname{tr}\\left(U_{k,m}{\\rho }_{k-1}U_{k,m}^\\dagger {\\rho }_k\\right)$ , will be called dominant evolutions.",
"$\\color {black}{\\text{\\color {black}{}The \\emph {single-step entropy} is defined as}\\hspace*{10.0pt}\\Delta \\mathrm {\\mathbf {S}}_{k}\\!=-\\log \\mathrm {\\mathbf {P}}_{k}.\\hspace*{145.0pt}}$ Continuum approximation of (REF ) leads to the Lagrangian $\\color {black}{}\\mathcal {L}$ .",
"Taking the logarithm of the probability of the whole trajectory, $\\color {black}{}\\mathrm {\\mathbf {P}}_{0\\rightarrow {n}}=\\prod _{k=1}^n\\mathrm {\\mathbf {P}}_k$ , we arrive at the entropy of trajectory $\\color {black}{}\\mathrm {\\mathbf {S}}_{{0}\\rightarrow {n}}\\!=\\sum _{k=1}^n\\Delta \\mathrm {\\mathbf {S}}_{k}$ , the continuum approximation of which is the action $\\color {black}{}\\mathcal {S}=\\int \\!\\mathcal {L}dt$ ."
],
[
"Continuum approximation of discrete model",
"Continuum approximation of the above model requires the following simplifying assumptions: Sequence (REF ) should be replaced by a continuous time interval $\\color {black}{}\\left[t_0,t_n\\right]\\subseteq \\mathbb {R}$ .",
"Sequences (REF ) and (REF ) are to be replaced by continuous functions of time, $\\color {black}{}{\\rho }={\\rho }\\left(t\\right)$ and $\\color {black}{}U=U\\left(t\\right)$ .",
"The relation $\\color {black}{}\\operatorname{tr}\\left({\\rho }^2\\right)=1$ is necessary to ensure the continuity of probability.",
"This relation holds only for pure states $\\color {black}{}{\\rho }=\\left|\\psi \\right\\rangle \\!\\left\\langle \\psi \\right|$ .",
"So, we will consider $\\color {black}{}{\\psi }$ instead of $\\color {black}{}{\\rho }$ .",
"Assuming that $\\color {black}{}U$ belongs to a unitary representation of a Lie group, we use the Lie algebra approximation, $\\color {black}{}U\\approx \\operatorname{{1}}+\\mathrm {i}{}A$ , where $\\color {black}{}A=A\\left(t\\right)$ is a function whose values are Hermitian matrices.",
"We introduce derivatives and use the linear approximations $\\color {black}{}\\Delta {A}\\approx \\,\\dot{A}\\Delta {t}$ and $\\color {black}{}\\Delta {\\psi }\\approx \\,\\dot{\\psi }\\Delta {t}$ .",
"Applying these assumptions and approximations to the single-step entropy (REF ) and taking the infinitesimal limit we obtain the Lagrangian: $\\color {black}{\\mathcal {L}=\\underbrace{\\left\\langle \\psi \\left|\\,\\dot{A}^2\\right|\\psi \\right\\rangle -\\left\\langle \\psi \\left|\\,\\dot{A}\\right|\\psi \\right\\rangle ^2}_{\\text{\\color {black}{}dispersion of}~\\,\\dot{A}~\\text{\\color {black}{}in state}~\\psi }-\\mathrm {i}\\bigg (\\!\\left\\langle \\,\\dot{\\psi }\\left|\\,\\dot{A}\\right|\\psi \\right\\rangle -\\left\\langle \\psi \\left|\\,\\dot{A}\\right|\\,\\dot{\\psi }\\right\\rangle +2\\left\\langle \\psi \\left|\\,\\dot{A}\\right|\\psi \\right\\rangle \\left\\langle \\psi \\mid \\,\\dot{\\psi }\\right\\rangle \\!\\bigg )-\\left\\langle \\psi \\!\\mid \\!\\,\\dot{\\psi }\\right\\rangle ^2.",
"}$"
],
[
"Dominant unitary evolutions in symmetric group",
"The dominant evolutions between the states represented by the vectors from the module $\\color {black}{}H=\\mathbb {N}^\\mathsf {N}$ for the group $\\color {black}{}\\mathsf {S}_{\\mathsf {N}}$ can be computed as follows.",
"Let $\\color {black}{}\\left|n\\right\\rangle =\\begin{pmatrix}n_1\\\\\\vdots \\\\n_\\mathsf {N}\\end{pmatrix},~\\left|m\\right\\rangle =\\begin{pmatrix}m_1\\\\\\vdots \\\\m_\\mathsf {N}\\end{pmatrix},~\\left|1\\right\\rangle =\\begin{pmatrix}1\\\\\\vdots \\\\1\\end{pmatrix}$ be $\\color {black}{}\\mathsf {N}$ -dimensional vectors with natural components.",
"The Born probabilities for the pair $\\color {black}{}\\left|n\\right\\rangle $ and $\\color {black}{}\\left|m\\right\\rangle $ are Pnat(|n,|m)=nm2nnmm — natural representation, Pstd(|n,|m)=(nm-1Nn11m)2 (nn-1Nn11n)(mm-1Nm11m) — standard representation.",
"Let $\\color {black}{}R_a$ denote the permutation (as well as its representation), that sorts the components of vector $\\color {black}{}\\left|a\\right\\rangle $ in some order.",
"It is not hard to show that the unitary operator $\\color {black}{}U=R_m^{-1}R_n$ maximizes the probability $\\color {black}{}\\mathrm {\\mathbf {P}}_\\mathrm {\\!",
"*}\\!\\left(U\\!\\left|n\\right\\rangle \\!,\\left|m\\right\\rangle \\right)$ , where the permutations $\\color {black}{}R_n$ and $\\color {black}{}R_m$ sort the vectors $\\color {black}{}\\left|n\\right\\rangle $ and $\\color {black}{}\\left|m\\right\\rangle $ identically in the case of natural representation, and either identically or oppositely — depending on the value of the numerator in (REF ) — in the case of standard representation."
],
[
"Energy of permutation",
"Planck's formula, $\\color {black}{}E={h}{\\nu }$ , relates energy to frequency.",
"This relation is reproduced by the quantum-mechanical definition of energy as an eigenvalue of the Hamiltonian, $\\color {black}{}H={\\mathrm {i}}{\\hbar }\\ln {}U$ , associated with a unitary transformation.",
"Consider the energy spectrum of a unitary operator defined by a permutation.",
"Let $\\color {black}{}p$ be a permutation of the cycle type $\\color {black}{}\\left\\lbrace {{\\ell }_1^{\\mathit {m}_1}},\\ldots ,{{\\ell }_k^{\\mathit {m}_k}},\\ldots ,{{\\ell }_K^{\\mathit {m}_K}}\\right\\rbrace $ , where $\\color {black}{}{\\ell }_k$ and $\\color {black}{}\\mathit {m}_k$ represent lengths and multiplicities of cycles in the decomposition of $\\color {black}{}p$ into disjoint cycles.",
"A short calculation shows that the Hamiltonian of the permutation $\\color {black}{}p$ has the following diagonal form $\\color {black}{{}H_p=\\begin{pmatrix}\\operatorname{{1}}_{\\mathit {m}_1}\\!\\otimes \\,H_{{\\ell }_1}&&\\\\&\\hspace{-25.0pt}\\ddots &\\\\&&\\hspace{-25.0pt}\\operatorname{{1}}_{\\mathit {m}_K}\\!\\otimes \\,H_{{\\ell }_K}\\end{pmatrix},~~~\\text{\\color {black}{}where}~~~H_{{\\ell }_k}={\\displaystyle \\frac{1}{{\\ell }_k}}\\begin{pmatrix}0&&&\\\\[-4pt]&\\hspace{-6.0pt}{1}&&\\\\[-4pt]&&\\hspace{-6.0pt}\\ddots &\\\\[-4pt]&&&\\hspace{-8.0pt}{\\ell }_k\\!-\\!1\\end{pmatrix}.",
"}$ We shall call the least nonzero energy of a permutation the base energy: $\\color {black}{\\displaystyle {\\varepsilon }=\\frac{1}{\\max \\left({\\ell }_1,\\ldots ,{\\ell }_K\\right)}\\,.",
"}$ Simulation shows that the base (“ground state”, “zero-point”, “vacuum”) energy is statistically more significant than other energy levels.",
"Figure: Dominant evolutions between randomly generated states.",
"Born probability vs time"
],
[
"Monte Carlo simulation of dominant evolutions",
"Figure REF shows several dominant evolutions for the standard representation of the groups $\\color {black}{}\\mathsf {S}_{100}$ and $\\color {black}{}\\mathsf {S}_{2000}$ .",
"Each graph represents the time dependencies of Born's probabilities for the dominant evolutions between four randomly generated pairs of natural vectors.",
"The dominant evolutions are marked by labeling their peaks with their base energies: $\\color {black}{}{\\varepsilon }\\in \\left\\lbrace \\frac{1}{61}, \\frac{1}{69}, \\frac{1}{72}, \\frac{1}{98}\\right\\rbrace $ and $\\color {black}{}{\\varepsilon }\\in \\left\\lbrace \\frac{1}{1416}, \\frac{1}{1789}, \\frac{1}{1939}, \\frac{1}{1972}\\right\\rbrace $ for $\\color {black}{}\\mathsf {S}_{100}$ and $\\color {black}{}\\mathsf {S}_{2000}$ , respectively.",
"We see that with increasing the group size, non-dominant evolutions become almost invisible against the sharp peaks of dominant evolutions."
],
[
"Summary",
" A constructive version of quantum formalism can be formulated in terms of projections of permutations of finite sets into invariant subspaces.",
"Quantum randomness is a consequence of the fundamental impossibility of tracing the individuality of indistinguishable entities in their evolution.",
"The natural number systems for quantum formalism are cyclotomic fields, and the field of complex numbers is just their non-constructive metric completion.",
"Observable behavior of quantum system is determined by the dominants among all possible quantum evolutions.",
"The principle of least action is a continuum approximation of the principle of selection of the most probable trajectories."
]
] | 1709.01831 |
[
[
"Radial Line Fourier Descriptor for Historical Handwritten Text\n Representation"
],
[
"Abstract Automatic recognition of historical handwritten manuscripts is a daunting task due to paper degradation over time.",
"Recognition-free retrieval or word spotting is popularly used for information retrieval and digitization of the historical handwritten documents.",
"However, the performance of word spotting algorithms depends heavily on feature detection and representation methods.",
"Although there exist popular feature descriptors such as Scale Invariant Feature Transform (SIFT) and Speeded Up Robust Features (SURF), the invariant properties of these descriptors amplify the noise in the degraded document images, rendering them more sensitive to noise and complex characteristics of historical manuscripts.",
"Therefore, an efficient and relaxed feature descriptor is required as handwritten words across different documents are indeed similar, but not identical.",
"This paper introduces a Radial Line Fourier (RLF) descriptor for handwritten word representation, with a short feature vector of 32 dimensions.",
"A segmentation-free and training-free handwritten word spotting method is studied herein that relies on the proposed RLF descriptor, takes into account different keypoint representations and uses a simple preconditioner-based feature matching algorithm.",
"The effectiveness of the RLF descriptor for segmentation-free handwritten word spotting is empirically evaluated on well-known historical handwritten datasets using standard evaluation measures."
],
[
"Introduction",
"Automatic recognition of poorly degraded handwritten text is challenging due to complex layouts and paper degradations over time.",
"Typically, an old manuscript suffers from degradations such as paper stains, faded ink and ink bleed-through.",
"There is variability in writing style, and the presence of text and symbols written in an unknown language.",
"This hampers the document readability, and renders the task of searching a word in a set of non-indexed documents i.e.",
"word spotting, to be more difficult.",
"In literature [14], word spotting approaches can either be segmentation-based where the search space consists of a set of segmented word images, or segmentation-free with the complete document image in the search space.",
"This paper focuses on segmentation-free word spotting, which is typically preferred over segmentation-based methods when dealing with heavily degraded document images [42].",
"However, the performance of word spotting algorithms significantly depends on the appropriate selection of feature detection and representation methods [14].",
"In general, feature descriptors represent a region with distinct feature in a document image, coded into a numerical feature vector, which is subsequently compared with the feature vector of a reference image to perform matching.",
"Efforts have been made in the recent past towards research on feature detection and representation methods.",
"Some popular methods include Scale Invariant Feature Transform (SIFT) [29], Speeded Up Robust Features (SURF) [5] and Histograms of oriented Gradients (HoG) [8].",
"SIFT and HoG contributed significantly towards the progress of several visual recognition systems in the last decade [15].",
"However, these local descriptors were mainly designed for the representation of natural scene images, that possess structurally different characteristics from the document images.",
"For example, the detection of the most important edges using pyramid scaling in SIFT creates local interest points between the text lines [42].",
"The invariant properties of these descriptors amplify the noise in the degraded document images, rendering them more sensitive to noise and complex characteristics of historical manuscripts [42].",
"The work by [27] analyzed that the rotation-invariant features are more sensitive to noise in a document image, and perform poorly as compared to rotation-dependent features.",
"Since the existing descriptors are found to be unsuitable for representing handwritten text with high levels of degradations [42], [27], it is important to design a descriptor to address this issue.",
"This paper introduces a Radial Line Fourier (RLF) descriptor which is tailor-made for word spotting applications with fast feature representation and robustness to degradations.",
"RLF is a fast and short-length feature vector of 32 dimensions, based on log-polar sampling followed by computing a few elements of the Discrete Fourier Transform (DFT) along each radial line.",
"It does not require any orientation information from the feature detectors, and simple feature detectors can be used without compromising the descriptor and word spotting performance.",
"This paper is organized as follows.",
"Section reviews the state-of-art methods used in word spotting pipeline, with main focus on interest point detection and feature representation methods.",
"Section presents the proposed method based on the RLF descriptor for segmentation-free handwritten word spotting.",
"Section demonstrates the efficacy of the proposed method on well-known historical datasets using standard evaluation measures.",
"Section concludes the paper."
],
[
"Related Work",
"Appropriate selection of interest points (keypoints) and feature descriptors is indispensable for the performance of a word spotting system.",
"This section discusses some popular interest point detection and feature representation methods with reference to word spotting systems.",
"It is important to note that the segmentation-free word spotting framework presented herein is training-free, therefore training-based methods such as deep learning are not considered."
],
[
"Interest Point Detection",
"Feature detection, or interest point detection refers to finding keypoints in an image that contain crucial information.",
"There exist several interest point detectors in literature.",
"For example, the Harris corner detector [16] is popularly used for corner points detection.",
"It computes a combination of eigenvalues of the structure tensor such that the corners are located in an image.",
"Shi-Tomasi corner detector [37] is a modified version of Harris detector.",
"The minimum of two eigenvalues is computed and a point is considered as a corner point if this minimum value exceeds a certain threshold.",
"The Maximally Stable Extremal Regions (MSER) [30] detector detects keypoints such that all pixels inside the extremal region are either darker or brighter than all the outer boundary pixels.",
"Typically, interest point based feature matching is performed by using a single interest point detector type.",
"SIFT and SURF are the most popular detectors that capture the blob type of features in the image.",
"SIFT uses the Difference of Gaussians (DoG) that computes the difference between Gaussian blurred images using different values of $\\sigma $ , where $\\sigma $ defines the Gaussian blur from a continuous point of view.",
"SURF computes the Determinant of the Hessian (DoH) matrix, that defines the product of the eigenvalues.",
"In principle, any combination of different keypoint detectors can be selected depending upon the application.",
"This work uses a combination of four types of keypoint detectors for handwritten text representation, that consists of corner detectors, dark and bright blobs, saddle points, and the edges of text strokes."
],
[
"Feature Representation",
"After a set of interest points has been detected, a suitable representation of their values has to be defined to perform word matching.",
"In general, a feature descriptor is constructed from the pixels in the local neighborhood of each interest point.",
"Fixed length feature descriptors are most commonly used that generate a fixed length feature vector, which can be easily compared using standard distance metrics (e.g.",
"the Euclidean distance).",
"Sometimes, fixed length feature vectors are computed directly from the extracted features without the need of a learning step [14].",
"Gradient-based feature descriptors tend to be superior, and include SIFT [29], HoG [8] and SURF [5] descriptors.",
"The 128-dimensional SIFT descriptor is formed from histograms of local gradients.",
"SIFT is both scale and rotation invariant, and includes an intricate underlying framework to ensure this.",
"Similarly, HoG computes a histogram of gradient orientations in a certain local region.",
"An important difference between SIFT and HoG is that HoG normalizes the histograms in overlapping blocks, and creates a redundant expression.",
"SURF descriptor is generally faster than SIFT, and is created by concatenating Haar wavelet responses in sub-regions of an oriented square window.",
"SIFT and SURF are invariant to both scale and rotation changes.",
"There are several variants of these descriptors that have been employed for word spotting [33], [14].",
"Many feature descriptors use local image content in square areas around each interest point to form a feature vector [21].",
"Both scale and rotation invariance can be obtained in different ways [13].",
"The Fourier transform has been used to compute descriptors that is illumination and rotation invariant, and scale-invariant to a certain extent [6], [7].",
"In order to overcome dimensionality issues that may arise in a high-dimensional space, binary descriptors are introduced that are faster, but less precise, for example the Binary Robust Invariant Scalable Keypoints (BRISK) descriptor [25] and Fast Retina Keypoint (FREAK) descriptor [1].",
"However, these descriptors with strict invariance properties are not suitable for handwritten document representation.",
"This is mainly because the invariance property renders them more sensitive to noise in a degraded document, as has been carefully studied in [42], [27].",
"A method for searching handwritten Arabic documents based on a set of binary shape features is presented in [38], where a correlation distance based matching technique has been employed.",
"However, it was argued by [13] that the features that are dependent on word shape characteristics are not effective in dealing with multi-writer document collections.",
"Instead, the texture information in a spatial context is considered more reliable than the shape information, as suggested in [13], [28].",
"In [26], the image zones representing the most informative parts in a document image are detected based on the gradient orientation computed by taking convolution of the image with the first and second derivatives of the Gaussian kernel.",
"However, this method was found to be inefficient for short words with less than four characters, and therefore an improved version was proposed in [27].",
"The feature matching algorithm in [27] was found to be very sensitive to variations in handwriting and font sizes, and the overall matching process was too slow for processing large datasets.",
"An interesting block-based document image descriptor was presented by [12] where the query image was scaled and rotated to produce different word instances, and for each instance, a different set of feature vectors was computed.",
"However, several versions of queries generated significant amount of noise in the final merging state, rendering the method inefficient for handling large writing style and font variations.",
"Inspired by Bag-of-Visual Words (BoVW) model, a patch-based framework that uses SIFT for local feature representation was presented in [36].",
"The codebook generation step of BoVW model is expensive, and this method is also found to be unsuitable for handling query font size and handwriting variations [42].",
"The performance of popular word descriptors in a BoVW context was evaluated in [28], and it was suggested that the statistical BoVW approach generates the best result, but with significant increase in overhead in terms of memory requirements to store the descriptors.",
"The winning algorithm, [24], for segmentation-free track of ICFHR 2014 Handwritten Keyword Spotting Competition [31], employed HoG and Local Binary Patterns (LBP) descriptors, and the word retrieval is performed using the nearest neighbour search, followed by a simple oppression of extra overlapping candidates.",
"The work by [42] outperformed the winning algorithms from ICFHR 2014 Handwritten Keyword Spotting Competition [31], and ICDAR2015 Handwritten Keyword Spotting Competition [32].",
"They proposed a new approach towards handwritten word spotting, where the spatial information representing the current location of a feature point is taken into account, and is based on the texture information.",
"However, it is unclear how well this method performs in challenging cases where a a word shares several letters with other different words.",
"The RLF descriptor based method proposed herewith handles this issue by dividing a word into several parts (depending upon the size of the word) to eliminate false-positives, and perform reliable keypoint-based feature matching.",
"Figure: Radial Line Fourier (RLF) descriptor for feature representation in a word spotting framework.",
"Each keypoint detected is represented using log-polar sampling scheme with 16 sampling points per ring.",
"Each radial line, originating in the center and traversing each ring, is used to obtain a square (16 x 16) transformed image representation.",
"In the next step, DFT is applied along each row (corresponding to radial lines) to compute the amplitude of a few elements for each row that constitute the feature vector.",
"Finally, the feature vector generated is presented where x-axis denotes the feature vector length (i.e.",
"32), and y-axis denotes the amplitude of DFT.The performance of different features for word spotting applications was evaluated using Dynamic Time Warping (DTW) [33] and Hidden Markov Models (HMMs) [34].",
"It was found that the local gradient histogram features outperform other geometrical or profile-based features.",
"These methods generally match features from evenly distributed locations over normalized words where no nearest neighbor search is necessary.",
"This is because each point in a word has its corresponding point in some other word located in the very same position.",
"Recently, a method based on feature matching of keypoints derived from the words was proposed [21], which requires a nearest neighbor search.",
"In this case, a relaxed descriptor is required that is not over-precise, since the handwritten words are not normalized.",
"This is due to complex characteristic of handwritten words, unlike simple Optical character recognition (OCR) text.",
"Handwritten words across different documents are similar, but not identical due to variability in writing styles.",
"In an endeavor to address the issues discussed above, this work proposes the RLF descriptor, which is tailor-made for handwritten words representation.",
"The main highlights of this work are as follows: (a) a segmentation-free and training word spotting approach is studied; (b) the proposed method uses a combination of different keypoint detectors to capture different characteristics in a handwritten document, which consists of both lines, corners and blobs; (c) the RLF descriptor is designed, which is a fast and short-length feature vector of 32 dimensions with several advantages; (d) a simple preconditioner-based feature matching algorithm is presented.",
"Advantages of RLF descriptor include faster word spotting (due to short length of feature vector), robustness to degradations, flexibility to be employed with existing feature detectors, efficient memory utilization, and no increase in overhead for feature orientation estimation.",
"The proposed methodology is discussed as follows."
],
[
"Methodology",
"The pipeline of the word spotting framework is as follows.",
"For an input document image, preprocessing is performed to remove background noise using two band-pass filtering approach [40].",
"This is followed by keypoints detection, feature representation using RLF descriptor, and preconditioner-based feature matching.",
"The framework of the proposed approach is pictorially described in Figure REF ."
],
[
"Preprocessing",
"Preprocessing is the initial step of the word spotting algorithm where the background noise is removed using a simple two band-pass filtering approach, as proposed in [40].",
"A high frequency band-pass filter is used to separate the fine detailed text from the background, and a low frequency band-pass filter is used for masking and noise removal.",
"The background removal is performed in such a way that the gray-level information crucial for the feature extraction is not affected.",
"This allows the keypoint detector and the RLF descriptor to be more informative."
],
[
"Keypoint Detection",
"To begin with, keypoints are detected for the document image and the query word.",
"A combination of four different types of keypoint detectors is used to capture a variety of features that represent a handwritten document, and consists of lines, corners and blobs.",
"Figure REF presents the keypoint detectors used herein using an example image of a smoothed query word, $Bentham$ .",
"$Blue$ * represents the Harris corner detector [16], $green$ + represents the result of using the square of the Determinant of Hessian (DoH), which captures both dark and bright blobs, $red$ $\\Delta $ represents negative of DoH (-DoH) and finds the saddle points, and $cyan$ + represents the result of an edge detector ($Assymetric^2$ ) [20].",
"Figure: An example of a query word BenthamBentham depicting four different types of keypoints.",
"Blue * is a corner detector, green + finds the dark and bright blobs, red Δ\\Delta finds the saddle points, and cyan + finds the edges of the text strokes."
],
[
"Radial Line Fourier Descriptor",
"Radial Line Fourier (RLF) descriptor is a short-length feature vector of 32 dimensions, presented in this work for representation of handwritten words.",
"RLF is inspired from a variant of Scale Invariant Descriptor (SID) [23], known as SID-Rot [39], and the idea is to perform log-polar sampling in a circular neighborhood around each keypoint.",
"SID is a scale and rotation invariant descriptor, whereas SID-Rot is scale-invariant but rotation-sensitive descriptor.",
"Typically, the Fourier transform can be applied over scales only to obtain a scale-invariant and rotation-dependent descriptor, or a rotation-invariant and scale-sensitive descriptor.",
"The Fourier transformation over scales render the SID-Rot to be rotation-sensitive, and the scale invariance is achieved by sampling over a large radius with a descriptor length of 3360.",
"This method works well in representing natural scene images with scale changes and no rotations.",
"However, strict invariance properties amplify noise in degraded document images [42], and may lead to loss of useful information.",
"Therefore, a relaxed feature descriptor, such as RLF, is required.",
"RLF descriptor computes a feature vector representation of an image feature, and is based on log-polar sampling followed by computing a few elements of the DFT along each radial line.",
"It characterizes an image region as a whole using a single feature vector of fixed size, and no learning step is involved.",
"Figure REF presents the general framework of the RLF descriptor for feature representation in a word spotting pipeline, and discussed in detail as follows.",
"After the keypoints representing a document image have been detected, log-polar sampling is performed at each keypoint, where each radial line (going from the center, traversing each ring around the center along a line) is transformed into a square representation, as highlighted in Figure REF .",
"The log-polar transform resampling resolution is set to 16 sample points per ring to obtain a square (16 x 16) transformed image.",
"When sampling is done in a log-polar fashion, certain interpolation is required as the pixel coordinates are seldom in the center.",
"One could for instance use a bilinear interpolation to achieve higher accuracy.",
"In this work, sub-pixel sampling is computed using the Gaussian interpolation in a 3x3 neighborhood.",
"In the next step, DFT is used to compute the amplitude of a few elements that constitute the feature vector.",
"The Fast Fourier Transform (FFT) performed efficiently in [19] for creating descriptors that are relaxed.",
"However, it was found to be impractical for high level applications with large amount of data [21].",
"This is because the FFT is rather slow in computations, such as computing the distance measures (i.e.",
"phase correlation).",
"In general, the FFT requires $O\\left(N ~ log(N)\\right)$ computations for a discrete series $f(n)$ with $N$ elements.",
"Therefore, this work improves and simplifies the computations needed to generate a faster feature representation, still benefiting from the advantages of the Fourier transform.",
"We propose to use just a few elements from DFT of the sampled elements $f(n)$ along the radial line, and the computation required (using Euler's formula) is $\\small \\mathcal {F}[f(n)](k)=\\sum _{n=0}^{N-1}{f(n) \\cos (2 \\pi n k /N)- i (f(n) \\sin (2 \\pi n k /N))}.$ The value of $k$ determines the frequency used to compute the Fourier element, where $k \\in {0, 2, 4..}$ .",
"Typically, noise in a document image has higher frequency as compared to the main text in the document image, therefore the second ($k=2$ ) and third ($k=4$ ) elements of the Fourier transform are selected to form the feature descriptor.",
"DFT requires only $O\\left(N\\right)$ computations per element.",
"Note that the Discrete Cosine (DC) component is obtained for $k=0$ and is less informative.",
"The trigonometric functions in the DFT do not have to be computed for each step, and the computation requires simple mathematical operations using the Chebyshev recurrence relation, same as the original Fourier Transform.",
"The RLF descriptor is thus constructed by computing the amplitude of a few elements of DFT: $\\left| \\mathcal {F}[f(n)](k) \\right|= \\sqrt{\\Re {\\left(\\mathcal {F}[f(n)](k)\\right)}^2 + \\Im {\\left(\\mathcal {F}[f(n)](k)\\right)}^2}.$ The descriptor computation using only $k=2$ suffices well for handwritten word representation under the test settings, and most importantly the descriptor is very short (length 32) with fast feature representation.",
"However, experimentally it was found that by adding a second element for $k=4$ , the quality of the subsequent matching improved, even though the feature vector thus generated is twice as long.",
"The advantage is that it makes it possible to sample in a smaller neighborhood, while still getting the same number of corresponding matches, with better accuracy.",
"Nevertheless, adding a third element for $k=6$ did not improve the accuracy significantly, and is found to be not worth the extra computational effort.",
"This work uses RLF descriptor with length 32 for experimental analysis, taking into account the trade-off between computational cost and accuracy.",
"The RLF feature vector thus generated is presented in Figure REF , where x-axis denotes the feature vector length (i.e.",
"32 dimensions), and y-axis denote the frequency amplitudes of DFT.",
"The advantages of the RLF descriptor are many-fold.",
"The RLF descriptor computes a fast and short-length feature vector, to be able to perform quick feature matching in the nearest neighbor search.",
"The RLF descriptor emphasizes on the pixels closer to the feature center, making it less sensitive to erroneous feature size estimation.",
"It is resistant to high frequency changes, such as due to residuals from neighboring words, as it is based on the low frequency content in the local neighborhood.",
"Nevertheless, it is insensitive to small differences in form and shape, as long as they are almost same, i.e.",
"the low frequencies are sufficiently similar."
],
[
"Feature Matching",
"A segmentation-free and training-free word spotting method based on the proposed RLF descriptor is studied herein.",
"In general, no prior information is available about the potential word in the document that is to be matched with the query word.",
"By using the RLF descriptor, the word matching problem is reduced to a much faster search problem.",
"In this work, a simple preconditioner-based feature matching algorithm is employed.",
"To begin with, words are partitioned into several parts in order to avoid confusion between similar words and reduce false positives.",
"This is to overcome the drawback of keypoint-based matching techniques [42], where parts of the retrieved words may be very similar to some part of the query word, or where a word shares several letters with other different words, hence generate false positives.",
"In the experiments, words are divided into several parts depending upon the length of the query word.",
"For example, in Figure REF , a sample query word reberé and its corresponding retrieved word are divided into three parts, and the preconditioner-based matching is performed in the respective three different parts of both the words.",
"After the partitioning step, a nearest neighbor part-based search is performed in an optimal sliding window within the subgroups of the detected keypoints.",
"The keypoint matching algorithm computes the extent of the the matching points in a word, and therefore is able to capture words that are partially outside the sliding window.",
"Consequently, the matched points are removed from the set of points when a word is found, to avoid finding the same word again.",
"The resultant correspondences between the query word and the retrieved word in the sliding window obtained after a simple keypoint matching consists of many outliers and needs further refinement.",
"A common approach is to use Random sample consensus (RANSAC) [11] to learn transformations between the words.",
"However, it is important to have a relaxed transformation instead, because the same word at different locations in a document can differ with small variations in font sizes, or even larger variations in a multi-writer scenario.",
"Therefore, a deterministic preconditioner (inspired from [17]) is used in this work that eliminates the need to use RANSAC and helps in removing the false matches.",
"In [21], preconditioner had been used along with Putative Match Analysis (PUMA) [18], which is found to be computationally expensive and increases overhead in computing false positives.",
"To keep the matching algorithm simple yet effective, this work uses a matching algorithm that is solely based on the preconditioner.",
"The preconditioner creates a cluster of corresponding matches in a two-dimensional space as positional vectors.",
"This means that the correspondences between the query word and the retrieved word in the sliding window with same length and direction are potential inliers, that forms a two-dimensional cluster.",
"However, the clusters are expected to be slightly scattered due to complex characteristics of words (e.g.",
"words can differ in font and style), therefore the threshold must be relaxed or loosely set.",
"The preconditioner finds the inliers efficiently and removes the outliers with fast computation speed.",
"Figure REF represents the matched points obtained from the proposed method, where the matching keypoints or inliers are highlighted in green and the outliers discarded by the preconditioner are in red.",
"The preconditioner-based matching efficiently captures complex variations in handwriting by estimating the core text dimensions on-the-fly.",
"The effectiveness of the proposed method has been experimentally demonstrated in the next section."
],
[
"Experimental Results",
"This section describes the datasets used in the experiments, and empirically evaluates the proposed method."
],
[
"Datasets",
"For experimental analysis, the Barcelona Historical Handwritten Marriages dataset, and the Bentham dataset in two variants are taken into account.",
"The former is heavily degraded, posing challenges for the word spotter, and the latter in both variants demonstrate multi-writer handwriting variations to a certain extent, along with document degradations.",
"The datasets are discussed as follows: Barcelona Historical Handwritten Marriages Dataset (BH2M): It consists of historical handwritten marriage records stored in the archives of Barcelona cathedral, written between 1617 and 1619 by a single writer in old Catalan.",
"The reader is referred to [10] for a deeper understanding of the dataset.",
"Bentham Dataset: It consists of handwritten document pages from the Bentham collection, which have been prepared in the $tranScriptorium$ project.",
"The Bentham collection consists of manuscripts on law and moral philosophy handwritten by Jeremy Bentham (1748-1832) over a period of 60 years, and some handwritten documents from his secretarial staff.",
"This dataset in first variant was used in ICFHR 2014 Handwritten Keyword Spotting Competition [31], and the second variant in ICDAR 2015 Handwritten Keyword Spotting Competition [32].",
"For the experiments, all pages from both variants of Bentham dataset used in the competitions are employed, which have been written by different authors in different styles, font-sizes, and contains crossed-out words."
],
[
"Results",
"The performance of the proposed method is empirically evaluated against the winning algorithms of ICFHR 2014 Handwritten Keyword Spotting Competition [31], and ICDAR2015 Handwritten Keyword Spotting Competition [32], along with the other state-of-the-art methods such as [42], [27].",
"The evaluation measure used is the classic mean Average Precision (mAP) metric popularly used in document word spotting.",
"In general, the retrieved regions of all the document pages are combined and re-ranked according to the score obtained.",
"If a region overlaps more than 50% of the area of the ground truth corpora, it is classified as a positive region.",
"The Precision and Recall values are first computed, and since a single value is preferable for comparison across different methods, the mAP of each method is calculated as the final result.",
"A higher value of mAP is more desirable.",
"Table: Experimental results for BH2M Dataset.Table: Experimental results for Bentham Dataset used in ICFHR 2014 competition.Table: Experimental results for Bentham Dataset used in ICDAR 2015 competition.Tables REF -REF present the segmentation-free handwritten word spotting results for various methods.",
"In Table REF , the performance of the proposed method is evaluated on the BH2M dataset against the methods proposed in [4] and [42].",
"The method proposed in [4] is based on exemplar-SVM framework for word spotting, and the method presented in [42] is based on Document-oriented Local Features (DoLF).",
"It is observed from Table REF that the proposed method achieves higher mAP as compared to [4] and [42].",
"This is mainly because the performance of [4] and [42] is found to be weaker for challenging cases where a a word shares several letters with other different words.",
"Typically, a higher mAP is achieved when search is performed on a long query word (e.g.",
"$habitant$ ), as there is less possibility of finding the query word as part of other similar word.",
"However, in an ideal scenario it is highly possible for a query word to share several characters with other words, even with a longer word.",
"A simple example of a query word from the BH2M dataset is $donsella$ , where some characters are common with query words $fill$ and $filla$ .",
"A much challenging case observed is the sequence of overlapping characters in the query words $fill$ and $filla$ , where $fill$ is retrieved while searching for $filla$ .",
"The proposed method handles this effectively by dividing a word into several parts depending upon the length of the word, and then perform part-based keypoint matching.",
"This simple approach reduces the false-positives by a significant margin, as is evident from the results in Table REF .",
"Table REF presents the results obtained using different methods on the Bentham dataset from ICFHR 2014 Handwritten Keyword Spotting Competition [31].",
"The performance of the proposed method is empirically evaluated against the state-of-the-art methods such as [27], [22], [24] (i.e.",
"winner of ICFHR 2014 competition), and [42].",
"It is observed from Table REF that the proposed method achieves higher mAP as compared to [27], [22] and [24], and performs comparable against [42] for all test images under the experimental settings.",
"This is mainly because the relaxed nature of RLF allows it to capture more details in a degraded document image as compared to descriptors with stricter invariance properties that render them more sensitive to noise.",
"This is important as the same query word at different locations in a document can differ with small variations in font sizes, or even larger variations in a multi-writer scenario.",
"However, even though the proposed approach is observed to perform significantly in comparison with other methods discussed in Table REF , a mAP of 0.490 suggests further investigation.",
"It is observed that the document images in the Bentham dataset from ICFHR 2014 competition consists of handwritten text from two or more authors, where the core text size in a document page differs across different locations in the same document page.",
"This pose challenges for the algorithm in estimating the average core text size for each document page, as the normalization of text size might result in loss of information.",
"The authors aim at investigating this issue further and working towards the improvement of the proposed algorithm as future work.",
"Table REF evaluates the performance of the proposed method on the second variant of Bentham dataset introduced in the ICDAR 2015 Handwritten Keyword Spotting Competition [32].",
"Unlike the first variant of the Bentham dataset discussed above, this dataset does not significantly suffer from the problem of highly variable core text size across a document page.",
"This is evident from the higher mAP value achieved in Table REF .",
"It is observed that the proposed method achieves higher accuracy in comparison with the winner algorithms from the competition, as well as a recent method [42].",
"The RLF descriptor with relaxed feature description takes into account the handwriting variations to a considerable extent, and the standard core text size is estimated for each document page without significant errors.",
"Table: Performance evaluation of feature representation methods on Bentham Dataset used in ICFHR 2014 competition.Table: Performance evaluation of feature representation methods on BH2M Dataset.In order to highlight the importance of the proposed RLF descriptor, a comparison is done with the existing feature representation methods such as SIFT [29], SURF [5], BRISK [25], Oriented FAST and Rotated BRIEF (ORB) [35], KAZE [2], DoLF [42], HoG [3], Loci features [9], graph-based [41] and FFT [21].",
"Table REF presents the experimental results to evaluate the feature representation methods used in the word spotting framework for the Bentham dataset (ICFHR 2014 competition), as an example.",
"This is with reference to the mAP values published in a recent work [42] under the given experimental set up.",
"It is observed from Table REF that the RLF descriptor achieves higher mAP in comparison with SIFT, SURF, BRISK, ORB and KAZE, and performs comparable against DoLF.",
"Table REF validates the performance of the RLF descriptor with respect to the BH2M dataset, and the experiments are performed under the same test settings where the matching algorithm is same for all feature representation methods.",
"The RLF descriptor performs significantly in comparison with other methods, because of the advantages inherited from relaxed feature representation and efficient algorithm design.",
"Nevertheless, with reference to the three historical handwritten datasets used in the experiments, the proposed method is observed to be most consistent and stable with high mAP."
],
[
"Conclusion",
"This paper presented a fast and robust Radial Line Fourier descriptor, with a short feature vector of 32 dimensions, for segmentation-free and training-free handwritten word spotting.",
"A simple preconditioner-based feature matching algorithm is employed, and the experimental results on a variety of historical document images from well-known datasets demonstrate the effectiveness of the proposed method.",
"Under the experimental settings, the proposed RLF descriptor based method outperformed the state-of-the-art methods, including the winners of the popular keyword spotting competitions.",
"As future work, the ideas presented herein will be scaled to aid word feature representation for heavily degraded archival databases with improvements using query expansion."
]
] | 1709.01788 |
[
[
"Heavy tail and light tail of Cox-Ingersoll-Ross processes with\n regime-switching"
],
[
"Abstract This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching.",
"One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed.",
"Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence.",
"In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with regime-switching, which is applied to study the long term returns of interest rates.",
"Then under the existence of the stationary distribution, we provide a criterion to justify whether its stationary distribution is heavy-tailed or not."
],
[
"Introduction",
"Modelling the term structure of interest rates is a long-standing topic in financial economics.",
"Many stochastic interest rates models have been proposed in the past several decades in order to provide a realistic and tractable method to describe the term structure.",
"Some early contributions include Vasieck [24], Dothan [10], Cox et al.",
"[7] and Hull and White [17], amongst others.",
"Single-factor term structure models have been extended to multi-factor ones in the literatures, for instance, Longstaff and Schwartz [19], Duffie and Kan [11].",
"Regime-switching models have emerged in many research fields such as biological, ecological, mathematical finance, economics, etc.",
"Early applications of regime-switching models to economics include Hamilton [16], and Garcia and Perron [14].",
"Regime-switching behavior of interest rate models have been used in interest rates modelling.",
"Empirical evidence provided in the finance literatures Aug and Bekaert [1], [2] suggests that the switching of regimes in interest rates matches well with business cycles.",
"In addition to the statistical evidence, there are economic reasons as well to believe that the regime shifts are important to understand the behavior of entire process.",
"The standard Cox-Ingersoll-Ross model does not consider the possibility of changes in regime.",
"As shown in Brown and Dybvig [6], based on the empirical data of US.",
"treasure yields, the poor empirical performance of CIR model may well suggest the existence of regime shifts.",
"Hence, Gray in [15] models the short interest rate as a discrete regime-switching process.",
"Elliott and Siu [12] proposed a model of term structure of interest rates in a Markovian, regime-switching Health-Jarrow-Morton framework.",
"Zhang et al.",
"[26] showed the existence of stationary distribution of CIR process with Markov switching under certain conditions.",
"Besides, Bao and Yuan [4] generalized the study of long term return of CIR-type models of Deelstra and Delbaen [8], Zhao [27] to the situation with Markov switching.",
"In this paper, we develop the study of CIR process with regime-switching in the aspect of analysis of tail property of its stationary distribution after providing sharp conditions to justify its existence and uniqueness.",
"The existence of regime-switching can cause an essential difference in the tail behavior of the stationary distributions of interest rates models.",
"The stationary distribution could be heavy-tailed for CIR process with regime-switching, however, the stationary distribution of CIR process without regime-switching should be always light-tailed.",
"We refer to the monograph of Foss et al.",
"[13] on the study of heavy-tailed distributions in probability theory.",
"It is useful to mention that heavy-tailed property is closely related to the long-tailed property of a distribution.",
"According to the works [9] and [3], the Ornstein-Uhlenbeck process with regime-switching also presents the phenomenon that its stationary distribution could be light-tailed or heavy-tailed.",
"[9] used the method based on the renewal theory and explicit expression of Ornstein-Uhlenbeck process.",
"[3] used the stochastic analysis method.",
"We adopt the idea of [3] to study the long time behavior of CIR process with regime-switching in this work.",
"To study its recurrent property, we apply the criteria established by us in [20] and [23].",
"We refer to [20], [23], [25] and references therein on the recent development in the study of general regime-switching diffusion processes.",
"The CIR process with regime-switching investigated in current work is determined by the following stochastic differential equation (SDE): $\\text{\\rm {d}}r_t=a_{\\Lambda _t}(b_{\\Lambda _t}- r_t)\\text{\\rm {d}}t+2\\sigma _{\\Lambda _t}\\sqrt{r_t}\\text{\\rm {d}}B_t,\\ \\ r_0=x>0,$ where $a_i,\\,b_i,\\,\\sigma _i\\in \\mathbb {R}$ , $\\sigma _i\\ne 0$ , and $(\\Lambda _t)$ is a continuous time Markov chain on the state space $\\mathcal {S}=\\lbrace 1,2,\\ldots ,N\\rbrace $ with $2\\le N\\le \\infty $ which is independent of Brownian motion $(B_t)$ .",
"The transition rate matrix of $(\\Lambda _t)$ is denoted by $Q=(q_{ij})$ , which is assumed to be irreducible and conservative.",
"To provide a precise expression of our results, we present the following theorem, which is a partial collection of our Theorems REF and REF below.",
"Theorem 1.1 Assume $a_ib_i\\ge 2\\sigma _i^2$ for every $i\\in \\mathcal {S}$ , $N<\\infty $ .",
"Let $(\\mu _i)$ be the invariant probability measure of $(\\Lambda _t)$ .",
"Assume $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i>0$ .",
"Then $(r_t,\\Lambda _t)$ is positive recurrent.",
"Denote by $\\pi $ its stationary distribution.",
"Let $\\kappa $ be defined by (REF ), and assume $\\kappa >1$ .",
"(i) If $a_{\\min }:=\\min _{i\\in \\mathcal {S}} a_i>0$ , then there exists some $\\delta >0$ such that $\\int _{\\mathbb {R}_+\\times \\mathcal {S}} \\text{\\rm {e}}^{\\delta y}\\text{\\rm {d}}\\pi <\\infty .$ (ii) If $a_{\\min }<0$ , then $\\int _{\\mathbb {R}_+\\times \\mathcal {S}}y^p\\text{\\rm {d}}\\pi <\\infty $ if and only if $p\\in (0,\\kappa )$ .",
"The assertion (i) and (ii) of previous theorem tells us that the fact $a_{\\min }>0$ or $a_{\\min }<0$ determines the stationary distribution $\\pi $ of $(r_t,\\Lambda _t)$ being light-tailed or heavy-tailed under some conditions.",
"Recall the fact on the CIR model without switching: $\\text{\\rm {d}}r_t=a(b-r_t)\\text{\\rm {d}}t+2\\sigma \\sqrt{r_t}\\text{\\rm {d}}B_t.$ When the coefficient $a>0$ , this system owns a light-tailed stationary distribution; when the coefficient $a<0$ , the system has no stationary distribution.",
"Therefore, the existence of switching can derive the equilibrium between the existence of stationary distribution and tail behavior of stationary distribution.",
"Moreover, for the CIR model (REF ), the heavy-tail behaviour of the stationary distribution must imply the existence of regime-switching.",
"This provides a theoretical evidence of existence of regime-switching for interest rates model besides the empirical evidence given in [1], [2], [6].",
"The paper is organized as follows.",
"In Section 2, the criteria on justifying the transience and recurrence of CIR process with regime-switching is presented.",
"They are given separately according to the number of states in $\\mathcal {S}$ being finite or infinite, and the transition rate matrix being state-independent or state-dependent.",
"The tail behaviour of the stationary distribution of $(r_t,\\Lambda _t)$ is investigated in Section 3."
],
[
"Recurrent property of CIR process with regime-switching",
"In this work, we are interested in the situation that the process $(r_t)$ stays strictly positive.",
"Via the connection with the Bessel process, it is known that if $a_i b_i\\ge 2\\sigma _i^2$ , the CIR process in this fixed environment $i\\in \\mathcal {S}$ is always strictly positive (cf.",
"[18]).",
"We assume that (H1) $a_ib_i\\ge 2\\sigma _i^2,\\qquad \\forall \\,i\\in \\mathcal {S}.$ Under the condition (H1), it is easy to see that the process $(r_t)$ defined by (REF ) always stays strictly positive.",
"We refer the readers to [18] for more details on the Bessel process and CIR process without switching.",
"Let $R_t=\\sqrt{r_t}$ , then under the condition (H1), $R_t$ satisfies the following SDE: $\\text{\\rm {d}}R_t=\\frac{1}{2R_t} \\big (a_{\\Lambda _t}b_{\\Lambda _t}-\\sigma _{\\Lambda _t}^2-a_{\\Lambda _t} R_t^2\\big )\\text{\\rm {d}}t+\\sigma _{\\Lambda _t}\\text{\\rm {d}}B_t.$ The recurrent property of $(R_t,\\Lambda _t)$ is clearly equivalent to that of $(r_t,\\Lambda _t)$ , but the diffusion coefficient is not degenerated.",
"In the following we shall use the Lyapunov condition on recurrence established in [20] for regime-switching diffusion processes to study the recurrent property of $(R_t,\\Lambda _t)$ .",
"We shall consider first the case $N<\\infty $ then the case $N=\\infty $ .",
"Note that $(R_t,\\Lambda _t)$ is a Markov process, but $(R_t)$ itself is not a Markov process.",
"The infinitesimal generator of $(R_t,\\Lambda _t)$ is given by: $\\begin{split}A f(x,i)&=L^{(i)} f(\\cdot ,i)(x)+Qf(x,\\cdot )(i)\\\\&=\\frac{1}{2}\\sigma _i^2\\frac{\\text{\\rm {d}}^2}{\\text{\\rm {d}}x^2} f(x,i)+\\frac{1}{2x} \\big (a_ib_i-\\sigma _i^2-a_ix^2\\big )\\frac{\\text{\\rm {d}}}{\\text{\\rm {d}}x} f(x,i)+ \\sum _{j\\in \\mathcal {S}} q_{ij}f(x,j)\\end{split}$ for smooth function $f$ on $\\mathbb {R}_+\\times \\mathcal {S}$ .",
"Theorem 2.1 Assume (H1) holds and $N<\\infty $ .",
"Let $(\\mu _i)$ be the invariant probability measure of $(\\Lambda _t)$ .",
"Then (i) if $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i>0$ , $(R_t,\\Lambda _t)$ (hence $(r_t,\\Lambda _t)$ ) is positive recurrent; (ii) if $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i<0$ , $(R_t,\\Lambda _t)$ (hence $(r_t,\\Lambda _t)$ ) is transient.",
"Proof.",
"Take $h(x)=x^p$ for $p\\ne 1$ .",
"Then for any $\\varepsilon >0$ there exists $M>0$ such that for any $x\\ge M$ , $L^{(i)} h(x)&=\\frac{p}{2} x^p\\big [\\big ((p-1)\\sigma _i^4+a_ib_i-\\sigma _i^2\\big )/x^2-a_i\\big ]\\\\&\\le \\frac{p}{2}\\big (-a_i+\\mathrm {sgn}(p)\\varepsilon \\big )h(x).$ If $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i>0$ , we take $p>0,\\,p\\ne 1$ in the definition of $h(x)$ .",
"Then there exists $\\varepsilon >0$ so that $\\sum _{i\\in \\mathcal {S}}\\mu _i(-a_i+\\varepsilon )<0$ , and hence $\\sum _{i\\in \\mathcal {S}}\\frac{p}{2}\\mu _i(-a_i+\\varepsilon )<0$ .",
"According to [20], since $h(x)\\rightarrow \\infty $ as $x\\rightarrow \\infty $ , $(R_t,\\Lambda _t)$ is positive recurrent.",
"This implies that $(r_t, \\Lambda _t)$ is also positive recurrent in this case.",
"If $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i<0$ , we take $p<0$ in the definition of $h(x)$ .",
"Then there exists $\\varepsilon >0$ such that $\\sum _{i\\in \\mathcal {S}} \\mu _i (-a_i-\\varepsilon )>0$ .",
"By [20], $(R_t,\\Lambda _t)$ and hence $(r_t,\\Lambda _t)$ is transient due to the fact $h(x)\\rightarrow 0$ as $x\\rightarrow \\infty $ .",
"Remark 2.2 Applying previous theorem to the case $N=1$ , i.e.",
"a CIR process with no switching, we can get that when $a_i>0$ , the corresponding process $(r_t)$ is positive recurrent; when $a_i<0$ , $(r_t)$ is transient.",
"Invoking our discussion in next section on the tail behavior of CIR process with switching, the divergence of the process $(r_t)$ at some environment $i\\in \\mathcal {S}$ causes the heavy tail property of its stationary distribution.",
"Also the divergence must be restricted by the condition that $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i>0$ .",
"If not, the process $(r_t)$ will not admit stationary distribution.",
"Next, we proceed to providing two extension of the CIR process (REF ) with regime-switching.",
"The model $(r_t,\\Lambda _t)$ defined by (REF ) have a meaningful extension that the change of environment $(\\Lambda _t)$ can be impacted by the value of $(r_t)$ .",
"Namely, the switching rate of the process $(\\Lambda _t)$ could be impacted by the process $(r_t)$ .",
"Precisely, $\\mathbb {P}(\\Lambda _{t+\\Delta }=j|\\Lambda _t=i, r_t=x)={\\left\\lbrace \\begin{array}{ll}q_{ij}(x)\\Delta +o(\\Delta ),\\ \\ & i\\ne j,\\\\1+q_{ii}(x)\\Delta +o(\\Delta ),\\ \\ &i=j,\\end{array}\\right.",
"}$ provided $\\Delta >0$ small enough.",
"Assume that $x\\mapsto q_{ij}(x)$ is Lipschitz continuous and $\\sup _{x\\in \\mathbb {R}_+}\\sum _{j\\ne i} q_{ij}(x)<\\infty $ for each $i\\in \\mathcal {S}$ .",
"Then, there is a strong solution $(r_t,\\Lambda _t)$ satisfying (REF ) and (REF ) (cf.",
"[22]).",
"$(r_t,\\Lambda _t)$ is called a state-dependent regime-switching process, whose recurrent property has been studied in [20] as well.",
"Via the non-singular M-matrix theory, a criterion was established to study the recurrence of CIR process with state-dependent regime-switching.",
"Before presenting the result on state-dependent regime-switching CIR process, let us recall some useful notation.",
"Let $B$ be a matrix or vector.",
"By $B\\ge 0$ we mean that all elements of $B$ are non-negative.",
"By $B\\gg 0$ we mean that all elements of $B$ are positive.",
"Definition 2.3 (M-Matrix) A square matrix $A=(a_{ij})$ is called an M-matrix if $A$ can be expressed in the form $A=s I-B$ with some $B\\ge 0$ and $s\\ge \\mathrm {Ria}(B)$ , where $I$ is the identity matrix and $\\mathrm {Ria}(B)$ the spectral radius of $B$ .",
"When $s>\\mathrm {Ria}(B)$ , $A$ is called a non-singular M-matrix.",
"There are 50 equivalent conditions on the non-singular M-matrix given in the book [5].",
"We collect some easily verified conditions below.",
"Proposition 2.4 ([5]) The following statements are equivalent.",
"$A$ is a non-singular $n\\times n$ M-matrix.",
"All of the principal minors of $A$ are positive, that is $\\begin{vmatrix}a_{11}&\\ldots &a_{1k}\\\\ \\vdots & &\\vdots \\\\ a_{k1}&\\ldots &a_{kk}\\end{vmatrix} >0 \\ \\ \\text{for every $k=1,2,\\ldots ,n$}.$ Every real eigenvalue of $A$ is positive.",
"We construct an auxiliary Markov chain $(\\tilde{\\Lambda }_t)$ on $\\mathcal {S}$ with a conservative $Q$ -matrix defined by: $\\tilde{q}_{ik}={\\left\\lbrace \\begin{array}{ll}\\sup _{x\\in \\mathbb {R}_+} q_{ik}(x)\\ \\ &\\text{if}\\ k<i,\\\\\\inf _{x\\in \\mathbb {R}_+} q_{ik}(x)\\ \\ &\\text{if}\\ k>i,\\end{array}\\right.}",
"\\quad \\text{and}\\ \\tilde{q}_{ii}=-\\sum _{k\\ne i} \\tilde{q}_{ik}.$ This auxiliary Markov chain $(\\tilde{\\Lambda }_t)$ helps us to control the switching process $(\\Lambda _t)$ , which is associated with the matrix $H$ defined below.",
"One can reorder the set $\\mathcal {S}$ and use above definition to obtain alternative auxiliary Markov chain.",
"Theorem 2.5 Let $(r_t,\\Lambda _t)$ be defined by (REF ) and (REF ) with $N<\\infty $ .",
"Assume (H1) holds.",
"Set $\\mathrm {diag}(a_1,\\ldots ,a_N)$ the diagonal matrix with diagonal generated by the vector $(a_1,\\ldots ,a_N)$ .",
"Define the $N\\times N$ matrix $H=\\begin{pmatrix}1& 1&1&\\ldots &1\\\\0&1&1&\\ldots &1\\\\\\vdots &\\vdots &\\vdots &\\ldots &\\vdots \\\\0&0&0&\\ldots &1\\end{pmatrix}.$ Then, $(R_t,\\Lambda _t)$ (and hence $(r_t,\\Lambda _t)$ ) is positive recurrent if there exists some $p>0$ such that the matrix $-\\big (\\tilde{Q}-(p/2)\\mathrm {diag}(a_1,\\ldots ,a_N)\\big )H$ is a non-singular M-matrix; is transient if there exists some $p<0$ such that the matrix $-\\big (\\tilde{Q}-(p/2)\\mathrm {diag}(a_1,\\ldots ,a_N)\\big )H$ is a non-singular M-matrix.",
"Proof.",
"We can follow the procedure of [20] to prove this theorem by taking $V(x)=x^p$ , $\\beta _i=\\frac{p}{2}(-a_i+\\mathrm {sgn}(p)\\varepsilon )$ and applying the arbitrariness of $\\varepsilon $ .",
"The recurrent property of Markov chain in finite state space and infinite state space has essential difference.",
"Under the condition that the Makov chain is irreducible, a Markov chain in a finite state space is always recurrent, but the one in the infinite state space may be recurrent or not.",
"The criterion provided in [20] depends on the Perron-Frobenius theory of finite order matrix, which has no simple extension to deal with infinite order matrix.",
"To deal with the regime-switching processes with switching in an infinite state space, J. Shao has raised two methods in [20] and [21]: finite dimensional partition method and principle eigenvalue method, which have been extended to deal with the stability problem in [23].",
"The finite partition method is based on the theory of non-negative M-matrix, and see [20] for more details.",
"Now we proceed to studying the recurrent property of $(R_t,\\Lambda _t)$ when $(\\Lambda _t)$ is a Markov chain in an infinite state space using the method of principal eigenvalue method.",
"Suppose that $(\\Lambda _t)$ is reversible with invariant probability measure $(\\mu _i)$ .",
"Set $L^2(\\mu )=\\lbrace f; \\sum _{i=1}^\\infty \\mu _if_i^2<\\infty \\rbrace $ , and denote by $\\Vert \\cdot \\Vert $ and $\\langle \\cdot ,\\cdot \\rangle $ its associated norm and inner product.",
"Set $D(f)=\\frac{1}{2}\\sum _{i,j=1}^\\infty \\mu _i q_{ij}(f_j-f_i)^2+\\sum _{i=1}^\\infty \\mu _i a_i f_i^2,\\quad f\\in L^2(\\mu ),$ and $\\tilde{D}(f)=\\frac{1}{2}\\sum _{i,j=1}^\\infty \\mu _iq_{ij}(f_j-f_i)^2+\\frac{1}{2}\\sum _{i=1}^\\infty \\mu _i a_i f_i^2, \\quad f\\in L^2(\\mu ).$ Note that $D(f)$ and $\\tilde{D}(f)$ are not necessary Dirichlet forms since $a_i$ may be negative for some $i\\in \\mathcal {S}$ .",
"The principle eigenvalues $\\lambda _0$ and $\\tilde{\\lambda }_0$ corresponding to $D(f)$ and $\\tilde{D}(f)$ respectively are defined by $\\lambda _0=\\inf \\lbrace D(f);\\,\\Vert f\\Vert =1\\rbrace ,\\quad \\tilde{\\lambda }_0=\\inf \\lbrace \\tilde{D}(f);\\,\\Vert f\\Vert =1\\rbrace .$ Theorem 2.6 Assume (H1) holds and $N=\\infty $ .",
"Suppose $(\\Lambda _t)$ is reversible and positive recurrent with invariant probability measure $(\\mu _i)$ .",
"(i) Suppose there exists a bounded function $(g_i)_{i\\in \\mathcal {S}}$ such that $D(g)=\\lambda _0\\Vert g\\Vert ^2$ , $\\liminf _{i\\rightarrow \\infty } g_i\\ne 0$ .",
"If $\\lambda _0>0$ , then $(R_t,\\Lambda _t)$ is recurrent.",
"(ii) Assume that $\\tilde{\\lambda }_0>0$ and $\\tilde{\\lambda }_0$ is attainable, i.e.",
"there exists $g\\in L^2(\\mu )$ , $g\\lnot \\equiv 0$ so that $\\tilde{D}(g)=\\tilde{\\lambda }_0 \\Vert g\\Vert ^2$ , then $(R_t,\\Lambda _t)$ is transient.",
"Proof.",
"(i) The idea of this argument comes from [23].",
"By variational method (cf.",
"[23]), it holds that $Qg(i)=-\\lambda _0 g_i$ , and $g_i>0$ for each $i\\in \\mathcal {S}$ .",
"Let $V(x,i)=g_i x^2$ , then there exists $M_0>0$ such that $\\displaystyle \\frac{\\sigma _i^4+a_ib_i-\\sigma _i^2}{x^2}<\\frac{\\lambda _0}{2}$ for any $x>M_0$ .",
"Moreover, $\\begin{split}A V(x,i)&\\le \\Big ( Qg(i) -a_ig_i+\\frac{\\sigma _i^4+a_ib_i-\\sigma _i^2}{x^2} g_i\\Big ) x^2\\\\&\\le (-\\lambda _0+\\frac{\\lambda _0}{2}) g_i x^2=-\\frac{\\lambda _0}{2} V(x,i)<0.\\end{split}$ To study the recurrence of $(R_t,\\Lambda _t)$ , we only need to consider the situation $R_0=x_0>M_0$ .",
"Set $\\tau =\\inf \\lbrace t>0; (R_t,\\Lambda _t)\\in \\lbrace x:\\,x\\le K\\rbrace \\times \\lbrace 1,\\ldots ,m_0\\rbrace \\rbrace ,$ where constant $K$ satisfies $x_0>K>M_0$ and $m_0\\in \\mathbb {N}$ satisfies $\\Lambda _0=\\ell >m_0$ .",
"Put $\\tau _K=\\inf \\lbrace t>0; R_t\\ge K\\rbrace .$ Applying Itô's formula to $(R_t,\\Lambda _t)$ , by (REF ), we get $\\mathbb {E}V(R_{t\\wedge \\tau \\wedge \\tau _K},\\Lambda _{t\\wedge \\tau \\wedge \\tau _K})= V(x_0,\\ell )+\\mathbb {E}\\int _0^{t\\wedge \\tau \\wedge \\tau _K}A V(R_s,\\Lambda _s)\\text{\\rm {d}}s\\le V(x_0,\\ell ),$ which yields $\\mathbb {P}(\\tau >\\tau _K)\\le \\frac{V(x_0,\\ell )}{K^2\\inf _{i\\in \\mathcal {S}} g_i}.$ Letting $K\\rightarrow \\infty $ , it follows immediately from $\\lim _{K\\rightarrow \\infty }\\tau _K= \\infty $ a.e.",
"that $\\mathbb {P}(\\tau =\\infty )=0$ .",
"Consequently, $\\mathbb {P}(\\tau <\\infty )=1$ and $(R_t,\\Lambda _t)$ is recurrent.",
"The proof of (ii) is similar to that of [23], and hence is omitted.",
"As an application of Theorem REF , we go to present a result on the long term returns of interest rates, which develops the corresponding results of Deelstra and Delbaen [8] on an extended CIR model without regime-switching.",
"Following [8], the work [26] studied extended CIR model with regime-switching under a strong condition.",
"Restricted to classical CIR model (REF ), their condition means that $a_i>0$ for all $i\\in \\mathcal {S}$ , i.e.",
"the corresponding CIR process in each fixed environment $i$ is recurrent.",
"Especially, under the conditions of [26], there is no heavy-tailed phenomenon appeared for the stationary distribution.",
"Our method and corresponding results can be easily used to study extended CIR process considered in [26].",
"Under the existence of stationary distribution given in Theorem REF , according to the strong ergodicity theorem (see, for instance, [25]), the following assertion holds.",
"Corollary 2.7 Assume that $(r_t,\\Lambda _t)$ is positive recurrent.",
"Then for any measurable function $f$ on $\\mathbb {R}_+\\times \\mathcal {S}$ such that $\\sum _{i\\in \\mathcal {S}} \\int _{\\mathbb {R}_+} |f(x,i)|\\pi (\\text{\\rm {d}}x,i)<\\infty ,$ it holds $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\int _0^t f(r_s,\\Lambda _s)\\text{\\rm {d}}s=\\sum _{i\\in \\mathcal {S}} \\int _{\\mathbb {R}_+} f(x,i)\\pi (\\text{\\rm {d}}x,i),$ where $\\pi $ denotes the stationary distribution of $(r_t,\\Lambda _t)$ .",
"It is well known that CIR process is closely related to the Bessel process (cf.",
"[18]).",
"In the following, we are also interested to establish a connection between CIR process with regime-switching and squared Bessel process with regime-switching.",
"Proposition 2.8 Let $(r_t,\\Lambda _t)$ be the solution of (REF ), then $(r_t)_{t\\ge 0}=\\Big (\\frac{1}{\\ell (t)}\\rho \\Big (\\int _0^t\\ell (s)\\text{\\rm {d}}s\\Big )\\Big )_{t\\ge 0},$ in the sense that both sides admit the same distribution, where $\\ell (t)=\\exp \\big (\\int _0^t a_{\\Lambda _u}\\text{\\rm {d}}u\\big )$ , and $(\\rho _t)$ is a squared Bessel process with regime-switching satisfying the SDE: $\\text{\\rm {d}}\\rho _t=a_{\\Lambda (A_t)} b_{\\Lambda (A_t)}\\text{\\rm {d}}t+\\sigma _{\\Lambda (A_t)} \\sqrt{\\rho _t}\\text{\\rm {d}}W_t,$ where $A_t=\\inf \\lbrace u:\\int _0^u\\ell (s)\\text{\\rm {d}}s=t\\rbrace $ .",
"Proof.",
"We follow the idea of [18] to prove this proposition.",
"Set $C(t)=\\int _0^t \\ell (s)\\text{\\rm {d}}s$ , $t>0$ , then $t\\mapsto C(t)$ is a strictly increasing process and so its inverse $A(t)=\\inf \\lbrace u: C(u)=t\\rbrace $ is well defined.",
"Let $Z_t=r_t\\ell (t)$ , then Itô's formula yields that $\\text{\\rm {d}}Z_t= a_{\\Lambda _t}b_{\\Lambda _t}\\ell (t)\\text{\\rm {d}}t+ \\sigma _{\\Lambda _t}\\sqrt{\\ell (t)}\\sqrt{Z_t}\\text{\\rm {d}}B_t.$ Set $0=\\tau _0<\\tau _1<\\ldots <\\tau _k<\\ldots $ be the sequence of jumping times of the Markov chain $(\\Lambda _t)$ .",
"Put $\\zeta _k=C(\\tau _k)$ , $k\\ge 0$ , then $A_{\\zeta _k}=\\tau _k$ and there is no jumps for $(\\Lambda _t)$ in the interval $[A_{\\zeta _k},A_{\\zeta _{k+1}})$ .",
"Set $n(t)=\\inf \\lbrace k;\\,\\zeta _k\\le t<\\zeta _{k+1}\\rbrace $ for $t\\ge 0$ , and for the convenience of notation, denote by $\\zeta _{n(t)+1}=t$ .",
"By (REF ), $Z_{A(\\zeta _{k+1})}&=Z_{A(\\zeta _k)}+\\int _{A(\\zeta _{k})}^{A(\\zeta _{k+1})} a_{\\Lambda _s}b_{\\Lambda _s}\\ell (s)\\text{\\rm {d}}s +\\int _{A(\\zeta _{k})}^{A(\\zeta _{k+1})} \\sigma _{\\Lambda _s}\\sqrt{\\ell (s)Z_s}\\text{\\rm {d}}B_s\\\\&=Z_{A(\\zeta _k)}+a_{\\Lambda _{A(\\zeta _k)}}b_{\\Lambda _{A(\\zeta _k)}}(\\zeta _{k+1}-\\zeta _k) +\\sigma _{\\Lambda _{A(\\zeta _k)}}\\int _{ A(\\zeta _{k})}^{A(\\zeta _{k+1})}\\sqrt{\\ell (s)Z_s}\\text{\\rm {d}}B_s.$ Note that $\\int _0^t\\sqrt{\\ell (s)Z_s}\\text{\\rm {d}}B_s$ is a local martingale with quadratic variation $\\int _0^{A(t)}\\ell (s)Z_s\\text{\\rm {d}}s=\\int _0^tZ_{A(u)}\\text{\\rm {d}}u$ .",
"Thus, there is a Brownian motion $(W_t)$ such that $\\int _0^{A(t)}\\sqrt{\\ell (s)Z_s}\\text{\\rm {d}}B_s=\\int _0^t \\sqrt{Z_{A(s)}}\\text{\\rm {d}}W_s.$ Consequently, $Z_{A(t)}&=Z_{A(0)}+\\sum _{k=0}^{n(t)}\\big (Z_{A(\\zeta _{k+1})}-Z_{A(\\zeta _k)}\\big )\\\\&=Z_{A(0)}+\\sum _{k=0}^{n(t)}\\int _{\\zeta _k}^{\\zeta _{k+1}} a_{\\Lambda _{A(s)}}b_{\\Lambda _{A(s)}}\\text{\\rm {d}}s+\\sum _{k=0}^{n(t)}\\int _{\\zeta _k}^{\\zeta _{k+1}}\\sigma _{\\Lambda _{A(s)}}\\sqrt{Z_{A(s)}}\\text{\\rm {d}}W_s\\\\&=Z_{A(0)}+\\int _0^ta_{\\Lambda _{A(s)}}b_{\\Lambda _{A(s)}}\\text{\\rm {d}}s+\\int _0^t\\sigma _{\\Lambda _{A(s)}}\\sqrt{Z_{A(s)}}\\text{\\rm {d}}W_s.$ Let $\\rho _t=Z_{A(t)}$ , then $(\\rho _t)_{t\\ge 0}$ satisfies the SDE (REF ) and $r_t=\\frac{1}{\\ell (t)} Z_t=\\frac{1}{\\ell (t)} \\rho _{C(t)}$ , which is the desired result."
],
[
"Long time behavior of CIR process with switching",
"In this section, we only consider the CIR process with switching in a finite state space, i.e.",
"$N<\\infty $ .",
"We focus on difference of the tail behavior of the stationary distribution if it exists caused by the existence of regime-switching.",
"Our result reveals from the theoretical point of view the existence of regime-switching in interest rates as shown in [1], [2], [6] by empirical evidence.",
"Denote $Q_p$ the matrix $Q-p\\mathrm {diag}(a_1,\\ldots ,a_N)$ , where $\\mathrm {diag}(a_1,\\ldots ,a_N)$ stands for the diagonal matrix with diagonal $(a_1,\\ldots ,a_N)$ .",
"Set $\\eta _p=-\\max _{\\gamma \\in \\mathrm {spec}(Q_p)} \\mathrm {Re}\\, \\gamma ,$ where $\\mathrm {spec}(Q_p)$ denotes the spectrum of operator $Q_p$ .",
"According to Propositions 4.1 and 4.2 in [3], it holds: Lemma 3.1 (i) For any $p>0$ , there exist $0<C_1(p)<C_2(p)<\\infty $ such that, for any initial distribution $\\nu _0$ on $\\mathcal {S}$ , any $t>0$ , $C_1(p)\\text{\\rm {e}}^{-\\eta _p t}\\le \\mathbb {E}_{\\nu _0} \\Big [\\text{\\rm {e}}^{-\\int _0^t p a_{\\Lambda _s} \\text{\\rm {d}}s}\\Big ]\\le C_2(p)\\text{\\rm {e}}^{-\\eta _pt}.$ (ii) Set $a_{\\min }=\\min \\lbrace a_i; i\\in \\mathcal {S}\\rbrace $ .",
"If $a_{\\min }\\ge 0$ , then $\\eta _p>0$ for all $p>0$ ; if $a_{\\min }<0$ , there exists $\\kappa \\in (0, \\min \\lbrace -q_i/a_i; a_i<0\\rbrace )$ such that $\\eta _p>0$ for $p<\\kappa $ and $\\eta _p<0$ for $p>\\kappa $ .",
"Theorem 3.2 Assume that (H1) holds, $\\sum _{i\\in \\mathcal {S}} \\mu _i a_i>0$ , and $N<\\infty $ .",
"Let $\\pi $ be the stationary distribution of $(r_t,\\Lambda _t)$ on $\\mathbb {R}_+\\times \\mathcal {S}$ .",
"Set $\\kappa =\\sup \\lbrace p>0; \\eta _p>0\\rbrace \\in (0,\\infty ].$ Assume $\\kappa >1$ .",
"Then (i) If $a_{\\min }>0$ , then for some $\\delta >0$ , $\\displaystyle \\int _{\\mathbb {R}_+} \\text{\\rm {e}}^{\\delta y}\\pi (\\text{\\rm {d}}y, \\mathcal {S})<\\infty $ .",
"(ii) If $a_{\\min }<0$ , then the $p^{th}$ moment of $\\pi $ is finite if and only if $0<p<\\kappa $ .",
"Proof.",
"(i) Set $\\alpha =\\max _{i\\in \\mathcal {S}} \\sigma _i^2/a_i$ .",
"For $0<\\delta <1/(2\\alpha )$ , $\\text{\\rm {d}}\\text{\\rm {e}}^{\\delta r_t}= \\delta \\text{\\rm {e}}^{\\delta r_t}\\Big [a_{\\Lambda _t}b_{\\Lambda _t}-a_{\\Lambda _t}\\big (1-2\\delta \\frac{\\sigma _{\\Lambda _t}^2}{a_{\\Lambda _t}}\\big )r_t\\Big ]\\text{\\rm {d}}t+2\\delta \\text{\\rm {e}}^{\\delta r_t}\\sigma _{\\Lambda _t}\\sqrt{r_t}\\text{\\rm {d}}B_t.$ For any $c>0$ , there exists a constant $M>0$ such that, for any $x\\in \\mathbb {R}_+$ , any $i\\in \\mathcal {S}$ , $a_ib_i-a_{\\min }(1-2\\delta \\alpha )x\\le -c +M\\text{\\rm {e}}^{-\\delta x}.$ Therefore, taking expectation in both sides of (REF ) deduces that $\\frac{\\text{\\rm {d}}\\mathbb {E}[\\text{\\rm {e}}^{\\delta r_t}]}{\\text{\\rm {d}}t}\\le M\\delta -c\\delta \\mathbb {E}[\\text{\\rm {e}}^{\\delta r_t}],$ which implies that $\\mathbb {E}[\\text{\\rm {e}}^{\\delta r_t}]\\le \\mathbb {E}[\\text{\\rm {e}}^{\\delta r_0}]\\text{\\rm {e}}^{-c\\delta t}+M\\delta \\int _0^t\\text{\\rm {e}}^{-c\\delta (t-u)}\\text{\\rm {d}}u.$ Hence, $\\sup _{t>0}\\mathbb {E}[\\text{\\rm {e}}^{\\delta r_t}] \\ \\text{is finite when $\\displaystyle \\mathbb {E}[\\text{\\rm {e}}^{\\delta r_0}]$ is finite},$ which implies that $\\int _{\\mathbb {R}_+} \\text{\\rm {e}}^{\\delta y}\\pi (\\text{\\rm {d}}y, \\mathcal {S})<\\infty \\ \\text{for $\\delta <1/(2\\alpha )$.",
"}$ This means that when $a_{\\min }>0$ , the stationary distribution $\\pi $ of $(r_t,\\Lambda _t)$ is light-tailed.",
"(ii) For the case $p\\in (1,\\kappa )$ , $\\text{\\rm {d}}r_t^p=\\big [ -pa_{\\Lambda _t} r_t^p+p(a_{\\Lambda _t}b_{\\Lambda _t}+2(p-1)\\sigma _{\\Lambda _t}^2)r_t^{p-1}\\big ]\\text{\\rm {d}}t+2p\\sigma _{\\Lambda _t} r_t^{p-1}\\sqrt{r_t} \\text{\\rm {d}}B_t.$ Denote by $F_T^{\\Lambda }=\\sigma \\big \\lbrace \\Lambda _s; 0\\le s\\le T\\big \\rbrace $ for $T>0$ .",
"By the independence of $(\\Lambda _t)$ and $(B_t)$ , taking expectation in (REF ) conditioning on $F_T^{\\Lambda }$ leads to $\\alpha _p^{\\prime }(t)=-pa_{\\Lambda _t}\\alpha _p(t)+p(a_{\\Lambda _t}b_{\\Lambda _t}+2(p-1)\\sigma _{\\Lambda _t}^2)\\alpha _{p-1}(t), \\quad t\\in (0,T].$ For any $\\varepsilon >0$ there exists $c>0$ such that $\\alpha _p^{\\prime }(t)\\le (-pa_{\\Lambda _t}+\\varepsilon )\\alpha _p(t)+c,$ which deduces that $\\alpha _p(t)\\le \\alpha _p(0)\\text{\\rm {e}}^{\\int _0^t (-pa_{\\Lambda _s}+\\varepsilon )\\text{\\rm {d}}s}+c\\int _0^t\\text{\\rm {e}}^{\\int _u^t (-pa_{\\Lambda _s}+\\varepsilon )\\text{\\rm {d}}s}\\text{\\rm {d}}u.$ Taking expectation in both sides of (REF ) and using Lemma REF , we obtain $\\mathbb {E}r_t^p\\le \\mathbb {E}r_0^p C_2(p)\\text{\\rm {e}}^{(-\\eta _p+\\varepsilon )t}+cC_2(p)\\int _0^t \\text{\\rm {e}}^{(-\\eta _p+\\varepsilon )(t-u)}\\text{\\rm {d}}u.$ When $\\eta _p>\\varepsilon >0$ , this implies $\\sup _{t>0} \\mathbb {E}r_t^p<\\infty , \\quad \\text{if $\\mathbb {E}r_0^p<\\infty $}.$ Put $P_t$ the semigroup associated with the Markov process $(r_t,\\Lambda _t)$ .",
"By Theorem REF , the condition $\\sum _{i\\in \\mathcal {S}}\\mu _i a_i>0$ yields that for $\\delta _{(x,i)}P_t$ converges weakly to $\\pi $ as $t\\rightarrow \\infty $ .",
"Namely, the distribution of $(r_t,\\Lambda _t)$ with $(r_0,\\Lambda _0)=(x,i)$ converges weakly to its stationary distribution $\\pi $ as $t\\rightarrow \\infty $ .",
"Thus, (REF ) implies that $\\int _{\\mathbb {R}_+}y^p\\pi (\\text{\\rm {d}}y,\\mathcal {S})\\le \\liminf _{t\\rightarrow \\infty }\\int _{\\mathbb {R}_+\\times \\mathcal {S}} y^p \\text{\\rm {d}}\\big (\\delta _{(x,i)}P_t\\big )=\\liminf _{t\\rightarrow \\infty } \\mathbb {E}r_t^p<\\infty ,$ if $\\eta _p>0$ for $p>0$ .",
"Combining with Lemma REF , if $a_{\\min }\\ge 0$ , then $\\eta _p>0$ for all $p>0$ , and further $\\pi $ owns finite moment of all orders; if $a_{\\min }<0$ , then for $1\\le p<\\kappa $ , $\\eta _p>0$ and further $\\pi $ owns finite $p^{th}$ moment.",
"To show the moment of order $\\kappa $ of $\\pi $ is infinite, we use the proof by contradiction.",
"Assume $\\int _{\\mathbb {R}_+} y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})<\\infty $ , then we take the initial distribution of $(r_0,\\Lambda _0)$ to be the stationary distribution $\\pi $ , which implies that the distribution of $(r_t,\\Lambda _t)$ is also $\\pi $ for $t>0$ .",
"However, Itô's formula yields that $\\alpha _\\kappa (t)&=\\int _0^t\\text{\\rm {e}}^{-\\kappa \\int _u^t a_{\\Lambda _s}\\text{\\rm {d}}s}\\kappa \\big (a_{\\Lambda _u}b_{\\Lambda _u}+2(\\kappa -1)\\sigma _{\\Lambda _u}^2\\big )\\alpha _{\\kappa -1}(u)\\text{\\rm {d}}u+\\alpha _\\kappa (0)\\text{\\rm {e}}^{-\\kappa \\int _0^t a_{\\Lambda _s}\\text{\\rm {d}}s}\\\\&\\ge \\int _0^t\\text{\\rm {e}}^{-\\kappa \\int _u^t a_{\\Lambda _s}\\text{\\rm {d}}s} \\kappa \\min _{i\\in \\mathcal {S}}\\lbrace a_ib_i+2(\\kappa -1)\\sigma _i^2\\rbrace \\alpha _{\\kappa -1}(u)\\text{\\rm {d}}u.$ Since $\\eta _\\kappa =0$ , Lemma REF shows that $\\text{\\rm {e}}^{-p\\int _0^ta_{\\Lambda _s}}\\text{\\rm {d}}s\\ge C_1(p).$ Hence, $\\int _{\\mathbb {R}_+}y^\\kappa \\pi (\\text{\\rm {d}}y, \\mathcal {S})=\\mathbb {E}_{\\pi } r_t^p=\\mathbb {E}\\alpha _p(t)\\ge \\int _0^t C_1(p)p\\min _{i\\in \\mathcal {S}}\\lbrace a_ib_i+2(p-1)\\sigma _i^2\\rbrace \\mathbb {E}_{\\pi } r_s^{p-1}\\text{\\rm {d}}s,\\quad t>0,$ where we use $\\mathbb {E}_{\\pi }$ to emphasize the initial distribution of $(r_t,\\Lambda _t)$ is $\\pi $ .",
"Note that $\\mathbb {E}_{\\pi } r_t^{\\kappa -1}=\\int _{R_+} y^{\\kappa -1}\\pi (\\text{\\rm {d}}y, \\mathcal {S})\\in (0,\\infty )$ , then letting $t$ go to $+\\infty $ in previous inequality leads to $\\int _{\\mathbb {R}_+} y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})=\\infty $ , which is contradict to our assumption.",
"Therefore, the $\\kappa $ -th moment of $\\pi $ is infinite.",
"Remark 3.3 In the last step of previous argument, note that merely using the fact $\\delta _{(x,i)}P_t$ weakly converges to $\\pi $ , one can not derive $\\int _{\\mathbb {R}_+}y^p\\pi (\\text{\\rm {d}}y,\\mathcal {S})=\\infty $ from that $\\lim _{r\\rightarrow \\infty } \\mathbb {E}r_t^p =\\infty $ .",
"We present a simple example below.",
"Let $\\mu _n(\\text{\\rm {d}}x)={\\left\\lbrace \\begin{array}{ll}1-\\frac{1}{\\sqrt{n}}, & x\\in \\big [0,\\frac{\\sqrt{n- n^{-1}}}{\\sqrt{n}-1}\\big ],\\\\\\frac{1}{\\sqrt{n}}, &\\big [n,n+\\sqrt{n}-\\sqrt{n-n^{-1}}\\big ],\\end{array}\\right.",
"}$ and $\\mu (\\text{\\rm {d}}x)=1 $ if $x\\in [0,1]$ ; $\\mu (\\text{\\rm {d}}x)=0$ , otherwise.",
"It is easy to check that for any bounded continuous function $f$ on $[0,\\infty )$ , $\\lim _{n\\rightarrow \\infty }\\int _{\\mathbb {R}_+} f(x)\\mu _n(\\text{\\rm {d}}x)=\\int _{\\mathbb {R}_+} f(x)\\mu (\\text{\\rm {d}}x),$ which means that $\\mu _n$ weakly converges to $\\mu $ as $n\\rightarrow \\infty $ .",
"On the contrary, $&\\lim _{n\\rightarrow \\infty } \\int _{\\mathbb {R}_+}x^3\\mu _n(\\text{\\rm {d}}x)\\\\&=\\lim _{n\\rightarrow \\infty }\\frac{1}{4\\sqrt{n}} \\big ((n\\!+\\!\\sqrt{n}\\!-\\!\\sqrt{n\\!-\\!n^{-1}})^4-n^4\\big )+\\frac{1}{4} \\big (1\\!-\\!\\frac{1}{\\sqrt{n}}\\big )\\Big (\\frac{\\sqrt{n-n^{-1}}}{\\sqrt{n}-1}\\Big )^4=\\infty .$ But $ \\int _{\\mathbb {R}_+} x^3\\mu (\\text{\\rm {d}}x)<\\infty .$ To deal with the case $\\kappa \\in (0,1]$ , the methodology used in [3] by replacing the function $y\\mapsto |y|^p$ by $f: y\\mapsto \\frac{|y|^{p+2}}{1+y^2}$ to exploit the case $p\\in (0,2)$ therein is actually not applicable.",
"The reason is that the signs of coefficients of $f^{\\prime }(Y_t)$ vary according to the state $i$ in $\\mathcal {S}$ , and this causes the difficulty to control the quantity $Y_t\\, f^{\\prime }(Y_t)$ by $f(Y_t)$ .",
"In the following, we exploit the long time behaviour of $(r_t,\\Lambda _t)$ when $\\kappa \\in (0,1)$ under the boundedness of $\\mathbb {E}r_t^{-1}$ for $t>0$ .",
"Theorem 3.4 Suppose $N<\\infty $ , $\\sum _{i\\in \\mathcal {S}} \\mu _i a_i>0$ , $a_{\\min }<0$ and $\\kappa \\in (0,1]$ .",
"Set $\\tilde{Q}_1=Q-\\mathrm {diag}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N)$ where $\\tilde{a}_i=-a_i$ , $i\\in \\mathcal {S}$ .",
"Put $\\tilde{\\eta }_1=-\\max _{\\gamma \\in \\mathrm {spec}(\\tilde{Q}_1)} \\mathrm {Re}\\, \\gamma .$ Assume $a_ib_i\\ge 4\\sigma _i^2$ for all $i\\in \\mathcal {S}$ and $\\tilde{\\eta }_1>0$ .",
"Then the following assertions hold.",
"(i) For any $p\\in (0,\\kappa )$ , $\\sup _{t>0} \\mathbb {E}r_t^p<\\infty $ if $\\mathbb {E}r_0^p<\\infty $ , and further $\\int _{\\mathbb {R}_+} y^p\\pi (\\text{\\rm {d}}y,\\mathcal {S})<\\infty $ .",
"(ii) For any $p\\ge \\kappa $ , $\\sup _{t>0}\\mathbb {E}r_t^p=\\infty $ and $\\int _{\\mathbb {R}_+} y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})=\\infty $ .",
"Proof.",
"(i) When $\\kappa \\in (0,1]$ , instead of the condition (H1), we need to assume a stronger condition that $a_ib_i\\ge 4\\sigma _i^2$ for all $i\\in \\mathcal {S}$ .",
"Then, for $p\\in (0,\\kappa )$ , we obtain $\\text{\\rm {d}}r_t^p=\\big [-pa_{\\Lambda _t}r_t^p+p(a_{\\Lambda _t}b_{\\Lambda _t}+2(p-1)\\sigma _{\\Lambda _t}^2)r_t^{p-1}\\big ]\\text{\\rm {d}}t+2p\\sigma _{\\Lambda _t}r_t^{p-1}\\sqrt{r_t}\\text{\\rm {d}}B_t.$ We need to justify the finiteness of $\\mathbb {E}r_t^{p-1}$ for $p\\in (0,\\kappa )$ .",
"For this purpose, we only need to show the boundedness of $\\mathbb {E}r_t^{-1}$ .",
"It holds $\\text{\\rm {d}}r_t^{-1}&=[a_{\\Lambda _t} r_t^{-1} -(a_{\\Lambda _t}b_{\\Lambda _t}-4\\sigma _{\\Lambda _t}^2)r_t^{-2}]\\text{\\rm {d}}t-2\\sigma _{\\Lambda _t} r_t^{-2}\\sqrt{r_t}\\text{\\rm {d}}B_t\\\\&\\le -\\tilde{a}_{\\Lambda _t} r_t^{-1}\\text{\\rm {d}}t-2\\sigma _{\\Lambda _t} r_t^{-2}\\sqrt{r_t}\\text{\\rm {d}}B_t.$ By Lemma REF and $\\tilde{\\eta }_1>0$ , $\\mathbb {E}r_t^{-1}\\le \\mathbb {E}r_0^{-1}\\text{\\rm {e}}^{-\\int _0^t \\tilde{a}_{\\Lambda _s}\\text{\\rm {d}}s} \\le \\mathbb {E}r_0^{-1}C_2(1)\\text{\\rm {e}}^{-\\tilde{\\eta }_1 t},$ and hence $\\sup _{t>0} \\mathbb {E}r_t^{-1} <\\infty $ if $\\mathbb {E}r_0^{-1}<\\infty $ .",
"Furthermore, setting $(r_0,\\Lambda _0)=(x,i)\\in (0,\\infty )\\times \\mathcal {S}$ , it follows immediately that $\\int _{\\mathbb {R}_+} y^{-1}\\pi (\\text{\\rm {d}}y, \\mathcal {S})\\le \\liminf _{t\\rightarrow \\infty } \\mathbb {E}r_t^{-1}<\\infty .$ Invoking (REF ), there exists a constant $c>0$ such that $\\alpha _p^{\\prime }(t) \\le -p a_{\\Lambda _t} \\alpha _p(t)+ c,$ which yields that $\\alpha _p(t)\\le \\alpha _p(0)\\text{\\rm {e}}^{\\int _0^t -p a_{\\Lambda _s}\\text{\\rm {d}}s} +c\\int _0^t\\text{\\rm {e}}^{\\int _u^t-pa_{\\Lambda _s}\\text{\\rm {d}}s}\\text{\\rm {d}}u.$ Similar to (REF ) and (REF ), the fact $\\eta _p>0$ for $p\\in (0,\\kappa )$ implies that $\\sup _{t>0} \\mathbb {E}r_t^p<\\infty ,\\ \\text{if}\\ \\mathbb {E}r_0^p<\\infty .$ Hence $\\int _{R_+} y^p\\pi (\\text{\\rm {d}}y, \\mathcal {S})\\le \\liminf _{t\\rightarrow \\infty } \\mathbb {E}r_t^p<\\infty .$ (ii) When $p=\\kappa $ , the equation (REF ) implies that $\\mathbb {E}r_t^\\kappa \\ge \\mathbb {E}\\int _0^t\\text{\\rm {e}}^{-\\kappa \\int _u^t a_{\\Lambda _s}\\text{\\rm {d}}s} \\kappa \\min _{i\\in \\mathcal {S}} \\lbrace a_i b_i+2(\\kappa -1)\\sigma _i^2\\rbrace \\mathbb {E}[r_u^{\\kappa -1}\\big |F_T^{\\Lambda }]\\text{\\rm {d}}u.$ Due to Lemma REF , (REF ), if $\\int _{\\mathbb {R}_+} y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})<\\infty $ , we take the initial distribution of $(r_t,\\Lambda _t)$ to be $\\pi $ to yield that $\\int _{\\mathbb {R}_+}y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})=\\mathbb {E}_{\\pi } r_t^\\kappa \\ge \\kappa C_1(\\kappa )\\min _{i\\in \\mathcal {S}} \\lbrace a_i b_i+2(\\kappa -1)\\sigma _i^2\\rbrace t \\int _{\\mathbb {R}_+} y^{\\kappa -1} \\pi (\\text{\\rm {d}}y,\\mathcal {S}),$ and hence $\\int _{\\mathbb {R}_+} y^\\kappa \\pi (\\text{\\rm {d}}y,\\mathcal {S})=\\infty $ by letting $t\\rightarrow \\infty $ in previous inequality.",
"This is contradict to our assumption that $\\int _{\\mathbb {R}_+} y^{\\kappa } \\pi (\\text{\\rm {d}}y,\\mathcal {S})<\\infty $ , and we get the desired conclusion."
],
[
"Conclusion",
"We mainly focus on the tail behavior of stationary distributions of the CIR processes with regime-switching.",
"We provide an explicit connection between the heavy-tailed property of its stationary distribution and the coefficients of CIR process with regime-switching.",
"According to this result, heavy-tailed property of interest rates implies that the application of CIR model with regime-switching is more suitable in practice than CIR model without regime-switching.",
"Further investigation on the quantitative property, such as first passage probability, of CIR process with regime-switching will be considered to learn more impact on the existence of regime-switching in the interest rates models.",
"Besides, there is difficulty to estimate the development of the functional $\\int _0^t f(\\Lambda _s)\\text{\\rm {d}}s$ for $f:\\mathcal {S}\\rightarrow \\mathbb {R}$ when $(\\Lambda _t)$ is a state-dependent switching process.",
"Hence, at present stage, we cannot provide more information on the tail behavior of the stationary distribution for state-dependent regime-switching CIR process.",
"More work is needed in this direction."
]
] | 1709.01691 |
[
[
"Equivalent norms with an extremely nonlineable set of norm attaining\n functionals"
],
[
"Abstract We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read and Rmoutil.",
"Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $NA(X)$ for the original norm is not \"too large\".",
"The construction can be applied to every Banach space containing $c_0$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_0$.",
"We also provide some geometric properties of the norms we have constructed."
],
[
"Introduction",
"A subset $Y$ of a (real) Banach space $X$ is said to be proximinal if for every $x \\in X$ there is a $y \\in Y$ such that $\\Vert x - y\\Vert =\\operatorname{dist}(x,Y)$ .",
"The classical Bishop-Phelps theorem implies that every infinite-dimensional Banach space contains a one-codimensional proximinal subspace.",
"More than 40 years ago, Ivan Singer [32] asked whether every infinite-dimensional Banach space contains proximinal subspaces of codimension 2.",
"Recently Charles J.",
"Read [28] answered this question in the negative.",
"The corresponding space $\\mathcal {R}$ is $c_0$ equipped with a special equivalent norm $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|$ ingeniously constructed by Read.",
"In [29], Martin Rmoutil demonstrates that the same space $\\mathcal {R}$ gives the negative solution to another (at that time open) problem by Gilles Godefroy [20]: is it true that for every infinite-dimensional Banach space the set of those functionals in the dual space which attain their norm contains a two-dimensional linear subspace?",
"Recall that a subset $S$ of a vector space is called lineable if $S\\cup \\lbrace 0\\rbrace $ contains an infinite-dimensional linear subspace, and we call it extremely nonlineable if $S\\cup \\lbrace 0\\rbrace $ does not even contain a two-dimensional subspace.",
"So by Rmoutil's work, the set of norm attaining functionals on $\\mathcal {R}$ is extremely nonlineable.",
"We note that there is a general statement saying that if $X$ contains proximinal subspaces of finite codimension at least two, then the set of norm attaining functionals contains a two-dimensional linear subspace (see [20]).",
"Motivated by these facts, let us say that an equivalent norm $p$ on a Banach space $X$ is a Read norm if the set of norm attaining functionals for this norm does not contain two-dimensional linear subspaces, so the space $X$ endowed with the norm $p$ does not contain proximinal subspaces of finite codimension greater than one.",
"In this paper we present a clear geometric idea which enables us to simplify substantially Read's original construction of a Read norm on $c_0$ , and to extend the construction to some other spaces.",
"In particular, we show that every Banach space having a countable norming system of functionals and containing a copy of $c_0$ admits an equivalent Read norm.",
"We further provide some geometric properties of the constructed Read norms which extend the ones given in [22] for Read's original space $\\mathcal {R}$ .",
"To this end, we introduce the concept of modesty and weak-star modesty of subspaces (see Definition REF ) and show that a Read norm can be constructed whenever the linear span of the set of norm attaining functionals is weak-star modest.",
"The outline of the paper is as follows.",
"We finish this introduction with a subsection which collects all the notation and terminology used in the paper.",
"We devote Section to preliminaries: we provide properties of two kinds of renorming of a Banach space which will be used throughout the paper, we introduce the concept of modest and weak-star modest subspace, and we give some needed results.",
"The main part of the paper is contained in Section , where we show that a Banach space admits an equivalent Read norm if the linear span of the set of norm attaining functionals for the given norm is weak-star modest in the dual space, recovering in particular the original results of Read and Rmoutil.",
"We also show that the constructed Read norms are always strictly convex.",
"Section contains the main application of the previous result: if a Banach space $X$ has a countable norming system of functionals and contains an isomorphic copy of $c_0$ , then it admits an equivalent Read norm; in particular, this is so if $X$ is separable and contains a copy of $c_0$ .",
"We also show that for every $0<\\varepsilon <2$ , an equivalent Read norm can be chosen in such a way that all convex combinations of slices of its unit ball have diameter greater than $2-\\varepsilon $ , so its dual norm is $(2-\\varepsilon )$ -rough; in the case when $X$ is separable, it is possible to get a Read norm which is strictly convex and smooth and whose dual norm is strictly convex and rough; if moreover $X^*$ is separable, then in addition to the above properties the bidual norm is strictly convex.",
"Finally, we discuss in Section some limitations of our construction as, for instance, that no Banach space with the Radon-Nikodým property admits an equivalent norm for which the linear span of the norm attaining functionals is weak-star modest."
],
[
"Notation and terminology",
"Throughout the paper, the letters $X$ , $Y$ , $Z$ will stand for real Banach spaces.",
"For a Banach space $X$ , $X^*$ denotes its topological dual, $B_X$ and $S_X$ are, respectively, the closed unit ball and the unit sphere of $X$ , and we write $J_X\\colon X\\longrightarrow X^{**}$ to represent the canonical isometric inclusion of $X$ into its bidual.",
"We write $\\operatorname{NA}(X)$ to denote the subset of $X^*$ of all functionals attaining their norm, that is, those functionals $f\\in X^*$ such that $\\Vert f\\Vert =|f(x)|$ for some $x\\in S_X$ .",
"If necessary, we will write $\\operatorname{NA}(X,\\Vert \\cdot \\Vert )$ to make clear that we are considering the space $X$ endowed with the norm $\\Vert \\cdot \\Vert $ .",
"A Banach space $X$ (or its norm) is said to be strictly convex if $S_X$ does not contain any non-trivial segment or, equivalently, if $\\Vert x+y\\Vert <2$ whenever $x,y\\in B_X$ , $x\\ne y$ .",
"The space $X$ is said to be smooth if its norm is Gâteaux differentiable at every non-zero element.",
"A norm of a Banach space $X$ is said to be $\\rho $ -rough ($0<\\rho \\leqslant 2$ ) if $\\limsup _{\\Vert h\\Vert \\rightarrow 0}\\frac{\\Vert x+h\\Vert +\\Vert x-h\\Vert -2\\Vert x\\Vert }{\\Vert h\\Vert }\\geqslant \\rho $ for every $x\\in X$ .",
"We refer the reader to the classical books [15] and [17] for more information and background on the geometry of Banach spaces.",
"Finally, we will denote by $\\lbrace e_n\\rbrace $ the canonical basis of $c_0$ or $\\ell _1$ , that is, the $k$ -th coordinate of $e_n$ equals 0 for $n \\ne k$ and equals 1 for $n = k$ ."
],
[
"Preliminaries",
"Our first goal in this section is to present the properties of two types of equivalent renormings of a Banach space.",
"In the first one, we add to the original norm of each element the norm of its image under the action of a fixed operator.",
"This kind of renorming is well known in Banach space theory, see e.g.",
"[21], and it was also used by Read to produce his counterexample [28].",
"Lemma 2.1 Let $X$ , $Y$ be Banach spaces, and let $R\\colon X\\longrightarrow Y$ be a bounded linear operator.",
"Define an equivalent norm on $X$ by $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=\\Vert x\\Vert _X + \\Vert R(x)\\Vert _Y \\qquad (x\\in X).$ Then $\\displaystyle B_{(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)^*}=B_{(X,\\Vert \\cdot \\Vert )^*}+R^*(B_{Y^*})$ ; $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x^{**}|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=\\Vert x^{**}\\Vert _{X^{**}} + \\Vert R^{**}x^{**}\\Vert _{Y^{**}}$ for every $x^{**}\\in X^{**}$ ; if $x\\in X$ and $x^*\\in X^*$ satisfy $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x^*|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=1$ and $x^*(x)=|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|$ , then $x^*=\\tilde{x}^* + R^*y^*$ where $\\tilde{x}^*\\in S_{(X,\\Vert \\cdot \\Vert )^*}$ with $\\tilde{x}^*(x)=\\Vert x\\Vert _X$ and $y^*\\in S_{Y^*}$ with $y^*(Rx)=\\Vert Rx\\Vert _Y$ ; if $R(X)$ is strictly convex and $R$ is one-to-one, then $(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)$ is strictly convex; if $X$ is $\\rho $ -rough for some $0<\\rho \\leqslant 2$ , then $(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)$ is $\\rho (1+\\Vert R\\Vert )^{-1}$ -rough.",
"(a) Write $D=B_{(X,\\Vert \\cdot \\Vert )^*}+R^*(B_{Y^*})$ .",
"First, it is clear that $\\sup _{x^* \\in D} x^*(x) = |\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|$ for every $x \\in X$ .",
"This means that $B_{(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)}$ is the polar set of $D$ .",
"Consequently, $B_{(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)^*}$ is the bipolar of $D$ .",
"So, it remains to demonstrate that $D$ is weak-star closed, a fact which follows from the fact that both $B_{(X,\\Vert \\cdot \\Vert )^*}$ and $R^*(B_{Y^*})$ are weak-star compact.",
"(b) This is just [22].",
"Remark that this fact can be deduced much more easily directly from (a).",
"(c) This is immediate from (a).",
"(d) This fact is widely used in the theory of equivalent renormings; it was first remarked by Victor Klee, see the proof of [16].",
"Finally, (e) follows immediately from the definition of roughness.",
"In the second type of renorming, the new unit ball is the sum of the given unit ball and the image of a weakly compact unit ball by a bounded linear operator.",
"This kind of renorming was used in [14] to study properties of the set of norm attaining functionals.",
"Lemma 2.2 Let $X$ be a Banach space, let $Z$ be a reflexive space and let $S\\colon Z\\longrightarrow X$ be a bounded linear operator.",
"Then there is an equivalent norm $|\\cdot |$ on $X$ whose unit ball is the set $B_X + S(B_Z)$ , and the following assertions hold: $|x^*|=\\Vert x^*\\Vert _{X^*} + \\Vert S^*x^*\\Vert _{Z^*}$ for every $x^*\\in X^*$ ; $\\displaystyle B_{(X,|\\cdot |)^{**}} = B_{(X,\\Vert \\cdot \\Vert )^{**}} + J_X( S(B_{Z}))$ ; if $X$ and $Z$ are strictly convex, then $(X,|\\cdot |)$ is strictly convex; $\\operatorname{NA}(X,|\\cdot |)=\\operatorname{NA}(X,\\Vert \\cdot \\Vert )$ .",
"First, as $Z$ is reflexive and $S$ is weakly continuous, the set $S(B_Z)$ is weakly compact, so $B_X + S(B_Z)$ is closed.",
"As it is also bounded, balanced and solid, it is the unit ball of an equivalent norm $|\\cdot |$ on $X$ .",
"(a) is elementary and it is shown in the proof of [14].",
"(b) follows from (a) of this lemma and (a) of the previous Lemma REF .",
"(c) Consider $\\widetilde{x},\\widetilde{y}\\in S_{(X,|\\cdot |)}$ such that $|\\widetilde{x} + \\widetilde{y}|=2$ .",
"Write $\\widetilde{x}=x+T(u)$ , $\\widetilde{y}=y+T(v)$ with $x,y\\in B_X$ and $u,v\\in B_Z$ and consider $f\\in X^*$ with $|f|=1 \\quad \\text{and} \\quad |f(\\widetilde{y} + \\widetilde{z})|=2.$ As we have that $2=|f(\\widetilde{x}+\\widetilde{y})|& =|f(x+y)+[T^*f](u+v)| \\\\ &\\leqslant \\Vert f\\Vert _{X^*}\\Vert x+y\\Vert +\\Vert T^*f\\Vert _Y\\Vert u+v\\Vert \\\\&\\leqslant 2(\\Vert f\\Vert _{X^*}+\\Vert T^*f\\Vert )=2,$ it follows that $\\Vert x+y\\Vert =2$ and $\\Vert u+v\\Vert =2$ .",
"Since $X$ and $Z$ are both strictly convex, it follows that $x=y$ and $u=v$ , so $\\widetilde{x}=\\widetilde{y}$ .",
"(d) is proved in [14]: a bounded linear functional attains its supremum on $B_{(X,|\\cdot |)}$ if and only if it attains its supremum on both $B_X$ and $S(B_Z)$ , but all functionals attain their maxima on the weakly compact set $S(B_Z)$ .",
"The second goal in this section is to introduce the concepts of modesty and weak-star modesty of subspaces of a Banach space and to present some properties which will be important in our further discussion.",
"Definition 2.3 A linear subspace $Y$ of a Banach space $X$ is said to be an operator range if there is an infinite-dimensional Banach space $E$ and a bounded injective operator $T\\colon E \\longrightarrow X$ such that $T(E) = Y$ .",
"A linear subspace $Z \\subset X$ is said to be modest if there is a separable dense operator range $Y \\subset X$ such that $Z \\cap Y = \\lbrace 0\\rbrace $ .",
"If $X$ is a dual space, a linear subspace $Z \\subset X$ is said to be weak-star modest if there is a separable weak-star dense operator range $Y \\subset X$ such that $Z \\cap Y = \\lbrace 0\\rbrace $ .",
"The study of dense operator ranges in Hilbert spaces goes back to Dixmier, and many results were given by Fillmore and Williams (see [18]).",
"The extension of this study to operator ranges in Banach spaces has attracted the attention of many mathematicians since the domain of a closed operator between Banach spaces is an operator range and every operator range is the domain of some closed linear operator.",
"We refer to the paper [12] (and references therein) for a detailed account of the known results about operator ranges and also for references and background.",
"We would like to emphasize some remarks.",
"Let $X$ be a Banach space and let $Y$ be a linear subspace.",
"First, $Y$ is an operator range if and only if there is a complete norm on $Y$ which is stronger than the restriction of the given norm of $X$ to $Y$ , see [11]; if $Y$ is dense, $Y$ is contained in a non-closed dense operator range if and only if it is non-barrelled, see [33]; finally, the injectivity of $T$ in the definition of operator range can be substituted by the condition $\\dim Y = \\infty $ , because for every non-injective $T\\colon E \\longrightarrow X$ there is an injective $\\widetilde{T}\\colon E/ \\ker T \\longrightarrow X$ with the same range.",
"Next, we would like to make some remarks about modest and weak-star modest subspaces.",
"The first observation is that in the definition of modest (and weak-star modest) subspace, the space $E$ which is the domain of $T$ can be supposed to be separable (just consider the closed linear span of the inverse image of a dense subset of $Y$ ).",
"Actually, any infinite-dimensional separable Banach space can be chosen to be the domain of the dense (or weak-star dense) operator range, because every separable infinite-dimensional Banach space can be densely and injectively embedded into any other separable infinite-dimensional Banach space, and we may even suppose that the operator $T$ is nuclear, see [12] for both results.",
"We will often apply this remark in that the (weak-star) modesty of $Z\\subset X$ can be witnessed by an operator range $Y=T(\\ell _1)$ .",
"Remark also that, obviously, if a subspace is modest (or weak-star modest) then all smaller subspaces are also modest (or weak-star modest).",
"Here is the key example of a modest subspace.",
"Proposition 2.4 The subspace $\\operatorname{span}\\lbrace e_n\\rbrace \\subset \\ell _1$ consisting of all sequences with finite support is modest.",
"This results immediately from the following lemma which will also be useful later on.",
"Lemma 2.5 There is a dense operator range $Y \\subset \\ell _1$ such that every non-zero element of $Y$ has a finite number of zero coordinates.",
"Let $ be the closed unit disc, and let $ A($ be the disc algebraconsisting of all continuous functions on $ that are analytic on the interior of $, viewed as a real Banach space.",
"We let $ Ar( A($ be the closed real subspace consisting of those $ f$ that take real values on the real axis, and denote $ tn = 2-n$ for every $ nN$.",
"We define $ TAr(1$ by$$Tf = \\left(f(t_1), \\frac{1}{2} f(t_2), \\frac{1}{4} f(t_3), \\ldots \\right)\\qquad \\bigl (f\\in A_r(\\bigr ).$$Then, the identity theorem for analytic functionsimplies that in $ Y := T(Ar()$ every non-zero element has a finitenumber of zero coordinates (if any).",
"It remains to demonstrate thedensity of $ Y$ in $ 1$.",
"To this end it is sufficient to show thatevery element $ em$ of the canonical basis of $ 1$ belongs to theclosure of $ Y$.",
"Indeed, for a fixed $ mN$, consider the function$ f(z) = 14(4 - (z - tm)2)$ for every $ z.",
"This $f \\in A_r($ takes the value 1 at $t_m$ and $ 0 < f(t_k) < 1 $ for all $k \\ne m$ .",
"Denote $f_n = f^n \\in A_r($ .",
"Then, $\\lim _{n\\rightarrow \\infty }f_n(t_m) = 1$ , and $\\lim _{n\\rightarrow \\infty }f_n(t_k) = 0$ for $k \\ne m$ , indeed $\\lim _{n\\rightarrow \\infty } \\sup _{k\\ne m} |f_n(t_k)| =0$ , hence $\\lim _{n\\rightarrow \\infty }Tf_n = e_m/2^{m-1}$ , and $e_m$ is in the closure of $Y$ .",
"In fact, Proposition REF can be generalised.",
"Proposition 2.6 For every separable Banach space $X$ every subspace with a countable Hamel basis is modest.",
"For every subspace $W$ with a countable Hamel basis, there is a dense subspace $W_1 \\supset W$ with a countable Hamel basis, say $\\lbrace w_n\\colon n\\in \\mathbb {N}\\rbrace $ .",
"The construction of [25] provides us with sequences $(v_n)$ in $X$ and $(v_n^*)$ in $X^*$ such that $\\operatorname{span}\\lbrace v_1,\\dots ,v_n\\rbrace = \\operatorname{span}\\lbrace w_1,\\dots ,w_n\\rbrace $ and $v_n^*(v_m) =\\delta _{m,n}$ for all $m$ and $n$ .",
"Upon replacing $v_n$ by $v_n/\\Vert v_n\\Vert $ and $v_n^*$ by $\\Vert v_n\\Vert v_n^*$ we may assume that $(v_n)$ is bounded.",
"Consider now the bounded linear operator $S\\colon \\ell _1\\longrightarrow X$ defined by $S(x)=\\sum _{n=1}^\\infty x(n) v_n$ for every $x\\in \\ell _1$ and let $T\\colon A_r(\\longrightarrow \\ell _1$ be the operator defined in Lemma REF .",
"Then, $\\widetilde{T}=S\\circ T\\colon A_r(\\longrightarrow X$ is bounded, has dense range since $T(A_r()$ is dense in $\\ell _1$ and $S(\\ell _1)\\supset W_1$ is dense in $X$ .",
"Finally, $\\widetilde{T}(A_r() \\cap W_1=\\lbrace 0\\rbrace $ .",
"Indeed, if $w= \\sum _{k=1}^n \\alpha _k v_k \\in W_1$ has the form $\\sum _{k=1}^\\infty \\frac{f(t_k)}{2^{k-1}} v_k$ , then apply $v_l^*$ to these series to see that $\\alpha _l= f(t_l)/2^{l-1}$ for $l\\leqslant n$ and $0= f(t_l)/2^{l-1}$ for $l> n$ .",
"The latter implies $f=0$ since $f$ is analytic, therefore $w=0$ .",
"This shows that $W_1$ is modest and so is the smaller subspace $W$ .",
"We would like to comment that the gist of the construction of a Markushevich basis in [25] alluded to above is the Gram-Schmidt orthogonalisation.",
"Indeed, let $J\\colon X\\longrightarrow H$ be an injective bounded linear operator into a Hilbert space with dense range; for example, embed $X$ isometrically into $C[0,1]$ and further continuously into $L_2[0,1]$ , and let $H\\subset L_2[0,1]$ be the closure of the image of $X$ in $L_2[0,1]$ .",
"Then perform the Gram-Schmidt procedure on the linearly independent sequence $(J(w_n))$ to obtain an orthogonal basis $(h_n)\\subset J(X)$ for $H=H^*$ .",
"Finally, put $v_n= J^{-1}(h_n)$ and $v_n^*=J^*(h_n)$ , i.e., $v_n^*(x)= \\langle Jx, h_n\\rangle _H$ .",
"We next present a known result about operator ranges which we will use later on.",
"Proposition 2.7 ([11]) In every separable infinite-dimensional Banach space $X$ there are two dense operator ranges with trivial intersection.",
"The main property of operator ranges which we will need in the paper is the following one.",
"Proposition 2.8 Let $Y \\subset X$ be a separable operator range.",
"Then, there is an injective norm-one linear operator $T\\colon \\ell _1 \\longrightarrow Y$ such that the set $\\left\\lbrace \\frac{Te_n}{\\Vert Te_n\\Vert }\\colon n \\in \\mathbb {N}\\right\\rbrace $ is dense in $S_Y$ .",
"We need the following technical result to provide the proof of the proposition.",
"Lemma 2.9 Let $X$ be a Banach space and let $Y \\subset X$ be an operator range.",
"Then, for every sequence $\\lbrace u_n\\rbrace $ in $Y$ , there is a sequence of positive reals $\\lbrace s_n\\rbrace $ in $(0,1]$ such that for every $x=(x_1, x_2, \\ldots ) \\in \\ell _1$ we have $\\sum _n s_n x_n u_n \\in Y$ .",
"By definition, there is a Banach space $E$ and a bounded bijective linear operator $T\\colon E \\longrightarrow Y$ .",
"To complete the proof it is sufficient to take $s_n = \\min \\lbrace 1, \\Vert T^{-1}u_n\\Vert ^{-1}\\rbrace $ and remark that the series $\\sum _n s_n x_n T^{-1} u_n$ converges absolutely for each $x\\in \\ell _1$ , say to $e \\in E$ , so $\\sum _n s_n x_n u_n = T(e) \\in Y$ .",
"By the remarks after Definition REF , there is an infinite-dimensional separable Banach space $E$ and a bounded injective linear operator $T_1\\colon E \\longrightarrow Y$ with dense range.",
"Applying Proposition REF , we can find two dense operator ranges $E_1, E_2 \\subset E$ with trivial intersection.",
"Without loss of generality, we may assume the existence of injective $U_1, U_2\\colon \\ell _1\\longrightarrow E$ such that $U_i(\\ell _1) = E_i$ , $i=1,2$ (see the remarks following Definition REF ).",
"Fix a countable dense subset $\\lbrace w_n\\rbrace _{n \\in \\mathbb {N}} \\subset S_{T_1(E_1)}$ , then $\\lbrace w_n\\rbrace _{n \\in \\mathbb {N}}$ is dense in $S_Y$ as well.",
"Denote $u_n =\\frac{T_1^{-1}(w_n)}{\\Vert T_1^{-1}(w_n)\\Vert } \\in E_1$ , select the corresponding sequence $\\lbrace s_n\\rbrace $ from Lemma REF and define the requested operator $T\\colon \\ell _1 \\longrightarrow Y$ as follows: $T(e_n) = s_n T_1\\left( u_n + \\varepsilon _n U_2 e_n\\right),$ where the $\\varepsilon _n \\in (0,1)$ are small enough to ensure that $\\left\\Vert T_1^{-1}(w_n) \\right\\Vert \\varepsilon _n \\longrightarrow 0$ .",
"Then $\\left\\Vert \\frac{Te_n}{\\Vert Te_n\\Vert } - w_n \\right\\Vert & =\\left\\Vert \\frac{ T_1\\left( u_n + \\varepsilon _n U_2 e_n\\right)}{\\Vert T_1\\left( u_n + \\varepsilon _n U_2 e_n\\right)\\Vert } - w_n \\right\\Vert \\\\ &=\\left\\Vert \\frac{ w_n +\\varepsilon _n \\Vert T_1^{-1}(w_n)\\Vert T_1(U_2 e_n)}{\\bigl \\Vert w_n+\\varepsilon _n \\Vert T_1^{-1}(w_n)\\Vert T_1(U_2 e_n) \\bigr \\Vert } - w_n \\right\\Vert \\xrightarrow[n\\rightarrow \\infty ]{\\,} 0.$ So, $\\left\\lbrace \\frac{Te_n}{\\Vert Te_n\\Vert }\\colon n \\in \\mathbb {N}\\right\\rbrace $ is dense in $S_Y$ .",
"It remains to demonstrate that $T$ is injective.",
"Assume that for some $x = (x_1, x_2, \\ldots ) \\in \\ell _1$ $Tx = \\sum _{n \\in \\mathbb {N}} x_n Te_n = \\sum _{n \\in \\mathbb {N}} T_1\\left(x_n s_n u_n\\right) + T_1U_2 \\left(\\sum _{n \\in \\mathbb {N}} x_n s_n \\varepsilon _n e_n\\right) = 0.$ Then, $\\sum _{n \\in \\mathbb {N}} T_1\\left(x_n s_n u_n\\right) = - T_1U_2 \\left(\\sum _{n \\in \\mathbb {N}} x_n s_n \\varepsilon _n e_n\\right) ,$ and by the injectivity of $T_1$ $\\sum _{n \\in \\mathbb {N}} x_n s_n u_n = - U_2 \\left(\\sum _{n \\in \\mathbb {N}} x_n s_n \\varepsilon _n e_n\\right).$ But the left hand side of the last equation belongs to $E_1$ and the right hand side belongs to $E_2$ , so both of them are equal to 0.",
"Since $\\lbrace e_n\\rbrace _{n \\in \\mathbb {N}}$ forms a basis of $\\ell _1$ and $U_2$ is injective, this implies that $x = 0$ .",
"Finally, the fact that $\\Vert T\\Vert =1$ can be obtained just by dividing by its norm.",
"Our last result in this section allows us to extend a modest subspace from a complemented subspace to the whole space, in some cases.",
"Proposition 2.10 Let $X$ be a Banach space such that $X = X_1 \\oplus X_2$ for suitable closed subspaces $X_1$ and $X_2$ .",
"Writing $X^*$ in its canonical form $X^* = X_1^* \\oplus X_2^*$ we have the following.",
"If $X_1^*$ is weak-star separable and $F_2 \\subset X_2^*$ is weak-star modest in $X_2^*$ , then $X_1^* \\oplus F_2$ is weak-star modest in $X^*$ .",
"If $X_1^*$ is norm separable and $F_2 \\subset X_2^*$ is modest in $X_2^*$ , then $X_1^* \\oplus F_2$ is modest in $X^*$ .",
"Let $P_1$ , $P_2$ be the natural projections of $X^*$ onto $X_1^*$ and $X_2^*$ , respectively.",
"For (a), take in $S_{X_1^*}$ a countable subset $\\lbrace y_n^*\\rbrace _{n \\in \\mathbb {N}}$ whose linear span is weak-star dense in $X_1^*$ ; for (b), take in $S_{X_1^*}$ a countable subset $\\lbrace y_n^*\\rbrace _{n \\in \\mathbb {N}}$ whose linear span is norm dense in $X_1^*$ .",
"Let $T\\colon \\ell _1\\longrightarrow X_2^*$ be an injective operator whose image is weak-star dense for the case (a) and norm dense for the case (b) in $X_2^*$ and $F_2 \\cap T(\\ell _1) = \\lbrace 0\\rbrace $ .",
"Without loss of generality we may assume that $\\Vert T(e_n)\\Vert \\longrightarrow 0 $ , where $e_n$ are the elements of the canonical basis of $\\ell _1$ (indeed, if not, just compose $T$ with the operator $T_1 \\colon \\ell _1 \\longrightarrow \\ell _1$ that maps $e_k$ to $e_k/k$ for $k =1,2, \\dots $ ).",
"Also, fix a partition of $\\mathbb {N}$ , $\\mathbb {N}=\\bigsqcup _{n \\in \\mathbb {N}} A_n$ , into a countable family of disjoint infinite subsets.",
"Now let us define the requested operator $\\widetilde{T}\\colon \\ell _1 \\longrightarrow X^*$ as follows: $\\widetilde{T}(x) = \\sum _{n \\in \\mathbb {N}} \\sum _{k \\in A_n} x_k (y_n^* +T(e_k)) \\qquad \\bigl ( x=(x_n)_n \\bigr ),$ i.e., $\\widetilde{T}(e_k) = y_n^* + T(e_k)$ for all $k \\in A_n$ .",
"Then the closure of $\\widetilde{T}(\\ell _1)$ contains all the functionals $y_n^*$ , and consequently it contains also all $T(e_k)$ , so $\\overline{\\widetilde{T}(\\ell _1)} \\supset \\operatorname{span}\\lbrace y_n^*\\colon n \\in \\mathbb {N}\\rbrace \\oplus T(\\ell _1)$ which in the case (a) is weak-star dense in $X^*$ and norm dense in the case (b).",
"Injectivity of $\\widetilde{T}$ follows from injectivity of $T$ .",
"It remains to demonstrate that $\\widetilde{T}(\\ell _1)$ has trivial intersection with $X_1^* \\oplus F_2$ .",
"Indeed, let $x_1^* + f_2 = \\sum _{n \\in \\mathbb {N}} \\sum _{k \\in A_n} x_k (y_n^* + T(e_k))$ for some $x = (x_1, x_2, \\ldots ) \\in \\ell _1$ with $x_1^* \\in X_1^*$ and $f_2 \\in F_2$ .",
"Applying $P_2$ , we obtain $f_2 = \\sum _{n \\in \\mathbb {N}} \\sum _{k \\in A_n} x_k T(e_k) = Tx$ , which means that $x = 0$ ."
],
[
"The main construction",
"Our goal here is to present a general argument providing Read norms.",
"We also present some geometric properties of the norms constructed in this way.",
"We denote the dual norm to an equivalent norm $p$ by $p^*$ .",
"Theorem 3.1 Let $X$ be a Banach space such that $\\operatorname{span}(\\operatorname{NA}(X))$ is a weak-star modest subspace of $X^*$ .",
"Then $X$ possesses an equivalent Read norm $p$ .",
"Moreover $p$ can be chosen in such a way that, given two linearly independent functionals $x^*, z^* \\in \\operatorname{NA}(X,p)$ with $p^*(x^*) = p^*(z^*) =1$ , one has $x^* +z^* \\notin \\operatorname{NA}(X,p)$ or $x^* - z^* \\notin \\operatorname{NA}(X,p)$ .",
"Let $Y \\subset X^*$ be a separable weak-star dense operator range with $\\operatorname{span}(\\operatorname{NA}(X))\\cap Y = \\lbrace 0\\rbrace $ according to Definition REF .",
"By Proposition REF , we may assume that $Y = T(\\ell _1)$ , where $T\\colon \\ell _1 \\longrightarrow X^*$ is an injective bounded linear operator such that the set $\\left\\lbrace \\frac{Te_n}{\\Vert Te_n\\Vert }\\colon n \\in \\mathbb {N}\\right\\rbrace $ is dense in $S_Y$ .",
"Take a sequence $\\lbrace r_n\\rbrace $ of positive reals such that $\\sum _{k \\in \\mathbb {N}} r_k <\\infty $ , denote $v_n^* = T(e_n)$ , and consider the operator $R\\colon X\\longrightarrow \\ell _1$ given by $[R(x)](n)=r_n v_n^*(x)$ for every $n\\in \\mathbb {N}$ and every $x\\in X$ .",
"Then, we define an equivalent norm on $X$ by $p(x)=\\Vert x\\Vert + \\Vert Rx\\Vert _{\\ell _1} \\qquad (x\\in X).$ The adjoint operator $R^* \\colon \\ell _\\infty \\longrightarrow X^*$ acts as follows: $R^*\\bigl (\\lbrace t_n\\rbrace _{n \\in \\mathbb {N}}\\bigr ) = \\sum _{n \\in \\mathbb {N}} t_n r_n v_n^*$ .",
"Consequently, according to Lemma REF (a), we have that $B_{(X,p)^*} = B_{X^*} + R^*(B_{\\ell _\\infty })=B_{X^*}+\\sum _{n \\in \\mathbb {N}}r_n[-v_n^*, v_n^*].$ Consider two linearly independent functionals $x^*, z^* \\in \\operatorname{NA}(X,p)$ with $p^*(x^*) = p^*(z^*) =1$ , and let $x, z \\in X$ with $p(x)=p(z)=1$ such that $x^*(x)=z^*(z)=1$ .",
"Due to Lemma REF (c), there are representations $ x^* = x_0^* + \\sum _{n \\in \\mathbb {N}} t_n r_n v_n^*, \\quad z^* = z_0^* + \\sum _{n \\in \\mathbb {N}} \\tau _n r_n v_n^*.$ with $t_k, \\tau _k \\in [-1,1]$ such that $x_0^*, z_0^* \\in S_{X^*} \\cap \\operatorname{NA}(X)$ , for every $n \\in \\mathbb {N}$ where $v_n^*(x) \\ne 0$ one has $t_n = \\operatorname{sign}v_n^*(x)$ , and for every $n \\in \\mathbb {N}$ where $ v_n^*(z) \\ne 0$ one has $\\tau _n = \\operatorname{sign}v_n^*(z)$ .",
"Let $\\theta = \\pm 1$ be a sign such that $x \\ne \\theta z$ .",
"First, remark that, by weak-star density of $Y$ , the set of restrictions of functionals from $Y$ to the linear span of $x$ and $z$ is the whole $(\\operatorname{span}\\lbrace x,z\\rbrace )^*$ .",
"So, there is $y_0^* \\in S_Y$ such that $y_0^*(x) < 0$ and $y_0^*(\\theta z) >0$ .",
"Consequently, there is a neighbourhood $U_0$ of $y_0^*$ in $S_Y$ such that for all $y^* \\in U_0$ , we have $y^*(x) < 0$ and $y^*(\\theta z) > 0$ .",
"Then, for all those $n \\in \\mathbb {N}$ for which $\\frac{v_n^*}{\\Vert v_n^*\\Vert }\\in U_0$ , we have that $ t_n + \\theta \\tau _n = \\operatorname{sign}v_n^*(x) + \\theta \\operatorname{sign}v_n^*(z) = 0.$ We are going to demonstrate that $x^* + \\theta z^* \\notin \\operatorname{NA}(X,p)$ .",
"Assume to the contrary that there is $e \\in X$ with $p(e)=1$ at which $x^* + \\theta z^* $ attains its norm, that is $(x^* + \\theta z^*)(e) =p^*(x^* + \\theta z^*)$ .",
"Lemma REF (c) says that one can write $ \\frac{x^* + \\theta z^*}{ p^*(x^* + \\theta z^*)} = h_0^* + \\sum _{n \\in \\mathbb {N}} s_n r_n v_n^* ,$ with $s_k \\in [-1,1]$ , $h_0^* \\in \\operatorname{NA}(X)$ , and for every $n \\in \\mathbb {N}$ where $v_n^*(e) \\ne 0$ , one has $s_n = \\operatorname{sign}v_n^*(e)$ .",
"Since $Y$ is weak-star dense, it cannot be contained in a weak-star closed hyperplane.",
"Consequently, the set $S_Y \\cap \\lbrace h^* \\in X^*\\colon h^*(e) = 0 \\rbrace = S_Y \\cap \\lbrace h^* \\in Y\\colon h^*(e) = 0 \\rbrace $ is nowhere dense in $S_Y$ .",
"This implies that there is a non-empty relatively open subset $U_1 \\subset U_0$ of $S_Y$ which does not intersect the hyperplane $ \\lbrace h^* \\in Y\\colon h^*(e) = 0 \\rbrace $ .",
"Denote $N_1 = \\left\\lbrace n\\in \\mathbb {N}\\colon \\frac{v_n^*}{\\Vert v_n^*\\Vert } \\in U_1\\right\\rbrace $ , which is non-empty by density of $\\lbrace \\frac{v_n^*}{\\Vert v_n^*\\Vert } \\colon n\\in \\mathbb {N}\\rbrace $ in $S_Y$ .",
"Then, for every $n \\in N_1$ the conditions (REF ) and the fact that $s_n = \\operatorname{sign}v_n^*(e)$ hold true at the same time.",
"Now, from equations (REF ) and (REF ) we get $0 &= x^* + \\theta z^* - p^*(x^* + \\theta z^*) \\frac{x^* + \\theta z^*}{ p^*(x^* + \\theta z^*)} \\\\&= (x_0^*+ \\theta z_0^* - p^*(x^* + \\theta z^*)h_0^* )+ \\sum _{n \\in \\mathbb {N}} (t_n + \\theta \\tau _n - p^*(x^* + \\theta z^*)s_n) r_n v_n^*.$ In other words, $x_0^*+ \\theta z_0^* - p^*(x^* + \\theta z^*)h_0^*= - T\\left(\\sum _{n \\in \\mathbb {N}} (t_n + \\theta \\tau _n - p^*(x^* + \\theta z^*)s_n) r_n e_n\\right).$ The left hand side belongs to $\\operatorname{span}(\\operatorname{NA}(X))$ , the right hand side belongs to $Y$ , so both of them are equal to zero.",
"Since $T$ is injective, and $\\lbrace e_n\\rbrace _{n \\in \\mathbb {N}}$ forms a basis of $\\ell _1$ , this means that all $t_n + \\theta \\tau _n - p^*(x^* + \\theta z^*)s_n$ are equal to zero.",
"On the other hand, as we remarked before, for every $n \\in N_1$ we have $t_n + \\theta \\tau _n = 0$ and $s_n = \\operatorname{sign}v_n^*(e)\\ne 0$ , so $t_n + \\theta \\tau _n - p^*(x^* + \\theta z^*)s_n \\ne 0$ .",
"This contradiction completes the proof.",
"Observe that $\\operatorname{span}(\\operatorname{NA}(c_0))=\\operatorname{NA}(c_0)$ consists on those elements of $\\ell _1$ that have finite support, so it is modest by Proposition REF .",
"Therefore, Theorem REF applies, giving Read's [28] and Rmoutil's [29] results.",
"Corollary 3.2 ([28], [29]) There exists an equivalent norm $p$ on $c_0$ such that $\\operatorname{NA}(c_0,p)$ does not contain two-dimensional subspaces and, therefore, $(c_0,p)$ does not contain finite-codimensional proximinal subspaces of codimension greater than 1.",
"Although we extensively use Read's ideas in our construction, his original construction is not a particular case of ours.",
"Namely, Read's norm on $c_0$ is defined by a very similar formula, but his choice of corresponding functionals $v_n^*$ is quite different; in Read's choice they belong to $\\operatorname{NA}(c_0)$ whereas our $v_n^*$ are sort of “orthogonal” to this set.",
"We will provide further examples in the next section.",
"Next, we would like to present some geometric properties of the Read norms we have constructed here, extending some of the results of [22].",
"First, we need to expound in detail the norms constructed in Theorem REF .",
"Remark 3.3 Let $X$ be a Banach space.",
"If $\\operatorname{span}(\\operatorname{NA}(X))$ is a weak-star modest subspace of $X^*$ , then there is a sequence $\\lbrace v_n^*\\rbrace _{n\\in \\mathbb {N}}$ in $B_{X^*}$ for which $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ is weak-star dense in $S_{X^*}$ , such that given a sequence $\\lbrace r_n\\rbrace _{n\\in \\mathbb {N}}$ of positive reals with $\\rho =\\sum _{k\\in \\mathbb {N}} r_k <\\infty $ and defining the bounded linear operator $R\\colon X\\longrightarrow \\ell _1$ by $[R(x)](n)=r_n v_n^*(x) \\qquad \\bigl (n\\in \\mathbb {N},\\ x\\in X\\bigr ),$ the norm $p(x)=\\Vert x\\Vert _X + \\Vert R(x)\\Vert _{\\ell _1} \\qquad (x\\in X)$ is a Read norm.",
"If moreover $\\operatorname{span}(\\operatorname{NA}(X))$ is modest, we get that the sequence $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ can be selected to be norm-dense in $S_{X^*}$ .",
"Let us mention that it is clear that $\\Vert R\\Vert \\leqslant \\rho $ and that $R$ is compact since $\\Vert P_n R - R\\Vert \\leqslant \\sum _{k>n} r_k \\longrightarrow 0$ , where $P_n$ projects $\\ell _1$ onto $\\operatorname{span}\\lbrace e_1,\\dots ,e_n\\rbrace $ .",
"We are now ready to present some geometric properties of our Read norms.",
"Proposition 3.4 Let $X$ be a Banach space.",
"If $\\operatorname{span}(\\operatorname{NA}(X))$ is a weak-star modest subspace of $X^*$ , then the Read norm $p$ defined in (REF ) is strictly convex.",
"Moreover, if $\\operatorname{span}(\\operatorname{NA}(X))$ is actually a modest subspace of $X^*$ , then $p$ can be built in such a way that $(X,p)^{**}$ is strictly convex and so $(X,p)^*$ is smooth.",
"For the first part, we only have to show that the operator $R$ given in (REF ) is one-to-one and that $R(X)$ is strictly convex, and then apply Lemma REF (d).",
"Both assertions are a consequence of the fact that the sequence $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ is weak-star dense in $S_{X^*}$ , the first one being immediate.",
"For the strict convexity of $R(X)$ , consider $x,y\\in X$ such that $R(x)\\ne \\alpha R(y)$ for every $\\alpha >0$ .",
"Then, $x\\ne \\alpha y$ for every $\\alpha >0$ , so by the Hahn-Banach theorem, there is $x^*\\in S_{X^*}$ such that $x^*(x)<0<x^*(y)$ and by weak-star density of $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ in $S_{X^*}$ , we get that there is $n\\in \\mathbb {N}$ such that $v_n^*(x)<0<v_n^*(y)$ , so $|v_n^*(x+y)|<|v_n^*(x)|+|v_n^*(y)|$ .",
"From here, it is immediate that $\\Vert R(x) + R(y)\\Vert _{\\ell _1} < \\Vert R(x)\\Vert _{\\ell _1} + \\Vert R(y)\\Vert _{\\ell _1}$ , showing the strict convexity or $R(X)$ .",
"For the moreover part, we first use the modesty of $\\operatorname{span}(\\operatorname{NA}(X))$ in order to get that $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ is norm-dense in $S_{X^*}$ .",
"By Lemma REF (b), we know that the bidual norm of $p$ is given by $p(x^{**})=\\Vert x^{**}\\Vert _{X^{**}} + \\Vert R^{**}(x^{**})\\Vert _{\\ell _1^{**}} \\qquad (x^{**}\\in X^{**}).$ As $R$ is compact, $R^{**}(X^{**})\\subset J_{\\ell _1}(R(X))$ , so to get the strict convexity of the bidual norm we only need to show that $R^{**}$ is one-to-one, but this is consequence of the fact that now the sequence $\\lbrace v_n^*/\\Vert v_n^*\\Vert \\rbrace _{n\\in \\mathbb {N}}$ is norm-dense in $S_{X^*}$ , as this implies that $R^*(\\ell _\\infty )$ is norm dense in $X^*$ .",
"We do not know if for separable Banach spaces, the result above can be improved to get that the Read norm is actually weakly locally uniformly rotund, as it happens for the original Read norm of $c_0$ [22].",
"We finish the section with the following result which appears in [22]: given a Read norm on a separable Banach space, there is another equivalent Read norm which is smooth.",
"One obtains this fact just applying the renorming sketched in Lemma REF ."
],
[
"Applicability of the main construction",
"The aim of this section is to demonstrate that Theorem REF is applicable (after making an appropriate renorming) to all those Banach spaces that contain an isomorphic copy of $c_0$ and have a countable norming system of functionals.",
"A countable norming system of functionals of a Banach space $X$ is a bounded subset $\\lbrace x_n^*\\colon n\\in \\mathbb {N}\\rbrace $ of $X^*$ for which there is a constant $K\\geqslant 0$ such that $\\Vert x\\Vert \\leqslant K\\,\\sup _{n\\in \\mathbb {N}}\\bigl |x_n^*(x)\\bigr | \\qquad (x\\in X).$ Banach spaces with a countable norming system of functionals are those for which there is a bounded subset of the dual with non-empty interior which is weak-star separable or, equivalently, those which are isomorphic to closed subspaces of $\\ell _\\infty $ , see [13] for instance.",
"Our next result shows that the construction of the previous section is applicable to all Banach spaces which are isomorphic to a closed subspace of $\\ell _\\infty $ and contain a copy of $c_0$ ; in particular, it is applicable to separable spaces containing a copy of $c_0$ .",
"Theorem 4.1 Let $X$ be a Banach space containing an isomorphic copy of $c_0$ and possessing a countable norming system of functionals.",
"Then $X$ is isomorphic to a space $\\widetilde{X}$ such that $\\operatorname{span}(\\operatorname{NA}(\\widetilde{X}))$ is weak-star modest in $\\widetilde{X}^*$ .",
"Therefore, we can apply Theorem REF to get that the norm given by (REF ) originating from the norm of $\\widetilde{X}$ is a Read norm.",
"We need a preliminary technical result.",
"Lemma 4.2 Let $X$ be a Banach space containing an isomorphic copy of $c_0$ and possessing a countable norming system of functionals.",
"Then $X$ is isomorphic to a closed subspace $X_1$ of $\\ell _\\infty $ containing the canonical copy of $c_0$ inside $\\ell _\\infty $ .",
"As $X$ is isomorphic to a closed subspace of $\\ell _\\infty $ , we can assume that $X$ itself is a closed subspace of $\\ell _\\infty $ .",
"Denote by $Y_1$ a closed subspace of $X$ that is isomorphic to $c_0$ .",
"According to the Lindenstrauss-Rosenthal theorem [25], for arbitrary isomorphic closed subspaces $Y_1, Y_2$ of $\\ell _\\infty $ such that both $\\ell _\\infty /Y_1$ , $\\ell _\\infty /Y_2$ are non-reflexive, every bijective isomorphism $T \\colon Y_1 \\longrightarrow Y_2$ extends to an automorphism $\\widetilde{T}\\colon \\ell _\\infty \\longrightarrow \\ell _\\infty $ .",
"If we apply this result to our $Y_1$ , to $Y_2 = c_0$ , and to an arbitrary bijective isomorphism $T\\colon Y_1 \\longrightarrow c_0$ (which is possible by [25]), the resulting $X_1 = \\widetilde{T}(X)$ will be the subspace we are looking for.",
"By Lemma REF , we may assume without loss of generality that $c_0 \\subset X \\subset \\ell _\\infty $ .",
"Consider a non-trivial ultrafilter $\\mathfrak {U}$ on $\\mathbb {N}$ and denote by $u$ the linear functional on $\\ell _\\infty $ that assigns to each $x = (x_n)_{n \\in \\mathbb {N}} \\in \\ell _\\infty $ the $\\mathfrak {U}$ -limit of its coordinates: $u(x) = \\lim _\\mathfrak {U}x_n.$ There are two cases: (1) for some non-trivial ultrafilter $\\mathfrak {U}$ our space $X$ lies in the corresponding $\\ker u$ , and (2) $X\\lnot \\subset \\ker u$ for any $\\mathfrak {U}$ .",
"Let us demonstrate that the second case can be reduced to the first one.",
"Indeed, in the second case denote by $R_1\\colon \\ell _\\infty \\longrightarrow \\ell _\\infty $ the right shift operator: $R_1((x_1, x_2, \\ldots )) = (0, x_1, x_2, \\ldots )$ .",
"Then always $R_1(X) \\lnot \\subset \\ker u$ (otherwise $X$ lies in the kernel of the limit with respect to the shifted ultrafilter).",
"Consequently, $R_1(X) \\cap \\ker u$ is a one-codimensional subspace of $R_1(X)\\cong X$ , so $\\widetilde{X}:= \\mathbb {R}e_1 \\oplus (R_1(X) \\cap \\ker u)$ is isomorphic to $X$ .",
"Since $c_0 \\subset \\widetilde{X} \\subset \\ker u$ , the reduction to the first case is completed.",
"So the picture that we are considering is $c_0 \\subset X \\subset \\ker u$ .",
"Since $c_0$ forms an $M$ -ideal of $\\ell _\\infty $ , $c_0$ is also an $M$ -ideal of $X$ [21], that is, $X^* = (c_0)^\\bot \\oplus _1 \\ell _1$ .",
"Then $\\operatorname{NA}(X) \\subset \\bigl [(c_0)^\\bot \\cap \\operatorname{NA}(X)\\bigr ] \\oplus _1 \\bigl [\\ell _1 \\cap \\operatorname{NA}(X)\\bigr ] \\subset (c_0)^\\bot \\oplus _1 \\bigl [\\ell _1 \\cap \\operatorname{NA}(X)\\bigr ],$ where in the first inclusion we use the elementary fact that if $f +g$ with $\\Vert f+g\\Vert = \\Vert f\\Vert + \\Vert g\\Vert $ attains its norm, then both $f$ and $g$ attain their norms.",
"If a non-zero element $f = (f_1, f_2, \\ldots ) \\in \\ell _1$ attains its norm at some $x = (x_1, x_2, \\ldots ) \\in S_X$ , then for all $n$ where $f_n \\ne 0$ we have $|x_n| = 1$ .",
"Since $\\lim _\\mathfrak {U}x_n = 0$ , this means that for every element $f \\in \\ell _1 \\cap \\operatorname{NA}(X)$ the set $\\lbrace n\\in \\mathbb {N}\\colon f_n =0 \\rbrace $ belongs to $\\mathfrak {U}$ .",
"Any linear combination of elements of $\\ell _1 \\cap \\operatorname{NA}(X)$ will have the same property.",
"Let $Y \\subset \\ell _1$ be the dense operator range from Lemma REF .",
"Since $\\ell _1$ is weak-star dense in $X^*$ , this $Y$ is also weak-star dense in $X^*$ .",
"Every non-zero element of $Y$ has a finite number of zero coordinates, but for $f\\in Y \\cap \\operatorname{span}(\\operatorname{NA}(X))$ , the number of zero coordinates is infinite by the above discussion.",
"Consequently $Y \\cap \\operatorname{span}(\\operatorname{NA}(X)) \\subset Y \\cap \\operatorname{span}(\\ell _1 \\cap \\operatorname{NA}(X)) = \\lbrace 0\\rbrace $ .",
"This demonstrates that $\\operatorname{span}(\\operatorname{NA}(X))$ is weak-star modest in $X^*$ .",
"If $X$ is actually separable, things may be done in an easier fashion; and in the case when $X^*$ is separable we get a stronger result.",
"We state the result here.",
"Proposition 4.3 Let $X$ be a separable Banach space containing $c_0$ .",
"Then, there is an equivalent norm $q$ on $X$ such that, in this norm, $(X, q)=c_0 \\oplus _\\infty Z$ for some $Z$ and $\\operatorname{span}(\\operatorname{NA}((X, q))) \\subset \\operatorname{NA}(c_0) \\oplus Z^*$ is weak-star modest.",
"If moreover $X^*$ is separable, then $\\operatorname{span}(\\operatorname{NA}((X, q)))$ is actually modest.",
"Therefore, we can apply Theorem REF to get that the norm given by (REF ) is a Read norm.",
"This is just a consequence of Sobczyk's theorem (see [4]) and Proposition REF .",
"Our next aim is to give geometric properties of the Read norms that we have constructed in this section, which extends those results given in [22] for the original Read space.",
"The first result contains all the geometric properties of the Read norms in Theorem REF and Proposition REF we know about.",
"Proposition 4.4 Let $X$ be a Banach space containing $c_0$ and having a countable norming system of functionals.",
"Then, for every $0<\\varepsilon <2$ , there is an equivalent Read norm $p_\\varepsilon $ on $X$ satisfying the following: $(X,p_\\varepsilon )$ is strictly convex; every convex combination of slices of the unit ball of $(X,p_\\varepsilon )$ has diameter $\\geqslant 2-\\varepsilon $ , so every relatively weakly open subset of the unit ball of $(X,p_\\varepsilon )$ has diameter $\\geqslant 2-\\varepsilon $ ; the norm of $(X,p_\\varepsilon )^*$ is $(2-\\varepsilon )$ -rough.",
"Moreover, if $X^*$ is separable, then $(X,p_\\varepsilon )^{**}$ is strictly convex, so $(X,p_\\varepsilon )^*$ is smooth.",
"We need a couple of preliminary results for the proof which have their own interest.",
"The first is surely known, but we include an elementary proof for the sake of completeness.",
"Lemma 4.5 Let $X$ be a closed subspace of $\\ell _\\infty $ which contains the canonical copy of $c_0$ .",
"Then, given $x\\in B_X$ there are two sequences $\\lbrace y_n\\rbrace $ , $\\lbrace z_n\\rbrace $ in $S_X$ that both converge weakly to $x$ and such that $e_n^*(y_n-z_n)=2$ for every $n\\in \\mathbb {N}$ , where $e_n^*$ denotes the $n$ -th coordinate functional on $X$ .",
"Let $X$ be a Banach space such that $X=c_0\\oplus _\\infty Z$ for some closed subspace $Z$ .",
"Then there is a sequence $\\lbrace f_n\\rbrace $ in $S_{X^*}$ such that given $x\\in B_X$ there are two sequences $\\lbrace y_n\\rbrace $ , $\\lbrace z_n\\rbrace $ in $S_X$ which converge weakly to $x$ and such that $f_n(y_n-z_n)=2$ for every $n\\in \\mathbb {N}$ .",
"For the first part, just define $y_n= x + (1-x(n))e_n$ and $z_n = x - (1 + x(n))e_n$ for every $n\\in \\mathbb {N}$ , where $e_n$ is the $n$ -th element of the canonical basis of $c_0$ .",
"Then, $\\lbrace y_n\\rbrace $ , $\\lbrace z_n\\rbrace $ are contained in $S_X$ , both converge weakly to $x$ , and $e_n^*(y_n-z_n)=2$ for every $n\\in \\mathbb {N}$ .",
"The second part is equally easy: consider $f_n=(e_n^*,0)\\in X^*$ for every $n\\in \\mathbb {N}$ .",
"Given $x=(u,z)$ with $u\\in B_{c_0}$ and $z\\in B_Z$ , the sequences $\\lbrace (u + (1-u(n))e_n,z)\\rbrace \\qquad \\text{and} \\qquad \\lbrace (u - (1+u(n))e_n,z)\\rbrace $ fulfill all of our requirements.",
"The next preliminary result allows to transfer properties of a given norm to the norm constructed by (REF ).",
"Lemma 4.6 Let $X$ be a Banach space and suppose that there is a sequence $\\lbrace f_n\\rbrace $ in $S_{X^*}$ such that for every $x\\in B_X$ , there are two sequences $\\lbrace y_n\\rbrace $ , $\\lbrace z_n\\rbrace $ in $S_X$ which converge weakly to $x$ and such that $f_n(y_n-z_n)=2$ for every $n\\in \\mathbb {N}$ .",
"Let $R\\colon X\\longrightarrow Y$ be a compact operator from $X$ to some Banach space $Y$ and define an equivalent norm on $X$ by $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|= \\Vert x\\Vert _X + \\Vert R(x)\\Vert _Y \\qquad (x\\in X).$ Then, there is a sequence $\\lbrace g_n\\rbrace $ in the unit sphere of $(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)^*$ such that given $x\\in X$ with $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=1$ , there exist two sequences $\\lbrace \\tilde{y}_n\\rbrace $ , $\\lbrace \\tilde{z}_n\\rbrace $ in the unit ball of $(X,|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\cdot |\\hspace{-1.111pt}|\\hspace{-1.111pt}|)$ that both converge weakly to $x$ and such that $\\lim _n g_n(\\tilde{y}_n - \\tilde{z}_n)\\geqslant 2(1+\\Vert R\\Vert )^{-1}$ .",
"We have that $1=|\\hspace{-1.111pt}|\\hspace{-1.111pt}|x|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=\\Vert x\\Vert _X + \\Vert R(x)\\Vert _Y \\leqslant (1 +\\Vert R\\Vert )\\Vert x\\Vert _X,$ so $\\Vert x\\Vert _X\\geqslant (1+\\Vert R\\Vert )^{-1}$ .",
"By hypothesis, we may take two sequences $\\lbrace y_n\\rbrace $ , $\\lbrace z_n\\rbrace $ in $X$ both converging weakly to $x$ and a sequence $\\lbrace f_n\\rbrace $ in $S_{X^*}$ such that $f_n(y_n-z_n)=2\\Vert x\\Vert $ , $\\Vert y_n\\Vert =\\Vert z_n\\Vert =\\Vert x\\Vert $ and $\\Vert y_n-z_n\\Vert =2\\Vert x\\Vert $ for every $n\\in \\mathbb {N}$ .",
"As $R$ is compact, we have that $\\lim R y_n=\\lim R z_n=Rx$ , so $\\lim \\Vert R y_n\\Vert = \\lim \\Vert R z_n\\Vert =\\Vert Rx\\Vert \\quad \\text{and} \\quad \\lim \\Vert R(y_n - z_n)\\Vert =0.$ Therefore, $\\lim |\\hspace{-1.111pt}|\\hspace{-1.111pt}|y_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=\\lim |\\hspace{-1.111pt}|\\hspace{-1.111pt}|z_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|=1$ and $\\lim |\\hspace{-1.111pt}|\\hspace{-1.111pt}|y_n - z_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|\\geqslant 2\\Vert x\\Vert $ .",
"Also, $|\\hspace{-1.111pt}|\\hspace{-1.111pt}|f_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|^*\\leqslant \\Vert f_n\\Vert ^*=1$ .",
"Finally, the sequences $\\tilde{y}_n =|\\hspace{-1.111pt}|\\hspace{-1.111pt}|y_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|^{-1}y_n$ , $\\tilde{z}_n = |\\hspace{-1.111pt}|\\hspace{-1.111pt}|z_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|^{-1}z_n$ and $g_n=|\\hspace{-1.111pt}|\\hspace{-1.111pt}|f_n|\\hspace{-1.111pt}|\\hspace{-1.111pt}|^{-1}f_n$ fulfill all of our requirements.",
"We are now ready to give the proof of Proposition REF .",
"We start by using Lemma REF and (the proof of) Theorem REF to get an equivalent norm on $X$ such that $c_0\\subset X \\subset \\ell _\\infty $ isometrically, where $c_0$ is the canonical copy, and such that $\\operatorname{span}(\\operatorname{NA}(X))$ is weak-star modest.",
"Next, for $0<\\varepsilon <2$ , we consider an operator $R_\\varepsilon $ defined by (REF ) from Remark REF with $\\Vert R_\\varepsilon \\Vert <\\frac{\\varepsilon }{2-\\varepsilon }$ , and consider the norm $p_\\varepsilon (x)=\\Vert x\\Vert + \\Vert R_\\varepsilon (x)\\Vert _{\\ell _1} \\qquad (x\\in X),$ which is a Read norm.",
"By Proposition REF , $(X,p_\\varepsilon )$ is strictly convex, so this gives (a).",
"To get (b), we just have to apply Lemmas REF and REF .",
"Indeed, let $\\lbrace g_n\\rbrace $ be the sequence in the unit sphere of $(X, p_\\varepsilon )^*$ given by Lemma REF .",
"Consider $C=\\sum _{i=1}^Nt_iS_i$ , a convex combination of slices in the unit ball of $(X,p_\\varepsilon )$ , and $x_0\\in C$ .",
"We write $x_0=\\sum _{i=1}^N t_i x_i$ where $x_i\\in S_i$ for every $i$ .",
"There is no loss of generality if we assume that $p_\\varepsilon (x_i)\\geqslant 1-\\delta $ for every $i$ , where $\\delta $ is a positive number as small as we want.",
"By using Lemma REF again, we get that for every $i$ there are sequences $\\lbrace y_n^{i}\\rbrace $ and $\\lbrace z_n^{i}\\rbrace $ in the unit ball of $(X, p_\\varepsilon )$ both weakly converging to $x_i$ and such that $\\lim _ng_n(y_n^{i}-z_n^{i})\\geqslant (1-\\delta )(2-\\varepsilon ) $ .",
"Therefore, for large enough $n$ , we have that $\\sum _{i=1}^N t_i y_n^{i}$ , $\\sum _{i=1}^N t_i z_n^{i}$ are elements in $C$ with distance, at least, $(1-2\\delta )(2-\\varepsilon )$ .",
"As $\\delta $ is arbitrary, we conclude that the diameter of $C$ is, at least, $2-\\varepsilon $ .",
"Finally, every relatively weakly open subset of a unit ball contains a convex combination of slices (a result due to Bourgain, see [9]), and this gives the last part of (b).",
"Item (c) is a consequence of (b) by using [15].",
"If $X^*$ is separable, we may suppose that $X=c_0\\oplus _\\infty Z$ for some Banach space $Z$ and we use Proposition REF to get that this norm makes $\\operatorname{span}(\\operatorname{NA}(X))$ modest.",
"Now, for $0<\\varepsilon <2$ , we follow the same process as before to construct the norm $p_\\varepsilon $ .",
"Again, Proposition REF gives (a) and Lemmas REF and REF give (b), and [15] gives (c) from (b).",
"Finally, (d) is a consequence of Proposition REF since now $\\operatorname{span}(\\operatorname{NA}(X))$ is actually modest.",
"In the separable case, we may get Read norms with better properties by using a convenient renorming from [14] which was used in [22] for the original Read norm.",
"Proposition 4.7 Let $X$ be a separable Banach space containing $c_0$ .",
"Then, for every $0<\\varepsilon <2$ , there is an equivalent Read norm $q_\\varepsilon $ on $X$ such that: $(X,q_\\varepsilon )$ is strictly convex; $(X,q_\\varepsilon )^*$ is strictly convex, so $(X, q_\\varepsilon )$ is smooth; $(X,q_\\varepsilon )^*$ is $(2-\\varepsilon )$ -rough, equivalently, every slice of the unit ball of $(X,q_\\varepsilon )$ has diameter $\\geqslant 2-\\varepsilon $ .",
"Moreover, if $X^*$ is separable, then $(X,q_\\varepsilon )^{**}$ is strictly convex, so $(X,q_\\varepsilon )^*$ is smooth.",
"We fix a dense subset $\\lbrace x_n\\colon n\\in \\mathbb {N}\\rbrace $ of $B_X$ , and for every $0<\\rho <2$ , we define the bounded linear operator $S_\\rho \\colon \\ell _2\\longrightarrow X$ by $S_\\rho (\\lbrace a_n\\rbrace )=\\rho \\sum _{n=1}^\\infty \\tfrac{a_n}{2^n}\\,x_n$ for every $\\lbrace a_n\\rbrace \\in \\ell _2$ , which satisfies that $\\Vert S_\\rho \\Vert \\leqslant \\rho $ .",
"For $0<\\varepsilon <2$ , we take $0<\\varepsilon ^{\\prime }<\\varepsilon $ and $\\rho >0$ such that $(2-\\varepsilon ^{\\prime })(1+\\rho )^{-1}>2-\\varepsilon $ , we consider the norm $p_{\\varepsilon ^{\\prime }}$ from Proposition REF , and we define the equivalent norm $q_\\varepsilon $ on $X$ to be the one for which $B_{(X,q_\\varepsilon )}= B_{(X,p_{\\varepsilon ^{\\prime }})} + S_\\rho (B_{\\ell _2}).$ First, Lemma REF (d) gives that $\\operatorname{NA}(X,q_\\varepsilon )=\\operatorname{NA}(X,p_{\\varepsilon ^{\\prime }})$ and so $q_\\varepsilon $ is a Read norm.",
"It follows from Lemma REF (a) that $q_\\varepsilon (f)=p_{\\varepsilon ^{\\prime }}(f) + \\Vert S_\\rho ^*(f)\\Vert _2$ for every $f\\in X^*$ .",
"As $\\ell _2$ is strictly convex and $T^*$ is one-to-one, it follows from Lemma REF (d) that $(X,q_\\varepsilon )^*$ is strictly convex, so $(X,q_\\varepsilon )$ is smooth, giving (b).",
"Lemma REF (c) gives that $(X,q_\\varepsilon )$ is strictly convex since both $(X,p_{\\varepsilon ^{\\prime }})$ and $\\ell _2$ are; hence (a) holds.",
"Finally, we know from Proposition REF that $(X,p_{\\varepsilon ^{\\prime }})^*$ is $(2-\\varepsilon ^{\\prime })$ -rough, and then Lemma REF (e) gives that $(X,q_\\varepsilon )$ is $(2-\\varepsilon ^{\\prime })(1+\\rho )^{-1}$ -rough, which gives the first part of (c) due to the way in which we have chosen the constants $\\varepsilon ^{\\prime }$ and $\\rho $ .",
"Finally, the second part of (c) is equivalent to the first one by [15].",
"If moreover $X^*$ is separable, as $B_{(X,q_\\varepsilon )^{**}}= B_{(X,p_{\\varepsilon ^{\\prime }})^{**}} + J_X(S_\\rho (B_{\\ell _2}))$ by Lemma REF (b) and $(X,p_{\\varepsilon ^{\\prime }})^{**}$ is strictly convex by Proposition REF , the strict convexity of $(X,q_{\\varepsilon })^{**}$ follows from Lemma REF (c)."
],
[
"Limits of our construction",
"The main open problem related to Read norms is the following one.",
"Problem 5.1 Does every non-reflexive separable Banach space admit an equivalent norm such that the set of norm attaining functionals contains no linear subspaces of dimension two?",
"Remark that for non-reflexive non-separable Banach spaces the answer to the above problem is negative.",
"Indeed, every renorming $E$ of $\\ell _\\infty (\\Gamma )$ with uncountable $\\Gamma $ contains an isometric copy of $\\ell _\\infty (\\mathbb {N})$ [27].",
"This copy is one-complemented, so $\\operatorname{NA}(E) \\supset \\operatorname{NA}(\\ell _\\infty (\\mathbb {N}))$ , which in turn contains an infinite-dimensional linear subspace, viz., $\\ell _1(\\mathbb {N})$ .",
"Taking into account that $\\ell _\\infty (\\Gamma )$ is a $C(K)$ space, it is natural to ask the following question.",
"Problem 5.2 What is the description of those compacts $K$ for which the corresponding $C(K)$ admits an equivalent norm in which the set of norm attaining functionals contains no linear subspaces of dimension two?",
"We do not know whether the answer to Problem REF is positive, but we would like to discuss the reasons why our construction cannot provide such a positive answer.",
"Observe that the key in our construction is that $X^*\\setminus \\operatorname{span}(\\operatorname{NA}(X))$ is big enough to contain a weak-star dense separable operator range.",
"It is known that this is not possible for Banach spaces with the Radon-Nikodým property or with an almost LUR norm, as the following result of Bandyopadhyay and Godefroy shows.",
"Proposition 5.3 ([6]) Let $X$ be a Banach space with the Radon-Nikodým property or with an almost LUR norm.",
"Then $\\operatorname{span}(\\operatorname{NA}(X))=X^*$ .",
"Therefore, the main open question related to our construction is the following one.",
"Problem 5.4 Does every Banach space with weak-star separable dual and failing the Radon-Nikodým property admit an equivalent norm for which the linear span of the set of norm attaining functionals is weak-star modest in the dual space?",
"We don't even know the answer for the space $L_1[0,1]$ .",
"Let us comment that the proof of Proposition REF is a consequence of the fact that $\\operatorname{NA}(X)$ contains a dense $G_\\delta $ subset of $X^*$ when $X$ has the Radon-Nikodým property (see Theorem 8 in [8], for instance) or $X$ has an almost LUR norm ([6]), so $\\operatorname{NA}(X)$ is residual in this case.",
"Actually, this latter hypothesis is sufficient to get that $\\operatorname{NA}(X)-\\operatorname{NA}(X)=X^*$ from the Baire category theorem.",
"We include the next result, which is contained in the proof of [6], for completeness.",
"Proposition 5.5 ([6]) Let $X$ be a Banach space.",
"If $B$ is a residual subset of $X$ , then $B-B=X$ and so $\\operatorname{span}(B)=X$ .",
"In particular, if $\\operatorname{NA}(X)$ is residual in $X^*$ then $\\operatorname{span}(\\operatorname{NA}(X))=X^*$ .",
"We just have to note that for every $x\\in X$ , $\\bigl (x+B\\bigr )\\cap B$ is not empty since, otherwise, the second category set $x+B$ would be contained in the first category set $X\\setminus B$ , which is impossible.",
"Let us comment that the converse result to the above one is not true: for $X=L_1[0,1]$ , $\\operatorname{NA}(X)$ is of the Baire first category (so it cannot be residual), but $\\operatorname{span}(\\operatorname{NA}(X))=X^*$ (a description of $\\operatorname{NA}(L_1[0,1])$ can be found in [3]).",
"Therefore, our construction is not applicable to $X=L_1[0,1]$ in its usual norm, but we do not know whether it could be the case in some renorming.",
"Actually, it is known that every separable Banach space failing the Radon-Nikodým property can be renormed in such a way that the set of norm attaining functionals is of the first Baire category (see the proof of [10]) but, as the previous example shows, this does not imply that the linear span of the set of norm attaining functionals is also of the first Baire category.",
"Remark also that a similar argument to the proof of Proposition REF can give us the following curious result.",
"Proposition 5.6 Let $X$ be an infinite-dimensional Banach space.",
"If $B$ is a residual subset of $X$ such that $tx\\in B$ for every $x\\in B$ and $t\\in \\mathbb {Q}$ , then $B$ contains an infinite sequence of linearly independent elements whose linear span over the field $\\mathbb {Q}$ lies in $B$ .",
"In particular, if $\\operatorname{NA}(X)$ is residual in $X^*$ , then $\\operatorname{NA}(X)$ contains the $\\mathbb {Q}$ -linear span of an $\\mathbb {R}$ -linearly independent infinite sequence.",
"Take $0\\ne x_1\\in B$ .",
"Assume, inductively, that linearly independent elements $x_1,\\ldots ,x_n\\in B$ have been constructed so that the set $\\lbrace x_1,\\ldots ,x_n\\rbrace $ is linearly independent and the $\\mathbb {Q}$ -linear span of the set $\\lbrace x_1,\\ldots ,x_n\\rbrace $ lies in $B$ .",
"Consider $E = \\bigcap _{r_1,\\ldots ,r_n \\in \\mathbb {Q}}\\Bigl (B -\\sum _{i=1}^n r_i x_i^*\\Bigr ).$ This is a residual subset of $B$ , so it contains an element $x_{n+1}\\in B$ which is linearly independent from the set $\\lbrace x_1,\\ldots ,x_n\\rbrace $ .",
"Indeed, if not then $Z:=\\operatorname{span}(x_1,\\ldots ,x_n)$ contains $E$ and is hence residual; but $Z$ is a nowhere dense set, being a closed and proper subspace of $X$ , which is impossible by the Baire category theorem.",
"According to the definition of $E$ , the condition $x_{n+1}\\in E$ implies that $x_{n+1}+\\sum _{i=1}^n r_i x_n\\in B$ for every $r_1,\\ldots ,r_n\\in \\mathbb {Q}$ .",
"Thus we get the required infinite sequence $\\lbrace x_n\\rbrace $ of linearly independent elements in $B$ .",
"With the above result in mind, which can be applied to $\\operatorname{NA}(X)$ for Banach spaces $X$ with the Radon-Nikodým property or with an almost LUR norm, it would be nice to know if there is some Banach space $X$ so that $\\operatorname{NA}(X)$ is residual, but still $\\operatorname{NA}(X)$ does not contain two-dimensional subspaces.",
"Remark also that as a consequence of Proposition REF in combination with Theorem REF , for a general Banach space $X$ , if $\\operatorname{span}(\\operatorname{NA}(X))$ is weak-star modest then we again get that $\\operatorname{NA}(X)$ is not residual.",
"Also, the following result of Fonf and Lindenstrauss [19] is worth mentioning: for every non-reflexive Banach space $X$ , $X^*\\setminus \\operatorname{NA}(X)$ is not a subset of a proper operator range, equivalently, $\\operatorname{span}\\bigl (X^*\\setminus \\operatorname{NA}(X)\\bigr )$ is dense and barrelled (use [26] for the equivalence).",
"On the other hand, for separable Banach spaces, if $\\operatorname{span}(\\operatorname{NA}(X))$ is modest (or weak-star modest), then $\\operatorname{span}(\\operatorname{NA}(X))$ has to be of the first Baire category, as we may prove using a theorem of Banach.",
"Proposition 5.7 Let $X$ be a separable Banach space.",
"If $\\operatorname{span}(\\operatorname{NA}(X))$ is of the second Baire category in $X^*$ , then $\\operatorname{span}(\\operatorname{NA}(X))=X^*$ .",
"The argument relies on notions and results from descriptive set theory that we'll recall in the course of the proof.",
"A Polish space is a completely metrisable separable topological space, and an analytic set is a subset of a topological space which is a continuous image of some Polish space.",
"Since $X$ is separable, $\\operatorname{NA}(X)$ is an analytic subset of $X^*$ equipped with the weak-star topology; see [23].",
"We shall argue that $\\operatorname{span}(\\operatorname{NA}(X))$ is analytic as well.",
"For $n\\in \\mathbb {N}$ define $f_n\\colon \\operatorname{NA}(X)^n \\times \\mathbb {R}^n\\rightarrow X^*$ by $f_n(x_1^*,\\dots ,x_n^*, t_1,\\dots ,t_n) = \\sum _{k=1}^n t_k x_k^*;$ then $\\operatorname{span}(\\operatorname{NA}(X)) = \\bigcup _{n\\in \\mathbb {N}} f_n( \\operatorname{NA}(X)^n \\times \\mathbb {R}^n ).$ Now the class of analytic sets is closed under taking finite (even countable) products, continuous images, and countable unions; therefore $\\operatorname{span}(\\operatorname{NA}(X))$ is indeed analytic for the weak-star topology.",
"In a Hausdorff topological space, analytic sets are known to be $F$ -Souslin [7], that is, they can be represented as $\\bigcup _\\sigma \\bigcap _{n=1}^\\infty F_{\\sigma _1,\\dots ,\\sigma _n}$ for closed sets $F_{\\sigma _1,\\dots ,\\sigma _n} $ where the union is taken over all sequences $\\sigma = (\\sigma _1,\\sigma _2, \\dots )$ of positive integers.",
"Hence $\\operatorname{span}(\\operatorname{NA}(X))$ is $F$ -Souslin for the weak-star topology and therefore also for the norm topology.",
"The next piece of information that we need concerns the Baire property.",
"A subset of a topological space has the Baire property if it differs from an open set by a set of the first category; that is, if it can be written in the form $G \\mathbin {\\Delta } M$ with $G$ open and $M$ of the first category where $\\Delta $ denotes the symmetric difference.",
"In a Hausdorff space, every $F$ -Souslin set has the Baire property; see [30].",
"Consequently, $\\operatorname{span}(\\operatorname{NA}(X))$ has the Baire property for the norm topology.",
"Finally we apply a theorem due to Banach ([5] or [24]) which assures that, in a Banach space, a linear subspace of the second Baire category which satisfies the Baire property is the whole space.",
"We are grateful to W. Moors for indicating the above argument to us; in a preliminary version of this paper we had to make the far stronger assumption that $X^*$ is separable.",
"We do not know whether separability can be dropped from Proposition REF .",
"As a consequence of Proposition REF , if $X$ is separable and $\\operatorname{span}(\\operatorname{NA}(X))$ is weak-star modest, then $\\operatorname{span}(\\operatorname{NA}(X))$ has to be of the first Baire category.",
"We do not know whether the converse is also true, but there is a partial answer.",
"Proposition 5.8 Let $X$ be a Banach space such that $X^*$ is separable.",
"If $\\operatorname{span}(\\operatorname{NA}(X))$ is not barrelled, then $\\operatorname{span}(\\operatorname{NA}(X))$ is modest (and so, $X$ admits an equivalent Read norm).",
"By [33], it follows that $\\operatorname{span}(\\operatorname{NA}(X))$ is contained in a (dense) proper operator range.",
"Now, [31] shows that there is a dense operator range $Y$ in $X^*$ such that $Y\\cap \\operatorname{span}(\\operatorname{NA}(X))=\\lbrace 0\\rbrace $ , that is, $\\operatorname{span}(\\operatorname{NA}(X))$ is modest.",
"We have to mention that this result does not produce new examples of spaces which admit Read norms, as the following result by Fonf shows: if $\\operatorname{span}(\\operatorname{NA}(X))$ is not barrelled, then $X$ contains a copy of $ c_0$ (see [26] for a version of Fonf's result using this language).",
"Let us note that the task to find a Banach space $X$ with $X^*$ separable such that $\\operatorname{span}(\\operatorname{NA}(X))$ is weak-star modest and $X$ does not contain $c_0$ , requires to find a Banach space $X$ such that $\\operatorname{span}(\\operatorname{NA}(X))$ is of the first Baire category and barrelled.",
"Finally, it would be interesting to find examples of Banach spaces $X$ for which $\\operatorname{span}(\\operatorname{NA}(X))$ is modest (or weak-star modest) in their usual norm, as it happens with $c_0$ .",
"Another example of this kind is given in the papers [1], [2] by Acosta: let $w\\in \\ell _2\\setminus \\ell _1$ with $0<w_n<1$ for all $n$ and consider the space $Z$ of sequences $z$ of scalars for which $\\Vert z\\Vert :=\\Vert (1-w)z\\Vert _\\infty \\,+\\,\\Vert wz\\Vert _{\\ell _1} <\\infty $ endowed with this function as norm.",
"Then, the sequence $\\lbrace e_n\\rbrace $ of unit vectors is a 1-unconditional basic sequence of $Z^*$ whose closed linear span $X(w)$ is an isometric predual of $Z$ for which $\\lbrace e_n\\rbrace $ is a 1-unconditional basis whose biorthogonal basis $\\lbrace e_n^*\\rbrace $ is again the canonical unit vector basis [1].",
"Then, $\\operatorname{span}(\\operatorname{NA}(X(w)))$ is modest in $X^*$ .",
"Indeed, it is shown in [2] that if $x^*\\in X(w)^*$ belongs to $\\operatorname{NA}(X(w)^*)$ , then $w\\chi _{\\operatorname{supp}(x^*)}\\in \\ell _1$ ; if we consider the bounded linear operator $T\\colon \\ell _1 \\longrightarrow X(w)^*$ given by $T(e_n)=e_n^*$ for every $n\\in \\mathbb {N}$ , it follows, as $w\\notin \\ell _1$ , that $T(Y)\\cap \\operatorname{span}(\\operatorname{NA}(X))=\\lbrace 0\\rbrace $ where $Y$ is the operator range of $\\ell _1$ given in Lemma REF .",
"Let us observe that $X(w)$ contains a copy of $c_0$ (since the basis is unconditional and shrinking and the space is not reflexive), so we already know from Section that it admits an equivalent Read norm."
]
] | 1709.01756 |
[
[
"Electric fields at finite temperature"
],
[
"Abstract Partial differential equations for the electric potential at finite temperature, taking into account the thermal Euler-Heisenberg contribution to the electromagnetic Lagrangian are derived.",
"This complete temperature dependence introduces quantum corrections to several well known equations such as the Thomas-Fermi and the Poisson-Boltzmann equation.",
"Our unified approach allows at the same time to derive other similar equations which take into account the effect of the surrounding heat bath on electric fields.",
"We vary our approach by considering a neutral plasma as well as the screening caused by electrons only.",
"The effects of changing the statistics from Fermi-Dirac to the Tsallis statistics and including the presence of a magnetic field are also investigated.",
"Some useful applications of the above formalism are presented."
],
[
"Introduction",
"A class of nonlinear Poisson equations of the form $\\mathbf {\\mathbf {\\nabla }}^2\\Phi =F(\\Phi ,T, \\mathbf {r})$ (with $F$ a function) which take into account the effects (like the temperature dependence $T$ ) of the surrounding matter on the electric potential $\\Phi $ play an important role in many branches of physics.",
"We mention here the Thomas-Fermi equation which finds its applications in atomic physics [1], astrophysics [2] and solid states physics [3] and Poisson-Boltzmann equation applied in plasma physics [4] and solutions [5].",
"The derivation of these equations is seemingly unrelated and yet, as shown in this work, they are based on one and the same principle.",
"Feynman, Metropolis and Teller [6] have have postulated a self-consistent Poisson-like equation of the form $\\mathbf {\\mathbf {\\nabla }}^2 \\Phi =\\int d^3p FD(\\Phi , T, \\mathbf {p})$ , where $FD$ stands for the Fermi-Dirac distribution, from which the Thomas-Fermi, Poisson-Boltzmann and other similar equations can be derived.",
"We use this unifying principle to calculate quantum corrections to these nonlinear Poisson equations.",
"These corrections arise when we use the Quantum Electrodynamics (QED) at finite temperature to calculate the first quantum corrections to the classical electrodynamics known as the Euler-Heisenberg theory (in our case at finite $T$ ).",
"The Euler-Heisenberg theory at finite $T$ brings yet another temperature dependence of the electric potential.",
"To be specific, the QED effective Lagrangian in the presence of a thermal bath and arbitrary slowly varying electromagnetic field can be written as $\\mathcal {L}=\\frac{E^{2}-B^{2}}{8\\pi }+\\mathcal {L}_{EH}^{0}(\\mathbf {E},\\mathbf {B})+\\mathcal {L}_{EH}^{T}(\\mathbf {E},\\mathbf {B};T),$ where $\\mathcal {L}_{EH}^{0}(\\mathbf {E},\\mathbf {B})$ is the zero temperature effective Lagrangian of QED [7], [8], [9] giving rise to new effects like vacuum birefringence [10], [11], [12], vacuum dichroism [15], corrections to the Lorentz force [16], corrections to the field and energy of point charges [13], [14] among others (see [17], [18] for comprehensive reviews); and $\\mathcal {L}_{EH}^{T}(\\mathbf {E},\\mathbf {B};T)$ is the contribution from the thermal bath to the effective Lagrangian [19].",
"The Lagrangian (REF ) gives rise to modifications of the Maxwell's equations that can be used to study electromagnetic phenomena that occur beyond the classical electrodynamics [20].",
"The initial investigation about the finite temperature effective Lagrangian was done by Dittrich [21].",
"Further developments were made in [22], [23], [24], [25], [26], [27], [28], [29].",
"In particular, Ref [27] was the first one to show that, at temperatures below $m_{e}$ (mass of the electron), the two loop contributions dominate over the one loop term.",
"A review and an expanded bibliography can be found in [19].",
"Among the applications of the finite temperature Lagrangian, we can find the study of thermally induced photon splitting, [31], thermally induced pair production [32], [33], [34], and the velocity shift of light in thermalized media [35], [36], [37] (see [19] for more references).",
"Classical (or semi-classical) methods have been developed for the study of matter at laboratory conditions or plasmas in stars, supernovas, and even the electron-positron plasma at an early stage of the big-bang [38].",
"We refine these methods by including the effects of the QED effective Lagrangian (REF ).",
"We do so by implementing the effects of the Euler-Heisenberg theory via the modified Gauss law in the Poisson-like equations.",
"The set of the latter encompasses known equations (like the Thomas-Fermi or Poisson-Boltzmann, now equipped with quantum corrections) as well as new equations which will be derived in this work.",
"The paper is organized as follows.",
"In the next section we present the low temperature and high temperature approximation for the Euler-Heisenberg effective Lagrangian and we will discuss the way of incorporating the effective Lagrangian into the equations of classical electrodynamics.",
"In section III we calculate the correction to the electrostatic potential of point-like and extended charged objects when the charge density is given.",
"An explicit solution of an electric field at finite temperature due to the Euler-Heisenberg theory is given.",
"This solution neglects the fact that the particles surrounding the charge whose potential we wish to calculate can also be in a heat bath.",
"However, the solution is part of a more general treatment where it appears in the boundary conditions.",
"In section IV we focus on the temperature dependent charge densities.",
"We shall write the charge density with two separated terms as $\\rho =\\rho _{c}+\\rho _{m}$ where $\\rho _{c}$ is the density of the object whose effective electrostatic potential we want to compute and $\\rho _{m}$ is the charge density of the surrounding media.",
"This results in the Feynman-Metropolis-Teller equation.",
"We discuss several limiting cases of this master equation treating the degenerate and non-degenerate cases and carefully distinguishing between the relativistic and non-relativistic situation and the high and low temperature cases.",
"Taking into account the corrections from the Euler-Heisenberg theory we derive several nonlinear Poisson-like equations at finite temperature.",
"Section V is devoted to the “relatives” of the Thomas-Fermi equation, namely equations derived under a change of assumptions.",
"In the first case we change the Fermi-Dirac distribution for the Tsallis statistics and the second case considers a Thomas-Fermi equation in the presence of a magnetic field.",
"In the section VI we discuss two possible applications, one connected with tunneling in the presence of a surrounding heat bath and the second one treating an electron-positron neutral plasma.",
"In the last section we draw our conclusions."
],
[
"Euler Heisenberg Lagrangian for Low and High Temperature",
"The full expression of both the zero temperature and the thermal Euler-Heisenberg Lagrangian is very complex.",
"In this work we shall concentrate on some special cases where the effective Lagrangian can be approximated by more manageable expressions.",
"First, we shall deal only with electromagnetic fields that are weak compared to the so called critical field $B_{c}=e^{2}/m_{e}$ .",
"Secondly, all the fields are considered to be slowly varying compared to the scales of the problem, i.e., the fields obey $\\left|\\partial _{a}F_{\\mu \\nu }\\right|/\\left|F_{\\mu \\nu }\\right|^{2}\\ll \\left|2F_{\\mu \\nu }\\right|^{1/2}$ , $m_{e}$ , $T$ and $\\eta ^{1/3}$ , where $\\eta $ is the particle density.",
"Finally, for the thermal Lagrangian, we will only consider the two limiting cases of temperature much bigger or much smaller than the electron mass.",
"With the above restrictions the zero temperature Lagrangian can be written as [7], [9], [8], $\\mathcal {L}_{EH}^{0}=a\\left(4\\mathcal {F}^{2}+7\\mathcal {G}^{2}\\right),$ where $a=\\frac{e^{4}}{360\\pi ^{2}m_{e}^{4}},$ and the two relativistic invariants of the electromagnetic fields are given by $\\mathcal {F} & = & -\\frac{E^{2}-B^{2}}{2},\\\\\\mathcal {G} & = & \\mathbf {E\\cdot B}.$ As mentioned in the introduction, for temperatures below the electron mass $(T\\ll m_{e})$ , the dominant contribution in the thermal Lagrangian comes from the two loop term [27].",
"To quadratic order in the field invariants, the weak field expansion for the two loop Lagrangian is [27], $\\mathcal {L}_{EH}^{T}(T\\ll m_{e}) & = & b\\left(\\mathcal {F}+\\mathcal {E}\\right)-c\\mathcal {F}\\left(\\mathcal {F}+\\mathcal {E}\\right)\\nonumber \\\\& & +k\\left(2\\mathcal {F}^{2}+6\\mathcal {F}\\mathcal {E}+3\\mathcal {E}^{2}-\\mathcal {G}^{2}\\right).$ The coefficients appearing in (REF ) are $b & = & \\frac{44\\alpha ^{2}\\pi ^{2}}{2025}\\frac{T^{4}}{m_{e}^{4}},\\\\c & = & \\frac{2^{6}\\times 37\\alpha ^{3}\\pi ^{3}}{3^{4}\\times 5^{2}\\times 7}\\frac{T^{4}}{m_{e}^{8}},\\\\k & = & \\frac{2^{13}\\alpha ^{3}\\pi ^{5}}{3^{6}\\times 5\\times 7^{2}}\\frac{T^{6}}{m_{e}^{10}},$ and $\\mathcal {E}$ is a term involving the relative velocity of the thermal bath.",
"We will work only in the reference frame where the bath is at rest, and in that special frame we have $\\mathcal {E}=E^{2}$ .",
"In the high temperature limit $(T\\gg m_{e})$ the one loop correction is the dominating term and the thermal correction takes the form [28], [29] $\\mathcal {L}_{EH}^{T}(T\\gg m_{e})=-\\frac{2\\alpha }{3\\pi }\\mathcal {F}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{\\alpha }{6\\pi }\\mathcal {E}-\\mathcal {L}_{EH}^{0}.$ It can be seen from the above equation (REF ) that, for temperatures above the electron mass, the thermal bath cancels the vacuum polarization effects from the zero temperature Euler-Heisenberg Lagrangian.",
"In this paper, the interest in the effective Lagrangian comes from the fact that it can be related to a modification of the Maxwell's equations at the purely classical level.",
"Faraday's and magnetic Gauss's laws remain unchanged $\\mathbf {\\mathbf {\\nabla }}\\cdot \\mathbf {B} & = & 0,\\\\\\mathbf {\\mathbf {\\nabla }}\\times \\mathbf {E} & = & -\\frac{\\partial \\mathbf {B}}{\\partial t}.$ We see from (REF ) and () that electromagnetic potentials are still defined as in classical electrodynamics i.e, $\\mathbf {E}=-\\mathbf {\\nabla }\\phi -\\frac{\\partial \\mathbf {A}}{\\partial t}$ and $\\mathbf {B}=\\mathbf {\\nabla }\\times \\mathbf {A}$ .",
"The electric Gauss's and Ampere-Maxwell's law are modified by the use of the effective Lagrangian.",
"They now resembles the the form of the Maxwell's equation in matter [39], namely, $\\mathbf {\\nabla }\\cdot \\mathbf {D} & = & 4\\pi \\rho ,\\\\\\mathbf {\\nabla }\\times \\mathbf {H} & = & \\frac{\\partial \\mathbf {D}}{\\partial t},$ where $\\rho $ is the charge density and the auxiliary fields $\\mathbf {D}$ and $\\mathbf {H}$ are given by, $\\mathbf {D} & = & \\mathbf {E}+4\\pi \\frac{\\partial \\mathcal {L}_{EH}}{\\partial \\mathbf {E}},\\\\\\mathbf {H} & = & \\mathbf {B}-4\\pi \\frac{\\partial \\mathcal {L}_{EH}}{\\partial \\mathbf {B}}.$ In classical electrodynamics the Gauss law $\\mathbf {\\nabla }\\cdot \\mathbf {E}=\\rho $ or Laplace equation $-\\mathbf {\\nabla }^{2}\\phi =\\rho $ is solved in order to find the field created by a given charge density distribution.",
"Here we shall tackle the problem of finding the effective electric field of a spherical charge distribution produced by the modified Gauss Law (REF )."
],
[
"Temperature Independent charge density",
"To begin with, we shall derive the fields in the limit of low and high temperatures for the case of temperature independent charge densities."
],
[
"Low temperature",
"For pure electric field ($\\mathbf {B}=0$ ) and in the plasma rest frame, the effective Lagrangian takes the form $\\mathcal {L}_{Maxwell}+\\mathcal {L}_{EH}^{0}+\\mathcal {L}_{EH}^{T} & = & (\\frac{1}{8\\pi }+\\frac{b}{2})E^{2}+(a+\\frac{k}{2}-\\frac{c}{4})E^{4}.$ With this Lagrangian the Gauss law reads $\\mathbf {\\nabla }\\cdot (A(T)E^{2}\\mathbf {E}+B(T)\\mathbf {E})=4\\pi \\rho (\\mathbf {r})$ with $A(T) & = & 16\\pi \\left(a+\\frac{k}{2}-\\frac{c}{4}\\right),\\\\B(T) & = & 1+4\\pi b.$ The rest of this sections follows closely the works [13], [14].",
"In general, The charge distribution can be written as $4\\pi \\rho (\\mathbf {r})=\\mathbf {\\nabla }\\cdot \\mathbf {E}_{c}$ where $\\mathbf {E}_{c}$ is the field that would be produced by $\\rho $ in Maxwell's theory.",
"The field $\\mathbf {E}$ given by (REF ) has to approach $\\mathbf {E}_{c}$ in the limit that the Euler-Heisenberg coefficients vanish.",
"We can then write from (REF ) and (REF ) the following algebraic equation $A(T)E^{3}+B(T)E=E_{c}.$ This equation is a cubic equation which has only one real solution and is given by Cardano's formula, $E=\\@root 3 \\of {-\\frac{E_{c}}{A(T)}+\\sqrt{\\frac{E_{c}^{2}}{A^{2}(T)}+\\frac{B^{3}(T)}{A^{3}(T)}}}+\\@root 3 \\of {-\\frac{E_{c}}{A(T)}-\\sqrt{\\frac{E_{c}^{2}}{A^{2}(T)}+\\frac{B^{3}(T)}{A^{3}(T)}}}.$ Let's note that at large distances, the behaviour of $E$ is $E\\sim \\frac{e}{B(T)r^{2}}$ while for short distances it is $E\\sim \\left[\\frac{e}{A(T)r^{2}}.\\right]^{1/3}.$ In particular, we will later need the form of the potential at short distances.",
"The behaviour of the potential for small $r$ follows from (REF ) to be $\\phi \\sim -\\frac{1}{3}\\left[\\frac{e}{A(T)}\\right]^{1/3}r^{1/3}+\\phi (0)$ where $\\phi (0)$ is a positive constant given by $\\phi (0)=-\\int _{0}^{\\infty }Edr.$ Equation (REF ) would be a complete result if we could neglect screening effects and neglect the temperature dependence of the surrounding matter.",
"Nevertheless the result (REF ) is important for later, more exhaustive considerations when we will take into account the temperature dependence of the matter.",
"Then the boundary condition of the resulting partial differential equations will be formulated with the help of (REF ), i.e., at small distances the potential should follow (REF )."
],
[
"High Temperature",
"The high temperature case is mathematically easier.",
"$\\mathbf {\\nabla }\\cdot \\mathbf {E}_{c}=\\rho (\\mathbf {r}) & = & \\mathbf {\\nabla }\\cdot \\mathbf {D}=\\mathbf {\\nabla }\\cdot (\\mathbf {E}+4\\pi \\mathbf {P})\\\\& = & \\left[\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1\\right]\\mathbf {E}$ The solution for $\\mathbf {E}$ is trivial if $\\mathbf {E}_{c}$ is known and is given by $\\mathbf {E}=\\frac{1}{\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1}\\mathbf {E}_{c}$"
],
[
"Temperature Dependent charge density",
"We will work out the effective electric field of a point charge that is submerged in a neutral plasma consisting of electrons and positive charges which can consist of protons or positrons.",
"As is customary, statistical methods are needed when the charge density depends on temperature.",
"The density of fermions is governed by the Fermi-Dirac distribution and is given by $\\eta (\\mathbf {r},T)=4(2\\pi )^{4}\\int _{0}^{\\infty }\\frac{\\left(p^{2}/\\hbar ^{3}\\right)dp}{e^{\\beta \\left(K+q\\phi +\\mu \\right)}+1} \\,,$ where $\\mu $ stands for the chemical potential, $\\hbar $ denotes the Planck's constant, we use $c = 1$ and $K$ is the kinetic energy $K={\\left\\lbrace \\begin{array}{ll}p^{2}/2m & non-relativistic,\\\\\\sqrt{p^{2}+m^{2}} & relativistic,\\\\p & ultra-relativistic.\\end{array}\\right.",
"}$ The electron charge density is then given by $-e\\eta _{e}(\\mathbf {r},T)$ and an analogous analysis holds for the proton or positron charge density.",
"The equation for the effective field at $\\mathbf {r}$ created by a charge distribution is the modified Poisson's equation $\\mathbf {\\nabla }\\cdot \\left(\\mathbf {\\mathbf {E}+4\\pi \\frac{\\partial \\mathcal {L}_{EH}}{\\partial \\mathbf {E}}}\\right)=\\rho (\\mathbf {r},T)_{total},$ where the term $\\rho (\\mathbf {r},T)_{total}$ means the density taking into account all the electric charges and it depends on the system considered.",
"For example, consider the densities given by $\\rho (\\mathbf {r},T)_{total} & = & -e\\eta _{e}(\\mathbf {r},T),\\\\\\rho (\\mathbf {r},T)_{total} & = & \\rho (\\mathbf {r})-e\\eta _{e}(\\mathbf {r},T),\\\\\\rho (\\mathbf {r},T)_{total} & = & \\rho (\\mathbf {r})+4\\pi e\\eta _{p}(\\mathbf {r},T)-4\\pi e\\eta _{e}(\\mathbf {r},T).$ The above densities stand respectively for (a) a cloud of electron, (b) a given particle with charge density $\\rho (\\mathbf {r}$ ) that is surrounded by a cloud of electrons and (c) a charge $\\rho (\\mathbf {r}$ ) surrounded by a cloud of electrons and protons.",
"We always consider the charges to have spherical symmetry.",
"Neglecting the Euler-Heisenberg contribution in (REF ) and introducing an electric potential $\\phi $ , the resulting equation $\\mathbf {\\nabla }^2 \\phi = \\rho (\\mathbf {r},T)_{total}\\, ,$ is referred to as the Feynman-Metropolis-Teller (FMT) equation.",
"Using equation (REF ) may prove to be too difficult as stated.",
"It is convenient to look for special cases where the density (REF ) can be reduced to a simpler expression.",
"The way to simplify (REF ) depends on the relation between the temperature and the chemical potential in a given situation.",
"At this point it is important to make some clarifications about what we mean by the temperature regime.",
"The problem at hand has two natural scales of temperatures and they enter in different ways in the modified Poisson's equation (REF ).",
"First, we can compare $T$ with $m_{e}$ , and as we have seen this affects the form of the effective Lagrangian e.g, for $T\\gg m_{e}$ we use (REF ) in the left hand side of (REF ) and for $T\\ll m_{e}$ we use the Lagrangian (REF ) instead.",
"On the other hand, for the right hand side of (REF ) the temperature has to be compared with the chemical potential $\\mu $ in order to know what simplification can be done to the charge density.",
"We have a degenerate system when $T\\ll \\mu $ and a dilute system when $T\\gg \\mu $ , both with different approximate expansions (the form of the kinetic energy also affects the approximation of (REF )).",
"In the appendix we summarize different ways to approximate equation (REF )"
],
[
"Low temperature",
"We deal first with the systems whose temperatures are below $m_{e}$ and are composed of electrons and protons at equilibrium.",
"The equation of interest is $\\mathbf {\\nabla }\\cdot (A(T)E^{2}\\mathbf {E}+B(T)\\mathbf {E}) & = &4\\pi \\rho (\\mathbf {r},T)_{total},$ Since the charge density is related to the electric potential, it is better to work with the potential instead of the electric field.",
"To accomplish the above we first mention that the usual relation between the electric field and the potential is maintained i.e, $\\mathbf {E}=-\\mathbf {\\nabla }\\phi $ .",
"Then, it is just a matter of replacing the potential for electric field into (REF ).",
"Taking into account the spherical symmetry of the problem, the differential equation for the potential of a charge distribution can be written as $-\\left(\\mathbf {\\nabla }^{2}+\\mathbf {\\mathbf {\\mathfrak {D}}}_{EH}^{2}\\right)\\phi & =4\\pi \\rho (\\mathbf {r},T)_{total} & ,$ where the non-linear differential operator $\\mathbf {\\mathfrak {D}}_{EH}^{2}$ stands for $\\mathbf {\\mathbf {\\mathfrak {D}}}_{EH}^{2}\\bullet =A(T)\\frac{1}{r^{2}}\\left(\\frac{d\\bullet }{dr}\\right)^{2}\\frac{d}{dr}\\left(r^{2}\\frac{d\\bullet }{dr}\\right)+A(T)\\left(\\frac{d\\bullet }{dr}\\right)^{2}\\frac{d^{2}\\bullet }{dr^{2}}+4\\pi b\\mathbf {\\nabla }^{2}\\bullet ,$ with $A(T)$ and $B(T)$ given by (REF ) and ().",
"Let's note that $-\\mathbf {\\mathfrak {D}}^{2}\\rightarrow 0$ when $\\alpha \\rightarrow 0$ .",
"Before we embark on the derivation of the Thomas-Fermi like differential equations we note that for astrophysical applications a rough distinction between degenerate and non-degenerate matter can be made according to the mass density $\\rho $ [40].",
"For non-degenerate matter we should have $ \\rho \\le 8.49 \\times 10^{-17}\\left(\\frac{T^3}{1K}\\right) \\frac{\\rm g}{\\rm cm^3}$ whereas the degenerate matter should satisfy $ \\rho \\ge 4.2 \\times 10^{-10}\\left(\\frac{T^3}{1K}\\right)\\frac{\\rm g}{\\rm cm^3}\\, .$ We now present the differential equations for the potentials in the case where the temperature is low compared to the chemical potential.",
"For the degenerate case we can use the particle density (REF ) (given in the appendix) to write the non-relativistic equation as $& -\\left(\\mathbf {\\nabla }^{2}+\\mathbf {\\mathfrak {D}}_{EH}^{2}\\right)\\phi =\\rho (\\mathbf {r})\\nonumber \\\\& +\\frac{4(2\\pi )^{5}e(2m_{p}T){}^{3/2}}{\\hbar ^{3}}\\left(\\frac{2}{3}\\left(\\frac{\\mu _{p}-e\\phi }{T}\\right)^{3/2}+\\frac{\\pi ^{2}}{12}\\left(\\frac{\\mu _{p}-e\\phi }{T}\\right)^{-1/2}\\right)\\nonumber \\\\& -\\frac{4(2\\pi )^{5}e(2m_{e}T){}^{3/2}}{\\hbar ^{3}}\\left(\\frac{2}{3}\\left(\\frac{\\mu _{e}+e\\phi }{T}\\right)^{3/2}+\\frac{\\pi ^{2}}{12}\\left(\\frac{\\mu _{e}+e\\phi }{T}\\right)^{-1/2}\\right).$ Using (REF ) (from the appendix) the ultra-relativistic equation is $& -\\left(\\mathbf {\\nabla }^{2}+\\mathbf {\\mathfrak {D}}_{EH}^{2}\\right)\\phi =\\rho (\\mathbf {r})\\nonumber \\\\& +\\frac{8(2\\pi )^{5}e}{\\hbar ^{3}}\\left(\\mu _{p}-e\\phi \\right)^{3}\\left(1+\\frac{3}{4}\\pi ^{2}\\left(\\frac{T}{\\mu _{p}-e\\phi }\\right)^{2}\\right)\\nonumber \\\\& -\\frac{8(2\\pi )^{5}e}{\\hbar ^{3}}\\left(e\\phi +\\mu _{e}\\right)^{3}\\left(1+\\frac{3}{4}\\pi ^{2}\\left(\\frac{T}{e\\phi +\\mu _{e}}\\right)^{2}\\right).$ Equations (REF ) and (REF ) are the low $T$ Euler-Heisenberg generalization to the non-relativistic and ultra-relativistic Thomas-Fermi equations with the first thermal correction and a contribution from positive charges.",
"As already mentioned in section III we can drop the source $\\rho $ from these equations (and the other equations derived below) by incorporating it into the boundary conditions.",
"This is to say, we demand that at small distances the potential behaves as (REF ) which amounts to saying that the effect of matter is negligible.",
"In the standard Thomas-Fermi-like equations for spherical charge distributions and without the Euler-Heisenberg corrections this is equivalent to demanding that the Coulomb law be valid at short distances.",
"For the sake of comparison we now present the standard Thomas-Fermi equations for the non-relativistic and the ultra-relativistic cases.",
"Considering a point charge $Z_{1}e$ surrounded by a cloud of electrons, the non-relativistic Thomas-Fermi equation with the first thermal term is [41] $\\mathbf {\\nabla }^{2}\\phi =\\frac{8e(2\\pi )^{5}(2m_{e}){}^{3/2}}{3\\hbar ^{3}}\\left(\\mu _{e}+e\\phi \\right)^{3/2}\\left(1+\\frac{\\pi ^{2}}{8}\\left(\\frac{\\mu _{e}+e\\phi }{T}\\right)^{-2}\\right).$ The ultra-relativistic Thomas-Fermi equation reads $\\mathbf {\\nabla }^{2}\\phi =\\frac{4(2\\pi )^{4}e}{\\hbar ^{3}}\\left(\\mu +e\\phi \\right)^{3}\\left(1+\\frac{3}{4}\\pi ^{2}\\left(\\frac{T}{\\mu +e\\phi }\\right)^{2}\\right).$ It is possible to rewrite the Thomas-Fermi equation (REF ) using the following change of variables $\\mu _{e}+e\\phi & = & \\frac{Z_{1}e^{2}}{r}\\psi (s),\\\\r & = & Cs,\\\\C & = & \\frac{1}{2}\\left(\\frac{3}{8\\left(2\\pi \\right)^{5}}\\right)^{2/3}Z_{1}^{-1/3}a_{0},$ where $a_{0}=\\hbar ^{2}/e^{2}m_{e}$ is the Bohr radius and $s$ is a dimensionless quantity.",
"After using (REF ), () and (), the Thomas-Fermi equation (REF ) becomes $\\frac{d^{2}\\psi (s)}{ds^{2}}=\\frac{\\psi (s)^{3/2}}{\\sqrt{s}}\\left(1+\\frac{\\gamma T^{2}s^{2}}{\\psi ^2(s)}\\right),$ where $\\gamma =\\frac{\\pi ^{2}}{8}\\left(\\frac{3}{8\\left(2\\pi \\right)^{5}}\\right)^{2/3}Z_{1}^{-1/3}\\frac{a_{0}^{2}}{e^{4}}$ .",
"Another customary way to rewrite equation (REF ) is by using the change of variables [6], $\\frac{\\mu _{e}+e\\phi }{T} & = & \\frac{\\psi }{s},\\\\C^{\\prime } & = & \\sqrt{\\frac{\\hbar ^{3}}{8e^{2}(2\\pi )^{5}(2m_{e}T){}^{1/2}}}.$ In this case $\\psi $ obeys $\\frac{d^{2}\\psi (s)}{ds^{2}}=\\frac{\\psi (s)^{3/2}}{\\sqrt{s}}\\left(1+\\frac{\\pi ^{2}s^{2}}{8\\psi (s)^{2}}\\right).$ With the definitions (REF ), (), and (), we can write the low T Euler-Heisenberg generalization to the Thomas-Fermi equation (REF ) in the form $\\frac{d^{2}\\psi (s)}{ds^{2}}+\\frac{1}{C^{4}}\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\psi (s)=\\frac{\\psi (s)^{3/2}}{\\sqrt{s}}\\left(1+\\frac{\\gamma T^{2}s^{2}}{\\psi (s)^{2}}\\right),$ where $\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}$ in terms of $\\psi $ and $s$ reads $\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\bullet & = & A(T)\\left[\\frac{1}{s}\\frac{d\\bullet }{ds}-\\frac{\\bullet }{s^{2}}\\right]^{2}\\nonumber \\\\& & \\times \\left[2\\frac{d^{2}\\bullet }{ds^{2}}-2\\frac{1}{s}\\frac{d\\bullet }{ds}+2\\frac{\\bullet }{s^{2}}\\right].$ The equation (REF ) that considers both a cloud of negative and positive charges can also be rewritten using the change of variables (REF ) and (), with the result $\\frac{d^{2}\\psi (s)}{ds^{2}}&+&\\frac{1}{C^{4}}\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\psi (s) = -s\\rho (Cs)\\nonumber \\\\& +&\\frac{\\psi (s)^{3/2}}{\\sqrt{s}}\\left(1+\\frac{\\gamma T^{2}s^{2}}{\\psi (s)^{2}}\\right)\\nonumber \\\\& +&\\frac{8e(2\\pi )^{5}(2m_{e}){}^{3/2}C^{2}}{3\\hbar ^{3}}s\\nonumber \\\\& \\times &\\left(\\frac{e^{2}\\psi (s)}{Cs}+\\mu _{+}-\\mu _{e}\\right)^{3/2}\\left(1+\\frac{\\pi ^{2}T^{2}}{8}\\left(\\frac{e^{2}\\psi (s)}{Cs}+\\mu _{+}-\\mu _{e}\\right)^{-2}\\right).$ We can see that the whole effect of the low $T$ Euler-Heisenberg is contained in the term $\\frac{1}{C^{4}}\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\psi $ .",
"A similar treatment can be given for the ultra-relativistic equations.",
"The change of variables is the same as (REF ) and () but with $C=\\sqrt{\\frac{\\hbar ^{3}}{4(2\\pi )^{4}e}}$ (though in this case $s$ is not dimensionless).",
"With this change of variables equation (REF ) becomes $\\frac{d^{2}\\psi (s)}{ds^{2}}=\\frac{\\psi (s)^{3}}{s^{2}}\\left(1+\\frac{3\\pi ^{2}T^{2}s^{2}}{4\\psi (s)^{2}}\\right).$ The Euler-Heisenberg generalization to (REF ) now reads $\\frac{d^{2}\\psi (s)}{ds^{2}}+\\frac{1}{C^{4}}\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\psi (s)=\\frac{\\psi (s)^{3}}{s^{2}}\\left(1+\\frac{3\\pi ^{2}T^{2}s^{2}}{4\\psi (s)^{2}}\\right).$ With the inclusion of the test charge and a cloud of positive charges, the equation (REF ) becomes $& \\frac{d^{2}\\psi (s)}{ds^{2}}+\\frac{1}{C^{4}}\\mathbf {\\mathfrak {D}}^{\\prime }{}_{EH}^{2}\\psi (s)=-s\\rho (Cs)+\\frac{\\psi (s)^{3}}{s^{2}}\\left(1+\\frac{3\\pi ^{2}}{4}\\frac{s^{2}}{\\psi (s)^{2}}\\right)-\\nonumber \\\\& \\frac{1}{s^{2}}\\left(\\frac{e^{2}\\psi (s)}{s}+\\mu _{+}-\\mu _{e}\\right)^{3}\\left(1+\\frac{3}{4}\\pi ^{2}T^{2}\\left(\\frac{e^{2}\\psi (s)}{s}+\\mu _{+}-\\mu _{e}\\right)^{-2}\\right).$ In the standard Thomas-Fermi theory, the function $\\psi (r)$ has to obey the following boundary conditions $\\psi (0) & = & 1,\\\\\\psi (\\infty ) & = & 0,$ where we have ignored the size of the charged object.",
"Condition (REF ) ensures that we recover Coulomb electrostatic energy at short distances.",
"Condition () ensures the right behaviour at large distances.",
"However, for the Euler-Heisenberg generalization of Thomas-Fermi equations, the potential has to reduce to its Euler-Heisenberg form (REF ) at short distances.",
"Therefore, for small $r$ , $\\psi $ has to behave like $\\psi (s)\\sim -\\frac{1}{3e}\\left[\\frac{e}{A(T)}\\right]^{1/3}s^{7/3}+\\phi (0)\\frac{s^{2}}{e}.$"
],
[
"Dilute matter",
"For dilute matter, the Maxwell's-Boltzmann distribution (REF ) given in the appendix, can be used to write the density of particles: $n_{e} & = & n_{e0}e^{\\frac{e\\phi }{T}},\\\\n_{p} & = & n_{p0}e^{-\\frac{e\\phi }{T}},$ where $n_{e0}$ and $n_{i0}$ are the concentrations of electrons and protons respectively.",
"Assuming the concentration of negative and positive charges to be equal to $n_{0}$ , we can write the equation $-\\left(\\mathbf {\\mathbf {\\mathbf {\\nabla }}}^{2}+\\mathbf {\\mathfrak {D}}_{EH}^{2}\\right)\\phi & = & \\rho (\\mathbf {r})-en_{0}e^{\\frac{e\\phi }{T}}+en_{0}e^{-\\frac{e\\phi }{T}}.$ Equation (REF ) is a generalization of the Poisson-Boltzmann equation.",
"For regions where the perturbed potential obeys $\\phi \\ll T$ , the exponentials in (REF ) can be expanded and keeping only the first term, we get, $-\\left(\\mathbf {\\nabla }^{2}+\\mathbf {\\mathfrak {D}}_{EH}^{2}\\right)\\phi +\\kappa ^{2}\\phi & = & \\rho (\\mathbf {r})$ with, $\\kappa $ , the Debye parameter, given by $\\kappa ^{2}=\\frac{2e^{2}n_{0}}{T}.$ Equation (REF ) is a generalization of the linearized version of the Poisson-Boltzmann or Debye-Hückel equation [42].",
"Without considering the Euler-Heisenberg term, the Debye-Hückel equation is $-\\mathbf {\\nabla }^{2}\\phi +\\kappa ^{2}\\phi =\\rho (\\mathbf {r}).$ For for point charges $\\rho (\\mathbf {r})=q\\delta (\\mathbf {r})$ , equation (REF ) has the solution $\\phi =\\frac{q}{r}e^{-\\kappa r}.$ The screening effect is evident."
],
[
"Dilute matter",
"In the high temperature regime the particles move ultra-relativistically with kinetic energy $E=p$ and for the non-degenerate case, charge densities are given by the distribution (REF ).",
"We assume the electric interaction between charges to be small as compared to the temperature so that we can write the differential equation for the screened potential for a point-charge as $\\left[\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1\\right]\\mathbf {\\nabla }^{2}\\phi & = & q\\delta (\\mathbf {r})+n_{0}e^{\\frac{-e\\phi }{T}}-n_{0}e^{\\frac{e\\phi }{T}}\\\\& \\approx & q\\delta (\\mathbf {r})-\\kappa ^{2}\\phi .$ Equation (REF ) can easily be rewritten as $-\\mathbf {\\nabla }^{2}\\phi +\\kappa _{EH}^{2}\\phi =q_{EH}(T)\\delta (\\mathbf {r}),$ where we have defined $\\kappa _{EH}^{2} & = & \\frac{\\kappa ^{2}}{\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1},\\\\q_{EH}(T) & = & \\frac{q}{\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1}.$ We see that for dilute gases at temperatures above the electron mass, the effects of the Euler-Heisenberg Lagrangian are the renormalization of the Debye parameter (REF ) and the electric charge ().",
"The solution of (REF ) is given by [42] $\\phi =\\frac{q_{EH}(T)}{r}e^{-\\kappa _{EH}r}\\, .$ This represents an analytical solution of a Poisson-Boltzmann problem with Euler-Heisenberg corrections."
],
[
"Related equations",
"Having discussed the Thomas-Fermi and other equations at different temperatures and densities, we now consider the variants for different statistics and the equations obtained in the presence of magnetic fields."
],
[
"Tsallis Statistics",
"Tsallis statistics have been with us for about 30 years now [43].",
"It has been applied to physical situations like Euler turbulence [44], gravitating systems [45], ferrofluid-like systems [46] and neutron stars [47], among others.",
"Recently, it has been suggested that the Tsallis statistics could eventually explain the Lithium anomaly of early nucleosynthesis [48].",
"In the Tsallis statistics for Fermi particles the occupation number is given by $\\eta =4(2\\pi )^{4}\\int _{0}^{\\infty }\\frac{\\left(p^{2}/\\hbar ^{3}\\right)dp}{e_{q}(\\beta \\left(K+q\\phi -\\mu \\right)+1}$ where $q$ is a real number and $e_{q}(x)=\\left[1+(1-q)x\\right]^{\\frac{1}{1-q}}$ is a generalization of the standard exponential function, which is recovered in the limit $q\\rightarrow 1$ .",
"Density (REF ) has been used in literature to form a non-extensive generalization of the Thomas-Fermi equations; in the nonrelatistic case by [49], and in the relativistic case by [50].",
"The relativistic Poisson equation reads $\\mathbf {\\nabla }^{2}\\phi & = & \\frac{em_{e}^{3}}{3\\pi ^{2}\\hbar ^{2}}\\left[\\frac{\\left(\\mu +m_{e}+\\phi \\right)}{m_{e}^{2}}-1\\right]^{3/2}\\times \\left\\lbrace 1+\\frac{3TI_{1}^{(q)}}{m_{e}}\\left[\\frac{\\left(\\mu +m_{e}+\\phi \\right)}{m_{e}^{2}}-1\\right]^{-1}\\right.\\nonumber \\\\& & \\left.+\\frac{3TI_{2}^{(q)}}{m_{e}}\\left[\\frac{\\left(\\mu +m_{e}+\\phi \\right)}{m_{e}^{2}}-1\\right]^{-2}+\\ldots \\right\\rbrace ,$ where the q-generalized Fermi-Dirac integral $I_{k}^{(q)}$ is defined by $I_{k}^{(q)}=q\\int _{-\\infty }^{\\infty }\\frac{z^{n}\\left[1+(q-1)z\\right]^{1/(q-1)}dz}{\\left\\lbrace 1+\\left[1+(q-1)z\\right]^{q/(q-1)}\\right\\rbrace ^{2}}.$ Numerical evaluation of (REF ) for different q can be found in [51], [49].",
"With the following change of variables $\\mu _{e}+e\\phi & = & \\frac{Z_{1}e^{2}}{r}\\psi (s),\\\\r & = & Cs,\\\\C & = & \\left(\\frac{9\\pi ^{2}}{128}\\right)^{1/3}Z_{1}^{-1/3}a_{0},$ equation (REF ) transform into the non-extensive relativistic generalization of the Thomas-Fermi equation $\\frac{d^{2}\\psi }{ds^{2}} & = & \\frac{\\psi {}^{3/2}}{\\sqrt{s}}\\left(1+\\frac{\\gamma T^{2}s^{2}}{\\psi }\\right)\\nonumber \\\\& & \\times \\left\\lbrace 1+\\chi _{1}\\frac{Ts}{\\psi }\\left[1+\\gamma \\frac{s}{\\psi }\\right]^{-1}+\\chi _{2}\\frac{T^{2}s^{2}}{\\psi ^{2}}\\left[1+\\gamma \\frac{s}{\\psi }\\right]^{-2}+\\ldots \\right\\rbrace $ where $\\gamma =\\left[\\frac{4Z_{1}^{2}}{3\\pi }\\right]^{2/3}\\frac{e^{4}}{\\hbar ^{2}}, & \\chi _{1}=\\frac{3C}{2e^{2}Z_{1}}I_{1}^{(q)}, & \\chi _{2}=\\frac{3C^{2}}{8e^{4}Z_{1}^{2}}I_{2}^{(q)}.$ In the limit where the relativistic contribution is neglected ($\\gamma \\rightarrow 0$ ), equation (REF ) reduces to the following non-relativistic expression (originally obtained in [49]) $\\frac{d^{2}\\psi }{ds^{2}}=\\frac{\\psi {}^{3/2}}{\\sqrt{s}}\\left[1+\\chi _{1}\\frac{Ts}{\\psi }+\\chi _{2}\\frac{T^{2}s^{2}}{\\psi ^{2}}\\right].$"
],
[
"Thomas-Fermi equations in presence of magnetic fields",
"In the context of nuclear astrophysics, there are cases where the process of interest occurs in presence of magnetic fields.",
"When the magnetic field is intense enough, the quantum nature of the motion of the charged particle can not be ignored.",
"The first investigation of the modification of the Thomas-Fermi equation due to a magnetic field was done in [52].",
"Further developments were done in [53], [54], [55].",
"We follow the procedure of [56], where the discretization of the transverse motion into Landau levels is taken into account.",
"The motion of electrons perpendicular to the magnetic field is quantized into the discrete Landau Levels $\\nu B$ , with $\\nu =0,1,2,...$ .",
"The degeneracy of the levels, per unit area, is $\\frac{B}{2\\pi }$ for $\\nu =0$ , but, due to the electron spin, the degeneracy is twice as high for the higher $\\nu $ .",
"Along the direction of the field the motion is not quantized, and the degeneracy of states is $D(\\varepsilon )=\\varepsilon ^{-1/2}/(2^{1/2}\\pi )$ , where $\\varepsilon $ is the energy of the translational motion.",
"Taking the above into consideration, it follows that the density of electrons at temperature $T$ and electrical potential $-e\\phi $ is given by $\\eta & = & \\frac{B}{2\\pi }\\frac{1}{2^{1/2}\\pi }\\int _{0}^{\\infty }\\left[\\int _{0}^{\\infty }\\frac{\\varepsilon ^{-1/2}}{e^{(\\varepsilon -\\mu -e\\phi )/T}+1}d\\varepsilon +2\\sum _{v=1}^{\\infty }\\int _{0}^{\\infty }\\frac{\\varepsilon ^{-1/2}}{e^{(\\varepsilon +\\nu B-\\mu -e\\phi )/T}+1}d\\varepsilon \\right]\\nonumber \\\\& = & \\frac{BT^{1/2}}{2^{3/2}\\pi ^{2}}\\left[I_{-1/2}\\left(\\frac{\\mu +e\\phi }{T}\\right)+2\\sum _{v=1}^{\\infty }I_{-1/2}\\left(\\frac{\\mu +e\\phi -\\nu B}{T}\\right)\\right]$ where the Fermi-Dirac integral for $k>-1$ is defined by $I_{k}(x)=\\int _{0}^{\\infty }\\frac{y^{k}}{e^{y-x}+1}dy.$ Combining (REF ) with the Poisson's equation yields $\\mathbf {\\nabla }^{2}\\phi =4\\frac{BT^{1/2}}{2^{3/2}\\pi }\\left[I_{-1/2}\\left(\\frac{\\mu +e\\phi }{T}\\right)+2\\sum _{v=1}^{\\infty }I_{-1/2}\\left(\\frac{\\mu +e\\phi -\\nu B}{T}\\right)\\right].$ In the case where only the lowest Landau level is taken into account, equation (78) reduces to the one that can be found in [53], [55], namely, $\\mathbf {\\nabla }^{2}\\phi =4\\frac{BT^{1/2}}{2^{3/2}\\pi }I_{-1/2}\\left(\\frac{\\mu +e\\phi }{T}\\right).$ Using the relation $\\frac{d}{dx}I_{k}(x)=kI_{k-1}(x)$ we can obtain a low $T$ expression for (REF ).",
"Indeed, for low $T$ we can write $I_{-1/2}(x) & = & 2\\frac{d}{dx}I_{1/2}(x)\\approx 2\\frac{d}{dx}\\left[\\frac{2}{3}x^{3/2}\\left\\lbrace 1+\\frac{3}{8x^{2}}\\right\\rbrace \\right]\\nonumber \\\\& = & 2x^{1/2}\\left\\lbrace 1+\\frac{3}{8x^{2}}\\right\\rbrace -\\frac{3}{2x^{3/2}}.$ Then, at low T, equation (REF ) can be expanded as $\\mathbf {\\nabla }^{2}\\phi =4\\frac{B}{2^{3/2}\\pi }\\left[2(\\mu +e\\phi )^{1/2}\\left\\lbrace 1+\\frac{3T^{2}}{8(\\mu +e\\phi )^{2}}\\right\\rbrace -\\frac{3T^{2}}{2(\\mu +e\\phi )^{3/2}}\\right].$ To calculate the low-T Euler-Heisenberg correction to equation (REF ) we have to consider the Lagrangian (REF ), this time taking into account a magnetic term of the form $\\mathbf {B}=B\\widehat{\\mathbf {k}}$ in the electromagnetic invariants (REF ) and ().",
"With the magnetic terms included, the Gauss's law now reads $\\nabla \\cdot (A(T)E^{2}\\mathbf {E}+\\mathcal {B}(T)\\mathbf {E}-E_{z}B^{2}\\widehat{\\mathbf {k}})=4\\pi \\rho ,$ where $\\mathcal {B}(T)=B(T)+4\\pi (4k-6c)B^{2}.$ From the form of the Gauss's law (REF ), the modified Poisson's equation $\\nabla ^{2}\\phi +\\mathfrak {D}_{EH-B}^{2}\\phi =4\\frac{BT^{1/2}}{2^{3/2}\\pi }I_{-1/2}\\left(\\frac{\\mu +e\\phi }{T}\\right),$ with $\\mathfrak {D}_{EH-B}^{2}\\bullet & = &A(T)(\\nabla \\bullet )^{2}\\nabla ^{2}\\bullet +2A(T)\\left(\\nabla \\bullet \\right)\\cdot \\left[\\left(\\nabla \\bullet \\right)\\cdot \\nabla \\right]\\nabla \\bullet \\nonumber \\\\& & +4\\pi (b+4kB^{2}-6cB^{2})\\nabla ^{2}\\bullet +kB^{2}\\frac{d^{2}\\bullet }{dz^{2}}.$ We can see that, due to existence of the magnetic field, the operator $\\mathfrak {D}_{EH-B}^{2}$ is not spherically symmetric.",
"In the procedure above there is a subtlety that we have to mention.",
"When substituting into the electromagnetic invariants we have considered the magnetic field to be of the form $\\mathbf {B}=B\\widehat{\\mathbf {k}}$ .",
"However, an external magnetic field can induce the electric charges to produce a magnetic field of their own [57], [58].",
"So, in reality, the Gauss's law has to take into account this induced field as well.",
"However, we have ignored the induced field since it will be much smaller than the original external one."
],
[
"Applications",
"In this section we remind the reader of some applications.",
"We will explicitly examine the details of an electric potential in a neutral electron-positron plasma under conditions encountered in the beginning of the universe.",
"Secondly, we will recall how screening of charges affects the alpha decay."
],
[
"Ultra relativistic degenerate electron-positron gas",
"The electron-positron plasma at an early stage of the Big-Bang presents a situation where the thermal Euler-Heisenberg Lagrangian might prove of great relevance.",
"It is believed that the early pre-stellar period of the evolution of the Universe was dominated by electrons and positrons having ultra relativistic temperatures [59].",
"In the time between $10^{-6}s$ and $10s$ after the big bang, the universe reached temperatures between $10^{9}K$ and $10^{13}K$ and was composed mainly of electrons, positrons, and photons in thermodynamic equilibrium.",
"Furthermore, statistical mechanics states that for an electron-positron plasma in an electrostatic field which is in equilibrium, the chemical potential of the positrons and electrons must be the same in magnitude at all points [60], [61].",
"Furthermore, in thermodynamic equilibrium the mean particle numbers will change via the creation and annihilation processes, therefore the total density $\\eta _{-}-\\eta _{+}$ will remain a constant.",
"The total charge density was calculated in [38] and can be written as $e\\eta _{-}-e\\eta _{+} & = & \\frac{\\left(e\\phi +\\mu \\right)}{3\\hbar ^{3}}\\left[T^{2}+\\frac{\\left(e\\phi +\\mu \\right)^{2}}{\\pi ^{2}}\\right].$ With the charge density (REF ) and the Euler-Heisenberg contribution we can write for the potential the following equation $\\left[\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1\\right]\\mathbf {\\nabla }^{2}\\phi =4\\pi e\\frac{\\left(e\\phi +\\mu \\right)T^{2}}{3\\hbar ^{3}}\\left[1+\\frac{\\left(e\\phi +\\mu \\right)^{2}}{\\pi ^{2}T^{2}}\\right].$ With the change of variable $\\Phi =\\frac{e\\phi +\\mu }{T}$ , the equation (REF ) can be written as $\\mathbf {\\nabla }^{2}\\Phi =\\frac{\\Phi }{r_{EH}^{2}}\\left[1+\\Phi ^{2}\\right],$ where $r_{EH}^{2}=\\frac{(3/4\\pi )(\\hbar ^{3}/e^{2}T^{2})}{\\left[\\frac{8\\alpha }{3}\\ln \\left(\\frac{T}{m_{e}}\\right)+\\frac{4\\alpha }{3}+1\\right]}.$"
],
[
"Effect on tunneling probability",
"The original Thomas-Fermi equation was derived for bound electrons.",
"The derivation presented in this work shows that it is equally valid if the screening happens in a gas of free electrons.",
"We will use the Thomas-Fermi equation (REF ) in its simplest form, i.e., without the term proportional to $T^2$ and without Euler-Heisenberg corrections.",
"It is evident that in equation (REF ) the length scale is given by the atomic Bohr radius whereas the important quantities entering the tunneling probability of an alpha particle have to do with the much smaller nuclear scale.",
"Following [62], [63] one can expand the solution $\\psi $ of the Thomas-Fermi equation which simplifies the calculations.",
"According to (REF ) we can write the interaction potential between two positive charges (characterized by $Z_1$ and $Z_2$ ) as $ V(r)=\\frac{Z_1Z_2 \\alpha }{r}\\psi (s)\\,$ where $r = C s$ as used before.",
"We will look for solutions of $\\psi $ which at the lowest order behave linearly, i.e., $ \\psi _i(s) \\simeq 1-d_i s\\,.$ One such solution with a linear behaviour at the origin, which is one of the first attempts to derive a semi-analytical solution of the Thomas Fermi equation, is given by [64] with $d_0$ =1.588558.",
"Other semi-analytical solutions [65], [66], [67], [68], [69] have been attempted and we list below some of them in the order in which they are cited: $ \\psi _1(s)&=&(1+\\eta \\sqrt{s})e^{-\\eta \\sqrt{s}}\\simeq =1-d_1s, \\,\\, d_1=3.6229 \\nonumber \\\\\\psi _2(s)&=& (a_0e^{-\\alpha _0 x} +b_0e^{-\\beta _0 x})^2 \\simeq 1-d_2x, \\,\\, d_2=1.2357 \\nonumber \\\\\\psi _3(s)&=&(ae^{-\\alpha x} +be^{-\\beta x} + ce^{-\\gamma x})^2 \\simeq 1-d_3s, \\,\\, d_3=1.4042 \\nonumber \\\\\\psi _4(s)&=&(1+A\\sqrt{x} +Bxe^{-D\\sqrt{x}})^2e^{-2A\\sqrt{x}}\\simeq 1-d_4s, \\,\\, d_4=1.45612 \\nonumber \\\\\\psi _5(s)&=&\\frac{1}{(1+A_0x)^2}\\simeq 1-d_5s, \\, \\, d_5=0.9615$ The potential for the alpha tunneling is in the first approximation given by a potential well modeling the nuclear interaction plus the Coulomb or the modified Coulomb potential given in (REF ).",
"In the semiclassical JWKB approximation, the tunneling probability is simply given by [62], $ P &\\propto & e^{-\\gamma } \\nonumber \\\\\\gamma (E, r_1)&=&2\\sqrt{2m}I(E, r_1)=2\\sqrt{2m}\\int _{r_1}^{r_2}\\sqrt{[V(r)-E]}dr$ where $m$ is the reduced mass, $r_1$ the first turning point given in our simple model by the radius of the nucleus and $r_2$ the second turning point determined by $V(r_2)=E$ , with $E$ being the energy of the tunneling particle.",
"In passing we note that we have omitted some other approximate solutions which exist in the literature [71], [72], [73].",
"The integral $I$ with the Coulomb potential can be solved analytically to be [62], $ I(E,r_1)=2\\sqrt{2m}\\frac{Z_1Z_2\\alpha }{\\sqrt{E}}\\left[\\cos ^{-1}(x^{1/2}) -x^{1/2}(1-x)^{1/2}\\right]$ with $x=(Er_1)/(Z_1Z_2\\alpha )$ .",
"Since the modification of the electromagnetic interaction brought by the Thomas-Fermi equation can be approximated by $1-d_i s$ , the correction to the potential is simply a constant.",
"The integral for the modified Coulomb problem is then $I(E^*, r_1)$ with $E^*=E+Z_1Z_2d_i\\alpha /C$ .",
"Correspondingly, we have $\\gamma ^*=\\gamma (E^*, r_1)$ .",
"We have chosen the few examples (with exerimental $Q$ -values [74] denoted above as $E$ ) with some of them being the same as in [62].",
"The nuclear radii are taken from [75].",
"In table 1 we summarize the effects in the form of the ratio of half-lives, $\\tau /\\tau ^*$ for the decays, $^{106}_{52}$ Te $\\rightarrow \\, ^4$ He + $^{102}_{50}$ Sn, $^{148}_{62}$ Sm $\\rightarrow \\, ^{4}_{2}$ He + $^{144}_{60}$ Nd, $^{222}_{86}$ Rn $\\rightarrow \\,^{4}_2$ He + $^{218}_{84}$ Po, and $^{240}_{96}$ Cm $\\rightarrow \\, ^4_2$ He + $^{236}_{94}$ Pu.",
"Though the exact values of half-lives (and hence also the screening effects) are sensitive to the Q-values [76], the increase in the half-life due to screening seems to be quite sizable in some of the cases considered.",
"The results prompt us to consider a more sophisticated calculation, with the following points in future: (i) Inclusion of the $T^2$ term in the Thomas-Fermi equation for different gas temperatures, (ii) including the Euler-Heisenberg corrections and (iii) improving the nuclear model such that the first turning point is also sensitive to the nuclear potential.",
"In passing we note that the Gamow factor $e^{-\\gamma }$ appears also in stellar reaction rates $R$ defined by $ R \\propto \\int _0^{\\infty }\\, e^{-\\gamma }S(E) e^{-E/kT}dE$ where $S(E)$ is the astrophysical S-factor [77] which is sometimes approximated by a constant.",
"It would be interesting to study the screening effects in the reaction rates (which eventually affect the abundance of elements) in the fusion reactions in stars within a more refined model as mentioned above.",
"Table: The effect of electron gas on alpha tunneling.τ * \\tau ^* is the half-life of the decaying nucleus within the electron medium."
],
[
"Conclusions",
"The effect of surrounding matter at finite temperature on the electric potential of an object is encoded in the Feynman-Metropolis-Teller equation (REF ).",
"From this equation, various equations can be derived imposing different conditions on the matter.",
"Among the well known equations which emerge are the Thomas-Fermi and Poisson-Boltzmann equations.",
"Other, new equations like the relativistic Thomas-Fermi equation have been derived in the present work.",
"We have stressed the importance and the universal applicability of these equations.",
"Therefore, it appears timely to consider quantum corrections to these equations.",
"We have calculated these corrections using the Euler-Heisenberg theory at finite temperature.",
"For non-degenerate matter and high temperature analytical solutions have been presented.",
"Although our emphasis was on the derivations of these equations we have touched upon two examples where it can be applied.",
"One example concerns the electron-positron neutral plasma under the Big-Bang conditions in the early universe.",
"The other was a reminder of the state of art of screening charges in astrophysics and its effect on alpha tunneling.",
"The size of the effect makes us think that a more detailed investigation including temperature effects and the quantum corrections is in order.",
"This will be attempted in a future publication.",
"As we already mentioned the applicability of the equations resulting from the Feynman-Metropolis-Teller is manifold and not limited to the examples we presented here.",
"Apart from atomic physics [78], plasma physics [79] and biological applications [80], one can also find Thomas-Fermi like equations in gravitational physics [81].",
"Future projects could probe into such equations replacing the Fermi-Dirac distribution by the corresponding Bose-Einstein for bosons.",
"Regarding the novel aspects where Thomas-Fermi equations could be used we mention graphene where the electrons are treated relativistically [82].",
"With the inclusion of the quantum corrections we obtain a complete picture of the electric fields at finite temperature from which the electromagnetic force can be easily calculated.",
"Forces at finite temperature, of a different nature than the electromagnetic one, can, in general, be treated within quantum field theory at finite temperature (see [83] for an example)."
],
[
"Appendix: Expansions for the charge density",
"We review the form of the particle density for the limiting cases of both non-relativistic and ultra relativistic particles.",
"The special case of ultra relativistic electron-positron plasma is shown at the end.",
"The quantity of interest is $\\eta (\\mathbf {r},T) & = & 2\\frac{(2\\pi )^{3}}{\\hbar ^{3}}\\int _{0}^{\\infty }\\frac{d\\mathbf {p}}{e^{\\beta \\left(K+q\\phi +\\mu \\right)}+1},$ where $K$ is the kinetic energy of the particles.",
"For high temperatures the $+1$ in the denominator of (REF ) can be ignored.",
"Under this consideration of non-degeneracy, the equation (REF ) simplifies to $\\eta (\\mathbf {r},T) & \\approx & 2\\frac{(2\\pi )^{3}}{\\hbar ^{3}}e^{-\\beta \\left(q\\phi +\\mu \\right)}\\int _{0}^{\\infty }p^{2}e^{-\\beta K}dp.$ Equation (REF ) can be simplified further by taking into account the normalization condition $N & = & \\int \\eta (\\mathbf {r},T)d\\mathbf {V}=2\\frac{(2\\pi )^{3}}{\\hbar ^{3}}e^{-\\beta \\mu }\\int e^{-\\beta (K+q\\phi )}d\\mathbf {p}d\\mathbf {v},$ where $N$ is the total number of particles.",
"From the above we can write for the chemical potential $e^{-\\beta \\mu }=\\frac{N}{2\\frac{(2\\pi )^{3}}{\\hbar ^{3}}\\int e^{-\\beta (K+q\\phi )}d\\mathbf {p}d\\mathbf {v}}.$ Replacing (REF ) into (REF ) we get $\\eta (\\mathbf {r},T)=\\frac{Ne^{-q\\phi \\beta }\\int e^{-\\beta K}d\\mathbf {p}}{\\int e^{-\\beta (K+q\\phi )}d\\mathbf {p}d\\mathbf {v}}\\approx \\left(\\frac{N}{V}\\right)e^{-q\\phi \\beta }.$ In the last step of (REF ) we have made the final approximation $\\int e^{-q\\phi \\beta }d\\mathbf {v}\\approx V$ , the total volume.",
"The justification is based on the assumption that for high $T$ the exponential $e^{-q\\phi \\beta }$ will be small for almost all the volume considered.",
"The approximation for the degenerate case involves a Sommerfeld's expansion in power series of $\\frac{\\mu +q\\phi }{T}$ for the equation $\\eta (\\mathbf {r},T)=\\frac{4(2\\pi )^{4}}{\\hbar ^{3}}\\int _{0}^{\\infty }\\frac{p^{2}dp}{e^{\\beta \\left(K+q\\phi -\\mu \\right)}+1}.$ The first two terms for the non-relativistic cases read [6], $\\eta (\\mathbf {r},T)\\approx \\frac{2(2\\pi )^{4}(2mT)^{3/2}}{\\hbar ^{3}}\\left(\\frac{2}{3}\\left(\\frac{q\\phi -\\mu }{T}\\right)^{3/2}+\\frac{\\pi ^{2}}{12}\\left(\\frac{q\\phi -\\mu }{T}\\right)^{-1/2}\\right).$ The ultra relativistic expansion is given by $\\eta (\\mathbf {r},T)\\approx \\frac{4(2\\pi )^{4}}{\\hbar ^{3}}\\left(q\\phi -\\mu \\right)^{3}\\left(1+\\frac{3}{4}\\pi ^{2}\\left(\\frac{T}{q\\phi -\\mu }\\right)^{2}\\right).$ A special case is the electron-positron plasma [38].",
"Due to the relation between their chemical potentials, the exact total charge density $e\\eta _{-}-e\\eta _{+}$ can be written without any simplifying assumption as $e\\eta _{-}-e\\eta _{+} & = & \\frac{4(2\\pi )^{4}}{\\hbar ^{3}}\\int _{0}^{\\infty }dp\\, p^{2}\\left[\\frac{1}{e^{\\beta \\left(p-e\\phi -\\mu \\right)}+1}-\\frac{1}{e^{\\beta \\left(p+e\\phi +\\mu \\right)}+1}\\right]\\\\& = & (2\\pi )^{3}\\frac{\\left(e\\phi +\\mu \\right)}{3\\hbar ^{3}}\\left[T^{2}+\\frac{\\left(e\\phi +\\mu \\right)^{2}}{\\pi ^{2}}\\right].$"
]
] | 1709.01615 |
[
[
"Supernova remnants in the very-high-energy sky: prospects for the\n Cherenkov Telescope Array"
],
[
"Abstract The Cherenkov Telescope Array is expected to lead to the detection of many new supernova remnants in the TeV and multi-TeV range.",
"In addition to the individual study of each SNR, the study of these objects as a population can help constraining the parameters describing the acceleration of particles and increasing our understanding of the mechanisms involved.",
"We present Monte Carlo simulations of the population of Galactic SNRs emitting TeV gamma rays.",
"We also discuss how the simulated population can be confronted with future observations to provide a novel test for the SNR hypothesis of cosmic ray origins."
],
[
"Introduction",
"The ever increasing number of supernova remnants (SNRs) detected in the TeV range constitutes a major asset for the study of the acceleration mechanisms happening at strong shocks [8].",
"Indeed, TeV gamma rays are produced when very–high–energy particles interact with their environment, and the observation of excesses of TeV gamma rays at several astrophysical objects testify of efficient acceleration mechanisms.",
"The case of SNRs is especially interesting, because they have been widely pointed as the most probable sources of Galactic cosmic rays (CRs) (i.e.",
"at least up to energies of the knee $\\sim 1$ PeV) [3].",
"The SNR hypothesis is supported by several strong arguments, such as energy considerations, showing that converting a fraction of the order of 10% of the total explosion energy of supernovae into CRs can explain the measured level of CRs at the Earth.",
"Another strong argument is the diffusive shock acceleration mechanism, operating at SNR shocks, capable of explaining the power–law slope of accelerated particles, compatible with local measurements of CRs.",
"However, these supporting arguments are not enough to make the SNR hypothesis a definitive answer to the question of the origin of Galactic CRs.",
"CRs are mainly protons, and the sources of CRs are therefore expected to demonstrate that they can efficiently accelerate protons up to the knee.",
"The observation of gamma rays in the TeV range from SNRs attest of particle acceleration, but can often be explained by accelerated electrons as well as accelerated protons.",
"Indeed, accelerated electrons can undergo inverse Compton scattering on soft photons of the CMB, and protons can interact with the interstellar medium to produce gamma rays through pion decay.",
"In the TeV range, the situation is therefore often unclear, and motivates further testing of the SNR hypothesis.",
"The actual population of SNRs in the TeV range comes from targeted observations and from systematic Galactic surveys.",
"The case-by-case study of all these SNRs has greatly improved the understanding of the community on acceleration mechanisms, and the modeling efforts have in many cases allowed satisfying interpretation of the origin of their TeV emission.",
"But the growing number of detections motivates a study of the entire population as such.",
"In this context, it is possible to simulate the expected SNR population and compare it with actual observations from current TeV instruments, therefore providing a test for the SNR hypothesis.",
"The demonstrated efficiency of systematic surveys and the perspective of deeper all–sky survey, such as the one proposed by CTA suggest the detection in the coming years of a SNR population significantly larger than the current one [2], [7].",
"This population, confronted with theoretical simulations will help test again the role of SNRs, and constrain the parameters governing particle acceleration at SNR shocks."
],
[
"Method",
"We rely on Monte Carlo methods to simulate the population of SNRs potentially detectable by CTA.",
"For repeated realizations ($10^3$ ), we simulate the time and location of supernova explosions in the Galaxy, assuming a rate of 3 SN per century, and a spatial distribution described as in [12].",
"Two mechanisms, via four types of progenitors, are considered: thermonuclear (type Ia) and core–collapse (types Ib/c, IIP, IIb).",
"The relative rates and typical parameters associated to each type, such as the total supernova explosion energy, the velocity of the wind and the mass of the ejecta, are adopted as in [22], so that every simulated supernova is assigned a type and corresponding parameters.",
"At the location of each supernova, the typical value of the interstellar medium (ISM) is derived from surveys of atomic and molecular hydrogen [15], [16].",
"The evolution of the shock radius $R_{\\rm sh}$ and velocity $u_{\\rm sh}$ is computed using analytical and semi–analytical description of [5], [19].",
"Finally, the gamma–ray luminosity of each SNR is computed.",
"The contribution of protons and electrons is taken into account.",
"At the shock, the particles are assumed to be accelerated with a slope following a power–law in momentum $n(p) \\propto p^{-\\alpha }$ , where $\\alpha $ is treated as a parameter in the range $4.1 - 4.4$ .",
"At the shock, we assumed that a fraction $\\xi _{\\rm CR}$ of the ram pressure of the shock expanding through the ISM is converted into CRs, where $\\xi _{\\rm CR}~\\approx 0.1$ , and the shock compression factor is $\\sigma =4$ .",
"The distribution of CRs inside the SNR is computed by solving a transport equation and the structure of the interior of the SNR is derived by solving the gas continuity equation, as in [18], [19].",
"The maximum momentum reached by protons is computed by assuming that protons escape the shock when their diffusion length equates a fraction $\\zeta \\approx 0.1$ of the shock radius, adopting a Bohm diffusion coefficient, this leads to $p_{\\max } \\propto R_{\\rm sh} u_{\\rm sh}~B_{\\rm down}$ , where $B_{\\rm down}$ is the magnetic field downstream of the shock.",
"In order to account for magnetic field amplification downstream of the shock, and without making any assumption on the type of mechanism involved in the amplification, we describe $B_{\\rm down} = \\sigma B_{0}\\sqrt{({u_{\\rm sh}}/{v_{\\rm d}})^{2}+1}$ , where $v_{\\rm d}$ is explicited in [23] .",
"The hadronic contribution to the gamma–ray spectrum is then calculated following the approach of [13], weighted by a factor 1.8 to take into account nuclei heavier than hydrogen.",
"We follow by computing the leptonic component.",
"The spectrum of electrons is parametrized adopting the same spectral shape as protons $\\propto p^{-\\alpha }$ , weighted by a factor $K_{\\rm ep}$ , for momenta $p$ < $p_{\\rm break}$ , where $p_{break}$ accounts for radiative losses.",
"above $p_{\\rm break}$ the electron spectrum steepens by one order and follows $\\propto p^{-\\alpha -1}$ [14].",
"The maximum momentum of electrons is reached when the synchrotron loss time is of the order of the acceleration rate.",
"The gamma–ray luminosity from inverse Compton scattering of electrons on the cosmic microwave background is computed following the description proposed by [4].",
"The approach presented here is described more in detail in [6], [7] and was used to provide a statistical test for the SNR hypothesis of the origin of CRs."
],
[
"Results",
"Two strategies have been proposed for the Galactic survey of CTA: an all–sky survey where a typical integrated sensitivity of $\\approx 3$ mCrab could be reached, and a Galactic plane survey (GPS) centered on the Galactic center ($| l | < 60^{\\circ }$ , $| b | < 2^{\\circ }$ ) where a sensitivity of $\\approx 1$ mCrab could be reached [2].",
"Using the method described in the previous paragraph, we simulate the population that CTA could expect to detect in the GPS while performing an all–sky survey and plot in Fig.",
"REF the number of number of simulated SNRs with integral gamma–ray flux above F($>$ 1TeV) greater than 1 mCrab.",
"In order to take into account the extension of the simulated SNRs, the sensitivity of CTA was degraded linearly by the sources apparent size when this becomes larger the typical point spread function of the instrument, i.e.",
"$\\approx 3$ arcmin at 1 TeV.",
"The range of parameters adopted for Fig.",
"REF of $\\alpha =4.1-4.4$ and $K_{\\rm ep}=10^{-2}-10^{-5}$ have been proposed by theoretical studies [20], [17], [21], [11], [14].",
"The most optimistic situation represented, where $\\alpha =4.1$ and $K_{\\rm ep}=10^{-2}$ lead to a number of $\\approx 190^{+20}_{-20}$ potentially detectable SNRs.",
"The most pessimistic situation, where $\\alpha =4.4$ and $K_{\\rm ep}=10^{-5}$ lead to $\\approx 18^{+6}_{-5}$ potential detection.",
"Other effects should also be taken into account, such as for example the issue of source confusion: with the improved sensitivity, many of the new sources could overlap making the identification of SNRs problematic [9].",
"The main take–away message is that the different sets of parameters can lead to remarkably different populations, and that in the TeV range, CTA should be able to further constrain these parameters.",
"A more detailed description of the characteristics (age, distance, size) of the simulated populations, and a discussion on the robustness of our approach, can be found in [7].",
"Considering integral gamma–ray fluxes above 10 TeV and a sensitivity of 10 mCrab, the number of detection is $\\approx 30^{+8}_{-7}$ and $\\approx 4^{+2}_{-2}$ in the extreme situations.",
"In Fig.",
"REF , it thus becomes obvious that it will be more difficult to constrain the parameter $K_{\\rm ep}$ at 10 TeV than at 1 TeV.",
"This is expected, given the fact that in the multi–TeV range, the leptonic contribution to the gamma–ray emission becomes less important.",
"Figure: SNRs in the simulated Galactic plane survey of CTA with integral gamma–ray flux F(>> 1TeV) ≥\\ge 1 mCrab, as a function of the parameter α\\alpha .",
"The blue (solid) and black (dashed) curve correspond respectively to K ep =10 -2 K_{\\rm ep }= 10^{-2} and K ep =10 -5 K_{\\rm ep }= 10^{-5}.",
"In each case the +/- standard deviation is shown.Figure: Situation analogous to Fig.",
"with integral flux F(>> 10 TeV) ≥\\ge 10 mCrab."
],
[
"Conclusions",
"Next generation instruments operating in the TeV range, such as CTA, are expected to lead to many new SNR detections, especially thanks to systematic Galactic surveys.",
"In the most optimistic scenarios, the SNR population accessible by CTA could somewhat be comparable to the one detected at other wavelength, such as in the GHz range, where $\\lesssim 300$ SNRs have been reported [10].",
"More than predictions on what CTA should achieve, our work show that the parameters governing particle acceleration should lead to very different situations in terms of detection of SNR population, and that CTA should be able to discriminate between these situations.",
"This is not the case with the results of current TeV instruments, where the low number of detections can not at this stage efficiently been used for such an analysis.",
"The population detected by CTA should therefore be confronted to our simulations in order to provide a novel consistency test of the SNR hypothesis, and improve our understanding of the role of SNRs in the acceleration of very–high–energy particles."
],
[
"Acknowledgements",
"This work was conducted in the context of the CTA Consortium.",
"The authors gratefully acknowledge financial support from the agencies and organizations listed here: http://www.cta-observatory.org/consortium_acknowledgments, and thank the CTA consortium.",
"SG acknowledges support from the Programme National Hautes Energies (CNRS).",
"SG and RT acknowledge support from the Observatoire de Paris (Action Fédératrice Preparation à CTA).",
"PC acknowledges support from the Columbia University Frontiers of Science fellowship."
]
] | 1709.01900 |
[
[
"Optimization of the Brillouin operator on the KNL architecture"
],
[
"Abstract Experiences with optimizing the matrix-times-vector application of the Brillouin operator on the Intel KNL processor are reported.",
"Without adjustments to the memory layout, performance figures of 360 Gflop/s in single and 270 Gflop/s in double precision are observed.",
"This is with N_c=3 colors, N_v=12 right-hand-sides, N_{thr}=256 threads, on lattices of size 32^3*64, using exclusively OMP pragmas.",
"Interestingly, the same routine performs quite well on Intel Core i7 architectures, too.",
"Some observations on the much harder Wilson fermion matrix-times-vector optimization problem are added."
],
[
"Introduction",
"Conceptually the Brillouin operator $D_\\mathrm {B}$ is a sibling of the Wilson Dirac operator $D_\\mathrm {W}$ , since $D_\\mathrm {W}(x,y)&=&\\sum _\\mu \\gamma _\\mu \\nabla _\\mu ^\\mathrm {std}(x,y)-\\frac{a}{2}\\triangle ^\\mathrm {std}(x,y)+m_0\\delta _{x,y}-\\frac{c_{{}_\\mathrm {SW}}}{2}\\sum _{\\mu <\\nu }\\sigma _{\\mu \\nu }F_{\\mu \\nu }\\delta _{x,y}\\\\D_\\mathrm {B}(x,y)&=&\\sum _\\mu \\gamma _\\mu \\nabla _\\mu ^\\mathrm {iso}(x,y)-\\frac{a}{2}\\triangle ^\\mathrm {bri}(x,y)+m_0\\delta _{x,y}-\\frac{c_{{}_\\mathrm {SW}}}{2}\\sum _{\\mu <\\nu }\\sigma _{\\mu \\nu }F_{\\mu \\nu }\\delta _{x,y}$ share the same structure$m_0$ and $c_{{}_\\mathrm {SW}}$ must be tuned separately in (REF ) and () to establish vanishing pion mass and absence of $O(a)$ cut-off effects.",
"with $\\sigma _{\\mu \\nu }\\!=\\!\\frac{\\mathrm {i}}{2}[\\gamma _\\mu ,\\gamma _\\nu ]$ and $F_{\\mu \\nu }$ the hermitean clover-leaf field-strength tensor.",
"The main difference is that the isotropic derivative $\\nabla _\\mu ^\\mathrm {iso}$ and the Brillouin Laplacian $\\triangle ^\\mathrm {bri}$ both include 80 neighbors (all within the $[-1,+1]^4$ hypercube around a given lattice point $x$ , i.e.",
"up to 4 hops away from $x$ , but no two hops may be in the same direction), rather than the 8 neighbors present in the standard derivative $\\nabla _\\mu ^\\mathrm {std}$ and the standard Laplacian $\\triangle ^\\mathrm {std}$ .",
"Given that the Brillouin operator has a larger “footprint” and hence more operations per site than the Wilson operator, a natural question to ask is whether $D_\\mathrm {B}$ is more suitable for modern architectures (which typically involve lots of cores, but are limited by memory bandwidth) than $D_\\mathrm {W}$ .",
"As a first step to address this question, I decided to come up with a simple implementation of the Brillouin operator on the Intel KNL architecture.",
"More specifically, this boundary condition is meant to imply that ($i$ ) only shared memory parallelization via OpenMP pragmas on a single CPU is used (i.e.",
"no distributed memory parallelization with MPI), ($ii$ ) the memory layout is unchanged from the generic layout used throughout my code suite (see below for details), and ($iii$ ) no single-thread performance tuning is attempted beyond adding straightforward SIMD hints (again via OpenMP pragmas).",
"To the expert these constraints may look unnecessarily tight, but it turns out that nonetheless sustained performance figuresIn the meantime (i.e.",
"after the conference) a slight increase to 380 Gflop/s was achieved.",
"of 360 Gflop/s in single precision (sp) arithmetics can be achieved."
],
[
"Brillouin operator in a nutshell",
"Let $(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4)\\equiv (-240,8,4,2,1)/64$ ; then the free $\\triangle ^\\mathrm {bri}$ in () takes the form $a^2\\triangle ^\\mathrm {bri}(x,y)&=&\\lambda _0\\,\\delta _{x,y}+ \\lambda _1\\sum \\nolimits _{\\mu }\\delta _{x+\\hat{\\mu },y}+ \\lambda _2\\sum \\nolimits _{\\ne (\\nu ,\\mu )}\\delta _{x+\\hat{\\mu }+\\hat{\\nu },y}\\nonumber \\\\&+&\\lambda _3\\sum \\nolimits _{\\ne (\\rho ,\\nu ,\\mu )}\\delta _{x+\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho },y}+ \\lambda _4\\sum \\nolimits _{\\ne (\\sigma ,\\rho ,\\nu ,\\mu )}\\delta _{x+\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho }+\\hat{\\sigma },y}$ where $\\ne \\!",
"(\\rho ,\\nu ,\\mu )$ means that $\\rho $ , $\\nu $ and $\\mu $ are summed over, subject to the constraint that no two elements are equal.",
"Similarly, with $(\\rho _1,\\rho _2,\\rho _3,\\rho _4)\\equiv (64,16,4,1)/432$ the free $\\nabla _\\mu ^\\mathrm {iso}$ in () takes the form $a\\nabla _\\mu ^\\mathrm {iso}(x,y)&=&\\rho _1\\,[\\delta _{x+\\hat{\\mu },y}-\\delta _{x-\\hat{\\mu },y}]\\nonumber \\\\&+&\\rho _2\\sum \\nolimits _{\\ne (\\nu ;\\mu )}[\\delta _{x+\\hat{\\mu }+\\hat{\\nu },y}-\\delta _{x-\\hat{\\mu }+\\hat{\\nu },y}]\\nonumber \\\\&+&\\rho _3\\sum \\nolimits _{\\ne (\\rho ,\\nu ;\\mu )}[\\delta _{x+\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho },y}-\\delta _{x-\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho },y}]\\nonumber \\\\&+&\\rho _4\\sum \\nolimits _{\\ne (\\sigma ,\\rho ,\\nu ;\\mu )}[\\delta _{x+\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho }+\\hat{\\sigma },y}-\\delta _{x-\\hat{\\mu }+\\hat{\\nu }+\\hat{\\rho }+\\hat{\\sigma },y}]$ where $\\ne \\!",
"(\\rho ,\\nu ;\\mu )$ means that only $\\rho $ and $\\nu $ are summed over, while still no two out of the three indices may be equal.",
"In the interacting theory, these stencils are to be gauged in the obvious way.",
"For instance, considering (REF ) we see 8 terms ($\\propto \\!\\lambda _1$ ) with 1 hop; they are dressed with $U_\\mu (x)$ or $U_\\mu (x\\!-\\hat{\\mu })^\\dagger $ , for positive or negative $\\mu $ , respectively.",
"Similarly, there are 24 terms with 2 hops; they are dressed with off-axis links of the form $\\frac{1}{2}[U_\\mu (x)U_\\nu (x\\!+\\!\\hat{\\mu })+U_\\nu (x)U_\\mu (x\\!+\\!\\hat{\\nu })]$ .",
"Next, there are 32 terms with 3 hops; they are dressed with off-axis links which are the average of 6 products of three factors each.",
"And finally, there are 16 terms with 4 hops; they are dressed with hyperdiagonal links built as the average of 24 products of four factors each.",
"Further details of the operator can be found in [1], [2]."
],
[
"Code suite overall guidelines",
"The overall guidelines of the code suite are best illustrated by taking a look at the Wilson operator $D_\\mathrm {W}(x,y)=\\frac{1}{2}\\sum _\\mu \\Big \\lbrace (\\gamma _\\mu \\!-\\!I) \\; U_\\mu (x) \\; \\delta _{x+\\hat{\\mu },y}-(\\gamma _\\mu \\!+\\!I) \\; U_\\mu ^\\dagger (x\\!-\\!\\hat{\\mu }) \\; \\delta _{x-\\hat{\\mu },y}\\Big \\rbrace +(4\\!+\\!m_0) \\; \\delta _{x,y}$ which is said to operate on a vector of length $N_{\\!c}4 N_x N_y N_z N_t$ , where $N_{\\!c}$ is the number of colors.",
"From a computational viewpoint it is extremely convenient to declare the source and sink vectors as complex arrays of size (1:Nc,1:4,1:Nv,1:Nx*Ny*Nz*Nt).",
"Here the ordering and the stride notation common to Matlab and Fortran are used; with the default counting from 1 we could write (Nc,4,Nv,Nx*Ny*Nz*Nt).",
"A special feature is that we have one slot to address the right-hand-side (rhs), i.e.",
"$N_{\\!v}$ columns (in the mathematical setup terminology) can be processed simultaneously.",
"The underlying philosophy is that index computations are done by the compiler, except for the site index n=(l-1)*Nx*Ny*Nz+(k-1)*Nx*Ny+(j-1)*Nx+i, where we want to keep some freedom.",
"Figure: Overall structure of the Wilson routine; accumulation of the 1+8 contributions in the thread-privatevariable site(1:Nc,1:4,1:Nv) in the dotted blocks is specified in Fig.",
".In the actual routine COLLAPSE(2) is added to the !$OMP pragma, and the the statementsfor l_plu and l_min are transferred to the next line.Figure: Detail of the fourth block in Fig.",
"; the variable full(1:Nc,1:4) contains old(1:Nc,1:4,idx,nsh)after left-multiplication with 1 2\\frac{1}{2} times the link-variable, but before right-multiplication with (-I-γ 1 ) trsp (-I-\\gamma _1)^\\mathrm {trsp}.The working of these guidelines, on the basis of the Wilson operator (REF ), is spelled out in the code assembled in Figs.",
"REF and REF .",
"The arrangement of the loops over the $x,y,z,t$ coordinates (denoted i,j,k,l, respectively) makes sure the innermost loop belongs to the fastest index.",
"A noteworthy mathematical detail is how the $N_{\\!c}4\\times N_{\\!c}4$ matrix $\\frac{1}{2}(\\gamma _\\mu \\!-\\!I) \\otimes U_\\mu (x)$ acts on the $N_{\\!c}4\\times 1$ vector old(:,:,idx,n) with given rhs and site indices.",
"This product is realized by reshaping it into a $N_{\\!c}\\times 4$ matrix (which it already is in our setup), multiplying it with $U_\\mu (x)$ from the left, and with $\\frac{1}{2}(\\gamma _\\mu \\!-\\!I)^\\mathrm {trsp}$ from the right.",
"The full-spinor variable full(Nc,4) contains the result of the former multiplication; the $\\gamma $ -operation just reorders its columns (modulo some signs and factors of $\\mathrm {i}$ ).",
"The name of the gauge variable in Figs.",
"REF and REF is supposed to remind us that in most practical applications the smeared gauge field $V_\\mu (x)$ is used rather than the original gauge field $U_\\mu (x)$ .",
"In case of clover improvement, it is practical to precompute the field-strength tensor $F_{\\mu \\nu }(x)$ once, and to store it in a complex array F(Nc,Nc,6,Nx,Ny,Nz,Nt) which is then passed to the clover routine.",
"One might notice that in Fig.",
"REF only the linear combinations full(:,1)-i*full(:,4) and full(:,2)-i*full(:,3) are used.",
"Hence it would have been sufficient to form these linear combinations prior to left-multiplying with the gauge field.",
"This “shrink-expand-trick” can be used for all eight directions, regardless of the $\\gamma $ -representation chosen (Fig.",
"REF uses the chiral one)."
],
[
"Brillouin kernel details",
"The Brillouin matrix-times-vector routine, built according to the guidelines laid out in the previous section, is portrayed in Figs.",
"REF and REF .",
"The four-fold loop structure over the out-vector is OMP-parallelized in the a straightforward way (the COLLAPSE(2) statement makes sure that up to $N_zN_t$ threads can be launched).",
"The most important difference to the Wilson routine is that 40 out of the 81 directions of the off-axis links $W$ , build from the smeared gauge field $V$ , are assembled in the complex array W(Nc,Nc,40,Nx,Ny,Nz,Nt); the remaining ones are the identity or the hermitean conjugate of $W$ in a hypercube related point (i.e.",
"up to four hops away).",
"Similar to the Wilson routine, the SIMD hints are given as pragmas to the loop over the rhs-index idx.",
"Within this loop, the spinor and color operations are explicitly or implicitly unrolled (e.g.",
"by forall constructs, stride notation).",
"The complex numbers formed from fac_i,fac_j,fac_k,fac_l implement the right-multiplication with $\\gamma _1^\\mathrm {trsp},...,\\gamma _4^\\mathrm {trsp}$ in the chiral representation.",
"All 81 contributions are accumulated in the thread-private variable site(Nc,4,Nv); since this variable is written once to the respective site in new(:,:,:,n) there cannot be any thread-write-collision by construction."
],
[
"Brillouin operator timings",
"Timings are done on a node containing a single KNL chip with 64 cores.",
"All results for the Brillouin operator are converted into Gflop/s, based on a flop count of $2560N_{\\!c}^2+2376N_{\\!c}$ per site (i.e.",
"30168 for QCD, see [2] for details).",
"As a default setup we shall use a $32^3\\times 64$ lattice, with $N_{\\!c}=3$ and $N_{\\!v}=4N_{\\!c}$ .",
"This geometry is chosen such that a sp-field W, a sp-field F, and the dp-fields old,new fit into the high-bandwidth MCDRAM of 16 GB.",
"For any number of threads, memory allocation of these objects to the individual cores is done by a first-touch policy.",
"The static thread scheduling makes sure every thread gets exactly the same fraction of the out vector to work on.",
"Compilation is done with ifort version 17.2, with the flags -qopenmp -O2 -xmic-avx512 -align array64byte.",
"Figure: Scaling in the number of threads of the matrix-times-vector performance of the Brillouin operatoron a 32 3 ×6432^3\\times 64 lattice in sp (left) and dp (right), using a KNL chip with 64 cores.The scaling in the number of threads is shown in Fig.",
"REF .",
"We see an almost perfectly linear behavior up to 64 threads; this means that there is essentially no scheduling overhead.",
"Beyond 64 threads we see load balancing issues among the various threads, except for multiples of 64, where each core is kept busy with exactly the same number of threads.",
"Table: Performance in Gflop/s of the Brillouin matrix-times-vector operation on a 32 3 ×6432^3\\times 64 lattice,with N c =3N_{\\!c}=3 and N v =4N c N_{\\!v}=4N_{\\!c}, versus the number of threads.",
"The panels refer to a KNL and a Core i7 (Broadwell), respectively.Table: Performance in Gflop/s of the Brillouin matrix-times-vector operation on the KNL architecture in sp arithmetics.In the first panel the volume dependence is displayed (with N c =3N_{\\!c}=3, N v =4N c N_{\\!v}=4N_{\\!c}, and T=2LT=2L),in the second panel the scaling in the number of right-hand-sizes is shown (with N c =3N_{\\!c}=3 and 24 3 ×4824^3\\times 48 volume),and in the third panel the dependence on N c N_{\\!c} is considered (with N v =4N c N_{\\!v}=4N_{\\!c} and 24 3 ×4824^3\\times 48 volume).Some more details are presented in Tab.",
"REF .",
"We see some mild improvement when going from 2 to 4 threads per core; beyond 256 threads the performance plateaus.",
"When comparing sp to dp figures, one should keep in mind that the objects W and F are always in single precision (occupying 5760 MB and 864 MB, respectively), only the vectors old and new change from sp to dp.",
"Perhaps the most surprising observation is that the same routine (compiled with -xcore-avx2 instead of -xmic-avx512) performs well on a standard Core i7 architecture (Broadwell with 6 cores).",
"Here the plateauing effect sets in after each physical core is occupied with 2 threads.",
"Some more experiments on the KNL architecture in sp arithmetics are reported in Tab.",
"REF .",
"The first panel demonstrates that the $\\sim \\!360$ Gflop/s are more or less independent of the volume, i.e.",
"we do not see any peculiar cache size effects.",
"The second panel shows that using more than 12 right-hand-sides (for $N_{\\!c}=3$ ) improves the performance, but beyond 24 right-hand-sides benefits become marginal.",
"Finally, increasing the number of colors beyond 3 is found to be particularly beneficial.",
"My personal guess is that with $N_{\\!c}=4$ (or a multiple thereof) the colormatrix-times-spinor multiplication in the SIMD loop in Fig.",
"REF becomes particularly efficient due to a better filling of the SIMD pipeline."
],
[
"Wilson operator timings",
"Timings are done on a node containing a single KNL chip with 64 cores.",
"All results for the Wilson operator are converted into Gflop/s, based on a flop count of $128N_{\\!c}^2+72N_{\\!c}$ per site (i.e.",
"1368 for QCD, using the “shrink-expand-trick”, see e.g.",
"[2] for details).",
"The default setup is again a $32^3\\times 64$ lattice, with $N_{\\!c}=3$ and $N_{\\!v}=4N_{\\!c}$ .",
"This time a sp-field V, a sp-field F, and the dp-fields old,new fit well into the high-bandwidth MCDRAM.",
"For any given number of threads, all arrays are allocated afresh, and a first touch policy is used as in the Brillouin case.",
"Figure: Scaling in the number of threads of the matrix-times-vector performance of the Wilson operatoron a 32 3 ×6432^3\\times 64 lattice in sp (left) and dp (right), using a KNL chip with 64 cores.The scaling in the number of threads is shown in Fig.",
"REF .",
"Again, we see a linear behavior up to 64 threads; and beyond that local maxima are seen for multiples of 64 threads where each core is kept busy with exactly the same number of threads.",
"Table: Performance in Gflop/s of the Wilson matrix-times-vector operation on a 32 3 ×6432^3\\times 64 lattice,with N c =3N_{\\!c}=3 and N v =4N c N_{\\!v}=4N_{\\!c}, versus the number of threads.",
"The panels refer to a KNL and a Core i7 (Broadwell), respectively.Some more details are presented in Tab.",
"REF .",
"On the KNL some some mild improvement is seen when going from 2 to 4 threads per core; beyond 256 threads the performance plateaus.",
"When comparing sp to dp figures, one should keep in mind that the objects V and F are always in single precision (occupying 576 MB and 864 MB, respectively), only the vectors old and new change from sp to dp.",
"From the second panel we learn that also the Wilson routine performs (without any change) quite well an the standard Core i7 architecture (Broadwell with 6 cores).",
"The main difference to the KNL case is that optimum performance is reached with 1 to 2 threads per core rather than 4."
],
[
"Summary",
"The goal of this contribution was to explore whether acceptable performance figures for the Brillouin and Wilson matrix-times-vector applicationsAfter the conference a staggered routine, built with the same guidelines, was found to yield over 280 Gflop/s on the KNL.",
"on one KNL chip can be obtained, if we refrain from using advanced optimization techniques (for an overview see the recent plenary talks [3], [4]).",
"Table: Conversion of the performance measurements into sustained percentage figures, based on a peak performance of 5.2/2.6 [sp/dp] Tflop/s on the KNL and 690/345 Gflop/s on the Core i7 (Broadwell) architecture.Pertinent results are summarized in Tabs.",
"REF and REF for the two operators, respectively.",
"Not just beauty, also judgement of such figures is in the eye of the beholder.",
"To me it appears that these are acceptable figures – especially in view of the simplicity of the shared memory parallelization and SIMD encouragement strategies used (both with OMP pragmas only).",
"Perhaps the most surprising finding is that these routines (unchanged, just recompiled) perform quite well on the standard Core i7 architecture, too.",
"The loss in performance, compared to the KNL architecture, is a factor 2.2 for the Brillouin operator, and a factor 3.1 for the Wilson operator.",
"It is instructive to convert these figures into sustained performance ratios.",
"The KNL chip operates at 1.269 GHz; with 64 cores and 64/32 flop/s per cycle it has a peak performance of 5.2/2.6 Tflop/s in sp/dp arithmetics.",
"The Broadwell chip operates at 3.6 GHz; with 6 cores and 32/16 flop/s per cycle it has a peak performance of 690/345 Gflop/s in sp/dp arithmetics.",
"With these numbers in hand, the performance figures of the Brillouin and Wilson operators can be converted into sustained performance ratios.",
"The results, collected in Tab.",
"REF , indicate that (a) the efficiency of the Brillouin operator is generically higher than the efficiency of the Wilson operator, and (b) the efficiency on the Broadwell architecture is generically higher than the efficiency of the KNL.",
"An explanation of (a) is easily found.",
"For SU(3) gauge group the Brillouin-to-Wilson flop-count ratio is 22.1.",
"At the same time the Brillouin-to-Wilson memory-traffic ratio is 8.9 [2].",
"Taken together, this means that the computational intensity of the Brillouin operator is higher by a factor 2.5 [2].",
"We know that modern architectures tend to have plenty of CPU capability, and such prerequisites favor applications with high computational intensity.",
"As for (b) the overall time (as compared to e.g.",
"a scalar product) and the huge performance difference between SU(3) and SU(4) gauge group suggest that incomplete filling of the SIMD pipeline in the idx-loops in Figs.",
"REF and REF likely represents the actual bottleneck (at least on the KNL architecture which operates at 512-bit width).",
"The main lesson is that in Lattice QCD it is easy to get a reasonable (i.e.",
"non-excellent) performance, while maintaining full portability, if the compiler acts on code whose structure is simpleI thank Eric Gregory and Christian Hoelbling for discussion; I acknowledge partial funding by DFG through SFB TR-55.."
]
] | 1709.01828 |
[
[
"Investigation of the inner structures around HD169142 with VLT/SPHERE"
],
[
"Abstract We present observations of the Herbig Ae star HD169142 with VLT/SPHERE instruments InfraRed Dual-band Imager and Spectrograph (IRDIS) ($K1K2$ and $H2H3$ bands) and the Integral Field Spectrograph (IFS) ($Y$, $J$ and $H$ bands).",
"We detect several bright blobs at $\\sim$180 mas separation from the star, and a faint arc-like structure in the IFS data.",
"Our reference differential imaging (RDI) data analysis also finds a bright ring at the same separation.",
"We show, using a simulation based on polarized light data, that these blobs are actually part of the ring at 180 mas.",
"These results demonstrate that the earlier detections of blobs in the $H$ and $K_S$ bands at these separations in Biller et al.",
"as potential planet/substellar companions are actually tracing a bright ring with a Keplerian motion.",
"Moreover, we detect in the images an additional bright structure at $\\sim$93 mas separation and position angle of 355$^{\\circ}$, at a location very close to previous detections.",
"It appears point-like in the $YJ$ and $K$ bands but is more extended in the $H$ band.",
"We also marginally detect an inner ring in the RDI data at $\\sim$100 mas.",
"Follow-up observations are necessary to confirm the detection and the nature of this source and structure."
],
[
"Introduction",
"Young stellar objects are surrounded by circumstellar material, making them ideal targets to study planetary formation.",
"Transitional disks are particularly interesting as they may constitute the intermediate step between gas-rich protoplanetary disks where planets are supposed to form, and dusty debris disks.",
"Direct observations of companions and disk structures are necessary to bring constraints on planetary formation.",
"A few targets have already been identified as interesting cases to study this phenomenon, such as HD 100546 , , , , HD 142527 , or LkCa 15 , .",
"These examples show that determining the origin of disk structures is a difficult task, and overall, the risk to confuse them with forming planets is quite high .",
"HD 169142 is a well studied Herbig Ae star at 117 pc , hosting a nearly face-on disk often categorized as pre-transitional since it shows dust emissions both at close and large separations from the star separated by several gaps , .",
"The disk has first been spatially resolved by with polarimetry and studied by with spectroscopy, and later confirmed by .",
"reported polarimetric observations with NaCo in $H$ band, revealing a bright irregular ring at 170 milliarseconds (mas), that is 20 AU, and an annular gap from 270 to 480 mas (32-56 AU), the surface brightness smoothly decreasing after 550 mas (66 AU).",
"confirm a double ring structure using the Gemini Planet Imager (GPI) in $H$ band, showing a surface brightness enhancement at 180 mas (21 AU) and one at 510 mas.",
"Interestingly, ALMA (Atacama Large Millimetre/Sub-millimeter Array) observations revealed two rings at 170-300 mas and 479-709 mas, and an empty cavity (R<171 mas) at the center of the dust disk but filled with gas .",
"This might indicate the presence of multiple planets carving out the gaps and cavities , , or alternatively that magneto-rotational instability (MRI) effects combined with magnetohydrodynamical (MHD) winds shape the disk density structure.",
"Additional hints of planet formation have been identified around HD 169142. and performed observations of HD 169142 with NaCo in the $L^{\\prime }$ band.",
"detected a faint marginally resolved point-like feature in the data from July 2013, located at a position angle (PA) of 0$\\pm 14^\\circ $ and a separation of $\\rho $ =110$\\pm $ 30 mas (13$\\pm $ 3.5 AU), with $\\Delta mag$ =6.4$\\pm $ 0.2.",
"If this emission was photospheric, it would correspond to a 60-80 $M_{\\rm Jup}$ brown dwarf companion.",
"However, this companion was not confirmed by shorter wavelength follow-up observations performed with the adaptive optics system at the Magellan Clay Telescope (MagAO/MCT) in $H$ , $K_S$ and $z_p$ bands (where it should have been easily detected if it was a 60-80 M$_{\\rm Jup}$ companion), nor at 3.9$\\mu m$ , though at lower sensitivity.",
"This suggests that the object found in 2013 might be a part of the disk possibly showing planetary formation with an unknown heating source.",
"also detected a point-souce in NaCo data from June 2013.",
"This emission source was at $\\rho =$ 156$\\pm $ 32 mas (18$\\pm $ 3.8 AU) and PA=7.4$\\pm $ 1.3$^\\circ $ .",
"It has a $\\Delta mag$ =6.5$\\pm $ 0.5, and an apparent magnitude of 12.2$\\pm $ 0.5 mag.",
"They suggest that this could come from the photosphere of a 28-32 $M_{\\rm Jup}$ companion, or from an accreting lower-mass forming planet in the gap.",
"Additional observations were carried out with the GPI instrument in the $J$ band in April 2014, where this hypothetical companion was not retrieved, again suggesting that it was not a 28-32 M$_{\\rm Jup}$ object, as this should have been relatively easily detected in $J$ band.",
"Finally, additionally find with the Expanded Very Large Array (EVLA) 7 mm observations a knot of emission at 350 mas (41 AU at 117 pc), that could correspond to an object of 0.6 M$_{\\rm Jup}$ .",
"The follow-up MagAO/MCT observations performed by led to another, low signal-to-noise ratio (S/N) detection at $\\rho =180$ mas (21 AU) and PA=33$^\\circ $ .",
"If a real companion, this structure would correspond to a 8-15 $M_{\\rm Jup}$ substellar companion, but it was not found in the initial 2013 NaCo data in $L^{\\prime }$ band.",
"All these results demonstrate the complexity of this system which is even more critical given the lack of consistency (possibly because of observational limitations) between the results.",
"Figure REF summarises the different point-like structures identified around HD169142 so far.",
"Figure: Diagram of the HD169142 system.",
"The red parts represent the two rings and the white parts are the gaps.",
"The crosses represent the positions of the point-like structures discovered by , and .",
"We also show the structure around 100 mas detected with SPHERE in this work (L17, see Sec. ).",
"The dimension of the crosses represent the error bars (on scale), except for B14b and O14 where the error bars are given arbitrarily.",
"The inclination of the disk is not shown in this diagram.In this paper, we investigate the innermost structures (<300 mas) previously detected around HD 169142 to confirm the presence of the candidate companions detected by and and investigate their nature.",
"We present new near infrared (NIR) observations of HD 169142 obtained with SPHERE/VLT , as part of the Guaranteed Time Observations (GTO) dedicated to exoplanet search (SpHere INfrared survey for Exoplanets or SHINE, Chauvin et al.",
"in prep.).",
"SPHERE has primarily been designed to image and characterize exoplanets, but it is also a powerful instrument for probing the dusty surface of protoplanetary disks.",
"The observations are described in Sec.",
"and the data analysis in Sec. .",
"We report in Sec.",
"the detection of bright blobs at 180 mas that are actually part of the inner ring, and we show in Sec.",
"the marginal detection of a bright structure located at similar position to the object found by and .",
"We conclude in Sec. .",
"Table: Parameters of HD169142."
],
[
"Observations and data reduction",
"Observations of HD 169142 were performed from 2015 to 2017 (see Tab.",
"REF ).",
"The data were obtained in the IRDIFS or in the IRDIFS$\\_$ EXT modes, using simultaneously the DBI mode.",
"For the IRDIFS mode, the Integral Field Spectrograph was operating in the wavelength range between 0.95 $\\mu $ m and 1.35 $\\mu $ m ($YJ$ ) at a spectral resolution of R=50, and the InfraRed Dual-band Imager and Spectrograph in the $H$ band with the H23 filter pair ($\\lambda _{\\mathrm {H2}} = 0.055$ $\\mu $ m, $\\lambda _{\\mathrm {H3}} = 1.667$ $\\mu $ m).",
"For the IRDIFS$\\_$ EXT mode, the IFS was used between 0.95 $\\mu $ m and 1.65 $\\mu $ m ($YJH$ ) (R=30) and IRDIS in the $K$ band with the K12 filter pair ($\\lambda _{\\mathrm {K1}} = 2.110$ $\\mu $ m, $\\lambda _{\\mathrm {K2}} = 2.251$ $\\mu $ m).",
"Due to very small angular separation of previously reported detections, the two most recent observations were done without coronagraph, but the core of the stellar point-spread function (PSF) was saturated over a radius of $\\sim $ 1 $\\lambda /D$ and observations were performed in pupil-stabilized mode to enable angular differential imaging .",
"To check the consistency of the results, different pipelines were used to reduce and analyse the IFS and IRDIS data.",
"We used the LAM-ADI pipeline , pipeline and the SPHERE Data Reduction and Handling (DRH) automated pipeline for IRDIFS data, and the pipeline described in for the IFS data.",
"Even though the observing conditions were good, there were some temporal variations so we performed a frame selection on the data sets.",
"We used the sortframe routine developed by the SPHERE Data Center (DC) to select the good frames when using the DC and pipelines.",
"The minimum fraction of selected frames is about 80$\\%$ and the maximum one is near 100$\\%$ , depending on weather conditions.",
"For the LAM-ADI pipeline, we calculated the moving average of the flux in an annulus centered on the star.",
"We then excluded the frames presenting a flux above or below 1.5$\\sigma $ of the mean flux.",
"This method follows a Gaussian behavior and corresponds to $\\sim $ 14$\\%$ of the frames removed using the 1.5$\\sigma $ criterion.",
"This allows to keep enough frame to have a correct S/N while removing the very bad frames that could induce artifacts in the images (this case only applies to the images shown in Fig REF ).",
"Finally, the SHINE data were astrometrically calibrated following the analysis in .",
"To improve the S/N, and to show the different structures appearing in the images, the selected IFS and IRDIS data were collapsed to broad-band images equivalent to $K$ , $H$ and $YJ$ bands."
],
[
"PCA reduction",
"The data were first analysed using principal component analysis (PCA) based on the formalism described in .",
"The modes were calculated over the full sequence at separations up to 500 mas.",
"A variable number of modes were subtracted, up to $\\sim 10\\%$ of the total number of modes for IRDIS ($\\sim $ 50), and up to 50 modes for the IFS, before rotating the images to a common orientation and combining them with an average.",
"The resulting images obtained in $YJ$ , $K$ and $H$ band for periods 2015 June 7 (best quality image for 2015), 2016 June 27 and 2017 April 30 (best quality images for the 2016 and 2017 periods) are presented in Fig.",
"REF .",
"Figure REF shows extended and point-like surface brightness enhancements depending on the band, and a faint arc-like feature in IFS data appearing on the East part of the disk, in particular in $YJ$ band (refered to as spiral).",
"The bright structures are located at separations of $\\sim $ 180-200 mas, especially at PA = 20$^{\\circ }$ (structure A), 90$^{\\circ }$ , and 310$^{\\circ }$ (structure B).",
"Other structures are detected at $\\sim 150$ mas (PA= 320$^{\\circ }$ ) and at $\\sim 100$ mas (PA = 355$^{\\circ }$ ; structure C) from the central star.",
"In IRDIS data, the bright structures are still visible but appear fainter.",
"The structures appear point-like in the $YJ$ and $K$ band and more extended in the $H$ band.",
"They are persistent whatever the number of subtracted modes in IFS and IRDIS data.",
"In Appendix , we show the S/N maps of epochs 2017 April 30 and 2016 June 27.",
"The maps show bright and dark structures with positive and negative values of the S/N.",
"The S/N is calculated as the normalized difference in intensity between a considered feature and two neighbouring areas at the same separation .",
"Thus, bright peaks are significant features, just as much as dark peaks since they indicate darker features than the surrounding background.",
"It is important to note that the calculation of the S/N can be modulated by the background which is inhomogeneous.",
"In particular, the dark peak at $\\sim 40^{\\circ }$ in the S/N map is not dark in Fig.",
"REF , but is surrounded by two bright structures.",
"The positions of the structures showing a high signal (S/N$\\sim $ 3) are consistent with the structures appearing in Fig.",
"REF , in particular structure A and B.",
"Structure C appears with a lower S/N.",
"Several features have already been discovered at the separations of structures A, B and C (a bright ring and candidate companions, see Sec.",
"), but they appeared point-like.",
"Since we detect bright spots at similar positions in our images, we try to investigate whether or not these detections are the same as the previous ones.",
"In the next section, we analyse the structures found around $\\sim $ 180-200 mas, and in Sec., we focus on the detection at $\\sim $ 100 mas.",
"Figure: Result of the PCA analysis on the IFS with 50 subtracted modes, and IRDIS with 6, 50 and 20 subtracted modes for 2015, 2016 and 2017 data, respectively.",
"The square-pattern central circle shows the position of the coronagrah in the 2015 data set; the star is at the center.",
"The bright structures are indicated with blue arrows.",
"Letter A indicates the structure at PA = 20 ∘ ^{\\circ }, and letter B indicates the one at 310 ∘ ^{\\circ }, both being at separation ∼\\sim 180 mas.",
"Letter C shows the structure at ∼\\sim 100 mas and PA = 355 ∘ ^{\\circ }.",
"The two white dashed circles have a radius of 100 mas and 180 mas, respectively.",
"North is up and east is left."
],
[
"RDI reduction",
"We performed Reference Differential Imaging which consists in subtracting the reference image of one or several stars to the target image.",
"This technique allows subtracting the speckle pattern, while limiting the self-subtraction effects usually affecting ADI data, in particular in the case of extended structures like disks .",
"To select the reference images, we searched in the complete database of SPHERE GTO observations the reduced images that have the best correlation with our target images.",
"This means that we calculated the correlation coefficient between the data sets taken in a similar band as the considered target observation, and the considered data set of our target.",
"The best correlation coefficient (that is > 0.90 in general) designates the data set that is used as reference image.",
"For images taken without coronagrah, the correlation coefficient is lower (around 0.50) than with coronagraph because there are fewer images taken without the coronagraph in the SPHERE database.",
"Similarly, there are many more $Y-J$ images than $Y-H$ images, leading to lower coefficients for the latter mode.",
"Figure REF shows the RDI IFS images of HD169142 from June 2015 and April 2016 with a subtracted image of the same night each time.",
"We see a possible double ring structure, with one being located at $\\sim $ 180 mas, and possibly another one at $\\sim $ 100 mas, that is, close to the bright blobs detected with PCA analysis.",
"Both rings are inhomogeneous; in particular, the one at 180 mas shows a decrease in the brighness around PA=$45^\\circ $ compared to the surrounding ring signal (at $20^\\circ $ and $80^\\circ $ ), and there are several brightness enhancements in the North-West and South-West directions.",
"The ring appears more clearly in the April 2016 image, possibly due to a better quality of the data and a larger rotation field.",
"The inner ring at 100 mas is quite bright with a brighter region in the North West direction in the 2015 image.",
"However, it is not detected in each reduction (in particular, it is hardly seen in RDI data without coronagraph), and its appearance depends on the scale used (see Sec. ).",
"It appears much less bright in the 2016 April image, although we still detect a signal.",
"Figure: Result of the RDI analysis of the IFS data from 2015 June and 2016 April.",
"We clearly see an inhomogenous bright ring at ∼\\sim 180 mas, and possibly another inner ring, althought its position close to the star makes it less trustable.",
"North is up and east is left."
],
[
"Simulation of cADI reduction with PDI data",
"An important indication for understanding the nature of the detected structures would be to know if the scattered light is polarized.",
"Indeed, planets are usually considered to not emit polarized light, contrary to protoplanetary disks .",
"Polarized light due to reflection from hot Jupiter planets could also be detected in the optical (UBV bands).",
"However, the produced signal would be low , which might be impossible to detect when the planet is embedded in a disk, which produces polarized scattered light.",
"To investigate the nature of the blobs at $\\sim $ 180 mas, we use IRDIS PDI (Polarimetric Differential Imaging) data that were acquired on 2015-05-02 with the ALC$\\_$ YJ$\\_$ S apodized-pupil Lyot coronagraph (145 mas in diameter) in the $J$ band and reduced following .",
"A full analysis and modelling of the PDI data will be presented in a forthcoming paper .",
"The left and middle panels of Fig.",
"REF present a comparison between the IFS $J$ band data and the IRDIS PDI data of the very central region around the star ($\\pm $ 300 mas).",
"The ring at 180 mas in the IRDIS polarized intensity image is detected at extremely high significance and the use of a small coronagraphic mask allows to unambiguously confirm the existence of the cavity inside of the ring.",
"The polarized intensity image also clearly shows a variation of the ring brightness as a function of the position angle, with an increase of the brightness at positions angle of $\\sim $ 20$^{\\mathrm {o}}$ , $\\sim $ 90$^{\\mathrm {o}}$ , $\\sim $ 180$^{\\mathrm {o}}$ and to a lesser extent at $\\sim $ 310$^{\\mathrm {o}}$ .",
"The brightness of the structure has been measured by , and higher signals at these same PA are clearly visible.",
"Interestingly, these regions of increased brightness seem to correspond to position angles where the IFS $J$ band image shows extended bright structures at a significance of 2.5–3$\\sigma $ above the surrounding residualsNote that the signal-to-noise maps were calculated on our images assuming that we were looking for point sources, which does not translate directly when considering extended structures such as disks.",
"However, the structures visible in the data are clearly identifiable above the surrounding background..",
"In particular, the bright structures at PA$\\approx 20^\\circ $ and $310^\\circ $ in PDI data seem to correspond to the structures shown by the blue arrows in Fig.",
"REF (structures A and B), and a structure at PA=380$^{\\circ }$ and $\\rho $ =100 mas seems to correspond to structure C. A bright extended structure also appears between 180$^{\\circ }$ and 210$^{\\circ }$ in the cADI simulation images, which is clearly visible in the PDI image but appears fainter in the IFS $J$ band image.",
"Finally, we notice the strong similarity between the ring appearing in the PDI data and in the RDI data from 2016 April (Fig.",
"REF ).",
"To confirm that the residual structures seen in the IRDIS and IFS data are in fact ring structures filtered by the ADI processing, we perform a simulation of ADI reduction using the IRDIS PDI polarized intensity image.",
"First, we create a data cube with 1709 copies of the PDI image (because this is the number of frames after selection using the LAM-ADI pipeline, see Sec.",
"REF ), corresponding to each of the IFS images, with each of the images being rotated to match the pupil offset rotation and the position angle of the observations.",
"Then, the median image of the cube along the temporal dimension is calculated and subtracted to each of the images in the cube (classical ADI).",
"Finally, all the images are rotated back to a common orientation and mean-combined.",
"The result of this simulation is presented in the right panel of Fig.",
"REF .",
"Visually, we see a strong correlation between the main structures identified in the IFS $J$ band image and the bright ring in the disk that have been spatially filtered by the ADI analysis.",
"The effect of ADI processing on disks has already been studied by and they have identified that ADI can have a strong impact on the flux and morphology of disks, up to the point of creating artificial features.",
"This effect has also been encountered in the case of HD 100546 and T Cha .",
"The ring of HD 169142 seen almost face-on is an extreme case: all azimuthally symmetric structures of the ring are completely filtered out by ADI, leaving only the signature of the features brighter or fainter than average.",
"In the simulation, the shape of the features at $\\sim $ 20$^{\\mathrm {o}}$ and $\\sim $ 90$^{\\mathrm {o}}$ is almost identical to that in the IFS image.",
"The same bright spot at a position angle of $\\sim $ 310$^{\\mathrm {o}}$ is also clearly visible.",
"For a more quantitative assessment, we compare in Fig.",
"REF azimuthal cuts of the IFS $J$ band data with 50 PCA modes and the cADI reduction simulation, measured at different separations from the star.",
"The signal is averaged in annuli of 20 mas of radial extension to smooth the small pixel-to-pixel variations in the data.",
"These cuts show a very strong correlation between some of the main features seen in the IFS data and the cADI reduction simulation: the Pearson correlation coefficients between the two data sets are equal to 0.47, 0.64 and 0.80 for the 150–170 mas, 170–190 mas and 190–210 mas azimuthal cuts respectively.",
"This correlation is calculated on the full ring, but it would be even higher if we considered more local correlations centered on the main features."
],
[
"Interpretation of the results",
"In the NIR, we are not only sensitive to the thermal emission from point sources but also to stellar light scattering of the protoplanetary dust.",
"The detected point source at 180 mas lies on the surface of the ring which is optically thick in the NIR (as the whole disk).",
"Thus, the signal of a point source in the midplane of the disk will be dominated by scattered light and the planet emission would be strongly absorbed by the dust, making impossible its detection.",
"It would be possible to detect a very massive companion at that location, but in that case we would expect it to have opened a deep and broad gap.",
"Instead, we find a ring.",
"Since the detection of polarized light from a planet is not expected, we conclude that the blobs that we detect both in PDI and non-polarized data are part of the same structure: the ring.",
"Hence, we can conclude with a high confidence level that our images show disk features rather than planetary companions for the structures A and B.",
"We can thus exclude the thermal emission from giant planets being consistent with the blobs signal, but we cannot exclude clumps in an early stage of planet formation.",
"Multiple blobs are found on the same orbit at $\\rho \\approx 180$ mas.",
"Of particular interest is the structure at PA$\\approx 20^\\circ $ (structure A in Fig.",
"REF ) because it is bright and appears both in PDI and non-polarized data.",
"We try to investigate if it corresponds to an object candidate, and in particular, to the one detected by at PA=33$^\\circ $ and $\\rho $ =180 mas (see Sec. )",
"because it is close to our current detection's position.",
"Considering the stellar distance ($117.3^{+3.8}_{-4.1}$ pc) and mass (1.65 M$_\\odot $ ) of HD 169142 (see Tab.",
"REF ), and the inclination of the disk , 's candidate should have an orbit of $\\sim $ 78.5 yr if in the disk plane with no eccentricity, and therefore should have moved of $\\sim $ 13.7$^{\\circ }$ from June 2013 to June 2016 .",
"This would bring it to a PA=19.3$^{\\circ }$ if in a clockwise motion, which is in very good agreement with the measured PA of our structure (20$^{\\circ }$ ).",
"The movement of the blobs (structures A and B) according to the different epochs of observations are shown in Fig.",
"REF .",
"We also include the separation and PA obtained by (Fig.",
"REF , green triangle).",
"For each structure, we calculate the average separation over all the epochs , and the Keplerian speed corresponding to this average separation.",
"We then plot the Keplerian speed on the PA figure.",
"We can see that the structure motions are compatible with Keplerian speeds.",
"We note that 's position is indicated without error bars, but they give a rough estimation of the position in their paper.",
"We already showed that our detections could be related to blobs in the disk.",
"We thus conclude that the PA and separation evolution of the blob are consistent with a Keplerian motion.",
"The blobs trace the bright ring in the disk, and rotates in the clockwise direction with a Keplerian velocity.",
"Moreover, ALMA data provide the direction of rotation of the disk (the Northern part of the disk is moving faster toward us than the local rest frame, while the Southern part is moving slower) and its closest side to the observer (the Western side), which are compatible with the blob clockwise motion.",
"The exact nature of the blobs and the origin of the bright disk rings and gaps in general remain to be investigated.",
"We only make a hypothesis here which scenarii could be compatible with our observations.",
"We also refer the reader to the upcoming work by for a detailed modeling study of the disk around HD169142 including planet-disk interaction processes and dust evolution dynamics.",
"The first possibility invokes intrinsic disk variations in density and temperature.",
"Indeed, the dust concentration in the ring might be a tracer of a maximum density in the gas profile.",
"This jump in the surface density could trigger the formation of vortices by the Rossby Wave Instability (RWI) which concentrates dust azimuthally.",
"Our observations look like figs.",
"3 and 5 of who display simulations of Rossby vortices with several irregular blobs of enhanced dust density on the same orbit.",
"[2] show that these vortices could be favourable places to initiate planets formation.",
"If this is the case, HD169142 could be the site of ongoing planet formation, at an earlier stage than previously expected.",
"Although multiple vortices are non-permanent states, the mass ratio between the disk and star does not make the disk gravitationaly unstable, and woud allow self-gravity to improve the stability of multiple vortices .",
"If the blobs in our image indeed trace vortices, we would only observe their signatures in the upper disk layers in our SPHERE observations.",
"While ALMA observations, that trace the midplane layers, have already been interpreted as vortices in protoplanetary disk , recent observations of HD 169142 with this instrument did not show any asymmetrical features at a resolution of 0.2-0.3\".",
"In addition, the spatial distribution of particles inside a vortex depends on their size , and such structures would thus appear differently in the sub-millimeter regime.",
"However, the currently available ALMA observations would not be able to resolve the various structures shown in this paper if they have the same or smaller spatial extent.",
"The second scenario involves illumination variations because of azimuthally asymmetric optical depth variations through an inner disk closer to the star.",
"Even if much less plausible, this scenario has already been raised in previous studies concerning HD169142 , .",
"The observed brightness variations at 180 mas are relatively small [ suggest an azimuthal brightness variation of 25$\\%$ in the PDI data] and could be caused by such variations.",
"Besides, the inner disk at $\\sim 0.3$ au is known to present a variable spectral energy distribution (SED) in the NIR.",
"propose several scenarios to explain the variations of the SED of HD169142, but they invoke stable shadowing effect, otherwise an anti-correlated variability in the emission of the inner and outer disks should be observed in the SED, which is not the case.",
"If additional material exists within our inner working angle, at high altitude, it could shadow the ring at 180 mas, but this remains to be investigated.",
"In any case, our discovery of the Keplerian movement of the structures at 180 mas strongly suggests that the origin of their intensity variation does not come from an inner structure."
],
[
"A point-like structure at 100 mas",
"In the data we also detect a point-like structure North of (PA$\\approx 4^\\circ $ ) and close to ($\\rho $ =105$\\pm $ 6 mas) the star (structure C in Fig.",
"REF ).",
"This position was determined using the ASDI-PCA algorithm .",
"This structure is persistent in both IRDIS and IFS data, and is visible with large range of PCA reductions subtracting 12 to 200 modes.",
"The separation of this structure from the star is similar to the separation of the object detected by with NaCo in $L^{\\prime }$ band, and is slightly offset, but consistent with 's detection.",
"To confirm the robustness of our detection, we first split the IRDIS and IFS data into two sub-sets using the LAM-ADI pipeline, which were analysed following the same procedure as described in Sec.",
"REF .",
"In each of the resulting images the structure was still visible at a S/N higher than the surrounding background.",
"This reduction shows a structure at PA=355$\\pm 3^{\\circ }$ and $\\rho =93\\pm 6$ mas in average over all wavelengths (S/N of 3.3 in $H$ band), which is consistent with the estimation provided with the ASDI-PCA pipeline.",
"The IRDIS data were also analysed with the PYNPOINT pipeline [1].",
"In this analysis the structure is marginally detected, as it only appears between 8 and 15 PCA modes (over 50).",
"The structure at $\\sim $ 100 mas appears in most data set as a somewhat extended structure (see Fig.",
"REF ), in particular in the $H$ and $K$ bands.",
"This was not detected previously in and analysis, where it appeared point-like at $L^{\\prime }$ band and was not detected in lower sensitivity, short wavelength observations .",
"The structure is partially visible in the PDI image, and in the simulated image of cADI reduction of PDI data (Fig.",
"REF ).",
"This means that its signal is polarized, which indicates light scattered by dust than the emission from a planet photosphere.",
"The position of this structure in the simulation of cADI reduction is $\\rho $ =82$\\pm $ 3 mas at PA=355$\\pm 2^{\\circ }$ , which is very similar to the PA estimate from the LAM-ADI pipeline.",
"The separation measured in the simulation remains within the error bars of the estimate position in the IFS $J$ band ($\\rho $ =90.5$\\pm $ 2.5 mas, PA=357.9$\\pm 3.0^{\\circ }$ ), but the separation is smaller than the estimate obtained with the ASDI-PCA algorithm.",
"This could be explained by the measured positions on the real images which are made in average over all wavelenghts, as for the LAM-ADI pipeline.",
"As seen in Sec.",
", the average brightness of a disk can be filtered out by the cADI reduction.",
"It is thus possible that this structure coming out of the simulation of cADI reduction of PDI data actually traces a yet undiscovered ring, and that this structure is a bright part of this ring.",
"Moreover, the structure lies close to the edge of the mask, so it is likely attenuated in the PDI image.",
"This may explain why this hypothetical ring is not detected in the PDI image.",
"The RDI image also shows a ring at $\\sim $ 100 mas, that is, very close to the coronagraph (Fig.",
"REF ).",
"Moreover, the ring is not retrieved in each detection, appearing sometimes in mean-scaled images, other times in median-scale images.",
"These results tend toward a marginal detection of a bright ring at a separation of $\\sim $ 100 mas, that our current observations unfortunately cannot confirm.",
"Additional PDI observations closer to the star without coronagraph would certainly bring precious information."
],
[
"Summary and Conclusion",
"We performed observations of the Herbig Ae star HD 169142 using SPHERE/VLT in the NIR domain with and without coronagraph to investigate the inner parts of the system (<300 mas).",
"We observed this star at five different epochs, leading to several new results : The ADI analysis show bright structures both in IRDIS ans IFS data.",
"These structures appear more extended in the $H$ band than in the $YJ$ and $K$ bands.",
"They are mainly located at separations of $\\sim $ 180-200 mas and $\\sim $ 100 mas.",
"The RDI reduction clearly shows a bright ring at 180-200 mas.",
"It also shows a hint of another inner ring located at $\\sim $ 100 mas.",
"However, it is very close to the edge of the coronagraph, and does not appear identical in every data treatment.",
"Thus, it cannot be confirmed.",
"To assess the origin of the structures seen in ADI reductions, we performed a cADI simulation using the image of the ring detected in PDI at 180 mas.",
"While the main component of the ring is filtered out we still observed residual structures that appear to be common to both PDI and ADI reductions.",
"We therefore conclude that these structures are actually bright parts of the disk.",
"Given that (i) the bright blobs seen in PCA analysis, in particular structures A and B, and the ring detected with RDI analysis are located at the same separation (180 mas), and (ii) these blobs and the polarized data are actually part of the same structure: the ring, we conclude that the bright blobs trace this bright ring in the disk.",
"From the previous result, and considering the stellar parameters, we demonstrate that the structure A follows a Keplerian motion along the ring.",
"Considering this movement, structure A is very likely to be the same structure as the one detected by at PA=33$^{\\circ }$ and $\\rho =180$ mas.",
"It is likely that actually detected a bright structure in this ring, and that the ring brightness was averaged following the same process as for our PCA treatment.",
"The structure B also shows a Keplerian movement and also traces the bright ring in the disk.",
"The latter thus rotates in a clockwise direction with a Keplerian velocity, with the Western side closer to us and the Eastern farther.",
"The ring at 180 mas shows an inhomogeneous brightness.",
"One explanation could involve Rossby vortices before they merge into one bigger vortex.",
"These vortices are ideal place to trigger planetary formation at an early stage.",
"If the inner ring is real, another explanation could be illumination effects from this inner ring.",
"The irregularity of this ring could produce azimuthally optical depth variations of the ring at 180 mas, but the angular velocity does not match this hypothesis.",
"The structure located at 100 mas (structure C) appears to be point-like at shorter and longer wavelengths but extended in the $H$ band, and its position is consistent with previous $L^{\\prime }$ band detections.",
"The RDI images show a possible inner ring at the same separation.",
"Thus, although marginally detected, it could also trace a yet undetected ring that is even closer to the star.",
"The PCA treatment could easily make it appear point-like, as it does for structures A and B. HD 169142 is a very interesting case to study planet formation as it is a pre-transitional disk showing a succession of bright rings, gaps and a a ring/gap alternation.",
"To confirm the inner ring, additional observations would be needed, but the resolution of actual (and even future) direct-imaging instruments would hardly allow such a discovery."
],
[
"Acknowledgements",
"R. L. thanks CNES for financial support through its post-doctoral programme.",
"This work has been carried out in part within the frame of the National Competence in Research (NCIR) \"PlanetS\", supported by the Swiss National Science Fundation (SNSF).",
"The authors thank the ESO Paranal Staff for support for conducting the observations.",
"We also warmly thank H. Méheut, for the useful discussions about vortices.",
"We acknowledge financial support from the Programme National de Planétologie (PNP) and the Programme National de Physique Stellaire (PNPS) of CNRS-INSU.",
"This work has also been supported by a grant from the French Labex OSUG@2020 (Investissements d'avenir - ANR10 LABX56).",
"The project is supported by CNRS, by the Agence Nationale de la Recherche (ANR-14-CE33-0018).",
"MB acknowledges funding from ANR of France under contract number ANR-16-CE31-0013 (Planet Forming Disks).",
"This work has made use of the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/LAM/CeSAM (Marseille), OCA/Lagrange (Nice) and Observatoire de Paris/LESIA (Paris).",
"We thank P. Delorme and E. Lagadec (SPHERE Data Centre) for their efficient help during the data reduction process.",
"SPHERE is an instrument designed and built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF-Osservatorio di Padova (Italy), Observatoire astronomique de l'Université de Genève (Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and ASTRON (Netherlands) in collaboration with ESO.",
"SPHERE was funded by ESO, with additional contributions from CNRS (France), MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands).",
"SPHERE also received funding from the European Commission Sixth and Seventh Framework Programmes as part of the Optical Infrared Coordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), grant number 226604 for FP7 (2009-2012) and grant number 312430 for FP7 (2013-2016).",
"We acknowledge the use of the electronic database from CDS, Strasbourg and electronic bibliography maintained by the NASA/ADS system.",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement."
]
] | 1709.01734 |
[
[
"2D Weyl Fermi gas model of Superconductivity in the Surface state of a\n Topological Insulator at High Magnetic fields"
],
[
"Abstract The Nambu-Gorkov Green's function approach is applied to strongly type-II superconductivity in a 2D spin-momentum locked (Weyl) Fermi gas model at high perpendicular magnetic fields.",
"When the chemical potential is sufficiently close to the branching (Dirac) point, such that the cyclotron effective mass, $m^{\\ast }$, is a very small fraction of the free electron mass, $m_{e}$, relatively large portion of the $H-T$ phase diagram is exposed to magneto-quantum oscillation effects.",
"This model system is realized in the 2D superconducting state, observed recently on the surface of the topological insulator Sb$_{2}$Te$_{3} $, for which high field measurements were reported at low carrier densities with $m^{\\ast}=0.065m_{e}$.",
"Calculations of the pairing condensation energy in such a system, as a function of $H$ and $T$, using both the Weyl model and a reference standard model, that exploits a simple quadratic dispersion law, are found to yield indistinguishable results in comparison with the experimental data.",
"Significant deviations from the predictions of the standard model are found only for very small carrier densities, when the cyclotron energy becomes very large, the Landau level filling factors are smaller than unity, and the Fermi energy shrinks below the cutoff energy."
],
[
"Introduction",
"The recent discoveries of surface and interface superconductivity with exceptionally high superconducting (SC) transition temperatures in several material structures [1],[2], [3] have drawn much attention to the phenomenon of strong type-II superconductivity in two-dimensional (2D) electron systems, in which the application of high magnetic fields can lead to exotic phenomena both in the normal and SC states[4].",
"Of special interest is the unique situation of the 2D superconductivity realized in surface states of topological insulators, e.g.",
"Sb$_{2}$ Te$_{3}$ [5], where the chemical potential $E_{F}$ is close to a Dirac point [6] (with Fermi velocity $v$ ) and the cyclotron effective mass, $m^{\\ast }=E_{F}/v^{2}$[7] is a small fraction (e.g.",
"0.065 in Sb$_{2}$ Te$_{3}$ , see also [8]) of the free electron mass $m_{e}$ , resulting in a dramatic enhancement of the cyclotron frequency, $\\omega _{c}=eH/m^{\\ast }c$ , and the corresponding Landau level (LL) energy spacing.",
"In a recent paper [9] we have exploited a standard electron gas model, with a quadratic energy-momentum dispersion and an effective band mass $m^{\\ast }={0.065}m_{e}$ , in a systematic investigation of the quasi-particle states and the SC pair-potential in the vortex lattice state of this system under high perpendicular magnetic fields, by solving self-consistently the corresponding Bogoliubov de Gennes (BdG) equations.",
"The results account reasonably well for the 2D SC state observed on the surface of Sb$_{2}$ Te$_{3}$ under magnetic fields of up to 3 T [5], revealing a strong type-II superconductivity at unusually low carrier density and small cyclotron effective mass, which can be realized only in the strong coupling ($\\lambda \\sim 1$ ) superconductor limit.",
"This unique situation is due to the proximity of the Fermi energy to a Dirac point, which implies that other materials in the emerging field of surface superconductivity, with metallic surface states and Dirac dispersion law around the Fermi energy, can show similar features.",
"It should be noted, however, that the use of the standard LL spectrum, arising from a parabolic band-structure, in the self-consistent BdG theory, presented in Ref.",
"[9], has been done heuristically, without actual derivation from the effective 2D Weyl Hamiltonian describing the helical surface states observed in these topological insulators [10], [5].",
"Such a derivation is particularly necessary for the spin-momentum locked model under study here, since SC pairing involves certain spin-orbital correlations.",
"Our purpose in the present paper is, therefore, two-fold: First, to develop the formal framework for solving the self consistency equation for the SC order parameter in the 2D Weyl model Hamiltonian under a strong perpendicular magnetic field, and then exploit the developed formalism in a study of the transition to superconductivity in comparison with the well known results of the standard model[4].",
"The SC transition in helical surface states of topological insulators, such as those reported, e.g.",
"in Ref.",
"[5], is then comparatively studied with respect to both models.",
"It is found that, similar to the well known solution of the linearized self consistency equation, derived by Helfand and Werthamer for the standard model [11], [9] the desired solution of the corresponding integral equation in the 2D Weyl model is greatly facilitated by initially finding analytical solutions of the eigenvalue equation for the SC order parameter.",
"Furthermore, the calculated $H-T$ phase diagram for the Weyl model in the semiclassical limit (i.e.",
"for LL filling factors $n_{F}>1$ ) can be directly mapped onto that found for the standard model, having the same Fermi surface parameters $E_{F}$ and $v$ , and a cyclotron effective mass equal to $m^{\\ast }=E_{F}/2v^{2}$ .",
"Significant deviations from the predicted mapping are found only for very small carrier densities, when the cyclotron energy becomes very large, the LL filling factors are smaller than unity, and the Fermi energy shrinks below the cutoff energy."
],
[
"The 2D spin-momentum locked (Weyl) Fermion gas model",
"To describe the underlying normal surface state electron, with charge $-e$ , in a topological insulator under a magnetic field $\\mathbf {H=}\\left(0,0,H\\right) $ (vector potential in the Landau gauge, $\\mathbf {A=}\\left(-Hy,0,0\\right) $ ), we exploit the Weyl Hamiltonian: [10] $\\widehat{h}\\left( \\mathbf {r}\\right) =\\hbar v\\left( \\widehat{\\sigma }_{x}\\widetilde{p}_{x}+\\widehat{\\sigma }_{y}\\widetilde{p}_{y}\\right) -E_{F}\\widehat{\\sigma }_{0} $ with the Pauli matrices: $\\ \\widehat{\\sigma }_{x}=\\left(\\begin{array}{cc}0 & 1 \\\\1 & 0\\end{array}\\right) ;\\widehat{\\sigma }_{y}=\\left(\\begin{array}{cc}0 & -i \\\\i & 0\\end{array}\\right) ;\\widehat{\\sigma }_{0}=\\left(\\begin{array}{cc}1 & 0 \\\\0 & 1\\end{array}\\right) $ , and the gauge invariant momentum $\\widetilde{\\mathbf {p}}\\mathbf {\\equiv }\\left( -i \\mathbf {\\nabla +}\\left(e/\\hbar c\\right)\\mathbf {A}\\right) $ , such that: $\\widehat{h}\\left( \\mathbf {r}\\right) =\\left(\\begin{array}{cc}0 & -\\hbar v\\frac{\\partial }{\\partial y}-\\hbar v\\left( i\\frac{\\partial }{\\partial x}+\\frac{y}{a_{H}^{2}}\\right) \\\\\\hbar v\\frac{\\partial }{\\partial y}-\\hbar v\\left( i\\frac{\\partial }{\\partial x}+\\frac{y}{a_{H}^{2}}\\right) & 0\\end{array}\\right) .$ In these equations $v$ is the Fermi velocity and $a_{H}\\equiv \\sqrt{\\frac{\\hbar c}{eH}}$ is the magnetic length.",
"Note that Zeeman spin-splitting is neglected with respect to the cyclotron energy in Eq.REF due to the very small cyclotron effective mass considered here.",
"The corresponding Weyl equation for the spinor $\\left(\\begin{array}{c}\\psi _{\\uparrow }\\left( \\mathbf {r}\\right) \\\\\\psi _{\\downarrow }\\left( \\mathbf {r}\\right)\\end{array}\\right) $ takes the form: $\\ \\left(\\begin{array}{cc}-E_{F} & -\\hbar v\\frac{\\partial }{\\partial y}-\\hbar v\\left( i\\frac{\\partial }{\\partial x}+\\frac{y}{a_{H}^{2}}\\right) \\\\\\hbar v\\frac{\\partial }{\\partial y}-\\hbar v\\left( i\\frac{\\partial }{\\partial x}+\\frac{y}{a_{H}^{2}}\\right) & -E_{F}\\end{array}\\right) \\left(\\begin{array}{c}\\psi _{\\uparrow }\\left( \\mathbf {r}\\right) \\\\\\psi _{\\downarrow }\\left( \\mathbf {r}\\right)\\end{array}\\right) =E\\left(\\begin{array}{c}\\psi _{\\uparrow }\\left( \\mathbf {r}\\right) \\\\\\psi _{\\downarrow }\\left( \\mathbf {r}\\right)\\end{array}\\right) $ Expressing all length variables in units of the magnetic length $a_{H}$ , and introducing the dimensionless energy variable $\\mu \\equiv \\frac{E_{F}a_{H}}{\\sqrt{2}\\hbar v}$ , where all other energy symbols refer in what follows to quantities measured in units of the cyclotron energy $\\hbar \\omega _{c}\\equiv \\frac{\\sqrt{2}\\hbar v}{a_{H}}$ , we write the corresponding mean-field Hamiltonian for singlet pairing in Nambu representation: $\\widehat{H}=\\left(\\begin{array}{cc}\\frac{1}{\\sqrt{2}}\\widehat{\\mathbf {\\sigma }}\\cdot \\widetilde{\\mathbf {p}}-\\mu & i\\widehat{\\sigma }_{y}\\Delta \\left( \\mathbf {r}\\right) \\\\-i\\widehat{\\sigma }_{y}\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) & -\\frac{1}{\\sqrt{2}} \\widehat{\\mathbf {\\sigma }}^{\\ast }\\cdot \\widetilde{\\mathbf {p}}^{\\ast }+\\mu \\end{array}\\right) $ where the spin-singlet order parameter is defined by: $\\Delta ^{\\ast }\\left(\\mathbf {r}\\right) \\equiv -\\left(\\left|V\\right|/\\hbar \\omega _{c}\\right) \\left\\langle \\psi _{\\downarrow }^{\\dagger }\\left( \\mathbf {r}\\right)\\psi _{\\uparrow }^{\\dagger }\\left( \\mathbf {r}\\right) \\right\\rangle \\equiv \\Delta _{\\uparrow \\downarrow }^{\\ast }\\left( \\mathbf {r}\\right) =-\\Delta _{\\downarrow \\uparrow }^{\\ast }\\left( \\mathbf {r}\\right) $ [12] and $\\widehat{\\mathbf {\\sigma }}\\equiv \\left(\\sigma _{x},\\sigma _{y}\\right)$ .",
"The corresponding Nambu field operators: $\\widehat{\\Psi }\\left( \\mathbf {r};t\\right) \\equiv \\left[\\begin{array}{c}\\psi _{\\uparrow }\\left( \\mathbf {r};t\\right) \\\\\\psi _{\\downarrow }\\left( \\mathbf {r};t\\right) \\\\\\psi _{\\uparrow }^{\\dagger }\\left( \\mathbf {r};t\\right) \\\\\\psi _{\\downarrow }^{\\dagger }\\left( \\mathbf {r};t\\right)\\end{array}\\right] ,\\widehat{\\Psi }^{\\dagger }\\left( \\mathbf {r};t\\right) \\equiv \\left[\\begin{array}{cccc}\\psi _{\\uparrow }^{\\dagger }\\left( \\mathbf {r};t\\right) & \\psi _{\\downarrow }^{\\dagger }\\left( \\mathbf {r};t\\right) & \\psi _{\\uparrow }\\left( \\mathbf {r};t\\right) & \\psi _{\\downarrow }\\left( \\mathbf {r};t\\right)\\end{array}\\right]$ satisfy the equation of motion $i\\partial _{t}\\widehat{\\Psi }\\left( \\mathbf {r};t\\right) =\\widehat{H}\\widehat{\\Psi }\\left( \\mathbf {r};t\\right) $ , resulting in the corresponding equations for the Nambu-Gorkov time-ordered Green's functions $4\\times 4$ matrix, $\\widehat{G}\\left( \\mathbf {r,r}^{\\prime };t-t^{\\prime }\\right) \\equiv -i\\left\\langle T\\widehat{\\Psi }\\left( \\mathbf {r};t\\right) \\widehat{\\Psi }^{\\dagger }\\left( \\mathbf {r}^{\\prime };t^{\\prime }\\right) \\right\\rangle $ : $\\left[ i\\partial _{t}-\\left(\\begin{array}{cc}\\frac{1}{\\sqrt{2}}\\widehat{\\mathbf {\\sigma }}\\cdot \\widetilde{\\mathbf {p}}-\\mu & i\\widehat{\\sigma }_{y}\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) \\\\-i\\widehat{\\sigma }_{y}\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) & -\\frac{1}{\\sqrt{2}}\\widehat{\\mathbf {\\sigma }}^{\\ast }\\cdot \\widetilde{\\mathbf {p}}^{\\ast }+\\mu \\end{array}\\right) \\right] \\left(\\begin{array}{cc}\\widehat{G}_{11}\\left( \\mathbf {r,r}^{\\prime };t-t^{\\prime }\\right) &\\widehat{G}_{12}\\left( \\mathbf {r,r}^{\\prime };t-t^{\\prime }\\right) \\\\\\widehat{G}_{21}\\left( \\mathbf {r,r}^{\\prime };t-t^{\\prime }\\right) &\\widehat{G}_{22}\\left( \\mathbf {r,r}^{\\prime };t-t^{\\prime }\\right)\\end{array}\\right) =\\delta \\left( t-t^{\\prime }\\right) \\delta \\left( \\mathbf {r}-\\mathbf {r}^{\\prime }\\right) $ Time-Fourier transforming with frequency $\\omega $ and rewriting Eq.REF in its integral form, the relevant parts of these equations for our purpose here is written in the form: $\\widehat{G}_{11}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) &=&\\widehat{G}_{11}^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) +\\int d\\mathbf {r}^{\\prime \\prime }\\widehat{G}_{11}^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime \\prime };\\omega \\right) i\\widehat{\\sigma }_{y}\\Delta \\left(\\mathbf {r}^{\\prime \\prime }\\right) \\widehat{G}_{21}\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right) , \\\\\\widehat{G}_{21}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) &=&\\int d\\mathbf {r}^{\\prime \\prime }\\widehat{G}_{11}^{\\left( 0\\right) T}\\left(\\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) i\\widehat{\\sigma }_{y}\\Delta ^{\\ast }\\left( \\mathbf {r}^{\\prime \\prime }\\right) \\widehat{G}_{11}\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right)$ where the upper-left block of the Normal state $2\\times 2$ Green's function matrix: $\\widehat{G}_{11}^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) \\equiv \\left(\\begin{array}{cc}G_{\\uparrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) & G_{\\uparrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) \\\\G_{\\downarrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) & G_{\\downarrow \\downarrow }^{\\left( 0\\right) }\\left(\\mathbf {r,r}^{\\prime };\\omega \\right)\\end{array}\\right) $ , satisfies the equation: $\\left( \\omega +\\mu -\\frac{1}{\\sqrt{2}}\\widehat{\\mathbf {\\sigma }}\\cdot \\widetilde{\\mathbf {p}}\\right) \\widehat{G}_{11}^{\\left( 0\\right) }\\left(\\mathbf {r,r}^{\\prime };\\omega \\right) =\\delta \\left( \\mathbf {r}-\\mathbf {r}^{\\prime }\\right) $ and its transpose with frequency $-\\omega $ , $\\widehat{G}_{11}^{\\left(0\\right) T}\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};-\\omega \\right) $ , satisfies the dual equation: $\\left( \\omega -\\mu +\\frac{1}{\\sqrt{2}}\\mathbf {\\sigma }^{\\ast }\\cdot \\widetilde{\\mathbf {p}}^{\\ast }\\right) \\widehat{G}_{11}^{\\left( 0\\right)T}\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};-\\omega \\right) =-\\delta \\left(\\mathbf {r}-\\mathbf {r}^{\\prime }\\right) $ Expanding the above normal state Green's functions in terms of the complete set of solutions, $\\varphi _{n}\\left( y-k_{x}\\right) =\\frac{1}{\\pi ^{1/4}\\sqrt{2^{n}n!",
"}}e^{-\\frac{1}{2}\\left( y-k_{x}\\right) ^{2}}H_{n}\\left(y-k_{x}\\right) $ , of the eigenstate equation:$\\frac{1}{2}\\left[ -\\partial _{y}^{2}+\\left( y-k_{x}\\right) ^{2}-1\\right] \\varphi _{n}\\left(y-k_{x}\\right) =n\\varphi _{n}\\left( y-k_{x}\\right) $ , where $H_{n}\\left(y\\right) $ is Hermite polynomial of order $n=0,1,2,...,$ we find: $G_{\\uparrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) = \\frac{1}{L_{x}}\\sum \\limits _{k_{x}}e^{ik_{x}\\left(x-x^{\\prime }\\right) }\\sum \\limits _{n=0}^{\\infty }\\frac{\\left( \\omega +\\mu \\right) \\varphi _{n}\\left( y-k_{x}\\right) \\varphi _{n}\\left( y^{\\prime }-k_{x}\\right) }{\\left( \\omega +\\mu \\right) ^{2}-\\left( n+1\\right) }$ $G_{\\uparrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) = \\frac{1}{L_{x}}\\sum \\limits _{k_{x}}e^{ik_{x}\\left(x-x^{\\prime }\\right) }\\sum \\limits _{n=0}^{\\infty }\\frac{\\left( -\\sqrt{n}\\right) \\varphi _{n-1}\\left( y-k_{x}\\right) \\varphi _{n}\\left( y^{\\prime }-k_{x}\\right) }{\\left( \\omega +\\mu \\right) ^{2}-n} $ $G_{\\downarrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) = \\frac{1}{L_{x}}\\sum \\limits _{k_{x}}e^{ik_{x}\\left(x-x^{\\prime }\\right) }\\sum \\limits _{n=0}^{\\infty }\\frac{\\left( -\\sqrt{n+1}\\right) \\varphi _{n+1}\\left( y-k_{x}\\right) \\varphi _{n}\\left( y^{\\prime }-k_{x}\\right) }{\\left( \\omega +\\mu \\right) ^{2}-\\left( n+1\\right) }$ $G_{\\downarrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) = \\frac{1}{L_{x}}\\sum \\limits _{k_{x}}e^{ik_{x}\\left(x-x^{\\prime }\\right) }\\sum \\limits _{n=0}^{\\infty }\\frac{\\left( \\omega +\\mu \\right) \\varphi _{n}\\left( y-k_{x}\\right) \\varphi _{n}\\left( y^{\\prime }-k_{x}\\right) }{\\left( \\omega +\\mu \\right) ^{2}-n} $ where $L_{x}$ is the surface size along the x-axis (measured in units of $a_{H}$ ).",
"Leading order expansion of the integral equations Eq.",
"in the order parameter $\\Delta \\left( \\mathbf {r}\\right) $ yields: $\\widehat{G}_{21}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) \\equiv \\left(\\begin{array}{cc}F_{\\uparrow \\uparrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) &F_{\\uparrow \\downarrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) \\\\F_{\\downarrow \\uparrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) &F_{\\downarrow \\downarrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right)\\end{array}\\right) =\\int d\\mathbf {r}^{\\prime \\prime }\\widehat{G}_{11}^{\\left( 0\\right)T}\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) i\\sigma _{y}\\Delta ^{\\ast }\\left( \\mathbf {r}^{\\prime \\prime }\\right) \\widehat{G}_{11}^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right)$ so that for the anomalous Green's functions $F_{\\downarrow \\uparrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) $ and $F_{\\uparrow \\downarrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right) $ we find, respectively: $F_{\\downarrow \\uparrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right)=\\int d\\mathbf {r}^{\\prime \\prime }\\Delta ^{\\ast }\\left( \\mathbf {r}^{\\prime \\prime }\\right) \\left[ G_{\\uparrow \\downarrow }^{\\left( 0\\right) }\\left(\\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) G_{\\downarrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right) -G_{\\downarrow \\downarrow }^{\\left( 0\\right)}\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) G_{\\uparrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right) \\right] $ $F_{\\uparrow \\downarrow }^{+}\\left( \\mathbf {r,r}^{\\prime };\\omega \\right)=\\int d\\mathbf {r}^{\\prime \\prime }\\Delta ^{\\ast }\\left( \\mathbf {r}^{\\prime \\prime }\\right) \\left[ G_{\\uparrow \\uparrow }^{\\left( 0\\right) }\\left(\\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) G_{\\downarrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right) -G_{\\downarrow \\uparrow }^{\\left( 0\\right) }\\left(\\mathbf {r}^{\\prime \\prime }\\mathbf {,r};-\\omega \\right) G_{\\uparrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime \\prime }\\mathbf {,r}^{\\prime };\\omega \\right) \\right] $ The self-consistency condition for the singlet SC order parameter, in the imaginary (Matsubara) frequency representation, $\\omega _{\\nu }=\\left( 2\\nu +1\\right) \\pi \\tau \\left(\\tau \\equiv k_{B}T/\\hbar \\omega _{c}\\right),\\nu =0,\\pm 1,\\pm 2,...$ : $\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) =-\\left( \\left|V\\right|/\\hbar \\omega _{c}\\right) \\left\\langle \\widehat{\\psi }_{\\downarrow }^{\\dagger }\\left( \\mathbf {r};\\tau \\right) \\widehat{\\psi }_{\\uparrow }^{\\dagger }\\left( \\mathbf {r};\\tau \\right) \\right\\rangle = \\left( \\left|V\\right|/\\hbar \\omega _{c} \\right) \\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }F_{\\downarrow \\uparrow }^{+}\\left( \\mathbf {r,r};\\omega _{\\nu }\\right) ,\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) \\equiv \\Delta _{\\uparrow \\downarrow }^{\\ast }\\left( \\mathbf {r}\\right) =-\\Delta _{\\downarrow \\uparrow }^{\\ast }\\left( \\mathbf {r}\\right) $ where $V$ is the effective electron-electron interaction potential responsible for the pairing instability.",
"Note that, unlike the dimensionless quantity $\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) $ , both $V$ and $k_{B}T$ are expressed here in their absolute energy dimensions.",
"Thus, to leading order in $\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) $ Eq.REF takes the form: $\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) =\\left( \\left|V\\right|/\\hbar \\omega _{c}\\right) \\int d\\mathbf {r}^{\\prime }\\Delta ^{\\ast }\\left(\\mathbf {r}^{\\prime }\\right) Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r}\\right)$ where the kernel $Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r}\\right) $ is given by: $Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r}\\right) &=&\\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) , \\\\Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right)&=&Q_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) -Q_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right)$ with: $Q_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) &=&G_{\\downarrow \\uparrow }^{\\left(0\\right) }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};-\\omega _{\\nu }\\right)G_{\\uparrow \\downarrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) , \\\\Q_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) &=&G_{\\downarrow \\downarrow }^{\\left(0\\right) }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};-\\omega _{\\nu }\\right)G_{\\uparrow \\uparrow }^{\\left( 0\\right) }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right)$ Exploiting Eqs.REF -REF for the normal state Green's functions we derive the following explicit expressions for the kernels: $Q_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) &=&\\left( \\frac{1}{2\\pi }\\right) ^{2}\\sum \\limits _{n,n^{\\prime }=0}^{\\infty }\\frac{\\left( -i\\omega _{\\nu }+\\mu \\right)\\left( i\\omega _{\\nu }+\\mu \\right) }{\\left[ \\left( -i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] \\left[ \\left( i\\omega _{\\nu }+\\mu \\right) ^{2}-\\left(n^{\\prime }+1\\right) \\right] }\\times \\\\&&e^{i\\left( x^{\\prime }-x\\right) \\left( y+y^{\\prime }\\right) }e^{-\\frac{1}{2}\\rho ^{2}}L_{n^{\\prime }}\\left( \\frac{1}{2}\\rho ^{2}\\right) L_{n}\\left(\\frac{1}{2}\\rho ^{2}\\right) $ $Q_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right) &=&-\\frac{1}{2}\\left( \\frac{1}{2\\pi }\\right) ^{2}\\sum \\limits _{n,n^{\\prime }=1}^{\\infty }\\frac{1}{\\left[ \\left(-i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] }\\frac{1}{\\left[ \\left( i\\omega _{\\nu }+\\mu \\right) ^{2}-n^{\\prime }\\right] }\\times \\\\&&e^{i\\left( x^{\\prime }-x\\right) \\left( y+y^{\\prime }\\right) }e^{-\\frac{1}{2}\\rho ^{2}}\\rho ^{2}L_{n}^{\\prime }\\left( \\frac{1}{2}\\rho ^{2}\\right)L_{n^{\\prime }}^{\\prime }\\left( \\frac{1}{2}\\rho ^{2}\\right), $ where $\\rho = |\\mathbf {r}-\\mathbf {r^{\\prime }}|$ .",
"Similar to the situation in the standard, single band (with quadratic energy dispersion) 2D electron system, the order parameter of the Landau orbital form $\\Delta ^{\\ast }\\left( \\mathbf {r}\\right) \\propto e^{-2iq_{x}x}\\varphi _{m}\\left[ \\sqrt{2}\\left( y-q_{x}\\right) \\right] ,m=0,1,2,...$ , is an eigenfunction of the integral operator $\\int d\\mathbf {r}^{\\prime }Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r}\\right) ...,$ that is: $\\int d\\mathbf {r}^{\\prime }Q\\left( \\mathbf {r}^{\\prime }\\mathbf {,r}\\right)\\Delta ^{\\ast }\\left( \\mathbf {r}^{\\prime }\\right) =A\\Delta ^{\\ast }\\left(\\mathbf {r}\\right) $ This important result is obtained by showing that the above Landau orbital is also an eigenfunction of the integral operators $k_{B}T\\sum \\limits _{\\nu =-\\infty }^{\\infty }\\int d\\mathbf {r}^{\\prime }Q_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right)... , k_{B}T\\sum \\limits _{\\nu =-\\infty }^{\\infty }\\int d\\mathbf {r}^{\\prime }Q_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }\\left( \\mathbf {r}^{\\prime }\\mathbf {,r};\\omega _{\\nu }\\right)... $ , so that, by exploiting Eqs.REF ,REF , the eigenvalue $A$ in Eq.REF can be written in terms of the respective eigenvalues, $A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow },A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }$ , as: $A=A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }-A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow } $ where: $A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow } &=&-\\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }\\sum \\limits _{n,n^{\\prime }=1}^{\\infty }\\frac{1}{\\left[\\left( -i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] }\\frac{1}{\\left[ \\left(i\\omega _{\\nu }+\\mu \\right) ^{2}-n^{\\prime }\\right] }\\times \\\\&&\\frac{1}{2}\\left( \\frac{1}{2\\pi }\\right) \\int _{0}^{\\infty }\\rho ^{3}d\\rho e^{-\\rho ^{2}}L_{n^{\\prime }}^{\\prime }\\left( \\frac{1}{2}\\rho ^{2}\\right)L_{n}^{\\prime }\\left( \\frac{1}{2}\\rho ^{2}\\right) $ $A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow } &=&\\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }\\sum \\limits _{n,n^{\\prime }=0}^{\\infty }\\frac{\\left(-i\\omega _{\\nu }+\\mu \\right) \\left( i\\omega _{\\nu }+\\mu \\right) }{\\left[\\left( -i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] \\left[ \\left( i\\omega _{\\nu }+\\mu \\right) ^{2}-\\left( n^{\\prime }+1\\right) \\right] }\\times \\\\&&\\left( \\frac{1}{2\\pi }\\right) \\int _{0}^{\\infty }\\rho d\\rho e^{-\\rho ^{2}}L_{n^{\\prime }}\\left( \\frac{1}{2}\\rho ^{2}\\right) L_{n}\\left( \\frac{1}{2}\\rho ^{2}\\right) $ and $\\ \\tau \\equiv \\frac{k_{B}T}{\\hbar \\omega _{c}}$ .",
"Performing the integration over $\\rho $ in both Eqs.",
"REF and REF our results for the eigenvalues $A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow },A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }$ take the forms: $A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }\\equiv A_{0}=-\\left( \\frac{1}{2\\pi }\\right) \\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }\\sum \\limits _{n,n^{\\prime }=1}^{\\infty }\\frac{\\left( n+n^{\\prime }\\right) !",
"}{2^{n+n^{\\prime }}n^{\\prime }!n!",
"}\\frac{\\left( \\frac{n^{\\prime }n}{n+n^{\\prime }}\\right) }{\\left[ \\left( -i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] \\left[\\left( i\\omega _{\\nu }+\\mu \\right) ^{2}-n^{\\prime }\\right] } $ $A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow } &=&A_{1}+\\Delta A_{1}, \\\\A_{1} &=&\\left( \\frac{1}{4\\pi }\\right) \\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }\\sum \\limits _{n,n^{\\prime }=1}^{\\infty }\\frac{\\left( n+n^{\\prime }\\right) !",
"}{2^{n+n^{\\prime }}n!n^{\\prime }!",
"}\\frac{\\left( -i\\omega _{\\nu }+\\mu \\right) \\left( i\\omega _{\\nu }+\\mu \\right) }{\\left[ \\left( -i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] \\left[ \\left( i\\omega _{\\nu }+\\mu \\right)^{2}-n^{\\prime }\\right] }, \\\\\\Delta A_{1} &=&\\left( \\frac{1}{4\\pi }\\right) \\tau \\sum \\limits _{\\nu =-\\infty }^{\\infty }\\sum \\limits _{n=1}^{\\infty }\\frac{1}{2^{n}}\\left\\lbrace \\frac{\\left( i\\omega _{\\nu }+\\mu \\right) }{\\left( -i\\omega _{\\nu }+\\mu \\right) \\left[ \\left( i\\omega _{\\nu }+\\mu \\right) ^{2}-n\\right] }+CC\\right\\rbrace $ Note that the pairing correlations of electrons, preserving their spin-up projection, with electrons preserving their spin-down projection, as expressed by $A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }$ in Eq.REF , include contributions of correlations of the zero LL with nonzero LLs, as reflected by $\\Delta A_{1}$ in Eq.. On the other hand, the pairing correlations of electrons, flipping their spin-up to spin-down projections, due to spin orbit coupling, with electrons flipping their spin-down to spin-up projections, as expressed by $A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }$ in Eq.REF , do not include any contributions involving zero LL states.",
"Combining the self-consistency integral equation, Eq.REF , with the eigenvalue equation (for $A=A_{\\downarrow \\uparrow }^{\\uparrow \\downarrow }-A_{\\downarrow \\downarrow }^{\\uparrow \\uparrow }$ ), Eq.REF , the former reduces to the simple algebraic equation, $1=\\left|V\\right|A $ .",
"Performing the summation over the Matsubara frequencies $\\omega _{\\nu }$ we arrive at the following expression for $\\left|V\\right|A$ : $\\left|V\\right|A &=&\\frac{1}{32}\\frac{\\lambda }{\\sqrt{n_{F}}}\\sum \\limits _{i,j=1}^{2}\\sum \\limits _{n=N_{l}^{\\left( i\\right) }}^{N_{u}^{\\left(i\\right) }}\\sum \\limits _{m=N_{l}^{\\left( j\\right) }}^{N_{u}^{\\left( j\\right)}}\\frac{\\left( m+n\\right) !",
"}{2^{m+n}n!m!",
"}I_{nm}^{\\left( ij\\right) }, \\\\I_{nm}^{\\left( ij\\right) } &=&\\frac{\\left[ \\left( -1\\right) ^{j}\\sqrt{m}+\\left( -1\\right) ^{i}\\sqrt{n}\\right] ^{2}}{n+m}\\frac{\\tanh \\left( \\frac{\\mu +\\left( -1\\right) ^{j}\\sqrt{m}}{2\\tau }\\right) +\\tanh \\left( \\frac{\\mu +\\left( -1\\right) ^{i}\\sqrt{n}}{2\\tau }\\right) }{2\\mu +\\left( -1\\right) ^{j}\\sqrt{m}+\\left( -1\\right) ^{i}\\sqrt{n}},\\ \\ I_{00}^{\\left( ij\\right) }=0$ where: $\\lambda =\\frac{\\sqrt{n_{F}}\\left|V\\right|}{\\pi a_{H}^{2}\\hbar \\omega _{c}}=\\left|V\\right|N\\left( E_{F}\\right) =\\left|V\\right|\\left( \\frac{m^{\\ast }}{2\\pi \\hbar ^{2}}\\right) ,n_{F}\\equiv \\mu ^{2}$ with $N\\left( E_{F}\\right) =\\frac{E_{F}}{2\\pi \\left( \\hbar v\\right) ^{2}}$ being the single electron density of states per spin projection per unit area and $m^{\\ast }=\\frac{E_{F}}{v^{2}}$ the effective cyclotron mass at the Fermi energy.",
"The different cutoff LL indices $N_{u}^{i}$ ($N_{l}^{i}$ ), indicated in Eqs.REF , refer to the different branches, i.e.",
"the conduction (positive), or valence (negative) energy bands of the Weyl model contributing to the pairing correlation.",
"The different values arise due to the fact that the cutoff is introduced to the electron energy, by the mediating electron-phonon interaction, relative to the Fermi energy, rather than to the branching point (zero) energy of the Weyl bands structure.",
"Thus, assuming a (Debye) cutoff energy $\\hbar \\omega _{D}$ , we should distinguish between two different situations.",
"In the usual situation where $\\hbar \\omega _{D}<E_{F}$ , pairing takes place only in a single band, so that, e.g.",
"for a positive chemical potential, we find: $N_{u}^{\\left( 1\\right) }=\\left[n_{F}\\left( 1+\\gamma \\right) ^{2}\\right] $ , $N_{l}^{\\left( 1\\right) }=\\left[n_{F}\\left( 1-\\gamma \\right) ^{2}\\right] $ , $\\ N_{u}^{\\left( 2\\right)}=N_{l}^{\\left( 2\\right) }=0$ , where $\\gamma =\\hbar \\omega _{D}/E_{F}$ .",
"In the unusual situation where the cutoff energy, $\\hbar \\omega _{D}>E_{F}$ , both inter and intra band pairing take place, so that the cutoff LL indices are different for energies in the valence (V) and conduction (C) bands.",
"Thus, for CB pairing (corresponding to the energy denominator $2\\mu -\\sqrt{m}-\\sqrt{n}$ in Eq.REF ), we have: $N_{u}^{\\left( 1\\right) }=\\left[n_{F}\\left( 1+\\gamma \\right) ^{2}\\right] ,N_{l}^{\\left( 1\\right) }=0$ .",
"For the interband pairing (energy denominators $2\\mu -\\sqrt{m}+\\sqrt{n}$ , or $2\\mu +\\sqrt{m}-\\sqrt{n}$ in Eq.REF ) the cutoff indices are: $N_{u}^{\\left( 1\\right) }=\\left[ n_{F}\\left( 1+\\gamma \\right) ^{2}\\right],N_{l}^{\\left( 1\\right) }=0$ , or: $N_{u}^{\\left( 2\\right) }=\\left[n_{F}\\left( \\gamma -1\\right) ^{2}\\right] ,N_{l}^{\\left( 2\\right) }=0$ , respectively.",
"Figure: Schematic illustration of the Weyl model bands structure for apositive chemical potential, smaller than the cutoff energy, showing a pairof Landau levels in both subbands at the cutoff energy measured from theFermi energy."
],
[
"Comparison with the standard (nonrelativistic) electron gas model:\nThe semiclassical approximation",
"A useful reference model, for comparison with the 2D Weyl model developed above, starts with a nonrelativistic electron gas, characterized by a quadratic single-electron energy-momentum dispersion, $E=\\frac{\\hbar ^{2}k^{2}}{2m_{S}^{\\ast }}$ , with band effective mass, $m_{S}^{\\ast }$ , set equal to $\\frac{1}{2}m_{0}^{\\ast }=\\frac{E_{F0}}{2v^{2}}$ , and $E_{F0}$ - the Fermi energy in the Weyl model at a certain doping level, to be determined in reference to a concrete experiment.",
"Under these assumptions both the Fermi energy, $E_{F0}$ , and the Fermi wave number, $k_{F0}$ , are the same in both models: $E_{F0}=\\frac{\\hbar ^{2}k_{F0}^{2}}{2m_{S}^{\\ast }}=\\hbar vk_{F0} $ And in a perpendicular magnetic field $\\mathbf {H=}\\left( 0,0,H\\right) $ the cyclotron frequency, $\\omega _{c}^{S}\\equiv \\left( \\frac{eH}{m_{S}^{\\ast }c}\\right) $ , is related to the Weyl cyclotron frequency, $\\omega _{c}^{W}\\equiv \\frac{\\sqrt{2}v}{a_{H}}$ , via: $\\omega _{c}^{W}=2\\sqrt{n_{F0}}\\left( \\frac{eH}{m_{0}^{\\ast }c}\\right) =\\sqrt{n_{F0}}\\omega _{c}^{S} $ where in both models $n_{F0}\\equiv \\frac{E_{F0}}{\\hbar \\omega _{c}^{S}}\\equiv \\left( \\frac{E_{F0}}{\\hbar \\omega _{c}^{W}}\\right) ^{2}=\\frac{\\left( a_{H}k_{F0}\\right) ^{2}}{2}$ Using the set of parameters defined above, the well known expression for the pairing energy eigenvalue obtained in the standard model takes the form: $\\left|V\\right|A_{S}=\\frac{1}{4}\\lambda _{S}\\sum \\limits _{m,n=n_{F0}\\left( 1-\\gamma _{0}\\right) }^{n_{F0}\\left( 1+\\gamma _{0}\\right) }\\frac{\\left( m+n\\right) !",
"}{2^{m+n}n!m!",
"}\\frac{\\tanh \\left( \\frac{\\mu _{0}-n-1/2}{2\\tau _{S}}\\right) +\\tanh \\left( \\frac{\\mu _{0}-m-1/2}{2\\tau _{S}}\\right) }{2\\mu _{0}-n-m-1} $ where $\\mu _{0}=n_{F0},\\tau _{S}=\\frac{k_{B}T}{\\hbar \\omega _{c}^{S}},\\lambda _{S}\\equiv \\left|V\\right|\\left( \\frac{m_{S}^{\\ast }}{2\\pi \\hbar ^{2}}\\right) $ , and $\\gamma _{0}=\\hbar \\omega _{D}/E_{F0}$ .",
"The semiclassical limit of our theory in the Weyl model is basically established at sufficiently small magnetic fields for which the LL index at the Fermi energy, $n_{F}$ , is sufficiently large compared to unity.",
"Thus, assuming that $n_{F}\\gg 1$ , we may expand the CB energy appearing in the dominant contribution to $A$ (i.e.",
"$I_{nm}^{\\left( 11\\right) }$ ) in Eq.REF around $m=n_{F}$ , or $n=n_{F}$ , e.g.",
": $\\ \\sqrt{m}\\approx \\mu +\\frac{1}{2\\sqrt{n_{F}}}\\left( m-n_{F}\\right) $ , $\\sqrt{n}\\approx \\mu +\\frac{1}{2\\sqrt{n_{F}}}\\left( n-n_{F}\\right) $ , such that to leading order: $I_{nm}^{\\left( 11\\right) }\\approx 4\\sqrt{n_{F}}\\frac{\\tanh \\left( \\frac{n_{F}-m}{2\\left( 2\\tau \\sqrt{n_{F}}\\right) }\\right) +\\tanh \\left( \\frac{n_{F}-n}{2\\left( 2\\tau \\sqrt{n_{F}}\\right) }\\right) }{2n_{F}-m-n}$ , and: $\\left|V\\right|A_{W}\\approx \\left|V\\right|A_{W}^{SC}=\\frac{1}{8}\\lambda \\sum \\limits _{m,n=N_{l}^{\\left( 1\\right)}}^{N_{u}^{\\left( 1\\right) }}\\frac{\\left( m+n\\right) !",
"}{2^{m+n}n!m!",
"}\\frac{\\tanh \\left( \\frac{n_{F}-m}{2\\left( 2\\sqrt{n_{F}}\\tau _{W}\\right) }\\right)+\\tanh \\left( \\frac{n_{F}-n}{2\\left( 2\\sqrt{n_{F}}\\tau _{W}\\right) }\\right)}{2n_{F}-m-n} $ Note that, for $E_{F}=E_{F0}$ , $A_{W}$ in Eq.REF is seen to be close to $\\frac{1}{2}A_{S}$ in Eq.REF , provided the dimensionless temperature scale $\\tau _{W}\\equiv \\frac{k_{B}T}{\\hbar \\omega _{c}^{W}}$ is rescaled by the factor $2\\sqrt{n_{F0}}$ .",
"In fact, the rescaled value, $2\\sqrt{n_{F0}}\\tau _{W}=\\frac{k_{B}T}{\\hbar eH/m_{0}^{\\ast }c}=2\\left( \\frac{k_{B}T}{\\hbar \\omega _{c}^{S}}\\right) =2\\tau _{S}$ , is consistent with Eq.REF .",
"It should be noted that the dimensionless zero-point energy, $1/2$ in Eq.REF , characterizing the standard model, does not make any difference since it can always be absorbed into the chemical potential $\\mu $ .",
"The factor of $\\frac{1}{2}$ between the expressions REF and REF is due to the spin-momentum locking, inherent to the Weyl model, and the consequent splitting of its spectrum into positive and negative energy subbands, as compared to the single band of the standard spectrum.",
"There is, however, an essential difference between the two models, and that is the cyclotron effective mass in the Weyl model is a function of the Fermi energy, whereas in the standard model it is a constant.",
"The above comparison is, therefore, drastically modified in the ultimate quantum limit, when together with the doping level, the Fermi energy tends to zero, and the prefactor $\\frac{\\lambda }{\\sqrt{n_{F}}}$ in Eq.REF nominally diverges as $n_{F}\\rightarrow 0$ .",
"The vanishing of $\\lambda $ in the Weyl model with $E_{F}$ through the cyclotron effective mass, evidently removes this divergency, yielding: $\\frac{\\lambda }{\\sqrt{n_{F}}}\\rightarrow \\frac{1}{\\sqrt{2}\\pi }\\frac{\\left|V\\right|/\\hbar v}{a_{H}},E_{F}\\rightarrow 0$ .",
"It will be, therefore, helpful to extend the reference model expressed in Eq.REF for varying values of $E_{F}$ , to account for the dependence of the parameters $n_{F}$ and $m^{\\ast }$ in the Weyl model on $E_{F}$ .",
"This can be done by replacing $n_{F0}=\\mu _{0}$ in Eq.REF with $n_{F}$ , defined in the Weyl model by: $n_{F}\\equiv \\left( \\frac{E_{F}}{\\hbar \\omega _{c}^{W}}\\right) ^{2}=\\frac{1}{2}\\left( a_{H}k_{F}\\right) ^{2} $ so that: $\\left|V\\right|A_{S}\\rightarrow \\frac{1}{4}\\lambda _{S}\\sum \\limits _{m,n=n_{F}\\left( 1-\\gamma \\right) }^{n_{F}\\left( 1+\\gamma \\right) }\\frac{\\left( m+n\\right) !",
"}{2^{m+n}n!m!",
"}\\frac{\\tanh \\left( \\frac{n_{F}-n-1/2}{2\\tau _{S}}\\right) +\\tanh \\left( \\frac{n_{F}-m-1/2}{2\\tau _{S}}\\right) }{2n_{F}-n-m-1} $ where $\\tau _{S}=\\sqrt{n_{F0}}\\tau _{W}$ and $\\gamma =\\hbar \\omega _{D}/E_{F}$ .",
"The standard coupling constant, $\\lambda _{S}$ , is defined by fixing the value of the cyclotron mass at $m_{S}^{\\ast }$ (i.e.",
"at a certain value of the Fermi energy $E_{F0}$ ): $\\lambda _{S}\\equiv \\left|V\\right|\\left( m_{S}^{\\ast }/2\\pi \\hbar ^{2}\\right) =\\frac{1}{2}\\lambda _{0}$ , so that the prefactor in Eq.REF is rewritten in a form showing its independence of $E_{F}$ : $\\frac{\\lambda }{\\sqrt{n_{F}}}=\\frac{\\lambda _{0}}{\\sqrt{n_{F0}}}=2\\frac{\\lambda _{S}}{\\sqrt{n_{F0}}} $ Using this expression in Eq.REF , together with the semiclassical approximation that yields Expression REF , the pre-factor, $\\lambda /8$ , in the latter becomes: $\\left( \\lambda _{S}/4\\right) \\left( \\frac{k_{F}}{k_{F0}}\\right) $ , in full agreement with Eq.REF at the reference point $k_{F}=k_{F0}$ .",
"For doping levels away from the reference point, i.e.",
"for $k_{F}\\ne k_{F0}$ , one finds the simple relation: $A_{W}^{SC}=\\left( \\frac{k_{F}}{k_{F0}}\\right) A_{S} $ Figure: Pairing condensation energy eigenvalue, AA, as a function offield, h≡H/H 0 h\\equiv H/H_{0} , at temperature t≡T/T 0 =0.01t\\equiv T/T_{0}=0.01, calculatedfor the Weyl model, Eq.",
"(red curves), and for the extendedstandard model, Eq.",
"(blue curves), at various values of n ˜ F \\widetilde{n}_{F\\text{ }}(=15=15 (a),10 (b), 5 (c), 0.50.5 (d)).",
"Thereference parameters, H 0 ,T 0 H_{0},T_{0} , were selected in accord with theexperiment, as discussed in the text.",
"The cutoff was selected at: ℏω D /E F0 =0.5\\hbar \\protect \\omega _{D}/E_{F0}=0.5.",
"Note that for n ˜ F <5\\widetilde{n}_{F\\text{ }}<5the cutoff energy ℏω D >E F \\hbar \\protect \\omega _{D}>E_{F}."
],
[
"Mapping between the two models and their comparison with experiment",
"Experimental evidence for the existence of strong type-II superconductivity in a surface state of a topological insulator under a strong magnetic field can be found in results of transport, magnetic susceptibility, de Haas van Alphen (dHvA) oscillations and scanning tunnelling spectroscopy measurements, reported recently on Sb$_{2}$ Te$_{3}$ [5].",
"Using a simple s-wave BCS model, similar to the standard model described in Sec.3, with the experimentally observed dHvA frequency, $F_{0}=36.5$ T (implying $n_{F0}\\left( H\\right) =\\frac{F_{0}}{H}$ ), and cyclotron mass $m_{0}^{\\ast }=0.065m_{e}$ , it was shown in [9] that such an unusual SC state can exist only in the strong coupling superconductor limit.",
"In particular, the zero field limit of the self-consistent order parameter amplitude, $\\Delta _{SC}\\left( n_{F0}\\rightarrow \\infty \\right) \\rightarrow \\hbar \\omega _{D}/\\sinh \\left( 1/\\lambda _{0}\\right) $ , calculated in [9], was found, for $\\lambda _{0}=1$ and $\\hbar \\omega _{D}=0.25E_{F0}$ , to basically agree with the spatially average SC energy gap, derived from the STS measurements (i.e.",
"$\\simeq 13$ meV) [5], whereas the LL filling factor, calculated at the semiclassical $H_{c2}$ ( $n_{F0}\\approx 14$ ), was found to agree with the experimentally determined field of the resistivity onset downshift $H_{R}$ ($\\sim 2.5$ T, $n_{F0}\\sim 14$ ) [5].",
"Such an agreement, between the standard model, outlined in Sec.3, and the experiment reported in [5], seems to imply that the peculiar features of the helical surface state bands structure distinguishing the 2D Weyl Fermion gas model from the standard model, are irrelevant in constructing its high-fields SC state, except for a single parameter:- its unusual cyclotron effective mass, which can be dramatically modified upon variation of the chemical potential (e.g.",
"by doping or by changing the gate voltage).",
"The analysis presented in Sec.3 supports this conclusion for carrier densities and magnetic fields in the semiclassical limit.",
"Here we study the relationships between the Weyl model and the extended standard model, described above, in the general parameters range, finding conditions for a complete mapping between the two models, and searching for physical situations in which they are qualitatively distinguishable.",
"In Fig.2 we plot results of the pairing eigenvalue $A$ , calculated within both the Weyl and the extended standard models, as a function of the reduced magnetic field, $h\\equiv H/H_{0}$ , for various values of Fermi energy $E_{F}$ .",
"The temperature was selected sufficiently small to unfold the quantum oscillations associated with the Landau quantization.",
"Figure: H-T Phase diagram, obtained by solving the self-consistencyequation for both models at the reference point: n ˜ F =n ˜ F0 =10\\widetilde{n}_{F}=\\widetilde{n}_{F0}=10, on the basis of the reference parameters H 0 ,T 0 H_{0},T_{0} , as discussed in the text.The cutoff was selected at: ℏω D /E F0 =0.5\\hbar \\protect \\omega _{D}/E_{F0}=0.5.",
"Deviations are seen only around the darkblue area, where the Weyl phase boundary is slightly above the standard one.",
"Mutual reentrances of the SC and N phases, due to strongmagneto-oscillations effect, are seen around the upper-left corner of thephase diagram.Selecting for the reference parameters the values extracted from the transport and magneto-oscillations measurements [5], as described above: $F_{0}=36.5T,H_{0}=2.5T$ , $m_{0}^{\\ast }=0.065m_{e}$ , and from the magnetic susceptibility measurements [5] the value: $T_{0}=100K$ , we define the dimensionless reference parameters: $\\widetilde{\\tau }_{S}\\equiv \\left( k_{B}T_{0}/\\hbar \\widetilde{\\omega }_{c}^{S}\\right) $ and $\\widetilde{\\tau }_{W}\\equiv \\left( k_{B}T_{0}/\\hbar \\widetilde{\\omega }_{c}^{W}\\right) $ , where $\\widetilde{\\omega }_{c}^{S}\\equiv \\left(eH_{0}/m_{S}^{\\ast }c\\right) $ and $\\widetilde{\\omega }_{c}^{W}\\equiv \\left(\\sqrt{2}v/a_{H_{0}}\\right) $ , so that: $\\tau _{S}=\\widetilde{\\tau }_{S}\\left( t/h\\right) $ and $\\tau _{W}=\\widetilde{\\tau }_{W}\\left( t/\\sqrt{h}\\right) $ .",
"The two scales are therefore related via: $\\widetilde{\\tau }_{S}=\\left( \\widetilde{n}_{F0}\\right) ^{1/2}\\widetilde{\\tau }_{W}$ , where $\\widetilde{n}_{F0}\\equiv \\left( a_{H_{0}}k_{F0}\\right) ^{2}/2\\approx 10$ .",
"The eigenvalues $A_{W},A_{S}$ , plotted in Fig.2 as functions of $h$ , for various values of $\\widetilde{n}_{F}\\equiv \\left( a_{H_{0}}k_{F}\\right)^{2}/2$ , show at $\\widetilde{n}_{F}=\\widetilde{n}_{F0}$ complete agreement between the two models, including the fine structure of the quantum oscillations, provided $\\widetilde{\\tau }_{S}$ is re-scaled to $2\\times \\left( \\widetilde{n}_{F0}\\right) ^{1/2}\\widetilde{\\tau }_{W}$ , as found in the semiclassical approximation, Eq.REF .",
"Under these conditions, solutions of the self consistency equation, $1=\\left|V\\right|A$ , for both models, yield nearly identical results for the H-T phase diagrams, as shown in Fig.3, except for a small deviation in the low fields region, due to the different ultraviolet divergency predicted by the two models.",
"The two intersection points of the phase boundary with the axes, shown in Fig.3, are seen to be close to $t=1$ and $h=1$ , thus indicating that the calculated $H_{c2}\\left( T\\rightarrow 0\\right) $ and $T_{c}\\left( H\\rightarrow 0\\right)$ values are close to the values of $H_{0}$ and $T_{0}$ , respectively.",
"For values of $\\widetilde{n}_{F}$ away from $\\widetilde{n}_{F0}$ the baseline of $A_{W}$ is shifted with respect to that of $A_{S}$ , depending on wether $\\widetilde{n}_{F}>\\widetilde{n}_{F0}$ (shift up), or $\\widetilde{n}_{F}<\\widetilde{n}_{F0}$ (shift down), thus reflecting the dependence of the pairing correlation in the Weyl model on the carrier density.",
"This behavior is consistent with the relation REF derived in the semiclassical limit.",
"The oscillatory patterns remain nearly the same, except for slight relative narrowing of the Weyl peaks upon decreasing $\\widetilde{n}_{F}$ , which becomes quite significant in the quantum limit, e.g.",
"at $\\widetilde{n}_{F}=0.5$ in Fig.2d.",
"It is also remarkable that in the ultimate quantum limit, i.e.",
"when $\\widetilde{n}_{F}\\rightarrow 0$ , the pairing correlation in the Weyl model, despite its vanishing normal electron density of states at the Fermi energy, does not vanish."
],
[
"Conclusion",
"In this paper we have developed a Nambu-Gorkov Green's function approach to strongly type-II superconductivity in a 2D spin-momentum locked (Weyl) Fermi gas model at high perpendicular magnetic fields in order to study the transition to high field surface superconductivity observed recently on the topological insulator Sb$_{2}$ Te$_{3}$[5].",
"We have found that, for LL filling factors larger than unity, superconductivity in such a 2D Weyl Fermion gas can be mapped onto the standard 2D electron (or hole) gas model, having the same Fermi surface parameters, but with a cyclotron effective mass, $m^{\\ast }=E_{F}/2v^{2}$ , which could be dramatically reduced below the free electron mass, $m_{e}$ , by manipulating the doping level, or the gate voltage.",
"Our calculations for Sb$_{2}$ Te$_{3}$ show that the SC helical surface state reported in [5] was in the moderate semiclassical range ($n_{F}\\ge 10$ ), so justifying the mapping with the standard model.",
"They reveal a very unusual, strong type-II superconductivity at low carrier density and small cyclotron effective mass, $m^{\\ast }={0.065}m_{e}$ , which can be realized only in the strong coupling ($\\lambda \\sim 1$ ) superconductor limit[9].",
"Further reduction of the carrier density in such a system could yield an effective cyclotron energy comparable to or larger than the Fermi energy, LL filling factors smaller than unity, and cutoff energy larger than the chemical potential, resulting in significant deviations from the predictions of the standard model.",
"Note, however, that for such a dilute fermion gas system the simple mean field BCS theoretical framework of superconductivity, exploited in this paper, should be drastically revised, particularly due to the neglect of both phase and amplitude fluctuations of the SC order parameter [13], and to the breakdown of the adiabatic approximation in the electron phonon system [14].",
"Several recent reports on superconductivity in very dilute fermion gas systems, such as that found in compensated semimetallic FeSe [15], or in the large-gap semiconductor SrTiO$_{3}$ [16], have drawn much attention to fluctuation superconductivity beyond the Gaussian approximation, which could lead to crossover between weak-coupling BCS and strong-coupling Bose-Einstein condensate limits [17].",
"In the presence of strong magnetic fields the situation is further complicated due to complex interplay between vortex and SC amplitude fluctuations [18]."
]
] | 1709.01869 |
[
[
"The low-rank hurdle model"
],
[
"Abstract A composite loss framework is proposed for low-rank modeling of data consisting of interesting and common values, such as excess zeros or missing values.",
"The methodology is motivated by the generalized low-rank framework and the hurdle method which is commonly used to analyze zero-inflated counts.",
"The model is demonstrated on a manufacturing data set and applied to the problem of missing value imputation."
],
[
"Introduction",
"Principal component analysis (PCA) is a popular data science tool used for tasks such as dimensionality reduction, feature extraction, missing value imputation, denoising, and data compression.",
"PCA originated from the works of Pearson [15] and Hotelling [5], [6] and a detailed review of the subject is provided by Jolliffe [8].",
"Eckart and Young [3] described PCA as finding the best approximation of a numeric matrix $\\mbox{$A$}$ using a lower rank matrix $\\mbox{$Z$}$ , where the quality of the approximation is measured using least squares or quadratic loss.",
"Numerous authors have extended the concepts of PCA by changing the loss function and adding regularization to the low-rank matrix approximation problem.",
"Notably, Collins et al.",
"[1] proposed using exponential family loss functions and Gordon [4] used matching link-loss function pairs to construct procedures based on Bregman divergence.",
"These contributions generalized PCA and factor analysis similar to how generalized linear models [11] extended the concepts of regression.",
"Regularization has been used to construct low-rank approximations which account for data characteristics such as sparseness [20] and non-negativity [10].",
"Udell et al.",
"[19] summarized many of the major contributions using the generalized low-rank model framework.",
"This paper is focused on the task of constructing a low-rank approximation when some of the measured variables contain interesting values which occur frequently.",
"Examples include missing, censored, or truncated values; as well as zero-inflated data.",
"The zero-inflated case is commonly encountered when measuring manufacturing defect counts.",
"For regression analysis settings, Mullahy [13] proposed using the hurdle model, Lambert [9] described the zero-inflated model, and Min and Agresti [12] provided enhancements which included random effects.",
"For dimensionality reduction, Pierson and Yau [16] developed the zero-inflated factor analysis (ZIFA) model for analyzing single cell RNA sequencing data suffering from gene expression dropout.",
"The ZIFA model follows the probabilistic PCA approach of Tipping and Bishop [18] and optimization is carried out via the EM algorithm [2].",
"The ZIFA model can be expressed as a special case of the low-rank reduced hurdle model presented in Section .",
"The case of performing PCA in the presence of missing data has been examined previously, with Ilin and Raiko [7] providing a review of existing procedures.",
"The low-rank hurdle model offers a new representation which can be leveraged to gain additional data insights not directly available from competing PCA missing data methods.",
"The remaining contents are organized as follows.",
"Section describes the generalized low-rank framework.",
"The hurdle model is motivation in Section , along with details for proper implementation.",
"In Section the hurdle approach is used to analyze a zero-inflated manufacturing data set and investigate missing value imputation.",
"Lastly, Section contains some concluding remarks."
],
[
"The generalized low-rank model",
"The following notation is used throughout.",
"Matrices are denoted by bold uppercase letters or Greek symbols (e.g.",
"$\\mbox{$A$}$ , $\\mbox{$\\Sigma $}$ ) , vectors are represented by bold lowercase letters or Greek symbols (e.g.",
"$\\mbox{$a$}$ , $\\mbox{$\\mu $}$ ), and scalars are not bold (e.g.",
"$a_{ij}$ , $\\mu _j$ ).",
"Additionally, matrices with dimensions $m \\times d_j$ and vectors of length $d_j$ are denoted as matrices and vectors; respectively, even if $d_j = 1$ occurs for some $j$ .",
"Here we present a generalized framework for low-rank modeling, summarizing the methodology highlighted by Udell et al.",
"[19].",
"Let $\\mbox{$A$}$ be an $n \\times p$ data table where the rows represent $n$ observations consisting of measurements collected on $p$ variables.",
"Then for $i = 1,\\ldots ,n$ and $j=1,\\ldots ,p$ , $a_{ij}$ represents the $j^{th}$ variable value for the $i^{th}$ observation.",
"The domain for each column variable is denoted by $\\mathcal {F}_j$ , which is not restricted to $\\mathbb {R}$ , but includes discrete and non-numeric domains to facilitate abstract data types such as count, Boolean, categorical, and ordinal variables.",
"We will approximate abstract data types by representing $a_{ij} \\in \\mathcal {F}_j$ with numerical embeddings $\\mbox{$z$}_{ij} \\in \\mathbb {R}^{d_j}$ , where $d_j$ is the embedding dimension of the $j^{th}$ variable.",
"The resulting embedded dimension of the model is $d = \\sum _j d_j$ .",
"The loss incurred from using $\\mbox{$z$}_{ij}$ to describe $a_{ij}$ is measured using an appropriately selected loss function $L_{ij} : \\mathbb {R}^{d_j} \\times \\mathcal {F}_j \\rightarrow [0, \\infty )$ .",
"Essential to this analysis is the construction of a low-rank matrix $\\mbox{$Z$}\\in \\mathbb {R}^{n \\times d}$ which approximates our data table with minimal loss.",
"A rank-$k$ approximation can be found by specifying $\\mbox{$Z$}= \\mbox{$X$}\\mbox{$Y$}$ where $k < d$ , $\\mbox{$X$}\\in \\mathbb {R}^{n \\times k}$ , and $\\mbox{$Y$}\\in \\mathbb {R}^{k \\times d}$ .",
"Notice this decomposition is not unique since $\\mbox{$Z$}= \\mbox{$X$}\\mbox{$Y$}= \\mbox{$X$}\\mbox{$G$}^{-1}\\mbox{$G$}\\mbox{$Y$}$ for any non-singular $k \\times k$ matrix $\\mbox{$G$}$ .",
"An optimal rank-$k$ matrix decomposition can be found by minimizing the following optimization problem.",
"Let $\\mbox{$x$}_i \\in \\mathbb {R}^{1 \\times k}$ denote the $i^{th}$ row of $\\mbox{$X$}$ , and $\\mbox{$Y$}= \\left[\\mbox{$Y_1$}\\cdots \\mbox{$Y_p$}\\right]$ such that $\\mbox{$Y_j$}\\in \\mathbb {R}^{k \\times d_j}$ denotes the embedded columns associated with the $j^{th}$ variable of $\\mbox{$A$}$ , then the generalized low-rank model for $\\mbox{$A$}$ is found using $\\mbox{minimize } \\sum _{(i,j)\\in \\Omega } L_{ij}(\\mbox{$x$}_i \\mbox{$Y_j$}, a_{ij}) + \\sum _i r_i(\\mbox{$x$}_i) + \\sum _j \\tilde{r}_j(\\mbox{$Y_j$}),$ where $r_i : \\mathbb {R}^{1\\times k} \\rightarrow [0, \\infty )$ and $\\tilde{r}_j : \\mathbb {R}^{k\\times d_j} \\rightarrow [0, \\infty )$ are appropriately selected regularizers, and $\\Omega \\subseteq \\lbrace 1,\\ldots ,n\\rbrace \\times \\lbrace 1,\\ldots ,p\\rbrace $ represents the set of indices $(i,j)$ such that $a_{ij}$ is observed.",
"An appealing feature of the above generalized structure is the ability to combine different loss functions and regularizers to address different variable characteristics observed in the data table.",
"Many data reduction methods can be described in terms of equation (REF ).",
"For example, if unregularized quadratic loss is chosen for a numeric data table $\\mbox{$A$}$ with no missing values, then the optimization problem is solved using standard PCA [3].",
"This motivates the interpretation of the matrix $\\mbox{$X$}$ as a low-dimensional representation of $\\mbox{$A$}$ , with $\\mbox{$Y$}$ representing a mapping of $\\mbox{$X$}$ back into the original embedded data space.",
"Other special cases described by the general framework include robust and sparse PCA, exponential family PCA, non-negative matrix factorization, and matrix completion [19].",
"The task of optimizing equation (REF ) is simplified for convex loss functions and regularizers.",
"Under these conditions (REF ) becomes a biconvex minimization problem, which is commonly solved iteratively by alternating between convex updates in one argument while fixing the other.",
"Using the above notation, we alternate minimization over the rows of $\\mbox{$X$}$ while fixing $\\mbox{$Y$}$ , and minimization over the columns of $\\mbox{$Y$}$ while fixing $\\mbox{$X$}$ .",
"These updates can be parallelized over the rows of $\\mbox{$X$}$ and the columns of $\\mbox{$Y$}$ which may significantly improve computing times.",
"In general this alternating approach does not guarantee convergence to the global minimizer, and care may be required to avoid poor solutions.",
"In many applications, the usefulness of the sub-optimal solution is used to justify its adoption.",
"Variable scaling is a well known issue in multivariate analysis, and commonly data is normalized prior to performing methods such as PCA.",
"The concepts of offset and scaling can be generalized by replacing the loss functions in (REF ) by $L_{ij}(\\mbox{$x$}_i \\mbox{$Y_j$}+ \\mbox{$\\mu $}_j, a_{ij})/\\sigma ^2_j$ , where $\\mbox{$\\mu $}_j = \\operatornamewithlimits{arg\\,min}_{\\mbox{$\\mu $}\\in \\mathbb {R}^{d_j}} \\sum _{(i,j)\\in \\Omega } L_{ij}(\\mbox{$\\mu $}, a_{ij}), \\quad \\quad \\sigma ^2_j = \\frac{1}{n_j - 1} \\sum _{(i,j)\\in \\Omega } L_{ij}(\\mbox{$\\mu $}_j, a_{ij}),$ and $n_j$ is the number of non-missing values for the $j^{th}$ variable.",
"Using the above expressions, the loss contribution for the $j^{th}$ variable is equal to $n_j - 1$ under the offset only model.",
"This motivates the use of $\\sum _j (n_j - 1)$ as the total loss of the scaled model, which has a similar interpretation as total variation from standard PCA analysis.",
"It is important to note the offset and scaling adjustments are applied to the loss functions, and not directly to the data table itself.",
"This ensures aspects of the data table are maintained, such as sparseness or non-negativity."
],
[
"The hurdle model",
"Suppose data table $\\mbox{$A$}$ contains variable $\\mbox{$a$}_j$ with elements $a_{ij} \\in \\mathcal {F}_j$ which periodically take on the value $\\nu \\in \\mathcal {F}_j$ .",
"Assume the occurrence of $a_{ij} = \\nu $ is interesting because it potentially signifies a different generating process as compared to when $a_{ij} \\ne \\nu $ .",
"For example, defect counts are observed during the manufacturing of hard disc drives.",
"Normal counts are typically zero and appear to be governed by a process which differs from non-zero defect counts; where differences are observable across the data table variables.",
"Regression analysis techniques have been proposed for this paradigm, including the hurdle model from Mullahy [13] and the zero-inflated model from Lambert [9].",
"The hurdle model contains two components, where the first component represents the probability of observing $\\nu $ and the second describes the conditional behavior of the data provided $\\nu $ is not observed.",
"Explicitly, $\\Pr [a_{ij} = \\nu ] & = p_{ij},\\\\f_{a_j/\\nu }(a_{ij}\\,; \\mu _{ij}) & = (1 - p_{ij})g(a_{ij}\\,; \\mu _{ij}) \\quad \\mbox{for } a_{ij} \\ne \\nu ,$ where $f_{a_j/\\nu }(\\cdot \\, ; \\mu _{ij})$ represents the probability density or mass function when $\\nu $ is not observed, and $g(\\cdot \\, ; \\mu _{ij})$ is the possibly $\\nu $ -truncated density or mass function with mean parameter $\\mu _{ij}$ .",
"Following the generalized linear model framework [11], appropriate mean functions $(\\eta _1,\\eta _2)$ can be defined such that $p_{ij} = \\eta _1(\\mbox{$x$}_{i1}\\mbox{$\\beta $}_1) \\quad \\mbox{and} \\quad \\mu _{ij} = \\eta _2(\\mbox{$x$}_{i2}\\mbox{$\\beta $}_2),$ where $(\\mbox{$x$}_{i1}, \\mbox{$x$}_{i2})$ represent predictor row vectors and $(\\mbox{$\\beta $}_1, \\mbox{$\\beta $}_2)$ represent parameter column vectors.",
"Under the typical assumption of logit link for the probabilities $p_{ij}$ , maximum likelihood estimation is performed using the following equation: $\\operatornamewithlimits{arg\\,max}_{\\mbox{$\\beta $}_1, \\mbox{$\\beta $}_2} \\, \\prod _{i=1}^{n}\\left[\\frac{\\exp (\\mathbb {I}{(a_{ij} = \\nu )}\\mbox{$x$}_{i1}\\mbox{$\\beta $}_1)}{1+\\exp (\\mbox{$x$}_{i1}\\mbox{$\\beta $}_1)}\\,\\,g\\left(a_{ij}\\,; \\eta _2(\\mbox{$x$}_{i2}\\mbox{$\\beta $}_2)\\right)^{1 - \\mathbb {I}{(a_{ij} = \\nu )}}\\right].$ Equation (REF ) can be expressed as a minimization problem by examining the negative log of the likelihood function, which yields $\\operatornamewithlimits{arg\\,min}_{\\mbox{$\\beta $}_1, \\mbox{$\\beta $}_2} \\, \\sum _{i=1}^{n}\\log \\left[1 + \\exp (-a_{ij}^*\\mbox{$x$}_{i1}\\mbox{$\\beta $}_1)\\right] - \\sum _{i:a_{ij}\\ne \\nu } \\log \\left[g\\left(a_{ij}\\,; \\eta _2(\\mbox{$x$}_{i2}\\mbox{$\\beta $}_2)\\right)\\right],$ where $a_{ij}^* = 2*\\mathbb {I}{(a_{ij} = \\nu ) - 1}$ is an embedded indicator variable.",
"Previous authors have examined low-rank procedures motivated by loss functions based on the negative log likelihood; notably Collins et al.",
"[1] in the case of exponential family models.",
"In the context of the generalized low-rank model presented earlier (REF ), denoting $\\mbox{$Y_j$}= (\\mbox{$y$}_{j,1}, \\mbox{$y$}_{j,2}) \\in \\mathbb {R}^{k \\times 2}$ and replacing $(\\mbox{$x$}_{i1}\\mbox{$\\beta $}_1,\\mbox{$x$}_{i2}\\mbox{$\\beta $}_2)$ with $(\\mbox{$x$}_{i}\\mbox{$y$}_{j,1} + \\mu _{j,1},\\mbox{$x$}_{i}\\mbox{$y$}_{j,2} + \\mu _{j,2})$ in (REF ) yields the following equivalent low-rank expression for fixed $j$ : $\\sum _{i=1}^{n}L_{ij}(\\mbox{$x$}_i \\mbox{$Y_j$}+ \\mbox{$\\mu $}_j, a_{ij}) &= \\sum _{i=1}^{n}\\log \\left[1+\\exp (-a_{ij}^{*}(\\mbox{$x$}_{i}\\mbox{$y$}_{j,1}+\\mu _{j,1}))\\right]\\\\& \\quad \\quad \\quad + \\sum _{i:a_{ij}\\ne \\nu } \\log \\left[\\frac{g\\left(a_{ij}\\,; \\eta _2(m_{ij})\\right)}{g\\left(a_{ij}\\,; \\eta _2(\\mbox{$x$}_{i}\\mbox{$y$}_{j,2} + \\mu _{j,2})\\right)}\\right] \\\\= \\sum _{i=1}^{n}L_{\\ell ,ij}(\\mbox{$x$}_{i}\\mbox{$y$}_{j,1} & + \\mu _{j,1}, a_{ij}^{*}) + \\mathbb {I}{(a_{ij} \\ne \\nu )}L_{g,ij}(\\mbox{$x$}_{i}\\mbox{$y$}_{j,2} + \\mu _{j,2},a_{ij}),$ where $L_{\\ell ,ij}$ denotes logistic loss, $L_{g,ij}$ represents a $g(\\cdot ;\\cdot )$ derived loss, and $m_{ij} = \\operatornamewithlimits{arg\\,max}_c g\\left(a_{ij}\\,; \\eta _2(c)\\right)$ is a normalizing constant to ensure $L_{g,ij}$ is non-negative.",
"The derived equation in () can be further generalized for arbitrary data structures by making use of the subsequent composite loss definitions.",
"Definition (hurdle loss).",
"Let $\\mbox{$z$}= (z_1, z_2) \\in \\mathbb {R}^{2}$ , $a \\in \\mathcal {F}$ , $\\nu \\in \\mathcal {F}$ , $\\lambda _1, \\lambda _2 > 0$ , and $a^{*}$ be a binary variable indicating whether $a = \\nu $ has occurred, then full hurdle loss $L_{fh} : \\mathbb {R}^{2} \\times \\mathcal {F} \\rightarrow [0, \\infty )$ is specified by $L_{fh}(\\mbox{$z$}, a) = \\lambda _1 L_b(z_1, a^{*}) + \\mathbb {I}{(a \\ne \\nu )}\\lambda _2 L_g(z_2, a),$ where $L_b$ denotes a non-negative binary loss, and $L_g$ is an appropriate non-negative loss for describing the $\\nu $ -truncated data.",
"Furthermore let $z \\in \\mathbb {R}$ , then reduced hurdle loss $L_{rh} : \\mathbb {R} \\times \\mathcal {F} \\rightarrow [0, \\infty )$ is defined as $L_{rh}(z, a) = \\lambda _1 L_b(z, a^{*}) + \\mathbb {I}{(a \\ne \\nu )}\\lambda _2 L_g(z, a).$ The weights $\\lambda _{1}, \\lambda _{2}$ assign relative importance to the two model components, with larger weights implying higher importance on the resulting reduced representation.",
"The choice of weights will also affect the aggregated total loss, but this can be corrected using the formulas for offset and scaling appearing in equations (REF ).",
"One strategy is to assign weights proportional to total loss contributions resulting from the two hurdle components, where total loss is found using the offset only model.",
"Specifically for the $j^{th}$ variable, allow $n_{j,\\nu }$ to represent the number of $\\nu $ occurrences, $n_j - n_{j,\\nu }$ the number of non-$\\nu $ occurrences, and offsets $\\mu _b$ and $\\mu _g$ are found using (REF ), then weights $\\lambda _{j,1}, \\lambda _{j,2}$ which solve the below system of equations will yield a total loss of $n_j -1$ and ensure the binary loss contributes $c$ times the non-$\\nu $ loss: $\\begin{bmatrix}1 & 1\\\\1 & -c\\\\\\end{bmatrix}\\begin{bmatrix}\\sum _{i \\in \\Omega } L_{b,ij}(\\mu _b, a_{ij}^*) & 0\\\\0 & \\sum _{i \\in \\Omega } \\mathbb {I}{(a_{ij} \\ne \\nu )} L_{g,ij}(\\mu _g, a_{ij})\\end{bmatrix}\\begin{bmatrix}\\lambda _{j,1}\\\\\\lambda _{j,2}\\end{bmatrix}=\\begin{bmatrix}n_j - 1\\\\0\\end{bmatrix}$ where $c$ , $\\lambda _{j,1}$ , $\\lambda _{j,2} > 0$ .",
"Several intuitive choices for the multiplier are $c = 1$ or $c = n_{j,\\nu } / (n_j - n_{j,\\nu })$ .",
"In practice, data analysts are required to make several decisions in order for hurdle loss to be implemented within the generalized low-rank framework described in Section .",
"Logistic loss is likely a default choice for the binary loss $L_b$ , while selecting a form for $L_g$ may depend more heavily on the underlying data characteristics.",
"For example, quadratic and $\\ell _1$ losses are reasonable choices for continuous data, while Poisson loss is useful for count variables.",
"As previously mentioned, selecting convex loss functions and regularizers allows alternating minimization to be employed which may further guide the decision.",
"A more detailed overview of possible loss functions and regularizers is provided in Udell et al.",
"[19].",
"Reduced hurdle loss (REF ) simplifies the representation by setting $\\mbox{$y$}_{j,1} = \\mbox{$y$}_{j,2} = \\mbox{$y$}_j$ and $\\mu _{j,1} = \\mu _{j,2} = \\mu _j$ in ().",
"In the special case of the ZIFA model [16], quadratic loss is selected for $L_g$ and $L_b$ is based on the binomial probabilities $p_{ij} = \\exp (-\\xi _j (\\mbox{$x$}_i \\mbox{$y$}_j + \\mu _j)^2)$ where $\\xi _j$ is a positive decay coefficient.",
"The ZIFA model substitutions were justified in the context of the gene expression data problem and may be unsuitable for other subject domains.",
"In applications such as matrix completion and data reconstruction, mapping the reduced representation back into the original domain of $a_{ij}\\in \\mathcal {F}_j$ is required.",
"In general, low-rank models approximate $a_{ij}$ using some function of the vector $\\mbox{$z$}_{ij} = \\mbox{$x$}_i\\mbox{$Y_j$}+ \\mbox{$\\mu $}_j$ .",
"For quadratic loss this is simply $\\hat{a}_{ij} = z_{ij}$ since this model corresponds to the generalized linear model with identity link.",
"Under hurdle loss the original variable is encoded using both an indicator function and the identity function, where the latter function is applied only when non-$\\nu $ values occur.",
"These encodings are then approximated using a vector $\\mbox{$z$}\\in \\mathbb {R}^{2}$ in the full model setting, or $z \\in \\mathbb {R}$ for reduced models such as ZIFA.",
"Hence, the reverse mapping under hurdle loss can be found as follows.",
"Let $\\tilde{a}_{ij} = \\operatornamewithlimits{arg\\,min}_{a\\in \\mathcal {F}_j/\\nu }L_{g,ij}(\\mbox{$x$}_i\\mbox{$y$}_{j,2} + \\mu _{j,2}, a)$ and $a_{ij}^* = \\mathbb {I}{(a_{ij} = \\nu )}$ , then the reconstructed value $\\hat{a}_{ij}$ is determined by $\\hat{a}_{ij} &= \\operatornamewithlimits{arg\\,min}_a L_{h,ij}(\\mbox{$x$}_i\\mbox{$Y_j$}+ \\mbox{$\\mu $}_j, a)\\\\&= {\\left\\lbrace \\begin{array}{ll} \\nu & \\mbox{if}\\quad \\dfrac{\\lambda _{j,1} L_{b,ij}(\\mbox{$x$}_i\\mbox{$y$}_{j,1} + \\mu _{j,1}, 0) + \\lambda _{j,2} L_{g,ij}(\\mbox{$x$}_i\\mbox{$y$}_{j,2} + \\mu _{j,2}, \\tilde{a}_{ij})}{\\lambda _{j,1} L_{b,ij}(\\mbox{$x$}_i\\mbox{$y$}_{j,1} + \\mu _{j,1}, 1)} >1 ,\\\\\\tilde{a}_{ij} & \\mbox{otherwise.}\\end{array}\\right.",
"}$ The above piecewise condition is often simplified since $\\tilde{a}_{ij} = \\mbox{$x$}_i\\mbox{$y$}_{j,2} + \\mu _{j,2}$ and $L_{g}(z,z) = 0$ for commonly used loss functions, such as quadratic and Poisson losses.",
"Hence the condition in (REF ) becomes a ratio of only the binary loss components.",
"In the case of missing data, $\\tilde{a}_{ij}$ is still a reasonable imputed value even when the binary loss components suggest missingness is likely.",
"This is further demonstrated in Subsection REF .",
"The foundations for hurdle loss were developed following a likelihood model explanation, but additional intuition and advantages are worth mentioning.",
"First, in the full model setting each of the hurdle components receives a different principal vector in $\\mbox{$Y_j$}= (\\mbox{$y$}_{j,1},\\mbox{$y$}_{j,2})$ which allows for differing dependencies on the other columns of $\\mbox{$A$}$ .",
"This added flexibility mirrors the modeling complexities available in the hurdle and zero-inflated regression frameworks.",
"Low-rank applications concerned with data similar to those which motivated the regression models may find using the hurdle approach a suitable alternative to competing dimension reduction methods.",
"Second, since the low-dimensional representation retains information related to the likelihood of $\\nu $ values, we may extract probability type scores using $1/\\left[1 + \\exp (- \\mbox{$x$}_i\\mbox{$y$}_{j,1} - \\mu _{j,1})\\right]$ and measure associations with other variables by examining the cosine similarity between $\\mbox{$y$}_{j,1}$ and the remaining columns of $\\mbox{$Y$}$ .",
"These metrics inform the analyst about the quality of the low-rank representation with respect to discriminating $\\nu $ values, without the need to conduct additional analysis.",
"Lastly, employing the composite hurdle loss provides an additional degree of freedom when determining the offset and scaling for the underlying variable.",
"These values can be strongly influenced by $\\nu $ -inflation when using a single loss function, potentially obscuring meaningful representations of the underlying processes."
],
[
"Zero-inflated model",
"The first example investigates a factory data set which contains various defect count variables related to the manufacturing of hard disc drives.",
"In general, defect count variables tend to exhibit high degrees of zero-inflation.",
"This particular data set contains 14 different count variables measured on 2200 unique storage devices.",
"The observed zero-inflation varies across variables and ranges from $5\\%$ to $99\\%$ , with an aggregated value of about $60\\%$ .",
"The distribution of non-zero values displays a long tail with an overall median and mean of 2 and $13.3$ defects, respectively.",
"The generalized low-rank model (REF ) was used to analyze the defect data set.",
"Unregularized hurdle loss was chosen for all 14 count variables, with binary and non-zero components selected to be the following logistic and Poisson loss functions $L_\\ell (z, a^{*}) &= \\log \\left[1+\\exp (-a^{*}z)\\right],\\\\L_p(z, a) &= \\exp (z) -az + a \\log (a) - a,$ where $a^* = 2*\\mathbb {I}{(a = 0) - 1}$ as before.",
"Note that the likelihood motivated expressions from (REF - ) would suggest using the loss function derived from the zero-truncated Poisson $L_{tp}(z,a) = \\log \\left[\\exp (\\exp (z)) - 1\\right] - az + a\\log [g(a)] - \\log \\left[\\exp (g(a)) - 1\\right],$ where $g(a) = \\operatornamewithlimits{arg\\,max}_c a\\log (c) - \\log [\\exp (c) - 1]$ .",
"However, convergence tends to be slower and numerically unstable when employing the truncated version.",
"Additionally, differences between the ordinary Poisson loss and the zero-truncated version converge quickly to zero as $(z,a)$ increase.",
"All loss functions were centered and scaled according to (REF ).",
"Specifically, offset terms for logistic and Poisson losses are $\\log \\left[n_{j,\\nu } / (n_j - n_{j,\\nu })\\right]$ and $\\log \\left(\\bar{\\mbox{$a$}}_j\\right)$ , where $\\bar{\\mbox{$a$}}_j$ is the sample column mean.",
"Additionally, hurdle loss components were weighted and scaled using (REF ) with $c = n_{j,\\nu } / (n_j - n_{j,\\nu })$ .",
"Ordinary PCA and the ZIFA model were also considered for comparative purposes.",
"For the ZIFA model, initial values were based on PCA and the decay parameter $\\lambda _j$ was allowed to vary across variables.",
"Figure: Solid lines depict the hurdle model, dashed lines represent PCA, and dotted-dashed lines denote ZIFA.",
"Left: proportion of loss explained.",
"Middle: weighted reconstruction SSE.",
"Right: zero misclassification rate.Figure REF contains three plots comparing the full hurdle, ZIFA, and PCA approaches.",
"The left plot displays proportion of total loss explained as a function of the model dimension $k$ .",
"Recall total loss is calculated under the offset only model and equals $\\sum _j (n_j - 1)$ when the loss functions are appropriately scaled.",
"The hurdle model achieves a quicker rate of model loss reduction with respect to its model loss space, followed by PCA.",
"The middle plot compares element-wise weighted sum of squared reconstruction errors, where the weights are the sample standard deviations of the target variables.",
"The hurdle model performs similarly to PCA and shows improvement over dimensions 4 through 11.",
"The right plot displays zero misclassification rates for the three methods.",
"Simple threshold decision rules are used to map the reduced rank representations into zero/non-zero responses.",
"Specifically, PCA reports a zero outcome whenever a reconstructed value is less than 0.5, while the hurdle and ZIFA models assign a zero value whenever a reconstructed probability score exceeds 0.5.",
"The plot shows the full hurdle model performs noticeable better than PCA and ZIFA, which both performed similarly.",
"Overall, the ZIFA model either performs similar or worse than PCA.",
"The full hurdle model framework includes an additional column in the representation $\\mbox{$Y_j$}$ , which may provide advantages when optimizing the potential trade-offs between the competing composite losses.",
"This added flexibility is absent in reduced hurdle models and may explain the degraded ZIFA performance on this data set.",
"Missing in the analysis of Figure REF is the computational speed advantages of ordinary PCA.",
"A parallelized alternating second order gradient descent procedure was used to fit the hurdle model, and the EM algorithm was used for ZIFA.",
"In general, optimizing the generalized model is slower than ordinary PCA and care needs to be taken to avoid poor local minimums.",
"For applications which require fast implementations, stable representations for $\\mbox{$Y$}$ can be found offline and held fixed for efficient scoring of new data."
],
[
"Missing value model",
"Performing PCA in the presence of missing values is a well studied problem.",
"Ilin and Raiko [7] provide a review of common practical approaches to PCA with incomplete data.",
"For our purposes the problem can be reformulated in the context of the hurdle model.",
"Specifically, assume a logistic loss for the occurrence of missingness and quadratic loss for the observed data.",
"This approach is investigated by simulating 30 data sets each containing 5000 observations and 10 variables.",
"Each $10 \\times 1$ observation vector $\\mbox{$a$}_i$ is generated using the following low-rank sampling scheme: $\\mbox{$z$}_i &\\sim \\mbox{N}_4 (0, \\mbox{$I$}_4),\\\\\\mbox{$e$}_i &\\sim \\mbox{N}_{10} (0, \\mbox{$\\Sigma $})\\\\\\mbox{$\\mu $}& = \\left(1,2,3,4,5,6,7,8,9,10\\right)^T\\\\\\mbox{$a$}_i &= \\mbox{$W$}\\mbox{$z$}_i + \\mbox{$\\mu $}+ \\mbox{$e$}_i,$ where for each data set $\\mbox{$\\Sigma $}$ is a diagonal matrix sampled uniformly from (0.9, 1.1), and $\\mbox{$W$}$ is a $10 \\times 4$ matrix with entries $w_{k\\ell }$ generated from a standard normal distribution.",
"Missingness is induced using two alternative methods applied to the same generated data set.",
"The first approach assumes data is missing completely at random (MCAR), where as the second assumes data is missing at random (MAR) by correlating selection with the observed data.",
"Only the first entries $a_{i1}$ in $\\mbox{$a$}_i$ suffer from missingness with exclusions based on the following selection probabilities: $\\mbox{(MCAR)}\\quad \\quad \\mbox{Pr}\\left[a_{i1} \\mbox{ is missing} \\right] & = \\left[1 +\\exp (1.7)\\right]^{-1},\\\\\\mbox{(MAR)}\\quad \\quad \\mbox{Pr}\\left[a_{i1} \\mbox{ is missing} \\right] & = \\left[1 +\\exp (\\alpha + a_{i2} + a_{i3})\\right]^{-1}.$ The value of $\\alpha $ is recalculated for each data set so that the rate of missingness is approximately the same under both MCAR and MAR cases; yet under MAR, missingness is directly associated with the observed values of the second and third measured variables.",
"Under the zero-inflated model, offset terms for the $\\nu $ -truncated loss were found using (REF ).",
"For quadratic loss this suggests using the sample mean.",
"However under missing data the sample mean is known to be a biased estimate for the offset term [7], especially when considering MAR type missingness.",
"To account for bias, the offset term for the first variable was updated between alternating minimization steps using $\\mu _{1,2} = \\frac{1}{n_1} \\sum _{i\\in \\Omega }(a_{i1} - \\mbox{$x$}_i\\mbox{$y$}_{1,2}).$ Scaling for the first variable's hurdle loss components followed (REF ) with $c = n_{1,\\nu } / (n_1 - n_{1,\\nu })$ .",
"The remaining nine variables were modeled using only quadratic loss with offset and scaling terms found using (REF ).",
"Regularization was included in the low-rank model to reduce over-fitting and improve data imputation.",
"Quadratic regularizers $r(\\mbox{$x$}) = \\gamma _x ||\\mbox{$x$}||^2_2$ and $\\tilde{r}(\\mbox{$y$}) = \\gamma _y ||\\mbox{$y$}||^2_2$ were selected, and for simplicity $\\gamma = \\gamma _x = \\gamma _y$ was assumed.",
"In order to choose the regularization parameter $\\gamma $ , missing data was omitted from the generated data sets and new missing values were randomly created using a MCAR scheme with a similar selection rate as observed in the generated data.",
"The new missing values were imputed using a range of $\\gamma $ values and the mean squared imputation error was used to find optimal values.",
"The regularized full hurdle model was applied to the MCAR and MAR data sets, along with four additional models for comparative purposes: Bayesian PCA (BPCA) [14], Probabilistic PCA (PPCA) [17], Nonlinear Iterative Partial Least Squares (NIPALS) [21], and imputation using the sample mean of the observed values.",
"All the data reduction techniques assumed a $k=4$ reduced representation.",
"The performance of the various methods was measured based on imputation and offset mean squared errors, and average performance is reported in Table REF .",
"In both cases the low-rank models significantly improve upon the sample mean approach.",
"In the MCAR setting, adding the hurdle structure is unnecessary causing performance to be slightly worse than the BPCA and PPCA approaches.",
"The hurdle model reports the overall best performance for the MAR data sets.",
"Under the MAR setting, missing values provide additional information regarding the underlying data structure which the hurdle model more accurately represents.",
"This point is further expressed by the left most plot in Figure REF .",
"For each of the MAR data sets, the probability $\\rho $ of the observed $a_{i1}$ values exceeding the unobserved $a_{i1}$ missing values was recorded.",
"Data sets with high separation between observed and missing distributions exhibit small $\\rho (1-\\rho )$ values.",
"This separation measure was compared to the percentage improvement in MSE for the hurdle model over BPCA, where positive values indicate better performance for the hurdle model.",
"The plot clearly reveals the hurdle model becomes more preferable as the overlap between the observed and missing data decreases.",
"Table: Missing data imputationFigure: Left: hurdle model MSE improvement plotted against data separation.",
"Middle: hurdle ROC curve for the first MCAR data set.",
"Right: hurdle ROC curve for the first MAR data set.The hurdle model representation provides several diagnostics for missing data which are not directly obtainable using other approaches.",
"The first is based on the missingness probability score found using the sigmoid expression $1/\\left[1 + \\exp (- \\mbox{$x$}_i\\mbox{$y$}_{1,1} - \\mu _{1,1})\\right]$ which represents the fitted value for the Boolean portion of the hurdle data.",
"These scores can be used to construct ROC curves to measure how well the low-rank representation can discriminate between missing and non-missing occurrences.",
"Figure REF contains ROC curves for both the first simulated MCAR and MAR data sets.",
"In the MCAR data sets, missingness is unexplained by the observed data and the resulting average area under the ROC curve (AUC) was 0.53.",
"The MAR data sets had an average AUC of 0.88 which correctly suggests missingness is not likely to be completely at random.",
"The interpretability of the AUC value is dependent on the degree of the low-rank model.",
"Higher rank models which explain close to $100\\%$ of the total loss lack interpretability since their representation will over-fit observed noise.",
"In both MCAR and MAR cases the total loss reductions were near $80\\%$ over the offset only models.",
"This suggests missingness is difficult to explain for the MCAR example and remains as noise, whereas missingness is easily represented in the MAR case and does not remain as a contributor to unexplained loss.",
"The latter finding suggests the hurdle model is a useful representation for the underlying MAR data structure.",
"The second diagnostic is relevant when missingness is easily explained by the model.",
"Variables associated with missingness can be identified by inspecting the cosine similarity between the vector $\\mbox{$y$}_{1,1}$ and the other columns $\\mbox{$y$}_j$ in $\\mbox{$Y$}$ : $\\theta _{j} = 1 - \\frac{1}{\\pi } \\cos ^{-1}\\left[\\frac{\\mbox{$y$}_{1,1} \\cdot \\mbox{$y$}_j}{\\Vert \\mbox{$y$}_{1,1}\\Vert \\Vert \\mbox{$y$}_j\\Vert }\\right].$ The cosine similarities can be converted into distances using $d_{j} = 1 - 2|\\theta _{j} - 0.5|$ , where $d_j \\approx 0$ implies a high degree of dependence and $d_j \\approx 1$ suggests no association.",
"The similarities and distances for the first MAR data set are summarized in Table REF .",
"The distance measures for columns $\\mbox{$y$}_2$ and $\\mbox{$y$}_3$ are small, which correctly suggests the values $a_{i2}$ and $a_{i3}$ are related to missingness.",
"Interestingly columns $\\mbox{$y$}_9$ and $\\mbox{$y$}_8$ also report small distances.",
"Upon inspection, simulated entries $a_{i9}$ and $a_{i8}$ were moderate to highly correlated with $a_{i2}$ and $a_{i3}$ , indicating the reduced representation is distributing the observed influences across the collection of correlated variables.",
"This outcome seems somewhat expected given the nature of low-rank models.",
"Overall $87\\%$ of the simulated MAR data sets had at least one of the two influential variables in the top two distance scores, and this increased to $100\\%$ when considering the top three.",
"Table: Variables associated with missingness"
],
[
"Summary",
"This paper described the low-rank hurdle model which falls under the generalized low-rank framework.",
"Previous authors have proposed the ZIFA model which is a special case of the reduced hurdle model.",
"The methodology is particularly applicable to dimensionality reduction problems which exhibit characteristics similar to hurdle or zero-inflated regression problems.",
"In addition to providing a more natural loss approximation, the hurdle model's design allows practitioners to examine aspects of the low-rank representation not readily available when using alternative procedures.",
"This may be particularly useful in the case of missing data which was demonstrated in the applications."
]
] | 1709.01860 |
[
[
"The derived category of the projective line"
],
[
"Abstract We examine the localizing subcategories of the derived category of quasi-coherent sheaves on the projective line over a field.",
"We provide a complete classification of all such subcategories which arise as the kernel of a cohomological functor to a Grothendieck category."
],
[
"Introduction",
"Ostensibly, this article is about the projective line over a field, but secretly it is an invitation to a discussion of some open questions in the study of derived categories.",
"More specifically, we are thinking of localizing subcategories and to what extent one can hope for a complete classification.",
"The case of affine schemes is by now quite well understood, having been settled by Neeman in his celebrated chromatic tower paper [15].",
"However, surprisingly little is known in the simplest non-affine case, namely the projective line over a field.",
"We seek to begin to rectify this state of affairs and to advertise this and similar problems.",
"Let us start by recalling what is known.",
"We write $\\operatorname{QCoh}\\mathbb {P}^1_k$ for the category of quasi-coherent sheaves on the projective line $\\mathbb {P}^1_k$ over a field $k$ , and $\\operatorname{Coh}\\mathbb {P}^1_k$ denotes the full subcategory of coherent sheaves.",
"There is a complete description of the objects of $\\operatorname{Coh}\\mathbb {P}^1_k$ , due to Grothendieck [6].",
"The localizing subcategories of $\\operatorname{QCoh}\\mathbb {P}^1_k$ are known by work of Gabriel [5], and are parametrized by specialization closed collections of points of $\\mathbb {P}^1_k$ .",
"When one passes to the derived category $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ , the situation becomes considerably more complicated.",
"Several new localizations appear as a result of the fact that one can no longer non-trivially talk about subobjects and it remains a challenge to provide a complete classification of localizing subcategories.",
"An enticing aspect of this problem is that it not only represents the first stumbling block for those coming from algebraic geometry, but also for the representation theorists.",
"There is an equivalence of triangulated categories $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\stackrel{\\sim }{\\longrightarrow }\\mathsf {D}(\\operatorname{Mod}A)$ where $\\operatorname{Mod}A$ denotes the module category of $A=\\left[{\\begin{matrix}k&k^2\\\\0&k\\end{matrix}}\\right].$ The algebra $A$ is isomorphic to the path algebra of the Kronecker quiver $\\cdot \\,\\genfrac{}{}{0.0pt}{}{\\raisebox {-1.75pt}{\\longrightarrow }}{\\raisebox {1.75pt}{\\longrightarrow }}\\,\\cdot $ and is known to be of tame representation type.",
"The ring $A$ is one of the simplest non-representation finite algebras and so understanding its derived category is also a key question from the point of view of representation theory.",
"Of particular note, it is known by work of Ringel [16], [17] that $\\operatorname{Mod}A$ , the category of all representations, is wild and so it is very natural to ask if, as in the case of commutative noetherian rings, localizations can nonetheless be classified.",
"In this article we make a contribution toward this challenge in two different ways.",
"First of all, one of the main points of this work is to highlight this problem, provide some appropriate background, and set out what is known.",
"To this end the first part of the article discusses the various types of localization one might consider in a compactly generated triangulated category and sketches the localizations of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ which are known.",
"Our second contribution is to provide new perspective and new tools.",
"The main new result is that the subcategories we understand admit a natural intrinsic characterization: it is shown in Theorem REF that they are precisely the cohomological ones.",
"In the final section we provide a discussion of the various restrictions that would have to be met by a non-cohomological localizing subcategory.",
"Here our main results are that such subcategories come in $\\mathbb {Z}$ -families and consist of objects with full support on $\\mathbb {P}^1$ ."
],
[
"Preliminaries",
"This section contains some background on localizations, localizing subcategories, purity, and the projective line.",
"Also it serves to fix notation and may be safely skipped, especially by experts, and referred back to as needed."
],
[
"Localizing subcategories and localizations",
"Let $\\mathsf {T}$ be a triangulated category with all small coproducts and products.",
"The case we have in mind is that $\\mathsf {T}$ is either well-generated or compactly generated.",
"Definition 2.1.1 A full subcategory $\\mathsf {L}$ of $\\mathsf {T}$ is localizing if it is closed under suspensions, cones, and coproducts.",
"This is equivalent to saying that $\\mathsf {L}$ is a coproduct closed triangulated subcategory of $\\mathsf {T}$ .",
"Remark 2.1.2 It is a consequence of closure under (countable) coproducts that $\\mathsf {L}$ is closed under direct summands and hence thick (which means closed under finite sums, summands, suspensions, and cones).",
"Given a collection of objects $\\mathcal {X}$ of $\\mathsf {T}$ we denote by $\\operatorname{Loc}(\\mathcal {X})$ the localizing subcategory generated by $\\mathcal {X}$ , i.e.",
"the smallest localizing subcategory of $\\mathsf {T}$ containing $\\mathcal {X}$ .",
"The collection of localizing subcategories is partially ordered by inclusion, and forms a lattice (with the caveat it might not be a set) with meet given by intersection.",
"We next present the most basic reasonableness condition a localizing subcategory can satisfy.",
"Definition 2.1.3 A localizing subcategory $\\mathsf {L}$ of $\\mathsf {T}$ is said to be strictly localizing if the inclusion $i_*\\colon \\mathsf {L}\\rightarrow \\mathsf {T}$ admits a right adjoint $i^!$ , i.e.",
"if $\\mathsf {L}$ is coreflective.",
"Some remarks on this are in order.",
"First of all, it follows that $i^!$ is a Verdier quotient, and that there is a localization sequence $ { \\mathsf {L}[r]<0.5ex>^-{i_*}@{<-}[r]<-0.5ex>_-{i^!}",
"& \\mathsf {T}[r]<0.5ex>^-{j^*}@{<-}[r]<-0.5ex>_-{j_*} & \\mathsf {T}/\\mathsf {L}}$ inducing a canonical equivalence $ \\mathsf {L}^\\perp := \\lbrace X\\in \\mathsf {T}\\mid \\operatorname{Hom}(\\mathsf {L},X) =0\\rbrace \\xrightarrow{} \\mathsf {T}/\\mathsf {L}.$ Next we note that in nature localizing subcategories tend to be strictly localizing.",
"This is, almost uniformly, a consequence of Brown representability; if $\\mathsf {T}$ is well-generated and $\\mathsf {L}$ has a generating set of objects then $\\mathsf {L}$ is strictly localizing.",
"Now let us return to the localization sequence above.",
"From it we obtain two endofunctors of $\\mathsf {T}$ , namely $ i_*i^!",
"\\quad \\text{and} \\quad j_*j^*$ which we refer to as the associated acyclization and localization respectively.",
"They come together with a counit and a unit which endow them with the structure of an idempotent comonoid and monoid respectively.",
"The localization (or acyclization) is equivalent information to $\\mathsf {L}$ .",
"One can give an abstract definition of a localization functor on $\\mathsf {T}$ (or in fact any category) and then work backward from such a functor to a strictly localizing subcategory.",
"Further details can be found in [12].",
"We will use the language of (strictly) localizing subcategories and localizations interchangeably."
],
[
"Purity",
"Let $\\mathsf {T}$ be a compactly generated triangulated category and let $\\mathsf {T}^c$ denote the thick subcategory of compact objects.",
"We denote by $\\operatorname{Mod}\\mathsf {T}^c$ the Grothendieck category of modules over $\\mathsf {T}^c$ , i.e.",
"the category of contravariant additive functors $\\mathsf {T}^c \\rightarrow \\mathrm {Ab}$ .",
"There is a restricted Yoneda functor $H\\colon \\mathsf {T}\\longrightarrow \\operatorname{Mod}\\mathsf {T}^c \\text{ definedby } HX = \\operatorname{Hom}(-,X)|_{\\mathsf {T}^c},$ which is cohomological, conservative, and preserves both products and coproducts.",
"Definition 2.2.1 A morphism $f\\colon X\\rightarrow Y$ in $\\mathsf {T}$ is a pure-monomorphism (resp.",
"pure-epimorphism) if $Hf$ is a monomorphism (resp.",
"epimorphism).",
"An object $I\\in \\mathsf {T}$ is pure-injective if every pure-monomorphism $I\\rightarrow X$ is split, i.e.",
"it is injective with respect to pure-monomorphisms.",
"It is clear from the definition that if $I\\in \\mathsf {T}$ with $HI$ injective then $I$ is pure-injective.",
"It turns out that the converse is true and so $I$ is pure-injective if and only if $HI$ is injective.",
"Moreover, Brown representability allows one to lift any injective object of $\\operatorname{Mod}\\mathsf {T}^c$ uniquely to $\\mathsf {T}$ and thus one obtains an equivalence of categories $ \\lbrace \\text{pure-injectives in}\\;\\mathsf {T}\\rbrace \\xrightarrow{}\\lbrace \\text{injectives in}\\;\\operatorname{Mod}\\mathsf {T}^c\\rbrace .$ Further details on purity, together with proofs and references for the above facts can be found, for instance, in [11]."
],
[
"The projective line",
"Throughout we will work over a fixed base field $k$ which will be supressed from the notation.",
"For instance, $\\mathbb {P}^1$ denotes the projective line $\\mathbb {P}^1_k$ over $k$ .",
"We will denote by $\\eta $ the generic point of $\\mathbb {P}^1$ .",
"The points of $\\mathbb {P}^1$ that are different from $\\eta $ are closed.",
"A subset $\\mathcal {V}\\subseteq \\mathbb {P}^1$ is specialization closed if it is the union of the closures of its points.",
"In our situation this just says that $\\mathcal {V}$ is specialization closed if $\\eta \\in \\mathcal {V}$ implies $\\mathcal {V}=\\mathbb {P}^1$ .",
"As usual $\\operatorname{QCoh}\\mathbb {P}^1$ is the Grothendieck category of quasi-coherent sheaves on $\\mathbb {P}^1$ and $\\operatorname{Coh}\\mathbb {P}^1$ is the full abelian subcategory of coherent sheaves.",
"We use standard notation for the usual `distinguished' objects of $\\operatorname{QCoh}\\mathbb {P}^1$ .",
"The $i$ th twisting sheaf is denoted $\\mathcal {O}(i)$ and for a point $x\\in \\mathbb {P}^1$ we let $k(x)$ denote the residue field at $x$ .",
"In particular, $k(\\eta )$ is the sheaf of rational functions on $\\mathbb {P}^1$ .",
"For an object $X\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ or a localizing subcategory $\\mathsf {L}$ we will often write $X(i)$ and $\\mathsf {L}(i)$ for $X\\otimes \\mathcal {O}(i)$ and $\\mathsf {L}\\otimes \\mathcal {O}(i)$ respectively.",
"All functors, unless explicitly mentioned otherwise, are derived.",
"In particular, $\\otimes $ denotes the left derived tensor product of quasi-coherent sheaves and $\\operatorname{\\mathcal {H}\\!\\!\\;\\mathit {om}}$ the right derived functor of the internal hom in $\\operatorname{QCoh}\\mathbb {P}^1$ .",
"For an object $X\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ we set $ \\operatorname{supp}X = \\lbrace x\\in \\mathbb {P}^1\\;\\vert \\; k(x)\\otimes X \\ne 0\\rbrace .$ This agrees with the notion of support one gets as in [3] by allowing $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ to act on itself; the localizing subcategories generated by $k(x)$ and $\\Gamma _x\\mathcal {O}$ coincide.",
"Remark 2.3.1 Let $\\mathsf {A}$ be a hereditary abelian category, for example $\\operatorname{QCoh}\\mathbb {P}^1$ .",
"Then $\\operatorname{Ext}^n(X,Y)$ vanishes for all $n>1$ and therefore every object of the derived category $\\mathsf {D}(\\mathsf {A})$ decomposes into complexes that are concentrated in a single degree.",
"It follows that the functor $H^0\\colon \\mathsf {D}(\\mathsf {A})\\rightarrow \\mathsf {A}$ induces a bijection between the localizing subcategories of $\\mathsf {D}(\\mathsf {A})$ and the full subcategories of $\\mathsf {A}$ that are closed under kernels, cokernels, extensions, and coproducts."
],
[
"Types of localization",
"In this section we give a further review of the notions of localization, or equivalently localizing subcategory, that naturally arise and that we treat in this article.",
"These come in various strengths and what is known in general varies accordingly.",
"We take advantage of this review to give a whirlwind tour of certain aspects of the subject and to expose some technical results that are absent from the literature.",
"Unless otherwise specified we will denote by $\\mathsf {T}$ a compactly generated triangulated category.",
"One also can, and should, consider the well-generated case and it arises naturally even when one starts with a compactly generated category.",
"However, our focus will, eventually, be on those categories controlled by pure-injectives which more or less binds us to the compactly generated case."
],
[
"Smashing localizations",
"In this section we make some brief recollections on the most well understood class of localizing subcategories.",
"Definition 3.1.1 A localizing subcategory $\\mathsf {L}$ of $\\mathsf {T}$ is smashing if it is strictly localizing and satisfies one, and hence all, of the following equivalent conditions: the subcategory $\\mathsf {L}^\\perp $ is localizing; the corresponding localization functor preserves coproducts, i.e.",
"the right adjoint to $\\mathsf {T}\\rightarrow \\mathsf {T}/\\mathsf {L}$ preserves coproducts; the quotient functor $\\mathsf {T}\\rightarrow \\mathsf {T}/\\mathsf {L}$ preserves compactness; the corresponding acyclization functor preserves coproducts, i.e.",
"the right adjoint to $\\mathsf {L}\\rightarrow \\mathsf {T}$ preserves coproducts.",
"The smashing subcategories always form a set.",
"Amongst the smashing subcategories there is a potentially smaller distinguished set of localizing subcategories.",
"Unfortunately, there is not a standard way to refer to such categories; the snappy nomenclature only exists for the corresponding localizations.",
"Definition 3.1.2 A localization is finite if its kernel is generated by objects of $\\mathsf {T}^c$ , i.e.",
"the corresponding localizing subcategory is generated by objects which are compact in $\\mathsf {T}$ .",
"If $\\mathsf {L}$ is the kernel of a finite localization then it is smashing.",
"It is also compactly generated, although there are in general many localizing subcategories of $\\mathsf {T}$ which are, as abstract triangulated categories, compactly generated but are not generated by objects compact in $\\mathsf {T}$ .",
"The smashing conjecture for $\\mathsf {T}$ asserts that every smashing localization is a finite localization.",
"This is true in many situations, for instance it holds for the derived category $\\mathsf {D}(\\operatorname{Mod}A)$ of a ring $A$ when it is commutative noetherian [15] or hereditary [13].",
"On the other hand it is known to fail for certain rings (see for instance [10]) and is open in many cases of interest, for example the stable homotopy category."
],
[
"Cohomological localizations",
"We now come to the next types of localizing subcategories in our hierarchy, which are defined by certain orthogonality conditions.",
"This gives a significantly weaker hierarchy of notions than being smashing.",
"First a couple of reminders.",
"An abelian category $\\mathsf {A}$ is said to be (AB5) if it is cocomplete and filtered colimits are exact.",
"If in addition $\\mathsf {A}$ has a generator then it is a Grothendieck category.",
"An additive functor $H\\colon \\mathsf {T}\\rightarrow \\mathsf {A}$ is cohomological if it sends triangles to long exact sequences i.e.",
"given a triangle $ { X [r]^-f & Y [r]^-g & Z[r]^-{h} & \\Sigma X }$ the sequence $ @C=1.7em{ \\cdots [r] & H(\\Sigma ^{-1}Z)[r]^-{\\Sigma ^{-1}h} & H(X) [r]^-{H(f)} & H(Y) [r]^-{H(g)} &H(Z) [r]^-{H(h)} & H(\\Sigma X) [r] & \\cdots }$ is exact in $\\mathsf {A}$ .",
"Definition 3.2.1 A localizing subcategory $\\mathsf {L}\\subseteq \\mathsf {T}$ is cohomological if there exists a cohomological functor $H\\colon \\mathsf {T}\\rightarrow \\mathsf {A}$ into an (AB5) abelian category such that $H$ preserves all coproducts and $ \\mathsf {L}=\\lbrace X\\in \\mathsf {T}\\mid H(\\Sigma ^n X)=0\\text{ forall }n\\in \\mathbb {Z}\\rbrace ,$ that is $\\mathsf {L}$ is the kernel of $H^*$ .",
"We can extend this definition to an analogue for arbitrary regular cardinals, with Definition REF being the $\\aleph _0$ or `base' case.",
"The idea is to relax the exactness condition on the target abelian category.",
"This requires a little terminological preparation.",
"Let $\\mathsf {J}$ be a small category and $\\alpha $ a regular cardinal.",
"We say that $\\mathsf {J}$ is $\\alpha $-filtered if for every category $\\mathsf {I}$ with $\\vert I \\vert < \\alpha $ , i.e.",
"$\\mathsf {I}$ has fewer than $\\alpha $ arrows, every functor $F\\colon \\mathsf {I}\\rightarrow \\mathsf {J}$ has a cocone.",
"For instance, this implies that any collection of fewer than $\\alpha $ objects of $\\mathsf {J}$ has an upper bound and any collection of fewer than $\\alpha $ parallel arrows has a weak coequalizer.",
"If $\\alpha =\\aleph _0$ we just get the usual notion of a filtered category.",
"Let $\\mathsf {A}$ be an abelian category.",
"We say it satisfies (AB$5^\\alpha $ ) if it is cocomplete and has exact $\\alpha $ -filtered colimits.",
"Definition 3.2.2 A localizing subcategory $\\mathsf {L}\\subseteq \\mathsf {T}$ is $\\alpha $-cohomological if there exists an (AB$5^\\alpha $ ) abelian category $\\mathsf {A}$ and a coproduct preserving cohomological functor $H\\colon \\mathsf {T}\\rightarrow \\mathsf {A}$ such that $ \\mathsf {L}=\\lbrace X\\in \\mathsf {T}\\mid H(\\Sigma ^n X)=0\\text{ forall }n\\in \\mathbb {Z}\\rbrace ,$ that is $\\mathsf {L}$ is the kernel of $H^*$ .",
"If $\\mathsf {L}$ is $\\alpha $ -cohomological then it is clearly $\\beta $ -cohomological for all $\\beta \\ge \\alpha $ .",
"Remark 3.2.3 An $\\aleph _0$ -cohomological localizing subcategory is just a cohomological localizing subcategory.",
"We will usually stick to the shorter terminology for the sake of brevity and to avoid a proliferation of $\\aleph $ 's.",
"We now make a few observations on $\\alpha $ -cohomological localizing subcategories and then make some further remarks on the case $\\alpha =\\aleph _0$ .",
"Lemma 3.2.4 Smashing subcategories are cohomological.",
"Suppose $\\mathsf {L}$ is smashing.",
"Then $\\mathsf {T}/\\mathsf {L}$ is compactly generated and for $H$ we can take the composite $ \\mathsf {T}\\longrightarrow \\mathsf {T}/\\mathsf {L}\\longrightarrow \\operatorname{Mod}(\\mathsf {T}/\\mathsf {L})^c$ where the latter functor is the restricted Yoneda functor (REF ).",
"Theorem 3.2.5 Let $\\mathsf {L}$ be an $\\alpha $ -cohomological localizing subcategory.",
"Then $\\mathsf {L}$ is generated by a set of objects and so it is, in particular, strictly localizing.",
"This follows by applying [12] and then [12].",
"Corollary 3.2.6 A localizing subcategory $\\mathsf {L}$ is generated by a set of objects of $\\mathsf {T}$ if and only if there exists an $\\alpha $ such that $\\mathsf {L}$ is $\\alpha $ -cohomological.",
"We have just seen that an $\\alpha $ -cohomological localizing subcategory has a generating set.",
"On the other hand if $\\mathsf {L}$ is generated by a set of objects then $\\mathsf {L}$ is well-generated, and so is strictly localizing, and the quotient $\\mathsf {T}/\\mathsf {L}$ is also well-generated (see [12]).",
"One can then compose the quotient $\\mathsf {T}\\rightarrow \\mathsf {T}/\\mathsf {L}$ with the universal functor from $\\mathsf {T}/\\mathsf {L}$ to an (AB$5^\\alpha $ ) abelian category, for a sufficiently large $\\alpha $ , to get the required cohomological functor.",
"Let us now restrict to cohomological localizations and make the connection to purity in triangulated categories.",
"Proposition 3.2.7 A localizing subcategory $\\mathsf {L}\\subseteq \\mathsf {T}$ is cohomological if and only if there is a suspension stable collection of pure-injective objects $(Y_i)_{i\\in I}$ in $\\mathsf {T}$ such that $\\mathsf {L}=\\lbrace X\\in \\mathsf {T}\\mid \\operatorname{Hom}(X,Y_i)=0\\text{ for all }i\\in I\\rbrace $ .",
"Recall from (REF ) the restricted Yoneda functor which we denote by $H_\\mathsf {T}$ , for clarity, for the duration of the proof.",
"This functor identifies the full subcategory of pure-injective objects in $\\mathsf {T}$ with the full subcategory of injective objects in $\\operatorname{Mod}\\mathsf {T}^c$ as noted earlier (see [11] for details).",
"A cohomological functor $H\\colon \\mathsf {T}\\rightarrow \\mathsf {A}$ that preserves coproducts admits a factorisation $H=\\bar{H}\\circ H_\\mathsf {T}$ such that $\\bar{H}\\colon \\operatorname{Mod}\\mathsf {T}^c\\rightarrow \\mathsf {A}$ is exact and preserves coproducts; see [11].",
"The full subcategory $\\operatorname{Ker}\\bar{H}=\\lbrace M\\in \\operatorname{Mod}\\mathsf {T}^c\\mid \\bar{H}(M)=0\\rbrace $ is a localizing subcategory, so of the form $\\lbrace M\\in \\operatorname{Mod}\\mathsf {T}^c\\mid \\operatorname{Hom}(M,N_i)=0 \\text{ for all }i\\in I\\rbrace $ for a collection of injective objects $(N_i)_{i\\in I}$ in $\\operatorname{Mod}\\mathsf {T}^c$ .",
"Now choose pure-injective objects $(Y_i)_{i\\in I}$ in $\\mathsf {T}$ such that $H_\\mathsf {T}(Y_i)\\cong N_i$ for all $i\\in I$ ."
],
[
"When things are sets",
"As has been alluded to in the previous sections, it is a significant subtlety that one does not usually know the class of all localizing subcategories forms a set.",
"In fact there is no example where one knows that there are a set of localizing subcategories by `abstract means'; all of the examples come from classification results.",
"If one does know there are a set of localizing subcategories then life is much easier.",
"The purpose of this section is to give some indication of this, and record some other simple observations.",
"Everything here should be known to experts, but these observations have not yet found a home in the literature.",
"Let $\\mathsf {T}$ be a well-generated triangulated category.",
"Lemma 3.3.1 If the localising subcategories of $\\mathsf {T}$ form a set then every localizing subcategory is generated by a set of objects (and hence by a single object).",
"Suppose, for a contradiction, that $\\mathsf {L}$ is a localizing subcategory of $\\mathsf {T}$ which is not generated by a set of objects.",
"We define a proper chain of proper localizing subcategories $ \\mathsf {L}_0 \\subsetneq \\mathsf {L}_1 \\subsetneq \\cdots \\subsetneq \\mathsf {L}_\\alpha \\subsetneq \\mathsf {L}_{\\alpha +1} \\subsetneq \\cdots \\subsetneq \\mathsf {L},$ each of which is generated by a set of objects, by transfinite induction.",
"For the base case pick any object $X_0$ of $\\mathsf {L}$ and set $\\mathsf {L}_0 = \\operatorname{Loc}(X_0)$ .",
"This is evidently generated by a set of objects, namely $\\lbrace X_0\\rbrace $ .",
"By assumption $\\mathsf {L}$ is not generated by a set of objects so $\\mathsf {L}_0 \\subsetneq \\mathsf {L}$ .",
"Suppose we have defined a proper localizing subcategory $\\mathsf {L}_\\alpha $ of $\\mathsf {L}$ which is generated by a set of objects.",
"Since $\\mathsf {L}_\\alpha $ is proper we may pick an object $X_{\\alpha +1}$ in $\\mathsf {L}$ but not in $\\mathsf {L}_\\alpha $ and set $ \\mathsf {L}_{\\alpha +1} = \\operatorname{Loc}(\\mathsf {L}_\\alpha , X_{\\alpha +1})\\supsetneq \\mathsf {L}_\\alpha .$ This is clearly still generated by a set of objects and hence is still a proper subcategory of $\\mathsf {L}$ .",
"For a limit ordinal $\\lambda $ we set $ \\mathsf {L}_{\\lambda } = \\operatorname{Loc}(\\mathsf {L}_\\kappa \\mid \\kappa <\\lambda ).$ Again this is generated by a set of objects (and so is still not all of $\\mathsf {L}$ ).",
"This gives an ordinal indexed chain of distinct localizing subcategories of $\\mathsf {T}$ .",
"However, this is absurd since the collection of ordinals is not a set and so cannot be embedded into the set of all localising subcategories of $\\mathsf {T}$ .",
"Hence $\\mathsf {L}$ must have a generating set (i.e.",
"the above construction must terminate).",
"Remark 3.3.2 The above argument does not use that $\\mathsf {T}$ is well-generated.",
"One then deduces that all localizations are cohomological for an appropriate cardinal.",
"Lemma 3.3.3 If the localising subcategories of $\\mathsf {T}$ form a set then every localizing subcategory of $\\mathsf {T}$ is $\\alpha $ -cohomological for some regular cardinal $\\alpha $ .",
"By the previous lemma the hypothesis imply that every localizing subcategory of $\\mathsf {T}$ is generated by a set of objects.",
"It then follows from Corollary REF that they are all cohomological.",
"One can, to some extent, also work in the other direction.",
"Lemma 3.3.4 If the collection $ \\bigcup _{\\alpha \\in \\mathsf {Card}} \\lbrace \\mathsf {L}\\mid \\mathsf {L}\\text{ is $\\alpha $-cohomological}\\rbrace $ forms a set then the collection of all localising subcategories of $\\mathsf {T}$ also forms a set.",
"We have seen in Corollary REF that being $\\alpha $ -cohomological for some $\\alpha $ is the same as being generated by a set of objects.",
"Thus the hypothesis asserts that there are a set of localizing subcategories which have generating sets.",
"From this perspective it is clear we can pick a regular cardinal $\\kappa $ such that every localizing subcategory of $\\mathsf {T}$ which is generated by a set is generated by $\\kappa $ -compact objects.",
"Moreover, since the union in the statement of the lemma is both a set and indexed by a class, we conclude that the chain stabilises and so, taking $\\kappa $ larger if necessary, we may also assume every $\\alpha $ -cohomological localizing subcategory of $\\mathsf {T}$ is $\\kappa $ -cohomological.",
"If the localising subcategories of $\\mathsf {T}$ do not form a set then, as there are a set of $\\kappa $ -cohomological localizing subcategories, there must be a localizing subcategory $\\mathsf {L}$ which is not generated by a set of objects.",
"In particular $ \\mathsf {L}^{\\prime } = \\operatorname{Loc}(\\mathsf {L}\\cap \\mathsf {T}^\\kappa ) \\subsetneq \\mathsf {L}.$ But this is nonsense.",
"Since $\\mathsf {L}^{\\prime }$ is a proper localizing subcategory of $\\mathsf {L}$ we can find some object $X$ in $\\mathsf {L}$ but not in $\\mathsf {L}^{\\prime }$ and consider $\\mathsf {L}^{\\prime \\prime } = \\operatorname{Loc}(\\mathsf {L}^{\\prime }, X)$ .",
"Clearly $\\mathsf {L}^{\\prime \\prime }$ is still contained in $\\mathsf {L}$ , it properly contains $\\mathsf {L}^{\\prime }$ , it is generated by a set and hence $\\kappa $ -cohomological, and it contains the $\\kappa $ -compact objects of $\\mathsf {L}$ .",
"These are not compatible statements: we have assumed $\\kappa $ large enough so that $\\mathsf {L}^{\\prime \\prime }$ must be generated by the $\\kappa $ -compact objects it contains but this contradicts $\\mathsf {L}^{\\prime } \\subsetneq \\mathsf {L}^{\\prime \\prime }$ ."
],
[
"Cohomological localizations for the projective line",
"We now turn to the example we have in mind, namely $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ the unbounded derived category of quasi-coherent sheaves on $\\mathbb {P}^1$ .",
"We first describe the thick subcategories of $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ , the bounded derived category of coherent sheaves on $\\mathbb {P}^1$ .",
"We then recall the classifications of smashing subcategories and of tensor ideals in $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Finally, we classify the ($\\aleph _0$ -)cohomological localizing subcategories—there are no surprises and they are exactly the ones which have been understood for some time.",
"It is of course possible that there are $\\alpha $ -cohomological localizing subcategories for $\\alpha > \\aleph _0$ which we are not aware of.",
"It is in some sense tempting to guess that this is not the case, i.e.",
"that our list is already a complete list of localizing subcategories, but there is no real evidence for this.",
"We close by making some remarks on the hurdles that such an `exotic' localization would have to clear.",
"Before getting on with this let us recapitulate the connection with representation theory.",
"By a result of Beilinson [4] there is a tilting object $T\\in \\operatorname{Coh}\\mathbb {P}^1$ which induces an exact equivalence $\\operatorname{\\mathbf {R}Hom}(T,-)\\colon \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\stackrel{\\sim }{\\longrightarrow }\\mathsf {D}(\\operatorname{Mod}A)$ where $\\operatorname{Mod}A$ denotes the module category of $A=\\operatorname{End}(T)\\cong \\left[{\\begin{matrix}k&k^2\\\\0&k\\end{matrix}}\\right].$ Note that $A$ is isomorphic to the path algebra of the Kronecker quiver $\\cdot \\,\\genfrac{}{}{0.0pt}{}{\\raisebox {-1.75pt}{\\longrightarrow }}{\\raisebox {1.75pt}{\\longrightarrow }}\\,\\cdot $ and this algebra is known to be of tame representation type.",
"In fact, the representation theory of this algebra amounts to the classification of pairs of $k$ -linear maps, up to simultaneous conjugation.",
"The finite dimensional representations were already known to Kronecker [14]."
],
[
"Thick subcategories of the bounded derived category",
"The structure of the lattice of thick subcategories of $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ , which we recall in this section, has been known for some time; it can be computed by hand using the fact that $\\operatorname{Coh}\\mathbb {P}^1$ is tame and hereditary.",
"The structure of the coherent sheaves on $\\mathbb {P}^1$ is well known: there is a $\\mathbb {Z}$ -indexed family of indecomposable vector bundles and a 1-parameter family of torsion sheaves for each point on $\\mathbb {P}^1$ .",
"For each $i\\in \\mathbb {Z}$ one has a thick subcategory $ \\operatorname{Thick}(\\mathcal {O}(i)) = \\operatorname{add}(\\Sigma ^j\\mathcal {O}(i)\\mid j\\in \\mathbb {Z}) \\cong \\mathsf {D}^\\mathrm {b}(k)$ where the identifications follow from the computation of the cohomology of $\\mathbb {P}^1$ .",
"These are the only proper non-trivial thick subcategories which are generated by vector bundles and are also the only thick subcategories which are not tensor ideals.",
"Thus we have a lattice isomorphism $\\lbrace \\text{thick subcategories of }\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) \\text{ generated by vector bundles}\\rbrace \\xrightarrow{} \\mathbb {Z}$ where $\\mathbb {Z}$ denotes the lattice given by the following Hasse diagram: $@=1em{&&\\bullet @{-}[dll]@{-}[dl]@{-}[dr]@{-}[drr]@{-}[d]\\\\\\cdots @{-}[drr]&\\bullet @{-}[dr]&\\bullet @{-}[d]&\\bullet @{-}[dl]&\\cdots @{-}[dll]\\\\&&\\bullet }$ This is a special case of a general result because the indecomposable vector bundles are precisely the exceptional objects of $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ .",
"For any hereditary artin algebra $A$ the thick subcategories of $\\mathsf {D}^\\mathrm {b}(\\operatorname{mod}A)$ that are generated by exceptional objects form a poset which is isomorphic to the poset of non-crossing partitions given by the Weyl group $W(A)$ ; see [7], [8].",
"Note that $W(A)$ is an affine Coxeter group of type $\\tilde{A}_1$ for the Kronecker algebra $A=\\left[{\\begin{matrix}k&k^2\\\\0&k\\end{matrix}}\\right]$ , keeping in mind the derived equivalence $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\xrightarrow{} \\mathsf {D}^\\mathrm {b}(\\operatorname{mod}A).$ The thick tensor ideals are classified by $\\operatorname{Spc}\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\cong \\mathbb {P}^1$ , where the space $\\operatorname{Spc}\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ is meant in the sense of Balmer [2], and its computation is a special case of a general result of Thomason [20].",
"What all this boils down to is that for any set of closed points $\\mathcal {V}$ of $\\mathbb {P}^1$ there is a thick tensor ideal $ \\mathsf {D}_\\mathcal {V}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) := \\lbrace E \\mid \\operatorname{supp}E\\subseteq \\mathcal {V}\\rbrace = \\operatorname{Thick}(k(x)\\mid x\\in \\mathcal {V})$ consisting of complexes of torsion sheaves supported on $\\mathcal {V}$ .",
"Moreover, together with 0 and $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ this is a complete list of thick tensor ideals.",
"One can make this uniform by considering subsets of $\\mathbb {P}^1$ which are specialization closed.",
"In this language, by extending the above notation to allow $\\mathsf {D}_\\varnothing ^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) = 0$ and $\\mathsf {D}_{\\mathbb {P}^1}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) = \\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ , we have a lattice isomorphism $\\lbrace \\text{thick tensor ideals of }\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\rbrace \\xrightarrow{}\\lbrace \\text{spc subsets of }\\mathbb {P}^1\\rbrace ,$ where `spc' is an abbreviation for `specialization closed', which is given by $\\mathsf {I}\\mapsto \\operatorname{supp}\\mathsf {I}= \\bigcup _{E\\in \\mathsf {I}}\\operatorname{supp}E \\quad \\text{and}\\quad \\mathcal {V}\\mapsto \\mathsf {D}_\\mathcal {V}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ for $\\mathsf {I}$ a thick tensor ideal and $\\mathcal {V}$ a specialization closed subset.",
"We know every object of $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ is a direct sum of shifts of line bundles and torsion sheaves and so one can readily combine these classifications to obtain a lattice isomorphismLet $L^{\\prime },L^{\\prime \\prime }$ be a pair of lattices with smallest elements $0^{\\prime },0^{\\prime \\prime }$ and greatest elements $1^{\\prime },1^{\\prime \\prime }$ .",
"Then $L^{\\prime }\\amalg L^{\\prime \\prime }$ denotes the new lattice which is obtained from the disjoint union $L^{\\prime }\\cup L^{\\prime \\prime }$ (viewed as sum of posets) by identifying $0^{\\prime }=0^{\\prime \\prime }$ and $1^{\\prime }=1^{\\prime \\prime }$ .",
"$\\lbrace \\text{thick subcategories of }\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\rbrace \\xrightarrow{}\\lbrace \\text{spc subsets of }\\mathbb {P}^1\\rbrace \\amalg \\mathbb {Z}.$ The verification that the evident bijection is indeed a lattice map as claimed is elementary: the twisting sheaves are supported everywhere so are not contained in any proper ideal, and any twisting sheaf and a torsion sheaf, or any pair of distinct twisting sheaves, generate the category.",
"Thus for $i\\ne j$ and $\\mathcal {V}$ proper non-empty and specialization closed in $\\mathbb {P}^1$ we have $ \\mathsf {D}_\\mathcal {V}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\vee \\operatorname{Thick}(\\mathcal {O}(i)) = \\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) = \\operatorname{Thick}(\\mathcal {O}(i)) \\vee \\operatorname{Thick}(\\mathcal {O}(j))$ and $ \\mathsf {D}_\\mathcal {V}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1) \\wedge \\operatorname{Thick}(\\mathcal {O}(i)) = 0 = \\operatorname{Thick}(\\mathcal {O}(i)) \\wedge \\operatorname{Thick}(\\mathcal {O}(j)).$"
],
[
"Ideals and smashing subcategories",
"We now describe the localizing subcategories that one easily constructs from our understanding of the compact objects $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ in $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"By [13] the smashing conjecture holds for $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ (our computations will also essentially give a direct proof of this fact).",
"Thus the finite localizations one obtains by inflating the thick subcategories listed above exhaust the smashing localizations i.e.",
"$\\lbrace \\text{thick subcategories of }\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)\\rbrace \\xrightarrow{} \\lbrace \\text{smashing subcategories of }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\rbrace $ The localizing ideals are also understood.",
"Again this is known more generally (there is such a classification for any locally noetherian scheme, see [1]) but can easily be computed by hand for $\\mathbb {P}^1$ .",
"The precise statement is that there is a lattice isomorphism $\\lbrace \\text{localizing tensor ideals of }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\rbrace \\xrightarrow{} 2^{\\mathbb {P}^1}$ where $2^{\\mathbb {P}^1}$ denotes the powerset of $\\mathbb {P}^1$ with the obvious lattice structure.",
"The bijection is given by the assignments $ \\mathsf {L}\\mapsto \\lbrace x\\in \\mathbb {P}^1 \\mid k(x) \\otimes \\mathsf {L}\\ne 0\\rbrace $ for a localizing ideal $\\mathsf {L}$ and $ \\mathcal {V}\\mapsto \\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1):=\\lbrace X \\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\mid X\\otimes k(y) \\cong 0 \\text{ for } y\\notin \\mathcal {V}\\rbrace $ for a subset $\\mathcal {V}$ of points on $\\mathbb {P}^1$ .",
"This agrees with the classification of smashing subcategories in the sense that the smashing ideals are precisely those inflated from the compacts, i.e.",
"those corresponding to specialization closed subsets of points.",
"Since $\\mathbb {P}^1$ is 1-dimensional the only new localizing ideals that occur are obtained by throwing the residue field of the generic point, $k(\\eta )$ , into a finite localization.",
"Thus we have identified a sublattice consisting of a copy of $\\mathbb {Z}$ and the powerset of $\\mathbb {P}^1$ inside the lattice of localizing subcategories of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"The lattice structure extends that of the lattice of thick subcategories of $\\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)$ in the expected way.",
"The naive guess is that this is, in fact, the whole lattice.",
"While we do not know if this is the case, we can give an intrinsic definition of the localizations we have stumbled into so far.",
"This description is the goal of the next two subsections."
],
[
"An aside on continuous pure-injectives",
"In order to describe the localizations we have listed so far a word on continuous pure-injectives is required.",
"Definition 4.3.1 A pure-injective object $I$ is continuous (or superdecomposable) if it has no indecomposable direct summands.",
"We say that $\\mathsf {T}$ has no continuous pure-injective objects if every non-zero pure-injective object has an indecomposable direct summand or, in other words, if there are no continous pure-injectives.",
"An equivalent condition is that every pure-injective object is the pure-injective envelope of a coproduct of indecomposable pure-injective objects.",
"Proposition 4.3.2 The category $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ has no continuous pure-injective objects.",
"Let $A=\\left[{\\begin{matrix}k&k^2\\\\0&k\\end{matrix}}\\right]$ denote the Kronecker algebra.",
"We use the derived equivalence $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\xrightarrow{}\\mathsf {D}(\\operatorname{Mod}A)$ .",
"Let $X$ be a pure-injective object in $\\mathsf {D}(\\operatorname{Mod}A)$ .",
"Observe that $X$ decomposes into a coproduct $X=\\coprod _{n\\in \\mathbb {Z}}X_n$ of complexes with cohomology concentrated in a single degree, since $A$ is a hereditary algebra.",
"Thus we may assume that $X$ is concentrated in degree zero and identifies with a pure-injective $A$ -module.",
"Now the assertion follows from the description of the pure-injective $A$ -modules in [9].",
"Corollary 4.3.3 A localizing subcategory $\\mathsf {L}\\subseteq \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ is cohomological if and only if there is a collection of indecomposable pure-injective objects $(Y_i)_{i\\in I}$ in $\\operatorname{QCoh}\\mathbb {P}^1$ such that $\\mathsf {L}=\\lbrace X\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\mid \\operatorname{Hom}(X,\\Sigma ^jY_i)=0\\text{ for all }i\\in I, j\\in \\mathbb {Z}\\rbrace $ .",
"By Proposition REF being cohomological is equivalent to being the left perpendicular of a collection of pure-injective objects.",
"By the last proposition $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ has no continuous pure-injectives and so we may replace such a collection of pure-injectives with the collection of its indecomposable summands without changing the left perpendicular.",
"These are all honest sheaves since $\\operatorname{QCoh}\\mathbb {P}^1$ is hereditary."
],
[
"Classifying cohomological localizations",
"In this section we give a classification of the cohomological localizing subcategories of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"As we will show in Theorem REF they are precisely the subcategories described in Section REF .",
"Our strategy is to use Corollary REF and the classification of indecomposable pure-injectives for $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ to compute everything explicitly; we can compute the set of indecomposable pure-injectives associated to each of the localizing subcategories described in Section REF and show any suspension stable set of pure-injectives has the same left perpendicular as one of these.",
"To this end we first recall the description of the indecomposable pure-injective objects of $\\operatorname{QCoh}\\mathbb {P}^1$ .",
"Let us set up a little notation: given a closed point $x\\in \\mathbb {P}^1$ we can consider the corresponding map of schemes $ i_x\\colon \\operatorname{Spec}\\mathcal {O}_{\\mathbb {P}^1,x} \\longrightarrow \\mathbb {P}^1.$ We denote the maximal ideal of $\\mathcal {O}_{\\mathbb {P}^1,x}$ by $\\mathfrak {m}_x$ and the residue field $\\mathcal {O}_{\\mathbb {P}^1,x}/\\mathfrak {m}_x$ by $k(x)$ .",
"Let $E(x)$ be the injective envelope of the residue field $k(x)$ , and $A(x)$ the $\\mathfrak {m}_x$ -adic completion of $\\mathcal {O}_{\\mathbb {P}^1,x}$ , which is the Matlis dual of $E(x)$ .",
"Pushing these forward along $i_x$ gives objects in $\\operatorname{QCoh}\\mathbb {P}^1$ which we denote by $ \\mathcal {E}(x) = {i_x}_*E(x) \\quad \\text{and} \\quad \\mathcal {A}(x) = {i_x}_*A(x).$ Proposition 4.4.1 The indecomposable pure-injective quasi-coherent sheaves are given by the following disjoint classes of sheaves: the indecomposable coherent sheaves; the Prüfer sheaves $\\mathcal {E}(x)$ for $x\\in \\mathbb {P}^1$ closed; the adic sheaves $\\mathcal {A}(x)$ for $x\\in \\mathbb {P}^1$ closed; the sheaf of rational functions $k(\\eta )$ .",
"The indecomposable pure-injective quasi-coherent sheaves correspond to the indecomposable pure-injective modules over the Kronecker algebra via the derived equivalence $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\xrightarrow{}\\mathsf {D}(\\operatorname{Mod}A)$ .",
"The latter have beeen classified in [9].",
"Remark 4.4.2 The Prüfer sheaves and $k(\\eta )$ are precisely the indecomposable injective quasi-coherent sheaves.",
"Having recalled the indecomposable pure-injective sheaves we now determine how they interact with the localizations described in Section REF .",
"Let us begin by recording their supports.",
"Lemma 4.4.3 We have $ \\operatorname{supp}\\mathcal {E}(x) = \\lbrace x\\rbrace , \\; \\operatorname{supp}k(\\eta ) =\\lbrace \\eta \\rbrace , \\;\\text{and}\\; \\operatorname{supp}\\mathcal {A}(x) = \\lbrace x,\\eta \\rbrace .$ All of these sheaves are pushforwards along the inclusions of the spectra of local rings at points, and so their supports are contained in the relevant subset $\\operatorname{Spec}\\mathcal {O}_{\\mathbb {P}^1,x}$ .",
"Having reduced to computing the support over $\\mathcal {O}_{\\mathbb {P}^1,x}$ this is then a standard computation.",
"As one would expect the localizations $\\operatorname{Loc}(\\mathcal {O}(i))$ are particularly simple.",
"Lemma 4.4.4 The only indecomposable pure-injective quasi-coherent sheaf in $\\operatorname{Loc}(\\mathcal {O}(i))^\\perp $ is $\\mathcal {O}(i-1)$ .",
"There is a localization sequence for the compacts $ { \\operatorname{Thick}(\\mathcal {O}(i)) [r]<0.5ex>@{<-}[r]<-0.5ex> [d]_-\\wr & \\mathsf {D}^\\mathrm {b}(\\operatorname{Coh}\\mathbb {P}^1)[r]<0.5ex> @{<-}[r]<-0.5ex> & \\operatorname{Thick}(\\mathcal {O}(i-1)) [d]^-\\wr \\\\\\mathsf {D}^\\mathrm {b}(k) & & \\mathsf {D}^\\mathrm {b}(k) }$ where the adjoints exist since $\\mathcal {O}(i)$ is exceptional and the computation of the right orthogonal follows from the computation of the cohomology of $\\mathbb {P}^1$ .",
"Applying Thomason's localization theorem shows that $ \\operatorname{Loc}(\\mathcal {O}(i))^\\perp = \\operatorname{Loc}(\\mathcal {O}(i-1)) =\\operatorname{Add}(\\Sigma ^j\\mathcal {O}(i-1)\\mid j\\in \\mathbb {Z})$ and the claim is then immediate.",
"We next compute the pure-injectives lying in the right perpendicular of the localizing ideals.",
"Lemma 4.4.5 Let $\\mathcal {V}$ be a set of closed points with complement $\\mathcal {U}$ .",
"Then the indecomposable pure-injective sheaves in $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)^\\perp $ are: the indecomposable coherent sheaves supported at closed points in $\\mathcal {U}$ ; the Prüfer sheaves $\\mathcal {E}(x)$ for $x\\in \\mathcal {U}$ ; the adic sheaves $\\mathcal {A}(x)$ for $x\\in \\mathcal {U}$ ; the sheaf of rational functions $k(\\eta )$ .",
"By the classification of localizing ideals of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ we know that the category $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)^\\perp $ consists of precisely those objects supported on $\\mathcal {U}$ .",
"Since $\\mathcal {V}$ consists of closed points we know $\\mathcal {U}$ contains the generic point $\\eta $ .",
"The list is then an immediate consequence of Lemma REF .",
"Lemma 4.4.6 Let $\\mathcal {V}$ be a subset of $\\mathbb {P}^1$ containing the generic point and denote its complement by $\\mathcal {U}$ .",
"Then the indecomposable pure-injective sheaves in $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)^\\perp $ are: the indecomposable coherent sheaves supported at closed points in $\\mathcal {U}$ ; the adic sheaves $\\mathcal {A}(x)$ for $x\\in \\mathcal {U}$ .",
"The sheaf of rational functions $k(\\eta )$ has a map to every indecomposable injective sheaf and so no $\\mathcal {E}(x)$ is contained in the right perpendicular category (and clearly $k(\\eta )$ is not).",
"The category $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ contains the torsion and adic sheaves for points in $\\mathcal {V}$ so the only indecomposable pure-injective sheaves which could lie in the right perpendicular are those indicated; it remains to check they really don't receive maps from objects of $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"This is clear for the residue fields $k(x)$ for $x\\in \\mathcal {U}$ , as they cannot receive a map from any of the residue fields generating $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Since the right perpendicular is thick it thus contains all the indecomposable coherent sheaves supported on $\\mathcal {U}$ .",
"Moreover, the right perpendicular is closed under homotopy limits and so contains the corresponding adic sheaves $\\mathcal {A}(x)$ .",
"We now know which subsets of indecomposable pure-injectives occur in the right perpendiculars of the localizing subcategories we understand.",
"It's natural to ask for the minimal set giving rise to one of these categories, as in Corollary REF .",
"Let us make the convention that for an object $E\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ $ {}^\\perp E = \\lbrace F\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1) \\mid \\operatorname{Hom}(F,\\Sigma ^jE) = 0 \\;\\; \\forall j\\in \\mathbb {Z}\\rbrace .$ We can, without too much difficulty, compute all of the left perpendiculars of the indecomposable pure-injectives.",
"Lemma 4.4.7 The left perpendicular categories to the suspension closures of the indecomposable pure-injectives are as follows: ${}^\\perp F = \\Gamma _{\\mathbb {P}^1\\setminus \\lbrace x\\rbrace }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ for any $F\\in \\operatorname{Coh}\\mathbb {P}^1$ supported at $x\\in \\mathbb {P}^1$ ; ${}^\\perp \\mathcal {O}(i) = \\operatorname{Loc}(\\mathcal {O}(i+1))$ ; ${}^\\perp \\mathcal {E}(x) =\\Gamma _{\\mathbb {P}^1\\setminus \\lbrace x,\\eta \\rbrace }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ ; ${}^\\perp \\mathcal {A}(x) =\\Gamma _{\\mathbb {P}^1\\setminus \\lbrace x\\rbrace }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ ; ${}^\\perp k(\\eta ) = \\Gamma _{\\mathbb {P}^1\\setminus \\lbrace \\eta \\rbrace }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"These are all (more or less) straightforward computations.",
"Knowing this it is not hard to write down minimal sets of pure-injectives determining the ideals.",
"Corollary 4.4.8 Let $\\mathcal {V}$ be a subset of $\\mathbb {P}^1$ .",
"Then we have $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)= {}^\\perp \\lbrace k(x)\\mid x\\notin \\mathcal {V}\\rbrace .$ We also now have enough information to confirm that we have a complete list of the cohomological localizing subcategories.",
"Theorem 4.4.9 There is a lattice isomorphism $\\lbrace \\text{cohomological localizing subcategoriesof }\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\rbrace \\xrightarrow{} 2^{\\mathbb {P}^1}\\amalg \\mathbb {Z},$ where $2^{\\mathbb {P}^1}$ is the powerset of $\\mathbb {P}^1$ , with inverse defined by $ \\mathcal {V}\\mapsto \\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1) \\quad \\text{and} \\quad i \\mapsto \\operatorname{Loc}(\\mathcal {O}(i)).$ That is, the cohomological localizing subcategories are precisely the localizing ideals and the $\\operatorname{Loc}(\\mathcal {O}(i))$ for $i\\in \\mathbb {Z}$ .",
"By Corollary REF the cohomological localizing subcategories are precisely the localizing subcategories which are left perpendicular to a set of indecomposable pure-injectives.",
"Taking the left perpendicular of a set of pure-injectives corresponds to intersecting the corresponding left perpendiculars.",
"By Lemma REF we thus see that any such localizing subcategory is of the form claimed.",
"Remark 4.4.10 Denote by $\\operatorname{Ind}\\mathbb {P}^1$ the set of isomorphism classes of indecomposable pure-injective sheaves.",
"The subsets of the form $\\mathsf {L}^\\perp \\cap \\operatorname{Ind}\\mathbb {P}^1$ for some cohomological localizing subcategory $\\mathsf {L}$ are listed in Lemmas REF , REF , and REF ."
],
[
"Exotic localizations",
"As noted in Section REF we have a classification both of ideals and of smashing localizations for $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Moreover, we have just shown in Theorem REF that together these are precisely the cohomological localizations.",
"It is obvious to ask if there are non-cohomological localizations; we do not know the answer to this question and don't hazard a guess.",
"In this section we at least provide some foundation for future work in this direction by presenting some criteria to guarantee a localizing subcategory is an ideal.",
"This is relevant as any non-cohomological localization could not be an ideal—we proved that all ideals are cohomological.",
"As we shall see this dramatically restricts the possible form of a potential `exotic' localizing subcategory."
],
[
"A restriction on supports",
"We begin by analysing support theoretic conditions that ensure a localizing subcategory is an ideal.",
"Since $\\mathbb {P}^1$ is 1-dimensional the consequences we obtain are quite strong.",
"However, the ideas present in the arguments should be of more general interest.",
"The first observation is that if the support of an object does not contain some closed point then that object generates an ideal.",
"Lemma 5.1.1 Let $y$ be a closed point of $\\mathbb {P}^1$ and let $X\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ be such that $y\\notin \\operatorname{supp}X$ .",
"Then $\\mathsf {L}= \\operatorname{Loc}(X)$ is an ideal.",
"By definition we have $k(y)\\otimes X\\cong 0$ .",
"Since $y$ is a closed point the torsion sheaf $k(y)$ is compact, and hence rigid, so we deduce that $ \\operatorname{\\mathcal {H}\\!\\!\\;\\mathit {om}}(k(y), X)\\cong 0.$ In particular, $X\\in \\operatorname{Loc}(k(y))^\\perp \\cong \\mathsf {D}(\\operatorname{QCoh}\\mathbb {A}^1)$ , where we have made an identification of $\\mathbb {P}^1\\setminus \\lbrace y\\rbrace $ with the affine line.",
"Since $k(y)$ is compact the subcategory $\\operatorname{Loc}(k(y))^\\perp $ is localizing and so $ \\mathsf {L}\\subseteq \\operatorname{Loc}(k(y))^\\perp \\cong \\mathsf {D}(\\operatorname{QCoh}\\mathbb {A}^1).$ It just remains to note that every localizing subcategory of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {A}^1)$ is an ideal and that $\\operatorname{Loc}(k(y))^\\perp $ is itself an ideal, from which it is immediate that $\\mathsf {L}$ is an ideal in $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Let $\\mathcal {V} = \\mathbb {P}^1\\setminus \\lbrace \\eta \\rbrace $ denote the set of closed points of $\\mathbb {P}^1$ .",
"Corresponding to this Thomason subset there is a smashing subcategory $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ which comes with a natural action of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ , in the sense of [18], via the corresponding acyclization functor.",
"Moreover, $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ is a tensor triangulated category in its own right, with tensor unit $\\Gamma _\\mathcal {V}\\mathcal {O}$ (which is, however, not compact).",
"Lemma 5.1.2 The category $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ is generated by its tensor unit and hence every localizing subcategory contained in it is an ideal in it, and thus a submodule for the $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ action.",
"In particular, every localizing subcategory of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ contained in $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ is an ideal of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"The subset $\\mathcal {V}$ is discrete, in the sense that there are no specialization relations between any distinct pair of points in it.",
"It follows from [19] that $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ decomposes as $ \\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1) \\cong \\prod _{x\\in \\mathcal {V}} \\Gamma _x\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1).$ With respect to this decomposition the monoidal unit $\\Gamma _\\mathcal {V}\\mathcal {O}$ is just $\\bigoplus _{x\\in \\mathcal {V}}\\Gamma _x\\mathcal {O}$ , which clearly generates.",
"It follows that every localizing subcategory of $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ is an ideal (see for instance [18]) and from this the remaining statements are clear.",
"As a particular consequence we get the following statement, which is more in the spirit of Lemma REF .",
"Lemma 5.1.3 Let $X\\in \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ be an object such that $\\eta \\notin \\operatorname{supp}X$ .",
"Then $\\operatorname{Loc}(X)$ is an ideal.",
"Since $\\eta \\notin \\operatorname{supp}X$ we have $X\\in \\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Thus $\\operatorname{Loc}(X)$ is contained in $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ and therefore an ideal by the previous lemma.",
"We have shown that for any object $X$ with proper support the category $\\operatorname{Loc}(X)$ is an ideal.",
"Next we will show that any localizing subcategory containing such an object is automatically an ideal.",
"This requires the following technical lemma.",
"Lemma 5.1.4 If $\\mathsf {L}$ is a non-zero localizing ideal of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ then the quotient $\\mathsf {T}=\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)/\\mathsf {L}$ is generated by the tensor unit.",
"Since the property of being generated by the tensor unit is preserved under taking quotients it is enough to verify the statement when $\\mathsf {L}$ has support a single point.",
"If $\\operatorname{supp}\\mathsf {L}$ is a closed point then we can identify $\\mathsf {T}$ with the derived category of the open complement, which is isomorphic to $\\mathbb {A}^1$ .",
"Having made this observation the conclusion follows immediately.",
"It remains to verify the lemma in the case that $\\operatorname{supp}\\mathsf {L}=\\lbrace \\eta \\rbrace $ .",
"In this situation there is a recollement $ { \\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)[r]<1ex> @{<-}[r] [r]<-1ex> & \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1) [r]<1ex>@{<-}[r] [r]<-1ex> & \\mathsf {L}}$ where, as above, $\\mathcal {V}$ denotes the set of closed points of $\\mathbb {P}^1$ .",
"The bottom four arrows identify $\\Gamma _\\mathcal {V}\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ with the quotient $\\mathsf {T}$ and the desired conclusion is given by Lemma REF .",
"Combining all of this we arrive at the following proposition.",
"Proposition 5.1.5 If $\\mathsf {L}$ is a localizing subcategory of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ such that there is a non-zero $X\\in \\mathsf {L}$ with $\\operatorname{supp}X \\subsetneq \\mathbb {P}^1$ then $\\mathsf {L}$ is an ideal.",
"Let $X\\in \\mathsf {L}$ as in the statement of the proposition.",
"The object $X$ generates a non-zero localizing subcategory $\\operatorname{Loc}(X)\\subseteq \\mathsf {L}$ .",
"Since the support of $X$ is proper and non-empty it fails to contain some point of $\\mathbb {P}^1$ and so, by one of Lemma REF or REF , it is an ideal.",
"We thus have a monoidal quotient functor $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)\\xrightarrow{} \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)/\\operatorname{Loc}(X)$ and an induced localizing subcategory $\\mathsf {L}/\\operatorname{Loc}(X)$ in the quotient.",
"By Lemma REF the quotient $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)/\\operatorname{Loc}(X)$ is generated by the tensor unit and so $\\mathsf {L}/\\operatorname{Loc}(X)$ is a tensor ideal in it.",
"But then $\\mathsf {L}= \\pi ^{-1}(\\mathsf {L}/\\operatorname{Loc}(X))$ is also an ideal, which completes the proof.",
"Example 5.1.6 The non-ideals we know, namely the $\\operatorname{Loc}(\\mathcal {O}(i))$ , are of course compatible with the proposition: every object of $\\operatorname{Loc}(\\mathcal {O}(i))$ is just a sum of suspensions of copies of $\\mathcal {O}(i)$ , and these are all supported everywhere.",
"The following interpretation is the most striking in our context.",
"Corollary 5.1.7 If $\\mathsf {L}$ is a localizing subcategory which is not cohomological then every non-zero object of $\\mathsf {L}$ is supported everywhere."
],
[
"Twisting sheaves and avoiding compacts",
"We next make a few comments concerning the interactions between localizing subcategories and the twisting sheaves.",
"Lemma 5.2.1 If $\\mathsf {L}$ is a localizing subcategory which is not an ideal then $ \\mathsf {L}\\cap \\mathsf {L}(i) = 0$ for all $i\\in \\mathbb {Z}\\setminus \\lbrace 0\\rbrace $ .",
"Without loss of generality we may assume $i>0$ .",
"Suppose, for a contradiction, that $X\\in \\mathsf {L}\\cap \\mathsf {L}(i)$ is non-zero.",
"Pick a closed point $y$ and consider a triangle $ \\mathcal {O}(-i) \\longrightarrow \\mathcal {O}\\longrightarrow Z(y) \\longrightarrow \\Sigma \\mathcal {O}(-i)$ where $Z(y)$ is the cyclic torsion sheaf of length $i$ supported at $y$ .",
"We can tensor with $X$ to get a new triangle $ X(-i) \\longrightarrow X \\longrightarrow X\\otimes Z(y) \\longrightarrow \\Sigma X(-i),$ where both $X$ and $X(-i)$ lie in $\\mathsf {L}$ by hypothesis.",
"Thus, since $\\mathsf {L}$ is localizing, we see that $X\\otimes Z(y)$ lies in $\\mathsf {L}$ .",
"By Proposition REF we know that $X$ is supported everywhere and so $X\\otimes Z(y)\\ne 0$ .",
"But on the other hand, $X\\otimes Z(y)$ is supported only at $y$ which, by the same Proposition, implies that $\\mathsf {L}$ is an ideal yielding a contradiction.",
"Remark 5.2.2 The lemma implies that non-cohomological localizing subcategories would have to come in $\\mathbb {Z}$ -indexed families.",
"Lemma 5.2.3 If $\\mathsf {L}$ is a localizing subcategory such that $ \\operatorname{Loc}(\\mathcal {O}(i)) \\subsetneq \\mathsf {L}$ for some $i\\in \\mathbb {Z}$ then $\\mathsf {L}= \\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)$ .",
"Localizing subcategories containing $\\operatorname{Loc}(\\mathcal {O}(i))$ are in bijection with localizing subcategories of $\\mathsf {D}(\\operatorname{QCoh}\\mathbb {P}^1)/\\operatorname{Loc}(\\mathcal {O}(i))$ .",
"This quotient is just $\\mathsf {D}(k)$ and so, since we have asked for a proper containment, the result follows.",
"We can now conclude that any non-cohomological localizing subcategory must intersect the compact objects trivially.",
"Proposition 5.2.4 If $\\mathsf {L}$ is a localizing subcategory which is not cohomological then $\\mathsf {L}$ contains no non-zero compact object.",
"The indecomposable compact objects are just the indecomposable torsion sheaves at each point and the twisting sheaves.",
"By Lemma REF we know $\\mathsf {L}$ cannot contain a torsion sheaf and by the last lemma it cannot contain a twisting sheaf."
]
] | 1709.01717 |
[
[
"Wavefront retrieval through random pupil plane phase probes:\n Gerchberg-Saxton approach"
],
[
"Abstract A pupil plane wavefront reconstruction procedure is proposed based on analysis of a sequence of focal plane images corresponding to a sequence of random pupil plane phase probes.",
"The developed method provides the unique nontrivial solution of wavefront retrieval problem and shows global convergence to this solution demonstrated using a Gerchberg-Saxton implementation.",
"The method is general and can be used in any optical system that includes deformable mirrors for active/adaptive wavefront correction.",
"The presented numerical simulation and lab experimental results show low noise sensitivity, high reliability and robustness of the proposed approach for high quality optical wavefront restoration.",
"Laboratory experiments have shown $\\lambda$/14 rms accuracy in retrieval of a poked DM actuator fiducial pattern with spatial resolution of 20-30$~\\mu$m that is comparable with accuracy of direct high-resolution interferometric measurements."
],
[
"Introduction",
"Optical applications often require knowledge of both phase and amplitude of an optical wavefront.",
"Similar problems arise in various fields such as electron microscopy [1], X-ray crystallography [2], astronomy [3], optical imaging [4], etc.",
"The particular interest of the authors of this paper is inspired by the importance of optical phase measurements for direct exoplanet imaging, where wavefront reconstruction is needed both for accurate modeling of the instrument, as well as for wavefront control methods such as Electric Field Conjugation (EFC) [5], [6].",
"Since it is not always possible to measure the desired wavefront directly, different methods have been developed that allow its reconstruction based on intensity measurements only.",
"Phase information is absent from intensity-only measurements [7] and cannot be reconstructed unless some technique is used that encodes phase information in a sequence of intensity images in some controlled fashion.",
"It was found in earlier works [8], [9], [10], [11] that for almost every two-dimensional pupil-plane wavefront, a unique solution exists if the focal-plane intensities are known together with some constraints applied to the pupil aperture.",
"In this case the wavefront reconstruction problem can be formulated as the problem of a complex-valued signal reconstruction from the modulus of its Fourier transform.",
"However, even if a unique solution exists, it is not always possible to find this solution, due to convergence issues associated with wavefront reconstruction algorithms [12], [13], [14].",
"The solution uniqueness is not absolute, because a few trivial ambiguities still remain unsolved and can affect algorithm convergence [15] not counting the non-convexity of the Fourier magnitude constraint [12], [16].",
"Figure: Schematic of phase retrieval algorithms: Gerchberg-Saxton iterative loopis shown with solid lines, the random phase probes modification is indicated with dashed lines.",
"Direct and inverse Fourier transforms aremarked as “FT” and “FT -1 ^{-1}”.",
"Correlated random pupil-plane probes can be produced by changing the DM shape.In this paper we propose a pupil-plane wavefront reconstruction method based on analysis of a sequence of randomly aberrated focal-plane images (Section ) produced by the Deformable Mirror (DM) or other means.",
"Random phase aberrations in the pupil plane provide a unique (Section ) and globally converging solution (Section ) for measuring pupil-plane wavefront as demonstrated through simulations and experimental results described in Sections and .",
"The low noise sensitivity, fast convergence of the applied algorithms to the global solution in the presence of both static and dynamic pupil-plane aberrations (as described in Section ), high reliability and robustness, insensitivity to phase discontinuities and non-common path errors, make the proposed approach useful in a wide range of optical applications where high quality wavefront retrieval is needed (as discussed in Section )."
],
[
"Method",
"Historically, the first successful phase retrieval method was proposed by Gerchberg and Saxton [17].",
"The algorithm was originally developed to restore the pupil-plane wavefront assuming that both pupil-plane intensities $I({\\bf u})=|E({\\bf u)}|^2$ and focal-plane intensities $i({\\bf r})=|e({\\bf r})|^2$ are known.",
"The pupil-plane (complex-valued) electric field E(u) and the focal-plane electric field $e({\\bf r})$ are related by the Fourier transform $e({\\bf r})=~|e({\\bf r})|\\exp [i\\varphi ({\\bf r})]~~~=\\int E({\\bf u}) \\exp [i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf u}, \\nonumber \\\\E({\\bf u})=|E({\\bf u})|\\exp [i\\Phi ({\\bf u})]=\\int e({\\bf r}) \\exp [-i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf r},$ where $|E({\\bf u)}|$ and $|e({\\bf r})|$ are wavefront amplitudes, $\\Phi ({\\bf u})$ and $\\varphi ({\\bf r})$ are wavefront phases, and ${\\bf u}$ and ${\\bf r}$ are radius-vectors in the pupil plane and the focal plane respectively.",
"A constant phase factor as well as a normalizing factor $1/\\lambda f$ , where $\\lambda $ is the wavelength and $f$ is the system focal length, are not included in Eq.",
"REF for simplicity.",
"The Gerchberg-Saxton iterative loop switches between pupil and focal plane performing the following sequence of steps in the $(k+1)$ -th iteration (the schematic of all described algorithms is shown in Fig.",
"REF ): Fourier transform the current wavefront estimate $E_k({\\bf u})$ $ e_k({\\bf r})=|e_k({\\bf r})|\\exp [i\\varphi _k({\\bf r})]=\\int E_k({\\bf u}) \\exp [i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf u}.$ Replace the amplitude $|e_k({\\bf r})|$ of the resulting Fourier transform with the measured focal-plane amplitude $|e({\\bf r})|$ $g_k({\\bf r})=|e({\\bf r})|\\exp [i\\varphi _k({\\bf r})].$ Inverse Fourier transform the current focal-plane wavefront estimation $g_k({\\bf r})$ $ G_k({\\bf u})=|G_k({\\bf u})|\\exp [i\\Phi _k({\\bf u})]=\\int g_k({\\bf r}) \\exp [-i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf u}.$ Obtain the next pupil-plane wavefront estimate by replacing the amplitude $|G_k({\\bf u})|$ with the measured pupil-plane amplitude $|E({\\bf u})|$ $E_{k+1}({\\bf u})=|E({\\bf u})|\\exp [i\\Phi _k({\\bf u})].$ Although pupil-plane and focal-plane intensity distributions used in an iterative loop as constraints can provide efficient locally converging algorithms [17], [18], [12].",
"they cannot guarantee global convergence to a unique solution even when such a solution exists [19], [12].",
"As a result projection algorithms, which include the Gerchberg-Saxton method, stagnate near the closest local minimum instead (see examples in [20], [21] ).",
"The main reasons for the stagnation are the existence of ambiguous (non-unique) pupil-plane wavefronts [19] and the non-convexity of the Fourier magnitude constrain [12], [16].",
"Even trivial ambiguities such as global tip/tilt or conjugate inversion of the wavefront in combination with loose or symmetric wavefront support can cause algorithm stagnation [13], [22], [15], [21] near a solution where the shifted wavefront appears combined with the recovered wavefront.",
"Modifications of the Gerchberg-Saxton algorithm such as basic input-output and hybrid input-output algorithms by Fienup [23], [18] improve convergence to the global solution in some cases but still demonstrate stagnation in other cases [13], [19], [21].",
"Another group of wavefront retrieval algorithms are “diversity” algorithms, which analyze focal-plane (or some intermediate plane) intensity variations caused by predetermined changes of the pupil plane to measure the pupil-plane wavefront.",
"The desired “diversity” can be produced by either pupil-plane phase [24], [25], [26], [27] or amplitude [28], [29], [30] changes that provide the unique solution for the wavefront retrieval.",
"It has been shown [31] that analytically unique (up to the global phase) 2-dimensional pupil-plane wavefront solution can be found by using three different exponential pupil-plane amplitude probes.",
"Similar conclusions have also been made for phase retrieval by using phase probes [32], [15].",
"Pupil-plane wavefront diversity removes the main reason for wavefront retrieval stagnation, namely non-uniqueness of the solution.",
"Though non-convexity of the constraints can still be theoretically relevant to some open-loop phase wavefront diversity applications, the related numerical problems can be easily solved by changing the predetermined diversities.",
"As a result, diversity algorithms are faster and show better global convergence compared with Gerchberg-Saxton based algorithms.",
"The simplest phase diversity may be produced by defocusing (similar to the classical Hartmann optical test) that introduces a quadratic phase variation across the pupil.",
"Such diversity, however, is not very sensitive to non-symmetric phase aberrations [33].",
"More complicated diversity methods often need additional optics which may cause non-common path errors.",
"Phase diversities need to be well calibrated [34] and often require perfect knowledge of optical system parameters [35].",
"Phase diversity can be combined with projective algorithms to improve the quality of wavefront reconstruction.",
"Although this approach is often useful in stable laboratory conditions [36], it cannot be easily adapted to optical aberrations randomly changing with time.",
"A classical example of such aberrations is introduced by the Earth-turbulent atmosphere, which serves as an inspiration for our approach.",
"We consider the problem of restoration of the pupil-plane phase caused by the atmosphere turbulence in more detail.",
"We assume that the pupil-plane wavefront $E({\\bf u},t)=|E({\\bf u},t)|\\exp [i\\Phi ({\\bf u},t)]$ is temporally changing under the influence of the atmosphere turbulence and we try to estimate $E({\\bf u},t_k)$ , where $\\lbrace t_k\\rbrace _{k=1}^\\infty $ is a uniform time sequence, by using the measured sequence of focal-plane intensities $i({\\bf r},t_{k})=|e({\\bf r},t_{k})|^2$ .",
"We also assume the pupil-plane amplitude $|E({\\bf u},t)|=|E({\\bf u})|$ is not changing with time being constant across the telescope pupil and equal to 0 outside.",
"Consider the following modification of the Gerchberg-Saxton algorithm: Fourier transform the current wavefront estimate obtained in the time moment $t_k$ $ e_k({\\bf r},t_k)=|e_k({\\bf r},t)|\\exp [i\\varphi _k({\\bf r},t_k)]=\\nonumber \\\\=\\int E_k({\\bf u},t_k) \\exp [i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf u}.$ Replace the amplitude $|e_k({\\bf r},t_k)|$ of the resulting Fourier transform with the new focal-plane amplitude $|e({\\bf r},t_{k+1})|$ that is changing with time $g_k({\\bf r},t_{k+1})=|e({\\bf r},t_{k+1})|\\exp [i\\varphi _k({\\bf r},t_k)].$ Inverse Fourier transform the current focal-plane wavefront estimation $g_k({\\bf r},t_{k+1})$ $ G_k({\\bf u}, t_{k+1})=|G_k({\\bf u},t_{k+1})|\\exp [i\\Phi _k({\\bf u},t_{k+1})]=\\nonumber \\\\=\\int g_k({\\bf r},t_{k+1}) \\exp [-i2\\pi {\\bf u}\\cdot {\\bf r}] d{\\bf u}.~~~~~~~$ Obtain the next pupil-plane wavefront estimate by replacing the amplitude $|G_k({\\bf u}),t_{k+1}|$ with the telescope pupil-plane amplitude $|E({\\bf u})|$ $E_{k+1}({\\bf u},t_{k+1})=|E({\\bf u})|\\exp [i\\Phi _k({\\bf u},t_{k+1})].$ It is clear that in the case of completely uncorrelated atmospheric wavefronts algorithm never converges.",
"However, if the successive wavefront change is small enough to provide continuous wavefront transformation, one can expect the sequence $E_{k}({\\bf u},t_{k})$ to converge to some temporally changing solution.",
"This solution is expected to be negligibly different from the actual pupil-plane wavefront $E({\\bf u},t_{k})$ .",
"The accuracy of such restoration should be limited by the rms of the successive wavefronts phase difference."
],
[
"Uniqueness Theorem",
"The described modification of the Gerchberg-Saxton algorithm can only converge to a unique, non-trivial solution.",
"This solution is a continuous time sequence of reconstructed pupil-plane phase distributions, each of which coincides with the corresponding atmospheric wavefront.",
"By non-trivial solutions we mean all solutions except for those whose difference is caused by the global wavefront tip/tilt and piston.",
"The phase solution ambiguity caused by the “conjugation” symmetry of the wavefront $E({\\bf u})$ and its complex conjugate $E^\\ast ({-\\bf u})$ is taken into account in our future discussion.",
"The solution uniqueness directly follows from the following lemmas.",
"Lemma 1: Let $E_1({\\bf u})$ and $E_2({\\bf u})$ be two different pupil-plane wavefronts that form the identical intensity distribution in the focal plane.",
"Any small enough pupil-plane phase perturbation $\\Delta \\Phi ({\\bf u})$ applied to $E_1({\\bf u})$ and $E_2({\\bf u})$ transforms corresponding focal-plane intensities $I_1({\\bf r})$ and $I_2({\\bf r})$ such that they no longer coincide with each other except for some special cases of cross-symmetry between the wavefronts.",
"Proof: The m-th intensity distribution $I_m({\\bf r})$ is related to the focal-plane electric field $e_m({\\bf r})$ as $I_m({\\bf r})=|e_m({\\bf r})|$ .",
"According to the lemma assumption, $I_1({\\bf r})=I_2({\\bf r})$ .",
"The perturbation $\\Delta \\Phi ({\\bf u})$ applied to the m-th ($m=1,2$ ) pupil-plane wavefront transforms $E_m({\\bf u})$ into $E_m({\\bf u})\\exp [i\\Delta \\Phi ({\\bf u})]$ , $\\,\\,e_m({\\bf r})$ into $e_{\\Delta \\Phi ,m}\\,({\\bf r})$ , and $I_m({\\bf r})$ into $I_{\\Delta \\Phi ,m}\\,({\\bf r})$ .",
"In discrete form, the focal-plane intensity change caused by this perturbation can be expressed as a linear combination of $\\Delta \\Phi ({\\bf u}_k)$ values taken at the $k$ -th sampling point in the pupil plane $\\Delta I_{\\Delta \\Phi ,m}({\\bf r}_j) = I_{\\Delta \\Phi ,m}({\\bf r}_j)-I_l({\\bf r}_j)=\\sum _k B^{(m)}_{jk}\\Delta \\Phi ({\\bf u}_k),$ where ${\\bf r_j}$ is the $j$ -th sampling point in the focal plane, elements of matrix $B^{(m)}=[\\,B^{(m)}_{jk}\\,]$ are equal to $\\partial I_m({\\bf r}_j)/{\\partial \\Phi _m({\\bf u}_k)}$ , $\\Phi _m({\\bf u}_k)=\\arg [E_m({\\bf u}_k)]$ , and the summation is performed for each sampling point in the pupil.",
"Let $K$ be the total number of sampling points over the pupil aperture.",
"To make intensities $I_{\\Delta \\Phi ,1}\\,({\\bf r})$ and $I_{\\Delta \\Phi ,2}\\,({\\bf r})$ equal to each other the perturbation $\\Delta \\Phi ({\\bf u})$ should be a solution of the following system of linear equations $\\sum _{k=1}^K (B^{(1)}_{jk}-B^{(2)}_{jk})\\, \\Delta \\Phi ({\\bf u}_k)=0, \\mbox{ } j=1, 2, ... K,$ written for $K$ independent sampling points in the image plane.",
"Let $[L_{jk}]=[B^{(1)}_{jk}-B^{(2)}_{jk}]$ be the matrix of the linear system (REF ).",
"Two different cases are possible.",
"In case $\\det \\, [L_{jk}]\\ne 0$ , the system (REF ) has a unique solution $\\Delta \\Phi ({\\bf u})=0$ that makes equality $I_{\\Delta \\Phi ,1}\\,({\\bf r})=I_{\\Delta \\Phi ,2}\\,({\\bf r})$ impossible in some continuous area of small arbitrary changes of the perturbation $\\Delta \\Phi ({\\bf u})$ except for $\\Delta \\Phi ({\\bf u})=0$ .",
"The case $\\det \\, [L_{jk}] = 0$ corresponds to infinitely many solutions of the system (REF ).",
"All these solutions belong to some continuous area of changes of perturbation $\\Delta \\Phi ({\\bf u})$ , but they are not arbitrary.",
"“Cross-symmetry” between $E_1({\\bf u})$ and $E_2({\\bf u})$ occurs when $\\det \\ [L_{jk}] = 0$ and is thus associated with an infinite set of solutions.",
"Thus, Lemma 1 is considered proven.",
"All possible cases of cross-symmetry between $E_1({\\bf u})$ and $E_2({\\bf u})$ are analyzed later.",
"$\\Box $ Corollary 1: One direct consequence immediately follows from Lemma 1.",
"Let two different pupil-plane wavefronts $E_1({\\bf u})$ and $E_2({\\bf u})$ be influenced by a continuous phase perturbation $\\Delta \\Phi _\\gamma ({\\bf u})$ such that $\\det [L_{jk}^\\gamma ]\\ne 0$ at any point of the transformation.",
"The perturbation $\\Delta \\Phi _\\gamma ({\\bf u})$ is given here in the form of a continuous “transformation line” parametrized by $\\gamma \\in [ \\, 0,1 ] \\,$ and the matrix $[L_{jk}]$ at the point $\\gamma $ is denoted as $[L_{jk}^\\gamma ]$ .",
"In this case, pupil-plane wavefronts $E_1({\\bf u})\\exp [i\\Delta \\Phi _\\gamma ({\\bf u})]$ and $E_2({\\bf u})\\exp [i\\Delta \\Phi _\\gamma ({\\bf u})]$ produce different focal-plane images $I_{\\gamma ,1}({\\bf r})\\ne I_{\\gamma ,2}({\\bf r})$ at any point $\\gamma $ of the transformation line except for a possible finite set of points $\\lbrace \\gamma _n\\rbrace $ where $I_{\\gamma _n\\, ,1}({\\bf r})=I_{\\gamma _n\\, ,2}({\\bf r})$ .",
"Proof: The assumption that the set $\\lbrace \\gamma _n\\rbrace $ is infinite means that there exists a converging subset $\\lbrace \\gamma _{n_l}\\rbrace $ such that linear system (REF ) has only one solution for each element of the subset $\\lbrace \\gamma _{n_l}\\rbrace $ .",
"Let $\\gamma _c$ be the point of convergence of $\\lbrace \\gamma _{n_l}\\rbrace $ .",
"This implies that $\\gamma _c\\in [0,1]$ and in an arbitrary small neighborhood around $\\gamma _c$ indefinitely many solutions of the linear system (REF ) exist.",
"Therefore, $\\det [L_{jk}^\\gamma ]=0$ contradicts the assumption $\\det [L_{jk}^\\gamma ]\\ne 0$ .",
"The obtained contradiction proves the corollary.$\\Box $ The linear system (REF ) always has at least one solution.",
"The analysis for the case of infinitely many solutions is given below.",
"It is generally accepted that all non-trivial ambiguous phase solutions fall under one of the following three cases of pupil-plane wavefront symmetry: Case 1.",
"The case of the “convolution” symmetry for which the focal-plane wavefront $e({\\bf r})$ can be presented as a factor of two entire functions $e({\\bf r})=f({\\bf r})g({\\bf r})$ [10].",
"The corresponding pupil-plane wavefront $E({\\bf u})$ is a convolution of two functions $E({\\bf u})=F({\\bf u}) \\otimes G({\\bf u})$ , where $F({\\bf u})$ and $G({\\bf u})$ are Fourier transforms of functions $f({\\bf r})$ and $g({\\bf r})$ , and symbol “$\\otimes $ ” is used to denote the convolution operation.",
"In this case the pupil-plane wavefronts $F({\\bf u}) \\otimes G({\\bf u}), \\, F({\\bf u}) \\otimes G^\\ast ({-\\bf u}), F^\\ast (-{\\bf u}) \\otimes G({\\bf u}) $ , and $F^\\ast (-{\\bf u}) \\otimes G^\\ast (-{\\bf u}) $ form identical focal-plane images coinciding with the focal-plane intensity $I({\\bf r})$ .",
"Case 2.",
"In the case of the conjugation symmetry the pupil-plane wavefront $E_2({\\bf u})=E_1^\\ast (-{\\bf u})$ always produces the focal-plane intensity $I_2({\\bf r})$ that is equal to the focal-plane intensity $I_1({\\bf r})$ produced by the wavefront $E_1({\\bf u})$ .",
"Case 3.",
"The case of separable variables for which the pupil-plane wavefront $E({\\bf u})$ can be presented as a product of two factors $E({\\bf u})=F({u_x}) \\, G(u_y)$ , $F({u_x})$ and $G({u_y})$ are functions of one variable, and $u_x$ and $u_y$ are components of the vector ${\\bf u}$ .",
"In this case the pupil-plane wavefronts $F({u_x}) \\, G(u_y)$ , $F({u_x}) \\, G^\\ast (-u_y), F^\\ast ({-u_x}) \\, G(u_y) $ , and $F^\\ast ({-u_x}) \\, G^\\ast (-u_y) $ also produce identical focal-plane intensities coinciding with the focal-plane intensity $I({\\bf r})$ .",
"We call all solutions described in Cases 1–3 “conjugated” wavefront solutions because they are all related to the complex conjugation of one of the function which describes the wavefront.",
"It is clear that any continuous pupil-plane wavefront transformation, which preserves its symmetry, that may keep multiple solutions.",
"We consider such transformations in detail next.",
"For simplicity we do not analyze cases of mixed symmetry.",
"Lemma 2: Let a pupil-plane wavefront $E_0({\\bf u})$ be a convolution of functions $F_0({\\bf u})$ and $G_0({\\bf u})$ , and a pupil-plane wavefront $E_1({\\bf u})$ be a convolution of functions $F_1({\\bf u})$ and $G_1({\\bf u})$ .",
"Consider all possible continuous phase transformations of the pupil $\\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u})]$ that convert $E_0({\\bf u})=F_0({\\bf u}) \\otimes G_0({\\bf u})$ into $E_1({\\bf u})=F_1({\\bf u}) \\otimes G_1({\\bf u})$ such that $E_\\gamma ({\\bf u})$ can be presented as a convolution at any point $\\gamma $ of the transformation line, i.e.",
"$E_\\gamma ({\\bf u})=E_0 ({\\bf u}) \\, \\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u})]=F_\\gamma ({\\bf u}) \\otimes G_\\gamma ({\\bf u}),$ where $F_\\gamma ({\\bf u})$ and $G_\\gamma ({\\bf u})$ are some functions.",
"Lemma 2 asserts that such transformations does not exist.",
"Proof: To preserve the continuity of $E_\\gamma ({\\bf u})$ along the transformation line both functions $F_\\gamma ({\\bf u})$ and $G_\\gamma ({\\bf u})$ must be continuous.",
"Arbitrary continuous transformations connecting $F_0({\\bf u})$ with $F_1({\\bf u})$ , and $G_0({\\bf u})$ with $G_1({\\bf u})$ can be expressed in a polynomial form as F(u)=(1-)n F0(u)+k=1n-1k (1-)n-k fk(u) + n F1(u), G(u)=(1-)n G0(u)+k=1n-1k (1-)n-k gk(u) + n G1(u), 2 where $\\lbrace f_n({\\bf u})\\rbrace $ and $\\lbrace g_n({\\bf u})\\rbrace $ are arbitrary sets of functions, and $n$ is the degree of approximating polynomials.",
"To simplify expressions we confine ourselves to the case $F_\\gamma ({\\bf u})=(1-\\gamma ) F_0({\\bf u})+\\gamma F_1({\\bf u}),~ \\nonumber \\\\G_\\gamma ({\\bf u})=(1-\\gamma ) G_0({\\bf u})+\\gamma G_1({\\bf u}).$ We also suggest that $|E_\\gamma ({\\bf u})|^2=1$ for any value of $\\gamma $ .",
"With the simplifications adopted the equation $|E_\\gamma ({\\bf u})|^2=1$ can be written down in the form $|(1-\\gamma )^2 \\, F_0 ({\\bf u}) \\otimes G_0({\\bf u})+\\gamma (1-\\gamma )\\times [F_1 ({\\bf u}) \\otimes G_0({\\bf u})\\nonumber + \\\\+F_0 ({\\bf u}) \\otimes G_1({\\bf u})] +\\gamma ^2 \\, F_1 ({\\bf u}) \\otimes G_1({\\bf u}) |^2=1.~~~~~~~~~~~~~~~~~~~~~~$ Eq.",
"REF reflects the independence of the wavefront amplitude $|E_\\gamma ({\\bf u})|$ from the pupil-plane phase perturbation.",
"This is an algebraic equation (with respect to $\\gamma $ ) of 4-th order with real coefficients.",
"The equation has 2 or 4 real roots only two of which can belong to the interval $(0,1)$ .",
"This means that there are no more than two points $\\gamma \\in (0,1)$ in which $E_\\gamma ({\\bf u})$ can be presented as a convolution of two functions $F_\\gamma ({\\bf u}) \\otimes G_\\gamma ({\\bf u})$ .",
"An increase in the degree $n$ increases the degree of the polynomial (REF ).",
"This in turn increases the number of polynomial roots that can be responsible for the convolution caused solution ambiguity along the transformation line.",
"The set $\\lbrace \\gamma _k\\rbrace _{k=1}^{4n-2}$ of such roots, however, is always countable and cannot cover a continual set $[0,1]$ .",
"In a limit where $n\\rightarrow \\infty $ polynomial (REF ) converts into a function that allows analytical extension into the whole complex plane $\\bf C$ .",
"This is a holomorphic function that can have an infinite number of countable zeros $\\lbrace \\gamma _k\\rbrace _{k=1}^\\infty $ on a real axis.",
"However, only a finite number of these zeros can be located on the segment $[0,1]$ as it follows from the uniqueness of the analytical extension theorem.",
"As a result any continuous pupil-plane phase transformation $E_\\gamma ({\\bf u})$ contains only a finite number of isolated points $\\lbrace \\gamma _k\\rbrace $ where $E_{\\gamma _k} ({\\bf u})=F_{\\gamma _k} ({\\bf u}) \\otimes G_{\\gamma _k} ({\\bf u})$ .",
"The changes that must be included in the proof in the case of transformation line derivative discontinuities are obvious.",
"$\\Box $ Lemmas 1 and 2 make it possible to formulate and prove a uniqueness theorem.",
"Uniqueness theorem: Let $E_\\gamma ({\\bf u})$ be a continuous sequence of pupil-plane wavefronts (parametrized by $\\gamma $ , $\\gamma \\in [0,1]$ ) that produces a continuous sequence of focal-plane intensities $I_\\gamma ({\\bf u})$ .",
"Let $|E_\\gamma ({\\bf u})|$ be constant with the change of $\\gamma $ .",
"Assuming no piston term, let the focal-plane wavefront $e_\\gamma ({\\bf r})$ not be a real function.",
"In this case the problem of restoration of the $E_\\gamma ({\\bf u})$ sequence from the $I_\\gamma ({\\bf u})$ sequence has: (1) infinite number of different non-trivial (no tip/tilt and piston) solutions in the case where $E_\\gamma ({\\bf u})$ can be presented as a product of three factors $E_\\gamma ({\\bf u})=G(u_y) \\, F({u_x}) \\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u}_x)]$ or $E_\\gamma ({\\bf u})=F({u_x}) \\, G(u_y)\\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u}_y)]$ ; (2) four different non-trivial solutions in the case where $E_\\gamma ({\\bf u})$ can be presented as a product of two factors $E_{\\gamma } ({\\bf u})=F_{\\gamma }({u_x}) \\, G_{\\gamma }(u_y)$ and does not allow factorization described in Case 1.",
"(3) two different solutions in all other cases.",
"Proof: All non-trivial multiple wavefront solutions can only be caused by conjugation related wavefront symmetries described earlier as Cases 1–3.",
"It is easy to prove analytically that the same phase perturbation $\\Delta \\Phi _\\gamma ({\\bf u})$ applied to the conjugated wavefronts $E_1({\\bf u})=E_\\gamma ({\\bf u})$ and $E_2({\\bf u})=E_\\gamma ^\\ast ({-\\bf u})$ transforms the corresponding focal-plane images such that $I_{\\Delta \\Phi _\\gamma ,1}({\\bf r})\\ne I_{\\Delta \\Phi _\\gamma ,2}({\\bf r})$ , which is equivalent to the fulfillment of conditions $\\det [L^\\gamma _{jk}]\\ne 0$ for any point $\\gamma \\in [0,1]$ for the matrix $[L^\\gamma _{jk}]=[B^{(E_\\gamma )}_{jk}-B^{(E_\\gamma ^\\ast )}_{jk}]$ (Lemma 1).",
"The condition $\\det [L^\\gamma _{jk}]\\ne 0$ means that the wavefront conjugation at any point $\\gamma $ should be consistent with the conjugation of all other points of the transformation (Corollary 1).",
"The actual solution at any transformation point can be distinguished between conjugated local (related to any single point $\\gamma $ ) solutions if the conjugation of the actual solution is known in at least one point of the transformation.",
"If the conjugation of the actual solution is unknown, the conjugation \"sign\" can be arbitrarily chosen at some point of the transformation that sets conjugation of solutions at all other points.",
"As a result the total number of possible wavefront solutions that provide the observed continuous transformation of the focal-plane intensity is equal to the minimal number of conjugated local wavefront solutions.",
"Lemma 1 can be applied to all points where $\\det [L^\\gamma _{jk}]\\ne 0$ .",
"At points where $\\det [L^\\gamma _{jk}]= 0$ , a unique continuous extension of the solution becomes impossible.",
"In the case where variables are not separable, $\\det [L^\\gamma _{jk}]$ is equal to 0 only if $E_{\\gamma } ({\\bf u})=E_{\\gamma }^\\ast ({-\\bf u})$ which is equivalent to the constraint $e_\\gamma ({\\bf r})$ not be a real function.",
"We now consider each of the three theorem cases.",
"1.",
"In the case where at any point $E_\\gamma ({\\bf u})=G(u_y) \\, F({u_x}) \\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u}_x)]$ the problem can be reduced to two one-dimensional problems, namely, the restoration of the function $F({u_x}) \\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u}_x)]$ from the modulus of the corresponding Fourier transform $|f_\\gamma ({x})|$ , and the restoration of the function $G({u_y})$ from the modulus of the corresponding Fourier transform $|g_({y})|$ .",
"Note that in this case $F({u_x}) \\exp [i \\Delta \\Phi _{\\gamma } ({\\bf u}_x)]$ and $|f_\\gamma ({x})|$ are continuously changing while $G(u_y)$ and $|g_({y})|$ do not change during the wavefront transformation.",
"The follows from the third case of the theorem the first problem has exactly two solutions.",
"The second problem has infinite number of solutions [8], [37] which determines the total number of solutions in the first case of the theorem.",
"2.",
"In contrast to the first case, in the second case $E_{\\gamma } ({\\bf u})=F_{\\gamma }({u_x}) \\, G_{\\gamma }(u_y)$ , and both $F_{\\gamma }({u_x})$ and $G_{\\gamma }({u_x})$ are continuously changing along the transformation line.",
"In accordance with the Case 3 description it means that this case has exactly 4 solutions.",
"They are $F_{\\gamma }({u_x}) \\, G_{\\gamma }(u_y), \\, F_{\\gamma }({u_x}) \\, G_{\\gamma }^\\ast (-u_y), F_{\\gamma }^\\ast ({-u_x}) \\, G_{\\gamma }(u_y) $ , and $F_{\\gamma }^\\ast ({-u_x}) \\, G_{\\gamma }^\\ast (-u_y)$ .",
"As it was discussed above if the actual conjugation is known at an arbitrary point $\\gamma _0$ of the phase transformation this conjugation can be expanded to all other points of the phase transformation.",
"3.",
"There are two different reasons that are responsible for multiple solutions in the third case.",
"The first one is the “convolution” symmetry discussed in detail in the Case 3 description and the Lemma 2.",
"To retain such multiple solutions along the transformation line the “convolution” symmetry should be preserved at every point of the transformation.",
"However, this is impossible in according to Lemma 2.",
"Thus, only the conjugation symmetry of the wavefront causes multiple solutions in this case which mean existence only two solutions, namely, $E_{\\gamma } ({\\bf u})$ and $E_{\\gamma }^\\ast ({-\\bf u})$ .",
"$\\Box $ Note, that from a practical point of view the first and second theorem cases do not really matter.",
"Even a small change of the pupil aperture or non-separable pupil-plane wavefronts statistics which is common resolve the separability issue.",
"Thus, we will assume that the problem always has only two conjugated wavefront solutions $E_{\\gamma } ({\\bf u})$ and $E_{\\gamma }^\\ast ({-\\bf u})$ .",
"As discussed above the conjugation sign of the solution can be established if it is known at an arbitrary point $\\gamma _0$ .",
"This fact allows determining the conjugation state of the measured wavefront by using either the asymmetry of the pupil aperture or introducing in the system an arbitrary defocus of known sign.",
"The image defocus not only provides the unique solution, but also improves the global convergence of the wavefront reconstruction algorithm, as discussed in Section REF .",
"Further, we consider a pair of conjugated wavefront solutions as one solution, meaning that their conjugation state can be recognized if necessary.",
"Figure: Restoration of a correlated sequence of random atmospheric disturbances(Kolmogorov turbulence, D/r 0 D/r_0=30, σ ΔΦ \\sigma _{\\Delta \\Phi }=λ/46\\lambda /46, r I r_I=0.99).The image defocus parameter aa is λ/20\\lambda /20.The actual and restored pupil-plane phases are shown together with the measured and corrected focal-plane intensities obtained after 1, 2, 21, 101 and 121 iterations.Figure: The algorithm convergence speed as a function of focal plane image defocus (Kolmogorov turbulence, D/r 0 D/r_0=30, σ ΔΦ \\sigma _{\\Delta \\Phi }=λ/46\\lambda /46, r I r_I=0.99).Figure: Simulations of the algorithm convergence.",
"The Strehl ratio StSt as function of the iteration number is shownfor r I r_I equal to 0.91 and 0.99 (top),and 0.94 (bottom) (Kolmogorov turbulence,D/r 0 D/r_0=30, a=λ/20a=\\lambda /20).Figure: The accuracy of the pupil-plane phase restoration.",
"The median value of Strehl ratios St m St_m as function of the correlation coefficient r I r_I and the differential phase rms σ ΔΦ \\sigma _{\\Delta \\Phi } is shown(Kolmogorov turbulence, D/r 0 D/r_0=30, a=λ/20a=\\lambda /20)."
],
[
"Global convergence",
"Existence of a unique wavefront solution does not mean convergence of the modified Gerchberg-Saxton algorithm to this solution.",
"To provide global convergence of the algorithm, the following two conditions must be met: Condition 1: Successive focal-plane images must demonstrate sufficiently high correlation.",
"Rationale: Local convergence provided by the Gerchberg-Saxton algorithm must be preserved during the continuous phase transformation from the proposed modification.",
"Hence, for any continuous pupil-plane phase change an iterative solution is always captured by the nearest local minimum or stagnation point [19].",
"The local minima “profile” is being continuously deformed by the corresponding pupil-plane phase changes.",
"Pupil-plane phase changes of sufficient magnitude can switch the iterative solution between different minima not connected by a continuous deformation.",
"As a result, the algorithm may not necessarily be locally converging.",
"However, in the case of sufficiently small sequential phase changes corresponding to the highly correlated successive focal-plane images, the captured iterative solution remains under the influence of a particular local minimum as long as needed for convergence.",
"Furthermore, the iterative solution remains trapped by the minimum while this minimum exists.",
"To satisfy the local convergence condition, sequential focal-plane images should have negligibly small differences corresponding to high correlation between sequential pupil-plane phases.",
"Condition 2: In a sequence of focal-plane images, there must exist sets of completely uncorrelated images.",
"Rationale: Local minima preclude convergence of the iterative solution to the global minimum.",
"To provide global convergence, an efficient method to escape the vicinity of the local minima must exist.",
"It should be taken into account that ambiguous multiple solutions produce identical focal-plane images.",
"Hence, in accordance with Lemma 1, in a small area where an ambiguous solution creates a local minimum, differences in morphology of corresponding focal-plane images produced by the global and local solutions should be negligibly small.",
"However, the discrepancy between these solutions is growing as the pupil plane changes.",
"At some moment, focal-plane intensities related to the global and local solutions become completely incompatible.",
"Thus, the iterative solution can no longer be trapped by the previous local minimum and the iterative solution will enter the vicinity of another minimum (either global or local).",
"The destruction of local minima is naturally associated with significant change in the morphology of focal-plane images.",
"In a case where the temporal pupil-plane phase variations are strong enough to produce significant change in the morphology of focal-plane images only two outcomes are possible: (a) convergence to the global solution (the global minimum can be never destroyed) or (b) no convergence.",
"Numerical simulations and lab experiments related to the algorithm convergence problem (meaning global convergence) are discussed in Section .",
"We only note here that there is a modification of the method where such convergence is always observed even in the presence of strong photon noise and background noise as described in Sections REF and REF .",
"Finally, the proposed approach can be used to recover not only a dynamic pupil-plane wavefront, but can be also generalized for the case of static pupil-plane aberrations.",
"In many cases correlated random pupil-plane phase aberrations can be created as probes, for example, with a DM.",
"The Gerchberg-Saxton loop can be used to recover these probes combined with the static aberrations of the optical system.",
"As soon as the iterative algorithm has found the global solution, gradual reduction of the amplitude of random DM probes forces the iterative solution to converge to the static aberration of the optical system.",
"Figure: Restoration of a correlated sequence of random atmospheric disturbances(Kolmogorov turbulence, D/r 0 D/r_0=30, r I r_I=0.99, a=λ/20a=\\lambda /20) in presence of photon noise.The actual pupil-plane phase and its unwrapped estimate after 105 iterations are shown together with themeasured photon limited focal-plane image (n ph =2n_{ph}=2 photons per speckle) and the corrected PSF (StSt=0.526).",
"The selected part of the photon limited image is shown with full resolution in the image upper left corner."
],
[
"Numerical simulations",
"Conditions 1 and 2 discussed in the previous section are necessary but insufficient for algorithm convergence.",
"For guaranteed convergence additional modifications of the method should be considered.",
"From a practical point of view, it is also important to estimate the algorithm convergence speed and its noise sensitivity in cases where such convergence is provided.",
"Numerical simulations described in this section address these questions.",
"Although the statistics of the pupil-plane phase aberrations does not affect the solution uniqueness, it can influence algorithm convergence.",
"Hence, we limited our simulations to two cases of practical importance.",
"The first case (Model I) is the case of large dynamically changing phase aberrations caused by atmospheric turbulence.",
"We utilized an atmosphere model with spatial phase variations satisfying Kolmogorov statistics and with temporal variations described by a “boiling” turbulence model [38].",
"In accordance with this model, temporal variations are set by the inhomogeneities disintegration time $T_L$ that depends on the inhomogeneities size (scale) $L$ as $T_L\\propto L^{2/3}$ .",
"The chosen turbulence strength ($D/r_0$ =30, $D$ is the entrance pupil size, and $r_0$ is Fried parameter) creates atmospheric wavefront disturbances typical for a 3 m telescope under 1\" seeing.",
"The temporal sampling for Model I is chosen such that the disintegration time of the smallest spatial inhomogeneities is equal to the temporal sampling step (i.e., the smallest inhomogeneities are completely uncorrelated for sequential wavefronts).",
"The selected parameters provide sequential focal-plane image correlation of 0.4-0.5 – consistent with cross-correlation measurements of speckle-interferometric images at exposures comparable with the atmospheric coherence time [39], [40].",
"The second case (Model II) is the case of a moderate static pupil-plane aberration $\\Phi _{stat}({\\bf u})$ .",
"In this case, a sequence of uncorrelated phase screens with the spatial autocorrelation function $\\propto \\exp (-u/2\\sigma ^2_s)$ is used as random phase probes $\\Phi _k({\\bf u})$ ($k=1, 2, 3\\dots $ ) that provide continuous transformation of the pupil-plane wavefront $\\exp (i[\\Phi _{stat}({\\bf u})+\\Phi _k({\\bf u})])$ .",
"In our simulations, we chose the spatial correlation radius $\\sigma _s$ equal to $D/12$ and the phase rms value of $\\lambda /5$ .",
"Unfortunately, neither Model I nor Model II data sets satisfy the local convergence condition, so a linear wavefront interpolation procedure is used to satisfy it.",
"In this procedure, pupil-plane phases $\\Phi _{i*N}$ with sequence numbers $i*N$ are generated according to Model I or II.",
"We refer to these pupil-plane phases as “nodal probes”.",
"The gaps between the nodal probes are filled with phases $\\Phi _{k+i*N}=k*\\Phi _{i*N}+(N-k)*\\Phi _{(i+1)*N}/N$ where $i, k, N$ are natural numbers, and $k<N$ .",
"The correlation between two sequential pupil-plane phases can be arbitrarily set by choice of $N$ , satisfying Condition 1.",
"In our numerical simulations $N$ is set to 32 for Model I and 100 for Model II, which is enough to provide global convergence though does not guarantee optimal algorithm convergence speed.",
"To describe correlation between sequential focal-plane images and their corresponding pupil-plane wavefronts we use two parameters.",
"One is the sequential focal-plane intensities correlation coefficient $r_I$ and the second is the sequential pupil-plane phase difference rms $\\sigma _{\\Delta \\Phi }$ .",
"After each iteration, the aberrated wavefront can be corrected by applying the new wavefront estimate.",
"The optical quality of the uncorrected and corrected point spread functions (PSF) can be characterized by their corresponding Strehl ratios $St_I$ and $St$ .",
"To avoid wavefront comparison difficulties caused by wavefront unwrapping we use these parameters to evaluate algorithm performance.",
"In some cases another evaluation metric we use is the median value of the Strehl ratios $St_m$ calculated for a sequence of corrected PSFs.",
"The flat pupil-plane phase is used as an initial pupil-plane phase guess in all simulations.",
"Figure: The accuracy of the pupil-plane phase restoration in the presence of photon noise.",
"The median value of Strehl ratios St m St_m as function of number of photons per speckle n ph n_{ph} is shown (Kolmogorov turbulence,D/r 0 D/r_0=30, r I r_I=0.99, a=λ/20a=\\lambda /20).Figure: Restoration of a correlated sequence of random atmospheric disturbances(Kolmogorov turbulence, D/r 0 D/r_0=30, σ ΔΦ =λ/11\\sigma _{\\Delta \\Phi }=\\lambda /11, r I =0.86r_I= 0.86, a=λ/20a=\\lambda /20) with r I r_I that does not provide algorithm convergence in the noiseless case.",
"The restoration result after 130 iterations in the presence of photon noise (n ph =2n_{ph}=2 photons/speckle, left) and the best restoration result in a noiseless run (300 iterations, right) are shown.",
"All restored phases are unwrapped."
],
[
"Algorithm convergence",
"In the case of large phase aberrations (similar to Model I) it is difficult to achieve algorithm convergence.",
"The absence of convergence can be explained by the wavefront degeneracy presented in the focal-plane where the convergence of the optical beam is replaced by the beam divergence.",
"Each local or global solution forms an area in which iterative approximations are attracted by the solution.",
"Due to the degeneracy, in a small region around the focus attraction areas formed by conjugated wavefront solutions $E({\\bf u},t)$ and $E^\\ast ({-\\bf u},t)$ overlap each other.",
"The overlapping causes the appearance of saddle points in the vicinity of which algorithm convergence slows down.",
"In the neighborhood of a saddle point the convergence speed is determined by the conditionality of the matrix $B_{jk}$ defined in Section .",
"The matrix conditionality rapidly decreases with increasing degrees of freedom ($\\propto 2^K/K!$ ).",
"The number of free parameters $K$ describing a wavefront is a quantity of order of number of bright speckles observed in the PSF.",
"This number is directly related to the pupil-plane phase rms.",
"As aberrations become stronger the convergence rate can slow down and quickly enter into stagnation, resulting in algorithm convergence failure.",
"To solve the convergence problem a combination of the Gerchberg-Saxton modification with the defocus based phase diversity is used.",
"In this procedure each iteration includes three successive Gerchberg-Saxton steps which use different focal-plane images.",
"These three images are differently defocused by adding a small static defocus term $\\Phi _{f}({\\bf u})=a (8*u_x^2+8*u_y^2-D^2)/D^2$ to the pupil plane phase, where $a$ is a defocus parameter.",
"The first the image is in the focus.",
"Two other images have the defocus identical in modulus but opposite in sign.",
"Sufficient defocus $a$ completely removes ambiguity caused by conjugated the solutions $E({\\bf u},t)$ and $E^\\ast ({-\\bf u},t)$ and recovers fast convergence in the vicinity of conjugated global solutions.",
"One example of such reconstruction performed for a Model I case is presented in (Fig.",
"REF ).",
"The example shows that a defocus as small as $\\lambda /20$ is sufficient to ensure global convergence (including correct conjugation) after only 121 iterations, though without the defocus based phase diversity, convergence cannot be attained.",
"For smaller phase aberrations the iterative process converges even without the phase diversity use.",
"However, in this case convergence speed is slow and it is impossible to predict which of two conjugated solutions will be reached.",
"As for the case of large aberrations the convergence rate for small aberrations improves with suitable image defocus diversity.",
"For example, the global solution for Model I with $D/r_0=10$ can be obtained without image defocus after 800-1000 Gerchberg-Saxton iterations.",
"The same solution is achieved in just 40-60 iterations if the image defocus diversity is used.",
"Fig.",
"REF shows the dependence of the algorithm convergence speed on the defocus parameter for the case of large phase aberrations (Model I).",
"The number of iterations needed for algorithm convergence is determined as the number of the first iteration for which $St>St_I$ and $St>St_m$ .",
"Global convergence of the algorithm is achieved as soon as the focal-plane degeneracy is removed by a sufficiently large defocus diversity.",
"The algorithm convergence is associated with sequential correlation of focal-plane images used for phase restoration.",
"In Fig.",
"REF the evolution of the phase retrieval quality $St$ is presented for three different cases simulated for Model I with the defocus parameter $a=\\lambda /20$ .",
"In the first case $r_I=0.91$ and the algorithm does not demonstrate convergence because local convergence is broken by low sequential correlation.",
"In the case with $r_I=0.99$ the iterative solution is easily captured by the global minimum and remains there for an indefinitely large number of iterations.",
"In an intermediate case with $r_I=0.94$ there is a high chance for the iterative solution to find the global solution.",
"However, due to continuity issues caused by insufficient sequential correlation the global solution cannot hold the iterative solution for a large number of iterations.",
"Sequential correlation also affects retrieval accuracy.",
"The average quality of the phase retrieval $St_m$ depends on the correlation coefficient $r_I$ as shown in Fig.",
"REF .",
"The algorithm does not converge until the correlation coefficient $r_I$ is less than 0.9 ($\\sigma _{\\Delta \\Phi }>\\lambda /15$ ).",
"The algorithm converges only (meaning that it forms the Airy-like focal-plane PSFs) if $r_I$ is larger than 0.94 ($\\sigma _{\\Delta \\Phi }<\\lambda /18$ ).",
"Once $r_I$ reaches 0.97 ($\\sigma _{\\Delta \\Phi }=\\lambda /25$ ) the Strehl ratio becomes larger than 0.95 for more than 50% of the corrected focal-plane images.",
"For $\\sigma _{\\Delta \\Phi }=\\lambda /46$ more than 90% of the corrected images have $St$ larger than 0.95, though 5% of them still show $St$ as low as 0.2 being, probably, affected by close local solutions (Fig.",
"REF , case with $r_I=0.99$ ).",
"The algorithm convergence is restored immediately after destruction of local minima associated with these ambiguous solutions.",
"The image defocus based phase diversity and high sequential image correlation provide fast algorithm convergence to the global solution.",
"In our simulations the global convergence is always observed in a few $N$ iteration cycles where $N$ is the number of correlated pupil-plane phase probes that fill the gap between two consecutive nodal probes."
],
[
"Photon noise",
"Simulations have shown good performance in the noiseless cases considered above, however, in the presence of photon noise algorithm performance is expected to degrade in terms of algorithm convergence.",
"A surprising result from simulated case studies in which photon noise is included is that initial algorithm convergence speed actually improves in low flux cases.",
"This convergence speed improvement comes at the cost of estimation accuracy as measured by an ultimately lower Strehl ratio.",
"A simulated case shown in Figure REF with photon flux as low as 2 photons per speckle (corresponding to approximately 5000 photons per frame) demonstrates convergence with a retrieved Strehl ratio $St$ of about $0.5$ after only 105 iterations.",
"This convergence occurs from an initial wavefront with rms on the order of $\\lambda $ and a corresponding $St$ of approximately $0.002$ .",
"To study the convergence of the wavefront retrieval method, the results of a set of simulated cases obtained for atmospheric wavefront disturbances (Model I) with sequential correlation $r_I=0.99$ are shown as a function of applied photon noise in Figure REF .",
"$St$ strongly depends on the number of photons per speckle.",
"Wavefront restoration accuracy improves with increasing flux, as expected, with a recovered Strehl ratio of $0.99$ for high photon flux levels.",
"Operation under low light flux levels led to two unexpected benefits.",
"One benefit is that for flux levels as low as 1-2 photons per speckle the modified Gerchberg-Saxton algorithm restores a moderate Strehl ratio $St$ even in cases where the sequential correlation is relatively low.",
"This can provide algorithm convergence outperforming the “no photon noise” case.",
"An example of such wavefront restoration is presented in Fig.",
"REF , where low flux has provided the algorithm convergence for as low $r_I$ value as $0.86$ after 130 iterations while a the “no photon noise” case shows no convergence.",
"A second benefit is apparent from the $St$ dependence on the number of iterations in Fig.",
"REF .",
"Algorithm convergence speed in the low photon rate mode is faster by a factor of 2-3 in comparison with “no photon noise” case.",
"This faster convergence can be explained as follows.",
"Assuming that a long exposure PSF can be roughly expressed as $\\sim \\exp [-r^2/2(\\lambda /r_0)^2]$ and the sampling interval $\\Delta x=\\lambda /D$ , number of sampling points $N_p$ needed for adequate pupil-plane phase description can be estimated as $(3D/r_0)^2\\pi /4$ .",
"At the same time, the number of speckles in the focal plane $N_s=2.299(D/r_0)^2$ [41] is about 3 times less than $N_p$ .",
"At a photon rate of 1 photon per speckle, with high probability the photons will be detected in the centers of the brightest speckles.",
"The pupil-plane wavefront can be roughly presented as a superposition of spherical wavefronts whose centers coincide with the centers of detected photons but with relative phases unknown.",
"The number of unknown phases is equal to the number of detected photons and is approximately equal to $N_s$ .",
"As a result the photon limited phase retrieval problem has about 3 times fewer unknowns in comparison with the “no photon noise” case.",
"The reduced number of degrees of freedom simplifies the related optimization problem improving algorithm convergence but produces less accurate solutions.",
"Taking into account the observed convergence acceleration for the low photon rate case, a “photonization” algorithm modification can be formulated to improve convergence speed in the “no photon noise” case.",
"This modification works as follows: the phase retrieval procedure applies with low-photon rate images before switching to the “no photon noise” case in subsequent iterations.",
"This procedure can be implemented by programmatic “photonization” of focal-plane images with a photon noise generator.",
"An example of this “photonization” procedure is presented in Fig.",
"REF where the algorithm with the “photonization” procedure converges to the global solution 3 times faster than the “no photon noise” case running on the same data set.",
"Another phenomenon apparent in Fig.",
"REF is the periodic dipping in the $St$ curve with increasing iteration number.",
"These dips are caused by the periodicity in the interpolation procedure used to generate correlated phase probes between uncorrelated nodal probes.",
"Thus, every N iterations ($N=32$ for Model I) one of the old nodal probes is replaced with a new probe.",
"Simultaneously, the local minima associated with the replaced probe are destroyed (as discussed in Section ) resulting in the observed periodic phenomenon as the iterative solution is released from its previous local minimum trap and continues convergence to the global minimum.",
"There are moments when the iterative solution diverges to the global solution due to convergence to a new local minimum or due to slow convergence that does not allow the iterative solution to keep pace with the changing probes.",
"The following nodal probe change starts the cycle again both releasing the new local minimum trap and adjusting convergence speed.",
"Thus, the observed periodicity demonstrates the process of rejection of local solutions.",
"The periodical divergence can be deeper in the presence of photon noise which creates expected inconsistency between the global and iterative solutions.",
"Figure: Comparison of algorithm convergence for restoration in both the absence and presence of photon noise.",
"The pupil-plane phase restoration quality StSt as function of the iteration number is shown for the “photon noise” case(n ph ≈2n_{ph}\\approx 2) and for the noiseless case running on the same data set that model Kolmogorov turbulence (D/r 0 D/r_0=30, r I r_I=0.99, a=λ/20a=\\lambda /20).",
"The “photon noise” related convergence curve is a result of data averaging obtained in 9 independent simulation runs.Figure: The algorithm acceleration with artificial “photonization” procedure.",
"Pupil-plane phase restoration results for runs with artificial “photonization” (top, after 15 iterations, in first 10 iterations n ph =n_{ph}= 2 photons/speckle) and without artificial “photonization”(bottom, after 15 and 44 iterations) are shown.",
"In both cases the same correlated sequence of random atmospheric disturbances(Kolmogorov turbulence, D/r 0 D/r_0=10, r I r_I=0.99, a=λ/20a=\\lambda /20) is used.",
"All restored phases are unwrapped."
],
[
"Phase retrieval with DM probes",
"In previous sections we have shown how the Gerchberg-Saxton approach can be used to restore random, dynamically changing pupil-plane phases that satisfy Kolomogorov statistics.",
"We will now show how the same approach works for different pupil-plane phase statistics described earlier as Model II.",
"The key requirement that provides algorithm convergence to the global solution is maintaining high sequential wavefront correlation.",
"Thus, for random pupil-plane phases with rms asymptotically approaching 0, the Gerchberg-Saxton algorithm should also converge to 0 phase or equivalently to the static aberration of the optical system.",
"To simplify discussion we will not apply the defocus based phase diversity and “photonization” procedure improvements discussed above.",
"Note that without these procedures, convergence speed of the algorithm is reduced and two conjugated phase solutions $\\Phi ({\\bf u})$ and $-\\Phi ({-\\bf u})$ may appear that will be transformed to the same orientation and polarity for comparison.",
"A numerical simulation case demonstrating restoration of a static pupil-plane wavefront is presented in Fig.",
"REF .",
"The first sample of the Model II sequence uncorrelated with remaining samples plays the role of a static aberration $\\Phi _{stat}({\\bf u})$ of the optical system.",
"This static aberration is subsequently restored using a sequence of nodal probes and correlated Model II samples.",
"The amplitude of phase probes with rms is about $\\lambda /5$ was not changing during the initial 500 iterations.",
"After catching the global solution the amplitude of phase probes was reduced between successive iterations until it became negligible.",
"In 100 iterations the amplitude of the phase probes was reduced from an rms level $\\lambda /5$ to $\\lambda /5000$ , which is below the phase restoration error of $\\lambda /150$ .",
"The achieved phase retrieval accuracy is appropriate for most optical applications, and the approach can be used to determine static aberrations for any optical system where random, correlated probe phases can be created with a DM, for example .",
"Unlike other phase probing methods, our solution does not rely on knowing what those probes are, so an accurate calibration of either the DM or the optical system is not a requirement.",
"Additionally, geometrical calibrations needed to match pupil-plane and focal-plane wavefronts can be extracted from focal-plane intensity measurements and the procedure can be used as a calibration tool for other applications [5], [6], [42].",
"In addition to pupil-plane phase retrieval, we note that the random phase probe approach can also be used to restore pupil-plane amplitudes.",
"An in-depth discussion of this application is, however, outside the scope of the current paper.",
"Figure: Restoration of a static optical aberration.",
"The actual aberration (left), the restored pupil-plane phase after reaching the global solution (center) and the final pupil-plane phase solution (right) are shown.",
"It took 500 iterations to find the global phase solution and 100 iterations more to estimate the actual pupil-plane aberration with rms of about λ/150\\lambda /150.Figure: Additive 1% background noise in a simulated focal-plane image before (left) and after image filtering (right).",
"The noise amplitude is measured relative to the average intensity of the brightest image speckles.",
"This image also gives an estimate of the magnitude of aberrations needed to ensure the global algorithm convergence in the initialstage of our laboratory demonstration (Section ).Figure: The additive background noise averaging in the pupil-plane phase reconstruction procedure.",
"The actual (left) and restored pupil-plane phases without (center) and with (right) background noise averaging are shown for two different cases: the phase retrieval without (top) and with focal-plane image filtering (bottom)."
],
[
"Background noise",
"Wavefront retrieval can also be affected by background image noise such as focal-plane detector read noise.",
"An example of focal-plane image with additive $1\\%$ rms background noise is shown in the left panel of Fig.",
"REF .",
"Results of wavefront retrieval simulations for a static pupil-plane aberrations with $1\\%$ background noise are presented in Fig.",
"REF .",
"The background noise results in slower convergence compared with the “no noise” case and also causes white noise that degrades the pupil-plane phase solution as clearly seen in the top, center panel of Fig.",
"REF .",
"An averaging procedure can be formulated to improve the accuracy of the phase estimate and mitigate the induced white noise: Figure: The experiment layout.$E({\\bf u})=|E({\\bf u})|\\exp [i\\Phi ({\\bf u})]=\\langle E_i({\\bf u})\\rangle .$ In Eq.",
"REF the final pupil-plane phase estimate $\\Phi ({\\bf u})=\\arg [\\langle E_i({\\bf u})\\rangle ]$ , and $\\langle ...\\rangle $ represent averaged sequential wavefront estimates $E_i({\\bf u})$ obtained after the wavefront retrieval procedure has converged to the static optical aberration.",
"Although the averaging procedure can improve the final pupil-plane wavefront estimate, it requires simple image filtering to be applied to every image used in reconstruction (Fig.",
"REF ).",
"Each image should be low-pass filtered in the frequency domain by removing all frequencies beyond the pupil cut-off frequency as shown in the right panel of Fig.",
"REF .",
"This simple filtering procedure shows significant improvements in wavefront reconstruction quality in the presence of background noise with Strehl ratio increasing from 0.68 (without filtering) to 0.91 (after filtering).",
"When the averaging procedure is also applied to the filtered reconstruction, a final Strehl ratio of 0.998 is obtained after averaging of 1000 wavefront estimates."
],
[
"Experimental results",
"The main results of the numerical simulations have been confirmed in a set of laboratory experiments performed at the NASA Ames Coronagraph Testbed [43].",
"During these experiments the phase in the entrance pupil of the PIAA (Phase Induced Amplitude Apodization) coronagraph [44] was restored using a set of random phase probes created by the DM located in the entrance pupil."
],
[
"Experiment description",
"Except for a few minor differences the PIAA coronagraph setup at NASA Ames is similar to the setup used in our EXCEDE (EXoplanetary Circumstellar Environments and Disk Explorer) demonstration [45].The front-end of this setup that feeds into the PIAA coronagraph was used in the wavefront retrieval experiments.",
"The experiment optical layout is shown in Fig.",
"REF .",
"The spherical wave formed by the point source S1 (655nm laser) passes through the diffractive pupil (DP) [46] and is focused in the front-end focus F1 at a distance of 531 mm from the diffractive pupil.",
"The distance between the point source and the diffractive pupil is 473 mm.",
"The diffractive pupil is a spherical mirror with curvature radius of 500 mm and features a low frequency diffractive grating on its surface.",
"The beam is collimated farther by an off-axis parabolic mirror OAP1, and is reflected by the fold mirror M1 and the deformable mirror DM.",
"Finally, the collimated beam is focused by the second off-axis parabola OAP2 in the first focus of the PIAA coronagraph.",
"The focal lengths of OAP1 and OAP2 are 305 mm and 127 mm respectively.",
"The diffractive pupil is optically conjugated with the DM and both are optically conjugated with the first PIAA mirror.",
"A circular pupil stop PA with the diameter of 9.3 mm is used to restrict the beam size.",
"The pupil stop is located upstream of the DM at the minimal distance that prevents beam vignetting.",
"The focal-plane images are focused on the Basler acA3800-14um camera with a pixel size of 1.67 $\\mu $ m that is small enough to provide an appropriate image sampling (the system $\\lambda f/D$ =9.2 $\\mu $ m).",
"The camera was operated in video mode with 14 twelve-bit frames/sec.",
"After averaging 100 frames, each 600$\\times $ 600 raw focal-plane image was clipped down to 512$\\times $ 512 before usage in the wavefront retrieval procedure.",
"Frame averaging was needed to reduce camera background noise and increase dynamic range.",
"Our experiments used a Boston Micromachines MEMS (Micro-Electro-Mechanical) DM featuring $32 \\times 32$ actuators.",
"Image sampling set a spatial resolution of 86 pixels across the pupil ($\\approx $ 2.5 pixels/DM actuator) as measured from the image power spectrum.",
"The fiducial pattern applied to the DM was the flat DM surface with three actuators poked in the same “positive” direction and one actuator poked in the opposite direction.",
"The related DM voltage map [47] and focal-plane image are presented in Fig.",
"REF .",
"The maximal amplitude of poked actuators reaches 140 nm resulting in no more than 0.02 degradation in Strehl ratio (in comparison with the flat DM case).",
"The wavefront retrieval algorithm was run as follows.",
"First, a set of independent nodal probes was generated similar to Model II with a correlation radius equal to 12 DM inter-actuator distance.",
"The corresponding DM voltages were calculated based on a quadratic model of the DM deflection curve with amplitude normalization chosen experimentally to be sufficiently strong such that the main lobe of the focal-plane image (Fig.",
"REF ) was destroyed (Fig.",
"REF ).",
"Second, the iterative retrieval algorithm was run with $N$ partially correlated phase probes between the nodal probes.",
"For the first 600 iterations, $N$ was set to 200 and the amplitude of probes was fixed until convergence to the global phase solution.",
"Subsequently, the phase amplitude of the nodal probes was exponentially reduced with the successive amplitude ratio of 0.9 until the probes became negligible after 2000 iterations (in total).",
"To speed-up convergence during this step, $N$ was gradually reduced to 100, 70, and 50.",
"Third, the final phase solution was obtained by applying the averaging procedure to 1000 successive wavefront estimates obtained with negligible phase probes."
],
[
"Phase retrieval results",
"The restored pupil-plane phase is presented in Fig.",
"REF .",
"Two separate wavefront retrieval run results are shown, one using a flat DM and a second using the fiducial DM pattern.",
"Since the DM is conjugate to the entrance pupil, DM feature details can be clearly seen in the estimated phase results.",
"First, the poked actuators are recognizable in locations and with polarities matching the applied fiducial pattern.",
"Recovered actuator poke amplitudes match the commanded signal (see discussion in Section REF ).",
"Ignoring small tip/tilt and defocus terms, the retrieved phase map appears flat except across the right edge of the pupil where a phase growth of $0.7 \\lambda $ is observed and two (known a priori) malfunctioning actuators marked as “5” and “6”.",
"There are also a few DM areas with weak phase depression (in relation to the flat DM map) likely caused by DM flattening errors or aberrations produced by other optical elements.",
"Finally, there is a poorly recognizable grid-like structure that is likely caused by the quilting pattern formed by DM actuators [48].",
"All the above features, except the DM quilting pattern, are highly repeatable and consistent between different wavefront retrieval runs.",
"The periodic DM quilting pattern cannot be firmly recognized because the spatial resolution of the retrieved pupil-plane phase is insufficient to resolve it.",
"The phase map resolution across the pupil plane is limited by two main factors: (1) sampling and (2) dynamic range of detected focal-plane images.",
"To allow restoration of the amplitude and location of the DM grating the camera dynamic range should be sufficient to linearly detect weak focal-plane speckles with contrast of a few times $10^{-5}$ near the diffractive peaks produced by the DM grating (contrast of $10^{-3}$ ).",
"These speckles interfere with the diffractive orders and are the most sensitive to the spatial shift of the grating.",
"To expand the dynamic range a high dynamic imaging tool was implemented that allows to combine a set of images taken with different exposure times in one image with dynamic range spanning 5 orders of magnitude.",
"We also doubled pupil-plane spatial resolution by increasing the size of collected images to 1200$\\times $ 1200 and clipping them down to 1024$\\times $ 1024.",
"Figure: The laboratory experiment results.",
"The restored pupil-plane phase is shown togetherwith a focal-plane image and the DM applied voltage.",
"The fiducial poked actuators (1, 2, 3, and 4)and malfunctioning DM actuators (5 and 6) are marked.",
"Actuators 1, 2 and 3 have positive phase while the malfunctioning actuators and actuator 4 have negative phase.The restored flat DM phase (unaveraged) and the difference between thepoked actuators phase map and flat DM phase are also shown.Figure: The pupil-plane phase restoration with implemented high dynamic range (HDR) imaging procedure.",
"The restored phase maps obtained in two independentphase retrieval runs are shown.",
"The phase difference between these two maps without lateral shift and with1 pixel of horizontal and 3 pixels of verticalrelative shift are also shown.The direct interferometric measurement of the flat DMsurface is presented for comparison .The following DM produced features are marked:poked actuators (1, 2, 3 and 4), malfunctioning actuators (5 and 6)the DM grid related structure (a),vertical structure associated with OAPs diamond turning errors, the DM boundaryand the pupil aperture.Better sampling and imaging dynamic range improve the resolution of the phase solution.",
"Two independent pupil-plane phase estimates with increased resolution of 5 pixels/actuator (172 pixels across the pupil) are shown in Fig.",
"REF .",
"We analyze these phase maps for consistency in Section REF ."
],
[
"Phase retrieval reliability",
"In comparison with the 2.5 pixels/actuator resolution map, the DM surface grating with amplitude of about $\\lambda /30$ nm ($\\sim $ 20 nm) is clearly visible in higher resolution maps.",
"However, a more detailed analysis indicates some systematic differences in the restored phases.",
"Following is an analysis of the observed differences: (1) Low order modes of the two estimated phase maps are practically identical, except for tip/tilt differences between different trials.",
"Since we did not set a certain focal-plane image position in the frame during the phase restoration, the observed tip/tilt simply reflects the changes of the source location caused by the long term instability of the optical setup.",
"(2) Fiducial actuator pokes are well-matched both in location and amplitude being completely subtracted in the phase difference map (Fig.",
"REF ).",
"Relative amplitudes of poked actuators (Table REF ) are consistent with amplitude ratios derived from direct DM deflection measurements [47].",
"The absolute poke amplitudes are calculated based on the quadratic DM deflection model with the maximal DM stroke estimated as the average of the known deflection based value and the inter-actuator maximal stroke from [49].",
"(3) A systematic pattern is seen in the phase difference map (Fig.",
"REF ) that consists of two components.",
"The first component is a regular grid-like structure produced by the DM surface grating.",
"Setting a 1 pixel horizontal shift and a 3 pixel vertical shift in one of the phase maps results in the DM grid artifact disappearing when subtracted.",
"This strongly suggests an ambiguity in the spatial location of the grid pattern.",
"(4) The second systematic component appears as irregular, mainly vertical stripes across the pupil.",
"These strips are, probably, introduced by the optics and variability in the stripes position can be explained by slow drift in the optical setup.",
"The source of this structure may be errors in the surface of OAPs produced by the diamond turning process.",
"These diamond turning errors are detectable in reflected light at large incidence angles and are also responsible for the tilted horizontal band visible in the focal-plane PSF (marked with “b” in Fig.",
"REF ).",
"(5) The phase difference rms of $\\lambda /14$ is dominated by the discussed systematic pattern.",
"Table: Amplitude of poked actuators.The first systematic component of the phase error indicates an ambiguity of the phase solution that should be explained.",
"This kind of ambiguity is observed when the focal-plane image can be decomposed in two or more components that either do not interfere with each other or the interference terms are negligible.",
"It happens, for example, in a practically interesting case where one of the components is produced by a multiplicative pupil-plane phase grating.",
"The diffractive orders formed by the grating interfere with background speckles produced by another component.",
"However, in many cases, the brightness of the background speckles is negligible in comparison with the diffractive orders brightness, so the interference term can be considered as equal to zero.",
"As a result, all phase solutions where the phase grating is arbitrarily shifted in the pupil plane produce undistinguished focal-plane intensity distributions meaning that the phase retrieval have an infinite number of ambiguous solutions.",
"The interferometric term, of course, is not equal to zero precisely.",
"However, any random registration noise makes solutions with arbitrary location of the grating in the pupil indistinguishable.",
"The phase ambiguity problem is additionally complicated by factors such as: (a) conjugated phase solutions; (b) small amplitude of DM diffractive peaks.",
"The relative contrast of the DM diffraction peaks is less that $10^{-3}$ .",
"These peaks can reduce the Strehl ratio of the focal-plane PSF no more than by 0.02 which is comparable with Strehl ratio degradation due to the fiducial DM pattern.",
"The local convergence condition discussed in Section is equivalent to the continuity condition for sequential pupil-plane phase probes.",
"Failure to converge with respect to the DM grating position could mean that phase probe changes that successfuly satisfy the continuity condition for the fiducial DM pattern case do not necessarily satisfy the continuity condition with respect to the DM surface grating.",
"The number of degrees of freedom required to describe the DM grating is $\\sim 200$ times larger than needed for a fiducial pattern that produces the same Strehl ratio degradation.",
"Thus, with respect to the DM grating the applied sequential pupil-plane phases are about $\\sim 14$ times less “continuous” than with respect to the fiducial actuators pattern.",
"Consequently, a higher correlation rate between sequential random phase probes and a larger total number of iterations is required to recover phase features responsible for diffractive order formation.",
"(c) Despite the high thermal and mechanical stability of the optical setup [43], slight long-term drift changes relative position of the optical elements.",
"The drift produces beam-walk that results in temporal variations in pupil-plane phase affecting the phase retrieval algorithm.",
"This drift thus causes uncertainty in the position of the observed irregular vertical phase structure (associated with OAP polishing features).",
"(d) Optical beams that form the PSF central lobe and the DM diffractive orders are reflected from different parts of OAP2 which produces non-common path errors.Together with thermal and mechanical temporal instabilities these non-common path errors can affect the parameters of the recovered DM surface grid.",
"(e) Finally, an uncertainty in the size of the pupil can be responsible for slightly different wavefront location across the pupil in different retrieval runs.",
"All these factors contribute in the final phase difference map, but it is challenging to estimate their individual contributions.",
"Though the amplitude of the DM phase grating matches the amplitude measured interferometrically, the relative position of the DM grating remains uncertain.",
"This uncertainty can probably be solved using the previously discussed defocus based diversity procedure."
],
[
"Discussion",
"The proposed random phase probes approach can be used to test any optical system in which random correlated phase aberrations can be produced.",
"Such probes can be created, for example, by a DM, slow air turbulence or convection, temperature gradients in the system, or random relative motion of the different optical elements.",
"Convergence of the algorithm solution to the real system wavefront is expected if the amplitude of the aberration creation process can be gradually decreased to 0.",
"The method can be used for both large (as large as a few wavelength) and small optical aberrations.",
"It is especially promising in the case when optical system performance is limited by non-common path propagation errors.",
"The method can provide accurate, high resolution wavefront sensing not influenced by additional optical elements and does not require well-calibrated camera travel along the focus direction as with most phase diversity methods.",
"A list of possible method applications includes: (1) Coronagraphy.",
"For a complex optical system with active wavefront control, the wavefront at different planes of the optical layout can be directly measured by using the random phase probe approach.",
"As a result the complete, high-accuracy system model can be produced.",
"The direct imaging of exoplanets requires an extraordinary high-contrast capability in the imaging system.",
"Current starlight suppression systems based on a combination of coronagraphy and wavefront control are able to reach the $10^{-9}$ –$10^{-10}$ broadband imaging contrast needed for direct exoplanet studies.",
"State of the art wavefront control algorithms, such as EFC [5], [42], provide good local convergence.",
"However, they need a highly accurate model of the starlight suppression system to reach the necessary system performance.",
"Random phase probes based algorithms could provide such a model.",
"These algorithms are fast, accurate and free from non-common path errors that can affect the system performance.",
"They do not need to reconfigure the system for measurements and can be run remotely to perform system calibration with an existing optical setup during space missions.",
"(2) Adaptive optics.",
"Obtained pupil-plane phase estimation can be used for adaptive wavefront correction, particularly to correct wavefront aberrations caused by atmosphere turbulence.",
"Even in low flux conditions as low as 1-2 photons/speckle the discussed approach allows diffraction-limited wavefront correction with Strehl ratios of 0.2-0.5.",
"Taking into account the presented simulation results and assuming an atmosphere coherence time of 20 milliseconds (in visual spectral range) [39], [50], [40] the imaging frequency needed for the adaptive wavefront correction can be roughly estimated in the range of several hundred to one thousand Hz.",
"(3) Segment co-phasing.",
"The random phase probe approach can be applied to both continuous and discontinuous pupil-plane phase distributions.",
"Thus, the proposed wavefront reconstruction method can be used to co-phase sub-apertures of large segmented mirrors [51].",
"A simulated demonstration of applicability of the wavefront reconstruction method to segment co-phasing is shown in Fig.",
"REF .",
"The sample aperture consists of 15 different rectangular static segments with the piston rms of $\\lambda /2$ .",
"In the presence of 1% focal-plane background noise, the restored pupil-plane phase rms is about $\\lambda /20$ corresponding to wavefront correction with Strehl ratio of about 0.92.",
"For static aberrations the averaging procedure using 20 independent phase estimates can additionally improve the Strehl ratio to 0.993.",
"In the case where the segments are randomly moving relative to each other the procedure can provide the signal needed to dynamically correct the segment positions.",
"This position, of course, will be affected by integer-wave ambiguities visible in the restored wavefronts (Fig.",
"REF ), that cannot be resolved in monochromatic Gerchberg-Saxton implementation of the considered approach.",
"However, random phase probing can be combined with any known broadband estimator (for example, with a gradient descent broadband estimator).",
"In such combination the random phase probing would be responsible for the algorithm convergence while the broadband estimator would resolve the integer-wave ambiguities issue.",
"Development of such estimators is beyond the scope of this paper.",
"Figure: The pupil-plane phase restoration in the case of segmented aperture.",
"The reconstructed pupil-plane phase maps for the cases of dynamically changing (top) and static (bottom) phase aberrations are shown.",
"To visualize phase restoration errors for dynamically changing aberrations, a bias of 3σ\\sigma is added to the unwrapped error map such that the darkest and brightest pixels correspond to ±3σ\\pm 3\\sigma levels."
],
[
"Conclusion",
"Our method allows wavefront estimation in a pupil plane of an optical instrument by analyzing the effects of random (unknown) wavefront perturbations on a focal-plane image.",
"Unlike other pupil-plane diversity methods, it has the advantage that it does not require knowledge of the applied pupil-plane perturbations, which makes it robust to model errors.",
"In fact, the method can estimate the perturbations themselves.",
"We have considered the possibility of the appearance of ambiguous solutions and have proven the uniqueness theorem.",
"We have also discussed the problem of global convergence that can be solved the defocus based phase divergence that provided the global solutions in all our simulations.",
"For phase retrieval, we have demonstrated low sensitivity to photon noise and registration noise.",
"To increase accuracy of the final phase estimates an averaging procedure has been proposed.",
"The considered algorithms provide fast and reliable pupil-plane phase measurements for both large (on the order of a few $\\lambda $ ) and small (optical surface structures as shallow as 30-40 nm can be measured) phase aberrations.",
"The algorithm can be applied to both continuous and discontinuous pupil-plane phase distributions and can be used to measure either static and or dynamically changing pupil-plane phases.",
"Similar to other focal-plane wavefront sensing methods, the random phase probe estimate method uses common-path and can thus be used by applications whose performance is limited by non-common path errors.",
"The phase retrieval solutions can be affected by systematic errors relevant especially to periodical high frequency pupil-plane structures whose position cannot be efficiently detected by applied phase probes.",
"In this case an appropriate solution can be obtained by increasing defocus term in the used phase diversity procedure that increases the related interferometric term.",
"For implementation of the random phase probe method, the following requirements should be met: – the focal-plane imaging procedure should provide an adequate image sampling; – a capability to create random pupil-plane phase aberrations and control their amplitudes using (for example) a DM.",
"Note, that the particular location of the DM is not important.",
"The DM can be placed anywhere in the optical system except at the system focus.",
"– the imaging frequency should be high enough to maintain high sequential correlation of the focal-plane images in the case when the random phase probes are created by a rapid dynamical process such as the atmosphere turbulence or convection.",
"– an appropriate propagation model should be used, that assumes that the procedure can be used in the case of Fresnel diffraction for example.",
"The described procedure can be implemented for a wide class of optical applications, including those that use the DM for adaptive wavefront correction.",
"Finally, the system noise not only limits the system performance, sometimes it helps to improve the performance.",
"This work was funded by the NASA Ames FY17 Center Innovation Fund (CIF) program.",
"We thank the two referees and the editor for constructive comments that significantly improved the manuscript."
]
] | 1709.01571 |
[
[
"Enriched Galerkin methods for two-phase flow in porous media with\n capillary pressure"
],
[
"Abstract In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media.",
"The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver.",
"To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations.",
"Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains.",
"The iterative and sequential IMplicit Pressure and Explicit Saturation (IMPES) algorithms are treated in time.",
"Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG."
],
[
"Introduction",
"We consider a two-phase flow system in porous media which has been widely employed in petroleum reservoir modeling and environmental engineering for the past several decades [7], [16], [22], [60], [62], [71].",
"The conventional two-phase flow system is formulated by coupling Darcy's law for multiphase flow with the saturation transport equation [50], [73].",
"An incomplete list of numerical approximations such as finite difference, mixed finite elements, and finite volume methods [2], [4], [7], [19], [20], [22], [26], [27], [62], [65], [68], [69], [76] have been successfully utilized in multiphase flow reservoir simulators.",
"Recent interest has centered on multiscale extensions to finite element methods [3], [23], [24], [34], [39], [42], [56], [63].",
"In all of these works, it was observed that local conservation was required for accurately solving the saturation transport equations [44], [70].",
"However, only several of these references considered capillary pressure effects for two-phase flow systems [5], [9], [25], [29], [41], [46], [67], [74].",
"For many problems such as CO$_2$ sequestration, the latter is crucial for realistic heterogeneous media.",
"In this paper, we focus on extensions of enriched Galerkin approximations (EG) to two-phase flow in porous media with capillary pressure.",
"Our objective is to demonstrate that high order spatial approximations for saturations can be computed efficiently using EG.",
"EG provides locally and globally conservative fluxes and preserves local mass balance for transport [51], [52], [55].",
"EG is constructed by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions [11], [72], with the same bilinear forms as the interior penalty DG schemes.",
"However, EG has substantially fewer degrees of freedom in comparison with DG and a fast effective high order solver for pressure whose cost is roughly that of CG [51].",
"EG has been successfully employed to realistic multiscale and multi-physics applications [55], [53], [54].",
"An additional advantage of EG is that only those subdomains that require local conservation need be enriched with a treatment of high order non-matching grids.",
"Local conservation of the flux is crucial for flow and saturation stabilization is critical for avoiding overshooting, undershooting, and spurious oscillations [48].",
"Our high order EG transport system is coupled with an entropy viscosity residual stabilization method introduced in [38] to avoid spurious oscillations near the interface of saturation fronts.",
"Instead of using limiters and non-oscillatory reconstructions, this method adds nonlinear dissipation to the numerical discretization [35], [36], [37].",
"The numerical diffusion is constructed by the local residual of an entropy residual.",
"Moreover, the entropy residual is employed for dynamic adaptive mesh refinement to capture the moving interface between the immiscible fluids [43], [45].",
"It is shown in [1], [64] that the entropy residual can be used as an a posteriori error indicator.",
"To take advantage of high order in space, each time derivative in the flow and transport system is discretized by second order backward difference formula (BDF2) and extrapolations are employed.",
"For the coupling solution algorithm, a sequential time-stepping scheme (IMPES) is applied for efficient computation [31].",
"First, we solve the pressure equation implicitly assuming saturation values are obtained by extrapolation in time and the transport equation is solved explicitly [17], [32], [46], [47], [58], [75].",
"In addition, we employ H(div) flux reconstruction to the incompressible flow to enhance the performance as applied for DG in [10], [30], [57]."
],
[
"Mathematical Model",
"In this section, a mathematical model for the slightly compressible two-phase Darcy flow and saturation system in a heterogeneous media is presented.",
"Let $\\Omega \\subset {d}$ be a bounded polygon (for $d=2$ ) or polyhedron (for $d=3$ ) with Lipschitz boundary $\\partial \\Omega $ , and $(0,\\mathbb {T}]$ the computational time interval with $\\mathbb {T} > 0$ .",
"The mass conservation equation for saturation equation is defined by $\\dfrac{\\partial }{\\partial t} (\\phi \\rho _i s_i )+ \\nabla \\cdot ( \\rho _i _i ) = \\rho _i f_i, \\; i \\in \\lbrace w,n\\rbrace ,$ where $\\phi $ is the porosity of the porous media, $\\rho _i$ is the density, $s_i : \\Omega \\times (0,\\mathbb {T}] \\rightarrow \\mathbb {R}$ is the saturation, and $i \\in \\lbrace w,n\\rbrace $ indicates wetting$(w)$ or non-wetting$(n)$ phases, respectively.",
"Here, $f_i := \\tilde{s}_i q_i$ , where $\\tilde{s}_i, q_i$ are the saturation injection/production term and flow injection/production, respectively.",
"If $q_i>0$ , $\\tilde{s}_i$ is the injected saturation of the fluid and if $q_i<0$ , $\\tilde{s}_i$ is the produced saturation.",
"Here $_i : \\Omega \\times (0,\\mathbb {T}] \\rightarrow \\mathbb {R}^d$ is the Darcy velocity for each phase i, given by $_i := - \\dfrac{k_i}{\\mu _i} \\left( \\nabla p_i - \\rho _i \\right),$ in which $k_i$ is the relative permeability, $:=()$ is the absolute permeability tensor of the porous media, $\\mu _i$ is the viscosity, $p_i : \\Omega \\times (0,\\mathbb {T}] \\rightarrow \\mathbb {R}$ is the pressure for each phase, and $$ is the gravity acceleration.",
"Relative permeability is a given function of saturation which is defined as $k_i := k_i(s_w).$ Here we define the capillary pressure, $p_c := p_c(s_w) = p_n - p_w,$ which is the pressure difference between the wetting and non-wetting phase [18].",
"Since, we assume that all pores are filled with fluid, we have $s_w + s_n = 1\\ \\text{ and }\\ \\tilde{s}_w + \\tilde{s}_n = 1.$ To derive a pressure equation, we sum the saturation equations (REF ) to get $\\phi \\dfrac{\\partial }{\\partial t} \\left(\\rho _w s_w +\\rho _n s_n \\right)+ \\nabla \\cdot ({\\rho _w}_w +{\\rho _n}_n) = \\rho _w f_w + \\rho _n f_n,$ where we consider a slightly compressible fluid satisfying $\\rho _i(p_i)\\approx \\rho _i^0 \\exp ^{c_i^F(p_i-p_i^0)}\\approx \\rho _i^0(1+c_i^F(p_i-p_i^0)),$ with a small compressibility coefficient, $c_i^F \\ll 1$ .",
"Here we assume the reference pressure $p_i^0$ is zero, and porosity $\\phi $ and reference density $\\rho _i^0$ are constants.",
"Thus, we can rewrite (REF ) and obtain $\\phi \\dfrac{\\partial }{\\partial t} \\left(c_w^F \\rho _w^0 p_w s_w + c_n^F \\rho _n^0 p_n s_n \\right)+ \\nabla \\cdot ({\\rho _w}_w +{\\rho _n}_n) = \\rho _w f_w + \\rho _n f_n.$ For the incompressible case, we set $c_i^F=0$ and have $\\nabla \\cdot ({\\rho _w}_w +{\\rho _n}_n) = \\rho _w f_w + \\rho _n f_n.$"
],
[
"Choice of primary variables",
"Throughout the paper, we set the wetting phase pressure $p_w$ and saturation $s_w$ as the primary variables.",
"Different choices and effects are illustrated in [5].",
"We rewrite the incompressible flow equation by combining the relations (REF ), (REF ), (REF ), and continuity of phase fluxes to obtain $-\\nabla \\cdot (\\lambda _t ( \\nabla p_w- \\rho _w )+ \\lambda _n (\\nabla p_c + (\\rho _w - \\rho _n)))= (\\rho f)_t,$ which is equivalent with $-\\nabla \\cdot ((\\lambda _t \\nabla p_w- (\\rho \\lambda )_t )+ \\lambda _n \\nabla p_c) = (\\rho f)_t,$ where $\\lambda _i &:= \\lambda _i(s_w) = \\rho _i\\dfrac{k_i(s_w)}{\\mu _i} , \\; \\text{phase mobility}\\\\\\lambda _t &:= \\lambda _t(s_w) = \\lambda _w(s_w) + \\lambda _n(s_w), \\; \\text{total mobility}\\\\(\\rho \\lambda )_t &:= (\\rho \\lambda (s_w))_t = \\rho _w \\lambda _w(s_w) + \\rho _n \\lambda _n(s_w), \\\\(\\rho f)_t &:=\\rho _w f_w + \\rho _n f_n.$ For the slightly compressible flow equations, we get the pressure equation $\\phi \\dfrac{\\partial }{\\partial t}\\left( c_w^F \\rho _w^0 s_w p_w+ c_n^F \\rho _n^0 (1-s_w) p_w + c_n^F \\rho _n^0 (1-s_w) p_c \\right)-\\nabla \\cdot ((\\lambda _t \\nabla p_w- (\\rho ^0 \\lambda )_t )+ \\lambda _n \\nabla p_c) = (\\rho ^0 f)_t,$ where $(\\rho ^0 \\lambda )_t &:= \\rho ^0_w \\lambda _w + \\rho ^0_n \\lambda _n, \\\\(\\rho ^0 f)_t &:=\\rho ^0_w f_w + \\rho ^0_n f_n.$ For the saturation equation, we solve $\\dfrac{\\partial }{\\partial t} (\\phi \\rho ^0_w s_w )+ \\nabla \\cdot ({\\rho ^0_w} _w ) = \\rho ^0_w f_w,$ and $s_w +s_n =1$ .",
"The boundary of $\\Omega $ is decomposed into three disjoint sets $\\Gamma _\\textsf {in}$ , $\\Gamma _\\textsf {out}$ and $\\Gamma _N$ so that $\\overline{\\partial \\Omega } =\\overline{\\Gamma }_\\textsf {in}\\cup \\overline{\\Gamma }_\\textsf {out}\\cup \\overline{\\Gamma }_N$ For the flow problem, we impose $p_w (\\text{ or } p_n) = p_\\textsf {in}\\mbox{ on } \\Gamma _\\textsf {in}&\\times (0,\\mathbb {T}], \\quad \\\\p_w (\\text{ or } p_n) = p_\\textsf {out}\\mbox{ on } \\Gamma _\\textsf {out}&\\times (0,\\mathbb {T}], \\quad \\\\(_w + _n) \\cdot = _{N}\\mbox{ on } \\Gamma _N&\\times (0,\\mathbb {T}],$ where $p_\\textsf {in} \\in L^2(\\Gamma _\\textsf {in})$ , $p_\\textsf {out} \\in L^2(\\Gamma _\\textsf {out})$ and $_N \\in L^2(\\Gamma _N)$ are the each Dirichlet and Neumann boundary conditions, respectively.",
"Thus we define ${\\Gamma }_D := \\Gamma _\\textsf {in} \\cup \\Gamma _\\textsf {out}$ .",
"Here inflow and outflow boundaries are defined as $\\Gamma _{\\rm in} := \\lbrace \\in \\partial \\Omega : _w\\cdot < 0\\rbrace \\ \\text{ and }\\Gamma _{\\rm out} := \\lbrace \\in \\partial \\Omega : _w\\cdot > 0\\rbrace .$ For the saturation system, we impose $s_w (\\text{ or } s_n) =s_\\textsf {in},\\mbox{ on } \\Gamma _\\textsf {in} \\times (0,\\mathbb {T}]$ where $s_\\textsf {in}$ is a given boundary value for saturation.",
"Finally, the above systems are supplemented by initial conditions $s_w(,0) = s_w^0(), \\mbox{ and }p_w(,0) = p_w^0(), \\quad \\forall \\in \\Omega .$"
],
[
"Numerical Method",
"Let $\\mathcal {T}_h$ be the shape-regular (in the sense of Ciarlet) triangulation by a family of partitions of $Ø$ into $d$ -simplices $$ (triangles/squares in $d=2$ or tetrahedra/cubes in $d=3$ ).",
"We denote by $h_{}$ the diameter of $$ and we set $h=\\max _{\\in } h_{}$ .",
"Also we denote by $$ the set of all edges and by $$ and $$ the collection of all interior and boundary edges, respectively.",
"In the following notation, we assume edges for two dimension but the results hold analogously for faces in three dimensional case.",
"For the flow problem, the boundary edges $$ can be further decomposed into $= \\mathcal {E}_h^{D,\\partial } \\cup \\mathcal {E}_{h}^{N,\\partial }$ , where $\\mathcal {E}_h^{D,\\partial }$ is the collection of edges where the Dirichlet boundary condition is imposed (i.e $\\mathcal {E}_h^{D,\\partial } :=\\mathcal {E}_h^{\\textsf {in},\\partial } \\cup \\mathcal {E}_{h}^{\\textsf {out},\\partial }$ ), while $\\mathcal {E}_h^{N,\\partial }$ is the collection of edges where the Neumann boundary condition is imposed.",
"In addition, we let $\\mathcal {E}_h^{1} := \\cup \\mathcal {E}_h^{D,\\partial }$ and $\\mathcal {E}_h^{2} := \\cup \\mathcal {E}_h^{N,\\partial }$ .",
"For the transport problem, the boundary edges $$ decompose into $= \\mathcal {E}_h^{\\text{in}} \\cup \\mathcal {E}_{h}^{\\text{out}}$ , where $\\mathcal {E}_h^{\\text{in}}$ is the collection of edges where the inflow boundary condition is imposed, while $\\mathcal {E}_h^{\\text{out}}$ is the collection of edges where the outflow boundary condition is imposed.",
"The space $H^{s}()$ $(s\\in {})$ is the set of element-wise $H^{s}$ functions on $\\mathcal {T}_h$ , and $L^{2}()$ refers to the set of functions whose traces on the elements of $$ are square integrable.",
"Let $\\mathbb {Q}_l()$ denote the space of polynomials of partial degree at most $l$ .",
"Regarding the time discretization, given an integer $N \\ge 2$ , we define a partition of the time interval $0 =: t^0 < t^1 < \\cdots < t^N:= \\mathbb {T}$ and denote $\\Delta t := t^k - t^{k-1}$ for the uniform time step.",
"Throughout the paper, we use the standard notation for Sobolev spaces and their norms.",
"For example, let $E \\subseteq \\Omega $ , then $\\Vert \\cdot \\Vert _{1,E}$ and $|\\cdot |_{1,E}$ denote the $H^1(E)$ norm and seminorm, respectively.",
"For simplicity, we eliminate the subscripts on the norms if $E = \\Omega $ .",
"For any vector space $$ , $^d$ will denote the vector space of size d, whose components belong to $$ and $^{d\\times d}$ will denote the $d \\times d$ matrix whose components belong to $$ .",
"We introduce the space of piecewise discontinuous polynomials of degree $l$ as $M^l(\\mathcal {T}_h) := \\left\\lbrace \\psi \\in L^2(\\Omega ) | \\ \\psi _{|_{}} \\in \\mathbb {Q}_l(), \\ \\forall \\in \\mathcal {T}_h \\right\\rbrace ,$ and let $M_0^l(\\mathcal {T}_h)$ be the subspace of $M^l(\\mathcal {T}_h)$ consisting of continuous piecewise polynomials; $M_0^l(\\mathcal {T}_h) = M^l(\\mathcal {T}_h) \\cap \\mathbb {C}_0(\\Omega ).$ The enriched Galerkin finite element space, denoted by $V_{h,l}^{\\textsf {EG}}$ is defined as $V_{h,l}^{\\textsf {EG}}(\\mathcal {T}_h) := M^l_0(\\mathcal {T}_h) + M^0(\\mathcal {T}_h),$ where $l \\ge 1$ , also see [11], [51], [52], [55], [72] for more details.",
"Remark 1 We remark that the degrees of freedom for $V_{h,l}^{\\textsf {EG}}(\\mathcal {T}_h) $ when $l = 1$ , is approximately one half and one fourth the degrees of freedom of the linear DG space, in two and three space dimensions, respectively.",
"See Figure REF .",
"Figure: A sketch of the degrees of freedom forenriched Galerkin ina two-dimensional Cartesian grid (ℚ\\mathbb {Q}) with l=1l=1.Four circles (∘\\circ ) are the degrees of freedom for continous Galerkin (M l (𝒯 h ))(M^l(\\mathcal {T}_h)) and (▵\\triangle ) is the discontinuous constant (M 0 (𝒯 h ))(M^0(\\mathcal {T}_h)).We define the coefficient $_T$ by $_T := |_T, \\quad \\forall T \\in \\mathcal {T}_h.$ For any $e \\in $ , let $^{+}$ and $^{-}$ be two neighboring elements such that $e = \\partial ^{+}\\cap \\partial ^{-}$ .",
"We denote by $h_{e}$ the length of the edge $e$ .",
"Let $^{+}$ and $^{-}$ be the outward normal unit vectors to $\\partial T^+$ and $\\partial T^-$ , respectively ($^{\\pm } :=_{|^{\\pm }}$ ).",
"For any given function $\\xi $ and vector function $\\bf {\\xi }$ , defined on the triangulation $\\mathcal {T}_h$ , we denote $\\xi ^{\\pm }$ and $^{\\pm }$ by the restrictions of $\\xi $ and $$ to $T^\\pm $ , respectively.",
"We define the average ${\\cdot }$ as follows: for $\\zeta \\in L^2(\\mathcal {T}_h)$ and $\\in L^2(\\mathcal {T}_h)^d$ , ${\\zeta } := \\frac{1}{2} (\\zeta ^+ + \\zeta ^- )\\quad \\mbox{ and } \\quad {} := \\frac{1}{2} (^+ + ^-) \\quad \\mbox{on } e\\in .$ On the other hand, for $e \\in $ , we set ${\\zeta } := \\zeta $ and ${} := $ .",
"The jump across the interior edge will be defined as usual: ${\\zeta } = \\zeta ^+^++\\zeta ^-^- \\quad \\mbox{ and } \\quad = ^+\\cdot ^+ + ^-\\cdot ^- \\quad \\mbox{on } e\\in .$ For inner products, we use the notations: $&(v,w)_{}:=\\sum _{\\in } \\int _{} v\\, w dx, \\quad \\forall \\,\\, v ,w \\in L^{2} (\\mathcal {T}_h), \\\\&\\langle v, w\\rangle _{}:=\\sum _{e\\in } \\int _{e} v\\, w \\,d\\gamma , \\quad \\forall \\, v, w \\in L^{2}().$ For example, a function in $\\psi _{\\textsf {EG}} \\in V_{h,l}^{\\textsf {EG}}(\\mathcal {T}_h)$ can be decomposed into $\\psi _{\\textsf {EG}} = \\psi _{\\textsf {CG}}+ \\psi _{\\textsf {DG}}$ , where $\\psi _{\\textsf {CG}} \\in M^l_0(\\mathcal {T}_h)$ and $\\psi _{\\textsf {DG}} \\in M^0(\\mathcal {T}_h)$ .",
"Thus the inner product $(\\psi _{\\textsf {EG}} ,\\psi _{\\textsf {EG}} ) =(\\psi _{\\textsf {CG}} ,\\psi _{\\textsf {CG}} ) + (\\psi _{\\textsf {CG}} ,\\psi _{\\textsf {DG}} ) + (\\psi _{\\textsf {DG}} ,\\psi _{\\textsf {CG}} ) + (\\psi _{\\textsf {DG}} ,\\psi _{\\textsf {DG}} )$ creates a matrix as $\\begin{pmatrix}\\psi _{\\textsf {CG}} \\psi _{\\textsf {CG}} & \\psi _{\\textsf {CG}} \\psi _{\\textsf {DG}} \\\\\\psi _{\\textsf {DG}} \\psi _{\\textsf {CG}} & \\psi _{\\textsf {DG}} \\psi _{\\textsf {DG}}\\end{pmatrix}.$ Finally, we introduce the interpolation operator $\\Pi _h$ for the space $V_{h,l}^{\\textsf {EG}}$ as $\\Pi _h v = \\Pi _0^l v + Q^0 ( v - \\Pi _0^l v),$ where $\\Pi _0^l$ is a continuous interpolation operator onto the space $M_0^l(\\mathcal {T}_h)$ , and $Q^0$ is the $L^2$ projection onto the space $M^0(\\mathcal {T}_h)$ .",
"See [51] for more details."
],
[
"Temporal Approximation",
"The time discretization is carried out by choosing $N\\in \\mathbb {N}$ , the number of time steps.",
"To simplify the discussion, we assume uniform time steps, let $\\Delta t = \\mathbb {T}/N$ .",
"We set $t^k = k \\Delta t$ and for a time dependent function we denote $\\varphi ^k = \\varphi (t^k)$ .",
"Over these sequences we define the operators ${\\textup {\\textsf {BDF}}_{m}({\\varphi ^{k+1}})} :={\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\Delta t}(\\varphi ^{k+1}-\\varphi ^{k}) & m = 1, \\\\\\frac{1}{2\\Delta t} \\left( 3 \\varphi ^{k+1} - 4\\varphi ^k + \\varphi ^{k-1} \\right) & m = 2,\\end{array}\\right.",
"}$ for the backward Euler time discretization order 1 and order 2.",
"In this paper, we employ BDF2 (second order backward difference formula) with $m=2$ to discretize the time derivatives.",
"Thus we obtain the following time discretized formulation $\\phi c_w^F \\rho _w^0 {\\textup {\\textsf {BDF}}_{m}({s_w^{k+1} p^{k+1}_w})}-\\nabla \\cdot \\left((\\lambda _t(s_w^{k+1}) \\nabla p_w^{k+1}- (\\rho ^0 \\lambda (s_w^{k+1}))_t ) \\right) \\\\-\\nabla \\cdot \\left( \\lambda _n(s_w^{k+1}) \\nabla p_c(s_w^{k+1}) \\right) = (\\rho ^0 f^{k+1})_t,$ As frequently done in modeling slightly compressible two-phase flow, we neglect the terms involving small compressibility $c_n^F$ in (REF ) with the exception of $c_w^F$ .",
"Here $c_w^F$ is included as a regularization term for the solver.",
"Next, the saturation system is discretized by $\\phi \\rho ^0_w {\\textup {\\textsf {BDF}}_{m}({s^{k+1}_w})}+ \\nabla \\cdot \\left(-\\rho ^0_w \\dfrac{k_w(s_w^{k+1}) }{\\mu _w} ( \\nabla p_w^{k+1} - \\rho _w ) \\right)= \\rho ^0_w f_w^{k+1},$ The above system is fully coupled and nonlinear.",
"We propose the following iterative decoupled scheme.",
"The implicit pressure and explicit saturation algorithm (IMPES) is frequently applied as an efficient algorithm for decoupling and sequentially solving the system [18].",
"For uniform time steps, to approximate the time dependent terms we define the extrapolation of $\\varphi ^{k+1,*}$ by $\\varphi ^{k+1,*} :=\\varphi ^k + ( \\varphi ^k - \\varphi ^{k-1}).$ The IMPES algorithm solves the system as follows: Initial conditions at time $t^{k-1}, t^k$ are given.",
"Solve $p^{k+1}_w$ at time $t^{k+1}$ by using the previous saturation to compute $\\lambda _i(s_w^{k+1,*})$ and $p_c(s_w^{k+1,*})$ .",
"$\\phi c_w^F \\rho _w^0 {\\textup {\\textsf {BDF}}_{m}({s_w^{k+1,*} p^{k+1}_w})}-\\nabla \\cdot \\left(\\lambda _t(s_w^{k+1,*}) \\nabla p_w^{k+1}\\right) \\\\= (\\rho ^0 f)_t-\\nabla \\cdot \\left((\\rho ^0 \\lambda (s_w^{k+1,*}))_t ) \\right)+\\nabla \\cdot \\left( \\lambda _n(s_w^{k+1,*}) \\nabla p_c(s_w^{k+1,*})\\right)$ Compute the velocity $_w^{k+1,*}$ by using $p_w^{k+1}$ and the saturation.",
"Compute $s_w^{k+1}$ using an explicit time stepping.",
"$\\phi \\rho ^0_w {\\textup {\\textsf {BDF}}_{m}({s^{k+1}_w})}= \\rho ^0_w f_w^{k+1}+ \\nabla \\cdot \\left(\\rho ^0_w \\dfrac{k_w(s_w^{k+1,*}) }{\\mu _w} \\left( \\nabla p_w^{k+1} - \\rho ^0_w \\right) \\right)$"
],
[
"Iterative IMPES algorithm",
"An iterative IMPES algorithm is to solve the following equations sequentially for iterations $j=1,\\cdots $ until it converges to a given tolerance or a fixed number of iterations has been reached.",
"For example, at each time step $t^k$ : For $j=0$ , set $s_w^{k+1,j} = s_w^k$ and $s_w^{k+1,j-1} = s_w^{k-1}$ .",
"Solve for $p_w^{k+1,j+1}$ satisfying $\\phi c_w^F \\rho _w^0 {\\textup {\\textsf {BDF}}_{m}({s_w^{k+1,*,j} p^{k+1}_w})}-\\nabla \\cdot \\left(\\lambda _t(s_w^{k+1,*,j}) \\nabla p_w^{k+1,j+1}\\right) \\\\= (\\rho ^0 f)_t-\\nabla \\cdot \\left((\\rho ^0 \\lambda (s_w^{k+1,*,j}))_t ) \\right)+\\nabla \\cdot \\left( \\lambda _n(s_w^{k+1,*,j}) \\nabla p_c(s_w^{k+1,*,j})\\right) ,$ where $s_w^{k+1,*,j} = s_w^{k+1,j} + (s_w^{k+1,j} - s_w^{k+1,j-1})$ .",
"Given $s_w^{k+1,*,j}$ and $p_w^{k+1,j+1}$ , solve for $s_w^{k+1,j+1}$ satisfying $\\phi \\rho ^0_w {\\textup {\\textsf {BDF}}_{m}({s^{k+1,j+1}_w})}= \\rho ^0_w f_w^{k+1}+ \\nabla \\cdot \\left(\\rho ^0_w \\dfrac{k_w(s_w^{k+1,*,j}) }{\\mu _w} \\left( \\nabla p_w^{k+1,j+1} - \\rho ^0_w \\right) \\right).$ Iteration continues until $\\Vert s_w^{k+1,j+1} - s_w^{k+1,j} \\Vert \\le \\varepsilon _{I}$ .",
"The locally conservative EG is selected for the space approximation of the pressure system (REF ).",
"Here we apply the discontinuous Galerkin (DG) IIPG (incomplete interior penalty Galerkin) method for the flow problem to satisfy the discrete sum compatibility condition [21], [55], [70].",
"Mathematical stability and error convergence of EG for a single phase system is discussed in [51], [52], [55] The EG finite element space approximation of the wetting phase pressure $p_w(,t)$ is denoted by $P_w(,t) \\in V^{\\textsf {EG}}_{h,l}(\\mathcal {T}_h) $ and we let $P_w^k := P_w(,t^k)$ for time discretization, $0 \\le k \\le N$ .",
"We set an initial condition for the pressure as $P_w^0 := \\Pi _h p_w(\\cdot ,0)$ .",
"Let $p_\\textsf {in}^{k+1}, p_\\textsf {out}^{k+1}, _N^{k+1}$ and $f^{k+1}$ are approximations of $p_\\textsf {in}(\\cdot , t^{k+1}), p_\\textsf {out}(\\cdot , t^{k+1}), _N(\\cdot , t^{k+1})$ and $f(\\cdot , t^{k+1})$ on $\\Gamma _D$ , $\\Gamma _N$ and $\\Omega $ , respectively at time $t^{k+1}$ .",
"Assuming $s_w(\\cdot ,t^{k+1})$ is known, and employing time lagged/extrapolated values for simplicity, the time stepping algorithm reads as follows: Given $P_w^{k-1}$ , $P_w^{k}$ , find $P_w^{k+1} \\in V_{h,l}^{\\textsf {EG}}(\\mathcal {T}_h) \\mbox{ such that } (P_w^{k+1},\\omega ) = \\mathcal {F}(\\omega ), \\quad \\forall \\, \\omega \\in V_{h,l}^{\\textsf {EG}}(\\mathcal {T}_h) , \\,$ where $$ and $\\mathcal {F}$ are the bilinear form and linear functional, respectively, are defined as $(P_w^{k+1},\\omega ) :=\\left( (\\phi \\rho _w^0 c_w^F s_w^{k+1,*})\\dfrac{ 3 }{2 \\Delta t}P_w^{k+1}, \\omega \\right)_{\\mathcal {T}_h}+\\left(\\lambda _t(s_w^{k+1,*}) \\nabla P_w^{k+1},\\nabla \\omega \\right)_{} \\\\- \\left\\langle {\\lambda _t(s_w^{k+1,*}) \\nabla P_w^{k+1} }, {\\omega } \\right\\rangle _{\\mathcal {E}_h^{1}}+ {\\dfrac{\\alpha }{h_{e}} }{\\lambda _t(s_w^{k+1,*})} \\left\\langle {P_w^{k+1}},{\\omega } \\right\\rangle _{\\mathcal {E}_h^{1}},$ and $\\mathcal {F}(\\omega ) :=\\left((\\phi \\rho ^0_w c^F_w s_w^{k} )(\\dfrac{ 2 }{\\Delta t}P_w^{k})-(\\phi \\rho ^0_w c^F_w s_w^{k-1} )(\\dfrac{1}{2 \\Delta t}P_w^{k-1} ), \\omega \\right)_{\\mathcal {T}_h}+\\left( (\\rho ^0 f^{k+1})_t,\\omega \\right)_{\\mathcal {T}_h} \\\\- \\left(\\lambda _n(s_w^{k+1,*}) \\nabla p_c(s_w^{k+1,*})- (\\rho ^0 \\lambda (s_w^{k+1,*}))_t , \\nabla \\omega \\right)_{\\mathcal {T}_h}+\\left<{\\lambda _n(s_w^{k+1,*}) \\nabla p_c(s_w^{k+1,*})-(\\rho ^0 \\lambda (s_w^{k+1,*}))_t },{\\omega }\\right>_{\\mathcal {E}_h^{1}} \\\\{- \\dfrac{\\alpha _c}{h_{e}}{ \\lambda _n(s_w^{k+1,*})} \\left\\langle {p_c(s_w^{k+1,*})},{\\omega } \\right\\rangle _{\\mathcal {E}_h^{1}},} \\\\+{\\dfrac{\\alpha }{h_{e}}}{ \\lambda _t(s_w^{k+1,*})} \\left\\langle p_\\textsf {in}^{k+1},{\\omega }\\right\\rangle _{\\mathcal {E}_h^{{\\textsf {in}},\\partial }}+{\\dfrac{\\alpha }{h_{e}}}{ \\lambda _t(s_w^{k+1,*})} \\left\\langle p_\\textsf {out}^{k+1},{\\omega }\\right\\rangle _{\\mathcal {E}_h^{{\\textsf {out}},\\partial }}- \\left\\langle {{}^{k+1}_N},{\\omega } \\right\\rangle _{\\mathcal {E}_h^{N,\\partial }}.$ Here $h_e$ denotes the maximum length of the edge $e \\in $ and $\\alpha , \\alpha _c$ are penalty parameters for pressure and capillary pressure, respectively.",
"For adaptive mesh refinement with hanging nodes, we make the usual assumption to set the $h_{e} = \\min (h^+,h^-)$ for $e=\\partial ^{+}\\cap \\partial ^{-}$ over the edges on a mesh $T$ ."
],
[
"Locally conservative flux",
"Conservative flux variables are described in [51], [72] with details for convergence analyses.",
"With slight modifications to the latter single phase case, we define the two-phase wetting phase velocity as $_w^{k+1,*}$ since it depends on the previous saturation value $s_w^{k+1,*}$ .",
"Let $P_w^{k+1}$ be the wetting phase solution to (REF ), then we define the globally and locally conservative flux variables $_w^{k+1,*}$ at time step $t^{k+1}$ by the following : $_w^{k+1,*} |_{T} &:= -\\dfrac{k_w(s_w^{k+1,*})}{\\mu _w} \\left( \\nabla P_w^{k+1} - \\rho _w^0 \\right), \\; \\forall T \\in \\mathcal {T}_h \\vspace*{14.45377pt} \\\\_w^{k+1,*} \\cdot |_{e} &:= -{ \\dfrac{k_w(s_w^{k+1,*})}{\\mu _w} \\left( \\nabla P_w^{k+1} - \\rho _w^0 \\right)} \\cdot + {\\dfrac{\\alpha }{h_e} \\dfrac{k_w(s_w^{k+1,*})}{\\mu _w} }{P_w^{k+1}},\\; \\forall e \\in \\mathcal {E}_h^I, \\\\_w^{k+1,*} \\cdot |_{e} &:= _{Nw}^{k+1}, \\; \\forall e \\in \\mathcal {E}_h^{N,\\partial }, \\\\_w^{k+1,*} \\cdot |_{e} &:= - \\dfrac{k_w(s_w^{k+1,*})}{\\mu _w} \\left(\\nabla P_w^{k+1} - \\rho _w^0 \\right) \\cdot + { \\dfrac{\\alpha }{h_e}\\dfrac{ k_w(s_w^{k+1,*})}{\\mu _w} }\\left( P_w^{k+1} - p_\\textsf {in/out}^{k+1} \\right), \\; \\forall e \\in \\mathcal {E}_{h_e}^{D,\\partial },$ where $$ is the unit normal vector of the boundary edge $e$ of $T$ and $_{Nw}^{k+1} := (_{N}^{k+1} - \\lambda ^{k+1}_n\\nabla p_c^{k+1})(\\lambda _w^{k+1}/(\\lambda _t^{k+1} \\rho ^0_w))$ ."
],
[
"H(div) reconstruction of the flux",
"For incompressible flow, it is frequently useful to project the velocity (flux) into a $H$ (div) space for high order approximation to a transport system, see [5], [10], [28], [29], [30] for more details.",
"We illustrate below, the reconstruction of the EG flux (REF )-() in a $H$ (div) space for quadrilateral elements [52], [57].",
"The flux is projected into the Raviart-Thomas (RT$_l$ ) space [12], [66], $\\mathcal {H} := \\lbrace v \\in H(\\text{div}) :v|_E \\in \\mathbb {Q}_{l+l,l}(T) \\times \\mathbb {Q}_{l,l+1}(T) ,\\ \\forall T \\in \\mathcal {T}_h \\rbrace ,$ where $\\mathbb {Q}_{a,b}(T) :=\\left\\lbrace v \\ : \\ v() = \\sum _{i=0}^a\\sum _{j=0}^b \\omega _{i,j} _1^i_2^j, \\ \\in T, \\omega _{i,j} \\in \\mathbb {R} \\right\\rbrace $ with polynomial order $l$ .",
"Let ${}^{div} \\in \\mathcal {H}$ be the reconstructed flux defined on each element $T$ as $({}^{div} , v)_T= (, v)_T,$ where $v \\in \\mathbb {Q}_{l-1,l}(T) \\times \\mathbb {Q}_{l,l-1}(T)$ and $\\langle {}^{div}\\cdot , w \\rangle _e= \\langle \\cdot , w \\rangle _e, \\ \\forall e \\in \\partial T, \\ w \\in \\mathbb {P}_l(e).$ We note that the polynomial order of the post-processed space $\\mathcal {H}$ is chosen consistently with the order of the pressure space $l$ .",
"The performance of the projection is illustrated in [52]."
],
[
"Spatial Approximation of the Saturation System",
"The bilinear form of EG coupled with an entropy residual stabilization is employed for modeling the transport system (REF ) with high order approximations [55].",
"Here, again we apply DG IIPG method although other interior penalty methods can be utilized.",
"Stability and error convergence analyses for the approximation are provided in [52].",
"The EG finite element space approximation of the wetting phase saturation $s_w(,t)$ is denoted by $S_w(,t) \\in V^{\\textsf {EG}}_{h,s}(\\mathcal {T}_h) $ and we let $S_w^k := S_w(,t^k)$ for time discretization, $0 \\le k \\le N$ .",
"We set an initial condition for the saturation as $S_w^0 := \\Pi _h s_w(\\cdot ,0)$ .",
"With $P_w^{k+1}$ computed by the system (REF ) and locally conservative fluxes (REF ), the time stepping algorithm reads as follows: Given $S_w^{k-1}$ ,$S_w^{k}$ , find $S_w^{k+1} \\in V_{h,{s}}^{\\textsf {EG}}(\\mathcal {T}_h) \\mbox{ such that } (S_w^{k+1},\\psi ) = \\mathcal {G}(\\psi ), \\quad \\forall \\, \\psi \\in V_{h,s}^{\\textsf {EG}}(\\mathcal {T}_h) , \\,$ where, $(S_w^{k+1},\\psi ) =\\left( \\phi \\rho _w^{0}\\dfrac{3}{2\\Delta t} S_w^{k+1}, \\psi \\right)_{\\mathcal {T}_h}{- (\\rho _w^{0} S_w^{k+1} (f_w^{k+1})^-, \\psi )_{\\mathcal {T}_h} }$ and $\\mathcal {G}(\\psi )&=\\left( \\phi \\dfrac{2\\rho _w^{0}}{\\Delta t} S_w^{k}-\\phi \\dfrac{\\rho _w^{0}}{2\\Delta t} S_w^{k-1}, \\psi \\right)_{\\mathcal {T}_h}{+( \\rho _w^{0} (f^{k+1}_w)^+, \\psi )_{\\mathcal {T}_h}}- ({\\rho _w^{0}} \\nabla \\cdot _w^{k+1,*}, \\psi )_{\\mathcal {T}_h} \\nonumber \\\\&=\\left( \\phi \\dfrac{2\\rho _w^{0}}{\\Delta t} S_w^{k}-\\phi \\dfrac{\\rho _w^{0}}{2\\Delta t} S_w^{k-1}, \\psi \\right)_{\\mathcal {T}_h}+(\\rho _w^{0} { (f^{k+1}_w)^+ }, \\psi )_{\\mathcal {T}_h}+ ({\\rho _w^{0}}_w^{k+1,*}, \\nabla \\psi )_{\\mathcal {T}_h}- \\left< {\\rho _w^{0}} _w^{k+1,*} \\cdot , {\\psi } \\right>_{}$ The injection/production term $f_w^{k+1}:= \\tilde{s}_w^{k+1} q^{k+1}_w $ splits by $(f_w^{k+1})^+ = \\max (0,f_w^{k+1}) \\quad \\text{ and } \\quad (f_w^{k+1})^- = \\min (0,f_w^{k+1}).$ Recall that $\\tilde{s}_w^{k+1}$ is the injected saturation if $q_w^{k+1} > 0$ and is the resident saturation if $q_w^{k+1} < 0$ .",
"The computed locally conservative numerical fluxes in the section REF are applied here."
],
[
"Entropy residual stabilization",
"Elimination of spurious numerical oscillations due to sharp gradients in the solution requires stabilizations for the high order approximation to the transport system ($s \\ge 1$ ).",
"In this section, we describe an entropy viscosity stabilization technique to avoid oscillations in the EG formulation (REF ).",
"This method was introduced in [38] and mathematical stability properties are discussed in [14] for CG and in [77] for DG.",
"Recently, it was employed for EG single phase miscible displacement problems [55] by the authors.",
"Here, we provide an extension to two-phase flow saturation equation.",
"We redefine the velocity term for the two-phase flow system by separating the relative permeability which is a function of saturation, as is frequently referred to as expanded mixed form [6].",
"We let $_i &= - \\dfrac{k_i(s_w) }{\\mu _i} \\left( \\nabla p_i - \\rho _i \\right) \\\\&= k_i(s_w) \\hat{}_i,$ where $\\hat{}_i := - \\dfrac{}{\\mu _i} \\left( \\nabla p_i - \\rho _i \\right), \\; \\; i\\in \\lbrace n,w\\rbrace .$ Now, we introduce a numerical dissipation term ${\\mathcal {E}}(S_w^{k+1}, \\psi )$ in (REF ) to obtain, $\\mathcal {M}(S_w^{k+1}, \\psi ) +\\mathcal {E}(S_w^{k+1}, \\psi )= {\\mathcal {G}}(\\psi ), \\quad \\forall \\psi \\in V_{h,s}^{\\textsf {EG}}(\\mathcal {T}_h),$ where ${\\mathcal {E}}(S_w^{k+1}, \\psi ):=\\left( {\\rho ^0_w} {\\mu }^{k+1}_{\\text{Stab}}(S_w,\\hat{}_{i}) _{|T} \\nabla S_w^{k+1}, \\nabla \\psi \\right)_{\\mathcal {T}_h} \\\\-\\left\\langle {\\rho ^0_w {\\mu }^{k+1}_{\\text{Stab}}(S_w,\\hat{}_{i}) _{|T} \\nabla S_w^{k+1} }, {\\psi } \\right\\rangle _{\\mathcal {E}_h^{I} }+\\left\\langle { \\dfrac{\\alpha _T}{h_{e}} \\rho ^0_w {\\mu }^{k+1}_{\\text{Stab}}(S_w,\\hat{}_{i}) _{|T}} {S_w^{k+1}}, {\\psi } \\right\\rangle _{\\mathcal {E}_h^{I}},$ and $\\alpha _T$ is a penalty parameter.",
"Here ${\\mu }^{k+1}_{\\text{Stab}}(S_w,\\hat{}_{i}) _{|T}:\\Omega \\times [0,\\mathbb {T}] \\rightarrow \\mathbb {R}$ is the stabilization coefficient, which is piecewise constant over the mesh $T$ .",
"It is defined on each $T \\in \\mathcal {T}_h$ by $\\rho ^0_w{\\mu }^{k+1}_{\\text{Stab}}(S_w,\\hat{}_{i}) _{|T} :=\\min (\\rho ^0_w \\mu ^{k+1}_{{\\textsf {Lin}}}(S_w,\\hat{}_{i})_{|T} ,\\rho ^0_w \\mu ^{k+1}_{{\\textsf {Ent}}}(S_w,\\hat{}_w)_{|T} ).$ The main idea of the entropy residual stabilization is to split the stabilization terms into $\\mu ^{k+1}_{{\\textsf {Lin}}}$ and $\\mu ^{k+1}_{{\\textsf {Ent}}}$ .",
"If $S_w(\\cdot , t)$ is smooth, the entropy viscosity stabilization $\\mu ^{k+1}_{{\\textsf {Ent}}}(S_w,\\hat{}_w)_{|T}$ will be activated, since $\\mu ^{k+1}_{{\\textsf {Ent}}}$ is small.",
"However, the linear viscosity ${\\mu }^{k+1}_{{\\textsf {Lin}}}(S_w,\\hat{}_{i})_{|T}$ is activated where $S_w(\\cdot , t)$ is not smooth.",
"The first order linear viscosity is defined by, ${\\mu }^{k+1}_{{\\textsf {Lin}}}(S_w,\\hat{}_w)_{|T} := \\lambda _{{\\textsf {Lin}}} h_T\\Vert \\max _{i\\in \\lbrace n,w\\rbrace }(k^{\\prime }_i({S^{k+1,*}_w}) \\ \\hat{}_{i}^{k+1}) \\Vert _{L^{\\infty }(T)},\\quad \\forall T \\in \\mathcal {T}_h ,$ where $h_T$ is the mesh size and $\\lambda _{{\\textsf {Lin}}}$ is a positive constant.",
"We note that $s_w$ is transported by $\\hat{}_w$ and $s_n = 1-s_w$ is transported by $\\hat{}_n$ .",
"Next, we describe the entropy viscosity stabilization.",
"Recall that it is known that the scalar-valued conservation equation ${\\partial _t} (\\phi \\rho _w s_w) + \\nabla \\cdot {v}(s_w) = \\rho _w {f_w}$ may have one weak solution in the sense of distributions satisfying the additional inequality ${\\partial _t} (\\phi \\rho _w E(s_w)) +\\nabla \\cdot {F}(s_w)- E^{\\prime }(s_w) \\rho _w {f_w} \\le 0,$ for any convex function $E \\in \\mathcal {C}^0(\\Omega ;\\mathbb {R})$ which is called entropy and ${F}^{\\prime }(s_w):= E^{\\prime }(s_w) {v}^{\\prime }(s_w)$ , the associated entropy flux [49], [61].",
"The equality holds for smooth solutions.",
"For the two-phase flow system, we redefined the velocity in (REF ) to split the relative permeability.",
"Thus, we set ${v}(s_w) := {\\rho ^0_w} k_w(s_w) \\hat{}_w$ .",
"Then we obtain ${F}^{\\prime }(s_w) = ({\\rho ^0_w} k_w^{\\prime }(s_w) \\hat{}_w) \\cdot E^{\\prime }(s_w)$ and $\\nabla \\cdot {F}(s_w) = {F}^{\\prime }(s_w) \\cdot \\nabla s_w$ .",
"Note that we can rewrite $\\nabla E(s_w) = E^{\\prime }(s_w)\\nabla s_w$ .",
"We define the entropy residual which is a reliable indicator of the regularity of $s_w$ as $R_{\\textsf {Ent}}^{k+1}(S_w,\\hat{}_w) :={\\textup {\\textsf {BDF}}_{m}({\\phi \\rho _w E(S_w^k)})}+ {\\rho ^0_w} k_w^{\\prime }({S_w^{k+1,*}}){\\hat{}_w^{k+1}} E^{\\prime }(S_w^{k+1,*})\\nabla (S_w^{k+1,*}) - E^{\\prime }(S_w^{k+1,*}) \\rho _w f_w,$ which is large when $S_w$ is not smooth.",
"In this paper, we chose $E(S_w^{k+1,*}) = \\dfrac{1}{b} |S_w^{k+1,*}|^b, \\ b \\text{ is a positive even number}$ with $b=10$ or $E(S_w^{k+1,*}) = - \\log ( | S_w^{k+1,*} (1-S_w^{k+1,*})| + \\varepsilon )$ with $\\varepsilon < 1$ as chosen in [13], [36], [55].",
"Finally, the local entropy viscosity for each step is defined as $\\mu ^{k+1}_{{\\textsf {Ent}}}(S_w, \\hat{}_w)_{|T} := \\lambda _{\\textsf {Ent}} h_T^2\\dfrac{{ER_{\\textsf {Ent}}^{k+1}}_{|T} }{\\Vert E(S_w^{k+1,*}) - \\bar{E}^{k+1,*} \\Vert _{L^{\\infty }(\\Omega )}},\\quad \\forall T \\in \\mathcal {T}_h,$ where ${ER_{\\textsf {Ent}}^{k+1}}_{|T} :=\\max (\\Vert R_{\\textsf {Ent}}^{k+1}\\Vert _{L^{\\infty }(T)}, \\Vert J_{\\textsf {Ent}}^{k+1}\\Vert _{L^{\\infty }(\\partial T)}).$ Here $\\lambda _{\\textsf {Ent}}$ is a positive constant to be chosen with the average $\\bar{E}^{k+1,*} := \\frac{1}{|\\Omega |} \\int _{\\Omega }E(S_w^{k+1,*}) \\ d$ .",
"We define the residual term calculated on the faces by $J_{\\textsf {Ent}}^{k+1}(S_w,\\hat{}_w) :=h^{-1}_{T} {\\hat{}_w^{k+1}} \\cdot {E(S_w^{k+1,*})}.$ The entropy stability with above residuals for discontinuous case is given with more details in [77].",
"Also, readers are referred to [38] for tuning the constants ($\\lambda _{\\textsf {Ent}}, \\lambda _{\\textsf {Lin}}$ )."
],
[
"Adaptive Mesh Refinement",
"In this section, we propose a refinement strategy by increasing the mesh resolution in the cells where the entropy residual values (REF ) are locally larger than others.",
"It is shown in [1], [64] that the entropy residual can be used as a posteriori error indicator.",
"The general residual of the system (REF ) could also be utilized as an error indicator, but this residual goes to zero as $h \\rightarrow 0$ due to consistency.",
"However, as discussed in [38], the entropy residual (REF ) converges to a Dirac measure supported in the neighborhood of shocks.",
"In this sense, the entropy residual is a robust indicator and also efficient since it is been computed for a stabilization.",
"Figure: Adaptive mesh refinement levels.",
"𝖱𝖾𝖿 T \\textsf {Ref}_T is the refinement level and ∘\\circ denotes the hanging nodes.",
"The mesh refines until 𝖱𝖾𝖿 T <R max \\textsf {Ref}_T < R_{\\max }.We denote the refinement level, $\\textsf {Ref}_T$ (see Figure REF ), to be the number of times a cell($T$ ) from the initial subdivision has been refined to produce the current cell.",
"Here, a cell $T$ is refined if its corresponding $\\textsf {Ref}_T$ is smaller than a given number $R_{\\max }$ and if $|{ER_{\\textsf {Ent}}^{k+1}}_{|T}(_T,t)| \\ge C_R\\max _{\\in } |{ER_{\\textsf {Ent}}^{k+1}}_{|T}(_T,t)|,$ where $_T$ is the barycenter of $T$ and $C_R \\in [0,1]$ .",
"The purpose of the parameter $R_{\\max }$ is to control the total number of cells, which is set to be two more than the initial $\\textsf {Ref}_T$ .",
"A cell $T$ is coarsened if $|{ER_{\\textsf {Ent}}^{k+1}}_{|T}(_T,t)| \\le C_C\\max _{\\in } |{ER_{\\textsf {Ent}}^{k+1}}_{|T}(_T,t)|,$ where $C_C \\in [0,1]$ .",
"However, a cell is not coarsened if the $\\textsf {Ref}_T$ is smaller than a given number $R_{\\min }$ .",
"Here $R_{\\min }$ is set to be two less than the initial $\\textsf {Ref}_T$ .",
"In addition, a cell is not refined more if the total number of cells are more than $\\textsf {Cell}_{\\max }$ .",
"The subdivisions are accomplished with at most one hanging node per face.",
"During mesh refinement, to initialize or remove nodal values, standard interpolations and restrictions are employed, respectively.",
"We take advantage of the dynamic mesh adaptivity feature with hanging nodes in deal.II [8] in which subdivision and mesh distribution are implemented using the p4est library [15]."
],
[
"Global Algorithm and Solvers",
"block00 = [rectangle, draw, text centered, rounded corners, text width= 4em, minimum height=4.em, node distance= 2.cm] block10 = [rectangle, draw, text centered, rounded corners, text width= 4em, minimum height=4.em, node distance= 2.2cm] block11 = [rectangle, draw, text centered, rounded corners, text width= 6em, minimum height=4em, node distance= 2.5cm] block12 = [rectangle, draw, text centered, rounded corners, text width=6em, minimum height=4em, node distance= 2.8cm] block13 = [rectangle, draw, text centered, rounded corners, text width=4em, minimum height=4em, node distance= 2.45cm] line11 = [draw, -latex'] line = [draw, -latex'] Figure: Flowchart of global solution algorithm.We present our global algorithm in Figure REF for modeling the two-phase flow problem.",
"An efficient solver developed in [51] is applied to solve the EG pressure and saturation system separately.",
"The current solver is GMRES Algebraic Multigrid(AMG) block diagonal preconditioner.",
"$H$ (div) projection is activated only for incompressible cases.",
"The entropy residuals are employed when solving the transport system as well as refining the mesh.",
"The authors created the EG two-phase flow code to compute the following numerical examples based on the open-source finite element package deal.II [8] which is coupled with the parallel MPI library [33] and Trilinos solver [40]."
],
[
"Numerical Examples",
"This section verifies and demonstrates the performance of our proposed EG algorithm.",
"First, the convergence of the spatial errors are shown for the two-phase EG flow system for decoupled, sequential and iterative IMPES.",
"Next, several numerical examples with capillary pressure, gravity and dynamic mesh adaptivity including a benchmark test are provided."
],
[
"Example 1. Convergence Tests - decoupled case with entropy residual stabilization.",
"Here we consider the two-phase flow problem with exact solution given by $p_w = \\cos (t+x-y), \\; \\; \\; s_w = \\sin (t+x-y+1)$ in the domain $\\Omega = (0,{1}{})^2$ .",
"A Dirichlet boundary condition is applied for the pressure system.",
"Figure: Example 1.",
"Given capillary pressure () valuesandrelative permeabilities ().The capillary pressure is defined as $p_c(s_w) := \\dfrac{B_c}{\\sqrt{K}} \\log ({s}_w+\\varepsilon _s),$ where $K$ is the absolute permeability in Darcy scale (i.e ${1}{D} = {9.869233e-13}{m^2})$ and $K = K_D I$ with $K_D={e-5}{D}$ , where $I$ is an identity matrix, $B_c = -{0.0001}$ and $\\varepsilon _s = 0.01$ to avoid zero singularity (see Figure REF ).",
"If $s_w + \\varepsilon _s \\ge 1$ then we set to $s_w + \\varepsilon _s=1$ .",
"Relative permeabilities are given as a function of the wetting phase saturation, $k_w(s_w) := s_w^2, \\ \\text{ and } \\ k_n(s_w) := (1-s_w)^2 ;$ see Figure REF for more details.",
"In addition, we define following the parameters: $\\mu _w = {1}{cp}$ , $\\mu _n = {2}{cp}$ , $\\rho _w = \\rho _n = {1000}{kg/m^3}$ , $=[0,{-9.8}{m/s^2}]/101325$ (scaling with pressure (atm) ${1}{atm} = {101325}{}$ ), $c_w^F={e-12}$ , and $\\phi = 0.8$ .",
"We illustrate the convergence of EG flow (REF ) and EG saturation (REF ), separately for the two-phase flow system with capillary pressure.",
"In this case, exact values of $s_w(t^{k})$ and $s_w(t^{k-1})$ are provided to compute $P_w^{k+1}$ , and exact values of $p_w(t^{k})$ and $p_w(t^{k-1})$ are provided to compute each $S_w^{k+1}$ .",
"The entropy residual stabilization term (REF ) discussed in Section REF is included with $\\lambda _{\\textsf {Ent}} =\\lambda _{\\textsf {Lin}} ={e-2}$ and entropy function (REF ) chosen with $\\varepsilon = {e-4}$ .",
"The penalty coefficients are set as $\\alpha = 100$ and $\\alpha _T = 0.01$ .",
"For each of the flow and transport equations, respectively, five computations on uniform meshes were computed where the mesh size $h$ is divided by two for each cycle.",
"The time discretization is chosen fine enough not to influence the spatial errors and the time step $\\Delta t$ is divided by two for each cycle.",
"Each cycle has $100,200,400,800$ and 1600 time steps and the errors are computed at the final time $\\mathbb {T}=0.1$ .",
"Figure: Example 1.",
"Decoupled case.",
"Error convergence rates for pressure and saturation in semi-H 1 H^1 norm and L 2 L^2 norm, respectively.Optimal order of convergences are observed for both linear and quadratic order cases.The behavior of the $H^1(\\Omega )$ semi norm errors for the approximated pressure solution versus the mesh size $h$ are depicted in Figure REF .",
"Next, the $L^2(\\Omega )$ error for the approximated saturation solutions versus the mesh size is illustrated in Figure REF .",
"Both linear and quadratic orders ($l,s=1,2$ ) were tested and the optimal order of convergences as discussed in [51] are observed."
],
[
"Example 2. Convergence Tests - coupled case",
"In this section, we solve the same problem as in the previous example but with a pressure and saturation system coupled.",
"Here, two different algorithms were tested and compared: sequential IMPES (Section REF ) and iterative IMPES (Section REF ).",
"The convergences of the errors for the pressure and the saturation are provided in Figures REF and REF .",
"We observed that the optimal rates of convergence for the high order cases ($l=2, s=2$ ) are obtained for both the sequential and iterative IMPES scheme.",
"Here the tolerance was set to $\\varepsilon _{I}= {e-10}$ and 3-4 iterations were required for the convergence at each time step for iterative IMPES.",
"Figure: Example 2.",
"Coupled case (sequential IMPES).",
"Error convergence rates for pressure and saturation in semi-H 1 H^1 norm and L 2 L^2 norm, respectively.Figure: Example 2.",
"Coupled case (iterative IMPES).",
"Error convergence rates for pressure and saturation in H 1 H^1 semi norm and L 2 L^2 norm, respectively."
],
[
"Example 3. A homogeneous channel.",
"In this example, we illustrate the computational features of our algorithms including entropy viscosity stabilization and dynamic mesh adaptivity with zero capillary pressure.",
"The computational domain is $\\Omega =({0},{0}{})\\times ({1.25}{},{0.5}{})$ and the domain is saturated with a non-wetting phase, residing fluid ($s^0_n = 1$ and $s^0_w = 0$ ).",
"A wetting phase fluid is injected at the left-hand side of the domain, thus $p_{w,\\textsf {in}} = {1}{atm},\\ s_{w,\\textsf {in}}=1 \\ \\text{ on } \\ x={0}{}.$ On the right hand side, we impose $p_{w,\\textsf {out}} = {0}{atm} \\ \\text{ on } \\ x={1.25}{},$ and no-flow boundary conditions on the top and the bottom of the domain.",
"Fluid and rock properties are given as $\\mu _w = {1}{cP}$ , $\\mu _n = {3}{cP}$ , $\\rho _w = {1000}{kg/m^3}$ , $\\rho _n={830}{kg/m^3}$ , $K_D = {1}{D}$ , $c_w^F = {e-8}$ and $\\phi = 0.2$ .",
"Relative permeabilities are given as a function of the wetting phase saturation (REF ), and the capillary pressure is set to zero for this case.",
"The penalty coefficients are set as $\\alpha = 100$ and $\\alpha _T = 100$ .",
"Figure REF illustrates the wetting phase saturation ($s_w$ ) at the time step number $k=50$ with the entropy stabilization coefficients ($\\lambda _{\\textsf {Ent}} = 0.1$ , $\\lambda _{\\textsf {Lin}}=1$ ) and entropy function (REF ) chosen with $\\varepsilon = {e-4}$ .",
"Dynamic mesh adaptivity is employed with initial refinement level $\\textsf {Ref}_T =4$ , maximum refinement level $R_{\\max }=6$ and minimum refinement level $R_{\\min }=2$ .",
"Here $C_R$ is chosen to mark and refine the cells which represent the top 20$\\%$ of the values (REF ) over the domain and $C_C$ is chosen to mark and coarsen the cells which represent the bottom 5$\\%$ of the values (REF ) over the domain.",
"The initial number of cells was approximately 2000 and maximum cell number was approximately 6000 with a minimum mesh size $h_{\\min }={1.1e-02}$ .",
"The uniform time step size was chosen as $\\Delta t = {5e-3}$ (CFL constant around $0.5$ ).",
"Figure REF plots the values of $S_w$ over the fixed line $y={0.25}{}$ .",
"We observe a saturation front without any spurious oscillations.",
"In addition, Figure REF presents the adaptive mesh refinements and entropy residual values (REF ) at the time step number $k=50$ .",
"This choice of stabilization (REF ) performs as expected; see Figure REF .",
"We note that the linear viscosity (REF ) is chosen where the entropy residual values are larger."
],
[
"Example 4. A layered three dimensional domain",
"This example presents a three dimensional computation in $\\Omega =(0,{1}{})^3$ with a given heterogeneous domain, see Figure REF for details and boundary conditions.",
"Permeabilities are defined as $K_D= \\max ( \\exp (-d_1^2/0.01), 0.01) $ , where $d_1 = | y -0.75 - 0.1 * \\sin (10 x) |$ for $y>0.5$ and $K_D= \\max ( \\exp (-d_2^2/0.01), 0.01) $ , where $d_2 = | y -0.25 - 0.2 * \\sin (x) |$ for $y<0.5$ .",
"All other physical parameters are the same as in the previous example.",
"Figure: Example 4.",
"Setup with a given permeability (K D K_D values).Figure REF illustrates the wetting phase saturation ($S_w$ ) at the time step number $k=10, 50, 200, $ and 300 with the entropy stabilization coefficients ($\\lambda _{\\textsf {Ent}} = 0.25$ , $\\lambda _{\\textsf {Lin}}=0.5$ ) and entropy function (REF ) chosen with $\\varepsilon = {e-3}$ .",
"Dynamic mesh adaptivity is employed with $\\textsf {Ref}_T =4$ , $R_{\\max }=6$ and $R_{\\min }=2$ .",
"The number of cells at $k=300$ is around 262100 and the minimum mesh size is $h_{\\min }={0.027}$ with a time step size $\\Delta t = 0.006$ (CFL constant is 1).",
"See figures REF -REF for adaptive mesh refinements for different time steps.",
"The adaptive mesh refinement strategy becomes very efficient for large-scale three dimensional problems using parallelization.",
"Figure: Example 4.The wetting phase saturation (S w S_w) at each time step number with adaptive mesh refinements."
],
[
"Example 5. A benchmark: effects of capillary pressure",
"In this example, we emphasize the effects of capillary pressure in a heterogeneous media as shown in [41], [74].",
"Here, we impose layers of different permeabilities in the computational domain $\\Omega =({0}{},{0}{})\\times ({1.25}{},{0.875}{})$ .",
"See Figure REF .",
"Figure: Example 5.",
"Two dimensional domain with heterogeneous permeabilities.",
"Layered setup to test the effect of the capillary pressure.",
"Permeabilities are defined as K D =0.01DK_D={0.01}{D} for the dark region and K D =1DK_D={1}{D} for the white region.Figure: Example 5.",
"Wetting phase saturation values at each time step number.",
"The left column (a),(c), and (e) are the values with the capillary pressure and the right column (b),(d), and (f) are the values without the capillary pressure.The domain is saturated with a non-wetting phase (oil), i.e $s^0_n = 1$ and $s^0_w = 0$ .",
"A wetting phase fluid is injected at the left-hand side of the domain, thus $p_{w,\\textsf {in}} = {0.1}{atm},\\ s_{w,\\textsf {in}}=1 \\ \\text{ on } \\ x={0}{}.$ On the right hand side, we impose $p_{w,\\textsf {out}} = {0}{atm} \\ \\text{ on } \\ x={1.25}{},$ and no-flow boundary conditions on the top and the bottom of the domain.",
"Fluid properties are set as $\\mu _w = {1}{cP}$ , $\\mu _n = {0.45}{cP}$ , $\\rho _w = {1000}{kg/m^3}$ , $\\rho _n={660}{kg/m^3}$ , $c_w^F = {e-8}$ , $\\phi = 0.2$ , and $K_D = {1}{D}$ or $K_D = {0.01}{D}$ as illustrated in Figure REF .",
"Relative permeabilities are given as a function of the wetting phase saturation (REF ), and the penalty coefficients are set as $\\alpha = 1$ , $\\alpha _c = 1$ and $\\alpha _T = 1000$ .",
"The entropy stabilization coefficients are $\\lambda _{\\textsf {Ent}} = 1$ and $\\lambda _{\\textsf {Lin}}=1$ .",
"Dynamic mesh adaptivity is employed as same as the example 3 and the minimum mesh size is $h_{\\min }={0.0027}$ .",
"The uniform time step size is taken as $\\Delta t = {0.005}$ .",
"The capillary pressure (REF ) is given with $B_c = -0.01$ and $\\varepsilon _s=0.1$ .",
"Here two tests are performed, one with the capillary pressure ($B_c = -0.01$ ) and a second with zero capillary pressure ($B_c = 0$ ).",
"The differences and effects of capillary pressure are depicted at Figure REF for different time steps.",
"The injected wetting phase water flows faster in the high permeability layers but is more diffused in the case with capillary pressure as shown in previous results [41], [74].",
"One can observe the capillary pressure is a non-linear diffusion source term for the residing non-wetting phase.",
"This causes more uniformed movement of the injected fluid."
],
[
"Example 6. A random heterogeneous domain with different relative permeability",
"This example considers well injection and production in a random heterogeneous domain $\\Omega = ({0},{1}{})^2$ .",
"Wells are specified at the corners with injection at $(0,0)$ and production at $({1}{},{1}{})$ .",
"See Figure REF for the setup.",
"We test and compare two different non-wetting phase relative permeabilities such as $\\text{i) }k^1_n(s_w) := {(1-s_w)^2}\\ \\text{ and } \\ \\text{ii) }k^2_n(s_w) := \\dfrac{(1-s_w)^2}{f_w} ,$ where the latter is often referred as the case with foam in a porous media [59].",
"Here, $f_w := 1 + R (0.5 + \\dfrac{1}{\\pi }\\arctan (\\kappa (s_w - s_w^*)))$ is a mobility reduction factor with a constant positive parameters set to $R=10$ , $\\kappa = 100$ , and a limiting water saturation $S_w^*=0.3$ .",
"Figure REF illustrates two different non-wetting phase relative permeabilities ($k^1_n, k^2_n$ ).",
"The wetting phase relative permeability ($k_w$ ) is identical with the previous examples.",
"Figure: Example 6.",
"Setup with a random absolute permeabilities,wetting phase relative permeability (k w )(k_w), andtwo different non-wetting phase relative permeabilities (k n 1 ,k n 2 k^1_n, k^2_n).",
"We note k n 2 k^2_n represents rough relative permeability which often referred as the case with foam in a porous media .Figure: Example 6.",
"S w S_w values for each time in a heterogeneous media with a non-wetting phase relative permeability k n 1 (s w )k^1_n(s_w).Figure: Example 6.",
"S w S_w values for each time in a heterogeneous media with a non-wetting phase relative permeability k n 2 (s w )k^2_n(s_w).We assume the domain is saturated with a non-wetting phase, i.e $s^0_n = 1$ and $s^0_w = 0$ and a wetting phase fluid is injected.",
"Fluid and rock properties are given as $\\mu _w = {1}{cP}$ , $\\mu _n = {3}{cP}$ , $\\rho _w = {1000}{kg/m^3}$ , $\\rho _n={830}{kg/m^3}$ , $c_w^F = {e-10}$ , $f_w^+ = {100}{m/s}$ , $f_w^-= -{100}{m/s}$ , $f_n = 0$ , and $\\phi = 0.2$ .",
"The capillary pressure and the gravity is neglected to emphasize the effects of heterogeneity and different non-wetting phase relative permeability.",
"Here the numerical parameters are chosen as $h_{\\min }={1.1e-02}$ and $\\Delta t = {3.8e-03}$ .",
"Due to the dynamic mesh refinement ($\\textsf {R}_{\\max } = 7$ and $\\textsf {R}_{\\min } = 2$ ), the number of degrees of freedom for EG transport and the maximum number of cells are 32158, 15934, respectively at the final time $\\mathbb {T}=15$ .",
"The entropy stabilization coefficients are set to $\\lambda _{\\textsf {Ent}} = 0.1$ and $\\lambda _{\\textsf {Lin}}=0.25$ , where the entropy function (REF ) is chosen with $\\varepsilon = {e-3}$ .",
"The penalty coefficients are set as $\\alpha = 1$ and $\\alpha _T = 1000$ .",
"Figure REF illustrates the EG-$_1$ solution of $S_w$ values for each time in a heterogeneous media with a non-wetting phase relative permeability $k^1_n(s_w)$ .",
"Next, Figure REF is the case with $k^2_n(s_w)$ .",
"We note that wetting phase saturation values above $S_w^*$ are restricted for the latter case due to the relative permeability, $k^2_n(s_w)$ ."
],
[
"Example 7. A three dimensional random heterogeneous domain",
"In this example, we simply extend the previous example to a three dimensional domain $\\Omega = ({0},{1}{})^3$ with absolute permeabilities given as figure REF .",
"Wells are specified at the corners with injection at $(0,0,0)$ and production at $({1}{},{1}{},{1}{})$ .",
"The numerical parameters are chosen as $h_{\\min }={5.4e-02}$ and $\\Delta t = {3.4e-03}$ .",
"All the other physical parameters and boundary conditions are the same as in the previous example.",
"Figure REF illustrates the contour value of $S_w = 0.3$ for each time step.",
"Here the maximum EG-$_1$ degrees of freedom for wetting phase saturation at the final time step is around $70,000$ and this example is computed by employing four multiple parallel processors (MPI).",
"Figure: Example 7.",
"Contour value S w =0.3S_w = 0.3 for each time step."
],
[
"Example 8. Well injections with gravity and a capillary pressure",
"Figure REF illustrates an example of an existing reservoir where we have sliced a computational domain vertically, $\\Omega = ({0}{},{50}{})^2$ as shown in Figure REF .",
"Wells are rate specified at the corners with injection at $(0,0)$ and production at $({50}{},{50}{})$ .",
"A high permeability zone representing long sediments is located at ($y \\ge 0.16x^2 - 7.78x + 112.22$ ), where $K_D={10}{D}$ and $K_D={1}{D}$ otherwise.",
"We assume the domain is saturated with a non-wetting phase, i.e $s^0_n = 1$ and $s^0_w = 0$ and a wetting phase fluid is injected.",
"Fluid and rock properties are given as $\\mu _w = {1}{cP}$ , $\\mu _n = {3}{cP}$ , $\\rho _w = {1000}{kg/m^3}$ , $\\rho _n={830}{kg/m^3}$ , $c_w^F = {e-10}$ , $f_w^+ = {2.5}{m/s}$ , $f_w^- = -{2.5}{m/s}$ , $f_n = 0$ , and $\\phi = 0.2$ .",
"Relative permeabilities are given as functions of the wetting phase saturation (REF ), and the capillary pressure is set with $B_c = -0.001$ and $\\varepsilon _s=0.1$ .",
"The penalty coefficients are set as $\\alpha = 1$ , $\\alpha _c = 1$ and $\\alpha _T = 1000$ and the time step is set by $\\Delta t= {0.18}$ .",
"Here, we employ the gravity $=[0,{-9.8}{m/s^2}$ ], and for the same scaling with pressure (atm), we divide it by 101325 (${1}{atm} = {101325}{}$ ).",
"Figure REF illustrates the injected wetting phase saturation values for each time step number.",
"We observe the effect of the gravity.",
"Figure: Example 6.",
"Entropy choices for early and later time.The entropy stabilization coefficients are set as $\\lambda _{\\textsf {Ent}} = 40$ and $\\lambda _{\\textsf {Lin}}=1$ , where the entropy function (REF ) is chosen with $\\varepsilon = {e-3}$ .",
"Figure REF illustrates the choice for stabilization.",
"Dynamic mesh adaptivity is employed with initial refinement level $\\textsf {Ref}_T =4$ , $R_{\\max }=7$ and $R_{\\min }=3$ with a minimum mesh size is $h_{\\min }={0.4}$ .",
"In addition, Figure REF presents the production data.",
"The oil saturation values (non-wetting phase $S_n$ ) over the time are plotted with the accumulative oil production rate ($\\sum _{k=0}^\\mathbb {T} |S_n {f}^-|$ ).",
"Figure: Example 6.",
"Production data"
],
[
"Conclusion",
"In this paper, we present enriched Galerkin (EG) approximations for two-phase flow problems in porous media with capillary pressure.",
"EG preserves local and global conservation for fluxes and has fewer degrees of freedom compared to DG.",
"For a high order EG transport system, entropy residual stabilization is applied to avoid spurious oscillations.",
"In addition, dynamic mesh adaptivity employing entropy residual as an error indicator reduces computational costs for large-scale computations.",
"Several examples in two and three dimensions including error convergences and a well known capillary pressure benchmark problem are shown in order to verify and demonstrate the performance of the algorithm.",
"Additional challenging effects arising from gravity and rough relative permeabilities for foam are presented."
],
[
"Acknowledgments",
"The research by S. Lee and M. F. Wheeler was partially supported by a DOE grant DE-FG02-04ER25617 and Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences, DOE Project $\\#$ DE-SC0001114.",
"M. F. Wheeler was also partially supported by Moncrief Grand Challenge Faculty Awards from The Institute for Computational Engineering and Sciences (ICES), the University of Texas at Austin."
]
] | 1709.01644 |
[
[
"Multiplicity theorem of singular Spectrum for general Anderson type\n Hamiltonian"
],
[
"Abstract In this work, we focus on the multiplicity of singular spectrum for operators of the form $A^\\omega=A+\\sum_{n}\\omega_n C_n$ on a separable Hilbert space $\\mathcal{H}$, for a self-adjoint operator $A$ and a countable collection $\\{C_n\\}_{n}$ of non-negative finite rank operators.",
"When $\\{\\omega_n\\}_n$ are independent real random variables with absolutely continuous distributions, we show that the multiplicity of singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of eigenvalues of $\\sqrt{C_n}(A^\\omega-z)^{-1}\\sqrt{C_n}$ for all $n$ and almost all $(z,\\omega)$.",
"The result is optimal in the sense that there are operators where the bound is achieved.",
"Using this, we also provide effective bounds on multiplicity of singular spectrum for some special cases."
],
[
"Introduction",
"Spectral theory of random operators is an important field of study, and within it, the Anderson tight binding model and random Schrödinger operator have gained significant attention.",
"Over the years much attention has been given to the nature of their spectrum.",
"But to completely characterize the structure of the operator, information on the multiplicity is also important.",
"Here we pay attention to the multiplicity of the singular spectrum for certain class of random operators.",
"One of the well studied class of random operators is the Anderson tight binding model.",
"Many results about its spectrum are known, for example: the existence of pure point spectrum is known for Anderson tight binding model over integer lattice [1], [4], [10], [16].",
"Absolutely continuous spectrum is known to exist for Anderson tight binding model over Bethe lattice [9], [17].",
"Other models where the pure point spectrum is known to exist includes random Schrödinger operator [3], [7], [11], [20], multi-particle Anderson model [2], [5], [19] and magnetic Schrödinger operators [8], [32].",
"There are important results which also concentrate on the multiplicity of the singular spectrum.",
"For the Anderson tight binding model, Simon [31], Klein-Molchanov [18] have shown the simplicity of pure point spectrum.",
"For Anderson type models when the randomness acts as rank one perturbations, Jakšić-Last [13], [15] showed that the singular spectrum is simple.",
"For random Schrödinger operator, in the regime of exponential decay of Green's function, Combes-Germinet-Klein [6] showed that the spectrum is simple.",
"Other work includes [29], where Sadel and Schulz-Baldes provided multiplicity result for absolute continuous spectrum for random Dirac operators with time reversal symmetry.",
"But general results concerning the multiplicity of the spectrum are not known.",
"One of the difficulties involving multiplicity results for random Schrödinger operator or multi-particle Anderson model is that the randomness acts as perturbation over an infinite rank operator.",
"Randomness acting through perturbation by a finite rank operator is an intermediate step between Anderson tight binding model and random Schrödinger operator.",
"Some example of such a random operator is Anderson dimer/polymer model, Toeplitz/Hankel random matrix, and random conductance model.",
"Here we will deal with Anderson type operators and provide multiplicity result for the singular spectrum when the randomness acts through perturbation by a finite rank non-negative operator.",
"This work is similar to the work done by Jakšić-Last [13], [15] and is a generalization and extension of the work done by Mallick [23].",
"Though this work does not answer the question about the multiplicity of singular spectrum for random Schrödinger operator but it is a step towards it.",
"The technique involved in the proof does not distinguish between point spectrum and singular continuous spectrum, so stated results are true for whole of the singular spectrum.",
"For a densely defined self-adjoint operator $A$ with domain $\\mathcal {D}(A)$ on a separable Hilbert space ${H}$ and a countable collection of finite rank non-negative operator $\\lbrace C_n\\rbrace _{n\\in \\mathcal {N}}$ , define the random operator $A^\\omega =A+\\sum _{n\\in \\mathcal {N}} \\omega _n C_n,$ where $\\lbrace \\omega _n\\rbrace _{n\\in \\mathcal {N}}$ are independent real random variables with absolutely continuous distribution.",
"Let $(\\Omega ,\\mathcal {B},\\mathbb {P})$ denote the probability space such that $\\omega _n$ are random variables over $\\Omega $ .",
"We will assume that $A^\\cdot :\\Omega \\rightarrow \\mathcal {S}({H})$ is an essentially self adjoint operator valued random variable.",
"This is a necessary assumption because otherwise there can be multiple self-adjoint extensions for the symmetric operator $A^\\omega $ .",
"The assumption itself is not too restrictive and a large class of operators satisfy this condition.",
"For example, if $A$ is bounded self-adjoint, $\\lbrace C_n\\rbrace _n$ are finite rank non-negative operators satisfying $C_nC_m=C_mC_n=0$ for any $n\\ne m$ and the distributions of the random variables $\\omega _n$ are supported in some fixed compact set $[-K,K]$ , then the operator $A^\\omega $ is almost surely bounded and self adjoint.",
"Anderson polymer/dimer model falls into this category of operators.",
"For the main result we need to focus on the linear maps $G^\\omega _{n,n}(z):=P_n(A^\\omega -z)^{-1}P_n:P_n{H}\\rightarrow P_n{H}$ for $z\\in \\mathbb {C}\\setminus \\mathbb {R}$ , where $P_n$ is projection onto the range of $C_n$ .",
"Using functional calculus, it is easy to see that the linear operator $G^\\omega _{n,n}(z)$ can be viewed as a matrix over $P_n{H}$ (after fixing a basis of $P_n{H}$ ), which belongs to the set of matrix valued Herglotz functions.",
"Using the representation of matrix-valued Herglotz functions (see [12]), we can extract all the properties of the spectral measure over the minimal closed $A^\\omega $ -invariant subspace containing $P_n{H}$ .",
"We will use the notation $Mult^\\omega _n(z)$ to denote the maximum multiplicity of the root of the polynomial $\\det (C_n G^\\omega _{n,n}(z)-xI)$ in $x$ , for $z\\in \\mathbb {C}\\setminus \\mathbb {R}$ , where $C_n$ and $G^\\omega _{n,n}(z)$ are viewed as a linear operator on $P_n{H}$ and so $I$ denotes the identity operator on $P_n{H}$ .",
"Since $C_n>0$ on $P_n{H}$ we have $\\det (C_n G^\\omega _{n,n}(z)-xI)=\\det (\\sqrt{C_n} G^\\omega _{n,n}(z)\\sqrt{C_n}-xI),$ because similarity transformation preserves determinant.",
"This is the reason why algebraic multiplicity of $\\sqrt{C_n}(A^\\omega -z)^{-1}\\sqrt{C_n}$ can also be used instead of $C_n G^\\omega _{n,n}(z)$ .",
"With these notations, we state our main result: Theorem 1.1 Let $A$ be a densely defined self-adjoint operator with domain $\\mathcal {D}(A)$ on a separable Hilbert space ${H}$ and $\\lbrace C_n\\rbrace _{n\\in \\mathcal {N}}$ be a countable collection of finite rank non-negative operator.",
"Denote $P_n$ to be the projection onto the range of $C_n$ and let $\\sum _n P_n=I$ .",
"Let $\\lbrace \\omega _n\\rbrace _{n\\in \\mathcal {N}}$ be a sequence of independent real random variables on the probability space $(\\Omega ,\\mathcal {B},\\mathbb {P})$ with absolutely continuous distribution.",
"Let $A^\\omega $ given by (REF ) be a family of essentially self adjoint operators.",
"Then For any $n\\in \\mathcal {N}$ $\\mathop {\\text{ess-sup}}_{z\\in \\mathbb {C}\\setminus \\mathbb {R}}Mult^\\omega _n(z)$ is constant for almost all $\\omega $ , which will be denoted by $\\mathcal {M}_n$ .",
"If $\\sup _{n\\in \\mathcal {N}} \\mathcal {M}_n<\\infty $ , then the multiplicity of singular spectrum for $A^\\omega $ is upper bounded by $\\sup _{n\\in \\mathcal {N}} \\mathcal {M}_n$ , for almost all $\\omega $ .",
"Remark 1.2 There are few observations to be made: Note that if $range(C_n)\\subset \\mathcal {D}(A)$ for all $n$ , then the subspace $\\mathcal {D}:=\\left\\lbrace \\sum _{i=1}^N \\phi _i:\\phi _i\\in range(C_{n_i}), n_i\\in \\mathcal {N}~\\forall 1\\le i\\le N,\\forall N\\in \\mathbb {N}\\right\\rbrace ,$ is dense and is the domain of $A^\\omega $ .",
"If either $A$ is bounded or $\\sup _n |\\omega _n|\\left\\Vert C_n\\right\\Vert $ is finite, then it is easy to show that $A^\\omega $ is essentially self adjoint.",
"Note that although $\\lbrace C_n\\rbrace _n$ are finite rank operators, there may not be a universal upper bound on their ranks.",
"An easy example of such an operator is $H^\\omega =\\Delta +\\sum _{n=0}^\\infty \\omega _n \\chi _{\\lbrace x:\\left\\Vert x\\right\\Vert _\\infty =n\\rbrace },$ defined on the Hilbert space $\\ell ^2(\\mathbb {Z}^d)$ , where $\\Delta $ is the discrete Laplacian and $\\chi _{\\lbrace x:\\left\\Vert x\\right\\Vert _\\infty =n\\rbrace }$ is projection onto the subspace $\\ell ^2(\\lbrace x\\in \\mathbb {Z}^d:\\left\\Vert x\\right\\Vert _\\infty =n\\rbrace )$ .",
"We need $\\sum _n P_n=I$ so that the subspace $\\sum _n {H}^\\omega _{P_n}$ is dense in ${H}$ .",
"Here we denote ${H}^\\omega _{P_n}=\\overline{\\left\\langle f(A^\\omega )\\phi :f\\in C_c(\\mathbb {R}),\\phi \\in P_n{H} \\right\\rangle },$ where $\\overline{\\left\\langle S \\right\\rangle }$ denotes the closure of finite linear combination of elements of the set $S$ .",
"Without this condition infinite multiplicity could easily be achieved.",
"For example consider the Hilbert space $\\oplus ^2\\ell ^2(\\mathbb {Z})$ , and define the operator $H^\\omega =\\left(\\Delta +\\sum _{n\\in \\mathbb {Z}} \\omega _n\\chi _{\\lbrace nN,\\cdots ,(n+1)N-1\\rbrace }\\right)\\oplus \\left(\\sum _{n\\in \\mathbb {Z}}x_n\\left|\\delta _n\\right\\rangle \\left\\langle \\delta _n\\right|\\right)$ where $\\lbrace x_n\\rbrace _{n\\in \\mathbb {Z}}$ is a fixed sequence and $\\lbrace \\omega _n\\rbrace _{n\\in \\mathbb {Z}}$ are independent real random variables with absolutely continuous distribution.",
"Notice that first operator is Anderson like operator with simple point spectrum but the second operator can have arbitrary multiplicity depending upon the sequence $\\lbrace x_n\\rbrace _n$ .",
"Remark 1.3 To understand the conclusion of the theorem, consider the following examples: On the Hilbert space $\\ell ^2(\\mathbb {Z}\\times \\lbrace 0,\\cdots ,N\\rbrace )$ , consider the operator $H^\\omega =\\tilde{\\Delta }+\\sum _{n\\in \\mathbb {Z}}\\omega _{n}P_{n},$ where $(\\tilde{\\Delta }u)(x,y)=u(x+1,y)+u(x-1,y)\\qquad \\forall (x,y)\\in \\mathbb {Z}\\times \\lbrace 0,\\cdots ,N\\rbrace $ and the sequence of projections $P_{n}$ is given by $(P_{n}u)(x,y)=\\left\\lbrace \\begin{matrix} u(x,y) & x=n\\\\ 0 & x\\ne n\\end{matrix}\\right..$ Figure: The operator described in the remark is visualized here for N=3N=3.The operator Δ ˜\\tilde{\\Delta } is the adjacency operator over the graph ℤ×{0,⋯,3}\\mathbb {Z}\\times \\lbrace 0,\\cdots ,3\\rbrace where the edges are denoted by the black lines.The shaded region denotes the support of the projections.First observe that ${H}_k=\\lbrace u\\in \\ell ^2(\\mathbb {Z}\\times \\lbrace 0,\\cdots ,N\\rbrace ): u(x,y)=0 ~\\forall x\\in \\mathbb {Z},y\\ne k\\rbrace $ is $H^\\omega $ invariant and $\\lbrace (H^\\omega ,{H}_k)\\rbrace _{k=0}^N$ are all unitarily equivalent.",
"So any singular spectrum has multiplicity $N$ .",
"When $\\lbrace \\omega _{n}\\rbrace _{n}$ are i.i.d, there are results [21], [22], [30] which shows that $(H^\\omega ,{H}_0)$ has pure point spectrum (hence singular spectrum).",
"It is easy to show that the matrix $G^\\omega _{n,n}(z)$ is of the form $f(z)I$ , where $f$ is a Herglotz function and $I$ is identity on $\\mathbb {C}^N$ .",
"On the Hilbert space $\\ell ^2(\\mathbb {N}\\times \\mathbb {N})$ consider the operator $H^\\omega =\\tilde{\\Delta }+\\sum _{(n,m)\\in \\mathbb {N}^2} \\omega _{(n,m)}P_{(n,m)}$ where $(\\tilde{\\Delta }u)(x,y)=\\left\\lbrace \\begin{matrix} u(2,y) & x=1 \\\\ u(x+1,y)+u(x-1,y) & x\\ne 1\\end{matrix}\\right.\\qquad \\forall (x,y)\\in \\mathbb {N}\\times \\mathbb {N}$ and the projections $P_{(n,m)}$ are given by $P_{(n,m)}=\\sum _{k=2^{n(m-1)}}^{2^{nm}-1} \\left|\\delta _{(n,k)}\\right\\rangle \\left\\langle \\delta _{(n,k)}\\right|.$ Figure: The operator described in the remark is visualized here.The operator Δ ˜\\tilde{\\Delta } is the adjacency operator over the graph ℕ 2 \\mathbb {N}^2 where the edges are denoted by the black lines.",
"The shaded region denotes the support of the projections.In this example $P_{(n,m)}(H^\\omega -z)^{-1}P_{(n,m)}$ is diagonal (w.r.t.",
"the Dirac basis $\\lbrace \\delta _{(n,m)}:n,m\\in \\mathbb {N}\\rbrace $ ), and it is easy to see that $\\sup _{(n,m)\\in \\mathbb {N}} \\mathcal {M}_{(n,m)}=\\infty .$ Similar to previous example the subspace ${H}_k=\\lbrace u\\in \\ell ^2(\\mathbb {N}\\times \\mathbb {N}): u(x,y)=0~\\forall x\\in \\mathbb {N},y\\ne k\\rbrace \\qquad \\forall k\\in \\mathbb {N},$ are invariant under the action of $H^\\omega $ .",
"Notice that $\\lbrace (H^\\omega ,{H}_k)\\rbrace _{k=2^m}^{2^{m+1}-1}$ are unitarily equivalent with each other for any $m\\in \\mathbb {N}$ .",
"So the singular spectrum of $H^\\omega $ has infinite multiplicity.",
"So the conclusion of the theorem is optimal in the sense that there are random operators $A^\\omega $ such that the multiplicity of singular spectrum is $\\sup _{n\\in \\mathcal {N}} \\mathcal {M}_{n}$ .",
"The main technique in the proof involves studying the behavior of singular spectrum because of perturbation by single non-negative operator.",
"This is done through resolvent identity and so properties of matrix valued Herglotz functions plays an essential role.",
"The steps involved in the proof will be further explained in section REF .",
"In general these kind of results fails to hold without perturbation and spectral averaging [7] plays an important role.",
"Since matrix valued Herglotz functions are the primary tool, Poltoratskii's theorem [27] is used to obtain and characterize the singular measure.",
"It should be noted that our result (Theorem REF ) extends the work of Jakšić-Last [13], [15], Naboko-Nichols-Stolz [25] and Mallick [24] in the following way: in case of Jakšić-Last[13], [15], since the rank of each $P_n$ are one, above theorem gives the simplicity of singular spectrum.",
"Naboko-Nichols-Stolz [25] showed simplicity of the point spectrum for certain classes of Anderson type operator on $\\mathbb {Z}^d$ and Mallick [24] provided a bound on the multiplicity of the singular spectrum for a similar class of Anderson type operator on $\\mathbb {Z}^d$ .",
"In general it is not possible to compute $G^\\omega _{n,n}(z)$ , and so other methods has to be devised to get $\\mathcal {M}_n$ .",
"The following corollary is a possible way to bound $\\mathcal {M}_n$ for certain classes of random operators.",
"Corollary 1.4 On a separable Hilbert space ${H}$ , let $A^\\omega $ defined by (REF ) satisfy the hypothesis of theorem REF .",
"Let $range(C_n)\\subset \\mathcal {D}(A)$ for all $n\\in \\mathcal {N}$ , and let $M\\in \\mathbb {R}$ be such that $\\sigma (A)$ and $\\sigma (A^\\omega )$ are subset of $(M,\\infty )$ for almost all $\\omega $ .",
"Then If $C_n$ is a finite rank projection for all $n$ , then the multiplicity of singular spectrum for $A^\\omega $ is bounded above by $\\max _{n\\in \\mathcal {N}} \\max _{x\\in \\sigma (C_nAC_n)} dim(ker(C_nAC_n-xI)),$ where $C_nAC_n$ is viewed as a linear operator on $P_n{H}$ .",
"If $C_n$ is a non-negative finite rank operator for all $n$ , then multiplicity of singular spectrum for $A^\\omega $ is bounded above by $\\max _{n\\in \\mathcal {N}} \\max _{x\\in \\sigma (C_n)} dim(ker(C_n-xI)),$ where $C_n$ is viewed as a linear operator on $P_n{H}$ .",
"Remark 1.5 It should be noted that the above bound is not optimal, but in many cases can be computed easily.",
"As an example, for the case of remark REF (1), all we have to do is count the eigenvalue multiplicity of $\\chi _{S_r}\\Delta \\chi _{S_r}$ , where $S_r=\\lbrace x\\in \\mathbb {Z}^d:\\left\\Vert x\\right\\Vert _\\infty =r\\rbrace $ .",
"For $d=2$ , this operator is same as the Laplacian on a set of $8n$ points arranged on a circle.",
"So the multiplicity of the operator can be at most two.",
"Another simple example is for the case when $C_n$ has simple spectrum, then the singular spectrum of $A^\\omega $ is almost surely simple.",
"The corollary should be considered as a generalization of the technique developed in Naboko-Nichols-Stolz [25].",
"There the authors used the simplicity of $P_n\\Delta P_n$ to conclude the simplicity of pure point spectrum for certain type of Anderson operators on $\\ell ^2(\\mathbb {Z}^d)$ .",
"Another similar work is [24], where the author bounded $\\mathcal {M}_n$ by considering first few terms of Neumann series while keeping track of the perturbation.",
"Using an approach similar to [24], we can show that the singular spectrum for Anderson type operator on Bethe lattice is simple.",
"Let $\\mathcal {B}=(V,E)$ denote the infinite tree with root $e$ where each vertex has $K$ neighbors.",
"Set $K> 2$ so that the tree is not isomorphic to $\\mathbb {Z}$ .",
"Define the class of random operators $H^\\omega =\\Delta _{\\mathcal {B}}+\\sum _{x\\in J} \\omega _x \\chi _{\\tilde{\\Lambda }(x)}$ where $\\Delta _{\\mathcal {B}}$ is the adjacency operator on $\\mathcal {B}$ , and $\\tilde{\\Lambda }(x)=\\lbrace y\\in V: d(e,x)\\le d(e,y)~\\&~d(x,y)< l_x\\rbrace ,$ for some $l_\\cdot :V\\rightarrow \\mathbb {N}$ .",
"Finally the indexing set $J\\subset V$ be such that $\\cup _{x\\in J}\\tilde{\\Lambda }(x)=V$ and $\\tilde{\\Lambda }(x)\\cap \\tilde{\\Lambda }(y)=\\phi \\qquad \\forall x\\ne y \\in J.$ The random variables $\\lbrace \\omega _x\\rbrace _{x\\in J}$ are independent real valued with absolutely continuous distribution.",
"With these notation we have: Theorem 1.6 On a Bethe lattice $\\mathcal {B}$ with $K>2$ , consider a family of random operators $H^\\omega $ given by (REF ), where $\\lbrace \\omega _x\\rbrace _{x\\in J}$ are i.i.d random variables following absolutely continuous distribution with bounded support.",
"Then singular spectrum of $H^\\omega $ is almost surely simple.",
"It can be seen that the spectrum of $\\chi _{\\tilde{\\Lambda }(x)}\\Delta _{\\mathcal {B}}\\chi _{\\tilde{\\Lambda }(x)}$ has non-trivial multiplicity (is exponential in terms of the diameter of $\\tilde{\\Lambda }(x)$ ).",
"So, the above result is not a consequence of previous corollary."
],
[
"Structure of the Proof",
"Rest of the article is divided into four parts.",
"In section , we setup the notations and collect the results that will be used throughout.",
"Section deals with single perturbation results.",
"Section contains the proof of Theorem REF , which is divided into Lemma REF and Lemma REF .",
"Finally in section , we prove the Corollary REF and Theorem REF .",
"The proof of Theorem REF is divided into three parts.",
"First we concentrate on the operator $H_\\lambda :=H+\\lambda C$ , where $H$ is a densely defined essentially self adjoint operator and $C$ is a finite rank non-negative operator.",
"Since all the results are obtained through properties of Borel-Stieltjes transform, there is a set $S\\subset \\mathbb {R}$ , independent of $\\lambda $ , of full Lebesgue measure where all the analysis will be done.",
"As a consequence of spectral averaging (see Lemma REF ), it is enough to concentrate on $S$ as long as we are working on the subspace ${H}^{\\lambda }_{C}=\\overline{\\left\\langle f(H_\\lambda )\\phi :f\\in C_c(\\mathbb {R})~\\&~\\phi \\in C{H} \\right\\rangle }.$ By spectral averaging, the spectrum of $H_\\lambda $ restricted to ${H}^{\\lambda }_{C}$ is contained in $S$ for almost all $\\lambda $ .",
"In section , we establish a certain inclusion relation between singular subspaces.",
"We show that for any finite rank projection $Q$ , the closed $H_\\lambda $ -invariant Hilbert subspace $\\tilde{{H}}^\\lambda _Q\\subseteq {H}^\\lambda _Q$ , such that the spectrum of $H_\\lambda $ restricted to $\\tilde{{H}}^\\lambda _Q$ is singular and is contained in $S$ , is a subset of singular subspace of ${H}^\\lambda _C$ .",
"This inclusion is shown in Lemma REF .",
"This is the reason that the multiplicity of the singular subspace for $ {H}^\\omega _{\\sum _{i=1}^N P_{n_i}}$ does not depend on $N$ .",
"Lemma REF uses this fact to get a bound on the multiplicity of singular spectrum for $ {H}^\\omega _{\\sum _{i=1}^N P_{n_i}}$ for any finite collection of $\\lbrace n_i\\rbrace _i$ .",
"Finally global bound on the multiplicity of singular spectrum is obtained by observing the fact that $\\cup _{N\\in \\mathbb {N}}{H}^\\omega _{\\sum _{i=1}^N P_{n_i}}$ is dense for any enumeration of $\\mathcal {N}$ .",
"Lemma REF provides the first conclusion of the theorem and also provides the relationship between $\\mathcal {M}_n$ and multiplicity of singular spectrum for ${H}^\\omega _{P_n}$ .",
"The proof is mostly a consequence of properties of polynomial algebra where the coefficients of the polynomial under consideration are holomorphic function on $\\mathbb {C}\\setminus \\mathbb {R}$ .",
"Part of the work is to establish a relation between multiplicity of singular spectrum and multiplicity of $\\sqrt{C_n}G^\\omega _{n,n}(z)\\sqrt{C_n}$ , which is achieved through resolvent equation.",
"After choosing a basis, we end up with matrix equations over function which are holomorphic on $\\mathbb {C}\\setminus \\mathbb {R}$ .",
"Since we are only dealing with matrices, multiplicity of $\\sqrt{C_n}G^\\omega _{n,n}(z)\\sqrt{C_n}$ can be computed through its characteristic equation and so we have polynomial equations where the coefficients are polynomials of the matrix elements.",
"Most of the work is to show that it is independent of a single perturbation.",
"Above argument also proves the independence from $z$ , this is because the matrix elements are holomorphic functions on $\\mathbb {C}\\setminus \\mathbb {R}$ , and so any non-zero polynomial can be zero only on a Lebesgue measure zero set.",
"Then by induction we show that $Mult^\\omega _n(z)$ is independent of any finite collection of random variables $\\lbrace \\omega _{p_i}\\rbrace _i$ .",
"Then Kolmogorov 0-1 law provides the stated result.",
"Finally in section , we prove Corollary REF and Theorem REF .",
"This is mostly done by writing the matrix $G^\\omega _{n,n}(z)$ into a particular form.",
"For the corollary, using the fact that $range(C_n)\\subset \\mathcal {D}(A)$ , the matrix $C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}$ is well defined over $P_n{H}$ , and we have to estimate the number of eigenvalues of $C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}+\\mu C^{-1}$ which are at most $O(1/\\mu )$ distance away from each other, for $\\mu \\gg 1$ .",
"The corollary just deals with two extreme cases.",
"For Theorem REF , most of the work is to show that for a tree (of finite depth), the adjacency operator perturbed at all the leaf nodes has simple spectrum.",
"Then the particular structure of $G^\\omega _{n,n}(z)$ provides the conclusion.",
"Even though $G^\\omega _{n,m}(z)$ are defined over $\\mathbb {C}\\setminus \\mathbb {R}$ , part of the proof of Lemma REF is done on $\\mathbb {C}^{+}$ itself.",
"The main problem that can arise on restricting to $\\mathbb {C}^{+}$ is because of F. and R. Riesz theorem [28].",
"It states that if the Borel-Stieltjes transform of a measure is zero on $\\mathbb {C}^{+}$ then the measure is equivalent to Lebesgue measure (see [15] for a proof).",
"This problem is avoided by using the fact that in case $G^\\omega _{n,m}(z)$ is zero for $z\\in \\mathbb {C}^{+}$ , we can repeat the proof by switching to $z\\in \\mathbb {C}^{-}$ and can replace $E+\\iota \\epsilon $ by $E-\\iota \\epsilon $ whenever necessary."
],
[
"Preliminaries",
"In this section we setup the notations and results used in the rest of the work.",
"Mostly we will deal with the linear operator $G^\\omega _{n,m}(z):= P_n(A^\\omega -z)^{-1}P_m: P_m{H}\\rightarrow P_n{H}\\qquad \\forall n,m\\in \\mathcal {N},$ which is well defined because of the assumption that $A^\\omega $ is essentially self adjoint.",
"Here $P_n$ denotes the orthogonal projection onto the range of $C_n$ .",
"We will denote ${H}^\\omega _{P_n}:=\\overline{\\left\\langle f(A^\\omega )\\phi : f\\in C_c(\\mathbb {R})~\\&~\\phi \\in P_n{H} \\right\\rangle }$ to be the minimal closed $A^\\omega $ -invariant subspace containing $P_n{H}$ .",
"All the results are stated in a basis independent form, but sometimes explicit basis is fixed so that $G^\\omega _{n,m}(z)$ can be viewed as a matrix valued functions.",
"We mostly focus on a single perturbation, which will be done as follows.",
"For $p\\in \\mathcal {N}$ set $A^{\\omega ,\\lambda }_p=A^\\omega +\\lambda C_p$ and define $G^{\\omega ,\\lambda }_{p,n,m}(z)=P_n(A^{\\omega ,\\lambda }_p-z)^{-1}P_m$ as before.",
"Using resolvent equation we have $G^{\\omega ,\\lambda }_{p,p,p}(z)&=G^\\omega _{p,p}(z)(I+\\lambda C_pG^\\omega _{p,p}(z))^{-1},\\\\G^{\\omega ,\\lambda }_{p,n,m}(z)&=G^\\omega _{n,m}(z)-\\lambda G^\\omega _{n,p}(z)(I+\\lambda C_p G^\\omega _{p,p}(z))^{-1}C_pG^\\omega _{p,m}(z).$ Another way to write above equations is $&(I-\\lambda C_pG^{\\omega ,\\lambda }_{p,p,p}(z))(I+\\lambda C_p G^\\omega _{p,p}(z))=I,\\\\&G^{\\omega ,\\lambda }_{p,n,m}(z)=G^\\omega _{n,m}(z)-\\lambda G^\\omega _{n,p}(z)C_pG^\\omega _{p,m}(z)\\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad +\\lambda ^2G^\\omega _{n,p}(z)C_pG^{\\omega ,\\lambda }_{p,p,p}(z)C_pG^\\omega _{p,m}(z).",
"$ Either of them will be used depending on the situation.",
"It should be noted that the identity operator in equations (REF ), () and (REF ) is the identity map on $P_p{H}$ .",
"For a fixed basis of each of $P_n{H}$ , using [13] (which follows from the property of Borel-Stieltjes transform) for each matrix elements of $G^\\omega _{n,m}(z)$ , we have that $G^\\omega _{n,m}(E\\pm \\iota 0):=\\lim _{\\epsilon \\downarrow 0}G^\\omega _{n,m}(E\\pm \\iota \\epsilon )$ exists for almost all $E$ with respect to Lebesgue measure and for any $n,m\\in \\mathcal {N}$ .",
"So the linear operator $G^\\omega _{n,m}(E\\pm \\iota 0)$ is well defined for almost all $E$ and any $n,m\\in \\mathcal {N}$ .",
"Using (REF ) we observe that for any $E\\in \\mathbb {R}$ such that $G^\\omega _{p,p}(E\\pm \\iota 0)$ exists and for $f:(0,\\infty )\\rightarrow \\mathbb {C}$ satisfying $\\lim _{\\epsilon \\downarrow 0}f(\\epsilon )=0$ , we have $\\lim _{\\epsilon \\downarrow 0}f(\\epsilon )(I-\\lambda C_p G^{\\omega ,\\lambda }_{p,p,p}(E\\pm \\iota \\epsilon ))(I+\\lambda C_p G^\\omega _{p,p}(E\\pm \\iota \\epsilon ))=0\\\\\\Rightarrow \\qquad \\left(\\lim _{\\epsilon \\downarrow 0}f(\\epsilon ) C_p G^{\\omega ,\\lambda }_{p,p,p}(E\\pm \\iota \\epsilon )C_p\\right)(C_p^{-1}+\\lambda G^\\omega _{p,p}(E\\pm \\iota 0))=0,$ and similarly $(C_p^{-1}+\\lambda G^\\omega _{p,p}(E\\pm \\iota 0)) \\left(\\lim _{\\epsilon \\downarrow 0} f(\\epsilon ) C_p G^{\\omega ,\\lambda }_{p,p,p}(E\\pm \\iota \\epsilon )C_p \\right)=0.$ The above equation implies $range\\left( \\left(\\lim _{\\epsilon \\downarrow 0}f(\\epsilon ) C_p G^{\\omega ,\\lambda }_{p,p,p}(E\\pm \\iota \\epsilon )C_p\\right)\\right)&\\subseteq ker(C_p^{-1}+\\lambda G^\\omega _{p,p}(E\\pm \\iota 0))\\nonumber \\\\&\\subseteq ker(\\Im G^\\omega _{p,p}(E\\pm \\iota 0) ),$ which is used to determine the singular spectrum.",
"One of the consequences of $\\pm \\Im G_{p,p}^\\omega (E\\pm \\iota 0)\\ge 0$ is $G^\\omega _{k,p}(E\\pm \\iota 0)\\phi =G^\\omega _{p,k}(E\\pm \\iota 0)^\\ast \\phi \\qquad \\forall \\phi \\in ker(\\pm \\Im G_{p,p}^\\omega (E\\pm \\iota 0)),$ which plays an important role in the proof of Lemma REF .",
"Since most of the analysis is done using a single perturbation, one of the important results needed is the spectral averaging; we refer to [7] for its proof.",
"Here we will use the following version: Lemma 2.1 Let $E_\\lambda (\\cdot )$ be the spectral family for the operator $A_\\lambda =A+\\lambda C$ , where $A$ is a self adjoint operator and $C$ is a non-negative compact operator.",
"For any $M\\subset \\mathbb {R}$ with zero Lebesgue measure, we have $\\sqrt{C}E_\\lambda (M)\\sqrt{C}=0$ for almost all $\\lambda $ , with respect to Lebesgue measure.",
"Since the set of $E$ where $\\lim _{\\epsilon \\downarrow 0}G^\\omega _{n,m}(E\\pm \\iota \\epsilon )$ does not exists for some $n,m\\in \\mathcal {N}$ , is a Lebesgue measure zero set, above lemma guarantees that we can leave this set from our analysis as long as we are only focusing on $A_p^{\\omega ,\\lambda }$ -invariant subspace containing $P_p{H}$ .",
"Another important result is Lemma 2.2 For a $\\sigma $ -finite positive measure space $(X,{B},m)$ and a collection of ${B}$ -measurable functions $a_i:X\\rightarrow \\mathbb {C}$ and $b_i:X\\rightarrow \\mathbb {C}$ , define $f(\\lambda )=\\frac{1+\\sum _{n=1}^N a_n(x)\\lambda ^n}{1+\\sum _{n=1}^N b_n(x)\\lambda ^n},$ Then the set $\\Lambda _f=\\lbrace \\lambda \\in \\mathbb {C}: m(x\\in X: f(\\lambda ,x)=0)>0\\rbrace $ is countable.",
"Its proof can be found in [23].",
"This lemma ensures that the linear operator $G^{\\omega ,\\lambda }_{p,p,p}(z)$ is well defined for almost all $\\lambda $ .",
"This is the case because $G^{\\omega ,\\lambda }_{p,p,p}(z)$ and $G^{\\omega }_{p,p}(z)$ are related through the equation (REF ), and so the set $\\lbrace E:\\det (I+\\lambda C_p G^\\omega _{p,p}(E\\pm \\iota 0))=0\\rbrace $ should have zero Lebesgue measure, otherwise the analysis will fail.",
"This is also the set in which the singular spectrum of $A^{\\omega ,\\lambda }_p$ restricted to ${H}^{\\omega }_{P_p}$ (it is easy to see that the space${H}^\\omega _{P_p}$ is invariant under action of $A^{\\omega ,\\lambda }_p$ ) belongs.",
"Next result is Poltoratskii's theorem and is the main tool through which singular part of the spectrum is handled.",
"Since we only deal with finite measures, we will denote the Borel-Stieltjes transform $F_\\mu :\\mathbb {C}^{+}\\rightarrow \\mathbb {C}^{+}$ for the Borel measure $\\mu $ by $F_\\mu (z)=\\int \\frac{d\\mu (x)}{x-z}.$ For $f\\in L^1(\\mathbb {R},d\\mu )$ , let $f\\mu $ be the unique measure associated with the linear functional $C_c(\\mathbb {R})\\ni g\\mapsto \\int g(x)f(x)d\\mu (x)$ .",
"The version of the Poltoratskii's theorem we will use is: Lemma 2.3 For any complex valued Borel measure $\\mu $ on $\\mathbb {R}$ , let $f\\in L^1(\\mathbb {R},d\\mu )$ , then $\\lim _{\\epsilon \\downarrow 0}\\frac{F_{f\\mu }(E+\\iota \\epsilon )}{F_\\mu (E+\\iota \\epsilon )}=f(E)$ for a.e $E$ with respect to $\\mu $ -singular.",
"The proof of this can be found in [14].",
"With these results in hand, we can now prove our results."
],
[
"Single Perturbation Results",
"This section will concentrate on a single perturbation.",
"Lemma REF will play an important role for proving the main result.",
"For this section a different notation will be followed, because it is not necessary to keep track of all the random variables $\\lbrace \\omega _n\\rbrace _n$ .",
"Let $H$ be a densely defined self adjoint operator on a separable Hilbert space ${H}$ and $C_1$ be a finite rank non-negative operator.",
"Set $H_\\lambda =H+\\lambda C_1$ and let $P_1$ be the orthogonal projection onto the range of $C_1$ .",
"For any projection $Q$ define ${H}^\\lambda _{Q}:=\\overline{\\left\\langle f(H_\\lambda )\\psi : \\psi \\in Q{H}~\\&~f\\in C_c(\\mathbb {R}) \\right\\rangle },$ to be the minimal closed $H_\\lambda $ -invariant subspace containing the range of $Q$ .",
"Let $\\sigma ^\\lambda _1$ denote the trace measure $tr(P_1 E^{H_\\lambda }(\\cdot ))$ , where $E^{H_\\lambda }(\\cdot )$ is the spectral projection for the operator $H_\\lambda $ .",
"The subscript $sing$ will be used to denote the singular part of the measure whenever necessary.",
"The main result of this section is the following: Lemma 3.1 Let $Q$ be a finite rank projection and set $\\lbrace e_i\\rbrace _{i}$ to be an orthonormal basis of $Q{H}+P_1{H}$ .",
"Define the set $S=\\lbrace E\\in \\mathbb {R}: \\left\\langle e_i,(H-E\\mp \\iota 0)^{-1}e_j\\right\\rangle \\text{ exists and finite}\\rbrace ,$ and denote $E^\\lambda _{sing }$ to be the spectral measure onto the singular part of spectrum of $H_\\lambda $ , then $E^\\mu _{sing }(S){H}^\\lambda _{Q}\\subseteq E^\\mu _{sing }(S){H}^\\lambda _{P_1}$ for almost all $\\lambda $ with respect to the Lebesgue measure.",
"Remark 3.2 Spectral averaging result (Lemma REF ) gives $\\sigma ^\\lambda _1(\\mathbb {R}\\setminus S)=0$ for a.a. $\\lambda $ w.r.t.",
"Lebesgue measure, so it is actually not necessary to write $S$ on RHS of above equation.",
"But $E^\\lambda _{sing}(\\mathbb {R}\\setminus S){H}^\\lambda _{Q}$ can be non-trivial.",
"In view of Lemma REF , it is enough to show $E^\\lambda _{sing }(S){H}^\\lambda _{e_i}\\subseteq E^\\lambda _{sing }(S){H}^\\lambda _{P_1},$ where ${H}^\\lambda _{e_i}$ is the minimal closed $H_\\lambda $ -invariant subspaces containing $e_i$ .",
"This is because applying Lemma REF for the operator $E^\\lambda _{sing }(S) H_\\lambda $ will give the singular subspaces in the conclusion of the lemma.",
"Using the resolvent equation $(H_\\lambda -z)^{-1}-(H-z)^{-1}=-\\lambda (H_\\lambda -z)^{-1}C_1(H-z)^{-1}$ and similarly $(H_\\lambda -z)^{-1}&=(H-z)^{-1}-\\lambda (H-z)^{-1}C_1(H_\\lambda -z)^{-1}\\\\&=(H-z)^{-1}-\\lambda (H-z)^{-1}C_1(H-z)^{-1}\\\\&\\qquad +\\lambda ^2 (H-z)^{-1}C_1(H_\\lambda -z)^{-1}C_1(H_\\lambda -z)^{-1},$ we have $\\left\\langle e_i,(H_\\lambda -z)^{-1}e_i\\right\\rangle &=\\left\\langle e_i,(H-z)^{-1}e_i\\right\\rangle -\\lambda \\left\\langle e_i,(H-z)^{-1}C_1(H-z)^{-1}e_i\\right\\rangle \\nonumber \\\\&\\qquad +\\lambda ^2 \\left\\langle e_i,(H-z)^{-1}C_1(H_\\lambda -z)^{-1}C_1(H-z)^{-1}e_i\\right\\rangle .$ Let $\\lbrace e_{1i}\\rbrace _{i=1}^{r_1}$ , where $r_1=dim(P_1{H})$ , be an orthonormal basis of $P_1{H}$ (so that they are linear combinations of $\\lbrace e_i\\rbrace _i$ ); hence $G^\\lambda _{1,1}(z)=P_1(H_\\lambda -z)^{-1}P_1$ is a matrix for this basis and also set $G_{i,1}(z)=\\left(\\begin{matrix} \\left\\langle e_i,(H-z)^{-1}e_{11}\\right\\rangle \\\\ \\left\\langle e_i,(H-z)^{-1}e_{12}\\right\\rangle \\\\ \\vdots \\\\ \\left\\langle e_i,(H-z)^{-1}e_{1r_1}\\right\\rangle \\end{matrix}\\right)^t~~\\&~~ G_{1,i}(z)=\\left(\\begin{matrix} \\left\\langle e_{11},(H-z)^{-1}e_i\\right\\rangle \\\\\\left\\langle e_{12},(H-z)^{-1}e_i\\right\\rangle \\\\ \\vdots \\\\ \\left\\langle e_{1r_1},(H-z)^{-1}e_i\\right\\rangle \\end{matrix}\\right) .$ Then the equation (REF ) can be written as $\\left\\langle e_i,(H_\\lambda -z)^{-1}e_i\\right\\rangle &=\\left\\langle e_i,(H-z)^{-1}e_i\\right\\rangle -\\lambda G_{i,1}(z)C_1 G_{1,i}(z)\\\\&\\qquad +\\lambda ^2 G_{i,1}(z)C_1 G_{1,1}^\\lambda (z)C_1G_{1,i}(z).$ Using the fact that LHS is the Borel-Stieltjes transform of the measure $\\langle e_i E^{H_\\lambda }( \\cdot )e_i\\rangle $ , the support of singular part lies in the set of $E\\in \\mathbb {R}$ where $\\lim _{\\epsilon \\downarrow 0} \\left(\\left\\langle e_i,(H_\\lambda -E-\\iota \\epsilon )^{-1}e_i\\right\\rangle \\right)^{-1}=0.$ We don't need to consider the case $\\left\\langle e_i,(H_\\lambda -z)^{-1}e_i\\right\\rangle = 0$ for all $z\\in \\mathbb {C}^{+}$ because by F. and R. Riesz theorem [28], the measure $\\left\\langle e_i,E^{H_\\lambda }(\\cdot )e_i\\right\\rangle $ is absolutely continuous.",
"But by definition of the set $S$ , we have $G_{i,1}(E\\pm \\iota 0),G_{1,i}(E\\pm \\iota 0)$ and $\\left\\langle e_i,(H-E\\mp \\iota 0)^{-1}e_i\\right\\rangle $ exist for each $E\\in S$ .",
"So singular part of $\\left\\langle e_i,E^{H_\\lambda }(\\cdot )e_i\\right\\rangle $ can lie on $\\mathbb {R}\\setminus S$ or on the set of $E\\in S$ where $\\lim _{\\epsilon \\downarrow 0}(tr(G_{1,1}^\\lambda (E+\\iota \\epsilon )))^{-1}=0$ .",
"For $E\\in S$ where $\\lim _{\\epsilon \\downarrow 0}(tr(G_{1,1}^\\lambda (E+\\iota \\epsilon )))^{-1}=0$ , note that $&\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle e_i,(H_\\lambda -E-\\iota \\epsilon )^{-1}e_i\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\\\&\\qquad =\\lambda ^2 G_{i,1}(E+\\iota 0)C_1\\left(\\lim _{\\epsilon \\downarrow 0} \\frac{ G_{1,1}^\\lambda (E+\\iota \\epsilon )}{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\right)C_1G_{1,i}(E+\\iota 0).$ Using (REF ), we have $&\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle e_i,(H_\\lambda -E-\\iota \\epsilon )^{-1}e_i\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\nonumber \\\\&\\qquad =\\lambda ^2 [C_1 G_{1,i}(E+\\iota 0)]^\\ast \\left(\\lim _{\\epsilon \\downarrow 0} \\frac{ G_{1,1}^\\lambda (E+\\iota \\epsilon )}{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\right)[C_1G_{1,i}(E+\\iota 0)].$ Since $G_{1,1}^\\lambda (\\cdot )$ is a matrix valued Herglotz function for a positive operator valued measure (it is the Borel transform of $P_1 E^{H_\\lambda }(\\cdot )P_1$ ), there exists a matrix valued function $M_1^\\lambda \\in L^1(\\mathbb {R},\\sigma _1^\\lambda ,M_{rank(P_1)}(\\mathbb {C}))$ , (using the Herglotz representation theorem for matrix valued measures, see [12]) such that we have $G_{1,1}^\\lambda (z)=\\int \\frac{1}{x-z}M_1^\\lambda (x)d\\sigma ^\\lambda _1(x),$ for $z\\in \\mathbb {C}\\setminus \\mathbb {R}$ .",
"Using Poltoratskii's theorem (lemma REF ) we have $\\lim _{\\epsilon \\downarrow 0}\\frac{1}{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}G_{1,1}^\\lambda (E+\\iota \\epsilon )=M_1^\\lambda (E)$ for almost all $E$ with respect to $\\sigma ^\\lambda _{1,sing}$ .",
"Since the measure $P_1E^{H_\\lambda }(\\cdot )P_1$ is non-negative, the matrix valued function $M_1^\\lambda (E)\\ge 0$ for almost all $E$ with respect to $\\sigma ^\\lambda _1$ .",
"Let $U_1^\\lambda (E)$ be the unitary matrix which diagonalizes $M_1^\\lambda (E)$ , i.e $U_1^\\lambda (E)M_1^\\lambda (E)U_1^\\lambda (E)^\\ast =diag(f^\\lambda _j;1\\le j\\le r_1),$ where some of the $f^\\lambda _j$ can be zero.",
"Using Hahn-Hellinger theorem (see [26]), the function $U_i^\\lambda $ can be chosen to be a Borel measurable Unitary matrix valued function.",
"Since we only focus on singular part, set $U_1^\\lambda (E)=0$ for $E$ not in support of $\\sigma ^\\lambda _{1,sing}$ and define $\\psi ^\\lambda _j=U_1^\\lambda (H_\\lambda )^\\ast e_{1j}$ .",
"Now observe that $&\\left\\langle \\psi ^\\lambda _k,(H_\\lambda -z)^{-1}\\psi ^\\lambda _l\\right\\rangle =\\int \\frac{1}{x-z} \\left\\langle \\psi ^\\lambda _k,E^{H_\\lambda }(dx)\\psi ^\\lambda _l\\right\\rangle \\\\&\\qquad =\\int \\frac{1}{x-z} \\left\\langle U_1^\\lambda (x)^\\ast e_{1k},E^{H_\\lambda }(dx)U_1^\\lambda (x)^\\ast e_{1l}\\right\\rangle \\\\&\\qquad =\\int \\frac{1}{x-z} \\sum _{p,q}\\left\\langle U_1^\\lambda (x)^\\ast e_{1k},e_{1p}\\right\\rangle \\left\\langle e_{1q},U_1^\\lambda (x)^\\ast e_{1l}\\right\\rangle \\left\\langle e_{1p},E^{H_\\lambda }(dx) e_{1q}\\right\\rangle \\\\&\\qquad =\\sum _{p,q}\\int \\frac{1}{x-z}\\left\\langle U_1^\\lambda (x)^\\ast e_{1k},e_{1p}\\right\\rangle \\left\\langle e_{1q},U_1^\\lambda (x)^\\ast e_{1l}\\right\\rangle \\left\\langle e_{1p},E^{H_\\lambda }(dx) e_{1q}\\right\\rangle ,$ and so using Poltoratskii's theorem (lemma REF ) we get $&\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle \\psi ^\\lambda _k,(H_\\lambda -E-\\iota \\epsilon )^{-1}\\psi ^\\lambda _l\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\\\&\\qquad =\\sum _{p,q}\\left\\langle U_1^\\lambda (E)^\\ast e_{1k},e_{1p}\\right\\rangle \\left\\langle e_{1q},U_1^\\lambda (E)^\\ast e_{1l}\\right\\rangle \\left(\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle e_{1p},(H_\\lambda -E-\\iota \\epsilon )^{-1}e_{1q}\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\right)\\\\&\\qquad =\\left\\langle e_{1k},U_1^\\lambda (E)M_1^\\lambda (E)U_1^\\lambda (E)^\\ast e_{1l}\\right\\rangle =f^\\lambda _k(E)\\delta _{k,l}$ for almost all $E$ with respect to $\\sigma ^\\lambda _{1,sing}$ .",
"By construction of $\\psi ^\\lambda _j$ , the spectral measure $\\left\\langle \\psi ^\\lambda _j,E^{H_\\lambda }(\\cdot )\\psi ^\\lambda _j\\right\\rangle $ is purely singular with respect to the Lebesgue measure, so above computation implies $\\left\\langle \\psi ^\\lambda _k,(H_\\lambda -z)^{-1}\\psi ^\\lambda _l\\right\\rangle =0$ for all $z$ for $k\\ne l$ , which implies that the measure $\\left\\langle \\psi ^\\lambda _k,E^{H_\\lambda }(\\cdot )\\psi ^\\lambda _l\\right\\rangle $ is zero, and in particular we have ${H}^\\lambda _{\\psi ^\\lambda _k}\\perp {H}^\\lambda _{\\psi ^\\lambda _l}$ for $k\\ne l$ .",
"Next, using the resolvent equation, we obtain $&\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle \\psi ^\\lambda _k,(H_\\lambda -E-\\iota \\epsilon )^{-1}e_i\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\\\&\\qquad =\\lim _{\\epsilon \\downarrow 0}-\\lambda \\frac{\\left\\langle \\psi ^\\lambda _k,(H_\\lambda -E-\\iota \\epsilon )^{-1}C_1(H-E-\\iota \\epsilon )^{-1} e_i\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}\\nonumber \\\\&\\qquad = -\\lambda f^\\lambda _k(E)\\left\\langle e_{1k},U_1^\\lambda (E)C_1 G_{1,i}(E+\\iota 0)\\right\\rangle ,\\nonumber $ for a.e.",
"$E$ w.r.t.",
"$\\sigma ^\\lambda _{1,sing}$ .",
"Using Lemma REF and the above equation (REF ) on equation (REF ), we conclude $\\lim _{\\epsilon \\downarrow 0}\\frac{\\left\\langle e_i,(H_\\lambda -E-\\iota \\epsilon )^{-1}e_i\\right\\rangle }{tr(G_{1,1}^\\lambda (E+\\iota \\epsilon ))}=\\sum _j \\left|(Q_{\\psi ^\\lambda _j}^\\lambda e_i)(E)\\right|^2 f_j^\\lambda (E)$ for a.e.",
"$E$ w.r.t $\\sigma ^\\lambda _{1,sing}$ , where $Q_{\\psi ^\\lambda _j}^\\lambda e_i$ is the projection of $e_i$ on the Hilbert subspace ${H}^\\mu _{\\psi ^\\lambda _j}$ .",
"So for $g\\in C_c(\\mathbb {R})$ , we can write $\\left\\langle e_i,E^\\lambda _{sing}(S) g(H_\\lambda )e_i\\right\\rangle &=\\sum _j \\int g(E)\\left|(Q_{\\psi ^\\lambda _j}^\\lambda e_i)(E)\\right|^2 f_j^\\lambda (E) d\\sigma ^\\lambda _{1,sing}(E),$ which implies that the projection of $E^\\lambda _{sing}(S)e_i$ onto ${H}^\\lambda _{P_1}$ is isometry, hence $E^\\lambda _{sing}(S){H}_{e_i}^\\lambda \\subseteq E^\\lambda _{sing}(S){H}^\\lambda _{P_1}.$ The lemma follows by an application of Lemma REF ."
],
[
"Proof of Theorem ",
"The proof of the main result is divided into Lemma REF and Lemma REF .",
"It should be noted that the conclusion of Lemma REF is similar to the conclusion reached by combining [24] and [24].",
"This section deals with $A^\\omega $ itself and so the notations established in section are followed.",
"Following the notations from previous section, set ${H}^\\omega _P$ to be the minimal closed $A^\\omega $ -invariant subspace containing the range of the projection $P$ .",
"Lemma 4.1 For any $n\\in \\mathcal {N}$ , $\\mathcal {M}^\\omega _n:=\\mathop {\\text{ess-sup}}_{z\\in \\mathbb {C}\\setminus \\mathbb {R}} Mult^\\omega _n(z)$ is almost surely constant; denote it by $\\mathcal {M}_n$ .",
"The multiplicity of singular spectrum for ${H}^\\omega _{P_n}$ is bounded above by $\\mathcal {M}_n$ .",
"First we prove that $\\mathcal {M}_n^\\omega $ is independent of $\\omega $ .",
"This is done using Kolmogorov 0-1 law.",
"So first step is to show that $\\mathcal {M}_n^\\omega $ is independent of any finite collection of random variables $\\lbrace \\omega _{p_i}\\rbrace _{i}$ .",
"Following the notations from section , set $A^{\\omega ,\\lambda }_p=A^\\omega +\\lambda C_p$ for $p\\in \\mathcal {N}\\setminus \\lbrace n\\rbrace $ , we have the equation () $G^{\\omega ,\\lambda }_{p,n,n}(z)=G^\\omega _{n,n}(z)-\\lambda G^\\omega _{n,p}(z)(I+\\lambda C_p G^\\omega _{p,p}(z))^{-1}C_p G^\\omega _{p,n}(z).$ Looking at $G^\\omega _{i,j}(z)$ as a matrix, observe that $\\tilde{g}^\\omega _{\\lambda ,z}(x)&=\\det (C_n G^{\\omega ,\\lambda }_{p,n,n}(z)-xI)\\\\&=\\det (C_n G^\\omega _{n,n}(z)-\\lambda C_n G^\\omega _{n,p}(z)(I+\\lambda C_p G^\\omega _{p,p}(z))^{-1}C_p G^\\omega _{p,n}(z)-xI)\\\\&=\\frac{p_l^\\omega (z,\\lambda )x^l+p_{l-1}^\\omega (z,\\lambda )x^{l-1}+\\cdots +p_0^\\omega (z,\\lambda ) }{\\det (C_p^{-1}+\\lambda G^\\omega _{n,n}(z))},$ where $l=rank(P_n)$ .",
"Here $\\lbrace p_i^\\omega (z,\\lambda )\\rbrace _{i=0}^l$ are polynomials in the elements of the matrices $\\lbrace G^\\omega _{i,j}(z)\\rbrace _{i,j\\in \\lbrace n,p\\rbrace }$ and $\\lambda $ .",
"We don't need to focus on the denominator, so set $g^\\omega _{\\lambda ,z}(x)=p_l^\\omega (z,\\lambda )x^l+p_{l-1}^\\omega (z,\\lambda )x^{l-1}+\\cdots +p_0^\\omega (z,\\lambda ).$ The maximum algebraic multiplicity of $G^{\\omega ,\\lambda }_{p,n,n}(z)$ is $k$ if the function $\\mathcal {F}^\\omega _{\\lambda ,z}(x)=gcd\\left(g^\\omega _{\\lambda ,z}(x),\\frac{dg^\\omega _{\\lambda ,z}}{dx}(x),\\cdots ,\\frac{d^kg^\\omega _{\\lambda ,z}}{dx^k}(x)\\right)$ is constant with respect to $x$ .",
"Using the fact that $gcd(f_1(x),\\cdots ,f_m(x))=gcd(f_1(x),\\cdots ,f_{m-2}(x),gcd(f_{m-1}(x),f_m(x)))$ and Euclid's algorithm for polynomials, we get $\\mathcal {F}^\\omega _{\\lambda ,z}(x)=q^\\omega _0(\\lambda ,z)+q^\\omega _1(\\lambda ,z)x+\\cdots +q^\\omega _s(\\lambda ,z)x^s$ where $\\lbrace q^\\omega _i(\\lambda ,z)\\rbrace _{i=0}^s$ are rational polynomials of $\\lbrace p_i^\\omega (z,\\lambda )\\rbrace _i$ .",
"We need to consider the numerators of $q_i^\\omega $ , which are denoted by $\\tilde{q}_i^\\omega $ .",
"Since $\\lbrace \\tilde{q}_i^\\omega \\rbrace $ are polynomials of the matrix elements $\\lbrace G^\\omega _{i,j}(z)\\rbrace _{i,j\\in \\lbrace n,p\\rbrace }$ and $\\lambda $ , write $\\tilde{q}^\\omega _i(\\lambda ,z)=\\sum _j a^\\omega _{ij}(z)\\lambda ^j$ where $\\lbrace a^\\omega _{ij}\\rbrace _{i,j}$ are holomorphic functions on $\\mathbb {C}\\setminus \\mathbb {R}$ .",
"So $\\lbrace \\tilde{q}^\\omega _i\\rbrace $ are well defined over $(\\lambda ,z)\\in \\mathbb {R}\\times (\\mathbb {C}\\setminus \\mathbb {R})$ for each $i$ .",
"Now suppose $\\mathcal {M}^\\omega _n=k$ , then $q^\\omega _0(0,\\cdot )\\ne 0$ and $q^\\omega _i(0,\\cdot )=0$ identically, which implies $a^\\omega _{i0}(\\cdot )=0$ for $i\\ne 0$ .",
"This implies $G^{\\omega ,\\lambda }_{p,n,n}(z)$ can have multiplicity greater than $k$ .",
"Setting $\\tilde{\\omega }^p$ to be such that $\\tilde{\\omega }^p_k=\\omega _k$ for $k\\ne p$ and $\\tilde{\\omega }^p_p=\\omega _p+\\lambda $ , gives $\\mathcal {M}_n^\\omega \\le \\mathcal {M}_n^{\\tilde{\\omega }^p}$ .",
"Since $\\mathcal {M}_n^{\\tilde{\\omega }^p}$ can be at most $rank(P_n)$ , this implies $\\mathcal {M}_n^{\\tilde{\\omega }^p}$ is independent of $\\lambda $ .",
"Now repeating the proof inductively for a collection of sites $\\lbrace p_i\\rbrace _{i=1}^N$ proves the independence of $\\mathcal {M}_n^{\\omega }$ from the random variables $\\lbrace \\omega _{p_i}\\rbrace _{i=1}^N$ .",
"Hence, using Kolmogorov 0-1 law, $\\mathcal {M}_n^\\omega $ is independent of $\\omega $ .",
"Assume that $\\mathcal {M}_n=k$ , which implies that the maximum multiplicity for the matrix $G^\\omega _{n,n}(z)$ is $k$ for almost every $z$ .",
"Using above argument for the polynomial $g^\\omega _z(x)=\\det (C_n G^\\omega _{n,n}(z)-xI)=(-x)^l+(-x)^{l-1}p_{l-1}^\\omega (z)+\\cdots +p_0^\\omega (z),$ we get that the function $gcd\\left(g^\\omega _{z}(x),\\frac{dg^\\omega _{z}}{dx}(x),\\cdots ,\\frac{d^kg^\\omega _{z}}{dx^k}(x)\\right)$ is a rational polynomial of matrix elements of $G^\\omega _{n,n}(z)$ and so the numerator is holomorphic on $\\mathbb {C}\\setminus \\mathbb {R}$ .",
"Since it is non-zero for a positive Lebesgue measure set, it is non-zero for almost all $z\\in \\mathbb {C}\\setminus \\mathbb {R}$ , which implies $k=\\mathop {\\text{ess-sup}}_{E\\in \\mathbb {R}} \\lbrace \\text{Maximum multiplicity of roots of }\\nonumber \\\\\\det (C_n G_{n,n}^\\omega (E\\pm \\iota 0)-xI)\\rbrace .$ Now focus on the second conclusion of the Lemma, i.e.",
"multiplicity of singular spectrum on ${H}^\\omega _{P_n}$ is bounded by $\\mathcal {M}_n$ .",
"Denote $S=\\lbrace E\\in \\mathbb {R}: \\text{Maximum multiplicity of roots of }\\nonumber \\\\\\det (C_n G_{n,n}^\\omega (E\\pm \\iota 0)-xI) \\text{ is }k\\rbrace ,$ which by above has full Lebesgue measure.",
"Using Spectral theorem (see [23]) for the operator $A^{\\omega ,\\lambda }_n=A^\\omega +\\lambda C_n$ gives $({H}^{\\omega ,\\lambda ,n}_{P_n},A^{\\omega ,\\lambda }_n)\\cong (L^2(\\mathbb {R},P_n E^{A^{\\omega ,\\lambda }_n}(\\cdot )P_n,P_n{H}),M_{Id}).$ Here $E^{A^{\\omega ,\\lambda }_n}$ is the spectral measure for $A^{\\omega ,\\lambda }_n$ and ${H}^{\\omega ,\\lambda ,n}_{Q}$ is the minimal closed $A^{\\omega ,\\lambda }_n$ -invariant space containing the subspace $Q{H}$ for a projection $Q$ .",
"Since the measure $P_n E^{A^{\\omega ,\\lambda }_n}(\\cdot )P_n$ is absolutely continuous with respect to the trace measure $\\sigma _n^{\\omega ,\\lambda }(\\cdot )=tr(P_n E^{A^{\\omega ,\\lambda }_n}(\\cdot )P_n)$ , after a choice of basis, there exists a non-negative $M_n^{\\omega ,\\lambda }\\in L^1(\\mathbb {R},\\sigma _n^{\\omega ,\\lambda },M_{rank(P_n)}(\\mathbb {C}))$ such that $P_n E^{A^{\\omega ,\\lambda }_n}(dx)P_n=M_n^{\\omega ,\\lambda }(x)\\sigma _n^{\\omega ,\\lambda }(dx),$ and Poltoratskii's theorem (lemma REF ) gives us that $\\lim _{\\epsilon \\downarrow 0}\\frac{1}{tr(G^{\\omega ,\\lambda }_{n,n,n}(E+\\iota \\epsilon ))}G^{\\omega ,\\lambda }_{n,n,n}(E+\\iota \\epsilon )=M_n^{\\omega ,\\lambda }(E)$ for almost all $E$ with respect to $\\sigma _n^{\\omega ,\\lambda }$ -singular.",
"Here we are assuming that $\\sigma _n^{\\omega ,\\lambda }$ has a non-trivial singular component, so $G^{\\omega ,\\lambda }_{n,n,n}(z)\\ne 0$ for almost all $z\\in \\mathbb {C}^{+}$ .",
"Just as in (REF ) we also have $(I+\\lambda C_n G^\\omega _{n,n}(z))(I-\\lambda C_n G^{\\omega ,\\lambda }_{n,n,n}(z))=I,$ which gives (using steps involved for obtaining (REF )) $(I+\\lambda C_n G^\\omega _{n,n}(E+\\iota 0)) \\left[C_n\\lim _{\\epsilon \\downarrow 0}\\frac{1}{tr(G^{\\omega ,\\lambda }_{n,n,n}(E+\\iota \\epsilon ))} G^{\\omega ,\\lambda }_{n,n,n}(E+\\iota \\epsilon )\\right]=0,$ for $E$ whenever $\\lim _{\\epsilon \\downarrow 0}\\frac{1}{tr(G^{\\omega ,\\lambda }_{n,n,n}(E+\\iota \\epsilon ))}=0$ .",
"So $(I+\\lambda C_n G^\\omega _{n,n}(E+\\iota 0)) C_n M_n^{\\omega ,\\lambda }(E)=0$ for almost all $E$ with respect to $\\sigma _n^{\\omega ,\\lambda }$ -singular.",
"Using the fact that $\\sigma _n^{\\omega ,\\lambda }(\\mathbb {R}\\setminus S)=0$ for almost all $\\lambda $ and the above equation, which implies that the rank of $M_n^{\\omega ,\\lambda }(E)$ is upper bounded by dimension of the kernel $(I+\\lambda C_n G^\\omega _{n,n}(E+\\iota 0))$ which in turn is upper bounded by $k$ over the set $S$ (follows from (REF )), we get that the multiplicity of the singular spectrum for $A^{\\omega ,\\lambda }_n$ is bounded above by $k$ over ${H}^{\\omega ,\\lambda ,n}_{P_n}$ .",
"This completes the proof as the above statement is true for almost all $(\\omega ,\\lambda )$ .",
"Note that, in the above lemma bound for the multiplicity of singular spectrum is given for the subspace ${H}^\\omega _{P_n}$ and not on the entire Hilbert space.",
"Lemma REF is used to obtain the final result, which is as follows: Lemma 4.2 Assuming the hypothesis of Theorem REF and that $\\mathcal {M}_n\\le K$ for all $n\\in \\mathcal {N}$ .",
"Then the multiplicity of singular spectrum for $A^\\omega $ is bounded above by $K$ almost surely.",
"The proof is done in two steps.",
"First we show that for any finite collections of $\\lbrace p_i\\rbrace _{i=1}^N\\subset \\mathcal {N}$ , the multiplicity of singular spectrum restricted to ${H}^\\omega _{\\sum _{i=1}^N P_{p_i}}$ is bounded by $K$ .",
"Then the proof is completed using the density of $\\cup _{N=1}^\\infty {H}^\\omega _{\\sum _{i=1}^N P_{p_i}}$ .",
"First part is through induction, so let $\\lbrace p_i\\rbrace _{i\\in \\mathbb {N}}$ be an enumeration of the set $\\mathcal {N}$ .",
"The induction is done over the statement $\\mathcal {S}_N$ which is: Multiplicity of Singular spectrum for $A^\\omega $ restricted to the subspace ${H}^\\omega _{\\sum _{i=1}^N P_{p_i}}$ is at most $K$.",
"For the case $N=1$ , the conclusion follows from the Lemma REF , i.e the multiplicity of the singular spectrum over ${H}^\\omega _{P_{p_1}}$ is at most $K$ .",
"For the induction step assume $\\mathcal {S}_N$ is true, i.e the multiplicity of the singular spectrum over ${H}^\\omega _{\\sum _{i=1}^N P_{p_i}}$ is bounded by $K$ .",
"Before going in to prove $\\mathcal {S}_{N+1}$ , note that ${H}^\\omega _{\\sum _{i=1}^{N+1} P_{p_i}}= {H}^\\omega _{\\sum _{i=1}^N P_{p_i}}+{H}^\\omega _{P_{p_{N+1}}},$ RHS is a subset of LHS is obvious, and for the other inclusion observe that RHS is dense and closed in LHS.",
"Now consider the operator $A^{\\omega ,\\lambda }_{p_{N+1}}=A^\\omega +\\lambda C_{p_{N+1}}$ .",
"By Lemma REF , the multiplicity of singular spectrum for $A^{\\omega ,\\lambda }_{p_{N+1}}$ over ${H}^{\\omega ,\\lambda ,p_{N+1}}_{ P_{p_{N+1}}}$ is bounded by $K$ .",
"By induction hypothesis, the multiplicity of singular spectrum for $\\left({H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^N P_{p_i}},A^{\\omega ,\\lambda }_{p_{N+1}}\\right)$ is at most $K$ .",
"Using Lemma REF , there exists a full Lebesgue measure set $S^\\omega $ such that $E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}(S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^N P_{p_i}}\\subseteq E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}(S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{ P_{p_{N+1}}}.$ From spectral averaging we have $E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}(\\mathbb {R}\\setminus S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{ P_{p_{N+1}}}=\\lbrace 0\\rbrace $ for almost all $\\lambda $ (w.r.t Lebesgue measure).",
"Now the decomposition ${H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^N P_{p_i}}=E^{A^{\\omega ,\\lambda }_{p_{N+1}}}(S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^N P_{p_i}}\\oplus E^{A^{\\omega ,\\lambda }_{p_{N+1}}}(\\mathbb {R}\\setminus S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^N P_{p_i}},$ gives $& E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}{H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^{N+1} P_{p_i}}=E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}{H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^{N} P_{p_i}}+E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}{H}^{\\omega ,\\lambda ,p_{N+1}}_{ P_{p_{N+1}}}\\\\&\\qquad \\qquad =E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}(\\mathbb {R}\\setminus S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{\\sum _{i=1}^{N} P_{p_i}}\\oplus E^{A^{\\omega ,\\lambda }_{p_{N+1}}}_{sing}(S^\\omega ){H}^{\\omega ,\\lambda ,p_{N+1}}_{ P_{p_{N+1}}},$ where both the subspaces have multiplicity at most $K$ .",
"The supports of the singular spectrum of $A^{\\omega ,\\lambda }_{p_{N+1}}$ restricted over the two subspaces are disjoint and this proves the induction hypothesis.",
"So this completes the first part of the proof.",
"With the induction completed, note that ${H}^\\omega _{\\sum _{i=1}^{N}P_{p_i}}\\subseteq {H}^\\omega _{\\sum _{i=1}^{N+1}P_{p_i}}\\qquad \\forall N\\in \\mathbb {N},$ which implies $\\tilde{{H}}^\\omega :=\\cup _{n\\in \\mathbb {N}}{H}^\\omega _{\\sum _{i=1}^{N}P_{p_i}}$ is a linear subspace of ${H}$ , and it is dense because $\\sum _{p\\in \\mathcal {N}}P_p=I$ .",
"Clearly the space $\\tilde{{H}}^\\omega $ is invariant under the action of $A^\\omega $ .",
"For any finite collection $\\lbrace \\phi _i\\rbrace _{i=1}^N\\in \\tilde{{H}}^\\omega $ , there exists $M\\in \\mathbb {N}$ such that $\\phi _i\\in {H}^\\omega _{\\sum _{j=1}^M P_{p_j}}$ for all $i$ .",
"So the multiplicity of the singular spectrum for $\\tilde{{H}}^\\omega $ is bounded by $K$ .",
"Hence using the density of $\\tilde{{H}}^\\omega $ in ${H}$ , we get that the multiplicity of the singular spectrum is bounded by $K$ ."
],
[
"Application",
"For proving the Corollary REF or Theorem REF , we need to obtain results about the multiplicity of the matrix $\\sqrt{C_n}G^\\omega _{n,n}(z)\\sqrt{C_n}$ .",
"This is done by using resolvent equation for a special decomposition of $A^\\omega $ .",
"Let $n\\in \\mathcal {N}$ be fixed, then using the fact that $range(C_n)\\subset \\mathcal {D}(A)$ the operators $P_nAP_n$ , $(I-P_n)AP_n$ and $P_nA(I-P_n)$ are well defined, and since they are finite rank operators, they are bounded.",
"Hence using the resolvent equation between $A^\\omega $ and $\\tilde{A}^\\omega =P_n AP_n+(I-P_n)A(I-P_n)+\\sum _{m\\in \\mathcal {N}}\\omega _m C_m,$ we obtain $G^\\omega _{n,n}(z)=\\left[ P_nAP_n+\\omega _nC_n-z P_n-P_nA(I-P_n)(\\tilde{A}^\\omega -z)^{-1}(I-P_n)AP_n\\right]^{-1},$ where the operator on RHS is viewed as a linear operator on $P_n{H}$ .",
"So the maximum algebraic multiplicity of eigenvalues of $\\sqrt{C_n}G^\\omega _{n,n}(z)\\sqrt{C_n}$ is same as the maximum algebraic multiplicity of eigenvalues of $C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}-zC_n^{-1}-C_n^{-\\frac{1}{2}} A(I-P_n)(\\tilde{A}^\\omega -z)^{-1}(I-P_n)A C_n^{-\\frac{1}{2}}.$ Notice that above equation is independent of $\\omega _n$ .",
"The basic difference between the proof of Corollary REF and Theorem REF is how the term $C_n^{-\\frac{1}{2}} A(I-P_n)(\\tilde{A}^\\omega -z)^{-1}(I-P_n)A C_n^{-\\frac{1}{2}}$ is handled.",
"Since the norm of above operator is $O((\\Im z)^{-1})$ , it is clear that we can ignore this term by choosing $\\Im z$ large enough, but this term provides the simplicity of the spectrum in Theorem REF .",
"We will be using the following lemma: Lemma 5.1 Consider the operator $A^\\omega $ and $A$ satisfying the hypothesis of corollary REF .",
"Let $I$ be a bounded interval contained in $(-\\infty ,M)$ such that maximum algebraic multiplicity of eigenvalues of $\\sqrt{C_n}G^\\omega _{n,n}(E)\\sqrt{C_n}$ is bounded by $K$ , for $E\\in I$ .",
"Then for almost all $z$ the maximum algebraic multiplicity of $\\sqrt{C_n}G^\\omega _{n,n}(z)\\sqrt{C_n}$ is bounded by $K$ .",
"Remark 5.2 The main advantage of this lemma is that instead of looking for a bound in $\\mathbb {C}\\setminus (M,\\infty )$ , we can work with $z\\in \\mathbb {R}\\setminus (\\sigma (A^\\omega )\\cup \\sigma (A))$ and so the operator $P_n(A^\\omega -E)^{-1}P_n=\\lim _{\\epsilon \\downarrow 0} P_n(A^\\omega -E-\\iota \\epsilon )^{-1}P_n$ is self adjoint, hence the algebraic and geometric multiplicities coincides.",
"The proof follows same steps as the proof of Lemma REF and so we are omitting it here.",
"Now we are ready to prove the other two results."
],
[
"Proof of Corollary ",
"Using Lemma REF and the fact that the algebraic multiplicity of $\\sqrt{C_n} G^\\omega _{n,n}(E) \\sqrt{C_n}$ is same as algebraic multiplicity of $C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}-EC_n^{-1}-C_n^{-\\frac{1}{2}} A(I-P_n)\\left(\\tilde{A}^\\omega -E\\right)^{-1}(I-P_n)A C_n^{-\\frac{1}{2}},$ bounding the multiplicity of above equation for $E\\ll M$ is enough.",
"First we handle the case when $C_n$ are projections.",
"The maximum algebraic multiplicity of (REF ) is same as $P_nAP_n-P_n A(I-P_n)\\left(\\tilde{A}^\\omega -E\\right)^{-1}(I-P_n)A P_n,$ we can ignore the $EC_n^{-1}$ term because it is the identity operator, and so does not affect the multiplicity.",
"Let $\\delta =\\min _{\\begin{array}{c}x,y\\in \\sigma (P_nAP_n)\\\\ x\\ne y\\end{array}} |x-y|,$ then for $E<-M-\\frac{3}{\\delta }\\left\\Vert P_nA(I-P_n)\\right\\Vert ^2$ we have $\\left\\Vert P_n A(I-P_n)\\left(\\tilde{A}^\\omega -E\\right)^{-1}(I-P_n)A P_n\\right\\Vert <\\frac{\\delta }{3}.$ So viewing $P_n A(I-P_n)\\left(\\tilde{A}^\\omega -E\\right)^{-1}(I-P_n)A P_n$ as a perturbation, we get that any eigenvalue of (REF ) is in $\\frac{\\delta }{3}$ neighborhood of eigenvalues of $P_nAP_n$ .",
"So the multiplicity of any eigenvalue of (REF ) cannot exceed the multiplicity of the eigenvalues of $P_nAP_n$ .",
"This completes the proof for the case of projection.",
"For general $C_n$ , the maximum algebraic multiplicity of (REF ) is same as the maximum algebraic multiplicity of $-C_n^{-1}+\\frac{1}{E}\\left(C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}-C_n^{-\\frac{1}{2}} A(I-P_n)\\left(\\tilde{A}^\\omega -E\\right)^{-1}(I-P_n)A C_n^{-\\frac{1}{2}}\\right),$ so setting $\\delta =\\min _{\\begin{array}{c}x,y\\in \\sigma (C_n^{-1})\\\\ x\\ne y\\end{array}} |x-y|$ and choosing $E<-2M-\\frac{3}{\\delta }\\left(\\left\\Vert C_n^{-\\frac{1}{2}}AC_n^{-\\frac{1}{2}}\\right\\Vert +\\left\\Vert C_n^{-\\frac{1}{2}} A(I-P_n)\\right\\Vert ^2\\right),$ we get that the eigenvalues of (REF ) are in $\\frac{\\delta }{3}$ neighborhood of $C_n^{-1}$ .",
"So following the argument for projection case we get that the multiplicity of any eigenvalue of (REF ) is upper bounded by the multiplicity of the eigenvalues of $C_n^{-1}$ ."
],
[
"Proof of Theorem ",
"Since $P_n\\Delta _{\\mathcal {B}}P_n$ has a non-trivial multiplicity, previous argument does not give us the desired result.",
"So we have to concentrate on (REF ), which in this case is $P_n\\Delta _{\\mathcal {B}}P_n-P_n \\Delta _{\\mathcal {B}}(I-P_n)\\left(\\tilde{H}^\\omega -E\\right)^{-1}(I-P_n)\\Delta _{\\mathcal {B}} P_n,$ where $\\tilde{H}^\\omega =P_n\\Delta _{\\mathcal {B}}P_n+(I-P_n)\\Delta _{\\mathcal {B}}(I-P_n)+\\sum _{x\\in J}\\omega _x P_x.$ Here we denote $P_x=\\chi _{\\tilde{\\Lambda }(x)}$ .",
"For simplicity of notation let us denote $\\partial \\tilde{\\Lambda }(x)=\\lbrace (p,q)\\in \\tilde{\\Lambda }(x)\\times \\tilde{\\Lambda }(x)^c: d(p,q)=1\\rbrace ,$ i.e we pair all the leaf nodes of the tree $\\tilde{\\Lambda }(x)$ with its neighbors outside the tree.",
"Figure: A representation of the rooted tree with three neighbors.",
"Observe that removing the sub-tree Λ ˜(x)\\tilde{\\Lambda }(x) divides the graphs into nine connected components.Following Dirac notation, observe that $&P_n \\Delta _{\\mathcal {B}}(I-P_n)\\left(\\tilde{H}^\\omega -E\\right)^{-1}(I-P_n)\\Delta _{\\mathcal {B}} P_n\\\\&\\qquad =\\sum _{(p,q)\\in \\partial \\tilde{\\Lambda }(x)} \\left|\\delta _p\\right\\rangle \\left\\langle \\delta _p\\right| \\left\\langle \\delta _q,(\\tilde{H}^\\omega -E)^{-1}\\delta _q\\right\\rangle ,$ this follows because $\\left\\langle \\delta _q,(I-P_n)\\Delta _{\\mathcal {B}}P_n \\delta _p\\right\\rangle =\\left\\lbrace \\begin{matrix} 1 & (p,q)\\in \\partial \\tilde{\\Lambda }(n)\\\\ 0 & otherwise \\end{matrix}\\right.,$ and $\\left\\langle \\delta _{q_1},(\\tilde{H}^\\omega )^k \\delta _{q_2}\\right\\rangle =0\\qquad \\forall k\\in \\mathbb {N}$ for $(p_1,q_1),(p_2,q_2)\\in \\partial \\tilde{\\Lambda }(n)$ and $q_1\\ne q_2$ .",
"This is also the reason why the random variables $\\left\\lbrace \\left\\langle \\delta _q,(\\tilde{H}^\\omega -E)^{-1}\\delta _q\\right\\rangle \\right\\rbrace _{(p,q)\\in \\partial \\tilde{\\Lambda }(x)}$ are independent of each other.",
"The random variable $\\left\\langle \\delta _q,(\\tilde{H}^\\omega -E)^{-1}\\delta _q\\right\\rangle $ is real for $E\\in \\mathbb {R}$ , and has absolutely continuous distribution, which follows from the following expression $&\\left\\langle \\delta _q,(\\tilde{H}^\\omega -E)^{-1}\\delta _q\\right\\rangle \\\\&\\qquad \\qquad =\\cfrac{1}{\\omega _q-E-\\sum _{x_1\\in N_{q}}\\cfrac{1}{\\omega _{x_1}-E-\\sum _{x_2\\in N_{x_1}} \\cfrac{1}{\\ddots -\\sum _{x_l\\in N_{x_{l-1}}} a^\\omega _{x_l}(E)}}},$ where $\\lbrace a^\\omega _{x_l}(E)\\rbrace $ are independent of $\\omega _q$ , and the distribution of $\\omega _q$ is absolutely continuous with respect to the Lebesgue measure.",
"Now Theorem REF follows from Theorem REF .",
"But first few notations are needed.",
"Denote $\\mathcal {T}_L$ to be a rooted tree with root $0_L$ and every vertex have $K+1$ neighbors except root $0_L$ (which has $K$ neighbors) and vertices in the boundary $\\partial \\mathcal {T}_L:=\\lbrace x\\in \\mathcal {T}_L: d(0_L,x)=L\\rbrace $ which have one neighbor each.",
"Theorem 5.3 Let $\\Delta _{\\mathcal {T}_L}$ denote the adjacency matrix over $\\mathcal {T}_L$ and set $B_\\tau =\\sum _{x\\in \\partial \\mathcal {T}_L}t_x\\left|\\delta _x\\right\\rangle \\left\\langle \\delta _x\\right|$ for $\\tau =\\lbrace t_x\\rbrace _{x\\in \\partial \\mathcal {T}_L}\\in \\mathbb {R}^{\\partial \\mathcal {T}_L}$ .",
"Then for almost all $\\tau $ w.r.t.",
"the Lebesgue measure, the spectrum of $H_\\tau =\\Delta _{\\mathcal {T}_L}+B_\\tau $ is simple.",
"The proof is done by induction on $L$ .",
"For the proof denote $H_{\\tau ,l}$ to be the operator $H_{\\tau ,l}=\\Delta _{\\mathcal {T}_l}+\\sum _{x\\in \\partial \\mathcal {T}_l}\\tau _x \\left|\\delta _x\\right\\rangle \\left\\langle \\delta _x\\right|$ where $\\Delta _{\\mathcal {T}_l}$ is the adjacency operator on the rooted tree $\\mathcal {T}_l$ with root $0_l$ .",
"The induction is done over the statement For almost all $\\tau $ , $H_{\\tau ,l}$ has simple spectrum with the property that all the eigenfunctions are non-zero at root, and $\\sigma (H_{\\tau ,l})\\cap \\sigma (H_{\\omega ,l})=\\phi $ for almost all $\\omega $.",
"For $l=0$ , the statement is trivial because $H_{\\tau ,0}$ is the operator on $\\mathbb {C}$ which is multiplication by the random variable $\\tau _{0_l}$ .",
"For the induction step suppose that the statement holds for all $l=N-1$ .",
"Observe that $H_{\\tau ,N}=\\sum _{x:d(0_N,x)=1}(|\\delta _{0_N}\\rangle \\langle \\delta _x|+|\\delta _x\\rangle \\langle \\delta _{0_N}|)+\\sum _{x:d(0_N,x)=1}H_{\\tau ,x},$ where $H_{\\tau ,x}:=\\chi _{\\mathcal {T}_{x}} H_{\\tau ,l}\\chi _{\\mathcal {T}_{x}}$ for the sub-tree $\\mathcal {T}_{x}:=\\lbrace y\\in \\mathcal {T}_l: d(0_N,y)=d(0_N,x)+d(x,y)\\rbrace $ .",
"Figure: The tree 𝒯 l \\mathcal {T}_l can be viewed as a union of KK disjoint trees {𝒯 x i } i \\lbrace \\mathcal {T}_{x_i}\\rbrace _i which are connected through their roots {x 1 ,⋯,x K }\\lbrace x_1,\\cdots ,x_K\\rbrace to a separate node 0 l 0_l.First notice that $H_{\\tau ,x}$ is unitarily equivalent to $H_{\\tilde{\\tau },N-1}$ where $\\tilde{\\tau }$ is restriction of $\\tau $ onto the $\\partial \\mathcal {T}_x$ .",
"Next note that $\\lbrace \\tau _y\\rbrace _y$ that appear in $H_{\\tau ,x_i}$ are disjoint for two subtrees $\\mathcal {T}_{x_1}$ and $\\mathcal {T}_{x_2}$ for $x_1\\ne x_2$ .",
"Hence by induction hypothesis we have $\\sigma (H_{\\tau ,x})\\cap \\sigma (H_{\\tau ,y})=\\phi $ for $x\\ne y$ and the spectrum of $H_{\\tau ,x}$ is simple with the property that the eigenfunctions corresponding to the eigenvalues are non-zero at the root, for each $x$ .",
"Since we are working on tree graphs, we have $\\left\\langle \\delta _{0_N},(H_{\\tau ,N}-z)^{-1}\\delta _{0_N}\\right\\rangle &=\\frac{1}{-z-\\sum _{x:d(0_N,x)=1}\\left\\langle \\delta _{x},(H_{\\tau ,x}-z)^{-1}\\delta _{x}\\right\\rangle }\\nonumber \\\\&=\\frac{1}{-z-\\sum _{x:d(0_N,x)=1}\\sum _{E\\in \\sigma (H_{\\tau ,x})}\\frac{\\left|\\left\\langle \\psi _{\\tau ,x,E},\\delta _x\\right\\rangle \\right|^2}{E-z}}$ where $\\psi _{\\tau ,x,E}$ is the eigenfunction of $H_{\\tau ,x}$ for the eigenvalue $E$ .",
"By the induction hypothesis we have $\\left\\langle \\psi _{\\tau ,x,E},\\delta _x\\right\\rangle \\ne 0$ for each $E\\in \\sigma (H_{\\tau ,x})$ and $x$ a neighbor of $0_N$ .",
"Next using the fact that $\\sigma (H_{\\tau ,x})\\cap \\sigma (H_{\\tau ,y})=\\phi $ for $x\\ne y$ , we get that $z+\\sum _{x:d(0_N,x)=1}\\sum _{E\\in \\sigma (H_{\\tau ,x})}\\frac{\\left|\\left\\langle \\psi _{\\tau ,x,E},\\delta _x\\right\\rangle \\right|^2}{E-z}$ has $\\sum _{x:d(0_N,x)=1}\\#\\sigma (H_{\\tau ,x})$ many poles and so the equation (REF ) has $1+\\sum _{x:d(0_N,x)=1}\\#\\sigma (H_{\\tau ,x})$ many roots, which is equal to $|\\mathcal {T}_N|$ .",
"But using functional calculus we also have $\\left\\langle \\delta _{0_N},(H_{\\tau ,N}-z)^{-1}\\delta _{0_N}\\right\\rangle =\\sum _{E\\in \\sigma (H_{\\tau ,N})}\\frac{|\\left\\langle \\psi _{\\tau ,N,E},\\delta _{0_N}\\right\\rangle |^2}{E-z}$ where $\\psi _{\\tau ,N,E}$ is the eigenfunction corresponding to the eigenvalue $E$ for the matrix $H_{\\tau ,N}$ .",
"So each pole $\\left\\langle \\delta _{0_N},(H_{\\tau ,N}-z)^{-1}\\delta _{0_N}\\right\\rangle $ corresponds to an eigenvalue, and previous computation shows that there are $|\\mathcal {T}_N|$ many poles, which gives the simplicity of the spectrum of $H_{\\tau ,N}$ .",
"Finally, the eigenfunction $\\psi _{\\tau ,N,E}$ is non-zero at the root $0_N$ because of the fact that if $\\left\\langle \\psi _{\\tau ,N,E},\\delta _{0_N}\\right\\rangle =0$ , then the pole corresponding to $E$ will not be present in the above expression.",
"Finally we have to prove $\\sigma (H_{\\tau ,l})\\cap \\sigma (H_{\\omega ,l})=\\phi $ for almost all $\\tau ,\\omega $ .",
"But first we need the following claim: Claim: For any solution $\\psi \\in \\mathbb {C}^{\\mathcal {T}_l}\\setminus \\lbrace 0\\rbrace $ of $H_{\\tau ,l}\\psi =E\\psi $ for $E\\in \\mathbb {R}$ , there exists $x\\in \\partial \\mathcal {T}_l$ such that $\\psi _x\\ne 0$ .",
"proof: If for some $E\\in \\mathbb {R}$ there exists $\\psi \\in \\mathbb {C}^{\\mathcal {T}_l}$ such that $H_{\\tau ,l}\\psi =E\\psi $ and $\\psi _x=0\\qquad \\forall x\\in \\partial \\mathcal {T}_l,$ then for any $x\\in \\partial \\mathcal {T}_l$ $& (H_{\\tau ,l}\\psi )_x=E\\psi _x=0\\\\\\Rightarrow \\qquad & \\psi _{Px}+t_x\\psi _x=0\\\\\\Rightarrow \\qquad & \\psi _{Px}=0,$ where $Px$ is the unique neighbor of $x$ satisfying $d(0_l,x)=d(0_l,Px)+1$ .",
"So we get that $\\psi _x=0$ for all $x\\in \\mathcal {T}_l$ such that $d(0,x)=l-1$ .",
"Repeating the above argument for $x$ satisfying $d(0,x)=l-1$ will give $\\psi _x=0$ for all $x$ such that $d(0_l,x)=l-2$ .",
"Repeating the last step recursively gives $\\psi \\equiv 0$ giving contradiction, which completes the proof of the claim.",
"Now to prove $\\sigma (H_{\\tau ,l})\\cap \\sigma (H_{\\omega ,l})=\\phi $ for almost all $\\tau ,\\omega $ .",
"Denote $\\tau =\\lbrace \\tau _x\\rbrace _{x\\in \\partial \\mathcal {T}_l}$ , $\\omega =\\lbrace \\omega _x\\rbrace _{x\\in \\partial \\mathcal {T}_l}$ , set $\\lbrace E^\\tau _{i}\\rbrace _i$ and $\\lbrace \\psi ^\\tau _i\\rbrace $ to be the eigenvalues and the corresponding eigenfunctions for $H_{\\tau ,l}$ and similarly for $H_{\\omega ,l}$ .",
"Using Feynman-Hellmann theorem for rank one perturbation, we have $\\frac{dE^\\tau _i}{d\\tau _x}=|\\left\\langle \\psi ^\\tau _{i},\\delta _x\\right\\rangle |^2\\qquad \\forall x\\in \\partial \\mathcal {T}_l,\\forall i,$ and similarly $\\frac{dE^\\omega _i}{d\\omega _x}=|\\left\\langle \\psi ^\\omega _{i},\\delta _x\\right\\rangle |^2\\qquad \\forall x\\in \\partial \\mathcal {T}_l,\\forall i.$ For each $i$ , using the previous claim, there exists $x^\\tau _i\\in \\partial \\mathcal {T}_l$ such that $\\left\\langle \\psi ^\\tau _{i},\\delta _{x^\\tau _i}\\right\\rangle \\ne 0$ , and similarly for $\\omega $ .",
"Now using Implicit Function Theorem over $E^\\tau _{i}-E^\\omega _j=0$ , the manifold $\\lbrace (\\tau ,\\omega )\\in \\mathbb {R}^{\\partial \\mathcal {T}_l}\\times \\mathbb {R}^{\\partial \\mathcal {T}_l} : E^\\tau _{i}=E^\\omega _j\\rbrace $ has lower dimension than $2|\\partial \\mathcal {T}_l|$ .",
"So in particular $Leb\\left(\\lbrace (\\tau ,\\omega )\\in \\mathbb {R}^{\\partial \\mathcal {T}_l}\\times \\mathbb {R}^{\\partial \\mathcal {T}_l} : E^\\tau _{i}=E^\\omega _j\\rbrace \\right)=0$ which completes the proof of the induction step.",
"Acknowledgement: The author, Dhriti Ranjan Dolai is supported by from the J. C. Bose Fellowship grant of Prof. B. V. Rajarama Bhat."
],
[
"Appendix",
"Lemma A.1 On a separable Hilbert space ${H}$ , let $H$ be a self adjoint operator, and for $\\phi ,\\psi \\in {H}$ set $\\sigma _\\phi (\\cdot )=\\left\\langle \\phi ,E_H(\\cdot )\\phi \\right\\rangle $ and $\\sigma _{\\phi ,\\psi }(\\cdot )=\\left\\langle \\phi ,E_H(\\cdot )\\psi \\right\\rangle $ .",
"Let $f$ be the Radon-Nikodym derivative of $\\sigma _{\\phi ,\\psi }$ w.r.t $\\sigma _\\phi $ , then $f(H)\\phi $ is the projection of $\\psi $ onto the minimal closed $H$ -invariant subspace containing $\\phi $ .",
"Let ${H}_\\phi $ denote the minimal closed $H$ -invariant subspace containing $\\phi $ , then $({H}_\\phi ,H)$ is unitarily equivalent to $(L^2(\\mathbb {R},\\sigma _\\phi ),M_{Id})$ where $M_{Id}$ is multiplication with the identity map on $\\mathbb {R}$ .",
"We have the linear functional $g\\mapsto \\left\\langle g(H)\\phi ,\\psi -f(H)\\phi \\right\\rangle $ for $g\\in L^2(\\mathbb {R},\\sigma _\\phi )$ .",
"Observe that $\\left\\langle g(H)\\phi ,\\psi -f(H)\\phi \\right\\rangle =\\left\\langle g(H)\\phi ,\\psi \\right\\rangle -\\left\\langle g(H),f(H)\\phi \\right\\rangle \\\\=\\int g(x)d\\sigma _{\\phi ,\\psi }(x)-\\int g(x)f(x)d\\sigma _{\\phi }(x)=0,$ because $f$ is Radon-Nikodym derivative of $\\sigma _{\\phi ,\\psi }$ with respect to $\\sigma _\\phi $ .",
"Since $g(H)\\phi $ are dense in ${H}_\\phi $ for $\\phi \\in L^2(\\mathbb {R},\\sigma _\\phi )$ , we have $\\psi -f(H)\\phi \\perp {H}_\\phi ,$ hence $f(H)\\phi $ is the projection of $\\psi $ on to ${H}_\\phi $ .",
"Lemma A.2 On a separable Hilbert space ${H}$ let $H$ be a self-adjoint operator and $Q$ be a finite ranked projection.",
"Let $\\lbrace e_i\\rbrace _{i\\in \\mathbb {N}}$ be a orthonormal basis for the subspace $Q{H}$ and denote ${H}_i=\\overline{\\left\\langle f(H)e_i:~f\\in C_c(\\mathbb {R}) \\right\\rangle },$ and ${H}_Q=\\overline{\\left\\langle f(H)\\phi :~f\\in C_c(\\mathbb {R})~\\&~\\phi \\in Q{H} \\right\\rangle }.$ Then ${H}_Q=\\sum _i {H}_i,$ where $\\sum _i {H}_i$ denotes the closure of finite linear combinations of elements of ${H}_i$ .",
"Since ${H}_i\\subseteq {H}_Q$ for any $i$ , we always have $\\sum _i {H}_i\\subseteq {H}_Q.$ For the other way round note that we only have to show $f(H)\\phi \\in \\sum _i {H}_i$ , for $\\phi \\in Q{H}$ .",
"Since $\\lbrace e_i\\rbrace _i$ is a basis, we have $\\phi =\\sum _i a_ie_i.$ Using it, define $\\psi _N=\\sum _{i=1}^N a_i f(H)e_i,$ which satisfies $\\psi _N\\in \\sum _i{H}_i$ for any $N\\in \\mathbb {N}$ .",
"Now the conclusion of the lemma holds, since $\\sum _i{H}_i$ is closed."
]
] | 1709.01774 |
[
[
"Non-hyperbolic solutions to tangle equations involving composite links"
],
[
"Abstract Solving tangle equations is deeply connected with studying enzyme action on DNA.",
"The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2 \\# b_3$, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$ nontrivial.",
"We solve this system of equations under the assumption $\\widetilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e.$O$ is not $\\pi$-hyperbolic.",
"Besides, we also deal with tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle."
],
[
"Introduction and main theorem",
"In order to study enzyme action on DNA, Ernst and Sumners [11] introduced tangle model, which models DNA as a knot/link and regards the enzyme action on the DNA knot/link as a tangle replacement.",
"Fortunately, most of the DNA knot/link belong to the mathematically well-known class of 2-bridge knots/links (4-plat knots/links).",
"Besides, based on biological experiments, it is possible to obtain a composite knot/link as a product of some enzyme action in certain situations.",
"Therefore, the central purpose of this paper is to solve the following system of tangle equations, and the main theorem is given as follows: Theorem 1 Suppose N(O+X1)=b1 N(O+X2)=b2 # b3, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$ , with $b_2$ and $b_3$ nontrivial.",
"Suppose $\\widetilde{O}$ is irreducible toroidal but not Seifert fibered.",
"then the system of tangle equations has solutions if and only if one of the following holds: (i) There exist 3 pairs of relatively prime integers $(a,b)$ , $(p_1,q_1)$ , and $(p,q)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty ), p>1,p_1>1,|aq_1-bp_1|>1$ , and $q-pp_1q_1=1$ such that $b_1=b(a,b)$ and $b_2 \\# b_3 =b(p,q) \\# b(a+ap_1q_1p-bp_1^2p, b+aq_1^2p-bp_1q_1p)$ .",
"Solutions up to equivalence are shown as the following: Figure: O=O= the tangle in (a) or (b), where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1} (or A=ad-be aq 1 -bp 1 A=\\frac{ad-be}{aq_1-bp_1}, B=-e p 1 B=\\frac{-e}{p_1}), and p 1 d-q 1 e=1p_1d-q_1e=1 with d,e∈ℤd,e\\in \\mathbb {Z}.",
"X 1 =0X_1=0-tangle and X 2 =∞X_2=\\infty -tangle.",
"(Note that choosing different dd and ee such that p 1 d-q 1 e=1p_1d-q_1e=1 has no effect on the tangle OO.",
")(ii) There exist 3 pairs of relatively prime integers $(a,b)$ , $(p_1,q_1)$ , and $(p,q)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty ), p>1,p_1>1,|aq_1-bp_1|>1$ , and $q-pp_1q_1=-1$ such that $b_1=b(a,b)$ and $b_2 \\# b_3 =b(p,q) \\# b(a-ap_1q_1p+bp_1^2p, b-aq_1^2p+bp_1q_1p)$ .",
"Solutions up to equivalence are shown as the following: Figure: O=O= the tangle in (a) or (b), where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1} (or A=ad-be aq 1 -bp 1 A=\\frac{ad-be}{aq_1-bp_1}, B=-e p 1 B=\\frac{-e}{p_1}), and p 1 d-q 1 e=1p_1d-q_1e=1 with d,e∈ℤd,e\\in \\mathbb {Z}.",
"X 1 =0X_1=0-tangle and X 2 =∞X_2=\\infty -tangle.",
"(Note that choosing different dd and ee such that p 1 d-q 1 e=1p_1d-q_1e=1 has no effect on the tangle OO.",
")As a matter of fact, the key point to solve tangle equations is lifting to the double branched covers.",
"When we take the \"sum\" of two tangles $O$ and $X$ , it results in a link $K$ .",
"Lifting to their double branched covers, this operation on tangles $O$ and $X$ induces a gluing of the boundaries of their respective double branched covers, which produces a 3-manifold $\\widetilde{K}$ , the double cover of $S^3$ branched over $K$ .",
"The double branched cover of a rational tangle is a solid torus, so a rational tangle replacement induces a Dehn surgery on a 3-manifold.",
"Here the 3-manifold is a lens space or a connected sum of two lens spaces since the double cover of $S^3$ branched over a 2-bridge link is a lens space.",
"Thus, our problem turns out to be finding knots in a lens space admitting a surgery to give a connected sum of two lens spaces.",
"Kenneth L. Baker [3] proposed a lens space version of cabling conjecture stating that if a knot in a lens space admits a surgery to a non-prime 3-manifold then the knot is either lying in a ball, or a knot with Seifert fibered exterior, or a cabled knot, or hyperbolic.",
"He proved the conjecture when the knot is non-hyperbolic in [3].",
"Buck and Mauricio's paper [5] addresses the problem in two cases, (1)the double branched cover of the tangle $O$ , denoted by $\\widetilde{O}$ , is a Seifert fiber space and (2)$\\widetilde{O}$ is reducible, namely $\\widetilde{O}$ is the complement of a knot with Seifert fibered exterior and $\\widetilde{O}$ is the complement of a knot in a ball.",
"This paper mainly solves the aforementioned system of equations when $\\widetilde{O}$ is the complement of a cabled knot, i.e.",
"$\\widetilde{O}$ is irreducible toroidal but not Seifert fibered.",
"We also give solutions for the Seifert fibered case, which comprise a special case excluded in [5].",
"In fact, we have a different definition of tangle from [5], since we allow tangle to have circles embedded in, so more solutions are given in this paper.",
"The following outlines the proof of Theorem REF .",
"First of all, we show that the cabled knot is a cable of a torus knot lying in one of the solid tori in the lens space, and the complement of this cabled knot is a graph manifold with only one essential torus.",
"Then by studying Dehn surgeries along this cabled knot, we obtain all possible products $b_2 \\# b_3$ when $b_1$ is given.",
"Split the graph manifold along the essential torus into two pieces, both of which are Seifert fiber spaces.",
"We can easily find two tangles whose double branched covers are the two pieces respectively, since it is not hard to find a Montesinos tangle (or a Montesinos pair) with this type of Seifert fiber spaces as the double branched cover.",
"By gluing the two tangles together, we get a tangle whose double branched cover is this graph manifold.",
"Finally, by studying the involutions on this graph manifold, we prove that any tangle whose double branched cover is this graph manifold is homeomorphic to the tangle we construct.",
"Connecting with the analysis about Dehn fillings on this graph manifold, we give all the solutions $(O,X_1,X_2)$ of the tangle equations above when $\\widetilde{O}$ is the complement of a cabled knot.",
"This paper is structured as follows: Section provides some preliminaries about tangles, 2-bridge link and their double branched covers.",
"Section gives the complete process of solving tangle equations above.",
"In Section , we solve the system of tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle by using the same method as solving the aforementioned tangle equations."
],
[
"Acknowledgements",
"toc tocsectionAcknowledgementsI would like to express my gratitude to my supervisor Professor Zhongtao Wu for his professional guidance and helpful discussion.",
"He introduced this subject to me and gave me many valuable suggestions.",
"I would also like to thank Erica Flapan for introducing me algebraic tangle and Bonahon-Siebenmann theory.",
"This work was partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No.",
"CUHK 14309016)."
],
[
"Tangles and 2-bridge link",
"Definition 2.1 A tangle is a pair $(M,t)$ where $M$ is the complement of disjoint 3-balls in $S^3$ and $t$ is a properly embedded 1-manifold which intersects each boundary component of $M$ in 4 points.",
"Definition 2.2 A marked tangle is a tangle $(M,t)$ with $k$ boundary components $S_1, \\dots , S_k$ parameterized by a family of orientation-preserving homeomorphisms $\\Phi =\\cup _i \\Phi _i: (\\partial S_i, \\partial S_i \\cap t) \\rightarrow (S^2, P)$ , where $S^2$ is the unit sphere in $\\mathbb {R}^3$ and $P=\\lbrace NE=(e^{\\frac{\\pi }{4}i},0),SE=(e^{\\frac{-\\pi }{4}i},0),SW=(e^{\\frac{-3\\pi }{4}i},0),NW=(e^{\\frac{3\\pi }{4}i},0)\\rbrace $ .",
"We denote a marked tangle as a triple $(M,t,\\Phi )$ .",
"Definition 2.3 Two tangles $X=(M,t,\\Phi )$ and $Y=(M^{\\prime },t^{\\prime },\\Phi ^{\\prime })$ are isomorphic if there exists a homeomorphism $H:(M,t)\\rightarrow (M^{\\prime },t^{\\prime })$ such that $\\Phi =\\Phi ^{\\prime }H$ .",
"$X$ and $Y$ are homeomorphic if there exists a homeomorphism $H:(M,t)\\rightarrow (M^{\\prime },t^{\\prime })$ .",
"In this paper, we mainly deal with tangles in a 3-ball with a few exceptions.",
"The following are some operations and useful results about tangles in a 3-ball.",
"Definition 2.4 Given two tangles $A$ and $B$ in a 3-ball, tangle addition is defined as shown in Figure REF , denoted by $A+B$ .",
"tangle multiplication is defined as shown in Figure REF , denoted by $A \\times B$ .",
"Figure: Tangle addition and multiplicationDefinition 2.5 Given a tangle $A$ in a 3-ball, the numerator closure and denominator closure are defined as shown in Figure REF and Figure REF respectively, denoted by $N(A)$ and $D(A)$ .",
"Figure: Numerator closure and denominator closureRemark $D(A+B)=D(A) \\# D(B)$ .",
"Definition 2.6 The circle product of a tangle $A$ in a 3-ball and an integer vector $C=(c_1, c_2, \\dots , c_n)$ is shown in Figure REF , denoted by $A \\circ C$ or $A \\circ (c_1, c_2, \\dots , c_n)$ .",
"Figure: Circle productRemark When $c_i>0$ , it represents $|c_i|$ positive half twists.",
"Conversely, $c_i<0$ represents $|c_i|$ negative half twists.",
"Positive and negative twists are shown in the following pictures.",
"Figure: Half twistDefinition 2.7 A tangle $X=(B^3,t)$ is rational if it is homeomorphic to the trivial tangle $(D^2 \\times I,\\lbrace x,y\\rbrace \\times I)$ , where $D^2$ is the unite 2-ball in $\\mathbb {R}^2$ and $\\lbrace x,y\\rbrace $ are two points interior to $D^2$ .",
"See Figure REF .",
"Each rational tangle is isomorphic to a so-called \"basic vertical tangle\" which is constructed by taking the circle product of the $\\infty $ -tangle shown in Figure REF and an integer vector $C=(c_1,\\dots , c_n)$ with $n$ even, i.e.$\\infty \\circ C$ .",
"The following rational tangle classification theorem tells us that rational tangles are classified, up to isomorphism, by their continued fraction: $\\frac{\\beta }{\\alpha }=c_n+\\frac{1}{c_{n-1}+\\frac{1}{\\dots +\\frac{1}{c_1}}}$ .",
"Theorem 2.8 (Rational Tangle Classification Theorem [6]) There exists a 1-1 correspondence between isomorphism classes of rational tangles and the extended rational numbers $\\beta / \\alpha \\in \\mathbb {Q} \\cup \\lbrace 1/0=\\infty \\rbrace $ , where $\\alpha \\in \\mathbb {N}, \\,\\, \\beta \\in \\mathbb {Z}$ , and $gcd(\\alpha , \\beta )=1$ .",
"Figure: Rational tanglesA 2-bridge knot/link (4-plat knot/link or rational knot/link) is a knot/link obtained by taking numerator or denominator closure of rational tangles.",
"More precisely, $D(\\frac{\\beta }{\\alpha })$ gives a 2-bridge knot/link, denoted by $b(\\alpha , \\beta )$ .",
"$N(\\frac{\\beta }{\\alpha })$ gives the 2-bridge knot/link $b(\\beta ,-\\alpha )$ .",
"In fact, the numerator or denominator closure of rational tangles also produce the unknot.",
"For convenience, the unknot is contained in the 2-bridge knot/link class in this paper, and it is denoted by $b(1,1)$ .",
"Theorem 2.9 (2-Bridge Link Classification Theorem [21]) The 2-bridge knot/link $b(\\alpha , \\beta )$ with $\\alpha >0$ is equivalent to the 2-bridge knot/link $b(\\alpha ^{\\prime }, \\beta ^{\\prime })$ with $\\alpha ^{\\prime }>0$ if and only if $\\alpha =\\alpha ^{\\prime }$ and $\\beta ^{\\pm 1}\\equiv \\beta ^{\\prime } (mod \\,\\alpha )$ .",
"Adding two rational tangles and then taking the numerator closure also gives a 2-bridge knot/link.",
"See the following lemma.",
"Lemma 2.10 [11] Given two rational tangles $X_1=\\beta _1 / \\alpha _1$ and $X_2=\\beta _2 / \\alpha _2$ , then $N(X_1+X_2)$ is the 2-bridge knot/link $b(\\alpha _1 \\beta _2 + \\alpha _2 \\beta _1, \\alpha _1 \\beta ^{\\prime }_2 + \\alpha ^{\\prime }_2\\beta _1)$ , where $\\beta _2 \\alpha ^{\\prime }_2 - \\alpha _2 \\beta ^{\\prime }_2=1$ .",
"Definition 2.11 The Montesinos tangle $\\left( \\frac{\\beta _1}{\\alpha _1}, \\dots , \\frac{\\beta _i}{\\alpha _i}, \\dots , \\frac{\\beta _n}{\\alpha _n} \\right)$ , where $n \\ge 2$ , is obtained from adding $n$ rational tangles $\\beta _i / \\alpha _i$ together, that is $\\left(\\frac{\\beta _1}{\\alpha _1}, \\dots , \\frac{\\beta _i}{\\alpha _i}, \\dots , \\frac{\\beta _n}{\\alpha _n} \\right)=\\frac{\\beta _1}{\\alpha _1}+ \\dots +\\frac{\\beta _i}{\\alpha _i}+ \\dots +\\frac{\\beta _n}{\\alpha _n}$ .",
"See Figure REF .",
"Figure: Montesinos tangleDefinition 2.12 A Generalized Montesinos tangle is a tangle obtained by taking the circle product of a Montesinos tangle $(A_1, \\dots , A_n)$ and an integer entry vector $C=(c_1, c_2, \\dots , c_m)$ , where $A_i$ is a rational tangle for $i=1,\\dots ,n$ , and $m \\in \\mathbb {Z}^{+}$ , denoted by $(A_1+ \\dots +A_n) \\circ C$ .",
"See Figure REF .",
"Figure: Generalized Montesinos tangleDefinition 2.13 An algebraic tangle is a tangle obtained by preforming the operations of tangle addition and multiplication on rational tangles.",
"Remark Algebraic tangle includes all of rational tangles, Montesinos tangles and generalized Montesinos tangles.",
"Definition 2.14 A tangle $X=(B^3,t)$ is locally knotted if there exists a local knot in one of the strands.",
"More precisely, there exists a 2-sphere in $X$ intersecting $t$ transversely in 2 points, such that the 3-ball it bounds in $X$ meets $t$ in a knotted arc.",
"Now we introduce a special type of tangle which is not necessarily in a 3-ball.",
"Following Bonahon and Siebenmann [4], we call it a Montesinos pair instead of a tangle, since there is a specifying definition of a Montesinos tangle.",
"Note that the previous Montesinos tangles and generalized Montesinos tangles are included in Montesinos pair.",
"Definition 2.15 A Montesinos pair is homeomorphic to a tangle constructed from a tangle of the type (a) or (b) shown in Figure REF by plugging some of the holes with rational tangles.",
"If a Montesinos pair $M$ is built from the tangle of the type (a) with rational tangle $\\frac{\\alpha _i}{\\beta _i}$ plugging in from left to right, and with $k$ boundary components(i.e.$k$ holes with no rational tangle plugging in), then we denote $M$ as $M=(0,k;\\frac{\\alpha _1}{\\beta _1}, \\dots , \\frac{\\alpha _N}{\\beta _N})$ .",
"If there is no rational tangle plugging in some of the holes, then we fill in the symbol $\\varnothing $ instead of a rational number.",
"If a Montesinos pair $M$ is built from the tangle of the type (b) with rational tangle $\\frac{\\alpha _i}{\\beta _i}$ plugging in from left to right, and with $k$ boundary components, then we denote $M$ as $M=(-1,k;\\frac{\\alpha _1}{\\beta _1}, \\dots , \\frac{\\alpha _N}{\\beta _N})$ .",
"If there is no rational tangle plugging in some of the holes, then we fill in the symbol $\\varnothing $ instead of a rational number.",
"Figure: Montesinos pairs built from the type (a) (resp.",
"(b)) contain no ring tangle (resp.",
"one ring tangle).",
"The ring tangle is a tangle shown in (c).Definition 2.16 A Conway sphere in a tangle $(M,t)$ is a 2-sphere in int$M$ which intersects $t$ transversely in 4 points."
],
[
"Double branched covers",
"The key point to solve tangle equations is considering their corresponding double branched covers.",
"When we perform tangle addition and then take the numerator closure on two tangles $O$ and $X$ , it gives a knot $K$ .",
"Lifting to their double branched covers, these operations induce a gluing of the boundaries of their respective double branched covers, $\\widetilde{O}$ and $\\widetilde{X}$ .",
"It yields a 3-manifold $\\widetilde{K}$ , the double cover of $S^3$ branched over $K$ : $N(O+X)=K\\Longrightarrow \\widetilde{O} \\cup _h \\widetilde{X}=\\widetilde{K}$ where $h: \\partial \\widetilde{O} \\rightarrow \\partial \\widetilde{X}$ is the gluing map.",
"In particular, when $X$ is a rational tangle, this tangle equation corresponds to a Dehn filling on $\\widetilde{O}$ .",
"A slant on $(S^2, P=4 \\, points)$ is the isotopy class of essential simple closed curves in $S^2-P$ .",
"Let $f: T^2 \\rightarrow S^2$ be the double covering map branched over $P$ (It is well known that the double cover of $S^2$ branched over $P$ is a torus).",
"For each slant $c$ , $f^{-1}(c)$ are two parallel simple closed curves in $T^2$ , and we denote an arbitrary one as $\\widetilde{c}$ .",
"Let $m$ , $l$ be the slants on $(S^2, P)$ shown as the following figure: Figure: NO_CAPTIONOrient $\\widetilde{m}$ and $\\widetilde{l}$ so that $\\widetilde{m} \\cdot \\widetilde{l}=1$ with respect to the orientation of the torus.",
"Then $[\\widetilde{m}]$ and $[\\widetilde{l}]$ form a basis of $H_1(T^2)$ .",
"In fact, there are bijections {slants on $(S^2,P)$ } $\\leftrightarrow $ {slopes on $T^2$ } $\\leftrightarrow $ {$\\mathbb {Q} \\cup \\infty $ }.",
"For example, the slant $l$ corresponds to the rational number 0, and $m$ corresponds to $\\infty $ .",
"For a $\\frac{q}{p}$ slant on $(S^2,P)$ , we can construct it by firstly taking $p$ parallel copies of $l$ and then performing a $\\frac{\\pi q}{p}$ twist along $m$ .",
"This $\\frac{q}{p}$ slant lifts to a $p[\\widetilde{l}]+q[\\widetilde{m}]$ slope on $T^2$ .",
"For each rational tangle, there exists a properly embedded disk that separates the tangle into two 3-balls each containing an unknotted arc.",
"We call this disk a meridional disk, and it can be shown that meridian disk is unique up to isotopy.",
"The boundary of the meridional disk for a rational tangle is a slant on the boundary of the tangle, called the meridian of this rational tangle.",
"Also the meridian of a rational tangle is unique up to isotopy.",
"In fact the fraction corresponding to the meridian for a given rational tangle coincides with its continued fraction which we use to classify rational tangles in Theorem REF .",
"Also It is well known that the double branched cover of a rational tangle is a solid torus.",
"So there is a 1-1 correspondence between plugging one boundary component of a tangle in the rational tangle $q / p$ and the $q / p$ Dehn filling (under $[\\widetilde{m}]$ and $[\\widetilde{l}]$ basis) of the corresponding torus boundary component in the double branched cover of the tangle.",
"Now we give some useful results about double branched covers.",
"X: a tangle (B3,t) X: the double branched cover of B3 with branched set t. K: a link in S3 K: the double branched cover of S3 with branched set K. Y: a tangle (M,t) Y: the double branched cover of M with branched set t. A tangle $X$ is rational if and only if $\\widetilde{X}$ is a solid torus.",
"A knot/link $K$ is a 2-bridge knot/link if and only if $\\widetilde{K}$ is a lens space.",
"The double branched cover of the 2-bridge knot/link $b(p,q)$ is the lens space $L(p,q)$ .",
"The double branched cover of $K_1 \\# K_2$ is $\\widetilde{K_1} \\# \\widetilde{K_2}$ , where $K_i$ is a knot/link for $i=1,2$ [5].",
"The double branched cover of a locally knotted tangle is a reducible manifold.",
"The double branched cover of an algebraic tangle is a graph manifold with one torus boundary (a graph manifold is a 3-manifold which is obtained by gluing some circle bundles) [4].",
"The double branched cover of a Montesinos pair $Y=(M,t)$ is a generalized Seifert fiber space with orbit surface of genus 0 or -1.",
"If $Y=(0,k;\\frac{\\beta _1}{\\alpha _1}, \\dots , \\frac{\\beta _N}{\\alpha _N})$ , then $\\widetilde{Y}=M(0,k;(\\alpha _1,\\beta _1), \\dots , (\\alpha _N,\\beta _N))$ .",
"If $Y=(-1,k;\\frac{\\beta _1}{\\alpha _1}, \\dots , \\frac{\\beta _N}{\\alpha _N})$ , then $\\widetilde{Y}=M(-1,k;(\\alpha _1,\\beta _1), \\dots , (\\alpha _N,\\beta _N))$ [4].",
"A Seifert fiber space over a disk with $n$ exceptional fibers is the double branched cover of a tangle, then the tangle is a generalized Montesinos tangle with $n$ rational tangle summands [10].",
"Generalized Seifert fiber space is defined similarly to Seifert fiber space, and it comprises all true Seifert fiber spaces.",
"The only difference is that the exceptional fiber of a generalized Seifert fiber space may have the coefficient $(0,1)$ .",
"For more details, please see [17].",
"Each double branched covering map is induced by an involution.",
"We call the involution of a torus with 4 fixed points the standard involution.",
"As mentioned above, $M(0,k;(\\alpha _1,\\beta _1), \\dots , (\\alpha _N,\\beta _N))$ is the double branched cover of the Montesinos pair $Y=(0,k;\\frac{\\beta _1}{\\alpha _1}, \\dots , \\frac{\\beta _N}{\\alpha _N})$ .",
"We call the non-trivial covering transformation corresponding to this double branched cover the standard involution on $M(0,k;(\\alpha _1,\\beta _1), \\dots , (\\alpha _N,\\beta _N))$ ."
],
[
"Solving tangle equations",
"The main goal of this paper is to solve the following tangle equations: N(O+X1)=b1 N(O+X2)=b2 # b3, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$ , with $b_2$ and $b_3$ nontrivial.",
"Given the above tangle equations, we also say that the knot/link $b_2 \\# b_3$ can be obtained from $b_1$ by an $(X_1, X_2)$ -move.",
"Let $X_i$ and $X^{\\prime }_i$ be rational tangles for $i=1,2$ .",
"An $(X_1,X_2)$ -move is equivalent to an $(X^{\\prime }_1,X^{\\prime }_2)$ -move if for any two knots/links $K_1$ and $K_2$ , there exists a tangle $O$ satisfying tangle equations $N(O+X_1)=K_1$ and $N(O+X_2)=K_2$ , if and only if there exists a tangle $O^{\\prime }$ such that $N(O^{\\prime }+X^{\\prime }_1)=K_1$ and $N(O^{\\prime }+X^{\\prime }_2)=K_2$ .",
"As discussed in [8], any solution $(O,X_1,X_2)$ of the above tangle equations is equivalent to a solution $(O^{\\prime },X^{\\prime }_1=0,X^{\\prime }_2)$ .",
"If one solution $(O,X_1,X_2)$ satisfying $X_1=0$ -tangle is given, there is a standard algorithm to give all equivalent solutions.",
"For more details, please see [8].",
"So we only need to solve the above tangle equations assuming $X_1=0$ -tangle.",
"Lifting to the double branched covers, the system of tangle equations can be translated to O()= the lens space b1 O()= the connnected sum of two lens spaces b2 and b3 b2 # b3, where $\\widetilde{O}$ (resp.$\\widetilde{b_i}$ ) denotes the double branched cover of $O$ (resp.$b_i$ ), and $\\theta $ (resp.$\\eta $ ) is the induced Dehn filling slope by adding rational tangle $X_1$ (resp.$X_2$ ) since $\\widetilde{X_i}$ is a solid torus.",
"Therefore, the problem turns out to be finding knots in the lens space $\\widetilde{b_1}$ which admit a surgery to a connected sum of two lens spaces.",
"Here $\\widetilde{X_1}$ is a tubular neighborhood of the knot in $\\widetilde{b_1}$ , and $\\widetilde{O}$ is the complement of the knot.",
"In fact, Kenneth L.Baker gave a lens space version of cabling conjecture.",
"Conjecture 3.1 (The Lens Space Cabling Conjecture [3]) Assume a knot $K$ in a lens space $L$ admits a surgery to a non-prime 3-manifold $Y$ .",
"If $K$ is hyperbolic, then $Y=L(r,1)\\#L(s,1)$ .",
"Otherwise either $K$ is a torus knot, a Klein bottle knot, or a cabled knot and the surgery is along the boundary slope of an essential annulus in the exterior of $K$ , or $K$ is contained in a ball.",
"The non-hyperbolic case has been proved by Baker in [3], although Fyodor Gainullin [13] constructed a counterexample to this conjecture for the hyperbolic case.",
"So we can use this conjecture to solve the equations under the assumption that $\\widetilde{O}$ is not hyperbolic, i.e.$O$ is not $\\pi $ -hyperbolic.",
"Definition 3.2 A tangle is $\\pi $ -hyperbolic if its double branched cover admits a hyperbolic structure and the covering translation is an isometry.",
"According to the lens space cabling conjecture, there are three cases except hyperbolic case: when $\\widetilde{O}$ is reducible, $\\widetilde{O}$ is the complement of a knot contained in a ball in a non-trivial lens space $\\widetilde{b_1}$ .",
"when $\\widetilde{O}$ is a Seifert fiber space, $\\widetilde{O}$ is the complement of a torus knot or a Klein bottle knot in $\\widetilde{b_1}$ .",
"when $\\widetilde{O}$ is irreducible toroidal but not Seifert fibered, $\\widetilde{O}$ is the complement of a cabled knot in $\\widetilde{b_1}$ .",
"In fact, (2)(3) make up the case that $\\widetilde{O}$ is irreducible but not hyperbolic."
],
[
"Case I: $\\widetilde{O}$ is reducible",
"If $\\widetilde{O}$ is reducible, then $\\widetilde{O}$ contains essential 2-spheres.",
"$\\widetilde{O}$ is the double branched cover of a tangle $O=(B^3,t)$ with an associated involution $\\sigma $ , namely $\\widetilde{O}/ \\sigma =O$ .",
"The essential 2-spheres in $\\widetilde{O}$ either do not intersect the fixed points of the involution $\\sigma $ , denoted by $fix(\\sigma )$ , or intersect $fix(\\sigma )$ transversely in 2 points.",
"The essential 2-spheres which do not intersect $fix(\\sigma )$ are mapped to essential 2-spheres in the complement of $O$ (i.e.$B^3-t$ ) by the covering map induced by $\\sigma $ .",
"The essential 2-spheres in $O$ remain essential after adding the tangle $X_1$ and taking the numerator closure.",
"In other words, $N(O+X_1)=b_1$ is split.",
"Since $b_1$ is a 2-bridge link, $b_1$ is split if and only if $b_1=b(0,1)$ (i.e.the unlink).",
"So there is at most one essential 2-sphere in $O$ which splits from $O$ the unknot.",
"For any essential 2-sphere $S$ which meets $fix(\\sigma )$ in 2 points, we can assume that it is invariant under $\\sigma $ , then $S/\\sigma $ is a 2-sphere intersecting $t$ in 2 points and the 3-ball it bounds in $O$ meets $t$ in a knotted arc.",
"There is at most one local knot in $O$ since $b_1$ is a 2-bridge link which is prime.",
"In addition, it is impossible for $O$ to contain both locally knotted arc and a splittable unknot.",
"The locally knotted case has been discussed in Buck and Mauricio's paper [5].",
"In [5], by excising the knotted arc in $O$ , the tangle equations and reduce to the following equations: N(O'+X1)=b(1,1) (the unknot) N(O'+X2)=b3, where $O^{\\prime }$ is the tangle we obtain after excising the knotted arc in $O$ .",
"In the tangle equations and , there exists $i=2$ or 3 such that $b_i=b_1$ , both of which contain the information of the knotted arc.",
"Otherwise there does not exist any locally knotted solution.",
"Without loss of generality, we assume $i=2$ .",
"Once we get a solution $O^{\\prime }$ , then we can recover the original $O$ by letting $O=O^{\\prime } \\# b_1$ .",
"When $O$ contains a splittable unknot, both $b_1$ and $b_2 \\# b_3$ contain one.",
"It implies one of $b_2$ and $b_3$ is $b(0,1)$ , without loss of generality, $b_2=b(0,1)$ .",
"Also we can remove the splittable unknot, then the tangle equations and reduce to the system of equations above.",
"Once we get a solution $O^{\\prime }$ , then we can recover the original $O$ by putting the splittable unknot back.",
"Lifting to the double branched covers, we have O'()=S3 O'()=the lens space b3, Therefore $\\widetilde{O^{\\prime }}$ is the complement of a knot $K$ in $S^3$ .",
"The problem turns out to be finding the knot $K$ in $S^3$ which admits lens space surgeries.",
"This is still an open question.",
"But for some knot, this question is well-understood.",
"If $K$ is the unknot, then $\\widetilde{O^{\\prime }}$ is a solid torus, which means $O^{\\prime }$ is a rational tangle, and this case can be solved by using Lemma REF .",
"If $K$ is a torus knot, then $\\widetilde{O^{\\prime }}$ is a Seifert fiber space over a disk, which means $O^{\\prime }$ is a generalized Montesinos tangle, and this case is solved in [9] and [8].",
"If $K$ is neither the unknot nor a torus knot, then this case can be related to Berge knots which is a family of knots constructed by John Berge with lens space surgeries.",
"Berge conjecture states that Berge knots contain all the knots in $S^3$ admitting lens space surgeries.",
"Besides, Joshua Greene [16] proved that the lens spaces obtained by doing surgeries along a knot in $S^3$ are precisely the lens spaces obtained by doing surgeries along Berge knots.",
"It implies that only if $b_3$ satisfies that $\\widetilde{b_3}$ is a lens space arising from surgeries along Berge knots, then the solution of the system of tangle equations REF and REF exists, and equivalently the solution of the system of tangle equations and exists when $\\widetilde{O}$ is reducible.",
"Meanwhile, Baker's papers[1][2] give surgery descriptions of Berge knots on some chain links and also give tangle descriptions of Berge knots, which can help us to find some solutions in this case."
],
[
"Case II: $\\widetilde{O}$ is irreducible but not hyperbolic",
"Here, $O$ can not be a rational tangle, for otherwise $N(O+X_2)$ is a 2-bridge link by Lemma REF , which is not a composite link.",
"Then $\\widetilde{O}$ is not a solid torus, thus $\\widetilde{O}$ is irreducible and $\\partial $ -irreducible.",
"If $\\widetilde{O}$ is non-hyperbolic, then this case consists of two cases: (1)$\\widetilde{O}$ is irreducible toroidal but not Seifert fibered and (2)$\\widetilde{O}$ is a Seifert fiber space.",
"Before our analysis, we will give some useful results of Gordon.",
"Here are some notation we will use.",
"Let $J$ be a knot in int $S^1 \\times D^2$ with winding number $w \\ge 0$ .",
"Let $\\alpha =[S^1 \\times *] \\in H_1(S^1 \\times \\partial D^2)$ , $* \\in \\partial D^2$ , $\\beta =[* \\times \\partial D^2] \\in H_1(S^1 \\times \\partial D^2)$ , $* \\in S^1$ .",
"Let $Y=S^1 \\times D^2 \\setminus N(J)$ , then $H_1(Y)=\\mathbb {Z}\\alpha \\bigoplus \\mathbb {Z} \\mu $ where $\\mu $ is the class of a meridian of $N(J)$ .",
"There is a homeomorphism $h: S^1 \\times D^2 \\rightarrow N(J)$ such that $[h(S^1 \\times *)]=w\\alpha \\in H_1(Y)$ , $* \\in \\partial D^2$ .",
"Then $\\lambda =[h(S^1 \\times *)]$ and $\\mu $ is a longitude-meridian basis for $J$ .",
"We use $J(r)$ to denote performing $r=m/n$ Dehn surgery on $S^1 \\times D^2$ along $J$ .",
"Lemma 3.3 [14] Let $J$ and $r$ be defined as above.",
"Then the kernel of $H_1(\\partial J(r)) \\rightarrow H_1(J(r))$ is the cyclic group generated by { nw2gcd(w,m)+ mgcd(w,m), if w 0; if w=0.",
".",
"Lemma 3.4 [14] Let $J$ and $r$ be defined as above, and let $J$ be a $(p,q)$ -torus knot with $p \\ge 2$ .",
"$J(r) \\cong \\left\\lbrace \\begin{matrix}&S^1 \\times D^2 \\# L(p,q) &if& \\, &r=pq\\,& \\\\&S^1 \\times D^2 &if& \\, &r=m/n& \\,\\, and \\quad m=npq \\pm 1\\end{matrix}\\right.$ and otherwise is a Seifert fiber space with incompressible boundary.",
"The following are some useful results about lens space and Seifert fiber space.",
"Let $T_i$ be a solid torus with a meridian $M_i$ and a longitude $L_i$ , for $i=1,2$ .",
"Definition 3.5 The lens space $L(a,b)=T_1 \\cup _h T_2$ where $h:\\partial T_2 \\rightarrow \\partial T_1$ is an orientation-preserving homeomorphism and $h(M_2)=aL_1+bM_1$ with $a, b \\in \\mathbb {Z}$ and $gcd(a,b)=1$ .",
"Lemma 3.6 [17] (1) If $L(a,b)$ is a generalized Seifert fiber space with orientable orbit surface, then there is a fiber preserving homeomorphism such that: $L(a,b) \\cong M(0,0;(\\alpha _1,\\beta _1), (\\alpha _2,\\beta _2)) \\cong T_1 \\cup T_2$ where $a=det\\begin{pmatrix}\\alpha _1 &\\alpha _2 \\\\-\\beta _1&\\beta _2\\end{pmatrix}$ , $b=-det\\begin{pmatrix}\\alpha _1 &\\alpha ^{\\prime }_2 \\\\-\\beta _1&\\beta ^{\\prime }_2\\end{pmatrix}$ and $det\\begin{pmatrix}\\alpha _2 &\\alpha ^{\\prime }_2 \\\\\\beta _2&\\beta ^{\\prime }_2\\end{pmatrix}=1$ (2) If $L(a,b)$ is a generalized Seifert fiber space with non-orientable orbit surface, then there is a fiber preserving homeomorphism such that $L(a,b) \\cong M(-1,0; (\\alpha ,\\pm 1)) \\cong T_1 \\cup _g (S^1 \\widetilde{\\times } M\\ddot{o}bius \\,\\, band) $ In this case $L(a,b) \\cong L(4\\alpha , \\pm 1 -2\\alpha )$ .",
"Corollary 3.7 [9] If $L(a,b)=T_1 \\cup T_2$ and $T_1$ is fibered by $H \\cong pL_1+qM_1$ , then $L(a,b) \\cong M(0,0;(p,-e), (aq-bp,ad-be))$ where $pd-qe=1$ .",
"If $L(a,b)=T_1 \\cup (S^1 \\widetilde{\\times } M\\ddot{o}bius \\,\\, band)$ and $T_1$ is fibered by $H \\cong pL_1+qM_1$ , then $L(a,b) \\cong M(-1,0;(p, \\pm 1))$ and $q \\cong \\pm 1 \\,mod \\, p$ .",
"Lemma 3.8 [9] If $\\mathcal {A}=Y(s/t)$ where $Y=L(a,b) \\setminus N(T_{p,q})$ , $L(a,b)=T_1 \\cup T_2$ , and $T_{p,q}$ is a $(p,q)$ -torus knot in $T_1$ , then $\\mathcal {A}=M(0,0;(p,-e), (aq-bp, ad-be), (s-tpq,t))$ , where $pd-qe=1$ .",
"Lemma 3.9 [9] If $\\mathcal {A}=Y(s/t)$ where $Y=L(a,b) \\setminus N(T_{p,q})$ , $L(a,b)=T_1 \\cup _g (S^1 \\widetilde{\\times } M\\ddot{o}bius \\,\\, band)$ , and $T_{p,q}$ is a $(p,q)$ -torus knot in $T_1$ , then $\\mathcal {A}=M(-1,0;(p,\\pm 1), (s-tpq,t))$ ."
],
[
"$\\widetilde{O}$ is irreducible toroidal but not Seifert fibered",
"According to Baker's lens space cabling conjecture, $\\widetilde{O}$ is the complement of a cabled knot in the lens space $\\widetilde{b_1}$ if $\\widetilde{O}$ is irreducible toroidal but not Seifert fibered, and to obtain a non-prime 3-manifold, the surgery is along the boundary slope of an essential annulus in $\\widetilde{O}$ .",
"Assume that $b_1$ is the 2-bridge link $b(a,b)$ for a pair of relatively prime integers $(a,b)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ , then $\\widetilde{b_1}=L(a,b)$ .",
"Let $K$ be the cabled knot lying in the lens space $L(a,b)$ .",
"Then there exists a knot $K^{\\prime }$ in $L(a,b)$ and a homeomorphism $f:S^1 \\times D^2 \\rightarrow N(K^{\\prime })$ , such that $K=f(J)$ where $J=T_{p,q}$ is a $(p,q)$ -torus knot in $S^1 \\times D^2$ for some $p>1$ , $q \\in \\mathbb {Z}$ .",
"$\\widetilde{O}=L(a,b)\\setminus N(K)$ .",
"We choose $\\alpha ^{\\prime }=\\left( f|_{S^1 \\times \\partial D^2} \\right)_* (\\alpha )$ and $\\beta ^{\\prime }=\\left( f|_{S^1 \\times \\partial D^2} \\right)_* (\\beta )$ as a longitude-meridian basis for $\\partial N(K^{\\prime })$ .",
"$\\lambda ^{\\prime }=\\left( f|_{\\partial (N(J))} \\right)_*(\\lambda )$ , $\\mu ^{\\prime }=\\left( f|_{\\partial (N(J))} \\right)_*(\\mu )$ is a longitude-meridian basis for $K$ .",
"Actually, $T=f(S^1 \\times \\partial D^2)$ is an incompressible torus in $\\widetilde{O}$ .",
"If $T$ is compressible, then $T$ either bounds a solid torus or lies in a ball in $\\widetilde{O}$ since $\\widetilde{O}$ is irreducible.",
"The latter case can be excluded easily.",
"If $T$ bounds a solid torus $V$ in $\\widetilde{O}$ , then $\\widetilde{O}=V \\cup (S^1 \\times D^2 \\setminus N(T_{p,q}))$ is atoroidal, so contradicts our assumption.",
"Splitting $\\widetilde{O}$ along the incompressible torus $T$ , we obtain two pieces, a cable space $C_{p,q}=S^1 \\times D^2 \\setminus N(T_{p,q})$ and another piece $L(a,b) \\setminus N(K^{\\prime })$ denoted by $M$ .",
"Now we prove that $M$ is a Seifert fiber space.",
"Proposition 3.10 $M$ is a Seifert fiber space over a disk with exact 2 exceptional fibers.",
"$\\infty $ -surgery along $K$ , i.e.not doing surgery, produces the lens space $L(a,b)$ , which also means $\\beta ^{\\prime }$ -Dehn filling (i.e.$\\infty $ -Dehn filling) on $M$ produces $L(a,b)$ .",
"The boundary slope of an essential annulus in $\\widetilde{O}$ is actually the slope $r=pq$ along $K$ .",
"Performing $pq$ -surgery along $K$ , by Lemma REF , we have that $N(K^{\\prime })$ transforms into $S^1 \\times D^2 \\# L(p,q)$ .",
"According to Lemma REF , $p \\alpha ^{\\prime }+ q \\beta ^{\\prime }$ bounds a disk in $S^1 \\times D^2 \\# L(p,q)$ .",
"It means that $pq$ -surgery along $K$ produces $M(q/p) \\# L(p,q)$ .",
"According to our equations, we expect that $M(q/p)$ is a lens space and $L(p,q) \\ne S^3$ (i.e.$p \\ne 1$ ).",
"So now we have $M(1/0)$ and $M(q/p)$ are two lens spaces with $p \\ne 1$ .",
"Obviously $\\triangle (1/0,q/p)>1$ , by Cyclic Surgery Theorem [7], $M$ must be a Seifert fiber space.",
"$M$ is a Seifert fiber space admitting two Dehn filings to produce lens space.",
"By Lemma REF , $M$ can be a Seifert fiber space over a disk with at most two exceptional fibers or a Seifert fiber space over a $M\\ddot{o}bius \\, band$ with at most one exceptional fiber.",
"If $M$ is a Seifert fiber space over a disk, then $M$ must have exact two exceptional fibers.",
"Because otherwise $\\partial M$ is compressible in $M$ , which contradicts our previous analysis.",
"If $M$ is over a $M\\ddot{o}bius \\, band$ without exceptional fiber, then we can choose another fibration of $M$ such that $M$ is a Seifert fiber space over a disk with 2 exceptional fibers, since $M(-1,1;)=M(0,1;(2,1),(2,-1))$ .",
"Suppose $M=M(-1,1; (a,b))$ with $a>1$ .",
"There are two Dehn fillings with $\\triangle >1$ on $M$ producing lens spaces as our analysis above.",
"Assume that the two fillings add two fibers $(a_1,b_1)$ and $(a_2,b_2)$ to $M$ respectively ($a_i=1$ means adding an ordinary fiber; $a_i>1$ means adding an exceptional fiber).",
"Then we obtain $M(-1,0;(a,b),(a_1,b_1))$ and $M(-1,0; (a,b),(a_2,b_2))$ .",
"To be lens spaces, by Lemma REF , we must have $a_1=a_2=1$ , and then $M(-1,0;(a,b),(a_1,b_1))=M(-1,0;(a,b+ab_1))$ and $M(-1,0;(a,b),(a_2,b_2))=M(-1,0;(a,b+ab_2))$ satisfying one of the four systems of equations: $\\left\\lbrace \\begin{matrix}b+ab_1=1\\\\b+ab_2=1\\end{matrix}\\right.$ $\\left\\lbrace \\begin{matrix}b+ab_1=-1\\\\b+ab_2=-1\\end{matrix}\\right.$ $\\left\\lbrace \\begin{matrix}b+ab_1=-1\\\\b+ab_2=1\\end{matrix}\\right.$ $\\left\\lbrace \\begin{matrix}b+ab_1=1\\\\b+ab_2=-1\\end{matrix}\\right.$ .",
"The first two systems of equations are impossible.",
"Because if one holds, then $a(b_1-b_2)=0$ , so $b_1=b_2$ , which contradicts $\\triangle >1$ .",
"The last two systems of equations are also impossible.",
"Because if one holds, then $a(b_1-b_2)=\\pm 2$ and $a>1$ , thus $b_1-b_2=\\pm 1$ .",
"It implies $\\triangle (b_1/a_1, b_2/a_2)=1$ , which also contradicts our analysis.",
"Therefore $M$ is a Seifert fiber space over a disk with exact two exceptional fibers.",
"Now, we have $M=L(a,b)\\setminus N(K^{\\prime })$ is a Seifert fiber space over a disk with 2 exceptional fibers, so $K^{\\prime }$ is isotopic to a fiber in some generalized Seifert fibration of $L(a,b)$ over a 2-sphere.",
"Moreover, the fiber is an ordinary fiber, for otherwise $M=L(a,b) \\setminus N(K^{\\prime })$ has at most one exceptional fiber.",
"In fact, we can regard $L(a,b)$ as a union of two solid tori, $T_1$ and $T_2$ , and $K^{\\prime }$ is isotopic to a $(p_1,q_1)$ -torus knot in $T_1$ for some $p_1 >1, q_1 \\in \\mathbb {Z}$ .",
"Because if $p_1=1$ , then $M=L(a,b)\\setminus N(K^{\\prime })$ is a solid torus instead of a Seifert fiber space with 2 exceptional fibers; if $p_1=0$ , then $K$ is lying in a ball, thus either $\\widetilde{O}=L(a,b)\\setminus N(K)$ is reducible which contradicts our assumption, or $L(a,b)=S^3$ .",
"$L(a,b)=S^3$ is impossible, since if it is and $p_1=0$ , then $M=L(a,b)\\setminus N(K^{\\prime })=S^3\\setminus N(K^{\\prime })$ is a solid torus.",
"We chose a longitude-meridian basis $(\\lambda _1, \\mu _1)$ for $\\partial N(K^{\\prime })$ using the same principle as choosing basis for $J$ in $S^1 \\times D^2$ .",
"Now we redefine the homeomorphism $f: S^1 \\times D^2 \\rightarrow N(K^{\\prime })$ such that $(f|_{S^1 \\times \\partial D^2})_*(\\alpha ) =\\lambda _1$ and $(f|_{S^1 \\times \\partial D^2})_*(\\beta ) =\\mu _1$ , then $K=f(J)$ where $J=T_{p,q}$ for some $p>1$ , $q \\in \\mathbb {Z}$ .",
"So now, $K$ is a $(p,q)$ -cable of $(p_1,q_1)$ -torus knot lying in $T_1$ of $L(a,b)$ with $p>1$ and $p_1>1$ .",
"Proposition 3.11 $K$ is a $(p,q)$ -cable of $(p_1,q_1)$ -torus knot lying in $T_1$ of $L(a,b)=T_1 \\cup T_2$ with $q=pp_1q_1 \\pm 1$ and $|aq_1-bp_1|>1$ , where $p>1$ , $p_1>1$ , $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ , and $T_i$ is a solid torus, for $i=1,2$ .",
"When the surgery slope is $r=pq$ , the manifold obtained is $L(p,q) \\# L(a \\pm ap_1q_1p \\mp bp_1^2p, b \\pm aq_1^2p \\mp bp_1q_1p)$ .",
"By Lemma REF and Lemma REF , $K(pq)=[L(a,b) \\setminus N(T_{p_1,q_1})](q / p) \\# L(p, q),$ where $K(r)$ denotes $r$ -surgery along $K$ .",
"By Lemma REF , $[L(a,b) \\setminus N(T_{p_1,q_1})](q / p) = M(0,0;(p_1,-e),(aq_1-bp_1,ad-be),(q-pp_1q_1, p)),$ where $p_1d-q_1e=1$ .",
"We want it to be a lens space, and that happens if and only if one of $|q-pp_1q_1|$ , $|aq_1-bp_1|$ and $|p_1|$ equals 1.",
"In fact, $|aq_1-bp_1|>1$ , for the same reason as $p_1>1$ since a $(p_1,q_1)$ -torus knot in $T_1$ of $L(a,b)$ is also a torus knot with winding number $|aq_1-bp_1|$ in $T_2$ .",
"So we have $|q-pp_1q_1|=1$ .",
"There are two cases.",
"(i) $q-pp_1q_1=1$ [L(a,b) N(Tp1,q1)](q / p) = M(0,0;(p1,-e),(aq1-bp1,ad-be),(1,p)) =M(0,0;(p1,-e),(aq1-bp1,ad-be+aq1p-bp1p)).",
"Using the formula in Lemma REF , then M(0,0;(p1,-e),(aq1-bp1,ad-be+aq1p-bp1p)) =L(a+ap1q1p-bp12p, b+aq12p-bp1q1p).",
"(ii) $q-pp_1q_1=-1$ [L(a,b) N(Tp1,q1)](q / p) =M(0,0;(p1,-e),(aq1-bp1,ad-be),(1,-p)) =M(0,0;(p1,-e),(aq1-bp1,ad-be-aq1p+bp1p)).",
"Using the formula in Lemma REF , M(0,0;(p1,-e),(aq1-bp1,ad-be-aq1p+bp1p)) =L(a-ap1q1p+bp12p, b-aq12p+bp1q1p).",
"Before proving the main theorem, we will give a useful proposition about the mapping class group of four-times-punctured sphere denoted by $S_{0,4}$ .",
"We use $Mod$ to denote mapping class group.",
"Proposition 3.12 [12] $Mod(S_{0,4}) \\cong PSL(2,\\mathbb {Z}) \\ltimes (\\mathbb {Z}_2 \\times \\mathbb {Z}_2),$ Remark The subgroup $\\mathbb {Z}_2 \\times \\mathbb {Z}_2$ is generated by two elements of order 2, $c_1$ and $c_2$ shown as Figure REF .",
"The subgroup $PSL(2,\\mathbb {Z})$ is generated by two half Dehn twists $\\bar{\\alpha }$ and $\\bar{\\beta }$ shown in Figure REF .",
"$Mod(S_{0,4})=<\\bar{\\alpha }, \\bar{\\beta }, c_1, c_2>$ .",
"Figure: The generators of Mod(S 0,4 )Mod(S_{0,4})Definition 3.13 Given a tangle $X=(M,t)$ and a Conway sphere $S$ in it, split the tangle $X$ along $S$ into two pieces $M_1$ and $M_2$ .",
"Let $h: S \\rightarrow S^2$ be a homeomorphism such that $h(S \\cap t)=P=4 \\,\\, points$ .",
"The tangle $X^{\\prime }=M_1 \\cup _{h^{-1}gh} M_2$ , where $g: (S^2, P) \\rightarrow (S^2, P)$ such that $[g] \\in \\langle c_1,c_2 \\rangle - \\lbrace 1\\rbrace \\in Mod(S_{0,4})$ , is called a mutant of $X$ , and the operation of replacing $X$ by $X^{\\prime }$ is called mutation of $X$ along $S$ .",
"[proof of Theorem REF ] Assume that $b_1=b(a,b)$ for a pair of relatively prime integers $(a,b)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ .",
"As discussed above, the solution of this system of tangle equations exists only if there exist two pairs of relatively prime integers $(p_1,q_1)$ and $(p,q)$ satisfying that $p>1,p_1>1,|aq_1-bp_1|>1$ and $q=pp_1q_1\\pm 1$ such that $\\widetilde{b_2} \\# \\widetilde{b_3}=L(p,q) \\# L(a \\pm ap_1q_1p \\mp bp_1^2p, b \\pm aq_1^2p \\mp bp_1q_1p)$ .",
"In this case, $O$ should be a tangle whose double branched cover is $\\widetilde{O}$ , and $\\widetilde{O}=L(a,b) \\setminus N(K)$ where $K$ is a $(p,q)$ -cable of $K^{\\prime }=T_{p_1,q_1}$ lying in $T_1$ of $L(a,b)=T_1 \\cup T_2$ .",
"We first construct a tangle satisfying that its double branched cover is $\\widetilde{O}$ .",
"Then we will show that any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to the tangle we construct.",
"As discussed in Proposition REF , there are two cases.",
"Case (i): $q=pp_1q_1+1$ .",
"Let $f:S^1 \\times D^2 \\rightarrow N(K^{\\prime })$ be the homeomorphism defined as above.",
"Then $T=f(S^1 \\times \\partial D^2)$ is an essential torus in $\\widetilde{O}$ and also the only one essential torus in $\\widetilde{O}$ .",
"Splitting $\\widetilde{O}$ along the essential torus $T$ , we obtain two manifolds $M=L(a,b) \\setminus N(K^{\\prime })$ and $C_{p,q}$ .",
"By Corollary REF , $M=M(0,1;(p_1,-e),(aq_1-bp_1,ad-be))$ where $p_1d-q_1e=1$ .",
"The cable space $C_{p,q}$ is the Seifert fiber space $M(0,2;(p,1))$ .",
"According to the results about double branched covers listed in Section REF , $M$ is the double branched cover of a tangle $Q$ shown as Figure REF .",
"We denote the associated standard involution of $M$ by $\\sigma _1$ .",
"Meanwhile, $C_{p,q}=M(0,2;(p,1))$ is the double branched cover of a tangle $P$ shown in Figure REF , which is a Montesinos pair in $S^2 \\times I$ .",
"The associated standard involution of $C_{p,q}=M(0,2;(p,1))$ is denoted by $\\sigma _2$ .",
"Restrict $\\sigma _1$ (resp.$\\sigma _2$ ) on the torus boundary of $M$ (resp.$C_{p,q}$ ), it is the so-called standard involution of a torus.",
"In fact, given an isotopy class of homeomorphisms of the torus, there exists a representative which commutes with the standard involution, so we can extend the involutions $\\sigma _1$ and $\\sigma _2$ of $M$ and $C_{p,q}$ to an involution $\\sigma $ of $\\widetilde{O}$ .",
"In other words, there is a tangle $O_1$ which is obtained by gluing $Q$ and $P$ together satisfying that its double branched cover is $\\widetilde{O}$ .",
"Let $f:\\partial M \\rightarrow \\partial C_{p,q}$ be the gluing map to obtain $\\widetilde{O}$ , then the gluing map $\\bar{f}: \\partial Q \\rightarrow \\partial P$ to give $O_1$ is induced by $f$ .",
"Here we just give a tangle $O_1$ shown in Figure REF , and one can easily check $O_1$ 's double branched cover is homeomorphic to $\\widetilde{O}$ by carefully studying the two gluing maps $f$ and $\\bar{f}$ .",
"Figure: The tangle QQ and PPAs discussed above, performing $\\infty $ -filling on $C_{p,q}$ (i.e.$\\infty $ -surgery along $K$ ) gives the original lens space $L(a,b)$ , and this is equivalent to filling a $(1,0)$ -fiber in $M(0,2;(p,1))$ , namely $C_{p,q}(\\infty )=M(0,1;(p,1),(1,0))$ .",
"Performing $pq$ -filling on $C_{p,q}$ (i.e.$pq$ -surgery along $K$ ) gives the connected sum of a solid torus and $L(p,q)$ , which is equivalent to filling a $(0,1)$ -fiber in $M(0,2;(p,1))$ , namely $C_{p,q}(pq)=M(0,1;(p,1),(0,1))$ .",
"The two Dehn filling slopes on $\\partial C_{p,q}$ are mapped respectively to two slants $\\frac{0}{1}$ and $\\frac{1}{0}$ on the corresponding boundary component of the tangle $P$ by the covering map induced by $\\sigma _2$ .",
"In Figure REF , the thick curve stands for the $\\frac{0}{1}$ slant, and the thin curve is the $\\frac{1}{0}$ slant.",
"After gluing $P$ and $Q$ together, the two corresponding slants on $\\partial O_1$ are shown in Figure REF .",
"It means adding rational tangles which have the two slants as meridians respectively to $O_1$ give the 2-bridge link and the connected sum of two 2-bridge links we want.",
"There is no other slant satisfying this since there is no other Dehn surgery slope along $K$ giving the original $L(a,b)$ or a non-prime manifold.",
"In fact, only the Dehn filling slope $r=pq$ gives a non-prime manifold.",
"To obtain the original $L(a,b)$ , it probably happens only in the case that the slope $r=\\frac{m}{n}$ and $m=npq \\pm 1$ , by Lemma REF , since the other two cases in Lemma REF produce a non-prime manifold or a toroidal manifold.",
"Using the formula in Lemma REF , $K(r)=L(a,b)\\setminus N(T_{p_1,q_1})(m/np^2).$ According to Lemma REF , $L(a,b) \\setminus N(T_{p_1,q_1})(m/np^2)=M(0,0;(p_1,-e),(aq_1-bp_1,ad-de),(m-np^2p_1q_1,np^2))$ where $p_1d-q_1e=1$ .",
"$M(0,0;(p_1,-e),(aq_1-bp_1,ad-de),(m-np^2p_1q_1,np^2))=L(a,b)$ if and only if $m-np^2p_1q_1=1$ and $np^2=0$ since $p_1>1$ and $|aq_1-bp_1|>1$ .",
"Thus $r=\\frac{m}{n}=\\frac{1}{0}=\\infty $ since $p>1$ .",
"Therefore, as shown in Figure REF , only adding 0-tangle and $\\infty $ -tangle give the links we want.",
"One can easily check that $N(O_1+0)=b(a,b)$ and $N(O_1+\\infty )=b(p,q) \\# b(a+ap_1q_1p-bp_1^2p, b+aq_1^2p-bp_1q_1p)$ .",
"This gives a pair of solutions $(X_1,X_2)$ when $O=O_1$ for the given tangle equations.",
"Figure: The tangle O 1 O_1 and the corresponding pair of solutions (X 1 ,X 2 )(X_1,X_2), where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1}, and p 1 d-q 1 e=1p_1d-q_1e=1Now we denote $M$ (resp.$C_{p,q}$ ) by $M_1$ (resp.$M_2$ ).",
"$\\sigma _i$ is still the standard involution on $M_i$ .",
"$\\sigma $ is the involution on $\\widetilde{O}$ which is extended by $\\sigma _1$ and $\\sigma _2$ .",
"As discussed above, $\\widetilde{O}/\\sigma =O_1=M_1/\\sigma _1 \\cup _{\\bar{f}} M_2/\\sigma _2=Q\\cup _{\\bar{f}}P$ .",
"Suppose $O^{\\prime }$ is a tangle whose double branched cover is homeomorphic to $\\widetilde{O}$ , and $\\iota : \\widetilde{O} \\rightarrow \\widetilde{O}$ is the associated involution, namely $\\widetilde{O} / \\iota =O^{\\prime }$ .",
"Since $T$ is the only one essential torus in $\\widetilde{O}$ , up to isotopy, we can assume that $T$ is invariant under $\\iota $ by Theorem 8.6 in [19].",
"$M_1$ is not homeomorphic to $M_2$ , so $\\iota $ preserves $T$ , $M_1$ and $M_2$ .",
"Let $\\iota _i$ be the restriction of $\\iota $ to $M_i$ .",
"$M_2$ is a Seifert fiber space over an annulus with one exceptional fiber.",
"Restrict $\\iota _2$ to $\\partial \\widetilde{O} \\subset \\partial M_2$ , then it is the standard involution on this tours boundary since $\\widetilde{O} / \\iota $ is a tangle.",
"Lemma 3.8 in [15] tells us there exists a homeomorphism $\\phi _2 : M_2 \\rightarrow M_2$ isotopic to the identity such that $\\iota _2=\\phi ^{-1}_2 \\sigma _2 \\phi _2$ .",
"Then $\\iota $ restricted to $T$ is also the standard involution for $T$ as a torus.",
"Therefore $T / (\\iota \\mid _{T})$ is a Conway sphere in $O^{\\prime }$ .",
"Then $M_1$ is the double branched cover of the tangle $O^{\\prime }- M_2 / \\iota _2$ , i.e.",
"the tangle inside the Conway sphere $T / (\\iota \\mid _{T})$ in $O^{\\prime }$ .",
"According to Proposition 2.8 in [20], there is only one involution up to conjugation on $M_1$ satisfying that $M_1 /$ (the involution) is a tangle, and $\\sigma _1$ is such an involution.",
"So there exists a homeomorphism $\\phi _1 : M_1 \\rightarrow M_1$ such that $\\iota _1=\\phi ^{-1}_1 \\sigma _1 \\phi _1$ .",
"It will be shown that we can choose a $\\phi _1$ such that $\\partial \\phi _1 : \\partial M_1 \\rightarrow \\partial M_1$ is isotopic to the identity.",
"We can assume that $\\phi _1$ is orientation-preserving, since we can easily find an orientation-reversing homeomorphism which commutes with $\\sigma _1$ .",
"Also we assume $\\phi _1$ preserves the orientation of fiber by multiplying with $\\sigma _1$ or not.",
"By Proposition 25.3 in [18], the mapping class group of Seifert fiber space with orientable orbit surface except some special cases is the semidirect product of \"vertical subgroup\" and the extended mapping class group of the orbit surface.",
"The \"vertical subgroup\" is generated by some Dehn twists along vertical annulus or torus and acts trivially on the orbit surface denoted by $F$ .",
"Besides, by the extended mapping class group of the orbit surface $F$ we mean the group of homeomorphisms of $F$ which send exceptional points to exceptional points with the same coefficients, modulo isotopies which are constant on the exceptional points.",
"Another useful result is that the \"vertical subgroup\" is isomorphic to $H_1(F, \\partial F)$ .",
"For $M_1$ , the \"vertical subgroup\" is trivial since the first relative homology group of its orbit surface is trivial.",
"Thus, the mapping class group of $M_1$ is isomorphic to the extended mapping class group of its orbit surface.",
"We already have assumed $\\phi _1$ preserves the orientations of $M_1$ and fiber, then $\\partial \\phi _1$ must be isotopic to the identity.",
"As our assumption, $\\widetilde{O_1} / \\iota = M_1/\\iota _1 \\cup _h M_2/\\iota _2$ for some $h: \\partial (M_1/\\iota _1) \\rightarrow \\partial (M_2/\\iota _2)$ satisfying $f: \\partial M_1 \\rightarrow \\partial M_2$ is a lift of $h$ .",
"There exists $\\phi _i: M_i \\rightarrow M_i$ such that $\\iota _i=\\phi ^{-1}_i \\sigma _i \\phi _i$ with $\\partial \\phi _i$ isotopic to the identity, for $i=1,2$ .",
"Then $\\phi _i$ induces a homeomorphism $\\bar{\\phi _i} :(M_i/\\iota _i) \\rightarrow (M_i/\\sigma _i)$ .",
"Restrict $\\bar{\\phi _i}$ to the boundary, we have the following commutative diagram: ${\\partial (M_1/\\iota _1) [d]^{\\partial \\bar{\\phi _1}} [rr]^{h} && \\partial (M_2/\\iota _2) [d]^{\\partial \\bar{\\phi _2}}\\\\\\partial (M_2/\\sigma _2) [rr]^{\\partial \\bar{\\phi _2} \\circ h \\circ \\partial \\bar{\\phi _1}^{-1}} && \\partial (M_2/\\sigma _2).",
"}$ This induces a homeomorphism: $\\bar{\\phi }=\\bar{\\phi _1} \\cup \\bar{\\phi _2}: M_1/\\iota _1 \\cup _h M_2/\\iota _2 \\rightarrow M_1/\\sigma _1 \\cup _{\\partial \\bar{\\phi _2} \\circ h \\circ \\partial \\bar{\\phi _1}^{-1}} M_2/\\sigma _2.$ $\\partial \\bar{\\phi _2} \\circ h \\circ \\partial \\bar{\\phi _1}^{-1}$ lifts to $\\partial \\phi _2 \\circ f \\circ \\partial \\phi ^{-1}_1$ which is isotopic to $f$ since $\\partial \\phi _i$ is isotopic to the identity.",
"In fact, only the lift of $\\bar{f}k$ is isotopic to $f$ , where $k \\in \\langle c_1,c_2 \\rangle \\in Mod(\\partial {Q})$ and $Mod(\\partial {Q})$ represents the mapping class group of $\\partial Q$ as a four-times-punctured sphere.",
"Thus $\\partial \\bar{\\phi _2} \\circ h \\circ \\partial \\bar{\\phi _1}^{-1}=\\bar{f}k$ .",
"Then, O'=O1/=M1/1 h M2/2 M1/1 fk M2/2=Q fk P. In fact, gluing the tangle $Q$ and $P$ together by the gluing map $\\bar{f}k:\\partial Q \\rightarrow \\partial P$ is equivalent to performing mutations on $O_1$ along the dotted Conway sphere $S$ shown as Figure REF .",
"The tangle in the Conway sphere $S$ in $O_1$ is a Montesinos tangle which is invariant under some rotations in $\\langle c_1,c_2 \\rangle $ .",
"Thus we only obtain two tangles by mutations, $O_1$ itself and $O_2$ shown in Figure REF .",
"We can easily show that $O_2$ is homeomorphic to $O_1$ , by extending the operation $c_1 \\in Mod(\\partial O_1)$ to the whole $O_1$ .",
"Thus $O^{\\prime } \\cong O_1$ .",
"Figure: The tangle O 1 O_1 and O 2 O_2All the homeomorphisms on $O_1$ can be induced by homeomorphisms on boundary of $O_1$ .",
"These homeomorphisms on $\\partial O_1$ also induce a new pair of slants corresponding to the new pair of solutions $(X_1,X_2)$ for the new tangle obtained.",
"This pair of solutions $(X_1,X_2)$ is unique, for otherwise there exist other Dehn fillings giving the manifolds we want.",
"Actually, we only choose orientation-preserving homeomorphisms.",
"Besides, we only want the pair of solutions with $X_1=0$ as mentioned at the beginning of this section, thus the homeomorphisms we can perform on $\\partial O_1$ are limited and can be easily worked out.",
"In fact, only $\\bar{\\beta }^n k \\, \\in Mod(\\partial O_1)$ could preserve the meridian of $X_1=0$ -tangle, where $k\\, \\in \\langle c_1,c_2 \\rangle $ , and $n \\in \\mathbb {Z}$ .",
"Therefore, all the solutions up to equivalence are O=k(O1) (n,0) X1=0-tangle, X2=(-n,0)-tangle, where $k(\\ast )$ means extending the operation $k$ on the boundary of the tangle $\\ast $ to the whole $\\ast $ .",
"In [8], it is shown that the solutions $(O \\circ (n,0), X_1=0, X_2=\\infty \\circ (-n,0))$ is equivalent to $(O, X_1=0, X_2=\\infty )$ .",
"Thus $(O=k(O_1), X_1=0, X_2=\\infty )$ give all the solutions, up to equivalence, as shown in the case (i) of this theorem.",
"Obviously, $N(k(O_1)+0)=b(a,b)$ and $N(k(O_1)+\\infty )=b(p,q) \\# b(a+ap_1q_1p-bp_1^2p, b+aq_1^2p-bp_1q_1p)$ .",
"Case (ii): $q=pp_1q_1-1$ .",
"This case is similar to the case (i).",
"Split $\\widetilde{O}$ along the only one essential torus $T$ , we obtain two pieces $M=L(a,b)\\setminus N(K^{\\prime })$ and $C_{p,q}$ .",
"$M$ is still the Seifert fiber space $M(0,1;(p_1,e),(aq_1-bp_1,ad-be))$ , which is the double branched cover of the tangle $Q$ in Figure REF .",
"$C_{p,q}=M(0,2;(p,-1))$ is the double branched cover of the tangle $P^{\\prime }$ shown in Figure REF .",
"Then we construct a tangle $O^{\\prime }_1$ whose double branched cover is $\\widetilde{O}$ , by gluing $P^{\\prime }$ and $Q$ together.",
"The pair of solutions $(X_1,X_2)$ when $O=O^{\\prime }_1$ is given in Figure REF , similarly by studying the surgeries on $\\widetilde{O}$ and $C_{p,q}$ .",
"Besides, when $O=O^{\\prime }_1$ the pair of solutions $(X_1,X_2)$ is unique.",
"The same method as in case (i) can be used to prove any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to $O^{\\prime }_1$ .",
"Similarly, $(O=k(O^{\\prime }_1),X_1=0,X_2=\\infty )$ give all the solutions up to equivalence as shown in the case (ii) of this theorem.",
"One can check that $N(k(O^{\\prime }_1)+0)=b(a,b)$ and $N(k(O^{\\prime }_1)+\\infty )=b(p,q) \\# b(a-ap_1q_1p+bp_1^2p, b-aq_1^2p+bp_1q_1p)$ .",
"Figure: The tangle O 1 ' O^{\\prime }_1 and the corresponding pair of solutions (X 1 ,X 2 )(X_1,X_2), where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1}, and p 1 d-q 1 e=1p_1d-q_1e=1.Remark In [15], Gordon proved that any tangle whose double branched cover is homeomorphic to that of EM-tangle is homeomorphic to the EM-tangle by a similar method, where EM-tangle has a similar structure as our tangle $O_1$ .",
"Besides, Paoluzzi's method in [20] can be used to prove that there are at most 4 tangles whose double branched cover are $\\widetilde{O}$ and the 4 tangles are obtained by mutations of the tangle $O_1$ ."
],
[
"$\\widetilde{O}$ is a Seifert fiber space",
"Buck and Mauricio's paper [5] also includes this case, while it assumes that neither of $b_2$ and $b_3$ is $b(0,1)$ (i.e.the unlink).",
"Besides, our definition of tangle is different from [5], since we allow tangle to have circles embedded in.",
"Here we will give a theorem without such an assumption.",
"Theorem 2 Suppose N(O+X1)=b1 N(O+X2)=b2 # b3, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$ , with $b_2$ and $b_3$ nontrivial.",
"Suppose $\\widetilde{O}$ is a Seifert fiber space, then the system of tangle equations has solutions if and only if one of the following holds: (i) There exist 2 pairs of relatively prime integers $(a,b)$ and $(p,q)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty ), p>1$ , and $|aq-bp|>1$ such that $b_1=b(a,b)$ and $b_2 \\# b_3=b(p,-e) \\# b(aq-bp, ad-be)$ , where $pd-qe=1$ (Note that choosing different $d$ and $e$ such that $pd-qe=1$ has no effect on results).",
"Solutions up to equivalence are shown as the following: Figure: OO = the tangle in (a), where A=-e pA=\\frac{-e}{p}, B=ad-be aq-bpB=\\frac{ad-be}{aq-bp} (or A=ad-be aq-bpA=\\frac{ad-be}{aq-bp}, B=-e pB=\\frac{-e}{p}).",
"X 1 =0X_1=0-tangle and X 2 =∞X_2=\\infty -tangle.",
"(ii)There exists an integer $p$ satisfying $|p|>1$ such that $b_1=b(4p, 1-2p)$ and $b_2 \\# b_3=b(0,1) \\# b(p,1)$ .",
"Solutions up to equivalence are shown as the following: Figure: O=R+1 pO=R + \\frac{1}{p} the tangle shown in (a), where RR is the ring tangle.",
"X 1 =0X_1=0-tangle and X 2 =∞X_2=\\infty -tangle.Suppose that $b_1=b(a,b)$ for a pair of relatively prime integers $(a,b)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ , then $\\widetilde{b_1}=L(a,b)$ .",
"$\\widetilde{X_1}$ is a solid torus lying in $L(a,b)$ , and we regard it as a tubular neighborhood of a knot $K$ in $L(a,b)$ .",
"Since we assume that $\\widetilde{O}=L(a,b)\\setminus N(K)$ is a Seifert fiber space, $K$ is isotopic to a fiber in some generalized Seifert fibration of $L(a,b)$ .",
"According to Lemma REF , there are two types of fibration for a lens space.",
"(i)$L(a,b)$ is fibered over $S^2$ .",
"In fact, $K$ is isotopic to an ordinary fiber of this type of fibration, since otherwise $L(a,b)\\setminus N(K)=\\widetilde{O}$ is a solid torus, but there dose not exist rational tangle solution for $O$ .",
"Let $L(a,b)=T_1 \\cup T_2$ where $T_i$ is a solid torus, for $i=1,2$ .",
"$K$ can be regarded as a $(p,q)$ -torus knot in one of the solid tori of $L(a,b)$ , without loss of generality $T_1$ .",
"In fact, $p\\ne 0$ because otherwise either $\\widetilde{O}=L(a,b)\\setminus N(K)$ is reducible which contradicts our assumption, or $L(a,b)=S^3$ .",
"$L(a,b)=S^3$ is also impossible since if it is and $p=0$ , then $L(a,b)\\setminus N(K)=$ solid torus.",
"Besides, $p \\ne 1$ for otherwise $L(a,b)\\setminus N(K)=$ solid torus.",
"Thus $p>1$ .",
"By Corollary REF , $L(a,b)=M(0,0;(p,-e), (aq-bp,ad-be)),$ where $pd-qe=1$ .",
"$|aq-bp|>1$ for the same reason as $p>1$ since a $(p,q)$ -torus knot in $T_1$ of $L(a,b)$ is also a torus knot with winding number $|aq-bp|$ in $T_2$ of $L(a,b)$ .",
"Since $K$ is isotopy to an ordinary fiber, $\\widetilde{O}=L(a,b)\\setminus N(K)=M(0,1;(p,-e), (aq-bp,ad-be)).$ We choose a longitude-meridian basis for $K$ using the same principle as choosing basis for $J$ in $S^1 \\times D^2$ .",
"According to Lemma REF , only doing $\\infty $ -surgery (resp.$pq$ -surgery) along $K$ produces $L(a,b)$ (resp.",
"a non-prime manifold), and $\\infty $ -surgery (resp.$pq$ -surgery) is equivalent to filling a $(1,0)$ -fiber (resp.$(0,1)$ -fiber) in $M(0,1;(p,-e), (aq-bp,ad-be))$ .",
"That is, $K(\\infty )=M(0,0;(p,-e), (aq-bp,ad-be), (1,0))=L(a,b)$ $(resp.",
"K(pq)=M(0,0;(p,-e), (aq-bp,ad-be), (0,1))=L(p,-e) \\# L(aq-bp,ad-be)).$ According to the results about double branched covers listed in Section REF , the tangle $O_1$ shown in Figure REF satisfies that its double branched cover is $\\widetilde{O}$ .",
"Let $v$ denote the associated standard involution on $\\widetilde{O}$ , then $\\widetilde{O}/v=O_1$ .",
"As our analysis about surgeries above, filling along two slopes $\\frac{0}{1}$ and $\\frac{1}{0}$ (i.e.filling $(1,0)$ -fiber and $(0,1)$ -fiber in $M(0,1;(p,-e), (aq-bp,ad-be))$ ) produce the manifolds we want.",
"The two Dehn filling slopes are mapped to $\\frac{0}{1}$ and $\\frac{1}{0}$ slants respectively on $\\partial O_1$ by the covering map induced by $v$ .",
"It means adding the tangle $X_1=0$ -tangle and $X_2=\\infty $ -tangle give the 2-bridge link and the connected sum of two 2-bridge links we want.",
"One can check that $N(O_1+0)=b(a,b)$ and $N(O_1+\\infty )=b(p,-e) \\# b(aq-bp, ad-be)$ .",
"There is no other slant satisfying this, since there is no other surgery slope along $K$ giving the original $L(a,b)$ or a non-prime manifold.",
"Figure: The tangle O 1 =A+BO_1=A+B and the corresponding pair of solutions (X 1 ,X 2 )(X_1,X_2), where A=-e pA=\\frac{-e}{p}, B=ad-be aq-bpB=\\frac{ad-be}{aq-bp} and pd-qe=1pd-qe=1,By Proposition 2.8 in [20], there is only one involution on $\\widetilde{O}$ (i.e.$v$ ), up to conjugation, satisfying $\\widetilde{O} / (the \\, involution)$ is a tangle.",
"Therefore, any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to the tangle $O_1$ .",
"Since we expect $X_1=0$ -tangle, $O=k(O_1), \\, X_1=0, \\, X_2=\\infty ,$ where $k \\in \\langle c_1,c_2 \\rangle \\in Mod(\\partial O_1)$ , give all the solutions up to equivalence, like analysis in Theorem REF .",
"Obviously, $N(k(O_1)+0)=b(a,b)$ and $N(k(O_1)+\\infty )=b(p,-e) \\# b(aq-bp, ad-be)$ .",
"We know that performing $pq$ -surgery gives a $L(p,q)$ summand.",
"Actually, $b(p,-e)=b(p,q)$ since $pd-qe=1$ .",
"(ii)$L(a,b)$ is fibered over $\\mathbb {R}P^2$ .",
"Let $L(a,b)=T_1 \\cup _g (S^1 \\widetilde{\\times } M\\ddot{o}bius \\,\\, band)$ where $T_1$ is a solid torus.",
"If $K$ is isotopic to an exceptional fiber, then $L(a,b)\\setminus N(K)$ has no exceptional fiber, so we can choose another fibration of $L(a,b)$ over $S^2$ such that $K$ is isotopic to an ordinary fiber of the new fibration, which has been discussed in the previous case.",
"Therefore we assume that $K$ is isotopic to an ordinary fiber.",
"We can regard $K$ as a $(p,q)$ -torus knot lying in $T_1$ with $|p|>1$ .",
"Because if $|p|=0$ , then $K$ is a knot in a ball, thus $L(a,b)\\setminus N(K)$ is reducible; if $|p|=1$ , then we can also refiber $L(a,b)$ over $S^2$ such that $K$ is isotopic to an ordinary fiber.",
"Here we choose a longitude-meridian basis for $K$ by using the same principle as choosing basis for $J$ in $S^1 \\times D^2$ .",
"By Corollary REF and Lemma REF , $L(a,b)=M(-1,0;(p,1))=L(4p,1-2p) \\,\\, and \\,\\, q \\cong 1 \\,\\, mod \\, p.$ In fact, $M(-1,0;(p,1)) \\cong M(-1,0;(p,-1))$ with different orientation.",
"Since $K$ is isotopic to an ordinary fiber, then $\\widetilde{O}=L(a,b)\\setminus N(K)= M(-1,1;(p,1)).$ Using the formula in Lemma REF , only doing $\\infty $ -surgery (resp.$pq$ -surgery) along $K$ gives $L(a,b)=L(4p,1-2p)$ (resp.a non-prime manifold), and $\\infty $ -surgery (resp.$pq$ -surgery) is equivalent to filling a $(1,0)$ -fiber (resp.$(0,1)$ -fiber) in this given fibration of $L(a,b)$ .",
"That is, $K(\\infty )=M(-1,0;(p,1), (1,0))=L(4p,1-2p)$ $(resp.",
"K(pq)=M(-1,0;(p,1),(0,1))=S^1 \\times S^2 \\#L(p,1)).$ As shown in Section REF , the double branched cover of the tangle $O_2$ shown in Figure REF is $\\widetilde{O}$ .",
"Fillings along the two slopes $\\frac{0}{1}$ and $\\frac{1}{0}$ on the boundary of $M(-1,1;(p,1))$ (i.e.filling $(1,0)$ - fiber and $(0,1)$ -fiber in $M(-1,1,(p,1))$ respectively) give the manifolds we want as discussion above.",
"The two Dehn filling slopes are mapped to two slants $\\frac{0}{1}$ and $\\frac{1}{0}$ respectively on $\\partial O_2$ by the covering map, which give the pair of solutions $(X_1=0, X_2=\\infty )$ when $O=O_2$ .",
"Besides, when $O=O_2$ the pair of solutions is unique by the analysis about surgeries above.",
"One can easily check that $N(O_2+0)=b(4p,1-2p)$ and $N(O_2+\\infty )=b(0,1) \\# b(p,1)$ .",
"Figure: The tangle O 2 O_2 and the corresponding pair of solutions (X 1 ,X 2 )(X_1,X_2).Figure: Splitting the tangle O 2 O_2 as U∪MU \\cup MNow we show that any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to $O_2$ .",
"Split the tangle $O_2$ into two tangles.",
"One is the tangle, denoted by $U$ , in the dotted Conway sphere $S$ in $O_2$ shown in Figure REF , which is actually the ring tangle.",
"Another one is the tangle outside $U$ in $O_2$ , denoted by $M$ , shown in Figure REF , which is a Montesinos pair in $S^2 \\times I$ .",
"According to Section REF , $\\widetilde{U}=M(-1,1;) \\cong M(0,1; (2,1),(2,-1))$ , denoted by $M_1$ , which can be regarded as a Seifert fiber space over a disk with two exceptional fibers, and $\\widetilde{M}=M(0,2;(p,1))$ , denoted by $M_2$ , which is a Seifert fiber space over an annulus with one exceptional fiber.",
"Then $\\widetilde{O_2}=\\widetilde{O}=M_1 \\cup M_2$ .",
"The lift of the Conway sphere $S$ , i.e.$M_1 \\cap M_2$ , is the only one essential torus in $\\widetilde{O}$ up to isotopy.",
"This is very similar to the situation in Theorem REF .",
"Therefore we can use the same method to show that any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to $O_2$ or a mutant of $O_2$ along the Conway sphere $S$ .",
"The tangle $U$ is so special such that $U=k(U)$ for any $k \\in \\langle c_1,c_2 \\rangle \\in Mod(\\partial U)$ , namely any mutant of $O_2$ is homeomorphic to $O_2$ .",
"Then any tangle whose double branched cover is $\\widetilde{O}$ is homeomorphic to the tangle $O_2$ .",
"In addition, $O_2$ is invariant under any homeomorphism extended by $k \\in \\langle c_1,c_2 \\rangle \\in Mod(\\partial O_2)$ , then $O=O_2, \\, X_1=0, \\, X_2=\\infty $ give all the solutions up to equivalence."
],
[
"Some other tangle equations",
"We can also solve the following system of tangle equations: N(U+X1)=b1 N(U+X2)=b2, where $X_1$ and $X_2$ are rational tangles, $U$ is an algebraic tangle but not a generalized Montesinos tangle, $b_1$ and $b_2$ are 2-bridge links with $b_1 \\ne b_2$ .",
"In fact, the system of tangle equations has been discussed in many papers, like [9] and [8].",
"But they solve the equations under the assumption that $U$ is a generalized Montesinos tangle or $d(X_1,X_2)>1$ (if $d(X_1,X_2)>1$ , then we have that $\\widetilde{U}$ is a Seifert fiber space by Cyclic Surgery Theorem [7].",
"It can be shown that $\\widetilde{U}$ is a Seifert fiber space over a disk, thus $U$ is a generalized Montesinos tangle).",
"Also lifting to the double branched covers, the system of tangle equations is translated to U()= the lens space b1 U()= the lens space b2, where $\\widetilde{U}$ (resp.$\\widetilde{b_i}$ ) denotes the double branched cover of $U$ (resp.$b_i$ ) and $\\alpha $ (resp.$\\beta $ ) is the induced Dehn filling slope by adding rational tangle $X_1$ (resp.$X_2$ ).",
"Therefore, the problem turns out to be finding knots in the lens space $\\widetilde{b_1}$ which admits a surgery to another lens space.",
"According to Section REF , the double branched cover of an algebraic tangle is a graph manifold.",
"Obviously the algebraic tangle $U$ is locally unknotted, since $b_1 \\ne b_2$ , both of which are prime.",
"Also it is impossible for $U$ to contain a splittable unknot.",
"Therefore, $\\widetilde{U}$ is an irreducible graph manifold, but not a Seifert fiber space since $U$ is not a generalized Montesinos tangle.",
"Now we split $\\widetilde{U}$ along its incompressible tori to study $\\widetilde{U}$ and this is the idea from Buck and Mauricio [5].",
"In fact, that all the tori in $\\widetilde{U}$ are separating.",
"Because if not, a non-separating torus is still non-separating after Dehn filling, which contradicts the fact that there is no non-separating torus in a lens space.",
"Let $T$ be a collection of disjoint non-parallel incompressible tori such that each component of $\\widetilde{U}|T$ is atoroidal.",
"Here $\\widetilde{U}$ is an irreducible graph manifold.",
"After cutting it along $T$ , we only have atoroidal Seifert fiber spaces (i.e.small Seifert fiber spaces) left.",
"Definition 4.1 A splitting graph of $\\widetilde{U}$ along $T$ is a graph $G$ which uses edges to represent the incompressible tori in $T$ and use vertices to represent the connected components of this decomposition.",
"An edge connects two vertices if and only if the incompressible torus corresponding to the edge separates the two components corresponding to these two vertices.",
"In fact, the splitting graph of $\\widetilde{U}$ along $T$ is a tree, since all the tori in $T$ are separating.",
"Choose the vertex whose corresponding component contains $\\partial \\widetilde{U}$ to be the root of this graph, and denote it by $v_0$ .",
"We define the level of a vertex to be the minimum number of edges of a path which connects this vertex and the root, and the level of an edge is defined to be the same as the level of the adjacent vertex which is closer to the root.",
"Then we have the splitting graph of $\\widetilde{U}$ is like the following: Figure: The splitting graph of U ˜\\widetilde{U} along TTWe can verify the following lemma by almost the same argument as Proposition 5.8 in [5] and the formula of Lemma REF .",
"Lemma 4.2 $v_0$ is a Seifert fiber space over an annulus with exact one exceptional fiber.",
"$v_0(\\alpha )$ is a solid torus.",
"Proposition 4.3 The splitting graph of $\\widetilde{U}$ along $T$ is a linear tree as Figure REF shown.",
"Each component is a Seifert fiber space over an annulus with exact one exceptional fiber, except the component at the $n$ th level which is a Seifert fiber space over a disk with exact two exceptional fibers, and $n \\ge 1$ .",
"By Lemma REF , $v_0(\\alpha )$ is a solid torus, which means $\\alpha $ filling on $v_0$ induces a Dehn filling on $v_1$ (i.e.the only one vertex on the 1-level since $v_0$ has only two boundary components).",
"This is the same situation when we discussed about $v_0$ , so we can use Lemma REF inductively to verify that $v_{i}$ ($i=1,\\dots ,n-1$ ) is a Seifert fiber space over an annulus with exact 1 exceptional fiber, and then the splitting graph is a linear tree shown as Figure REF .",
"Besides, $v_{i} \\cup \\dots \\cup v_0 \\cup \\widetilde{X_1}$ is a solid torus, for $i=1,\\dots ,n-1$ .",
"Now we only need to work out the end piece $v_n$ .",
"Figure: The splitting graph of U ˜\\widetilde{U}: v i v_i is a Seifert fiber space over an annulus with exact one exceptional fiber, for i=0,⋯,n-1i=0,\\dots ,n-1.",
"v n v_n is a Seifert fiber space over a disk with exact two exceptional fibers.$v_n$ is a small Seifert fiber space with one boundary component.",
"Only small Seifert fiber spaces $M(0,1;(\\alpha _1,\\beta _1),(\\alpha _2,\\beta _2))$ and $M(-1,1;)$ have one boundary component.",
"Actually, $M(-1,1;)=M(0,1; (2,1), (2,-1))$ .",
"Therefore we can assume $v_n$ is a Seifert fiber space over a disk.",
"It's impossible for $v_n$ to have only one exceptional fiber, for otherwise $v_n$ is a solid torus, and $e_{n-1}$ is not incompressible.",
"So $v_n$ is a Seifert fiber space over a disk with exact two exceptional fibers.",
"Obviously, $n \\ge 1$ since $\\widetilde{U}$ is not a Seifert fiber space.",
"Definition 4.4 The $(p_0, q_0)$ -cable of the $(p_1, q_1)$ -cable of ...$(p_k, q_k)$ -torus knot is called an iterated knot, denoted by $[p_0,q_0; p_1, q_1; \\dots ; p_k, q_k]$ , where $p_i \\ge 2$ , for $i=0,\\dots ,k$ .",
"Proposition 4.5 Let the lens space $\\widetilde{b_1}=T_1 \\cup T_2$ , where $T_i$ is a solid torus, for $i=1,2$ .",
"Then $\\widetilde{U}=\\widetilde{b_1} \\setminus N(K)$ , where $K$ is an iterated knot $[p_0,q_0; p_1, q_1; \\dots ; p_n,q_n]$ in $T_1$ where $p_i \\ge 2$ for $i=0,\\dots , n$ , and $n \\ge 1$ .",
"We already have that $v_n$ is a Seifert fiber space over a disk with exact 2 exceptional fibers, and $v_i$ is a Seifert fiber space over an annulus with exact 1 exceptional fiber, for $i=1,\\dots ,n-1$ .",
"$\\widetilde{X_1}$ is a solid torus lying in $\\widetilde{b_1}$ , and it can be regarded as a tubular neighborhood of a knot $K$ in $\\widetilde{b_1}$ .",
"$\\widetilde{U}=\\widetilde{b_1}\\setminus N(K)$ .",
"Let $r_i$ be the corresponding slope of the Dehn filling on $v_i$ induced by $r_{i-1}$ filling on $v_{i-1}$ where $i=0,\\dots ,n$ and $r_0=\\alpha $ .",
"First of all, $v_{n-1}(r_{n-1})$ is a solid torus lying in the lens space $\\widetilde{b_1}$ such that $\\widetilde{b_1} \\setminus v_{n-1}(r_{n-1})= v_n$ , which is a Seifert fiber space over a disk with exact 2 exceptional fibers.",
"Then $v_{n-1}(r_{n-1})$ is isotopic to a tubular neighborhood of a fiber, denoted by $F$ , of some generalized Seifert fibration of the lens space $\\widetilde{b_1}$ over $S^2$ .",
"The fiber $F$ is an ordinary fiber, for otherwise $v_{n}$ has at most 1 exceptional fiber which contradicts Proposition REF .",
"We regard $v_{n-1}(r_{n-1})$ as a tubular neighborhood of a $(p_n,q_n)$ -torus knot in one of the solid tori of $\\widetilde{b_1}$ , without loss of generality $T_1$ .",
"We claim that $p_n \\ge 2$ .",
"If $p_n=0$ , then either $\\widetilde{U}=\\widetilde{b_1} \\setminus N(K)$ is reducible, or $\\widetilde{b_1}=S^3$ , in which case $\\widetilde{b_1} \\setminus v_{n-1}(r_{n-1})=v_n$ is a solid torus instead of a Seifert fiber space over a disk with 2 exceptional fibers.",
"If $p_n=1$ , then $\\widetilde{b_1} \\setminus v_{n-1}(r_{n-1})=v_n$ is also a solid torus.",
"$v_{n-2}(r_{n-2})$ is a solid torus lying in $v_{n-1}(r_{n-1})$ such that $v_{n-1}(r_{n-1}) \\setminus v_{n-2}(r_{n-2})=v_{n-1}$ is a Seifert fiber space over an annulus with exact 1 exceptional fiber.",
"So $v_{n-2}(r_{n-2})$ must lie in $v_{n-1}(r_{n-1})$ as a tubular neighborhood of a $(p_{n-1},q_{n-1})$ -torus knot with $p_{n-1} \\ge 2$ .",
"$p_{n-1} \\ne 0$ for the same reason as $p_n \\ne 0$ , and if $p_{n-1}=1$ then $v_{n-1}$ has no exceptional fiber.",
"Hence, $v_{n-2}(r_{n-2})$ lies in $T_1$ of $\\widetilde{b_1}$ as a tubular neighborhood of a $(p_{n-1},q_{n-1})$ -cable of $(p_{n},q_{n})$ -torus knot with $p_n$ , $p_{n-1} \\ge 2$ .",
"By induction, $v_0(\\alpha )$ lies in $T_1$ of $\\widetilde{b_1}$ as a tubular neighborhood of an iterated knot.",
"Also the solid torus $\\widetilde{X_1}$ lies in $T_1$ of $\\widetilde{b_1}$ as a tubular neighborhood of $K=[p_0,q_0; p_1,q_1; \\dots ; \\\\ p_n,q_n]$ with $p_i \\ge 2$ , for $i=0, \\dots , n$ and $n \\ge 1$ .",
"$\\widetilde{U}=\\widetilde{b_1} \\setminus N(K)$ is the complement of an iterated knot in $T_1$ of $\\widetilde{b_1}$ .",
"Now we want to find out what kind of iterated knots lying in a lens space admit a surgery to give another lens space.",
"Actually, Gordon [14] has deeply studied on surgeries along an iterated knot in $S^3$ by using Lemma REF and REF .",
"Here we just use the lemmas of Gordon and results about Seifert fiber space to study Surgeries along the iterated knot $K$ in the lens space $\\widetilde{b_1}$ .",
"Proposition 4.6 Let $\\widetilde{b_1}=L(a,b)=T_1 \\cup T_2$ , where $(a,b)$ is a pair of relatively prime integers satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ , and $T_i$ is a solid torus, for $i=1,2$ .",
"Then $\\widetilde{U}=L(a,b)\\setminus N(K)$ , where $K=[p_0, q_0; p_1, q_1]$ in $T_1$ of $\\widetilde{b_1}$ with $q_0=2p_1q_1 \\pm 1, p_0=2, p_1>1, |aq_1-bp_1|>1$ .",
"Only $r=4p_1q_1 \\pm 1$ surgery along $K$ can produce another lens space and the manifold obtained by this Dehn surgery is $L(a \\pm 4ap_1q_1 \\mp 4b p^2_1, b \\pm 4aq^2_1 \\mp 4b p_1q_1)$ .",
"According to Proposition REF , when $n=1$ , the knot $K=[p_0, q_0; p_1, q_1]$ which is a $(p_0, q_0)$ -cable of $(p_1, q_1)$ -torus knot $T_{p_1,q_1}$ lying in $T_1$ of $L(a,b)$ with $p_1 \\ge 2$ , $p_0 \\ge 2$ .",
"By Lemma REF , there are three cases.",
"(1)The surgery slope $r \\ne p_0q_0$ and $\\ne m/n, \\, m=np_0q_0 \\pm 1$ .",
"$K(r)$ contains an essential torus, which can not be a lens space.",
"(2)The surgery slope $r=p_0q_0$ .",
"We get a non-prime manifold.",
"(3)The surgery slope $r=m/n, \\, m=np_0q_0 \\pm 1$ .",
"By Lemma REF and REF , $K(r)=[L(a,b) \\setminus N(T_{p_1,q_1})](m/(np^2_0)).$ By Lemma REF , $[L(a,b) \\setminus N(T_{p_1,q_1})](m/np_0^2) = M(0,0;(p_1, -e), (aq_1-bp_1,ad-be), (m-np^2_0p_1q_1,np^2_0)),$ where $p_1d-q_1e=1$ .",
"We want it to be a lens space, and that happens if and only if one of $|m-np^2_0p_1q_1|$ , $|aq_1-bp_1|$ and $|p_1|$ equals 1.",
"In fact, $|aq_1-bp_1|>1$ for the same reason as $p_n=p_1>1$ since a $(p_1,q_1)$ -torus knot in $T_1$ of $L(a,b)$ is also a torus knot with winding number $|aq_1-bp_1|$ in $T_2$ of $L(a,b)$ .",
"So we have $|m-np^2_0p_1q_1|=1$ , and there are two cases.",
"(i) $m-np^2_0p_1q_1=1$ .",
"According to the assumption, we also have $m=np_0q_0 \\pm 1$ .",
"If $m=np_0q_0+1$ , then $np_0(q_0-p_0p_1q_1)=0$ .",
"If $n=0$ , then $m=1$ and $K(r)=L(a,b)$ , while we want it to be another lens space.",
"Besides, $p_0 \\ge 2$ , and $q_0-p_0p_1q_1 \\ne 0$ since $gcd(p_0,q_0)=1$ .",
"Thus it is impossible.",
"If $m=np_0q_0-1$ , it implies that $np_0(q_0-p_0p_1q_1)=2.$ Therefore $p_0=2$ (since $p_0 \\ge 2$ ), $n=\\pm 1$ , $q_0-p_0p_1q_1=\\pm 1$ so $q_0=2p_1q_1 \\pm 1$ .",
"$m=\\pm p_0q_0-1=\\pm 4p_1q_1+1$ , then $r=\\frac{m}{n}=4p_1q_1 \\pm 1$ .",
"Then Using the formula in Lemma REF , [L(a,b) N(Tp1,q1)](m/np02) =M(0,0; (p1,-e), (aq1-bp1,ad-be), (1,p20)) =M(0,0;(p1,-e), (aq1-bp1,ad-be p20(aq1-bp1))) =L(a 4ap1q1 4b p21, b 4aq21 4b p1q1).",
"(ii) $m-np^2_0p_1q_1=-1$ .",
"As the argument above, if $m=np_0q_0-1$ , then $np_0(q_0-p_0p_1q_1)=0$ which is impossible.",
"Thus $m=np_0q_0+1$ , it implies that $np_0(p_0p_1q_1-q_0)=2.$ Therefore $p_0=2$ (since $p_0 \\ge 2$ ), $n=\\pm 1$ , $p_0p_1q_1-q_0=\\pm 1$ so $q_0=2p_1q_1 \\mp 1$ .",
"$m=\\pm p_0q_0+1=\\pm 4p_1q_1-1$ , then $r=\\frac{m}{n}=4p_1q_1 \\mp 1$ .",
"Then Using the formula in Lemma REF , M(0,0;(p1,-e), (aq1-bp1,ad-be), (1,p20)) =M(0,0;(p1, -e), (aq1-bp1, ad-be p20(aq1-bp1))) =L(a 4ap1q1 4b p21, b 4aq21 4b p1q1).",
"When $n=2$ , the knot $K=[p_0, q_0;p_1, q_1; p_2, q_2]$ lying in $T_1$ of $L(a,b)$ with $p_i \\ge 2$ , for $i=0,1,2$ .",
"Let $K_1=[p_1, q_1; p_2, q_2]$ .",
"Also by Lemma REF , there are three cases.",
"(1)The surgery slope $r \\ne p_0q_0$ and $\\ne m/n, \\, m=np_0q_0 \\pm 1$ .",
"$K(r)$ contains an essential torus, which can not be a lens space.",
"(2)The surgery slope $r=p_0q_0$ .",
"We obtain a non-prime manifold.",
"(3)The surgery slope $r=m/n, \\, m=np_0q_0 \\pm 1$ .",
"By Lemma REF and REF , $K(r)=[L(a,b) \\setminus N(K_1)](m/(np^2_0)).$ We want it to be a lens space, and that may happen when $m=np^2_0p_1q_1 \\pm 1$ by the same argument when $n=1$ .",
"Then, by Lemma REF and REF , $K(r)=[L(a,b) \\setminus N(K_1)](m/(np^2_0))=[L(a,b) \\setminus N(T_{p_2,q_2})](m/(np^2_0p^2_1)).$ By Lemma REF , $[L(a,b) \\setminus N(T_{p_2,q_2})](m/(np^2_0p^2_1))\\\\= M(0,0;(p_2,-e), (aq_2-bp_2, ad-be), (m-np^2_0p^2_1p_2q_2, np^2_0p^2_1)),$ where $p_2d-q_2e=1$ .",
"It can be a lens space if and only if $|m-np^2_0p^2_1p_2q_2|=1$ , since if one of $|aq_2-bp_2|$ and $|p_2|$ equals 1, then $v_n=v_2$ has only one exceptional fiber.",
"So far we have 3 equations shown as the following: m=np0q0 1 m=np20p1q1 1 m=np20p21p2q2 1 By Equation () and (), we have $np^2_0p_1(q_1-p_1p_2q_2)=0$ or $\\pm 2$ .",
"It is impossible since $p_0, p_1 \\ge 2$ , $gcd(p_1,q_1)=1$ and $n \\ne 0$ for otherwise $K(r)=L(a,b)$ .",
"Therefore, when $n=2$ , surgeries along $K$ can not produce another lens space.",
"Using the same argument as $n=2$ , we have $n=3,4,\\dots $ is also impossible.",
"Then it concludes the proposition.",
"It not hard to find that $\\widetilde{U}$ satisfying the equations and is contained in $\\widetilde{O}$ satisfying the equations and when $\\widetilde{O}$ is irreducible toroidal but not Seifert fibered.",
"Actually $\\widetilde{U}$ is a special case of the previous $\\widetilde{O}$ (i.e.$p=2$ ).",
"So we can easily get the following theorem.",
"Theorem 3 Suppose N(U+X1)=b1 N(U+X2)=b2, where $X_1$ and $X_2$ are rational tangles, $U$ is an algebraic tangle but not a generalized Montesinos tangle, $b_1$ and $b_2$ are 2-bridge links with $b_1 \\ne b_2$ .",
"The system of tangle equations has solutions if and only if one of the following holds: (1) There exist 2 pairs of relatively prime integers $(a,b)$ , $(p_1,q_1)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty ), p_1>1$ , and $|aq_1-bp_1|>1$ such that $b_1=b(a,b)$ and $b_2=b(a+4ap_1q_1-4b p^2_1, b+4aq^2_1-4b p_1q_1)$ .",
"Solutions up to equivalence are shown as the following: Figure: U=U= the tangle in (a) or (b) where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1} (or A=ad-be aq 1 -bp 1 A=\\frac{ad-be}{aq_1-bp_1}, B=-e p 1 B=\\frac{-e}{p_1}), and p 1 d-q 1 e=1p_1d-q_1e=1 with d,e∈ℤd,e\\in \\mathbb {Z}.",
"X 1 =0X_1=0-tangle and X 2 =-1X_2=-1-tangle.",
"(Note that choosing different dd and ee such that p 1 d-q 1 e=1p_1d-q_1e=1 has no effect on the tangle UU.",
")(2) There exist 2 pairs of relatively prime integers $(a,b)$ , $(p_1,q_1)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty ), p_1>1$ and $|aq_1-bp_1|>1$ such that $b_1=b(a,b)$ and $b_2=b(a-4ap_1q_1+4b p^2_1, b-4aq^2_1+4b p_1q_1)$ .",
"Solutions up to equivalence are shown as the following: Figure: U=U= the tangle in (a) or (b) where A=-e p 1 A=\\frac{-e}{p_1}, B=ad-be aq 1 -bp 1 B=\\frac{ad-be}{aq_1-bp_1} (or A=ad-be aq 1 -bp 1 A=\\frac{ad-be}{aq_1-bp_1}, B=-e p 1 B=\\frac{-e}{p_1}), and p 1 d-q 1 e=1p_1d-q_1e=1 with d,e∈ℤd,e\\in \\mathbb {Z}.",
"X 1 =0X_1=0-tangle and X 2 =1X_2=1-tangle.",
"(Note that choosing different dd and ee such that p 1 d-q 1 e=1p_1d-q_1e=1 has no effect on the tangle UU.",
")We just use the same method as in Theorem REF .",
"Suppose $b_1=b(a,b)$ for a pair of relatively prime integers $(a,b)$ satisfying $0<\\frac{b}{a}\\le 1 (or \\,\\, \\frac{b}{a}=\\frac{1}{0}=\\infty )$ .",
"According to Proposition REF , $\\widetilde{U}$ is the complement of the iterated knot $K=[p_0, q_0; p_1, q_1]$ with $q_0=2p_1q_1 \\pm 1, p_0=2$ , $p_1>1, |aq_1-bp_1|>1$ lying in one of the solid torus of the lens space $L(a,b)$ .",
"This is a special case of $\\widetilde{O}$ in Theorem REF , i.e.$p=2$ for $\\widetilde{O}$ .",
"Therefore, we can find all the tangles whose double branched cover are $\\widetilde{U}$ , by letting $p=2$ in all the solutions for $O$ in Theorem REF .",
"The only difference is the surgery slopes on $K$ , since here we want to obtain a lens space instead of a connected sum of two lens spaces.",
"Obviously, only $\\infty $ -surgery along $K$ gives the original $L(a,b)$ .",
"By Proposition REF , only $r=4p_1q_1 \\pm 1$ -surgery produces another lens space.",
"By carefully studying the images of these slopes under the covering maps, we obtain the pairs of rational tangle solutions up to equivalence as above."
]
] | 1709.01785 |
[
[
"Quantum Privacy and Schur Product Channels"
],
[
"Abstract We investigate the quantum privacy properties of an important class of quantum chan-nels, by making use of a connection with Schur product matrix operations and associated correlationmatrix structures.",
"For channels implemented by mutually commuting unitaries, which cannot priva-tise qubits encoded directly into subspaces, we nevertheless identify private algebras and subsystemsthat can be privatised by the channels.",
"We also obtain further results by combining our analysiswith tools from the theory of quasiorthogonal operator algebras and graph theory."
],
[
"Introduction",
"Private algebras are fundamental objects of study in the theory of quantum privacy.",
"They are the mathematical manifestation of physically motivated techniques that have been developed to hide qubits from observers in quantum systems in a variety of settings, and also referred to as private quantum channels, private quantum codes and private subsystems [1], [2], [3], [4], [7], [8], [9], [10], [12], [13].",
"One of the basic challenges in the subject is to identify private codes for specific quantum channels or classes of channels.",
"Only recently has the base of examples begun to expand significantly, starting with a surprisingly simple physical model in [12] and expanding to general Pauli channels in [17].",
"In this paper, we begin with the observation that the privacy properties of a large class of quantum channels can be investigated via the matrix theoretic notions of Schur products and correlation matrices.",
"Specifically, all random unitary channels defined by mutually commuting unitaries can be implemented as a Schur product completely positive map with a correlation matrix [16], [18], [20].",
"It is known that these channels cannot privatise qubits encoded directly into subspaces of the underlying Hilbert space for the channel.",
"However, as initially discovered in [12], such channels can nevertheless privatise quantum information, via more delicate subsystem encodings, which from the algebra perspective correspond to more general `ampliated' subalgebras of the full C$^*$ -algebra of operators on the system Hilbert space.",
"Here we present a comprehensive analysis of the privacy properties for this class of channels, identifying when private algebras exist for them.",
"Our analysis is based on the underlying correlation matrix graph structure.",
"We then combine this analysis with a connection between operator systems, another graph, and quasiorthonal operator algebra techniques [17] to obtain further general privatisation results on the channels.",
"This paper is organised as follows.",
"In the next section we present preliminary notions and a motivating example.",
"In Section 3 we make the connection with Schur products and we use associated correlation matrix and graph structures to identify private algebras and subsystem codes for random unitary channels implemented with mutually commuting operators.",
"In Section 4 we expand our analysis from a different perspective, making use of an operator system and quasiorthogonal algebra approach."
],
[
"Quantum Channels and Private Algebras",
"Let $\\Phi :M_n(\\rightarrow M_n($ be a quantum channel: a trace-preserving, completely positive linear map on the $n\\times n$ complex matrices.",
"Then $\\Phi (\\rho ) = \\sum _i A_i \\rho A_i^*$ for some matrices $A_i$ , the Kraus operators of the channel.",
"That $\\Phi $ is trace preserving is equivalent to the condition that $\\sum _i A_i^*A_i = I.$ Recall that every $*$ -subalgebra $\\mathcal {A}$ of $M_n($ is $*$ -isomorphic to an orthogonal direct sum of full matrix algebras $M_{n_1}\\oplus \\ldots \\oplus M_{n_r}$ and is unitarily equivalent to a corresponding direct sum of simple irreducible algebras: $\\mathcal {A} \\cong \\oplus _{k=1}^r (I_{m_k}\\otimes M_{n_k}()$ , where $I_m$ is the identity operator on $M_m($ .",
"Definition 1 [7], [9], [17] A channel $\\Phi $ privatises a $*$ -subalgebra $\\mathcal {A}$ of $M_n($ , if there exists a fixed density operator $\\rho _0$ such that $\\Phi (\\rho )= \\rho _0,$ for all density operators $\\rho \\in \\mathcal {A}$ .",
"In many instances that naturally arise we have $\\rho _0 = \\frac{1}{n} I_n$ .",
"As this will be our primary focal point in the paper, we shall say the channel privatises to the unit in this case.",
"Remark 1 Notice that although Definition REF defines privatisation of a $*$ -subalgebra, the definition of privatisation can be modified in the obvious way to encompass privatisation of any subset of density operators $S$ : $\\Phi $ privatises $S$ if there exists a fixed $\\rho _0$ such that for all $\\rho \\in S$ , $\\Phi (\\rho )=\\rho _0$ .",
"Observe that if $\\mathcal {A}\\simeq M_2($ then $\\Phi $ can privatise a logical qubit of information, and if $\\mathcal {A} \\simeq M_{2^k}($ then $k$ qubits can be privatised.",
"We also remark that privatisation to the unit is a seemingly natural type of privatisation to consider, especially when we are interested in channels that privatise $*$ -subalgebras.",
"For instance, privatisation was first considered in the context of random unitary channels, where it was noted in [1] that for unital channels, if the privatised set $S$ contains the maximally mixed state, $\\rho _0$ must be the identity.",
"For Schur product channels, which is our primary focus, $\\Phi $ is unital if and only if $\\Phi $ is trace-preserving, so in order to investigate Schur product channels that privatise unital $*$ -subalgebras, we must work with privatisation to the unit.",
"Many other interesting and tractable classes of channels, such as random unitary channels are also necessarily unital, and so to study privatisation of $*$ -subalgebras in this setting, we again are forced to use privatisation to the unit.",
"It should be possible to extend our analysis to more general privacy for unital channels, using the structure theory for such channels as in [15], but we leave this for consideration elsewhere.",
"Here we are primarily interested in studying a class of quantum channels that, at first glance, do not appear to have necessary privatising features.",
"A map $\\Phi $ is a random unitary channel if its Kraus operators are positive multiples of unitary operators; $\\Phi (\\rho ) = \\sum _i p_i U_i \\rho U_i^*$ , with the $p_i$ forming a probability distribution.",
"These channels are centrally important in quantum privacy and quantum error correction, and many privatise quantum information; for instance, the simple complete depolarizing channel, with (unnormalized) Kraus operators given by the Pauli operators $I,X,Y,Z$ , satisfies $\\Phi (\\rho ) = \\frac{1}{2} I$ for all single qubit $\\rho $ .",
"However, if the unitaries $U_i$ are mutually commuting one can verify [12], [13] that $\\Phi $ cannot privatise any (unamplified) matrix algebras of the form $M_k($ ; that is, $\\Phi $ has no private subspaces.",
"Nevertheless, as we now understand, random unitary channels determined by mutually commuting unitaries can still privatise algebras, but only in the form of so-called private subsystems, which translates to algebras $I_m\\otimes M_n($ with $m>1$ .",
"Consider the following motivating example from [12] that illustrates these points, and which we will return to following the analysis of the next section.",
"Example 1 Let $\\Phi : M_2(\\rightarrow M_2($ be the channel whose Kraus operators are $\\frac{1}{\\sqrt{2}}I$ and $\\frac{1}{\\sqrt{2}}Z$ , where $Z$ is the Pauli $Z$ matrix.",
"Then $\\Phi (\\rho ) = \\frac{1}{2}\\bigl ( \\rho + Z\\rho Z\\bigr )$ acts by projecting $\\rho $ down to its diagonal.",
"If $\\Phi (\\rho ) = \\frac{\\mathrm {tr}}{2}(\\rho )I$ then necessarily $\\rho $ has constant diagonal.",
"The space of such matrices is of dimension 3, and so cannot accommodate a qubit $*$ -subalgebra.",
"In particular, $\\Phi $ does not privatise any qubits.",
"However, while $\\Phi $ itself does not privatise any qubits, the channel $\\Phi ^{\\otimes 2} = \\Phi \\otimes \\Phi $ does privatise a qubit.",
"Indeed, consider the channel $\\Phi \\otimes \\Phi : M_4(\\rightarrow M_4($ .",
"The Kraus operators of $\\Phi \\otimes \\Phi $ are $\\lbrace \\frac{1}{2}I\\otimes I,\\frac{1}{2} I\\otimes Z, \\frac{1}{2}Z\\otimes I,\\frac{1}{2}Z\\otimes Z\\rbrace $ and so $\\Phi \\otimes \\Phi (\\rho )= \\frac{1}{4}\\biggl ( \\rho + (I\\otimes Z)\\rho (I\\otimes Z)+(Z\\otimes I)\\rho (Z\\otimes I)+(Z\\otimes Z)\\rho (Z\\otimes Z)\\biggr ),$ which again is the projection onto the diagonal.",
"Thus, as before, for $\\Phi \\otimes \\Phi (\\rho ) = \\frac{\\mathrm {tr}(\\rho )}{4}I$ we require that $\\rho $ have constant diagonal.",
"But now, there is an algebra isomorphic to $M_2($ that satisfies this: the algebra generated by $\\lbrace I\\otimes X, Y\\otimes Z\\rbrace $ , which is equal to $\\mathrm {span}\\lbrace I\\otimes I, I\\otimes X, Y\\otimes Y, Y\\otimes Z\\rbrace $ .",
"Note that $I\\otimes X$ , $Y\\otimes Y$ , $Y\\otimes Z$ have no non-zero on-diagonal entries, and so the diagonal of $a_1 I\\otimes I + a_2 I\\otimes X + a_3 Y\\otimes Y + a_4 Y\\otimes Z$ is $a_1$ .",
"Hence, in addition to the basic fact that random unitary channels with commuting unitaries cannot privatise a subspace (i.e., corresponding to unampliated matrix algebras $M_n($ ), we observe that in some cases a tensor power of the channel may yet privatise an ampliated matrix algebra, in this case, an algebra unitarily equivalent to $I_2\\otimes M_2($ .",
"These points for this particular class of channels are the focus of our analysis in the next section."
],
[
"Schur Product Channels and Privacy",
"Let $\\Phi $ be a channel whose Kraus operators $A_i$ are scalar multiples of mutually commuting unitaries.",
"We begin with an observation that connects private codes for such channels with a fundamental operation in matrix theory.",
"By applying the spectral theorem we may simultaneously diagonalize all the $A_i$ ; that is, we can find a common basis $\\lbrace | k \\rangle \\rbrace $ such that the matrix representation for each $A_i$ in this basis is diagonal.",
"We shall write $A_i$ both for the operator and for this diagonal matrix representation, and put $A_i = \\operatorname{diag}(A^{(i)}_{k})$ with $A_k^{(i)}\\in \\mathbb {C}$ as the diagonal entries of $A_i$ .",
"It follows that the matrix entries in this basis of an output state $\\Phi (\\rho )= (\\Phi (\\rho )_{kl})$ satisfy: $\\Phi (\\rho )_{kl} = \\sum _i \\rho _{kl} A^{(i)}_{k} \\overline{A^{(i)}_{l}} = \\rho _{kl} \\sum _i A^{(i)}_{k} \\overline{A^{(i)}_{l}}.$ Now, if $C = (C_{kl})$ is the matrix given by $C_{kl} = \\sum _i A^{(i)}_{k} \\overline{A^{(i)}_{l}},$ then we conclude that $\\Phi (\\rho ) = C\\circ \\rho $ where $\\circ $ denotes the entrywise Schur (also called Hadamard) product of two matrices.",
"Moreover, in order for $\\Phi $ of this form to be completely positive and trace preserving, necessarily $C$ is positive semidefinite and has all 1's down the diagonal.",
"Such a matrix is called a correlation matrix.",
"For more on the structure of Schur product channels and correlation matrices, see [16], [18], [20].",
"Let us formulate this observation as a result.",
"Proposition 1 Let $\\Phi $ be a channel with Kraus operators that are scalar multiples of mutually commuting unitaries.",
"Then there is a correlation matrix $C$ such that, up to a unitary change of basis, $\\Phi (\\rho ) = C \\circ \\rho $ for all $\\rho $ .",
"Notice that the channel in Example REF is of the form $(\\Phi \\otimes \\Phi )(\\rho ) = I_4 \\circ \\rho $ for all $\\rho $ , and in particular $\\Phi (\\rho ) = \\frac{1}{4} I_4$ for all density matrices $\\rho $ with constant diagonal.",
"Note that the algebra generated by $\\lbrace I\\otimes X, Y\\otimes Z\\rbrace $ consists entirely of matrices with constant diagonal.",
"Let us consider the Schur product form in more detail from the perspective of private quantum codes.",
"We begin with a simple observation and do not concern ourselves with algebra structures for the moment.",
"Proposition 2 Suppose that $\\Phi $ is a channel and $C$ is a correlation matrix such that $\\Phi (\\rho )=\\rho \\circ C$ for all $\\rho $ .",
"Then a set of density operators $\\mathcal {S}$ is privatised by $\\Phi $ to the unit if and only if every $\\rho \\in \\mathcal {S}$ has constant diagonal and $\\rho _{ij}c_{ij}=0$ for all $1\\le i\\ne j\\le n$ .",
"It is clear then, that the zero pattern of $C$ determines what is privatised by $\\Phi $ ; namely, a $*$ -subalgebra $\\mathcal {A} \\subseteq M_n($ is privatised to the unit for $\\Phi $ if and only if for all $\\rho \\in \\mathcal {A}$ , $\\rho $ has constant diagonal and $\\rho _{ij}c_{ij}=0$ for all $i\\ne j$ .",
"We next inject into the analysis the graph naturally associated with a correlation matrix.",
"Definition 2 Let $C \\in M_n($ be a correlation matrix.",
"The graph of $C$ is the graph $G_C$ whose vertices are $\\lbrace 1,2,\\ldots , n\\rbrace $ and whose edge set is $\\lbrace (i,j):i\\ne j$ and $c_{ij}\\ne 0\\rbrace $ .",
"Clearly, $G_C$ contains all information about the location of non-zero entries of $C$ .",
"We say that a matrix is “irreducible” if its underlying graph is connected.",
"Proposition 3 Let $C\\in M_n($ be a correlation matrix with graph $G_C$ .",
"If $G$ has $m$ connected components $\\lbrace G_i\\rbrace _{i=1}^m$ each on $k_i$ vertices, then $C$ has $m$ irreducible components $C_i$ corresponding to the $G_i$ , and by a permutation may be brought into the form $C=\\bigoplus _{i=1}^m C_i$ where each $C_i$ is a $k_i\\times k_i$ irreducible correlation matrix.",
"The following graph product first introduced in [23] will be useful in our analysis.",
"Definition 3 Let $G,H$ be two graphs.",
"The strong product of $G$ and $H$ , $G\\boxtimes H$ has vertex set $V(G)\\times V(H)$ , and $((i,j),(k,l))\\in E(G\\boxtimes H)$ if and only if one of the following holds: $i=k,(j,l)\\in E(H)$ $j=l,(i,k)\\in E(G)$ $(i,k)\\in E(G),(j,l)\\in E(H)$ Let $C$ be a correlation matrix.",
"Then one can show that the graph of $C\\otimes C$ is $G_C\\boxtimes G_C$ .",
"This follows from the fact that the definition of $G\\boxtimes G$ recapitulates exactly when an entry of $C\\otimes C$ is non-zero: a diagonal entry of $C$ times a non-diagonal non-zero entry is non-zero a non-diagonal non-zero entry of $C$ times another non-diagonal non-zero entry of $C$ is non-zero.",
"Observe that if $\\Phi $ is a Schur product channel such that $\\Phi (\\rho )=\\rho \\circ \\, C$ for a correlation matrix $C\\in M_n($ , then the tensor product channel satisfies $\\Phi ^{\\otimes k}(\\rho ) = \\rho \\circ \\, C^{\\otimes k}$ for all $k \\ge 1$ .",
"Hence the privacy features of a tensored Schur product channel $\\Phi ^{\\otimes k}$ can be understood in terms of the non-zero entry structure of $C^{\\otimes k}$ , which in turn is captured by the graph $G_C^{\\boxtimes k}$ .",
"Consider the most restrictive case for privacy.",
"Let $K_n$ be the complete graph on $n$ vertices, and note that $K_n^{\\boxtimes k} = K_{kn}$ .",
"As a direct consequence of Proposition REF we have the following.",
"Proposition 4 Let $\\Phi (\\rho )=\\rho \\circ \\, C$ for a correlation matrix $C$ whose graph $G_C$ is the complete graph on $n$ vertices.",
"Then neither $\\Phi $ nor any tensor power $\\Phi ^{\\otimes k}$ can privatise a non-scalar subalgebra to the unit.",
"More generally, let $G$ be a graph with disjoint connected components $G_i$ : $G=\\biguplus _i G_i$ .",
"Then it follows that $G\\boxtimes G = \\biguplus _{i,j} (G_i\\boxtimes G_j).$ In particular, if $G = \\biguplus _{i=1}^m K_{k_i}$ is the disjoint union of complete graphs $K_{k_i}$ , then we have $G\\boxtimes G = \\biguplus _{i,j=1}^m K_{k_ik_j}.$ Hence, for any correlation matrix $C$ whose graph $G_C$ is the disjoint union of complete graphs, there exists some permutation of $C^{\\otimes N}$ such that $C^{\\otimes N}$ has the form $ C^{\\otimes N} = \\bigoplus _{i_1,\\ldots ,i_N=1}^m T_{k_{i_1}\\ldots k_{i_N}},$ where $T_k$ is a $k\\times k$ matrix with all non-zero entries.",
"Lemma 1 Let $C$ be a correlation matrix whose graph $G_C$ is the disjoint union of $m$ complete graphs, so that $C^{\\otimes N}$ has a direct sum decomposition of the form of Equation REF .",
"Then $C^{\\otimes N}$ has a principal submatrix of size $m^N$ that is just the identity matrix.",
"For each element $I=(i_1,\\ldots ,i_N)\\in \\lbrace 1,\\ldots ,m\\rbrace ^N$ denote by $|I|$ the product $\\Pi _{i\\in I} k_i$ .",
"Also for each such $I$ there is a component of the direct sum of $C$ indexed by $I$ : $T_I:=T_{k_{i_1}\\ldots k_{i_N}}=T_{|I|}$ appearing as a diagonal block of $C$ .",
"Arrange the elements of $\\lbrace 1,\\ldots ,m\\rbrace ^N$ in lexicographic order so that $I_1 = \\lbrace 1,1,\\ldots ,1)$ , $I_2 = (1,1,\\ldots ,2)$ , $\\cdots , I_{m^N} = (m,m,\\ldots ,m)$ , and permute $C$ so that its $j^{th}$ diagonal block is $T_{I_j}$ .",
"Then we can choose the principal submatrix indexed by $J = \\lbrace 1,|I_1|+1,|I_1|+|I_2|+1,\\ldots ,\\sum _{j=1}^{m^N-1} |I_j| +1 \\rbrace .$ Clearly, for any $j\\in J$ , $C^{\\otimes N}_{jj} = 1$ , and for $i\\ne j\\in J$ we have that $C^{\\otimes N}_{ij}$ is not in block $I$ for any $I\\in \\lbrace 1,\\ldots ,m\\rbrace ^N$ .",
"Hence such an entry is equal to 0.",
"This approach yields an alternate proof of Theorem 2 from [17].",
"Lemma 2 Let $I$ be the $2^n\\times 2^n$ identity matrix.",
"Then the channel $\\Phi :M_n(\\rightarrow M_n($ defined by $\\Phi (X) = I\\circ X$ privatises to the unit a $*$ -subalgebra $\\mathcal {A}$ unitarily equivalent to $M_{2^{\\left\\lfloor {n/2}\\right\\rfloor }}($ .",
"Moving beyond this motivating special case, we may utilize the Schur product approach to establish the following general result that applies to all random unitary channels with mutually commuting Kraus operators.",
"Theorem 1 Let $C \\in M_n($ be a correlation matrix whose graph $G_C$ has $m>1$ connected components.",
"Consider the channel $\\Phi (\\rho ) = \\rho \\, \\circ \\, C$ .",
"Then $\\Phi ^{\\otimes N}$ can privatise to the unit a $*$ -subalgebra $\\mathcal {A}$ unitarily equivalent to $M_{2^{\\left\\lfloor {N/2}\\right\\rfloor }}($ .",
"It suffices to prove the result in the case that all the connected components of $G_C$ are complete graphs, as adding more zeros to a correlation matrix only makes it simpler to privatise an algebra.",
"Recall that $\\Phi ^{\\otimes N}(Y) = Y\\circ C^{\\otimes N}$ .",
"By Lemma REF there exists a submatrix $C[J]$ of $C^{\\otimes N}$ of size $m^N$ that is just the $m^N\\times m^N$ identity matrix.",
"Restricted to this submatrix, $\\Phi ^{\\otimes N}$ is just the map $Y \\mapsto I_{m^N}\\circ Y$ .",
"Since $m\\ge 2$ , $m^N\\ge 2^N$ and so restricting further, we have the map $Y\\mapsto I_{2^N}\\circ Y$ .",
"By Lemma REF this map can privatise an algebra $\\widehat{\\mathcal {A}}\\subseteq M_{2^N}$ unitarily equivalent to $M_{2^{\\left\\lfloor {N/2}\\right\\rfloor }}($ .",
"Embedding this algebra in the obvious way first into the $m^N\\times m^N$ matrices by $\\widehat{\\mathcal {A}}\\oplus I_{m^N-2^N}$ and from there in the $m^N\\times m^N$ submatrix indexed by $[J]$ , we obtain a subalgebra $\\mathcal {A}$ unitarily equivalent to $M_{2^{\\left\\lfloor {N/2}\\right\\rfloor }}($ such that $\\Phi ^{\\otimes N}$ restricted to this subalgebra is just Schur product with the identity, and so $\\mathcal {A}$ is privatised by $\\Phi ^{\\otimes N}$ .",
"Corollary 1 For a Schur product channel $\\Phi (\\rho )=\\rho \\,\\circ \\, C$ where $C$ has two connected components, $\\Phi ^{\\otimes 2}$ can privatise a qubit.",
"This supplies another proof that the channel from Example REF privatises a qubit when tensored with itself: the channel is $\\Phi (\\rho ) = I_2\\circ \\rho $ and clearly $G_{I_2}$ has two connected components.",
"Theorem REF and its corollary can be compared to the examples given in [25] of a channel $\\Phi $ which has zero one-shot quantum zero-error capacity but whose tensor product $\\Phi \\otimes \\Phi $ has positive one-shot quantum zero-error capacity.",
"Example 2 As another example, let $\\Phi $ be the two-qubit channel $\\Phi (\\rho ) = \\frac{1}{2} ( \\rho + U \\rho U^* )$ , where $U$ is the unitary CNOT gate; $U| i \\rangle | j \\rangle = | i \\rangle | i\\oplus j \\rangle $ .",
"Consider $U$ in its diagonal matrix form $U = \\operatorname{diag}(1,1,1,-1)$ .",
"Then the corresponding correlation matrix for $\\Phi (\\rho ) = C \\, \\circ \\, \\rho $ is $C = \\begin{bmatrix} 1 & 1 & 1 & 0 \\\\ 1 & 1 & 1 & 0 \\\\ 1 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix}.$ By Theorem REF , we know that $\\Phi ^{\\otimes 2}$ privatises a qubit as $G_C$ is made up of two complete components.",
"Let us observe this directly.",
"It is not hard to see that $\\Phi $ is a conditional expectation onto the algebra $M_3\\oplus \\mathbb {C}$ .",
"Thus, $\\Phi ^{\\otimes 2}(\\rho ) = \\rho \\,\\circ \\,(C\\otimes C)$ , which has correlation matrix permutationally equivalent to $C\\otimes C \\simeq P_9 \\oplus P_3 \\oplus P_3 \\oplus 1$ where $P_k$ is the $k\\times k$ all 1's matrix.",
"Consider the principal submatrix of $C\\otimes C$ supported on the indices 1, 10, 13, 16.",
"For this set of indices, $C_{ii}=1$ obviously, but $C_{ij}=0$ for $i\\ne j$ since each index is from a different block and only indices from within the same block will yield a non-zero entry.",
"Let $\\mathcal {A}$ be the algebra, which is unitarily equivalent to $M_4\\oplus \\mathbb {C}I_{12}$ , supported on the same indices.",
"Then there is a subalgebra of this which is unitarily equivalent to $\\mathrm {Alg}\\lbrace I\\otimes X, Y\\otimes Y, Y\\otimes Z\\rbrace \\oplus \\mathbb {C} I_{12}\\simeq (I_2\\otimes M_2)\\oplus \\mathbb {C} I_{12}$ , which we know is privatised by the map to the diagonal.",
"However, this algebra has non-trivial support only on the submatrix where $C\\otimes C$ is the identity, so $\\Phi $ restricted to this algebra is just the algebra $\\mathrm {Alg}\\lbrace I\\otimes X,Y\\otimes Y, Y\\otimes Z\\rbrace $ Schur producted with the $4\\times 4$ identity; in other words, it is privatised by $\\Phi $ .",
"We conclude this section by noting there are connections between our analysis and some classical graph theoretical concepts.",
"Definition 4 Let $G=(V,E)$ be a graph.",
"Then an $S\\subset V$ is an independent vertex set of $G$ if $(i,j)\\notin E$ for all $i,j\\in S$ .",
"The independence number of the graph $G$ (denoted by $\\alpha (G)$ ) is the cardinality of the largest independent vertex set in $G$ .",
"Proposition 5 Let $C$ be a correlation matrix with graph $G_C$ .",
"Then the largest principal submatrix of $C$ which is an identity matrix is $\\alpha (G_C)$ by $\\alpha (G_C)$ .",
"If $S$ and $T$ are independent vertex sets of the graphs $G$ and $H$ respectively, then $S\\boxtimes T$ is an independent vertex set of $G\\boxtimes H$ .",
"It follows from this observation that the independence number is supermultiplicative with respect to the strong product: $\\alpha (G\\boxtimes H) \\ge \\alpha (G)\\alpha (H)$ .",
"We now introduce the Shannon capacity of a graph, a concept first introduced by Shannon in [24] and then extensively studied by Lovász in [19].",
"Definition 5 Let $G$ be a graph.",
"Then the Shannon capacity of $G$ is denoted as $\\Theta (G)$ and is defined as $\\Theta (G)=\\sup _{n\\in \\mathbb {N}} \\alpha (G^{\\boxtimes k})^\\frac{1}{k}$ .",
"Remark 2 We note that the graph theoretic interpretation of Lemma REF is essentially the fact that any disjoint union of $m$ complete graphs has Shannon capacity $m$ .",
"From the above analysis, graphs with relatively large Shannon capacity may be useful in quantum privacy.",
"We also note the work [11], which makes use of graph theory in quantum information as well, and suggest there are likely connections with quantum privacy worth exploring."
],
[
"Privacy, Operator Systems, and Quasiorthgonality",
"We begin this section by recalling that an operator system $S$ is a subspace of $M_n($ that is $*$ -closed, and contains the identity $I\\in S$ .",
"Further, if $\\Phi :M_n(\\rightarrow M_m($ is a completely positive map with Kraus operators $\\lbrace A_i\\rbrace _{i=1}^p$ , we define the adjoint channel, $\\Phi ^{\\dagger }$ in terms of the (Hilbert-Schmidt) trace inner product: $ \\mathrm {Tr}(X\\Phi (Y)) = \\mathrm {Tr}(\\Phi ^{\\dagger }(X)Y).$ It is straightforward from this definition that $ \\Phi ^{\\dagger }(X) = \\sum _{i=1}^p A_i^*X A_i.$ If $\\Phi $ is trace-preserving, then $\\Phi ^{\\dagger }$ is unital, and $\\mathrm {range}(\\Phi ^{\\dagger })$ is an operator system, $S\\subseteq M_n($ .",
"One more idea we need is the following: we say that a $*$ -subalgebra $\\mathcal {A}\\subseteq M_n($ is quasiorthogonal to an operator system $S\\subseteq M_n($ if $n\\mathrm {Tr}(sa) = \\mathrm {Tr}(s)\\mathrm {Tr}(a)$ for all $s\\in S$ , $a \\in \\mathcal {A}$ .",
"For more details on quasiorthogonality see [17] and the references therein.",
"The following result brings these concepts together.",
"Lemma 3 Let $\\Phi : M_n(\\rightarrow M_m($ be a trace-preserving completely positive map, so that $S = \\mathrm {range}(\\Phi ^{\\dagger })$ is an operator system.",
"If $\\mathcal {A}$ is a $*$ -subalgebra of $M_n($ quasiorthogonal to $S$ , then $\\mathcal {A}$ is privatised by $\\Phi $ .",
"For all $a\\in \\mathcal {A}$ and $x\\in M_n($ we have $ \\mathrm {Tr}(x\\Phi (a)) & = \\mathrm {Tr}(\\Phi ^{\\dagger }(x)a) \\\\& = \\frac{1}{n}\\mathrm {Tr}(\\Phi ^{\\dagger }(x))\\mathrm {Tr}(a) \\\\& = \\mathrm {Tr}(x\\Phi (I)\\frac{\\mathrm {Tr}(a)}{n})$ and so $\\Phi (a) = \\frac{\\mathrm {Tr}(a)}{n}\\Phi (I)$ .",
"Lemma REF is convenient for us, as it will allow us to state a simple and necessary condition on algebras privatised by a given arbitrary unital channel.",
"Proposition 6 Let $\\Phi :M_n(\\rightarrow M_m($ be a unital quantum channel, with $S = \\mathrm {range}(\\Phi ^{\\dagger })$ the operator system that is the range of the adjoint.",
"Let $\\mathcal {A}$ be a $*$ -subalgebra contained in $S$ .",
"If a $*$ -subalgebra $\\mathcal {B}$ is private for $\\Phi $ , necessarily $\\mathcal {A}$ and $\\mathcal {B}$ are quasiorthogonal algebras.",
"Apply Lemma REF and the fact that $S$ contains $\\mathcal {A}$ .",
"For the rest of this section we focus on Schur product channels.",
"Observe that in the case that $\\Phi (X) = X\\circ C$ for some correlation matrix $C$ , it is easy to see that $\\Phi ^{\\dagger }(X) = \\overline{C}\\circ X$ , where $\\overline{C}$ is the matrix whose entries are the complex conjugates of $C$ .",
"Thus, it follows that the range of $\\Phi ^{\\dagger }$ is the operator system spanned by the matrix units $E_{ij}=|i\\rangle \\!\\langle j |$ such that $C_{ij} \\ne 0$ .",
"Definition 6 Let $G$ be a graph with vertex set $V = \\lbrace 1,2,\\ldots , n\\rbrace $ and edge set $E$ .",
"The operator system of $G$ , $S_G$ , is the subspace of $M_n($ given by $S_G = \\mathrm {span}\\lbrace E_{ij}: (i,j)\\in E(G)\\rbrace \\cup \\lbrace E_{ii}: 1\\le i \\le n\\rbrace .$ Let $C$ be an $n\\times n$ correlation matrix.",
"As above, let $G_C$ be the graph on $n$ vertices with an edge $(i,j)$ whenever $C_{ij} \\ne 0$ .",
"Then the operator system $\\mathrm {range} (X\\circ \\overline{C})$ is the operator system $S_{G_C}$ .",
"If $\\Phi (X)=C\\circ X$ , then $\\Phi ^{\\dagger }(X)=\\overline{C}\\circ X$ , so the operator system $S =\\mathrm {range}(\\Phi ^{\\dagger })$ is the same as the operator system $\\mathrm {range}(\\Phi )$ .",
"We can prove a general privacy result for Schur product channels as follows.",
"Lemma 4 If $\\mathcal {A}$ is a $*$ -subalgebra privatised by a Schur product channel $\\Phi :M_n(\\rightarrow M_n($ , then $\\mathcal {A}$ is quasiorthogonal to the diagonal algebra on $n\\times n$ matrices, $\\Delta _n$ .",
"Note first that for a diagonal matrix $D$ , we have $\\Phi (D)=C\\circ D = D$ , and so the full diagonal algebra is contained in the range of $\\Phi $ , and hence inside $S=\\mathrm {range}(\\Phi ^{\\dagger })$ .",
"Given this fact, the result follows as a consequence of Proposition REF .",
"Let $\\mathcal {A}$ be a $*$ -subalgebra of $M_n($ .",
"We recall that a separating vector for $\\mathcal {A}$ is a vector $v\\in \\mathbb {C}^n$ such that $av = 0$ implies $a=0$ for all $a\\in \\mathcal {A}$ .",
"Given the decomposition of $\\mathcal {A}$ up to unitary equivalence, $\\mathcal {A} \\simeq \\bigoplus _{k=1}^m I_{i_k}\\otimes M_{j_k}$ , one can show that $\\mathcal {A}$ admits a separating vector if and only if $i_k \\ge j_k$ for all $1\\le k \\le m$ .",
"Moreover, it is known [22] that a $*$ -subalgebra is quasiorthogonal to $\\Delta _n$ if and only if it admits a separating vector.",
"Combining these facts with Lemma REF allows us to prove the following theorem.",
"Theorem 2 A necessary condition for an algebra $\\mathcal {A}$ to be be privatised by a Schur product channel $\\Phi :M_n(\\rightarrow M_n($ is that $\\mathcal {A}$ admits a separating vector.",
"We can strengthen Theorem REF in the case that we can find more algebras inside the range of $\\Phi $ .",
"In order to show how we can obtain stronger results in the case, we first need to recall some notions from matrix and operator theory.",
"Let $C$ be an $n\\times n$ correlation matrix and $\\Phi (X)=X\\circ C$ .",
"Then $\\Phi ^{\\dagger }(X) = \\overline{C}\\circ X$ is also a unital and trace-preserving quantum channel.",
"Any unital channel has an associated multiplicative domain; the set $\\lbrace X \\in M_n(: \\Phi (X)\\Phi (X^*) = \\Phi (XX^*)\\rbrace ,$ which is necessarily an algebra, and $X$ in the multiplicative domain satisfies $\\Phi (XA)=\\Phi (X)\\Phi (A)$ for all $A\\in M_n($ .",
"The multiplicative domain is a standard structure of interest in the study of completely positive maps, and more recently it has found applications in quantum error correction [6], [14].",
"Now let $\\mathcal {A}$ be the multiplicative domain of $\\Phi ^{\\dagger }$ , then $ \\mathrm {Tr}(XY\\Phi (I)) = \\mathrm {Tr}(\\Phi ^{\\dagger }(XY))= \\mathrm {Tr}(\\Phi ^{\\dagger }(X)\\Phi ^{\\dagger }(Y))= \\mathrm {Tr}((\\Phi \\circ \\Phi ^{\\dagger })(X)Y) $ for all $Y$ , and so $(\\Phi \\circ \\Phi ^{\\dagger })(X)=X\\Phi (I)$ .",
"Thus when $\\Phi $ is unital, $X$ is a fixed point of $\\Phi \\circ \\Phi ^{\\dagger }$ .",
"In general the fixed point set of a channel is an operator system, but in the case of a unital channel more is true, it is known to be an algebra [15].",
"Clearly, the fixed point algebra $\\mathrm {Fix}(\\Phi \\circ \\Phi ^{\\dagger })\\subseteq \\mathrm {range}(\\Phi )$ , and so following Proposition REF , if a $*$ -subalgebra $\\mathcal {A}$ is privatised by $\\Phi $ , then $\\mathcal {A}$ is necessarily quasiorthogonal to $\\mathrm {Fix}(\\Phi \\circ \\Phi ^{\\dagger })$ , and hence also to the multiplicative domain of $\\Phi $ .",
"For a Schur product channel, the multiplicative domain algebra is a subset of the fixed point algebra of the map $X\\mapsto \\overline{C}\\circ (C\\circ X) = (\\overline{C}\\circ C)\\circ X.$ An operator $X$ is such a fixed point so long as $X_{ij}=0$ for all $(i,j)$ such that $|C_{ij}|^2 \\ne 1$ .",
"Notice that Lemma REF can be re-characterized in light of the above as the trivial observation that, for a correlation matrix, $|C_{ii}|^2 =1$ for any correlation matrix.",
"For a correlation matrix $C$ with other entries of modulus 1, we can strengthen the claims of Lemma REF and Theorem REF ; namely, any $*$ -subalgebra $\\mathcal {A}$ privatised by the channel $\\Phi (X) = C\\circ X$ must be quasiorthogonal to any algebra which has non-zero entries at $(i,j)$ only if $|C_{ij}|^2 =1$ ."
],
[
"Unitary Graphs of Correlation Matrices",
"We conclude by returning to a graph-theoretic perspective and considering the following notion.",
"Definition 7 Let $C=(C_{ij})$ be an $n\\times n$ correlation matrix.",
"The unitary graph $UG_C$ of $C$ is the graph whose vertices are labelled by $1,2,\\ldots ,n$ with an edge from $i$ to $j$ if and only if $|C_{ij}|^2 = 1$ .",
"Theorem 3 Let $UG_C$ be the unitary graph of a correlation matrix $C$ .",
"Then any connected component of $UG_C$ having more than 1 vertex is a complete graph.",
"We proceed by induction.",
"The first non-trivial case is a connected component with three vertices.",
"Denote the vertices $i,j,k$ with edges $(i,j)$ , $(j,k)$ .",
"Then the submatrix of $C$ indexed by $i,j,k$ is $ C[i,j,k] = \\begin{pmatrix} 1 & C_{ij} & C_{ik} \\\\ \\overline{C}_{ij} & 1 & C_{jk} \\\\ \\overline{C}_{ik} & \\overline{C}_{jk} & 1 \\end{pmatrix}$ which, as a principle submatrix of a positive semidefinite matrix must be positive itself.",
"Hence $ \\begin{pmatrix} 1 & C_{jk} \\\\ \\overline{C}_{jk} & 1 \\end{pmatrix} - \\begin{bmatrix}\\overline{C}_{ij} \\\\ \\overline{C}_{ik} \\end{bmatrix}\\begin{bmatrix} C_{ij} & C_{ik} \\end{bmatrix} = \\begin{pmatrix} 0 & C_{jk} - \\overline{C}_{ij}C_{ik} \\\\ \\overline{C}_{jk}-C_{ij}\\overline{C}_{ik} & 1-|C_{ik}|^2\\end{pmatrix}\\ge 0$ and hence $|C_{ik}| = |C_{jk}|/|C_{ij}| = 1$ and $(i,k)$ is an edge as well.",
"Now, assume that it is true that for all connected components on $n$ vertices, the unitary graph must be complete.",
"Then for a graph of size $n+1$ , each subgraph of size $n$ must be complete, and we are done.",
"Thus $UG_C$ is a subgraph of $G_C$ consisting of disconnected components $\\lbrace G_i\\rbrace _{i=1}^m$ , each of which is a complete graph on $k_i$ vertices, $\\sum _{i=1}^m k_i = n$ .",
"So the operator system generated by $UG_C$ , $S_{UG_C}$ , is (up to unitary equivalence) $ S_{UG_C} \\cong \\bigoplus _{i=1}^m M_{k_i}(.$ This is, of course, a $*$ -subalgebra, and if $\\mathcal {A}$ is privatised by $\\Phi $ , then necessarily $\\mathcal {A}$ is quasiorthogonal to the algebra $S_{UG_C}$ .",
"In a sense, this gives us a lower bound on the size of algebras that a private algebra must be quasiorthogonal to.",
"We have already seen that any algebra privatised by a Schur product channel must be quasiorthogonal to the diagonal algebra, and hence must contain a separating vector.",
"Our preceding remarks strengthen this: the algebra defined by having non-zero entries wherever $C$ has an entry of modulus 1 is a larger algebra, necessarily containing the diagonal algebra, such that any algebra privatised by the channel must be quasiorthogonal to this algebra.",
"This “lower bound\" algebra is, up to unitary equivalence, simply a direct sum of matrix algebras.",
"As a consequence of von Neumann's double commutant theorem, we have that the $*$ -algebra generated by $S$ is $S^{\\prime \\prime }$ , the commutant of the commutant of $S$ .",
"Given a graph $G$ , the let $S_G^{\\prime \\prime }$ be the algebra generated by the operator system of $G$ , $S_G$ .",
"If we define $G^*$ to be the graph with the same vertex set as $G$ and with an edge between two vertices if and only if they are in the same connected component of $G$ , then $S_G^{\\prime \\prime }=S_{G^*}$ .",
"It is easy to see that if an algebra $\\mathcal {B}$ is quasiorthogonal to $S_G^{\\prime \\prime }$ then $\\mathcal {B}$ is privatised by any Schur product channel $\\Phi (X) = X\\circ C$ for which the graph of $C$ is $G$ .",
"Hence, given a Schur product channel, $\\Phi (X)=X\\circ C$ , we have two algebras $S_{UG_C}$ and $S_{G_C}^{\\prime \\prime }$ , which define necessary and sufficient conditions respectively for an algebra to be privatised by $\\Phi $ .",
"Obviously, if these two algebras are equal, which occurs if and only if $C$ is permutationally similar to a direct sum of rank one correlation matrices, we can completely characterize the algebras privatised by $\\Phi $ .",
"We conclude with an example that serves to illustrate how these necessary and sufficient conditions are, in general, not strong enough to completely characterize the algebras that are privatised by $\\Phi $ .",
"Example 3 Let $\\Phi :M_3(\\rightarrow M_3($ be given by $\\Phi (X) = X\\circ C$ with $C=\\begin{pmatrix} 1 & 0 & \\frac{1}{2} \\\\ 0& 1 & \\frac{1}{2} \\\\ \\frac{1}{2} & \\frac{1}{2} & 1 \\end{pmatrix}.$ The graph of $C$ is $ G_C =\\begin{tikzpicture}[thick, scale=0.8]\\node [shape=circle,draw=black](1) at (1,0) {1};\\node [shape=circle,draw=black](2) at (3,0){2};\\node [shape=circle,draw=black](3) at (5,0){3};(3.4,0) to (4.6,0);(1.4,0) to[out=60,in=135] (4.6,0);\\end{tikzpicture},$ while the unitary graph is $UG_C = \\begin{tikzpicture}[thick, scale=0.8]\\node [shape=circle,draw=black] (1) at (1,0) {1};\\node [shape=circle,draw=black] (2) at (3,0){2};\\node [shape=circle,draw=black] (3) at (5,0){3};\\end{tikzpicture},$ the completely disconnected graph.",
"Hence, $\\mathcal {A}_{G_C} = M_3($ and $S_{UG_C} = \\Delta _3$ .",
"However, being quasiorthogonal to $M_3($ and $\\Delta _3$ are not necessary or sufficient respectively, for an algebra to be privatised by $\\Phi $ ; for instance, the algebra $\\mathcal {A} = \\left\\lbrace \\begin{pmatrix} a & b & 0 \\\\ b & a & 0 \\\\ 0 & 0 & a \\end{pmatrix}: \\ a,b\\in \\rbrace \\right.$ is privatised by $\\Phi $ , despite not being quasiorthogonal to $M_3($ , and the circulant algebra $\\mathcal {C} = \\left\\lbrace \\begin{pmatrix} a & b & c \\\\ c & a & b \\\\ b & c & a \\end{pmatrix}: \\ a,b,c\\in \\rbrace \\right.$ is quasiorthogonal to $\\Delta _3$ , but is not privatised by $\\Phi $ .",
"Remark 3 We conclude with a note on our use of graph theory here and related work.",
"A number of papers have commented on the fact that operator systems behave in many ways like 'non-commutative' or 'quantum' versions of graphs.",
"In [11] it was shown that certain properties of an operator system can be regarded as quantum analogues of the independence number and Lovasz number of a graph, while in [26] quantum cliques have been considered, where an analogue of Ramsey theory for operator systems was established.",
"Other authors have focused on defining quantum chromatic numbers, and other graph parameters [5], [21].",
"The study of these `quantum' graph parameters is an active and interesting area of research in pure mathematics and quantum information.",
"What the present analysis aims to do is to understand how certain operator systems associated to a channel control the ability of that channel to privatise information.",
"Schur product channels are the channels whose operator systems are the simplest as they are isomorphic to graphs, and so studying privatisation by means of Schur product channels is the simplest place to start such an analysis.",
"Our work shows that a certain graph parameter, the independence number, controls how copies of a channel can privatise information; we expect that some quantum version of the independence number should play an analogous role for channels whose ranges are operator systems more complicated than those arising directly from graphs.",
"One final point of note is that not one, but two graphs are important for understanding how a Schur product channel privatises a $\\ast $ -subalgebra: the graph whose associated operator system is the range of the channel, and a graph that encodes the multiplicative domain of the channel.",
"This seems like a hint that in order to understand how operator systems of channels allow privatisation, the multiplicative domain will be an important object.",
"Acknowledgements.",
"J.L.",
"was supported by an AIMS-University of Guelph Postdoctoral Fellowship.",
"D.W.K.",
"was supported by NSERC and a University Research Chair at Guelph.",
"R.P.",
"was supported by NSERC.",
"We are grateful to the referees for helpful comments."
]
] | 1709.01752 |
[
[
"Information Theory and the Length Distribution of all Discrete Systems"
],
[
"Abstract We begin with the extraordinary observation that the length distribution of 80 million proteins in UniProt, the Universal Protein Resource, measured in amino acids, is qualitatively identical to the length distribution of large collections of computer functions measured in programming language tokens, at all scales.",
"That two such disparate discrete systems share important structural properties suggests that yet other apparently unrelated discrete systems might share the same properties, and certainly invites an explanation.",
"We demonstrate that this is inevitable for all discrete systems of components built from tokens or symbols.",
"Departing from existing work by embedding the Conservation of Hartley-Shannon information (CoHSI) in a classical statistical mechanics framework, we identify two kinds of discrete system, heterogeneous and homogeneous.",
"Heterogeneous systems contain components built from a unique alphabet of tokens and yield an implicit CoHSI distribution with a sharp unimodal peak asymptoting to a power-law.",
"Homogeneous systems contain components each built from just one kind of token unique to that component and yield a CoHSI distribution corresponding to Zipf's law.",
"This theory is applied to heterogeneous systems, (proteome, computer software, music); homogeneous systems (language texts, abundance of the elements); and to systems in which both heterogeneous and homogeneous behaviour co-exist (word frequencies and word length frequencies in language texts).",
"In each case, the predictions of the theory are tested and supported to high levels of statistical significance.",
"We also show that in the same heterogeneous system, different but consistent alphabets must be related by a power-law.",
"We demonstrate this on a large body of music by excluding and including note duration in the definition of the unique alphabet of notes."
],
[
"Abstract",
"We begin with the extraordinary observation that the length distribution of 80 million proteins in UniProt, the Universal Protein Resource, measured in amino acids, is qualitatively identical to the length distribution of large collections of computer functions measured in programming language tokens, at all scales.",
"That two such disparate discrete systems share important structural properties suggests that yet other apparently unrelated discrete systems might share the same properties, and certainly invites an explanation.",
"We demonstrate that this is inevitable for all discrete systems of components built from tokens or symbols.",
"Departing from existing work by embedding the Conservation of Hartley-Shannon information (CoHSI) in a classical statistical mechanics framework, we identify two kinds of discrete system, heterogeneous and homogeneous.",
"Heterogeneous systems contain components built from a unique alphabet of tokens and yield an implicit CoHSI distribution with a sharp unimodal peak asymptoting to a power-law.",
"Homogeneous systems contain components each built from just one kind of token unique to that component and yield a CoHSI distribution corresponding to Zipf's law.",
"This theory is applied to heterogeneous systems, (proteome, computer software, music); homogeneous systems (language texts, abundance of the elements); and to systems in which both heterogeneous and homogeneous behaviour co-exist (word frequencies and word length frequencies in language texts).",
"In each case, the predictions of the theory are tested and supported to high levels of statistical significance.",
"We also show that in the same heterogeneous system, different but consistent alphabets must be related by a power-law.",
"We demonstrate this on a large body of music by excluding and including note duration in the definition of the unique alphabet of notes.",
"This paper adheres to the transparency and reproducibility principles espoused by [44], [72], [9], [24], [14], [27] and includes references to all methods and source code necessary to reproduce the results presented.",
"These are referred to here as the reproducibility deliverables and are available in several packageshttp://leshatton.org/index_RE.html, one for each published paper [23], [25] and another covering the work presented here.",
"All data used are openly available, thanks to the efforts of scientists everywhere.",
"Each reproducibility deliverable allows all results, tables and diagrams to be reproduced individually for that paper, as well as performing verification checks on machine environment, availability of essential open source packages, quality of arithmetic and regression testing of the outputs [26].",
"Note that these packages are designed to run on Linux machines for no other reason than to guarantee the absence of any closed source and therefore potentially opaque contributions to these results.",
"We start with an interesting observation.",
"If we measure in large populations, how often proteins of different lengths occur (measured in amino acids), and how often computer functions of different lengths occur (measured in programming tokens), we find the following frequency distributions, Figs.",
"REF , REF .",
"Figure: The frequency distributions of component lengths in two radically different discrete systems - the known proteome as represented by version 17-03 of the TrEMBL distribution (A), and a large collection of computer programs (B) .",
"In each case the y-axis is the frequency of occurrence and the x-axis the length.We observe the following:- Fig.",
"REF is derived from the European Protein Database TrEMBL version 17-03 (March 2017), [64].",
"It is a very large and rapidly growing dataset currently containing around 80 million proteins built from almost 27 billion amino acids.",
"Fig.",
"REF is derived from an analysis of 80 million lines of computer programs written in the programming language C, corresponding to around 500 million programming tokens, around 99% of which is open source, downloaded from various online archives, [23].",
"The two systems arose over very different timescales - proteins first appeared perhaps as long ago as 4 billion years, and have been evolving since under natural selection, whilst the C software is definitely less than 40 years old (the C language did not exist much before this).",
"They are of very different sizes.",
"The protein data are around 50 times larger than the software data.",
"They arose through very different processes; proteins are the result of natural selection - nature's “Blind Watchmaker” [13], whilst computer programs are the result of deliberate human intellectual endeavour.",
"Protein data are inherently less accurate than software data [4].",
"Protein data sequencinghttps://en.wikipedia.org/wiki/Protein_sequencing, accessed 03-Jun-2017 is subject to experimental error whereas software data sequencing (or tokenization) is precisely defined by programming language standards (ISO/IEC 9899:2011 in the case of C [28]) and is therefore repeatable.",
"The frequency distributions of Figs.",
"REF and REF are quite extraordinarily alike.",
"Both have a sharp unimodal peak with almost linear slopes away from the peaks and from the peak onwards towards longer components, both obey an astonishingly accurate power-law as we shall see."
],
[
"So, why are they so alike ?",
"Note that it is not simply a serendipitous choice of datasets, because we will see this pattern appearing again and again at different levels of aggregation and in entirely different systems.",
"Instead, as we will show here, this actually arises from the operation of a conservation principle for discrete systems - Conservation of Hartley-Shannon Information (referred to generically as CoHSI here to distinguish from other uses of the word “information”) - acting at a deeper level than either natural selection or human volition.",
"Introduction Such uncanny similarity in very disparate systems strongly suggests the action of an external principle independent of any particular system, so to explore the concepts, let us first consider some examples of discrete systems.",
"Here a discrete system is considered to be a set of components, each of which is built from a unique alphabet of discrete choices or tokens.",
"Table REF illustrates this nomenclature and its equivalents in various kinds of system.",
"Table: Comparable entities in discrete systems considered in this paper.The first thing to notice is that this seems a very coarse taxonomy.",
"In the case of proteins, there is no mention of domain of life or species or any other kind of aggregation.",
"Similarly with computer programs, we do not include the language in which they were written or the application area.",
"The reason for this as will be seen later is that these considerations are irrelevant.",
"We will expand on each system later as we apply the theory and discuss its ramifications for each, but the most important concept to grasp now is that there are two measures, total length and unique alphabet, which will turn out to be fundamental across all such systems.",
"To illustrate, consider two hypothetical components consisting of the strings of letters shown as rows in Table REF .",
"Table: Two simple strings of letters.The total length in letters of the first string in Table REF is 25.",
"This is made up of six occurrences of A, 6 of B, 1 of C, 1 of D, 3 of F, 2 of G and 6 of Q.",
"We therefore define the unique alphabet of this component to be of size 7, corresponding to ABCDFGQ.",
"In other words, the string is built up from one or more occurrences of each and every letter in its unique alphabet.",
"The second string is made up of 20 characters consisting of 2 of X, 4 of Y, 4 of Z, 4 of T, 2 of W, 2 of I and 2 of S. Its unique alphabet is therefore XYZTWIS.",
"In other words, these two strings have different lengths but they have the same size unique alphabet.",
"As we will see by considering the information content shortly, this property will turn out to be fundamental - the actual letters making up the unique alphabet will turn out to be irrelevant and the two strings inextricably linked in an information theoretic sense.",
"Methodology The methodology we use combines two disparate but long-established methodologies - Statistical Mechanics and Information Theory in a novel way using the simplest possible definition of Information originally defined by Hartley [20].",
"We will show that this alone is sufficient to predict all the observed features of Figs.",
"REF and REF and why indeed they are so similar.",
"Statistical Mechanics can be used to predict component distributions of general systems made from discrete tokens subject to restrictions known as constraints.",
"Its classical origins can be found in the Kinetic Theory of Gases [59] (p.217-) wherein constraints are applied by fixing the total number of particles and the total energy [17].",
"However, the methodology is very general and can equally be used with different constraints on collections of proteins (made from amino acids), software (made from programming language tokens) and, as we shall see, simple boxes containing coloured beads.",
"Hartley-Shannon Information theory is the result of the pioneering works of Ralph Hartley [20] as developed later by Claude Shannon [54], [55].",
"It forms the backbone of modern digital communication theory and is also astonishingly versatile.",
"The Hartley-Shannon Information Content of a component, in the sense we use here is simply defined to be the natural logarithm of the total number of distinct ways of arranging the tokens of that component, without any regard for what those tokens actually mean.",
"The motivation behind the choice of this form of information is that Figs.",
"REF and REF derive from systems with little if anything in common, but Hartley's definition of information is token-agnostic; in other words the meaning of the tokens is irrelevant.",
"Furthermore, its use favours the ergodic nature of classical Statistical Mechanics with token choice equally likely.",
"Combining Statistical Mechanics and some form of Information Theory is not new.",
"For example, building on the maximum entropy framework of [30] rooted in probability theory, Frank demonstrates that by combining Shannon Information [54] in a maximum entropy context, the common patterns of nature - Gaussian, exponential, power-law - as predicted by neutral generative processes, naturally emerge [15].",
"Here a neutral generative process assumes that each microscopic process follows random stochastic fluctuations.",
"Frank's use of information can be interpreted as additional knowledge about a system constraining the possible patterns which might result.",
"For example, amongst other things, he demonstrates that simply from an assumption about the measurement scale and knowledge about the geometric mean, a power-law arises.",
"Frank [15] also stresses the need to distinguish between the generation of patterns by purely random or neutral process on the one hand, and the generation of patterns by aggregation of non-neutral processes in which non-neutral fluctuations cancel in the aggregate.",
"Power-laws Power-laws (a.k.a the Pareto distribution) are ubiquitous in nature and are emphatically present in all of the datasets analysed in this paper.",
"In essence they have a probability distribution which depends on a power $b > 1$ of the independent variable.",
"$p(x) \\sim x^{-b}$ It is important to note that there are numerous known processes which lead to power-laws [42] and indeed the literature abounds with studies, from the original empirical work of Zipf [73], and the earliest generative models such as preferential attachment [57] onwards.",
"It is also important to note that other statistical distributions notably lognormal frequently occur in natural phenomena [38].",
"In earlier work [22], [23], [25] using the original and arguably the most parsimonious definition of information, [20], embedded as a constraint directly within the classical Statistical Mechanics framework, we demonstrated with compelling support from measurement, that for large components, this alone was enough to generate the extraordinarily precise power-laws observed not only in the length distribution of proteins and software, but also in the distributions of the alphabet of unique tokens.",
"However given the ubiquity of power-laws (and indeed the reasons for this [15]), perhaps the most compelling reason for accepting the power-law generation inherent in CoHSI as opposed to other generative mechanisms, is to realise that conservation principles cannot be applied selectively.",
"They are inherently global and either apply everywhere or nowhere and therefore our use of CoHSI must explain satisfactorily all of the observed properties of the length distributions which appear as Figs.",
"REF , REF , including the sharply unimodal behaviour for smaller values of the independent variable.",
"Our novel contributions to the existing body of work are:- We show that the token-agnostic and scale-independent CoHSI does indeed predict all the qualitative features of the distributions of Figs.",
"REF and REF with no other assumptions apart from the conventional use of Stirling's theorem to approximate factorials (Appendix A p. REF ).",
"This is particularly significant because distributions of the nature of Figs.",
"REF and REF are often treated by combining two separate distributions, such as lognormal with a power-law tail [39].",
"As we show, a single implicit distribution which naturally follows from CoHSI is sufficient, thereby emphasizing the parsimony of this approach, CoHSI naturally leads to an alternative proof of Zipf's law (Appendix A p. REF ), as we might expect for this Conservation principle, We enlarge on the results originally derived asymptotically, [25] that average component lengths (protein, software function ...) are highly conserved across aggregations.",
"We also point out why very long components naturally must appear quite frequently without any obvious domain-based reason, (Appendix B p. REF ), We show that the asymptotic duality first reported in [25] between length distribution and alphabet size distribution, naturally implies that different but consistent alphabets for the same system must also be related by power-law, (Appendix C p. REF ).",
"It also follows that the maximum size of a unique alphabet is intimately related to the total number of components through the slope of the corresponding power-law, We give experimental confirmation to high levels of significance in multiple disparate datasets at different levels of aggregation for these predictions including systems which contain both heterogeneous and homogeneous behaviour.",
"We stress we are not data-fitting here, and we are not explicitly applying constraints on knowledge of types of mean or variance.",
"Instead, both the sharp unimodal peak and the very precise power-law tail of Figs.",
"REF and REF naturally emerge from the single Conservation principle, just as the Maxwell-Boltzmann distribution naturally emerges from the Conservation of Energy in Kinetic Theory [59].",
"Why conserve information ?",
"To understand this seemingly ad hoc assumption, we must delve into ergodicity and consider exactly what happens in classical statistical mechanics when we apply the constraints of total size and total energy to find the most likely distribution of particles amongst energy levels.",
"Perhaps the most important thing to realise about the statistical mechanics methodology is that it is simply a mathematical technique.",
"The fact that it is energy (a physical quantity) which is being conserved along with the total number of contributing particles in kinetic theory is irrelevant - conventionally, anything additive can be conserved, however abstract.",
"The real world only intrudes into the statistical mechanics of Kinetic Theory via Clausius' entropic version of the Second Law of Thermodynamics.",
"Without this, statistical mechanics simply answers the mathematical question of the most likely distribution of particles or tokens when their total number and their total payload (in this case energy) is conserved.",
"In other words, amongst all the possible systems with that number of particles and that total payload (the ergodic ensemble), then presented with one of them, it is most likely to follow the distribution predicted by statistical mechanics for the ensemble.",
"It doesn't have to, but it is overwhelmingly likely that it does.",
"In kinetic theory, the payload happens to be a physical additive quantity, the energy, and the corresponding distribution of particles is then the Maxwell-Boltzmann distribution, but to say something about a system, statistical mechanics does not need to be rooted in tangibly defined entities in the physical world.",
"We simply have to interpret the result appropriately.",
"Now it so happens that Hartley-Shannon information content, like energy, is also additive for independent sub-systems.",
"The total energy $E$ of two sub-systems with individual energies $E_{1}$ and $E_{2}$ is $E = E_{1} + E_{2}$ .",
"Similarly, by virtue of its logarithmic definition, the total Hartley-Shannon information content $I$ of two sub-systems with individual information content $I_{1}$ and $I_{2}$ is $I = I_{1} + I_{2}$ .",
"The difference between the two is that energy is a physical quantity, whereas Hartley-Shannon information content is just the $log$ of the total number of ways of arranging something [8].",
"Mathematically it resembles entropy but we should be hesitant about reading too much into this [33], p. 144.",
"However, we may still use the formalism of statistical mechanics, which we do here.",
"It is in this sense that we assert that Conservation of Hartley-Shannon Information underlies the length distribution of discrete systems whatever their provenance.",
"It is a natural consequence of statistical mechanics that if we are presented with a system with a total number of tokens and a total information payload, then it is overwhelmingly likely to follow a certain size distribution as described in what follows.",
"The scale-independence of the results follows from the fact that, given any system, it is the properties of the ergodic system of the same parameters which defines the most likely distributions to occur in any one of its constituents.",
"This paper contains many examples of real systems at all levels of aggregation where precisely this size distribution is found, just as we expect from the theory.",
"Statistical Mechanics Statistical Mechanics connects the minutiae of systems of large numbers of small particles to macroscopic properties of those systems [17].",
"Like Hartley-Shannon Information, it is quite astonishingly versatile and arose originally in the Kinetic Theory of Gases, leading eventually in the hands of James Clerk Maxwell and later Ludwig Boltzmann in the 19th century, to the statistical distribution of velocities and on to the concept of Entropyhttps://en.wikipedia.org/wiki/Kinetic_theory_of_gases, accessed 25-May-2017..",
"The methodology of Statistical Mechanics leads naturally to links with probability distributions and energy in the case of gases.",
"Here we use the same methodology but by embedding CoHSI as a constraint in Statistical Mechanics rather than Conservation of Energy, we demonstrate that this links Hartley-Shannon Information directly to probability distributions of component length and unique alphabet size in discrete systems.",
"We distinguish between two fundamental types of system which lead naturally to two different definitions of information.",
"Both contain components made from discrete tokens as described above but with one fundamental difference.",
"Heterogeneous We define heterogeneous systems here as systems wherein a component has more than one kind of distinct token.",
"This would include systems as disparate as the proteome, software and digital representations of music.",
"Appendix A p. REF contains a detailed development of these systems.",
"Homogeneous We define homogeneous systems here as systems wherein a component has only one kind of distinct token and each distinct token is unique to one component.",
"This would include textual documents and word counts as well as the distribution of elements in the universe.",
"In such systems, a heterogeneous definition of information would be degenerate and a different definition is necessary.",
"Appendix A p. REF contains the detailed development for this kind of system leading directly to an alternative proof of Zipf's law.",
"However, this is irrelevant as far as statistical mechanics goes because for a given definition of Hartley-Shannon Information, the methodology simply tells us the most likely, or canonical distribution for ergodic systems with the same fixed size and fixed information content, howsoever defined.",
"For heterogeneous systems, we will refer to the resulting distributions as the heterogeneous CoHSI distribution.",
"The corresponding distribution for homogeneous systems is simply Zipf's law.",
"The heterogeneous CoHSI distribution The theory described in Appendix A p. REF predicts that the length distribution of a heterogeneous discrete system such as the proteome or software systems, at all scales with total number of tokens $T$ and total Hartley-Shannon Information $I$ is the solution $(t_{i},a_{i})$ of the implicit pdf corresponding to $\\log t_{i} = -\\alpha -\\beta ( \\frac{d}{dt_{i}} \\log N(t_{i}, a_{i}; a_{i} ) ), $ with $T = \\sum _{i=1}^{M} t_{i}$ and $I = \\sum _{i=1}^{M} I_{i}$ where $t_{i}, a_{i}, I_{i}$ are the length in tokens, the size of unique alphabet of tokens and the Hartley-Shannon Information content, respectively, of the $i^{th}$ component of a system containing $M$ components in all.",
"$\\alpha $ and $\\beta $ are Lagrange undetermined multipliers.",
"$N(t_{i}, a_{i}; a_{i})$ is the total number of ways of choosing $t_{i}$ tokens at random, choosing from a replaceable unique set of tokens $a_{i}$ .",
"We write $N(t_{i}, a_{i}; a_{i})$ in this special form to remind us of the recursive nature of its construction.",
"Here, $t_{i}$ is the independent variable and $a_{i}$ plays a dual role acting also as the scaled frequency of occurrence.",
"In addition, for components which are much longer than their unique alphabet, $t_{i} \\gg a_{i}$ , the full solution (REF ) tends to the asymptotic pdf (probability distribution function) [23] given by $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{a_{i}^{-\\beta }}{\\sum _{i=1}^{M} a_{i}^{-\\beta }}, $ which has an algebraic dual pdf [25] given by $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }}, $ where $A = \\sum _{i=1}^{M} a_{i}$ (REF ) and (REF ) show that the tails of both unique alphabet size distributions and length distributions respectively of the M components will be asymptotically power-law as emphatically confirmed in [25].",
"A typical solution of (REF ) is shown as Fig.",
"REF .",
"Figure: A typical solution of () shown as a pdf.",
"Both the sharp unimodal peak and power-law tail can be seen clearly.Before applying this theory predictively to various systems so that we may test it, we make two comments.",
"The Lagrange multipliers $\\alpha , \\beta $ are undetermined by the methodology of statistical mechanics.",
"$\\alpha $ parameterises the total size of the system and therefore emerges naturally as a normalisation condition so that a pdf results.",
"$\\beta $ is more interesting.",
"It parameterises the total payload.",
"The payload in our theory is Hartley-Shannon information which as described in Appendix A p. REF , depends on the size of the alphabet we use to categorise a discrete system.",
"Small alphabets correspond to large $\\beta $ and vice versa.",
"The implication of this indeterminism is that the range of values of $\\beta $ which emerge via information theory can be much wider than those tied to physical systems, (which are mostly in the range 1.5-4).",
"(In the Maxwell-Boltzmann distribution where the payload is energy, Appendix A p. REF , it is fixed by being closely linked by Boltzmann's constant to the temperature, through the Second Law of Thermodynamics.)",
"Dual regime behaviour is often identified and modelled as lognormal transitioning to power-law [39], [37].",
"We stress here that no such juxtaposition of distinct pdfs is necessary with the theory presented here.",
"Instead the sharp unimodal peak and the power-law regime of Figs.",
"REF , REF naturally emerge according to the implicit solution of (REF ) as $t_{i} \\rightarrow 1$ from large values as can be seen in Fig.",
"REF .",
"This is a direct consequence of CoHSI in an ergodic system as described in Appendix A p. REF .",
"The observed shape of the sharply unimodal regime of Fig.",
"REF simply lends itself to a lognormal fit.",
"We also note that the unusual implicit behaviour inherent in (REF ) is also a feature of modified entropy definitions such as Tsallis entropy [67], [66] constructed to account for non-additive entropy.",
"Tsallis entropy is a modification of Shannon entropy with an additional parameter.",
"In our development, the implicit behaviour emerges naturally from the one assumption of CoHSI.",
"We now apply these conclusions predictively to various systems.",
"Results Proteins Proteins are constructed as strings of amino acids corresponding to the heterogeneous model we describe here, Appendix A p. REF .",
"They are represented in exactly the same way as the strings of letters in Table REF but the unique alphabet from which the letters are chosen is the 22 unmodified amino acids which are coded directly from DNA [60], [16], supplemented with modified versions produced by a process known as Post-Translational Modification, of which there are already thousands known [31], [46], [5].",
"Table REF shows two small proteins, one from an Archaean and the other from a virus [64] along with their sequences in their single letter abbreviations for compactness [25] Table: Sequences of two small proteins.Biologically, these two proteins differ significantly.",
"They have different lengths, (32 and 44 amino acids respectively); are built using different amino acids; and they have very distinct structures and functions.",
"FLA1_METHU is built from the unique amino acid alphabet FSGLEAIVLYMGT (13) and VE5_PAPVR is built from the unique amino acid alphabet MNHPGLFTAVQWDCRI (16).",
"Following our earlier argument about unique alphabets, it does not matter if an amino acid is present once or more often in the sequence.",
"If it is present at all, then it contributes a count of 1 to the unique amino acid count.",
"While these two numbers are clearly independent of any physicochemical properties of the amino acids, they are fundamental in determining the length distribution of any aggregation of proteins.",
"The protein sequences are collected in public databases from which they can be downloaded and analysed [64].",
"There are currently more than 80 million proteins in TrEMBL version 17-03 built from almost 27 billion amino acids, (the most recent version analysed before writing this paper).",
"The proteins vary in length from just four amino acids to over 36,000 amino acids but their average length is only around 300 amino acids.",
"The reason for the existence of such long proteins is directly predicted by the development of theory which follows later in this paper, Appendix B, p. REF .",
"The length of a protein is of course one of the factors which determines its folding properties and therefore its functionality [34], [29], [32].",
"Power-law tails of alphabet and length As originally shown in [25], the tails of the alphabet and length distributions are both power-law to a high degree of precision.",
"The protein alphabets are analysed in Appendix C p. REF .",
"The ccdf (complementary cumulative distribution function) of the length distribution corresponding to Fig.",
"REF is shown as Fig REF .",
"Figure: The data of Fig.",
"shown as a log-log\\log -\\log ccdf.",
"The linearity in the tail of Fig.",
"REF provides striking confirmation of equation (REF ).",
"The R lm() function reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $300.0-30000.0$ , with an adjusted R-squared value of $0.9942$ .",
"The slope is $-3.13 \\pm 0.20$ .",
"Aggregations by domain and species As stated earlier, the canonical shape of Figs.",
"REF , REF occurs at each level of aggregation in these systems.",
"This we now show for both the domains of life and also down to individual species.",
"In contrast to TrEMBL, the SwissProt database [63] provides a smaller but well annotated set of data suitable for the extraction and analysis of data from taxa at diverse levels of aggregation, from the highest taxonomic classification shown (the three domains of life) down to individual species.",
"In Figs.",
"REF -REF : are Archaea (Figs REF : 19,063 species), Bacteria Figs (REF : 329,526 species) and Eukarya Figs REF : 177,020 species).",
"Included for comparison are the viruses Figs (REF : 16,423 species).",
"In every case, the characteristic qualitative signature of Fig.",
"REF is evident in the domain of life (with variations inevitably depending on the sample size), and even (in the case of viruses), a dataset outside the domains of life.",
"The pdfs are scaled separately to show the qualitative similarity whilst Fig.",
"REF shows the matching absolutely-scaled ccdfs and the emergence of the power-law tail in each collection.",
"Figure: The frequency distributions of protein lengths in the three domains of life, (A): Archaea, (B): Bacteria and (C): Eukarya, along with (D): Viruses.Figure: The three domains of life and viruses shown as a log-log\\log -\\log ccdf.",
"The significantly shallower slope in Viruses is notable.The only remaining approximation of CoHSI is that the number of components (i.e.",
"proteins in this case) be reasonably large, so a critical test case is the analysis of individual species, where the protein databases allow us to analyze small sets of proteins naturally defined by species.",
"To demonstrate the resilience of CoHSI, we consider species with very different numbers of unique proteins.",
"Figs.",
"REF -REF show the length distributions of proteins in (Fig.",
"REF ) humans (126,468 proteins); (Fig.",
"REF ) maize (85,311 proteins); (Fig.",
"REF ) fruit fly (18,966 proteins); and (Fig.",
"REF ) Haloarcula marismortui (3,892 proteins) respectively.",
"Even in the smallest of these datasets, H. marismortui, a halophilic red Archaeon found in the extreme environment of the Dead Sea, the canonical shape of Fig.",
"REF is apparent.",
"The pdfs are again scaled separately to show the qualitative similarity whilst the corresponding ccdfs are shown absolutely scaled in Fig.",
"REF .",
"Figure: The frequency distributions of protein lengths in four species, (A): Human, (B): Maize and (C): Fruit fly, along with (D): Haloarcula marismortui.Figure: Four species shown together as a log-log\\log -\\log ccdf.",
"Computer programs Computer programs are an invention of the human mind following the ground-breaking work of Alan Turing.",
"In the 50 or so years since they first appeared, many programming languages have arisen, from which computer programs of almost limitless functionality are built.",
"The individual bases or alphabet of a programming language are called tokens and may take two forms; the fixed tokens of the language as provided by the language designers, and the variable tokens.",
"Fixed tokens include (in the languages C and C++ for example) keywords such as if, else, while, {, }.",
"These can not be changed, the programmer can only choose to use them or not.",
"Variable tokens, with some small lexical restrictions (such as the common requirement for identifiers to begin with a letter), can be arbitrarily invented by the programmer whilst constructing their program.",
"These might be names such as numberOfCandidateCollisions or lengthOfGene or constants such as 3.14159265.",
"There are many programming languages but all obey the same principles and every form of software system evolves from such tokens.",
"They are therefore another example of the heterogeneous model we describe here, Appendix A p. REF .",
"It should be noted that classifying programs in terms of fixed and variable tokens is not new and appeared at least as early as 1977 in the influential work of Halstead who called them operators and operands, [19].",
"He developed his work to define various dependent concepts such as software volume and effort and tested them against programs of the time.",
"This was further elaborated by Shooman [56].",
"A different approach is used here which borrows from the methods of variational calculus.",
"Computer programs are often huge.",
"The software deployed in the search for the recently discovered Higg's boson comprises around 4 million lines of code [50].",
"At an average of around six tokens per line of code, this corresponds to some 20 million tokens, although this is still less than 1% of the human genome in which the tokens are the four bases adenine, cytosine, guanine and thymine.",
"The largest systems in use today appear to be around 100 million lines of source code [40], corresponding to perhaps 15% of the number of tokens of the human genome.",
"The (largely) open systems used to test the model described here total almost 100 million lines, (specifically 98,476,765 lines), totalling some 600 million tokens.",
"(If around 6 tokens per line seems a little low, it should be recalled that lines of code include comment lines here in line with common practice, whilst token counts do not.)",
"As an example of the nomenclature used here, consider the following simple sorting algorithm written in C, for example [52].",
"void bubble( int a[], int N ) { int i, j, t; for( i = N; i >= 1; i--) { for( j = 2; j <= i; j++) { if ( a[j-1] > a[j] ) { t = a[j-1];a[j-1] = a[j];a[j] = t; } } } } This algorithm contains 94 tokens in all based on 18 of the fixed tokens and 8 of the variable tokens of ISO C, so the size of the unique alphabet for this component is $18+8 = 26$ .",
"Note that extracting the tokens of programming languages to assemble these measures requires the development of compiler front-end tools [1], [51].",
"These are included in the reproducibility materials, notably associated with [23].",
"Power-law tails of alphabet and length We have from (REF ) and (REF ) that power-laws in both the unique alphabet distributions and length distributions are overwhelmingly likely to appear in the tails of the distributions.",
"Figs.",
"REF and REF show the $\\log -\\log $ ccdf plots for the unique alphabet and length distributions respectively of 100 million lines of source code in seven different programming languages [23].",
"Figure: The unique alphabet a i a_{i} (A) and length distributions t i t_{i} (B) of 100 million lines of source code in 7 different programming languages shown as ccdfs.",
"The linearity in each tail is striking confirmation of (REF ) and (REF ).",
"For Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $80.0-3500.0$ , with an adjusted R-squared value of $0.9975$ .",
"The slope is $-2.15 \\pm 0.08$ .",
"For Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $200.0-68000.0$ , with an adjusted R-squared value of $0.9995$ .",
"The slope is $-1.47 \\pm 0.03$ .",
"We note in passing that (REF ) and (REF ) suggest that the slopes are reciprocals of each other, ($\\beta $ and $1/\\beta $ ).",
"They are clearly not so here but this raises an interesting question concerning the choice of alphabets which is explained in the Appendix C p. REF .",
"Aggregations by language and package Although the collections of software available for analysis are many fewer than for proteins, we can still identify, in software, collections equivalent to the domains of life on the basis of software written in different programming languages.",
"Figs.",
"REF -REF illustrate the length distributions of collections of components (software functions) in four programming languages (Fig.",
"REF C++ 22,628 components; Fig.",
"REF Java 32,552 components; Fig.",
"REF Fortran 14,028 components and Fig.",
"REF Ada 12,680 components).",
"Despite the disparity in their sizes, each of these collections again shows striking similarity to the canonical form of length distributions of Figs.",
"REF and REF , as predicted by (REF ) and manifest in proteins, as we have already seen.",
"In each case, the values in the x-axis scale are the same whilst the y-axis is scaled according to the size of the packages to make the peaks of approximately the same vertical extent.",
"Figure: The length distributions of functions in large collections of software in four programming languages, (A): C++, (B): Java (C): Fortran and (D): Ada.Again, there are considerable differences in scale compared with that of proteins, but we can still identify in software collections equivalent to species on the basis of software written for individual applications.",
"Figs.",
"REF -REF illustrates the length distributions of collections of software functions in four applications of very different sizes, in four different programming languages (Fig.",
"REF , The Gimp image manipulation program (ISO C) 18,693 components; Fig.",
"REF , The KDE desktop libraries (C++) 16,241 components; Fig.",
"REF , The Eclipse interactive Development Environment (Java) 9,588 components and Fig.",
"REF , The gcc Ada compiler (Ada) 3,765 components).",
"Once again, despite the disparity in their sizes, each of these collections again shows striking similarity to the canonical form of length distributions exhibited by Figs.",
"REF , REF .",
"In this case, the x-axis scale is the same whilst the y-axis is again scaled according to the size of the packages to normalise the size of the peaks approximately.",
"Figure: The length distributions of functions in four packages each in different programming languages, (A): Gimp (ISO C), (B): kdelibs (C++), (C): Java (Eclipse) and (D): Ada (GCC).",
"Music CoHSI also predicts the length distributions of musical compositions.",
"Much of the theory and discussion is deferred to Appendix C p. REF .",
"We will simply point out here that modern digital formats for musical annotation such as MusicXMLhttps://en.wikipedia.org/wiki/MusicXML, accessed 07-Jul-2017 allow us to apply the heterogeneous theory described in Appendix A p. REF , to yet another distinct discrete system where, in this case, the components are pieces of music and the unique alphabet comprises of notes as shown in Table REF .",
"Extracting the appropriate data is fortunately relatively simple compared with the daunting task of extracting possibly post-translationally modified amino acids in proteins or programming language tokens in computer programs, as the following XML snippethttps://hymnary.org/media/fetch/99378, accessed 07-Jul-2017 taken from “Nun danket alle Gott”, (Words Rinkart 1636, Music Crüger, 1647) shows.",
"<part id=\"P1\"> <measure number=\"1\"> ... <note> <pitch> <step>E</step> <alter>-1</alter> <octave>4</octave> </pitch> <duration>480</duration> <voice>1</voice> ...",
"This snippet refers to the note Eb in the 4th octave (middle C is annotated C4, so this is a minor third above middle C).",
"The duration must be determined from other parameters in the XML but this note actually corresponds to a 1/4 note or crotchet.",
"Arguably the most beneficial aspect of studying music from the point of view of this paper, however, is that it provides a simple example of when there are multiple candidate unique alphabets, for example, whether or not to include musical note duration as well as pitch in defining the alphabet.",
"When we first considered this aspect, the potential ambiguity worried us until we eventually realised that it led naturally and elegantly to the important conclusion, proved in Appendix C p. REF and verified experimentally, that all consistent unique alphabets are themselves related by a power-law.",
"In this study, we used 883 pieces of music, mostly classical but a very eclectic mix of chorales, piano concertos, horn duets, blue-grass music and indeed almost anything in an XML format we could get our hands on.",
"This process was not as simple as accumulating large amounts of open source code unfortunately and took some considerable time and manual effort.",
"As a result, this is by far the smallest system we consider and does not therefore allow us much scope to demonstrate different sized aggregations.",
"Even so, the length distribution of these 883 pieces of music is still gratifyingly suggestive of the presence of the predicted canonical distribution as shown in Figs.",
"REF -REF .",
"Figure: The length distribution of the 883 pieces of music analysed shown as a pdf (A) and a ccdf (B).",
"An R lm() analysis on the tail of Fig REF reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $100.0-10000.0$ , with an adjusted R-squared value of $0.9936$ .",
"The slope is $-1.66 \\pm 0.08$ .",
"The written word The pioneering work which first suggested the ubiquity of power-laws in texts was that by George K. Zipf [73].",
"Zipf showed empirically that if the frequency of occurrence of words in a text were plotted in rank order on a $\\log -\\log $ ccdf, a power-law in frequency was observed.",
"This is an example of a system of homogeneous boxes, Appendix A p. REF , where we give a proof of Zipf's law using the methodology of this paper.",
"Archetypal examples of this at different scales are shown in Figs.",
"REF -REF .",
"These show respectively, (Fig.",
"REF The Mitre Common Vulnerabilities database 2,410,350 wordshttps://cve.mitre.org/, accessed 01-May-2015; Fig.",
"REF The complete works of Shakespeare 948,516 words; Fig.",
"REF The King James Bible in Swedish 807,969 words and Fig.",
"REF the classic English text “Three Men in a Boat” published by Jerome K. Jerome in 1889 67,435 words)https://www.gutenberg.org/, accessed 01-Jul-2017.",
"The classic straight line signature of the power-law is evident in each case even though the datasets are different in size by a factor of 40 from largest to smallest.",
"Figure: The rank ordered word distributions in, (A): The Common Vulnerabilities database, (B): The complete works of Shakespeare, (C): The King James Bible in Swedish and (D): Jerome K. Jerome's “Three Men in a Boat”.Before leaving this section, we point out that some systems can in fact be characterised both by using the homogeneous model, Appendix A p. REF , and also using the heterogeneous model, Appendix A p. REF .",
"It would therefore provide substantial additional support for the generality of the information model we propose in this paper if both the homogeneous model predictions and the heterogeneous model predictions held for a system in which both could be used.",
"There is no conflict between information measures here even though they are different, provided they are consistently applied.",
"Treating the words of a text as indivisible as we have done above yields the homogeneous model where word frequency follows the predicted Zipf power-law in rank, and we have already seen that the homogeneous model predicts exactly this long-established behaviour, Appendix A p. REF .",
"It is also possible to consider the individual words of a text as being further sub-divided into their letters as a heterogeneous model, just as if each word were a protein built from a unique alphabet, which in English, is 26 letters.",
"Word length has been studied extensively for various languages, for example [58], however for our purposes, we expect by analogy with our protein studies, that applying the heterogeneous model to word-length frequency will yield the canonical distribution seen in Fig.",
"REF for example.",
"Fig.",
"REF shows the word-length frequency for the text “Three Men in a Boat” whose word frequency is shown in Fig.",
"REF .",
"Even though the x-axis is limited to a maximum word length of around 40 in this novel (it includes single- and, unusually, double- hyphenated words such as the archaic currency reference “two-pounds-ten”), the canonical shape is once again evident with a sharp unimodal peak and as can be seen in Fig.",
"REF , good evidence of the predicted power-law tail.",
"The steep slope is associated with a relatively small alphabet as described in Appendix D, p. REF .",
"An R lm() analysis on the tail of Fig.",
"REF reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $2.303 \\times e^{-15}$ over the range starting at the mode $5.0-30.0$ , with an adjusted R-squared value of $0.9853$ .",
"The slope is $-6.40 \\pm 0.32$ .",
"Figure: The text “Three Men in a Boat” with each word considered as a component and the frequency of occurrence of length of word plotted as a pdf (A) and as a ccdf (B).",
"We believe this adds considerable weight to the information-theoretic arguments of this paper.",
"In this case, the same system, treated using two different models of information (the homogeneous case of word frequencies and the heterogeneous case of letter frequencies and word lengths) obeys the predictions of both models, showing that consistent but different measures of information within the same system lead to valid predictions for the different distributions.",
"In both cases, conserving Hartley-Shannon information in an ergodic system is the underlying mechanism.",
"The atomic elements The distribution of atomic elements in the universe is a similar system to that of word frequencies and we use the homogeneous model, Appendix p. REF .",
"Again, the components each consist of one type only, in this case atoms of each element and, intrigued by its apparent preponderance we chose to include current estimates of dark material, i.e.",
"energy and matter.",
"The frequencies of occurrence, Fig.",
"REF , have been taken from NASAhttps://map.gsfc.nasa.gov/universe/uni_matter.html, accessed 29-Jun-2017 and Wikipedia,https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements, accessed 29-Jun-2017.",
"Figure: The frequency of occurrence of the elements in the universe supplemented by estimates of dark energy (data point 1) and dark matter (data point 2) shown as a log-log\\log -\\log ccdf.The distribution of elements fits well on the predicted power-law distribution for homogeneous boxes, where the rank ordering turns out to follow the atomic number.",
"(The relationship with atomic number is outwith the theory described here.)",
"An R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $1.0-85.0$ , with an adjusted R-squared value of $0.9779$ .",
"The slope is $-6.80 \\pm 0.94$ .",
"Intriguingly, the observed amount of dark energy (first point Fig.",
"REF ) and dark matter (second point Fig.",
"REF ) fit the predicted homogeneous box distribution well, suggesting that from the point of view of CoHSI, dark matter corresponds to something with an atomic number of zero (a re-generating sea of neutrons ?)",
"and dark energy corresponds to something with an atomic number of -1 (?",
"); so that's something for the theorists to chew on.",
"On a much smaller scale than Fig.",
"REF , the characteristic straight line signature is again visible in the distribution of the elements in seawaterhttps://en.wikipedia.org/wiki/Abundances_of_the_elements_(data_page), accessed 29-Jun-2017, Fig.",
"REF , not in this case including dark material.",
"Figure: The frequency of occurrence of the elements in sea water shown as a log-log\\log -\\log ccdf.",
"Even though the tail is short, an R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $10.0-72.0$ , with an adjusted R-squared value of $0.9963$ .",
"The slope is $-8.82 \\pm 0.29$ .",
"Conclusions This paper, through theory and testing against multiple datasets of different provenance and levels of aggregation, makes the case that a conservation principle derived from information theory (Conservation of Hartley-Shannon Information) operates within all discrete systems to impose important and common structural properties on length and unique alphabet size distributions.",
"By development of a statistical mechanics argument in which we consider ergodic ensembles with a fixed number of tokens and a fixed total H-S information content, independent of the meaning of the tokens chosen without bias, we demonstrate that the length and unique token alphabet for components of discrete systems are inextricably linked by a canonical distribution - the heterogeneous CoHSI distribution (REF ) - visible in all aggregations and at all scales where numbers are sufficiently large for statistical mechanics to operate.",
"The two Lagrange multipliers which naturally emerge $\\alpha , \\beta $ are undetermined and simply parameterise the range of possible solutions.",
"This has a number of interesting implications which we will divide into levels of confidence based on the data analysis, distinguishing in each case whether this is a heterogeneous system, Appendix A p. REF (which has a canonical distribution (REF )) or a homogeneous system, Appendix p. REF (which has a canonical distribution (REF ) corresponding to Zipf's law) or both, recalling that such systems differ only in the relevant definition of Hartley-Shannon information.",
"Very confident These conclusions are strongly supported by statistical analysis using R and documented individually in the body of the paper.",
"Where linear analysis was done on a power-law tail, this is noted below as (R).",
"All 14 such analyses gave an adjusted $R^{2}$ within the range $0.970 - 0.999$ with values of $p < e^{-14}$ .",
"All heterogeneous systems will tend to the canonical frequency distribution (REF ) as total size grows, with a sharp unimodal peak and a power-law tail.",
"Justification Equation (REF ).",
"Development starting Appendix A p. REF .",
"Evidence Fig.",
"REF (Proteins); Fig.",
"REF (Software); Fig.",
"REF (Music); Fig.",
"REF (Texts) (R)).",
"All heterogeneous discrete systems made up from components, themselves comprising indivisible tokens chosen from an alphabet, will exhibit a precise power-law tail in both their length distribution and in the distribution of their unique alphabet.",
"Justification Length: Eq.",
"(REF ), Alphabet: Eq.",
"(REF ).",
"Development starting at Appendix A p. REF and also p. REF .",
"Evidence Length: Fig.",
"REF (Proteins); Fig.",
"REF (Software); Fig.",
"REF (Music), (R).",
"Alphabet: Fig.",
"REF (Proteins); Fig REF (Software); Fig.",
"REF (Music), (R).",
"The canonical CoHSI distribution (REF ) tends to the asymptotic solution found in [25].",
"Justification Appendix A p. REF onwards.",
"Evidence Figs.",
"REF , REF , example solutions of equation (REF ).",
"The canonical frequency distribution will appear in all aggregations of a system.",
"Justification The theory for both heterogeneous systems and homogeneous systems is scale independent.",
"It finds the most likely length distribution for a given total size $T$ and a given total heterogeneous or homogeneous Information $I$ , but other than requiring a reasonably large system in the general sense of Statistical Mechanics [17], does not depend on the values of $T$ or $I$ .",
"Appendix A p. REF onwards.",
"Evidence Heterogeneous systems, Figs.",
"REF -REF , Figs.",
"REF -REF (Proteins); Homogeneous systems, Figs.",
"REF -REF (Texts); Figs.",
"REF , REF (Atomic elements).",
"The canonical frequency distribution will appear in all qualifying discrete systems independently of their provenance.",
"Justification At the heart of the definition of Hartley-Shannon Information is the original prescient advice from Ralph Hartley that the meaning of the tokens is irrelevant.",
"Evidence Heterogeneous systems, Fig.",
"REF (Proteins), Fig.",
"REF (Software), Fig.",
"REF (Music), Fig.",
"REF (Text); Homogeneous systems, Figs.",
"REF -REF (Text), Figs.",
"REF , REF (Atomic Elements).",
"If a system can be considered as both a heterogeneous system and a homogeneous system, then the predictions for both kinds of system will appear.",
"Justification Only a different definition of information is needed.",
"The mechanism of statistical mechanics then automatically generates the relevant length distribution, Appendix A p. REF and p. REF .",
"Evidence Heterogeneous system, Fig.",
"REF (Lengths of words in a text); Homogeneous systems, Fig.",
"REF (Rank-ordered word frequency in text) (R).",
"The alphabet used to categorise a heterogeneous system is irrelevant provided it is consistent.",
"Justification Consistent alphabets are power-laws of each other asymptotically for large components, Appendix C p. REF onwards.",
"Evidence Figs.",
"REF , REF (Duration and no-duration notes in Music) (R).",
"Average component length is highly preserved across collections in heterogeneous systems.",
"Justification This is a direct consequence of the sharply unimodal shape of the canonical Distribution for heterogeneous systems.",
"Evidence Figs.",
"REF , REF (Proteins) (R), from [25]).",
"This is also reported for software [23].",
"Unusually long components are inevitable in heterogeneous systems.",
"Justification Such components are inevitable because of the presence of the ubiquitous power-law tail, (noting the comments of [11]).",
"Evidence Fig.",
"REF (Proteins) which shows for example, approximately 10,000 proteins longer than 10 times the average length.).",
"This phenomenon has also been reported for software [43].",
"Confident We expect to see the numbers of known amino acids to expand with time to preserve the power-law tail already evident in Fig.",
"REF .",
"There were around 800 structurally distinct Post Translationally Modified amino acids known in SwissProt 13-11.",
"There are already suspected to be thousands more [31].",
"Our information argument strongly supports this thesis and may indeed help to quantify it.",
"Speculative We find it intriguing that the power-law slope of virus protein lengths is quite different from those observed in the three domains of life.",
"We speculate that this relates to their unique alphabet.",
"The asymptotic behaviour for large components (REF ) in heterogeneous systems, implies that tokens carry a payload of the average information content of the component in which they appear.",
"For example, in proteins a particular amino acid might carry a different information payload in different proteins by virtue of the company it keeps.",
"We do not know if this has any useful physical interpretation.",
"We (wildly) speculate that since both dark energy and dark matter lie on the same information theoretic distribution as the elements in order of atomic number, there is an undiscovered but intimate relationship between dark material and atomic number.",
"It is clear from the above results that important structural features of discrete systems are well-predicted by a single conservation principle applied to ergodic systems at all levels of aggregation and of all kinds.",
"Nothing more is asked of the theory than that the size of the systems should be sufficiently large that the methodology of Statistical Mechanics can be applied.",
"That the above features need not depend for explanation on any mechanism of natural selection in the case of proteins or anything to do with human volition in the case of either music or software, we find remarkable.",
"Instead, they simply manifest themelves as emergent properties of heterogeneous or homogeneous large systems of components, revealed when we consider an ergodic ensemble of the same size and H-S information content, from which we seek the most likely distribution of its component lengths using the methodology of this paper.",
"Acknowledgements Competing Interests: The authors declare that they have no competing financial interests.",
"Correspondence: Correspondence and requests for materials should be addressed to Les Hatton (email: [email protected]).",
"This material is based on work supported in part by the National Science Foundation.",
"Any opinion, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.",
"Although some parts of this paper are published [22], [23], [25], the main body of the paper, the lower and upper bounds on the implicit canonical pdf (the heterogeneous CoHSI distribution), the relationship between alphabets and most of the empirical work, appear here for the first time both to clarify and extend the theory with new results.",
"During review processes for various journals, the general thesis of this paper that a simple global principle appears to be responsible for some important aspects of protein evolution was a bridge too far for most of the reviewers given these restrictions.",
"This resistance has been noted before in biology research [15] and publishing inter-disciplinary papers in general remains much more difficult than it should be.",
"However, there were some insightful responses for which we are both very grateful.",
"One anonymous reviewer pointed out an error with the combinatorics leading up to the chocolate box argument of Appendix p. REF , which opened up a real can of worms (the role of additive partitions) which was finally resolved by the recursive argument of Appendix A p. REF .",
"Yet another reviewer made us really think about the application of this apparently abstract principle to real systems we can touch and feel, as well as suggesting we clarify the emergent role of $A$ in the dual distribution, which we hopefully have in Appendix A p. REF .",
"We would particularly like to thank Dan Rothman for a whole string of insights which really helped clarify some of the more opaque sections; Ken Larner who went through the document with a fine toothcomb suggesting numerous clarifications; Leslie Valiant for bringing to our attention Frank's excellent paper [15]; Alex Potanin and colleagues in New Zealand for useful comments after an early seminar on this topic.",
"Any mistakes which remain are our responsibility of course but we hope that the theory is now sufficiently well explained and its novelty delineated and that the associated reproducibility packages will help others to verify the computational aspects and extend it in new directions.",
"Appendix A: The Conservation of Hartley-Shannon Information: From Statistical Mechanics to the Canonical CoHSI Distribution Statistical mechanics is a methodology for predicting component distributions of general systems made from discrete pieces or components subject to restrictions known as constraints.",
"Such systems include gases (made from molecules), proteins (made from amino acids), software (made from programming language tokens) and even boxes containing beads.",
"Conventionally, constraints are applied by fixing the total number of pieces and/or the total energy [17].",
"Statistical Mechanics: Classical use in physics To illustrate the method, we describe a classical problem of determining the most likely distribution of particles amongst energy levels.",
"To see this, the following variational methodology is borrowed from the world of statistical physics, ([59] (p.217-); and for an excellent introduction, see [17]).",
"In the kinetic theory of gases, a standard application is to find the most common arrangement of molecules amongst energy levels in a gas subject to various constraints such as a fixed total number of molecules and fixed total energy.",
"For this, imagine that there are M energy levels, where the number of particles with energy level $\\varepsilon _{i}$ is $t_{i}, i=1,..,M$ .",
"For this system, the total number of ways $W$ of organising the particles amongst the M energy levels is given by:- $W=\\frac{T!}{t_{1}!t_{2}!..t_{M}!",
"}, $ where $T = \\sum _{i=1}^{M} t_{i} $ The total amount of energy in this system is just the sum of all the particle energies and is given by $E = \\sum _{i=1}^{M} t_{i} \\varepsilon _{i} $ In a physical system, E corresponds to the total internal energy and the variational method to follow constrains this value to be fixed; i.e.",
"solutions are sought in which energy is conserved.",
"Using the method of Lagrangian multipliers and Stirling's approximation as described in [17], will give the most likely distribution satisfying equation (REF ) subject to the constraints in equations (REF ) and (REF ).",
"This is equivalent to maximising the following variational derived by taking the natural log of (REF ).",
"Just as in maximum likelihood theory, taking the log dramatically simplifies the proceedings, in this case the factorials, and allows the use of Stirling's theorem for large numbers.",
"Also, since it is monotonic, a maximum in log W is coincident with a maximum in W. This leads to $\\log W = T \\log T - \\sum _{i=1}^{M} t_{i} \\log (t_{i}) + \\lambda \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace + \\beta \\lbrace U - \\sum _{i=1}^{M} t_{i} \\varepsilon _{i} \\rbrace $ where $\\lambda $ and $\\beta $ are the multipliers [17].",
"In essence, the variational process envisages varying the contents $t_{i}$ of each of the components until a maximum of $\\log W$ is found.",
"The maximum is indicated by taking $\\delta (\\log W) = 0$ , (analogous to finding maxima in differential calculus).",
"Noting that the variational operator $\\delta $ acts on pure constants such as $T \\log T$ , $\\lambda T$ and $\\beta U$ to produce zero just as when differentiating a constant, the product rule of differentiation gives $\\delta (t_{i} \\log (t_{i})) = \\delta t_{i} \\log (t_{i}) + t_{i} \\delta (\\log (t_{i}) = \\delta t_{i} (1 + \\log (t_{i}))$ , $\\varepsilon _{i}$ is independent of the variation by assumption, T and the $t_{i}$ are $\\gg 1$ (to satisfy Stirling's theorem, although it is surprisingly accurate even for relatively small values).",
"This leads to $0 = - \\sum _{i=1}^{M} \\delta t_{i} \\lbrace \\log (t_{i}) + \\alpha + \\beta \\varepsilon _{i} \\rbrace $ where $\\alpha = 1 + \\lambda $ .",
"(Further elaboration of this standard technique can be found in Glazer and Wark [17].)",
"Finally, (REF ) must be true for all variations to the occupancies $\\delta t_{i}$ and therefore implies $\\log (t_{i}) = - \\alpha - \\beta \\varepsilon _{i} $ for all $i$.",
"Using equation (REF ) to replace $\\alpha $ , this can be manipulated into the most likely, i.e.",
"the equilibrium distribution, of particles amongst the M components.",
"$t_{i} = \\frac{T e^{-\\beta \\varepsilon _{i}}}{\\sum _{i=1}^{M} e^{-\\beta \\varepsilon _{i}}} $ Defining $p_{i} = \\frac{t_{i}}{T}$ means that $p_{i}$ can be interpreted as a probability density function since it is non-negative everywhere and its sum everywhere is equal to 1.",
"Then (REF ) yields $p_{i} = \\frac{e^{-\\beta \\varepsilon _{i}}}{\\sum _{i=1}^{M} e^{-\\beta \\varepsilon _{i}}} $ This is a result with a profound interpretation in physics.",
"It states If we consider all possible systems which share the same number of particles T and the same total energy U (i.e.",
"energy is conserved), then given any one example of such a system, it is overwhelmingly likely to obey (REF ).",
"In other words by considering all possible systems with these parameters and constraining them to have the same T and U, probability distribution (REF ) is overwhelmingly likely, provided the $t_{i}$ are large enough for Stirling's approximation to hold.",
"In this case, the distribution is exponential and exactly as is found in nature - exponentially fewer particles occupy higher energy levels.",
"All the above has been known for decades and is extremely successful at explaining classical systems such as gases and even quantum mechanical systems; the methodology of statistical mechanics however is exceedingly versatile, so let us consider a simple model just consisting of boxes of coloured beads.",
"Conservation of Hartley-Shannon Information Identical boxes of beads Let us put some flesh on the meaning of “overwhelmingly likely” as used earlier in this paper.",
"Consider now a system of M boxes of identical beads, where the $i^{th}$ box contains $t_{i}$ beads and M is reasonably large.",
"If we have T beads in total numbered by sequence, where $T = \\sum _{i=1}^{M} t_{i}$ , so that they are distinguishable by their order, the number of possible ways of arranging them in each of the M boxes is given by $\\Omega = \\frac{T!",
"}{\\prod _{i=1}^{M} (t_{i}!)}",
"$ Suppose there are $M=10$ boxes and $T=100$ beads and we simply assign them one by one to a randomly chosen box.",
"We would be very surprised if the first box contained all the beads with the others empty, and the number of ways this can happen according to (REF ) is $100!/(100!\\times 0!\\times 0!..0!)",
"= 1$ .",
"If each box contains 10 beads however as shown in Fig.",
"REF , this situation can happen in $100!/(10!\\times 10!\\times ..10!",
")$ ways, which is approximately $10^{100}$ , a gigantic number.",
"Figure: Boxes containing exactly the same number of exactly the same bead.In other words, we are overwhelmingly more likely to see equal box populations than 1 single filled box.",
"In fact statistical mechanics allows us to prove that, in this case, equal population is by far the most likely distribution of contents simply by finding the maximum of (REF ) subject to a fixed number of T beads in the form $\\log \\Omega = T \\log T - T - \\sum _{i=1}^{M} \\lbrace t_{i} \\log (t_{i}) - t_{i} \\rbrace \\\\+ \\alpha \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace $ where the constraint on fixing T is controlled by the Lagrangian parameter $\\alpha $ .",
"Finding the maximum of (REF ) using the standard $\\delta ()$ method [17], gives the solution $t_{i} \\sim constant$ , corresponding to equal box populations.",
"Frank also demonstrates this in his maximum entropy formulation [15], p. 9.",
"Heterogeneous boxes of beads We now describe an extension which is directly relevant to systems such as the known proteome or computer programs.",
"Consider Fig.",
"REF .",
"Figure: The heterogeneous case where each box contains mixed types and different numbers of beads.",
"This is relevant to proteins, computer program functions and the length distribution of words in texts.Here the boxes contain differently coloured beads.",
"We envisage this as the $i^{th}$ box containing $t_{i}$ beads selected randomly from a unique alphabet of $a_{i}$ colours, ordered by sequence.",
"For the proteome, the “colours” correspond to different amino acids and for software functions they correspond to different programming language tokens.",
"Now we utilise the great generality of statistical mechanics by generalizing the payload to be Hartley-Shannon (H-S) information content instead of energy [23].",
"The H-S information content of the $i^{th}$ box $I_{i}$ , (Appendix p. REF ) is simply the log of the number of ways of arranging the beads in that box, so that it is guaranteed to contain at least one of each of the $a_{i}$ colours.",
"H-S information is completely agnostic about what the colours actually mean, indeed Hartley specifically advised against attaching any meaning to a token [20].",
"The only thing that matters is that beads change colour, so the actual colour is irrelevant and the total H-S information is just the sum of the information for each box.",
"Presented with such a system, we can ask what is the most likely distribution of contents for systems for which both the total number of beads and the total H-S information are conserved ?",
"We must also recall that proteins and software are both constructed sequentially so we are considering systems where beads are distinguishable by the order in which they appear, but the actual order is irrelevant.",
"The relevant variational form we must solve is therefore $\\log \\Omega = T \\log T - T - \\sum _{i=1}^{M} \\lbrace t_{i} \\log (t_{i}) - t_{i} \\rbrace \\\\+ \\alpha \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace + \\beta \\lbrace I - \\sum _{i=1}^{M} I_{i} \\rbrace $ The only term which is different in this formulation from the classical solution derived above (REF ), is the last term on the right hand side of (REF ).",
"In the variational methodology, each term has the $\\delta ()$ operation applied in order to vary the $t_{i}$ and derive the distribution in (REF ), so we are interested specifically in $\\delta \\big (\\beta \\lbrace I - \\sum _{i=1}^{M} I_{i} \\rbrace \\big ) = - \\beta \\sum _{i=1}^{M} \\delta ( I_{i} ) = - \\beta \\sum _{i=1}^{M} \\frac{d I_{i}}{d t_{i}} \\delta t_{i}, $ since I is being held constant.",
"Now consider what happens when boxes are very large compared with their unique alphabet, i.e.",
"$t_{i} \\gg a_{i}$ .",
"In this case, [23], the information content is $I_{i} = \\log (a_{i}\\times a_{i}\\times ... \\times a_{i}) = \\log (a_{i}^{t_{i}}) = t_{i} \\log a_{i}$ In other words, we select $t_{i}$ times from a choice of $a_{i}$ colours secure in the knowledge that since $t_{i} \\gg a_{i}$ , it is very unlikely that any of the $a_{i}$ colours would be missed out and we therefore meet the requirement of having exactly $a_{i}$ unique colours.",
"In this case, (REF ) becomes $- \\beta \\sum _{i=1}^{M} \\frac{d I_{i}}{d t_{i}} \\delta t_{i} = - \\beta \\sum _{i=1}^{M} \\frac{d (t_{i} \\log a_{i})}{d t_{i}} \\delta t_{i} = - \\beta \\sum _{i=1}^{M} (\\log a_{i}) \\delta t_{i} $ (REF ) fits perfectly into the variational methodology leading to (REF ), modifying (REF ) to give $\\log (t_{i}) = - \\alpha - \\beta \\log a_{i} $ The analogue of (REF ) is therefore $p_{i} \\equiv \\frac{t_{i}}{T}= \\frac{a_{i}^{-\\beta }}{\\sum _{i=1}^{M} a_{i}^{-\\beta }} $ To summarize, maximising (REF ) subject to a fixed total number of beads T AND a fixed total H-S information $I = \\sum _{i=1}^{M} I_{i}$ is directly analogous to maximising (REF ) with $\\log a_{i}$ replacing $\\epsilon _{i}$ .",
"Like its classical equivalent (REF ), (REF ) is also fundamental.",
"It states In any discrete system satisfying the model described here, the tail (i.e.",
"large $t_{i}$ ) of the distribution of unique alphabets is overwhelmingly likely to obey a power-law.",
"Note that by analogy with (REF ), we can interpret $t_{i} \\log a_{i}$ as each bead carrying a payload of $\\log a_{i}$ , so that even though H-S information is token agnostic, the beads in a particular box still carry a box-dependent payload which is a function of the unique alphabet of colours in that box, $a_{i}$ .",
"This is exactly analogous to $t_{i} \\varepsilon _{i}$ being interpreted as each particle carrying an energy $\\varepsilon _{i}$ in classical statistical mechanics.",
"In other words, each box behaves as if it had a fixed information level $\\log a_{i}$ determined by its unique alphabet.",
"In a protein for example, this has the intriguing implication that even though H-S information is token-agnostic, a particular amino acid in one protein may carry a different information payload than when present in another protein, simply because its neighbours are different.",
"The asymptotic dual distribution As pointed out by [25], (REF ) has a dual solution.",
"With some algebra, it can be shown that $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }}, $ where $A = \\sum _{i=1}^{M} a_{i}$ Note here that $A$ emerges naturally as the sum of the unique alphabets of each component.",
"It is not the size of the unique alphabet across all components.",
"This is simply another manifestation of the token-agnosticism of Hartley-Shannon information - system-wide uniqueness of the alphabet simply does not emerge as a requirement.",
"The only requirements for a pdf are that it be positive definite and normalisable so this in no way detracts from the fact that (REF ) is also a power-law.",
"In other words, The length distribution of large proteins or software functions for which $t_{i} \\gg a_{i}$ will also be a power-law.",
"Note also the natural appearance of the reciprocal slope $1/\\beta $ .",
"This value is not found in the datasets here but this difference is discussed and we think resolved in the discussion of alphabets in music in the Appendices p. REF .",
"The chocolate box analogy and additive partitions: the CoHSI distribution For smaller boxes containing fewer beads, the above value of $I_{i}$ (REF ) is not correct.",
"If $t_{i}$ is closer in size to $a_{i}$ , (it cannot be smaller since the length must be at least equal to the unique alphabet), there is an increasingly high probability that we might miss out one of the colours in the unique alphabet as we select our $t_{i}$ beads, negating the fundamental assumption that each box contains a unique alphabet of exactly $a_{i}$ .",
"We must therefore make different provisions as the boxes get smaller.",
"Figure: A box of 22 chocolates chosen from 12 different types as shown on the left.The situation is akin to boxes of mixed chocolates, Fig.",
"REF .",
"Such boxes are constructed from a fixed set of chocolates advertised on the lid, and every box must contain at least one of each.",
"Larger boxes simply contain more than one of some kinds.",
"In how many ways can such boxes be created ?",
"Note that it is simple to find an algorithm to guarantee that the unique alphabet is exactly $a_{i}$ .",
"All that is necessary is to fill any $a_{i}$ places with one chocolate of each type and then fill the remaining $t_{i} - a_{i}$ at random from the available types.",
"The number of ways of doing this is $\\big ((a_{i}!)",
".",
"({}^{t_{i}}C_{a_{i}})) .",
"(a_{i}^{(t_{i} - a_{i})}\\big ), $ where ${}^{n}C_{r} = n!/((n-r)!r!",
")$ is the combination operator.",
"This however, is not the same as counting all the possible ways of filling the box such that it contains exactly $a_{i}$ chocolates.",
"We are trying to find the number of different ways of filling the $i^{th}$ box with $t_{i}$ chocolates chosen from a unique set of exactly $a_{i}$ chocolates and we must do this in a way which fits into the statistical mechanical framework so we can use its methodology.",
"To explore this, suppose we have a box of $t_{i} = 5$ chocolates such that it contains exactly $a_{i} = 2$ different chocolates of types A and B.",
"The total number of ways this can be done $N(t_{i},a_{i})$ , is given by $N(5,2) = \\frac{5!}{1!4!}",
"+ \\frac{5!}{4!1!}",
"+ \\frac{5!}{3!2!}",
"+ \\frac{5!}{2!3!}",
"$ Note The first term on the right hand side of (REF ) is the total number of ways of selecting 5 chocolates by using 1 chocolate of type A and 4 chocolates of type B.",
"This is equal to 5 (ABBBB, BABBB, BBABB, BBBAB, BBBBA).",
"The second term corresponds to 4 chocolates of type A and 1 of B and is also equal to 5 (BAAAA, ABAAA, AABAA, AAABA, AAAAB).",
"The third term corresponds to taking 3 of type A and 2 of type B.",
"This is equal to 10, (AAABB, AABAB, AABBA, ABAAB, ABABA, ABBAA, BBAAA, BABAA, BAABA, BAAAB).",
"The fourth term corresponds to taking 2 of type A and 3 of type B.",
"This is also equal to 10, (BBBAA, BBABA, BBAAB, BABBA, BABAB, BAABB, AABBB, ABABB, ABBAB, ABBBA).",
"There are no other ways of arranging the box such that there are exactly 2 colours and exactly 5 chocolates altogether.",
"There are therefore 5 + 5 + 10 + 10 = 30 different such boxes in total.",
"Note that (REF ) gives $(2!)",
".",
"({}^{5}C_{2})) .",
"(2^{(5 - 2)}) = 160$ boxes.",
"This over-counting is because a box such as ABBAB could be generated several times by that algorithm, for example, by filling the first two places with AB and then the rest at random or by filling the first and third places with AB and the rest at random.).",
"The denominators of (REF ) correspond to elements of the additive compositionshttps://en.wikipedia.org/wiki/Partition_(number_theory), accessed 02-Jun-2017.",
"of size 2 of the number 5.",
"These are $5 = 1 + 4; 5 = 4 + 1; 5 = 3 + 2; 5 = 2 + 3$ There are other additive compositions such as $2+2+1$ , but this corresponds to three different kinds of chocolate so must be excluded.",
"The fact that the compositions are additive presents a real complication when merging with the methodology of statistical mechanics because it breaks the steps leading from (REF )-(REF ) by introducing the log of a recursive definition as we shall see.",
"Prior to discovery of this recursive method, the solution was simply trapped between a lower and upper bound.",
"The lower bound consisted of just one of the terms leading to the recursive definition and the upper bound was the pure power-law (REF ).",
"The recursive method is however far more compelling.",
"First we slightly modify the definition in (REF ) by letting $N(t_{i}, a_{i}; a^{\\prime }_{i})$ be the number of ways of producing a chocolate box with $t_{i}$ chocolates containing exactly $a^{\\prime }_{i}$ unique types chosen from a total unique number of types of $a_{i}$ .",
"In this notation, for example, $N(5,2;1) = 2$ and $N(5,2;2) = 30$ .",
"The distinction between $a_{i}$ and $a^{\\prime }_{i}$ is to make way for the use of recursion.",
"It can be easily verified that the following recursion then generates the desired total number of ways $N(t_{i}, a_{i}; a_{i})$ of generating a chocolate box of $t_{i}$ chocolates from a unique set of chocolates $a_{i}$ .",
"$N(t_{i}, 1; 1) = 1; \\hspace{14.22636pt} N(t_{i}, a_{i}; i) = {}^{a_{i}}C_{i} N(t_{i}, i; i), i = 1,..,a_{i} - 1, a_{i} = 1,..,t_{i}$ completed by $N(t_{i}, a_{i}; a_{i}) = a_{i}^{t_{i}} - \\sum _{i=1}^{a_{i}-1} N(t_{i},a_{i}; i)$ The corresponding Hartley-Shannon information content for a box containing $t_{i}$ chocolates chosen from a unique alphabet of $a_{i}$ chocolates is therefore given by $I_{i} = \\log \\big ( N(t_{i}, a_{i}; a_{i}) \\big ) $ In contrast, the equivalent form for the pure power-law (REF ) is $I_{i} \\vert _{P} = \\log \\big ( a_{i}^{t_{i}} \\big ) = \\log \\big ( t_{i} \\log a_{i} \\big ) $ We can now see the problem posed by (REF ) when we apply the $\\delta ()$ operator to $I_{i}$ in the statistical mechanical framework leading from (REF )-(REF ).",
"The presence of the recursion prevents the clean separation of factors by the $\\log $ operation.",
"If we simply solved this computationally, that would ordinarily be no problem but (REF ) is computationally difficult for the large factorial values which arise even for modest values of $(t_{i},a_{i})$ .",
"(Whatever method we choose, however, it must have the property of producing the power-law form (REF ) in the asymptotic limit $t_{i} \\gg a_{i}$ .)",
"Applying the $\\delta ()$ operator to (REF ) using (REF ) leads to the pure power-law equation $\\log t_{i} = -\\alpha -\\beta ( \\log a_{i} ), $ whereas applying the $\\delta ()$ operator to (REF ) using (REF ), (REF ) leads to the full equation $\\log t_{i} = -\\alpha -\\beta ( \\frac{d}{dt_{i}} \\log N(t_{i}, a_{i}; a_{i} ) ), $ Here, the unique alphabet $a_{i}$ is playing a dual role as the frequency in a pdf by analogy with (REF ) using (REF ).",
"To complete our chocolate box analogy, if we are presented with a system of boxes with a total number of chocolates $T$ chosen from a fixed alphabet of chocolates and a total H-S information $I$ , then by far the most likely distribution of numbers of chocolates in any box will be given by a pdf which is the solution of (REF ).",
"Finally we note that following the argument that led up to (REF ), for $t_{i} \\gg a_{i}$ , $\\log N(t_{i}, a_{i}; a_{i} ) \\rightarrow t_{i} \\log a_{i},$ so the full solution correctly asymptotes to the pure power-law.",
"This will be confirmed during the computation with both forms being displayed together.",
"Computational aspects of the CoHSI distribution Before we proceed with this, there is a technical limitation to overcome since the equation for the pdf which results from applying the variational method to (REF ) is implicit.",
"As we pointed out in the text, there is a precedent for this in the definition of Tsallis entropy [67], [66], although in our case, the implicit nature of the pdf arises naturally from CoHSI.",
"(In Tsallis entropy, the entropy term is adjusted using an additional parameter and this adjustment can lead to an implicit pdf.)",
"We must therefore generalise the argument from integer values of ($t_{i}, a_{i}$ ), to the real line.",
"This will not affect our computation of factorials however, which are done at integer values of $t_{i}, a_{i}$ , with interpolation for non-integer values.",
"(REF ) defines the canonical implicit pdf with solutions $(t_{i},a_{i})$ which a) conserves H-S information and b) asymptotes to the pure power-law (REF ) for $t_{i} \\gg a_{i}$ as required for any heterogeneous system at all scales.",
"Solving (REF ) proved challenging.",
"The one thing we do know, however, is that it asymptotes to the simple explicit solution $I_{i} \\sim t_{i} \\log a_{i}$ leading to (REF ) when $t_{i} \\gg a_{i}$ .",
"This suggested the following procedure.",
"We start with large $t_{i}$ solving (REF ) explicitly for $a_{i}$ , followed by the use of this value as the starting value for the full solution of (REF ).",
"This was found by searching a pre-computed grid of integer $(t_{i},a_{i})$ values for the appropriate value of $d/dt_{i} (\\log N(t_{i}, a_{i}; a_{i}))$ interpolating as necessary.",
"When the solutions are found, we decrement $t_{i}$ and start again.",
"This process is somewhat akin to shooting methods in boundary layer solutions in fluid dynamics, [68], [21].",
"All code used is in the reproducibility package.",
"Figs.",
"REF , REF indicate the behaviour we expect to find if CoHSI is indeed controlling these distributions.",
"The power-law behaviour for larger components is already modelled to a high degree of precision by the large component approximation (REF ).",
"We focus now on the behaviour for all component sizes.",
"The full solution - the solution of (REF ), and the pure power-law solution - the solution of (REF ), using the same parameters, are shown together at two different scales as Figs.",
"REF and REF .",
"The shaded zone corresponds to the region where the full solution departs from the pure power-law solution.",
"Note that the the first order approximation for numerical differentiation means that the first few points for the full CoHSI solution do not converge.",
"The behaviour is clear from the remaining points however.",
"Figure: The length distributions using the same modelling parameters for (A) the full CoHSI solution (), and the pure (asymptotic) power-law () for components smaller than 100 tokens and (B), the same data for components up to 2,000 tokens long.We make the following observations about Figs.",
"REF and REF with respect to (REF ) and (REF ).",
"In (REF ) as the left hand side decreases with decreasing $t_{i}$ , the value of $a_{i}$ must increase to give a solution.",
"This gives the pure power-law behaviour shown which continues to increase as $t_{i}$ decreases as shown clearly in Fig.",
"REF .",
"In (REF ), $d/dt_{i}(N(t_{i}, a_{i}; a_{i})$ naturally decreases without having to keep increasing $a_{i}$ as was the case in (REF ).",
"This is a natural consequence of CoHSI.",
"As can be seen in Figs.",
"REF , REF , the qualitative behaviour of the full CoHSI solution around the unimodal peak remains sharp but is more rounded than the pure power-law solution and is qualitatively similar to the software data close up of Fig.",
"REF shown as Fig.",
"REF .",
"The sharpness of the peak is related to the boundary condition naturally emerging in this theory that $t_{i} \\ge a_{i}$ , i.e., no component can be shorter than its unique alphabet.",
"The peak of the full solution differs slightly in position as well as their amplitudes as the power-law parameter $\\beta $ changes.",
"A value of $\\beta =1.8$ was used.",
"As $\\beta $ increases, the peak moves left and the amplitude diminishes.",
"The full solution naturally asymptotes to the pure power-law behaviour as required.",
"We can compare the behaviour around the peak with a close-up of the dataset of Fig.",
"REF , as shown as Fig.",
"REF .",
"Even on observed data, the transition from power-law to near linearity is abrupt, taking place over perhaps 10 tokens, and is qualitatively very similar to the full CoHSI solution.",
"As was noted in the body of the paper, the juxtaposition of dual regions, one matched by a lognormal distribution and the other by a power-law has been described in the past, [39], [37].",
"In our theory, this transition emerges completely naturally as the implicit solution of (REF ).",
"Figure: A close-up around the peak of the measured dataset shown as Fig.",
".To summarise, these results strongly support the thesis of this paper that the Conservation of Hartley-Shannon Information (CoHSI) acts as a constraint on how the length and alphabet size distributions of systems of a given size $T$ and total Hartley-Shannon information $I$ , can evolve at all scales giving an excellent qualitative match which does not require juxtaposing existing pdfs of known properties.",
"Approximate properties of the heterogeneous CoHSI distribution From the shape of Figs.",
"REF , REF and the theory which led up to Figs.",
"REF , REF , we can approximate the distribution satisfactorily by glueing together a right-angled triangle up to the modal value $a_{max}$ at $t = t_{max}$ , say and a power-law afterward because the solution corresponding to (REF ) transitions from power-law to almost linear behaviour so quickly.",
"In other words, we can define the approximate canonical distribution $c(t)$ as follows $c(t) = \\left\\lbrace \\begin{array} {ll}(\\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))}) t & \\mbox{$0 \\le t \\le t_{max}$} \\\\(\\frac{2(\\beta - 1)}{(t_{max} (\\beta + 1))}) (\\frac{t}{t_{max}})^{-\\beta } & \\mbox{$t_{max} < t < \\infty $}\\end{array}\\right.$ We require $\\beta >1$ for this to be positive definite.",
"This has been normalised so as to integrate to unity over it's support $[0,\\infty ]$ .",
"This approximation will allow us to make useful inferences.",
"First we will calculate the mean location and spread of this distribution.",
"The mean location is given by $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\int _{s=0}^{t_{max}} s^{2} ds + t_{max} (\\frac{1}{t_{max}})^{-\\beta } \\int _{s=t_{max}}^{\\infty } s^{-\\beta + 1} ds \\bigg ],$ which is $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\big [ \\frac{s^{3}}{3} \\big ]_{s=0}^{t_{max}} + t_{max} (\\frac{1}{t_{max}})^{-\\beta } \\big [ \\frac{s^{-\\beta + 2}}{-\\beta + 2} \\big ]_{s=t_{max}}^{\\infty } \\bigg ],$ and, provided $\\beta > 2$ , gives $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\big [ \\frac{t_{max}^{3}}{3} \\big ] + ((t_{max})^{\\beta + 1} \\big [ - \\frac{t_{max}^{-\\beta + 2}}{-\\beta + 2} \\big ] \\bigg ],$ and finally $\\langle c \\rangle = \\frac{2 (\\beta - 1)t_{max}}{(\\beta + 1)} \\bigg [ \\frac{1}{3} + \\frac{1}{\\beta - 2} \\bigg ] = \\frac{2 (\\beta - 1)t_{max}}{3(\\beta - 2)} ; \\hspace{5.69046pt} \\beta > 2$ There is little point in computing higher moments because they place even greater constraints on the value of $\\beta $ (they diverge unless the support for the distribution is a finite interval), and will not apply to our examples for which $2 \\le \\beta \\le 4.5$ , (recall that if the pdf has slope $-\\beta $ , the ccdf will have slope $-\\beta + 1$ and we are measuring from the ccdf following [42].).",
"Applying the above estimate for $\\langle c \\rangle $ to the full Trembl distribution, Fig.",
"REF , for which $\\beta = 4.14$ suggests that $\\langle c \\rangle / t_{max} \\approx 1$ , where as analysis of the data itself in R gives a ratio of around $1.5$ which is reasonable given the nature of the approximation.",
"The actual values from Trembl are given in Table REF .",
"Table: Different measures of average length of proteins in amino acids in the full Trembl 17-03 distribution, Fig.",
"Homogeneous boxes of beads: CoHSI and Zipf's law There are some kinds of discrete system for which the above information model does not apply.",
"Consider the case of homogeneous components.",
"Here, each bead carries a payload such that each box contains only beads with the same payload, unique to that box.",
"We represent this by assembling beads of the same colour in the appropriate box, Fig.",
"REF .",
"Figure: The homogeneous case.",
"In each box, all the beads are the same but different boxes contain different types and numbers of beads.",
"This is relevant to the distribution of atomic elements and to the rank ordering of frequency occurrence of words in texts.We could of course simply set $a_{i} = 1$ in the heterogeneous case above.",
"However, this immediately causes problems because the asymptotic Hartley-Shannon information content of any box in this case would be $t{_i} \\log 1 = 0$ and is simply degenerate.",
"However, because Hartley-Shannon information is simply the $\\log $ of the number of ways of arranging the beads of a box, in the absence of an alphabet of choices in each box we can still find a suitable non-degenerate definition as follows.",
"Suppose we have a unique alphabet of beads $a^{\\prime }_{i}, i=1,..,M$ for the system as a whole.",
"This is in contrast to the heterogeneous case where the unique alphabet $a_{i}$ was relevant only to the $i^{th}$ box.",
"Suppose from this system-wide alphabet, we seek to fill the M boxes each with $t_{i}$ of the $a^{\\prime }_{i}$ beads such that each box contains only one type.",
"The total population of the $M$ boxes is as before $T = \\sum _{i=1}^{M} t_{i}$ .",
"We will renumber them without loss of generality so that $t_{1} \\ge t_{2} \\ge .. \\ge t_{M}$ .",
"We proceed as follows.",
"Select any box and then fill it by selecting $t_{1}$ beads of the same colour.",
"Since we are selecting from $M$ different beads, the probability that we will achieve this selecting at random is $( 1/M )^{t_{1}}$ .",
"For the second box, we then have an alphabet available of $M-1$ , so the probability of filling this box with only one colour of the remaining colours is $( 1/(M-1) )^{t_{2}}$ and so on.",
"The total number of ways $N_{h}$ this can be done is then given by this probability multiplied by the total number of ways in which $T$ beads can be selected, which is $T!$ .",
"$N_{h} = T!",
"\\bigg [ \\big ( \\frac{1}{M} \\big )^{t_{1}} \\times \\big ( \\frac{1}{M-1} \\big )^{t_{2}} \\times .. \\times \\big ( \\frac{1}{1} \\big )^{t_{M}} \\big ] = T!",
"\\prod _{i=1}^{M} \\big ( \\frac{1}{i} \\big )^{t_{i}}$ Rewriting (REF ) then, the information content of this system is $\\log N_{h} = \\log T!",
"+ \\sum _{i=1}^{M} t_{i} \\log \\big ( \\frac{1}{i} \\big ) = \\log T!",
"- \\sum _{i=1}^{M} t_{i} \\log i$ The development (REF )-(REF ) then follows but with $\\log i$ replacing $\\log a_{i}$ .",
"The end result is the equivalent of (REF ) and amounts to $t_{i} \\sim i^{-\\eta },$ where $\\eta $ is some constant.",
"(REF ) states that if we organise these homogeneous boxes in rank order of contents, (i.e.",
"fullest first), then it is overwhelmingly likely that they will be distributed as a power-law in that rank.",
"This is a famous law known as Zipf's law [73].",
"Zipf's law is empirical although others have produced statistical derivations [57], [35].",
"The above derivation therefore serves as an alternative theoretical justification which places it nicely amongst those distributions which can be explained by the approach taken in this paper.",
"Appendix B: CoHSI and Implications for Average Component Length and Long Components Average component length It has been observed experimentally on several occasions [69], [70], [25] that proteins appear to preserve their average length across aggregations within relatively tight bounds.",
"The sharply unimodal peak of Figs.",
"REF , REF as predicted by the theoretical development in this paper suggests that we should not be surprised at this.",
"Indeed at all scales and ensembles the estimates of average protein length will be highly conserved within collections as a result, even though the position of the peak may move a little.",
"In some aggregations, the degree to which the average length is preserved is quite remarkable, for example in Bacteria (Fig.",
"REF ), whilst in Eukarya, there is evidence of some fine structure [25] which invites further analysis Fig.",
"REF .",
"Figure: A plot of the total concatenated length of proteins against the total number of proteins for each species in (A) Bacteria and (B) Eukaryota.",
"Each data point is a species.",
"The gradient of the linearity evident in both plots effectively defines the average protein length for that collection, from .We also note that preservation of the average component length has also been reported for software [23].",
"Measuring average protein length The skewed nature of the distribution of Figs.",
"REF , REF suggests that the use of the mean as a measure of average length alone may be misleading and should be accompanied by other more robust measures such as the median and mode.",
"Table REF demonstrates this by calculating them for the three domains of Archaea, Bacteria, Eukaryota, along with Viruses, as shown in Figs.",
"REF -REF .",
"As expected, the more robust measure of median is less affected by the skew and the medians are therefore considerably less spread out than the means.",
"This is particularly true of viruses which although they have an anomalously large mean in comparison, their median is much closer aligned with those of Archaea and Bacteria.",
"The modes are subject to considerable noise.",
"Table: Different Measures of average length of proteins in the domains of life and virusesTable REF shows that the mean is around $1-3$ times the modal value but we do not compare this using the approximate distribution, Appendix p. REF , as the data are rather noisier than for the full Trembl distribution (compare Fig.",
"REF with Fig.",
"REF ).",
"Long components One of the most important features of power-laws compared with any kind of exponential distribution such as the normal distribution is that “events that are effectively 'impossible' (negligible probability under an exponential distribution) become practically commonplace under a power-law distribution.” [11].",
"The emphatic power-law in both the protein lengths and in software function lengths leads to large ratios when comparing the longest components with the average.",
"For example, proteins of around $36,000$ amino acids have been found and this is $100 \\times $ the average.",
"In terms of the theory developed here, there is no need for any biological reason for very long proteins - they exist simply because of the naturally emerging power-law resulting from consideration of Information-conserving ergodic systems.",
"We note that precisely the same thing has been observed in software [43].",
"Appendix C: CoHSI and Token Alphabets The definition of alphabets, i.e., unique sets of tokens from which choices can be made, poses interesting questions.",
"First of all, we must point out that there is generally no obvious definitive unique alphabet for any system.",
"Alphabets are partly subjective and partly objective because at their heart, they are about how humans categorise systems.",
"Take a simple example of a normally sighted person and a colour blind person both counting the number of differently coloured beads in a collection.",
"Barring counting errors, they will both find the same number of beads in total, however, they will not necessarily agree on the number of beads of each colour.",
"In particular, red-green confusion is likelyhttps://en.wikipedia.org/wiki/Color_blindness.",
"How does this affect the theory we describe here ?",
"It might be thought that by linearly increasing the size of the alphabet, the distribution of the two alphabets are themselves linearly related, i.e.",
"$alphabet1_{i} = constant \\times alphabet2_{i}$ However, this turns out to be not the case and to understand what is happening, we must return to the duality of the asymptotic behaviour (REF ) and (REF ), which we repeat here, $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{a_{i}^{-\\beta }}{Q(\\beta )} $ and its algebraic dual given by $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }} $ Since our normally-sighted person and our colour-blind person will count the same numbers but with different alphabets, we can say that for the normally sighted person, $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{(a^{\\prime }_{i})^{-\\beta ^{\\prime }}}{Q(\\beta ^{\\prime })} $ and for our colour blind person $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{(a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime }}}{Q(\\beta ^{\\prime \\prime })} $ where $a^{\\prime }_{i}, a^{\\prime \\prime }_{i}$ are the two unique alphabets they use and $\\beta ^{\\prime }, \\beta ^{\\prime \\prime }$ their slopes.",
"Since the lengths are unchanged, we can see straight away from (REF ) and (REF ), that the two unique alphabets will themselves be power-law related asymptotically $(a^{\\prime }_{i})^{-\\beta ^{\\prime }} \\sim (a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime }} \\Rightarrow a^{\\prime }_{i} \\sim (a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime \\prime }}, $ where $\\beta ^{\\prime \\prime \\prime } = - \\beta ^{\\prime \\prime } / \\beta ^{\\prime }$ .",
"This leads us to predict a general rule In any consistent categorisation of the same system with different unique alphabets, the distributions of the unique alphabets will also be related by a power-law.",
"Music alphabets Consider an example from the world of music.",
"Music is also a system of discrete components in the sense described here, Table REF .",
"In recent years, discrete formats representing the notes and structure of a musical composition have appeared, for example MusicXML as referenced in the main text.",
"If we consider the 88 notes of a full-scale piano as defining the possible notes in the equal-tempered scale used in the vast majority of published music, then we have a candidate unique alphabet $a^{\\prime }_{i}$ of 88 (no-duration alphabet).",
"However, we can subdivide this alphabet quite naturally and consistently into notes and duration.",
"The standard durations are divided into fractions of a whole note as breve (2), semi-breve (1), minim (1/2), crotchet (1/4), quaver (1/8), semiquaver (1/16) and demisemiquaver (1/32).",
"There are others defined off either end of this list but they are obviously rare as there were no occurrences in the body of music studied here.",
"This gives seven flavours of each note and expands the unique alphabet considerably to $88 \\times 7 = 616$ items, (duration alphabet).",
"Figs.",
"REF shows the distribution of the two alphabets no-duration and duration, measured on the same body of music.",
"As expected from (REF ) they both exhibit power-law behaviour.",
"For the no-duration alphabet R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $40.0-400.0$ , with an adjusted R-squared value of $0.9937$ .",
"The slope is $-1.66 \\pm 0.01$ .",
"For the duration alphabet R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $40.0-4000.0$ , with an adjusted R-squared value of $0.9951$ .",
"The slope is $-1.44 \\pm 0.02$ .",
"These are emphatic results.",
"Figure: The log-log\\log -\\log ccdf of the duration and no-duration alphabets measured on the same body of music used in this study (A), and a comparison of the two alphabets as log-log\\log -\\log showing their clear linear power-law relationship (B).",
"Moreover in Fig.",
"REF which compares the two alphabets directly on a $\\log -\\log $ scale, the predicted power-law relationship of (REF ) is clearly visible.",
"R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $10.0-500.0$ , with an adjusted R-squared value of $0.9879$ .",
"The slope is $1.181 \\pm 0.002$ , also consistent with (REF ).",
"This too is an emphatic result.",
"We believe that this throws some light (but does not necessarily explain) why the predicted reciprocal relationship between the power-law slope of the unique alphabet distribution and that of the length distribution is not adhered to closely in our data.",
"There are a potentially infinite number of alphabets related themselves by power-laws, but only one length distribution.",
"Protein alphabets For proteins, with increasing sophistication we are able to recognise not just the 22 amino acids transcribed directly from DNA but also the increasingly large number of known post-translational modifications (PTM) which dramatically extend and continue to extend the size of the unique amino acid alphabet that allows us to categorise proteins.",
"Will this process of discovery stop ?",
"We argue that it cannot as it is intimately linked to the total number of proteins known, and this continues to grow apace.",
"We can gain insight into the growth of the unique protein alphabet by studying collections, such as the SwissProt database, over different revisions [61], [63], [64] as it incorporates PTM information from the Selene project [53].",
"Figure: The classic linear signature of a power-law distributions of unique amino acid alphabets for SwissProt 13-11 and the more up to date SwissProt 15-07 on a log-log\\log -\\log ccdf.Fig.",
"REF shows the frequencies of the unique amino acid alphabet recorded in the proteins of SwissProt release 13-11 and SwissProt release 15-07 as ccdfs in $\\log -\\log $ form.",
"Although the maximum unique alphabet for amino acids is 22 for those decoded directly from DNA, we should note that finding a protein with all 22 would be unlikely as both the 21st and 22nd amino acids, selenocysteine and pyrollysine, are rare in proteins.",
"Pyrrolysine is found in methanogenic archaea and bacteria and is encoded by a re-purposed stop codon (UAG), requiring the action of additional gene products to accomplish its incorporation [47].",
"Thus it is not easy to annotate pyrrolysine from the gene sequence alone, and direct chemical analysis of proteins would be more informative.",
"Selenocysteine is found in all domains of life, but the selenoproteome is small [49] and an additional concern is misannotation in the databases, because a stop codon (UGA) is re-purposed from “halt translation” to “incorporate seleocysteine” by additional sequences downstream of the gene as well as other trans-acting factors [36].",
"As a result, any unique amino acid count beyond 21 must contain post-translationally modified amino acids and we note the following: Almost the whole of the tail of Fig.",
"REF consists of proteins in which there must be post-translationally modified amino acids, effectively doubling the unique alphabet derived directly from DNA.",
"The increase in numbers between 20 and 26 unique amino acids can be seen by the slightly displaced points upwards in the SwissProt 15-07 dataset compared with the SwissProt 13-11 dataset.",
"The remainder of the tail significantly straightens with the more comprehensively annotated SwissProt 15-07 presumably due to reduction in noise and increasing numbers.",
"The linearity in each tail strongly supports equation (REF ) even though the range is less than 1 decade because the slope is so steep arising from the current paucity of the unique alphabet.",
"For SwissProt 13-11 of Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $8.124 \\times e^{-13}$ over the range $19 - 35$ , with an adjusted R-squared value of $0.9698$ .",
"The slope is $-15.91 \\pm 0.56$ .",
"For SwissProt 15-07 of Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $19-33$ , with an adjusted R-squared value of $0.9968$ .",
"The slope is $-18.56 \\pm 0.19$ .",
"We should also clear up a potential point of confusion here.",
"One reviewer stated that the distribution of number of proteins against unique amino acid count had a “tiny power-law tail” and that the distribution was uniform.",
"The reviewer reasoned that this explained why the average length of proteins was highly conserved in contrast to our explanation in the Appendix p. REF .",
"It is indeed true at the present rate of knowledge that the distribution of unique amino acids has a tiny power-law tail but the distribution is anything but uniform as we can see in SwissProt 13-11 by considering two different kinds of plot.",
"Fig.",
"REF plots the logarithmic frequency of proteins against their unique amino acid alphabet.",
"Whatever this distribution is, it is certainly not uniform, although we know the tail from $19-35$ amino acids is an accurate power-law from the analysis of the data shown in Fig.",
"REF .",
"Figure: The frequency of proteins plotted against the unique amino acid count for SwissProt 13-11 on a log-linear\\log -linear plot (A), and the frequency at which each amino acid occurs including PTM amino acids plotted in rank order on a log-log\\log -\\log ccdf (B).In contrast, Fig.",
"REF plots the occurrence rate of each unique amino acid including post-translational modification across the entire SwissProt 13-11 distribution, of which there are more than 800 recorded by the Selene project.",
"In other words it shows in how many proteins each amino acid appears, organised in rank order.",
"This matches the homogeneous model discussed in Appendix A p. REF , and a power-law in the tail is evident as expected.",
"We note in passing a possible intriguing relationship between the overhang in Fig.",
"REF between around 10 and 30 on the x-axis and the contemporary question of PTM undercounting [65].",
"R lm() reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $22.0-800.0$ , with an adjusted R-squared value of $0.9778$ .",
"The slope is $-2.63 \\pm 0.31$ .",
"This too is an emphatic result.",
"Appendix D: Power-laws, Statistical Rigour and Rules of Thumb Power-laws are ubiquitous in nature and are generated by a number of mechanisms, [42].",
"In essence, power-law behaviour can be represented by the pdf (probability density function) p(s) of entities of size s appearing in some process, given by a relationship like $p(s) = \\frac{k}{s^b} $ where $k, b$ are constants.",
"On a $\\log p - \\log s$ scale the pdf is a straight line with negative slope $- b$ .",
"It can easily be verified that the equivalent cdf (cumulative density function) $c^{\\prime }(s)$ derived by integrating (REF ) also obeys a power-law $\\sim s^{-b+1}$ , (for $b \\ne 1$ ).",
"The classic linear signature of a power-law tail in a ccdf (complementary cumulative distribution function) is usually shown as in Fig.",
"REF which displays $c(s) = 1 - c^{\\prime }(s)$ .",
"Figure: The classic linear signature of a power-law in the tail of a log-log\\log -\\log ccdf.For noisy data, the ccdf form is used most often because of its fundamental property of reducing noise present in the pdf, as noted by [42].",
"This effect is because the ccdf is obtained by integration.",
"This reduces noise inherent in the pdf preserving any power-law behaviour while allowing any linearity to be measured more accurately.",
"The benefit of this can be clearly seen in data extracted from software systems as shown in Figs.",
"REF (the pdf) and REF (the corresponding ccdf).",
"The effect is even more pronounced in the inherently more noisy protein data.",
"Figure: The pdf (A) and the ccdf (B) of the length distributions of the same large population of software.Whilst on the subject of significance, a rule of thumb often used to determine the existence of a power-law is that it should appear over two or more decades in the x-axis of the ccdf.",
"This is useful only as a rule of thumb when the slope is not too steep.",
"Since the scale of the y-axis on the $\\log -\\log $ ccdf is effectively the scale of the x-axis times the slope of the power-law, then a steep slope would require a large scale of y-axis frequency measurements to provide the rule of thumb of 2-3 decades in the x-axis.",
"For example, a slope of around 3 would require y-axis frequency measurement only over some 6-9 decades to give the rule of thumb of 2-3 decades in the x-axis, which is reasonable.",
"On the other hand, if the slope is 10, y-axis frequency measurement over some unreasonably large 20-30 decades would be required to give the rule of thumb of 2-3 decades in the x-axis.",
"This is an important point for the protein studies considered here wherein we are investigating the predicted power-law in unique alphabet.",
"Here, the x-axis is the unique alphabet of amino acids.",
"The size of this alphabet is small at the current state of knowledge, leading to a steep power-law slope.",
"In such situations, we fall back on normal procedures of statistical inference to replace subjective belief with objective perception and therefore all that is required is that there is statistically significant linearity in the tail of the distribution of the $\\log -\\log $ ccdf for the number of measurement points used.",
"A rule of thumb guides but does not replace normal statistical inference whereby a result is either significant at some level or it is not for a given model and data.",
"This effect can be seen in Fig.",
"REF , a $\\log -\\log $ ccdf of the occurrence rate in the size of the unique alphabet in SwissProt version 13-11 [61], which is merged with the Selene post-translational modification data [53], [62].",
"These represent amongst the best annotated protein data including the rapidly growing field of post-translational modification (PTM), a process whereby nature alters some of the amino acids by covalent processes such as glycosylation, phosphorylation, methylation, acylation, etc., thereby extending the unique alphabet beyond the 22 amino acids directly coded from DNA [2], [31], [71], [46], [6].",
"Figure: The highly linear tail of the occurrence frequency of unique alphabet sizes in the SwissProt 13-11 protein distribution merged with the Selene 2013 post-translational modification annotation, extending the range of the natural unique alphabet of 22 amino acids directly coded from DNA to just over 30 in this dataset.As can be seen, the tail of SwissProt 13-11 in Fig.",
"REF , covers only a range up to just over 30 even though there are thousands of PTM known or predicted by existing research.",
"As a result there are only nine data points for unique alphabets of size greater than 20, although each point is an aggregate of a large number of observations.",
"An R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $6.576 \\times e^{-12}$ over the range $21.0-30.0$ , with an adjusted R-squared value of $0.9951$ .",
"The slope is $-22.9 \\pm 0.2$ .",
"This is a statistically emphatic result for the existence of a power-law, even though the size of the slope is steep because the x-axis is restricted.",
"Later versions of SwissProt with Selene annotations increase the PTM alphabet, Appendix C p. REF .",
"Power-law behaviour has been studied in a wide variety of environments starting with the pioneering work of [73] (linguistics) and followed by [48] (economic systems) and the excellent reviews by [38] and [42].",
"In software systems significant activity, much of it recent, [10], [37], [41], [7], [18], [45], [3], [12], [43] and [22] has addressed power-law behaviour in various contexts.",
"To give some idea of the scope of these, Mitzenmacher [37] considers the distributions of file sizes in general filing systems and observed that such file sizes were typically distributed with a lognormal body and a Pareto (i.e.",
"power-law) tail.",
"Gorshenev and Pis'mak [18] studied the version control records of a number of open source systems with particular reference to the number of lines added and deleted at each revision cycle.",
"Louridas et.",
"al.",
"[43] show evidence that power laws appear in software at the class and function level and that distributions with long, fat tails in software are much more pervasive than previously established.",
"Appendix E: Hartley-Shannon Information, parsimony and token-agnosticism Information theory has its roots in the work of Hartley [20] who showed that a message of N signs (i.e.",
"tokens) chosen from an alphabet or code book of S signs has $S^{N}$ possibilities and that the quantity of information is most reasonably defined as the logarithm of the number of possibilities or choices $\\log S^{N} = N \\log S$ .",
"To gain insight into the reason why the logarithm makes sense, consider Fig.",
"REF .",
"The number of choices necessary to reach any of the 16 possible targets is the number of levels which is $\\log _{2}$ (number of possibilities).",
"The base of the logarithm is not important here.",
"Figure: A binary tree.",
"Each level proceeding down can either go left or right.",
"There are four levels leading down to one of 2 4 2^{4} = 16 possibilities.",
"Only four choices are needed to reach any of the possibilities.",
"We note that log 2 (16)=4\\log _{2}(16) = 4.",
"Here the number 7 has been singled out by the choices left, right, left, right as the tree is descended.Information theory was developed substantially by the pioneering work of Shannon [54], [55] and many researchers since but we have remained with Hartley's original clear vision and most importantly its token-agnosticism.",
"We re-iterate that is important not to conflate information content with functionality or meaning and Cherry [8] specifically cautions against this noting that the concept of information based on alphabets as extended by Shannon and Wiener amongst others, relates only to the symbols themselves and not their meaning.",
"Indeed, Hartley in his original work, defined information as the successive selection of signs, rejecting all meaning as a mere subjective factor.",
"In the sense used here therefore, Conservation of H-S Information will be synonymous with Conservation of Choice, not meaning.",
"This turns out to be enough to predict the important system properties detailed in this paper.",
"In other words, those properties depend only on the alphabet and not on what combining tokens of the alphabet might mean in any human sense.",
"We believe CoHSI therefore represents the most parsimonious theory capable of explaining all the observed features of the numerous disparate datasets analysed in this paper."
],
[
"Introduction",
"Such uncanny similarity in very disparate systems strongly suggests the action of an external principle independent of any particular system, so to explore the concepts, let us first consider some examples of discrete systems.",
"Here a discrete system is considered to be a set of components, each of which is built from a unique alphabet of discrete choices or tokens.",
"Table REF illustrates this nomenclature and its equivalents in various kinds of system.",
"Table: Comparable entities in discrete systems considered in this paper.The first thing to notice is that this seems a very coarse taxonomy.",
"In the case of proteins, there is no mention of domain of life or species or any other kind of aggregation.",
"Similarly with computer programs, we do not include the language in which they were written or the application area.",
"The reason for this as will be seen later is that these considerations are irrelevant.",
"We will expand on each system later as we apply the theory and discuss its ramifications for each, but the most important concept to grasp now is that there are two measures, total length and unique alphabet, which will turn out to be fundamental across all such systems.",
"To illustrate, consider two hypothetical components consisting of the strings of letters shown as rows in Table REF .",
"Table: Two simple strings of letters.The total length in letters of the first string in Table REF is 25.",
"This is made up of six occurrences of A, 6 of B, 1 of C, 1 of D, 3 of F, 2 of G and 6 of Q.",
"We therefore define the unique alphabet of this component to be of size 7, corresponding to ABCDFGQ.",
"In other words, the string is built up from one or more occurrences of each and every letter in its unique alphabet.",
"The second string is made up of 20 characters consisting of 2 of X, 4 of Y, 4 of Z, 4 of T, 2 of W, 2 of I and 2 of S. Its unique alphabet is therefore XYZTWIS.",
"In other words, these two strings have different lengths but they have the same size unique alphabet.",
"As we will see by considering the information content shortly, this property will turn out to be fundamental - the actual letters making up the unique alphabet will turn out to be irrelevant and the two strings inextricably linked in an information theoretic sense.",
"The methodology we use combines two disparate but long-established methodologies - Statistical Mechanics and Information Theory in a novel way using the simplest possible definition of Information originally defined by Hartley [20].",
"We will show that this alone is sufficient to predict all the observed features of Figs.",
"REF and REF and why indeed they are so similar.",
"Statistical Mechanics can be used to predict component distributions of general systems made from discrete tokens subject to restrictions known as constraints.",
"Its classical origins can be found in the Kinetic Theory of Gases [59] (p.217-) wherein constraints are applied by fixing the total number of particles and the total energy [17].",
"However, the methodology is very general and can equally be used with different constraints on collections of proteins (made from amino acids), software (made from programming language tokens) and, as we shall see, simple boxes containing coloured beads.",
"Hartley-Shannon Information theory is the result of the pioneering works of Ralph Hartley [20] as developed later by Claude Shannon [54], [55].",
"It forms the backbone of modern digital communication theory and is also astonishingly versatile."
],
[
"The Hartley-Shannon Information Content of a component, in the sense we use here is simply defined to be the natural logarithm of the total number of distinct ways of arranging the tokens of that component, without any regard for what those tokens actually mean."
],
[
"The motivation behind the choice of this form of information is that Figs.",
"REF and REF derive from systems with little if anything in common, but Hartley's definition of information is token-agnostic; in other words the meaning of the tokens is irrelevant.",
"Furthermore, its use favours the ergodic nature of classical Statistical Mechanics with token choice equally likely.",
"Combining Statistical Mechanics and some form of Information Theory is not new.",
"For example, building on the maximum entropy framework of [30] rooted in probability theory, Frank demonstrates that by combining Shannon Information [54] in a maximum entropy context, the common patterns of nature - Gaussian, exponential, power-law - as predicted by neutral generative processes, naturally emerge [15].",
"Here a neutral generative process assumes that each microscopic process follows random stochastic fluctuations.",
"Frank's use of information can be interpreted as additional knowledge about a system constraining the possible patterns which might result.",
"For example, amongst other things, he demonstrates that simply from an assumption about the measurement scale and knowledge about the geometric mean, a power-law arises.",
"Frank [15] also stresses the need to distinguish between the generation of patterns by purely random or neutral process on the one hand, and the generation of patterns by aggregation of non-neutral processes in which non-neutral fluctuations cancel in the aggregate."
],
[
"Power-laws",
"Power-laws (a.k.a the Pareto distribution) are ubiquitous in nature and are emphatically present in all of the datasets analysed in this paper.",
"In essence they have a probability distribution which depends on a power $b > 1$ of the independent variable.",
"$p(x) \\sim x^{-b}$ It is important to note that there are numerous known processes which lead to power-laws [42] and indeed the literature abounds with studies, from the original empirical work of Zipf [73], and the earliest generative models such as preferential attachment [57] onwards.",
"It is also important to note that other statistical distributions notably lognormal frequently occur in natural phenomena [38].",
"In earlier work [22], [23], [25] using the original and arguably the most parsimonious definition of information, [20], embedded as a constraint directly within the classical Statistical Mechanics framework, we demonstrated with compelling support from measurement, that for large components, this alone was enough to generate the extraordinarily precise power-laws observed not only in the length distribution of proteins and software, but also in the distributions of the alphabet of unique tokens.",
"However given the ubiquity of power-laws (and indeed the reasons for this [15]), perhaps the most compelling reason for accepting the power-law generation inherent in CoHSI as opposed to other generative mechanisms, is to realise that conservation principles cannot be applied selectively.",
"They are inherently global and either apply everywhere or nowhere and therefore our use of CoHSI must explain satisfactorily all of the observed properties of the length distributions which appear as Figs.",
"REF , REF , including the sharply unimodal behaviour for smaller values of the independent variable.",
"Our novel contributions to the existing body of work are:- We show that the token-agnostic and scale-independent CoHSI does indeed predict all the qualitative features of the distributions of Figs.",
"REF and REF with no other assumptions apart from the conventional use of Stirling's theorem to approximate factorials (Appendix A p. REF ).",
"This is particularly significant because distributions of the nature of Figs.",
"REF and REF are often treated by combining two separate distributions, such as lognormal with a power-law tail [39].",
"As we show, a single implicit distribution which naturally follows from CoHSI is sufficient, thereby emphasizing the parsimony of this approach, CoHSI naturally leads to an alternative proof of Zipf's law (Appendix A p. REF ), as we might expect for this Conservation principle, We enlarge on the results originally derived asymptotically, [25] that average component lengths (protein, software function ...) are highly conserved across aggregations.",
"We also point out why very long components naturally must appear quite frequently without any obvious domain-based reason, (Appendix B p. REF ), We show that the asymptotic duality first reported in [25] between length distribution and alphabet size distribution, naturally implies that different but consistent alphabets for the same system must also be related by power-law, (Appendix C p. REF ).",
"It also follows that the maximum size of a unique alphabet is intimately related to the total number of components through the slope of the corresponding power-law, We give experimental confirmation to high levels of significance in multiple disparate datasets at different levels of aggregation for these predictions including systems which contain both heterogeneous and homogeneous behaviour.",
"We stress we are not data-fitting here, and we are not explicitly applying constraints on knowledge of types of mean or variance.",
"Instead, both the sharp unimodal peak and the very precise power-law tail of Figs.",
"REF and REF naturally emerge from the single Conservation principle, just as the Maxwell-Boltzmann distribution naturally emerges from the Conservation of Energy in Kinetic Theory [59].",
"To understand this seemingly ad hoc assumption, we must delve into ergodicity and consider exactly what happens in classical statistical mechanics when we apply the constraints of total size and total energy to find the most likely distribution of particles amongst energy levels.",
"Perhaps the most important thing to realise about the statistical mechanics methodology is that it is simply a mathematical technique.",
"The fact that it is energy (a physical quantity) which is being conserved along with the total number of contributing particles in kinetic theory is irrelevant - conventionally, anything additive can be conserved, however abstract.",
"The real world only intrudes into the statistical mechanics of Kinetic Theory via Clausius' entropic version of the Second Law of Thermodynamics.",
"Without this, statistical mechanics simply answers the mathematical question of the most likely distribution of particles or tokens when their total number and their total payload (in this case energy) is conserved.",
"In other words, amongst all the possible systems with that number of particles and that total payload (the ergodic ensemble), then presented with one of them, it is most likely to follow the distribution predicted by statistical mechanics for the ensemble.",
"It doesn't have to, but it is overwhelmingly likely that it does.",
"In kinetic theory, the payload happens to be a physical additive quantity, the energy, and the corresponding distribution of particles is then the Maxwell-Boltzmann distribution, but to say something about a system, statistical mechanics does not need to be rooted in tangibly defined entities in the physical world.",
"We simply have to interpret the result appropriately.",
"Now it so happens that Hartley-Shannon information content, like energy, is also additive for independent sub-systems.",
"The total energy $E$ of two sub-systems with individual energies $E_{1}$ and $E_{2}$ is $E = E_{1} + E_{2}$ .",
"Similarly, by virtue of its logarithmic definition, the total Hartley-Shannon information content $I$ of two sub-systems with individual information content $I_{1}$ and $I_{2}$ is $I = I_{1} + I_{2}$ .",
"The difference between the two is that energy is a physical quantity, whereas Hartley-Shannon information content is just the $log$ of the total number of ways of arranging something [8].",
"Mathematically it resembles entropy but we should be hesitant about reading too much into this [33], p. 144.",
"However, we may still use the formalism of statistical mechanics, which we do here."
],
[
"It is in this sense that we assert that Conservation of Hartley-Shannon Information underlies the length distribution of discrete systems whatever their provenance.",
"It is a natural consequence of statistical mechanics that if we are presented with a system with a total number of tokens and a total information payload, then it is overwhelmingly likely to follow a certain size distribution as described in what follows.",
"The scale-independence of the results follows from the fact that, given any system, it is the properties of the ergodic system of the same parameters which defines the most likely distributions to occur in any one of its constituents.",
"This paper contains many examples of real systems at all levels of aggregation where precisely this size distribution is found, just as we expect from the theory."
],
[
"Statistical Mechanics",
"Statistical Mechanics connects the minutiae of systems of large numbers of small particles to macroscopic properties of those systems [17].",
"Like Hartley-Shannon Information, it is quite astonishingly versatile and arose originally in the Kinetic Theory of Gases, leading eventually in the hands of James Clerk Maxwell and later Ludwig Boltzmann in the 19th century, to the statistical distribution of velocities and on to the concept of Entropyhttps://en.wikipedia.org/wiki/Kinetic_theory_of_gases, accessed 25-May-2017..",
"The methodology of Statistical Mechanics leads naturally to links with probability distributions and energy in the case of gases.",
"Here we use the same methodology but by embedding CoHSI as a constraint in Statistical Mechanics rather than Conservation of Energy, we demonstrate that this links Hartley-Shannon Information directly to probability distributions of component length and unique alphabet size in discrete systems.",
"We distinguish between two fundamental types of system which lead naturally to two different definitions of information.",
"Both contain components made from discrete tokens as described above but with one fundamental difference.",
"Heterogeneous We define heterogeneous systems here as systems wherein a component has more than one kind of distinct token.",
"This would include systems as disparate as the proteome, software and digital representations of music.",
"Appendix A p. REF contains a detailed development of these systems.",
"Homogeneous We define homogeneous systems here as systems wherein a component has only one kind of distinct token and each distinct token is unique to one component.",
"This would include textual documents and word counts as well as the distribution of elements in the universe.",
"In such systems, a heterogeneous definition of information would be degenerate and a different definition is necessary.",
"Appendix A p. REF contains the detailed development for this kind of system leading directly to an alternative proof of Zipf's law.",
"However, this is irrelevant as far as statistical mechanics goes because for a given definition of Hartley-Shannon Information, the methodology simply tells us the most likely, or canonical distribution for ergodic systems with the same fixed size and fixed information content, howsoever defined.",
"For heterogeneous systems, we will refer to the resulting distributions as the heterogeneous CoHSI distribution.",
"The corresponding distribution for homogeneous systems is simply Zipf's law.",
"The theory described in Appendix A p. REF predicts that the length distribution of a heterogeneous discrete system such as the proteome or software systems, at all scales with total number of tokens $T$ and total Hartley-Shannon Information $I$ is the solution $(t_{i},a_{i})$ of the implicit pdf corresponding to $\\log t_{i} = -\\alpha -\\beta ( \\frac{d}{dt_{i}} \\log N(t_{i}, a_{i}; a_{i} ) ), $ with $T = \\sum _{i=1}^{M} t_{i}$ and $I = \\sum _{i=1}^{M} I_{i}$ where $t_{i}, a_{i}, I_{i}$ are the length in tokens, the size of unique alphabet of tokens and the Hartley-Shannon Information content, respectively, of the $i^{th}$ component of a system containing $M$ components in all.",
"$\\alpha $ and $\\beta $ are Lagrange undetermined multipliers.",
"$N(t_{i}, a_{i}; a_{i})$ is the total number of ways of choosing $t_{i}$ tokens at random, choosing from a replaceable unique set of tokens $a_{i}$ .",
"We write $N(t_{i}, a_{i}; a_{i})$ in this special form to remind us of the recursive nature of its construction.",
"Here, $t_{i}$ is the independent variable and $a_{i}$ plays a dual role acting also as the scaled frequency of occurrence.",
"In addition, for components which are much longer than their unique alphabet, $t_{i} \\gg a_{i}$ , the full solution (REF ) tends to the asymptotic pdf (probability distribution function) [23] given by $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{a_{i}^{-\\beta }}{\\sum _{i=1}^{M} a_{i}^{-\\beta }}, $ which has an algebraic dual pdf [25] given by $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }}, $ where $A = \\sum _{i=1}^{M} a_{i}$ (REF ) and (REF ) show that the tails of both unique alphabet size distributions and length distributions respectively of the M components will be asymptotically power-law as emphatically confirmed in [25].",
"A typical solution of (REF ) is shown as Fig.",
"REF .",
"Figure: A typical solution of () shown as a pdf.",
"Both the sharp unimodal peak and power-law tail can be seen clearly.Before applying this theory predictively to various systems so that we may test it, we make two comments.",
"The Lagrange multipliers $\\alpha , \\beta $ are undetermined by the methodology of statistical mechanics.",
"$\\alpha $ parameterises the total size of the system and therefore emerges naturally as a normalisation condition so that a pdf results.",
"$\\beta $ is more interesting.",
"It parameterises the total payload.",
"The payload in our theory is Hartley-Shannon information which as described in Appendix A p. REF , depends on the size of the alphabet we use to categorise a discrete system.",
"Small alphabets correspond to large $\\beta $ and vice versa.",
"The implication of this indeterminism is that the range of values of $\\beta $ which emerge via information theory can be much wider than those tied to physical systems, (which are mostly in the range 1.5-4).",
"(In the Maxwell-Boltzmann distribution where the payload is energy, Appendix A p. REF , it is fixed by being closely linked by Boltzmann's constant to the temperature, through the Second Law of Thermodynamics.)",
"Dual regime behaviour is often identified and modelled as lognormal transitioning to power-law [39], [37].",
"We stress here that no such juxtaposition of distinct pdfs is necessary with the theory presented here.",
"Instead the sharp unimodal peak and the power-law regime of Figs.",
"REF , REF naturally emerge according to the implicit solution of (REF ) as $t_{i} \\rightarrow 1$ from large values as can be seen in Fig.",
"REF .",
"This is a direct consequence of CoHSI in an ergodic system as described in Appendix A p. REF .",
"The observed shape of the sharply unimodal regime of Fig.",
"REF simply lends itself to a lognormal fit.",
"We also note that the unusual implicit behaviour inherent in (REF ) is also a feature of modified entropy definitions such as Tsallis entropy [67], [66] constructed to account for non-additive entropy.",
"Tsallis entropy is a modification of Shannon entropy with an additional parameter.",
"In our development, the implicit behaviour emerges naturally from the one assumption of CoHSI.",
"We now apply these conclusions predictively to various systems.",
"Proteins are constructed as strings of amino acids corresponding to the heterogeneous model we describe here, Appendix A p. REF .",
"They are represented in exactly the same way as the strings of letters in Table REF but the unique alphabet from which the letters are chosen is the 22 unmodified amino acids which are coded directly from DNA [60], [16], supplemented with modified versions produced by a process known as Post-Translational Modification, of which there are already thousands known [31], [46], [5].",
"Table REF shows two small proteins, one from an Archaean and the other from a virus [64] along with their sequences in their single letter abbreviations for compactness [25] Table: Sequences of two small proteins.Biologically, these two proteins differ significantly.",
"They have different lengths, (32 and 44 amino acids respectively); are built using different amino acids; and they have very distinct structures and functions.",
"FLA1_METHU is built from the unique amino acid alphabet FSGLEAIVLYMGT (13) and VE5_PAPVR is built from the unique amino acid alphabet MNHPGLFTAVQWDCRI (16).",
"Following our earlier argument about unique alphabets, it does not matter if an amino acid is present once or more often in the sequence.",
"If it is present at all, then it contributes a count of 1 to the unique amino acid count.",
"While these two numbers are clearly independent of any physicochemical properties of the amino acids, they are fundamental in determining the length distribution of any aggregation of proteins.",
"The protein sequences are collected in public databases from which they can be downloaded and analysed [64].",
"There are currently more than 80 million proteins in TrEMBL version 17-03 built from almost 27 billion amino acids, (the most recent version analysed before writing this paper).",
"The proteins vary in length from just four amino acids to over 36,000 amino acids but their average length is only around 300 amino acids.",
"The reason for the existence of such long proteins is directly predicted by the development of theory which follows later in this paper, Appendix B, p. REF .",
"The length of a protein is of course one of the factors which determines its folding properties and therefore its functionality [34], [29], [32].",
"As originally shown in [25], the tails of the alphabet and length distributions are both power-law to a high degree of precision.",
"The protein alphabets are analysed in Appendix C p. REF .",
"The ccdf (complementary cumulative distribution function) of the length distribution corresponding to Fig.",
"REF is shown as Fig REF .",
"Figure: The data of Fig.",
"shown as a log-log\\log -\\log ccdf."
],
[
"The linearity in the tail of Fig.",
"REF provides striking confirmation of equation (REF ).",
"The R lm() function reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $300.0-30000.0$ , with an adjusted R-squared value of $0.9942$ .",
"The slope is $-3.13 \\pm 0.20$ ."
],
[
"Aggregations by domain and species",
"As stated earlier, the canonical shape of Figs.",
"REF , REF occurs at each level of aggregation in these systems.",
"This we now show for both the domains of life and also down to individual species.",
"In contrast to TrEMBL, the SwissProt database [63] provides a smaller but well annotated set of data suitable for the extraction and analysis of data from taxa at diverse levels of aggregation, from the highest taxonomic classification shown (the three domains of life) down to individual species.",
"In Figs.",
"REF -REF : are Archaea (Figs REF : 19,063 species), Bacteria Figs (REF : 329,526 species) and Eukarya Figs REF : 177,020 species).",
"Included for comparison are the viruses Figs (REF : 16,423 species).",
"In every case, the characteristic qualitative signature of Fig.",
"REF is evident in the domain of life (with variations inevitably depending on the sample size), and even (in the case of viruses), a dataset outside the domains of life.",
"The pdfs are scaled separately to show the qualitative similarity whilst Fig.",
"REF shows the matching absolutely-scaled ccdfs and the emergence of the power-law tail in each collection.",
"Figure: The frequency distributions of protein lengths in the three domains of life, (A): Archaea, (B): Bacteria and (C): Eukarya, along with (D): Viruses.Figure: The three domains of life and viruses shown as a log-log\\log -\\log ccdf.",
"The significantly shallower slope in Viruses is notable.The only remaining approximation of CoHSI is that the number of components (i.e.",
"proteins in this case) be reasonably large, so a critical test case is the analysis of individual species, where the protein databases allow us to analyze small sets of proteins naturally defined by species.",
"To demonstrate the resilience of CoHSI, we consider species with very different numbers of unique proteins.",
"Figs.",
"REF -REF show the length distributions of proteins in (Fig.",
"REF ) humans (126,468 proteins); (Fig.",
"REF ) maize (85,311 proteins); (Fig.",
"REF ) fruit fly (18,966 proteins); and (Fig.",
"REF ) Haloarcula marismortui (3,892 proteins) respectively.",
"Even in the smallest of these datasets, H. marismortui, a halophilic red Archaeon found in the extreme environment of the Dead Sea, the canonical shape of Fig.",
"REF is apparent.",
"The pdfs are again scaled separately to show the qualitative similarity whilst the corresponding ccdfs are shown absolutely scaled in Fig.",
"REF .",
"Figure: The frequency distributions of protein lengths in four species, (A): Human, (B): Maize and (C): Fruit fly, along with (D): Haloarcula marismortui.Figure: Four species shown together as a log-log\\log -\\log ccdf.Computer programs are an invention of the human mind following the ground-breaking work of Alan Turing.",
"In the 50 or so years since they first appeared, many programming languages have arisen, from which computer programs of almost limitless functionality are built.",
"The individual bases or alphabet of a programming language are called tokens and may take two forms; the fixed tokens of the language as provided by the language designers, and the variable tokens.",
"Fixed tokens include (in the languages C and C++ for example) keywords such as if, else, while, {, }.",
"These can not be changed, the programmer can only choose to use them or not.",
"Variable tokens, with some small lexical restrictions (such as the common requirement for identifiers to begin with a letter), can be arbitrarily invented by the programmer whilst constructing their program.",
"These might be names such as numberOfCandidateCollisions or lengthOfGene or constants such as 3.14159265.",
"There are many programming languages but all obey the same principles and every form of software system evolves from such tokens.",
"They are therefore another example of the heterogeneous model we describe here, Appendix A p. REF .",
"It should be noted that classifying programs in terms of fixed and variable tokens is not new and appeared at least as early as 1977 in the influential work of Halstead who called them operators and operands, [19].",
"He developed his work to define various dependent concepts such as software volume and effort and tested them against programs of the time.",
"This was further elaborated by Shooman [56].",
"A different approach is used here which borrows from the methods of variational calculus.",
"Computer programs are often huge.",
"The software deployed in the search for the recently discovered Higg's boson comprises around 4 million lines of code [50].",
"At an average of around six tokens per line of code, this corresponds to some 20 million tokens, although this is still less than 1% of the human genome in which the tokens are the four bases adenine, cytosine, guanine and thymine.",
"The largest systems in use today appear to be around 100 million lines of source code [40], corresponding to perhaps 15% of the number of tokens of the human genome.",
"The (largely) open systems used to test the model described here total almost 100 million lines, (specifically 98,476,765 lines), totalling some 600 million tokens.",
"(If around 6 tokens per line seems a little low, it should be recalled that lines of code include comment lines here in line with common practice, whilst token counts do not.)",
"As an example of the nomenclature used here, consider the following simple sorting algorithm written in C, for example [52].",
"void bubble( int a[], int N ) { int i, j, t; for( i = N; i >= 1; i--) { for( j = 2; j <= i; j++) { if ( a[j-1] > a[j] ) { t = a[j-1];a[j-1] = a[j];a[j] = t; } } } } This algorithm contains 94 tokens in all based on 18 of the fixed tokens and 8 of the variable tokens of ISO C, so the size of the unique alphabet for this component is $18+8 = 26$ .",
"Note that extracting the tokens of programming languages to assemble these measures requires the development of compiler front-end tools [1], [51].",
"These are included in the reproducibility materials, notably associated with [23].",
"We have from (REF ) and (REF ) that power-laws in both the unique alphabet distributions and length distributions are overwhelmingly likely to appear in the tails of the distributions.",
"Figs.",
"REF and REF show the $\\log -\\log $ ccdf plots for the unique alphabet and length distributions respectively of 100 million lines of source code in seven different programming languages [23].",
"Figure: The unique alphabet a i a_{i} (A) and length distributions t i t_{i} (B) of 100 million lines of source code in 7 different programming languages shown as ccdfs."
],
[
"The linearity in each tail is striking confirmation of (REF ) and (REF ).",
"For Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $80.0-3500.0$ , with an adjusted R-squared value of $0.9975$ .",
"The slope is $-2.15 \\pm 0.08$ .",
"For Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $200.0-68000.0$ , with an adjusted R-squared value of $0.9995$ .",
"The slope is $-1.47 \\pm 0.03$ ."
],
[
"We note in passing that (REF ) and (REF ) suggest that the slopes are reciprocals of each other, ($\\beta $ and $1/\\beta $ ).",
"They are clearly not so here but this raises an interesting question concerning the choice of alphabets which is explained in the Appendix C p. REF ."
],
[
"Aggregations by language and package",
"Although the collections of software available for analysis are many fewer than for proteins, we can still identify, in software, collections equivalent to the domains of life on the basis of software written in different programming languages.",
"Figs.",
"REF -REF illustrate the length distributions of collections of components (software functions) in four programming languages (Fig.",
"REF C++ 22,628 components; Fig.",
"REF Java 32,552 components; Fig.",
"REF Fortran 14,028 components and Fig.",
"REF Ada 12,680 components).",
"Despite the disparity in their sizes, each of these collections again shows striking similarity to the canonical form of length distributions of Figs.",
"REF and REF , as predicted by (REF ) and manifest in proteins, as we have already seen.",
"In each case, the values in the x-axis scale are the same whilst the y-axis is scaled according to the size of the packages to make the peaks of approximately the same vertical extent.",
"Figure: The length distributions of functions in large collections of software in four programming languages, (A): C++, (B): Java (C): Fortran and (D): Ada.Again, there are considerable differences in scale compared with that of proteins, but we can still identify in software collections equivalent to species on the basis of software written for individual applications.",
"Figs.",
"REF -REF illustrates the length distributions of collections of software functions in four applications of very different sizes, in four different programming languages (Fig.",
"REF , The Gimp image manipulation program (ISO C) 18,693 components; Fig.",
"REF , The KDE desktop libraries (C++) 16,241 components; Fig.",
"REF , The Eclipse interactive Development Environment (Java) 9,588 components and Fig.",
"REF , The gcc Ada compiler (Ada) 3,765 components).",
"Once again, despite the disparity in their sizes, each of these collections again shows striking similarity to the canonical form of length distributions exhibited by Figs.",
"REF , REF .",
"In this case, the x-axis scale is the same whilst the y-axis is again scaled according to the size of the packages to normalise the size of the peaks approximately.",
"Figure: The length distributions of functions in four packages each in different programming languages, (A): Gimp (ISO C), (B): kdelibs (C++), (C): Java (Eclipse) and (D): Ada (GCC).CoHSI also predicts the length distributions of musical compositions.",
"Much of the theory and discussion is deferred to Appendix C p. REF .",
"We will simply point out here that modern digital formats for musical annotation such as MusicXMLhttps://en.wikipedia.org/wiki/MusicXML, accessed 07-Jul-2017 allow us to apply the heterogeneous theory described in Appendix A p. REF , to yet another distinct discrete system where, in this case, the components are pieces of music and the unique alphabet comprises of notes as shown in Table REF .",
"Extracting the appropriate data is fortunately relatively simple compared with the daunting task of extracting possibly post-translationally modified amino acids in proteins or programming language tokens in computer programs, as the following XML snippethttps://hymnary.org/media/fetch/99378, accessed 07-Jul-2017 taken from “Nun danket alle Gott”, (Words Rinkart 1636, Music Crüger, 1647) shows.",
"<part id=\"P1\"> <measure number=\"1\"> ... <note> <pitch> <step>E</step> <alter>-1</alter> <octave>4</octave> </pitch> <duration>480</duration> <voice>1</voice> ...",
"This snippet refers to the note Eb in the 4th octave (middle C is annotated C4, so this is a minor third above middle C).",
"The duration must be determined from other parameters in the XML but this note actually corresponds to a 1/4 note or crotchet.",
"Arguably the most beneficial aspect of studying music from the point of view of this paper, however, is that it provides a simple example of when there are multiple candidate unique alphabets, for example, whether or not to include musical note duration as well as pitch in defining the alphabet.",
"When we first considered this aspect, the potential ambiguity worried us until we eventually realised that it led naturally and elegantly to the important conclusion, proved in Appendix C p. REF and verified experimentally, that all consistent unique alphabets are themselves related by a power-law.",
"In this study, we used 883 pieces of music, mostly classical but a very eclectic mix of chorales, piano concertos, horn duets, blue-grass music and indeed almost anything in an XML format we could get our hands on.",
"This process was not as simple as accumulating large amounts of open source code unfortunately and took some considerable time and manual effort.",
"As a result, this is by far the smallest system we consider and does not therefore allow us much scope to demonstrate different sized aggregations.",
"Even so, the length distribution of these 883 pieces of music is still gratifyingly suggestive of the presence of the predicted canonical distribution as shown in Figs.",
"REF -REF .",
"Figure: The length distribution of the 883 pieces of music analysed shown as a pdf (A) and a ccdf (B)."
],
[
"An R lm() analysis on the tail of Fig REF reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $100.0-10000.0$ , with an adjusted R-squared value of $0.9936$ .",
"The slope is $-1.66 \\pm 0.08$ ."
],
[
"The written word",
"The pioneering work which first suggested the ubiquity of power-laws in texts was that by George K. Zipf [73].",
"Zipf showed empirically that if the frequency of occurrence of words in a text were plotted in rank order on a $\\log -\\log $ ccdf, a power-law in frequency was observed.",
"This is an example of a system of homogeneous boxes, Appendix A p. REF , where we give a proof of Zipf's law using the methodology of this paper.",
"Archetypal examples of this at different scales are shown in Figs.",
"REF -REF .",
"These show respectively, (Fig.",
"REF The Mitre Common Vulnerabilities database 2,410,350 wordshttps://cve.mitre.org/, accessed 01-May-2015; Fig.",
"REF The complete works of Shakespeare 948,516 words; Fig.",
"REF The King James Bible in Swedish 807,969 words and Fig.",
"REF the classic English text “Three Men in a Boat” published by Jerome K. Jerome in 1889 67,435 words)https://www.gutenberg.org/, accessed 01-Jul-2017.",
"The classic straight line signature of the power-law is evident in each case even though the datasets are different in size by a factor of 40 from largest to smallest.",
"Figure: The rank ordered word distributions in, (A): The Common Vulnerabilities database, (B): The complete works of Shakespeare, (C): The King James Bible in Swedish and (D): Jerome K. Jerome's “Three Men in a Boat”.Before leaving this section, we point out that some systems can in fact be characterised both by using the homogeneous model, Appendix A p. REF , and also using the heterogeneous model, Appendix A p. REF .",
"It would therefore provide substantial additional support for the generality of the information model we propose in this paper if both the homogeneous model predictions and the heterogeneous model predictions held for a system in which both could be used.",
"There is no conflict between information measures here even though they are different, provided they are consistently applied.",
"Treating the words of a text as indivisible as we have done above yields the homogeneous model where word frequency follows the predicted Zipf power-law in rank, and we have already seen that the homogeneous model predicts exactly this long-established behaviour, Appendix A p. REF .",
"It is also possible to consider the individual words of a text as being further sub-divided into their letters as a heterogeneous model, just as if each word were a protein built from a unique alphabet, which in English, is 26 letters.",
"Word length has been studied extensively for various languages, for example [58], however for our purposes, we expect by analogy with our protein studies, that applying the heterogeneous model to word-length frequency will yield the canonical distribution seen in Fig.",
"REF for example.",
"Fig.",
"REF shows the word-length frequency for the text “Three Men in a Boat” whose word frequency is shown in Fig.",
"REF .",
"Even though the x-axis is limited to a maximum word length of around 40 in this novel (it includes single- and, unusually, double- hyphenated words such as the archaic currency reference “two-pounds-ten”), the canonical shape is once again evident with a sharp unimodal peak and as can be seen in Fig.",
"REF , good evidence of the predicted power-law tail.",
"The steep slope is associated with a relatively small alphabet as described in Appendix D, p. REF ."
],
[
"An R lm() analysis on the tail of Fig.",
"REF reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $2.303 \\times e^{-15}$ over the range starting at the mode $5.0-30.0$ , with an adjusted R-squared value of $0.9853$ .",
"The slope is $-6.40 \\pm 0.32$ ."
],
[
"We believe this adds considerable weight to the information-theoretic arguments of this paper.",
"In this case, the same system, treated using two different models of information (the homogeneous case of word frequencies and the heterogeneous case of letter frequencies and word lengths) obeys the predictions of both models, showing that consistent but different measures of information within the same system lead to valid predictions for the different distributions.",
"In both cases, conserving Hartley-Shannon information in an ergodic system is the underlying mechanism."
],
[
"The atomic elements",
"The distribution of atomic elements in the universe is a similar system to that of word frequencies and we use the homogeneous model, Appendix p. REF .",
"Again, the components each consist of one type only, in this case atoms of each element and, intrigued by its apparent preponderance we chose to include current estimates of dark material, i.e.",
"energy and matter.",
"The frequencies of occurrence, Fig.",
"REF , have been taken from NASAhttps://map.gsfc.nasa.gov/universe/uni_matter.html, accessed 29-Jun-2017 and Wikipedia,https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements, accessed 29-Jun-2017.",
"Figure: The frequency of occurrence of the elements in the universe supplemented by estimates of dark energy (data point 1) and dark matter (data point 2) shown as a log-log\\log -\\log ccdf.The distribution of elements fits well on the predicted power-law distribution for homogeneous boxes, where the rank ordering turns out to follow the atomic number.",
"(The relationship with atomic number is outwith the theory described here.)"
],
[
"An R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $1.0-85.0$ , with an adjusted R-squared value of $0.9779$ .",
"The slope is $-6.80 \\pm 0.94$ ."
],
[
"Intriguingly, the observed amount of dark energy (first point Fig.",
"REF ) and dark matter (second point Fig.",
"REF ) fit the predicted homogeneous box distribution well, suggesting that from the point of view of CoHSI, dark matter corresponds to something with an atomic number of zero (a re-generating sea of neutrons ?)",
"and dark energy corresponds to something with an atomic number of -1 (?",
"); so that's something for the theorists to chew on.",
"On a much smaller scale than Fig.",
"REF , the characteristic straight line signature is again visible in the distribution of the elements in seawaterhttps://en.wikipedia.org/wiki/Abundances_of_the_elements_(data_page), accessed 29-Jun-2017, Fig.",
"REF , not in this case including dark material.",
"Figure: The frequency of occurrence of the elements in sea water shown as a log-log\\log -\\log ccdf."
],
[
"Even though the tail is short, an R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< 2.2 \\times e^{-16}$ over the range $10.0-72.0$ , with an adjusted R-squared value of $0.9963$ .",
"The slope is $-8.82 \\pm 0.29$ ."
],
[
"Conclusions",
"This paper, through theory and testing against multiple datasets of different provenance and levels of aggregation, makes the case that a conservation principle derived from information theory (Conservation of Hartley-Shannon Information) operates within all discrete systems to impose important and common structural properties on length and unique alphabet size distributions.",
"By development of a statistical mechanics argument in which we consider ergodic ensembles with a fixed number of tokens and a fixed total H-S information content, independent of the meaning of the tokens chosen without bias, we demonstrate that the length and unique token alphabet for components of discrete systems are inextricably linked by a canonical distribution - the heterogeneous CoHSI distribution (REF ) - visible in all aggregations and at all scales where numbers are sufficiently large for statistical mechanics to operate.",
"The two Lagrange multipliers which naturally emerge $\\alpha , \\beta $ are undetermined and simply parameterise the range of possible solutions.",
"This has a number of interesting implications which we will divide into levels of confidence based on the data analysis, distinguishing in each case whether this is a heterogeneous system, Appendix A p. REF (which has a canonical distribution (REF )) or a homogeneous system, Appendix p. REF (which has a canonical distribution (REF ) corresponding to Zipf's law) or both, recalling that such systems differ only in the relevant definition of Hartley-Shannon information.",
"These conclusions are strongly supported by statistical analysis using R and documented individually in the body of the paper.",
"Where linear analysis was done on a power-law tail, this is noted below as (R).",
"All 14 such analyses gave an adjusted $R^{2}$ within the range $0.970 - 0.999$ with values of $p < e^{-14}$ .",
"All heterogeneous systems will tend to the canonical frequency distribution (REF ) as total size grows, with a sharp unimodal peak and a power-law tail.",
"Justification Equation (REF ).",
"Development starting Appendix A p. REF .",
"Evidence Fig.",
"REF (Proteins); Fig.",
"REF (Software); Fig.",
"REF (Music); Fig.",
"REF (Texts) (R)).",
"All heterogeneous discrete systems made up from components, themselves comprising indivisible tokens chosen from an alphabet, will exhibit a precise power-law tail in both their length distribution and in the distribution of their unique alphabet.",
"Justification Length: Eq.",
"(REF ), Alphabet: Eq.",
"(REF ).",
"Development starting at Appendix A p. REF and also p. REF .",
"Evidence Length: Fig.",
"REF (Proteins); Fig.",
"REF (Software); Fig.",
"REF (Music), (R).",
"Alphabet: Fig.",
"REF (Proteins); Fig REF (Software); Fig.",
"REF (Music), (R).",
"The canonical CoHSI distribution (REF ) tends to the asymptotic solution found in [25].",
"Justification Appendix A p. REF onwards.",
"Evidence Figs.",
"REF , REF , example solutions of equation (REF ).",
"The canonical frequency distribution will appear in all aggregations of a system.",
"Justification The theory for both heterogeneous systems and homogeneous systems is scale independent.",
"It finds the most likely length distribution for a given total size $T$ and a given total heterogeneous or homogeneous Information $I$ , but other than requiring a reasonably large system in the general sense of Statistical Mechanics [17], does not depend on the values of $T$ or $I$ .",
"Appendix A p. REF onwards.",
"Evidence Heterogeneous systems, Figs.",
"REF -REF , Figs.",
"REF -REF (Proteins); Homogeneous systems, Figs.",
"REF -REF (Texts); Figs.",
"REF , REF (Atomic elements).",
"The canonical frequency distribution will appear in all qualifying discrete systems independently of their provenance.",
"Justification At the heart of the definition of Hartley-Shannon Information is the original prescient advice from Ralph Hartley that the meaning of the tokens is irrelevant.",
"Evidence Heterogeneous systems, Fig.",
"REF (Proteins), Fig.",
"REF (Software), Fig.",
"REF (Music), Fig.",
"REF (Text); Homogeneous systems, Figs.",
"REF -REF (Text), Figs.",
"REF , REF (Atomic Elements).",
"If a system can be considered as both a heterogeneous system and a homogeneous system, then the predictions for both kinds of system will appear.",
"Justification Only a different definition of information is needed.",
"The mechanism of statistical mechanics then automatically generates the relevant length distribution, Appendix A p. REF and p. REF .",
"Evidence Heterogeneous system, Fig.",
"REF (Lengths of words in a text); Homogeneous systems, Fig.",
"REF (Rank-ordered word frequency in text) (R).",
"The alphabet used to categorise a heterogeneous system is irrelevant provided it is consistent.",
"Justification Consistent alphabets are power-laws of each other asymptotically for large components, Appendix C p. REF onwards.",
"Evidence Figs.",
"REF , REF (Duration and no-duration notes in Music) (R).",
"Average component length is highly preserved across collections in heterogeneous systems.",
"Justification This is a direct consequence of the sharply unimodal shape of the canonical Distribution for heterogeneous systems.",
"Evidence Figs.",
"REF , REF (Proteins) (R), from [25]).",
"This is also reported for software [23].",
"Unusually long components are inevitable in heterogeneous systems.",
"Justification Such components are inevitable because of the presence of the ubiquitous power-law tail, (noting the comments of [11]).",
"Evidence Fig.",
"REF (Proteins) which shows for example, approximately 10,000 proteins longer than 10 times the average length.).",
"This phenomenon has also been reported for software [43].",
"We expect to see the numbers of known amino acids to expand with time to preserve the power-law tail already evident in Fig.",
"REF .",
"There were around 800 structurally distinct Post Translationally Modified amino acids known in SwissProt 13-11.",
"There are already suspected to be thousands more [31].",
"Our information argument strongly supports this thesis and may indeed help to quantify it.",
"We find it intriguing that the power-law slope of virus protein lengths is quite different from those observed in the three domains of life.",
"We speculate that this relates to their unique alphabet.",
"The asymptotic behaviour for large components (REF ) in heterogeneous systems, implies that tokens carry a payload of the average information content of the component in which they appear.",
"For example, in proteins a particular amino acid might carry a different information payload in different proteins by virtue of the company it keeps.",
"We do not know if this has any useful physical interpretation.",
"We (wildly) speculate that since both dark energy and dark matter lie on the same information theoretic distribution as the elements in order of atomic number, there is an undiscovered but intimate relationship between dark material and atomic number.",
"It is clear from the above results that important structural features of discrete systems are well-predicted by a single conservation principle applied to ergodic systems at all levels of aggregation and of all kinds.",
"Nothing more is asked of the theory than that the size of the systems should be sufficiently large that the methodology of Statistical Mechanics can be applied.",
"That the above features need not depend for explanation on any mechanism of natural selection in the case of proteins or anything to do with human volition in the case of either music or software, we find remarkable.",
"Instead, they simply manifest themelves as emergent properties of heterogeneous or homogeneous large systems of components, revealed when we consider an ergodic ensemble of the same size and H-S information content, from which we seek the most likely distribution of its component lengths using the methodology of this paper.",
"Competing Interests: The authors declare that they have no competing financial interests.",
"Correspondence: Correspondence and requests for materials should be addressed to Les Hatton (email: [email protected]).",
"This material is based on work supported in part by the National Science Foundation.",
"Any opinion, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.",
"Although some parts of this paper are published [22], [23], [25], the main body of the paper, the lower and upper bounds on the implicit canonical pdf (the heterogeneous CoHSI distribution), the relationship between alphabets and most of the empirical work, appear here for the first time both to clarify and extend the theory with new results.",
"During review processes for various journals, the general thesis of this paper that a simple global principle appears to be responsible for some important aspects of protein evolution was a bridge too far for most of the reviewers given these restrictions.",
"This resistance has been noted before in biology research [15] and publishing inter-disciplinary papers in general remains much more difficult than it should be.",
"However, there were some insightful responses for which we are both very grateful.",
"One anonymous reviewer pointed out an error with the combinatorics leading up to the chocolate box argument of Appendix p. REF , which opened up a real can of worms (the role of additive partitions) which was finally resolved by the recursive argument of Appendix A p. REF .",
"Yet another reviewer made us really think about the application of this apparently abstract principle to real systems we can touch and feel, as well as suggesting we clarify the emergent role of $A$ in the dual distribution, which we hopefully have in Appendix A p. REF .",
"We would particularly like to thank Dan Rothman for a whole string of insights which really helped clarify some of the more opaque sections; Ken Larner who went through the document with a fine toothcomb suggesting numerous clarifications; Leslie Valiant for bringing to our attention Frank's excellent paper [15]; Alex Potanin and colleagues in New Zealand for useful comments after an early seminar on this topic.",
"Any mistakes which remain are our responsibility of course but we hope that the theory is now sufficiently well explained and its novelty delineated and that the associated reproducibility packages will help others to verify the computational aspects and extend it in new directions.",
"Statistical mechanics is a methodology for predicting component distributions of general systems made from discrete pieces or components subject to restrictions known as constraints.",
"Such systems include gases (made from molecules), proteins (made from amino acids), software (made from programming language tokens) and even boxes containing beads.",
"Conventionally, constraints are applied by fixing the total number of pieces and/or the total energy [17].",
"To illustrate the method, we describe a classical problem of determining the most likely distribution of particles amongst energy levels.",
"To see this, the following variational methodology is borrowed from the world of statistical physics, ([59] (p.217-); and for an excellent introduction, see [17]).",
"In the kinetic theory of gases, a standard application is to find the most common arrangement of molecules amongst energy levels in a gas subject to various constraints such as a fixed total number of molecules and fixed total energy.",
"For this, imagine that there are M energy levels, where the number of particles with energy level $\\varepsilon _{i}$ is $t_{i}, i=1,..,M$ .",
"For this system, the total number of ways $W$ of organising the particles amongst the M energy levels is given by:- $W=\\frac{T!}{t_{1}!t_{2}!..t_{M}!",
"}, $ where $T = \\sum _{i=1}^{M} t_{i} $ The total amount of energy in this system is just the sum of all the particle energies and is given by $E = \\sum _{i=1}^{M} t_{i} \\varepsilon _{i} $ In a physical system, E corresponds to the total internal energy and the variational method to follow constrains this value to be fixed; i.e.",
"solutions are sought in which energy is conserved.",
"Using the method of Lagrangian multipliers and Stirling's approximation as described in [17], will give the most likely distribution satisfying equation (REF ) subject to the constraints in equations (REF ) and (REF ).",
"This is equivalent to maximising the following variational derived by taking the natural log of (REF ).",
"Just as in maximum likelihood theory, taking the log dramatically simplifies the proceedings, in this case the factorials, and allows the use of Stirling's theorem for large numbers.",
"Also, since it is monotonic, a maximum in log W is coincident with a maximum in W. This leads to $\\log W = T \\log T - \\sum _{i=1}^{M} t_{i} \\log (t_{i}) + \\lambda \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace + \\beta \\lbrace U - \\sum _{i=1}^{M} t_{i} \\varepsilon _{i} \\rbrace $ where $\\lambda $ and $\\beta $ are the multipliers [17].",
"In essence, the variational process envisages varying the contents $t_{i}$ of each of the components until a maximum of $\\log W$ is found.",
"The maximum is indicated by taking $\\delta (\\log W) = 0$ , (analogous to finding maxima in differential calculus).",
"Noting that the variational operator $\\delta $ acts on pure constants such as $T \\log T$ , $\\lambda T$ and $\\beta U$ to produce zero just as when differentiating a constant, the product rule of differentiation gives $\\delta (t_{i} \\log (t_{i})) = \\delta t_{i} \\log (t_{i}) + t_{i} \\delta (\\log (t_{i}) = \\delta t_{i} (1 + \\log (t_{i}))$ , $\\varepsilon _{i}$ is independent of the variation by assumption, T and the $t_{i}$ are $\\gg 1$ (to satisfy Stirling's theorem, although it is surprisingly accurate even for relatively small values).",
"This leads to $0 = - \\sum _{i=1}^{M} \\delta t_{i} \\lbrace \\log (t_{i}) + \\alpha + \\beta \\varepsilon _{i} \\rbrace $ where $\\alpha = 1 + \\lambda $ .",
"(Further elaboration of this standard technique can be found in Glazer and Wark [17].)",
"Finally, (REF ) must be true for all variations to the occupancies $\\delta t_{i}$ and therefore implies $\\log (t_{i}) = - \\alpha - \\beta \\varepsilon _{i} $ for all $i$.",
"Using equation (REF ) to replace $\\alpha $ , this can be manipulated into the most likely, i.e.",
"the equilibrium distribution, of particles amongst the M components.",
"$t_{i} = \\frac{T e^{-\\beta \\varepsilon _{i}}}{\\sum _{i=1}^{M} e^{-\\beta \\varepsilon _{i}}} $ Defining $p_{i} = \\frac{t_{i}}{T}$ means that $p_{i}$ can be interpreted as a probability density function since it is non-negative everywhere and its sum everywhere is equal to 1.",
"Then (REF ) yields $p_{i} = \\frac{e^{-\\beta \\varepsilon _{i}}}{\\sum _{i=1}^{M} e^{-\\beta \\varepsilon _{i}}} $ This is a result with a profound interpretation in physics.",
"It states"
],
[
"If we consider all possible systems which share the same number of particles T and the same total energy U (i.e.",
"energy is conserved), then given any one example of such a system, it is overwhelmingly likely to obey (REF ).",
"In other words by considering all possible systems with these parameters and constraining them to have the same T and U, probability distribution (REF ) is overwhelmingly likely, provided the $t_{i}$ are large enough for Stirling's approximation to hold."
],
[
"In this case, the distribution is exponential and exactly as is found in nature - exponentially fewer particles occupy higher energy levels.",
"All the above has been known for decades and is extremely successful at explaining classical systems such as gases and even quantum mechanical systems; the methodology of statistical mechanics however is exceedingly versatile, so let us consider a simple model just consisting of boxes of coloured beads."
],
[
"Identical boxes of beads",
"Let us put some flesh on the meaning of “overwhelmingly likely” as used earlier in this paper.",
"Consider now a system of M boxes of identical beads, where the $i^{th}$ box contains $t_{i}$ beads and M is reasonably large.",
"If we have T beads in total numbered by sequence, where $T = \\sum _{i=1}^{M} t_{i}$ , so that they are distinguishable by their order, the number of possible ways of arranging them in each of the M boxes is given by $\\Omega = \\frac{T!",
"}{\\prod _{i=1}^{M} (t_{i}!)}",
"$ Suppose there are $M=10$ boxes and $T=100$ beads and we simply assign them one by one to a randomly chosen box.",
"We would be very surprised if the first box contained all the beads with the others empty, and the number of ways this can happen according to (REF ) is $100!/(100!\\times 0!\\times 0!..0!)",
"= 1$ .",
"If each box contains 10 beads however as shown in Fig.",
"REF , this situation can happen in $100!/(10!\\times 10!\\times ..10!",
")$ ways, which is approximately $10^{100}$ , a gigantic number.",
"Figure: Boxes containing exactly the same number of exactly the same bead.In other words, we are overwhelmingly more likely to see equal box populations than 1 single filled box.",
"In fact statistical mechanics allows us to prove that, in this case, equal population is by far the most likely distribution of contents simply by finding the maximum of (REF ) subject to a fixed number of T beads in the form $\\log \\Omega = T \\log T - T - \\sum _{i=1}^{M} \\lbrace t_{i} \\log (t_{i}) - t_{i} \\rbrace \\\\+ \\alpha \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace $ where the constraint on fixing T is controlled by the Lagrangian parameter $\\alpha $ .",
"Finding the maximum of (REF ) using the standard $\\delta ()$ method [17], gives the solution $t_{i} \\sim constant$ , corresponding to equal box populations.",
"Frank also demonstrates this in his maximum entropy formulation [15], p. 9.",
"We now describe an extension which is directly relevant to systems such as the known proteome or computer programs.",
"Consider Fig.",
"REF .",
"Figure: The heterogeneous case where each box contains mixed types and different numbers of beads.",
"This is relevant to proteins, computer program functions and the length distribution of words in texts.Here the boxes contain differently coloured beads.",
"We envisage this as the $i^{th}$ box containing $t_{i}$ beads selected randomly from a unique alphabet of $a_{i}$ colours, ordered by sequence.",
"For the proteome, the “colours” correspond to different amino acids and for software functions they correspond to different programming language tokens.",
"Now we utilise the great generality of statistical mechanics by generalizing the payload to be Hartley-Shannon (H-S) information content instead of energy [23].",
"The H-S information content of the $i^{th}$ box $I_{i}$ , (Appendix p. REF ) is simply the log of the number of ways of arranging the beads in that box, so that it is guaranteed to contain at least one of each of the $a_{i}$ colours.",
"H-S information is completely agnostic about what the colours actually mean, indeed Hartley specifically advised against attaching any meaning to a token [20].",
"The only thing that matters is that beads change colour, so the actual colour is irrelevant and the total H-S information is just the sum of the information for each box.",
"Presented with such a system, we can ask what is the most likely distribution of contents for systems for which both the total number of beads and the total H-S information are conserved ?",
"We must also recall that proteins and software are both constructed sequentially so we are considering systems where beads are distinguishable by the order in which they appear, but the actual order is irrelevant.",
"The relevant variational form we must solve is therefore $\\log \\Omega = T \\log T - T - \\sum _{i=1}^{M} \\lbrace t_{i} \\log (t_{i}) - t_{i} \\rbrace \\\\+ \\alpha \\lbrace T - \\sum _{i=1}^{M}t_{i} \\rbrace + \\beta \\lbrace I - \\sum _{i=1}^{M} I_{i} \\rbrace $ The only term which is different in this formulation from the classical solution derived above (REF ), is the last term on the right hand side of (REF ).",
"In the variational methodology, each term has the $\\delta ()$ operation applied in order to vary the $t_{i}$ and derive the distribution in (REF ), so we are interested specifically in $\\delta \\big (\\beta \\lbrace I - \\sum _{i=1}^{M} I_{i} \\rbrace \\big ) = - \\beta \\sum _{i=1}^{M} \\delta ( I_{i} ) = - \\beta \\sum _{i=1}^{M} \\frac{d I_{i}}{d t_{i}} \\delta t_{i}, $ since I is being held constant.",
"Now consider what happens when boxes are very large compared with their unique alphabet, i.e.",
"$t_{i} \\gg a_{i}$ .",
"In this case, [23], the information content is $I_{i} = \\log (a_{i}\\times a_{i}\\times ... \\times a_{i}) = \\log (a_{i}^{t_{i}}) = t_{i} \\log a_{i}$ In other words, we select $t_{i}$ times from a choice of $a_{i}$ colours secure in the knowledge that since $t_{i} \\gg a_{i}$ , it is very unlikely that any of the $a_{i}$ colours would be missed out and we therefore meet the requirement of having exactly $a_{i}$ unique colours.",
"In this case, (REF ) becomes $- \\beta \\sum _{i=1}^{M} \\frac{d I_{i}}{d t_{i}} \\delta t_{i} = - \\beta \\sum _{i=1}^{M} \\frac{d (t_{i} \\log a_{i})}{d t_{i}} \\delta t_{i} = - \\beta \\sum _{i=1}^{M} (\\log a_{i}) \\delta t_{i} $ (REF ) fits perfectly into the variational methodology leading to (REF ), modifying (REF ) to give $\\log (t_{i}) = - \\alpha - \\beta \\log a_{i} $ The analogue of (REF ) is therefore $p_{i} \\equiv \\frac{t_{i}}{T}= \\frac{a_{i}^{-\\beta }}{\\sum _{i=1}^{M} a_{i}^{-\\beta }} $ To summarize, maximising (REF ) subject to a fixed total number of beads T AND a fixed total H-S information $I = \\sum _{i=1}^{M} I_{i}$ is directly analogous to maximising (REF ) with $\\log a_{i}$ replacing $\\epsilon _{i}$ .",
"Like its classical equivalent (REF ), (REF ) is also fundamental.",
"It states"
],
[
"In any discrete system satisfying the model described here, the tail (i.e.",
"large $t_{i}$ ) of the distribution of unique alphabets is overwhelmingly likely to obey a power-law."
],
[
"Note that by analogy with (REF ), we can interpret $t_{i} \\log a_{i}$ as each bead carrying a payload of $\\log a_{i}$ , so that even though H-S information is token agnostic, the beads in a particular box still carry a box-dependent payload which is a function of the unique alphabet of colours in that box, $a_{i}$ .",
"This is exactly analogous to $t_{i} \\varepsilon _{i}$ being interpreted as each particle carrying an energy $\\varepsilon _{i}$ in classical statistical mechanics.",
"In other words, each box behaves as if it had a fixed information level $\\log a_{i}$ determined by its unique alphabet.",
"In a protein for example, this has the intriguing implication that even though H-S information is token-agnostic, a particular amino acid in one protein may carry a different information payload than when present in another protein, simply because its neighbours are different."
],
[
"The asymptotic dual distribution",
"As pointed out by [25], (REF ) has a dual solution.",
"With some algebra, it can be shown that $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }}, $ where $A = \\sum _{i=1}^{M} a_{i}$ Note here that $A$ emerges naturally as the sum of the unique alphabets of each component.",
"It is not the size of the unique alphabet across all components.",
"This is simply another manifestation of the token-agnosticism of Hartley-Shannon information - system-wide uniqueness of the alphabet simply does not emerge as a requirement.",
"The only requirements for a pdf are that it be positive definite and normalisable so this in no way detracts from the fact that (REF ) is also a power-law.",
"In other words,"
],
[
"The length distribution of large proteins or software functions for which $t_{i} \\gg a_{i}$ will also be a power-law."
],
[
"Note also the natural appearance of the reciprocal slope $1/\\beta $ .",
"This value is not found in the datasets here but this difference is discussed and we think resolved in the discussion of alphabets in music in the Appendices p. REF ."
],
[
"The chocolate box analogy and additive partitions: the CoHSI distribution",
"For smaller boxes containing fewer beads, the above value of $I_{i}$ (REF ) is not correct.",
"If $t_{i}$ is closer in size to $a_{i}$ , (it cannot be smaller since the length must be at least equal to the unique alphabet), there is an increasingly high probability that we might miss out one of the colours in the unique alphabet as we select our $t_{i}$ beads, negating the fundamental assumption that each box contains a unique alphabet of exactly $a_{i}$ .",
"We must therefore make different provisions as the boxes get smaller.",
"Figure: A box of 22 chocolates chosen from 12 different types as shown on the left.The situation is akin to boxes of mixed chocolates, Fig.",
"REF .",
"Such boxes are constructed from a fixed set of chocolates advertised on the lid, and every box must contain at least one of each.",
"Larger boxes simply contain more than one of some kinds.",
"In how many ways can such boxes be created ?",
"Note that it is simple to find an algorithm to guarantee that the unique alphabet is exactly $a_{i}$ .",
"All that is necessary is to fill any $a_{i}$ places with one chocolate of each type and then fill the remaining $t_{i} - a_{i}$ at random from the available types.",
"The number of ways of doing this is $\\big ((a_{i}!)",
".",
"({}^{t_{i}}C_{a_{i}})) .",
"(a_{i}^{(t_{i} - a_{i})}\\big ), $ where ${}^{n}C_{r} = n!/((n-r)!r!",
")$ is the combination operator.",
"This however, is not the same as counting all the possible ways of filling the box such that it contains exactly $a_{i}$ chocolates.",
"We are trying to find the number of different ways of filling the $i^{th}$ box with $t_{i}$ chocolates chosen from a unique set of exactly $a_{i}$ chocolates and we must do this in a way which fits into the statistical mechanical framework so we can use its methodology.",
"To explore this, suppose we have a box of $t_{i} = 5$ chocolates such that it contains exactly $a_{i} = 2$ different chocolates of types A and B.",
"The total number of ways this can be done $N(t_{i},a_{i})$ , is given by $N(5,2) = \\frac{5!}{1!4!}",
"+ \\frac{5!}{4!1!}",
"+ \\frac{5!}{3!2!}",
"+ \\frac{5!}{2!3!}",
"$ Note The first term on the right hand side of (REF ) is the total number of ways of selecting 5 chocolates by using 1 chocolate of type A and 4 chocolates of type B.",
"This is equal to 5 (ABBBB, BABBB, BBABB, BBBAB, BBBBA).",
"The second term corresponds to 4 chocolates of type A and 1 of B and is also equal to 5 (BAAAA, ABAAA, AABAA, AAABA, AAAAB).",
"The third term corresponds to taking 3 of type A and 2 of type B.",
"This is equal to 10, (AAABB, AABAB, AABBA, ABAAB, ABABA, ABBAA, BBAAA, BABAA, BAABA, BAAAB).",
"The fourth term corresponds to taking 2 of type A and 3 of type B.",
"This is also equal to 10, (BBBAA, BBABA, BBAAB, BABBA, BABAB, BAABB, AABBB, ABABB, ABBAB, ABBBA).",
"There are no other ways of arranging the box such that there are exactly 2 colours and exactly 5 chocolates altogether.",
"There are therefore 5 + 5 + 10 + 10 = 30 different such boxes in total.",
"Note that (REF ) gives $(2!)",
".",
"({}^{5}C_{2})) .",
"(2^{(5 - 2)}) = 160$ boxes.",
"This over-counting is because a box such as ABBAB could be generated several times by that algorithm, for example, by filling the first two places with AB and then the rest at random or by filling the first and third places with AB and the rest at random.).",
"The denominators of (REF ) correspond to elements of the additive compositionshttps://en.wikipedia.org/wiki/Partition_(number_theory), accessed 02-Jun-2017.",
"of size 2 of the number 5.",
"These are $5 = 1 + 4; 5 = 4 + 1; 5 = 3 + 2; 5 = 2 + 3$ There are other additive compositions such as $2+2+1$ , but this corresponds to three different kinds of chocolate so must be excluded.",
"The fact that the compositions are additive presents a real complication when merging with the methodology of statistical mechanics because it breaks the steps leading from (REF )-(REF ) by introducing the log of a recursive definition as we shall see.",
"Prior to discovery of this recursive method, the solution was simply trapped between a lower and upper bound.",
"The lower bound consisted of just one of the terms leading to the recursive definition and the upper bound was the pure power-law (REF ).",
"The recursive method is however far more compelling.",
"First we slightly modify the definition in (REF ) by letting $N(t_{i}, a_{i}; a^{\\prime }_{i})$ be the number of ways of producing a chocolate box with $t_{i}$ chocolates containing exactly $a^{\\prime }_{i}$ unique types chosen from a total unique number of types of $a_{i}$ .",
"In this notation, for example, $N(5,2;1) = 2$ and $N(5,2;2) = 30$ .",
"The distinction between $a_{i}$ and $a^{\\prime }_{i}$ is to make way for the use of recursion.",
"It can be easily verified that the following recursion then generates the desired total number of ways $N(t_{i}, a_{i}; a_{i})$ of generating a chocolate box of $t_{i}$ chocolates from a unique set of chocolates $a_{i}$ .",
"$N(t_{i}, 1; 1) = 1; \\hspace{14.22636pt} N(t_{i}, a_{i}; i) = {}^{a_{i}}C_{i} N(t_{i}, i; i), i = 1,..,a_{i} - 1, a_{i} = 1,..,t_{i}$ completed by $N(t_{i}, a_{i}; a_{i}) = a_{i}^{t_{i}} - \\sum _{i=1}^{a_{i}-1} N(t_{i},a_{i}; i)$ The corresponding Hartley-Shannon information content for a box containing $t_{i}$ chocolates chosen from a unique alphabet of $a_{i}$ chocolates is therefore given by $I_{i} = \\log \\big ( N(t_{i}, a_{i}; a_{i}) \\big ) $ In contrast, the equivalent form for the pure power-law (REF ) is $I_{i} \\vert _{P} = \\log \\big ( a_{i}^{t_{i}} \\big ) = \\log \\big ( t_{i} \\log a_{i} \\big ) $ We can now see the problem posed by (REF ) when we apply the $\\delta ()$ operator to $I_{i}$ in the statistical mechanical framework leading from (REF )-(REF ).",
"The presence of the recursion prevents the clean separation of factors by the $\\log $ operation.",
"If we simply solved this computationally, that would ordinarily be no problem but (REF ) is computationally difficult for the large factorial values which arise even for modest values of $(t_{i},a_{i})$ .",
"(Whatever method we choose, however, it must have the property of producing the power-law form (REF ) in the asymptotic limit $t_{i} \\gg a_{i}$ .)",
"Applying the $\\delta ()$ operator to (REF ) using (REF ) leads to the pure power-law equation $\\log t_{i} = -\\alpha -\\beta ( \\log a_{i} ), $ whereas applying the $\\delta ()$ operator to (REF ) using (REF ), (REF ) leads to the full equation $\\log t_{i} = -\\alpha -\\beta ( \\frac{d}{dt_{i}} \\log N(t_{i}, a_{i}; a_{i} ) ), $ Here, the unique alphabet $a_{i}$ is playing a dual role as the frequency in a pdf by analogy with (REF ) using (REF ).",
"To complete our chocolate box analogy, if we are presented with a system of boxes with a total number of chocolates $T$ chosen from a fixed alphabet of chocolates and a total H-S information $I$ , then by far the most likely distribution of numbers of chocolates in any box will be given by a pdf which is the solution of (REF ).",
"Finally we note that following the argument that led up to (REF ), for $t_{i} \\gg a_{i}$ , $\\log N(t_{i}, a_{i}; a_{i} ) \\rightarrow t_{i} \\log a_{i},$ so the full solution correctly asymptotes to the pure power-law.",
"This will be confirmed during the computation with both forms being displayed together.",
"Before we proceed with this, there is a technical limitation to overcome since the equation for the pdf which results from applying the variational method to (REF ) is implicit.",
"As we pointed out in the text, there is a precedent for this in the definition of Tsallis entropy [67], [66], although in our case, the implicit nature of the pdf arises naturally from CoHSI.",
"(In Tsallis entropy, the entropy term is adjusted using an additional parameter and this adjustment can lead to an implicit pdf.)",
"We must therefore generalise the argument from integer values of ($t_{i}, a_{i}$ ), to the real line.",
"This will not affect our computation of factorials however, which are done at integer values of $t_{i}, a_{i}$ , with interpolation for non-integer values."
],
[
"(REF ) defines the canonical implicit pdf with solutions $(t_{i},a_{i})$ which a) conserves H-S information and b) asymptotes to the pure power-law (REF ) for $t_{i} \\gg a_{i}$ as required for any heterogeneous system at all scales."
],
[
"Solving (REF ) proved challenging.",
"The one thing we do know, however, is that it asymptotes to the simple explicit solution $I_{i} \\sim t_{i} \\log a_{i}$ leading to (REF ) when $t_{i} \\gg a_{i}$ .",
"This suggested the following procedure.",
"We start with large $t_{i}$ solving (REF ) explicitly for $a_{i}$ , followed by the use of this value as the starting value for the full solution of (REF ).",
"This was found by searching a pre-computed grid of integer $(t_{i},a_{i})$ values for the appropriate value of $d/dt_{i} (\\log N(t_{i}, a_{i}; a_{i}))$ interpolating as necessary.",
"When the solutions are found, we decrement $t_{i}$ and start again.",
"This process is somewhat akin to shooting methods in boundary layer solutions in fluid dynamics, [68], [21].",
"All code used is in the reproducibility package.",
"Figs.",
"REF , REF indicate the behaviour we expect to find if CoHSI is indeed controlling these distributions.",
"The power-law behaviour for larger components is already modelled to a high degree of precision by the large component approximation (REF ).",
"We focus now on the behaviour for all component sizes.",
"The full solution - the solution of (REF ), and the pure power-law solution - the solution of (REF ), using the same parameters, are shown together at two different scales as Figs.",
"REF and REF .",
"The shaded zone corresponds to the region where the full solution departs from the pure power-law solution.",
"Note that the the first order approximation for numerical differentiation means that the first few points for the full CoHSI solution do not converge.",
"The behaviour is clear from the remaining points however.",
"Figure: The length distributions using the same modelling parameters for (A) the full CoHSI solution (), and the pure (asymptotic) power-law () for components smaller than 100 tokens and (B), the same data for components up to 2,000 tokens long.We make the following observations about Figs.",
"REF and REF with respect to (REF ) and (REF ).",
"In (REF ) as the left hand side decreases with decreasing $t_{i}$ , the value of $a_{i}$ must increase to give a solution.",
"This gives the pure power-law behaviour shown which continues to increase as $t_{i}$ decreases as shown clearly in Fig.",
"REF .",
"In (REF ), $d/dt_{i}(N(t_{i}, a_{i}; a_{i})$ naturally decreases without having to keep increasing $a_{i}$ as was the case in (REF ).",
"This is a natural consequence of CoHSI.",
"As can be seen in Figs.",
"REF , REF , the qualitative behaviour of the full CoHSI solution around the unimodal peak remains sharp but is more rounded than the pure power-law solution and is qualitatively similar to the software data close up of Fig.",
"REF shown as Fig.",
"REF .",
"The sharpness of the peak is related to the boundary condition naturally emerging in this theory that $t_{i} \\ge a_{i}$ , i.e., no component can be shorter than its unique alphabet.",
"The peak of the full solution differs slightly in position as well as their amplitudes as the power-law parameter $\\beta $ changes.",
"A value of $\\beta =1.8$ was used.",
"As $\\beta $ increases, the peak moves left and the amplitude diminishes.",
"The full solution naturally asymptotes to the pure power-law behaviour as required.",
"We can compare the behaviour around the peak with a close-up of the dataset of Fig.",
"REF , as shown as Fig.",
"REF .",
"Even on observed data, the transition from power-law to near linearity is abrupt, taking place over perhaps 10 tokens, and is qualitatively very similar to the full CoHSI solution.",
"As was noted in the body of the paper, the juxtaposition of dual regions, one matched by a lognormal distribution and the other by a power-law has been described in the past, [39], [37].",
"In our theory, this transition emerges completely naturally as the implicit solution of (REF ).",
"Figure: A close-up around the peak of the measured dataset shown as Fig.",
".To summarise, these results strongly support the thesis of this paper that the Conservation of Hartley-Shannon Information (CoHSI) acts as a constraint on how the length and alphabet size distributions of systems of a given size $T$ and total Hartley-Shannon information $I$ , can evolve at all scales giving an excellent qualitative match which does not require juxtaposing existing pdfs of known properties."
],
[
"Approximate properties of the heterogeneous CoHSI distribution",
"From the shape of Figs.",
"REF , REF and the theory which led up to Figs.",
"REF , REF , we can approximate the distribution satisfactorily by glueing together a right-angled triangle up to the modal value $a_{max}$ at $t = t_{max}$ , say and a power-law afterward because the solution corresponding to (REF ) transitions from power-law to almost linear behaviour so quickly.",
"In other words, we can define the approximate canonical distribution $c(t)$ as follows $c(t) = \\left\\lbrace \\begin{array} {ll}(\\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))}) t & \\mbox{$0 \\le t \\le t_{max}$} \\\\(\\frac{2(\\beta - 1)}{(t_{max} (\\beta + 1))}) (\\frac{t}{t_{max}})^{-\\beta } & \\mbox{$t_{max} < t < \\infty $}\\end{array}\\right.$ We require $\\beta >1$ for this to be positive definite.",
"This has been normalised so as to integrate to unity over it's support $[0,\\infty ]$ .",
"This approximation will allow us to make useful inferences.",
"First we will calculate the mean location and spread of this distribution.",
"The mean location is given by $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\int _{s=0}^{t_{max}} s^{2} ds + t_{max} (\\frac{1}{t_{max}})^{-\\beta } \\int _{s=t_{max}}^{\\infty } s^{-\\beta + 1} ds \\bigg ],$ which is $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\big [ \\frac{s^{3}}{3} \\big ]_{s=0}^{t_{max}} + t_{max} (\\frac{1}{t_{max}})^{-\\beta } \\big [ \\frac{s^{-\\beta + 2}}{-\\beta + 2} \\big ]_{s=t_{max}}^{\\infty } \\bigg ],$ and, provided $\\beta > 2$ , gives $\\langle c \\rangle = \\frac{2(\\beta - 1)}{(t_{max}^{2} (\\beta + 1))} \\bigg [ \\big [ \\frac{t_{max}^{3}}{3} \\big ] + ((t_{max})^{\\beta + 1} \\big [ - \\frac{t_{max}^{-\\beta + 2}}{-\\beta + 2} \\big ] \\bigg ],$ and finally $\\langle c \\rangle = \\frac{2 (\\beta - 1)t_{max}}{(\\beta + 1)} \\bigg [ \\frac{1}{3} + \\frac{1}{\\beta - 2} \\bigg ] = \\frac{2 (\\beta - 1)t_{max}}{3(\\beta - 2)} ; \\hspace{5.69046pt} \\beta > 2$ There is little point in computing higher moments because they place even greater constraints on the value of $\\beta $ (they diverge unless the support for the distribution is a finite interval), and will not apply to our examples for which $2 \\le \\beta \\le 4.5$ , (recall that if the pdf has slope $-\\beta $ , the ccdf will have slope $-\\beta + 1$ and we are measuring from the ccdf following [42].).",
"Applying the above estimate for $\\langle c \\rangle $ to the full Trembl distribution, Fig.",
"REF , for which $\\beta = 4.14$ suggests that $\\langle c \\rangle / t_{max} \\approx 1$ , where as analysis of the data itself in R gives a ratio of around $1.5$ which is reasonable given the nature of the approximation.",
"The actual values from Trembl are given in Table REF .",
"Table: Different measures of average length of proteins in amino acids in the full Trembl 17-03 distribution, Fig.",
"There are some kinds of discrete system for which the above information model does not apply.",
"Consider the case of homogeneous components.",
"Here, each bead carries a payload such that each box contains only beads with the same payload, unique to that box.",
"We represent this by assembling beads of the same colour in the appropriate box, Fig.",
"REF .",
"Figure: The homogeneous case.",
"In each box, all the beads are the same but different boxes contain different types and numbers of beads.",
"This is relevant to the distribution of atomic elements and to the rank ordering of frequency occurrence of words in texts.We could of course simply set $a_{i} = 1$ in the heterogeneous case above.",
"However, this immediately causes problems because the asymptotic Hartley-Shannon information content of any box in this case would be $t{_i} \\log 1 = 0$ and is simply degenerate.",
"However, because Hartley-Shannon information is simply the $\\log $ of the number of ways of arranging the beads of a box, in the absence of an alphabet of choices in each box we can still find a suitable non-degenerate definition as follows.",
"Suppose we have a unique alphabet of beads $a^{\\prime }_{i}, i=1,..,M$ for the system as a whole.",
"This is in contrast to the heterogeneous case where the unique alphabet $a_{i}$ was relevant only to the $i^{th}$ box.",
"Suppose from this system-wide alphabet, we seek to fill the M boxes each with $t_{i}$ of the $a^{\\prime }_{i}$ beads such that each box contains only one type.",
"The total population of the $M$ boxes is as before $T = \\sum _{i=1}^{M} t_{i}$ .",
"We will renumber them without loss of generality so that $t_{1} \\ge t_{2} \\ge .. \\ge t_{M}$ .",
"We proceed as follows.",
"Select any box and then fill it by selecting $t_{1}$ beads of the same colour.",
"Since we are selecting from $M$ different beads, the probability that we will achieve this selecting at random is $( 1/M )^{t_{1}}$ .",
"For the second box, we then have an alphabet available of $M-1$ , so the probability of filling this box with only one colour of the remaining colours is $( 1/(M-1) )^{t_{2}}$ and so on.",
"The total number of ways $N_{h}$ this can be done is then given by this probability multiplied by the total number of ways in which $T$ beads can be selected, which is $T!$ .",
"$N_{h} = T!",
"\\bigg [ \\big ( \\frac{1}{M} \\big )^{t_{1}} \\times \\big ( \\frac{1}{M-1} \\big )^{t_{2}} \\times .. \\times \\big ( \\frac{1}{1} \\big )^{t_{M}} \\big ] = T!",
"\\prod _{i=1}^{M} \\big ( \\frac{1}{i} \\big )^{t_{i}}$ Rewriting (REF ) then, the information content of this system is $\\log N_{h} = \\log T!",
"+ \\sum _{i=1}^{M} t_{i} \\log \\big ( \\frac{1}{i} \\big ) = \\log T!",
"- \\sum _{i=1}^{M} t_{i} \\log i$ The development (REF )-(REF ) then follows but with $\\log i$ replacing $\\log a_{i}$ .",
"The end result is the equivalent of (REF ) and amounts to $t_{i} \\sim i^{-\\eta },$ where $\\eta $ is some constant."
],
[
"(REF ) states that if we organise these homogeneous boxes in rank order of contents, (i.e.",
"fullest first), then it is overwhelmingly likely that they will be distributed as a power-law in that rank.",
"This is a famous law known as Zipf's law [73].",
"Zipf's law is empirical although others have produced statistical derivations [57], [35].",
"The above derivation therefore serves as an alternative theoretical justification which places it nicely amongst those distributions which can be explained by the approach taken in this paper."
],
[
"Average component length",
"It has been observed experimentally on several occasions [69], [70], [25] that proteins appear to preserve their average length across aggregations within relatively tight bounds.",
"The sharply unimodal peak of Figs.",
"REF , REF as predicted by the theoretical development in this paper suggests that we should not be surprised at this.",
"Indeed at all scales and ensembles the estimates of average protein length will be highly conserved within collections as a result, even though the position of the peak may move a little.",
"In some aggregations, the degree to which the average length is preserved is quite remarkable, for example in Bacteria (Fig.",
"REF ), whilst in Eukarya, there is evidence of some fine structure [25] which invites further analysis Fig.",
"REF .",
"Figure: A plot of the total concatenated length of proteins against the total number of proteins for each species in (A) Bacteria and (B) Eukaryota.",
"Each data point is a species.",
"The gradient of the linearity evident in both plots effectively defines the average protein length for that collection, from .We also note that preservation of the average component length has also been reported for software [23].",
"The skewed nature of the distribution of Figs.",
"REF , REF suggests that the use of the mean as a measure of average length alone may be misleading and should be accompanied by other more robust measures such as the median and mode.",
"Table REF demonstrates this by calculating them for the three domains of Archaea, Bacteria, Eukaryota, along with Viruses, as shown in Figs.",
"REF -REF .",
"As expected, the more robust measure of median is less affected by the skew and the medians are therefore considerably less spread out than the means.",
"This is particularly true of viruses which although they have an anomalously large mean in comparison, their median is much closer aligned with those of Archaea and Bacteria.",
"The modes are subject to considerable noise.",
"Table: Different Measures of average length of proteins in the domains of life and virusesTable REF shows that the mean is around $1-3$ times the modal value but we do not compare this using the approximate distribution, Appendix p. REF , as the data are rather noisier than for the full Trembl distribution (compare Fig.",
"REF with Fig.",
"REF ).",
"One of the most important features of power-laws compared with any kind of exponential distribution such as the normal distribution is that “events that are effectively 'impossible' (negligible probability under an exponential distribution) become practically commonplace under a power-law distribution.” [11].",
"The emphatic power-law in both the protein lengths and in software function lengths leads to large ratios when comparing the longest components with the average.",
"For example, proteins of around $36,000$ amino acids have been found and this is $100 \\times $ the average."
],
[
"In terms of the theory developed here, there is no need for any biological reason for very long proteins - they exist simply because of the naturally emerging power-law resulting from consideration of Information-conserving ergodic systems."
],
[
"We note that precisely the same thing has been observed in software [43]."
],
[
"Appendix C: CoHSI and Token Alphabets",
"The definition of alphabets, i.e., unique sets of tokens from which choices can be made, poses interesting questions.",
"First of all, we must point out that there is generally no obvious definitive unique alphabet for any system.",
"Alphabets are partly subjective and partly objective because at their heart, they are about how humans categorise systems.",
"Take a simple example of a normally sighted person and a colour blind person both counting the number of differently coloured beads in a collection.",
"Barring counting errors, they will both find the same number of beads in total, however, they will not necessarily agree on the number of beads of each colour.",
"In particular, red-green confusion is likelyhttps://en.wikipedia.org/wiki/Color_blindness.",
"How does this affect the theory we describe here ?",
"It might be thought that by linearly increasing the size of the alphabet, the distribution of the two alphabets are themselves linearly related, i.e.",
"$alphabet1_{i} = constant \\times alphabet2_{i}$ However, this turns out to be not the case and to understand what is happening, we must return to the duality of the asymptotic behaviour (REF ) and (REF ), which we repeat here, $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{a_{i}^{-\\beta }}{Q(\\beta )} $ and its algebraic dual given by $q_{i} \\equiv \\frac{a_{i}}{A} = \\frac{t_{i}^{-1/\\beta }}{\\sum _{i=1}^{M} t_{i}^{-1/\\beta }} $ Since our normally-sighted person and our colour-blind person will count the same numbers but with different alphabets, we can say that for the normally sighted person, $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{(a^{\\prime }_{i})^{-\\beta ^{\\prime }}}{Q(\\beta ^{\\prime })} $ and for our colour blind person $p_{i} \\equiv \\frac{t_{i}}{T} = \\frac{(a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime }}}{Q(\\beta ^{\\prime \\prime })} $ where $a^{\\prime }_{i}, a^{\\prime \\prime }_{i}$ are the two unique alphabets they use and $\\beta ^{\\prime }, \\beta ^{\\prime \\prime }$ their slopes.",
"Since the lengths are unchanged, we can see straight away from (REF ) and (REF ), that the two unique alphabets will themselves be power-law related asymptotically $(a^{\\prime }_{i})^{-\\beta ^{\\prime }} \\sim (a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime }} \\Rightarrow a^{\\prime }_{i} \\sim (a^{\\prime \\prime }_{i})^{-\\beta ^{\\prime \\prime \\prime }}, $ where $\\beta ^{\\prime \\prime \\prime } = - \\beta ^{\\prime \\prime } / \\beta ^{\\prime }$ .",
"This leads us to predict a general rule"
],
[
"In any consistent categorisation of the same system with different unique alphabets, the distributions of the unique alphabets will also be related by a power-law."
],
[
"Music alphabets",
"Consider an example from the world of music.",
"Music is also a system of discrete components in the sense described here, Table REF .",
"In recent years, discrete formats representing the notes and structure of a musical composition have appeared, for example MusicXML as referenced in the main text.",
"If we consider the 88 notes of a full-scale piano as defining the possible notes in the equal-tempered scale used in the vast majority of published music, then we have a candidate unique alphabet $a^{\\prime }_{i}$ of 88 (no-duration alphabet).",
"However, we can subdivide this alphabet quite naturally and consistently into notes and duration.",
"The standard durations are divided into fractions of a whole note as breve (2), semi-breve (1), minim (1/2), crotchet (1/4), quaver (1/8), semiquaver (1/16) and demisemiquaver (1/32).",
"There are others defined off either end of this list but they are obviously rare as there were no occurrences in the body of music studied here.",
"This gives seven flavours of each note and expands the unique alphabet considerably to $88 \\times 7 = 616$ items, (duration alphabet).",
"Figs.",
"REF shows the distribution of the two alphabets no-duration and duration, measured on the same body of music.",
"As expected from (REF ) they both exhibit power-law behaviour."
],
[
"For the no-duration alphabet R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $40.0-400.0$ , with an adjusted R-squared value of $0.9937$ .",
"The slope is $-1.66 \\pm 0.01$ .",
"For the duration alphabet R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $40.0-4000.0$ , with an adjusted R-squared value of $0.9951$ .",
"The slope is $-1.44 \\pm 0.02$ ."
],
[
"These are emphatic results.",
"Figure: The log-log\\log -\\log ccdf of the duration and no-duration alphabets measured on the same body of music used in this study (A), and a comparison of the two alphabets as log-log\\log -\\log showing their clear linear power-law relationship (B)."
],
[
"Moreover in Fig.",
"REF which compares the two alphabets directly on a $\\log -\\log $ scale, the predicted power-law relationship of (REF ) is clearly visible.",
"R reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $10.0-500.0$ , with an adjusted R-squared value of $0.9879$ .",
"The slope is $1.181 \\pm 0.002$ , also consistent with (REF ).",
"This too is an emphatic result."
],
[
"We believe that this throws some light (but does not necessarily explain) why the predicted reciprocal relationship between the power-law slope of the unique alphabet distribution and that of the length distribution is not adhered to closely in our data.",
"There are a potentially infinite number of alphabets related themselves by power-laws, but only one length distribution."
],
[
"Protein alphabets",
"For proteins, with increasing sophistication we are able to recognise not just the 22 amino acids transcribed directly from DNA but also the increasingly large number of known post-translational modifications (PTM) which dramatically extend and continue to extend the size of the unique amino acid alphabet that allows us to categorise proteins.",
"Will this process of discovery stop ?",
"We argue that it cannot as it is intimately linked to the total number of proteins known, and this continues to grow apace.",
"We can gain insight into the growth of the unique protein alphabet by studying collections, such as the SwissProt database, over different revisions [61], [63], [64] as it incorporates PTM information from the Selene project [53].",
"Figure: The classic linear signature of a power-law distributions of unique amino acid alphabets for SwissProt 13-11 and the more up to date SwissProt 15-07 on a log-log\\log -\\log ccdf.Fig.",
"REF shows the frequencies of the unique amino acid alphabet recorded in the proteins of SwissProt release 13-11 and SwissProt release 15-07 as ccdfs in $\\log -\\log $ form.",
"Although the maximum unique alphabet for amino acids is 22 for those decoded directly from DNA, we should note that finding a protein with all 22 would be unlikely as both the 21st and 22nd amino acids, selenocysteine and pyrollysine, are rare in proteins.",
"Pyrrolysine is found in methanogenic archaea and bacteria and is encoded by a re-purposed stop codon (UAG), requiring the action of additional gene products to accomplish its incorporation [47].",
"Thus it is not easy to annotate pyrrolysine from the gene sequence alone, and direct chemical analysis of proteins would be more informative.",
"Selenocysteine is found in all domains of life, but the selenoproteome is small [49] and an additional concern is misannotation in the databases, because a stop codon (UGA) is re-purposed from “halt translation” to “incorporate seleocysteine” by additional sequences downstream of the gene as well as other trans-acting factors [36].",
"As a result, any unique amino acid count beyond 21 must contain post-translationally modified amino acids and we note the following: Almost the whole of the tail of Fig.",
"REF consists of proteins in which there must be post-translationally modified amino acids, effectively doubling the unique alphabet derived directly from DNA.",
"The increase in numbers between 20 and 26 unique amino acids can be seen by the slightly displaced points upwards in the SwissProt 15-07 dataset compared with the SwissProt 13-11 dataset.",
"The remainder of the tail significantly straightens with the more comprehensively annotated SwissProt 15-07 presumably due to reduction in noise and increasing numbers."
],
[
"The linearity in each tail strongly supports equation (REF ) even though the range is less than 1 decade because the slope is so steep arising from the current paucity of the unique alphabet.",
"For SwissProt 13-11 of Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $8.124 \\times e^{-13}$ over the range $19 - 35$ , with an adjusted R-squared value of $0.9698$ .",
"The slope is $-15.91 \\pm 0.56$ .",
"For SwissProt 15-07 of Fig.",
"REF , R lm() reports that the associated p-value matching the power-law tail linearity is $< 2.2 \\times e^{-16}$ over the range $19-33$ , with an adjusted R-squared value of $0.9968$ .",
"The slope is $-18.56 \\pm 0.19$ ."
],
[
"We should also clear up a potential point of confusion here.",
"One reviewer stated that the distribution of number of proteins against unique amino acid count had a “tiny power-law tail” and that the distribution was uniform.",
"The reviewer reasoned that this explained why the average length of proteins was highly conserved in contrast to our explanation in the Appendix p. REF .",
"It is indeed true at the present rate of knowledge that the distribution of unique amino acids has a tiny power-law tail but the distribution is anything but uniform as we can see in SwissProt 13-11 by considering two different kinds of plot.",
"Fig.",
"REF plots the logarithmic frequency of proteins against their unique amino acid alphabet.",
"Whatever this distribution is, it is certainly not uniform, although we know the tail from $19-35$ amino acids is an accurate power-law from the analysis of the data shown in Fig.",
"REF .",
"Figure: The frequency of proteins plotted against the unique amino acid count for SwissProt 13-11 on a log-linear\\log -linear plot (A), and the frequency at which each amino acid occurs including PTM amino acids plotted in rank order on a log-log\\log -\\log ccdf (B).In contrast, Fig.",
"REF plots the occurrence rate of each unique amino acid including post-translational modification across the entire SwissProt 13-11 distribution, of which there are more than 800 recorded by the Selene project.",
"In other words it shows in how many proteins each amino acid appears, organised in rank order.",
"This matches the homogeneous model discussed in Appendix A p. REF , and a power-law in the tail is evident as expected.",
"We note in passing a possible intriguing relationship between the overhang in Fig.",
"REF between around 10 and 30 on the x-axis and the contemporary question of PTM undercounting [65]."
],
[
"R lm() reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $< (2.2) \\times e^{-16}$ over the range $22.0-800.0$ , with an adjusted R-squared value of $0.9778$ .",
"The slope is $-2.63 \\pm 0.31$ .",
"This too is an emphatic result."
],
[
"Appendix D: Power-laws, Statistical Rigour and Rules of Thumb",
"Power-laws are ubiquitous in nature and are generated by a number of mechanisms, [42].",
"In essence, power-law behaviour can be represented by the pdf (probability density function) p(s) of entities of size s appearing in some process, given by a relationship like $p(s) = \\frac{k}{s^b} $ where $k, b$ are constants.",
"On a $\\log p - \\log s$ scale the pdf is a straight line with negative slope $- b$ .",
"It can easily be verified that the equivalent cdf (cumulative density function) $c^{\\prime }(s)$ derived by integrating (REF ) also obeys a power-law $\\sim s^{-b+1}$ , (for $b \\ne 1$ ).",
"The classic linear signature of a power-law tail in a ccdf (complementary cumulative distribution function) is usually shown as in Fig.",
"REF which displays $c(s) = 1 - c^{\\prime }(s)$ .",
"Figure: The classic linear signature of a power-law in the tail of a log-log\\log -\\log ccdf.For noisy data, the ccdf form is used most often because of its fundamental property of reducing noise present in the pdf, as noted by [42].",
"This effect is because the ccdf is obtained by integration.",
"This reduces noise inherent in the pdf preserving any power-law behaviour while allowing any linearity to be measured more accurately.",
"The benefit of this can be clearly seen in data extracted from software systems as shown in Figs.",
"REF (the pdf) and REF (the corresponding ccdf).",
"The effect is even more pronounced in the inherently more noisy protein data.",
"Figure: The pdf (A) and the ccdf (B) of the length distributions of the same large population of software.Whilst on the subject of significance, a rule of thumb often used to determine the existence of a power-law is that it should appear over two or more decades in the x-axis of the ccdf.",
"This is useful only as a rule of thumb when the slope is not too steep.",
"Since the scale of the y-axis on the $\\log -\\log $ ccdf is effectively the scale of the x-axis times the slope of the power-law, then a steep slope would require a large scale of y-axis frequency measurements to provide the rule of thumb of 2-3 decades in the x-axis.",
"For example, a slope of around 3 would require y-axis frequency measurement only over some 6-9 decades to give the rule of thumb of 2-3 decades in the x-axis, which is reasonable.",
"On the other hand, if the slope is 10, y-axis frequency measurement over some unreasonably large 20-30 decades would be required to give the rule of thumb of 2-3 decades in the x-axis.",
"This is an important point for the protein studies considered here wherein we are investigating the predicted power-law in unique alphabet.",
"Here, the x-axis is the unique alphabet of amino acids.",
"The size of this alphabet is small at the current state of knowledge, leading to a steep power-law slope.",
"In such situations, we fall back on normal procedures of statistical inference to replace subjective belief with objective perception and therefore all that is required is that there is statistically significant linearity in the tail of the distribution of the $\\log -\\log $ ccdf for the number of measurement points used.",
"A rule of thumb guides but does not replace normal statistical inference whereby a result is either significant at some level or it is not for a given model and data.",
"This effect can be seen in Fig.",
"REF , a $\\log -\\log $ ccdf of the occurrence rate in the size of the unique alphabet in SwissProt version 13-11 [61], which is merged with the Selene post-translational modification data [53], [62].",
"These represent amongst the best annotated protein data including the rapidly growing field of post-translational modification (PTM), a process whereby nature alters some of the amino acids by covalent processes such as glycosylation, phosphorylation, methylation, acylation, etc., thereby extending the unique alphabet beyond the 22 amino acids directly coded from DNA [2], [31], [71], [46], [6].",
"Figure: The highly linear tail of the occurrence frequency of unique alphabet sizes in the SwissProt 13-11 protein distribution merged with the Selene 2013 post-translational modification annotation, extending the range of the natural unique alphabet of 22 amino acids directly coded from DNA to just over 30 in this dataset.As can be seen, the tail of SwissProt 13-11 in Fig.",
"REF , covers only a range up to just over 30 even though there are thousands of PTM known or predicted by existing research.",
"As a result there are only nine data points for unique alphabets of size greater than 20, although each point is an aggregate of a large number of observations."
],
[
"An R lm() analysis on this tail reports that the associated p-value matching the power-law tail linearity in the ccdf of Fig.",
"REF is $6.576 \\times e^{-12}$ over the range $21.0-30.0$ , with an adjusted R-squared value of $0.9951$ .",
"The slope is $-22.9 \\pm 0.2$ ."
],
[
"This is a statistically emphatic result for the existence of a power-law, even though the size of the slope is steep because the x-axis is restricted.",
"Later versions of SwissProt with Selene annotations increase the PTM alphabet, Appendix C p. REF .",
"Power-law behaviour has been studied in a wide variety of environments starting with the pioneering work of [73] (linguistics) and followed by [48] (economic systems) and the excellent reviews by [38] and [42].",
"In software systems significant activity, much of it recent, [10], [37], [41], [7], [18], [45], [3], [12], [43] and [22] has addressed power-law behaviour in various contexts.",
"To give some idea of the scope of these, Mitzenmacher [37] considers the distributions of file sizes in general filing systems and observed that such file sizes were typically distributed with a lognormal body and a Pareto (i.e.",
"power-law) tail.",
"Gorshenev and Pis'mak [18] studied the version control records of a number of open source systems with particular reference to the number of lines added and deleted at each revision cycle.",
"Louridas et.",
"al.",
"[43] show evidence that power laws appear in software at the class and function level and that distributions with long, fat tails in software are much more pervasive than previously established."
],
[
"Appendix E: Hartley-Shannon Information, parsimony and token-agnosticism",
"Information theory has its roots in the work of Hartley [20] who showed that a message of N signs (i.e.",
"tokens) chosen from an alphabet or code book of S signs has $S^{N}$ possibilities and that the quantity of information is most reasonably defined as the logarithm of the number of possibilities or choices $\\log S^{N} = N \\log S$ .",
"To gain insight into the reason why the logarithm makes sense, consider Fig.",
"REF .",
"The number of choices necessary to reach any of the 16 possible targets is the number of levels which is $\\log _{2}$ (number of possibilities).",
"The base of the logarithm is not important here.",
"Figure: A binary tree.",
"Each level proceeding down can either go left or right.",
"There are four levels leading down to one of 2 4 2^{4} = 16 possibilities.",
"Only four choices are needed to reach any of the possibilities.",
"We note that log 2 (16)=4\\log _{2}(16) = 4.",
"Here the number 7 has been singled out by the choices left, right, left, right as the tree is descended.Information theory was developed substantially by the pioneering work of Shannon [54], [55] and many researchers since but we have remained with Hartley's original clear vision and most importantly its token-agnosticism.",
"We re-iterate that is important not to conflate information content with functionality or meaning and Cherry [8] specifically cautions against this noting that the concept of information based on alphabets as extended by Shannon and Wiener amongst others, relates only to the symbols themselves and not their meaning.",
"Indeed, Hartley in his original work, defined information as the successive selection of signs, rejecting all meaning as a mere subjective factor.",
"In the sense used here therefore, Conservation of H-S Information will be synonymous with Conservation of Choice, not meaning.",
"This turns out to be enough to predict the important system properties detailed in this paper.",
"In other words, those properties depend only on the alphabet and not on what combining tokens of the alphabet might mean in any human sense.",
"We believe CoHSI therefore represents the most parsimonious theory capable of explaining all the observed features of the numerous disparate datasets analysed in this paper."
]
] | 1709.01712 |
[
[
"Solar Flares Complex Networks"
],
[
"Abstract We investigate the characteristics of the solar flares complex network.",
"The limited predictability, non-linearity, and self-organized criticality of the flares allow us to study systems of flares in the field of the complex systems.",
"Both the occurrence time and the location of flares detected from January 1, 2006 to July 21, 2016 are used to design the growing flares network.",
"The solar surface is divided into cells with equal areas.",
"The cells, which include flare(s), are considered as nodes of the network.",
"The related links are equivalent to sympathetic flaring.",
"The extracted features present that the network of flares follows quantitative measures of complexity.",
"The power-law nature of the connectivity distribution with a degree exponent greater than three reveals that flares form a scale-free and small-world network.",
"The great value of the clustering coefficient, small characteristic path length, and slowly change of the diameter are all characteristics of the flares network.",
"We show that the degree correlation of the flares network has the characteristics of a disassortative network.",
"About 11% of the large energetic flares (M and X types in GOES classification) that occurred in the network hubs cover 3% of the solar surface."
],
[
"INTRODUCTION",
"Since space weather is undeniably influenced by solar activities, investigation of the dynamic variations in the solar atmosphere presents an interesting field of study for researchers.",
"Among large-scale solar phenomena, flares are influential events releasing a huge amount of energy of up to 10$^{27}$ J [47], [20] and affecting the space weather [42], [74].",
"The solar corona is dynamically exposed to the effects of energetic flares [37] which frequently occur over active regions (ARs) manifesting as radiation in the extreme ultraviolet and shorter wavelengths.",
"Generally, the accumulated energy of the freezing plasma in a twisted case of magnetic fields appear as ephemeral disturbances while magnetic lines are reconnected leading to flares in ARs.",
"Solar flares have direct results in increasing the complexity of evolving magnetic fields in ARs , .",
"The accelerated particles of flares can cause disturbance on satellites and electrical power source.",
"So, studying the statistical properties of flares, simulations, and their prediction has been the subject of many scientific articles [60], [4], [76], [20], [17], [58].",
"It has been accepted that these flare events are rooted in the solar interior magneto-convection [48], [70].",
"The sudden flash of the flares generates waves within the solar atmosphere that are similar to the seismic waves produced during earthquakes.",
"Both solar flares and earthquakes locally occur with the intensive release of energy and momentum with temporary fluctuations in their time series.",
"The energy frequency of both flares and earthquakes follows the power-law distribution [29].",
"To characterize the behavior of solar flares and earthquakes, commonly accepted evidence shows that both follow the same empirical laws [32].",
"For solar flares, some of the most important laws exhibit scale invariance and self-organized criticality [7], [10], [33].",
"By analogy of Omori's law for seismic sequences, the power-law distribution is obtained for the main flares and after-flare sequences [33].",
"The study of complex systems requires the analysis of network theory.",
"This helps to investigate the procedure of changes occurring in the system and to maybe extract a pattern for prediction.",
"Therefore, to analyze the flares complex system, we employed a graph theory to construct the complex network.",
"A network (graph) consists of nodes (vertices) and edges (links).",
"Generally, it can be considered as a simple, directed or undirected, and weighted or unweighted graph.",
"Several networks of interest are regular, complete, scale free, and small world indicating many physical descriptions of the system.",
"By comparing each network property with the equivalent characteristics of the random network, firstly, the network type must be identified.",
"Some characteristics (e.g., degree distribution, clustering coefficient, characteristic path length, and diameter) in the network are obtained to determine the network type.",
"The values of these parameters help us to analyze the behavior of the system.",
"It is usual to construct two main complex networks (i.e., scale-free and small-world networks) to conduct a survey about physical systems [1], [67], [31].",
"In a recent study, Daei et al.",
"(2017) constructed a complex network for solar ARs.",
"They obtained that the ARs network follows regimes that govern the scale-free and small-world networks.",
"It was shown that the probability of flare occurrence increases where ARs act as hubs all over the network.",
"Here, we investigate the conditions of the flares system as a complex system using a detrended fluctuation analysis applied to the time series of flares, as well as their non-linearity, limited predictability and so on.",
"To do this, we construct a network of 14395 flares with regard to their locations and occurrence times.",
"Then, we computed the degree distribution of the nodes, clustering coefficient, characteristic path length, diameter, and degree correlation of the flares network.",
"The paper is organized as follows: In Section , the description of the solar flares data set is introduced.",
"In Section , we survey the complexity characteristics for the solar flares system.",
"In Section , the flares network is constructed.",
"In Sections and , we discuss about the properties of the random, scale-free, small-world, and regular networks, respectively.",
"In Section , we describe assortative, disassortative, and neutral networks by employing degree correlation.",
"In Sections and , the results and conclusions are presented, respectively."
],
[
"FLARE DATA SETS",
"We used the information of the 14395 solar flares taken from January 1, 2006 to July 21, 2016 which is available at $http://www.lmsal.com/solarsoft/latest\\_events\\_archive.html$ .",
"This site, which is associated with the Lockheed Martin Solar and Astrophysics Laboratory (LMSAL), provides information about the properties of solar features and updates its data center with the help of solar physics teams at the National Aeronautics and Space Administration (NASA) and Stanford University.",
"The other data center is the Solar Monitor System which is already known as the Active Region Monitor [42].",
"This site is supported by the National Oceanic and Atmospheric Administration (NOAA) to make solar data (e.g., solar flares, and ARs) publicly available in an updated list.",
"The flare information consists of an event number, EName (e.g., $\\texttt {gev}\\_20101114\\_1020$ ), flares start, stop, and peak times, X-ray (GOES) classification (X, M, C, B, and A), event type, and position on the Sun (Table 1).",
"The occurrence (start) times, classification types, and locations (latitude and longitude) of flares on the Sun are used to construct the network.",
"Bad data (e.g., wrong information about locations) is removed from the analysis.",
"Using the diff$\\_$ rot function in the SunPy software, the location (longitude) of the flares is rotated with respect to January 1, 2006 (the occurrence time of the first flare in our data set).",
"The longitudes and latitudes of the flares on the solar sphere surface are restricted to $-180^{\\circ }$ to $180^{\\circ }$ and $-90^{\\circ }$ to $90^{\\circ }$ , respectively (Figure REF ).",
"The scattering of the flares positions in the solar latitudes is presented in Figure REF .",
"ccccccc A small part of solar flares data EName YYYY/MM/DD Start time GOES Class Latitude Longitude gev_20020926_11402002/09/2611:40:00C1.7 N19W47 gev_20020927_14322002/09/27 14:32:00C1.6N13E40 gev_20020927_19032002/09/27 19:03:00C8.6N13E37 gev_20020928_00402002/09/28 00:40:00C3.4N11E36 gev_200209228_04362002/09/28 04:36:00C1.0N12E35 gev_200209228_05192002/09/28 05:19:00C1.0N12E35 Table 1 is published in its entirety in the electronic edition of the Astrophysical Journal.",
"A portion is shown here for guidance regarding its form and content."
],
[
"DO FLARES FORM A COMPLEX SYSTEM?",
"Complex system studies focus on the collective behavior of a system characterized by the relationship of elements and interactions with the environment.",
"Many systems in nature, economy, biology, power network, traffic, brain, the World Wide Web, astrophysics, and ecology are classified into groups of complex systems [19], , , , , .",
"Some common characteristics of the complex systems are: emergence treatment, non-linearity, limited predictability, and self-organized criticality , , , .",
"In this section, we survey the complexity characteristics of the solar flares system.",
"During the 11 years of our flares data set, the mean daily number of flares emergence within the solar atmosphere is about $3.7$ .",
"In Figure REF , the time series of the number of flares during January 1, 2006 to July 21, 2016 is presented.",
"One may ask whether the large numbers of emerged flares in the time series are related to the other large numbers?",
"In other words, dose the time series of the number of flares have a long-temporal correlation (self-affinity)?",
"To address this question, we used DFA.",
"In DFA, the value of the Hurst exponent (H) is used to explain the correlation of time series [56], , , , .",
"If $H$ takes the values in the ranges of $(0.5,1)$ and $(0,0.5)$ , we can say that the time series has a long-term correlation in its correlated or anti-correlated behavior, respectively.",
"In the case of $H = 0.5$ , there is an uncorrelated signal in the time series.",
"We applied DFA to the time series of the number of emerged flares on each day.",
"The value of the Hurst exponent is obtained at about $0.86$ .",
"This shows that the time series of the flares has a long-temporal correlation.",
"The key characteristic suggests that solar flares are governed by self-organized criticality [53], [38], [24], [34], [3], [17].",
"The prediction of the solar flares is important for space weather and communication.",
"Several attempts have been made to predict the solar flares occurrence based on flare statistics [74], magnetic properties of ARs [50], [16], [2], [23], [17], [65], and cellular automaton avalanche models [12], , , , , , .",
"The results of recent studies show that the flares system has a limited predictability.",
"The recently developed method based on the properties of ARs magnetograms can predict flares only over 48 hours before the flare occurrence [23], [17].",
"The avalanche model of cellular automaton based on the reconnection of magnetic fields has been developed for the solar flares [52], [54], [71].",
"This progressed model is in the category of non-linear and self-organized critical systems [9].",
"The above-mentioned features (i.e., limited predictability, non-linearity, and self-organized criticality) confirm that the solar flares system builds up a complex system.",
"In the rest of this paper, the complexity properties of the flares system are investigated using the complex network approach."
],
[
"CONSTRUCTING THE SOLAR FLARES COMPLEX NETWORK",
"The occurrence time and location of the flares on the solar surface are employed to construct the growing flares graph (network).",
"The solar spherical surface is divided into $n\\times n$ cells with equal areas considering the spherical coordinates ($\\theta $ , $\\phi $ ) as $S_{ij}=4\\pi R_{\\odot }^2/n^2$ ($i, j = 1, 2, ..., N$ ), where the parameter $R_{\\odot }$ is the solar radius, in the same manner as in the earthquake network developed by [1].",
"The angles $\\theta $ and $\\phi $ for each equal area (cell) are given by $~~~~~~ \\phi _{i+1} = \\phi _i +\\frac{2\\pi }{n}, ~\\phi _1=-180^{\\circ },~ -180^{\\circ }<\\phi <180^{\\circ },$ (j+1) = (j) -2n, 1=0, -90<0, (j+1) = (j) +2n, 1=0, 0<90, where $\\theta $ is an angle measured from the solar equator.",
"We construct the flares network with edges (links) and loops defined based on the flares interactions.",
"It should be noted that links and loops are representative of the correlation between sympathetic flaring [62], [25], [57].",
"Each cell is regarded as a vertex (node) if the emerged flare(s) is (are) located in it (Figure REF ).",
"The edges are defined as a relation between two successive flares.",
"If two successive flares occur in the same cell, we will have a loop.",
"By using this approach, we can map the flares information to a growing graph.",
"We note that the solar flares network naturally is a directed graph.",
"A small part of the connectivity distribution of the 12 nodes and 21 flares with ENames (e.g., $\\texttt {\\texttt {gev}}\\_ 20110411\\_2211$ ) of the solar flares network with loops and multiple edges is presented in Figure REF .",
"The nodes and edges of the flares network are shown in Figure REF .",
"The variety and number of connections demonstrates the complexity of the flares system.",
"Each line presents a link between two successive flares (nodes).",
"Since there is the mutual influential interaction between two hemispheres, lots of connections are made by all consecutive flares over two hemispheres (see the caption in Figure REF ).",
"A simple graph (unweighted and undirected) is obtained by removing the loops, and directions, and replacing multiple edges with single links.",
"An important point, which requires emphasis when constructing the flares network, is estimating the cell size.",
"Here, we used an arbitrary cell size to construct the network.",
"Also, we converted a directed graph to an undirected one to study the small-world presentation.",
"In other words, we use the simple graph to present an illustration for a small-world network."
],
[
"RANDOM AND SCALE-FREE NETWORKS",
"A graph -consisting of vertices and edges- is a geometrical representation of a network.",
"In general, graphs can be classified as directed, undirected, weighted, and unweighted graphs depending on their vertices and edges.",
"A graph is called undirected if the links are bi-directional.",
"A graph with different number labeled to links is known as a weighted network.",
"The unweighted graph is a weighted one when all the weights are set to one.",
"Every node is not in relationship with itself; in other words, the elements lying on the main diagonal of the matrix take the value zero.",
"In the complex network approach, the topological properties (local and global scales) taken from the related graph lie on the adjacency matrix [28], [69].",
"The simplest way to study the network is based on the properties extracted from the adjacency matrix $A$ .",
"The adjacency matrix for a network with $N$ nodes is a square matrix of order $N$ .",
"The adjacency matrix for a directed network with $N$ nodes is defined as $A_{ij} = 1$ , if node $j$ is linked to node $i$ ($i,j=1, 2, 3, ..., N$ ); the component $A_{ij}$ equals to 0 if there is no link between the $j$ th node toward the $i$ th node.",
"For a weighted network, the value of $A_{ij}$ can take an arbitrary value $A_{ij} = W_{ij}$ .",
"For undirected networks, the adjacency matrix is symmetric (i.e., $A_{ij}=A_{ji}$ and $A_{ii} = 0$ ).",
"The degree of the $i$ th node $k_{i}$ in an undirected network that can be extracted from the adjacency matrix is $k_i = \\sum _{j=1}^{N}A_{ij} = \\sum _{i=1}^{N}A_{ij}.$ For a directed network, we have $k_i^{in} = \\sum _{j=1}^{N}A_{ij} ,~~ k_i^{out} = \\sum _{i=1}^{N}A_{ij},$ where $k_i^{in}$ and $k_i^{out}$ are the incoming and outgoing degree of the node $i$ .",
"The degree of the $i$ th node is obtained as $k_i = k_i^{in} + k_i^{out}.$ To describe a network, the average of the nodes, $\\langle k\\rangle $ , plays a key role.",
"The average degree can be written as $\\langle k\\rangle = \\frac{2L}{N},$ where $L$ is the number of links.",
"The several known and applicable networks are random, scale free, complete, regular and small world.",
"These networks are distinguishable from each other by their degree distributions.",
"Degree distribution is an important characteristics of complex networks.",
"A random network is constructed by $N$ labeled nodes where each pair is linked with the same probability $P$ .",
"Two ways to generate the random network with $N$ nodes, $L$ edges, and a probability $P$ are explained by [39], .",
"For a random network, degree distribution follows a Poisson distribution [36], $P(k)=\\frac{e^{-\\lambda } {\\lambda }^{-k}}{k!",
"},$ where parameters the $k$ and $\\lambda $ are the degree of node and a positive constant, respectively.",
"Indeed, the probability of the node, $P(k)$ , with a $k$ th degree shows the degree of the node that can be selected randomly.",
"The degree distribution of a scale-free network is characterized by a power-law distribution $P(k)\\sim k^{-\\gamma },$ where $\\gamma $ is a positive constant called the degree exponent.",
"The basic difference between a random and a scale-free network is appears in the hubs (high-$k$ region).",
"For example, in the World Wide Web, which is a scale-free network with approximately $10^{12}$ nodes (e.g., $https://venturebeat.com/2013/03/01/$ or $https://googleblog.blogspot.com/2008/07/$ ), the probability of having a node with $k=100$ is about $P(100)\\approx 10^{-94}$ in a Poisson distribution; meanwhile it is about $P(100)\\approx 10^{-4}$ in a power-law distribution.",
"In a random network, the average degree $\\langle k\\rangle $ is comparable with lots of degrees.",
"In a random network, the difference between two degrees is in the order of $\\langle k\\rangle $ , which results in: (a) the degree of nodes is comparable with average degree $\\langle k\\rangle $ and (b) highly connected nodes (hubs) are not possible.",
"These points are the keys to distinguishing a random network from a scale-free network.",
"In a random network, a hub is effectively forbidden whereas in a scale-free network, a hub is absolutely necessary.",
"For a scale-free network, there is a limit on the degree of the largest hub.",
"The upper limit on the degrees of the largest hub is called the cutoff maximum degree $k_{cut}$ or the natural cutoff of the degree distribution.",
"The degree exponent with a natural cutoff for a scale-free network is estimated as [35] $\\gamma _{est}\\approx 1 + {\\frac{\\ln N}{\\ln \\emph {k}_{cut}}},$ where $N$ is the number of nodes.",
"Following Eq.",
"(REF ), if $\\gamma $ takes sufficiently high values, scale-free and random networks are hardly distinguishable.",
"It seems that distinguishing the power-law distribution from the Poisson distribution is crucial.",
"If the ratio of $k_{max}/\\langle k\\rangle $ is large enough, the network would be categorized in the group of scale-free networks.",
"In this case, the parameter $k_{max}$ is a node with the highest degree."
],
[
"SMALL-WORLD AND REGULAR NETWORKS",
"We computed the values of the clustering coefficient, characteristic path length, and diameter parameters of the network to describe a small-world network.",
"The clustering coefficient is a key parameter for studying most of the networks.",
"In graph theory, the clustering coefficient represents the tendency of neighbors to cluster around each other in an undirected simple graph [73].",
"Mathematically, it is defined as $c_i=\\frac{2t_i}{k_i(k_i-1)},$ where $c_i$ and $k_i$ are the local clustering coefficient and the number of neighbors, respectively.",
"The parameter $t_i$ is the number of edges linked between the neighbors of the $i$ th vertex.",
"Indeed, $k_i(k_i-1)/2$ is the maximum number of links that could exist between the neighbors.",
"The clustering coefficient is given by $C=\\frac{1}{N}\\sum ^{N}_{i=1}c_i,$ where $N$ is the network size.",
"The values defined for the clustering coefficient of a complete graph (all nodes have connections with each other) $C_{comp}$ and a random graph $C_{rand}$ are unity and much smaller than unity, respectively.",
"In the network science, the regular network is a network where all nodes have the same degrees.",
"The clustering coefficient for random and regular network are respectively given by [14], $C_{rand}\\simeq \\frac{\\langle k\\rangle }{N},$ $C_{reg} = \\frac{3(\\langle k\\rangle -1)}{4(\\langle k\\rangle -2)}.$ The clustering coefficient for the most of the networks depends on the degree of nodes.",
"For a random and a regular network, the clustering coefficient is not related to the degree of nodes.",
"One way to distinguish a random network from a scale-free one is by using the average local clustering coefficient of the nodes with the same degree, which is called the $C(k)$ function.",
"The function $C(k)$ for a random network is constant for all degrees of the nodes (Eq.",
"(REF )).",
"The path in a connected graph (e.g., flares network) is a finite sequence of edges defined for every two connected vertices.",
"Sometimes, there are several paths for each pair.",
"The average shortest path $d_{i,j}$ between all pairs of nodes is an important parameter for analyzing the network.",
"The average shortest paths for all pairs is called the characteristic path length $\\Lambda $ and is defined as $\\Lambda =\\frac{1}{N(N-1)}\\sum ^{N}_{i,j=1, i\\ne j}d_{i,j}.$ The characteristic path lengths of a random and a regular networks are respectively expressed as [21], [41] $\\Lambda _{rand}\\sim \\frac{\\ln N}{\\ln (\\langle k\\rangle - 1)},$ $\\Lambda _{reg}\\sim \\frac{N}{2 \\langle k\\rangle }.$ The other key parameter in the constructed network is the longest path length or network diameter $D$ .",
"As explained, in a simple graph, a path is an edge that connects vertices.",
"The average path length of a random graph is smaller than that defined for a regular graph $\\Lambda _{reg}>\\Lambda _{rand}$ .",
"In addition, the clustering coefficient of the regular graph is larger than that assigned for its equivalent random graph $C_{reg}\\gg C_{rand}$ .",
"In the small-world networks, a typical path between two arbitrary nodes is peculiarly short.",
"In comparing $C, C_{rand}$ , and $C_{reg}$ with the same network size (the same number of nodes, links, and equal average degree of nodes), the clustering coefficient of the small-world network takes the greater and smaller than that of defined for random and regular network, respectively (i.e., $C_{reg}>C> C_{rand}$ ) [73].",
"For the small-world networks, there is a relation between $N$ and $\\Lambda $ as follows [22], [27] $\\Lambda \\sim \\log N.$ The degree exponent is extracted from the power-law distribution to give a better description of a network.",
"If the degree exponent of the scale-free network takes a value greater than three, the network is a small-world one [27].",
"The relationships between the characteristic path length $\\Lambda $ and the degree exponent $\\gamma $ can be expressed as [22], [27] Constant if = 2, ((N))(-1) if 2 < < 3, (N)((N)) if = 3, (N) if >3.",
"In the case of $\\gamma $ = 2 (anomalous regime), the average path length has no relation to $N$ .",
"In this regime, when the system size increases, the hub with the highest degree grows linearly.",
"If $\\gamma $ ranges between two and three (ultra-small world), the characteristic path length is proportional to $\\ln (\\ln (N))$ .",
"It has a considerably slower regime than the $\\ln (N)$ , which is determined for random networks.",
"When $\\gamma $ = 3 (critical point), the characteristic path length takes values slightly smaller than that obtained for the random network because of the presence of $\\ln (\\ln (N))$ .",
"Finally, in the case of $\\gamma >$ 3 (small world), the hubs do not have a meaningful influence on the characteristic path length [22]."
],
[
"ASSORTATIVE, DISASSORTATIVE, AND NEUTRAL NETWORKS",
"Degree correlations are indicative of the relation between the degrees of nodes that are linked to each other.",
"Using the adjacency matrix $(A)$ , the average degree of the neighbors $(k_{nn})$ for the $i$ th node is given by $k_{nn}(k_i)=\\frac{1}{k_{i}}\\sum _j^NA_{ij}k_j.$ The degree correlation function for nodes with degree $k$ is obtained as $k_{nn}(k)=\\frac{1}{N_k}\\sum _{i/k_i=k} k_{nn}(k_i),$ where $N_k$ is the number of nodes with the degree $k$ .",
"The degree correlation function has the following relation [61] $k_{nn}(k)\\propto k^{\\mu },$ where the parameter $\\mu $ is a correlation exponent.",
"For assortative networks, the correlation exponent is positive ($\\mu >0$ ) and for disassortative networks, the correlation exponent is negative ($\\mu <0$ ).",
"In the case of $\\mu =0$ $k_{nn}(k)$ is independent of $k$ .",
"In a such a case, no correlation is found in the network (neutral network).",
"In the assortative networks, hubs tend to connect to other hubs.",
"Thus, in this kind of networks, the nodes with approximately same degree have a tendency to connect with each other.",
"Indeed, in assortative(disassortative) networks, the parameter $k_{nn}(k)$ increases (decreases) with increasing $k$ ."
],
[
"RESULTS",
"We constructed the flares complex network using the position and the occurrence time of 14395 flares.",
"On the basis of solar differential rotation, the positions (longitudes and latitudes) on the solar sphere were rotated with respect to the position of the first flare ( January 1, 2006).",
"We divided the solar surface into cells with equal areas, as presented in Figure REF .",
"The number of cells ($n^2$ ) ranged between 1936 and 7744.",
"The birth positions of the flares are set to assigned cells.",
"The filling factor of nodes $(N/n^2)$ over the solar surface varies from $0.59$ to $0.45$ (Table REF ).",
"As seen in Figure REF , when the aggregation of the number of flares in one of the solar hemispheres increases over several years, it decreases in the other hemisphere.",
"During the years 2006 to 2009, the number of flares in the southern hemisphere is noticeably more than in the northern hemisphere.",
"In the vicinity of the southern pole (latitudes$<-80$ ), a smaller number of flares were detected.",
"About $47\\%$ and $53\\%$ of the flares occurred at the northern and southern solar hemisphere, respectively.",
"The DFA method is applied on the time series of the occurrence flares and the result of this analysis is obtained to be 0.85.",
"As noted, if the value of Hurst exponent is ranged in (0.5 1), there is a long-temporal correlation over the time series.",
"The probability distribution function (PDF) for the degree of nodes is shown in Figure REF .",
"[11] showed that the thresholded power-law distribution is a suitable function for describing of the solar and stellar flares size (energy) distributions.",
"The thresholded power-law function is given by $p(k) \\propto (k+k_0)^{-\\gamma },$ where $k_0$ and $\\gamma $ are the thresholded value and the power-law exponent.",
"In the fitting process, we used the key steps are prescribed by [11].",
"The uncertainty of the power-law exponent is $\\sigma _k =\\gamma / \\sqrt{n}$ [8].",
"As we see in the figure, the values of the degree exponent for the different network sizes are greater than three.",
"Following Eq.",
"(REF ), if we use $k_{max}$ instead of $k_{cut}$ , the estimated power-law exponent ($\\gamma _{est}$ ) will be in good agreement with the values given in Table REF (Columns 8 and 9).",
"The ratio of the maximum to the average degree of nodes ($k_{max}/\\langle k\\rangle $ ) in the flares network for different sizes of networks is obtained to be greater than $3.5$ (Table REF , Column 7).",
"This indicates that the flares network is not a random network.",
"In Figure REF , two \"flares belts\" ($-29<$ latitudes$<-4 $ and $1<$ latitudes$<29$ ) are exhibited.",
"As seen, we found that more than $65\\%$ of the flares were only generated at $15\\%$ of the solar surface.",
"The positions of the 118 hubs (high-connectivity regions) are demonstrated in Figure REF .",
"About $3\\%$ of the solar surface is assigned to regions consisting of hubs and about $11\\%$ of the generated flares were located at these positions.",
"The occurrence rates of the flares (M and X) are three times as much as that computed for the hubs.",
"In Figure REF , the degree correlation $k_{nn}(k)$ versus the degree of nodes for different network sizes is presented.",
"The negative value obtained for the slope of the fitted straight line shows that the network is disassortative.",
"A similar behavior was found for \"arxiv.org\" network [49].",
"The average of the clustering coefficient for the same degree of nodes $C(k)$ is presented in Figure REF .",
"The values of the power-law exponent $(\\alpha \\approx 0.5)$ are approximately constant for different sizes of the networks.",
"The power-law behavior of $C(k) \\sim k^{- \\alpha }$ ensures that the flares network is a scale-free network.",
"In some scale-free networks (e.g., the World Wide Web, semantic web, etc.",
"), the probability of getting a new link to a new node increases by increasing the connectivity of a node [14], [36], [66].",
"This is generic property of hierarchial networks.",
"The explanation of the hierarchial network is given by [49].",
"They showed that, the power-law exponent of $C(k)$ remains approximately constant for the scale-free networks with the degree exponents fall in the range 3 to 5 (See Figure 9 therein).",
"The clustering coefficient of the hubs for the flares network takes small values.",
"By decreasing the degrees of nodes, the clustering coefficient increases.",
"As shown in Figure REF , the clustering coefficient of the constructed network ($C$ ) and its equivalent random network ($C_{rand }$ ) is presented.",
"When the cell size is small (i.e., the network resolution increases), the ratio of the flares clustering coefficient to the random one ($C/C_{rand}$ ) takes the larger values (see Table REF and Figure REF ).",
"It means that the flares network becomes completely distinguishable from its equivalent random network.",
"In Figure REF , the behavior of the characteristic path length versus the network size is displayed.",
"The characteristic path length has a logarithmic relation with the network size as $\\Lambda \\sim 2.58 \\log (N)$ .",
"Furthermore, when the network size grows from 1137 to 3487, the diameter of the flares network changes slightly from 10 to 14 (Table REF , Column 8)."
],
[
"CONCLUSIONS",
"In this work, the characteristics of the solar flares network are studied to extract laws governing flare occurrence over the solar surface.",
"To do this, the complex network is constructed using a flares data set (including positions and occurrence times) recorded during January 1, 2006 to July 21, 2016.",
"Since the system of flares is a limited, predictable, self-organized with long temporal correlation, non-linear, and scale-free system, it is concluded that the flares system is a complex one.",
"We constructed the complex network of the flares system using their positions and occurrence times on the solar surface in the same way [1] proposed as constructing the earthquake networks.",
"We divided the solar surface into cells with equal areas where the number of cells increases from 1936 to 7744.",
"Because the length of cells along the solar latitudes is non-uniform (Eq. )",
"and the recorded positions of the flares are in degree form (integer), constructing a network with small cell sizes ($< 1 ^{\\circ }$ ) is crucial with the present data.",
"By increasing the spatial resolution of the flares position, designing a flares network with of a larger size is possible.",
"The power-law nature of the PDF degree confirms shows that the flares network is a scale-free network.",
"At the positions of the network hubs, the flaring probability is higher than at other nodes.",
"We found out that over the flares networks, hubs do not have a tendency to form links with the other hubs.",
"There is a tendency to create a link between small degree of nodes and hubs.",
"Our results show that the probability of the occurrence of large flares (M and X) over regions generating flares covering only 15% of the solar surface is about twice as much as in other regions.",
"Also, we found that the flares occurring over one of the hemispheres has a certain effect on flare occurrence emerged in the other hemisphere.",
"Our results show that the flares network is not a random network because the degree distribution does not follow the Poisson distribution.",
"In the flares network, there are several special nodes with large values of degree (large $k$ ) where the nodes become hubs characterizing the scale-free network.",
"The degree exponents of the nodes for undirected, incoming, and outgoing networks are the same.",
"Furthermore, the ratio of $k_{max}/\\langle k\\rangle $ ensures that the flares network is scale-free, and so, hubs are naturally generated.",
"Also, the power-law behavior of degrees with $\\gamma >3$ expresses that all flares networks construct a small-world network [27].",
"Since the degree correlation exponents take the negative values, the flares network is categorized in the group of disassortative networks.",
"We found that in the flares networks, the hubs are not correlated to the other hubs; they are only correlated with nodes including smaller degrees.",
"In other words, although some of the hubs are neighbors on the solar surface, there do not tendency to interact directly with each other.",
"Computing the filling factors of hubs in a different temporal range of our data set shows that the hubs always covers about 3 % of the solar surface.",
"The scale-free and small-world behavior of flares confirms that there is universality in the characteristic of the solar flares system.",
"Given the low resolution (spatial, temporal, and energy band) of early solar instruments, the lack of full-covering solar surface by telescopes, and the computational algorithmic errors for the identification of small events, the number of low-energy flares (A type) with certain positions is thinly populated in the solar flare data set.",
"Furthermore, the number of high-energy flares (X type) intrinsically occurs at a lower rate.",
"Although, the flares data set provides parameters for constructing flares network; it is not yet adequate for investigating time evolution of the system.",
"We acknowledge the Lockheed Martin Solar and Astrophysics Laboratory (lmsal) team for making the data publicly available.",
"This research makes use of SunPy, an open-source and free community-developed solar data analysis package written in Python [72].",
"Table: The properties of the scale-free network extracted from the complex flares network.Table: The properties of the small world extracted from the complex flares networks.Figure: The solar surface (latitudes and longitudes) is divided in 88×8888\\times 88 cells with equal areas.",
"The location of the flares is placed into cells (nodes) and the empty cells are removed from the flares network analysis.",
"About 45%45\\% of cells are considered as the nodes of the flares network.Figure: The scattering of the flares over the solar latitudes is presented.",
"During the years 2006 to 2009, the number of flares over the southern hemisphere is noticeably more than in the northern hemisphere.",
"Over the solar latitudes <-80< -80, just a small number of flares appeared.",
"About 47%47\\% and 53%53\\% of the flares occurred in the northern and southern solar hemispheres, respectively.Figure: A time series of the daily solar flares (black line) with its smoothed monthly (yellow) curve from January 1, 2006 to July 21, 2016 including the number of 14395 flares.Figure: A small part of the flares network with its connectivity distribution for 12 nodes and 21 flares with ENames (e.g., 𝚐𝚎𝚟_20110411_2211\\texttt {\\texttt {gev}}\\_ 20110411\\_2211) are presented.",
"For example, within the node 420, the flare with EName 𝚐𝚎𝚟_20110411_1652\\texttt {\\texttt {gev}}\\_ 20110411\\_1652 appeared and connected with the flare with EName 𝚐𝚎𝚟_20110411_2025\\texttt {\\texttt {gev}}\\_ 20110411\\_2025 which is occurred in node 1461.",
"A loop connects two successive flares (𝚐𝚎𝚟_20140207_1441\\texttt {\\texttt {gev}}\\_ 20140207\\_1441 and 𝚐𝚎𝚟_20140208_0028\\texttt {\\texttt {gev}}\\_ 20140208\\_0028), which appeared at the same node (3049).Figure: The nodes (circles) and edges (line) of a flares network are depicted.",
"Each line presents a connection between sympathetic flaring (nodes).",
"As we see, the flaring belts over the northern and southern hemispheres are completely separated; but, because of the mutual influential interaction between the two hemispheres, lots of connections are made by all consecutive flares.",
"Therefore, the solar equatorial region was filled with lots of links while a smaller number of flares occurred over the solar equator region.Figure: The PDF for the degree distribution of the flares networks are plotted in a log-log scale for the network size (a)1137,(b)2018,(c)2681,(a)~1137, (b)~2018, (c) ~2681, and (d)3487 (d)~3487 .",
"The degree exponents for the power-law fits for different sizes of networks are obtained to be greater than 3.Figure: Two \"flares belts\" (-29<-29 < latitudes<-4 < -4 and 1<1 < latitudes <29< 29) are shown.",
"We see that these two belts cover more than 65%65\\% of the flares generated at 15%15\\% of the solar surface.",
"The probability of large flares (M and X) occurring over these regions determined by the belts is about twice as much as that in other regions.Figure: The positions of the 118 hubs (high-connectivity regions) are presented.",
"It is discovered that about 3% of the solar surface covers by hubs regions and 11% of the flares were generated at these positions.",
"A similar results are obtained for smaller networks.",
"The occurrence rates of flares M and X within the cells consisting of hubs are three times as much as those that emerged in the other nodes.Figure: The degree correlation function presents that in the flares network (a) 1137, there is no degree correlation between the nodes μ∼0\\mu \\sim 0.",
"The degree correlation function k nn (k)k_{nn}(k) of the flares networks for different network sizes (b)2018,(c)2681,(b) ~2018, (c) ~2681, and (d)3487(d)~3487, with μ<0\\mu < 0 shows that these flares networks are disassortative.Figure: The average of the clustering coefficient for nodes (circles) with the same degree in the undirected flaresnetwork is presented.",
"The power-law fits (solid lines) for different network sizes (N),(a)1137,(b)2018,(c)2681,(N),~ (a) ~1137, (b) ~2018, (c)~ 2681, and (d)3487(d)~ 3487 are presented.Figure: The behavior of the clustering coefficient versus both the size of the flares network (circle) and its equivalent random network (square) is shown.",
"The ratio of the C/C rand C/C_{rand} (triangle) becomes larger when the cell size decreases or the network resolution increases.Figure: The characteristic path length of the flares networks versus the network size (N)(N) and a fitted straight line as Λ∼2.58log(N)\\Lambda \\sim 2.58 ~\\log (N) are displayed."
]
] | 1709.01677 |
[
[
"The interior angular momentum of core hydrogen burning stars from\n gravity-mode oscillations"
],
[
"Abstract A major uncertainty in the theory of stellar evolution is the angular momentum distribution inside stars and its change during stellar life.",
"We compose a sample of 67 stars in the core-hydrogen burning phase with a $\\log\\,g$ value from high-resolution spectroscopy, as well as an asteroseismic estimate of the near-core rotation rate derived from gravity-mode oscillations detected in space photometry.",
"This assembly includes 8 B-type stars and 59 AF-type stars, covering a mass range from 1.4 to 5\\,M$_\\odot$, i.e., it concerns intermediate-mass stars born with a well-developed convective core.",
"The sample covers projected surface rotation velocities $v\\sin\\,i \\in[9,242]\\,$km\\,s$^{-1}$ and core rotation rates up to $26\\mu$Hz, which corresponds to 50\\% of the critical rotation frequency.",
"We find deviations from rigid rotation to be moderate in the single stars of this sample.",
"We place the near-core rotation rates in an evolutionary context and find that the core rotation must drop drastically before or during the short phase between the end of the core-hydrogen burning and the onset of core-helium burning.",
"We compute the spin parameter, which is the ratio of twice the rotation rate to the mode frequency (also known as the inverse Rossby number), for 1682 gravity modes and find the majority (95\\%) to occur in the sub-inertial regime.",
"The ten stars with Rossby modes have spin parameters between 14 and 30, while the gravito-inertial modes cover the range from 1 to 15."
],
[
"Introduction",
"Rotation has a significant impact on the life of a star [21].",
"It induces a multitude of hydrodynamical processes that affect stellar structure but have remained poorly calibrated by data.",
"While surface abundances and surface rotation give some constraints for stellar models, one needs calibrations of the interior rotation and chemical mixing profiles of stars to evaluate the theoretical concepts that have been developed to describe transport phenomena and their interplay [18], [23].",
"Here, we are concerned with the interior rotation rate of stars in the core-hydrogen burning phase of their life (aka the main sequence).",
"Asteroseismology based on long-term high-precision monitoring of nonradial oscillations is the best way to deduce the interior rotation of stars, following the case of helioseismology [44].",
"Early estimates of the interior rotation of main sequence stars were achieved from rotational splitting of a few pressure modes in $\\beta \\,$ Cep stars from ground-based data [6], [32], [10].",
"However, it concerned only very rough estimates of the ratio of the near-core to envelope rotation rates, with values from 1 to $\\sim 4$ for three stars with a mass between 8 and 10 M$_\\odot $ [1].",
"A major breakthrough in the derivation of interior rotation rates was achieved thanks to the NASA Kepler mission, once it had delivered light curves with a duration longer than two years.",
"The early and so far majority of interior rotation rates were obtained from the rotational splitting of dipole mixed modes in low- and intermediate-mass evolved stars [8], [28], [16], [15], [14].",
"These seismic results constituted an unanticipated result, since the cyclic rotation frequency $\\Omega /2\\pi $ of the stellar core (denoted as $\\Omega _{\\rm core}$ hereafter) of these evolved stars turned out to be two orders of magnitude lower than theoretical predictions.",
"This pointed to stronger coupling between the stellar core and envelope during and/or after the main sequence, irrespective of the star having undergone a helium flash or not.",
"This major shortcoming of the theory of angular momentum transport remains unsolved [43], [11], [17].",
"In Sect.",
"2, we shed new light on this topic by considering the interior rotation rates derived so far from Kepler and BRITE photometry of red-giant progenitors, i.e., intermediate-mass gravity-mode pulsators on the main sequence.",
"In Sect.",
"3, we provide the observational information to evaluate the assumptions made in the theory of angular momentum transport by waves.",
"A promising physical ingredient that can explain the observed behavior of the core-to-surface rotation of main sequence stars is the transport of angular momentum by low-frequency internal gravity waves created at the interface between the convective core and the radiative envelope, observed from space photometry [4], [38].",
"The theory of the interaction between such waves and (differential) rotation inside stars requires proper treatment of the various forces at play.",
"Based on our findings for the rotation, we provide the spin parameters (inverse Rossby numbers) of the gravity modes as input for angular momentum computations.",
"We end the paper with a discussion in Sect.",
"4."
],
[
"Core rotation from gravity-mode oscillations",
"Gravity-mode oscillations of main sequence stars have periods from 0.5 to 3 d [2].",
"Recent space photometry revealed period spacing patterns of such modes in B-, A- and F-type stars covering the entire main sequence [13], [33], [20], [40], [48], [29], [31], [35], [50], just as predicted by theory [24], [9].",
"Such modes have their dominant mode energy in the near-core region of the star and are therefore excellent probes of the chemical gradient left behind during the main sequence shrinkage of the convective core.",
"The detected gravity modes have rotational kernels that are typically 5 to 10 times larger in a narrow near-core region than in the extended radiative envelope [46], [47].",
"For stars that reveal both gravity and pressure modes with rotational splitting, derivation of $\\Omega _{\\rm core}$ and $\\Omega _{\\rm env}$ is possible in an almost model-independent way.",
"This was first achieved by [20] for the A-type Kepler target KIC 11145123 and soon followed by [40] for the F-type star KIC 9244992.",
"More so-called hybrid pulsators with rotational splitting in both types of modes have been found meanwhile.",
"The majority of intermediate-mass gravity-mode pulsators found in the Kepler data reveal quantitative information on the near-core rotation but not on the envelope rotation.",
"As derived by [47] and [31], the slope of the period spacing pattern of prograde, zonal, or retrograde modes of consecutive radial order delivers a direct probe of $\\Omega _{\\rm core}$ , even in the absence of rotational splitting.",
"A value of $\\Omega _{\\rm env}$ can be deduced from rotational splitting of pressure modes.",
"In the absence of identified pressure modes, the surface rotation can still be derived with high precision from the detection of rotational modulation when this phenomenon occurs together with gravity modes in the light curves.",
"Otherwise, only a lower limit for $\\Omega _{\\rm env}$ can be deduced from a spectroscopic measurement of the projected surface rotation velocity $\\Omega _{\\rm env}\\,R\\sin \\,i$ [29], [47].",
"This can be achieved by adopting a reasonable value for the stellar radius while paying attention to line-profile broadening induced by all the tangential velocity fields due to gravity modes [5].",
"Figure: Core rotation rates (circles) as a function ofspectroscopically-derived gravity for core-hydrogen burning stars with a massbetween 1.4 and 2.0 M ⊙ _\\odot (green) and 3 to ∼5\\sim \\,5 M ⊙ _\\odot (blue)derived from dipole prograde gravito-inertial modesin main sequence stars.",
"Surface rotation rates(triangles) are deduced from pressure modes or from rotationalmodulation.",
"Errors on the rotation rates are smaller than the symbol size,while the errors on the gravity are indicated by dotted lines.Asteroseismically derived rotation rates and gravities for evolved stars withsolar-like oscillations in the mass range 1.41.4\\,M ⊙ <_\\odot <M<3.3<3.3\\,M ⊙ _\\odot have been added (errors smaller than symbol sizes for both quantities): RGBstars (red), red clump stars (gray) and secondary clump stars (orange).Binary components are indicated with an extra ++ sign inside the circles.",
"Theerrors of logg\\log \\,g have been omitted in the zoomed window for clarity.Figure REF includes 67 main sequence stars from [20], [40], [46], [26], [29], [47], [41], [31], [35], [42], [19], and Guo et al.",
"(submitted), covering spectral types from mid B to early F. All these studies rely on Kepler or BRITE space photometry.",
"Their gravity modes revealed $\\Omega _{\\rm core}$ through asteroseismology and their $\\log \\,g$ resulted from high-resolution spectroscopy (except for the four stars in [31], for which we took $\\log \\,g$ from the Kepler input catalogue and estimated its error from [48], their Fig. 2).",
"For 18 of the stars, the surface rotation rate is also available, either from asteroseismology of pressure modes or from rotational modulation.",
"Twelve main sequence stars are member of a spectroscopic binary.",
"The 59 green stars in Fig.",
"REF cover $1.4\\,$ M$_\\odot <$ M$<2.0\\,$ M$_\\odot $ and $v\\sin \\,i\\in [9,170]\\,$ km s$^{-1}$ .",
"According to [49], stars in this mass range rotate on average with $v\\sin \\,i=148\\,$ km s$^{-1}$ , with a dispersion of $54\\,$ km s$^{-1}$ .",
"Given that none of the $v\\sin \\,i$ studies of large ensembles take into account pulsational line broadening, while this overestimates $v\\sin \\,i$ [5], our sample of AF stars is representative in terms of surface rotation for this mass range.",
"On the other hand, our sample contains only 8 B stars with a mass in $3\\,$ M$_\\odot <$ M$<5\\,$ M$_\\odot $ .",
"While they cover a broad range of $v\\sin \\,i\\in [18,242]\\,$ km s$^{-1}$ , they are not representative of the bimodal distribution for this mass range [49].",
"For both the B and AF stars in Fig.",
"REF , the highest core rotation frequency corresponds to $\\sim \\,50\\%$ of the critical value in the Roche formalism.",
"At present, core rotation from gravity modes for stars rotating faster than half critical are not available from seismic modeling due to lack of mode identification.",
"We deduce from Fig.",
"REF that the deviation from rigid rotation in our sample of 67 stars is low to moderate, pointing to strong core-to-envelope coupling during core hydrogen burning, irrespective of the value of the rotation rate.",
"Unfortunately, the errors for the spectroscopic $\\log \\,g$ are too large for this quantity to serve as a proxy of the evolutionary stage of the sample stars.",
"Ideally, we want to use the central hydrogen fraction $X_c$ from seismic modeling on the $x-$ axis rather than $\\log \\,g$ .",
"At present this time-consuming task was only done for two B stars, revealing both to be in the first part of the main sequence with $X_c=0.63\\pm 0.01$ and 0.50$\\pm 0.01$ , in agreement with the spectroscopic $\\log \\,g$ of the former being 0.2 dex higher than the one of the latter [25], [26].",
"For five of the eight F stars in the inset in Fig.",
"REF , a seismic estimate of the evolutionary stage is available from comparison of the morphology of the gravity-mode period spacings: $X_c\\in [0.01,0.22]$ .",
"Differences in the core-to-surface rotation, even if moderate, are significant for several stars in Fig.",
"REF .",
"Both faster cores than surfaces and vice versa occur.",
"Large nonrigidity occurs in some close binaries, where tidal forces are active.",
"Even though we cannot pinpoint $X_c$ values for all stars at present, the results in Fig.",
"REF require an angular momentum redistribution mechanism that can explain the diversity of the measured core-to-envelope rotation during core hydrogen burning.",
"One such mechanism is discussed in the following section.",
"To interpret the results in Fig.",
"REF in a more global evolutionary context, we also show the core and surface rotation rates of successors of these 67 stars, i.e., red giants on the RGB (red), in the red clump (gray), and in the secondary clump (orange), limiting to the 152 stars with masses above 1.4 M$_\\odot $ from [28], [14] and one binary from [7].",
"For all these evolved stars, the gravity in the figure is derived from scaling relations of solar-like oscillations (errors are smaller than the symbol size; B. Mosser kindly made the masses and radii of the stars in [28] available to us).",
"As argued by [27], stars born with M$>2.1\\,$ M$_\\odot $ avoid a helium flash after the hydrogen shell burning.",
"Their core contraction during hydrogen shell burning occurs on a short timescale (the so-called Hertzsprung gap) and the onset of core helium burning happens quietly in non-degenerate matter.",
"These stars are hard to catch in their shell burning phase but reveal themselves as secondary clump stars (mass range up to 3.3 M$_\\odot $ in Fig.",
"REF ).",
"On the other hand, the successors of the stars born with $1.4\\,$ M$_\\odot <$ M$<2.1\\,$ M$_\\odot $ can readily be observed while burning hydrogen in a shell surrounding their helium core and start helium burning violently in degenerate matter.",
"The importance of the stellar mass and the occurrence (or not) of a convective core and/or $\\mu -$ gradient zone for the evolution of the angular momentum was already emphasized by [43] and [17].",
"This becomes even more prominent from our sample of intermediate-mass gravity-mode pulsators: efficient slow-down of the stellar core happens already on the main sequence for stars with a convective core.",
"The three slowest rotators among the B stars have masses between 3.0 and 3.3 M$_\\odot $ and are thus progenitors of the most massive orange stars; two of these are mid main sequence while the one with the counter-rotating envelope is young with $X_c=0.63$ ."
],
[
"Angular momentum transport by waves",
"2D simulations of angular momentum transport by internal gravity waves of a 3 M$_\\odot $ star at birth are able to explain the observed core-to-envelope rotation in Fig.",
"REF in a qualitative sense [38].",
"Such simulations are computationally too demanding to be coupled to stellar evolution computations.",
"The latter thus must resort to analytical treatments of angular momentum transport.",
"These rely on approximations [22].",
"Observed sub- and super-inertial waves can only propagate in a mode cavity set by the Brüntt-Väisälä frequency $N$ .",
"Nonlinear wave interactions for the low-frequency regime can at first instance be ignored because the mode frequencies (denoted as $\\nu $ ) fulfill $\\nu <N$ .",
"The displacement vector of the observed coherent heat-driven gravity modes excited in the partial ionization zone of iron-like elements are dominantly horizontal.",
"Indeed, multicolor photometry and high-precision line-profile variations in spectroscopy reveal a typical ratio of 10 to 100 between the horizontal and vertical component of the displacement vector of the gravity modes [12].",
"There is no reason why this would not be the case for convectively driven internal gravity waves.",
"This property simplifies the theoretical treatment of the wave transport [22].",
"The adiabatic and Cowling approximations are appropriate to describe the low-frequency gravity waves in the near-core region of our sample stars.",
"Figure: Spin parameters for 47 main sequence stars in Fig.",
"1 for which weidentified the mode wavenumbers (same color convention).",
"For each star,the range in frequencies of detected mode series of consecutive radial orderis indicated by a line, from the circle that represents the mode with thelowest frequency in the corotating frame to the triangle representing the modewith the highest frequency.",
"For all the F-type stars with |s|>15|s|>15, the highspin parameters are caused by low-frequency retrograde Rossby modes.Ignoring the Lorentz and tidal forces, this brings us to the key parameter to evaluate the approximations that can or cannot be made in the treatment of the angular momentum transport by waves: the interior rotation frequency of the star with respect to the wave frequencies.",
"For the stars on the main sequence indicated in blue and green in Fig.",
"REF , we provide an estimate of the importance of the Coriolis force for their transport of angular momentum.",
"This importance is expressed by the spin parameter, also known as the inverse of the Rossby number and defined here as $s=2\\,\\Omega _{\\rm core}/\\nu $ , where $\\nu $ is the cyclic frequency of the gravity wave under study in a reference frame of the star corotating with $\\Omega _{\\rm core}$ .",
"Defining a corotating reference frame is not obvious in the case of nonrigid rotation; we choose to define $s$ with respect to the value of $\\Omega (r)/2\\pi $ that is most robustly determined.",
"Following the properties of the rotational kernels of the gravity modes, this is the value in the near-core $\\mu $ -gradient region [46], [47].",
"In Fig.",
"REF we show the computed spin parameters based on the asteroseismic $\\Omega _{\\rm core}$ and on the frequencies of the identified gravity-mode series with consecutive radial order for the 7 B-type stars in [33], [34], [35] and 40 F-type stars in [48], all of which are included in Fig.",
"REF .",
"For each star, we connect the lowest- and highest-frequency mode (in the corotating frame) to visualize the range of $s$ for each series in each star and compare it to a Rossby number of 1 (horizontal black line).",
"Figure REF represents 1682 modes in these 47 stars, all of which have been detected and identified from 4-year Kepler space photometry.",
"We find that most of the stars have gravito-inertial modes rather than pure gravity modes: 1607 modes have frequencies in the sub-inertial regime.",
"Among the detected modes are 273 (17%) retrograde Rossby modes occuring in 10 F-type stars in the sample.",
"The slowest rotators reveal 75 (i.e., $<5\\%$ ) pure gravity modes in the super-inertial regime.",
"About 12% of the modes in the 7 B stars and 4% of the modes in the 40 AF stars occur in the super-inertial regime.",
"We conclude that only the modes in the slowest rotators fulfil the formal mathematical condition on the spin parameter to apply the so-called Traditional Approximation (TA) for the Coriolis force [45], [22].",
"The detected Rossby modes have by far the highest spin parameters, covering $s\\in [14,30]$ , while the gravito-inertial modes have $s\\in [1,15]$ .",
"We point out that the core rotation rates in Fig.",
"REF were obtained by adopting the TA and assuming rigid rotation for the observed gravito-inertial modes of the stars.",
"The assumption of rigidity is not a drawback, because these modes have by far their largest mode energy in the near-core region while the mode kernels and mass distribution in the stellar envelope hardly contribute to the core rotation values [20], [47].",
"On the other hand, it was shown by [31], who considered both 1D and 2D treatments of the modes for static equilibrium models, that the TA still provides a good approximation in the case of zonal and prograde gravito-inertial modes for the frequency regimes and spin parameters treated here and used to derive Fig.",
"REF .",
"However, the TA is less justified for the 20 retrograde gravito-inertial modes among the 1607 sub-inertial modes.",
"Modelling the 293 retrograde gravito-inertial and Rossby modes will benefit from a 2D non-perturbative treatment of the rotation as developed by [30]."
],
[
"Discussion",
"An important aspect to consider along with simulations of angular momentum transport by internal gravity waves is the chemical mixing induced by these waves and its comparison with mixing prescriptions adopted in 1D stellar evolution codes.",
"Hints that pulsational mixing might be dominant over rotational mixing were found from a sample of pulsating magnetic and non-magnetic OB-type stars [3].",
"A quantitative evaluation of the level of mixing in the near-core region and in the radiative envelope can be obtained from gravity-mode asteroseismology [25], [26].",
"This opens new perspectives to include seismically calibrated prescriptions for mixing and angular momentum transport based on multi-D numerical simulations of waves in stellar evolution computations.",
"First attempts to compute chemical mixing due to gravity waves look promising when compared with asteroseismic results [39].",
"More gravity-mode pulsators than those treated here are under study, but they either have too few unambiguously identified modes to derive $\\Omega _{\\rm core}$ [50] or they have not been observed in spectroscopy to derive their $v\\sin \\,i$ and $\\log \\,g$ [31].",
"Including all those with identified modes but without a spectroscopic $\\log \\,g$ in Fig.",
"REF requires forward modeling to derive a seismic $X_c$ and $\\log \\,g$ .",
"Progress to populate Fig.",
"REF more densely from deeper exploitation of the Kepler and BRITE data and to transform this figure into a true “evolutionary” diagram is expected in the near future.",
"Unfortunately, we currently lack $\\Omega _{\\rm core}$ from gravity modes for stars born with $>5\\,$ M$_\\odot $ , and in particular of core-collapse supernova progenitors.",
"A major breakthrough is expected from large samples of OB-type stars in the Milky Way and in the Large Magellanic Cloud to be observed during almost one year with the TESS mission [37] and for the long pointings of the PLATO mission [36].",
"Only after deducing the interior rotation of single and binary OB-type stars from asteroseismology covering the hydrogen and helium core and shell burning stages will we be able to understand angular momentum evolution and confront it with the one of white dwarfs and neutron stars.",
"CA and TVR are grateful for the kind hospitality and opportunity to perform part of this research at the Kavli Institute of Theoretical Physics, University of California at Santa Barbara, USA.",
"The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ $ 670519: MAMSIE), from the Research Foundation Flanders (FWO, grant agreements G.0B69.13 and V4.272.17N) and from the National Science Foundation of the United States under Grant NSF PHY11–25915."
]
] | 1709.01874 |
[
[
"Cascading failures in interdependent systems under a flow redistribution\n model"
],
[
"Abstract Robustness and cascading failures in interdependent systems has been an active research field in the past decade.",
"However, most existing works use percolation-based models where only the largest component of each network remains functional throughout the cascade.",
"Although suitable for communication networks, this assumption fails to capture the dependencies in systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines.",
"Here, we consider a model consisting of systems $A$ and $B$ with initial line loads and capacities given by $\\{L_{A,i},C_{A,i}\\}_{i=1}^{n}$ and $\\{L_{B,i},C_{B,i}\\}_{i=1}^{n}$, respectively.",
"When a line fails in system $A$, $a$-fraction of its load is redistributed to alive lines in $B$, while remaining $(1-a)$-fraction is redistributed equally among all functional lines in $A$; a line failure in $B$ is treated similarly with $b$ giving the fraction to be redistributed to $A$.",
"We give a thorough analysis of cascading failures of this model initiated by a random attack targeting $p_1$-fraction of lines in $A$ and $p_2$-fraction in $B$.",
"We show that (i) the model captures the real-world phenomenon of unexpected large scale cascades and exhibits interesting transition behavior: the final collapse is always first-order, but it can be preceded by a sequence of first and second-order transitions; (ii) network robustness tightly depends on the coupling coefficients $a$ and $b$, and robustness is maximized at non-trivial $a,b$ values in general; (iii) unlike existing models, interdependence has a multi-faceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network."
],
[
"Introduction",
"With the development of modern technology, networks emerge as the new form of how things work in every aspect of our life, from online social media to cyber-physical systems, from intelligent highways to aerospace systems.",
"Soon we will expect computing and communication capabilities to be embedded in all physical objects and structures and more complex networks to appear [1].",
"Recently, researchers have become increasingly aware of the fact that most systems do not live in isolation, and that they exhibit significant inter-dependencies with each other.",
"In particular, it has been shown that interdependence and coupling among networks lead to dramatic changes in network dynamics, with studies focusing on cascading failure and robustness [2], [3], [4], [5], [6], [7], [8], information and influence propagation [9], [10], [11], [12], [13], percolation [14], [15], [16], [17], [18], etc.",
"One of the most widely studied network dynamics is the cascade (or, spread) of failures.",
"Due to the coupling between diverse infrastructures such as water supply, transportation, fuel and power stations, interdependent networks are tend to be extremely vulnerable [19], because the failure of a small fraction of nodes from one network can produce an iterative cascade of failures in several interdependent networks.",
"Blackouts are typical examples of cascading failures catalyzed by the dependencies between networks: the September 28, 2003 blackout in Italy resulted in a widespread failure of the railway network, health care systems, and financial services and, in addition, severely influenced communication networks.",
"As a result, the partial failure of the communication system in turn further impaired the power grid management system.",
"Robustness of interdependent networks has been an active research field after the seminal paper of Buldyrev et al.",
"[3], with the key result being interdependent networks are more vulnerable than their isolated counterparts.",
"However, existing works on cascading failures in interdependent networks focus extensively on percolation-based models [3], [14], [20], [21], [22], [23], where a node can function only if it belongs to the largest connected (i.e., giant) component of its own network; nodes that lose their connection to this giant core are deemed non-functional.",
"While such models are suitable for communication networks, they fail to accurately capture the dynamics of cascading failures in many real-world systems that are tasked with transporting physical commodities; e.g., power networks, traffic networks, etc.",
"In such flow networks, failure of nodes (or, lines) lead to redistribution of their load to functional nodes, potentially overloading and failing them.",
"As a result, the dynamics of failures is governed primarily by load redistribution rather than the structural changes in the network.",
"A real-world example to this phenomenon took place on July 21, 2012, when a heavy rain shut down a metro line and caused 100 bus routes to detour, dump stop, or stop operation completely in Beijing [24].",
"Figure: Possible transition behaviors under the load redistribution based cascade model.",
"We see that final system collapse is always first order, which may be preceded with one or more first- or, second-order transitions.In this paper, we initiate a study on robustness of interdependent networks under a load redistribution based cascading failure model.",
"Our approach is inspired by the fiber-bundle model that has been extensively used to investigate the fracture and breakdown of a broad class of disordered systems; e.g., magnets driven by an applied field [25], earthquakes [26], [27], power system failure [28], social phenomena [29].",
"This model has already been demonstrated to exhibit rich transition behavior in a single network setting under random attacks of varying size, while being able to capture some key characteristics of real-world cascades [30], [28]; e.g., see Figure REF .",
"In particular, it was shown that the transition point where the system has a total breakdown is always discontinuous, reminiscent of the real-world phenomena of unexpected large-scale system collapses; i.e., cases where seemingly identical attacks leading to entirely different consequences.",
"While this breakdown can take place abruptly without any indicators at smaller attack sizes (as in the middle curve in Figure REF ), it may also be preceded with one or more first-order or second-order transitions (as seen in the other two curves of Figure REF ) that can be taken as early warning signs of a catastrophic cascade.",
"We extend the fiber-bundle-like cascading failure model to interdependent networks as follows.",
"Assume that the system consists of $n$ coupled networks each with a given number of transmission lines.",
"Every line is given an initial load $L$ and a capacity $C$ defined as the maximum load it can tolerate; if the load on the line exceeds its capacity (for any reason) the line is assumed to fail.",
"The main ingredient of the model is the load redistribution rule: upon failure of a line in any network, the load it was carrying before the failure will be redistributed among all networks in the system, with the proportion received by each network being determined by the coupling coefficients across networks; see Section for precise details.",
"Within each network, we adopt the fiber-bundle-like model [30], [28] and distribute this received load equally among all functional lines.",
"We give a thorough analysis of cascading failures (based on the model described above) in a system of two interdependent networks initiated by a random attack.",
"We show that in addition to providing a more realistic model of cascading failures for interdependent systems (as compared to percolation-based models), the model described above gives rise to interesting and novel transition behavior, and challenges the widely accepted notion that interdependence (or, coupling, or, inter-connectivity) is always detrimental for system robustness.",
"In particular, we show that (i) the model captures the real-world phenomenon of unexpected large scale cascades: final collapse is always first-order, but it can be preceded by a sequence of first and second-order transitions; to the best of our knowledge such behavior has not been observed before in any model.",
"(ii) network robustness tightly depends on the coupling coefficients and robustness is maximized at non-trivial coupling levels in general; (iii) unlike existing models, interdependence has a multi-faceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network.",
"We reiterate that although extensive, the literature on cascading failures in interdependent networks is limited to percolation-based models that fail to capture many real-world settings.",
"Load redistribution models on the other hand have mostly been constrained to single-network settings; e.g., [31], [32], [33].",
"The closest work to our paper is by Brummitt et al.",
"[6] where a sandpile model was studied for two inter-connected networks (each being a random regular graph).",
"Although a similar observation regarding the impact of inter-connectivity was made (that it can sometimes help improve robustness), their work is limited to cascades triggered by increased initial load on the system (imitating the sand dropping process) instead of random failures or attacks considered here; as such, [6] does not provide any insight regarding the transition behavior of the system against attacks and how that behavior is affected by the level of inter-connectivity In addition, [6] considers a specific load-capacity relation, while our work covers more general settings.. To the best of our knowledge, the only other relevant work is by Scala et.",
"al.",
"[35] who studied cascades in coupled distribution grids, but again under a load growth model instead of external attacks.",
"The rest of the paper is organized as follows: we formally define the load redistribution model and analysis tools used in Section .",
"Our analytic results are presented in Section , including the solutions for steady-state system sizes.",
"Numerical results are given in section , and we conclude our work in section ."
],
[
"Model Definitions",
"We consider a system composed of $n$ networks that interact with each other.",
"Let $\\mathcal {N}=\\lbrace 1,\\ldots ,n\\rbrace $ denote the set of all networks in the system.",
"For each $i \\in \\mathcal {N}$ , we assume that network $i$ has $N_i$ lines $\\mathcal {L}_{1,i}, \\ldots , \\mathcal {L}_{N_i,i}$ with initial loads $L_{1,i}, \\ldots , L_{N_i,i}$ .",
"Each of these lines is associated with a capacity $C_{1,i}, \\ldots , C_{N_i,i}$ above which the line will be tripped.",
"In other words, $C_{k,i}$ defines the maximum flow that line $k$ in network $i$ can sustain and is given by $C_{k,i} = L_{k,i} + S_{k,i}, \\qquad i \\in \\mathcal {N},\\quad k = 1, \\ldots , N_i\\nonumber $ where $S_{k,i}$ denotes the free space on line $k$ in network $i$ , i.e., the maximum amount of extra load it can take.",
"The load-free space pairs $\\lbrace L_{k,i},S_{k,i}\\rbrace _{k=1}^{N_i}$ are independently and identically distributed with $P_{L_iS_i}(x,y):={\\mathbb {P}}\\left[{L_{k,i} \\le x,~ S_{k,i} \\le y}\\right], \\quad k=1, \\ldots , N_i$ for each $i \\in \\mathcal {N}$ .",
"The corresponding joint probability density function is given by $p_{L_iS_i}(x,y)=\\frac{\\partial ^2}{\\partial x \\partial y} P_{L_iS_i}(x,y)$ .",
"In order to avoid trivial cases, we assume that $S_{k,i}>0$ and $L_{k,i}>0$ with probability one for each $i \\in \\mathcal {N}$ and each $k=1, \\ldots , N_i$ .",
"Finally, we assume that the marginal densities $p_{L_i}(x)$ and $p_{S_i}(y)$ are continuous on their support.",
"Initially, $p_i$ -fraction of lines are attacked (or failed) randomly in network $i$ , where $p_i \\in [0,1]$ .",
"The load on failed lines will be redistributed within the original network and/or shed to other coupled networks depending on the underlying redistribution rules governing the system.",
"Further failures may then take place within the initially attacked network or in the coupled ones due to lines undertaking extra load exceeding their capacity; this in turn leads to further redistribution in all constituent network, potentially leading to a cascade of failures.",
"The cascade of failures taking place simultaneously within and across networks leads to an interesting dynamical behavior and an intricate relationship between the level of coupling and the system's overall robustness.",
"The cascade process is monotone (once failed, a line remains so forever), and thus it will eventually stop, potentially when all lines in all networks have failed.",
"Otherwise a positive fraction of lines may survive the cascade in one or more of the constituent networks.",
"One of our main goals in this paper is to characterize the fraction of alive lines in each network at that `steady state'; i.e., at the point where cascades stop.",
"To that end, we provide a mean-field analysis of dynamical process of cascading failures.",
"Under this approach, it is assumed that when a line fails, its flow will be redistributed to its own network as well as to other networks with the proportion redistributed to each network determined by coupling coefficients among the networks; more on this later.",
"Each network will then distribute its own share of the failed load equally and globally among all of its remaining lines.",
"Although simple, the equal load redistribution model is able to capture the long-range nature of failure propagation in physical systems (e.g., Kirchhoff's law for power networks), at least in the mean-field sense, as opposed to the topological models [36], [32] where failed load is redistributed only locally among neighboring lines.",
"In our case, it also enables focusing on how coupling and interdependence of two arbitrary networks affect their overall robustness, even if individual network topologies might be unknown.",
"As mentioned before, the flow of a failed line in a network will not only be redistributed internally, but will also be shed to other coupled networks.",
"The proportion of load to be shed from a failed line in network $i$ to network $j$ is determined by the coupling coefficient $a_{ij}$ , where we have $\\sum _{j \\in \\mathcal {N}}a_{ij}=1$ for all $i$ in $\\mathcal {N}$ ; thus, $1-\\sum _{j \\in \\mathcal {N}-\\lbrace i\\rbrace }a_{ij}$ gives the fraction of the load that will be redistributed internally in network $i$ .",
"The load received in each network is then shared equally among all of its functional lines.",
"Upon redistribution of flows, the load on each alive line will be updated potentially leading to some lines having more load than their capacity, and thus failing.",
"Subsequently, the load of those additionally failed lines will be redistributed in the same manner, which in turn may cause further failures, possibly leading to a cascade of failures in both the initiating networks and their coupled networks.",
"This phenomenon imitates the interdependent systems in real world where the failure in one network, such as power network, can affect the behavior of another network, such as water system and financial systems.",
"Figure: Illustration of a two-network system.",
"When failures happen in network BB, bb-portion of the failed loads goes to network AA and (1-b)(1-b)-portion stays in BB.",
"Similarly in network AA, (1-a)(1-a)-portion stays and aa-portion goes to BB.",
"Failed loads will be redistributed equally and globally among the remaining lines in each network.For the ease of exposition, we consider a two-network system in the rest of the paper, although our results can be extended trivially to arbitrary number of networks.",
"Consider a system composed of networks $A$ and $B$ that are interdependent in the following manner Of course, there are other ways for two networks to be “interdependent\" with each other.",
"Here, we use this term with its general meaning, i.e., that failures in one network may lead to failures in the other and vice versa, potentially leading to a cascade of failures.",
"Our model constitutes a special case where interdependence emerges from the inter-connectivity between the two networks: when a failure happens in network $A$ , $a$ fraction of the failed load is transferred to network $B$ , while the remaining $1-a$ fraction being redistributed internally in $A$ .",
"Similarly upon failures in network $B$ , $b$ fraction of the failed load will be shed to network $A$ ; here $a,b \\in [0,1]$ are system defined constants.",
"An illustration of the system can be found in Figure REF .",
"We assume that initially $p_1$ -fraction of lines in network $A$ and $p_2$ -fraction of lines in network $B$ fail randomly.",
"The initial attacks may cause cascading failures, and if one of the network collapses (i.e., if all of its lines fail) during this process, the other network will take over the rest of the load in it and function as a single network from that point on.",
"With appropriate meanings of load and capacity, this type of load oriented models can capture the dependencies in a wide range of physical systems; e.g., two smart-grid operators coupled to provide better service [38], two banks highly correlated for collective risk shifting [39], or two interacting transportation networks [21].",
"In what follows, we provide an analytic solution for the dynamics of cascading failures in the model described above."
],
[
"Analytic Results",
"We now provide the mean-field analysis of cascading failures in the two-network interdependent system.",
"Without loss of generality, we assume that both networks have the same number of lines, i.e., $N_A = N_B = N$ .",
"We assume that time is divided into discrete steps, $t=1, 2, \\ldots $ .",
"For each time stage $t$ , and with $X \\in \\lbrace A,B\\rbrace $ , we use the following notation: $f_{t,X}$ : fraction of failed lines until $t$ ; $F_{t,X}$ : total load from lines that fail exactly at time $t$ within network $X$ ; $Q_{t,X}$ : extra load to be redistribution at $t$ per alive line in $X$ ; $N_{t,X}$ : number of alive lines at $t$ in $X$ before redistribution.",
"In what follows, we occasionally provide expressions only for the quantities regarding network $A$ , while the corresponding expressions for network $B$ (that are omitted in the text for brevity) can be obtained similarly.",
"Initially, $p_1$ -fraction of lines in network $A$ and $p_2$ -fraction of lines in network $B$ are attacked (or failed) randomly.",
"Thus, the fraction of failed lines within each network at $t=0$ is given by $f_{0,A} = p_1, \\quad f_{0,B} = p_2$ , while the number of alive lines satisfy $\\nonumber N_{0,A} &= (1 - f_{0,A})N = (1 - p_1)N\\\\ \\nonumber N_{0,B} &= (1 - f_{0,B})N = (1 - p_2)N$ Because the initially attacked lines are selected uniformly at random, the total load from failed lines (in the mean-field sense) satisfy $\\nonumber F_{0,A} &= {\\mathbb {E}}\\left[{L_A}\\right] \\cdot f_{0,A} \\cdot N = {\\mathbb {E}}\\left[{L_A}\\right] \\cdot p_1 \\cdot N\\\\ \\nonumber F_{0,B} &= {\\mathbb {E}}\\left[{L_B}\\right] \\cdot f_{0,B} \\cdot N = {\\mathbb {E}}\\left[{L_B}\\right] \\cdot p_2 \\cdot N$ Based on the equal redistribution rule and the load shedding rule between the two interdependent networks, the extra load per alive line in network $A$ at $t=0$ is: $Q_{0,A} &= \\frac{(1-a) \\cdot F_{0,A} + b \\cdot F_{0,B} }{(1 - f_{0,A})N} \\\\&= \\frac{(1-a) \\cdot {\\mathbb {E}}\\left[{L_A}\\right] \\cdot p_1 + b \\cdot {\\mathbb {E}}\\left[{L_B}\\right] \\cdot p_2 }{1 - p_1}$ and similarly for network $B$ : $Q_{0,B} = \\frac{a \\cdot {\\mathbb {E}}\\left[{L_A}\\right] \\cdot p_1 + (1-b) \\cdot {\\mathbb {E}}\\left[{L_B}\\right] \\cdot p_2 }{1 - p_2}$ At stage $t=1$ , line $k$ in network $A$ that survives the initial attack will fail if and only if the updated loads exceed its capacity, i.e., if $L_{k,A} + Q_{0,A} \\ge L_{k,A} + S_{k,A}$ , or equivalently, if $S_{k,A} \\le Q_{0,A}$ .",
"Based on this condition, the fraction of failed lines at $t=1$ is given by $f_{1,A} &= f_{0,A} + (1-f_{0,A}) \\cdot {\\mathbb {P}}\\left[{S_{A} \\le Q_{0,A}}\\right] \\\\&= 1 - (1 - f_{0,A}){\\mathbb {P}}\\left[{S_{A} > Q_{0,A}}\\right]$ To compute the extra load per alive line in each network at $t=1$ , we need to know the lines that fail exactly at this stage in each network (so that their load can be appropriately redistributed to both networks according to the coupling coefficients).",
"Namely, we need to find the lines that survive the initial attack, but have smaller free space than the redistributed load $Q_{0,A}$ or $Q_{0,B}$ from the previous stage.",
"Let $\\mathcal {A}$ and $\\mathcal {B}$ be the initial set of lines that are attacked or failed initially in network $A$ and $B$ , respectively.",
"Then, the total load on these failed lines in network $A$ at $t=1$ can be derived as $F_{1,A}&= {\\mathbb {E}}\\left[{\\sum _{i \\notin \\mathcal {A},S_{i,A} \\le Q_{0,A}}(L_{i,A}+ Q_{0,A})}\\right] \\nonumber \\\\&= {\\mathbb {E}}\\left[{\\sum _{i \\notin \\mathcal {A}}(L_{i,A}+ Q_{0,A}) \\cdot {\\bf 1}\\left[S_{i,A} \\le Q_{0,A}\\right]}\\right] \\nonumber \\\\&= (1-p_1)N{\\mathbb {E}}\\left[{(L_{A}+ Q_{0,A}) \\cdot {\\bf 1}\\left[S_A \\le Q_{0,A}\\right]}\\right]\\nonumber $ where ${\\bf 1}\\left[\\cdot \\right]$ is the indicator function Let $E$ be an event.",
"Then, ${\\bf 1}\\left[E\\right]$ is a Binomial random variable that takes the value of 1 if $E$ takes place, and 0 otherwise; here we used the fact that for each line $i$ in $A$ , $L_i,S_i$ follow the same distribution $p_{L_A,S_A}$ .",
"Similarly for network $B$ , we have $F_{1,B} &= {\\mathbb {E}}\\left[{\\sum _{i \\notin \\mathcal {B},S_{i,B} \\le Q_{0,B}}(L_{i,B}+ Q_{0,B})}\\right] \\nonumber \\\\&= (1-p_2)N{\\mathbb {E}}\\left[{(L_{B}+ Q_{0,B}) \\cdot {\\bf 1}\\left[S_B \\le Q_{0,B}\\right]}\\right]\\nonumber $ The load of these lines failed at stage 1 will then be redistributed internally and across network, based on the aforementioned coupling coefficients.",
"This leads to the extra load per alive line in network $A$ at $t=1$ being given by $&Q_{1,A}\\nonumber \\\\&= Q_{0,A} + \\frac{(1-a) \\cdot F_{1,A} + b \\cdot F_{1,B}}{N(1-f_{1,A})} \\nonumber \\\\&= Q_{0,A} + \\nonumber \\\\& \\quad \\dfrac{ {(1-a)(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{0,A}) \\cdot {\\bf 1}\\left[S_A \\le Q_{0,A}\\right] }\\right] }{ + b(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{0,B}) \\cdot {\\bf 1}\\left[S_B \\le Q_{0,B}\\right]}\\right] }}{1-f_{1,A}}\\nonumber $ $Q_{1,B}$ can be written in a similar manner.",
"At $t=2$ , more lines will fail because of the redistribution in the previous stage.",
"The condition for a line to fail exactly at $t=2$ is: (i) it doesn't belong to the initial attack set $\\lbrace \\mathcal {A}$ , $\\mathcal {B}\\rbrace $ ; (ii) it survived the redistribution in the previous stage $t=1$ ; and (iii) its capacity is less than the updated total load after redistribution at $t=2$ .",
"From this we can derive the fraction of failed lines till $t=2$ as $&f_{2,A} = 1-(1-f_{1,A}){\\mathbb {P}}\\left[{S_A > Q_{1,A} ~|~ S_A > Q_{0,A}}\\right] \\nonumber \\\\&f_{2,B} = 1-(1-f_{1,B}){\\mathbb {P}}\\left[{S_B > Q_{1,B} ~|~ S_B > Q_{0,B}}\\right]\\nonumber $ Then, the total load from lines that fail exactly at $t=2$ in network $A$ is given by $&F_{2,A}\\nonumber \\\\&= {\\mathbb {E}}\\left[{\\sum _{i \\notin \\mathcal {A},Q_{0,A}<S_{i,A} \\le Q_{1,A}}(L_{i,A}+ Q_{1,A})}\\right] \\nonumber \\\\&= (1-p_1)N {\\mathbb {E}}\\left[{(L_{A}+ Q_{1,A}) {\\bf 1}\\left[Q_{0,A}<S_A \\le Q_{1,A}\\right]}\\right] \\nonumber $ Similarly in network $B$ , we have $& F_{2,B}\\nonumber \\\\&= {\\mathbb {E}}\\left[{\\sum _{i \\notin \\mathcal {B},Q_{0,B}<S_{i,B} \\le Q_{1,B}}(L_{i,B}+ Q_{1,B})}\\right] \\nonumber \\\\&= (1-p_2)N {\\mathbb {E}}\\left[{(L_{B}+ Q_{1,B}) {\\bf 1}\\left[Q_{0,B}<S_B \\le Q_{1,B}\\right]}\\right]\\nonumber $ With the total loads on failed lines $F_{2,A}$ , $F_{2,B}$ and the fraction of failed lines $f_{2,A}$ , $f_{2,B}$ in each network, the extra load per alive line in network $A$ at stage $t=2$ can be calculated as $&Q_{2,A}\\nonumber \\\\& = Q_{1,A} + \\frac{(1-a)F_{2,A} + bF_{2,B}}{N(1-f_{2,A})} \\nonumber \\\\&= Q_{1,A} +\\nonumber \\\\&\\dfrac{ {(1-a)(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{1,A}) \\cdot {\\bf 1}\\left[Q_{0,A}<S_A \\le Q_{1,A}\\right] }\\right] }{ + b(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{1,B}) \\cdot {\\bf 1}\\left[Q_{0,B}<S_B \\le Q_{1,B}\\right]}\\right] }}{1-f_{2,A}}\\nonumber $ A similar expression gives $Q_{2,B}$ .",
"In light of the above derivation, the form of the recursive equations is now clear: for each time stage $t=0,1,\\ldots ,$ we have $&f_{t+1,A} = 1-(1-f_{t,A}){\\mathbb {P}}\\left[{S_A > Q_{t,A} ~|~ S_A > Q_{t-1,A}}\\right] \\nonumber \\\\\\nonumber \\\\&N_{t+1,A} = (1-f_{t+1,A})N\\nonumber \\\\ \\\\&Q_{t+1,A} = Q_{t,A} + \\dfrac{ {(1-a)(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{t,A})\\cdot {\\bf 1}\\left[Q_{t-1,A}<S_A \\le Q_{t,A}\\right] }\\right]}{+ b(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{t,B}) \\cdot {\\bf 1}\\left[Q_{t-1,B}<S_B \\le Q_{t,B}\\right]}\\right] }}{1-f_{t+1,A}}, \\nonumber $ and similarly for network B.",
"From (REF ) we can see that the cascade of failures will stop and the steady state will be reached only when the number of alive lines doesn't change in both networks, i.e., $N_{t+2,A}=N_{t+1,A}$ , $N_{t+2,B}=N_{t+1,B}$ .",
"This is equivalent to having $&{\\mathbb {P}}\\left[{S_A > Q_{t+1,A} ~|~ S_A > Q_{t,A}}\\right] = 1, ~\\text{and} \\nonumber \\\\&{\\mathbb {P}}\\left[{S_B > Q_{t+1,B} ~|~ S_B > Q_{t,B}}\\right] = 1$ In other words, whenever we have finite $Q_{t+1,A}$ , $Q_{t,A}$ , $Q_{t+1,B}$ and $Q_{t,B}$ values that satisfy (REF ), cascading failures will stop and the system will reach the steady state.",
"The recursive expressions (REF ) can be simplified further in a way that will make computing the final system sizes (i.e., fraction of alive lines at steady-state) much easier.",
"Firstly, we use the first expression in (REF ) repeatedly for each $t=0, 1, \\ldots $ to get $\\nonumber \\begin{array}{ll}1-f_{t+1,A} &= (1-f_{t,A}) {\\mathbb {P}}\\left[{S_A > {Q_{t,A}} ~|~ S_A > {Q_{t-1,A}}}\\right] \\\\1-f_{t,A} &= (1-f_{t-1,A}) {\\mathbb {P}}\\left[{S_A > {Q_{t-1,A}} ~|~ S_A > {Q_{t-2,A}}}\\right] \\\\~~~\\vdots & \\\\1-f_{1,A} &= (1-f_{0,A}) {\\mathbb {P}}\\left[{S_A > Q_{0,A}}\\right]\\end{array}$ Multiplying these equations together, we obtain $1-f_{t+1,A} = (1-f_{0,A}) \\prod _{\\ell = 0}^{t} {\\mathbb {P}}\\left[{S_A > {Q_{\\ell ,A}} ~\\big |~ S_A > Q_{\\ell -1,A}}\\right],$ where we set $Q_{-1,A} = 0$ for convenience.",
"Using the fact that $Q_{t,A}$ is non-decreasing in $t$ , i.e., $Q_{t+1,A}\\ge Q_{t,A}$ for all $t$ , we then get $&1-f_{t+1,A}\\nonumber \\\\&= (1-f_{0,A})\\nonumber \\\\&~~~ \\cdot \\frac{{\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right]}{{\\mathbb {P}}\\left[{S_A > Q_{t-1,A}}\\right]}\\cdots \\frac{{\\mathbb {P}}\\left[{S_A > Q_{1,A}}\\right]}{{\\mathbb {P}}\\left[{S_A > Q_{0,A}}\\right]} \\cdot {\\mathbb {P}}\\left[{S_A > Q_{0,A}}\\right] \\nonumber \\\\&= (1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right]$ as we recall that $f_{0,A}=p_1$ .",
"Using the simplified result (REF ) in (REF ), we now get $&f_{t+1,A} = 1 - (1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right] \\nonumber \\\\\\nonumber \\\\&N_{t+1,A} = (1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right] N\\nonumber \\\\\\\\&Q_{t+1,A} = Q_{t,A} + \\dfrac{ {(1-a)(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{t,A})\\cdot {\\bf 1}\\left[Q_{t-1,A}<S_A \\le Q_{t,A}\\right] }\\right] }{ + b(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{t,B}) \\cdot {\\bf 1}\\left[Q_{t-1,B}<S_B \\le Q_{t,B}\\right]}\\right] }}{(1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right] }\\nonumber $ leading to a much more intuitive expression than before.",
"To see why (REF ) makes sense realize that for a line to survive stage $t+1$ without failing, it is necessary and sufficient that it survives the initial attack (which happens with probability $1-p_1$ for line in network $A$ ) and its free-space is greater than the total additional load $Q_{t,A}$ that has been shed on it (which happens with probability ${\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right]$ .",
"This explains the first and second expressions in (REF ).",
"For the last equation that computes $Q_{t+1,A}$ , the extra load per alive line at the end of stage $t+1$ (to be redistributed at stage $t+2$ ), we write it as the previous extra load $Q_{t,A}$ plus the extra load from lines that fail precisely at stage $t+1$ .",
"For a line in network $A$ , failing precisely at stage $t+1$ implies that the line was not in the initial attack (happens with probability $1-p_1$ ) and its free space falls in $(Q_{t-1,A}, Q_{t,A}]$ so that it survived the previous load shedding stage but not the current one.",
"Arguing similarly for lines in network $B$ and recalling the redistribution rule based on coupling coefficients, we can see that the nominator in the second term of $Q_{t+1,A}$ (in (REF )) gives the additional new load that will be shed on the alive lines of $A$ .",
"The whole expression is now understood upon recalling that $(1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right]$ gives the fraction of lines from $A$ that survive stage $t+1$ to take this extra load.",
"It is now easy to realize that the dynamics of cascading failures is fully governed and understood by the recursions on $Q_{t,A}, Q_{t,B}$ given by $&Q_{t+1,A} = Q_{t,A} + \\dfrac{ {(1-a)(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{t,A})\\cdot {\\bf 1}\\left[Q_{t-1,A}<S_A \\le Q_{t,A}\\right] }\\right] }{ + b(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{t,B}) \\cdot {\\bf 1}\\left[Q_{t-1,B}<S_B \\le Q_{t,B}\\right]}\\right] }}{(1-p_1){\\mathbb {P}}\\left[{S_A > Q_{t,A}}\\right] }\\\\\\nonumber \\\\&Q_{t+1,B} = Q_{t,B} + \\dfrac{ {a(1-p_1) {\\mathbb {E}}\\left[{(L_{A}+Q_{t,A})\\cdot {\\bf 1}\\left[Q_{t-1,A}<S_A \\le Q_{t,A}\\right] }\\right] }{ + (1-b)(1-p_2) {\\mathbb {E}}\\left[{(L_{B}+Q_{t,B}) \\cdot {\\bf 1}\\left[Q_{t-1,B}<S_B \\le Q_{t,B}\\right]}\\right] }}{(1-p_2){\\mathbb {P}}\\left[{S_B > Q_{t,B}}\\right] }$ with the conditions for reaching the steady-state still being (REF ).",
"Put differently, in order to find the final system sizes, we need to iterate (REF )-() for each $t=0,1, \\ldots $ until the stop condition (REF ) is satisfied.",
"Let $t^{\\star }$ be the stage steady-state is reached and $Q_A^{\\star },Q_B^{\\star }$ be the corresponding values at that point.",
"The final system sizes $n_{\\infty ,A}$ and $n_{\\infty ,B}$ , defined as the fraction of alive lines in network $A$ and $B$ at the steady state, respectively, can then be computed simply from (viz.",
"(REF )) $ \\begin{split}&n_{\\infty ,A} = 1 - f_{\\infty ,A} = (1-p_1){\\mathbb {P}}\\left[{S_A > Q_A^{\\star }}\\right] \\\\&n_{\\infty ,B} = 1 - f_{\\infty ,B} = (1-p_2){\\mathbb {P}}\\left[{S_B > Q_B^{\\star }}\\right].\\end{split}$ The expressions given above for the steady-state of cascading failures in interdependent systems constitute a non-deterministic, nonlinear system of equations, which often do not have to closed-form solution; contrast this with the single network [28] case, where it is possible to provide a closed form solution to the final system size.",
"Therefore, in the interdependent network case, we solve $\\lbrace Q_A^{\\star }, ~ Q_B^{\\star } \\rbrace $ by numerically iterating over (REF )-().",
"The difficulty of obtaining a closed-form expression for final system sizes arises due to the recursive shedding of load across the two networks.",
"At each stage of the cascade, both networks send a portion of the load from its failed lines to the other network, while receiving a portion of load from the lines failed in the coupled network.",
"Furthermore, the load a line was carrying right before failure depends directly on the extra load per alive line (which decide who fails in the next stage) at the time of its failure.",
"This is why we need to keep track of the set of lines that fail precisely at a particular stage to be able to obtain an exact account of these loads This is also evident from (REF ) where we see that $Q_{t+1}$ depends not only on $Q_t$ but also on $Q_{t-1}$.",
"As a result, the final system size can only be obtained by running over the iterations and identifying the first stage at which the stop conditions (REF ) are satisfied."
],
[
"Final system size under different load-free space distributions and coupling coefficients",
"To verify our analysis with simulations, we choose different load-free space distributions under various coupling coefficients.",
"Throughout, we consider three commonly used families of distributions: i) Uniform, ii) Pareto, and iii) Weibull.",
"These distributions are chosen here because they cover a wide range of commonly used and representative cases.",
"In particular, uniform distribution provides an intuitive baseline.",
"Distributions belonging to the Pareto family are also known as a power-law distributions and have been observed in many real-world networks including the Internet, the citation network, as well as power systems [42].",
"Weibull distribution is widely used in engineering problems involving reliability and survival analysis, and contains several classical distributions as special cases; e.g., Exponential, Rayleigh, and Dirac-delta.",
"The corresponding probability density functions of these distributions are given below for a generic variable $L$ .",
"Uniform Distribution: $L \\sim U(L_{\\textrm {min}},L_{\\textrm {max}})$ .",
"$p_L(x)=\\frac{1}{ L_{\\textrm {max}} - L_{\\textrm {min}} } \\cdot {\\bf 1}\\left[ L_{\\textrm {min}} \\le x \\le L_{\\textrm {max}} \\right]$ Pareto Distribution: $L \\sim {Pareto}(L_{\\textrm {min}}, \\beta )$ .",
"With $L_{\\textrm {min}}>0$ and $\\beta >0$ , the density is given by $p_L(x) = L_{\\textrm {min}}^{\\beta } \\beta x^{-\\beta -1} {\\bf 1}\\left[x \\ge L_{\\textrm {min}}\\right].$ Weibull Distribution: $L \\sim Weibull(L_{\\textrm {min}},\\lambda , k)$ .",
"With $\\lambda , k, L_{\\textrm {min}}>0$ , the density is given by $&p_L(x)\\nonumber \\\\ \\nonumber &= \\frac{k}{\\lambda } \\left(\\frac{x-L_{\\textrm {min}}}{\\lambda } \\right)^{k-1} e^{-\\left(\\frac{x-L_{\\textrm {min}}}{\\lambda } \\right)^{k}}{\\bf 1}\\left[x \\ge L_{\\textrm {min}}\\right]$ The case $k=1$ corresponds to the exponential distribution, and $k=2$ corresponds to Rayleigh distribution.",
"Figure: Final system size under different load-free space distributions and coupling coefficients.",
"We observe interesting transition behaviors under different load-free space distributions and coupling level, and the simulation represented in symbol matches with the analytical results represented in lines.In all simulations, we fix the network size at $N=10^7$ , and for each set of parameters being considered we run 20 independent experiments.",
"The results are shown in Fig.",
"REF where symbols represent the empirical value of the final system size $n_{\\infty ,A}$ of network $A$ (obtained by averaging over 20 independent runs for each data point), and lines represent the analytical results computed from (REF ).",
"We see that theoretical results match the simulations very well in all cases.",
"The specific distributions used in Fig.",
"REF are, from left to right, (i) $L_A \\sim Pareto(10,2)$ , $S_A=0.7 L_A$ , $L_B \\sim Pareto(15,1.5)$ , $S_B=0.4 L$ , and initial attacks are set to $p_2=p_1$ ; (ii) $L_A, L_B \\sim Weibull( 10, 100, k=0.6)$ , $S_A=1.74 L_A$ , $S_B=1.5 L_B$ , and $p_2=0$ ; (iii) $L_A \\sim U[10,30]$ , $S_A \\sim U[5,20]$ , $L_B \\sim U[20,40]$ , $S_B \\sim U[20,75]$ , and $p_1=p_2$ ; (iv) $L_A,L_B \\sim Pareto(10,2)$ , $S_A =0.7 L_A$ , $S_B= 0.7 L_B$ , and $p_1=p_2$ ; (v) $L_A,L_B \\sim U[10,30]$ , $S_A,S_B \\sim U[10,65]$ , and $p_2=0$ .",
"The plots in Fig.",
"REF demonstrate the effect of the load-free space distribution as well as coupling level on the robustness of the resulting interdependent system.",
"We see that both the family that the distribution belongs to (e.g., Uniform, Weibull, or Pareto) as well as the specific parameters of the family affect the behavior of $n_{\\infty ,A}(p)$ .",
"For instance, the curves representing the two cases where load and free space in both networks follow a Uniform distribution demonstrate that both abrupt ruptures and ruptures with a preceding divergence are possible in this setting, depending on the parameters.",
"Both cases on Pareto networks give an abrupt breakdown at the final point, and we see that Weibull distribution gives rise to a richer set of possibilities for the transition of final system size $n_{\\infty ,A}(p)$ .",
"Namely, we see that not only we can observe an abrupt rupture, or a rupture with preceding divergence (i.e., a second-order transition followed by a first-order breakdown), it is also possible that $n_{\\infty ,A}(p)$ goes through a first-order transition (that does not breakdown the system) followed by a second-order transition that is followed by an ultimate first-order breakdown; see the behavior of the purple circled line in Fig.",
"REF .",
"Thus in the next section, we will use Weibull distribution to explore the interesting transition behaviors observed in interdependent systems composed of two identical networks."
],
[
"Transition behavior for two identical networks",
"To explore the effect of coupling and interdependency on the robustness of networks, we couple two (statistically) identical networks.",
"Put differently, we consider networks $A$ and $B$ where the load and capacity of each of their lines are drawn independently from the same distribution.",
"We also assume that they are coupled together in a symmetric way, i.e., that $a=b$ .",
"This is a commonly seen case of an interdependent systems where networks of similar characteristics establish a coupling for mutual benefit; e.g., two grid distributors or financial institutions with similar characteristics.",
"More importantly, this will help us understand the affect of coupling with another identical system on the robustness of a given system; the seminal results of Buldyrev et al.",
"[3] suggest that coupling leads to increased vulnerability under percolation based models.",
"With these motivations in mind, we let the initial loads in both networks follow a Weibull distribution, with shape parameter $k=0.4$ , scale parameter $\\lambda = 100$ , and minimum initial load $L_{min}=10$ .",
"The free space is assigned proportional to the initial load on each line with a tolerance factor $\\alpha $ , i.e.",
"$S=\\alpha L$ where $\\alpha =0.6$ .",
"The network size is fixed at $N=10^8$ .",
"We attack $p$ -fraction of lines randomly in network $A$ , and observe the dynamics of failures driven by the load redistribution across and within the two networks.",
"We then compute the final (i.e., steady-state) size of network $A$ as a function of initial attack sizes $p$ under different values of the coupling coefficient $a$ .",
"The results are depicted in Fig.",
"REF , where symbols represent simulation results averaged over 20 independent runs, while lines correspond to our analytical results; in all parameter settings, we observed little to no variance in the final system size across the 20 independent experiments We believe this is because the network size $N$ is taken to be very large in the experiments and the random variable $n_{\\infty ,A}(p_1)$ converges almost surely to its mean (e.g., by virtue of Strong Law of Large Numbers); though it is beyond the scope of this paper to prove this..",
"Figure: Effect of coupling on the robustness of a single system.We see that contrary to percolation-based models, robustness can indeed be improved by having non-zero coupling between the constituent networks.Inset.",
"The critical point p ☆ p_{\\star } defined as the smallest p 1 p_1 at which n ∞,A (p 1 )n_{\\infty ,A}(p_1) deviates from 1-p 1 1-p_1.",
"The optimal (i.e., largest) p ☆ p_{\\star } is attained at a non-trivial coupling level a=b=≃0.53a = b = \\simeq 0.53.A number of interesting observations can be made from Fig.",
"REF .",
"First, we see that coupling level can lead to significant changes in the robustness against random attacks.",
"In particular, the inset in Fig.",
"REF plots the critical attack size $p_{\\star }$ at which the final network size deviates from the $1-p$ line; given attack size $p$ , the final system size can be at most $1-p$ , which happens when the initial attack does not lead to any further failures.",
"The network can be deemed to be more robust when $p_{\\star }$ is larger.",
"An interesting observation is that unlike the traditional percolation-based models, here coupling with another network might lead to a network to become more robust against failures.",
"To the best of our knowledge, the only other model where coupling can improve robustness is studied by Brummitt et al.",
"[6], which constitutes an extension of the sandpile model.",
"Perhaps more interestingly, we also see that the optimal robustness (i.e., largest $p_{\\star }$ ) is attained at a non-trivial coupling level $a = \\simeq 0.53$ .",
"This suggests that coupling has a multi-faceted impact on robustness and that systems are most robust when they are coupled in a specific, non-trivial way; in Section REF we provide some concrete ways to identify such optimum coupling levels.",
"In addition to affecting the system robustness in non-trivial ways, we see from Figure REF that changing the coupling level can also give rise to different (and, sometimes very interesting) transition behaviors.",
"In particular, we see that network $A$ can go through any one of the transitions demonstrated in previous work [28], [30] for single networks (see Figure REF ) depending on its coupling level with network $B$ .",
"More interestingly, when coupled to network $B$ at a specific level, i.e., with $a=b=0.37$ , it is seen to go through a type of transitions that was not seen in the case when it operates as an isolated network.",
"This behavior can be described as a sequence of first, second, first, second, and first order transitions, and to the best of our knowledge was not seen before in any model We note that the behavior demonstrated here is fundamentally different from the few other cases in the literature where multiple transitions have been reported; e.g., see [8], [18].",
"There, the type or the number of transitions do not change with the level of coupling across the networks.",
"Instead, multiple transitions arise only when networks with different robustness levels are coupled together, and their total (or, average) size is plotted against the size of the attack that is applied to all networks involved..",
"In this case, the network stabilizes twice after a sudden drop in the network size during the cascading process, before going through an abrupt final breakdown.",
"To further explore the transition behavior during the cascading failure process, we plot the number of iterations (i.e., the number of load redistribution steps) needed for the system to reach steady-state.",
"The divergence of the number of iterations is considered to be a good indicator of the onset of large failures, and often suggested as a marker of transition points in simulations; e.g., see [29], [45].",
"We see that this is indeed the case for our model as well.",
"In Fig.",
"REF , we plot the final system size together with the number of iterations taken to reach that final size.",
"The solid lines represent final system size under different coupling coefficients, and the symbols represent the number of iterations needed (divided by the maximum iterations number, 1000) in each case.",
"We see that the number of iterations needed is piece-wise stable with discontinuous jumps corresponding to the transition points, and it diverges near the final breakdown of the network.",
"In Appendix , we provide a more detailed discussion on the possible correlations between the type and number of transitions a network exhibits with the distribution of its load and free-space.",
"Figure: Number of steps needed to reach steady state for identical networks (a=ba=b), for various aa values.",
"For the case when a=0.37a=0.37, we observe a novel, unforeseen transition behavior.For a clearer explanation, let us focus on the case when $a=0.37$ (purple asterisks).",
"We see that both discontinuous drops in the final system size coincide with a discontinuous increase in the number of iterations.",
"As the attack size $p_1$ increases further from that second jump, we see a continuous increase in the number of iterations coinciding with the continuous decrease in final system size.",
"This eventually leads to the number of iterations diverging, and as would be expected coincides with the system breaking down entirely.",
"In Fig.",
"REF , the final system size of network $A$ and $B$ are depicted together (for the case $a=0.37$ ), showing clearly the effect of interdependence on transition behaviors.",
"Up until $p_1=0.0287$ , there are no failed lines in network $B$ although network $A$ already experiences cascading failures; this indicates that all lines in $B$ are able take the extra load from network $A$ even though $A$ loses a significant fraction of its lines at $p_1=0.0271$ .",
"When some lines start failing in network $B$ at $p_1=0.0287$ , a large cascade of failures take place causing a significant number of lines fail from both networks marked by discontinuous drop in the final size of both networks.",
"After this point, the remaining system is able to sustain higher initial attacks (because the lines that survive until this point tend to have larger free-space than average).",
"However, when we reach $p_1=0.0314$ , another large cascade takes place that collapses both networks.",
"This final breakdown is observed almost simultaneously in networks $A$ and $B$ , primarily because once a network collapses, the other network will need to take over all the load in the system, and in most cases will not be able survive on its own.",
"Figure: Final system size in two networks when only network AA has been attacked initially.",
"The two networks are statistically identical with a=b=0.36a=b=0.36.",
"Their loads follow a Weibull distribution with k=0.4k=0.4, λ=100\\lambda =100, L min =10L_{min}=10, and S=0.6LS=0.6 L"
],
[
"Optimizing the robustness of an interdependent system",
"Final breakdown point and critical deviation size are good indicators of robustness, but only when we focus on a single network or a specific network in an interdependent system.",
"We now discuss how the robustness of an entire interdependent system can be quantified, with an eye towards identifying optimal coupling levels that maximize system robustness.",
"Assume that initially $p_1$ fraction of lines from $A$ and $p_2$ fraction of lines from $B$ are attacked randomly.",
"The $p_1,p_2 \\in [0,1]$ plane is naturally divided into four survival regions [35].",
"where i) $S_{12}$ represents the initial attack pair $(p_1,p_2)$ under which both networks survive, i.e., have positive fraction of functional lines when steady state is reached; ii) $S_1$ represents the case where only network $A$ survives; iii) $S_2$ represents the case where only network $B$ survives; and iv) $S_0$ represents the region where no network survives, i.e., the entire system fails with no alive lines.",
"It is then tempting to study the affect of network coupling on these four regions.",
"Figure: Survival regions of the coupled system under load-redistribution based model.",
"When coupling is introduced, regions where both networks survive or collapse (S 12 S_{12} and S 0 S_0, respectively) get larger, while regions where only one network survives (S 1 S_1 and S 2 S_2) shrink significantly.To provide a concrete example, let network $A$ have $L_A \\sim U[10,30]$ , $S_A \\sim U[40,100]$ , and network $B$ have $L_B \\sim U[20,40]$ , $S_B \\sim U[30,85]$ , with $U$ denoting uniform distribution.",
"The initial load distribution and free space distribution are assumed to be independent in each network.",
"We see from Fig.",
"REF that when there is no coupling ($a=b=0$ ), both networks operate in isolation and the survival of $A$ and $B$ are independent from each other; as we would expect, the two dashed lines (in red color) mark the critical attack sizes for $A$ and $B$ when they are in isolation [28].",
"When we introduce coupling to the system, e.g., with $a=0.33$ and $b=0.37$ , we see an interesting phenomenon indicating a multi-faceted impact of coupling on system robustness.",
"The region $S_{12}$ where both networks survives enlarges, while $S_1$ , $S_2$ where only one network survives shrink dramatically.",
"Meanwhile, $S_0$ where both networks collapse also enlarges.",
"In a nutshell, when coupled together, the two networks are able to help each other to survive larger attack sizes as compared to the case when they are isolated; however, this comes at the expense of also failing together at smaller attack sizes than before.",
"Figure: Color map of the critical attack size under different coupling coefficients aa and bb.",
"Darker colors indicate larger p sys ☆ p^{\\star }_{sys} values, meaning that the interdependent system is more robust.To further quantify the effect of coupling on system robustness, we consider the setting above while varying the coupling coefficients $a$ and $b$ .",
"For both networks, we deploy the same initial attack, i.e., $p_1=p_2=p$ , and define the critical system attack size $p^{\\star }_{sys}$ as the minimum $p$ that collapses at least one network in the system when cascading failures stop; i.e., $p^{\\star }_{sys}$ marks the intersection of the $p_1=p_2$ line and the boundary of the $S_{12}$ region in Figure REF .",
"The metric $p^{\\star }_{sys}$ proposed here provides a simple and useful way to quantify the robustness of the overall system.",
"For example, aside from being the smallest attack size needed to be launched on both networks to fail at least one of them completely, it gives a good indication of the area of the $S_{12}$ region where both networks are functional at steady-state.",
"In Fig.",
"REF we show the value of $p^{\\star }_{sys}$ for different coupling coefficients $(a,b)$ using a color map; the darker the graph, the larger is the $p^{\\star }_{sys}$ value.",
"Using this, one can design an interdependent system to have the optimum coupling levels $(a,b)$ so that robustness of the overall system is maximized (in the sense of maximizing $p^{\\star }_{sys}$ ).",
"We see that the optimum $(a,b)$ is not unique, but instead contain in a certain strip of the $[0,1]^2$ plane.",
"This indicates that the robustness of the interdependent system can be optimized even under certain application-specific constraints on the coupling levels $a$ and $b$ ; e.g., one might need to have $a=b$ for fairness to both networks, or $a+b=1$ to bound the total load transfer across networks, etc."
],
[
"Conclusion",
"In this paper, we studied the robustness of interdependent systems under a flow-redistribution based model.",
"In contrast to percolation-based models that most existing works are based on, our model is suitable for systems carrying a flow (e.g., power systems, road transportation networks), where cascading failures are often triggered by redistribution of flows leading to overloading of lines.",
"We give a thorough analysis of cascading failures in a system of two interdependent networks initiated by a random attack.",
"We show that (i) the model captures the real-world phenomenon of unexpected large scale cascades: final collapse is always first-order, but it can be preceded by a sequence of several first and second-order transitions; (ii) network robustness tightly depends on the coupling coefficients, and robustness is maximized at non-trivial coupling levels in general; (iii) unlike existing models, interdependence has a multi-faceted impact on system robustness in that interdependency can lead to an improved robustness for each individual network."
],
[
"Acknowledgement",
"This research was supported by the National Science Foundation through grant CCF # 1422165, and by the Department of ECE at Carnegie Mellon University.",
"A.A. acknowledges financial support by the Spanish government through grant FIS2015-38266, ICREA Academia and James S. McDonnell Foundation."
],
[
"Explanation on Multiple Continuous/Discontinuous Transitions",
"In this Section, we will explore in more details the underlying reasons for a network to undergo multiple continuous/discontinuous transitions under the flow redistribution model studied in this paper.",
"First of all, we note that whether a line survives or fails a particular stage of cascading failure depends on the the extra load per alive line at that iteration, i.e., $Q_{t,A}$ or $Q_{t,B}$ .",
"With this in mind, in Figure REF we plot $Q_{t,A}$ as a function of the iteration step $t$ under the setting of Figure REF (i.e., when network $A$ experiences multiple transitions).",
"In all cases, we vary attack size $p_1$ over a range with small increments, so that a single curve in Figure REF represents the change of $Q_{t,A}$ vs. $t$ under a specific attack size $p_1$ .",
"Figure: Extra load per alive line Q t,A Q_{t,A} is shown (at different attack sizes p 1 p_1 on Network AA) as a function of cascade step t=0,1,...t=0,1,\\ldots , forthe setting considered in Figure .",
"The jumps in the transitions divide the final system curve into four regions (marked with circled numbers), which correspond to four clusters in the Q t,A Q_{t,A} plots (distinguished by four colors).We observe that each $p_1$ value leads to a variation of $Q_{t,A}$ that belongs to one of the four clusters, distinguished by different colors in Figure REF .",
"For example, as $p_1$ increases from $0.0250$ to $0.0271$ , the corresponding $Q_{t,A}$ curves move up smoothly forming the blue cluster.",
"At $p_1=0.0272$ , $Q_{t,A}$ experiences a jump, but as $p_1$ increases further, $Q_{t,A}$ curves move up continuously until $p_1=0.0287$ , forming the red cluster.",
"The jump between the blue and red clusters at $p_1=0.0271$ coincides with the first jump in the transition in Figure REF .",
"Similarly, at $p_1=0.0287$ we observe a second jump in $Q_{t,A}$ curves between the red and black clusters, which corresponds to the second jump in Figure REF .",
"When attack size $p_1$ further increases, $Q_{t,A}$ curves keep moving up smoothly until $p_1=0.0314$ after which $Q_{t,A}$ goes to infinity as $t \\rightarrow \\infty $ , meaning that network $A$ collapses completely without any alive lines; the corresponding $Q_{t,A}$ curves for $p_1 \\ge 0.0315$ form the fourth cluster show by dotted green lines.",
"Not surprisingly, $p_1=0.0314$ corresponds to the final breakdown point observed in Figure REF .",
"Another way to read these figures is that after the extra load per non-failed line $Q_{t,A}$ (resp.",
"$Q_{t,B}$ ) reaches a certain value, the network $A$ (resp.",
"$B$ ) goes through a sequence of failures after which it either stabilizes with a large fraction of failed lines, or it can not stabilize and goes through a complete breakdown.",
"These critical values of $Q_{t,A},Q_{t,B}$ and their connection to the emergence of multiple transitions can be understood better in the case of a single network.",
"In [28], we have provided a detailed analysis of the global redistribution model in single networks and demonstrated that the critical transition values are determined by the inequality: $ g(x):=\\mathbb {P}[S>x]( x+\\mathbb {E}[L \\mid S>x]) \\ge \\frac{\\mathbb {E}[L]}{1-p}, \\quad x \\in (0, \\infty )$ With $x^{\\star }$ denoting the smallest solution of (REF ), the final system size is given by $n_{\\infty }(p)= (1-p){\\mathbb {P}}\\left[{S>x^{\\star }}\\right].$ Here $x$ represents candidate values for the extra load per alive line at the steady-state; i.e., it represents potential solutions to $Q_{\\infty }$ .",
"To see this better, we can rewrite the inequality (REF ) as $x \\ge \\frac{p {\\mathbb {E}}\\left[{L}\\right] + (1-p) {\\mathbb {E}}\\left[{L {\\bf 1}\\left[S \\le x\\right]}\\right]}{(1-p)\\mathbb {P}[S>x]}.$ We can now realize that for any $p$ and $x$ for which this inequality holds, the alternative attack that kills i) $p$ -fraction of the lines randomly; and ii) all remaining lines whose free-space is less than $x$ (i.e., that satisfy $S \\le x$ ), is a stable one that does not lead to any single additional line failure.",
"To see this, note that the term $(1-p)\\mathbb {P}[S>x] $ in (REF ) gives the fraction of lines that survive the alternative attack, where each surviving line having at least $x$ amount of free-space, while $p {\\mathbb {E}}\\left[{L}\\right] + (1-p) {\\mathbb {E}}\\left[{L {\\bf 1}\\left[S \\le x\\right]}\\right]$ gives the total load failed initially as a result of the alternative attack.",
"Thus, for a given attack size $p$ , the smallest $x$ satisfying inequality (REF ) or (REF ) will give us the steady-state extra load per alive line $Q_{\\infty }$ .",
"With these in mind, we now explore the underlying reasons for the final system size $n_{\\infty }(p)$ to exhibit (potentially multiple) discontinuous transitions.",
"From Figure REF and the discussion that follows, we expect discontinuous transitions in $n_{\\infty }(p)$ to appear simultaneously with discontinuous jumps in the behavior of $Q_t$ as $p$ varies.",
"We now show that our results given at (REF )-(REF ) confirm this intuition.",
"To visualize the implications of (REF )-(REF ) better, we should plot $g(x)$ as a function of $x$ , and find the leftmost intersection of this curve and the horizontal line drawn at $\\frac{\\mathbb {E}[L]}{1-p}$ .",
"Let this leftmost intersection be denoted by $x^{\\star }(p)$ (with the notation making the dependence of $x^{\\star }$ on the attack size $p$ explicit).",
"The final system size is given from (REF ) as $n_{\\infty }(p)= (1-p){\\mathbb {P}}\\left[{S>x^{\\star }(p)}\\right]$ .",
"Assuming that the tail of the distribution of $S$ is continuous, we see that $n_{\\infty }(p)$ will exhibit a discontinuous jump if (and only at the points where) $x^{\\star }(p)$ , which is analogous to the steady-state extra-load per alive line $Q_{\\infty }$ , exhibits a discontinuous jump.",
"This confirms the intuition stated above.",
"Figure: Multiple transitions in a single network and the corresponding function g(x)g(x) (defined at ()) is plotted when LL follows Weibull distribution with k=0.4k=0.4, λ=100\\lambda =100, L min =10L_{min}=10, and S=αLS=\\alpha L where α=1.74\\alpha = 1.74.",
"The Inset zooms in to the region where g(x)g(x) has a local maximum.Recall that $x^{\\star }(p)$ is the leftmost intersection of $g(x)$ and ${\\mathbb {E}}\\left[{L}\\right]/(1-p)$ , and assume that ${\\mathbb {E}}\\left[{L ~|~ S > x}\\right]$ is continuous, so that $g(x)$ is continuous.",
"Then, $x^{\\star }(p)$ (and thus the final system size $n_{\\infty }(p)$ ) will exhibit one discontinuous jump for every local and the global maxima of $g(x)$ .",
"This last statement explains why certain $L,S$ distributions lead only to a single discontinuous jump (since the corresponding $g(x)$ has a single maxima) while others give two (or, potentially more) discontinuous transitions.",
"An example for the latter case is given in Figure REF .",
"We see that the corresponding function $g(x)$ (Figure REF ) exhibits a local maxima at $x=17.4$ .",
"As a result, when we search for the leftmost intersection of $g(x)$ and ${\\mathbb {E}}\\left[{L}\\right]/(1-p)$ as $p$ varies from zero to one, we see that at a certain $p$ value, the leftmost solution $x^{\\star }(p)$ jumps from $x=17.4$ to $x=29.3$ , creating a first-order transition in the final system size $n_{\\infty }(p)= (1-p){\\mathbb {P}}\\left[{S>x^{\\star }(p)}\\right]$ .",
"After this point, as $p$ increases further, the (leftmost) intersection points increase smoothly, leading to the continuous transition seen in Figure REF , until the global maxima of $g(x)$ is reached.",
"At that $p$ value, the leftmost intersection of $g(x)$ and ${\\mathbb {E}}\\left[{L}\\right]/(1-p)$ jumps from a finite value to infinity (indicating that there is no $x$ satisfying inequality (REF )), and the system goes through a discontinuous transition leading to its complete break down."
],
[
"Simulation Results under Global-Local Combined Redistribution Model",
"The main problem considered in this paper, concerning the cascade of failures in two interdependent flow networks, would be expected to depend on the network connectivity patterns in practical scenarios.",
"However, the approach used in this paper offers physical insight by proposing a mean field approach on the setup presented.",
"In fact, the abstraction used in this paper is equivalent in spirit to the determination of percolation properties based on degree distributions, mean-field, heterogeneous mean-field, and generating function approaches, etc.",
"In addition, merely topology-based models where the failed load is redistributed solely in the local neighborhood of the failed line (e.g., as in [31], [32], [33]) suffers from two main issues.",
"First of all, it is often not possible to obtain complete analytic results under topology-based redistribution models, even within the single network framework.",
"Thus, unlike the detailed analytical results given in this paper for interdependent networks, one would most likely be constrained to simulation results if a topology-based redistribution model was used.",
"Secondly, models where the failed flow gets redistributed only locally according to a topology cannot capture the long-range behavior of failures that are observed in most real-world cascades [35].",
"With these in mind, we believe our paper exercises a reasonable trade-off of capturing key aspects of real-world cascades while being able to obtain complete analytic results.",
"Nevertheless, we find it useful to complement our analytical results with simulations that demonstrate how network topology affects the robustness properties of interdependent networks.",
"To this end, we consider a model that combines the global redistribution model described in Section and the local redistribution model used in [31].",
"In particular, assume that upon failures in a network, a $\\gamma $ -fraction of the failed flow is redistributed solely in the local neighborhood of the failed line, while the rest gets redistributed among all functional lines.",
"In the case of interdependent networks studied here, we only focus on the intra-topology of networks $A$ and $B$ and still couple them through parameters $a$ and $b$ ; i.e., when a line in $A$ fails, $a$ -fraction of the failed flow gets redistributed equally among all functional lines of $B$ , while $(1-a)\\gamma $ -fraction gets redistributed locally in $A$ among the neighbors of the failed line, and the remaining $(1-a)(1-\\gamma )$ -fraction gets redistributed among all functional lines of $A$ .",
"Figure: Effect of parameter γ\\gamma , which controls the fraction of failed load that will be redistributed locally according to network topology, on the robustness of interdependent systems.With this approach, we recover the model analyzed in our paper when $\\gamma =0$ , while setting $\\gamma =1$ gives a merely topology-based model.",
"We now present a simulation result that shows the robustness of an interdependent system under different $\\gamma $ values.",
"For convenience, we consider the same set-up used in Fig.",
"REF , i.e.",
"the two networks are statistically identical with coupling coefficient $a=b=0.36$ , and their loads follow a Weibull distribution with $k=0.4$ , $\\lambda =100$ , $L_{\\textrm {min}}=10$ , and $S=0.6L$ .",
"For simplicity, we assume that the topologies of both networks are generated by the Erdős-Rényi model with 9000 nodes and link probability 0.2, leading to a mean number $N$ of links around $8.1 \\times 10^6$ .",
"The results are depicted in Figure REF .",
"As would be expected, as $\\gamma $ decreases from one (purely topology-based model) to zero (the model analyzed in our paper), the robustness of network $A$ increases.",
"In other words, the more fraction of failed flow gets shared globally instead of locally, the more robust the network becomes.",
"This is intuitive since when failed flow is shared globally, the additional load per functional line decreases, leading to a lower chance of triggering cascading failures.",
"Nevertheless, the qualitative behavior of the robustness of network $A$ as the attack size $p_1$ increases remains relatively unchanged at different $\\gamma $ values; e.g., in all cases, we observe multiple discontinuous transitions, with continuous transitions in between.",
"This suggests that the mean-field approach used in our analysis (i.e., the case with ($\\gamma =0$ )) is able to capture very well the qualitative behavior of final system size for all $\\gamma $ values."
]
] | 1709.01651 |
[
[
"Photometric Properties of Network and Faculae Derived from HMI Data\n Compensated for Scattered-Light"
],
[
"Abstract We report on the photometric properties of faculae and network as observed in full-disk, scattered-light corrected images from the Helioseismic Magnetic Imager.",
"We use a Lucy-Richardson deconvolution routine that corrects an image in less than one second.",
"Faculae are distinguished from network through proximity to active regions.",
"This is the first report that full-disk observations, including center-to-limb variations, reproduce the photometric properties of faculae and network observed previously only in sub-arcsecond resolution, small field-of-view studies, i.e.",
"that network, as defined by distance from active regions, exhibit higher photometric contrasts.",
"Specifically, for magnetic flux values larger than approximately 300 G, the network is brighter than faculae and the contrast differences increases toward the limb, where the network contrast is about twice the facular one.",
"For lower magnetic flux values, network appear darker than faculae.",
"Contrary to reports from previous full-disk observations, we also found that network exhibits a higher center-to-limb variation.",
"Our results are in agreement with reports from simulations that indicate magnetic flux alone is a poor proxy of the photometric properties of magnetic features.",
"We estimate that the contribution of faculae and network to Total Solar Irradiance variability of the current Cycle 24 is overestimated by at least 11\\% due to the photometric properties of network and faculae not being recognized as different.",
"This estimate is specific to the method employed in this study to reconstruct irradiance variations, so caution should be paid when extending it to other techniques."
],
[
"Introduction",
"Over the past few decades, much effort has been dedicated to measuring solar irradiance variations and understanding and modeling the physical processes that drive them.",
"Motivating this research is the impact that irradiance variations have on the Earth's atmosphere and climate, especially at the eleven-years solar cycle and longer time scale [28], [70].",
"Variations of solar irradiance at temporal scales longer than one day are modulated by the area and position occupied over the disk by photospheric magnetic structures.",
"Accordingly, various techniques have been developed to reproduce variations of both the Total Solar Irradiance (TSI, the irradiance integrated over the whole solar spectrum) and the Spectral Solar Irradiance (SSI, irradiance integrated over finite spectral bands) using direct measurements or estimates (through proxies) of varying population of magnetic features on the solar disk [23], [28].",
"While it is relatively well understood how sunspots dominate on a daily (sunspot evolution) or monthly (solar rotation) time-scale, the contribution of faculae and network, which dominate at the eleven-year solar cycle and longer temporal scales, is still uncertain.",
"This is in spite of numerous investigations of their photometric properties [37], [59], [25], [4], [27], [84], [57], [95], [83], and attempts to reconcile observed properties of faculae and network with those predicted by models [37], [82], [25], [27].",
"As a result, while TSI reconstructions of the last and current cycles agree up to the 96$\\%$ level, SSI reconstructions, especially in the UV, and reconstructions of the past cycles obtained by different techniques, present discrepancies which are too large to accurately assess the effects of solar irradiance variations on the Earth atmosphere [28], [48], [2].",
"Reconstructions, in most cases, only partially take into account the variety of physical conditions which, as also inferred from numerical modeling [74], [61], [75], [17], [19], determine the radiative emission of magnetic features.",
"In particular, results obtained from the analysis of high spatial-resolution observations [73], [42], [47], [64], [29] showed that small-size magnetic elements located in quiet regions are characterized by a higher photometric contrast in photospheric continua, than magnetic elements located in active regions.",
"[13] employed three-dimensional magneto hydrodynamic (3D-MHD) simulations of the solar photosphere to show that such differences are caused by suppression of convection in active regions, which induces a decrease of the plasma temperature within and around magnetic elements.",
"Table: A summary, albeit incomplete, of intensity contrast studies from both observations and simulations from the past fifteen years.The columns correspond to authors, year of publication, whether faculae or network were found to have a higher contrast at most disk positions and field strengths,type of observation, the instrument used and its spatial pixel scale and the criteria used to discriminate between network and faculae.The bottom two rows correspond to simulation efforts.",
"Therefore, no instrument is specified, instead a source code is named.The high spatial-resolution results characterizing network as higher contrast features than faculae are apparently at odd with previous studies employing full-disk observations.",
"For instance, [59] analyzed data acquired with the Michelson Doppler Imager (MDI) aboard the Solar and Heliospheric mission [67] and concluded that the contrast of network pixels is smaller and presents a rather modest center-to-limb variation with respect to the contrast of faculae.",
"Similar results were found by [95], who analyzed data acquired with the Helioseismic and Magnetic Imager (HMI) aboard the Solar Dynamics Observatory [69], and by [24] and [25], who analyzed photospheric solar images acquired with the Precision Solar Photometric Telescope [10].",
"However, [59] and [95] also noted that network elements are characterized by a higher 'intrinsic' contrast, thus suggesting that the lower contrast observed for network is mostly consequence of spatial resolution effects.",
"Table 1 provides a summary of photometric contrast studies published in the last 15 years.",
"This table is incomplete but gives the reader a comprehensive view of the findings of a variety of observational and numerical efforts at-a-glance, including the lead author, year of publication, whether network or faculae were found to be brighter over the majority of field strengths and disk positions, whether or not the study was full-disk or limited in its field of view, the instrument used, its pixel size, and criteria for distinguishing between faculae and network.",
"Different irradiance reconstruction techniques employ different approaches to take into account the contribution of magnetic features, but the majority of these techniques, included the two most employed in Earth-atmosphere studies, i.e.",
"the Naval Research Laboratory models [51], [8] and the Spectral and Total Irradiance REconstructions [50], [94], do not typically take into account the observational evidence of network elements being brighter than facular ones.",
"In particular, NRL-TSI and NRL-SSI typically rely on measurements of the MgII and sunspot indeces to estimate total and spectral irradiance variations through multivariate analysis; in these models the contribution of network is therefore only indirectly accounted for through the derived correlation coefficients.",
"SATIRE models distinguish between the contribution of diferent magnetic structures, and the radiative emission of bright magnetic elements is assumed to increase linearly with the magnetic flux, up to a saturation value [50], without taking into consideration the pixel's proximity to active regions.",
"imilarly, other models not taking into account of a network brighter than faculae have been suggested in the literature [21], [71], [56], [77], [6], [93].",
"In particular, in SRPM [31], [32], OAR [26] and COSI [41], various types of magnetic features (which include different types of network and facular features) are classified according to their emission in chromospheric images (typically CaIIK), but the modeled network contrast is lower than the facular one, especially toward the limb [27].",
"Simplifications in implementing an irradiance reconstruction that accounts for all network and facular properties arise because reported differences are dependent on the methods of feature-discrimination used, and the intensity contrasts are a function of wavelength, magnetic field strength, spatial resolution of the instrument, activity levels and center-to-limb position [72].",
"We were inspired by [91] who showed that the contrast-magnetic flux relation derived at disk-center using data compensated for the instrumental Point Spread Function (PSF) presents characteristics so far observed only using sub-arcsecond spatial resolution observations and simulations.",
"We extend the [91] work to include full-disk analysis of HMI data compensated for the PSF for dates that sample a variety of magnetic activity on the disk.",
"This paper adds to the existing literature on photometric contrasts of network and faculae as pertains to irradiance modeling in the following way: This is the first full-disk analysis to report on photometric contrast compensated for scattered-light of faculae and network defined by their proximity to active regions instead of their magnetic flux only.",
"We utilize HMI data corrected for scattered-light using a fast deconvolution routine already in the HMI pipeline HMI data compensated for the PSF can be found within the HMI JSOC environment by searching for data series appended by '$\\_$ dcon'.",
"For example, HMI continuum intensity data obtained at a 45s cadence, normally designated as hmi.",
"Ic$\\_$ 45s, that have been corrected, are found in the data series named hmi.Ic$\\_$ 45s_dcon.",
"Several time periods of data are already available.",
"The HMI team is working towards supplying data on a daily and continuous basis but the efforts are dependent upon funding outcomes.",
"Requests for scattered-light corrections for specific data periods and observables are welcome and should be addressed to [email protected]$ ..",
"The PSF was developed to account for both short- and long-range scattering range (see Sec. ).",
"The deconvolution is fast (less than 1 s per full-disk image) and could easily supply daily data for irradiance reconstruction purposes.",
"Within this paper, we analyze original and PSF corrected full-disk intensity and magnetograms from ten different days between 2013 and 2015.",
"By utilizing a pre-existing HMI data-product, i.e.",
"the HMI Active Region Patch data, we can distinguish between faculae and network in one step (see Sec. ).",
"This simple methodology could easily be incorporated into ongoing irradiance reconstruction efforts.",
"The analysis that we performed is similar to that of [59] and [95], who analyzed data acquired with MDI and HMI, respectively.",
"These previous studies did not differentiate between pixels located near AR, except that [95] excluded some pixels very close to active regions using a magnetic extension analysis, although after that exclusion there was no further distinction between, or separate analysis of, pixels closer or further from AR.",
"It is even noted in the conclusions of [95] that 'while the largest effect is produced by the removal of magnetic signal adjoining to sunspot and pores...there remains a fair representation of active region faculae in the measured constrasts'.",
"Instead, network and facular regions were distinguished using the assumption that network and facular pixels are characterized by low/high magnetic flux values, respectively.",
"Moreover, the data employed by [59] were characterized by a different spatial resolution (about four times worse) and in both studies the data were affected by scattered-light.",
"Therefore, we were inspired by [59] and [95] to conduct a study on full-disk data while distinguishing faculae and network by their spatial proximity to AR."
],
[
"Observations and Data Analysis",
"We analyzed a set of 45 s Intensitygrams and Magnetograms acquired in ten different days between 2013 and 2015 with the HMI.",
"The solar disk is imaged on a 4096$\\times $ 4096 pixels detector, with a pixel/scale of 0.5 arcsec and a spatial resolution of 1 arc sec [85].",
"The HMI samples the Fe I 6173.3 nm photospheric absorption line at six wavelength positions in two orthogonal circular polarization states.",
"The acquired filtergrams are then combined through an algorithm (the MDI-like algorithm) to produce estimates of the line-of-sight magnetic flux, Doppler velocity, and Fe I 6173.34 Å nearby continuum intensity, line-depth and line-width [11], [9].",
"A full description of the HMI-pipeline is provided in [12].",
"Estimates of the HMI observables are known to suffer from uncertainties resulting from various factors which include the assumption of a Gaussian shaped Fe I line profile, saturation of the line in the presence of strong magnetic fields, line-shifts induced by plasma motion, solar rotation and the orbital velocity of the spacecraft, stability of the tunable-filters and other optical components [30], , , .",
"The effects of these uncertainties on our results are discussed in Sec.",
".",
"Scattered-light is known to affect photometric studies [78], [54], [15], [91].",
"The PSF used for correcting HMI data for scattered-light is described in detail in Sec.",
"3.6 in [12], so we limit our description to the basics.",
"The form of the PSF is an Airy function convolved with a Lorentzian.",
"The parameters are bound by observational ground-based testing of the instrument conducted prior to launch [85], and by using post-launch, in flight data off the limb, during the transit of Venus and also during a partial lunar eclipse.",
"The PSF employed herein is distinctly different from the PSF used by [91], which takes the form of a sum of Gaussian derived from the transit of Venus data.",
"The PSF used by [91] does not account for the large-angle, or long-distance scattering, since the shadow of Venus is too small to effectively measure the long-distance scattering.",
"In addition, using the sum of simple functions such as Gaussian does not describe properly the diffraction-limited case, thus potentially introducing artifacts on restored images [89].",
"These can be avoided by introducing constraints on the parameters describing the width of the PSF [91].",
"For completeness, it must be mentioned that the PSF description as sum of Gaussian and Lorentzian functions is still proper for some applications, as for Earth-based observations dominated by seeing [78], [15].",
"To discriminate between pixels located in active and in network regions we employed the HMI Active Region Patches (HARPs) available for download from the Joint Science Operations Center (JSOC) website (http://jsoc.stanford.edu/ajax/lookdata.html), derived from HMI magnetograms following the procedure described in [80].",
"Note that the HARP mask locations do not change between the original and data compensated for scattered-light.",
"We discarded pixels belonging to sunspots umbrae and penumbrae by using an intensity contrast criterion.",
"For each image we first estimated the average quiet-sun limb darkening as a function of the cosine of the heliocentric angle $\\mu $ , by computing the intensity histograms at 60 different $\\mu $ values and averaging those pixels whose intensity values are within $\\pm 3 \\sigma $ the median value of the intensity distribution.",
"We created a contrast map to be the ratio of the intensitygram and the limb darkening image.",
"We defined pixels as belonging to sunspots in places where the contrast is lower than four times the standard deviation of the contrast image.",
"Projection effects were reduced by applying a 6-pixel kernel smooth on the obtained sunspot masks [95].",
"We also discarded from our analysis those pixels where the magnetic flux value is below three times the magnetogram noise level.",
"This last quantity varied quadratically with $\\mu $ and ranged from about 9 G at disk-center to about 12 G at the limb, as described in [53].",
"Finally, due to the uncertainties toward the limb, we restricted the analysis to pixels located at $\\mu >$ 0.2.",
"Examples of a contrast map, a magnetogram (showing only pixels exceeding the noise level), the HARPs and sunspot masks are shown in Fig.",
"REF for the original and the restored data.",
"The images in Fig.",
"REF (bottom panels) show that the method applied to select sunspot regions also includes pores and micropores, and a small fraction of dark intergranular lanes.",
"These last type of pixels typically have magnetic flux lower than the magnetogram noise level and would have been discarded by the analysis anyways.",
"Note that the difference in identifying dark features, as shown in the bottom panels comparing the left and right columns, arises from the standard deviation being lower in the original data compared to the compensated data.",
"Meaning, more pixels in the original data were discarded as being lower than 4 sigma level.",
"We refer to pixels located within HARP regions as “facular pixels”, while magnetic pixels located everywhere else on the disk will be referred to as “network pixels”.",
"We use this notation because the first category of regions is more likely to include facular regions, whereas the second mostly include intranetwork and quiet and active network regions, but also because the method is straightforward to implement both in our efforts as well as in future irradiance reconstructions.",
"Figure: Examples of the analyzed data acquired on December 13th 2014.",
"Intensitygram compensated for the limb darkening function (top), magnetogramsaturated at±\\pm 100 G (center) and masks (bottom) obtained on original (left)and restored (right) data.",
"The grey pixels on the magnetograms have line-of-sight magnetic flux below the noise threshold.The mask images show the pixels belonging to faculae (derived from HARP masks, see text) in white color, while black pixels on the disk belong to regions identifiedas sunspotumbrae, penumbrae and pores.Figure: Top: Detail of the contrast image shown in Fig.",
"for the original (left) and restored (right) data.",
"Images show detail of AR12236 which,on the day of the observation, was located at μ≃0.4\\mu \\simeq ~0.4.",
"Bottom: original (left) and restored (right) contrast images.",
"The black contours in the top imagesenclose regions belonging to sunspots and pores, excluded by our analyses.",
"In all images, red contours enclose the HARP regions, while blue and green lines enclosepixels where |B|/μ>|B|/\\mu > 800 G and that appear brighter and darker, respectively, with respect to the quiet background.It is important to note that [59] and [95] considered only pixels where the magnetic flux compensated for line-of-sight effects ($|B|/\\mu $ ) is lower than 800 G, the number of large magnetic flux pixels being statistically not significant in their data.",
"We decided instead to extend the analysis to higher magnetic flux values, as the restoration increases the number of large magnetic flux pixels (see Sec. ).",
"[95] also argued that high magnetic flux values, especially toward the limb, most likely result from horizontal fields, and that these are typically associated with sunspots and pores.",
"Inspection of our data confirms that statistically this is the case for pixels located at the extreme limb ($\\mu <0.1$ ), which are discarded from our analysis, but we do not find a clear association of 'dark' pixels within locations of sunspots and big pores at other positions over the disk.",
"Figure REF shows for instance, in blue and green color, pixels with magnetic flux larger than 800 G, which appear brighter and darker, respectively, with respect to the average quiet sun intensity, belonging to a HARP Region located at $\\mu \\simeq ~0.4$ .",
"On the contrary, the bottom panel of Fig.",
"REF shows that, especially on restored images, these pixels seem to be distributed everywhere over the disk, with a preference in active regions and in the activity belt, but not exclusively at disk-center."
],
[
"Results",
"For each pixel $i$ over the disk we defined the continuum intensity contrast as $C=\\frac{I_i}{I_q} -1$ , where $I_i$ is the continuum intensity of the pixel, and $I_q$ is the average quiet sun intensity of pixels located at similar angular distance from disk-center, estimated as described in Sec. .",
"We investigated the dependence of the contrast on the magnetic flux, B, and on the cosine of the heliocentric angle, $\\mu $ .",
"We considered 50 G bins of magnetic flux values compensated for line-of-sight effects ($|B|/\\mu $ ) of pixels located at 16 different radial distances from disk-center, in $\\Delta \\mu =0.025$ intervals."
],
[
"Comparison between results obtained on original and restored data",
"In this section we discuss the effects of restoration on the determination of the dependence of the intensity contrast on the magnetic flux and on the cosine of the heliocentric angle.",
"We refer the reader to [91] for an additional description of the effects of the compensation for the instrumental Point Spread Function on the HMI observables.",
"Fig.",
"REF shows how the intensity contrast depends on the magnetic flux value at eight different heliocentric angles for original and restored data.",
"Fig.",
"REF shows, instead, the variation of the intensity contrast with $\\mu $ for eight magnetic flux ranges.",
"In both plots, data points correspond to average intensity contrast values computed over the corresponding bins while the error bars represent the standard deviations of values in each bin.",
"Fig.",
"REF shows that the contrast, towards disk-center, is negative for magnetic flux values smaller than about 200 G, but the contrast increases to reach a maximum between 300-400 G and then decreases again.",
"This “fish hook” trend has been observed in the analysis of sub-arc second spatial resolution observations [68], [46], [47], [45], and was also obtained by [91] on HMI data compensated for the instrumental PSF (a detailed comparison with the results obtained by these authors is given in Sec. ).",
"The “fish hook” trend is not seen in the original data, and is partially visible only when increasing the magnetic flux bin size (see Sec. ).",
"[63] employed 3D MHD simulations to interpret the physical origin of the contrast-magnetic field dependence observed at disk-center.",
"They showed that the decrease of contrast for low magnetic flux values is consequence of the accumulation of the flux within intergranular lanes, while the decrease of contrast at large magnetic flux values is an effect of reduced spatial resolution in observations, which decreases both the magnetic flux and the contrast of the bright edges of micropore structures.",
"This also explains why the restoration (Fig.",
"REF and Fig.",
"REF ), contrary to what typically expected, toward disk-center produces a (small) enhancement of the average contrast only at the lower magnetic flux values, while at higher magnetic fluxes the contrast of restored pixels is up to 4-5 times smaller.",
"Toward the limb the restoration increases the contrast for $|B|/\\mu >$ 300 G, as expected by the fact that the contrast at small values of $\\mu $ is determined by the ”hot wall“ effect.",
"The amount of contrast variation induced by the restoration is function of both the angular distance and the magnetic flux, generally increasing with the magnetic flux and toward the limb, where the contrast of restored pixels is up to 4-5 times larger than the original ones.",
"On the other hand, the restoration seems to have little effect on the angular distance at which the contrast reaches the maximum, while it shifts toward disk-center the value of $\\mu $ at which magnetic elements become brighter than the background.",
"This result is explained by the fact that the restoration compensates for scattered-light effects, as previous studies indicate that the position of the maximum contrast is sensitive to the spatial resolution [17], while the angular position at which features become brighter is less so [95].",
"Finally, as noted in previous studies [15], the standard deviation in each bin is enhanced by the restoration.",
"In particular, as also noted in Sec.",
"(Fig.",
"REF ), although on average the restoration causes highest magnetic flux features to appear “brighter”, there is still a considerable amount of features that instead appear “darker”.",
"Figure: Variation of the intensity contrast with the magnetic flux for pixels located at various radial distances from disk-center derived by original (black)and restored data (red).",
"Continuous lines: cross-sections of the surface fits to the data (see Sec.",
"); dotted lines:extrapolation of the fit.Figure: Variation of the intensity contrast with the cosine of the heliocentric angle for various magnetic flux ranges derived by original (black)and restored (red) data.",
"Continuous lines: cross-sections of the surface fits to the data (see Sec.",
"); dotted lines:extrapolation of the fit.Figure: Variation of the intensity contrast with the magnetic flux for pixels located at various radial distances from disk-center in facular (red)and network (black) regions singled out on restored data.",
"Continuous lines: cross-sections of the surface fits to the data (see Sec.",
"); dotted lines:extrapolation of the fit.Figure: Variation of the intensity contrast with the cosine of the heliocentric angle infacular (red) and network (black) regions singled out on restored data.",
"Continuous lines: cross-sections of the surface fits to the data (see Sec.",
"); dotted lines:extrapolation of the fit."
],
[
"Comparison between Network and Faculae",
"The differences between network and facular regions are illustrated in Fig.",
"REF and Fig.",
"REF for results obtained from restored data only.",
"Results obtained from original data show little or no difference of contrast between pixels located in different magnetic activity regions, especially at disk-center, and are shown in Appendix .",
"On the contrary, results obtained from restored data show that pixels located in intergranular lanes, or at low magnetic flux values (about less than 300 G), appear always darker in facular regions rather than in network ones.",
"Pixels with higher magnetic flux are instead always brighter in network regions.",
"We also note that the contrast differences are smaller toward the disk-center but increase toward the limb, where the network contrast at higher magnetic fluxes can be up to almost twice the facular one.",
"Finally, inspection of Fig.",
"REF reveals that the angular position of the maximum contrast is similar for pixels located in different regions, while the angular positions at which the contrast changes sign occur closer to disk-center for network rather than for faculae.",
"These results are in qualitative agreement with results obtained by the analysis of sub-arc second spatial resolution observations at disk-center [42], [46], [47], [64], [29].",
"As explained in [13] the different contrasts obtained in active and in quiet regions are consequence of the decrease of temperature in photospheric layers induced by the suppression of convective motions within active regions.",
"This reduces the contrast of granulation (both granules and intergranular lanes) and consequently of small-size magnetic flux concentrations whose temperature stratification is determined by the temperature of the surrounding plasma.",
"Similarly, toward the limb the observed higher contrast in network regions can be explained by the reduction of the 'hot wall' temperature in facular regions.",
"Finally, a comparison of plots in Fig.",
"REF and REF with those in Fig.",
"REF and REF reveals that the effects of restoration are larger for network pixels, where, for the largest magnetic fluxes, the contrast variations can be up to twice the one obtained for faculae."
],
[
"Surface fits",
"Following [95], we fitted our results with cubic surface functions.",
"The analytical form of the fit and the values of the fit coefficients are reported in Appendix .",
"The fits were evaluated using data binned over 16 equally spaced in $\\mu $ values and 10 G magnetic flux intervals.",
"Bins where the number of elements was lower than 100 were excluded from the analysis by imposing a null weight during the fit.",
"The results are illustrated in Figs.",
"REF , REF , REF and REF .",
"The agreement of the fits with the observational data points is in general very good at most of the flux values and angular positions, especially at the data points employed to produce the fits.",
"The agreement is worse at small magnetic flux values on restored data, where the fits do not reproduce the 'fish hook' shape.",
"Recently, [68] modeled the contrast-magnetic field relation obtained from the analysis of SST [66] and HINODE [49] data as a sum of exponential functions, while [45], who analyzed SUNRISE data [3], employed a logarithmic function, although in this last case the fit reproduced the contrast at magnetic flux values larger than approximately 80 G (that is the dimming at small magnetic flux values was not reproduced).",
"We found that the contrast-magnetic flux relation obtained from restored data is best reproduced when using a 10-th order polynomial in $|B|/\\mu $ .",
"Figure REF shows indeed that this function reproduces the observations at both low and high magnetic flux values at various angular positions.",
"Results from the fit are reported in the Appendix .",
"Figure: Comparison between results obtained from restored data (plus signs) and 10th-order polynomial fit to the data (continuous lines).",
"Black: μ\\mu =0.975; red: μ\\mu =0.725;green: μ\\mu =0.525; blue: μ\\mu =0.325."
],
[
"HMI uncertainties",
"Previous investigations showed that HMI data-products are affected by uncertainties stemming from the pipeline employed to estimate the data-products as well as by instrumental effects (see Sec ).",
"[9] employed results from numerical synthesis of the Fe I 617.3 nm line to show that the error in the estimate of the continuum intensity introduced by approximations in the MDI-like algorithm are function of the magnetic field and the line-of-sight, and, as second order effects, of Doppler shifts.",
"Specifically, their Fig.",
"3 shows that the uncertainty for a facular region with associated a magnetic field of 1000 G is lower than 2% at disk-center and decreases rapidly toward the limb, while the uncertainty for quiet regions increases toward the limb up to about 1%.",
"Therefore uncertainties in the estimates of the contrast induced by approximations in the MDI-like algorithm are below 2% and show little dependence with $\\mu $ .",
"These uncertainties are typically below the amplitude of the error bars in our plots, and we therefore conclude that they have negligible effects on our results.",
"Because we used the contrast to characterize the photometric properties of magnetic elements, the increase of opacity of the entrance window during the first years of operations of the SDO, also reported in [9], have no influence on our results.",
"Uncertainties stemming from variations of the properties of the filter transmission profiles are instead more difficult to assess, as there is no direct measurement available.",
"[16] estimated uncertainties lower than 10% introduced by variations of the transmission profiles for continuum measurements provided by the MDI, which, similarly to HMI, combines filtergrams acquired with a tunable filter.",
"In the case of HMI we expect uncertainties due to these effects to be smaller, as the filters are periodically retuned.",
"Moreover, the data analyzed were acquired in a relatively short temporal frame (about three years), so that effects introduced by variations of the positions and shapes of the filters are most likely negligible on our results.",
"In addition, [9] showed that HMI estimates of the Fe I 617.3 nm line core intensity are affected by large uncertainties (several tens of percent).",
"For this reason we refrained from using those HMI data-products, although such measurements would have provided extremely valuable information about the different temperature stratification within network and facular structures.",
"It is also important to mention that similar studies carried out on magnetograms usually employed data averaged over longer temporal range [95] to reduce noise and p-mode oscillation effects.",
"In our analysis we decided to employ 45 s data.",
"To investigate how much of a difference it might make to use 720 s data instead of 45 s data, we repeated the analysis on a subset of three 720 s images and found that the curves describing the dependence of the contrast on the magnetic flux agree within the 2% level (and are therefore not reported).",
"Hence, we conclude that our results are not affected by the use of different types of HMI data.",
"Finally, the Lucy-Richardson algorithm is known to potentially enhance noise in restored data [90].",
"Inspection of the restored data (e.g.",
"Fig.",
"REF ) suggests this effect to be negligible.",
"An analysis of the power spectra of both intensity images and magnetograms reveals indeed a small enhancement of power beyond the frequency cutoff of the telescope, but the cumulative power beyond this frequency is below 2%.",
"To investigate the effect of this enhancement on our results we first estimated the noise level on a subset of original and restored magnetograms employing a method similar to the one described in [53] (we removed active pixels from the analysis while [53] analyzed data acquired during low magnetic activity), and found that the restoration typically doubles the noise level.",
"We therefore repeated the analysis increasing the noise level threshold by a factor of two and found that the maximum relative difference between results obtained with the two noise thresholds is below 1.8%."
],
[
"Features classification and quiet Sun definition",
"It is very well known that identification methods employed to singled out features on solar images can affect the derived properties of such features [25], [44], [95], [81], [1].",
"We employ masks produced with the method suggested in [80] to distinguish between facular and network regions.",
"As explained in Sec.",
", the HARP regions employed mostly include faculae and probably part of features that other methods might have classified as active network.",
"Magnetic pixels not-belonging to HARPs include intranetwork, network and active network and no effort was made to discriminate between these latter types of features.",
"On the other hand, the adopted classification is sufficient for the purpose of this study, which is to investigate the effects of the level of activity of the surrounding plasma on the photometric contrast of magnetic features.",
"It is important to notice that in order to further reduce the effects of noise, some authors apply a minimum size threshold (typical between 1 to 10 pixels) to the structures analyzed [59], [43], [95], [14].",
"To investigate the effect of isolated pixels on the estimated average contrasts, we then repeated the analysis applying an “opening” operation with a 2-pixels kernel to the pixels exceeding the noise level of the magnetograms [14].",
"We found that the application of such threshold produces a modest increase of the average contrast of pixels with low magnetic flux located toward the limb, with the largest effects found on restored data.",
"In particular, for pixels with $|B|/\\mu \\le 300~G$ and $\\mu \\le 0.5$ singled out on original data the average and maximum relative increase of contrast are 0.6% and 2.6%, respectively, while for pixels singled-out on restored data are 1.7% and 3.2%, respectively.",
"These differences are well below the statistical uncertainties of our measurements, so that we can conclude that the application of a minimum size threshold on our data does not affect our results.",
"Photometric contrast is also known to be affected by the arbitrary definition of quiet sun regions [60].",
"We therefore compared our results with those obtained defining as quiet pixels those that are below the noise level on magnetograms.",
"The contrast relative differences found with this method and the one described in Sec.",
"is below 0.6%, so that we conclude that our results are not significantly affected by the criteria adopted to define quiet sun pixels."
],
[
"Comparison with previous studies",
"The investigation of the dependence of magnetic elements' continuum contrast with the heliocentric angle and the magnetic flux has been the subject of several studies (see Sec. ).",
"Qualitatively in agreement with previous analysis [39], [79], [65], [59], [87], [58], [95], we found that highest magnetic flux features appear dark at disk-center and bright toward the limb, the value of $\\mu $ at which the contrast changes sign depending on the value of the magnetic flux considered.",
"However, a quantitative comparison of our results with these studies is hampered by the different observing conditions, as spatial resolution, scattered-light, wavelength and methods employed to identify the magnetic features play an important role in determining the dependence of contrast on magnetic flux and line-of-sight [17].",
"The studies by [59] and [95] were conducted with the MDI and the HMI, respectively and are those that allow a more direct comparison with our investigation.",
"Figure REF shows the contrasts obtained at three angular positions from the cubic fits to our original data (Eq.",
"REF ) and those presented in [95] (their Eq. 3).",
"For magnetic flux values smaller than 800 G, the curves present a very good agreement, with differences being within the error of the measurements.",
"On the contrary, at larger magnetic flux values the fits presented in [95] do not seem to represent our measurements, most likely because those authors did not include high magnetic flux pixels in their analysis (see Sec.",
").",
"The agreement is remarkable if we consider the difference in the type of HMI images employed (but see discussion in Sec.",
"REF ) and the different data reduction strategies employed in the two studies.",
"In particular, [95] employed data obtained averaging original HMI 45 s data acquired over a 315-s interval; they also employed a different method to estimate the limb darkening shape of quiet regions, and different criteria to select sunspots and pores.",
"It is worth to note that [95] reported the presence of residual patterns in their intensity images compensated for the limb darkening.",
"Following a procedure similar to the one described by those authors, we also estimated the residual intensity on our continuum contrast images, and we found an average value of about $3\\cdot 10^{-5}$ , which is about two orders of magnitude smaller than the value reported by [91], so that our data were not compensated for this effect.",
"Figure: Comparison of contrasts obtained from cubic surface fits on original data (continuous), results presented in (dashed),and restored data (diamonds).",
"Black: μ\\mu =0.975.Red: μ\\mu =0.625.",
"Green: μ\\mu =0.325.",
"The bar shows the average standard deviation of the values in the bins.The comparison of results obtained by [59] using MDI data with those obtained using HMI are largely discussed in [95], and we do not repeat it here.",
"We only note that, as showed by [53], the MDI magnetic flux values are 1.3 - 1.4 times larger that the HMI magnetic flux values, so that HMI contrasts obtained at certain magnetic flux ranges should be compared with MDI contrasts obtained at higher magnetic flux ranges.",
"Nevertheless, as shown by a comparison of results in Fig.",
"REF and Fig.",
"REF , with the ones reported in Fig.",
"3 and Fig.",
"4 of [59], this scaling factor is not sufficient to explain the different contrasts obtained with the two instruments.",
"We therefore confirm that the different magnetic flux and angular dependences of the continuum contrast obtained by our analysis and the one by [95] on one side, and [59] on the other, must be ascribed to the different spatial resolution and the different definition of sunspot regions adopted.",
"In Fig.",
"REF we compare our results with those obtained by [91], who also compensated HMI data for scattered-light effects, but employing a different procedure (see Sec.",
"), and restricting their analysis of photometric contrast only to disk center.",
"The plot shows that the original data produce very similar results.",
"The restoration produces in both cases an enhancement of the contrast at low magnetic flux values ( $|B|/\\mu < $ 200 G), and an enhancement of the brightness at higher magnetic flux values, as expected from simulations [63], but values present some discrepancies.",
"These differences must be attributed to the limited sample of data employed by [91] (one day), to the use of one of the filtergrams (-344 m$Å$ from line center) instead of the intensitygram data-product, to the different criteria employed to define sunspots ([91] employed a -0.11 contrast threshold and a 3-pixel kernel) and, finally, to the different algorithms employed to derive the instrumental PSF.",
"It is difficult to discern which of these effects plays the largest role, as, for instance, the small differences in the original data might have been amplified by the restoration.",
"A detailed comparison of the performance of the two codes employed for the restoration goes beyond the purpose of this paper.",
"Here we note that the differences are overall within the error bars of our measurements.",
"Figure: Comparison with results obtained close to disk-center from this study (continuous) and results reported inFig.",
"15 of (dashed).",
"Black: original.",
"Red: restored.",
"Data were averaged on 10 G magnetic flux bins.",
"The error bar shows theaverage standard deviation value over the bins obtained for restored data.To the best of our knowledge, the effects of scattered-light compensation on the center-to-limb variation of magnetic features contrast cannot be directly compared with any previous studies.",
"Most of the studies performed on high spatial resolution data or on MHD simulations were in fact conducted at or close to disk-center [55], [63], [22], [20], while results obtained on full-disk data mostly focused on sunspot properties [86], [54], [15] or facular properties but at different wavelength ranges [86], [15].",
"Here we note that contrast variations are on average larger for network pixels rather than for facular ones, as a result of the different effects that image degradation has on magnetic features of different sizes [18], [15], [84].",
"Figure: Magnetic flux distributions of pixels located in network (black) and facular (red) regions obtained from original (continuous) and restored data (dashed).The network and facular contrast differences estimated on restored images are small, especially at disk center, and within the error bars.",
"On the other hand, these differences are in qualitative agreement with previous studies obtained at high spatial resolution close to disk-center (see Sec.",
"), but a quantitative comparison is hampered by the different type of data employed.",
"Our results are also in qualitative agreement with measurements carried out by [59] and [95], which showed that the specific contrast of low-magnetic flux pixels is larger than the specific contrast of high-magnetic flux pixels.",
"On the other hand, our results do not agree with previous observations carried out using full-disk observations, which produced a network contrast lower than the facular one, and almost constant over the disk.",
"As previously suggested in [59], these differences are mostly caused by spatial resolution and scattered-light effects.",
"Indeed, we found that network and facular regions are characterized by different contrasts only when using restored data."
],
[
"Implications for solar irradiance studies",
"In this study we employed spatial proximity criteria to discriminate between network and facular regions.",
"Some previous studies discriminated between the two features assuming that statistically lower magnetic flux pixels belong to network, while higher magnetic flux pixels belong to faculae and active network regions.",
"In the following we evaluate how these different criteria affect the estimate of magnetic features contrast.",
"Figure REF shows the magnetic field distribution of network and facular regions derived from original and restored data.",
"Original data shows that for $|B|/\\mu <$ 200 G pixels are more likely to be located in network than in facular regions.",
"This value is in near-agreement with the 130 G values employed by [59] and [95] to distinguish between network and faculae.",
"The threshold increases to about 400 G when analyzing restored data, and the difference between the two populations is up to one order of magnitude.",
"Results in Fig.",
"REF and in Fig.",
"REF show that for $|B|/\\mu <$ 200 G and $|B|/\\mu <$ 400 G, respectively, the contrast differences between network and facular pixels are small, up to approximately 0.02 at the limb, so that the criteria employed in previous studies to separate network from faculae allow reasonable estimates of network properties.",
"At larger magnetic flux values, instead, the contrast differences between network and faculae are larger, especially toward the limb, so that we expect facular contrasts derived using the sole magnetic flux to discriminate between the two classes of features to be statistically affected by the network.",
"This is confirmed by results in Fig.",
"REF , which show as an example the contrasts of network and facular pixels with magnetic fluxes of $\\approx $ 200 G (red lines) and $\\approx $ 600 G (black lines), together with the contrast derived without distinguishing between the two features, obtained from original (top) and restored (bottom) data.",
"Both panels in Fig.",
"REF show that the contrast derived for $|B|/\\mu \\approx $ 200 G without distinguishing between network and faculae, closely follow the results obtained for the network.",
"For $|B|/\\mu \\approx $ 600 G, instead, the contrast derived without distinguishing between the two features is in between the values found for network and faculae, thus showing that the network statistically affects the estimates of facular contrast if faculae are singled out only according to the magnetic flux value.",
"The effects are larger toward the limb and on restored images, where the network and facular contrast differences are larger.",
"Figure: Center-to-limb variation of the continuum contrast derived from original (top) and restored (bottom) data.Black symbols correspond to pixels where 600 G<|B|/μ<<|B|/\\mu < 650 G and red to pixels where 200 G<|B|/μ<<|B|/\\mu < 250 G. Triangles denote network pixels,diamonds facular pixelsand stars-continuous lines denotes results obtained without discrimination.",
"The error bar corresponds tothe typical standard deviation over the bins.To evaluate the effects of discriminating between network and faculae on the estimate of solar irradiance variations over the solar activity cycle, we computed the daily facular and network contribution to TSI [52], [38]: $\\frac{\\Delta F}{F} & = & \\sum \\limits _{k} \\sum \\limits _{j} \\frac{5 \\mu _j N(\\mu _j,B_k)C(\\mu _j,B_k)\\Psi (\\mu )}{2}$ where the two sums run over the magnetic flux and the cosine of the heliocentric angle, $N(\\mu _j,B_k)$ is the area of pixels at position $\\mu _j$ and magnetic flux $B_k$ normalized to the surface of the solar hemisphere, $C(\\mu _j,B_k)$ is the contrast as derived by the bi-cubic fits to our data and $\\Psi (\\mu )=(3\\mu +2)/5$ is the quiet Sun limb-darkening function.",
"It is important to note that this model is raher simple, especially if compared to modern irradiance reconstruction techniques, as it lacks of detailed knowledge of the radiometric contribution of magnetic features to the disk integrated irradiance.",
"In particular, because the bolometric contrast is knwon to be larger than the one measured in the red continuum [36], our computations underestimate the bolometric facular/network contribution to TSI.",
"The daily facular/network coverage $N(\\mu _j,B_k)$ was estimated employing original 45 s HMI data acquired between April 2010 and October 2015.",
"Coherently with the analysis presented above, network and facular regions were discriminated using the HARP masks.",
"The facular excess was then computed using Eq.",
"REF first employing the $C(\\mu _j,B_k)$ curves derived from the whole dataset (Model A, in the following), second using the contrast curves derived for network and faculae separately (Model B, in the following).",
"The minimum value of the magnetic flux considered is 300 G. This value was chosen for two reasons.",
"First, the bi-cubic fits seem to reproduce best the observations for magnetic flux values above this threshold (for smaller magnetic flux values, toward the limb the fits overestimate the contrast up to 0.03).",
"Second, as discussed in [68], at magnetic flux values smaller than this threshold the contrast of small, unresolved magnetic features is largely underestimated because of the brightness contribution of the dark lanes they are embedded in.",
"Results presented in Fig.",
"REF show that Model A overestimates the TSI excess, mostly because, as shown in Fig.",
"REF , the facular contribution during the periods of high activity is overestimated.",
"The variability measured between the periods of largest and lowest activity ($\\Delta F/F _{Max} - \\Delta F/F_{Min}$ ) is $1.25\\cdot 10^{-4}$ and $1.12\\cdot 10^{-4}$ for Model A and Model B, respectively, which corresponds to a difference of about 11%.",
"This estimate must be considered a lower limit for several reasons.",
"The most important, is that in our analysis we discarded pixels with magnetic flux smaller than 300 G, which occupy the majority (more than 98%) of the solar magnetized surface.",
"Indeed, [68] employed high-spatial resolution data acquired in a red continuum to show that the contribution of magnetic pixels with flux larger than 300 G to the network brightness is only $0.2\\cdot 10^{-4}$ , compared to the excess brightness of about $1.1\\cdot 10^{-3}$ found when including all magnetized network pixels.",
"We note that $0.2\\cdot 10^{-4}$ is in reasonable agreement with the $0.16\\cdot 10^{-4}$ value that we found for the network during 2010 (not shown), the period of lowest magnetic activity in our observations.",
"Spatial resolution effects, (which reduced the estimate of contrasts, magnetic fluxes, and the contrast difference between network and faculae) and the fact that our observations did not include a period of minimum (the HMI operations started during the rising phase of cycle 24), also contribute to underestimate both $\\Delta F/F$ and the differences between Modelels A and B.",
"Finally, the use of original data instead of restored ones to derive the coverage of magnetic pixels is another source of uncertainty, as the restoration changes the distribution of magnetic pixels.",
"In particular, Fig.",
"REF shows that at magnetic flux values larger than approximately 500 G the difference between the number of facular and network pixels is smaller in restored than in original data, which might have lead to an overestimate of the variability difference produced by the two models.",
"It is also important to note that the facular/network contributions to irradiance variations were estimated using contrasts averaged in magnetic field bins.",
"Our results show that the restoration largely increases the scatter of the contrast in each bin, thus pointing to the necessity of employing criteria to discriminate between various types of magnetic features other than the magnetic flux alone.",
"In particular, as already noted in Sec.",
", part of the pixels with $|B|/\\mu >$ 800 G on restored images present a negative contrast at all angular positions.",
"Although on average the brightness increases after restoration, the fraction of negative contrast pixels at these magnetic flux ranges is more than 50%, thus suggesting a different temperature stratification than pixels characterized by a positive contrast.",
"If, as suggested in [35] [33], [39], the coverage of such features increases with the activity, then available reconstruction techniques making use of facular coverage as derived by chromospheric emission [51], [34] overestimate the contribution of faculae to irradiance, especially during the strongest cycles.",
"A detailed study of properties of dark faculae is under investigation.",
"Finally, it is important to mention that most recent irradiance reconstruction models do not make use of direct measurement of photometric contrast (if not in some cases as a proxy e.g.",
"[5]) and that we expect the amount of uncertainty introduced by not taking into account network and faculae separately to vary for different techniques.",
"Nonetheless, we note that the uncertainty value of $\\approx $ 10% in the contribution of network and faculae to TSI variations is smaller than or of the same order of uncertainties reported for some TSI and SSI reconstructions [21], [92], [7].",
"Figure: Temporal variation of the network and facular contribution to irradiance for Model A (black) and Model B (red)."
],
[
"Conclusions",
"We employed HMI data-products compensated for the instrumental PSF to compare the center-to-limb variation of the contrast of network and facular regions as determined by their proximity to active regions in addition to continuum intensity and magnetogram threshold.",
"Contrary to previous results obtained on full-disk observations, we found that for magnetic flux values above 300 G the network is brighter than faculae and presents a stronger contrast center-to-limb variation.",
"This finding for full-disk data must be attributed to the increased photometric contrast resulting from correcting for scattered-light.",
"Note that the 300 G thresold is function of the spatial-resolution of the employed data (see also discussion in Sec. ).",
"Our results are in qualitative agreement with those obtained at or close to disk-center by the analysis of sub-arcsecond observations and MHD simulations.",
"We extend the analysis to full-disk to report the photometric contrasts of network and faculae, as determined based on proximity to active regions, as a function of magnetic flux and line-of-sight observing angle.",
"We employed a simple model to estimate the contribution of magnetic pixels to TSI variations ($\\Delta TSI$ ) and found that if the contribution of network and faculae is not taken into account separately, or is only based on the magnetic flux of a pixel, $\\Delta TSI$ is overestimated of about 11%.",
"This value is a lower bound for error for the following reasons.",
"First, the contrast measured using HMI images is still affected by limited spatial resolution.",
"Second, in our analysis we discarded pixels located in dark lanes, so that the contribution of unresolved bright structures located in these regions is not taken into account.",
"Third, comparison of results obtained on original and restored data shows that for each magnetic flux bin the restoration increases the scatter of results, even when taking into account faculae and network separately.",
"This suggests that classification of features according to the magnetic flux value and spatial aggregation might not be enough to properly characterize the contribution of magnetic elements to irradiance variations, if irradiance reconstruction techniques just employ the above parameters as input data.",
"Our report on the photometric contrasts of faculae and network, defined using spatial proximity to active regions, as a function of center-to-limb angles can be used to further improve existing irradiance reconstruction techniques.",
"A majority of irradiance reconstruction techniques either do not explicitly differentiate between features, or assume a lower contrast for the network based on previous studies.",
"The continuity and frequency of full-disk HMI data, combined with the ease by which HARP masks allow identification of active region locations, plus the availability of a fast routine for removal of scattered-light, means that a daily, corrected HMI images of similar quality could easily be implemented for irradiance modeling.",
"Finally, our results support the conclusion of [40] that the contribution of network during the Maunder and Spörer minima might be underestimated by irradiance reconstruction models [21], [76], [88], [51], that assume a linear relation between magnetic flux and/or plage coverage and irradiance.",
"This work was carried out through the National Solar Observatory Research Experiences for Undergraduate (REU) site program, which is co-funded by the Department of Defense in partnership with the NSF REU Program.",
"The National Solar Observatory is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.",
"S.C. is grateful to Dr. Peter Foukal for reading the paper and providing useful comments, and to Dr. Odele Coddington for the interesting discussions about the NRL reconstructions."
],
[
"Polynomial surface fit coefficients",
"The analytical formula is the following: $C\\left(\\mu ,\\frac{B}{\\mu }\\right)& = &\\left[\\begin{array}{c}\\end{array}10^{-2} \\left(\\frac{B}{\\mu }\\right)^0 \\\\10^{-3} \\left(\\frac{B}{\\mu }\\right)^1 \\\\10^{-6} \\left(\\frac{B}{\\mu }\\right)^2 \\\\10^{-9} \\left(\\frac{B}{\\mu }\\right)^3 \\\\\\right.$ ]T [ M ] [ c 0 1 2 3 ] Fit coefficients original-data: $\\mathcal {M} = \\left[ \\begin{array}{cccc}-6.77 & 28 & -43 & 21\\\\0.73 & -1.46 & 2.07 & -1.18\\\\-1.22 & 4.37 & -7.99 & 4.35\\\\0.73 & -3.31 & 5.72 & -2.91\\end{array} \\right]$ Fit coefficients for deconvolved-data: $\\mathcal {M} = \\left[ \\begin{array}{cccc}-16.62 & 44.",
"& -54.6 & 24 \\\\0.47 & 0.71 & -1.55 & 0.56 \\\\-0.25 & -1.02 & 1.05 & -0.083\\\\0.082 & 0.085 & 0.095 & -0.16\\end{array}\\right.",
"$ Fit coefficients for Network deconvolved-data: $\\mathcal {M} = \\left[ \\begin{array}{cccc}-15.068 & 33.757 & -36.765 & 14.03\\\\-00.289 & 4.333 & -6.918 & 3.115 \\\\1.452 & -8.771 & 12.349 & -5.373\\\\-0.739 & 3.9512 & -5.670 & 2.570\\end{array}\\right.",
"$ Fit coefficients for Faculae deconvolved-data: $\\mathcal {M} = \\left[ \\begin{array}{cccc}-0.948 & 2.56 & -10.96 & 7.41 \\\\0.239 & 0.184 & 0.044 & -0.346\\\\-0.7036 & 2.747 & -5.168 & 2.919\\\\0.5589 & -2.601 & 4.116 & -2.020\\end{array}\\right.",
"$"
],
[
"10-th order polynomial fit coefficients",
"The magnetic flux dependence of the contrast was fitted separately for different positions over the disk $\\mu $ : $C\\left(\\frac{B}{\\mu }\\right)=\\sum \\limits _{k=0}^{k=10} a_k \\cdot \\left(\\frac{B}{\\mu }\\right)^k$ The coefficients of the fits are given for different $\\mu $ intervals in Table REF Table: Coefficients derived fitting the restored data with a 10-th order polynomial."
]
] | 1709.01593 |
[
[
"An application of the measurement of expectation values for the photon\n annihilation and creation operators"
],
[
"Abstract Motivated by the readout scheme in interferometric gravitational-wave detectors, we consider the device which measures the expectation value of the photon annihilation and creation operators for output optical field from the main interferometer.",
"As the result, the eight-port homodyne detection is rediscovered as such a device.",
"We evaluate the noise spectral density in this measurement.",
"We also briefly discuss on the application of our results to the readout scheme of gravitational-wave detectors.",
"We call this measurement scheme to measure these expectation values as \"double balanced homodyne detection.\""
],
[
"Introduction",
"In quantum theory, an “observable” is represented by a self-adjoint operator acting on the Hilbert space as an axiom.",
"Besides of the origin of the terminology of “observable,” it is sometime discussed which variable is measurable in quantum theory.",
"The photon phase difference is a typical example and there are many literature since Dirac [1], which discuss the self-adjoint operator corresponding to the phase difference.",
"Although there are many theoretical arguments on the self-adjoint operator which corresponds to the phase difference, a huge number of experiments to measure the photon phase difference have also been carried out.",
"The most recent impressive example is the gravitational-wave detector which finally succeed direct observations of gravitational wave through the measurement of the photon phase difference [2].",
"The fundamental principle of the interferometric gravitational-wave detectors is to imprint gravitational-wave signals to the phase difference of photons which propagate different paths and to measure the phase difference between them.",
"Current gravitational-wave detectors use the DC readout scheme, in which the output photon power is directly measured.",
"On the other hand, homodyne detections are regarded as one of candidates of the readout scheme in the future gravitational-wave detectors.",
"From the proposal by Vyatchanin, Matsko, and Zubova [3] in 1993, it has been believed in the gravitational-wave community that “we can measure the output quadrature $\\hat{b}_{\\theta }$ defined by $\\hat{b}_{\\theta }:=\\cos \\theta \\hat{b}_{1} + \\sin \\theta \\hat{b}_{2}$ by the balanced homodyne detection” [5], where $\\theta $ is the homodyne angle and $\\hat{b}_{1,2}$ are the amplitude and phase quadratures in the two-photon formulation [4], respectively.",
"Note that these operators $\\hat{b}_{1,2}$ are defined as linear combinations of the annihilation and creation operators for the output optical field from the interferometer.",
"In the case of the interferometric gravitational-wave detectors, the output quadrature $\\hat{b}_{\\theta }$ includes gravitational-wave signal.",
"Apart from the leakage of the classical carrier field, we can formally write this output quadrature as $\\hat{b}_{\\theta }(\\Omega )=R(\\Omega ,\\theta ) \\left(\\hat{h}_{n}(\\Omega ,\\theta ) + h(\\Omega )\\right),$ where $h(\\Omega )$ is a classical gravitational-wave signal in the frequency domain, $\\hat{h}_{n}(\\Omega )$ is the noise operator which is given by the linear combination of the photon annihilation and creation operators for the optical fields which are injected to the main interferometer.",
"Furthermore, it was pointed out that if we can prepare an appropriate frequency-dependent homodyne angle $\\theta =\\theta (\\Omega )$ , we can reduce the quantum noise produced by the noise operator $\\hat{h}_{n}(\\Omega ,\\theta )$ [5].",
"Therefore, in interferometric gravitational-wave detectors, it is important to develop technique of the homodyne detection, theoretically and experimentally, for the extraction of the information of the gravitational-wave signal $h(\\Omega )$ through the measurement of some expectation values of some quantum operators which are related to the operator $\\hat{b}_{\\theta }$ .",
"On the other hand, in quantum measurement theory, homodyne detections are known as the measurement scheme of a linear combination of the photon annihilation and creation operators [6].",
"As noted above, the operator $\\hat{b}_{\\theta }$ is constructed from linear combinations of the photon annihilation and creation operators $\\hat{b}$ and $\\hat{b}^{\\dagger }$ .",
"This means that if we can measure the both expectation values of the photon annihilation operator $\\hat{b}$ and the creation operator $\\hat{b}^{\\dagger }$ themselves, we can calculate the expectation value of the operator $\\hat{b}_{\\theta }$ from these expectation values.",
"In this Letter, we report our rediscovery of the eight-port homodyne detection [7] as the measurements of the expectation values of the photon annihilation and creation operator themselves.",
"We explicitly show that we can calculate the expectation value of the operator $\\hat{b}_{\\theta }$ through the expectation values of the photon numbers of the output from the eight-port homodyne detection.",
"We also briefly discuss the noise spectral density in the case where we apply our results to a readout scheme of interferometric gravitational-wave detectors."
],
[
"Balanced homodyne detections in the Heisenberg picture",
"First, we briefly review a quantum mechanical description of the balanced homodyne detection [6] in the Heisenberg picture depicted in Fig.",
"REF .",
"Figure: Configuration of the interferometer for the balanced homodynedetection.The beam splitter BS is 50:50.The notations of the quadratures a ^\\hat{a}, b ^\\hat{b},c ^ o \\hat{c}_{o}, c ^ i \\hat{c}_{i}, d ^ o \\hat{d}_{o}, d ^ i \\hat{d}_{i},l ^ o \\hat{l}_{o}, and l ^ i \\hat{l}_{i} are also given in this figure.As well known, in interferometers, the electric field operator associated with the annihilation operator $\\hat{a}(\\omega )$ at time $t$ and the length of the propagation direction $z$ in interferometers is described by $\\hat{E}_{a}(t-z)&=&\\int _{-\\infty }^{+\\infty }\\frac{d\\omega }{2\\pi }\\sqrt{\\frac{2\\pi \\hbar |\\omega |}{{\\cal A}c}}e^{-i\\omega (t-z)}\\nonumber \\\\&& \\quad \\quad \\times \\left\\lbrace \\hat{a}(\\omega ) \\Theta (\\omega )+\\hat{a}^{\\dagger }(-\\omega ) \\Theta (-\\omega )\\right\\rbrace ,$ where ${\\cal A}$ is the cross-sectional area of the optical beam, $\\Theta (\\omega )$ is the Heaviside step function, and the annihilation operator $\\hat{a}(\\omega )$ satisfies the usual commutation relation $\\left[\\hat{a}(\\omega ),\\hat{a}^{\\dagger }(\\omega )\\right]$ $=$ $2 \\pi \\delta (\\omega -\\omega ^{\\prime })$ .",
"Throughout this letter, we denote the quadrature $\\hat{a}$ as that for the input to the main interferometer.",
"On the other hand, we denote the output quadrature from the main interferometer by $\\hat{b}$ .",
"Furthermore, in the homodyne detections, we use the electric field whose state is a coherent state, which comes from the local oscillator.",
"The quadrature associated with the electric field from the local oscillator is denoted by $\\hat{l}_{i}$ and the state for the quadrature $\\hat{l}_{i}$ is the coherent state $|\\gamma \\rangle _{l_{i}}$ which satisfies $\\hat{l}_{i}(\\omega ) |\\gamma \\rangle _{l_{i}}=\\gamma (\\omega ) |\\gamma \\rangle _{l_{i}}.$ Here, $\\gamma =\\gamma (\\omega )$ is the complex eigenvalue for the coherent state $|\\gamma \\rangle _{l_{i}}$ .",
"Through the notation of the electric fields as (REF ), we consider the balanced homodyne detection depicted in Fig.",
"REF .",
"We assign the notation of the photon annihilation operators $\\hat{a}$ , $\\hat{b}$ , $\\hat{c}_{o}$ , $\\hat{c}_{i}$ , $\\hat{d}_{o}$ , $\\hat{d}_{i}$ , $\\hat{l}_{o}$ , and $\\hat{l}_{i}$ as in Fig.",
"REF .",
"In the balanced homodyne detection, we detect the photon numbers $\\hat{n}_{c_{o}}:=\\hat{c}_{o}^{\\dagger }\\hat{c}_{o}$ and $\\hat{n}_{d_{o}}:=\\hat{d}_{o}^{\\dagger }\\hat{d}_{o}$ through the photodetectors D1 and D2, respectively.",
"We also assume the transmissivity of the beam splitter is 50:50.",
"From the field junction conditions at the beamsplitter in Fig.",
"REF , we obtain the relations of the quadratures $\\hat{c}_{o}$ , $\\hat{d}_{o}$ , $\\hat{b}$ , and $\\hat{l}_{i}$ as $\\hat{c}_{o}=\\frac{\\hat{b} + \\hat{l}_{i}}{\\sqrt{2}},\\quad \\hat{d}_{o}=\\frac{\\hat{b} - \\hat{l}_{i}}{\\sqrt{2}}$ and the difference of these photon-number expectation values yields the expectation value of a linear combination of the output quadrature $\\hat{b}$ as $\\langle \\hat{n}_{c_{o}}\\rangle -\\langle \\hat{n}_{d_{o}}\\rangle =\\left\\langle \\gamma ^{*}\\hat{b}+\\gamma \\hat{b}^{\\dagger }\\right\\rangle .$ Here, we used the state for the field from the local oscillator is in the coherent state (REF ).",
"This corresponds to the measurement of the expectation value of the operator $\\hat{s}$ defined by $\\hat{s}:=\\hat{n}_{c_{o}}-\\hat{n}_{d_{o}}=\\hat{l}_{i}^{\\dagger } \\hat{b}+\\hat{b}^{\\dagger } \\hat{l}_{i}.$ Note that the linear combination (REF ) does not directly yield the expectation value of the output field quadrature $\\hat{b}$ itself, but the phase of the right-hand side in Eq.",
"(REF ) yields the cosine of the relative phase between the output field and the coherent state from the local oscillator.",
"From the view point of the measurement of the quadrature (REF ), we want to measure both cosine and sine parts of the phase of the quadrature $\\hat{b}$ with a fixed phase of the coherent state from the local oscillator.",
"This is accomplished by the measurements of the expectation values of the photon annihilation and creation operators themselves through the “eight-port homodyne detection” discussed in Refs.",
"[7]."
],
[
"Expectation values of photon annihilation and creation operators",
"The interferometer configuration of the eight-port homodyne detection is depicted in Fig.",
"REF .",
"Here, we assume that we already knew both of the amplitude and the phase of the complex amplitude $\\gamma $ for the coherent state from the local oscillator.",
"We also assume that all beam splitters in Fig.",
"REF are 50:50 in this letter.",
"Figure: A realization of the measurement process of the expectation valuesof the signal photon annihilation operator b ^\\hat{b} and creationoperator b ^ † \\hat{b}^{\\dagger } themselves.In this figure, “BS” is the beam splitter, “PR” is thephase rotator.We assume that all beamsplitters are 50:50.To carry out two balanced homodyne detections, we separate thesignal photon field associated with the quadrature b ^\\hat{b} fromthe main interferometer and the photon field associated with thequadrature l ^ i \\hat{l}_{i} from the local oscillator through thebeam splitters BS1 and BS3, respectively.One of these two paths is used for the usual balanced homodynedetection through the beam splitter BS2 and the photodetectors D1and D2.We introduce PR on the path between BS3 and BS4 to addπ/2\\pi /2-phase offset to the coherent state from the localoscillator and we perform the usual balanced homodyne detectionafter this phase addition through the beam splitter B4 and thephotodetector D3 and D2.The notation of the quadratures for the photon fields are alsodescribed in this figure.As depicted in Fig.",
"REF , at the beam splitter 1 (BS1), the output signal $\\hat{b}$ from the main interferometer is separated into two parts, which we denote $\\hat{b}_{(1)}$ and $\\hat{b}_{(2)}$ , respectively.",
"In addition to the output quadrature $\\hat{b}$ , the additional noise source may be inserted, whose quadrature is denoted by $\\hat{e}_{i}$ and assume that the state for $\\hat{e}_{i}$ is the vacuum state.",
"Then, the junction conditions for the quadratures at BS1 yield $\\hat{b}_{(1)}=\\frac{\\hat{b} - \\hat{e}_{i}}{\\sqrt{2}},\\quad \\hat{b}_{(2)}=\\frac{\\hat{b} + \\hat{e}_{i}}{\\sqrt{2}}.$ On the other hand, at the beam splitter 3 (BS3), the incident electric field from the local oscillator is in the coherent state and its quadrature is denoted by $\\hat{l}_{i}$ .",
"Further, from the configuration depicted in Fig.",
"REF , another incident field to BS3 should be taken into account.",
"We denote the quadrature for this additional field as $\\hat{f}_{i}$ and assume that the state for $\\hat{f}_{i}$ is the vacuum state.",
"The beam splitter BS3 separate the electric field into two paths.",
"We denote the quadrature associated with this electric field which goes from BS3 to BS2 by $\\hat{l}_{(0)i}$ .",
"The electric field along another path from BS3 is towards the beam splitter 4 (BS4) and we denote the quadrature for this field by $\\hat{l}_{(1)i}$ .",
"By the beam splitter condition, quadratures $\\hat{l}_{(0)i}$ and $\\hat{l}_{(1)i}$ are determined by the equation $\\hat{l}_{(0)i}=\\frac{\\hat{l}_{i}-\\hat{f}_{i}}{\\sqrt{2}}, \\quad \\hat{l}_{(1)i}=\\frac{\\hat{l}_{i}+\\hat{f}_{i}}{\\sqrt{2}}.$ The field associated with the quadrature $\\hat{l}_{(0)i}$ is used the balanced homodyne detection through the beam splitter 2 (BS2).",
"On the other hand, the field associated with the quadrature $\\hat{l}_{(1)i}$ is used the balanced homodyne detection through the beam splitter 4 (BS4) after introducing the phase offset $\\pi /2$ .",
"This phase offset is introduced by the phase rotator (PR) between BS3 and BS4.",
"Due to this phase rotator, the quadrature $\\hat{l}_{(1)i}$ is changed into the quadrature $\\hat{l}_{(1/4)i}$ as $\\hat{l}_{(1/4)i}=i\\hat{l}_{(1)i}.$ This quadurature $\\hat{l}_{(1/4)i}$ is directly used the balanced homodyne detection through BS4.",
"In the balanced homodyne detection through the beam splitter BS2, the output quadratures $\\hat{c}_{(1)o}$ and $\\hat{d}_{(1)o}$ are related to the input quadratures $\\hat{b}_{(1)}$ and $\\hat{l}_{(0)i}$ as $\\hat{c}_{(1)o}=\\frac{\\hat{b}_{(1)}+\\hat{l}_{(0)i}}{\\sqrt{2}},\\quad \\hat{d}_{(1)o}=\\frac{\\hat{l}_{(0)i}-\\hat{b}_{(1)}}{\\sqrt{2}}.$ The photon numbers $\\hat{n}_{c_{(1)o}}:=\\hat{c}_{(1)o}^{\\dagger }\\hat{c}_{(1)o}$ and $\\hat{n}_{d_{(1)o}}:=\\hat{d}_{(1)o}^{\\dagger }\\hat{d}_{(1)o}$ of the output fields associated with the quadratures $\\hat{c}_{(1)o}$ and $\\hat{d}_{(1)o}$ are detected through the photodetector D1 and D2 in Fig.",
"REF , respectively.",
"These photon numbers are given in terms of the quadrature $\\hat{b}$ , $\\hat{l}_{i}$ , $\\hat{e}_{i}$ , and $\\hat{f}_{i}$ through Eqs.",
"(REF ), (REF ), and (REF ).",
"The balanced homodyne detection from the photodetector D1 and D2 yields the expectation value $2\\left(\\langle \\hat{n}_{c_{(1)o}}\\rangle -\\langle \\hat{n}_{d_{(1)o}}\\rangle \\right)=\\left\\langle \\gamma ^{*}\\hat{b}+\\gamma \\hat{b}^{\\dagger }\\right\\rangle ,$ as Eq.",
"(REF ), which is also regarded as the expectation value of the operator $\\hat{s}_{D1D2}&:=&2\\left(\\hat{n}_{c_{(1)o}}-\\hat{n}_{d_{(1)o}}\\right)\\\\&=&\\hat{b} \\hat{l}_{i}^{\\dagger }+ \\hat{b}^{\\dagger } \\hat{l}_{i}\\nonumber \\\\&&- \\hat{b} \\hat{f}_{i}^{\\dagger }- \\hat{b}^{\\dagger } \\hat{f}_{i}+ \\hat{e}_{i} \\hat{f}_{i}^{\\dagger }+ \\hat{e}_{i}^{\\dagger } \\hat{f}_{i}- \\hat{e}_{i} \\hat{l}_{i}^{\\dagger }- \\hat{e}_{i}^{\\dagger } \\hat{l}_{i}.$ In the right-hand side of Eq.",
"(), the first line gives Eq.",
"(REF ), the second line is the vacuum contributions.",
"Similarly, the balanced homodyne detection through the beam splitter BS4 and photodetectors D3 and D4 yields the expectation value $2i\\left(\\langle \\hat{n}_{d_{(2)o}}\\rangle -\\langle \\hat{n}_{c_{(2)o}}\\rangle \\right)=\\left\\langle \\gamma ^{*}\\hat{b}-\\gamma \\hat{b}^{\\dagger }\\right\\rangle ,$ which is regarded as the expectation value of the operator $\\hat{s}_{D3D4}&:=&2i\\left(\\hat{n}_{d_{(2)o}}-\\hat{n}_{c_{(2)o}}\\right)\\\\&=&\\hat{l}_{i}^{\\dagger }\\hat{b}-\\hat{b}^{\\dagger }\\hat{l}_{i}\\nonumber \\\\&&- \\hat{b}^{\\dagger } \\hat{f}_{i}+ \\hat{b} \\hat{f}_{i}^{\\dagger }+ \\hat{l}_{i}^{\\dagger } \\hat{e}_{i}+ \\hat{f}_{i}^{\\dagger } \\hat{e}_{i}- \\hat{e}_{i}^{\\dagger } \\hat{l}_{i}- \\hat{e}_{i}^{\\dagger } \\hat{f}_{i}.$ In the right-hand side of Eq.",
"(), the first line gives Eq.",
"(REF ) and the second line is the vacuum contributions.",
"Here, we have to emphasize that the overall factors in Eqs.",
"(REF ) and () are purely imaginary which break the self-adjointness of our result.",
"Since we assumed that we already knew the complex amplitude $\\gamma $ , from Eqs.",
"(REF ) and (REF ), we can calculate the expectation values of operators $\\hat{b}$ and $\\hat{b}^{\\dagger }$ as $&&\\frac{1}{2\\gamma ^{*}}\\left(\\left\\langle \\hat{s}_{D1D2}\\right\\rangle +\\left\\langle \\hat{s}_{D3D4}\\right\\rangle \\right)=\\left\\langle \\hat{b}\\right\\rangle ,\\\\&&\\frac{1}{2\\gamma }\\left(\\left\\langle \\hat{s}_{D1D2}\\right\\rangle -\\left\\langle \\hat{s}_{D3D4}\\right\\rangle \\right)=\\left\\langle \\hat{b}^{\\dagger }\\right\\rangle .$ Here, we have to emphasize that the expectation value of the operators $\\hat{s}_{D1D2}$ and $\\hat{s}_{D3D4}$ are given through the measurement of the expectation values of photon numbers at the photodetector D1, D2, D3, D4, and the complex amplitude $\\gamma $ for the coherent state from the local oscillator.",
"Similar formulae were also derived in Refs.",
"[8] in the context of the characterization of the nonclassicality of the system.",
"The noise spectral density $S_{\\hat{Q}}(\\omega )$ defined by $\\frac{1}{2} S_{\\hat{Q}}(\\omega ) 2 \\pi \\delta (\\omega -\\omega ^{\\prime }):=\\frac{1}{2} \\left\\langle \\hat{Q}\\hat{Q}^{^{\\prime }\\dagger }+\\hat{Q}^{^{\\prime }\\dagger }\\hat{Q}\\right\\rangle ,$ for the operator $\\hat{Q}$ with its expectation value $\\langle \\hat{Q}\\rangle =0$ is commonly used in the gravitational-wave community to evaluate quantum fluctuations in the measurement of the operator $\\hat{Q}$ [5], [9].",
"Here, $\\hat{Q}=\\hat{Q}(\\omega )$ and $\\hat{Q}^{^{\\prime }\\dagger }=\\hat{Q}^{\\dagger }(\\omega ^{\\prime })$ .",
"In our case, we define the operators $\\hat{t}_{b+}&:=&\\frac{1}{2\\gamma ^{*}}\\left(\\hat{s}_{D1D2}+\\hat{s}_{D3D4}\\right)\\nonumber \\\\&=&\\hat{b} \\frac{\\hat{l}_{i}^{\\dagger }}{\\gamma ^{*}}+\\frac{1}{\\gamma ^{*}}\\left(- \\hat{b}^{\\dagger } \\hat{f}_{i}+ \\hat{e}_{i} \\hat{f}_{i}^{\\dagger }- \\hat{e}_{i}^{\\dagger } \\hat{l}_{i}\\right),\\\\\\hat{t}_{b-}&:=&\\frac{1}{2\\gamma }\\left(\\hat{s}_{D1D2}-\\hat{s}_{D3D4}\\right)=\\hat{t}_{b+}^{\\dagger }.$ Eqs.",
"(REF ) and () are regarded as the expectation values of these operators $\\hat{t}_{b+}$ and $\\hat{t}_{b-}$ , respectively.",
"We also define the noise operators $\\hat{t}_{b+}^{(n)}$ , $\\hat{t}_{b-}^{(n)}$ , and $\\hat{b}^{(n)}$ by $\\hat{t}_{b+}&=:&\\langle \\hat{b}\\rangle + \\hat{t}_{b+}^{(n)},\\quad \\langle \\hat{t}_{b+}^{(n)}\\rangle =0,\\\\\\hat{t}_{b-}&=:&\\langle \\hat{b}^{\\dagger }\\rangle + \\hat{t}_{b-}^{(n)},\\quad \\langle \\hat{t}_{b-}^{(n)}\\rangle =0,\\\\\\hat{b}&=:&\\langle \\hat{b}\\rangle + \\hat{b}^{(n)},\\quad \\langle \\hat{b}^{(n)}\\rangle =0.$ Further, from the interferometer setup in Fig.",
"REF , we should regard that the commutators $\\left[\\hat{e}_{i},\\hat{f}_{i}\\right]$ , $\\left[\\hat{e}_{i},\\hat{f}_{i}^{\\dagger }\\right]$ , $\\left[\\hat{f}_{i},\\hat{l}_{i}\\right]$ , $\\left[\\hat{f}_{i},\\hat{l}_{i}^{\\dagger }\\right]$ , $\\left[\\hat{l}_{i},\\hat{e}_{i}\\right]$ , and $\\left[\\hat{l}_{i},\\hat{e}_{i}^{\\dagger }\\right]$ vanish.",
"Moreover, we can easily check that the commutators $\\left[\\hat{b}, \\hat{e}_{i}\\right]$ , $\\left[\\hat{b},\\hat{e}_{i}^{\\dagger }\\right]$ , $\\left[\\hat{b},\\hat{f}_{i}\\right]$ , $\\left[\\hat{b},\\hat{f}_{i}^{\\dagger }\\right]$ , $\\left[\\hat{b},\\hat{l}_{i}\\right]$ , and $\\left[\\hat{b},\\hat{l}_{i}^{\\dagger }\\right]$ also vanish even if the output quadrature $\\hat{b}$ depends on the input quadrature $\\hat{a}$ .",
"Equations (REF )–() and the above commutation relations lead us to the noise spectral densities $S_{\\hat{t}_{b+}^{(n)}}(\\omega )=S_{\\hat{t}_{b-}^{(n)}}(\\omega )=S_{\\hat{b}^{(n)}}(\\omega )+\\frac{2}{|\\gamma |^{2}}\\langle \\hat{n}_{b}\\rangle +1.$ This noise spectral density indicates that in addition to the noise spectral density $S_{\\hat{b}^{(n)}}(\\omega )$ , we have the additional fluctuations in our measurement of $\\langle \\hat{b}\\rangle $ through the measurement of the operator $\\hat{t}_{b+}$ .",
"We note that the term $2\\langle \\hat{n}_{b}\\rangle /|\\gamma ^{2}|$ will be negligible if $\\langle \\hat{n}_{b}\\rangle \\ll |\\gamma ^{2}|$ .",
"On the other hand, the the last term 1 in Eq.",
"(REF ), which comes from the shot noise from the additional input vacuum fields, is not controllable."
],
[
"Expectation value of the operator $\\hat{b}_{\\theta }$ and its noise",
"In the case of gravitational-wave detectors, the two-photon formulation [4] is always used.",
"In this formulation, we consider the sideband fluctuations around the classical carrier field which proportional to $\\cos \\omega _{0}t$ .",
"The sideband fluctuations are described by the quadrature $\\hat{b}_{\\pm }(\\Omega ):=\\hat{b}(\\omega _{0}\\pm \\Omega )$ and we introduce quadratures $\\hat{b}_{1,2}$ by $\\hat{b}_{1}:=\\frac{1}{\\sqrt{2}} \\left(\\hat{b}_{+}+\\hat{b}_{-}^{\\dagger }\\right), \\quad \\hat{b}_{2}:=\\frac{1}{\\sqrt{2}i} \\left(\\hat{b}_{+}-\\hat{b}_{-}^{\\dagger }\\right).$ These are the definitions of the amplitude (phase) quadrature $\\hat{b}_{1}$ ($\\hat{b}_{2}$ ) in Eq.",
"(REF ).",
"We also consider the sideband quadratures of the local oscillator $\\hat{l}_{i\\pm }=\\hat{l}_{i}(\\omega _{0}\\pm \\Omega )$ and assume that the state of the local oscillator is the coherent state (REF ) in which the eigenvalues of operators $\\hat{l}_{i\\pm }$ are given by $\\gamma _{\\pm }:=\\gamma (\\omega _{0}\\pm \\Omega )$ , respectively.",
"In this situation, we can obtain the information of the expectation values $\\langle \\hat{s}_{\\pm }\\rangle $ $=$ $\\left\\langle \\gamma _{\\pm }^{*}\\hat{b}_{\\pm } + \\gamma _{\\pm }\\hat{b}_{\\pm }^{\\dagger }\\right\\rangle $ through the usual balanced homodyne detection, where we used the definition (REF ) and $\\hat{s}_{\\pm }:=\\hat{s}(\\omega _{0}\\pm \\Omega )$ .",
"Within the linear level of quadratures, we can obtain the expectation value of any linear combination $&& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\sqrt{2}\\left(\\alpha \\left\\langle \\hat{s}_{+}\\right\\rangle +\\beta \\left\\langle \\hat{s}_{-}\\right\\rangle \\right)\\nonumber \\\\&=&\\left(\\alpha \\gamma _{+}^{*}+\\beta \\gamma _{-}\\right)\\langle \\hat{b}_{1}\\rangle +i\\left(\\alpha \\gamma _{+}^{*}-\\beta \\gamma _{-}\\right)\\langle \\hat{b}_{2}\\rangle \\nonumber \\\\&&+\\left(\\alpha \\gamma _{+}+\\beta \\gamma _{-}^{*}\\right)\\langle \\hat{b}_{1}^{\\dagger }\\rangle +i\\left(-\\alpha \\gamma _{+}+\\beta \\gamma _{-}^{*}\\right)\\langle \\hat{b}_{2}^{\\dagger }\\rangle .$ with complex coefficients $\\alpha $ and $\\beta $ .",
"The problem whether or not we can measure the expectation value of the operator $\\hat{b}_{\\theta }$ defined by (REF ) through the conventional balanced homodyne detection is reduced to the problem whether or not the linear combination (REF ) gives the linear combination of the expectation values $\\langle \\hat{b}_{1}\\rangle $ and $\\langle \\hat{b}_{2}\\rangle $ through an appropriate choice of $\\alpha $ , $\\beta $ , and $\\gamma _{\\pm }$ .",
"Within the linear optics, Eq.",
"(REF ) yields the expectation value of the linear combination of $\\langle \\hat{b}_{1}\\rangle $ and $\\langle \\hat{b}_{2}\\rangle $ if and only if there exists a nontrivial solution with $\\alpha \\beta \\gamma _{+}\\gamma _{-}\\ne 0$ of the matrix equation $\\left(\\begin{array}{cc}\\gamma _{+} & \\gamma _{-}^{*} \\\\- \\gamma _{+} & \\gamma _{-}^{*}\\end{array}\\right)\\left(\\begin{array}{c}\\alpha \\\\\\beta \\end{array}\\right)=\\left(\\begin{array}{c}0 \\\\0\\end{array}\\right).$ However, we can easily show that Eq.",
"(REF ) has no nontrivial solution satisfies the condition $\\alpha \\beta \\gamma _{+}\\gamma _{-}\\ne 0$ .",
"This means that any choice of $\\alpha $ , $\\beta $ and $\\gamma _{\\pm }$ never yields the linear combination (REF ).",
"Thus, we cannot measure the expectation value of the operator $\\hat{b}_{\\theta }$ through the conventional balanced homodyne detection depicted in Fig.",
"REF [10].",
"On the other hand, the measurement of the expectation value of $\\hat{b}_{\\theta }$ is possible through the eight-port homodyne detection depicted in Fig.",
"REF .",
"In this interferometer setup, we obtain the expectation values of the operators $\\hat{s}_{D1D2}$ and $\\hat{s}_{D3D4}$ defined by Eqs.",
"(REF ) and (REF ), respectively.",
"Since we consider the sideband fluctuations around the classical carrier with the frequency $\\omega _{0}$ , we have four expectation values $\\langle \\hat{s}_{D1D2\\pm }\\rangle := \\langle \\hat{s}_{D1D2}(\\omega _{0}\\pm \\Omega )\\rangle =\\left\\langle \\gamma _{\\pm }^{*}\\hat{b}_{\\pm } + \\gamma _{\\pm }\\hat{b}_{\\pm }^{\\dagger }\\right\\rangle , \\\\\\langle \\hat{s}_{D3D4\\pm }\\rangle := \\langle \\hat{s}_{D3D4}(\\omega _{0}\\pm \\Omega )\\rangle =\\left\\langle \\gamma _{\\pm }^{*}\\hat{b}_{\\pm } - \\gamma _{\\pm }\\hat{b}_{\\pm }^{\\dagger }\\right\\rangle .$ Inspecting the expectation values (REF ) and (), we first consider the operators $\\hat{t}_{D1D2+}&:=&\\frac{1}{\\sqrt{2}} \\left(\\frac{\\hat{s}_{D1D2+}}{|\\gamma _{+}|}+\\frac{\\hat{s}_{D1D2-}}{|\\gamma _{-}|}\\right),\\\\\\hat{t}_{D3D4-}&:=&\\frac{1}{\\sqrt{2}}\\left(\\frac{\\hat{s}_{D3D4+}}{|\\gamma _{+}|}-\\frac{\\hat{s}_{D3D4-}}{|\\gamma _{-}|}\\right),$ where $\\gamma _{\\pm }=:|\\gamma _{\\pm }|e^{i\\theta _{\\pm }}$ .",
"Here, we choose the phase $\\theta _{\\pm }$ of the coherent amplitude $\\gamma _{\\pm }$ so that $\\theta _{\\pm }=\\theta $ .",
"Furthermore, we assume that $|\\gamma _{\\pm }|=|\\gamma |$ , for simplicity.",
"Due to this phase choice, we easily obtain the expectation value of the operator $\\hat{t}_{\\theta }:=(\\hat{t}_{D1D2+}+\\hat{t}_{D3D4-})/2$ as $\\langle \\hat{t}_{\\theta }\\rangle =\\left\\langle \\cos \\theta \\hat{b}_{1} + \\sin \\theta \\hat{b}_{2}\\right\\rangle =\\langle \\hat{b}_{\\theta }\\rangle .$ Thus, we can obtain the expectation value of the operator $\\hat{b}_{\\theta }$ as the direct output of the eight-port homodyne detection.",
"We have to emphasis that our measurement of the expectation value of the operator $\\hat{b}_{\\theta }$ is an indirect measurement of the expectation value of operator $\\hat{b}_{\\theta }$ .",
"To obtain the expectation value of the operator $\\hat{b}_{\\theta }$ , we just calculate the linear combination of the expectation values of the output photon number operators with the complex coefficients.",
"Since we directly measure the photon number operators which are self-adjoint operators, our analyses and results do not contradict to the axiom of the quantum theory.",
"The fact that our measurement of the expectation value $\\hat{b}_{\\theta }$ is an indirect measurement directly leads the fact that the noise in our measurement cannot be given by the expectation value of the square of the operators $\\hat{b}_{\\theta }$ .",
"Actually, through the separation $\\hat{t}_{\\theta }:=\\langle \\hat{b}_{\\theta }\\rangle +\\hat{t}^{(n)}_{\\theta },\\quad \\hat{b}_{\\theta }:=\\langle \\hat{b}_{\\theta }\\rangle +\\hat{b}^{(n)}_{\\theta },$ we can also evaluate the noise-spectral density of this measurement through the similar derivation to Eq.",
"(REF ) as $S_{\\hat{t}_{\\theta }^{(n)}}(\\Omega )=\\hat{S}_{\\hat{b}_{\\theta }^{(n)}}(\\Omega )+\\frac{\\langle \\hat{n}_{b_{-}}+\\hat{n}_{b_{+}}\\rangle }{|\\gamma |^{2}} + 1.$ This noise spectral density indicates a noise level in our measurement of the operator $\\hat{b}_{\\theta }$ ."
],
[
"Gravitational-wave signal referred noise",
"The input-output relation of gravitational-wave detectors are given by Eq.",
"(REF ).",
"Through the above eight-port homodyne detection, we can measure the expectation value of the operator $\\hat{t}_{\\theta }$ to obtain the expectation value $\\langle \\hat{b}_{\\theta }\\rangle $ which includes gravitational-wave signals $h(\\Omega )$ .",
"To evaluate the noise spectral density from the input-output relation (REF ), we can directly obtain the signal-referred noise spectral density as $\\frac{1}{|R|^{2}} S_{\\hat{t}_{\\theta }^{(n)}}=S_{\\hat{h}_{(n)}}+\\frac{1}{|R|^{2}}\\left(\\frac{\\langle \\hat{n}_{-}+\\hat{n}_{+}\\rangle }{|\\gamma |^{2}}+1\\right).$ This relation between noise-spectral densities yields that the second term in Eq.",
"(REF ), which corresponds to the additional noise due to our indirect measurement, is negligible if the response function $R(\\Omega )$ is sufficiently large.",
"In this case, the signal-referred noise spectral density $S_{\\hat{t}_{\\theta }^{(n)}}/|R|^{2}$ in our indirect measurement coincides with the signal-referred noise spectral density $S_{\\hat{h}^{(n)}}$ ."
],
[
"Summary",
"In summary, we rediscovered so-called the eight-port homodyne detection as the device to measure the expectation values of the photon annihilation and creation operator for the output optical field from the main interferometer and discuss its application to a readout scheme of the gravitational-wave detectors.",
"As we emphasized above, we just proposed an indirect measurement which yields the expectation value $\\langle \\hat{b}_{\\theta }\\rangle $ through the calculation by the linear combination of four photon-number expectation values with complex coefficients.",
"This indirect measurement of the operator $\\hat{b}_{\\theta }$ leads the additional fluctuations in the measurement as in Eq.",
"(REF ).",
"Furthermore, we explained the outline of our proof for the assertion that we cannot measure the expectation value of the operator $\\hat{b}_{\\theta }$ in Eq.",
"(REF ) by the balanced homodyne detection, but it is possible by the eight-port homodyne detection if the coherent state from the local oscillator is appropriately prepared.",
"We also evaluated the noise spectral densities of these measurements and discussed the noise level when we apply our result to the readout scheme of interferometric gravitational-wave detectors.",
"We call this measurement scheme to measure the expectation values of the photon annihilation and creation operators, or operator $\\hat{b}_{\\theta }$ defined by Eq.",
"(REF ) using the eight-port homodyne detection as the “double balanced homodyne detection.” More detailed analysis of our double balanced homodyne detection will be explained in Ref.",
"[10]."
],
[
"Acknowledgments",
"K.N.",
"acknowledges to Dr. Tomotada Akutsu and the other members of the GWPO in NAOJ for their continuous encouragement to our research and also appreciate Prof. Akio Hosoya, Prof. Izumi Tsutsui, and Dr. Hiroyuki Takahashi for their supports and encouragement."
]
] | 1709.01697 |
[
[
"Neural Networks Regularization Through Class-wise Invariant\n Representation Learning"
],
[
"Abstract Training deep neural networks is known to require a large number of training samples.",
"However, in many applications only few training samples are available.",
"In this work, we tackle the issue of training neural networks for classification task when few training samples are available.",
"We attempt to solve this issue by proposing a new regularization term that constrains the hidden layers of a network to learn class-wise invariant representations.",
"In our regularization framework, learning invariant representations is generalized to the class membership where samples with the same class should have the same representation.",
"Numerical experiments over MNIST and its variants showed that our proposal helps improving the generalization of neural network particularly when trained with few samples.",
"We provide the source code of our framework https://github.com/sbelharbi/learning-class-invariant-features ."
],
[
"Introduction",
"For a long time, it has been understood in the field of deep learning that building a model by stacking multiple levels of non-linearity is an efficient way to achieve good performance on complicated artificial intelligence tasks such as vision [24], [41], [43], [19] or natural language processing [12], [46], [22], [16].",
"The rationale behind this statement is the hierarchical learned representations throughout the depth of the network which circumvent the need of extracting handcrafted features.",
"For many years, the non-convex optimization problem of learning a neural network has prevented going beyond one or two hidden layers.",
"In the last decade, deep learning has seen a breakthrough with efficient training strategies of deeper architectures[21], [33], [6], and a race toward deeper models has began[24], [41], [43], [19].",
"This urge to deeper architectures was due to (i) large progress in optimization, (ii) the powerful computation resources brought by GPUsGraphical Processing Units.",
"and (iii) the availability of huge datasets such as ImageNet [14] for computer vision problems.",
"However, in real applications, few training samples are usually available which makes the training of deep architectures difficult.",
"Therefore, it becomes necessary to provide new learning schemes for deep networks to perform better using few training samples.",
"A common strategy to circumvent the lack of annotated data is to exploit extra informations related to the data, the model or the application domain, in order to guide the learning process.",
"This is typically carried out through regularization which can rely for instance on data augmentation, $L_2$ regularization [44], dropout[42], unsupervised training [21], [33], [6], [36], [35], [5], shared parameters[26], [34], [15], etc.",
"Our research direction in this work is to provide a new regularization framework to guide the training process in a supervised classification context.",
"The framework relies on the exploitation of prior knowledge which has already been used in the literature to train and improve models performance when few training samples are available [29], [37], [20], [31], [25], [48], [47], [49].",
"Indeed, prior knowledge can offer the advantage of more consistency, better generalization and fast convergence using less training data by guiding the learning process [29].",
"By using prior knowledge about the target function, the learner has a better chance to generalize from sparse data [29], [1], [2], [3].",
"For instance, in object localization such as part of the face, knowing that the eyes are located above the nose and the mouth can be helpful.",
"One can exploit this prior structure about the data representation: to constrain the model architecture, to guide the learning process, or to post-process the model's decision.",
"In classification task, although it is difficult to define what makes a representation good, two properties are inherent to the task: Discrimination i.e.",
"representations must allow to separate samples of distinct classes.",
"Invariance i.e.",
"representations must allow to obtain robust decision despite some variations of input samples.",
"Formally, given two samples $x_1$ and $x_2$ , a representation function ${\\Gamma (\\cdot )}$ and a decision function $\\Psi (\\cdot )$ ; when ${x_1 \\approx x_2}$ , we seek invariant representations that provide ${\\Gamma (x_1) \\approx \\Gamma (x_2)}$ , leading to smooth decision ${\\Psi (\\Gamma (x_1)) \\approx \\Psi (\\Gamma (x_2))}$ .",
"In this work, we are interested in the invariance aspect of the representations.",
"This definition can be extended to more elaborated transformations such as rotation, scaling, translation, etc.",
"However, in real life there are many other transformations which are difficult to formalize or even enumerate.",
"Therefore, we extend in this work the definition of the invariant representations to the class membership, where samples within the same class should have the same representation.",
"At a representation level, this should generate homogeneous and tighter clusters per class.",
"In the training of neural networks, while the output layer is guided by the provided target, the hidden layers are left to the effect of the propagated error from the output layer without a specific target.",
"Nevertheless, once the network trained, examples may form (many) modes on hidden representations, i.e.",
"outputs of hidden layers, conditionally to their classes.",
"Most notably, on the penultimate representation before the decision stage, examples should agglomerate in distinct clusters according to their label as seen on Figure REF .",
"From the aforementioned prior perspective about the hidden representations, we aim in this work to provide a learning scheme that promotes the hidden layers to build representations which are class-invariant and thus agglomerate in restricted number of modes.",
"By doing so, we constrain the network to build invariant intermediate representations per class with respect to the variations in the input samples without explicitly specifying these variations nor the transformations that caused them.",
"Figure: Input/Hidden representations of samples from an artificial dataset along 4 layers of a MLP.",
"Each representation is projected into a 2D space.We express this class-invariance prior as an explicit criterion combined with the classification training criterion.",
"It is formulated as a dissimilarity between the representations of each pair of samples within the same class.",
"The average dissimilarity over all the pairs of all the classes is considered to be minimized.",
"To the best of our knowledge, none has used this class membership to build invariant representations.",
"Our motivation in using this prior knowledge, as a form of regularization, is to be able to train deep neural networks and obtain better generalization error using less training data.",
"We have conducted different experiments over MNIST benchmarck using two models (multilayer perceptrons and convolutional networks) for different classification tasks.",
"We have obtained results that show important improvements of the model's generalization error particularly when trained with few samples.",
"The rest of the paper is organized as follows: Section presents related works for invariance learning in neural networks.",
"We present our learning framework in Section followed by a discussion of the obtained results in Section ."
],
[
"Related Work",
"Learning general invariance, particularly in deep architectures, is an attractive subject where different approaches have been proposed.",
"The rational behind this framework is to ensure the invariance of the learned model toward the variations of the input data.",
"In this section, we describe three kinds of approaches of learning invariance within neural networks.",
"Some of these methods were not necessarily designed to learn invariance however we present them from the invariance perspective.",
"For this description, $f$ is the target function to be learned.",
"Invariance through data transformations: It is well known that generalization performance can be improved by using larger quantity of training samples.",
"Enlarging the number of samples can be achieved by generating new samples through the application of small random transformations such as rotation, scaling, random noise, etc [4], [10], [40] to the original examples.",
"Incorporating such transformed data within the learning process has shown to be helpful in generalization [31].",
"[1] proposes the use of prior information about the behavior of $f$ over perturbed examples using different transformations where $f$ is constrained to be invariant over all the samples generated using these transformations.",
"While data transformations successfully incorporate certain invariance into the learned model, they remain limited to some predefined and well known transformations.",
"Indeed, there are many other transformations which are either unknown or difficult to formalize.",
"Invariance through model architectures: In some neural network models, the architecture implicitly builds a certain type of invariance.",
"For instance, in convolutional networks [26], [34], [15], combining layers of feature extractors using weight sharing with local pooling of the feature maps introduces some degree of translation invariance [32], [28].",
"These models are currently state of the art strategies for achieving invariance in computer vision tasks.",
"However, it is unclear how to explicitly incorporate in these models more complicated invariances such as large angle rotation and complex illumination.",
"Moreover, convolutional and max-pooling techniques are somewhat specialized to visual and audio processing, while deep architectures are generally task independent.",
"Invariance through analytical constraints: Analytical invariance consists in adding an explicit penalty term to the training objective function in order to reduce the variations of $f$ or its sub-parts when the input varies.",
"This penalty is generally based on the derivatives of a criterion related to $f$ with respect to the input.",
"For instance, in unsupervised representation learning, [36] introduces a penalty for training auto-encoders which encourages the intermediate representation to be robust to small changes of the input around the training samples, referred to as contractive auto-encoders.",
"This penalty is based on the Frobenius norm of the first order derivative of the hidden representation of the auto-encoder with respect to the input.",
"Later, [35] extended the contractive auto-encoders by adding another penalty using the norm of an approximation of the second order derivative of the hidden representation with respect to the input.",
"The added term penalizes curvatures and thus favors smooth manifolds.",
"[17] exploit the idea that solving adversarial examples is equivalent to increase the attention of the network to small perturbation for each example.",
"Therefore, they propose a layer-wise penalty which creates flat invariance regions around the input data using the contractive penalty proposed in [36].",
"[38], [39] penalize the derivatives of $f$ with respect to perturbed inputs using simple distortions in order to ensure local invariance to these transformations.",
"Learning invariant representations through the penalization of the derivatives of the representation function $\\Gamma (\\cdot )$ is a strong mathematical tool.",
"However, its main drawback is that the learned invariance is local and is generally robust toward small variations.",
"Learning invariance through explicit analytical constraints can also be found in metric learning.",
"For instance, [9], [18] use a contrastive loss which constrains the projection in the output space as follows: input samples annotated as similar must have close (adjacent) projections and samples annotated as dissimilar must have far projections.",
"In the same way, Siamese networks[7] proceed in learning similarity by projecting input points annotated as similar to be adjacent in the output space.",
"This approach of analytical constraints is our main inspiration in this work, where we provide a penalty that constrains the representation function $\\Gamma (\\cdot )$ to build similar representation for samples from the same class, i.e.",
"in a supervised way.",
"In the following section, we present our proposal with more details."
],
[
"Proposed Method",
"In deep neural networks, high layers tend to learn abstract representations that we have assumed to be closer and closer for the same class along the layers.",
"We would like to promote this behavior.",
"In order to do so, we add a penalty to the training criterion of the network to constrain intermediate representations to be class-invariant.",
"We first describe a model decomposition, the general training framework and then more specific implementation details."
],
[
"Model Decomposition",
"Let us consider a parametric mapping function for classification: ${\\mathcal {M}(.",
"; \\theta ): \\mathbf {X} \\rightarrow \\mathbf {Y}}$ , represented here by a neural network model, where $\\mathbf {X}$ is the input space and $\\mathbf {Y}$ is the label space.",
"This neural network is arbitrarily decomposed into two parametric sub-functions: ${\\Gamma (\\cdot ;\\theta _{\\Gamma }): \\mathbf {X} \\rightarrow \\mathbf {Z}}$ , a representation function parameterized with the set $\\theta _{\\Gamma }$ .",
"This sub-function projects an input sample $x$ into a representation space $\\mathbf {Z}$ .",
"${\\Psi (\\cdot ;\\theta _{\\Psi }): \\mathbf {Z} \\rightarrow \\mathbf {Y}}$ , a decision function parameterized with the set $\\theta _{\\Psi }$ .",
"It performs the classification decision over the representation space $\\mathbf {Z}$ .",
"The network decision function can be written as follows: $\\mathcal {M}(x_i;\\theta ) = \\Psi (\\Gamma (x_i;\\theta _{\\Gamma }); \\theta _{\\Psi })$ where ${\\theta =\\lbrace \\theta _{\\Gamma },\\theta _{\\Psi }\\rbrace }$ .",
"Such a possible decomposition of a neural network with $K=4$ layers is presented in Fig.REF .",
"Here, the decision function $\\Psi (\\cdot )$ is composed of solely the output layer while the rest of the hidden layers form the representation function $\\Gamma (\\cdot )$ .",
"Figure: Decomposition of the neural network ℳ(·)\\mathcal {M}(\\cdot ) into a representation function Γ(·)\\Gamma (\\cdot ) and a decision function Ψ(·)\\Psi (\\cdot ).General Training Framework In order to constrain the intermediate representations $\\Gamma (\\cdot )$ to form clusters over all the samples within the same class we modify the training loss by adding a regularization term.",
"Thus, the training criterion $J$ is composed of the sum of two terms.",
"The first term $\\mathbf {J_{sup}}$ is a standard supervised term which aims at reducing the classification error.",
"The second and proposed regularization term $\\mathbf {J_{H}}$ is a hint penalty that aims at constraining the intermediate representations of samples within the same class to be similar.",
"By doing so, we constrain $\\Gamma (\\cdot )$ to lean invariant representations with respect to the class membership of the input sample.",
"Proposed Hint Penalty Let ${\\mathcal {D} = \\lbrace (x_i, y_i)\\rbrace }$ be a training set for classification task with $S$ classes and $N$ samples; $(x_i, y_i)$ denotes an input sample and its label.",
"Let $\\mathcal {D}_s$ be the sub-set of $\\mathcal {D}$ that consists in all the examples of class $s$ , i.e.",
"$\\mathcal {D}_s=\\lbrace (x,y) \\in \\mathcal {D}\\hspace{5.0pt}s.t.",
"\\hspace{5.0pt}y=s\\rbrace $ .",
"By definition, $\\mathcal {D}=\\bigcup \\limits _{s=1}^S\\mathcal {D}_s$ .",
"For the sake of simplicity, even if $\\mathcal {D}$ and $\\mathcal {D}_s$ contains tuples of (feature,target), $x$ represents only the feature part in the notation $x \\in \\mathcal {D}$ .",
"Let $x_i$ be an input sample.",
"We want to reduce the dissimilarity over the space $\\mathbf {Z}$ between the projection of $x_i$ and the projection of every sample ${x_j \\in \\mathcal {D}_s}$ with $j\\ne {i}$ .",
"For this sample $x_i$ , our hint penalty can be written as follows: $J_{h}(x_i; \\theta _{\\Gamma }) = \\frac{1}{|\\mathcal {D}_s|-1}\\sum _{\\begin{array}{c}{x_j \\in \\mathcal {D}_s}\\\\{j \\ne i}\\end{array}} \\mathcal {C}_h(\\Gamma (x_i;\\theta _{\\Gamma }), \\Gamma (x_j; \\theta _{\\Gamma }))$ where $\\mathcal {C}_{h}(\\cdot , \\cdot )$ is a loss function that measures how much two projections in $\\mathbf {Z}$ are dissimilar and $|\\mathcal {D}_s|$ is the number of samples in $\\mathcal {D}_s$ .",
"Fig.REF illustrates the procedure to measure the dissimilarity in the intermediate representation space $\\mathbf {Z}$ between two input samples $x_i$ and $x_j$ with the same label.",
"Here, we constrained only one hidden layer to be invariant.",
"Extending this procedure for multiple layers is straightforward.",
"It can be done by applying a similar constraint over each concerned layer.",
"Figure: Constraining the intermediate learned representations to be similar over a decomposed network ℳ(·)\\mathcal {M}(\\cdot ) during the training phase.Regularized Training Loss The full training loss can be formulated as follows: $J(\\mathcal {D};\\theta ) &= \\underbrace{ \\frac{\\gamma }{N} \\sum _{(x_i,y_i) \\in \\mathcal {D}} \\mathcal {C}_{sup}(\\Psi (\\Gamma (x_i; \\theta _{\\Gamma }); \\theta _{\\Psi }), y_i)}_{\\text{Supervised loss } \\mathbf {J_{sup}}}+ \\underbrace{ \\frac{\\lambda }{S} \\sum _{s=1}^{S} \\frac{1}{|\\mathcal {D}_s|} \\sum _{x_i \\in \\mathcal {D}_s} J_{h}(x_i; \\theta _{\\Gamma })}_{\\text{Hint penalty }\\mathbf {J_{H}}}$ where $\\gamma $ and $\\lambda $ are regularization weights, $\\mathcal {C}_{sup}(\\cdot , \\cdot )$ the classification loss function.",
"If one use a dissimilarity measure $\\mathcal {C}_h(\\cdot , \\cdot )$ in $J_h$ that is symmetrical such as typically a distance, summations in the term $\\mathbf {J_H}$ could be rewritten to prevent the same sample couple to appear twice.",
"Eq.REF shares a similarity with the contrastive loss [9], [18], [7].",
"This last one is composed of two terms.",
"One term constrains the learned model to project similar inputs to be closer in the output space.",
"In Eq.REF , this is represented by the hint term.",
"In [9], [18], [7], to avoid collapsing all the inputs into one single output point, the contrastive loss uses a second term which projects dissimilar points far from each other by at least a minimal distance.",
"In Eq.REF , the supervised term prevents, implicitly, this collapsing by constraining the extracted representations to be discriminative with respect to each class in order to minimize the classification training error.",
"Implementation and Optimization Details In the present work, we have chosen the cross-entropy as the classification loss $\\mathcal {C}_{sup}(\\cdot , \\cdot )$ .",
"In order to quantify how much two representation vectors in $\\mathbf {Z}$ are dissimilar we proceed using a distance based approach for $\\mathcal {C}_h(\\cdot , \\cdot )$ .",
"We study three different measures: the squared Euclidean distance (SED), $\\mathcal {C}_h(a, b) = \\Vert {a} - {b}\\Vert ^2_2 = \\sum _{v=1}^V ({a}_{ v} - {b}_{v})^2\\hspace{5.0pt},$ the normalized Manhattan distance (NMD), $\\mathcal {C}_h({a}, {b}) = \\frac{1}{V}\\sum _{v=1}^V |{a}_{v} - {b}_{v}|\\hspace{5.0pt},$ and the angular similarity (AS), $\\mathcal {C}_h({a}, {b}) = \\arccos \\left(\\frac{\\langle \\, {a}, {b}\\rangle }{\\Vert {a}\\Vert _2 \\; \\Vert {b}\\Vert _2}\\right) \\hspace{5.0pt}.$ Minimizing the loss function of Eq.REF is achieved using Stochastic Gradient Descent (SGD).",
"Eq.REF can be seen as multi-tasking where two tasks represented by the supervised term and the hint term are in concurrence.",
"One way to minimize Eq.REF is to perform a parallel optimization of both tasks by adding their gradient.",
"Summing up the gradient of both tasks can lead to issues mainly because both tasks have different objectives that do not steer necessarily in the same direction.",
"In order to avoid these issues, we propose to separate the gradients by alternating between the two terms at each mini-batch which showed to work well in practice [8], [45], [11], [5].",
"Moreover, we use two separate optimizers where each term has its own optimizer.",
"By doing so, we make sure that both gradients are separated.",
"On a large dataset, computing all the dissimilarity measures in $\\mathbf {J_H}$ in Eq.REF over the whole training dataset is computationally expensive due to the large number of pairs.",
"Therefore, we propose to compute it only over the mini-batch presented to the network.",
"Consequently, we need to shuffle the training set $\\mathcal {D}$ periodically in order to ensure that the network has seen almost all the possible combinations of the pairs.",
"We describe our implementation in Alg.REF .",
"[h!]",
"Our training strategy [1] $\\mathcal {D}$ is the training set.",
"$B_s$ a mini-batch.",
"$B_r$ a mini-batch of all the possible pairs in $B_s$ (Eq.REF ).",
"$OP_s$ an optimizer of the supervised term.",
"$OP_r$ an optimizer of the dissimilarity term.",
"max_epochs: maximum epochs.",
"$\\gamma , \\lambda $ are regularization weights.",
"i=1..max_epoch Shuffle $\\mathcal {D}$ .",
"Then, split it into mini-batches.",
"$(B_s, B_r)$ in $\\mathcal {D}$ Make a gradient step toward $\\mathbf {J_{sup}}$ using $B_s$ and $OP_s$ .",
"(Eq.REF ) Make a gradient step toward $\\mathbf {J_H}$ using $B_h$ and $OP_r$ .",
"(Eq.REF ) Experiments In this section, we evaluate our regularization framework for training deep networks on a classification task as described in Section .",
"In order to show the effect of using our regularization on the generalization performance, we will mainly compare the generalization error of a network trained with and without our regularizer on different benchmarks of classification problems.",
"Classification Problems and Experimental Methodology In our experiments, we consider three classification problems.",
"We start by the standard MNIST digit dataset.",
"Then, we complicate the classification task by adding different types of noise.",
"We consider the three following problems: The standard MNIST digit classification problem with $\\mathit {50000}$ , $\\mathit {10000}$ and $\\mathit {10000}$ training, validation and test set.",
"We refer to this benchmark as mnist-std.",
"(Fig.REF , top row).",
"MNIST digit classification problem where we use a background mask composed of a random noise followed by a uniform filter.",
"The dataset is composed of $\\mathit {100000}$ , $\\mathit {20000}$ and $\\mathit {50000}$ samples for train, validation and test set.",
"Each set is generated from the corresponding set in the benchmark mnist-std.",
"We refer to this benchmark as mnist-noise.",
"(Fig.REF , middle row).",
"MNIST digit classification problem where we use a background mask composed of a random picture taken from CIFAR-10 dataset [23].",
"This benchmark is composed of $\\mathit {100000}$ samples for training built upon $\\mathit {40000}$ training samples of CIFAR-10 training set, $\\mathit {20000}$ samples for validation built upon the rest of CIFAR-10 training set (i.e.",
"$\\mathit {10000}$ samples) and $\\mathit {50000}$ samples for test built upon the $\\mathit {10000}$ test samples of CIFAR-10.",
"We refer to this benchmark as mnist-img.",
"(Fig.REF , bottom row).",
"Figure: Samples from training set of each benchmark.",
"Top row: mnist-std benchmark.",
"Middle row: mnist-noise benchmark.",
"Bottom row: mnist-img benchmark.All the images are $28 \\times 28$ gray-scale values scaled to $[0, 1]$ .",
"In order to study the behavior of our proposal where we have few training samples, we use different configurations for the training set size.",
"We consider four configurations where we take only $\\mathit {1000}$ , $\\mathit {3000}$ , $\\mathit {5000}$ , $\\mathit {50000}$ or $\\mathit {100000}$ training samples from the whole available training set.",
"We refer to each configuration by $\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ , $\\mathit {50k}$ and $\\mathit {100k}$ respectively.",
"For the benchmark mnist-std, only the configurations $\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ and $\\mathit {50k}$ are considered.",
"For all the experiments, we consider the two following neural network architectures: Multilayer perceptron with 3 hidden layers followed by a classification output layer.",
"We use the same architecture as in [13] which is $1200-1200-200$ .",
"This model is referred to as mlp.",
"LeNet convolutional network [27], which is well known in computer vision tasks, (with similar architecture to LeNet-4) with 2 convolution layers with 20 and 50 filters of size $5 \\times 5$ , followed by a dense layer of size 500, followed by a classification output layer.",
"This model is referred to as lenet.",
"Each model has three hidden layers, we refer to each layer from the input toward the output layer by: $h_1, h_2$ and $h_3$ respectively.",
"The output layer is referred to as $h_4$ .",
"When using our hint term, we refer to the model by mlp + hint and lenet + hint for the mlp and lenet models respectively.",
"Each experiment is repeated 7 times.",
"The best and the worst test classification error cases are discarded.",
"We report the mean $\\pm $ standard deviation of the validation (vl) and the test (tst) classification error of each benchmark.",
"Models without regularization are trained for $\\mathit {2000}$ epochs.",
"All the models regularized with our proposal are trained for $\\mathit {400}$ epochs which we found enough to converge and find a better model over the validation set.",
"All the trainings are performed using stochastic gradient descent with an adaptive learning rate applied using AdaDelta [50], with a batch size of $\\mathit {100}$ .",
"Technical Details: We found that layers with bounded activation functions such as the logistic sigmoid or the hyperbolic tangent function are more suitable when applying our hint term.",
"Applying the regularization term over a layer with unbounded activation function such as the Relu [30] did not show an improvement.",
"In practice, we found that setting $\\gamma =1, \\lambda =1$ works well.",
"The source code of our implementation is freely available https://github.com/sbelharbi/learning-class-invariant-features.",
"Results As we have described in Sec., our hint term can be applied at any hidden layer of the network.",
"In this section, we perform a set of experiments in order to have an idea which one is more adequate to use our regularization.",
"To do so, we trained the mlp model for classification task over the benchmark mnist-std using different configurations with and without regularization.",
"The regularization is applied for one hidden layer at a time $h_1, h_2$ or $h_3$ .",
"We used the squared Euclidean distance (Eq.REF ) as a dissimilarity measure.",
"The obtained results are presented in Tab.REF .",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmark mnist-std using the model mlp and the SED as dissimilarity measure over the different hidden layers: h 1 ,h 2 ,h 3 h_1, h_2, h_3.",
"(bold font indicates lowest error.",
")From Tab.REF , one can see that regularizing low layers $h_1, h_2$ did not help improving the performance error but it did increase it in the configuration $1k$ , for instance.",
"This may be explained by the fact that low layers in neural networks tend to learn low representations which are shared among high representations.",
"This means that these representations are not ready yet to discriminate between the classes.",
"Therefore, they can not be used to describe each class separately.",
"This makes our regularization inadequate at these levels because we aim at constraining the representations to be similar within each class while these layers are incapable to deliver such representations.",
"Therefore, regularizing these layers may hamper their learning.",
"As a future work, we think that it would be beneficial to use at low layers a regularization term that constrains the representations of samples within different classes be dissimilar such as the one in the contrastive loss [9], [18], [7].",
"In the case of regularizing the last hidden layer $h_3$ , we notice from Tab.REF an important improvement in the classification error over the validation and the test set in most configurations.",
"This may be explained by the fact that the representations at this layer are more abstract, therefore, they are able to discriminate the classes.",
"Our regularization term constrains these representations to be tighter by re-enforcing their invariance which helps in generalization.",
"Therefore, applying our hint term over the last hidden layer makes more sense and supports the idea that high layers in neural networks learn more abstract representations.",
"Making these discriminative representations invariant helps the linear output layer in the classification task.",
"For all the following experiments, we apply hint term over the last hidden layer.",
"Moreover, one can notice that our regularization has less impact when adding more training samples.",
"For instance, we reduced the classification test error by: $1.74\\%$ , $0.92\\%$ and $0.58\\%$ in the configurations $1k$ , $3k$ and $5k$ .",
"This suggests that our proposal is more efficient in the case where few training samples are available.",
"However, this does not exclude using it for large training datasets as we will see later (Tab.REF , REF ).",
"We believe that this behavior depends mostly on the model's capacity to learn invariant representations.",
"For instance, from the invariance perspective, convolutional networks are more adapted, conceptually, to process visual content than multilayers perceptrons.",
"In another experimental setup, we investigated the effect of the measure used to compute the dissimilarity between two feature vectors as described in Section.REF .",
"To do so, we applied our hint term over the last hidden layer $h_3$ using the measures SED, NMD and AS over the benchmark mnist-std.",
"The obtained results are presented in Tab.REF .",
"These results show that the squared Euclidean distance performs significantly better than the other measures and has more stability when changing the number of training samples ($\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ , $\\mathit {50k}$ ) or the model (mlp, lenet).",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmark mnist-std using different dissimilarity measures (SED, NMD, AS) over the layer h 3 h_3.",
"(bold font indicates lowest error.",
")In another experiment, we evaluated the benchmarks mnist-noise and mnist-img, which are more difficult compared to mnist-std, using the model lenet which is more suitable to process visual content.",
"Similarly to the previous experiments, we applied our regularization term over the last hidden layer $h_3$ using the SED measure.",
"The results depicted in Tab.REF show again that using our proposal improves the generalization error of the network particularly when only few training samples are available.",
"For example, our regularization allows to reduce the classification error over the test set by $2.98\\%$ and by $4.16\\%$ over the benchmark mnist-noise and mnist-img, respectively when using only $\\mathit {1k}$ training samples.",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmarks mnist-noise and mnist-img using lenet model (regularization applied over the layer h 3 h_3).",
"(bold font indicates lowest error.",
")Based on the above results, we conclude that using our hint term in the context of classification task using neural networks is helpful in improving their generalization error particularly when only few training samples are available.",
"This generalization improvement came at the price of an extra computational cost due the dissimilarity measures between pair of samples.",
"Our experiments showed that regularizing the last hidden layer using the squared Euclidean distance give better results.",
"More generally, the obtained results confirm that guiding the learning process of the intermediate representations of a neural network can be helpful to improve its generalization.",
"On Learning Invariance within Neural Networks We show in this section an intriguing property of the learned representations at each layer of a neural network from the invariance perspective.",
"For this purpose and for the sake of simplicity, we consider a binary classification case of the two digits “1” and “7”.",
"Furthermore, we consider the mlp model over the lenet in order to be able to measure the features invariances over all the layers.",
"We trained the mlp model over the benchmark mnist-std where we used all the available training samples of both digits.",
"The model is trained without our regularization.",
"However, we tracked, at each layer and at the same time, the value of the hint term $\\mathbf {J_H}$ in Eq.REF over the training set using the normalized Manhattan distance as a dissimilarity measure.",
"This particular dissimilarity measure allows comparing the representations invariance between the different layers due to the normalization of the measure by the representations dimension.",
"The obtained results are depicted in Fig.REF where the x-axis represents the number of mini-batches already processed and the y-axis represents the value of the hint term $\\mathbf {J_H}$ at each layer.",
"Low value of $\\mathbf {J_H}$ means high invariance (better case) whereas high value of $\\mathbf {J_H}$ means low invariance.",
"Figure: Measuring the hint term 𝐉 𝐇 \\mathbf {J_H} of Eq.",
"over the training set within each layer (simultaneously) of the mlp over the train set of mnist-std benchmark for a binary classification task: the digit “1” against the digit “7”.In Fig.REF , we note two main observations: The value of the hint term $\\mathbf {J_H}$ is reduced through the depth of the network which means that the network learns more invariant representations at each layer in this order: layer 1, 2, 3, 4.",
"This result supports the idea that abstract representations, which are known to be more invariant, are learned toward the top layers.",
"At each layer, the network does not seem to learn to improve the invariance of the learned representations by reducing $\\mathbf {J_H}$ .",
"It appears that the representations invariance is kept steady all along the training process.",
"Only the output layer has learned to reduce the value of $\\mathbf {J_H}$ term because minimizing the classification term $\\mathbf {J_{sup}}$ reduces automatically our hint term $\\mathbf {J_H}$ .",
"This shows a flaw in the back-propagation procedure with respect to learning intermediate representations.",
"Assisting the propagated error through regularization can be helpful to guide the hidden layers to learn more suitable representations.",
"These results show that relying on the classification error propagated from the output layer does not necessarily constrain the hidden layers to learn better representations for classification task.",
"Therefore, one would like to use different prior knowledge to guide the internal layers to learn better representations which is our future work.",
"Using these guidelines can help improving neural networks generalization especially when trained with few samples.",
"Conclusion We have presented in this work a new regularization framework for training neural networks for classification task.",
"Our regularization constrains the hidden layers of the network to learn class-wise invariant representations where samples of the same class have the same representation.",
"Empirical results over MNIST dataset and its variants showed that the proposed regularization helps neural networks to generalize better particularly when few training samples are available which is the case in many real world applications.",
"Another result based on tracking the representation invariance within the network layers confirms that neural networks tend to learn invariant representations throughout staking multiple layers.",
"However, an intriguing observation is that the invariance level does not seem to be improved, within the same layer, through learning.",
"We found that the hidden layers tend to maintain a certain level of invariance through the training process.",
"All the results found in this work suggest that guiding the learning process of the internal representations of a neural network can be helpful to train them and improve their generalization particularly when few training samples are available.",
"Furthermore, this shows that the classification error propagated from the output layer does not necessarily train the hidden layers to provide better representations.",
"This encourages us to explore other directions to incorporate different prior knowledge to constrain the hidden layers to learn better representations in order to improve the generalization of the network and be able to train it with less data.",
"Acknowledgments This work has been partly supported by the grant ANR-16-CE23-0006 “Deep in France” and benefited from computational means from CRIANN, the contributions of which are greatly appreciated."
],
[
"Experiments",
"In this section, we evaluate our regularization framework for training deep networks on a classification task as described in Section .",
"In order to show the effect of using our regularization on the generalization performance, we will mainly compare the generalization error of a network trained with and without our regularizer on different benchmarks of classification problems."
],
[
"Classification Problems and Experimental Methodology",
"In our experiments, we consider three classification problems.",
"We start by the standard MNIST digit dataset.",
"Then, we complicate the classification task by adding different types of noise.",
"We consider the three following problems: The standard MNIST digit classification problem with $\\mathit {50000}$ , $\\mathit {10000}$ and $\\mathit {10000}$ training, validation and test set.",
"We refer to this benchmark as mnist-std.",
"(Fig.REF , top row).",
"MNIST digit classification problem where we use a background mask composed of a random noise followed by a uniform filter.",
"The dataset is composed of $\\mathit {100000}$ , $\\mathit {20000}$ and $\\mathit {50000}$ samples for train, validation and test set.",
"Each set is generated from the corresponding set in the benchmark mnist-std.",
"We refer to this benchmark as mnist-noise.",
"(Fig.REF , middle row).",
"MNIST digit classification problem where we use a background mask composed of a random picture taken from CIFAR-10 dataset [23].",
"This benchmark is composed of $\\mathit {100000}$ samples for training built upon $\\mathit {40000}$ training samples of CIFAR-10 training set, $\\mathit {20000}$ samples for validation built upon the rest of CIFAR-10 training set (i.e.",
"$\\mathit {10000}$ samples) and $\\mathit {50000}$ samples for test built upon the $\\mathit {10000}$ test samples of CIFAR-10.",
"We refer to this benchmark as mnist-img.",
"(Fig.REF , bottom row).",
"Figure: Samples from training set of each benchmark.",
"Top row: mnist-std benchmark.",
"Middle row: mnist-noise benchmark.",
"Bottom row: mnist-img benchmark.All the images are $28 \\times 28$ gray-scale values scaled to $[0, 1]$ .",
"In order to study the behavior of our proposal where we have few training samples, we use different configurations for the training set size.",
"We consider four configurations where we take only $\\mathit {1000}$ , $\\mathit {3000}$ , $\\mathit {5000}$ , $\\mathit {50000}$ or $\\mathit {100000}$ training samples from the whole available training set.",
"We refer to each configuration by $\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ , $\\mathit {50k}$ and $\\mathit {100k}$ respectively.",
"For the benchmark mnist-std, only the configurations $\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ and $\\mathit {50k}$ are considered.",
"For all the experiments, we consider the two following neural network architectures: Multilayer perceptron with 3 hidden layers followed by a classification output layer.",
"We use the same architecture as in [13] which is $1200-1200-200$ .",
"This model is referred to as mlp.",
"LeNet convolutional network [27], which is well known in computer vision tasks, (with similar architecture to LeNet-4) with 2 convolution layers with 20 and 50 filters of size $5 \\times 5$ , followed by a dense layer of size 500, followed by a classification output layer.",
"This model is referred to as lenet.",
"Each model has three hidden layers, we refer to each layer from the input toward the output layer by: $h_1, h_2$ and $h_3$ respectively.",
"The output layer is referred to as $h_4$ .",
"When using our hint term, we refer to the model by mlp + hint and lenet + hint for the mlp and lenet models respectively.",
"Each experiment is repeated 7 times.",
"The best and the worst test classification error cases are discarded.",
"We report the mean $\\pm $ standard deviation of the validation (vl) and the test (tst) classification error of each benchmark.",
"Models without regularization are trained for $\\mathit {2000}$ epochs.",
"All the models regularized with our proposal are trained for $\\mathit {400}$ epochs which we found enough to converge and find a better model over the validation set.",
"All the trainings are performed using stochastic gradient descent with an adaptive learning rate applied using AdaDelta [50], with a batch size of $\\mathit {100}$ .",
"Technical Details: We found that layers with bounded activation functions such as the logistic sigmoid or the hyperbolic tangent function are more suitable when applying our hint term.",
"Applying the regularization term over a layer with unbounded activation function such as the Relu [30] did not show an improvement.",
"In practice, we found that setting $\\gamma =1, \\lambda =1$ works well.",
"The source code of our implementation is freely available https://github.com/sbelharbi/learning-class-invariant-features."
],
[
"Results",
"As we have described in Sec., our hint term can be applied at any hidden layer of the network.",
"In this section, we perform a set of experiments in order to have an idea which one is more adequate to use our regularization.",
"To do so, we trained the mlp model for classification task over the benchmark mnist-std using different configurations with and without regularization.",
"The regularization is applied for one hidden layer at a time $h_1, h_2$ or $h_3$ .",
"We used the squared Euclidean distance (Eq.REF ) as a dissimilarity measure.",
"The obtained results are presented in Tab.REF .",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmark mnist-std using the model mlp and the SED as dissimilarity measure over the different hidden layers: h 1 ,h 2 ,h 3 h_1, h_2, h_3.",
"(bold font indicates lowest error.",
")From Tab.REF , one can see that regularizing low layers $h_1, h_2$ did not help improving the performance error but it did increase it in the configuration $1k$ , for instance.",
"This may be explained by the fact that low layers in neural networks tend to learn low representations which are shared among high representations.",
"This means that these representations are not ready yet to discriminate between the classes.",
"Therefore, they can not be used to describe each class separately.",
"This makes our regularization inadequate at these levels because we aim at constraining the representations to be similar within each class while these layers are incapable to deliver such representations.",
"Therefore, regularizing these layers may hamper their learning.",
"As a future work, we think that it would be beneficial to use at low layers a regularization term that constrains the representations of samples within different classes be dissimilar such as the one in the contrastive loss [9], [18], [7].",
"In the case of regularizing the last hidden layer $h_3$ , we notice from Tab.REF an important improvement in the classification error over the validation and the test set in most configurations.",
"This may be explained by the fact that the representations at this layer are more abstract, therefore, they are able to discriminate the classes.",
"Our regularization term constrains these representations to be tighter by re-enforcing their invariance which helps in generalization.",
"Therefore, applying our hint term over the last hidden layer makes more sense and supports the idea that high layers in neural networks learn more abstract representations.",
"Making these discriminative representations invariant helps the linear output layer in the classification task.",
"For all the following experiments, we apply hint term over the last hidden layer.",
"Moreover, one can notice that our regularization has less impact when adding more training samples.",
"For instance, we reduced the classification test error by: $1.74\\%$ , $0.92\\%$ and $0.58\\%$ in the configurations $1k$ , $3k$ and $5k$ .",
"This suggests that our proposal is more efficient in the case where few training samples are available.",
"However, this does not exclude using it for large training datasets as we will see later (Tab.REF , REF ).",
"We believe that this behavior depends mostly on the model's capacity to learn invariant representations.",
"For instance, from the invariance perspective, convolutional networks are more adapted, conceptually, to process visual content than multilayers perceptrons.",
"In another experimental setup, we investigated the effect of the measure used to compute the dissimilarity between two feature vectors as described in Section.REF .",
"To do so, we applied our hint term over the last hidden layer $h_3$ using the measures SED, NMD and AS over the benchmark mnist-std.",
"The obtained results are presented in Tab.REF .",
"These results show that the squared Euclidean distance performs significantly better than the other measures and has more stability when changing the number of training samples ($\\mathit {1k}$ , $\\mathit {3k}$ , $\\mathit {5k}$ , $\\mathit {50k}$ ) or the model (mlp, lenet).",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmark mnist-std using different dissimilarity measures (SED, NMD, AS) over the layer h 3 h_3.",
"(bold font indicates lowest error.",
")In another experiment, we evaluated the benchmarks mnist-noise and mnist-img, which are more difficult compared to mnist-std, using the model lenet which is more suitable to process visual content.",
"Similarly to the previous experiments, we applied our regularization term over the last hidden layer $h_3$ using the SED measure.",
"The results depicted in Tab.REF show again that using our proposal improves the generalization error of the network particularly when only few training samples are available.",
"For example, our regularization allows to reduce the classification error over the test set by $2.98\\%$ and by $4.16\\%$ over the benchmark mnist-noise and mnist-img, respectively when using only $\\mathit {1k}$ training samples.",
"Table: Mean ±\\pm standard deviation error over validation and test set of the benchmarks mnist-noise and mnist-img using lenet model (regularization applied over the layer h 3 h_3).",
"(bold font indicates lowest error.",
")Based on the above results, we conclude that using our hint term in the context of classification task using neural networks is helpful in improving their generalization error particularly when only few training samples are available.",
"This generalization improvement came at the price of an extra computational cost due the dissimilarity measures between pair of samples.",
"Our experiments showed that regularizing the last hidden layer using the squared Euclidean distance give better results.",
"More generally, the obtained results confirm that guiding the learning process of the intermediate representations of a neural network can be helpful to improve its generalization."
],
[
"On Learning Invariance within Neural Networks",
"We show in this section an intriguing property of the learned representations at each layer of a neural network from the invariance perspective.",
"For this purpose and for the sake of simplicity, we consider a binary classification case of the two digits “1” and “7”.",
"Furthermore, we consider the mlp model over the lenet in order to be able to measure the features invariances over all the layers.",
"We trained the mlp model over the benchmark mnist-std where we used all the available training samples of both digits.",
"The model is trained without our regularization.",
"However, we tracked, at each layer and at the same time, the value of the hint term $\\mathbf {J_H}$ in Eq.REF over the training set using the normalized Manhattan distance as a dissimilarity measure.",
"This particular dissimilarity measure allows comparing the representations invariance between the different layers due to the normalization of the measure by the representations dimension.",
"The obtained results are depicted in Fig.REF where the x-axis represents the number of mini-batches already processed and the y-axis represents the value of the hint term $\\mathbf {J_H}$ at each layer.",
"Low value of $\\mathbf {J_H}$ means high invariance (better case) whereas high value of $\\mathbf {J_H}$ means low invariance.",
"Figure: Measuring the hint term 𝐉 𝐇 \\mathbf {J_H} of Eq.",
"over the training set within each layer (simultaneously) of the mlp over the train set of mnist-std benchmark for a binary classification task: the digit “1” against the digit “7”.In Fig.REF , we note two main observations: The value of the hint term $\\mathbf {J_H}$ is reduced through the depth of the network which means that the network learns more invariant representations at each layer in this order: layer 1, 2, 3, 4.",
"This result supports the idea that abstract representations, which are known to be more invariant, are learned toward the top layers.",
"At each layer, the network does not seem to learn to improve the invariance of the learned representations by reducing $\\mathbf {J_H}$ .",
"It appears that the representations invariance is kept steady all along the training process.",
"Only the output layer has learned to reduce the value of $\\mathbf {J_H}$ term because minimizing the classification term $\\mathbf {J_{sup}}$ reduces automatically our hint term $\\mathbf {J_H}$ .",
"This shows a flaw in the back-propagation procedure with respect to learning intermediate representations.",
"Assisting the propagated error through regularization can be helpful to guide the hidden layers to learn more suitable representations.",
"These results show that relying on the classification error propagated from the output layer does not necessarily constrain the hidden layers to learn better representations for classification task.",
"Therefore, one would like to use different prior knowledge to guide the internal layers to learn better representations which is our future work.",
"Using these guidelines can help improving neural networks generalization especially when trained with few samples."
],
[
"Conclusion",
"We have presented in this work a new regularization framework for training neural networks for classification task.",
"Our regularization constrains the hidden layers of the network to learn class-wise invariant representations where samples of the same class have the same representation.",
"Empirical results over MNIST dataset and its variants showed that the proposed regularization helps neural networks to generalize better particularly when few training samples are available which is the case in many real world applications.",
"Another result based on tracking the representation invariance within the network layers confirms that neural networks tend to learn invariant representations throughout staking multiple layers.",
"However, an intriguing observation is that the invariance level does not seem to be improved, within the same layer, through learning.",
"We found that the hidden layers tend to maintain a certain level of invariance through the training process.",
"All the results found in this work suggest that guiding the learning process of the internal representations of a neural network can be helpful to train them and improve their generalization particularly when few training samples are available.",
"Furthermore, this shows that the classification error propagated from the output layer does not necessarily train the hidden layers to provide better representations.",
"This encourages us to explore other directions to incorporate different prior knowledge to constrain the hidden layers to learn better representations in order to improve the generalization of the network and be able to train it with less data."
],
[
"Acknowledgments",
"This work has been partly supported by the grant ANR-16-CE23-0006 “Deep in France” and benefited from computational means from CRIANN, the contributions of which are greatly appreciated."
]
] | 1709.01867 |
[
[
"Complex THz and DC inverse spin Hall effect in YIG/Cu$_{1-x}$Ir$_{x}$\n bilayers across a wide concentration range"
],
[
"Abstract We measure the inverse spin Hall effect of Cu$_{1-x}$Ir$_{x}$ thin films on yttrium iron garnet over a wide range of Ir concentrations ($0.05 \\leqslant x \\leqslant 0.7$).",
"Spin currents are triggered through the spin Seebeck effect, either by a DC temperature gradient or by ultrafast optical heating of the metal layer.",
"The spin Hall current is detected by, respectively, electrical contacts or measurement of the emitted THz radiation.",
"With both approaches, we reveal the same Ir concentration dependence that follows a novel complex, non-monotonous behavior as compared to previous studies.",
"For small Ir concentrations a signal minimum is observed, while a pronounced maximum appears near the equiatomic composition.",
"We identify this behavior as originating from the interplay of different spin Hall mechanisms as well as a concentration-dependent variation of the integrated spin current density in Cu$_{1-x}$Ir$_{x}$.",
"The coinciding results obtained for DC and ultrafast stimuli show that the studied material allows for efficient spin-to-charge conversion even on ultrafast timescales, thus enabling a transfer of established spintronic measurement schemes into the terahertz regime."
],
[
"Introduction",
"Spin currents are a promising ingredient for the implementation of next-generation, energy-efficient spintronic applications.",
"Instead of exploiting the electronic charge, transfer as well as processing of information is mediated by spin angular momentum.",
"Crucial steps towards the realization of spintronic devices are the efficient generation, manipulation and detection of spin currents at highest speeds possible.",
"Here, the spin Hall effect (SHE) and its inverse (ISHE) are in the focus of current research [1] as they allow for an interconversion of spin and charge currents in heavy metals with strong spin-orbit interaction (SOI).",
"The efficiency of this conversion is quantified by the spin Hall angle $\\theta _{\\mathrm {SH}}$ .",
"In general, the SHE has intrinsic as well as extrinsic spin-dependent contributions.",
"The intrinsic SHE results from a momentum-space Berry phase effect and can, amongst others, be observed in 4$d$ and 5$d$ transition metals [2], [3], [1].",
"The extrinsic SHE, on the other hand, is a consequence of skew and side-jump scattering off impurities or defects [4].",
"It occurs in (dilute) alloys of normal metals with strong SOI impurity scatterers [5], [6], [7], [8], but can also be prominent in pure metals in the superclean regime [9].",
"As a consequence, the type of employed metals and the alloy composition are handles to adjust and maximize the SHE.",
"Remarkably, it was recently shown that the SHE in alloys of two heavy metals (e.g.",
"AuPt) can even exceed the SHE observed for the single alloy partners [10].",
"Pioneering work within this research field covered the extrinsic SHE by skew scattering in copper-iridium alloys [5].",
"However, previously the iridium concentration was limited to 12 effective doping of Cu with dilute Ir.",
"The evolution of the SHE in the alloy regime for large concentration thus remains an open question and the achievable maximum by an optimized alloying strategy is unknown.",
"The potential of a metal for spintronic applications (i.e.",
"$\\theta _{\\mathrm {SH}}$ ) can be quantified by injecting a spin current and measuring the resulting charge response.",
"This can be accomplished by, for instance, coherent spin pumping through ferromagnetic resonance [11], [12], [13] or the spin Seebeck effect (SSE) [14], [15].",
"The SSE describes the generation of a magnon spin current along a temperature gradient within a magnetic material.",
"Typically, such experiments involve a heterostructure composed of a magnetic insulator, such as yttrium iron garnet (YIG), and the ISHE metal under study [see Fig.",
"REF (a)].",
"A DC temperature gradient in the YIG bulk is induced by heating the sample from one side.",
"On the femtosecond timescale, however, a temperature difference and thus a spin current across the YIG-metal interface can be induced by heating the metal layer with an optical laser pulse [Fig.",
"REF (b)] [16], [17], [18], [19].",
"This interfacial SSE has been shown to dominate the spin current in the metal on timescales below $\\sim 300$ ns [16].",
"For ultrafast laser excitation, the resulting sub-picosecond ISHE current leads to the emission of electromagnetic pulses at frequencies extending into the terahertz (THz) range, which can be detected by optical means [20].",
"Therefore, femtosecond laser excitation offers the remarkable benefit of contact-free measurements of the ISHE current without any need of micro-structuring the sample.",
"The all-optical generation as well as detection of ultrafast electron spin currents [20], [21] is a key requirement for transferring spintronic concepts into the THz range [22].",
"So far, however, characterization of the ISHE was conducted by experiments including DC spin current signals as, for instance, the bulk SSE [Fig.",
"REF (a)].",
"For the use in ultrafast applications, it thus remains to be shown whether alloying yields the same notable changes of the spin-to-charge conversion efficiency in THz interfacial SSE experiments [Fig.",
"REF (b)] and whether alloys can provide an efficient spin-to-charge conversion even at the ultrafast timescale.",
"In this work, we study the compositional dependence of the ISHE in YIG/Cu$_{1-x}$ Ir$_{x}$ bilayers over a wide concentration range ($0.05 \\leqslant x \\leqslant 0.7$ ), exceeding the dilute doping phase investigated in previous studies [5].",
"The ISHE response of Cu$_{1-x}$ Ir$_{x}$ is measured as a function of $x$ , for which both DC bulk and THz interfacial SSE are employed.",
"Eventually, we compare the spin-to-charge conversion efficiency in the two highly distinct regimes of DC and terahertz dynamics across a wide alloying range."
],
[
"Experiment",
"The YIG samples used for this study are of 870 thickness, grown epitaxially on (111)-oriented Gd3Ga5O12 (GGG) substrates by liquid-phase-epitaxy.",
"After cleaving the GGG/YIG into samples of dimension 2.5 x 10 x 0.5, Cu$_{1-x}$ Ir$_{x}$ thin films (thickness $d_{\\mathrm {CuIr}} = {4}{}$ ) of varying composition ($x = 0.05, 0.1, 0.2, 0.3, 0.5$ and 0.7) are deposited by multi-source magnetron sputtering.",
"To prevent oxidation of the metal film, a 3 Al capping layer is deposited, which, when exposed to air, forms an AlO$_{x}$ protection layer.",
"For the contact-free ultrafast SSE measurements, patterning of the Cu$_{1-x}$ Ir$_{x}$ films into defined nanostructures is not necessary.",
"In the case of DC SSE measurements, the unpatterned film is contacted for the detection of the thermal voltage.",
"Figure: (a) Scheme of the setup used for DC SSE measurements.",
"The out-of-plane temperature gradient isgenerated by two copper blocks set to individual temperatures T 1 T_1 and T 2 T_2.",
"An external magneticfield is applied in the sample plane.",
"The resulting thermovoltage V t V_{\\mathrm {t}} is recorded by a nanovoltmeter.",
"(b) Scheme of the contact-free ultrafast SSE/ISHE THz emission approach.",
"The in-plane magnetizedsample is illuminated by a femtosecond laser pulse, inducing a step-like temperature gradientacross the YIG/Cu 1-x _{1-x}Ir x _{x} interface.",
"The SSE-induced THz spin current in the CuIr layer issubsequently converted into a sub-picosecond in-plane charge current by the ISHE, thereby leadingto the emission of a THz electromagnetic pulse into the optical far-field.The DC SSE measurements are performed at room temperature in the conventional longitudinal configuration [15].",
"While an external magnetic field is applied in the sample plane, two copper blocks, which can be set to individual temperatures, generate a static out-of-plane temperature gradient, see Fig.",
"REF (a).",
"This thermal perturbation results in a magnonic spin current in the YIG layer [23], thereby transferring angular momentum into the Cu$_{1-x}$ Ir$_{x}$ .",
"A spin accumulation builds up, diffuses as a pure spin current and is eventually converted into a transverse charge current by means of the ISHE, yielding a measurable voltage signal.",
"The spin current and consequently the thermal voltage change sign when the YIG magnetization is reversed.",
"The SSE voltage $V$SSE is defined as the difference between the voltage signals obtained for positive and negative magnetic field divided by 2.",
"Since $V$SSE is the result of the continuous conversion of a steady spin current, it can, applying the notation of conventional electronics, be considered as a DC signal.",
"For the THz SSE measurements, the same in-plane magnetized YIG/Cu$_{1-x}$ Ir$_{x}$ samples are illuminated at room temperature by femtosecond laser pulses (energy of 2.5, duration of 10, center wavelength of ${800}{}$ corresponding to a photon energy of ${1.55}{}$ , repetition rate of 80) of a Ti:sapphire laser oscillator.",
"Owing to its large bandgap of ${2.6}{}$ [24], YIG is transparent for these laser pulses.",
"They are, however, partially (about 50) absorbed by the electrons of the Cu$_{1-x}$ Ir$_{x}$ layer.",
"The spatially step-like temperature gradient across the YIG/metal interface leads to an ultrafast spin current in the metal layer polarized parallel to the sample magnetization [19].",
"Subsequently, this spin current is converted into a transverse sub-picosecond charge current through the ISHE, resulting in the emission of a THz electromagnetic pulse into the optical far-field.",
"The THz electric field is sampled using a standard electrooptical detection scheme employing a 1 thick ZnTe detection crystal [25].",
"The magnetic response of the system is quantified by the root mean square (RMS) of half the THz signal difference $S$ for positive and negative magnetic fields."
],
[
"Results",
"Figure REF (a)-(f) shows DC SSE hysteresis loops measured for YIG/Cu$_{1-x}$ Ir$_{x}$ /AlOx multilayers with varying Ir concentration $x$ .",
"The temperature difference between sample top and bottom is fixed to $\\Delta T = {10}{}$ with a base temperature of $T = {288.15}{}$ .",
"In the Cu-rich phase, we observe an increase of the thermal voltage signal with increasing $x$ , exhibiting a maximum at $x = 0.3$ .",
"Interestingly, upon further increasing the Ir content $V_{\\mathrm {SSE}}$ reduces again.",
"This behavior is easily visible in Fig.",
"REF (g), in which the SSE coefficient $V_{\\mathrm {SSE}} / \\Delta T$ is plotted as a function of $x$ .",
"The measured concentration dependence shows that $V_{\\mathrm {SSE}}/\\Delta T$ exhibits a clear maximum in the range from $x=0.3$ to $0.5$ .",
"Thus, as a first key result the maximum spin Hall effect is obtained for the previously neglected alloying regime beyond the dilute doping.",
"For comparison, the resistivity $\\sigma ^{-1}$ of the metal film is also shown in Fig.",
"REF (g).",
"We see that the resistivity of the Cu$_{1-x}$ Ir$_{x}$ layer follows a similar trend as the DC SSE signal.",
"Figure: (a)-(f) Measured DC SSE voltage in YIG/Cu 1-x _{1-x}Ir x _{x}/AlOx stacks for different Irconcentrations xx in ascending order.",
"The temperature difference between sample top and bottom isfixed to ΔT=10\\Delta T = {10}{}.",
"(g) SSE coefficient V SSE /ΔTV_{\\mathrm {SSE}} / \\Delta T (redsquares) and resistivity σ -1 \\sigma ^{-1} (blue circles) as a function of Ir concentration xx.Typical THz emission signals from the YIG/Cu$_{1-x}$ Ir$_{x}$ /AlOx samples are depicted in Figs.",
"REF (a)-(f).",
"The THz transients were low-pass filtered in the frequency domain with a Gaussian centered at zero frequency and a full width at half maximum of 20.",
"The RMS of the THz signal odd in sample magnetization is plotted in Fig.",
"REF (g) as a function of $x$ .",
"After an initial signal drop in the Cu-rich phase, the THz signal increases with increasing Ir concentration, indicating a signal maximum in the range between $x = {0.3}$ and ${0.5}$ .",
"Further increase of the Ir content leads to a second reduction of the THz signal strength.",
"Figure: (a)-(f) Signal waveforms (odd in the sample magnetization) of the THz pulses emitted fromYIG/Cu 1-x _{1-x}Ir x _{x}/AlOx stacks for different Ir concentrations xx in ascending order.",
"(g) THzsignal strength (RMS) as a function of Ir concentration xx."
],
[
"Discussion",
"In the following, a direct comparison of the signals obtained from the DC and the ultrafast THz measurements is established.",
"To begin with, the emitted THz electric field right behind the sample is described by a generalized Ohm's law, which in the thin-film limit (film is much thinner than the wavelength and attenuation length of the THz wave in the sample) is in the frequency domain given by [21] $\\tilde{E}(\\omega ) \\propto \\theta _{\\mathrm {SH}}Z(\\omega )\\int _{0}^{d}\\mathrm {d}z \\,j_{\\mathrm {s}}(z,\\omega ) ,$ where $\\omega $ is the angular frequency.",
"The spin-current density $j_{\\mathrm {s}}(z,\\omega )$ is integrated over the full thickness $d$ of the metal film.",
"The total impedance $Z(\\omega )$ can be understood as the impedance of an equivalent parallel circuit comprising the metal film (Cu$_{1-x}$ Ir$_{x}$ ) and the surrounding substrate (GGG/YIG) and air half-spaces, $\\frac{1}{Z(\\omega )} = \\frac{n_1( \\omega ) + n_2 ( \\omega )}{Z_0} + G(\\omega ).$ Here, $n_1$ and $n_2 \\approx 1$ are the refractive indices of substrate and air, respectively, $Z_0= {377}{}$ is the vacuum impedance, and $G(\\omega )$ is the THz sheet conductance of the Cu$_{1-x}$ Ir$_{x}$ films.",
"Considering the Drude model and a velocity relaxation rate of ${28}{}$ for pure Cu at room temperature as lower boundary [26], the values of $G(\\omega )$ vary only slightly over the detected frequency range from 1 to 5 THz (as given by the ZnTe detector crystal).",
"Therefore, the frequency dependence of the conductance can be neglected, i.e.",
"$G(\\omega ) \\approx G(\\omega =0)$ .",
"Importantly, the metal-film conductance ($G\\approx {8e-3}{}$ ) is much smaller than the shunt conductance ($\\left[n_1(\\omega ) +n_2(\\omega )\\right]/Z_0 \\approx {4e-2}{}$ ) for the investigated metal film thickness ($d= {4}{}$ ) and can be thus neglected.",
"Therefore, the Ir-concentration influences the THz emission strength only directly through the ISHE-induced in-plane charge current flowing inside the NM layer.",
"The measured DC SSE voltage, on the other hand, is given by an analogous expression related to the underlying in-plane charge current by the standard Ohm's law, $\\frac{V_{\\mathrm {SSE}}}{\\Delta T} \\propto \\theta _{\\mathrm {SH}}R \\int _{0}^{d}\\mathrm {d}z \\,j_{\\mathrm {s}}(z).",
"$ Here, $R$ is the Ohmic resistance of the metal layer between the electrodes, which is inversely proportional to the metal resistivity $\\sigma $ , and $j_{\\mathrm {s}}(z)$ is the DC spin current density.",
"Therefore, in contrast to the THz data, the impact of alloying on $V_{\\mathrm {SSE}}$ through $\\sigma ^{-1}$ is significant.",
"For a direct comparison with the THz measurements, we thus contrast the RMS of the THz signal waveform with the DC SSE current density $j_{\\mathrm {SSE}} = V_{\\mathrm {SSE}} \\cdot \\sigma /\\Delta T$ .",
"In Fig.",
"REF , the respective amplitudes are plotted as a function of the Ir concentration.",
"Remarkably, DC and THz SSE/ISHE measurements exhibit the very same concentration dependence.",
"This agreement suggests that the ISHE retains its functionality from DC up to THz frequencies, which vindicates the findings and interpretations of previous experiments [21].",
"Small discrepancies may originate from a varying optical absorptance of the near-infrared pump light, which is, however, expected to depend monotonically on $x$ and to only vary by a few percent [21].",
"Furthermore, as discussed below, these findings imply that for DC and THz spin currents comparable concentration dependences of spin-relaxation lengths may be expected.",
"Figure: Ir concentration dependence of the thermal DC spin current (red squares) and the RMS of the THzsignal (green diamonds).To discuss the concentration dependence of the DC and THz SSE signals (Fig.",
"REF ), we consider Eqs.",
"(REF ) and (REF ).",
"According to these relationships, the THz signal and the SSE voltage normalized by the metal resistivity result from a competition of (i) the spin Hall angle $\\theta _{\\mathrm {SH}}$ and (ii) the integrated spin-current density $\\int _{0}^{d}\\mathrm {d}z j_{\\mathrm {s}}(z,\\omega )$ .",
"At first, we consider the local spin signal minimum at small, increasing Ir concentration $x$ (dilute regime) that appears for both $j_{\\mathrm {SSE}}$ and the THz signal.",
"In fact, with regard to (i) $\\theta _{\\mathrm {SH}}$ one would expect the opposite behavior as for the dilute regime the skew scattering mechanism has been predicted [4] and experimentally shown [5] to yield the dominant ISHE contribution.",
"With increasing SOI scattering center density ($\\rho _{\\mathrm {imp}} \\propto \\sigma ^{-1}$ ), a linear increase of the spin signal should appear.",
"In this work, this trend is observed for $V_{\\mathrm {SSE}}$ [Fig.",
"REF (g)].",
"The significantly deviating signal shapes of $j_{\\mathrm {SSE}}$ and the THz signal, however, suggest that the converted in-plane charge current is notably governed by additional effects.",
"An explanation can be given by (ii), considering a spatial variation of the spin current density that, as we discuss below, can be influenced by both electron momentum- and spin-relaxation.",
"The initial electron momenta and spin information of a directional spin current become randomized over length scales characterized by the mean free path $\\ell $ and the spin diffusion length $\\lambda _{\\mathrm {sd}}$ , yielding a reduction of the spin current density.",
"For spin-relaxation, the integrated spin current density is given by [27]: $\\int _{0}^{\\mathrm {d_{\\mathrm {CuIr}}}} dz j_{\\mathrm {s}} (z) \\propto \\lambda _{\\mathrm {sd}} \\tanh \\left( \\frac{d_{\\mathrm {CuIr}}}{2\\lambda _{\\mathrm {sd}}} \\right) j_{\\mathrm {s}}^0$ with $d_{\\mathrm {CuIr}}$ being the thickness of the Cu1-xIrx layer.",
"According to Niimi et al.",
"[5] the spin-diffusion length $\\lambda _{\\mathrm {sd}}$ decreases exponentially from $\\lambda _{\\mathrm {sd}}\\approx {30}{}$ for $x=0.01$ to $\\lambda _{\\mathrm {sd}}\\approx {5}{}$ for $x=0.12$ .",
"This exponential decay implies that the integrated spin current density is nearly constant for both small and large $x$ , but undergoes a significant decline in the concentration region where $\\lambda _{\\mathrm {sd}} \\approx d_{\\mathrm {CuIr}}$ .",
"This effect possibly explains the observed reduction of the signal amplitude from $x = 0.05$ to $x=0.2$ .",
"Furthermore, we interpret the fact that for DC and THz SSE signals similar trends are observed as an indication of similar concentration dependences of $\\lambda _{\\mathrm {sd}}$ in the distinct DC and THz regimes.",
"This appears reasonable when considering that spin-dependent scattering rates are of the same order of magnitude as the momentum scattering [28] (e.g.",
"$\\Gamma _{\\mathrm {Cu}}^{\\mathrm {mom.}}",
"= {1/36}{} \\approx {28}{}$ [26]) and thus above the experimentally covered bandwidth.",
"In addition to spin-relaxation, the integrated spin current density is influenced by momentum scattering.",
"As shown in Fig.",
"REF , alloying introduces impurities and lattice defects in the dilute phase, such that enhanced momentum scattering rates occur.",
"Assuming that the latter increase more rapidly than $\\theta _{\\mathrm {SH}}$ , the appearance of the previously unexpected local minimum near $x \\approx 0.2$ can be thus explained.",
"We now focus on the subsequent increase of the spin signal at higher $x$ (concentrated phase).",
"It can be explained by a further increase of extrinsic ISHE as well as intrinsic ISHE contributions, as pure Ir itself exhibits a sizeable intrinsic spin Hall effect [2], [3].",
"A quantitative explanation of the intrinsic ISHE, however, requires knowledge of the electron band structure (obtainable by algorithms based on the tight-binding model [2] or the density functional theory [29]), which is beyond the scope of this work.",
"The decrease of $j_{\\mathrm {SSE}}$ and the THz Signal at $x=0.7$ may then be ascribed to an increase of atomic order and thus a decrease of the extrinsic ISHE.",
"In conclusion, we compare the spin-to-charge conversion of steady state and THz spin currents in copper-iridium alloys as a function of the iridium concentration.",
"We find a clear maximum of the spin Hall effect for alloys of around 40 Ir concentration, far beyond the previously probed dilute doping regime.",
"While the detected DC spin Seebeck voltage exhibits a concentration dependence different from the raw THz signal, very good qualitative agreement between the DC spin Seebeck current and the THz emission signal is observed, which is well understood within our model for THz emission.",
"Ultimately, our results show that tuning the spin Hall effect by alloying delivers an unexpected, complex concentration dependence that is equal for spin-to-charge conversion at DC and THz frequencies and allows us to conclude that the large spin Hall effect in CuIr can be used for spintronic applications on ultrafast timescales."
],
[
"Acknowledgments",
"This work was supported by Deutsche Forschungsgemeinschaft (DFG) (SPP 1538 “Spin Caloric Transport”, SFB/TRR 173 \"SPIN+X\"), the Graduate School of Excellence Materials Science in Mainz (DFG/GSC 266), and the EU projects IFOX, NMP3-LA-2012246102, INSPIN FP7-ICT-2013-X 612759, TERAMAG H2020 681917."
]
] | 1709.01890 |
[
[
"Top Quark Rare Decays via Loop-Induced FCNC Interactions in Extended\n Mirror Fermion Model"
],
[
"Abstract Flavor changing neutral current (FCNC) interactions for a top quark $t$ decays into $Xq$ with $X$ represents a neutral gauge or Higgs boson, and $q$ a up- or charm-quark are highly suppressed in the Standard Model (SM) due to the Glashow-Iliopoulos-Miami mechanism.",
"Whilst current limits on the branching ratios of these processes have been established at the order of $10^{-4}$ from the Large Hadron Collider experiments, SM predictions are at least nine orders of magnitude below.",
"In this work, we study some of these FCNC processes in the context of an extended mirror fermion model, originally proposed to implement the electroweak scale seesaw mechanism for non-sterile right-handed neutrinos.",
"We show that one can probe the process $t \\to Zc$ for a wide range of parameter space with branching ratios varying from $10^{-6}$ to $10^{-8}$, comparable with various new physics models including the general two Higgs doublet model with or without flavor violations at tree level, minimal supersymmetric standard model with or without $R$-parity, and extra dimension model."
],
[
"Introduction",
"Absence of flavor changing neutral current (FCNC) interactions in the Standard Model (SM) at tree level is quite a unique property due to the special quantum numbers of the three generations of fermions (quarks and leptons) and one Higgs doublet under the gauge group of $SU(3)_C \\times SU(2)_L \\times U(1)_Y$ .",
"FCNC interactions can nevertheless be induced at the quantum loop level and therefore are suppressed by the GIM mechanism [1].",
"Experimental results for various FCNC processes in the kaon, $D$ and $B$ meson systems are all in line with the SM expectations.",
"For the heavy top quark $t$ , the story is quite different.",
"Since there is no time for the heavy top quark to form bound states, we can discuss its free decay, like the dominant decay mode $t \\rightarrow W^+ b$ at tree level or its rare FCNC decays.",
"The SM branching ratios ${\\mathcal {B}}(t \\rightarrow X q)$ where $X$ denotes one of the following neutral particle $Z, \\gamma , g$ or $h$ in SM, and $q$ denotes the light $u$ or $c$ quark, vary in the range $10^{-17} -10^{-12}$ [2], which are unobservable at the present technology.",
"However, in many models beyond the SM, branching ratios for some of these processes of order up to $10^{-3}$ can be achieved.",
"Observations of these rare top quark FCNC decays at the Large Hadron Collider (LHC) with significant larger branching ratios than the SM predictions would then be clear signals, albeit indirect, of new physics.",
"Indeed LHC can be considered as a top quark factory, estimated to produce $10^8$ $t \\bar{t}$ pair with an integrated luminosity of 100 inverse femtobarn.",
"For an updated review on top quark properties at the LHC, see [3].",
"The current limits for $t \\rightarrow Zq$ [4], [5], [6], [7] are ${\\mathcal {B}}(t \\rightarrow Z u) \\le \\left\\lbrace \\begin{array}{l}2.2 \\times 10^{-4} \\; [{\\rm CMS}] \\; , \\\\1.7 \\times 10^{-4} \\; [{\\rm ATLAS}] \\; ,\\end{array}\\right.$ ${\\mathcal {B}}(t \\rightarrow Z c) \\le \\left\\lbrace \\begin{array}{l}4.9 \\times 10^{-4} \\; [{\\rm CMS}] \\; , \\\\2.3 \\times 10^{-4} \\; [{\\rm ATLAS}] \\; ;\\end{array}\\right.$ and for $t \\rightarrow \\gamma q$ , we have the limits from CMS [8] $\\begin{aligned}{\\mathcal {B}}(t \\rightarrow \\gamma u) & \\le 1.3 \\times 10^{-4} \\; , \\\\{\\mathcal {B}}(t \\rightarrow \\gamma c) & \\le 1.7 \\times 10^{-3} \\; .\\end{aligned}$ Projected limits for the above as well as other FCNC processes for the top quark are expected to be improved constantly in the future at the LHC.",
"Thus searching for or discovery of any one of these FCNC rare top decays $t \\rightarrow X q$ at the LHC would be providing interesting constraints or discriminations of various new physics models in the future.",
"Over the years FCNC top quark decays had been studied intensively in the literature for many new physics models, like the minimal supersymmetric standard model (MSSM) with [9] or without [10], [11] $R$ -parity, flavor conserving [12] or flavor violating [13], [14], [15], [16] two Higgs doublet model (2HDM), aligned two Higgs doublet model (A2HDM) [17], warped extra-dimensions [18], [19], and effective Lagrangian framework [20], etc.",
"Branching ratios for FCNC top quark decays in all these models are typically many orders of magnitude above the SM and some of them may lead to detectable signals at the LHC.",
"In this work, we compute the FCNC decays of $t \\rightarrow Vq$ $(V=Z,\\gamma ; \\, q=u,c)$ in an extension of mirror fermion model [21] originally proposed by one of us [22].",
"In contrast with various left-right symmetric models, the model in [22] did not include the gauge group $SU(2)_R$ while adding the mirror partners of the SM fermions.",
"Despite having the same SM gauge group, the scalar sector must be enlarged.",
"In additional to employ the bi-triplets in the Georgi-Machacek (GM) model [23], [24] and a Higgs singlet to implement the electroweak scale seesaw mechanism for the non-sterile right-handed neutrino masses [22], one needs to add a mirror Higgs doublet [25] in the scalar sector so as to make consistency with the various signal strengths of the 125 GeV Higgs measured at the LHC.",
"We will briefly review this class of mirror fermion model and its further extension with a horizontal $A_4$ symmetry in the following section.",
"We layout the paper as follows.",
"In Section , after giving a brief highlight on some of the salient features of the model, we present the relevant interaction Lagrangian.",
"Our calculation and analysis are presented in Section and respectively.",
"We finally summarize our results in Section .",
"Analytical expressions for the loop functions are collected in the Appendix.",
"As already eluded to in the Introduction, the mirror fermion model in [22] was devised to implement the so-called electroweak scale seesaw mechanism for the neutrino masses.",
"We first list the particle content of the model in Table REF for further discussions.",
"One special feature of this mirror model is to treat the right-handed neutrino in each generation to be non-sterile by grouping it with a new heavy mirror right-handed charged lepton into a weak doublet $l^M_{Ri}$ , regarded as the mirror of the SM doublet $l_{Li}$ with $i$ labelling the generation.",
"When the Higgs singlet $\\phi _{0S}$ develops a small vacuum expectation value (VEV) of order $10^5$ eV, through its Yukawa couplings between the SM lepton doublets and their mirror partners, it can provide a small Dirac mass term for the neutrinos.",
"On the other hand, when the triplet $\\tilde{\\chi }$ field with hypercharge $Y/2 = 1$ in Table REF develops a VEV of order $v_{\\rm SM} = 246 $ GeV, a Majorana mass term of electroweak scale can be generated through its Yukawa couplings among these new mirror lepton doublets.",
"Details of this electroweak scale seesaw mechanism in the mirror fermion model can be found in [22].",
"Table: The Standard Model quantum numbers of the fermion and scalar sectors in the extended mirror model together with their assignments under the horizontal A 4 A_4 symmetry.Note that the other triplet $\\xi $ which has zero hypercharge is grouped with $\\tilde{\\chi }$ to form the bi-triplets in the GM model to maintain the custodial symmetry and therefore the $\\rho $ parameter equals unity at tree level.",
"In [26], the potential dangerous contributions from the GM triplets to the $S$ and $T$ oblique parameters are shown to be partially cancelled by the opposite contributions from mirror fermions such that the model is still healthy against electroweak precision tests.",
"Mirrors of other SM fermions, both leptons and quarks, can be introduced in the same way as listed in Table REF .",
"Searches for these heavy mirror fermions at the LHC were presented in [27] and [28] for the mirror quarks and mirror leptons respectively.",
"In order to reproduce the signal strengths of $h_{125} \\rightarrow \\gamma \\gamma $ and $h_{125} \\rightarrow Z \\gamma $ for the 125 GeV Higgs observed at the LHC, a mirror Higgs doublet $\\Phi _M$ of the SM one $\\Phi $ was introduced [25].",
"We note that mixing effects among the two doublets as well as with the triplet $\\xi $ must be taken into account in order to satisfy the LHC results.",
"A global $U(1) \\times U(1)$ symmetry was also enforced in the Yukawa interactions so that the SM Higgs doublet only couples to the SM fermions and the mirror Higgs doublet only couples to the mirror fermions.",
"Thus there is no FCNC Higgs interactions at tree level in the model.",
"Processes like $h \\rightarrow \\tau \\mu $ [29] and $t \\rightarrow h q$ can only occur at the quantum loop level.",
"To address the issues of neutrino and charged lepton masses and mixings, the original mirror model was extended in [21] by introducing a horizontal family symmetry of the tetrahedral group $A_4$ .",
"The $A_4$ assignments of all the scalars and fermions as well as their SM quantum numbers are shown at the last column in Table REF .",
"The lone singlet $\\phi _{0S}$ is now accompanied with a $A_4$ triplet $\\vec{\\phi }_{S}=(\\phi _{1S},\\phi _{2S},\\phi _{3S})$ .",
"Both $\\phi _{0S}$ and $\\vec{\\phi }_{S}$ are electroweak singlets and they are the only fields communicating the SM sector with the mirror sector through the Yukawa couplings, which must be invariant under both gauge symmetry and $A_4$ .",
"Other scalars are $A_4$ singlets.",
"Phenomenological implications of the extended mirror fermion model with the $A_4$ symmetry have been explored for the charged lepton flavor violating (CLFV) processes $\\mu \\rightarrow e \\gamma $ [30], $\\mu - e$ conversion [31] and $h_{125} \\rightarrow \\tau \\mu $ [29], as well as for the electron electric dipole moment [32].",
"Here we will explore its implication in the rare FCNC top decays.",
"Implications of the $A_4$ symmetry for the quark masses and mixings will be given in [33]."
],
[
"Interaction Lagrangian for Quarks and Their Mirrors",
"Here we will write down the interactions for the quarks and their mirrors that are relevant to the FCNC processes $t \\rightarrow Vq$ that we are studying.",
"Since the result for $t \\rightarrow gq$ can be easily obtained from that of $t \\rightarrow \\gamma q$ , we will not present detailed formulas for the former process.",
"As for $t \\rightarrow h q$ , one must consider the mixing effects from the more complicated Higgs sector in the extended mirror model.",
"We will leave it for future work."
],
[
"Quark Yukawa Couplings with $A_4$ Symmetry",
"Recall that the tetrahedron symmetry group $A_4$ has four irreducible representations $\\bf 1$ , $\\bf 1^{\\prime }$ , $\\bf 1^{\\prime \\prime }$ , and $\\bf 3$ with the following multiplication rule ${\\bf 3} \\times {\\bf 3} & = & {\\bf 3_1} (23, 31, 12) + {\\bf 3_2} (32, 13, 21) \\nonumber \\\\&+& {\\bf 1} (11+ 22 + 33) + {\\bf 1^{\\prime }} (11 + \\omega ^2 22 + \\omega 33)+ {\\bf 1^{\\prime \\prime }} (11 + \\omega 22 + \\omega ^2 33)$ where $\\omega = e^{2 \\pi i/3} = -\\frac{1}{2} + i \\frac{\\sqrt{3}}{2}$ .",
"Using the above $A_4$ multiplication rules one can construct new Yukawa couplings in the leptonic sector, which are both gauge invariant and $A_4$ symmetric, to implement small Dirac neutrino masses in electroweak seesaw and to discuss charged lepton mixings [21].",
"In the same vein, one can write down the following new Yukawa couplings for the quarks and their mirrors (both in the flavor basis with subscripts “0\") with the scalar singlets, which are both gauge invariant and $A_4$ symmetric, $- {\\cal L}_Y& \\supset & g^Q_{0S} \\phi _{0S} (\\overline{q_{L,0}} q^{M}_{R,0})_{\\bf 1}+ g^Q_{1S} \\vec{\\phi }_S \\cdot (\\overline{q_{L,0}} \\times q_{R,0}^{M})_{\\bf 3_1}+ g^Q_{2S} \\vec{\\phi }_S \\cdot (\\overline{q_{L,0}} \\times q_{R,0}^{M})_{\\bf 3_2}\\nonumber \\\\&+& g^{u}_{0S} \\phi _{0S} (\\overline{u_{R,0}} u^{M}_{L,0})_{\\bf 1}+ g^{u}_{1S} \\vec{\\phi }_S \\cdot (\\overline{u_{R,0}} \\times u_{L,0}^{M})_{\\bf 3_1}+ g^{u}_{2S} \\vec{\\phi }_S \\cdot (\\overline{u_{R,0}} \\times u_{L,0}^{M})_{\\bf 3_2}\\\\&+& g^{d}_{0S} \\phi _{0S} (\\overline{d_{R,0}} d^{M}_{L,0})_{\\bf 1}+ g^{d}_{1S} \\vec{\\phi }_S \\cdot (\\overline{d_{R,0}} \\times d_{L,0}^{M})_{\\bf 3_1}+ g^{d}_{2S} \\vec{\\phi }_S \\cdot (\\overline{d_{R,0}} \\times d_{L,0}^{M})_{\\bf 3_2}+ {\\rm H.c.} \\nonumber $ where $g^{Q,u,d}_{0S}$ , $g^{Q,u,d}_{1S}$ and $g^{Q,u,d}_{2S}$ are in general complex coupling constants.",
"Implications of the above Yukawa interactions on the quark mixings will be presented in [33].",
"Next we move to the physical basis by making the following unitary transformations on the left-handed fields $u_{L,0} = V^u_L u_L, \\;d_{L,0} = V^d_L d_L, \\;u^M_{L,0} = V^{u^M}_L u^M_L, \\;d^M_{L,0} = V^{d^M}_L d^M_L,$ and similarly for the right-handed fields $u_{R,0} = V^u_R u_R, \\;d_{R,0} = V^d_R d_R, \\;u^M_{R,0} = V^{u^M}_R u^M_R, \\;d^M_{R,0} = V^{d^M}_R d^M_R.$ We can then recast the Yukawa interactions in the following form $\\begin{aligned}{ \\mathcal {L} }_{Y} \\supset &-\\bar{ u } \\left({ { V }_{ L }^{ u } }^{ \\dagger }{ M }_{ S }^{ Q }(\\phi ){ V }_{ R }^{ u^M }P_R+{ { V }_{ R }^{ u } }^{ \\dagger }{ M }_{ S }^{ u }(\\phi ){ V }_{ L }^{ u^M }P_L\\right){ u }^{ M }\\\\&-\\bar{ d } \\left({ { V }_{ L }^{ d } }^{ \\dagger }{ M }_{ S }^{ Q }(\\phi ){ V }_{ R }^{ d^M }P_R+{ { V }_{ R }^{ d } }^{ \\dagger }{ M }_{ S }^{ d }(\\phi ){ V }_{ L }^{ d^M }P_L\\right){ d }^{ M }+{\\rm H.c.}\\end{aligned}$ with $P_{L,R}=(1\\mp \\gamma _5)/2$ .",
"Here $M_S^{Q,u,d}(\\phi )$ are three field-dependent three by three matrices which can be decomposed in terms of the four scalar fields according to ${ M }_{ S }^{ Q }(\\phi )={ M }^{ Q,0 }{ \\phi }_{ 0S }+{ M }^{ Q,1 }{ \\phi }_{ 1S }+{ M }^{ Q,2 }{ \\phi }_{ 2S }+{ { M } }^{ Q,3 }{ \\phi }_{ 3S }\\;,$ where $\\begin{aligned}&{ M }^{ Q,0 }=\\begin{pmatrix} { g }_{ 0S }^{ Q } & 0 & 0 \\\\ 0 & { g }_{ 0S }^{ Q } & 0 \\\\ 0 & 0 & { g }_{ 0S }^{ Q } \\end{pmatrix} \\quad , \\quad { M }^{ Q,1 }=\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & { g }_{ 1S }^{ Q } \\\\ 0 & { g }_{ 2S }^{ Q } & 0 \\end{pmatrix} \\; ,\\\\&{ M }^{ Q,2 }=\\begin{pmatrix} 0 & 0 & { g }_{ 2S }^{ Q } \\\\ 0 & 0 & 0 \\\\ { g }_{ 1S }^{ Q } & 0 & 0 \\end{pmatrix}\\quad , \\quad { M }^{ Q,3 }=\\begin{pmatrix} 0 & { g }_{ 1S }^{ Q } & 0 \\\\ { g }_{ 2S }^{ Q } & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\; ,\\end{aligned}$ and similar decompositions for $M^{u}_S(\\phi )$ and $M^{d}_S(\\phi )$ with $M^{u,k}$ and $M^{d,k}$ obtained by the substitutions of $g_{iS}^Q \\rightarrow $ $g_{iS}^u$ and $g_{iS}^d$ respectively in Eq.",
"(REF ).",
"Introducing the following combinations of the coupling matrices ${ M }^{ Q,k }$ and ${ M }^{ q,k }$ $(k=0,1,2,3)$ with the fermion mixing matrices $\\begin{aligned}{V^{q,k}_L} & \\equiv { { V }_{ L }^{ q } }^{ \\dagger }{ M }^{ Q,k }{ V }_{ R }^{ q^M }\\;, \\\\{V^{q,k}_R} & \\equiv { { V }_{ R }^{ q } }^{ \\dagger }{ M }^{ q,k }{ V }_{ L }^{ q^M }\\;,\\end{aligned}$ we arrive at the final form of the Yukawa interactions ${ \\mathcal {L} }_{ Y } \\supset -{ \\sum _{ k=0 }^{ 3 }{ \\sum _{ i,j=1 }^{ 3 }{ \\bar{ { u} }_{ i} \\left\\lbrace { { V }_{ L }^{ u,k } }_{ ij }P_R+{ { V }_{ R }^{ u,k } }_{ ij }P_L \\right\\rbrace { u }_{ j }^{ M }{ \\phi }_{ kS } + (u \\leftrightarrow d)+{\\rm H.c.} } } }$ We have combined the four scalars $\\phi _{0S}$ and $\\vec{\\phi }_{S}$ into $\\phi _{kS}$ with $k=0,1,2,3$ .",
"For the FCNC rare top decays that we are studying, only ${ { V }_{ (L,R) }^{ u,k } }$ are relevant."
],
[
"Neutral Currents",
"We also need the neutral current interactions for the SM $Z$ boson and photon couple to the quarks and their mirrors.",
"${\\mathcal {L}}_{\\rm NC} \\supset g Z^\\mu J_\\mu ^Z + e A^\\mu J^{\\rm EM}_\\mu $ with $\\begin{aligned}J_\\mu ^Z = \\frac{1}{\\cos \\theta _W} &\\left[\\bar{q}_L \\gamma _\\mu \\left( T^3 - Q_q \\sin ^2\\theta _W \\right) q_L- \\bar{q}_R \\gamma _\\mu Q_q\\sin ^2 \\theta _W q_R\\right.",
"\\\\&+\\left.\\bar{q}^M_R \\gamma _\\mu \\left( T^3 - Q_q \\sin ^2\\theta _W \\right) q^M_R- \\bar{q}^M_L \\gamma _\\mu Q_q\\sin ^2 \\theta _W q_L^M\\right] \\; ,\\end{aligned}$ $J^{\\rm EM}_\\mu = Q_q \\left( \\bar{q} \\gamma _\\mu q+ \\bar{q}^M \\gamma _\\mu q^M \\right) \\; .$ The above neutral current interactions in SM and the new Yukawa couplings can induce FCNC decay $t \\rightarrow Vq$ at one-loop level as depicted by the three Feynman diagrams in Fig.",
"REF .",
"Figure: Feynman diagrams contributing to t→Vqt \\rightarrow Vq."
],
[
"FCNC Top Decays $t \\rightarrow Vq$",
"The effective Lagrangian for $t \\rightarrow Z q$ and $t \\rightarrow \\gamma q$ can be expressed as ${\\mathcal {L}}_{\\rm eff} & = & - \\bar{q} \\gamma _\\mu ( C_L P_L + C_R P_R ) t Z^\\mu - \\frac{1}{m_t} \\bar{q} \\sigma _{\\mu \\nu } ( A_L P_L + A_R P_R ) t Z^{\\mu \\nu } \\nonumber \\\\& & \\;\\;\\;\\;\\;\\;\\; -\\frac{1}{m_t} \\bar{q} \\sigma _{\\mu \\nu } ( A^\\prime _L P_L + A^\\prime _R P_R ) t F^{\\mu \\nu }+ {\\rm H.c.}$ where $q=(u,c)$ ; $Z^{\\mu \\nu } = \\partial ^\\mu Z^\\nu - \\partial ^\\nu Z^\\mu $ and $F^{\\mu \\nu } = \\partial ^\\mu A^\\nu - \\partial ^\\nu A^\\mu $ ; and $A_{L,R}$ , $A^\\prime _{L,R}$ and $C_{L,R}$ are dimensionless quantities.",
"In terms of the dimensionless mass ratios $r_q \\equiv m_q/m_t \\; \\; , \\quad \\quad r_Z \\equiv m_Z/m_t$ the partial decay rate for $t \\rightarrow Zq$ is given by $\\Gamma ( t \\rightarrow q Z ) = \\frac{1}{16 \\pi } \\frac{1}{m_t}\\lambda ^{\\frac{1}{2}}\\left(1, r_q^2, r_Z^2\\right)\\left\\langle \\sum \\vert {\\mathcal {M}} \\vert ^2 \\right\\rangle \\; ,$ where $\\lambda (1,y,z)=(1- (\\sqrt{y} + \\sqrt{z})^2)(1- (\\sqrt{y} - \\sqrt{z})^2)$ and $\\left\\langle \\sum \\vert {\\mathcal {M}} \\vert ^2 \\right\\rangle &=&\\frac{m_t^2}{2} \\biggl \\lbrace +2 \\left( \\vert C_L \\vert ^2 + \\vert C_R \\vert ^2 \\right) \\left( 1 + r_q^2 - r_Z^2 \\right) \\biggr .\\nonumber \\\\&+& 4 \\left( \\vert A_L \\vert ^2 + \\vert A_R \\vert ^2 \\right)\\left[ 2 \\left( 1 - r_q^2 \\right)^2 - \\left( 1 + r_q^2 \\right) r_Z^2 - r_Z^4 \\right] \\nonumber \\\\&-& 16 \\, {\\rm Re} \\left(C_L C_R^* \\right) r_q - 48 \\, {\\rm Re} \\left(A_L A_R^* \\right) r_q r_Z^2 \\nonumber \\\\&-&12 \\, {\\rm Re} \\left( C_L A_L^* + C_R A_R^* \\right) r_q \\left( 1 - r_q^2 + r_Z^2 \\right) \\\\&+&12 \\, {\\rm Re} \\left( C_L A_R^* + C_R A_L^* \\right) \\left( 1 - r_q^2 - r_Z^2 \\right) \\nonumber \\\\&+&\\frac{1}{r_Z^2} \\biggl .",
"\\left[\\left( \\vert C_L \\vert ^2 + \\vert C_R \\vert ^2 \\right)\\left( \\left( 1 - r_q^2 \\right)^2 - \\left(1 + r_q^2 \\right) r_Z^2 \\right)+ 4 \\, {\\rm Re} \\left( C_L C_R^* \\right) r_q r_Z^2\\right] \\biggr \\rbrace \\, .\\nonumber $ For a given model, the dimensionless quantities $A_{L,R}$ and $C_{L,R}$ can be determined.",
"In the mirror fermion model, these quantities are induced at one loop level, as depicted by the Feynman diagrams in Fig.",
"REF .",
"Their analytical expressions are given in the Appendix.",
"Similarly, for $t \\rightarrow \\gamma q$ we have $\\Gamma ( t \\rightarrow q \\gamma ) = \\frac{1}{16 \\pi } \\frac{1}{m_t}\\lambda ^{\\frac{1}{2}}\\left(1, r_q^2, 0\\right)\\left\\langle \\sum \\vert {\\mathcal {M}} \\vert ^2 \\right\\rangle $ with $\\left\\langle \\sum \\vert {\\mathcal {M}} \\vert ^2 \\right\\rangle =4m_t^2(1-r_q^2)^2( \\vert A^\\prime _L \\vert ^2 + \\vert A^\\prime _R \\vert ^2)$ .",
"The expressions for $A^\\prime _L$ and $A^\\prime _R$ are also given in the Appendix."
],
[
"Analysis",
"In our numerical analysis, we will make the following assumptions on the parameter space of the model.",
"(1) First, we will take all the unknown Yukawa couplings to be real and assume $g_{iS}^q=g_{iS}^Q$ for $q=(u,d)$ and $i=0,1,2$ .",
"We will explore how our results depend on the couplings $g_{iS}^Q$ .",
"We note that it has been shown recently [34] in the extended mirror fermion model [25] the complex values of some of these Yukawa couplings, combined with the electroweak scale seesaw mechanism generating the minuscule neutrino masses, one can provide a solution to the strong CP problem without introducing axion.",
"(2) Since only the product $V_{\\rm CKM}=(V^u_L)^\\dagger V_L^d$ are known experimentally, we will study the following scenarios for illustrative purpose.",
"Scenario 1: $\\begin{aligned}V_L^u &= V_{\\rm CKM}^\\dagger \\; , \\\\V_R^u = V_L^{u^M} = V_R^{u^M} & = 1 \\; .\\end{aligned}$ Scenario 2: $\\begin{aligned}V_L^u = V_L^{u^M} & = V_{\\rm CKM}^\\dagger \\; , \\\\V_R^u = V_R^{u^M} & = 1 \\; .\\end{aligned}$ (3) For the three generation of mirror quark masses, we assume $m_{q^M_1} : m_{q^M_2} : m_{q^M_3} = M : M + 10 \\; {\\rm GeV} : M+ 20 \\; {\\rm GeV} \\; ,$ and vary the common mirror quark mass $M$ from 150 to 800 GeV.",
"We note that mirror fermions in this class of electroweak scale mirror fermion model are expected to have masses of electroweak scale to satisfy unitarity [27].",
"(4) For the scalars $\\phi _{kS}$ , their masses are necessarily small since they are link to the Dirac neutrino masses [21], [22].",
"We set their masses $m_{kS}$ all equal 10 MeV.",
"Our numerical results are not sensitive to this choice as long as $m_{kS} \\ll m_{q^M_m}$ .",
"(5) For the SM parameters, we use [35] $m_t & = &173.21\\, {\\rm GeV} \\; , \\; m_c = 1.275 \\, {\\rm GeV} \\; , \\; m_u = 2.3 \\, {\\rm MeV} \\; , \\nonumber \\\\\\sin ^2 \\theta _W & = & 0.23126 \\; , \\; \\alpha = 1/127.944 \\; , \\\\\\Gamma _t & = & 1.41\\, {\\rm GeV} \\; , \\; {\\cal B}\\left( t \\rightarrow W^+b \\right) = 0.957 \\; .",
"\\nonumber $ Figure: Branching ratios of t→γut \\rightarrow \\gamma u (left) and t→γct \\rightarrow \\gamma c (right)versus the logarithmic of Yukawa couplings g 0S Q g^Q_{0S} and g 1S Q g^Q_{1S} withg 2S Q =10 -3 g^Q_{2S} = 10^{-3} and M=150M = 150 GeV in Scenario 1.Figure: Branching ratios of t→Zut \\rightarrow Z u (left) and t→Zct \\rightarrow Z c (right)versus the logarithmic of Yukawa couplings g 0S Q g^Q_{0S} and g 1S Q g^Q_{1S} withg 2S Q =10 -3 g^Q_{2S} =10^{-3} and M=150M = 150 GeV in Scenario 1.In Fig.",
"(REF ), we show the contour plots of $\\log {\\cal B}(t \\rightarrow \\gamma u)$ (left panel) and $\\log {\\cal B}(t \\rightarrow \\gamma c)$ (right panel) on the $(\\log _{10} (g_{0S}^Q), \\log _{10} (g_{1S}^Q))$ plane in the case of Scenario 1 with $g^Q_{2S}$ set to be $10^{-3}$ .",
"Fig.",
"(REF ) is similar as Fig.",
"(REF ) but for $t \\rightarrow Zq$ .",
"Figs.",
"(REF ) and (REF ) are the same as Figs.",
"(REF ) and (REF ) respectively but for Scenario 2.",
"The common mirror fermion mass $M$ is set to be 150 GeV in these 4 figures.",
"Figure: Same as Fig.",
"() in Scenario 2.Figure: Same as Fig.",
"() in Scenario 2.Mirror quarks may be pair produced at the LHC [27].",
"Once produced the heavier mirror fermions may cascade into lighter ones by emitting an on-shell or off-shell SM $W$ -boson, depending on the detail mass spectrum of the mirror fermions.",
"The lightest mirror quark will then decay into its SM partner with any one of the scalar singlets $\\phi _{kS}$ via the new Yukawa interactions which are responsible to the FCNC decays of the top quark studied here.",
"In the mirror lepton case, the corresponding Yukawa couplings $g_{iS}^L$ are necessarily small since they are responsible for providing small Dirac masses to the neutrinos in the electroweak scale seesaw mechanism.",
"Assuming the lightest mirror fermion is $u^M$ .",
"In Fig.",
"(REF ), we plot the contours of the decay length of $u^M$ in the $(M, \\log _{10} (g_{0S}^Q))$ plane.",
"We take all the Yukawa couplings to be the same just for illustrations.",
"One can see that for very small Yukawa couplings $<10^{-6}$ , can the decay length reach a few mm for a displaced vertex.",
"Search strategies for the mirror fermions would then be quite different from the usual cases, involving not merely the missing energies but displaced vertices as well [27].",
"Current experiments at the LHC have the capability to perform such kind of searches.",
"Further studies of this issue are warranted.",
"Nevertheless, for the mirror quarks, there is no a priori reason that these new Yukawa couplings have to be very small except that there are stringent constraints from the mixings between SM fermions and their mirrors.",
"The mixing angle is roughly of order $g^Q_{iS} \\langle \\phi _{kS} \\rangle / M$ .",
"For $g^Q_{iS} \\sim 1$ , $\\langle \\phi _{kS} \\rangle \\sim 1$ MeV and $M \\sim $ 500 GeV, this mixing angle is about $2 \\times 10^{-6}$ .",
"A full analysis taking into the account of the mixing effects is beyond the scope of this paper.",
"Figure: Decay length of the lightest mirror quark versuslog 10 (g 0S Q )\\log _{10} (g^Q_{0S}) assuming all the unknown Yukawa couplings equal to each other.In Figs.",
"(REF ) and (REF ), we show the scatter plots for the logarithmic of branching ratios of the 4 processes, $t \\rightarrow \\gamma u$ and $t \\rightarrow \\gamma c$ in the left panel and $t \\rightarrow Z u$ and $t \\rightarrow Z c$ in the right panel, versus $\\log _{10} (g_{0S}^Q)$ for Scenarios 1 and 2 respectively.",
"We have set all the Yukawa couplings equal to each other in these plots.",
"The different colors in the scatter plots represent different values of the common mirror fermion mass $M$ varied from 150 to 800 GeV as indicated by the color palettes on the top of each plot.",
"Current experimental limits of these processes are also shown in these plots by the horizontal red dashed lines, while the black dashed lines are the SM predictions.",
"It is clear from these plots that the mirror quarks in this class of model with mass less than 800 GeV could play an important role in FCNC decays of the top quark, provided that the Yukawa couplings are of the same size as the top quark Yukawa coupling in SM.",
"However if the Yukawa couplings are very small to allow for a displaced vertex for the lightest mirror fermion, all these FCNC top decays are beyond the reach at LHC.",
"Figure: Scatter plots for the branching ratios of t→Vqt \\rightarrow Vq in Scenario 1.Figure: Same as Fig.",
"() in Scenario 2.Table: Comparisons of theoretical predictions for thebranching ratios of FCNC rare top decay t→Vqt \\rightarrow Vq in various models.The numbers in brackets in 2HDM and MSSM are for 2HDM with tree level flavor violations andMSSM with RR-parity violation respectively."
],
[
"Conclusion",
"In Table REF , we summarize our numerical results as well as those from SM and other three popular new physics models taken from [36] for comparisons.",
"The numbers in brackets in the 2HDM and MSSM columns are for 2HDM with tree level flavor violation and MSSM with $R$ -parity violation respectively.",
"Our results shown in the last column are taken from Figs.",
"(REF ) and (REF ) for $g_{0S}^Q=y_t =\\sqrt{2} m_t/v_{\\rm SM} \\sim 1$ and the mirror mass $M$ varying from 150 to 800 GeV.",
"On the other hand, if $\\vert g_{0S}^Q \\vert $ turns out to be small, of order $10^{-4}$ or less as suggested by the new solution to the strong CP problem discussed in [34], all FCNC top decays in the model would be unobservable.",
"While the experimental results of many FCNC processes in the kaon, $D$ and $B$ meson systems, accumulated over the past several decades, had been mostly consistent with the SM expectations, theoretical predictions for FCNC processes involving the top quark and/or the Higgs boson have not been challenged by high energy experiments until recent years.",
"In this work, we have computed the FCNC processes $t \\rightarrow Zq$ and $\\gamma q$ in a class of mirror fermion model equipped with a horizontal $A_4$ symmetry in the fermion and scalar sectors.",
"We found that branching ratio for $t \\rightarrow Zc$ is typically of order $10^{-8} - 10^{-6}$ as mirror quark masses are varying in the range from 800 to 150 GeV and new Yukawa couplings are of the same size of the top Yukawa coupling in SM.",
"At 14 TeV the total cross section for $t \\bar{t}$ production at the LHC is about 598 pb.",
"With a luminosity of 300 (1000) fb$^{-1}$ , we thus expect 180 (6) events of $t \\rightarrow Zc$ for a branching ratio of $10^{-6}$ ($10^{-8}$ ) before any experimental cuts.",
"For the other processes $t \\rightarrow Zu$ and $t \\rightarrow \\gamma q$ , their branching ratios are typically smaller by $1-2$ orders of magnitude.",
"For the gluon mode $t \\rightarrow g q$ its partial width is about 42 times larger than that of the photon mode $t \\rightarrow \\gamma q$ .",
"The LHC limits for the branching ratios of $t \\rightarrow g u$ and $t \\rightarrow g c$ are $2.0 \\times 10^{-5}$ and $4.1 \\times 10^{-4}$ respectively from CMS [37], and $4.0 \\times 10^{-5}$ and $20 \\times 10^{-5}$ respectively from ATLAS [38].",
"These branching ratios are extracted from the single top production via FCNC interactions from gluon plus up- or charm-quark initial states.",
"They are still $1-2$ orders of magnitude above our theoretical predictions.",
"It is also interesting to consider FCNC processes involving both the heavy top quark and the 125 GeV Higgs, the two heaviest particles in the SM.",
"One particular important process is $t \\rightarrow h q$ , which LHC has obtained the following limits [39], [40] ${\\mathcal {B}}(t \\rightarrow hu) \\le \\left\\lbrace \\begin{array}{l}4.5 \\times 10^{-3} \\; [{\\rm ATLAS}] \\; , \\\\5.5 \\times 10^{-3} \\; [{\\rm CMS}] \\; ;\\end{array}\\right.$ ${\\mathcal {B}}(t \\rightarrow hc) \\le \\left\\lbrace \\begin{array}{l}4.6 \\times 10^{-3} [{\\rm ATLAS}] \\; ,\\\\4.0\\times 10^{-3} [{\\rm CMS}] \\; .\\end{array}\\right.$ In the mirror fermion model, the mirror Higgs as well as the GM triplets could be an imposter for the 125 GeV Higgs due to mixing effects, which must be taken into account.",
"This work will be reported elsewhere [41].",
"LHC has unique opportunity for probing the top quark FCNC decays in new physics models since the SM contributions are at least nine orders of magnitude below the current limits of these processes.",
"With its high luminosity upgrade in the second phase, HL-LHC can impose powerful constraints on any underlying new physics responsible for the FCNC interactions."
],
[
"Acknowledgments",
"We would like to thank Chuan-Ren Chen for stimulating discussions.",
"This work is supported by the Ministry of Science and Technology (MoST) of Taiwan under Grant Number MOST-104-2112-M-001-001-MY3.",
"All three Feynman diagrams (A), (B) and (C) in Fig.",
"(REF ) contribute to the form factors $C_L$ and $C_R$ : $C_{(L,R)}=C_{(L,R)}^A+C_{(L,R)}^B+C_{(L,R)}^C \\; .$ To minimize cluttering in our expressions given below, we define $a=\\frac{ g }{ 4\\cos { \\theta }_{ W } }\\left(1-\\frac{ 8 }{ 3 } { \\sin }^{ 2 }{ \\theta }_{ W }\\right) \\; , \\;\\;\\ \\;\\;\\; b=\\frac{ g }{ 4\\cos { \\theta }_{ W } } \\; ,$ and $V^{L,k}_{ij} = \\left( V^{u,k}_{L} \\right)_{ij} \\; , \\;\\;\\ \\;\\;\\; V^{R,k}_{ij} = \\left( V^{u,k}_{R} \\right)_{ij} \\; ,$ where $V^{u,k}_{(L,R)}$ are given by Eq.",
"(REF ).",
"The individual contributions from each diagrams can be computed using the automated tools FormCalc and LoopTools in FeynArts [42].",
"The results are listed as follows.",
"$\\begin{aligned}{ C }_{ L }^{ A }=&\\frac{ 1 }{ { ( { m }_{ t } }^{ 2 }-{ { m }_{ q } }^{ 2 } ) } \\frac{ ( a+b ) }{ 16{ \\pi }^{ 2 } } \\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace { { m }_{ q } }^{ 2 } { V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\bigl [ B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\bigr .",
"\\biggr .\\\\&+B_1({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ]+{ m }_{ t }{ m }_{ q } { V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+B_1({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ]+{ m }_{ q }{ m }_{ { q }_{ m }^{ M } } { V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .",
"{ m }_{ t }{ m }_{ { q }_{ m }^{ M } } { V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr \\rbrace \\; ,\\end{aligned}$ $\\begin{aligned}{ C }_{ L }^{ B }=&\\frac{ -1 }{ { ( { m }_{ t } }^{ 2 }-{ { m }_{ q } }^{ 2 } ) } \\frac{ ( a+b ) }{ 16{ \\pi }^{ 2 } } \\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace { { m }_{ t } }^{ 2 } { V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr .\\\\&+B_1({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ]+{ m }_{ t }{ m }_{ q } { V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+B_1({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ]+{ m }_{ q }{ m }_{ { q }_{ m }^{ M } } { V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .",
"{ m }_{ t }{ m }_{ { q }_{ m }^{ M } } { V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr \\rbrace \\; ,\\end{aligned}$ and $\\begin{aligned}{ C }_{ L }^{ C }=&\\frac{ -1 }{ 16{ \\pi }^{ 2 } }\\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace (a+b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [\\frac{ 1 }{ 2 } -2C_{00}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ] \\biggr .\\\\& +{ { m }_{ { q }_{ m }^{ M } } }^{ 2 } (a-b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+{ { m }_{ Z } }^{ 2 } (a+b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\\\&+{ m }_{ t }{ m }_{ q } (a-b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [ 2C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\\\&+2C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+2C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_{11}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_{22}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+ C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ]\\\\&+{ m }_{ t }{ m }_{ { q }_{ m }^{ M } } (a-b){ V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\left[ C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\right.",
"\\\\&+C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ] \\\\&+{ m }_{ q }{ m }_{ { q }_{ m }^{ M } } (a-b){ V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\left[ C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\right.",
"\\\\&+C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .",
"C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ] \\biggr \\rbrace \\; ;\\end{aligned}$ $\\begin{aligned}{ C }_{ R }^{ A }=&\\frac{ 1 }{ { ( { m }_{ t } }^{ 2 }-{ { m }_{ q } }^{ 2 } ) } \\frac{ ( a-b ) }{ 16{ \\pi }^{ 2 } } \\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace { { m }_{ q } }^{ 2 } { V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [ B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .\\\\&+B_1({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ] +{ m }_{ t }{ m }_{ q } { V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [ B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+B_1({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ] +{ m }_{ q }{ m }_{ { q }_{ m }^{ M } } { V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .",
"{ m }_{ t }{ m }_{ { q }_{ m }^{ M } } { V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } B_0({ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr \\rbrace \\; ,\\end{aligned}$ $\\begin{aligned}{ C }_{ R }^{ B}=&\\frac{ -1 }{ { ( { m }_{ t } }^{ 2 }-{ { m }_{ q } }^{ 2 }) } \\frac{ ( a-b ) }{ 16{ \\pi }^{ 2 } }\\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace { { m }_{ t } }^{ 2 } { V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr .\\\\&+B_1({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ]+{ m }_{ t }{ m }_{ q } { V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+B_1({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ]+{ m }_{ q }{ m }_{ { q }_{ m }^{ M } }{ V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .",
"{ m }_{ t }{ m }_{ { q }_{ m }^{ M } } { V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } B_0({ { m }_{ t } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr \\rbrace \\; ,\\end{aligned}$ and $\\begin{aligned}{ C }_{ R }^{ C }=&\\frac{ -1 }{ 16{ \\pi }^{ 2 } }\\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace (a-b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [\\frac{ 1 }{ 2 } -2C_{00}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ]\\biggr .\\\\&+{ { m }_{ { q }_{ m }^{ M } } }^{ 2 } (a+b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+{ { m }_{ Z } }^{ 2 } (a-b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+{ m }_{ t }{ m }_{ q }(a+b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [ 2C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+2C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+2C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_{11}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_{22}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ] \\\\&+{ m }_{ t }{ m }_{ { q }_{ m }^{ M } }(a+b){ V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\big [ C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\big ] \\\\&+{ m }_{ q }{ m }_{ { q }_{ m }^{ M } } (a+b){ V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\big [ C_0({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\\\&+\\biggl .C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\big ] \\biggr \\rbrace \\; .\\end{aligned}$ Each of the above contributions $C^A_{L,R}$ , $C^B_{L,R}$ and $C^C_{L,R}$ are ultraviolet divergent.",
"However by using the divergent parts of the Passarino-Veltman (PV) functions $\\begin{aligned}{\\rm Div}[B_0] & = +\\Delta _\\epsilon \\; ,\\\\{\\rm Div}[B_1] & = -\\frac{1}{2}\\Delta _\\epsilon \\; ,\\\\{\\rm Div}[C_{00}] & = +\\frac{1}{4}\\Delta _\\epsilon \\; ,\\end{aligned}$ where $\\Delta _\\epsilon = 2/\\epsilon - \\gamma _E + \\ln 4 \\pi $ with $\\epsilon = 4-d$ is the regulator in dimensional regularization and $\\gamma _E$ being the Euler's constant, one can easily verify that the divergences in the three diagrams summed up to nil leading to finite results for $C_L$ and $C_R$ .",
"Only Diagram (C) contributes to the dipole form factors $A_L$ and $A_R$ .",
"They are given by $\\begin{aligned}{ A }_{ L }=&\\frac{ 1 }{ 16{ \\pi }^{ 2 } }\\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace \\frac{ { { m }_{ t } }^{ 2 } }{ 2 } (a-b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * }\\biggl [ C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr .",
"\\biggr .\\\\&+C_{11}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })+\\biggl .",
"C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ]\\\\&+\\frac{ { m }_{ t }{ m }_{ q } }{ 2 } (a+b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * } \\biggl [ C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .",
"\\\\&+C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })+\\biggl .",
"C_{22}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ]\\\\&+\\frac{ { m }_{ t }{ m }_{ { q }_{ m }^{ M } } }{ 2 } { V }^{ R,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * }\\biggl [ (a-b)C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .",
"\\\\&\\biggl .",
"\\biggl .",
"+(a+b)C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ] \\biggr \\rbrace \\; ,\\end{aligned}$ and $\\begin{aligned}{ A }_{ R }=&\\frac{ 1 }{ 16{ \\pi }^{ 2 } }\\sum _{ k=0 }^{ 3 }\\sum _{ m=1 }^{ 3 }\\biggl \\lbrace \\frac{ { { m }_{ t } }^{ 2 } }{ 2 } (a+b){ V }^{ L,k }_{qm}{ { V }^{ L,k }_{tm} }^{ * }\\biggl [ C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .",
"\\biggr .",
"\\\\&+C_{11}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })+\\biggl .",
"C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ]\\\\&+\\frac{ { m }_{ t }{ m }_{ q } }{ 2 } (a-b){ V }^{ R,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * } \\biggl [ C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .",
"\\\\&+C_{12}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })+\\biggl .",
"C_{22}({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ]\\\\&+\\frac{ { m }_{ t }{ m }_{ { q }_{ m }^{ M } } }{ 2 } { V }^{ L,k }_{qm}{ { V }^{ R,k }_{tm} }^{ * }\\biggl [ (a+b)C_1({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 }) \\biggr .",
"\\\\&+ \\biggl .",
"\\biggl .",
"(a-b)C_2({ { m }_{ t } }^{ 2 },{ { m }_{ Z } }^{ 2 },{ { m }_{ q } }^{ 2 },{ { m }_{ {\\phi }_{kS} } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 },{ { m }_{ { q }_{ m }^{ M } } }^{ 2 })\\biggr ] \\biggr \\rbrace \\; .\\end{aligned}$ Since the PV functions $C_1,C_2,C_{11},C_{12}$ and $C_{22}$ do not have ultraviolet divergences, $A_L$ and $A_R$ are finite, as one should expect for they are the coefficients of the non-renormalizable magnetic and electric dipole operators.",
"$A^{\\prime }_L$ and $A^{\\prime }_R$ can be obtained from the above $A_L$ and $A_R$ respectively by replacing $m_Z^2 \\rightarrow 0 \\; , \\;\\;\\;\\;a \\rightarrow \\frac{2}{3}e \\; , \\;\\;\\;\\;b \\rightarrow 0 \\; .$ The decay rate for $t \\rightarrow g q$ can be obtained from that of $t \\rightarrow \\gamma q$ simply by replacing the top quark electric charge $\\frac{2}{3}e$ by the strong coupling $g_s$ and multiply the final result by an overall color factor $(N_C^2-1)/2N_C$ where $N_C$ is the number of color.",
"Thus $\\frac{\\Gamma (t \\rightarrow g q)}{\\Gamma (t \\rightarrow \\gamma q)} = \\frac{9}{4} \\cdot \\frac{N_C^2 - 1}{2 N_C} \\cdot \\frac{\\alpha _s}{\\alpha _{\\rm em}}.$ Taking $N_C=3$ , $\\alpha _s = 0.11$ and $\\alpha ^{-1}_{\\rm em} = 128$ , this ratio is about 42.",
"Next-to-leading order QCD corrections to the processes $t\\rightarrow \\gamma q$ , $t \\rightarrow Z q$ and $t \\rightarrow g q$ can be found in [43], [44].",
"Moreover, the next-to-leading order and next-to-next-to-leading order QCD corrections for the dominant SM top quark decay mode $t \\rightarrow W^+b$ had been computed in [45], [46] and [47] respectively."
]
] | 1709.01690 |
[
[
"Convergence estimates for the Magnus expansion I. Banach algebras"
],
[
"Abstract We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates.",
"Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made.",
"In this Part I, we consider the general Banach algebraic setting.",
"We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\\mathrm C_2=2.89847930\\ldots$."
],
[
"Introduction",
"The Baker–Campbell–Hausdorff expansion and its continuous generalization, the Magnus expansion, attract attention from time to time; this is probably partly due to their aesthetically pleasing nature.",
"There are many rediscoveries in this area.",
"For example, related to the Magnus expansion, see Magnus [18] and Chen [7]; Mielnik, Plebański [21] and Strichartz [32] and Vinokurov [36]; Goldberg [14] and Helmstetter [16].",
"Here we review and provide simplified proofs related to the basic properties of the Magnus expansion in the Banach algebra and Banach–Lie algebra settings, and improve various convergence estimates.",
"We will not discuss the Magnus expansion for vector fields or PDEs, where there is much to do yet.",
"See Blanes, Casas, Oteo, Ros [3] for some other aspects of Magnus expansion.",
"Some observations and improvements concerning the Baker–Campbell–Hausdorff expansion are also made.",
"We refer to Bourbaki [5], Reutenauer [30], and Bonfiglioli, Fulci [4] for general background in Lie theory, including the Baker–Campbell–Hausdorff expansion, the Poincaré–Birkhoff–Witt theorem, the Dynkin–Specht–Wever lemma, and the Dynkin formula, for which we do not make particular references.",
"Especially, [4] can be useful: It discusses the above mentioned topics in rather great detail, and it provides a guide to further literature, but the detailed discussion ends about the same place where our discussion begins.",
"Dunford, Schwartz [12] still provides a reasonably good reference for functional analysis.",
"Furthermore, we cite Flajolet, Sedgewick [13] for generating function techniques, including a discussion of the role of Pringsheim's theorem, etc.",
"We refer to Coddington, Levinson [9], Hsieh, Sibuya [17], Teschl [33] for the theory of ordinary differential equations, including existence and uniqueness theorems, and some standard solution techniques.",
"Section provides an introduction to the Magnus expansion in the setting of Banach algebras.",
"The emphasis is on recovering some known results, organized efficiently.",
"Section uses the resolvent technique to refine these results.",
"Section provides an introduction the conformal range of operators on Hilbert spaces, something we use in the next three sections.",
"In Section we give explicit growth estimates for the Magnus expansion in setting of Hilbert space operators.",
"In Section we consider some carefully selected examples.",
"These are used to test our earlier estimates but also help to understand the $2\\times 2$ real case better.",
"Section develops an analysis of the $2\\times 2$ real case.",
"As a main tool, we introduce some normal forms for $2\\times 2$ real matrices based on time-ordered exponentials, which are not ordinary exponentials.",
"Section provides a discussion of the Magnus expansion in the setting of Banach–Lie algebras.",
"Section discusses improvements of the standard convergence bound $\\delta = 2.1737374\\ldots $ .",
"Some remarks on the notation are as follows.",
"On the complex plane, $\\operatorname{D}(z_0,r)$ denotes the (possibly degenerate) closed disk with center $z_0$ and radius $r$ .",
"$\\operatorname{\\mathring{D}}(z_0,r)$ denotes the corresponding open disk.",
"However, sometimes $\\mathbb {C}$ is identified with $\\mathbb {R}^2$ , so one should not be surprised by the notation $\\operatorname{D}((a,b),r)$ .",
"$\\mathbf {1}$ denotes the Lebesgue measure.",
"$\\Sigma _S$ denotes the permutations of the set $S$ , $\\Sigma _n$ denotes the permutations of $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"Several definitions will be given in statements (or even in proofs), but this practice is perhaps acceptable in the sense that later usage of these definitions requires the understanding of these places in question.",
"In general, our formalism is more on the algebraic side than on the ODE side, but there are plenty of analytical considerations.",
"When we consider generating functions, we may understand them either as formal, real function theoretic, or in analytic sense.",
"For example, the series $\\sum _{n=1}^\\infty \\frac{(2n)!}{4^nn!(n+1)!}",
"x^n$ can be understood as a formal power series; or as a real function which is equal to $\\frac{2}{1+\\sqrt{1-x}}$ for $0\\le x\\le 1$ and equal to $+\\infty $ for $x>2$ ; or as the analytic $\\frac{2}{1+\\sqrt{1-x}}$ function defined at least on $\\operatorname{\\mathring{D}}(0,1)$ , but rather on $\\mathbb {C}\\setminus [1,\\infty ) $ .",
"We try to be clear in what sense we understand the generating function at hand, but, of course, much of the power of generating functions comes from change between various viewpoints.",
"Most original results were stated for matrices and/or in the (piecewise) continuous setting, and then extended (and often simplified) by other authors to more general cases.",
"Sometimes the very same phenomenon is viewed in different formalisms.",
"Thus, proper attribution of results may be somewhat intricate but I try to list the significant contributions.",
"Apart from the general background material, we try to be quite self-contained."
],
[
"The Magnus expansion in the Banach algebra setting",
"Let $\\mathfrak {A}$ be a Banach algebra (real or complex).",
"For $X_1,\\ldots ,X_k\\in \\mathfrak {A}$ , we define the Magnus commutator by $\\mu _k(X_1,\\ldots ,X_k)=\\frac{\\partial ^k}{\\partial t_1\\cdot \\ldots \\cdot \\partial t_k}\\log (\\exp (t_1X_1)\\cdot \\ldots \\cdot \\exp (t_kX_k))\\Bigl |_{t_1=\\ldots =t_k=0}; $ where $\\exp $ is defined as usual, and $\\log $ can be defined either by power series around 1, or with spectral calculus, the complex plane cut along the negative real axis.",
"Algebraically, in terms of formal power series, $\\mu _k(X_1,\\ldots ,X_k)=\\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k))_{\\text{the variables }X_1\\ldots X_k\\text{ has multiplicity }1}, $ i. e. the part of $\\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k))$ where every variable $X_i$ has multiplicity 1.",
"This implies that, in terms of formal power series, $\\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k))=\\sum _{\\begin{array}{c}\\lbrace i_1,\\ldots , i_l\\rbrace \\subset \\lbrace 1,\\ldots ,k\\rbrace \\\\ i_1<\\ldots <i_l \\end{array}}\\mu _l(X_{i_1},\\ldots ,X_{i_l}) +H(X_1,\\ldots ,X_k), $ where $H(X_1,\\ldots ,X_k)$ collects the terms where some variable have multiplicity more than one.",
"Taking the exponential, and detecting the terms where every variable has multiplicity 1, this yields $X_1\\cdot \\ldots \\cdot X_k=\\sum _{\\begin{array}{c}I_1\\dot{\\cup }\\ldots \\dot{\\cup }I_s=\\lbrace 1,\\ldots ,k\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}\\frac{1}{s!",
"}\\cdot \\mu _{l_1}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}})\\cdot \\ldots \\cdot \\mu _{l_s}(X_{i_{s,1}},\\ldots ,X_{i_{s,l_s}}), $ where we sum over ordered partitions.",
"A continuous recast of the identities (REF ) is as follows.",
"Suppose that $I\\subset \\mathbb {R} $ is an interval, $h$ is an $\\mathfrak {A}$ -valued, say, Lebesgue-Bochner integrable function on $I$ .",
"Let $\\phi $ be the measure which is $h$ times the Lebesgue measure restricted to $I$ .",
"In what follows, whenever we use $\\mathfrak {A}$ or $\\mathfrak {g}$ valued measures, we will consider measures like above.",
"Theorem 1.1 (Magnus [18], 1954) Let $\\phi $ be a $\\mathfrak {A}$ valued measure as above.",
"If $\\sum _{k=1}^\\infty \\left|\\int _{t_1\\le \\ldots \\le t_k\\in I}\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k))\\right|<+\\infty , $ then $1+\\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_k\\in I}\\phi (t_1)\\cdot \\ldots \\cdot \\phi (t_k)=\\exp \\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_k\\in I}\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k)).",
"$ Take the exponential on the RHS, and contract the terms of order $k$ according to (REF ).",
"More precisely, Magnus has the information of this amount, however he is more interested in the commutator recursion for $\\mu _k$ than starting with (REF ) directly.",
"In a more compact terminology, the right Magnus expansion of $\\phi $ is $\\mu _{\\mathrm {R}}(\\phi )=\\sum _{k=1}^\\infty \\mu _{\\mathrm {R}[k]}(\\phi ), $ where $\\mu _{\\mathrm {R}[k]}(\\phi )=\\int _{t_1\\le \\ldots \\le t_k\\in I}\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k));$ and the theorem says that it exponentiates to the right time-ordered exponential $\\exp _{\\mathrm {R}}(\\phi )=1+\\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_k\\in I}\\phi (t_1)\\cdot \\ldots \\cdot \\phi (t_k)$ of $\\phi $ , as long as the Magnus expansion (REF ) is absolute convergent as a series.",
"One can, of course, formulate similar statements in terms of the left Magnus expansion $\\mu _{\\mathrm {L}}(\\phi )$ and left time-ordered exponential $\\exp _{\\mathrm {L}}(\\phi )$ by reversing the ordering on $I$ .",
"The two formalisms have the same capabilities (we just have to reverse the measures $\\phi \\leftrightarrow \\phi ^\\dag $ ), but the `$\\mathrm {R}$ ' formalism is better suited for purely algebraic manipulations, and the `$\\mathrm {L}$ ' formalism is better suited to the analytic viewpoint of differential equations.",
"One has a direct expression for $\\mu _k$ .",
"In order to have it, let us introduce some terminology.",
"If $\\sigma =(\\sigma (1),\\ldots ,\\sigma (k) )$ is a finite sequences of real numbers, let $\\operatorname{asc}(\\sigma )$ denote the number of its ascents, i. e. the number of pairs such that $\\sigma (i)<\\sigma (i+1)$ ; and let $\\operatorname{des}(\\sigma )$ denote the number of its descents, i. e. the number of pairs such that $\\sigma (i)>\\sigma (i+1)$ .",
"This naturally applies in the special case when $\\sigma $ is a permutation from the symmetric group $\\Sigma _k$ .",
"Then $\\operatorname{asc}(\\sigma )+\\operatorname{des}(\\sigma )=k-1$ .",
"Theorem 1.2 (Mielnik, Plebański [21], 1970) $\\mu _k(X_1,\\ldots ,X_k)=\\sum _{\\sigma \\in \\Sigma _k}(-1)^{\\operatorname{des}(\\sigma )}\\frac{\\operatorname{des}(\\sigma )!\\operatorname{asc}(\\sigma )!}{k!",
"}X_{\\sigma (1)}\\cdot \\ldots \\cdot X_{\\sigma (k)} .",
"$ (After Strichartz [32], 1987.)",
"Considering (REF ), and the power series of $\\log $ , one can see that $\\mu _k(X_1,\\ldots ,X_k)$ is a sum of terms $\\frac{(-1)^{j-1}}{j}X_{\\sigma (1)}\\cdot \\ldots \\cdot X_{\\sigma (l_1)}\\mid X_{\\sigma (l_1+1)}\\cdot \\ldots \\ldots \\cdot X_{\\sigma (l_{j-1})}\\mid X_{\\sigma (l_{j-1}+1)}\\cdot \\ldots \\cdot X_{\\sigma (k)},$ where separators show the ascendingly indexed components which enter into power series of $\\log $ .",
"Considering a given permutation $\\sigma $ , the placement of the separators is either necessary (in case $\\sigma (i)>\\sigma (i+1)$ ), or optional (in case $\\sigma (i)<\\sigma (i+1)$ ).",
"Summing over the $2^{\\operatorname{asc}(\\sigma )}$ many optional possibilities, the coefficient of $X_{\\sigma (1)}\\cdot \\ldots \\cdot X_{\\sigma (k)}$ is $\\sum _{p=0}^{\\operatorname{asc}(\\sigma )}\\frac{(-1)^{\\operatorname{des}(\\sigma )+p}}{\\operatorname{des}(\\sigma )+1+p}\\begin{pmatrix}\\operatorname{asc}(\\sigma )\\\\p\\end{pmatrix}.$ This simplifies according to the combinatorial identity $\\sum _{p=0}^{r}\\frac{(-1)^{p}}{d+1+p}\\binom{r}{p}=\\frac{d!r!}{(d+1+r)!",
"}$ .",
"(But see the approach of Mielnik and Plebański later.)",
"Actually, the RHS of (REF ) appears already in Solomon [31], 1968, but not directly in the context of Magnus expansion; however, see Theorem REF later.",
"We can give an estimate for the convergence of the Magnus expansion as follows: Let $\\Theta _k$ be $\\frac{1}{k!",
"}$ times the sum of the absolute value of the coefficients in the RHS of (REF ) for a fixed $k$ .",
"For $x\\ge 0$ we define the absolute Magnus characteristic $\\Theta $ by $\\Theta (x)=\\sum _{k=1}^\\infty \\Theta _kx^k,$ i. e. it is the real function-theoretic exponential generating function associated to the sum of the absolute value of the coefficients in the RHS of (REF ).",
"Then $\\int _{t_1\\le \\ldots \\le t_k\\in I}\\left|\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k))\\right|\\le \\Theta _k\\cdot \\left(\\int |\\mu | \\right)^k,$ and, consequently, $\\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_k\\in I}\\left|\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k))\\right|\\le \\Theta \\left(\\int |\\mu |\\right).",
"$ Thus, if we estimate $\\Theta $ , then we get not only an estimate on the LHS of (REF ), the convergence of the Magnus expansion as a series but an estimate on the LHS of (REF ), the Magnus expansion as an integral on a time-ordered measure space.",
"The following is very classical, cf.",
"Comtet [8], Graham, Knuth, Patashnik [15], or Petersen [28]: Theorem 1.3 (Euler, 1755) Let $A(n,m)$ denote the number of permutations $\\sigma \\in \\Sigma _n$ such that $\\operatorname{des}(\\sigma )=m$ .",
"(These are the Eulerian numbers.)",
"Consider the exponential generating function $\\sum _{0\\le m< n}^\\infty \\frac{A(n,m)}{n!}",
"u^mv^{n-1-m}.",
"$ Then the following hold: (a) The exponential generating function is analytically equal to, i. e. it has the same Taylor expansion at $(0,0)$ as, the meromorphic function $\\widetilde{G}(u,v)={\\left\\lbrace \\begin{array}{ll}\\frac{\\mathrm {e}^u-\\mathrm {e}^v}{u\\mathrm {e}^v-v\\mathrm {e}^u}&u\\ne v,\\\\\\frac{1}{1-u}&u= v.\\end{array}\\right.",
"}$ This function is analytic around $(0,0)$ , with poles in the real quadrant $u,v>0$ at $u+v={\\left\\lbrace \\begin{array}{ll}\\frac{\\frac{u}{v}+1}{\\frac{u}{v}-1}\\log \\frac{u}{v}&\\frac{u}{v}\\ne 1,\\\\2&\\frac{u}{v}=1,\\end{array}\\right.",
"}$ using the `rational polar' coordinates $u+v,\\frac{u}{v}$ .",
"(b) Regarding absolute convergence, for $u,v\\ge 0$ , the generating function gives the real function $G(u,v)$ , which is finite and equal to $\\widetilde{G}(u,v)$ if $u=0 \\quad \\text{or}\\quad v=0\\quad \\text{or}\\quad u+v<{\\left\\lbrace \\begin{array}{ll}\\frac{\\frac{u}{v}+1}{\\frac{u}{v}-1}\\log \\frac{u}{v}&\\frac{u}{v}\\ne 1,\\\\2&\\frac{u}{v}=1,\\end{array}\\right.",
"}$ and $ G(u,v)=+\\infty $ otherwise.",
"Based on the image of 1 in the permutations, one can develop a recursion for the Eulerian numbers, which, in terms of the generating function, formally reads as the radial differential equation $u\\frac{\\partial }{\\partial u}f(u,v)+v\\frac{\\partial }{\\partial v}f(u,v)+f(u,v)=1+(u+v)f(u,v)+uvf(u,v)^2, $ $f(0,0)=1.",
"$ (In fact, analiticity around $(0,0)$ and (REF ) implies (REF ).)",
"The function given in the statement solves this equation.",
"The nature of the poles follows from elementary considerations.",
"The statement about absolute convergence follows from restricting in radial directions and considering positive analiticity.",
"(Originally, Euler computes $1+t\\widetilde{G}(t,tz)=\\frac{1-z}{\\mathrm {e}^{t(z-1)}-z} $ , but the idea is the same.)",
"Strangely, Mielnik and Plebański, who have the generating function, are not interested in metric estimates, and miss to give the following estimate.",
"Theorem 1.4 The absolute Magnus characteristic is given by $\\Theta (x)=\\int _{t=0}^x G(t,x-t)\\,\\mathrm {d}t=x\\int _{t=0}^1 G(tx,(1-t)x)\\,\\mathrm {d}t.$ In particular, $\\Theta (x)<+\\infty $ if $0\\le x<2$ ; and $\\Theta (x)=+\\infty $ if $2\\le x$ .",
"According to (REF ), $\\Theta (x)=\\sum _{0\\le m< n}^\\infty \\frac{A(n,m)}{n!}",
"\\cdot \\frac{m!(n-1-m)!}{n!}",
"x^n.$ Then the equality in the statement follows form the beta function identity $ \\frac{m!(n-1-m)!}{n!",
"}x^n=\\int _{t=0}^x t^m (x-t)^{n-1-m}\\,\\mathrm {d}t.$ Now, $\\theta (x)<+\\infty $ if $0\\le x<2$ , because the integrand is finite and continuous (it is easy to see that for the poles $u+v\\ge 2$ ).",
"On the other hand, a simple function estimate shows $\\Theta (2)=+\\infty $ .",
"Corollary 1.5 (Moan, Oteo [26] ) In the Banach algebra setting, the Magnus expansion converges absolutely (even as a time-ordered integral, in the sense of (REF )) if $\\int |\\phi |<2.$ $\\Box $ In fact, Moan and Oteo extend the work of Thompson [33] (see the special case of the Goldberg presentation later), but using Mielnik, Plebański [21].",
"Thus, they formulate convergence on the unit interval, for $L^\\infty $ norm $<2$ ; but a reparametrization of the measure implies the $L^1$ norm $<2$ convergence statement.",
"For later reference, we give here some terms: $\\Theta (x)=x+\\frac{1}{2}x^2+\\frac{2}{9}x^3+\\frac{7}{72}x^4+\\frac{13}{300}x^5+O(x^6).",
"$ A special case of the Magnus expansion is when $\\phi =X\\mathbf {1}_{[0,1]}.Y\\mathbf {1}_{[0,1]}$ , i. e. when we take constant $X$ function on one unit interval, take the constant $Y$ function on another one, and concatenate them.",
"(We have the following convention to concatenation: If $\\phi _1$ and $\\phi _2$ are measures on the intervals $[a_1,b_1]$ and $[a_2,b_2]$ ; then $\\phi _1.\\phi _2$ is a measure supported on the interval $[a_1+a_2,b_1+b_2]$ such that $(\\phi _1.\\phi _2)(\\theta )={\\left\\lbrace \\begin{array}{ll}\\phi _1(\\theta -b_1)&\\text{if }\\theta \\le a_2+b_1\\\\\\phi _2(\\theta -a_2)&\\text{if }\\theta \\ge a_2+b_1.\\end{array}\\right.",
"}$ This convention has the advantage of being always well-defined and associative.)",
"Then the Magnus integral of order $n$ immediately specifies to $\\Delta _n(X,Y)=\\sum _{j=0}^n\\frac{1}{j!(n-j)!",
"}\\mu _n(\\underbrace{X,\\ldots ,X}_{j\\,\\text{ terms}},\\underbrace{Y,\\ldots ,Y}_{n-j\\text{ terms}}).",
"$ Thus, from the Magnus formula we obtain $\\exp (X)\\exp (Y)=\\exp \\left(\\sum _{n=1}^\\infty \\Delta _n(X,Y) \\right),$ which is valid, as long as the sum in the RHS converges absolutely.",
"This situation can also be adapted to formal variables $X,Y$ ; or, just take certain nilpotent elements in truncated associative free algebras to see that $\\sum _{n=1}^\\infty \\Delta _n(X,Y)=\\log (\\exp (X)\\exp (Y))$ in formal sense.",
"$\\Delta _n(X,Y)$ collects exactly the terms of degree $n$ .",
"From the explicit form of Magnus expansion one can obtain Theorem 1.6 (Goldberg [14], 1956) The coefficient of monomial $M=(X \\vee Y)^{k_1}\\cdot \\ldots \\cdot X^{k_i}Y^{k_{i+1}}\\cdot \\ldots \\cdot (X \\vee Y)^{k_p}$ ($X$ and $Y$ alternating) in $\\log (\\exp (X)\\exp (Y))$ is $c_M=\\int _{t=0}^1 (t-1)^{\\operatorname{des}(M)}t^{\\operatorname{asc}(M)} G_{k_1}(t-1,t)\\cdot \\ldots \\cdot G_{k_p}(t-1,t)\\,\\mathrm {d}t $ where $\\operatorname{des}(M)$ is number of consecutive $YX$ pairs in $M$ , $\\operatorname{asc}(M)$ is number of consecutive $XY$ pairs in $M$ , and $G_n(u,v)$ is the $n$ -summand in (REF ), i. e. an Eulerian polynomial (in bivariate form) divided by $n!$ .",
"(After Helmstetter, [16].)",
"Let $\\deg _X(M)$ be sum of the exponents $k_i$ belonging to $X$ , and let $\\deg _Y(M)$ be sum of the exponents $k_i$ belonging to $Y$ .",
"Consider (REF ) in the case when the the first $\\deg _X(M)$ many variables are substituted by $X$ , and the remaining $\\deg _X(M)$ many variables are substituted by $Y$ .",
"Examine those permutations which lead to $M$ , and compute the generating polynomial of their descents ($u$ ) and ascents ($v$ ).",
"$M$ itself introduces ordered partitions of the variables.",
"There are $\\frac{\\deg _X(M)!\\deg _Y(M)!",
"}{k_1!\\cdot \\ldots \\cdot k_p!",
"}$ many possible partitions.",
"Inside each partition coming from $(X\\vee Y)^{k_i}$ , the generating polynomial is $k_i!G_{k_i}(u,v)$ ; and there are $u^{\\operatorname{des}(M)}$ and $v^{\\operatorname{asc}(M)}$ coming from the internal boundaries.",
"Thus the generating polynomial is $\\deg _X(M)!\\deg _Y(M)!\\, u^{\\operatorname{des}(M)}v^{\\operatorname{asc}(M)}G_{k_1}(u,v)\\cdot \\ldots \\cdot G_{k_p}(u,v).$ We get the coefficient $c_M$ by replacing $u^av^b$ with $(-1)^a\\frac{a!b!}{(a+1+b)!",
"}$ .",
"According to the beta function identity, this corresponds exactly to integration as in the statement.",
"Yet, according to (REF ), we have to divide by $\\deg _X(M)!\\deg _Y(M)!",
"$ .",
"For $x,y\\ge 0$ , we can define the absolute Goldberg characteristic as $\\Gamma (x,y)=\\sum _{M\\,:\\,X,Y\\text{-monomial}}|c_M|x^{\\deg _X(M)}y^{\\deg _Y(M)},$ the function-theoretic generating function associated to the sum of the absolute value of coefficients of appropriate $X,Y$ -degree in the formal expansion of $\\log (\\exp (X)\\exp (Y))$ .",
"(Or, alternatively, it is obtained by turning the coefficients in the power series expansion of $\\log (\\exp (X)\\exp (Y))$ nonnegative, and then turning the variables commutative.)",
"The specialization from the Magnus expansion yields $\\Gamma (x,y)\\le \\Theta (x+y).$ In particular, the formal expansion of $\\log (\\exp (X)\\exp (Y))$ converges absolutely in the Banach algebra setting if $|X|+|Y|<2$ .",
"Considering the explicit nature of (REF ), and the fact that Goldberg [14] also computes the generating functions of the coefficients (with fixed $p$ ), in theory, one could obtain much sharper estimates for $\\Gamma (x,y)$ .",
"But, in practice, this is not so straightforward.",
"However, in next section, we will show that the bound 2 is not sharp in the BCH case."
],
[
"The resolvent approach",
"We have seen, in the Banach algebraic setting, that the Magnus expansion can be treated quite directly.",
"In particular, the clever arguments of Mielnik, Plebański [21], and their very remarkable symbolic functional calculus for time-ordered products, are not needed in their full power.",
"However, their ideas come handy when we inquire about finer analytical details.",
"In Banach algebras, we can define the logarithm of $A\\in \\mathfrak {A}$ by $\\log A=\\int _{\\lambda =0}^1 \\frac{A-1}{\\lambda +(1-\\lambda )A }\\,d\\lambda =\\int ^{0}_{s=-\\infty }\\frac{A-1}{(1-s)(A-s)}\\, \\mathrm {d}s, $ which is well-defined if and only if $\\operatorname{sp}(A)$ is disjoint from the closed negative real axis.",
"For the sake of simplicity, we call $A$ $\\log $ -able, if it has this spectral property.",
"It did not really matter before, but here we clarify that we consider (REF ) as our official definition of $\\log $ .",
"This definition also works perfectly well in the formal setting.",
"Notice the resolvent expression $\\mathcal {R}(A,\\lambda ):=\\frac{A-1}{\\lambda +(1-\\lambda )A}.$ From the power series form (cf.",
"Lemma REF later), or directly, it is easy to see that the resolvent expression satisfies the formal noncommutative differential rule $\\mathcal {R}(A(1+\\varepsilon ),\\lambda )-\\mathcal {R}(A,\\lambda )=(1+\\lambda \\mathcal {R}(A,\\lambda ))\\varepsilon (1+(\\lambda -1) \\mathcal {R}(A,\\lambda ))+O(\\varepsilon )^2.",
"$ Applying this to $A=\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k-1})$ and $\\varepsilon =\\exp (X_k)-1$ , we find $\\mathcal {R}\\,&(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k),\\lambda )=\\\\=\\,&\\mathcal {R}(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k-1}),\\lambda )\\\\&+X_k\\\\&+\\lambda \\mathcal {R}(\\exp (X_1)\\ldots \\cdot \\exp (X_{k-1}),\\lambda )X_k \\\\&+(\\lambda -1)\\cdot X_k\\mathcal {R}(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k-1}),\\lambda )\\\\&+\\lambda (\\lambda -1)\\mathcal {R}(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k-1}),\\lambda ) X_k\\mathcal {R}(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k-1}),\\lambda )\\\\&+H(X_1,\\ldots ,X_k),$ where $H(X_1,\\ldots ,X_k)$ contains some terms where some variables $X_i$ have multiplicity more than 1.",
"Using this as an induction step, one can prove that, in terms of formal variables, $\\mathcal {R}(\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k),\\lambda )=\\\\=\\sum _{\\begin{array}{c}\\mathbf {i}=(i_1,\\ldots , i_l)\\in \\lbrace 1,\\ldots ,k\\rbrace ^l\\\\ i_a\\ne i_b,\\,l\\ge 1 \\end{array}}\\lambda ^{\\operatorname{asc}(\\mathbf {i})}(\\lambda -1)^{\\operatorname{des}(\\mathbf {i})} X_{i_1}\\cdot \\ldots \\cdot X_{i_l} +H(X_1,\\ldots ,X_k), $ where $H(X_1,\\ldots ,X_k)$ collects the terms with multiplicities in the variables.",
"This is the starting point of the arguments of Mielnik, Plebański [21].",
"If we integrate (REF ) in $\\lambda \\in [0,1]$ , then the beta function identity yields $\\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_k),\\lambda )=\\\\=\\sum _{\\begin{array}{c}\\mathbf {i}=(i_1,\\ldots , i_l)\\in \\lbrace 1,\\ldots ,k\\rbrace ^l\\\\ i_a\\ne i_b,\\,l\\ge 1 \\end{array}}(-1)^{\\operatorname{des}(\\mathbf {i})}\\frac{\\operatorname{asc}(\\mathbf {i})!\\operatorname{des}(\\mathbf {i})!}{l!}",
"X_{i_1}\\cdot \\ldots \\cdot X_{i_l} +H(X_1,\\ldots ,X_k).",
"$ Comparing this to (REF ), gives a proof of (REF ).",
"Their approach also provides the generator function naturally.",
"Let $R^{(\\lambda )}(X_1,\\ldots ,X_n)=\\sum _{\\sigma \\in \\Sigma _n} \\lambda ^{\\operatorname{asc}(\\sigma )}(\\lambda -1)^{\\operatorname{des}(\\sigma )} X_{\\sigma (1)}\\cdot \\ldots \\cdot X_{\\sigma (n)} .$ Simple inspection yields the recursions $R^{(\\lambda )}(X_1,\\ldots ,X_n)=R^{(\\lambda )}(X_1,\\ldots ,X_{n-1}) \\\\+\\lambda R^{(\\lambda )}(X_1,\\ldots ,X_{n-1})\\cdot X_n+(\\lambda -1)X_n\\cdot R^{(\\lambda )}(X_1,\\ldots ,X_{n-1})\\\\+\\lambda (\\lambda -1)\\!\\!\\!\\!\\!\\sum _{\\begin{array}{c}I_1 \\dot{\\cup }I_2=\\lbrace 1,\\ldots ,n-1\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}\\!\\!\\!\\!\\!R^{(\\lambda )}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}})\\cdot X_n \\cdot R^{(\\lambda )}(X_{i_{2,1}},\\ldots ,X_{i_{2,l_2}}) ,$ and $R^{(\\lambda )}(X_1,\\ldots ,X_n)=R^{(\\lambda )}(X_2,\\ldots ,X_{n}) \\\\+(\\lambda -1) R^{(\\lambda )}(X_2,\\ldots ,X_{n})\\cdot X_1+\\lambda X_1\\cdot R^{(\\lambda )}(X_2,\\ldots ,X_{n})\\\\+\\lambda (\\lambda -1)\\!\\!\\!\\!\\!\\sum _{\\begin{array}{c}I_1 \\dot{\\cup }I_2=\\lbrace 2,\\ldots ,n\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}\\!\\!\\!\\!\\!R^{(\\lambda )}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}})\\cdot X_1 \\cdot R^{(\\lambda )}(X_{i_{2,1}},\\ldots ,X_{i_{2,l_2}});$ but, for example, (REF ) can also be interpreted as the translation (REF ), etc.",
"One can compute the formal exponential generating function $G^{(\\lambda )}(x,u,v)=\\sum _{0\\le m< n}^\\infty A(n,m)(\\lambda -1)^{m}\\lambda ^{n-1-m}u^{m}v^{n-1-m} \\frac{x^n}{n!",
"}$ for the coefficient scheme of $R^{(\\lambda )}$ by the differential equation $G^{(\\lambda )\\prime }(x)=(1+\\lambda vG^{(\\lambda )}(x,u,v))(1+(\\lambda -1)uG^{(\\lambda )}(x,u,v) )$ $G^{(\\lambda )}(0,u,v)=0,$ (cf.",
"(REF ) / (REF )).",
"This gives $G^{(\\lambda )}(x,u,v)=\\frac{\\mathrm {e}^{x(v\\lambda +u(1-\\lambda ))}-1}{v\\lambda +u(1-\\lambda )\\mathrm {e}^{x(v\\lambda +u(1-\\lambda ))}}.$ for $v\\lambda \\ne u(1-\\lambda )$ .",
"Substituting $u(\\lambda -1)\\mapsto u$ , $v\\lambda \\mapsto v$ , $x\\mapsto 1$ formally yields $G(u,v)=\\frac{\\mathrm {e}^{v-u}-1}{v-u\\mathrm {e}^{v-u}}, $ in agreement to the previous formula.",
"(In fact, $G^{(\\lambda )}(x,u,v)=xG(x(\\lambda -1)u,x\\lambda v )$ .)",
"This is very much the same as the proof of Theorem REF , but individual expressions has interpretation with resolvents.",
"Theorem 2.1 (The resolvent formula of Mielnik, Plebański [21], analytic version.)",
"Let $\\phi $ be an $\\mathfrak {A}$ -valued measure.",
"If $\\sum _{k=1}^\\infty \\left|\\int _{\\mathbf {t}=(t_1,\\ldots ,t_k)\\in I^k}\\lambda ^{\\operatorname{asc}(\\mathbf {t})}(\\lambda -1)^{\\operatorname{des}(\\mathbf {t})}\\phi (t_1)\\ldots \\phi (t_k)\\right| <+\\infty , $ then $\\mathcal {R}(\\operatorname{exp_{R}}(\\phi ),\\lambda )=\\sum _{k=1}^\\infty \\int _{\\mathbf {t}=(t_1,\\ldots ,t_k)\\in I^k}\\lambda ^{\\operatorname{asc}(\\mathbf {t})}(\\lambda -1)^{\\operatorname{des}(\\mathbf {t})}\\phi (t_1)\\ldots \\phi (t_k).",
"$ Indeed, (REF ) supplies sufficiently many identities to prove that $(1-\\lambda )+\\lambda \\operatorname{exp_{R}}(\\phi )$ times the RHS of (REF ) contracts to $\\operatorname{exp_{R}}(\\phi )-1$ .",
"And this implies the statement.",
"Theorem 2.2 (The logarithmic version of the Magnus formula.)",
"Let $\\phi $ be an $\\mathfrak {A}$ -valued measure.",
"If $\\sum _{k=1}^\\infty \\left|\\int _{\\mathbf {t}=(t_1,\\ldots ,t_k)\\in I^k}\\lambda ^{\\operatorname{asc}(\\mathbf {t})}(\\lambda -1)^{\\operatorname{des}(\\mathbf {t})}\\phi (t_1)\\ldots \\phi (t_k)\\right| $ is bounded on $\\lambda \\in [0,1]$ , then $\\sum _{n=1}^\\infty |\\mu _{\\mathrm {R}[n]}(\\phi )|<+\\infty $ holds, $\\operatorname{exp_{R}}(\\phi )$ is $\\log $ -able, and $\\mu _{\\mathrm {R}}(\\phi )=\\log \\operatorname{exp_{R}}(\\phi ).",
"$ The statement (REF ) looks innocent, but it tells something stronger about the spectral properties of the objects in question.",
"It implies $\\operatorname{sp}(\\operatorname{exp_{R}}(\\phi ))\\cap (-\\infty ,0] =\\emptyset $ and $\\operatorname{sp}(\\mu _{\\mathrm {R}}(\\phi ))\\subset \\lbrace z\\in \\mathbb {C}\\,:\\,|\\operatorname{Im}z|<\\pi \\rbrace .$ Integrating (REF ) on $\\lambda \\in [0,1]$ , and bringing this outer integral into $|\\cdot |$ yields $\\sum _{n=1}^\\infty |\\mu _{\\mathrm {R}[n]}(\\phi )|<+\\infty $ immediately.",
"The requirements of the previous theorem hold, and our statement follows from integrating (REF ) on $\\lambda \\in [0,1]$ .",
"We say that the $\\mathfrak {A}$ -valued measure is spectrally short, if for any $t\\in \\operatorname{D}(0,1)$ , the spectrum of $\\operatorname{exp_{R}}(t\\cdot \\phi )$ does not intersect the $(-\\infty ,0]$ .",
"(The algebra $\\mathfrak {A}$ can be complex, or it can be real but then complexified.)",
"With this terminology, we can state: Theorem 2.3 If $\\phi $ is an $\\mathfrak {A}$ -valued measure, and $\\int |\\phi |<2$ , then $\\phi $ is spectrally short.",
"(a) We can estimate (REF ) by $\\sum _{k=1}^\\infty \\int _{\\mathbf {t}=(t_1,\\ldots ,t_k)\\in I^k}\\lambda ^{\\operatorname{asc}(\\mathbf {t})}(1-\\lambda )^{\\operatorname{des}(\\mathbf {t})}\\left|\\phi (t_1)|\\cdot \\ldots \\cdot |\\phi (t_k)\\right|,$ which, in turn, can be estimated by $G\\left((1-\\lambda ) \\int |\\phi |, \\lambda \\int |\\phi | \\right)\\int |\\phi |,$ which will be finite.",
"This and the previous theorem shows that $\\operatorname{exp_{R}}(\\phi )$ is $\\log $ -able.",
"The estimate remains valid if we multiply $\\phi $ by $t\\in \\operatorname{D}(0,1)$ .",
"In what follows, we use the notation $R^{(\\lambda )}_{\\mathrm {R}[k]}(\\phi )=\\int _{\\mathbf {t}=(t_1,\\ldots ,t_k)\\in I^k}\\lambda ^{\\operatorname{asc}(\\mathbf {t})}(\\lambda -1)^{\\operatorname{des}(\\mathbf {t})}\\phi (t_1)\\ldots \\phi (t_k),$ analogously to the terms of the Magnus expansion.",
"Then $\\mu _{\\mathrm {R}[k]}(\\phi )=\\int _{\\lambda =0}^1R^{(\\lambda )}_{\\mathrm {R}[k]}(\\phi )\\,\\mathrm {d}\\lambda .$ Remark 2.4 We see that that in the critical case $\\smallint |\\phi |=2$ , the divergence of the Magnus expansion comes from around $G(1,1)$ ; corresponding to the Cayley transform $\\mathcal {R}(\\operatorname{exp_{R}}(\\phi ),\\frac{1}{2})$ .",
"if we manage to provide absolute convergence on that area, the (logarithmic) Magnus formula will converge better.",
"The spectral shortness is a quite strong property.",
"In the setting of the previous theorem we already know that that the Magnus expansion is absolutely convergent (even in time-ordered sense), but we can also derive the absolute convergence from the spectral shortness itself: Theorem 2.5 (Essentially, Mityagin [18], Moan, Niesen [25], Casas [6]) If $\\phi $ is a spectrally short $\\mathfrak {A}$ -valued measure, then $\\log \\operatorname{exp_{R}}(t\\phi )$ is well-defined, and analytic for $t$ in a disk $\\operatorname{\\mathring{D}}(0,R)$ , with $R>1$ .",
"On such a such a disk $\\operatorname{\\mathring{D}}(0,R)$ ($R$ can be infinite), $\\log (\\operatorname{exp_{R}}(t\\phi )) =\\sum _{k=1}^{\\infty } \\mu _{\\mathrm {R}[k]}(\\phi ) t^k $ holds.",
"In particular, the convergence radius of the series is larger than 1, and the Magnus expansion converges absolutely.",
"The elements $\\lambda +(1-\\lambda )\\operatorname{exp_{R}}(t\\phi )$ are invertible for $(\\lambda ,t)\\in [0,1]\\times \\operatorname{D}(0,1)$ , and, due to continuity, even in a neighborhood of $[0,1]\\times \\operatorname{D}(0,1)$ .",
"This proves that $f(t)=\\log \\operatorname{exp_{R}}(t\\phi )$ is well-defined, and analytic for $t$ in a neighborhood of $\\operatorname{D}(0,1)$ .",
"We know that the power series expansion of $f(t)$ is given by $f_k=\\mu _{\\mathrm {R}[k]}(\\phi )$ around 0.",
"Then, a standard application of the generalized Cauchy formula shows that the growth of the coefficients is limited by the analytic radius; which we know to be larger than 1.",
"For $A\\in \\mathfrak {A}$ , we define its Magnus exponent as $\\operatorname{\\mathcal {M}}(A):=\\inf \\left\\lbrace \\int |\\phi |\\,:\\,\\operatorname{exp_{R}}(\\phi )=A \\right\\rbrace .",
"$ Theorem 2.6 If $\\operatorname{\\mathcal {M}}(A)<2$ , then $A$ is $\\log $ -able, and $|\\log A|\\le \\Theta (\\operatorname{\\mathcal {M}}(A)).$ Whenever $A=\\operatorname{exp_{R}}(\\phi )$ , $\\int |\\phi |<2$ , by Theorems REF and REF , we know that $|\\log A|=|\\mu _{\\mathrm {R}}(\\phi )|\\le \\Theta (\\int |\\phi |)$ holds.",
"The resolvent approach is also enlightening in the sense that it shows that the integrals in Theorems REF and REF are due to no accidental trickery but come from the integral presentation of the logarithm.",
"Let us consider the sharpness of the bound 2.",
"We can consider the measures $\\Xi _{n}= X_{1}\\underbrace{\\mathbf {1}_{[0,1/2^n]}.\\ldots .",
"X_{2^n}\\mathbf {1}_{[0,1/2^n]}}_{2^n\\text{ terms}},$ where the $X_j$ are formal noncommutative variables with $\\ell _1$ norm.",
"If $c\\ge 0$ , then $\\smallint |c\\Xi _{n}|=c$ , and it is easy to show that $\\lim _{n\\rightarrow \\infty }|\\mu (c\\Xi _{n})|=\\Theta (c)$ .",
"This shows that one cannot get a better universal growth estimate than the one provided by $\\Theta $ .",
"This is, however, not the same thing as convergence behaviour.",
"Our estimates were based on the 'worst case scenario' for the measure $\\phi $ .",
"This worst case scenario, however, is not realized, as Lebesgue-Bochner functions retain some remnants of continuity.",
"First, take another look to the time-ordered resolvent expression.",
"Let $\\Theta ^{(\\lambda )}_n$ be $1/n!$ times the sum of the absolute value of the coefficients in $R^{(\\lambda )}(X_1,\\ldots ,X_n)$ ; and let $\\Theta ^{(\\lambda )}(x)=\\sum _{n=1}^{\\infty }\\Theta ^{(\\lambda )}_n(x)$ be the associated generating function.",
"Then, due to (REF ), this can majorized by (in every Taylor term) by the solution $G^{(\\lambda )}(x)$ of the differential equation (IVP) $G^{(\\lambda )\\prime }(x)=(1+|\\lambda | G^{(\\lambda )}(x) )(1+|1-\\lambda | G^{(\\lambda )}(x) ), $ $G^{(\\lambda )}(0)=0.$ From the discussion of the generating function earlier we find $\\Theta ^{(\\lambda )}(x)=G^{(\\lambda )}(x)=xG(|1-\\lambda |x,|\\lambda |x).$ In particular, $G^{(\\lambda )}(x)$ not only majorizes $\\Theta ^{(\\lambda )}(x)$ , but they are equal.",
"Now, the recursive nature of $R^{(\\lambda )}(X_1,\\ldots )$ is also expressed by the phenomenon that it allows a continuation form.",
"Indeed, from the recursion formula (REF ), it is easy to show that there are noncommutative polynomials $\\widehat{R}^{(\\lambda )}(E,Y_1,\\ldots ,Y_n)$ , without multiplicies in the $Y_i$ such that $R^{(\\lambda )}(X_1,\\ldots ,X_k,Y_1,\\ldots ,Y_n) =\\sum _{\\begin{array}{c}I_1\\dot{\\cup }\\ldots \\dot{\\cup }I_s=\\lbrace 1,\\ldots ,k\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}} \\\\\\underbrace{\\ldots \\cdot \\overbrace{R^{(\\lambda )}_{l_1}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}})}^{E\\rightarrow }\\cdot \\ldots \\cdot \\overbrace{R^{(\\lambda )}_{l_j}(X_{i_{j,1}},\\ldots ,X_{i_{j,l_j}})}^{E\\rightarrow }\\cdot \\ldots \\cdot \\overbrace{R^{(\\lambda )}_{l_s}(X_{i_{s,1}},\\ldots ,X_{i_{s,l_s}})}^{E\\rightarrow }\\cdot \\ldots }_{\\text{the part of $\\widehat{R}^{(\\lambda )}(E,Y_1,\\ldots ,Y_n)$ where the multiplicity of $E$ is $s$; but the $E$'s are replaced as indicated.",
"}}$ Let $\\widehat{\\Theta }^{(\\lambda )}_{s,n}$ be $1/n!$ times the sum of the coefficients in that part $\\widehat{R}^{(\\lambda )}(E,Y_1,\\ldots ,Y_n)$ , where the multiplicity of $E$ is $s$ .",
"Let $\\widehat{\\Theta }^{(\\lambda )}(x,y)=\\sum _{n=0}^\\infty \\sum _{s=0}^\\infty \\widehat{\\Theta }^{(\\lambda )}_{s,n}x^sy^n$ .",
"This $\\widehat{\\Theta }^{(\\lambda )}(x,y)$ is majorized by the solution of the differential equation $\\frac{\\partial }{\\partial y}\\widehat{G}^{(\\lambda )}(x,y)=(1+|\\lambda | \\widehat{G}^{(\\lambda )}(x,y) )(1+|1-\\lambda | \\widehat{G}^{(\\lambda )}(x,y) ), $ $\\widehat{G}^{(\\lambda )}(x,0)=x.$ Due to the sharpness of $\\Theta ^{(\\lambda )}(x)=G^{(\\lambda )}(x)$ , we actually know that the solution satiesfies $\\widehat{\\Theta }^{(\\lambda )}(x,y)=\\widehat{G}^{(\\lambda )}(x,y)=G^{(\\lambda )}\\left(\\left(G^{(\\lambda )}\\right)^{-1}(x)+y\\right).$ Similar statement holds if the roles of $X_j$ and $Y_j$ are interchanged.",
"Theorem 2.7 Suppose that $\\phi _1,\\phi _2,\\phi _3$ are $\\mathfrak {A}$ -valued measures.",
"Then $\\sum _{n=1}^\\infty |R^{\\lambda }_{\\mathrm {R}[n]}(\\phi _1.\\phi _2.\\phi _3)|\\le \\widehat{G}^{(\\lambda )}\\left(\\sum _{n=1}^\\infty |R^{\\lambda }_{\\mathrm {R}[n]}(\\phi _2)|,\\int |\\phi _1|+\\int |\\phi _3|\\right).$ If $\\phi _1$ is missing, then this is just (REF ) integrated.",
"However, it can be augmented in the other direction, using the same principle.",
"Theorem 2.8 Suppose that $\\phi $ is a $\\mathfrak {A}$ -valued measure, which, we remind, is a Lebesgue-Bochner integrable function times the Lebesgue measure on an interval.",
"Suppose that $\\smallint |\\phi |=2$ .",
"Then the boundedness of (REF ) holds, thus the (logarithmic) Magnus formula holds.",
"In particular, the Magnus series is absolute convergent.",
"Suppose that $\\phi (t)=h(t)\\mathbf {1}_I$ .",
"Let $\\varepsilon >0$ .",
"Then $h(t)$ can be approximated by step-functions (in $L^1$ norm) arbitrarily well.",
"This implies that there is a nontrivial subinterval $I^{\\prime }\\subset I$ and a nonzero element $a\\in \\mathfrak {A}$ such that $\\int _{I^{\\prime }} |h(t)-a|\\le \\varepsilon |a|\\cdot |I^{\\prime }|$ .",
"Let $k(t)=h(t)-a$ .",
"Note that $R^{(\\lambda )}(X_1,X_2)=\\lambda X_1X_2-(1-\\lambda )X_2X_1.$ Thus $\\Bigl |\\int _{t_1\\le t_2\\in I^{\\prime }}&R^{(\\lambda )}(h(t_1),h(t_2))\\,\\mathrm {d}t_1\\,\\mathrm {d}t_2\\Bigr |\\le \\\\\\le &\\min (\\lambda ,1-\\lambda )\\int _{t_1,t_2\\in I^{\\prime }}|a|\\,|k(t_2)|+|k(t_1)|\\,|a|+|k(t_1)|\\,|k(t_2)| \\,\\mathrm {d}t_1\\,\\mathrm {d}t_2\\\\&+\\left|\\frac{1}{2}-\\lambda \\right|\\int _{t_1,t_2\\in I^{\\prime }} |a|^2+|a|\\,|k(t_2)|+|k(t_1)|\\,|a|+|k(t_1)|\\,|k(t_2)| \\,\\mathrm {d}t_1\\,\\mathrm {d}t_2\\\\\\le &\\frac{1}{2}|a|^2|I^{\\prime }|^2\\left(2\\left|\\frac{1}{2}-\\lambda \\right|+2\\varepsilon +\\varepsilon ^2\\right).$ On the other hand, in the formal estimate we count at least $\\int _{t_1,t_2\\in I^{\\prime }} |a|^2-|a|\\,|k(t_2)|-|k(t_1)|\\,|a|-|k(t_1)|\\,|k(t_2)|\\,\\mathrm {d}t_1\\,\\mathrm {d}t_2\\ge \\frac{1}{2}|a|^2|I^{\\prime }|^2\\left(1-2\\varepsilon -\\varepsilon ^2\\right).",
"$ Between the two estimates there is a gap $\\frac{1}{2}|a|^2|I^{\\prime }|^2(1-2\\varepsilon -\\varepsilon ^2)-\\frac{1}{2}|a|^2|I^{\\prime }|^2\\left(2\\left|\\frac{1}{2}-\\lambda \\right|+2\\varepsilon +\\varepsilon ^2\\right).$ If, say, $\\left|\\frac{1}{2}-\\lambda \\right|\\le \\frac{1}{8}$ and $\\varepsilon =\\frac{1}{8}$ , then this gap is at least $\\frac{7}{64}|a|^2|I^{\\prime }|^2$ .",
"Thus $\\sum _{n=1}^\\infty |R^{\\lambda }_{\\mathrm {R}[n]}(\\phi |_{I^{\\prime }})|\\le G^{(\\lambda )}(\\smallint |\\phi |_{I^{\\prime }}|)-\\frac{7}{64}|a|^2|I^{\\prime }|^2.$ This fixed delay, however, according to Theorem REF , allows uniform absolute convergence in a neighborhood of the Cayley transform, thus on the whole resolvent segment.",
"Using the resolvent approach, we show how to improve the convergence estimate in the Baker–Campbell–Hausdorff case.",
"Lemma 2.9 If $X,Y$ are formal variables, then $\\mathcal {R}((\\exp X)(\\exp Y),\\lambda )=\\sum _{k=0}^\\infty \\biggl (&(\\lambda -1)^k\\lambda ^k\\mathcal {R}(\\exp Y,\\lambda )(\\mathcal {R}(\\exp X,\\lambda )\\mathcal {R}(\\exp Y,\\lambda ))^{k}\\\\+&(\\lambda -1)^k\\lambda ^k\\mathcal {R}(\\exp X,\\lambda )(\\mathcal {R}(\\exp Y,\\lambda )\\mathcal {R}(\\exp X,\\lambda ))^{k}\\\\+&(\\lambda -1)^{k}\\lambda ^{k+1}(\\mathcal {R}(\\exp X,\\lambda )\\mathcal {R}(\\exp Y,\\lambda ))^{k+1} \\\\+&(\\lambda -1)^{k+1}\\lambda ^{k}(\\mathcal {R}(\\exp Y,\\lambda )\\mathcal {R}(\\exp X,\\lambda ))^{k+1} \\qquad \\,\\,\\biggr ).$ One can replace $\\exp X$ and $\\exp Y$ by other formal perturbations of 1.",
"Let $x=1-(\\exp X)$ , $y=1-(\\exp -Y)$ .",
"Then $\\mathcal {R}((\\exp X)&(\\exp Y),\\lambda )=\\\\&=\\frac{(\\exp X)(\\exp Y)-1}{\\lambda +(1-\\lambda )(\\exp X)(\\exp Y)}\\\\&=((\\exp X)(\\exp Y)-1)(\\exp Y )^{-1}(\\exp Y)(\\lambda +(1-\\lambda )(\\exp X)(\\exp Y))^{-1}\\\\&=((\\exp X)-(\\exp -Y))(\\lambda (\\exp -Y)+(1-\\lambda )(\\exp X))^{-1}\\\\&=(y-x)(1-(1-\\lambda )x-\\lambda y)^{-1}.$ Using the formal expansion $\\frac{1}{1-\\tilde{x}-\\tilde{y}}=1+\\sum _{k=1}^\\infty \\biggl (\\left(\\frac{\\tilde{x}}{1-\\tilde{x}}\\frac{\\tilde{y}}{1-\\tilde{y}}\\right)^k+\\left(\\frac{\\tilde{y}}{1-\\tilde{y}}\\frac{\\tilde{x}}{1-\\tilde{x}}\\right)^k+\\\\+\\frac{\\tilde{y}}{1-\\tilde{y}}\\left(\\frac{\\tilde{x}}{1-\\tilde{x}}\\frac{\\tilde{y}}{1-\\tilde{y}}\\right)^{k-1}+\\frac{\\tilde{x}}{1-\\tilde{x}}\\left(\\frac{\\tilde{y}}{1-\\tilde{y}}\\frac{\\tilde{x}}{1-\\tilde{x}}\\right)^{k-1}\\biggr ),$ we find $\\mathcal {R}((\\exp X)&(\\exp Y),\\lambda )=\\\\&=y\\left(1+ \\frac{\\lambda y}{1-\\lambda y}\\right)\\left(1+\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}+\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}\\frac{\\lambda y}{1-\\lambda y}+\\ldots \\right)\\\\&-x\\left(1+ \\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}\\right)\\left(1+\\frac{\\lambda y}{1-\\lambda y}+\\frac{\\lambda y}{1-\\lambda y}\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}+\\ldots \\right)\\\\&=\\frac{y}{1-\\lambda y}\\left(1+\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}+\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}\\frac{\\lambda y}{1-\\lambda y}+\\ldots \\right)\\\\&-\\frac{ x}{1-(1-\\lambda ) x}\\left(1+\\frac{\\lambda y}{1-\\lambda y}+\\frac{\\lambda y}{1-\\lambda y}\\frac{(1-\\lambda ) x}{1-(1-\\lambda ) x}+\\ldots \\right).$ Notice that $\\frac{y}{1-\\lambda y}=\\mathcal {R}(\\exp Y,\\lambda ),\\qquad \\frac{x}{1-(1-\\lambda ) x}=-\\mathcal {R}(\\exp X,\\lambda );$ yielding the equality in the statement.",
"The replaceability of the exponentials follows from simple change of variables.",
"For $\\lambda \\in [0,1]$ , let $\\rho _\\lambda (x)$ be the power series expansion of $\\mathcal {R}(\\exp x,\\lambda )$ around $x=0$ .",
"$\\mathcal {R}(\\exp x,\\lambda )$ satisfies the recursion / IVP $y^{\\prime }_\\lambda (x)=1+(1-2\\lambda )y_\\lambda (x)+\\lambda (1-\\lambda )y_\\lambda (x)^2,$ $y^{\\prime }_\\lambda (0)=0.$ Thus, we can estimate $\\rho _\\lambda (x)$ by the solution of the IVP $g^{\\prime }_\\lambda (x)=1+|1-2\\lambda |g_\\lambda (x)+\\lambda (1-\\lambda )g_\\lambda (x)^2,$ $g^{\\prime }_\\lambda (0)=0.$ The solution is given by $g_\\lambda (x)={\\left\\lbrace \\begin{array}{ll}\\dfrac{2}{-|1-2\\lambda |+\\sqrt{-8\\lambda ^2+8\\lambda -1}\\cot \\left(\\dfrac{x}{2}\\sqrt{-8\\lambda ^2+8\\lambda -1}\\right)}&\\text{if }8\\lambda ^2-8\\lambda +1<0,\\\\\\dfrac{4x}{4-\\sqrt{2}x}&\\text{if }8\\lambda ^2-8\\lambda +1=0,\\\\\\dfrac{2}{-|1-2\\lambda |+\\sqrt{8\\lambda ^2-8\\lambda +1}\\coth \\left(\\dfrac{x}{2}\\sqrt{8\\lambda ^2-8\\lambda +1}\\right)}&\\text{if }8\\lambda ^2-8\\lambda +1>0.\\end{array}\\right.",
"}$ In fact, the coefficients of $g_\\lambda $ majorize the coefficients of $\\tilde{\\rho }_\\lambda $ , where the latter is $\\rho _\\lambda (x)$ but the coefficients are turned into nonnegative.",
"We remark that this is exact at $\\lambda =1/2$ , $\\tilde{\\rho }_{1/2}(x)=g_{1/2}(x)=2\\tan \\dfrac{x}{2}, $ cf.",
"(REF ) and (REF ) in Remark REF .",
"Take the power series expansion of $\\mathcal {R}((\\exp X)(\\exp Y),\\lambda )$ , turn all coefficients nonnegative, and change the variables to commutative.",
"In this way we obtain $\\Gamma ^{(\\lambda )}(x,y)$ .",
"The coefficients in the resulted series are majorized by the coefficients in the power series expansion of $\\frac{g_\\lambda (x)+g_\\lambda (y) +g_\\lambda (x)g_\\lambda (y) }{1-\\lambda (1-\\lambda )g_\\lambda (x)g_\\lambda (y) }.$ Absolute convergence is ensured as long as $\\lambda (1-\\lambda )g_\\lambda (|x|)g_\\lambda (|y|)<1.$ Using standard calculus, one can check that $g_\\lambda (x)$ is logarithmically convex on the interval $[0,h(\\lambda )]$ , where $h(\\lambda )={\\left\\lbrace \\begin{array}{ll}\\dfrac{2\\arctan \\left(\\dfrac{\\sqrt{-8\\lambda ^2+8\\lambda -1}}{|1-2\\lambda |}\\right)}{\\sqrt{-8\\lambda ^2+8\\lambda -1}}&\\text{if }8\\lambda ^2-8\\lambda +1<0,\\\\2\\sqrt{2}&\\text{if }8\\lambda ^2-8\\lambda +1=0,\\\\\\dfrac{2\\operatorname{artanh}\\left(\\dfrac{\\sqrt{8\\lambda ^2-8\\lambda +1}}{|1-2\\lambda |}\\right)}{\\sqrt{8\\lambda ^2-8\\lambda +1}}&\\text{if }8\\lambda ^2-8\\lambda +1>0.\\end{array}\\right.",
"}$ Thus, if $x\\in [0,h(\\lambda )]$ , then $\\lambda (1-\\lambda )g_\\lambda \\left(\\dfrac{|x|+|y|}{2}\\right)^2<1$ is sufficient for absolute convergence.",
"It can also be checked that, for $x\\in [0,h(\\lambda )/2]$ , $\\lambda (1-\\lambda )g_\\lambda \\left(x\\right)^2<1.$ This shows that absolute convergence for the resolvent expansion holds, as long as $|X|+|Y|<C_1,$ where $C_1=\\min _{x\\in [0,1]} h(\\lambda )=\\min _{x\\in [0,1/2]} h(\\lambda ).$ Now, $C_1=2.7014\\ldots ,$ which is a significant improvement compared to 2.",
"Theorem 2.10 Let $\\phi =X\\mathbf {1}_{[0,1]}.",
"Y\\mathbf {1}_{[0,1]}$ , and $\\lambda \\in [0,1]$ .",
"Then $\\sum _{n=1}^\\infty |R^\\lambda _{\\mathrm {R}[n]}(\\phi )|&\\le \\Gamma ^{(\\lambda )}(|X|,|Y|)\\\\&\\le \\frac{\\tilde{\\rho }_\\lambda (|X|)+\\tilde{\\rho }_\\lambda (|Y|) +\\tilde{\\rho }_\\lambda (|X|)\\tilde{\\rho }_\\lambda (|Y|) }{1-\\lambda (1-\\lambda )\\tilde{\\rho }_\\lambda (|X|)\\tilde{\\rho }_\\lambda (|Y|) }\\\\&\\le \\frac{g_\\lambda (|X|)+g_\\lambda (|Y|) +g_\\lambda (|X|)g_\\lambda (|Y|) }{1-\\lambda (1-\\lambda )g_\\lambda (|X|)g_\\lambda (|Y|) }$ which are well-defined and finite for $|X|+|Y|<C_1$ .",
"Thus, if $|X|+|Y|<C_1$ holds, then the conditions of Theorem REF are satiesfied, and the (stronger) logarithmic version of Baker-Campbell-Hausdorff formula holds.",
"This a consequence of the previous discussion.",
"Remark 2.11 In estimating $\\tilde{\\rho }_\\lambda (x)$ , one can also deal with coefficients of low order directly, and use estimates only for coefficients of higher order.",
"Such a more computational approach allows to push convergence to $|X|+|Y|<C_2$ , with $C_2=2.8984\\ldots ,$ which may be quite close to the real convergence radius, as long as the logarithmic case is considered.",
"We do not give a proof here, as a careful description would be too complicated.",
"What is strange about the values $C_1$ , $C_2$ is that they do not come from critical behaviour around $\\lambda =1/2$ .",
"In particular, the Cayley transform $\\mathcal {R}((\\exp X)(\\exp Y) ,1/2)$ converges better.",
"Formula (REF ) yields convergence for $|X|+|Y|<\\pi $ .",
"It is not clear that the ordinary (non-logarithmic) version of the BCH formula allows convergence like that or not.",
"However, one certainly cannot push the convergence bound beyond $\\pi $ , cf.",
"Example REF .",
"In the case of the BCH expansion, there is a single Banach algebra, the algebra of noncommutative polynomials with two variables endowed with the $\\ell ^1$ norm, where the convergence behaviour of some very specific expressions is the worst possible one.",
"As, we have seen, in the case of the Magnus expansion, the structure of Lebesgue-Bochner integrable functions is more complicated.",
"Although we have a best universal norm estimate, it is not clear what critical convergence behaviour is like.",
"This makes more important to consider specific Banach algebras in the study of the Magnus expansion.",
"Regarding that, apart from the (quasi)nilpotent case, very little is known but one rather important case, which is as follows.",
"Theorem REF suggests a way to deal with the problem, using spectral arguments, but controlling spectral behaviour is difficult in general.",
"However, there is a line arguments due to Mityagin [18], Moan [24], Moan, Niesen [25], Casas [6] that this can be done if $\\mathfrak {A}=\\mathcal {B}(\\mathfrak {H})$ , the algebra of bounded operators on a Hilbert space with the usual operator $\\sup $ -norm.",
"Their argument is essentially geometric.",
"We present a version augmented by some spectral and norm estimates."
],
[
"The conformal range",
"This section gives an introduction to the conformal range of Hilbert space operators.",
"For the general estimates we need very little from this section: essentially Lemma REF and formula (REF ), and only in the complex case.",
"For the analysis of $2\\times 2$ matrices a bit more information is needed, which provide here.",
"In that we try to be fairly thorough.",
"Nevertheless, as we want to avoid the impression that the understanding of Bolyai–Lobachevsky geometry is needed to the convergence estimates, we refrain from using its terminology.",
"The experienced reader will surely recognize it, anyway.",
"(However, see [1] for a standard account of geometry, if interested.)",
"In what follows, $\\mathfrak {H}$ will be a real or complex Hilbert space.",
"In order to avoid confusion, we denote the norm on $\\mathfrak {H}$ by $|\\cdot |_2$ , and the operator sup-norm by $\\Vert \\cdot \\Vert _2$ .",
"For $\\mathbf {x},\\mathbf {y} \\in \\mathfrak {H}\\setminus \\lbrace 0\\rbrace $ let $\\sphericalangle (\\mathbf {x},\\mathbf {y})$ be denote their angle.",
"This can already be obtained from the underlying real scalar product $\\langle \\cdot ,\\cdot \\cdot \\rangle _{\\mathrm {real}}=\\operatorname{Re}\\,\\langle \\cdot ,\\cdot \\cdot \\rangle $ .",
"For $\\mathbf {x},\\mathbf {y}\\in \\mathfrak {H}$ , $\\mathbf {x}\\ne 0$ , let $\\mathbf {y}:\\mathbf {x}=\\frac{\\langle \\mathbf {y},\\mathbf {x}\\rangle _{\\mathrm {real}}}{|\\mathbf {x}|_2^2}+\\mathrm {i}\\left|\\frac{\\mathbf {y}}{|\\mathbf {x}|_2}- \\frac{\\langle \\mathbf {y},\\mathbf {x}\\rangle _{\\mathrm {real}}}{|\\mathbf {x}|_2^2}\\frac{\\mathbf {x}}{|\\mathbf {x}|_2}\\right|_2.$ (This is the metric information of the real orthogonal decomposition of $\\mathbf {y}$ with respect to $\\mathbf {x}$ .)",
"Note that $|\\mathbf {y}:\\mathbf {x} |= |\\mathbf {y}|_2:|\\mathbf {x} |_2.",
"$ For $A\\in \\mathcal {B}(\\mathfrak {H})$ , we define the conformal range as $\\operatorname{CR}(A)=\\lbrace A\\mathbf {x}:\\mathbf {x}, \\,\\overline{(A\\mathbf {x}:\\mathbf {x})}\\,:\\, \\mathbf {x}\\in \\mathfrak {H}\\setminus \\lbrace 0\\rbrace \\rbrace .$ Lemma 3.1 (Conformal invariance.)",
"Suppose that $g(x)=\\frac{ax+b}{cx+d}$ is a real rational function, $ad-bc\\ne 0$ .",
"Assume $A\\in \\mathcal {B}(\\mathfrak {H})$ and that $cA+d\\operatorname{Id}$ is invertible.",
"(a) If $\\mathbf {x}\\in \\mathfrak {H}\\setminus 0$ and $\\mathbf {y}= (cA+d\\operatorname{Id})^{-1}\\mathbf {x}$ , then $g(A)\\mathbf {x}:\\mathbf {x}= g(A\\mathbf {y}:\\mathbf {y} )^{\\textrm {conjugated if } ad-bc<0}.$ (b) In general, $\\operatorname{CR}(g(A))=g(\\operatorname{CR}(A)).$ (a) The elementary rules $\\alpha \\mathbf {y}:\\mathbf {x}&=\\alpha \\cdot (\\mathbf {y}:\\mathbf {x})^{\\textrm {conjugated if } \\alpha <0}&&(\\alpha \\in \\mathbb {R}),\\\\(\\mathbf {y}+\\beta \\mathbf {x}):\\mathbf {x}&=\\mathbf {y}:\\mathbf {x}+\\beta &&(\\beta \\in \\mathbb {R}),\\\\\\gamma \\mathbf {y}:\\gamma \\mathbf {x}&=\\mathbf {y}:\\mathbf {x}&&(\\gamma \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace ),\\\\\\mathbf {y}:\\mathbf {x} &= \\overline{(\\mathbf {x}:\\mathbf {y})}^{-1}&&(\\mathbf {y}\\ne 0)$ are easy to check.",
"If $g$ is linear ($c=0$ ), then the statement follows from from the first three rules.",
"If $g$ is not linear ($c\\ne 0$ ), then $g(x)=\\frac{a}{c}-\\frac{ad-bc}{c^2}\\left(x+\\frac{d}{c}\\right)^{-1}$ , and $g(A)\\mathbf {x}:\\mathbf {x}&=\\frac{a}{c}-\\frac{ad-bc}{c^2}\\left( \\mathbf {x}:\\left(A+\\frac{d}{c}\\operatorname{Id}\\right)^{-1}\\mathbf {x}\\right)^{-1,\\textrm { conjugated if } ad-bc<0} \\\\&=\\frac{a}{c}-\\frac{ad-bc}{c^2}\\left( \\left(A+\\frac{d}{c}\\operatorname{Id}\\right)\\mathbf {y}: \\mathbf {y}\\right)^{-1,\\textrm { conjugated if } ad-bc<0}\\\\&=\\left(\\frac{a}{c}-\\frac{ad-bc}{c^2} \\left( \\left(A\\mathbf {y}: \\mathbf {y}\\right)+\\frac{d}{c}\\right)^{-1}\\right)^{\\textrm { conjugated if } ad-bc<0}\\\\&=g \\left(A\\mathbf {y}: \\mathbf {y}\\right)^{\\textrm { conjugated if } ad-bc<0}.$ (b) This follows from the previous part and the conjugational symmetry of $\\operatorname{CR}(A)$ .",
"The following lemma is not needed for our estimates, but it tells much about the nature of $\\operatorname{CR}$ .",
"Let $z_1,z_2\\in \\mathbb {C}$ such that $\\operatorname{Im}z_1,\\operatorname{Im}z_2\\ge 0$ .",
"We say that the $h$ -segment $[z_1,z_2]_h$ is the circular or straight segment connecting $z_1$ and $z_2$ , whose circle or line is perpendicular to the real axis, and lies in the upper half plane $\\overline{\\mathbb {C}}^{+}=\\lbrace z\\in \\mathbb {C}\\,:\\,\\operatorname{Im}z\\ge 0\\rbrace $ .",
"Lemma 3.2 ($h$ -Convexity.)",
"Suppose that $A\\in \\mathcal {B}(\\mathfrak {H})$ , and $\\dim _{\\mathbb {R}}\\ne 2$ .",
"Then $\\operatorname{CR}(A)\\cap \\overline{\\mathbb {C}}^{+}$ is $h$ -convex, i. e. $z_1,z_2\\in \\operatorname{CR}(A)\\cap \\overline{\\mathbb {C}}^{+}$ implies $[z_1,z_2]_h\\subset \\operatorname{CR}(A)\\cap \\overline{\\mathbb {C}}^{+}$ .",
"We can suppose that $z_1\\ne z_2$ .",
"Applying linear conformal transformations to $A$ , we can assume that $\\operatorname{Re}z_1=\\operatorname{Re}z_2=0$ (lineal case) or $|z_1|=|z_2|=1$ (circular case).",
"Assume that $ A\\mathbf {x}_1:\\mathbf {x}_1=z_1$ , $ A\\mathbf {x}_2:\\mathbf {x}_2=z_2$ .",
"Extend the span of $\\lbrace \\mathbf {x}_1,\\mathbf {x}_2\\rbrace $ to a 3-dimensional space $V\\subset \\mathfrak {H}$ .",
"Consider the quadratic form defined by $q(\\mathbf {x})={\\left\\lbrace \\begin{array}{ll} \\langle A\\mathbf {x},\\mathbf {x}\\rangle _{\\mathrm {real}}&\\text{(lineal case)}\\\\\\langle A\\mathbf {x},A\\mathbf {x}\\rangle _{\\mathrm {real}}-\\langle \\mathbf {x},\\mathbf {x}\\rangle _{\\mathrm {real}}&\\text{(circular case)}.\\end{array}\\right.",
"}$ The nullset $V_q$ of $q$ on $V$ is either $V$ , a plane, or a double cone (cf.",
"$\\mathbf {x}_1,\\mathbf {x}_2\\in V_q$ ).",
"In any case, $\\lbrace (A\\mathbf {x}:\\mathbf {x})\\,:\\, \\mathbf {x}\\in V_q\\setminus \\lbrace 0\\rbrace \\rbrace $ is a connected set (cf.",
"$(A\\mathbf {x}:\\mathbf {x})=(A(-\\mathbf {x}):(-\\mathbf {x}))$ ), which is contained in $L_h={\\left\\lbrace \\begin{array}{ll} \\lbrace z\\in \\overline{\\mathbb {C}}^+\\,:\\,\\operatorname{Re}z=0\\rbrace &\\text{(lineal case)}\\\\\\lbrace z\\in \\overline{\\mathbb {C}}^+\\,:|z|=1\\rbrace &\\text{(circular case)}.\\end{array}\\right.",
"}$ The connectedness implies $[z_1,z_2]_h\\subset L_h$ .",
"Lemma 3.3 (a) Suppose that $A_1\\in \\mathcal {B}(\\mathfrak {H}_1)$ , $A_2\\in \\mathcal {B}(\\mathfrak {H}_2)$ .",
"Let us consider the direct sum $A_1\\oplus A_2\\in \\mathcal {B}(\\mathfrak {H}_1\\oplus \\mathfrak {H}_2)$ .",
"Then $\\operatorname{CR}(A_1\\oplus A_2)\\cap \\overline{\\mathbb {C}}^+=\\bigcup \\lbrace [z_1,z_2]_h \\,:\\,z_1 \\in \\operatorname{CR}(A_1)\\cap \\overline{\\mathbb {C}}^+,z_2\\in \\operatorname{CR}(A_2)\\cap \\overline{\\mathbb {C}}^+\\rbrace .$ (b) If $\\dim _{\\mathbb {R}}\\mathfrak {H}\\ne 2$ , then complexification of $A$ does not change $\\operatorname{CR}(A)$ .",
"(a) Suppose that $\\mathbf {x}_1\\in \\mathfrak {H}_1$ , $\\mathbf {x}_2\\in \\mathfrak {H}_2$ , $A\\mathbf {x}_i:\\mathbf {x}_i=z_i$ .",
"Let $b$ be a real number such that $\\operatorname{Re}z_1+b=\\operatorname{Re}z_2+b=0$ or $|z_1+b|=|z_2+b|$ .",
"Then $(A+b\\operatorname{Id})\\mathbf {x}_i:\\mathbf {x}_i=z_i+b$ .",
"Now, it is simple geometry that $(A+b\\operatorname{Id})(\\sqrt{1-t^2}\\mathbf {x}_1+ t\\mathbf {x}_2):(\\sqrt{1-t^2}\\mathbf {x}_1+ t\\mathbf {x}_2)$ runs along $[z_1+b,z_2+b]_h$ for $t\\in [0,1]$ .",
"This implies that $A(\\sqrt{1-t^2}\\mathbf {x}_1+ t\\mathbf {x}_2):(\\sqrt{1-t^2}\\mathbf {x}_1+ t\\mathbf {x}_2)$ runs along $[z_1,z_2]_h$ .",
"(b) $\\operatorname{CR}(A^{\\mathbb {C}})\\cap \\overline{\\mathbb {C}}^+=\\operatorname{CR}(A\\oplus A)\\cap \\overline{\\mathbb {C}}^+$ , but $\\operatorname{CR}(A)\\cap \\overline{\\mathbb {C}}^+$ is already $h$ -convex.",
"From (REF ), it is immediate that $\\Vert A\\Vert _2=\\sup \\lbrace |\\omega |\\,:\\,\\omega \\in \\operatorname{CR}(A)\\rbrace .",
"$ If $\\dim \\mathfrak {H}<\\infty $ , then $\\operatorname{CR}(A)$ is compact (as it is a continuous image of the compact unit sphere), and $\\operatorname{sp}(A)\\cap \\mathbb {R}=\\operatorname{CR}(A)\\cap \\mathbb {R}$ ; but not in general (cf.",
"Example REF ).",
"Assume, for now, that $\\mathfrak {H}$ is complex.",
"Then, for $\\lambda \\in \\mathbb {C}$ , $|(A-\\lambda \\operatorname{Id})\\mathbf {x}|_2\\ge \\operatorname{dist}(\\lambda , \\operatorname{CR}(A)) |\\mathbf {x}|_2.",
"$ Thus, for $\\lambda \\in \\mathbb {C}\\setminus \\overline{\\operatorname{CR}(A)}$ , the operator $A-\\lambda \\operatorname{Id}$ is invertible on its (closed) range.",
"This range is $\\mathfrak {H}$ if $\\ker A^*-\\bar{\\lambda }\\operatorname{Id}=0$ .",
"Consequently, for the spectrum, $\\operatorname{sp}(A)\\subset \\overline{\\operatorname{CR}(A)}\\cup \\operatorname{CR}(A^*).",
"$ It, however, might be more practical to use Lemma 3.4 $ \\operatorname{sp}(A)\\subset \\operatorname{sc}(\\overline{\\operatorname{CR}(A)}), $ where $ \\operatorname{sc}(\\overline{\\operatorname{CR}(A)})$ denotes the simply connected closure of $\\overline{\\operatorname{CR}(A)}$ , i. e. the complement of infinite component of $\\mathbb {C}\\setminus \\overline{\\operatorname{CR}(A)}$ .",
"Indeed, indirectly, suppose that $C$ is a polygonal chain from $\\infty $ to $\\xi $ in the complement $\\mathbb {C}\\setminus \\overline{\\operatorname{CR}(A)}$ .",
"It can be assumed that $\\xi $ is the first and (last) element of $C$ such that $A-\\xi \\operatorname{Id}$ is not invertible.",
"According to (REF ), the inverse $(A-\\lambda \\operatorname{Id})^{-1}$ is bounded by $\\operatorname{dist}(C,\\operatorname{CR}(A) )^{-1}$ for $\\lambda \\in C\\setminus \\lbrace \\xi \\rbrace $ .",
"Hence, its derivative $(A-\\lambda \\operatorname{Id})^{-2}A$ is bounded by $\\operatorname{dist}((-\\infty ,0],\\operatorname{CR}(A) )^{-2}\\Vert A\\Vert _2$ for $\\lambda \\in C\\setminus \\lbrace \\xi \\rbrace $ .",
"This, however, implies that the inverse extends to $A-\\xi \\operatorname{Id}$ ; which is a contradiction.",
"Remark 3.5 In fact, $\\operatorname{sp}(A)\\subset \\overline{\\operatorname{CR}(A)}\\cup \\operatorname{CR}(A^*)\\subset \\operatorname{sc}(\\overline{\\operatorname{CR}(A)} )=\\operatorname{sc}(\\overline{\\operatorname{CR}(A^*)} ) $ holds.",
"This follows from the characterization $\\operatorname{sc}(\\overline{\\operatorname{CR}(A)} )=\\lbrace z\\in \\mathbb {C} \\,:\\, |z-\\lambda |\\le \\Vert A-\\lambda \\operatorname{Id}\\Vert _2\\text{ for all }\\lambda \\in \\mathbb {R};\\text{ and }\\\\|z-\\lambda |\\ge \\Vert (A-\\lambda \\operatorname{Id})^{-1}\\Vert _2^{-1}\\text{ for all }\\lambda \\in \\mathbb {R}\\setminus (\\text{ the convex hull of } \\mathbb {R}\\cap \\operatorname{sp}(A) )\\rbrace .",
"$ This, in turn, follows from the $h$ -convexity of $\\operatorname{CR}(A)\\cap \\mathbb {C}^+$ .",
"Also, $\\operatorname{CR}(A)=\\operatorname{CR}(A^*)\\qquad \\text{if}\\quad \\dim \\mathfrak {H}<\\infty .",
"$ This follows from the characterization ${\\operatorname{CR}(A)}=\\lbrace z\\in \\mathbb {C} \\,:\\, |z-\\lambda |\\le \\Vert A-\\lambda \\operatorname{Id}\\Vert _2\\text{ for all }\\lambda \\in \\mathbb {R};\\text{ and }\\\\|z-\\lambda |\\ge \\Vert (A-\\lambda \\operatorname{Id})^{-1}\\Vert _2^{-1}\\text{ for all }\\lambda \\in \\mathbb {R}\\setminus \\operatorname{sp}(A) )\\rbrace \\qquad \\text{if}\\quad \\dim \\mathfrak {H}<\\infty .",
"$ Example 3.6 (a) Let $\\mathfrak {H}=\\ell ^2(\\mathbb {N};\\mathbb {C})$ , and let $A$ be the unilateral shift $A\\mathbf {e}_n=\\mathbf {e}_{n+1}$ .",
"Then $\\operatorname{CR}(A)=\\partial \\operatorname{D}(0,1)\\setminus \\lbrace -1,1\\rbrace ,$ $\\operatorname{CR}(A^*)=\\operatorname{D}(0,1)\\setminus \\lbrace -1,1\\rbrace ,$ $\\operatorname{sp}(A)=\\operatorname{D}(0,1),$ $\\operatorname{sc}(\\overline{\\operatorname{CR}(A)})= \\operatorname{D}(0,1).$ (b) If $\\mathfrak {H}=\\ell ^2(\\mathbb {Z};\\mathbb {C})$ , and let $T$ be the unilateral shift defined similarly.",
"Then $\\operatorname{CR}(A)=\\partial \\operatorname{D}(0,1)\\setminus \\lbrace -1,1\\rbrace ,$ $\\operatorname{CR}(A^*)=\\partial \\operatorname{D}(0,1)\\setminus \\lbrace -1,1\\rbrace ,$ $\\operatorname{sp}(A)=\\partial \\operatorname{D}(0,1),$ $\\operatorname{sc}(\\overline{\\operatorname{CR}(A)})=\\operatorname{D}(0,1).$ The preceding discussions can also be applied in the real case after complexification.",
"If $\\dim \\mathfrak {H}\\ne 2$ , then complexification does not change the conformal range (nor the spectrum), all the formulas (REF )–(REF ) remain valid.",
"If $\\dim \\mathfrak {H}=2$ , then $\\operatorname{CR}(A)=\\operatorname{CR}(A^*)$ should be replaced $\\operatorname{CR}(A^{\\mathbb {C}})=\\operatorname{CR}((A^{\\mathbb {C}})^*)$ , which is already closed.",
"However, this case is really easy to overview: Lemma 3.7 Consider the real matrix $A=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}.",
"$ (a) For $A$ acting on $\\mathbb {R}^2$ , $\\operatorname{CR}(A^{\\mathbb {R}})=\\partial \\operatorname{D}\\left(\\tfrac{a+d}{2}+\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right)\\cup \\partial \\operatorname{D}\\left(\\tfrac{a+d}{2}-\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right).$ (b) For $A$ acting on $\\mathbb {C}^2$ , $\\operatorname{CR}(A^{\\mathbb {C}})=& \\operatorname{D}\\left(\\tfrac{a+d}{2}+\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right)\\setminus \\operatorname{\\mathring{D}}\\left(\\tfrac{a+d}{2}-\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right)\\\\&\\cup \\operatorname{D}\\left(\\tfrac{a+d}{2}-\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right)\\setminus \\operatorname{\\mathring{D}}\\left(\\tfrac{a+d}{2}+\\tfrac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\tfrac{a-d}{2}\\right)^2+\\left(\\tfrac{b+c}{2}\\right)^2} \\right).$ This is $\\operatorname{CR}(A^{\\mathbb {R}})$ but with the components of $\\mathbb {C}\\setminus \\operatorname{CR}(A^{\\mathbb {R}})$ disjoint from $\\mathbb {R}$ filled in.",
"(a) $\\mathbb {R}^2$ can be identified $\\mathbb {C}$ .",
"One can check that for $|w|=1$ , $\\frac{Aw}{w}=\\left(\\frac{a+d}{2}+\\frac{c-b}{2}\\mathrm {i}\\right)+\\frac{1}{w^2}\\left(\\frac{a-d}{2}+\\frac{b+c}{2}\\mathrm {i}\\right).$ The statement is an immediate consequence of this formula.",
"(b) This is a consequence of $\\operatorname{CR}(A^{\\mathbb {C}})\\cap \\mathbb {C}^+=\\operatorname{CR}(A^{\\mathbb {R}}\\oplus A^{\\mathbb {R}})\\cap \\mathbb {C}^+$ .",
"We see, for $\\mathbb {R}\\times \\mathbb {R}$ , that the information encoded in $\\operatorname{CR}(A)$ is the same as the one in the principal disk $\\operatorname{PD}(A):=\\operatorname{D}\\left(\\frac{a+d}{2}+\\frac{|c-b|}{2}\\mathrm {i},\\sqrt{\\left(\\frac{a-d}{2}\\right)^2+\\left(\\frac{b+c}{2}\\right)^2} \\right).$ The principal disk is a point if $A$ has the effect of a complex multiplication.",
"In general, matrices $A$ fall into three categories: elliptic, parabolic, hyperbolic; such that the principal disk are disjoint, tangent or secant to the real axis, respectively.",
"This is refined by the chiral disk $\\operatorname{CD}(A):=\\operatorname{D}\\left(\\frac{a+d}{2}+\\frac{c-b}{2}\\mathrm {i},\\sqrt{\\left(\\frac{a-d}{2}\\right)^2+\\left(\\frac{b+c}{2}\\right)^2} \\right).$ The additional data in the chiral disk is the chirality, which is the sign of the twisted trace, $\\operatorname{sgn}(c-b)=\\operatorname{sgn}\\operatorname{tr}\\begin{bmatrix}&1\\\\-1&\\end{bmatrix}A.$ This chirality is, in fact, understood with respect to a fixed orientation of $\\mathbb {R}^2$ .",
"It does not change if we conjugate $A$ by a rotation, but it changes sign if we conjugate $A$ by a reflection.",
"One can read off many data from the disks.",
"For example, if $\\operatorname{PD}(A)=\\operatorname{D}((\\tilde{a},\\tilde{b}),r)$ , then $\\det A=\\tilde{a}^2+\\tilde{b}^2-r^2$ .",
"In fact, Lemma 3.8 $\\operatorname{CD}$ makes a bijective correspondence between possibly degenerated disks in $\\mathbb {C}$ and the orbits of $\\mathrm {M}_2(\\mathbb {R})$ with respect to conjugacy by special orthogonal matrices (i. e. rotations).",
"$\\operatorname{PD}$ makes a bijective correspondence between possibly degenerated disks with center in $\\mathbb {C}^+$ and the orbits of $\\mathrm {M}_2(\\mathbb {R})$ with respect to conjugacy by orthogonal matrices.",
"One can write $A\\in \\mathrm {M}_2(\\mathbb {R})$ in skew-quaternionic form $A= \\tilde{a}\\operatorname{Id}+\\tilde{b}\\tilde{I}+\\tilde{c}\\tilde{J}+\\tilde{d}\\tilde{K}\\equiv \\tilde{a}\\begin{bmatrix} 1&\\\\&1\\end{bmatrix}+\\tilde{b}\\begin{bmatrix} &-1\\\\1&\\end{bmatrix}+\\tilde{c}\\begin{bmatrix} 1&\\\\&-1\\end{bmatrix}+\\tilde{d}\\begin{bmatrix} &1\\\\1&\\end{bmatrix}.$ The principal disk of this matrix is $\\operatorname{D}(\\tilde{a}+\\tilde{b} \\mathrm {i},\\sqrt{\\tilde{c}^2+\\tilde{d}^2})$ , every possibly degenerated disk occurs.",
"On the other hand, conjugation by $\\begin{bmatrix} \\cos \\alpha &-\\sin \\alpha \\\\\\sin \\alpha &\\cos \\alpha \\end{bmatrix}$ takes $A$ into $\\tilde{a}\\operatorname{Id}+\\tilde{b}\\tilde{I}+(\\tilde{c}\\cos 2\\alpha -\\tilde{d}\\sin 2\\alpha )\\tilde{J}+(\\tilde{c}\\sin 2\\alpha +\\tilde{d}\\cos 2\\alpha )\\tilde{K}$ .",
"This shows that the orbit data is the same as the principal disk data.",
"Conjugation by $\\begin{bmatrix} 1&\\\\&-1\\end{bmatrix}$ takes $A$ into $\\tilde{a}\\operatorname{Id}-\\tilde{b}\\tilde{I}+\\tilde{c}\\tilde{J}-\\tilde{d}\\tilde{K}$ .",
"This shows the second part.",
"The case of complex $2\\times 2$ matrices is treatable but much more complicated.",
"Geometrically, apart from $A=0_2$ , up to conformal and orthogonal equivalence, it is sufficient to consider the cases $S_\\beta =\\begin{bmatrix} 0&\\cos \\beta \\\\0&\\mathrm {i}\\sin \\beta \\end{bmatrix} \\qquad \\beta \\in \\left[0,\\frac{\\pi }{2}\\right] $ and $L_{\\alpha ,t}=\\begin{bmatrix} \\cos \\alpha +\\mathrm {i}\\sin \\alpha & t\\\\&-\\cos \\alpha +\\mathrm {i}\\sin \\alpha \\end{bmatrix} \\qquad \\alpha \\in \\left[0,\\frac{\\pi }{2}\\right], t\\ge 0.",
"$ Here the zero matrix and $\\beta =0$ correspond to the real parabolic case, $\\alpha =0$ to the real hyperbolic case; $\\alpha =\\pi /2$ to the real elliptic case.",
"(Note that in the families above, changing a single occurrence of $\\mathrm {i}$ to $-\\mathrm {i}$ still produces an orthogonally equivalent version.)",
"For example, in the first case, $\\beta =0$ gives a disk (real case), $\\beta =\\pi /2$ gives a segment between 0 and $\\mathrm {i}$ (direct sum case) for $\\operatorname{CR}(S_\\beta )\\cap \\mathbb {C}^+$ .",
"They deform into each other as $\\beta $ changes, but 0 and $\\mathrm {i}$ are continually elements of the conformal range.",
"In order to have this kind of behaviour, $h$ -cycles (i. e. lines and circles) are not sufficient anymore.",
"See Remark REF for further information.",
"Lemma 3.9 (a) Let $A=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$ be a real matrix.",
"Then $\\left\\Vert A \\right\\Vert _2=\\frac{\\sqrt{(a+d)^2+(c-b)^2}+\\sqrt{(a-d)^2+(b+c)^2}}{2}.",
"$ On the other hand, $\\left\\Vert A^{-1} \\right\\Vert _2^{-1}=\\left|\\frac{\\sqrt{(a+d)^2+(c-b)^2}-\\sqrt{(a-d)^2+(b+c)^2}}{2}\\right|, $ where the LHS is considered to be 0 for non-invertible matrices.",
"It is true that $\\operatorname{sgn}\\det A= \\operatorname{sgn}\\frac{\\sqrt{(a+d)^2+(c-b)^2}-\\sqrt{(a-d)^2+(b+c)^2}}{2}.",
"$ (b) If $A$ were a complex matrix, then $\\left\\Vert A \\right\\Vert _2=\\frac{\\sqrt{\\operatorname{tr}(A^*A)+|\\det A|}+\\sqrt{\\operatorname{tr}(A^*A)-|\\det A|} }{2}$ $=\\frac{\\sqrt{|a|^2+|b|^2+|c|^2+|d|^2+2|ad-bc| }+\\sqrt{|a|^2+|b|^2+|c|^2+|d|^2-2|ad-bc|}}{2};$ and $\\left\\Vert A^{-1} \\right\\Vert _2^{-1}=\\frac{\\sqrt{\\operatorname{tr}(A^*A)+|\\det A|}-\\sqrt{\\operatorname{tr}(A^*A)-|\\det A|} }{2}$ $=\\frac{\\sqrt{|a|^2+|b|^2+|c|^2+|d|^2+2|ad-bc| }-\\sqrt{|a|^2+|b|^2+|c|^2+|d|^2-2|ad-bc|}}{2}.$ (Similar 0-convention applies.)",
"(a) $\\operatorname{CR}(A^{\\mathbb {R}})$ is constituted of circles.",
"The farthest distance from the origin gives the norm; and the closest distance from the origin gives the, say, co-norm.",
"These distances, however, can immediately be read off from the center and the radius.",
"(But they are also a corollaries of the complex case.)",
"The sign formula is an easy exercise.",
"(b) This can be computed from $\\Vert A\\Vert _2^2=\\max \\operatorname{sp}(A^*A)$ .",
"Motivated by (REF )–(REF ), for a real matrix $A=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$ , we define its signed co-norm by $\\left\\lfloor A\\right\\rfloor _2 =\\operatorname{sgn}(\\det A )\\left\\Vert A^{-1} \\right\\Vert _2^{-1}=\\frac{\\sqrt{(a+d)^2+(c-b)^2}-\\sqrt{(a-d)^2+(b+c)^2}}{2}.",
"$ Remark 3.10 In theory, we can determine the closure of the conformal range using norms and co-norms.",
"Let $N(\\cdot )$ denote the square of the norm or the co-norm.",
"Then $\\partial \\operatorname{CR}(A)$ is the enveloping curve of the circles $(x-\\lambda )^2+y^2=N(A+\\lambda \\operatorname{Id}).$ This curve can be computed as $\\lambda \\mapsto \\left(\\lambda -\\frac{1}{2} \\frac{\\mathrm {d} N(A+\\lambda \\operatorname{Id})}{\\mathrm {d}\\lambda }\\right)+\\mathrm {i} \\sqrt{ N(A+\\lambda \\operatorname{Id})^2- \\left(\\frac{1}{2}\\frac{\\mathrm {d} N(A+\\lambda \\operatorname{Id})}{\\mathrm {d}\\lambda }\\right)^2}.$ The norm produces the upper part, the co-norm produces the lower part.",
"(The joins correspond to $\\lambda =\\pm \\infty $ .)",
"The expression is defined almost everywhere, but large discontinuities can occur, which should be supplemented by $h$ -segments.",
"Cf.",
"$A=\\begin{bmatrix}1&\\\\&-1\\end{bmatrix}$ : Generally, in the real case, $A= \\tilde{a}\\operatorname{Id}+\\tilde{b}\\tilde{I}+\\tilde{c}\\tilde{J}+\\tilde{d}\\tilde{K}$ gives the norm branch $\\lambda \\mapsto \\tilde{a}+\\tilde{b}\\mathrm {i}+\\frac{\\sqrt{\\tilde{c}^2+\\tilde{d}^2}}{\\sqrt{(\\tilde{a}-\\lambda )^2+\\tilde{b}^2}}\\left((\\tilde{a}-\\lambda )+\\tilde{b}\\mathrm {i}\\right)\\quad \\text{possibly conjugated into $\\mathbb {C}^+$.",
"}$ We see that for $\\tilde{a}=\\tilde{b}=\\tilde{d}=0$ , $\\tilde{c}=1$ this degenerates to $\\lambda \\mapsto -\\operatorname{sgn}\\lambda $ ; and almost the whole conformal range comes from a discontinuity (the co-norm case is not different).",
"As we can also compute with the complex $2\\times 2$ norms, this method can be applied to the complex case, and especially to the representative types (REF ) and (REF ).",
"Nevertheless, the curves resulted so are quite unwieldy.",
"We can do much better, if we apply the map $\\tfrac{\\mathrm {CKB}}{\\mathrm {PHP}}: \\qquad (u_1,u_2)\\mapsto \\left(\\frac{2u_1}{1+u_1^2+u_2^2} ,-\\frac{1-u_1^2-u_2^2}{1+u_1^2+u_2^2}\\right)$ to $\\operatorname{CR}(A)\\cap \\mathbb {C}^+$ .",
"(This is a conversion map from the Poincaré half plane to the Cayley-Klein-Beltrami disk.)",
"In this way, $S_\\beta $ yields ellipses with axes $ \\left[-\\frac{\\sqrt{2}}{2}\\cos \\beta ,\\frac{\\sqrt{2}}{2}\\cos \\beta \\right]\\times \\left\\lbrace -\\frac{1}{2}\\right\\rbrace \\qquad \\text{and}\\qquad \\lbrace 0\\rbrace \\times [-1,0];$ and $L_{\\alpha ,t}$ yields ellipses with axes $ \\left[-\\frac{\\sqrt{4(\\cos \\alpha )^2+t^2 }}{\\sqrt{4+t^2}} ,\\frac{\\sqrt{4(\\cos \\alpha )^2+t^2 }}{\\sqrt{4+t^2}} \\right]\\times \\lbrace 0\\rbrace \\qquad \\text{and}\\qquad \\lbrace 0\\rbrace \\times \\left[-\\frac{t}{\\sqrt{4+t^2}},\\frac{t}{\\sqrt{4+t^2}}\\right];$ (the ellipses may be degenerate;) the zero matrix yields the point ellipse $\\lbrace (0,-1)\\rbrace $ .",
"From this, one can conclude, in general, that $\\tfrac{\\mathrm {CKB}}{\\mathrm {PHP}}(\\operatorname{CR}(A)\\cap \\mathbb {C}^+)$ yields ellipses in the unit disk but which do not contain the point $(0,1)$ .",
"Hence, they can be identified as possibly degenerate $h$ -ellipses.",
"From the norm formula and the enveloping construction, one can see that these ellipses depend on the `five data' $\\det A$ (complex), $\\operatorname{tr}A$ (complex), $\\operatorname{tr}(A^*A)$ (real), with some minor degeneracy.",
"(These are `three data' in the real case without chirality.)",
"Actually, due to this dependency, it is sufficient to compute with a very few Taylor terms of the enveloping curves.",
"Later we compute much with logarithms of $2\\times 2$ matrices.",
"According to the definition (REF ), $\\log A$ is well-defined if and only if the segment $(1-t)\\operatorname{Id}+tA$ ($t\\in [0,1]$ ) contains only invertible operators; or, equivalently, if $\\operatorname{sp}(A)\\cap (-\\infty ,0]=0$ .",
"Lemma 3.11 Let $A$ be a $\\log $ -able real $2\\times 2$ matrix.",
"Then $\\det A>0$ ; $\\dfrac{\\operatorname{tr}A}{2\\sqrt{\\det A}}>-1$ ; and $\\log A= \\frac{\\log \\det A}{2}\\operatorname{Id}+\\frac{\\operatorname{AC}\\left(\\dfrac{\\operatorname{tr}A}{2\\sqrt{\\det A}}\\right)}{\\sqrt{\\det A}}\\left(A- \\frac{\\operatorname{tr}A}{2}\\operatorname{Id}\\right), $ where $\\operatorname{AC}(x)={\\left\\lbrace \\begin{array}{ll}\\dfrac{\\arccos x}{\\sqrt{1-x^2}}&\\text{if }-1< x<1\\\\1&\\text{if }x=1\\\\\\dfrac{\\operatorname{arcosh}x}{\\sqrt{x^2-1}}\\qquad &\\text{if } 1<x.\\\\\\end{array}\\right.",
"}$ $\\det A>0$ is easy, and left to the reader.",
"Due to the nature of the other expressions, the determinant can be normalized to 1, through multiplication by a positive number.",
"In general, all expressions involved are also conjugation invariant.",
"Hence, apart from the identity, it is sufficient to check the statement for the orbit types $\\begin{bmatrix}\\cos \\alpha &-\\sin \\alpha \\\\\\sin \\alpha &\\cos \\alpha \\end{bmatrix}$ ($\\alpha \\in (0,\\pi /2]$ ), $\\begin{bmatrix}1&1\\\\&1\\end{bmatrix}$ , $\\begin{bmatrix}\\mathrm {e}^\\beta &\\\\& \\mathrm {e}^{-\\beta }\\end{bmatrix}$ ($\\beta >0$ ) of $\\operatorname{SL}_2(\\mathbb {R})$ ; the not $\\log $ -able orbit types $\\begin{bmatrix}-1&\\\\&-1\\end{bmatrix}$ , $\\begin{bmatrix}-1&1\\\\&-1\\end{bmatrix}$ , $\\begin{bmatrix}-\\mathrm {e}^\\beta &\\\\& -\\mathrm {e}^{-\\beta }\\end{bmatrix}$ ($\\beta >0$ ) do not play role here.",
"As the proof shows, we compute $\\operatorname{AC}$ by $\\arccos $ for elliptic matrices, by $\\operatorname{arcosh}$ for hyperbolic matrices, and as 1 for parabolic matrices.",
"From the properties of the twisted trace, it is also easy too see that $\\log $ respects chirality.",
"Lemma 3.12 (a) The function $\\operatorname{AC}$ extends to $\\mathbb {C}\\setminus (-\\infty ,-1]$ analytically.",
"$\\operatorname{AC}$ is monotone decreasing on $(-1,\\infty )$ with range $(0,\\infty )$ .",
"It also satisfies the functional equation $\\operatorname{AC}(x)^{\\prime }=\\frac{1-x\\operatorname{AC}(x)}{1-x^2}.$ (b) The function $\\operatorname{AS}(x)=\\sqrt{\\frac{\\operatorname{AC}(x)^2-1}{1-x^2}}$ is analytic on $(-1,\\infty )$ .",
"$\\operatorname{AS}$ is also monotone decreasing on $(-1,\\infty )$ with range $(0,\\infty )$ .",
"(c) The function $\\operatorname{At}(x)=\\frac{\\operatorname{AC}(x)-1}{\\operatorname{AS}(x)}$ is analytic on $(-1,\\infty )$ .",
"The function $x\\mapsto x+\\operatorname{At}(x)$ is monotone increasing, a bijection from $(-1,1]$ to itself.",
"Analyticity of $\\operatorname{AC}$ , and analytic extendibility on the indicated domain is guaranteed by the formula $\\operatorname{AC}(z)=\\frac{1}{2}\\operatorname{tr}\\left(\\begin{bmatrix}&1\\\\-1&\\end{bmatrix}\\log \\begin{bmatrix}z&z-1\\\\z+1&z\\end{bmatrix}\\right).$ Indeed, the eigenvalues of the matrix under the $\\log $ are $z\\pm \\sqrt{z^2-1}$ .",
"The equation $z\\pm \\sqrt{z^2-1}=r\\le 0$ , however, solves to $z=\\frac{r+1/r}{2}<0$ , excluded by assumption.",
"The rest is simple function calculus.",
"Lemma 3.13 Suppose that $A$ is $2\\times 2$ complex matrix which is $\\log $ -able.",
"Let $\\sqrt{\\det A}$ denote that value of the standard branch of the square root of the determinant on $\\log $ -able elements.",
"(It can be realized as $\\sqrt{\\det A}=\\exp \\frac{1}{2}\\int _{t=0}^1 \\operatorname{tr}\\frac{\\mathrm {d}((1-t)\\operatorname{Id}+tA )}{(1-t)\\operatorname{Id}+tA},$ or as $\\sqrt{\\varepsilon _1}\\sqrt{\\varepsilon _2}$ , where $\\varepsilon _i$ are the eigenvalues of $A$ , and the square root is of $\\mathbb {C}\\setminus (-\\infty ,0]$ .)",
"Then $\\det A\\in \\mathbb {C}\\setminus (\\infty ,0]$ , $\\dfrac{\\operatorname{tr}A}{2\\sqrt{\\det A}}\\in \\mathbb {C}\\setminus (\\infty ,-1]$ , and formula (REF ) holds.",
"Then $\\varepsilon _i=\\mathrm {e}^{\\alpha _i}$ , with $-\\pi < \\operatorname{Re}\\alpha _i <\\pi $ .",
"Hence, $\\det A=\\mathrm {e}^{\\frac{\\alpha _1+\\alpha _2}{2}}$ , and $\\left|\\operatorname{Re}\\frac{\\alpha _1+\\alpha _2}{2}\\right|<\\pi $ is transparent.",
"Indirectly, $\\frac{\\operatorname{tr}A}{2\\sqrt{\\det A}}=\\frac{\\mathrm {e}^{\\frac{\\alpha _1-\\alpha _2}{2}}+ \\mathrm {e}^{-\\frac{\\alpha _1-\\alpha _2}{2}} }{2}=r\\le -1$ solves to $\\mathrm {e}^{\\pm \\frac{\\alpha _1-\\alpha _2}{2}}=r\\pm \\sqrt{r^2-1}\\le 0.$ But this contradicts $\\left|\\operatorname{Re}\\frac{\\alpha _1-\\alpha _2}{2}\\right|<\\pi $ .",
"The logarithm formula extends analytically.",
"For finite matrices $\\operatorname{sp}(A)\\cap \\mathbb {R}=\\operatorname{CR}(A)^{\\mathrm {real}}\\cap \\mathbb {R}$ .",
"Consequently, $A$ is $\\log $ -able if and only if $\\operatorname{CR}(A)^{\\mathrm {real}}\\cap (\\infty ,0]=\\emptyset $ .",
"Or, in terms of the principal disk, if and only if $\\operatorname{PD}(A)^{\\mathrm {real}}\\cap (\\infty ,0]=\\emptyset $ .",
"For the sake of the next statements using $\\tilde{a},\\tilde{b}$ instead of $a,b$ would be more appropriate, but it is probably better to keep the notation simple.",
"Lemma 3.14 Suppose that $A$ is a $\\log $ -able real $2\\times 2$ matrix with principal disk $\\operatorname{PD}(A)=\\operatorname{D}(a+\\mathrm {i}b,r).$ In that case, $\\Vert \\log A\\Vert _2=f_{\\mathrm {CA}}(a,b,r)+f_{\\mathrm {RD}}(a,b,r), $ and $\\left\\lfloor \\log A\\right\\rfloor _2=f_{\\mathrm {CA}}(a,b,r)-f_{\\mathrm {RD}}(a,b,r), $ where $f_{\\mathrm {CA}}(a,b,r)= \\sqrt{\\Bigl (\\log \\sqrt{a^2+b^2-r^2} \\Bigr )^2+\\left(\\frac{b\\operatorname{AC}\\left(\\dfrac{a}{\\sqrt{a^2+b^2-r^2}}\\right)}{\\sqrt{a^2+b^2-r^2}}\\right)^2}$ and $f_{\\mathrm {RD}}(a,b,r)=\\frac{r\\operatorname{AC}\\left(\\dfrac{a}{\\sqrt{a^2+b^2-r^2}}\\right)}{\\sqrt{a^2+b^2-r^2}}.$ In particular, if $\\det A=1$ , then $a^2+b^2-r^2=1$ , and $f_{\\mathrm {CA}}(a,b,r)=\\operatorname{AC}(a)b$ , $f_{\\mathrm {RD}}(a,b,r)=\\operatorname{AC}(a)r$ .",
"This is just the combination of (REF ) and (REF )–(REF ), computed explicitly.",
"Theorem 3.15 Suppose that $A_1,A_2$ are $\\log $ -able real $2\\times 2$ matrices such that $\\operatorname{PD}(A_1)\\subset \\operatorname{PD}(A_2).$ Then $\\Vert \\log A_1\\Vert _2\\le \\Vert \\log A_2\\Vert _2.",
"$ and $\\lfloor \\log A_1\\rfloor _2\\ge \\lfloor \\log A_2\\rfloor _2.",
"$ The monotonicity of $\\Vert \\cdot \\Vert _2$ is strict, except if $\\operatorname{PD}(A_1)$ and $ \\operatorname{PD}(A_2)$ are centered on the real line and $\\sup \\lbrace |\\log x|\\,:\\,x\\in \\mathbb {R}\\cap \\operatorname{PD}(A_1)\\rbrace =\\sup \\lbrace |\\log x|\\,:\\,x\\in \\mathbb {R}\\cap \\operatorname{PD}(A_2)\\rbrace $ .",
"The monotonicity of $\\lfloor \\cdot \\rfloor _2$ is strict, except if $\\operatorname{PD}(A_1)$ and $ \\operatorname{PD}(A_2)$ are centered on the real line and $\\inf \\lbrace |\\log x|\\,:\\,x\\in \\mathbb {R}\\cap \\operatorname{PD}(A_1)\\rbrace =\\inf \\lbrace |\\log x|\\,:\\,x\\in \\mathbb {R}\\cap \\operatorname{PD}(A_2)\\rbrace $ .",
"Let $f(a,b,r)$ denote the functional expression on the right side of (REF ).",
"Then it is a straightforward but long computation to check the identity $\\left(\\frac{\\partial f(a,b,r)}{\\partial r}\\right)^2-\\left(\\frac{\\partial f(a,b,r)}{\\partial a}\\right)^2-\\left(\\frac{\\partial f(a,b,r)}{\\partial b}\\right)^2= \\left(\\frac{f(a,b,r)}{f_{\\mathrm {CA}}(a,b,r)} \\frac{b\\operatorname{AS}\\left(\\dfrac{a}{\\sqrt{a^2+b^2-r^2}}\\right)}{{a^2+b^2-r^2}}\\right)^2 .",
"$ This is valid, except if $b=0$ and $a=\\sqrt{1+r^2}$ , the exceptional configurations.",
"In particular, if $b>0$ , then $ \\left(\\frac{\\partial f(a,b,r)}{\\partial r}\\right)^2-\\left(\\frac{\\partial f(a,b,r)}{\\partial a}\\right)^2-\\left(\\frac{\\partial f(a,b,r)}{\\partial b}\\right)^2>0 .$ The principal disks with $b>0$ form a connected set, consequently $ \\frac{\\partial f(a,b,r)}{\\partial r}>\\sqrt{\\left(\\frac{\\partial f(a,b,r)}{\\partial a}\\right)^2+\\left(\\frac{\\partial f(a,b,r)}{\\partial b}\\right)^2} $ or $ \\frac{\\partial f(a,b,r)}{\\partial r}<-\\sqrt{\\left(\\frac{\\partial f(a,b,r)}{\\partial a}\\right)^2+\\left(\\frac{\\partial f(a,b,r)}{\\partial b}\\right)^2} $ should hold globally for $b>0$ .",
"The question is: which one?",
"It is sufficient to check the sign $\\frac{\\partial f(a,b,r)}{\\partial r}$ at a single place.",
"Now, it is not hard to check that $\\frac{\\partial f(a,b,r)}{\\partial r}\\Bigl |_{r=0}=\\frac{\\operatorname{AC}\\left(\\frac{a}{\\sqrt{a^2+b^2}}\\right)}{\\sqrt{a^2+b^2}}$ (except if $a=1,b=0$ ), which shows that (REF ) holds.",
"The meaning of (REF ) is that expanding principal disks smoothly with non-real centers leads to growth in the norm of the logarithm.",
"Let us return to principal disks $D_i=\\operatorname{PD}(A_i)$ in the statement.",
"If $b_1,b_2>0$ , then we can expand the smaller one to the bigger one with non-real centers.",
"(Indeed, magnify $D_2$ from its lowest point, until the perimeters touch, and then magnify from the touching point.)",
"This proves the (REF ) for $b_1,b_2>0$ .",
"The general statement follows from the continuity of the norm of the logarithm.",
"Notice that the norm grows if we can expand through $b>0$ .",
"Regarding (REF ): Let $f_{\\mathrm {co}}(a,b,r)$ denote the functional expression on the right side of (REF ).",
"It satisfies the very same equation (REF ) but with $f(a,b,r)$ replaced by $f_{\\mathrm {co}}(a,b,r)$ throughout.",
"However, $\\frac{\\partial f_{\\mathrm {co}}(a,b,r)}{\\partial r}\\Bigl |_{r=0}=-\\frac{\\operatorname{AC}\\left(\\frac{a}{\\sqrt{a^2+b^2}}\\right)}{\\sqrt{a^2+b^2}}.$ The rest is analogous.",
"Lemma 3.16 Suppose that $A_1,A_2$ are $2\\times 2$ matrices.",
"Then $\\operatorname{PD}(A_1)\\subset \\operatorname{PD}(A_2)$ holds if and only if $\\Vert A_1+\\lambda \\operatorname{Id}\\Vert _2\\le \\Vert A_2+\\lambda \\operatorname{Id}\\Vert _2 \\qquad \\text{ for all }\\lambda \\in \\mathbb {R}$ and $\\lfloor A_1+\\lambda \\operatorname{Id}\\rfloor _2\\ge \\lfloor A_2+\\lambda \\operatorname{Id}\\rfloor _2 \\qquad \\text{ for all }\\lambda \\in \\mathbb {R}.$ The norms and co-norms can be read off from the principal disk immediately.",
"Hence the statement is simple geometry.",
"Theorem 3.17 Suppose that $A_1,A_2$ are $\\log $ -able $2\\times 2$ matrices.",
"If $\\operatorname{PD}(A_1)\\subset \\operatorname{PD}(A_2),$ then $\\operatorname{PD}(\\log A_1)\\subset \\operatorname{PD}(\\log A_2).",
"$ The monotonicity is strict.",
"Similar statement applies to $\\operatorname{CD}$ .",
"In this case, the matrices $\\mathrm {e}^{\\lambda }A_i$ will also be $\\log $ -able.",
"Moreover, $\\operatorname{PD}(\\mathrm {e}^{\\lambda }A_1)\\subset \\operatorname{PD}(\\mathrm {e}^{\\lambda }A_2)$ holds.",
"Now, $\\log (\\mathrm {e}^{\\lambda }A_i)=\\log A_i+\\lambda \\operatorname{Id}$ .",
"By the previous theorem, $\\Vert \\log A_1+\\lambda \\operatorname{Id}\\Vert _2\\le \\Vert \\log A_2+\\lambda \\operatorname{Id}\\Vert _2$ and $\\lfloor A_1+\\lambda \\operatorname{Id}\\rfloor _2\\ge \\lfloor A_2+\\lambda \\operatorname{Id}\\rfloor _2$ holds for every $\\lambda \\in \\mathbb {R}$ .",
"According to the previous lemma, this implies the main statement.",
"The monotonicity is transparent in this case, as both $\\log $ and $\\exp $ are compatible with conjugation by orthogonal matrices, hence the orbit correspondence is one-to-one.",
"$\\log $ respects chirality, hence the statement can also be transferred to chiral disks."
],
[
"The case of Hilbert space operators",
"Theorem 4.1 Suppose that $\\mathbf {z}:[a,b]\\rightarrow \\mathfrak {H}$ is continuous.",
"Then $\\sqrt{ \\left(\\log \\frac{|\\mathbf {z}(b)|_2}{|\\mathbf {z}(a)|_2}\\right)^2+\\sphericalangle (\\mathbf {z}(a),\\mathbf {z} (b))^2}\\le \\int _{t\\in [a,b]}\\frac{|\\mathrm {d}\\mathbf {z}(t)|_2}{|\\mathbf {z}(t)|_2}.$ (Balázs Csikós, [10].)",
"The statement is non-vacuous only if the logarithmic variation $\\int _{t\\in [a,b]}\\frac{|\\mathrm {d}\\mathbf {z}(t)|_2}{|\\mathbf {z}(t)|_2}$ is finite.",
"This, however, implies that the (smaller) angular variation $\\int _{t\\in [a,b]}\\left|\\mathrm {d}\\frac{\\mathbf {z}(t)}{|\\mathbf {z}(t)|_2}\\right|_2$ is finite.",
"This allows to define a continuous map $\\tilde{\\mathbf {z}}:[a,b]\\rightarrow \\widetilde{\\mathbb {C}}$ by $\\tilde{\\mathbf {z}}(t)=\\left(|\\mathbf {z}(t)|_2, \\int _{s\\in [a,t]}\\left|\\mathrm {d}\\frac{\\mathbf {z}(s)}{|\\mathbf {z}(s)|_2}\\right|_2 \\right)_{\\mathrm {polar}},$ where $\\widetilde{\\mathbb {C}}$ is the universal covering space of $\\mathbb {C}\\setminus \\lbrace 0\\rbrace $ .",
"The intuitive idea is that one can consider the cone over $\\mathbf {z}$ , which is a developable surface, which we unfold to $\\widetilde{\\mathbb {C}}$ .",
"The curves ${\\mathbf {z}}$ and $\\tilde{\\mathbf {z}}$ look quite different but their (log)variations are the same because the their (log)radial and angular variations are the same, and the (log)variations can be assembled from them in the same manner.",
"Then $\\sqrt{ \\left(\\log \\frac{|\\mathbf {z}(b)|_2}{|\\mathbf {z}(a)|_2}\\right)^2+\\sphericalangle (\\mathbf {z}(a),\\mathbf {z} (b))^2}&\\le |\\log \\tilde{\\mathbf {z}}(b) -\\log \\tilde{\\mathbf {z}}(b) |=\\left| \\int _{t\\in [a,b]}\\frac{\\mathrm {d}\\tilde{\\mathbf {z}}(t)}{\\tilde{\\mathbf {z}}(t)}\\right|\\\\&\\le \\int _{t\\in [a,b]}\\frac{|\\mathrm {d}\\tilde{\\mathbf {z}}(t)|}{|\\tilde{\\mathbf {z}}(t)|}=\\int _{t\\in [a,b]}\\frac{|\\mathrm {d}\\mathbf {z}(t)|_2}{|\\mathbf {z}(t)|_2}$ shows the statement.",
"Theorem 4.2 If $\\phi $ is $\\mathcal {B}(\\mathfrak {H})$ -valued, then $ \\operatorname{CR}(\\operatorname{exp_{L}}\\phi )\\subset \\exp \\operatorname{D}(0,\\textstyle {\\int \\Vert \\phi \\Vert _2}), $ and $ \\operatorname{sp}(\\operatorname{exp_{L}}\\phi )\\subset \\exp \\operatorname{D}(0,\\textstyle {\\int \\Vert \\phi \\Vert _2}).",
"$ In particular, if $\\int \\Vert \\phi \\Vert _2<\\pi $ , then $\\log \\operatorname{exp_{L}}\\phi $ is well-defined, and for its spectral radius $ \\mathrm {r}(\\log \\operatorname{exp_{L}}\\phi ) \\le \\textstyle {\\int \\Vert \\phi \\Vert _2}.",
"$ Let $\\mathbf {x}\\in \\mathfrak {H}$ , $|\\mathbf {x}|_2=1$ .",
"Let us define $\\mathbf {z}:[a,b]\\rightarrow \\mathfrak {H}$ by $\\mathbf {z}(t)=\\operatorname{exp_{L}}(\\phi |_{[a,t]}) \\mathbf {x}.$ Apply Theorem REF .",
"Due to $\\mathbf {z}(a)= \\mathbf {x} $ , $\\mathbf {z}(b)= \\operatorname{exp_{L}}(\\phi )\\mathbf {x}$ , and the estimate $\\int _{t\\in [a,b]}\\frac{|\\mathrm {d}\\mathbf {z}(t)|_2}{|\\mathbf {z}(t)|_2}\\le \\int \\Vert \\phi \\Vert _2,$ we obtain (REF ) immediately.",
"If we replace $\\phi $ by $(\\phi ^*)^\\dag $ , i.e.",
"adjoined and order-reversed, then it yields $\\operatorname{CR}((\\operatorname{exp_{L}}\\phi )^*)\\subset \\exp \\operatorname{D}(0,\\textstyle {\\int \\Vert \\phi \\Vert _2})$ .",
"Then (REF ) implies (REF ).",
"An immediate consequence is Theorem 4.3 (Moan, Niesen [25], Casas [6].)",
"If $\\phi $ is a $\\mathcal {B}(\\mathfrak {H})$ -valued measure, $\\int \\Vert \\phi \\Vert _2<\\pi $ , then the Magnus expansion $\\sum _{k=1}^\\infty \\mu _{\\mathrm {L}[k]}(\\phi )$ is absolute convergent.",
"This follows Theorem REF and the spectral properties established above.",
"Next, we give some growth estimates.",
"Theorem 4.4 If $\\operatorname{CR}(A)\\subset \\exp \\operatorname{D}(0,p)$ , $0< p<\\pi $ , then $\\Vert (\\log A)\\Vert _2\\le H(p), $ where $H(p)=p-2\\log \\left(2\\cosh \\frac{p}{2}-\\frac{2}{p}\\sinh \\frac{p}{2}\\right)+\\int _{t=0}^\\pi HH(p,t)\\,\\mathrm {d}t $ with $HH(p,t)=\\frac{(\\sin (p\\sin t)-(p \\sin t) \\cos (p \\sin t))(\\mathrm {e}^{p\\cos t} +\\mathrm {e}^{-p\\cos t}-2\\cos (p \\sin t))}{(\\sin (p\\sin t))(2\\sin t+\\mathrm {e}^{p\\cos t}\\sin (-t+p\\sin t)-\\mathrm {e}^{-p\\cos t}\\sin (t+p\\sin t))}.",
"$ $H(p)$ and $HH(p,t)$ are positive and finite for $0<p<\\pi $ .",
"The statement (trivially) extends to $p=0$ with $H(p)=0$ .",
"The expression of $HH(p,t)$ looks complicated.",
"However, it can be rewritten as $HH(p,t)=\\frac{(p^2\\sin t)\\left(\\frac{1}{p^3\\sin t} \\int _{q=0}^p q\\sin (q\\sin t)\\,\\mathrm {d}q \\right)\\cdot \\left(\\frac{1}{p^2}(\\cosh (p\\cos t)-\\cos (p \\sin t))\\right)}{\\left(\\frac{\\sin (p\\sin t)}{p\\sin t}\\right)\\left( \\frac{1}{p^2\\sin t}\\int _{q=0}^p \\cosh (q\\cos t)\\sin (q\\sin t)\\,\\mathrm {d}q \\right)}.$ From the power series expansion, it is easy to see that the expressions in the big parentheses are actually entire functions of $p$ and $t$ .",
"Moreover, one can see that these entire functions are positive for $(p,t)\\in [0,\\pi )\\times [0,\\pi ]$ .",
"In fact, what prevents the smooth extension to $(p,t)\\in [0,\\pi ]\\times [0,\\pi ]$ is only the singularity in $\\frac{\\sin (p\\sin t)}{p\\sin t}$ .",
"Assume that $|\\mathbf {x}|_2=1$ .",
"According to the Lemma REF , $\\left|\\frac{A-\\operatorname{Id}}{A-\\lambda \\operatorname{Id}}\\mathbf {x}\\right|_2\\in \\left\\lbrace \\frac{|\\omega -1|}{|\\omega -\\lambda |} :\\omega \\in \\operatorname{CR}(A)\\right\\rbrace .",
"$ So, we can estimate $\\left|\\frac{A-\\operatorname{Id}}{A-\\lambda \\operatorname{Id}}\\mathbf {x}\\right|_2$ as follows.",
"Take the Apollonian circles relative to $\\lambda $ and 1, and take the closest one to $\\lambda $ but which still touches $\\operatorname{D}(0,p)$ .",
"Then the characteristic ratio of this Apollonian circle provides an upper estimate.",
"This leads to considering circles (and lines) which are tangent to the curve $\\gamma _p(t)=\\mathrm {e}^{p\\cos (t)}\\cos (p\\sin t)+\\mathrm {i} \\mathrm {e}^{p\\cos (t)}\\sin (p\\sin t) )$ ($t\\in [0,\\pi ]$ ), and their center is on the real axis (or in the infinity).",
"If $t\\in (0,\\pi )$ , then the normal line at $\\gamma _p(t)$ intersects the real axis at $C_p(t)=\\frac{\\mathrm {e}^{p\\cos t}\\sin t }{\\sin (t+p\\sin t)},$ the center of the circle.",
"This leads to radius $r_p(t)= \\frac{\\mathrm {e}^{p\\cos t}\\sin (p\\sin t) }{\\sin (t+p\\sin t)}.$ (The sign counts the touching orientation to $\\gamma _p$ .)",
"Taking the inverse of 1, relative to the circle above, leads to the Apollonian pole $f_p(t)=-\\frac{\\sin t-\\mathrm {e}^{p\\cos t}\\sin (t-p\\sin t)}{\\sin t-\\mathrm {e}^{-p\\cos t}\\sin (t+p\\sin t)}$ conjugate to 1.",
"The functions $C_p$ and $r_p$ are singular, but $f_p$ is not.",
"This can be seen from $ \\sin t-\\mathrm {e}^{p\\cos t}\\sin (t-p\\sin t)=\\int _{q=0}^p \\mathrm {e}^{q\\cos t}\\sin (q\\sin t)\\,\\mathrm {d}q>0,$ $\\sin t-\\mathrm {e}^{-p\\cos t}\\sin (t+p\\sin t)=\\int _{q=0}^p \\mathrm {e}^{-q\\cos t}\\sin (q\\sin t)\\,\\mathrm {d}q>0.$ In fact, $f_p$ is strictly increasing.",
"Indeed, $f^{\\prime }_p(t)=\\frac{(\\sin (p\\sin t)-(p \\sin t) \\cos (p \\sin t))(\\mathrm {e}^{p\\cos t} +\\mathrm {e}^{-p\\cos t}-2\\cos (p \\sin t))}{(\\sin t-\\mathrm {e}^{-p\\cos t}\\sin (t+p\\sin t))^2}=$ $=\\frac{(\\sin t)^2\\left( \\int _{q=0}^p q\\sin (q\\sin t)\\,\\mathrm {d}q \\right)\\cdot 2(\\cosh (p\\cos t)-\\cos (p \\sin t))}{\\left( \\int _{q=0}^p \\mathrm {e}^{-q\\cos t}\\sin (q\\sin t)\\,\\mathrm {d}q \\right)^2}>0.$ It is easy to see that the range of $f_p$ is $(f_p(0+),f_p(\\pi +))=\\left(-\\frac{1-\\mathrm {e}^{p}(1-p)}{1-\\mathrm {e}^{-p}(1+p)},-\\frac{1-\\mathrm {e}^{-p}(1+p)}{1-\\mathrm {e}^{p}(1-p)}\\right).$ The characteristic ratio belonging to the relevant Apollonian circle is $\\chi _p(t)=\\frac{|\\gamma _p(1)-1|}{|\\gamma _p(t)-f_p(t)|}=\\frac{\\sin t-\\mathrm {e}^{-p\\cos t}\\sin (t+p\\sin t)}{\\sin (p\\sin t)}.$ The values $t=0$ and $t=\\pi $ exceptional, because tangent circles there always have their centers on the real axis.",
"Let $s\\in (-\\infty ,0]$ .",
"Consider the Apollonian circles between $s$ and 1, and consider the one closest to $s$ but still touching $\\gamma _p$ .",
"From geometrical considerations (the injectivity of $f_p$ ) we can devise that closest touching circle touches at $\\gamma _p(0)&\\quad \\text{if}\\quad s\\in (-\\infty ,f_p(0+)],\\\\\\gamma _p(t)&\\quad \\text{if}\\quad s=f_p(t)\\in (f_p(0+),f_p(\\pi -)),\\\\\\gamma _p(\\pi )&\\quad \\text{if}\\quad s\\in [f_p(\\pi -),0].$ This provides the estimate $|(\\log A)\\mathbf {x}|_2\\le &\\int _{s=-\\infty }^{f_p(0+)} \\frac{|\\gamma _p(0)-1|}{(1-s)|\\gamma _p(0)-s|} \\,\\mathrm {d}s+\\int _{t=0}^\\pi \\frac{\\chi _p(t)}{1-f_p(t)}\\,\\mathrm {d}f_p(t)+\\\\&+\\int _{s=f_p(\\pi -)}^0 \\frac{|\\gamma _p(\\pi )-1|}{(1-s)|\\gamma _p(\\pi )-s|} \\,\\mathrm {d}s.$ The first and third integrals expands as $\\int _{s=-\\infty }^{f_p(0+)}{\\frac{{{\\mathrm {e}}^{p}}-1}{\\left( 1-s \\right) \\left( {{\\mathrm {e}}^{p}}-s \\right) }}\\,\\mathrm {d}s=\\left[\\log \\left( {\\frac{{{\\rm e}^{p}}-s}{1-s}} \\right)\\right]_{s=-\\infty }^{f_p(0+)}=\\log \\frac{p}{p-1+\\mathrm {e}^{-p}(p+1)},$ $\\int _{s=f_p(\\pi -)}^0 {\\frac{ 1-{{\\rm e}^{-p}}}{ \\left( 1-s \\right) \\left( {{\\rm e}^{-p}}-s \\right) }} \\,\\mathrm {d}s=\\left[\\log \\left( {\\frac{1-s}{{{\\rm e}^{-p}}-s}} \\right) \\right]_{s=f_p(\\pi -)}^0=\\log \\frac{p}{p-1+\\mathrm {e}^{-p}(p+1)}.$ Note that $\\log \\frac{p}{p-1+\\mathrm {e}^{-p}(p+1)}=\\frac{p}{2}-\\log \\frac{\\mathrm {e}^{\\frac{p}{2}}(p-1)+\\mathrm {e}^{-\\frac{p}{2}}(p+1)}{p}.$ The integrand in the second integral expands as indicated in (REF ).",
"The estimate (REF ) is certainly not sharp.",
"For example, in the proof, we estimated $|A^{-1}\\mathbf {x}|_2$ by $\\mathrm {e}^{p}$ , which belongs to $A(A^{-1}\\mathbf {x}): A^{-1}\\mathbf {x}=\\mathrm {e}^{-p}$ , i. e. $A^{-1}\\mathbf {x}=\\mathrm {e}^{p}\\mathbf {x}$ .",
"But then $|(\\log A)\\mathbf {x}|_2=|-p\\mathbf {x}|_2=p<H(p)$ would hold.",
"In general, there is a penalty or gain (depending on the viewpoint) for approaching the real axis in $\\operatorname{CR}(A)$ , for which we have not accounted.",
"Formulating this numerically, we can obtain a stronger estimate than $H(p)$ , but making the argument more technical.",
"Theorem 4.5 (a) As $p\\searrow 0$ , $H(p)=p+\\frac{1}{4}{p}^{2}+{\\frac{23}{864}}{p}^{4}+\\mathrm {O}(p^6).",
"$ (b) As $p\\nearrow \\pi $ $H(p)=\\frac{2\\pi ^2}{\\sqrt{\\pi ^2-p^2}}+H_\\pi +o(1)=\\frac{\\sqrt{2}\\pi ^{3/2}}{\\sqrt{\\pi -p}}+H_\\pi +o(1)=p\\sqrt{\\frac{\\pi +p}{\\pi -p}}+H_\\pi +o(1),$ where $H_\\pi =\\pi -2\\log \\left(2\\cosh \\frac{\\pi }{2}-\\frac{2}{\\pi }\\sinh \\frac{\\pi }{2}\\right)+\\int _{t=0}^\\pi \\left(HH(\\pi ,t)-\\frac{2}{\\cos ^2 t} \\right)\\,\\mathrm {d}t$ (and the integrand is actually a smooth function of $t$ ).",
"Numerically, $H_\\pi =-2.513\\ldots $ (c) In general, the crude estimate $H(p)\\le (1+o(1))\\, p\\sqrt{\\frac{\\pi +p}{\\pi -p}}$ holds, where $o(1)$ is understood as $p\\searrow 0$ or $p\\nearrow \\pi $ .",
"$(1+o(1))$ can be replaced by 1, thus yielding an absolute estimate; but the computation is tedious.",
"Consider (REF ).",
"One finds $p-2\\log \\left(2\\cosh \\frac{p}{2}-\\frac{2}{p}\\sinh \\frac{p}{2}\\right)=p-{\\frac{5}{12}}{p}^{2}+{\\frac{49}{1440}}{p}^{4}+O \\left( {p}^{6}\\right).",
"$ Regarding $HH(p,t)$ , one can see that $\\frac{1}{p^3\\sin t} \\int _{q=0}^p q\\sin (q\\sin t)\\,\\mathrm {d}q&=\\frac{1}{3}-\\frac{\\sin ^{2} t }{30}\\,{p}^{2}+O(p^4),\\\\\\frac{\\cosh (p\\cos t)-\\cos (p \\sin t)}{p^2}&=\\frac{1}{2}+ \\frac{ \\cos ^{2} t - \\sin ^{2} t }{24} \\,{p}^{2}+O(p^4),\\\\\\frac{\\sin (p\\sin t)}{p\\sin t}&=1-\\frac{\\sin ^{2}t}{6}\\,{p}^{2}+O(p^4),\\\\\\frac{1}{p^2\\sin t}\\int _{q=0}^p \\cosh (q\\cos t)\\sin (q\\sin t)\\,\\mathrm {d}q&=\\frac{1}{2}+{\\frac{3\\, \\cos ^{2} t - \\sin ^{2} t }{24 }}\\,{p}^{2}+O(p^4).$ Consequently, $HH(p,t)=\\frac{\\sin t}{3}p^2+{\\frac{(2\\sin ^{2} t-5 \\cos ^{2} t )\\sin t }{90}}{p}^{4}+O(p^6).$ Integrating this for $t\\in [0,\\pi ]$ , it gives $\\int _{t=0}^\\pi HH(p,t)\\,\\mathrm {d}t=\\frac{2}{3}\\,{p}^{2}-{\\frac{1}{135}}{p}^{4}+O(p^6).",
"$ Adding (REF ) and (REF ) yields (REF ).",
"(b) Notice that $\\frac{\\sin x}{(\\pi ^2-x^2)}$ is analytic function, which is positive on $x\\in [-\\pi ,\\pi ]$ .",
"Consequently, $\\frac{\\sin (p\\sin t)}{(\\pi ^2- p^2\\sin ^2 t)p\\sin t }$ is an entire function of $p,t$ such that it is positive for $(p,t)\\in [0,\\pi ]\\times [0,\\pi ]$ .",
"Hence $HH(p,t)=\\frac{1}{\\pi ^2- p^2\\sin ^2 t } \\widetilde{HH}(p,t),$ where $\\widetilde{HH}(p,t)$ is smooth on $(p,t)\\in [0,\\pi ]\\times [0,\\pi ]$ .",
"Due to symmetry for $t\\leftrightarrow \\pi -t$ , $\\widetilde{HH}(p,t)- \\widetilde{HH}(p,\\pi /2)$ not only vanishes at $t=\\pi /2$ but $\\cos ^2 t$ can be factored out.",
"Thus $\\widehat{HH}(p,t)=\\frac{\\widetilde{HH}(p,t)- \\widetilde{HH}(p,\\pi /2)}{\\pi ^2\\cos ^2 t}$ can also be considered as a smooth function on $(p,t)\\in [0,\\pi ]\\times [0,\\pi ]$ .",
"Now we have $HH(p,t)=\\frac{1}{\\pi ^2- p^2\\sin ^2 t } \\widetilde{HH}(p,\\pi /2)+\\frac{\\pi ^2\\cos ^2 t}{\\pi ^2- p^2\\sin ^2 t }\\widehat{HH}(p,t).$ For a fixed $p$ the first summand integrates to $\\int _{t=0}^\\pi \\frac{1}{\\pi ^2- p^2\\sin ^2 t } \\widetilde{HH}(p,\\pi /2)\\,\\mathrm {d}t=\\frac{\\widetilde{HH}(p,\\pi /2)}{\\sqrt{\\pi ^2-p^2}}=\\sqrt{\\pi ^2-p^2}\\,\\frac{\\sin p-p\\cos p}{\\sin p}=\\\\=\\frac{2\\pi ^2}{\\sqrt{\\pi ^2-p^2}}+o(1)=\\frac{\\sqrt{2}\\pi ^{3/2}}{\\sqrt{\\pi -p}}+H_\\pi +o(1)=p\\sqrt{\\frac{\\pi +p}{\\pi -p}}+H_\\pi +o(1).$ The function $\\frac{\\pi ^2\\cos ^2 t}{\\pi ^2- p^2\\sin ^2 t }=\\frac{\\pi ^2- \\pi ^2\\sin ^2 t}{\\pi ^2- p^2\\sin ^2 t } $ is uniformly bounded by 0 and 1, and, in fact $\\lim _{ p \\nearrow \\pi }\\frac{\\cos ^2 t}{\\pi ^2- p^2\\sin ^2 t }=1 \\qquad \\text{ for }t\\in [0,\\pi ]\\setminus \\left\\lbrace \\frac{\\pi }{2}\\right\\rbrace $ pointwise.",
"Thus, by Lebesgue's dominated convergence theorem, the integral of the second summand is $\\int _{t=0}^\\pi \\widehat{HH}(\\pi ,t)\\,\\mathrm {d}t+o(1).$ Notice that $\\widehat{HH}(\\pi ,t)$ is a smooth function.",
"Taking limit with $p\\nearrow \\pi $ we find that $\\widehat{HH}(\\pi ,t)=HH(\\pi ,t)-\\frac{2}{\\cos ^2 t}.$ The numerical evaluation of $H_\\pi $ can be realized by various methods.",
"(c) This immediately follows from the power series expansion $p\\sqrt{\\frac{\\pi +p}{\\pi -p}}=p+\\frac{1}{\\pi }p^2+O(p^3)$ as $p\\searrow 0$ ; and from the asymptotic behaviour as $p\\nearrow \\pi $ , what we have seen.",
"As a corollary, we obtain Theorem 4.6 If $\\phi $ is $\\mathcal {B}(\\mathfrak {H})$ -valued, and $\\int \\Vert \\phi \\Vert _2<\\pi $ , then the following hold: (a) Regarding the norm of the Magnus expansion, $\\Vert \\mu _{\\mathrm {R}}(\\phi )\\Vert _2\\equiv \\left\\Vert \\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_k\\in I}\\phi (t_1)\\cdot \\ldots \\cdot \\phi (t_k)\\right\\Vert _2\\le H\\left(\\int \\Vert \\phi \\Vert _2\\right).$ (b) Regarding the $k$ th term of the Magnus expansion, $\\Vert \\mu _{\\mathrm {R}[k]}(\\phi )\\Vert _2\\equiv \\left\\Vert \\int _{t_1\\le \\ldots \\le t_k\\in I}\\mu _k(\\phi (t_1),\\ldots ,\\phi (t_k))\\right\\Vert _2\\le (1+o(1))\\,\\pi ^{-k+1}2\\sqrt{\\mathrm {e} k}\\left(\\int \\Vert \\phi \\Vert _2\\right)^k;$ where $(1+o(1))$ is understood in absolute sense, it does not depend on $\\phi $ .",
"$(1+o(1))$ can be replaced by 1.",
"(a) This follows from Theorems REF and REF .",
"(b) $\\int \\Vert \\phi \\Vert _2>0$ can be assumed.",
"Consider the operator valued function $\\eta $ given by $\\eta (z)=\\log \\operatorname{exp_{R}}\\left(\\frac{z}{\\smallint \\Vert \\phi \\Vert _2}\\phi \\right).$ This is analytic in $\\operatorname{D}(0,\\pi )$ , moreover, $\\Vert \\eta (z)\\Vert _2\\le H(|z|).$ Applying the generalized Cauchy theorem with $\\partial \\operatorname{D}\\left(0,\\pi -\\frac{1}{2k}\\pi \\right)$ , we estimate the $k$ th power series coefficient $\\eta _k$ of $\\eta $ at $z=0$ , by $\\Vert \\eta _k\\Vert _2&\\le \\left(\\pi -\\frac{1}{2k}\\pi \\right)^{-k}H\\left(\\pi -\\frac{1}{2k}\\right)\\\\&\\le \\left(\\pi -\\frac{1}{2k}\\pi \\right)^{-k}(1+o(1))\\left(\\pi -\\frac{1}{2k}\\pi \\right)\\sqrt{\\frac{2\\pi -\\frac{1}{2k}\\pi }{\\frac{1}{2k}\\pi }}\\\\&=(1+o(1))\\pi ^{-k+1}\\left(1-\\frac{1}{2k}\\right)^{-k+1}2\\sqrt{k-\\frac{1}{4}}\\\\&\\le (1+o(1))\\,\\pi ^{-k+1}2\\sqrt{\\mathrm {e} k}.$ On the other hand, $\\eta _k=\\left(\\int \\Vert \\phi \\Vert _2\\right)^{-k}\\mu _{\\mathrm {R}[k]}(\\phi ).$ This proves the statement.",
"Remark 4.7 Compared to the expansion (REF ) of $\\Theta $ , the expansion (REF ) of $H$ is strikingly different.",
"It shows that the naive estimate algebraic expansion does not deal with geometry very well."
],
[
"Some examples from $\\operatorname{SL}_2(\\mathbb {R})$",
"Example 5.1 (Skew-loxodromic composition.)",
"Consider the matrices $\\tilde{J}=\\begin{bmatrix}1&\\\\&-1\\end{bmatrix},\\qquad \\tilde{I}=\\begin{bmatrix}0&-1\\\\1&0\\end{bmatrix}.$ For $\\alpha ,\\beta \\in \\mathbb {C}$ , let $\\Upsilon _{\\alpha ,\\beta }=\\alpha \\tilde{J}\\mathbf {1}.\\beta \\tilde{I}\\mathbf {1}.",
"$ Then $\\int \\Vert \\Upsilon _{\\alpha ,\\beta }\\Vert _2=|\\alpha |+|\\beta |.$ For $|\\alpha |+|\\beta |<\\pi $ , we can consider $\\mu _{\\mathrm {L}}(\\Upsilon _{\\alpha ,\\beta })&=\\log (\\exp _{\\mathrm {L}}(\\Upsilon _{\\alpha ,\\beta }))\\\\&= \\log (\\exp (\\beta \\tilde{I})\\exp (\\alpha \\tilde{J}))\\\\&=\\log \\begin{bmatrix}\\mathrm {e}^\\alpha \\cos \\beta &-\\mathrm {e}^{-\\alpha }\\sin \\beta \\\\\\mathrm {e}^{\\alpha }\\sin \\beta &\\mathrm {e}^{-\\alpha }\\cos \\beta \\end{bmatrix}\\\\&=\\operatorname{AC}(\\cosh \\alpha \\cos \\beta )\\begin{bmatrix}\\sinh \\alpha \\cos \\beta &-\\mathrm {e}^{-\\alpha }\\sin \\beta \\\\\\mathrm {e}^{\\alpha }\\sin \\beta &-\\sinh \\alpha \\cos \\beta \\end{bmatrix}.$ If $\\alpha ,\\beta \\ge 0$ , then $\\Vert \\mu _{\\mathrm {L}}(\\Upsilon _{\\alpha ,\\beta })\\Vert _2=\\operatorname{AC}(\\cosh \\alpha \\cos \\beta )\\cdot (\\sinh \\alpha +\\cosh \\alpha \\sin \\beta ).$ Now, for $p\\in [0,\\pi )$ , let $\\tilde{\\alpha }(p)=p-\\pi +\\@root 3 \\of {\\pi ^2(\\pi -p)},$ $\\tilde{\\beta }(p)=\\pi -\\@root 3 \\of {\\pi ^2(\\pi -p)}.$ Then $\\tilde{\\alpha }(p),\\tilde{\\beta }(p)\\ge 0$ , and $\\tilde{\\alpha }(p)+\\tilde{\\beta }(p)=p.$ Thus, $\\int \\Vert \\Upsilon _{\\tilde{\\alpha }(p),\\tilde{\\beta }(p)}\\Vert _2=p.",
"$ As $p\\nearrow \\pi $ , we see that $\\tilde{\\alpha }(p)\\searrow 0$ (eventually) and $\\tilde{\\beta }(p)\\nearrow \\pi $ .",
"Consequently $\\lim _{p\\rightarrow \\pi } \\cosh \\tilde{\\alpha }(p)\\cos \\tilde{\\beta }(p)=-1.",
"$ In that (elliptic) domain $\\operatorname{AC}$ is computed by $\\arccos $ .",
"Now, elementary function calculus shows that as $p\\nearrow \\pi $ , $\\Vert \\mu _{\\mathrm {L}}(\\Upsilon _{\\tilde{\\alpha }(p),\\tilde{\\beta }(p)})\\Vert _2&\\stackrel{\\rightarrow }{=}\\frac{\\arccos (\\cosh \\tilde{\\alpha }(p)\\cos \\tilde{\\beta }(p))}{\\sqrt{1-\\cosh ^2\\tilde{\\alpha }(p)\\cos ^2\\tilde{\\beta }(p)}}(\\sinh \\tilde{\\alpha }(p)+\\cosh \\tilde{\\alpha }(p)\\sin \\tilde{\\beta }(p))\\\\&=\\sqrt{\\frac{12\\pi ^{8/3}}{\\pi ^2+6}} (\\pi -p)^{-1/3}+O((\\pi -p)^{1/3}).$ We see that in Baker–Campbell–Hausdorff setting we can produce the asymptotics $O((\\pi -p)^{-1/3})$ , although having exponent $-1/3$ instead of $-1/2$ is strange.",
"It is interesting to see that in the setting of the present example, one cannot do much better.",
"If we try to optimize $\\Vert \\mu _{\\mathrm {L}}(\\Upsilon _{\\alpha ,\\beta })\\Vert _2$ for $\\alpha +\\beta $ $(\\alpha ,\\beta \\ge 0)$ , then, after some computation, it turns out that the best approach is along a well-defined ridge.",
"This ridge starts hyperbolic, but turns elliptic.",
"Its elliptic part is part is parametrized by $x\\in (-1,1]$ , and $\\hat{\\alpha }(x)=\\operatorname{arcosh}\\left(\\frac{\\operatorname{AC}(x)+\\sqrt{\\operatorname{AC}(x)^2-4x(1-x\\operatorname{AS}(x))\\operatorname{AS}(x)}}{2(1-x\\operatorname{AS}(x))}\\right);$ $\\hat{\\beta }(x)=\\arccos \\left(\\frac{\\operatorname{AC}(x)-\\sqrt{\\operatorname{AC}(x)^2-4x(1-x\\operatorname{AS}(x))\\operatorname{AS}(x)}}{2\\operatorname{AS}(x)}\\right).$ Then $\\cosh \\hat{\\alpha }(x)\\cos \\hat{\\beta }(x)=x.$ Actually, $x=1$ gives a parabolic $\\operatorname{exp_{L}}(\\Upsilon _{ \\hat{\\alpha }(x),\\hat{\\beta }(x)})$ , but for $x\\in (-1,1)$ it is elliptic.",
"Then $\\hat{\\alpha }(x),\\hat{\\beta }(x)\\ge 0$ .",
"As $y\\searrow -1$ , one can see that $\\alpha \\searrow 0$ (eventually) and $\\beta \\nearrow \\pi $ ; and, more importantly, $\\hat{\\alpha }(x)+\\hat{\\beta }(x)\\nearrow \\pi .",
"$ Now, as $x\\searrow -1$ , $\\frac{\\arccos x}{\\sqrt{1-x^2}}(\\sinh \\hat{\\alpha }(x)+\\cosh \\hat{\\alpha }(x)\\sin \\hat{\\beta }(x))=\\pi 2^{3/4}(x+1)^{-1/4}+O((x+1)^{1/4}),$ and $\\pi -\\hat{\\alpha }(x)-\\hat{\\beta }(x)=\\frac{1}{3} 2^{3/4}(x+1)^{3/4}+O((x+1)^{5/4}).$ Hence, using the notation $\\hat{p}(x)=\\hat{\\alpha }(x)+\\hat{\\beta }(x)$ , we find $\\Vert \\mu _{\\mathrm {L}}(\\Upsilon _{ \\hat{\\alpha }(x),\\hat{\\beta }(x)})\\Vert _2=2\\pi 3^{-1/3}(\\pi - \\hat{p}(x))^{-1/3}+O((\\pi - \\hat{p}(x))^{1/3}).$ This $2\\pi 3^{-1/3}=4.356\\ldots $ is just slightly better than $\\sqrt{\\frac{12\\pi ^{8/3}}{\\pi ^2+6}}=4.001\\ldots $ .",
"Example 5.2 (Skew-loxodromic divergence) Suppose that $\\alpha >0$ .",
"Then $\\int \\Vert \\Upsilon _{\\alpha ,\\pi }\\Vert _2=\\alpha +\\pi .$ Now, we claim, $\\sum _{n=1}^\\infty \\mu _{\\mathrm {L}[n]}(\\Upsilon _{ \\alpha ,\\pi })\\qquad \\text{is divergent.",
"}$ Indeed, consider $\\operatorname{exp_{L}}(\\Upsilon _{ t\\alpha ,t\\pi })$ for $t\\in \\mathbb {C}$ .",
"For $t=1$ , $\\operatorname{exp_{L}}(\\Upsilon _{ \\alpha ,\\pi })=-\\begin{bmatrix}\\cosh \\alpha &\\sinh \\alpha \\\\\\sinh \\alpha &\\cosh \\alpha \\end{bmatrix}$ , which has two distinct real roots, $-\\mathrm {e}^{\\pm \\alpha }$ .",
"This implies that $\\operatorname{exp_{L}}(\\Upsilon _{t\\alpha ,t\\pi })$ is not an exponential of a real $2\\times 2$ matrix for $t\\in (1-\\varepsilon ,1]$ , with some $\\varepsilon >0$ .",
"Consequently, the convergence radius of the germ of $\\log \\operatorname{exp_{L}}(\\Upsilon _{ t\\alpha ,t\\pi })$ around $t=0$ is at most $1-\\varepsilon $ .",
"But this implies divergence at $t=1$ .",
"More quantitatively, consider the function $t\\mapsto \\log (\\exp _{\\mathrm {L}}(\\Upsilon _{\\alpha t,\\pi t}))=\\operatorname{AC}(\\cosh \\alpha t\\cos \\pi t)\\begin{bmatrix}\\sinh \\alpha t\\cos \\pi t&-\\mathrm {e}^{-\\alpha t}\\sin \\pi t\\\\\\mathrm {e}^{\\alpha t}\\sin \\pi t &-\\sinh \\alpha t\\cos \\pi t \\end{bmatrix},$ and try to extend it analytically from around $t=0$ along $[0,+\\infty )$ .",
"Then we see that it develops a singularity corresponding to $\\cosh \\alpha t\\cos \\pi t=-1$ before $t=1$ .",
"Example 5.3 (Skew-elliptic composition.)",
"Consider the matrices $\\tilde{P}=\\begin{bmatrix}0&-1\\\\&0\\end{bmatrix},\\qquad \\tilde{I}=\\begin{bmatrix}&-1\\\\1&\\end{bmatrix}.$ For $\\alpha ,\\beta \\in \\mathbb {C}$ , let $\\tilde{\\Upsilon }_{\\alpha ,\\beta }=\\alpha \\tilde{P}\\mathbf {1}.\\beta \\tilde{I}\\mathbf {1}.",
"$ Then $\\int \\Vert \\tilde{\\Upsilon }_{\\alpha ,\\beta }\\Vert _2=|\\alpha |+|\\beta |.$ For $|\\alpha |+|\\beta |<\\pi $ , we can consider $\\mu _{\\mathrm {L}}(\\tilde{\\Upsilon }_{\\alpha ,\\beta })&=\\log (\\exp _{\\mathrm {L}}(\\tilde{\\Upsilon }_{\\alpha ,\\beta }))\\\\&= \\log (\\exp (\\beta \\tilde{I})\\exp (\\alpha \\tilde{P}))\\\\&=\\log \\begin{bmatrix}\\cos \\beta &-\\alpha \\cos \\beta -\\sin \\beta \\\\\\sin \\beta &-\\alpha \\sin \\beta +\\cos \\beta \\end{bmatrix}\\\\&=\\operatorname{AC}\\left(\\cos \\beta -\\frac{\\alpha }{2}\\sin \\beta \\right)\\begin{bmatrix}\\frac{\\alpha }{2}\\sin \\beta &\\alpha \\cos \\beta -\\sin \\beta \\\\\\sin \\beta &-\\frac{\\alpha }{2}\\sin \\beta \\end{bmatrix}.$ If $\\alpha ,\\beta \\ge 0$ , then $\\Vert \\mu _{\\mathrm {L}}(\\tilde{\\Upsilon }_{\\alpha ,\\beta })\\Vert _2=\\operatorname{AC}\\left(\\cos \\beta -\\frac{\\alpha }{2}\\sin \\beta \\right)\\cdot \\left(\\sin \\beta +\\frac{\\alpha }{2}\\cos \\beta +\\frac{\\alpha }{2}\\right).$ For optimal approach, consider $x\\in (-1,1]$ , and let $\\hat{\\alpha }(x)=\\frac{2\\operatorname{At}(x)}{\\sqrt{1-(x+\\operatorname{At}(x))^2}};\\qquad \\hat{\\beta }(x)=\\arccos \\left(x+\\operatorname{At}(x)\\right).$ Then $\\cos \\hat{\\beta }(x)-\\frac{\\hat{\\alpha }(x)}{2}\\sin \\beta (x)=x.$ As $x\\searrow -1$ , we have $\\alpha \\searrow 0$ (eventually) and $\\beta \\nearrow \\pi $ ; and, $\\hat{\\alpha }(x)+\\hat{\\beta }(x)\\nearrow \\pi .$ Now, as $x\\searrow -1$ , $\\Vert \\mu _{\\mathrm {L}}(\\tilde{\\Upsilon }_{ \\hat{\\alpha }(x),\\hat{\\beta }(x)})\\Vert _2&=\\frac{\\arccos x}{\\sqrt{1-x^2}}\\left(\\sin \\hat{\\beta }(x)+\\frac{\\hat{\\alpha }(x)}{2}\\cos \\hat{\\beta }(x)+\\frac{\\hat{\\alpha }(x)}{2}\\right)\\\\&=2^{1/4}\\pi (t+1)^{-1/4}+O((t+1)^{1/4}),$ and $\\pi -\\hat{\\alpha }(x)-\\hat{\\beta }(x)=\\frac{2}{3} 2^{1/4}(x+1)^{3/4}+O((x+1)^{5/4}).$ Hence, using the notation $\\hat{p}(x)=\\hat{\\alpha }(x)+\\hat{\\beta }(x)$ , we find $\\Vert \\mu _{\\mathrm {L}}(\\tilde{\\Upsilon }_{ \\hat{\\alpha }(x),\\hat{\\beta }(x)})\\Vert _2=\\pi (4/3)^{1/3}(\\pi - \\hat{p}(x))^{-1/3}+O((\\pi - \\hat{p}(x))^{1/3}).$ This leading coefficient $\\pi (4/3)^{1/3}=1.100\\ldots $ is worse than the previous ones.",
"A similar analysis of divergence can be carried out.",
"The previous two examples are usual subjects of convergence estimates of the Baker–Campbell–Hausdorff formula.",
"For example, the latter one already appears in Wei [37] (without asymptotics).",
"More sophisticated investigations start with Michel [20] (he uses Frobenius norm).",
"The following two examples (variants of each other), were already used by Moan [24] in order obtain $\\pi $ as the upper bound for the convergence radius of the Magnus expansion.",
"Lemma 5.4 The solution of the ordinary differential equation $\\frac{\\mathrm {d}A(\\theta )}{\\mathrm {d}t}A(\\theta )^{-1}=a\\begin{bmatrix}-\\sin 2b\\theta & \\cos 2b\\theta \\\\\\cos 2b\\theta &\\sin 2b\\theta \\end{bmatrix}\\equiv \\exp (b\\theta \\tilde{I}) a\\tilde{K} \\exp (-b\\theta \\tilde{I}),$ $A(0)=\\begin{bmatrix}1& \\\\&1\\end{bmatrix} \\equiv \\operatorname{Id}_2,$ is given by $A(\\theta )=W(a\\theta ,b\\theta );$ where $W(p,w)=\\begin{bmatrix}\\cos w&-\\sin w\\\\\\sin w&\\cos w\\end{bmatrix}\\begin{bmatrix}\\operatorname{\\lnot Cosh}(p^2-w^2)&(p+w)\\operatorname{\\lnot Sinh}(p^2-w^2) \\\\(p-w)\\operatorname{\\lnot Sinh}(p^2-w^2)&\\operatorname{\\lnot Cosh}(p^2-w^2)\\end{bmatrix}$ $\\equiv \\exp (w\\tilde{I})(\\operatorname{\\lnot Cosh}(p^2-w^2 ) \\operatorname{Id}+\\operatorname{\\lnot Sinh}(p^2-w^2) (-w\\tilde{I}+p\\tilde{K}) ) ;$ such that the functions $\\operatorname{\\lnot Cosh}$ and $\\operatorname{\\lnot Sinh}$ are given by $\\operatorname{\\lnot Cosh}(x)={\\left\\lbrace \\begin{array}{ll}\\cos \\sqrt{-x}&\\text{if }x<0 \\\\1&\\text{if }x=0\\\\\\cosh \\sqrt{x}&\\text{if }x>0,\\end{array}\\right.",
"}$ $\\operatorname{\\lnot Sinh}(x)={\\left\\lbrace \\begin{array}{ll}\\frac{\\sin \\sqrt{-x}}{\\sqrt{-x}}&\\text{if }x<0 \\\\1&\\text{if }x=0\\\\\\frac{\\sinh \\sqrt{x}}{\\sqrt{x}}&\\text{if }x>0,\\end{array}\\right.",
"}$ on the real domain, but they are, in fact, entire functions on the complex plane.",
"This can be checked by direct computation.",
"Example 5.5 (Moan's example / Magnus critical development.)",
"On the interval $[0,\\pi ]$ , consider the measure $\\Phi $ , such that $\\Phi (\\theta )=\\begin{bmatrix}-\\sin 2\\theta & \\cos 2\\theta \\\\\\cos 2\\theta &\\sin 2\\theta \\end{bmatrix}\\,\\mathrm {d}\\theta |_{[0,\\pi ]}.$ Then, $\\int \\Vert \\Phi \\Vert _2=\\pi .$ For $t\\in \\operatorname{\\mathring{D}}(0,\\pi )$ , we can consider $\\mu _{\\mathrm {L}}(t\\cdot \\Phi )=\\log \\operatorname{exp_{L}}(t\\cdot \\Phi ).$ We know that it is analytic on $\\operatorname{\\mathring{D}}(0,\\pi )$ , but it can also be computed explicitly.",
"$\\operatorname{exp_{L}}(t\\cdot \\Phi )&=\\operatorname{exp_{L}}\\left( t\\begin{bmatrix}-\\sin 2\\theta & \\cos 2\\theta \\\\\\cos 2\\theta &\\sin 2\\theta \\end{bmatrix}\\mathrm {d}\\theta |_{[0,\\pi ]}\\right)\\\\&=W(\\pi t,\\pi )\\\\&=-\\begin{bmatrix}\\cos ( \\pi \\sqrt{1-t^2})& \\frac{\\sin ( \\pi \\sqrt{1-t^2})}{\\sqrt{1-t^2}}(t+1)\\\\\\frac{\\sin ( \\pi \\sqrt{1-t^2})}{\\sqrt{1-t^2}}(t-1)&\\cos ( \\pi \\sqrt{1-t^2})\\end{bmatrix}.$ So, $\\mu _{\\mathrm {L}}(t\\cdot \\Phi )&=\\log \\operatorname{exp_{L}}(t\\cdot \\Phi )\\\\&=\\frac{\\operatorname{AC}(\\cos (-\\pi \\sqrt{1-t^2}))\\sin (\\pi \\sqrt{1-t^2}) }{\\sqrt{1-t^2}}\\begin{bmatrix}& -t-1\\\\-t+1&\\end{bmatrix}\\\\&=\\pi \\left(\\frac{1}{\\sqrt{1-t^2}}-1\\right)\\begin{bmatrix}& -t-1\\\\-t+1&\\end{bmatrix}.$ Consequently, if $t\\in [0,1]$ , then $\\Vert \\mu _{\\mathrm {L}}(t\\cdot \\Phi )\\Vert _2&=\\Vert \\log \\operatorname{exp_{L}}(t\\cdot \\Phi )\\Vert _2\\\\&=\\pi \\left(\\frac{1}{\\sqrt{1-t^2}}-1\\right)(1+t)\\\\&=\\sqrt{2}\\pi (t-1)^{-1/2}-2\\pi -\\frac{\\sqrt{2}}{4}\\pi (t-1)^{1/2}+O(t-1),$ as $t\\nearrow 1$ .",
"Or using the notation $p=\\pi t$ , we find $\\int \\Vert p/\\pi \\cdot \\Phi \\Vert _2=p$ and $\\mu _{\\mathrm {L}}(p/\\pi \\cdot \\Phi )=\\sqrt{2}\\pi ^{3/2} (\\pi -p)^{-1/2}-2\\pi -\\frac{\\sqrt{2}}{4}\\pi ^{1/2} (\\pi -p)^{1/2}+O(\\pi -p), $ as $p\\nearrow \\pi $ .",
"This is asymptotically the same as the general estimate in Theorems REF and REF , which, henceforth, turn out to be not so bad after all.",
"In terms of the Magnus expansion, we see that $\\mu _{\\mathrm {L}[n]}(\\Phi )={\\left\\lbrace \\begin{array}{ll}0 & \\text{if }n=1\\\\(-1)^{\\lfloor n/2\\rfloor }\\binom{-1/2}{\\lfloor n/2\\rfloor }\\pi \\tilde{I}&\\text{if $n$ is even, $n\\ge 2$}\\\\(-1)^{\\lfloor n/2\\rfloor }\\binom{-1/2}{\\lfloor n/2\\rfloor }\\pi (-\\tilde{K})&\\text{if $n$ is odd, $n\\ge 2$}\\end{array}\\right.",
"}$ Now, for any integer $n$ , $(-1)^{n}\\binom{-1/2}{n}=\\frac{(2n)!}{2^{2n}(n!",
")^2}; $ and a simple application of Stirling's formula shows that $\\Vert \\mu _{\\mathrm {L}[n]}(\\Phi ) \\Vert _2=\\sqrt{\\frac{2\\pi }{n}}+o(1),$ as $n\\rightarrow \\infty $ .",
"This is smaller by a linear factor than the crude estimate of Theorem REF .b, but, considering essential monotonicity, we cannot expect better.",
"Nevertheless, in this case we explicitly see that $\\sum _{n=1}^\\infty \\mu _{\\mathrm {L}[n]}(\\Phi )$ is divergent.",
"Example 5.6 (Moan's example / Magnus parabolic development.)",
"On the interval $[0,\\pi ]$ , consider again the measure $\\Phi $ , such that $\\Phi (\\theta )=\\begin{bmatrix}-\\sin 2\\theta & \\cos 2\\theta \\\\\\cos 2\\theta &\\sin 2\\theta \\end{bmatrix}\\,\\mathrm {d}\\theta |_{[0,\\pi ]}.$ Then, for $p\\in [0,\\pi )$ , $\\int \\Vert \\Phi |_{[0,p]}\\Vert _2=p.$ Here $\\operatorname{exp_{L}}(\\Phi |_{[0,p]})=W(p,p)=\\begin{bmatrix}\\cos p&2p\\cos p -\\sin p\\\\\\sin p&2p\\sin p+\\cos p\\end{bmatrix}=(\\cos p\\operatorname{Id}+\\sin p\\tilde{I})(\\operatorname{Id}_2-p\\tilde{I}+p\\tilde{K}).$ Thus $\\mu _{\\mathrm {L}}(\\Phi |_{[0,p]})=\\log \\operatorname{exp_{L}}(\\Phi |_{[0,p]})=\\operatorname{AC}(\\cos p+p\\sin p)\\begin{bmatrix}-p\\sin p&2p\\cos p -\\sin p\\\\\\sin p&p\\sin p\\end{bmatrix}.$ Consequently, $\\Vert \\mu _{\\mathrm {L}}(\\Phi |_{[0,p]})\\Vert _2=\\operatorname{AC}(\\cos p+p\\sin p)\\cdot (\\sin p-p\\cos p+p).$ As $p\\nearrow \\pi $ , $\\Vert \\mu _{\\mathrm {L}}(\\Phi |_{[0,p]})\\Vert _2=\\sqrt{2}\\pi ^{3/2} (\\pi -p)^{-1/2}-2\\pi +\\frac{\\sqrt{2\\pi }(\\pi ^2-1) }{4} (\\pi -p)^{1/2}+O(\\pi -p).",
"$ This is not only better than (REF ), but it has the advantage that it can be interpreted in terms of the solution of a differential equation blowing up.",
"Example 5.7 (Magnus elliptic development.)",
"Let $h\\in [0,1]$ be a parameter.",
"On the interval $[0,\\pi ]$ , consider the measure $\\widehat{\\Psi }_h$ such that $\\widehat{\\Psi }_h= (1-h)\\begin{bmatrix}& -1\\\\1&\\end{bmatrix}+h\\begin{bmatrix}-\\sin 2\\theta & \\cos 2\\theta \\\\\\cos 2\\theta &\\sin 2\\theta \\end{bmatrix}\\,\\mathrm {d}\\theta |_{[0,\\pi ]}.",
"$ Then, for $p\\in [0,\\pi )$ $\\int \\Vert \\widehat{\\Phi }_{h}|_{[0,p]}\\Vert _2=p.$ It is easy to see that $\\operatorname{exp_{L}}(\\widehat{\\Phi }_{h}|_{[0,p]})=E(p,ph),$ where $E(p,w)=\\begin{bmatrix}\\cos p&2w\\cos p -\\sin p\\\\\\sin p&2w\\sin p+\\cos p\\end{bmatrix}=(\\cos p\\operatorname{Id}+\\sin p\\tilde{I})(\\operatorname{Id}_2-w\\tilde{I}+w\\tilde{K}).$ Here $\\widehat{\\Phi }_1=\\Phi $ .",
"We find that $\\Vert \\mu _{\\mathrm {L}}(\\widehat{\\Phi }_{h}|_{[0,p]})\\Vert =\\Vert \\log \\operatorname{exp_{L}}(\\widehat{\\Phi }_{h}|_{[0,p]})\\Vert =\\operatorname{AC}(\\cos p+hp\\sin p)\\cdot (\\sin p-hp\\cos p+hp).$ Thus, if $h\\ne 0$ , then $\\lim _{p\\nearrow \\pi }\\Vert \\mu _{\\mathrm {L}}(\\widehat{\\Phi }_{h}|_{[0,p]})\\Vert _2=+\\infty .$ It is notable that $\\operatorname{CD}(\\operatorname{exp_{L}}(\\widehat{\\Phi }_{h}|_{[0,p]})) =\\operatorname{D}( \\mathrm {e}^{\\mathrm {i}p}-\\mathrm {i}\\mathrm {e}^{\\mathrm {i}p}ph , ph) ,$ which is $\\operatorname{CD}(\\operatorname{exp_{L}}(\\Phi |_{[0,p]}))$ contracted from the boundary point $\\mathrm {e}^{\\mathrm {i}p}$ by factor $h$ .",
"Example 5.8 (Magnus hyperbolic development.)",
"More generally, let $t$ be a real parameter.",
"On the interval $[0,\\pi ]$ consider the measure $\\Phi _{\\sin t}$ , such that $\\Phi _{\\sin t}(\\theta )=\\begin{bmatrix}-\\sin 2(\\theta \\sin t)& \\cos 2(\\theta \\sin t)\\\\\\cos 2(\\theta \\sin t)&\\sin 2(\\theta \\sin t)\\end{bmatrix}\\,\\mathrm {d}\\theta |_{[0,\\pi ]}.$ Then, for $p\\in [0,\\pi )$ $\\int \\Vert \\Phi _{\\sin t}|_{[0,p]}\\Vert _2=p.$ $\\Phi _1$ is the same as $\\Phi $ , and $\\Phi _{-1}=\\tilde{K}\\cdot \\Phi _1\\cdot \\tilde{K}$ .",
"If $t\\in (-\\pi /2,\\pi /2)$ , then $\\operatorname{exp_{L}}(\\Phi _{\\sin t}|_{[0,p]})=\\operatorname{exp_{L}}\\left( \\begin{bmatrix}-\\sin 2(\\theta \\sin t)& \\cos 2(\\theta \\sin t)\\\\\\cos 2(\\theta \\sin t)&\\sin 2(\\theta \\sin t)\\end{bmatrix}\\,\\mathrm {d}\\theta |_{[0,p]} \\right)=W(p,p\\sin t)\\\\=(\\cos (p\\sin t)\\operatorname{Id}+\\sin (p\\sin t)\\tilde{I})\\cdot \\left(\\cosh (p\\cos t)\\operatorname{Id}_2+\\frac{\\sinh (p\\cos t)}{\\cos t}\\Bigl (-\\sin t\\tilde{I}+\\tilde{K}\\Bigr )\\right).$ Consequently, $\\Vert \\mu _{\\mathrm {L}}(\\Phi _{\\sin t}|_{[0,p]} )\\Vert _2=\\operatorname{AC}\\left({\\cosh \\left( p\\cos t \\right) \\cos \\left( p\\sin t \\right) +\\frac{\\sinh \\left( p\\cos t \\right)}{\\cos \\left( t \\right) }\\sin \\left( p\\sin t \\right) \\sin t}\\right)\\\\\\cdot \\left(\\left|{\\cosh \\left( p\\cos t \\right) \\sin \\left( p\\sin t \\right) -\\frac{\\sinh \\left( p\\cos t \\right)}{\\cos t }\\cos \\left( p\\sin t \\right) \\sin t}\\right|+\\frac{\\sinh \\left( p\\cos t \\right)}{\\cos t }\\right).$ Now, in the special case $p/\\pi =\\sin t$ , we see that $\\int \\Vert \\Phi _{p/\\pi }|_{[0,p]}\\Vert _2=p, $ and $\\Vert \\mu _{\\mathrm {L}}(\\Phi _{p/\\pi }|_{[0,p]})\\Vert _2=\\sqrt{2}\\pi ^{3/2} (\\pi -p)^{-1/2}-2\\pi +\\frac{\\sqrt{2\\pi }(4\\pi ^2-3) }{12} (\\pi -p)^{1/2}+O(\\pi -p).",
"$ This shows that (REF ) is not optimal, either.",
"In what follows, whenever we use the terms `Magnus elliptic development' and `Magnus hyperbolic development', we understand that they allow the case of the Magnus parabolic development.",
"If we want to exclude it, we say `strictly elliptic' or `strictly hyperbolic' development."
],
[
"An analysis of the $\\operatorname{GL}_2^+(\\mathbb {R})$ case",
"Theorem 6.1 Let $p\\in (0,\\pi )$ .",
"Consider the family of disk parameterized by $t\\in [-\\pi /2,\\pi /2]$ , such that the centers and radii are $\\Omega _p(t)=\\mathrm {e}^{\\mathrm {i} p\\sin t}\\left(\\cosh (p\\sin t) -\\mathrm {i}\\frac{\\sinh (p\\cos t)\\sin t}{\\cos t}\\right),$ $\\omega _p(t)=\\frac{\\sinh (p\\cos t)}{\\cos t},$ for $t\\ne \\pm \\pi /2$ ; and $\\Omega _p(\\pm \\pi /2)=(\\cos p+p\\sin p )\\pm \\mathrm {i}(\\sin p-p\\cos p),$ $\\omega _p(\\pm \\pi /2)=p.$ (a) The circle $\\partial \\operatorname{D}(\\Omega _p(t),\\omega _p(t))$ is tangent to $\\partial \\exp \\operatorname{D}(0,p)$ at $\\gamma _p(t)=\\mathrm {e}^{p\\cos t+\\mathrm {i}p\\sin t}\\qquad \\text{and}\\qquad \\gamma _p(\\pi -t\\!\\!\\!\\!\\mod {2}\\pi )=\\mathrm {e}^{-p\\cos t+\\mathrm {i} p\\sin t}.$ These points are inverses of each other relative to the unit circle.",
"If the points are equal ($t=\\pm \\pi /2$ ), then the disk is the osculating disk at $\\gamma _p(t)$ .",
"The disks themselves are orthogonal to the unit circle.",
"The disks are distinct from each other.",
"Extending $t\\in [-\\pi ,\\pi ]$ , we have $\\Omega _p(t)=\\Omega _p(\\pi -t\\!\\!\\mod {2}\\pi )$ , $\\omega _p(t)=\\omega _p(\\pi -t\\!\\!\\mod {2}\\pi )$ .",
"(b) $\\operatorname{CD}(\\operatorname{exp_{L}}(\\Phi _{\\sin t}|_{[0,p]})=\\operatorname{CD}(W(p,p\\sin t))=\\operatorname{D}(\\Omega _p(t),\\omega _p(t)).$ (c) The disks $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))$ are the maximal disks in $\\exp \\operatorname{D}(0,p)$ .",
"The maximal disk $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))$ touches $\\partial \\exp \\operatorname{D}(0,p)$ only at $\\gamma _p(t)$ , $\\gamma _p(\\pi -t\\!\\!\\mod {2}\\pi )$ .",
"(a) The disks are distinct because, the centers are distinct: For $t\\in (-\\pi /2,\\pi /2)$ , $\\frac{\\mathrm {\\arg }\\Omega _p(t)}{\\mathrm {d}t}=\\operatorname{Im}\\frac{\\mathrm {d}\\log \\Omega _p(t)}{\\mathrm {d}t}=\\frac{(p\\sin (t)\\cosh (p\\sin t)-\\cosh (p\\cos t))\\cosh (p\\sin t)}{\\cosh (p\\sin t)^2-\\sin ^2 t}>0.$ (Cf.",
"$\\int _0^x y\\sinh y\\mathrm {d}y=x\\cosh x-\\sinh x.$ ) The rest can easily be checked using the observation $\\Omega _p(t)=\\mathrm {e}^{p\\cos t+\\mathrm {i}p\\sin t}-\\frac{\\sinh (p\\cos t)}{\\cos t}\\mathrm {e}^{\\mathrm {i}(t+p\\sin t)}=\\mathrm {e}^{-p\\cos t+\\mathrm {i}p\\sin t}+\\frac{\\sinh (p\\cos t)}{\\cos t}\\mathrm {e}^{\\mathrm {i}(-t+p\\sin t)}.$ (b) This is direct computation.",
"(c) In general, maximal disks touch the boundary curve $\\gamma _p$ , and any such touching point determines the maximal disk.",
"(But a maximal disk might belong to different points.)",
"Due to the double tangent / osculating property the given disks are surely the maximal disks, once we prove that they are indeed contained in $\\exp \\operatorname{D}(0,p)$ .",
"However, $\\operatorname{CD}(\\operatorname{exp_{L}}(\\Phi _{\\sin t}|_{[0,p]})=\\operatorname{D}(\\Omega _p(t),\\omega _p(t))$ together with Theorem REF implies that $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))\\subset \\exp \\operatorname{D}(0,p)$ .",
"The distinctness of the circles implies that they touch the boundary only at the indicated points.",
"Here we give a purely differential geometric argument.",
"One can see that the given disks $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))$ are characterized by the following properties: ($\\alpha $ ) If $\\gamma _p(t)\\ne \\gamma _p(\\pi -t\\!\\!\\mod {2}\\pi )$ , then the disk is tangent to $\\gamma _p$ at the these points.",
"($\\beta $ ) If $\\gamma _p(t)\\ne \\gamma _p(\\pi -t\\!\\!\\mod {2}\\pi )$ , i. e. $t=\\pm \\pi $ , then the disk is the osculating disk at $\\gamma _p(\\pm \\pi /2)$ .",
"Now, we prove that $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))\\subset \\exp \\operatorname{D}(0,p)$ .",
"First, we show that $\\operatorname{D}(\\Omega _p(0),\\omega _p(0))\\subset \\exp \\operatorname{D}(0,p)$ .",
"Indeed, $\\operatorname{D}(\\Omega _p(0),\\omega _p(0))=\\operatorname{PD}\\left(\\begin{bmatrix}\\mathrm {e}^p&\\\\&\\mathrm {e}^{-p}\\end{bmatrix}\\right);$ hence, by Theorem REF , the $\\log $ of any element of $\\operatorname{D}(\\Omega _p(0),\\omega _p(0))$ is contained in $\\operatorname{PD}\\left(\\log \\begin{bmatrix}\\mathrm {e}^p&\\\\&\\mathrm {e}^{-p}\\end{bmatrix}\\right)=\\operatorname{PD}\\left( \\begin{bmatrix}p&\\\\&{-p}\\end{bmatrix}\\right)=\\operatorname{D}(0,p).$ Let $L$ be the maximal real number such that $\\operatorname{D}(\\Omega _p(t),\\omega _p(t))\\subset \\exp \\operatorname{D}(0,p) $ for any $t\\in [-L,L]$ , and $L<\\pi /2$ .",
"(Due to continuity, there is a maximum.)",
"Indirectly, assume that $L<\\pi /2$ .",
"Then one of following should happen: (i) Besides $\\gamma _p(L)$ and $ \\gamma _p(\\pi -L\\!\\!\\mod {2}\\pi )$ there is another pair (due to inversion symmetry) of distinct points $\\gamma _p(\\tilde{L})$ and $ \\gamma _p(\\pi -\\tilde{L}\\!\\!\\mod {2}\\pi )$ , where $\\operatorname{D}(\\Omega _p(L),\\omega _p(L))$ touches the boundary of $\\exp \\operatorname{D}(0,p)$ .",
"(ii) $\\operatorname{D}(\\Omega _p(L),\\omega _p(L))$ touches the boundary at $\\gamma _p(\\pi /2)$ or $\\gamma _p(-\\pi /2)$ .",
"(iii) $\\operatorname{D}(\\Omega _p(L),\\omega _p(L))$ is osculating at $\\gamma _p(L)$ or at $ \\gamma _p(\\pi -L\\!\\!\\mod {2}\\pi )$ .",
"(Symmetry implies that $t=\\pm L$ are equally bad.)",
"Case (i) is impossible, because the given circles are distinct and the characterising properties hold.",
"Case (ii) is impossible, because, due to $\\omega _p(L)>p$ , and the extremality of $\\arg \\gamma _p(\\pm \\pi /2)$ , the situation would imply that that $\\operatorname{D}(\\Omega _p(L),\\omega _p(L))$ strictly contains the osculating disk at $\\gamma _p(\\pi /2)$ or $\\gamma _p(-\\pi /2)$ , which is a contradiction to $\\operatorname{D}(\\Omega _p(L),\\omega _p(L))\\subset \\exp \\operatorname{D}(0,p) $ .",
"Case (iii) is impossible, because for oriented plane curvature of $\\gamma _p$ , $\\varkappa _{\\gamma _p}(t)=\\frac{1+p\\cos t}{p\\mathrm {e}^{p\\cos t}}<\\frac{1}{\\omega _p(t)}=\\frac{\\cos t}{\\sinh (p\\cos t) }$ if $\\cos t\\ne 0$ .",
"(In general, $\\frac{1+x}{\\mathrm {e}^x}<\\frac{x}{\\sinh x}$ for $x\\ne 0$ .)",
"This implies $L=\\pi /2$ , proving the statement.",
"In what follows, we will not make much issue out of expressions like $\\frac{\\sinh px}{x}$ when $x=0$ ; we just assume that they are equal to $p$ , in the spirit of continuity.",
"Theorem 6.2 Suppose that $p\\in (0,\\pi )$ .",
"Suppose that $D$ is a disk in $\\exp \\operatorname{D}(0,p)$ , which touches $\\partial \\exp \\operatorname{D}(0,p)$ at $\\gamma _p(t)=\\mathrm {e}^{p\\cos t+\\mathrm {i}p\\sin t}$ .",
"Then for an appropriate nonnegative decomposition $p=p_1+p_2$ , $D=\\operatorname{CD}\\left(\\exp ( p_1(\\operatorname{Id}\\cos t+\\tilde{I}\\sin t) ) \\cdot W(p_2,p_2\\sin t) \\right).$ The bigger the $p_2$ is, the bigger the corresponding disk is.",
"$p_2=p$ corresponds to the maximal disk, $p_2=0$ corresponds to the point disk.",
"Let $W_{p_1,p_2,t} $ denote the argument of $\\operatorname{CD}$ .",
"Then its first component is Magnus exponentiable by norm $p_1$ , and its second component is Magnus exponentiable by norm $p_2$ .",
"Thus the principal disk must lie in $\\exp \\operatorname{D}(0,p)$ .",
"One can compute the center and the radius of the chiral disk (cf.",
"the Remark), and find that that $\\gamma _p(t)$ is on the boundary of the disk.",
"So, $\\operatorname{CD}(W_{p_1,p_2,t})$ must be the maximal $\\operatorname{CD}(W_{0,p_1+p_2,t})$ contracted from $\\gamma _p(t)$ .",
"One, in particular, finds that the radius of $\\operatorname{CD}(W_{p_1,p_2,t})$ is $\\frac{\\mathrm {e}^{p_1+p_2}- \\mathrm {e}^{p_1-p_2}}{2\\cos t}=\\frac{\\mathrm {e}^{p}}{\\cos t}(1-\\mathrm {e}^{-2p_2}).$ This shows that bigger $p_2$ leads to bigger disk.",
"It is easy to see that, for $p=p_1+p_2$ , $&\\exp ( p_1(\\operatorname{Id}\\cos t+\\tilde{I}\\sin t) ) \\cdot W(p_2,p_2\\sin t)=\\\\&=\\mathrm {e}^{p_1\\cos t}\\exp ((p_1+p_2)\\sin t\\tilde{I})\\cdot \\left(\\cosh (p_2\\cos t)\\operatorname{Id}_2+\\frac{\\sinh (p_2\\cos t)}{\\cos t}\\Bigl (-\\sin t\\tilde{I}+\\tilde{K}\\Bigr )\\right)\\\\&=\\operatorname{exp_{L}}\\left(\\frac{p_1}{p}\\begin{bmatrix}\\cos t& -\\sin t\\\\\\sin t&\\cos t\\end{bmatrix}+\\frac{p_2}{p}\\begin{bmatrix}-\\sin (2\\theta \\sin t)& \\cos (2\\theta \\sin t)\\\\\\cos (2\\theta \\sin t)&\\sin (2\\theta \\sin t)\\end{bmatrix} \\mathrm {d}\\theta |_{[0,p]}\\right)\\\\&=\\operatorname{exp_{L}}\\left(p_1\\begin{bmatrix}\\cos t& -\\sin t\\\\\\sin t&\\cos t\\end{bmatrix}+p_2\\begin{bmatrix}-\\sin (2p\\theta \\sin t)& \\cos (2p\\theta \\sin t)\\\\\\cos (2p\\theta \\sin t)&\\sin (2p\\theta \\sin t)\\end{bmatrix}\\mathrm {d}\\theta |_{[0,1]}\\right).$ This immediately implies the existence of a certain normal form.",
"For the sake of compact notation, let $\\tilde{\\mathbb {K}}:=\\lbrace -\\sin \\beta \\tilde{J}+\\cos \\beta \\tilde{K}\\,:\\,\\beta \\in [0,2\\pi ) \\rbrace ,$ which is the set conjugates of $\\tilde{K}$ by orthogonal matrices.",
"Theorem 6.3 Suppose that $A\\in \\mathrm {M}_2(\\mathbb {R})$ such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{\\mathring{D}}(0,\\pi )$ .",
"Assume that $p$ is the smallest real number such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{D}(0,p)$ , and $\\operatorname{CD}(A)$ touches $\\exp \\partial \\operatorname{D}(0,p)$ at $\\mathrm {e}^{p(\\cos t +\\mathrm {i}\\sin t)}$ .",
"Then there is an nonnegative decomposition $p=p_1+p_2$ , and a matrix $\\tilde{F}\\in \\tilde{\\mathbb {K}}$ , such that $A=\\mathrm {e}^{p_1\\cos t}&\\exp (p\\sin t\\tilde{I})\\cdot \\left(\\cosh (p_2\\cos t)\\operatorname{Id}_2-\\frac{\\sinh (p_2\\cos t)}{\\cos t}\\sin t\\tilde{I}\\right)+\\frac{\\sinh (p_2\\cos t)}{\\cos t}\\tilde{F} \\\\&=\\operatorname{exp_{L}}\\left( p_1\\exp (t\\tilde{I})+p_2\\exp (2p\\theta \\sin t\\tilde{I})\\cdot \\tilde{F} \\,\\,\\mathrm {d}\\theta |_{[-1/2,1/2]} \\right) \\\\&=\\operatorname{exp_{L}}(\\exp (t\\tilde{I}) \\,\\mathrm {d}\\theta |_{[0,p_1]})\\operatorname{exp_{L}}\\left( \\exp ((2\\theta -p_1-p_2)\\sin t \\tilde{I})\\tilde{F} \\, \\mathrm {d}\\theta |_{[0,p_2]} \\right) \\\\&=\\operatorname{exp_{L}}\\left( \\exp ((2\\theta +p_1-p_2)\\sin t \\tilde{I})\\tilde{F} \\, \\mathrm {d}\\theta |_{[0,p_2]} \\right) \\operatorname{exp_{L}}( \\exp (t\\tilde{I}) \\,\\mathrm {d}\\theta |_{[0,p_1]}).",
"$ The case $p_1=p_2=0$ corresponds to $A=\\operatorname{Id}_2$ .",
"The case $p_1>0,p_2=0$ corresponds to point disk case, the expression does not depend on $\\tilde{F}$ .",
"The case $p_1=0,p_2>0$ corresponds to the maximal disk case, it has degeneracy $t\\leftrightarrow \\pi - t\\mod {2}\\pi $ .",
"In the general case $p_1,p_2>0$ , the presentation is unique in terms of $p_1,p_2,t\\!\\!\\mod {2}\\pi ,\\tilde{F}$ .",
"This is an immediate consequence of the previous statement and the observation $(\\cos \\alpha +\\tilde{I}\\sin \\alpha )\\tilde{K}(\\cos \\alpha +\\tilde{I}\\sin \\alpha )^{-1}=(\\cos 2\\alpha +\\tilde{I}\\sin 2\\alpha )\\tilde{K}= -\\tilde{J}\\sin 2\\alpha +\\tilde{K}\\cos 2\\alpha .$ In what follows, we use the notation $\\operatorname{N}(p_1,p_2,t,\\tilde{F})$ to denote the arithmetic expression on the RHS of (REF ).",
"The statement above offers three ways to imagine the matrix in question as a left-exponential: () is sufficiently nice and compact with norm density $p$ on an interval of unit length.",
"() and () are concatenations of intervals of length $p_1$ and $p_2$ with norm density 1.",
"One part is essentially a complex exponential, relatively uninteresting; the other part is the Magnus parabolic or hyperbolic development of Examples REF and REF , but up to conjugation by a special orthogonal matrix, which is the same to say as `up to phase'.",
"Theorem 6.4 Suppose that $A\\in \\mathrm {M}_2(\\mathbb {R})$ such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{\\mathring{D}}(0,\\pi )$ .",
"Then $\\operatorname{\\mathcal {M}}(A)=\\inf \\lbrace \\lambda \\in [0,\\pi )\\,:\\, \\operatorname{CD}(A)\\subset \\exp \\operatorname{D}(0,\\lambda )\\rbrace .$ Or, in other words, $\\operatorname{\\mathcal {M}}(A)=\\sup \\lbrace |\\log z|\\,:\\, z\\in \\operatorname{CD}(A) \\rbrace .",
"$ Assume that $p$ is the smallest real number such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{D}(0,p)$ .",
"By Theorem REF , $\\operatorname{\\mathcal {M}}(A)$ is at least $p$ , while the left-exponentials of Theorem REF does indeed Magnus-exponentiate them with norm $p$ .",
"Suppose that $A\\in \\mathrm {M}_2(\\mathbb {R})$ such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{\\mathring{D}}(0,\\pi )$ , $A\\ne \\operatorname{Id}_2$ , $p=\\operatorname{\\mathcal {M}}(A)$ .",
"If $\\det A=1$ , then $A$ can be of the three kinds: Magnus elliptic, when $\\operatorname{CD}(A)$ touches $\\exp \\partial \\operatorname{D}(0,p)$ at $\\mathrm {e}^{\\mathrm {i}p}$ or $\\mathrm {e}^{-\\mathrm {i}p}$ , but it is not an osculating disk; Magnus parabolic, when $\\operatorname{CD}(A)$ touches $\\exp \\partial \\operatorname{D}(0,p)$ at $\\mathrm {e}^{\\mathrm {i}p}$ or $\\mathrm {e}^{-\\mathrm {i}p}$ , and it is an osculating disk; or Magnus hyperbolic when $\\operatorname{CD}(A)$ touches $\\exp \\partial \\operatorname{D}(0,p)$ at two distinct points.",
"If $\\det A\\ne 1$ then $\\operatorname{CD}(A)$ touches $\\exp \\partial \\operatorname{D}(0,p)$ at a single point, asymmetrically; we can call these Magnus loxodromic.",
"We see that Examples REF , REF , and REF , cover all the Magnus parabolic, hyperbolic and elliptic cases up to conjugation by an orthogonal matrix.",
"In general, if $A$ is not Magnus hyperbolic, then it determines a unique Magnus direction $\\cos t+\\mathrm {i}\\sin t$ (in the notation Theorem REF ).",
"It is the direction of the farthest point of $\\lbrace \\log z\\,:\\,z\\in \\operatorname{CD}(A)\\rbrace $ from the origin.",
"If $A$ is Magnus hyperbolic, then this direction is determined only up to sign in the real part.",
"Lemma 6.5 Suppose $A\\in \\mathrm {M}_2(\\mathbb {R})$ such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{\\mathring{D}}(0,\\pi )$ , $A\\ne \\operatorname{Id}_2$ , $\\det A=1$ , $\\operatorname{CD}(A)=\\operatorname{D}((a,b),r)$ .",
"Then $a^2+b^2=r^2+1$ and $a+1>0$ .",
"We claim that $A$ is Magnus hyperbolic or parabolic if and only if $2\\arctan \\frac{r+|b|}{a+1}\\le r.$ If $A$ is Magnus elliptic or parabolic, then $\\operatorname{\\mathcal {M}}(A)=2\\arctan \\frac{r+|b|}{a+1}.$ $\\partial \\operatorname{D}((a,b),r)$ intersects the unit circle at $(\\cos \\varphi _\\pm ,\\sin \\varphi _\\pm ): \\left(\\dfrac{a\\pm br}{a^2+b^2},\\dfrac{b\\mp ar}{a^2+b^2}\\right),$ $\\varphi _\\pm \\in (-\\pi ,\\pi )$ .",
"In particular, $\\dfrac{a\\pm br}{a^2+b^2}+1>0$ ; multiplying them, we get $a+1>0$ .",
"Then $\\phi _\\pm =2\\arctan \\frac{r\\pm b}{a+1}$ .",
"If one them is equal to $r$ , then it is a Magnus parabolic case; if those are smaller than $r$ , then it is Magnus hyperbolic case; if one of them is bigger than $r$ , this it must be a Magnus elliptic case.",
"(Cf.",
"the size of the chiral disk in Theorem REF .)",
"We say that the measure $\\phi $ is a minimal Magnus presentation for $A$ , if $\\operatorname{exp_{L}}(\\phi )=A$ and $\\int \\Vert \\phi \\Vert _2=\\operatorname{\\mathcal {M}}(A)$ .",
"Lemma 6.6 Any element $A\\in \\operatorname{GL}^+_2(\\mathbb {R})$ has at least one minimal Magnus presentation.",
"$\\operatorname{GL}^+_2(\\mathbb {R})$ is connected, which implies that any element $A$ has at least one Magnus presentation $\\psi $ .",
"If $\\int \\Vert \\phi \\Vert _2$ is small enough, then we can divide the supporting interval of $\\phi $ into $\\lfloor \\operatorname{\\mathcal {M}}(A)/\\pi \\rfloor $ many subintervals, such that the variation of $\\phi $ on any of them is less than $\\pi $ .",
"Replace $\\phi $ by a normal form on every such subinterval.",
"By this we have managed to get a presentation of variation at most $\\int \\Vert \\phi \\Vert _2$ by a data from $([0,\\pi ]\\times [0,\\pi ]\\times [0,2\\pi ]\\times \\mathbb {K}) ^{\\lfloor \\operatorname{\\mathcal {M}}(A)/\\pi \\rfloor }$ .",
"Conversely, such a data always gives a presentation, whose $\\operatorname{exp_{L}}$ depends continuously on the data.",
"Then the statement follows from a standard compactness argument.",
"Lemma 6.7 Suppose that $A_\\lambda \\rightarrow \\operatorname{Id}$ , such that $A_\\lambda $ is Magnus hyperbolic, but $A_\\lambda \\ne \\operatorname{Id}$ for any $\\lambda $ .",
"Suppose that $\\operatorname{CD}(A_\\lambda )=\\operatorname{D}((1+a_\\lambda ,b_\\lambda ),r_\\lambda )$ .",
"Then, as the sequence converges, $\\operatorname{\\mathcal {M}}(A_\\lambda )^2=2 a_\\lambda +O(\\mathrm {itself}^2);$ or more precisely, $\\operatorname{\\mathcal {M}}(A_\\lambda )^2=2 a_\\lambda -\\frac{1}{3}a_\\lambda ^2+\\frac{3}{2}\\frac{b_\\lambda ^2}{a_\\lambda } +O(\\mathrm {itself}^3).$ We can assume that $A_\\lambda =W(p_\\lambda ,p_\\lambda \\sin t_\\lambda )$ .",
"From the formula of $W(p,p\\sin t)$ one can see that $\\operatorname{CD}(W(p,p\\sin t))$ is an entire function of $x=p\\cos t, y=p\\sin t$ .",
"One actually finds that the center is $(1+\\hat{a}(x,y),\\hat{b}(x,y))=&\\biggl (1+\\frac{x^2+y^2}{2}+\\frac{(x^2-y^2)(x^2+y^2)}{24 }\\\\&+\\frac{(x^4-10x^2y^2+5y^4)(x^2+y^2)}{720}+O(x,y)^8, \\\\&\\frac{y(x^2+y^2)}{3} + \\frac{y(x^2+y^2)(x^2-y^2)}{30} +O(x,y)^7\\biggr ).$ (One can check that in the expansion $\\hat{a}(x,y)$ , every term is divisible by $(x^2+y^2)$ ; in the expansion $\\hat{b}(x,y)$ , every term is divisible by $y(x^2+y^2)$ .)",
"Eventually, one finds that $p^2=x^2+y^2= 2\\hat{a}(x,y)+O(x,y)^4$ and $p^2=x^2+y^2= 2\\hat{a}(x,y)- \\frac{1}{3}\\hat{a}(x,y)^2+\\frac{3}{2} \\frac{\\hat{b}(x,y)^2}{\\hat{a}(x,y)} +O(x,y)^6.$ The hyperbolic developments $p\\mapsto W(p,p\\sin t) $ are uniform motions in the sense that the increments $ W((p+\\varepsilon ),(p+\\varepsilon )\\sin t) W(p,p\\sin t) ^{-1}$ differ from each other by conjugation by orthogonal matrices as $p$ changes.",
"In fact, they are locally characterized by the speed $\\sin t$ , and a phase, i. e. conjugation by rotations.",
"Lemma 6.8 Assume that $0< p_1,p_2$ ; $p_1+p_2<\\pi $ ; $t_1,t_2\\in [-\\pi /2,\\pi /2]$ ; $\\varepsilon \\in (-\\pi /2,\\pi /2]$ .",
"On the interval $[-p_1,p_2]$ , consider the measure $\\phi $ given by $\\phi (\\theta )=\\eta (\\theta )\\,\\mathrm {d}\\theta |_{[-p_1,p_2]}, $ where $\\eta (\\theta )={\\left\\lbrace \\begin{array}{ll}\\begin{bmatrix}-\\sin 2(\\theta \\sin t_2)& \\cos 2(\\theta \\sin t_2)\\\\\\cos 2(\\theta \\sin t_2)&\\sin 2(\\theta \\sin t_2)\\end{bmatrix}&\\text{if }\\theta \\ge 0\\\\\\begin{bmatrix}\\cos \\varepsilon &-\\sin \\varepsilon \\\\\\sin \\varepsilon &\\cos \\varepsilon \\end{bmatrix}\\begin{bmatrix}-\\sin 2(\\theta \\sin t_1)& \\cos 2(\\theta \\sin t_1)\\\\\\cos 2(\\theta \\sin t_1)&\\sin 2(\\theta \\sin t_1)\\end{bmatrix}\\begin{bmatrix}\\cos \\varepsilon &\\sin \\varepsilon \\\\-\\sin \\varepsilon &\\cos \\varepsilon \\end{bmatrix}&\\text{if }\\theta \\le 0.\\end{array}\\right.}",
"$ Then $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi ))<p_1+p_2$ unless $\\varepsilon =0$ and $v_1=v_2$ .",
"It is sufficient to prove this for a small subinterval around 0.",
"So let us take the choice $p_1=p_2=p/2$ , $p\\searrow 0$ .",
"Then $\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]})=W\\left(\\frac{p}{2},\\frac{p}{2}\\sin t_2\\right)\\begin{bmatrix}\\cos \\varepsilon &-\\sin \\varepsilon \\\\\\sin \\varepsilon &\\cos \\varepsilon \\end{bmatrix}W\\left(-\\frac{p}{2},-\\frac{p}{2} \\sin t_1\\right)^{-1}\\begin{bmatrix}\\cos \\varepsilon &\\sin \\varepsilon \\\\-\\sin \\varepsilon &\\cos \\varepsilon \\end{bmatrix}.$ Let $\\operatorname{D}((a_p,b_p),r_p)=\\operatorname{CD}(\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]}) ).$ (i) If $\\varepsilon \\in (-\\pi /2,0)\\cup (0,\\pi /2)$ , then $2\\arctan \\frac{r_p\\pm b_p}{a_p+1}-r_p=\\mp \\frac{1}{4}\\sin (2\\varepsilon )p^2+O(p^3).$ This shows that $\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]})$ gets Magnus elliptic.",
"However, $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]}))=2\\arctan \\frac{r_p\\pm b_r}{a_p+1}=p\\cos (\\varepsilon ) +O(p^2)$ shows Magnus non-minimality.",
"(ii) If $\\varepsilon =\\pi /2$ , $\\sin t_1+\\sin t_2\\ne 0$ , then $2\\arctan \\frac{r_p\\pm b_r}{a_p+1}-r_p=\\mp \\frac{1}{12}(\\sin t_1+\\sin t_2 )p^3+O(p^4).$ This also shows Magnus ellipticity, and $2\\arctan \\frac{r_p\\pm b_r}{a_p+1}=\\frac{1}{4}|\\sin t_1+\\sin t_2| p^2+O(p^3)$ shows Magnus non-minimality.",
"(iii) If $\\varepsilon =\\pi /2$ , $\\sin t_1+\\sin t_2=0$ , then $\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]})=\\operatorname{Id}_2$ .",
"Hence, full cancellation occurs, this is not Magnus minimal.",
"(iv) If $\\varepsilon =0$ , $\\sin t_1\\ne \\sin t_2$ , then $\\sin t_1+\\sin t_2<2$ , and $2\\arctan \\frac{r_p\\pm b_p}{a_p+1}-r_p=\\frac{1}{6}(\\pm (\\sin t_1+\\sin t_2 )-2)p^3+O(p^4).$ this shows that $\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]})$ gets Magnus hyperbolic.",
"Then assuming Magnus minimality and using the previous lemma, we get a contradiction by $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi |_{[-p/2,p/2]}))^2=p^2- \\frac{1}{48}p^4(\\sin t_2-\\sin t_1)^2+O(\\text{itself}^3)<p^2.",
"$ This proves the statement.",
"Lemma 6.9 Assume that $0< p_1,p_2$ ; $p_1+p_2<\\pi $ ; $t_1\\in [-\\pi /2,\\pi /2)$ ; .",
"On the interval $[-p_1,p_2]$ , consider the measure $\\phi $ given by $\\phi (\\theta )=\\eta (\\theta )\\,\\mathrm {d}\\theta , $ where $\\eta (\\theta )={\\left\\lbrace \\begin{array}{ll}\\tilde{I} =\\begin{bmatrix}& -1\\\\1&\\end{bmatrix}&\\text{if }\\theta \\ge 0\\\\\\begin{bmatrix}-\\sin 2(\\theta \\sin t)& \\cos 2(\\theta \\sin t)\\\\\\cos 2(\\theta \\sin t)&\\sin 2(\\theta \\sin t)\\end{bmatrix}&\\text{if }\\theta \\le 0.\\end{array}\\right.}",
"$ Then $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi ))<p_1+p_2.$ Again, it is sufficient to show it for a small subinterval around 0.",
"(i) Suppose $t\\in (-\\pi /2,\\pi /2)$ .",
"As $p\\searrow 0$ , restrict to the interval $\\mathcal {I}_p=\\left[-p,\\frac{\\sinh p\\cos t}{\\cos t}-p\\right].$ Then $ \\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_p})=\\exp \\left(\\tilde{I}\\left(\\sin \\frac{\\sinh p\\cos t}{\\cos t }-p\\right) \\right)W(-p,-p\\sin t )^{-1}.$ Let $\\operatorname{D}((a_p,b_p),r_p)=\\operatorname{CD}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_p}) ).$ If we assume Magnus minimality, then $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_p}))=\\frac{\\sinh p\\cos t}{\\cos t}=r_p.$ Thus, $\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_p})$ is Magnus parabolic.",
"By direct computation, we find $2\\arctan \\frac{r_p+| b_p|}{a_p+1}=p+\\frac{1}{3}p^3\\max (\\cos ^2t+\\sin t-1,-1-\\sin t )+O(p^4),$ in contradiction to $\\frac{\\sinh p\\cos t}{\\cos t}=p+\\frac{1}{6}p^3(\\cos ^2t )+O(p^4), $ which is another way to express $\\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_p}))$ from the density.",
"(The coefficients of $p^3$ differ for $t\\in (-\\pi /2,\\pi /2)$ .)",
"(ii) Consider now the case $t=-\\pi /2$ .",
"$2\\arctan \\frac{r_p\\pm b_p}{a_p+1}=\\pm \\frac{1}{2}p+O(p^2)$ shows Magnus ellipticity, and $2\\arctan \\frac{r_p+|b_p|}{a_p+1}=p-\\frac{1}{12}p^3+O(p^4)$ shows non-minimality.",
"Now we deal with the unicity of the normal forms as left exponentials.",
"In the context of Theorem $\\ref {th:normalform}$ we call $\\operatorname{ell}(A):=p_1(\\cos t+\\tilde{I}\\sin t)$ the elliptic component of $A$ , and we call $\\operatorname{hyp}(A):=p_2$ the hyperbolic length of $A$ .",
"Theorem 6.10 Suppose that $A\\in \\mathrm {M}_2(\\mathbb {R})$ such that $\\operatorname{CD}(A)\\subset \\exp \\operatorname{\\mathring{D}}(0,\\pi )$ , and $\\phi $ is a minimal Magnus presentation for $A$ supported on $[a,b]$ .",
"Then, restricted to any subinterval $\\mathcal {I}$ , the value $\\operatorname{ell}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}}))$ is a multiple of $\\operatorname{ell}(A)$ by a nonnegative real number.",
"Furthermore the interval functions $\\mathcal {I}\\mapsto \\operatorname{\\mathcal {M}}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}}))=\\smallint \\Vert \\phi |_{\\mathcal {I}}\\Vert _2,$ $\\mathcal {I}\\mapsto \\operatorname{ell}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}})),$ $\\mathcal {I}\\mapsto \\operatorname{hyp}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}}))$ are additive.",
"In particular, if $A$ is Magnus hyperbolic or parabolic, then $\\operatorname{ell}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}}))$ is always 0.",
"Divide supporting interval of $\\phi $ into smaller intervals $\\mathcal {I}_1,\\ldots ,\\mathcal {I}_s$ .",
"On these intervals replace $\\phi |_{\\mathcal {I}_k}$ by a left-complex normal form.",
"Thus we obtain $\\phi ^{\\prime }=\\Phi ^{(1)}_{\\mathcal {K}_1}.",
"(\\cos t_1+\\tilde{I}\\sin t_1)\\mathbf {1}_{\\mathcal {J}_1}.\\ldots .\\Phi ^{(s)}_{\\mathcal {K}_s}.",
"(\\cos t_s+\\tilde{I}\\sin t_s)\\mathbf {1}_{\\mathcal {J}_s},$ where $\\mathcal {J}_j$ are $\\mathcal {K}_j$ are some intervals, and $\\Phi ^{(j)}_{\\mathcal {K}_j}$ are hyperbolic developments (up to conjugation).",
"(They can be parabolic but for the sake simplicity let us call them hyperbolic.)",
"Further, rearrange this as $\\phi ^{\\prime \\prime }=\\Phi ^{\\prime (1)}_{\\mathcal {K}_1}.\\ldots .\\Phi ^{\\prime (s)}_{\\mathcal {K}_s}.",
"(\\cos t_1+\\tilde{I}\\sin t_1)\\mathbf {1}_{\\mathcal {J}_1}.\\ldots .",
"(\\cos t_s+\\tilde{I}\\sin t_s)\\mathbf {1}_{\\mathcal {J}_s},$ where the hyperbolic developments suffer some special orthogonal conjugation but they remain hyperbolic developments.",
"Now, the elliptic parts $\\operatorname{ell}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_j}) )= |\\mathcal {J}_j|(\\cos t_j+\\tilde{I}\\sin t_j)$ must be nonnegatively proportional to each other, otherwise cancelation would occur when the elliptic parts are contracted, in contradiction to the minimality of the presentation.",
"By this, we have proved that in a minimal presentation elliptic parts of disjoint intervals are nonnegatively proportional to each other.",
"Suppose that in a division $|\\mathcal {J}_j|\\cos t_j\\ne 0$ occurs.",
"Contract the elliptic parts in $\\phi ^{\\prime \\prime }$ but immediately divide them into two equal parts: $\\phi ^{\\prime \\prime \\prime }=\\Phi ^{\\prime (1)}_{\\mathcal {K}_1}.\\ldots .\\Phi ^{\\prime (s)}_{\\mathcal {K}_s}.",
"(\\cos t_j+\\tilde{I}\\sin t_j)\\mathbf {1}_{\\mathcal {J}}.",
"(\\cos t_j+\\tilde{I}\\sin t_j)\\mathbf {1}_{\\mathcal {J}}.$ Now replace everything but the last term by a normal form $\\phi ^{\\prime \\prime \\prime \\prime }=\\Phi ^{\\prime (0)}_{\\mathcal {K}_0}.",
"(\\cos t_0+\\tilde{I}\\sin t_0)\\mathbf {1}_{\\mathcal {J}_0}.",
"(\\cos t_j+\\tilde{I}\\sin t_j)\\mathbf {1}_{\\mathcal {J}}.$ Taking the determinant of the various left-exponential term we find $\\mathrm {e}^{ |\\mathcal {J}_0|\\cos t_0+|\\mathcal {J}|\\cos t_j }= \\mathrm {e}^{ 2|\\mathcal {J}|\\cos t_j }.$ Thus $|\\mathcal {J}_0|\\cos t_0\\ne 0$ , hence, by minimality $t_j=t_0\\mod {2}\\pi $ , moreover $|\\mathcal {J}_0|=|\\mathcal {J}|$ .",
"However, the $\\phi ^{\\prime \\prime \\prime }$ constitutes a normal form (prolonged in the elliptic part), which in this form is unique, thus, eventually $\\operatorname{ell}(\\operatorname{exp_{L}}(\\phi ))=\\sum _{j=1}^s \\operatorname{ell}(\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_j})) $ must hold.",
"Suppose now that $\\sin t_k=1$ or $\\sin t_k=-1$ occurs with $|\\mathcal {J}_k|\\ne 0$ .",
"Consider $\\phi ^{\\prime \\prime }$ .",
"By Magnus minimality and Lemma REF , the hyperbolic development must fit into single hyperbolic development $\\Psi _{\\mathcal {K}}$ (without phase or speed change).",
"Furthermore, by Lemma REF , $\\Psi _{\\mathcal {K}}$ must be parabolic fitting properly to the elliptic parts.",
"Thus $\\phi ^{\\prime \\prime }$ , in fact, yields a normal form $\\Psi _{\\mathcal {K}}.",
"(\\sin t_k)\\mathbf {1}_{\\mathcal {J}}$ .",
"Then (REF ) holds.",
"The third possibility in $\\phi ^{\\prime \\prime }$ is that all the intervals $\\mathcal {J}_j$ are of zero length.",
"Then the hyperbolic developments fit into a single development $\\Psi _{\\mathcal {K}}$ , but (REF ) also holds.",
"Thus (REF ) is proven.",
"It implies nonnegative proportionality relative to the total $\\operatorname{ell}(\\operatorname{exp_{L}}(\\phi ))$ .",
"Now, subintervals of minimal presentations also yield minimal presentations, therefore additivity holds in full generality.",
"Regarding the interval functions, the additivity of $\\operatorname{\\mathcal {M}}$ is trivial, the additivity of $\\operatorname{ell}$ is just demonstrated, and $\\operatorname{hyp}$ is just the $\\operatorname{\\mathcal {M}}$ minus the absolute value (norm) of $\\operatorname{ell}$ .",
"Remark 6.11 Suppose that $\\phi :\\mathcal {I}\\rightarrow \\mathcal {B}(\\mathfrak {H})$ is a measure.",
"Assume that $\\mathcal {I}_1\\subset \\mathcal {I}$ is a subinterval such that $\\smallint \\Vert \\phi |_{\\mathcal {I}_1}\\Vert _2<\\pi $ .",
"Let us replace $\\phi |_{\\mathcal {I}_1}$ by a Magnus minimal presentation of $\\operatorname{exp_{L}}(\\phi |_{\\mathcal {I}_1})$ , in order to obtain an other measure $\\phi _1$ .",
"Then we call $\\phi _1$ a semilocal contraction of $\\phi $ .",
"We call $\\phi $ semilocally Magnus minimal, if finitely many application of semilocal contractions does not decrease $\\smallint \\Vert \\phi \\Vert _2$ .",
"(In this case, the semilocal contractions will not really be contractions, as they are reversible.)",
"We call $\\phi $ locally Magnus minimal, if any application of a semilocal contraction does not decrease $\\smallint \\Vert \\phi \\Vert _2$ .",
"It is easy to see that $\\text{(Magnus minimal) $\\Rightarrow $(semilocally Magnus minimal) $\\Rightarrow $(locally Magnus minimal)}.$ The arrows do not hold in the other directions.",
"For example, $\\tilde{I}\\mathbf {1}_{[0,2\\pi ]}$ is semilocally minimal, but not Magnus minimal.",
"Also, $(-\\mathbf {1}_{[0,1]}).",
"\\Psi _0.\\mathbf {1}_{[0,1]}$ is locally Magnus minimal but not semilocally Magnus minimal: Using semilocal contraction we can move $(-\\mathbf {1}_{[0,1]})$ and $\\mathbf {1}_{[0,1]}$ beside each other, and then there is a proper cancellation.",
"The proper local generalization of Magnus minimality is semilocal Magnus minimality.",
"If $\\phi $ is locally Magnus minimal, the we can define $\\operatorname{ell}(\\phi )$ and $\\operatorname{hyp}(\\phi )$ by taking a finite division of $\\lbrace \\mathcal {I}_j\\rbrace $ of $\\mathcal {I}$ to intervals of variation less than $\\pi $ , and simply adding $\\operatorname{ell}(\\phi _j)$ and $\\operatorname{hyp}(\\phi _j)$ .",
"What semilocality is needed for is to show that $\\operatorname{ell}(\\phi _{\\mathcal {I}})$ is nonnegatively proportional to $\\operatorname{ell}(\\phi )$ , and to a proper definition of the Magnus direction of $\\phi $ .",
"Having that, semilocally Magnus minimal presentations up to semilocal contractions behave like Magnus minimal presentations.",
"They can also be classified as Magnus elliptic, parabolic, hyperbolic, or loxodromic.",
"(But they are not elements of $\\operatorname{GL}_2^+(\\mathbb {R})$ anymore but presentations.)",
"In fact, semilocally Magnus minimal presentations up to semilocal contractions have a very geometrical interpretation, cf.",
"Remark REF .",
"(Interpreted as elements of $\\widetilde{\\operatorname{GL}_2^+(\\mathbb {R})}$ .)",
"As Theorem REF suggests, hyperbolic developments are rather rigid, while in other cases there is some wiggling of elliptic parts.",
"Theorem 6.12 Suppose that $A\\ne \\operatorname{Id}_2$ , $p=\\operatorname{\\mathcal {M}}(A)<\\pi $ , and $\\phi $ is a minimal presentation to $A$ supported on the interval $[a,b]$ .",
"(a) Suppose that $A$ is Magnus hyperbolic or parabolic.",
"Then there are unique elements $t\\in [-\\pi /2,\\pi /2]$ and $\\tilde{F}\\in \\tilde{\\mathbb {K}}$ such that $\\operatorname{exp_{L}}(\\phi |_{[a,x]})=W\\left(0,\\int \\Vert \\phi |_{[a,x]}\\Vert _2, t, \\tilde{F}\\right).$ Thus, minimal presentations for Magnus hyperbolic and parabolic matrices are unique, up to reparametrization of the measure.",
"(b) Suppose that $\\operatorname{CD}(A)$ is point disk.",
"Then there is a unique element $t\\in [0,2\\pi )$ such that $\\operatorname{exp_{L}}(\\phi |_{[a,x]})=\\exp \\left((\\operatorname{Id}_2\\cos t+\\tilde{I}\\sin t) \\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right).$ Thus, minimal presentations for quasicomplex matrices are unique, up to reparametrization of the measure.",
"(c) Suppose that $A$ is not of the cases above.",
"Then there are unique elements $t\\in [0,2\\pi )$ , $p_1$ , $p_2>0$ , $\\tilde{F}\\in \\tilde{\\mathbb {K}}$ and surjective monotone increasing function $\\varpi _i:[a,b]\\rightarrow [0,p_i]$ such that $\\varpi _1(x)+\\varpi _2(x)=x-a$ and $\\operatorname{exp_{L}}(\\phi |_{[a,x]})=W\\left(\\varpi _1\\left(\\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right) ,\\varpi _2\\left(\\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right), t, \\tilde{F} \\right).$ Thus, minimal presentations in the general case are unique, up to displacement of elliptic parts.",
"Divide $[a,b]$ to $[a,x]$ and $[x,b]$ , and replace the minimal presentation by norma parts.",
"They must fit in accordance to minimality.",
"The statement can easily be generalized to semilocally Magnus minimal presentations.",
"Theorem REF says that certain minimal Magnus presentations are essentially unique.",
"Theorems REF and REF will give some explanation to the fact that it is not easy to give examples for the Magnus expansion blowing up in the critical case $\\smallint \\Vert \\phi \\Vert _2=\\pi $ .",
"Theorem 6.13 Suppose that $A\\ne \\operatorname{Id}_2$ , $p=\\operatorname{\\mathcal {M}}(A)<\\pi $ , and $\\phi $ is a minimal presentation to $A$ supported on the interval $[a,b]$ .",
"If $\\phi $ is of shape $\\operatorname{exp_{L}}(\\phi |_{[a,x]})=\\exp \\left(S \\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right)$ with some matrix $S$ (i. e., it is essentially an exponential), then $S$ is of shape $\\operatorname{Id}_2\\cos t+\\tilde{I} \\sin t$ , (i. e. it is the quasicomplex case, Theorem REF .b).",
"Due to homogeneity, $\\operatorname{ell}(\\Phi |_{\\mathcal {I}})$ and $\\operatorname{hyp}(\\Phi |_{\\mathcal {I}})$ must be proportional to $\\operatorname{\\mathcal {M}}(\\Phi |_{\\mathcal {I}})$ .",
"But it is easy to see that (up to parametrization) only the homogeneous normal densities () have this property, and they are locally constant only if the Magnus non-elliptic component vanishes.",
"In particular, the Baker-Campbell-Hausdorff setting (for $2\\times 2$ real matrices) is never Magnus minimal except in the degenerate quasicomplex case.",
"Theorem 6.14 Suppose that $\\phi $ is a measure, $\\int \\Vert \\phi \\Vert _2=\\pi ,$ but $\\log \\operatorname{exp_{L}}(\\phi )$ does not exist.",
"Then there are uniquely determined elements $t\\in \\lbrace -\\pi ,\\pi \\rbrace $ and $\\tilde{F}\\in \\tilde{\\mathbb {K}}$ , a nonnegative decomposition $\\pi =p_1+p_2$ , with $p_2>0$ , and surjective monotone increasing functions $\\varpi _i:[a,b]\\rightarrow [0,p_i]$ such that $\\varpi _1(x)+\\varpi _2(x)=x-a$ and $\\operatorname{exp_{L}}(\\phi |_{[a,x]})=W\\left(\\varpi _1\\left(\\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right) ,\\varpi _2\\left(\\int \\Vert \\phi |_{[a,x]}\\Vert _2\\right), t, \\tilde{F} \\right).$ Thus, critical cases with $\\log $ blowing up are the Magnus elliptic and parabolic (but not quasicomplex) developments up to reparametrization and rearrangement of elliptic parts.",
"The presentation must be Magnus minimal, otherwise the $\\log $ would be OK. Divide $[a,b]$ to $[a,x]$ and $[x,b]$ , and replace the minimal presentation by normal parts.",
"They must fit in accordance to minimality.",
"It is easy to see that in the Magnus hyperbolic / loxodromic cases $\\operatorname{CD}(\\operatorname{exp_{L}}(\\phi |_{[a,x]}))$ has no chance to reach $(-\\infty ,0]$ .",
"The disks are the largest in the Magnus hyperbolic cases, and the chiral disks $\\operatorname{CD}(W(\\pi ,\\pi \\sin t))$ of Magnus strictly hyperbolic developments do not reach the negative axis.",
"So the Magnus elliptic and parabolic cases remain but the quasicomplex is ruled out.",
"Thus even critical cases with $\\int \\Vert \\phi \\Vert _2=\\pi $ are scarce.",
"Remark 6.15 We started this section by investigating matrices $A$ with $\\operatorname{CD}(A)\\subset \\operatorname{\\mathring{D}}(0,\\pi )$ .",
"It is a natural question to ask whether the treatment extends to matrices $A$ with, say, $\\operatorname{CD}(A)\\cap (\\infty ,0]=\\emptyset $ .",
"The answer is affirmative.",
"However, if we consider this question, then it is advisable to take an even bolder step: Extend the statements for $A\\in \\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ , the universal cover of ${\\operatorname{GL}^+_2}(\\mathbb {R})$ .",
"This of course, implies that we have to use the covering exponential $\\widetilde{\\exp }:\\mathrm {M}_2(\\mathbb {R})\\rightarrow \\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R}),$ and $\\operatorname{exp_{L}}$ should also be replaced by $\\widetilde{\\operatorname{exp_{L}}}$ .",
"Now, the chiral disks of elements of $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ live in $\\widetilde{\\mathbb {C}}$ , the universal cover of $\\mathbb {C}\\setminus \\lbrace 0\\rbrace $ .",
"Mutatis mutandis, Theorems REF , REF , REF extend in a straightforward manner.",
"Remarkably, Theorems REF and REF have versions in this case, however we do not really need them that much, because chiral disks can be traced directly to prove a variant of Theorem REF .",
"Elements of $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ also have minimal Magnus presentations.",
"In our previous terminology, they are semilocally Magnus minimal presentations.",
"In fact, semilocally Magnus minimal presentations up to semilocal contractions will correspond to elements of $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ .",
"They classification Magnus hyperbolic, elliptic, parabolic, loxodromic, quasicomplex elements extends to $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ .",
"This picture of $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ helps to understand ${\\operatorname{GL}^+_2}(\\mathbb {R})$ .",
"Indeed, we see that every element of ${\\operatorname{GL}^+_2}(\\mathbb {R})$ have countably many semilocally Magnus minimal presentations up to semilocal contractions, and among those one or two (conjugates) are minimal.",
"The Magnus exponent of an element of ${\\operatorname{GL}^+_2}(\\mathbb {R})$ is the minimal Magnus exponent of its lifts to $\\widetilde{\\operatorname{GL}^+_2}(\\mathbb {R})$ .",
"Example 6.16 Let $z=4.493\\ldots $ be the solution of $\\tan z=z$ on the interval $[\\pi ,2\\pi ]$ .",
"Consider $Z=\\begin{bmatrix}-\\sqrt{1+z^2}-z&\\\\ &-\\sqrt{1+z^2}+z\\end{bmatrix}.$ The determinant of the matrix is 1, we want to compute its Magnus exponent.",
"The optimistic suggestion is $\\sqrt{\\pi ^2+\\log (z+\\sqrt{1+z^2})^2}=3.839\\ldots $ .",
"Indeed, in the complex case, or in the doubled real case, this is realizable from $Z=\\exp \\begin{bmatrix}\\log (z+\\sqrt{1+z^2})+\\pi \\mathrm {i}&\\\\ &-\\log (z+\\sqrt{1+z^2})+\\pi \\mathrm {i}\\end{bmatrix}.$ However, in the real case, there is `not enough space' to do this.",
"The pessimistic suggestion is $\\pi +|\\log (z+\\sqrt{1+z^2})|=5.349\\ldots $ .",
"Indeed, we can change sign by an elliptic exponential, and then continue by a hyperbolic exponential.",
"This, we know, cannot be optimal.",
"In reality, the answer is $\\operatorname{\\mathcal {M}}(Z)=z=4.493\\ldots $ .",
"In fact, $Z$ is Magnus parabolic, one can check that $Z\\sim W(z,z)$ .",
"This is easy to from the chiral disk.",
"In this case there are two Magnus minimal representations, because of the conjugational symmetry."
],
[
"Magnus expansion in the Banach–Lie algebra setting",
"Still in the setting of Banach algebras, let $\\operatorname{ad}X$ denote the operator $\\operatorname{ad}X\\,:\\,\\mathfrak {A}\\rightarrow \\mathfrak {A},$ given by $Y\\mapsto [X,Y]$ .",
"Consider the meromorphic function $\\beta (x)=\\frac{x}{\\mathrm {e}^x-1}=\\sum _{j=0}^\\infty \\beta _j x^j.$ The function has poles at $2\\pi \\mathrm {i}(\\mathbb {N}\\setminus \\lbrace 0\\rbrace )$ , so the indicated power series expansion converges on $\\operatorname{D}(0,2\\pi )$ .",
"In particular, if $|X|<\\pi $ (or $\\operatorname{sp}(X)\\subset \\lbrace z\\in \\mathbb {C}\\,:\\,|\\operatorname{Re}z|<\\pi \\rbrace $ ), then $\\Vert \\operatorname{ad}X\\Vert <2\\pi $ (or $\\operatorname{sp}(\\operatorname{ad}X)\\subset \\lbrace z\\in \\mathbb {C}\\,:\\,|\\operatorname{Re}z|<2\\pi \\rbrace $ ) and $\\beta (\\operatorname{ad}X):\\mathfrak {A}\\rightarrow \\mathfrak {A}$ makes sense as absolute convergent power series (or homomorphic function) of $\\operatorname{ad}X$ .",
"In this setting Schur's formulae hold, i. e. for $Y\\in \\mathfrak {A}$ $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\log (\\exp (tY)\\exp (X))\\Bigl |_{t=0}=\\beta (\\operatorname{ad}X)Y$ and $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\log (\\exp (X)\\exp (tY))\\Bigl |_{t=0}=\\beta (-\\operatorname{ad}X)Y;$ with the usual $\\log $ branch cut along the negative $x$ -axis.",
"These formulae also extends, mutatis mutandis, for Banach–Lie groups.",
"However, what we really need is not something more but something less; the consequences in the formal case: If $X$ and $Y$ are formal noncommutative variables, then $\\log (\\exp (Y)\\exp (X))_{\\text{the multiplicity of $Y$ is $1$}}=\\beta (\\operatorname{ad}X)Y$ and $\\log (\\exp (X)\\exp (Y))_{\\text{the multiplicity of $Y$ is $1$}}=\\beta (-\\operatorname{ad}X)Y;$ where $\\beta (\\operatorname{ad}X)$ is understood in the sense of formal power series.",
"An immediate consequence is the Magnus recursion theorem.",
"Theorem 7.1 (Magnus,[18], 1954) $\\mu _k$ satisfies the recursions $&\\mu _k(X_1,\\ldots ,X_k)=\\\\&=\\sum _{\\begin{array}{c}I_1\\dot{\\cup }\\ldots \\dot{\\cup }I_s=\\lbrace 2,\\ldots ,k\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}\\beta _s\\cdot (\\operatorname{ad}\\mu _{l_1}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}}))\\ldots (\\operatorname{ad}\\mu _{l_s}(X_{i_{s,1}},\\ldots ,X_{i_{s,l_s}}))X_1$ and $&\\mu _k(X_1,\\ldots ,X_k)=\\\\&= \\sum _{\\begin{array}{c}I_1\\dot{\\cup }\\ldots \\dot{\\cup }I_s=\\lbrace 2,\\ldots ,k-1\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}(-1)^s\\beta _s\\cdot (\\operatorname{ad}\\mu _{l_1}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}}))\\ldots (\\operatorname{ad}\\mu _{l_s}(X_{i_{s,1}},\\ldots ,X_{i_{s,l_s}}))X_k.$ In particular, we find that $\\mu _k(X_1,\\ldots ,X_k)$ is a commutator polynomial of its variables.",
"Consider the first equation.",
"Let us apply the first formal Schur formula with $X=\\log (\\exp (X_2)\\cdot \\ldots \\cdots \\exp (X_{k}))$ and $Y=X_1$ , and select the terms where every variable $X_i$ has multiplicity 1.",
"Considering (REF ) yields the formula immediately.",
"The second equation is an analogous.",
"More precisely, this is combinatorially equivalent to the original version formulated in more ODE looking setting.",
"Remark 7.2 The coefficients $\\beta _i$ are very well-known: Let us recollect some standard information about Bernoulli numbers.",
"The Bernoulli numbers are defined by the expansion $\\beta (x)=\\text{``$\\frac{x}{\\mathrm {e}^x-1}$''}=\\sum _{j=0}^\\infty \\frac{B_j}{j!}",
"x^j.$ Then $B_j=\\sum _{k=0}^j\\begin{pmatrix} j\\\\ k\\end{pmatrix}B_k\\qquad (j\\ge 2) $ $B_j=-\\begin{pmatrix} j\\\\ k\\end{pmatrix}\\sum _{k=0}^{j-1}\\frac{B_k}{j+1-k}\\qquad (j\\ge 2) $ $B_{2j+1}=0\\qquad (j\\ge 1) $ $\\tan x=\\sum _{j=1}^\\infty (-1)^{j+1} 2^{2j}(2^{2j}-1)\\frac{B_{2j}}{(2j)!",
"}x^{2j-1} \\quad (|x|<\\pi ) $ $\\operatorname{sgn}B_{2j}=(-1)^{j+1}\\qquad (j\\ge 1) $ $\\tanh x=\\sum _{j=1}^\\infty 2^{2j}(2^{2j}-1)\\frac{B_{2j}}{(2j)!",
"}x^{2j-1} \\quad (|x|<\\pi ) $ $B_0=1,\\quad \\!",
"B_1=-\\frac{1}{2},\\quad \\!",
"B_2=\\frac{1}{6},\\quad \\!",
"B_4=-\\frac{1}{30},\\quad \\!",
"B_6=\\frac{1}{42},\\quad \\!",
"B_8=-\\frac{1}{30},\\quad \\!",
"B_{10}=\\frac{5}{66} $ $B_{2j}=(-1)^{j+1}(2j)!",
"(2\\pi )^{-2j}2\\zeta (2j)=(-1)^{j+1}(2j)!",
"(2\\pi )^{-2j}2\\sum _{N=1}^\\infty \\frac{1}{N^{2j}}\\quad (j\\ge 1).",
"$ Properties (REF –REF ) are relatively straightforward to prove using elementary analysis, while (REF ) can be proven from Euler's formula $\\pi \\cot \\pi x=\\frac{1}{x} +\\sum _{\\begin{array}{c}N=-\\infty \\\\N\\ne 0\\end{array}}^\\infty \\left( \\frac{1}{x-N}+\\frac{1}{N} \\right)$ .",
"In our terminology, it is $\\beta _j=\\frac{B_j}{j!",
"}$ .",
"As $\\mu _k(X_1,\\ldots ,X_k)$ is a commutator expression, we can associate a Lie-polynomial $\\mu _k^{\\operatorname{Lie}}(X_1,\\ldots ,X_k)$ .",
"According to the Poincaré–Birkhoff–Witt theorem, the Lie-polynomial itself is independent of which commutator expression is used.",
"More generally, if $\\mathfrak {g}$ is Lie-algebra over a field of characteristic $k$ , then we have natural maps ${\\odot \\mathfrak {g}[r]_\\iota &@/_/[l]_\\varpi \\otimes \\mathfrak {g}[r]_U&\\mathcal {U}\\mathfrak {g}}.$ Here $\\iota (X_1\\odot \\ldots \\odot X_n)=\\frac{1}{n!",
"}\\sum _{\\sigma \\in \\Sigma _n}X_{\\sigma (1)}\\otimes \\ldots \\otimes X_{\\sigma (n)}$ , and $U$ is the factorization generated by $X_1\\otimes X_2-X_2\\otimes X_1=[X_1,X_2]$ .",
"According to the Poincaré–Birkhoff–Witt theorem, $U\\circ \\iota $ is a linear isomorphism, and we set $\\varpi =(\\iota \\circ U)^{-1}\\circ U$ .",
"I. e. $\\iota ,\\varpi $ are the natural splitting maps, and $U$ is the natural factorization.",
"Let $\\varpi _k$ denote the further projection of $\\varpi $ to $\\odot ^k\\mathfrak {g}$ ; this is the $k$ -th canonical projection.",
"Theorem 7.3 (Solomon [31], 1968) The first canonical projection is given by Magnus commutators: $(U\\circ \\iota )\\circ \\varpi _1(X_1\\otimes \\ldots \\otimes X_n)=\\mu _n(X_1,\\ldots ,X_n)_{\\,\\mathrm {in}\\,\\mathcal {U}\\mathfrak {g}},$ and $\\varpi _1(X_1\\otimes \\ldots \\otimes X_n)=\\mu _n^{\\operatorname{Lie}}(X_1,\\ldots ,X_n).$ (After Helmstetter [16], 1989.)",
"Formula (REF ) decomposes any product to symmetric products of commutator expressions.",
"The part of symmetric degree 1 is exactly the Magnus commutator.",
"More accurately, Solomon [31] computes $(U\\circ \\iota )\\circ \\varpi _1$ directly to the RHS of (REF ).",
"Helmstetter [16] understands the connection to $\\log \\Pi \\exp $ -structure, but does not care about the Magnus expansion.",
"Reutenauer [30] has the full picture algebraically.",
"Equation (REF ) also shows how to express the higher canonical projections with Magnus commutators.",
"From the viewpoint of the canonical projections, it is easy to prove Corollary 7.4 (Generalized Magnus recursion) For $1\\le h_1,h_2\\le k$ , $h_1+h_2\\le k$ , $\\mu _k(X_1,\\ldots ,X_k)=\\sum _{\\begin{array}{c}I_1\\dot{\\cup }\\ldots \\dot{\\cup }I_s=\\lbrace h_1+1,\\ldots ,k-h_2\\rbrace \\\\I_j=\\lbrace i_{j,1},\\ldots ,i_{j,l_j}\\rbrace \\ne \\emptyset \\\\i_{j,1}<\\ldots <i_{j,l_j}\\end{array}}\\\\\\mu _{s}(X_1,\\ldots ,X_{h_1},\\mu _{l_1}(X_{i_{1,1}},\\ldots ,X_{i_{1,l_1}}),\\ldots , \\mu _{l_s}(X_{i_{s,1}},\\ldots ,X_{i_{s,l_s}}),X_{k-h_2+1},\\ldots ,X_{k}).$ This follows from applying the first canonical projection to $\\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{k}))= \\log (\\exp (X_1)\\cdot \\ldots \\cdot \\exp (X_{h_1})\\cdot \\\\\\exp (\\log (\\exp (X_{h_1+1})\\cdot \\ldots \\cdot \\exp (X_{k-h_1}))\\cdot \\exp (X_{k-h_2+1})\\cdot \\ldots \\cdot \\exp (X_{k}) ).",
"$ One can obtain many other similar identities in this way.",
"Corollary REF specializes to the classical Magnus recursion in the cases $h_1+h_2=1$ according to Lemma 7.5 (Generalized Schur identity) $\\sum _{\\sigma \\in \\Sigma _{\\lbrace 2,\\ldots ,n\\rbrace }}\\mu (X_1,X_{\\sigma (2)},\\ldots ,X_{\\sigma (n)})=\\sum _{\\sigma \\in \\Sigma _{\\lbrace 2,\\ldots ,n\\rbrace }}\\beta _{n-1} (\\operatorname{ad}X_{\\sigma (2)}) \\ldots (\\operatorname{ad}X_{\\sigma (n)})X_1,$ $\\sum _{\\sigma \\in \\Sigma _{\\lbrace 1,\\ldots ,n-1\\rbrace }}\\mu (X_{\\sigma (1)},\\ldots ,X_{\\sigma (n-1)},X_n)=\\sum _{\\sigma \\in \\Sigma _{\\lbrace 1,\\ldots ,n-1\\rbrace }}\\beta _{n-1} (\\operatorname{ad}X_{\\sigma (1)}) \\ldots (\\operatorname{ad}X_{\\sigma (n-1)})X_n.$ The first identity follows from the canonical projection applied to the first Schur identity of $\\frac{\\mathrm {d}}{\\mathrm {d}t}\\log (\\exp (tX_1)\\exp (X_2+\\ldots X_n) )$ ; but it is also a consequence of the Magnus recursion formulas.",
"The second one is similar.",
"Using Theorem REF , one can compute $\\mu ^{\\operatorname{Lie}}_k$ effectively.",
"Other possibility is to use the Dynkin–Specht–Wever lemma in order to turn (REF ) into an explicit Lie-polynomial, as it was already done in Mielnik, Plebański, [21].",
"Unfortunately, this latter approach is not as effective analytically as one would like.",
"The convergence of Magnus expansion in the setting of Banach–Lie-algebras, is customarily examined in terms of Banach–Lie norms.",
"In what follows, let $\\mathfrak {g}$ be a Banach–Lie-algebra, i. e. Banach space endowed with a norm-compatible Lie algebra structure $\\Vert \\cdot \\Vert $ such that $\\Vert [X,Y]\\Vert \\le \\Vert X\\Vert \\cdot \\Vert Y\\Vert $ holds.",
"This is not exactly compatible with the Banach-algebra settings.",
"If $\\mathfrak {A}$ is a Banach algebra with norm $|\\cdot |$ , then it becomes a Banach–Lie algebra with the norm $\\Vert A\\Vert =2|A|.$ (Indeed, in this case $\\Vert [A,B]\\Vert =2|[A,B]|\\le 4|A|\\cdot |B|=\\Vert A\\Vert \\cdot \\Vert B\\Vert $ .)",
"Our objective is to examine of the convergence of $\\sum _{n=1}^\\infty \\mu _n^{\\operatorname{Lie}}(\\phi )$ depending on $\\int {\\Vert \\phi \\Vert }$ .",
"First of all, there is a case we already know, the linear case.",
"This applies to the large Lie algebra $\\operatorname{\\mathfrak {gl}}(\\mathfrak {H})$ ($\\dim \\mathfrak {H}\\ge 2$ ) and the small $\\operatorname{\\mathfrak {sl}}_2(\\mathbb {R})$ alike: Using the notation $\\Vert \\cdot \\Vert =2\\Vert \\cdot \\Vert _2$ , we know that $\\int {\\Vert \\phi \\Vert }<2\\pi $ implies the convergence of the Magnus series; while Example REF yields a case with $\\int {\\Vert \\phi \\Vert }=2\\pi $ such that the Magnus expansion is divergent.",
"One can proceed with the general case as follows.",
"Let $\\Gamma _k^{\\operatorname{Lie}}$ be the set of all Lie-monomials of $X_1,\\ldots ,X_k$ , where every variable is with multiplicity 1.",
"Let $\\Theta _k^{\\operatorname{Lie}}:=\\frac{1}{k!",
"}\\inf \\left\\lbrace \\sum _{\\gamma \\in \\Gamma _k^{\\operatorname{Lie}}} |\\theta _\\gamma |\\,:\\,\\mu ^{\\operatorname{Lie}}_k(X_1,\\ldots ,X_k)=\\sum _{\\gamma \\in \\Gamma _k^{\\operatorname{Lie}}} \\theta _\\gamma \\cdot \\gamma (X_1,\\ldots ,X_k)\\,,\\, \\theta _\\gamma \\in \\mathbb {R} \\right\\rbrace ,$ that is $1/k!$ times the minimal sum of the absolute value of the coefficients of the presentations of $\\mu ^{\\operatorname{Lie}}_k(X_1,\\ldots ,X_k)$ .",
"We use the term `minimal presentation' where this sum is minimal.",
"For practical reason, we will not consider two Lie-monomials different, if they can be obtained from each other by switching the order of the brackets in them.",
"We prefer lexicographically minimal presentations in the order of variables.",
"E. g., we prefer $-[[X_1,X_3],X_2]$ to $[X_2,[X_1,X_3]]$ .",
"For the sake of compactness, we will use notation $X_{[[1,3],2]}\\equiv [[X_1,X_3],X_2]$ , etc.",
"Computing $\\Theta _k^{\\operatorname{Lie}}$ can be posed as a straightforward problem in rational linear programming, which, however, quickly grows intractable due to its size.",
"We present some examples, but we omit the details: standard methods apply.",
"Example 7.6 $\\Theta _1^{\\operatorname{Lie}}=1,\\qquad \\Theta _2^{\\operatorname{Lie}}=\\frac{1}{2!",
"}\\cdot \\frac{1}{2},\\qquad \\Theta _3^{\\operatorname{Lie}}=\\frac{1}{3!",
"}\\cdot \\frac{1}{3}; $ due to the unique minimal presentations $\\mu _1^{\\operatorname{Lie}}(X_1)=X_1,\\qquad \\mu _2^{\\operatorname{Lie}}(X_1,X_2)=\\frac{1}{2}X_{[1,2]},\\qquad \\mu _3^{\\operatorname{Lie}}(X_1,X_2,X_3)=\\frac{1}{6}X_{[[1,2],3]}+\\frac{1}{6}X_{[1,[2,3]]}.$ Example 7.7 $\\Theta _4^{\\operatorname{Lie}}=\\frac{1}{4!",
"}\\cdot \\frac{1}{3}, $ but the minimal presentation is not unique.",
"The minimal presentations form a $\\Delta _4=4$ dimensional simplex.",
"They are of form $\\mu _4^{\\operatorname{Lie}}(X_1,X_2,X_3,X_4,X_5)=\\frac{1}{12}\\Bigl (&-X_{{[[[1,4],2],3]}}\\lambda _1+X_{{[1,[2,[3,4]]]}} \\left(\\lambda _5+\\lambda _1+\\lambda _2 \\right)\\\\&+X_{{[[1,[2,4]],3]}}\\lambda _2+X_{{[[1,[2,3]],4]}} \\left(\\lambda _1+\\lambda _2+\\lambda _3 \\right)\\\\&+X_{{[[1,3],[2,4]]}}\\lambda _3+X_{{[[1,2],[3,4]]}} \\left(\\lambda _2+\\lambda _3+\\lambda _4 \\right)\\\\&-X_{{[[[1,3],4],2]}}\\lambda _4+X_{{[1,[[2,3],4]]}} \\left(\\lambda _3+\\lambda _4+\\lambda _5 \\right)\\\\&-X_{{[[[1,4],3],2]}}\\lambda _5+X_{{[[[1,2],3],4]}} \\left(\\lambda _4+\\lambda _5+\\lambda _1 \\right)\\Bigr ),$ where $\\lambda _i\\ge 0,\\qquad \\lambda _1+\\lambda _2+\\lambda _3+\\lambda _4+\\lambda _5=1.$ The presentation which is most economical and symmetrical at the same time, belongs to $(\\lambda _1,\\lambda _2,\\lambda _3,\\lambda _4,\\lambda _5)=(0,0,1,0,0)$ .",
"It yields $\\mu _4^{\\operatorname{Lie}}(X_1,X_2,X_3,X_4)=\\frac{1}{12}\\Bigl (X_{{[[1,[2,3]],4]}}+X_{{[[1,3],[2,4]]}}+X_{{[[1,2],[3,4]]}}+X_{{[1,[[2,3],4]]}}\\Bigr ).$ Example 7.8 $\\Theta _5^{\\operatorname{Lie}}=\\frac{1}{5!",
"}\\cdot \\frac{2}{5}.",
"$ The minimal presentations form a nontrivial $\\Delta _5=32$ dimensional polytope.",
"A sort of most economical and symmetrical presentation is given by $\\mu _5^{\\operatorname{Lie}}(X_1,&X_2,X_3,X_4,X_5)=\\\\=\\frac{1}{120}\\Bigl (&+4\\,X_{{[[1,2],[3,[4,5]]]}}+4\\,X_{{[[[1,2],[3,4]],5]}}+4\\,X_{{[[[1,2],3],[4,5]]}}+4\\,X_{{[1,[[2,3],[4,5]]]}}\\\\&+4\\,X_{{[[1,[2,[3,5]]],4]}}-4\\,X_{{[[[[1,3],4],5],2]}}+4\\,X_{{[[1,[[2,3],4]],5]}}+4\\,X_{{[1,[[2,[3,4]],5]]}}\\\\&+2\\,X_{{[[1,3],[[2,4],5]]}}+2\\,X_{{[[1,4],[[2,5],3]]}}-2\\,X_{{[[[1,4],3],[2,5]]}}+2\\,X_{{[[1,[2,4]],[3,5]]}}\\\\&+2\\,X_{{[[[[1,5],4],3],2]}}-2\\,X_{{[[[[1,5],2],3],4]}}+2\\,X_{{[[[1,2],[3,5]],4]}}-2\\,X_{{[[[1,3],[4,5]],2]}}\\Bigl ).$ (This is shorter than the formula given in Prato, Lamberti [29].)",
"Example 7.9 $\\Theta _6^{\\operatorname{Lie}}=\\frac{1}{6!",
"}\\cdot \\frac{37}{60}.",
"$ The minimal presentations form a nontrivial $\\Delta _5=370$ dimensional polytope.",
"A possible presentation is $\\mu _6^{\\operatorname{Lie}}(X_1,&X_2,X_3,X_4,X_5,X_6)=\\\\=\\frac{1}{240}\\Bigl (&+4\\,X_{{[[1,[3,5]],[[2,4],6]]}}+4\\,X_{{[[1,[4,5]],[[2,3],6]]}}-4\\,X_{{[[[1,4],5],[2,[3,6]]]}}+4\\,X_{{[[1,[2,3]],[[4,5],6]]}}\\\\&+4\\,X_{{[[[1,2],3],[4,[5,6]]]}}+4\\,X_{{[[1,[2,4]],[[3,5],6]]}}+4\\,X_{{[[1,[2,5]],[[3,4],6]]}}+4\\,X_{{[[1,[3,4]],[[2,5],6]]}}\\\\&+4\\,X_{{[1,[[2,[3,[4,5]]],6]]}}+4\\,X_{{[1,[[[2,[3,4]],5],6]]}}-4\\,X_{{[1,[[[[2,5],3],4],6]]}}+4\\,X_{{[[1,3],[[2,[4,5]],6]]}}\\\\&+4\\,X_{{[[1,[2,[[3,4],5]]],6]}}+4\\,X_{{[[1,[[[2,3],4],5]],6]}}-4\\,X_{{[[1,[[[2,5],4],3]],6]}}+4\\,X_{{[[1,[[2,3],5]],[4,6]]}}\\\\&+4\\,X_{{[[1,4],[[2,[3,5]],6]]}}+4\\,X_{{[[1,5],[[2,[3,4]],6]]}}+4\\,X_{{[[1,[[2,4],5]],[3,6]]}}+4\\,X_{{[[1,[[3,4],5]],[2,6]]}}\\\\&-2\\,X_{{[[[[1,3],[4,5]],6],2]}}-2\\,X_{{[[[[[1,3],4],5],6],2]}}+2\\,X_{{[[[[[1,3],6],5],4],2]}}+4\\,X_{{[[[[[1,4],5],6],3],2]}}\\\\&+2\\,X_{{[[[[[1,2],6],5],4],3]}}-2\\,X_{{[[[[1,2],6],[4,5]],3]}}-2\\,X_{{[[[[[1,2],4],5],6],3]}}-4\\,X_{{[[[1,[4,[5,6]]],2],3]}}\\\\&+2\\,X_{{[[1,[2,[3,[5,6]]]],4]}}-2\\,X_{{[[[[1,[5,6]],2],3],4]}}-2\\,X_{{[[[1,[5,6]],[2,3]],4]}}-4\\,X_{{[[[[[1,2],3],6],5],4]}}\\\\&+2\\,X_{{[[1,[2,[3,[4,6]]]],5]}}+2\\,X_{{[[1,[[2,3],[4,6]]],5]}}-2\\,X_{{[[[[1,[4,6]],2],3],5]}}+4\\,X_{{[[[1,[2,[3,6]]],4],5]}}\\\\&+2\\,X_{{[[1,2],[[3,[4,6]],5]]}}+2\\,X_{{[[1,2],[3,[[4,5],6]]]}}+2\\,X_{{[[1,2],[[3,4],[5,6]]]}}+4\\,X_{{[[1,2],[[3,[4,5]],6]]}}\\\\&+2\\,X_{{[[[1,2],[3,4]],[5,6]]}}+2\\,X_{{[[[1,[2,3]],4],[5,6]]}}-2\\,X_{{[[[[1,3],4],2],[5,6]]}}+4\\,X_{{[[1,[[2,3],4]],[5,6]]}} \\\\&+2\\,X_{{[[1,3],[[[2,4],5],6]]}}-2\\,X_{{[[1,3],[[[2,6],4],5]]}}+2\\,X_{{[[1,[2,[3,5]]],[4,6]]}}-2\\,X_{{[[[[1,5],3],2],[4,6]]}}\\Bigl );$ but similar ones exist.",
"(For example, $+2\\,X_{{[[1,[2,[3,4]]],[5,6]]}}+2\\,X_{{[[1,[[2,3],4]],[5,6]]}}-2\\,X_{{[[[[1,4],2],3],[5,6]]}}+4\\,X_{{[[[1,[2,3]],4],[5,6]]}}$ can replace line (REF ), but many other possibilities exist.)",
"For $x\\ge 0$ we define the absolute Magnus characteristic $\\Theta $ by $\\Theta ^{\\operatorname{Lie}}(x)=\\sum _{k=1}^\\infty \\Theta _k^{\\operatorname{Lie}}x^k.$ Then, in Banach–Lie algebras, $\\int _{t_1\\le \\ldots \\le t_n\\in I}\\left|\\mu _k^{\\operatorname{Lie}}(\\phi (t_1),\\ldots ,\\phi (t_k))\\right|\\le \\Theta _k^{\\operatorname{Lie}}\\cdot \\left(\\int |\\mu | \\right)^k,$ and, consequently, $\\sum _{k=1}^\\infty \\int _{t_1\\le \\ldots \\le t_n\\in I}\\left|\\mu _k^{\\operatorname{Lie}}(\\phi (t_1),\\ldots ,\\phi (t_k))\\right|\\le \\Theta ^{\\operatorname{Lie}}\\left(\\int |\\mu |\\right).",
"$ Regarding the convergence radius of $\\Theta ^{\\operatorname{Lie}}$ , we immediately have the following estimates.",
"From the commutator expansion, it immediately follows that $\\Theta _n\\le 2^{n-1}\\Theta _n^{\\operatorname{Lie}},$ hence $\\Theta (x/2)/2\\le \\Theta ^{\\operatorname{Lie}}(x).$ Consequently, the convergence radius of $\\Theta ^{\\operatorname{Lie}}$ is at most 4.",
"On the other hand, the Dynkin–Specht–Wever lemma implies $\\Theta _n^{\\operatorname{Lie}}\\le \\frac{1}{n} \\Theta _n, $ thus $\\Theta ^{\\operatorname{Lie}}(x)\\le \\int _{t=0}^x\\frac{\\Theta (t)}{t}\\,\\mathrm {d}t=\\int _{t=0}^1\\frac{\\Theta (xt)}{t}\\,\\mathrm {d}t.$ Consequently, the convergence radius of $\\Theta ^{\\operatorname{Lie}}$ is at least 2.",
"This lower estimate however, can be improved to the Varadarajan–Mérigot–Newman–So–Thompson number $\\delta =\\int _{x=0}^{2\\pi } \\frac{\\mathrm {d}y}{2+\\frac{x}{2}-\\frac{x}{2}\\cot \\frac{x}{2} }\\approx 2.1737374\\ldots .$ This bound was first established in the setting of Baker–Campbell–Hausdorff formula due to the work of Varadarajan [35], Mérigot [19], Michel [20], Newman, So, Thompson [27], etc., and later it was extended to the Magnus expansion setting by Blanes, Casas, Oteo, Ros [2], Moan [23].",
"It is the bound generally cited in the literature, thus we call it as the `standard estimate'.",
"It is, however, but the trivial estimate which can be obtained from the Magnus recursion formulas.",
"Furthermore, it is very easy to improve, even if just by a little bit.",
"We explain this in the next section.",
"(In fact, some convergence improvement can be realized already in the resolvent approach but making use of the commutators is a bit cumbersome there.)",
"It must be said, however, that the Dynkin–Specht–Wever lemma can be used well to transpose Theorem REF to the Banach–Lie setting.",
"Indeed, $\\Gamma ^{\\operatorname{Lie}}(x,y)$ can be defined analogously to $\\Gamma (x,y)$ , and a crude application of Dynkin–Specht–Wever lemma yields $\\Gamma ^{\\operatorname{Lie}}(x,y)\\le \\int _{t=0}^1\\frac{\\Gamma (xt,yt)}{t}\\,\\mathrm {d}t.$ Theorem 7.10 .",
"$\\Gamma ^{\\operatorname{Lie}}(x,y)$ is absolute convergent and finite if $|x|+|y|<C_1$ .",
"In particular, the Baker–Campbell–Hausdorff expansion of the pair $X,Y$ is absolutely convergent in the Banach–Lie setting if $\\Vert X\\Vert +\\Vert Y\\Vert <C_1=2.7014\\ldots $ .",
"$C_1$ can be replaced by $C_2=2.8984\\ldots $ .",
"Apply the Dynkin–Specht–Wever lemma using estimates of Theorem REF .",
"What is missing in our discussion, up to now, is the Lie-Banach algebraic version of the Magnus expansion formula, Theorem REF itself.",
"It is not completely obvious what it should be.",
"The most uncomplicated answer is Theorem 7.11 Suppose that $\\phi $ is a $\\mathfrak {g}$ -valued measure, $Y\\in \\mathfrak {g}$ .",
"If $\\sum _{k=1}^\\infty \\left\\Vert \\mu _{\\mathrm {L}[k]}^{\\operatorname{Lie}}(\\phi )\\right\\Vert <+\\infty , $ then $\\exp (\\mu _{\\mathrm {L}}(\\phi ))Y=\\operatorname{exp_{L}}(\\operatorname{ad}\\phi )Y.",
"$ This a corollary of the Theorem REF applied to the adjoint representation.",
"This can be adapted to the purpose of more general representations.",
"It is, however, but a poor algebraic substitute in the lack of Lie-theoretic exponentials.",
"Another viewpoint is in the opposite direction: We can use the Magnus expansion formula to establish a local Lie group structure.",
"This involves theorems like Theorem 7.12 Suppose that $\\phi _1,\\phi _2,\\phi _3$ are a $\\mathfrak {g}$ -valued measures.",
"If $\\Theta ^{\\operatorname{Lie}}( \\smallint \\Vert \\phi _1\\Vert +\\Theta ^{\\operatorname{Lie}}( \\smallint \\Vert \\phi _2\\Vert ) + \\smallint \\Vert \\phi _3\\Vert ) <+\\infty ,$ then $\\mu _{\\mathrm {R}}(\\phi _1.",
"\\phi _2 .\\phi _2 )=\\mu _{\\mathrm {R}}(\\phi _1.",
"\\mu _{\\mathrm {R}}(\\phi _2)\\mathbf {1}_{[0,1]} .\\phi _3 ).$ This is Corollary REF integrated and contracted.",
"Similar statements can be devised for $\\mu _{\\mathrm {R}}(\\phi _1.",
"\\mu _{\\mathrm {R}}(\\phi _2)\\mathbf {1}_{[0,1]} .\\phi _2.",
"\\mu _{\\mathrm {R}}(\\phi _4)\\mathbf {1}_{[0,1]} .\\phi _5 )$ , etc.",
"The third viewpoint involves a fully established local Lie group theory (at least), and considering Lie-group theoretic time ordered exponentials.",
"Then one can study convergence in (local) Lie groups and invariance with respect to the Maurer-Cartan equation, etc.",
"In any case, the subject of this paper, the discussion of the convergence of Magnus series makes sense in itself, and probably should be part of any approach."
],
[
"The standard estimate and its improvements",
"The Magnus recursion formulas imply $k!\\Theta ^{\\operatorname{Lie}}_{k}=\\sum _{l_1+\\ldots +l_s=k-1} |\\beta _s|\\frac{(k-1)!",
"}{l_1!\\cdot \\ldots \\cdot l_s!",
"}(l_1!\\Theta ^{\\operatorname{Lie}}_{l_1})\\cdot \\ldots \\cdot (l_s!\\Theta ^{\\operatorname{Lie}}_{l_s}) ;$ which can be rewritten as $k \\Theta ^{\\operatorname{Lie}}_{k}=\\sum _{l_1+\\ldots +l_s=k-1} |\\beta _s| \\Theta ^{\\operatorname{Lie}}_{l_1}\\cdot \\ldots \\cdot \\Theta ^{\\operatorname{Lie}}_{l_s}.",
"$ This can be turned into an estimate as follows.",
"Let us define the numbers $\\psi _k$ $(k\\in \\mathbb {N})$ by the recursion $\\psi _0=0$ and $k \\psi _{k}=\\sum _{l_1+\\ldots +l_s=k-1} |\\beta _s| \\psi _{l_1}\\cdot \\ldots \\cdot \\psi _{l_s}.",
"$ We also consider the formal generating function $\\tilde{\\beta }(x):=\\sum _{j=0}^\\infty |\\beta _j| x^j.$ Then it is immediate that $\\Theta _k^{\\operatorname{Lie}}\\le \\psi _k;$ and the formal generator function $\\psi (x)=\\sum _{k=0}^\\infty \\psi _k x^k$ satisfies the formal IVP $\\psi (0)=0 $ $\\psi ^{\\prime }(x)=\\tilde{\\beta }(\\psi (x)).",
"$ Theorem 8.1 (a) Interpreted as an analytic function around 0, $\\tilde{\\beta }(x)=2+\\frac{x}{2}-\\frac{x}{2}\\cot \\frac{x}{2};$ and it is convergent in $\\operatorname{\\mathring{D}}(0,\\pi )$ .",
"(b) The analytic version of IVP (REF )–(REF ) has a solution around 0 with convergence radius $\\delta =\\int _{y=0}^{2\\pi } \\frac{\\mathrm {d}y}{\\tilde{\\beta }(y)}\\approx 2.1737374\\ldots .$ As $x\\nearrow \\delta $ , we have $\\psi (x)\\nearrow 2\\pi $ and $\\psi ^{\\prime }(x)\\nearrow +\\infty $ .",
"(So, in real function theoretic sense, $\\psi (\\delta )=2\\pi $ and $\\psi ^{\\prime }(\\delta )=+\\infty $ .)",
"Thus, $\\delta $ is the convergence radius of $\\psi $ .",
"(a) It follows from $\\frac{\\beta (\\mathrm {i}x)+\\beta (-\\mathrm {i}x)}{2}=\\frac{x}{2}\\cot \\frac{x}{2}$ and the information on the signs of the Bernoulli numbers $B_j$ for $j\\ge 2$ .",
"(b) By Pringsheim's theorem it is sufficient to consider the development for $x\\ge 0$ .",
"Then the standard method of separation of variables can be applied, $\\frac{\\mathrm {d}\\psi }{\\tilde{\\beta }(\\psi )}=\\mathrm {d}x.$ Thus the $\\psi $ will be the inverse function of $y\\mapsto \\chi (y)=\\int _{t=0}^y \\frac{\\mathrm {d}t}{\\beta (t)}$ , as long as $\\tilde{\\beta }$ is positive, and develops a singularity when $\\tilde{\\beta }$ becomes infinite.",
"(We know that the solution $\\psi $ is convex.)",
"This happens when $t=2\\pi $ , which means that the range of the nonsingular $\\chi $ is $\\left[0,\\int _{t=0}^{2\\pi } \\frac{\\mathrm {d}t}{\\beta (t)}\\right)$ .",
"Thus, so is the (nonnegative) domain of the nonsingular $\\psi $ , and the behaviour around $x\\sim \\delta $ follows from the inverse function picture.",
"Corollary 8.2 (The standard estimate, see attributions from earlier, algebraic form) In real function theoretic sense, $\\Theta ^{\\operatorname{Lie}}(x)\\le \\psi (x),$ showing, in particular, that the convergence radius of $\\Theta ^{\\operatorname{Lie}}(x)$ is at least $\\delta $ .",
"$\\Box $ This algebraic version gives the standard estimate through (REF ).",
"Actually, in the context of the Baker–Campbell–Hausdorff formula, a somewhat finer estimate is used, the $(1/4)$ -commutative version, which we sketch.",
"Let us define $\\Theta _{k,l}^{\\operatorname{Lie}}$ , as the minimal possible sum of the absolute values of the coefficients of the Lie-presentations of $\\mu _{k+l}(X_1,\\ldots ,X_k, Y_1,\\ldots ,Y_l)$ but with the additional assumption that the variables $Y_i$ commute with each other.",
"Then $ \\Theta _{k,l}^{\\operatorname{Lie}}\\le \\Theta _{k+l}^{\\operatorname{Lie}}$ .",
"Consider the formal generating function $\\Theta ^{\\operatorname{Lie}}(x,y)=\\sum _{k+l\\ge 1}\\Theta _{k,l}^{\\operatorname{Lie}}x^ky^l.",
"$ Its coefficients can be estimated by the coefficients of the solution of the formal IVP $\\psi (0,y)=y $ $\\frac{\\partial }{\\partial x}\\psi (x,y)=\\tilde{\\beta }(\\psi (x,y)).",
"$ By similar arguments as before, $\\psi (x,y)$ will be finite in real analytical sense for $x,y\\ge 0$ if $x=0$ , or $y\\le 2\\pi $ and $x\\le \\int _{t=y}^{2\\pi } \\frac{\\mathrm {d}t}{\\tilde{\\beta }(t)}.$ In fact, in the second case, $\\psi (x,y)=\\psi (x+\\chi (y)).$ Standard convergence estimates in the BCH setting are typically based on the estimates above.",
"See the already cited sources and Day, So, Thompson [11] for some consequences.",
"We see that the situation is more complicated than in the plain Magnus case, but the same principles apply.",
"In what follows we refrain from discussing the BCH case, but the arguments can be adapted to it.",
"Also, one should notice the estimate of Theorem REF is already quite good.",
"Let us return to the Magnus expansion.",
"By the discussion above, in terms of the convergence radius of $\\Theta ^{\\operatorname{Lie}}$ , we have a gap between $\\delta $ and 4.",
"Closing this gap likely requires some deeper insight.",
"However, an advantage of the algebraic formalism is that it offers several ways to improve the standard estimate a bit.",
"We show some methods.",
"We are less interested in numerical constants but that they show greater convergence domains.",
"The methods can be combined for stronger bounds.",
"Method 8.3 (Forced coefficients) One can observe that $\\psi (x)=x+\\frac{1}{4}\\,{x}^{2}+{\\frac{5}{72}}\\,{x}^{3}+{\\frac{11}{576}}\\,{x}^{4}+{\\frac{479}{86400}}\\,{x}^{5}+{\\frac{1769}{1036800}}\\,{x}^{6}+\\ldots $ in contrast to $\\Theta ^{\\operatorname{Lie}}(x)=x+\\frac{1}{4}\\,{x}^{2}+\\frac{1}{18}\\,{x}^{3}+{\\frac{1}{72}}{x}^{4}+{\\frac{1}{300}}{x}^{5}+{\\frac{37}{43200}}\\,{x}^{6}+\\ldots .$ Now, $\\frac{\\mathrm {d}\\Theta ^{\\operatorname{Lie}}(x)}{\\mathrm {d}x}-\\tilde{\\beta }(\\Theta ^{\\operatorname{Lie}}(x))=-\\frac{1}{24}\\,{x}^{2}-{\\frac{1}{72}}{x}^{3}-{\\frac{53}{8640}}\\,{x}^{4}-{\\frac{11}{4320}}\\,{x}^{5}+\\ldots .$ Thus, when we solve the IVP $\\hat{\\psi }(0)=0 $ $\\hat{\\psi }^{\\prime }(x)=\\tilde{\\beta }(\\hat{\\psi }(x))\\underbrace{-\\frac{1}{24}\\,{x}^{2}-{\\frac{1}{72}}{x}^{3}-{\\frac{53}{8640}}\\,{x}^{4}-{\\frac{11}{4320}}\\,{x}^{5}}_{-\\Delta _6(x)}; $ we find that $\\hat{\\psi }(x)$ has the same coefficients as $\\Theta ^{\\operatorname{Lie}}(x)$ up to order 6, but after that the majorizing property relative to $\\Theta ^{\\operatorname{Lie}}(x)$ still holds; $\\Theta _k^{\\operatorname{Lie}}\\le \\hat{\\psi }_k\\le \\psi _k$ .",
"In order to demonstrate the larger convergence radius, let us compare $\\hat{\\psi }$ to $\\psi $ by a crude estimate.",
"For $x\\in [0,\\delta ]$ , we see that $\\hat{\\psi }^{\\prime }(x)-\\psi ^{\\prime }(x)=\\tilde{\\beta }(\\hat{\\psi }(x))-\\Delta _6(x)-\\tilde{\\beta }(\\psi (x))\\le -\\Delta _6(x).",
"$ Integrating in $x$ , we find $\\hat{\\psi }(x)-\\psi (x)\\le -\\int _{t=0}^{x} \\Delta _6(t) \\mathrm {d}t.$ Thus, in particular, $\\hat{\\psi }(\\delta )\\le \\psi (\\delta ) -\\int _{t=0}^{\\delta } \\Delta _6(t) \\mathrm {d}t.$ In particular, $\\hat{\\psi }(\\delta )$ is strictly smaller than $\\psi (\\delta )$ .",
"At this point, even if we continue with the slower (REF ), that would give an extra length $\\hat{L}=\\delta - \\chi \\left(\\psi (\\delta ) -\\int _{t=0}^{\\delta } \\Delta _6(t) \\mathrm {d}t\\right)$ for further development.",
"(But we know that it gives even more.)",
"So, we know that the convergence radius of $\\hat{\\phi }$ is bigger than $\\delta +\\hat{L}$ .",
"In fact, we have the estimate $\\hat{\\psi }(x)\\le {\\left\\lbrace \\begin{array}{ll}\\psi (x) -\\int _{t=0}^{x} \\Delta _6(t) \\mathrm {d}t&\\text{if }x\\in [0,\\delta ]\\\\\\psi (x-\\hat{L})&\\text{if }x\\in [\\delta ,\\delta +\\hat{L}].\\end{array}\\right.",
"}$ Numerically $\\hat{L}=0.0074001\\ldots $ , thus $\\delta +\\hat{L}=2.1811375\\ldots $ is obtained for a larger convergence radius.",
"Our estimates above were really simplistic, though; more precise numerical results show that the convergence radius of $\\hat{\\psi }$ is around $2.2762\\ldots $ What hinders the previous method is that we use the same naive recursion mechanism based on the Magnus recursion as originally.",
"We can achieve better results if we use recursion of higher order.",
"The next discussion will be essentially based on the identity $\\sum _{j=1}^kP(X_1,\\ldots ,X_{j-1},[X_j,Y],\\ldots ,X_{j+1},\\ldots ,X_{k} )=[P(X_1,\\ldots ,X_{k} ),Y],$ where $P$ is a Lie polynomial.",
"This allows to reduce the size of some expressions.",
"In order to compactify our formulas, let us introduce some notation.",
"Instead of explaining it in advance, we show how the Magnus recursion can be expressed in this notation.",
"Expansion in the first variable ($Z_1$ ): $\\mu =Z_1-\\frac{1}{2}[\\mu ,Z_1]+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1].$ expansion in the last variable ($Z_n$ ): $\\mu =Z_n+\\frac{1}{2}[\\mu ,Z_n]+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n].$ What happens is that we consider Lie polynomials in $Z_1,\\ldots ,Z_n$ , and terms in the expressions are understood so that whenever we have $\\mu $ 's with some unspecified variables, then the unnoted variables are distributed among them with no multiplicities, and ascendingly in every $\\mu $ .",
"The higher commutators should be resolved as $[X_1,\\ldots ,X_{k-1},X_k]=(\\operatorname{ad}X_1)\\ldots (\\operatorname{ad}X_{k-1})X_k.$ Clearly, one should be careful with this notation, but it has the advantage of being short.",
"We also use the notation $\\tilde{\\tilde{\\beta }}(x)=\\tilde{\\beta }(x)-1-\\frac{1}{2}x.$ Method 8.4 (Magnus recursion of second order) In the standard approach it did not matter if we used expansion by the first or last variable.",
"Here we take expansion by two variables, and, in order to gain a little additional improvement, we combine this with symmetrization.",
"If we expand in $Z_1$ , and later in $Z_n$ , then we find $\\mu =&Z_1\\\\&+\\frac{1}{2}[Z_1,Z_n]+\\frac{1}{4}[Z_1,[\\mu ,Z_n]]+\\sum _{k=1}^\\infty \\beta _{2k}\\frac{1}{2}[Z_1,[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},Z_n]]\\\\&+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,Z_n,\\ldots ,\\mu }_{2j},Z_1]\\\\&+\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],Z_n]-\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},[Z_1,Z_n]]\\\\&+\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_n]]\\\\&-\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1,\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_n].$ If we expand in $Z_n$ , and later in $Z_1$ , then we find $\\mu =&Z_n\\\\&+\\frac{1}{2}[Z_1,Z_n]+\\frac{1}{4}[[Z_1,\\mu ],Z_n]+\\sum _{k=1}^\\infty \\beta _{2k}\\frac{1}{2}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},Z_1],Z_n]\\\\&+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,Z_1,\\ldots ,\\mu }_{2j},Z_n]\\\\&+\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[Z_1,[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n]]-\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},[Z_1,Z_n]]\\\\&+\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n],[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_1]]\\\\&-\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n,\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_1].$ The averaged (symmetrized) expression is $\\mu =&Z+\\frac{1}{2}[Z_1,Z_n]\\\\&+\\frac{1}{8}[Z_1,[\\mu ,Z_n]]+\\frac{1}{8}[[Z_1,\\mu ],Z_n]\\\\&+\\frac{1}{2}\\sum _{j=2}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,Z_n,\\ldots ,\\mu }_{2j-1},\\underbrace{\\mu }_1,Z_1]+\\frac{1}{2}\\sum _{j=2}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,Z_1,\\ldots ,\\mu }_{2j-1},\\underbrace{\\mu }_1,Z_n]\\\\&+\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],Z_n]+\\sum _{k=1}^\\infty \\beta _{2k}\\frac{1}{2}[Z_1,[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},Z_n]]\\\\& -\\frac{1}{2}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},[Z_1,Z_n]]\\\\&+\\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_n]]\\\\&+\\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n],[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_1]]\\\\&-\\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1,\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_n]\\\\&-\\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_n,\\underbrace{\\mu ,\\ldots ,\\mu }_{2k-1},Z_1].$ Using these we can develop a majorizing series $\\psi $ for $\\Theta ^{\\operatorname{Lie}}$ by the recursion (formal IVP) $\\check{\\psi }(0)=0 $ $\\check{\\psi }^{\\prime }(0)=1 $ $\\check{\\psi }^{\\prime \\prime }(x)=f(\\psi (x)), $ where $f(x)=&\\frac{1}{2}+\\frac{1}{4}x+\\tilde{\\tilde{\\beta }}(x)^{\\prime }-\\frac{1}{x}\\tilde{\\tilde{\\beta }}(x)+\\frac{3}{2}\\tilde{\\tilde{\\beta }}(x)+\\frac{2}{x}\\tilde{\\tilde{\\beta }}(x)^2\\\\=&2+\\frac{x}{2}+\\frac{1}{x}-2\\,\\cot \\left( \\frac{x}{2} \\right)-\\frac{3}{4}\\,x\\cot \\left( \\frac{x}{2} \\right)+\\frac{3}{4}\\,x \\left(\\cot \\left(\\frac{x}{2}\\right) \\right) ^{2}.$ The IVP (REF )–(REF ) is one of the classically treatable ones, and its leads to convergence radius $\\delta _2=\\int _{u=0}^{2\\pi }\\frac{\\mathrm {d}u}{\\sqrt{1+2\\int _{t=0}^uf(t)\\,\\mathrm {d}t }}\\approx 2.281\\ldots $ This improvement is still small, but better than in the case of the previous method.",
"In general, we are interested in the size of $\\mu $ .",
"However, in its estimation, the size of $\\beta (\\operatorname{ad}\\mu )$ played a role, which we estimated from the size of $\\mu $ naively.",
"Perhaps we can do better keeping a separate check on the size of $\\beta (\\operatorname{ad}\\mu )$ .",
"More precisely, we will keep a check on the size of $\\tilde{\\tilde{\\beta }}(\\operatorname{ad}\\mu )$ .",
"Method 8.5 (A simplest compartmentalization.)",
"Consider the equation $\\mu =Z_1-\\frac{1}{2}[\\mu ,Z_1]+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1]$ and it consequence $\\sum _{k=1}^\\infty \\beta _{2k}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W]=&\\sum _{k=1}^\\infty \\beta _{2k}[\\underbrace{\\mu ,\\ldots ,Z_1+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],\\ldots \\mu }_{2k},W] \\\\&-\\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W],Z_1]+ \\frac{1}{2}\\sum _{k=1}^\\infty \\beta _{2k}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},[W,Z_1]].$ This leads to the majorizing sytem (formal IVP) $\\psi (0)=0,\\qquad \\tilde{\\tilde{\\psi }}(0)=0$ $\\psi ^{\\prime }(x)=1+\\frac{1}{2}\\psi (x)+\\tilde{\\tilde{\\psi }}(x),$ $\\tilde{\\tilde{\\psi }}^{\\prime }(x)=\\tilde{\\tilde{\\beta }}^{\\prime }(\\psi (x))(1+\\tilde{\\tilde{\\psi }}(x)) +\\tilde{\\tilde{\\psi }}(x).$ In this present form, this differential equation blows up around $x=2.204\\ldots $ , which leads to a quite modest lower estimate for the convergence radius.",
"However, compartmentalization schemes like that, in general, allow to separate various algebraic patterns in the Magnus expansion.",
"We do not pursue this direction in its full power now but we slightly improve this example.",
"The disadvantage of the previous method is that in the RHS of line (REF ) we still use exponential estimates.",
"The ideal thing would be keeping a check on the size $(\\operatorname{ad}\\mu )^k$ for every $k$ , but this is just too complicated for us to do here.",
"However, we will do this partially.",
"Let us use the notation $\\beta ^{(e)}(x)=\\frac{x^2}{4\\pi ^2-x^2}=\\sum _{j=1}^\\infty \\left(\\frac{x}{2\\pi }\\right)^{2j}$ $\\beta ^{(o)}(x)=\\frac{x^3}{2\\pi (4\\pi ^2-x^2)}=\\sum _{j=1}^\\infty \\left(\\frac{x}{2\\pi }\\right)^{2j+1}$ $\\mathring{\\beta }(x)=\\sum _{j=1}^\\infty \\left(\\frac{x}{2\\pi }\\right)^{2j}2\\sum _{N=2}^\\infty \\frac{1}{N^{2j}}$ Then $\\tilde{\\tilde{\\beta }}(x)=2\\beta ^{(e)}(x)+ \\mathring{\\beta }(x).",
"$ Method 8.6 (A slightly more sophisticated compartmentalization.)",
"Here we keep track on the size of $\\mu $ , $\\beta ^{(e)}(\\operatorname{ad}\\mu )$ , $\\beta ^{(o)}(\\operatorname{ad}\\mu )$ , $\\mathring{\\beta }(\\operatorname{ad}\\mu )$ .",
"The relevant equations are $\\mu =Z_1-\\frac{1}{2}[\\mu ,Z_1]+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1]$ $\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k}}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W]=\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k}}[\\underbrace{\\mu ,\\ldots ,Z_1+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],\\ldots \\mu }_{2k},W]\\\\-\\frac{1}{2}\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k}}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W],Z_1]+ \\frac{1}{2}\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k}}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},[W,Z_1]],$ $\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k+1}}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k+1},W]=\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k+1}}[\\underbrace{\\mu ,\\ldots ,Z_1+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],\\ldots \\mu }_{2k+1},W]\\\\-\\frac{1}{2}\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k+1}}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k+1},W],Z_1]+ \\frac{1}{2}\\sum _{k=1}^\\infty \\frac{1}{(2\\pi )^{2k+1}}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k+1},[W,Z_1]],$ $\\sum _{k=1}^\\infty \\mathring{\\beta }_{2k}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W]=\\sum _{k=1}^\\infty \\mathring{\\beta }_{2k}[\\underbrace{\\mu ,\\ldots ,Z_1+\\sum _{j=1}^\\infty \\beta _{2j}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2j},Z_1],\\ldots \\mu }_{2k},W]\\\\-\\frac{1}{2}\\sum _{k=1}^\\infty \\mathring{\\beta }_{2k}[[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},W],Z_1]+ \\frac{1}{2}\\sum _{k=1}^\\infty \\mathring{\\beta }_{2k}[\\underbrace{\\mu ,\\ldots ,\\mu }_{2k},[W,Z_1]].$ This leads to the IVP $\\psi (0)=0,\\quad \\psi ^{(e)}(0)=0,\\quad \\psi ^{(o)}(0)=0,\\quad \\mathring{\\psi }(0)=0,\\quad $ $\\psi ^{\\prime }(x)=1+\\frac{1}{2}\\psi (x)+2 \\psi ^{(e)}(x)+\\mathring{\\psi }(x),$ $\\psi ^{(e)\\prime }(x)=\\frac{1}{2\\pi }2\\left(\\frac{\\psi (x)}{2\\pi }+\\psi ^{(o)}(x)\\right)(1+\\psi ^{(e)}(x) )(1+2 \\psi ^{(e)}(x)+\\mathring{\\psi }(x))+\\psi ^{(e)}(x),$ $\\psi ^{(o)\\prime }(x)=\\frac{1}{2\\pi }\\left(2\\psi ^{(e)}(x) +\\psi ^{(e)}(x)^2+\\left(\\frac{\\psi (x)}{2\\pi }+\\psi ^{(o)}(x)\\right)^2 \\right)(1+2 \\psi ^{(e)}(x)+\\mathring{\\psi }(x))+\\psi ^{(o)}(x),$ $\\mathring{\\psi }^{\\prime }(x)=\\mathring{\\beta }^{\\prime }(\\psi (x))(1+2 \\psi ^{(e)}(x)+\\mathring{\\psi }(x)) +\\mathring{\\psi }(x).$ Numerical results show that this IVP blows up at $x=2.297\\ldots .$ This is our best lower bound for the convergence radius up to now.",
"(Making a slightly more complicated version, it is $x=2.298\\ldots .$ ) We will not make this result more precise; one can simply obtain slightly better ones, anyway.",
"Somebody with a good knowledge in robust algorithms can make the numerical bounds entirely precise.",
"Method 8.7 (A variant of the previous method) Let $\\beta ^{(ee)}(x)=\\sum _{j=1}^\\infty \\left(\\frac{x}{2\\cdot 2\\pi }\\right)^{2j},\\qquad \\beta ^{(oo)}(x)=\\sum _{j=1}^\\infty \\left(\\frac{x}{2\\cdot 2\\pi }\\right)^{2j+1}.$ According to this we have the analogue $\\tilde{\\tilde{\\beta }}(x)=2\\beta ^{(e)}(x)+2\\beta ^{(ee)}(x)+ \\ddot{\\beta }(x)$ of (REF ).",
"The sizes of the expressions $\\mu ,\\beta ^{(e)}(\\operatorname{ad}\\mu ),\\ldots ,\\ddot{\\beta }(\\operatorname{ad}\\mu )$ are described by the series $\\Theta ^{\\operatorname{Lie}}(x),$ $\\Theta ^{(e)}(x),\\ldots ,\\ddot{\\Theta }(x)$ .",
"The appropriate recursion relations imply $\\Theta (0)=0,\\quad \\Theta ^{(e)}(0)=0,\\quad \\Theta ^{(o)}(0)=0,\\quad \\Theta ^{(ee)}(0)=0,\\quad \\Theta ^{(oo)}(0)=0,\\quad \\ddot{\\Theta }(0)=0, $ $\\Theta ^{\\operatorname{Lie}\\prime }(x)\\le \\Delta \\Theta (x)+\\frac{1}{2}\\Theta ^{\\operatorname{Lie}}(x),$ $\\Theta ^{(e)\\prime }(x)\\le \\frac{1}{2\\pi }2\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\pi }+\\Theta ^{(o)}(x)\\right)(1+\\Theta ^{(e)}(x) )\\Delta \\Theta (x)+\\Theta ^{(e)}(x),$ $\\Theta ^{(o)\\prime }(x)\\le \\frac{1}{2\\pi }\\left(2\\Theta ^{(e)}(x) +\\Theta ^{(e)}(x)^2+\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\pi }+\\Theta ^{(o)}(x)\\right)^2 \\right)\\Delta \\Theta (x)+\\Theta ^{(o)}(x),$ $\\Theta ^{(ee)\\prime }(x)\\le \\frac{1}{2\\cdot 2\\pi }2\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\cdot 2\\pi }+\\Theta ^{(oo)}(x)\\right)(1+\\Theta ^{(ee)}(x) )\\Delta \\Theta (x)+\\Theta ^{(ee)}(x),$ $\\Theta ^{(oo)\\prime }(x)\\le \\frac{1}{2\\cdot 2\\pi }\\left(2\\Theta ^{(ee)}(x) +\\Theta ^{(ee)}(x)^2+\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\cdot 2\\pi }+\\Theta ^{(oo)}(x)\\right)^2 \\right)\\Delta \\Theta (x)+\\Theta ^{(oo)}(x),$ $\\ddot{\\Theta }^{\\prime }(x)\\le \\ddot{\\beta }^{\\prime }(\\Theta ^{\\operatorname{Lie}}(x))\\Delta \\Theta (x) +\\ddot{\\Theta }(x).$ where $\\Delta \\Theta (x)\\equiv 1+2 \\Theta ^{(e)}(x)+2 \\Theta ^{(ee)}(x)+\\ddot{\\Theta }(x)$ However, from the series expansion we also know that $\\ddot{\\Theta }(x)\\le 2\\cdot 2^2\\sum _{N=3}^\\infty \\frac{1}{N^2}\\cdot \\Theta ^{(ee)}(x).$ Thus we also have $\\Theta (0)=0,\\quad \\Theta ^{(e)}(0)=0,\\quad \\Theta ^{(o)}(0)=0,\\quad \\Theta ^{(ee)}(0)=0,\\quad \\Theta ^{(oo)}(0)=0, $ $\\Theta ^{\\operatorname{Lie}\\prime }(x)\\le \\Delta \\Theta (x)+\\frac{1}{2}\\Theta ^{\\operatorname{Lie}}(x),$ $\\Theta ^{(e)\\prime }(x)\\le \\frac{1}{2\\pi }2\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\pi }+\\Theta ^{(o)}(x)\\right)(1+\\Theta ^{(e)}(x) )\\Delta \\Theta (x)+\\Theta ^{(e)}(x),$ $\\Theta ^{(o)\\prime }(x)\\le \\frac{1}{2\\pi }\\left(2\\Theta ^{(e)}(x) +\\Theta ^{(e)}(x)^2+\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\pi }+\\Theta ^{(o)}(x)\\right)^2 \\right)\\Delta \\Theta (x)+\\Theta ^{(o)}(x),$ $\\Theta ^{(ee)\\prime }(x)\\le \\frac{1}{2\\cdot 2\\pi }2\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\cdot 2\\pi }+\\Theta ^{(oo)}(x)\\right)(1+\\Theta ^{(ee)}(x) )\\Delta \\Theta (x)+\\Theta ^{(ee)}(x),$ $\\Theta ^{(oo)\\prime }(x)\\le \\frac{1}{2\\cdot 2\\pi }\\left(2\\Theta ^{(ee)}(x) +\\Theta ^{(ee)}(x)^2+\\left(\\frac{\\Theta ^{\\operatorname{Lie}}(x)}{2\\cdot 2\\pi }+\\Theta ^{(oo)}(x)\\right)^2 \\right)\\Delta \\Theta (x)+\\Theta ^{(oo)}(x),$ where $\\Delta \\Theta (x)\\equiv 1+2 \\Theta ^{(e)}(x)+2\\left(2^2\\sum _{N=2}^\\infty \\frac{1}{N^2}\\right) \\Theta ^{(ee)}(x).$ We can draw a formal IVP for a majoring series upon these inequalities, too.",
"It turns out that this system blows up at $x=2.293\\ldots $ which is not so good as in the case of the previous method.",
"Nevertheless, the system itself is polynomial, which offers some technical advantages.",
"Now, various methods can be combined and refined.",
"The interested reader is invited to make his own lower bound for the convergence radius of the Magnus expansion."
]
] | 1709.01791 |
[
[
"Chandra studies of the globular cluster 47 Tucanae: A deeper X-ray\n source catalogue, five new X-ray counterparts to millisecond radio pulsars,\n and new constraints to r-mode instability window"
],
[
"Abstract We combined Chandra ACIS observations of the globular cluster 47 Tucanae (hereafter, 47 Tuc) from 2000, 2002, and 2014-15 to create a deeper X-ray source list, and study some of the faint radio millisecond pulsars (MSPs) present in this cluster.",
"We have detected 370 X-ray sources within the half-mass radius (2$'$.79) of the cluster, 81 of which are newly identified, by including new data and using improved source detection techniques.",
"The majority of the newly identified sources are in the crowded core region, indicating cluster membership.",
"We associate five of the new X-ray sources with chromospherically active BY Dra or W UMa variables identified by Albrow et al.",
"(2001).",
"We present alternative positions derived from two methods, centroiding and image reconstruction, for faint, crowded sources.",
"We are able to extract X-ray spectra of the recently discovered MSPs 47 Tuc aa, 47 Tuc ab, the newly timed MSP 47 Tuc Z, and the newly resolved MSPs 47 Tuc S and 47 Tuc F. Generally, they are well fit by black body or neutron star atmosphere models, with temperatures, luminosities and emitting radii similar to those of other known MSPs in 47 Tuc, though 47 Tuc aa and 47 Tuc ab reach lower X-ray luminosities.",
"We limit X-ray emission from the full surface of the rapidly spinning (542 Hz) MSP 47 Tuc aa, and use this limit to put an upper bound for amplitude of r-mode oscillations in this pulsar as $\\alpha<2.5\\times 10^{-9}$ and constrain the shape of the r-mode instability window."
],
[
"Introduction",
"In the cores of globular clusters (GCs), the very high stellar densities result in an extremely high rate of dynamical interactions.",
"These produce a large number and variety of close binary systems, many of them unique to this environment, including low-mass X-ray binaries (LMXBs; [30]), radio millisecond pulsars (MSPs; [74]), X-ray active binaries (ABs; [8]; [36]; [53] [53]), and cataclysmic variables (CVs; [93] [93]), which can engage in mass transfer.",
"Such compact binaries in globular clusters have been effectively discovered by recent X-ray studies, especially using the Chandra X-Ray Observatory (reviews include [107] [107], and [66] [66]).",
"Detailed analyses of such sources have also made it possible to probe deeper into the formation and evolution of such remarkable objects [73], [72].",
"MSPs are thought to have been produced from LMXBs when a low-mass companion star spins up the neutron star (NS) to millisecond periods by transferring angular momentum [12], [89].",
"MSPs can produce X-ray radiation of both thermal origin–blackbody-like radiation from a portion of the NS surface around the magnetic poles, heated by a flow of relativistic particles in the pulsar magnetosphere [61]–and non-thermal origin, generally highly beamed and sharply pulsed emission attributed to the pulsar magnetosphere, typically described by a power law with a photon index $\\scriptstyle \\sim $ 1.1-1.2 [10], [112], or non-pulsed emission from a shock between the pulsar wind and a flow of matter from a nondegenerate companion, as seen in “redback” or “black widow” MSPs [106], [20], [50], [102].",
"Hydrogen atmosphere models have been quite competent at describing X-ray spectra and rotation-induced pulsations of the nearby MSPs that exhibit thermal radiation [114], [18], [17].",
"The high density of MSPs in GCs, and their well-known distances and reddening make them ideal targets to study the relation of thermal radiation from MSPs to other pulsar parameters [75].",
"Additionally, a more complete study of MSPs is possible from X-ray observations, as compared to other wavelengths, because X-rays from surface hot spots of these highly compact MSPs will be bent by gravity to allow observers to see $\\scriptstyle \\sim $ 75% of the neutron star surface [92], [11], [19], so that a significant number of MSPs whose radio beams do not intercept the Earth should still be detectable in X-rays.",
"X-ray studies of MSPs also allow interesting constraints on the thermal physics of their cores.",
"In addition to external return current heating, MSPs can be heated up by internal heating mechanisms, namely, superfluid vortex creep [2], rotochemical heating [98], rotation-induced deep crustal heating [56], and heating produced by the dissipation of unstable oscillation modes (e.g., r-modes, [4], [97], [29], [105]).",
"Assuming that an MSP is in thermal balance (which is reasonable, because the thermal evolution timescale is much shorter than the spin down time-scale for MSPs), the total power of heating should be compensated by cooling, which depends on the MSP temperature.",
"Thus estimates of MSP temperatures (even the upper limits) can be used to constrain heating processes (see, e.g., [105], [29], [83]).",
"47 Tuc is a massive (M $\\scriptstyle \\sim $ 10$^6$ $M_{}$ , [96] [96]) globular cluster with a relatively high stellar concentration, although it is not core-collapsed [62].",
"It is considered to possess a significant population of binaries whose properties have been altered by close encounters with other stars or binaries, by virtue of being one of the clusters with the highest predicted close encounter frequencies [109], [93].",
"We use a cluster absorbing column of 3.5$\\times $ 10$^{20}$ cm$-^2$ [62], a distance of 4.53 kpc [14], [60], and metal abundances compiled in [67].",
"Measuring the various binary populations of clusters like 47 Tuc is crucial for understanding and modelling the dynamical encounters between binaries that produce X-ray sources in globular clusters (e.g.",
"[108]).",
"Observations with the ACIS instrument of the Chandra X-Ray Observatory in 2000 and 2002 have resulted in the identification of 300 X-ray sources within the half-mass radius of 47 Tuc [53], [68].",
"47 Tuc is also a subject of intense scrutiny due to the large number of MSPs residing within the cluster.",
"There have been 25 MSPs discovered so far using the 64-m Parkes Radio Telescope [86], [85], [103], [26], [88], [44], with radio timing positions known for 23 of them [46], [45], [88], [99], [44].",
"Identification of optical counterparts to X-ray sources in 47 Tuc has relied on the exquisite angular resolution of the Hubble Space Telescope (HST).",
"Three X-ray sources with ROSAT positions were identified with HST counterparts [91], [90], [104], but large numbers of identifications became possible with the subarcsecond angular resolution of Chandra.",
"The deep HST program GO-8267, searching for photometric variability [51], enabled [1] to assemble a large catalogue of binaries exhibiting variability, largely dominated by BY Dra variables, with admixtures of short-period eclipsing variables, W UMa contact binaries, “red stragglers”, and other variables.",
"BY Dra variables are pairs of main-sequence stars that have enhanced chromospheric variability compared to other stars of the same age, due to their more rapid rotation, produced by tidal locking; these stars also tend to be X-ray sources (e.g.",
"[35]).",
"The combination of Chandra positions and HST data have enabled the detection of scores of optical/ultraviolet counterparts to X-ray sources, including 42 cataclysmic variables [53], [37], [38], [100], 61 chromospherically active binaries [53], [37], [38], [68], [76], 6 companions to radio millisecond pulsars [41], [39], [37], [101], [25], a quiescent neutron star low-mass X-ray binary [40], and a candidate black hole binary [7].",
"Identifications of optical counterparts to X-ray sources typically rely on variability and/or unusual colors.",
"Cataclysmic variables are generally bluer than the main sequence (especially in the ultraviolet), with strong H-$\\alpha $ emission and variability; two of these three properties suffices to clearly identify a counterpart.",
"Chromospherically active binaries lie up to 0.75 magnitudes above the main sequence (due to combining light from two stars), and generally show weak H-$\\alpha $ emission and variability.",
"Millisecond pulsars and quiescent low-mass X-ray binaries show faint blue, variable counterparts, sometimes with H-$\\alpha $ emission, and typically require extra information from X-ray spectra or detection of radio pulsations to distinguish from cataclysmic variables.",
"These methods have been used to identify numerous optical/UV counterparts in 47 Tuc (references above) and in numerous other clusters [54], [94], [77], [9], [81], [31], [32].",
"We combined the 2014-15 Chandra ACIS observations [14] with those made in 2000 and 2002, to obtain a deeper image of 47 Tuc with improved angular resolution.",
"In this paper, we describe the X-ray analysis we used to create a larger source catalogue with accurate source positions, and focus on the X-ray properties of the MSPs 47 Tuc F, S, Z, aa and ab, whose X-ray counterparts we have identified in this work.",
"We have also identified new X-ray counterparts to five chromospherically active binaries previously identified by A01.",
"We have left searches of our X-ray error circles for additional optical/UV counterparts to future works.",
"Finally, we used X-ray spectral fitting of the fast-spinning and X-ray dim MSP 47 Tuc aa to place tight constraints on r-mode heating processes."
],
[
"Observations and data reduction",
"We used data from the 2000, 2002, and 2014-15 Chandra ACIS observations of the globular cluster 47 TucWe omitted the 2005-6 Chandra observations of 47 Tuc taken with the HRC-S camera, since these have significantly higher background..",
"While the five 2000 observations (described in [53]) were carried out with the ACIS-I CCD array at telescope focus, the eight 2002 observations (described in H05) and the six 2014-15 observations (described in [14]) were acquired with the ACIS-S CCD array.",
"All the 2014-15 observations, as well as short observations in 2000 and 2002, were taken using a subarray, which reduced the frame time, and thus reduced “pile-up”–the (incorrect) recording of two photons which landed on nearby pixels during one frame as a single event.",
"Subarray observations, however, only read out a portion of the CCD, encompassing the core of 47 Tuc.",
"All the observations are summarized in Table REF .",
"The total Chandra exposure time of 47 Tuc is 540 ks.",
"Table: Summary of Chandra observationsFigure: Combined 0.5-6 keV image of all Chandra ACIS observations of 47 Tuc, binned to half a pixel.",
"The centres of the circles represent the positions, while the labels indicate the W numbers of the X-ray sources.",
"The sources shown in red were already identified by H05 while those in blue have been identified in this work.",
"The sources within the inset box are shown in Fig.",
".Figure: Same as that of Fig.",
"but within the inset box.The data were reduced using CIAO version 4.8http://cxc.cfa.harvard.edu/ciao/ and CALDB version 4.7.1, in accordance with standard CIAO science threadshttp://cxc.harvard.edu/ciao/threads/index.html.",
"All the observations were reprocessed from the original level 2 event files following the default ACIS reprocessing steps.",
"The reprocessing applies the sub-pixel event-repositioning algorithm EDSERhttp://cxc.harvard.edu/ciao/why/acissubpix.html [79], which improves on-axis image resolution, thereby improving the resolution for the crowded centre of 47 Tuc.",
"The intrinsic on-axis PSF of the Chandra mirrors for a typical spectrum is reasonably represented by a Gaussian with FWHM of $0.4^{\\prime \\prime }$ (Chandra POGhttp://cxc.cfa.harvard.edu/proposer/POG/html/, chapter 7).",
"The ACIS detector pixels are $0.492^{\\prime \\prime }$ in size, so oversampling them is clearly beneficial.",
"The effects of using the EDSER algorithm on the Chandra PSF are yet to be calibrated.",
"We limited the energy range to 0.5-6 keV and covered the cluster out to 2.79 arcminutes, the half-mass radius used by Heinke et al.",
"2005, in each observation.",
"(We use this limiting radius for compatibility with Heinke et al.",
"2005 and other Chandra studies of globular clusters, motivated by the dominance of non-cluster sources outside that radius.)",
"We matched the astrometry of all the observations to ObsID 2735.",
"Pile-up at the level of 10-15 % (the maximum seen for any source in 47 Tuc) has a very small effect on astrometric corrections, as proven by the lack of detectable differences in alignment with their optical counterparts between the brightest X-ray sources and fainter X-ray sources in 47 Tuc [37].",
"We merged all the observations to obtain a deeper event file and subsequently produced images binned to a quarter of an arcsecond (half a pixel), for source detection.",
"We also created exposure maps and aspect histograms for all the CCDs, for each observation, to use with the ACIS-EXTRACT (AE) packagehttp://www2.astro.psu.edu/xray/docs/TARA/ae_users_guide.html, version 2016feb1, as discussed later."
],
[
"Source Detection",
"We employed two source detection algorithms, CIAO's wavdetecthttp://cxc.harvard.edu/ciao/threads/wavdetect/ algorithm [43], and the independent pwdetecthttp://www.astropa.unipa.it/progetti_ricerca/PWDetect/ algorithm [33].",
"The wavdetect tool employs a Mexican-Hat wavelet-based source detection algorithm that detects probable sources within a dataset using significant correlations of source pixels with wavelets of different scales.",
"For this, we created images binned to half a pixel in the 0.5-6 keV, 0.5-2 keV and 2-6 keV energy bands, using scales of 1.414, 2.0, 2.828, 4, 5.656 and 8.0 pixels, with a source detection significance threshold of 10$^{-6}$ , which should result in one false detection per ACIS chip.",
"We chose to use larger source detection scales in order to permit the detection of point sources further away from the aimpoint but within the half-mass radius.",
"The half-pixel binning of the images ensured that even in the crowded centre of 47 Tuc, wavdetect could separate close sources in spite of the larger source detection scales.",
"pwdetect is also a wavelet-based source detection algorithm which performs a multiscale analysis of the data, thus allowing the detection of both pointlike and moderately extended sources in the entire field of view.",
"Compared to wavdetect, it is more effective in the detection of faint sources close to brighter ones (as seen by [69] and [42]).",
"To use pwdetect, we created images binned to one pixel in the aforementioned energy bands, using scales from 0.5$^{\\prime \\prime }$ to 1$^{\\prime \\prime }$ , and a final detection threshold of 5.1$\\sigma $ , which should also result in one false detection per ACIS chip.",
"We also created a merged event file consisting exclusively of split-pixel events, as this is known [115] to improve Chandra's angular resolution at the cost of $\\scriptstyle \\sim $ 25% of the total counts.",
"Although, in our case, we lost about $\\scriptstyle \\sim $ 33% of the total counts.",
"We found an improved performance for wavdetect as compared to previous reports (e.g.",
"H05) in separating sources in the crowded cluster centre, which we attribute to the EDSER algorithm and use of the half-pixel binned image.",
"We also corroborated the performance of pwdetect in detecting faint sources close to bright ones.",
"Despite the 12 detection runs, an additional 7 possible sources near the crowded cluster centre can still be clearly identified by eye.",
"We created a combined source list with all the sources identified from the detection runs and by eye.",
"Subsequent comparison with the source list from H05 leads us to identify 20 additional sources.",
"We add these to our source catalogue as well, and refine our combined source catalogue further using the AE package, detailed by [23].",
"Spectra and background were extracted for each source in the catalog using the AE package (explained in detail in §REF ).",
"Subsequently, the positions of these sources were refined by calculating the centroid of the data within a preliminary extraction region.",
"When the probability of the extracted counts being produced by fluctuations in the background (PROB_NO_SOURCE) was above a threshold value of $\\scriptstyle \\sim $ 10% [110], the source was removed from the catalogue, followed by a refinement of the positions.",
"This process was repeated till no further pruning of the catalogue was necessary.",
"Table: 47 Tuc basic X-ray source properties, combined data-setThis pruning left us with 370 X-ray sources detected within the cluster half-mass radius, as catalogued in Table REF (a portion of this table is shown here for guidance, the full table is available in the electronic edition of the Journal).",
"The source positions are centroid positions which were further corrected by matching the astrometry with that of the previously known MSPs in 47 Tuc (detailed in §REF ).",
"For the positional error of each source, we quote the radius of the 95% error circle (P$_{\\rm err}$ ), calculated using the empirical formula derived by [71] from applying wavdetect on simulated data.",
"Here the source positions were ordered by decreasing luminosity in the 0.5-6 keV band, and have been labelled accordingly.",
"We retained the W numbering scheme of [53], as extended in H05 to cover all the sources previously identified from the 2000 and 2002 observations, extending it further to cover the additional ones identified in this work.",
"The W numbers for 5 sources which have been resolved into multiple sources, and 6 sources which were previously identified but were not detected in our analysis, have been omitted.",
"Sources with PROB_NO_SOURCE greater than $\\scriptstyle \\sim $ 1.5% but less than $\\scriptstyle \\sim $ 10% have been retained in our final catalogue, but since their detection is marginal, they have been marked as c in Table REF .",
"The sources identified by eye have been marked as m. Fig.",
"REF identifies all the sources found within the cluster half-mass radius, while Fig.",
"REF identifies those within the crowded core, both plotted on an image produced by combining data from the 19 ACIS observations binned to half a pixel in the 0.5-6 keV energy band.",
"The sources shown in red were already identified by H05 while those in blue, numbering 81, have been newly identified in this work.",
"The newly identified sources are either fainter than the previously identified ones, or are located close to bright sources.",
"Fig.",
"REF shows a merged exposure map (in units of cm$^2$ s, encoding the telescope effective area and the amount of time each location was imaged) covering the half-mass radius, illustrating that we obtain the highest sensitivity in the core.",
"Fig.",
"REF shows a representative true-colour image of the combined data obtained from the 19 merged observations made from the 0.2-1.5, 1.5-2.5, and 2.5-8 keV data.",
"Figure: Combined exposure map from the 19 merged observations of 47 Tuc in units of cm 2 ^2s.",
"The core (0.36') and half-mass (2.79', as used by H05) radii are marked for reference.",
"Features in the exposure map include reduced exposure due to a chip gap at the east edge of the half-mass radius from the 2002 observations, chip gaps crossing the interior from the 2000 observations, and regions of increased exposure due to the positions of the 2014-2015 subarray observations (all of which covered the cluster core).Figure: True colour image for the 19 merged observations of 47 Tuc within the half-mass radius.",
"It was constructed from a 0.2-1.5 keV image (red), a 1.5-2.5 keV image (green), and a 2.5-8 keV image (blue).",
"All images were binned to 0.25 pixels, smoothed using ds9 with a gaussian kernel of radius 3 pixels, and then combined."
],
[
"Extraction and photometry",
"From the positions defined in our initial source catalogue, source and background spectra were extracted.",
"Events were selected from each observation, for each source, from within a region that encompassed 90 percent of the Point Spread Function (PSF) centred on each catalogue position, or a region of reduced size if the sources were too crowded.",
"The typical on-axis extraction radius was $\\sim 0.9^{\\prime \\prime }$ .",
"From these events, source and background spectra were extracted by the ae_standard_extraction tool, which also generated effective area files (ARFs) and response matrices (RMFs) by calling the appropriate CIAO tools.",
"Background extractions included at least 50 counts, involving masks designed to accurately assess the local background due to neighbouring point sources as well as the instrumental background, and sampled pixels from areas outside all source extraction regions [23].",
"After extracting photons, AE runs the CHECK_POSITIONS stage, which can produce estimates of the source position by several methods, including performing image reconstruction on a crowded field with a maximum likelihood method [23].",
"We obtained these image reconstruction positions for faint sources that suffered significant crowding in the core region; these objects have been listed in Table REF .",
"The average shift between these positions and the centroid positions quoted in Table REF for the corresponding sources are 0.34$^{\\prime \\prime }$ in RA and 0.13$^{\\prime \\prime }$ in DEC.",
"These extractions were repeated until the positions of sources were well-determined and no further pruning was needed.",
"Background-subtracted photometry was calculated in several bands.",
"We also determined the number of counts for each catalogue source in the 0.5-2 and 2-6 keV bands, and also photon fluxes (quoted as “FLUX2” in AE; that is, net counts divided by the mean effective area in the band, and by the exposure time) in the 0.5-1, 1-2, 2-4 and 4-6 keV bands.",
"Table: Alternative positions of crowded sources, obtained from image reconstruction.We calculated the total luminosity in the 0.5-6 keV band, using XSPEC version 12.9http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/.",
"We calculated the conversion from observed photon fluxes to (unabsorbed) energy fluxes using the XSPEC VMEKAL model, with a typical temperature of 2 keV, and accounting for Galactic absorption with the TBABS model [111], to the combined spectrum.",
"We chose the VMEKAL model since we expect these faint sources to be dominated by chromospherically active binaries and CVs, both of which have X-ray spectra well represented by MEKAL models (e.g.",
"H05).",
"Using the known cluster absorbing column, distance and metal abundances, we computed a photon cm$^{-2}$ s$^{-1}$ to erg cm$^{-2}$ s$^{-1}$ conversion for the different flux bands, and subsequently used these conversions to calculate luminosity in the 0.5-6 keV band, for all the sources (as shown in Table REF )."
],
[
"Astrometric corrections",
"The final refined positions of our sources were further corrected to the known radio-timing positions of 19 out of the 23 MSPs with well-constrained positions in 47 Tuc.",
"The known positions for 16 of these MSPs were obtained from [45] while that for the MSP 47 Tuc ab was obtained from [88], for 47 Tuc W from [99], and for 47 Tuc Y from [44].",
"The astrometric corrections were carried out using CIAO with a matching radius of 1$^{\\prime \\prime }$ .",
"Disregarding the counterparts of 47 Tuc S and 47 Tuc F for reasons explained in §, and 47 Tuc L, whose counterpart is affected by being too close to a bright source, we have constrained the positions of our sources to a standard deviation of $\\scriptstyle \\sim $ 0.16$^{\\prime \\prime }$ in RA and $\\scriptstyle \\sim $ 0.08$^{\\prime \\prime }$ in DEC, of the other MSPs.",
"The deviations for individual MSPs are listed in Table REF .",
"This table also includes the positions for MSPs 47 Tuc R and Z, whose positions have been reported in [44], and 47 Tuc aa, whose position will be reported in Freire & Ridolfi (2017, in preparation).",
"Only three of the 22 X-ray counterparts lie farther than the [71] 95% error circle from the radio position, suggesting that most of the X-ray counterparts are correct counterparts.",
"Two of the three counterparts which are farther than expected from their radio positions (R and L) lie in the wings (1.5” and 2.6” distance, respectively) of brighter sources, which is likely to affect their estimated X-ray positions.",
"We shifted the radio positions of all the MSPs by $5^{\\prime \\prime }$ in four directions, and searched for X-ray sources within $0.5\"$ of these shifted positions, thus estimating the probability of chance superposition with unrelated sources.",
"This false match probability per source per trial was found to be $0.0375$ , indicating that of order one MSP is expected to coincide with another X-ray source within 47 Tuc.",
"This is consistent with the calculation in H05 of a 3.5% chance for any given source above 20 counts to fall within 0.5$\"$ of another such source.",
"Any MSPs where this situation occurs would be detected as the sum of the X-ray emission of the MSP and the other source.",
"H05 speculates that this may explain the detection of X-ray variability from 47 Tuc O, as the X-ray emission from 47 Tuc O is not expected to be variable on these timescales, while it would not be surprising, for instance, from a chromospherically active binary.",
"Table: Deviations in X-ray positions of individual MSPs from their known radio positions."
],
[
"Optical Counterparts",
"We identified candidate optical counterparts from the lists of variable stars given by A01 for 5 of the newly identified X-ray sources (Table REF ).",
"Astrometric corrections to the HST positions of the A01 binaries (0.269$^{\\prime \\prime }$ in RA and 0.085$^{\\prime \\prime }$ in DEC) were made by aligning HST counterparts to the five brightest X-ray sources with identifications by [37] to their X-ray positions derived in this work.",
"These X-ray sources were found to have a rms offset of $\\scriptstyle \\sim $ 0.025$^{\\prime \\prime }$ in RA and $\\scriptstyle \\sim $ 0.035$^{\\prime \\prime }$ in DEC with their corresponding astrometrically corrected HST positions.",
"We identified variable stars from A01 as candidate optical counterparts to our newly identified X-ray sources within thrice the rms offsets in RA and DEC from the X-ray positions (0.13; Table REF ).",
"All the candidate optical counterparts identified in this work, except WF2-V19, were suggested by H05 to be faint and/or confused X-ray sources, which we were able to identify with our deeper X-ray source list.",
"Table: New Optical Counterpart IdentificationsFigure: Distribution of luminosity vs. radial distance from the centre of 47 Tuc, plotted in units of arcmin 2 ^2, for sources identified by this work.",
"Outside θ 2 =2\\theta ^2=2arcmin 2 ^2, the flat distribution suggests a uniform spatial distribution."
],
[
"Radial Distribution",
"In Fig.",
"REF , we plotted the radial distance from the centre of 47 Tuc, $\\alpha $ = $00^{h}24^{m}05^{s}.29$ , $\\delta $ = $-72^{\\circ }04^{\\prime }52^{\\prime \\prime }.3$ [34], in units of arcmin$^2$ , for the 81 sources newly identified by this work.",
"The sources in the core are most likely cluster members, while sources near the outskirts are probably dominated by background extragalactic sources.",
"The distribution of expected background extragalactic objects should be homogeneous over the half-mass radius of 47 Tuc, so following H05, it appears that the sources in the central 2 arcmin$^2$ are dominated by cluster members, while those in the remainder of the field are not primarily cluster members.",
"As the remainder of the field contains 26 new sources, we estimated (assuming that new non-cluster members in the central region are added at the same rate) that of order 35 new detections are not cluster members, while roughly 46 new cluster members have been added.",
"Figure: The radio-identified spatial positions of the MSPs studied in this work are shown in green.",
"Their identified X-ray counterparts are indicated in magenta.",
"The background image is a combined 0.5-6 keV image of all Chandra ACIS observations of 47 Tuc, binned to half a pixel.Table: X-ray spectral properties of MSPs.Figure: NSATMOS spectral fits to the X-ray counterparts of MSPs 47 Tuc aa and 47 Tuc Z.",
"The spectra were modelled with an absorbed neutron star hydrogen atmosphere (top panels are data and model, lower panels are residuals).",
"The data and spectral fit are shown in black, red and green for the 2000, 2002 and 2014-15 observations respectively, all of which were fit simultaneously."
],
[
"Spectral Analysis",
"In this paper, we concentrated only on spectral analysis of the newly identified X-ray counterparts to MSPs.",
"Spectra were obtained for the X-ray counterparts of MSPs 47 Tuc F, S, Z, aa and ab separately for the 2000, 2002 and 2014-15 observations using AE (2014-15 observations were not used to fit the X-ray spectra of 47 Tuc aa).",
"Their positions are shown in Fig.",
"REF .",
"For the MSPs 47 Tuc F, Z, and aa, we used the C-statistic, to perform spectral fitting with few photons [27], with bins of 5 counts.",
"On the other hand, the spectra of 47 Tuc S and ab (where we could bin to 10 counts per bin) were fit using the $\\chi ^2$ statistic.",
"Each were first fit to an XSPEC power law, PEGPWRLW, then to the XSPEC blackbody model, BBODYRAD, and finally to a NS hydrogen atmosphere model [67], keeping the hydrogen column density frozen to the cluster value in each case [111], as it could not be reasonably constrained by spectral fits.",
"The steep photon indices, over 2.2 (3.02$^{+0.48}_{-0.46}$ , 3.09$^{+0.44}_{-0.39}$ , 2.89$^{+0.28}_{-0.27}$ , 2.29$^{+0.59}_{-0.55}$ and 2.92$^{+0.43}_{-0.38}$ for 47 Tuc F, S, Z, aa and ab respectively), obtained for all five MSPs with power law model fits implied that the spectra are too steep to be produced by typical pulsar magnetospheres, and are likely dominated by thermal emission from the NS surface.",
"For the NSATMOS model, the NS mass and radius were fixed to 1.4 $M_{}$ and 11 km, respectively, and the distance to 4.53 kpc, while the normalization was left free (physically interpreted as a variable portion of the surface radiating), as used by e.g.",
"[19].",
"The results of both model fits for the temperature, radius, and luminosity (0.5-6 keV) are given in Table REF .",
"Our fitted temperatures are generally consistent with those found for 47 Tuc MSPs by [19], who found average blackbody/NSATMOS temperatures of 0.18/0.10 keV (range 0.13–0.24/0.07–0.16 keV).",
"Our fitted radii are also generally consistent ([19] found average blackbody/NSATMOS radii of 0.17/0.81 km, ranges of 0.08-0.29/0.28-1.75 km), though 47 Tuc aa has the smallest inferred radius, and the smallest inferred luminosity, of 47 Tuc MSPs.",
"NSATMOS fits are shown in Figs.",
"REF and REF .",
"We used the XSPEC command `goodness 1000', which generates 1000 Monte Carlo simulations of the chosen model to see what fraction have a lower fitting statistic than the actual data, to test whether a model is a good fit [42]."
],
[
"MSP 47 Tuc aa",
"47 Tuc aa was discovered by [88], and its timing solution will be presented in Freire & Ridolfi (2017, in preparation).",
"We identified W96 as its X-ray counterpart.",
"The possibility of W96 being an MSP was previously suggested by [38], based on the lack of an optical counterpart, and H05 based on its X-ray properties.",
"It is one of the fainter detected sources, with only 33.5 counts.",
"We did not use the 2014-15 observations to fit the X-ray spectra of 47 Tuc aa, since they lay on the edge of the ACIS subarray.",
"The fit is relatively poor (the fraction of simulated spectra with a worse C statistic than the data is only 5.7%), apparently due to excess emission at high energies (see Fig.",
"6), but adding a power-law component (with spectral index between 1 and 2) does not improve the fit.",
"The fitted BBODYRAD temperature of 0.20$^{+0.06}_{-0.02}$ keV is consistent with other MSPs in 47 Tuc [19], but the inferred effective radius of 0.07$^{+0.06}_{-0.07}$ or 0.20$^{+0.53}_{-0.20}$ km (for blackbody or NSATMOS fits respectively) is the smallest in 47 Tuc.",
"Between $0.5-6$ keV, it has a luminosity of 0.7$^{+0.3}_{-0.2} \\times 10^{30}$ ergs s$^{-1}$ (for a blackbody fit, or 0.9$^{+0.7}_{-0.8}\\times 10^{30}$ erg s$^{-1}$ for the NSATMOS model), making it also the faintest MSP yet identified in 47 Tuc.",
"Figure: Same as Fig.",
"but for 47 Tuc ab, 47 Tuc S and 47 Tuc F.We also performed fits to 47 Tuc aa assuming emission from the entire surface, in order to calculate constraints on r-mode heating of the interior.",
"In these fits, we used the model TBABS*(NSATMOS+NSATMOS), where one of the NSATMOS models used a fixed normalization of 1 (corresponding to emission from the entire surface) while the normalization of the other was allowed free (corresponding to emission from the poles).",
"For these fits only, we explored several neutron star radii; 11, 12, and 12.5 km.",
"Larger stars have weaker constraints on the total luminosity, so we focus on the 12.5 km radius case, selected as roughly the top end of the range of the most plausible neutron star radius estimates (see e.g.",
"[78]).",
"The 90% confidence limit on the (unredshifted) surface temperature is $2.80\\times 10^5$ K, $2.68\\times 10^5$ K, and $2.62\\times 10^5$ K for 11, 12, or 12.5 km respectively, leading to (bolometric, calculated as 0.01-10 keV) unredshifted luminosity limits of $3.27\\times 10^{30}$ erg/s, $3.43\\times 10^{30}$ erg/s, or $3.50\\times 10^{30}$ erg/s, respectively."
],
[
"MSP 47 Tuc ab",
"47 Tuc ab was recently discovered by [88] who published its timing solution, this was recently updated by [44].",
"We identify W265 as its X-ray counterpart, although, as noted by H05, W265 appears to consist of multiple confused sources.",
"W265 is well-fit by blackbody or NSATMOS models with inferred temperature and radius consistent with other MSPs in 47 Tuc, suggesting that the observed X-ray emission at this location is indeed mostly due to 47 Tuc ab.",
"Its $0.5-6$ keV (blackbody fit) luminosity of 1.6$^{+0.3}_{-0.4} \\times 10^{30}$ ergs s$^{-1}$ makes it the second-most X-ray faint MSP identified in 47 Tuc, after 47 Tuc aa."
],
[
"MSP 47 Tuc Z",
"[44] have found a timing solution for MSP 47 Tuc Z.",
"We identify W28 as its X-ray counterpart.",
"[38] and H05 both previously suggested that W28 was a possible MSP.",
"The spectral fit to blackbody or NSATMOS models is good, and the inferred parameters are typical of MSPs in 47 Tuc."
],
[
"MSP 47 Tuc S",
"X-ray emission from MSPs 47 Tuc S and 47 Tuc F was previously studied together, as they had not been resolved [19].",
"With the increased resolution in this work, however, we were able to analyse them separately, identifying W352 as the X-ray counterpart of MSP S. The positional discrepancy, 0.25$^{\\prime \\prime }$ in RA and 0.03$^{\\prime \\prime }$ in DEC from its known radio position, is slightly larger than typical, which is likely due to crowding.",
"The spectral fits to blackbody or NSATMOS models are good, and the inferred parameters are typical of MSPs in 47 Tuc."
],
[
"MSP 47 Tuc F",
"We identify W353 as the X-ray counterpart to 47 Tuc F, at 0.27$^{\\prime \\prime }$ in RA and 0.15$^{\\prime \\prime }$ in DEC from its known radio position.",
"As for 47 Tuc S, the positional discrepancy is larger than average, which we attribute to crowding in this region.",
"The spectral fits to blackbody or NSATMOS models are acceptable, and the inferred parameters are typical of 47 Tuc MSPs.",
"We note that the temperatures for 47 Tuc F and 47 Tuc S individually are consistent with the average temperature for the merged source quoted by [19]."
],
[
"Source catalog",
"By merging all observations taken with the ACIS instrument of the Chandra X-ray Observatory, using the EDSER algorithm, and overbinning the final image, we were able to obtain significantly higher resolution.",
"This, in conjunction with detection algorithms like wavdetect and pwdetect, and algorithms and tools from AE, allowed us to identify 81 new sources in 47 Tuc, over half of which are present in and near the crowded core region.",
"We thus find a total of 370 sources within the half-mass radius of 47 Tuc.",
"While the newly detected sources in the outskirts of 47 Tuc are probably dominated by background extragalactic sources (considering their relatively uniform spatial distribution), those near the crowded core are likely to be a mixture of ABs and CVs in the cluster, almost certainly dominated by ABs (see the luminosity functions in Fig.",
"13 of H05).",
"There may also be a few MSPs and quiescent LMXBs, although new MSPs are likely to be among sources that are newly resolved (e.g.",
"47 Tuc F and S), rather than sources at the detection limit, based on the known X-ray luminosity function of globular cluster MSPs (see H05).",
"MSPs generally have luminosities (0.5-2.5 keV) in the range $10^{30}-10^{31}$ erg s$^{-1}$ , while quiescent LMXBs range from $\\sim 10^{31}-10^{33}$ erg s$^{-1}$ .",
"H05 was already complete for uncrowded sources with luminosities above $8\\times 10^{29}$ ergs s$^{-1}$ and this work only pushes the detection limit marginally.",
"So the increased sources we detect are largely due to improvement in the angular resolution and detection efficiency.",
"Figure: Comparison of luminosities in the 0.5-6 keV band derived by this work with that obtained by H05.",
"The top panel shows all the sources identified in both works while the bottom panel shows those sources with luminosities less than 100×10 30 \\times 10^{30} ergs s -1 ^{-1}.The luminosities for these sources, computed from the photometry as explained in §REF , do not always match those obtained by H05 as illustrated in Fig.",
"REF .",
"Sources with higher counts in the 2-6 keV range, notably W42 (X9), seem to show an increased luminosity in this work, whereas sources with lesser counts in this hard band, notably W46 (X7), seem to show a decreased luminosity (though spectral fitting shows that W46/X7 has demonstrated that it is not actually variable; [14]).",
"A likely explanation is the reduced sensitivity of the ACIS-S detector in the lower energy bands, which combined with a single assumed spectral model can produce such apparent variations.",
"(For example, the expected 0.5-6 keV countrate for an on-axis source described by a power-law with photon index 1.8 and the cluster $N_H$ decreases by 28% from the 2002 to 2014 ACIS-S observations; for a 2 keV VMEKAL model, the decrease is 30%.)",
"It is to be noted that the individual luminosities of these sources must be obtained more accurately by fitting different spectral models suitable for different objects.",
"W42 (X9) has been comprehensively studied by [7] who confirm this source as an ultracompact X-ray binary with a C/O white dwarf donor and a possible black hole primary."
],
[
"MSP polar caps",
"From the spectral fits for the recently discovered MSPs 47 Tuc aa, 47 Tuc ab and 47 Tuc Z, and freshly resolved MSPs 47 Tuc S and 47 Tuc F, we obtained their temperatures and luminosities, which were found to be consistent with those of most other MSPs in 47 Tuc, analysed by [19].",
"In comparison to the BBODYRAD model, the NSATMOS hydrogen atmosphere model gives lower estimates of the temperatures (Table REF ), while the unabsorbed luminosity estimates from the two models are consistent.",
"Since the NS surface is almost certainly covered with a hydrogen atmosphere in these MSPs [114], [18], the (larger) NSATMOS model estimates of the emitting radii are more realistic.",
"As discussed by [19], these estimates of the emitting radius will vary from the true polar cap sizes, as high-quality spectra of the nearby MSP PSR J0437$-$ 4715 show that at least two, probably three, thermal components are required [113], [15], [55], likely from different parts of the polar caps.",
"Figure: Fitted (BBODYRAD) polar cap radius against spin period for MSPs in 47 Tuc (red), NGC 6397 (black) and NGC 6752 (blue).",
"The best fitting power law is indicated, with a best fitting slope -0.41±0.27-0.41 \\pm 0.27, consistent with the predicted index of -0.5-0.5.It has long been predicted that the size of the polar cap region of a radio pulsar, $R_{\\rm pc}=(2 \\pi R_{\\rm NS}/(cP))^{1/2}R_{\\rm NS}$ [82], where $R_{\\rm NS}$ is the radius of the NS having period P, depends inversely on the spin period.",
"It thus follows that MSPs in 47 Tuc, having shorter periods on average than those in NGC 6752, should have larger polar caps (given similar luminosities).",
"By fitting the effective radii measurements of MSPs in 47 Tuc, NGC 6752 and NGC 6397, with a power law in spin period, [42] found a best-fitting index of $-0.65 \\pm 0.40$ ($1\\sigma $ error bars), consistent with the predicted index of $-0.5$ .",
"We plotted the spin periods against the inferred MSP effective radii (from BBODYRAD) for all the MSPs in 47 Tuc, NGC 6752 and NGC 6397, adding six new pulsars to 47 Tuc (those studied here, and 47 Tuc X from [99]; Fig.",
"REF ).",
"Fitting the effective radii measurements with a power law in spin period, we found a best-fitting index of $-0.41 \\pm 0.27$ ($1\\sigma $ error bars).",
"This slightly increases the evidence for a correlation by reducing the $1\\sigma $ uncertainty, and is still consistent with the theoretically predicted index of $-0.5$ .",
"Dispersion into this correlation is expected from the (unknown) differences in geometries of the pulsars, and by variations in the strength of unmodelled non-thermal radiation."
],
[
"Upper limit for surface temperature of MSP 47 Tuc aa and constraints on r-modes",
"An intriguing consequence of the low X-ray luminosity of 47 Tuc aa is the constraint it allows us to impose upon internal heating in rapidly spinning neutron stars.",
"MSPs should be in thermal balance, because their thermal evolution timescale is much shorter than their spin down time-scale.",
"Thus the total heating power should be compensated by cooling, which depends on the MSP temperature.",
"The surface temperature of 47 Tuc aa is low enough (see §REF ) to exclude strong neutrino emission from the bulk of the star (see e.g.",
"[29] for more detailed discussion).",
"Thus, the total cooling power of this source can be estimated almost directly from observations: it equals the thermal emission from the entire surface, excluding hot spots (i.e.",
"from the component with a fixed normalization of 1 in §REF ), Here we use the least constraining limit on the bolometric luminosity from the entire surface of 47 Tuc aa from §REF , $L\\le 3.50\\times 10^{30}$ erg/s for an assumed radius of 12.5 km.",
"$L_\\mathrm {cool}\\approx L\\le 3.5\\times 10^{30}\\,\\mathrm {erg\\,s}^{-1}, $ providing the same constraint to the total heating power, which can be presented as a sum of internal heating mechanisms (namely, superfluid vortex creep [2], rotochemical heating [98], rotation-induced deep crustal heating [56]), and possible heating produced by the dissipation of unstable oscillation modes (e.g., r-modes, [4], [97], [29], [105]).",
"We start from the internal heating.",
"Its power $Q$ is generally proportional to the (intrinsic) spin down rate ([52], [56]) which can be bounded as $\\dot{\\nu }\\ge -5\\times 10^{-15}\\, \\mathrm {s}^{-2}$ on the basis of the accurate timing solution (Freire & Ridolfi 2017, in preparation) and accounting for the strongest possible negative acceleration in the cluster gravitational field ([44]).",
"The typical spin-down power of other MSPs with X-ray luminosity $\\sim 10^{30}$ erg s$^{-1}$ (see, e.g., Fig.",
"8 in [42]) suggests that real spin-down rate of 47Tuc aa is slower, but for estimates below we apply robust bound from the radio observations.",
"Following [29], to estimate the internal heating power we appeal to the rotation-induced deep crustal heating ([56]), which is associated with the same physics as deep crustal heating in accreting NSs ([24]).",
"Namely, compression of the accreted material (due to subsequent accretion for accreting NSs, or due to spin-down in case of MSPs) leads to nuclear reactions in the crust and production of heat.",
"This mechanism does not depend on the uncertain parameters of superfluid transition in the star and thus it is rather robust.",
"For the parameters of 47 Tuc aa it gives $Q&\\gtrsim & Q_\\mathrm {DCH}\\approx 1.8\\times 10^{30} \\frac{\\mathrm {erg}}{\\mathrm {s}}\\left(\\frac{R}{12.5\\,\\mathrm {km}}\\right)^7\\,\\left(\\frac{M}{1.4\\, M_\\odot }\\right)^{-2}\\,\\nonumber \\\\&\\times & \\frac{\\nu }{542\\,\\mathrm {Hz}}\\frac{\\left|\\dot{\\nu }\\right|}{5\\times 10^{-15}\\,\\mathrm {s}^{-2}}\\sum _i \\frac{P_i}{10^{31}\\, \\mathrm {erg\\,\\, cm}^{-3}}\\frac{ q_i}{\\mathrm {MeV}},$ where Eq.",
"(7) from [56] was applied (parameter $a$ taken to be $a=0.5$ ).",
"Here $\\left|\\dot{\\nu }\\right|=-\\dot{\\nu }$ is absolute value of the intrinsic spin-down rate; $i$ enumerates the different reactions in the crust; $P_i$ , and $q_i$ are, respectively, the threshold pressure and energy production for each reaction.",
"For all models of the accreted crust discussed by [59] $\\sum _i P_i q_i$ lies in the range $(0.9-1.5)\\times 10^{31} \\,\\mathrm {erg\\, MeV\\, cm}^{-3}$ , making $Q_\\mathrm {DCH}$ well constrained, and close to the observational upper bound (REF ).",
"Note, $Q_\\mathrm {DCH}$ is strongly increasing with increase of $R$ ($Q_\\mathrm {DCH}\\propto R^7$ ), thus for larger radius or if other internal heating mechanisms are competitive with $Q_\\mathrm {DCH}$ , the total heating power can exceed constraint (REF ) for $\\dot{\\nu }=-5\\times 10^{-15}\\, \\mathrm {s}^{-2}$ , suggesting thus that the intrinsic spin-down rate of MSP 47 Tuc aa should be lower.",
"However, we leave detailed analysis of such constraints beyond the scope of the paper because they are model dependent.",
"Figure: Examples of r-mode instability windows in the standard model [panel (a)] and theminimally constrained model [panel (b), see text for details].The stability region is shaded in grey;in the white region the r-mode is unstable.Temperatures and frequenciesof NSs observed in LMXBs are shown by filled circles,while error bars show uncertainties due to the unknown NS envelope composition.",
"The upper limit for the internal temperature of MSP 47 Tuc aa is also marked.Ticks on the left side of ν\\nu axis show measured MSP frequencies.The heating by unstable modes can take place in rapidly rotating neutron stars due to the Chandrasekhar-Friedman-Schutz (CFS; [28], [48], [49], [5], [47]) instability, driven by emission of gravitational waves.",
"In the absence of dissipation, the CFS instability takes place at an arbitrary rotation rate and results in the exponential growth of a certain class of oscillation modes.",
"The most unstable among them are r-modes (predominantly toroidal modes, which are similar to Rossby waves and controlled by the Coriolis force).",
"Dissipation suppresses the instability only up to a threshold spin frequency (which depends on the internal NS temperature; see Fig.",
"REF and e.g.",
"[63] for a recent review).",
"The standard model of r-mode instability (suggested by [80]; [87], see also [57] for discussion of recent microphysical updates) assumes a hadronic composition of the NS core and dissipation by shear and bulk viscosity.",
"It stabilizes the NS in the grey region in Fig.",
"REF (a), the unstable (white) region is referred to as the `instability window'.",
"As shown by [70], [65], the observations of transiently accreting neutron stars (their spin frequencies and internal temperatures are shown in Fig.",
"REF by dots with error bars associated with uncertainty in thermal insulating envelope composition; data are taken from [58], [29]) reveal the inadequacy of the standard model: additional dissipation of r-modes is required to stabilize many observed NSs [i.e., all NSs in the white region of REF (a)].",
"[29] analyze the formation of MSPs via the recycling scenario ([13]; [3], [12], [89]) and suggest `minimal' constraints to the instability windows, which allow them to explain the observations of transiently accreting neutron stars, and the formation of high-frequency MSPs, simultaneously.",
"The corresponding `minimally constrained' instability window is shown in Fig.",
"REF (b).",
"As indicated by [29], strong upper bounds on thermal emission from the entire surface of rapidly rotating MSPs can lead to even stronger constraints, and here we apply the upper limit (REF ) to this aim.",
"The first constraint comes from the thermal equilibrium in the current state of the MSP.",
"As discussed by [29], [105], [83], the upper limit for the MSP surface temperature gives an upper bound to the heating by r-modes, and thus to the r-mode amplitude.",
"Substituting the upper limit on the surface temperature from §REF into Eq.",
"(12) from [29], we get the constraint $\\alpha \\lesssim 2.5\\times 10^{-9}$ for the r-mode amplitude (defined as in [80]) in MSP 47 Tuc aa.",
"The numerical value corresponds to a radius $R=11$ km, which gives the least constraining bound (assuming $R=12$ km we come to $\\alpha \\lesssim 2\\times 10^{-9}$ ).",
"It is worthwhile to note, that Eq.",
"(REF ) is the strongest constraint available for MSPs ([105], [29], [83])[105] suggest a comparable constraint for PSR J1023+0038 ($\\nu =592.4$ Hz), but in this paper the surface temperature of PSR J1023+0038 was taken to be $T^\\infty _\\mathrm {eff}\\lesssim 3\\times 10^5$ K (see pulsar at $\\nu =592$ Hz in Figure 1 of that paper), without detailed spectral fitting.",
"However, such a strong upper limit is not justified for J1023+0038 (e.g., the joint fit of X-ray spectra by [16] suggests a higher surface temperature $T^\\infty _\\mathrm {eff}\\sim 4\\times 10^5$ K, leading thus to weaker constraints on the r-mode amplitude).",
"and accreting neutron stars ([84]).",
"As long as non-linear saturation of the r-mode instability predicted by state-of-art models ([6], [22], [21], [64]) takes place at much larger amplitudes, the most natural explanation of the bound (REF ) is that MSP 47 Tuc aa is stable (at least with respect to CFS instability of r-modes).",
"Indeed, the upper limit for the surface temperature of MSP 47 Tuc aa allows us to constrain the redshifted internal temperature as $T^\\infty \\le 10^7$ K even for the iron thermally insulating envelope model by [95] [a layer of light (accreted) elements with mass $\\Delta M>10^{-13}M_\\odot $ reduce this constraint to $T^\\infty \\le 5\\times 10^6$ K], which is almost enough to guarantee stability of 47 Tuc aa in the standard r-mode instability model [see Fig.",
"REF , panel (a)].",
"The second constraint, developed by [29], appeals not only to the current state of this MSP, but also to its formation.",
"Namely, it comes from the requirement that the r-mode instability does not prevent cooling of the MSP to the observed temperature, after the end of accretion (and spinup) during the LMXB stage.",
"[29] apply $T^\\infty \\lesssim 2\\times 10^7$ K as a fiducial upper limit for internal temperatures of MSPs and demonstrate that it leads to a `minimally' constrained instability window [see Fig.",
"REF (b)].",
"Since MSP 47 Tuc aa has a lower internal temperature, it puts a stronger constraint on the shape of the instability window.",
"For the minimally constrained instability window from [29], the evolution of the MSP progenitor during the LMXB stage is unaffected by the r-mode instability, and the evolutionary trajectory is shown schematically by the thick solid line in Fig.",
"REF (b).",
"After the end of accretion, the newly born MSP loses accretion-induced heating, cools down and evolves along the low temperature boundary of the stability peak, where r-modes force spin-down of the MSP and keep it heated (dotted [blue] line in the plot).",
"It is easy to see that this line cannot explain the temperature and spin of MSP 47 Tuc aa (namely, this path predicts its internal temperature to be $\\sim 2\\times 10^7$ K, corresponding to $T_\\mathrm {eff}^\\infty \\sim 6\\times 10^5$ K (an accreted thermal insulating envelope is assumed), which is well above the upper limit obtained in §REF ).",
"To allow MSP 47 Tuc aa to cool down enough to be in agreement with observations, the r-mode instability should be suppressed at $(1-2)\\times 10^7$ K at least for the frequency of this pulsar ($\\nu \\sim 542$ Hz, [88]).",
"Assuming that there are no isolated regions of r-mode instability, r-modes must be stable in the shaded region in Fig.",
"REF (b).",
"In this case, the evolution of MSPs can follow the dashed (red) line in this figure, which is in agreement with observations of MSP 47 Tuc aa."
],
[
"Conclusions",
"We combined 180 ks of new Chandra ACIS data on 47 Tuc with 370 ks of archival data, and used improved algorithms to generate a new source catalog, finding 81 new sources for a total of 370 within the half-mass region.",
"Roughly half of the new sources are likely associated with the cluster, and half are background AGN.",
"We resolved the X-ray emission from MSPs 47 Tuc F and 47 Tuc S, and use recent pulsar timing solutions to identify X-ray emission from the MSPs 47 Tuc aa, 47 Tuc ab, and 47 Tuc Z.",
"In general, their X-ray emission is consistent with that of other MSPs in 47 Tuc, though 47 Tuc aa is the X-ray faintest MSP yet measured in 47 Tuc.",
"Comparing the fitted blackbody radii of millisecond pulsar polar caps with their spin rates, we find modest evidence for the expected anticorrelation.",
"Finally, we use our upper limit on the temperature of the surface of the fast-spinning (542 Hz) MSP 47 Tuc aa to constrain the heating of this neutron star by r-modes.",
"We find a constraint on the amplitude of r-modes in 47 Tuc aa of $\\lesssim 2.5\\times 10^{-9}$ , the most constraining yet for MSPs.",
"We also use the temperature and rotation frequency of 47 Tuc aa to place a constraint on the shape of the r-mode instability window in neutron stars."
],
[
"Acknowledgements",
"S. Bhattacharya acknowledges support from MITACS for sponsoring his stay at the University of Alberta.",
"COH acknowledges support from an NSERC Discovery Grant, and an NSERC Discovery Accelerator Supplement.",
"AR and PCCF gratefully acknowledge financial support by the European Research Council for the ERC Starting grant BEACON under contract no.",
"279702.",
"AR is a member of the International Max Planck research school for Astronomy and Astrophysics at the Universities of Bonn and Cologne and acknowledges partial support through the Bonn-Cologne Graduate School of Physics and Astronomy.",
"This work was funded in part by NASA Chandra grant GO4-15029A awarded through Columbia University and issued by the Chandra X-ray Observatory Center (CXC), which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS803060.",
"This research has made use of the NASA Astrophysics Data System (ADS) and software provided by the CXC in the application package CIAO.",
"pwdetect has been developed by scientists at Osservatorio Astronomico di Palermo G. S. Vaiana thanks to Italian CNAA and MURST(COFIN) grants."
]
] | 1709.01807 |
[
[
"SPECTRE: Supporting Consumption Policies in Window-Based Parallel\n Complex Event Processing"
],
[
"Abstract Distributed Complex Event Processing (DCEP) is a paradigm to infer the occurrence of complex situations in the surrounding world from basic events like sensor readings.",
"In doing so, DCEP operators detect event patterns on their incoming event streams.",
"To yield high operator throughput, data parallelization frameworks divide the incoming event streams of an operator into overlapping windows that are processed in parallel by a number of operator instances.",
"In doing so, the basic assumption is that the different windows can be processed independently from each other.",
"However, consumption policies enforce that events can only be part of one pattern instance; then, they are consumed, i.e., removed from further pattern detection.",
"That implies that the constituent events of a pattern instance detected in one window are excluded from all other windows as well, which breaks the data parallelism between different windows.",
"In this paper, we tackle this problem by means of speculation: Based on the likelihood of an event's consumption in a window, subsequent windows may speculatively suppress that event.",
"We propose the SPECTRE framework for speculative processing of multiple dependent windows in parallel.",
"Our evaluations show an up to linear scalability of SPECTRE with the number of CPU cores."
],
[
"Introduction",
"Distributed Complex Event Processing (DCEP) [19], [29] is a paradigm applied in many different application areas like logistics, traffic monitoring, and algorithmic trading, to infer the occurrence of complex situations in the surrounding world from basic events like sensor readings or stock quotes.",
"Such situations can be, for instance, the delayed delivery of a packet, traffic jams or accidents and leading market signals.",
"In order to stepwise infer their occurrence from the sensor streams, a distributed network of interconnected DCEP operators, the operator graph, is deployed.",
"Each operator processes incoming event streams and detects a designated part of an event pattern that corresponds to a situation of interest.",
"If such a pattern is detected, a new (complex) event is produced and emitted to successor operators or to a consumer, i.e., an entity interested in the corresponding situation.",
"In doing so, operators face increasingly high event loads from their incoming event streams.",
"In order to be capable of processing high load, the parallelization of DCEP operators has been proposed.",
"In this regard, data parallelization has proven to be a powerful technique to parallelize operators [18], [5], [25], [27], [26], [15].",
"Data-parallel DCEP systems split the incoming event streams into independently processable windows that capture the temporal relations between single events posed by the queried event pattern.",
"The windows are processed in parallel by a number of identical operator instances.",
"An event can be part of different windows, so that windows may overlap.",
"A crucial question in overlapping windows is whether an event can be used in multiple pattern instances or not.",
"In many cases, it is preferable to consume an event once it is part of a pattern instance.",
"In particular, this means to not use the same event for the detection of further pattern instances in other windows.",
"This way, semantic ambiguities and inconsistencies in the complex events that are emitted can be resolved or prevented.",
"The problem tackled in this paper is that event consumptions impose dependencies between the different windows and thus, prevent their parallel processing.",
"When the same event is processed in parallel in two different windows, consuming it in the first window also consumes it from the second window; hence, there is a dependency between both windows, which can hinder their parallel processing.",
"Understanding that problem, it is no surprise that existing parallel implementations of DCEP systems [13], [5], [25] do not support event consumptions, whereas sequential systems often do [10], [1], [12].",
"This limits the scalability of operators that impose event consumptions.",
"Moreover, it even impedes event consumptions from their further development in academia and industry, as in times of Big Data and Internet of Things, parallel DCEP systems are becoming the gold standard.",
"In this paper, we propose a speculative processing method that allows for parallel processing of window-based DCEP operators in case of event consumptions.",
"The basic idea is to speculate in each window which events are consumed in the previous windows—instead of waiting until the previous windows are completely processed.",
"This way, multiple overlapping windows can be processed in parallel despite inter-window dependencies.",
"To this end, we propose the SPECTRE (SPECulaTive Runtime Environment) framework, comprising the following contributions: (1) A speculative processing concept that allows the execution of multiple versions of multiple windows using different event sets in parallel.",
"(2) A probabilistic model to process always those window versions that have the highest probability to be correct.",
"(3) Extensive evaluations that show the scalability with a growing number of CPU cores."
],
[
"Background and Problem Analysis",
"To solve the problem of parallel event processing in face of event consumptions, we first discuss a common DCEP model in Section REF .",
"In Section REF , we analyze existing DCEP operator parallelization methods and highlight the properties of window-based data parallelization as an expressive and scalable parallelization method [25], [27], [15].",
"Finally, in Section REF we explain the challenges on parallel processing imposed by event consumptions."
],
[
"DCEP Systems",
"A DCEP system is modeled as an operator graph which inter-connects event sources, operators and consumers by event streams.",
"An event $e$ consists of attribute-value pairs containing meta-data, such as event type, sequence numbers or timestamps, and the event payload, such as sensor readings, stock quotes, etc.",
"Based on the event meta-data, events from different streams arriving at an operator have a well-defined global ordering (e.g., by timestamps and tie-breaker rules).",
"Each operator $\\omega $ processes events in-order on its incoming streams, detecting event patterns according to a pattern specification.",
"If a pattern instance is detected, the operator emits a (complex) event to its successor in the operator graph.",
"Event patterns are specified in an event specification language such as Snoop [10], Amit [1], SASE [31], or Tesla [11].",
"Those languages involve operators like event sequences, conjunctions, and negations, in order to define the event patterns to be detected.",
"To express the set of relevant events in pattern detection, the pattern specification imposes a sliding window of valid events [3], [15].",
"This can depend on time or the number of events [31], [11], [12], but also on more complex predicates, e.g., on (combinations of) specific event occurrences that mark the beginning and end of a window [25].",
"In this paper, we denote the valid window at a specific point in time as $w_i$ .",
"When the window slides, the subsequent valid windows are denoted as $w_{i+1}$ , $w_{i+2}$ etc.",
"Depending on the sliding semantics, different subsequent windows can overlap, i.e., events are part of multiple different windows.",
"Example: In intra-day stock trading, an operator $\\omega $ receives an event stream containing live stock quote changes of stock $A$ and $B$ throughout the trading day.",
"An analyst wants to detect correlations between a change in $A$ and a change in $B$ .",
"To this end, he formulates a query in the Tesla [11] event specification language: $[Q_E]\\begin{aligned}\\ & \\mathtt {define\\ Influence(Factor)} \\\\\\ & \\mathtt {from\\ B()\\ and } \\\\\\ & \\mathtt {A()\\ within\\ 1min\\ from\\ B }\\\\\\ & \\mathtt {where\\ Factor = B:change\\ /\\ A:change }\\end{aligned}$ This pattern can be detected by opening a window with a scope of 1 minute whenever an A event occurs; when a B event is detected in a window opened by an A event, a complex event can be created.",
"Suppose the events $A_1$ , $A_2$ , $B_1$ , $B_2$ and $B_3$ occur in the event stream in that order, i.e., $A_i$ denotes the $i$ -th occurrence of an event of type $A$ in the stream (cf.",
"Figure REF ).",
"Let us assume that the first $A$ in a window is correlated with every $B$ in the same window—this can be defined in a so-called selection policy.",
"As shown in Figure REF , 5 complex events are detected: $_{Y}^{X}$ denotes a complex event created from incoming events $X$ and $Y$ .",
"$_{B_1}^{A_1}$ , $_{B_2}^{A_1}$ , $_{B_1}^{A_2}$ , $_{B_2}^{A_2}$ , and $_{B_3}^{A_2}$ .",
"Notice, that all events are correlated multiple times, i.e., they are not consumed after building a complex event.",
"Generally, such multiple correlations of the same event can be problematic.",
"If there is a many-to-one relation between incoming events and detected situations, i.e., many events build a pattern instance but a single event can only be part of one pattern instance, contradicting complex events are produced when events are not consumed.",
"Many-to-one or one-to-one relations are a common case in situation detections.",
"Therefore, many event specification languages allow for the specification of a consumption policy [10], [34], [1], [11].",
"The consumption policy defines which selected events are consumed after they have participated in a complex event detection: It might be none, all or some of them—e.g., depending on the event type or other parameters.",
"A detailed discussion on consumption policies supported in event specification languages is provided in Section .",
"In the example in Figure REF , selected events of type B are consumed when a complex event is detected, referred to as consumption policy “selected B”.",
"Now, only 3 complex events are produced: $_{B_1}^{A_1}$ , $_{B_2}^{A_1}$ , and $_{B_3}^{A_2}$ .",
"In that case, $B_1$ and $B_2$ are not re-used after being correlated with $A_1$ in the first window w$_1$.",
"When a complex event is detected, all constituent events of the event pattern are checked against the consumption policy.",
"Then, all events defined by the consumption policy are consumed as a whole.",
"This implies that events are not consumed while they only build a partial match, but only when the match is completed and a complex event is produced.",
"This inherent property is independent of the concrete selection and consumption policy.",
"Figure: Query Q E Q_E with different consumption policies (CP)."
],
[
"Operator Parallelization",
"The paradigm of data parallelization is very powerful in increasing operator throughput.",
"The incoming event stream is split and processed by an elastic number of identical copies of the operator—called operator instances.",
"This paradigm has been applied to a wide range of parallel CEP and stream processing systems [25], [6], [32], [17], [9], [16], [22], [27], [26], [15].",
"We assume a shared memory (multi-core) architecture, where the splitter and operator instances are executed by independent threads running on dedicated CPU cores.",
"We assume that the underlying system can provide $k+1$ threads, so that 1 thread is pinned to the splitter and $k$ threads are pinned to the operator instances.",
"In the rest of this paper, we do not differentiate between operator instances (i.e., instances of the pattern detection logic) and the threads that execute them—we simply refer to both as operator instances.",
"As mentioned above, we follow a window-based data parallelization approach.",
"The incoming event streams are partitioned into windows that capture (temporal) relations defined in the queried pattern.",
"They can naturally be processed by operator instances, as DCEP operators in their core typically work on a sliding window on the event stream [31], [11], [12], [21], [25], [27], [15].",
"The windows can be based on time, event count or logical predicates that evaluate whether arbitrary window start and end conditions are fulfilled—a more detailed analysis of window-based data parallelization is provided in [25] and in [15].",
"For instance, for the time-based window definition of example query $Q_E$ , a new window is opened on each event of type $A$ , whereas an open window is closed after 1 minute based on the events' timestamps.",
"The windows are assigned with increasing window IDs and their boundaries are stored in the shared memory (e.g., “w$_{i}$ from event $X$ to event $Y$ ”).",
"The splitter periodically schedules to each operator instance a specific window for processing.",
"The operator instances can hold local state of the processing in shared memory, e.g., partial pattern matches detected in the assigned window.",
"This allows a specific window to be processed by any operator instance at any time; in particular, the processing of a window can be interrupted for some time and resumed later by the same or a different operator instance."
],
[
"Challenges and Goal",
"In systems without consumptions, processing of a window cannot impact the events within another window, i.e.",
"in principle each pair of windows can be processed in parallel.",
"However, event consumptions impose a dependency between the windows, restricting parallelism, as we discuss in the following.",
"Recall the example in Figure REF .",
"The Selection Policy is “first A, each B” and the Consumption Policy is “selected B”.",
"In the first window w$_1$, $A_1$ and $B_1$ build a complex event $_{B_1}^{A_1}$ , such that $B_1$ is consumed; furthermore, $A_1$ and $B_2$ build a complex event $_{B_2}^{A_1}$ , such that $B_2$ is consumed.",
"If w$_1$ and w$_2$ are processed in parallel, the consumption of $B_1$ and $B_2$ in w$_1$ might not be known in w$_2$, so that $B_1$ and $B_2$ are erroneously processed in w$_2$, too, leading to inconsistent results.",
"To prevent anomalies due to concurrent processing, w$_2$ can only be processed after the consumptions in w$_1$ are known.",
"When the event patterns are more complex than in the given minimal working examples, the dependencies become hard to control.",
"For instance, if the pattern requires 3 rising stock quotes of $B$ in a sequence, the completion of the pattern in w$_1$—and hence, the event consumptions—might be unsure until w$_1$ is completely processed.",
"If 2 events of type $B$ with rising quotes have already been detected in w$_1$, the completion of the pattern depends on whether a third $B$ occurs; this might only be known at the end of w$_1$.",
"The standard procedure to deal with data dependencies is to wait with processing w$_2$ until w$_1$ is completely processed and hence, all consumptions in w$_1$ are known.",
"This, however, impedes the parallel processing of overlapping windows.",
"Figure: Data parallelization framework.In this paper, we aim to develop a framework to enable parallel processing of all DCEP operators, regardless of their selection and consumption policy.",
"To this end, we develop a speculative processing method that overcomes the data dependencies imposed by event consumptions, so that data-parallel processing becomes possible.",
"The framework shall deliver exactly those complex events that would be produced in sequential processing; in particular, no false-positive and false-negatives shall occur.",
"Figure: Consumption Problem: (a) Structural View.",
"(b) Processing View.",
"(c) Management View.To tackle the dependencies between different windows imposed by event consumptions, we propose the SPECTRE (SPECulaTive Runtime Environment) system, a highly parallel framework for DCEP operators.",
"SPECTRE aims to detect the dependencies between different windows and to resolve them by means of speculative execution.",
"This section is organized as follows.",
"In Section REF , we introduce the speculative processing approach we follow in SPECTRE.",
"It is based on creating multiple speculative window versions in order to resolve inter-dependencies between windows.",
"Based on that concept, in Section REF , we explain how SPECTRE determines and schedules the $k$ “best” window versions to $k$ operator instances for parallel processing.",
"Finally, in Section REF , we provide details on how the $k$ operator instances perform the parallel processing of the assigned window versions."
],
[
"Speculation Approach",
"As pointed out above, operators process their incoming data stream based on windows.",
"In particular, operators search for queried patterns to occur in the sequence of events comprised by a window.",
"Windows can overlap, i.e.",
"a pair of windows might have a sequence of events in common.",
"The windows of an operator are totally ordered according to their start events.",
"We call a window, say w$_{j}$, a successor of another window, w$_{i}$, iff the start event of w$_{i}$ occurs before the starting event of w$_{j}$ in the corresponding event stream.",
"For example, in Figure REF (a), w$_{1}$ starts earlier than w$_{2}$; hence, w$_{2}$ is a successor of w$_{1}$.",
"In the same way, w$_{3}$ is a successor both of w$_{2}$ and of w$_{1}$.",
"Now, we can define a consumption dependency (or dependency for short) between windows.",
"Roughly speaking, a window w$_{j}$ depends on another window w$_{i}$, if the consumption of some events in w$_{i}$ might affect the processing of window w$_{j}$.",
"Formally, we define that w$_{j}$ depends on w$_{i}$ iff w$_{j}$ is a successor of w$_{i}$ and w$_{j}$ overlaps with w$_{i}$.",
"For example, in Figure REF (a), w$_{2}$ depends on w$_{1}$, and w$_{3}$ depends both on w$_{2}$ and on w$_{1}$.",
"Now, we will introduce the concept of a consumption group.",
"A consumption group is maintained for each partial match of a search pattern found in a window.",
"It records all events of this window that need to be consumed if the partial match becomes a total match, i.e.",
"the corresponding search pattern is eventually detected in the window.",
"Let's assume that an operator is acting on some window w. Whenever the operator processes an event starting a new partial match of some search pattern, it creates a new consumption group associated with w. When it processes an event that completes a pattern, it completes the corresponding consumption group.",
"On the other hand, a consumption group is abandoned if the corresponding pattern cannot be completed anymore.",
"Consequently, while processing the events of a window, multiple consumption groups can be created that are associated with w. However, all of them will be completed or abandoned at the latest when processing of w is finished.",
"While acting upon w, the operator adds events to be potentially consumed to the consumption groups associated with w, in conformance with the specified consumption policy.",
"When a consumption group is completed, all events contained in this group are consumed together.",
"If the consumption group is abandoned instead, it is just dropped and no events are consumed.",
"For example, let us assume that a query for pattern of a sequence of three events of type $A$ , $B$ and $C$ in a window of time scope 1 minute, is processed by an operator.",
"Let us further assume the consumption policy is set to consume all participating events in case of a pattern match.",
"When detecting an event of type $A$ , say $A_1$ , in a window, the operator creates a new consumption group.",
"The first event of type $B$ , $B_1$ , is added to the consumption group.",
"If the window ends (i.e., 1 minute has passed) and no event of type $C$ is detected, the consumption group is abandoned and no events are consumed in the window.",
"If an event of type $C$ , say $C_1$ , occurs after $B_1$ and within the window scope, the consumption group is completed, and all three events participating in the pattern match, $A_1$ , $B_1$ and $C_1$ , are consumed together.",
"At the time a consumption group is created that is associated with window w, it is unknown whether the corresponding pattern will eventually be completed in w. Clearly, the outcome of the consumption group (complete or abandon) might affect events of all windows that depend on w. One way to handle this uncertainty is to defer the processing of all depending windows until the consumption group terminates (completed or abandoned).",
"However, in general this amounts to processing all windows sequentially.",
"The approach that we follow in SPECTRE is to generate two window versions for each window depending on w, one version assuming that the consumption group will be completed and the other one assuming the consumption group will be abandoned.",
"These window versions can then be processed in parallel to w. Once the outcome of the consumption group is known, i.e., completed or abandoned, processing continues on the corresponding window versions that assume the correct outcome while the other window versions that assume the wrong outcome are just dropped.",
"Obviously, this approach allows for processing dependent windows in parallel even in the presence of event consumptions.",
"With this approach, windows that depend on other windows may have multiple versions that depend on the outcome of the associated consumption groups.",
"In principle there is a window version for any combination of the complete and abandon case of the consumption groups that a window depends upon.",
"When one of these consumption group is abandoned, all window versions assuming this consumption group to complete can be dropped, and vice versa.",
"To capture the dependency between consumption groups and window versions, we introduce the concept of a dependency tree.",
"There exists an individual dependency tree for each independent window, i.e., each window that does not depend on any other window according to our definition above.",
"The vertices of the dependency tree are window versions or consumption groups, while the directed edges of the tree specify the dependencies between them.",
"The root of the dependency tree is the only version of an independent window—by definition, there is only one version of an independent window.",
"The vertex of a window version $\\mathit {WV}$ , say v($\\mathit {WV}$ ), has at most one child.",
"The sub-hierarchy rooted by this child includes all versions of windows depending on $\\mathit {WV}$ , if any.",
"We will denote this sub-hierarchy as v($\\mathit {WV}$ )'s subtree.",
"The subtree is rooted by a consumption group if a consumption group is associated with v($\\mathit {WV}$ ).",
"Otherwise the root of the subtree is a window version directly dependent on v($\\mathit {WV}$ ), if any.",
"A vertex representing a consumption group $\\mathit {CG}$ , say v($\\mathit {CG}$ ), always has two children, one for each possible outcome of $\\mathit {CG}$ (completed or abandoned).",
"The so-called completion edge of v($\\mathit {CG}$ ) links the subtree of window versions for which completion of $\\mathit {CG}$ is assumed, whereas the so-called abandon edge of v($\\mathit {CG})$ links the subtree of window versions which assume $\\mathit {CG}$ to be abandoned.",
"That is, all window versions that can be reached via v($\\mathit {CG}$ )'s completion edge do not include any event included in $\\mathit {CG}$ , while events in $\\mathit {CG}$ have no effect on window versions linked by v($\\mathit {CG}$ )'s abandon edge.",
"When a consumption group $\\mathit {CG}$ associated with a window version $\\mathit {WV}$ is created, the following is performed: v($\\mathit {CG}$ ) is added as a new child of v($\\mathit {WV}$ ) to the dependency tree.",
"The old subtree of v($\\mathit {WV}$ ) is linked by v($\\mathit {CG}$ )'s abandon edge, while a modified copy of the subtree is linked by v($\\mathit {CG}$ )'s completion edge.",
"The modification makes sure that no events included in $\\mathit {CG}$ occur in the window versions of the subtree linked by v($\\mathit {CG}$ )'s completion edge.",
"In other words, for each window version existing in v($\\mathit {WV}$ )'s old dependent versions subtree, a copy that suppresses all events listed in $\\mathit {CG}$ is added.",
"Therefore, each new consumption group associated with v($\\mathit {WV}$ ) doubles the window versions in v($\\mathit {WV}$ )'s subtree.",
"Examples and Algorithms: In the following, a set of examples on the management of the dependency tree is provided along with a formalization of the associated management algorithms.",
"We discuss the following cases: (1) a new dependent window is opened, (2) a new consumption group associated to a window version is created, (3) an existing consumption group is completed or abandoned.",
"New dependent window.",
"When a new window w$_{\\mathit {new}}$ is opened that depends on another window w$_{x}$, for every leaf vertex of the dependency tree rooted by the window version of w$_{x}$, new window versions are created as child vertices (Figure REF , lines 1–10).",
"For example, in Figure REF , at the start of w$_{3}$, new window versions ($\\mathit {WV}_6$ to $\\mathit {WV}_{10}$ ) of w$_{3}$ are created and the corresponding vertices ( v($\\mathit {WV}_6$ ) to v($\\mathit {WV}_{10}$ ) ) are attached to all leaf nodes of the dependency tree rooted by the window version of w$_{1}$.",
"If a leaf vertex is a consumption group $\\mathit {CG}$ , two window versions of w$_{3}$ are created and attached (a version for completion of $\\mathit {CG}$ , and a version for abandoning of $\\mathit {CG}$ ); if a leaf vertex is a window version, one window version of w$_{3}$ is created and attached.",
"Consumption group created.",
"Recall, that when a consumption group $\\mathit {CG}$ associated with a window version $\\mathit {WV}$ is created, the old subtree of v($\\mathit {WV}$ ) is linked by v($\\mathit {CG}$ )'s abandon edge, while a modified copy of the subtree is linked by v($\\mathit {CG}$ )'s completion edge (Figure REF , lines 12–16).",
"In the example in Figure REF , $\\mathit {WV}_2$ creates $\\mathit {CG}_3$ .",
"Then, v($\\mathit {CG}_3$ ) is attached as a new child to v($\\mathit {WV}_2$ ), and the former child, v($\\mathit {WV}_{6}$ ), becomes the root of the unmodified subtree of v($\\mathit {CG}_3$ ).",
"For all window versions in the unmodified subtree of v($\\mathit {CG}_3$ ), a new alternative version is created that assumes that $\\mathit {CG}_3$ will be completed.",
"Suppose $\\mathit {CG}_3$ contains event $E_4$ .",
"Then, window version $\\mathit {WV}_{6}$ (from the unmodified subtree) contains event $E_4$ , whereas the alternative window version $\\mathit {WV}_{7}$ (from the modified subtree) suppresses event $E_4$ .",
"Consumption group completed / abandoned.",
"When a consumption group is completed or abandoned, the respective opposite abandon or completion path of that consumption group is removed from the dependency tree.",
"There are two different reasons why a consumption group is abandoned: (1) Due to the termination of the corresponding window version/end of window, or (2) due to a condition from a negation statement being fulfilled.",
"For instance, a pattern specification of a sequence of events of type $A$ and $B$ can define that no event of type $C$ shall occur between the $A$ and $B$ events.",
"If a consumption group is opened with an $A$ event, the occurrence of a $C$ event would trigger the consumption group to be abandoned as the pattern instance cannot be completed any more, even if a $B$ event would occur later.",
"The algorithms for subtree removal are listed in Figure REF , lines 18–26.",
"Discussion: To be able to process $k$ window versions in parallel we obviously need $k$ operator instances.",
"That means, that typically only a small fraction of all possible window versions can be considered for speculative processing.",
"To be able select the $k$ most promising window versions, we need a method for predicting the probability of possible window versions to survive (i.e., not to be dropped).",
"In Section REF , we propose a scheme for scheduling the $k$ most promising window versions on a collection of $k$ operator instances.",
"Figure: Algorithms for managing the dependency tree."
],
[
"Selecting and Scheduling the Top-k Window Versions",
"The intuition behind SPECTRE is to predict the $k$ “best” speculative window versions and schedule them for parallel processing on $k$ operator instances.",
"To determine the top-$k$ window versions, SPECTRE periodically determines the $k$ window versions with the highest probability to survive in the entire dependency tree.",
"In other words, SPECTRE does not create and schedule windows, as assumed in Section REF , but window versions; in doing so, multiple versions of the same window can be scheduled to different operator instances in parallel.",
"Whether or not a window version $\\mathit {WV}$ survives depends on the outcome of the preceding consumption groups, i.e.",
"the consumption groups on the path from $\\mathit {WV}$ to the root of the dependency tree.",
"In the following, we will denote this path as $\\mathit {WV}$ 's root path.",
"Remember, each vertex representing a consumption group has two outgoing edges, a complete and an abandon edge.",
"We say that the complete or abandon edge of a consumption group, say $\\mathit {CG}$ , becomes valid when $\\mathit {CG}$ is completed or abandoned, respectively.",
"Once one of these edges becomes valid, the other one turns invalid.",
"Consequently, $\\mathit {WV}$ survives only if all abandon and complete edges on its root path eventually become valid, i.e., $\\mathit {WV}$ is dropped if at least one of these edges turn invalid.",
"The probability of $\\mathit {WV}$ to survive depends on the completion probabilities of the consumption groups on $\\mathit {WV}$ 's root path.",
"The survival probability of $\\mathit {WV}$ , denoted as $\\mathit {SP(WV)}$ is determined as follows: Let $\\mathit {P(CG)}$ be the probability that $\\mathit {CG}$ is completed.",
"Moreover, let $\\mathit {CG}_c$ and $\\mathit {CG}_a$ be the set of consumption groups that contribute a complete and abandon edge to $\\mathit {WV}$ 's root path, respectively.",
"ThenNote that this calculation bases on the assumption that the different consumption groups are completed or abandoned independently from each other.",
"If there are dependencies between different occurrences of a pattern and, hence, between the completion of different consumption groups, this can be incorporated in the probability calculation by using dependent / conditional probabilities.",
"However, for the sake of simplicity of the presentation of technical concepts and algorithms, we use the formula for independent probabilities here., $\\mathit {SP(WV)} = \\prod _{c \\in \\mathit {CG}_c} P(c) \\times \\prod _{c^{\\prime } \\in \\mathit {CG}_a}(1 - P(c^{\\prime }))$ ."
],
[
"Prediction Model",
"Now, we discuss how we predict the completion probability of a consumption group.",
"Generally, we observe that the probability that a consumption group is completed equals to the probability that the underlying partial match for a search pattern is completed.",
"Our scheme for predicting the completion probability $\\mathit {P(CG)}$ of a consumption group $\\mathit {CG}$ at a given time takes into account two factors: (1) The inverse degree of completion, i.e., how many more events are at least required in order to complete the pattern—denoted by $\\delta $ —and (2) the expected number of events left in the window, denoted by $n$ .",
"If $\\delta $ is low and many events are still expected to occur in the window, the probability of completion is high.",
"On the other hand, if $\\delta $ is high and only very few events are still expected in the window, the probability of completion is low.",
"In the following, we describe how the probabilistic model is built and updated at system run-time.",
"The dynamic process of pattern completion while processing events is modeled as a discrete-time Markov process.",
"The state of the Markov process is spanned from $\\delta $ to 0.",
"For instance, if a pattern instance consists of at least 3 events (e.g., a sequence of 3 events, or a set of 3 events), the state-space has the elements “3”, “2”, “1” and “0”, with “0” representing the state of total pattern completion.",
"Based on statistics monitored at system run-time, a stochastic matrix $T_1$ is built that describes the transition probabilities between the states of the Markov process when processing one event.",
"To this end, window versions of independent windows gather statistics about the probability of changing from $\\delta _{\\mathit {old}}$ to $\\delta _{\\mathit {new}}$ when an event is processed.",
"The transition probabilities between any pair of $\\delta _{\\mathit {old}}$ and $\\delta _{\\mathit {new}}$ are captured in a matrix $T_1^{new}$ .",
"After $\\rho $ new measurements are available, an updated $T_1$ is computed from the old $T_1^{old}$ and the newly calculated $T_1^{new}$ as $T_1 = (1 - \\alpha ) * T_1^{old} + \\alpha * T_1^{new}$ (exponential smoothing).",
"$\\alpha \\in [0,1]$ is a system parameter to control the impact of recent and of old statistics on $T_1$ .",
"Figure: Calculation of completion probability of a consumption group.Now, the probability of state transitions when processing $n$ events can be computed by raising $T_1$ to the $n$ -th power: $T_n = (T_1)^n$ .",
"The initial state is modeled as a row vector $v_0 = (0, ..., 0, 1, 0, ..., 0)$ —the $\\delta $ -th unit vector, where the $\\delta $ -th position is 1 and all other positions are 0.",
"The probabilities of reaching the different states in $n$ steps can be computed as $v_n = T_n * v_0$ .",
"The last entry of $v_n$ , referring to state “0”, is the probability to complete the pattern in $n$ steps starting from state $v_0$ .",
"To reduce the number of matrix multiplications, each time when $T_1$ is updated, a set of predefined “step sizes” is precomputed, e.g., $T_{10}$ , $T_{20}$ , $T_{30}$ , etc., providing transition probabilities when 10, 20, 30, ... events are processed.",
"If the number of expected events $n$ is in between two precomputed steps, the transition probabilities are linearly interpolated, e.g., $T_{14} = 0.6 * T_{10} + 0.4 * T_{20} $ .",
"The step size, denoted as $\\ell $ , is a system parameter.",
"Figure REF formalizes the described methods in an algorithm.",
"The expected number of events left in the window, $n$ , is calculated from the average window size monitored in the splitter and the position of the last processed event in the window (line 2).",
"The probability matrix $T_n$ is calculated by linear interpolation of precomputed matrices (line 6).",
"$\\delta $ is obtained directly from $\\mathit {CG}$ (line 7), and used in order to build $v_0$ (line 8); $v_n$ is calculated according to the description above (line 9).",
"The resulting completion probability (transition to state “ 0” / pattern completed) is returned (line 10)."
],
[
"Scheduling",
"Here, we describe how SPECTRE periodically selects and schedules the k window versions with the highest survival probability.",
"Notice that the survival probability of window versions is decreasing in a root-to-leaf direction in the dependency tree, i.e.",
"in a window version's subtree there exist only window versions that have the same or a lower survival probability.",
"Therefore, window versions are already sorted by their survival probability in the dependency tree, so that it already represents a max-heap, which simplifies the selection of the top-$k$ versions substantially.",
"From top to the bottom, window versions are added to the top-$k$ list as detailed in the algorithm in Figure REF .",
"The algorithm works with two data structures: (1) a set storing the resulting top-$k$ versions (line 2), and (2) a priority queue storing candidates for being added to the top-$k$ versions (line 3).",
"The priority queue sorts the contained versions by their probability, highest probability first.",
"Until $k$ versions are found, the highest version from the candidate list is added to the result set (lines 4–6).",
"The children of that version are also added as candidates (lines 7–9).",
"This way, the top-$k$ window versions are determined with only visiting the minimal number of vertices in the dependency tree.",
"The scheduling algorithm, listed in Figure REF , does not re-schedule window versions that are already scheduled to avoid unnecessary operations and to increase memory and cache locality of operator instances.",
"Hence, the to-be-scheduled versions are determined (lines 7–9).",
"Further, “free” operator instances are determined that will get a new window version scheduled (lines 10–11).",
"Then, every window version that needs to be scheduled is scheduled to one of the free operator instances (lines 14–17).",
"Figure: Top-k window version selection algorithm.Figure: Splitter: Scheduling algorithm."
],
[
"Parallel Processing of Window Versions",
"Here, we describe how operator instances process their assigned window version according to the dependencies in the dependency tree.",
"In particular, we describe how events are processed and suppressed, and how consumption groups are updated when sub-patterns are detected in a window version.",
"Figure: Operator Instances: Event Processing.The scheduled window versions are processed in parallel by the associated operator instances.",
"This means, that an operator instance processes or suppresses events according to the dependencies of the window version.",
"In particular, when the root path of the window version meets the completion edge of a consumption group, events in that consumption group are not processed: they are suppressed.",
"Complex events produced when processing a speculative window version are kept buffered until the window version either becomes valid—then, the complex events are emitted—or is dropped—then, the complex events are dropped, too.",
"Further, when an event is processed, updates of the consumption groups can occur (creation, completion or abandoning a consumption group, or adding the event to an existing consumption group).",
"In the following, we detail the underlying algorithms.",
"Figure REF lists the algorithm for event processing in the operator instances.",
"In the beginning of a processing cycle, the operator instance checks whether the splitter has scheduled a new window version (lines 7–9).",
"Then, the next event of the currently scheduled window version is processed (lines 11–29).",
"The operator instance checks whether the event is part of any consumption group that shall be suppressed (line 13).",
"If this is the case, the event is suppressed, i.e., its processing is skipped.",
"If the event is not suppressed, it is processed according to the operator logic (line 14).",
"In doing so, there can be four different actions triggered based on feedback the operator logic provides.",
"(1) The processed event can complete one or multiple partial matches: This induces the creation of one or multiple complex events and the completion of the associated consumption groups.",
"In that case, the emitted complex events are buffered, and the dependency tree is updated, calling the consumptionGroupCompleted function (cf.",
"Section REF ).",
"(2) The processed event can lead to the abandoning of consumption groups, either by closing the window, or by invalidating the underlying partial match.",
"In this case, the dependency tree is updated, calling the consumptionGroupAbandoned function (cf.",
"Section REF ).",
"(3) The processed event can lead to the creation of a new consumption group by initiating a new partial match.",
"In this case, the dependency tree is updated, calling the consumptionGroupCreated function (cf.",
"Section REF ).",
"(4) The processed event can become part of one or several existing partial matches, possibly adding the event to the associated consumption groups.",
"In this case, the affected consumption groups are updated directly without changing the structure of the dependency tree.",
"Note, that in the implementation of SPECTRE, the function calls of the operator instances on the dependency tree are buffered—they are actually executed on the dependency tree in a batch at each new scheduling cycle of the splitter.",
"The $k$ scheduled window versions are processed concurrently by the $k$ operator instances, without synchronizing the processing progress of the different window versions.",
"This can lead to a situation where an update on an existing consumption group is propagated too late, causing inconsistencies.",
"For instance, when an event is added to a consumption group $\\mathit {CG}$ in one window version $\\mathit {WV_a}$ after it has been processed in another window version $\\mathit {WV_b}$ adjacent to $\\mathit {CG}$ 's completion edge, an inconsistency can be induced in $\\mathit {WV_b}$ (i.e., an event is processed that should be suppressed).",
"To detect such situations, SPECTRE employs periodic consistency checks; the underlying algorithm is sketched in lines 31 – 45.",
"For every consumption group to be suppressed in the currently processed window version, the algorithm checks whether an update has occurred since the last consistency check.",
"If this is the case, the algorithm checks whether in the current window version, any event in the updated consumption group has been erroneously processed.",
"If yes, then an inconsistency has been detected: The event should have been suppressed, but has actually been processed.",
"If an inconsistency is detected, the state of the window version is rolled back to the start, i.e., the window version is reprocessed from the start.",
"Instead of reprocessing a window version from the start in case of an inconsistency, it could also be recovered from an intermediate checkpoint.",
"However, when implementing that approach, we realized that the overhead in periodically checkpointing all window versions is much higher than the gain from recovering from checkpoints."
],
[
"Evaluations",
"In this section, we evaluate the performance of SPECTRE under different real-world and synthetic workloads and varying queries in the setting of an algorithmic trading scenario.",
"We analyze the scalability of SPECTRE with a growing number of operator instances and the overhead involved in speculation and dependency management."
],
[
"Experimental Setup",
"Here, we describe the evaluation platform, the SPECTRE implementation and the datasets and queries used in the evaluations.",
"Evaluation Platform.",
"We run SPECTRE on a shared memory multi-core machine with 2x10 CPU cores (Intel Xeon E5-2687WV3 3.1 GHz) that support hyper-threading (i.e., 40 hardware threads).",
"The total available memory in the machine is 128 GB and the operating system is CentOS 7.3.",
"Implementation.",
"SPECTRE is implemented using C++.",
"The pattern detection and window splitting logic of the queries in these evaluations are implemented as a user-defined function (UDF) inside SPECTRE.",
"Further, we provide a client program that reads events from a source file and sends them to SPECTRE over a TCP connection.",
"Our implementation of SPECTRE is open sourcehttps://github.com/spectreCEP.",
"Datasets.",
"We employ two different datasets centered around an algorithmic trading scenario.",
"First, a real-world stock quotes stream originating from the New York Stock Exchange (NYSE).",
"This dataset contains real intra-day quotes of around 3000 stock symbols from NYSE collected over two months from Google Financehttps://www.google.com/finance; in total, it contains more than 24 million stock quotes.",
"The quotes have a resolution of 1 quote per minute for each stock symbol.",
"We refer to this dataset as the NYSE Stock Quotes dataset, denoted as NYSE.",
"NYSE represents realistic data for stock market pattern analytics.",
"Second, we generated a random sequence of 3 million events consisting of 300 different stock symbols; the probability of each stock symbol is equally distributed in the sequence.",
"We refer to this dataset as the Random Stock Symbols dataset, denoted as RAND.",
"Figure: Queries.Queries.",
"We employ three different queries, Q1 to Q3, in the evaluations (cf.",
"Figure REF ).",
"The queries are listed in the MATCH-RECOGNIZE notation [33], which is concise and easy to understand.",
"Note, that we extended the MATCH-RECOGNIZE notation by two additional constructs stemming from the Tesla language [11]: WITHIN ... FROM to specify a window size and window start condition, and CONSUME to specify consumption policies.",
"Q1 detects a complex event when the first $q$ rising or the first $q$ falling stock quotes of any stock symbol (defined as $\\mathtt {RE}$ or $\\mathtt {FE}$ , respectively) are detected within $\\mathit {ws}$ minutes from a rising or falling quote of a leading stock symbol (defined as $\\mathtt {MLE}$ ).",
"The leading stock symbols are composed of a list of 16 technology blue chip companies.",
"In the listing of Q1, we show only the stock rising pattern; the falling pattern is constructed accordingly.",
"In case a complex event is detected, all constituent incoming events are consumed.",
"Note, that this query always has a fixed pattern length of $q$ , and each matching event moves the pattern detection to a higher completion stage.",
"Q2 is a query from related work (Balkesen and Tatbul [5], Query 9) that we extended by a window size of $\\mathit {ws}$ events, a window slide of $s$ events and a consumption policy.",
"It detects a complex event when specific changes occur in the price of a stock symbol between defined upper and lower limits.",
"As in Q1, all constituent incoming events are consumed when a complex event is detected.",
"We use the lower and upper limits to control the average pattern size.",
"A small lower and a large upper limit results in a larger average pattern size, and vice versa.",
"In contrast to Q1, Q2 has a variable length even for a fixed lower and upper limit.",
"A matching event might or might not influence the pattern completion: the Kleene$^+$ implies that many events can match while the pattern completion does not progress.",
"Q3 detects a set of $n$ specific stock symbols following stock symbol $A$ .",
"In contrast to the other queries, the ordering of those $n$ symbols is not important.",
"The pattern length $n$ , window size $\\mathit {ws}$ , and window slide $s$ can be freely varied.",
"All constituent events are consumed when a complex event is detected."
],
[
"Performance Evaluation",
"In this section, we evaluate the throughput and scalability of SPECTRE.",
"First of all, we evaluate how SPECTRE performs with a growing number of parallel operator instances and with different consumption group completion probabilities.",
"After that, we provide a detailed analysis of the Markov model SPECTRE uses to predict the completion probability of consumption groups.",
"Finally, we discuss a comparison to the CEP engine T-REX [12].",
"If not noted otherwise, we employ the following settings.",
"The number of created consumption groups is limited to one per window version.",
"The Markov model is employed with the parameters $\\alpha = 0.7$ and $\\ell = 10$ .",
"To measure the system throughput, we streamed the datasets as fast as possible to the system.",
"Each experiment was repeated 10 times.",
"The figures show the 0th, 25th, 50th, 75th and 100th percentiles of the experiment results in a “candlesticks” representation."
],
[
"Scalability",
"Here, we evaluate the scalability of SPECTRE.",
"To this end, we analyze the system throughput, i.e., the number of events processed per second, with a growing number of operator instances.",
"The following questions are addressed: (1) How does the scalability depend on the completion probability of the consumption groups?",
"(2) How much computational and memory overhead is induced by maintaining the dependency tree and determining the top-$k$ window versions?",
"We expect that the completion probability of consumption groups influences the system throughput.",
"To make that clear, regard two extreme cases: All consumption groups are abandoned, or all consumption groups are completed.",
"In the first case, SPECTRE should only schedule window versions on the left-most path of the dependency tree.",
"In the second case, SPECTRE should only schedule window versions on the right-most path of the dependency tree.",
"In both cases, the scheduling algorithm should traverse the dependency tree in depth; i.e., it should schedule $k$ window versions from $k$ different windows.",
"Further, none of the scheduled window versions should be dropped; all of them should survive.",
"Hence, the throughput should be maximal.",
"On the other hand, suppose that the completion probability of all consumption groups is constantly at 50 %.",
"In that case, SPECTRE should traverse the dependency tree in breadth; i.e., it should schedule 1 window version of the first window, 2 window versions of the second window, 4 window versions of the third window, etc.",
"However, only 1 window version of each window can survive; all others will be dropped.",
"Hence, the higher $k$ is, the more futile processing is performed, as the probability to predict the correct window version drops exponentially with $k$ .",
"In the following, we analyze whether SPECTRE shows the expected behavior and discuss implications.",
"To this end, we run a set of experiments with queries Q1 and Q2, using the NYSE dataset.",
"In both queries, there are parameters that can be changed such that the average completion probability of consumption groups is manipulated.",
"In Q1, we achieve this by directly setting the pattern size $q$ , such that the ratio between pattern size and window size changes.",
"Larger patterns are less likely to complete.",
"In Q2, we cannot directly set the pattern size.",
"However, we influence the average pattern size—and thus, the average completion probability—by changing the upper and lower limit parameters in the pattern definition.",
"In Q1, we employ a sliding window with a window size $\\mathit {ws}$ of 8,000 events, setting pattern sizes $q$ of 40, 80, 160, 320, 640, 1280, and 2560 events.",
"We calculate a “ground truth” value of the completion probability of consumption groups by performing a sequential pass without speculations: The number of created consumption groups divided by the number of produced complex events provides the ground truth value.",
"The system throughput employing 1, 2, 4, 8, 16, and 32 operator instances, is depicted in Figure REF (a).",
"The corresponding ground truth probabilities are depicted in Figure REF (d).",
"At a ratio of pattern size to window size of 40 / 8,000 (i.e., 0.005), the ground truth of consumption group completion probability is at 100 %, i.e., all partial matches are completed.",
"The throughput scales almost linearly with a growing number of operator instances, from 10,800 events/second at 1 operator instance to 154,000 events/second at 16 operator instances (scaling factor 14.3) and 218,000 events/second at 32 operator instances (scaling factor 20.2).",
"Increasing the pattern size decreases the completion probability of consumption groups.",
"At a ratio of pattern size to window size of 640 / 8,000 (i.e., 0.08), the ground truth of consumption group completion probability is at 56 %, i.e., half of partial matches are completed and the other half are abandoned.",
"The throughput scales from 9,200 events/second at 1 operator instance to 35,000 events/second at 8 operator instances (scaling factor 3.8).",
"However, employing more than 8 operator instances does not increase the throughput further: With 16 and 32 operator instances, it is comparable to 8 operator instances.",
"Further increasing the pattern size, we reach a ground truth of consumption group completion probability of 13 % at a ratio of pattern size to window size of 2560 / 8,000 (i.e., 0.32).",
"Here, the throughput scales better, from 8,700 events/second at 1 operator instance to 131,900 events/second at 16 operator instances (scaling factor 15.2).",
"Here, 32 operator instances do not improve the throughput further compared to 16 operator instances.",
"In Q2, we employ a sliding window with a window size $\\mathit {ws}$ of 8,000 events and a sliding factor $s$ of 1,000 events.",
"We arranged the lower and upper limit parameters in the pattern definition such that the corresponding average pattern sizes were 180, 226, 496, 560, 839, 1261, 1653, and 2223 events, plus one setting that made it impossible for a pattern to be completed.",
"The system throughput employing 1, 2, 4, 8, 16, and 32 operator instances, is depicted in Figure REF (b).",
"The corresponding ground truth probabilities are depicted in Figure REF (e).",
"At a ratio of pattern size to window size of 180 / 8,000 (i.e., 0.02), the ground truth of consumption group completion probability is at 100 %, i.e., all partial matches are completed.",
"The throughput scales almost linearly with a growing number of operator instances, from 10,300 events/second at 1 operator instance to 139,800 events/second at 16 operator instances (scaling factor 13.8) and 200,400 events/second at 32 operator instances (scaling factor 19.5).",
"At a ratio of pattern size to window size of 560 / 8,000 (i.e., 0.07), the ground truth of consumption group completion probability is at 50 %, i.e., half of partial matches are completed and the other half are abandoned.",
"The throughput scales from 10,900 events/second at 1 operator instance to 64,900 events/second at 8 operator instances (scaling factor 6.0).",
"Employing more than 8 operator instances does not increase the throughput further: With 16 and 32 operator instances, it is comparable to 8 operator instances.",
"When none of the partial matches can complete (denoted by “0 cplx”), the throughput scales from 10,400 events/second at 1 operator instance to 108,400 events/second at 16 operator instances (scaling factor 10.4) and 174,300 events/second at 32 operator instances (scaling factor 16.8).",
"Discussion of the results.",
"We draw the following conclusions from the results.",
"First of all, our assumptions on the system behavior are backed by the measurements.",
"Further, the different queries impose “throughput profiles” that have a similar shape.",
"The scaling behavior in SPECTRE, using the speculation approach, is very different from other event processing systems that have been analyzed in related work.",
"In SPECTRE, the parallelization-to-throughput ratio largely depends on the completion probability of partial matches.",
"This new factor leads to interesting implications when adapting the parallelization degree (i.e., elasticity), which is typically done based on event rates [14], [25], [24] or CPU utilization [2], [9].",
"Existing elasticity mechanisms do not take into account the completion probability to determine the optimal resource provisioning.",
"Using the described throughput curves, SPECTRE could adapt the number of operator instances based on the current pattern completion probability.",
"Figure: Evaluation of Markov Model."
],
[
"Overhead of Speculation",
"Here, we analyze the computational and memory overhead of maintaining the dependency tree in the splitter and scheduling the top-$k$ window versions.",
"In a first experiment (Q1, NYSE dataset, $q$ = 80, window size = 8,000), we measure how often the splitter can perform a complete cycle of tree maintenance and top-$k$ scheduling per second.",
"The cycle is described as follows: (a) Maintenance: performing all updates on the dependency tree that have been issued since the last maintenance, i.e., creating new consumption groups and window versions and delete dropped ones, and (b) scheduling: schedule the new top-$k$ window versions to the $k$ operator instances according to the updated dependency tree.",
"In Figure REF (c), the results are depicted.",
"With 1 operator instance, SPECTRE achieves a maintenance and scheduling frequency of 4 million cycles per second.",
"With increasing number of operator instances, the scheduling frequency decreases but is still considerably high, where SPECTRE achieves a scheduling frequency of $650,000$ and $450,000$ times per second with 16 and 32 operator instances, respectively.",
"We conclude that there is some overhead involved in the management of the dependency tree and the scheduling algorithm, but there are no indications that this would become a bottleneck in the system.",
"Another concern about the dependency tree might be its growth and size in memory.",
"To this end, we measured the maximal number of window versions maintained in the dependency tree at the same time (Q1, NYSE dataset, $q$ = 80, window size = 8,000).",
"The results of the experiments are depicted in Figure REF (f).",
"With 1 operator instance, the maximal tree size was at 41 window versions, growing up to 4,332 at 16 operator instances and 6,730 window versions at 32 operator instances.",
"This is not a serious issue in terms of memory consumption.",
"Indeed, the importance of a suitable top-$k$ window version selection becomes obvious here: Determining the $k$ window versions that will survive out of a large number of window versions that will eventually be dropped is a huge challenge, which SPECTRE could handle reasonably well in the performed experiments.",
"After we have discussed the overall system throughput and different factors that impact it, we go into a more detailed analysis of the completion probability model of consumption groups.",
"In particular, we want to know how well the proposed Markov model behaves when the probabilities of complex events are changing.",
"To this end, we perform two different experiments of query Q3 with different ratios of pattern size to window size: A ratio of 0.002 that has a high consumption group completion probability and a ratio of 0.1 that has a lower consumption group completion probability.",
"We employed 32 operator instances and the window size $\\mathit {ws}$ was set to 1000 events where a new window is opened every 100 events ($s$ = 100).",
"We compare the proposed Markov model with a probability model that assigns each consumption group a fixed completion probability.",
"The results of the two experiments are depicted in Figure REF (a) and (b), respectively.",
"At a ratio 0.002, the completion probability of a consumption group was at $100\\%$ .",
"Accordingly, assigning a fixed probability of $100\\%$ to the consumption groups yielded a throughput of 279,000 events per second, which was significantly better than other fixed probabilities.",
"The Markov model with a throughput of 277,000 events per second proved to be competitive with the best fixed model.",
"At a ratio of 0.1, the probability of a complex event was at of $32\\%$ .",
"Accordingly, assigning a fixed probability of $20\\%$ to the consumption groups yielded a throughput of 86,000 events per second, which was significantly better than other fixed probabilities.",
"The Markov model with a throughput of 79,000 events per second performed almost as good as the best fixed model.",
"From those results, we draw two conclusions.",
"First, the Markov model is able to automatically learn suitable consumption group probabilities in different settings.",
"Second, we can see that wrong probability predictions can cause a large throughput penalty."
],
[
"Comparison to T-REX",
"We have also implemented query Q1 in the T-REX event processing engine [12].",
"In total numbers, T-REX performed much worse than SPECTRE, reaching a throughput of only about 1,000 events per second.",
"While this shows that the throughput of SPECTRE is competitive, it is worth to mention that both systems are different.",
"T-REX is a general-purpose event processing engine that automatically translates queries into state machines, whereas SPECTRE employs user-defined functions to implement queries which allows for more code optimizations.",
"T-REX does not support event consumptions in parallel processing, while SPECTRE can utilize multi-core machines to scale the throughput."
],
[
"Related Work",
"In the past decades, a number of different Complex Event Processing systems and languages has been proposed.",
"Besides CEP languages that do not support event consumptions, such as SASE [31], the concept of event consumption gained growing importance.",
"Based on practical use cases, Snoop [10] defined 4 different so-called parameter contexts, which are predefined combinations of Selection and Consumption Policies.",
"Building on a more systematic analysis of the problem, Zimmer and Unland [34] proposed an event algebra that differentiated between 5 different Selection and 3 different Consumption Policies that can be combined.",
"Picking up and extending that work, the Amit system [1] allowed for distinct specifications of the Selection and Consumption Policy.",
"Finally, Tesla [11] and its implementation T-REX [12] introduced a formal definition of its supported policies.",
"The proposed speculation methods and the SPECTRE framework are applicable to any combination of selection and consumption policies.",
"The crucial question in exploiting data parallelism in a DCEP operator is how to split the incoming event streams, such that the different partitions, assigned to different operator instances, can be processed in parallel.",
"Besides window-based splitting, as used in SPECTRE, other splitting methods have been proposed.",
"However, they lack the expressiveness to capture temporal relations between events that many DCEP queries expose.",
"In key-based splitting [26], [32], [17], [9], [16], the event stream is split by a key that is encoded in the events, e.g., a stock symbol in algorithmic trading [17] or a post ID in social network analysis [26].",
"Different key value ranges are assigned to different operator instances.",
"However, the parallelism is restricted to the number of different key values; moreover, not all pattern definitions exhibit key-based data parallelism.",
"For instance, in example query $Q_E$ (cf.",
"Section REF ), events of both stock symbols $A$ and $B$ have to be correlated, so that key-based splitting cannot be applied.",
"Pane-based splitting has been proposed in stream processing systems [6], [22].",
"For instance, when the max or median value of a window of 1 minute shall be computed, that window is split into 6 fragments of 10 seconds, the fragments' max or median values are computed in parallel, and the global window's value is aggregated from the fragments' results.",
"This parallel aggregation procedure bases on the idea of pane-based aggregations [23].",
"However, DCEP patterns often impose a temporal dependency between the events of a window that hinders the vertical splitting, e.g., when a sequence of events $A$ and $B$ is queried as in example query $Q_E$ (cf.",
"Section REF ).",
"Furthermore, additional constraints on the events can be formulated, e.g., $A$ and $B$ have a parameter $x$ , such that $A:x > B:x$ (e.g., to detect chart patterns in stock markets [17]).",
"If the events are scattered among different vertical windows, such dependencies and constraints cannot be analyzed.",
"Besides data parallelization, intra-operator parallelization, also known as pipelining, has been proposed.",
"Internal processing steps that can be run in parallel are identified by deriving operator states and transitions from the query (e.g., state-based approach in [5]).",
"According to the identified processing steps, the operator logic is split and the processing steps are executed in parallel.",
"This offers only a limited achievable parallelization degree depending on the number of processing steps in the query.",
"For instance, in example query $Q_E$ (cf.",
"Section REF ), only 2 processing steps, detecting A and detecting B, are available, leading to a maximum parallelization degree of 2.",
"A common variant of intra-operator parallelization uses lazy evaluation techniques on event sequence patterns to increase the operator throughput [13], [20].",
"Those techniques check the event stream for terminator events, i.e., the last event of the event sequence in a pattern, and only evaluate preceding events when such a terminator event is found.",
"The underlying assumption is that a terminator event can be determined independently from other events, e.g., solely based on its event type.",
"However, often, sequence patterns depend on the comparison of the events' payload, e.g., a stock quote increasing 3 times in a row; whether a quote is the third in a row that is increasing can only be determined when the two preceding quotes are analyzed.",
"Hence, such techniques are only addressing a subset of possible event patterns.",
"Speculation has been widely applied to deal with out-of-order events in stream processing.",
"Mutschler and Philippsen [28] propose an adaptive buffering mechanism to sort the events before processing them, introducing a slack time.",
"When an event arrives outside of the slack time, results are recomputed.",
"However, slack times cannot be used to overcome window dependencies in the event consumption problem: If one window is processed later, all depending windows would also need to be deferred.",
"Brito et al.",
"[8] as well as Wester et al.",
"[30] propose transaction-based systems to roll-back processing when out-of-order events arrive.",
"Their systems are not parallel, meaning that they only employ one speculation path for each operator.",
"We also roll-back when window versions reach an inconsistent state.",
"However, we propose a highly parallel multi-path speculation method (not only one path) and employ a probabilistic model to schedule the most promising window versions; hence, our system scales with an increasing number of CPU cores.",
"Balazinska et al.",
"[4] propose a system that quickly emits approximate results that are later refined when out-of-order events arrive.",
"Our model would generally allow to be extended toward supporting probabilistic approximations, as a survival probability is given on the window versions.",
"However, in this paper, we focus on consistent event detection (no false-positives, no false-negatives) and leave approximate applications of our model to the future work.",
"Brito et al.",
"[7] propose for non-deterministic stream processing operators to mark events as speculative before logs have been committed to disc for consistent recovery.",
"The speculative events can be forwarded to successor operators in the operator graph that treat them specifically.",
"In SPECTRE, speculative complex events are kept buffered until the window version is confirmed.",
"We focus on providing deterministic event streams to the successor operators; in particular, we do not assume that subsequent operators or event consumers can handle events that are marked as speculative."
],
[
"Conclusion",
"The SPECTRE system uses window-based data parallelization and optimized speculative execution of interdependent windows to scale the throughput of DCEP operators that impose consumption policies.",
"The novel speculation approach employs a probabilistic consumption model that allows for processing the $k$ most promising window versions by $k$ operator instances in parallel on a multi-core machine.",
"Evaluations of the system show good scalability at a moderate overhead for speculation management."
],
[
"Acknowledgments",
"This work was funded by DFG grant RO 1086/19-1 (PRECEPT)."
]
] | 1709.01821 |
[
[
"Causal properties of nonlinear gravitational waves in modified gravity"
],
[
"Abstract Some exact, nonlinear, vacuum gravitational wave solutions are derived for certain polynomial $f(R)$ gravities.",
"We show that the boundaries of the gravitational domain of dependence, associated with events in polynomial $f(R)$ gravity, are not null as they are in general relativity.",
"The implication is that electromagnetic and gravitational causality separate into distinct notions in modified gravity, which may have observable astrophysical consequences.",
"The linear theory predicts that tachyonic instabilities occur, when the quadratic coefficient $a_{2}$ of the Taylor expansion of $f(R)$ is negative, while the exact, nonlinear, cylindrical wave solutions presented here can be superluminal for all values of $a_{2}$.",
"Anisotropic solutions are found, whose wave-fronts trace out time- or space-like hypersurfaces with complicated geometric properties.",
"We show that the solutions exist in $f(R)$ theories that are consistent with Solar System and pulsar timing experiments."
],
[
"Introduction",
"When a field develops a localised perturbation, information about the disturbance is communicated at some finite speed to the surrounding universe.",
"Gravitational waves (GWs) act as the energy and information transport mechanism for time-varying gravitational fields [1], [2], [3].",
"Within the theory of general relativity (GR), GWs propagate in the linearised regime outside the near zone in even the most compact relativistic sources (e.g.",
"[4], [5]).",
"Linearisation schemes are convenient because tools such as multipole expansions exist for calculating amplitudes and polarisations straightforwardly given a model of the source [6], [7], [8].",
"At least within GR, it is well known that the phase speed of GWs is precisely the speed of lightWe adopt natural units throughout with $G=c=1$ , although the constant $c$ is occasionally written explicitly for emphasis.",
"in both the linear and nonlinear theories [9], [10].",
"The waves propagate along null hypersurfaces in vacuum and thus the notions of (Maxwellian) electromagnetic and general relativistic causality coincide.",
"Fundamental inconsistencies between quantum field theories and GR suggest that a quantum theory of gravity will modify the geometry-matter relations of GR [11].",
"Within bosonic string theories, for example, the quantization of the Polyakov action introduces scalar potentials (graviton-dilaton couplings) into the Einstein action which modifies the gravitational dynamics [12], [13].",
"Transforming into the Jordan frame shows that these dilaton-tensor theories behave like higher-order curvature theories [such as the $f(R)$ theories considered in this paper; see below], and that GR correction terms are large when the curvatures are large [14], [15].",
"Classically speaking, therefore, GWs in string-inspired or other gravity theories may propagate differently to their GR counterparts in the vicinity of strong sources or elsewhere [16], [17].",
"In particular, the wave-fronts may trace out hypersurfaces, which are not null, indicating that notions of causality may differ between electromagnetic and gravitational events in modified theories of gravity.",
"Theories with massive gravitons, for example, predict that the wave-fronts are frequency dependent, propagate slower than light, and trace out time-like hypersurfaces [18].",
"A modification of the phase speed represents the simplest kind of topological adjustment that can occur in the causal structure [19].",
"Other, exotic kinds of topological structures can also occur in wave-fronts in $f(R)$ gravity.",
"For example, there exist choices of $f$ such that the gravitational past and future of some event can have a non-empty intersection, thereby violating chronology protection [20], [21].",
"The linearised $f(R)$ theory predicts an exact dispersion relation for GWs [22].",
"However, dispersion relations in linear and nonlinear theories can have very different physical characters.",
"Consider a scalar field theory whose equation of motion reads $ 0 = \\phi _{,tt} - \\nabla ^2 \\phi + V^{\\prime }(\\phi ),$ with scalar field $\\phi $ and potential function $V$ .",
"Linearisation of equation (REF ) returns either the Klein-Gordon or massless wave equation depending on the coefficient of the linear term in $V^{\\prime }(\\phi )$ .",
"Both the Klein-Gordon and massless wave equations admit propagating solutions with fixed propagation speeds (e.g.",
"[23]).",
"However, depending on the form of $V$ , the nonlinear dispersion relation can be modified by self-interaction [24].",
"For example, there exist choices of $V$ such that equation (REF ) admits soliton-like solutions with arbitrary phase speeds (such as $V \\propto \\phi ^3$ ), while other choices of $V$ preserve the Klein-Gordon character of the dispersion relation (such as $V \\propto \\cos \\phi $ ) [25], [26], [27], [28].",
"Given the well-studied equivalence between $f(R)$ and scalar-tensor theories of gravity, it is reasonable to expect a similar phenomenon to occur in $f(R)$ gravity depending on the particulars of the function $f$ [14].",
"Hence, one must be careful when drawing conclusions about nonlinear GWs from analysis of the corresponding linearised field equations [29], [30].",
"This phenomenon is related to the Vainshtein mechanism [31].",
"The purpose of this short paper is to demonstrate, by explicit construction, some topological properties of nonlinear GWs in $f(R)$ theories of gravity.",
"We show that the predictions offered by the linear and nonlinear theories may differ significantly.",
"In Section II we define some general notions of causality that are used throughout the paper.",
"In Section III we present the $f(R)$ field equations and recall some results concerning phase speeds of GWs in the linear theory.",
"In Section IV we show, by constructing two exact solutions, that these relations may fail to describe the propagation speed of nonlinear GWs, that exotic topological properties can occur in the GW-front defining the causal structure, and that the theories considered are consistent with Solar System and pulsar timing constraints.",
"However, the analytic solutions exhibit certain artificial properties, which are likely to be avoided in more general, numerical solutions, a topic for future work.",
"Some brief, additional discussion regarding $f(R)$ theories and causality is presented in Section V."
],
[
"Causality in modified gravity",
"In any physical theory where information propagates at a finite speed, a notion of causality emerges.",
"Given an event $E_{1}$ , a second event $E_{2}$ is causally connected through electromagnetic signals to $E_{1}$ provided that it lies within the null cone originating at $E_{1}$ .",
"The same two events are causally connected gravitationally, if there exists a curve joining $E_{1}$ and $E_{2}$ that is contained within the domain of dependence, defined by the hypersurface traced out by the GW-fronts emanating from $E_{1}$ (see e.g.",
"Hawking and Ellis [10] for formal definitions).",
"In vacuum GR, the domain of dependence coincides exactly with the null cone for any event, and an unambiguous notion of causality emerges.",
"The domain of dependence, however, depends on the structure of the field equations (since it depends on the properties of GWs) and need not coincide with the null cones in modified gravity.",
"Throughout this work we use the phrase `causal' to refer to gravitational causality unless otherwise stated.",
"Consider a universe where GW-fronts propagate isotropically with phase speed $v$ in vacuum, and suppose some perturbation event occurs at $P$ .",
"Fig.",
"REF illustrates three kinds of causal connection that can occur in such a universe.",
"The domain of events which could be influenced by (influence) $P$ is known as the future (past) domain of dependence and is denoted by $D^{+}(P)$ [$D^{-}(P)$ ].",
"The set $D(P) = D^{+}(P) \\cup D^{-}(P)$ represents the causal domain of the event $P$ .",
"For $v<c$ , there exist observers in Lorentz-boosted frames who see the wave travel at non-zero speeds less than $c$ .",
"The domain $D^{+}(P)$ [$D^{-}(P)$ ] extends to future (past) time-like infinity $i^{+}$ ($i^{-}$ ).",
"In such a universe, events exist that are electromagnetically but not gravitationally connected, i.e.",
"events which lie within the null cone originating at $P$ but not in $D(P)$ .",
"For $v=c$ , gravitational events are seen at the same time as electromagnetic ones by all observers, and the domain $D^{+}(P)$ [$D^{-}(P)$ ] extends to future (past) null infinity $\\mathcal {J}^{+}$ ($\\mathcal {J}^{-}$ ).",
"This is the case in vacuum GR.",
"If GWs are superluminal with speeds $v>c$ , boosted-frame observers exist who see the waves travel at arbitrarily high speeds, the domains $D^{\\pm }(P)$ extend to space-like infinity $i^{0}$ , and events exist that are gravitationally but not electromagnetically connected.",
"See e.g.",
"Refs.",
"[25], [33] for a discussion on physical consequences.",
"Fig.",
"REF represents the causal domain of the event $P$ in a universe where GW wave-fronts no longer trace out two cones (future and past) joined at $P$ but rather some other topological surface, i.e.",
"GW propagation is not isotropic.",
"Note that in this particular illustration we have that $D^{+}(P) \\cap D^{-}(P) = \\emptyset $ .",
"There is no reason to assume a priori that this holds for general theories of gravity, i.e.",
"closed time-like curves can exist in $D^{+}(P) \\cap D^{-}(P)$ in general.This scenario can also occur in GR for universes filled with exotic matter, e.g.",
"the Gödel solution [1].",
"In this work we consider vacuum spacetimes only.",
"Figure: Cross section of the causal domains for an event PP for a universe where GWs propagate at speed vv.",
"The shaded blue region represents the causal domain of PP, when GWs are subluminal (v<c)(v<c).",
"For v<cv<c, the domain D(P)=D + (P)∪D - (P)D(P) = D^{+}(P) \\cup D^{-}(P) is strictly contained within the null cone (v=c)(v=c) represented by the union of the red region, which extends to future (past) null infinity 𝒥 + \\mathcal {J}^{+} (𝒥 - )(\\mathcal {J}^{-}), and the blue region, which extends to future (past) time-like infinity i + i^{+} (i - )(i^{-}).",
"The causal domain for superluminal GWs (v>c)(v>c) is represented by the union of the grey, red, and blue regions and extends to space-like infinity i 0 i^{0}.",
"Note that, although drawn as conical structures here for simplicity, the global shape of D(P)D(P) will be warped by the metric coefficients in general (see section 12.6 of ).Figure: Cross section of the causal domain for an event PP within a universe where GW-fronts propagate anisotropically and subluminally.",
"The domain D(P)=D + (P)∪D - (P)D(P) = D^{+}(P) \\cup D^{-}(P) has some non-trivial topological structure, i.e.",
"∂D(P)\\partial D(P) is complicated.",
"In this particular example, D(P)D(P) is completely confined within the null cone, and as such extends to future (past) time-like infinity i + i^{+} (i - i^{-}).Below we show that, in $f(R)$ gravity, both of the situations depicted in Figs.",
"REF and REF may occur for nonlinear GWs."
],
[
"Field equations",
"In an $f(R)$ theory of gravity, the Ricci scalar, $R$ , is replaced by an arbitrary function of this quantity, $f(R)$ , in the Einstein-Hilbert action.",
"The vacuum field equations read (e.g.",
"[14]) $ 0 = f^{\\prime }(R) R_{\\mu \\nu } - \\frac{f(R)}{2} g_{\\mu \\nu } + \\left( g_{\\mu \\nu } \\square - \\nabla _{\\mu } \\nabla _{\\nu } \\right) f^{\\prime }(R),$ where $R_{\\mu \\nu } = R^{\\alpha }_{\\mu \\alpha \\nu }$ is the Ricci tensor, $g_{\\mu \\nu }$ is the metric tensor, and $\\square = \\nabla _{\\mu } \\nabla ^{\\mu }$ symbolises the d'Alembert operator."
],
[
"Linear theory",
"Following Berry and Gair [22] we consider $f$ to be an analytic function about $R=0$ so that it can be expressed as a power series, $ f(R) = a_{0} + a_{1} R + \\frac{a_{2}}{2!}",
"R^2 + \\frac{a_{3}}{3!}",
"R^3 + \\cdots ,$ where the $a_{i}$ are the Maclaurin coefficients [11], [14].",
"We set $a_{0} = 0$ to expand about a Minkowski background, though some of the results carry over to other backgrounds as well, e.g.",
"(anti-) de Sitter.",
"Perturbing the metric according to $ g_{\\mu \\nu } = \\eta _{\\mu \\nu } + h_{\\mu \\nu },$ we find (REF ) reduces to $ 0 = \\square \\bar{h}_{\\mu \\nu }$ to linear order.",
"We introduce the trace-reversed potential $\\bar{h}_{\\mu \\nu } = a_{1} \\left( h_{\\mu \\nu } - \\frac{1}{2} h_{\\sigma \\rho } \\eta ^{\\sigma \\rho } \\eta _{\\mu \\nu } \\right) - a_{2} R^{(1)} \\eta _{\\mu \\nu } ,$ enforce the generalised de Donder gauge $\\nabla ^{\\mu } \\bar{h}_{\\mu \\nu } = 0$ , and write the linearised Ricci scalar [to order $\\mathcal {O}(h)$ ] as $R^{(1)}$ .",
"Equation (REF ) implies the existence of two tensor polarisation modes for general $f(R)$ theories [22], just like in GR.",
"The trace of (REF ) shows that the linearised Ricci scalar satisfies a Klein-Gordon equation of the form $ 0 = 3 a_{2} \\square R^{(1)} - a_{1} R^{(1)},$ indicating that there is also a propagating scalar mode for $a_{2} \\ne 0$ (massive for $a_{1} \\ne 0$ ) in addition to the two tensor modes of GR [34].",
"An important feature, for our purposes, is that equation (REF ) predicts the existence of scalar modes with group velocityNote that there is a misplaced minus sign in equation (30) in Ref.",
"[22]; Berry and Gair's no-tachyon condition should read $\\Upsilon ^2 < 0$ to be consistent with the usual Starobinsky $[f^{\\prime }(R) > 0]$ and Dolgov-Kawasaki $[f^{\\prime \\prime }(R) \\ge 0]$ conditions [37], [38].",
"$(a_{2} \\ne 0)$ $ c_{\\textrm {g}} = \\frac{\\sqrt{\\omega ^2 - a_{1} \\left(3 a_{2} \\right)^{-1} }}{\\omega },$ where $\\omega $ is the wave frequency [23], [22].",
"As such, the velocity of a linear GW in $f(R)$ gravity is uniquely determined by the value of the coefficients $a_{2}$ and $a_{1}$ given (REF ).",
"In particular, expression (REF ) demands $a_{1} a_{2} > 0$ to ensure $c_{\\textrm {g}} < 1$ , so that tachyonic instabilities are avoided [35], [36]."
],
[
"Exact solutions",
"In this section we construct three explicit examples of nonlinear wave solutions to (REF ).",
"In Sec.",
"IV.",
"A, we present a class of solutions which admit an arbitrary phase speed independent of the value of $a_{2}$ , the situation depicted in Fig.",
"REF .",
"We also derive a class of generalised Peres waves which propagate anisotropically in Sec.",
"IV.",
"B, the situation depicted in Fig.",
"REF .",
"In both cases, we work with the function $ f(R) = R + \\frac{a_{2}}{2!}",
"R^2 + \\frac{a_{3}}{3!}",
"R^3 + \\frac{a_{k}}{\\Gamma (k+1)} R^k,$ where $\\Gamma (\\xi ) = \\int ^{\\infty }_{0} d \\tau \\tau ^{\\xi -1} e^{-\\tau }$ , is the usual gamma function, $k$ is an integer greater than three, and at least two of the $a_{2}$ , $a_{3}$ , and $a_{k}$ are non-zero.",
"Functions of the form (REF ) have been considered in the literature as geometric models of dark energy (e.g.",
"[39]).",
"In this context, the parameters appearing in (REF ) have been constrained through Solar System experiments [40], supernova Ia luminosity distance data [41], and stochastic gravitational wave background limits [42].",
"While in dark energy models one typically sets $0 \\le k < 1$ , the Maclaurin expansion (REF ) does not exist for $k$ in this range, since $f^{\\prime }(0)$ diverges and feeds into equation (REF ).",
"Hence, we consider $k$ to be a positive integer here.",
"In fact the exact solutions presented below exist formally for all real $k$ (though not for any $a_{k}$ ), so some models may admit tachyonic gravitational waves, even when the theory cannot be linearised about a Minkowski background.",
"A discussion of astrophysical constraints on theories of the form (REF ) is presented in Sec.",
"IV.",
"C."
],
[
"Arbitrary phase speed",
"We construct an exact solution that is cylindrically symmetric.",
"Such solutions can be described by the Jordan-Ehlers-Kompaneets line element in Weyl coordinates $(t,\\rho ,\\phi ,z)$ [43], $ ds^2 = e^{-2 \\psi } \\left[ e^{2 \\gamma } \\left( -d t^2 + d \\rho ^2 \\right) + \\rho ^2 d \\phi ^2 \\right] + e^{2\\psi } dz^{2} ,$ where $\\psi $ and $\\gamma $ are functions of $t$ and $\\rho $ [1], [29].",
"In GR, vacuum GW solutions, represented by (REF ) or otherwise, must necessarily have unit propagation speed (see Theorem 8.8 of [32]).",
"Many exact, cylindrical GW solutions are known [44], [45], [46], [47].",
"While non-cylindrical GWs exist (e.g.",
"in Ref.",
"[48] or any multipole with nonzero azimuthal wavenumber; see also below), cylindrical GWs suffice to demonstrate the points considered here.",
"Consider the metric (REF ) for the choices $ \\psi = 0,$ and $ \\gamma = \\frac{1}{2} \\ln \\left\\lbrace A \\left( 1 - v^2 \\right) \\operatorname{csch}\\left[ \\delta + \\omega \\left( t - v \\rho \\right) \\right]^{2} \\right\\rbrace ,$ where $\\operatorname{csch}(\\xi ) = 2/\\left(e^{\\xi } - e^{-\\xi }\\right)$ is the hyperbolic cosecant function (which is singular at $\\xi = 0$ ), $\\omega $ represents a frequency, $A \\ne 0$ is an amplitude factorNote that for $v>1$ $(v<1)$ we require $A<0$ $(A>0)$ to ensure that the metric (REF ) has a Lorentzian signature., $\\delta $ represents a phase shift, and $v \\ne 1$ is a phase velocity.",
"In the zero-frequency limit $\\omega \\rightarrow 0$ we recover the Minkowski spacetime.",
"It can be verified by direct computation that the metric given through (REF ) and (REF ) is a solution to (REF ) for $f$ given by (REF ), provided that (i) the wave satisfies an amplitude-frequency relation, as for any nonlinear wave [49], of the form $(a_{3} \\ne 0)$ $ \\frac{\\omega ^2}{A} = \\frac{\\alpha - 3 a_{2} \\left(k -2 \\right)}{4 a_{3} \\left(k - 3 \\right)};$ and (ii) that the coefficient $a_{k}$ is given by $a_{k} =& \\frac{2^{k-2} \\Gamma (k+1) \\left[ \\alpha - 3 a_{2} \\left( k -2 \\right) \\right]^{-k}}{a_{3}^{2-k} \\left( k -3 \\right)^{3-k} } \\nonumber \\\\& \\times \\Big [ 4 a_{3} \\alpha \\left( k -3 \\right) - 3 a_{2}^2 \\alpha \\left( k - 2 \\right) \\nonumber \\\\& + 9 a_{2}^3 \\left(k -2 \\right)^{2} - 12 a_{2} a_{3} \\left( k -3 \\right) \\left( 2 k -3 \\right) \\Big ] ,$ where $ \\alpha = \\left[ 9 a_{2}^2 \\left( k - 2 \\right)^{2} - 24 a_{3} \\left( k -3 \\right) \\left( k - 1 \\right) \\right]^{1/2}.$ The parameter $v$ , which takes any value except unityThe metric given by (REF ) and (REF ) is genuinely singular for $v=1$ since the Kretschmann invariant, $\\mathcal {K}=R_{\\mu \\nu \\alpha \\beta } R^{\\mu \\nu \\alpha \\beta }$ , diverges there [50]., is the phase speed of the solitonic GW described by (REF )–(REF ).",
"Therefore, tachyonic GWs may exist regardless of the sign or value of $a_{2}$ , contrary to the prediction (REF ) from the linear theory outlined in Sec III.B.",
"Furthermore, the metric is discontinuous for $\\delta = 0$ in Weyl coordinates along the curve $t = v \\rho $ .",
"The causal domain for an event occurring at the origin, which emits GWs described by (REF ) and (REF ), is represented by Fig.",
"REF except that the case $v=c$ is not permitted.",
"In particular, both sub- and super-luminal nonlinear modes exist regardless of the value of $a_{2}$ .",
"To the authors' knowledge, the metric given by (REF ) and (REF ) is reported here for the first time.",
"It should be noted that for an arbitrary value of $k$ , the solution given by (REF ) and (REF ) only exists in the special case, where $a_{k}$ is given by (REF ).",
"There is no reason a priori to favour or disfavour theories that satisfy (REF ).",
"The main purpose of the solution is to demonstrate that the linear criterion $a_{2} > 0$ does not guarantee the absence of tachyonic GWs.",
"Incidentally, we show in Sec.",
"IV.",
"C that the constraint (REF ) is consistent with various astrophysical tests for a variety of values of $a_{2}$ .",
"As a side remark, in GR, it is well known that the gravitational collapse of stars with mass beyond the Tolman-Oppenheimer-Volkoff limit strips away information concerning the collapsing stellar remnant due to the no-hair theorems [51], [52].",
"Information is removed by the formation of horizons, which causally separate regions within the spacetime [53].",
"In $f(R)$ theories of the form (REF ), which permit the existence of superluminal GWs, it is possible that gravitational information can leak beyond the electromagnetic event horizons, which traditionally define black hole boundaries [19] (cf.",
"[54])."
],
[
"Anisotropic propagation",
"We show that arbitrary domains of dependence exist in theories of the form (REF ) by considering a class of GWs that are not cylindrically symmetric.",
"We consider a class of generalised Peres waves, which are described by the line element [55], $ ds^2 =&\\,\\, U(\\rho ) \\left(-dt^2 + dz^2 + d \\rho ^2 + \\rho ^2 d \\phi ^2 \\right) \\nonumber \\\\&\\,\\,+ 2 \\lambda (\\rho ,\\phi ,z+t) \\left( dt + dz \\right)^2$ in Weyl coordinates for some functions $U$ and $\\lambda $ .",
"In GR, the Peres waves (REF ) are defined with $U=1$ and represent a subclass of the well-studied pp-waves [1].",
"A Peres wave represents a GW whose source is electromagnetic in origin.",
"A perturbation of the Faraday tensor in some region of spacetime defines initial conditions, which induce GWs of the form (REF ) [56].",
"In GR, the Einstein equations reduce to the requirement that $\\lambda $ be harmonic in $\\rho $ and $\\phi $ , i.e.",
"$ 0 = \\lambda _{,\\rho \\rho } + \\rho ^{-1} \\lambda _{,\\rho } + \\rho ^{-2} \\lambda _{,\\phi \\phi }.$ However, for certain special choices of $a_{k}$ and $U$ , the metric (REF ) is an exact solution to the field equations (REF ) for any function $\\lambda $ .",
"Explicitly, we find that the metric (REF ) is a solution to (REF ) for $ U(\\rho ) = \\frac{A}{\\omega ^2 \\rho ^2 },$ $ \\frac{A}{\\omega ^2} = \\frac{\\alpha + 3 a_{2} \\left( k -2 \\right)}{2 \\left( k -1 \\right)},$ and $ a_{k} = \\frac{2^{3-2k} 3^{1-k} \\Gamma (k+1) \\left( \\alpha - 3 a_{2} \\right) \\left[ \\alpha + 3 a_{2} \\left( k -2 \\right) \\right]^{k-2} }{\\left( 1 - k \\right)^{k-1} \\left( k - 3 \\right)} ,$ where $A$ and $\\omega $ are constants, the parameter $\\alpha $ is defined through (REF ), $a_{2}$ and $a_{3}$ are arbitrary, and the function $\\lambda $ is arbitrary.",
"To the authors' knowledge, the metric given by (REF ) with (REF )–(REF ) is reported here for the first time.",
"The domain of dependence associated with a generalised Peres wave is arbitrary, because the function $\\lambda $ is arbitrary.",
"Consider, for example, the case $\\lambda ={\\left\\lbrace \\begin{array}{ll}\\exp \\left\\lbrace - A \\left[(t+z)^2 - u(\\rho )^2 \\right]^{-1}\\right\\rbrace \\textrm { for } (t+z)^2 > u(\\rho )^2, \\\\0 \\hspace*{132.02077pt} \\textrm { otherwise}, \\\\\\end{array}\\right.",
"}$ for some function $u$ and amplitude $A>0$ .",
"For $z=0$ , the function $\\lambda $ tends to zero along the curve $t^2 = u(\\rho )^2$ .",
"Hence the metric (REF ) continuously tends to the (conformal) Minkowski spacetime outside this domain, but may have discontinuous derivatives along this boundary [57], [58].",
"Hence the causal domain for an event occurring at the origin is defined as the region $u(\\rho )^2 \\le \\left(t + z \\right)^2$ , which is arbitrary since $u$ is arbitrary.",
"This situation is represented by Fig.",
"REF , where we have $\\partial D^{\\pm }(O) = \\lbrace (t,\\rho ): u(\\rho ) = \\pm |t|\\rbrace $ for $z=0$ .",
"Choices of $u$ exist that yield $D^{+}(O) \\cap D^{-}(O) \\ne \\emptyset $ , indicating that the notions of past and future can become conflated when generalised Peres waves with general $\\lambda $ are permitted."
],
[
"Astrophysical constraints",
"In this section we review briefly, for completeness, astrophysical constraints on polynomial $f(R)$ theories given by (REF ).",
"To this end we introduce the Parametersied Post-Newtonian (PPN) Eddington parameters $\\gamma ^{\\text{PPN}}$ and $\\beta ^{\\text{PPN}}$ , which may be written as [59], [60] $ \\gamma ^{\\text{PPN}} -1 = -\\frac{f^{\\prime \\prime }(R)^2}{f^{\\prime }(R) + 2 f^{\\prime \\prime }(R)},$ and $ \\beta ^{\\text{PPN}} - 1 = \\frac{f^{\\prime }(R) f^{\\prime \\prime }(R)}{8 f^{\\prime }(R) + 12 f^{\\prime \\prime }(R)^{2}} \\frac{d \\gamma ^{\\text{PPN}}}{d R},$ in a general $f(R)$ theory.",
"Table REF presents a summary of data collected from recent Solar System and pulsar timing experiments when interpreted as constraints on the parameters $\\gamma ^{\\text{PPN}}$ and $\\beta ^{\\text{PPN}}$ [61], [62], [63], [64], [65].",
"The PPN parameters (REF ) and (REF ) are evaluated at the measured value of the background scalar curvature $R_{0}$ , which is determined through the Friedmann-Lemaître-Robertson-Walker relationship $R_{0} = 12 c^{-2} H_{0}^2$ where $H_{0}$ is the Hubble constant [66].",
"We assume that the Hubble radius takes the value $c H_{0}^{-1} = 4.0 \\times 10^{3} \\text{ Mpc}$ .",
"Table: Selected Solar System and pulsar timing constraints on the PPN Eddington parameters γ PPN \\gamma ^{\\text{PPN}} and β PPN \\beta ^{\\text{PPN}}.Using the experimental bounds on the parameters $\\gamma ^{\\text{PPN}}$ and $\\beta ^{\\text{PPN}}$ described in Table REF , one can place constraints on the parameters $a_{2}$ , $a_{3}$ , and $a_{k}$ appearing in the function (REF ), and consequently constrain the set of allowed amplitude-frequency relationships (REF ).",
"We focus on the case presented in Sec.",
"IV.",
"A, where we assume that the parameter $a_{k}$ is given by (REF ).",
"In Figure REF we present values of $\\omega ^2 / A$ and $a_{2}$ , which share the same units of length squared, consistent with the data presented in Table REF , for the illustrative choice $k=9$ .",
"Figure REF demonstrates that tachyonic GWs with amplitude-frequency relation $\\omega ^2/A = \\mu $ for some $\\mu $ are permitted within astrophysically-constrained $f(R)$ theories given by (REF ) provided that $a_{2}$ takes a value within the shaded region.",
"In particular, theories for which the linear theory does (does not) predict tachyons according to (REF ), are shown in the blue (red) region.",
"Figure: Allowed values of a 2 a_{2} and ω 2 /|A|\\omega ^2/|A| (shaded region) in arbitrary units for the f(R)f(R) theory given by () consistent with the data presented in Table , under the assumption that a k a_{k} is given by () for k=9k=9.",
"Theories for which the linear theory does (does not) predict tachyons are shaded in blue (red)."
],
[
"Discussion",
"In this paper we study the causal properties of some nonlinear GWs in vacuum $f(R)$ theories of gravity.",
"It is found that the causal domains admit certain exotic features.",
"The phase speeds of the waves can also be arbitrary for a wide range of functions $f(R)$ , a result which does not hold in the linear regime [22].",
"The results suggest that the notion of causality is sensitive to the particulars of the modified theory of gravity under investigation [19].",
"For example, we show that the restrictions on $f(R)$ derived previously to avoid the existence of linear superluminal GW modes [22], [35], [36] must be augmented to avoid the existence of such modes in the nonlinear regime.",
"We emphasise that it is unclear whether the exact, nonlinear solutions discussed here can actually be emitted by a realistic source with a time-dependent quadrupole moment (cf.",
"the discussion in Ref.",
"[67]).",
"A full investigation of causality in this context relies on solving the initial boundary-value problem for a particular experiment, something falling outside the scope of this work.",
"Can the ideas in this paper be tested observationally?",
"In principle, yes, although any astrophysical tests are likely to be confounded by systematic uncertainties introduced by complicated electromagnetic emission physics in the source.",
"For example, if an event occurs which emits electromagnetic and gravitational radiation, some observers may witness the electromagnetic pulses but not the gravitational ones (if $v<c$ ) or vice-versa (if $v>c$ ).",
"Likewise, observers at rest equidistant from the source in different planes may or may not experience the gravitational radiation if the propagation is anisotropic (see Fig.",
"REF ).",
"Moreover, in relativistic systems like a black hole surrounded by an accretion disk, electromagnetic and gravitational wave modes induce backreactions on the disk which may not be felt simultaneously if the phase speeds are different, because of how the metric and Faraday tensors enter into the various magnetohydrodynamic couplings [68], [70], [71], [69], [72].",
"Again, designing a “clean” experiment of this sort is a major challenge in an astrophysical context.",
"The recent detection of GWs by the Laser Interferometer Gravitational-Wave Observatory (LIGO) has opened up new avenues for experimentally determining the phase speed of GWs [73], [74], [75].",
"Given the small number of operational interferometers at present, which makes localisation and real-time electromagnetic follow-up a difficult task, direct bounds placed on the phase speed of GWs are fairly weak at this stage [76], [77], [78] (however see Table 5 of Ref.",
"[79]).",
"To the authors' knowledge, no LIGO-related experimental bounds on the structure of $D^{\\pm }$ exist.",
"Detection of an inherently nonlinear property of GWs, such as the Christodoulou memory [80], [81], would be useful in this direction.",
"It is interesting to compare predictions of the amplitude of the nonlinear memory for various theories of gravity; one can have different memory amplitudes for different GW polarisations; see e.g.",
"equations (7) and (8) of [82]."
],
[
"Acknowledgements",
"We thank the anonymous referee for their carefully considered suggestions, which improved the quality of the manuscript.",
"This work was supported in part by an Australian Postgraduate Award, the Albert Shimmins fund, and the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav) (grant number CE170100004)."
]
] | 1709.01628 |
[
[
"Quasi-neutral limit of Euler-Poisson system of compressible fluids\n coupled to a magnetic field"
],
[
"Abstract In this paper, we consider the quasi-neutral limit of a three dimensional Euler-Poisson system of compressible fluids coupled to a magnetic field.",
"We prove that, as Debye length tends to zero, periodic initial-value problems of the model have unique smooth solutions existing in the time interval where the ideal incompressible magnetohydrodynamic equations has smooth solution.",
"Meanwhile, it is proved that smooth solutions converge to solutions of incompressible magnetohydrodynamic equations with a sharp convergence rate in the process of quasi-neutral limit."
],
[
" Introduction ",
"The main objective of this paper is to study the quasi-neutral limit of the following Euler-Poisson system of compressible fluids coupled to a magnetic field [6], [21]: $&\\partial _tn^{\\lambda }+\\mbox{div}(n^{\\lambda }u^{\\lambda })=0 \\:\\:\\mbox{in}\\:\\:{\\mathbb {T}}^3, t>0,\\\\&\\displaystyle \\partial _t(n^{\\lambda }u^{\\lambda })+\\mbox{div}(n^{\\lambda }u^{\\lambda }\\otimes u^{\\lambda })+\\nabla p(n^{\\lambda })=n^{\\lambda }\\nabla \\phi ^{\\lambda }+\\mbox{curl\\,}B^{\\lambda }\\times B^{\\lambda }, \\\\&\\partial _tB^{\\lambda }-\\mbox{curl\\,}(u^{\\lambda }\\times B^{\\lambda })=0,\\\\&\\mbox{div}B^{\\lambda }=0,\\\\&\\lambda ^2\\Delta \\phi ^{\\lambda }=n^{\\lambda }-1,$ with initial conditions: $n^{\\lambda }(\\cdot ,0)=n^{\\lambda }_{0}, u^{\\lambda }(\\cdot ,0)=u^{\\lambda }_{0}, B^{\\lambda }(\\cdot ,0)=B^{\\lambda }_{0}.$ In the above equations, ${\\mathbb {T}}^3$ is 3-dimensional torus and $\\lambda >0$ is the (scaled) Debye length.",
"The unknown functions are the density $n^{\\lambda }$ , the velocity $u^{\\lambda } =(u^{\\lambda }_1,u^{\\lambda }_2,u^{\\lambda }_3)$ , the magnetic field $B^{\\lambda } = (B^{\\lambda }_1,B^{\\lambda }_2,B^{\\lambda }_3)$ and the gravitational potential $\\phi ^{\\lambda }$ .",
"Throughout this paper, we assume that the pressure function $p(n^{\\lambda })$ satisfies the usual $\\gamma -$ law, $p(n^{\\lambda })=\\frac{(n^{\\lambda })^\\gamma }{\\gamma },n^\\lambda >0,$ for some constant $\\gamma >1$ .",
"It is obvious that equations() is redundant with equations (), as soon as they are satisfied by the initial conditions $\\mbox{div}B^{\\lambda }_0=0$ .",
"System (REF )-() is used to model the evolution of a magnetic stars [3].",
"The effects of magnetic fields arise in some physically interesting and important phenomena in astrophysics; e.g.",
"solar flares.",
"Without taking magnetic effects into account, system (REF )-() reduces to the Euler-Poisson equations.",
"In recent years, the quasi-neutral limit ($\\lambda \\rightarrow 0$ ) of various models has attracted much attention.",
"In particular, the limit $\\lambda \\rightarrow 0$ has been performed in Vlasov-Poisson system by Brenier [1], Grenier [11], [12], [13] and Masmoudi [19], in drift-diffusion equations by Gasser et al.",
"[7], [8] and Jüngel and Peng [9], and in the one dimensional and isothermal Euler-Poisson system by Cordier and Grenier [2], in more general isentropic models by Wang [23], in non-isentropic Euler-Poisson equations by Peng et al.",
"[20] and Li [15], in Euler-Monge-Ampère systems by Loeper [17], in Navier-Stokes-Poisson system by Wang and Jiang [24], Donatelli and Marcati [5] and Ju et al.",
"[10], in quantum hydrodynamics equations [14], in Navier-Stokes-Fourier-Poisson system by Li et al.",
"[16], etc.",
"As far as we know, there is no result on quasi-neutral limit of the Euler-Poisson system coupled to a magnetic field (REF )-().",
"In this paper, we will study the quasi-neutral limit for the smooth solution of the system (REF )-() in the framework of the convergence-stability principle developed in [26].",
"Formally, taking the (scaled) Debye length $\\lambda \\rightarrow 0$ in (), we obtain the following ideal incompressible magnetohydrodynamic equations $&\\partial _tu^0+(u^0\\cdot \\nabla )u^0+\\nabla p^0=\\mbox{curl\\,}B^0\\times B^0,\\\\& \\partial _tB^0-\\mbox{curl\\,}(u^0\\times B^0)=0,\\\\& \\mbox{div}u^0=\\mbox{div}B^0=0.$ The objective of this paper is to make this limit rigorous.",
"Our proof requires the (local) existence of a smooth solution to (REF )-(), which is shown in next section.",
"The proof of our result is based on the convergence-stability principle developed by Yong [25], [26] for singular limit problems of symmetrizable hyperbolic systems.",
"In contrast with the results in [1], [2], [5], [10], [14], [15], [16], [17], [23], [24], where the limiting equations are incompressible Euler equations or the incompressible Navier-Stokes equations, our limiting equations are the incompressible magnetohydrodynamic equations (REF )-().",
"In our case where the Euler-Poisson equations are coupled to a magnetic field, the problem becomes more challenging.",
"Because of the magnetic field and non-linearity terms, some elaborated energy analysis are required to obtain the desired convergence results.",
"This paper is organized as follows.",
"In section 2, we rewrite the system (REF )-() as a symmetrizable hyperbolic system to obtain the local-in-time existence result, and present our main results.",
"The proof of Theorem REF is obtained in section 3.",
"Before ending the introduction, we give the notation and Lemma used throughout the current paper.",
"The letters $C$ and $C_T$ denote various positive constants independent of $\\lambda $ , which can be different from one line to another one, but $C_T$ may depend on $T$ .",
"The symbol “:”means summation over both matrix indices.",
"$|U|$ denotes some norm of a vector or matrix $U$ .",
"Also, we denote $\\Vert \\cdot \\Vert =\\Vert \\cdot \\Vert _{L^2({\\mathbb {T}}^3)},\\,\\,\\Vert \\cdot \\Vert _{\\infty }=\\Vert \\cdot \\Vert _{L^\\infty ({\\mathbb {T}}^3)},\\,\\,\\Vert \\cdot \\Vert _{k}=\\Vert \\cdot \\Vert _{H^k({\\mathbb {T}}^3)}, \\,\\,k\\in \\mathbb {N}^{\\ast }.$ Lemma 1.1 (see, e.g.[18]).",
"Let $s, s_1$ , and $s_2$ be three nonnegative integers and $s_0=\\left[\\frac{d}{2}\\right] + 1$ .",
"1.",
"If $f, g\\in H^s\\cap L^\\infty $ and any nonnegative multi index $\\beta , |\\beta |\\le s$ , then we have $\\Vert D^\\beta (fg)\\Vert \\le C_s (\\Vert f\\Vert _{L^\\infty }\\Vert D^\\beta g\\Vert +\\Vert g\\Vert _{L^\\infty }\\Vert D^\\beta f\\Vert )\\le c_s\\Vert f\\Vert _s\\Vert g\\Vert _s.$ 2.",
"If $f\\in H^s$ , $Df\\in L^\\infty $ ,$g\\in H^{s-1}\\cap L^\\infty $ , then we have $\\Vert D^\\beta (fg)-fD^\\beta g\\Vert \\le C_s (\\Vert Df\\Vert _{L^\\infty }\\Vert D_x^{\\beta ^{\\prime }}g\\Vert +\\Vert g\\Vert _{L^\\infty }\\Vert D^\\beta f\\Vert ),\\:|\\beta ^{\\prime }|=|\\beta |- 1.",
"$ 3.",
"Let $s_3=\\min \\lbrace s_1,s_2,s_1+s_2-s_0\\rbrace \\ge 0$ , then $H^{s_1}H^{s_2}\\subset H^{s_3}$ .",
"Here the inclusion symbol $\\subset $ implies the continuity of the embedding.",
"4.",
"Suppose $s\\ge s_0$ , $A\\in C_b^s(G)$ , and $U\\in H^s(\\Omega ,G)$ .",
"Then $A(U(\\cdot ))\\in H^s$ and $\\Vert A(U(\\cdot ))\\Vert _s\\le C_s|A|_s(1+\\Vert U\\Vert ^s_s).$ Here and below, $C_s$ denotes a generic constant depending only on $s$ and $d$ , and $|A|_s$ stands for $\\sup \\limits _{\\lbrace U\\in G,|\\alpha |\\le s\\rbrace }|\\partial ^{\\alpha }_{U}A(U)|$ ."
],
[
"Main Results",
"First, we consider the local existence of smooth solution of the system (REF )-() for any fixed $\\lambda >0$ .",
"By Green's formulation, it follows from (REF ) and () that $\\nabla \\phi ^\\lambda =\\frac{1}{\\lambda ^2}\\left(\\nabla \\Delta ^{-1}(n^\\lambda _0-1)-\\nabla \\Delta ^{-1}\\mbox{div}\\int _0^t(n^\\lambda u^\\lambda )(x,\\tau )d\\tau \\right).$ Using (REF ) and the following equality $\\mbox{div}(n^\\lambda u^\\lambda \\otimes u^\\lambda )=n^\\lambda (u^\\lambda \\cdot \\nabla )u^\\lambda +u^\\lambda \\mbox{div}(n^\\lambda u^\\lambda ),$ we can rewrite () as $\\partial _tu^\\lambda +(u^\\lambda \\cdot \\nabla )u^\\lambda +\\nabla h(n^\\lambda )=\\nabla \\phi ^\\lambda +\\frac{1}{n^\\lambda }\\mbox{curl\\,}B^\\lambda \\times B^\\lambda ,$ where the enthalpy $h(n^\\lambda ) > 0$ is defined by $h^{\\prime }(n^\\lambda )=\\frac{p^{\\prime }(n^\\lambda )}{n^\\lambda }$ for $n^\\lambda >0$ .",
"Set $ W^\\lambda =\\left(\\begin{array}{c}n^\\lambda \\\\u^\\lambda \\\\B^\\lambda \\\\\\end{array}\\right), W^\\lambda _0=\\left(\\begin{array}{c}n^\\lambda _0 \\\\u^\\lambda _0\\\\B^\\lambda _0 \\\\\\end{array}\\right),$ $A_i(W^\\lambda )=\\left(\\begin{array}{ccccccc}u^\\lambda _i & n^\\lambda e_i^T & 0 \\\\h^{\\prime }(n^\\lambda )e_i & u^\\lambda _i\\textbf {I}_{3\\times 3} & \\frac{(G^\\lambda _i)^T}{n^\\lambda } \\\\0 & \\frac{G^\\lambda _i}{n^\\lambda } & u^\\lambda _i\\textbf {I}_{3\\times 3}\\\\\\end{array}\\right), $ $F^\\lambda =\\left(\\begin{array}{c}0 \\\\\\nabla \\Delta ^{-1}(n^\\lambda _0-1)-\\nabla \\Delta ^{-1}\\mbox{div}\\int _0^t(n^\\lambda u^\\lambda )(x,\\tau )d\\tau \\\\0 \\\\\\end{array}\\right),$ where $(e_1,e_2,e_3)$ is the canonical base of $\\mathbb {R}^3$ , $\\textbf {I}_{3\\times 3}$ is a unit matrix, $y_i$ denotes the $i$ th component of $y\\in \\mathbb {R}^3$ and $G^\\lambda _1=\\left(\\begin{array}{ccc}0 & 0 & 0 \\\\B^\\lambda _2 & -B^\\lambda _1 & 0 \\\\B^\\lambda _3 & 0 & -B^\\lambda _1 \\\\\\end{array}\\right),G^\\lambda _2=\\left(\\begin{array}{ccc}-B^\\lambda _2 & B^\\lambda _1 & 0 \\\\0 & 0 & 0 \\\\0 & B^\\lambda _3 & -B^\\lambda _2 \\\\\\end{array}\\right),$ $G^\\lambda _3=\\left(\\begin{array}{ccc}-B^\\lambda _3 & 0 & B^\\lambda _1 \\\\0 & -B^\\lambda _3 & B^\\lambda _2 \\\\0 & 0 & 0 \\\\\\end{array}\\right).$ Thus the problem (REF )-(REF ) for the unknown $W$ can be rewritten as $&\\partial _tW^\\lambda +\\sum \\limits _{i=1}^{3}A_i(W^\\lambda )\\partial _{x_i}W^\\lambda =\\frac{1}{\\lambda ^2}F^\\lambda ,\\\\&W^\\lambda |_{t=0}=W^\\lambda _0.$ It is not difficult to see that the equations of $W^\\lambda $ in (REF )-() are symmetrizable hyperbolic, i.e.",
"if we introduce $A_0(n^\\lambda )=\\left(\\begin{array}{ccccccc}h^{\\prime }(n^\\lambda ) & 0 & 0 \\\\0 & n^\\lambda \\textbf {I}_{3\\times 3} & 0 \\\\0 & 0 &n^\\lambda \\textbf {I}_{3\\times 3} \\\\\\end{array}\\right),$ which is positively definite when $n^\\lambda >\\delta >0$ , then $\\widehat{A}_i(W^\\lambda )=A_0(n^\\lambda )A_i(W^\\lambda )$ are symmetric for all $1\\le i\\le 3$ .",
"Thus, to solve the system (REF )-(REF ), it suffices to solve the system (REF )-().",
"Since the non-local source term $\\nabla \\Delta ^{-1}\\mbox{div}\\int _0^t(n^\\lambda u^\\lambda )(x,\\tau )d\\tau $ is a sum of products of Riesz transforms of $\\int _0^t(n^\\lambda u^\\lambda )(x,\\tau )d\\tau $ , we have, by the $L^2$ boundedness of the Riesz transformation (see [22]), $\\left\\Vert \\nabla \\Delta ^{-1}\\mbox{div}\\int _{0}^{t}(n^\\lambda u^\\lambda )(x,\\tau )d\\tau \\right\\Vert _{s}\\le C\\left\\Vert \\int _{0}^{t}(n^\\lambda u^\\lambda )(x,\\mu )d\\mu \\right\\Vert _{s},$ for some constant $C > 0$ independent of $t$ .",
"Moreover, we recall the other elementary fact which can be easily proven by using Fourier series.",
"Lemma 2.1 $\\nabla \\Delta ^{-1}$ is a bounded linear operator from $V=\\lbrace v\\in L^2({\\mathbb {T}}^3)| \\textbf {m}(v)=0\\rbrace $ into $H^1({\\mathbb {T}}^3)$ .",
"So one gets $\\Vert \\nabla \\Delta ^{-1}(n^\\lambda _{0}-1)\\Vert _{s+1}\\le C\\Vert (n^\\lambda _{0}-1)\\Vert _s$ for some constant $C > 0$ .",
"Based on the above crucial facts, using the standard iteration techniques of local existence theory for symmetrizable hyperbolic systems (see [18]), we have Proposition 2.2 Assume that the initial conditions $(n^\\lambda _0,u^\\lambda _0,B^\\lambda _0)\\in H^s({\\mathbb {T}}^3)$ , $s>\\frac{5}{2}, n_0>\\delta >0, \\emph {\\mbox{div}} B^\\lambda _0=0$ and $\\int _{\\mathbb {T}}^3(n^\\lambda _0-1)dx=0$ .",
"Then for any fixed $\\lambda >0$ , there exists a positive constant $T$ (may depend on $\\lambda $ ) such that the periodic problem (REF )-() has a unique smooth solution $(n^\\lambda , u^\\lambda , B^\\lambda )\\in C([0,T];H^s({\\mathbb {T}}^3))$ , well-defined on ${\\mathbb {T}}^3\\times [0,T]$ .",
"Hence the nonlinear periodic problem (REF )-(REF ) admits a unique solution $(n^\\lambda , u^\\lambda ,\\nabla \\phi ^\\lambda ,B^\\lambda )$ satisfying $(n^\\lambda , u^\\lambda ,\\nabla \\phi ^\\lambda , B^\\lambda )\\in C([0,T];H^s({\\mathbb {T}}^3)).$ According to Proposition REF , for each fixed $\\lambda >0$ in (REF )-(REF ), there exists a time interval $[0,T]$ such that system (REF )-(REF ) has a unique solution $(n^\\lambda , u^\\lambda ,\\nabla \\phi ^\\lambda ,B^\\lambda )$ satisfying (REF ).",
"Define $\\begin{split} T_\\lambda &=\\sup \\,\\Big \\lbrace T>0: (n^\\lambda , u^\\lambda ,\\nabla \\phi ^\\lambda , B^\\lambda )\\in C([0,T];H^s({\\mathbb {T}}^3)),\\\\& \\frac{1}{2}\\le n^\\lambda \\le \\frac{3}{2}, \\forall (x,t)\\in {\\mathbb {T}}^3\\times [0,T]\\Big \\rbrace .\\end{split}$ Namely, $[0, T_\\lambda ]$ is the maximal time interval of $H^s-$ existence.",
"Note that $T_\\lambda $ may tend to zero as $\\lambda $ goes to 0.",
"In order to show that $\\liminf \\limits _{\\lambda \\rightarrow 0}T_\\lambda > 0$ , we follow the convergence-stability principe [26] and seek a formal approximation of $(n^\\lambda , u^\\lambda , B^\\lambda )$ .",
"To this end, we consider the initial-value problem of the ideal incompressible magnetohydrodynamic equation (REF )-() with initial data $u^{0}(\\cdot ,0)=u^{0}_{0}, B^{0}(\\cdot ,0)=B^{0}_{0}.$ Let us recall the local existence of a strong solution to the ideal incompressible magnetohydrodynamic equations (REF )-().",
"The proof can be found in [4].",
"Proposition 2.3 (see [4].)",
"Let $s >\\frac{5}{2}$ be an integer.",
"Assume that the initial data $(u^0(x, t),B^0(x, t))|_{t=0} = (u^0_0(x),B^0_0(x))$ satisfy $(u^0_0,B^0_0)\\in H^{s+1}({\\mathbb {T}}^3) \\:\\:\\:\\emph {\\mbox{and}}\\:\\:\\:\\emph {\\mbox{div}} u^0_0 = 0, \\emph {\\mbox{div}} B^0_0 = 0.$ Then, there exist a $T_\\ast \\in (0,+\\infty )$ and a unique solution $(u^0,B^0)\\in L^\\infty ([0, T_\\ast ),H^{s+1}({\\mathbb {T}}^3))$ to the ideal incompressible magnetohydrodynamic equations (REF )-() satisfying, for any $0 < T < T_\\ast $ , $\\emph {\\mbox{div}} u^0 = 0, \\emph {\\mbox{div}} B^0 = 0$ and $\\sup _{0\\le t\\le T}\\Big (\\Vert (u^0,B^0)\\Vert _{s+1}+\\Vert (\\partial _t{u^0},\\partial _tB^0)\\Vert _{s}+\\Vert \\nabla p^0\\Vert _{s+1} +\\Vert \\partial _t\\nabla p^0\\Vert _{s}\\Big )\\le C_T $ for some positive constant $C_T$ .",
"Now the main result of this paper reads as follows.",
"Theorem 2.4 Let $s > \\frac{3}{2}+ 1$ be an integer.",
"Suppose $\\emph {\\mbox{div}} u^0_0=0,$ $\\emph {\\mbox{div}} B^0_0=0$ , and incompressible magnetohydrodynamic equations (REF )-() with the initial data $(u^0_0,B^0_0)$ has a solution $(u^0,B^0)\\in L^\\infty ([0, T_\\ast ),H^{s+1}({\\mathbb {T}}^3))$ .",
"Then, for $\\lambda $ sufficiently small, there exists a $\\lambda -$ independent positive number $T_{\\ast \\ast }<T\\ast $ , such that the model (REF )-() with periodic initial data $(n^\\lambda _0,u^\\lambda _0,B^\\lambda _0)$ satisfying $n^\\lambda _0=1, u^\\lambda _0=u^0_0, B^\\lambda _0=B^0_0,$ has a unique solution $(n^\\lambda ,u^\\lambda ,B^\\lambda ,\\nabla \\phi ^\\lambda )\\in C([0,T_{\\ast \\ast }];H^s({\\mathbb {T}}^3))$ .",
"Moreover, there exists a $\\lambda -$ independent constant $M>0$ such that $\\sup _{0\\le t\\le T}(\\Vert n^\\lambda -1\\Vert _s+\\Vert u^\\lambda -u^0\\Vert _s+\\Vert B^\\lambda -B^0\\Vert _s)\\le M\\lambda .$ Remark 2.1 The initial data $n^\\lambda _0=1, u^\\lambda _0=u^0_0, B^\\lambda _0=B^0_0$ can be relaxed as $n^\\lambda _0=1+O(\\lambda ^2), u^\\lambda _0=u^0_0+O(\\lambda ),B^\\lambda _0=B^0_0+O(\\lambda ),$ without changing our arguments.",
"Remark 2.2 Theorem REF describes the quasi-neutral limit $\\lambda \\rightarrow 0$ of the system (REF )-() with well-prepared initial data, avoiding the presence of the initial time layer.",
"We will discuss the case of general initial data (ill-prepared initial data) allowing the presence of the fast singular oscillation in the future."
],
[
"Proof of Theorem ",
"Thanks to the convergence-stability principle developed in [25], [26], it suffices to prove the error estimate in (REF ) for $t\\in [0,\\min \\lbrace T_\\lambda ,T_{\\ast \\ast }\\rbrace ]$ with $T_{\\ast \\ast }<T_{\\ast }$ independent of $\\lambda $ and to be determined.",
"Thus we directly make the error estimate (REF ) in the time interval $[0,\\min \\lbrace T_\\lambda ,T_{\\ast \\ast }\\rbrace ]$ .",
"Now we rewrite (REF ) as the following form $&\\partial _tW^\\lambda +\\sum \\limits _{i=1}^{3}A_i(W^\\lambda )\\partial _{x_i}W^\\lambda =\\left(\\begin{array}{c}0 \\\\\\nabla \\phi ^\\lambda \\\\0 \\\\\\end{array}\\right),\\\\&\\lambda ^2\\Delta \\phi ^{\\lambda }=n^\\lambda -1,\\mbox{div}B^\\lambda =0.$ We note that with $(u^0,p^0,B^0)$ constructed in Proposition REF , $(n_\\lambda ,u_\\lambda ,\\phi _\\lambda ,B_\\lambda )=(1,u^0,-p^0,B^0)$ satisfies $&\\partial _tn_{\\lambda }+\\mbox{div}(n_{\\lambda }u_{\\lambda })=0,\\\\&\\displaystyle \\partial _tu^{\\lambda }+(u_{\\lambda }\\cdot \\nabla )u_{\\lambda }+\\nabla h^{\\prime }(n_{\\lambda })=\\nabla \\phi _{\\lambda }+\\frac{1}{n_\\lambda }\\mbox{curl\\,}B_{\\lambda }\\times B_{\\lambda }, \\\\&\\partial _tB_{\\lambda }-\\mbox{curl\\,}(u_{\\lambda }\\times B_{\\lambda })=0,\\\\&\\mbox{div}B_{\\lambda }=0,\\\\&\\lambda ^2\\Delta \\phi _{\\lambda }=n_{\\lambda }-1-\\lambda ^2\\Delta Q,$ with $Q=-p^0.$ From Proposition REF , we have $\\sup _{t\\in [0,T_\\ast ]}(\\Vert \\nabla Q(\\cdot ,t)\\Vert _{s+1}+\\Vert \\partial _t\\nabla Q(\\cdot ,t)\\Vert _s)<+\\infty .$ So, we can rewrite (REF )-() as $&\\partial _tW_\\lambda +\\sum \\limits _{i=1}^{3}A_i(W_\\lambda )\\partial _{x_i}W_\\lambda =\\left(\\begin{array}{c}0 \\\\\\nabla \\phi _\\lambda \\\\0 \\\\\\end{array}\\right),\\\\&\\lambda ^2\\Delta \\phi _{\\lambda }=n_\\lambda -1-\\lambda ^2\\Delta Q,\\mbox{div}B^\\lambda =0.$ Set $E=W^\\lambda -W_\\lambda =\\left(\\begin{array}{c}N \\\\U \\\\H \\\\\\end{array}\\right)=\\left(\\begin{array}{c}n^\\lambda -n_\\lambda \\\\u^\\lambda -u_\\lambda \\\\B^\\lambda -B_\\lambda \\\\\\end{array}\\right), \\Phi =\\lambda (\\phi ^\\lambda -\\phi _\\lambda ).$ We deduce from (REF )-() and (REF )-() that $&\\partial _tE+\\sum \\limits _{i=1}^{3}A_i(W^\\lambda )\\partial _{x_i}E=\\frac{1}{\\lambda }G+\\sum \\limits _{i=1}^{3}(A_i(W^\\lambda )-A_i(W_\\lambda ))\\partial _{x_i}W_\\lambda ,\\\\&\\lambda \\Delta \\Phi =N+\\lambda ^2\\Delta Q,$ where $G=\\left(\\begin{array}{c}0 \\\\\\nabla \\Phi \\\\0 \\\\\\end{array}\\right).$ We differentiate (REF ) with $\\partial ^{\\alpha }_{x}$ for a multi-index $\\alpha $ satisfying $|\\alpha |\\le s$ with $s>\\frac{5}{2}$ to get $&\\partial _tE_\\alpha +\\sum \\limits _{i=1}^{3}A_i(W^\\lambda )\\partial _{x_i}E_\\alpha =\\frac{1}{\\lambda }G_\\alpha +R^1_\\alpha +R^2_\\alpha $ with $\\partial ^{\\alpha }_{x}f=f_\\alpha ,$ where $R^1_\\alpha =\\sum \\limits _{i=1}^{3}[(A_i(W^\\lambda )-A_i(W_\\lambda ))\\partial _{x_i}W_\\lambda ]_\\alpha ,$ $R^2_\\alpha =\\sum \\limits _{i=1}^{3}[A_i(W^\\lambda )\\partial _{x_i}E_\\alpha -(A_i(W^\\lambda )\\partial _{x_i}E)_\\alpha ].$ For the sake of clarity, we divide the following arguments into lemmas.",
"Lemma 3.1 Under the conditions of Theorem REF, we have $&\\frac{d}{dt}\\int _{{\\mathbb {T}}^3}(E^T_\\alpha A_0(n^\\lambda )E_\\alpha +|\\nabla \\Phi _\\alpha |^2)dx\\\\&\\qquad \\le C\\Vert \\nabla \\Phi \\Vert _s^2+\\frac{C}{\\lambda }\\Vert U\\Vert _s\\Vert N\\Vert _s\\Vert \\nabla \\Phi _\\alpha \\Vert +\\frac{2}{\\lambda }\\Vert E\\Vert _s\\Vert U_\\alpha \\Vert \\Vert \\nabla \\Phi _\\alpha \\Vert \\\\&\\quad \\quad \\quad +C\\Vert E_\\alpha \\Vert \\Vert R^1_\\alpha \\Vert +C\\Vert E_\\alpha \\Vert \\Vert R^2_\\alpha \\Vert +C(1+\\Vert E\\Vert _s)\\Vert E_\\alpha \\Vert ^2+\\lambda ^2,$ where $C$ is a generic constant depending only on the range $(\\frac{1}{2},\\frac{3}{2})$ of $n^\\lambda $ .",
"Taking the $L^2$ inner product of (REF ) with $D_0(n^\\lambda )E_\\alpha $ , one gets, by integration by parts, that $\\frac{d}{dt}\\int _{{\\mathbb {T}}^3}E^T_\\alpha A_0(n^\\lambda )E_\\alpha dx&=\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}E^T_\\alpha A_0(n^\\lambda )G_\\alpha dx+2\\int _{{\\mathbb {T}}^3}E^T_\\alpha A_0(n^\\lambda )R^1_\\alpha dx\\\\&\\quad +2\\int _{{\\mathbb {T}}^3}E^T_\\alpha A_0(n^\\lambda )R^2_\\alpha dx\\\\&\\quad +\\int _{{\\mathbb {T}}^3}E^T_\\alpha \\mbox{div}A(W^\\lambda )E_\\alpha dx\\\\&=\\mathcal {I}^1_\\alpha +\\mathcal {I}^2_\\alpha +\\mathcal {I}^3_\\alpha +\\mathcal {I}^4_\\alpha $ with $\\mbox{div}A(W^\\lambda )=\\partial _tA_0(n^\\lambda )+\\sum \\limits _{i=1}^{3}\\partial _{x_{i}}(A_0(n^\\lambda )A_i(W^\\lambda )).$ Recalling that $A_0(n^\\lambda )=\\left(\\begin{array}{ccccccc}h^{\\prime }(n^\\lambda ) & 0 & 0 \\\\0 & n^\\lambda \\textbf {I}_{3\\times 3} & 0 \\\\0 & 0 & n^\\lambda \\textbf {I}_{3\\times 3} \\\\\\end{array}\\right)$ and $n^\\lambda =n_\\lambda +N=1+N$ , it is obvious that $\\mathcal {I}^1_\\alpha &=\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}n^\\lambda E^T_\\alpha G_\\alpha dx\\\\&= \\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}n^\\lambda U_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&= \\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}(1+N)U_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&=-\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}\\mbox{div}U_\\alpha \\cdot \\Phi _\\alpha dx+\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}NU_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&\\le -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}\\mbox{div}U_\\alpha \\cdot \\Phi _\\alpha dx+\\frac{2}{\\lambda }\\Vert N\\Vert _\\infty \\int _{{\\mathbb {T}}^3}|U_\\alpha ||\\nabla \\Phi _\\alpha |dx\\\\&\\le -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}\\mbox{div}U_\\alpha \\cdot \\Phi _\\alpha dx+\\frac{2}{\\lambda }\\Vert N\\Vert _s\\Vert U_\\alpha \\Vert \\Vert \\nabla \\Phi _\\alpha \\Vert .$ Recalling $n_\\lambda =1$ and $N=n^\\lambda -n_\\lambda $ , from (REF ) and (REF ), we have $\\mbox{div}U=-\\partial _tN-\\mbox{div}(u^\\lambda N).$ Then, from () we have $-\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3}\\mbox{div}U_\\alpha \\cdot \\Phi _\\alpha dx&=\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} \\partial _tN_\\alpha \\cdot \\Phi _\\alpha dx-\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u^\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&= \\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (\\lambda \\partial _t\\Delta \\Phi _\\alpha -\\lambda ^2\\partial _t\\Delta Q_\\alpha )\\cdot \\Phi _\\alpha dx-\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u^\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&=-\\frac{d}{dt}\\int _{{\\mathbb {T}}^3}|\\nabla \\Phi _\\alpha |^2dx+2\\lambda \\int _{{\\mathbb {T}}^3} \\partial _t\\nabla Q_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&\\quad -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (U N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u_\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&\\le -\\frac{d}{dt}\\int _{{\\mathbb {T}}^3} |\\nabla \\Phi _\\alpha |^2dx+2\\lambda \\Vert \\partial _t\\nabla Q_\\alpha \\Vert \\Vert \\nabla \\Phi _\\alpha \\Vert \\\\&\\quad +\\frac{C}{\\lambda }\\Vert U\\Vert _s\\Vert N\\Vert _s\\Vert \\nabla \\Phi _\\alpha \\Vert -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u_\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&\\le -\\frac{d}{dt}\\int _{{\\mathbb {T}}^3} |\\nabla \\Phi _\\alpha |^2dx+C\\Vert \\nabla \\Phi _\\alpha \\Vert ^2+\\lambda ^2+\\frac{C}{\\lambda }\\Vert U\\Vert _s\\Vert N\\Vert _s\\Vert \\nabla \\Phi _\\alpha \\Vert \\\\&\\quad -\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u_\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx.$ Using () and $\\mbox{div}u_\\lambda =0$ , we have, by part by integrate, that $-\\frac{2}{\\lambda }\\int _{{\\mathbb {T}}^3} (u_\\lambda N)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx&=-2\\int _{{\\mathbb {T}}^3} (u_\\lambda \\Delta \\Phi )_\\alpha \\cdot \\nabla \\Phi _\\alpha dx-2\\lambda \\int _{{\\mathbb {T}}^3} (u_\\lambda \\Delta Q)_\\alpha \\cdot \\nabla \\Phi _\\alpha dx\\\\&\\le -2\\int _{{\\mathbb {T}}^3} (u_\\lambda \\Delta \\Phi )_\\alpha \\cdot \\nabla \\Phi _\\alpha dx+C\\Vert \\nabla \\Phi _\\alpha \\Vert ^2+\\lambda ^2\\\\&=-2\\int _{{\\mathbb {T}}^3} u_\\lambda \\Delta \\Phi _\\alpha \\cdot \\nabla \\Phi _\\alpha dx+C\\Vert \\nabla \\Phi _\\alpha \\Vert ^2+\\lambda ^2\\\\&\\quad -2\\int _{{\\mathbb {T}}^3}[ (u_\\lambda \\Delta \\Phi )_\\alpha -u_\\lambda \\Delta \\Phi _\\alpha ]\\cdot \\nabla \\Phi _\\alpha dx\\\\&=2\\int _{{\\mathbb {T}}^3}\\nabla u_\\lambda :(\\nabla \\Phi _\\alpha \\otimes \\nabla \\Phi _\\alpha ) dx+C\\Vert \\nabla \\Phi _\\alpha \\Vert ^2+\\lambda ^2\\\\&\\quad -2\\int _{{\\mathbb {T}}^3}[ (u_\\lambda \\Delta \\Phi )_\\alpha -u_\\lambda \\Delta \\Phi _\\alpha ]\\cdot \\nabla \\Phi _\\alpha dx\\\\&\\le C\\Vert \\nabla \\Phi \\Vert _s^2+\\lambda ^2.$ where we have used the formulation $\\Delta \\Phi _\\alpha \\cdot \\nabla \\Phi _\\alpha =\\mbox{div}(\\nabla \\Phi _\\alpha \\otimes \\nabla \\Phi _\\alpha )-\\frac{1}{2}\\nabla |\\nabla \\Phi _\\alpha |^2.$ Then we can show that $\\mathcal {I}^1_\\alpha &\\le -\\frac{d}{dt}\\int _{{\\mathbb {T}}^3} |\\nabla \\Phi _\\alpha |^2dx+C\\Vert \\nabla \\Phi \\Vert _s^2+\\frac{C}{\\lambda }\\Vert U\\Vert _s\\Vert N\\Vert _s\\Vert \\nabla \\Phi _\\alpha \\Vert \\\\&\\quad +\\frac{2}{\\lambda }\\Vert E\\Vert _s\\Vert U_\\alpha \\Vert \\Vert \\nabla \\Phi _\\alpha \\Vert +\\lambda ^2.$ For $\\mathcal {I}^2_\\alpha $ and $\\mathcal {I}^3_\\alpha $ , they are simply estimated as $\\mathcal {I}^2_\\alpha &=C\\int _{{\\mathbb {T}}^3}|E_\\alpha ||R^1_\\alpha |dx\\le C\\Vert E_\\alpha \\Vert \\Vert R^1_\\alpha \\Vert ,\\\\\\mathcal {I}^3_\\alpha &=C\\int _{{\\mathbb {T}}^3}|E_\\alpha ||R^1_\\alpha | dx\\le C\\Vert E_\\alpha \\Vert \\Vert R^2_\\alpha \\Vert .$ Moreover, we have $|\\mbox{div}A(W^\\lambda )|\\le C(1+\\Vert E\\Vert _s).$ Then $\\mathcal {I}^4_\\alpha $ can be estimated as $\\mathcal {I}^4_\\alpha \\le \\Vert \\mbox{div}A(W^\\lambda )\\Vert _\\infty \\int _{{\\mathbb {T}}^3}E^T_\\alpha E_\\alpha dx\\le C(1+\\Vert E\\Vert _s)\\Vert E_\\alpha \\Vert ^2.$ Now, substituting the inequalities (REF )-(REF ) into (REF ) gives (REF ).",
"Set $\\mathcal {D}=\\mathcal {D}(t)=\\frac{\\Vert E\\Vert _s+\\Vert \\nabla \\Phi \\Vert _s}{\\lambda }.$ Then, for the inequality in Lemma REF , we have the following claim.",
"Lemma 3.2 For any $ \\lambda \\in (0,1)$ , we have $&\\frac{d}{dt}\\int _{{\\mathbb {T}}^3}(E^T_\\alpha A_0(n^\\lambda )E_\\alpha +|\\nabla \\Phi _\\alpha |^2)dx\\le C(1+\\mathcal {D}^s)(\\Vert E\\Vert ^2_{s}+\\Vert \\nabla \\Phi \\Vert ^2_s)+\\lambda ^2.$ It is obviously that $&\\frac{C}{\\lambda }\\Vert U\\Vert _s\\Vert N\\Vert _s\\Vert \\nabla \\Phi _\\alpha \\Vert \\le C\\mathcal {D}(\\Vert E\\Vert _s^2+\\Vert \\nabla \\Phi \\Vert ^2_s),\\\\&\\frac{2}{\\lambda }\\Vert E\\Vert _s\\Vert U_\\alpha \\Vert \\Vert \\nabla \\Phi _\\alpha \\Vert \\le C\\mathcal {D}(\\Vert E\\Vert _s^2+\\Vert \\nabla \\Phi \\Vert ^2_s),\\\\&(1+\\Vert E\\Vert _s)\\Vert E_\\alpha \\Vert ^2\\le C(1+\\Vert \\mathcal {D}\\Vert ^s_s)\\Vert E\\Vert ^2_s.$ Next we estimate $\\Vert E_\\alpha \\Vert \\Vert R^1_\\alpha \\Vert $ .",
"We use the boundedness of $\\Vert (n_\\lambda ,u_\\lambda ,B_\\lambda )\\Vert _{s+1}=\\Vert (1,u^0,B^0)\\Vert _{s+1}$ indicated in Proposition REF to conclude that $\\Vert R^1_\\alpha \\Vert &\\le C\\sum ^{3}_{i=1}\\Vert u^\\lambda _i-u_{\\lambda i}\\Vert _{|\\alpha |}\\Vert \\partial _{x_{i}}W_\\lambda \\Vert _s+C\\sum ^{3}_{i=1}\\Vert h^{\\prime }(n^\\lambda )-h^{\\prime }(n_{\\lambda })\\Vert _{|\\alpha |}\\Vert \\partial _{x_{i}}W_\\lambda \\Vert _s\\\\&\\quad +C\\sum ^{3}_{i=1}\\Vert n^\\lambda -n_{\\lambda }\\Vert _{|\\alpha |}\\Vert \\partial _{x_{i}}W_\\lambda \\Vert _s+C\\sum ^{3}_{i=1}\\left\\Vert \\frac{G^\\lambda _i}{n^\\lambda }-\\frac{G_{\\lambda i}}{n_\\lambda }\\right\\Vert _{|\\alpha |}\\Vert \\partial _{x_{i}}W_\\lambda \\Vert _s\\\\&\\le C\\Vert U\\Vert _{|\\alpha |}+C(1+\\mathcal {D}^s)\\Vert N\\Vert _{|\\alpha |}+C\\Vert N\\Vert _{|\\alpha |}+C(1+\\mathcal {D}^s)(\\Vert N\\Vert _{|\\alpha |}+\\Vert B\\Vert _{|\\alpha |})\\\\&\\le C(1+\\mathcal {D}^s)\\Vert E\\Vert _s.$ In a similar spirit, $\\Vert R^2_\\alpha \\Vert $ is estimated as $\\Vert R^2_\\alpha \\Vert &\\le C\\sum ^{3}_{i=1}\\Vert u^\\lambda _i\\partial _{x_{i}}E_\\alpha -(u_{\\lambda i}\\partial _{x_{i}}E)_\\alpha \\Vert \\\\&\\quad +C\\sum ^{3}_{i=1}\\Vert h^{\\prime }(n^\\lambda )\\partial _{x_{i}}N_\\alpha -(h^{\\prime }(n^{\\lambda })\\partial _{x_{i}}N)_\\alpha \\Vert \\\\&\\quad +C\\sum ^{3}_{i=1}\\Vert n^\\lambda \\partial _{x_{i}}U_\\alpha -(n^{\\lambda }\\partial _{x_{i}}U)_\\alpha \\Vert \\\\&\\quad +C\\sum ^{3}_{i=1}\\left\\Vert \\frac{(G^\\lambda _i)^T}{n^\\lambda }\\partial _{x_{i}}H_\\alpha -\\left(\\frac{(G^{\\lambda }_{i} )^T}{n_\\lambda }\\partial _{x_{i}}H\\right)_\\alpha \\right\\Vert \\\\&\\quad +C\\sum ^{3}_{i=1}\\left\\Vert \\frac{G^\\lambda _i}{n^\\lambda }\\partial _{x_{i}}U_\\alpha -\\left(\\frac{G^{\\lambda }_{i}}{n_\\lambda }\\partial _{x_{i}}U\\right)_\\alpha \\right\\Vert \\\\&\\le C\\sum ^{3}_{i=1}\\Vert u^\\lambda _i\\Vert _s\\Vert \\partial _{x_{i}}E\\Vert _{|\\alpha |-1}+C\\sum ^{3}_{i=1}\\Vert h^{\\prime }(n^\\lambda )\\Vert _s\\Vert \\partial _{x_{i}}N\\Vert _{|\\alpha |-1}\\\\&\\quad +C\\sum ^{3}_{i=1}\\Vert n^\\lambda \\Vert _s\\Vert \\partial _{x_{i}}U\\Vert _{|\\alpha |-1}+C\\sum ^{3}_{i=1}\\left\\Vert \\frac{(G^\\lambda _i)^T}{n^\\lambda }\\right\\Vert _s\\Vert \\partial _{x_{I}}H\\Vert _{|\\alpha |-1}\\\\&\\quad +C\\sum ^{3}_{i=1}\\left\\Vert \\frac{G^\\lambda _i}{n^\\lambda }\\right\\Vert _s\\Vert \\partial _{x_{i}}U\\Vert _{|\\alpha |-1}\\\\&\\le C\\sum ^{3}_{i=1}(\\Vert u^\\lambda _i-u_{\\lambda i}\\Vert _s+\\Vert u_{\\lambda i}\\Vert _s)\\Vert E\\Vert _{|\\alpha |}+C(1+\\Vert N\\Vert _s^s)\\Vert N\\Vert _{|\\alpha |}\\\\&\\quad +C(\\Vert n^\\lambda -n_\\lambda \\Vert _s+\\Vert n_\\lambda \\Vert _s)\\Vert U\\Vert _{|\\alpha |}+C(1+\\Vert E\\Vert _s^s)\\Vert E\\Vert _{|\\alpha |}\\\\&\\le C(1+\\mathcal {D}^s)\\Vert E\\Vert _{s}.$ This completes the proof of Lemma REF .",
"Note that $C^{-1}\\Vert E_\\alpha \\Vert ^2\\le \\int _{{\\mathbb {T}}^3} E^T_\\alpha A_0(n^\\lambda )E_\\alpha dx\\le C \\Vert E_\\alpha \\Vert ^2$ .",
"We integrate (REF ) from 0 to $t$ with $[0,t]\\subset [0,\\min \\lbrace T_\\lambda ,T_{\\ast \\ast }\\rbrace ]$ to obtain $\\Vert E_\\alpha (t)\\Vert ^2+\\Vert \\nabla \\Phi _{\\alpha }(t)\\Vert ^2\\le C\\int _0^t(1+\\mathcal {D}^s)(\\Vert E\\Vert ^2_{s}+\\Vert \\nabla \\Phi \\Vert ^2_s)d\\tau +C\\lambda ^2.$ Here we have used the fact that the initial data constructed in Theorem REF .",
"Summing up the last inequality over all ¦Á satisfying $|\\alpha |\\le s$ , we get $\\Vert E(t)\\Vert ^2_s+\\Vert \\nabla \\Phi (t)\\Vert ^2_s\\le C\\int _0^t(1+\\mathcal {D}^s)(\\Vert E\\Vert ^2_{s}+\\Vert \\nabla \\Phi \\Vert ^2_s)d\\tau +CT_{\\ast \\ast }\\lambda ^2.$ Applying Gronwall's lemma to (REF ),we obtain $\\Vert E(t)\\Vert ^2_s+\\Vert \\nabla \\Phi (t)\\Vert ^2_s\\le CT_{\\ast \\ast }\\lambda ^2e^{C\\int _0^t(1+\\mathcal {D}^s)d\\tau }.$ In view of $\\Vert E\\Vert _s+\\Vert \\nabla \\Phi \\Vert _s=\\lambda \\mathcal {D}$ , it follows from (REF ) that $D^2(t)\\le CT_{\\ast \\ast }e^{C\\int _0^t(1+\\mathcal {D}^s)d\\tau }\\equiv \\Gamma (t).$ Thus, it holds that $\\Gamma ^{\\prime }(t)= C(1+\\mathcal {D}^s)\\Gamma (t)\\le C\\Gamma (t)+C\\Gamma ^{\\frac{s+2}{2}}(t).$ Applying the nonlinear Gronwall-type inequality in [25] to the last inequality yields $\\Gamma (t)\\le e^{CT_{\\ast \\ast }}$ for $t\\in [0,\\min \\lbrace T_\\lambda ,T_{\\ast \\ast }\\rbrace ]$ if we choose $T_{\\ast \\ast }>0$ (independent of $\\lambda $ ) so small that $\\Gamma (0)=CT_{\\ast \\ast }<e^{-CT_{\\ast \\ast }}.$ Then, because of (REF ), there exists a positive constant $M$ , independent of $\\lambda $ , such that $\\mathcal {D}(t)\\le M$ for $t\\in [0,\\lbrace T_\\lambda ,T_{\\ast \\ast }\\rbrace ]$ .",
"Finally, from (REF ),(REF ) and the definition of $(E,\\nabla \\Phi )$ , we conclude the proof of Theorem REF .",
"Acknowledgments : J. Yang's research was partially supported by the Joint Funds of the National Natural Science Foundation of China (Grant No.",
"U1204103)."
]
] | 1709.01627 |
[
[
"Crowded Field Galaxy Photometry: Precision Colors in the CLASH Clusters"
],
[
"Abstract We present a new method for photometering objects in galaxy clusters.",
"We introduce a mode-filtering technique for removing spatially variable backgrounds, improving both detection and photometric accuracy (roughly halving the scatter in the red sequence compared to previous catalogs of the same clusters).",
"This method is based on robustly determining the distribution of background pixel values and should provide comparable improvement in photometric analysis of any crowded fields.",
"We produce new multiwavelength catalogs for the 25 CLASH cluster fields in all 16 bandpasses from the UV through the near IR, as well as rest-frame magnitudes.",
"A comparison with spectroscopic values from the literature finds a ~30% decrease in the redshift deviation from previously-released CLASH photometry.",
"This improvement in redshift precision, in combination with a detection scheme designed to maximize purity, yields a substantial upgrade in cluster member identification over the previous CLASH galaxy catalog.",
"We construct luminosity functions for each cluster, reliably reaching depths of at least 4.5 mag below M* in every case, and deeper still in several clusters.",
"We measure M* , $\\alpha$, and their redshift evolution, assuming the cluster populations are coeval, and find little to no evolution of $\\alpha$, $-0.9\\lesssim\\langle\\alpha\\rangle\\lesssim -0.8$, and M* values consistent with passive evolution.",
"We present a catalog of galaxy photometry, photometric and spectroscopic redshifts, and rest-frame photometry for the full fields of view of all 25 CLASH clusters.",
"Not only will our new photometric catalogs enable new studies of the properties of CLASH clusters, but mode-filtering techniques, such as those presented here, should greatly enhance the data quality of future photometric surveys of crowded fields."
],
[
"Introduction",
"Current models of structure formation predict that rich clusters of galaxies lie in the most significant overdensities in the cosmic web.",
"The stellar components of individual galaxies make up only a small fraction of a cluster's mass [69], but they are relatively easily observed and trace the conditions of the cluster through cosmic history.",
"In addition to being used to identify clusters [67], [83], [84], photometric observations of cluster galaxies have been used to constrain the processes that drive galaxy evolution [29], [30], [85], [53], [43], [65], [89] and to characterize the properties of their entire host systems [105], [11], [152], [13], [7].",
"Figure: A portion of the MACS J0717 field, seen in F125W (red), F814W (green), and F555W (blue).",
"Objects detected in this work are outlined in white ellipses corresponding to the region used for photometry.",
"Also shown are objects classified as stars, which are marked by blue circles.",
"This image is approximately 70 '' 70^{\\prime \\prime } wide.Since its launch in 1990, the Hubble Space Telescope (HST) has enabled high-quality photometric observations of clusters.",
"Early works focused on single pointings, usually with one filter [39], [54], [106].",
"Mosaicked exposures allowed more complete observations of individual clusters, such as CL 1358+62 [140], [81] and MS 1054$-$ 03 [139], [138].",
"With the installation of the Advanced Camera for Surveys [62], more detailed surveys of clusters were begun, including the ACS Intermediate Redshift Survey [21], [22], [61], [112], as well as deep surveys of the closest clusters, including the Virgo [38], Coma [72], and Fornax [78] clusters.",
"However, until recently, there has been no survey of a large number of clusters with comprehensive photometric coverage.",
"To study the evolution of faint cluster galaxies, we desire a survey that has deep imaging of a sample of clusters covering a range of redshifts.",
"In addition, this ideal cluster survey would have comprehensive, multi-wavelength photometry, allowing for quality photometric redshift estimates and better analysis of stellar population parameters.",
"Increased metallicity, dust content, age, and redshift can all redden a galaxy, but these properties affect its spectrum in different ways.",
"Multi-wavelength photometry is therefore needed to break degeneracies in stellar populations, which cannot be done with few-band colors [146], [147].",
"The Cluster Lensing and Supernova Survey with Hubble [114] is a deep cluster survey with broad wavelength coverage; it has 16-band (UV to IR) observations of 25 massive galaxy clusters at redshifts $0.2 \\lesssim z \\lesssim 0.9$ obtained with the HST.",
"We show in Figure REF a sample region of one CLASH cluster.",
"Because of the crowding and spatially variable backgrounds, detailed studies of cluster galaxy properties with CLASH, aside from the central brightest cluster galaxies [113], [49], [47], [60], have been limited due to the difficulty in obtaining reliable photometry [79].",
"At this observational depth, the light from cluster galaxies is observed co-spatially with the intracluster light (ICL) and the extended wings of massive galaxies such as the BCGs; this effect is apparent in Figure REF .",
"Disentangling the flux contributions from individual galaxies and the scale- and wavelength-variant background light distribution is a significant challenge for utilizing not only the CLASH data set, but also any deep survey of clusters of galaxies.",
"Previous works have dealt with the complex light distribution of clusters by parametrically modeling large galaxies and the ICL [72], [113], [97].",
"Here, we propose an alternative method: using the mode of nearby pixels on a range of physical scales to determine the distribution of background light for each pixel.",
"The mode is a difficult quantity to determine for even moderately well sampled data, and so several approximations exist, such as those of [108] and [17].",
"These approximations are not robust, however, and are not suitable for precision photometry.",
"We therefore propose a new technique to not only measure the modal value of a background distribution, but also to measure the overall shape of the distribution itself.",
"Through the use of our new modal estimation techniques, we model each galaxy's individual photometric background pixel-by-pixel, recovering the local structure in the backgrounds on the broad range of physical scales that can contaminate photometry, and thereby accurately measure the flux of galaxies in the CLASH clusters.",
"The photometric catalog produced in this way traces the cluster population down to ${\\sim } M^*+4.5$ for all clusters and down to ${\\sim } M^* + 7$ for the closest ($z \\sim 0.2$ ) clusters, as measured by the magnitude at which 90% of the galaxies expected by our luminosity function are present.",
"We compute photometric redshifts for the galaxies in these cluster fields; in comparison to previous results for the CLASH clusters, we reduce the median absolute deviation in offset between photometric and spectroscopic redshifts by ${\\sim }30$ % and find significant improvement in the purity of our sample for cluster science.",
"Finally, we fit spectral energy distributions (SEDs) to our photometry to provide a catalog of stellar properties for the galaxies in the CLASH fields.",
"Throughout this work, we assume a $\\Lambda $ CDM cosmology, with $\\Omega _{\\rm M} = 0.3$ , $\\Omega _\\Lambda = 0.7$ , and ${\\rm H}_0 = 70\\ {\\rm km}\\ {\\rm s}^{-1}\\ {\\rm Mpc}^{-1}$ ."
],
[
"Data Set",
"Imaging data in this work come from CLASH.",
"This HST Multi-cycle Treasury Program imaged 25 galaxy cluster in 16 filters, covering the UV to the IR.",
"CLASH clusters were chosen from the Massive Cluster Survey [56], [55], [57] and the Abell catalog [1], [2].",
"Of the 25 clusters, 20 [4] were chosen due to their dynamically relaxed X-ray morphology and five chosen due to their strength as strong lenses.",
"With an average of 20 orbits per cluster, the HST component of CLASH provides us with an unprecedented look into the environments of galaxy clusters.",
"The CLASH dataset combines observations from multiple instruments, which entails differing coverage for each filter.",
"The ACS field-of-view is approximately $202^{\\prime \\prime } \\times 202^{\\prime \\prime }$ while the WFC-IR field covers roughly 40% of that area, so many objects away from the cluster center lack IR coverage.",
"HST filters used in this work have the naming convention that a filter of central wavelength NNN is labeled as either “FNNNW” or “FNNNLP,” depending on whether it is a wide or long-pass filter, respectively (see Figure 10 in [114] for the spectral responses of filters used in this work).",
"For central wavelengths longer than 1 $\\mu {\\rm m}$ , NNN is listed in hundredths of $\\mu {\\rm m}$ ; below 1 $\\mu {\\rm m}$ , NNN is in nm (for these filters, NNN is always greater than 200).",
"Filter properties are listed in Table .",
"For some clusters we also used archival F555W images.",
"Throughout this paper we use the public full-depth CLASH HST mosaicshttps://archive.stsci.edu/prepds/clash/, which were produced using the approaches described in [82] and references therein.",
"We only use CLASH HST mosaics binned to 0065 per pixel.",
"lrrrr Filter Parameters 0pt Filter Name Instrument ${\\rm zp}_{\\rm AB}$ $\\lambda _{\\rm pivot}$ $\\Delta \\lambda $ (mag) (Å) (Å) F225W WFC3-UVIS 24.0966 2359a 467a F275W WFC3-UVIS 24.1742 2704a 398a F336W WFC3-UVIS 24.6453 3355a 511a F390W WFC3-UVIS 25.3714 3921a 896a F435W ACS-WFC 25.6578 4328 1018b F475W ACS-WFC 26.0593 4747 1422b F555W ACS-WFC 25.7347 5361 1263b F606W ACS-WFC 26.4912 5921 2236b F625W ACS-WFC 25.9067 6311 1396b F775W ACS-WFC 25.6651 7692 1493b F814W ACS-WFC 25.9593 8057 2338b F850LP ACS-WFC 24.8425 9033 2063b F105W WFC3-IR 26.2707 10552a 2650a F110W WFC3-IR 26.8251 11534a 4430a F120W WFC3-IR 26.2474 12486a 2845a F140W WFC3-IR 26.4645 13923a 3840a F160W WFC3-IR 25.9559 15369a 2683a aTaken from the WFC3 Instrument Handbook [50].",
"b95% cumulative throughput width As part of the public release of CLASH data, [114] released a catalog of photometry for objects in all 25 CLASH filters, not optimized for cluster members.",
"In Figure REF we show a color-magnitude diagram for all objects detected in the fields of four CLASH clusters at redshifts $\\langle z \\rangle = 0.350 \\pm 0.005$ : MACS 1115, MACS 1931, RXJ 1532, and RXJ 2248.",
"Data from [114] are shown on the right; data we will present in this work are shown on the left.",
"Our new work not only has greater purity for cluster studies, but we find a red sequenceRed sequence fitting was performed using the LTS_LINEFIT program described in [31], based on the methods of [118].",
"for these clusters with decreased scatter ($\\sigma _{int} = 0.18$ mag with our data, compared to $\\sigma _{int} = 0.32$ mag with the previous release).",
"Our measured red sequence slope and scatter are more consistent with previous studies (slope: e.g., [66], [43], [133], [137]; scatter: e.g., [26], [130], [136], [95], [121]).",
"As derived properties from the CLASH dataset – photometric redshifts and SED fits – are based on the set of colors in all 16 CLASH filters, our new work will provide us with the ability to reliably infer cluster galaxy properties in a way the original work did not."
],
[
"Statistical Background Light Estimators",
"One of the fundamental assumptions of this work is that, for a pixel containing the light from a galaxy, the observed flux in that pixel is the sum of light from the galaxy and from the background light drawn from some unknown distribution.",
"This background distribution may include contributions from physical structure (such as the ICL and other galaxies), contaminating light (such as star spikes and scattered light), and the sky (including instrumental effects).",
"Lacking a complete understanding of the background light due to the limitations of our telescope's optics and the finite observation time, our best solution is to model the background light distribution from nearby pixels.",
"Accurately describing the background brings three specific challenges: determining a nominal measure of the expected value of the background light, determining the range of that distribution, and performing this characterization with a limited sample of pixels, some of which may be outliers from a separate distribution (such as from the wings of a nearby galaxy) Galaxy clusters are particularly challenging environments for background estimation.",
"While a number of works have considered the problem of crowded-field photometry for stellar clusters, galaxies have significant angular extents, ensuring overlapping profiles and extreme difficulty in estimating local photometric background levels.",
"And unlike stars, which are point sources described by their point spread function (PSF), galaxies are extended surface brightness distributions convolved with the PSF.",
"Furthermore, clusters are filled with ICL, stellar emission associated with the cluster but not with any individual galaxy [141], [98], [45].",
"Accurately counting the flux of the ICL in an aperture can impact the observed colors of galaxies [41], [155], [145], [119], but improperly accounting for a galaxy's full extent due to the ICL can impact the total measured magnitude.",
"Any cluster photometry routine must be able to deal with a spatially and chromatically varying background.",
"A number of routines have been introduced to deal with the complexities of cluster photometry, many of which utilize parametric modeling of cluster galaxies through tools such as GALFIT [109], [110] and techniques including shapelets [94] and Chebyshev Rational Functions [77].",
"[64] simultaneously fit GALFIT galaxy light profiles and a diffuse background model for CL0024+17.",
"[97] modeled a limited set of galaxies and the ICL independently for two Frontier Fields clusters, and then ran detection and photometry routines using the model-subtracted images.",
"[99] modeled all of the galaxies in the CLASH fields with CHEFs, and subtracted off the residual light to compute background-free photometry.",
"Maps of the Frontier Fields ICL were produced by [101] by combining the measured background from postage stamp GALFIT models pixel-by-pixel.",
"[87] mapped and subtracted ICL in the Frontier Fields through the use of wavelet decomposition.",
"The common theme of these works is that background light is computed as the residual light after modeling galaxies.",
"We have implemented a method that robustly and locally computes the mode over scales that capture the spatially variable background, which we use to calculate a pixel-by-pixel background on a galaxy-by-galaxy basis, without needing to model individual galaxies.",
"Estimating a common background level in a set of pixels has a storied history [19], [132], [76], [40], [17], [104], [154], [23].",
"The simplest approach is to estimate the first moment, or mean, of the distribution, but this estimator is notoriously biased [115], [14].",
"Beyond that, for regions containing multiple flux distributions (e.g., sky and a diffraction spike from a bright star), simple characterizations of the background will be inherently inaccurate.",
"In this work, we treat the set of pixels as a distribution of distinct observations of the background, where some fraction of pixels may be contaminated by unknown interloping sources of flux.",
"As such, the most frequent occurrence in the set, the mode, ought to represent an unbiased estimate of the photometric background.",
"Each pixel also has an associated random measurement error, which we treat as symmetric about zero.",
"We discuss below two techniques for estimating the mode: a computationally fast but less robust technique for detecting galaxies, and a more robust yet computationally intensive method to measure the peak and distribution of background pixels for photometry."
],
[
"Detection Images",
"Detecting objects with angular size scales ranging over two orders of magnitude requires a multiresolution analysis of the images.",
"To identify small galaxies, we need an image where small-scale structure is visible, while to identify large galaxies we require a map of the large-scale structure.",
"We create these images by using the method of steepest ascent (described below) to measure the mode in a sliding box across each image, where we use different box sizes to create unbiased maps of the spatially varying background over the broad range of scales needed to detect both small and large objects.",
"In this work, we use boxes with sizes in seven scales separated logarithmically between 4 and 256 pixels (026 and 1664), inclusive.",
"Three of the background-subtracted images for Abell 383 are shown in the top panel of Figure REF .",
"Figure: Schematic representation of the method of steepest ascent for finding the mode.",
"On the left, we present a sample of points drawn from two gaussian distributions (orange); the peak of this distribution is the mode, marked by a vertical orange line.",
"By plotting the measured value as a function of ranked order (middle), we see that the inflection point corresponds to the mode (orange).",
"The transpose of this (right), which shows the cumulative count of each value, is modeled by a cubic spline.To create the background-subtracted maps, we determine a local background for each pixel by using the method of steepest ascent to calculate the mode of the set of background pixel values (this technique has also been adapted by [107]).",
"The mode corresponds to the value at which most sample points are expected to lie; in other words, it is the flux with the highest density of points in the sample.",
"To determine this value, we first sort each pixel flux in a sample of $N$ pixels from smallest to largest, ${\\rm F}_1\\le {\\rm F}_2 \\le ... \\le {\\rm F}_{{\\rm N}-1} \\le {\\rm F}_{\\rm N}$ .",
"For a well-sampled distribution, this ordered set will have a range of values such that $F_{i + j} - F_{i - j} \\approx 0$ , where $i$ is the index of the mode and $j$ is a sampling range.",
"The transpose of this distribution is the number of points with flux less than $f$ , ${\\rm N}_{f \\le 0} \\le {\\rm N}_{f \\le i} \\le ... \\le {\\rm N}_{f \\le F-i} \\le {\\rm N}_{f \\le F}$ .",
"In this distribution, the mode is therefore the region of steepest ascent, where ${\\rm N}_{f \\le i + j} - {\\rm N}_{f \\le i - j}$ is maximized.",
"To calculate the point of steepest ascent, we fit the transpose function with a cubic spline, so that the steepest ascent is calculated numerically.",
"This technique is shown schematically in Figure REF .",
"While this technique allows for rapid computation of the mode, it is limited in its accuracy in the presence of complex backgrounds (which may distort a polynomial fit) and for small samples.",
"Due to these concerns, we use this method for computational expediency in estimating the background for purposes of identification and detection of sources, but not for flux estimation.",
"Since our science does not depend on teasing out detections of the faintest, smallest-background sources, small errors in the mode for individual pixels do not affect detection and source definition for our purposes here.",
"One issue that hampers detection of objects in cluster fields is source confusion, whereby two nearby galaxies are detected as one, creating a pseudo-object that includes both of them but is concentric with neither.",
"To mitigate this issue, we produce a separate suite of detection images (shown in the bottom panel of Figure REF ) from the previously generated background-subtracted images (shown in the top panel of Figure REF ); here, the new detection image is the difference of the background-subtracted images at neighboring scales (e.g., the image with an 8 pixel background scale was subtracted from that with a 16 pixel background scale).",
"In essence, we are performing an á trous wavelet transform [87] on the original images, but with the critical difference that the dual is not uniform across the field due to the nonlinear nature in which the mode is calculated.",
"This strategy removes smaller sources from the detection areas of larger galaxies.",
"Our method of determining the mode does not perform well at small scales when no clear background can be determined, such as at the center of a massive galaxy.",
"In this regime, the background selection region does not include any background pixels, so the background subtraction routine subtracts real signal.",
"The effect of performing background subtraction without having any background can be seen in the upper right panel of Figure REF .",
"To properly photometer these galaxies, we need to be able to subtract a background in their central regions consistent with the background at the outer regions.",
"Because the generated background-subtracted images lack that flexibility, we only use these images for detection and not for photometry."
],
[
"Source Detection",
"lccccc Source Extractor Detection Parameters 0pt Detectiona CLEAN_PARAMb DEBLEND_MINCONTb DEBLEND_NTHRESHb DETECT_MINAREAb Max Offset (pix) (pix) 004 0.2 0.40 50 15 008$-$ 004 0.2 0.20 60 16 6 016$-$ 008 0.3 0.10 60 18 6 032$-$ 016 0.4 0.10 60 20 8 064$-$ 032 0.5 0.10 60 20 8 128$-$ 064 0.6 0.10 60 20 12 256$-$ 128 0.2 0.10 60 20 12 Stars 0.4 0.10 60 20 aNumerical detection images are either the background size of mode filtering (004) or the backgound sizes of the subtracted images (e.g., 008$-$ 004) bCLEAN_PARAM is the efficiency of cleaning artifacts of bright sources from the detection list; a lower value of CLEAN_PARAM fits more extended structure to bright sources, resulting in a more aggressive cleaning.",
"DEBLEND_MINCONT and DEBLEND_NTHRESH determine whether Source Extractor separates detections into multiple objects.",
"For DEBLEND_NTHRESH logarithmically spaced flux bins, objects are deblended into two objects if they both have DEBLEND_MINCONT of the flux of their combined flux measure.",
"DETECT_MINAREA is the minimum required number of pixels above the detection threshold for a galaxy to be detected by Source Extractor.",
"We have nonparametric models of backgrounds mapped on scales from 8 to 256 pixels, as well as images with subtracted backgrounds of 4 pixels, for the non-UVIS filters for each cluster.",
"We run Source Extractor [17] on each, with parameters listed in Table REF .",
"We use an RMS map based on the weight map produced by MosaicDrizzle for each filter and cluster combination, which is itself an inverse variance image based on the input exposures that contributed to each pixel.",
"For each detection, Source Extractor outputs the geometric properties of the detection ellipse, specifically WCS and pixel coordinates, semi-major and -minor axes, position angle, and the KRON_RADIUSThe Kron radius [86] is a radius selected to capture more than 90% of a galaxy's flux.",
"See [70] for a discussion of its usage in Source Extractor.. We assemble a detection catalog for each individual filter by working from the detection images with the largest background regions to the smallest (from left to right in Figure REF ).",
"For every object detected in one image, we check to see if there was a match in the next-smallest detection image within a small offset, as specified in Table REF .",
"Those that had matches, as well as unmatched objects from the smaller catalog, are passed on to the next scale.",
"For all but the smallest scale, the actual detection is performed on a subtraction image (the bottom row of Figure REF ); therefore, our technique has the result of detecting the full extent of galaxies and propagating those sizes down to the small-background images.",
"After collecting all the detections from each filter into one catalog, we then create a master source list for each cluster field based on the multi-wavelength detection suite.",
"We merge the detection catalogs filter-by-filter, combining objects with peak flux values located near each other.",
"As the geometric properties differed between detections in different filters, we reduce each object to the properties from one filter.",
"These properties are from the detection with the fourth-largest size – or, for objects detected in fewer than eight filters, the median size – where the size was the sum of semi-major and minor axes.",
"The choice to not use the largest detection of an object was motivated by source confusion lumping multiple objects into excessively large apertures; the choice of fourth-largest detection was motivated by experimentation.",
"When objects were detected in multiple filters, the aperture sizes would quickly converge to the same size, but several detections could be excessively large if multiple nearby structures blended together at one background scale.",
"After trying multiple methods of selecting an appropriately sized aperture from the detections in every filter, we found that skipping the three largest detections would remove the unphysically large apertures while not causing measured galaxy sizes to otherwise shrink (as aperture sizes converge toward the same value).",
"We trim this catalog to only those objects detected in at least three filters.",
"To avoid detecting diffraction spikes from stars, we also only include those objects with semimajor axis no more than eight times the length of the semi-minor axis or with a semi-minor axis of at least 5 pixels, which excludes long, thin detections without removing larger elliptical objects, such as edge-on spirals.",
"To remove stars from our catalog, we run Source Extractor on the original images for each filter, using the parameters given in Table REF .",
"Detections in each filter are combined in the same manner as before; to only include stars, we exclude any object with a CLASS_STAR value below 0.9.",
"We match this star catalog with our previous detection catalog, and those objects included in both are marked as stars.",
"We generate a circular mask aperture with a radius equal to the semi-minor axis from our original detection image for each star.",
"llll Star Masks, $r \\ge 10$ 0pt $\\alpha _{2000}$ $\\delta _{2000}$ $r$ Cluster ($^{\\prime \\prime }$ ) 01:31:57.74 $-$ 13:34:43.5 1.26 Abell 209 01:31:53.94 $-$ 13:35:58.1 3.42 Abell 209 01:31:50.73 $-$ 13:37:23.2 1.39 Abell 209 01:31:47.74 $-$ 13:37:24.0 1.47 Abell 209 02:47:55.00 $-$ 03:30:41.4 2.46 Abell 383 02:48:05.48 $-$ 03:30:59.0 2.21 Abell 383 02:48:05.69 $-$ 03:31:16.9 1.50 Abell 383 02:48:09.28 $-$ 03:31:32.9 2.94 Abell 383 02:48:02.83 $-$ 03:31:32.8 1.08 Abell 383 02:48:00.14 $-$ 03:31:34.7 2.03 Abell 383 This table is available in its entirety in machine-readable form.",
"After creating the master detection images for each cluster, we inspect them by eye.",
"We verify the accuracy of our star masks, reclassifying easily identifiable objects that our pipeline had misidentified as stars (an average of four objects per field were reclassified).",
"To ensure repeatability, we provide the coordinates and radii of the star masks used in this work with mask radii of $r > 2^{\\prime \\prime }$ in Table REF .",
"Second, we clean the detection catalog of over-detections (such as in a spiral galaxy being detected at both the galaxy level and as individual knots) and adjust a small number of mis-proportioned ellipses to circles with size given by either their semimajor or semiminor axes (usually caused by extended structure from other, nearby galaxies).",
"We also had to create a new aperture for the BCG of Abell 2261, which has an unusually flat central surface brightness profile [113], making it difficult for our technique to link large-scale detections to a central point at small background scales.",
"As an example of our detection efficiency, the detection regions for a section of MACS J0717 are shown in Figure REF .",
"As with the creation of any photometric catalog, we tailored our detection strategy toward a specific science goal; in this case, the detection and photometry of cluster galaxies.",
"Because of this decision, we did not prioritize the detection of strongly lensed arcs [150] or lensed galaxies [156], [36].",
"By requiring detections in at least three filters, we minimized false detections, but also excluded drop-out galaxies at too large of a redshift to be seen in many filters.",
"We also linked emission at multiple angular scales through their shared centers, making our technique unsuited for the complex morphologies of strongly lensed arcs.",
"The primary challenge of photometring crowded fields is assigning the entirety of the observed flux across individual sources and the background.",
"Here, we describe a new technique for accurately accounting for the flux of objects in a cluster field.",
"Starting with the smallest galaxies, we measure a background at the outer edges of each object and work our way inward, replacing pixels with their measured background as we go.",
"This replacement allows us to not only provide a background measurement at the inner parts of larger galaxies, but also to effectively peel off smaller galaxies from larger ones, letting us distribute the flux between overlapping objects accurately.",
"As part of this measurement, we characterize the distribution of background light, which in turn gives us a per-pixel estimate of the expected variation in the background light; this preserves the noise properties of the background.",
"We detail this process below.",
"We use the method of steepest ascent to determine the mode to estimate and subtract a background for source detection; this technique calculates a mode using individual flux measurements of background pixels.",
"One additional input to consider when determining the background is the contribution of measurement uncertainty.",
"Each pixel has both a flux and a flux uncertainty; this can be thought of as each pixel being a normalized probability distribution of fluxes.",
"In this scheme, the mode will correspond to the flux value with the most total probability from the combined probability distributions of all the pixels in the background.",
"Specifically, this means that the mode is the peak of a probability density function, which we estimate through a technique similar to a kernel density estimation, although where each pixel has a unique kernel.",
"Each background pixel is defined by two values: $f_i$ and $\\sigma _i$ , the flux and uncertainty, respectively.",
"By convolving the flux in each pixel in the background region, $B$ , with a Gaussian kernel of width $\\sigma _i$ , we create a probability distribution for the fluxes of all the pixels in the background region.",
"The mode is then found as the peak of this distribution, which is given by $P(x) = \\sum _{i \\in {\\rm B}} \\frac{1}{\\sigma _i}\\ e^{ - (x - f_i)^2 / (2 \\sigma _i^2)}.$ To find the maximum of this function in a computationally expedient manner over a large number of points, we identify the zeros of its derivative, given by $dP(x)/dx = \\sum _{i \\in {\\rm B}} \\frac{ (f_i - x)}{\\sigma _i^3}\\ e^{ - (x - f_i)^2 / (2 \\sigma _i^2)}.$ Due to the limits of computational efficiency, we only employ the more accurate kernel density estimation using Equations REF and REF when measuring flux.",
"For a given pixel, the measured flux, $f_i$ , is itself drawn from a probability distribution; the drizzling step combines multiple samples of that probability distribution to calculate a measurement and uncertainty for that pixel.",
"Thus, $f_i$ has already been convolved with the noise before we convolve it with a gaussian kernel.",
"We performed numerical simulations to analyze how well our technique could recover the original distribution from a sample of $N$ points averaged from $X$ draws (analogous to $N$ pixels made from $X$ exposures).",
"For a normal distribution, finding the mode through our technique is an order of magnitude more precise at finding the central moment of the original distribution than through using the median or mean.",
"For large $X$ ($X \\gtrsim 25$ ) the drizzling step induces kurtosis, causing the estimate of $\\sigma $ to be biased high, but our mode performs better at this test than the median or mean for more reasonable values of $X$ .",
"For skew normal probability distributions, the average of $X$ random draws (pixel values after drizzling) is offset from the peak of the skew normal distribution, so the recovered mode is systematically offset from the peak of original probability.",
"The severity of this offset, which is toward the average of the probability distribution, increases with increasing $X$ .",
"Again, for the skew normal distribution, the mode is the most precise measurement.",
"Our simulations show that, while drizzling leaves an imprint on the noise characteristics of a region, our mode technique can still precisely recover a distribution's central value, particularly for small numbers of drizzled exposures.",
"Photometry is performed on each galaxy, beginning with the smallest and working to the largest (as ranked by semi-major axis), on a pixel-by-pixel basis.",
"Our technique involves eroding each galaxy from the outside in, replacing pixels with the value of the background around them, thereby propagating the conditions outside of the galaxy inward.",
"For small galaxies overlapping with larger galaxies, this local background replacement preserves the structure of the larger galaxy.",
"A schematic representation of how a background is measured is shown in Figure REF ; the measured background in this example would then replace the value of the pixel highlighted in orange, and the photometry routine would advance to the next pixel.",
"The width of the distribution of background pixels is propagated into the variance map, maintaining the statistical properties of the measured background.",
"Our implementation of this technique for the CLASH observations is presented below.",
"Figure: Schematic representation of the photometry technique.",
"On the left, we show an example of how the background for a pixel in a galaxy is selected.",
"A background region (light red) is created around a pixel of interest (orange), avoiding any pixels in the aperture of the galaxy (purple).",
"The middle panel shows the distribution of flux in those pixels; for the set of pixels ordered by flux, orange lines trace the uncertainties and the nominal values are shown in purple.",
"For legibility, errorbars are only shown for every 20th pixel.",
"Each of those flux measurements is convolved with a Gaussian kernel with σ\\sigma given by the pixel's variance from the weight map; the sum of those kernels is shown in purple in the right panel.",
"The peak of this distribution (pink) is the background value for this pixel.",
"In contrast, the mode found using the technique of , the median, and the mean are shown with dotted, dashed, and dotted-dashed lines, respectively.",
"The uncertainty in this background determination is set by the flux values at which the probability has dropped to e -1/2 e^{-1/2} times the maximum.",
"The excess flux above this background for the pixel of interest is assigned to the galaxy, the flux of that pixel is set to the determined background value, and the uncertainty in the background is used to set a new value in the weight image for that pixel.",
"To photometer the entire galaxy, this process is repeated for each pixel, working from the outer boundaries in.",
"The background region in this example is slightly enlarged for demonstrative purposes.When photometering each galaxy, we create an elliptical aperture based on the Source Extractor detection parameters.",
"This region is blocky; pixels are either in the aperture or out of it, with no partial associations.",
"We then assign an order of photometry by taking a one-pixel-wide annulus with outer radius equal to the galaxy's semi-major axis, and finding all of the galaxy pixels inside that ring.",
"We shrink the annular radius pixel-by-pixel, noting the order of galaxy pixels to fall within it, until we have reached the centermost pixel.",
"We then photometer the galaxy following that order.",
"To measure the flux in a pixel, we must first compute the background value.",
"We consider a circular aperture around that pixel, with radius equal to $1.5 \\times b$ , where $b$ is the semi-minor axis of the galaxy.",
"This value is constrained to lie within 3 and 12 pixels (0195 and 078, respectively), the former to ensure enough background pixels can be found and the latter to keep the background local to the galaxy.",
"We exclude from this aperture any pixels contained within the galaxy itself that have not yet been photometered.",
"For the remaining pixels, we pass their measured fluxes and uncertainties to our background measuring routine.",
"This routine, shown in Figure REF , convolves each flux measurement with a Gaussian kernel that has a standard deviation given by that flux measurement's uncertainty (drawn from the variance map of the image, which is itself updated for pixels that are replaced with background measurements).",
"These distributions are then summed to create a single probability frequency curve for the background flux (as given in Equation REF and shown on the right of Figure REF ).",
"The nominal location of the background intensity is determined by finding the peak of this distribution; to do this, we employ a root finder on the derivative (given in Equation REF ) to find all maxima.",
"As these distributions can be multi-modal, it is important to find all maxima.",
"We therefore step through the ordered range of flux measurements.",
"If the sign of the derivative changes from positive to negative, we use these bounds to find a root.",
"If the derivative is zero at any flux measurement, we add it to the list of roots.",
"We ignore any changes from negative derivative to positive, as those mark minima.",
"One implicit assumption is that maxima cannot occur bounded by two flux measurements at which the derivatives are the same sign – these maxima would not be found by our root finder, which requires boundary values of opposite sign to function.",
"We use the interval bisection method of [27] as implemented in the scipy brentq algorithim to find the roots, given that the maxima are bounded.",
"In the event of an error in this process, our code will find the mode using the Newton–Raphson method; as this method is unbounded, we seed it with an initial guess of the median of the background flux distribution.",
"In both cases, the uncertainty on the background value is found by finding the flux value in both directions at which the probability distribution given by Equation REF is equal to $e^{-1/2} \\times P( \\mu )$ , where $P( \\mu )$ is the probability at the measured mode; this value is the relative height of a Gaussian distribution at $1 \\sigma $ .",
"It is important to note that we are not looking for a quantity equivalent to the standard error in the mean, which would be inversely proportional to the square root of the number of pixels in the background; that quantity would be a measure of the accuracy of the mode.",
"Instead, we characterize the intrinsic spread in the background, as each pixel has an additive flux component drawn from this distribution, and even with perfect knowledge of the background distribution, we cannot know the background flux for a certain pixel better than this level.",
"Figure: Effect of our background modeling and subtraction technique for Abell 209.",
"Images are from the F814W filter: the original (left), after every galaxy has been photometered (center), and after the subtracted image has been adjusted by a random draw from the final uncertainty model (right).Having determined the background value for a pixel, the excess flux is assigned to the galaxy, while the pixel value is replaced with the background value and the width of the background probability distribution is incorporated into the variance map.",
"We work from the outside of the galaxy in, so that the background measurement for the center of each galaxy is based on the backgrounds measured all around it.",
"After the entire field has been photometered, we produce an image made by adjusting each pixel's flux by a random draw from the variance map; this step maintains the noise properties of the galaxy-subtracted image.",
"We show the before, after, and randomly drawn images of a section of Abell 209 in Figure REF .",
"In the galaxy-subtracted images of Figure REF , the ICL is clearly visible.",
"As [45] and [46] have already performed a careful analysis of the ICL in CLASH clusters, we will not be discussing the residual ICL in this work, although the techniques presented here do offer a new route for studying ICL.",
"Also visible in these images are some residual structures; as discussed in Section REF , we did not prioritize detecting strongly lensed arcs.",
"Although we do not obtain photometry for a complete sample of lensed objects, our photometric technique was designed such that undetected structures such as these arcs would not affect the background measurements.",
"crrr Photometric Properties 0pt Cluster Filter ${\\rm t}_{\\rm exp}$ ${\\rm A}_{\\lambda }$ (s) (mag) Abell 209 F225W 7316.0 0.135 Abell 209 F275W 7464.0 0.106 Abell 209 F336W 4752.0 0.086 Abell 209 F390W 4894.0 0.075 Abell 209 F435W 4136.0 0.070 Abell 209 F475W 4128.0 0.063 Abell 209 F606W 4096.0 0.048 Abell 209 F625W 4066.0 0.043 Abell 209 F775W 4126.0 0.032 Abell 209 F814W 8080.0 0.030 This table is available in its entirety in machine-readable form.",
"We provide the photometric parameters used for each cluster and filter in Table .",
"As each processed image was reduced to a count rate, exposure times double as gain values.",
"We assume Galactic extinction values taken from the NASA/IPAC Extragalactic Database (NED)NED is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration., which uses the [126] recalibration of the [127] infrared-based dust maps utilizing a [59] reddening law with $\\textrm {R}_{\\rm v} = 3.1$ ."
],
[
"Photometric Redshifts",
"We derive redshift probability distribution functions for each galaxy we observed using the Bayesian photometric redshift code presented by [15] [16], [34].",
"BPZ fits the broadband flux measurements with a set of empirical templates in a grid of redshifts and computes a probability distribtion function from those fits.",
"This code was used to produce estimates of photometric redshifts for the default CLASH source catalog [114].",
"Our determination of $P(z)$ for each galaxy covers the range of redshifts from $z = 0.01$ to $z = 12.0$ in steps of $\\Delta _z = 0.001$ .",
"We use 11 template galaxy spectra, including both early-type and late-type galaxy spectral templates, and we interpolate between these 11 to make an additional 90 templates.",
"To account for zero-point uncertainties, we set a minimum photometric uncertainty of 0.02 mag [25], [24].",
"To characterize the accuracy of our redshifts, we compare a sample of galaxies with spectroscopically derived redshifts to our values.",
"These values come from the CLASH-VLT collaboration [20], [12], [100], the Sloan Digital Sky Survey Data Release 13 [128], as well as works by [37], [96], [71], [74], [116], [131], [35], [68], [117], [58], and [63].",
"Additionally, we use a sample of unpublished VLT-VIMOS redshifts for four cluster fields (P. Rosati & M. Nonino 2017, private communication) and a sample of unpublished redshifts for nine clusters from the IMACS-GISMO instrument on Magellan (D. D. Kelson 2017, private communication).",
"Figure: Comparison of photometric redshifts measured in this work to spectroscopic redshifts.",
"The thin band traces the region where |(z p -z s )/(1+z)|<0.05| (z_{\\rm p} - z_{\\rm s}) / (1 + z)| < 0.05.",
"The full redshift coverage from z=0z=0 to z=3.0z=3.0 is shown on the left panel; a zoom-in to just z=0z=0 to z=1.0z=1.0 is shown on the right.",
"Points are binned into hexagons, with the total density of points scaled logarithmically from 1 to 100 counts per hex, as indicated by the colorbar.lrr Photometric Redshift Accuracy and Sample Purity 0pt This Work [114]a Spectroscopic Matches 1716 1942 Well Matchedb 1149 (66.96%) 1136 (58.50%) Minor Outliersc 325 (18.94%) 409 (21.06%) Substantial Outliersd 108 (6.29%) 170 (8.75%) Catastrophic Outlierse 134 (7.80%) 227 (11.69%) Median Absolute Deviationf, All Galaxies 0.030 0.038 75th Percentilef, All Galaxies 0.053 0.073 Median Absolute Deviationf, $z_{\\rm s} \\le 1.0$ 0.027 0.035 75th Percentilef, $z_{\\rm s} \\le 1.0$ 0.045 0.063 Median Absolute Deviationf, Excluding Catastrophic Outliers 0.027 0.031 75th Percentilef, Excluding Catastrophic Outliers 0.045 0.054 $|z_{\\rm p} - z_{\\rm c}| / (1 + z_{\\rm c}) \\le 0.05$ and $m_{\\rm F814W} \\le 22$ 2109 / 4139 (51.0%) 2554 / 10684 (23.9%) $|z_{\\rm p} - z_{\\rm c}| / (1 + z_{\\rm c}) \\le 0.05$ and $m_{\\rm F814W} \\le 24$ 3612 / 10563 (34.3%) 4140 / 20263 (20.4%) $|z_{\\rm p} - z_{\\rm c}| / (1 + z_{\\rm c}) \\le 0.05$ and $m_{\\rm F814W} \\le 26$ 4574 /20930 (21.9%) 6347 / 46338 (13.7%) aPhotometric redshifts re-derived using the same techniques and parameters as in this work b$|(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} ) \\le 0.05$ c$0.15 \\ge |(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} ) > 0.05$ d$0.5 \\ge |(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} ) > 0.15$ e$|(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} ) > 0.5$ f$|(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} )$ After combining the spectral redshift catalogs and removing duplicates (any two objects with positions within 1$^{\\prime \\prime }$ of each other), we cross-match this catalog with our own catalog of detected objects.",
"For each spectroscopic redshift, we match it with any object within 15 of the reported coordinates.",
"In the event of multiple objects within this region, we match with the brightest object; if multiple objects are within 0.5 mag of this object, we match to the object closest to the spectroscopic position.",
"From this matched catalog, we report 1716 objects with spectroscopic counterparts.",
"For all galaxies, the median absolute redshift deviation between photometric and spectroscopic redshifts occurs at $\\frac{|(z_{\\rm p} - z_{\\rm s})| }{ (1 + z_{\\rm s} )} = 0.030$ .",
"In Table REF we provide the median absolute deviations and 75th percentile deviations for the full galaxy sample, as well as the sample without catastrophic outliers and the sample with only galaxies with redshift $z_{\\rm s} \\le 1.0$ .",
"The complete comparison of photometric and spectroscopic redshifts is shown in Figure REF ."
],
[
"Rest-Frame Magnitudes",
"We fit SEDs to every galaxy in our catalog with ${\\rm F814W} \\le 25.5$ mag (AB) or, for those galaxies outside of the F814W field of view, with apparent magnitude ${\\rm m} \\le 25.5$ mag (AB) in the closest available filter.",
"Stellar population models were fit to the data using iSEDfit [102], an IDL-based Bayesian inference routine for extracting physical parameters from broadband photometry.",
"iSEDfit is a widely used code for estimating galaxy properties at all redshifts [156], [3], [28], [153], [60].",
"For all galaxies, we assume they are at the redshift of the cluster they are near.",
"To enable comparisons between clusters and with existing literature, we obtain rest-frame ugriz, UBVRI [18], and JHK (from 2MASS filter curves) magnitudes for every object.",
"We denote rest-frame magnitudes in this work with a leading superscript, e.g., ${}^{0.0}r$ .",
"To derive these values, iSEDfit begins by determining the closest matched filter to each target filter at the redshift of the galaxy.",
"It then computes a magnitude offset based on the best-fit SED and returns the original photometry corrected by that offset.",
"This procedure retains the photometric errors on the original filter.",
"For galaxies lacking photometry in the best-matched filter, the code will synthesize a magnitude from the SED fit but will not return a photometric error, as those are limited to observed errors.",
"Synthetic magnitudes are needed when galaxies are outside of the field of view of certain filters and therefore are not observed in the closest-matched filter."
],
[
"Comparison to Previous CLASH Studies",
"One way to identify potential systematic issues with our technique is to compare our results to other photometric studies of these clusters.",
"However, the primary previous study using CLASH photometry [114] was not tailored for optimizing cluster galaxy photometry; rather it was a general-purpose attempt to measure everything in the field, particularly background galaxies.",
"Nevertheless, it provides an important first check of our results.",
"We match the publicly available photometric catalogs [114] to spectroscopic redshifts following the same technique as we used to match our new results.",
"To avoid biasing our results, we re-fit photometric redshifts to the public catalog using the same methods and parameters described in Section REF .",
"Due to the differing methodology in producing a detection catalog, there are slightly more galaxies with spectroscopic matches in the [114] catalog than in ours.",
"Using the same standards as in Section REF , we compare the 1942 total matches to their spectroscopic counterparts in Table REF .",
"In addition to an almost $10\\%$ drop in the percentage of well-matched galaxies, the old catalog also has ${\\sim } 20\\%$ to ${\\sim } 40\\%$ larger deviation between spectroscopic and photometric redshifts compared to our new work.",
"The colors we present here provide a substantial increase in the accuracy of photometric redshifts for cluster members, although we note that [114] presented accurate photometric redshifts for arcs, which are not in this analysis.",
"In comparison to the earlier catalog, our improved detection routine combined with more accurate photometric redshifts should produce a sample with greater purity for cluster identification.",
"To test this claim, we consider the photometric redshifts of every object in these 25 fields, within three magnitude constraints.",
"The fraction of galaxies with photometric redshifts close to the cluster redshift, such that $|(z_{\\rm p} - z_{\\rm c})| / (1 + z_{\\rm c} ) \\le 0.05$ , is significantly improved in our new catalog over the previous work; the exact fractions in three magnitude cuts are provided in Table REF .",
"As we do not need to segment objects to detect faint structure overlapping with extended halos, we produce a significantly less cluttered catalog.",
"Recent work by [99] produced another catalog of galaxies using the HST CLASH images, also demonstrating greatly improved photometric redshift accuracy over prior work by [79].",
"Their improvement is a result of optimizing BPZ inputs.",
"They also subtracted a parametric background model from the HST images to minimize the effects of ICL on photometry.",
"However, the analysis of [99] is limited to the restrictive field of view of WFC3-IR, which only covers ${\\sim } 40$ % of the area covered by CLASH ACS imaging.",
"For cluster population studies, the known correlations between galaxy properties and local galaxy density ([51]; also see [105], [80], [111]) or clustercentric radius [143], [144], [103] imply that the significantly smaller fields of view of the [99] catalogs limit their utility for any analysis of cluster galaxies.",
"Additionally, that work uses Source Extractor for detection and photometry at only one scale, whereas in this work we perform a multi-scale, multi-color detection and compute galaxy-size-dependent local backgrounds/foregrounds."
],
[
"Comparison to the Hubble Frontier Fields",
"Another comparison for our work is that of the Hubble Frontier Fields (HFF) catalogs produced by the ASTRODEEP collaboration [32], [97].",
"They analyzed HST imaging of MACS-J0416 (as well as Abell 2744, which is outside the scope of the CLASH observations) using HFF HST data as well as ground-based K and Spitzer IRAC observations.",
"HFF observations achieve a substantial depth, nominally ${\\sim } 2$ mag fainter than CLASH data.",
"The ASTRODEEP catalogs are created through a series of steps involving masking bright objects, fitting the ICL, and fitting bright galaxies, with intermediate steps involving subtraction of either ICL or galaxies.",
"They make an excellent comparison sample for our work; their observations are deeper, their results have been refereed, and their catalog is compiled under different assumptions (galaxy and ICL model-based vs. model-agnostic).",
"To compare our results for MACS 0416 with those of ASTRODEEP, we first match our catalogs galaxy-by-galaxy.",
"After sorting our catalog from brightest to faintest (using F814W magnitudes), we find a best match for each galaxy using the following process: if only one galaxy in the ASTRODEEP catalog is within 065 (10 pixels) of our target, that galaxy is matched with ours.",
"If more than one possible match is within that angular radius, we consider only those galaxies with F814W magnitudes $\\textrm {m} \\le \\textrm {m}_\\textrm {b} + 0.5$ , where $\\textrm {m}_\\textrm {b}$ is the brightest galaxy in that angular range.",
"We take the galaxy with smallest angular offset to the target in that subset to be the match.",
"Figure: Comparison between F814W magnitudes measured by the ASTRODEEP Collaboration and this work for MACS 0416.",
"Points are binned into hexagons, with the total density of points scaled logarithmically from 1 to 30 counts per hex.",
"Details of the fit are provided in the text.",
"A comparison of the offsets is provided in the lower panel.Our first cross-check between the two samples is to compare the measured magnitudes for each galaxy.",
"This comparison is sensitive to both the ability of each method to capture the entire flux as well as how the background subtraction affects the faintest galaxies.",
"This comparison is shown in Figure REF .",
"Most galaxies have similar magnitudes reported in both studies, but we find slightly fainter magnitudes for the faintest galaxies; for magnitudes F814W $\\ge 23$ , the galaxies in this work are a median 0.23 mag fainter.",
"Figure: Comparison between measured colors for MACS 0416 in the ASTRODEEP catalog and this work.",
"Counts are binned into hexagons, with the total density of points scaled logarithmically.",
"Note that the hex size is increased for bluer colors.We next consider how galaxy colors differ between the two techniques.",
"While the HFF observations are deeper than those of CLASH, they only have seven HST filters, so the color comparison is limited.",
"Figure REF shows the difference between measured colors between this work and ASTRODEEP.",
"We see a slight trend for galaxies in our sample to be bluer in F435W-F606W than in the ASTRODEEP catalog, but that color discrepancy is diminished for all of the redder filters.",
"Numerically, the median offset (defined such that a negative offset corresponds to a bluer object in our catalog) between colors is $\\langle \\Delta _{({\\rm F435W} - {\\rm F606W})}\\rangle = -0.137 \\pm 0.396$ $\\langle \\Delta _{({\\rm F606W} - {\\rm F814W})}\\rangle = -0.060 \\pm 0.129$ $\\langle \\Delta _{({\\rm F814W} - {\\rm F105W})}\\rangle = -0.037 \\pm 0.112$ $\\langle \\Delta _{({\\rm F105W} - {\\rm F140W})}\\rangle = -0.026 \\pm 0.068$ Here the reported uncertainties are 1.4286$\\times $ the median absolute deviation and all values are in magnitudes.",
"For galaxies with photometric redshift offset from the cluster $\\vert z_p - z_c \\vert $ <0.1 and red filter magnitude < 25 mag (AB), we report the same set of values, as well as the median color uncertainty for our photometry alone: $\\langle \\Delta _{({\\rm F435W} - {\\rm F606W})}\\rangle = -0.162 \\pm 0.442$ , $\\langle \\sigma \\rangle = 0.171$ $\\langle \\Delta _{({\\rm F606W} - {\\rm F814W})}\\rangle = -0.036 \\pm 0.079$ , $\\langle \\sigma \\rangle = 0.033 $ $\\langle \\Delta _{({\\rm F814W} - {\\rm F105W})}\\rangle = +0.004 \\pm 0.057$ , $\\langle \\sigma \\rangle = 0.018 $ $\\langle \\Delta _{({\\rm F105W} - {\\rm F140W})}\\rangle = -0.012 \\pm 0.026$ , $\\langle \\sigma \\rangle = 0.013 $ Again, all values are in magnitudes.",
"There is no significant offset in colors between these two works, although the deviation is larger than expected from measurement errors alone.",
"One possible source of the scatter is from different galaxy apertures, which would influence the colors of galaxies that have a color gradient.",
"While ASTRODEEP did not publish their detection areas, we investigate the effects of size differences by limiting our comparison to galaxies with measured brightnesses within 0.1 mag of each other in one filter.",
"While we were able to reduce measured deviations to the order of the calculated uncertainties for the bluest two colors through magnitude constraints, there remained a larger spread in color offsets for the redder colors, particularly among cluster galaxies, than could be explained by color uncertainties alone.",
"The color differences appear to therefore be real, and caused by the different methodologies for calculating a photometric background.",
"Nevertheless, as the largest color differences involve the F435W and F606W filters, we do not expect the differing colors to cause significant variations in SED fitting between the two reductions – at $z = 0.397$ (the redshift of MACS 0416), these filters cover the range below the $4,000\\ {\\rm Å}$ break for cluster galaxies.",
"Figure: Comparison between redshifts measured by the ASTRODEEP Collaboration and this work for MACS 0416.",
"Points are binned into hexagons, with the total density of points scaled logarithmically from 1 to 30 counts per hex, as indicated by the colorbar.",
"Details of the fit are provided in the text.",
"A zoom-in to just matches below redshift 1.0 is provided in the right panel.",
"The cluster redshift is indicated by the vertical and horizontal lines; the diagonal line is the identity line.One way to quantify the importance of these color offsets is to cross-check the photometric redshifts reported here and by ASTRODEEP.",
"The latter used multiple methods for redshift estimation, including using spectroscopic redshifts where available, and combined them all for a final answer, while we only consider our redshift estimates from BPZ.",
"We show in Figure REF a comparison between our reported photometric redshifts and the photometric and spectroscopic redshifts reported by the ASTRODEEP collaboration.",
"When considering the uncertainties on these measurements, we find good agreement between our reported redshifts and the comparison sample.",
"For galaxies with photometric redshifts $z_p < 1.0$ in either of our samples and brighter than 25th magnitude in the F814W filter, the median redshift offset between our catalogs is 0.004, with an uncertainty ($1.4826 \\times $ the median absolute deviation) of $\\pm $ 0.188.",
"We have compared the results of our mode-measuring background calculation technique to the results of a more traditional background-modeling photometry routine for one cluster.",
"Despite comparing CLASH observations to deeper HFF observations, we see no significant offset in photometry.",
"While further work is still needed to understand the systematic issues inherent to these and other photometric techniques, the comparisons presented here suggest that there is no significant offset between parametric structure modeling and local modal background estimation."
],
[
"Cluster Membership",
"Our next step is to define membership criteria for selection of cluster members.",
"Determining cluster membership is necessary for, among other things, measuring cluster luminosity functions.",
"We have already calculated a redshift probability distribution and an SED goodness-of-fit at the cluster redshift for all of our galaxies, and we supplement these calculations with measured spectroscopic redshifts where available.",
"We describe how we use this information to select cluster members below; we then compare how the sample completeness of cluster galaxies differs between our criteria and through cluster membership selection of only galaxies along the red sequence.",
"For the sake of homogeneity and uniform data quality, we investigate membership, and the properties of candidate cluster members, down to ${\\rm F814W} \\le 25.5$ mag (AB).",
"We consider the photometry, photometric redshift parameters, SED fit values, rest-frame magnitudes, and spectroscopic redshift measurements for all of these galaxies in our membership selection.",
"For each galaxy, we use the discrete probability distributions produced by BPZ to determine a total probability of that galaxy being within some redshift range of the nominal cluster redshift.",
"Here, we consider $|\\Delta _{\\rm z}| < 0.03$ , $|\\Delta _{\\rm z}| < 0.05$ , $|\\Delta _{\\rm z}|/(1 + {\\rm z}_{\\rm c}) < 0.03$ , and $|\\Delta _{\\rm z}|/(1 + {\\rm z}_{\\rm c}) < 0.05$ ; we label the summed probabilities within those ranges ${\\rm P}_{03}$ , ${\\rm P}_{05}$ , ${\\rm P}_{103}$ , and ${\\rm P}_{105}$ , respectively.",
"To determine membership, we step each galaxy through a series of screens; the result of each step is that a galaxy is classified as a member, classified as a non-member, or passed on to the next step.",
"The first sieve is to select cluster galaxies by their spectroscopic redshifts.",
"Any galaxy with $|\\Delta _{\\rm z}|/(1 + {\\rm z}_{\\rm c}) < 0.03$ is considered a cluster member; those with $|\\Delta _{\\rm z}|/(1 + {\\rm z}_{\\rm c}) > 0.10$ are considered non-members.",
"The rest of the galaxies – either those with indeterminate or no spectroscopic redshifts – are then characterized by their photometric redshift probabilities.",
"As a first pass, those galaxies with total probability ${\\rm P}_{03} > 0.8$ are assigned as members, while those with ${\\rm P}_{105} < 0.1$ are classified as non-members.",
"We next screen for cluster members by selecting those with a well-fit SED or a best-fit photometric redshift solution consistent with being in a cluster.",
"Galaxies with $\\chi ^2_\\nu < 1.5$ and $\\chi ^2_\\nu > 0.7$ (to avoid selecting galaxies with poorly constrained fits through this cut) in the SED fit at the cluster redshift are classified as cluster members, as are those with a most likely or best redshift determination from BPZ within $|\\Delta _{\\rm z}|/(1 + {\\rm z}_{\\rm c}) < 0.05$ .",
"For the remaining objects, we examine the distributions of ${\\rm P}_{03}$ , ${\\rm P}_{05}$ , ${\\rm P}_{103}$ , and ${\\rm P}_{105}$ ; we only classify those remaining galaxies with ${\\rm P}_{103} > 0.2$ and ${\\rm P}_{105} > 0.6$ as members, while the rest are classified as non-members.",
"An alternative way to select cluster members is to select only those galaxies with colors that align with the identified red sequence in a cluster [67], [84], [73], [122], [123].",
"Red sequence selection is a powerful technique for many applications but is problematic for others.",
"For example, the color selection criteria can be overly restrictive and exclude galaxies with recent star formation, such as, e.g., Butcher–Oemler [29] or Dressler–Gunn [52] galaxies.",
"Therefore, we quantify the sample completeness as functions of magnitude and color below.",
"Figure: Rest-frame 0.0 (g-r)^{0.0}(g-r) CMD for all 25 clusters.",
"Shown on the left are those galaxies we call members, as presented in Section , while those classified as non-members are plotted on the right.",
"A potential red sequence selection region is shown as a shaded box.",
"Hex bins are scaled logarithmically with the number of galaxies contained inside.In Figure REF we show a plot of $^{0.0}(g-r)$ colors using rest-frame magnitudes, as described in Section REF .",
"Here, we define a selection region, counting those galaxies with $0.5 < {}^{0.0}(g-r) < 0.875$ .",
"For galaxies brighter than $^{0.0}r < -16$ , 70.4% of cluster members are inside this color region, but so is an additional population of non-members with size equal to 75.2% of the total member population above that brightness threshold.",
"Increasing the magnitude cut to $^{0.0}r < -20$ , 89.1% of members fall within that color region, while the contaminant population is only equal to 20.0% of the total cluster population in that luminosity range.",
"Based on these results, selecting galaxies using the red sequence is well suited for selecting galaxies at the tip of the relation.",
"However, this selection not only fails to account for the entire cluster population at fainter magnitudes, it also becomes significantly affected by contamination."
],
[
"Luminosity Function",
"Having assembled a catalog of cluster galaxies with magnitudes adjusted to the same band via SED-fitting, we investigate the variations in cluster populations by comparing their luminosity functions.",
"Here, we only consider the luminosity functions of the entire cluster populations (we do not fit luminosity functions to just the red or blue populations) in only one band.",
"Likewise, we do not correct for the different metric fields of view for each cluster.",
"We will present a more comprehensive analysis of the CLASH luminosity functions in a future work.",
"Figure: Rest-frame 0.0 i^{0.0}i-magnitude luminosity functions for all 25 CLASH clusters.",
"Clusters are shown binned in half-magnitude intervals.",
"The best-fit Schechter luminosity function is shown in orange (purple) for X-ray-selected (high-magnification) clusters, while the binned galaxy counts are shown in purple (orange).",
"Clusters are plotted in order of increasing redshift.",
"Best-fit values of α\\alpha , M * M^*, and φ * \\phi ^* are provided in that order on each plot.",
"Details of the fits are provided in the text.lrrrr 0.6 CLASH Luminosity Function Fit 0pt Cluster Name $\\alpha $ ${\\rm M}_i^*$ $\\phi ^*$ Det.",
"Limit (mag) (mag) Abell 383 $-0.94^{+ 0.05}_{- 0.05}$ $-22.17^{+ 0.26}_{- 0.35}$ $ 26.6^{+ 6.2}_{- 5.9}$ $M^* + 7.32$ Abell 209 $-0.89^{+ 0.05}_{- 0.05}$ $-22.02^{+ 0.24}_{- 0.27}$ $ 33.2^{+ 7.1}_{- 6.6}$ $M^* + 6.97$ Abell 1423 $-0.87^{+ 0.06}_{- 0.05}$ $-21.85^{+ 0.25}_{- 0.26}$ $ 31.9^{+ 7.6}_{- 5.9}$ $M^* + 7.38$ Abell 2261 $-0.86^{+ 0.04}_{- 0.04}$ $-21.86^{+ 0.20}_{- 0.21}$ $ 53.3^{+ 9.2}_{- 8.4}$ $M^* + 6.83$ RXJ 2129 $-0.97^{+ 0.05}_{- 0.05}$ $-22.35^{+ 0.27}_{- 0.34}$ $ 27.0^{+ 6.9}_{- 5.7}$ $M^* + 6.80$ Abell 611 $-0.75^{+ 0.08}_{- 0.05}$ $-21.79^{+ 0.22}_{- 0.18}$ $ 60.8^{+ 14.6}_{- 8.5}$ $M^* + 5.58$ MS 2137 $-0.86^{+ 0.06}_{- 0.07}$ $-21.97^{+ 0.22}_{- 0.33}$ $ 37.0^{+ 8.3}_{- 8.5}$ $M^* + 6.50$ RXJ 1532 $-0.86^{+ 0.06}_{- 0.07}$ $-21.92^{+ 0.20}_{- 0.22}$ $ 51.0^{+ 10.6}_{- 9.5}$ $M^* + 5.85$ RXJ 2248 $-0.75^{+ 0.04}_{- 0.06}$ $-21.87^{+ 0.14}_{- 0.21}$ $ 93.3^{+ 11.7}_{- 15.5}$ $M^* + 5.76$ MACS 1931 $-1.03^{+ 0.06}_{- 0.05}$ $-21.72^{+ 0.21}_{- 0.24}$ $ 45.3^{+ 11.4}_{- 7.9}$ $M^* + 5.06$ MACS 1115 $-0.79^{+ 0.05}_{- 0.08}$ $-21.83^{+ 0.17}_{- 0.23}$ $ 69.7^{+ 10.8}_{- 12.3}$ $M^* + 4.88$ MACS 1720 $-0.85^{+ 0.07}_{- 0.05}$ $-22.43^{+ 0.21}_{- 0.22}$ $ 53.5^{+ 12.2}_{- 8.4}$ $M^* + 6.02$ MACS 0416 $-0.74^{+ 0.05}_{- 0.07}$ $-22.07^{+ 0.15}_{- 0.21}$ $ 93.5^{+ 13.0}_{- 15.3}$ $M^* + 5.71$ MACS 0429 $-0.71^{+ 0.06}_{- 0.10}$ $-21.32^{+ 0.16}_{- 0.31}$ $ 71.5^{+ 10.7}_{- 16.0}$ $M^* + 5.08$ MACS 1206 $-0.78^{+ 0.05}_{- 0.05}$ $-22.21^{+ 0.14}_{- 0.16}$ $113.2^{+ 15.0}_{- 16.0}$ $M^* + 5.83$ MACS 0329 $-0.82^{+ 0.05}_{- 0.06}$ $-22.40^{+ 0.17}_{- 0.19}$ $ 81.7^{+ 11.8}_{- 14.0}$ $M^* + 5.98$ RXJ 1347 $-0.87^{+ 0.07}_{- 0.04}$ $-22.22^{+ 0.17}_{- 0.17}$ $ 86.0^{+ 17.2}_{- 11.2}$ $M^* + 5.50$ MACS 1311 $-1.04^{+ 0.05}_{- 0.06}$ $-22.73^{+ 0.24}_{- 0.41}$ $ 35.4^{+ 7.9}_{- 8.4}$ $M^* + 5.88$ MACS 1149 $-0.71^{+ 0.04}_{- 0.06}$ $-22.18^{+ 0.12}_{- 0.18}$ $151.3^{+ 15.6}_{- 21.9}$ $M^* + 5.34$ MACS 1423 $-0.97^{+ 0.06}_{- 0.06}$ $-22.85^{+ 0.24}_{- 0.39}$ $ 43.5^{+ 10.0}_{- 9.4}$ $M^* + 5.83$ MACS 0717 $-0.82^{+ 0.05}_{- 0.05}$ $-22.56^{+ 0.14}_{- 0.16}$ $133.1^{+ 17.6}_{- 16.8}$ $M^* + 5.04$ MACS 2129 $-0.87^{+ 0.06}_{- 0.06}$ $-22.11^{+ 0.16}_{- 0.18}$ $ 82.5^{+ 14.7}_{- 14.0}$ $M^* + 4.95$ MACS 0647 $-0.92^{+ 0.07}_{- 0.05}$ $-22.55^{+ 0.20}_{- 0.21}$ $ 62.5^{+ 14.1}_{- 9.4}$ $M^* + 5.58$ MACS 0744 $-1.01^{+ 0.06}_{- 0.06}$ $-22.88^{+ 0.21}_{- 0.28}$ $ 57.7^{+ 12.5}_{- 12.9}$ $M^* + 5.10$ CLJ 1226 $-1.05^{+ 0.07}_{- 0.06}$ $-22.86^{+ 0.33}_{- 0.25}$ $ 53.1^{+ 13.4}_{- 11.2}$ $M^* + 4.52$ $0.0 < z < 0.32 \\rule {0pt}{3ex} $ $-0.90^{+ 0.02}_{- 0.04}$ $-22.04^{+ 0.09}_{- 0.13}$ $ 35.8^{+ 2.5}_{- 4.2}$ -16.00 $0.32 < z < 0.4 \\rule {0pt}{3ex} $ $-0.87^{+ 0.04}_{- 0.02}$ $-22.05^{+ 0.10}_{- 0.07}$ $ 58.9^{+ 6.8}_{- 3.0}$ -16.50 $0.4 < z < 0.55 \\rule {0pt}{3ex} $ $-0.84^{+ 0.01}_{- 0.04}$ $-22.41^{+ 0.05}_{- 0.10}$ $ 91.0^{+ 3.6}_{- 9.3}$ -17.00 We fit our data with a Schechter luminosity function [125], of the form $\\begin{split}\\phi (M)\\ dM = &0.4\\ \\phi ^* \\ln (10) \\times 10^{0.4 (M^* - M) \\times (1 + \\alpha )}\\\\& \\times \\exp ( -10^{0.4 (M^* - M)})\\ dM,\\end{split}$ where $\\phi ^*$ is the number of galaxies per magnitude, $M^*$ is the characteristic magnitude of the function, $M$ is the absolute magnitude of a galaxy, and $\\alpha $ is the faint-end slope.",
"We remind the reader that, in this form, a “flat slope” occurs at $\\alpha = -1$ .",
"As we have made no attempt to adjust for the volumes sampled by each cluster, $\\phi ^*$ is a normalization and not a density.",
"For each cluster, we determine the parameters of the Schechter function using unbinned luminosity data, by following the maximum likelihood technique described in [91] and [48].",
"Using interpolated rest-frame $^{0.0}i$ -band magnitudes, we find the minimum of a likelihood function defined as $\\begin{split}S &= -2 \\ln (\\mathcal {L}) \\\\&= -2 \\sum \\limits _{\\rm i}^{\\rm N} \\ln (\\ \\phi (\\ {\\rm M}_{\\rm i})) + 2 \\int \\limits _{ {\\rm M}_{\\rm faint}} \\phi ({\\rm M})\\ dM\\end{split}$ in a grid of values for $\\alpha $ , $M^*$ and $\\phi ^*$ .",
"While $S$ itself does not provide a goodness-of-fit statistic in the same way as $\\chi ^2$ , the errors in individual parameters are still bound by contours in $\\Delta S$ in the same way as they would be for $\\Delta \\chi ^2$ .",
"${\\rm M}_{\\rm faint}$ is set for each cluster to be the faintest object classified as a member.",
"We use a 50x50x25 grid uniformly covering $\\alpha {=} [-1.3, -0.5]$ , $M^* {=} [-23, -18]$ , and $\\phi ^* {=} [10^{-1}, 10^{4}]$ , where $\\phi ^*$ is in units of galaxies per magnitude per cluster field (within the observed field of view).",
"After finding the best value, we examine a sub-grid covering the 9x9x9 box centered on the best parameters.",
"We repeat the sub-sampling step a second time, for a total of three resolution levels for the three luminosity function parameters of interest.",
"To determine the $1 \\sigma $ uncertainties we find the largest and smallest value of each parameter in the $\\alpha $ , $M^*$ , $\\phi ^*$ parameter space in which $\\Delta (S) \\le 1.00$ .",
"We present our best-fit values for each of the CLASH clusters in Table .",
"Uncertainties listed are $1 \\sigma $ values.",
"We plot our luminosity functions, as well as histograms of cluster members, in Figure REF .",
"$\\phi ^*$ has been divided by 2 in this figure to match the half-magnitude-wide bins.",
"We also fit the combined galaxy sample in three redshift bins, each containing seven clusters.",
"The results of those fits, as well as the values of ${\\rm M}_{\\rm faint}$ used in the fits, are shown in Table .",
"For the combined samples, the values of $\\phi ^*$ presented in Table have been divided by 7.",
"Figure: Left: measured values of M * M^* for i-band luminosity functions of CLASH clusters fit individually (purple) and in redshift bins (black).",
"The dotted black line traces the best fit of the redshift evolution of M * M^*, as described in the text.",
"Shown in blue and orange are fits for ii-band M * M^* values for all (blue) and only red (orange) galaxies, with evolution parameters from .",
"Right: measured values of α\\alpha for ii-band luminosity functions of CLASH clusters fit individually (purple) and in redshift bins (black).In Figure REF we plot our measured values of $M^*$ in $i$ -band for all clusters (purple) as a function of redshift, as well as for the three binned fits to the luminosity function, set to the average redshift of the binned clusters (black).",
"There is a trend for increasing $M^*$ brightness with increasing redshift, which is not unexpected [149].",
"We perform a linear regression of our measured values of $M^*$ to a line of the form $M^*(z) = a \\times z + M^*_0.$ The best-fit values of this regression are $a = -1.65$ and $M^*_0 = -21.45$ .",
"This fit is shown in Figure REF by the dotted black line.",
"We also plot the best-fit evolutionary parameters from the Galaxy and Mass Assembly (GAMA) k-corrected $i$ -band luminosity fuctions presented by [88], after adjusting their reported values of $M^*$ to an $H_0 = 70\\ {\\rm km} \\ {\\rm s}^{-1}$ cosmology.",
"Figure REF shows the evolution of $M^*$ both for all galaxies (blue line) and only red galaxies (orange line) reported by [88].",
"When integrating the Schechter function to find the likelihood as described by Equation REF , we set ${\\rm M}_{\\rm faint}$ equal to the faintest cluster member.",
"However, we were also interested in the limit at which the Schechter function no longer described the observed galaxy counts in terms of ${\\rm M} - {\\rm M}^* $ .",
"To determine this limit, we found the faintest half-magnitude-wide bin in which the number of galaxies was within 90% of that expected by the cluster's luminosity function.",
"These values are reported for each cluster in Table .",
"We find that the luminosity functions hold for ${\\sim } 5$ mag fainter than $M^*$ for the entire sample and to ${\\sim } M^* + 7$ for the lowest-redshift clusters.",
"Previous works that have probed the luminosity function of clusters to this depth have been mostly limited to low redshift [124], [33], [129], [151], with the notable exception of [44], who obtained a similar depth in three combined samples containing fewer total clusters than studied here.",
"For our sample, we find best-fit $\\alpha $ values in the range $-1.0 \\lesssim \\alpha \\lesssim -0.7$ and $M^*$ values in the range $-23 \\lesssim {\\rm M}^* \\lesssim -21.5$ ; when considering the binned samples, we find that $\\alpha $ declines from -0.90 to -0.84 and $M^*$ brightens from -22.0 to -22.4 in the redshift range sampled here.",
"We compare these values to those of other works, beginning with [120].",
"They fit SDSS clusters drawn from the sample presented in [142] to a maximum redshift of $z \\leqslant 0.06$ .",
"For $i$ -band observations, their best fit was ${\\rm M}^{*}_{i} = -21.46^{+0.03}_{-0.04}$ , $\\alpha = -0.75^{+0.02}_{-0.01}$ .",
"That work also examined the luminosity function of 16 clusters in the ESO Distant Cluster Survey (EDisCS) spanning redshifts $0.4 < z < 0.8$ .",
"When only including red sequence members, the EDisCS clusters together were best fit by an i-band luminosity function of ${\\rm M}^{*}_{i} = -21.80^{+0.22}_{-0.17}$ and $\\alpha = -0.34^{+0.16}_{-0.10}$ .",
"[44] studied 11 merging clusters at $0.2 \\lesssim z \\lesssim 0.6$ , finding best-fit Schechter slopes of $\\alpha \\approx -1$ .",
"[134] derived values of $\\alpha = -0.91 \\pm 0.02$ , ${\\rm M}_{\\rm V} = -21.39 \\pm 0.05$ for 10 MACS [56] clusters at $z \\sim 0.5$ , including several considered in this work.",
"[93] presented measurements of luminosity functions for $0.4 \\le z < 0.9$ at rest-frame I and R; their red-sequence-selected results are $\\alpha _{\\rm R} = -0.80 \\pm 0.14$ , ${\\rm M}^*_{\\rm R} = -22.4 \\pm 0.2$ and $\\alpha _{\\rm I} = -0.37 \\pm 0.18$ , ${\\rm M}^*_{\\rm I} = -22.0 \\pm 0.2$ .",
"[92] looked at a subset of CLASH clusters using only two filters of HST imaging and selecting all galaxies along the red sequence to a shallower depth than explored here; in two redshift bins, they find a steepening slope ($\\alpha = -0.85 \\pm 0.13$ at $z = 0.289$ to $\\alpha = -0.63 \\pm 0.17$ at $z = 0.512$ ) for HST data.",
"Using ground-based imaging of two CLASH clusters to measure the stellar mass function of all galaxies beyond $r_{200}$ , [9], [10] found slopes of $\\alpha = -0.85 \\pm 0.04$ and $\\alpha = -1.17 \\pm 0.02$ .",
"The results in these works are consistent with what is presented here.",
"We see a passive evolution in $M^*$ with redshift and no statistically significant change in $\\alpha $ with redshift.",
"Several works report a much steeper slope (lower value of $\\alpha $ ), namely [92], [93], and [120], but these fits only considered red sequence galaxies.",
"While several other studies have found steeper faint-end slopes for red sequence-only samples (e.g., [42]; but see also [5], [6], [8]) works that do not select only red sequence members find $\\alpha $ values more consistent with what we present here [135], [90]."
],
[
"Summary",
"We have presented a new framework for detection and photometry of galaxies in crowded fields.",
"Using the computational power of a desktop computer and the maps of observed variance in pixel fluxes from HST observations of the CLASH fields, we characterize the background local to each galaxy on a scale appropriate to that galaxy's size.",
"As that local background may include the flux of a nearby galaxy, we work galaxy-by-galaxy, leaving behind the measured background as we go.",
"This allows us to accurately photometer overlapping galaxies.",
"We also present a technique for detecting galaxies in a cluster by mapping structure at all scales in an image.",
"By connecting the large-scale structure in an image with the peaks of flux distributions, we can detect galaxies at their full extent; however, by looking for small-scale structure, we can identify smaller galaxies that are otherwise buried in the wings of larger, brighter galaxies.",
"This technique results in a significantly improved fraction of cluster members in the sample of detected galaxies, as small galaxies can be recovered without needing to fragment large galaxies.",
"We provide one caution for using the same methods to detect sources in other data.",
"As the same statistical invariance across large scales that causes light from BCGs and the ICL to be removed around other galaxies also exists in resolved spiral galaxies, detection catalogs that successfully identify small cluster members will also detect individual knots of stars in late-type galaxies.",
"Without filtering, a detection catalog will over-count foreground galaxy populations and also not correctly photometer the resolved foreground spiral galaxies that actually do exist.",
"As future observatories come online, and as new deep surveys of cluster environments are undertaken, it will be necessary for photometric pipelines to account for the issues of crowded environments in order to maximize the return on investment for these programs.",
"With the CLASH images, we have demonstrated the ability of mode-based background determination to detect and photometer galaxies of all sizes across deep, multi-wavelength datasets.",
"We briefly summarize the catalog of CLASH galaxies we have produced and the initial scientific results of that survey.",
"We detect and photometer 20,930 objects in the fields of 25 massive galaxy clusters, with a median of 658 objects per cluster field brighter than F814W = 25 mag (AB).",
"A median of 463 of these galaxies per cluster are well detected (above $3 \\sigma $ in at least 8 filters; 327 in 12, and 207 in 14, spanning the UV to the near-IR.",
"Out photometry is given in the Appendix.",
"Thanks to the optimized background measurement, source detection, and photometry techniques described in this paper, we obtain an approximately $30\\%$ increase in the accuracy of photometric redshifts when compared with the previous CLASH photometry, with median absolute offsets between spectroscopic and photometric redshifts of $|(z_{\\rm p} - z_{\\rm s})| / (1 + z_{\\rm s} ) = 0.030$ .",
"This photometric improvement is in addition to obtaining a much purer sample of cluster galaxies.",
"We are able to detect galaxies to $M^* + 4.5$ for all clusters and to $M^* + 7.5$ for the nearest clusters.",
"The depth to which we can measure the luminosity function, combined with the large number of available filters with observations for each cluster, presents us with an unprecedented look at the growth of clusters across redshift.",
"We see a passively evolving value of $M^*$ with redshift and no significant evolution in the population of faint galaxies (as measured by the slope of the luminosity function).",
"TC acknowledges support from a fellowship from the Michigan State Unversity College of Natural Science.",
"T.C.",
"and M.D.",
"were supported by NASA/STScI grant HST-GO-12065.07-A.",
"K.U.",
"acknowledges support from the Ministry of Science and Technology of Taiwan through the grant MOST 106-2628-M-001-003-MY3.",
"Results are partially based on ESO LP186.A-0798.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"We used the cosmological calculator presented in [148] in this work.",
"Our work with Source Extractor was greatly aided by the guide from [75].",
"T.C.",
"thanks Thomas Hettinger and Ravi Jagasia for discussions on software implementation.",
"All of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).",
"STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.",
"Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.",
"Facilities: Hubble Space Telescope/ACS, WFC3 Our catalogs of observed photometry for all detected objects and rest-frame photometry for cluster members are provided online.",
"We summarize the available photometry and photometric and spectroscopic redshift data for all galaxies in Table .",
"For those galaxies we classified as members in Section , we present rest-frame photometry derived using iSEDfit in Table .",
"cll Galaxy Properties 0pt Column Number Column Name Column Description 1 Object ID 2 Cluster Cluster Name 3 $\\alpha _{2000}$ Right Ascension (J2000) 4 $\\delta _{2000}$ Declination (J2000) 5 $X$ $X$ pixel coordinate 6 $Y$ $Y$ pixel coordinate 7 $a$ Semimajor axis length (pixels) 8 $b$ Semiminor axis length (pixels) 9 PA Position angle (degrees) 10 F225W Mag Magnitude in F225W (mag) 11 F225W Mag Error Error in F225W magnitude (mag) 12 F275W Mag Magnitude in F275W (mag) 13 F275W Mag Error Error in F275W magnitude (mag) 14 F336W Mag Magnitude in F336W (mag) 15 F336W Mag Error Error in F390W magnitude (mag) 16 F390W Mag Magnitude in F390W (mag) 17 F390W Mag Error Error in F390W magnitude (mag) 18 F435W Mag Magnitude in F435W (mag) 19 F435W Mag Error Error in F435W magnitude (mag) 20 F475W Mag Magnitude in F475W (mag) 21 F475W Mag Error Error in F475W magnitude (mag) 22 F555W Mag Magnitude in F555W (mag) 23 F555W Mag Error Error in F555W magnitude (mag) 24 F606W Mag Magnitude in F606W (mag) 25 F606W Mag Error Error in F606W magnitude (mag) 26 F625W Mag Magnitude in F625W (mag) 27 F625W Mag Error Error in F625W magnitude (mag) 28 F775W Mag Magnitude in F775W (mag) 29 F775W Mag Error Error in F775W magnitude (mag) 30 F814W Mag Magnitude in F814W (mag) 31 F814W Mag Error Error in F814W magnitude (mag) 32 F850LP Mag Magnitude in F850LP (mag) 33 F850LP Mag Error Error in F850LP magnitude (mag) 34 F105W Mag Magnitude in F105W (mag) 35 F105W Mag Error Error in F105W magnitude (mag) 36 F110W Mag Magnitude in F110W (mag) 37 F110W Mag Error Error in F110W magnitude (mag) 38 F125W Mag Magnitude in F125W (mag) 39 F125W Mag Error Error in F125W magnitude (mag) 40 F140W Mag Magnitude in F140W (mag) 41 F140W Mag Error Error in F140W magnitude (mag) 42 F160W Mag Magnitude in F140W (mag) 43 F160W Mag Error Error in F160W magnitude (mag) 44 ${\\rm z}_{\\rm b}$ BPZ zb 45 ${\\rm z}_{\\rm b}$ min BPZ zbmin 46 ${\\rm z}_{\\rm b}$ max BPZ zbmax 47 ${\\rm z}_{\\rm mk}$ BPZ zml 48 odds BPZ Odds 49 chisq BPZ Chi-squared 50 ${\\rm z}_{\\rm spec}$ Spectroscopic redshift 51 $\\sigma {\\rm z}_{\\rm spec}$ Spectroscopic redshift uncertainty 52 ${\\rm z}_{\\rm spec}$ source Source of spectroscopic redshift This table is available in its entirety in machine-readable form.",
"cll Rest-frame Photometry 0pt Column Number Column Name Column Description 1 Object ID 2 ${}^{0.0}u$ ${}^{0.0}u$ magnitude 3 $\\sigma ({}^{0.0}u)$ ${}^{0.0}u$ magnitude uncertainty 4 ${}^{0.0}g$ ${}^{0.0}g$ magnitude 5 $\\sigma ({}^{0.0}g)$ ${}^{0.0}g$ magnitude uncertainty 6 ${}^{0.0}r$ ${}^{0.0}r$ magnitude 7 $\\sigma ({}^{0.0}r)$ ${}^{0.0}r$ magnitude uncertainty 8 ${}^{0.0}i$ ${}^{0.0}i$ magnitude 9 $\\sigma ({}^{0.0}i)$ ${}^{0.0}i$ magnitude uncertainty 10 ${}^{0.0}z$ ${}^{0.0}z$ magnitude 11 $\\sigma ({}^{0.0}z)$ ${}^{0.0}z$ magnitude uncertainty 12 ${}^{0.0}$ U ${}^{0.0}$ U magnitude 13 $\\sigma ({}^{0.0}$ U) ${}^{0.0}$ U magnitude uncertainty 14 ${}^{0.0}$ B ${}^{0.0}$ B magnitude 15 $\\sigma ({}^{0.0}$ B) ${}^{0.0}$ B magnitude uncertainty 16 ${}^{0.0}$ V ${}^{0.0}$ V magnitude 17 $\\sigma ({}^{0.0}$ V) ${}^{0.0}$ V magnitude uncertainty 18 ${}^{0.0}$ R ${}^{0.0}$ R magnitude 19 $\\sigma ({}^{0.0}$ R) ${}^{0.0}$ R magnitude uncertainty 20 ${}^{0.0}$ I ${}^{0.0}$ I magnitude 21 $\\sigma ({}^{0.0}$ I) ${}^{0.0}$ I magnitude uncertainty 22 ${}^{0.0}$ J ${}^{0.0}$ J magnitude 23 $\\sigma ({}^{0.0}$ J) ${}^{0.0}$ J magnitude uncertainty 24 ${}^{0.0}$ H ${}^{0.0}$ H magnitude 25 $\\sigma ({}^{0.0}$ H) ${}^{0.0}$ H magnitude uncertainty 26 ${}^{0.0}$ K ${}^{0.0}$ K magnitude 27 $\\sigma ({}^{0.0}$ K) ${}^{0.0}$ K magnitude uncertainty This table is available in its entirety in machine-readable form."
]
] | 1709.01925 |
[
[
"A second order primal-dual method for nonsmooth convex composite\n optimization"
],
[
"Abstract We develop a second order primal-dual method for optimization problems in which the objective function is given by the sum of a strongly convex twice differentiable term and a possibly nondifferentiable convex regularizer.",
"After introducing an auxiliary variable, we utilize the proximal operator of the nonsmooth regularizer to transform the associated augmented Lagrangian into a function that is once, but not twice, continuously differentiable.",
"The saddle point of this function corresponds to the solution of the original optimization problem.",
"We employ a generalization of the Hessian to define second order updates on this function and prove global exponential stability of the corresponding differential inclusion.",
"Furthermore, we develop a globally convergent customized algorithm that utilizes the primal-dual augmented Lagrangian as a merit function.",
"We show that the search direction can be computed efficiently and prove quadratic/superlinear asymptotic convergence.",
"We use the $\\ell_1$-regularized model predictive control problem and the problem of designing a distributed controller for a spatially-invariant system to demonstrate the merits and the effectiveness of our method."
],
[
"Introduction",
"We study a class of composite optimization problems in which the objective function is given by the sum of a differentiable strongly convex component and a nondifferentiable convex component.",
"Problems of this form are encountered in diverse fields including compressive sensing, machine learning, statistics, image processing, and control [1], [2], [3], [4], [5].",
"They often arise in structured feedback synthesis where it is desired to balance controller performance (e.g., the closed-loop ${\\cal H}_2$ or ${\\cal H}_\\infty $ norm) with structural complexity [5], [6].",
"The lack of differentiability in the regularization term precludes the use of standard descent methods for smooth objective functions.",
"Proximal gradient methods [7], [8], [9] and their accelerated variants [10] generalize gradient descent, but typically require the nonsmooth term to be separable.",
"An alternative approach introduces an auxiliary variable to split the smooth and nonsmooth components of the objective function.",
"The reformulated problem facilitates the use of splitting methods such as the alternating direction method of multipliers (ADMM) [11].",
"This augmented-Lagrangian-based method divides the optimization problem into simpler subproblems, allows for a broader class of regularizers than proximal gradient, and it is convenient for distributed implementation.",
"In [12], we exploited the structure of proximal operators associated with nonsmooth regularizers and introduced the proximal augmented Lagrangian.",
"This continuously differentiable function enables the use of the standard method of multipliers (MM) for nonsmooth optimization and it is convenient for distributed implementation via the Arrow-Hurwicz-Uzawa gradient flow dynamics.",
"Recent work has extended MM to incorporate second order updates of the primal and dual variables [13], [14], [15] for nonconvex problems with twice continuously differentiable objective functions.",
"Since first order approaches tend to converge slowly to high-accuracy solutions, much work has focused on developing second order methods for nonsmooth composite optimization.",
"A generalization of Newton's method was developed in [16], [17], [18], [19] and it requires solving a regularized quadratic subproblem to determine a search direction.",
"Related ideas have been utilized for sparse inverse covariance estimation in graphical models [20], [21] and topology design in consensus networks [22].",
"Generalized Newton updates for identifying stationary points of (strongly) semismooth gradient mappings were first considered in [23], [24], [25].",
"In [26], [27], [28], the authors introduce the once-continuously differentiable Forward-Backward Envelope (FBE) and solve composite problems by minimizing the FBE using line search, quasi-Newton methods, or second order updates based on an approximation of the generalized Hessian.",
"We develop a second order primal-dual algorithm for nonsmooth composite optimization by leveraging these advances.",
"Second order updates for the once continuously differentiable proximal augmented Lagrangian are formed using a generalized Hessian.",
"We employ the merit function introduced in [13] to assess progress towards the optimal solution and develop a globally convergent customized algorithm with fast asymptotic convergence rate.",
"When the proximal operator associated with a nonsmooth regularizer is (strongly) semismooth, our algorithm exhibits local (quadratic) superlinear convergence.",
"Our presentation is organized as follows.",
"In Section , we formulate the problem and provide necessary background material.",
"In Section , we define second order updates to find the saddle points of the proximal augmented Lagrangian.",
"In Section , we prove global exponential stability of a continuous-time differential inclusion.",
"In Section , we develop a customized algorithm that converges globally to the saddle point and exhibits superlinear or quadratic asymptotic convergence rates.",
"In Section , we provide examples to illustrate the utility of our method.",
"We discuss interpretations of our approach and connections to the alternative algorithms in Section and conclude the paper in Section ."
],
[
"Problem formulation and background",
"We consider the problem of minimizing the sum of two convex functions over an optimization variable $x \\in \\mathbb {R}^n$ , $\\operatornamewithlimits{minimize}\\limits _x ~~ f(x) \\;+\\; g(Tx)$ where $T \\in \\mathbb {R}^{m \\times n}$ .",
"Problem (REF ) was originally formulated in the context of compressive sensing to incorporate structural considerations into traditional sensing or regression [1], [2], [3].",
"As specified in Assumption REF , the differentiable part of the objective function $f$ , which quantifies loss or performance, is strictly convex with a Lipschitz continuous gradient.",
"In contrast, the regularization function $g$ may be nondifferentiable and is used to incorporate structural requirements on the optimization variable.",
"In structured feedback synthesis [5], [6], $f$ typically quantifies the closed-loop performance, e.g., the $\\mathcal {H}_2$ norm, and $g$ imposes structural requirements on the controller, e.g., by penalizing the amount of network traffic [29], [30], [31], [32].",
"Although our strongest results require strong convexity of $f$ , our theory and techniques are applicable as long as the Hessian of $f$ is positive definite.",
"Assumption 1 The function $f$ is twice continuously differentiable, has an $L_f$ Lipschitz continuous gradient $\\nabla f$ , and is strictly convex with $\\nabla ^2 f \\succ 0$ ; the function $g$ is proper, lower semicontinuous, and convex; and the matrix $T$ has full row rank.",
"The matrix $T$ is important when the desired structure has a simple representation in the co-domain of $T$ , but it makes the problem more challenging.",
"One approach is to reformulate (REF ) by introducing an auxiliary optimization variable $z \\in \\mathbb {R}^m$ , $\\begin{array}{rl}\\operatornamewithlimits{minimize}\\limits _{x, \\, z} & f(x) \\;+\\; g(z)\\\\[0.1cm]\\operatornamewithlimits{subject~to}& Tx \\;-\\; z \\;=\\; 0.\\end{array}$ Problem (REF ) is convenient for constrained optimization algorithms based on the augmented Lagrangian, $\\nonumber \\mathcal {L}_\\mu (x,z;y)\\; \\mathrel {\\mathop :}=\\;f(x) \\; + \\; g(z) \\; + \\; y^T(Tx -z) \\; + \\; \\tfrac{1}{2\\mu } \\, \\Vert Tx \\, - \\, z \\Vert ^2,$ where $y \\in \\mathbb {R}^m$ is the Lagrange multiplier and $\\mu $ is a positive parameter.",
"Relative to the standard Lagrangian, $\\mathcal {L}_\\mu $ contains an additional quadratic penalty on the linear constraint in (REF ).",
"In the remainder of this section, we provide background on proximal operators and describe generalizations of the gradient for nondifferentiable functions.",
"We also briefly overview existing approaches for solving (REF )."
],
[
"Proximal operators",
"The proximal operator of the function $g$ is the minimizer of the sum of $g$ and a proximal term, $\\mathbf {prox}_{\\mu g}(v)\\;\\mathrel {\\mathop :}=\\;\\operatornamewithlimits{argmin}_z~g(z)\\;+\\;\\tfrac{1}{2\\mu } \\, \\Vert z \\,-\\, v \\Vert ^2$ where $\\mu $ is a positive parameter and $v$ is a given vector.",
"When $g$ is convex, its proximal operator is Lipschitz continuous with parameter 1, differentiable almost everywhere, and firmly non-expansive [9].",
"The value function associated with (REF ) specifies the Moreau envelope of $g$ , $\\!\\!\\!\\!\\begin{array}{rrl}M_{\\mu g} (v)& \\!\\!\\!",
"\\mathrel {\\mathop :}=\\!\\!\\!",
"&\\inf \\limits _{z} \\; g (z)\\, + \\,\\tfrac{1}{2\\mu } \\, \\Vert z \\, - \\, v \\Vert ^2\\\\[0.15cm]&\\!\\!\\!",
"=\\!\\!\\!",
"&g (\\mathbf {prox}_{\\mu g}(v))\\, + \\,\\tfrac{1}{2\\mu } \\, \\Vert \\mathbf {prox}_{\\mu g}(v) - v \\Vert ^2.\\end{array}$ The Moreau envelope is continuously differentiable, even when $g$ is not, and its gradient $\\nabla M_{\\mu g}(v)\\;=\\;\\tfrac{1}{\\mu }\\left(v \\;-\\; \\mathbf {prox}_{\\mu g}(v) \\right)$ is Lipschitz continuous with parameter $1/\\mu $ .",
"For example, the proximal operator associated with the $\\ell _1$ norm, $g (z) = \\sum |z_i|$ , is determined by soft-thresholding, ${\\cal S}_{\\mu } (v_i)\\;\\mathrel {\\mathop :}=\\;\\mathrm {sign}(v_i)\\max \\lbrace |v_i| \\,-\\, \\mu ,\\; 0\\rbrace ,$ the associated Moreau envelope is the Huber function, $M_{\\mu g}(v_i)\\; = \\;\\left\\lbrace \\begin{array}{lr}\\tfrac{1}{2\\mu }v_i^2,&|v_i|\\le \\mu \\\\[0.15cm]|v_i| - \\tfrac{\\mu }{2},&|v_i|\\ge \\mu \\end{array}\\right.$ and its gradient is the saturation function, $\\nabla M_{\\mu g}(v_i)\\;=\\;\\mathrm {sign}(v_i) \\min \\lbrace |v_i|/\\mu , \\,1 \\rbrace .\\nonumber $ Other regularizers with efficiently computable proximal operators include indicator functions of simple convex sets as well as the nuclear and Frobenius norms of matricial variables.",
"Such regularizers can enforce bounds on $x$ , promote low rank solutions, and enhance group sparsity, respectively."
],
[
"Generalization of the gradient and Jacobian",
"Although $\\mathbf {prox}_{\\mu g}$ is typically not differentiable, it is Lipschitz continuous and therefore differentiable almost everywhere [33].",
"One generalization of the gradient for such functions is given by the $B$ -subdifferential set [34], which applies to locally Lipschitz continuous functions $h$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}$ .",
"Let $C_h$ be a set at which $h$ is differentiable.",
"Each element in the set $\\partial _B h (\\bar{z})$ is the limit point of a sequence of gradients $\\lbrace \\nabla h(z_k)\\rbrace $ evaluated at a sequence of points $\\lbrace z_k\\rbrace \\subset C_h$ whose limit is $\\bar{z}$ , $\\partial _B h(\\bar{z})\\; \\mathrel {\\mathop :}=\\,\\left\\lbrace J\\,|~\\exists \\lbrace z_k\\rbrace \\subset C_h,~z_k \\rightarrow \\bar{z},~\\nabla h(z_k) \\rightarrow J\\right\\rbrace .$ If $h$ is continuously differentiable in the neighborhood of a point $z$ , the $B$ -subdifferential set becomes single valued and it is given by the gradient, $\\partial _B h (z) = \\nabla h (z)$ .",
"In general, $\\partial _B h(\\bar{z})$ is not a convex set; e.g., if $h (z) = | z |$ , $\\partial _B h(0) = \\lbrace -1, 1 \\rbrace $ .",
"The Clarke subdifferential set of $h$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}$ at $\\bar{z}$ is the convex hull of the $B$ -subdifferential set [35], $\\partial _C h(\\bar{z})\\;\\mathrel {\\mathop :}=\\;\\mbox{conv}\\,(\\partial _B h(\\bar{z})).$ When $h$ is a convex function, the Clarke subdifferential set is equal to the subdifferential set $\\partial h(\\bar{z})$ which defines the supporting hyperplanes of $h$ at $\\bar{z}$ [36].",
"For a function $G$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}^n$ , the $B$ -generalization of the Jacobian at a point $\\bar{z}$ is given by $\\partial _B G(\\bar{z})\\;\\mathrel {\\mathop :}=\\;\\left[\\begin{array}{ccc}J_1^T & \\dots & J_n^T\\end{array} \\right]^T$ where each $J_i \\in \\partial _B G_i(\\bar{z})$ is a member of the $B$ -subdifferential set of the $i$ th component of $G$ evaluated at $\\bar{z}$ .",
"The Clarke generalization of the Jacobian at a point $\\bar{z}$ , $\\partial _CG(\\bar{z})$ , has the same structure where each $J_i \\in \\partial _C G_i(\\bar{z})$ is a member of the Clarke subdifferential of $G_i(\\bar{z})$ ."
],
[
"Semismoothness",
"The mapping $G$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}^n$ is semismooth at $\\bar{z}$ if for any sequence $z_k \\rightarrow \\bar{z}$ , the sequence of Clarke generalized Jacobians $J_{G_k} \\in \\partial _C G(z_k)$ provides a first order approximation of $G$ , $\\Vert G(z_k) \\,-\\, G(\\bar{z}) \\,+\\, J_{G_k} (\\bar{z} \\,-\\, z_k) \\Vert \\;=\\;o(\\Vert z_k \\,-\\, \\bar{z} \\Vert ).$ where $\\phi (k) = o(\\psi (k))$ denotes that $\\phi (k)/\\psi (k) \\rightarrow 0$ as $k$ tends to infinity [37].",
"The function $G$ is strongly semismooth if this approximation satisfies the stronger condition, $\\Vert G(z_k) \\,-\\, G(\\bar{z}) \\,+\\, J_{G_k} (\\bar{z} \\,-\\, z_k) \\Vert \\;=\\;O(\\Vert z_k \\,-\\, \\bar{z} \\Vert ^2),$ where $\\phi (k) = O(\\psi (k))$ signifies that $|\\phi (k)| \\le L \\psi (k)$ for some positive constant $L$ and positive $\\psi (k)$ [37].",
"Remark 1 (Strong) semismoothness of the proximal operator leads to fast asymptotic convergence of the differential inclusion (see Section ) and the efficient algorithm (see Section ).",
"Proximal operators associated with many typical regularization functions (e.g., the $\\ell _1$ and nuclear norms [38], piecewise quadratic functions [39], and indicator functions of affine convex sets [39]) are strongly semismooth.",
"In general, semismoothness of $\\mathbf {prox}_{\\mu g}$ follows from semismoothness of the projection onto the epigraph of $g$ [39].",
"However, there are convex sets onto which projection is not directionally differentiable [40].",
"The indicator functions associated with such sets or functions whose epigraph is described by such sets may induce proximal operators which are not semismooth."
],
[
"Existing methods",
"Problem (REF ) is encountered in a host of applications and it has been the subject of extensive study.",
"Herein, we provide a brief overview of existing approaches to solving it."
],
[
"First order methods",
"When $T$ is identity or a diagonal matrix, the proximal gradient method, which generalizes gradient descent to certain classes of nonsmooth composite optimization problems [10], [9], can be used to solve (REF ), $x^{k+1}\\;=\\;\\mathbf {prox}_{\\alpha ^k g}\\!\\left(x^k \\,-\\, \\alpha ^k \\nabla f(x^k)\\right)$ where $x^k$ is the current iterate and $\\alpha ^k$ is the step size.",
"When $g= 0$ , we recover gradient descent, when $g$ is the indicator function $I_{\\cal C}(x)$ of the convex set ${\\cal C}$ , it simplifies to projected gradient descent, and when $g$ is the $\\ell _1$ norm, it corresponds to the Iterative Soft-Thresholding Algorithm (ISTA).",
"Nesterov-style techniques can also be employed for acceleration [10].",
"When the matrix $T$ is not diagonal, the alternating direction method of multipliers (ADMM) provides an appealing option for solving (REF ) via (REF ) by alternating between minimization of $\\mathcal {L}_\\mu (x,z;y)$ over $x$ (a continuously differentiable problem), minimization of $\\mathcal {L}_\\mu (x,z;y)$ over $z$ (amounts to evaluating $\\mathbf {prox}_{\\mu g}$ ), and a gradient ascent step in $y$ [11], $\\begin{array}{rcl}x^{k+1}&\\!\\!=\\!\\!&\\displaystyle \\operatornamewithlimits{argmin}_x \\mathcal {L}_\\mu (x,z^k;y^k)\\\\[0.15cm]z^{k+1}&\\!\\!=\\!\\!&\\mathbf {prox}_{\\mu g} (Tx^{k+1} \\,+\\, \\mu y^k)\\\\[0.15cm]y^{k+1}&\\!\\!=\\!\\!&y^k \\,+\\, \\tfrac{1}{\\mu } \\, (Tx^{k+1} \\,-\\, z^{k+1}).\\end{array}$ Although each step in ADMM is conveniently computable, its convergence rate is strongly influenced by the parameter $\\mu $ ."
],
[
"Second order methods",
"The slow convergence of first order methods to high-accuracy solutions motivates the development of second order methods for solving (REF ).",
"A generalization of Newton's method for nonsmooth problems (REF ) with $T = I$ was developed in [17], [16], [19], [18].",
"A sequential quadratic approximation of the smooth part of the objective function is utilized and a search direction $\\tilde{x}$ is obtained as the solution of a regularized quadratic subproblem, $\\operatornamewithlimits{minimize}_{\\tilde{x}}~~\\tfrac{1}{2}\\,{\\tilde{x}}^TH {\\tilde{x}}\\;+\\;\\nabla f(x^k)^T{\\tilde{x}}\\;+\\;g(x^k + \\, {\\tilde{x}})$ where $x^k$ is the current iterate and $H$ is the Hessian of $f$ .",
"This method generalizes the projected Newton method [41] to a broader class of regularizers.",
"For example, when $g$ is the $\\ell _1$ norm, this amounts to solving a LASSO problem [42], which can be a challenging task.",
"Coordinate descent is often used to solve this subproblem [19] and it has been observed to perform well in practice [20], [21], [22].",
"The Forward-Backward Envelope (FBE) was introduced in [26], [27], [28].",
"FBE is once-continuously differentiable nonconvex function of $x$ and its minimum corresponds to the solution of (REF ) with $T = I$ .",
"As demonstrated in Section , FBE can be obtained from the proximal augmented Lagrangian (that we introduce in Section ).",
"Since the generalized Hessian of FBE involves third-order derivatives of $f$ (which may be expensive to compute), references [26], [27], [28] employ either truncated- or quasi-Newton methods to obtain a second order update to $x$ ."
],
[
"The proximal augmented Lagrangian and second order updates",
"In this section, we transform $\\mathcal {L}_\\mu (x,z;y)$ into a form that is once but not twice continuously differentiable.",
"For the resulting function, which we call the proximal augmented Lagrangian, we define second order updates to find its saddle points, show that they are always well defined, and prove that they are locally (quadratically) superlinearly convergent when $\\mathbf {prox}_{\\mu g}$ is (strongly) semismooth."
],
[
"Proximal augmented Lagrangian",
"The continuously differentiable proximal augmented Lagrangian was recently introduced in [12].",
"This was done by rewriting $\\mathcal {L}_\\mu (x,z;y)$ via completion of squares, $\\mathcal {L}_\\mu (x,z;y)\\, = \\,f(x)\\, + \\,g(z)\\, + \\,\\tfrac{1}{2\\mu } \\, \\Vert z \\, - \\, (Tx + \\mu y) \\Vert ^2\\, - \\,\\tfrac{\\mu }{2} \\, \\Vert y \\Vert ^2$ and restricting it to the manifold that corresponds to explicit minimization over the auxiliary variable $z$ , $\\mathcal {L}_\\mu (x; y)\\; \\mathrel {\\mathop :}=\\;\\mathcal {L}_\\mu (x, z_\\mu ^\\star (x,y); y)\\nonumber $ where the minimizer of $\\mathcal {L}_\\mu (x,z;y)$ over $z$ is determined by the proximal operator of the function $g$ , $z_\\mu ^\\star (x,y)\\; = \\;\\operatornamewithlimits{argmin}\\limits _z\\,\\mathcal {L}_\\mu (x, z; y)\\; = \\;\\mathbf {prox}_{\\mu g}(Tx \\, + \\, \\mu y)\\nonumber $ and it defines the aforementioned manifold.",
"Theorem 1 (Theorem 1 in [12]) Let Assumption REF hold.",
"Then, minimization of the augmented Lagrangian $\\mathcal {L}_\\mu (x,z; y)$ associated with problem (REF ) over $(x,z)$ is equivalent to minimization of the proximal augmented Lagrangian $\\mathcal {L}_\\mu (x;y)\\;=\\;f(x)\\; + \\;M_{\\mu g}(Tx \\, + \\, \\mu y)\\; - \\;\\tfrac{\\mu }{2} \\, \\Vert y \\Vert ^2$ over $x$ .",
"Moreover, $\\mathcal {L}_\\mu (x;y)$ is continuously differentiable over $x$ and $y$ and its gradient $\\nabla \\mathcal {L}_\\mu (x; y)$ , $\\begin{array}{rcl}\\nabla \\mathcal {L}_\\mu (x;y)& \\!\\!\\!",
"= \\!\\!\\!",
"&\\left[\\begin{array}{c}\\nabla _x \\mathcal {L}_\\mu (x;y) \\\\ \\nabla _y \\mathcal {L}_\\mu (x;y)\\end{array} \\right]\\; = \\;\\left[\\begin{array}{c}\\nabla f(x) \\, + \\, T^T\\nabla M_{\\mu g}(T x + \\mu y) \\\\ \\mu \\nabla M_{\\mu g}(T x + \\mu y) \\,-\\, \\mu y\\end{array} \\right]\\end{array}$ is Lipschitz continuous.",
"The proximal augmented Lagrangian $\\mathcal {L}_\\mu (x;y)$ contains the Moreau envelope of $g$ and its introduction allows the use of the method of multipliers (MM) to solve problem (REF ).",
"MM requires joint minimization of $\\mathcal {L}_\\mu (x,z;y)$ over $x$ and $z$ which is, in general, challenging because the $(x,z)$ -minimization subproblem is nondifferentiable.",
"However, Theorem REF enables an equivalent implementation of MM $\\begin{array}{rcl}x^{k+1}& \\!\\!",
"= \\!\\!",
"&\\operatornamewithlimits{argmin}\\limits _{x}\\,\\mathcal {L}_\\mu (x; y^k)\\\\[0.1cm] y^{k+1}& \\!\\!",
"= \\!\\!",
"&y^k~+~\\tfrac{1}{\\mu } \\,(Tx^{k+1} \\, - \\, z^\\star _\\mu (x^{k+1},y^k) )\\end{array}$ which improves performance relative to ADMM and has guaranteed convergence to a local minimum even when $f$ is nonconvex [12].",
"Continuous differentiability of $\\mathcal {L}_\\mu (x;y)$ also enables a joint update of $x$ and $y$ via the primal-dual Arrow-Hurwicz-Uzawa gradient flow dynamics, $\\begin{array}{rcl}\\dot{x}&\\!\\!=\\!\\!&- \\,\\nabla _x \\mathcal {L}_\\mu (x;y)\\\\[0.05cm]\\dot{y}&\\!\\!=\\!\\!&\\phantom{-} \\,\\nabla _y \\mathcal {L}_\\mu (x;y)\\end{array}$ where $\\nabla _x \\mathcal {L}_\\mu $ and $\\nabla _y \\mathcal {L}_\\mu $ are given by (REF ).",
"When $\\nabla f$ and $T$ are sparse mappings, this method is convenient for distributed implementation and it is guaranteed to converge at an exponential rate for strongly convex $f$ and sufficiently large $\\mu $ [12].",
"In what follows, we extend the primal-dual algorithm to incorporate second order information of $\\mathcal {L}_\\mu (x;y)$ and thereby achieve fast convergence to high-accuracy solutions."
],
[
"Second order updates",
"Even though Newton's method is primarily used for solving minimization problems in modern optimization, it was originally formulated as a root-finding technique and it has long been employed for finding stationary points [43].",
"In [23], a generalized Jacobian was used to extend Newton's method to semismooth problems.",
"We employ this generalization of Newton's method to $\\nabla \\mathcal {L}_\\mu (x;y)$ in order to compute the saddle point of the proximal augmented Lagrangian.",
"The unique saddle point of $\\mathcal {L}_\\mu (x;y)$ is given by the optimal primal-dual pair ($x^\\star ,y^\\star $ ) and it thus provides the solution to (REF )."
],
[
"Generalized Newton updates",
"Let $H \\mathrel {\\mathop :}=\\nabla ^2 f(x)$ .",
"We use the $B$ -generalized Jacobian of the proximal operator $\\mathbf {prox}_{\\mu g}$ , $\\mathbb {P}_B \\mathrel {\\mathop :}=\\partial _B \\, \\mathbf {prox}_{\\mu g}(Tx + \\mu y)$ , to define the set of $B$ -generalized Hessians of the proximal augmented Lagrangian, $\\partial _B^2\\mathcal {L}_\\mu \\; \\mathrel {\\mathop :}=\\;\\left\\lbrace \\left[\\begin{array}{cc}\\!\\!H+ \\frac{1}{\\mu }\\,T^T(I \\,-\\, P)T\\!",
"& \\!T^T(I \\,-\\, P)\\!\\!",
"\\\\ \\!\\!(I\\,-\\,P)T\\!",
"& \\!-\\mu P\\!\\!\\end{array} \\right],P \\in \\mathbb {P}_B\\right\\rbrace $ and the Clarke generalized Jacobian $\\mathbb {P}_C \\mathrel {\\mathop :}=\\partial _C \\, \\mathbf {prox}_{\\mu g}(Tx + \\mu y)$ to define the set of Clarke generalized Hessians of the proximal augmented Lagrangian, $\\partial _C^2\\mathcal {L}_\\mu \\; \\mathrel {\\mathop :}=\\;\\left\\lbrace \\left[\\begin{array}{cc}\\!\\!H+ \\frac{1}{\\mu }\\,T^T(I \\,-\\, P)T\\!",
"& \\!T^T(I \\,-\\, P)\\!\\!",
"\\\\ \\!\\!(I\\,-\\,P)T\\!",
"& \\!-\\mu P\\!\\!\\end{array} \\right],P \\in \\mathbb {P}_C\\right\\rbrace .$ Note that $\\partial _B^2\\mathcal {L}_\\mu (x;y) \\subset \\partial _C^2\\mathcal {L}_\\mu (x;y)$ because $\\mathbb {P}_B \\subset \\mathbb {P}_C$ .",
"In the rest of the paper, we introduce the composite variable, $w\\mathrel {\\mathop :}=[\\, x^T ~ y^T \\,]^T,$ use $\\mathcal {L}_\\mu (w)$ interchangeably with $\\mathcal {L}_\\mu (x;y)$ , and suppress the dependance of $H$ and $P$ on $w$ to reduce notational clutter.",
"For simplicity of exposition, we assume that $\\mathbf {prox}_{\\mu g}$ is semismooth and state the results for the Clarke generalized Hessian (REF ), i.e., $\\partial ^2\\mathcal {L}_\\mu (w) = \\partial _C^2\\mathcal {L}_\\mu (w)$ .",
"As described in Remark REF in Section , analogous convergence results for non-semismooth $\\mathbf {prox}_{\\mu g}$ can be obtained for the $B$ -generalized Hessian (REF ), i.e., $\\partial ^2\\mathcal {L}_\\mu (w) = \\partial _B^2\\mathcal {L}_\\mu (w)$ .",
"We use the Clarke generalized Hessian (REF ) to obtain a second order update $\\tilde{w}$ by linearizing the stationarity condition $\\nabla \\mathcal {L}_\\mu (w) = 0$ around the current iterate $w^k$ , $\\partial _C^2\\mathcal {L}_\\mu (w^k) \\, \\tilde{w}^k\\; = \\; -\\nabla \\mathcal {L}_\\mu (w^k).$ Since $\\mathbf {prox}_{\\mu g}$ is firmly nonexpansive, $0 \\preceq P \\preceq I$ .",
"In Lemma REF we use this fact to prove that the second order update ${\\tilde{w}}$ is well-defined for any generalized Hessian (REF ) of the proximal augmented Lagrangian $\\mathcal {L}_\\mu (x;y)$ as long as $f$ is strictly convex with $\\nabla ^2 f(x) \\succ 0$ for all $x \\in \\mathbb {R}^n$ .",
"Lemma 2 Let $H \\in \\mathbb {R}^{n \\times n}$ be symmetric positive definite, $H \\succ 0$ , let $P \\in \\mathbb {R}^{m \\times m}$ be symmetric positive semidefinite with eigenvalues less than one, $0 \\preceq P \\preceq I$ , let $T \\in \\mathbb {R}^{m \\times n}$ be full row rank, and let $\\mu > 0$ .",
"Then, the matrix $\\left[\\begin{array}{cc}H + \\tfrac{1}{\\mu } \\, T^T(I - P)T & T^T(I - P) \\\\ (I - P)T & -\\mu P\\end{array} \\right]$ is invertible and it has $n$ positive and $m$ negative eigenvalues.",
"The Haynsworth inertia additivity formula [44] implies that the inertia of matrix (REF ) is determined by the sum of the inertias of matrices, $H \\;+\\; \\tfrac{1}{\\mu } \\, T^T(I \\,-\\, P)T$ and $-\\mu P\\, - \\,(I - P) \\, T\\left(H + \\tfrac{1}{\\mu } \\, T^T(I - P)T\\right)^{-1}\\!T^T (I - P).$ Matrix (REF ) is positive definite because $H \\succ 0$ and both $P$ and $I - P$ are positive semidefinite.",
"Matrix (REF ) is negative definite because the kernels of $P$ and $I - P$ have no nontrivial intersection and $T$ has full row rank."
],
[
"Fast local convergence",
"The use of generalized Newton updates for solving the nonlinear equation $G(x) = 0$ for nondifferentiable $G$ was studied in [23].",
"We apply this framework to the stationarity condition $\\nabla \\mathcal {L}_\\mu (w) = 0$ when $\\mathbf {prox}_{\\mu g}$ is (strongly) semismooth and show that second order updates (REF ) converge (quadratically) superlinearly within a neighborhood of the optimal primal-dual pair.",
"Proposition 3 Let $\\mathbf {prox}_{\\mu g}$ be (strongly) semismooth, and let ${\\tilde{w}}^k$ be defined by (REF ).",
"Then, there is a neighborhood of the optimal solution $w^\\star $ in which the second order iterates $w^{k+1}=w^k+{\\tilde{w}}^k$ converge (quadratically) superlinearly to $w^\\star $ .",
"Lemma REF establishes that $\\partial _C^2\\,\\mathcal {L}_\\mu (w)$ is nonsingular for any $P \\in \\mathbb {P}_C$ .",
"Since the gradient $\\nabla \\mathcal {L}_\\mu (w)$ of the proximal augmented Lagrangian is Lipschitz continuous by Theorem REF , nonsingularity of $\\partial _C^2\\mathcal {L}_\\mu (w)$ and (strong) semismoothness of the proximal operator guarantee (quadratic) superlinear convergence of the iterates by [23]."
],
[
"A globally convergent differential inclusion",
"Since we apply a generalization of Newton's method to a saddle point problem and the second order updates are set valued, convergence to the optimal point is not immediate.",
"Although we showed local convergence rates in Proposition REF by leveraging the results of [23], proof of the global convergence is more subtle and it is established next.",
"To justify the development of a discrete-time algorithm based on the search direction resulting from (REF ), we first examine the corresponding differential inclusion, $\\dot{ w}\\;\\in \\;- (\\partial _C^2\\mathcal {L}_\\mu (w))^{-1} \\, \\nabla \\mathcal {L}_\\mu (w)$ where $\\partial _C^2 \\mathcal {L}_\\mu $ is the Clarke generalized Hessian (REF ) of $\\mathcal {L}_\\mu $ .",
"We assume existence of a solution and prove asymptotic stability of (REF ) under Assumption REF and global exponential stability under an additional assumption that $f$ is strongly convex.",
"Assumption 2 Differential inclusion (REF ) has a solution."
],
[
"Asymptotic stability",
"We first establish asymptotic stability of differential inclusion (REF ).",
"Theorem 4 Let Assumptions REF and REF hold and let $\\mathbf {prox}_{\\mu g}$ be semismooth.",
"Then, differential inclusion (REF ) is asymptotically stable.",
"Moreover, $V(w)\\;\\mathrel {\\mathop :}=\\;\\tfrac{1}{2} \\, \\Vert \\nabla \\mathcal {L}_\\mu (w) \\Vert ^2$ provides a Lyapunov function and $\\dot{V} (t) \\;=\\; - \\, 2 V (t).$ Lyapunov function candidate (REF ) is a positive function of $w$ everywhere apart from the optimal primal-dual pair $w^\\star $ where it is zero.",
"It remains to show that $V$ is decreasing along the solutions $w(t)$ of (REF ), i.e., that $\\dot{V}$ is strictly negative for all $w(t) \\ne w^\\star $ , $\\dot{V} (t)\\;\\mathrel {\\mathop :}=\\;\\tfrac{\\mathrm {d}}{\\mathrm {d}t}V(w(t))\\;=\\;- \\, 2V(w(t))$ For Lyapunov function candidates $\\hat{V}(w)$ which are differentiable with respect to $w$ , $\\dot{\\hat{V}} = \\dot{w}^T \\nabla \\hat{V}$ .",
"Although (REF ) is not differentiable with respect to $w$ , we show that $V(w(t))$ is differentiable along the solutions of (REF ).",
"Instead of employing the chain rule, we use the limit that defines the derivative, $\\dot{V} (t)\\, \\mathrel {\\mathop :}=\\,\\tfrac{\\mathrm {d}}{\\mathrm {d}t}V(w(t))\\, = \\,\\lim _{s \\, \\rightarrow \\, 0}\\dfrac{V(w(t) \\,+\\, s {\\tilde{w}}(t)) \\,-\\, V(w(t))}{s}$ to show that $\\dot{V}$ exists and is negative along the solutions of (REF ).",
"Here, ${\\tilde{w}}\\in - (\\partial _C^2\\mathcal {L}_\\mu (w))^{-1} \\, \\nabla \\mathcal {L}_\\mu (w)$ is determined by the dynamics (REF ).",
"We first introduce $h_s (t)\\; \\mathrel {\\mathop :}=\\;\\dfrac{V(w(t) \\,+\\, s {\\tilde{w}}(t)) \\,-\\, V(w(t))}{s}$ which gives $\\dot{V}$ in the limit $s \\rightarrow 0$ .",
"We then rewrite $h_s (t)$ as the limit point of a sequence of functions $\\lbrace h_{s,k} (t)\\rbrace $ so that $\\dot{V} (t)\\;=\\;\\lim _{s \\, \\rightarrow \\, 0}\\;h_s (t)\\;=\\;\\lim _{s \\, \\rightarrow \\, 0}\\;\\lim _{k \\, \\rightarrow \\, \\infty }\\;h_{s,k} (t)$ and use the Moore-Osgood theorem [45] to exchange the order of the limits and establish that $\\dot{V} = -2 V$ .",
"Let $C_g$ denote a subset of $\\mathbb {R}^{n + m}$ over which $\\mathbf {prox}_{\\mu g}(Tx + \\mu y)$ is differentiable (and therefore $V$ is differentiable with respect to $w$ ) and let $\\lbrace w_k\\rbrace $ be a sequence of points in $C_g$ that converges to $w$ .",
"We define the sequence of functions $\\lbrace h_{s,k} (t) \\rbrace $ , $h_{s,k} (t)\\;\\mathrel {\\mathop :}=\\;\\dfrac{V(w_k(t) \\,+\\, s {\\tilde{w}}_k (t)) \\;-\\; V(w_k(t))}{s}$ where ${\\tilde{w}}_k (t) \\in -( \\partial _C^2 \\mathcal {L}_\\mu (w(t)) )^{-1} \\nabla \\mathcal {L}_\\mu (w_k (t))$ , as we establish below, converges to ${\\tilde{w}}$ .",
"To employ the Moore-Osgood theorem, it remains to show that $h_{s,k} (t)$ converges pointwise (for any $k$ ) as $s \\rightarrow 0$ and that $h_{s,k}(t)$ converges uniformly on some interval $s \\in [0,\\bar{s}]$ as $k \\rightarrow \\infty $ .",
"Since $\\lbrace w_k\\rbrace \\subset C_g$ , $V(w_k)$ is differentiable for every $k \\in \\mathbb {Z}_{+}$ and $\\nabla V(w_k) = \\nabla ^2 \\mathcal {L}_\\mu (w_k) = \\partial _C^2\\mathcal {L}_\\mu (w_k)$ .",
"It thus follows that $\\lim _{s \\, \\rightarrow \\, 0} \\; h_{s,k} (t)\\; = \\;{\\tilde{w}}_k^T (t)\\,\\partial _C^2\\mathcal {L}_\\mu (w_k (t))\\nabla \\mathcal {L}_\\mu (w_k (t))$ pointwise (for any $k$ ).",
"We now show that the sequence $\\lbrace h_{s,k} (t)\\rbrace $ converges uniformly to $h_s (t)$ as $k \\rightarrow \\infty $ implying that we can exchange the order of the limits in (REF ).",
"Since $\\mathbf {prox}_{\\mu g}$ is semismooth, $\\nabla \\mathcal {L}_\\mu $ is semismooth.",
"By (REF ), $\\nabla \\mathcal {L}_\\mu (w_k)$ can be written as, $\\nabla \\mathcal {L}_\\mu (w_k)\\;=\\;\\nabla \\mathcal {L}_\\mu (w)\\;-\\;\\partial _C^2\\mathcal {L}_\\mu (w_k)(w \\,-\\, w_k)\\;+\\;R_k$ for sufficiently large $k$ , where $\\Vert R_k \\Vert = o(\\Vert w_k - w \\Vert )$ .",
"Lemma REF and [23] imply that $\\partial _C^2\\mathcal {L}_\\mu (w_k)$ is bounded within some neighborhood of $w$ and thus that ${\\tilde{w}}_k$ can be written as ${\\tilde{w}}_k={\\tilde{w}}+\\hat{R}_k$ where $\\Vert \\hat{R}_k \\Vert = O(\\Vert w_k - w \\Vert )$ .",
"This implies convergence of $\\tilde{w}_k$ to $\\tilde{w}$ and, combined with local Lipschitz continuity of $V$ with respect to $w_k$ , uniform convergence of $h_{s,k} (t)$ to $h_s (t)$ on $s \\in (0,\\bar{s} \\,]$ where $\\bar{s} >0$ .",
"Therefore, the Moore-Osgood theorem on exchanging limits [45] in conjunction with (REF ) imply $\\begin{array}{rcl}\\dot{V} (t)& \\!\\!\\!",
"= \\!\\!\\!",
"&\\displaystyle \\lim _{s \\, \\rightarrow \\, 0}h_s (t)\\\\[0.25cm]& \\!\\!\\!",
"= \\!\\!\\!",
"&\\displaystyle \\lim _{s \\, \\rightarrow \\, 0}\\,\\displaystyle \\lim _{k \\, \\rightarrow \\, \\infty }h_{s,k} (t)\\, = \\,\\displaystyle \\lim _{k \\, \\rightarrow \\, \\infty }\\,\\displaystyle \\lim _{s\\, \\rightarrow \\, 0}h_{s,k} (t)\\\\[0.25cm]& \\!\\!\\!",
"= \\!\\!\\!",
"& \\displaystyle \\lim _{k \\, \\rightarrow \\, \\infty }{\\tilde{w}}_k^T (t) \\, \\partial _C^2 \\mathcal {L}_\\mu (w_k (t))\\nabla \\mathcal {L}_\\mu (w_k (t))\\\\[0.25cm]& \\!\\!\\!",
"= \\!\\!\\!",
"&-\\Vert \\nabla \\mathcal {L}_\\mu (w (t)) \\Vert ^2\\, = \\,-2 V(t)\\end{array}\\nonumber $ which establishes (REF ) and thereby completes the proof."
],
[
"Global exponential stability",
"To establish global asymptotic stability, we show that the Lyapunov function (REF ) is radially unbounded, and to prove exponential stability we bound it with quadratic functions.",
"We first provide two lemmas that characterize the mappings $\\mathbf {prox}_{\\mu g}$ and $\\nabla f$ in terms of the spectral properties of matrices that describe the corresponding input-output relations at given points.",
"Lemma 5 (Lemma 2 in [12]) Let $g$ be a proper, lower semicontinuous, convex function and let $\\mathbf {prox}_{\\mu g}$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}^m$ be the corresponding proximal operator.",
"Then, for any $a,b \\in \\mathbb {R}^m$ , there exists a symmetric matrix $D_{a,b}$ satisfying $0 \\preceq D_{a,b} \\preceq I$ such that $\\mathbf {prox}_{\\mu g}(a) \\,-\\, \\mathbf {prox}_{\\mu g}(b) \\; = \\; D_{a,b} \\, (a \\, - \\, b).\\nonumber $ Lemma 6 Let $f$ be strongly convex with parameter $m_f$ and let its gradient $\\nabla f$ be Lipschitz continuous with parameter $L_f$ .",
"Then, for any $a,b \\in \\mathbb {R}^n$ there exists a symmetric matrix $G_{a,b}$ satisfying $m_f I \\preceq G_{a,b} \\preceq L_f I $ such that $\\nabla {f} (a)\\, - \\,\\nabla {f} (b)\\; = \\;G_{a,b} \\, (a \\, - \\, b).\\nonumber $ Let $c \\mathrel {\\mathop :}=a - b$ , $d \\mathrel {\\mathop :}=\\nabla {f} (a) - \\nabla {f} (b)$ , $e \\mathrel {\\mathop :}=d - m_f c$ , and rcl Ga,b := {eeT/(eTc), e 0; 0, otherwise} Ga,b := Ga,b + mfI .",
"Clearly, by construction, $\\hat{G}_{a,b} = \\hat{G}_{a,b}^T \\succeq 0$ .",
"It is also readily verified that $G_{a,b} \\, c = d$ when $e^Tc \\ne 0$ .",
"It thus remains to show that (i) $G_{a,b} \\, c = d$ when $e^Tc = 0$ ; and (ii) $\\hat{G}_{a,b} \\preceq (L_f - m_f)I$ .",
"(i) Since $f$ is $m_f$ strongly convex and $\\nabla f$ is $L_f$ Lipschitz continuous, $h(x) \\mathrel {\\mathop :}=f(x) - \\tfrac{m_f}{2} \\Vert x \\Vert ^2$ is convex and $\\nabla h(x) = \\nabla f(x) - m_f x$ is $L_f - m_f$ Lipschitz continuous.",
"Furthermore, we have $e = \\nabla h(a) - \\nabla h(b)$ , and [46] implies $e^Tc\\; \\ge \\;\\tfrac{1}{L_f \\, - \\, m_f} \\, \\Vert e \\Vert ^2,~~\\mbox{for~all}~c \\, \\in \\, \\mathbb {R}^n.$ This shows that $e^Tc = 0$ only if $e \\mathrel {\\mathop :}=d - m_f c = 0$ and, thus, $d = m_f c = G_{a,b} \\, c$ when $e^Tc = 0$ .",
"Therefore, there always exist a symmetric matrix $G_{a,b}$ such that $G_{a,b} \\, c = d$ .",
"(ii) When $e^Tc \\ne 0$ , $\\hat{G}_{a,b}$ is a rank one matrix and its only nonzero eigenvalue is $\\Vert e \\Vert ^2/(e^Tc)$ ; this follows from $\\hat{G}_{a,b} \\, e = (\\Vert e \\Vert ^2/(e^Tc) ) \\, e$ .",
"In this case, inequality (REF ) implies $e^Tc > 0$ and (REF ) is equivalent to $1/(e^Tc) \\le (L_f - m_f)/\\Vert e \\Vert ^2$ .",
"Thus, $\\Vert e \\Vert ^2/(e^Tc) \\le L_f - m_f$ and $\\hat{G}_{a,b} \\preceq (L_f - m_f) I$ when $e^Tc \\ne 0$ .",
"Since $\\hat{G}_{a,b} = 0$ when $e^Tc = 0$ , $\\hat{G}_{a,b} \\preceq (L_f - m_f) I$ for all $a$ and $b$ .",
"Finally, $\\hat{G}_{a,b} \\succeq 0$ and (REF ) imply $m_fI \\preceq G_{a,b} \\preceq L_fI$ .",
"Remark 2 Although matrices $D_{a,b}$ and $G_{a,b}$ in Lemmas REF and REF depend on the operating point, their spectral properties, $0 \\preceq D_{a,b} \\preceq I$ and $m_f I \\preceq G_{a,b} \\preceq L_f I$ , hold for all $a$ and $b$ .",
"These lemmas can be interpreted as a combination between a generalization of the mean value theorem [45] to vector-valued functions and spectral bounds on the operators $\\mathbf {prox}_{\\mu g}$ : $\\mathbb {R}^m \\rightarrow \\mathbb {R}^m$ and $\\nabla f$ : $\\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ arising from firm nonexpansiveness of $\\mathbf {prox}_{\\mu g}$ , strong convexity of $f$ , and Lipschitz continuity of $\\nabla f$ .",
"We now combine Lemmas REF and REF to establish quadratic upper and lower bounds for Lyapunov function (REF ) and thereby prove global exponential stability of differential inclusion (REF ) for strongly convex $f$ .",
"Theorem 7 Let Assumptions REF and REF hold, let $\\mathbf {prox}_{\\mu g}$ be semismooth, and let $f$ be $m_f$ strongly convex.",
"Then, differential inclusion (REF ) is globally exponentially stable, i.e., there exists $\\kappa > 0$ such that $\\Vert w(t) - w^\\star \\Vert \\le \\kappa \\, \\mathrm {e}^{-t} \\, \\Vert w(0) - w^\\star \\Vert $ .",
"Given the assumptions, Theorem REF establishes asymptotic stability of (REF ) with the dissipation rate $\\dot{V}(w) \\;=\\; - \\, 2 V(w).$ It remains to show the existence of positive constants $\\kappa _1$ and $\\kappa _2$ such that Lyapunov function (REF ) satisfies c 12 w 2 V(w) 22 w 2 where ${\\tilde{w}}\\mathrel {\\mathop :}=w - w^\\star $ and $w^\\star \\mathrel {\\mathop :}=(x^\\star ,y^\\star )$ is the optimal primal-dual pair.",
"The upper bound in (REF ) follows from Lipschitz continuity of $\\nabla \\mathcal {L}_\\mu (w)$ (see Theorem REF ), with $\\kappa _2$ determined by the Lipschitz constant of $\\nabla \\mathcal {L}_\\mu (w)$ .",
"To show the lower bound in (REF ), and thus establish radial unboundedness of $V(w)$ , we construct matrices that relate $V(w)$ to ${\\tilde{w}}$ .",
"Lemmas REF and REF imply the existence of symmetric matrices $D_{\\tilde{w}}$ and $G_{\\tilde{w}}$ such that $0 \\preceq D_{\\tilde{w}}\\preceq I$ , $m_f I \\preceq G_{\\tilde{w}}\\preceq L_f I$ , and $\\begin{array}{rcl}\\mathbf {prox}_{\\mu g}(Tx + \\mu y) - \\mathbf {prox}_{\\mu g}(Tx^\\star + \\mu y^\\star )& \\!\\!\\!",
"= \\!\\!\\!",
"&D_{\\tilde{w}}\\,(T {\\tilde{x}}+ \\mu {\\tilde{y}})\\\\f(x) - f(x^\\star )& \\!\\!\\!",
"= \\!\\!\\!",
"&G_{\\tilde{w}}\\, {\\tilde{x}}.\\end{array}\\nonumber $ As noted in Remark REF , although $D_{\\tilde{w}}$ and $G_{\\tilde{w}}$ depend on the operating point, their spectral properties hold for all ${\\tilde{w}}$ .",
"Since $\\nabla \\mathcal {L}_\\mu (w^\\star ) = 0$ , we can write $\\nabla \\mathcal {L}_\\mu (w)\\; = \\;\\nabla \\mathcal {L}_\\mu (w)\\,-\\,\\nabla \\mathcal {L}_\\mu (w^\\star )\\; = \\;Q_{\\tilde{w}}\\, {\\tilde{w}}\\nonumber $ and express Lyapunov function (REF ) as $V(w)\\; = \\;\\tfrac{1}{2} \\,{\\tilde{w}}^T Q_{\\tilde{w}}^T \\, Q_{\\tilde{w}}\\, {\\tilde{w}}\\nonumber $ where $Q_{\\tilde{w}}\\; \\mathrel {\\mathop :}=\\;\\left[\\begin{array}{cc}G_{\\tilde{w}}+ \\tfrac{1}{\\mu } \\, T^T(I - D_{\\tilde{w}})T & T^T(I - D_{\\tilde{w}}) \\\\ (I - D_{\\tilde{w}})T & -\\mu D_{\\tilde{w}}\\end{array} \\right]\\nonumber $ for some $(D_{\\tilde{w}},G_{\\tilde{w}}) \\in \\Omega _{\\tilde{w}}$ , $\\Omega _{\\tilde{w}}\\, \\mathrel {\\mathop :}=\\,\\left\\lbrace (D_{\\tilde{w}},G_{\\tilde{w}})~|~0 \\, \\preceq \\, D_{\\tilde{w}}\\, \\preceq \\, I,~m_f I \\, \\preceq \\, G_{\\tilde{w}}\\, \\preceq \\, L_f I\\right\\rbrace .$ The set $\\Omega _{\\tilde{w}}$ is closed and bounded and the minimum eigenvalue of $Q_{\\tilde{w}}^T \\, Q_{\\tilde{w}}$ is a continuous function of $G_{\\tilde{w}}$ and $D_{\\tilde{w}}$ .",
"Thus, the extreme value theorem [45] implies that its infimum over $\\Omega _{\\tilde{w}}$ , $\\kappa _1\\;=\\;\\displaystyle \\inf \\limits _{(D_{\\tilde{w}},G_{\\tilde{w}}) \\,\\in \\, \\Omega _{\\tilde{w}}}\\lambda _{\\min }\\left(Q_{\\tilde{w}}^T \\, Q_{\\tilde{w}}\\right)$ is achieved.",
"By Lemma REF , $Q_{\\tilde{w}}$ is a full rank matrix, which implies that $Q_{\\tilde{w}}^T \\, Q_{\\tilde{w}}\\succ 0$ for all ${\\tilde{w}}$ and therefore that $\\kappa _1$ is positive.",
"Thus, $V(w) \\ge \\tfrac{\\kappa _1}{2} \\, \\Vert {\\tilde{w}} \\Vert ^2$ , establishing condition (REF ).",
"Condition (REF ) and [47] imply $V(w(t)) = \\mathrm {e}^{-2t} V(w(0))$ .",
"It then follows from (REF ) that $\\Vert w(t) - w^\\star \\Vert ^2\\;\\le \\;(\\kappa _2/\\kappa _1) \\, \\mathrm {e}^{-2t} \\, \\Vert w(0) - w^\\star \\Vert ^2.$ Taking the square root completes the proof and provides an upper bound for the constant $\\kappa $ , $\\kappa \\le \\sqrt{\\kappa _2/\\kappa _1}$ .",
"Remark 3 The rate of exponential convergence established by Theorem REF is independent of $m_f$ , $L_f$ , and $\\mu $ .",
"This is a consequence of insensitivity of Newton-like methods to poor conditioning.",
"In contrast, the first order primal-dual method considered in [12] requires a sufficiently large $\\mu $ for exponential convergence.",
"In our second order primal-dual method, problem conditioning and parameter selection affect the multiplicative constant $\\kappa $ but not the rate of convergence.",
"Remark 4 When differential inclusion (REF ) is defined with the $B$ -generalized Hessian (REF ), Theorems REF and REF hold even for proximal operators which are not semismooth.",
"This follows from defining ${\\tilde{w}}_k = {\\tilde{w}}\\in - (\\partial ^2_B\\mathcal {L}_\\mu (w))^{-1}\\nabla \\mathcal {L}_\\mu (w)$ and choosing $\\lbrace w_k\\rbrace \\subset C_g$ such that $\\partial ^2_B\\mathcal {L}_\\mu (w) = \\lim _{k \\rightarrow \\infty } \\nabla ^2\\mathcal {L}_\\mu (w_k)$ in the proof of Theorem REF .",
"Such a choice of $\\lbrace w_k\\rbrace $ is possible by the definition of the $B$ -subdifferential (REF ); since the Clarke subgradient is the convex hull of the $B$ -subdifferential, it contains points outside of $\\partial _B^2\\mathcal {L}_\\mu (w)$ and thus such a sequence $\\lbrace w_k\\rbrace \\subset C_g$ is not guaranteed to exist for any $\\partial _C^2\\mathcal {L}_\\mu (w)$ .",
"When defined in this manner, ${\\tilde{w}}_k$ is constant with respect to $w_k$ and thus uniform convergence of $h_k(t,s)$ to $h(t,s)$ is immediate."
],
[
"A second order primal-dual algorithm",
"An algorithm based on the second order updates (REF ) requires step size selection to ensure global convergence.",
"This is challenging for saddle point problems because standard notions, such as sufficient descent, cannot be applied to assess the progress of the iterates.",
"Instead, it is necessary to identify a merit function whose minimum lies at the stationary point and whose sufficient descent can be used to evaluate progress towards the saddle point.",
"An approach based on discretization of differential inclusion (REF ) and Lyapunov function (REF ) as a merit function leads to Algorithm REF in Appendix REF .",
"However, such a merit function is nonconvex and nondifferentiable in general which makes the utility of backtracking (e.g., the Armijo rule) unclear.",
"Moreover, Algorithm REF employs a fixed penalty parameter $\\mu $ .",
"A priori selection of this parameter is difficult and it has a large effect on the convergence speed.",
"Instead, we employ the primal-dual augmented Lagrangian introduced in [13] as a merit function and incorporate an adaptive $\\mu $ update.",
"This merit function is convex in both $x$ and $y$ and it facilitates an implementation with outstanding practical performance.",
"Drawing upon recent advancements for constrained optimization of twice differentiable functions [14], [15], we show that our algorithm converges to the solution of (REF ).",
"Finally, our algorithm exhibits local (quadratic) superlinear convergence for (strongly) semismooth $\\mathbf {prox}_{\\mu g}$ ."
],
[
"Merit function",
"The primal-dual augmented Lagrangian, $\\nonumber {\\cal V}_\\mu (x,z;y,\\lambda )\\, \\mathrel {\\mathop :}=\\,\\mathcal {L}_\\mu (x,z;\\lambda )\\, + \\,\\tfrac{1}{2\\mu } \\, \\Vert Tx - z + \\mu \\, (\\lambda - y) \\Vert ^2$ was introduced in [13], where $\\lambda $ is an estimate of the optimal Lagrange multiplier $y^\\star $ .",
"Following [13], it can be shown that the optimal primal-dual pair $(x^\\star ,z^\\star ; y^\\star )$ of optimization problem (REF ) is a stationary point of ${\\cal V}_\\mu (x,z;y,y^\\star )$ .",
"Furthermore, for any fixed $\\lambda $ , ${\\cal V}_\\mu $ is a convex function of $x$ , $z$ , and $y$ and it has a unique global minimizer.",
"In contrast to [13], we study problems in which a component of the objective function is not differentiable.",
"As in Theorem REF , the Moreau envelope associated with the nondifferentiable component $g$ allows us to eliminate the dependence of the primal-dual augmented Lagrangian ${\\cal V}_\\mu $ on $z$ , $\\begin{array}{rcl}\\hat{z}_\\mu ^\\star (x;y,\\lambda )& \\!\\!\\!",
"= \\!\\!\\!",
"&\\operatornamewithlimits{argmin}\\limits _z\\,{\\cal V}_\\mu (x, z; y, \\lambda )\\\\[0.1cm]& \\!\\!\\!",
"= \\!\\!\\!",
"&\\mathbf {prox}_{\\frac{\\mu }{2}g}(Tx \\,+\\, \\tfrac{\\mu }{2}(2\\lambda \\,-\\, y))\\end{array}$ and to express ${\\cal V}_\\mu $ as a continuously differentiable function, $\\begin{array}{l}{\\cal V}_\\mu (x;y,\\lambda )\\; \\mathrel {\\mathop :}=\\;{\\cal V}_\\mu (x,\\hat{z}_\\mu ^\\star (x;y,\\lambda );y,\\lambda )~ = ~f(x)\\, + \\,M_{\\frac{\\mu }{2} g}\\!\\left(Tx \\,+\\, \\tfrac{\\mu }{2}(2\\lambda \\,-\\, y)\\right)\\, + \\,\\tfrac{\\mu }{4} \\, \\Vert y \\Vert ^2\\, - \\,\\tfrac{\\mu }{2} \\, \\Vert \\lambda \\Vert ^2.\\end{array}\\nonumber $ For notational compactness, we suppress the dependence on $\\lambda $ and write ${\\cal V}_\\mu (w)$ when $\\lambda $ is fixed.",
"Remark 5 The primal-dual augmented Lagrangian is not a Lyapunov function unless $\\lambda = y^\\star $ .",
"We establish convergence by minimizing ${\\cal V}_\\mu (x;y,\\lambda )$ over $(x;y)$ – a convex problem – while adaptively updating the Lagrange multiplier estimate $\\lambda $ .",
"In [13], [15], the authors obtain a search direction using the Hessian of the merit function, $\\nabla ^2 {\\cal V}_\\mu .$ Instead of implementing an analogous update using generalized Hessian $\\partial ^2{\\cal V}_\\mu $ , we take advantage of the efficient inversion of $\\partial ^2 \\mathcal {L}_\\mu $ (see Section REF ) to define the update $\\partial ^2\\mathcal {L}_\\mu (w^k)\\,{\\tilde{w}}\\;=\\;-\\,\\operatornamewithlimits{blkdiag}(I,-I)\\,\\nabla {\\cal V}_{2\\mu }(w^k)$ where the identity matrices are sized conformably with the dimensions of $x$ and $y$ , and $\\nabla {\\cal V}_{2\\mu }(w)=\\left[\\begin{array}{c}\\nabla f(x)+T^T\\nabla M_{\\mu g}\\!\\left(Tx + \\mu (2\\lambda - y)\\right)\\\\\\mu (y -\\nabla M_{\\mu g}\\!\\left(Tx + \\mu (2\\lambda - y)\\right))\\end{array}\\right].$ Multiplication by $\\operatornamewithlimits{blkdiag}(I,-I)$ is used to ensure descent in the dual direction and ${\\cal V}_{2\\mu }$ is employed because $\\hat{z}^\\star _\\mu (x;y,\\lambda )$ is determined by the proximal operator associated with $(\\mu /2) g$ .",
"When $\\lambda = y$ , $\\nabla _x {\\cal V}_{2\\mu } = \\nabla _x \\mathcal {L}_{\\mu }$ , $\\nabla _y {\\cal V}_{2\\mu } = - \\nabla _y \\mathcal {L}_{\\mu }$ , and (REF ) becomes equivalent to the second order update (REF ).",
"Lemma 8 Let ${\\tilde{w}}$ solve (REF ).",
"Then, for the fixed value of the Lagrange multiplier estimate $\\lambda $ and any $\\sigma \\in (0,1]$ , $d\\; \\mathrel {\\mathop :}=\\;(1 \\, - \\, \\sigma ) \\, {\\tilde{w}}\\;-\\;\\sigma \\, \\nabla {\\cal V}_{2\\mu }(w,\\lambda )$ is a descent direction of the merit function ${\\cal V}_{2\\mu }(w,\\lambda )$ .",
"By multiplying (REF ) with the nonsingular matrix $\\Pi \\;\\mathrel {\\mathop :}=\\;\\left[\\begin{array}{cc}I & -\\,\\tfrac{1}{\\mu }\\,T^T \\\\ 0 & I\\end{array} \\right]$ we can express it as $\\left[\\begin{array}{cc}\\!\\!H\\!\\!",
"& \\!\\!T^T\\!\\!",
"\\\\ \\!\\!",
"(I \\,-\\, P)T\\!\\!",
"& \\!\\!-\\mu P\\!\\!\\end{array} \\right]\\left[\\begin{array}{c}{\\tilde{x}} \\\\ {\\tilde{y}}\\end{array} \\right]=\\left[\\begin{array}{c}- (\\nabla f(x) + T^Ty) \\\\ \\nabla _y {\\cal V}_{2 \\mu } (x; y, \\lambda )\\end{array} \\right]$ where $H \\mathrel {\\mathop :}=\\nabla ^2 f (x) \\succ 0$ .",
"Using (REF ) and (REF ), $\\nabla {\\cal V}_{2\\mu }(w)$ can be expressed as, $\\nabla {\\cal V}_{2\\mu }(w)=\\left[\\begin{array}{c}-(H+ \\tfrac{1}{\\mu }T^T(I-P)T ) {\\tilde{x}}\\, - \\, T^T(I - P) {\\tilde{y}} \\\\ (I\\,-\\,P)T{\\tilde{x}}\\;-\\; \\mu P {\\tilde{y}}\\end{array} \\right].\\nonumber $ Thus, ${{\\tilde{w}}}^T {\\nabla {\\cal V}_{2\\mu }(w)}\\, = \\,- {\\tilde{x}}^T (H+ \\tfrac{1}{\\mu }T^T(I\\,-\\,P)T) {\\tilde{x}}\\, - \\, \\mu {\\tilde{y}}^TP{\\tilde{y}}$ is negative semidefinite, and the inner product $\\begin{array}{c}{d}^T {\\nabla {\\cal V}_{2\\mu }(w)}\\, = \\,(1-\\sigma )\\,{{\\tilde{w}}}^T {\\nabla {\\cal V}_{2\\mu }(w)}\\,-\\,\\sigma \\Vert \\nabla {\\cal V}_{2\\mu } (w) \\Vert ^2\\end{array}$ is negative definite when $\\nabla {\\cal V}_{2\\mu }$ is nonzero."
],
[
"Second order primal-dual algorithm",
"We now develop a customized algorithm that alternates between minimizing the merit function ${\\cal V}_\\mu (x;y,\\lambda )$ over $(x;y)$ and updating $\\lambda $ .",
"Near the optimal solution, the algorithm approaches second order updates (REF ) with unit step size, leading to local (quadratic) superlinear convergence for (strongly) semismooth $\\mathbf {prox}_{\\mu g}$ .",
"Our approach builds on the sequential quadratic programming method described in [13], [14], [15] and it uses the primal-dual augmented Lagrangian as a merit function to assess progress of iterates to the optimal solution.",
"Inspired by [48], we ensure sufficient progress with damped second order updates.",
"The following two quantities $\\begin{array}{rcl}r& \\!\\!\\!",
"\\mathrel {\\mathop :}=\\!\\!\\!",
"&Tx \\, - \\, \\mathbf {prox}_{\\mu g}\\!\\left(Tx \\, + \\, \\mu y \\right)\\\\ s& \\!\\!\\!",
"\\mathrel {\\mathop :}=\\!\\!\\!",
"&Tx \\, - \\, \\mathbf {prox}_{\\mu g}\\!\\left(Tx \\, + \\, \\mu \\, (2\\lambda \\, - \\, y) \\right)\\end{array}$ appear in the proof of global convergence.",
"Note that $r$ is the primal residual of optimization problem (REF ) and that $\\nabla {\\cal V}_{2\\mu }$ can be equivalently expressed as $\\nabla {\\cal V}_{2\\mu }(w)\\;=\\;\\left[\\begin{array}{c}\\nabla f(x) \\, + \\, \\tfrac{1}{\\mu } \\, T^T (s + \\mu (2\\lambda - y))\\\\-(s \\, + \\, 2\\mu (\\lambda - y))\\end{array}\\right].$"
],
[
"Global convergence",
"We now establish global convergence of Algorithm REF under an assumption that the sequence of gradients generated by the algorithm, $\\nabla f(x^k)$ , is bounded.",
"This assumption is standard for augmented Lagrangian based methods [15], [49] and it does not lead to a loss of generality when $f$ is strongly convex.",
"Theorem 9 Let Assumption REF hold and let the sequence $\\lbrace \\nabla f(x^k)\\rbrace $ resulting from Algorithm REF be bounded.",
"Then, the sequence of iterates $\\left\\lbrace w^k\\right\\rbrace $ converges to the optimal primal-dual point of problem (REF ) and the Lagrange multiplier estimates $\\lbrace \\lambda ^k \\rbrace $ converge to the optimal Lagrange multiplier.",
"Since ${\\cal V}_{2\\mu }(w,\\lambda )$ is convex in $w$ for any fixed $\\lambda $ , condition (REF ) in Algorithm REF will be satisfied after finite number of iterations.",
"Combining (REF ) and (REF ) shows that $s^k + 2\\mu ^k(\\lambda ^k - y^k) \\rightarrow 0$ and $\\nabla f(x^k) + \\tfrac{1}{\\mu ^k} \\, T^T (s^k + \\mu (2\\lambda ^k - y^k)) \\rightarrow 0$ .",
"Together, these statements imply that the dual residual $\\nabla f(x^k) + T^Ty^k$ of (REF ) converges to zero.",
"To show that the primal residual $r^k$ converges to zero, we first show that $s^k \\rightarrow 0$ .",
"If Step $2a$ in Algorithm REF is executed infinitely often, $s^k \\rightarrow 0$ since it satisfies (REF ) at every iteration and $\\eta \\in (0,1)$ .",
"If Step $2a$ is executed finitely often, there is $k_0$ after which $\\lambda ^k = \\lambda ^{k_0}$ .",
"By adding and subtracting $2\\mu ^k \\nabla f(x^k) + T^Ts^k + 4\\mu ^kT^T(\\lambda ^{k_0} - y^k)$ and rearranging terms, we can write $T^Ts^k\\, = \\,2\\mu ^k(\\nabla f(x^k) \\,+\\, \\tfrac{1}{\\mu ^k}T^T(s^k + \\mu ^k(2\\lambda ^{k_0} - y^k)))\\,-\\,2\\mu ^k \\nabla f(x^k)\\,-\\,T^T(s^k + 2\\mu ^k(\\lambda ^{k_0} - y^k))\\,-\\,2\\mu ^k T^T\\lambda ^{k_0}.$ Taking the norm of each side and applying the triangle inequality, (REF ) and (REF ) yields $\\Vert T^Ts^k \\Vert \\, \\le \\,{2\\mu ^k}\\epsilon ^k+2\\mu ^k\\Vert \\nabla f(x^k) \\Vert +\\Vert T^T \\Vert \\epsilon _k+2\\mu ^k\\Vert T^T\\lambda ^{k_0} \\Vert .$ This inequality implies that $T^Ts^k \\rightarrow 0$ because $\\nabla f(x^k)$ is bounded, $\\epsilon ^k \\rightarrow 0$ , and $\\mu ^k \\rightarrow 0$ .",
"Since $T$ has full row rank, $T^T$ has full column rank and it follows that $s^k \\rightarrow 0$ .",
"Substituting $s^k \\rightarrow 0$ and $\\nabla f(x^k) + T^Ty^k \\rightarrow 0$ into the first row of (REF ) and applying (REF ) implies $\\lambda ^k \\rightarrow y^k$ .",
"Thus, $s^k \\rightarrow r^k$ , implying that the iterates asymptotically drive the primal residual $r^k$ to zero, thereby completing the proof.",
"Remark 6 Despite the assumption that $\\lbrace \\nabla f(x^k)\\rbrace $ is bounded, Theorem REF can be used to ensure global convergence whenever $f$ is strongly convex.",
"We show in Lemma REF in Appendix REF that a bounded set ${\\cal C}_f$ containing the optimal point can be identified a priori.",
"One can thus artificially bound $\\nabla f(x)$ for all $x \\notin {\\cal C}_f$ to satisfy the conditions of Theorem REF and guarantee global convergence to the solution of (REF ).",
"Second order primal-dual algorithm for nonsmooth composite optimization.",
"input: Initial point $w^0 = (x^0,y^0)$ , and parameters ${\\eta } \\in (0,1)$ , $\\beta \\in (0,1)$ , $\\tau _a,\\tau _b \\in (0,1)$ , $\\epsilon ^k \\ge 0$ such that $\\epsilon ^k \\rightarrow 0$ .",
"initialize: Set $\\lambda ^0 = y^0$ .",
"Step 1: If $\\Vert s^k \\Vert \\;\\le \\;{\\eta }\\Vert s^{k-1} \\Vert $ go to Step $2a$ .",
"If not, go to Step $2b$ .",
"Step $2a$ : Set $\\begin{array}{rclrcl}\\mu ^{k+1}&\\!\\!\\!",
"= \\!\\!\\!&\\tau _a \\mu ^k,&\\hspace{14.22636pt}\\lambda ^{k+1}&\\!\\!\\!",
"= \\!\\!\\!&y^k\\end{array}$ Step $2b$ : Set $\\begin{array}{rclrcl}\\mu ^{k+1}&\\!\\!\\!",
"= \\!\\!\\!&\\tau _b \\mu ^k,&\\hspace{14.22636pt}\\lambda ^{k+1}&\\!\\!\\!",
"= \\!\\!\\!&\\lambda ^k\\end{array}$ Step 3: Using a backtracking line search, perform a sequence of inner iterations to choose $w^{k+1}$ until $\\Vert \\nabla {\\cal V}_{2\\mu ^{k+1}}(w^{k+1},\\lambda ^{k+1}) \\Vert \\;\\le \\;\\epsilon ^k$ where the search direction $d$ is obtained using (REF )–(REF ) with lr = 0, (wk)T V2k+1(wk) V2k+1(wk) 2 -, (0,1], otherwise."
],
[
"Asymptotic convergence rate",
"The invertibility of the generalized Hessian $\\partial ^2 \\mathcal {L}_\\mu (w)$ allows us to establish local convergence rates for the second order updates (REF ) when $\\mathbf {prox}_{\\mu g}$ is (strongly) semismooth.",
"We now show that the updates in Algorithm REF are equivalent to the second order updates (REF ) as $k \\rightarrow \\infty $ .",
"Thus, if $\\mathbf {prox}_{\\mu g}$ is (strongly) semismooth, the sequence of iterates generated by Algorithm REF converges (quadratically) superlinearly to the optimal point in some neighborhood of it.",
"Theorem 10 Let the conditions of Theorem REF hold, let $\\mathbf {prox}_{\\mu g}$ be (strongly) semismooth, and let $\\epsilon ^k$ be such that $\\Vert w^k - w^\\star \\Vert = O(\\epsilon ^k)$ .",
"Then, in a neighborhood of the optimal point $w^\\star $ , the iterates $w^k$ converge (quadratically) superlinearly to $w^\\star $ .",
"From Theorem REF , $\\lambda ^k \\rightarrow y^k$ and thus $\\nabla {\\cal V}_{2\\mu }(w^k) \\rightarrow \\nabla \\mathcal {L}_\\mu (w^k)$ .",
"Descent of the Lyapunov function in Theorem REF therefore implies that the update in Step 3 of Algorithm REF is given by (REF ), which is equivalent to (REF ) because $\\lambda = y$ .",
"The assumption on $\\lbrace \\epsilon ^k\\rbrace $ in conjunction with Proposition REF , and [50] imply that this update asymptotically satisfies (REF ) in one iteration with a unit step size.",
"Therefore, Step 3 reduces to (REF ) in some neighborhood of the optimal solution and Proposition REF implies that $w^k$ converges to $w^\\star $ (quadratically) superlinearly."
],
[
"Efficient computation of the Newton direction",
"When $g$ is (block) separable, the matrix $P$ in (REF ) is (block) diagonal.",
"We next demonstrate that the solution to (REF ) can be efficiently computed when $T = I$ and $P$ is a sparse diagonal matrix whose entries are either 0 or 1.",
"The extensions to a low rank $P$ , to a $P$ with entries between 0 and 1, or to a general diagonal $T$ follow from similar arguments.",
"These conditions occur, for example, when $g(z) = \\gamma \\Vert z \\Vert _1$ .",
"The matrix $P$ is sparse when $\\mathbf {prox}_{\\mu g}(x + \\mu y) = {\\cal S}_{\\gamma \\mu }(x + \\mu y)$ is sparse.",
"Larger values of $\\gamma $ are more likely to produce a sequence of iterates $w^k$ for which $P$ is sparse and thus the second order search directions (REF ) are cheaper to compute.",
"We can write (REF ) as $\\left[\\begin{array}{cc}H & I \\\\ I-P & -\\mu P\\end{array} \\right]\\left[\\begin{array}{c}{\\tilde{x}} \\\\ {\\tilde{y}}\\end{array} \\right]\\;=\\;\\left[\\begin{array}{c}\\vartheta \\\\ \\theta \\end{array} \\right],$ permute it according to the entries of $P$ which are 1 and 0, respectively, and partition the matrices $H$ , $P$ , and $I - P$ conformably such that $\\begin{array}{l}H\\;=\\;\\left[\\begin{array}{cc}H_{11} & H_{12} \\\\ H_{12}^T & H_{22}\\end{array} \\right],~~~P\\;=\\;\\left[\\begin{array}{cc}I & \\\\ & 0\\end{array} \\right].\\end{array}$ Let $v$ denote either the primal variable $x$ or the dual variable $y$ .",
"We use $v_1$ to denote the subvector of $v$ corresponding to the entries of $P$ which are equal to 1 and $v_2$ to denote the subvector corresponding to the zero diagonal entries of $P$ .",
"Note that $(I - P)v = 0$ when $v_2 = 0$ and $Pv = 0$ when $v_1 = 0$ .",
"As a result, ${\\tilde{x}}_2$ and ${\\tilde{y}}_1$ are explicitly determined by the bottom row of the system of equations (REF ), $\\left[\\begin{array}{cc}0 & -\\mu I \\\\ I & 0\\end{array} \\right]\\left[\\begin{array}{c}{\\tilde{x}}_2 \\\\ {\\tilde{y}}_1\\end{array} \\right]\\;=\\;\\left[\\begin{array}{c}\\theta _1 \\\\ \\theta _2\\end{array} \\right]$ Substitution of the subvectors ${\\tilde{x}}_2$ and ${\\tilde{y}}_1$ into (REF ) yields, $H_{11}{\\tilde{x}}_1\\;=\\;\\vartheta _1 \\,+\\, H_{12}{\\tilde{x}}_2 \\,+\\, {\\tilde{y}}_1$ which must be solved for ${\\tilde{x}}_1$ .",
"Finally, the computation of ${\\tilde{y}}_2$ requires only matrix-vector products, ${\\tilde{y}}_2\\;=\\;- \\left( \\vartheta _2 \\,+\\, H_{21}{\\tilde{x}}_1 \\,+\\, H_{22}{\\tilde{x}}_2\\right).$ Thus, the major computational burden in solving (REF ) lies in performing a Cholesky factorization to solve (REF ), where $H_{11}$ is a matrix of a much smaller size than $H$ ."
],
[
"Computational experiments",
"In this section, we illustrate the merits and the effectiveness of our approach.",
"We first apply our algorithm to the $\\ell _1$ -regularized least squares problem and then study a system theoretic problem of controlling a spatially-invariant system."
],
[
"$\\ell _1$ -regularized least squares",
"The LASSO problem (REF ) regularizes a least squares objective with a $\\gamma $ -weighted $\\ell _1$ penalty, $\\operatornamewithlimits{minimize}\\limits _x~\\,\\tfrac{1}{2} \\, \\Vert Ax \\,-\\, b \\Vert ^2\\;+\\;\\gamma \\,\\Vert x \\Vert _1.$ As described in Section REF , the associated proximal operator is given by soft-thresholding ${\\cal S}_{\\gamma \\mu }$ , the Moreau envelope is the Huber function, and its gradient is the saturation function.",
"Thus, $P \\in \\mathbb {P}$ is diagonal and $P_{ii}$ is 0 when $| x_i + \\mu y_i | < \\gamma \\mu $ , 1 outside this interval, and between 0 and 1 on the boundary.",
"Larger values of the regularization parameter $\\gamma $ induce sparser solutions for which one can expect a sparser sequence of iterates.",
"Note that we require strong convexity of the least squares penalty; i.e., that $A^TA$ is positive definite.",
"In Fig.",
"REF , we show the distance of the iterates from the optimal for the standard proximal gradient algorithm ISTA, its accelerated version FISTA, and our customized second order primal-dual algorithm for a problem where $A^TA$ has condition number $3.26 \\times 10^4$ .",
"We plot distance from the optimal point as a function of both iteration number and solve time.",
"Although our method always requires much fewer iterations, it is most effective when $\\gamma $ is large.",
"In this case the most computationally demanding step (REF ) required to determine the second order search direction (REF ) involves a smaller matrix inversion; see Section REF for details.",
"In Fig.",
"REF , we show the solve times for $n=1000$ as the sparsity-promoting parameter $\\gamma $ ranges from 0 to $\\gamma _{\\max } = \\Vert A^Tb \\Vert $ , where $\\gamma _{\\max }$ yields a zero solution.",
"All numerical experiments consist of 20 averaged trials.",
"In Fig.",
"REF , we compare the performance of our algorithm with the LASSO function in Matlab (a coordinate descent method [51]), SpaRSA [52], an interior point method [53], and YALL1 [54].",
"Problem instances were randomly generated with $A \\in \\mathbb {R}^{m \\times n}$ , $n$ ranging from 100 to 2000, $m = 3n$ , and $\\gamma = 0.15\\gamma _{\\max }$ or $0.85\\gamma _{\\max }$ .",
"The solve times and scaling of our algorithm is competitive with these state-of-the-art methods.",
"For larger values of $\\gamma $ , the second order search direction (REF ) is cheaper to compute and our algorithm is the fastest."
],
[
"Distributed control of a spatially-invariant system",
"We now apply our algorithm to a structured control design problem aimed at balancing closed-loop $\\mathcal {H}_2$ performance with spatial support of a state-feedback controller.",
"Following the problem formulation of [5], ADMM was used in [32], [55] to design sparse feedback gains for spatially-invariant systems.",
"Herein, we demonstrate that our algorithm provides significant computational advantage over both the ADMM algorithm and a proximal Newton scheme."
],
[
"Spatially-invariant systems",
"Let us consider $\\begin{array}{rcl}\\dot{\\psi }&\\!\\!=\\!\\!&A\\,\\psi \\;+\\;u\\;+\\;d\\\\[0.1cm]\\zeta &\\!\\!=\\!\\!&\\left[\\begin{array}{c}Q^{1/2}\\,\\psi \\\\ R^{1/2}\\,u\\end{array} \\right]\\end{array}$ where $\\psi $ , $u$ , $d$ , and $\\zeta $ are the system state, control input, white stochastic disturbance, and performance output and $A$ , $Q \\succeq 0$ , and $R \\succ 0$ are $n \\times n$ circulant matrices.",
"Such systems evolve over a discrete spatially-periodic domain; they can be used to model spatially-invariant vehicular platoons [56] and can result from a spatial discretization of fluid flows [57].",
"Any circulant matrix can be diagonalized via the discrete Fourier transform (DFT).",
"Thus, the coordinate transformation $\\psi \\mathrel {\\mathop :}=T \\hat{\\psi }$ , $u \\mathrel {\\mathop :}=T \\hat{u}$ , $d \\mathrel {\\mathop :}=T \\hat{d}$ , where $T^{-1}$ is the DFT matrix, brings the state equation in (REF ) into, $\\dot{\\hat{\\psi }}\\; = \\;\\hat{A} \\, \\hat{\\psi }\\;+\\;\\hat{u}\\;+\\;\\hat{d}.$ Here, $\\hat{A} \\mathrel {\\mathop :}=T^{-1} A T$ is a diagonal matrix whose main diagonal $\\hat{a}$ is determined by the DFT of the first row $a$ of the matrix $A$ , $\\hat{a}_k\\; \\mathrel {\\mathop :}=\\;\\sum _{i \\, = \\, 0}^{n-1}\\,a_i\\,\\mathrm {e}^{- \\mathrm {j}\\tfrac{2 \\pi i k}{n}},~~k\\, \\in \\,\\lbrace 0, \\ldots , n - 1 \\rbrace .\\nonumber $ We are interested in designing a structured state-feedback controller, $u = - Z \\psi $ , that minimizes the closed-loop $\\mathcal {H}_2$ norm, i.e., the variance amplification from the disturbance $d$ to the regulated output $\\zeta $ .",
"Since the optimal unstructured $Z$ for spatially-invariant system (REF ) is a circulant matrix [58], we restrict our attention to circulant feedback gains $Z$ .",
"Thus, $Z$ can also be diagonalized via a DFT and we equivalently take $x \\mathrel {\\mathop :}=\\hat{z}$ as our optimization variable where $T^{-1} Z T = \\mathrm {diag}\\, (\\hat{z})$ .",
"For simplicity, we assume that $A$ and $Z$ are symmetric.",
"In this case, $\\hat{a}$ and $\\hat{z}$ are real vectors and the closed-loop $\\mathcal {H}_2$ norm of system (REF ) takes separable form $f (x) = \\sum _{k \\, = \\, 0}^{n - 1} f_k (x_k)$ , $f_k (x_k)\\;=\\;\\left\\lbrace \\begin{array}{rl}\\dfrac{\\hat{q}_k \\, + \\, \\hat{r}_k x_k^2}{2 \\, (x_k \\, - \\, \\hat{a}_k)},&x_k \\,>\\, \\hat{a}_k\\\\[0.25cm]\\infty ,&\\mbox{otherwise}\\end{array}\\right.$ where $x_k > \\hat{a}_k$ guarantees closed-loop stability.",
"To promote sparsity of $Z$ , we consider a regularized optimization problem (REF ), $\\begin{array}{rl}\\operatornamewithlimits{minimize}\\limits _{x, \\, z}&f (x)\\;+\\;\\gamma \\, \\Vert z \\Vert _1\\\\[0.125cm]\\operatornamewithlimits{subject~to}&T x \\, - \\, z\\;=\\;0\\end{array}$ where $\\gamma $ is a positive regularization parameter, $T$ is the inverse DFT matrix, and $z \\in \\mathbb {R}^n$ denotes the first row of the symmetric circulant matrix $Z$ .",
"Formulation (REF ) signifies that while it is convenient to quantify the $\\mathcal {H}_2$ norm in the spatial frequency domain, sparsity has to be promoted in the physical space.",
"By solving (REF ) over a range of $\\gamma $ , we identify distributed controller structures which are specified by the sparsity pattern of the solutions $z^\\star _\\gamma $ to (REF ) at different values of $\\gamma $ .",
"After selecting a controller structure associated with a particular value of $\\gamma $ , we solve a `polishing' or `debiasing' problem, $\\begin{array}{rl}\\displaystyle \\operatornamewithlimits{minimize}_{x,z}&f (x)\\;+\\;I_{\\text{sp}(z^\\star _\\gamma )}(z)\\\\[0.125cm]\\operatornamewithlimits{subject~to}&T x \\, - \\, z\\;=\\;0\\end{array}$ where $I_{\\text{sp}(z^\\star _\\gamma )}(z)$ is the indicator function associated with the sparsity pattern of $z^\\star _\\gamma $ .",
"The solution to this problem is the optimal controller for system (REF ) with the desired structure, i.e., the same sparsity pattern as $z^\\star _\\gamma $ .",
"This step is necessary because the $\\ell _1$ norm in (REF ) imposes an additional penalty on $z$ that compromises closed-loop performance."
],
[
"Implementation",
"The elements of the gradient of $f$ are $\\dfrac{\\mathrm {d}f_k (x_k)}{ \\mathrm {d}x_k}\\; = \\;\\dfrac{\\hat{r}_k x_k^2\\, - \\,2\\hat{a}_k\\hat{r}_kx_k\\, - \\,\\hat{q}_k}{2 \\, (x_k \\, - \\, \\hat{a}_k)^2}$ the Hessian is a diagonal matrix with non-zero entries, $\\dfrac{\\mathrm {d}^2 f_k (x_k)}{ \\mathrm {d}x_k^2}\\; = \\;\\dfrac{\\hat{q}_k \\, + \\, \\hat{r}_k \\hat{a}_k^2}{ (x_k \\, - \\, \\hat{a}_k)^3}$ and the proximal operator associated with the nonsmooth regularizer in (REF ) is given by soft-thresholding ${\\cal S}_{\\gamma \\mu }$ .",
"While the optimal unstructured controller can be obtained by solving $n$ uncoupled scalar quadratic equations for $x_k$ , sparsity-promoting problem (REF ) is not in a separable form (because of the linear constraint) and computing the second order update (REF ) requires solving a system of equations $\\left[\\begin{array}{cc}H & T^* \\\\ (I - P)T & -\\mu P\\end{array} \\right]\\left[\\begin{array}{c}\\tilde{x} \\\\ \\tilde{y}\\end{array} \\right]\\;=\\;\\left[\\begin{array}{c}\\vartheta \\\\ \\theta \\end{array} \\right]$ Pre-multiplying by the nonsingular matrix $\\operatornamewithlimits{blkdiag}(T,I)$ and changing variables to solve for $\\tilde{z} \\mathrel {\\mathop :}=T\\tilde{x}$ brings (REF ) into $\\left[\\begin{array}{cc}T \\, H \\, T^{-1} & (1/n) \\, I \\\\ I - P & -\\mu P\\end{array} \\right]\\left[\\begin{array}{c}\\tilde{z} \\\\ \\tilde{y}\\end{array} \\right]\\;=\\;\\left[\\begin{array}{c}T\\vartheta \\\\ \\theta \\end{array} \\right]$ which is of the same form as (REF ).",
"This equation can be solved efficiently when $P$ is sparse (i.e., $\\gamma \\mu $ is large; cf.",
"Section REF ) and $\\tilde{x}$ can be recovered from $\\tilde{z}$ via FFT.",
"Since the Hessian $H$ of a separable function $f$ is a diagonal matrix, the search direction can also be efficiently computed when $I - P$ is sparse (i.e., $\\gamma \\mu $ is small).",
"As in Section REF , the component of ${\\tilde{y}}$ in the support of $P$ is determined from the bottom row of (REF ).",
"The top row of (REF ) implies $\\tilde{x} = H^{-1}(\\vartheta - T^* {\\tilde{y}})$ and substitution into the bottom row yields $(I - P) \\, T \\, H^{-1} \\, T^* \\, (I - P) \\, \\tilde{y}\\;=\\;\\tilde{\\theta } $ where $\\tilde{\\theta } \\mathrel {\\mathop :}=(I - P) (TH^{-1}(\\vartheta - T^* P \\tilde{y}) - \\theta )$ is a known vector.",
"Thus, the component of $\\tilde{y}$ in the support of $I - P$ can be determined by inverting a matrix whose size is determined by the support of $I - P$ and $\\tilde{ x}$ is readily obtained from $\\tilde{y}$ and $\\vartheta $ .",
"The operations involving $T$ and $T^*$ can be performed via FFT; since $H$ is diagonal, multiplication by these matrices is cheap and the computational burden in solving (REF ) again arises from a limited matrix inversion.",
"In contrast to Section REF , the computation of the search direction using this approach is efficient when $I - P$ is sparse, i.e., $\\gamma \\mu $ is small."
],
[
"Swift-Hohenberg equation",
"We consider the linearized Swift-Hohenberg equation [59], $\\partial _t \\psi (t, \\xi )\\, = \\,(c I\\,-\\,(I\\,+\\,\\partial _{\\xi \\xi })^2)\\,\\psi (t,\\xi )\\,+\\,u(t,\\xi )\\,+\\,d(t,\\xi )$ with periodic boundary conditions on a spatial domain $\\xi \\in [-\\pi ,\\pi ]$ .",
"Finite-dimensional approximation and diagonalization via the DFT (with an even number of Fourier modes $n$ ) yields (REF ) with $\\hat{a}_k=c - (1 - k^2)^2$ where $k = \\lbrace -n/2 +1, \\ldots , n/2 \\rbrace $ is a spatial wavenumber.",
"Figure REF shows the optimal centralized controller and solutions to (REF ) for $c = -0.01$ , $n = 64$ , $Q = R = I$ , and $\\gamma = 4 \\times 10^{-4}$ , $ 4 \\times 10^{-3}$ , and 4.",
"As further illustrated in Fig.",
"REF , the optimal solutions to (REF ) become sparser as $\\gamma $ is increased.",
"In Fig.",
"REF , we demonstrate the utility of using regularized problems to navigate the tradeoff between controller performance and structure, an approach pioneered by [5].",
"The polished optimal structured controllers (—$\\circ $ —) were designed by first solving (REF ) to identify an optimal structure and then solving (REF ) to further improve the closed-loop performance.",
"To illustrate the importance of polishing step (REF ), we also show the closed-loop performance of unpolished optimal structured controllers (- -$\\times $ - -) resulting from (REF ).",
"Finally, to evaluate the controller structures identified by (REF ), we show the closed-loop performance of polished `reference' structured controllers ($\\cdots + \\cdots $).",
"Instead of solving (REF ), the reference structures are a priori specified as nearest neighbor symmetric controllers of the same cardinality as controllers resulting from (REF ).",
"Among controllers with the same number of nonzero entries, the polished optimal structured controller consistently achieves the best closed-loop performance.",
"We compare the computational efficiency of our approach with the proximal Newton method [19] and ADMM [55], [32].",
"The proximal Newton method requires solving a LASSO subproblem (REF ), for which we employ SpaRSA [52].",
"Since $A$ is circulant, the $x$ -minimization step in ADMM (REF ) requires solving $n$ uncoupled cubic scalar equations.",
"In general, when $A$ is block circulant, the DFT only block diagonalizes the dynamics and thus the $x$ -minimization step has to be solved via an iterative procedure [55], [32].",
"Figure REF shows the time to solve (REF ) with $\\gamma = 0.004$ using our method, proximal Newton, and ADMM.",
"Our algorithm and ADMM were stopped when the primal and dual residuals were below $1 \\times 10^{-8}$ .",
"The proximal Newton method was stopped when the norm of the difference between two consecutive iterates was smaller than $1 \\times 10^{-8}$ .",
"In Fig.",
"REF , we show the per iteration cost and the total number of iterations required to find the optimal solution using each method.",
"Our algorithm clearly outperforms proximal Newton and ADMM.",
"Although proximal Newton requires a similar number of iterations, the LASSO subproblem (REF ) that determines its search direction is much more expensive; this increases the computation cost of each iteration and slows the overall algorithm.",
"Moreover, for larger problem sizes, the proximal Newton method struggles with finding a stabilizing search direction because $\\nabla ^2 f$ seems to bring it away from the set of stabilizing feedback gains.",
"It appears that our method circumvents this issue because its iterates lie in a larger lifted space in which stability is easier to enforce via backtracking.",
"On the other hand, while the $x$ - and $z$ - minimization steps in ADMM are quite efficient, as a first order method ADMM requires a large number of iterations to reach high-accuracy solutions.",
"Our algorithm achieves better performance because its use of second order information leads to relatively few iterations and the structured matrix inversion leads to efficient computation of the search direction.",
"Figure: (a) The middle row of the circulant feedback gain matrix ZZ; and (b) the sparsity level of z γ ☆ z^\\star _\\gamma (relative to the sparsity level of the optimal centralized controller z 0 ☆ z^\\star _0) resulting from the solutions to () for the linearized Swift-Hohenberg equation with n=64n = 64 Fourier modes and c=-0.01c = -0.01.Figure: Performance degradation (in percents) of structured controllers relative to the optimal centralized controller: polished optimal structured controller obtained by solving () and () (—∘\\circ —); unpolished optimal structured controller obtained by solving only () (- -×\\times - -); and optimal structured controller obtained by solving () for an a priori specified nearest neighbor reference structure (⋯+⋯\\cdots +\\cdots ).Figure: Total time to compute the solution to () with γ=0.004\\gamma = 0.004 using our algorithm (2ndMM), proximal Newton, and ADMM.Figure: Comparison of (a) times to compute an iteration (averaged over all iterations); and (b) numbers of iterations required to solve () with γ=0.004\\gamma = 0.004."
],
[
"Connections and discussion",
"The proximal augmented Lagrangian $\\mathcal {L}_\\mu (x;y)$ is obtained by constraining $\\mathcal {L}_\\mu (x,z;y)$ to the manifold $\\begin{array}{rrl}{\\cal Z}&\\!\\!\\!",
"\\mathrel {\\mathop :}=\\!\\!\\!&\\lbrace (x,z^{\\star }_\\mu ;y)~|~z^\\star _\\mu \\;=\\;\\displaystyle \\operatornamewithlimits{argmin}_{z}\\;\\mathcal {L}_\\mu (x,z;y)\\rbrace \\\\[0.15cm]&\\!\\!\\!",
"= \\!\\!\\!&\\lbrace (x,z^{\\star }_\\mu ;y)~|~Tx \\, + \\, \\mu y\\, \\in \\,z^\\star _\\mu \\, + \\,\\mu \\,\\partial _C g (z^\\star _\\mu )\\rbrace \\end{array}$ which results from the explicit minimization over the auxiliary variable $z$ .",
"Herein, we interpret the second order search direction as a linearized update to the KKT conditions for problem (REF ) and discuss connections to the alternative algorithms."
],
[
"Second order updates as linearized KKT corrections",
"The second order update (REF ) can be viewed as a first order correction to the KKT conditions for optimization problem (REF ), $\\begin{array}{rcl}0&\\!\\!=\\!\\!&\\nabla f(x)\\,+\\,T^Ty\\\\[0.05cm]0&\\!\\!=\\!\\!&Tx\\,-\\,z\\\\[0.05cm]0&\\!\\!",
"\\in \\!\\!&\\partial _C g(z)\\,-\\,y.\\end{array}$ Substitution of $z^\\star _\\mu $ into (REF ) makes the last two conditions redundant; when combined with the definition of the manifold $\\cal Z$ , $Tx = z$ implies $y \\in \\partial _C g(z)$ and $y \\in \\partial _C g(z)$ implies $Tx = z$ .",
"Multiplying equation (REF ) for the second order update with the nonsingular matrix (REF ) recovers the equivalent expression $\\left[\\begin{array}{cc}\\!\\!H\\!\\!",
"& \\!\\!T^T\\!\\!",
"\\\\ \\!\\!",
"(I \\,-\\, P)T\\!\\!",
"& \\!\\!-\\mu P\\!\\!\\end{array} \\right]\\left[\\begin{array}{c}{\\tilde{x}}^k \\\\ {\\tilde{y}}^k\\end{array} \\right]=-\\left[\\begin{array}{c}\\nabla f(x^k) + T^Ty^k \\\\ r^k\\end{array} \\right]$ where $r^k \\mathrel {\\mathop :}=Tx^k - z^\\star _\\mu (x^k;y^k) = Tx^k - \\mathbf {prox}_{\\mu g} (Tx^k + \\mu y^k)$ is the primal residual of (REF ).",
"Thus, (REF ) describes a first order correction to the first and second KKT conditions in (REF )."
],
[
"Connections with other methods",
"We now discuss broader implications of our framework and draw connections to the existing methods for solving versions of (REF ).",
"Many techniques for solving composite minimization problems of the form (REF ) can be expressed in terms of functions embedded in the augmented Lagrangian; see Table REF .",
"Trivially, the objective function in (REF ) corresponds to $\\mathcal {L}_\\mu (x,z;y)$ over the manifold $z = Tx$ .",
"The Lagrange dual of a problem equivalent to (REF ), $\\begin{array}{rl}\\operatornamewithlimits{minimize}\\limits _{x, \\, z}&f(x) \\;+\\; g(z) \\;+\\; \\tfrac{1}{2\\mu } \\, \\Vert Tx \\,-\\, z \\Vert ^2\\\\\\operatornamewithlimits{subject~to}&Tx \\,-\\, z\\;=\\;0\\end{array}$ is recovered by collapsing $\\mathcal {L}_\\mu (x,z;y)$ onto the intersection of the $z$ -minimization manifold $\\cal Z$ and the $x$ -minimization manifold, $\\begin{array}{rrl}{\\cal X}&\\!\\!\\!",
"\\mathrel {\\mathop :}=\\!\\!\\!&\\lbrace (x^\\star ,z;y)~|~x^\\star \\, = \\,\\displaystyle \\operatornamewithlimits{argmin}_{x}\\;\\mathcal {L}_\\mu (x,z;y)\\rbrace \\\\[-0.cm]&\\!\\!\\!",
"= \\!\\!\\!&\\lbrace (x^\\star ,z;y)~|~\\mu \\,(\\nabla f (x^\\star )\\, + \\,T^T y)\\, = \\,T^T(z \\,-\\, Tx^\\star )\\rbrace .\\end{array}$ The Lagrange dual of (REF ) is recovered from the Lagrange dual of (REF ) in the limit $\\mu \\rightarrow \\infty $ ."
],
[
"MM and ADMM",
"MM implements gradient ascent on the dual of (REF ) by collapsing $\\mathcal {L}_\\mu (x,z;y)$ computationally onto $\\cal X \\cap \\cal Z$ .",
"The joint $(x,z)$ -minimization step in (REF ) evaluates the Lagrange dual at discrete iterates $y^k$ by finding the corresponding $(x,z)$ -pair on $\\cal X \\cap \\cal Z$ ; i.e., the iterate $(x^{k+1},z^{k+1};y^k)$ generated by (REF ) lies on the manifold $\\cal X \\cap \\cal Z$ .",
"Note that, in this form, joint $(x,z)$ -minimization is a challenging nondifferentiable optimization problem.",
"ADMM avoids this challenge by collapsing $\\mathcal {L}_\\mu (x,z;y)$ onto $\\cal X$ and $\\cal Z$ separately.",
"While the underlying $x$ - and $z$ -minimization subproblems in ADMM are relatively simple, the iterate $(x^{k+1},z^{k+1}; y^k)$ generated by (REF ) does not typically lie on the $\\cal X \\cap \\cal Z$ manifold.",
"Thus ADMM does not represent gradient ascent on the dual of (REF ), causing looser theoretical guarantees and often worse practical performance.",
"Table: Summary of different functions embedded in the augmented Lagrangian of () and methods for solving () based on these functions.By collapsing the augmented Lagrangian onto $\\cal Z$ , the proximal augmented Lagrangian (REF ) allows us to express the $(x,z)$ -minimization step in MM as a tractable continuously differentiable problem (cf.",
"Theorem REF ).",
"This avoids challenges associated with ADMM and it does not increase computational complexity in the $x$ -minimization step in (REF ) relative to ADMM when using first order methods.",
"We finally note that unlike the Rockafellar's proximal method of multipliers [61] which applies the proximal point algorithm to the primal-dual optimality conditions, our framework reformulates the standard method of multipliers and develops second order algorithm to solve nonsmooth composite optimization problems."
],
[
"Second order methods",
"We identify the saddle point of $\\mathcal {L}_\\mu (x,z;y)$ by forming second order updates to $x$ and $y$ along the $\\cal Z$ manifold.",
"Just as Newton's method approximates an objective function with a convex quadratic function, we approximate $\\mathcal {L}_{\\mu }(x;y)$ with a quadratic saddle function.",
"Constraining the dual variable itself yields connections with other methods.",
"When $T = I$ , (REF ) implies that the optimal dual variable is given by $y^\\star = -\\nabla f(x^\\star )$ , so it is natural to collapse the augmented Lagrangian onto the manifold ${\\cal Y}\\; \\mathrel {\\mathop :}=\\;\\lbrace (x,z;y^\\star )~|~y^\\star \\, = \\,-\\,\\nabla f(x)\\rbrace .$ The augmented Lagrangian over the manifold $\\cal Z \\cap \\cal Y$ corresponds to the Forward-Backward Envelope (FBE) introduced in [26].",
"The proximal gradient algorithm with step size $\\mu $ can be recovered from a variable metric gradient iteration on the FBE [26].",
"In [26], [27], [28], the approximate line search and quasi-Newton methods based on the FBE were developed to solve (REF ) with $T = I$ .",
"Since the Hessian of the FBE involves third order derivatives of $f$ , these techniques employ either truncated- or quasi-Newton methods.",
"The approach advanced in the current paper applies a second order method to the augmented Lagrangian that is constrained over the larger manifold $\\cal Z$ .",
"Relative to alternatives, our framework offers several advantages.",
"First, while the FBE is in general nonconvex function of $x$ , $\\mathcal {L}_\\mu (x;y)$ is always convex in $x$ and concave in $y$ .",
"Furthermore, we can compute the Hessian exactly using only second order derivatives of $f$ and its structure allows for efficient computation of the search direction.",
"Finally, our formulation allows us to leverage recent advances in second order methods for augmented Lagrangian methods, e.g., [13], [14], [15]."
],
[
"Concluding remarks",
"We have developed a second order primal-dual method for nonsmooth convex composite optimization.",
"We establish global exponential convergence of the corresponding differential inclusion in continuous-time and use the primal-dual augmented Lagrangian as a merit function in a discrete-time implementation.",
"Our globally convergent customized algorithm exhibits asymptotic (quadratic) superlinear convergence rate when the proximal operator associated with nonsmooth regularizer is (strongly) semismooth.",
"We use the $\\ell _1$ -regularized least squares and spatially-invariant control problems to demonstrate competitive performance of our algorithm relative to the available state-of-the-art alternatives.",
"Future research will focus on nonconvex problems, the development of inexact second order methods, and application of our framework to problems in which the codomain of the regularizer has a larger dimensions than the domain of the optimization variable.",
"In particular, inexact but structured approximations of the Hessian can lead to efficient methods that may even be convenient for distributed implementation."
],
[
"Algorithm based on $V(w)$ as a merit function",
"Since Theorem REF establishes global convergence of the differential inclusion, one algorithmic approach is to implement a Forward Euler discretization of differential inclusion (REF ).",
"A natural choice of merit function is the Lyapunov function $V$ defined in (REF ).",
"A simple corollary of Theorem REF shows that the second order update (REF ) is a descent direction for $V$ .",
"Corollary 11 The second order update (REF ) is a descent direction for the merit function $V$ defined in (REF ).",
"Follows from (REF ) in Theorem REF .",
"Corollary REF enables the use of a backtracking Armijo rule for step size selection.",
"A natural choice of stopping criterion for such an algorithm is a condition on the size of the primal and dual residuals.",
"Moreover, proposition REF suggests fast asymptotic convergence when $\\mathbf {prox}_{\\mu g}$ is semismooth.",
"The LASSO example in Fig.",
"REF verifies this intuition when solving LASSO to a threshold of $1 \\times 10^{-8}$ for the primal and dual residuals.",
"Second order primal-dual algorithm for nonsmooth composite optimization based on discretizing (REF ).",
"input: Initial point $x_0$ , $y_0$ , backtracking constant $\\alpha \\in (0,1)$ , Armijo parameter $\\sigma \\in (0,1)$ , and stopping tolerances $\\varepsilon _1$ , $\\varepsilon _2$ .",
"While: $\\Vert Tx^k - \\mathbf {prox}_{\\mu g}(Tx^k + \\mu y^k) \\Vert > \\varepsilon _1$ or While: $\\Vert \\nabla f (x^k) \\,-\\, T^Ty^k \\Vert > \\varepsilon _2$ Step 1: Compute ${\\tilde{w}}^k$ as defined in (REF ) Step 2: Choose the smallest $j \\in \\mathbb {Z}_{+}$ such that $V(w^k + \\alpha ^j {\\tilde{w}}^k)\\;\\le \\;V(w^k)\\,-\\,\\sigma \\,\\alpha ^j\\,\\Vert \\nabla \\mathcal {L}_\\mu (w^k) \\Vert ^2$ Step 3: Update the primal and dual variables $w^{k+1}\\, = \\,w^k\\, + \\,\\alpha ^j {\\tilde{w}}^k$ However, such an implementation would require a fixed penalty parameter $\\mu $ , which typically has a large effect on the convergence speed of augmented Lagrangian algorithms and is difficult to select a priori.",
"Moreover, stability of the solution to a differential equation does not always imply stability of its discretization."
],
[
"An example",
"We implement Algorithm REF to solve the LASSO problem (REF ) studied in Section .",
"The LASSO problem was randomly generated with $n = 500$ , $m = 1000$ , and $\\gamma = 0.85\\gamma _{\\max }$ .",
"Figure REF illustrates the quadratic asymptotic convergence of Algorithm REF and a strong influence of $\\mu $ .",
"Figure: Distance from the optimal solution as a function of iteration number when solving LASSO using Algorithm for different values of μ\\mu ."
],
[
"Bounding $\\nabla f(x)$ for Algorithm ",
"A bounded set ${\\cal C}_f$ containing the solution to (REF ) can always be identified from any point $\\bar{x}$ , $\\nabla f(\\bar{x})$ , any element of the subgradient $\\partial g(\\bar{x})$ , and a lower bound on the parameter of strong convexity.",
"Lemma 12 Let Assumption REF hold and let $f$ be $m_f$ -strongly convex.",
"Then, for any $\\bar{x}$ , the optimal solution to (REF ) lies within the ball of radius $\\tfrac{2}{m_f}\\Vert \\nabla f(\\bar{x}) \\,+\\, T^T\\partial g(T\\bar{x}) \\Vert $ centered at $\\bar{x}$ .",
"Given any point $\\bar{x}$ , strong convexity of $f$ and convexity of $g$ imply that, $f(x) \\, + \\, g(Tx)\\,-\\,f(\\bar{x}) \\, - \\, g(T\\bar{x})\\; \\ge \\;\\left\\langle \\nabla f(\\bar{x}) \\,+\\, T^T\\partial g(T\\bar{x}),x \\,-\\, \\bar{x} \\right\\rangle \\, + \\,\\tfrac{m_f}{2}\\Vert x \\,-\\, \\bar{x} \\Vert ^2$ for all $x$ where $\\partial g(T\\bar{x})$ is any member of the subgradient of $g(T\\bar{x})$ .",
"For any $x$ with $\\Vert x - \\bar{x} \\Vert \\ge \\tfrac{2}{m_f}\\Vert \\nabla f(\\bar{x}) + T^T\\partial g(T\\bar{x}) \\Vert ,$ the right-hand side of (REF ) must be nonnegative which implies that $f(x) + g(Tx) \\ge f(\\bar{x}) + g(T\\bar{x})$ and thus that $x$ cannot solve (REF ).",
"For any strongly convex function $f$ and convex function $g$ , a function $\\hat{f}$ can be selected such that $\\operatornamewithlimits{argmin}\\, f(x) \\,+\\, g(Tx)\\;=\\;\\operatornamewithlimits{argmin}\\, \\hat{f}(x) \\,+\\, g(Tx).$ Here, $\\hat{f}$ is identical to $f$ in some closed set ${\\cal C}_f$ containing $x^\\star $ , $\\hat{f}(x)\\;\\mathrel {\\mathop :}=\\;\\left\\lbrace \\begin{array}{rl}f(x),&x \\,\\in \\, {\\cal C}_f\\\\[0.1cm]\\tilde{f}(x),&x \\,\\notin \\, {\\cal C}_f\\end{array}\\right.$ and $\\tilde{f}(x)$ is chosen such that $\\hat{f}(x)$ is convex and twice differentiable, $\\nabla \\hat{f}(x)$ is uniformly bounded, and $\\nabla ^2 \\hat{f}(x) \\succ 0$ .",
"Lemma REF implies that a set ${\\cal C}_f$ that contains the optimal solution $x^\\star $ of (REF ) can be identified a priori from any given point $\\bar{x}$ .",
"Since $\\hat{f}(x)$ satisfies all the conditions of Theorem REF , Algorithm REF can be used to solve (REF ) and, since ${\\cal C}_f$ contains $x^\\star $ , its optimal solution will also solve (REF )."
],
[
"An example",
"When $x \\in \\mathbb {R}$ , $f(x) = \\tfrac{1}{2}x^2$ , and ${\\cal C}_f = [-1, 1]$ , a potential choice of $\\tilde{f}(x)$ is given by, $\\tilde{f}(x)\\;=\\;\\left\\lbrace \\begin{array}{rl}-2x \\,+\\, \\mathrm {e}^{x + 1} \\,-\\, 2.5,&x \\, \\le \\, -1\\\\2x \\,+\\, \\mathrm {e}^{-x + 1} \\,-\\, 2.5,&x \\, \\ge \\, \\phantom{-}1.\\end{array}\\right.$ For this choice of $\\tilde{f}$ , the gradient of $\\hat{f}$ is continuous and bounded, $\\nabla \\hat{f}(x)\\;=\\;\\left\\lbrace \\begin{array}{rl}-2 + \\mathrm {e}^{x + 1},&x \\, \\le \\, -1\\\\[0.1cm]x,&x \\, \\in \\, [-1,1]\\\\[0.1cm]2 - \\mathrm {e}^{-x + 1},&x \\, \\ge \\, 1\\end{array}\\right.$ and the Hessian of $\\hat{f}$ is determined by $\\nabla ^2 \\hat{f}(x)\\;=\\;\\left\\lbrace \\begin{array}{rl}\\mathrm {e}^{x + 1},&x \\, \\le \\,-1\\\\1,&x \\, \\in \\, [-1,1]\\\\[0.1cm]\\mathrm {e}^{-x + 1},&x \\, \\ge \\, 1.\\end{array}\\right.$"
]
] | 1709.01610 |
[
[
"Distributed Optimal Frequency Control Considering a Nonlinear\n Network-Preserving Model"
],
[
"Abstract This paper addresses the distributed optimal frequency control of power systems considering a network-preserving model with nonlinear power flows and excitation voltage dynamics.",
"Salient features of the proposed distributed control strategy are fourfold: i) nonlinearity is considered to cope with large disturbances; ii) only a part of generators are controllable; iii) no load measurement is required; iv) communication connectivity is required only for the controllable generators.",
"To this end, benefiting from the concept of 'virtual load demand', we first design the distributed controller for the controllable generators by leveraging the primal-dual decomposition technique.",
"We then propose a method to estimate the virtual load demand of each controllable generator based on local frequencies.",
"We derive incremental passivity conditions for the uncontrollable generators.",
"Finally, we prove that the closed-loop system is asymptotically stable and its equilibrium attains the optimal solution to the associated economic dispatch problem.",
"Simulations, including small and large-disturbance scenarios, are carried on the New England system, demonstrating the effectiveness of our design."
],
[
"Introduction",
"Frequency restoration and economic dispatch (ED) are two important problems in power system operation.",
"Conventionally, they are implemented hierarchically in a centralized fashion, where the former is addressed in a fast time scale while the latter in a slow time scale[1], [2].",
"While this centralized hierarchy works well for the traditional power system, it may not be able to keep pace with the fast development of our power system due to: 1) slow response, 2) insufficient flexibility, 3) low privacy, 4) intense communication, and 5) single point of failure issue.",
"In this regard, the idea of breaking such a hierarchy is proposed in [3], [4].",
"In [5], an intrinsic connection between the asymptotic stability of the dynamical frequency control system with the ED problem is proposed.",
"It leads to a so-called inverse-engineering methodology for designing optimal frequency controllers, where the (partial) primal-dual gradient algorithm plays an essential role [6], [1].",
"When designing optimal frequency controllers, in choice of power flow models, including the linear model (usually associated with DC power flow, e.g.",
"[5], [7], [8], [9], [10], [11], [12], [13], [14], [15]) and the nonlinear model (usually associated with AC power flow, e.g.",
"[16], [17], [18], [19], [20], [21]), is crucial.",
"While the closed-loop system can be interpreted in a linear model as carrying out a primal-dual algorithm for solving ED, this interpretation of frequency control may not hold in a nonlinear model.",
"In addition to nonlinear power flow, excitation voltage dynamics are considered in [16], [17], [18], making the model more realistic.",
"The aforementioned idea is further developed to enable the design of distributed optimal frequency controllers.",
"Roughly speaking, the works of distributed optimal frequency control can be divided into two categories in terms of different power system models: network-reduced models e.g.",
"[5], [13], [16], [12], [18], [14], [15] and network-preserving models e.g.",
"[7], [8], [9], [17], [10], [11].",
"In network-reduced models, generators and/or loads are aggregated and treated as one bus or control area, which are connected to each other through tie lines.",
"In [5], [13], aggregated generators in each area are driven by automatic generation control (AGC) to restore system frequency.",
"[16], [12], [18], [14], [15] further consider both the aggregated generators and load demands in frequency control.",
"In network-preserving models, generator and load buses are separately handled with different dynamic models and coupled by power flows, rendering a set of differential algebraic equations (DAEs).",
"In [7], an optimal load control (OLC) problem is formulated and a primary load-side control is derived as a partial primal-dual gradient algorithm for solving the OLC problem.",
"The design approach is extended to secondary frequency control (SFC) that restores nominal frequency in [8].",
"It is generalized in [9], where passivity condition guaranteeing stability is proposed for each local bus.",
"Then, a unified framework combining load and generator control is advocated in [10].",
"A similar model is also utilized in [11], where only limited control coverage is needed.",
"Similar to [18], the Hamiltonian method is used to analyze the network-preserving model in [17].",
"Compared with the network-reduced model, the network-preserving model describes power systems more precisely and appear more suitable for analyzing interactions between different control areas.",
"Therefore, we specifically consider the network-preserving model in this work.",
"Almost all of aforementioned works assume that all buses are controllable and load demands at all buses are accurately measurable, especially for those proposing secondary frequency control.",
"Moreover, it is usually assumed that the communication network has the same topology of the power grid.",
"These assumptions are strong and arguably unrealistic for practice.",
"First, only a part of generators and controllable load buses can participate in frequency control in practice.",
"Second, the communication network may not be identical to the topology of the power grid.",
"Third, it is difficult to accurately measure the load injection on every bus.",
"In extreme cases, even the number of load buses is unknown in practice.",
"These problems highlight some reasons why theoretical work in this area is hardly applied in practical systems.",
"In this context, a novel distributed frequency recovery controller is proposed that only needs to be implemented on controllable generator buses.",
"To this end, a network-preserving power system model is adopted.",
"This work is an extension of our former work [11].",
"However, in this paper, we design a totally different controller considering a third-order nonlinear generator model with excitation voltage dynamics and nonlinear power flow.",
"It is also motivated partly by [17], which adopts a similar model, although our results are significantly different from those in [17].",
"Differing from [11], the controller avoids load measurement, which greatly facilitates implementation.",
"By using LaSalle's invariance principle, it is proved that the closed-loop system converges to an equilibrium point that solves the economic dispatch problem.",
"The salient features of the proposed distributed optimal frequency controller are: Model: The network-preserving model of power system is used, including excitation voltage dynamics and nonlinear power flow.",
"This, unlike work on the linear model, returns a valid controller even under large disturbances.",
"Controllability: We allow an arbitrary subset of generator buses to be controllable.",
"Controller: The distributed controller achieves the optimal solution to economic dispatch while restoring the nominal frequency, provided that certain sufficient conditions on active power dynamics of uncontrollable generators and excitation voltage dynamics of all generators are satisfied.",
"Communication: Communication is required between neighboring controllable generators only, and the communication network can be arbitrary as long as it remains connected.",
"Measurement: No load measurement is needed, and the controller is adaptive to unknown load changes.",
"The rest of this paper is organized as follows.",
"In Section II, we introduce the power system model.",
"Section III formulates the optimal economic dispatch problem.",
"The distributed controller is proposed in Section IV, and we further prove the optimality and stability of the corresponding equilibrium point in Section V. The load estimation method is proposed in Sectin VI.",
"We confirm the performance of controllers via simulations on a detailed power system in Section VII.",
"Section VIII concludes the paper.",
"Power systems are composed of many generators and loads, which are integrated in different buses and connected by power lines, forming a power network.",
"Buses are divided into three types, controllable generator buses, uncontrollable generator buses and pure load buses.",
"Denote controllable generator buses as $\\mathcal {N}_{CG}=\\lbrace 1, 2, \\dots , n_{CG}\\rbrace $ , uncontrollable generator buses as $\\mathcal {N}_{UG}=\\lbrace n_{CG}+1, n_{CG}+2, \\dots , n_{CG}+n_{UG}\\rbrace $ , and pure load buses as $\\mathcal {N}_{L}=\\lbrace n_{CG}+n_{UG}+1, \\dots , n_{CG}+n_{UG}+n_{L}\\rbrace $ .",
"Then the set of generator buses is $\\mathcal {N}_{G}=\\mathcal {N}_{CG} \\cup \\mathcal {N}_{UG}$ and set of all the buses is $\\mathcal {N}=\\mathcal {N}_{G}\\cup \\mathcal {N}_{L}$ .",
"It should be noted that load can be connected to any bus besides pure load buses.",
"Let $\\mathcal {E}\\subseteq \\mathcal {N}\\times \\mathcal {N}$ be the set of lines, where $(i,j)\\in \\mathcal {E}$ if buses $i$ and $j$ are connected directly.",
"Then the whole system is modeled as a connected graph $\\mathcal {G}=(\\mathcal {N},\\mathcal {E})$ .",
"The admittance of each line is $Y_{ij}:=G_{ij}+\\sqrt{-1}B_{ij}$ with $G_{ij}=0$ for every line.",
"Denote the bus voltage by $V_i\\angle \\theta _i$ , where $V_i$ is the amplitude and $\\theta _i$ is the voltage phase angle.",
"The active and reactive power $P_{ij},Q_{ij}$ from bus $i$ to bus $j$ is $P_{ij}&={{V_i}{V_j}{B_{ij}}\\sin \\left( {{\\theta _i} - {\\theta _j}} \\right)} \\\\Q_{ij}&={B_{ij}V_i^2 - {V_i}{V_j}B_{ij}\\cos \\left( {{\\theta _i} - {\\theta _j}} \\right)}$ Figure: Summary of notationsFor convenience, most notations are summarized in Fig.REF ."
],
[
"Synchronous Generators ",
"For $i\\in \\mathcal {N}_{G}$ , we use the standard third-order generator model (e.g.",
"[18], [22], [23]) (REF )-().",
"Here () is the simplified governor-turbine model, and () is the excitation voltage control model: $\\dot{\\delta }_i & = \\omega _i\\\\\\dot{\\omega }_i & = (P^g_i -D_i \\omega _i- P_{ei} )/{M_i}\\\\\\dot{E}^{^{\\prime }}_{qi} & = - {E_{qi}}/{T^{^{\\prime }}_{d0i}} + {E_{fi}}/{T^{^{\\prime }}_{d0i}}\\\\\\dot{P}_i^g &= - {P_i^g}/{T_i} + u_i^g\\\\\\dot{E}_{fi} & = h(E_{fi},E_{qi})$ In this model, $M_i$ is the moment of inertia; $D_i$ the damping constant; $T^{^{\\prime }}_{d0i}$ the $d$ -axis transient time constant; ${T_i}$ the time constant of turbine; $\\delta _i$ the power angle of generator $i$ ; $\\omega _i$ the generator frequency deviation compared to an steady state value; $P^g_i$ the mechanical power input; $p_j$ the active load demand; $P_{ei}$ the active power injected to network; $E^{^{\\prime }}_{qi}$ the $q$ -axis transient internal voltage; $E_{qi}$ the $q$ -axis internal voltage; $E_{fi}$ the excitation voltage.",
"$E_{qi}$ is given by $E_{qi}=\\frac{x_{di}}{x^{^{\\prime }}_{di}}E^{^{\\prime }}_{qi}-\\frac{x_{di}-x_{di}^{^{\\prime }}}{x^{^{\\prime }}_{di}}V_i\\cos (\\delta _i-\\theta _i)$ where $x_{di}$ is the $d$ -axis synchronous reactance, and $x_{di}^{^{\\prime }}$ is the $d$ -axis transient reactance.",
"The active and reactive power (denoted by $Q_{ei}$ ) injection to the network are $&P_{ei}=\\frac{E^{^{\\prime }}_{qi}V_i}{x_{di}^{^{\\prime }}}\\sin (\\delta _i-\\theta _i)\\\\&Q_{ei}=\\frac{V_i^2}{x^{^{\\prime }}_{di}}-\\frac{E^{^{\\prime }}_{qi}V_i}{x_{di}^{^{\\prime }}}\\cos (\\delta _i-\\theta _i)$ For controllable generators $i\\in \\mathcal {N}_{CG}$ , the capacity limits are $\\underline{P}_i^g \\le {P}_i^g \\le \\overline{P}_i^g$ where $\\underline{P}_i^g, \\overline{P}_i^g$ are lower and upper limits of ${P}_i^g$ ."
],
[
"Dynamics of Voltage Phase Angles",
"To build a network-preserving power system model, relation between generators and power network should be explicitly established.",
"In this paper, loads for bus $i\\in \\mathcal {N}$ are simply modeled as constant active and reactive power injections.",
"Then the following equations are used to dictate the power balance and voltage phase-angle dynamics at each bus: $\\dot{\\theta }_{i}&=\\tilde{\\omega }_i,\\ \\ i\\in \\mathcal {N} \\\\0&=P_{ei}-\\tilde{D}_i\\tilde{\\omega }_i-p_{i}-\\sum \\nolimits _{j \\in N_i} {P_{ij} }, \\ \\ i\\in \\mathcal {N}_{G}\\\\0&=-\\tilde{D}_i\\tilde{\\omega }_i-p_{i}-\\sum \\nolimits _{j \\in N_i} {P_{ij} }, \\ \\ i\\in \\mathcal {N}_{L} \\\\0&=Q_{ei}-q_{i}-{ \\sum \\nolimits _{j \\in N_i}Q_{ij} }, \\ \\ i\\in \\mathcal {N}_{G} \\\\0&=-q_{i}-{ \\sum \\nolimits _{j \\in N_i}Q_{ij} } , \\ \\ i\\in \\mathcal {N}_{L}$ where, $p_{i}, q_{i}$ are active and reactive load demands, respectively; $\\tilde{\\omega }_i$ the frequency deviation at bus $i$ ; $N_i$ the set of buses connected directly to bus $i$ ; $\\tilde{D}_i$ the damping constant at bus $i$ ; $\\tilde{D}_i\\tilde{\\omega }_i$ the change of frequency-sensitive load [7].",
"In power system, line power flows are mainly related to power angle difference between two buses rather than the power angles independently.",
"Then, we define new variables to denote angle differences as $\\eta _{ii}:=\\delta _i-\\theta _i, i \\in {\\cal N}_{G}$ and $\\eta _{ij}:=\\theta _i-\\theta _j, i, j \\in {\\cal N}$ .",
"The time derivative of $\\eta _{ii}$ and $\\eta _{ij}$ are $\\dot{\\eta }_{ii} & = \\omega _i - \\tilde{\\omega }_i, \\ i \\in {\\cal N}_{G}\\\\\\dot{\\eta }_{ij} & = \\tilde{\\omega }_i - \\tilde{\\omega }_j, \\ i, j \\in {\\cal N}, i\\ne j$ respectively.",
"In the following analysis, we use $\\eta _{ii}$ and $\\eta _{ij}$ as new state variables instead of $\\delta _i$ and $\\theta _{i}$ .",
"To summarize, $(\\ref {line power}) - (\\ref {eq:Model.PQ}), (\\ref {load activepower})-(\\ref {reactive_power}), (\\ref {eq:closedloop.1a1}), (\\ref {eq:closedloop.1a2})$ constitute the nonlinear network preserving model of power systems, which is in a form of differential-algebraic equations (DAE)."
],
[
"Communication Network",
"In this paper, we consider a communication graph among the buses of controllable generators only.",
"Denote $E\\subseteq \\mathcal {N}_{CG} \\times \\mathcal {N}_{CG}$ as the set of communication links.",
"If generator $i$ and $j$ can communicate directly to each other, we denote $(i,j)\\in E$ .",
"Obviously, edges in the communication graph $E$ can be different from lines in the power network $\\mathcal {E}$ .",
"For the communication network, we make the following assumption: A1: The communication graph $E$ is undirected and connected."
],
[
"Optimal Power-Sharing Problem in Frequency Control",
"The purpose of optimal frequency control is to let all the controllable generators share power mismatch economically when restoring frequency.",
"Then we have the following optimization formulation, denoted by SFC.",
"$\\text{SFC:}\\ \\ \\ &\\mathop {\\min }\\limits _{P_i^g, i\\in {\\cal {N}}_{CG}}\\ \\ \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} f_i(P_i^g) \\\\&\\text{s.t.",
"}\\quad \\sum \\limits _{i\\in {\\cal {N}}_{CG}} P_i^g = \\sum \\limits _{i\\in \\cal {N}} p_i - \\sum \\limits _{i\\in {\\cal {N}}_{UG}} P_i^{g*}\\\\& \\quad \\quad \\ \\ \\underline{P}_i^g \\le {P}_i^g \\le \\overline{P}_i^g, \\quad i\\in {\\cal {N}}_{CG}$ where $P_i^{g*}$ is the mechanical power of uncontrollable generator in the steady state.",
"In (REF ), $f_i(P_i^g)$ concerns the controllable generation $P_i^g$ , satisfying the following assumption: A2: The objective $f_i(P_i^g)$ is second-order continuously differentiable, strongly convex and $f^{^{\\prime }}_i(P_i^g)$ is Lipschitz continuous with Lipschitz constant $l_i>0$ .",
"i.e.",
"$\\exists \\ \\alpha _i >0, \\alpha _i\\le f_i^{^{\\prime \\prime }}(P_i^g)\\le l_i $ .",
"To ensure the feasibility of the optimization problem, we make an additional assumption.",
"A3: The system satisfies $\\sum \\limits _{i\\in {\\cal {N}}_{CG}} \\underline{P}_i^g \\le \\sum \\limits _{i\\in \\cal {N}} p_i - \\sum \\limits _{i\\in {\\cal {N}}_{UG}} P_i^{g*} \\le \\sum \\limits _{i\\in {\\cal {N}}_{CG}} \\overline{P}_i^g$ Specifically, we say A3 is strictly satisfied if all the inequalities in (REF ) strictly hold."
],
[
"Equivalent Optimization Model with Virtual Load Demands",
"In (), load demands are injected to every bus, which sometimes cannot be measured accurately if at all.",
"As a consequence, the values of $p_i$ may be unknown to both the controllable generators $i$ , $i\\in {\\cal {N}}_{CG}$ and the uncontrollable generators $i$ , $i\\in {\\cal {N}}_{UG}$ .",
"To circumvent such an obstacle in design, we introduce a set of new variables, $\\hat{p}_i$ , to re-formulate SFC as the following equivalent problem: $\\text{ESFC:}\\ \\ \\ &\\mathop {\\min }\\limits _{P_i^g, i\\in {\\cal {N}}_{CG}}\\ \\ \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} f_i(P_i^g) \\\\&\\text{s.t.",
"}\\quad \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} P_i^g = \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i \\\\& \\quad \\quad \\ \\ \\underline{P}_i^g \\le {P}_i^g \\le \\overline{P}_i^g, \\quad i\\in {\\cal {N}}_{CG}$ where $\\hat{p}_i$ is the virtual load demand supplied by generator $i$ in the steady state, which is a constant, satisfying $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i = \\sum \\nolimits _{i\\in \\cal {N}} p_i- \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*}$ .",
"Obviously, the number of virtual loads should be equal to that of the controllable generators.",
"Note that the power balance constraint () only requires that all the generators supply all the loads while it is not necessary to figure out which loads are supplied exactly by which generators.",
"Hence we treat virtual load demands $\\hat{p}_i$ as the effective demands supplied by generator $i$ for dealing with the issue that only a part of generators are controllable.",
"Simply letting $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i = \\sum \\nolimits _{i\\in \\cal {N}} p_i- \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*}$ , we immediately have the following Lemma: Lemma 1 The problems SFC (REF ) and ESFC (REF ) have the same optimal solutions."
],
[
"Controller design based on primal-dual gradient algorithm",
"Invoking the primal-dual gradient algorithm, the Lagrangian of the ESFC (REF ) is given by $L=&\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} f_i(P_i^{g}) + \\mu \\left(\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} P_i^g - \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i\\right) \\nonumber \\\\& + \\gamma _i^-(\\underline{P}_i^g - {P}_i^g ) + \\gamma _i^+ ( {P}_i^g - \\overline{P}_i^g), \\quad \\quad i\\in {\\cal {N}}_{CG}$ where $\\mu , \\gamma _i^-, \\gamma _i^+$ are Lagrangian multipliers.",
"Based on primal-dual update, the controller for $i\\in {\\cal {N}}_{CG}$ is designed as $u^g_i & ={P^g_i}/{T_i} -{k_{P_i^g}}\\left(\\omega _i + (f_i^{^{\\prime }}(P_i^g)+\\mu -\\gamma _i^- + \\gamma _i^+)\\right) \\\\\\dot{\\mu }& = k_{\\mu } \\bigg (\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} P_i^g - \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i \\bigg ) \\\\\\dot{\\gamma }_i^- & = k_{\\gamma _i} \\left[\\underline{P}_i^g - {P}_i^g \\right]_{\\gamma _i^-}^+ \\\\\\dot{\\gamma }_i^+ & = k_{\\gamma _i}\\left[{P}_i^g - \\overline{P}_i^g \\right]_{\\gamma _i^+}^+$ where $k_{P_i^g}, k_{\\mu }, k_{\\gamma _i}$ are positive constants; $u^g_i$ the control input; $f_i^{^{\\prime }}(P_i^g)$ the marginal cost at $P_i^g$ .",
"For any $x_i$ , $a_i \\in \\mathbb {R}$ , the operator is defined as: $[x_i]^+_{a_i}=x_i$ if $a_i>0\\ \\text{or}\\ x_i>0$ ; and $[x_i]^+_{a_i}=0$ otherwise."
],
[
"Estimating $\\mu $ by second-order consensus",
"In (), $\\mu $ is a global variable, which is a function of mechanical powers and loads of the entire system.",
"To circumvent the obstacle in implementation, a second-order consensus based method is utilized to estimate $\\mu $ locally by using neighboring information only.",
"Specifically, for $i\\in \\mathcal {N}_{CG}$ , the controller is revised to: $u^g_i & ={P^g_i}/{T_i} -{k_{P_i^g}}\\left(\\omega _i + f_i^{^{\\prime }}(P_i^g)+\\mu _i-\\gamma _i^- + \\gamma _i^+\\right) \\\\\\dot{\\mu }_i & = k_{\\mu _i} \\bigg (P_i^g - \\hat{p}_i - \\sum \\limits _{j \\in {N_{ci}}} {\\left( {{{ \\mu }_i} - {{\\mu }_j}} \\right)} - \\sum \\limits _{j \\in {N_{ci}}} {{z_{ij}}} \\bigg ) \\\\\\dot{z}_{ij} &= k_{z_i} (\\mu _i-\\mu _j) \\\\\\dot{\\gamma }_i^- & = k_{\\gamma _i} \\left[\\underline{P}_i^g - {P}_i^g \\right]_{\\gamma _i^-}^+ \\\\\\dot{\\gamma }_i^+ & = k_{\\gamma _i}\\left[{P}_i^g - \\overline{P}_i^g \\right]_{\\gamma _i^+}^+$ where, $ k_{\\mu _i}, k_{z_i}$ are positive constants; $N_{ci}$ the set of neighbors of bus $i$ in the communication graph; $\\mu _i$ the local estimation of $\\mu $ .",
"Here, () and () are used to estimate $\\mu $ locally, where only neighboring information is needed.",
"$z_{ij}$ is an auxiliary variable to guarantee the consistency of all $\\mu _i$ .",
"For the Larangian multiplier $\\mu $ , $-\\mu $ is often regarded as the marginal cost of generation.",
"Theoretically, $-\\mu _i$ should reach consensus for all the generators in the steady state.",
"Since $\\dot{\\mu }_i = 0$ holds in the steady state, we have $P_i^g - \\hat{p}_i - \\sum \\nolimits _{j \\in {N_{ci}}} {{z_{ij}}} =0$ .",
"Hence, $z_{ij}$ can be regarded as the virtual line power flow of edge $(i,j)$ in the communication graph.",
"To guarantee system stability, a sufficient condition is given for the active power dynamics of uncontrollable generators.",
"C1: The active power dynamics of uncontrollable generators are strictly incrementally output passive in terms of the input $-\\omega _i$ and output $P_{i}^g$ , i.e., there exists a continuously differentiable, positive semidefinite function $S_{\\omega _i}$ such that $\\dot{S}_{\\omega _i} \\le \\left(-\\omega _i-(-\\omega _i^*)\\right)\\left(P_{i}^g-P_{i}^{g*}\\right) - \\phi _{\\omega _i}\\ (P_{i}^g-P_{i}^{g*}) \\nonumber $ where $\\phi _{\\omega _i}$ is a positive definite function, and $\\phi _{\\omega _i}=0$ holds only when $P_{i}^g=P_{i}^{g*}$ .",
"The condition C1 on the active power dynamics of uncontrollable generators is easy to verify.",
"As an example, the commonly-used primary frequency controller $u_i^g=-\\omega _i+\\omega _i^*-k_{\\omega _i}(P_i^g-P_i^{g*})+{P^g_i}/{T_i}$ satisfies C1 whenever $k_{\\omega _i}>0$ .",
"In this case, we have $S_{\\omega _i}=\\frac{k_1}{2}(P_i^g-P_i^{g*})^2$ with $k_1>0$ and $\\phi =k_2(P_i^g-P_i^{g*})^2$ with $0<k_2\\le k_{\\omega _i} \\cdot k_1$ ."
],
[
"Excitation Voltage Dynamics of All Generators",
"Similar to the uncontrollable generators, the following sufficient condition on excitation voltage dynamics of all generators is needed to guarantee system stability, since we do not design specific excitation voltage controllers here.",
"C2: The excitation voltage dynamics are strictly incrementally output passive in terms of the input $-E_{qi}$ and output $E_{fi}$ , i.e., there exists continuously differentiable, positive semidefinite function $S_{E_i}$ such that $\\dot{S}_{E_i} \\le \\left(-E_{qi}-(-E^*_{qi})\\right)\\left(E_{fi}-E^*_{fi}\\right) - \\phi _{E_i}\\ (E_{fi}-E^*_{fi}) \\nonumber $ where $\\phi _{E_i}$ is a positive definite function, and $\\phi _{E_i}=0$ holds only when $E_{fi}=E^*_{fi}$ .",
"C2 is also easy to satisfy.",
"As an example, it can be verified that the controller given in [23] $h(E_{fi},E_{qi})=-E_{fi}+E^*_{fi}-k_{E_i}(E_{qi}-E^*_{qi})$ with $k_{E_i}>0$ satisfies C2.",
"In this case, $S_{E_i}=\\frac{k_3}{2}(E_{fi}-E^*_{fi})^2$ with $k_3>0$ and $\\phi =k_4(E_{fi}-E^*_{fi})^2$ with $0<k_4\\le k_{E_i} \\cdot k_3$ ."
],
[
"Optimality and Stability ",
"After implementing the controller on the physical power system, the closed-loop system reads $\\left\\lbrace \\begin{array}{l}(\\ref {line power}) - (\\ref {eq:Model.PQ}),(\\ref {load activepower})-(\\ref {reactive_power}),(\\ref {eq:closedloop.1a1}),\\ (\\ref {eq:closedloop.1a2}) \\\\(\\ref {eq:controlmodel})-(\\ref {controller_e})\\end{array} \\right.$ In this section, we prove the optimality and stability of the closed-loop system (REF )."
],
[
"Optimality",
"Denote the trajectory of closed-loop system as $v(t)=\\left(\\eta (t), \\omega (t), \\tilde{\\omega }(t), P^{g}(t), \\mu (t), z(t), \\gamma ^-(t), \\gamma ^+(t), E_{q}^{^{\\prime }}(t), V(t)\\right)$ .",
"Define the equilibrium set of (REF ) as ${\\cal V}:=\\lbrace v^*|v^* \\text{is an equilibrium of}\\ (\\ref {eq:closedloop}) \\rbrace $ We first present the following Theorem.",
"Theorem 2 Suppose assumptions A1, A2 and A3 hold.",
"In equilibrium of (REF ), following assertions are true.",
"The mechanical powers $P_i^g$ satisfy $\\underline{P}_i^g \\le {P}_i^{g*} \\le \\overline{P}_i^g $ , $\\forall i \\in \\mathcal {N}_{CG}$ .",
"System frequency recovers to the nominal value, i.e.",
"$\\omega _i^*=0, \\forall i \\in \\mathcal {N}_{CG} \\cup \\mathcal {N}_{UG}$ , and $\\tilde{\\omega }_i^*=0, \\forall i \\in \\cal N$ .",
"The marginal generation costs satisfy $f_i^{^{\\prime }}(P_i^{g*}) - \\gamma _i^{-*} + \\gamma _i^{+*} = f_j^{^{\\prime }}(P_j^{g*}) - \\gamma _j^{-*} + \\gamma _j^{+*}, i,j \\in {\\cal N}_{CG}$ .",
"$P_i^{g*}$ is the unique optimal solution of SFC problem (REF ).",
"$\\mu _i^*$ is unique if A3 is strictly satisfied.",
"The detailed proof of Theorem REF is given in Appendix.A of this paper.",
"It shows that the nominal frequency is recovered at equilibrium, and marginal generation costs are identical for all controllable generators, implying the optimality of any equilibrium."
],
[
"Stability",
"In this section, the stability of the closed-loop system (REF ) is proved.",
"First we define a function as $\\hat{L}&: = \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} f_i(P_i^{g*}) + \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} \\mu _i (P_i^g - \\hat{p}_i ) \\nonumber \\\\&- \\sum \\nolimits _{i \\in {{\\cal {N}}_{CG}}} {{{ \\mu }_i}{z_{ij}}} - \\frac{1}{2}\\sum \\nolimits _{i \\in {\\cal {N}}_{CG}} \\left({ \\mu }_i\\sum \\nolimits _{j \\in {N_i}} {\\left( {{{ \\mu }_i} - {{ \\mu }_j}} \\right)} \\right) \\nonumber \\\\& + \\sum \\limits _{i\\in {\\cal {N}}_{CG}} \\gamma _i^-(\\underline{P}_i^g - {P}_i^g ) + \\sum \\limits _{i\\in {\\cal {N}}_{CG}} \\gamma _i^+ ( {P}_i^g - \\overline{P}_i^g)$ Denote $x_1 := ( P^g)$ , $x_2 := ( \\mu , z, \\gamma _i^{-}, \\gamma _i^{+})$ , $x := (x_1, x_2)$ .",
"Then $\\hat{L}(x_1, x_2)$ is convex in $x_1$ and concave in $x_2$ .",
"Before giving the main result, we construct a Lyapunov candidate function composed of four parts: the quadratic part, the potential energy part, conditions C1 and C2 related parts, as we now explain.",
"For $i\\in {\\cal N}_G$ , the quadratic part is given by $W_k(\\omega , x)=&\\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{1}{2}M_i(\\omega _i-\\omega _i^*)^2 + \\frac{1 }{2} (x-x^*)^TK^{-1}(x-x^*)$ where $K=\\text{diag}(k_{P_i^g}, k_{\\mu _i}, k_{z_i}, k_{\\gamma _i})$ is a diagonal positive definite matrix.",
"Denoting $x_p=(E_{qi}^{^{\\prime }},V_i,\\delta _i,\\theta _i)$ , the potential energy part is $W_p(x_p)=\\tilde{W}_p(x_p)-(x_p-x_p^*)^T\\nabla _{x_p}\\tilde{W}_p(x_p^*)-\\tilde{W}_p(x_p^*)$ where, $&\\tilde{W}_p(E_{qi}^{^{\\prime }},E_{i},V_i,\\delta _i,\\theta _i)=\\sum \\nolimits _{i \\in {\\cal N}} \\frac{1}{2}B_{ii}V_i^2 + \\sum \\nolimits _{i \\in \\cal {\\cal N}} p_i\\theta _i \\nonumber \\\\&\\ - \\sum \\limits _{i \\in {\\cal {N}}} q_i\\text{ln}\\ V_i - \\frac{1}{2}\\sum \\limits _{i \\in {\\cal N}}\\sum \\limits _{j \\in N_i} {{V_i}{V_j}{B_{ij}} \\cos \\left( {{\\theta _i} - {\\theta _j}} \\right)} \\\\&\\ -\\sum \\limits _{i \\in {\\cal N}_{G}} \\frac{E^{^{\\prime }}_{qi}V_i}{x_{di}^{^{\\prime }}}\\cos (\\delta _i-\\theta _i) +\\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{x_{di}}{2x^{^{\\prime }}_{di}(x_{di}-x_{di}^{^{\\prime }})}\\left(E^{^{\\prime }}_{qi}\\right)^2 \\nonumber $ Conditions C1 and C2 related parts are $\\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} S_{\\omega _i}$ and $\\sum \\nolimits _{i\\in {\\cal {N}}_{G}}\\frac{1}{{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}} S_{E_i}$ respectively.",
"The Lyapunov function is defined as $W=&W_k+W_p +\\sum \\limits _{i\\in {\\cal {N}}_{UG}} S_{\\omega _i} + \\sum \\limits _{i\\in {\\cal {N}}_{G}}\\frac{S_{E_i}}{{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}}$ Then, we give the following assumption.",
"A4: The Hessian of $W_p$ satisfies $\\nabla ^2_v W_p(v)>0$ at desired equilbirium.",
"Since the voltage phase deviation between two neighboring buses is not large in practice, A4 is usually satisfied.",
"Detailed explanations can be found in Appendix.B of this paper.",
"The following stability result can be obtained.",
"Theorem 3 Suppose A1–A4 and C1, C2 hold.",
"For every $v^*$ , there exists a neighborhood $\\cal S$ around $v^*$ where all trajectories $v(t)$ satisfying (REF ) starting in $\\cal S$ converge to the set ${\\cal V}$ .",
"In addition, each trajectory converges to an equilibrium point.",
"We can further prove that $\\nabla ^2 W>0$ and $\\dot{W} \\le 0$ .",
"Moreover, $\\dot{W} = 0$ holds only at equilibrium point.",
"Then the theorem can be proved using the LaSalle's invariance principle [24].",
"Details of the proof are given in Appendix.B."
],
[
"Estimation and Optimality",
"Note that virtual load demands $\\hat{p}_i$ used in the controller (REF ) are difficult to directly measure or estimate in practice.",
"Lemma REF implies that any $\\hat{p}_i$ are valid as long as $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i = \\sum \\nolimits _{i\\in \\cal {N}} p_i - \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*}$ .",
"Noticing that the power imbalance is very small in normal operation, we have $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}P_{ei} \\approx \\sum \\nolimits _{i\\in \\cal {N}} \\hat{p}_i$ .",
"In fact, they are identical in steady state.",
"Hence, we specify $\\hat{p}_i=P_{ei}$ , which implies $P_i^g-\\hat{p}_i= P_i^g-P_{ei}=M_i\\dot{\\omega }_i+D_i\\omega _i$ .",
"That leads to an estimation algorithm of $\\mu _i$ $\\dot{\\mu }_i & = k_{\\mu _i} \\big (- \\sum \\limits _{j \\in {N_i}} {\\left( {{{ \\mu }_i} - {{\\mu }_j}} \\right)} - \\sum \\limits _{j \\in {N_i}} {{z_{ij}}} + M_i\\dot{\\omega }_i+D_i\\omega _i \\nonumber \\\\&\\qquad \\qquad +\\tau _i (-\\mu _i - f_i^{^{\\prime }}(P_i^g)+\\gamma _i^- - \\gamma _i^+)\\big )$ where $0<\\tau _i<4/l_i$ .",
"This way, we only need to measure frequencies $\\omega _i$ at each bus locally, other than the global load demands.",
"Since the controller only needs $\\mu _i$ of neighboring buses in the communication graph, it is easy to implement.",
"Now, we reconstruct the closed-loop system by replacing () with (REF ) in (REF ), which is $\\left\\lbrace \\begin{array}{l}(\\ref {line power}) - (\\ref {eq:Model.PQ}),(\\ref {load activepower})-(\\ref {reactive_power}),(\\ref {eq:closedloop.1a1}),\\ (\\ref {eq:closedloop.1a2}) \\\\(\\ref {eq:controlmodel}), (\\ref {controller_c}) -(\\ref {controller_e}), (\\ref {load estimation})\\end{array} \\right.$ We have the following lemma.",
"Lemma 4 Assertions 1)-5) in Theorem REF still hold for the equilibrium of (REF ).",
"The proof of Lemma REF is given in Appendix.C."
],
[
"Discussion on Stability",
"Recall (), then (REF ) is derived to $\\dot{\\mu }_i & = k_{\\mu _i} \\big ( P_i^g -\\hat{p}_i + \\hat{p}_i - P_{ei} - \\sum \\limits _{j \\in {N_i}} {\\left( {{{ \\mu }_i} - {{\\mu }_j}} \\right)} - \\sum \\limits _{j \\in {N_i}} {{z_{ij}}} \\nonumber \\\\&\\qquad \\qquad +\\tau _i (-\\mu _i - f_i^{^{\\prime }}(P_i^g)+\\gamma _i^- - \\gamma _i^+)\\big )$ Denote $\\rho _i = \\hat{p}_i - P_{ei}= P_{ei}^* - P_{ei}$ , which is the difference of electric power and its value in the steady state.",
"We have the following assumption A5: The disturbance can be written as $\\rho _i=\\beta _i(t)\\omega _i$ , where $|\\beta _i(t)|\\le \\bar{\\beta }_i$ and $\\bar{\\beta }_i$ is a positive constant.",
"In addition, the set $\\lbrace \\ t<\\infty \\ |\\ \\omega _i(t)=\\omega _i^*\\ \\rbrace $ has a measure zero.",
"Whenever $\\omega _i\\ne \\omega _i^*$ , there alway exists such $\\beta _i(t)$ .",
"A5 argues that $\\omega _i(t)=\\omega _i^*$ only happens at isolated points except equilibrium.",
"Generally, this is reasonable in power system.",
"Denote the state variables of (REF ) and its equilibrium set are $\\tilde{v}$ and $\\tilde{\\cal V}$ respectively.",
"We have following stability result Theorem 5 Suppose A1–A5, C1, C2 hold and (REF ) is not binding.",
"For every $\\tilde{v}^*$ , there exists a neighborhood $\\cal S$ around $\\tilde{v}^*$ where all trajectories $\\tilde{v}(t)$ satisfying (REF ) starting in $\\cal S$ converge to the set $\\tilde{\\cal V}$ whenever $\\bar{\\beta }_i<\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}.$ Moreover, the convergence of each such trajectory is to a point.",
"The proof of Theorem REF is given in Appendix.D.",
"In fact, the range of $\\beta (t)$ can be very large as long as $l_i$ is small enough.",
"For example, set $\\tau _i=3/l_i$ , then $\\bar{\\beta }_i(t)<\\sqrt{3D_i/l_i}$ .",
"As we know, if we change the objective function to $k\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} f_i(P_i^g), k>0$ , the optimal solution will not change.",
"Thus, $l_i$ can be very small as long as $k$ is small enough.",
"In this regard, (REF ) is not conservative.",
"Moreover, we will illustrate in Section that even (REF ) is binding, our controller still works."
],
[
"Test System",
"To test the proposed controller, the New England 39-bus system with 10 generators as shown in Fig.REF , is utilized.",
"In the simulation, we apply (REF ) to estimate the virtual load demands.",
"All simulations are implemented in the commercial power system simulation software PSCAD [25], and are carried on a notebook with 8GB memory and 2.39 GHz CPU.",
"We control only a subset of these generators, namely G32, G36, G38, G39, while the remaining are equipped with the primary frequency control given in (REF ).",
"In particular, we apply the controller (REF ) derived based on a simple model to a much more realistic and complicated model in PSCAD.",
"The detailed electromagnetic transient model of three-phase synchronous machines (sixth-order model) is adopted to simulate generators with governors and exciters.",
"All the lines and transformers take both resistance and reactance into account.",
"The loads are modeled as fixed loads in PSCAD.",
"The communication graph is undirected and set as $G32\\leftrightarrow {}{}G36\\leftrightarrow {}{}G38\\leftrightarrow {}{}G39\\leftrightarrow {}{}G32$ .",
"The objective function is set as $f_i=\\frac{1}{2}a_i(P_i^g)^2+b_iP_i^g$ , which is the generation cost of generator $i$ .",
"Capacity limits of $P_i^g$ and parameters $a_i, b_i$ are given in Table REF .",
"The closed-loop system is shown in Fig.REF , where each generator only needs to measure local frequency, mechanical power, voltage and phase angle to compute its control command.",
"Note that there is no load measurement and only $\\mu _i$ are communicated between neighboring controllable generators.",
"Figure: The New England 39-bus systemFigure: Diagram of the closed-loop systemTable: Capacity limits of generators"
],
[
"Results under Small Disturbances",
"We consider the following scenario: 1) at $t=10$ s, there is a step change of 60MW load demands at each of buses 3, 15, 23, 24, 25; 2) at $t=70$ s, there is another step change of 120MW load at bus 23.",
"Neither the original load demands nor their changes are known to the generators."
],
[
"Equilibrium",
"In steady states, the nominal frequency is well recovered.",
"The optimal mechanical powers are given in Table REF , which are identical to the optimal solution of (REF ) computed by centralized optimization.",
"Stage 1 is for the period from 10s to 70s, and Stage 2 from 70s to 130s.",
"The values in Table REF are generations at the end of each stage.",
"In Stage 1, no generation reaches its limit, while in Stage 2 both G38 and G39 reach their upper limits.",
"At the end of each stage, the marginal generation cost $-\\mu _i$ of generator $i$ , converges identically (see Fig.",
"REF ), implying the optimality of the results.",
"The test results confirm the theoretical analyses and demonstrate that our controller can automatically attain optimal operation points even in the more realistic and sophisicated model.",
"Table: Equilibrium points"
],
[
"Dynamic Performance",
"In this subsection, we analyze the dynamic performance of the closed-loop system.",
"For comparison, automatic generation control (AGC) is tested in the same scenario.",
"In the AGC implementation, the signal of area control error (ACE) is given by $ACE=K_f\\omega +P_{ij}$ [26], where, $K_f$ is the frequency response coefficient; $\\omega $ the frequency deviation; $P_{ij}$ the deviation of tie line power.",
"In the case studies, we can treat the whole system as one control area, implying $P_{ij}=0$ .",
"Hence, the control center computes $ACE=K_f\\omega $ and allocates it to AGC generators, G32, G36, G38 and G39.",
"In this situation, the control command of each generator is $–r_i\\cdot ACE$ , where $\\sum _i r_i=1$ .",
"In this paper, we set $r_i=0.25$ for $i=1,2,3,4$ .",
"The trajectories of frequencies are given in Fig.REF , where the left one stands for the proposed controller and the right one for the AGC.",
"Figure: Dynamics of frequenciesIt is shown in Fig.REF that the frequencies are recovered to the nominal value under both controls.",
"The frequency drops under two controls are very similar while the recovery time under the proposed control is much less than that under the conventional AGC.",
"Although a number of studies have been devoted to distributed frequency control of power systems, most of them assume that all the nodes are controllable except [11].",
"To make a fair comparison, the controller proposed in [11] is adopted as a rival in our tests.",
"As shown in Eq.",
"(8) of [11], each controller needs to predict the load it should supply in steady state.",
"However, it is hard to acquire an accurate prediction in practice, which could lead to steady-state frequency error.",
"Figure: Dynamics of frequencies with the controller in In the next case, we compare the two controls in the same scenario as that in Section VII.B.",
"The dynamics of frequencies with the controller given in [11] are shown in Fig.REF .",
"It is observed that there is a frequency deviation in steady state, as the prediction is inaccurate.",
"Although the deviation is usually quite small, it is difficult to completely eliminate.",
"In contrast, when the proposed method is adopted, there is no frequency deviation in steady state, as is shown in Fig.REF .",
"This result is perfectly in coincidence with the indication given by Theorems REF , REF and REF and Lemma REF in this paper.",
"Mechanical power dynamics under the AGC and the proposed controller are shown in Fig.REF and Fig.REF , respectively.",
"The left parts show mechanical powers of G32, G36, G38, G39, while the right parts show mechanical powers of other generators adopting conventional controller (REF ).",
"With both controls, mechanical powers of the generators adopting (REF ) remain identical in the steady state.",
"However, there are two problems when adopting the AGC: 1) mechanical powers are not optimal; 2) mechanical power of G39 violates the capacity limit.",
"In contrast, the proposed control can avoid these problems.",
"In Stage 1 of Fig.REF , no generator reaches capacity limits.",
"In Stage 2, both G38 and G39 reach their upper limits.",
"Then, G38 and G39 stop increasing their outputs while G32 and G36 raise their outputs to balance the load demands.",
"In addition, the steady states of in both stages are optimal, which are the same as shown in Table REF .",
"Figure: Dynamics of mechanical powers under AGCFigure: Dynamics of mechanical powers under the proposed controlFigure: Dynamics of bus voltagesWe also illustrate in Fig.REF the dynamics of voltage at buses 3, 15, 23, 24, 25.",
"The voltages converge rapidly, and only experience small drops when loads increase.",
"This result validates the effectiveness of the voltage control.",
"The marginal generation cost of generator $i$ , $-\\mu _i$ , are shown in the left part of Fig.REF .",
"They converge in both stages and the steady-state values in Stage 2 are slightly bigger than that in Stage 1, as the load changes lead to an increase in the marginal generation cost.",
"Dynamics of $z_{ij}, (i,j)\\in E$ are illustrated in the right part of Fig.REF , which demonstrate that the steady state values do not change in the two stages.",
"In addition, the variation of $z$ in transient is very small as the deviation of $\\mu _i$ is very small.",
"Figure: Dynamics of -μ-\\mu and zz"
],
[
"Performance under Large Disturbances",
"In this subsection, two scenarios of large disturbances are considered.",
"One is a generator tripping and the other is a short-circuit fault followed by a line tripping."
],
[
"Generator tripping",
"At $t=10s$ , G32 is tripped, followed by occurrence of certain power imbalance.",
"Note that the communication graph still remains connected.",
"The output of G32 drops to zero.",
"Frequency and mechanical powers change accordingly.",
"System dynamics are illustrated in Fig.REF .",
"The left part of Fig.REF shows the frequency dynamics, and the right shows the mechanical power dynamics of controllable generators.",
"It is observed that the system frequency experiences a big drop at first, and then recovers to the nominal value quickly as other controllable generators increase outputs to balance the power mismatch.",
"These results confirm that our controller can adapt to generator tripping autonomously even if the tripped generator is contributing to frequency control.",
"Figure: Dynamics of frequencies and mechanical powers with generator trip"
],
[
"Short-circuit fault",
"At $t=10s$ , there occurs a three-phase short-circuit fault on line $(4,14)$ .",
"At $t=10.05$ s, this line is tripped by breakers.",
"At $t=60$ s, the fault is removed and line $(4,14)$ is re-closed.",
"Frequency dynamics and voltage dynamics of buses 4 and 14 are given in Fig.REF , where the left part shows frequency dynamics and the right shows voltage dynamics.",
"It can be seen that the frequency experiences violent oscillations after the fault happens.",
"And then it is stabilized quickly once the line is tripped.",
"Small frequency oscillation occurs when the line is re-closed.",
"At the same time, the voltages of buses 4 and 14 drop to nearly zero when the fault happens.",
"The voltages are then stabilized to a new steady-state value in around 10s after the fault line is tripped.",
"They are slightly different from their initial values because the system’s operating point has changed due to line tripping.",
"When the tripped line is re-closed, the voltages recover to the initial values quickly.",
"Figure: Dynamics of frequencies and voltages with line tripFigure: Dynamics of -μ-\\mu and mechanical powers with line tripWhen line $(4,14)$ is tripped, the power flow across the power network varies accordingly.",
"As a consequence, the line loss also changes, causing variations of mechanical powers, as shown in Fig.REF .",
"In the left part of Fig.REF , the inset is the dynamics of mechanical power of G39 from 30s to 110s.",
"Similarly, the inset in the right part is dynamics of $-\\mu $ of all generators.",
"Mechanical powers and their marginal costs all increase when the line is tripped.",
"However, the proposed controller compensates the loss change autonomously.",
"These simulation results demonstrate that the proposed distributed optimal frequency controller can cope with large disturbances such as generator tripping and short-circuit fault."
],
[
"Conclusion",
"In this paper, we have designed a distributed optimal frequency control using a nonlinear network-preserving model, where only a subset of generator buses is controlled.",
"We have also simplified the implementation by relaxing the requirements of load measurements and communication topology.",
"Since nonlinearity of power flow model and dynamics of excitation voltage has been taken into account, our controllers can cope with large disturbances.",
"We have proved that the closed-loop system asymptotically converges to the optimal solution of the economic dispatch problem.",
"We have also carried out substantial simulations to verify the good performance of our controller under both small and large disturbances.",
"In this work, we have not considered the constraints on line flows, since the controllable generators are selected arbitrarily and may not suffice for congestion management.",
"An interesting problem is how to find out the minimal set of controllable generators to fulfill the requirement of congestion management, which is our future work."
],
[
"Proof of Theorem ",
"From $\\dot{\\gamma }_i^-=\\dot{\\gamma }_i^+=0$ in (), it follows that $\\underline{P}_i^g \\le {P}_i^{g*} \\le \\overline{P}_i^g $ , which is the first assertion.",
"From $\\dot{z}_{ij}=0$ in (), we get $\\mu _i^*=\\mu _j^*=\\mu _0$ .",
"Set $\\dot{\\mu }_i=0$ , add () for $i\\in {\\cal N}_{CG}$ , and recall $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}\\hat{p}_i = \\sum \\nolimits _{i\\in \\cal {N}} p_i- \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*}$ , we have $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} P_i^{g*} - \\sum \\nolimits _{i\\in \\cal {N}} p_i + \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*} = 0$ The right sides of (REF ) and () vanish in the equilibrium points, which implies $\\omega _i^*=\\tilde{\\omega }_i^*=\\tilde{\\omega }_j^*=\\omega _0$ .",
"Set $\\dot{\\omega }_i = 0$ and add (), (REF ) and ().",
"We have $\\omega _0\\sum \\limits _{i\\in {\\cal {N}}} \\tilde{D}_i = \\sum \\limits _{i\\in {\\cal {N}}_{CG}} P_i^{g*} - \\sum \\limits _{i\\in \\cal {N}} p_i + \\sum \\limits _{i\\in {\\cal {N}}_{UG}} P_i^{g*} = 0$ This implies that $\\omega _0=0$ due to $\\tilde{D}_i>0$ , which is the second assertion.",
"Combine (REF ), () and $\\dot{P}_i^g=0$ .",
"We have $f_i^{^{\\prime }}(P_i^{g*}) - \\gamma _i^{-*} + \\gamma _i^{+*} +\\omega _i^*+\\mu _i^*=0$ Since $\\omega _0=0$ , $\\mu _i^*=\\mu _j^*=\\mu _0$ , we can obtain the third assertion.",
"Next we will prove assertion 4).",
"Since all the constraints of SFC are linear, A3 implies that Slater's condition holds [27].",
"Moreover, the objective function is strictly convex.",
"We only need to prove that $(P_i^{g*}, \\mu _0, \\gamma _i^{-*}, \\gamma _i^{+*})$ satisfies the KKT condition of SFC in order to prove the fourth assertion.",
"The KKT conditions of SFC problem (REF ) are $& f_i^{^{\\prime }}(P_i^{g*}) - \\gamma _i^{-*} + \\gamma _i^{+*} + \\mu _0=0 \\\\& \\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} P_i^{g*} - \\sum \\nolimits _{i\\in \\cal {N}} p_i + \\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} P_i^{g*} = 0 \\\\& \\underline{P}_i^g \\le {P}_i^{g*} \\le \\overline{P}_i^g \\\\& \\gamma _i^{-*} \\ge 0, \\gamma _i^{+*} \\ge 0 \\\\& \\gamma _i^{-*}(\\underline{P}_i^g - {P}_i^{g*} )=0, \\gamma _i^{+*} ( {P}_i^{g*} - \\overline{P}_i^g) =0$ From $\\dot{\\gamma }_i^- = \\dot{\\gamma }_i^+ = 0$ , we have (), () and ().",
"From the third assertion, we have (REF ).",
"From $\\dot{\\omega }=0$ and the second assertion, we have ().",
"Therefore, the equilibrium points of the closed-loop system (REF ) satisfy the KKT conditions (REF ).",
"This implies the fourth assertion.",
"If A3 is strictly satisfied, we know $\\exists \\ i\\in {\\cal N}_{CG}$ that $\\gamma _i^{-*} = \\gamma _i^{+*} =0$ .",
"Then, $ \\mu _i^*=-f_i^{^{\\prime }}(P_i^{g*}) $ is uniquely determined by $P_i^{g*}$ , implying the last assertion.",
"This completes the proof."
],
[
"Proof of Theorem ",
"[Proof of Theorem REF ] Recall (REF ), and dynamics of (REF ) - () are rewritten as $\\dot{P}^g_i & = -{k_{P_i^g}}\\cdot \\left(\\omega _i + {\\partial \\hat{L}(x_1, x_2)}/{\\partial P_i^g}\\right) \\\\\\dot{\\mu }_i & = k_{\\mu _i}\\cdot {\\partial \\hat{L}(x_1, x_2)}/{\\partial \\mu _i} \\\\\\dot{z}_{ij} &= k_{z_i}\\cdot {\\partial \\hat{L}(x_1, x_2)}/{\\partial z_{ij}} \\\\ \\dot{\\gamma }_i^- & = k_{\\gamma _i}\\cdot \\left[{\\partial \\hat{L}(x_1, x_2)}/{\\partial \\gamma _i^-} \\right]_{\\gamma _i^-}^+ \\\\\\dot{\\gamma }_i^+ & = k_{\\gamma _i} \\cdot \\left[{\\partial \\hat{L}(x_1, x_2)}/{\\partial \\gamma _i^+} \\right]_{\\gamma _i^+}^+$ With regard to the closed-loop system, we first define two sets, $\\sigma ^+$ and $\\sigma ^-$ , as follows [6].",
"$\\sigma ^+ &:= \\lbrace i\\in {\\cal N}_{CG} \\, | \\, \\gamma ^+_{i}=0, \\, P_i^g-\\overline{P}_i^g<0\\rbrace \\\\\\sigma ^- &:= \\lbrace i\\in {\\cal N}_{CG} \\, | \\, \\gamma ^-_{i}=0, \\, \\underline{P}_i^g-P_i^g<0\\rbrace $ Then () and () are equivalent to $\\dot{\\gamma }^+_{i} &= \\left\\lbrace \\begin{array}{ll}k_{\\gamma _i}(P_i^g-\\overline{P}_i^g),& \\text{if}\\ i \\notin \\sigma ^+ ;\\\\0,& \\text{if}\\ i \\in \\sigma ^+ .",
"\\end{array}\\right.\\\\\\dot{\\gamma }^-_{i} &= \\left\\lbrace \\begin{array}{ll}k_{\\gamma _i}(\\underline{P}_i^g-P_i^g),& \\text{if}\\ i \\notin \\sigma ^- ;\\\\0,& \\text{if}\\ i \\in \\sigma ^- .",
"\\end{array}\\right.$ The derivative of $W_k$ is $&\\dot{W}_k = \\sum \\limits _{i \\in {\\cal N}_{G}}M_i(\\omega _i-\\omega _i^*)\\dot{\\omega }_i + (x-x^*)^T\\cdot K^{-1} \\dot{x} \\nonumber \\\\ &\\le \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\omega _i^*) (P^g_i -D_i \\omega _i- P_{ei}) - \\sum \\limits _{i \\in {\\cal N}_{CG}}(P_i^g-P_i^{g*})\\omega _i \\nonumber \\\\&\\ \\ - (x_1-x^*_1)^T\\cdot \\nabla _{x_1}\\hat{L} + (x_2-x^*_2)^T\\cdot \\nabla _{x_2}\\hat{L} \\nonumber \\\\&= \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\omega _i^*) (P^g_i - P^{g*}_i -D_i (\\omega _i - \\omega _i^*)- (P_{ei}-P_{ei}^*)) \\nonumber \\\\&\\ \\ - \\sum \\limits _{i \\in {\\cal N}_{CG}}(P_i^g-P_i^{g*})\\cdot (\\omega _i- \\omega _i^* ) \\nonumber \\\\&\\ \\ - (x_1-x^*_1)^T\\cdot \\nabla _{x_1}\\hat{L} + (x_2-x^*_2)^T\\cdot \\nabla _{x_2}\\hat{L} \\nonumber \\\\&= - (x_1-x^*_1)^T\\cdot \\nabla _{x_1}\\hat{L} + (x_2-x^*_2)^T\\cdot \\nabla _{x_2}\\hat{L} \\nonumber \\\\&\\ \\ -\\sum \\limits _{i \\in {\\cal N}_{G}}D_i(\\omega _i-\\omega _i^*)^2 - \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\omega _i^*) (P_{ei}-P_{ei}^*) \\nonumber \\\\&\\ \\ + \\sum \\limits _{i \\in {\\cal N}_{UG}}(P_i^g-P_i^{g*})(\\omega _i-\\omega _i^*)$ where the inequality is due to $(\\gamma ^--\\gamma ^{-*})^T[\\underline{P}^g-P^g]^+_{\\gamma ^-}&\\le (\\gamma ^--\\gamma ^{-*})^T(\\underline{P}^g-P^g)\\nonumber \\\\&=(\\gamma ^--\\gamma ^{-*})^T\\nabla _{\\gamma ^-} \\hat{L}\\nonumber $ Here the inequality holds since $\\gamma ^-_i=0\\le \\gamma _i^{-*}$ and $\\underline{P}^g_i-P^g_i < 0$ for $i\\in \\sigma ^-$ , i.e., $(\\gamma ^-_i - \\gamma _i^{-*}) \\cdot (\\underline{P}^g_i-P^g_i)\\ge 0$ .",
"Similarly for $i\\in \\sigma ^+$ , we have $(\\gamma ^+-\\gamma ^{+*})^T[ P^g-\\overline{P}^g]^+_{\\gamma ^+}&\\le (\\gamma ^+-\\gamma ^{+*})^T(P^g-\\overline{P}^g)\\nonumber \\\\&=(\\gamma ^+-\\gamma ^{+*})^T\\nabla _{\\gamma ^+} \\hat{L}\\nonumber $ From (REF ) and (), it yields $0&=\\sum \\limits _{i \\in {\\cal N}_{G}} (\\tilde{\\omega }_i- \\tilde{\\omega }_i^*) \\bigg ((P_{ei}-P_{ei}^*)-\\sum \\limits _{j \\in {N}_{i}} (P_{ij}-P_{ij}^*) \\bigg ) \\\\&\\ \\ - \\sum \\limits _{i \\in {\\cal N}}\\tilde{D}_i(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)^2 - \\sum \\limits _{i \\in {\\cal N}_{L}}(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)\\sum \\limits _{j \\in {N}_{i}} (P_{ij}-P_{ij}^*) \\nonumber $ Add (REF ) to (REF ), then we have $&\\dot{W}_k\\le -\\sum \\limits _{i \\in {\\cal N}_{G}}D_i(\\omega _i-\\omega _i^*)^2 - \\sum \\limits _{i \\in {\\cal N}}\\tilde{D}_i(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)^2 \\nonumber \\\\& + \\sum \\limits _{i \\in {\\cal N}_{G}}(\\tilde{\\omega }_i-\\omega _i) (P_{ei}-P_{ei}^*) - \\sum \\limits _{(i,j) \\in {\\cal E}}(\\tilde{\\omega }_i -\\tilde{\\omega }_j) (P_{ij}-P_{ij}^*) \\nonumber \\\\& + \\sum \\limits _{i \\in {\\cal N}_{UG}}(P_i^g-P_i^{g*})(\\omega _i-\\omega _i^*) \\nonumber \\\\& - (x_1-x^*_1)^T\\cdot \\nabla _{x_1}\\hat{L} + (x_2-x^*_2)^T\\cdot \\nabla _{x_2}\\hat{L}$ Since $\\hat{L}$ is a convex function of $x_1$ and a concave function of $x_2$ , we have $- & (x_1-x^*_1)^T\\cdot \\nabla _{x_1}\\hat{L} (x_1,x_2) + (x_2-x^*_2)^T\\cdot \\nabla _{x_2}\\hat{L} (x_1,x_2) \\nonumber \\\\\\le & \\hat{L}(x_1^*,x_2) - \\hat{L}(x_1,x_2) +\\hat{L}(x_1,x_2) - \\hat{L}(x_1,x^*_2) \\nonumber \\\\= & \\hat{L}(x_1^*,x_2) - \\hat{L}(x_1^*,x_2^*) +\\hat{L}(x_1^*,x_2^*) - \\hat{L}(x_1,x^*_2) \\nonumber \\\\\\le & \\ 0 $ where the first inequality follows because $\\hat{L}$ is convex in $x_1$ and concave in $x_2$ and the second inequality follows because $(x^*_1, x^*_2)$ is a saddle point.",
"Hence, we have $&\\dot{W}_k\\le -\\sum \\limits _{i \\in {\\cal N}_{G}}D_i(\\omega _i-\\omega _i^*)^2 - \\sum \\limits _{i \\in {\\cal N}}\\tilde{D}_i(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)^2 \\nonumber \\\\& - \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\tilde{\\omega }_i) (P_{ei}-P_{ei}^*) - \\sum \\limits _{(i,j) \\in {\\cal E}}(\\tilde{\\omega }_i -\\tilde{\\omega }_j) (P_{ij}-P_{ij}^*) \\nonumber \\\\& + \\sum \\nolimits _{i \\in {\\cal N}_{UG}}(P_i^g-P_i^{g*})(\\omega _i-\\omega _i^*)$ The partial of $W_p(x_p)$ is $\\nabla _{E_{qi}^{^{\\prime }}} W_p&=\\frac{1}{x_{di}-x_{di}^{^{\\prime }}}(E_{qi}-E^*_{qi}) \\\\\\nabla _{V_{i}} W_p&=0 \\\\\\nabla _{\\delta _{i}} W_p&=P_{ei}-P^*_{ei} \\\\ \\nabla _{\\theta _{i}} W_p&=\\sum \\limits _{(i,j) \\in {\\cal E}} (P_{ij}-P_{ij}^*) - \\sum \\limits _{i \\in {\\cal N}_{G}}(P_{ei}-P^*_{ei})$ The derivative of $W_p$ is $\\dot{W}_p & = \\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{(E_{qi}-E^*_{qi})(E_{fi}-E^*_{fi})}{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})} - \\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{(E_{qi}-E^*_{qi})^2}{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})} \\nonumber \\\\&+\\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\tilde{\\omega }_i) (P_{ei}-P_{ei}^*) + \\sum \\limits _{(i,j) \\in {\\cal E}}(\\tilde{\\omega }_i -\\tilde{\\omega }_j) (P_{ij}-P_{ij}^*)$ The Lyapunov function is defined as $W=W_k+W_p+\\sum \\limits _{i\\in {\\cal {N}}_{UG}} S_{\\omega _i} + \\sum \\limits _{i\\in {\\cal {N}}_{G}} \\frac{1}{{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}}S_{E_i}$ , and its derivative is $\\dot{W} & = \\dot{W}_k+\\dot{W}_p +\\sum \\nolimits _{i\\in {\\cal {N}}_{UG}} \\dot{S}_{\\omega _i} + \\sum \\nolimits _{i\\in {\\cal {N}}_{G}}\\frac{\\dot{S}_{E_i}}{{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}} \\nonumber \\\\&\\le -\\sum \\limits _{i \\in {\\cal N}_{G}}D_i(\\omega _i-\\omega _i^*)^2 - \\sum \\limits _{i \\in {\\cal N}}\\tilde{D}_i(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)^2 \\nonumber \\\\& - \\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{(E_{qi}-E^*_{qi})^2}{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}\\nonumber \\\\& + \\sum \\limits _{i \\in {\\cal N}_{UG}}(P_i^g-P_i^{g*})(\\omega _i-\\omega _i^*) +\\sum \\limits _{i\\in {\\cal {N}}_{UG}} \\dot{S}_{\\omega _i} \\nonumber \\\\& + \\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{(E_{qi}-E^*_{qi})(E_{fi}-E^*_{fi})}{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})} + \\sum \\limits _{i\\in {\\cal {N}}_{G}}\\frac{\\dot{S}_{E_i}}{{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})}} \\nonumber \\\\&\\le -\\sum \\limits _{i \\in {\\cal N}_{G}}D_i(\\omega _i-\\omega _i^*)^2 - \\sum \\limits _{i \\in {\\cal N}}\\tilde{D}_i(\\tilde{\\omega }_i -\\tilde{\\omega }_i^*)^2 \\nonumber \\\\& - \\sum \\limits _{i \\in {\\cal N}_{G}}\\frac{(E_{qi}-E^*_{qi})^2}{T_{d0i}^{^{\\prime }}(x_{di}-x_{di}^{^{\\prime }})} - \\sum \\limits _{i \\in {\\cal N}_{UG}} \\phi _{\\omega _i} - \\sum \\limits _{i \\in {\\cal N}_{G}} \\phi _{E_i}\\nonumber \\\\& \\le 0$ The last two inequalities are due to assumption A4.",
"To prove the locally asymptotic stability of the closed-loop system, we also need to prove that $W>0, \\forall v\\in S \\backslash v^*$ .",
"Equivalently, we need $\\nabla ^2_v W>0, \\forall \\ v\\in S \\backslash v^*$ , i.e.",
"A4.",
"Consequently, there exists a neighborhood set $\\lbrace \\ v:W(v)\\le \\epsilon \\ \\rbrace $ for all sufficiently small $\\epsilon >0$ so that $\\nabla ^2_v W(v)>0$ .",
"Hence, there is a compact set $\\cal S$ around $v^*$ contained in such neighborhood, which is forward invariant.",
"Let $Z_1:= \\lbrace \\ v :\\dot{W}(v)=0\\ \\rbrace $ .",
"By LaSalle's invariance principle, the each of trajectories $v(t)$ starting from $\\cal S$ converges to the largest invariant set $Z^+$ contained in ${\\cal S} \\cap Z_1$ .",
"From above analysis, if $\\dot{W}(v)=0$ , $v$ is an equilibrium point of the closed-loop system (REF ).",
"Hence, $v$ converges to $Z^+ \\in \\cal V$ .",
"Finally, we will prove the convergence of each $v(t)$ starting from $\\cal V$ is to a point by following the proof of Theorem 1 in [18].",
"Since $v(t)$ is bounded, its $\\omega $ -limit set $\\Omega (v)\\ne \\emptyset $ .",
"By contradiction, suppose there exist two different points in $\\Omega (v)$ , i.e., $v_1^*, v_2^* \\in \\Omega (v), v_1^*\\ne v_2^*$ .",
"Since the Hessian of $W_1(v), W_2(v)$ is positive definite in $\\cal S$ , there exist $W_1(v), W_2(v)$ defined by (REF ) with respect to $v_1^*, v_2^*$ and scalars $c_1>0, c_2>0$ such that two sets $W_1^{-1}(\\le c_1):=\\lbrace v|W_1(v)\\le c_1 \\rbrace , W_2^{-1}(\\le c_2):=\\lbrace v|W_2(v)\\le c_2 \\rbrace $ are disjoint (i.e.",
"$W_1^{-1}(\\le c_1)\\cap W_2^{-1}(\\le c_2)=\\emptyset $ ) and compact.",
"In addition, $W_1^{-1}(\\le c_1), W_2^{-1}(\\le c_2)$ are forward invariant.",
"By (REF ), there exists a finite time $t_1>0$ such that $v(t)\\in W_1^{-1}(\\le c_1)$ for $\\forall t\\ge t_1$ .",
"Similarly, there exists a finite time $t_2>0$ such that $v(t)\\in W_2^{-1}(\\le c_2)$ for $\\forall t\\ge t_2$ .",
"This implies that $v(t)\\in W_1^{-1}(\\le c_1)\\cap W_2^{-1}(\\le c_2)$ for $\\forall t\\ge \\max (t_1, t_2)$ , which is a contradiction.",
"This completes the proof.",
"In this part, we discuss the reasonableness of Assumption A4 by referring to [28].",
"Reference [28] investigates the control of inverter-based microgrids based on a network-preserving model, while we extend some results to more complicated synchronous-generator based bulk power systems.",
"For simplicity, we first present some notations following [28].",
"Comparing (REF ) and (REF ), $P_{ei},Q_{ei}$ have same structures with $P_{ij}, Q_{ij}$ , respectively.",
"We can treat the reactance of generator as a line with admittance as $1/{x_{di}^{^{\\prime }}}$ , which connects $i\\in \\mathcal {N}_{G}$ and inner node of the generator.",
"We denote the inner nodes of generators as $\\mathcal {N}_{G}^{^{\\prime }}$ .",
"Then, we can get a augmented power network, the incidence matrix of which is denoted as $\\hat{C}$ .",
"The set of nodes in the augmented power network is denoted as $\\mathcal {N}^{^{\\prime }}=\\mathcal {N}\\cup \\mathcal {N}_{G}^{^{\\prime }}$ .",
"Denote $\\hat{V}=(E_{q}^{^{\\prime }},V)$ , $\\hat{\\theta }= (\\delta ,\\theta )$ .",
"Let $|\\hat{C}|$ denote the matrix obtained from $\\hat{C}$ by replacing all the elements $c_{ij}$ with $|c_{ij}|$ .",
"Define $\\Gamma (\\hat{V}):=\\text{diag}(|B_{ij}|V_iV_j), i,j \\in \\mathcal {N}^{^{\\prime }}$ .",
"Define $A$ as $A_{ij}=\\left\\lbrace \\begin{array}{ll}-|B_{ij}|\\cos (\\theta _i-\\theta _j),& i\\ne j, i,j\\in \\mathcal {N} ;\\\\\\text{diag}(|B_{ii}|),& i= j, i,j\\in \\mathcal {N};\\\\-{\\cos (\\delta _i-\\theta _j)}/{x_{di}^{^{\\prime }}},& i \\in \\mathcal {N}_{G}^{^{\\prime }}, j \\in \\mathcal {N}_{G}.",
"\\end{array}\\right.$ For simplicity, we use the following notation.",
"For an $n$ -dimensional vector $r:=\\lbrace r_1, r_2, \\cdots , r_n\\rbrace $ , the diagonal matrix $\\text{diag}(r_1, r_2,\\cdots , r_n)$ is denoted in short by $[r]_\\mathcal {D}$ .",
"And $\\textbf {cos}(\\cdot ),\\textbf {sin}(\\cdot )$ are defined component-wise.",
"From the definition of $W$ in (REF ), $\\nabla ^2_v W(v)>0$ if and only if $\\nabla ^2_v W_p(v)>0$ , i.e.",
"the matrix $\\left[ {\\begin{array}{*{20}{c}}\\Gamma (\\hat{V})[\\textbf {cos}(\\hat{C}^T {\\hat{\\theta }})]_\\mathcal {D} &[\\textbf {sin}(\\hat{C}^T {\\hat{\\theta }})]_\\mathcal {D}\\Gamma (\\hat{V})|\\hat{C}|^T[\\hat{V}]_\\mathcal {D}^{-1}\\\\{[\\hat{V}]_\\mathcal {D}^{-1}}|\\hat{C}|\\Gamma (\\hat{V})[\\textbf {sin}(\\hat{C}^T {\\hat{\\theta }})]_\\mathcal {D}&A+H(\\hat{V})\\end{array}} \\right]$ is positive definite, where $H(\\hat{V})=\\left[ \\begin{array}{*{20}{c}}\\left[\\frac{x_{di}}{2x^{^{\\prime }}_{di}(x_{di}-x_{di}^{^{\\prime }})}\\right]_\\mathcal {D} & 0\\\\0 & [{q_i}/{V_i^{2}}]_\\mathcal {D}\\end{array} \\right]$ In any steady state of power system (i.e.",
"), the phase-angle difference between two neighboring nodes is usually small.",
"In addition, the difference between $\\delta _i$ and $\\theta _i$ is also small.",
"This implies that the matrix in (REF ) is diagonal dominant as well as its positive definiteness.",
"Therefore, Assumption A4 is usually satisfied and makes sense."
],
[
"Proof of Lemma ",
"From $\\dot{z}_{ij}=0$ , we get $\\mu _i^*=\\mu _j^*=\\mu _0$ .",
"Set $\\dot{\\mu }_i=0$ and add (REF ) for $i\\in {\\cal N}_{CG}$ , we have $\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}M_i\\dot{\\omega }_i+\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}}(D_i+\\tau _i)\\omega _i = 0$ The right sides of (REF ) and () vanish in the equilibrium points, which implies $\\omega _i^*=\\tilde{\\omega }_i^*=\\tilde{\\omega }_j^*=\\omega _0$ .",
"Set $\\dot{\\omega }=0$ in (REF ) and we have $\\omega _0\\sum \\nolimits _{i\\in {\\cal {N}}_{CG}} (D_i+\\tau _i) =0$ This implies that $\\omega _0=0$ due to $D_i+\\tau _i > 0$ .",
"Other proofs are same with that in Theorem REF , which are omitted."
],
[
"Proof of Theorem ",
"[Proof of Theorem REF ] We still use the Lyapunov function (REF ) to analyze the stability of the closed-loop system (REF ).",
"Denote $y=(\\omega _i^T,x_1^T,x_2^T)^T, i\\in {\\cal N}_{CG}$ , and define the following function $\\tilde{f}\\left( y \\right) = \\left[ {\\begin{array}{*{20}{c}}-D_i \\omega _i\\\\-\\left( f_i^{^{\\prime }}(P_i^g)+\\mu _i-\\gamma _i^- + \\gamma _i^+\\right)\\\\f_{\\mu _i} \\\\\\mu _i-\\mu _j\\\\\\underline{P}_i^g - {P}_i^g \\\\{P}_i^g - \\overline{P}_i^g\\end{array}} \\right], i\\in {\\cal N}_{CG}$ where $f_{\\mu _i}=P_i^g - \\hat{p}_i - \\sum \\limits _{j \\in {N_{ci}}} {\\left( {{{ \\mu }_i} - {{\\mu }_j}} \\right)} - \\sum \\limits _{j \\in {N_{ci}}} {{z_{ij}}} -\\tau _i\\mu _i - \\tau _if_i^{^{\\prime }}(P_i^g) +\\beta _i\\omega _i$Sometimes, we also use $\\beta _i$ instead of $\\beta _i(t)$ for simplification.",
"In addition, if (REF ) is not binding, we can omit $\\gamma _i^-, \\gamma _i^+$ in neighborhoods of equilibrium points.",
"The derivative of $W_k$ is $&\\dot{W}_k = \\sum \\limits _{i \\in {\\cal N}_{G}}M_i(\\omega _i-\\omega _i^*)\\dot{\\omega }_i + (x-x^*)^T\\cdot K^{-1} \\dot{x} \\nonumber \\\\ &= \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\omega _i^*) (P^g_i - P^{g*}_i -D_i (\\omega _i - \\omega _i^*)- (P_{ei}-P_{ei}^*)) \\nonumber \\\\& + (x-x^*)^T\\cdot K^{-1}\\dot{x} \\nonumber \\\\& \\le \\sum \\limits _{i \\in {\\cal N}_{G}}(\\omega _i-\\omega _i^*) (P^g_i - P^{g*}_i - (P_{ei}-P_{ei}^*)) \\nonumber \\\\& -\\sum \\limits _{i \\in {\\cal N}_{UG}} D_i (\\omega _i - \\omega _i^*)^2 + (y-y^*)^T\\tilde{f}(y)$ where the inequality is due to the same reason for that in (REF ).",
"Divide $\\dot{W}_k$ into two parts, and $\\dot{W}^1_k=(y-y^*)^T \\tilde{f}(y)$ , $\\dot{W}^2_k=\\dot{W}_k-\\dot{W}^1_k$ .",
"Then we will analyze the sign of $\\dot{W}^1_k$ .",
"$&\\dot{W}^1_k = (y-y^*)^T\\tilde{f}(y) \\nonumber \\\\& = \\int _0^1 (y-y^*)^T \\frac{\\partial }{\\partial \\tilde{z}}\\tilde{f}(\\tilde{z}(s)) (y-y^*) ds + (y-y^*)^T \\tilde{f}(y^*)\\nonumber \\\\& \\le \\frac{1}{2}\\int _0^1 (y-y^*)^T \\left[\\frac{\\partial ^T }{\\partial \\tilde{z}} \\tilde{f}(\\tilde{z}(s)) + \\frac{\\partial }{\\partial \\tilde{z}} \\tilde{f}(\\tilde{z}(s)) \\right] (y-y^*) ds \\nonumber \\\\& = \\int _0^1 (y-y^*)^T \\left[H(\\tilde{z}(s)) \\right] (y-y^*) ds$ where $\\tilde{z}(s)=y^*+s(y-y^*)$ .",
"The second equation is from the fact that $\\tilde{f}(y)-\\tilde{f}(y^*) = \\int _0^1 \\frac{\\partial }{\\partial \\tilde{z}} \\tilde{f}(\\tilde{z}(s)) (y-y^*) ds$ .",
"The inequality is due to either $\\tilde{f}(y^*) =0$ or $\\tilde{f}(y^*) < 0, y_i\\ge 0$ , i.e.",
"$(y-y^*)^T \\tilde{f}(y^*)\\le 0$ .",
"$\\frac{\\partial \\tilde{f}(y) }{\\partial y} = -\\left[ {\\begin{array}{*{20}{c}}D& 0& 0& 0& 0& 0\\\\0& \\nabla ^2_{P^g} f(P^g)& I& 0& -I& I\\\\\\beta & \\tau \\nabla ^2_{P^g} f(P^g)-I& \\tau +L_c& C& 0& 0\\\\0& 0& -C^T& 0& 0& 0\\\\0& I& 0& 0& 0& 0\\\\0& -I& 0& 0& 0& 0\\end{array}} \\right]$ where $D=\\text{diag}(D_i)$ , $\\beta =\\text{diag}(\\beta _i)$ , $\\tau =\\text{diag}(\\tau _i)$ , $I$ is the identity matrix with dimension $n_{CG}$ , $C$ is the incidence matrix of the communication graph, $L_c$ is the Laplacian matrix of the communication graph.",
"Finally, $H$ in (REF ) is $H&=\\frac{1}{2}\\left[\\frac{\\partial ^T }{\\partial y} \\tilde{f}(y) + \\frac{\\partial }{\\partial y} \\tilde{f}(y) \\right] \\nonumber \\\\&= \\left[ {\\begin{array}{*{20}{c}}-D& 0& -\\frac{1}{2}\\beta & 0& 0& 0\\\\0& -\\nabla ^2_{P^g} f(P^g)& -\\frac{\\tau }{2}\\nabla ^2_{P^g} f(P^g)& 0& 0& 0\\\\-\\frac{1}{2}\\beta & -\\frac{\\tau }{2}\\nabla ^2_{P^g} f(P^g)& -\\tau -L_c& 0& 0& 0\\\\0& 0& 0& 0& 0& 0\\\\0& 0& 0& 0& 0& 0\\\\0& 0& 0& 0& 0& 0\\end{array}} \\right]$ $H\\le 0$ , if $\\left[ {\\begin{array}{*{20}{c}}-D& 0& -\\frac{1}{2}\\beta \\\\0& -\\nabla ^2_{P^g} f(P^g)& -\\frac{\\tau }{2}\\nabla ^2_{P^g} f(P^g)\\\\-\\frac{1}{2}\\beta & -\\frac{\\tau }{2}\\nabla ^2_{P^g} f(P^g)& -\\tau \\end{array}} \\right]<0$ By Schur complement [29], we only need $-\\tau _i - \\left[ {\\begin{array}{*{20}{c}}-\\frac{1}{2}\\beta _i& -\\frac{\\tau _i}{2}c_i\\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}}-D_i& 0\\\\0& -c_i\\end{array}} \\right]^{-1}\\left[ {\\begin{array}{*{20}{c}}-\\frac{1}{2}\\beta _i\\\\ -\\frac{\\tau _i}{2}c_i\\end{array}} \\right] <0$ where $c_i=\\nabla ^2_{P^g_i} f(P^g_i)$ .",
"Solving (REF ), we can get $-\\sqrt{\\tau _i D_i(4-\\tau _i c_i)}<\\beta _i<\\sqrt{\\tau _i D_i(4-\\tau _i c_i)}$ By A2, we know $c_i\\le l_i$ , thus $\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}\\le \\sqrt{\\tau _i D_i(4-\\tau _i c_i)},$ $-\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}\\ge -\\sqrt{\\tau _i D_i(4-\\tau _i c_i)}.$ Here we need $\\tau _i>0,\\ 4-\\tau _i l_i>0$ , i.e.",
"$0<\\tau _i<4/l_i$ .",
"Finally, we have $-\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}<\\beta _i<\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}$ i.e.",
"$\\bar{\\beta }_i<\\sqrt{\\tau _i D_i(4-\\tau _i l_i)}$ , implying (REF ).",
"Analysis of $\\dot{W}_k^2$ and $\\dot{W}_p$ as well as the convergence to a point are same as those in the proof of Theorem REF , which are omitted here."
]
] | 1709.01543 |
[
[
"Auto-G-Computation of Causal Effects on a Network"
],
[
"Abstract Methods for inferring average causal effects have traditionally relied on two key assumptions: (i) the intervention received by one unit cannot causally influence the outcome of another; and (ii) units can be organized into non-overlapping groups such that outcomes of units in separate groups are independent.",
"In this paper, we develop new statistical methods for causal inference based on a single realization of a network of connected units for which neither assumption (i) nor (ii) holds.",
"The proposed approach allows both for arbitrary forms of interference, whereby the outcome of a unit may depend on interventions received by other units with whom a network path through connected units exists; and long range dependence, whereby outcomes for any two units likewise connected by a path in the network may be dependent.",
"Under network versions of consistency and no unobserved confounding, inference is made tractable by an assumption that the network's outcome, treatment and covariate vectors are a single realization of a certain chain graph model.",
"This assumption allows inferences about various network causal effects via the auto-g-computation algorithm, a network generalization of Robins' well-known g-computation algorithm previously described for causal inference under assumptions (i) and (ii)."
],
[
"1. INTRODUCTION",
"Statistical methods for inferring average causal effects in a population of units have traditionally assumed (i) that the outcome of one unit cannot be influenced by an intervention received by another, also known as the no-interference assumption [7], [38]; and (ii) that units can be organized into non-overlapping groups, blocks or clusters such that outcomes of units in separate groups are independent and the number of groups grows with sample size.",
"Only fairly recently has causal inference literature formally considered settings where assumption (i) does not necessarily hold [40], [37], [17], [16], [12], [28], [43].",
"Early work on relaxing assumption (i) considered blocks of non-overlapping units, where assumptions (i) and (ii) held across blocks, but not necessarily within blocks.",
"This setting is known as partial interference [40], [16], [17], [43], [24], [26], [9].",
"More recent literature has sought to further relax the assumption of partial interference by allowing the pattern of interference to be somewhat arbitrary [46], [2], [25], [41], while still restricting a unit's set of interfering units to be a small set defined by spacial proximity or network ties, as well as severely limiting the degree of outcome dependence in order to facilitate inference.",
"A separate strand of work has primarily focused on detection of specific forms of spillover effects in the context of an experimental design in which the intervention assignment process is known to the analyst [1], [5], [3].",
"In much of this work, outcome dependence across units can be left fairly arbitrary, therefore relaxing (ii), without compromising validity of randomization tests for spillover effects.",
"Similar methods for non-experimental data, such as observational studies, are not currently available.",
"Another area of research which has recently received increased interest in the interference literature concerns the task of effect decomposition of the spillover effect of an intervention on an outcome known to spread over a given network into so-called contagion and infectiousness components [44].",
"The first quantifies the extent to which an intervention received by one person may prevent another person's outcome from occurring because the intervention prevents the first from experiencing the outcome and thus somehow from transmitting it to another [44], [30], [39].",
"The second quantifies the extent to which even if a person experiences the outcome, the intervention may impair his or her ability to transmit the outcome to another.",
"A prominent example of such queries corresponds to vaccine studies for an infectious disease [44], [30], [39].",
"In this latter strand of work, it is typically assumed that interference and outcome dependence occur only within non-overlapping groups, and that the number of independent groups is large.",
"We refer the reader to [43], [45], and [14] for extensive overviews of the fast growing literature on interference and spillover effects.",
"An important gap remains in the current literature: no general approach exists which can be used to facilitate the evaluation of spillover effects on a single network in settings where treatment outcome relationships are confounded, unit interference may be due not only to immediate network ties but also from indirect connections (friend of a friend, and so on) in a network, and non-trivial dependence between outcomes may exist for units connected via long range indirect relationships in a network.",
"The current paper aims to fill this important gap in the literature.",
"Specifically, in this paper, the outcome experienced by a given unit could in principle be influenced by an intervention received by a unit with whom no direct network tie exists, provided there is a path of connected units linking the two.",
"Furthermore, the approach developed in this paper respects a fundamental feature of outcomes measured on a network, by allowing for an association of outcomes for any two units connected by a path on the network.",
"Although network causal effects are shown to in principle be nonparametrically identified by a network version of the g-formula [34] under standard assumptions of consistency and no unmeasured confounding adapted to the network setting, statistical inference is however intractable given the single realization of data observed on the network and lack of partial interference assumption.",
"Nonetheless, progress is made by an assumption that network data admit a representation as a graphical model corresponding to chain graphs [22].",
"This graphical representation of network data generalizes that introduced in [39] for the purpose of interrogating causal effects under partial interference and it is particularly fruitful in the setting of a single network as it implies, under fairly mild positivity conditions, that the outcomes observed on the network may be viewed as a single realization of a certain conditional Markov random field (MRF); and that the set of confounders likewise constitute a single realization of an MRF.",
"By leveraging the local Markov property associated with the resulting chain graph which we encode in non-lattice versions of Besag's auto-models [4], we develop a certain Gibbs sampling algorithm which we call the auto-g-computation algorithm as a general approach to evaluate network effects such as direct and spillover effects.",
"Furthermore, we describe corresponding statistical techniques to draw inference which appropriately account for interference and complex outcome dependence across the network.",
"Auto-g-computation may be viewed as a network generalization of Robins' well-known g-computation algorithm previously described for causal inference under no-interference and i.i.d data [34].",
"We also note that while MRFs have a longstanding history as models for network data starting with [4] (see also [20] for a textbook treatment and summary of this literature), a general chain graph representation of network data appears not to have previously been used in the context of interference and this paper appears to be the first instance of their use in conjunction with g-computation in a formal counterfactual framework for inferring causal effects from observational network data.",
"[31] have recently proposed in parallel to this work, an alternative approach for evaluating causal effects on a single realization of a network, which is based on traditional causal directed acyclic graphs (DAG) and their algebraic representation as causal structural equation models.",
"As discussed in [22], such alternative representation as a DAG will generally be incompatible with our chain graph representation and therefore the respective contribution of these two manuscripts present little to no overlap.",
"Specifically, similar to our setting, [31] allow for a single realization of the network which is fully observed; however, they assume (i) an underlying nonparametric structural equation model with independent error terms [32] compatible with a certain DAG generated the network data.",
"This assumption implies a large number of cross-world counterfactual independences which are largely unnecessary for identification but inherent to their model [33].",
"Furthermore, (ii) their approach precludes any dependence between outcomes not directly connected on the network nor does it allow for interference between units which are not network ties.",
"Finally, (iii) inferences are primarily based on an assumption that outcome errors for the network are conditionally independent given baseline characteristics.",
"Our proposed approach do not require any of assumptions (i)-(iii).",
"The remainder of this paper is organized as followed.",
"In Section 2 we present notation used throughout.",
"In Section 3 we review notions of direct and spillover effects which arise in the presence of interference.",
"In this same section, we review sufficient conditions for identification of network causal effects by a network version of the g-formula, assuming the knowledge of the observed data distribution, or (alternatively) infinitely many realizations from this distribution.",
"We then argue that the network g-formula cannot be empirically identified nonparametrically in more realistic settings where a single realization of the network is observed.",
"To remedy this difficulty, we leverage information encoding network ties (which we assume is both available and accurate) to obtain a chain graph representation of observed variables for units of the network.",
"This chain graph is then shown to induce conditional independences which allow versions of coding and pseudo maximum likelihood estimators due to [4] to be used to make inferences about the parameters of the joint distribution of the observed data sample.",
"These estimators are described in Section 4, for parametric auto-models of [4].",
"The resulting parametrization is then used to make inferences about network causal effects via a specialized Gibbs sampling algorithm we have called the auto-g-computation algorithm, also described in Section 4.",
"In Section 5, we describe results from a simulation study evaluating the performance of the proposed approach.",
"Finally, in Section 6, we offer some concluding remarks and directions for future research.",
"Suppose one has observed data on a population of $N$ interconnected units.",
"Specifically, for each $i\\in \\lbrace 1,\\ldots N\\rbrace $ one has observed $(A_{i},Y_{i})$ , where $A_{i}$ denotes the binary treatment or intervention received by unit $i$ , and $Y_{i}$ is the corresponding outcome.",
"Let $\\mathbf {A}\\equiv (A_{1},\\ldots ,A_{N})$ denote the vector of treatments all individuals received, which takes values in the set $\\lbrace 0,1\\rbrace ^{N},$ and $\\mathbf {A}_{-j}\\equiv (A_{1},\\ldots A_{N})\\backslash A_{j}\\equiv (A_{1},\\ldots ,A_{j-1},A_{j+1},\\ldots A_{N})$ denote the $N-1$ subvector of $\\mathbf {A}$ with the $jth$ entry deleted.",
"In general, for any vector $\\mathbf {X=}\\left( X_{i},...,X_{N}\\right) ,$ $\\mathbf {X}_{-j}=(X_{1},...,X_{N})\\backslash X_{j}=(X_{1},...,X_{j-1},X_{j+1},...,X_{N}).$ Likewise if $X_{i}=(X_{1,i},...,X_{p,i})$ is a vector with $p$ components, $X_{\\backslash s,i}=(X_{1,i},...,X_{s-1,i},X_{s+1,i},...,X_{p,i}).$ Following [40] and [17], we refer to $\\mathbf {A}$ as an intervention, treatment or allocation program, to distinguish it from the individual treatment $A_{i}.\\ $ Furthermore, for $n=1,2,\\ldots ,$ we define $\\mathcal {A(}n)$ as the set of vectors of possible treatment allocations of length $n$ ; for instance $\\mathcal {A(}2)\\equiv \\left\\lbrace (0,0),(0,1),(1,0),(1,1)\\right\\rbrace .$ Therefore, $\\mathbf {A}$ takes one of $2^{N}$ possible values in $\\mathcal {A(}N)$ , while $\\mathbf {A}_{-j}$ takes values in $\\mathcal {A(}N-1)$ for all $j$ .",
"As standard in causal inference, we assume the existence of counterfactual (potential outcome) data $\\mathbf {Y}(\\mathbf {\\cdot })=\\lbrace Y_{i}(\\mathbf {a}):\\mathbf {a}\\in \\mathcal {A(}N)\\mathcal {\\rbrace }$ where $\\mathbf {Y}(\\mathbf {a})=\\lbrace Y_{1}\\left( \\mathbf {a}\\right) ,\\ldots ,Y_{N}(\\mathbf {a})\\rbrace $ , $Y_{i}\\left( \\mathbf {a}\\right) $ is unit $i^{\\prime }s$ response under treatment allocation $\\mathbf {a}$ ; and that the observed outcome $Y_{i}$ for unit $i$ is equal to his counterfactual outcome $Y_{i}\\left( \\mathbf {A}\\right) $ under the realized treatment allocation $\\mathbf {A;}$ more formally, we assume the network version of the consistency assumption in causal inference: $\\mathbf {Y}\\left( \\mathbf {A}\\right) =\\mathbf {Y}\\text{ a.e.}",
"$ Notation for the random variable $Y_{i}(\\mathbf {a})$ makes explicit the possibility of the potential outcome for unit $i$ depending on treatment values of other units, that is the possibility of interference.",
"The standard no-interference assumption [7], [38] made in the causal inference literature, namely that for all $j$ if $\\mathbf {a}$ and $\\mathbf {a}^{\\prime }$ are such that $a_{j}=a_{j}^{\\prime }$ then $Y_{j}\\left(\\mathbf {a}\\right) =Y_{j}\\left( \\mathbf {a}^{\\prime }\\right) $ a.e., implies that the counterfactual outcomes for individual $j$ can be written in a simplified form as $\\left\\lbrace Y_{j}\\left( a\\right) :a\\in \\lbrace 0,1\\rbrace \\right\\rbrace $ .",
"The partial interference assumption [40], [17], [43], which weakens the no-interference assumption, assumes that the $N$ units can be partitioned into $K$ blocks of units, such that interference may occur within a block but not between blocks.",
"Under partial interference, $Y_{i}\\left( \\mathbf {a}\\right) =Y_{i}\\left(\\mathbf {a}^{\\prime }\\right) $ a.s. only if $a_{j}=a_{j}^{\\prime }$ for all $j$ in the same block as unit $i.$ The assumption of partial interference is particularly appropriate when the observed blocks are well separated by space or time such as in certain group randomized studies in the social sciences, or community-randomized vaccine trials.",
"[2] relaxed the requirement of non-overlapping blocks, and allowed for more complex patterns of interference across the network.",
"Obtaining identification required a priori knowledge of the “interference set,” that is for each unit $i$ , the knowledge of the set of units $\\left\\lbrace j:Y_{i}\\left( \\mathbf {a}\\right) \\ne Y_{i}\\left( \\mathbf {a}^{\\prime }\\right)\\text{ a.s. if }a_{k}=a_{k}^{\\prime }\\text{ and }a_{j}\\ne a_{j}^{\\prime }\\text{ for all }k\\ne j\\right\\rbrace $ .",
"In addition, the number of units interfering with any given unit had to be negligible relative to the size of the network.",
"See [25] for closely related assumptions.",
"In contrast to existing approaches, our approach allows full rather than partial interference in settings where treatments are also not necessarily randomly assigned.",
"The assumptions that we make can be separated into two parts: network versions of standard causal inference assumptions, given below, and independence restrictions placed on the observed data distribution which can be described by a graphical model, described in more detail later.",
"We assume that for each $\\mathbf {a}\\in \\mathcal {A(}N)$ the vector of potential outcomes $\\mathbf {Y}(\\mathbf {a})$ is a single realization of a random field.",
"In addition to treatment and outcome data, we suppose that one has also observed a realization of a (multivariate) random field $\\mathbf {L}=\\left(L_{1},\\ldots ,L_{N}\\right) ,$ where $L_{i}$ denotes pre-treatment covariates for unit $i$ .",
"For identification purposes, we take advantage of a network version of the conditional ignorability assumption about treatment allocation which is analogous to the standard assumption often made in causal inference settings; specifically, we assume that: $\\mathbf {A}\\perp \\!\\!\\!\\perp \\mathbf {Y}(\\mathbf {a})|\\mathbf {L}\\text{ for all}\\mathbf {a}\\in \\mathcal {A(}N),\\text{ } $ This assumption basically states that all relevant information used in generating the treatment allocation whether by a researcher in an experiment or by \"nature\" in an observational setting, is contained in $\\mathbf {L.}$ Network ignorability can be enforced in an experimental design where treatment allocation is under the researcher's control.",
"On the other hand, the assumption cannot be ensured to hold in an observational study since treatment allocation is no longer under experimental control, in which case credibility of the assumption depends crucially on subject matter grounds.",
"Equation $\\left( \\ref {NetIgn}\\right) $ simplifies to the standard assumption of no unmeasured confounding in the case of no interference and i.i.d.",
"unit data, in which case $A_{i}\\perp \\!\\!\\!\\perp Y_{i}\\left( a\\right) |L_{i}$ for all $a\\in \\left\\lbrace 0,1\\right\\rbrace $ .",
"$\\ $ Finally, we make the following positivity assumption at the network treatment allocation level: $f\\left( \\mathbf {a}|\\mathbf {L}\\right) >\\sigma >0\\text{ a.e.",
"for all}\\mathbf {a}\\in \\mathcal {A(}N).",
"$ We will consider a variety of network causal effects that are expressed in terms of unit potential outcome expectations $\\psi _{i}\\left( \\mathbf {a}\\right) =E\\left( Y_{i}\\left( \\mathbf {a}\\right) \\right) ,$ $i=1,...,N.$ Let $\\psi _{i}\\left( \\mathbf {a}_{-i},a_{i}\\right) =E\\left( Y_{i}\\left(\\mathbf {a}_{-i},a_{i}\\right) \\right) $ The following definitions are motivated by analogous definitions for fixed counterfactuals given in [17].",
"The first definition gives the average direct causal effect for unit $i$ upon changing the unit's treatment status from inactive ($a=0)$ to active ($a=1)$ while setting the treatment received by other units to $\\mathbf {a}_{-i}:$ $DE_{i}\\left( \\mathbf {a}_{-i}\\right) \\equiv \\psi _{i}\\left( \\mathbf {a}_{-i},a_{i}=1\\right) -\\psi _{i}\\left( \\mathbf {a}_{-i},a_{i}=0\\right) ;$ The second definition gives the average spillover (or “indirect\") causal effect experienced by unit $i$ upon setting the unit's treatment inactive, while changing the treatment of other units from inactive to $\\mathbf {a}_{-i}:$ $IE_{i}\\left( \\mathbf {a}_{-i}\\right) \\equiv \\psi _{i}\\left( \\mathbf {a}_{-i},a_{i}=0\\right) -\\psi _{i}\\left( \\mathbf {a}_{-i}=\\mathbf {0},a_{i}=0\\right) ;$ Similar to [17] these effects can be averaged over a hypothetical allocation regime $\\pi _{i}\\left( \\mathbf {a}_{-i};\\alpha \\right)$ indexed by $\\alpha $ to obtain allocation-specific unit average direct and spillover effects $DE_{i}\\left( \\alpha \\right) =\\sum _{\\mathbf {a}_{-i}\\in \\mathcal {A(}N)}\\pi _{i}\\left( \\mathbf {a}_{-i};\\alpha \\right) DE_{i}\\left(\\mathbf {a}_{-i}\\right) $ and $IE_{i}\\left( \\alpha \\right)=\\sum _{\\mathbf {a}_{-i}\\in \\mathcal {A(}N)}\\pi _{i}\\left( \\mathbf {a}_{-i};\\alpha \\right) IE_{i}\\left( \\mathbf {a}_{-i}\\right) ,$ respectively.",
"One may further average over the units in the network to obtain allocation-specific network average direct and spillover effects $DE\\left( \\alpha \\right) =$ $N^{-1}\\sum _{i}DE_{i}\\left( \\alpha \\right) $ and $IE\\left( \\alpha \\right) =$ $N^{-1}\\sum _{i}IE_{i}\\left( \\alpha \\right) $ , respectively.",
"These quantities can further be used to obtain other related network effects such as average total and overall effects at the unit or network level analogous to [17] and [43].",
"Identification of these effects follow from identification of $\\psi _{i}\\left(\\mathbf {a}\\right) $ for each $i=1,...,N.$ In fact, under assumptions $\\left(\\ref {NetCons}\\right) $ -$\\left( \\ref {positivity}\\right) ,$ it is straightforward to show that $\\psi _{i}\\left( \\mathbf {a}\\right) $ is given by a network version of Robins' g-formula: $\\psi _{i}\\left( \\mathbf {a}\\right)=\\beta _{i}\\left( \\mathbf {a}\\right) $ where $\\beta _{i}\\left( \\mathbf {a}\\right) \\equiv \\sum _{\\mathbf {l}}E\\left( Y_{i}|\\mathbf {A=a,L=l}\\right)f\\left( \\mathbf {l}\\right) \\mathbf {,}$ $f\\left( \\mathbf {l}\\right) $ is the density of $\\mathbf {l,}$ and $\\sum $ may be interpreted as integral when appropriate.",
"Although $\\psi _{i}\\left( \\mathbf {a}\\right) $ can be expressed as the functional $\\beta _{i}\\left( \\mathbf {a}\\right) $ of the observed data law, $\\beta _{i}\\left( \\mathbf {a}\\right) $ cannot be identified nonparametrically from a single realization $(\\mathbf {Y,A,L)}$ drawn from this law without imposing additional assumptions.",
"In the absence of interference, it is standard to rely on the additional assumption that $(Y_{i},A_{i},L_{i}\\mathbf {)}$ , $i=1,...N~$ are i.i.d., in which case the above g-formula reduces to the standard g-formula $\\beta _{i}\\left( \\mathbf {a}\\right)=\\beta \\left( a_{i}\\right) =\\sum _{l}E\\left( Y_{i}|A_{i}=a_{i},L_{i}=l\\right) f(l\\mathbf {)\\,\\ }$ which is nonparametrically identified [34].",
"Since we consider a sample of interconnected units in a network, the i.i.d.",
"assumption is unrealistic.",
"Below, we consider assumptions on the observed data law that are much weaker, but still allow inferences about network effects to be made.",
"We first introduce a convenient representation of $E\\left( Y_{i}|\\mathbf {A=a,L=l}\\right) $ , and describe a corresponding Gibbs sampling algorithm which could in principle be used to compute the network g-formula under the unrealistic assumption that the observed data law is known.",
"First, note that $\\beta _{i}\\left( \\mathbf {a}\\right) =\\sum _{\\mathbf {y,l}}y_{i}f\\left( \\mathbf {y|A=a,L=l}\\right) f\\left( \\mathbf {l}\\right) .$ Suppose that one has available the conditional densities (also referred to as Gibbs factors) $f\\left( Y_{i}\\mathbf {|Y}_{-i}=\\mathbf {y}_{-i},\\mathbf {a,l}\\right) $ and $f\\left( L_{i}\\mathbf {|L}_{-i}\\mathbf {=l}_{-i}\\right) $ , $i=1,...,N,\\,$ and that it is straightforward to sample from these densities.",
"Then, evaluation of the above formula for $\\beta _{i}\\left( \\mathbf {a}\\right)$ can be achieved with the following Gibbs sampling algorithm.",
"Gibbs Sampler I: $\\text{for }m & =0,\\text{let }\\left( \\mathbf {L}^{(0)},\\mathbf {Y}^{(0)}\\right) \\text{ denote initial values ;}\\\\\\text{for }m & =0,...,M\\\\& \\text{let }i=(m\\mod {N})+1;\\\\& \\text{draw }L_{i}^{(m+1)}\\text{ from }f\\left( L_{i}\\mathbf {|L}_{-i}^{(m)}\\right) \\text{ and }Y_{i}^{(m+1)}\\text{ from }f\\left( Y_{i}\\mathbf {|Y}_{-i}^{(m)},\\mathbf {a,L}^{(m)}\\right) ;\\\\& \\text{let }\\mathbf {L}_{-i}^{(m+1)}\\left.",
"=\\right.",
"\\mathbf {L}_{-i}^{(m)}\\text{ and }\\mathbf {Y}_{-i}^{(m+1)}\\left.",
"=\\right.",
"\\mathbf {Y}_{-i}^{(m)}.\\\\&$ The sequence $\\left( \\mathbf {L}^{(0)},\\mathbf {Y}^{(0)}\\right) ,\\left(\\mathbf {L}^{(1)},\\mathbf {Y}^{(1)}\\right) ,\\ldots ,\\left( \\mathbf {L}^{(m)},\\mathbf {Y}^{(m)}\\right) $ forms a Markov chain, which under appropriate regularity conditions converges to the stationary distribution $f\\left( \\mathbf {Y|a,L}\\right) \\times f\\left( \\mathbf {L}\\right) $ [23].",
"Specifically, we assume $M$ is an integer larger than the number of transitions necessary for the appropriate Markov chain to reach equilibrium from the starting state.",
"Thus, for sufficiently large $m^{\\ast }$ and $K,$ $\\beta _{i}\\left( \\mathbf {a}\\right) \\approx K^{-1}\\sum _{k=0}^{K}Y_{i}^{(m^{\\ast }+k)}.$ Thus, if Gibbs factors $f\\left( Y_{i}\\mathbf {|Y}_{-i}=\\mathbf {y}_{-i},\\mathbf {a,l}\\right) $ and $f\\left( L_{i}\\mathbf {|L}_{-i}\\mathbf {=l}_{-i}\\right) $ are available for every $i$ , all networks causal effects can be computed.",
"This approach to evaluating the g-formula is the network analogue of Monte Carlo sampling approaches to evaluating functionals arising from the g-computation algorithm in the sequentially ignorable model, see for instance [47].",
"Unfortunately these factors are not identified from a single realization of the observed data law, without additional assumptions.",
"In the following section we describe additional assumptions which will imply identification.",
"To motivate our approach, we introduce a representation for network data proposed by [39] and based on chain graphs.",
"A chain graph (CG) [21] is a mixed graph containing undirected ($-$ ) and directed ($\\rightarrow $ ) edges with the property that it is impossible to add orientations to undirected edges in such a way as to create a directed cycle.",
"A chain graph without undirected edges is called a directed acyclic graph (DAG).",
"A statistical model associated with a CG $\\mathcal {G}$ with a vertex set $\\mathbf {O}$ is a set of densities that obey the following two level factorization: $p(\\mathbf {O}) = \\prod _{\\mathbf {B} \\in \\mathcal {B}(\\mathcal {G})} p(\\mathbf {B}\\mid \\text{pa}_{\\mathcal {G}}(\\mathbf {B})),$ where $\\mathcal {B}(\\mathcal {G})$ is the partition of vertices in $\\mathcal {G}$ into blocks, or sets of connected components via undirected edges, and $\\text{pa}_{\\mathcal {G}}(\\mathbf {B})$ is the set $\\lbrace W : W \\rightarrow B\\in \\mathbf {B} \\text{ exists in }\\mathcal {G} \\rbrace $ .",
"This outer factorization resembles the Markov factorization of DAG models.",
"Furthermore, each factor $p(\\mathbf {B} \\mid \\text{pa}_{\\mathcal {G}}(\\mathbf {B}))$ obeys the following inner factorization, which is a clique factorization for a conditional Markov random field: $p(\\mathbf {B} \\mid \\text{pa}_{\\mathcal {G}}(\\mathbf {B})) = \\frac{1}{Z(\\text{pa}_{\\mathcal {G}}(\\mathbf {B}))} \\prod _{\\mathbf {C} \\in \\mathcal {C}(\\mathcal {G}^{a}_{\\mathbf {B} \\cup \\text{pa}_{\\mathcal {G}}(\\mathbf {B}))});\\mathbf {C} \\lnot \\subseteq \\text{pa}_{\\mathcal {G}}(\\mathbf {B})} \\phi _{\\mathbf {C}}(\\mathbf {C}),$ where $Z(\\text{pa}_{\\mathcal {G}}(\\mathbf {B}))$ is a normalizing function which ensures a valid conditional density, $\\mathcal {C}(\\mathcal {G})$ is a set of maximal pairwise connected components (cliques) in an undirected graph $\\mathcal {G}$ , $\\phi _{\\mathcal {C}}(\\mathbf {C})$ is a mapping from values of $\\mathbf {C}$ to real numbers, and $\\mathcal {G}^{a}_{\\mathbf {B} \\cup \\text{pa}_{\\mathcal {G}}(\\mathbf {B}))}$ is an undirected graph with vertices $\\mathbf {B} \\cup \\text{pa}_{\\mathcal {G}}(\\mathbf {B})$ and an edge between any pair in $\\text{pa}_{\\mathcal {G}}(\\mathbf {B})$ and any pair in $\\mathbf {B}\\cup \\text{pa}_{\\mathcal {G}}(\\mathbf {B})$ adjacent in $\\mathcal {G}$ .",
"A density $p(\\mathbf {O})$ that obeys the two level factorization given by (REF ) and (REF ) with respect to a CG $\\mathcal {G}$ is said to be Markov relative to $\\mathcal {G}$ .",
"This factorization implies a number of Markov properties relating conditional independences in $p(\\mathbf {O})$ and missing edges in $\\mathcal {G}$ .",
"Conversely, these Markov properties imply the factorization under an appropriate version of the Hammersley-Clifford theorem, which does not hold for all densities, but does hold for wide classes of densities, which includes positive densities [15].",
"Special cases of these Markov properties are described further below.",
"Details can be found in [21].",
"Observed data distributions entailed by causal models of a DAG do not necessarily yield a good representation of network data.",
"This is because DAGs impose an ordering on variables that is natural in temporally ordered longitudinal studies but not necessarily in network settings.",
"As we now show the Markov property associated with CGs accommodates both dependences associated with causal or temporal orderings of variables, but also symmetric dependences induced by the network.",
"Let $\\mathcal {E}$ denote the set of neighboring pairs of units in the network; that is $(i,j)\\in \\mathcal {E}$ only if units $i$ and $j$ are directly connected on the network.",
"We represent data $\\mathbf {O}$ drawn from a joint distribution associated with a network with neighboring pairs $\\mathcal {E}$ as a CG $\\mathcal {G}_{\\mathcal {E}}$ in which each variable corresponds to a vertex, and directed and undirected edges of $\\mathcal {G}_{\\mathcal {E}}$ are defined as follows.",
"For each pair of units $(i,j)\\in \\mathcal {E}$ , variables $L_{i}$ and $L_{j}$ are connected by an undirected edge in $\\mathcal {G}_{\\mathcal {E}}$ .",
"We use an undirected edge to represent the fact that $L_{i}$ and $L_{j}$ are associated, but this association is not in general due to unobserved common causes, nor as the variables are contemporaneous can they be ordered temporally or causally [39].",
"Vertices for $A_{i}$ and $A_{j},$ and $Y_{i}$ and $Y_{j}\\,$ are likewise connected by an undirected edge in $\\mathcal {G}_{\\mathcal {E}}$ if and only if $(i,j)\\in $ $\\mathcal {E}$ .",
"Furthermore, for each $(i,j)\\in $ $\\mathcal {E}$ , a directed edge connects $L_{i}$ to both $A_{i}$ and $A_{j}$ encoding the fact that covariates of a given unit may be direct causes of the unit's treatment but also of the neighbor treatments, i.e.",
"$L_{i}\\rightarrow $ $\\left\\lbrace A_{i},A_{j}\\right\\rbrace ;$ edges $L_{i}\\rightarrow $ $\\left\\lbrace Y_{i},Y_{j}\\right\\rbrace $ and $A_{i}\\rightarrow $ $\\left\\lbrace Y_{i},Y_{j}\\right\\rbrace $ should be added to the chain graph for a similar reason.",
"As an illustration, the CG in Figure 1 corresponds to a three-unit network where $\\mathcal {E=}\\left\\lbrace \\left( 1,2\\right) ,\\left( 2,3\\right) \\right\\rbrace $ .",
"Figure: Chain graph representation of data from a network of three unitsWe will assume the observed data distribution on $\\mathbf {O}$ associated with our network causal model is Markov relative to the CG constructed from unit connections in a network via the above two level factorization [21].",
"This implies the observed data distribution obeys certain conditional independence restrictions that one might intuitively expect to hold in a network, and which serve as the basis of the proposed approach.",
"Let $\\mathcal {N}_{i}$ denote the set of neighbors of unit $i,$ i.e.",
"$\\mathcal {N}_{i}=\\left\\lbrace j:\\left( i,j\\right) \\in \\mathcal {E}\\right\\rbrace $ , and let $\\mathcal {O}_{i}=\\left\\lbrace \\mathbf {O}_{j},j\\in \\mathcal {N}_{i}\\right\\rbrace $ denote data observed on all neighbors of unit $i.$ Given a CG $\\mathcal {G}_{\\mathcal {E}}$ with associated neighboring pairs $\\mathcal {E}$ , the following conditional independences follow by the global Markov property associated with CGs [21]: $Y_{i} & \\perp \\!\\!\\!\\perp \\lbrace Y_{k},A_{k},L_{k}\\rbrace |(A_{i},L_{i},\\mathcal {O}_{i})\\ \\text{for all }i\\text{ and }k, \\text{ }k\\ne i;\\text{ }\\\\\\text{ }L_{i} & \\perp \\!\\!\\!\\perp \\mathbf {L}_{-i}\\setminus \\mathcal {O}_{i}|\\mathbf {L}_{-i}\\cap \\mathcal {O}_{i}\\text{ for all }i\\text{ and }k\\notin \\mathcal {N}_{i},\\text{ }k\\ne i.",
"$ In words, equation (REF ) states that the outcome of a given unit can be screened-off (i.e.",
"made independent) from the variables of all non-neighboring units by conditioning on the unit's treatment and covariates as well as on all data observed on its neighboring units, where the neighborhood structure is determined by $\\mathcal {G}_{\\mathcal {E}}$ .",
"That is $(A_{i},L_{i},\\mathcal {O}_{i})$ is the Markov blanket of $Y_{i}$ in CG $\\mathcal {G}_{\\mathcal {E}}$ .",
"This assumption, coupled with a sparse network structure leads to extensive dimension reduction of the model specification for $\\mathbf {Y|A,L}$ .",
"In particular, the conditional density of $Y_{i}|\\left\\lbrace \\mathbf {O\\backslash }Y_{i}\\right\\rbrace $ only depends on $\\left(A_{i},L_{i}\\right) $ and on neighbors' data $\\mathcal {O}_{i}.$ Similarly, $\\mathbf {L}_{-i}\\cap \\mathcal {O}_{i}$ is the Markov blanket of $L_{i}$ in CG $\\mathcal {G}_{\\mathcal {E}}$ .",
"Suppose that instead of $\\left( \\ref {positivity}\\right) $ , the following stronger positivity condition holds: $\\mathbb {P}\\left( \\mathbf {O=o}\\right) >0,\\text{ for all possible values}\\mathbf {o.}",
"$ Since $\\left( \\ref {Markov}\\right) $ holds for the conditional law of $\\mathbf {Y}$ given $\\mathbf {A},\\mathbf {L}$ , it lies in the conditional MRF (CMRF) model associated with the induced undirected graph $\\mathcal {G}_{\\mathcal {E}}^{a}$ .",
"In addition, since $\\left( \\ref {positivie}\\right) $ holds, the conditional MRF version of the Hammersley-Clifford (H-C) theorem and $\\left( \\ref {Markov}\\right) $ imply the following version of the clique factorization in (REF ), $f\\left( \\mathbf {y|a,l}\\right) =\\left( \\frac{1}{\\kappa \\left( \\mathbf {a,l}\\right) }\\right) \\exp \\left\\lbrace U\\left( \\mathbf {y;a,l}\\right) \\right\\rbrace ,$ where $\\kappa \\left( \\mathbf {a,l}\\right) =\\sum _{\\mathbf {y}}\\exp \\left\\lbrace U\\left( \\mathbf {y;a,l}\\right) \\right\\rbrace ,$ and $U\\left( \\mathbf {y;a,l}\\right) $ is a conditional energy function which can be decomposed into a sum of terms called conditional clique potentials, with a term for every maximal clique in the graph $\\mathcal {G}_{\\mathcal {E}}^{a}$ [4].Conditional clique potentials offer a natural way to specify a CMRF using only terms that depend on a small set of variables.",
"Specifically, $& f\\left( Y_{i}=y_{i}|\\mathbf {Y}_{-i}=\\mathbf {y}_{-i},\\mathbf {a,l}\\right)\\nonumber \\\\& =f\\left( Y_{i}=y_{i}|\\mathbf {Y}_{-i}=\\mathbf {y}_{-i},\\left\\lbrace a_{j}\\mathbf {,}l_{j}:j\\in \\mathcal {N}_{i}\\right\\rbrace \\text{ }\\right) \\nonumber \\\\& =\\frac{\\exp \\left\\lbrace \\sum _{c\\in \\mathcal {C}_{i}}U_{c}\\left( \\mathbf {y;a,l}\\right) \\right\\rbrace }{\\sum _{\\mathbf {y}^{\\prime }\\mathbf {:y}_{-i}^{\\prime }=\\mathbf {y}_{-i}}\\exp \\left\\lbrace \\sum _{c\\in \\mathcal {C}_{i}}U_{c}\\left(\\mathbf {y}^{\\prime }\\mathbf {;a,l}\\right) \\right\\rbrace }, $ where $\\mathcal {C}_{i}$ are all maximal cliques of $\\mathcal {G}_{\\mathcal {E}}^{a}$ that involve $Y_{i}$ .",
"Gibbs densities specified as in $\\left( \\ref {gibbs factor}\\right) $ is a rich class of densities, and are often regularized in practice by setting to zero conditional clique potentials for cliques of size greater than a pre-specified cut-off.",
"This type of regularization corresponds to setting higher order interactions terms to zero in log-linear models.",
"For instance, closely following [4], one may introduce conditions (a) only cliques $c\\in \\mathcal {C}$ of size one or two have non-zero potential functions $U_{c},$ and (b) the conditional probabilities in $\\left( \\ref {gibbs factor}\\right) $ have an exponential family form.",
"Under these additional conditions, given $\\mathbf {a,l,}$ the energy function takes the form $U\\left( \\mathbf {y;a,l}\\right) =\\sum _{i\\in \\mathcal {G}_{\\mathcal {E}}}y_{i}G_{i}\\left( y_{i}\\mathbf {;a,l}\\right) +\\sum _{\\left\\lbrace i,j\\right\\rbrace \\in \\mathcal {E}}y_{i}y_{j}\\theta _{ij}\\left( \\mathbf {a,l}\\right) ,$ for some functions $G_{i}\\left( \\cdot \\mathbf {;a,l}\\right) $ and coefficients $\\theta _{ij}\\left( \\mathbf {a,l}\\right) .$ Note that in order to be consistent with local Markov conditions $\\left( \\ref {Markov})\\text{ and(}\\ref {Markovii}\\right) ,G_{i}\\left( \\cdot \\mathbf {;a,l}\\right) $ can only depend on $\\left\\lbrace \\left( a_{s},l_{s}\\right) :s\\in \\mathcal {N}_{j}\\right\\rbrace ,$ while because of symmetry $\\theta _{ij}\\left( \\mathbf {a,l}\\right)$ can depend at most on $\\left\\lbrace \\left( a_{s},l_{s}\\right):s\\in \\mathcal {N}_{j}\\cap \\mathcal {N}_{i}\\right\\rbrace $ .",
"Following [4], we call the resulting class of models conditional auto-models.",
"Conditions $\\left( \\ref {Markovii}\\right) $ and $\\left( \\ref {positivie}\\right) $ imply that $\\mathbf {L}$ is an MRF; standard Hammersley-Clifford theorem further implies that the joint density of $\\mathbf {L}$ can be written as $f\\left( \\mathbf {l}\\right) =\\left( \\frac{1}{\\nu }\\right) \\exp \\left\\lbrace W\\left( \\mathbf {l}\\right) \\right\\rbrace $ where $\\nu =\\sum _{\\mathbf {l}^{\\prime }}\\exp \\left\\lbrace W\\left( \\mathbf {l}^{\\prime }\\right) \\right\\rbrace $ , and $W\\left( \\mathbf {l}\\right) $ is an energy function which can be decomposed as a sum over cliques in the induced undirected graph $(\\mathcal {G}_{\\mathcal {E}})_{\\mathbf {L}}$ .",
"Analogous to the conditional auto-model described above, we restrict attention to densities of $\\mathbf {L}$ of the form: $W\\left( \\mathbf {L}\\right) =\\sum _{i\\in \\mathcal {G}_{\\mathcal {E}}}\\left\\lbrace \\sum _{k=1}^{p}L_{k,i}H_{k,i}\\left( L_{k,i}\\right) +\\sum _{k\\ne s}\\rho _{k,s,i}L_{k,i}L_{s,i}\\right\\rbrace +\\sum _{\\left\\lbrace i,j\\right\\rbrace \\in \\mathcal {E}}\\sum _{k=1}^{p}\\sum _{s=1}^{p}\\omega _{k,s,i,j}L_{k,i}L_{s,j},$ for some functions $H_{k,i}\\left( L_{k,i}\\right) $ and coefficients $\\rho _{k,s,i},\\omega _{k,s,i,j}.$ Note that $\\rho _{k,s,i}$ encodes the association between covariate $L_{k,i}$ and covariate $L_{s,i}$ observed on unit $i,$ while $\\omega _{k,s,i,j}$ captures the association between $L_{k,i}$ observed on unit $i$ and $L_{s,j}$ observed on unit $j.$ A prominent auto-regression model for binary outcomes is the so-called auto-logistic regression first proposed by [4].",
"Note that as $\\left(\\mathbf {a,l}\\right) $ is likely to be high dimensional, identification and inference about $G_{i}$ and $\\theta _{ij}$ requires one to further restrict heterogeneity by specifying simple low dimensional parametric models for these functions of the form$:$ $G_{i}\\left( y_{i}\\mathbf {;a,l}\\right) & =\\widetilde{G}_{i}\\left(\\mathbf {a,l}\\right) =\\mathrm {log}\\frac{\\Pr \\left( Y_{i}=1|\\mathbf {a,l,Y}_{-i}=0\\right) }{\\Pr \\left( Y_{i}=0|\\mathbf {a,l,Y}_{-i}=0\\right) }\\\\& =\\beta _{0}+\\beta _{1}a_{i}+\\beta _{2}^{\\prime }l_{i}+\\beta _{3}\\sum _{j\\in \\mathcal {N}_{i}}w_{ij}^{a}a_{j}+\\beta _{4}^{\\prime }\\sum _{j\\in \\mathcal {N}_{i}}w_{ij}^{l}l_{j};\\\\\\theta _{ij} & =w_{ij}^{y}\\theta ,$ where $w_{ij}^{a}$ , $w_{ij}^{l}$ , $w_{ij}^{y}$ are user specified weights which may depend on network features associated with units $i$ and $j$ , with $\\sum _{j}w_{ij}^{a}=\\sum _{j}w_{ij}^{l}=\\sum _{j}w_{ij}^{y}=1;$ e.g.",
"$w_{ij}^{a}=1/\\mathrm {card}\\left( \\mathcal {N}_{i}\\right) $ standardizes the regression coefficient by the size of a unit's neighborhood.",
"We assume model parameters $\\tau =\\left( \\beta _{0},\\beta _{1},\\beta _{2}^{\\prime },\\beta _{3},\\beta _{4}^{\\prime },\\theta \\right) $ are shared across units in a network.",
"In addition, network features can be incorporated into the auto-models as model parameters, which may be desirable in settings where network features are confounders for the relationship between exposure and outcome.",
"For example, one could further adjust for a unit's degree (i.e.",
"number of ties).",
"For a continuous outcome, an auto-Gaussian model may be specified as followed: $G_{i}\\left( y_{i}\\mathbf {;a,l}\\right) & =-\\left( \\frac{1}{2\\sigma _{y}^{2}}\\right) (y_{i}-2\\mu _{y,i}\\left( \\mathbf {a,l}\\right) );\\\\\\mu _{y,i}\\left( \\mathbf {a,l}\\right) & =\\beta _{0}+\\beta _{1}a_{i}+\\beta _{2}^{\\prime }l_{i}+\\beta _{3}\\sum _{j\\in \\mathcal {N}_{i}}w_{ij}^{a}a_{j}+\\beta _{4}^{\\prime }\\sum _{j\\in \\mathcal {N}_{i}}w_{ij}^{l}l_{j};\\\\\\theta _{ij} & =w_{ij}^{y}\\theta ,$ where $\\mu _{y,i}\\left( \\mathbf {a,l}\\right) =E\\left( Y_{i}|\\mathbf {a,l,Y}_{-i}=0\\right) $ , and $\\sigma _{y}^{2}=\\mathrm {var}\\left( Y_{i}|\\mathbf {a,l,Y}_{-i}=0\\right) $ .",
"Similarly, model parameters $\\tau _{Y}=\\left( \\beta _{0},\\beta _{1},\\beta _{2}^{\\prime },\\beta _{3},\\beta _{4}^{\\prime },\\sigma _{y}^{2},\\theta \\right) $ are shared across units in the network.",
"Other auto-models within the exponential family can likewise be conditionally specified, e.g.",
"the auto-Poisson model.",
"Auto-model density of $\\mathbf {L}$ is specified similarly.",
"For example, fix parameters in (REF ) $\\rho _{k,s,i} & =\\rho _{k,s},\\\\\\omega _{k,s,i,j} & =\\widetilde{\\omega }_{k,s}v_{i,j},$ where $v_{i,j}$ is a user-specified weight which satisfies $\\sum _{j}v_{i,j}=1$ .",
"For $L_{k}$ binary, one might take $H_{k,i}\\left( L_{k,i};\\tau _{k}\\right) =\\tau _{k}=\\mathrm {log}\\frac{\\Pr \\left( L_{k,i}=1|L_{\\backslash k,i}=0\\mathbf {,L}_{-i}=0\\right) }{\\Pr \\left( L_{k,i}=0|L_{\\backslash k,i}=0\\mathbf {,L}_{-i}=0\\right) },$ corresponding to a logistic auto-model for $L_{k,i}|L_{\\backslash k,i}=0\\mathbf {,L}_{-i}=0,$ while for continuous $L_{k}$ $H_{k,i}\\left( L_{k,i};\\tau _{k}=\\left( \\sigma _{k}^{2},\\mu _{k}\\right)\\right) =-\\left( \\frac{1}{2\\sigma _{k}^{2}}\\right) (L_{k,i}-2\\mu _{k}),$ corresponding to a Gaussian auto-model for $L_{k,i}|L_{\\backslash k,i}=0\\mathbf {,L}_{-i}=0.$ As before, model parameters $\\tau _{L}=(\\tau _{1}^{\\prime },...,\\tau _{p}^{\\prime })$ are shared across units in the network.",
"Suppose that one has specified auto-models for $\\mathbf {Y}$ and $\\mathbf {L}$ as in the previous section with unknown parameters $\\tau _{Y}$ and $\\tau _{L}$ respectively.",
"To estimate these parameters, one could in principle attempt to maximize the corresponding joint likelihood function.",
"However, such task is well-known to be computationally daunting as it requires a normalization step which involves evaluating a high dimensional sum or integral which, outside relatively simple auto-Gaussian models is generally not available in closed form.",
"For example, to evaluate the conditional likelihood of $\\mathbf {Y|A,L}$ for binary $Y$ requires evaluating a sum of $2^{N}$ terms in order to compute $\\kappa \\left( \\mathbf {A,L}\\right) \\mathbf {.",
"}$ Fortunately, less computationally intensive strategies for estimating auto-models exist including pseudo-likelihood estimation and so called-coding estimators [4], which may be adopted here.",
"We first consider coding-type estimators, mainly because unlike pseudo-likelihood estimation, standard asymptotic theory applies.",
"To describe these estimators in more detail requires additional definitions.",
"We define a stable set or independent set on $\\mathcal {G}_{\\mathcal {E}}$ as the set of nodes, $\\mathcal {S}\\left(\\mathcal {G}_{\\mathcal {E}}\\right)$ , such that $ (i,j) \\notin \\mathcal {E} \\ \\forall (i,j) \\in \\mathcal {S}\\left(\\mathcal {G}_{\\mathcal {E}}\\right)$ That is, a stable set is a set of nodes with the property that no two nodes in the set have an edge connecting them in the network.",
"The size of a stable set is the number of units it contains.",
"A maximal stable set is a stable set such that no unit in $\\mathcal {G}_{\\mathcal {E}}$ can be added without violating the independence condition.",
"A maximum stable set $\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ is a maximal stable set of largest possible size for $\\mathcal {G}_{\\mathcal {E}}$ .",
"This size is called the stable number or independence number of $\\mathcal {G}_{\\mathcal {E}}$ , which we denote $n_{1,N}=n_{1}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ .",
"A maximum stable set is not necessarily unique in a given graph, and finding one such set and enumerating them all is challenging but a well-studied problem of computer science.",
"In fact, finding a maximum stable set is a well-known NP-complete problem.",
"Nevertheless, both exact and approximate algorithms exist that are computationally more efficient than an exhaustive search.",
"Exact algorithms which identify all maximum stable sets were described in [36], [27], [10].",
"Unfortunately, exact algorithms for finding maximum stable sets quickly become computationally prohibitive with moderate to large networks.",
"In fact, the maximum stable set problem is known not to have an efficient approximation algorithm unless P=NP [48].",
"A practical approach we take in this paper is to simply use an enumeration algorithm that lists a collection of maximal stable sets [29], and pick the largest of the maximal sets found.",
"Let $\\Xi _{1}=\\left\\lbrace \\mathcal {S}_{\\max }\\left(\\mathcal {G}_{\\mathcal {E}}\\right) :\\mathrm {card}\\left( \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) \\right) =\\mathrm {n}_{1}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) \\right\\rbrace $ denote the collection of all maximum (or largest identified maximal) stable sets for $\\mathcal {G}_{\\mathcal {E}}$ .",
"The Markov property associated with $\\mathcal {G}_{\\mathcal {E}}$ implies that outcomes of units within such sets are mutually conditionally independent given their Markov blankets.",
"This implies the (partial) conditional likelihood function which only involves units in the stable set factorizes, suggesting that tools from maximum likelihood estimation may apply.",
"In the Appendix, we establish that this is in fact the case, in the sense that under certain regularity conditions, coding maximum likelihood estimators of $\\tau $ based on maximum (or largest identified maximal) stable sets are consistent and asymptotically normal (CAN).",
"Consider the coding likelihood functions for $\\tau _{Y}$ and $\\tau _{L}$ based on a stable set $\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) \\in $ $\\Xi _{1}$ : $\\mathcal {CL}_{Y}\\left( \\tau _{Y}\\right) & ={\\displaystyle \\prod \\limits _{i\\in \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }}\\mathcal {L}_{Y,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right),i}\\left( \\tau _{Y}\\right) ={\\displaystyle \\prod \\limits _{i\\in \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }}f\\left( Y_{i}|\\mathcal {O}_{i},A_{i},L_{i};\\tau _{Y}\\right) ;\\\\\\mathcal {CL}_{L}\\left( \\tau _{L}\\right) & ={\\displaystyle \\prod \\limits _{i\\in \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }}\\mathcal {L}_{L,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right),i}\\left( \\tau _{L}\\right) ={\\displaystyle \\prod \\limits _{i\\in \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }}f\\left( L_{i}|\\left\\lbrace L_{j}:j\\in \\mathcal {N}_{i}\\right\\rbrace ;\\tau _{L}\\right) .$ The estimators $\\widehat{\\tau }_{Y}=\\arg \\max _{\\tau _{Y}}\\log \\mathcal {CL}_{Y}\\left( \\tau _{Y}\\right) $ and $\\widehat{\\tau }_{L}=\\arg \\max _{\\tau _{L}}\\log \\mathcal {CL}_{L}\\left( \\tau _{L}\\right) $ are analogous to Besag's coding maximum likelihood estimators.",
"Consider a network asymptotic theory according to which $\\lbrace \\mathcal {G}_{\\mathcal {E}_{N}}: N\\rbrace $ is a sequence of chain graphs as $N\\rightarrow \\infty ,$ with vertices $(\\mathbf {A}_{\\mathcal {E}}\\mathbf {,L}_{\\mathcal {E}}\\mathbf {,Y}_{\\mathcal {E}}\\mathbf {)}$ that follow correctly specified auto-models with unknown parameters $\\left( \\tau _{Y},\\tau _{L}\\right) $ , and with edges defined according to a sequence of networks $\\mathcal {E}_{N}$ , $N = 1, 2, \\ldots $ of increasing size.",
"We establish the following result in the Appendix Result 1: Suppose that$~n_{1,N}\\rightarrow \\infty $ as $N\\rightarrow \\infty $ then under conditions 1-6 given in the Appendix, $& \\widehat{\\tau }_{L}\\underset{N\\longrightarrow \\infty }{\\longrightarrow }\\tau \\text{ \\textit {in probability;}}\\widehat{\\tau }_{Y}\\underset{N\\longrightarrow \\infty }{\\longrightarrow }\\tau \\text{ \\textit {inprobability.}",
"}\\\\& \\sqrt{n_{1,N}}\\Gamma _{n_{1,N}}^{1/2}\\left( \\widehat{\\tau }_{L}-\\tau _{L}\\right) \\underset{N\\longrightarrow \\infty }{\\longrightarrow }N\\left(0,I\\right) ;\\\\& \\sqrt{n_{1,N}}\\Omega _{n_{1,N}}^{1/2}\\left( \\widehat{\\tau }_{Y}-\\tau _{Y}\\right) \\underset{N\\longrightarrow \\infty }{\\longrightarrow }N\\left(0,I\\right) ;\\\\\\Gamma _{n_{1,N}} & =\\frac{1}{n_{1,N}}\\sum _{i\\in \\mathcal {S}_{\\max }\\left(\\mathcal {G}_{\\mathcal {E}_{N}}\\right) }\\left\\lbrace \\frac{\\partial \\log \\mathcal {CL}_{L,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}_{N}}\\right) ,i}\\left( \\tau _{L}\\right) }{\\partial \\tau _{L}}\\right\\rbrace ^{\\otimes 2},\\\\\\Omega _{n_{1,N}} & =\\frac{1}{n_{1,N}}\\sum _{i\\in \\mathcal {S}_{\\max }\\left(\\mathcal {G}_{\\mathcal {E}_{N}}\\right) }\\left\\lbrace \\frac{\\partial \\log \\mathcal {CL}_{Y,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}_{N}}\\right) ,i}\\left( \\tau _{Y}\\right) }{\\partial \\tau _{Y}}\\right\\rbrace ^{\\otimes 2}.$ Note that by the information equality, $\\Gamma _{n_{1,N}}$ and $\\Omega _{n_{1,N}}$ can be replaced by the standardized (by $n_{1,N}$ ) negative second derivative matrix of corresponding coding log likelihood functions.",
"Note also that condition $n_{1,N}\\rightarrow \\infty $ as $N\\rightarrow \\infty $ essentially rules out the presence of an ever-growing hub on the network as it expands with $N$ , thus ensuring that there is no small set of units in which majority of connections are concentrated asymptotically.",
"Suppose that each unit on a network of size $N$ is connected to no more than $C_{\\max }<N,$ then according to Brooks' Theorem, the stable number $n_{1,N}$ satisfies the inequalities [6]: $\\frac{N}{C_{\\max }+1}\\le n_{1,N}\\le N.$ This implies that in a network of bounded degree, $n_{1,N}=O\\left( N\\right)$ is guaranteed to be of the same order as the size of the network$;$ however $n_{1,N}$ may grow at substantially slower rates $(n_{1,N}=o(N))$ if $C_{\\max }$ is unbounded.",
"Note that because $L_{i}$ is likely multivariate, further computational simplification can be achieved by replacing $f\\left( L_{i}|\\left\\lbrace L_{j}:j\\in \\mathcal {N}_{i}\\right\\rbrace ;\\tau _{L}\\right) $ with the pseudo-likelihood (PL) function ${\\displaystyle \\prod \\limits _{s=1}^{p}}f\\left( L_{s,i}|L_{\\backslash s,i},\\left\\lbrace L_{j}:j\\in \\mathcal {N}_{i}\\right\\rbrace ;\\tau _{L}\\right)$ in equation $\\left( \\ref {likL}\\right) .$ This substitution is computationally more efficient as it obviates the need to evaluate a multivariate integral in order to normalize the joint law of $L_{i}$ .",
"Let $\\widetilde{\\tau }$ denote the estimator which maximizes the log of the resulting modified coding likelihood function $\\mathcal {L}_{L,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,i}^{\\ast }\\left( \\tau _{L}\\right) .$ It is straightforward using the proof of Result 1 to establish that its covariance may be approximated by the sandwich formula $\\Phi _{L,n_{1,N}}^{-1}\\Gamma _{L,n_{1,N}}\\Phi _{L,n_{1,N}}^{-1}$ [13], where $\\Gamma _{L,n_{1,N}} & =\\frac{1}{n_{1,N}}\\sum _{i\\in \\mathcal {S}_{\\max }\\left(\\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\frac{\\partial \\log \\mathcal {L}_{L,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,i}^{\\ast }\\left( \\tau _{L}\\right) }{\\partial \\tau _{L}}\\right\\rbrace ^{\\otimes 2},\\\\\\Phi _{L,n_{1,N}} & =\\frac{1}{n_{1,N}}\\sum _{i\\in \\mathcal {S}_{\\max }\\left(\\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\frac{\\partial ^{2}\\log \\mathcal {L}_{L,\\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right),i}^{\\ast }\\left( \\tau _{L}\\right) }{\\partial \\tau _{L}\\partial \\tau _{L}^{T}}\\right\\rbrace .$ As later illustrated in extensive simulation studies, coding estimators can be inefficient, since the partial conditional likelihood function associated with coding estimators disregards contributions of units $i\\notin \\mathcal {S}_{\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) .$ Substantial information may be recovered by combining multiple coding estimators each obtained from a separate approximate maximum stable set, however accounting for dependence between the different estimators can be challenging.",
"Pseudo-likelihood (PL) estimation offers a simple alternative approach which is potentially more efficient than either approach described above.",
"PL estimators maximize the log-PLs $\\log \\left\\lbrace \\mathcal {PL}_{Y}\\left( \\tau _{Y}\\right) \\right\\rbrace & ={\\displaystyle \\sum \\limits _{i\\in \\mathcal {G}_{\\mathcal {E}}}}\\log f\\left( Y_{i}|\\mathcal {O}_{i},A_{i},L_{i};\\tau _{Y}\\right) ;\\\\\\log \\left\\lbrace \\mathcal {PL}_{L}\\left( \\tau _{L}\\right) \\right\\rbrace & ={\\displaystyle \\sum \\limits _{i\\in \\mathcal {G}_{\\mathcal {E}}}}\\log f\\left( L_{k,i}| \\left\\lbrace L_{s,i}:s \\in \\lbrace 1,..,p\\rbrace \\setminus k \\right\\rbrace ,\\left\\lbrace L_{s,j}: s \\in \\lbrace 1,..,p\\rbrace , j\\in \\mathcal {N}_{i}\\right\\rbrace ;\\tau _{L}\\right) .$ Denote corresponding estimators $\\check{\\tau }_{Y}$ and $\\check{\\tau }_{L}$ , $\\ $ which are shown to be consistent in the Appendix.",
"There however is generally no guarantee that their asymptotic distribution follows a Gaussian distribution due to complex dependence between units on the network prohibiting application of the central limit theorem.",
"As a consequence, for inference, we recommend using the parametric bootstrap, whereby algorithm Gibbs sampler I of Section 2.2 may be used to generate multiple bootstrap samples from the observed data likelihood evaluated at $\\left( \\check{\\tau }_{Y},\\check{\\tau }_{L}\\right) ,$ which in turn can be used to obtain a bootstrap distribution for $\\left( \\check{\\tau }_{Y},\\check{\\tau }_{L}\\right)$ and corresponding inferences such as bootstrap quantile confidence intervals.",
"We now return to the main goal of the paper, which is to obtain valid inferences about $\\beta _{i}\\left( \\mathbf {a}\\right) .$ The auto-G-computation algorithm entails evaluating $\\widehat{\\beta }_{i}\\left( \\mathbf {a}\\right) \\approx K^{-1}\\sum _{k=0}^{K}\\widehat{Y}_{i}^{(m^{\\ast }+k)},$ where $\\widehat{Y}_{i}^{(m)}$ are generated by Gibbs Sampler I algorithm under posited auto-models with estimated parameters $\\left( \\hat{\\tau }_{Y},\\hat{\\tau }_{L}\\right) .$ An analogous estimator $\\breve{\\beta }_{i}\\left(\\mathbf {a}\\right) $ can be obtained using $\\left( \\check{\\tau }_{Y},\\check{\\tau }_{L}\\right) $ instead of $\\left( \\hat{\\tau }_{Y},\\hat{\\tau }_{L}\\right) .$ In either case, the parametric bootstrap may be used in conjunction with Gibbs Sampler I in order to generate the corresponding bootstrap distribution of estimators of $\\beta _{i}\\left( \\mathbf {a}\\right) $ conditional on either $\\widehat{\\beta }_{i}\\left( \\mathbf {a}\\right) $ or $\\breve{\\beta }_{i}\\left( \\mathbf {a}\\right) $ .",
"Alternatively, a less computationally intensive approach first generates i.i.d.",
"samples $\\tau _{Y}^{(j)}$ and $\\tau _{L}^{(j)}$ $,$ $j=1,...J$ from $N\\left( \\hat{\\tau }_{Y},\\widehat{\\Gamma }_{n_{1,N}}\\right) $ and $N\\left( \\hat{\\tau }_{L},\\widehat{\\Omega }_{n_{1,N}}\\right) $ respectively, conditional on the observed data, where $\\widehat{\\Gamma }_{n_{1,N}}=\\Gamma _{n_{1,N}}\\left(\\widehat{\\tau }_{L}\\right) $ and $\\widehat{\\Omega }_{n_{1,N}}=\\Omega _{n_{1,N}}\\left( \\widehat{\\tau }_{Y}\\right) $ estimate $\\Gamma _{n_{1,N}}$ and $\\Omega _{n_{1,N}}.$ .",
"Next, one computes corresponding estimators $\\widehat{\\beta }_{i}^{(j)}\\left( \\mathbf {a}\\right) $ based on simulated data generated using Gibbs Sampler I algorithm under $\\tau _{Y}^{(j)}$ and $\\tau _{L}^{(j)},$ $j=1,...,J.$ The empirical distribution of $\\left\\lbrace \\widehat{\\beta }_{i}^{(j)}\\left( \\mathbf {a}\\right) :j\\right\\rbrace $ may be used to obtain standard errors for $\\widehat{\\beta }_{i}\\left( \\mathbf {a}\\right)$ , and corresponding Wald type or quantile-based confidence intervals for direct and spillover causal effects.",
"We performed an extensive simulation study to evaluate the performance of the proposed methods on networks of varying density and size.",
"Specifically, we investigated the properties of the coding-type and pseudo-likelihood estimators of unknown parameters $\\tau _{Y}$ and $\\tau _{L}$ indexing the joint observed data likelihood.",
"Additionally, we evaluated the performance of proposed estimators of the network counterfactual mean $\\beta (\\alpha )=N^{-1}\\sum _{i=1}^{N}\\sum _{\\mathbf {a}_{-i}\\in \\mathcal {A(}N)}\\pi _{i}\\left(\\mathbf {a}_{-i};\\alpha \\right) E\\left( Y_{i}\\left( \\mathbf {a}\\right)\\right) ,$ as well as for the direct effect $DE(\\alpha ),$ and the spillover effect $IE(\\alpha )$ , where $\\alpha $ is a specified treatment allocation law described below.",
"We simulated three networks of size 800 with varying densities: low (each node has either 2, 3, or 4 neighbors), medium (each node has either 5, 6, or 7 neighbors), and high (each node has either 8, 9, or 10 neighbors).",
"For reference, a depiction of the low density network of size 800 is given in Figure 2.",
"Additionally, we simulated low density networks of size 200, 400, and 1,000.",
"The network graphs were all simulated in Wolfram Mathematica 10 using the RandomGraph function.",
"For each network, we obtained an (approximate) maximum stable set.",
"The stable sets for the 800 node networks were of size $n_{1,low}=375$ , $n_{1,med}=275$ , $n_{1,high}=224$ .",
"Figure: Network of size 800 with low density For units $i=1,...,N$ , we generated using Gibbs Sampler I$\\ $ a vector of binary confounders $\\lbrace L_{1i},L_{2i},L_{3i}\\rbrace $ , a binary treatment assignment $A_{i}$ , and a binary outcome $Y_{i}$ from the following auto-models consistent with the chain graph induced by the simulated network: $Pr(L_{1,i}=1\\mid \\mathbf {L}_{\\setminus 1,i},\\lbrace \\mathbf {L}_{1,j}:j\\in \\mathcal {N}_{i}\\rbrace ) & =\\mathrm {expit}\\bigg (\\tau _{1}+\\rho _{12}L_{2,i}+\\rho _{13}L_{3,i}+\\nu _{11}\\sum _{j\\in \\mathcal {N}_{i}}L_{1,j}+\\nu _{12}\\sum _{j\\in \\mathcal {N}_{i}}L_{2,j}+\\nu _{13}\\sum _{j\\in \\mathcal {N}_{i}}L_{3,j}\\bigg )\\\\Pr(L_{2,i}=1\\mid \\mathbf {L}_{\\setminus 2,i},\\lbrace \\mathbf {L}_{j}:j\\in \\mathcal {N}_{i}\\rbrace ) & =\\mathrm {expit}\\bigg (\\tau _{2}+\\rho _{12}L_{1,i}+\\rho _{23}L_{3,i}+\\nu _{21}\\sum _{j\\in \\mathcal {N}_{i}}L_{1,j}+\\nu _{22}\\sum _{j\\in \\mathcal {N}_{i}}L_{2,j}+\\nu _{23}\\sum _{j\\in \\mathcal {N}_{i}}L_{3,j}\\bigg )\\\\Pr(L_{3,i}=1\\mid \\mid \\mathbf {L}_{\\setminus 3,i},\\lbrace \\mathbf {L}_{j}:j\\in \\mathcal {N}_{i}\\rbrace ) & =\\mathrm {expit}\\bigg (\\tau _{3}+\\rho _{13}L_{1,i}+\\rho _{23}L_{2,i}+\\nu _{31}\\sum _{j\\in \\mathcal {N}_{i}}L_{1,j}+\\nu _{32}\\sum _{j\\in \\mathcal {N}_{i}}L_{2,j}+\\nu _{33}\\sum _{j\\in \\mathcal {N}_{i}}L_{3,j}\\bigg )$ $Pr(A_{i}=1\\mid L_{i},\\lbrace A_{j},\\mathbf {L}_{j}:j\\in \\mathcal {N}_{i}\\rbrace ) &=\\mathrm {expit}\\bigg (\\gamma _{0}+\\gamma _{1}L_{1,i}+\\gamma _{2}\\sum _{j\\in \\mathcal {N}_{i}}L_{1,j}+\\gamma _{3}L_{2,i}\\\\& \\hspace{56.9055pt}+\\gamma _{4}\\sum _{j\\in \\mathcal {N}_{i}}L_{2,j}+\\gamma _{5}L_{3,i}+\\gamma _{6}\\sum _{j\\in \\mathcal {N}_{i}}L_{3,j}+\\gamma _{7}\\sum _{j\\in \\mathcal {N}_{i}}A_{j}\\bigg )$ $Pr(Y_{i}=1\\mid A_{i},L_{i},\\mathcal {O}_{i}) & =\\mathrm {expit}\\bigg (\\beta _{0}+\\beta _{1}A_{i}+\\beta _{2}\\sum _{j\\in \\mathcal {N}_{i}}A_{j}+\\beta _{3}L_{1,i}+\\beta _{4}\\sum _{j\\in \\mathcal {N}_{i}}L_{1,j}\\\\& \\hspace{56.9055pt}+\\beta _{5}L_{2,i}+\\beta _{6}\\sum _{j\\in \\mathcal {N}_{i}}L_{2,j}+\\beta _{7}L_{3,i}+\\beta _{8}\\sum _{j\\in \\mathcal {N}_{i}}L_{3,j}+\\beta _{9}\\sum _{j\\in \\mathcal {N}_{i}}Y_{j}\\bigg )$ where $\\mathrm {expit}\\left( x\\right) =(1+\\exp \\left( -x\\right))^{-1},\\tau _{L}=\\lbrace \\tau _{1},\\tau _{2},\\tau _{3},\\rho _{12},\\rho _{13},\\rho _{23},\\nu _{11},\\nu _{12},\\nu _{13},\\nu _{22},\\nu _{21},\\nu _{23},\\nu _{33},\\nu _{31},\\nu _{32}\\rbrace $ , $\\tau _{A}=\\lbrace \\gamma _{0},...,\\gamma _{7}\\rbrace $ , and $\\tau _{Y}=\\lbrace \\beta _{0},...,\\beta _{9}\\rbrace $ .",
"We evaluated network average direct and spillover effects via the Gibbs Sampler I algorithm under true parameter values $\\tau _{Y}$ and $\\tau _{L}$ and a treatment allocation, $\\alpha $ given by a binomial distribution with event probability equal to $0.7$ .",
"All parameter values are summarized in Table 1.",
"We generated $S=1,000$ simulations of the chain graph for each of the 4 simulated network structures.",
"For each simulation $s$ , data were generated by running the Gibbs sampler I algorithm $4,000$ times with the first $1,000$ iterations as burn-in.",
"Additionally, we thinned the chain by retaining every third realization to reduce autocorrelation.",
"Table: True parameter values for simulation studyFor each realization of the chain graph, $\\mathcal {G}_{\\mathcal {E}_{N},s}$ , we estimated $\\tau _{Y}$ via coding-type maximum likelihood estimation and $\\tau _{L}$ via the modified coding estimator.",
"Both sets of parameters were also estimated via maximum pseudo-likelihood estimation.",
"For each estimator we computed corresponding causal effect estimators, their standard errors and $95\\%$ Wald confidence intervals as outlined in previous Sections.",
"The estimation of the auto-model parameters was computed in R using functions optim() and glm()(R Core Team, 2013).",
"The network average causal effects were estimated using Gibbs Sampler I using the agcEffect function in the autognet R package by plugging in estimates for $(\\tau _{L},\\tau _{Y})$ using $K=50$ iterations and a burn-in of $m^{\\ast }=10$ iterations.",
"For variance estimation of the coding-type estimator, 200 bootstrap replications were used.",
"Simulation results for the various density networks of size 800 are summarized in Tables 2 and 3 for the following parameters: the network average counterfactual $\\beta (\\alpha ),$ the network average direct effect, and the network average spillover effect.",
"Both coding and pseudo-likelihood estimators had small bias in estimating $\\beta (\\alpha )$ regardless of network density (absolute bias $<0.01$ ).",
"Coverage of the coding estimator ranged between $93.1\\%$ and $94.5\\%$ .",
"Biases were also small for both spillover and direct effects: the bias slightly increased with network density, but still stayed below an absolute bias of $0.01$ .",
"Coverage of coding-based confidence intervals for direct effects ranged from $92.5\\%$ to $95.6\\%$ , while the coverage for spillover effects decreased slightly with network density from $93.7\\%$ to $92.2\\%$ .",
"It is important to note that as the network structure changes with network size and density, the corresponding estimated parameters likewise vary and therefore it is not necessarily straightforward to compare performance of the methodology across network structure.",
"Table 3 gives the MC variance for the pseudo-likelihood estimator which confirms greater efficiency compared to the coding estimator given the significantly larger effective sample size used by pseudo-likelihood.",
"Appendix Tables 1-3 report bias and coverage for the network causal effect parameters for low density networks of size 200, 400, and 1,000.",
"Additionally, Appendix Figures 1 and 2 report bias and coverage for all 25 auto-model parameters in the low-density network of size 800.",
"As predicted by theory, coding-type and pseudo-likelihood estimators exhibit small bias.",
"Additionally, coding-type estimators had approximately correct coverage, while pseudo-likelihood estimators had coverage substantially lower than the nominal level for a number of auto-model parameters.",
"These results confirm the anticipated failure of pseudo-likelihood estimators to be asymptotically Gaussian.",
"Most notably, the coverage for the outcome auto-model coefficient capturing dependence on neighbors' outcomes $\\beta _{9}$ was $81\\%$ , while coverage of the coding-type Wald CI for this coefficient was $94\\%$ .",
"Although not shown here, the coverage results for the auto-model parameters are consistent across all simulations.",
"We also assessed the performance of auto-g-computation in small, dense networks and in the presence of missing network edges.",
"For the first, we generated one network of size 100 ($n_{1,100} = 25$ ) and an additional network of size 200 ($n_{1,200} = 57$ ).",
"For the network of size 100, coding estimation of auto-model parameters in 437 of the 1,000 simulated samples had convergence issues due to the small size of the maximal independent set.",
"Excluding results with convergence issues, the causal estimates were biased and did not have correct coverage (see Appendix Figure 3a).",
"The performance for the network of size 200 was much improved across these endpoints, though oftentimes the confidence intervals were too wide to be informative.",
"In both cases, the pseudo-likelihood estimator exhibited less bias than the coding estimator.",
"In the previously described dense network of size 800, we randomly removed 564 (14%) of edges.",
"The estimated parameters from the auto-models were unbiased and had correct coverage (see Appendix Figure 4).",
"However, the causal estimates for both the coding and pseudo-likelihood estimators exhibited bias, and the coding estimator had coverage slightly below the nominal level with the estimated spillover effect shifted towards null (see Appendix Figure 5).",
"Table: Simulation results of coding based estimators of network causaleffects for networks of size 800 by densityTable: Simulation results of pseudo-likelihood based estimators of networkcausal effects for networks of size 800 by densityWe consider an application of the auto-g-computation algorithm to the Networks, Norms, and HIV/STI Risk Among Youth (NNAHRAY) study to assess the effect of past incarceration on infection with HIV, STI, or Hepatitis C accounting for the network structure [18].",
"The NNAHRAY study was conducted in a New York neighborhood with epidemic HIV and widespread drug use from 2002-2005 [11].",
"Through in-person interviews, information was collected regarding the respondents' demographic characteristics, incarceration history, sexual partnerships and histories, and past drug use.",
"At the time of the interviews, respondents were also tested for HIV, gonorrhea, chlamydia, Herpes Simple Virus (HSV) 2, Hepatitis C virus (HCV), and syphilis.",
"The study population we consider includes all interviewed persons with recorded results from their HIV, STI, and HCV tests ($n=8$ persons missing) for a total sample size of $N=457$ persons.",
"We assume that HIV/STI/HCV status is missing completely at random.",
"We defined a network tie (i.e.",
"edge) as a sexual and/or injection drug use partnership in the past three months if at least one of the partners reported the relationship.",
"The network structure is given in Figure 3.",
"The number of partners (i.e.",
"neighbors) for each respondent varied from none to 10 resulting in a maximal independent set of $n_1 = 274$ .",
"We estimated the network-level spillover and direct effect of past incarceration on infection with HIV, STI, or Hepatitis C (HCV) under a Bernoulli allocation strategy with treatment probability equal to 0.50.",
"Past incarceration was defined as any amount of jail time in the respondents' history.",
"We accounted for confounding by Latino/a ethnicity, age, education, and past illicit drug use.",
"The same models and estimation procedure detailed in the simulation section were utilized; note that $\\nu _{ij}$ where $i \\ne j$ were assumed to be 0.",
"For comparison, the auto-model parameters were estimated using the coding-type and pseudolikelihood estimators.",
"Network average spillover and direct effects were restricted to persons with at least one network tie.",
"Table 4 gives the outcome auto-model parameter point estimates for the coding and pseudolikelihood estimators with 95% confidence intervals for the coding estimators excluding the covariate terms.",
"Due to scaling by number of network ties, the outcome and exposure influence of network ties can be interpreted as the effect of average covariate value among network ties.",
"Individuals who experienced prior incarceration had 2.12 [95% CI: 1.07-4.21] times the odds of infection with HIV/STI/HCV compared to those without prior incarceration.",
"However, the incarceration status of network ties was not significantly associated with a person's risk of HIV/STI/HCV (OR= 1.21 [95% CI: 0.52-2.84]) conditional on the neighbors' outcomes.",
"Individual's with a greater proportion of their ties infected with HIV, STI, and/or HCV were much more likely to be infected with HIV, STI, and/or HCV (OR=3.07 [95% CI: 1.33-7.09]).",
"The pseudolikelihood point estimates were similar to the coding results.",
"The full results for auto-model parameters from both the covariate and outcome model are given in Appendix Figure 6.",
"The network average direct effect is 0.14 [95% CI: 0.02-0.28] when the proportion of persons with prior history of incarceration is 0.50.",
"There was no significant evidence of a spillover effect of incarceration on HIV/STI/HCV risk over the network, as increasing the proportion of persons with a history of incarceration from 0 to 0.50 resulted in a negligible increase in average HIV/STI/HCV risk of a person with no prior incarceration [$\\widehat{DE}$ =0.04; 95% CI: -0.06-0.14].",
"In the Appendix, we have included two alternate outcome auto-model specifications that incorporate the number of sexual and injection drug use partners for each person in the network.",
"In an infection disease setting, the number of partners should in principle be accounted for in the analysis as it is likely a confounder for the effect of incarceration (both individual and neighbors' status) on infection status [19].",
"As shown in the Appendix, adjusting for the number of network ties (e.g sexual and injection drug partners) did not change our conclusions.",
"Following a reviewer's recommendation, we performed a simulation study based on the NNAHRAY network under under the sharp null and verified that we have valid inference in this setting.",
"Results are provided in the Appendix Table 7.",
"Figure: Network graph from the NNAHRAY data (N=457) with individuals in the maximal independent set (n 1 =274n_1 = 274) in blue.Table: Outcome auto-model parameters estimates for coding and pseudolikelihood estimators (excluding covariates) on the odds ratio scaleWe have described a new approach for evaluating causal effects on a network of connected units.",
"Our methodology relies on the crucial assumption that accurate information on network ties between observed units is available to the analyst, which may not always be the case in practice.",
"In fact, as demonstrated in our simulation study, bias may ensue if information about the network is incomplete, and therefore omits to account for all existing ties.",
"In future work, we plan to further develop our methods to appropriately account for uncertainty about the underlying network structure.",
"Another limitation of the proposed approach is that it relies heavily on parametric assumptions and as a result may be open to bias due to model mis-specification.",
"Although this limitation also applies to standard g-computation for i.i.d settings which nevertheless has gained prominence in epidemiology [42], [35], [8], our parametric auto-models which are inherently non-i.i.d may be substantially more complex, as they must appropriately account both for outcome and covariate dependence, as well as for interference.",
"Developing appropriate goodness-of-fit tests for auto-models is clearly a priority for future research.",
"In addition, to further alleviate concerns about modeling bias, we plan in future work to extend semiparametric models such as structural nested models to the network context.",
"Such developments may offer a real opportunity for more robust inference about network causal effects.",
"Acknowledgments: We are grateful to Dr. Samuel R. Friedman at National Development and Research Institutes, Inc. for access to the Networks, Norms, and HIV/STI Risk Among Youth study data and contributions to the data application section.",
"Appendix: Theorems, detailed proofs, and additional simulation results (.zip file) Code: Code for estimation and inference of network causal effects.",
"To download, please visit: https://isabelfulcher.github.io/autoGnetworks/ (R) Appendix Throughout, we assume that we observe a vector of Random Fields $\\left(\\mathbf {Y}_{N},\\mathbf {A}_{N},\\mathbf {L}_{N}\\right) ,$ such that $\\mathbf {Y}_{N}$ is a conditional Markov Random Field (MRF) given $(\\mathbf {A}_{N},\\mathbf {L}_{N})$ on the sequence of Chain Graphs (CG) $\\mathcal {G}_{\\mathcal {E}}$ associated with an increasing sequence of networks $\\mathcal {E}_{N},$ with distribution uniquely specified by the parametric model for its Gibbs factors $f\\left( Y_{i}|\\partial _{i};\\tau _{Y}\\right) $ where $\\partial _{i}=\\left( \\mathcal {O}_{i},A_{i},L_{i}\\right) $ for all $i=1,...,N.$ Let $\\mathcal {N}_{i}^{(k)}$ denote the $kth$ order neighborhood of unit $i,$ defined as followed: $\\mathcal {N}_{i}^{(1)}=$ $\\mathcal {N}_{i},$ $\\mathcal {N}_{i}^{(2)}={\\displaystyle \\bigcup \\limits _{j\\in \\mathcal {N}_{i}^{(1)}}}\\mathcal {N}_{j}^{(1)}\\backslash \\left( \\mathcal {N}_{i}^{(1)}\\cup \\lbrace i\\rbrace \\right),...,\\mathcal {N}_{i}^{(k)}={\\displaystyle \\bigcup \\limits _{j\\in \\mathcal {N}_{i}^{(k-1)}}}\\mathcal {N}_{j}^{(1)}\\backslash \\left({\\displaystyle \\bigcup \\limits _{s\\le k-1}}\\mathcal {N}_{i}^{(s)}\\cup \\lbrace i\\rbrace \\right) .$ A k-stable set $\\mathcal {S}^{(k)}\\left( \\mathcal {G}_{\\mathcal {E}},k\\right) $ of $\\mathcal {G}_{\\mathcal {E}}$ is a set of units $\\left( i,j\\right) $ in $\\mathcal {G}_{\\mathcal {E}}$ such that ${\\displaystyle \\bigcup \\limits _{s\\le k}}\\mathcal {N}_{i}^{(s)}$ and ${\\displaystyle \\bigcup \\limits _{s\\le k}}\\mathcal {N}_{j}^{(s)}$ are not neighbors$.$ The size of a k-stable set is the number of units it contains.",
"A maximal k-stable set is a k-stable set such that no unit in $\\mathcal {G}_{\\mathcal {E}}$ can be added without violating the independence condition.",
"A maximum k-stable set $\\mathcal {S}_{k,\\max }\\left(\\mathcal {G}_{\\mathcal {E}}\\right) $ is a maximal k-stable set of largest possible size for $\\mathcal {G}_{\\mathcal {E}}$ .",
"This size is called the k-stable number of $\\mathcal {G}_{\\mathcal {E}}$ , which we denote $n_{k,N}=n_{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ .",
"Let $\\Xi _{k}=\\left\\lbrace \\mathcal {S}_{k,\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) :\\mathrm {card}\\left( \\mathcal {S}_{k,\\max }\\left( \\mathcal {G}_{\\mathcal {E}}\\right) \\right)=\\mathrm {s}_{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) \\right\\rbrace $ denote the collection of all (approximate) maximum k-stable sets for $\\mathcal {G}_{\\mathcal {E}}$ .",
"The unknown parameter $\\tau _{Y}$ is in the interior of a compact $\\Theta \\subset \\mathbb {R}^{p}.$ Let $\\partial \\partial _{i}=\\left\\lbrace \\mathcal {O}_{j},A_{j},L_{j}:j\\in \\mathcal {N}_{i}^{(2)}\\right\\rbrace \\cup \\left\\lbrace A_{j},L_{j}:j\\in \\mathcal {N}_{i}^{(3)}\\right\\rbrace \\cup \\left\\lbrace L_{j}:j\\in \\mathcal {N}_{i}^{4)}\\right\\rbrace .$ Let $\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;\\partial _{i}\\right) =-E\\left\\lbrace \\log \\frac{f\\left( Y_{i}|\\mathcal {O}_{i},A_{i},L_{i};t\\right) }{f\\left(Y_{i}|\\mathcal {O}_{i},A_{i},L_{i};\\tau _{Y}\\right) }|\\partial _{i}\\right\\rbrace \\ge 0;$ We make the following assumptions: Regularity conditions: Suppose that $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ is a 1-stable set.",
"$\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ can be partitioned into $K$ 4-stable subsets $\\left\\lbrace \\mathcal {S}^{k}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) :k=1,...,K\\right\\rbrace $ such that $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ={\\displaystyle \\bigcup \\limits _{k=1}^{K}}\\mathcal {S}^{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) .$ Let $n^{(k)}$ denote the number of units in $\\mathcal {S}^{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ and $n=\\sum _{k}n^{(k)}$ denote the total number of units in $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) .$ We further assume that there is a $k_{0}$ such that: $\\lim \\inf _{\\mathcal {E}}$ $n^{(k_{0})}/n>0$ and $\\lim \\inf _{\\mathcal {E}}$ $n/N>0.$ Let $\\mathcal {D}_{i}$ denote the support $\\partial _{i}=\\left(\\mathcal {O}_{i},A_{i},L_{i}\\right) .$ The joint support $\\mathcal {D=}{\\displaystyle \\prod \\limits _{j\\in \\mathcal {N}_{i}}}\\mathcal {D}_{j}$ is a fixed space of neighborhood configurations when $i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) .$ $\\exists c>0$ such that for all $i\\in \\mathcal {S}^{k_{0}}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) ,$ we have for all values y$_{i},$ $\\partial _{i},\\partial \\partial _{i}$ and $t\\in \\Theta ,$ $f\\left( y_{i}|\\partial _{i};t_{y}\\right) & >c\\\\f\\left( \\partial _{i}|\\partial \\partial _{i},t_{y}\\right) & >c$ w.here $f\\left( y_{i}|\\partial _{i};t_{y}\\right) $ and $f\\left( \\partial _{i}|\\partial \\partial _{i},t_{y}\\right) $ are density functions where $f\\left( y_{i}|\\partial _{i};t_{y}\\right) $ is uniformly (in $i,y_{i},\\partial _{i})$ continuous in $t_{y.",
"}$ There exists $\\mathcal {M}_{y}\\left( \\tau _{Y},t;z\\right) \\ge 0$ with $\\left( t,z\\right) \\in \\Theta \\times \\mathcal {D}$ , $\\mu -$ integrable for all $t$ such that: $\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;z\\right) \\ge \\mathcal {M}_{Y}\\left( \\tau _{Y},t;z\\right) $ if $i\\in \\mathcal {S}^{k_{0}}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) .$ $t\\rightarrow N\\left( \\tau _{Y},t\\right) =\\int _{\\mathcal {D}}\\mathcal {M}_{Y}\\left( \\tau _{Y},t;z\\right) \\mu \\left( dz\\right) $ is continuous and has a unique minimum at $t=\\tau _{Y}.$ For all $i\\in \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,$ $f\\left( y_{i}|\\partial _{i};t_{y}\\right) $ admits three continuous derivatives at $t_{y}$ in a neighborhood of $\\tau _{Y}$ , and $1/f\\left(y_{i}|\\partial _{i};t_{y}\\right) $ and the $vth$ derivatives $f^{(v)}\\left(y_{i}|\\partial _{i};t_{y}\\right) $ $v=1,2,3$ are uniformly bounded in $i,y_{i},\\partial _{i}$ and $t_{y}$ in a neighborhood of $\\tau _{Y}.$ There exist a positive-definite symmetric non-random $p\\times p$ matrix $I\\left( \\tau _{Y}\\right) $ such that $\\lim \\inf _{\\mathcal {E}}\\Omega _{n_{1,N}}\\ge I\\left( \\tau _{Y}\\right)$ Theorem 1 Under assumptions 1-4, the maximum coding likelihood estimator on $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ is consistent as $N\\rightarrow \\infty ,$ that is $\\hat{\\tau }_{Y}\\rightarrow \\tau _{Y}\\text{ in probability }$ where $\\hat{\\tau }_{Y}=\\arg \\max _{t}\\sum _{i\\in \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\log f\\left( Y_{i}|\\mathcal {O}_{i},A_{i},L_{i};t\\right)$ Proof.",
"Define $U_{\\mathcal {E}}^{k_{0}}\\left( t\\right) =-\\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\log f\\left( Y_{i}|\\partial _{i};t\\right)$ and write $Z_{\\mathcal {E}}=-\\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\log \\frac{f\\left( Y_{i}|\\partial _{i};t\\right) }{f\\left( Y_{i}|\\partial _{i};\\tau _{Y}\\right)}+\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;\\partial _{i}\\right) \\right\\rbrace +\\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;\\partial _{i}\\right)$ As the term in curly braces is the sum of centered random variables with bounded variance and conditionally independent given $\\left\\lbrace \\partial _{i}:i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right)\\right\\rbrace ,$ $\\lim _{N\\rightarrow \\infty }\\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\log \\frac{f\\left(Y_{i}|\\partial _{i};t\\right) }{f\\left( Y_{i}|\\partial _{i};\\tau _{Y}\\right)}+\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;\\partial _{i}\\right) \\right\\rbrace =0\\text{}a.s.$ Then we deduce the following sequence of inequalities almost surely: $\\lim \\inf _{\\mathcal {E}}Z_{\\mathcal {E}} & =\\lim \\inf _{\\mathcal {E}}\\left\\lbrace \\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\mathcal {M}_{Y,i}\\left( \\tau _{Y},t;\\partial _{i}\\right) \\right\\rbrace \\\\& \\ge \\lim \\inf _{\\mathcal {E}}\\int _{\\mathcal {D}}\\mathcal {M}_{Y}\\left( \\tau _{Y},t;z\\right) F_{n}\\left( \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,dz\\right)$ where $F_{n}\\left( \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right),dz\\right) =\\frac{1}{n^{(k_{0})}}\\sum _{i\\in \\mathcal {S}^{k_{0}}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) }1\\left( \\partial i\\in dz\\right)$ By assumptions 1-3 and Lemma 5.2.2 of Guyon (1995), then there is a positive constant $c^{\\ast }$ such that $\\lim \\inf _{\\mathcal {E}}F_{n}\\left( \\mathcal {S}^{k_{0}}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,dz\\right) \\ge c^{\\ast }\\lambda \\left( dz\\right)$ therefore $\\lim \\inf _{\\mathcal {E}}Z_{\\mathcal {E}} & \\ge c^{\\ast }\\int _{\\mathcal {D}}\\mathcal {M}_{Y}\\left( \\tau _{Y},t;z\\right) \\lambda \\left( dz\\right) \\\\& \\equiv c^{\\ast }N\\left( \\tau _{Y},t\\right)$ Then note that $U_{\\mathcal {E}}\\left( t\\right) & =-\\frac{1}{n}\\sum _{i\\in \\mathcal {S}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) }\\log f\\left( Y_{i}|\\partial _{i};t\\right)\\\\& =\\sum _{k=1}^{K}\\frac{n^{(k)}}{n}U_{\\mathcal {E}}^{k}\\left( t\\right)$ where $U_{\\mathcal {E}}^{k}\\left( t\\right) =-\\frac{1}{n^{(k)}}\\sum _{i\\in \\mathcal {S}^{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\log f\\left(Y_{i}|\\partial _{i};t\\right)$ Consistency of the coding maximum likelihood estimator then follows by assumption 1-4 and Corollary 3.4.1 of Guyon (1995).",
"Consistency of the pseudo maximum likelihood estimator follows from $U_{\\mathcal {E}}^{P}\\left( t\\right) & \\equiv -\\frac{1}{N}\\sum _{i}\\log f\\left( Y_{i}|\\partial _{i};t\\right) \\\\& =\\frac{n}{N}U_{\\mathcal {E}}\\left( t\\right) +\\frac{\\left( N-n\\right)}{N}\\overline{U}_{\\mathcal {E}}\\left( t\\right)$ $U_{\\mathcal {E}}^{P}\\left( t\\right) \\equiv \\left( N-n\\right) ^{-1}\\sum _{i\\notin \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\log f\\left( Y_{i}|\\partial _{i};t\\right) $ and a further application of Corollary 3.4.1 of Guyon (1995).",
"The next result establishes asymptotic normality of $\\hat{\\tau }_{Y}.$ Theorem 2 Under Assumptions 1-6, as $N\\rightarrow \\infty $, $& \\sqrt{n}\\Omega _{n}^{1/2}\\left( \\widehat{\\tau }_{Y}-\\tau _{Y}\\right)\\underset{N\\longrightarrow \\infty }{\\longrightarrow }N\\left( 0,I\\right) ;\\\\\\Omega _{n} & =\\frac{1}{n}\\sum _{i\\in \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\frac{\\partial \\log \\mathcal {CL}_{Y,\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,i}\\left( \\tau _{Y}\\right)}{\\partial \\tau _{Y}}\\right\\rbrace ^{\\otimes 2}.$ Proof.",
"$\\hat{\\tau }_{Y}$ solves $V_{n}\\left( \\hat{\\tau }_{Y}\\right) =\\frac{\\partial \\log \\mathcal {CL}_{Y}\\left( \\tau _{Y}\\right) }{\\partial \\tau _{Y}}|_{\\hat{\\tau }_{Y}}=0,$ therefore $0 & =\\sqrt{n}V_{n}\\left( \\tau _{Y}\\right) +\\dot{V}_{n}\\left( \\tau _{Y},\\hat{\\tau }_{Y}\\right) \\sqrt{n}\\left( \\hat{\\tau }_{Y}-\\tau _{Y}\\right) \\\\V_{n}\\left( \\tau _{Y}\\right) & =\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\mathcal {D}_{i}=\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) }\\left\\lbrace \\frac{\\partial \\log \\mathcal {CL}_{Y,\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,i}\\left( \\tau _{Y}\\right) }{\\partial \\tau _{Y}}\\right\\rbrace \\\\\\dot{V}_{n}\\left( \\tau _{Y},\\hat{\\tau }_{Y}\\right) & =\\int _{0}^{1}\\frac{\\partial ^{2}\\log \\mathcal {CL}_{Y}\\left( \\tau _{Y}\\right) }{\\partial \\tau _{Y}\\partial \\tau _{Y}^{T}}|_{t\\left( \\hat{\\tau }_{Y}-\\tau _{Y}\\right)+\\tau _{Y}}dt$ As the variables $\\mathcal {D}_{i}$ are centered, bounded and independent conditionally on $\\left\\lbrace \\partial _{i}:i\\in \\mathcal {S}^{k_{0}}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) \\right\\rbrace ,$ one may apply a central limit theorem for non-iid bounded variable (Breiman, 1992).",
"Under assumptions 5 and 6, it follows that $\\dot{V}_{n}\\left( \\tau _{Y},\\hat{\\tau }_{Y}\\right)+\\Omega _{n}\\left( \\tau _{Y}\\right) \\rightarrow ^{P}0_{p\\times p},$ proving the result.",
"Proofs of consistency of coding and pseudo maximum likelihood estimators of $\\tau _{L}$ , as well as asymptotic normality of coding estimator of $\\tau _{L}$ follow along the same lines as above, upon substituting $\\partial _{i}=\\left\\lbrace L_{j}:j\\in \\mathcal {N}_{i}\\right\\rbrace ,$ and replacing Assumption 2 with the assumption that $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) $ can be partitioned into $K$ 2-stable subsets $\\left\\lbrace \\mathcal {S}^{k}\\left(\\mathcal {G}_{\\mathcal {E}}\\right) :k=1,...,K\\right\\rbrace $ such that $\\mathcal {S}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ={\\displaystyle \\bigcup \\limits _{k=1}^{K}}\\mathcal {S}^{k}\\left( \\mathcal {G}_{\\mathcal {E}}\\right) ,$ and that there is a $k_{0}$ such that Assumptions 2.a.",
"and 2.b.",
"are satisfied.",
"We propose the two alternate specifications of the outcome auto-model that incorporate neighbor terms, $W_i$ , excluding covariate terms ($L_i$ , $\\sum _j L_j$ ).",
"Note that we accounted for covariates in estimation.",
"The auto-model parameter estimates for each option are given in Table 4.",
"The network causal effect estimates are given in Table 5.",
"$\\textrm {logit}( Pr[Y_i = 1 | W_i, A_i, A_j, Y_i, Y_j ]) & = \\beta _0 + \\beta _1 A_i + \\beta _2 \\sum _j A_j/W_i + \\beta _3 \\sum _j Y_j/W_i + \\beta _4 W_i \\\\\\textrm {logit}( Pr[Y_i = 1 | W_i, A_i, A_j, Y_i, Y_j ]) & = \\beta _0 + \\beta _1 A_i + \\beta _2 \\sum _j A_j + \\beta _3 \\sum _j Y_j + \\beta _4 W_i $ Table: Outcome auto-model parameters estimates for coding estimators (excluding covariates) on the odds ratio scaleTable: Network causal effect estimates for coding based estimators (excluding covariates)We conducted a simulation study to evaluate the operating characteristics of our proposed estimator under the sharp null in the NNAHRAY network.",
"The network structure for the simulation study was based on the NNAHRAY observed network structure excluding singletons (i.e.",
"persons with no ties), $N=412$ ($n_1 = 229$ ).",
"The true parameter values for $\\alpha $ and $\\beta $ are given in Table 6.",
"We generated 500 realizations of ($\\mathbf {Y},\\mathbf {A},\\mathbf {L}$ ) by sampling from the Gibbs factors at their true parameter values.",
"Table: NNAHRAY simulation parameter valuesWe then calculated the “true\" value of the direct and spillover effect using the same method described in the Simulation section 5.",
"For all effects, we considered treatment assignment to be a Bernoulli random variable with probability $\\gamma = 0.5$ .",
"The spillover effects compared network incarceration allocations of 50% to 0% and the direct effect was estimated at an incarceration allocation of 50%.",
"Under the sharp null, the true network direct and spillover effects were 0.",
"The results are as expected and given in Table 7.",
"Table: Simulation results of coding estimators of network causal effects in NNAHRAY network under sharp null"
]
] | 1709.01577 |
[
[
"Distribution law of the Dirac eigenmodes in QCD"
],
[
"Abstract The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks-Casher relation.",
"At the same time the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE).",
"Then it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS.",
"The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related $SU(2)_{CS}$ and $SU(2N_F)$ symmetries.",
"We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within $N_F=2$ dynamical simulations.",
"We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT.",
"This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is related to some more general property of QCD in confinement regime."
],
[
"Introduction",
"In QCD the $SU(N_F)_L \\times SU(N_F)_R$ chiral symmetry of the Lagrangian is broken spontaneously by the quark condensate of the vacuum down to $SU(N_F)_V$ .",
"A density of the near-zero modes of the Euclidean Dirac operator is connected via the Banks-Casher relation [1] with the quark condensate, that is an order parameter for spontaneous breaking of chiral symmetry (SBCS).",
"It is believed that in the low-energy domain around the chiral limit QCD can be described by an effective theory that involves the lowest excitations of the theory, the (pseudo) Goldstone bosons $U(x) = e^{i \\pi (x)/F_\\pi }.$ The effective low-energy Lagrangian $\\mathcal {L}_{eff}$ in an expansion in powers of derivatives of the field $U(x)$ and powers of quark masses is given as $\\mathcal {L}_{eff} =\\frac{F_{\\pi }^2}{4}tr\\left( \\partial _{\\mu } U(x)^{\\dagger }\\partial _{\\mu } U(x)\\right)- \\Sigma Re\\;\\left[ e^{i\\Theta /N_F}\\right]tr\\left( MU^{\\dagger } (x)\\right) + ...,$ where $M$ is the mass matrix in a theory with $N_F$ degenerate flavors $M = m I$ , $m$ is the mass of a single quark flavor and $I$ is a $N_F\\times N_F$ identity.",
"$\\Theta $ is the vacuum angle and $\\Sigma $ is the quark condensate.",
"Then the effective low-energy partition for Euclidean QCD in a finite box with the volume $V$ is given as $\\mathcal {Z}_{eff} = \\int D[U] \\exp \\left\\lbrace -\\int _V d^4 x\\; \\mathcal {L}_{eff} \\right\\rbrace .$ Thus, two constants $F_\\pi $ and $\\Sigma $ determine the leading term of the effective Lagrangian and the higher-derivative terms generate corrections, involving powers of $1/L^2$ , where $L$ is the linear size of the box, and powers of $M$ .",
"The interaction of Goldstone bosons is suppressed by their momenta.",
"Consequently, one can parametrize the $U(x)$ field as $U(x) = U_0U_1(x)$ , where $U_0$ describes the zero-momentum modes $p=0$ and is space-independent and $U_1(x)$ describes the modes with $p \\ne 0$ .",
"In the limit in which $\\frac{\\Sigma \\: m}{F_{\\pi }^2} \\ll \\frac{1}{\\sqrt{V}},$ the zero-momentum modes dominate and the effective partition function with the trivial theta-angle, $\\Theta =0$ , becomes $\\mathcal {Z}_{eff} =A \\int _{SU(N_F)} D[U_0] \\exp \\left\\lbrace V\\Sigma Re\\; tr \\left( MU_0^{\\dagger } \\right)\\right\\rbrace ,$ where $A$ is a normalization constant.",
"Thus, QCD in the low-energy domain near the chiral limit with spontaneous breaking of chiral symmetry can be effectively described within the $\\epsilon $ -regime ($\\Lambda _{QCD} L >> 1$ , $m_\\pi L << 1$ ) by a theory with the partition function above [2].",
"At the same time it is known that these low-energy chiral properties of QCD can be described within a model that relies on randomly distributed weakly interacting instantons in the QCD vacuum [3], [4].",
"The randomness of the instanton distribution in Euclidean space-time is reflected in the distribution of the near-zero modes of the Euclidean Dirac operator, because within this model the exact quark zero modes, which are due to a zero-mode solution of the Dirac equation for a massless quark in the field of an isolated instanton, in an ensemble of the (weakly) interacting instantons become the near-zero modes.",
"Motivated by these observations it was suggested in Ref.",
"bib:ShurVer that the low-energy domain of QCD, related to spontaneous breaking of chiral symmetry, can be described by the chiral random matrix theory (chRMT) with $\\mathcal {Z}_{eff} = \\int P(W)dW,$ where $W$ is some random matrix such that the density probability distribution of $W$ for $N_F$ degenerate flavors is given by $P(W)dW= \\mathcal {N} \\left( \\det (D + m)\\right)^{N_F} e^{-\\frac{N\\beta \\Sigma ^2}{4}tr(W^{\\dagger }W)} dW.$ Here $dW$ is Haar measure and $D$ is Euclidean Dirac operator, $D = \\gamma _{\\mu } (\\partial _{\\mu } + igA_{\\mu }(x)).$ Choosing the chiral representation for the $\\gamma $ -matrices $\\begin{split}\\gamma _k =\\left(\\begin{matrix}0 & i\\sigma _k\\\\-i\\sigma _k &0 \\\\\\end{matrix}\\right),\\; k = 1, 2, 3\\\\\\gamma _4 =\\left(\\begin{matrix}0 & 1 \\\\1 & 0 \\\\\\end{matrix}\\right),\\;\\gamma _5 =\\left(\\begin{matrix}1 & 0 \\\\0 & -1 \\\\\\end{matrix}\\right),\\end{split}$ the Dirac operator, if the mass is set to zero, has the following structure: $D =\\left(\\begin{matrix}0 & iW \\\\iW^{\\dagger } & 0\\\\\\end{matrix}\\right).$ The Dirac operator in a finite volume (i.e.",
"on the lattice) is a large $ N \\times N$ matrix that is determined by the lattice size.",
"If this matrix is random and recovers for $N \\rightarrow \\infty $ the Dirac operator in continuum, then the low-energy properties of QCD, related to SBCS, should be consistent with the chRMT.",
"In Eq.",
"(REF ) $\\mathcal {N}$ is the normalization constant, $\\Sigma $ is a parameter that it is not always related to the chiral condensate (not - if we are beyond the $\\epsilon $ regime), and $\\beta $ is the Dyson index which is determined by the symmetry properties of the matrix $W$ .",
"Different values of $\\beta $ correspond to different matrix ensembles.",
"If $\\beta =1$ we have the chiral Gaussian Orthogonal Ensemble (chGOE), if $\\beta =2$ the chiral Gaussian Unitary Ensemble (chGUE) and $\\beta = 4$ , the chiral Gaussian Symplectic Ensemble (chGSE).",
"In QCD $\\beta =2$ as was shown in Ref.",
"bib:Ver.",
"Subsequent lattice studies of distributions of the lowest-lying modes in QCD have confirmed that these distributions follow a universal behaviour imposed by the Wigner-Dyson Random Matrix Theory [7], [8], [9], [10], [11], [12].",
"As a result it has become an accepted paradigm that randomness of the low-lying modes is a consequence of SBCS.",
"The lowest-lying modes of the Dirac operator are strongly affected by SBCS.",
"At the same time the higher-lying modes are subject to confinement physics.",
"This was recently established on the lattice via truncation of the lowest modes of the overlap Dirac operator from the quark propagators [13], [14], [15], [16].",
"Hadrons (except for pion) survive this truncation and their mass remains large.",
"Not only $SU(2)_L \\times SU(2)_R$ and $U(1)_A$ chiral symmetries get restored, but actually some higher symmetries emerge.",
"These symmetries were established to be $SU(2)_{CS}$ (chiral-spin) and $SU(4)$ that contain chiral symmetries as subgroups and that are symmetries of confining chromo-electric interaction [17], [18].",
"Figure: Mass evolution of J=1J = 1 mesons on exclusion of the near-zero modes of the Dirac operator; kk is the number of truncated lowest-lying modes.",
"The value σ\\sigma denotes the energy gap in the spectrum of the Dirac operator.",
"Fig.",
"from Ref.",
"bib:Denissenya2.The mass evolution of the $J=1$ mesons upon truncation of $k$ lowest eigenmodes of the Dirac operator is shown on Fig.",
"REF [14].",
"It is obvious that information about $SU(2)_L \\times SU(2)_R$ and $U(1)_A$ breakings is contained in lowest 10-20 modes (given the lattice size $L \\sim 2$ fm).",
"The higher-lying modes reflect a $SU(2)_{CS}$ and $SU(4)$ symmetric regime and are not sensitive to SBCS.",
"Given success of chRMT for the lowest-lying modes of the Dirac operator it is natural to expect that the distribution law of the higher-lying modes should be different and should reflect confinement physics.",
"This motivates our study of the distribution of the lowest-lying and higher-lying modes of the Dirac operator and their comparison."
],
[
"Lattice Setup",
"We compute 200 lowest eigenmodes of the overlap Dirac operator (see Refs.",
"bib:Neuberger1,bib:Neuberger2) $D_{ov} (m) = \\left( \\rho + \\frac{m}{2} \\right) + \\left( \\rho - \\frac{m}{2} \\right) \\gamma ^5 sign \\left[ H(-\\rho ) \\right] ,$ where $H(-\\rho ) = \\gamma ^5 D(-\\rho )$ and $D(-\\rho )$ is the Wilson-Dirac operator; $m = 0.015$ is the valence quarks mass and $\\rho = 1.6$ is a simulation parameter.",
"The overlap operator is $\\gamma ^5$ -hermitian $D_{ov} (0)^{\\dagger }= \\gamma ^5 D_{ov} (0) \\gamma ^5$ and satisfies the Ginsparg-Wilson relation $\\lbrace \\gamma ^5,D_{ov} (0) \\rbrace = \\frac{1}{\\rho } D_{ov}(0)\\gamma ^5 D_{ov}(0).$ The eigenvalues of the overlap Dirac operator lie on a circle with radius $R= \\rho - \\frac{m}{2}$ , see Fig.",
"2, and come in pairs $(\\lambda _{ov}(m),\\lambda _{ov}^{*}(m))$ .",
"This is a consequence of Eq.",
"(REF ) and the $\\gamma ^5$ -hermiticity.",
"Hence the eigenvalues below the real axis bring the same informations as the eigenvalues above the real axis.",
"For this reason we consider, for our analysis, only the eigenmodes with $Im(\\lambda _{ov} (m)) \\ge 0$ .",
"In order to recover the eigenvalue $\\lambda $ of the massless Dirac operator in continuum theory we need to project our eigenvalues on the imaginary axis.",
"There is no unique way to define projection.",
"For this purpose we consider three different definitions.",
"All of these definitions are illustrated on Fig.",
"2.",
"We will study the sensitivity of our results on choice of projection definition.",
"For reasonably low eigenvalues and not large quark masses we don't expect a large variation in so defined projections.",
"We use 100 gauge field configurations in the zero global topological charge sector generated by JLQCD collaboration with $N_F = 2$ dynamical overlap fermions on a $L^3 \\times L_t = 16^3 \\times 32$ lattice with $\\beta = 2.30$ and lattice spacing $a \\sim 0.12fm$ .",
"The pion mass is $m_{\\pi } = 289(2)MeV$ , see Refs.",
"bib:JLQCD1,bib:JLQCD2.",
"Precisely the same gauge configurations have been used in truncation studies [13], [14], [15], [16].",
"The eigenvalues $\\lambda _{ov}(m)$ of $D_{ov} (m)$ are obtained calculating, at first, the sign function $ sign [H]$ .",
"We use the Chebyshev polynomials to approximate $ sign [H]$ with an accuracy of $ \\epsilon = 10^{-18} $ , and then compute 200 eigenvalues of $ D_{ov} (m) $ .",
"We notice that with this lattice setup it is not a priori obvious that chRMT should work, because we are beyond the $\\epsilon $ -regime, in our case $m_{\\pi }L \\simeq 3$ .",
"Figure: λ=ρ-m 2θ\\lambda = \\left( \\rho - \\frac{m}{2}\\right)\\theta"
],
[
"Lowest Eigenvalues",
"As we have seen we are out from $\\epsilon $ -regime and the full agreement with chRMT is not a priori expected.",
"Nevertheless we want to check whether chRMT can still describe the lowest eigenvalues in our system.",
"An important prediction of chRMT is the distribution of the lowest eigenvalues in the limit when the four dimensional volume $V\\rightarrow \\infty $ and the quantity $V\\Sigma m_{\\pi }$ is fixed, see Ref.",
"bib:Dam.",
"We order the projected eigenvalues such that $\\lambda _1 \\le \\lambda _2 \\le ... \\le \\lambda _N$ , then we define the variables $\\zeta _k = V\\Sigma \\lambda _k$ and get the distribution $p_k (\\zeta _k)$ of each $\\zeta _k$ (Ref.",
"bib:Dam).",
"In Table REF we show the ratios $\\langle \\lambda _k \\rangle /\\langle \\lambda _j \\rangle $ , for $1 \\le j < k \\le 4$ , where $\\langle \\lambda _i \\rangle $ is the average over all gauge configurations for the $i$ th projected eigenvalue.",
"Since we don't know the parameter $\\Sigma $ , we can use that $\\langle \\zeta _k \\rangle = V\\Sigma \\langle \\lambda _k \\rangle $ and we can compare our ratios with the predictions of chRMT.",
"Table: NO_CAPTIONFigure: Distribution of the lowest eigenvalue.",
"In this case ζ=VΣλ\\zeta = V\\Sigma \\lambda .We see that the ratios for the first 3 projected eigenvalues are in good agreement with the chRMT.",
"The ratios involving the 4-th projected eigenvalues have a larger discrepancy.",
"From the theoretical values of $\\langle \\zeta _k \\rangle $ and the observed values of $\\langle \\lambda _k \\rangle $ we can extract the parameter $\\Sigma $ .",
"We find $\\Sigma = (232.2 \\pm 0.9 MeV)^3$ .",
"We use this parameter to compare the distribution $p_1(\\zeta ) = dN_1/d\\zeta $ of the first lowest projected eigenvalue with the theoretical distribution given by chRMT, as we report in Fig.",
"REF .",
"$dN_1$ is the number of values assumed by the lowest projected eigenvalue of the Dirac operator, multiplied by $V\\Sigma $ , for different configurations in the interval $(\\zeta ,\\zeta + d\\zeta )$ .",
"We conclude this section noting that, even though we are not in the $\\epsilon $ -regime, for very low eigenvalues the predictions of chRMT are in good agreement with data."
],
[
"Nearest Neighbor Spacing Distribution",
"In this section we consider another important prediction of chRMT.",
"We first define the variable $s_n = \\xi _{n+1}-\\xi _n,$ where $\\xi _n = \\xi (\\lambda _n) = \\int _0^{\\lambda _n}R(\\lambda )d\\lambda $ and $R(\\lambda )$ is the probability to find an eigenvalue of the Dirac operator inside the interval $(\\lambda ,\\lambda +d\\lambda )$ .",
"$n$ indicates the number of the lowest projected eigenvalue, supposing we have ordered them in ascending order as described in the previous section.",
"The distribution of the variable in Eq.",
"(REF ) is called nearest neighbor spacing distribution (or NNS distribution).",
"In principle we don't have access to the theoretical distribution $R(\\lambda )$ and the calculation of $\\xi (\\lambda _n)$ is not trivial.",
"The procedure to map the set of variables $\\lbrace \\lambda _1, ...,\\lambda _N \\rbrace $ into the set $\\lbrace \\xi _1,...,\\xi _N \\rbrace $ is called unfolding and it is described in Ref.",
"bib:Guhr.",
"To unfold we introduce the following variable $\\begin{split}\\eta (\\lambda _n) =& \\int _{0}^{\\lambda _n} \\rho (\\lambda )d\\lambda = \\frac{1}{N} \\langle \\sum _k \\theta (\\lambda _n - \\lambda _k) \\rangle = \\\\=& \\frac{1}{N}\\frac{1}{M}\\sum _{i=1}^M \\sum _k \\theta (\\lambda _n - \\lambda _k^i),\\end{split}$ where $\\rho (\\lambda ) = \\frac{1}{N} \\langle \\sum _k \\delta (\\lambda - \\lambda _k) \\rangle $ is the spectral density of the Dirac operator averaged over all gauge field configurations, $\\lambda _k^i$ denotes the $k$ th lowest projected eigenvalue of the Dirac operator computed using the $i$ th gauge configuration and $M = 100$ is the number of gauge configurations.",
"It is shown in Ref.",
"bib:Guhr that we can decompose $\\eta (\\lambda )$ into a global smooth part $\\xi (\\lambda )$ and a local fluctuating part $\\eta _{fl}(\\lambda )$ : $\\eta (\\lambda ) = \\xi (\\lambda ) + \\eta _{fl}(\\lambda ).$ Figure: Unfolding procedure normalized to the total number of the calculated eigenvalues of the Dirac operator.The smooth part can be obtained by a polynomial fit of $\\eta (\\lambda )$ , as shown in Fig.",
"REF .",
"For different values of the Dyson index $\\beta $ we have different shapes of the NNS distribution.",
"We use the NNS distribution $p(s) = dN_s/ds$ , where $dN_s$ represents the number of the values $s_n$ inside the interval $(s,s+ds)$ , to study the lowest and the higher eigenvalues of the overlap Dirac operator.",
"The lowest eigenvalues contain the information about $SU(2)_L \\otimes SU(2)_R \\otimes U(1)_A$ breaking as is evident from Fig.",
"REF .",
"On the other hand the higher-lying eigenvalues, with $k > 10-20$ are not sensitive to SBCS and to breaking of $U(1)_A$ , but reflect physics of confinement and of $SU(2)_{CS}$ and $SU(4)$ symmetries.",
"The NNS distributions obtained with the lowest 10 eigenmodes and with the eigenmodes in the interval 81 - 100 are shown on Fig.",
"REF .",
"We see that the distribution of the lowest 10 eigenmodes is perfectly described by the Gaussian Unitary Ensemble, in agreement with chRMT.",
"However, the same Wigner distribution is observed for the higher-lying eigenmodes, which is unexpected.",
"This tells that the Wigner distribution is not a consequence of SBCS in QCD and has a more general root.",
"Figure: Range eigenvalues: 81 - 100Finally, in Fig.",
"REF we show NNS distributions of the lowest 100 modes calculated with three different definitions of projected eigenvalue, compare with Fig.",
"1.",
"It is clear that results for the distribution is not sensitive to definition of projected eigenvalue.",
"Figure: λ=ρ-m 2θ\\lambda = \\left( \\rho - \\frac{m}{2}\\right)\\theta"
],
[
"Discussion and Conclusions",
"In the past the distributions were studied typically for the low-lying modes, see e.g.",
"Ref.",
"bib:Universality-RMT,bib:Farchioni-Lang-Splittorff,bib:Luscher-Giusti,bib:Fukaya-JLQCD,bib:Edwards1,bib:Edwards2.",
"The observed Wigner distribution was linked to the SBCS phenomenon.",
"The NNS distribution for all eigenmodes has been investigated on a small $6^3 \\times 4$ lattice with the staggered fermions in Ref.",
"P. The Wigner surmise has been noticed in this study (see also Ref.",
"H).",
"On a small lattice it is difficult to distinguish the near-zero modes, responsible for SBCS and the bulk modes, however.",
"We have used a reasonably large lattice with the chirally-invariant Dirac operator.",
"More important, given our previous works [13], [14], [15], [16], we have a control over which modes can be considered as the near-zero modes, that are related with SBCS, and which are the bulk modes that are not affected by SBCS.",
"It is clear from the Fig.",
"REF that on our lattice the physics of SBCS is contained roughly in 10 lowest modes of the Dirac operator.The higher-lying modes are subject to confinement physics and related $SU(2)_{CS}$ and $SU(4)$ symmetries.",
"The higher-lying modes do not carry information about SBCS.",
"We have found that even beyond the $\\epsilon $ -regime RMT describes well the lowest eigenvalues of our system in agreement with previous results.",
"We have also found that the higher-lying modes, that are not sensitive to SBCS, follow the same Wigner distribution as the near-zero modes.",
"This observation means that the Wigner distribution seen both for the near-zero and higher-lying modes, while consistent with spontaneous breaking of chiral symmetry, is not a consequence of spontaneous breaking of chiral symmetry in QCD but has some more general origin in QCD in confinement regime.",
"An interesting question is what part of the QCD dynamics is primarily linked to randomness.",
"We can answer this question given the new $SU(2)_{CS}$ and $SU(4)$ symmetries and their connection to a specific part of the QCD dynamics [17], [18].",
"In particular, it is the chromo-electric part of the QCD dynamics that is a source of these symmetries.",
"At the same time the chromo-magnetic interaction breaks both symmetries.",
"This symmetry classification distinguishes different parts of the QCD dynamics.",
"The emergence of the $SU(2)_{CS}$ and $SU(4)$ symmetries upon truncation of the near-zero modes of the Dirac operator allows to claim that the effect of the chromo-magnetic interaction in QCD is located exclusively in the near-zero modes, while confining chromo-electric interaction is distributed among all modes of the Dirac operator.",
"Obviously some microscopic dynamics should be responsible for this.",
"Given our observation that both the near-zero and the bulk modes are subject to randomness, we can conclude that some unknown random dynamics in QCD is linked to the confining chromo-electric field.",
"This conclusion is reinforced by a recent study R of high temperature QCD where the near-zero modes of the Dirac operator are suppressed and where the chiral symmetry is restored.",
"There the same $SU(2)_{CS}$ and $SU(4)$ symmetries are observed and the results indicate that the notion of “trivial\" deconfinement (related to the Polyakov loop) has to be reconsidered."
],
[
"Acknowledgments",
"We are grateful to C.B.",
"Lang for numerous discussions and thank J. Verbaarschot for careful reading of the paper.",
"We also thank the JLQCD collaboration for supplying us with the overlap gauge configurations.",
"This work is supported by the Austrian Science Fund FWF through grants DK W1203-N16 and P26627-N27."
]
] | 1709.01886 |
[
[
"Enhancement of the Dark Matter Abundance Before Reheating: Applications\n to Gravitino Dark Matter"
],
[
"Abstract In the first stages of inflationary reheating, the temperature of the radiation produced by inflaton decays is typically higher than the commonly defined reheating temperature $T_{RH} \\sim (\\Gamma_\\phi M_P)^{1/2}$ where $\\Gamma_\\phi$ is the inflaton decay rate.",
"We consider the effect of particle production at temperatures at or near the maximum temperature attained during reheating.",
"We show that the impact of this early production on the final particle abundance depends strongly on the temperature dependence of the production cross section.",
"For $\\langle \\sigma v \\rangle \\sim T^n/M^{n+2}$, and for $n < 6$, any particle produced at $T_{\\rm max}$ is diluted by the later generation of entropy near $T_{RH}$.",
"This applies to cases such as gravitino production in low scale supersymmetric models ($n=0$) or NETDM models of dark matter ($n=2$).",
"However, for $n\\ge6$ the net abundance of particles produced during reheating is enhanced by over an order of magnitude, dominating over the dilution effect.",
"This applies, for instance to gravitino production in high scale supersymmetry models where $n=6$."
],
[
"Introduction",
"One of the key attributes of inflationary cosmology [1] is its independence of initial conditions.",
"Once inflation commences, all prior history is inflated away, and the universe begins afresh with new nearly homogeneous and isotropic initial conditions which depend primarily on the reheating process after inflation.",
"In its simplest form, reheating occurs as the inflaton settles to its minimum after inflation and the coherent scalar field oscillations of the inflaton decay.",
"If the decay products thermalize rapidly, a radiation temperature is established, and in the limit of instantaneous decay and reheating at $\\Gamma _\\phi \\sim H$ where $\\Gamma _\\phi $ is the inflaton decay rate and $H$ is the Hubble parameter, we can define a reheating temperature as $T_{RH} \\sim (\\Gamma _\\phi M_P)^{1/2}$ , where $M_P = (8 \\pi G_N)^{-1/2}$ is the reduced Planck mass [2], [3].",
"In reality, inflaton decay is not instantaneous, though thermalization may indeed be quite rapid [4], [5].",
"If thermalization is rapid, then the early inflaton decay products can achieve temperatures significantly higher than $T_{RH}$ [6], [7], [5].",
"In turn, this may significantly alter the production rate and abundance of particles which are weakly coupled to the thermal bath.",
"The gravitino is a prime example.",
"Gravitinos are produced during reheating and their abundance is typically proportional to the reheating temperature [3], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [5].",
"Although the rate for gravitino production is enhanced at temperatures above $T_{RH}$ , gravitinos produced at $T > T_{RH}$ are diluted by the bulk of the entropy produced in subsequent inflaton decays.",
"These (non)-results are, however, specific to the cross sections that characterize the particle production.",
"Here we consider particle production during reheating at temperatures $T > T_{RH}$ .",
"We consider a general form for the temperature dependence of the production cross section.",
"We then apply these results to three specific cases.",
"1) The gravitino, as discussed above in models of low energy supersymmetry.",
"2) Non-equilibrium thermal dark matter [21] models.",
"These are models where the dark matter candidate couples to the thermal bath through the exchange of some massive mediator.",
"As a result, they never attain thermal equilibrium, yet are produced from the thermal bath.",
"While similar to the gravitino, the details of the production mechanism differs.",
"3) We return to gravitinos in the case of high scale supersymmetry, where all superpartners (other than the gravitino) have masses above the inflaton mass [22], [23].",
"In this case, gravitinos can not be singly produced but rather can only be produced in pairs.",
"Once again, the details of the production mechanism differs from the previous two cases.",
"The paper is organized as follows.",
"In section 2, we write down the relevant equations for generalized particle production and describe the three specific models we use as examples.",
"In section 3, we derive the abundance of particles produced assuming instantaneous reheating and derive the effect of particle production at $T> T_{RH}$ in section 4.",
"In section 4, we also provide some numerical results to support the analytic approximations made.",
"Our conclusions are given in section 5."
],
[
"Dark matter production at reheating",
"For our analysis, we first need to compute the dark matter production at early stages of reheating.",
"We can define the thermally averaged cross section $\\langle \\sigma |v|\\rangle = \\frac{ \\lambda T^{n}}{\\pi M^{n+2}}\\,,$ for dark matter production, where we assumed a dark matter mass $m_\\chi \\ll T$ , and that $\\chi $ is coupled to the thermal bath by a heavy mediator of mass $m_X \\gg T$ .",
"In this case, the mass scale $M$ in (REF ) is parametrically related to the mediator mass, $M \\sim m_X$ .",
"For the case of the gravitino, one should associate the scale $M$ with the supersymmetry breaking scale, $F$ which may be related to the geometric mean of the Planck scale $M_P$ , and gravitino mass, $m_{3/2}$ for the production of longitudinal modes of the gravitino.",
"Reheating is a finite duration process that starts at the end of inflation, and is concluded with the formation of a dominant thermal bath due to inflaton decay.",
"Assuming instantaneous thermalization of the inflaton decay products [4], [5], this thermal bath reaches the maximum temperature $T_{\\rm max}$ shortly after inflation ends, when only a small fraction of the inflaton energy has decayed, and the energy density of the universe is still dominated by the inflaton mass.",
"This temperature may be orders of magnitude greater than the reheating temperature $T_{RH}$ , that is achieved later on, when most of the inflaton energy has decayed, and the thermal bath has become dominant [5], [6], [7].",
"Most computations of relic abundances from the early universe assume an instantaneous inflaton decay into a thermal bath of temperature $T_{RH}$ .",
"These computations ignore any production that took place during reheating (namely, while the thermal bath was subdominant, as its temperature decreased from $T_{\\rm max}$ to $T_{RH}$ ).",
"This approach is valid as long as the production rate in eq.",
"(REF ) is $not$ competitive with the dilution rate due to the inflaton decay, which is (as we will demonstrate) not always justified.",
"In this section, we propose to precisely quantify the validity of this assumption, by comparing the dark matter production obtained supposing an instantaneous reheating (subsection REF ) with the complete process, that accounts for the finite-time duration of the inflaton decay (subsection REF ).",
"We will see that the degree of accuracy depends on the specific value of the exponent $n$ in the temperature dependence $T^n$ of the thermally averaged cross-section (REF ).",
"We will then discuss three different microscopic/UV models, characterized by three different values of $n$ ."
],
[
"Instantaneous reheating",
"Under the assumption of instantaneous reheating, the inflaton instantaneously decays into a thermal bath of initial temperature [2], [3] $T_{RH} = \\left(\\frac{40}{g_{RH}\\pi ^2}\\right)^{1/4}\\left(\\frac{\\Gamma _{\\phi }M_P}{c}\\right)^{1/2}\\,,$ which dominates the energy density of the universe, where $\\Gamma _{\\phi }$ is the inflaton decay rate, $g_{RH} \\equiv g \\left( T_{RH} \\right)$ is the number of effective degrees of freedom in the thermal bath, and $c$ is an order one parameter that depends on when exactly the decay is assumed to take place.",
"For instance, $c=1$ if we set the decay time $t_{RH}=\\Gamma _{\\phi }^{-1}$ , or $c=2/3$ if we set the Hubble rate $H(T_{RH})=\\Gamma _{\\phi }$ .",
"Numerical solutions to reheating give $c \\approx 1.2$ [19], [5].",
"In what follows we will set $c=1$ for definiteness.",
"Consider for instance the process $\\gamma _1+\\gamma _2\\rightarrow \\chi _1+\\chi _2$ , where $\\gamma _{1,2}$ are constituents of the thermal plasma, and $\\chi _{1,2}$ denote the scattering products, out of which $\\chi _1$ or both $\\chi _{1,2}$ correspond to the dark matter particle; in this section we assume for simplicity that $\\chi _1=\\chi _2\\equiv \\chi $ .",
"If the scattering cross section is small enough to keep the dark matter number density, $n_{\\chi }$ , well below its thermal equilibrium value, $n_{\\chi }^{\\rm eq}$ , at all times, then the Boltzmann equation controlling the dark matter abundance $Y_\\chi \\left( T \\right) \\equiv \\frac{n_\\chi \\left( T \\right)}{n_{\\rm rad} \\left( T \\right)}$ is of the form Here for convenience, $n_{\\rm rad}$ is defined as the number density of a single bosonic relativistic degree of freedom in thermal equilibrium, $n_{\\rm rad}=\\zeta (3)T^3/\\pi ^2$ .",
"The final abundance $Y_\\chi $ can be immediately related to the dark matter relic fractional density through $\\Omega _{\\chi } = \\frac{m_{\\chi } n_\\chi }{\\rho _c} = \\frac{m_\\chi \\, Y_\\chi n_{\\rm rad}}{\\rho _c}$ , where $\\rho _c$ is the critical energy density of the universe.",
"$\\dot{Y}_{\\chi } + 3\\left(H + \\frac{\\dot{T}}{T}\\right) Y_{\\chi } = g_{\\chi }^2\\langle \\sigma |v|\\rangle n_{\\rm rad}\\,,$ where $H$ is the Hubble rate and $g_\\chi $ is the number of degrees of freedom of $\\chi $ (times 3/4 if $\\chi $ is a fermion).",
"This is solved by $Y_{\\chi }(T)=Y_{\\chi }(T_{RH})\\,\\frac{g(T)}{g_{RH}} - g(T)\\int _{T_{RH}}^T \\frac{g_{\\chi }^2\\langle \\sigma |v|\\rangle n_{\\rm rad}(\\tau )}{g(\\tau )H(\\tau )\\,\\tau }\\left[1+\\frac{\\tau }{3} \\frac{d\\ln g(\\tau )}{d\\tau }\\right]\\,d\\tau \\,,$ where $g \\left( T \\right)$ is the number of effective relativistic degrees of freedom in the thermal bath at temperature $T$ .",
"We have assumed entropy conservation so that $g T^3 a^3 =$ const., where $a$ is the cosmological scale factor.",
"We now use the thermal cross section (REF ), and assume a vanishing dark matter abundance at the beginning of reheating, $Y_{\\chi }(T_{RH})=0$ .",
"Accounting for the fact that $g$ and the coupling $\\lambda $ depend only weakly on the temperature, eq.",
"(REF ) integrates to $Y_{\\chi ,{\\rm instant.",
"}}(T) &\\simeq - \\frac{\\zeta (3)\\sqrt{90}\\, g_{\\chi }^2M_P}{\\pi ^4 M^{n+2}} g(T)\\int _{T_{RH}}^T \\frac{ \\lambda (\\tau ) \\tau ^n}{g(\\tau )^{3/2}} \\,d\\tau \\\\&\\simeq \\frac{\\zeta (3)\\sqrt{90}\\, g_{\\chi }^2 M_P}{(n+1) \\pi ^4 M^{n+2}} g(T) \\left[ \\frac{\\lambda (T_{RH}) T_{RH}^{n+1}}{g_{RH}^{3/2}} - \\frac{\\lambda (T) T^{n+1}}{g(T)^{3/2}} \\right] $ which asymptotes to the value $Y_{\\chi ,{\\rm instant.}}",
"\\simeq \\left(\\frac{90}{g_{RH}}\\right)^{1/2} \\left(\\frac{g(T)}{g_{RH}}\\right) \\frac{\\zeta (3) g_{\\chi }^2 \\lambda (T_{RH}) T_{RH}^{n+1} M_P}{(n+1) \\pi ^4 M^{n+2}} \\,,$ when $T < T_{RH}$ .",
"We have assumed $n > -1$ in eq.",
"(REF ) (so that the first term dominates in the square parenthesis of eq.",
"(REF )).",
"We can then define $R_{\\chi ,{\\rm instant.}}",
"(T) = \\frac{Y_{\\chi ,{\\rm instant.",
"}}(T)}{Y_{\\chi ,{\\rm instant.",
"}}}$ as the ratio of the temperature-dependent abundance relative to its asymptotic value.",
"In the next subsection we compare the result (REF ), obtained under the assumption of instantaneous reheating, against the abundance obtained if we more properly account for the finite duration of reheating."
],
[
"Instantaneous thermalization",
"Reheating after inflation is a continuous process, that dumps the energy density of the inflaton into the relativistic plasma, while diluting the previously created content of the universe.",
"Therefore, in order to track the relic dark matter density, one must solve the following system of equations $& \\dot{\\rho }_{\\phi } + 3H\\rho _{\\phi } + \\Gamma _{\\phi }\\rho _{\\phi } = 0\\\\ & \\dot{\\rho }_{\\gamma } + 4H\\rho _{\\gamma } - \\Gamma _{\\phi }\\rho _{\\phi } =0 \\\\ & \\dot{n}_{\\chi } + 3Hn_{\\chi } + \\langle \\sigma |v|\\rangle \\left[ n_{\\chi }^2 - (n_{\\chi }^{\\rm eq})^2 \\right] = 0 \\\\& \\rho _{\\phi }+\\rho _{\\gamma } = 3 \\, M_P^2 \\, H^2 $ where $\\rho _\\phi $ and $\\rho _\\gamma $ , are, respectively, the energy density of the inflaton and of the thermal bath formed by inflaton decay.",
"We stress that we are assuming that the dark matter is not directly coupled to the inflaton, and it is only produced by the thermal bath with the cross section (REF ).",
"We continue to assume instantaneous thermalization of the inflaton decay products, as justified in [4], [5].",
"Finally, we disregard the production of dark matter in the second and fourth equations ( and ), as we will work in the limit of small dark matter production, so that $\\rho _\\gamma $ and $H$ are negligibly modified by dark matter production.",
"Solving the first two equations of the system one finds that the thermal bath reaches a maximum temperature $T_{\\rm max}$ when only a small amount of the inflaton energy has decayed [6], [7], [5].",
"This temperature is much higher than the reheating temperature, defined to be the temperature of the thermal bath when it starts to dominate over the residual energy of the inflaton.",
"One finds (see for instance [5]) $T_{\\rm max} \\simeq 0.5\\left(\\frac{m_{\\phi }}{\\Gamma _{\\phi }}\\right)^{1/4}T_{RH}\\,,$ where $m_\\phi $ is the inflaton mass.",
"Perturbativity requires $\\Gamma _{\\phi } < m_\\phi $ , and it is not uncommon to have $\\Gamma _\\phi \\ll m_\\phi $ (this is for instance the case if the inflaton decays gravitationally).",
"Therefore $T_{\\rm max}$ can be many orders of magnitude greater than $T_{RH}$ , possibly leading to a larger production of dark matter.",
"This opens the question regarding the accuracy of the result (REF ), that assumes that the temperature was never above $T_{RH}$ .",
"On the other hand, most of the energy of the universe is still in the inflaton when $T = T_{\\rm max}$ , and the entropy generated by the subsequent decay of this energy dilutes the dark matter quanta produced at $T \\sim T_{\\rm max}$ .",
"Given these two contrasting arguments, only an explicit solution of the system (REF )-()-() can shed light on the accuracy of the instantaneous reheating result (REF ).",
"We assume that the inflaton performs coherent oscillations about the (quadratic) minimum of its potential at the end of inflaton.",
"This leads to an equation of state for the inflaton $w=p/\\rho =0$ , when averaged over a complete oscillation (the oscillations occur on a timescale $m_\\phi ^{-1}$ , which is much shorter than the other timescales of reheating, and taking $w=0$ for the inflaton is therefore a very accurate assumption).",
"The inflaton dominates the energy density until the very end of reheating, so it is a good approximation to set $w=0$ for the whole duration of reheating.",
"This will allow us to obtain an analytic result for the dark matter abundance, that we can compare with an exact numerical solution of the system (REF )-()-().",
"Under this assumption, the scale factor evolves as [5] $\\frac{a(t)}{a_{\\rm end}} \\simeq \\left(1+\\frac{v}{A}\\right)^{2/3} \\simeq \\left(\\frac{v}{A}\\right)^{2/3}\\,,$ with $v \\equiv \\Gamma _{\\phi } \\left( t-t_{\\rm end}\\right)$ (the suffix “end” indicating the end of inflation, when $w = -1/3$ ) and $A \\equiv \\frac{\\Gamma _{\\phi }}{m}\\left(\\frac{3}{4}\\frac{\\rho _{\\rm end}}{m^2M_P^2}\\right)^{-1/2}\\simeq \\mathcal {O}(1)\\, \\frac{\\Gamma _{\\phi }}{m} \\,,$ where the $\\mathcal {O}(1)$ factor in the second equality is approximately equal to 2.8 for Starobinsky inflation, and to 1.3 for a quadratic potential.",
"In the regime $A\\ll v\\ll 1$ , we obtain [5] $\\rho _{\\gamma }\\simeq \\rho _{\\rm end}A^2 v^{-8/3}\\gamma (5/3,v) \\simeq \\frac{3}{5}\\rho _{\\rm end}A^2 v^{-1} = \\frac{4}{5}(\\Gamma _{\\phi }M_P)^2 v^{-1}\\,,$ where $\\gamma $ denotes the lower incomplete gamma function.",
"This in turn, implies $T\\simeq \\left(\\frac{24}{\\pi ^2 g}\\right)^{1/4}(\\Gamma _{\\phi }M_P)^{1/2}v^{-1/4}\\,.$ With the scattering cross section given by (REF ), and $n_{\\chi }^{\\rm eq}= g_{\\chi } n_{\\rm rad}$ , we can readily rewrite () as $\\frac{d}{dT}\\left[n_{\\chi }\\left(\\frac{a}{a_{\\rm end}}\\right)^3\\right] &= \\frac{g_{\\chi }^2\\langle \\sigma |v|\\rangle n_{\\gamma }^2}{\\dot{T}}\\left(\\frac{a}{a_{\\rm end}}\\right)^3\\\\ & = g_{\\chi }^2\\left(\\frac{ \\lambda T^{n}}{\\pi M^{n+2}}\\right)\\left(\\frac{\\zeta (3)T^3}{\\pi ^2}\\right)^2\\left(-\\frac{96 \\Gamma _{\\phi }M_P^2}{g\\pi ^2 T^5}\\right) \\left(\\frac{24\\Gamma _{\\phi }^2M_P^2}{g\\pi ^2 T^4A}\\right)^{2} \\;,$ which is solved by $n_{\\chi }\\left(\\frac{a}{a_{\\rm end}}\\right)^3 = \\frac{55296\\zeta (3)^2 \\, g_{\\chi }^2\\lambda \\Gamma _{\\phi }^5M_P^6}{g^3\\pi ^{11}M^{n+2}A^2} \\times {\\left\\lbrace \\begin{array}{ll}\\dfrac{1}{n-6}\\left(T_{\\rm max}^{n-6} - T^{n-6} \\right)\\,, & n\\ne 6\\\\[5pt]\\ln \\left(\\dfrac{T_{\\rm max}}{T }\\right)\\,, & n=6\\end{array}\\right.",
"}\\,.$ Dividing this by $n_{\\rm rad}$ , we find, at the end of reheating, $Y_{\\chi }^{(n)} (T_{RH}) = \\frac{96\\zeta (3)\\, g_{\\chi }^2\\lambda M_P T_{RH}^{7}}{\\sqrt{40} g_{RH}^{1/2}\\pi ^{4}M^{n+2}} \\times {\\left\\lbrace \\begin{array}{ll}\\dfrac{1}{n-6}\\left(T_{\\rm max}^{n-6} - T_{RH}^{n-6} \\right)\\,, & n\\ne 6\\\\[5pt]\\ln \\left(\\dfrac{T_{\\rm max}}{T_{RH} }\\right)\\,, & n=6\\end{array}\\right.",
"}\\,.$ We can now compare this result with (REF ), obtained under the assumption of instantaneous reheating.",
"At $T \\ll T_{RH}$ we find $R_{\\chi }^{(n)}(T)\\equiv \\frac{Y_{\\chi }^{(n)} (T)}{Y_{\\chi ,{\\rm instant.}}",
"} \\simeq f(n){\\left\\lbrace \\begin{array}{ll}\\dfrac{8}{5}\\left(\\dfrac{n+1}{6-n}\\right)\\,, & n<6\\\\[15pt]\\dfrac{56}{5}\\ln \\left(\\dfrac{T_{\\rm max}}{T_{RH}}\\right)\\,, & n=6 \\\\[15pt]\\dfrac{8}{5}\\left(\\dfrac{n+1}{n-6}\\right)\\left(\\dfrac{T_{\\rm max}}{T_{RH}}\\right)^{n-6}\\,, & n>6\\end{array}\\right.}",
"\\,,$ where we have inserted a function $f(n)$ shown in Fig.",
"REF , which corrects the analytic result discussed above, with the exact numerical evaluation.",
"This correction is necessary as, around $v\\sim 1$ , the approximation (REF ) to the plasma temperature is not accurate, due to the shift of the equation of state parameter from $w\\approx 0$ to $w\\approx 1/3$ ; moreover, entropy production continues beyond $v=1$ , which further dilutes the dark matter yield below the analytical approximation.",
"Note that, nevertheless, the correction is not large, $0.4 \\lesssim f(n) \\lesssim 3.3$ for $n> 0$ .",
"Eq.",
"(REF ) is one of the main results of this paper.",
"Figure: Numerical correction to the analytical result () for the ratio of the exact dark matter yield to the instantaneous approximation, R χ (n) R_{\\chi }^{(n)}.",
"The function f(n)f(n) asymptotes to the value ∼0.4\\sim 0.4 for large nn .We see from eq.",
"(REF ) that as $n$ increases, the final result for the abundance is increasingly sensitive to the highest temperature, and the details of reheating are relevant.",
"In particular, physically different results are obtained for $n<6$ vs. $n\\ge 6$ , as already noted in [7] (that only focused on the $n<6$ case).",
"For $n<6$ the more accurate result (REF ) corrects the instantaneous reheating result by a factor of $\\mathcal {O}(1)$ .",
"For a steeper dependence of the cross section on the temperature, the final dark matter abundance can be significantly different from the naive expectation.",
"In particular, in terms of the inflaton decay rate, the enhancement for the $n=6$ case can be equivalently rewritten as $R_{\\chi } \\simeq 1.14\\ln \\left(\\dfrac{m_{\\phi }}{\\Gamma _{\\phi }}\\right) - 3.17 \\,.$"
],
[
"Representative examples",
"In this section we consider three representative cases characterized by the thermal cross section (REF ), with three different values of the coefficient $n$ .",
"These are: (1) Gravitino production in low scale supersymmetry models (with a single gravitino in the final state).",
"This is characterized by $n=0$ ; (2) Non-Equilibrium Thermal Dark Matter (NETDM), characterized by $n=2$ ; (3) Gravitino production in high scale supersymmetry where production occurs in processes having two gravitinos in final state, leading to $n=6$ ."
],
[
"Single Gravitino Production",
"In commonly studied models of weak scale supersymmetry, in the absence of direct inflaton to gravitino decays, the dominant scattering source for gravitino production is $X +{\\tilde{Y}} \\rightarrow {\\tilde{G}} + Z$ or $X + Y \\rightarrow {\\tilde{Z}} + {\\tilde{G}}$ where $X, Y, Z$ are standard model (SM) particles or their supersymmetric partners.",
"The cross section for the production of the transverse components of the gravitino is simply proportional to $(1/M_P^2)$ [3], [8], [9], [10], [13].",
"However, when the mass of the gravitino is less than the gaugino masses (and in particular the gluino mass), the cross section for the production of the longitudinal components is enhanced by a factor of $(m_{\\tilde{g}}/m_{3/2})^2$ [12], [15], [16], [17], [18], [19], [20], [5].",
"The thermally-averaged cross section for the Standard Model $SU(3)_c\\times SU(2)_L\\times U(1)_Y$ gauge group was calculated in [17], [18], [20].",
"The dominant contributions to the cross section can be parametrized as $\\langle \\sigma _{\\rm tot}v_{\\rm rel}\\rangle = \\langle \\sigma _{\\rm tot}v_{\\rm rel}\\rangle _{\\rm top}+ \\langle \\sigma _{\\rm tot}v_{\\rm rel}\\rangle _{\\rm gauge} \\,,$ with $\\langle \\sigma _{\\rm tot}v_{\\rm rel}\\rangle _{\\rm top} = 1.29\\,\\frac{|y_t|^2}{M_P^2}\\left[1+\\frac{A_t^2}{3m_{3/2}^2}\\right] \\,,$ where $A_t$ is the top-quark supersymmetry-breaking trilinear coupling, and $\\langle \\sigma _{\\rm tot}v_{\\rm rel}\\rangle _{\\rm gauge} & = & \\sum _{i=1}^3 \\frac{3\\pi c_ig_i^2}{16 \\zeta (3) M_P^2} \\left[1+\\frac{m_{\\tilde{g}_i}^2}{3m_{3/2}^2}\\right]\\ln \\left(\\frac{k_i}{g_i}\\right) \\nonumber \\\\& & \\!\\!\\!\\!",
"\\!\\!\\!\\!",
"\\!\\!\\!\\!",
"\\!\\!\\!\\!",
"\\!\\!\\!\\!= \\frac{26.24}{M_P^2} \\left[\\left(1+0.558\\,\\frac{m_{1/2}^2}{m_{3/2}^2}\\right) - 0.011 \\left(1+3.062\\,\\frac{m_{1/2}^2}{m_{3/2}^2}\\right) \\log \\left(\\frac{T}{10^{10}\\,{\\rm GeV}}\\right)\\right] \\, ,$ where the $m_{\\tilde{g}_i}$ are the gaugino masses and the constants $c_i,k_i$ depend on the gauge group, as shown in Table REF .",
"The second line of (REF ) was obtained in ref.",
"[5] from a fit to the result of [20] using the parametrization of [18], under the assumption of a unified gauge coupling $\\alpha =1/24$ and universal gaugino masses $m_{1/2}$ at the scale $M_{\\rm GUT} = 2\\times 10^{16}\\,$ GeV (see [5] for details).",
"Note that the first term in the gaugino mass-dependent factors $(1+m_{\\tilde{g}_i}^2/3m_{3/2}^2)$ corresponds to the production of the transversally polarized gravitino, while the second term is associated with the production of the longitudinal (Goldstino) component.",
"For $m_{3/2} \\ll m_{\\tilde{g_i}}$ , the production of the longitudinal components dominates.",
"Table: The values of the constants c i c_i and k i k_i in the parameterization ()for the Standard Model gauge groups U(1) Y U(1)_Y, SU(2) L SU(2)_L, and SU(3) c SU(3)_c.",
"See for details.Ignoring the logarithmic dependence in eq.",
"(REF ), the cross section is constant corresponding to $n=0$ in eq.",
"(REF ).",
"Figure REF shows the comparison between the fully numerical calculation (black, continuous), using $R_\\chi ^{(n)}$ in eq.",
"(REF ) with $n=0$ , and the instantaneous reheating result (orange, dotted), given by $R_{\\chi ,{\\rm instant.",
"}}(T)$ from eq.",
"(REF ).",
"The latter by definition asymptotes to 1 at late times (large $v$ ).",
"As it is clear, the instantaneous approximation slightly overestimates the true gravitino abundance by a factor of $\\sim 1.1$ , as expected from eq.",
"(REF ).",
"More importantly we see that gravitino production prior to the end of reheating can be ignored, as any production between $T_{RH}$ and $T_{\\rm max}$ is diluted by the bulk of the entropy produced in later inflaton decays.",
"Figure: Dark matter yield during and after reheating with n=0n=0; here Γ φ =10 -11 \\Gamma _{\\phi }=10^{-11} m φ m_\\phi .",
"The numerical result using R χ (n) R_\\chi ^{(n)} (eq.",
"()) with n=0n=0 is shown as the continuous black curve.",
"The orange dotted curve is the instantaneous reheating solution from R χ, instant .",
"(T)R_{\\chi ,{\\rm instant.",
"}}(T) (eq.",
"())."
],
[
"NETDM Production",
"In the standard gravitino production mechanism discussed above, the gravitino is produced from the thermal bath, but it never itself achieves thermal equilibrium with the bath.",
"Dark matter particles coupled to the thermal bath through a heavy mediator (such as an intermediate scale gauge boson) can also be produced from the thermal bath while never achieving thermal equilibrium.",
"Such NETDM candidates [21], [24], [25], may arise in non-supersymmetric grand unified theories such as SO(10) when a SM singlet component of either a 45, 54 or 210 representation of SO(10) is the dark matter [24], [26].",
"Here, we consider the production of a fermionic dark matter candidate, $\\chi $ , via a $2\\leftrightarrow 2$ process mediated by the exchange of a heavy gauge boson $X$ .",
"For definiteness, we assume that the parent SM particles (denoted by $f$ ) are also fermions, leading to the diagram depicted in Fig.",
"REF with matrix element squared $|\\mathcal {M}|^2 = \\frac{\\alpha _{f}^2\\alpha _{\\chi }^2s^2}{(s-m_X^2)^2}(1+\\cos ^2\\theta )\\,.$ Here $\\alpha _{f,\\chi }$ denote the gauge couplings, while $\\theta $ is the angle between the incoming and outgoing particles in the CM frame.",
"The same amplitude is obtained for a scalar mediator $X$ , with $\\alpha _{f,\\chi }$ playing the role of Yukawa couplings without the $\\cos ^2 \\theta $ .",
"The scattering cross section can be obtained in a straightforward way, $\\sigma _{\\chi \\chi \\leftrightarrow ff} = \\frac{\\alpha _{f}^2\\alpha _{\\chi }^2 s}{12\\pi (s-m_X^2)^2}\\,.$ The dark matter abundance follows eq.",
"(), with $n_{\\chi }^{\\rm eq}=g_{\\chi }n_{\\rm rad}$ .",
"The thermally averaged cross section can be computed in the ultrarelativistic limit $T\\gg m_{\\chi }$ as [27], [21] $\\langle \\sigma v \\rangle = \\frac{49}{18}\\frac{N_f \\alpha _f^2 \\alpha _\\chi ^2}{m_X^4 \\pi }\\left[ \\frac{\\zeta (4)}{\\zeta (3)}\\right]^2 T^2\\simeq 2.2~N_f~\\frac{\\alpha _f^2 \\alpha _\\chi ^2}{\\pi m_X^4} ~T^2\\, ,$ where $N_f$ the number of SM fermions coupling with the mediator $X$ and we have assumed $T\\gg m_{\\chi }$ and $T \\ll m_X$ .",
"The final expression is of the form (REF ), with $n=2$ , $\\lambda =\\alpha _{f}^2\\alpha _{\\chi }^2$ and $M=m_{X}$ .",
"We use this expression in the system of equations (REF )-()-(), which we integrate numerically, under the assumption, characteristic of NETDM, that the dark matter abundance is much smaller than the thermal equilibrium value, $n_{\\chi } \\ll n_{\\chi }^{\\rm eq}$ .",
"For generality we plot $R_{\\chi }$ , which scales out all model-dependent factors from the dark matter yield during reheating, and we have assumed a constant $g=g_{RH}$ .",
"Figure: Feynman diagram depicting the freeze in production of the dark matter χ\\chi through a heavy XX mediator.Figure: As in Fig.",
", for n=2n=2.Figure REF shows the comparison between the fully numerical calculation (black, continuous) using $R_\\chi ^{(n)}$ in eq.",
"(REF ) with $n=2$ , and the instantaneous reheating result (orange, dotted) given by $R_{\\chi ,{\\rm instant.",
"}}(T)$ from eq.",
"(REF ).",
"It can be seen that the instantaneous approximation minimally overshoots the exact solution by a mere $3\\%$ , in agreement with (REF )."
],
[
"High Scale Supersymmetry, with two gravitinos final state processes",
"Our final example is that of two-gravitino final state processes which are the dominant gravitino production mechanisms in high scale supersymmetry models where the only supersymmetric state below the inflationary scale is the gravitino [22], [23].",
"In this case, the process $X +{\\tilde{Y}} \\rightarrow {\\tilde{G}} + Z$ is not possible as there are no supersymmetric particles in the thermal bath and $X + Y \\rightarrow {\\tilde{Z}} + {\\tilde{G}}$ is kinematically forbidden.",
"Thus, the dominant process becomes $X + Y \\rightarrow {\\tilde{G}} + {\\tilde{G}}$ which is highly suppressed.",
"Since $m_{3/2} \\ll m_{\\tilde{g}}$ , we expect the cross section to longitudinal modes to dominate and when accounting for all possible SM initial states, the thermally averaged cross section can be written as [22] $\\langle \\sigma v \\rangle \\simeq 2000 \\frac{T^6}{\\pi F^4}\\,,$ where $F = \\sqrt{3} M_P m_{3/2}$ is the supersymmetry breaking order parameter.",
"The strong suppression ($\\propto F^4$ ) of the cross section would indicate that a relatively high reheating temperature and gravitino mass are required to produce a sufficient quantity of gravitinos to account for the observed relic density.",
"Indeed for a gravitino mass of 1 EeV, a reheating temperature of approximately $5 \\times 10^{10}$ GeV is needed [23], placing strong constraints on inflationary models and supersymmetry breaking [28].",
"Figure REF shows the exact and instantaneous results for $R_{\\chi }$ in the $n=6$ case.",
"In this case, one sees that the standard estimate of the dark matter abundance evaluated at $T_{RH}$ is not very accurate and the final ratio is $R_{\\chi } \\sim 25.7$ , consistent with the result (REF ).",
"From eq.",
"(REF ) we see that, in order to obtain the correct gravitino dark matter abundance, the reheating temperature should be decreased by a factor $\\sim \\frac{2}{3}$ with respect to that indicated by the naive assumption of instantaneous decay.",
"Figure: As in Fig.",
", for n=6n=6."
],
[
"Conclusions",
"Reheating after inflation is responsible for the entire matter and radiation content of the Universe.",
"Thus, understanding the details of this process is crucial to our ability to develop models incorporating entropy production, baryogenesis, and dark matter among many other important ideas in cosmology.",
"In many models of dark matter, including well studied models of supersymmetric dark matter, thermally produced dark matter particles come into thermal equilibrium, and their final abundance is often determined after they freeze out of the thermal bath.",
"On the other hand, there are many models in which the dark matter candidates never attain thermal equilibrium yet are produced from the thermal bath.",
"Gravitino dark matter is a good example of this situation, and early estimates of the final gravitino abundance [2], [3] relied on the instantaneous reheating approximation and the definition of the reheating temperature.",
"Reheating, however, is not an instantaneous process, but rather a continuous one.",
"The rapid thermalization of the particles produced in the earliest stages of reheating results in a thermal bath with temperatures potentially much higher than the classically defined reheating temperature.",
"Here, we have examined the effect of the high temperatures attained during reheating on the production of dark matter particles.",
"We computed the abundance of a particle produced from the thermal bath with thermally averaged cross section $\\langle \\sigma v \\rangle \\propto T^n$ .",
"Eq.",
"(REF ) provides a simple result for the discrepancy between the exact abundance, and the naive calculation based on instantaneous reheating.",
"This result can be immediately applied to obtain the exact abundance for a number of particle physics models.",
"We considered three specific examples, characterized by three different values of the exponent $n$ .",
"Two cases, singly produced gravitinos in low energy supersymmetric models, and NETDM candidates coupled to the SM through heavy mediators, have production cross sections with a relatively mild temperature dependence $(n=0$ and $n=2$ , respectively).",
"Even in the case of $n=2$ , the increased cross section at temperatures $T>T_{RH}$ , is not sufficient to overcome the dilution from inflaton decays when $T< T_{\\rm max}$ .",
"However, we also considered the case of gravitino production in high scale supersymmetric models when the only non-SM particle lighter than the inflaton is the gravitino.",
"In this case, gravitinos must be produced in pairs leading to an additional scale suppression of the cross section which in turn, leads to a larger temperature dependence.",
"Indeed, in this case, $n=6$ and the production of gravitinos near $T_{\\rm max}$ can not be neglected.",
"We found that the true gravitino abundance exceeds the naive calculation by a factor of $\\sim 25$ ."
],
[
"Acknowledgements",
"This work was supported by the France-US PICS no. 06482.",
"Y.M.",
"acknowledges partial support from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grants No.",
"690575 and No.",
"674896 and the ERC advanced grants Higgs@LHC.",
"The work of K.A.O.",
"and M.P.",
"was supported in part by DOE grant DE–SC0011842 at the University of Minnesota.",
"The work of M.A.G.G.",
"was partially completed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293."
]
] | 1709.01549 |
[
[
"Covers of Query Results"
],
[
"Abstract We introduce succinct lossless representations of query results called covers.",
"They are subsets of the query results that correspond to minimal edge covers in the hypergraphs of these results.",
"We first study covers whose structures are given by fractional hypertree decompositions of join queries.",
"For any decomposition of a query, we give asymptotically tight size bounds for the covers of the query result over that decomposition and show that such covers can be computed in worst-case optimal time up to a logarithmic factor in the database size.",
"For acyclic join queries, we can compute covers compositionally using query plans with a new operator called cover-join.",
"The tuples in the query result can be enumerated from any of its covers with linearithmic pre-computation time and constant delay.",
"We then generalize covers from joins to functional aggregate queries that express a host of computational problems such as aggregate-join queries, in-database optimization, matrix chain multiplication, and inference in probabilistic graphical models."
],
[
"Introduction",
"This paper introduces succinct lossless representations of query results called covers.",
"Given a database and a join query or, more generally, a functional aggregate query (FAQ) [17], a cover is a subset of the query result that, together with a (fractional hypertree) decomposition of the query [13], recovers the query result.",
"Covers enjoy desirable properties.",
"First, they can be more succinct than the listing representation of the query result.",
"For a join query $Q$ , database $\\mathbf {D}$ , and a decomposition $\\mathcal {T}$ of $Q$ with fractional hypertree width $w$ [20], a cover over $\\mathcal {T}$ has size ${\\mathcal {O}}(|\\mathbf {D}|^w)$ .",
"In contrast, there are arbitrarily large databases for which the listing representation of the query result has size $\\Omega (|\\mathbf {D}|^{\\rho ^*})$ , where $\\rho ^*$ is the fractional edge cover number of $Q$ [4].",
"The gap between the fractional hypertree width and the fractional edge cover number can be as large as the number of relation symbols in $Q$ .",
"For an FAQ (and the special case of a join query) $\\varphi $ , any cover of its result can be computed in time ${\\mathcal {O}}(|\\mathbf {D}|^w\\log |\\mathbf {D}|)$ , where $w$ is the FAQ-width [17] of $\\varphi $ .",
"FAQs can express aggregates over database joins [6], in-database optimization [24], [2], matrix chain multiplication, and inference in probabilistic graphical models.",
"Second, the tuples in the query result can be enumerated from one of its covers with linearithmic pre-computation time and constant delay.",
"This is not the case for the representation defined by the pair of database and join query (unless W[1]=FPT) [25].",
"The benefits of covers over the latter representation are less apparent for acyclic queries, for which both representations share the same linear-size bound and desirable enumeration complexity [5].",
"For acyclic joins, the question thus becomes why to succinctly represent a query result by one relation instead of the pair of a set of relations and the query.",
"We next highlight three practical benefits.",
"Covers readily provide a subset of the query result without the need to compute the join.",
"This improves cache locality for subsequent operations, e.g., aggregates, since we only need to read in tuple by tuple from the cover instead of reading tuples from different relations stored at different locations in memory and then joining them.",
"Similarly, covers provide access locality for disk operations since tuples from the cover are stored on the same disk page, whereas tuples from different relations are stored on different pages.",
"Furthermore, covers are samples of the query result that disregard the uninformative yet exhaustive pairings brought by Cartesian products.",
"In exploratory data analysis, the explicit listing of Cartesian products is overwhelming to the user since it may be very large.",
"An alternative approach that would present the user with many relations and the query, would have to rely on the user to figure out possible tuples in the query result, which is not desirable.",
"A cover, in contrast, is a compact relation that absolves the user from ad-hoc joining of relations and from re-discovering Cartesian products in a large listing of tuples.",
"Finally, processing following the in-database joins may require a single relation as input, as it is the case for machine learning over joins [24].",
"Indeed, instead of learning regression models over the result of a join we can instead learn them over one of its covers.",
"Third, covers use the standard listing representation.",
"Prior work introduced lossless representations of query results called factorized databases that achieve the same succinctness as covers, yet they are directed acyclic graphs that represent the query result as circuits whose nodes are data values or the relational operators Cartesian product and union [23].",
"The graph representation makes difficult their adoption as a data representation model by mainstream database systems that rely on relational storage (factorized computation is however used in relational systems [2]).",
"A relational alternative to factorized databases, as metamorphosed in covers, can prove useful in a variety of settings.",
"The intermediate results in query plans can be represented as covers.",
"In distributed query plans, covers can encode succinctly the otherwise expensive intermediate query results that are communicated among servers in each round [26] and can be processed as soon as each of their tuples is received.",
"The contributions of this paper are as follows: Section introduces covers of join query results and their correspondence to minimal edge covers in the hypergraphs of the query results.",
"We also give tight size bounds for covers and show that the tuples in the query result can be enumerated from any cover with linearithmic pre-computation time and constant delay.",
"Given a database and a join query, covers of its result can be computed in worst-case optimal time (modulo a log factor).",
"Section focuses on the compositionality of cover computation for acyclic join queries.",
"We introduce cover-join plans to compute covers in time linearithmic in their sizes and the size of the input database.",
"A cover-join plan is a binary plan that follows the structure of a join tree of the acyclic query.",
"It uses a cover-join operator that computes covers of the join of two relations, which may be input relations or covers for subqueries.",
"Different plans may lead to different sets of covers.",
"There are covers that cannot be obtained using binary plans.",
"Section generalizes our notion of covers from joins to functional aggregate queries by representing succinctly both tuples and aggregates in the query result.",
"We consider natural join queries where each relation is used at most once.",
"The appendix extends our results to arbitrary equi-join queries and provides further details, examples and proofs.",
"Related work.",
"There are three strands of directly related work: cores in databases and graph theory; succinct representations of query results; and normal forms for relational data.",
"Cores of graphs, queries, and universal solutions to data exchange problems revolve around smaller yet lossless representations that are homomorphically minimal subgraphs [16], subqueries [8], and universal solutions [11], respectively.",
"A further application of graph cores is in the context of the Semantic Web, where cores of RDF graphs are used to obtain minimal representations and normal forms of such graphs [15].",
"Our notion of covers is different.",
"Covers rely on query decompositions to achieve succinctness, and they only become lossless in conjunction with a decomposition.",
"If we ignore the decomposition, the covers become lossy as they are subsets of the result.",
"Whereas in data exchange all universal solutions have the same core (up to isomorphism), the result of a query may have exponentially many incomparable covers.",
"While not a defining component of cores in data exchange, generalized hypertree decompositions can help derive improved algorithms for computing the core of a relational instance with labeled nulls under different classes of dependencies [12].",
"Covers are relational encodings of d-representations, a lossless graph-based factorization of the query result [23].",
"The structure of d-representations is given by variable orders called d-trees, which are an alternative syntax for fractional hypertree decompositions.",
"Whereas d-representations are lossless on their own, covers need the decomposition to derive the missing tuples.",
"Decompositions are the data-independent price to pay for achieving the data-dependent succinctness of factorized representations using the listing representation.",
"Both d-representations and covers achieve succinctness by avoiding the materialization of Cartesian products.",
"Whereas the former encode the products symbolically and losslessly, the covers only keep a minimal subset of the product that is enough to reconstruct it entirely.",
"The goal of database design is to avoid redundancy in the input database.",
"Existing normal forms achieve this by decomposing one relation into several relations guided by functional and join dependencies [9].",
"Covers exploit the join dependencies to avoid redundancy in the query output.",
"They do not decompose the result back into the (now globally consistent) input database.",
"Like factorized representations, covers are a normal form for relations representing query results.",
"From a cover of a join result over a decomposition, we can obtain a decomposition of the join result in project-join normal form (5NF) [10] by taking one projection of the cover onto the attributes of each bag of the decomposition."
],
[
"Preliminaries",
"Databases.",
"We assume an ordered domain of data values.",
"A relation schema is a finite set of attributes.",
"For an attribute $A$ , we denote by $\\textsf {dom}(A)$ its domain.",
"A database schema is a finite set of relation symbols.",
"A tuple $t$ over a relation schema $S$ is a mapping from the attributes in $S$ to values in their respective domains.",
"A relation over a relation schema $S$ is a finite set of tuples over $S$ .",
"A database $\\mathbf {D}$ over a database schema $\\mathcal {S}$ contains for each relation symbol in $\\mathcal {S}$ , a relation over the same schema.",
"For a relation (symbol) $R$ and tuple $t$ , we use ${\\cal S}(R)$ and ${\\cal S}(t)$ to refer to their schemas and write $R(S)$ to express that the schema of $R$ is $S$ .",
"The tuples $t_1, \\ldots ,t_n$ are joinable if $\\pi _{S_{i,j}}t_i = \\pi _{S_{i,j}}t_j$ for all $i,j \\in [n]$ and $S_{i,j}={\\cal S}(t_i)\\cap {\\cal S}(t_j)$ .",
"The size $|R|$ of a relation $R$ is the number of its tuples.",
"The size $|\\mathbf {D}|$ of a database $\\mathbf {D}$ is the sum of the sizes of its relations.",
"Natural Join Queries.",
"We consider natural join queries of the form $Q = R_1(S_1) \\bowtie \\ldots \\bowtie R_n(S_n)$ , where each $R_i$ is a relation symbol over relation schema $S_i$ and refers to a database relation over the same schema.",
"Notation-wise we do not distinguish between a relation symbol and the corresponding relation.",
"The joins in $Q$ are expressed by sharing attributes across relation schemas.",
"The schema ${\\cal S}(Q)$ of $Q$ is the set of relation symbols in $Q$ : ${\\cal S}(Q)=\\lbrace R_i\\rbrace _{i\\in [n]}$ .",
"The set $\\mathit {att}(Q)$ of attributes of $Q$ is the union of the schemas of its relation symbols: $\\mathit {att}(Q) = \\bigcup _{i\\in [n]}S_i$ .",
"The size $|Q|$ of $Q$ is the number of its relation symbols: $|Q|=n$ .",
"A database is globally consistent with respect to a query $Q$ if there are no (dangling) tuples that do not contribute to the result of $Q$ [1].",
"Two relations $R_1$ and $R_2$ are called consistent if the database $\\lbrace R_1,R_2\\rbrace $ is globally consistent with respect to the query $R_1 \\bowtie R_2$ .",
"We assume that relation symbols in $Q$ are non-repeating and each relation symbol corresponds to a distinct relation.",
"Appendix lifts these restrictions and extends our contributions to arbitrary equi-join queries.",
"Hypergraphs.",
"Let $H$ be a multi-hypergraph (hypergraph for short) whose edge multiset $E$ may contain multiple hyperedges (edges for short) with the same node set.",
"A fractional edge cover for $H$ is a function $\\gamma $ mapping each edge in $H$ to a positive number such that $\\Sigma _{e \\ni v} \\gamma (e) \\ge 1$ for each node $v$ of $H$ , i.e., the sum of the function values for all edges incident to $v$ is at least 1.",
"We define the weight of a fractional edge cover $\\gamma $ as $\\mathit {weight}(\\gamma )=\\Sigma _{e \\in E} \\gamma (e)$ .",
"The fractional edge cover number $\\rho ^*(H)$ of $H$ is the minimum weight of fractional edge covers of $H$ .",
"It can be obtained from a fractional edge cover where the edge weights are rational numbers of bit-length polynomial in the size of $H$ [4].",
"We use hypergraphs for queries and for relations representing their results.",
"The hypergraph $H$ of a query $Q$ consists of one node $A$ for each attribute $A$ in $Q$ and one edge ${\\cal S}(R)$ for each relation symbol $R\\in {\\cal S}(Q)$ .",
"We define $\\rho ^*(Q) = \\rho ^*(H)$ .",
"Let $R$ be a relation and $\\mathcal {P}$ a set of (possibly overlapping) subsets of ${\\cal S}(R)$ such that $\\bigcup _{S \\in \\mathcal {P}}S = {\\cal S}(R)$ .",
"The hypergraph $H$ of $R$ over ${\\cal P}$ consists of one node for each distinct tuple in $\\pi _S R$ for each attribute set $S\\in {\\cal P}$ and one edge for each tuple in $R$ .",
"The edge for a tuple $t$ thus consists of all nodes for tuples $\\pi _S (t)$ with $S\\in \\mathcal {P}$ .",
"We use $\\mathit {tuple}_{}(v)$ to denote the tuple represented by a node or edge $v$ in $H$ .",
"Given a subset $M$ of the edges in $H$ , we define $\\mathit {rel}_{}(M)=\\lbrace \\mathit {tuple}_{}(e)\\rbrace _{e \\in M}$ as the relation represented by $M$ .",
"The set $M$ is an edge cover of $H$ if each node in $H$ is contained in at least one edge in $M$ .",
"The set $M$ is a minimal edge cover if it is an edge cover and any of its strict subsets is not.",
"Example 1 Consider the path query $Q=R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ .",
"Figure REF depicts in the top row a database of the three relations $R_1$ , $R_2$ and $R_3$ , the query result and a subset of it.",
"In the bottom row, the figure depicts the hypergraph of $Q$ (and its decomposition defined below), the hypergraph of its result over the attribute sets $\\lbrace \\lbrace A,B\\rbrace ,\\lbrace B,C\\rbrace ,\\lbrace C,D\\rbrace \\rbrace $ , and the hypergraph of a subset of the query result over the same attribute sets.",
"Figure: Top row: database 𝐃={R 1 ,R 2 ,R 3 }\\mathbf {D}=\\lbrace R_1,R_2,R_3\\rbrace , the result Q(𝐃)Q(\\mathbf {D}) of the path query QQ in Example , and a subset of Q(𝐃)Q(\\mathbf {D}); bottom row: the hypergraph of QQ, the tree of a decomposition 𝒯\\mathcal {T} of QQ, the hypergraph of Q(𝐃)Q(\\mathbf {D}) over attribute sets 𝒮(𝒯){\\cal S}(\\mathcal {T}), and a minimal edge cover MM of this hypergraph.Decompositions.",
"A hypertree decomposition $\\mathcal {T}$ of (the hypergraph $H$ of) a query $Q$ is a pair $(T,\\chi )$ , where $T$ is a tree and $\\chi $ a function mapping each node in $T$ to a subset of the nodes of $H$ .",
"For a node $t\\in T$ , the set $\\chi (t)$ is called a bag.",
"A hypertree decomposition satisfies two properties.",
"Coverage: For each edge $e$ in $H$ , there must be a node $t$ in $T$ with $e \\subseteq \\chi (t)$ .",
"Connectivity: For each node $v$ in $H$ , the set $\\lbrace t \\mid t \\in T, v\\in \\chi (t)\\rbrace $ must be non-empty and form a connected subtree in $T$ .",
"The schema of $\\mathcal {T}$ is the set of its bags: ${\\cal S}(\\mathcal {T})= \\lbrace \\chi (t)\\mid t\\in T\\rbrace $ .",
"The attributes of $\\mathcal {T}$ are defined by $\\mathit {att}(\\mathcal {T})=\\bigcup _{B\\in {\\cal S}(\\mathcal {T})} B$ .",
"A fractional hypertree decomposition [14] of (the hypergraph $H$ of) a query $Q$ is a triple $(T,\\chi ,\\lbrace \\gamma _t\\rbrace _{t \\in T})$ where $(T,\\chi )$ is a hypertree decomposition of $H$ and for each node $t\\in T$ , $\\gamma _t$ is a fractional edge cover of minimal weight for the subgraph of $H$ restricted to $\\chi (t)$ .",
"We define the fractional hypertree width of $\\mathcal {T}=(T,\\chi , \\lbrace \\gamma _t\\rbrace _{t \\in T})$ as $\\max _{t \\in T} \\lbrace \\mathit {weight}(\\gamma _t)\\rbrace $ and we denote it by $\\textsf {fhtw}(\\mathcal {T})$ .",
"The fractional hypertree width $\\textsf {fhtw}(H)$ of the hypergraph $H$ is the minimal possible such width of any fractional hypertree decomposition of $H$ .",
"The fractional hypertree width $\\textsf {fhtw}(Q)$ of a query $Q$ is the fractional hypertree width $\\textsf {fhtw}(H)$ of its hypergraph $H$ .",
"For simplicity, we use the terms decomposition and width in place of fractional hypertree decomposition and fractional hypertree width, respectively.",
"A hypergraph $H$ is $\\alpha $ -acyclic (acyclic for short) if it has a decomposition in which each bag is contained in an edge of $H$ [7].",
"A query whose hypergraph is acyclic is also called acyclic.",
"The width of any acyclic hypergraph or query is one.",
"A join tree of a query $Q$ is a labelled tree $(T,\\ell )$ where $T = ({\\cal S}(Q),E)$ is a tree and $\\ell $ is an edge labelling such that [(i)] each edge $e = (R,R^{\\prime }) \\in E$ is labelled by $\\ell (e) = {\\cal S}(R) \\cap {\\cal S}(R^{\\prime })$ and for every pair $R$ , $R^{\\prime }$ of distinct nodes and for each attribute $A \\in {\\cal S}(R) \\cap {\\cal S}(R^{\\prime })$ , the label of each edge along the unique path between $R$ and $R^{\\prime }$ includes $A$ (Section 6.4 in [1]).",
"A query is acyclic if and only if it admits a join tree (Theorem 6.4.5 in [1]).",
"The decomposition $\\mathcal {T}$ corresponding to the join tree $\\mathcal {J}$ of a query $Q$ is constructed as follows.",
"Each node in $\\mathcal {J}$ , which corresponds to a relation symbol $R$ , is mapped to a node in $\\mathcal {T}$ , which has the bag ${\\cal S}(R)$ .",
"For each node $t$ in $\\mathcal {T}$ with bag ${\\cal S}(R)$ , the function $\\gamma _t$ maps the hyperedge for $R$ to 1.",
"Example 2 Figure REF gives the hypergraph (left, bottom row) of the path query in Example REF along with one of its decompositions.",
"This decomposition has width one, since each bag is included in one edge of the hypergraph; the path query is acyclic.",
"The decomposition, where the top two bags are merged into one, has width two.",
"For queries with cycles, e.g., Loomis-Whitney queries [21], the width can be larger than one.",
"For instance, the width of the triangle query (Loomis-Whitney query over three relations) is $3/2$ [4].",
"Computational Model.",
"We use the uniform-cost RAM model [3] where data values as well as pointers to databases are of constant size.",
"Our analysis is with respect to data complexity where the query is assumed fixed.",
"We use $\\widetilde{\\mathcal {O}}$ to hide a $\\log |\\mathbf {D}|$ factor.",
"Result-preserving Transformation.",
"Let $(Q,\\mathcal {T},\\mathbf {D})$ denote a triple of a natural join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ .",
"Proposition 3 Given $(Q,\\mathcal {T},\\mathbf {D})$ , we can compute $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ with size ${\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ and in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ such that $Q^{\\prime }$ is an acyclic natural join query, $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ , $\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ and $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"Example 4 Consider the path query $Q$ , decomposition $\\mathcal {T}$ , and database $\\mathbf {D}$ in Example REF .",
"The application of Proposition REF leaves $Q$ unchanged, since $Q$ is already acyclic and $\\mathcal {T}$ corresponds to a join tree of $Q$ .",
"The database in Figure REF is not globally consistent with respect to $Q$ , since it contains tuples (under the thin lines) that do not contribute to the result.",
"We remove these dangling tuples to make it consistent.",
"Consider now the bowtie query $Q_\\bowtie = R_1(A,B) \\bowtie R_2(B,C) \\bowtie R_3(A,C) \\bowtie R_4(A,D)\\bowtie R_5(D,E) \\bowtie R_6(A,E)$ .",
"A decomposition $\\mathcal {T}_{\\bowtie }$ with the lowest width of $3/2$ has two bags $S_1=\\lbrace A, B,C\\rbrace $ and $S_2=\\lbrace A, D, E\\rbrace $ , one for each clique (triangle) in the query.",
"The application of Proposition REF constructs the acyclic query $Q^{\\prime } = B_1(A,B,C)\\bowtie B_2(A,D,E)$ .",
"The relations $B_1(A,B,C)$ and $B_2(A,D,E)$ are materializations of the two bags of $\\mathcal {T}_{\\bowtie }$ .",
"The database $\\mathbf {D}^{\\prime } =\\lbrace B_1(A,B,C), B_2(A,D,E)\\rbrace $ is globally consistent with respect to $Q^{\\prime }$ , i.e., each tuple in $B_1^{\\prime }$ has at least one joinable tuple in $B_2^{\\prime }$ and vice versa.",
"The decomposition $\\mathcal {T}_{\\bowtie }$ corresponds to a join tree of $Q^{\\prime }$ .",
"Covers for Join Queries In this section we introduce the notion of covers of join query results along with a characterization of their size bounds, the connection to minimal edge covers for hypergraphs of join query results, and the complexity for enumerating the tuples in the query result from a cover.",
"Let $(Q,\\mathcal {T},\\mathbf {D})$ denote a triple of a natural join query $Q$ , decomposition $\\mathcal {T}$ of $Q$ , and database $\\mathbf {D}$ .",
"For an instance $(Q,\\mathcal {T},\\mathbf {D})$ , covers of the query result $Q(\\mathbf {D})$ are relations that are minimal while preserving the information in the query result $Q(\\mathbf {D})$ in the following sense.",
"Definition 5 (Result Preservation) A relation $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if its schema ${\\cal S}(K)$ is $\\mathit {att}(Q)$ and $\\pi _B K = \\pi _B Q(\\mathbf {D})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"That is, for each bag $B$ in the decomposition $\\mathcal {T}$ of $Q$ , both the relation $K$ and the query result $Q(\\mathbf {D})$ have the same projection onto $B$ .",
"This also means that the natural join of these projections of $K$ is precisely $Q(\\mathbf {D})$ .",
"Proposition 6 Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ with schema $\\mathit {att}(Q)$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if and only if $\\bowtie _{B\\in {\\cal S}(\\mathcal {T})} \\pi _B K = Q(\\mathbf {D})$ .",
"We further say that the relation $K$ is minimal result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if it is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , yet this is not the case for any strict subset of it.",
"We can now define the notion of covers of query results.",
"Definition 7 (Covers) Given $(Q,\\mathcal {T},\\mathbf {D})$ , a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ is a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"Example 8 Figure REF gives the decomposition $\\mathcal {T}$ of a path query and one cover $\\mathit {rel}(M)$ of the query result over $\\mathcal {T}$ .",
"We give below four relations that are subsets of the query result.",
"The relations $K_1$ and $K_2$ are covers, while the relations $N_1$ and $N_2$ are not covers: Table: NO_CAPTION To check the minimal result-preservation property, we take projections onto the bags $B_1=\\lbrace A,B\\rbrace $ , $B_2=\\lbrace B,C\\rbrace $ , and $B_3=\\lbrace C,D\\rbrace $ .",
"The relation $N_1$ is not result-preserving, because $(a_2,b_2)\\notin \\pi _{B_1}N_1$ .",
"The same argument also applies to relation $N_2$ .",
"Consider now the coarser decomposition $\\mathcal {T}^{\\prime }$ with bags $B^{\\prime }_{1,2}=\\lbrace A,B,C\\rbrace $ and $B^{\\prime }_3=\\lbrace C,D\\rbrace $ .",
"The covers over $\\mathcal {T}$ discussed above are also covers over $\\mathcal {T}^{\\prime }$ .",
"The query result is the only cover over the coarsest decomposition $\\mathcal {T}^{\\prime \\prime }$ with only one bag.",
"Example 9 A query result may admit exponentially many covers over the same decomposition.",
"Consider for instance the product query $R_1(A) \\bowtie R_2(B)$ with relations $R_1$ and $R_2$ of size two and respectively $n>1$ .",
"The query result has size $2\\cdot n$ .",
"To compute a cover, we pair the first tuple in $R_1$ with any non-empty and strict subset of the $n$ tuples in $R_2$ , while the second tuple in $R_1$ is paired with the remaining tuples in $R_2$ .",
"There are $2^n-2$ possible covers.",
"The empty and the full sets are missing from the choice of a subset of $R_2$ as they would mean that one of the two tuples in $R_1$ would have to be paired with tuples in $R_2$ that are already paired with the other tuple in $R_1$ and that would violate the minimality criterion of the covers.",
"All covers have size $n$ and none is contained in another.",
"We next give a characterization of covers via the hypergraph of the query result.",
"Proposition 10 Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ has a minimal edge cover $M$ such that $\\mathit {rel}(M)=K$ .",
"Example 11 Figure REF gives a minimal edge cover $M$ and the cover $\\mathit {rel}(M)$ .",
"By removing any edge from $M$ , it is not anymore an edge cover.",
"By removing the tuple corresponding to that edge from $\\mathit {rel}(M)$ , it is not anymore a cover since it is not result preserving.",
"By adding an edge to $M$ or the corresponding tuple to $\\mathit {rel}(M)$ , they are not anymore minimal.",
"We now turn our investigation to sizes and first note the following immediate property.",
"Proposition 12 Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is a subset of $Q(\\mathbf {D})$ .",
"An implication of Proposition REF is that the covers cannot be larger than the query result.",
"However, they can be much more succinct.",
"We first give size bounds for covers using the sizes of projections of the query result onto the bags of the underlying decomposition.",
"Proposition 13 Given $(Q,\\mathcal {T},\\mathbf {D})$ , the size of each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ satisfies the inequalities $\\max _{B\\in {\\cal S}(\\mathcal {T})}\\lbrace \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid \\rbrace $ $\\le $ $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ .",
"We can now characterize the size of a cover using the width of the decomposition.",
"Theorem 14 Let $Q$ be a natural join query and $\\mathcal {T}$ a decomposition of $Q$ .",
"For any database $\\mathbf {D}$ , each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"There are arbitrarily large databases $\\mathbf {D}$ such that each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The size gaps between query results and their covers can be arbitrarily large.",
"For any join query $Q$ and database $\\mathbf {D}$ , it holds that $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ and there are arbitrarily large databases $\\mathbf {D}$ for which $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ [4].",
"For acyclic queries, the fractional edge cover number $\\rho ^*$ can be as large as $|Q|$ , while the fractional hypertree width is one.",
"Section shows that the same gap also holds for time complexity.",
"Example 15 We continue Example REF .",
"The decomposition $\\mathcal {T}$ has width one, which is minimal.",
"The covers over $\\mathcal {T}$ , such as $K_1$ and $K_2$ , have sizes upper bounded by the input database size.",
"The minimum size of a cover over $\\mathcal {T}$ is the maximum size of a relation used in the query (assuming the relations are globally consistent).",
"In contrast, there are arbitrarily large databases of size $N$ for which the query result has size $\\Omega (N^2)$ .",
"Proposition REF and Theorem REF give alternative equivalent characterizations of the size of a cover of a query result.",
"The former gives it as the size of a minimal edge cover of the hypergraph of the query result over the attribute sets given by the bags of a decomposition $\\mathcal {T}$ , while the latter states it using the fractional hypertree width of $\\mathcal {T}$ or equivalently the maximum fractional edge cover number over all the bags of $\\mathcal {T}$ .",
"Most notably, whereas the former is an integral number, the latter is a fractional number.",
"This size gap between query results and their covers is precisely the same as for query results and their factorized representations called d-representations [23].",
"In this sense, covers can be seen as relational encodings of factorized representations of query results.",
"We can easily translate covers into factorized representations.",
"Appendix gives a brief introduction to d-representations and a translation example.",
"Proposition 16 Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover $K$ of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ can be translated into a d-representation of $Q(\\mathbf {D})$ of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ .",
"The above translation allows us to extend the applicability of covers to known workloads over factorized representations, such as in-database optimization problems [2] and in particular learning regression models [22].",
"Nevertheless, it is practically desirable to process such workloads directly on covers, since this would avoid the indirection via factorized representations that comes with extra space cost and non-relational data representation.",
"Aggregates, which are at the core of such workloads, can be computed directly on covers by joint scans of the projections of the cover onto the bags of the decomposition; alternatively, they can be computed by expressing any cover as the natural join of its bag projections and then pushing the aggregates past the join.",
"Example 17 We consider the query $Q = R(A,B) \\bowtie S(B,C)$ and its decomposition $\\mathcal {T}$ with bags $\\lbrace A,B\\rbrace $ and $\\lbrace B,C\\rbrace $ .",
"To compute aggregates over the join result $Q(\\mathbf {D})$ , we can use any cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"The expression for counting the number of result tuples is $\\sum _{b\\in \\text{dom}(B)}\\sum _{a\\in \\text{dom}(A)}\\sum _{c\\in \\text{dom}(C)} {\\bf 1}_{R(a,b)}\\cdot {\\bf 1}_{S(b,c)}$ , where $1_E$ is the Kronecker delta that is evaluated to 1 if the event $E$ is satisfied and 0 otherwise.",
"We can compute it in one scan over $K$ if $K$ is sorted on $(B,A,C)$ or $(B,C,A)$ .",
"For each $B$ -value $b$ , we multiply the distinct numbers of $A$ -values and of $C$ -values paired with $b$ in $K$ , and we sum up these products over all $B$ -values.",
"We can rewrite this expression as follows: $\\sum _{b\\in \\text{dom}(B)}(\\sum _{a\\in \\text{dom}(A)}{\\bf 1}_{(a,b)\\in \\pi _{\\lbrace A,B\\rbrace } K})(\\sum _{c\\in \\text{dom}(C)} {\\bf 1}_{(b,c)\\in \\pi _{\\lbrace B,C\\rbrace } K})$ .",
"This expression only uses the pairs $(a,b)$ and $(b,c)$ in $K$ .",
"The pairs $(a,c)$ , which make the difference among covers and are the culprits for the explosion in the size of the query result, are not needed.",
"Despite their succinctness over the explicit listing of tuples in a query result, any cover of the query result can be used to enumerate the result tuples with constant delay and extra space (data complexity) following linear-time pre-computation.",
"In particular, the delay and the space are linear in the number of attributes of the query result which is as good as enumerating directly from the result.",
"This complexity follows from Proposition REF and the enumeration for factorized representations [23] with constant delay and extra space.",
"Corollary 18 (Proposition REF , Theorem 4.11 [23]) Given $(Q,\\mathcal {T},\\mathbf {D})$ , the tuples in the query result $Q(\\mathbf {D})$ can be enumerated from any cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space.",
"An alternative way to achieve constant-delay enumeration with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation is by noting that the acyclic join queries considered in this paper are free-connex and thus allow for enumeration with constant delay and $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|)$ pre-computation [5].",
"An acyclic conjunctive query is called free-connex if its extension by a new relation symbol covering all attributes of the result remains acyclic [25].",
"Moreover, given a cover $K$ over a decomposition $\\mathcal {T}$ , the natural join of the projections of $K$ onto the bags of $\\mathcal {T}$ is an acyclic query that computes the original query result (Proposition REF ).",
"Computing Covers for Join Queries using Cover-Join Plans Given an arbitrary join query and database, we can compute covers using a monolithic algorithm akin to known algorithms for computing factorized representations of query results [22].",
"However, is it possible to compute covers in a compositional way, by computing covers for one join at a time?",
"In this section, we answer this question in the affirmative for acyclic natural join queries $Q$ and globally consistent databases $\\mathbf {D}$ with respect to $Q$ .",
"For a triple $(Q,\\mathcal {J},\\mathbf {D})$ , where $Q$ is an acyclic natural join query, $\\mathcal {J}$ is a join tree of $Q$ , and $\\mathbf {D}$ is a database globally consistent with respect to $Q$ , we use so-called cover-join plans to compute covers of the query result $Q(\\mathbf {D})$ over the decomposition corresponding to the join tree $\\mathcal {J}$ .",
"Such plans follow the structure of the join tree $\\mathcal {J}$ and use a new binary join operator called cover-join.",
"The cover-join of two relations yields a cover of their natural join.",
"This approach is in the spirit of standard relational query evaluation.",
"It is compositional in the sense that to compute a cover of the query result, it suffices to repeatedly compute a cover of the join of two relations.",
"This is practical since it can be supported by existing query engines extended with the cover-join operator.",
"We also show that, due to the binary nature of the cover-join operator, the cover-join plans cannot recover all possible covers of the query result.",
"Furthermore, different plans may lead to different covers.",
"Plans that do not follow the structure of a join tree may be unsound as they do not necessarily construct covers.",
"To compute covers for an arbitrary join query and database, we proceed in two stages.",
"We first materialize the bags of a decomposition of the query so as to reduce it to an acyclic query $Q$ over an extended database $\\mathbf {D}$ that is now globally consistent with respect to $Q$ (Proposition REF ).",
"We then use a cover-join plan to compute covers of $Q(\\mathbf {D})$ .",
"The first step has a non-trivial time complexity overhead, whereas the second step is linearithmic.",
"Overall, this strategy is worst-case optimal for computing covers for arbitrary join queries and databases.",
"The Cover-Join Operator The building block of our approach to computing covers is the binary cover-join operator.",
"Definition 19 (Cover-Join) The cover-join of two relations $R_1$ and $R_2$ , denoted by $R_1\\mathring{}R_2$ , computes a cover of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"Following the alternative characterization of covers of a query result by minimal edge covers in the hypergraph of the query result (Proposition REF ), the cover-join defines the relation $\\mathit {rel}(M)$ of a minimal edge cover $M$ of the hypergraph $H$ of the result of the join $R_1 \\bowtie R_2$ over the attribute sets ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"The hypergraph $H$ is bipartite and consists of disjoint complete bipartite subgraphs.",
"Since a cover is a minimal edge cover, it corresponds to a bipartite subgraph with the same number of nodes but a subset of the edges, where all paths can only have one or two edges.",
"A cover cannot have unconnected nodes, since it would not be an edge cover.",
"A path of three (or more) edges violates the minimality of the edge cover: Such a path $a_1-b_1-a_2-b_2$ in a bipartite graph covers the four nodes, yet a minimal cover would only have the two edges $a_1-b_1$ and $a_2-b_2$ .",
"We can compute a cover of a join of two relations $R_1$ and $R_2$ in time $\\widetilde{\\mathcal {O}}(|R_1| + |R_2|)$ , since it amounts to computing a minimal edge cover in a collection of disjoint complete bipartite graphs that encode the join result.",
"The smallest size of a cover is given by the edge cover number of the bipartite graph representing the join result, which is the maximum of the sizes of the two sets of nodes in the graph [19].",
"The largest size can be achieved in case one of the two node sets has size one, in which case this is paired with all nodes in the second set.",
"In case both sets have more than one node, the largest size is achieved when we pair one node from one of the two node sets with all but one node in the second set and then the remaining node in the second set with all but the already used node in the first set.",
"For the analysis in this paper, we assume that our cover-join algorithm may return any cover of the natural join of two relations.",
"In practice, however, it makes sense to compute a cover of minimum size.",
"We choose this cover as follows: For each complete bipartite hypergraph in the join result with node sets $V_1$ and $V_2$ such that $|V_1|\\le |V_2|$ , we choose a minimum edge cover by pairing each node in $V_1$ with one distinct node in $V_2$ and all remaining nodes in $V_2$ with one node in $V_1$ .",
"Proposition 20 Given two consistent relations $R_1$ and $R_2$ , the cover-join computes a cover $K$ of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ in time $\\widetilde{\\mathcal {O}}(|R_1|+|R_2|)$ and with size $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ .",
"Example 21 Consider again the product $R_1(A)\\bowtie R_2(B)$ in Example REF , where $R_1=[2]$ and $R_2=[n]$ with $n > 1$ .",
"Examples of covers of size $n$ over the decomposition $\\mathcal {T}$ with bags $\\lbrace A\\rbrace $ and $\\lbrace B\\rbrace $ are: $\\lbrace (1,i)\\mid i\\in [n]-\\lbrace k\\rbrace \\rbrace \\cup \\lbrace (2,k)\\rbrace $ for any $k\\in [n]$ ; $\\lbrace (1,i)\\mid i\\in [k]\\rbrace \\cup \\lbrace (2,j+k)\\mid j\\in [n-k]\\rbrace $ for any $k\\in [n-1]$ .",
"If $R_1=[m]$ with $m>n$ , then examples of covers over $\\mathcal {T}$ of minimum size $m$ are: $\\lbrace (i,i)\\mid i\\in [k-1]\\rbrace \\cup \\lbrace (k-1+i , k + i) \\mid i \\in [n-k]\\rbrace \\cup \\lbrace (n-1+i , k) \\mid i \\in [m-n+1]\\rbrace $ for any $k\\in [n]$ .",
"A cover over $\\mathcal {T}$ of maximal size $n+m-2$ is: $\\lbrace (1,i)\\mid i\\in [n-1]\\rbrace \\cup \\lbrace (j+1,n)\\mid j\\in [m-1]\\rbrace $ .",
"Below are depictions of the complete bipartite graph corresponding to the query result for $n=4$ and $m=5$ , where the edges in a minimal edge cover are solid lines and all other edges are dotted.",
"The left minimal edge cover corresponds to a cover over $\\mathcal {T}$ of minimum size $m=5$ , while the right minimal edge cover corresponds to a cover over $\\mathcal {T}$ of maximum size $n+m-2=7$ .",
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"Before we define such plans, we need to introduce some notation.",
"For a join tree $\\mathcal {J}$ of a query $Q$ , we write $\\mathcal {J}= \\mathcal {J}_1\\circ \\mathcal {J}_2$ if $\\mathcal {J}$ can be split into two non-empty subtrees $\\mathcal {J}_1$ and $\\mathcal {J}_2$ that are connected by a single edge in $\\mathcal {J}$ .",
"Any subtree $\\mathcal {J}^{\\prime }$ of $\\mathcal {J}$ defines the subquery of $Q$ that is the natural join of all relation symbols that are nodes in $\\mathcal {J}^{\\prime }$ .",
"Definition 22 (Cover-Join Plan) Given $(Q,\\mathcal {J},\\mathbf {D})$ , a cover-join plan $\\varphi $ over the join tree $\\mathcal {J}$ is defined recursively as follows: If $\\mathcal {J}$ consists of one node $R$ , then $\\varphi = R$ .",
"The plan $\\varphi $ returns $R$ .",
"If $\\mathcal {J}= \\mathcal {J}_1\\circ \\mathcal {J}_2$ and $\\varphi _{i}$ is a cover-join plan over $\\mathcal {J}_i$ , then $\\varphi =\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ .",
"The plan $\\varphi $ returns the result of $R_1\\ \\mathring{}\\ R_2$ , where the relation $R_i$ is returned by the plan $\\varphi _{i}$ ($i\\in [2]$ ).",
"Lemma REF states next that a cover-join plan computes a cover of the query result over the decomposition corresponding to a given join tree of the query.",
"Lemma 23 Given $(Q,\\mathcal {J},\\mathbf {D})$ where $\\mathbf {D}=\\lbrace R_i\\rbrace _{i\\in [n]}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(|K|)$ and with size $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Lemma REF states three remarkable properties of cover-join plans.",
"First, they compute covers compositionally: To obtain a cover of the entire query result it is sufficient to compute covers of the results for subqueries.",
"More precisely, for a cover-join plan $\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ , the sub-plans $\\varphi _{1}$ and $\\varphi _{2}$ compute covers for the subqueries defined by the joins of the relations in the join trees $\\mathcal {J}_1$ and respectively $\\mathcal {J}_2$ .",
"Then, the plan $\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ computes a cover for the join of the relations in the join tree $\\mathcal {J}=\\mathcal {J}_1\\circ \\mathcal {J}_2$ .",
"Second, the output of a cover-join plan is always a cover, regardless which cover is picked at each cover-join operator in the plan.",
"Third, it does not matter which cover-join plan we choose for a given join tree, the resulting covers are computed with the same time guarantee.",
"Nevertheless, different plans for the same join tree may lead to different covers (Example REF ).",
"These properties rely on the global consistency of the database and on the fact that the plans follow the structure of the join tree.",
"For arbitrary databases, a cover-join operator may wrongly construct covers using dangling tuples at the expense of relevant tuples that are not anymore covered and therefore lost.",
"Furthermore, plans that do not follow the structure of a join tree may be unsound (Example REF ).",
"Although each cover-join operator computes a cover of minimum size for the join of its input relations, the overall cover computed by a cover-join plan may not be a cover of minimum size of the query result (Example REF in Appendix ).",
"Example 24 A join tree that admits several splits can define many plans.",
"For instance, the join tree for the query $R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ is the path $R_1-R_2-R_3$ and admits two possible splits that lead to the plans $\\varphi _1 = (R_1(A,B) \\mathring{}R_2(B,C)) \\mathring{}R_3(C,D)$ and $\\varphi _2 = R_1(A,B) \\mathring{}(R_2(B,C) \\mathring{}R_3(C,D))$ .",
"The relations are those in Figure REF , now calibrated.",
"For this database, the covers computed by the sub-plans $R_1(A,B) \\mathring{}R_2(B,C)$ and $R_2(B,C) \\mathring{}R_3(C,D)$ correspond to full join results, since all join values only occur once in the relations.",
"By taking any possible cover at each cover-join operator in the plans, both plans yield the same four possible covers of the query result: One of them is $\\mathit {rel}(M)$ in Figure REF and two of them are $K_1$ and $K_2$ in Example REF .",
"The last cover is not depicted: It is the same as $K_1$ with the change that the values $d_1$ and $d_2$ are swapped between the first two rows.",
"A corollary of Proposition REF and Lemma REF is that covers over decompositions of arbitrary natural join queries can be computed in time proportional to their sizes.",
"Theorem 25 (Proposition REF , Lemma REF ) Given a natural join query $Q$ , decomposition $\\mathcal {T}$ of $Q$ , and database $\\mathbf {D}$ , a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ and with size $\\mathcal {O}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ where $Q$ is an arbitrary natural join query and $\\mathbf {D}$ is an arbitrary database, we can compute a cover in four steps: construct $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ such that $Q^{\\prime }$ is an acyclic natural join query, $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ and $\\mathbf {D}^{\\prime }$ consists of materializations of the bags of $\\mathcal {T}$ ; turn $\\mathbf {D}^{\\prime }$ into a globally consistent database $\\mathbf {D}^{\\prime \\prime }$ with respect to $Q^{\\prime }$ ; turn $\\mathcal {T}$ into a join tree $\\mathcal {J}$ of $Q^{\\prime }$ by replacing each bag by the corresponding relation symbol in $Q^{\\prime }$ ; and execute on $\\mathbf {D}^{\\prime \\prime }$ a cover-join plan for $Q^{\\prime }$ over $\\mathcal {J}$ .",
"Since there are arbitrarily large databases for which the size bounds on covers are tight (Theorem REF ), the cover-join plans, together with a worst-case optimal algorithm for materializing bags [21], represent a worst-case optimal algorithm for computing covers.",
"We conclude this section with three insights into the ability of cover-join plans to compute covers.",
"We give an example of an unsound cover-join plan that does not follow the structure of a join tree.",
"We then note the incompleteness of our cover-join plans due to the binary nature of the cover-join operator.",
"We give an example of a cover that cannot be computed with our cover-join plans, but can be computed using a multi-way cover-join operator.",
"Finally, we give an example showing that distinct cover-join plans over the same (or also distinct) join trees can yield incomparable sets of covers.",
"Example 26 (Unsound plan) Consider the query $Q = R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ , the following database with relations $R_1$ , $R_2$ , and $R_3$ , and four relations computed by cover-joining two of the three relations: Table: NO_CAPTION Following Definition REF , the plan $(R_1(A,B) \\mathring{}R_3(C,D)) \\mathring{}R_2(B,C)$ would require a split $\\mathcal {J}_{1,3}\\circ \\mathcal {J}_2$ of a join tree, where the join tree $\\mathcal {J}_{1,3}$ has two nodes $R_1$ and $R_3$ while the join tree $\\mathcal {J}_{2}$ has one node $R_2$ .",
"However, there is no join tree that allows such a split.",
"The cover-join $R_1(A,B) \\mathring{}R_3(C,D)$ computes one of the two covers $K_{1,3}$ and $K^{\\prime }_{1,3}$ .",
"The result of the join of $K^{\\prime }_{1,3}$ and $R_2$ is empty and so is the cover-join.",
"This means that this plan does not always compute a cover, which makes it unsound.",
"This problem cannot occur with cover-join plans over join trees of $Q$ .",
"The only cover-join plans over join trees of $Q$ are (up to commutativity) $(R_1(A,B)$ $\\mathring{}$ $R_2(B,C))$ $\\mathring{}$ $R_3(C,D)$ and $R_1(A,B) \\mathring{}(R_2(B,C)\\mathring{}R_3(C,D))$ .",
"The only cover of $R_1(A,B) \\mathring{}R_2(B,C)$ is $K_{1,2}$ above, which can be cover-joined with $R_3$ .",
"The only cover of $R_2(B,C) \\mathring{}R_3(C,D)$ is $K_{2,3}$ above, which can be cover-joined with $R_1$ .",
"Example 27 (Cover-Join Incompleteness) Consider the product query $Q =R_1(A)\\bowtie R_2(B)\\bowtie R_3(C)$ , the following database $\\mathbf {D}$ with relations $R_1$ , $R_2$ , and $R_3$ and one cover $K$ of the query result over the decomposition with bags $\\lbrace A\\rbrace $ , $\\lbrace B\\rbrace $ , and $\\lbrace C\\rbrace $ : Table: NO_CAPTION A decomposition of $Q$ can have up to three bags which are not included in other bags.",
"In case of decompositions with three bags, each bag consists of exactly one attribute.",
"These decompositions correspond to the join trees that are permutations of the three relation symbols.",
"There are three possible cover-join plans (up to commutativity) over these join trees: $\\varphi _1= R_1(A) \\mathring{}(R_2(B) \\mathring{}R_3(C))$ , $\\varphi _2= R_2(B) \\mathring{}(R_1(A) \\mathring{}R_3(C))$ and $\\varphi _3= R_3(C) \\mathring{}(R_1(A) \\mathring{}R_2(B))$ .",
"None of these plans can yield the cover $K$ above.",
"As discussed after Definition REF , a minimal edge cover corresponding to a cover computed by a binary cover-join operator can only have paths of one or two edges.",
"For instance, $\\pi _{\\lbrace A,B\\rbrace }K$ , which should correspond to a cover of $R_1(A) \\mathring{}R_2(B)$ , has the path of three edges $b_2 - a_1 - b_1 - a_2$ .",
"The cover-join $R_1(A) \\mathring{}R_2(B)$ would not create this path since it corresponds to a non-minimal edge cover.",
"Similarly, $\\pi _{\\lbrace A,C\\rbrace }K$ and $\\pi _{\\lbrace B,C\\rbrace }K$ have paths of three edges.",
"For decompositions with two bags, two of the three attributes are in the same bag.",
"Without loss of generality, assume $A$ and $B$ are in the same bag.",
"Following Proposition REF , this bag is covered by a new relation $R_{1,2}$ that is the product of $R_1$ and $R_2$ .",
"This means that $K$ has to be the cover of $R_{1,2}(A,B)\\mathring{}R_3(C)$ , yet $\\pi _{\\lbrace A,B\\rbrace }K$ is not $R_{1,2}$ !",
"The decomposition with one bag consisting of all three attributes has this bag covered by a new relation that is the product of the three relations.",
"This relation is the Cartesian product of the three relations that is the full query result and different from $K=\\pi _{\\lbrace A,B,C\\rbrace }K$ .",
"We conclude that the cover $K$ cannot be computed using cover-join plans with binary cover-join operators.",
"Example 28 (Incomparable Sets of Covers) Consider the product query $Q=R_1(A)\\bowtie R_2(B)\\bowtie R_3(C)$ and the following database $\\lbrace R_1, R_2, R_3\\rbrace $ : Table: NO_CAPTION Let us consider the join tree $\\mathcal {J}=R_1-R_2-R_3$ of $Q$ .",
"There are (up to commutativity) two possible cover-join plans over $\\mathcal {J}$ : $\\varphi _1= R_1(A) \\mathring{}(R_2(B) \\mathring{}R_3(C))$ and $\\varphi _2=(R_1(A) \\mathring{}R_2(B)) \\mathring{}R_3(C)$ .",
"The above relation $K$ is a cover of the result of $Q$ and can be computed by $\\varphi _1$ , which cover-joins $R_1(A)$ and a cover of the join of $R_2(B)$ and $R_3(C)$ .",
"This cover cannot be computed by $\\varphi _2$ .",
"Indeed, $\\varphi _2$ first cover-joins $R_1(A)$ and $R_2(B)$ , yielding $K_{1,2}$ or $K^{\\prime }_{1,2}$ as the only possible covers.",
"Then, cover-joining any of them with $R_3(C)$ does not yield the cover $K$ since $\\pi _{\\lbrace A,B\\rbrace }K$ is different from both $K_{1,2}$ and $K^{\\prime }_{1,2}$ .",
"Similarly, $\\varphi _2$ computes covers that cannot be computed by $\\varphi _1$ .",
"Covers for Functional Aggregate Queries We first give a brief introduction to functional aggregate queries (FAQ) [17].",
"A detailed description can be found in the appendix.",
"Given an attribute set $S$ , we use ${\\textsf {a}}_S$ to indicate that tuple ${\\textsf {a}}$ has schema $S$ .",
"For $S^{\\prime } \\subseteq S$ , we denote by ${\\textsf {a}}_{S^{\\prime }}$ the restriction of ${\\textsf {a}}$ to $S^{\\prime }$ .",
"A functional aggregate query has the following form (slightly adapted to our notation): $\\varphi ({\\textsf {a}}_{\\lbrace A_1,\\ldots ,A_f\\rbrace }) = \\underset{a_{f+1} \\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n} \\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}}\\ \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes }\\ \\psi _S({\\textsf {a}}_S),\\text{ where:}$ $H=( \\mathcal {V}, \\mathcal {E})$ is the multi-hypergraph of the query with ${\\cal V}=\\lbrace A_i\\rbrace _{i\\in [n]}$ .",
"$\\textsf {Dom}$ is a fixed (output) domain, such as $\\lbrace $true,false$\\rbrace $ , $\\lbrace 0,1\\rbrace $ , or $\\mathbb {R}^+$ .",
"$\\mathcal {V}_{\\text{free}} = \\lbrace A_1,\\ldots ,A_f\\rbrace $ is the set of result or free attributes; all other attributes are bound.",
"For each attribute $A_i$ with $i>f$ , $\\oplus ^{(i)}$ is a binary (aggregate) operator on the domain $\\textsf {Dom}$ .",
"Different bound attributes may have different aggregate operators.",
"For each attribute $A_i$ with $i>f$ , either $\\oplus ^{(i)}$ is $\\otimes $ or $(\\textsf {Dom},\\oplus ^{(i)},\\otimes )$ forms a commutative semiring with the same additive identity ${\\bf 0}$ and multiplicative identity ${\\bf 1}$ for all semirings.",
"For every hyperedge $S$ in $\\cal E$ , $\\psi _S: \\prod _{A\\in S}\\textsf {dom}(A) \\rightarrow \\textsf {Dom}$ is an (input) function.",
"FAQs are a semiring generalization of aggregates over join queries, where the aggregates are the operators $\\oplus ^{(i)}$ and the natural join is expressed by $\\bigotimes _{S\\in \\mathcal {E}} \\psi _S({\\sf a}_S)$ .",
"The listing representation $R_{\\psi _S}$ of a function $\\psi _S$ is a relation over the schema $S \\cup \\lbrace \\psi _S(S)\\rbrace $ which consists of all input-output pairs for $\\psi _S$ where the output is non-zero, i.e., $R_{\\psi _S}$ contains a tuple ${\\textsf {a}}_{S \\cup \\lbrace \\psi _S(S)\\rbrace }$ if and only if $\\psi _S({\\textsf {a}}_S) = {\\textsf {a}}_{\\psi _S(S)} \\ne {\\bf 0}$ .",
"An input database for $\\varphi $ contains for each $\\psi _S$ its listing representation.",
"We say that $\\mathcal {T}$ is a decomposition of $\\varphi $ if $\\mathcal {T}$ is a decomposition of the hypergraph $H$ of $\\varphi $ .",
"Given an FAQ $\\varphi $ and database $\\mathbf {D}$ , the FAQ-problem is to compute the query result $\\varphi (\\mathbf {D})$ .",
"Each FAQ $\\varphi $ has an FAQ-width $\\textsf {faqw}(\\varphi )$ which is defined similarly to the fractional hypertree width of the hypergraph of $\\varphi $ .",
"For instance, in case where all attributes of $\\varphi $ are free, $\\textsf {faqw}(\\varphi )$ is equal to the fractional hypertree width of the hypergraph of $\\varphi $ .",
"Given an FAQ $\\varphi $ and a database $\\mathbf {D}$ , the InsideOut algorithm [17] solves the FAQ-problem as follows.",
"First, it eliminates all bound attributes along with their corresponding aggregate operators by performing equivalence-preserving transformations on $\\varphi $ .",
"Then, it computes the listing representation of the remaining query.",
"The algorithm runs in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )} + Z)$ where $Z$ is the size of the output, i.e., the listing representation of $\\varphi $ .",
"We can compute a cover of the result of a given FAQ $\\varphi $ in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )})$ , which does not depend on the size of the listing representation of $\\varphi $ .",
"Our strategy is as follows.",
"We first eliminate all bound attributes in $\\varphi $ by using InsideOut resulting in an FAQ $\\varphi ^{\\prime }$ .",
"We then take a decomposition $\\mathcal {T}$ of $\\varphi ^{\\prime }$ and compute bag functions $\\beta _B$ , $B \\in {\\cal S}(\\mathcal {T})$ , with $\\varphi ^{\\prime }({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) =\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B})$ .",
"Finally, we compute a cover of the join result of the listing representations of the bag functions over the extension of $\\mathcal {T}$ that contains, for each bag $B$ , the attribute $\\beta _B(B)$ for the values of the function $\\beta _B$ .",
"Keeping the $\\beta _B(B)$ -values of the bag functions in the cover is necessary for recovering the output values of $\\varphi $ when enumerating the result of $\\varphi $ from the cover.",
"Example 29 We consider the following FAQ $\\varphi $ over the sum-product semiring $(\\mathbb {N},+,\\cdot )$ (for simplicity we skip the explicit iteration over the domains of the attributes in $\\varphi $ ): $\\varphi (a,b,d) = \\sum _{c,e,f,g,h} \\psi _1(a,b,c)\\cdot \\psi _2(b,d,e)\\cdot \\psi _3(d,e,f)\\cdot \\psi _4(f,h)\\cdot \\psi _5(e,g), \\mbox{ where }$ $\\varphi $ , $\\psi _1$ , $\\psi _2$ , $\\psi _3$ , $\\psi _4$ and $\\psi _5$ are over $\\lbrace A,B,D\\rbrace $ , $\\lbrace A,B,C\\rbrace $ , $\\lbrace B,D,E\\rbrace $ , $\\lbrace D,E,F\\rbrace $ , $\\lbrace F,H\\rbrace $ and $\\lbrace E,G\\rbrace $ , respectively.",
"We first run InsideOut on $\\varphi $ to eliminate the bound attributes and obtain the following FAQ: $\\varphi ^{\\prime }(a,b,d) &= \\underbrace{\\big (\\sum _{c} \\psi _1(a,b,c)\\big )}_{\\psi _6(a,b)}\\cdot \\underbrace{\\sum _e \\big (\\psi _2(b,d,e)\\cdot \\underbrace{\\sum _f \\big (\\psi _3(d,e,f)\\cdot \\underbrace{\\sum _h \\psi _4(f,h)}_{\\psi _7(f)}\\big )}_{\\psi _{9}(d,e)}\\cdot \\underbrace{\\sum _g \\psi _5(e,g)}_{\\psi _8(e)}\\big )}_{\\psi _{10}(b,d)}.$ We consider the decomposition $\\mathcal {T}$ of $\\varphi ^{\\prime }$ with two bags $B_1 = \\lbrace A,B \\rbrace $ and $B_2 = \\lbrace B,D\\rbrace $ and bag functions $\\psi _6$ and respectively $\\psi _{10}$ .",
"Then, we execute the cover-join plan $R_{\\psi _6}\\ \\mathring{}\\ R_{\\psi _{10}}$ over the extended decomposition $\\mathcal {T}^{\\prime }$ with bags $\\lbrace A,B, \\psi _{6}(A,B)\\rbrace $ and $\\lbrace B,D, \\psi _{10}(B,D)\\rbrace $ .",
"While the computation of the result of $\\varphi ^{\\prime }$ can take quadratic time, the above cover-join plan takes linear time.",
"We exemplify the computation of the cover-join plan.",
"Assume the following tuples in $\\psi _6$ and $\\psi _{10}$ , where $\\gamma _1,\\ldots ,\\gamma _4,\\delta _1,\\ldots ,\\delta _3\\in \\mathbb {N}$ : Table: NO_CAPTION The relation $K$ is a possible cover computed by the cover-join plan.",
"The cover carries over the aggregates in columns $\\psi _{6}(A,B)$ and $\\psi _{10}(B,D)$ , one per bag of $\\mathcal {T}^{\\prime }$ .",
"The aggregate of the first tuple in $K$ is $\\gamma _1\\cdot \\delta _1$ (or $\\gamma _1\\otimes \\delta _1$ under a semiring with multiplication $\\otimes $ ).",
"The following theorem relies on Lemma REF and Theorem REF that give an upper bound on the time complexity for constructing covers of join results.",
"Theorem 30 For any FAQ $\\varphi $ and database $\\mathbf {D}$ , a cover of the query result $\\varphi (\\mathbf {D})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {faqw}(\\varphi )})$ .",
"Any enumeration algorithm for covers of join results can be used to enumerate the tuples of an FAQ result from one of its covers.",
"We thus have the following corollary: Corollary 31 (Corollary REF ) Given a cover $K$ of the result $\\varphi (\\mathbf {D})$ of an FAQ $\\varphi $ over a database $\\mathbf {D}$ , the tuples in the query result $\\varphi (\\mathbf {D})$ can be enumerated with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space.",
"Conclusion Results of join and functional aggregate queries entail redundancy in both their computation and representation.",
"In this paper we propose the notion of covers of query results to reduce such redundancy.",
"While covers can be more succinct than the query results, they nevertheless enjoy desirable properties such as listing representation and constant-delay enumeration of result tuples.",
"For a given database and a join or functional aggregate query, the query result can be normalized as a globally consistent database over an acyclic schema.",
"Covers represent one-relational, lossless, linear-size encodings of such normalized databases.",
"Definition 32 borged /bôrjd/ : Buy One Relation, Get Entire Database!",
"Acknowledgements.",
"This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement 682588.",
"The authors would like to thank Milos Nikolic, Max Schleich, and the anonymous reviewers for their feedback on drafts of this paper, and Yu Tang for inspiring discussions that led to the concept of cover as a relational alternative to factorized representations.",
"Further Preliminaries We introduce necessary notation for the proofs in the following sections.",
"Restrictions of Queries and Databases.",
"Given a set $X$ of attributes and a natural join query $Q= R_1 \\bowtie \\ldots \\bowtie R_n$ , the $X$ -restriction of $Q$ is defined as $Q_X = R_1^X \\bowtie \\ldots \\bowtie R_n^X$ where each $R_i^X$ results from $R$ by restricting its schema to $X$ .",
"Likewise, we obtain the $X$ -restriction $\\mathbf {D}_X$ of a database $\\mathbf {D}$ by projecting each relation in $\\mathbf {D}$ onto the attributes in $X$ .",
"From Covers to D-Representations We next give a brief introduction to d-representations; for a detailed description, we refer the reader to the literature [23].",
"We then discuss a translation from covers to d-representations.",
"Figure: Top row: database 𝐃={R 1 ,R 2 ,R 3 }\\mathbf {D}=\\lbrace R_1,R_2, R_3\\rbrace , the result Q(𝐃)Q(\\mathbf {D}) of the path query Q=R 1 ⋈R 2 ⋈R 3 Q = R_1\\bowtie R_2\\bowtie R_3, and a cover K⊆Q(𝐃)K\\subseteq Q(\\mathbf {D}) over the decomposition 𝒯\\mathcal {T}; bottom row: decomposition 𝒯\\mathcal {T} of QQ and an equivalent d-tree 𝒯 ' \\mathcal {T}^{\\prime }.D-Representations in a Nutshell D-representations are a succinct and lossless representation for relational data.",
"A d-representation is a set of named relational algebra expressions $\\lbrace N_1: =E_1, \\ldots , N_n:= E_n\\rbrace $ , where each $N_i$ is a unique name (or a pointer) and each $E_i$ is a relational algebra expression with unions, Cartesian products, singleton relations, i.e., unary relations with one tuple, and name references in place of singleton relations.",
"The size $\\mid \\hspace{-2.84526pt} E \\hspace{-2.84526pt} \\mid $ of a d-representation $E$ is the number of its singletons.",
"We consider a special class of d-representations that encode results of join queries and whose nesting structure is given by so-called d-trees.",
"In the literature, d-trees are defined as orderings on query variables.",
"We give here an alternative, equivalent definition that is in line with our notion of fractional hypertree decomposition.",
"Given a query $Q$ , a d-tree of $Q$ is a decomposition of $Q$ where each bag is partitioned into one attribute $A$ , called the bag attribute, and a set of attributes, called the key of $A$ and denoted by $\\mathit {key}(A)$ .",
"There is one bag per distinct attribute $A$ in $Q$ .",
"Each decomposition $\\mathcal {T}$ of a query $Q$ can be translated into a d-tree $\\mathcal {T}^{\\prime }$ of $Q$ with $\\textsf {fhtw}(\\mathcal {T}^{\\prime }) \\le \\textsf {fhtw}(\\mathcal {T})$ (Proposition 9.3 in [23]).",
"Given a query $Q$ , a d-tree $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , a d-representation $E$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ with size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ (Theorem 7.13 and Proposition 8.2 in [23]).",
"Example 33 We consider the path query $Q = R_1(A,B) \\bowtie R_2(B,C) \\bowtie R_3(C,D)$ .",
"Figure REF depicts a database with relations $R_1$ , $R_2$ and $R_3$ and the result of $Q$ over the input database $\\lbrace R_1, R_2, R_3\\rbrace $ .",
"It also shows a decomposition $\\mathcal {T}$ of $Q$ and a cover $K$ of the query result over $\\mathcal {T}$ .",
"Finally, it depicts a d-tree $\\mathcal {T}^{\\prime }$ (right below) derived from $\\mathcal {T}$ by using the translation in the proof of Proposition 9.3 in [23].",
"Figure: A d-representation encoded as a parse graph (left) and asa set of multimaps (right).D-representations admit encoding as parse graphs and sets of multi-maps.",
"Figure REF visualizes the d-representation of the query result from Figure REF over the d-tree $\\mathcal {T}^{\\prime }$ in the forms of a parse graph and of multi-maps.",
"The parse graph follows the structure of the d-tree.",
"At the top level we have a union of $B$ -values.",
"Then, given any $B$ -value, the $A$ -values are independent of the values for $C$ and $D$ .",
"Therefore, under each $B$ -value, the $A$ -values are represented in a different branch than the values for $C$ and $D$ .",
"Within the branches for $C$ and $D$ , the values are first grouped by $C$ and then by $D$ .",
"The information on keys is used to share subtrees across branches.",
"Since the key of attribute $D$ is $C$ only, all $C$ -nodes with the same value point to the same union of $D$ -values.",
"In our example, both $c_1$ -nodes point to the same set $\\lbrace d_1,d_2\\rbrace $ of $D$ -values.",
"The cover $K$ from Figure REF can be mapped immediately to the parse graph: Under each product node, we take a minimum number of combinations of its children to ensure that every value under the product node occurs in one of these combinations.",
"To enumerate the tuples in the query result, it suffices to choose in turn one branch of each union node and all branches of each product node.",
"For instance, the left product node represents the combinations of $\\lbrace a_1,a_2\\rbrace $ with $\\lbrace d_1,d_2\\rbrace $ , together with the values $b_1$ and $c_1$ .",
"There are four combinations, so four tuples in the result.",
"The first two tuples in the cover represent two of them, yet they are sufficient to recover all these tuples.",
"The multi-map encoding of a d-representation consists of one multi-map for each bag attribute: $m_A$ maps tuples over the attributes in $key(A)$ to (possibly several) values of $A$ .",
"Figure REF shows these maps as relations whose columns are distinctly separated into those for the key attributes (the map keys) and the column for the attribute $A$ itself (the map payload).",
"We have, for instance, $m_A(b_1) = a_1$ and $m_A(b_1) = a_2$ , whereas $m_C(b_1) = c_1$ .",
"Since $\\mathit {key}(A)=\\lbrace B\\rbrace $ and there are two $B$ -values in the d-representation leading to the sets $\\lbrace a_1,a_2\\rbrace $ and $\\lbrace a_3,a_4\\rbrace $ , respectively, $m_A$ maps the $B$ -value $ b_1$ to both $A$ -values $a_1$ and $a_2$ and the $B$ -value $b_2$ to both $A$ -values $a_3$ and $a_4$ .",
"Translating Covers into D-Representations Figure: Translating a cover KK over a decomposition 𝒯\\mathcal {T} into an equivalent d-representation.Figure REF gives an algorithm that constructs an equivalent d-representation from a cover over a decomposition.",
"Both the cover $K$ and the output d-representation are for the same query result $Q(\\mathbf {D})$ of a query $Q$ .",
"The decomposition $\\mathcal {T}$ is for the query $Q$ .",
"The algorithm creates a multi-map for each attribute $A$ and populates it with assignments of tuples over the keys of $A$ to the values of $A$ as encountered in the tuples of the cover.",
"Example 34 We consider the cover $K$ over the decomposition $\\mathcal {T}$ in Figure REF and the d-tree $\\mathcal {T}^{\\prime }$ equivalent to $\\mathcal {T}$ .",
"Following the algorithm in Figure REF , the cover $K$ is translated into a d-representation over $\\mathcal {T}^{\\prime }$ as follows.",
"After reading the first tuple $(a_1,b_1,c_1,d_1)$ , we add $() \\mapsto b_1$ to $m_B$ , $b_1 \\mapsto a_1$ to $m_A$ , $b_1 \\mapsto c_1$ to $m_C$ , and $c_1 \\mapsto d_1$ to $m_D$ , where $()$ means the empty tuple.",
"After processing the second tuple $(a_2,b_1,c_1,d_1)$ , we only change $m_A$ by adding $b_1 \\mapsto a_2$ to $m_A$ .",
"After the third tuple $(a_3,b_2,c_1,d_2)$ , we add the following new assignments: $() \\mapsto b_2$ to $m_B$ , $b_2 \\mapsto a_3$ to $m_A$ , $b_2 \\mapsto c_1$ to $m_C$ , and $c_1 \\mapsto d_2$ to $m_D$ .",
"After reading the last tuple $(a_4,b_2,c_1,d_2)$ , we add the new assignment $b_2 \\mapsto a_4$ to $m_A$ .",
"Cover-Join Plans Computing Covers of Non-Minimum Size Example 35 We consider the acyclic natural join query $Q=R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ , the database $\\mathbf {D}=\\lbrace R_1, R_2, R_3\\rbrace $ globally consistent with respect to $Q$ , and the join tree $\\mathcal {J}=R_1-R_2-R_3$ .",
"The relations $R_i$ are depicted below.",
"Table: NO_CAPTION The relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ corresponding to $\\mathcal {J}$ .",
"It follows from Proposition REF that every cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least three.",
"Hence, $K$ is a minimum-sized cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We take the cover-join plan $(R_1(A,B) \\mathring{}R_2(B,C)) \\mathring{}R_3(C,D)$ over $\\mathcal {J}$ and assume that the cover-join operator computes for each two input relations $R$ and $R^{\\prime }$ , a minimum-sized cover of $R \\bowtie R^{\\prime }$ over the decomposition with bags ${\\cal S}(R)$ and ${\\cal S}(R^{\\prime })$ .",
"Then, a possible output of the sub-plan $R_1(A,B) \\mathring{}R_2(B,C)$ is the relation $K_{1,2}$ .",
"A possible result of the cover-join of the latter relation with $R_3$ is the relation $K^{\\prime }$ .",
"Although $K^{\\prime }$ is a valid cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ , it is not a minimum-sized cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"Covers for Equi-Join Queries In this section, we extend the class of queries from natural join queries to arbitrary equi-join queries, whose relation symbols may map to the same database relation.",
"Equi-join Queries.",
"An equi-join query, aka full conjunctive query, has the form $Q = \\sigma _\\psi (R_1(S_1) \\times \\ldots \\times R_n(S_n))$ , where each $R_i$ is a relation symbol with schema $S_i$ and $\\psi $ is a conjunction of equalities of the form $A_1=A_2$ with attributes $A_1$ and $A_2$ .",
"We require that all relation symbols in the query as well as all attributes occurring in the schemas of the relation symbols are distinct.",
"We assume that each query comes with mappings $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ , called the signature mappings of $Q$ , where $\\lambda $ maps the relation symbols in $Q$ to relation symbols in the schema of the database and each $\\mu _{R_i}$ is a bijective mapping from the attributes of $R_i$ to the attributes of $\\lambda (R_i)$ .",
"Since we do not require $\\lambda $ to be injective, distinct relation symbols in $Q$ might refer to the same relation in the database (cf.",
"Example REF ).",
"The joins in equi-join queries are expressed by the equalities in $\\psi $ .",
"The transitive closure $\\psi ^+$ of $\\psi $ under the equality on attributes defines the attribute equivalence classes: The equivalence class ${\\cal A}$ of an attribute $A$ is the set consisting of $A$ and of all attributes equal to $A$ in $\\psi ^+$ .",
"For a set $S$ of attributes, $S^+$ denotes the set of attributes transitively equivalent to those in $S$ .",
"Hypergraphs and hypertree decompositions of equi-join queries are defined just like for natural join queries with the additional requirement that each hyperedge or bag includes all equivalent attributes for each contained attribute.",
"More formally, the hypergraph of an equi-jon query $Q$ consists of one node $A$ for each attribute $A$ in $Q$ and one edge ${\\cal S}(R)^+$ for each relation symbol $R\\in {\\cal S}(Q)$ .",
"Similarly, a hypertree decomposition $\\mathcal {T}$ (of the hypergraph $H$ ) of $Q$ is a pair $(T,\\chi )$ , where $T$ is a tree and $\\chi $ is a function mapping each node in $T$ to a set $V^+$ where $V$ is a subset of the nodes of $H$ .",
"All other notions and notations introduced in Section as well as the definitions of result preservation and covers in Section carry over to equi-join queries without any change.",
"Example 36 We consider the equi-join query $Q=\\sigma _\\psi (R_1(A_1,A_2)\\times R_2(A_3,A_4))$ , where $\\psi $ consists of the equality $A_2=A_3$ .",
"Let $(\\lambda , \\lbrace \\mu _{R_1}, \\mu _{R_2}\\rbrace )$ be the signature mappings of the query.",
"Assume that $\\lambda (R_1(A_1,A_2)) = \\lambda (R_2(A_3,A_4)) = R(A,B)$ , $\\mu _{R_1}(A_1) = \\mu _{R_2}(A_4) = A$ and $\\mu _{R_1}(A_2) = \\mu _{R_2}(A_3) = B$ , i.e., both relation symbols are mapped to the same relation symbol $R(A,B)$ , attributes $A_1$ and $A_4$ are mapped to attribute $A$ and attributes $A_2$ and $A_3$ are mapped to attribute $B$ .",
"Let $\\mathbf {D} = \\lbrace R\\rbrace $ where $R$ is defined as in Figure REF .",
"The figure depicts in the top row (besides $R$ ) the query result $Q(\\mathbf {D})$ , a cover $K$ of the query result over the decomposition $\\mathcal {T}$ depicted in the bottom row and two relations $R_1^{\\prime }, R_2^{\\prime }$ obtained from $R$ by the application of Proposition REF (given below).",
"The bottom row shows the hypergraph of $Q$ , the hypergraph of $Q(\\mathbf {D})$ over the attribute sets $\\lbrace \\lbrace A_1,A_2,A_3\\rbrace ,\\lbrace A_2,A_3,A_4\\rbrace \\rbrace $ , and a minimal edge cover $M$ of the latter hypergraph with $\\mathit {rel}(M) = K$ .",
"Figure: Top row: database 𝐃={R}\\mathbf {D}=\\lbrace R\\rbrace , the result Q(𝐃)Q(\\mathbf {D}) of the queryQQ in Example , a cover KK ofQ(𝐃)Q(\\mathbf {D}) over 𝒯\\mathcal {T}, and relations R 1 ' ,R 2 ' R_1^{\\prime },R_2^{\\prime } obtainedfrom RR by the application of Proposition ;bottom row: the hypergraph of QQ, a decomposition 𝒯\\mathcal {T} of QQ, the hypergraph of Q(𝐃)Q(\\mathbf {D}) over the attribute sets 𝒮(𝒯){\\cal S}(\\mathcal {T}), and a minimal edge cover MM of this hypergraph.Adaption of the results on covers to equi-join queries.",
"Due to the following two propositions, all results on covers in Sections and carry over to equi-join queries.",
"Proposition 37 Given an equi-join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , there exist a natural join query $Q^{\\prime }$ and a database $\\mathbf {D}^{\\prime }$ such that: $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , $Q^{\\prime }$ has the decomposition $\\mathcal {T}$ and can be constructed in time $\\mathcal {O}(|Q|)$ , and $\\mathbf {D}^{\\prime }$ can be constructed in time $\\mathcal {O}(|\\mathbf {D}|)$ .",
"We briefly explain the construction.",
"The query $Q^{\\prime }$ is obtained from $Q$ by replacing each relation symbol $R(S)$ in $Q$ by a relation symbol $R^{\\prime }(S^+)$ .",
"The database $\\mathbf {D}^{\\prime }$ contains, for each relation symbol $R^{\\prime }(S^+)$ in $Q^{\\prime }$ , a relation over the same schema that is obtained from relation $\\lambda (R(S))$ as follows: for each attribute $A$ contained in $S^+$ but not in $S$ , $\\lambda (R(S))$ is extended by a new $A$ -column that is a copy of any $B$ -column in $\\lambda (R(S))$ such that $A$ is equivalent to $B$ .",
"Figure REF gives in the top row two relations $R_1^{\\prime }$ and $R_2^{\\prime }$ that result from relation $R$ by the application of Proposition REF in case $Q$ is defined as in Example REF .",
"It follows from Proposition REF that, since $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , any relation $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if $K$ is a cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ .",
"Given the construction times for $Q^{\\prime }$ and $D^{\\prime }$ , all our results on natural join queries in Sections and , except the lower size bound on covers in Theorem REF (ii), hold for equi-join queries, too.",
"The following proposition is the counterpart of Theorem REF (ii) for equi-join queries.",
"Proposition 38 For any equi-join query $Q$ and any decomposition $\\mathcal {T}$ of $Q$ , there are arbitrarily large databases $\\mathbf {D}$ such that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"In Proposition REF , we first construct a natural join query $Q^{\\prime }$ from $Q$ as in Proposition REF .",
"By Theorem REF (ii), there are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that each cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given such a database $\\mathbf {D}^{\\prime }$ , it follows from Proposition REF , that $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })| =\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ , hence, $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace = \\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The database $\\mathbf {D}^{\\prime }$ can be converted into a database $\\mathbf {D}$ of size $\\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ such that $|\\pi _BQ(\\mathbf {D})| \\ge |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By Proposition REF (adapted to equi-join queries), each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ .",
"Since $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ $=$ $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ and $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ $\\ge $ $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ , we conclude that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is of size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Missing Proofs of Section Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , we can compute $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ with size ${\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ and in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ such that $Q^{\\prime }$ is an acyclic natural join query, $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ , $\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ and $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"The construction is standard in the literature [1], [24].",
"For convenience, we describe the main ideas.",
"Construction.",
"The construction comprises two transformation steps.",
"We first compute $R_B = Q_B(\\mathbf {D}_B)$ for each $B \\in {\\cal S}(\\mathcal {T})$ (recall that $Q_B$ and $\\mathbf {D}_B$ are $B$ -restrictions of $Q$ and $\\mathbf {D}$ , respectively).",
"Let $\\widehat{\\mathbf {D}}=\\lbrace R_B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ and $\\widehat{Q}=\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B$ .",
"In the second transformation step, we execute a semi-join programme on $\\widehat{\\mathbf {D}}$ to turn it into a database $\\mathbf {D}^{\\prime } = \\lbrace R_B^{\\prime }\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ that is pairwise consistent with respect to $\\widehat{Q}$ , i.e., $\\mathbf {D}^{\\prime }$ does not contain any pair of relations such that one of the two relations contains a tuple which cannot be joined with any tuple from the other relation.",
"To achieve pairwise consistency, it is not necessary to consider all pairs of relations in $\\widehat{\\mathbf {D}}$ .",
"It suffices to execute a bottom-up and a subsequent top-down traversal in $\\mathcal {T}$ [27].",
"During each traversal, we delete for each father-child pair $B_1, B_2$ of bags, all tuples in each of the two relations $R_{B_1}$ and $R_{B_2}$ which do not have any join partner in the other relation.",
"We define $Q^{\\prime }=\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B^{\\prime }$ .",
"$Q^{\\prime }$ is an acyclic natural join query and $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ .",
"By construction, we have a one-to-one correspondence between relation symbols $R_B^{\\prime } \\in {\\cal S}(Q^{\\prime })$ and bags $B \\in {\\cal S}(\\mathcal {T})$ with ${\\cal S}(R_B^{\\prime })=B$ .",
"Hence, $\\mathcal {T}$ corresponds to the join tree of $Q^{\\prime }$ that is obtained from $\\mathcal {T}$ by, basically, replacing each bag by the corresponding relation symbol in $Q$ .",
"Since $Q^{\\prime }$ has a join tree, it is acyclic.",
"$\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ .",
"The relations in $\\mathbf {D}^{\\prime }$ are pairwise consistent with respect to $Q^{\\prime }$ .",
"For acyclic queries, pairwise consistency implies global consistency (Theorem 6.4.5 of [1]).",
"Hence, $\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ .",
"$Q(\\mathbf {D}) = Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Since the second transformation step only deletes tuples in $\\widehat{\\mathbf {D}}$ which do not contribute to the result of $\\widehat{Q}(\\widehat{\\mathbf {D}})$ , it suffices to show that $Q(\\mathbf {D}) = \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"Let $Q = \\bowtie _{i \\in [n]}R_i$ .",
"We first show $Q(\\mathbf {D}) \\subseteq \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"Let $t \\in Q(\\mathbf {D})$ .",
"Since $\\pi _{B} Q(\\mathbf {D}) \\subseteq Q_B(\\mathbf {D}_B)$ , it follows that $\\pi _{B} t \\in Q_B(\\mathbf {D}_B)$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Hence, $\\pi _{B} t \\in R_B$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B t$ , we derive that $t \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B$ , thus, $t \\in \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"We now show $\\widehat{Q}(\\widehat{\\mathbf {D}}) \\subseteq Q(\\mathbf {D})$ .",
"Let $t \\in \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"By definition, $\\pi _{B}t \\in R_B$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By the fact that the attributes of each relation symbol in $Q$ are covered by at least one bag of $\\mathcal {T}$ and by the construction of the relations $R_B$ , it holds that $\\pi _{{\\cal S}(R_i)}t \\in R_i$ for each $i \\in [n]$ .",
"This implies $t \\in Q(\\mathbf {D})$ .",
"Construction size.",
"Each relation $R_B$ in $\\widehat{\\mathbf {D}}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [4].",
"Since $\\textsf {fhtw}(\\mathcal {T})=\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace \\rho ^*(Q_B)\\rbrace $ , it follows that the size of $\\widehat{\\mathbf {D}}$ is $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The semi-join program on $\\widehat{\\mathbf {D}}$ does not increase the size of the database.",
"The size of $Q^{\\prime }$ is $\\mathcal {O}(|Q|)$ .",
"Altogether, the size of $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ is $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Construction time.",
"Each relation $R_B$ in $\\widehat{\\mathbf {D}}$ is computable in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [21].",
"By $\\textsf {fhtw}(\\mathcal {T})=\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace \\rho ^*(Q_B)\\rbrace $ , we derive that the computation time for $\\widehat{\\mathbf {D}}$ is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"During the semi-join program on $\\widehat{\\mathbf {D}}$ , we can achieve consistency between each pair $R_{B_1}, R_{B_2}$ of father-child relations as follows.",
"We first sort both relations on the join attributes.",
"In a subsequent scan we delete in each of the relations each tuple with no join partner in the other relation.",
"Hence, the semi-join programme can be realised in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"It follows that the overall running time is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Missing Proofs of Section Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ with schema $\\mathit {att}(Q)$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if and only if $\\bowtie _{B\\in {\\cal S}(\\mathcal {T})} \\pi _B K = Q(\\mathbf {D})$ .",
"Proof of the “$\\Rightarrow $ ”-direction.",
"Assume that $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"We show in two steps that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K = Q(\\mathbf {D})$ .",
"$\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K \\subseteq Q(\\mathbf {D})$ : Let $t$ be an arbitrary tuple from $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"This means that $\\pi _{B}t \\in \\pi _{B}K$ for every $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , we derive that $\\pi _{B}t \\in \\pi _{B}Q(\\mathbf {D})$ for every $B \\in {\\cal S}(\\mathcal {T})$ .",
"By the definition of decompositions, for every relation symbol $R$ in $Q$ , there is at least one bag of $\\mathcal {T}$ containing all attributes of $R$ .",
"Hence, $\\pi _{{\\cal S}(R)}t \\in \\pi _{{\\cal S}(R)}Q(\\mathbf {D})$ for every $R \\in {\\cal S}(Q)$ .",
"It follows that $t$ is included in $Q(\\mathbf {D})$ .",
"Thus, $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K \\subseteq Q(\\mathbf {D})$ .",
"$Q(\\mathbf {D}) \\subseteq \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ : Let $t \\in Q(\\mathbf {D})$ .",
"It follows that $\\pi _B t \\in \\pi _B Q(\\mathbf {D})$ for every $B \\subseteq {\\cal S}(Q(\\mathbf {D}))$ , hence, in particular for every $B\\in {\\cal S}(\\mathcal {T})$ .",
"Due to result-preservation of $K$ with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ , this implies that $\\pi _B t \\in \\pi _B K$ for every $B\\in {\\cal S}(\\mathcal {T})$ which means that $t \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"Hence, $Q(\\mathbf {D}) \\subseteq \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"Proof of the “$\\Leftarrow $ ”-direction.",
"Assume that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B K = Q(\\mathbf {D})$ .",
"Given any $B \\in {\\cal S}(\\mathcal {T})$ , we show in two steps that $\\pi _B K = \\pi _B Q(\\mathbf {D})$ .",
"$\\pi _B K \\subseteq \\pi _B Q(\\mathbf {D})$ : Let $t$ be an arbitrary tuple from $\\pi _B K$ .",
"This means that there is a tuple $t^{\\prime } \\in K$ with $\\pi _B t^{\\prime }=t$ .",
"Since $\\pi _{B^{\\prime }} t^{\\prime } \\in \\pi _{B^{\\prime }}K$ for each $B^{\\prime } \\in {\\cal S}(\\mathcal {T})$ , we derive that $t^{\\prime }\\in \\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }}K$ .",
"Using our assumption $\\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }} K = Q(\\mathbf {D})$ , we get $t^{\\prime }\\in Q(\\mathbf {D})$ .",
"From the latter and the fact that $t = \\pi _B t^{\\prime }$ , it follows $t \\in \\pi _B Q(\\mathbf {D})$ .",
"Altogether, we conclude $\\pi _B K \\subseteq \\pi _B Q(\\mathbf {D})$ .",
"$\\pi _B Q(\\mathbf {D}) \\subseteq \\pi _B K$ : Let $t$ be an arbitrary tuple from $\\pi _B Q(\\mathbf {D})$ .",
"This means that there is a tuple $t^{\\prime } \\in Q(\\mathbf {D})$ with $\\pi _B t^{\\prime } = t$ .",
"By assumption, $t^{\\prime } \\in \\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }} K$ .",
"Since $B$ is an element of ${\\cal S}(\\mathcal {T})$ , the latter implies $\\pi _B t^{\\prime }= t \\in \\pi _B K$ .",
"Altogether, we get $\\pi _B Q(\\mathbf {D}) \\subseteq \\pi _B K$ .",
"Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ has a minimal edge cover $M$ such that $\\mathit {rel}(M)=K$ .",
"We first recall that $Q(\\mathbf {D})$ is obviously result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ and therefore, by Proposition REF , it holds $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D}) = Q(\\mathbf {D})$ .",
"Let $H=(V,E)$ be the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ .",
"Let $\\mathit {tuple}_V$ be a function mapping each node $v \\in V$ to its corresponding tuple in $\\bigcup _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Furthermore, let $\\mathit {tuple}_E$ be a function mapping each edge $e \\in E$ to $\\bowtie _{v \\in e}\\mathit {tuple}_V(v)$ .",
"The function $\\mathit {tuple}_V$ is a bijection from $V$ to $\\bigcup _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Since $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D}) = Q(\\mathbf {D})$ , $\\mathit {tuple}_E$ is a bijection from $E$ to $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Likewise, the function $\\mathit {rel}$ (as defined in Section ) is a bijection from subsets of $E$ to subsets of $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Proof of the “$\\Rightarrow $ ”-direction.",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We show that $\\mathit {rel}^-(K)$ is defined and a minimal edge cover of $H$ .",
"Let $t \\in K$ .",
"This means that for each $B \\in {\\cal S}(\\mathcal {T})$ , there is $t_B \\in \\pi _B K$ with $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})} t_B$ .",
"Since $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , each $t_B$ is included in $\\pi _B Q(\\mathbf {D})$ .",
"Hence, $\\mathit {tuple}_V^-(t_B)$ must be defined for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $Q(\\mathbf {D}) = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}Q(\\mathbf {D})$ , $t$ is included in $Q(\\mathbf {D})$ .",
"It follows that $\\mathit {tuple}_E^-(t) = \\lbrace \\mathit {tuple}_V^-(t_B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is defined.",
"Thus, $\\mathit {rel}^-(K) = \\lbrace \\mathit {tuple}_E^-(t)\\rbrace _{t \\in K}$ is defined.",
"It follows from $\\pi _BQ(\\mathbf {D}) = \\pi _BK$ , $B \\in {\\cal S}(\\mathcal {T})$ , that $\\mathit {rel}^-(K)$ is an edge cover of $H$ .",
"It remains to show that $\\mathit {rel}^-(K)$ is a minimal edge cover of $H$ .",
"For the sake of contradiction, assume that $\\mathit {rel}^-(K)$ is not a minimal edge cover of $H$ .",
"This implies that there is an edge $\\overline{e} \\in \\mathit {rel}^-(K)$ such that $\\mathit {rel}^-(K) \\backslash \\lbrace \\overline{e}\\rbrace $ is an edge cover of $H$ .",
"It follows that for each node $v \\in V$ , there is an edge $e \\in \\mathit {rel}^-(K) \\backslash \\lbrace \\overline{e}\\rbrace $ with $v \\in e$ .",
"This means that for each tuple $t_B \\in \\pi _B Q(\\mathbf {D})= \\pi _B K$ , there is a tuple $t \\in K \\backslash \\lbrace \\mathit {tuple}_E(\\overline{e})\\rbrace $ with $\\pi _B t = t_B$ .",
"We conclude that $K \\backslash \\lbrace \\mathit {tuple}_E(\\overline{e})\\rbrace $ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"The latter is, however, a contradiction to our assumption that $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and, therefore, a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"Proof of the “$\\Leftarrow $ ”-direction.",
"Let $M$ be a minimal edge cover of $H$ .",
"We show that $\\mathit {rel}(M)$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We first observe that ${\\cal S}(\\mathit {rel}(M)) = {\\cal S}(Q(\\mathbf {D})) = \\mathit {att}(Q)$ .",
"Since $Q(\\mathbf {D})$ is result-preserving with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ and $M$ is an edge cover of $H$ , $\\mathit {rel}(M)$ must also be result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"It remains to show that $\\mathit {rel}(M)$ is a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"For the sake of contradiction, assume that $\\mathit {rel}(M)$ is not minimal in that respect.",
"It follows that there is a tuple $\\overline{t}\\in \\mathit {rel}(M)$ such that $\\mathit {rel}(M) \\backslash \\lbrace \\overline{t}\\rbrace $ is result-preserving with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ .",
"This means that for each $B \\in {\\cal S}(\\mathcal {T})$ and each tuple $t_B \\in \\pi _BQ(\\mathbf {D})$ , there is a tuple $t \\in \\mathit {rel}(M) \\backslash \\lbrace \\overline{t}\\rbrace $ with $\\pi _Bt = t_B$ .",
"This implies that for each node $v \\in V$ , there is an edge $e \\in M \\backslash \\lbrace \\mathit {tuple}_E^-(\\overline{t})\\rbrace $ with $v \\in e$ .",
"We derive that $M \\backslash \\lbrace \\mathit {tuple}_E^-(\\overline{t})\\rbrace $ is a minimal edge cover of $H$ , a contradiction to the minimality of $M$ .",
"Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is a subset of $Q(\\mathbf {D})$ .",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and let $t\\in K$ be an arbitrary tuple from $K$ .",
"We show that $t$ must be included in $Q(\\mathbf {D})$ .",
"For each $B \\in {\\cal S}(\\mathcal {T})$ , let $t_B = \\pi _{B}t$ .",
"It holds that $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}t_B$ .",
"As $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , $t_B$ must be included in $\\pi _{B}Q(\\mathbf {D})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since by Proposition REF , $Q(\\mathbf {D}) = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _BQ(\\mathbf {D})$ , it follows that $t$ is included in $Q(\\mathbf {D})$ .",
"Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , the size of each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ satisfies the inequalities $\\max _{B\\in {\\cal S}(\\mathcal {T})}\\lbrace \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid \\rbrace $ $\\le $ $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ .",
"The first inequality holds due to $K$ being result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"The second inequality is implied by Proposition REF , since the hypergraph $H=(V,E)$ of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ must have a minimal edge cover $M$ with $\\mathit {rel}(M) = K$ .",
"Each hyperedge $e$ in $M$ must cover at least one node in $V$ which is not covered by any other hyperedge in $M$ .",
"Otherwise, $M\\backslash \\lbrace e\\rbrace $ would be an edge cover, which is a contradiction to the minimality of $M$ .",
"Hence, the total number of edges in $M$ is upper-bounded by $\\mid \\hspace{-2.84526pt} V \\hspace{-2.84526pt} \\mid $ .",
"As $\\mid \\hspace{-2.84526pt} V \\hspace{-2.84526pt} \\mid =\\Sigma _{B \\in {\\cal S}(\\mathcal {T})} \\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ and $\\mid \\hspace{-2.84526pt} M \\hspace{-2.84526pt} \\mid = \\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ , we derive that the number of tuples in $K$ is upper-bounded by $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ .",
"Proof of Theorem REF Theorem REF .",
"Let $Q$ be a natural join query and $\\mathcal {T}$ a decomposition of $Q$ .",
"For any database $\\mathbf {D}$ , each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"There are arbitrarily large databases $\\mathbf {D}$ such that each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Our proof relies on the results that for any natural join query $Q$ and database $\\mathbf {D}$ , it holds $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ and there are arbitrarily large databases $\\mathbf {D}$ with $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ [4].",
"Let $\\mathcal {T}=(T,\\chi ,\\lbrace \\gamma _t\\rbrace _{t \\in T})$ .",
"Given a node $t$ in $T$ with $\\chi (t) = B$ for some set $B$ , we recall that $\\mathit {weight}(\\gamma _{t}) = \\rho ^*(Q_{B})$ .",
"Moreover, if $\\mathit {weight}(\\gamma _{t})$ is maximal over all weight functions in $\\mathcal {T}$ , then $\\mathit {weight}(\\gamma _{t})$ $=\\textsf {fhtw}(\\mathcal {T})$ .",
"Proof of statement (i).",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and let $t$ be an arbitrary node of $T$ with $\\chi (t)=B$ for some set $B$ .",
"It holds $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [4], thus, $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Since $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $\\le $ $\\mid \\hspace{-2.84526pt} Q_B (\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ (Proposition 3.2 of [23]), it follows that $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Using Proposition REF , we conclude $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})} \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} {\\cal S}(\\mathcal {T}) \\hspace{-2.84526pt} \\mid \\cdot \\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Proof of statement (ii).",
"Let $t$ be a node in $T$ such that $\\gamma _t$ has maximal weight and let $\\chi (t)=B$ .",
"There are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}^{\\prime }) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ [4].",
"For each such database $\\mathbf {D}^{\\prime }$ , there exists a database $\\mathbf {D}$ with $\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid )$ and $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\Omega (\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}^{\\prime }) \\hspace{-2.84526pt} \\mid )= \\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ (Lemma 7.18 of [23]).",
"This means that there are arbitrarily large databases $\\mathbf {D}$ such that $\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Due to Proposition REF , each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must be at least of size $\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ , hence, $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid = \\Omega (\\mathbf {D}^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Proof of Proposition REF Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover $K$ of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ can be translated into a d-representation of $Q(\\mathbf {D})$ of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ .",
"Using the algorithm in Figure REF , we construct from $K$ and $\\mathcal {T}$ a d-representation of $Q(\\mathbf {D})$ encoded as a set M of maps.",
"Recall that the constructed d-representation is over a d-tree $\\mathcal {T}^{\\prime }$ equivalent to $\\mathcal {T}$ .",
"Correctness of the construction.",
"For each $m_A \\in M$ , we denote by $R_A$ the listing representation of $m_A$ as presented in Figure REF .",
"For each bag attribute $A$ in $\\mathcal {T}^{\\prime }$ , the set $\\lbrace A\\rbrace \\cup \\mathit {key}(A)$ constitutes a bag in the signature ${\\cal S}(\\mathcal {T}^{\\prime })$ of $\\mathcal {T}^{\\prime }$ .",
"We write $B_A$ to express that the bag attribute of $B_A$ is $A$ .",
"By the definition of d-representations, the query result represented by the map set $M$ is $R = \\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}R_A$ [23].",
"It remains to show that $R = Q(\\mathbf {D})$ .",
"By construction of the maps in $M$ , we have $R_A = \\pi _{B_A}K$ for each $B_A$ .",
"For each $B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })$ , there is a $B \\in {\\cal S}(\\mathcal {T})$ with $B_A \\subseteq B$ (proof of Proposition 9.3 in [23]).",
"Hence, by the definition of covers, we have $R_A = \\pi _{B_A}K = \\pi _{B_A} Q(\\mathbf {D})$ for each $B_A$ .",
"As $\\mathcal {T}^{\\prime }$ is a valid decomposition of $Q$ , it follows from Proposition REF that $\\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}\\pi _{B_A}K = Q(\\mathbf {D})$ .",
"Since for each $B_A$ , we have $\\pi _{B_A}K = R_A$ and $R = \\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}R_{A}$ , it follows $R = Q(\\mathbf {D})$ .",
"Construction size and translation time.",
"The number of the maps in $M$ is bounded by the number of attributes in $K$ .",
"We consider the cover $K$ sorted using a topological order of the decomposition $\\mathcal {T}^{\\prime }$ , so that inserts into the multimaps become appends (alternatively, inserts in sorted order would take logarithmic time in the number of entries).",
"For each tuple in $K$ we insert at most one tuple in the multimap of each attribute.",
"Thus, the overall size of the set of multimaps, and thus of the d-representation, is $\\mathcal {O}(|K|)$ with respect to data complexity (the linear factor in the number of attributes is ignored).",
"The data complexity of the overall translation time is thus $\\widetilde{\\mathcal {O}}(|K|)$ .",
"Missing Proofs of Section Proof of Proposition REF Proposition REF .",
"Given two consistent relations $R_1$ and $R_2$ , the cover-join computes a cover $K$ of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ in time $\\widetilde{\\mathcal {O}}(|R_1|+|R_2|)$ and with size $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ .",
"Let $Q = R_1 \\bowtie R_2$ , $\\mathbf {D} = \\lbrace R_1, R_2\\rbrace $ .",
"Moreover, let $\\mathcal {T}$ be the decomposition of $Q$ with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"By Proposition REF , a relation $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph $H$ of $Q(\\mathbf {D})$ over the attribute sets $\\lbrace {\\cal S}(R_1), {\\cal S}(R_2)\\rbrace $ has a minimal edge cover $M$ with $\\mathit {rel}(M)=K$ .",
"The hypergraph $H$ is a collection of disjoint complete bipartite subgraphs.",
"The set of nodes of each such subgraph corresponds to a maximal subset of tuples of the input relations agreeing on the join attributes.",
"A minimal edge cover of $H$ is a collection of minimal edge covers for these subgraphs.",
"We construct a cover $K$ of minimum size such that each maximal subset of tuples in $K$ agreeing on the join attributes corresponds to a minimal edge cover of one of the complete bipartite subgraphs of $H$ .",
"Construction.",
"Let $\\mathcal {A}$ be the set of common attributes of $R_1$ and $R_2$ .",
"For $i \\in \\lbrace 1,2\\rbrace $ and $t \\in \\pi _{\\mathcal {A}}R_i$ , we call $\\sigma _{\\mathcal {A}= t} R_i$ the $t$ -block in $R_i$ and denote its size by $n^t_i$ .",
"Since $R_1$ and $R_2$ are consistent, for each $t$ -block in $R_1$ , there must be a corresponding $t$ -block in $R_2$ , and vice-versa.",
"First, the algorithm sorts $R_1$ and $R_2$ with respect to the values of the attributes in $\\mathcal {A}$ .",
"After sorting, the $t$ -blocks occur in the same order in both relations.",
"The cover $K$ is constructed by performing the following procedure for each pair of corresponding $t$ -blocks in $R_1$ and $R_2$ .",
"Without loss of generality, assume $n_1^t \\ge n_2^t$ .",
"For each $j < n_{2}^t$ , the $j$ -th tuple $t^{\\prime }$ in the $t$ -block of $R_1$ is combined with the $j$ -th tuple $t^{\\prime \\prime }$ in the $t$ -block of $R_2$ resulting in a new tuple $t^{\\prime } \\bowtie t^{\\prime \\prime }$ .",
"Then, all remaining tuples in the $t$ -block of $R_1$ are combined with the $n_{2}^t$ -th tuple in the $t$ -block of $R_2$ .",
"All new tuples are added to $K$ .",
"Construction time.",
"The sorting phase can be realised in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} R_1 \\hspace{-2.84526pt} \\mid +\\mid \\hspace{-2.84526pt} R_2 \\hspace{-2.84526pt} \\mid )$ .",
"The phase for constructing the new tuples can be done in one pass over the sorted relations.",
"Hence, the overall running time of the described algorithm is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} R_1 \\hspace{-2.84526pt} \\mid +\\mid \\hspace{-2.84526pt} R_2 \\hspace{-2.84526pt} \\mid )$ .",
"Size of the Cover.",
"The size bounds $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ follow from Proposition REF and the assumption that $R_1$ and $R_2$ are consistent, so we have $\\pi _{{\\cal S}(R_i)} Q(\\mathbf {D}) = R_i$ for each $i \\in \\lbrace 1,2\\rbrace $ .",
"Our algorithm above constructs a specific cover.",
"Other covers can be constructed within the same time bounds.",
"We exemplify the construction of some further covers following different patterns.",
"In our construction above, after combining the first $n_2^t-1$ tuples in the $t$ -block of $R_1$ with the first $n_2^t-1$ tuples in the $t$ -block of $R_2$ , we combined the last tuple in the $t$ -block of $R_2$ with all remaining tuples in the $t$ -block of $R_1$ .",
"Alternatively, we can fix any tuple $t^{\\prime }$ in the $t$ -block of $R_2$ , combine the first $n_2^t-1$ tuples in the $t$ -block of $R_1$ with all tuples besides $t^{\\prime }$ in the $t$ -block of $R_2$ and then combine the remaining tuples in the $t$ -block of $R_1$ with $t^{\\prime }$ .",
"Proof of Lemma REF Lemma REF .",
"Given $(Q,\\mathcal {J},\\mathbf {D})$ where $\\mathbf {D}=\\lbrace R_i\\rbrace _{i\\in [n]}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(|K|)$ and with size $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Any cover-join plan over $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ .",
"We show by induction on the structure of cover-join plans that given $(Q,\\mathcal {J},\\mathbf {D})$ , where $\\mathbf {D}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ .",
"For the base case, assume that $\\varphi $ consists of a single relation symbol $R$ .",
"By Definition REF , $\\mathcal {J}$ consists of a single node $R$ , hence, $Q=R$ .",
"The decomposition $\\mathcal {T}$ corresponding to $\\mathcal {J}$ consists of a single bag ${\\cal S}(R)$ .",
"By Definition REF , $\\varphi $ returns the relation $R$ .",
"By Definition REF , $R$ is indeed the unique cover of $Q(\\lbrace R\\rbrace )$ over $\\mathcal {T}$ .",
"Assume now that $\\varphi $ is of the form $\\varphi _1 \\mathring{}\\varphi _2$ .",
"By definition of cover-join plans, there are subtrees $\\mathcal {J}_1$ and $\\mathcal {J}_2$ of $\\mathcal {J}$ such that $\\mathcal {J}= \\mathcal {J}_1 \\circ \\mathcal {J}_2$ and each $\\varphi _i$ is a cover-join plan over $\\mathcal {J}_i$ .",
"Let $\\mathcal {T}_1$ and $\\mathcal {T}_2$ be the decompositions corresponding to $\\mathcal {J}_1$ and $\\mathcal {J}_2$ , respectively.",
"The decomposition corresponding to $\\mathcal {J}$ is obtained by connecting $\\mathcal {T}_1$ and $\\mathcal {T}_2$ by the same tree edge connecting $\\mathcal {J}_1$ and $\\mathcal {J}_2$ in $\\mathcal {J}$ .",
"We have $Q = Q_1 \\bowtie Q_2$ where each $Q_i$ expresses the join of the relation symbols occurring in $\\mathcal {J}_i$ .",
"Moreover, $\\mathbf {D} = \\mathbf {D}_1 \\cup \\mathbf {D}_2$ where $\\mathbf {D}_i= \\lbrace R\\rbrace _{R \\in {\\cal S}(Q_i)}$ , $i \\in [2]$ .",
"Note that for each $i \\in [2]$ , $Q_i$ is acyclic, $\\mathcal {J}_i$ is a join tree of $Q_i$ and $\\mathbf {D}_i$ is globally consistent with respect to $Q_i$ .",
"The latter follows simply from the globally consistency of $\\mathbf {D}$ with respect to $Q$ .",
"Hence, by induction hypothesis, each $\\varphi _i$ returns a cover $K_i$ of $Q_i(\\mathbf {D}_i)$ over $\\mathcal {T}_i$ .",
"Due to Proposition REF , in case $K_1$ and $K_2$ are consistent, the cover-join operator computes a cover $K$ of $K_1 \\bowtie K_2$ over the decomposition with bags ${\\cal S}(K_1)$ and ${\\cal S}(K_2)$ .",
"Thus, by Definition REF , the plan $\\varphi $ returns $K$ .",
"We proceed as follows.",
"First, we show that $K_1$ and $K_2$ must be consistent.",
"Then, we prove that $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ , that is, $K$ is result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ and it is minimal in this respect.",
"$K_1$ and $K_2$ are consistent: Let $R_1$ and $R_2$ be the two relation symbols incident to the single edge connecting $\\mathcal {J}_1$ and $\\mathcal {J}_2$ in $\\mathcal {J}$ and let $\\mathcal {B}$ be the set of common attributes of these relation symbols.",
"Let $\\mathcal {A}$ be the set of common attributes of $K_1$ and $K_2$ .",
"We first show that $\\mathcal {A}\\subseteq \\mathcal {B}$ .",
"Let $A \\in \\mathcal {A}$ .",
"Since each $K_i$ is computed by the plan $\\mathcal {J}_i$ , there must be at least one relation symbol $R_1^{\\prime }$ in $\\mathcal {J}_1$ and at least one relation symbol $R_2^{\\prime }$ in $\\mathcal {J}_2$ containing $A$ in their schemas.",
"Due to the construction of join trees, $A$ must occur in the schemas of all relation symbols on the single path between $R_1^{\\prime }$ and $R_2^{\\prime }$ .",
"Since $R_1$ and $R_2$ are on this path, both must include $A$ .",
"Hence, $\\mathcal {A}\\subseteq \\mathcal {B}$ .",
"Since $\\mathbf {D}$ is globally consistent, the relations $R_1$ and $R_2$ must be consistent as well.",
"As each $K_i$ is result-preserving with respect to $(Q_i,\\mathcal {T}_i,\\mathbf {D}_i)$ , $\\pi _{{\\cal S}(R_i)} Q_i(\\mathbf {D}_i) = R_i$ (due to global consistency) and $\\mathcal {B}\\subseteq {\\cal S}(R_i)$ , it follows $\\pi _{\\mathcal {B}}K_1 = \\pi _{\\mathcal {B}}Q_1(\\mathbf {D}_1) = \\pi _{\\mathcal {B}}R_1= \\pi _{\\mathcal {B}}R_2 = \\pi _{\\mathcal {B}}Q_2(\\mathbf {D}_2) = \\pi _{\\mathcal {B}}K_2$ .",
"As $\\mathcal {A}\\subseteq \\mathcal {B}$ , the relations $K_1$ and $K_2$ must be consistent.",
"$K$ is result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ : Let $B$ be an arbitrary bag of $\\mathcal {T}$ .",
"Since $\\mathcal {T}$ corresponds to $\\mathcal {J}$ , the join tree $\\mathcal {J}$ must have a node $R$ with ${\\cal S}(R) = B$ .",
"Without loss of generality, assume that $R \\in \\mathbf {D}_1$ (the other case is handled along the same lines).",
"Since, by induction hypothesis, $K_1$ is result-preserving with respect to $(Q_1, \\mathcal {T}_1, \\mathbf {D}_1)$ and $\\mathbf {D}_1$ is globally consistent, we have $R= \\pi _{{\\cal S}(R)}K_1$ .",
"Since $\\pi _{{\\cal S}(K_1)}K = K_1$ and ${\\cal S}(R) \\subseteq {\\cal S}(K_1)$ , we get $R= \\pi _{{\\cal S}(R)}K$ .",
"Using the global consistency of $\\mathbf {D}$ with respect to $Q$ , we conclude $\\pi _{B}Q(\\mathbf {D}) = R= \\pi _{B}K$ .",
"$K$ is a minimal result-preserving relation with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ : For the sake of contradiction, assume that $K$ is not minimal in this respect.",
"This means that there is a tuple $t^- \\in K$ such that $K\\backslash \\lbrace t^-\\rbrace $ is still result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ .",
"It follows that $\\pi _{{\\cal S}(K_i)} (K\\backslash \\lbrace t^-\\rbrace )$ is result-preserving with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ for each $i \\in [2]$ .",
"Observe that the minimal edge cover $M$ with $\\mathit {rel}(M) = K$ in the hypergraph of $K_1 \\bowtie K_2$ over the attribute sets $\\lbrace {\\cal S}(K_1), {\\cal S}(K_2)\\rbrace $ must contain an edge $e^-$ connecting $\\pi _{{\\cal S}(K_1)}t^-$ and $\\pi _{{\\cal S}(K_2)}t^-$ .",
"This implies that $M$ cannot have two further edges $e_1$ and $e_2$ such that $e_1$ covers $\\pi _{{\\cal S}(K_1)}t^-$ and $e_2$ covers $\\pi _{{\\cal S}(K_2)}t^-$ .",
"Indeed, in this case, $M\\backslash \\lbrace e^-\\rbrace $ would be an edge cover, contradicting the minimality of $M$ .",
"Hence, there is no tuple $t \\ne t^-$ in $K$ with $\\pi _{{\\cal S}(K_1)}t^- =\\pi _{{\\cal S}(K_1)}t$ or there is no tuple $t \\ne t^-$ in $K$ with $\\pi _{{\\cal S}(K_2)}t^-= \\pi _{{\\cal S}(K_2)}t$ .",
"It follows that $\\pi _{{\\cal S}(K_1)} (K\\backslash \\lbrace t^-\\rbrace ) \\subset \\pi _{{\\cal S}(K_1)} K$ or $\\pi _{{\\cal S}(K_2)} (K\\backslash \\lbrace t^-\\rbrace ) \\subset \\pi _{{\\cal S}(K_2)} K$ .",
"Using the consistency of $K_1$ and $K_2$ , we obtain $\\pi _{{\\cal S}(K_1)}$ $(K\\backslash \\lbrace t^-\\rbrace )$ $\\subset \\pi _{{\\cal S}(K_1)} K$ $=$ $K_1$ or $\\pi _{{\\cal S}(K_2)}$ $(K\\backslash \\lbrace t^-\\rbrace )$ $\\subset \\pi _{{\\cal S}(K_2)} K$ $=$ $K_2$ .",
"However, as we noticed that $\\pi _{{\\cal S}(K_i)} (K\\backslash \\lbrace t^-\\rbrace )$ is result-preserving with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ for each $i \\in [2]$ , the statement of the last sentence contradicts the induction hypothesis that each $K_i$ is a minimal result-preserving relation with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ .",
"Size of $K$ .",
"From the global consistency of $\\mathbf {D}$ with respect to $Q$ and Proposition REF , it follows for any cover $K$ of $Q(\\mathbf {D})$ over the tree decomposition corresponding to $\\mathcal {J}$ that $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Computation time for $K$ .",
"By Proposition REF , we can design an algorithm for the cover-join operator which for every two input covers $K_1$ and $K_2$ , computes a cover-join result of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K_1 \\hspace{-2.84526pt} \\mid + \\mid \\hspace{-2.84526pt} K_2 \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K_1 \\hspace{-2.84526pt} \\mid + \\mid \\hspace{-2.84526pt} K_2 \\hspace{-2.84526pt} \\mid )$ .",
"Hence, given a triple $(Q,\\mathcal {J},\\mathbf {D})$ and a cover-join plan $\\varphi $ over $\\mathcal {J}$ , starting from the innermost expressions of $\\varphi $ , we can compute a cover $K$ of $Q(\\mathbf {D})$ over the tree decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ .",
"Missing Details and Proofs in Section Given the hypergraph $H$ of an FAQ and an attribute set $U$ , we denote by $H_U$ the hypergraph obtained from $H$ by restricting each hyperedge in $H$ to the attributes in $U$ .",
"For the rest of this section we fix an FAQ $\\varphi $ as written in (REF ).",
"Recap on FAQs Indicator projections are used in the InsideOut algorithm [17] solving the FAQ-problem.",
"They will also occur in our construction of FAQ-covers.",
"Definition 39 (Indicator projections) Given two attribute sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ and a function $\\psi _S$ , the function $\\psi _{S/T}: \\prod _{A\\in (S \\cap T)}\\textsf {dom}(A) \\rightarrow \\textsf {Dom}$ defined by $\\psi _{S/T}({\\textsf {a}}_{S \\cap T}) ={\\left\\lbrace \\begin{array}{ll}{\\bf 1}& \\quad \\exists {\\textsf {b}}_{S} \\text{ s.t. }",
"\\psi _S({\\textsf {b}}_S) \\ne 0 \\text{ and }{\\textsf {a}}_{S\\cap T} = {\\textsf {b}}_{S\\cap T}, \\\\{\\bf 0}& \\quad \\text{otherwise } \\\\\\end{array}\\right.",
"}$ is called the indicator projection of $\\psi _S$ onto $T$ .",
"In particular, if $S \\subseteq T$ , then $\\psi _{S/T}({\\textsf {a}}_{S}) = {\\bf 1}$ if and only if $\\psi _S({\\textsf {a}}_{S}) \\ne {\\bf 0}$ .",
"Equivalent attribute orderings.",
"A $\\varphi $ -equivalent attribute ordering $\\tau = \\tau (1), \\ldots , \\tau (n)$ is a permutation of the indices of the attributes in $\\mathcal {V}$ satisfying the following conditions: $\\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (f)}\\rbrace = \\lbrace A_1, \\ldots , A_f\\rbrace $ and $\\varphi ^{\\prime }({\\textsf {a}}_{\\lbrace A_{\\tau (1)},\\ldots ,A_{\\tau (f)}\\rbrace }) = \\underset{a_{\\tau (f+1)}\\in \\textsf {dom}(A_{\\tau (f+1)})}{\\bigoplus \\ ^{(\\tau (f+1))}} \\cdots \\underset{a_{\\tau (n)}\\in \\textsf {dom}(A_{\\tau (n)})}{\\bigoplus \\ ^{(\\tau (n))}}\\ \\underset{S\\in \\mathcal {E}}{\\bigotimes }\\ \\psi _S({\\sf a}_S)$ is equivalent to $\\varphi $ irrespective of the definition of the input functions $\\psi _S$ .",
"We denote by $\\textsf {EVO}(\\varphi )$ the set of all $\\varphi $ -equivalent attribute orderings.",
"The InsideOut algorithm Given an FAQ $\\varphi $ , a database $\\mathbf {D}$ and a $\\varphi $ -equivalent attribute ordering, the InsideOut algorithm computes the listing representation of $\\varphi (\\mathbf {D})$ .",
"The algorithm first rewrites the query according to the given attribute ordering and then processes the resulting query in two phases: bound attribute elimination and output computation.",
"We sketch the main steps of the algorithm on input $\\varphi $ , some database $\\mathbf {D}$ and the attribute ordering that corresponds to the identity permutation.",
"Thus, the initial rewriting step does not change the structure of $\\varphi $ .",
"In the bound attribute elimination phase, the algorithm eliminates attributes $A_{f+1},$ $\\ldots ,$ $A_{n}$ along with their corresponding aggregate operators in reverse order.",
"When eliminating an attribute $A_j$ it distinguishes between the cases whether $\\bigoplus ^{(j)}$ is different from $\\bigotimes $ or not.",
"We demonstrate the two cases in the elimination step for $A_n$ .",
"In case that $\\bigoplus ^{(n)}$ is different from $\\bigotimes $ , the algorithm first rewrites the query as follows $&\\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}} \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\sf a}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{S\\in \\mathcal {E}\\backslash \\partial (n)}{\\bigotimes } \\psi _S({\\textsf {a}}_S)\\otimes \\Big (\\underbrace{\\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigoplus \\ ^{(n)}} \\bigotimes _{S \\in \\partial (n)} \\psi _S({\\textsf {a}}_S)}_{\\delta }\\Big ),$ where $\\partial (n)= \\lbrace S \\in \\mathcal {E}\\mid A_n \\in S\\rbrace $ and $U_n = \\bigcup _{S\\in \\partial (n)} S$ .",
"The correctness of the rewriting follows from the distributivity of $\\otimes $ over $\\oplus ^{(n)}$ .",
"Then, the algorithm computes the listing representation of a function $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ such that replacing $\\delta $ by $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ does not change the semantics of $\\varphi $ .",
"Observe that the cartesian product of the domains of the attributes in $U_n \\backslash \\lbrace A_n\\rbrace $ can contain tuples ${\\textsf {a}}_{U_n \\backslash \\lbrace A_n\\rbrace }$ such that [(i)] there is a $\\psi _S$ with $S \\in \\mathcal {E}\\backslash \\partial (n)$ , $S \\cap (U_n \\backslash \\lbrace A_n\\rbrace ) \\ne \\emptyset $ and there is no ${\\textsf {b}}_S$ that agrees with ${\\textsf {a}}_{U_n \\backslash \\lbrace A_n\\rbrace }$ on the common attributes and $\\psi _S({\\textsf {b}}_S)\\ne 0$ .",
"Such tuples will not occur in the final result.",
"To rule them out in advance, indicator projections are used inside $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ .",
"The function $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ is defined as $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }({\\textsf {a}}_{U_n\\backslash \\lbrace A_n\\rbrace }) = \\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigoplus \\ ^{(n)}} \\bigg [ \\Big ( \\bigotimes _{S \\in \\partial (n)}\\psi _S({\\textsf {a}}_S) \\Big ) \\otimes \\Big (\\bigotimes _{\\begin{array}{c}S \\notin \\partial (n) \\\\S \\cap U_n \\ne \\emptyset \\end{array}} \\psi _{S/U_{n}} ({\\textsf {a}}_{S\\cap U_n}) \\Big )\\bigg ].$ The computation of the listing representation of this function requires the computation of the join of the listing representations of the functions $\\psi _S$ with $S \\in \\partial (n)$ and the indicator projections.",
"The computation time for this elimination step is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\rho ^{\\ast }(H_{U_n})})$ .",
"In case that $\\bigoplus ^{(n)}$ is equal to $\\bigotimes $ , the formula is rewritten as follows $&\\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}} \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigotimes } \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{S\\notin \\partial (n)}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S)^{|\\textsf {dom}(A_n)|}\\underset{S \\in \\partial (A_n)}{\\bigotimes } \\ \\underbrace{\\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S)}_{\\delta ^S},$ where $\\partial (n)$ is defined as above.",
"Then, the algorithm computes for each $S \\notin \\partial (n)$ , a function $\\psi _S^{\\prime }$ equivalent to $\\psi _S^{|\\textsf {dom}(A_n)|}$ and for each $S \\in \\partial (n)$ , a function $\\psi _{S\\backslash A_n}^{\\prime }$ equivalent to $\\delta ^S$ .",
"This elimination step can be realised in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|)$ .",
"After the elimination of all bound attributes we are left with a formula $\\varphi ^{\\prime }_{{\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }}$ without any bound attributes.",
"In the output computation phase the algorithm first computes (a factorized representation of) the set of tuples ${\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }$ for which $\\varphi ^{\\prime }_{{\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }}({\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace })\\ne {\\bf 0}$ and then reports the output.",
"Before giving the overall running time of InsideOut, we introduce elimination hypergraph sequences corresponding to attribute orderings.",
"Elimination hypergraph sequence Given a $\\varphi $ -equivalent attribute ordering $\\tau = \\tau (1),$ $\\ldots ,$ $\\tau (n)$ , we recursively define the elimination hypergraph sequence $H_n^{\\tau }, \\ldots ,H_1^{\\tau }$ associated with $\\tau $ .",
"For each $j$ with $n \\ge j \\ge 1$ , we additionally define two sets $U_j^{\\tau }$ and $\\partial ^{\\tau }(j)$ .",
"For the sake of readability, in the following we skip the superscript $\\tau $ in our notation.",
"We set $H_n = (\\mathcal {V}_n, \\mathcal {E}_n) = H$ and define $\\partial (n) = \\lbrace S \\in \\mathcal {E}_n \\mid A_{\\tau (n)} \\in S\\rbrace $ and $U_n = \\bigcup _{S \\in \\partial (n)}S.$ For each $j$ with $n-1 \\ge j \\ge 1$ , we define: If $\\bigoplus \\ ^{(\\tau (j+1))} = \\bigotimes $ , then, $\\mathcal {V}_j = \\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (j)}\\rbrace $ and $\\mathcal {E}_j$ is obtained from $\\mathcal {E}_{j+1}$ by removing $A_{\\tau (j+1)}$ from all edges in $\\mathcal {E}_{j+1}$ .",
"Otherwise, $\\mathcal {V}_j = \\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (j)}\\rbrace $ and $\\mathcal {E}_j = (\\mathcal {E}_{j+1} \\backslash \\partial (j+1) ) \\cup (U_{j+1} \\backslash \\lbrace A_{\\tau (j+1)}\\rbrace )$ .",
"We further set $\\partial (j) = \\lbrace S \\in \\mathcal {E}_j \\mid A_{\\tau (j)} \\in S\\rbrace $ and $U_j = \\bigcup _{S \\in \\partial (j)}S.$ Running time of InsideOut For a $\\varphi $ -equivalent attribute ordering $\\tau $ , let $K = [f] \\cup \\lbrace j \\mid j > f, \\oplus ^{(\\tau (j))} \\ne \\otimes \\rbrace $ .",
"The FAQ-width of $\\tau $ is defined as $\\textsf {faqw}(\\tau ) =\\max _{j \\in K}\\lbrace \\rho ^{\\ast }(H_{U_j^{\\tau }})\\rbrace $ .",
"For a given $\\tau $ , InsideOut runs in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )} + Z)$ where $Z$ is the size of the output.",
"The FAQ-width of $\\varphi $ is defined as $\\textsf {faqw}(\\varphi ) = \\min _{\\tau \\in \\textsf {EVO}(\\varphi )}\\lbrace \\textsf {faqw}(\\tau )\\rbrace $ .",
"Hence, given the best attribute ordering (i.e., with smallest FAQ-width), the running time of InsideOut is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )} + Z)$ .",
"From attribute orderings to decompositions We say that $\\mathcal {T}$ is a decomposition of $\\varphi $ if $\\mathcal {T}$ is a decomposition of the hypergraph $H$ of $\\varphi $ .",
"Proposition 40 ([18], Proposition C.2) For any FAQ $\\varphi $ without bound attributes and any $\\varphi $ -equivalent attribute ordering $\\tau $ , one can construct a decomposition $\\mathcal {T}$ of $\\varphi $ with $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau )$ .",
"Covers for FAQs Given two input functions $\\psi _S$ and $\\psi _T$ with $T \\subseteq S$ , we can always compute the function $\\psi ^{\\prime }_S = \\psi _S \\otimes \\psi _T$ in time $\\widetilde{\\mathcal {O}}(|R_{\\psi _S}| + |R_{\\psi _T}|)$ and replace $\\psi _S \\otimes \\psi _T$ by $\\psi ^{\\prime }_S$ without changing the semantics of the FAQ.",
"To do this, we first sort the listing representations $R_{\\psi _S}$ and $R_{\\psi _T}$ of $\\psi _S$ and $\\psi _T$ on the attributes in $T$ .",
"During a subsequent scan through both relations we add for each pair ${\\textsf {a}}_{S \\cup \\lbrace \\psi _S(S)\\rbrace } \\in R_{\\psi _S}$ and ${\\textsf {b}}_{T\\cup \\lbrace \\psi _T(T)\\rbrace } \\in R_{\\psi _T}$ with ${\\textsf {a}}_T = {\\textsf {b}}_T$ , the tuple ${\\textsf {c}}_{S\\cup \\lbrace \\psi ^{\\prime }_S(S)\\rbrace }$ with ${\\textsf {c}}_S = {\\textsf {a}}_S$ and ${\\textsf {c}}_{\\lbrace \\psi ^{\\prime }_S(S)\\rbrace }=\\psi _S({\\textsf {a}}_S) \\otimes \\psi _T({\\textsf {b}}_T)$ to the listing representation of $\\psi ^{\\prime }_S$ .",
"Hence, in the following we assume, without loss of generality, that $\\varphi $ does not contain any function whose attributes are included in the attribute set of another function.",
"Bag functions Given an FAQ $\\varphi $ without bound attributes and a decomposition of $\\varphi $ , we define bag functions which are the counterparts of bag relations in case of join queries.",
"Our goal is to define for each bag $B$ of $\\mathcal {T}$ , a function $\\beta _B$ such that $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) =\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B})$ .",
"While in case of join queries it is harmless to include all relations sharing attributes with $B$ into the join computing the bag relation of $B$ , in case of FAQs we have to be a bit careful.",
"Including the same input function into the computation of bag functions of several bags can violate the above equality.",
"Therefore, in the definition below we use a mapping from input functions to bags.",
"To keep the sizes of the bag functions small we also use indicator projections which achieve pairwise consistency between listing representations of bag functions sharing attributes.",
"Definition 41 (Bag functions) Given an FAQ $\\varphi $ without bound attributes and a decomposition $\\mathcal {T}$ of $\\varphi $ , a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is called a set of bag functions for $\\varphi $ and $\\mathcal {T}$ if there is a mapping $m: \\mathcal {E}\\rightarrow {\\cal S}(\\mathcal {T})$ such that $S \\subseteq m(S)$ for each $S \\in \\mathcal {E}$ and $\\beta _B$ is defined by $\\beta _{B}({\\textsf {a}}_B) =\\underset{{S \\in \\mathcal {E}:S \\cap B \\ne \\emptyset }}{\\bigotimes } \\psi _{S/B}({{\\textsf {a}}}_{B\\cap S}) \\ \\otimes \\underset{S \\in \\mathcal {E}: m(S) = B}{\\bigotimes } \\psi _S({\\textsf {a}}_{S})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"We define $\\mathcal {B}(\\varphi ,\\mathcal {T}) = \\lbrace \\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}\\mid $ $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is a set of bag functions for $\\varphi $ and $\\mathcal {T}\\rbrace $ .",
"Note that since each hyperedge in the hypergraph of $\\varphi $ must be included in at least one bag of the decomposition, one can always find a mapping $m$ meeting the condition given in the above definition.",
"Observe also that for bags $B$ to which no input function is mapped, the function $\\beta _B$ is just the product of indicator projections of all $\\psi _S$ sharing attributes with $B$ onto $B$ .",
"Observation 42 Given an FAQ $\\varphi $ without bound attributes, a decomposition $\\mathcal {T}$ of $\\varphi $ and a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ,\\mathcal {T})$ , it holds $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) = \\underset{B\\in {\\cal S}(\\mathcal {T})}{\\bigotimes }\\ \\beta _B({{\\textsf {a}}}_{B}).$ Given $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ,\\mathcal {T})$ , we denote by $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ the decomposition obtained from $\\mathcal {T}$ by adding into each bag $B$ the attribute $\\beta _B(B)$ .",
"Observe that if $\\mathcal {T}$ is a decomposition of $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) = \\bigotimes _{B\\in {\\cal S}(\\mathcal {T})}\\beta _B({{\\textsf {a}}}_{B})$ , then, $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ is a decomposition of the query joining the listing representations of the functions $\\beta _B$ .",
"Moreover, $\\mathcal {T}$ and $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ have the same fractional hypertree width.",
"Covers of FAQ results We turn towards the general case where FAQs can contain bound attributes also.",
"Let $\\tau = \\tau _1\\tau _2$ be a $\\varphi $ -equivalent attribute ordering where $\\tau _1$ consists of the free and $\\tau _2$ consists of the bound attributes in $\\varphi $ .",
"By $\\varphi ^{\\tau }_{\\textit {free}}$ we denote the FAQ constructed by the InsideOut algorithm after eliminating all bound attributes in $\\varphi $ according to the ordering $\\tau _2$ .",
"We write $(\\varphi ,\\tau ,\\mathcal {T}, \\mathbf {D})$ to express that $\\varphi $ is an FAQ, $\\tau $ is a $\\varphi $ -equivalent attribute ordering, $\\mathcal {T}$ is a decomposition of $\\varphi ^{\\tau }_{\\textit {free}}$ with $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau _1)$ and $\\mathbf {D}$ is an input database for $\\varphi $ .",
"Note that due to Proposition REF , for any $\\tau $ such a decomposition $\\mathcal {T}$ is always constructible.",
"Definition 43 (Covers of FAQ results) Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ if there is a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}\\in \\mathcal {B}(\\varphi ^{\\tau }_{\\textit {free}},\\mathcal {T})$ such that $K$ is a cover of the join of the relations $\\lbrace R_{\\beta _B}\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ over $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ .",
"We call $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ the set of bag functions underlying $K$ .",
"Observe that if $K$ is a cover of $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ with underlying bag functions $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ , then, $\\pi _{\\mathcal {V}_{\\text{free}}} K$ must be a cover of $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _BR_{\\beta _B}$ over $\\mathcal {T}$ .",
"The following Proposition relies on Lemma REF and Theorem REF which give an upper bound on the time complexity for constructing covers of join results.",
"Proposition 44 Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , a cover of the query result $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )})$ .",
"Construction.",
"Let $\\tau = \\tau _1\\tau _2$ where $\\tau _1$ consists of the free and $\\tau _2$ consists of the bound attributes in $\\varphi $ .",
"We first run InsideOut on $\\varphi $ according to the attribute ordering $\\tau _2$ until all bound attributes are eliminated and we obtain $\\varphi ^{\\tau }_{\\textit {free}}$ .",
"Then, we construct a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ^{\\tau }_{\\textit {free}},\\mathcal {T})$ of bag functions.",
"Finally, using a cover-join plan as introduced in Definition REF , we construct a cover $K$ of the join of the relations $\\lbrace R_{\\beta _B}\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ over $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ .",
"Construction time.",
"The FAQ $\\varphi ^{\\tau }_{\\textit {free}}$ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau _2)})$ [17].",
"The construction of the bag functions $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ can be realised via the computation of the bag relations of $\\mathcal {T}$ .",
"By Proposition REF , the size of the listing representations of these bag functions is $\\mathcal {O}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ and their computation time is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"By Theorem REF , $K$ can be computed in time $\\widetilde{\\mathcal {O}}(\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|R_{\\beta _B}|)$ .",
"Hence, the time for computing $K$ from $\\varphi ^{\\tau }_{\\textit {free}}$ is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Since $\\textsf {faqw}(\\tau ) = \\max _{1 \\le i \\le 2}\\lbrace \\textsf {faqw}(\\tau _i)\\rbrace $ and $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau _1)$ (by construction), the overall computation time is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )})$ .",
"Theorem REF is an immediate corollary: Theorem REF .",
"For any FAQ $\\varphi $ and database $\\mathbf {D}$ , a cover of the query result $\\varphi (\\mathbf {D})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {faqw}(\\varphi )})$ .",
"Enumeration of Tuples in FAQ Results using Covers Any enumeration algorithm on covers of join results can easily be turned into an enumeration algorithm on covers of FAQ-results.",
"Assume that $K$ is a cover of the result of the FAQ $\\varphi $ over some decomposition $\\mathcal {T}$ (induced by some attribute ordering).",
"Let $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ be the underlying set of bag functions.",
"Recall that the set of attributes of $K$ is $\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ and the set of attributes of the listing representation of $\\varphi $ must be $\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\varphi (\\mathcal {V}_{\\text{free}})\\rbrace $ .",
"To enumerate the listing representation of $\\varphi $ , we can run any enumeration algorithm on $K$ with respect to the decomposition $\\textit {ext}(\\mathcal {T}, \\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ and adapt its output as follows.",
"For each output tuple ${\\textsf {a}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}}$ , we output the tuple ${\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\varphi (\\mathcal {V}_{\\text{free}})\\rbrace }$ that agrees with ${\\textsf {a}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}}$ on $\\mathcal {V}_{\\text{free}}$ and where the $\\varphi (\\mathcal {V}_{\\text{free}})$ -value is defined by $\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})}{\\textsf {a}}_{\\lbrace \\beta _B(B)\\rbrace }$ .",
"The following proposition shows that by this strategy we indeed enumerate the listing representation of $\\varphi $ .",
"Proposition 45 Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , let $K$ be a cover of the query result of $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ and let $\\lbrace \\beta _B\\rbrace _{B \\in \\mathcal {T}}$ be the set of bag functions underlying $K$ .",
"It holds $\\varphi ({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) = v \\ne 0 \\text{ for some } v \\in \\textsf {Dom}$ if and only if $\\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K,\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and }\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{ \\lbrace \\beta _B(B)\\rbrace } = v.$ Let $\\varphi _{\\textit {free}}^{\\tau } = \\bigotimes _{S^{\\prime } \\in \\mathcal {E}^{\\prime }} \\psi _{S^{\\prime }}$ .",
"Then, $&\\ \\ \\ \\varphi ({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) = v \\ne {\\bf 0}\\\\\\overset{(1)}{\\Longleftrightarrow } & \\ \\ \\bigotimes _{S^{\\prime } \\in \\mathcal {E}^{\\prime }} \\psi _{S^{\\prime }}({\\textsf {a}}_{S^{\\prime }}) = v \\ne {\\bf 0}\\\\\\overset{(2)}{\\Longleftrightarrow } &\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B}) = v \\ne {\\bf 0}\\\\\\overset{(3)}{\\Longleftrightarrow } &\\ \\ \\ \\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_{\\beta _B},\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and }\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{\\lbrace \\beta _B(B)\\rbrace } = v \\\\\\overset{(4)}{\\Longleftrightarrow } &\\ \\ \\ \\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K,\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and } \\\\& \\ \\ \\ \\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{\\lbrace \\beta _B(B)\\rbrace } = v.$ Equivalence (1) holds by the correctness of the InsideOut algorithm.",
"The second equivalence holds by Observation REF .",
"Equivalence (3) follows from the simple observation that the product of functions corresponds to the join of their listing representations.",
"The last equivalence follows from Proposition REF which guarantees that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_{\\beta _B}$ is equal to $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K$ .",
"Thus, our enumeration result for covers of join results carries over to covers of FAQ-results.",
"Corollary REF .",
"(Corollary REF , Proposition REF ).",
"Given a cover $K$ of the result $\\varphi (\\mathbf {D})$ of an FAQ $\\varphi $ over a database $\\mathbf {D}$ , the tuples in the query result $\\varphi (\\mathbf {D})$ can be enumerated with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space.",
"Missing Proofs of Appendix In case the signature mappings of an equi-join query are not clear from the context, we write the signature mappings as a superscript to the query.",
"Moreover, for a relation symbol $R$ in an equi-join query with signature mappings $(\\lambda , \\lbrace \\mu _{R}\\rbrace _{R \\in {\\cal S}(Q)})$ and a database $\\mathbf {D}$ , we write $\\lambda (R)_{\\mathbf {D}}$ to denote the relation assigned to the relation symbol $\\lambda (R)$ in $\\mathbf {D}$ .",
"Proof of Proposition REF Proposition REF .",
"Given an equi-join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , there exist a natural join query $Q^{\\prime }$ and a database $\\mathbf {D}^{\\prime }$ such that: $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , $Q^{\\prime }$ has the decomposition $\\mathcal {T}$ and can be constructed in time $\\mathcal {O}(|Q|)$ , and $\\mathbf {D}^{\\prime }$ can be constructed in time $\\mathcal {O}(|\\mathbf {D}|)$ .",
"The query $Q$ has the form $\\sigma _\\psi (R_1\\times \\cdots \\times R_n)$ , where $\\psi $ is a conjunction of equality conditions.",
"The relation symbols as well as all attributes occurring in the schemas of the relation symbols are pairwise distinct.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Given an equivalence class ${\\cal A}$ of attributes in $Q$ , we let $\\phi _{\\cal A}=\\bigwedge _{A_i,A_j\\in {\\cal A}} A_i=A_j$ .",
"Then, given the set $\\lbrace {\\cal A}_j\\rbrace _{j\\in [l]}$ of all equivalence classes in $Q$ , the conjunction $\\bigwedge _{j\\in [l]}\\phi _{{\\cal A}_j}$ is the transitive closure $\\psi ^+$ of $\\psi $ in $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The query $Q^{\\prime }$ has one relation symbol $R^{\\prime }_i$ for each relation symbol $R_i$ in $Q$ such that ${\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"We thus have $Q^{\\prime }=R_1^{\\prime }\\bowtie \\cdots \\bowtie R_n^{\\prime }$ , where the equality conditions in the transitive closure of $\\psi $ are now expressed by natural joins in $Q^{\\prime }$ .",
"Construction of $\\mathbf {D}^{\\prime }$ .",
"For the sake of simplicity, we describe the construction of $\\mathbf {D}^{\\prime }$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : The database $\\mathbf {D}_1$ contains for each $R_i \\in {\\cal S}(Q)$ , a relation $R_i^1$ which results from $\\lambda (R_i)_{\\mathbf {D}}$ by replacing each attribute $A$ by the attribute $B$ with $\\mu _{R_i}(B) = A$ .",
"Construction of database $\\mathbf {D}_2$ : The database $\\mathbf {D}_2$ consists of the relations $R_1^2, \\ldots , R_n^2$ where each $R_i^2$ results from $R_i^1$ as follows.",
"For each equality $A=B$ in $\\psi ^+$ such that $A,B \\in {\\cal S}(R_i^1)$ , we delete in $R_i^1$ all tuples $t$ with $t(A) \\ne t(B)$ .",
"Note that such tuples $t$ cannot occur in the projection of $Q(\\mathbf {D})$ onto the schema of $t$ .",
"Construction of database $\\mathbf {D}^{\\prime }$ : We obtain the database $\\mathbf {D}^{\\prime }$ from $\\mathbf {D}_2$ by replacing each relation $R_i^2$ by a relation $R_i^{\\prime }$ defined as follows.",
"The relation $R_i^{\\prime }$ is a copy of $R_i^2$ extended with one new column for each attribute $A$ in ${\\cal S}(R_i^{\\prime }) \\backslash {\\cal S}(R_i)$ such that $\\pi _{A} R_i^{\\prime } =\\pi _{B} R_i^2$ for any attribute $B\\in {\\cal S}(R_i)$ transitively equal to $A$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"By construction, $Q$ and $Q^{\\prime }$ have the same set of attributes and thus the same equivalence classes of attributes.",
"Moreover, the transitive closures of the schemas of relation symbols are identical: For any pair of relation symbols $R_i \\in {\\cal S}(Q)$ and $R_i^{\\prime } \\in {\\cal S}(Q^{\\prime })$ , it holds that ${\\cal S}(R^{\\prime }_i)^+={\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"The hypergraphs of $Q^{\\prime }$ and $Q$ are thus the same as they have the same nodes, which are the attributes in $Q$ and $Q^{\\prime }$ respectively, and the same hyperedges, which are the transitive closures ${\\cal S}(R_i)^+$ and ${\\cal S}(R^{\\prime }_i)^+$ respectively.",
"This means that the decomposition $\\mathcal {T}$ of $Q$ is also a decomposition of $Q^{\\prime }$ .",
"$Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"We define two further signature mappings $(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})$ and $(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})$ for $Q$ .",
"The function $\\lambda ^1$ maps each relation symbol $R_i$ in $Q$ to $R_i^1$ .",
"Moreover, each $\\mu ^1_{R_i}$ is an identity mapping on the attributes of $R_i$ .",
"The function $\\lambda ^2$ maps each relation symbol $R_i$ in $Q$ to $R_i^2$ .",
"Finally, $\\mu ^1_{R_i} = \\mu ^2_{R_i}$ for each $R_i \\in {\\cal S}(Q)$ .",
"The Database $\\mathbf {D}_1$ results from $\\mathbf {D}$ by, basically, making for each relation $R$ as many copies as the number of relation symbols in $Q$ mapped to $R$ .",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by ruling out tuples which cannot be contained in (the projections of) the final result.",
"Hence, it easily follows $Q^{(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D})$ $=$ $Q^{(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_1)$ $=$ $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Thus, it remains to show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ $=$ $Q(\\mathbf {D})$ .",
"We first treat the special case when $Q$ is a Cartesian product, i.e., it does not contain any equality conditions.",
"Then, $Q^{\\prime }=Q$ and each relation in $\\mathbf {D}^{\\prime }$ is an exact copy of a relation in $\\mathbf {D}_1$ .",
"Hence, $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ holds trivially.",
"We next consider the case when $Q$ has equality conditions.",
"We first show $Q^{\\prime }(\\mathbf {D}^{\\prime })\\subseteq Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Assume there is a tuple $t$ that is contained in $Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Then, $t=\\bowtie _{i\\in [n]} t_i$ is the natural join of tuples $t_i\\in R^{\\prime }_i$ .",
"Let ${\\cal A}$ be any equivalence class of attributes in $Q^{\\prime }$ .",
"By construction, whenever one of these attributes occur in the schema of a relation $R^{\\prime }_i$ , so are the others.",
"Furthermore, their values are the same in any tuple of $R^{\\prime }_i$ .",
"Since $t$ is a join of tuples $t_i$ , it follows that all attributes in ${\\cal A}$ have the same value in $t$ and therefore $\\sigma _{\\phi _{\\cal A}}(t)=t$ .",
"This holds for all equivalence classes of attributes, so $\\sigma _{\\psi ^+}(t)=t$ and thus $\\sigma _{\\psi }(t)=t$ .",
"This means that $t\\in Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We now show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Assume there is a tuple $t$ that is in $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"This means that $t = _{i\\in [n]} t_i$ is a product of tuples $t_i\\in R_i^2$ , $\\sigma _{\\psi ^+}(t)=t$ and in particular $\\sigma _{\\phi _{\\cal A}}(t)=t$ for each equivalence class ${\\cal A}$ in $Q$ .",
"We extend each tuple $t_i$ with values for all attributes in the class ${\\cal A}$ whenever ${\\cal S}(t_i)\\cap {\\cal A}\\ne \\emptyset $ .",
"Let $t^{\\prime }_i$ be the extension of $t_i$ .",
"Then, $t = \\bowtie _{i\\in [n]} t^{\\prime }_i$ .",
"All attributes in ${\\cal A}$ thus have the same value in $t^{\\prime }_i$ .",
"Since, by construction, the relation $R^{\\prime }_i$ is an extension of $R_i^2$ with same-valued columns for all attributes in ${\\cal A}$ whenever ${\\cal S}(R_i^2)\\cap {\\cal A}\\ne \\emptyset $ , it follows that $t^{\\prime }_i\\in R^{\\prime }_i$ .",
"Thus, $t\\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Construction time.",
"The natural join query $Q^{\\prime }$ evolves from $Q$ by replacing the schema $S$ of each relation symbol by $S^+$ .",
"This can be done in time $\\mathcal {O}(|Q|)$ .",
"The database $\\mathbf {D}_1$ evolves from $\\mathbf {D}$ by duplicating each relation in $\\mathbf {D}$ at most $|Q|$ times.",
"Hence, $\\mathbf {D}_1$ can be constructed in linear time.",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by deleting in each relation $R_i^1$ in $\\mathbf {D}_1$ , each tuple tuple $t$ with $t(A)\\ne t(B)$ and $A= B \\in \\psi ^+$ .",
"This deletion procedure can be realised via a single pass through the relations in $\\mathbf {D}_1$ and requires, therefore, only linear time.",
"Likewise, each relation $R_i^{\\prime }$ in $\\mathbf {D}^{\\prime }$ can be constructed from $R_i^2$ in $\\mathbf {D}_2$ by a single pass through the tuples in $R_i^2$ .",
"For each tuple, we choose for each new attribute $A$ in $R_i^{\\prime }$ but not in $R_i^2$ , an equivalent attribute in $R_i^2$ and copy its value to the $A$ -column.",
"Thus, the transformation from $\\mathbf {D}_2$ to $\\mathbf {D}^{\\prime }$ can also be done in linear time.",
"Proof of Proposition REF Proposition REF .",
"For any equi-join query $Q$ and any decomposition $\\mathcal {T}$ of $Q$ , there are arbitrarily large databases $\\mathbf {D}$ such that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We will prove the following claim: Claim: Given an equi-join query $Q$ and a decomposition $\\mathcal {T}$ of $Q$ , there exist a natural join query $Q^{\\prime }$ that has the decomposition $\\mathcal {T}$ such that: $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ and for each database $\\mathbf {D}^{\\prime }$ there is a database $\\mathbf {D}$ of size $\\mathcal {O}(\\mathbf {D}^{\\prime })$ such that $|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B} Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Using this claim, the result of the proposition can be derived straightforwardly.",
"Given an equi-join query $Q$ , we first construct the natural join query as promised in the claim.",
"By Theorem REF (ii), there are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that each cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given such a database $\\mathbf {D}^{\\prime }$ , it follows from Proposition REF , that $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })| =\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ , hence, $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace = \\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"By our claim, the database $\\mathbf {D}^{\\prime }$ can be converted into a database $\\mathbf {D}$ of size $\\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ such that $|\\pi _BQ(\\mathbf {D})| \\ge |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By Proposition REF (adapted to equi-join queries), each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ .",
"Since $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ $=$ $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ and $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ $\\ge $ $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ , we conclude that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is of size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We turn towards the proof of our claim.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The natural join query $Q^{\\prime }$ is constructed exactly as in the proof of Proposition REF .",
"Construction of $\\mathbf {D}$ .",
"Given a database $\\mathbf {D}^{\\prime }$ , we describe the construction of $\\mathbf {D}$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : For each equivalence class $\\mathcal {A}\\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ , let $f_{\\mathcal {A}}$ be an injective function mapping tuples over $\\mathcal {A}$ to fresh data values not occurring in $\\mathbf {D}^{\\prime }$ .",
"Moreover, let $f$ be a function mapping tuples $t$ with ${\\cal S}(t) \\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ and ${\\cal S}(t) = {\\cal S}(t)^+$ to tuples $t^{\\prime }$ with ${\\cal S}(t^{\\prime }) = {\\cal S}(t)$ as follows.",
"For each attribute $A \\in {\\cal S}(t^{\\prime })$ from some equivalence class $\\mathcal {A}$ , it holds $t^{\\prime }(A) = f_{\\mathcal {A}}(\\pi _{\\mathcal {A}}t)$ .",
"From each relation $R_i^{\\prime } \\in \\mathbf {D}^{\\prime }$ , we construct a relation $R_i^1$ where each tuple $t$ is replaced by $f(t)$ .",
"We define $\\mathbf {D}_1 = \\lbrace R_{i}^1\\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}_2$ : From each relation $R_{i}^1 \\in \\mathbf {D}_1$ we design a relation $R_{i}^2$ by performing the following procedure.",
"We first project away all columns of attributes not included in ${\\cal S}(R_i)$ .",
"Then, we rename each attribute $A$ in the resulting relation by $\\mu _{R_i}(A)$ .",
"Let $\\mathbf {D}_2 = \\lbrace R_{i}^2 \\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}$ : We obtain database $\\mathbf {D}$ from $\\mathbf {D}_2$ as follows.",
"For each maximal set $\\lbrace R_{i_1}, \\ldots ,R_{i_k}\\rbrace \\subseteq {\\cal S}(Q)$ such that all $R_{i_j}$ are mapped to the same relation symbol $\\lambda (R_{i_j})=R$ , we replace the relations $R_{i_1}^2, \\ldots ,R_{i_k}^2$ by a single relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with relation symbol $R$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"This follows from the proof of Proposition REF .",
"$|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Our proof contains three steps.",
"We show that $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Let $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $B = B^+$ , the function $f$ is defined for all tuples in $\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Moreover, for two distinct tuples $t_B, t_B^{\\prime } \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , the tuples $f(t_B)$ and $f(t_B^{\\prime })$ are distinct, too.",
"Hence, it suffices to show that for each $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , we have $f(t_B) \\in \\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"It follows that there is a tuple $t \\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ with $t_B = \\pi _B t$ .",
"By definition of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ , it must hold $\\pi _{{\\cal S}(R_i^{\\prime })} t \\in R_i^{\\prime }$ for each $i \\in [n]$ .",
"We have ${\\cal S}(R_i^{\\prime }) = {\\cal S}(R_i^{\\prime })^+$ for each $i \\in [n]$ .",
"Thus, it holds $f(\\pi _{{\\cal S}(R_i^{\\prime })} t) \\in R_i^1$ for each $i \\in [n]$ .",
"This is equivalent to saying $\\pi _{{\\cal S}(R_i^1)} f(t) \\in R_i^1$ for each $i \\in [n]$ .",
"By definition of $\\bowtie _{i \\in [n]}R_i^1$ , this implies that $f(t) \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"Thus, $f(t_B) = \\pi _B f(t) \\in \\pi _B(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $\\lambda ^{\\prime }$ be a function that maps each relation symbol $R_i$ in ${\\cal S}(Q)$ to $R_i^2$ .",
"We show that $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"To this end, let $t \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"It follows that $\\pi _{{\\cal S}(R_i^1)}t \\in R_i^1$ for each $i \\in [n]$ .",
"By the construction of $\\mathbf {D}_2$ , it holds $\\pi _{{\\cal S}(R_i^2)}t \\in R_i^2$ for each $i \\in [n]$ .",
"Furthermore, by the construction of $\\mathbf {D}_1$ , for each equivalence class $\\mathcal {A}$ and all attributes $A,B \\in \\mathcal {A}$ , we have $t(A) = t(B)$ .",
"Thus, $t \\in Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We show that $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We recall that database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by replacing each maximal set $R_{i_1}^2, \\ldots ,R_{i_k}^2$ of relations with $\\lambda (R_{i_1}) = \\ldots = \\lambda (R_{i_k}) = R$ , by the relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with the relation symbol $\\lambda (R_{i_1})$ .",
"Observe that the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\lbrace R_{i_1}^2, \\ldots ,R_{i_k}^2\\rbrace )$ (under signature mappings $(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ) must be included in the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\bigcup _{j \\in [k]} R_{i_j}^2)$ (under signature mappings $(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ).",
"By generalising this insight, we obtain that every tuple from $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ must be included in $Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"By (1), $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By (2) and (3), $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We conclude that $|\\pi _{B}Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Construction time for $Q$ .",
"It follows from the proof of Proposition REF that $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ .",
"Size of $\\mathbf {D}$ .",
"Since $f$ is a bijective mapping and $\\mathbf {D}_1$ is obtained from $\\mathbf {D}^{\\prime }$ by replacing tuples $t$ by $f(t)$ , we have $|\\mathbf {D}_1| = |\\mathbf {D}^{\\prime }|$ .",
"As $\\mathbf {D}_2$ results from $\\mathbf {D}_1$ by taking projections of relations, the size of $\\mathbf {D}_2$ cannot be larger than the size of $\\mathbf {D}_1$ .",
"Database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by taking unions of relations.",
"Thus, the number of tuples in $\\mathbf {D}$ cannot be more than the number of tuples in $\\mathbf {D}_2$ .",
"Altogether, we have $|\\mathbf {D}| = \\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ ."
],
[
"Covers for Join Queries",
"In this section we introduce the notion of covers of join query results along with a characterization of their size bounds, the connection to minimal edge covers for hypergraphs of join query results, and the complexity for enumerating the tuples in the query result from a cover.",
"Let $(Q,\\mathcal {T},\\mathbf {D})$ denote a triple of a natural join query $Q$ , decomposition $\\mathcal {T}$ of $Q$ , and database $\\mathbf {D}$ .",
"For an instance $(Q,\\mathcal {T},\\mathbf {D})$ , covers of the query result $Q(\\mathbf {D})$ are relations that are minimal while preserving the information in the query result $Q(\\mathbf {D})$ in the following sense.",
"Definition 5 (Result Preservation) A relation $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if its schema ${\\cal S}(K)$ is $\\mathit {att}(Q)$ and $\\pi _B K = \\pi _B Q(\\mathbf {D})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"That is, for each bag $B$ in the decomposition $\\mathcal {T}$ of $Q$ , both the relation $K$ and the query result $Q(\\mathbf {D})$ have the same projection onto $B$ .",
"This also means that the natural join of these projections of $K$ is precisely $Q(\\mathbf {D})$ .",
"Proposition 6 Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ with schema $\\mathit {att}(Q)$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if and only if $\\bowtie _{B\\in {\\cal S}(\\mathcal {T})} \\pi _B K = Q(\\mathbf {D})$ .",
"We further say that the relation $K$ is minimal result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if it is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , yet this is not the case for any strict subset of it.",
"We can now define the notion of covers of query results.",
"Definition 7 (Covers) Given $(Q,\\mathcal {T},\\mathbf {D})$ , a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ is a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"Example 8 Figure REF gives the decomposition $\\mathcal {T}$ of a path query and one cover $\\mathit {rel}(M)$ of the query result over $\\mathcal {T}$ .",
"We give below four relations that are subsets of the query result.",
"The relations $K_1$ and $K_2$ are covers, while the relations $N_1$ and $N_2$ are not covers: Table: NO_CAPTIONTo check the minimal result-preservation property, we take projections onto the bags $B_1=\\lbrace A,B\\rbrace $ , $B_2=\\lbrace B,C\\rbrace $ , and $B_3=\\lbrace C,D\\rbrace $ .",
"The relation $N_1$ is not result-preserving, because $(a_2,b_2)\\notin \\pi _{B_1}N_1$ .",
"The same argument also applies to relation $N_2$ .",
"Consider now the coarser decomposition $\\mathcal {T}^{\\prime }$ with bags $B^{\\prime }_{1,2}=\\lbrace A,B,C\\rbrace $ and $B^{\\prime }_3=\\lbrace C,D\\rbrace $ .",
"The covers over $\\mathcal {T}$ discussed above are also covers over $\\mathcal {T}^{\\prime }$ .",
"The query result is the only cover over the coarsest decomposition $\\mathcal {T}^{\\prime \\prime }$ with only one bag.",
"Example 9 A query result may admit exponentially many covers over the same decomposition.",
"Consider for instance the product query $R_1(A) \\bowtie R_2(B)$ with relations $R_1$ and $R_2$ of size two and respectively $n>1$ .",
"The query result has size $2\\cdot n$ .",
"To compute a cover, we pair the first tuple in $R_1$ with any non-empty and strict subset of the $n$ tuples in $R_2$ , while the second tuple in $R_1$ is paired with the remaining tuples in $R_2$ .",
"There are $2^n-2$ possible covers.",
"The empty and the full sets are missing from the choice of a subset of $R_2$ as they would mean that one of the two tuples in $R_1$ would have to be paired with tuples in $R_2$ that are already paired with the other tuple in $R_1$ and that would violate the minimality criterion of the covers.",
"All covers have size $n$ and none is contained in another.",
"We next give a characterization of covers via the hypergraph of the query result.",
"Proposition 10 Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ has a minimal edge cover $M$ such that $\\mathit {rel}(M)=K$ .",
"Example 11 Figure REF gives a minimal edge cover $M$ and the cover $\\mathit {rel}(M)$ .",
"By removing any edge from $M$ , it is not anymore an edge cover.",
"By removing the tuple corresponding to that edge from $\\mathit {rel}(M)$ , it is not anymore a cover since it is not result preserving.",
"By adding an edge to $M$ or the corresponding tuple to $\\mathit {rel}(M)$ , they are not anymore minimal.",
"We now turn our investigation to sizes and first note the following immediate property.",
"Proposition 12 Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is a subset of $Q(\\mathbf {D})$ .",
"An implication of Proposition REF is that the covers cannot be larger than the query result.",
"However, they can be much more succinct.",
"We first give size bounds for covers using the sizes of projections of the query result onto the bags of the underlying decomposition.",
"Proposition 13 Given $(Q,\\mathcal {T},\\mathbf {D})$ , the size of each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ satisfies the inequalities $\\max _{B\\in {\\cal S}(\\mathcal {T})}\\lbrace \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid \\rbrace $ $\\le $ $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ .",
"We can now characterize the size of a cover using the width of the decomposition.",
"Theorem 14 Let $Q$ be a natural join query and $\\mathcal {T}$ a decomposition of $Q$ .",
"For any database $\\mathbf {D}$ , each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"There are arbitrarily large databases $\\mathbf {D}$ such that each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The size gaps between query results and their covers can be arbitrarily large.",
"For any join query $Q$ and database $\\mathbf {D}$ , it holds that $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ and there are arbitrarily large databases $\\mathbf {D}$ for which $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ [4].",
"For acyclic queries, the fractional edge cover number $\\rho ^*$ can be as large as $|Q|$ , while the fractional hypertree width is one.",
"Section shows that the same gap also holds for time complexity.",
"Example 15 We continue Example REF .",
"The decomposition $\\mathcal {T}$ has width one, which is minimal.",
"The covers over $\\mathcal {T}$ , such as $K_1$ and $K_2$ , have sizes upper bounded by the input database size.",
"The minimum size of a cover over $\\mathcal {T}$ is the maximum size of a relation used in the query (assuming the relations are globally consistent).",
"In contrast, there are arbitrarily large databases of size $N$ for which the query result has size $\\Omega (N^2)$ .",
"Proposition REF and Theorem REF give alternative equivalent characterizations of the size of a cover of a query result.",
"The former gives it as the size of a minimal edge cover of the hypergraph of the query result over the attribute sets given by the bags of a decomposition $\\mathcal {T}$ , while the latter states it using the fractional hypertree width of $\\mathcal {T}$ or equivalently the maximum fractional edge cover number over all the bags of $\\mathcal {T}$ .",
"Most notably, whereas the former is an integral number, the latter is a fractional number.",
"This size gap between query results and their covers is precisely the same as for query results and their factorized representations called d-representations [23].",
"In this sense, covers can be seen as relational encodings of factorized representations of query results.",
"We can easily translate covers into factorized representations.",
"Appendix gives a brief introduction to d-representations and a translation example.",
"Proposition 16 Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover $K$ of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ can be translated into a d-representation of $Q(\\mathbf {D})$ of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ .",
"The above translation allows us to extend the applicability of covers to known workloads over factorized representations, such as in-database optimization problems [2] and in particular learning regression models [22].",
"Nevertheless, it is practically desirable to process such workloads directly on covers, since this would avoid the indirection via factorized representations that comes with extra space cost and non-relational data representation.",
"Aggregates, which are at the core of such workloads, can be computed directly on covers by joint scans of the projections of the cover onto the bags of the decomposition; alternatively, they can be computed by expressing any cover as the natural join of its bag projections and then pushing the aggregates past the join.",
"Example 17 We consider the query $Q = R(A,B) \\bowtie S(B,C)$ and its decomposition $\\mathcal {T}$ with bags $\\lbrace A,B\\rbrace $ and $\\lbrace B,C\\rbrace $ .",
"To compute aggregates over the join result $Q(\\mathbf {D})$ , we can use any cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"The expression for counting the number of result tuples is $\\sum _{b\\in \\text{dom}(B)}\\sum _{a\\in \\text{dom}(A)}\\sum _{c\\in \\text{dom}(C)} {\\bf 1}_{R(a,b)}\\cdot {\\bf 1}_{S(b,c)}$ , where $1_E$ is the Kronecker delta that is evaluated to 1 if the event $E$ is satisfied and 0 otherwise.",
"We can compute it in one scan over $K$ if $K$ is sorted on $(B,A,C)$ or $(B,C,A)$ .",
"For each $B$ -value $b$ , we multiply the distinct numbers of $A$ -values and of $C$ -values paired with $b$ in $K$ , and we sum up these products over all $B$ -values.",
"We can rewrite this expression as follows: $\\sum _{b\\in \\text{dom}(B)}(\\sum _{a\\in \\text{dom}(A)}{\\bf 1}_{(a,b)\\in \\pi _{\\lbrace A,B\\rbrace } K})(\\sum _{c\\in \\text{dom}(C)} {\\bf 1}_{(b,c)\\in \\pi _{\\lbrace B,C\\rbrace } K})$ .",
"This expression only uses the pairs $(a,b)$ and $(b,c)$ in $K$ .",
"The pairs $(a,c)$ , which make the difference among covers and are the culprits for the explosion in the size of the query result, are not needed.",
"Despite their succinctness over the explicit listing of tuples in a query result, any cover of the query result can be used to enumerate the result tuples with constant delay and extra space (data complexity) following linear-time pre-computation.",
"In particular, the delay and the space are linear in the number of attributes of the query result which is as good as enumerating directly from the result.",
"This complexity follows from Proposition REF and the enumeration for factorized representations [23] with constant delay and extra space.",
"Corollary 18 (Proposition REF , Theorem 4.11 [23]) Given $(Q,\\mathcal {T},\\mathbf {D})$ , the tuples in the query result $Q(\\mathbf {D})$ can be enumerated from any cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space.",
"An alternative way to achieve constant-delay enumeration with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation is by noting that the acyclic join queries considered in this paper are free-connex and thus allow for enumeration with constant delay and $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|)$ pre-computation [5].",
"An acyclic conjunctive query is called free-connex if its extension by a new relation symbol covering all attributes of the result remains acyclic [25].",
"Moreover, given a cover $K$ over a decomposition $\\mathcal {T}$ , the natural join of the projections of $K$ onto the bags of $\\mathcal {T}$ is an acyclic query that computes the original query result (Proposition REF )."
],
[
"Computing Covers for Join Queries using Cover-Join Plans",
"Given an arbitrary join query and database, we can compute covers using a monolithic algorithm akin to known algorithms for computing factorized representations of query results [22].",
"However, is it possible to compute covers in a compositional way, by computing covers for one join at a time?",
"In this section, we answer this question in the affirmative for acyclic natural join queries $Q$ and globally consistent databases $\\mathbf {D}$ with respect to $Q$ .",
"For a triple $(Q,\\mathcal {J},\\mathbf {D})$ , where $Q$ is an acyclic natural join query, $\\mathcal {J}$ is a join tree of $Q$ , and $\\mathbf {D}$ is a database globally consistent with respect to $Q$ , we use so-called cover-join plans to compute covers of the query result $Q(\\mathbf {D})$ over the decomposition corresponding to the join tree $\\mathcal {J}$ .",
"Such plans follow the structure of the join tree $\\mathcal {J}$ and use a new binary join operator called cover-join.",
"The cover-join of two relations yields a cover of their natural join.",
"This approach is in the spirit of standard relational query evaluation.",
"It is compositional in the sense that to compute a cover of the query result, it suffices to repeatedly compute a cover of the join of two relations.",
"This is practical since it can be supported by existing query engines extended with the cover-join operator.",
"We also show that, due to the binary nature of the cover-join operator, the cover-join plans cannot recover all possible covers of the query result.",
"Furthermore, different plans may lead to different covers.",
"Plans that do not follow the structure of a join tree may be unsound as they do not necessarily construct covers.",
"To compute covers for an arbitrary join query and database, we proceed in two stages.",
"We first materialize the bags of a decomposition of the query so as to reduce it to an acyclic query $Q$ over an extended database $\\mathbf {D}$ that is now globally consistent with respect to $Q$ (Proposition REF ).",
"We then use a cover-join plan to compute covers of $Q(\\mathbf {D})$ .",
"The first step has a non-trivial time complexity overhead, whereas the second step is linearithmic.",
"Overall, this strategy is worst-case optimal for computing covers for arbitrary join queries and databases."
],
[
"The Cover-Join Operator",
"The building block of our approach to computing covers is the binary cover-join operator.",
"Definition 19 (Cover-Join) The cover-join of two relations $R_1$ and $R_2$ , denoted by $R_1\\mathring{}R_2$ , computes a cover of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"Following the alternative characterization of covers of a query result by minimal edge covers in the hypergraph of the query result (Proposition REF ), the cover-join defines the relation $\\mathit {rel}(M)$ of a minimal edge cover $M$ of the hypergraph $H$ of the result of the join $R_1 \\bowtie R_2$ over the attribute sets ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"The hypergraph $H$ is bipartite and consists of disjoint complete bipartite subgraphs.",
"Since a cover is a minimal edge cover, it corresponds to a bipartite subgraph with the same number of nodes but a subset of the edges, where all paths can only have one or two edges.",
"A cover cannot have unconnected nodes, since it would not be an edge cover.",
"A path of three (or more) edges violates the minimality of the edge cover: Such a path $a_1-b_1-a_2-b_2$ in a bipartite graph covers the four nodes, yet a minimal cover would only have the two edges $a_1-b_1$ and $a_2-b_2$ .",
"We can compute a cover of a join of two relations $R_1$ and $R_2$ in time $\\widetilde{\\mathcal {O}}(|R_1| + |R_2|)$ , since it amounts to computing a minimal edge cover in a collection of disjoint complete bipartite graphs that encode the join result.",
"The smallest size of a cover is given by the edge cover number of the bipartite graph representing the join result, which is the maximum of the sizes of the two sets of nodes in the graph [19].",
"The largest size can be achieved in case one of the two node sets has size one, in which case this is paired with all nodes in the second set.",
"In case both sets have more than one node, the largest size is achieved when we pair one node from one of the two node sets with all but one node in the second set and then the remaining node in the second set with all but the already used node in the first set.",
"For the analysis in this paper, we assume that our cover-join algorithm may return any cover of the natural join of two relations.",
"In practice, however, it makes sense to compute a cover of minimum size.",
"We choose this cover as follows: For each complete bipartite hypergraph in the join result with node sets $V_1$ and $V_2$ such that $|V_1|\\le |V_2|$ , we choose a minimum edge cover by pairing each node in $V_1$ with one distinct node in $V_2$ and all remaining nodes in $V_2$ with one node in $V_1$ .",
"Proposition 20 Given two consistent relations $R_1$ and $R_2$ , the cover-join computes a cover $K$ of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ in time $\\widetilde{\\mathcal {O}}(|R_1|+|R_2|)$ and with size $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ .",
"Example 21 Consider again the product $R_1(A)\\bowtie R_2(B)$ in Example REF , where $R_1=[2]$ and $R_2=[n]$ with $n > 1$ .",
"Examples of covers of size $n$ over the decomposition $\\mathcal {T}$ with bags $\\lbrace A\\rbrace $ and $\\lbrace B\\rbrace $ are: $\\lbrace (1,i)\\mid i\\in [n]-\\lbrace k\\rbrace \\rbrace \\cup \\lbrace (2,k)\\rbrace $ for any $k\\in [n]$ ; $\\lbrace (1,i)\\mid i\\in [k]\\rbrace \\cup \\lbrace (2,j+k)\\mid j\\in [n-k]\\rbrace $ for any $k\\in [n-1]$ .",
"If $R_1=[m]$ with $m>n$ , then examples of covers over $\\mathcal {T}$ of minimum size $m$ are: $\\lbrace (i,i)\\mid i\\in [k-1]\\rbrace \\cup \\lbrace (k-1+i , k + i) \\mid i \\in [n-k]\\rbrace \\cup \\lbrace (n-1+i , k) \\mid i \\in [m-n+1]\\rbrace $ for any $k\\in [n]$ .",
"A cover over $\\mathcal {T}$ of maximal size $n+m-2$ is: $\\lbrace (1,i)\\mid i\\in [n-1]\\rbrace \\cup \\lbrace (j+1,n)\\mid j\\in [m-1]\\rbrace $ .",
"Below are depictions of the complete bipartite graph corresponding to the query result for $n=4$ and $m=5$ , where the edges in a minimal edge cover are solid lines and all other edges are dotted.",
"The left minimal edge cover corresponds to a cover over $\\mathcal {T}$ of minimum size $m=5$ , while the right minimal edge cover corresponds to a cover over $\\mathcal {T}$ of maximum size $n+m-2=7$ .",
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],
[
"Cover-join Plans",
"We now compose cover-join operators into so-called cover-join plans to compute covers for acyclic natural join queries.",
"Before we define such plans, we need to introduce some notation.",
"For a join tree $\\mathcal {J}$ of a query $Q$ , we write $\\mathcal {J}= \\mathcal {J}_1\\circ \\mathcal {J}_2$ if $\\mathcal {J}$ can be split into two non-empty subtrees $\\mathcal {J}_1$ and $\\mathcal {J}_2$ that are connected by a single edge in $\\mathcal {J}$ .",
"Any subtree $\\mathcal {J}^{\\prime }$ of $\\mathcal {J}$ defines the subquery of $Q$ that is the natural join of all relation symbols that are nodes in $\\mathcal {J}^{\\prime }$ .",
"Definition 22 (Cover-Join Plan) Given $(Q,\\mathcal {J},\\mathbf {D})$ , a cover-join plan $\\varphi $ over the join tree $\\mathcal {J}$ is defined recursively as follows: If $\\mathcal {J}$ consists of one node $R$ , then $\\varphi = R$ .",
"The plan $\\varphi $ returns $R$ .",
"If $\\mathcal {J}= \\mathcal {J}_1\\circ \\mathcal {J}_2$ and $\\varphi _{i}$ is a cover-join plan over $\\mathcal {J}_i$ , then $\\varphi =\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ .",
"The plan $\\varphi $ returns the result of $R_1\\ \\mathring{}\\ R_2$ , where the relation $R_i$ is returned by the plan $\\varphi _{i}$ ($i\\in [2]$ ).",
"Lemma REF states next that a cover-join plan computes a cover of the query result over the decomposition corresponding to a given join tree of the query.",
"Lemma 23 Given $(Q,\\mathcal {J},\\mathbf {D})$ where $\\mathbf {D}=\\lbrace R_i\\rbrace _{i\\in [n]}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(|K|)$ and with size $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Lemma REF states three remarkable properties of cover-join plans.",
"First, they compute covers compositionally: To obtain a cover of the entire query result it is sufficient to compute covers of the results for subqueries.",
"More precisely, for a cover-join plan $\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ , the sub-plans $\\varphi _{1}$ and $\\varphi _{2}$ compute covers for the subqueries defined by the joins of the relations in the join trees $\\mathcal {J}_1$ and respectively $\\mathcal {J}_2$ .",
"Then, the plan $\\varphi _{1}\\ \\mathring{}\\ \\varphi _{2}$ computes a cover for the join of the relations in the join tree $\\mathcal {J}=\\mathcal {J}_1\\circ \\mathcal {J}_2$ .",
"Second, the output of a cover-join plan is always a cover, regardless which cover is picked at each cover-join operator in the plan.",
"Third, it does not matter which cover-join plan we choose for a given join tree, the resulting covers are computed with the same time guarantee.",
"Nevertheless, different plans for the same join tree may lead to different covers (Example REF ).",
"These properties rely on the global consistency of the database and on the fact that the plans follow the structure of the join tree.",
"For arbitrary databases, a cover-join operator may wrongly construct covers using dangling tuples at the expense of relevant tuples that are not anymore covered and therefore lost.",
"Furthermore, plans that do not follow the structure of a join tree may be unsound (Example REF ).",
"Although each cover-join operator computes a cover of minimum size for the join of its input relations, the overall cover computed by a cover-join plan may not be a cover of minimum size of the query result (Example REF in Appendix ).",
"Example 24 A join tree that admits several splits can define many plans.",
"For instance, the join tree for the query $R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ is the path $R_1-R_2-R_3$ and admits two possible splits that lead to the plans $\\varphi _1 = (R_1(A,B) \\mathring{}R_2(B,C)) \\mathring{}R_3(C,D)$ and $\\varphi _2 = R_1(A,B) \\mathring{}(R_2(B,C) \\mathring{}R_3(C,D))$ .",
"The relations are those in Figure REF , now calibrated.",
"For this database, the covers computed by the sub-plans $R_1(A,B) \\mathring{}R_2(B,C)$ and $R_2(B,C) \\mathring{}R_3(C,D)$ correspond to full join results, since all join values only occur once in the relations.",
"By taking any possible cover at each cover-join operator in the plans, both plans yield the same four possible covers of the query result: One of them is $\\mathit {rel}(M)$ in Figure REF and two of them are $K_1$ and $K_2$ in Example REF .",
"The last cover is not depicted: It is the same as $K_1$ with the change that the values $d_1$ and $d_2$ are swapped between the first two rows.",
"A corollary of Proposition REF and Lemma REF is that covers over decompositions of arbitrary natural join queries can be computed in time proportional to their sizes.",
"Theorem 25 (Proposition REF , Lemma REF ) Given a natural join query $Q$ , decomposition $\\mathcal {T}$ of $Q$ , and database $\\mathbf {D}$ , a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ and with size $\\mathcal {O}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ where $Q$ is an arbitrary natural join query and $\\mathbf {D}$ is an arbitrary database, we can compute a cover in four steps: construct $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ such that $Q^{\\prime }$ is an acyclic natural join query, $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ and $\\mathbf {D}^{\\prime }$ consists of materializations of the bags of $\\mathcal {T}$ ; turn $\\mathbf {D}^{\\prime }$ into a globally consistent database $\\mathbf {D}^{\\prime \\prime }$ with respect to $Q^{\\prime }$ ; turn $\\mathcal {T}$ into a join tree $\\mathcal {J}$ of $Q^{\\prime }$ by replacing each bag by the corresponding relation symbol in $Q^{\\prime }$ ; and execute on $\\mathbf {D}^{\\prime \\prime }$ a cover-join plan for $Q^{\\prime }$ over $\\mathcal {J}$ .",
"Since there are arbitrarily large databases for which the size bounds on covers are tight (Theorem REF ), the cover-join plans, together with a worst-case optimal algorithm for materializing bags [21], represent a worst-case optimal algorithm for computing covers.",
"We conclude this section with three insights into the ability of cover-join plans to compute covers.",
"We give an example of an unsound cover-join plan that does not follow the structure of a join tree.",
"We then note the incompleteness of our cover-join plans due to the binary nature of the cover-join operator.",
"We give an example of a cover that cannot be computed with our cover-join plans, but can be computed using a multi-way cover-join operator.",
"Finally, we give an example showing that distinct cover-join plans over the same (or also distinct) join trees can yield incomparable sets of covers.",
"Example 26 (Unsound plan) Consider the query $Q = R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ , the following database with relations $R_1$ , $R_2$ , and $R_3$ , and four relations computed by cover-joining two of the three relations: Table: NO_CAPTIONFollowing Definition REF , the plan $(R_1(A,B) \\mathring{}R_3(C,D)) \\mathring{}R_2(B,C)$ would require a split $\\mathcal {J}_{1,3}\\circ \\mathcal {J}_2$ of a join tree, where the join tree $\\mathcal {J}_{1,3}$ has two nodes $R_1$ and $R_3$ while the join tree $\\mathcal {J}_{2}$ has one node $R_2$ .",
"However, there is no join tree that allows such a split.",
"The cover-join $R_1(A,B) \\mathring{}R_3(C,D)$ computes one of the two covers $K_{1,3}$ and $K^{\\prime }_{1,3}$ .",
"The result of the join of $K^{\\prime }_{1,3}$ and $R_2$ is empty and so is the cover-join.",
"This means that this plan does not always compute a cover, which makes it unsound.",
"This problem cannot occur with cover-join plans over join trees of $Q$ .",
"The only cover-join plans over join trees of $Q$ are (up to commutativity) $(R_1(A,B)$ $\\mathring{}$ $R_2(B,C))$ $\\mathring{}$ $R_3(C,D)$ and $R_1(A,B) \\mathring{}(R_2(B,C)\\mathring{}R_3(C,D))$ .",
"The only cover of $R_1(A,B) \\mathring{}R_2(B,C)$ is $K_{1,2}$ above, which can be cover-joined with $R_3$ .",
"The only cover of $R_2(B,C) \\mathring{}R_3(C,D)$ is $K_{2,3}$ above, which can be cover-joined with $R_1$ .",
"Example 27 (Cover-Join Incompleteness) Consider the product query $Q =R_1(A)\\bowtie R_2(B)\\bowtie R_3(C)$ , the following database $\\mathbf {D}$ with relations $R_1$ , $R_2$ , and $R_3$ and one cover $K$ of the query result over the decomposition with bags $\\lbrace A\\rbrace $ , $\\lbrace B\\rbrace $ , and $\\lbrace C\\rbrace $ : Table: NO_CAPTIONA decomposition of $Q$ can have up to three bags which are not included in other bags.",
"In case of decompositions with three bags, each bag consists of exactly one attribute.",
"These decompositions correspond to the join trees that are permutations of the three relation symbols.",
"There are three possible cover-join plans (up to commutativity) over these join trees: $\\varphi _1= R_1(A) \\mathring{}(R_2(B) \\mathring{}R_3(C))$ , $\\varphi _2= R_2(B) \\mathring{}(R_1(A) \\mathring{}R_3(C))$ and $\\varphi _3= R_3(C) \\mathring{}(R_1(A) \\mathring{}R_2(B))$ .",
"None of these plans can yield the cover $K$ above.",
"As discussed after Definition REF , a minimal edge cover corresponding to a cover computed by a binary cover-join operator can only have paths of one or two edges.",
"For instance, $\\pi _{\\lbrace A,B\\rbrace }K$ , which should correspond to a cover of $R_1(A) \\mathring{}R_2(B)$ , has the path of three edges $b_2 - a_1 - b_1 - a_2$ .",
"The cover-join $R_1(A) \\mathring{}R_2(B)$ would not create this path since it corresponds to a non-minimal edge cover.",
"Similarly, $\\pi _{\\lbrace A,C\\rbrace }K$ and $\\pi _{\\lbrace B,C\\rbrace }K$ have paths of three edges.",
"For decompositions with two bags, two of the three attributes are in the same bag.",
"Without loss of generality, assume $A$ and $B$ are in the same bag.",
"Following Proposition REF , this bag is covered by a new relation $R_{1,2}$ that is the product of $R_1$ and $R_2$ .",
"This means that $K$ has to be the cover of $R_{1,2}(A,B)\\mathring{}R_3(C)$ , yet $\\pi _{\\lbrace A,B\\rbrace }K$ is not $R_{1,2}$ !",
"The decomposition with one bag consisting of all three attributes has this bag covered by a new relation that is the product of the three relations.",
"This relation is the Cartesian product of the three relations that is the full query result and different from $K=\\pi _{\\lbrace A,B,C\\rbrace }K$ .",
"We conclude that the cover $K$ cannot be computed using cover-join plans with binary cover-join operators.",
"Example 28 (Incomparable Sets of Covers) Consider the product query $Q=R_1(A)\\bowtie R_2(B)\\bowtie R_3(C)$ and the following database $\\lbrace R_1, R_2, R_3\\rbrace $ : Table: NO_CAPTIONLet us consider the join tree $\\mathcal {J}=R_1-R_2-R_3$ of $Q$ .",
"There are (up to commutativity) two possible cover-join plans over $\\mathcal {J}$ : $\\varphi _1= R_1(A) \\mathring{}(R_2(B) \\mathring{}R_3(C))$ and $\\varphi _2=(R_1(A) \\mathring{}R_2(B)) \\mathring{}R_3(C)$ .",
"The above relation $K$ is a cover of the result of $Q$ and can be computed by $\\varphi _1$ , which cover-joins $R_1(A)$ and a cover of the join of $R_2(B)$ and $R_3(C)$ .",
"This cover cannot be computed by $\\varphi _2$ .",
"Indeed, $\\varphi _2$ first cover-joins $R_1(A)$ and $R_2(B)$ , yielding $K_{1,2}$ or $K^{\\prime }_{1,2}$ as the only possible covers.",
"Then, cover-joining any of them with $R_3(C)$ does not yield the cover $K$ since $\\pi _{\\lbrace A,B\\rbrace }K$ is different from both $K_{1,2}$ and $K^{\\prime }_{1,2}$ .",
"Similarly, $\\varphi _2$ computes covers that cannot be computed by $\\varphi _1$ ."
],
[
"Covers for Functional Aggregate Queries",
"We first give a brief introduction to functional aggregate queries (FAQ) [17].",
"A detailed description can be found in the appendix.",
"Given an attribute set $S$ , we use ${\\textsf {a}}_S$ to indicate that tuple ${\\textsf {a}}$ has schema $S$ .",
"For $S^{\\prime } \\subseteq S$ , we denote by ${\\textsf {a}}_{S^{\\prime }}$ the restriction of ${\\textsf {a}}$ to $S^{\\prime }$ .",
"A functional aggregate query has the following form (slightly adapted to our notation): $\\varphi ({\\textsf {a}}_{\\lbrace A_1,\\ldots ,A_f\\rbrace }) = \\underset{a_{f+1} \\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n} \\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}}\\ \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes }\\ \\psi _S({\\textsf {a}}_S),\\text{ where:}$ $H=( \\mathcal {V}, \\mathcal {E})$ is the multi-hypergraph of the query with ${\\cal V}=\\lbrace A_i\\rbrace _{i\\in [n]}$ .",
"$\\textsf {Dom}$ is a fixed (output) domain, such as $\\lbrace $true,false$\\rbrace $ , $\\lbrace 0,1\\rbrace $ , or $\\mathbb {R}^+$ .",
"$\\mathcal {V}_{\\text{free}} = \\lbrace A_1,\\ldots ,A_f\\rbrace $ is the set of result or free attributes; all other attributes are bound.",
"For each attribute $A_i$ with $i>f$ , $\\oplus ^{(i)}$ is a binary (aggregate) operator on the domain $\\textsf {Dom}$ .",
"Different bound attributes may have different aggregate operators.",
"For each attribute $A_i$ with $i>f$ , either $\\oplus ^{(i)}$ is $\\otimes $ or $(\\textsf {Dom},\\oplus ^{(i)},\\otimes )$ forms a commutative semiring with the same additive identity ${\\bf 0}$ and multiplicative identity ${\\bf 1}$ for all semirings.",
"For every hyperedge $S$ in $\\cal E$ , $\\psi _S: \\prod _{A\\in S}\\textsf {dom}(A) \\rightarrow \\textsf {Dom}$ is an (input) function.",
"FAQs are a semiring generalization of aggregates over join queries, where the aggregates are the operators $\\oplus ^{(i)}$ and the natural join is expressed by $\\bigotimes _{S\\in \\mathcal {E}} \\psi _S({\\sf a}_S)$ .",
"The listing representation $R_{\\psi _S}$ of a function $\\psi _S$ is a relation over the schema $S \\cup \\lbrace \\psi _S(S)\\rbrace $ which consists of all input-output pairs for $\\psi _S$ where the output is non-zero, i.e., $R_{\\psi _S}$ contains a tuple ${\\textsf {a}}_{S \\cup \\lbrace \\psi _S(S)\\rbrace }$ if and only if $\\psi _S({\\textsf {a}}_S) = {\\textsf {a}}_{\\psi _S(S)} \\ne {\\bf 0}$ .",
"An input database for $\\varphi $ contains for each $\\psi _S$ its listing representation.",
"We say that $\\mathcal {T}$ is a decomposition of $\\varphi $ if $\\mathcal {T}$ is a decomposition of the hypergraph $H$ of $\\varphi $ .",
"Given an FAQ $\\varphi $ and database $\\mathbf {D}$ , the FAQ-problem is to compute the query result $\\varphi (\\mathbf {D})$ .",
"Each FAQ $\\varphi $ has an FAQ-width $\\textsf {faqw}(\\varphi )$ which is defined similarly to the fractional hypertree width of the hypergraph of $\\varphi $ .",
"For instance, in case where all attributes of $\\varphi $ are free, $\\textsf {faqw}(\\varphi )$ is equal to the fractional hypertree width of the hypergraph of $\\varphi $ .",
"Given an FAQ $\\varphi $ and a database $\\mathbf {D}$ , the InsideOut algorithm [17] solves the FAQ-problem as follows.",
"First, it eliminates all bound attributes along with their corresponding aggregate operators by performing equivalence-preserving transformations on $\\varphi $ .",
"Then, it computes the listing representation of the remaining query.",
"The algorithm runs in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )} + Z)$ where $Z$ is the size of the output, i.e., the listing representation of $\\varphi $ .",
"We can compute a cover of the result of a given FAQ $\\varphi $ in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )})$ , which does not depend on the size of the listing representation of $\\varphi $ .",
"Our strategy is as follows.",
"We first eliminate all bound attributes in $\\varphi $ by using InsideOut resulting in an FAQ $\\varphi ^{\\prime }$ .",
"We then take a decomposition $\\mathcal {T}$ of $\\varphi ^{\\prime }$ and compute bag functions $\\beta _B$ , $B \\in {\\cal S}(\\mathcal {T})$ , with $\\varphi ^{\\prime }({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) =\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B})$ .",
"Finally, we compute a cover of the join result of the listing representations of the bag functions over the extension of $\\mathcal {T}$ that contains, for each bag $B$ , the attribute $\\beta _B(B)$ for the values of the function $\\beta _B$ .",
"Keeping the $\\beta _B(B)$ -values of the bag functions in the cover is necessary for recovering the output values of $\\varphi $ when enumerating the result of $\\varphi $ from the cover.",
"Example 29 We consider the following FAQ $\\varphi $ over the sum-product semiring $(\\mathbb {N},+,\\cdot )$ (for simplicity we skip the explicit iteration over the domains of the attributes in $\\varphi $ ): $\\varphi (a,b,d) = \\sum _{c,e,f,g,h} \\psi _1(a,b,c)\\cdot \\psi _2(b,d,e)\\cdot \\psi _3(d,e,f)\\cdot \\psi _4(f,h)\\cdot \\psi _5(e,g), \\mbox{ where }$ $\\varphi $ , $\\psi _1$ , $\\psi _2$ , $\\psi _3$ , $\\psi _4$ and $\\psi _5$ are over $\\lbrace A,B,D\\rbrace $ , $\\lbrace A,B,C\\rbrace $ , $\\lbrace B,D,E\\rbrace $ , $\\lbrace D,E,F\\rbrace $ , $\\lbrace F,H\\rbrace $ and $\\lbrace E,G\\rbrace $ , respectively.",
"We first run InsideOut on $\\varphi $ to eliminate the bound attributes and obtain the following FAQ: $\\varphi ^{\\prime }(a,b,d) &= \\underbrace{\\big (\\sum _{c} \\psi _1(a,b,c)\\big )}_{\\psi _6(a,b)}\\cdot \\underbrace{\\sum _e \\big (\\psi _2(b,d,e)\\cdot \\underbrace{\\sum _f \\big (\\psi _3(d,e,f)\\cdot \\underbrace{\\sum _h \\psi _4(f,h)}_{\\psi _7(f)}\\big )}_{\\psi _{9}(d,e)}\\cdot \\underbrace{\\sum _g \\psi _5(e,g)}_{\\psi _8(e)}\\big )}_{\\psi _{10}(b,d)}.$ We consider the decomposition $\\mathcal {T}$ of $\\varphi ^{\\prime }$ with two bags $B_1 = \\lbrace A,B \\rbrace $ and $B_2 = \\lbrace B,D\\rbrace $ and bag functions $\\psi _6$ and respectively $\\psi _{10}$ .",
"Then, we execute the cover-join plan $R_{\\psi _6}\\ \\mathring{}\\ R_{\\psi _{10}}$ over the extended decomposition $\\mathcal {T}^{\\prime }$ with bags $\\lbrace A,B, \\psi _{6}(A,B)\\rbrace $ and $\\lbrace B,D, \\psi _{10}(B,D)\\rbrace $ .",
"While the computation of the result of $\\varphi ^{\\prime }$ can take quadratic time, the above cover-join plan takes linear time.",
"We exemplify the computation of the cover-join plan.",
"Assume the following tuples in $\\psi _6$ and $\\psi _{10}$ , where $\\gamma _1,\\ldots ,\\gamma _4,\\delta _1,\\ldots ,\\delta _3\\in \\mathbb {N}$ : Table: NO_CAPTIONThe relation $K$ is a possible cover computed by the cover-join plan.",
"The cover carries over the aggregates in columns $\\psi _{6}(A,B)$ and $\\psi _{10}(B,D)$ , one per bag of $\\mathcal {T}^{\\prime }$ .",
"The aggregate of the first tuple in $K$ is $\\gamma _1\\cdot \\delta _1$ (or $\\gamma _1\\otimes \\delta _1$ under a semiring with multiplication $\\otimes $ ).",
"The following theorem relies on Lemma REF and Theorem REF that give an upper bound on the time complexity for constructing covers of join results.",
"Theorem 30 For any FAQ $\\varphi $ and database $\\mathbf {D}$ , a cover of the query result $\\varphi (\\mathbf {D})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {faqw}(\\varphi )})$ .",
"Any enumeration algorithm for covers of join results can be used to enumerate the tuples of an FAQ result from one of its covers.",
"We thus have the following corollary: Corollary 31 (Corollary REF ) Given a cover $K$ of the result $\\varphi (\\mathbf {D})$ of an FAQ $\\varphi $ over a database $\\mathbf {D}$ , the tuples in the query result $\\varphi (\\mathbf {D})$ can be enumerated with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space."
],
[
"Conclusion",
"Results of join and functional aggregate queries entail redundancy in both their computation and representation.",
"In this paper we propose the notion of covers of query results to reduce such redundancy.",
"While covers can be more succinct than the query results, they nevertheless enjoy desirable properties such as listing representation and constant-delay enumeration of result tuples.",
"For a given database and a join or functional aggregate query, the query result can be normalized as a globally consistent database over an acyclic schema.",
"Covers represent one-relational, lossless, linear-size encodings of such normalized databases.",
"Definition 32 borged /bôrjd/ : Buy One Relation, Get Entire Database!"
],
[
"Acknowledgements.",
"This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement 682588.",
"The authors would like to thank Milos Nikolic, Max Schleich, and the anonymous reviewers for their feedback on drafts of this paper, and Yu Tang for inspiring discussions that led to the concept of cover as a relational alternative to factorized representations."
],
[
"Further Preliminaries",
"We introduce necessary notation for the proofs in the following sections.",
"Restrictions of Queries and Databases.",
"Given a set $X$ of attributes and a natural join query $Q= R_1 \\bowtie \\ldots \\bowtie R_n$ , the $X$ -restriction of $Q$ is defined as $Q_X = R_1^X \\bowtie \\ldots \\bowtie R_n^X$ where each $R_i^X$ results from $R$ by restricting its schema to $X$ .",
"Likewise, we obtain the $X$ -restriction $\\mathbf {D}_X$ of a database $\\mathbf {D}$ by projecting each relation in $\\mathbf {D}$ onto the attributes in $X$ ."
],
[
"From Covers to D-Representations",
"We next give a brief introduction to d-representations; for a detailed description, we refer the reader to the literature [23].",
"We then discuss a translation from covers to d-representations.",
"Figure: Top row: database 𝐃={R 1 ,R 2 ,R 3 }\\mathbf {D}=\\lbrace R_1,R_2, R_3\\rbrace , the result Q(𝐃)Q(\\mathbf {D}) of the path query Q=R 1 ⋈R 2 ⋈R 3 Q = R_1\\bowtie R_2\\bowtie R_3, and a cover K⊆Q(𝐃)K\\subseteq Q(\\mathbf {D}) over the decomposition 𝒯\\mathcal {T}; bottom row: decomposition 𝒯\\mathcal {T} of QQ and an equivalent d-tree 𝒯 ' \\mathcal {T}^{\\prime }."
],
[
"D-Representations in a Nutshell",
"D-representations are a succinct and lossless representation for relational data.",
"A d-representation is a set of named relational algebra expressions $\\lbrace N_1: =E_1, \\ldots , N_n:= E_n\\rbrace $ , where each $N_i$ is a unique name (or a pointer) and each $E_i$ is a relational algebra expression with unions, Cartesian products, singleton relations, i.e., unary relations with one tuple, and name references in place of singleton relations.",
"The size $\\mid \\hspace{-2.84526pt} E \\hspace{-2.84526pt} \\mid $ of a d-representation $E$ is the number of its singletons.",
"We consider a special class of d-representations that encode results of join queries and whose nesting structure is given by so-called d-trees.",
"In the literature, d-trees are defined as orderings on query variables.",
"We give here an alternative, equivalent definition that is in line with our notion of fractional hypertree decomposition.",
"Given a query $Q$ , a d-tree of $Q$ is a decomposition of $Q$ where each bag is partitioned into one attribute $A$ , called the bag attribute, and a set of attributes, called the key of $A$ and denoted by $\\mathit {key}(A)$ .",
"There is one bag per distinct attribute $A$ in $Q$ .",
"Each decomposition $\\mathcal {T}$ of a query $Q$ can be translated into a d-tree $\\mathcal {T}^{\\prime }$ of $Q$ with $\\textsf {fhtw}(\\mathcal {T}^{\\prime }) \\le \\textsf {fhtw}(\\mathcal {T})$ (Proposition 9.3 in [23]).",
"Given a query $Q$ , a d-tree $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , a d-representation $E$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ with size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ (Theorem 7.13 and Proposition 8.2 in [23]).",
"Example 33 We consider the path query $Q = R_1(A,B) \\bowtie R_2(B,C) \\bowtie R_3(C,D)$ .",
"Figure REF depicts a database with relations $R_1$ , $R_2$ and $R_3$ and the result of $Q$ over the input database $\\lbrace R_1, R_2, R_3\\rbrace $ .",
"It also shows a decomposition $\\mathcal {T}$ of $Q$ and a cover $K$ of the query result over $\\mathcal {T}$ .",
"Finally, it depicts a d-tree $\\mathcal {T}^{\\prime }$ (right below) derived from $\\mathcal {T}$ by using the translation in the proof of Proposition 9.3 in [23].",
"Figure: A d-representation encoded as a parse graph (left) and asa set of multimaps (right).D-representations admit encoding as parse graphs and sets of multi-maps.",
"Figure REF visualizes the d-representation of the query result from Figure REF over the d-tree $\\mathcal {T}^{\\prime }$ in the forms of a parse graph and of multi-maps.",
"The parse graph follows the structure of the d-tree.",
"At the top level we have a union of $B$ -values.",
"Then, given any $B$ -value, the $A$ -values are independent of the values for $C$ and $D$ .",
"Therefore, under each $B$ -value, the $A$ -values are represented in a different branch than the values for $C$ and $D$ .",
"Within the branches for $C$ and $D$ , the values are first grouped by $C$ and then by $D$ .",
"The information on keys is used to share subtrees across branches.",
"Since the key of attribute $D$ is $C$ only, all $C$ -nodes with the same value point to the same union of $D$ -values.",
"In our example, both $c_1$ -nodes point to the same set $\\lbrace d_1,d_2\\rbrace $ of $D$ -values.",
"The cover $K$ from Figure REF can be mapped immediately to the parse graph: Under each product node, we take a minimum number of combinations of its children to ensure that every value under the product node occurs in one of these combinations.",
"To enumerate the tuples in the query result, it suffices to choose in turn one branch of each union node and all branches of each product node.",
"For instance, the left product node represents the combinations of $\\lbrace a_1,a_2\\rbrace $ with $\\lbrace d_1,d_2\\rbrace $ , together with the values $b_1$ and $c_1$ .",
"There are four combinations, so four tuples in the result.",
"The first two tuples in the cover represent two of them, yet they are sufficient to recover all these tuples.",
"The multi-map encoding of a d-representation consists of one multi-map for each bag attribute: $m_A$ maps tuples over the attributes in $key(A)$ to (possibly several) values of $A$ .",
"Figure REF shows these maps as relations whose columns are distinctly separated into those for the key attributes (the map keys) and the column for the attribute $A$ itself (the map payload).",
"We have, for instance, $m_A(b_1) = a_1$ and $m_A(b_1) = a_2$ , whereas $m_C(b_1) = c_1$ .",
"Since $\\mathit {key}(A)=\\lbrace B\\rbrace $ and there are two $B$ -values in the d-representation leading to the sets $\\lbrace a_1,a_2\\rbrace $ and $\\lbrace a_3,a_4\\rbrace $ , respectively, $m_A$ maps the $B$ -value $ b_1$ to both $A$ -values $a_1$ and $a_2$ and the $B$ -value $b_2$ to both $A$ -values $a_3$ and $a_4$ ."
],
[
"Translating Covers into D-Representations",
"Figure REF gives an algorithm that constructs an equivalent d-representation from a cover over a decomposition.",
"Both the cover $K$ and the output d-representation are for the same query result $Q(\\mathbf {D})$ of a query $Q$ .",
"The decomposition $\\mathcal {T}$ is for the query $Q$ .",
"The algorithm creates a multi-map for each attribute $A$ and populates it with assignments of tuples over the keys of $A$ to the values of $A$ as encountered in the tuples of the cover.",
"Example 34 We consider the cover $K$ over the decomposition $\\mathcal {T}$ in Figure REF and the d-tree $\\mathcal {T}^{\\prime }$ equivalent to $\\mathcal {T}$ .",
"Following the algorithm in Figure REF , the cover $K$ is translated into a d-representation over $\\mathcal {T}^{\\prime }$ as follows.",
"After reading the first tuple $(a_1,b_1,c_1,d_1)$ , we add $() \\mapsto b_1$ to $m_B$ , $b_1 \\mapsto a_1$ to $m_A$ , $b_1 \\mapsto c_1$ to $m_C$ , and $c_1 \\mapsto d_1$ to $m_D$ , where $()$ means the empty tuple.",
"After processing the second tuple $(a_2,b_1,c_1,d_1)$ , we only change $m_A$ by adding $b_1 \\mapsto a_2$ to $m_A$ .",
"After the third tuple $(a_3,b_2,c_1,d_2)$ , we add the following new assignments: $() \\mapsto b_2$ to $m_B$ , $b_2 \\mapsto a_3$ to $m_A$ , $b_2 \\mapsto c_1$ to $m_C$ , and $c_1 \\mapsto d_2$ to $m_D$ .",
"After reading the last tuple $(a_4,b_2,c_1,d_2)$ , we add the new assignment $b_2 \\mapsto a_4$ to $m_A$ ."
],
[
"Cover-Join Plans Computing Covers of Non-Minimum Size",
"Example 35 We consider the acyclic natural join query $Q=R_1(A,B)\\bowtie R_2(B,C)\\bowtie R_3(C,D)$ , the database $\\mathbf {D}=\\lbrace R_1, R_2, R_3\\rbrace $ globally consistent with respect to $Q$ , and the join tree $\\mathcal {J}=R_1-R_2-R_3$ .",
"The relations $R_i$ are depicted below.",
"Table: NO_CAPTIONThe relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over the decomposition $\\mathcal {T}$ corresponding to $\\mathcal {J}$ .",
"It follows from Proposition REF that every cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least three.",
"Hence, $K$ is a minimum-sized cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We take the cover-join plan $(R_1(A,B) \\mathring{}R_2(B,C)) \\mathring{}R_3(C,D)$ over $\\mathcal {J}$ and assume that the cover-join operator computes for each two input relations $R$ and $R^{\\prime }$ , a minimum-sized cover of $R \\bowtie R^{\\prime }$ over the decomposition with bags ${\\cal S}(R)$ and ${\\cal S}(R^{\\prime })$ .",
"Then, a possible output of the sub-plan $R_1(A,B) \\mathring{}R_2(B,C)$ is the relation $K_{1,2}$ .",
"A possible result of the cover-join of the latter relation with $R_3$ is the relation $K^{\\prime }$ .",
"Although $K^{\\prime }$ is a valid cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ , it is not a minimum-sized cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ ."
],
[
"Covers for Equi-Join Queries",
"In this section, we extend the class of queries from natural join queries to arbitrary equi-join queries, whose relation symbols may map to the same database relation.",
"Equi-join Queries.",
"An equi-join query, aka full conjunctive query, has the form $Q = \\sigma _\\psi (R_1(S_1) \\times \\ldots \\times R_n(S_n))$ , where each $R_i$ is a relation symbol with schema $S_i$ and $\\psi $ is a conjunction of equalities of the form $A_1=A_2$ with attributes $A_1$ and $A_2$ .",
"We require that all relation symbols in the query as well as all attributes occurring in the schemas of the relation symbols are distinct.",
"We assume that each query comes with mappings $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ , called the signature mappings of $Q$ , where $\\lambda $ maps the relation symbols in $Q$ to relation symbols in the schema of the database and each $\\mu _{R_i}$ is a bijective mapping from the attributes of $R_i$ to the attributes of $\\lambda (R_i)$ .",
"Since we do not require $\\lambda $ to be injective, distinct relation symbols in $Q$ might refer to the same relation in the database (cf.",
"Example REF ).",
"The joins in equi-join queries are expressed by the equalities in $\\psi $ .",
"The transitive closure $\\psi ^+$ of $\\psi $ under the equality on attributes defines the attribute equivalence classes: The equivalence class ${\\cal A}$ of an attribute $A$ is the set consisting of $A$ and of all attributes equal to $A$ in $\\psi ^+$ .",
"For a set $S$ of attributes, $S^+$ denotes the set of attributes transitively equivalent to those in $S$ .",
"Hypergraphs and hypertree decompositions of equi-join queries are defined just like for natural join queries with the additional requirement that each hyperedge or bag includes all equivalent attributes for each contained attribute.",
"More formally, the hypergraph of an equi-jon query $Q$ consists of one node $A$ for each attribute $A$ in $Q$ and one edge ${\\cal S}(R)^+$ for each relation symbol $R\\in {\\cal S}(Q)$ .",
"Similarly, a hypertree decomposition $\\mathcal {T}$ (of the hypergraph $H$ ) of $Q$ is a pair $(T,\\chi )$ , where $T$ is a tree and $\\chi $ is a function mapping each node in $T$ to a set $V^+$ where $V$ is a subset of the nodes of $H$ .",
"All other notions and notations introduced in Section as well as the definitions of result preservation and covers in Section carry over to equi-join queries without any change.",
"Example 36 We consider the equi-join query $Q=\\sigma _\\psi (R_1(A_1,A_2)\\times R_2(A_3,A_4))$ , where $\\psi $ consists of the equality $A_2=A_3$ .",
"Let $(\\lambda , \\lbrace \\mu _{R_1}, \\mu _{R_2}\\rbrace )$ be the signature mappings of the query.",
"Assume that $\\lambda (R_1(A_1,A_2)) = \\lambda (R_2(A_3,A_4)) = R(A,B)$ , $\\mu _{R_1}(A_1) = \\mu _{R_2}(A_4) = A$ and $\\mu _{R_1}(A_2) = \\mu _{R_2}(A_3) = B$ , i.e., both relation symbols are mapped to the same relation symbol $R(A,B)$ , attributes $A_1$ and $A_4$ are mapped to attribute $A$ and attributes $A_2$ and $A_3$ are mapped to attribute $B$ .",
"Let $\\mathbf {D} = \\lbrace R\\rbrace $ where $R$ is defined as in Figure REF .",
"The figure depicts in the top row (besides $R$ ) the query result $Q(\\mathbf {D})$ , a cover $K$ of the query result over the decomposition $\\mathcal {T}$ depicted in the bottom row and two relations $R_1^{\\prime }, R_2^{\\prime }$ obtained from $R$ by the application of Proposition REF (given below).",
"The bottom row shows the hypergraph of $Q$ , the hypergraph of $Q(\\mathbf {D})$ over the attribute sets $\\lbrace \\lbrace A_1,A_2,A_3\\rbrace ,\\lbrace A_2,A_3,A_4\\rbrace \\rbrace $ , and a minimal edge cover $M$ of the latter hypergraph with $\\mathit {rel}(M) = K$ .",
"Figure: Top row: database 𝐃={R}\\mathbf {D}=\\lbrace R\\rbrace , the result Q(𝐃)Q(\\mathbf {D}) of the queryQQ in Example , a cover KK ofQ(𝐃)Q(\\mathbf {D}) over 𝒯\\mathcal {T}, and relations R 1 ' ,R 2 ' R_1^{\\prime },R_2^{\\prime } obtainedfrom RR by the application of Proposition ;bottom row: the hypergraph of QQ, a decomposition 𝒯\\mathcal {T} of QQ, the hypergraph of Q(𝐃)Q(\\mathbf {D}) over the attribute sets 𝒮(𝒯){\\cal S}(\\mathcal {T}), and a minimal edge cover MM of this hypergraph.Adaption of the results on covers to equi-join queries.",
"Due to the following two propositions, all results on covers in Sections and carry over to equi-join queries.",
"Proposition 37 Given an equi-join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , there exist a natural join query $Q^{\\prime }$ and a database $\\mathbf {D}^{\\prime }$ such that: $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , $Q^{\\prime }$ has the decomposition $\\mathcal {T}$ and can be constructed in time $\\mathcal {O}(|Q|)$ , and $\\mathbf {D}^{\\prime }$ can be constructed in time $\\mathcal {O}(|\\mathbf {D}|)$ .",
"We briefly explain the construction.",
"The query $Q^{\\prime }$ is obtained from $Q$ by replacing each relation symbol $R(S)$ in $Q$ by a relation symbol $R^{\\prime }(S^+)$ .",
"The database $\\mathbf {D}^{\\prime }$ contains, for each relation symbol $R^{\\prime }(S^+)$ in $Q^{\\prime }$ , a relation over the same schema that is obtained from relation $\\lambda (R(S))$ as follows: for each attribute $A$ contained in $S^+$ but not in $S$ , $\\lambda (R(S))$ is extended by a new $A$ -column that is a copy of any $B$ -column in $\\lambda (R(S))$ such that $A$ is equivalent to $B$ .",
"Figure REF gives in the top row two relations $R_1^{\\prime }$ and $R_2^{\\prime }$ that result from relation $R$ by the application of Proposition REF in case $Q$ is defined as in Example REF .",
"It follows from Proposition REF that, since $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , any relation $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if $K$ is a cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ .",
"Given the construction times for $Q^{\\prime }$ and $D^{\\prime }$ , all our results on natural join queries in Sections and , except the lower size bound on covers in Theorem REF (ii), hold for equi-join queries, too.",
"The following proposition is the counterpart of Theorem REF (ii) for equi-join queries.",
"Proposition 38 For any equi-join query $Q$ and any decomposition $\\mathcal {T}$ of $Q$ , there are arbitrarily large databases $\\mathbf {D}$ such that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"In Proposition REF , we first construct a natural join query $Q^{\\prime }$ from $Q$ as in Proposition REF .",
"By Theorem REF (ii), there are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that each cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given such a database $\\mathbf {D}^{\\prime }$ , it follows from Proposition REF , that $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })| =\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ , hence, $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace = \\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The database $\\mathbf {D}^{\\prime }$ can be converted into a database $\\mathbf {D}$ of size $\\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ such that $|\\pi _BQ(\\mathbf {D})| \\ge |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By Proposition REF (adapted to equi-join queries), each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ .",
"Since $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ $=$ $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ and $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ $\\ge $ $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ , we conclude that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is of size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , we can compute $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ with size ${\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ and in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ such that $Q^{\\prime }$ is an acyclic natural join query, $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ , $\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ and $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"The construction is standard in the literature [1], [24].",
"For convenience, we describe the main ideas.",
"Construction.",
"The construction comprises two transformation steps.",
"We first compute $R_B = Q_B(\\mathbf {D}_B)$ for each $B \\in {\\cal S}(\\mathcal {T})$ (recall that $Q_B$ and $\\mathbf {D}_B$ are $B$ -restrictions of $Q$ and $\\mathbf {D}$ , respectively).",
"Let $\\widehat{\\mathbf {D}}=\\lbrace R_B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ and $\\widehat{Q}=\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B$ .",
"In the second transformation step, we execute a semi-join programme on $\\widehat{\\mathbf {D}}$ to turn it into a database $\\mathbf {D}^{\\prime } = \\lbrace R_B^{\\prime }\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ that is pairwise consistent with respect to $\\widehat{Q}$ , i.e., $\\mathbf {D}^{\\prime }$ does not contain any pair of relations such that one of the two relations contains a tuple which cannot be joined with any tuple from the other relation.",
"To achieve pairwise consistency, it is not necessary to consider all pairs of relations in $\\widehat{\\mathbf {D}}$ .",
"It suffices to execute a bottom-up and a subsequent top-down traversal in $\\mathcal {T}$ [27].",
"During each traversal, we delete for each father-child pair $B_1, B_2$ of bags, all tuples in each of the two relations $R_{B_1}$ and $R_{B_2}$ which do not have any join partner in the other relation.",
"We define $Q^{\\prime }=\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B^{\\prime }$ .",
"$Q^{\\prime }$ is an acyclic natural join query and $\\mathcal {T}$ corresponds to a join tree of $Q^{\\prime }$ .",
"By construction, we have a one-to-one correspondence between relation symbols $R_B^{\\prime } \\in {\\cal S}(Q^{\\prime })$ and bags $B \\in {\\cal S}(\\mathcal {T})$ with ${\\cal S}(R_B^{\\prime })=B$ .",
"Hence, $\\mathcal {T}$ corresponds to the join tree of $Q^{\\prime }$ that is obtained from $\\mathcal {T}$ by, basically, replacing each bag by the corresponding relation symbol in $Q$ .",
"Since $Q^{\\prime }$ has a join tree, it is acyclic.",
"$\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ .",
"The relations in $\\mathbf {D}^{\\prime }$ are pairwise consistent with respect to $Q^{\\prime }$ .",
"For acyclic queries, pairwise consistency implies global consistency (Theorem 6.4.5 of [1]).",
"Hence, $\\mathbf {D}^{\\prime }$ is globally consistent with respect to $Q^{\\prime }$ .",
"$Q(\\mathbf {D}) = Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Since the second transformation step only deletes tuples in $\\widehat{\\mathbf {D}}$ which do not contribute to the result of $\\widehat{Q}(\\widehat{\\mathbf {D}})$ , it suffices to show that $Q(\\mathbf {D}) = \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"Let $Q = \\bowtie _{i \\in [n]}R_i$ .",
"We first show $Q(\\mathbf {D}) \\subseteq \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"Let $t \\in Q(\\mathbf {D})$ .",
"Since $\\pi _{B} Q(\\mathbf {D}) \\subseteq Q_B(\\mathbf {D}_B)$ , it follows that $\\pi _{B} t \\in Q_B(\\mathbf {D}_B)$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Hence, $\\pi _{B} t \\in R_B$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B t$ , we derive that $t \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_B$ , thus, $t \\in \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"We now show $\\widehat{Q}(\\widehat{\\mathbf {D}}) \\subseteq Q(\\mathbf {D})$ .",
"Let $t \\in \\widehat{Q}(\\widehat{\\mathbf {D}})$ .",
"By definition, $\\pi _{B}t \\in R_B$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By the fact that the attributes of each relation symbol in $Q$ are covered by at least one bag of $\\mathcal {T}$ and by the construction of the relations $R_B$ , it holds that $\\pi _{{\\cal S}(R_i)}t \\in R_i$ for each $i \\in [n]$ .",
"This implies $t \\in Q(\\mathbf {D})$ .",
"Construction size.",
"Each relation $R_B$ in $\\widehat{\\mathbf {D}}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [4].",
"Since $\\textsf {fhtw}(\\mathcal {T})=\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace \\rho ^*(Q_B)\\rbrace $ , it follows that the size of $\\widehat{\\mathbf {D}}$ is $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"The semi-join program on $\\widehat{\\mathbf {D}}$ does not increase the size of the database.",
"The size of $Q^{\\prime }$ is $\\mathcal {O}(|Q|)$ .",
"Altogether, the size of $(Q^{\\prime },\\mathcal {T},\\mathbf {D}^{\\prime })$ is $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Construction time.",
"Each relation $R_B$ in $\\widehat{\\mathbf {D}}$ is computable in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [21].",
"By $\\textsf {fhtw}(\\mathcal {T})=\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace \\rho ^*(Q_B)\\rbrace $ , we derive that the computation time for $\\widehat{\\mathbf {D}}$ is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"During the semi-join program on $\\widehat{\\mathbf {D}}$ , we can achieve consistency between each pair $R_{B_1}, R_{B_2}$ of father-child relations as follows.",
"We first sort both relations on the join attributes.",
"In a subsequent scan we delete in each of the relations each tuple with no join partner in the other relation.",
"Hence, the semi-join programme can be realised in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"It follows that the overall running time is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ with schema $\\mathit {att}(Q)$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ if and only if $\\bowtie _{B\\in {\\cal S}(\\mathcal {T})} \\pi _B K = Q(\\mathbf {D})$ .",
"Proof of the “$\\Rightarrow $ ”-direction.",
"Assume that $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"We show in two steps that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K = Q(\\mathbf {D})$ .",
"$\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K \\subseteq Q(\\mathbf {D})$ : Let $t$ be an arbitrary tuple from $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"This means that $\\pi _{B}t \\in \\pi _{B}K$ for every $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , we derive that $\\pi _{B}t \\in \\pi _{B}Q(\\mathbf {D})$ for every $B \\in {\\cal S}(\\mathcal {T})$ .",
"By the definition of decompositions, for every relation symbol $R$ in $Q$ , there is at least one bag of $\\mathcal {T}$ containing all attributes of $R$ .",
"Hence, $\\pi _{{\\cal S}(R)}t \\in \\pi _{{\\cal S}(R)}Q(\\mathbf {D})$ for every $R \\in {\\cal S}(Q)$ .",
"It follows that $t$ is included in $Q(\\mathbf {D})$ .",
"Thus, $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K \\subseteq Q(\\mathbf {D})$ .",
"$Q(\\mathbf {D}) \\subseteq \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ : Let $t \\in Q(\\mathbf {D})$ .",
"It follows that $\\pi _B t \\in \\pi _B Q(\\mathbf {D})$ for every $B \\subseteq {\\cal S}(Q(\\mathbf {D}))$ , hence, in particular for every $B\\in {\\cal S}(\\mathcal {T})$ .",
"Due to result-preservation of $K$ with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ , this implies that $\\pi _B t \\in \\pi _B K$ for every $B\\in {\\cal S}(\\mathcal {T})$ which means that $t \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"Hence, $Q(\\mathbf {D}) \\subseteq \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B}K$ .",
"Proof of the “$\\Leftarrow $ ”-direction.",
"Assume that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B K = Q(\\mathbf {D})$ .",
"Given any $B \\in {\\cal S}(\\mathcal {T})$ , we show in two steps that $\\pi _B K = \\pi _B Q(\\mathbf {D})$ .",
"$\\pi _B K \\subseteq \\pi _B Q(\\mathbf {D})$ : Let $t$ be an arbitrary tuple from $\\pi _B K$ .",
"This means that there is a tuple $t^{\\prime } \\in K$ with $\\pi _B t^{\\prime }=t$ .",
"Since $\\pi _{B^{\\prime }} t^{\\prime } \\in \\pi _{B^{\\prime }}K$ for each $B^{\\prime } \\in {\\cal S}(\\mathcal {T})$ , we derive that $t^{\\prime }\\in \\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }}K$ .",
"Using our assumption $\\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }} K = Q(\\mathbf {D})$ , we get $t^{\\prime }\\in Q(\\mathbf {D})$ .",
"From the latter and the fact that $t = \\pi _B t^{\\prime }$ , it follows $t \\in \\pi _B Q(\\mathbf {D})$ .",
"Altogether, we conclude $\\pi _B K \\subseteq \\pi _B Q(\\mathbf {D})$ .",
"$\\pi _B Q(\\mathbf {D}) \\subseteq \\pi _B K$ : Let $t$ be an arbitrary tuple from $\\pi _B Q(\\mathbf {D})$ .",
"This means that there is a tuple $t^{\\prime } \\in Q(\\mathbf {D})$ with $\\pi _B t^{\\prime } = t$ .",
"By assumption, $t^{\\prime } \\in \\bowtie _{B^{\\prime } \\in {\\cal S}(\\mathcal {T})}\\pi _{B^{\\prime }} K$ .",
"Since $B$ is an element of ${\\cal S}(\\mathcal {T})$ , the latter implies $\\pi _B t^{\\prime }= t \\in \\pi _B K$ .",
"Altogether, we get $\\pi _B Q(\\mathbf {D}) \\subseteq \\pi _B K$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ has a minimal edge cover $M$ such that $\\mathit {rel}(M)=K$ .",
"We first recall that $Q(\\mathbf {D})$ is obviously result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ and therefore, by Proposition REF , it holds $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D}) = Q(\\mathbf {D})$ .",
"Let $H=(V,E)$ be the hypergraph of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ .",
"Let $\\mathit {tuple}_V$ be a function mapping each node $v \\in V$ to its corresponding tuple in $\\bigcup _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Furthermore, let $\\mathit {tuple}_E$ be a function mapping each edge $e \\in E$ to $\\bowtie _{v \\in e}\\mathit {tuple}_V(v)$ .",
"The function $\\mathit {tuple}_V$ is a bijection from $V$ to $\\bigcup _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Since $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D}) = Q(\\mathbf {D})$ , $\\mathit {tuple}_E$ is a bijection from $E$ to $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Likewise, the function $\\mathit {rel}$ (as defined in Section ) is a bijection from subsets of $E$ to subsets of $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _B Q(\\mathbf {D})$ .",
"Proof of the “$\\Rightarrow $ ”-direction.",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We show that $\\mathit {rel}^-(K)$ is defined and a minimal edge cover of $H$ .",
"Let $t \\in K$ .",
"This means that for each $B \\in {\\cal S}(\\mathcal {T})$ , there is $t_B \\in \\pi _B K$ with $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})} t_B$ .",
"Since $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , each $t_B$ is included in $\\pi _B Q(\\mathbf {D})$ .",
"Hence, $\\mathit {tuple}_V^-(t_B)$ must be defined for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $Q(\\mathbf {D}) = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}Q(\\mathbf {D})$ , $t$ is included in $Q(\\mathbf {D})$ .",
"It follows that $\\mathit {tuple}_E^-(t) = \\lbrace \\mathit {tuple}_V^-(t_B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is defined.",
"Thus, $\\mathit {rel}^-(K) = \\lbrace \\mathit {tuple}_E^-(t)\\rbrace _{t \\in K}$ is defined.",
"It follows from $\\pi _BQ(\\mathbf {D}) = \\pi _BK$ , $B \\in {\\cal S}(\\mathcal {T})$ , that $\\mathit {rel}^-(K)$ is an edge cover of $H$ .",
"It remains to show that $\\mathit {rel}^-(K)$ is a minimal edge cover of $H$ .",
"For the sake of contradiction, assume that $\\mathit {rel}^-(K)$ is not a minimal edge cover of $H$ .",
"This implies that there is an edge $\\overline{e} \\in \\mathit {rel}^-(K)$ such that $\\mathit {rel}^-(K) \\backslash \\lbrace \\overline{e}\\rbrace $ is an edge cover of $H$ .",
"It follows that for each node $v \\in V$ , there is an edge $e \\in \\mathit {rel}^-(K) \\backslash \\lbrace \\overline{e}\\rbrace $ with $v \\in e$ .",
"This means that for each tuple $t_B \\in \\pi _B Q(\\mathbf {D})= \\pi _B K$ , there is a tuple $t \\in K \\backslash \\lbrace \\mathit {tuple}_E(\\overline{e})\\rbrace $ with $\\pi _B t = t_B$ .",
"We conclude that $K \\backslash \\lbrace \\mathit {tuple}_E(\\overline{e})\\rbrace $ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"The latter is, however, a contradiction to our assumption that $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and, therefore, a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"Proof of the “$\\Leftarrow $ ”-direction.",
"Let $M$ be a minimal edge cover of $H$ .",
"We show that $\\mathit {rel}(M)$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ .",
"We first observe that ${\\cal S}(\\mathit {rel}(M)) = {\\cal S}(Q(\\mathbf {D})) = \\mathit {att}(Q)$ .",
"Since $Q(\\mathbf {D})$ is result-preserving with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ and $M$ is an edge cover of $H$ , $\\mathit {rel}(M)$ must also be result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"It remains to show that $\\mathit {rel}(M)$ is a minimal result-preserving relation with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"For the sake of contradiction, assume that $\\mathit {rel}(M)$ is not minimal in that respect.",
"It follows that there is a tuple $\\overline{t}\\in \\mathit {rel}(M)$ such that $\\mathit {rel}(M) \\backslash \\lbrace \\overline{t}\\rbrace $ is result-preserving with respect to $(Q,\\mathcal {T}, \\mathbf {D})$ .",
"This means that for each $B \\in {\\cal S}(\\mathcal {T})$ and each tuple $t_B \\in \\pi _BQ(\\mathbf {D})$ , there is a tuple $t \\in \\mathit {rel}(M) \\backslash \\lbrace \\overline{t}\\rbrace $ with $\\pi _Bt = t_B$ .",
"This implies that for each node $v \\in V$ , there is an edge $e \\in M \\backslash \\lbrace \\mathit {tuple}_E^-(\\overline{t})\\rbrace $ with $v \\in e$ .",
"We derive that $M \\backslash \\lbrace \\mathit {tuple}_E^-(\\overline{t})\\rbrace $ is a minimal edge cover of $H$ , a contradiction to the minimality of $M$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is a subset of $Q(\\mathbf {D})$ .",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and let $t\\in K$ be an arbitrary tuple from $K$ .",
"We show that $t$ must be included in $Q(\\mathbf {D})$ .",
"For each $B \\in {\\cal S}(\\mathcal {T})$ , let $t_B = \\pi _{B}t$ .",
"It holds that $t = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}t_B$ .",
"As $K$ is result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ , $t_B$ must be included in $\\pi _{B}Q(\\mathbf {D})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since by Proposition REF , $Q(\\mathbf {D}) = \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _BQ(\\mathbf {D})$ , it follows that $t$ is included in $Q(\\mathbf {D})$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , the size of each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ satisfies the inequalities $\\max _{B\\in {\\cal S}(\\mathcal {T})}\\lbrace \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid \\rbrace $ $\\le $ $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ .",
"The first inequality holds due to $K$ being result-preserving with respect to $(Q,\\mathcal {T},\\mathbf {D})$ .",
"The second inequality is implied by Proposition REF , since the hypergraph $H=(V,E)$ of $Q(\\mathbf {D})$ over ${\\cal S}(\\mathcal {T})$ must have a minimal edge cover $M$ with $\\mathit {rel}(M) = K$ .",
"Each hyperedge $e$ in $M$ must cover at least one node in $V$ which is not covered by any other hyperedge in $M$ .",
"Otherwise, $M\\backslash \\lbrace e\\rbrace $ would be an edge cover, which is a contradiction to the minimality of $M$ .",
"Hence, the total number of edges in $M$ is upper-bounded by $\\mid \\hspace{-2.84526pt} V \\hspace{-2.84526pt} \\mid $ .",
"As $\\mid \\hspace{-2.84526pt} V \\hspace{-2.84526pt} \\mid =\\Sigma _{B \\in {\\cal S}(\\mathcal {T})} \\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ and $\\mid \\hspace{-2.84526pt} M \\hspace{-2.84526pt} \\mid = \\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ , we derive that the number of tuples in $K$ is upper-bounded by $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ ."
],
[
"Proof of Theorem ",
"Theorem REF .",
"Let $Q$ be a natural join query and $\\mathcal {T}$ a decomposition of $Q$ .",
"For any database $\\mathbf {D}$ , each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"There are arbitrarily large databases $\\mathbf {D}$ such that each cover of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Our proof relies on the results that for any natural join query $Q$ and database $\\mathbf {D}$ , it holds $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ and there are arbitrarily large databases $\\mathbf {D}$ with $\\mid \\hspace{-2.84526pt} Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q)})$ [4].",
"Let $\\mathcal {T}=(T,\\chi ,\\lbrace \\gamma _t\\rbrace _{t \\in T})$ .",
"Given a node $t$ in $T$ with $\\chi (t) = B$ for some set $B$ , we recall that $\\mathit {weight}(\\gamma _{t}) = \\rho ^*(Q_{B})$ .",
"Moreover, if $\\mathit {weight}(\\gamma _{t})$ is maximal over all weight functions in $\\mathcal {T}$ , then $\\mathit {weight}(\\gamma _{t})$ $=\\textsf {fhtw}(\\mathcal {T})$ .",
"Proof of statement (i).",
"Let $K$ be a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ and let $t$ be an arbitrary node of $T$ with $\\chi (t)=B$ for some set $B$ .",
"It holds $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ [4], thus, $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}_B \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Since $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $\\le $ $\\mid \\hspace{-2.84526pt} Q_B (\\mathbf {D}_B) \\hspace{-2.84526pt} \\mid $ (Proposition 3.2 of [23]), it follows that $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Using Proposition REF , we conclude $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid $ $\\le $ $\\Sigma _{B\\in {\\cal S}(\\mathcal {T})} \\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} {\\cal S}(\\mathcal {T}) \\hspace{-2.84526pt} \\mid \\cdot \\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Proof of statement (ii).",
"Let $t$ be a node in $T$ such that $\\gamma _t$ has maximal weight and let $\\chi (t)=B$ .",
"There are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that $\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}^{\\prime }) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\rho ^*(Q_B)})$ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ [4].",
"For each such database $\\mathbf {D}^{\\prime }$ , there exists a database $\\mathbf {D}$ with $\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid = \\mathcal {O}(\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid )$ and $\\mid \\hspace{-2.84526pt} \\pi _B Q(\\mathbf {D}) \\hspace{-2.84526pt} \\mid = \\Omega (\\mid \\hspace{-2.84526pt} Q_B(\\mathbf {D}^{\\prime }) \\hspace{-2.84526pt} \\mid )= \\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D}^{\\prime } \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ (Lemma 7.18 of [23]).",
"This means that there are arbitrarily large databases $\\mathbf {D}$ such that $\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ $=$ $\\Omega (\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Due to Proposition REF , each cover $K$ of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must be at least of size $\\mid \\hspace{-2.84526pt} \\pi _BQ(\\mathbf {D}) \\hspace{-2.84526pt} \\mid $ , hence, $\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid = \\Omega (\\mathbf {D}^{\\textsf {fhtw}(\\mathcal {T})})$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given $(Q,\\mathcal {T},\\mathbf {D})$ , each cover $K$ of the query result $Q(\\mathbf {D})$ over $\\mathcal {T}$ can be translated into a d-representation of $Q(\\mathbf {D})$ of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ .",
"Using the algorithm in Figure REF , we construct from $K$ and $\\mathcal {T}$ a d-representation of $Q(\\mathbf {D})$ encoded as a set M of maps.",
"Recall that the constructed d-representation is over a d-tree $\\mathcal {T}^{\\prime }$ equivalent to $\\mathcal {T}$ .",
"Correctness of the construction.",
"For each $m_A \\in M$ , we denote by $R_A$ the listing representation of $m_A$ as presented in Figure REF .",
"For each bag attribute $A$ in $\\mathcal {T}^{\\prime }$ , the set $\\lbrace A\\rbrace \\cup \\mathit {key}(A)$ constitutes a bag in the signature ${\\cal S}(\\mathcal {T}^{\\prime })$ of $\\mathcal {T}^{\\prime }$ .",
"We write $B_A$ to express that the bag attribute of $B_A$ is $A$ .",
"By the definition of d-representations, the query result represented by the map set $M$ is $R = \\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}R_A$ [23].",
"It remains to show that $R = Q(\\mathbf {D})$ .",
"By construction of the maps in $M$ , we have $R_A = \\pi _{B_A}K$ for each $B_A$ .",
"For each $B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })$ , there is a $B \\in {\\cal S}(\\mathcal {T})$ with $B_A \\subseteq B$ (proof of Proposition 9.3 in [23]).",
"Hence, by the definition of covers, we have $R_A = \\pi _{B_A}K = \\pi _{B_A} Q(\\mathbf {D})$ for each $B_A$ .",
"As $\\mathcal {T}^{\\prime }$ is a valid decomposition of $Q$ , it follows from Proposition REF that $\\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}\\pi _{B_A}K = Q(\\mathbf {D})$ .",
"Since for each $B_A$ , we have $\\pi _{B_A}K = R_A$ and $R = \\bowtie _{B_A \\in {\\cal S}(\\mathcal {T}^{\\prime })}R_{A}$ , it follows $R = Q(\\mathbf {D})$ .",
"Construction size and translation time.",
"The number of the maps in $M$ is bounded by the number of attributes in $K$ .",
"We consider the cover $K$ sorted using a topological order of the decomposition $\\mathcal {T}^{\\prime }$ , so that inserts into the multimaps become appends (alternatively, inserts in sorted order would take logarithmic time in the number of entries).",
"For each tuple in $K$ we insert at most one tuple in the multimap of each attribute.",
"Thus, the overall size of the set of multimaps, and thus of the d-representation, is $\\mathcal {O}(|K|)$ with respect to data complexity (the linear factor in the number of attributes is ignored).",
"The data complexity of the overall translation time is thus $\\widetilde{\\mathcal {O}}(|K|)$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given two consistent relations $R_1$ and $R_2$ , the cover-join computes a cover $K$ of their join result over the decomposition with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ in time $\\widetilde{\\mathcal {O}}(|R_1|+|R_2|)$ and with size $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ .",
"Let $Q = R_1 \\bowtie R_2$ , $\\mathbf {D} = \\lbrace R_1, R_2\\rbrace $ .",
"Moreover, let $\\mathcal {T}$ be the decomposition of $Q$ with bags ${\\cal S}(R_1)$ and ${\\cal S}(R_2)$ .",
"By Proposition REF , a relation $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ if and only if the hypergraph $H$ of $Q(\\mathbf {D})$ over the attribute sets $\\lbrace {\\cal S}(R_1), {\\cal S}(R_2)\\rbrace $ has a minimal edge cover $M$ with $\\mathit {rel}(M)=K$ .",
"The hypergraph $H$ is a collection of disjoint complete bipartite subgraphs.",
"The set of nodes of each such subgraph corresponds to a maximal subset of tuples of the input relations agreeing on the join attributes.",
"A minimal edge cover of $H$ is a collection of minimal edge covers for these subgraphs.",
"We construct a cover $K$ of minimum size such that each maximal subset of tuples in $K$ agreeing on the join attributes corresponds to a minimal edge cover of one of the complete bipartite subgraphs of $H$ .",
"Construction.",
"Let $\\mathcal {A}$ be the set of common attributes of $R_1$ and $R_2$ .",
"For $i \\in \\lbrace 1,2\\rbrace $ and $t \\in \\pi _{\\mathcal {A}}R_i$ , we call $\\sigma _{\\mathcal {A}= t} R_i$ the $t$ -block in $R_i$ and denote its size by $n^t_i$ .",
"Since $R_1$ and $R_2$ are consistent, for each $t$ -block in $R_1$ , there must be a corresponding $t$ -block in $R_2$ , and vice-versa.",
"First, the algorithm sorts $R_1$ and $R_2$ with respect to the values of the attributes in $\\mathcal {A}$ .",
"After sorting, the $t$ -blocks occur in the same order in both relations.",
"The cover $K$ is constructed by performing the following procedure for each pair of corresponding $t$ -blocks in $R_1$ and $R_2$ .",
"Without loss of generality, assume $n_1^t \\ge n_2^t$ .",
"For each $j < n_{2}^t$ , the $j$ -th tuple $t^{\\prime }$ in the $t$ -block of $R_1$ is combined with the $j$ -th tuple $t^{\\prime \\prime }$ in the $t$ -block of $R_2$ resulting in a new tuple $t^{\\prime } \\bowtie t^{\\prime \\prime }$ .",
"Then, all remaining tuples in the $t$ -block of $R_1$ are combined with the $n_{2}^t$ -th tuple in the $t$ -block of $R_2$ .",
"All new tuples are added to $K$ .",
"Construction time.",
"The sorting phase can be realised in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} R_1 \\hspace{-2.84526pt} \\mid +\\mid \\hspace{-2.84526pt} R_2 \\hspace{-2.84526pt} \\mid )$ .",
"The phase for constructing the new tuples can be done in one pass over the sorted relations.",
"Hence, the overall running time of the described algorithm is $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} R_1 \\hspace{-2.84526pt} \\mid +\\mid \\hspace{-2.84526pt} R_2 \\hspace{-2.84526pt} \\mid )$ .",
"Size of the Cover.",
"The size bounds $\\max \\lbrace |R_1|, |R_2|\\rbrace \\le |K|\\le |R_1|+|R_2|$ follow from Proposition REF and the assumption that $R_1$ and $R_2$ are consistent, so we have $\\pi _{{\\cal S}(R_i)} Q(\\mathbf {D}) = R_i$ for each $i \\in \\lbrace 1,2\\rbrace $ .",
"Our algorithm above constructs a specific cover.",
"Other covers can be constructed within the same time bounds.",
"We exemplify the construction of some further covers following different patterns.",
"In our construction above, after combining the first $n_2^t-1$ tuples in the $t$ -block of $R_1$ with the first $n_2^t-1$ tuples in the $t$ -block of $R_2$ , we combined the last tuple in the $t$ -block of $R_2$ with all remaining tuples in the $t$ -block of $R_1$ .",
"Alternatively, we can fix any tuple $t^{\\prime }$ in the $t$ -block of $R_2$ , combine the first $n_2^t-1$ tuples in the $t$ -block of $R_1$ with all tuples besides $t^{\\prime }$ in the $t$ -block of $R_2$ and then combine the remaining tuples in the $t$ -block of $R_1$ with $t^{\\prime }$ ."
],
[
"Proof of Lemma ",
"Lemma REF .",
"Given $(Q,\\mathcal {J},\\mathbf {D})$ where $\\mathbf {D}=\\lbrace R_i\\rbrace _{i\\in [n]}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(|K|)$ and with size $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Any cover-join plan over $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ .",
"We show by induction on the structure of cover-join plans that given $(Q,\\mathcal {J},\\mathbf {D})$ , where $\\mathbf {D}$ is globally consistent with respect to $Q$ , any cover-join plan over the join tree $\\mathcal {J}$ computes a cover $K$ of $Q(\\mathbf {D})$ over the decomposition corresponding to $\\mathcal {J}$ .",
"For the base case, assume that $\\varphi $ consists of a single relation symbol $R$ .",
"By Definition REF , $\\mathcal {J}$ consists of a single node $R$ , hence, $Q=R$ .",
"The decomposition $\\mathcal {T}$ corresponding to $\\mathcal {J}$ consists of a single bag ${\\cal S}(R)$ .",
"By Definition REF , $\\varphi $ returns the relation $R$ .",
"By Definition REF , $R$ is indeed the unique cover of $Q(\\lbrace R\\rbrace )$ over $\\mathcal {T}$ .",
"Assume now that $\\varphi $ is of the form $\\varphi _1 \\mathring{}\\varphi _2$ .",
"By definition of cover-join plans, there are subtrees $\\mathcal {J}_1$ and $\\mathcal {J}_2$ of $\\mathcal {J}$ such that $\\mathcal {J}= \\mathcal {J}_1 \\circ \\mathcal {J}_2$ and each $\\varphi _i$ is a cover-join plan over $\\mathcal {J}_i$ .",
"Let $\\mathcal {T}_1$ and $\\mathcal {T}_2$ be the decompositions corresponding to $\\mathcal {J}_1$ and $\\mathcal {J}_2$ , respectively.",
"The decomposition corresponding to $\\mathcal {J}$ is obtained by connecting $\\mathcal {T}_1$ and $\\mathcal {T}_2$ by the same tree edge connecting $\\mathcal {J}_1$ and $\\mathcal {J}_2$ in $\\mathcal {J}$ .",
"We have $Q = Q_1 \\bowtie Q_2$ where each $Q_i$ expresses the join of the relation symbols occurring in $\\mathcal {J}_i$ .",
"Moreover, $\\mathbf {D} = \\mathbf {D}_1 \\cup \\mathbf {D}_2$ where $\\mathbf {D}_i= \\lbrace R\\rbrace _{R \\in {\\cal S}(Q_i)}$ , $i \\in [2]$ .",
"Note that for each $i \\in [2]$ , $Q_i$ is acyclic, $\\mathcal {J}_i$ is a join tree of $Q_i$ and $\\mathbf {D}_i$ is globally consistent with respect to $Q_i$ .",
"The latter follows simply from the globally consistency of $\\mathbf {D}$ with respect to $Q$ .",
"Hence, by induction hypothesis, each $\\varphi _i$ returns a cover $K_i$ of $Q_i(\\mathbf {D}_i)$ over $\\mathcal {T}_i$ .",
"Due to Proposition REF , in case $K_1$ and $K_2$ are consistent, the cover-join operator computes a cover $K$ of $K_1 \\bowtie K_2$ over the decomposition with bags ${\\cal S}(K_1)$ and ${\\cal S}(K_2)$ .",
"Thus, by Definition REF , the plan $\\varphi $ returns $K$ .",
"We proceed as follows.",
"First, we show that $K_1$ and $K_2$ must be consistent.",
"Then, we prove that $K$ is a cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ , that is, $K$ is result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ and it is minimal in this respect.",
"$K_1$ and $K_2$ are consistent: Let $R_1$ and $R_2$ be the two relation symbols incident to the single edge connecting $\\mathcal {J}_1$ and $\\mathcal {J}_2$ in $\\mathcal {J}$ and let $\\mathcal {B}$ be the set of common attributes of these relation symbols.",
"Let $\\mathcal {A}$ be the set of common attributes of $K_1$ and $K_2$ .",
"We first show that $\\mathcal {A}\\subseteq \\mathcal {B}$ .",
"Let $A \\in \\mathcal {A}$ .",
"Since each $K_i$ is computed by the plan $\\mathcal {J}_i$ , there must be at least one relation symbol $R_1^{\\prime }$ in $\\mathcal {J}_1$ and at least one relation symbol $R_2^{\\prime }$ in $\\mathcal {J}_2$ containing $A$ in their schemas.",
"Due to the construction of join trees, $A$ must occur in the schemas of all relation symbols on the single path between $R_1^{\\prime }$ and $R_2^{\\prime }$ .",
"Since $R_1$ and $R_2$ are on this path, both must include $A$ .",
"Hence, $\\mathcal {A}\\subseteq \\mathcal {B}$ .",
"Since $\\mathbf {D}$ is globally consistent, the relations $R_1$ and $R_2$ must be consistent as well.",
"As each $K_i$ is result-preserving with respect to $(Q_i,\\mathcal {T}_i,\\mathbf {D}_i)$ , $\\pi _{{\\cal S}(R_i)} Q_i(\\mathbf {D}_i) = R_i$ (due to global consistency) and $\\mathcal {B}\\subseteq {\\cal S}(R_i)$ , it follows $\\pi _{\\mathcal {B}}K_1 = \\pi _{\\mathcal {B}}Q_1(\\mathbf {D}_1) = \\pi _{\\mathcal {B}}R_1= \\pi _{\\mathcal {B}}R_2 = \\pi _{\\mathcal {B}}Q_2(\\mathbf {D}_2) = \\pi _{\\mathcal {B}}K_2$ .",
"As $\\mathcal {A}\\subseteq \\mathcal {B}$ , the relations $K_1$ and $K_2$ must be consistent.",
"$K$ is result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ : Let $B$ be an arbitrary bag of $\\mathcal {T}$ .",
"Since $\\mathcal {T}$ corresponds to $\\mathcal {J}$ , the join tree $\\mathcal {J}$ must have a node $R$ with ${\\cal S}(R) = B$ .",
"Without loss of generality, assume that $R \\in \\mathbf {D}_1$ (the other case is handled along the same lines).",
"Since, by induction hypothesis, $K_1$ is result-preserving with respect to $(Q_1, \\mathcal {T}_1, \\mathbf {D}_1)$ and $\\mathbf {D}_1$ is globally consistent, we have $R= \\pi _{{\\cal S}(R)}K_1$ .",
"Since $\\pi _{{\\cal S}(K_1)}K = K_1$ and ${\\cal S}(R) \\subseteq {\\cal S}(K_1)$ , we get $R= \\pi _{{\\cal S}(R)}K$ .",
"Using the global consistency of $\\mathbf {D}$ with respect to $Q$ , we conclude $\\pi _{B}Q(\\mathbf {D}) = R= \\pi _{B}K$ .",
"$K$ is a minimal result-preserving relation with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ : For the sake of contradiction, assume that $K$ is not minimal in this respect.",
"This means that there is a tuple $t^- \\in K$ such that $K\\backslash \\lbrace t^-\\rbrace $ is still result-preserving with respect to $(Q, \\mathcal {T}, \\mathbf {D})$ .",
"It follows that $\\pi _{{\\cal S}(K_i)} (K\\backslash \\lbrace t^-\\rbrace )$ is result-preserving with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ for each $i \\in [2]$ .",
"Observe that the minimal edge cover $M$ with $\\mathit {rel}(M) = K$ in the hypergraph of $K_1 \\bowtie K_2$ over the attribute sets $\\lbrace {\\cal S}(K_1), {\\cal S}(K_2)\\rbrace $ must contain an edge $e^-$ connecting $\\pi _{{\\cal S}(K_1)}t^-$ and $\\pi _{{\\cal S}(K_2)}t^-$ .",
"This implies that $M$ cannot have two further edges $e_1$ and $e_2$ such that $e_1$ covers $\\pi _{{\\cal S}(K_1)}t^-$ and $e_2$ covers $\\pi _{{\\cal S}(K_2)}t^-$ .",
"Indeed, in this case, $M\\backslash \\lbrace e^-\\rbrace $ would be an edge cover, contradicting the minimality of $M$ .",
"Hence, there is no tuple $t \\ne t^-$ in $K$ with $\\pi _{{\\cal S}(K_1)}t^- =\\pi _{{\\cal S}(K_1)}t$ or there is no tuple $t \\ne t^-$ in $K$ with $\\pi _{{\\cal S}(K_2)}t^-= \\pi _{{\\cal S}(K_2)}t$ .",
"It follows that $\\pi _{{\\cal S}(K_1)} (K\\backslash \\lbrace t^-\\rbrace ) \\subset \\pi _{{\\cal S}(K_1)} K$ or $\\pi _{{\\cal S}(K_2)} (K\\backslash \\lbrace t^-\\rbrace ) \\subset \\pi _{{\\cal S}(K_2)} K$ .",
"Using the consistency of $K_1$ and $K_2$ , we obtain $\\pi _{{\\cal S}(K_1)}$ $(K\\backslash \\lbrace t^-\\rbrace )$ $\\subset \\pi _{{\\cal S}(K_1)} K$ $=$ $K_1$ or $\\pi _{{\\cal S}(K_2)}$ $(K\\backslash \\lbrace t^-\\rbrace )$ $\\subset \\pi _{{\\cal S}(K_2)} K$ $=$ $K_2$ .",
"However, as we noticed that $\\pi _{{\\cal S}(K_i)} (K\\backslash \\lbrace t^-\\rbrace )$ is result-preserving with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ for each $i \\in [2]$ , the statement of the last sentence contradicts the induction hypothesis that each $K_i$ is a minimal result-preserving relation with respect to $(Q_i, \\mathcal {T}_i, \\mathbf {D}_i)$ .",
"Size of $K$ .",
"From the global consistency of $\\mathbf {D}$ with respect to $Q$ and Proposition REF , it follows for any cover $K$ of $Q(\\mathbf {D})$ over the tree decomposition corresponding to $\\mathcal {J}$ that $\\max _{{i\\in [n]}}\\lbrace \\mid \\hspace{-2.84526pt} R_i \\hspace{-2.84526pt} \\mid \\rbrace \\le |K|\\le \\sum _{i\\in [n]}|R_i|$ .",
"Computation time for $K$ .",
"By Proposition REF , we can design an algorithm for the cover-join operator which for every two input covers $K_1$ and $K_2$ , computes a cover-join result of size $\\mathcal {O}(\\mid \\hspace{-2.84526pt} K_1 \\hspace{-2.84526pt} \\mid + \\mid \\hspace{-2.84526pt} K_2 \\hspace{-2.84526pt} \\mid )$ and in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K_1 \\hspace{-2.84526pt} \\mid + \\mid \\hspace{-2.84526pt} K_2 \\hspace{-2.84526pt} \\mid )$ .",
"Hence, given a triple $(Q,\\mathcal {J},\\mathbf {D})$ and a cover-join plan $\\varphi $ over $\\mathcal {J}$ , starting from the innermost expressions of $\\varphi $ , we can compute a cover $K$ of $Q(\\mathbf {D})$ over the tree decomposition corresponding to $\\mathcal {J}$ in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} K \\hspace{-2.84526pt} \\mid )$ ."
],
[
"Missing Details and Proofs in Section ",
"Given the hypergraph $H$ of an FAQ and an attribute set $U$ , we denote by $H_U$ the hypergraph obtained from $H$ by restricting each hyperedge in $H$ to the attributes in $U$ .",
"For the rest of this section we fix an FAQ $\\varphi $ as written in (REF )."
],
[
"Recap on FAQs",
"Indicator projections are used in the InsideOut algorithm [17] solving the FAQ-problem.",
"They will also occur in our construction of FAQ-covers.",
"Definition 39 (Indicator projections) Given two attribute sets $S$ and $T$ with $S \\cap T \\ne \\emptyset $ and a function $\\psi _S$ , the function $\\psi _{S/T}: \\prod _{A\\in (S \\cap T)}\\textsf {dom}(A) \\rightarrow \\textsf {Dom}$ defined by $\\psi _{S/T}({\\textsf {a}}_{S \\cap T}) ={\\left\\lbrace \\begin{array}{ll}{\\bf 1}& \\quad \\exists {\\textsf {b}}_{S} \\text{ s.t. }",
"\\psi _S({\\textsf {b}}_S) \\ne 0 \\text{ and }{\\textsf {a}}_{S\\cap T} = {\\textsf {b}}_{S\\cap T}, \\\\{\\bf 0}& \\quad \\text{otherwise } \\\\\\end{array}\\right.",
"}$ is called the indicator projection of $\\psi _S$ onto $T$ .",
"In particular, if $S \\subseteq T$ , then $\\psi _{S/T}({\\textsf {a}}_{S}) = {\\bf 1}$ if and only if $\\psi _S({\\textsf {a}}_{S}) \\ne {\\bf 0}$ .",
"Equivalent attribute orderings.",
"A $\\varphi $ -equivalent attribute ordering $\\tau = \\tau (1), \\ldots , \\tau (n)$ is a permutation of the indices of the attributes in $\\mathcal {V}$ satisfying the following conditions: $\\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (f)}\\rbrace = \\lbrace A_1, \\ldots , A_f\\rbrace $ and $\\varphi ^{\\prime }({\\textsf {a}}_{\\lbrace A_{\\tau (1)},\\ldots ,A_{\\tau (f)}\\rbrace }) = \\underset{a_{\\tau (f+1)}\\in \\textsf {dom}(A_{\\tau (f+1)})}{\\bigoplus \\ ^{(\\tau (f+1))}} \\cdots \\underset{a_{\\tau (n)}\\in \\textsf {dom}(A_{\\tau (n)})}{\\bigoplus \\ ^{(\\tau (n))}}\\ \\underset{S\\in \\mathcal {E}}{\\bigotimes }\\ \\psi _S({\\sf a}_S)$ is equivalent to $\\varphi $ irrespective of the definition of the input functions $\\psi _S$ .",
"We denote by $\\textsf {EVO}(\\varphi )$ the set of all $\\varphi $ -equivalent attribute orderings."
],
[
"The ",
"Given an FAQ $\\varphi $ , a database $\\mathbf {D}$ and a $\\varphi $ -equivalent attribute ordering, the InsideOut algorithm computes the listing representation of $\\varphi (\\mathbf {D})$ .",
"The algorithm first rewrites the query according to the given attribute ordering and then processes the resulting query in two phases: bound attribute elimination and output computation.",
"We sketch the main steps of the algorithm on input $\\varphi $ , some database $\\mathbf {D}$ and the attribute ordering that corresponds to the identity permutation.",
"Thus, the initial rewriting step does not change the structure of $\\varphi $ .",
"In the bound attribute elimination phase, the algorithm eliminates attributes $A_{f+1},$ $\\ldots ,$ $A_{n}$ along with their corresponding aggregate operators in reverse order.",
"When eliminating an attribute $A_j$ it distinguishes between the cases whether $\\bigoplus ^{(j)}$ is different from $\\bigotimes $ or not.",
"We demonstrate the two cases in the elimination step for $A_n$ .",
"In case that $\\bigoplus ^{(n)}$ is different from $\\bigotimes $ , the algorithm first rewrites the query as follows $&\\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}} \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\sf a}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{S\\in \\mathcal {E}\\backslash \\partial (n)}{\\bigotimes } \\psi _S({\\textsf {a}}_S)\\otimes \\Big (\\underbrace{\\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigoplus \\ ^{(n)}} \\bigotimes _{S \\in \\partial (n)} \\psi _S({\\textsf {a}}_S)}_{\\delta }\\Big ),$ where $\\partial (n)= \\lbrace S \\in \\mathcal {E}\\mid A_n \\in S\\rbrace $ and $U_n = \\bigcup _{S\\in \\partial (n)} S$ .",
"The correctness of the rewriting follows from the distributivity of $\\otimes $ over $\\oplus ^{(n)}$ .",
"Then, the algorithm computes the listing representation of a function $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ such that replacing $\\delta $ by $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ does not change the semantics of $\\varphi $ .",
"Observe that the cartesian product of the domains of the attributes in $U_n \\backslash \\lbrace A_n\\rbrace $ can contain tuples ${\\textsf {a}}_{U_n \\backslash \\lbrace A_n\\rbrace }$ such that [(i)] there is a $\\psi _S$ with $S \\in \\mathcal {E}\\backslash \\partial (n)$ , $S \\cap (U_n \\backslash \\lbrace A_n\\rbrace ) \\ne \\emptyset $ and there is no ${\\textsf {b}}_S$ that agrees with ${\\textsf {a}}_{U_n \\backslash \\lbrace A_n\\rbrace }$ on the common attributes and $\\psi _S({\\textsf {b}}_S)\\ne 0$ .",
"Such tuples will not occur in the final result.",
"To rule them out in advance, indicator projections are used inside $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ .",
"The function $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }$ is defined as $\\psi ^{\\prime }_{U_n\\backslash \\lbrace A_n\\rbrace }({\\textsf {a}}_{U_n\\backslash \\lbrace A_n\\rbrace }) = \\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigoplus \\ ^{(n)}} \\bigg [ \\Big ( \\bigotimes _{S \\in \\partial (n)}\\psi _S({\\textsf {a}}_S) \\Big ) \\otimes \\Big (\\bigotimes _{\\begin{array}{c}S \\notin \\partial (n) \\\\S \\cap U_n \\ne \\emptyset \\end{array}} \\psi _{S/U_{n}} ({\\textsf {a}}_{S\\cap U_n}) \\Big )\\bigg ].$ The computation of the listing representation of this function requires the computation of the join of the listing representations of the functions $\\psi _S$ with $S \\in \\partial (n)$ and the indicator projections.",
"The computation time for this elimination step is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\rho ^{\\ast }(H_{U_n})})$ .",
"In case that $\\bigoplus ^{(n)}$ is equal to $\\bigotimes $ , the formula is rewritten as follows $&\\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigoplus \\ ^{(n)}} \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{a_{n}\\in \\textsf {dom}(A_{n})}{\\bigotimes } \\ \\underset{S\\in \\mathcal {E}}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S) \\\\= & \\underset{a_{f+1}\\in \\textsf {dom}(A_{f+1})}{\\bigoplus \\ ^{(f+1)}} \\cdots \\underset{a_{n-1}\\in \\textsf {dom}(A_{n-1})}{\\bigoplus \\ ^{(n-1)}} \\ \\underset{S\\notin \\partial (n)}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S)^{|\\textsf {dom}(A_n)|}\\underset{S \\in \\partial (A_n)}{\\bigotimes } \\ \\underbrace{\\underset{a_n \\in \\textsf {dom}(A_n)}{\\bigotimes } \\ \\psi _S({\\textsf {a}}_S)}_{\\delta ^S},$ where $\\partial (n)$ is defined as above.",
"Then, the algorithm computes for each $S \\notin \\partial (n)$ , a function $\\psi _S^{\\prime }$ equivalent to $\\psi _S^{|\\textsf {dom}(A_n)|}$ and for each $S \\in \\partial (n)$ , a function $\\psi _{S\\backslash A_n}^{\\prime }$ equivalent to $\\delta ^S$ .",
"This elimination step can be realised in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|)$ .",
"After the elimination of all bound attributes we are left with a formula $\\varphi ^{\\prime }_{{\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }}$ without any bound attributes.",
"In the output computation phase the algorithm first computes (a factorized representation of) the set of tuples ${\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }$ for which $\\varphi ^{\\prime }_{{\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace }}({\\textsf {a}}_{\\lbrace A_1, \\ldots ,A_f\\rbrace })\\ne {\\bf 0}$ and then reports the output.",
"Before giving the overall running time of InsideOut, we introduce elimination hypergraph sequences corresponding to attribute orderings.",
"Elimination hypergraph sequence Given a $\\varphi $ -equivalent attribute ordering $\\tau = \\tau (1),$ $\\ldots ,$ $\\tau (n)$ , we recursively define the elimination hypergraph sequence $H_n^{\\tau }, \\ldots ,H_1^{\\tau }$ associated with $\\tau $ .",
"For each $j$ with $n \\ge j \\ge 1$ , we additionally define two sets $U_j^{\\tau }$ and $\\partial ^{\\tau }(j)$ .",
"For the sake of readability, in the following we skip the superscript $\\tau $ in our notation.",
"We set $H_n = (\\mathcal {V}_n, \\mathcal {E}_n) = H$ and define $\\partial (n) = \\lbrace S \\in \\mathcal {E}_n \\mid A_{\\tau (n)} \\in S\\rbrace $ and $U_n = \\bigcup _{S \\in \\partial (n)}S.$ For each $j$ with $n-1 \\ge j \\ge 1$ , we define: If $\\bigoplus \\ ^{(\\tau (j+1))} = \\bigotimes $ , then, $\\mathcal {V}_j = \\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (j)}\\rbrace $ and $\\mathcal {E}_j$ is obtained from $\\mathcal {E}_{j+1}$ by removing $A_{\\tau (j+1)}$ from all edges in $\\mathcal {E}_{j+1}$ .",
"Otherwise, $\\mathcal {V}_j = \\lbrace A_{\\tau (1)}, \\ldots , A_{\\tau (j)}\\rbrace $ and $\\mathcal {E}_j = (\\mathcal {E}_{j+1} \\backslash \\partial (j+1) ) \\cup (U_{j+1} \\backslash \\lbrace A_{\\tau (j+1)}\\rbrace )$ .",
"We further set $\\partial (j) = \\lbrace S \\in \\mathcal {E}_j \\mid A_{\\tau (j)} \\in S\\rbrace $ and $U_j = \\bigcup _{S \\in \\partial (j)}S.$ Running time of InsideOut For a $\\varphi $ -equivalent attribute ordering $\\tau $ , let $K = [f] \\cup \\lbrace j \\mid j > f, \\oplus ^{(\\tau (j))} \\ne \\otimes \\rbrace $ .",
"The FAQ-width of $\\tau $ is defined as $\\textsf {faqw}(\\tau ) =\\max _{j \\in K}\\lbrace \\rho ^{\\ast }(H_{U_j^{\\tau }})\\rbrace $ .",
"For a given $\\tau $ , InsideOut runs in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )} + Z)$ where $Z$ is the size of the output.",
"The FAQ-width of $\\varphi $ is defined as $\\textsf {faqw}(\\varphi ) = \\min _{\\tau \\in \\textsf {EVO}(\\varphi )}\\lbrace \\textsf {faqw}(\\tau )\\rbrace $ .",
"Hence, given the best attribute ordering (i.e., with smallest FAQ-width), the running time of InsideOut is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\varphi )} + Z)$ .",
"From attribute orderings to decompositions We say that $\\mathcal {T}$ is a decomposition of $\\varphi $ if $\\mathcal {T}$ is a decomposition of the hypergraph $H$ of $\\varphi $ .",
"Proposition 40 ([18], Proposition C.2) For any FAQ $\\varphi $ without bound attributes and any $\\varphi $ -equivalent attribute ordering $\\tau $ , one can construct a decomposition $\\mathcal {T}$ of $\\varphi $ with $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau )$ .",
"Covers for FAQs Given two input functions $\\psi _S$ and $\\psi _T$ with $T \\subseteq S$ , we can always compute the function $\\psi ^{\\prime }_S = \\psi _S \\otimes \\psi _T$ in time $\\widetilde{\\mathcal {O}}(|R_{\\psi _S}| + |R_{\\psi _T}|)$ and replace $\\psi _S \\otimes \\psi _T$ by $\\psi ^{\\prime }_S$ without changing the semantics of the FAQ.",
"To do this, we first sort the listing representations $R_{\\psi _S}$ and $R_{\\psi _T}$ of $\\psi _S$ and $\\psi _T$ on the attributes in $T$ .",
"During a subsequent scan through both relations we add for each pair ${\\textsf {a}}_{S \\cup \\lbrace \\psi _S(S)\\rbrace } \\in R_{\\psi _S}$ and ${\\textsf {b}}_{T\\cup \\lbrace \\psi _T(T)\\rbrace } \\in R_{\\psi _T}$ with ${\\textsf {a}}_T = {\\textsf {b}}_T$ , the tuple ${\\textsf {c}}_{S\\cup \\lbrace \\psi ^{\\prime }_S(S)\\rbrace }$ with ${\\textsf {c}}_S = {\\textsf {a}}_S$ and ${\\textsf {c}}_{\\lbrace \\psi ^{\\prime }_S(S)\\rbrace }=\\psi _S({\\textsf {a}}_S) \\otimes \\psi _T({\\textsf {b}}_T)$ to the listing representation of $\\psi ^{\\prime }_S$ .",
"Hence, in the following we assume, without loss of generality, that $\\varphi $ does not contain any function whose attributes are included in the attribute set of another function.",
"Bag functions Given an FAQ $\\varphi $ without bound attributes and a decomposition of $\\varphi $ , we define bag functions which are the counterparts of bag relations in case of join queries.",
"Our goal is to define for each bag $B$ of $\\mathcal {T}$ , a function $\\beta _B$ such that $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) =\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B})$ .",
"While in case of join queries it is harmless to include all relations sharing attributes with $B$ into the join computing the bag relation of $B$ , in case of FAQs we have to be a bit careful.",
"Including the same input function into the computation of bag functions of several bags can violate the above equality.",
"Therefore, in the definition below we use a mapping from input functions to bags.",
"To keep the sizes of the bag functions small we also use indicator projections which achieve pairwise consistency between listing representations of bag functions sharing attributes.",
"Definition 41 (Bag functions) Given an FAQ $\\varphi $ without bound attributes and a decomposition $\\mathcal {T}$ of $\\varphi $ , a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is called a set of bag functions for $\\varphi $ and $\\mathcal {T}$ if there is a mapping $m: \\mathcal {E}\\rightarrow {\\cal S}(\\mathcal {T})$ such that $S \\subseteq m(S)$ for each $S \\in \\mathcal {E}$ and $\\beta _B$ is defined by $\\beta _{B}({\\textsf {a}}_B) =\\underset{{S \\in \\mathcal {E}:S \\cap B \\ne \\emptyset }}{\\bigotimes } \\psi _{S/B}({{\\textsf {a}}}_{B\\cap S}) \\ \\otimes \\underset{S \\in \\mathcal {E}: m(S) = B}{\\bigotimes } \\psi _S({\\textsf {a}}_{S})$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"We define $\\mathcal {B}(\\varphi ,\\mathcal {T}) = \\lbrace \\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}\\mid $ $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ is a set of bag functions for $\\varphi $ and $\\mathcal {T}\\rbrace $ .",
"Note that since each hyperedge in the hypergraph of $\\varphi $ must be included in at least one bag of the decomposition, one can always find a mapping $m$ meeting the condition given in the above definition.",
"Observe also that for bags $B$ to which no input function is mapped, the function $\\beta _B$ is just the product of indicator projections of all $\\psi _S$ sharing attributes with $B$ onto $B$ .",
"Observation 42 Given an FAQ $\\varphi $ without bound attributes, a decomposition $\\mathcal {T}$ of $\\varphi $ and a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ,\\mathcal {T})$ , it holds $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) = \\underset{B\\in {\\cal S}(\\mathcal {T})}{\\bigotimes }\\ \\beta _B({{\\textsf {a}}}_{B}).$ Given $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ,\\mathcal {T})$ , we denote by $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ the decomposition obtained from $\\mathcal {T}$ by adding into each bag $B$ the attribute $\\beta _B(B)$ .",
"Observe that if $\\mathcal {T}$ is a decomposition of $\\varphi ({\\textsf {a}}_{\\mathcal {V}}) = \\bigotimes _{B\\in {\\cal S}(\\mathcal {T})}\\beta _B({{\\textsf {a}}}_{B})$ , then, $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ is a decomposition of the query joining the listing representations of the functions $\\beta _B$ .",
"Moreover, $\\mathcal {T}$ and $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ have the same fractional hypertree width.",
"Covers of FAQ results We turn towards the general case where FAQs can contain bound attributes also.",
"Let $\\tau = \\tau _1\\tau _2$ be a $\\varphi $ -equivalent attribute ordering where $\\tau _1$ consists of the free and $\\tau _2$ consists of the bound attributes in $\\varphi $ .",
"By $\\varphi ^{\\tau }_{\\textit {free}}$ we denote the FAQ constructed by the InsideOut algorithm after eliminating all bound attributes in $\\varphi $ according to the ordering $\\tau _2$ .",
"We write $(\\varphi ,\\tau ,\\mathcal {T}, \\mathbf {D})$ to express that $\\varphi $ is an FAQ, $\\tau $ is a $\\varphi $ -equivalent attribute ordering, $\\mathcal {T}$ is a decomposition of $\\varphi ^{\\tau }_{\\textit {free}}$ with $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau _1)$ and $\\mathbf {D}$ is an input database for $\\varphi $ .",
"Note that due to Proposition REF , for any $\\tau $ such a decomposition $\\mathcal {T}$ is always constructible.",
"Definition 43 (Covers of FAQ results) Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , a relation $K$ is a cover of the query result $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ if there is a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}\\in \\mathcal {B}(\\varphi ^{\\tau }_{\\textit {free}},\\mathcal {T})$ such that $K$ is a cover of the join of the relations $\\lbrace R_{\\beta _B}\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ over $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ .",
"We call $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ the set of bag functions underlying $K$ .",
"Observe that if $K$ is a cover of $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ with underlying bag functions $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ , then, $\\pi _{\\mathcal {V}_{\\text{free}}} K$ must be a cover of $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _BR_{\\beta _B}$ over $\\mathcal {T}$ .",
"The following Proposition relies on Lemma REF and Theorem REF which give an upper bound on the time complexity for constructing covers of join results.",
"Proposition 44 Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , a cover of the query result $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )})$ .",
"Construction.",
"Let $\\tau = \\tau _1\\tau _2$ where $\\tau _1$ consists of the free and $\\tau _2$ consists of the bound attributes in $\\varphi $ .",
"We first run InsideOut on $\\varphi $ according to the attribute ordering $\\tau _2$ until all bound attributes are eliminated and we obtain $\\varphi ^{\\tau }_{\\textit {free}}$ .",
"Then, we construct a set $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})} \\in \\mathcal {B}(\\varphi ^{\\tau }_{\\textit {free}},\\mathcal {T})$ of bag functions.",
"Finally, using a cover-join plan as introduced in Definition REF , we construct a cover $K$ of the join of the relations $\\lbrace R_{\\beta _B}\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ over $\\textit {ext}(\\mathcal {T},\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ .",
"Construction time.",
"The FAQ $\\varphi ^{\\tau }_{\\textit {free}}$ can be computed in time $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau _2)})$ [17].",
"The construction of the bag functions $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ can be realised via the computation of the bag relations of $\\mathcal {T}$ .",
"By Proposition REF , the size of the listing representations of these bag functions is $\\mathcal {O}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ and their computation time is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"By Theorem REF , $K$ can be computed in time $\\widetilde{\\mathcal {O}}(\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|R_{\\beta _B}|)$ .",
"Hence, the time for computing $K$ from $\\varphi ^{\\tau }_{\\textit {free}}$ is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Since $\\textsf {faqw}(\\tau ) = \\max _{1 \\le i \\le 2}\\lbrace \\textsf {faqw}(\\tau _i)\\rbrace $ and $\\textsf {fhtw}(\\mathcal {T}) \\le \\textsf {faqw}(\\tau _1)$ (by construction), the overall computation time is $\\widetilde{\\mathcal {O}}(|\\mathbf {D}|^{\\textsf {faqw}(\\tau )})$ .",
"Theorem REF is an immediate corollary: Theorem REF .",
"For any FAQ $\\varphi $ and database $\\mathbf {D}$ , a cover of the query result $\\varphi (\\mathbf {D})$ can be computed in time $\\widetilde{\\mathcal {O}}(\\mid \\hspace{-2.84526pt} \\mathbf {D} \\hspace{-2.84526pt} \\mid ^{\\textsf {faqw}(\\varphi )})$ .",
"Enumeration of Tuples in FAQ Results using Covers Any enumeration algorithm on covers of join results can easily be turned into an enumeration algorithm on covers of FAQ-results.",
"Assume that $K$ is a cover of the result of the FAQ $\\varphi $ over some decomposition $\\mathcal {T}$ (induced by some attribute ordering).",
"Let $\\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ be the underlying set of bag functions.",
"Recall that the set of attributes of $K$ is $\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}$ and the set of attributes of the listing representation of $\\varphi $ must be $\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\varphi (\\mathcal {V}_{\\text{free}})\\rbrace $ .",
"To enumerate the listing representation of $\\varphi $ , we can run any enumeration algorithm on $K$ with respect to the decomposition $\\textit {ext}(\\mathcal {T}, \\lbrace \\beta _B\\rbrace _{B \\in {\\cal S}(\\mathcal {T})})$ and adapt its output as follows.",
"For each output tuple ${\\textsf {a}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}}$ , we output the tuple ${\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\varphi (\\mathcal {V}_{\\text{free}})\\rbrace }$ that agrees with ${\\textsf {a}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}}$ on $\\mathcal {V}_{\\text{free}}$ and where the $\\varphi (\\mathcal {V}_{\\text{free}})$ -value is defined by $\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})}{\\textsf {a}}_{\\lbrace \\beta _B(B)\\rbrace }$ .",
"The following proposition shows that by this strategy we indeed enumerate the listing representation of $\\varphi $ .",
"Proposition 45 Given $(\\varphi ,\\tau , \\mathcal {T},\\mathbf {D})$ , let $K$ be a cover of the query result of $\\varphi (\\mathbf {D})$ over $\\mathcal {T}$ induced by $\\tau $ and let $\\lbrace \\beta _B\\rbrace _{B \\in \\mathcal {T}}$ be the set of bag functions underlying $K$ .",
"It holds $\\varphi ({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) = v \\ne 0 \\text{ for some } v \\in \\textsf {Dom}$ if and only if $\\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K,\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and }\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{ \\lbrace \\beta _B(B)\\rbrace } = v.$ Let $\\varphi _{\\textit {free}}^{\\tau } = \\bigotimes _{S^{\\prime } \\in \\mathcal {E}^{\\prime }} \\psi _{S^{\\prime }}$ .",
"Then, $&\\ \\ \\ \\varphi ({\\textsf {a}}_{\\mathcal {V}_{\\text{free}}}) = v \\ne {\\bf 0}\\\\\\overset{(1)}{\\Longleftrightarrow } & \\ \\ \\bigotimes _{S^{\\prime } \\in \\mathcal {E}^{\\prime }} \\psi _{S^{\\prime }}({\\textsf {a}}_{S^{\\prime }}) = v \\ne {\\bf 0}\\\\\\overset{(2)}{\\Longleftrightarrow } &\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} \\beta _B({\\textsf {a}}_{B}) = v \\ne {\\bf 0}\\\\\\overset{(3)}{\\Longleftrightarrow } &\\ \\ \\ \\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_{\\beta _B},\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and }\\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{\\lbrace \\beta _B(B)\\rbrace } = v \\\\\\overset{(4)}{\\Longleftrightarrow } &\\ \\ \\ \\exists {\\textsf {b}}_{\\mathcal {V}_{\\text{free}} \\cup \\lbrace \\beta _B(B)\\rbrace _{B \\in {\\cal S}(\\mathcal {T})}} \\in \\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K,\\ {\\textsf {a}}_{\\mathcal {V}_{\\text{free}}} = {\\textsf {b}}_{\\mathcal {V}_{\\text{free}}}\\text{ and } \\\\& \\ \\ \\ \\bigotimes _{B \\in {\\cal S}(\\mathcal {T})} {\\textsf {b}}_{\\lbrace \\beta _B(B)\\rbrace } = v.$ Equivalence (1) holds by the correctness of the InsideOut algorithm.",
"The second equivalence holds by Observation REF .",
"Equivalence (3) follows from the simple observation that the product of functions corresponds to the join of their listing representations.",
"The last equivalence follows from Proposition REF which guarantees that $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}R_{\\beta _B}$ is equal to $\\bowtie _{B \\in {\\cal S}(\\mathcal {T})}\\pi _{B \\cup \\lbrace \\beta _B(B) \\rbrace }K$ .",
"Thus, our enumeration result for covers of join results carries over to covers of FAQ-results.",
"Corollary REF .",
"(Corollary REF , Proposition REF ).",
"Given a cover $K$ of the result $\\varphi (\\mathbf {D})$ of an FAQ $\\varphi $ over a database $\\mathbf {D}$ , the tuples in the query result $\\varphi (\\mathbf {D})$ can be enumerated with $\\widetilde{\\mathcal {O}}(|K|)$ pre-computation time and $\\mathcal {O}(1)$ delay and extra space.",
"Missing Proofs of Appendix In case the signature mappings of an equi-join query are not clear from the context, we write the signature mappings as a superscript to the query.",
"Moreover, for a relation symbol $R$ in an equi-join query with signature mappings $(\\lambda , \\lbrace \\mu _{R}\\rbrace _{R \\in {\\cal S}(Q)})$ and a database $\\mathbf {D}$ , we write $\\lambda (R)_{\\mathbf {D}}$ to denote the relation assigned to the relation symbol $\\lambda (R)$ in $\\mathbf {D}$ .",
"Proof of Proposition REF Proposition REF .",
"Given an equi-join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , there exist a natural join query $Q^{\\prime }$ and a database $\\mathbf {D}^{\\prime }$ such that: $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , $Q^{\\prime }$ has the decomposition $\\mathcal {T}$ and can be constructed in time $\\mathcal {O}(|Q|)$ , and $\\mathbf {D}^{\\prime }$ can be constructed in time $\\mathcal {O}(|\\mathbf {D}|)$ .",
"The query $Q$ has the form $\\sigma _\\psi (R_1\\times \\cdots \\times R_n)$ , where $\\psi $ is a conjunction of equality conditions.",
"The relation symbols as well as all attributes occurring in the schemas of the relation symbols are pairwise distinct.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Given an equivalence class ${\\cal A}$ of attributes in $Q$ , we let $\\phi _{\\cal A}=\\bigwedge _{A_i,A_j\\in {\\cal A}} A_i=A_j$ .",
"Then, given the set $\\lbrace {\\cal A}_j\\rbrace _{j\\in [l]}$ of all equivalence classes in $Q$ , the conjunction $\\bigwedge _{j\\in [l]}\\phi _{{\\cal A}_j}$ is the transitive closure $\\psi ^+$ of $\\psi $ in $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The query $Q^{\\prime }$ has one relation symbol $R^{\\prime }_i$ for each relation symbol $R_i$ in $Q$ such that ${\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"We thus have $Q^{\\prime }=R_1^{\\prime }\\bowtie \\cdots \\bowtie R_n^{\\prime }$ , where the equality conditions in the transitive closure of $\\psi $ are now expressed by natural joins in $Q^{\\prime }$ .",
"Construction of $\\mathbf {D}^{\\prime }$ .",
"For the sake of simplicity, we describe the construction of $\\mathbf {D}^{\\prime }$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : The database $\\mathbf {D}_1$ contains for each $R_i \\in {\\cal S}(Q)$ , a relation $R_i^1$ which results from $\\lambda (R_i)_{\\mathbf {D}}$ by replacing each attribute $A$ by the attribute $B$ with $\\mu _{R_i}(B) = A$ .",
"Construction of database $\\mathbf {D}_2$ : The database $\\mathbf {D}_2$ consists of the relations $R_1^2, \\ldots , R_n^2$ where each $R_i^2$ results from $R_i^1$ as follows.",
"For each equality $A=B$ in $\\psi ^+$ such that $A,B \\in {\\cal S}(R_i^1)$ , we delete in $R_i^1$ all tuples $t$ with $t(A) \\ne t(B)$ .",
"Note that such tuples $t$ cannot occur in the projection of $Q(\\mathbf {D})$ onto the schema of $t$ .",
"Construction of database $\\mathbf {D}^{\\prime }$ : We obtain the database $\\mathbf {D}^{\\prime }$ from $\\mathbf {D}_2$ by replacing each relation $R_i^2$ by a relation $R_i^{\\prime }$ defined as follows.",
"The relation $R_i^{\\prime }$ is a copy of $R_i^2$ extended with one new column for each attribute $A$ in ${\\cal S}(R_i^{\\prime }) \\backslash {\\cal S}(R_i)$ such that $\\pi _{A} R_i^{\\prime } =\\pi _{B} R_i^2$ for any attribute $B\\in {\\cal S}(R_i)$ transitively equal to $A$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"By construction, $Q$ and $Q^{\\prime }$ have the same set of attributes and thus the same equivalence classes of attributes.",
"Moreover, the transitive closures of the schemas of relation symbols are identical: For any pair of relation symbols $R_i \\in {\\cal S}(Q)$ and $R_i^{\\prime } \\in {\\cal S}(Q^{\\prime })$ , it holds that ${\\cal S}(R^{\\prime }_i)^+={\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"The hypergraphs of $Q^{\\prime }$ and $Q$ are thus the same as they have the same nodes, which are the attributes in $Q$ and $Q^{\\prime }$ respectively, and the same hyperedges, which are the transitive closures ${\\cal S}(R_i)^+$ and ${\\cal S}(R^{\\prime }_i)^+$ respectively.",
"This means that the decomposition $\\mathcal {T}$ of $Q$ is also a decomposition of $Q^{\\prime }$ .",
"$Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"We define two further signature mappings $(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})$ and $(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})$ for $Q$ .",
"The function $\\lambda ^1$ maps each relation symbol $R_i$ in $Q$ to $R_i^1$ .",
"Moreover, each $\\mu ^1_{R_i}$ is an identity mapping on the attributes of $R_i$ .",
"The function $\\lambda ^2$ maps each relation symbol $R_i$ in $Q$ to $R_i^2$ .",
"Finally, $\\mu ^1_{R_i} = \\mu ^2_{R_i}$ for each $R_i \\in {\\cal S}(Q)$ .",
"The Database $\\mathbf {D}_1$ results from $\\mathbf {D}$ by, basically, making for each relation $R$ as many copies as the number of relation symbols in $Q$ mapped to $R$ .",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by ruling out tuples which cannot be contained in (the projections of) the final result.",
"Hence, it easily follows $Q^{(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D})$ $=$ $Q^{(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_1)$ $=$ $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Thus, it remains to show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ $=$ $Q(\\mathbf {D})$ .",
"We first treat the special case when $Q$ is a Cartesian product, i.e., it does not contain any equality conditions.",
"Then, $Q^{\\prime }=Q$ and each relation in $\\mathbf {D}^{\\prime }$ is an exact copy of a relation in $\\mathbf {D}_1$ .",
"Hence, $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ holds trivially.",
"We next consider the case when $Q$ has equality conditions.",
"We first show $Q^{\\prime }(\\mathbf {D}^{\\prime })\\subseteq Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Assume there is a tuple $t$ that is contained in $Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Then, $t=\\bowtie _{i\\in [n]} t_i$ is the natural join of tuples $t_i\\in R^{\\prime }_i$ .",
"Let ${\\cal A}$ be any equivalence class of attributes in $Q^{\\prime }$ .",
"By construction, whenever one of these attributes occur in the schema of a relation $R^{\\prime }_i$ , so are the others.",
"Furthermore, their values are the same in any tuple of $R^{\\prime }_i$ .",
"Since $t$ is a join of tuples $t_i$ , it follows that all attributes in ${\\cal A}$ have the same value in $t$ and therefore $\\sigma _{\\phi _{\\cal A}}(t)=t$ .",
"This holds for all equivalence classes of attributes, so $\\sigma _{\\psi ^+}(t)=t$ and thus $\\sigma _{\\psi }(t)=t$ .",
"This means that $t\\in Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We now show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Assume there is a tuple $t$ that is in $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"This means that $t = _{i\\in [n]} t_i$ is a product of tuples $t_i\\in R_i^2$ , $\\sigma _{\\psi ^+}(t)=t$ and in particular $\\sigma _{\\phi _{\\cal A}}(t)=t$ for each equivalence class ${\\cal A}$ in $Q$ .",
"We extend each tuple $t_i$ with values for all attributes in the class ${\\cal A}$ whenever ${\\cal S}(t_i)\\cap {\\cal A}\\ne \\emptyset $ .",
"Let $t^{\\prime }_i$ be the extension of $t_i$ .",
"Then, $t = \\bowtie _{i\\in [n]} t^{\\prime }_i$ .",
"All attributes in ${\\cal A}$ thus have the same value in $t^{\\prime }_i$ .",
"Since, by construction, the relation $R^{\\prime }_i$ is an extension of $R_i^2$ with same-valued columns for all attributes in ${\\cal A}$ whenever ${\\cal S}(R_i^2)\\cap {\\cal A}\\ne \\emptyset $ , it follows that $t^{\\prime }_i\\in R^{\\prime }_i$ .",
"Thus, $t\\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Construction time.",
"The natural join query $Q^{\\prime }$ evolves from $Q$ by replacing the schema $S$ of each relation symbol by $S^+$ .",
"This can be done in time $\\mathcal {O}(|Q|)$ .",
"The database $\\mathbf {D}_1$ evolves from $\\mathbf {D}$ by duplicating each relation in $\\mathbf {D}$ at most $|Q|$ times.",
"Hence, $\\mathbf {D}_1$ can be constructed in linear time.",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by deleting in each relation $R_i^1$ in $\\mathbf {D}_1$ , each tuple tuple $t$ with $t(A)\\ne t(B)$ and $A= B \\in \\psi ^+$ .",
"This deletion procedure can be realised via a single pass through the relations in $\\mathbf {D}_1$ and requires, therefore, only linear time.",
"Likewise, each relation $R_i^{\\prime }$ in $\\mathbf {D}^{\\prime }$ can be constructed from $R_i^2$ in $\\mathbf {D}_2$ by a single pass through the tuples in $R_i^2$ .",
"For each tuple, we choose for each new attribute $A$ in $R_i^{\\prime }$ but not in $R_i^2$ , an equivalent attribute in $R_i^2$ and copy its value to the $A$ -column.",
"Thus, the transformation from $\\mathbf {D}_2$ to $\\mathbf {D}^{\\prime }$ can also be done in linear time.",
"Proof of Proposition REF Proposition REF .",
"For any equi-join query $Q$ and any decomposition $\\mathcal {T}$ of $Q$ , there are arbitrarily large databases $\\mathbf {D}$ such that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We will prove the following claim: Claim: Given an equi-join query $Q$ and a decomposition $\\mathcal {T}$ of $Q$ , there exist a natural join query $Q^{\\prime }$ that has the decomposition $\\mathcal {T}$ such that: $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ and for each database $\\mathbf {D}^{\\prime }$ there is a database $\\mathbf {D}$ of size $\\mathcal {O}(\\mathbf {D}^{\\prime })$ such that $|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B} Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Using this claim, the result of the proposition can be derived straightforwardly.",
"Given an equi-join query $Q$ , we first construct the natural join query as promised in the claim.",
"By Theorem REF (ii), there are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that each cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given such a database $\\mathbf {D}^{\\prime }$ , it follows from Proposition REF , that $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })| =\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ , hence, $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace = \\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"By our claim, the database $\\mathbf {D}^{\\prime }$ can be converted into a database $\\mathbf {D}$ of size $\\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ such that $|\\pi _BQ(\\mathbf {D})| \\ge |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By Proposition REF (adapted to equi-join queries), each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ .",
"Since $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ $=$ $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ and $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ $\\ge $ $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ , we conclude that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is of size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We turn towards the proof of our claim.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The natural join query $Q^{\\prime }$ is constructed exactly as in the proof of Proposition REF .",
"Construction of $\\mathbf {D}$ .",
"Given a database $\\mathbf {D}^{\\prime }$ , we describe the construction of $\\mathbf {D}$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : For each equivalence class $\\mathcal {A}\\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ , let $f_{\\mathcal {A}}$ be an injective function mapping tuples over $\\mathcal {A}$ to fresh data values not occurring in $\\mathbf {D}^{\\prime }$ .",
"Moreover, let $f$ be a function mapping tuples $t$ with ${\\cal S}(t) \\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ and ${\\cal S}(t) = {\\cal S}(t)^+$ to tuples $t^{\\prime }$ with ${\\cal S}(t^{\\prime }) = {\\cal S}(t)$ as follows.",
"For each attribute $A \\in {\\cal S}(t^{\\prime })$ from some equivalence class $\\mathcal {A}$ , it holds $t^{\\prime }(A) = f_{\\mathcal {A}}(\\pi _{\\mathcal {A}}t)$ .",
"From each relation $R_i^{\\prime } \\in \\mathbf {D}^{\\prime }$ , we construct a relation $R_i^1$ where each tuple $t$ is replaced by $f(t)$ .",
"We define $\\mathbf {D}_1 = \\lbrace R_{i}^1\\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}_2$ : From each relation $R_{i}^1 \\in \\mathbf {D}_1$ we design a relation $R_{i}^2$ by performing the following procedure.",
"We first project away all columns of attributes not included in ${\\cal S}(R_i)$ .",
"Then, we rename each attribute $A$ in the resulting relation by $\\mu _{R_i}(A)$ .",
"Let $\\mathbf {D}_2 = \\lbrace R_{i}^2 \\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}$ : We obtain database $\\mathbf {D}$ from $\\mathbf {D}_2$ as follows.",
"For each maximal set $\\lbrace R_{i_1}, \\ldots ,R_{i_k}\\rbrace \\subseteq {\\cal S}(Q)$ such that all $R_{i_j}$ are mapped to the same relation symbol $\\lambda (R_{i_j})=R$ , we replace the relations $R_{i_1}^2, \\ldots ,R_{i_k}^2$ by a single relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with relation symbol $R$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"This follows from the proof of Proposition REF .",
"$|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Our proof contains three steps.",
"We show that $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Let $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $B = B^+$ , the function $f$ is defined for all tuples in $\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Moreover, for two distinct tuples $t_B, t_B^{\\prime } \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , the tuples $f(t_B)$ and $f(t_B^{\\prime })$ are distinct, too.",
"Hence, it suffices to show that for each $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , we have $f(t_B) \\in \\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"It follows that there is a tuple $t \\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ with $t_B = \\pi _B t$ .",
"By definition of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ , it must hold $\\pi _{{\\cal S}(R_i^{\\prime })} t \\in R_i^{\\prime }$ for each $i \\in [n]$ .",
"We have ${\\cal S}(R_i^{\\prime }) = {\\cal S}(R_i^{\\prime })^+$ for each $i \\in [n]$ .",
"Thus, it holds $f(\\pi _{{\\cal S}(R_i^{\\prime })} t) \\in R_i^1$ for each $i \\in [n]$ .",
"This is equivalent to saying $\\pi _{{\\cal S}(R_i^1)} f(t) \\in R_i^1$ for each $i \\in [n]$ .",
"By definition of $\\bowtie _{i \\in [n]}R_i^1$ , this implies that $f(t) \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"Thus, $f(t_B) = \\pi _B f(t) \\in \\pi _B(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $\\lambda ^{\\prime }$ be a function that maps each relation symbol $R_i$ in ${\\cal S}(Q)$ to $R_i^2$ .",
"We show that $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"To this end, let $t \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"It follows that $\\pi _{{\\cal S}(R_i^1)}t \\in R_i^1$ for each $i \\in [n]$ .",
"By the construction of $\\mathbf {D}_2$ , it holds $\\pi _{{\\cal S}(R_i^2)}t \\in R_i^2$ for each $i \\in [n]$ .",
"Furthermore, by the construction of $\\mathbf {D}_1$ , for each equivalence class $\\mathcal {A}$ and all attributes $A,B \\in \\mathcal {A}$ , we have $t(A) = t(B)$ .",
"Thus, $t \\in Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We show that $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We recall that database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by replacing each maximal set $R_{i_1}^2, \\ldots ,R_{i_k}^2$ of relations with $\\lambda (R_{i_1}) = \\ldots = \\lambda (R_{i_k}) = R$ , by the relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with the relation symbol $\\lambda (R_{i_1})$ .",
"Observe that the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\lbrace R_{i_1}^2, \\ldots ,R_{i_k}^2\\rbrace )$ (under signature mappings $(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ) must be included in the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\bigcup _{j \\in [k]} R_{i_j}^2)$ (under signature mappings $(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ).",
"By generalising this insight, we obtain that every tuple from $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ must be included in $Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"By (1), $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By (2) and (3), $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We conclude that $|\\pi _{B}Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Construction time for $Q$ .",
"It follows from the proof of Proposition REF that $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ .",
"Size of $\\mathbf {D}$ .",
"Since $f$ is a bijective mapping and $\\mathbf {D}_1$ is obtained from $\\mathbf {D}^{\\prime }$ by replacing tuples $t$ by $f(t)$ , we have $|\\mathbf {D}_1| = |\\mathbf {D}^{\\prime }|$ .",
"As $\\mathbf {D}_2$ results from $\\mathbf {D}_1$ by taking projections of relations, the size of $\\mathbf {D}_2$ cannot be larger than the size of $\\mathbf {D}_1$ .",
"Database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by taking unions of relations.",
"Thus, the number of tuples in $\\mathbf {D}$ cannot be more than the number of tuples in $\\mathbf {D}_2$ .",
"Altogether, we have $|\\mathbf {D}| = \\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ ."
],
[
"Missing Proofs of Appendix ",
"In case the signature mappings of an equi-join query are not clear from the context, we write the signature mappings as a superscript to the query.",
"Moreover, for a relation symbol $R$ in an equi-join query with signature mappings $(\\lambda , \\lbrace \\mu _{R}\\rbrace _{R \\in {\\cal S}(Q)})$ and a database $\\mathbf {D}$ , we write $\\lambda (R)_{\\mathbf {D}}$ to denote the relation assigned to the relation symbol $\\lambda (R)$ in $\\mathbf {D}$ ."
],
[
"Proof of Proposition ",
"Proposition REF .",
"Given an equi-join query $Q$ , a decomposition $\\mathcal {T}$ of $Q$ , and a database $\\mathbf {D}$ , there exist a natural join query $Q^{\\prime }$ and a database $\\mathbf {D}^{\\prime }$ such that: $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ , $Q^{\\prime }$ has the decomposition $\\mathcal {T}$ and can be constructed in time $\\mathcal {O}(|Q|)$ , and $\\mathbf {D}^{\\prime }$ can be constructed in time $\\mathcal {O}(|\\mathbf {D}|)$ .",
"The query $Q$ has the form $\\sigma _\\psi (R_1\\times \\cdots \\times R_n)$ , where $\\psi $ is a conjunction of equality conditions.",
"The relation symbols as well as all attributes occurring in the schemas of the relation symbols are pairwise distinct.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Given an equivalence class ${\\cal A}$ of attributes in $Q$ , we let $\\phi _{\\cal A}=\\bigwedge _{A_i,A_j\\in {\\cal A}} A_i=A_j$ .",
"Then, given the set $\\lbrace {\\cal A}_j\\rbrace _{j\\in [l]}$ of all equivalence classes in $Q$ , the conjunction $\\bigwedge _{j\\in [l]}\\phi _{{\\cal A}_j}$ is the transitive closure $\\psi ^+$ of $\\psi $ in $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The query $Q^{\\prime }$ has one relation symbol $R^{\\prime }_i$ for each relation symbol $R_i$ in $Q$ such that ${\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"We thus have $Q^{\\prime }=R_1^{\\prime }\\bowtie \\cdots \\bowtie R_n^{\\prime }$ , where the equality conditions in the transitive closure of $\\psi $ are now expressed by natural joins in $Q^{\\prime }$ .",
"Construction of $\\mathbf {D}^{\\prime }$ .",
"For the sake of simplicity, we describe the construction of $\\mathbf {D}^{\\prime }$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : The database $\\mathbf {D}_1$ contains for each $R_i \\in {\\cal S}(Q)$ , a relation $R_i^1$ which results from $\\lambda (R_i)_{\\mathbf {D}}$ by replacing each attribute $A$ by the attribute $B$ with $\\mu _{R_i}(B) = A$ .",
"Construction of database $\\mathbf {D}_2$ : The database $\\mathbf {D}_2$ consists of the relations $R_1^2, \\ldots , R_n^2$ where each $R_i^2$ results from $R_i^1$ as follows.",
"For each equality $A=B$ in $\\psi ^+$ such that $A,B \\in {\\cal S}(R_i^1)$ , we delete in $R_i^1$ all tuples $t$ with $t(A) \\ne t(B)$ .",
"Note that such tuples $t$ cannot occur in the projection of $Q(\\mathbf {D})$ onto the schema of $t$ .",
"Construction of database $\\mathbf {D}^{\\prime }$ : We obtain the database $\\mathbf {D}^{\\prime }$ from $\\mathbf {D}_2$ by replacing each relation $R_i^2$ by a relation $R_i^{\\prime }$ defined as follows.",
"The relation $R_i^{\\prime }$ is a copy of $R_i^2$ extended with one new column for each attribute $A$ in ${\\cal S}(R_i^{\\prime }) \\backslash {\\cal S}(R_i)$ such that $\\pi _{A} R_i^{\\prime } =\\pi _{B} R_i^2$ for any attribute $B\\in {\\cal S}(R_i)$ transitively equal to $A$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"By construction, $Q$ and $Q^{\\prime }$ have the same set of attributes and thus the same equivalence classes of attributes.",
"Moreover, the transitive closures of the schemas of relation symbols are identical: For any pair of relation symbols $R_i \\in {\\cal S}(Q)$ and $R_i^{\\prime } \\in {\\cal S}(Q^{\\prime })$ , it holds that ${\\cal S}(R^{\\prime }_i)^+={\\cal S}(R^{\\prime }_i)={\\cal S}(R_i)^+$ .",
"The hypergraphs of $Q^{\\prime }$ and $Q$ are thus the same as they have the same nodes, which are the attributes in $Q$ and $Q^{\\prime }$ respectively, and the same hyperedges, which are the transitive closures ${\\cal S}(R_i)^+$ and ${\\cal S}(R^{\\prime }_i)^+$ respectively.",
"This means that the decomposition $\\mathcal {T}$ of $Q$ is also a decomposition of $Q^{\\prime }$ .",
"$Q^{\\prime }(\\mathbf {D}^{\\prime })=Q(\\mathbf {D})$ .",
"We define two further signature mappings $(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})$ and $(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})$ for $Q$ .",
"The function $\\lambda ^1$ maps each relation symbol $R_i$ in $Q$ to $R_i^1$ .",
"Moreover, each $\\mu ^1_{R_i}$ is an identity mapping on the attributes of $R_i$ .",
"The function $\\lambda ^2$ maps each relation symbol $R_i$ in $Q$ to $R_i^2$ .",
"Finally, $\\mu ^1_{R_i} = \\mu ^2_{R_i}$ for each $R_i \\in {\\cal S}(Q)$ .",
"The Database $\\mathbf {D}_1$ results from $\\mathbf {D}$ by, basically, making for each relation $R$ as many copies as the number of relation symbols in $Q$ mapped to $R$ .",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by ruling out tuples which cannot be contained in (the projections of) the final result.",
"Hence, it easily follows $Q^{(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D})$ $=$ $Q^{(\\lambda ^1, \\lbrace \\mu ^1_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_1)$ $=$ $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Thus, it remains to show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ $=$ $Q(\\mathbf {D})$ .",
"We first treat the special case when $Q$ is a Cartesian product, i.e., it does not contain any equality conditions.",
"Then, $Q^{\\prime }=Q$ and each relation in $\\mathbf {D}^{\\prime }$ is an exact copy of a relation in $\\mathbf {D}_1$ .",
"Hence, $Q^{\\prime }(\\mathbf {D}^{\\prime })=Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ holds trivially.",
"We next consider the case when $Q$ has equality conditions.",
"We first show $Q^{\\prime }(\\mathbf {D}^{\\prime })\\subseteq Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"Assume there is a tuple $t$ that is contained in $Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Then, $t=\\bowtie _{i\\in [n]} t_i$ is the natural join of tuples $t_i\\in R^{\\prime }_i$ .",
"Let ${\\cal A}$ be any equivalence class of attributes in $Q^{\\prime }$ .",
"By construction, whenever one of these attributes occur in the schema of a relation $R^{\\prime }_i$ , so are the others.",
"Furthermore, their values are the same in any tuple of $R^{\\prime }_i$ .",
"Since $t$ is a join of tuples $t_i$ , it follows that all attributes in ${\\cal A}$ have the same value in $t$ and therefore $\\sigma _{\\phi _{\\cal A}}(t)=t$ .",
"This holds for all equivalence classes of attributes, so $\\sigma _{\\psi ^+}(t)=t$ and thus $\\sigma _{\\psi }(t)=t$ .",
"This means that $t\\in Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We now show $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Assume there is a tuple $t$ that is in $Q^{(\\lambda ^2, \\lbrace \\mu ^2_{R_i}\\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"This means that $t = _{i\\in [n]} t_i$ is a product of tuples $t_i\\in R_i^2$ , $\\sigma _{\\psi ^+}(t)=t$ and in particular $\\sigma _{\\phi _{\\cal A}}(t)=t$ for each equivalence class ${\\cal A}$ in $Q$ .",
"We extend each tuple $t_i$ with values for all attributes in the class ${\\cal A}$ whenever ${\\cal S}(t_i)\\cap {\\cal A}\\ne \\emptyset $ .",
"Let $t^{\\prime }_i$ be the extension of $t_i$ .",
"Then, $t = \\bowtie _{i\\in [n]} t^{\\prime }_i$ .",
"All attributes in ${\\cal A}$ thus have the same value in $t^{\\prime }_i$ .",
"Since, by construction, the relation $R^{\\prime }_i$ is an extension of $R_i^2$ with same-valued columns for all attributes in ${\\cal A}$ whenever ${\\cal S}(R_i^2)\\cap {\\cal A}\\ne \\emptyset $ , it follows that $t^{\\prime }_i\\in R^{\\prime }_i$ .",
"Thus, $t\\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Construction time.",
"The natural join query $Q^{\\prime }$ evolves from $Q$ by replacing the schema $S$ of each relation symbol by $S^+$ .",
"This can be done in time $\\mathcal {O}(|Q|)$ .",
"The database $\\mathbf {D}_1$ evolves from $\\mathbf {D}$ by duplicating each relation in $\\mathbf {D}$ at most $|Q|$ times.",
"Hence, $\\mathbf {D}_1$ can be constructed in linear time.",
"We obtain $\\mathbf {D}_2$ from $\\mathbf {D}_1$ by deleting in each relation $R_i^1$ in $\\mathbf {D}_1$ , each tuple tuple $t$ with $t(A)\\ne t(B)$ and $A= B \\in \\psi ^+$ .",
"This deletion procedure can be realised via a single pass through the relations in $\\mathbf {D}_1$ and requires, therefore, only linear time.",
"Likewise, each relation $R_i^{\\prime }$ in $\\mathbf {D}^{\\prime }$ can be constructed from $R_i^2$ in $\\mathbf {D}_2$ by a single pass through the tuples in $R_i^2$ .",
"For each tuple, we choose for each new attribute $A$ in $R_i^{\\prime }$ but not in $R_i^2$ , an equivalent attribute in $R_i^2$ and copy its value to the $A$ -column.",
"Thus, the transformation from $\\mathbf {D}_2$ to $\\mathbf {D}^{\\prime }$ can also be done in linear time."
],
[
"Proof of Proposition ",
"Proposition REF .",
"For any equi-join query $Q$ and any decomposition $\\mathcal {T}$ of $Q$ , there are arbitrarily large databases $\\mathbf {D}$ such that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We will prove the following claim: Claim: Given an equi-join query $Q$ and a decomposition $\\mathcal {T}$ of $Q$ , there exist a natural join query $Q^{\\prime }$ that has the decomposition $\\mathcal {T}$ such that: $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ and for each database $\\mathbf {D}^{\\prime }$ there is a database $\\mathbf {D}$ of size $\\mathcal {O}(\\mathbf {D}^{\\prime })$ such that $|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B} Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Using this claim, the result of the proposition can be derived straightforwardly.",
"Given an equi-join query $Q$ , we first construct the natural join query as promised in the claim.",
"By Theorem REF (ii), there are arbitrarily large databases $\\mathbf {D}^{\\prime }$ such that each cover of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ over $\\mathcal {T}$ has size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"Given such a database $\\mathbf {D}^{\\prime }$ , it follows from Proposition REF , that $\\Sigma _{B \\in {\\cal S}(\\mathcal {T})}|\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })| =\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ , hence, $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace = \\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"By our claim, the database $\\mathbf {D}^{\\prime }$ can be converted into a database $\\mathbf {D}$ of size $\\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ such that $|\\pi _BQ(\\mathbf {D})| \\ge |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By Proposition REF (adapted to equi-join queries), each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ must have size at least $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ .",
"Since $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ $=$ $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ and $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ(\\mathbf {D})|\\rbrace $ $\\ge $ $\\max _{B \\in {\\cal S}(\\mathcal {T})}\\lbrace |\\pi _BQ^{\\prime }(\\mathbf {D}^{\\prime })|\\rbrace $ , we conclude that each cover of $Q(\\mathbf {D})$ over $\\mathcal {T}$ is of size $\\Omega (|\\mathbf {D}^{\\prime }|^{\\textsf {fhtw}(\\mathcal {T})})$ $=$ $\\Omega (|\\mathbf {D}|^{\\textsf {fhtw}(\\mathcal {T})})$ .",
"We turn towards the proof of our claim.",
"Let $(\\lambda , \\lbrace \\mu _{R_i}\\rbrace _{i \\in [n]})$ be the signature mappings of $Q$ .",
"Construction of $Q^{\\prime }$ .",
"The natural join query $Q^{\\prime }$ is constructed exactly as in the proof of Proposition REF .",
"Construction of $\\mathbf {D}$ .",
"Given a database $\\mathbf {D}^{\\prime }$ , we describe the construction of $\\mathbf {D}$ in three steps.",
"Construction of database $\\mathbf {D}_1$ : For each equivalence class $\\mathcal {A}\\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ , let $f_{\\mathcal {A}}$ be an injective function mapping tuples over $\\mathcal {A}$ to fresh data values not occurring in $\\mathbf {D}^{\\prime }$ .",
"Moreover, let $f$ be a function mapping tuples $t$ with ${\\cal S}(t) \\subseteq \\bigcup _{i \\in [n]}{\\cal S}(R_i^{\\prime })$ and ${\\cal S}(t) = {\\cal S}(t)^+$ to tuples $t^{\\prime }$ with ${\\cal S}(t^{\\prime }) = {\\cal S}(t)$ as follows.",
"For each attribute $A \\in {\\cal S}(t^{\\prime })$ from some equivalence class $\\mathcal {A}$ , it holds $t^{\\prime }(A) = f_{\\mathcal {A}}(\\pi _{\\mathcal {A}}t)$ .",
"From each relation $R_i^{\\prime } \\in \\mathbf {D}^{\\prime }$ , we construct a relation $R_i^1$ where each tuple $t$ is replaced by $f(t)$ .",
"We define $\\mathbf {D}_1 = \\lbrace R_{i}^1\\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}_2$ : From each relation $R_{i}^1 \\in \\mathbf {D}_1$ we design a relation $R_{i}^2$ by performing the following procedure.",
"We first project away all columns of attributes not included in ${\\cal S}(R_i)$ .",
"Then, we rename each attribute $A$ in the resulting relation by $\\mu _{R_i}(A)$ .",
"Let $\\mathbf {D}_2 = \\lbrace R_{i}^2 \\rbrace _{i \\in [n]}$ .",
"Construction of database $\\mathbf {D}$ : We obtain database $\\mathbf {D}$ from $\\mathbf {D}_2$ as follows.",
"For each maximal set $\\lbrace R_{i_1}, \\ldots ,R_{i_k}\\rbrace \\subseteq {\\cal S}(Q)$ such that all $R_{i_j}$ are mapped to the same relation symbol $\\lambda (R_{i_j})=R$ , we replace the relations $R_{i_1}^2, \\ldots ,R_{i_k}^2$ by a single relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with relation symbol $R$ .",
"$\\mathcal {T}$ is a decomposition of $Q^{\\prime }$ .",
"This follows from the proof of Proposition REF .",
"$|\\pi _{B}Q(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Our proof contains three steps.",
"We show that $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Let $B \\in {\\cal S}(\\mathcal {T})$ .",
"Since $B = B^+$ , the function $f$ is defined for all tuples in $\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"Moreover, for two distinct tuples $t_B, t_B^{\\prime } \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , the tuples $f(t_B)$ and $f(t_B^{\\prime })$ are distinct, too.",
"Hence, it suffices to show that for each $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ , we have $f(t_B) \\in \\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $t_B \\in \\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })$ .",
"It follows that there is a tuple $t \\in Q^{\\prime }(\\mathbf {D}^{\\prime })$ with $t_B = \\pi _B t$ .",
"By definition of $Q^{\\prime }(\\mathbf {D}^{\\prime })$ , it must hold $\\pi _{{\\cal S}(R_i^{\\prime })} t \\in R_i^{\\prime }$ for each $i \\in [n]$ .",
"We have ${\\cal S}(R_i^{\\prime }) = {\\cal S}(R_i^{\\prime })^+$ for each $i \\in [n]$ .",
"Thus, it holds $f(\\pi _{{\\cal S}(R_i^{\\prime })} t) \\in R_i^1$ for each $i \\in [n]$ .",
"This is equivalent to saying $\\pi _{{\\cal S}(R_i^1)} f(t) \\in R_i^1$ for each $i \\in [n]$ .",
"By definition of $\\bowtie _{i \\in [n]}R_i^1$ , this implies that $f(t) \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"Thus, $f(t_B) = \\pi _B f(t) \\in \\pi _B(\\bowtie _{i \\in [n]}R_i^1)$ .",
"Let $\\lambda ^{\\prime }$ be a function that maps each relation symbol $R_i$ in ${\\cal S}(Q)$ to $R_i^2$ .",
"We show that $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"To this end, let $t \\in \\bowtie _{i \\in [n]}R_i^1$ .",
"It follows that $\\pi _{{\\cal S}(R_i^1)}t \\in R_i^1$ for each $i \\in [n]$ .",
"By the construction of $\\mathbf {D}_2$ , it holds $\\pi _{{\\cal S}(R_i^2)}t \\in R_i^2$ for each $i \\in [n]$ .",
"Furthermore, by the construction of $\\mathbf {D}_1$ , for each equivalence class $\\mathcal {A}$ and all attributes $A,B \\in \\mathcal {A}$ , we have $t(A) = t(B)$ .",
"Thus, $t \\in Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ .",
"We show that $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2) \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We recall that database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by replacing each maximal set $R_{i_1}^2, \\ldots ,R_{i_k}^2$ of relations with $\\lambda (R_{i_1}) = \\ldots = \\lambda (R_{i_k}) = R$ , by the relation $\\bigcup _{j \\in [k]} R_{i_j}^2$ with the relation symbol $\\lambda (R_{i_1})$ .",
"Observe that the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\lbrace R_{i_1}^2, \\ldots ,R_{i_k}^2\\rbrace )$ (under signature mappings $(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ) must be included in the result of $\\sigma _{\\psi }(R_{i_1} \\times \\ldots \\times R_{i_k})(\\bigcup _{j \\in [k]} R_{i_j}^2)$ (under signature mappings $(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})$ ).",
"By generalising this insight, we obtain that every tuple from $Q^{(\\lambda ^{\\prime },\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D}_2)$ must be included in $Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"By (1), $|\\pi _{B}(\\bowtie _{i \\in [n]}R_i^1)| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"By (2) and (3), $\\bowtie _{i \\in [n]}R_i^1 \\subseteq Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})$ .",
"We conclude that $|\\pi _{B}Q^{(\\lambda ,\\lbrace \\mu _{R_i} \\rbrace _{i \\in [n]})}(\\mathbf {D})| \\ge |\\pi _{B}Q^{\\prime }(\\mathbf {D}^{\\prime })|$ for each $B \\in {\\cal S}(\\mathcal {T})$ .",
"Construction time for $Q$ .",
"It follows from the proof of Proposition REF that $Q^{\\prime }$ can be constructed in time $\\mathcal {O}(|Q|)$ .",
"Size of $\\mathbf {D}$ .",
"Since $f$ is a bijective mapping and $\\mathbf {D}_1$ is obtained from $\\mathbf {D}^{\\prime }$ by replacing tuples $t$ by $f(t)$ , we have $|\\mathbf {D}_1| = |\\mathbf {D}^{\\prime }|$ .",
"As $\\mathbf {D}_2$ results from $\\mathbf {D}_1$ by taking projections of relations, the size of $\\mathbf {D}_2$ cannot be larger than the size of $\\mathbf {D}_1$ .",
"Database $\\mathbf {D}$ results from $\\mathbf {D}_2$ by taking unions of relations.",
"Thus, the number of tuples in $\\mathbf {D}$ cannot be more than the number of tuples in $\\mathbf {D}_2$ .",
"Altogether, we have $|\\mathbf {D}| = \\mathcal {O}(|\\mathbf {D}^{\\prime }|)$ ."
]
] | 1709.01600 |
[
[
"Scattering theory from artificial piezoelectric-like meta-atoms and\n molecules"
],
[
"Abstract Inspired by the natural piezoelectric effect, we introduce hybrid-wave electromechanical meta-atoms and meta-molecules that consist of coupled electrical and mechanical oscillators with similar resonance frequencies.",
"We propose an analytical model for the linearized electromechanical scattering process, and explore its properties based on first principles.",
"We demonstrate that by exploiting the linearized hybrid-wave interaction, one may enable functionalities that are forbidden otherwise, going beyond the limits of today's metamaterials.",
"As an example we show an electrically deep sub-wavelength dimer of meta-atoms with extremely sensitive response to the direction-of-arrival of an impinging electromagnetic wave.",
"This scheme of meta-atoms and molecules may open ways for metamaterials with a plethora of exciting dynamics and phenomena that have not been studied before with potential technological implications in radio-frequencies and acoustics."
],
[
"Derivation of the extracted electromechanical power",
"The power that an electromechanical wave ${\\bf {U}}=[{\\bf {E}},{\\cal P}]^T$ extracts to excite an electromechanical meta-atom with induced source ${\\bf {S}}=[{\\bf {p}}_e,{\\cal V}]^T$ is given by $P^{ext}=\\frac{1}{2}\\Re \\lbrace {\\bf {J}}^*\\cdot {\\bf {E}} + {\\cal U}^*{\\cal P} \\rbrace $ where ${\\bf {J}}=j\\omega {\\bf {p}}_e$ is the current source associated with the induced electric dipole on the scatterer, and ${\\cal U}=j\\omega {\\cal V}$ is the volume velocity associated with the monopole volume amplitude $\\cal V$ .",
"Then, we have, $P^{ext}=\\frac{\\omega }{2}\\Im \\lbrace {\\bf {p}}_e^*\\cdot {\\bf {E}} + {\\cal V}^*{\\cal P} \\rbrace =\\frac{\\omega }{2}{\\bf {S}}^{H}{\\bf {U}}.$ However, since ${\\bf {S}}=\\underline{\\underline{\\alpha }}{\\bf {U}}$ (Eq.",
"(1) in the main text), we can write $P^{ext}=\\frac{\\omega }{2}\\Im \\lbrace (\\underline{\\underline{\\alpha }}{\\bf {U}})^{H}{\\bf {U}}\\rbrace =\\frac{\\omega }{2}\\Im \\lbrace {\\bf {U}}^H\\underline{\\underline{\\alpha }}^H{\\bf {U}}\\rbrace .$ This completes the derivation of the extracted power given in the main text."
],
[
"Derivation of the total radiated power from an EMCL meta-atom",
"The total radiated power from an EMCL source ${\\bf {S}}$ in electromagnetically as well as acoustically homogeneous medium reads $P^{rad}=\\frac{\\rho \\omega ^4}{8\\pi c_a}{\\cal V}^*{\\cal V} + \\frac{\\mu \\omega ^4}{12\\pi c_e}{\\bf {p}}_e^*\\cdot {\\bf {p}}_e.$ For a general medium, however, although the coefficients in the equation above will be changed, the square dependence on ${\\cal V}^*{\\cal V}$ and ${\\bf {p}}_e^*\\cdot {\\bf {p}}_e$ will be maintained.",
"Therefore, we define (as in the main text) $P_a^{rad}$ and $P_e^{rad}$ as the total radiated power for acoustic monopole and electromagnetic dipole sources with unit amplitude, namely, with $|{\\cal V}|=1$ and $|{\\bf {p}}_e|=1$ .",
"Then, by defining $\\underline{\\underline{\\chi }}=\\mbox{diag}[P_e^{rad},P_e^{rad},P_e^{rad},P_a^{rad}] = \\mbox{diag}[P_e^{rad}\\underline{\\underline{I}}_{3\\times 3}, P_a^{rad}].$ we obtain the expression given in the main text for the radiated power, $P^{rad}={\\bf {U}}^H\\underline{\\underline{\\alpha }}^H\\underline{\\underline{\\chi }}\\underline{\\underline{\\alpha }}{\\bf {U}}.$"
],
[
"Lumped sources model for the parallel plates meta-atom",
"For the parallel plate meta-atom discussed in Fig.",
"2(a) of the main text the impinging electric and pressure fields are modelled using lumped sources.",
"The sources excitation scheme is shown in Fig.",
"REF below.",
"Figure: Lumped excitation model"
],
[
"Derivation of the parallel plates meta-atom dynamics",
"The meta-atom dynamics it captured by the Lagrangian ${\\cal L} = \\frac{1}{2}m\\dot{x}^2-\\frac{1}{2}kx^2 + \\frac{1}{2}L\\dot{q}^2 - \\frac{1}{2}\\frac{q^2}{C(x)},$ with source ${\\cal L}_s$ and dissipation $\\cal F$ functionals ${\\cal L}_s=qV+xf, \\quad {\\cal F}=\\frac{1}{2}R\\dot{q}^2 + \\frac{1}{2}B\\dot{x}^2$ where $V=V_0+v(t)$ , $x=-x_0+\\delta x$ ($x_0$ is taken such that $x_0>0$ ), and $q=q_0+\\delta q$ are the plate's deflection and charge accumulation, respectively.",
"The dissipation term $\\cal F$ includes two loss mechanisms, electromagnetic and mechanical, represented, respectively, by resistance $R$ and viscosity $B$ , where both consist of radiation as well as material loss channels.",
"Using Eqs.",
"(REF )-(REF ) in the Lagrange equations we obtain the relations between the static quantities, $x_0,q_0,V_0$ , as well as relations between the small signal terms $\\delta x, \\delta q, v(t), f(t)$ , that represent the temporal dynamics.",
"For the static terms we find $q_0=C_0V_0,\\quad kx_0=q_0^2/2\\epsilon A$ which are nothing but the voltage-charge relation on the capacitor, and equality of forces between the spring repulsion due to contraction of $x_0$ and the Coulomb attraction between the plates of capacitor charged with $q_0$ .",
"Assuming that $x_0\\ll d$ we may approximate $x_0\\approx \\epsilon A V_0/2kd^2$ and $q_0\\approx \\epsilon A V_0/d$ .",
"For the dynamic terms we find the following nonlinear system $\\ddot{\\delta q}+2\\tau _e^{-1}\\dot{\\delta q}+\\omega _e^2\\left(q_0\\frac{\\delta x}{d-x_0} + \\delta q +\\frac{\\delta x}{d-x_0}\\delta q \\right) = \\frac{1}{L}v(t),$ and $\\ddot{\\delta x}+2\\tau _m^{-1}\\dot{\\delta x} +\\omega _m^2\\delta x +\\frac{1}{m}\\frac{V_0}{d-x_0}\\delta q +\\frac{1}{2}\\frac{\\delta q^2}{C_0}\\frac{1}{m(d-x_0)}=\\frac{1}{m} f(t).$ Here $\\tau _e^{-1}=R/2L$ and $\\tau _m^{-1}=B/2m$ are the electromagnetic and mechanical damping rates, $\\omega _e$ and $\\omega _m$ are the electromagnetic and mechanical resonance frequencies in the absence of electromechanical coupling between the resonators, and $E_0=-V_0/(d-x_0)$ is the average electric field between the capacitor plates due to the static biasing.",
"Eq.",
"(9) in the main text is derived from the nonlinear system above under the assumption that $\\delta q\\ll q_0$ and $\\delta x\\ll x_0$ ."
],
[
"Electromagnetic and acoustic Green's function used ",
"The electromagnetic Green's function in electromagnetically homogeneous medium with permittivity and permeability $\\epsilon $ and $\\mu $ is used in the calculation of the dimer response in Eq.",
"(14) of the main text.",
"It reads $\\underline{\\underline{G}}_e({\\bf {r}},{\\bf {r}}^{\\prime })=\\frac{e^{-jk_er}}{4\\pi \\epsilon }\\left\\lbrace \\underline{\\underline{A}}\\frac{k_e^2}{r} + [3\\underline{\\underline{B}}-\\underline{\\underline{I}}_{3\\times 3}]\\left( \\frac{1}{r^3}+\\frac{jk_e}{r^2} \\right) \\right\\rbrace $ with $\\underline{\\underline{A}}=\\left[\\begin{array}{ccc}n_y^2+n_z^2 & -n_xn_y & -n_xn_z \\\\-n_xn_y & n_x^2+n_z^2 & -n_zn_y \\\\-n_zn_x & -n_zn_y & n_x^2+n_y^2\\end{array}\\right]$ and $\\underline{\\underline{B}}=\\left[\\begin{array}{ccc}n_x^2 & n_xn_y & n_xn_z \\\\n_xn_y & n_y^2 & n_zn_y \\\\n_zn_x & n_zn_y & n_z^2\\end{array}\\right]$ where $\\underline{\\underline{I}}_{3\\times 3}$ is the 3 by 3 unitary matrix, $r=|{\\bf {r}}-{\\bf {r}}^{\\prime }|$ is the distance between the source and observer points, and $\\hat{n}=(n_x,n_y,n_z)=({\\bf {r}}-{\\bf {r}}^{\\prime })/r$ is the unit vector pointing between the source and the observer.",
"Moreover, $k_e=\\omega /c_e$ where $c_e$ is the speed of light.",
"For the dimer problem considered in the last part of the paper we had $\\hat{n}=(0,\\pm 1,0)$ and ${\\bf {p}}_{1,2}=p_{1,2}\\hat{x}$ .",
"The acoustic Green's function we used in that example was the Green's function of small acoustic monopole source in a duct supporting a plane wave only at the operation frequency (fundamental mode of the duct).",
"In this case, since the driving frequency is below the cutoff of the higher order modes, the latter decays away from the source.",
"Therefore, sufficiently far from the source (in terms of acoustic wavelength $d\\gg \\lambda _a$ ), we can asymptotically approximate the acoustic Green's function by $G_a\\sim \\frac{-j\\omega \\rho c_a}{2A_d}e^{-jk_a r}$ where here $r$ is the distance between the source and the observer along the duct axis.",
"In the main text this is the $\\hat{y}$ axis and the distance between the two meta-atoms ranged between $d=20\\lambda _m$ and $d=30\\lambda _m$ , in any case satisfying the asymptotic assumption requirement.",
"Moreover $k_a=\\omega /c_a$ , where $c_a$ is the speed of sound in the duct and $A_d$ is the duct cross section area."
]
] | 1709.01909 |
[
[
"Numerical radius inequalities involving commutators of $G_{1}$ operators"
],
[
"Abstract We prove numerical radius inequalities involving commutators of $G_{1}$ operators and certain analytic functions.",
"Among other inequalities, it is shown that if $A$ and $X$ are bounded linear operators on a complex Hilbert space, then \\begin{equation*} w(f(A)X+X\\bar{f}(A))\\leq {\\frac{2}{d_{A}^{2}}}w(X-AXA^{\\ast }), \\end{equation*} where $A$ is a $G_{1}$ operator with $\\sigma (A)\\subset \\mathbb{D}$ and $f$ is analytic on the unit disk $\\mathbb{D}$ such that $\\textrm{{Re}}(f)>0$ and $f(0)=1$."
],
[
"Introduction",
"Let $({H},\\langle \\,\\cdot \\,,\\,\\cdot \\,\\rangle )$ be a complex Hilbert space and ${\\mathbb {B}}(H)$ denote the $C^{\\ast }$ -algebra of all bounded linear operators on ${H}$ with the identity $I$ .",
"In the case when dim${H}=n$ , we identify ${\\mathbb {B}}({H})$ with the matrix algebra $\\mathbb {M}_{n}$ of all $n\\times n$ matrices having entries in the complex field.",
"The numerical radius of $A\\in {\\mathbb {B}}({H})$ is defined by $w(A):=\\sup \\Big \\lbrace |\\langle Ax,x\\rangle |:x\\in {H},\\parallel x\\parallel =1\\Big \\rbrace .$ It is well known that $w(\\,\\cdot \\,)$ defines a norm on ${\\mathbb {B}}({H})$ , which is equivalent to the usual operator norm $\\Vert \\,\\cdot \\,\\Vert $ .",
"In fact, for any $A\\in {\\mathbb {B}}({H})$ , $\\frac{1}{2}\\Vert A\\Vert \\le w(A)\\le \\Vert A\\Vert $ (see [9]).",
"If $A^{2}=0 $ , then equality holds in the first inequality, and if $A$ is normal, then equality holds in the second inequality.",
"For further information about numerical radius inequalities, we refer the reader to [1], [2], [3], [12], [16], [17] and references therein.",
"An operator $A\\in {\\mathbb {B}}({H})$ is called a $G_{1}$ operator if the growth condition $\\Vert (z-A)^{-1}\\Vert ={\\frac{1}{\\text{dist}(z,\\sigma (A))}}$ holds for all $z$ not in the spectrum $\\sigma (A)$ of $A$ , where $\\text{dist}(z,\\sigma (A))$ denotes the distance between $z$ and $\\sigma (A)$ .",
"For simplicity, if $z$ is a complex number, we write $z$ instead of $zI$ .",
"It is known that hyponormal (in particular, normal) operators are $G_{1}$ operators (see, e.g., [15]).",
"Let $A\\in {\\mathbb {B}}({H})$ and $f$ be a function which is analytic on an open neighborhood $\\Omega $ of $\\sigma (A)$ in the complex plane.",
"Then $f(A)$ denotes the operator defined on ${H}$ by the Riesz-Dunford integral as $f(A)={\\frac{1}{2\\pi i}}\\int _{C}f(z)(z-A)^{-1}dz,$ where $C$ is a positively oriented simple closed rectifiable contour surrounding $\\sigma (A)$ in $\\Omega $ (see e.g., [8]).",
"The spectral mapping theorem asserts that $\\sigma (f(A))=f(\\sigma (A))$ .",
"Throughout this note, $\\mathbb {D}=\\left\\lbrace z\\in \\mathbb {C}:|z|<1\\right\\rbrace $ denotes the unit disk, $\\partial \\mathbb {D}$ stands for the boundary of $\\mathbb {D}$ and $d_{A}=\\text{dist}(\\partial \\mathbb {D},\\sigma (A))$ .",
"In addition, we denote $\\mathfrak {A}=\\left\\lbrace f:\\mathbb {D}\\rightarrow \\mathbb {C}:f\\,\\text{is analytic},\\text{Re}(f)>0\\,\\text{and}\\,f(0)=1\\right\\rbrace .$ The Sylvester type equations $AXB\\pm X=C$ have been investigated in matrix theory (see [4]).",
"Several perturbation bounds for the norms of sums or differences of operators have been presented in the literature by employing some integral representations of certain functions.",
"See [5], [13], [14] and references therein.",
"In this paper, we present some upper bounds for the numerical radii of the commutators and elementary operators of the form $f(A)X\\pm X\\bar{f}(A)$ , $f(A)X\\bar{f}(B)-f(B)X\\bar{f}(A)$ and $f(A)X\\bar{f}(B)+2X+f(B)X\\bar{f}(A)$ , where $A,B,X\\in {\\mathbb {B}}({H})$ and $f\\in \\mathfrak {A}$ ."
],
[
"main results",
"To prove our first result, the following lemma concerning numerical radius inequalities and an equality is required.",
"Lemma 2.1 [10], [11] Let $A, B, X, Y\\in {\\mathbb {B}}({H})$ .",
"Then $(a)\\,\\,w(A^{\\ast }XA)\\le \\Vert A\\Vert ^{2}w(X).$ $(b)\\,\\,w\\left( AX\\pm XA^{\\ast }\\right) \\le 2\\Vert A\\Vert w(X).$ $(c)\\,\\,w\\left( A^{\\ast }XB\\pm B^{\\ast }YA\\right) \\le 2\\Vert A\\Vert \\Vert B\\Vert \\,w\\left( \\left[\\begin{array}{cc}0 & X \\\\Y & 0\\end{array}\\right] \\right) .$ $(d)\\,\\,w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BYA^{\\ast } & 0\\end{array}\\right] \\right) \\le \\max \\lbrace ||A||^{2},||B||^{2}\\rbrace w\\left( \\left[\\begin{array}{cc}0 & X \\\\Y & 0\\end{array}\\right] \\right) .$ $(e)\\,\\,w\\left( \\left[\\begin{array}{cc}0 & X \\\\Y & 0\\end{array}\\right] \\right) \\le {\\frac{w(X+Y)+w(X-Y)}{2}}.$ $(f)$ $w\\left( \\left[\\begin{array}{cc}0 & X \\\\e^{i\\theta }X & 0\\end{array}\\right] \\right) =w(X)$ for $\\theta \\in \\mathbb {R}$ .",
"Since all parts, except part (d), have bee shown in [10], [11], we prove only part (d).",
"If we take $C=\\left[\\begin{array}{cc}A & 0 \\\\0 & B\\end{array}\\right] $ and $S=\\left[\\begin{array}{cc}0 & X \\\\Y & 0\\end{array}\\right] $ , then $CSC^{\\ast }=\\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BYA^{\\ast } & 0\\end{array}\\right] $ .",
"Now, using part (a), we have $w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BYA^{\\ast } & 0\\end{array}\\right] \\right) & =w(CSC^{\\ast }) \\\\& \\le \\Vert C\\Vert ^{2}w(S) \\\\& =\\max \\lbrace ||A||^{2},||B||^{2}\\rbrace w\\left( \\left[\\begin{array}{cc}0 & X \\\\Y & 0\\end{array}\\right] \\right) \\text{,}$ as required.",
"Now, we are in position to demonstrate the main results of this section by using some ideas from [13], [14].",
"Theorem 2.2 Let $A\\in {\\mathbb {B}}({H})$ be a $G_{1}$ operator with $\\sigma (A)\\subset \\mathbb {D}$ and $f\\in \\mathfrak {A}$ .",
"Then for every $X\\in {\\mathbb {B}}({H})$ , we have $w(f(A)X+X\\bar{f}(A))\\le {\\frac{2}{d_{A}^{2}}}w(X-AXA^{\\ast })$ and $w(f(A)X-X\\bar{f}(A))\\le {\\frac{4}{d_{A}^{2}}}\\Vert A\\Vert w(X).$ Using the Herglotz representation theorem (see e.g., [7]), we have $f(z)=\\int _{0}^{2\\pi }{\\frac{e^{i\\alpha }+z}{e^{i\\alpha }-z}}d\\mu (\\alpha )+i\\text{Im}\\,f(0)=\\int _{0}^{2\\pi }{\\frac{e^{i\\alpha }+z}{e^{i\\alpha }-z}}d\\mu (\\alpha ),$ where $\\mu $ is a positive Borel measure on the interval $[0,2\\pi ]$ with finite total mass $\\int _{0}^{2\\pi }d\\mu (\\alpha )=f(0)=1$ .",
"Hence, $\\bar{f}({z})=\\overline{{\\int _{0}^{2\\pi }{\\frac{e^{i\\alpha }+{z}}{e^{i\\alpha }-{z}}}d\\mu (\\alpha )}}=\\int _{0}^{2\\pi }{\\frac{e^{-i\\alpha }+\\bar{z}}{e^{-i\\alpha }-\\bar{z}}}d\\mu (\\alpha ),$ where $\\bar{f}$ is the conjugate function of $f$ .",
"So, $f(A)X+X\\bar{f}(A)& =\\int _{0}^{2\\pi }\\left[ \\left( e^{i\\alpha }+A\\right)\\left( e^{i\\alpha }-A\\right) ^{-1}X+X\\left( e^{-i\\alpha }+A^{\\ast }\\right)\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\right] d\\mu (\\alpha ) \\\\& =\\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}\\Big [\\left( e^{i\\alpha }+A\\right) X\\left( e^{-i\\alpha }-A^{\\ast }\\right) \\\\& \\qquad \\quad +\\left( e^{i\\alpha }-A\\right) X\\left( e^{-i\\alpha }+A^{\\ast }\\right) \\Big ]\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha ) \\\\& =2\\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}(X-AXA^{\\ast })\\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha ).$ Hence, $w(f(A)X& +X\\bar{f}(A)) \\\\& =w\\left( \\int _{0}^{2\\pi }\\left[ \\left( e^{i\\alpha }+A\\right) \\left(e^{i\\alpha }-A\\right) ^{-1}X+X\\left( e^{-i\\alpha }+A^{\\ast }\\right) \\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\right] d\\mu (\\alpha )\\right) \\\\& =2\\,w\\left( \\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}(X-AXA^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha )\\right) \\\\& \\le 2\\int _{0}^{2\\pi }w\\left( \\left( e^{i\\alpha }-A\\right)^{-1}(X-AXA^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\right) d\\mu (\\alpha ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{since}\\,w(\\,\\cdot \\,)\\,\\text{is a norm}) \\\\& \\le 2\\int _{0}^{2\\pi }\\Vert \\left( e^{i\\alpha }-A\\right) ^{-1}\\Vert ^{2}w\\left( X-AXA^{\\ast }\\right) d\\mu (\\alpha ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma}\\,\\ref {kho}(a)).$ Since $A$ is a $G_{1}$ operator, it follows that $\\left\\Vert \\left( e^{i\\alpha }-A\\right) ^{-1}\\right\\Vert ={\\frac{1}{\\text{dist}(e^{i\\alpha },\\sigma (A))}}\\le {\\frac{1}{\\text{dist}(\\partial \\mathbb {D},\\sigma (A))}}={\\frac{1}{d_{A}},}$ and so $w\\left( f(A)X+X\\bar{f}(A)\\right) & \\le \\left( {\\frac{2}{d_{A}^{2}}}\\int _{0}^{2\\pi }d\\mu (\\alpha )\\right) w(X-AXA^{\\ast }) \\\\& =\\left( {\\frac{2}{d_{A}^{2}}}f(0)\\right) w(X-AXA^{\\ast }) \\\\& ={\\frac{2}{d_{A}^{2}}}w(X-AXA^{\\ast }).$ This proves the first inequality.",
"Similarly, it follows from the equations $f(A)X-& X\\bar{f}(A) \\\\& =\\int _{0}^{2\\pi }\\left[ \\left( e^{i\\alpha }+A\\right) \\left( e^{i\\alpha }-A\\right) ^{-1}X-X\\left( e^{-i\\alpha }+A^{\\ast }\\right) \\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\right] d\\mu (\\alpha ) \\\\& =\\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}\\Big [\\left( e^{i\\alpha }+A\\right) X\\left( e^{-i\\alpha }-A^{\\ast }\\right) \\\\& \\qquad \\quad -\\left( e^{i\\alpha }-A\\right) X\\left( e^{-i\\alpha }+A^{\\ast }\\right) \\Big ]\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha ) \\\\& =2\\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}(e^{-i\\alpha }AX-e^{i\\alpha }XA^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha ) \\\\& =2\\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}\\left( \\left(e^{-i\\alpha }A\\right) X-X\\left( e^{-i\\alpha }A\\right) ^{\\ast }\\right) \\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha )$ that $w(& f(A)X-X\\bar{f}(A)) \\\\& =2w\\left( \\int _{0}^{2\\pi }\\left( e^{i\\alpha }-A\\right) ^{-1}\\left( \\left(e^{-i\\alpha }A\\right) X-X\\left( e^{-i\\alpha }A\\right) ^{\\ast }\\right) \\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}d\\mu (\\alpha )\\right) \\\\& \\le 2\\int _{0}^{2\\pi }w\\left( \\left( e^{i\\alpha }-A\\right) ^{-1}\\left(\\left( e^{-i\\alpha }A\\right) X-X\\left( e^{-i\\alpha }A\\right) ^{\\ast }\\right)\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\right) d\\mu (\\alpha ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{since}\\,w(\\,\\cdot \\,)\\,\\text{is a norm}) \\\\& \\le 2\\int _{0}^{2\\pi }\\left\\Vert \\left( e^{i\\alpha }-A\\right)^{-1}\\right\\Vert ^{2}w\\left( \\left( e^{-i\\alpha }A\\right) X-X\\left(e^{-i\\alpha }A\\right) ^{\\ast }\\right) d\\mu (\\alpha ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma\\,\\ref {kho} (a)}) \\\\& \\le 4\\int _{0}^{2\\pi }\\left\\Vert \\left( e^{i\\alpha }-A\\right)^{-1}\\right\\Vert ^{2}\\Vert e^{-i\\alpha }A\\Vert w(X)d\\mu (\\alpha ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma\\,\\ref {kho} (b)}) \\\\& \\le \\frac{4}{d_{A}^{2}}\\Vert A\\Vert w(X)\\int _{0}^{2\\pi }d\\mu (\\alpha ) \\\\& \\le \\frac{4}{d_{A}^{2}}\\Vert A\\Vert w(X).$ This proves the second inequality and completes the proof of the theorem.",
"If we take $X=I$ in Theorem REF , we get the following result.",
"Observe that $\\bar{f}(A)=\\left( f(A)\\right) ^{\\ast }$ .",
"Corollary 2.3 Let $A\\in {\\mathbb {B}}({H})$ be a $G_1$ operator with $\\sigma (A)\\subset \\mathbb {D}$ and $f\\in \\mathfrak {A}$ .",
"Then $\\Vert \\text{Re}(f(A))\\Vert \\le {\\frac{1}{d^2_A}}\\Vert I-AA^*\\Vert $ and $\\Vert \\text{Im} (f(A))\\Vert \\le {\\frac{2}{d^2_A}}\\Vert A\\Vert .$ Theorem 2.4 Let $A,B\\in {\\mathbb {B}}({H})$ be $G_{1}$ operators with $\\sigma (A)\\cup \\sigma (B)\\subset \\mathbb {D}$ and $f\\in \\mathfrak {A}$ .",
"Then for every $X\\in {\\mathbb {B}}({H})$ , we have $w(f(A)X\\bar{f}(B)& -f(B)X\\bar{f}(A)) \\\\& \\le {\\frac{2}{d_{A}d_{B}}}\\,\\left[ 2w\\left( X\\right) +w\\left( AXB^{\\ast }+BXA^{\\ast }\\right) +w\\left( AXB^{\\ast }-BXA^{\\ast }\\right) \\right]$ and $w(f(A)X\\bar{f}(B)& +2X+f(B)X\\bar{f}(A)) \\\\& \\le {\\frac{2}{d_{A}d_{B}}}\\,\\left[ 2w\\left( X\\right) +w\\left( AXB^{\\ast }+BXA^{\\ast }\\right) +w\\left( AXB^{\\ast }-BXA^{\\ast }\\right) \\right] .$ We have $f(A)X\\bar{f}(B)& -f(B)X\\bar{f}(A) \\\\& =\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\Big [\\left( e^{i\\alpha }-A\\right)^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1} \\\\& \\,\\,\\,\\,-\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\Big ]d\\mu (\\alpha )d\\mu (\\beta ).$ Using the equations $& \\left( e^{i\\alpha }-A\\right) ^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1} \\\\& \\,\\,\\,\\,-\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =\\left( e^{i\\alpha }-A\\right) ^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1}+X \\\\& \\,\\,\\,\\,-X-\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\beta }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =\\left( e^{i\\alpha }-A\\right) ^{-1}\\left[ (e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })+(e^{i\\alpha }-A)X(e^{-i\\beta }-B^{\\ast })\\right] \\left(e^{-i\\beta }-B^{\\ast }\\right) ^{-1} \\\\& \\,\\,-\\left( e^{i\\beta }-B\\right) ^{-1}\\left[ (e^{i\\beta }-B)X(e^{-i\\alpha }-A^{\\ast })+(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\right] \\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =2(e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1} \\\\& \\,\\,-2(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1},$ we have $w(f& (A)X\\bar{f}(B)-f(B)X\\bar{f}(A)) \\\\& =2w\\Big (\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }(e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1} \\\\& \\qquad -(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1}d\\mu (\\alpha )d\\mu (\\beta )\\Big ) \\\\& \\le 2\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }w\\Big ((e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1}\\\\& \\qquad -(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1}\\Big )d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{since}\\,w(\\,\\cdot \\,)\\,\\text{is a norm})$ $& \\le 4\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\Vert (e^{i\\alpha }-A)^{-1}\\Vert \\Vert (e^{i\\beta }-B)^{-1}\\Vert \\\\& \\qquad \\times w\\left( \\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast } \\\\e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast } & 0\\end{array}\\right] \\right) d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma \\,\\ref {kho} (c)}) \\\\& \\le {\\frac{4}{d_{A}d_{B}}}\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\left[ w\\left( \\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X \\\\e^{-i\\alpha }e^{i\\beta }X & 0\\end{array}\\right] \\right) \\right.",
"\\\\& \\qquad \\left.",
"+w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BXA^{\\ast } & 0\\end{array}\\right] \\right) \\right] d\\mu (\\alpha )d\\mu (\\beta ) \\\\& ={\\frac{4}{d_{A}d_{B}}}\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\left[ w\\left( \\left[\\begin{array}{cc}0 & X \\\\X & 0\\end{array}\\right] \\right) +w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BXA^{\\ast } & 0\\end{array}\\right] \\right) \\right] d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\le {\\frac{2}{d_{A}d_{B}}}\\left[ 2w\\left( X\\right) +w\\left( AXB^{\\ast }+BXA^{\\ast }\\right) +w\\left( AXB^{\\ast }-BXA^{\\ast }\\right) \\right] \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma \\,\\ref {kho}(e) and (f)}).$ This proves the first inequality.",
"Similarly, we have $f(A)X\\bar{f}(B)& +2X+f(B)X\\bar{f}(A) \\\\& =\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\Big [\\left( e^{i\\alpha }-A\\right)^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1}+2X \\\\& \\,\\,\\,\\,+\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1}\\Big ]d\\mu (\\alpha )d\\mu (\\beta ).$ Using the equations $& \\left( e^{i\\alpha }-A\\right) ^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1}+2X \\\\& \\,\\,\\,\\,+\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =\\left( e^{i\\alpha }-A\\right) ^{-1}(e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })\\left( e^{-i\\beta }-B^{\\ast }\\right) ^{-1}+X \\\\& \\,\\,\\,\\,+X+\\left( e^{i\\beta }-B\\right) ^{-1}(e^{i\\beta }+B)X(e^{-i\\beta }+A^{\\ast })\\left( e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =\\left( e^{i\\alpha }-A\\right) ^{-1}\\left[ (e^{i\\alpha }+A)X(e^{-i\\beta }+B^{\\ast })+(e^{i\\alpha }-A)X(e^{-i\\beta }-B^{\\ast })\\right] \\left(e^{-i\\beta }-B^{\\ast }\\right) ^{-1} \\\\& \\,\\,+\\left( e^{i\\beta }-B\\right) ^{-1}\\left[ (e^{i\\beta }-B)X(e^{-i\\alpha }-A^{\\ast })+(e^{i\\beta }+B)X(e^{-i\\alpha }+A^{\\ast })\\right] \\left(e^{-i\\alpha }-A^{\\ast }\\right) ^{-1} \\\\& =2(e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1} \\\\& \\,\\,+2(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1},$ we have $w(f& (A)X\\bar{f}(B)+2X+f(B)X\\bar{f}(A)) \\\\& =2w\\Big (\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }(e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1} \\\\& \\qquad +(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1}d\\mu (\\alpha )d\\mu (\\beta )\\Big ) \\\\& \\le 2\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }w\\Big ((e^{i\\alpha }-A)^{-1}(e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast })(e^{-i\\beta }-B^{\\ast })^{-1}\\\\& \\qquad +(e^{i\\beta }-B)^{-1}(e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast })(e^{-i\\alpha }-A^{\\ast })^{-1}\\Big )d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{since}\\,w(\\,\\cdot \\,)\\,\\text{is a norm}) \\\\& \\le 4\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\Vert (e^{i\\alpha }-A)^{-1}\\Vert \\Vert (e^{i\\beta }-B)^{-1}\\Vert \\\\& \\qquad \\times w\\left( \\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast } \\\\e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast } & 0\\end{array}\\right] \\right) d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma \\,\\ref {kho} (c)}) \\\\& \\le {\\frac{4}{d_{A}d_{B}}}\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\left[ w\\left( \\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X \\\\e^{-i\\alpha }e^{i\\beta }X & 0\\end{array}\\right] \\right) \\right.",
"\\\\& \\qquad \\left.",
"+w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BXA^{\\ast } & 0\\end{array}\\right] \\right) \\right] d\\mu (\\alpha )d\\mu (\\beta ) \\\\& ={\\frac{4}{d_{A}d_{B}}}\\int _{0}^{2\\pi }\\int _{0}^{2\\pi }\\left[ w\\left( \\left[\\begin{array}{cc}0 & X \\\\X & 0\\end{array}\\right] \\right) +w\\left( \\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BXA^{\\ast } & 0\\end{array}\\right] \\right) \\right] d\\mu (\\alpha )d\\mu (\\beta ) \\\\& \\le {\\frac{2}{d_{A}d_{B}}}\\left[ 2w\\left( X\\right) +w\\left( AXB^{\\ast }+BXA^{\\ast }\\right) +w\\left( AXB^{\\ast }-BXA^{\\ast }\\right) \\right] \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad (\\text{by Lemma \\,\\ref {kho}(e) and (f)}).$ This proves the second inequality and completes the proof of the theorem.",
"Remark 2.5 Under the assumptions of Theorem REF and the hypothesis that $X$ is self-adjoint, we have $\\Vert f(A)X\\bar{f}(B)& -f(B)X\\bar{f}(A)\\Vert \\\\& \\le {\\frac{4}{d_{A}d_{B}}}\\,\\max \\lbrace \\Vert \\,|X|\\,\\Vert +\\Vert \\, |AXB^{\\ast }|\\,\\Vert ,\\Vert \\,|X|\\,\\Vert +\\Vert \\,|BXA^{\\ast }|\\,\\Vert \\rbrace $ and $\\Vert f(A)X\\bar{f}(B)& +2X+f(B)X\\bar{f}(A)\\Vert \\\\& \\le {\\frac{4}{d_{A}d_{B}}}\\,\\max \\lbrace \\Vert \\,|X|\\,\\Vert +\\Vert \\,|AXB^{\\ast }|\\,\\Vert ,\\Vert \\, |X|\\,\\Vert +\\Vert \\,|BXA^{\\ast }|\\,\\Vert \\rbrace .$ To see this, first note that if $X$ is self-adjoint, then the operator matrix $T=\\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X+AXB^{\\ast } \\\\e^{-i\\alpha }e^{i\\beta }X+BXA^{\\ast } & 0\\end{array}\\right]$ is self-adjoint, whence $w(T)=\\Vert T\\Vert $ .",
"Moreover, $T=M+N$ , where $M=\\left[\\begin{array}{cc}0 & e^{i\\alpha }e^{-i\\beta }X \\\\e^{-i\\alpha }e^{i\\beta }X & 0\\end{array}\\right] ,\\qquad N=\\left[\\begin{array}{cc}0 & AXB^{\\ast } \\\\BXA^{\\ast } & 0\\end{array}\\right]$ are self-adjoint operators.",
"Using the fact that $\\Vert C+D\\Vert \\le \\Vert \\,|C|+|D|\\,\\Vert $ for any normal operators $C$ and $D$ (see [6]), we have $w(T)=\\Vert M+N\\Vert \\le \\Vert \\,|M|+|N|\\,\\Vert =\\max \\lbrace \\Vert \\,|X|\\,\\Vert +\\Vert \\,|AXB^{\\ast }|\\,\\Vert ,\\Vert \\,|X|\\,\\Vert +\\Vert \\,|BXA^{\\ast }|\\,\\Vert \\rbrace .$ Hence, we get the required inequalities by the same arguments as in the proof of Theorem REF .",
"If we take $X=I$ in Theorem REF , we get the following result.",
"Corollary 2.6 Let $A, B\\in {\\mathbb {B}}({H})$ be $G_1$ operators with $\\sigma (A)\\cup \\sigma (B)\\subset \\mathbb {D}$ and $f\\in \\mathfrak {A}$ .",
"Then $\\Vert \\text{Im}(f(A)\\bar{f}(B))\\Vert \\le {\\frac{2}{d_Ad_B}}\\,\\left(1+\\Vert AB^*\\Vert \\right)$ and $\\Vert \\text{Re}(f(A)\\bar{f}(B))+I\\Vert \\le {\\frac{2}{d_Ad_B}}\\,\\left(1+\\Vert AB^*\\Vert \\right).$ Remark 2.7 If instead of applying Lemma REF (c) we use Lemma REF (d) and (f) in the proof Theorem REF , we obtain the related inequalities $w(f(A)X\\bar{f}(B)-f(B)X\\bar{f}(A))\\le {\\frac{4}{d_{A}d_{B}}}\\,\\left[ 1+\\max \\lbrace \\Vert A\\Vert ^{2},\\Vert B\\Vert ^{2}\\rbrace \\right] w\\left( X\\right)$ and $w(f(A)X\\bar{f}(B)+2X+f(B)X\\bar{f}(A))\\le {\\frac{4}{d_{A}d_{B}}}\\,\\left[1+\\max \\lbrace \\Vert A\\Vert ^{2},\\Vert B\\Vert ^{2}\\rbrace \\right] w\\left( X\\right) .$ Acknowledgement.",
"The first author would like to thank the Tusi Mathematical Research Group (TMRG)."
]
] | 1709.01850 |
[
[
"Godeaux-Serre Varieties with Prescribed Arithmetic Fundamental Group"
],
[
"Abstract We show that for any given field $k$ and natural number $r\\geq2$, every continuous extension of the absolute Galois group $\\mathrm{Gal}_k$ by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective variety over $k$ of dimension $r$."
],
[
"Introduction",
"The difficult question which groups can occur as fundamental groups of smooth projective varieties over an algebraically closed field is still an open question.",
"As listed in [1], there are several classes of groups for which this question is answered positively, but also many that yield negative results.",
"For example, every finite group occurs as such a fundamental group.",
"In fact, Serre constructed in [6] a smooth projective variety which is a complete intersection of dimension at least 2 and on which a given finite group acts without fixed points.",
"Hence the quotient is a smooth projective variety with the given finite group as fundamental group, the Godeaux-Serre variety.",
"We are interested in the following arithmetic situation: Let $k$ be an arbitrary field.",
"It is known that for an arbitrary geometrically connected scheme $X$ of finite type over $k$ , we have the exact sequence $ 1 \\longrightarrow \\pi _1(X\\otimes _k{k_a}) \\longrightarrow \\pi _1(X) \\longrightarrow \\operatorname{Gal}_k \\longrightarrow 1,\\qquad \\mathrm {(\\ast )}$ where ${k_a}$ denotes an algebraic closure of $k$ , cf. [7].",
"Hence one might ask which continuous extensions of the absolute Galois group $\\operatorname{Gal}_k$ occur as arithmetic fundamental groups of smooth projective geometrically connected varieties over $k$ .",
"This question is even more difficult than the question for varieties over algebraically closed fields.",
"Here we restrict our attention to extensions of $\\operatorname{Gal}_k$ by a finite group, i.e.",
"the case $\\pi _1(X\\otimes _k{k_a})$ is finite.",
"Our main result can be formulated as follows: Theorem A (see Theorem REF ) Let $k$ be a field, $G$ a finite group, $r\\in {\\mathbb {N}}$ with $r\\ge 2$ and $1 \\longrightarrow G \\longrightarrow E \\longrightarrow \\operatorname{Gal}_k \\longrightarrow 1$ a continuous extension of profinite groups.",
"There exists a geometrically connected smooth projective variety $X$ over $k$ of dimension $r$ such that the sequence (REF ) is isomorphic to the given extension.",
"Also note that Harari and Stix constructed in [4] an example of a real projective variety $X$ with $\\pi _1(X)\\cong {\\mathbb {Z}}/4{\\mathbb {Z}}$ as special case of Theorem REF .",
"This provides an example of a real Godeaux-Serre variety without real points since ${\\mathbb {Z}}/4{\\mathbb {Z}}$ as extension of $\\operatorname{Gal}_{\\mathbb {R}}\\cong {\\mathbb {Z}}/2{\\mathbb {Z}}$ by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ does not split.",
"The construction in the general context is similar to that of Godeaux-Serre varieties but several modifications are needed: Being an extension of the absolute Galois group, the group $E$ as in Theorem REF is not finite in general.",
"Hence the action on a complete intersection we consider here is not given by $E$ but an appropriate finite quotient ${\\widetilde{E}}$ , compare Lemma REF .",
"Furthermore, as in the example given by Harari and Stix, we consider a complete intersection not over the given field $k$ but an appropriate field extension $k^{\\prime }|k$ .",
"The action of ${\\widetilde{E}}$ on this complete intersection is semilinear, compare §REF .",
"Also note that in the construction of a Godeaux-Serre variety over an algebraically closed field, we need to find a linear subspace of a certain projective space in general position.",
"This is not always possible if the ground field is finite, so that a version of Bertini's theorem for finite fields is needed, see for instance [5].",
"This paper is organized as follows: Section provides basic facts about admissible group actions on schemes in the sense of [7].",
"In particular, we are interested in those without fixed points as well as semilinear actions in connection with the étale fundamental groups.",
"The construction of our varieties is given in Section .",
"Here we consider an extension of a finite Galois group instead of the given extension of the absolute Galois group, which is possible due to Lemma REF .",
"Based on this extension, we construct a $k$ -form of a Godeaux-Serre variety as done in §REF , and derive the main result in §REF ."
],
[
"Throughout this paper, $\\Omega $ will always denote an algebraically closed field.",
"By a group extension of a group $G$ by a group $H$ , we mean a group $E$ fitting into an exact sequence $1 \\longrightarrow H \\longrightarrow E \\longrightarrow G \\longrightarrow 1.$ A group action on a scheme will always be from the right.",
"Hence an action of a group $G$ on a scheme $X$ is induced by a group homomorphism $G^{\\rm op}\\rightarrow \\operatorname{Aut}(X)$ .",
"The automorphism on $X$ induced by an element $g\\in G$ in this way will be denoted by $\\rho _g$ .",
"The category of finite sets will be denoted by ${\\sf sets}$ , and the one of finite étale coverings of a connected scheme $X$ by $\\text{\\sf fét}_X$ .",
"The fiber functor at the geometric point ${\\overline{x}}\\in X(\\Omega )$ is given by $F_{\\overline{x}}:\\text{\\sf fét}_X\\longrightarrow {\\sf sets}, \\quad Y\\longmapsto F_{\\overline{x}}(Y) := \\operatorname{Hom}_X(\\operatorname{Spec}\\Omega ,Y).$ For a morphism of connected schemes $\\phi :Y\\rightarrow Z$ and geometric point ${\\overline{y}}\\in Y(\\Omega )$ , the induced homomorphism between the fundamental groups will be denoted by $\\phi _\\ast :\\pi _1(Y,{\\overline{y}})\\rightarrow \\pi _1(Z,\\phi ({\\overline{y}}))$ .",
"A fixed algebraic closure of a field $k$ will be denoted by ${k_a}$ and the separable closure inside ${k_a}$ by $k_s$ .",
"The Galois group of a Galois extension $k^{\\prime }|k$ will be denoted by $\\operatorname{Gal}(k^{\\prime }|k)$ and the absolute Galois group of $k$ by $\\operatorname{Gal}_k:=\\operatorname{Gal}(k_s|k)$ .",
"Finally, if $X$ is a scheme over $k$ and $A$ is a $k$ -algebra, the fiber product $X\\times _{\\operatorname{Spec}k}\\operatorname{Spec}A$ will be denoted by $X\\otimes _kA$ or $X_A$ if the ground field $k$ is clear from the context.",
"The base change of a morphism $\\phi :X\\rightarrow Y$ between $k$ -schemes will be denoted by $\\phi _A:X_A\\rightarrow Y_A$ ."
],
[
"Admissible group actions",
"In what follows, let $G$ be a finite group and $X$ be a scheme of finite type over a fixed locally noetherian base scheme $S$ .",
"Recall that a group action of $G$ on $X$ is admissible if the categorical quotient $X/G$ exists and the quotient morphism $X\\rightarrow X/G$ is affine.",
"We are particularly interested in an admissible group action without fixed points, i.e.",
"an action of $G$ on $X$ such that ${\\overline{x}}g\\ne {\\overline{x}}$ for all $g\\in G\\setminus \\lbrace 1\\rbrace $ and geometric points ${\\overline{x}}\\in X(\\Omega )$ .",
"Proposition 1.1 Suppose that $X$ ist connected and $G$ acts on $X$ as an $S$ -scheme admissibly without fixed points.",
"Then the following holds: The quotient morphism $p:X\\rightarrow X/G$ is a finite étale Galois covering.",
"Let ${\\overline{x}}\\in X(\\Omega )$ and ${\\overline{z}}$ be its image on $X/G$ under $p$ .",
"Then the mapping $\\Phi =\\Phi _{G,{\\overline{x}}}:\\pi _1(X/G,{\\overline{z}})\\longrightarrow G, \\quad \\alpha \\longmapsto g_\\alpha \\quad \\text{if} \\quad \\alpha _X({\\overline{x}})={\\overline{x}}g_\\alpha \\in F_{\\overline{z}}(X)$ is a well-defined continuous surjective group homomorphism.",
"We have the following exact sequence: $1\\longrightarrow \\pi _1(X,{\\overline{x}})\\longrightarrow \\pi _1(X/G,{\\overline{z}}) \\stackrel{\\Phi }{\\longrightarrow } G\\longrightarrow 1.$ Observe that $p:X\\rightarrow X/G$ is finite and $X/G$ is of finite type over $S$ by [7].",
"By [7], $p$ is étale and $G$ is canonically isomorphic to $\\operatorname{Aut}(X|(X/G))^{\\rm op}$ .",
"Furthermore, [2] shows that $G$ acts on $F_{\\overline{z}}(X)$ transitively.",
"Hence $p$ is a Galois covering, which proves (1).",
"Assertion (2) holds since the projection $\\pi _1(X/G,{\\overline{z}})\\rightarrow \\operatorname{Aut}(X|(X/G))^{\\rm op}$ given by the geometric point ${\\overline{x}}\\in X(\\Omega )$ is a continuous surjective group homomorphism.",
"We now come to (3).",
"By [7], the map $\\pi _1(X,{\\overline{x}})\\rightarrow \\pi _1(X/G,{\\overline{z}})$ is injective and its image is the subgroup of those $\\alpha \\in \\pi _1(X/G,{\\overline{z}})$ such that $\\alpha _X({\\overline{x}})={\\overline{x}}$ , i.e.",
"exactly the kernel of $\\Phi $ .",
"Hence the whole sequence is exact.",
"The next proposition is about the functoriality between schemes with admissible group actions without fixed points and the group homomorphism $\\Phi _{G,{\\overline{x}}}$ from Proposition REF .",
"Proposition 1.2 Let $G$ , $H$ be finite groups acting admissibly without fixed points on connected $S$ -schemes $Y$ , $Z$ of finite type with quotient maps $p_G:Y\\rightarrow Y/G$ and $p_H:Z\\rightarrow Z/H$ respectively.",
"Let $f:G\\rightarrow H$ be a group homomorphism, $\\phi :Y\\rightarrow Z$ an $f$ -equivariant morphism, i.e.",
"$\\phi \\circ \\rho _g = \\rho _{f(g)}\\circ \\phi \\quad \\text{for all} \\quad g\\in G,$ and $\\bar{\\phi }:Y/G\\rightarrow Z/H$ the morphism induced by $\\phi $ .",
"Then for ${\\overline{y}}\\in Y(\\Omega )$ , ${\\overline{z}}:=\\phi ({\\overline{y}})\\in Z(\\Omega )$ , ${\\overline{y}}^{\\prime }:=p_G({\\overline{y}})\\in (Y/G)(\\Omega )$ and ${\\overline{z}}^{\\prime }:=p_H({\\overline{z}})\\in (Z/H)(\\Omega )$ , the following diagram is commutative: [description/.style=fill=white,inner sep=2pt] (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] 1(Y/G,y') 1(Z/H,z') G H. ; [->,font=] (m-1-1) edge node[auto] $ \\overline{\\phi }_{\\ast } $ (m-1-2) edge node[auto] $\\Phi _{G,{\\overline{y}}}$ (m-2-1) (m-1-2) edge node[auto] $\\Phi _{H,{\\overline{z}}}$ (m-2-2) (m-2-1) edge node[auto] $f$ (m-2-2); Let $\\alpha \\in \\pi _1(Y/G,{\\overline{y}}^{\\prime })$ and $g:=\\Phi _{G,{\\overline{y}}}(\\alpha )$ , i.e.",
"$\\alpha _Y({\\overline{y}}) = \\rho _g({\\overline{y}})$ .",
"Let $Y_0:=Z\\times _{Z/H}(Y/G)$ with canonical projections $\\operatorname{pr}_1:Y_0\\rightarrow Z$ and $\\operatorname{pr}_2:Y_0\\rightarrow Y/G$ .",
"Note that $\\operatorname{pr}_2\\in \\text{\\sf fét}_{Y/G}$ .",
"Furthermore, let $\\phi _0:=(\\phi ,p_G):Y\\rightarrow Y_0$ and ${\\overline{y}}_0:=\\phi _0({\\overline{y}})$ .",
"Since $\\operatorname{pr}_1({\\overline{y}}_0) = \\phi ({\\overline{y}}) = {\\overline{z}}$ , we have $(\\phi _\\ast \\alpha )_Z({\\overline{z}}) = \\operatorname{pr}_1(\\alpha _{Y_0}({\\overline{y}}_0)) = \\operatorname{pr}_1(\\phi _0(\\alpha _Y({\\overline{y}}))) = \\phi (\\alpha _Y({\\overline{y}})) = \\phi (\\rho _g({\\overline{y}})) = \\rho _{f(g)}(\\phi ({\\overline{y}})) = \\rho _{f(g)}({\\overline{z}}).$ This implies that $\\Phi _{H,{\\overline{z}}}(\\phi _\\ast \\alpha ) = f(g) = f(\\Phi _{G,{\\overline{y}}}(\\alpha ))$ .",
"Hence $\\Phi _{H,{\\overline{z}}}\\circ \\phi _\\ast = f\\circ \\Phi _{G,{\\overline{y}}}$ as desired."
],
[
"Semilinear actions",
"Given a Galois field extension $k^{\\prime }|k$ , we are interested in group actions on schemes over $k^{\\prime }$ given by automorphisms which may not be defined over $k^{\\prime }$ , but are in some sense compatible with $k$ -automorphisms of $k^{\\prime }$ .",
"This leads to the notion of a semilinear action.",
"Definition 1.3 Let $k^{\\prime }|k$ be a Galois field extension, $Y$ a scheme over $k^{\\prime }$ and $\\pi :E\\rightarrow \\operatorname{Gal}(k^{\\prime }|k)$ a group homomorphism.",
"A group action of $E$ on $Y$ is said to be semilinear with respect to $\\pi $ or $\\pi $ -semilinear if for each $g\\in E$ , the diagram $\\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}](m) [matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex]{ Y & Y \\\\ \\operatorname{Spec}{k^{\\prime }} & \\operatorname{Spec}{k^{\\prime }} \\\\ };[->,font=\\scriptsize ] (m-1-1) edge node[auto] { \\rho _g } (m-1-2) edge (m-2-1) (m-1-2) edge (m-2-2) (m-2-1) edge node[auto] { \\pi (g)^\\ast } (m-2-2);\\end{tikzpicture}$ is commutative, where $\\pi (g)^\\ast :\\operatorname{Spec}{k^{\\prime }}\\rightarrow \\operatorname{Spec}{k^{\\prime }}$ denotes the morphism induced by $\\pi (g)$ .",
"Remark 1.4 It is easy to check that in the situation of Definition REF , a geometric point on $Y$ can be fixed by $g\\in E$ only if $g\\in \\ker \\pi $ .",
"Proposition 1.5 Let $k^{\\prime }|k$ be a finite Galois extension and $1 \\longrightarrow G \\longrightarrow E \\xrightarrow{} \\operatorname{Gal}(k^{\\prime }|k) \\longrightarrow 1$ an exact sequence of finite groups.",
"Furthermore, let $\\psi :Y\\rightarrow \\operatorname{Spec}k^{\\prime }$ be a connected scheme of finite type over $k^{\\prime }$ with an admissible $\\pi $ -semilinear action of $E$ and quotient $X:=Y/E$ .",
"Then the morphisms $Y/G\\rightarrow X$ and $Y/G\\rightarrow \\operatorname{Spec}{k^{\\prime }}$ induce a $\\operatorname{Gal}(k^{\\prime }|k)$ -equivariant canonical isomorphism $Y/G\\cong X\\otimes _kk^{\\prime }.$ Observe first that both $Y/G\\rightarrow X$ and $X\\otimes _kk^{\\prime }\\rightarrow X$ are finite étale coverings.",
"Indeed, $Y/G$ is of finite type over $k$ by [7].",
"Since $E/G\\cong \\operatorname{Gal}(k^{\\prime }|k)$ acts on $Y/G$ without fixed points, the quotient morphism $Y/G\\rightarrow (Y/G)/\\operatorname{Gal}(k^{\\prime }|k)=X$ is a finite étale covering by Proposition REF .",
"On the other hand, $X\\otimes _kk^{\\prime }\\rightarrow X$ is obtained by base change from $\\operatorname{Spec}{k^{\\prime }}\\rightarrow \\operatorname{Spec}{k}$ , hence also a finite étale covering.",
"It is easily seen that the morphism $\\psi ^{\\prime }:Y\\rightarrow X\\otimes _kk^{\\prime }$ obtained by the morphisms $Y/G\\rightarrow X$ and $Y/G\\rightarrow \\operatorname{Spec}{k^{\\prime }}$ is $\\operatorname{Gal}(k^{\\prime }|k)$ -equivariant.",
"Now fix a geometric point ${\\overline{x}}\\in X(\\Omega )$ with fiber functor $F_{\\overline{x}}$ .",
"Since $F_{\\overline{x}}(\\psi ):F_{\\overline{x}}(Y/G)\\rightarrow F_{\\overline{x}}(X\\otimes _kk^{\\prime })$ is also $\\operatorname{Gal}(k^{\\prime }|k)$ -equivariant and $\\operatorname{Gal}(k^{\\prime }|k)$ acts on the fiber $F_{\\overline{x}}(X\\otimes _kk^{\\prime })$ transitively, $F_{\\overline{x}}(\\psi )$ is surjective.",
"Furthermore, the same argument as in Proposition REF shows that both fibers $F_{\\overline{x}}(Y/G)$ and $F_{\\overline{x}}(X\\otimes _kk^{\\prime })$ have the same cardinality as $\\operatorname{Gal}(k^{\\prime }|k)$ .",
"Hence $F_{\\overline{x}}(\\psi )$ is bijective.",
"But $F_{\\overline{x}}$ is a fiber functor of the Galois category $\\text{\\sf fét}_X$ .",
"Therefore, $\\psi :Y/G\\rightarrow X\\otimes _kk^{\\prime }$ is an isomorphism as desired.",
"Proposition 1.6 Let $k^{\\prime }|k$ , $G$ , $E$ , $\\pi $ , $\\psi :Y\\rightarrow \\operatorname{Spec}k^{\\prime }$ be as in Proposition REF and $X:=Y/E$ with structure morphism $\\phi :X\\rightarrow \\operatorname{Spec}{k}$ .",
"Suppose that $E$ acts on $Y$ without fixed points.",
"Fix a geometric point ${\\overline{y}}\\in Y(\\Omega )$ with its image ${\\overline{x}}\\in X(\\Omega )$ under the quotient map $Y\\rightarrow X$ .",
"Then the diagram $\\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}](m) [matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex]{ \\pi _1(X,{\\overline{x}}) & \\pi _1(\\operatorname{Spec}{k},\\phi ({\\overline{x}})) \\\\ E & \\operatorname{Gal}(k^{\\prime }|k) \\\\ };[->,font=\\scriptsize ] (m-1-1) edge node[auto] { \\phi _\\ast } (m-1-2) edge node[auto] { \\Phi } (m-2-1) (m-1-2) edge node[auto] { \\Psi } (m-2-2) (m-2-1) edge node[auto] {\\pi } (m-2-2);\\end{tikzpicture}$ is commutative, where $\\Phi =\\Phi _{E,{\\overline{y}}}:\\pi _1(X,{\\overline{x}})\\rightarrow E$ is the group homomorphism from Proposition REF and $\\Psi $ is defined by the projection $\\pi _1(\\operatorname{Spec}{k},\\phi ({\\overline{x}})) \\rightarrow \\operatorname{Aut}(\\operatorname{Spec}{k^{\\prime }}|\\operatorname{Spec}{k})^{\\rm op}$ given by the geometric point $\\psi ({\\overline{y}})\\in (\\operatorname{Spec}{k^{\\prime }})(\\Omega )$ .",
"This follows from Proposition REF since $\\psi :Y\\rightarrow \\operatorname{Spec}{k^{\\prime }}$ is $\\pi $ -equivariant by the definition of a $\\pi $ -semilinear action."
],
[
"A construction",
"We begin with a construction for a given finite group extension of a finite Galois group.",
"Proposition 2.1 Given $r\\in {\\mathbb {N}}$ with $r\\ge 2$ , a finite Galois extension $k^{\\prime }|k$ in a fixed algebraic closure $\\bar{k}$ and an extension of finite groups $1 \\longrightarrow G \\xrightarrow{} {\\widetilde{E}}\\xrightarrow{} \\operatorname{Gal}(k^{\\prime }|k) \\longrightarrow 1,$ there exists a smooth projective geometrically connected variety over $k^{\\prime }$ of dimension $r$ which is a complete intersection in ${\\mathbb {P}}_{k^{\\prime }}^n$ for some $n\\in {\\mathbb {N}}$ and on which $\\tilde{E}$ acts $\\pi $ -semilinearly and admissibly without fixed points.",
"We proceed in several steps.",
"Step 1: Define a semilinear action of $\\tilde{E}$ on ${\\mathbb {P}}_{k^{\\prime }}^n$ and consider its quotient.",
"Let $\\tau :{\\widetilde{E}}\\rightarrow \\operatorname{GL}_{n+1}(k)$ be a faithful linear representation such that $\\tau (g)$ is not a multiple of the identity matrix for all $g\\in {\\widetilde{E}}\\setminus \\lbrace 1\\rbrace $ (for example, the regular representation).",
"Define the semilinear action of ${\\widetilde{E}}$ on the homogeneous coordinate ring $k^{\\prime }[T_0,\\ldots ,T_n]$ by ${\\widetilde{E}}\\times k^{\\prime }[T_0,\\ldots ,T_n]&\\longrightarrow k^{\\prime }[T_0,\\ldots ,T_n],\\\\(g,f(T_0,\\ldots ,T_n))&\\longmapsto (gf)(T_0,\\ldots ,T_n) := \\pi (g)\\big (f\\big ((T_0,\\ldots ,T_n)\\!\\cdot \\!\\tau (g)\\big )\\big ).$ Since this defines a left action of ${\\widetilde{E}}$ on the graded ring $k^{\\prime }[T_0,\\ldots ,T_n]$ , we obtain the right action of ${\\widetilde{E}}$ on ${\\mathbb {P}}_{k^{\\prime }}^n = \\operatorname{Proj}k^{\\prime }[T_0,\\ldots ,T_n]$ .",
"This action is clearly $\\pi $ -semilinear and admissible with quotient ${\\mathbb {P}}_{k^{\\prime }}^n/{\\widetilde{E}}\\cong \\operatorname{Proj}{A}, \\quad \\text{where} \\quad A := k^{\\prime }[T_0,\\ldots ,T_n]^{\\widetilde{E}}.$ Since $A$ is a finitely generated algebra over $k$ by [2], there exist $d,s\\in {\\mathbb {N}}$ and $f_0,\\ldots ,f_s\\in A_d$ such that $A^{(d)}=k[f_0,\\ldots ,f_s]$ by [2].",
"Hence the quotient ${\\mathbb {P}}_{k}^n/{\\widetilde{E}}$ is a projective variety $Z\\subseteq {\\mathbb {P}}_k^s$ .",
"The quotient map will be denoted by $p:{\\mathbb {P}}_{k^{\\prime }}^n\\rightarrow Z$ .",
"Step 2: The closed subscheme of “bad points” and its complement in ${\\mathbb {P}}_{k^{\\prime }}^n$ .",
"Observe that for each $g\\in G\\setminus \\lbrace 1\\rbrace $ , the difference kernel $Q_g:=\\ker (\\operatorname{id},\\rho _g)$ is a proper closed subscheme of ${\\mathbb {P}}_{k^{\\prime }}^n$ defined over $k$ .",
"The finite union $Q := \\bigcup _{g\\in G\\setminus \\lbrace 1\\rbrace } Q_g \\subset {\\mathbb {P}}_{k^{\\prime }}^n$ is a proper closed subset.",
"Its image $Q_0:=p(Q)$ is a proper closed subset in $Z$ since $p$ is finite.",
"Thus $p^{-1}(Q_0)$ is also a proper closed subset in ${\\mathbb {P}}_{k^{\\prime }}^n$ .",
"Hence $W:={\\mathbb {P}}_{k^{\\prime }}^n\\setminus p^{-1}(Q_0)$ is a dense open subscheme of ${\\mathbb {P}}_{k^{\\prime }}^n$ with an admissible ${\\widetilde{E}}$ -action without fixed points with quotient $Z_0:=Z\\setminus Q_0$ , a dense open subscheme of $Z$ .",
"Furthermore, since $p|_W:W\\rightarrow Z_0$ is a finite étale covering by Proposition REF and $W$ is smooth over $k^{\\prime }$ and thus also over $k$ , $Z_0$ is also smooth over $k$ .",
"Step 3: Using Bertini.",
"Observe that we can assume without loss of generality that $r<n-\\dim {Q}$ .",
"Otherwise we can consider a representation $\\tilde{\\tau }:E\\rightarrow \\operatorname{GL}_{\\tilde{n}+1}(k)$ , where $\\tilde{n} := m(n+1)-1$ , obtained by $m$ copies of $\\tau $ for some $m\\in {\\mathbb {N}}$ .",
"Indeed, the closed subscheme $\\tilde{Q}$ of “bad points” obtained by $\\tilde{\\tau }$ has dimension $\\dim {\\tilde{Q}} = m(\\dim {Q}+1)-1$ and $\\tilde{n} - \\dim {\\tilde{Q}} = m(n-\\dim {Q})$ , i.e.",
"we can choose any $m>r$ .",
"Starting with $Z_0$ and $Q_0$ , we construct $Z_1,Q_1,\\ldots ,Z_{n-r},Q_{n-r}$ recursively as follows: For each $i=1,\\ldots ,n-r$ , use Bertini's theorem to find a hypersurface $L_i\\subseteq {\\mathbb {P}}_k^s$ given by a homogeneous polynomial $h_i\\in k[U_0,\\ldots ,U_s]$ such that $Z_i:=L_i\\cap Z_{i-1}$ is smooth over $k$ and for $Q_i:=Q_{i-1}\\cap L_i$ , we have $\\dim Q_i \\le \\dim Q_{i-1}-1$ (or $Q_i=\\emptyset $ if $\\dim Q_{i-1}=0$ or $Q_{i-1}=\\emptyset $ ).",
"Note that $h_i$ can be chosen to be linear if $k$ is infinite, but if $k$ is finite, one might need to choose $h_i$ defined over $k$ of higher degree, see [5].",
"An inductive argument shows that $Z_{n-r}$ is smooth over $k$ of dimension $r$ and $Q_{n-r}=\\emptyset $ .",
"This implies that $Z_{n-r} = Z\\cap L_1\\cap \\cdots \\cap L_{n-r}$ , i.e.",
"$Z^{\\prime }:=Z_{n-r}$ is closed in $Z$ and smooth over $k$ .",
"Step 4: The projective variety $Y$ .",
"Consider the projective variety $Y\\subseteq {\\mathbb {P}}_{k^{\\prime }}^n$ given by the polynomials $g_j:=h_j(f_0,\\ldots ,f_s)$ for $j=1,\\ldots ,n-r$ .",
"We are going to show that $Y$ is the variety we are looking for.",
"Observe first that $Y=p^{-1}(Z^{\\prime }) \\subseteq p^{-1}(Z_0)=W$ .",
"This implies that $Y_{{k_a}}:=Y\\otimes _{k^{\\prime }}{k_a}$ is contained in $W_{{k_a}}$ .",
"Hence for each $y\\in Y_{{k_a}}$ and $z:=p_{k_a}(y)\\in Z^{\\prime }_{k_a}$ , the ring homomorphism ${O}_{Z_{k_a},z}\\rightarrow {O}_{{\\mathbb {P}}_{k_a}^n,y}$ is étale.",
"Furthermore, $\\lbrace h_1,\\ldots ,h_{n-r}\\rbrace $ is a subset of a regular parameter system of ${O}_{Z_{k_a},z}$ since ${O}_{Z^{\\prime }_{k_a},z}={O}_{Z_{k_a},z}/(h_1,\\ldots ,h_{n-r})$ is regular of dimension $r$ .",
"Hence $\\lbrace g_1,\\ldots ,g_{n-r}\\rbrace $ is such a subset of ${O}_{{\\mathbb {P}}_{k_a}^n,y}$ , i.e.",
"${O}_{Y_{{k_a}},y} = {O}_{{\\mathbb {P}}_{k_a}^n,y}/(g_1,\\ldots ,g_{n-r})$ is regular of dimension $r$ .",
"Therefore, $Y$ is smooth over $k^{\\prime }$ and has dimension $r$ .",
"In particular, $Y$ is a complete intersection in ${\\mathbb {P}}_{k^{\\prime }}^n$ .",
"It is geometrically connected since $Y_{{k_a}}$ is again a complete intersection in ${\\mathbb {P}}_{k_a}^n$ .",
"Since $Y$ as subscheme of ${\\mathbb {P}}_{k^{\\prime }}^n$ is given by ${\\widetilde{E}}$ -invariant polynomials, the restriction of the action of ${\\widetilde{E}}$ to $Y\\subseteq {\\mathbb {P}}_{k^{\\prime }}^n$ is well-defined.",
"Furthermore, this action is admissible, $\\pi $ -semilinear and avoids fixed points since $Y$ is contained in $W$ .",
"Hence $Y$ has all the desired properties."
],
[
"The main result",
"We wish to construct a $k$ -form of a Godeaux-Serre variety for a given continuous extension of $\\operatorname{Gal}_k$ by a finite group.",
"The strategy is to reduce this extension to an extension of $\\operatorname{Gal}(k^{\\prime }|k)$ for some finite Galois extension $k^{\\prime }|k$ .",
"This is done in the following Lemma: Lemma 2.2 For a given finite group $G$ and continuous extension of profinite groups $1 \\longrightarrow G \\xrightarrow{} E \\xrightarrow{} \\Gamma \\longrightarrow 1,$ there exists an open normal subgroup $H\\unlhd E$ which is under $\\pi $ isomorphic to an open normal subgroup $H^{\\prime }\\unlhd \\Gamma $ .",
"In this case, we have $E\\cong (E/H)\\times _{\\Gamma /H^{\\prime }}\\Gamma $ .",
"Since $G$ is a finite subgroup in the profinite group $E$ , there exists an open normal subgroup $H\\unlhd E$ such that $G\\cap H =\\lbrace 1\\rbrace $ .",
"The image $H^{\\prime }:=\\pi (H)$ is isomorphic to $H$ since the restriction of $\\pi $ to $H$ is injective.",
"Furthermore, it is an open normal subgroup of $\\Gamma $ since $\\pi $ is surjective.",
"Hence we obtain the following commutative diagram with exact rows: $\\begin{tikzpicture}[descr/.style={fill=white,fill opacity=0.75,inner sep=0.5pt}](m) [matrix of math nodes, row sep=1.5em, column sep=2em, text height=1.5ex, text depth=0.25ex]{ 1 & G & E & \\Gamma & 1 \\\\ 1 & G & E/H & \\Gamma /H^{\\prime } & 1\\unknown.",
"{.",
"}\\\\};[->,font=\\scriptsize ] (m-1-1) edge (m-1-2) (m-1-2) edge node[auto] {\\iota } (m-1-3) (m-1-3) edge (m-2-3) edge node[auto] {\\pi } (m-1-4) (m-1-4) edge (m-2-4) edge (m-1-5) (m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge node[auto] {\\tilde{\\pi }} (m-2-4) (m-2-4) edge (m-2-5);[double distance=2pt] (m-1-2) edge [double] (m-2-2);\\end{tikzpicture}$ The right square of the diagram is cartesian since $\\pi $ and $\\tilde{\\pi }$ have the same kernel.",
"Therefore, $E\\cong (E/H)\\times _{\\Gamma /H^{\\prime }}\\Gamma $ .",
"We now come to the main result.",
"Theorem 2.3 Let $k$ be a field, $G$ a finite group, $r\\in {\\mathbb {N}}$ with $r\\ge 2$ and $1 \\longrightarrow G \\stackrel{\\iota }{\\longrightarrow } E \\stackrel{\\pi }{\\longrightarrow } \\operatorname{Gal}_k \\longrightarrow 1 $ a continuous extension of profinite groups.",
"There exists a geometrically integral smooth projective variety $X$ over $k$ of dimension $r$ such that the exact sequence $1 \\longrightarrow \\pi _1(X\\otimes _k{k_a},{\\overline{x}}^{\\prime }) \\longrightarrow \\pi _1(X,{\\overline{x}}) \\longrightarrow \\operatorname{Gal}_k \\longrightarrow 1,$ where ${\\overline{x}}^{\\prime }\\in (X\\otimes _k{k_a})(\\Omega )$ and ${\\overline{x}}\\in X(\\Omega )$ is the image of ${\\overline{x}}^{\\prime }$ , is isomorphic to (REF ).",
"By Lemma REF , there is an open normal subgroup $H\\unlhd E$ which is under $\\pi $ isomorphic to an open normal subgroup $H^{\\prime }$ of $\\operatorname{Gal}_k$ .",
"Let $k^{\\prime }\\subseteq {k_a}$ be the finite Galois extension of $k$ corresponding to $H^{\\prime }$ .",
"By setting ${\\widetilde{E}}:=E/H$ , we obtain the following commutative diagram with exact rows: $\\begin{tikzpicture}[descr/.style={fill=white,fill opacity=0.75,inner sep=0.5pt}](m) [matrix of math nodes, row sep=1.5em, column sep=2.5em, text height=1.75ex, text depth=0.25ex]{ 1 & G & E & \\operatorname{Gal}_k & 1 \\\\ 1 & G & {\\widetilde{E}}& \\operatorname{Gal}(k^{\\prime }|k) & 1\\unknown.",
"{.",
"}\\\\};[->,font=\\scriptsize ] (m-1-1) edge (m-1-2) (m-1-2) edge node[auto] {\\iota } (m-1-3) (m-1-3) edge (m-2-3) edge node[auto] {\\pi } (m-1-4) (m-1-4) edge (m-2-4) edge (m-1-5) (m-2-1) edge (m-2-2) (m-2-2) edge node [auto] {\\widetilde{\\iota }} (m-2-3) (m-2-3) edge node [auto] {\\widetilde{\\pi }} (m-2-4) (m-2-4) edge (m-2-5);[double distance=2pt] (m-1-2) edge [double] (m-2-2);\\end{tikzpicture}$ For the lower exact sequence, we can find by Proposition REF a geometrically connected smooth projective variety $Y$ of dimension $r$ which is a complete intersection in ${\\mathbb {P}}_{k^{\\prime }}^n$ , on which ${\\widetilde{E}}$ acts admissibly, $\\pi $ -semilinearly and without fixed points.",
"Then the quotient $X:=Y/{\\widetilde{E}}$ is a projective variety over $k$ .",
"It is geometrically connected and smooth over $k$ since $X\\otimes _k{k_a}\\cong (X\\otimes _kk^{\\prime })\\otimes _{k^{\\prime }}{k_a}\\cong (Y/G)\\otimes _{k^{\\prime }}{k_a}\\cong (Y\\otimes _{k^{\\prime }}{k_a})/G$ and $Y\\otimes _{k^{\\prime }}{k_a}$ is connected and regular.",
"Now let $\\phi :Y\\rightarrow \\operatorname{Spec}{k^{\\prime }}$ and $\\psi :X\\rightarrow \\operatorname{Spec}{k}$ be the structure morphism and ${\\overline{y}}^{\\prime }\\in (Y\\otimes _{k^{\\prime }}{k_a})(\\Omega )$ with images ${\\overline{y}},{\\overline{x}}^{\\prime },{\\overline{x}}$ in $Y,X\\otimes _k{k_a},X$ respectively.",
"Consider the following diagram: [descr/.style=fill=white,fill opacity=0.75,inner sep=0.5pt] (m) [matrix of math nodes, row sep=2em, column sep=1em, text height=1.75ex, text depth=0.25ex] 1 1(Xkka,x') 1(X,x) 1(Speck,(x)) 1 1 G E Galk 1 1 G E Gal(k'|k) 1. ; [->,font=] (m-1-1) edge (m-1-3) (m-1-3) edge node[auto] $\\operatorname{pr}_{1,\\ast }$ (m-1-5) edge node[above left] $\\Phi _{G,{\\overline{y}}^{\\prime }}$ (m-3-3) (m-1-3.335) edge (m-2-4.135) (m-1-5) edge node[auto] $\\psi _\\ast $ (m-1-7) edge node[above left] $\\Phi _{{\\widetilde{E}},{\\overline{y}}}$ (m-3-5) (m-1-7) edge (m-1-9) edge(m-3-7) (m-2-2) edge (m-2-4) (m-2-4) edge node[above right] $\\iota $ (m-2-6) (m-2-6) edge node[above right] $\\pi $ (m-2-8) (m-2-6.220) edge node[right=2.5pt] $p$ (m-3-5.45) (m-2-8) edge (m-2-10) (m-2-8.200) edge node[right=2.5pt] $q$ (m-3-7.30) (m-3-1) edge (m-3-3) (m-3-3) edge node[auto] $\\widetilde{\\iota }$ (m-3-5) (m-3-5) edge node[auto] $\\widetilde{\\pi }$ (m-3-7) (m-3-7) edge (m-3-9) (m-1-7.330) edge node[right=0pt] $\\cong $ (m-2-8.145); [double distance=2pt] (m-2-4.205) edge [double] (m-3-3.25); [->,dashed,font=] (m-1-5.300) edge (m-2-6.135); Here the isomorphism between $\\pi _1(\\operatorname{Spec}{k},\\psi ({\\overline{x}}))$ and $\\operatorname{Gal}_k$ is given by an embedding $k_s\\hookrightarrow \\Omega $ lying over $\\phi ({\\overline{y}})\\in (\\operatorname{Spec}{k^{\\prime }})(\\Omega )$ .",
"The first and the third rows are compatible with the vertical arrows by Propositions REF and REF , i.e.",
"the whole diagram up to the dashed arrow is commutative.",
"We now construct an isomorphism between $\\pi _1(X)$ and $E$ .",
"Since $\\widetilde{\\pi }\\circ \\Phi _{{\\widetilde{E}},{\\overline{y}}} = q\\circ \\psi _\\ast $ and $(E,p,\\pi )$ is the fiber product of ${\\widetilde{E}}\\xrightarrow{}\\operatorname{Gal}(k^{\\prime }|k)$ and $\\operatorname{Gal}_k\\xrightarrow{}\\operatorname{Gal}(k^{\\prime }|k)$ , there exists a unique profinite group homomorphism $\\varphi :\\pi _1(X,{\\overline{x}})\\rightarrow E$ such that $\\Phi _{{\\widetilde{E}},{\\overline{y}}}=p\\circ \\varphi $ and $\\psi _\\ast =\\pi \\circ \\varphi $ , i.e.",
"the upper right parallelogram is commutative.",
"To see that the left one also commutes, observe that $p\\circ \\varphi \\circ \\operatorname{pr}_{1,\\ast } = \\Phi _{{\\widetilde{E}},{\\overline{y}}}\\circ \\operatorname{pr}_{1,\\ast } = \\widetilde{\\iota }\\circ \\Phi _{G,{\\overline{y}}^{\\prime }} = p\\circ \\iota \\circ \\Phi _{G,{\\overline{y}}^{\\prime }} \\unknown.",
"{\\quad and}\\\\\\pi \\circ \\varphi \\circ \\operatorname{pr}_{1,\\ast } = \\psi _\\ast \\circ \\operatorname{pr}_{1,\\ast } = 1 = \\pi \\circ \\iota \\circ \\Phi _{G,{\\overline{y}}^{\\prime }}.$ Hence by the universal property of the fiber product, we have $\\varphi \\circ \\operatorname{pr}_{1,\\ast } = \\iota \\circ \\Phi _{G,{\\overline{y}}^{\\prime }}$ .",
"Therefore, the whole diagram above is commutative.",
"Since $Y\\otimes _{k^{\\prime }}{k_a}$ is a complete intersection in ${\\mathbb {P}}_{k_a}^n$ as shown in Proposition REF , $\\pi _1(Y\\otimes _{k^{\\prime }}{k_a},{\\overline{y}}^{\\prime })=1$ by the Lefschetz Hyperplane Theorem, see [3], and the fact that $\\pi _1({\\mathbb {P}}_{k_a}^n)=1$ , see [7].",
"Hence $\\Phi _{G,{\\overline{y}}^{\\prime }}:\\pi _1(X\\otimes _k{k_a},{\\overline{x}}^{\\prime })\\rightarrow G$ is an isomorphism by Proposition REF .",
"Since the left and right vertical arrows between the first two lines of the diagram above are isomorphisms, $\\varphi :\\pi _1(X,{\\overline{x}})\\rightarrow E$ is also an isomorphism by the (not necessarily commutative) five lemma and we are done.",
"Remark 2.4 The base change from $k$ to ${k_a}$ of the variety constructed in Theorem REF is indeed a Godeaux-Serre variety.",
"Hence what we have constructed is a $k$ -form of a Godeaux-Serre variety with prescribed arithmetic fundamental group."
]
] | 1709.01835 |
[
[
"Generalized twisted centralizer codes"
],
[
"Abstract An important code of length $n^2$ is obtained by taking centralizer of a square matrix over a finite field $\\mathbb{F}_q$.",
"Twisted centralizer codes, twisted by an element $a \\in \\mathbb{F}_q$, are also similar type of codes but different in nature.",
"The main results were embedded on dimension and minimum distance.",
"In this paper, we have defined a new family of twisted centralizer codes namely generalized twisted centralizer (GTC) codes by $\\mathcal{C}(A,D):= \\lbrace B \\in \\mathbb{F}_q^{n \\times n}|AB=BAD \\rbrace$ twisted by a matrix $D$ and investigated results on dimension and minimum distance.",
"Parity-check matrix and syndromes are also investigated.",
"Length of the centralizer codes is $n^2$ by construction but in this paper, we have constructed centralizer codes of length $(n^2-i)$, where $i$ is a positive integer.",
"In twisted centralizer codes, minimum distance can be at most $n$ when the field is binary whereas GTC codes can be constructed with minimum distance more than $n$."
],
[
"Introduction",
"Let $ \\mathbb {F}_q $ be a finite field with $ q $ elements.",
"The set of all square matrices of order $ n $ over $ \\mathbb {F}_q $ is denoted by $ \\mathbb {F}_q^{ n \\times n } $ .",
"Algebraic codes are important tools in data transmission.",
"Ability of a good code is that, it detects or corrects more errors of an encoded message when it is transmitted over a noisy channel.",
"An error correction capability of a code totally depends on its construction.",
"A fundamental problem in error correcting codes is to produce a code $ [n, k, d] $ with given $n$ and $k$ , find maximum possible minimum distance $d$ .",
"Centralizer codes are very special type linear codes of length $ n^2 $ .",
"The concept of the centralizer codes are beautifully constructed in [1].",
"For $A \\in \\mathbb {F}_q^{n \\times n}$ , the centralizer code is defined by $ \\mathcal {C}(A):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=BA\\rbrace $ .",
"The authors have computed bounds on dimension.",
"They have given an efficient encoding and decoding procedure.",
"It has shown that centralizer codes can locate a single error by looking at syndrome only.",
"If $ A $ is a non cyclic matrix then centralizer code $ \\mathcal {C}(A) $ has dimension greater than $n$ .",
"Non cyclic matrices are very rare according to [3] but the adjacency matrices of distance regular graphs of diameter less than $ n-1 $ , are not cyclic.",
"Thus authors relates automorphism groups of graphs with centralizer codes.",
"In 2017, this work is extended in [2], namely twisted centralizer codes, defined as $\\mathcal {C}(A,a):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=aBA \\rbrace $ , where $a \\in \\mathbb {F}_q$ .",
"It is clear from the definition that centralizer codes are special kinds of twisted centralizer codes for $ a = 1 $ .",
"It has been shown that dimensions of centralizer codes and twisted centralizer codes are equal if there is an invertible matrix in the code.",
"They have refined bounds of dimension and minimum distance in centralizer codes.",
"These codes have less computational complexity to decode a received codeword through noisy channel.",
"They have ability to correct single error only and also assert that if $ a \\ne 0, 1 $ then the minimal distance can be greater than $ n $ whereas in centralizer codes (for $ a = 1 $ ) the minimal distance is at most $ n $ .",
"In this paper, we define generalized twisted centralizer (GTC) codes, obtained from $A$ twisted by a matrix $D \\in \\mathbb {F}_q^{n \\times n}$ , defined as $ \\mathcal {C}(A,D):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=BAD \\rbrace $ .",
"It is clear that $\\mathcal {C}(A,D)$ is a $\\mathbb {F}$ -linear subspace of the vector space $\\mathbb {F}_q^{n \\times n}$ .",
"The centralizer codes defined in [1] are obtained from $\\mathcal {C}(A,D)$ when $D = I_n$ , identity matrix of order $n$ and twisted centralizer codes defined in [2] are obtained from $\\mathcal {C}(A,D)$ when $D = aI_n$ , scalar matrix of order $n$ .",
"$\\mathcal {C}(A,D)$ is considered to be a code by constructing codewords of length $n^2$ from matrices $B \\in \\mathcal {C}(A,D)$ by writing column-by-column.",
"We execute some salient results of twisted centralizer codes.",
"We give some idea on centralizer code of various length which is not of the form $n^2$ using the concept of puncture codes.",
"Some examples are given which are the witness on existence of generalized twisted centralizer codes.",
"We show that for a matrix $D \\in \\mathbb {F}_2^{ n \\times n}$ minimum distance of GTC codes can be larger than $ n $ .",
"The paper is organized as follows.",
"In Section 2, we give definition of GTC code and establish some basic results on parity check matrix and dimension.",
"In Section 3, we explain our main results.",
"Complete encoding and decoding procedure is discussed in Section 4.",
"In Section 5, we provide some examples on optimal GTC codes.",
"We provide GTC codes of length less than $ n^2 $ in Section 6.",
"In Section 7, we give conclusion with an open problem."
],
[
"Preliminaries",
"Definition 2.1 For any square matrix $A \\in \\mathbb {F}_q^{n \\times n}$ and any matrix $D \\in \\mathbb {F}_q^{n \\times n}$ , the subspace $\\mathcal {C}(A,D):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=BAD \\rbrace $ of $\\mathbb {F}_q^{n \\times n}$ is called generalized twisted centralizer code of $ A $ twisted by the matrix $D$ .",
"Proposition 2.1 Parity-check matrix for a GTC code $\\mathcal {C}(A,D)$ is given by $H=I_n \\otimes A - (D^t \\otimes I_n)(A^t \\otimes I_n)$ , where $\\otimes $ denotes the Kronecker product, $A^t$ is the transpose of the matrix $A$ , and $ I_n $ is the identity matrix of order $ n $ .",
"If we take $ B = AD $ and $ C= O $ in Theorem 27.5.1 of [4], the theorem follows easily.",
"Theorem 2.1 Let, $ A,D \\in \\mathbb {F}_q^{n \\times n}$ and $ O $ be the null matrix of order $ n $ .",
"Then the following are true: $A \\in \\mathcal {C}(A,D)$ if and only if $D=I_n$ or $A^2=O$ .",
"If $ D $ is invertible then $ B \\in \\mathcal {C}(A,D) \\Leftrightarrow A \\in \\mathcal {C}(B,D^{-1})$ .",
"For $ A \\ne O $ , we have $I_n \\in \\mathcal {C}(A,D) \\Leftrightarrow D=I_n$ .",
"Theorem 2.2 If $O \\ne A \\in \\mathbb {F}_q^{n \\times n}$ and $ D \\ne I_n $ , then the dimension $dim(\\mathcal {C}(A,D)) \\leqslant n^2-1$ .",
"It is clear from context of linear algebra that the dimension of $ \\mathcal {C}(A,D) $ is at most $ n^2 $ .",
"Now, if $ dim(\\mathcal {C}(A,D)) = n^2 $ , then every matrix B satisfies the relation $ AB=BAD $ .",
"But, if we take $ B=I_n $ then $ A=AD $ .",
"Which is not possible for any $ D \\ne I_n$ .",
"Hence, $ dim(\\mathcal {C}(A,D)) \\le n^2-1 $ .",
"Theorem 2.3 Let $ A, D \\in \\mathbb {F}^{n \\times n} $ , and the GTC code $ \\mathcal {C}(A,D) $ contains an invertible matrix, then $ dim(\\mathcal {C}(A,D)) = dim(\\mathcal {C}(A)) $ .",
"Let $ B $ be an invertible matrix in $ \\mathcal {C}(A,D) $ .",
"Then, $ AB = BAD \\Rightarrow A = BADB^{-1} $ .",
"Now, consider the linear mapping $ f_B : \\mathcal {C}(A) \\rightarrow \\mathcal {C}(A,D) $ such that $ f_B(X) = XB $ .",
"This mapping is closed since, $ X \\in \\mathcal {C}(A) \\Rightarrow AX = XA \\Rightarrow AXB = XAB \\Rightarrow A(XB) = (XB)AD~~(\\because AB = BAD) \\Rightarrow XB \\in \\mathcal {C}(A,D) $ .",
"Clearly, the mapping $ f_B $ is injective and hence we can conclude that $ dim(\\mathcal {C}(A)) \\le dim(\\mathcal {C}(A,D)) $ .",
"Again, the mapping $ \\phi _B : \\mathcal {C}(A,D) \\rightarrow \\mathcal {C}(A) $ such that $ \\Phi _B(Y) = YB^{-1} $ is closed since $ Y \\in \\mathcal {C}(A,D) \\Rightarrow AY = YAD \\Rightarrow AYB^{-1} = YADB^{-1} \\Rightarrow AYB^{-1} = YB^{-1}BADB^{-1} \\Rightarrow AYB^{-1} = YB^{-1}A ~~ ( \\because A = BADB^{-1}) \\Rightarrow YB^{-1} \\in \\mathcal {C}(A) $ .",
"The mapping $ \\Phi _B $ is also injective.",
"So, $ dim(\\mathcal {C}(A)) \\ge dim(\\mathcal {C}(A,D)) $ .",
"Combining both results we have $ dim(\\mathcal {C}(A,D)) = dim(\\mathcal {C}(A)) $ .",
"Theorem 2.4 For all $D \\in \\mathbb {F}_q$ , the code $\\mathcal {C}(A,D)$ contains the product code $Ker(A) \\otimes Ker(D^t A^t) $ .",
"If $Ker(A)$ , $Ker(D^t A^t)$ have respective parameters $[n, k, d]$ and $[n, k^{\\prime }, d^{\\prime }]$ , then $ \\mathcal {C}(A,D) $ has parameters $ [n^2,K,D] $ with $ K \\ge kk^{\\prime }$ and $ D \\le dd^{\\prime } $ .",
"Let, $ u \\in Ker(A) $ and $v \\in Ker(D^t A^t) $ .",
"Then $ A(u v^t) = (Au)v^t = O $ and $ uv^t AD = u (D^t A^t v) = O $ .",
"Which shows that $ B = u v^t \\in \\mathcal {C}(A,D) $ as $ AB = O = BAD $ .",
"So, $ Ker(A) \\otimes Ker(D^t A^t) \\subseteq \\mathcal {C}(A,D) $ .",
"The next part of the theorem will be established by a simple property of product code in [5]."
],
[
"Main results",
"Let $ E, A \\in \\mathbb {F}_q^{n \\times n} $ .",
"We define a set $ \\mathcal {T}_E := \\lbrace B \\in \\mathbb {F}^{n \\times n}: AB = BAD = E \\rbrace $ .",
"According to our definition, we have $ \\mathcal {C}(A,D) = \\bigcup \\limits _{E \\in \\mathbb {F}^{n \\times n}_q} \\lbrace B: AB = BAD = E \\rbrace = \\bigcup \\limits _{E \\in \\mathbb {F}^{n \\times n}_q} \\mathcal {T}_E $ .",
"Throughout this section we denote $ r_A $ as the rank of a matrix $ A $ .",
"Proposition 3.1 Dimension of $ \\mathcal {T}_O $ is less than or equal to $ n^2-n \\cdot r_A $ .",
"According to our notation $ \\mathcal {T}_O:= \\lbrace B: AB = BAD = O \\rbrace $ .",
"Now, $ AB = O \\Rightarrow B \\in Ker(A) $ .",
"Now let $\\mathcal {K}_{AD,O} := \\lbrace B: BAD = O \\rbrace $ .",
"Since $ BAD = O \\Rightarrow D^t A^t B^t = O \\Rightarrow B^t \\in Ker(D^t A^t) $ .",
"We take a mapping $ \\psi : \\mathcal {K}_{AD,O} \\rightarrow Ker(D^t A^t) $ such that $ \\psi (B) = B^t $ .",
"Clearly, this map is bijective.",
"So, $ |\\mathcal {K}_{AD,O}| =|Ker(D^t A^t)| \\Rightarrow dim(\\mathcal {K}_{AD,O}) = dim(Ker(D^t A^t)) \\Rightarrow dim(\\mathcal {K}_{AD,O}) = n^2 - n \\cdot r_{D^t A^t} = n^2 - n \\cdot r_{AD}$ .",
"Again $ dim(Ker(A)) = n^2 - n \\cdot r_A $ .",
"Now, $ dim(\\mathcal {T}_O) \\le min \\lbrace dim(Ker(A)),dim(k_{AD,O}) \\rbrace = min \\lbrace n^2 - n \\cdot r_A, n^2 - n \\cdot r_{AD} \\rbrace = n^2 - n \\cdot r_A$ .",
"Proposition 3.2 If $ \\mathcal {T}_E $ is non-empty then $ |\\mathcal {T}_E| = |\\mathcal {T}_O| $ .",
"Let $ B \\in \\mathcal {T}_O $ and $ S_1 \\in \\mathcal {T}_E $ for a fixed $ E \\ne O $ and $ S \\in \\mathbb {F}^{n \\times n} $ , then $ S_1 + B \\in \\mathcal {T}_E $ .",
"Which shows that the $ | \\mathcal {T}_E | \\ge | \\mathcal {T}_O| $ .",
"Let if possible $ | \\mathcal {T}_E | > | \\mathcal {T}_O| $ .",
"Then there exists an element $ S_2 \\ne S_1 $ which is not of the form $ S_1 + B $ for $ B \\in \\mathcal {T}_O $ .",
"Now, $ S_1, S_2 \\in \\mathcal {T}_E \\Rightarrow S_2-S_1 \\in \\mathcal {T}_O $ .",
"But, $ S_2 = S_1 + (S_2 - S_1) $ is in the form of $ S_1 + B $ , contradicts our assumption.",
"Hence $ |\\mathcal {T}_E| = |\\mathcal {T}_O| $ is proved.",
"Theorem 3.1 Let $ A, D \\in \\mathbb {F}^{n \\times n}_q $ , then the GTC code $ \\mathcal {C}(A,D) $ has the dimension less than or equal to $ n^2 - n \\cdot r_A + n \\cdot r_{AD}$ .",
"Let us consider $ \\mathcal {T}_E := \\lbrace B \\in \\mathbb {F}^{n \\times n}_q: AB = BAD = E \\rbrace $ .",
"Basically the set $ \\mathcal {T}_E $ is the common solutions of $ AB = E $ and $ BAD = E $ .",
"Now, $ AB = E $ is possible if columns of $ B \\in Columnspace(A) $ and $ BAD = E $ is possible if rows of $ B \\in Rowspace(AD) $ .",
"So, $ B \\in \\mathcal {T}_E $ if $ B \\in Columnspace(A) \\cap Rowspace(AD) $ .",
"We denote, $ G = Columnspace(A) \\cap Rowspace(AD) $ .",
"$ \\mathcal {C}(A,D) = \\bigcup \\limits _{E \\in G} \\lbrace B: AB = BAD = E \\rbrace = \\bigcup \\limits _{E \\in G} \\mathcal {T}_E $ .",
"By Proposition REF if $ \\mathcal {T}_E $ is non empty then, it has the same cardinality as $ \\mathcal {T}_O $ .",
"Let us assume $ \\mathcal {T}_E $ is solvable and non empty for each $ C \\in G $ .",
"Then, $ \\mathcal {C}(A,D) = \\bigcup \\limits _{E \\in G} \\mathcal {T}_E \\Rightarrow |\\mathcal {C}(A,D)| \\le |\\mathcal {T}_O| \\cdot |G|$ $ \\Rightarrow dim( \\mathcal {C}(A,D) ) \\le dim(\\mathcal {T}_O) + dim(G).",
"$ Now, $ dim(G) = n \\cdot dim(Columnspace(A) \\cap Rowspace(AD))$ $ \\le n \\cdot min \\lbrace dim(Columnspace(A)), dim(Rowspace(AD)) \\rbrace $ $= n \\cdot min \\lbrace r_A, r_{AD} \\rbrace = n \\cdot r_{AD}.$ Using Proposition REF we have $ dim(\\mathcal {C}(A,D)) \\le n^2 - n \\cdot r_A + n \\cdot r_{AD} $ .",
"Corollary 3.1 If $ AD = O $ in Theorem REF then $ dim(\\mathcal {C}(A,D)) \\le n^2 - n \\cdot r_A $ .",
"Let $\\Gamma $ be a graph with vertices $v_1, v_2, \\dots , v_n$ .",
"The adjacency matrix of $\\Gamma $ is a square matrix of order $n$ whose $(i,j)$ -entry is 1 if the vertices $v_i$ and $v_j$ are adjacent, otherwise the entry is 0.",
"Automorphism group of graph is the set of all automorphisms from the vertex set to itself of the graph which preserves adjacency.",
"It is denoted by $Aut(\\Gamma )$ .",
"Theorem 3.2 If $ A \\in {F}^{n \\times n}_q $ is the adjacency matrix of graph $ \\Gamma _{1} $ and $ G_1 = Aut(\\Gamma _{1}) $ and if $ AD \\in {F}^{n \\times n} $ is the adjacency matrix of graph $ \\Gamma _2 $ and $ G_2= Aut(\\Gamma _2) $ then the direct product $ G_1 \\times G_2 $ acts on the code $ \\mathcal {C}(A,D) $ by coordinate permutations.",
"It is known by [6] that a permutation matrix $ P $ lies in $ Aut(\\Gamma ) $ if and only if $ AP^{-1} = P^{-1}A $ .",
"Let $ (P,Q) \\in G_1 \\times G_2 $ and $ B \\in \\mathcal {C}(A,D) $ .",
"Then we have, $ AB = BAD $ , $ P^{-1}A = AP^{-1} $ , and $ QAD = ADQ $ .",
"So, $ AB = BAD \\Rightarrow P^{-1}AB = P^{-1}BAD \\Rightarrow P^{-1}ABQ = P^{-1}BADQ \\Rightarrow AP^{-1}BQ = P^{-1}BQAD \\Rightarrow P^{-1}BQ \\in \\mathcal {C}(A,D) $ .",
"So, $\\begin{array}{cccc} \\phi : & (G_1 \\times G_2) \\times \\mathcal {C}(A,D) & \\rightarrow & \\mathcal {C}(A,D) \\\\ & (P,Q) \\times B & \\mapsto & P^{-1}BQ \\end{array}$ is a group action.",
"Hence the theorem is proved.",
"In twisted centralizer codes [2], it was shown that for $a \\ne 0, 1$ , minimum distance can be larger than $n$ .",
"Here we show by few examples that GTC codes over binary fields have minimum distances greater than $n$ which is not possible to the twisted centralizer codes.",
"Examples of optimal binary GTC codes are given below whose minimum distances are larger than the order of $A$ .",
"Example 3.1 Suppose $ A = \\begin{bmatrix}1 & 1 & 0 \\\\0 & 1 & 1 \\\\1 & 1 & 0\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_2 $ and $ D = \\begin{bmatrix}1 & 1 & 0 \\\\1 & 1 & 1 \\\\0 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_2 $ .",
"Then an optimal binary GTC code $ [9, 2, 6] $ is acquired.",
"Example 3.2 An optimal GTC code $ [9, 3, 4] $ is obtained for $ A = \\begin{bmatrix}0 & 1 & 0 \\\\1 & 1 & 1 \\\\0 & 1 & 0\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_2 $ and $ D = \\begin{bmatrix}1 & 1 & 1 \\\\1 & 1 & 1 \\\\1 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_2 $ .",
"In both examples, the order of $A$ is 3.",
"In first example we have minimum distance 6 and in second it is 4."
],
[
"Encoding-decoding procedure",
"Let the generalized centralizer code $\\mathcal {C}(A,D)$ has length $ n^2 $ and dimension $ k $ .",
"$ \\mathcal {C}(A,D) $ is a vector space over $ \\mathbb {F}_q $ and it has a basis of dimension $ k $ .",
"Let $ \\lbrace B_1, B_2, \\dots , B_k \\rbrace $ is a basis of $ \\mathcal {C}(A,D) $ .",
"So, for an information message $ (a_1, a_2, \\dots , a_k) \\in \\mathbb {F}_{q}^{k}$ can be encoded as $ a_1 B_1+a_2 B_2+\\dots +a_k B_k $ .",
"The method of decoding is the reverse process of encoding.",
"A receiver should know the basis $ \\lbrace B_1, B_2, \\dots , B_k \\rbrace $ to decode the received message into information message.",
"We have already established that $ \\mathcal {C}(A,D) $ is a linear subspace of $ \\mathbb {F}_q^{n \\times n} $ and hence we can state that $ \\mathcal {C}(A,D) $ is an additive subgroup of $ \\mathbb {F}_q^{n \\times n} $ .",
"Then, cosets of $ \\mathcal {C}(A,D) $ is in $ \\mathbb {F}_q^{n \\times n} $ .",
"We can use $ A $ as a parity check matrix since $AB - BAD = O$ for every $ B \\in \\mathcal {C}(A,D) $ .",
"To decode the information message, we can use syndrome decoding.",
"Definition 4.1 Let $ \\mathcal {C}(A,D) $ be the non-empty generalized twisted centralizer code for a matrix $ A $ twisted by the matrix $ D $ and let $ B \\in \\mathcal {C}(A,D) $ .",
"The syndrome of B is defined a $S_A(B)= AB-BAD$ .",
"Theorem 4.1 Consider two matrices $B_1, B_2 \\in \\mathbb {F}_q^{n \\times n}$ .",
"Then $S_A(B_1) = S_A(B_2)$ iff $B_1$ and $B_2$ are in same coset of $\\mathcal {C}(A,D)$ .",
"Let $ S_A(B_1) = S_A(B_2)$ $\\Leftrightarrow A B_{1} - B_{1} A D = A B_{2} - B_{2} A D $ $\\Leftrightarrow A(B_1-B_2) = (B_1-B_2)AD $ $ \\Leftrightarrow B_1- B_2 \\in \\mathcal {C}(A,D).$ Therefore, $ B_1 $ and $ B_2 $ are in same coset of $ \\mathcal {C}(A,D) $ iff $ B_1- B_2 \\in \\mathcal {C}(A,D)$ .",
"Hence the theorem is proved.",
"The motive of defining syndrome is that syndrome computation is more easier than the computation by using $ n^2 \\times n^2 $ parity check matrix for authenticity.",
"By using the parity-check matrix we need $ O(n^4) $ multiplicative complexity but using $ A $ as a parity check matrix computation has $ O(n^m) $ complexity where $ m < 2.3729 $ by [7].",
"Also, the purpose of taking the code $\\mathcal {C}(A,D):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=BAD\\rbrace $ instead of $\\mathcal {C}(A,D):= \\lbrace B \\in \\mathbb {F}_q^{n \\times n}|AB=DBA)\\rbrace $ is due to less computational complexity of the syndrome because at that time we can consider a fixed matrix $C=AD$ .",
"The process of syndrome decoding is very similar with usual decoding process."
],
[
"Some optimal generalized twisted centralizer codes",
"Here we provide some examples of optimal twisted centralizer codes where the optimality is verified by [8].",
"Example 5.1 Let $ A = \\begin{bmatrix}1 & 1 & 1 \\\\1 & 1 & 1 \\\\1 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_2 $ .",
"Then, the matrices $ D_1$ , $D_2$ , $D_3 \\in \\mathbb {F}^{3 \\times 3}_2 $ where $ D_1 = \\begin{bmatrix}1 & 1 & 1 \\\\1 & 0 & 1 \\\\0 & 1 & 1\\end{bmatrix} $ , $ D_2 = \\begin{bmatrix}1 & 0 & 1 \\\\1 & 1 & 1 \\\\0 & 0 & 1\\end{bmatrix} $ , and $ D_3 = \\begin{bmatrix}1 & 1 & 0 \\\\0 & 0 & 1 \\\\1 & 1 & 1\\end{bmatrix} $ give optimal GTC codes $ [9, 5, 3] $ , $ [9, 4, 4] $ and $ [9, 6, 2] $ respectively.",
"Example 5.2 Consider $ A = \\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & 1 & 1 & 1 \\\\1 & 1 & 1 & 1 \\\\1 & 1 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_2 $ .",
"Optimal GTC codes $ [16, 9, 4] $ , $ [16, 10, 4] $ and $ [16, 12, 2]$ are obtained for $D_1$ , $D_2$ and $D_3 \\in \\mathbb {F}^{4 \\times 4}_2$ respectively, where $ D_1 = \\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & 1 & 0 & 1 \\\\0 & 1 & 1 & 1 \\\\1 & 0 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_2 $ , $ D_2 = \\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & 1 & 0 & 0 \\\\1 & 1 & 1 & 0 \\\\1 & 0 & 1 & 1\\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_2 $ and $ D_3 = \\begin{bmatrix}1 & 1 & 1 & 1 \\\\1 & 1 & 0 & 1 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 1\\end{bmatrix} $ .",
"Example 5.3 For $ A = \\begin{bmatrix} 1 & 2 & 2 \\\\ 2 & 1 & 1 \\\\ 2 & 1 & 1 \\end{bmatrix} $ and $ D = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 1 & 2 \\\\ 1 & 2 & 1 \\end{bmatrix} $ GTC code $ [9, 5, 4] $ , and for $ A = \\begin{bmatrix} 1 & 0 & 1 \\\\ 1 & 1 & 1 \\\\ 2 & 1 & 0 \\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_3 $ and $ D = \\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 1 & 1 \\\\ 2 & 2 & 2 \\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_3 $ GTC code $ [9, 3, 6] $ are attained.",
"Example 5.4 For $ A = \\begin{bmatrix} 2 & 2 & 2 & 2 \\\\ 2 & 2 & 2 & 2 \\\\ 2 & 2 & 2 & 2 \\\\ 2 & 2 & 2 & 2 \\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_3 $ and $ D = \\begin{bmatrix} 1 & 0 & 2 & 1 \\\\ 1 & 2 & 0 & 1 \\\\ 0 & 0 & 2 & 0 \\\\ 2 & 0 & 0 & 1 \\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_3 $ GTC code $ [16, 10, 4] $ , and for $ A = \\begin{bmatrix} 0 & 2 & 1 & 0 \\\\ 1 & 1 & 2 & 1 \\\\ 1 & 0 & 2 & 1 \\\\ 1 & 0 & 0 & 2 \\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_3 $ and $ D = \\begin{bmatrix} 1 & 0 & 0 & 1 \\\\ 0 & 2 & 0 & 2 \\\\ 0 & 0 & 2 & 0 \\\\ 2 & 0 & 0 & 1 \\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_3 $ GTC code $ [16, 3, 10] $ are obtained."
],
[
"Generalized centralizer codes of length less than $n^2$",
"Generalized twisted centralizer codes of length less than $n^2$ can be constructed by choosing $B$ with entries are 0 in fixed $i$ positions.",
"Clearly, that set is a subcode of $\\mathcal {C}(A,D)$ say $\\mathcal {S}(A,D)$ .",
"Puncturing those fixed $i$ number of entries of the subcode $\\mathcal {S}(A,D)$ , we get a new code of length $n^2 - i$ where $i$ is a positive integer in $1 \\le i <(n-1)^2$ .",
"Using this puncturing method, the multiplicative complexity is reduced as $ O((n-i)^m) $ where $ m <2.3729 $ .",
"This result works well when $ i $ is near to 1.",
"Example 6.1 For $ A = \\begin{bmatrix}1 & 1 & 1 \\\\1 & 2 & 2 \\\\0 & 0 & 1\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_3 $ and $ D = \\begin{bmatrix}0 & 0 & 2 \\\\1 & 1 & 0 \\\\1 & 2 & 2\\end{bmatrix} \\in \\mathbb {F}^{3 \\times 3}_3 $ an optimal code $ [7, 2, 5] $ is obtained by puncturing $(1,2)$ -entry and $(2,3)$ -entry in $B$ .",
"Example 6.2 For $ A = \\begin{bmatrix}0 & 0 & 1 & 1 \\\\0 & 1 & 0 & 0 \\\\1 & 1 & 1 & 0 \\\\1 & 0 & 0 & 1\\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_2 $ and $ D = \\begin{bmatrix}1 & 0 & 1 & 1 \\\\1 & 1 & 0 & 0 \\\\0 & 0 & 1 & 1 \\\\0 & 1 & 0 & 1\\end{bmatrix} \\in \\mathbb {F}^{4 \\times 4}_2 $ an optimal code $ [12, 3, 6] $ is obtained by puncturing all entries in 4th column of $B$ ."
],
[
"Conclusion",
"In this paper we have generalized the idea of twisted centralizer codes [2].",
"It has shown in Section 2 that the dimension of a twisted centralizer code is equal to GTC code.",
"In Section 3, an upper bound on dimension of GTC code has been derived.",
"Encoding and decoding procedure has been implemented to GTC codes.",
"Length of centralizer codes could be shorten by using concept of puncture codes.",
"In twisted centralizer codes, minimum distance can be at most $ n $ when the field is binary whereas we have constructed GTC code with minimum distance more than $ n $ when $ D \\in \\mathbb {F}_2^{n \\times n} $ .",
"An error can be corrected by simply looking at the syndrome.",
"But finding $t$ -errors ($t > 1$ ) in twisted centralizer codes or in GTC codes is still open.",
"Acknowledgements The authors Joydeb Pal is thankful to DST-INSPIRE and Pramod Kumar Maurya is thankful to MHRD for their financial support to pursue his research work."
]
] | 1709.01825 |
[
[
"On-the-fly Historical Handwritten Text Annotation"
],
[
"Abstract The performance of information retrieval algorithms depends upon the availability of ground truth labels annotated by experts.",
"This is an important prerequisite, and difficulties arise when the annotated ground truth labels are incorrect or incomplete due to high levels of degradation.",
"To address this problem, this paper presents a simple method to perform on-the-fly annotation of degraded historical handwritten text in ancient manuscripts.",
"The proposed method aims at quick generation of ground truth and correction of inaccurate annotations such that the bounding box perfectly encapsulates the word, and contains no added noise from the background or surroundings.",
"This method will potentially be of help to historians and researchers in generating and correcting word labels in a document dynamically.",
"The effectiveness of the annotation method is empirically evaluated on an archival manuscript collection from well-known publicly available datasets."
],
[
"Introduction",
"Libraries and cultural organisations contain valuable manuscripts from ancient times that are to be digitised for preservation and protection from degradation over time.",
"However, digitisation and automatic recognition of handwritten archival manuscripts is a challenging task.",
"Unlike modern machine-printed documents that have simple layouts and common fonts, ancient handwritten documents have complex layouts and paper degradation over time.",
"They commonly suffer from variability in writing behaviour and degradations such as ink bleed through, faded ink, wrinkles, stained paper, missing data, poor contrast, warping effects or even noise due to lighting variation during document scanning.",
"Such issues hamper manuscript readability and pose difficulties for experts in generating ground truth annotations that are essential for information retrieval algorithms.",
"Efforts have been made towards automatic generation of ground truth in the literature , , .",
"Some popular tools include TrueViz, PerfectDoc , PixLabeler , GEDI , Aletheia , WebGT and TEA .",
"However, most of these tools and methods are not suitable for annotating historical datasets.",
"For example, PixLabeler and TrueViz are useful for labeling documents with regular elements or OCR text, rather than historical documents with handwritten text.",
"Typically, these tools are hardware specific, with strict system requirements for configuration and installation.",
"For example, TrueViz is a Java based tool for editing and visualising ground truth for OCR, and uses labels in XML format.",
"GEDI is a highly configurable tool with XML-based metadata.",
"Most of these methods represent ground truth with imprecise and inaccurate bounding boxes, as discussed in , and are suitable for a certain application.",
"For example, PerfectDoc is commonly used for document correction instead of ground truth generation.",
"To address these issues, this paper presents a simple method for quickly annotating degraded historical handwritten text on-the-fly.",
"The proposed text annotation method allows a user to view a document page, select the desired word to be labeled with simple drag-and-drop feature, and generate the corresponding bounding box annotation.",
"In the next step, it automatically adjusts and corrects user generated bounding box annotation, such that the word perfectly fits in the bounding box and contains no added noise from the background or surroundings.",
"This method can help historians and computer scientists in generating and correcting ground truth corresponding to words in a document dynamically.",
"Also, the method can potentially benefit the research community working on document image analysis and recognition based applications where ground truth is indispensable for method evaluation.",
"For example, in a keyword spotting scenario , if a ground truth label corresponding to a query word is missing or is inaccurate, the proposed method can be used for quick generation or correction of annotations on-the-fly.",
"This paper is organized as follows.",
"Section discusses various document image annotation methods available in literature.",
"Section explains the proposed word annotation algorithm in detail.",
"Section demonstrate the efficacy of the proposed method on two well-known historical document datasets.",
"Section concludes the paper."
],
[
"Document annotation methods and tools",
"Several document image ground truth annotation methods and tools have been suggested in literature.",
"An approach for automatic generation of ground truth for image analysis systems is proposed in .",
"Generated ground truth contains information related to layout structure, formatting rules and reading order.",
"An XML-based page image representation framework PAGE (Page Analysis and Ground-truth Elements) was proposed in .",
"Problems related to ground truth design, representation and creation are discussed in .",
"However, these methods are not suitable for annotating degraded historical datasets with complex layouts.",
"Ground truth can be automatically generated using popular tools.",
"The earliest tool was Pink Panther that allows a user to annotate document image with simple mouse clicks.",
"TrueViz is a commonly used Java based tool for editing and visualising ground truth for OCR text, and is not suitable for handwritten historical documents annotation.",
"PerfectDoc is a tool used for document correction, with possible application to ground truth generation.",
"One of the most popular document labeling tools is PixLabeler .",
"It uses a Java based user interface for annotating documents at pixel level.",
"Like Pink Panther, TrueViz and PerfectDoc, PixLabeler works well on simple documents only and perform poorly on historical handwritten document images .",
"The tool GEDI supports multiple functionalities such as merging, splitting and ordering.",
"It is a highly configurable document annotation tool.",
"The advanced tool Aletheia is proposed in for accurate and cost effective ground truth generation of large collection of document images.",
"The first web-based annotation tool, WebGT , provides several semi-automatic tools for annotating degraded documents and has gained prominence recently.",
"An interesting prototype called Text Encoder and Annotator (TEA) is proposed in that annotates manuscripts using semantic web technologies.",
"Figure: Detailed view of the Esposalles Database .",
"Figure best viewed in color.However, these tools require specific system requirements for configuration and installation.",
"Most of these tools and methods are either not suitable for annotating historical handwritten datasets, or represent ground truth with imprecise and inaccurate bounding boxes .",
"For example, Fig.",
"REF shows a detailed view of the Esposalles Database .",
"It can be clearly observed that the annotation for word $pages$ (blue bounding box) is inaccurate and misses certain parts of characters $p$ and $g$ .",
"Similarly, the annotation for the word $Mas$ is imprecise and the database is missing annotations for several words.",
"This paper takes into account such issues, and proposes a simple and fast method for dynamically annotating degraded historical handwritten text on-the-fly."
],
[
"On-the-fly text annotation",
"The proposed on-the-fly text annotation method allows the user to generate word annotations dynamically with simple drag-and-drop gesture.",
"The algorithm finds the extent of the word and automatically adjusts the bounding box such that the word is perfectly encapsulated and is noise-free.",
"For example, Fig.",
"REF shows the user marked word in the red bounding box, and the green bounding box depicts the corrected annotation generated by the proposed method.",
"The general framework of the proposed method is described in Fig.",
"REF .",
"Figure: An example illustrating the annotation for the word reberé.",
"The red bounding box represents the user marked label.",
"The algorithm finds the extent of the word and automatically adjusts the bounding box (green) such that the word is perfectly encapsulated.",
"Figure best viewed in color.Figure: Example of connected components extraction and labeling of the word.",
"Figure best viewed in color.Figure: Example of a case with blue bounding box selection that lies ∼\\sim 5% inside the example word.",
"The red bounding box denotes the user annotated label.",
"Figure best viewed in color.Figure: General framework of the proposed on-the-fly text annotation method.",
"For an input document, the user labels the query word using a simple drag-and-drop gesture.",
"The user annotated red bounding box is then automatically adjusted and corrected.",
"The output green bounding box represents an accurate noise-free annotation for the query word.",
"Figure best viewed in color.The algorithm begins with the extraction of a larger bounding box area than the user selected region in order to be able to capture strokes that go beyond the selected borders.",
"The height of the selected word is chosen as a base for area calculation.",
"This is because the height of the selected word is approximately same across all the words in a line.",
"The width of the selected word is also taken into account as the number of characters may vary in different words.",
"The algorithm enhances the bounding box by one-third increase in height along the x-axis and two-third increase in height along the y-axis.",
"In order to separate the word from noisy background, Gaussian filtering based background removal is performed.",
"This is followed by connected components extraction from the word image, as highlighted in Fig.",
"REF for an example word reberé.",
"There is also some noise around the word that needs to be removed in order to avoid error in the calculation of bounding box for the extent of the word.",
"This is done by determining whether each individual component covers a large enough area inside the word.",
"Fig.",
"REF depicts a case where the blue bounding box is selected that lies $\\sim $ 5% inside the example word.",
"If a connected component covers more than 1% of the total area inside the rectangle region, it is considered to be a part of the word.",
"This is particularly important for the text with strokes that extend beyond the marked rectangle.",
"Strokes from the adjacent lines and words should not be included in the bounding box even if they are partially inside the labeled region.",
"One can observe that the rightmost component (eré) in Fig.",
"REF contains both foreground strokes and some bleed through artifacts.",
"The noise due to bleed through is efficiently removed by thresholding the final result before generating the corrected bounding box annotation."
],
[
"Experiments",
"This section describes the datasets used in the experiments and empirically evaluates the results obtained from the proposed word annotation algorithm."
],
[
"Dataset",
"The experiments were performed on the following two publicly available datasets of varying complexity.",
"A subset of the Barcelona Historical Handwritten Marriages (BH2M) database i.e.",
"the Esposalles dataset is used for experiments.",
"BH2M consists of 244 books with information on 550,000 marriages registered between 15th and 19th century.",
"The Esposalles dataset consists of historical handwritten marriages records stored in archives of Barcelona cathedral, written between 1617 and 1619 by a single writer in old Catalan.",
"In total, there are 174 pages handwritten by a single author corresponding to volume 69, out of which 50 pages are selected from 17th century.",
"The proposed method generates a document page query and allows the user to label the desired word with simple drag-and-drop gesture.",
"For each word annotated by the user, a bounding box corresponding to the corrected annotation is generated as ground truth dynamically."
],
[
"KWS-2015 Bentham dataset",
"The KWS-2015 Bentham dataset, prepared in the $tranScriptorium$ project , is employed to test the effectiveness of the proposed method.",
"It is a challenging dataset consisting of 70 handwritten document pages and 15,419 segmented word images from the Bentham collection.",
"The documents have been written by different authors in varying styles, font-sizes, with crossed-out words.",
"The Bentham collection contains historical manuscripts on law and moral philosophy handwritten by Jeremy Bentham (1748-1832) over a period of 60 years, and some handwritten documents from his secretarial staff.",
"This dataset was used in the ICDAR2015 competition on keyword spotting for handwritten documents (KWS-2015 competition) .",
"In order to evaluate the performance of the proposed word annotation method, bounding box labeling correction is calculated as the difference in area (in pixels) between the original ground truth available from dataset, and the method corrected bounding box.",
"Let the areas corresponding to the original bounding box and the corrected bounding box be $O$ and $C$ , respectively.",
"The labeling correction is defined as: $Correction = \\mid O - C \\mid $ In the ideal case, the correction is 0.",
"A higher value corresponding to the correction indicates inaccuracy and imprecision in the corresponding ground truth bounding box.",
"Since the proposed method allows users to generate missing ground truth annotations on-the-fly, correction in user generated labels is also taken into account.",
"Therefore, the correction can also be defined as the percentage of difference in area (in pixels) between the user annotated bounding box and the corrected bounding box, relative to the larger bounding box, as follows: $Correction = \\mid U - C \\mid $ $\\textit {Relative Correction (\\%)} = \\Bigg \\vert \\frac{U - C}{max(U,C)} \\Bigg \\vert \\times 100$ where, $U$ is the area corresponding to the user annotated bounding box.",
"The percentage of relative correction quantitatively signifies the word annotation correction achieved by the proposed method.",
"Table REF and Table REF present word labeling correction results corresponding to original ground truth from the dataset and user annotated ground truth.",
"In Table REF , it can be observed that ground truth is unavailable in the Esposalles dataset for the query words $Anlefa$ , $franca$ , $Candia$ , $popular$ , $parayre$ and $eleonor$ from page 2.",
"However, ground truth is dynamically generated for these words by user annotation on-the-fly with automatic correction using the proposed method.",
"Similarly, Table REF presents the efficacy of the proposed method with reference to the KWS-2015 Bentham dataset where ground truth is unavailable for the query words $power$ on page 6, $Law$ on page 10, $knowlege$ on page 13 and $demand$ on page 20.",
"Table REF and Table REF present rectangle bounding box coordinates, [x-coordinate y-coordinate width height], for ground truth from dataset, user annotated ground truth, and the proposed method corrected ground truth for test query words.",
"Table: Experimental results representing the ground truth bounding box generated on random page 2 from the Esposalles dataset with 4 test words.",
"BB refers to Bounding Box, denoted as [x-coordinate y-coordinate width height] of the rectangle.Table: Experimental results representing the ground truth bounding box (BB) generated for 6 test words from the KWS-2015 Bentham dataset.Word annotation results on both datasets are visualised in Fig.",
"REF and Fig.",
"REF .",
"The impact of the proposed word annotation method is demonstrated by the difference in area covered by the user annotated bounding box (in red) and the method corrected bounding box (in green).",
"The proposed method performs background removal on word images in order to capture a noise free accurate representation of the query word.",
"The proposed method performs efficiently for the document images from the Esposalles dataset, as can be seen in Fig.",
"REF .",
"However, the algorithm will fail if the user generated bounding box is too large, and captures multiple words and lines.",
"There are some test failure cases to be taken into account with reference to the KWS-2015 Bentham dataset.",
"To a certain extent, the proposed method struggles in identifying very fine and thin pen strokes on a degraded paper.",
"This is because thresholding is applied on the final result before generating the corrected bounding box annotation to remove the noise due to bleed through.",
"The thresholding also removes very thin segments of the word by mistaking it as noise.",
"Therefore, the proposed algorithm sometimes misses certain minor details of the word.",
"Some of the test failure cases are highlighted in Fig.",
"REF , using the preprocessed images for better visualisation of the challenging cases.",
"For example, in Fig.",
"REF (h), the proposed method missed a small segment of the character $e$ in the word $knowlege$ .",
"The authors believe that this issue can be taken into account by using automatic thresholding for noise removal.",
"Figure: Word annotation results on the Esposalles dataset with 10 sample query words.",
"Red bounding box corresponds to user annotated labels.",
"Green bounding box corresponds to method generated corrected labels.",
"The final output is a cleaned word with accurate annotation.",
"Figure best viewed in color.Figure: Word annotation results on preprocessed document images from the KWS-2015 Bentham dataset with 10 sample query words.",
"Red bounding box corresponds to user annotated labels.",
"Green bounding box corresponds to method generated corrected labels.",
"Figure best viewed in color."
],
[
"Conclusion",
"A simple and efficient method to perform on-the-fly annotation of degraded historical handwritten text is presented in this paper.",
"The novelty lies in dynamic generation of noise-free ground truth bounding box labels and automatic correction of inaccurate annotations for text in degraded historical documents.",
"The experimental results on well-known datasets containing high levels of degradation demonstrate the effectiveness of the proposed method.",
"As future work, automatic thresholding and parameter optimisation will be added to the algorithm, and the performance will be tested for the generation of character annotations.",
"The ideas presented herein will be scaled to aid generation and correction of ground truth annotations for heavily degraded archival databases."
],
[
"Acknowledgment",
"This work has been partially supported by the eSSENCE strategic collaboration on eScience and the Riksbankens Jubileumsfond (Dnr NHS14-2068:1)."
]
] | 1709.01775 |
[
[
"Effective computation of $\\mathrm{SO}(3)$ and $\\mathrm{O}(3)$ linear\n representations symmetry classes"
],
[
"Abstract We propose a general algorithm to compute all the symmetry classes of any $\\mathrm{SO}(3)$ or $\\mathrm{O}(3)$ linear representation.",
"This method relies on the introduction of a binary operator between sets of conjugacy classes of closed subgroups, called the clips.",
"We compute explicit tables for this operation which allows to solve definitively the problem."
],
[
"Introduction",
"The problem of finding the symmetry classes (also called isotropy classes) of a given Lie group linear representation is a difficult task in general, even for a compact group, where their number is known to be finite [26], [22].",
"It is only in 1996 that Forte–Vianello [13] were able to define clearly the symmetry classes of tensor spaces.",
"Such tensor spaces, with natural $\\mathrm {O}(3)$ and $\\mathrm {SO}(3)$ representations, appear in continuum mechanics via linear constitutive laws.",
"Thanks to this clarification, Forte–Vianello obtained for the first time the 8 symmetry classes of the $\\mathrm {SO}(3)$ reducible representation on the space of Elasticity tensors.",
"One goal was to clarify and correct all the attempts already done to model the notion of symmetry in continuum mechanics, as initiated by Curie [8], but strongly influenced by crystallography.",
"The results were contradictory - some authors announced nine different elasticity anisotropies [21] while others announced 10 of them [31], [19], [10], [17].",
"Following Forte–Vianello, similar results were obtained in piezoelectricity [33], photoelasticity [14] and flexoelasticity [20].",
"Besides these results on symmetry classes in continuum mechanics, the subject has also been active in the Mathematical community, especially due to its importance in Bifurcation theory.",
"For instance, Michel in [23], obtained the isotropy classes for irreducible $\\mathrm {SO}(3)$ representations.",
"These results were confirmed by Ihrig and Golubitsky [18] and completed by the symmetry classes for $\\mathrm {O}(3)$ .",
"Later, they were corrected by Chossat & all in [6].",
"Thereafter, Chossat–Guyard [4] calculated the symmetry classes of a direct sum of two irreducible representations of $\\mathrm {SO}(3)$.",
"In this paper, we propose an algorithm – already mentioned in [28], [29], [27] – to obtain the finite set of symmetry classes for any $\\mathrm {O}(3)$ or $\\mathrm {SO}(3)$ representation.",
"Such an algorithm uses a binary operation defined over the set of conjugacy classes of a given group $G$ and that we decided to call the clips operation.",
"This operation was almost formulated in [4], but with no specific name, and only computed for $\\mathrm {SO}(3)$ closed subgroups.",
"As mentioned in [5], the clips operation allows to compute the set of symmetry classes $\\mathfrak {I}(V)$ of a direct sum $V=V_{1}\\oplus V_{2}$ of linear representations of a group $G$ , if we know the symmetry classes for each individual representations $\\mathfrak {I}(V_{1})$ and $\\mathfrak {I}(V_{2})$ .",
"We compute clips tables for all conjugacy classes of closed subgroups of $\\mathrm {O}(3)$ and $\\mathrm {SO}(3)$ .",
"The clips table for $\\mathrm {SO}(3)$ was obtained in [4], but we give here more detail on the proof and extend the calculation to conjugacy classes of closed $\\mathrm {O}(3)$ subgroups.",
"These results allow to obtain, in a finite step process, the set of symmetry classes for any reducible $\\mathrm {O}(3)$ or $\\mathrm {SO}(3)$ -representation, so that for instance we directly obtain the 16 symmetry classes of Piezoelectric tensors.",
"The subject being particularly important for applications to tensorial properties of any order, the necessity to convince of the correctness of the Clips tables require a complete/correct full article with sound proofs.",
"Present article is therefore intended to be a final point to the theoretical problem and to the effective calculation of symmetry classes.",
"Of course, we try to show the direct interest in the mechanics of continuous media (for this references to [28], [29] are important), but we have no other choice than to insist on the mathematical formulation of the problem.",
"The paper is organized as follow.",
"In sec:clips-theory, which is close to [5], the theory of clips is introduced for a general group $G$ and applied in the context of symmetry classes where it is shown that isotropy classes of a direct sum corresponds to the clips of their respective isotropy classes.",
"In sec:closed-subgroups, we recall classical results on the classification of closed subgroups of $\\mathrm {SO}(3)$ and $\\mathrm {O}(3)$ up to conjugacy.",
"Models for irreducible representations of $\\mathrm {O}(3)$ and $\\mathrm {SO}(3)$ and their symmetry classes are recalled in sec:irreducible-representations.",
"We then provide in subsec:MechanicalProp some applications to tensorial mechanical properties, as the non classical example of Cosserat elasticity.",
"The clips tables for $\\mathrm {SO}(3)$ and $\\mathrm {O}(3)$ are presented in sec:clips-tables.",
"The details and proofs of how to obtain these tables are provided in sec:proofs-SO3 and sec:proofs-O3."
],
[
"A general theory of clips",
"Given a group $G$ and a subgroup $H$ of $G$ , the conjugacy class of $H$ $[H] := \\left\\lbrace gHg^{-1},\\quad g\\in G\\right\\rbrace $ is a subset of $\\mathcal {P}(G)$ .",
"We define $\\mathrm {Conj}(G)$ to be the set of all conjugacy classes of a given group $G$ : $\\mathrm {Conj}(G) := \\left\\lbrace [H],\\quad H\\subset G\\right\\rbrace .$ Recall that, on $\\mathrm {Conj}(G)$ , there is a pre-order relation induced by inclusion.",
"It is defined as follows: $[H_{1}] \\preceq [H_{2}] \\quad \\text{if $H_{1}$ is conjugate to a subgroup of $H_{2}$}.$ When restricted to the closed subgroups of a topological compact group, this pre-order relation becomes a partial order [3] and defines the poset (partial ordered set) of conjugacy classes of closed subgroups of $G$ .",
"We now define a binary operation called the clips operation on the set $\\mathrm {Conj}(G)$ .",
"Definition 2.1 Given two conjugacy classes $[H_{1}]$ and $[H_{2}]$ of a group $G$ , we define their clips as the following subset of conjugacy classes: $[H_{1}] \\circledcirc [H_{2}] := \\left\\lbrace [H_{1} \\cap gH_{2}g^{-1}],\\quad g \\in G\\right\\rbrace .$ This definition immediately extends to two families (finite or infinite) $\\mathcal {F}_{1}$ and $\\mathcal {F}_{2}$ of conjugacy classes: $\\mathcal {F}_{1} \\circledcirc \\mathcal {F}_{2} := \\bigcup _{[H_{i}] \\in \\mathcal {F}_{i}} [H_{1}] \\circledcirc [H_{2}].$ Remark 2.2 The clips operation was already introduced, with no specific name, in [4], the notation being $\\mathbf {P}(H_1,H_2)$ .",
"In this article, the author only focus on the $\\mathrm {SO}(3)$ case, with no meaning to deal with a general theory.",
"This clips operation defined thus a binary operation on the set $\\mathcal {P}(\\mathrm {Conj}(G))$ which is associative and commutative.",
"We have moreover $[{1}]\\circledcirc [H] = \\left\\lbrace [{1}] \\right\\rbrace \\text{ and } [G]\\circledcirc [H] = \\left\\lbrace [H]\\right\\rbrace ,$ for every conjugacy class $[H]$ , where ${1}:=\\left\\lbrace e\\right\\rbrace $ and $e$ is the identity element of $G$ .",
"Consider now a linear representation $(V,\\rho )$ of the group $G$ .",
"Given $\\mathbf {v}\\in V$ , its isotropy group (or symmetry group) is defined as $G_{\\mathbf {v}} := \\left\\lbrace g\\in G,\\quad g\\cdot \\mathbf {v}=\\mathbf {v}\\right\\rbrace $ and its isotropy class is the conjugacy class $[G_{\\mathbf {v}}]$ of its isotropy group.",
"The isotropy classes (or orbit types) of the representation $V$ is the family of all isotropy classes of vectors $\\mathbf {v}$ in $V$ : $\\mathfrak {I}(V) := \\left\\lbrace [G_{\\mathbf {v}}]; \\; \\mathbf {v}\\in V\\right\\rbrace .$ The central observation is that the isotropy classes of a direct sum of representations is obtained by the clips of their respective isotropy classes.",
"Lemma 2.3 Let $V_{1}$ and $V_{2}$ be two linear representations of $G$ .",
"Then $\\mathfrak {I}(V_{1}\\oplus V_{2})=\\mathfrak {I}(V_{1})\\circledcirc \\mathfrak {I}(V_{2}).$ Let $[G_{\\mathbf {v}}]$ be some isotropy class for $\\mathfrak {I}(V_{1}\\oplus V_{2})$ and write $\\mathbf {v}=\\mathbf {v}_{1}+\\mathbf {v}_{2}$ where $\\mathbf {v}_{i}\\in V_{i}$ .",
"Note first that $G_{\\mathbf {v}_{1}}\\cap G_{\\mathbf {v}_{2}}\\subset G_{\\mathbf {v}}$ .",
"Conversely given $g\\in G_{\\mathbf {v}}$ , we get $g\\cdot \\mathbf {v}=g\\cdot \\mathbf {v}_{1}+g\\cdot \\mathbf {v}_{2}=\\mathbf {v},\\quad g\\cdot \\mathbf {v}_{i}\\in V_{i},$ and thus $g\\cdot \\mathbf {v}_{i}= \\mathbf {v}_{i}$ .",
"This shows that $G_{\\mathbf {v}} = G_{\\mathbf {v}_{1}}\\cap G_{\\mathbf {v}_{2}}$ and therefore that $\\mathfrak {I}(V_{1}\\oplus V_{2})\\subset \\mathfrak {I}(V_{1})\\circledcirc \\mathfrak {I}(V_{2}).$ Conversely, let $[H]=[H_{1}\\cap gH_{2}g^{-1}]$ in $\\mathfrak {I}(V_{1})\\circledcirc \\mathfrak {I}(V_{2})$ where $H_{i} = G_{\\mathbf {v}_{i}}$ for some vectors $\\mathbf {v}_{i} \\in V_{i}$ .",
"Then, if we set $\\mathbf {v}=\\mathbf {v}_{1}+g\\cdot \\mathbf {v}_{2},$ we have $G_{\\mathbf {v}}=H_{1}\\cap gH_{2}g^{-1}$ , as before, which shows that $[H_{1}\\cap gH_{2}g^{-1}] \\in \\mathfrak {I}(V_{1}\\oplus V_{2})$ and achieves the proof.",
"Using this lemma, we deduce a general algorithm to obtain the isotropy classes $\\mathfrak {I}(V)$ of a finite dimensional representation of a reductive algebraic group $G$ , provided we know: a decomposition $V=\\bigoplus _{i} W_{i}$ into irreducible representations $W_{i}$ .",
"the isotropy classes $\\mathfrak {I}(W_{i})$ for the irreducible representations $W_{i}$ ; the tables of clips operations $[H_{1}]\\circledcirc [H_{2}]$ between conjugacy classes of closed subgroups $[H_{i}]$ of $G$ .",
"In the sequel of this paper, we will apply successfully this program to the linear representations of $\\mathrm {SO}(3)$ and $\\mathrm {O}(3)$ ."
],
[
"Closed subgroups of $\\mathrm {O}(3)$",
"Every closed subgroup of $\\mathrm {SO}(3)$ is conjugate to one of the following list [16] $\\mathrm {SO}(3),\\, \\mathrm {O}(2),\\, \\mathrm {SO}(2),\\, \\mathbb {D}_{n} (n \\ge 2),\\, \\mathbb {Z}_{n} (n \\ge 2),\\, \\mathbb {T},\\, \\mathbb {O},\\, \\mathbb {I},\\, \\text{and}\\, {1}$ where: $\\mathrm {O}(2)$ is the subgroup generated by all the rotations around the $z$ -axis and the order 2 rotation $r : (x,y,z)\\mapsto (x,-y,-z)$ around the $x$ -axis; $\\mathrm {SO}(2)$ is the subgroup of all the rotations around the $z$ -axis; $\\mathbb {Z}_{n}$ is the unique cyclic subgroup of order $n$ of $\\mathrm {SO}(2)$ ($\\mathbb {Z}_{1} = \\left\\lbrace I\\right\\rbrace $ ); $\\mathbb {D}_{n}$ is the dihedral group.",
"It is generated by $\\mathbb {Z}_{n}$ and $r :(x,y,z)\\mapsto (x,-y,-z)$ ($\\mathbb {D}_{1} = \\left\\lbrace I\\right\\rbrace $ ); $\\mathbb {T}$ is the tetrahedral group, the (orientation-preserving) symmetry group of the tetrahedron $\\mathcal {T}_{0}$ defined in fig:cube0.",
"It has order 12; $\\mathbb {O}$ is the octahedral group, the (orientation-preserving) symmetry group of the cube $\\mathcal {C}_{0}$ defined in fig:cube0.",
"It has order 24; $\\mathbb {I}$ is the icosahedral group, the (orientation-preserving) symmetry group of the dodecahedron $\\mathcal {D}_{0}$ in fig:dode.",
"It has order 60; 1 is the trivial subgroup, containing only the unit element.",
"The poset of conjugacy classes of closed subgroups of $\\mathrm {SO}(3)$ is completely described by the following inclusion of subgroups [16]: $& \\mathbb {Z}_{n} \\subset \\mathbb {D}_{n} \\subset \\mathrm {O}(2) \\qquad (n \\ge 2); \\\\& \\mathbb {Z}_{n} \\subset \\mathbb {Z}_{m} \\text{ and } \\mathbb {D}_{n} \\subset \\mathbb {D}_{m}, \\qquad (\\text{if $n$ divides $m$}); \\\\& \\mathbb {Z}_{n} \\subset \\mathrm {SO}(2) \\subset \\mathrm {O}(2) \\qquad (n \\ge 2);$ completed by $[\\mathbb {Z}_{2}]\\preceq [\\mathbb {D}_{n}]$ ($n\\ge 2$ ) and by the arrows in fig:SO3-lattice (note that an arrow between the classes $[H_{1}]$ and $[H_{2}]$ means that $[H_{1}]\\preceq [H_{2}]$ ), taking account of the exceptional subgroups $\\mathbb {O}, \\mathbb {T}, \\mathbb {I}$ .",
"Figure: Exceptional conjugacy classes of closed SO (3)\\mathrm {SO}(3) subgroupsClassification of $\\mathrm {O}(3)$ -closed subgroups is more involving [18], [32] and has been described using three types of subgroups.",
"Given a closed subgroup $\\Gamma $ of $\\mathrm {O}(3)$ this classification runs as follows.",
"Type I A subgroup $\\Gamma $ is of type I if it is a subgroup of $\\mathrm {SO}(3)$ ; Type II A subgroup $\\Gamma $ is of type II if $-I\\in \\Gamma $ .",
"In that case, $\\Gamma $ is generated by some subgroup $K$ of $\\mathrm {SO}(3)$ and $-I$ ; Type III A subgroup $\\Gamma $ is of type III if $-I\\notin \\Gamma $ and $\\Gamma $ is not a subgroup of $\\mathrm {SO}(3)$ .",
"The description of type III subgroups requires more details.",
"We will denote by $\\mathbf {Q}(\\mathbf {v};\\theta )\\in \\mathrm {SO}(3)$ the rotation around $\\mathbf {v}\\in \\mathbb {R}^3$ with angle $\\theta \\in [open right]{0}{2\\pi }$ and by $\\sigma _\\mathbf {v}\\in \\mathrm {O}(3)$ , the reflection through the plane normal to $\\mathbf {v}$ .",
"Finally, we fix an arbitrary orthonormal frame $(\\mathbf {i},\\mathbf {j},\\mathbf {k})$ , and we introduce the following definitions.",
"$\\mathbb {Z}^{-}_{2}$ is the order 2 reflection group generated by $\\sigma _\\mathbf {i}$ (where $\\mathbb {Z}_{1}^- = \\left\\lbrace {1}\\right\\rbrace $ ); $\\mathbb {Z}^{-}_{2n}$ ($n\\ge 2$ ) is the group of order $2n$ , generated by $\\mathbb {Z}_{n}$ and $-\\mathbf {Q}\\left(\\mathbf {k};\\displaystyle {\\frac{\\pi }{n}}\\right)$ (see (REF )); $\\mathbb {D}_{2n}^h$ ($n\\ge 2$ ) is the group of order $4n$ generated by $\\mathbb {D}_{n}$ and $-\\mathbf {Q}\\left(\\mathbf {k};\\displaystyle {\\frac{\\pi }{n}}\\right)$ (see (REF )); $\\mathbb {D}_{n}^{v}$ ($n\\ge 2$ ) is the group of order $2n$ generated by $\\mathbb {Z}_{n}$ and $\\sigma _\\mathbf {i}$ (where $\\mathbb {D}_{1}^{v} = \\left\\lbrace {1}\\right\\rbrace $ ); $\\mathrm {O}(2)^{-}$ is generated by $\\mathrm {SO}(2)$ and $\\sigma _\\mathbf {i}$ .",
"These planar subgroups are completed by the subgroup $\\mathbb {O}^{-}$ which is of order 24 (see subsubsec:OMoins and (REF ) for details).",
"The poset of conjugacy classes of closed subgroups of $\\mathrm {O}(3)$ is given in fig:cubic-sub-lattice for $\\mathbb {O}^{-}$ subgroups and in fig:O3-lattice for $\\mathrm {O}(3)$ subgroups.",
"Figure: Poset of closed 𝕆 - \\mathbb {O}^{-} subgroupsFigure: Poset of closed O(3)\\mathrm {O}(3) subgroups"
],
[
"Symmetry classes for irreducible representations",
"Let $\\mathcal {P}_{n}(\\mathbb {R}^{3})$ be the space of homogeneous polynomials of degree $n$ on $\\mathbb {R}^{3}$ .",
"We have two natural representations of $\\mathrm {O}(3)$ on $\\mathcal {P}_{n}(\\mathbb {R}^{3})$ .",
"The first one, noted $\\rho _{n}$ is given by $[\\rho _{n}(\\mathrm {p})](\\mathbf {x}):=\\mathrm {p}(g^{-1}\\mathbf {x}),\\quad g\\in G,\\quad \\mathbf {x}\\in \\mathbb {R}^{3}$ whereas the second one, noted $\\rho ^{*}_{n}$ , is given by $[\\rho ^{*}_{n}(\\mathrm {p})](\\mathbf {x}):=\\det (g)\\mathrm {p}(g^{-1}\\mathbf {x}),\\quad g\\in G,\\quad \\mathbf {x}\\in \\mathbb {R}^{3}.$ Note that both of them induce the same representation $\\rho _{n}$ of $\\mathrm {SO}(3)$ .",
"Let $\\mathcal {H}_{n}(\\mathbb {R}^{3})\\subset \\mathcal {P}_{n}(\\mathbb {R}^{3})$ be the subspace of homogeneous harmonic polynomials of degree $n$ (polynomials with null Laplacian).",
"It is a classical fact [16] that $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n})$ and $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n}^{*})$ ($n \\ge 0$ ) are irreducible $\\mathrm {O}(3)$ -representations, and each irreducible $\\mathrm {O}(3)$ -representation is isomorphic to one of them.",
"Models for irreducible representations of $\\mathrm {SO}(3)$ reduce to $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n})$ ($n \\ge 0$ ).",
"Remark 4.1 Other classical models for $\\mathrm {O}(3)$ and $\\mathrm {SO}(3)$ irreducible representations, used in mechanics [13], are given by spaces of harmonic tensors (i.e.",
"totally symmetric traceless tensors).",
"The isotropy classes for irreducible representations of $\\mathrm {SO}(3)$ was first obtained by Michel [23].",
"Same results were then obtained and completed in the $\\mathrm {O}(3)$ case by Ihrig-Golubistky [18] and then by Chossat and al [6].",
"Theorem 4.2 The isotropy classes for the $\\mathrm {SO}(3)$ -representation $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n})$ are: $[{1}]$ for $n\\ge 3$ ; $[\\mathbb {Z}_k]$ for $2 \\le k \\le n$ if $n$ is odd and $2 \\le k \\le n/2$ if $n$ is even; $[\\mathbb {D}_k]$ for $2 \\le k \\le n$ ; $[\\mathbb {T}]$ for $n=3,6$ , 7 or $n\\ge 9$ ; $[\\mathbb {O}]$ for $n\\ne 1,2,3,5,7,11$ ; $[\\mathbb {I}]$ for $n=6,10,12,15,16,18$ or $n\\ge 20$ and $n\\ne 23,29$ ; $[\\mathrm {SO}(2)]$ for $n$ odd; $[\\mathrm {O}(2)]$ for $n$ even; $[\\mathrm {SO}(3)]$ for any $n$ .",
"Remark 4.3 The list in Theorem REF is similar to the list in [23] and [6].",
"In [18] (for $\\mathrm {SO}(3)$ irreducible representations) : $[\\mathbb {T}]$ is an isotropy class for $n=6,7$ and $n\\ge 9$ ; $[\\mathbb {O}]$ is an isotropy class for $n\\ne 1,2,5,7,11$ .",
"Such lists are different from (REF ) and (REF ) in our Theorem REF .",
"But according to [18], $[\\mathbb {T}]$ is a maximum isotropy class for $n=3$ .",
"We have thus corrected this error in Theorem REF .",
"Theorem 4.4 The isotropy classes for the $\\mathrm {O}(3)$ -representations $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n})$ (for $n$ odd) and $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n}^{*})$ (for $n$ even) are: $[{1}]$ for $n\\ge 3$ ; $[\\mathbb {Z}_k]$ for $2\\le k\\le n/2$ ; $[\\mathbb {Z}_{2k}^{-}]$ for $k\\le \\dfrac{n}{3}$ ; $[\\mathbb {D}_k]$ for $2 \\le k \\le n/2$ if $n$ is odd and for $2 \\le k \\le n$ if $n$ is even.",
"$[\\mathbb {D}_k^{v}]$ for $2 \\le k \\le n$ if $n$ is odd and $2 \\le k \\le n/2$ if $n$ is even; $[\\mathbb {D}_{2k}^h]$ for $2 \\le k\\le n$ , except $\\mathbb {D}_4^h$ for $n=3$ ; $[\\mathbb {T}]$ for $n\\ne 1,2,3,5,7,8,11$ ; $[\\mathbb {O}]$ for $n\\ne 1,2,3,5,7,11$ ; $[\\mathbb {O}^{-}]$ for $n\\ne 1,2,4,5,8$ ; $[\\mathbb {I}]$ for $n=6,10,12,15,16,18$ or $n\\ge 20$ and $n\\ne 23,29$ ; $[\\mathrm {O}(2)]$ when $n$ is even ; $[\\mathrm {O}(2)^{-}]$ when $n$ is odd.",
"Remark 4.5 The list in Theorem REF for $n$ odd is similar to the list in [6].",
"In [18] (for $\\mathrm {O}(3)$ irreducible representations), $[\\mathbb {T}]$ is an isotropy class for $n\\ne 1,2,5,7,8,11$ (which is different from the list (REF ) in our Theorem REF above).",
"But according to [15], [33] and [6], $[\\mathbb {T}]$ is not an isotropy class in the case $n=3$ , and we corrected this error in the list (REF ) of Theorem REF ."
],
[
"$\\mathrm {SO}(3)$ closed subgroups",
"The resulting conjugacy classes for the clips operation of closed $\\mathrm {SO}(3)$ subgroups are given in tab:SO3-clips.",
"The following notations have been used: $d & := \\gcd (m,n), & d_{2} & := \\gcd (n,2), & k_{2} & := 3-d_{2},\\\\d_{3} & := \\gcd (n,3), & d_{5} & := \\gcd (n,5) \\\\dz & :=2, \\text{ if $m$ and $n$ even}, & dz & :=1, \\text{ otherwise}, \\\\d_4 & := 4, \\text{ if } 4 \\text{ divide } n, & d_4 & := 1, \\text{ otherwise}, \\\\\\mathbb {Z}_{1}&=\\mathbb {D}_{1}:={1}.$ Table: Clips operations for SO (3)\\mathrm {SO}(3)Remark 5.1 The clips operations $[\\mathbb {T}]\\circledcirc [\\mathbb {T}]$ and $[\\mathbb {T}]\\circledcirc [\\mathbb {O}]$ were wrong in [28], [27], since for instance the isotropy class $[\\mathbb {D}_{2}]$ was omitted.",
"Example 5.2 (Isotropy classes for a family of $n$ vectors) For one vector, we get $\\mathfrak {I}(\\mathcal {H}_{1}(\\mathbb {R}^{3})) = \\left\\lbrace [\\mathrm {SO}(2)],[\\mathrm {SO}(3)]\\right\\rbrace .$ From tab:SO3-clips, we deduce that the isotropy classes for a family of $n$ vectors ($n \\ge 2$ ) is $\\mathfrak {I}\\left(\\bigoplus _{k=1}^{n} \\mathcal {H}_{1}(\\mathbb {R}^{3})\\right) = \\left\\lbrace [{1}],[\\mathrm {SO}(2)],[\\mathrm {SO}(3)]\\right\\rbrace .$ Example 5.3 (Isotropy classes for a family of $n$ quadratic forms) The space of quadratic forms on $\\mathbb {R}^{3}$ , $\\mathrm {S}_2(\\mathbb {R}^{3})$ , decomposes into two irreducible components (deviatoric and spherical tensors for the mechanicians): $\\mathrm {S}_2(\\mathbb {R}^{3}) = \\mathcal {H}_{2}(\\mathbb {R}^{3}) \\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3}).$ We get thus $\\mathfrak {I}(\\mathrm {S}_2(\\mathbb {R}^{3})) = \\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3}))=\\left\\lbrace [\\mathbb {D}_{2}],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\right\\rbrace .$ The useful part of tab:SO3-clips, for our purpose, reads: Table: NO_CAPTIONWe deduce therefore that the set of isotropy classes for a family of $n$ quadratic forms ($n \\ge 2$ ) is $\\mathfrak {I}\\left( \\bigoplus _{k=1}^{n} \\mathrm {S}_2(\\mathbb {R}^{3}) \\right) = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\right\\rbrace .$"
],
[
"$\\mathrm {O}(3)$ closed subgroups",
"Let us first consider an $\\mathrm {O}(3)$ -representation where $-I$ act as $-\\mathrm {Id}$ (meaning that this representation doesn't reduce to some $\\mathrm {SO}(3)$ representation).",
"In such a case, only the null vector can be fixed by $-\\mathrm {Id}$ , and so type II subgroups never appear as isotropy subgroups.",
"In that case, we need only to focus on clips operations between type I and type III subgroups, and then between type III subgroups, since clips operations between type I subgroups have already been considered in tab:SO3-clips.",
"For type III subgroups as detailed in sec:proofs-O3 we have: Lemma 5.4 Let $H_{1}$ be some type III closed subgroup of $\\mathrm {O}(3)$ and $H_{2}$ be some type I closed subgroup of $\\mathrm {O}(3)$ .",
"Then we have $H_{1}\\cap H_{2}=(H_{1}\\cap \\mathrm {SO}(3))\\cap H_{2},$ and for every closed subgroup $H$ of $\\mathrm {SO}(3)$ , we get: $[\\mathbb {Z}_{2}^{-}]\\circledcirc [H] & =\\left\\lbrace [{1}]\\right\\rbrace , & [\\mathbb {Z}_{2n}^{-}]\\circledcirc [H] & =[\\mathbb {Z}_{n}]\\circledcirc [H], \\\\[\\mathbb {D}_{n}^{v}]\\circledcirc [H] & =[\\mathbb {Z}_{n}]\\circledcirc [H], & [\\mathbb {D}_{2n}^h]\\circledcirc [H] & =[\\mathbb {D}_{n}]\\circledcirc [H], \\\\[\\mathbb {O}^{-}]\\circledcirc [H] & =[\\mathbb {T}]\\circledcirc [H], & [\\mathrm {O}(2)^{-}]\\circledcirc [H] & =[\\mathrm {SO}(2)]\\circledcirc [H].$ The resulting conjugacy classes for the clips operation for type III subgroups are given in tab:otrglobal, where the following notations have been used: $d & := \\gcd (n,m), & d_{2}(n) & := \\gcd (n,2), \\\\d_{3}(n) & := \\gcd (n,3), & i(n) & := 3-\\gcd (2,n), \\\\\\mathbb {Z}_{1}^-&=\\mathbb {D}_{1}^v=\\mathbb {D}_{2}^h={1}.$ Table: Clips operations on type III O(3)\\mathrm {O}(3)-subgroupsRemark 5.5 One misprint in [27], [29] for clips operation $[\\mathrm {O}(2)^{-}]\\circledcirc [\\mathbb {D}_{2m}^h]$ has been corrected: the conjugacy class $[\\mathbb {Z}_{2}]$ appears for $m$ even (and not for $m$ odd).",
"Figure: NO_CAPTIONFigure: 𝔻 2n h \\mathbb {D}_{2n}^h and ℤ 2n - \\mathbb {Z}_{2n}^{-}Figure: 𝔻 2n h \\mathbb {D}_{2n}^h and 𝔻 n v \\mathbb {D}_{n}^{v}Figure: 𝔻 2n h \\mathbb {D}_{2n}^h and 𝔻 2m h \\mathbb {D}_{2m}^{h}Figure: NO_CAPTION"
],
[
"Application to tensorial mechanical properties",
"We propose here some direct applications of our results for many different tensorial spaces, each one being endowed with the natural $\\mathrm {O}(3)$ representation.",
"Such tensorial spaces occur for instance in the modeling of mechanical properties.",
"The main idea is to use the irreducible decomposition, also known as the harmonic decomposition [2], [13].",
"We then use clips operations given in Table REF and Table REF .",
"Note that we don't need to know explicit irreducible decomposition of those tensorial representations.",
"From now on, we define $\\mathcal {H}_{n}(\\mathbb {R}^{3})^*$ to be $\\mathrm {O}(3)$ representation given by $(\\mathcal {H}_{n}(\\mathbb {R}^{3}),\\rho _{n}^*)$ .",
"Furthermore, we define $\\mathcal {H}_{n}(\\mathbb {R}^{3})^{\\oplus k}:=\\bigoplus _{i=1}^{k} \\mathcal {H}_{n}(\\mathbb {R}^{3}),\\quad \\mathcal {H}_{n}(\\mathbb {R}^{3})^{\\oplus k*}:=\\bigoplus _{i=1}^{k} \\mathcal {H}_{n}(\\mathbb {R}^{3})^*.$ We propose here to give some specific irreducible decompositions, related to mechanical theory, without any further details.",
"We explain the use of clips operation in the classical case of Elasticity, all other examples being done in the same way."
],
[
"Classical results",
"We first give some classsical results we obtain here directly (1) Elasticity [13]: $\\mathrm {SO}(3)$ tensor space $\\mathbb {E}\\mathrm {la}\\simeq \\mathcal {H}_{4}(\\mathbb {R}^{3})\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 2}\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{\\oplus 2}.$ We have from Theorem REF $\\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3}))&=\\lbrace [\\mathbb {D}_2],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\rbrace , \\\\\\mathfrak {I}(\\mathcal {H}_{4}(\\mathbb {R}^{3}))&=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],[\\mathbb {O}],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\rbrace $ so, to obtain $\\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 2})$ we use the clips table Table: NO_CAPTIONso that $\\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 2})&=\\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3}))\\circledcirc \\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3})) \\\\&=\\left( [\\mathbb {D}_2]\\circledcirc [\\mathbb {D}_2]\\right) \\bigcup \\left( [\\mathbb {D}_2]\\circledcirc [\\mathrm {O}_2]\\right) \\bigcup \\left( [\\mathrm {O}_2]\\circledcirc [\\mathrm {O}_2]\\right) \\\\&=\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathrm {O}(2)]\\rbrace $ and now we conclude using the clips table coming from the clips operation $\\mathfrak {I}(\\mathbb {E}\\mathrm {la})=\\mathfrak {I}(\\mathcal {H}_{4}(\\mathbb {R}^{3}))\\circledcirc \\mathfrak {I}(\\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 2}).$ Finally we obtain 8 symmetry classes $\\mathfrak {I}(\\mathbb {E}\\mathrm {la})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],[\\mathbb {O}],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\rbrace .$ (2) Photoelasticity [14]: $\\mathrm {SO}(3)$ tensor space $\\mathbb {P}\\mathrm {la}\\simeq \\mathcal {H}_{4}(\\mathbb {R}^{3})\\oplus &\\mathcal {H}_{3}(\\mathbb {R}^{3})\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 3}\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{\\oplus 2}.$ 12 symmetry classes $\\mathfrak {I}(\\mathbb {P}\\mathrm {la})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {D}_2],[\\mathbb {Z}_3],[\\mathbb {D}_3],[\\mathbb {Z}_4],[\\mathbb {D}_4],[\\mathbb {T}],[\\mathbb {O}],\\\\[\\mathrm {SO}(2)],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\rbrace .$ (3) Piezoelecricity [15]: $\\mathrm {O}(3)$ tensor space $\\mathbb {P}\\mathrm {iez}\\simeq \\mathcal {H}_{3}(\\mathbb {R}^{3})\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^*\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})^{\\oplus 2}.$ 16 symmetry classes $\\mathfrak {I}(\\mathbb {P}\\mathrm {iez})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {Z}_{3}],[\\mathbb {D}^{v}_2],[\\mathbb {D}^{v}_{3}],[\\mathbb {Z}^{-}_2],[\\mathbb {Z}^{-}_{4}],[\\mathbb {D}_2],[\\mathbb {D}_{3}],[\\mathbb {D}^{h}_4],[\\mathbb {D}^{h}_{6}],\\\\[\\mathrm {SO}(2)],[\\mathrm {O}(2)],[\\mathrm {O}(2)^{-}],[\\mathbb {O}^-],[\\mathrm {O}(3)]\\rbrace .$ Remark 5.6 Note that the symmetry classes of the space $(\\mathbb {P}\\mathrm {iez},\\mathrm {O}(3))$ appear in many different works : in Weller phD thesis [33][Theorem 3.19, p.84], where 14 symmetry classes are announced, but 15 appeared in the poset, in Weller–Geymonat [15] where they establish 14 symmetry classes, in Olive–Auffray [29] (with a typo), in Zou & al [34] (without $[\\mathrm {O}(3)]$ symmetry class), and finally in Olive [27], with 16 symmetry classes."
],
[
"Non classical results",
"We now present some non classical tensor space, coming from Cosserat elasticity [7], [9], [11], [12] and Strain gradient elasticity [24], [25], [30], [1], with their harmonic decomposition and symmetry classes.",
"(1) Classical Cosserat elasticity: $\\mathrm {SO}(3)$ tensor space $\\mathbb {C}\\mathrm {os}\\simeq \\mathcal {H}_{4}(\\mathbb {R}^{3})\\oplus &\\mathcal {H}_{3}(\\mathbb {R}^{3})\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 4}\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})^{\\oplus 2}\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{\\oplus 3}.$ 12 symmetry classes $\\mathfrak {I}(\\mathbb {C}\\mathrm {os})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {Z}_3],[\\mathbb {Z}_4],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],[\\mathbb {T}],[\\mathbb {O}],\\\\[\\mathrm {SO}(2)],[\\mathrm {O}(2)],[\\mathrm {SO}(3)]\\rbrace .$ (2) Rotational Cosserat elasticity: $\\mathrm {O}(3)$ tensor space $\\mathbb {C}\\mathrm {hi}\\simeq \\mathcal {H}_{4}(\\mathbb {R}^{3})^*\\oplus &\\mathcal {H}_{3}(\\mathbb {R}^{3})^{\\oplus 3}\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 6*}\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})^{\\oplus 6}\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{\\oplus 3*}.$ 24 symmetry classes $\\mathfrak {I}(\\mathbb {C}\\mathrm {hi})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {Z}_3],[\\mathbb {Z}_4],[\\mathbb {Z}_2^-],[\\mathbb {Z}_4^-],[\\mathbb {Z}_6^-],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],\\\\[\\mathbb {D}_2^v],[\\mathbb {D}_3^v],[\\mathbb {D}_4^v],[\\mathbb {D}_4^h],[\\mathbb {D}_6^h],[\\mathbb {D}_8^h],[\\mathbb {T}],[\\mathbb {O}],[\\mathbb {O}^-],[\\mathrm {SO}(2)],\\\\[\\mathrm {O}(2)],[\\mathrm {O}(2)^-],[\\mathrm {SO}(3)],[\\mathrm {O}(3)]\\rbrace .$ (3) Fifth-order $\\mathrm {O}(3)$ tensor space in strain gradient elasticity, given by $\\mathbb {S}\\mathrm {ge}\\simeq \\mathcal {H}_{5}(\\mathbb {R}^{3})\\oplus &\\mathcal {H}_{4}(\\mathbb {R}^{3})^{\\oplus 2*}\\oplus \\mathcal {H}_{3}(\\mathbb {R}^{3})^{\\oplus 5}\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 5*}\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})^{\\oplus 6}\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{*}.$ 29 symmetry classes $\\mathfrak {I}(\\mathbb {S}\\mathrm {ge})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {Z}_3],[\\mathbb {Z}_4],[\\mathbb {Z}_5],[\\mathbb {Z}_2^-],[\\mathbb {Z}_4^-],\\\\[\\mathbb {Z}_6^-],[\\mathbb {Z}_8^-],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],[\\mathbb {D}_5],[\\mathbb {D}_2^v],[\\mathbb {D}_3^v],[\\mathbb {D}_4^v],\\\\[\\mathbb {D}_5^v],[\\mathbb {D}_4^h],[\\mathbb {D}_6^h],[\\mathbb {D}_8^h],[\\mathbb {D}_{10}^h],[\\mathbb {T}],[\\mathbb {O}],[\\mathbb {O}^-],\\\\[\\mathrm {SO}(2)],[\\mathrm {O}(2)],[\\mathrm {O}(2)^-],[\\mathrm {SO}(3)],[\\mathrm {O}(3)]\\rbrace .$ (4) Fifth order $\\mathrm {O}(3)$ tensor space of acoustical gyrotropic tensor [30] (reducing to fourth order tensor space), given by $\\mathbb {A}\\mathrm {gy}\\simeq \\mathcal {H}_{4}(\\mathbb {R}^{3})^{*}\\oplus &\\mathcal {H}_{3}(\\mathbb {R}^{3})^{\\oplus 2}\\oplus \\mathcal {H}_{2}(\\mathbb {R}^{3})^{\\oplus 3*}\\oplus \\mathcal {H}_{1}(\\mathbb {R}^{3})^{\\oplus 2}\\oplus \\mathcal {H}_{0}(\\mathbb {R}^{3})^{*}.$ 24 symmetry classes $\\mathfrak {I}(\\mathbb {A}\\mathrm {gy})=\\lbrace [{1}],[\\mathbb {Z}_2],[\\mathbb {Z}_3],[\\mathbb {Z}_4],[\\mathbb {Z}_2^-],[\\mathbb {Z}_4^-],[\\mathbb {Z}_6^-],[\\mathbb {D}_2],[\\mathbb {D}_3],[\\mathbb {D}_4],\\\\[\\mathbb {D}_2^v],[\\mathbb {D}_3^v],[\\mathbb {D}_4^v],[\\mathbb {D}_4^h],[\\mathbb {D}_6^h],[\\mathbb {D}_8^h],[\\mathbb {T}],[\\mathbb {O}],[\\mathbb {O}^-],[\\mathrm {SO}(2)],\\\\[\\mathrm {O}(2)],[\\mathrm {O}(2)^-],[\\mathrm {SO}(3)],[\\mathrm {O}(3)]\\rbrace .$"
],
[
"Proofs for $\\mathrm {SO}(3)$",
"In this section, we provide all the details required to obtain the results in tab:SO3-clips.",
"We will start by the following definition which was introduced in [16] and happens to be quite useful for this task.",
"Definition 1.1 Let $K_{1}, K_{2}, \\cdots , K_s$ be subgroups of a given group $G$ .",
"We say that $G$ is the direct union of the $K_{i}$ and we write $G=\\biguplus _{i=1}^s K_{i}$ if $G = \\bigcup _{i=1}^s K_{i} \\qquad \\text{and} \\qquad K_{i}\\cap K_{j} = \\left\\lbrace e\\right\\rbrace ,\\quad \\forall i\\ne j.$ In the following, we will have to identify repeatedly the conjugacy class of intersections such as $H_{1}\\cap \\left(g H_{2}g^{-1}\\right),$ where $H_{1}$ and $H_{2}$ are two closed subgroups of $\\mathrm {SO}(3)$ and $g\\in \\mathrm {SO}(3)$ .",
"A useful observation is that all closed $\\mathrm {SO}(3)$ subgroups have some characteristic axes and that intersection (REF ) depends only on the relative positions of these characteristic axes.",
"As detailed below, for any subgroup conjugate to $\\mathbb {Z}_{n}$ or $\\mathbb {D}_{n}$ ($n\\ge 3$ ), the axis of an $n$ -th order rotation (in this subgroup) is called its primary axis.",
"For subgroups conjugate to $\\mathbb {D}_{n}$ ($n\\ge 3$ ), axes of order two rotations are said to be secondary axes.",
"In the special case $n=2$ , the $z$ -axis is the primary axis of $\\mathbb {Z}_{2}$ , while any of the $x$ , $y$ or $z$ axis can be considered as a primary axis of $\\mathbb {D}_{2}$ ."
],
[
"Cyclic subgroup",
"For any axis $a$ of $\\mathbb {R}^{3}$ (throughout the origin), we denote by $\\mathbb {Z}_{n}^{a}$ , the unique cyclic subgroup of order $n$ around the $a$ -axis, which is its primary axis.",
"We have then: Lemma 1.2 Let $m,n\\ge 2$ be two integers and $d=\\gcd (n,m)$ .",
"Then $[\\mathbb {Z}_{n}]\\circledcirc [\\mathbb {Z}_{m}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{d}]\\right\\rbrace .$ We have to consider intersections, such as $\\mathbb {Z}_{n}\\cap (g\\mathbb {Z}_{m}g^{-1})=\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}^{a},$ for some axis $a$ , and only two cases occur: If $a\\ne (Oz)$ , then necessarily the intersection reduces to 1.",
"If $a=(Oz)$ , then the order $r$ of a rotation in $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}$ divides both $n$ and $m$ and thus divides $d=\\gcd (m,n)$ .",
"We get therefore: $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m} \\subset \\mathbb {Z}_{d}$ .",
"But obviously, $\\mathbb {Z}_{d} \\subset \\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}$ and thus $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}=\\mathbb {Z}_{d}$ ."
],
[
"Dihedral subgroup",
"Let $b_{1}$ be the $x$ -axis and $b_{k}$ ($k=2,\\cdots ,n$ ) be the axis recursively defined by $b_{k} := \\mathbf {Q}\\left(\\mathbf {k};\\displaystyle {\\frac{\\pi }{n}}\\right)b_{k-1}.$ Then, we have $\\mathbb {D}_{n} = \\mathbb {Z}_{n} \\biguplus \\mathbb {Z}_{2}^{b_{1}} \\biguplus \\cdots \\biguplus \\mathbb {Z}_{2}^{b_{n}},$ where the $z$ -axis (corresponding to a $n$ -th order rotation) is the primary axis and the $b_{k}$ -axes (corresponding to order two rotations) are the secondary axes of this dihedral group (see fig:dihedral-second-axis).",
"Figure: Secondary axis of the dihedral group 𝔻 6 \\mathbb {D}_{6}Lemma 1.3 Let $m,n\\ge 2$ be two integers.",
"Set $d:=\\gcd (n,m)$ and $d_{2}(m):=\\gcd (m,2)$ .",
"Then, we have $[\\mathbb {D}_{n}]\\circledcirc [\\mathbb {Z}_{m}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{d_{2}(m)}],[\\mathbb {Z}_{d}] \\right\\rbrace .$ Let $\\Gamma =\\mathbb {D}_{n} \\cap g\\mathbb {Z}_{m}g^{-1}$ for $g\\in \\mathrm {SO}(3)$ .",
"From decomposition (REF ), we have to consider intersections $\\mathbb {Z}_{n}\\cap g\\mathbb {Z}_{m}g^{-1},\\quad \\mathbb {Z}_{2}^{b_{j}}\\cap g\\mathbb {Z}_{m}g^{-1}.$ which thus reduce to Lemma REF .",
"Lemma 1.4 Let $m,n\\ge 2$ be two integers.",
"Set $d := \\gcd (n,m)$ and $dz :={\\left\\lbrace \\begin{array}{ll}2 \\quad \\text{if} \\quad m \\text{ and } n \\text{ even}, \\\\1 \\quad \\text{otherwise}.\\end{array}\\right.",
"}$ Then, we have $[\\mathbb {D}_{n}]\\circledcirc [\\mathbb {D}_{m}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{dz}],[\\mathbb {Z}_{d}],[\\mathbb {D}_{d}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {D}_{n} \\cap \\left(g\\mathbb {D}_{m} g^{-1}\\right)$ .",
"If both primary axes and one secondary axis match, $\\Gamma =\\mathbb {D}_{d}$ if $d\\ne 1$ and $\\Gamma \\in [\\mathbb {Z}_{2}]$ otherwise; if only the primary axes match, $\\Gamma =\\mathbb {Z}_{d}$ ; if the angle of primary axes is $\\dfrac{\\pi }{4}$ and a secondary axis match, then $\\Gamma \\in [\\mathbb {Z}_{2}]$ if the primary axis of $g\\mathbb {D}_{m}g^{-1}$ matches with the secondary axis $(Ox)$ of $\\mathbb {D}_{n}$ (or the converse), we obtain $\\Gamma \\in [\\mathbb {D}_{2}]$ for $n$ and $m$ even and a secondary axis of $g\\mathbb {D}_{m}g^{-1}$ is $(Oz)$ , otherwise we obtain $\\Gamma \\in [\\mathbb {Z}_{2}]$ in all other cases we have $\\Gamma ={1}$ ."
],
[
"Tetrahedral subgroup",
"The (orientation-preserving) symmetry group $\\mathbb {T}$ of the tetrahedron $\\mathcal {T}_{0}:=A_{1}A_{3}A_7A_{5}$ (see fig:cube0) decomposes as (see [18]): $\\mathbb {T}= \\biguplus _{i=1}^4 \\mathbb {Z}_{3}^{\\mathbf {vt}_{i}} \\biguplus _{j=1}^3 \\mathbb {Z}_{2}^{\\mathbf {et}_{j}}$ where $\\mathbf {vt}_{i}$ (resp.",
"$\\mathbf {et}_{j}$ ) are the vertices axes (resp.",
"edges axes) of the tetrahedron (see fig:cube0): $\\mathbf {vt}_{1} & := (OA_{1}), & \\mathbf {vt}_{2} & := (OA_{3}), & \\mathbf {vt}_{3} & := (OA_{5}), & \\mathbf {vt}_4 & := (OA_7),\\\\\\mathbf {et}_{1} & := (Ox), & \\mathbf {et}_{2} & :=(Oy) , & \\mathbf {et}_{3} & :=(Oz).",
"& &$ Figure: Cube 𝒞 0 \\mathcal {C}_{0} and tetrahedron 𝒯 0 :=A 1 A 3 A 7 A 5 \\mathcal {T}_{0}:=A_{1}A_{3}A_7A_{5}Corollary 1.5 Let $n\\ge 2$ be an integer.",
"Set $d_{2}(n):=\\gcd (n,2)$ and $d_{3}(n):=\\gcd (3,n)$ .",
"Then, we have $[\\mathbb {Z}_{n}] \\circledcirc [\\mathbb {T}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{d_{2}(n)}],[\\mathbb {Z}_{d_{3}(n)}]\\right\\rbrace .$ Consider $\\mathbb {T}\\cap \\mathbb {Z}_{n}^{a}$ for some axis $a$ .",
"As a consequence of Lemma REF , we need only to consider the case where $a$ is an edge axis or a face axis of the tetrahedron, reducing to the clips operations $[\\mathbb {Z}_{2}]\\circledcirc [\\mathbb {Z}_{n}],\\quad [\\mathbb {Z}_{3}]\\circledcirc [\\mathbb {Z}_{n}]$ which directly leads to the Lemma.",
"Corollary 1.6 Let $n\\ge 2$ be some integer.",
"Set $d_{2}(n):=\\gcd (n,2)$ and $d_{3}(n):=\\gcd (3,n)$ .",
"Then, we have $[\\mathbb {D}_{n}] \\circledcirc [\\mathbb {T}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {D}_{d_{2}(n)}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {T}\\cap \\left(g\\mathbb {D}_{n} g^{-1}\\right)$ .",
"From decomposition (REF ), we need only to consider intersections $\\mathbb {Z}_{3}^{\\mathbf {vt}_{i}}\\cap \\left(g\\mathbb {D}_{n} g^{-1}\\right) \\text{ and } \\mathbb {Z}_{2}^{\\mathbf {et}_{j}}\\cap \\left(g\\mathbb {D}_{n} g^{-1}\\right)$ which have already been studied (see Lemma REF ).",
"Lemma 1.7 We have $[\\mathbb {T}]\\circledcirc [\\mathbb {T}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {T}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {T}\\cap \\left(g\\mathbb {T}g^{-1}\\right)$ .",
"If no axes match, then $\\Gamma ={1}$ ; if only one edge axis (resp.",
"one face axis) from both configurations match, then $\\Gamma \\in [\\mathbb {Z}_{2}]$ (resp.",
"$[\\mathbb {Z}_{3}]$ ); if $g=\\mathbf {Q}\\left(\\mathbf {k},\\frac{\\pi }{2}\\right)$ , then $\\Gamma =\\mathbb {D}_{2}$ ."
],
[
"Octahedral subgroup",
"The group $\\mathbb {O}$ is the (orientation-preserving) symmetry group of the cube $\\mathcal {C}_{0}$ (see fig:cube0) with vertices $\\lbrace A_{i}\\rbrace _{i=1\\cdots 8} = {(\\pm 1,\\pm 1,\\pm 1)}.$ We have the decomposition (see [18]): $\\mathbb {O}=\\biguplus _{i=1}^3 \\mathbb {Z}_4^{\\mathbf {fc}_{i}} \\biguplus _{j=1}^4 \\mathbb {Z}_{3}^{\\mathbf {vc}_{j}} \\biguplus _{l=1}^6 \\mathbb {Z}_{2}^{\\mathbf {ec}_{l}}$ with vertices, edges and faces axes respectively denoted $\\mathbf {vc}_{i}$ , $\\mathbf {ec}_{j}$ and $\\mathbf {fc}_{j}$ .",
"For instance we have $\\mathbf {vc}_{1} := (OA_{1}),\\quad \\mathbf {ec}_{1} := (OM_{1}),\\quad \\mathbf {fc}_{1} := (OI).$ As an application of decomposition (REF ) and Lemma REF , we obtain the following corollary.",
"Corollary 1.8 Let $n\\ge 2$ be some integer.",
"Set $d_{2}(n) = \\gcd (n,2), \\quad d_{3}(n) = \\gcd (n,3),$ and $d_4(n)={\\left\\lbrace \\begin{array}{ll}4 \\text{ if } 4 \\text{ divide } n, \\\\1 \\text{ otherwise}.\\end{array}\\right.",
"}$ Then, we have $[\\mathbb {Z}_{n}]\\circledcirc [\\mathbb {O}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{d_{2}(n)}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {Z}_{d_4(n)}]\\right\\rbrace .$ Corollary 1.9 Let $n\\ge 2$ be some integer.",
"Set $d_{2}(n) = \\gcd (n,2), \\quad d_{3}(n) = \\gcd (n,3),$ and $d_4(n)={\\left\\lbrace \\begin{array}{ll}4 \\text{ if } 4\\mid n, \\\\1 \\text{ otherwise}.\\end{array}\\right.",
"}$ Then, we have $[\\mathbb {D}_{n}]\\circledcirc [\\mathbb {O}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {Z}_{d_4(n)}],[\\mathbb {D}_{d_{2}(n)}],[\\mathbb {D}_{d_{3}(n)}],[\\mathbb {D}_{d_4(n)}]\\right\\rbrace .$ Using decomposition (REF ), we have to consider intersections $\\mathbb {D}_{n}\\cap \\left(g\\mathbb {Z}_4^{\\mathbf {fc}_{i}}g^{-1}\\right),\\quad \\mathbb {D}_{n}\\cap \\left(g\\mathbb {Z}_{3}^{\\mathbf {vc}_{j}}g^{-1}\\right),\\quad \\mathbb {D}_{n}\\cap \\left(g\\mathbb {Z}_{2}^{\\mathbf {ec}_{l}}g^{-1}\\right)$ which have already been studied in Lemma REF .",
"Lemma 1.10 We have $[\\mathbb {T}]\\circledcirc [\\mathbb {O}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {T}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {O}\\cap \\left(g\\mathbb {T}g^{-1}\\right)$ .",
"From decompositions (REF )–(REF ) and Lemma REF , we only have to consider intersections $\\mathbb {Z}_{4}^{\\mathbf {fc}_{i}}\\cap \\left(g\\mathbb {Z}_{2}^{\\mathbf {et}_{j}}g^{-1}\\right),\\quad \\mathbb {Z}_{3}^{\\mathbf {vc}_{j}}\\cap \\left(g\\mathbb {Z}_{3}^{\\mathbf {vt}_{i}}g^{-1}\\right),\\quad \\mathbb {Z}_{2}^{\\mathbf {ec}_{l}}\\cap \\left(g\\mathbb {Z}_{2}^{\\mathbf {et}_{j}}g^{-1}\\right).$ Now, we always can find $g$ such that the intersection $\\Gamma $ reduces to some subgroup conjugate to ${1},\\mathbb {Z}_{2}$ or $\\mathbb {Z}_{3}$ and taking $g=\\mathbf {Q}\\left(\\mathbf {k},\\frac{\\pi }{4}\\right)$ , we get that $\\Gamma $ is conjugate to $\\mathbb {D}_{2}$ , which achieves the proof.",
"Lemma 1.11 We have $[\\mathbb {O}]\\circledcirc [\\mathbb {O}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {D}_{3}],[\\mathbb {Z}_4],[\\mathbb {D}_4],[\\mathbb {O}]\\right\\rbrace .$ Consider the subgroup $\\Gamma =\\mathbb {O}\\cap \\left(g\\mathbb {O}g^{-1}\\right)\\subset \\mathbb {O}$ .",
"From the poset in fig:SO3-lattice, we deduce that the conjugacy class $[\\Gamma ]$ belong to the following list $\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {D}_{3}],[\\mathbb {Z}_4],[\\mathbb {D}_4],[\\mathbb {T}],[\\mathbb {O}] \\right\\rbrace .$ Now: if $g$ fixes only one edge axis (resp.",
"one vertex axis), then $\\Gamma \\in [\\mathbb {Z}_{2}]$ (resp.",
"$\\Gamma \\in [\\mathbb {Z}_{3}]$ ); if $g=\\displaystyle {\\mathbf {Q}\\left( \\mathbf {i};\\frac{\\pi }{6}\\right)}$ , only one face axis is fixed by $g$ and $\\Gamma \\in [\\mathbb {Z}_4]$ ; if $g=\\displaystyle {\\mathbf {Q}\\left( \\mathbf {i};\\frac{\\pi }{4}\\right)}$ , $\\Gamma \\supset \\mathbb {Z}_4^{\\mathbf {i}}\\uplus \\mathbb {Z}_{2}^{\\mathbf {k}}$ and thus $\\Gamma \\in [\\mathbb {D}_4]$ ; if $g=\\displaystyle {\\mathbf {Q}\\left( \\mathbf {k};\\frac{\\pi }{4}\\right) \\circ \\mathbf {Q}\\left( \\mathbf {i};\\frac{\\pi }{4}\\right)}$ , $\\Gamma \\in [\\mathbb {D}_{2}]$ with characteristic axes $g\\mathbf {fc}_{3}=\\mathbf {ec}_6$ , $g\\mathbf {ec}_{1}=\\mathbf {fc}_{1}$ and $g\\mathbf {ec}_{2}=\\mathbf {ec}_{5}$ ; if $g=\\mathbf {Q}(\\mathbf {vc}_{1},\\pi )$ , $\\Gamma \\in [\\mathbb {D}_{3}]$ with $\\mathbf {vc}_{1}$ as the primary axis and $\\mathbf {ec}_{5}$ as the secondary axis of $\\Gamma $ ; if $\\Gamma \\supset \\mathbb {T}$ , then $g$ fixes the three edge axes of the tetrahedron, and $g$ fix the cube.",
"In that case, $\\Gamma =\\mathbb {O}$ ."
],
[
"Icosahedral subgroup",
"The group $\\mathbb {I}$ is the (orientation-preserving) symmetry group of the dodecahedron $\\mathcal {D}_{0}$ (fig:dode), where we have Twelve vertices: $(\\pm \\phi ,\\pm \\phi ^{-1},0),(\\pm \\phi ^{-1},0,\\pm \\phi ),(0,\\pm \\phi ,\\pm \\phi ^{-1})$ , $\\phi $ being the gold number.",
"Eight vertices: $(\\pm 1,\\pm 1,\\pm 1)$ of a cube.",
"Figure: Dodecahedron 𝒟 0 \\mathcal {D}_{0}We thus have the decomposition $\\mathbb {I}=\\biguplus _{i=1}^6\\mathbb {Z}_{5}^{\\mathbf {fd}_{i}}\\biguplus _{j=1}^{10}\\mathbb {Z}_{3}^{\\mathbf {vd}_{j}}\\biguplus _{l=1}^{15}\\mathbb {Z}_{2}^{\\mathbf {ed}_{l}}$ with vertices, edges and faces axes respectively denoted $\\mathbf {vd}_{i}$ , $\\mathbf {ed}_{j}$ and $\\mathbf {fd}_{j}$ .",
"For instance we have $\\mathbf {vd}_{1}:=(OA_{1}),\\quad \\mathbf {ed}_{1}:=(OI),\\quad \\mathbf {fd}_{1}:=(OM)$ where $M$ is the center of some face.",
"From decomposition (REF ) and from Lemma REF we obtain the following corollary.",
"Corollary 1.12 Let $n\\ge 2$ be some integer.",
"Set $d_{2}:=\\gcd (n,2)\\:; \\: d_{3}:=\\gcd (n,3)\\:; \\: d_{5}:=\\gcd (n,5).$ Then, we have $[\\mathbb {Z}_{n}]\\circledcirc [\\mathbb {I}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{d_{2}}],[\\mathbb {Z}_{d_{3}}],[\\mathbb {Z}_{d_{5}}]\\right\\rbrace .$ Using once again decomposition (REF ) and clips operation $[\\mathbb {D}_{n}]\\circledcirc [\\mathbb {Z}_{m}]$ in Lemma REF we get the following corollary.",
"Corollary 1.13 Let $n\\ge 2$ be some integer.",
"Set $d_{2}:=\\gcd (n,2)\\:; \\: d_{3}:=\\gcd (n,3)\\:; \\: d_{5}:=\\gcd (n,5).$ Then, we have $[\\mathbb {D}_{n}]\\circledcirc [\\mathbb {I}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{d_{3}}],[\\mathbb {Z}_{d_{5}}],[\\mathbb {D}_{d_{2}}],[\\mathbb {D}_{d_{3}}],[\\mathbb {D}_{d_{5}}]\\right\\rbrace .$ Lemma 1.14 We have $[\\mathbb {I}]\\circledcirc [\\mathbb {T}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {T}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {I}\\cap \\left(g\\mathbb {T}g^{-1}\\right)$ .",
"From decompositions (REF )–(REF ) and Lemma REF , we only have to consider intersections $\\mathbb {Z}_{3}^{\\mathbf {vd}_{j}}\\cap \\left(g\\mathbb {Z}_{3}^{\\mathbf {vt}_{i}}g^{-1}\\right),\\quad \\mathbb {Z}_{2}^{\\mathbf {ed}_{l}}\\cap \\left(g\\mathbb {Z}_{2}^{\\mathbf {et}_{j}}g^{-1}\\right).$ First, note that there always exists $g$ such that $\\Gamma $ reduces to a subgroup conjugate to ${1},\\mathbb {Z}_{2}$ or $\\mathbb {Z}_{3}$ .",
"Now, if $\\Gamma $ contains a subgroup conjugate to $\\mathbb {D}_{2}$ , then its three characteristic axes are edge axes of the dodecahedron: say $Ox$ , $Oy$ and $Oz$ .",
"In that case, $g$ fixes these three axes, and also the 8 vertices of the cube $\\mathcal {C}_{0}$ .",
"The subgroup $g\\mathbb {T}g^{-1}$ is thus the (orientation-preserving) symmetry group of a tetrahedron included in the dodecahedron $\\mathcal {D}_{0}$ .",
"Then, $\\Gamma \\in [\\mathbb {T}]$ .",
"The next two Lemmas are more involving.",
"Lemma 1.15 We have $[\\mathbb {O}]\\circledcirc [\\mathbb {I}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {D}_{3}],[\\mathbb {T}] \\right\\rbrace .$ Let $\\Gamma =\\mathbb {I}\\cap \\left(g\\mathbb {O}g^{-1}\\right)$ .",
"From the poset in Figure REF , we deduce that the conjugacy class $[\\Gamma ]$ belongs to the following list $\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {D}_{3}],[\\mathbb {T}]\\right\\rbrace .$ First, we can always find $g\\in \\mathrm {SO}(3)$ such that $\\Gamma \\in [\\mathbb {Z}_{3}]$ , $\\Gamma \\in [\\mathbb {Z}_{2}]$ or $\\Gamma ={1}$ .",
"Moreover, as in the proof of Lemma REF , if $\\Gamma $ contains a subgroup conjugate to $\\mathbb {D}_{2}$ , then $\\Gamma \\in [\\mathbb {T}]$ .",
"Finally, we will exhibit some $g\\in \\mathrm {SO}(3)$ such that $\\Gamma \\in [\\mathbb {D}_{3}]$ .",
"First, recall that $A_{1}(1,1,1), \\quad A_{2}(1,1,-1), \\quad A_4(1,-1,1), \\quad A_{5}(-1,-1,1)$ are common vertices of the cube $\\mathcal {C}_{0}$ and the dodecahedron $\\mathcal {D}_{0}$ .",
"Let now $B_{2}(\\phi ^{-1},0,-\\phi )$ be a vertex of the dodecahedron and $I_{2}$ (resp.",
"$I_4$ ) be the middle-point of $[B_{2}A_{2}]$ (resp.",
"$[A_{4}A_{5}]$ – see fig:dodeclipscube).",
"Figure: Rotation gg to obtain [𝔻 3 ][\\mathbb {D}_{3}] in [𝕆]⊚[𝕀][\\mathbb {O}]\\circledcirc [\\mathbb {I}].Then, $a_{1}=(OI_4)$ and $a_{2}=(OI_{2})$ are perpendicular axes to $a=(OA_{1})$ .",
"Choose $\\alpha $ such that $\\mathbf {Q}(\\mathbf {v},\\alpha )\\mathbf {v}_{1}=\\mathbf {v}_{2}$ , with (see fig:dodeclipscube): $\\mathbf {v}=\\overrightarrow{OA_{1}}, \\quad \\mathbf {v}_{1}=\\overrightarrow{OI_4}, \\quad \\mathbf {v}_{2}=\\overrightarrow{OI_{2}},$ and set $g=\\mathbf {Q}(\\mathbf {v},\\alpha )$ .",
"From decompositions (REF ) and (REF ), we deduce then, that $\\mathbb {I}\\cap \\left(g\\mathbb {O}g^{-1}\\right)$ contains the subgroups $\\mathbb {Z}_{2}^{a_{2}}\\cap \\left(g\\mathbb {Z}_{2}^{a_{1}}g^{-1}\\right) = \\mathbb {Z}_{2}^{a_{2}},\\quad \\mathbb {Z}_{3}^{a}\\cap , \\left(g\\mathbb {Z}_{3}^{a}g^{-1}\\right) = \\mathbb {Z}_{3}^{a}.$ Therefore, $\\Gamma $ contains a subgroup conjugate to $\\mathbb {D}_{3}$ .",
"Using a maximality argument (see poset in fig:SO3-lattice), we must have $\\Gamma \\in [\\mathbb {D}_{3}]$ , and this concludes the proof.",
"Lemma 1.16 We have $[\\mathbb {I}]\\circledcirc [\\mathbb {I}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{3}],[\\mathbb {D}_{3}],[\\mathbb {Z}_{5}],[\\mathbb {D}_{5}],[\\mathbb {I}]\\right\\rbrace .$ Let $\\Gamma =\\mathbb {I}\\cap \\left(g\\mathbb {I}g^{-1}\\right)$ .",
"Considering the subclasses of $[\\mathbb {I}]$ , we have to check the classes $[\\mathbb {T}], \\quad [\\mathbb {D}_{3}], \\quad [\\mathbb {D}_{5}], \\quad [\\mathbb {D}_{2}], \\quad [\\mathbb {Z}_{3}], \\quad [\\mathbb {Z}_{5}], \\quad [\\mathbb {Z}_{2}].$ Note first, that there exist rotations $g$ such that $\\Gamma \\in [\\mathbb {Z}_{2}]$ , $\\Gamma \\in [\\mathbb {Z}_{3}]$ , $\\Gamma \\in [\\mathbb {Z}_{5}]$ or $\\Gamma ={1}$ .",
"When $\\Gamma $ contains a subgroup conjugate to $\\mathbb {T}$ or $\\mathbb {D}_{2}$ , using the same argument as in the proof of Lemma REF , $g$ fixes all the dodecahedron vertices.",
"In that case, $\\Gamma =\\mathbb {I}$ .",
"We will now exhibit some $g\\in \\mathrm {SO}(3)$ such that $\\Gamma \\in [\\mathbb {D}_{3}]$ .",
"Consider the dodecahedron $\\mathcal {D}_{0}$ in fig:dode and the points $A_{3}(1,-1,-1)$ and $B_{3}(\\phi ,-\\phi ^{-1},0)$ .",
"Let $I_{3}$ be the middle-point of $[A_{3}B_{3}]$ and $g$ be the order two rotation around $a_{1}:=(OA_{1})$ (see fig:dodeclipsdode).",
"Let $b_{1}:=(OI_{3}),\\quad b_{2}:=\\mathbf {Q}\\left(a_{1},\\frac{2\\pi }{3}\\right)b_{1},\\quad b_{3}:=\\mathbf {Q}\\left(a_{1},\\frac{2\\pi }{3}\\right)b_{2}.$ We check directly that $a_{1},b_{i}$ ($i=1,\\cdots ,3)$ are the only $g$ -invariant characteristic axes of the dodecahedron.",
"We deduce then, from decomposition (REF ) that $\\Gamma $ reduces to $\\mathbb {Z}_{3}^{a_{1}}\\biguplus _{i=1}^{3} \\mathbb {Z}_{2}^{b_{i}}\\in [\\mathbb {D}_{3}].$ Figure: Rotation gg to obtain [𝔻 3 ][\\mathbb {D}_{3}] or [𝔻 5 ][\\mathbb {D}_{5}] in [𝕀]⊚[𝕀][\\mathbb {I}]\\circledcirc [\\mathbb {I}].In the same way, we can find $g\\in \\mathrm {SO}(3)$ , such that $\\Gamma \\in [\\mathbb {D}_{5}]$ .",
"Let $B_{1}(\\phi ,\\phi ^{-1},0), \\quad B_{5}(\\phi ^{-1},0,\\phi ), \\quad B_{6}(-\\phi ^{-1},0,\\phi ), \\quad A_{6}(-1,1,1)$ be vertices of the dodecahedron $\\mathcal {D}_{0}$ .",
"Let $G_{1}$ be the center of the pentagon $A_{1}B_{1}B_{3}A_{4}B_{5}$ (see fig:dodeclipsdode) and $I_{6}$ be the middle-point of $[B_{6}A_{6}]$ .",
"Let $g$ be the order two rotation around $f_{1}:=(OG_{1})$ and set $c_{1}:=(OI_{6}),\\quad c_{k+1}:=\\mathbf {Q}\\left(\\overrightarrow{OA_{1}},\\frac{2\\pi }{5}\\right)c_k,\\quad 1\\le k\\le 4.$ Then we can check that $f_{1}$ , $c_k$ ($k=1,\\cdots ,5$ ) are the only $g$ -invariant characteristic axes of the dodecahedron.",
"Using decomposition (REF ), we deduce then, that $\\Gamma \\in [\\mathbb {D}_{5}]$ , which concludes the proof."
],
[
"Infinite subgroups",
"The primary axis of both $\\mathrm {SO}(2)$ and $\\mathrm {O}(2)$ is defined as the $z$ -axis, while any perpendicular axis to $(Oz)$ is a secondary axis for $\\mathrm {O}(2)$ .",
"Clips operation between $\\mathrm {SO}(2)$ or $\\mathrm {O}(2)$ and finite subgroups are obtained using simple arguments on characteristic axes.",
"The same holds for the clips $[\\mathrm {SO}(2)]\\circledcirc [\\mathrm {SO}(2)]$ .",
"To compute $[\\mathrm {O}(2)]\\circledcirc [\\mathrm {O}(2)]$ , consider the subgroup $\\Gamma =\\mathrm {O}(2)\\cap \\left(g\\mathrm {O}(2) g^{-1}\\right)$ for some $g\\in \\mathrm {SO}(3)$ .",
"If both primary axes are the same, then $\\Gamma =\\mathrm {O}(2)$ ; if the primary axis of $g\\mathrm {O}(2) g^{-1}$ is in the $xy$ –plane, then $\\Gamma \\in [\\mathbb {D}_{2}]$ ; in all other cases, $\\Gamma \\in [\\mathbb {Z}_{2}]$ , where the primary axis of $\\Gamma $ is perpendicular to the primary axes of $\\mathrm {O}(2)$ and $g\\mathrm {O}(2) g^{-1}$ ."
],
[
"Proofs for $\\mathrm {O}(3)$",
"In this Appendix, we provide the details about clips operations between type III closed $\\mathrm {O}(3)$ subgroups.",
"The proofs follow the same ideas that has been used for $\\mathrm {SO}(3)$ closed subgroups (decomposition into simpler subgroups and discussion about their characteristic axes), but most of them are unfortunately more involving.",
"We first recall the general structure of type III subgroups $\\Gamma $ of $\\mathrm {O}(3)$ (see [18] for details).",
"For each such subgroup $\\Gamma $ , there exists a couple $L\\subset H$ of $\\mathrm {SO}(3)$ subgroups such that $H=\\pi (\\Gamma )$ , where $\\pi \\: : \\: g\\in \\mathrm {O}(3)\\mapsto \\det (g) g\\in \\mathrm {SO}(3)$ and $L=\\mathrm {SO}(3)\\cap \\Gamma $ is an indexed 2 subgroup of $H$ .",
"These characteristic couples are detailed in tab:ClassIII.",
"Note that, for a given couple $(L,H)$ , $\\Gamma $ can be recovered as $\\Gamma = L\\cap g L$ , where $-g\\in H\\setminus L$ .",
"Table: Characteristic couples for type III subgroupsIn the following, we shall use the following convention: $\\mathbb {Z}_{1}^{\\sigma } = \\mathbb {Z}_{1}^{-} = \\mathbb {D}_{1}^{v} = {1}.$"
],
[
"$\\mathbb {Z}_{2n}^{-}$ subgroups",
"Consider the couple $\\mathbb {Z}_{n} \\subset \\mathbb {Z}_{2n}$ ($n>1$ ) in tab:ClassIII, where $\\mathbb {Z}_{2n} = \\left\\lbrace I,\\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{n}\\right),\\mathbf {Q}\\left(\\mathbf {k};\\frac{2\\pi }{n}\\right),\\cdots \\right\\rbrace $ and let $\\mathbf {r}_{n} := \\displaystyle {\\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{n}\\right)} \\in \\mathbb {Z}_{2n}\\setminus \\mathbb {Z}_{n}$ .",
"Set $\\mathbb {Z}_{2n}^{-}:=\\mathbb {Z}_{n} \\cup (-\\mathbf {r}_{n}\\mathbb {Z}_{n}).$ The primary axis of the subgroup $\\mathbb {Z}_{2n}^{-}$ is defined as the $z$ -axis.",
"Remark 2.1 The subgroup $\\mathbb {Z}_{2}^{-}$ is generated by $-\\mathbf {Q}(\\mathbf {k},\\pi )$ which is the reflection through the $xy$ plane.",
"If $\\sigma _{b}$ is the reflection through the plane with normal axis $b$ , then $\\mathbb {Z}_{2}^{\\sigma _b} := \\left\\lbrace e,\\sigma _{b}\\right\\rbrace $ , which is conjugate to $\\mathbb {Z}_{2}^{-}$ .",
"We have the following lemma.",
"Lemma 2.2 Let $m,n\\ge 2$ be two integers.",
"Set $d:=\\gcd (n,m)$ and $\\mathbf {r}_{n} := \\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{n}\\right), \\quad \\mathbf {r}_{m} := \\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{m}\\right).$ The intersection $(-\\mathbf {r}_{n}\\mathbb {Z}_{n})\\cap (-\\mathbf {r}_{m}\\mathbb {Z}_{m})$ does not reduce to $\\emptyset $ if and only if $m/d$ and $n/d$ are odds.",
"In such a case, we have $(-\\mathbf {r}_{n}\\mathbb {Z}_{n})\\cap (-\\mathbf {r}_{m}\\mathbb {Z}_{m}) = -\\mathbf {r}_{d}\\mathbb {Z}_{d},\\quad \\mathbf {r}_{d} = \\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{d}\\right).$ The intersection $(-\\mathbf {r}_{n}\\mathbb {Z}_{n})\\cap (-\\mathbf {r}_{m}\\mathbb {Z}_{m})$ differs from $\\emptyset $ , if and only if, there exist integers $i,j$ such that $\\frac{2i+1}{n}\\pi = \\frac{2j+1}{m}\\pi ,\\quad 2i+1\\le 2n,\\quad 2j+1\\le 2m.$ Let $n=dn_{1}$ and $m=dm_{1}$ .",
"The preceding equation can then be recast as $(2i+1)m_{1}=(2j+1)n_{1}$ , so that $2i+1=pn_{1} \\text{ and } 2j+1=pm_{1}.$ Thus, $m_{1}$ and $n_{1}$ are necessarily odds, in which case (recall that $\\mathbb {Z}_{1} = {1}$ ) $(-\\mathbf {r}_{n}\\mathbb {Z}_{n})\\cap (-\\mathbf {r}_{n}\\mathbb {Z}_{m}) = -\\mathbf {r}_{d}\\mathbb {Z}_{d},\\quad \\mathbf {r}_{d} = \\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{d}\\right).$ Corollary 2.3 Let $m,n\\ge 1$ be two integers.",
"Set $d:=\\gcd (n,m)$ .",
"Then, we have $\\left[ \\mathbb {Z}_{2n}^{-} \\right] \\circledcirc \\left[ \\mathbb {Z}_{2m}^{-}\\right]={\\left\\lbrace \\begin{array}{ll}\\left\\lbrace [{1}],[\\mathbb {Z}_{2d}^-]\\right\\rbrace \\text{if $n/d$ and $m/d$ are odd} \\\\\\left\\lbrace [{1}],[\\mathbb {Z}_{d}]\\right\\rbrace \\text{otherwise}\\end{array}\\right.",
"}$ Note first that all intersections reduce to 1 when the characteristic axes don't match, so we have only to consider the situation where they match.",
"Now, by (REF ), we have only to consider the intersection $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2m}^{-} =(\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m})\\cup \\left((-\\mathbf {r}_{n}\\mathbb {Z}_{n})\\cap (-\\mathbf {r}_{m}\\mathbb {Z}_{m})\\right).$ By Lemma REF , $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}=\\mathbb {Z}_{d}$ and we directly conclude using Lemma REF ."
],
[
"$\\mathbb {D}_{n}^{v}$ subgroups",
"Consider the couple $\\mathbb {Z}_{n} \\subset \\mathbb {D}_{n}$ in tab:ClassIII.",
"Recall that $\\mathbb {D}_{n}$ contains $\\mathbb {Z}_{n}$ and all the second order rotations about the $b_{j}$ 's axes (see (REF ) and fig:dihedral-second-axis).",
"Set $\\mathbb {D}_{n}^{v} := \\mathbb {Z}_{n} \\biguplus _{j=1}^{n} \\mathbb {Z}_{2}^{\\sigma _{b_{j}}}.$ Given $g\\in \\mathrm {O}(3)$ , the primary axis of $g\\mathbb {D}_{n}^{v}g^{-1}$ is $g(Oz)$ , and its secondary axes are $gb_{j}$ .",
"Lemma 2.4 Let $n\\ge 2$ , $m\\ge 1$ be two integers.",
"Set $d=\\gcd (n,m)$ and $i(m):=3-\\gcd (2,m)={\\left\\lbrace \\begin{array}{ll}1, & \\text{ if $m$ is even}, \\\\2, & \\text{ if $m$ is odd}.\\end{array}\\right.",
"}$ Then, we have $\\left[ \\mathbb {D}_{n}^{v} \\right] \\circledcirc \\left[ \\mathbb {Z}_{2m}^{-}\\right] = \\left\\lbrace {1},\\left[ \\mathbb {Z}_{i(m)}^{-}\\right],\\left[ \\mathbb {Z}_{d} \\right]\\right\\rbrace .$ Let $\\Gamma := \\mathbb {D}_{n}^{v} \\cap \\left(g\\mathbb {Z}_{2m}^{-} g^{-1}\\right)$ and $\\mathbb {Z}_{2m}^{-}=\\mathbb {Z}_{m} \\cup (-\\mathbf {r}_{m}\\mathbb {Z}_{m}),\\quad \\mathbf {r}_{m}=\\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{m}\\right).$ If both primary axes of $\\mathbb {D}_{n}^{v}$ and $g\\mathbb {Z}_{2m}^{-} g^{-1}$ (generated by $g\\mathbf {k}$ ) match, then by decomposition (REF ) and Lemma REF , $\\Gamma $ reduces to $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}=\\mathbb {Z}_{d}$ .",
"If the primary axis of $g\\mathbb {Z}_{2m}^{-}g^{-1}$ matches with a secondary axis of $\\mathbb {D}_{n}^{v}$ , say $(Ox)$ , then $\\Gamma $ reduces to $\\mathbb {Z}_{2}^{\\sigma _{b_{0}}}\\cap \\left(g\\mathbb {Z}_{2m}^{-}g^{-1}\\right)$ .",
"Such an intersection has already be studied in the clips operation $[\\mathbb {Z}_{2}^{-}] \\circledcirc [\\mathbb {Z}_{2m}^{-}]$ (see Lemma REF ).",
"Otherwise, $\\Gamma ={1}$ , which concludes the proof.",
"Lemma 2.5 Let $m,n\\ge 2$ be two integers and $d=\\gcd (n,m)$ .",
"Then, we have $\\left[ \\mathbb {D}_{n}^{v} \\right] \\circledcirc \\left[ \\mathbb {D}_{m}^{v}\\right]=\\left\\lbrace {1},\\left[ \\mathbb {Z}_{2}^{-}\\right],\\left[ \\mathbb {D}_{d}^{v} \\right],\\left[ \\mathbb {Z}_{d} \\right]\\right\\rbrace .$ Only two cases need to be considered.",
"If the primary axes of $\\mathbb {D}_{n}^{v}$ and $g\\mathbb {D}_{m}^{v}g^{-1}$ do not match, then we get 1.",
"If they have the same primary axis, by decomposition (REF ), we have to consider the intersections $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m},\\quad \\mathbb {Z}_{2}^{\\sigma _{b_{j}}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_k}},$ which reduce to $\\mathbb {Z}_{d}\\biguplus \\mathbb {Z}_{2}^{\\sigma _{c_{l}}},$ where $c_{l}$ are the common secondary axis of the two subgroups.",
"Then, we get either $\\mathbb {Z}_{d}$ , $\\mathbb {D}_{d}^{v}$ or a subgroup conjugate to $\\mathbb {Z}_{2}^{-}$ (when $d=1$ and $b_{0}=b^{\\prime }_{0}$ ), which concludes the proof."
],
[
"$\\mathbb {D}_{2n}^h$ subgroups",
"Consider the couple $\\mathbb {D}_{n} \\subset \\mathbb {D}_{2n}$ in tab:ClassIII.",
"For $j=0,\\cdots , n-1$ , let $p_{j}$ be the axis generated by $\\mathbf {v}_{j} := \\mathbf {Q}\\left(\\mathbf {k};\\frac{j\\pi }{n}\\right)\\cdot \\mathbf {i},$ and $q_{j}$ , be the axis generated by $\\mathbf {w}_{j}:=\\mathbf {Q}\\left(\\mathbf {k};\\frac{(2j+1)\\pi }{2n}\\right)\\cdot \\mathbf {i}.$ Figure: NO_CAPTIONSet $\\mathbb {D}_{n}=\\left\\lbrace {1},\\mathbf {Q}\\left(\\mathbf {k};\\frac{2\\pi }{n}\\right),\\mathbf {Q}\\left(\\mathbf {k};\\frac{4\\pi }{n}\\right), \\cdots , \\mathbf {Q}\\left(\\mathbf {v}_{0};\\pi \\right),\\mathbf {Q}\\left(\\mathbf {v}_{1};\\pi \\right),\\cdots \\right\\rbrace ,$ and $-\\mathbf {r}_{n}\\mathbb {D}_{n}=\\left\\lbrace -\\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{n}\\right),-\\mathbf {Q}\\left(\\mathbf {k};\\frac{3\\pi }{n}\\right), \\cdots , -\\mathbf {Q}\\left(\\mathbf {w}_{0};\\pi \\right),-\\mathbf {Q}\\left(\\mathbf {w}_{1};\\pi \\right),\\cdots \\right\\rbrace ,$ where $\\mathbf {r}_{n}=\\displaystyle {\\mathbf {Q}\\left(\\mathbf {k};\\frac{\\pi }{n}\\right)}$ .",
"We define $\\mathbb {D}_{2n}^h := \\mathbb {D}_{n} \\cup \\left(-\\mathbf {r}_{n}\\mathbb {D}_{n} \\right),$ which decomposes as $\\mathbb {D}_{2n}^h=\\mathbb {Z}_{2n}^{-}\\biguplus _{j=0}^{n-1} \\mathbb {Z}_{2}^{p_{j}} \\biguplus _{j=0}^{n-1} \\mathbb {Z}_{2}^{\\sigma _{q_{j}}} .$ Note that in this decomposition, there are $n$ subgroups conjugate to $\\mathbb {Z}_{2}$ and $n$ others conjugate to $\\mathbb {Z}_{2}^{-}$ .",
"The $z$ -axis (resp $x$ -axis) is said to be the primary (resp.",
"secondary) axis of $\\mathbb {D}_{2n}^h$ .",
"For each $g\\in \\mathrm {O}(3)$ , the primary (resp.",
"secondary) axis of the subgroup $g\\mathbb {D}_{2n}^{h}g^{-1}$ is generated by $g\\mathbf {k}$ (resp.",
"by $g\\mathbf {i}$ ).",
"Lemma 2.6 Let $m,n\\ge 2$ be two integers.",
"Set $d=\\gcd (n,m)$ , $d_{2}(m)=\\gcd (m,2)$ and $i(m)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if $m$ is even}, \\\\2, & \\text{otherwise}.\\end{array}\\right.",
"}$ Then, If $\\cfrac{n}{d}$ or $\\cfrac{m}{d}$ is even, we have $[\\mathbb {D}_{2n}^h]\\circledcirc [\\mathbb {Z}_{2m}^{-}]=\\left\\lbrace {1},[\\mathbb {Z}_{d_{2}(m)}],[\\mathbb {Z}_{i(m)}^{-}],[\\mathbb {Z}_{d}]\\right\\rbrace ;$ If $\\cfrac{n}{d}$ and $\\cfrac{m}{d}$ are odd, we have $[\\mathbb {D}_{2n}^h]\\circledcirc [\\mathbb {Z}_{2m}^{-}]=\\left\\lbrace {1},[\\mathbb {Z}_{d_{2}(m)}],[\\mathbb {Z}_{i(m)}^{-}],[\\mathbb {Z}_{2d}^{-}]\\right\\rbrace .$ First of all, if no characteristic axes of $\\mathbb {D}_{2n}^h$ and $g\\mathbb {Z}_{2m}^{-}g^{-1}$ match, then their intersection reduces to 1.",
"We have now to consider three cases: $(1)$ The first case is when $\\mathbb {D}_{2n}^h$ and $g\\mathbb {Z}_{2m}^{-}g^{-1}$ have the same primary axis.",
"Then, using decompositions (REF ) and (REF ), we only have to consider the intersection $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2m}^{-}.$ This has already been studied in the clips operation $[\\mathbb {Z}_{2n}^{-}]\\circledcirc [\\mathbb {Z}_{2m}^{-}]$ in Lemma REF , leading to the conjugacy class $[\\mathbb {Z}_{d}]$ or $[\\mathbb {Z}_{2d}^{-}]$ .",
"$(2)$ The second one is when some secondary axis $p_{j}$ (say $p_{0}$ ) match with the primary axis of $g\\mathbb {Z}_{2m}^{-}g^{-1}$ , then we only have to consider intersection $\\mathbb {Z}_{2}^{p_{0}}\\cap \\left(g\\mathbb {Z}_{2m}^{-}g^{-1}\\right)=\\mathbb {Z}_{2}^{p_{0}}\\cap \\left(g\\mathbb {Z}_{m}g^{-1}\\right)$ leading to $\\mathbb {Z}_{2}^{p_{0}}$ if $m$ is even.",
"$(2)$ Finally, we have to consider the case when the primary axis of $g\\mathbb {Z}_{2m}^{-}g^{-1}$ is $q_{0}$ .",
"In that case the problem reduces to the intersection $\\mathbb {Z}_{2}^{\\sigma _{q_{0}}}\\cap \\left(g\\mathbb {Z}_{2m}^{-}g^{-1}\\right)$ leading to the conjugacy class $[\\mathbb {Z}_{2}^{-}]$ for odd $m$ (see Lemma REF ).",
"This concludes the proof.",
"The cases $[\\mathbb {D}_{2n}^{h}]\\circledcirc [\\mathbb {D}_{n}^{v}]$ and $[\\mathbb {D}_{2n}^{h}]\\circledcirc [\\mathbb {D}_{2m}^h]$ are more involving.",
"We start by formulating the following lemma, without proof (see fig:dihedral-second-axis for an example): Lemma 2.7 If $n$ is even then there exist $p_k\\perp p_{l}$ and $q_r\\perp q_s$ , where no axes $p_{i},q_{j}$ are perpendicular.",
"If $n$ is odd, there exist $p_{i}\\perp q_{j}$ and no axes $p_k,p_{l}$ , nor $q_r,q_s$ are perpendicular.",
"Lemma 2.8 Let $m,n\\ge 2$ be two integers.",
"Set $d_{2}(m):=\\gcd (m,2)$ , $i(m,n) :={\\left\\lbrace \\begin{array}{ll}2, & \\text{if $m$ is even and $n$ is odd}, \\\\1, & \\text{otherwise},\\end{array}\\right.",
"}$ and $i(m) :=3-\\gcd (2,m)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if $m$ is even}, \\\\2, & \\text{if $m$ is odd}.\\end{array}\\right.",
"}$ Then, we have $[\\mathbb {D}_{2n}^h]\\circledcirc [\\mathbb {D}_{m}^{v}]=\\left\\lbrace [{1}],\\left[\\mathbb {Z}_{i(m)}^{-}\\right],\\left[ \\mathbb {Z}_{d_{2}(m)}\\right],\\left[ \\mathbb {D}_{i(m,n)}^{v}\\right],\\left[ \\mathbb {Z}_{d}\\right],\\left[\\mathbb {D}_{d}^{v}\\right]\\right\\rbrace .$ The only non trivial cases are when $g\\mathbb {D}_{m}^{v}g^{-1}$ and $\\mathbb {D}_{2n}^h$ have no matching characteristic axes.",
"Now we have to distinguish wether the principal axis $a$ of $g\\mathbb {D}_{m}^{v}g^{-1}$ is $(Oz)$ or not: Let first suppose that $a=(Oz)$ .",
"In that case, we need to compute the intersection $\\mathbb {D}_{2n}^h \\cap \\mathbb {D}_{m}^{v}$ .",
"From (REF ) and (REF ), this reduces to study the following three intersections $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{m},\\quad \\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2}^{\\sigma _{b_{j}}},\\quad \\mathbb {Z}_{2}^{\\sigma _{q_k}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b_{j}}}.$ Now: The first intersection, $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{m}$ , reduces to $\\mathbb {Z}_{n}\\cap \\mathbb {Z}_{m}=\\mathbb {Z}_{d}$ (from (REF ) and Lemma REF ).",
"The second one, $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2}^{\\sigma _{b_{j}}}$ , reduces to 1, since primary axes of $\\mathbb {Z}_{2n}^{-}$ and $\\mathbb {Z}_{2}^{\\sigma _{b_{j}}}$ (conjugate to $\\mathbb {Z}_{2}^{-}$ ) do not match.",
"The last one, $\\mathbb {Z}_{2}^{\\sigma _{q_k}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b_{j}}}$ , can reduce to some $\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ if $b_{0}=q_{0}$ .",
"In that case, $\\mathbb {D}_{2n}^h \\cap \\mathbb {D}_{m}^{v}$ contains $\\mathbb {Z}_{d}$ and some $\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ , which generate $\\mathbb {D}_{d}^{v}$ (see (REF )).",
"This first case thus leads to $\\mathbb {Z}_d$ or $\\mathbb {D}_d^v$ .",
"Consider now the case when $a\\ne (Oz)$ .",
"Thus the intersections to be considered are $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_{0}}},\\quad \\mathbb {Z}_{2}^{p_{j}}\\cap \\mathbb {Z}_{m}^a,\\quad \\mathbb {Z}_{2}^{\\sigma _{q_k}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_{j}}}.$ where the $b^{\\prime }_j$ axes are the secondary axis of $g\\mathbb {D}_{m}^{v}g^{-1}$ .",
"Now: First suppose that $a=p_0$ (for instance) and all the other axis are different.",
"Then, $\\mathbb {Z}_{2}^{p_{j}}\\cap \\mathbb {Z}_{m}^a=\\mathbb {Z}_{d_2(m)}^a$ .",
"Suppose now that $a=p_0$ and $b^{\\prime }j=q_k$ for some couple $(k,j)$ , so that $\\mathbb {Z}_{2}^{\\sigma _{q_k}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_{j}}}=\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ .",
"As $q_k\\perp p_0$ , we deduce from Lemma REF that $n$ is odd.",
"All depend on $m$ parity: if $m$ is even then $\\Gamma $ contains $\\mathbb {Z}_{2}^{p_0}$ and $\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ , and we obtain some subgroup conjugate to $\\mathbb {D}_{2}^{v}$ .",
"If $m$ is odd, then $\\Gamma $ reduces to $\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ , which is conjugate to $\\mathbb {Z}_{2}^-$ .",
"Finally, suppose that $a\\ne p_j$ for all $j$ (and recall that $a\\ne (Oz)$ ), so that the intersections to be considered are $\\mathbb {Z}_{2n}^{-}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_{0}}},\\quad \\mathbb {Z}_{2}^{\\sigma _{q_k}}\\cap \\mathbb {Z}_{2}^{\\sigma _{b^{\\prime }_{j}}}.$ We thus only obtain some subgroups already considered in the previous cases, which conclude the proof.",
"Lemma 2.9 Let $m,n\\ge 2$ be two integers.",
"Set $d=\\gcd (n,m)$ and $\\Delta =[\\mathbb {D}_{2n}^h]\\circledcirc [\\mathbb {D}_{2m}^h].$ Then, $[{1}]\\subset \\Delta $ and: For every integer $d$ : $\\diamond $ If $m$ and $n$ are even, then $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{2}],[\\mathbb {D}_{2}]\\right\\rbrace $ ; $\\diamond $ If $m$ and $n$ are odds, then $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{2}^{-}]\\right\\rbrace $ ; $\\diamond $ Otherwise, $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{2}],[\\mathbb {D}_{2}^{v}]\\right\\rbrace $ ; If $d=1$ , then $\\diamond $ If $m$ and $n$ are odds, then $\\Delta \\supset \\left\\lbrace [\\mathbb {D}_{2}^{v}]\\right\\rbrace $ ; $\\diamond $ Otherwise $m$ or $n$ is even and $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{2}],[\\mathbb {Z}_{2}^{-}]\\right\\rbrace $ ; If $d\\ne 1$ , then $\\diamond $ If $\\dfrac{m}{d}$ and $\\dfrac{n}{d}$ are odds, then $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{2d}^{-}],[\\mathbb {D}_{2d}^h]\\right\\rbrace $ ; $\\diamond $ Otherwise, $\\dfrac{m}{d}$ or $\\dfrac{n}{d}$ is even and $\\Delta \\supset \\left\\lbrace [\\mathbb {Z}_{d}],[\\mathbb {D}_{d}],[\\mathbb {D}_{d}^{v}]\\right\\rbrace $ ; We consider decomposition (REF ).",
"If no characteristic axes $\\mathbb {D}_{2n}^{h}$ and $g\\mathbb {D}_{2m}^{h}g^{-1}$ match, then their intersection reduces to 1.",
"Otherwise, from (REF ) it reduces to $\\mathbb {Z}_{2n}^{-}\\cap \\left(g\\mathbb {Z}_{2m}^{-}g^{-1}\\right) & ,\\quad \\mathbb {Z}_{2n}^{-}\\cap \\left(g\\mathbb {Z}_{2}^{\\sigma _{q^{\\prime }_k}}g^{-1}\\right),\\quad \\mathbb {Z}_{2}^{p_{j}}\\cap \\left(g\\mathbb {Z}_{2}^{p^{\\prime }_k}g^{-1}\\right), \\\\\\mathbb {Z}_{2}^{\\sigma _{q_{j}}}\\cap \\left(g\\mathbb {Z}_{2m}^{-}g^{-1}\\right) & ,\\quad \\mathbb {Z}_{2}^{\\sigma _{q_{j}}}\\cap \\left(g\\mathbb {Z}_{2}^{\\sigma _{q^{\\prime }_k}}g^{-1}\\right),$ where all $\\mathbb {Z}_{2}^{\\sigma _{q_{j}}},\\mathbb {Z}_{2}^{\\sigma _{q^{\\prime }_k}}$ are subgroups conjugate to $\\mathbb {Z}_{2}^{-}$ .",
"Now all these intersections have already been studied in the clips operation $[\\mathbb {Z}_{2r}^{-}] \\circledcirc [\\mathbb {Z}_{2s}^{-}]$ .",
"We can thus use Lemma REF , argue on the characteristic axes and Lemma REF , to conclude the proof in each case."
],
[
"$\\mathbb {O}^{-}$ subgroup",
"Consider the couple $\\mathbb {T}\\subset \\mathbb {O}$ in tab:ClassIII and the following decompositions $\\mathbb {O}=\\biguplus _{i=1}^3 \\mathbb {Z}_4^{\\mathbf {fc}_{i}} \\biguplus _{j=1}^4 \\mathbb {Z}_{3}^{\\mathbf {vc}_{j}} \\biguplus _{l=1}^6 \\mathbb {Z}_{2}^{\\mathbf {ec}_{l}},$ and $\\mathbb {T}=\\biguplus _{j=1}^4 \\mathbb {Z}_{3}^{\\mathbf {vt}_{j}} \\biguplus \\mathbb {Z}_{2}^{\\mathbf {et}_{1}}\\biguplus \\mathbb {Z}_{2}^{\\mathbf {et}_{2}}\\biguplus \\mathbb {Z}_{2}^{\\mathbf {et}_{3}},\\quad \\mathbb {Z}_{2}^{\\mathbf {et}_{i}}\\subset \\mathbb {Z}_4^{\\mathbf {fc}_{i}},\\quad i=1,2,3.$ This leads (see [18] for details) to the decomposition $\\mathbb {O}^{-}:=\\biguplus _{i=1}^3(\\mathbb {Z}_4^{\\mathbf {fc}_{i}})^{-}\\biguplus _{j=1}^4 \\mathbb {Z}_{3}^{\\mathbf {vc}_{j}}\\biguplus _{l=1}^6 \\mathbb {Z}_{2}^{\\sigma _{\\mathbf {ec}_{l}}},$ where $(\\mathbb {Z}_4^{\\mathbf {fc}_{i}})^{-}$ is the subgroup conjugate to $\\mathbb {Z}_{4}^{-}$ with $\\mathbf {fc}_{i}$ as primary axis.",
"Note also that $\\mathbb {Z}_{2}^{\\sigma _{e_{l}}}$ are subgroups conjugate to $\\mathbb {Z}_{2}^{-}$ with $\\mathbf {ec}_{l}$ as primary axis.",
"Using this decomposition (REF ), and those of type III closed $\\mathrm {O}(3)$ subgroups previously mentioned directly lead to the following corollaries.",
"Corollary 2.10 Let $n\\ge 2$ be an integer.",
"Set $d_{2}(n)=\\gcd (n,2)$ and $d_{3}(n)=\\gcd (3,n)$ .",
"Then, if $n$ is odd, we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {Z}_{2n}^{-}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {Z}_{d_{3}(n)}]\\right\\rbrace ;$ if $n=2+4k$ for $k\\in \\mathbb {N}$ , we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {Z}_{2n}^{-}]=\\left\\lbrace [{1}],[\\mathbb {Z}_4^{-}],[\\mathbb {Z}_{d_{3}(n)}]\\right\\rbrace ;$ if $n$ is even and $4\\nmid n$ , we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {Z}_{2n}^{-}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{d_{3}(n)}]\\right\\rbrace .$ Moreover in all cases, we have $[\\mathbb {O}^{-}] \\circledcirc [\\mathbb {D}_{n}^{v}] = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {D}_{d_{3}(n)}^{v}],[\\mathbb {Z}_{d_{2}(n)}],[\\mathbb {D}_{d_{2}(n)}^{v}]\\right\\rbrace .$ Corollary 2.11 Let $n\\ge 2$ be an integer and $d_{3}(n):=\\gcd (n,3)$ .",
"If $n$ is even and $n=2+4k$ for $k\\in \\mathbb {N}$ , then we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {D}_{2n}^{h}]=\\left\\lbrace [{1}],[\\mathbb {Z}_4^{-}],[\\mathbb {D}_4^h],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {D}_{d_{3}(n)}^{v}]\\right\\rbrace ;$ if $n$ is even and $4\\mid n$ , then we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {D}_{2n}^{h}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {D}_{2}],[\\mathbb {D}_{2}^{v}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {D}_{d_{3}(n)}^{v}]\\right\\rbrace .$ if $n$ is odd, then we have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {D}_{2n}^{h}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {D}_{2}],[\\mathbb {D}_{2}^{v}],[\\mathbb {Z}_{d_{3}(n)}],[\\mathbb {D}_{d_{3}(n)}^{v}]\\right\\rbrace .$ Corollary 2.12 We have $[\\mathbb {O}^{-}]\\circledcirc [\\mathbb {O}^{-}]=\\left\\lbrace [{1}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {Z}_4^{-}],[\\mathbb {Z}_{3}]\\right\\rbrace .$"
],
[
"$\\mathrm {O}(2)^{-}$ subgroup",
"Consider the couple $\\mathrm {SO}(2) \\subset \\mathrm {O}(2)$ in tab:ClassIII and set $\\mathrm {O}(2)^{-} := \\mathrm {SO}(2)\\biguplus _{v\\subset xy\\text{-plane}} \\mathbb {Z}_{2}^{\\sigma _{v}}.$ As $\\mathbb {Z}_{2}^{\\sigma _{v}}$ are subgroups conjugate to $\\mathbb {Z}_{2}^{-}$ , previous results on clips operation of $[\\mathbb {Z}_{2}^{-}]$ and Type III subgroups except $[\\mathrm {O}(2)^{-}]$ leads to the following lemma.",
"Lemma 2.13 Let $n\\ge 2$ be some integer.",
"Set $d_{2}(n):=\\gcd (2,n)$ and $i(n):=3-\\gcd (2,n)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if $n$ is even}, \\\\2, & \\text{if $n$ is odd}.\\end{array}\\right.",
"}$ Then, we have $[\\mathrm {O}(2)^{-}] \\circledcirc [\\mathbb {Z}_{2n}^{-}] & = \\left\\lbrace [{1}],[\\mathbb {Z}_{i(n)}^{-}],[\\mathbb {Z}_{n}]\\right\\rbrace , \\\\[\\mathrm {O}(2)^{-}] \\circledcirc [\\mathbb {D}_{n}^{v}] & = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {D}_{n}^{v}]\\right\\rbrace , \\\\[\\mathrm {O}(2)^{-}] \\circledcirc [\\mathbb {D}_{2n}^h] & = \\left\\lbrace [{1}],[\\mathbb {Z}_{d_{2}(n)}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {D}_{i(n)}^{v}],[\\mathbb {D}_{n}^{v}]\\right\\rbrace , \\\\[\\mathrm {O}(2)^{-}]\\circledcirc [\\mathbb {O}^{-}] & = \\left\\lbrace [{1}],[\\mathbb {Z}_{2}^{-}],[\\mathbb {D}_{3}^{v}],[\\mathbb {D}_{2}^{v}]\\right\\rbrace , \\\\[\\mathrm {O}(2)^{-}]\\circledcirc [\\mathrm {O}(2)^{-}] & =\\left\\lbrace [\\mathbb {Z}_{2}^{-}],[\\mathrm {O}(2)^{-}]\\right\\rbrace .$ We will only focus on the clips operation $[\\mathrm {O}(2)^{-}]\\circledcirc [\\mathbb {D}_{2n}^h]$ and consider thus intersections $\\mathrm {O}(2)^{-}\\cap \\left(g\\mathbb {D}_{2n}^hg^{-1}\\right),\\quad g\\in \\mathrm {O}(3).$ There are only two non trivial cases to work on, whether characteristic axes match or not.",
"If primary axes match, then, by (REF ) and (REF ), we have to consider intersections $\\mathbb {Z}^{\\sigma _v}\\cap \\mathbb {Z}_{2n}^{-},\\quad \\mathrm {SO}(2)\\cap \\mathbb {Z}_{2}^{p_{j}},\\quad \\mathbb {Z}^{\\sigma _{b}}\\cap \\mathbb {Z}_{2}^{\\sigma _{q_{j}}}$ which reduce to $\\mathbb {D}_{n}^{v}$ (see decomposition (REF )).",
"Suppose, moreover, that $p_{0}=(Oz)$ , in which case $\\mathrm {SO}(2)\\cap \\mathbb {Z}_{2}^{p_{j}}=\\mathbb {Z}_{2}$ .",
"For $n$ odd, there exists some secondary axes $q_k$ in the $xy$ plane (see Lemma REF ) and thus $\\mathbb {Z}^{\\sigma _{b}}\\cap \\mathbb {Z}_{2}^{\\sigma _{q_{j}}}$ reduces to $\\mathbb {Z}_{2}^{\\sigma _{q_k}}$ .",
"Moreover, $\\mathbb {Z}^{\\sigma _v}\\cap \\mathbb {Z}_{2n}^{-}$ reduces to some $\\mathbb {Z}_{2}^{\\sigma _{v}}$ with $v$ perpendicular to $p_{0}$ and $q_{j}$ and the final result is a subgroup conjugate to $\\mathbb {D}_{2}^{v}$ .",
"For $n$ even, we obtain $\\mathbb {Z}_{2}$ .",
"If now the primary axis $a$ of $g\\mathbb {D}_{2n}^hg^{-1}$ is $(Ox)$ , and no other characteristic axes correspond to $(Oz)$ nor $(Oy)$ , then intersections (REF ) reduce to $\\mathbb {Z}_{2}^{\\sigma _a}\\cap \\left(g\\mathbb {Z}_{2n}^{-}g^{-1}\\right)$ which is conjugate to $\\mathbb {Z}_{2}^-$ ."
]
] | 1709.01776 |
[
[
"Unitarity Analyses of $\\pi N$ Elastic Scattering Amplitudes"
],
[
"Abstract The pion - nucleon scattering phase shifts in $s$ and $p$ waves are analyzed using PKU unitarization approach that can separate the phase shifts into different contributions from poles and branch cuts.",
"It is found that in $S_{11}$ and $P_{11}$ channels, there exist large and positive missing contributions when one compares the phase shift from known resonances plus branch cuts with the experimental data, which indicates that those two channels may contain sizable effects from $N^*(1535)$ and $N^*(1440)$ shadow poles.",
"Those results are obtained using tree level results of the $\\pi N$ amplitude."
],
[
"Acknowledgment",
"I would like to thank Han-Qing Zheng and De-Liang Yao for helpful advices.",
"This work is supported in part by National Nature Science Foundations of China (NSFC) under Contracts No.",
"10925522, No.",
"11021092, No.",
"11575052 and No.",
"11105038."
]
] | 1709.01647 |
[
[
"CacheShield: Protecting Legacy Processes Against Cache Attacks"
],
[
"Abstract Cache attacks pose a threat to any code whose execution flow or memory accesses depend on sensitive information.",
"Especially in public clouds, where caches are shared across several tenants, cache attacks remain an unsolved problem.",
"Cache attacks rely on evictions by the spy process, which alter the execution behavior of the victim process.",
"We show that hardware performance events of cryptographic routines reveal the presence of cache attacks.",
"Based on this observation, we propose CacheShield, a tool to protect legacy code by monitoring its execution and detecting the presence of cache attacks, thus providing the opportunity to take preventative measures.",
"CacheShield can be run by users and does not require alteration of the OS or hypervisor, while previously proposed software-based countermeasures require cooperation from the hypervisor.",
"Unlike methods that try to detect malicious processes, our approach is lean, as only a fraction of the system needs to be monitored.",
"It also integrates well into today's cloud infrastructure, as concerned users can opt to use CacheShield without support from the cloud service provider.",
"Our results show that CacheShield detects cache attacks fast, with high reliability, and with few false positives, even in the presence of strong noise."
],
[
"Introduction",
"Modern computing technologies like cloud computing build on shared hardware resources opaquely accessible by independent tenants, ensuring protection through sandboxing techniques.",
"However, although this isolation is solid at the logical level, ensuring tenants cannot access each others memory, hypervisors cannot properly prevent information leakage stemming from the shared hardware resources such as caches.",
"Compared to other resources like Branch Prediction Units or the DRAM, caches can be exploited to recover fine grain information from co-resident tenants in shared environments.",
"Cache attacks extract private information by setting up the cache memory, executing the victim process and observing effects related to sensitive data.",
"Time-based cache attacks measure the effect on the victim process execution time [5] while access-based cache attacks measure the effect on the attacker [32].",
"Practical access-based cache attacks have been published for cloud environments with different variants: Prime+Probe [34], [47], [48], [17], [9], Flush+Reload [13], [43], [2], [14], and Flush+Flush [11].",
"All of them have demonstrated to recover cryptographic keys, break security protocols or infer privacy related information.",
"They have shown that attacks can succeed in contemporary public cloud systems, with severe consequences to sensitive data of cloud customers.",
"To deter cache attacks, several techniques for detection and/or mitigation have been proposed.",
"Most of the proposed mitigation techniques succeed in stopping cache based attacks, but are not being adopted by cloud service providers.",
"Proposed hardware countermeasures require making modifications to the hardware that not only induce severe performance penalties but also take years to integrate and deploy into the infrastructure.",
"Cloud hypervisors, on the contrary, can implement any of the proposed hypervisor based countermeasures [19], [21], [39] by just making small modifications to the kernel configuration.",
"Despite the immediate fix that these countermeasures would provide, they are not being adopted by cloud providers, mainly due to the constant performance overhead that they add to their systems.",
"Other feasible mitigation proposals consider periodic VM migration to avoid long-term co-location.",
"VM migration, however, also introduces extra overhead whether there is an attack or not.",
"Other proposals suggest that just as the attacker uses a side-channel to obtain information, the VM can defend itself by using a side-channel to detect co-resident tenants with possibly malicious intentions [46] The current situation leaves tenants with little help from hardware and hypervisor designers or cloud service providers to protect themselves against cache attacks.",
"Thus, we observe the necessity of giving those tenants that voluntarily want to protect against cache attacks, tools to defend themselves.",
"So far, all known cache attacks have in common that they cause cache misses in the victim VM process.",
"Thus, detecting an anomaly in the number of cache misses in the victim can indicate an ongoing cache attack and thus trigger VM migration or other actions to mitigate the attack.",
"Cache misses can be obtained by reading the hardware performance counters found in all modern processors.",
"These hardware event counters track hardware events such as cache misses, and were originally intended to enable the detection of bottlenecks in executed software.",
"Optimization is not the only application of these counters, it has also been demonstrated that the hardware performance counters are also useful to detect malware and security breaches [40], [8], [3], [36].",
"Libraries such as PAPI (Performance Application Programming Interface), facilitate the task of configuring and reading those hardware counters.",
"There have been several attempts to detect cache attacks using the hardware counters [7], [31], [45], but they have strong drawbacks.",
"Some works require the hypervisor to periodically monitor all existing processes, which introduces a great overhead in CPU usage and depends on how efficiently an attacker can hide from the monitoring tool [45], [31].",
"Other works offer solutions applicable to multi-process environments, but not feasible in cloud environments [7], [11].",
"We propose to use a monitoring service inside the VM that detects anomalies in the cache miss hardware performance counter only in the victim side.",
"The monitoring service can be activated on demand inside the VM.",
"The performance counters must be exposed to the VM in order to be feasible.",
"Just changing the configuration of the hypervisor, it is possible to enable performance counters access inside the VM.",
"This access can be enabled only in the VMs that request the service, and as the hypervisor is responsible for the virtualization of the counters that can be read inside a VM, they refer uniquely to this VM.",
"That is, one VM can not read counters referring to another VM, even when they share the hardware.",
"Right now, most cloud service providers only expose them if the customer is renting the whole machine, probably due to their fear of utilization as a side channel in hardware shared by various tenants.",
"However, we believe they do not have much to worry about, as current attacking techniques exploiting shared hardware expose much more information than hardware counters would.",
"Our work demonstrates for the first time, that performance counter access for tenant VMs can indeed be utilized to improve security of the tenants.",
"We offer tenant VMs a new monitoring service, CacheShield, to detect cache attacks.",
"CacheShield can be activated before running sensitive processes.",
"CacheShield detects attacks quickly and with high reliability and low CPU overhead, due to the use of Page's cumulative sum method [30].",
"The CUSUM method is an unsupervised anomaly detection method, ensuring that even new attack techniques can be detected with high confidence.",
"CacheShield automatically turns off if the monitored process is idle by detecting the lack of activity, resulting in a significant reduction in CPU processing overheads.",
"In summary, our work [leftmargin=5ex] presents a performance counter based monitoring service that users can voluntarily activate to detect when they are under attack.",
"only monitors the victim process upon when active, i.e., the cloud service provider does not waste cycles continuously monitoring all processes.",
"only requires the hypervisor to enable VM access to the performance counters, a feature commonly supported by all major hypervisor systems, including KVM, VMware and Xen.",
"No other additional help from the underlying system is needed.",
"implements an efficient algorithm that maximises fast and reliable attack detection, while minimizing false positives and keeping the performance overhead minimal and restricted to the victim VM.",
"succeeds detecting all existing cache attacks, including stealthy attacks that are miss-detected by other solutions, e.g.",
"Flush+Flush, since our detection uses attack characteristics that are independent of attack and victim behavior.",
"The rest of the paper is organized as follows.",
"After discussing background and related work in Section , we show that monitoring performance events of a victim process is sufficient for reliable attack detection in Section .",
"CacheShield is developed in Section .",
"Section presents the performance evaluation in several relevant scenarios and in section we suggest different countermeasures.",
"Finally, Section discusses the conclusions of our work.",
"In the last years cache attacks have shown to pose a big threat in those systems in which the underlying hardware architecture is shared with a potential attacker.",
"Cache attacks monitor the utilization of the cache to retrieve information about a co-resident victim.",
"Indeed, if the utilization of such a hardware piece is directly correlated with a security-critical piece of information (e.g., a cryptographic key) the consequences of the attack can be as devastating as an impersonation of the victim.",
"Two main cache attack designs out-stand over the rest: the Flush+Reload and the Prime+Probe attacks.",
"The first was first introduced in [13], and was later extended to target the LLC to retrieve cryptographic keys, TLS protocol session messages or keyboard keystrokes across VMs [43], [12], [18].",
"Further, Zhang et al.",
"[48] showed that Flush+Reload is applicable in several commercial PaaS clouds.",
"Despite its popularity and resistance to micro-architectural noise, the Flush+Reload presents a main drawback, as it can only be applied in systems in which memory deduplication mechanisms are in place, and further, can only recover information coming from statically allocated data.",
"The Prime+Probe attack design, contrary to the Flush+Reload attack, is agnostic to special OS features in the system, and therefore it can not only be applied in virtually every system, but further, it can additionally recover information from dynamically allocated data.",
"This attack was first proposed for the L1 data cache in [29], while later was expanded to the L1 instruction cache [1].",
"Recently, it has been shown to also bypass several difficulties to target the LLC and recover cryptographic keys or keyboard typed keystrokes [9], [17], [22].",
"Even further, the Prime+Probe attack was used to retrieve a RSA key in the Amazon EC2 cloud [16].",
"Variations of both attacks have also been proposed to bypass specific difficulties found in some systems (e.g., lack of a flush instruction in the Instruction Set Architecture).",
"Perhaps the one that most directly influences this work is the design of the Flush+Flush attack, as it was proposed to be stealthy and bypass attack monitoring systems [11].",
"This attack retrieves information by measuring the execution time of the flush instruction, thus avoiding direct cache accesses.",
"As we will see, although this design might be effective against some of the proposed detection systems, ours correctly identifies when such an attack is being executed."
],
[
"Performance counters",
"The performance counters are special purpose hardware registers that count a broad spectrum of low-level hardware events related to code execution.",
"The selection of observable events is usually larger than the number of actual counters, hence, counters must be configured in advance.",
"All events associated with a counter are recorded in parallel.",
"As the PMU allows detailed insight into the state of the processor in real-time, it is a valuable tool for debugging applications and their performance.",
"The list of available events consequently focuses on waiting periods (e.g.",
"clock cycles the processor is stalled), memory or bus accesses (e.g.",
"cache misses or DRAM requests), and other performance-critical metrics like branch prediction or TLB events.",
"All main micro-processor architectures, i.e., Intel, AMD and ARM, include a bigger or a smaller number of these configurable registers.",
"However, while monitoring of these hardware events in Intel and AMD processors is usually possible from user mode (when referring to an application also being run in user mode), ARM devices require root rights to enable them.",
"Emulating the behavior in ARM devices, cloud providers might disable the utilization of performance counters from guest VMs.",
"Indeed we find two main reasons why they would do this [leftmargin=5ex] Performance counters might be utilized with malicious purposes, similarly to the way the thermal sensor was used in [25], and retrieve information from co-resident user hardware utilization [6], which in theory should not be possible as the hypervisor only gives information to each VM about itself.",
"As performance counters are hardware dependent, giving a guest VM access to benign utilization of performance counters might be problematic if guest VMs are migrated over different architectures, as customers would have to design code for different hardware architectures.",
"We do not believe that these facts should make cloud providers disable the usage of performance counters from guest VMs, specially when one can use them as a protection mechanism as we will see later in this work.",
"In fact, attackers have already found alternative ways to retrieve the same information performance counters give.",
"For instance, attackers can read the cycle counter or an incremental thread to know when TLB or cache misses occur.",
"Thus, disabling the counters does not entirely prevent the leakage of hardware events information.",
"As for the second claim, a possible solution could be to create clusters with the same hardware configuration, and migrate VMs within this cluster.",
"Thus, we do not believe the above concerns are strong enough arguments against the guest VM performance counter usage.",
"In this paper we will further show that such a usage can indeed offer more protection to cloud infrastructure customers."
],
[
"Detection, mitigation and other countermeasures",
"HPCs have been used to detect generic malware [36], [3], [35] as well as microarchitectural attacks [7], [45], [11], [31].",
"Their success mostly depends on the ability to correctly identify cache (and other resource) attack patterns monitoring the associated event in the HPC.",
"This approach is usually implemented at the OS or hypervisor level that has enough permissions to monitor what is running in the system.",
"However, we observe two main problems with these detection-based approaches: [leftmargin=5ex] Most of these detection approaches incur severe performance overheads that hypervisors or OSs do not seem willing to pay, as to the best of our knowledge no OS is implementing such a mechanism.",
"This leaves the user of the system with few resources to know whether her code will be executed in a safe environment.",
"As these detection countermeasures base their success on the monitoring of both the victim and the attacker processes, the attacker can vary patterns in a smart way to try to bypass the detection mechanisms.",
"These facts are observed, for instance in [45], [31], [7].",
"All three works incur significant overheads on all applications.",
"CloudRadar, for example, requires three dedicated cores for its detection [45].",
"In addition, they usually assume the ability to monitor the attacking process [11], [7], which is not possible across VM boundaries (except for the hypervisor), and usually not even possible for user-level processes.",
"Detection-based countermeasures are not the only possibility shown to prevent cache timing attacks.",
"Preemptive approaches can be taken at the hardware, software and application level.",
"The first usually requires changes in the hardware pieces such that collisions in the cache can not happen, or if they do, they do not carry information [41].",
"The second involves the utilization of specific software features (e.g., page allocation) to prevent two processes from colliding in the cache [19].",
"Finally, the latter is achieved by utilizing specific tools to ensure a security sensitive binary does not leak information, even if it is under attack [44].",
"Our objective is to build an attack detection tool that detects any abuse of the LLC without any modifications to the hypervisor, OS, or the CPU hardware.",
"Unlike previous approaches, we show that monitoring the behavior of a victim application is sufficient for the detection of cache attacks.",
"To that end, we first analyze the behavior of these victim applications by monitoring several critical hardware performance counters.",
"This behavior of critical applications is analyzed in the presence and absence of various cache attacks, and further analysis is performed to determine how well each counter serves as an indicator for ongoing attacks.",
"For the sake of simplicity, we base our analysis on cryptographic algorithms, which are the most popular target for cache attacks.",
"Our approach can also detect attacks on other security-critical pieces of code like SSL/TLS protocol stacks.",
"There are different types of cryptographic algorithms in use, which traditionally have been classified as symmetric cryptography an public-key cryptography.",
"[leftmargin=4ex] Symmetric cryptosystems are sometimes also called private key algorithms, and include algorithms for encryption, authentication as well as hashing.",
"Encryption and authentication schemes use single key for both the encryption/authentication and decryption/verification.",
"Popular algorithms include AES and DES for encryption, SHA-2 and SHA-3 for hashing and HMAC or GCM for authentication or authenticated encryption.",
"Symmetric primitives are usually heavily optimized for performance and feature constant execution flows.",
"However, some implementations make use of table look-ups, which often result in exploitable cache leakage.",
"One example is AES, the most widely used encryption algorithm.",
"For AES, table look-ups are difficult to avoid, unless hardware support such as AES-NI is available.",
"Public key cryptosystems use a public key for encryption or verification and private key for decryption or signing.",
"While public key cryptography can be used in more flexible ways, the used primitives are much more costly than for symmetric cryptography.",
"As a result, public key cryptography is mainly used for authentication and key exchange to establish a communication session, where payloads are protected using symmetric cryptography.",
"Another important offered service are certificates, which require digital signatures for generation and verification of certificates.",
"RSA, ECC and ElGamal are currently the prevailing schemes for public key cryptography.",
"As explained before, we only collect information about the victim processes, i.e.",
"the processes operating on sensitive data.",
"This approach avoids the need of monitoring other processes or VMs running in the same host.",
"This approach also avoids relying on the information gathered from an attacker who might try to hide changes on its behavior to avoid triggering an alarm.",
"Considering that each kind of algorithm presents different characteristics, we gather and analyze data of the execution of different algorithms in an initial scenario.",
"Next, we show that the main results obtained in this scenario can be extended to others."
],
[
"Analyzing Hardware Performance Events",
"Modern server CPUs make a large number hardware performance counters available, but only a limited number, typically 4 to 8, can be monitored in parallel.",
"We use the Performance API (PAPI) [27] to access the performance counters.",
"PAPI provides sufficient resolution to detect attacks while it also simplifies the task of collecting performance data.",
"In this preliminary step, we collect data from 30 accessible hardware event counters on our test platform, for sample public and private key algorithms, in the presence and the absence of cache attacks.",
"The PAPI interface provides instructions that allow us to read the counters for our process before and after each cryptographic operation, that is, we get detailed information about the variation of the counters for a single encryption or decryption execution.",
"Since the number of counters that can be read at the same time is limited, we collect the data for different groups of counters at different times.",
"We then join the data and compute the statistics.",
"Once we get all this data, we carry further study to determine and quantify which counters provide meaningful information to detect the attacks.",
"As sample victim algorithms for this analysis, we chose the software AES T-Table implementation and the RSA sliding window implementation (with flag RSA_FLAG_NO_CONSTTIME set) of OpenSSL 1.0.1f, which give representative results for public key and symmetric key cryptography.",
"As sample attacks we use the Flush+Reload against both implementations.",
"Flush+Reload tries to gain information from the execution of certain instructions or from the accesses to certain data which depend on the key.",
"For the used version of RSA, attacks target the instructions (depending on the implementation RSA can also be attacked considering accesses to data), while AES attacks are an example of cache attacks focused on the data.",
"Our experiments are performed on an Intel Core i7-4790 CPU 3.60 GHz machine with 8 MB of L3 cache and 8 GB of RAM, with Centos 7 OS.",
"For each counter we collect samples for 1 million encryption or decryption operations.",
"One noteworthy observation is that, whereas in the case of AES the values of the counters do not seem to depend on the key.",
"For the analyzed vulnerable RSA implementation, however, some of the parameters depend on the value of the key.",
"This behavior can be noticed, for example, in the number of instructions executed and in the decryption times.",
"In fact, the number of operations performed depends on the distribution of zeros and ones in the key.",
"However, while the number of instructions is not affected by the attacks, the decryption times are, as they include the extra times for cache misses.",
"We can select up to 5 or 6 counters which are representative of the attacks, as this is the maximum number of counters readable in parallel on our platform.",
"The number of counters that can be read at the same time also varies depending on which counters are used and the combination of them.",
"In order to decide which counters carry more information relative to the attacks, we use the WEKA tool [15].",
"This tool was designed with the aim of allowing researchers to easily access to state-of-the-art techniques in machine learning.",
"WEKA implements several algorithms to perform attribute selection.",
"As inputs for the tool, we select a subset among all the samples (otherwise the time it takes to perform the selection increases exponentially).",
"We randomly select 50000 instances of each of the groups, that is for AES attack and non-attack and for RSA attack and non-attack, so we obtain 200000 samples with information about 30 counters, each labeled with '1' for attacks and '0' for non-attacks.",
"We first use the infoGain function, which evaluates the worth of an attribute by measuring the information gain with respect to the class according with “InfoGain(Class,Attribute) = H(Class) - H(Class | Attribute)”, where H is the entropy.",
"Note that our experiments are balanced between attack and non-attack \"classes”, that is H(Class)$=1$ , thus an ideal attribute would gain 1 bit.",
"Values around 0.5 may indicate the attribute carries meaningful information, but only for one of the algorithms or one of the attacks.",
"Thus, L3 cache misses are not only the most meaningful predictions, but also work across the considered scenarios.",
"We have also evaluated the relief algorithm [20] for feature selection.",
"Unlike the InfoGain, which only evaluates information gained from each attribute individually, the relief algorithm outputs a score of the predictive value of an attribute relative to other attributes.",
"More positive weights indicate more predictability for this attribute.",
"To calculate the weight of an attribute, it iteratively first identifies the nearest neighbors from the same and different classes.",
"Then, weight increases if a change in the attribute leads to a change in the class and decreases when a change in the attribute value has no effect on the class.",
"Table REF presents a summary of the counters which give most relevant information for detection according to the selection algorithms, altogether with their mean values for the considered scenarios, and with the differences between attacks and the expected behavior.",
"Both tests indicate that L3 cache misses are most meaningful.",
"In fact, the relief algorithm scores all other attributes with very low scores, implying only little additional gain from using them."
],
[
"Concurrent Signal Assessment",
"Tracking hardware performance events for each cryptographic operation showed that victim-based attack detection is feasible and helped identifying relevant counters.",
"However, achieving fast detection with this approach, would require adding instructions in the middle of the code we want to protect.",
"Hence, it requires alteration of the target code, which adds unnecessary burden on the user and diminishes practicality.",
"Also, for more effective attack detection, it is preferable to read performance counters concurrently to the execution of the sensitive process.",
"This way, even attacks that succeed during the execution of a single call to the sensitive function, e.g.",
"the attacks presented in [42], [43], can be detected and prevented in time.",
"Figure: Mean LLC miss traces over time for AES and RSA executions in the presence and absence of cache attacks.",
"The numbers next to flush indicate the number of lines flushed at a time.",
"After the start up peaks, the misses go to zero in the absence of a cache attack, while under attack they remain high.During our initial experiments we have observed that all implementations feature a start up behavior, where data is loaded into the cache for the first time and the frequency of the CPU might be adjusted.",
"The subsequent executions feature a more constant behavior.",
"Regarding to the counters analyzed, for AES these start up executions show indistinguishable behavior under attack and without an attack.",
"For RSA, they can be distinguished, but it would be necessary to know exactly if the current sample belongs to the start-up group of normal executions.",
"However, if we switch to continuous monitoring, the differences between algorithms disappear and the start-up behavior is restricted to a short time at the beginning of the processes.",
"Figure REF represents the mean value of the L3 miss counter in our initial scenario setup, for Flush+Reload attacks as well as normal execution of the mentioned encryption processes.",
"The average is computed over 1000 encryptions and counters are read every 100 $\\mu s$ .",
"It can be clearly observed that after the initial transient state, the number of misses goes to zero in the absense of attacks (aes no attack and rsa no attack) for both crypto primitives.",
"It can also be observed that the mean number of misses in the case of an attack varies with the number of lines flushed each time aes 1 flush, aes 2 flush....",
"Thus, with concurrent monitoring, both algorithms behave similarly for the normal executions.",
"Switching to continuous monitoring of the counters implies that the information on total encryption times or reference cycles is no longer useful nor available.",
"To ensure the information of the other counters mentioned in Table REF is still optimal for attack detection, we performed a new analysis considering each sample collected at a period of 1 ms as an independent input to the selection attribute algorithms.",
"The results show that for the LLC misses counter the infoGain increases up to 0.92, while values for the other counters decreases.",
"Additionally, the relief algorithm output still gives better score for the L3 cache misses (0.18) and in this scenario, this value is still 5 times bigger than the weight of the next counter, indicating the L3_TCM is still the one counter of choice for cache attack detection.",
"We performed additional experiments to determine how well a cluster algorithm would distinguish between attacks and non-attacks with the periodically sampled data from several counters at once.",
"WEKA also includes clustering algorithms.",
"We tested EM and Self Organizing Maps, setting the number of clusters to two.",
"The most interesting result of this experiments is that while these algorithms were able to classify in the same cluster respectively 84% and 91% of the attack samples when using only the LLC misses counter, this number decreases to around 50-60% when adding other counters.",
"These results indicate that cache attacks can be detected, regardless of the algorithm the victim process runs, by only using information gathered from the L3 cache miss counter.",
"The algorithms feature zero misses after the initial warm-up, except if an attacker is forcing misses.",
"Additionally, as all known cache attacks, including Flush+Flush, cause cache misses on the victim process to obtain information, the results obtained here for the Flush+Reload attack are applicable for other attacks.",
"Thus, we decided to only use this one attribute, as it provides most information and, also allows us to keep the detection tool simple."
],
[
"Cache Shield",
"So far, techniques proposed to detect cache attacks imply monitoring the victim VM, the attacker VM, and any other VM running in the same host [7], [45].",
"Monitoring all VMs at rates which vary from 1 us to 5 ms result in huge overheads, and increases with each new virtual machine allocated in the same host.",
"As a consequence, cloud providers may not want to implement such a tool, as it increases overall system cost, while the benefit of preventing cache attacks might be a benefit only few customers are willing to pay for.",
"Yet, only the hypervisor, and thus the cloud service provider (CSP) has the ability to monitor all VMs on a system.",
"Indeed, as of now, we are not aware of any CSPs employing VM monitoring for microarchitectural attacks.",
"As a difference with previous approaches, our goal is to design CacheShield in such a way that we avoid monitoring all the other processes or VMs running in the same host, i.e., we only focus on our own process.",
"We assume that we have access to the performance counters within the VMs.",
"Although most cloud providers currently do not allow access to the performance counters, hypervisor systems such as VMware and KVM can be easily configured to permit reading the counters inside the VM.",
"Moreover, it is possible to decide which of the VMs allocated in a host would have access to the counters for their processes upon request.",
"Even when our approach can be implemented at the hypervisor, we believe that for cloud providers would be easier just to enable the counters for the VMs that require it, leaving the responsibility on them, than to take care of these attacks.",
"By leaving the choice of deciding which processes should be monitored and when in the hands of the user, the impact in performance of such monitoring is reduced to a minimum, as we only watch a possible victim when it is executing the protected task.",
"From the cloud provider's point of view, this way of facing detection also means no waste, as it only affects the implied VM and only when it is necessary.",
"Additionally, as the user decides when it is necessary to protect a process, we avoid the need to detect when a sensitive process is executed.",
"As a consequence, we also reduce the risk of not detecting the execution of this sensitive process and then the probability of missing an attack.",
"Figure: Overview of CacheShield.Figure REF presents a diagram of our proposed solution.",
"Whenever a user wants protection, he informs the CacheShield module, which utilizes the information gathered from the performance counters to decide whether the user is being attacked.",
"If CacheShield detects an attack, an appropriate response mechanism to prevent the information leakage is put in place.",
"Although we mainly focus on the detection phase, we discuss in Section some of the countermeasures that can be implemented to effectively prevent the attack from retrieving information, such as the utilization of a fake key or the addition of noise patterns in the cache."
],
[
"Detection Algorithm",
"One of our goals is obtaining a technique for attack detection no matter which algorithm is being attacked.",
"Additionally, we want to detect all types of cache attacks, even unknown attacks, for which the tool has not been trained to detect.",
"Supervised learning algorithms such as neural networks, have already been used to detect certain cache attacks.",
"As any supervised algorithm they have to be “trained” to detect the attack.",
"That is, they require a labeled data set including data from the different attacks we want to detect, so they can build models of them and identify their characteristic features.",
"The drawback of supervised learning is precisely that we need to train the algorithm for each situation, for each algorithm and each attack.",
"As a consequence new attacks, or attacks with different patterns would not be detected.",
"The alternative is using unsupervised techniques.",
"An unsupervised algorithm does not receive labeled data, by itself tries to cluster the received data into different groups or to find relationships between different inputs in order to put any new sample in the appropriate cluster.",
"We will briefly explore clustering techniques in the next section to select the counters which can identify an attack.",
"Other kind of unsupervised techniques are anomaly-based detection algorithms, which in theory could detect “zero-day\" attacks.",
"To the best of our knowledge no successful cache detection fully based on anomaly-detection techniques has been yet demonstrated.",
"Change-point detection methods are designed to deal with the problem of detecting abrupt changes in distributions.",
"Under the assumption that cache attacks have an effect in the performance of the protected algorithms, change-point detection algorithms stand as great candidates to detect LLC attacks.",
"We propose an algorithm based on change point-detection techniques which is self-learning so it adapts itself to detect different attack patterns, which allows us to fix the attack detection delay, and which is computationally simple so it respects the constraint of minimum impact in performance and can be implemented online."
],
[
"Cache Shield Design",
"CacheShield monitors the counters for LLC misses and for total cycles.",
"The former gives information about the use of the LLC of the protected processes while the latter gives information about when it is running or when it has finished.",
"Based on Figure REF , the CacheShield module needs the PID of the process we desire to protect and the process protected also needs to know the PID of the CacheShield process.",
"The reason is both processes need to communicate with each other (one needs to inform the other when to watch and other needs to inform the one when there is an attack going on), and that the counters can be attached to gather the data from a single process given its PID.",
"On Unix systems, the easiest way to use CacheShield is to use the fork operation, and then to use the exec system call to run the module and to give it the PID of the parent process.",
"The parent process then can execute the desired operation while being monitored.",
"In case that the parent process stops or waits for something, CacheShield automatically stops after noticing the parent has not been running for a while.",
"This means that when the parent runs again, it needs to send a “SIGCONT” signal to the CacheShield tool.",
"In a similar way, if the tool detects an attack, it can send a signal to inform the parent.",
"On Windows Systems the mechanisms for inter-process communications are slightly different, but the tool can be also adapted.",
"Change Point Detection: In order to effectively asses the detection task, we made use of change point detection theory (CPD) [4].",
"This theory can be used to construct the commonly known as quick detection algorithms, which have been successfully applied for quality control, signal-processing, anomaly or intrusion detection tasks among other problems [37], [26], [38], [28].",
"The assumption in these scenarios is that the parameters describing the monitored system do not change or change very slowly under normal conditions.",
"The parameters can, however, change at unknown time instants (including at startup) into anomalous conditions.",
"Thus, CPD algorithms are used to determine if there has been a significant change in the characteristic parameters of the monitored system, quickly and with high confidence.",
"The theory of change point detection leads to the development of efficient algorithms presenting certain optimality properties, in the sense that for a given false-alarm rate (FAR) they minimize the average time it takes to detect the change in the descriptive system's features [28].",
"CPD algorithms can be easily implemented, do not require too much memory and, as a consequence do not have significant computation overhead.",
"These methods belong to the “anomaly detection” class and are unsupervised techniques.",
"Hence, they are well-suited to detect new attacks.",
"All these properties made them very attractive for the attack-detection objective.",
"In the following, we describe the parameters of the algorithm and how we assess key issues, such as the choice of models or the use of prior information.",
"We denote the sequence of observations of the $N$ variables monitored in parallel as $X(t)=(X_{1}(t),...,X_{N}(t)), t \\ge 1$ .",
"Before a change occurs, the joint probability distribution (pdf) of the random variables $X_{1},...,X_{N}$ also known as prechange distribution, can be denoted as $p_0(X_{1},...,X_{N})$ .",
"If a change occurs at an unknown time instant $\\lambda $ , the observations will follow a different distribution $p_1(X_{1},...,X_{N})$ , also called postchange distribution.",
"That is, when $t < \\lambda $ the observations $X(t)$ will have conditional pdf $p_0(X(t)|X(1),...,X(t-1))$ , and pdf $p_1(X(t)|X(1),...,X(t-1))$ for $t \\ge \\lambda $ .",
"Under the hypothesis that a change has occurred, the stopping time $\\tau $ at which the alarm is triggered gives a measurement of the detection time.",
"It is typically defined as the first time the change sensitive statistic watching the system, exceeds a threshold.",
"Naming $E_0$ and $E_\\lambda $ the expectations for the sequence of observations prior and after the change at time $\\lambda $ , the average detection delay (ADD) is defined as: $ADD_\\lambda ( \\tau ) = E_\\lambda ( \\tau - \\lambda | \\tau \\ge \\lambda )$ On the other hand, considering that there has not occurred any alarm, the mean time between false alarms will be given by the expression $E_0 \\tau $ .",
"As a consequence of this definition, the average frequency of false alarms or false alarm rate (FAR) is defined as: $FAR(\\tau )=\\frac{1}{E_0 \\tau }$ For a good detection procedure it is expected low FAR and small values of the expected detection delay.",
"The design of CPD algorithms often involves a trade-off between these two parameters.",
"Page's cumulative sum (CUSUM) detection algorithm [30] is one of the most popular CPD algorithms: with a full-knowledge of the pre-change and post-change distributions it provides an optimal scheme minimizing the worst-case detection delay.",
"Page's CUSUM algorithm utilizes the log-likelihood ratio (LLR) to check the hypothesis that a change occurred, LLR is defined as: $s(t)= ln \\dfrac{p_1(X(t)|X(1),...,X(t-1))}{p_0(X(t)|X(1),...,X(t-1))}$ The key property of this ratio is that a change in the parameter under study will also cause a change in the sign of the log-likelihood ratio.",
"In other words, $s(t)$ shows a negative drift before change and a positive drift after change.",
"The relevant information for the detection task lies then in the difference between the value of $s$ and a minimum value.",
"The decision rule is based on a comparison with a threshold $h$ : $g_k= S_k - m_k \\ge h$ where $S_k &=\\sum _{t=1}^{k}s(t) & m_k &=\\min _{1\\le j \\le k} S_j$ This decision rule can be replaced by the following, which obeys the recursion and whose value for the initial observation is $k=0$ .",
"$g_k = \\max \\left\\lbrace 0,g_{k-1}+\\ln \\dfrac{p_1(X(k))}{p_0(X(k))}\\right\\rbrace \\ge h$ Then the detection time for the given threshold is $\\tau (h) = \\min \\left\\lbrace k \\ge 1 : g_k \\ge h \\right\\rbrace $ Although this first approach considers that both distributions are known, this assumption is usually not true, and as a consequence this proposal has to be adapted for each situation.",
"We may know one distribution in advance or none, so it may be necessary to estimate the parameters of the algorithm during the runtime.",
"As long as the estimators for the distributions and the real observation meet certain convergence conditions, we will be able to fix for example the desired detection delay or the FAR.",
"Change Point Detection in CacheShield: While facing the cache attack detection, the attack may start from the very beginning or it may start after a few “normal” transactions.",
"Both situations are efficiently managed by the proposed CUSUM algorithm.",
"We assume that each new sample can be classified into one of two different groups or clusters, namely “attack” and “non-attack”.",
"The “non-attack” cluster represents how we expect the protected process to behave under normal conditions.",
"Based on the information we can gain from the counters, this assumption is that after a few samples corresponding to the initialization of the protected process, the number of L3 cache misses will be around 0, then $\\mu _{na}=0$ .",
"On the other hand, when there is an attack, we have observed that the mean number of misses is $\\mu _a$ .",
"Then each new sample belonging to the “attack” cluster will be around $\\mu _a$ .",
"The value of $\\mu _a$ is unknown and depends on the attack so it needs to be computed and recalculated with each new sample.",
"If we denote as $miss_i$ each new sample that the CacheShield module gets referring to the protected process, we need to decide if it belongs to one cluster or to the other.",
"To do so, we compute the value of the \"probability\" that $miss_i$ belongs to each one making use of the distance metric, this way we define the distance from $miss_i$ to $\\mu _{na}$ as: $d_{na}(i) = miss_i -\\mu _{na}=miss_i$ Then, the distance with the \"attack\" cluster will be $d_{a}(i) = | miss_i -\\mu _{a} |$ As stated before the value of $\\mu _a$ is unknown when we start to monitor the process.",
"We select an arbitrary initial value, and whenever a new sample $miss_i$ is obtained, if $miss_i \\ge 0$ we update the value of $\\mu _{a}$ as follows: $\\mu _{a} = (1-\\beta ) * \\mu _{a} + \\beta * miss_i $ This method is known as exponentially weighted moving average, where the weight of the older datum decreases exponentially.",
"This way of estimating the mean of the \"attack cluster\" makes the the election of the initial arbitrary value irrelevant after collecting a few misses samples.",
"If the initial value is chosen too low, we may trigger false positives.",
"We recommend the election of an initial value higher than 10, in order to keep the rate of false positives low, while being able to detect the attack in a reasonable time.",
"We will further discuss the noise tolerance of the proposed detection algorithm in the next section.",
"In our experiments we set $ \\beta = 0.05$ and the initial value to 12.5.",
"Now we are in conditions to define the probability of belonging to each cluster: $p_{na}(miss_i ) &= \\dfrac{d_{a}(i)+1}{| d_{na}(i) |+| d_{a}(i) |}, && p_{a}(miss_i ) &= \\dfrac{d_{na}(i)+1}{| d_{na}(i) |+| d_{a}(i) |}$ The value 1 has been added to avoid divisions by 0 in the LLR calculation that has to be performed as part of the detection algorithm.",
"As a result, for every sample $k$ , $k \\le 1$ we can express the detection rule as follows: $g_k = \\max \\left\\lbrace 0,g_{k-1}+\\log \\dfrac{d_{na}(k)+1}{d_{a}(k)+1}\\right\\rbrace \\ge h$ As it can be easily derived from the previous equation and according to the properties of the LLR, when the number of misses is 0 or close to 0, the distance between the sample and the \"non-attack\" cluster $d_{na}(k)$ will be lower than the distance to the attack cluster $d_{a}(k)$ , so the value of the metric $g_k$ decreases or stays at zero.",
"On the other hand, readings from the LLC misses counter approaching to the attack cluster will increase the value $g_k$ .",
"The properties of this approach let us choose the threshold based on a minimum detection time we want to achieve.",
"Note that when the error in the estimation of the mean $\\epsilon $ approaches to zero, $d_{a}(i)=\\epsilon $ also tends to zero, then the increase in the value of $g_k$ is also limited $\\log \\dfrac{d_{na}(k)+1}{d_{a}(k)+1} \\le \\log ( \\mu _{a} +1) $ As a consequence, the minimum expected detection time for the given threshold $h$ is: $\\tau _{e}(h) \\ge \\dfrac{h}{\\log ( \\mu _{a} +1) }$ or reformulating this equation, the threshold h, for a minimum expected delay $\\tau _{e}$ $h_{\\tau } \\le \\tau _{e}*\\log ( \\mu _{a} +1) $ The unit of the $\\tau _{e}(h)$ is number of samples.",
"Given that the most effective cache attacks can potentially extract most of the key with just one execution of the victim, the sampling rate must be chosen lower than the execution time of the victim.",
"As the execution time of these algorithms is in the order of few milliseconds, a sampling rate of 100 $\\mu s$ seems sufficient to provide evidence of the attack.",
"This frequency can be increased at additional load for the system.",
"So, for an expected detection delay of 1 ms with a sampling rate of 100 $\\mu s$ we can define the threshold as $h=10*\\log ( \\mu _{a} +1) $ .",
"As a result of this selection of $h$ , when the $\\mu _{a}$ is recalculated, the threshold should be recalculated too.",
"The choice of the threshold $h$ also determines the tolerance to noisy frames, and as a consequence the false positive rate.",
"In practice, the false positive rate cannot be estimated and has to be measured.",
"Algorithm REF summarizes CacheShield implementation and an example of the values of the parameters considered in the detection process is given in Figure REF .",
"end if [IF]IfEndIf[1] 1 end for [FOR]ForAllEndFor[1] 1 CacheShield detection algorithm [0] Process PID Attack detected read_counters(misses,cpu_cycles); wait; victim_is_running read_counters(misses,cpu_cycles); misses$>0$ update $\\mu _{a}$ ; update $h$ ; calculate $g_k $ $g_k > h$ trigger_alarm; wait; return detected; Figure: Relevant parameters for the detection task, prime+probe attack on AES"
],
[
"Evaluation of CacheShield",
"Once we have defined the relevant parameters of the detection algorithm and described it in detail, we evaluate its performance.",
"To this end we ran several experiments in different environments and machines.",
"[leftmargin=4ex] Native Environment The experiments for non-virtualized environments were performed in an Intel Core i7-4790 CPU 3.60GHz machine with 8 MB of L3 cache and 8 GB of RAM, with Centos 7 OS.",
"KVM-based Hypervisor These experiments used the same hardware as above, but this time within a VM also with Centos 7 hosted in KVM as hypervisor.",
"VMware-based Cloud Server We have also executed experiments in a host managed with VMware, this machine is equipped with a Intel XeonE5-2670 v2 processor, 25Mb of L3 cache and 32GB of RAM.",
"The OS in these VMs was Ubuntu 12.04.",
"Table: Mean detection time (ms) per attack and scenario for the evaluated crypto algorithms.",
"Note that in all cases CacheShield has the same configuration and that detection times are much lower than the ones required for the attack to succeedWhen a user is executing the crypto algorithm in their own machine, they can get information about the utilization of such machine or other task running concurrently.",
"However, when executing the crypto algorithms in cloud environments they can not get any information about what their neighbors are doing.",
"In such scenarios, it becomes mandatory to study how the execution of different applications running in parallel with the protected process affects the behavior of CacheShield.",
"Note that as we use the \"total cycles” counter (to determine if the victim is executing or not) and the LLC misses counter to decide if there is an attack going on, applications consuming high amount of memory resources are the most likely to cause the LLC misses indicator to rise, and as a consequence, to trigger false positives.",
"For this reason, we have selected several worst case scenario applications with high memory activity to run in parallel with the victim and CacheShield: [leftmargin=4ex] Yahoo Cloud Serving Benchmark This benchmark was originally designed as a tool that provides a common evaluation framework and a set of common workloads to test the performance of different serving stores as elastic search, Cassandra, MongoDB among others [33].",
"It allows different configurations for the workloads and provides a set of example workload scenarios, together with a workload generator, which generates the load to test storage systems.",
"In our experiments, we use this benchmark with the Apache Cassandra database and the example workload named workloada.",
"Video Streaming Another kind of application that can generate cache misses is web-browsing or video streaming.",
"The video streaming VM continuously streams and plays back youtube videos on the firefox browser.",
"Randmem Benchmark This benchmark was originally intended to test the impact of burst reading and writings [24].",
"Depending on the configuration, the benchmark accesses data stored in an array either sequentially or in random order.",
"The tool also allows to configure the size of the memory it is going to use, by default it tries to use as much as possible, up to 2 Gb.",
"In our experiments, we launch each randmem instance with no memory limitation, which means 2 Gb of RAM memory are used by each instance.",
"To show the applicability of CacheShield to a broad range of implementations that require protection against cache attacks, we chose from a range of crypto primitives and implementations, though focusing on vulnerable ones, since such legacy implementations actually require protection.",
"The three crypto algorithms considered as victims are [leftmargin=4ex] AES as the most common symmetric encryption algorithm.",
"We consider the T-Table implementation of AES from Openssl 1.0.1f, which is fast, but also leaky.",
"RSA is the probably most widely used signature and public key encryption algorithm.",
"We analyzed the RSA implementation from Openssl 1.0.1f, with a 2048 bit key, and the RSA_FLAG_NO_CONSTTIME flag set.",
"ElGamal we chose the ElGamal implementation of libgcrypt 1.5.0 with a 4096 bits key.",
"Unlike AES and RSA, ElGamal was not considered during the design of CacheShield, and hence shows how CacheShield can be expected to perform for other types of algorithms.",
"These algorithms differ quite significantly in their particular implementation and usage of cache.",
"Many other potentially leaky codes might require protection, and we are confident that CacheShield will perform well.",
"Table: False positive rate for different scenarios and algorithms.",
"(Instances: Y - Yahoo Cloud Serving, V - Video Streaming; R - Randmem)To evaluate the effectiveness of CacheShield across different types of cache attacks, we implemented and performed three popular attacks, namely Flush+Reload, Flush+Flush and Prime+Probe.",
"We collected data for the above-mentioned algorithms under attack as well as from normal executions, as baseline behavior.",
"Under each configuration, we collect data for more than 1000 executions of the crypto primitives, and in the case of the AES attack we also consider different attack rates (number of lines flushed at a time), as the attacker may try to gain different amount of information from the T-tables per execution [10].",
"As stated in previous sections, the main characteristics defining the detection algorithm are the mean detection time, and the false positive rate.",
"Table REF presents the results for mean detection time under different configurations, for the different attacks and algorithms and table REF shows the results related with false positives in noisy environments.",
"Note that the attack requirements for Flush+Flush/Flush+Reload and Prime+Probe differ significantly.",
"While Flush+X attacks are faster and more precise, they require shared data, i.e.",
"deduplication between attacker and victim.",
"All the attacks performed in virtualized scenarios were across VMs so we enabled deduplication features (KSM and TPS for KVM and vmware respectively) to perform Flush+X attacks.",
"Prime+Probe attacks work across VMs even without deduplication, so we disabled deduplication and enabled huge pages.",
"Prime+Probe attacks require, prior to the information extraction, a profiling of the cache [9], [17], [16].",
"The profiling stage reveals the sets the victim process is accessing and that carry the necessary information to succeed in the attack.",
"In this situation, the detection tool will trigger an alarm whenever the set being tested by the attacker was actually used by the victim.",
"Fig.",
"REF visualizes the output of the detection algorithm,for the cache profiling stage of an 8 MB L3 cache when the target is the T-table implementation of AES.",
"The x-axis represents each set of the cache being evicted during the Prime+Probe profiling step; a 1 on the the y-axis indicates that an alarm has been triggered.",
"Thus, alarms are only triggered when the cache attack affects the target.",
"Figure: Output of CacheShield when the cache is profiled accessing each set.",
"\"1\" indicates a positive attack detection.For all evaluated attacks, the detection rates are 100%.",
"Note that the sampling rate is 100 $\\mu $ s and that we want to detect the attack before the end of each decryption (for public key cryptography).",
"If we wished to detect attacks against algorithms whose duration is below 5 or 6 ms, we will need to increase the sampling rate, since mean number of samples required to detect the attack cannot be lowered arbitrarily without increasing the FAR too much.",
"The duration of the decryption/encryption depends on frequency of the processor, and as a consequence on the machine.",
"For example, ElGamal encryption takes around 11 ms when being attacked on the i7 machine, while this time increases up to 24 ms on the Xeon machine.",
"Thus, we are able to detect attacks against ElGamal when less than the 37% of the encryption has been performed for the i7 machine, and 30% for the second machine in the worst case.",
"Regarding to RSA this mean execution times are around 18 ms for the i7 machine and around 37 ms for the other.",
"Then, in the worst case, on average we detect the attacks with less than 50% of the decryption performed in the first case and with about 37% of decryption in the second one.",
"Regarding to the existing differences between false positive rates for AES and public crypto algorithms, these are easy to explain.",
"While between AES encryptions exists some time in which the processor does nothing, the others execute uninterruptedly.",
"This fact increases the probability of other processes accessing the cache during the same interval.",
"For example, while the AES encryption in the period of 100 $\\mu $ s is only active during around 7000 cycles while the RSA process is active during about 30000 cycles for the VMware machine when there is no attack.",
"Fig.",
"REF depicts the LLC misses for one noisy RSA encryption, besides the initialization steps, it can be observed a high amount of cache misses during the whole decryption.",
"Similarly, Fig.",
"REF corresponds to one process performing AES encryptions.",
"Figure: LLC misses for a noisy RSA execution under randmem benckmark.Figure: Sample of a noisy execution of AES under randmem benchmark.The results also show that the tolerance to noise of the detection algorithm is more dependent on the hardware than on the virtualizing technology: While the results for the native and KVM scenarios are similar and the hardware is the same, the results are significantly better on the Xeon machine.",
"The Xeon machine did not trigger any false positives when there were one or two VMs generating \"noise\" concurrently, until we launched several more instances.",
"As this machine is more similar to the kind of machine cloud providers utilize, these results show that the tool is practical in these environments.",
"One approach to reduce the false positives in noisy environments could be considering the variance of the samples collected in the CUSUM algorithms proposed, as attacks present low variance compared with noise.",
"However, we could fail to detect attacks masquerading as memory activity by generating different number of misses each time.",
"Another consideration relative to memory utilization, and the false positives that are triggered when is high memory utilization is that Prime+Probe attacks need low memory activity to accurately locate the sets and to perform the attack, other way it renders much more difficult.",
"On the other hand, Flush+Flush attacks are more tolerant to noise, but memory activity degrade its performance.",
"So it is not likely that the attacker performs the attack in a situation where the memory is highly utilized.",
"Additionally, the level of utilization of public clouds is low [23], so the assumption of high memory utilization in the considered cloud scenarios may not be realistic.",
"As for using the tool in our controlled physical machine, we can get to know which is the level of utilization of the memory and then decide if it is worth it to change the parameters of the detection algorithm.",
"One last consideration about our tool is the amount of CPU it utilizes to monitor the victim and compute the detection algorithm.",
"Fig.",
"REF and REF show the mean CPU utilization of CacheShieldfor different sampling rates and for different situations, namely when the victim is attacked and when is not, because the amount of operations it has to do changes, and again for both architectures, depending on the sampling rate.",
"To obtain the CPU utilization we have measured the time it takes to read the counters and perform the calculations and the total time elapsed, then the utilization is given by its division.",
"Note that sampling rates of 10 $\\mu $ s are not always achievable as sometimes (around 10% of the time) it takes more time to read the counters and perform the calculations.",
"Note that in both cases total utilization of our tool is below 5% of CPU utilization.",
"Figure: CPU utilization of the i7 machine in native environment for different sampling rates in microseconds.",
"Highlighted the 100 us rate as it is the one we use in our experimentsFigure: CPU utilization of the Xeon machine in virtualized environment for different sampling rates in microseconds.",
"Highlighted the 100 us rate as it is the one we use in our experiments"
],
[
"Cache Attack Countermeasures",
"Once an attack has been detected, CacheShield needs to react in some way.",
"One way is to simply interrupt the monitored process and to purge used keys.",
"While this approach ensures high security, it decreases the usability, as any false positives will result a total cryptosystem shutdown.",
"An alternative is to continue execution, but to apply preventative measures to reduce or prevent the exploitability of the cache.",
"[leftmargin=4ex] Adding Noise A simple method to hinder cache attacks is making the channel noisier, e.g.",
"through frequently flushing cache lines used by the protected process, or by performing additional reads on data.",
"This approach works particularly well if critical data is known, e.g.",
"the tables of an AES implementation.",
"Dummy Operations An alternative approach is to perform dummy operations on meaningless secrets.",
"In practice this can mean to run the protected process with a newly generated secret.",
"the original process can either be paused, or be continued in parallel to the dummy process.",
"Parallel processing obfuscates the true leakage.",
"However, depending on the attack type, an attacker might still succeed with an increased number of observations.",
"Pausing has the advantage that the attacker might actually extract the dummy key and discontinue the attack.",
"The monitor can then restart the original process in the absence of the attack.",
"Either way, the performance degradation is not negligible, but it only is incurred in the presence of an attack.",
"Protected Implementations The main reason why leakage is still observed in security solutions is the performance overhead that pure constant time implementations present.",
"A way of avoiding such a scenario is to use protected implementations only when CacheShield detects an attack behavior.",
"When no attack is detected, faster (less secure) implementations can be used.",
"Other more sophisticated solutions are also possible, but might not be as universally applicable.",
"Since our focus is on the lightweight detectability of cache attacks at the user level, we do not explore these additional avenues of countermeasures."
],
[
"Conclusion",
"In this work we have introduced CacheShield, a tool that is able to detect all known types of cache attacks targeting cryptographic applications.",
"The analysis of various hardware performance counters revealed that the LLC miss counter by itself carries enough information to detect cache attacks.",
"We take advantage of change point detection algorithms and adapt them to our objective of cache attack detection.",
"CacheShield was designed to detect attacks based on the characteristics of two particular algorithms, AES and RSA.",
"The evaluation revealed that CacheShield can also be used for other algorithms (as shown for ElGamal) without further modification.",
"It is also effective against “unknown” attacks, as all known attacks force cache misses on the victim.",
"This behavior can be easily detected, since the number of L3 cache misses of crypto algorithms approaches zero after a brief initial warm-up.",
"In addition, we have shown that CacheShield tolerates considerably high amount of noise only triggering a few false positives in machines similar to the ones cloud providers use.",
"Previously proposed cache attack detection tools work at the hypervisor level and also need to continuously monitor all untrusted and concurrently running processes or VMs, resulting in huge performance overheads and often have questionable detection rates for novel attacks such as Flush+Flush.",
"CacheShield only needs access to the protected victim process, and only during its execution, greatly reducing the waste.",
"All major hypervisor systems support transparent access to hardware performance counters for guest VMs while ensuring proper isolation between VMs.",
"We urge Cloud Service Providers to enable these features in their systems and thus give their tenants finally the means to protect themselves against cache attacks with tools such as CacheShield."
],
[
"Acknowledgments",
"Visit of Samira Briongos to Vernam group at Worcester Polytechnic Institute has been supported by a collaboration fellowship of the European Network of Excellence on High Performance and Embedded Architecture and Compilation (HiPEAC).",
"This work was in part supported by the National Science Foundation under Grant No.",
"CNS-1618837 and by the Spanish Ministry of Economy and Competitiveness under contracts TIN-2015-65277-R, AYA2015-65973-C3-3-R and RTC-2016-5434-8."
]
] | 1709.01795 |
[
[
"Deep learning from crowds"
],
[
"Abstract Over the last few years, deep learning has revolutionized the field of machine learning by dramatically improving the state-of-the-art in various domains.",
"However, as the size of supervised artificial neural networks grows, typically so does the need for larger labeled datasets.",
"Recently, crowdsourcing has established itself as an efficient and cost-effective solution for labeling large sets of data in a scalable manner, but it often requires aggregating labels from multiple noisy contributors with different levels of expertise.",
"In this paper, we address the problem of learning deep neural networks from crowds.",
"We begin by describing an EM algorithm for jointly learning the parameters of the network and the reliabilities of the annotators.",
"Then, a novel general-purpose crowd layer is proposed, which allows us to train deep neural networks end-to-end, directly from the noisy labels of multiple annotators, using only backpropagation.",
"We empirically show that the proposed approach is able to internally capture the reliability and biases of different annotators and achieve new state-of-the-art results for various crowdsourced datasets across different settings, namely classification, regression and sequence labeling."
],
[
"Introduction",
"In the last decade, deep learning has made major advances in solving artificial intelligence problems in different domains such as speech recognition, visual object recognition, object detection and machine translation [21].",
"This success is often attributed to its ability to discover intricate structures in high-dimensional data [12], thereby making it particularly well suited for tackling complex tasks that are often regarded as characteristic of humans, such as vision, speech and natural language understanding.",
"However, typically, a key requirement for learning deep representations of complex high-dimensional data is large sets of labeled data.",
"Unfortunately, in many situations this data is not readily available, and humans are required to manually label large collections of data.",
"On the other hand, in recent years, crowdsourcing has established itself as a reliable solution to annotate large collections of data.",
"Indeed, crowdsourcing platforms like Amazon Mechanical Turkhttp://www.mturk.com and Crowdflowerhttp://crowdflower.com have proven to be an efficient and cost-effective way for obtaining labeled data [25], [2], especially for the kind of human-like tasks, such as vision, speech and natural language understanding, for which deep learning methods have been shown to excel.",
"Even in fields like medical imaging, crowdsourcing is being used to collect the large sets of labeled data that modern data-savvy deep learning methods enjoy [6], [1], [7].",
"However, while crowdsourcing is scalable enough to allow labeling datasets that would otherwise be impractical for a single annotator to handle, it is well known that the noise associated with the labels provided by the various annotators can compromise practical applications that make use of such type of data [22], [5].",
"Thus, it is not surprising that a large body of the recent machine learning literature is dedicated to mitigating the effects of the noise and biases inherent to such heterogeneous sources of data (e.g.",
"[29] Yan2014; [1] albarqouni2016aggnet; [7] guan2017said).",
"When learning deep neural networks from the labels of multiple annotators, typical approaches rely on some sort of label aggregation mechanisms prior to training.",
"In classification settings, the simplest and most common approach is to use majority voting, which naively assumes that all annotators are equally reliable.",
"More advanced approaches, such as the one proposed in [4] and other variants (e.g.",
"[9] ipeirotis2010quality; [28] whitehill2009whose) jointly model the unknown biases of the annotators and their answers as noisy versions of some latent ground truth.",
"Despite their improved ground truth estimates over majority voting, recent works have shown that jointly learning the classifier model and the annotators noise model using EM-style algorithms generally leads to improved results [15], [1].",
"In this paper, we begin by describing an EM algorithm for learning deep neural networks from crowds in multi-class classification settings, highlighting its limitations.",
"Then, a novel crowd layer is proposed, which allows us to train neural networks end-to-end, directly from the noisy labels of multiple annotators, using only backpropagation.",
"This alternative approach not only allows us to avoid the additional computational overhead of EM, but also leads to a general-purpose framework that generalizes trivially beyond classification settings.",
"Empirically, the proposed crowd layer is shown to be able to automatically distinguish the good from the unreliable annotators and capture their individual biases, thus achieving new state-of-the-art results in real data from Amazon Mechanical Turk for image classification, text regression and named entity recognition.",
"As our experiments show, when compared to the more complex EM-based approaches and other approaches from the state of the art, the crowd layer is able to achieve comparable or, in many cases, significantly superior results."
],
[
"Related work",
"The increasing popularity of crowdsourcing as a way to label large collections of data in an inexpensive and scalable manner has led to much interest of the machine learning community in developing methods to address the noise and trustworthiness issues associated with it.",
"In this direction, one of the key early contributions is the work of Dawid and Skene DawidSkeene1979, who proposed an EM algorithm to obtain point estimates of the error rates of patients given repeated but conflicting responses to medical questions.",
"This work was the basis for many other variants for aggregating labels from multiple annotators with different levels of expertise, such as the one proposed in [28], which further extends Dawid and Skene's model by also accounting for item difficulty in the context of image classification.",
"Similarly, Ipeirotis et al.",
"ipeirotis2010quality propose using Dawid and Skene's approach to extract a single quality score for each worker that allows to prune low-quality workers.",
"The approach proposed in our paper contrast with this line of work, by allowing neural networks to be trained directly on the noisy labels of multiple annotators, thereby avoiding the need to resort to prior label aggregation schemes.",
"Despite the generality of label aggregation approaches described above, which can be used in combination with any type of machine learning algorithm, they are sub-optimal when compared to approaches that also jointly learn the classifier itself.",
"One of the most prominent works in this direction is the one of Raykar et al.",
"Raykar2010, who proposed an EM algorithm for jointly learning the levels of expertise of different annotators and the parameters of a logistic regression classifier, by modeling the ground truth labels as latent variables.",
"This idea was later extended to other types of models such as Gaussian process classifiers [18], supervised latent Dirichlet allocation [16] and, recently, to convolutional neural networks with softmax outputs [1].",
"In this paper, we begin by describing a generalization of the approach in [1] to multi-class settings, highlighting some of the technical difficulties associated with it.",
"Then, a novel type of neural network layer is proposed, which allows the training of deep neural networks directly from the noisy labels of multiple annotators using pure backpropagation.",
"This contrasts with most of works in the literature, which rely on more complex iterative procedures based on EM.",
"Furthermore, the simplicity of the proposed approach allows for straightforward extensions to regression and structured prediction problems.",
"Recently, [7] guan2017said also proposed an approach for training deep neural networks that exploits information about the annotators.",
"The idea is to model the multiple experts individually in the neural network and then, while keeping their predictions fixed, independently learning averaging weights for combining them using backpropagation.",
"Like our proposed approach, this two-stage procedure does not require an EM algorithm to estimate the annotators weights.",
"However, while our approach has the ability to capture the biases of the different annotators (e.g.",
"confusing class 2 with class 4) and correct them, the approach in [7] only learns how to combine the predicted answers of multiple annotators by weighting them differently.",
"Moreover, its two-stage learning procedure increases the computation complexity of training, whereas in our proposed approach is kept the same.",
"Lastly, while the work in [7] focuses only on classification, we consider regression and structured prediction problems as well.",
"Regarding applications areas for multiple-annotator learning, some of the most popular ones are: image classification [24], [27], computer-aided diagnosis/radiology [15], [6], object detection [26], text classification [16], natural language processing [25] and speech-related tasks [13].",
"In this paper, we will use data from some of these areas to evaluate different approaches.",
"Given that these are precisely some of the areas that have seen the most dramatic improvements due to recent contributions in deep learning [12], [21], developing novel efficient algorithms for learning deep neural networks from crowds is of great importance to the field."
],
[
"EM algorithm for deep learning from crowds",
"Let $ \\lbrace \\textbf {x}_n, \\textbf {y}_n\\rbrace _{n=1}^N$ be a dataset of size $N$ , where for each input vector $\\textbf {x}_n \\in \\mathbb {R}^D$ we are given a vector of crowdsourced labels $\\textbf {y}_n = \\lbrace y_n^r\\rbrace _{r=1}^R$ , with $y_n^r$ representing the label provided by the $r^{th}$ annotator in a set of $R$ annotators.",
"Following the ideas in [15], [29], we shall assume the existence of a latent true class $z_n$ whose value is, in this particular case, determined by a softmax output layer of a deep neural network parameterized by $\\Theta $ , and that each annotator then provides a noisy version of $z_n$ according to $p(y_n^r|z_n, \\Pi ^r) = Multinomial(y_n^r|\\pi _{z_n}^r)$ .",
"This formulation corresponds to keeping a per-annotator confusion matrix $\\Pi ^r = (\\pi _1^r,\\dots ,\\pi _C^r)$ to model their expertise, where $C$ denotes the number of classes.",
"Further assuming that annotators provide labels independently of each other, we can write the complete-data likelihood as $p(\\textbf {z}|\\Theta ,\\lbrace \\Pi ^r\\rbrace _{r=1}^R) &= \\prod _{n=1}^N p(z_n | \\textbf {x}_n, \\Theta ) \\prod _{r=1}^R p(y_n^r|z_n, \\Pi ^r).\\nonumber $ Based on this formulation, we can derive an Expectation-Maximization (EM) algorithm for jointly learning the reliabilities of the annotators $\\Pi ^r$ and the parameters of the neural network $\\Theta $ .",
"The expected-value of the complete-data log-likelihood under a current estimate of the posterior distribution over latent variables $q(z_n)$ is given by $&\\mathbb {E}\\Big [\\ln p(\\textbf {z}|\\Theta ,\\Pi ^1,\\dots ,\\Pi ^R)\\Big ] = \\nonumber \\\\& \\sum _{n=1}^N \\sum _{z_n} q(z_n) \\ln \\bigg ( p(z_n | \\textbf {x}_n, \\Theta ) \\prod _{r=1}^R p(y_n^r|z_n, \\Pi ^r) \\bigg ),$ where the posterior $q(z_n)$ is obtained by making use of Bayes' theorem given the previous estimate of the model parameters $\\lbrace \\Theta _{\\mbox{\\tiny old}},\\Pi _{\\mbox{\\tiny old}}^1,\\dots ,\\Pi _{\\mbox{\\tiny old}}^R\\rbrace $ , yielding $q(z_n = c) &\\propto p(z_n = c | \\textbf {x}_n, \\Theta _{\\mbox{\\tiny old}}) \\prod _{r=1}^R p(y_n^r|z_n = c, \\Pi _{\\mbox{\\tiny old}}^r).\\nonumber $ This corresponds to the E-step of EM.",
"In the M-step, we find the new maximum likelihood for the model parameters.",
"The update for the annotators' reliability parameters is given by $\\pi _{c,l}^r = \\frac{\\sum _{n=1}^N q(z_n = c) \\, \\mathbb {I}(y_n^r = l)}{\\sum _{n=1}^N q(z_n = c)},\\nonumber $ where $\\mathbb {I}(y_n^r = l)$ is an indicator function that takes the value 1 when $y_n^r = l$ , and zero otherwise.",
"In practice, since crowd annotators typically only label a small portion of the data, it is particularly important to carefully impose Dirichlet priors on each $\\pi _c^r$ and compute MAP estimates instead, in order to avoid numerical issues.",
"As for estimating the parameters of the deep neural network $\\Theta $ , we follow the approach in [1] and use the noise-adjusted ground-truth estimates $q(z_n)$ to backpropagate the error through the network using standard stochastic optimization techniques such as stochastic gradient descent (SGD) or Adam [11].",
"Kindly notice how this raises the important question of how to schedule the EM steps.",
"If we perform one EM iteration per mini-batch, we risk not having enough evidence to estimate the annotators reliabilities.",
"On the other hand, if we run SGD or Adam until convergence, then the computational overhead of EM becomes very large.",
"In practice, we found that, typically, one EM iteration per training epoch provides good computational efficiency without compromising accuracy.",
"However, this seems to vary among different datasets, thus making it hard to tune in practice.",
"One key fundamental aspect for the development of this EM approach was the probabilistic interpretation of the softmax output layer of deep neural networks for classification.",
"Unfortunately, such probabilistic interpretation is typically not available when considering, for example, continuous output variables, thereby making it more difficult to generalize this approach to regression problems.",
"Furthermore, notice that if the target variable is a sequence (or any other structured prediction output), then the marginalization over the latent variables in (REF ) quickly become intractable, as the number of possible label sequences to sum over grows exponentially with the length of the sequence."
],
[
"Crowd layer",
"In this section, we propose the crowd layer: a special type of network layer that allows us to train deep neural networks directly from the noisy labels of multiple annotators, thereby avoiding some of the aforementioned limitations of EM-based approaches for learning from crowds.",
"The intuition is rather simple.",
"The crowd layer takes as input what would normally be the output layer of a deep neural network (e.g.",
"softmax for classification, or linear for regression), and learns an annotator-specific mapping from the output layer to the labels of the different annotators in the crowd that captures the annotator reliabilities and biases.",
"In this way, the former output layer becomes a bottleneck layer that is shared among the different annotators.",
"Figure REF illustrates this bottleneck structure in the context of a simple convolutional neural network for classification problems with 4 classes and R annotators.",
"Figure: Bottleneck structure for a CNN for classification with 4 classes and R annotators.The idea is then that when using the labels of a given annotator to propagate errors through the whole neural network, the crowd layer adjusts the gradients coming from the labels of that annotator according to his/her reliability by scaling them and adjusting their bias.",
"In doing so, the bottleneck layer of the network now receives adjusted gradients from the different annotators' labels, which it aggregates and backpropagates further through the rest of the network.",
"As it turns out, through this crowd layer, the network is able to account for unreliable annotators and even correct systematic biases in their labeling.",
"Moreover, all of that can be done naturally within the backpropagation framework.",
"Formally, let $\\sigma $ be the output of a deep neural network with an arbitrary structure.",
"Without loss of generality, we shall assume the vector $\\sigma $ to correspond to the output of a softmax layer, such that $\\sigma _c$ corresponds to the probability of the input instance belonging to class $c$ .",
"The activation of the crowd layer for each annotator $r$ is then defined as $\\textbf {a}^r = f_r(\\sigma )$ , where $f_r$ is an annotator-specific function, and the output of the crowd layer simply as the softmax of the activations $o_c^r = e^{a_c^r} / \\sum _{l=1}^C e^{a_l^r}$ .",
"The question is then how to define the function mapping $f_r(\\sigma )$ .",
"In the experiments section, we study different alternatives.",
"For classification problems a reasonable assumption is to consider a matrix transformation, such that $f_r(\\sigma ) = \\textbf {W}^r \\sigma $ , where $\\textbf {W}^r$ is an annotator-specific matrix.",
"Given a cost function $E(\\textbf {o}^r, y^r)$ between the expected output of the $r^{th}$ annotator and its actual label $y^r$ , we can compute the gradients $\\partial E / \\partial \\textbf {a}^r$ at the activation $\\textbf {a}^r$ for each annotator and backpropagate them to the bottleneck layer, leading to $\\frac{\\partial E}{\\partial \\sigma } = \\sum _{r=1}^R \\textbf {W}^r \\frac{\\partial E}{\\partial \\textbf {a}^r}.",
"\\nonumber $ The gradient vector at the bottleneck layer then naturally becomes a weighted sum of gradients according to the labels of the different annotators.",
"Moreover, if the annotator is likely to mislabel class $c$ as class $l$ (annotation bias), then the matrix $\\textbf {W}^r$ can actually adjust the gradients accordingly.",
"The problem of missing labels from some of the annotators can be easily addressed by setting their gradient contributions to zero.",
"As for estimating the annotator weights $\\lbrace \\textbf {W}^r\\rbrace _{r=1}^R$ , since they parameterize the mapping from the output of the bottleneck layer $\\sigma $ to the annotators labels $\\lbrace \\textbf {o}^r\\rbrace _{r=1}^R$ , they can be estimated using standard stochastic optimization techniques such as SGD or Adam [11].",
"Once the network is trained, the crowd layer can be removed, thus exposing the output of bottleneck layer $\\sigma $ , which can readily be used to make predictions for unseen instances.",
"An obvious concern with the approach described above is identifiability.",
"Therefore, it is important to not over-parametrize $f_r(\\sigma )$ , since adding parameters beyond necessary can make the output of the bottleneck layer $\\sigma $ lose its interpretability as a shared estimated ground truth.",
"Another important aspect is parameter initialization.",
"In our experiments, we found that the best practice is to initialize the crowd layer with identities, i.e.",
"zeros for additive parameters, ones for scalar parameters, identity matrix for multiplicative matrices, etc.",
"An alternative solution is to use regularization to force the parameters of the crowd layer to be close to identities.",
"However, in some cases this might be an undesirable property.",
"For example, if we consider a very biased annotator, then we do not wish to force the matrix $\\textbf {W}^r$ to be close to the identity matrix.",
"Based on our experiments, the initialization alternative provides the best results.",
"Lastly, it should be noted that, as with EM-based approaches, there is an implicit assumption that random or adversarial annotators do not constitute a vast majority (which generally holds in practice), in which case the crowd layer would not perform better than a random predictor.",
"A particularly important aspect to note, is that the framework described above is quite general.",
"For example, it can be straightforwardly applied to sequence labeling problems without further changes, or be adapted to regression problems by considering univariate scalar and bias parameters per annotator in the crowd layer."
],
[
"Experiments",
"The proposed crowd layer (CL) was implemented as a new type of layer in Keras [3], so that using it in practice requires only a single line of code.",
"Source code, datasets and demos for all experiments are provided at: http://www.fprodrigues.com/."
],
[
"Image classification",
"We begin by evaluating the proposed crowd layer in a more controlled setting, by using simulated annotators with different levels of expertise on a large image classification dataset consisting of 25000 images of dogs and cats from [10], where the goal is to distinguish between the two species.",
"Let the dog and cat classes be represented by 1 and 0, respectively.",
"Since this is a binary classification task, we can easily simulate annotators with different levels of expertise by assigning them individual sensitivities $\\alpha ^r$ and specificities $\\beta ^r$ , and sampling their answers from a Bernoulli distribution with parameter $\\alpha ^r$ if the true label is 1, and from a Bernoulli distribution with parameter $\\beta ^r$ otherwise.",
"Using this procedure, we simulated a challenging scenario with 5 annotators with the following values of $\\alpha ^r = [0.6, 0.9, 0.5, 0.9, 0.9]$ and $\\beta ^r = [0.7, 0.8, 0.5, 0.2, 0.9]$ .",
"For this particular problem we used a fairly standard CNN architecture with 4 convolutional layers with 3x3 patches, 2x2 max pooling and ReLU activations.",
"The output of the convolutional layers is then fed to a fully-connected (FC) layer with 128 ReLU units and finally goes to an output layer with a softmax activation.",
"We use batch normalization [8] and apply 50% dropout between the FC and output layers.",
"The proposed approach further adds a crowd layer on top of the softmax output layer during training.",
"The base architecture was selected from a set of possible configurations using the true labels by optimizing the accuracy on a validation set (consisting of 20% of the train set) through random search.",
"It is important to note that it is supposed to be representative of a set of typical approaches for image classification rather than being the single best possible architecture in the literature for this particular dataset.",
"Furthermore, our main interest in this paper is the contribution of the crowd layer to the training of the neural network.",
"The proposed CNN with a crowd layer (referred to as “DL-CL\") is compared with: the multi-annotator approach from [16] based on supervised latent Dirichlet allocation - “MA-sLDA\"; a CNN trained on the result of (hard) majority voting - “DL-MV\"; a CNN trained on the output of the label aggregation approach proposed by Dawid and Skene DawidSkeene1979 - “DL-DS\"; a CNN using the EM approach described earlier - “DL-EM\"; a CNN using the “Doctor Net\" approach from [7] - “DL-DN\", which consists on training a CNN to predict the labels of the multiple annotators and then combining their predictions using majority voting; and, lastly, a CNN using the “Weighted Doctor Net\" approach from [7] - “DL-WDN\", which is the best performing variant according to the original paper.",
"This approach is similar to “DL-DN\" but additionally learns how to weight the predictions of the different annotators.",
"Kindly see [7] for further details.",
"As a reference point, we also compare with a CNN trained on true labels - “DL-TRUE\".",
"We consider 3 variants of the proposed crowd layer (CL) with different annotator-specific functions $f_r$ with increasing number of parameters: a vector of per-class weights $\\textbf {w}^r$ , such that $f_r(\\sigma ) = \\textbf {w}^r \\odot \\sigma $ (referred to as “VW\"); a vector of per-class biases $\\textbf {b}^r$ , such that $f_r(\\sigma ) = \\sigma + \\textbf {b}^r$ (“VB\"); and a version with a matrix of weights $\\textbf {W}^r$ , such that $f_r(\\sigma ) = \\textbf {W}^r \\sigma $ (“MW\").",
"In our experiments, we found that for approaches with more parameters than MW, such as $f_r(\\sigma ) = \\textbf {W}^r \\sigma + \\textbf {b}^r$ , identifiability issues start to occur.",
"Table: Accuracy results for classification datasets: Dogs vs. Cats and LabelMe.Figure: Comparison between the true sensitivities and specificities of the annotators and the diagonal elements of their weight matrices 𝐖 r \\textbf {W}^r for the Dogs vs. Cats dataset.Figure: Comparison between the learned weight matrices 𝐖 r \\textbf {W}^r and the corresponding true confusion matrices.Table: Results for MovieReviews (MTurk) dataset.Of the 25000 images in the Dogs vs Cats dataset, 50% were used for training and the remaining for testing the different approaches.",
"In order to account for the effect of the random initialization that is used for most of the parameters in the network, we performed 30 executions of all approaches and report their average accuracies in Table REF .",
"We can immediately verify that both the EM-based and the crowd layer (CL) approaches significantly outperform the majority voting (DL-MV) and Dawid & Skene (DL-DS) baselines, thus demonstrating the gain of learning from the answers of multiple annotators directly rather than relying on aggregation schemes prior to training.",
"As for the DL-DN and DL-WDN approaches from [7], we can observe that, although they also outperform the DL-MV and DL-DS baselines, their accuracy is inferior to that of the proposed DL-CL, which can be explained by the fact that DL-DN and DL-WDN are unable to correct the annotators' biases (e.g.",
"confusing class 2 with class 4).",
"Furthermore, it important to recall that due to two-stage procedure of DL-WDN, its computational time can be significantly higher than DL-CL.",
"Regarding the different variants of the proposed crowd layer, we can verify that the MW approach is the one that gives the best average accuracy.",
"In order to better understand what the MW approach is doing, we inspected the weight matrices $\\textbf {W}^r$ of each annotator $r$ .",
"Figure REF shows the relationship between the diagonal elements of $\\textbf {W}^r$ and the true sensitivities and specificities of the corresponding annotators, highlighting the strong linear correlation between the two.",
"This evidences that the proposed crowd layer is able to internally represent the reliabilities of the annotators.",
"Having verified that the crowd layer was performing well for simulated annotators, we then moved on to evaluating it in real data from Amazon Mechanical Turk (AMT).",
"For this purpose, we used the image classification dataset from [16] adapted from part of the LabelMe data [19], whose goal is to classify images according to 8 classes: “highway\", “inside city\", “tall building\", “street\", “forest\", “coast\", “mountain\" or “open country\".",
"It consists of a total of 2688 images, where 1000 of them were used to obtain labels from multiple annotators from Amazon Mechanical Turk.",
"Each image was labeled by an average of 2.547 workers, with a mean accuracy of 69.2%.",
"The remaining 1688 images were using for evaluating the different approaches.",
"Since the training set is rather small, we use the pre-trained CNN layers of the VGG-16 deep neural network [23] and apply only one FC layer (with 128 units and ReLU activations) and one output layer on top, using 50% dropout.",
"The last column of Table REF shows the obtained results.",
"We can once more verify that DL-EM, DL-WDN and DL-CL approaches outperform the majority voting and Dawid & Skene baselines, and also the probabilistic approaches proposed in [16] based on supervised latent Dirichlet allocation (sLDA), being the proposed crowd layer (DL-CL) the approach that again gives the best results.",
"However, unlike for the Dogs vs. Cats dataset, the differences between the different function mappings $f_r$ for the crowd layer (CL) become more evident.",
"This can be justified by the ability of the MW version to be able to model the biases of the annotators.",
"Indeed, if we compare the learned weight matrices $\\textbf {W}^r$ with the respective true confusion matrices of the annotators, we can notice how they resemble each other.",
"Figure REF shows this comparison for 6 annotators, where the color intensity of the cells increases with the relative magnitude of the value, thus demonstrating that the crowd layer is able to learn the labeling patterns of the annotators."
],
[
"Text regression",
"As previously mentioned, one of the key advantages of the proposed crowd layer is its straightforward extension to other types of target variables.",
"In this section, we consider a regression problem based on the dataset also introduced in [16].",
"This dataset consists of 5006 movie reviews, where the goal is to predict the rating given to the movie (on a scale of 1 to 10) based on the text of the review.",
"Using AMT, the authors collected an average of 4.96 answers from a pool of 137 workers for 1500 movie reviews.",
"The remaining 3506 reviews were used for testing.",
"Letting the (continuous) output of the bottleneck layer be denoted $\\mu $ , we considered 3 variants of the proposed crowd layer with different annotator-specific functions $f_r$ : a per-annotator scale parameter $s^r$ , such that $f_r(\\mu ) = s^r \\mu $ (referred to as “S\"); a per-annotator bias parameter $b^r$ , such that $f_r(\\mu ) = \\mu + b^r$ (“B\"); and a version with both: $f_r(\\mu ) = s^r \\mu + b^r$ (“S+B\").",
"The base neural network architecture used for this problem consists of a 3x3 convolutional layer with 128 features and 5x5 max pooling, a 5x5 convolutional layer with 128 features and 5x5 max pooling, and a FC layer with 32 hidden units.",
"All layers, except for the output one, use ReLU activations.",
"The proposed DL-CL is compared with: a neural network trained on the mean answer of the annotators (DL-MEAN) and the approach from [16] based on supervised LDA.",
"In order to make the baselines even more competitive, we further propose a new variant of the EM algorithm described earlier that follows the same approach as the extension proposed in [15] for regression problems.",
"This approach assumes the following model for the annotators answers given the ground truth: $p(y_n^r|z_n) = \\mathcal {N}(y_n^r|z_n, 1/\\lambda ^r)$ .",
"Although the formulation in [15] relies on the probabilistic interpretation of the linear regression model to develop an EM algorithm for learning, we can nevertheless adapt the resultant EM algorithm by replacing the linear regression model with a deep neural network.",
"The final iterative procedure then alternates between computing the adjusted ground truth (E-step) and re-estimating the neural network and the annotators' parameters (M-step).",
"Finally, although Guan et al.",
"guan2017said do not discuss extensions to regression, we also developed variants of DL-DN and DL-WDN for continuous output variables.",
"For the DL-WDN approach, we considered different weighting functions for combining the answers of the multiple annotators, namely: a single weight per annotator, a single bias, or both.",
"We experimented with the different alternatives and found that using a per-annotator bias for combining the answers of the multiple annotators gives the best results.",
"Figure: Relationship between the learned b r b^r parameters and the true biases of the annotators.Table: Results for CoNLL-2003 NER (MTurk) dataset.Table REF shows the obtained results for 30 runs of the different approaches, where we verify that the proposed crowd layer, particularly the “B\" variant, significantly outperforms all the other methods.",
"In order to better understand what the crowd layer in the “B\" variant is doing, we plotted learned $b^r$ values in comparison with the true biases of the annotators, computed as the average difference between their answers and the ground truth.",
"Figure REF shows this comparison, in which we can verify that the learned values of $b^r$ are highly correlated with the true biases of the annotators, thus showing that crowd layer is able to account for annotator bias when learning from the noisy labels of multiple annotators."
],
[
"Named entity recognition",
"Lastly, we evaluated the proposed crowd layer on a named entity recognition (NER) task.",
"For this purpose, we used the AMT dataset introduced in [17] which is based on the 2003 CoNLL shared NER task [20], where the goal is to identify the named entities in the sentence and classify them as persons, locations, organizations or miscellaneous.",
"The dataset consists of 5985 labeled sentences using a pool of 47 workers.",
"The remaining 3250 sentences of the original dataset were used for testing.",
"The neural network architecture used for this problem consists of a layer of 300-dimensional word embeddings initialized with the pre-trained weights of GloVe [14], followed by a 5x5 convolutional layer with 512 features, whose output is then fed to a GRU cell with a 50-dimensional hidden state.",
"The individual hidden states of the GRU are then passed to a FC layer with a softmax activation.",
"The crowd layer uses the same annotator function mappings $f_r$ used for image classification.",
"The proposed crowd layer is compared with same baselines considered for the classification problems.",
"As previously explained, the EM approach is hard to generalize to sequence labelling problems due to marginalization over the latent ground truth sequences in Eq.",
"(REF ).",
"In order to make this marginalization tractable, we assume a fully factorized distribution of the posterior approximation $q(\\textbf {z}_n)$ , such that $q(\\textbf {z}_n) = \\prod _{t=1}^T q(z_{nt})$ , where $T$ denotes the length of the sequence.Please note that, while this makes EM tractable, the computational complexity of the E-step is now increased to $\\mathcal {O}(N T R)$ .",
"Although the focus of this paper is on deep learning approaches, for the sake of completeness, we also compare with the results of the multi-annotator approach from [17] based on conditional random fields (CRF-MA).",
"Table REF shows the obtained average results, which clearly demonstrate that the proposed approach significantly outperforms all the other methods, and provides similar results to those of CRF-MA, while reducing the training time by at least one order of magnitude when compared to the latter (minutes instead of several hours on a Core i7 with 32GB of RAM and a NVIDIA GTX 1070)."
],
[
"Conclusion",
"This paper proposed the crowd layer - a novel neural network layer that enables us to train deep neural networks end-to-end, directly from the labels of multiple annotators and crowds, using backpropagation.",
"Despite its simplicity, the crowd layer is able to capture the reliabilities and biases of the different annotators and adjust the error gradients that are backpropagated during training accordingly.",
"As our empirical evaluation shows, the proposed approach outperforms other approaches that rely on the aggregation of the annotators' answers prior to training, as well as other methods from the state-of-the-art which often rely on more complex, harder to setup and more computationally demanding EM-based approaches.",
"Furthermore, unlike the latter, the crowd layer is trivial to generalize beyond classification problems, which we empirically demonstrate using real data from Amazon Mechanical Turk for text regression and named entity recognition tasks."
],
[
"Acknowledgments",
"The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no.",
"609405 (COFUNDPostdocDTU), and from the European Union?s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Individual Fellowship H2020-MSCA-IF-2016, ID number 745673."
]
] | 1709.01779 |
[
[
"How Generic are the Robust Theoretical Aspects of Jamming in Hard Sphere\n Models?"
],
[
"Abstract In very recent work the mean field theory of the jamming transition in infinite dimensional hard spheres models was presented.",
"Surprisingly, this theory predicts quantitatively numerically determined characteristics of jamming in two and three dimensions.",
"This is a rare and unusual finding.",
"Here we argue that this agreement in non-generic: only for hard sphere models it happens that sufficiently close to jamming the effective interactions are in agreement with mean-field theory, justifying the truncation of many body interactions (which is the exact protocol in infinite dimensions).",
"Any softening of the bare hard sphere interactions results in effective interactions that are not mean-field all the way to jamming, making the discussed phenomenon non generic."
],
[
"Introduction",
"In this document we offer Supplementary Information to the main text.",
"In Sect.",
"we describe in detail the creation of jammed configurations of hard disks and their inflation to a wanted density away from jamming.",
"The next Sect.",
"provides similar details for the jamming of the soft disks.",
"In Sect.",
"we describe the manual procedure employed to guarantee that the mean positions of the particles do not change during our runs.",
"Sect.",
"explains the cleaning of the data from the consequences of infrequent collisions.",
"Sect.",
"examines whether computing the inter-particles separation using a scalar definition may change the main conclusions (it does not).",
"Finally in Sect.",
"we show that selecting particles with very close-by cage fluctuations does not lead to an elimination of the importance of contributions to the effective forces that cannot be approximated by the mean-field form."
],
[
" Hard Disks simulations ",
"To create jammed configurations of hard disks we follow the following steps: Create systems of harmonic disks at a packing fraction $\\phi = 0.86$ .",
"Implement the FIRE minimization algorithm [21] coupled to a Berendsen barostat to bring the pressure to $10^{-6} - 10^{-7}$ , depending on the system size.",
"Use 128-bit numerics from here: impose increments of expansive strain that are proportional to the current pressure, and follow each of these increments by a minimization using the FIRE algorithm (without the barostat, i.e.",
"at constant volume).",
"Repeat until the pressure approaches $10^{-10}$ or below.",
"Find the maximal overlap $-h_{ij}$ between any two particles, and expand L precisely to eliminate this maximal overlap.",
"Having determined the system volume as close to the jammed state as possible, we expand the system size from the jammed state by the desired $\\epsilon $ , cf.",
"Eq.",
"REF in the main text.",
"(Note that the value of $L_{\\rm in}$ fluctuated from realization to realization in our finite size samples).",
"Then the simulation is first equilibrated for $2\\times 10^6$ collisions and then run for $n_c=10^8$ collisions more.",
"As described in section , for each simulation only a section of $n_c/10$ collisions is used for computing the average positions.",
"This is in order to avoid transitions between meta-basins.",
"For each value of $\\epsilon $ the deviation of the force law from the 2-body putative interaction was averaged over 20-32 configurations, each taken from a different simulation starting from a different initial condition."
],
[
"Harmonic Disks simulations",
"We use velocity-Verlet algorithm with time steps $\\Delta t=0.001,0.0002$ for $V_0=1000,500000$ respectively to integrate the equation of motion.",
"The Nosé-Hoover chain thermostat was used to maintain the desired temperature."
],
[
"Setup of Jammed Configurations:",
"To create the jammed configurations, we follow the protocol as described in Refs.",
"[3], [18].",
"Starting with a random configuration in a square box at a temperature $T=0.01$ and low packing fraction of $\\phi =0.65$ the system is allowed to equilibrate.",
"Subsequently the system is quenched to a low temperature $T=10^{-12}$ , at a rate of $\\dot{T} = 10^{-4}$ .",
"Finally the energy is minimized (using the conjugate gradient technique).",
"After reaching the local minimum at initial low packing fraction $\\phi _i$ , we apply the “packing finder” algorithm [3], [18] to obtain the nearest static packing with infinitesimal particle overlaps.",
"The system is compressed or decompressed, followed by conjugate-gradient energy minimization at each step.",
"Compression is chosen when the total energy is zero after minimization while decompression is performed when the total energy is nonzero even after energy minimization, due to overlapping particles.",
"This procedure is terminated when the total potential energy per particle satisfies $U/N < 10^{-16}$ at which point we consider the configuration as jammed.",
"Note that the two methods described for hard and soft disks appear different but for all practical purposes are in fact equivalent and could be interchanged."
],
[
"Procedure",
"As described in section , for each simulation only a section of $\\Delta \\tau =\\tau _2/10$ is used for computing the average positions.",
"The aim is to avoid transitions between meta-basins.",
"For each value of $\\epsilon $ data was taken from 77-92 configurations each taken from a different simulation starting from a different initial condition."
],
[
" Testing changes in meta-basins",
"In order to get reliable average positions, we must guarantee that there are no transitions to different meta-basins during measurements.",
"This is achieved by considering three different time-correlation functions: (i) the self-intermediate scattering function , (ii) the mean square displacement, and (iii) the maximal distance traveled by any single particle.",
"The time-correlations are measured every 1000 collisions in the hard disks simulations and at each time-step in the harmonic disks simulation.",
"For hard disks, the values of these correlations functions mostly fluctuate around a constant value (apart from some initial decay/growth) with infrequent sharp drops/jumps (see Fig.",
"REF ).",
"Such a sharp drop/jump in one of these correlation functions indicates a transition between meta-basins.",
"We divide the simulation into 10 temporal sections (with equal number of collisions/ time-steps), and for each simulation analyze a single section in which such transitions were not observed.",
"All the average positions are computed within such transition-free sections.",
"This is a stricter criterion than the one used in Ref.",
"[13].",
"Figure: Self-intermediate scattering function measured for a simulation of hard disks at ϵ=10 -3 \\epsilon =10^{-3}.",
"Two transitions between meta-basins are clearly evident as sharp drops.",
"The simulation is divided into 10 temporal sections (delimited by red lines and numbered in the figure) in order to choose a section where no such transitions occur.Such sharp drops are observed mostly for the larger expansions $\\epsilon \\ge 10^{-4}$ .",
"For smaller expansions the simulation time is too short for transitions to occur.",
"For the soft harmonic disks, the values of the correlations functions can also change smoothly and we choose simulation sections where these values fluctuate around constant values without observable decay or growth."
],
[
" Cleaning the data from infrequent collisions",
"Hard Disks: Configurations of hard disks involve “rattlers\" that collide only infrequently compared to typical disks.",
"This intoduces errors in the effective force measurements.",
"To clean the data from such outliers we identify the range of $h_{ij}$ that can be trusted.",
"To this aim we plot a histogram of $log_{10}(h_{ij})$ and bin it into 50 bins, cf.",
"Fig.",
"REF .",
"The value of $h_{ij}$ in the bin with the highest weight is denoted as $h_{ij}^{freq}$ .",
"We then include only effective forces ${\\mathbf {f}}_{ij}$ for which $h_{ij}\\le 3 \\times h_{ij}^{freq}$ .",
"Figure: Histogram of log 10 (h ij )log_{10}(h_{ij}) for a hard disks configuration at expansion ϵ=10 -3 \\epsilon =10^{-3}.",
"Dashed cyan line indicates the most frequent bin h ij freq h_{ij}^{freq}.",
"The analysis employs effective forces 𝐟 ij {\\mathbf {f}}_{ij} for which h ij ≤3×h ij freq h_{ij} \\le 3 \\times h_{ij}^{freq} (solid cyan line).Harmonic Disks: In the case of harmonic disks the “gaps\" $h_{ij}$ can be negative, and the cleaning of the data is a bit more tricky.",
"Instead of using the gap $ h_{ij}=r_{ij}-\\sigma _i-\\sigma _j$ , we used $\\tilde{h}_{ij}=r_{ij}-r_{ij}^{min}$ where $r_{ij}^{min}$ , is the minimal $r_{ij}$ of the relevant interaction.",
"We then followed the same procedure as for the hard disks: We plotted a histogram (50 bins) of $log_{10}(\\tilde{h}_{ij})$ , found the largest bin $\\tilde{h}_{ij}^{freq}$ and considered effective forces associated with $\\tilde{h}_{ij}\\le 3 \\times \\tilde{h}_{ij}^{freq}$ .",
"Besides cleaning the data from pairs having very large values of $h_{ij}$ , one should also consider for both soft and hard disks some rare configurations that include particle pairs with extremely small and negative values of $h_{ij}$ that deviate strongly from the typical behavior, exhibiting abnormally small forces $f_{ij}$ .",
"These abnormally small $f_{ij}$ were not considered in the analysis.",
"This rare phenomenon disappears when the definition of $h_{ij}$ is changed in favor of a scalar average, and see Sect.",
"below.",
"At any rate these rare events do not change the general conclusions of the study, as is shown explicitly in Sect. .",
"To get an impression of the data before the clean-up of negative $h_{ij}$ we present in Fig.",
"REF some of the effective forces computed for the hard disk case as a function of $h_{ij}$ .",
"It is visually clear that the problematic points are rare.",
"Figure: The effective forces in the hard sphere case with ϵ=10 -3 \\epsilon =10^{-3}.",
"Thedata is shown only for small h ij h_{ij} to provide higher resolution around the rare events with negative h ij h_{ij}.",
"The few negative values of h ij h_{ij} are real, stemming from dynamics in which the difference in average positions are indeed negative.Figure: The standard deviation from the mean field binary force law for soft spheres as a function of the distance from jamming.",
"Here V 0 =1000V_0=1000.",
"Here we used the scalar definition of the distance between paires and particles and we show the results for eachtype of interaction (AA, BB and AB) separately for extra care.Figure: Upper panel: the histogram of values of K i K_i with coarse bins.",
"Middle panel: the same histogram with finer bins.",
"Lower panel: The standard deviation around a binary force law as a function of the bin size."
],
[
"analysis with scalar averaging of distances $r_{ij}$",
"Instead of using the definition of $h_{ij}$ in which $\\bar{r}_{ij}\\equiv |\\frac{1}{\\tau }\\int _0^\\tau dt ~{\\mathbf {r}}_{ij}(t) |$ which is computed as a vector average, one could employ a scalar definition of the mean distance between particles, $\\tilde{h}_{ij} = \\tilde{r}_{ij} -R_i - R_j \\ ; \\quad \\tilde{r}_{ij} \\equiv \\frac{1}{\\tau }\\int _0^\\tau dt~ r_{ij}(t) \\ .$ For the case of soft spheres we checked carefully whether this definition may lead to a different conclusion.",
"The answer is negative.",
"As an example we show in Fig.",
"REF the computed contribution of many body interactions as a function of $\\epsilon $ .",
"The overall order of magnitude of the standard deviation reduces compared to the vector definition of the distances, but still there is no indication for approaching the binary limit when $\\epsilon \\rightarrow 0$ ."
],
[
"Ruling out mean field effective forces in soft spheres",
"In order to determine whether in a given system the force-law conforms with mean-field expectations we need to determine the cage fluctuations $K_i$ .",
"The probability distribution function (pdf) of $K_i$ was measured for soft spheres using some 77-92 configurations (depending on $\\epsilon $ ).",
"Next, we selected pairs of particles from a bin of $K_i$ value with decreasing width of the bin.",
"If Eq.",
"REF of the main text pertains, we should expect that reducing the bin size and plotting the effective forces as a function of $h_{ij}$ must result in reducing the scatter around a functional behavior.",
"In Fig.",
"REF we show that this is not the case.",
"The data shown pertains to particle pairs whose $K_i\\approx K_j$ up to the bin width, selected from the bin with highest weight.",
"In the upper panel we show the histogram of $K_i$ with large bins, and in the middle panel with finer bins.",
"Finally, in the lower panel we show that the contribution of non-binary interaction does not reduce when the bins of the histogram get finer and finer.",
"The conclusion is that the mean-field expectation Eq.",
"REF of the main text is untenable in the case of harmonic spheres."
]
] | 1709.01607 |
[
[
"On vector-valued Siegel modular forms of degree 2 and weight (j,2)"
],
[
"Abstract We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it.",
"We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants.",
"Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes."
],
[
"Introduction",
"The usual methods for determining the dimensions of spaces of Siegel modular forms do not work for low weights.",
"For Siegel modular forms of degree 2 this means that we do not have formulas for the dimensions of the spaces of Siegel modular forms of weight $(j,k)$ , that is, corresponding to ${\\rm Sym}^j \\otimes \\det {}^k$ , in case $k<3$ .",
"In this paper we propose a description of the spaces of cusp forms of weight $(j,2)$ on the level 2 principal congruence subgroup $\\Gamma _2[2]=\\ker ({\\rm Sp}(4,{\\mathbb {Z}}) \\rightarrow {\\rm Sp}(4,{\\mathbb {Z}}/2{\\mathbb {Z}}))$ of $\\Gamma _2={\\rm Sp}(4,{\\mathbb {Z}})$ and we provide some evidence for this conjectural description.",
"Let $S_{j,k}(\\Gamma _2[2])$ be the space of cusp forms of weight $(j,k)$ , that is, corresponding to the factor of automorphy ${\\rm Sym}^j(c\\tau +d) \\det (c\\tau +d)^k$ on the group $ \\Gamma _2[2]$ .",
"Recall that the group ${\\rm Sp}(4,{\\mathbb {Z}}/2{\\mathbb {Z}})$ is isomorphic to the symmetric group $\\mathfrak {S}_6$ .",
"We fix an explicit isomorphism by identifying the symplectic lattice over ${\\mathbb {Z}}/2{\\mathbb {Z}}$ with the subspace $\\lbrace (a_1,\\ldots ,a_6) \\in ({\\mathbb {Z}}/2{\\mathbb {Z}})^6: \\sum a_i=0\\rbrace $ modulo the diagonally embedded ${\\mathbb {Z}}/2{\\mathbb {Z}}$ with form $\\sum _i a_ib_i$ as in [3]; it is given explicitly on generators of $\\mathfrak {S}_6$ in [10].",
"So $\\mathfrak {S}_6$ acts on the space of cusp forms $S_{j,k}(\\Gamma _2[2])$ and this space thus decomposes into isotypical components for the symmetric group $\\mathfrak {S}_6$ .",
"The irreducible representations of $\\mathfrak {S}_6$ correspond to the partitions of 6 and we thus have for each such partition $\\varpi $ a subspace $S_{j,k}(\\Gamma _2[2])^{s[\\varpi ]}$ of $S_{j,k}(\\Gamma _2[2])$ where $\\mathfrak {S}_6$ acts as $s[\\varpi ]$ .",
"Note that the case $s[6]$ corresponds to cusp forms on ${\\rm Sp}(4,{\\mathbb {Z}})$ , while the case $s[1^6]$ corresponds to modular forms of weight $(j,k)$ on ${\\rm Sp}(4,{\\mathbb {Z}})$ with a quadratic character: $S_{j,k}(\\Gamma _2[2])^{s[1^6]} = S_{j,k}({\\rm Sp}(4,{\\mathbb {Z}}),\\epsilon )$ with $\\epsilon $ the unique quadratic character of ${\\rm Sp}(4,{\\mathbb {Z}})$ .",
"Before we formulate our conjecture we recall that the group ${\\rm SL}(2,{\\mathbb {Z}}/2{\\mathbb {Z}})\\cong \\mathfrak {S}_3$ acts on the space $S_k(\\Gamma _1[2])$ of cusp forms on the principal congruence subgroup of level 2 $\\Gamma _1[2]=\\ker ({\\rm SL}(2,{\\mathbb {Z}})\\rightarrow {\\rm SL}(2,{\\mathbb {Z}}/2{\\mathbb {Z}}))$ .",
"We can thus decompose this space in isotypical components corresponding to the irreducible representations of $\\mathfrak {S}_3$ .",
"The map $f(z) \\mapsto f(2z)$ defines an isomorphism $S_k(\\Gamma _1[2]){\\sim \\over \\longrightarrow }S_k(\\Gamma _0(4))$ with $\\Gamma _0(4)$ the usual congruence subgroup of $\\Gamma _1={\\rm SL}(2,{\\mathbb {Z}})$ .",
"If we write its isotypical decomposition as $S_k(\\Gamma _1[2])= a_k\\, s[3]+b_k \\, s[2,1]+c_k \\, s[1^3]\\, ,$ then $\\dim S_k(\\Gamma _1)=a_k$ , $\\dim S_k(\\Gamma _0(2))^{\\rm new}=b_k-a_k$ and $\\dim S_k(\\Gamma _0(4))^{\\rm new}= c_k$ with their generating series given by $\\sum a_k t^k= t^{12}/(1-t^4)(1-t^6), \\quad \\sum b_k t^k=t^8/(1-t^2)(1-t^6)\\, ,\\quad \\sum c_k t^k=t^6/(1-t^4)(1-t^6)\\, .$ The Fricke involution $w_2: \\tau \\mapsto -1/2\\tau $ defines an involution on $S_k(\\Gamma _0(2))^{\\rm new}$ and this space splits into eigenspaces $S_k^{\\pm }(\\Gamma _0(2))^{\\rm new}$ and for $k>2$ we have $\\dim S_k^{+}(\\Gamma _0(2))^{\\rm new} - \\dim S_k^{-}(\\Gamma _0(2))^{\\rm new}={\\left\\lbrace \\begin{array}{ll} -1 & k\\equiv 2 \\, \\bmod \\, 8 \\\\0 & k \\equiv 4,6 \\, \\bmod \\, 8 \\\\1 & k \\equiv 0 \\, \\bmod \\, 8\\, .",
"\\\\\\end{array}\\right.",
"}$ We recall the notion of Yoshida type lifts.",
"Yoshida lifts are explained in [27]; see also [28], [26], [5], [21].",
"These are eigen forms associated to a pair of elliptic modular eigenforms whose spinor L-function is a product of the twisted L-functions of the elliptic modular cusp forms.",
"In [3] a number of conjectures on the existence of Yoshida lifts were made and these were proved by Rösner [20].",
"These conjectures deal with Siegel modular cusp forms of weight $(j,k)$ with $k\\ge 3$ .",
"It can be extended to the case of weight $(j,2)$ .",
"We denote the subspace of $S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}$ generated by Yoshida lifts by $YS_{j,2}^{s[\\varpi ]}$ .",
"Theorem 1.1 We have $YS_{j,2}^{s[w]} = 0$ unless we are in the following cases: (1) $w=[1^6]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ eigen-newforms of level $\\Gamma _0(2)$ of different sign.",
"In this case we have $\\dim YS_{j,2}^{s[w]} \\,= \\,\\dim \\,S^+_{j+2}(\\Gamma _0(2))^{\\rm new} \\otimes S^-_{j+2}(\\Gamma _0(2))^{\\rm new}.$ (2) $w=[2,1^4]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\Gamma _0(4)$ .",
"The multiplicity $\\mu (j)$ of $s[2,1^4]$ in $YS_{j,2}^{s[w]}$ is then $\\mu (j)\\, =\\, \\dim \\, \\Lambda ^2 S_{j+2}(\\Gamma _0(4))^{\\rm new}.$ (3) $w=[2^3]$ and $YS_{j,2}^{s[w]}$ is generated by the $Y(f,g)$ with $f$ and $g$ non proportional eigen-newforms on $\\Gamma _0(2)$ with the same sign.",
"The multiplicity $\\nu (j)$ of $s[2^3]$ is $\\nu (j) \\,= \\,\\dim \\Lambda ^2 S^+_{j+2}(\\Gamma _0(2))^{\\rm new} \\oplus \\Lambda ^2 S^-_{j+2}(\\Gamma _0(2))^{\\rm new}.$ The proof of this theorem follows from results of Rösner and Weissauer, in a way very similar to Rösner's proof of the Bergström-Faber-van der Geer conjecture in weight $k\\ge 3$ [20].",
"In the second appendix Chenevier explains how to derive it.",
"We now formulate our conjecture.",
"Conjecture 1.2 The space $S_{j,2}(\\Gamma _2[2])$ is generated by Yoshida type lifts.",
"Note that this implies that $S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}=(0)$ unless $\\varpi = [1^6], [2,1^4]$ or $[2^3]$ .",
"In particular, it implies that $S_{2,j}(\\Gamma _2)=(0)$ .",
"The evidence we have for the latter is the following.",
"Theorem 1.3 We have $\\dim S_{j,2}(\\Gamma _2)=0$ for $j\\le 52$ .",
"For $j\\le 20$ the vanishing of $S_{j,2}(\\Gamma _2)$ was proved by Ibukiyama, Wakatsuki and Uchida [16], [17], and [25].",
"The evidence we have for the $s[1^6]$ -part of the conjecture is the following.",
"Theorem 1.4 The dimension of $S_{j,2}(\\Gamma _2,\\epsilon )$ is given by the coefficient of $t^j$ in the expansion of $t^{12}/(1-t^6)(1-t^8)(1-t^{12})$ for $j\\le 30$ .",
"Modular forms in $S_{j,2}(\\Gamma _2,\\epsilon )$ can be constructed explicitly using covariants as explained in [7].",
"We prove this theorem by constructing a basis of the space $S_{j,7}(\\Gamma _2)$ using covariants of binary sextics (see [7]) and then by checking which forms are divisible by the cusp form $\\chi _5 \\in S_{0,5}(\\Gamma _2,\\epsilon )$ .",
"We thus give generators for these spaces $S_{j,2}(\\Gamma _2[2])^{s[1^6]}$ for $j\\le 30$ and we can calculate Hecke eigenvalues for these.",
"For $j>30$ this becomes quite laborious.",
"For all irreducible representations we have the following vanishing result: Proposition 1.5 For any $\\varpi $ we have $\\dim S_{j,2}(\\Gamma _2[2])^{s[\\varpi ]}=0$ for $j<12$ .",
"We end with some remarks on other `small' weights.",
"The vanishing of $S_{j,1}(\\Gamma _2)$ follows from work of Skoruppa [23].",
"In an appendix to this paper besides providing a different proof of the vanishing of $S_{j,2}(\\Gamma _2)$ for $j\\le 38$ , Chenevier gives a proof of the vanishing of $S_{j,1}(\\Gamma _2[2])$ .",
"For $k=3$ one knows that $S_{j,3}(\\Gamma _2)=(0)$ for $j<36$ .",
"But $S_{36,3}$ is 1-dimensional and using covariants we can construct a form of this weight in a relatively easy manner; cf.",
"the remarks in [17] on the difficulty of constructing such a form.",
"Acknowledgement.",
"We thank Gaëtan Chenevier warmly for asking the question on the existence of modular forms of weight $(j,2)$ on the full group ${\\rm Sp}(4,{\\mathbb {Z}})$ and for agreeing to add his results in the form of an appendix.",
"He also pointed out an error in an earlier version of our conjecture and provided a proof of the extension of Rösner's result on Yoshida lifts.",
"We thank Mirko Rösner for correspondence.",
"We also thank the Max-Planck Institut für Mathematik in Bonn for excellent working conditions."
],
[
"Modular Forms of Degree Two",
"For the definitions of Siegel modular forms and elementary properties we refer to [14].",
"We denote the Siegel upper half space of degree $g$ by $\\mathfrak {H}_g$ .",
"The Siegel modular group $\\Gamma _g={\\rm Sp}(2g,{\\mathbb {Z}})$ acts on $\\mathfrak {H}_g$ by fractional linear transformations $\\tau \\mapsto (a\\tau +b)(c\\tau +d)^{-1} \\qquad \\text{\\rm for $\\left(\\begin{matrix}a & b\\cr c & d \\cr \\end{matrix} \\right) \\in {\\rm Sp}(2g,{\\mathbb {Z}})$and $\\tau \\in \\mathfrak {H}_g$.",
"}$ If $\\rho : {\\rm GL}(g,{\\mathbb {C}}) \\rightarrow {\\rm GL}(W)$ is a finite-dimensional complex representation then a holomorphic map $f: \\mathfrak {H}_g \\rightarrow W$ is called a Siegel modular form of weight $\\rho $ if $f(a\\tau +b)(c\\tau +d)^{-1})=\\rho (c\\tau +d)f(\\tau )$ for all $(a,b;c,d) \\in \\Gamma _g$ .",
"The space of modular forms of weight $\\rho $ is finite-dimensional and denoted by $M_{\\rho }(\\Gamma _g)$ .",
"If $g=2$ then an irreducible representation of ${\\rm GL}(2,{\\mathbb {C}})$ is of the form ${\\rm Sym}^j({\\rm St}) \\otimes {\\det }({\\rm St})^k$ with ${\\rm St}$ the standard representation of ${\\rm GL}(2,{\\mathbb {C}})$ for some $j \\in {\\mathbb {Z}}_{\\ge 0}$ and $k \\in {\\mathbb {Z}}$ .",
"For $\\rho ={\\rm Sym}^j({\\rm St}) \\otimes {\\det }({\\rm St})^k$ we denote $M_{\\rho }$ by $M_{j,k}$ and we call $(j,k)$ the weight.",
"If $j=0$ we are dealing with scalar-valued modular forms.",
"The space of Siegel modular forms of degree 2 and weight $(j,k)$ is denoted by $M_{j,k}(\\Gamma _2)$ .",
"There is the Siegel operator $\\Phi _g$ that maps Siegel modular forms of degree $g$ to Siegel modular forms of degree $g-1$ .",
"The kernel of $\\Phi _2$ in $M_{j,k}(\\Gamma _2)$ is called the space of cusp forms of weight $(j,k)$ and denoted by $S_{j,k}(\\Gamma _2)$ .",
"Note that for $k=2$ we have $M_{j,2}(\\Gamma _2)=S_{j,2}(\\Gamma _2)$ , see [16].",
"For a finite index subgroup $\\Gamma $ of ${\\rm Sp}(4,{\\mathbb {Z}})$ we have similar notions.",
"Here we deal with the groups $\\Gamma _2$ and $\\Gamma _2[2]$ .",
"The quotient group $\\Gamma _2/\\Gamma _2[2]\\cong {\\rm Sp}(4,{\\mathbb {Z}}/2{\\mathbb {Z}})$ is identified with the symmetric group $\\mathfrak {S}_6$ as in the Introduction.",
"This group acts in a natural way on the space of cusp forms $S_{j,k}(\\Gamma _2[2])$ and we can decompose this space in isotypical components $S_{j,k}(\\Gamma _2[2])^{s[\\varpi ]}$ corresponding to the irreducible representations $s[\\varpi ]$ of $\\mathfrak {S}_6$ which in turn correspond bijectively to the partitions $\\varpi $ of 6.",
"The ring $R$ of scalar-valued Siegel modular forms on $\\Gamma _2$ was determined by Igusa in the 1960s, see [18].",
"In the 1980s Tsushima gave in [24] formulas for the dimensions of the spaces of vector-valued cusp forms on a subgroup between $\\Gamma _2[2]$ and $\\Gamma _2$ .",
"Bergström extended this to $\\Gamma _2[2]$ with the action of $\\mathfrak {S}_6$ , see [2].",
"We thus know the dimension of $S_{j,k}(\\Gamma _2[2])^{s[\\varpi ]}$ for all $j$ and $k\\ge 3$ .",
"The vector-valued modular forms of degree 2 form a ring $M=\\oplus _{j,k} M_{j,k}(\\Gamma _2)$ .",
"It is also a module over the ring $R$ .",
"For level 2 similar things hold.",
"A vector-valued Siegel modular form $f$ of weight $(j,k)$ on $\\Gamma _2$ has a Fourier-Jacobi expansion $f(\\tau )= \\sum _{m\\ge 0} \\varphi _m(\\tau _1,z) \\, e^{2\\pi i m \\tau _2}\\qquad \\text{ where $\\tau = \\left( \\begin{matrix} \\tau _1 & z \\\\z & \\tau _2 \\\\ \\end{matrix} \\right)$ }$ with $\\varphi _m: \\mathfrak {H}_1 \\times {\\mathbb {C}} \\rightarrow {\\rm Sym}^j({\\mathbb {C}}^2)$ a holomorphic map that satisfies certain functional equations under the action of the so-called Jacobi group ${\\rm SL}(2,{\\mathbb {Z}}) \\ltimes {\\mathbb {Z}}^2$ .",
"This group is embedded in $\\Gamma _2$ via $\\left( \\begin{matrix} a & b \\\\ c & d \\\\ \\end{matrix}\\right) \\mapsto \\left( \\begin{matrix} a & 0 & b & 0 \\\\0 & 1 & 0 & 0 \\\\c & 0 & d & 0 \\\\0 & 0 & 0 & 1 \\\\ \\end{matrix} \\right), \\qquad (\\lambda , \\mu ) \\mapsto \\left( \\begin{matrix} 1 & 0 & 0 & \\mu \\\\\\lambda & 1 & \\mu & 0 \\\\0 & 0 & 1 & -\\lambda \\\\0 & 0 & 0 & 1 \\\\ \\end{matrix} \\right)$ with action $\\tau \\mapsto \\left( \\begin{matrix} (a\\tau _1+b)/(c\\tau _1+d) & z/(c\\tau _1+d) \\\\z/(c\\tau _1+d) & \\tau _2-cz^2/(c\\tau _1+d)\\\\ \\end{matrix} \\right)$ and $\\tau \\mapsto \\left( \\begin{matrix} \\tau _1 & z+\\lambda \\tau _1+\\mu \\\\z+\\lambda \\tau _1+\\mu & \\tau _2+ \\lambda ^2\\tau _1+2\\lambda z +\\lambda \\mu \\\\ \\end{matrix} \\right) \\, .$ The fact that $f$ is a modular form of weight $(j,k)$ implies the corresponding functional equations $\\varphi _m(\\frac{a \\tau _1+b}{c\\tau _1+d}, \\frac{z}{c\\tau _1+d})e^{-2\\pi i m \\frac{c z^2}{c\\tau _1+d}}= {\\rm Sym}^j\\left( \\begin{matrix} c\\tau _1+d & c z \\\\ 0 & 1 \\\\ \\end{matrix} \\right) (c\\tau _1+d)^k\\varphi _m(\\tau _1,z)$ and $\\varphi _m(\\tau _1,z+\\lambda \\tau _1+\\mu ) e^{2\\pi i m (\\lambda ^2 \\tau _1 +2\\lambda z +\\lambda \\mu )} ={\\rm Sym}^j \\left( \\begin{matrix} 1 & -\\lambda \\\\ 0 & 1 \\\\ \\end{matrix} \\right) \\varphi _m(\\tau _1,z) \\, ,$ where we write $\\varphi _m$ as the transpose of the row vector $(\\varphi _m^{(0)}, \\ldots , \\varphi _m^{(j)})$ .",
"Corollary 2.1 If $f \\in M_{j,k}(\\Gamma _2)$ (resp.",
"$f \\in S_{j,k}(\\Gamma _2)$ ) then the last coordinate $\\varphi _m^{(j)}$ of the coefficient $\\varphi _m$ of $e^{2\\pi i m \\tau _2}$ in the Fourier-Jacobi expansion of $f$ is a Jacobi form (resp.",
"Jacobi cusp form) of weight $k$ and index $m$ .",
"We note that Jacobi cusp forms of weight 2 and index $m$ are zero for $m< 37$ , see [11].",
"This imposes strong conditions on forms of weight $(j,2)$ on $\\Gamma _2$ .",
"Since we shall compute the action of Hecke operators later we now describe formulas for the action of Hecke operators on forms on $\\Gamma _2$ .",
"For forms without character we refer to [9], so we deal with the case of forms with a character.",
"For $f \\in M_{j,k}(\\Gamma _2,\\epsilon )$ we write its Fourier expansion as $f(\\tau )= \\sum _{n\\ge 0} a(n) \\, e^{\\pi i {\\rm Tr}(n\\tau )} \\, ,$ where $n$ runs over the positive semi-definite half-integral symmetric matrices.",
"We will write $[n_1,n_2,n_3]$ for $\\left( {\\begin{matrix}n_1 & n_2/2 \\\\ n_2/2 & n_3 \\end{matrix}} \\right)$ .",
"For an odd prime $p$ we denote by $T_p$ the Hecke operator for $\\Gamma _2$ at $p$ .",
"Then we write the transform of $f$ under $T_p$ as $T_p(f)(\\tau )= \\sum _{n\\ge 0} a_p(n) \\, e^{\\pi i {\\rm Tr} (n\\tau )}\\, .$ Here for $p \\lnot \\equiv 1 \\bmod \\, 3$ the coefficient $a_p([1,1,1])$ is given by $a([p,p,p])$ , and for $p=3$ by $a([3,3,3])-3^{k-2}{\\mathrm {Sym}}^j\\left({\\begin{matrix}3 & -1 \\\\0 & 1 \\\\\\end{matrix}}\\right)a([1,3,3]) \\, ,$ while for $p \\equiv 1 \\bmod \\, 3$ by $a([p,p,p])+p^{k-2}\\sum _{i=1}^{2}(-1)^{m_i}{\\mathrm {Sym}}^j\\left({\\begin{matrix}p & -m_i \\\\0 & 1 \\\\\\end{matrix}}\\right)a([\\frac{1+m_i+m_i^2}{p},1+2m_i, p]) \\, ,$ where in the latter case $m_1$ and $m_2$ are the two roots of the polynomial $1+X+X^2$ over $\\mathbb {F}_p$ , which we view here as the set $\\left\\lbrace 0,\\ldots ,p-1\\right\\rbrace $ .",
"Similarly, the coefficient $a_{p^2}([1,1,1])$ of the transform of $f$ under the Hecke operator $T_{p^2}$ is given for $p \\lnot \\equiv 1 \\bmod \\, 3$ by $a([p^2,p^2,p^2])$ , and for $p=3$ by $a([9,9,9])-3^{k-2}{\\mathrm {Sym}}^j\\left({\\begin{matrix}3 & -1 \\\\0 & 1 \\\\\\end{matrix}}\\right)a([3,9,9]) \\, .$ As an example, consider the modular form $\\chi _5 \\in S_{0,5}(\\Gamma _2,\\epsilon )\\,,$ the product of the ten even theta characteristics and the square root of Igusa's cusp form $\\chi _{10}$ , that will play an important role in this paper.",
"It provides a check on these formulas for the Hecke operators.",
"Indeed, one knows $\\lambda _p(\\chi _5)=p^3+a_p(f)+p^4, \\qquad \\lambda _{p^2}(\\chi _5)=\\lambda _p(\\chi _5)^2-(p^4+p^3)\\lambda _p(\\chi _5)+p^{8}\\, ,$ where $f=q-8\\, q^2+12\\, q^3+64\\, q^4-210\\, q^5 +\\cdots $ is the normalized Hecke eigenform in $S_{8}^+(\\Gamma _0(2))^{\\text{new}}$ , which illustrates that $\\chi _5$ is a Saito-Kurokawa lift.",
"One can check that the above formulas agree with this."
],
[
"Restriction to the diagonal",
"In order to put restrictions on the existence of Siegel modular forms we restrict these to the `diagonal' given by the embedding $i: \\mathfrak {H}_1\\times \\mathfrak {H}_1 \\rightarrow \\mathfrak {H}_2,\\qquad (z_1,z_2) \\mapsto \\left(\\begin{matrix}z_1 & 0 \\cr 0 & z_2\\cr \\end{matrix}\\right)\\, .$ The stabilizer of $i(\\mathfrak {H}_1\\times \\mathfrak {H}_1)$ in ${\\rm Sp}(4,{\\mathbb {R}})$ is an extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ of the image of ${\\rm SL}(2,{\\mathbb {R}})^2$ under the embedding $(\\left({\\begin{matrix}a_1 & b_1 \\\\ c_1 & d_1\\end{matrix}}\\right),\\left({\\begin{matrix}a_2 & b_2 \\\\ c_2 & d_2\\end{matrix}}\\right)) \\, \\mapsto \\,\\left({\\begin{matrix}a_1 & 0 & b_1 & 0\\\\0 & a_2 & 0 & b_2 \\\\c_1 & 0 & d_1 & 0\\\\0 & c_2 & 0 & d_2 \\\\\\end{matrix}}\\right)$ The extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ corresponds to the involution that interchanges $\\tau _1$ and $\\tau _2$ in $\\tau =\\left({\\begin{matrix}\\tau _1 & \\tau _{12}\\cr \\tau _{12} & \\tau _2 \\cr \\end{matrix}} \\right) \\in \\mathfrak {H}_2$ (and $z_1$ and $z_2$ on $\\mathfrak {H}_1^2$ ).",
"This corresponds to the element $\\iota =\\left({\\begin{matrix} a & 0 \\\\ 0 & d \\\\ \\end{matrix}}\\right)$ in $\\Gamma _2$ with $a=d= \\left({\\begin{matrix} 0 & 1 \\\\ 1 & 0 \\\\ \\end{matrix}}\\right)$ .",
"The stabilizer inside $\\Gamma _2$ (resp.",
"inside $\\Gamma _2[2]$ ) is an extension by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ of ${\\rm SL}(2,{\\mathbb {Z}})\\times {\\rm SL}(2,{\\mathbb {Z}})$ (resp.",
"of $\\Gamma _1[2]\\times \\Gamma _1[2]$ ).",
"If $F=(F_0,\\ldots ,F_{j})^t$ is a Siegel modular form of weight $(j,k)$ of level 2, then its pullback under $i$ to $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ gives rise to an element of $(f_0,\\ldots , f_j)^t$ with $f_l \\in M_{j+k-l}(\\Gamma _1[2])\\otimes M_{k+l}(\\Gamma _1[2])$ .",
"By restricting a cusp form of level 1 we get cusp forms of level 1.",
"The action of $\\iota $ is given by a map $S_{j+k-i}(\\Gamma _1[2])\\otimes S_{k+i}(\\Gamma _1[2]) \\rightarrow S_{k+i}(\\Gamma _1[2])\\otimes S_{j+k-i}(\\Gamma _1[2]), \\qquad a\\otimes b \\mapsto (-1)^k b\\otimes a$ for $\\Gamma _1[2]$ and a similar one for $\\Gamma _1$ .",
"So for a form of level 1 without character we loose no information by looking at $\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\Gamma _1)\\otimes S_{k+i}(\\Gamma _1)\\bigoplus {\\left\\lbrace \\begin{array}{ll}{\\wedge }^2 S_{j/2+k}(\\Gamma _1) & \\text{for $k$ odd} \\\\{\\rm Sym}^2 S_{j/2+k}(\\Gamma _1) & \\text{for $k$ even.}",
"\\\\\\end{array}\\right.",
"}$ By multiplying with $\\chi _5$ we get an injective map $S_{j,2}(\\Gamma _2,\\epsilon ) \\rightarrow S_{j,7}(\\Gamma _2)$ .",
"The generating series for the dimensions is $\\sum _{j=2}^{\\infty } \\dim S_{j,7}(\\Gamma _2) \\, t^j ={t^{12} \\over (1-t)(1-t^3)(1-t^4)(1-t^6)} \\, .$ We observe that our conjecture on $S_{j,2}(\\Gamma _2,\\epsilon )$ implies that $\\dim S_{j,7}(\\Gamma _2) -\\dim S_{j,2}(\\Gamma _2,\\epsilon ) =\\sum _{i=0}^{j/2-1} \\dim S_{j+7-i}(\\Gamma _1) \\dim S_{7+i}(\\Gamma _1)+ \\dim \\wedge ^2 S_{j/2+7}(\\Gamma _1) \\, ,$ or equivalently, that the restriction $\\rho $ to the diagonal fits in an exact sequence $0 \\rightarrow S_{j,2}(\\Gamma _2,\\epsilon ) {\\cdot \\chi _5 \\over \\longrightarrow }S_{j,7}(\\Gamma _2){\\rho \\over \\longrightarrow } \\oplus _{i=0}^{j/2-1}S_{j+7-i}(\\Gamma _1)\\otimes S_{7+i}(\\Gamma _1)\\oplus \\wedge ^2 S_{j/2+7}(\\Gamma _1)\\rightarrow 0 \\, .$ If $F\\in S_{j,k}(\\Gamma _2,\\epsilon )$ we find by using $\\iota $ that we can restrict to $\\bigoplus _{i=0}^{j/2-1} S_{j+k-i}(\\Gamma _1[2])^{s[1^3]}\\otimes S_{k+i}(\\Gamma _1[2])^{s[1^3]}\\bigoplus {\\rm Sym}^2(S_{j/2+k}(\\Gamma _1[2])^{s[1^3]})\\, .$ Indeed, the group $\\mathfrak {S}_3={\\rm SL}(2,{\\mathbb {Z}}/2{\\mathbb {Z}})$ acts on $S_k(\\Gamma _1[2])$ and for a form on $\\Gamma _2$ with a character the components $f_i, f_i^{\\prime }$ of the restriction to $i(\\mathfrak {H}_1\\times \\mathfrak {H}_1)$ are modular forms on $\\Gamma _1[2]$ with a character, i.e.",
"they lie in the $s[1^3]$ -isotypical part of $S_k(\\Gamma _1[2])$ .",
"The module $\\oplus _k S_k(\\Gamma _1[2])^{s[1^3]}$ is a module over the ring ${\\mathbb {C}}[e_4,e_6]$ of modular forms on $\\Gamma _1$ and is generated by the cusp form $\\delta =\\eta ^{12}$ , a square root of $\\Delta \\in S_{12}(\\Gamma _1)$ .",
"The generating series for the dimensions is now $\\sum _{j=2}^{\\infty } \\dim S_{j,7}(\\Gamma _2,\\epsilon )\\, t^j ={t^6 \\over (1-t)(1-t^3)(1-t^4)(1-t^6)}\\, .$ Conjecturally we now find an exact sequence $\\begin{aligned}0 \\rightarrow S_{j,7}(\\Gamma _2,\\epsilon ) {\\rho \\over \\longrightarrow }\\oplus _{i=0}^{j/2-1} S_{j+7-i}(\\Gamma _0(4))^{n}\\otimes S_{7+i}(\\Gamma _0(4))^{n} \\,\\oplus {\\rm Sym}^2 (S_{j/2+7}(\\Gamma _0(4))^{n}) &\\\\\\rightarrow K \\rightarrow 0\\, , &\\\\\\end{aligned}$ where $S_k(\\Gamma _0(4))^{n}=S_k(\\Gamma _0(4))^{\\rm new}$ and the dimension of the cokernel $K$ is now predicted by minus the extrapolation to $k=2$ of the algorithms used in [3] to calculate the dimension of $S_{j,k}(\\Gamma _2)$ and which give negative numbers here."
],
[
"Constructing Modular Forms Using Covariants",
"In the paper [7] we explained how to use invariant theory to construct Siegel modular forms.",
"In this paper we shall make extensive use of the procedure.",
"Let $V$ be the standard representation space of ${\\rm GL}(2,{\\mathbb {C}})$ with basis $x_1,x_2$ .",
"We consider the space ${\\rm Sym}^6(V)$ of binary sextics, where we write an element as $f= \\sum _{i=0}^6 a_i \\binom{6}{i} x_1^{6-i}x_2^i \\, .$ Sometimes we call this expression the universal binary sextic.",
"For a description of invariants and covariants for the action of ${\\rm GL}(2,{\\mathbb {C}})$ we refer to [7].",
"An invariant can be viewed as a polynomial in the coefficients $a_i$ that is invariant under the action of ${\\rm SL}(2,{\\mathbb {C}})$ , while a covariant of degree $(a,b)$ can be viewed as a form of degree $a$ in the $a_i$ and degree $b$ in $x_1,x_2$ .",
"If $A[\\lambda _1,\\lambda _2]$ is an irreducible representation of highest weight $(\\lambda _1 \\ge \\lambda _2)$ of ${\\rm GL}(2,{\\mathbb {C}})$ embedded equivariantly in ${\\rm Sym}^d({\\rm Sym}^6(V))$ this defines a covariant of degree $(d,\\lambda _1-\\lambda _2)$ and it is unique up to a multiplicative non-zero constant.",
"We denote the ring of covariants by ${\\mathcal {C}}$ .",
"Clebsch and others constructed in the 19th century generators for this ring.",
"There are 26 generators, 5 invariants and 21 covariants, satisfying many relations.",
"They can be found in the book of Grace and Young [15].",
"For the convenience of the reader we reproduce these here.",
"In the following table $C_{a,b}$ denotes a generator of degree $(a,b)$ .",
"height2pt$a \\backslash b$ 0 2 4 6 8 10 12 1 $C_{1,6}$ 2 $C_{2,0}$ $C_{2,4}$ $C_{2,8}$ 3 $C_{3,2}$ $C_{3,6}$ $C_{3,8}$ $C_{3,12}$ 4 $C_{4,0}$ $C_{4,4}$ $C_{4,6}$ $C_{4,10}$ 5 $C_{5,2}$ $C_{5,4}$ $C_{5,8}$ 6 $C_{6,0}$ $C_{6,6}^{(1)}$ $C_{6,6}^{(2)}$ 7 $C_{7,2}$ $C_{7,4}$ 8 $C_{8,2}$ 9 $C_{9,4}$ 10 $C_{10,0}$ $C_{10,2}$ 12 $C_{12,2}$ 15 $C_{15,0}$ A theorem of Gordan says that all these covariants can be constructed explicitly by using so-called transvectants from the universal binary sextic.",
"If ${\\rm Sym}^m(V)$ denotes the space of binary quantics of degree $m$ then we define the $k$ th transvectant as follows.",
"It is a map ${\\rm Sym}^m(V) \\times {\\rm Sym}^n(V) \\rightarrow {\\rm Sym}^{m+n-2k}(V)$ that sends a pair $(f,g)$ to $(f,g)_k=\\frac{(m-k)!(n-k)!}{m!n!",
"}\\sum _{j=0}^k (-1)^j\\binom{k}{j}\\frac{\\partial ^k f}{\\partial x_1^{k-j}\\partial x_2^j}\\frac{\\partial ^k g}{\\partial x_1^{j}\\partial x_2^{k-j}}\\, .$ When $k=1$ , we omit the index: $(f,g)=(f,g)_1$ .",
"The next table gives the construction of the covariants in the preceding table.",
"height2pt 1 $C_{1,6}=f$ 2 $C_{2,0}=(f,f)_6$ $C_{2,4}=(f,f)_4$ $C_{2,8}=(f,f)_2$ 3 $C_{3,2}=(C_{1,6},C_{2,4})_4$ $C_{3,6}=(f,C_{2,4})_2$ $C_{3,8}=(f,C_{2,4})$ $C_{3,12}=(f,C_{2,8})$ 4 $C_{4,0}=(C_{2,4},C_{2,4})_4$ $C_{4,4}=(f,C_{3,2})_2$ $C_{4,6}=(f,C_{3,2})$ $C_{4,10}=(C_{2,8},C_{2,4})$ 5 $C_{5,2}=(C_{2,4},C_{3,2})_2$ $C_{5,4}=(C_{2,4},C_{3,2})$ $C_{5,8}=(C_{2,8},C_{3,2})$ 6 $C_{6,0}=(C_{3,2},C_{3,2})_2$ $C_{6,6}^{(1)}=(C_{3,6},C_{3,2})$ $C_{6,6}^{(2)}=(C_{3,8},C_{3,2})_2$ 7 $C_{7,2}=(f,C_{3,2}^2)_4$ $C_{7,4}=(f,C_{3,2}^2)_3$ 8 $C_{8,2}=(C_{2,4},C_{3,2}^2)_3$ 9 $C_{9,4}=(C_{3,8},C_{3,2}^2)_4$ 10 $C_{10,0}=(f,C_{3,2}^3)_6$ $C_{10,2}=(f,C_{3,2}^3)_5$ 12 $C_{12,2}=(C_{3,8},C_{3,2}^3)_6$ 15 $C_{15,0}=(C_{3,8},C_{3,2}^4)_8$ Let $M$ be the ring of vector-valued Siegel modular forms of degree 2.",
"It is a module over the ring $R$ of scalar-valued Siegel modular forms of degree 2.",
"In [7] we defined maps $M \\longrightarrow {\\mathcal {C}} {\\nu \\over \\longrightarrow }M_{\\chi _{10}}\\, ,$ where $M_{\\chi _{10}}$ is the localization of $M$ at $\\chi _{10}$ .",
"A modular form of weight $(j,k)$ maps to a covariant of degree $(j/2+k,j)$ and a covariant of degree $(a,b)$ is sent to a meromorphic modular form of weight $(b,a-b/2)$ .",
"Under the map $\\nu $ the universal binary sextic $f$ is mapped to $\\chi _{6,3}/\\chi _5$ of weight $(6,-2)$ .",
"Here $\\chi _{6,3}$ is a holomorphic form in $S_{6,3}(\\Gamma _2,\\epsilon )$ .",
"The beginning of its Fourier expansion is given in [7].",
"In practice instead of $\\nu $ often we use a slightly modified map $\\mu : {\\mathcal {C}} \\longrightarrow M\\oplus M_{\\epsilon }\\, ,$ where $M_{\\epsilon }= \\oplus M_{j,k}(\\Gamma _2,\\epsilon )$ , is the $R$ -module of modular forms with a character.",
"Under $\\mu $ the universal sextic $f$ is mapped to $\\chi _{6,3}$ .",
"Then a covariant maps of degree $(a,b)$ maps to a holomorphic Siegel modular form of weight $(b,6a-b/2)$ and character $\\epsilon ^a$ .",
"Remark 4.1 Since $\\chi _{6,3}$ vanishes simply at infinity the definition of $\\mu $ implies that the image under $\\mu $ of a covariant of degree $(a,b)$ vanishes at infinity with order $\\ge a$ .",
"Recall that the order of vanishing of $\\chi _5$ at infinity is 1.",
"For example, Igusa's generators $E_4,E_6,\\chi _{10}, \\chi _{12}$ and $\\chi _{35}$ of $R$ are up to a non-zero multiplicative constant obtained as $\\begin{aligned}&E_4=\\mu (75\\, C_{4,0} -8\\, C_{2,0}^2)/\\chi _{10}^2, \\quad E_6= \\mu (224\\, C_{2,0}^3 -1425\\, C_{2,0}C_{4,0} -1125 \\, C_{6,0})/\\chi _{10}^3 \\\\&\\chi _{10}^6 = \\mu (C_{\\chi _{10}}), \\quad \\chi _{12}= \\mu (C_{2,0}), \\quad \\chi _{35}= \\mu (C_{15,0})/\\chi _5^{11}, \\\\\\end{aligned}$ with $C_{\\chi _{10}}$ , up to a multiplicative constant equal to the discriminant, given by $768\\, C_{2,0}^5-7625 \\,C_{4,0}C_{2,0}^3-1875\\left(7 \\, C_{6,0}C_{2,0}^2 -10 \\, C_{4,0}^2C_{2,0}-30 \\, C_{6,0}C_{4,0}- 13860 \\, C_{10,0}\\right) \\, .$ Remark 4.2 The first scalar-valued cusp form on $\\Gamma _2$ with character is of weight 30 and can be obtained by dividing $\\mu (C_{15,0})$ by $\\chi _5^{12}$ .",
"Note that we have $M_{j,k}(\\Gamma _2,\\epsilon )=S_{j,k}(\\Gamma _2,\\epsilon ).$ (see [17])."
],
[
"Cusp forms of weight $(j,2)$ on {{formula:3c24a2d5-8cca-456e-8c8a-10623d91c714}} with a character",
"Our conjecture says that $S_{j,2}(\\Gamma _2,\\epsilon )=(0)$ for $j<12$ .",
"We begin by showing this.",
"Lemma 5.1 For $j=0, 2, 4, 6, 8$ and 10, we have $S_{j,2}(\\Gamma _2,\\epsilon )=(0).$ We know that $\\dim S_{2j,4}(\\Gamma _2)=0$ for $j=0,2,4,6,8,10$ .",
"Assume that for one of these values of $j$ there is a non-zero element $f\\in S_{j,2}(\\Gamma _2,\\epsilon )$ .",
"Then ${\\rm Sym}^2(f)\\in S_{2j,4}(\\Gamma _2)=(0)$ must be zero.",
"Using the fact that the ring of holomorphic functions on $\\mathfrak {H}_2$ is an integral domain, we get a contradiction.",
"The first case where $S_{j,2}(\\Gamma _2,\\epsilon )$ is predicted to be non-zero is for $j=12$ .",
"In the paper [7] we constructed a modular form $\\chi _{12,2}$ in this space using the covariant $C_{3,12}$ associated to $A[15,3]$ occurring in ${\\rm Sym}^3({\\rm Sym}^6(V))$ , after dividing by the cusp form $\\chi _{10}$ .",
"Its Fourier expansion starts with $\\chi _{12,2}(\\tau )=\\left({\\begin{matrix}0\\\\0\\\\0\\\\2(R-R^{-1})\\\\9(R+R^{-1})\\\\12(R-R^{-1})\\\\0\\\\-12(R-R^{-1})\\\\-9(R+R^{-1})\\\\-2(R-R^{-1})\\\\0\\\\0\\\\0\\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,$ where $Q_1= e^{\\pi i \\tau _1}$ , $Q_2=e^{\\pi i \\tau _2}$ and $R=e^{\\pi i \\tau _{12}}$ for $\\tau =\\left({\\begin{matrix} \\tau _1 & \\tau _{12} \\\\ \\tau _{12} & \\tau _2\\\\\\end{matrix}} \\right)$ .",
"By multiplication by $\\chi _{6,3}$ we get an injective map $S_{12,2}(\\Gamma _2,\\epsilon ) \\rightarrow S_{18,5}(\\Gamma _2)$ and this latter space is 1-dimensional.",
"Corollary 5.2 We have $\\dim S_{12,2}(\\Gamma _2,\\epsilon )=1$ and it is generated by $\\chi _{12,2}$ .",
"We compute a few Hecke eigenvalues as described in Section .",
"To compute these Hecke eigenvalues, we used the following Fourier coefficients: $a([1,1,1])^t&=[0, 0, 0, 2, 9, 12, 0, -12, -9, -2, 0, 0, 0]\\\\a([1,3,3])^t&=[0, 0, 0, -2, -27, -156, -504, -996, -1233, -934, -396, -72, 0]\\\\a([3,3,3])^t&=[0, 216, 1188, 258, -7749, -12708, 0, 12708, 7749, -258, -1188, -216, 0]\\\\a([5,5,5])^t&=[0, 0, 0, -106920, -481140, -641520, 0, 641520, 481140, 106920, 0, 0, 0]\\\\a([1,5,7])^t&=[0, 0, 0, 2, 45, 444, 2520, 9060, 21375, 33046, 32220, 17928, 4320]\\\\a([7,7,7])^t&=[0, -8208, -45144, -542204, -2101338, -2711496, 0, 2711496, 2101338, 542204, 45144, 8208, 0]\\\\a([3,9,7])^t&=[0, -72, -1188, -8854, -39339, -115764, -236880, -343884,\\\\& \\qquad \\qquad -354141, -253514, -120132, -33912, -4320]\\\\$ These and a few more (too big to be written here) yield the following eigenvalues: height2pt $p$ 3 5 7 11 13 17 height2pt $\\lambda _p$ $-600$ $-53460$ $-369200$ 4084344 $-2845700$ 131681700 together with $\\lambda _9=-1090791$ .",
"We find that for the operator $T_p$ these are indeed of the form $\\lambda _p(f^{+})+\\lambda _p(f^{-})$ with $f^{\\pm }$ generators of $S^{\\pm }_{14}(\\Gamma _0(2))^{\\rm new}$ , with $f^{+}(\\tau )&=q - 64\\, q^2 - 1836\\, q^3 + 4096\\, q^4 + 3990\\, q^5 +117504\\, q^6 +\\cdots \\\\f^{-}(\\tau )&=q + 64\\, q^2 + 1236\\, q^3 + 4096\\, q^4 - 57450\\, q^5 +79104\\, q^6 +\\cdots \\, ,$ while for $T_{p^2}$ we find $\\lambda _{p}(f^{+})^2+\\lambda _p(f^{+})\\lambda _{p}(f^{-})+\\lambda _p(f^{-})^2-2\\, (p+1)p^j$ .",
"This fits with being a Yoshida lift.",
"Lemma 5.3 We have $ S_{14,2}(\\Gamma _2,\\epsilon )=(0)=S_{16,2}(\\Gamma _2,\\epsilon )$ .",
"To see that $S_{14,2}(\\Gamma _2,\\epsilon )=(0)$ we multiply a form in $S_{14,2}(\\Gamma _2,\\epsilon )$ with $\\chi _5$ and we end up in $S_{14,7}(\\Gamma _2)$ and this space is generated by a form associated to $C_{1,6}C_{3,8}$ after division by $\\chi _5^2$ .",
"Restricting this form $\\chi _{14,7}$ to $\\mathfrak {H}_1^2$ gives $\\sum _{i=0}^{7}(f_i\\otimes f_i^{\\prime })$ and only the term $f_5 \\otimes f_5^{\\prime }$ in $S_{16}(\\Gamma _1)\\otimes S_{12}(\\Gamma _1)$ can be non-zero and it is equal to $56\\, e_4\\Delta \\otimes \\Delta $ .",
"So it does not vanish along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ , hence $\\chi _{14,7}$ is not divisible by $\\chi _5$ .",
"We conclude $S_{14,2}(\\Gamma _2[2],\\epsilon )=(0)$ .",
"For $S_{16,2}(\\Gamma _2,\\epsilon )$ we multiply by $\\chi _{6,3}$ and land in $S_{22,5}(\\Gamma _2)$ and this space is zero.",
"Next we deal with the case of weight $(18,2)$ .",
"Proposition 5.4 The space $S_{18,2}(\\Gamma _2,\\epsilon )$ has dimension 1.",
"First we construct a non-zero element in this space by using the covariant $C=135\\,C_{1,6}^{2}C_{4,6}+56\\,C_{1,6}C_{2,0}C_{3,12}-270\\,C_{2,8}C_{4,10}-930\\,C_{3,6}C_{3,12}.$ It occurs in $\\rm Sym^6(\\rm Sym^6(V))$ and provides a cusp form, $F_C$ , of weight $(18,27)$ on $\\Gamma _2$ .",
"The order of vanishing of $F_C$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 5, so we can divide it by $\\chi _{5}^5$ and we get by Remark REF a cusp form, denoted $\\chi _{18,2}$ , of weight $(18,2)$ on $\\Gamma _2$ with character.",
"Again we multiply by $\\chi _{5}$ and land in $S_{18,7}(\\Gamma _2)$ .",
"This space is 2-dimensional and we can construct a basis using the following covariants $C_1=C_{3,12}(8\\, C_{1,6}C_{2,0}-75\\, C_{3,6})\\quad \\text{ and} \\quad C_2= C_{1,6}^2 C_{4,6}-2\\, C_{2,8}C_{4,10}-3\\, C_{3,6}C_{3,12}.$ They occur in $\\rm Sym^6(\\rm Sym^6(V))$ and provide two cusp forms, $F_{C_i}$ , of weight $(18,27)$ on $\\Gamma _2$ .",
"Each cusp form $F_{C_i}$ vanishes with order 4 along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ , so we can divide it by $\\chi _{5}^4$ and get a cusp form, $\\chi _{18,7}^{(i)}$ , of weight $(18,7)$ on $\\Gamma _2$ .",
"The cusp forms $\\chi _{18,7}^{(1)}$ and $\\chi _{18,7}^{(2)}$ are $\\mathbb {C}$ -linearly independent as can be read off from the first terms of their Fourier expansions and the pullbacks to $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ are of the form $\\sum _{r=0}^9 f_r \\otimes f_r^{\\prime }$ with only non-zero terms for $r=5$ and these are $216\\,e_4^2\\Delta \\otimes \\Delta $ and $48\\, e_4^2\\Delta \\otimes \\Delta $ .",
"Up to a non-zero scalar there is only one non-trivial linear combination, that vanishes along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ and that gives a non-zero form in $S_{18,2}(\\Gamma _2,\\epsilon )$ after division by $\\chi _5$ .",
"Proposition 5.5 The space $S_{20,2}(\\Gamma _2,\\epsilon )$ has dimension 1.",
"We construct a non-zero form in this space by taking the covariant $\\begin{aligned}C=&224\\,C_{1,6}^{2}C_{5,8}+312\\,C_{1,6}C_{2,4}C_{4,10}-560\\,C_{1,6}C_{2,8}C_{4,6}\\\\&-108\\,C_{1,6}C_{3,6}C_{3,8}+728\\,C_{2,0}C_{2,8}C_{3,12}-1235\\,C_{2,4}^{2}C_{3,12}.\\end{aligned}$ occurring in ${\\rm Sym}^7({\\rm Sym}^6(V))$ and providing a cusp form, $F_C$ , of weight $(20,32)$ on $\\Gamma _2$ with character.",
"The order of vanishing of $F_C$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 6, so we can divide it by $\\chi _{5}^6$ and we get a cusp form, $\\chi _{20,2}$ , of weight $(20,2)$ on $\\Gamma _2$ with character.",
"In a similar way we construct a basis of the space $S_{20,7}(\\Gamma _2)$ by taking the covariants $\\begin{aligned}C_1&=480\\,C_{1,6}^{2}C_{5,8}-180\\,C_{1,6}C_{3,6}C_{3,8}+728\\,C_{2,0}C_{2,8}C_{3,12}-1315\\,C_{2,4}^{2}C_{3,12},\\\\C_2&=80\\,C_{1,6}^{2}C_{5,8}-80\\,C_{1,6}C_{2,8}C_{4,6}+104\\,C_{2,0}C_{2,8}C_{3,12}-125\\,C_{2,4}^{2}C_{3,12},\\\\C_3&=80\\,C_{1,6}^{2}C_{5,8}-40\\,C_{1,6}C_{2,4}C_{4,10}+56\\,C_{2,0}C_{2,8}C_{3,12}-55\\,C_{2,4}^{2}C_{3,12}.\\end{aligned}$ which provide cusp forms with character of weight $(20,32)$ and these are divisible by $\\chi _5^5$ and thus give cusp forms of weight $(20,7)$ generating $S_{20,7}(\\Gamma _2)$ .",
"By restriction to the diagonal one sees that there is just a 1-dimensional space of forms vanishing on the diagonal.",
"The case of weight $(24,2)$ is dealt with in a similar way.",
"Proposition 5.6 We have $\\dim S_{24,2}(\\Gamma _2[2],\\epsilon )=2$ .",
"We know that $\\dim S_{24,7}(\\Gamma _2)=5$ and we can construct a basis using the procedure described in Section .",
"In the case at hand we have $A[39,15]$ occurring in ${\\rm Sym}^9({\\rm Sym}^6(V))$ with multiplicity 13 and this gives a subspace of $S_{24,42}(\\Gamma _2,\\epsilon )$ of dimension 13.",
"One checks that there is a 5-dimensional subspace of forms vanishing with multiplicity 7 along the diagonal and dividing by $\\chi _5^7$ gives a 5-dimensional subspace of $S_{24,7}(\\Gamma _2)$ , hence the whole space.",
"Again one checks that there is a 2-dimensional space of forms vanishing on the diagonal and we can divide these forms by $\\chi _5$ .",
"So the two generators of $S_{24,2}(\\Gamma _2,\\epsilon )$ are defined by the covariants $C_1$ and $C_2$ given respectively by $\\begin{aligned}&-499408\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}-1505385\\,C_{1,6}^{3}C_{6,6}^{(1)}-14727825\\,C_{1,6}^{2}C_{2,4}C_{5,8}+6916455\\,C_{1,6}^{2}C_{2,8}C_{5,4}\\\\&-5728590\\,C_{1,6}^{2}C_{3,12}C_{4,0}+6972210\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}+4257120\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}\\\\&+2182950\\,C_{2,8}^{2}C_{5,8}+11708550\\,C_{2,8}C_{3,6}C_{4,10}+595350\\,C_{2,8}C_{3,12}C_{4,4}+35171325\\,C_{3,6}^{2}C_{3,12}\\\\&-400950\\,C_{3,8}^{3}\\\\\\end{aligned}$ and $\\begin{aligned}&-42235648\\,C_{1,6}^{2}C_{2,0}^{2}C_{3,12}+4434583545\\,C_{1,6}^{3}C_{6,6}^{(1)}+580982220\\,C_{1,6}^{3}C_{6,6}^{(2)}\\\\&+4919972400\\,C_{1,6}^{2}C_{2,4}C_{5,8}+4827362400\\,C_{1,6}^{2}C_{3,12}C_{4,0}-3504891600\\,C_{1,6}C_{2,0}C_{2,8}C_{4,10}\\\\&+1245336960\\,C_{1,6}C_{2,0}C_{3,6}C_{3,12}-4131252720\\,C_{2,8}^{2}C_{5,8}-24904998720\\,C_{2,8}C_{3,6}C_{4,10}\\\\&-281640240\\,C_{2,8}C_{3,12}C_{4,4}-58751907480\\,C_{3,6}^{2}C_{3,12}+1375354080\\,C_{3,8}^{3}\\, .\\end{aligned}$ The order of vanishing of $F_{C_i}$ along $\\mathfrak {H}_1\\times \\mathfrak {H}_1$ is 8, so we can divide it by $\\chi _{5}^8$ and we get two cusp forms, $\\chi _{24,2}^{(i)}$ , ($i=1,2$ ) of weight $(24,2)$ on $\\Gamma _2$ with character.",
"We set $\\chi _{24,2}^{(1)}=-12150\\, F_{C_1}/\\chi _5^8$ and $\\chi _{24,2}^{(2)}=-675(5368\\, F_{C_1} +5 \\, F_{C_2})/31528 \\chi _5^8$ .",
"Then their Fourier expansions are given by $\\chi _{24,2}^{(1)}=\\left({\\begin{matrix}0\\\\0\\\\0\\\\104(R-R^{-1})\\\\1092(R+R^{-1})\\\\3640(R-R^{-1})\\\\0\\\\-27678(R-R^{-1})\\\\-58905(R+R^{-1})\\\\-2916(R-R^{-1})\\\\148470(R+R^{-1})\\\\190778(R-R^{-1})\\\\0\\\\\\vdots \\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,\\quad \\chi _{24,2}^{2}(\\tau )=\\left({\\begin{matrix}0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\0\\\\2(R-R^{-1})\\\\17(R+R^{-1})\\\\60(R-R^{-1})\\\\110(R+R^{-1})\\\\98(R-R^{-1})\\\\0\\\\\\vdots \\end{matrix}}\\right)Q_1Q_2+\\cdots \\, ,$ where $Q_1=e^{i\\pi \\tau _1}$ , $Q_2=e^{i\\pi \\tau _2}$ , $R=e^{i\\pi \\tau _{12}}$ .",
"The action of $\\iota =({\\begin{matrix} a & 0 \\\\ 0 & d\\\\ \\end{matrix}})\\in \\Gamma _2$ with $a=d= ({\\begin{matrix} 0 & 1 \\\\ 1 & 0\\\\ \\end{matrix}})$ implies that the $i$ th coordinate is equal to $(-1)^{k+1}$ times the $(j+1-i)$ th coordinate, which gives the non-displayed coordinates .",
"A Hecke eigenbasis of the space $S_{24,2}(\\Gamma _2,\\epsilon )$ is: $F_1=&\\,439 \\chi _{24,2}^{(1)} +(114847+650\\sqrt{106705})\\, \\chi _{24,2}^{(2)}\\\\F_2=& \\, 439\\chi _{24,2}^{(1)} +(114847-650\\sqrt{106705})\\, \\chi _{24,2}^{(2)}$ with eigenvalues height2pt $p$ $\\lambda _p(F_1)$ $\\lambda _{p^2}(F_1)$ height2pt 3 $287880-4800\\sqrt{106705}$ $545747143689-2293459200\\sqrt{106705}$ 5 $711981900+1555200\\sqrt{106705}$ – 7 $-41070905840+92534400\\sqrt{106705}$ – 11 $ -10344705071976 + 4819953600\\sqrt{106705}$ – in perfect agreement with the eigenforms being Yoshida lifts.",
"Indeed a basis of the space $S_{26}(\\Gamma _0[2])^{\\text{new}}$ is given by $f&=q - 4096\\, q^2 + 97956\\, q^3 + 16777216\\, q^4 + 341005350\\, q^5- 401227776\\, q^6 +\\cdots \\\\g&=q + 4096\\, q^2 + (2048-a/2)\\, q^3 + 16777216\\, q^4 +(431848374+162\\, a)\\, q^5 +\\cdots \\\\g^{\\prime }&=q + 4096\\, q^2 + (2048+a/2)\\, q^3 + 16777216\\, q^4 + (431848374-162\\, a)\\, q^5 +\\cdots \\, ,$ where $ a=-375752+9600\\sqrt{106705} $ , and $f,g^{\\prime } \\in S^{-}_{26}$ and $g \\in S_{26}^{+}$ .",
"Then we check for example that $\\lambda _5(F_1)&=711981900+1555200\\sqrt{106705}=a_5(f)+a_5(g)\\\\\\lambda _5(F_2)&=711981900-1555200\\sqrt{106705}=a_5(f)+a_5(g^{\\prime }).$ Proposition 5.7 One has $\\dim S_{26,2}(\\Gamma _2,\\epsilon )=1=\\dim S_{28,2}(\\Gamma _2,\\epsilon )$ and $\\dim S_{30,2}(\\Gamma _2,\\epsilon )=2$ .",
"The proof of this proposition is similar to the above.",
"For the first statement we consider the space $S_{26,7}(\\Gamma _2)$ which has dimension 6 and construct a basis of this space using covariants associated to $A[43,17]$ in ${\\rm Sym}^{10}({\\rm Sym}^6(V))$ which occurs with multiplicity 17, thus giving rise to a 17-dimensional subspace of $S_{26,47}(\\Gamma _2)$ .",
"By restricting along the diagonal one checks that there is a 6-dimensional subspace of cusp forms divisible by $\\chi _5^8$ leading to the construction of $S_{26,7}(\\Gamma _2)$ .",
"Again by restricting to the diagonal one sees that there is exactly a 1-dimensional subspace of this space that vanish along the diagonal.",
"By dividing by $\\chi _5$ we thus find the space $S_{26,2}(\\Gamma _2,\\epsilon )$ .",
"For weight $(28,2)$ we now use the representation $A[47,19]$ that occurs with multiplicity 23 in ${\\rm Sym}^{11}({\\rm Sym}^6(V))$ and leading to a 23-dimensional subspace of $S_{28,52}(\\Gamma _2)$ in which we find a 7-dimensional subspace of forms divisible by $\\chi _5^9$ and division gives forms that generate $S_{28,7}(\\Gamma _2)$ .",
"In this space the subspace of forms divisible by $\\chi _5$ is of dimension 1, proving our claim.",
"In the case of weight $(30,7)$ the 9-dimensional space $S_{30,7}$ is constructed using covariants resulting from $A[51,21]$ that occurs with multiplicity 31 in ${\\rm Sym}^{12}({\\rm Sym}^6(V))$ leading to a space of dimension 31 of cusp forms of weight $(30,57)$ .",
"There is a 9-dimensional subspace of cusp forms divisible by $\\chi _5^{10}$ and we thus generate $S_{30,7}(\\Gamma _2)$ .",
"It turns out that there is a 2-dimensional subspace of forms divisible by $\\chi _5$ and this proves the claim.",
"For all the cases treated we can check our construction by verifying that the Hecke eigenvalues for $p=3,5,7,11,13, 17$ agree with the forms being Yoshida lifts like we indicated for $j=12$ and $j=24$ .",
"In a forthcoming paper ([8]) we shall use the relation with covariants to describe modules of forms with a character."
],
[
"Modular Forms of Weight $(j,2)$ on {{formula:8ac6fb49-ab1c-4eac-b5ae-da9c6611aa41}}",
"In this section we explain how we checked that $S_{j,2}(\\Gamma _2)=(0)$ for $j\\le 52$ .",
"We begin with a simple lemma.",
"Recall that we have maps $M \\rightarrow {\\mathcal {C}} {\\mu \\over \\longrightarrow } M$ .",
"Lemma 6.1 Let $f \\in M_{j,k}(\\Gamma _2)$ .",
"Then there exists a covariant $c_f$ of degree $(d,j)$ with $d\\le j/2+k$ such that $f=\\nu (c_f)=\\mu (c_f)/\\chi _5^d$ .",
"If $f$ is a cusp form then there is a covariant $c_f^{\\prime }$ of degree $\\le j/2+k-10$ such that $f=\\mu (c_f^{\\prime })/\\chi _5^r$ for some $r$ .",
"The first statement follows directly from [7].",
"If $f$ is a cusp form then the covariant it defines vanishes on the discriminant locus.",
"But then the covariant $c_f$ is divisible by the discriminant, and $\\mu (c_f)$ by $\\chi _5^{d+2}$ .",
"This makes it possible to check the existence of a non-zero form $f \\in S_{j,2}(\\Gamma _2)$ by checking whether the forms of weight $(j,2+5d)$ provided via $\\mu $ by the non-zero covariants of degree $(d,j)$ with $d\\le j/2-8$ are divisible by $\\chi _5^{d}$ .",
"We applied this for values of $j\\le 52$ using the covariants of degree $d\\le 18$ .",
"For smaller values of $j$ other methods of showing that $S_{j,2}(\\Gamma _2)=(0)$ are available.",
"We sketch some methods below.",
"In this way we checked that $S_{j,2}(\\Gamma _2)=(0)$ for $j\\le 52$ .",
"Another method is to construct a basis of $S_{j,7}(\\Gamma _2,\\epsilon )$ by using covariants.",
"We then check the divisibility by $\\chi _5$ of elements in $S_{j,7}(\\Gamma _2,\\epsilon )$ by restricting the modular forms in this space to the diagonal.",
"As an illustration we give the proof for the case $j=24$ .",
"We construct a basis of the 9-dimensional space $S_{24,7}(\\Gamma _2,\\epsilon )$ .",
"For this we use the covariants associated to the $A[54,30]$ -isotypical component of ${\\rm Sym}^{14}({\\rm Sym}^6(V))$ .",
"The representation $A[54,30]$ occurs with multiplicity 65 and leads to a 65-dimensional subspace of modular forms of weight $(24,72)$ on $\\Gamma _2$ .",
"By restricting to $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ we can check that there exists a 9-dimensional subspace of cusp forms that are divisible by $\\chi _5^{13}$ .",
"This leads to a basis of $S_{24,7}(\\Gamma _2,\\epsilon )$ .",
"We then check by restriction to $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ again that there is no non-trivial element in this space that is divisible by $\\chi _5$ .",
"This proves the result for $j=24$ .",
"We carried this out for all the cases $j\\le 52$ and thus proved Theorem REF .",
"Sometimes there are other and easier ways to eliminate cases.",
"For example, by restricting a modular form of weight $(j,2)$ to the diagonal we get an element $\\sum _{i=0}^{j/2} f_i \\otimes f_i^{\\prime }\\in \\oplus _{i=0}^{j/2} S_{j+2-i}(\\Gamma _1)\\otimes S_{2+i}(\\Gamma _1)$ .",
"If $j<24$ , $j\\ne 20$ the spaces in question are zero.",
"Therefore a form $f\\in S_{j,2}(\\Gamma _2)$ will vanish on $\\mathfrak {H}_1 \\times \\mathfrak {H}_1$ .",
"But then $f/\\chi _5$ will be a holomorphic modular form of weight $(j,-3)$ and this has to be zero.",
"So $f=0$ for $j\\le 18$ and $j=22$ .",
"As yet another example of eliminating cases we give a somewhat different argument for $j=26$ .",
"We write elements of $F\\in S_{j,k}(\\Gamma _2)$ as vectors $F=(F_0,\\ldots ,F_j)^t$ with the $F_i$ holomorphic functions on $\\mathcal {H}_2$ , that is, in a module of rank 27 over the ring ${\\mathcal {F}}$ of holomorphic functions on $\\mathcal {H}_2$ .",
"Take a basis $s_1,\\ldots ,s_3$ of $S_{26,6}(\\Gamma _2)$ and a basis $s_{4},\\ldots ,s_{12}$ of $S_{26,8}(\\Gamma _2)$ .",
"If there exists a non-zero form $f$ of weight $(26,2)$ then the vectors $E_4\\, f$ and $E_6\\, f$ are linearly dependent and thus the exterior product $s_1\\wedge \\cdots \\wedge s_{12}$ must vanish.",
"By calculating bases of $S_{26,6}(\\Gamma _2)$ and $S_{26,8}(\\Gamma _2)$ one can check that this exterior product does not vanish.",
"So $S_{26,2}(\\Gamma _2)=(0)$ ."
],
[
"Other Small Weights",
"We begin with an elementary argument that shows that $S_{j,k}(\\Gamma _2[2])=(0)$ for $j\\le 8$ and $k\\le 2$ .",
"Proposition 7.1 For $j\\le 8$ and $k\\le 2$ we have $\\dim S_{j,k}(\\Gamma _2[2])=0$ .",
"We need to deal with the cases $j$ even and $k=1$ and $k=2$ only since for other values $S_{j,k}(\\Gamma _2[2])$ vanishes.",
"We restrict to the ten components of the Humbert surface $H_1$ in $\\Gamma _2[2]\\backslash \\mathfrak {H}_2$ , one component of which is given by the diagonal $\\tau _{12}=0$ .",
"The group $\\mathfrak {S}_6$ acts transitively on these ten components.",
"The stabilizer inside $\\mathfrak {S}_6$ of a component of $H_1$ is an extension of $\\mathfrak {S}_3\\times \\mathfrak {S}_3$ by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ .",
"By restricting a modular form $f \\in S_{j,1}(\\Gamma _2[2])$ to a component we get an element of $\\oplus _{r=0}^{j} S_{j+1-r}(\\Gamma _1[2])\\otimes S_{1+r}(\\Gamma _1[2])$ and for $f\\in S_{j,2}(\\Gamma _2[2])$ we get an element of $\\oplus _{r=0}^{j} S_{j+2-r}(\\Gamma _1[2])\\otimes S_{2+r}(\\Gamma _1[2]) \\, .$ For $j\\le 8$ and $k=1$ and for $j<8$ and $k=2$ these spaces are zero.",
"Thus a form $f\\in S_{j,2}(\\Gamma _2[2])$ restricts to zero on all irreducible components of $H_1$ , hence is divisible by $\\chi _{5}$ , and so $f$ must be zero.",
"For $j=8$ and $k=2$ the restriction to $H_1$ gives an injective $\\mathfrak {S}_6$ -equivariant map $S_{8,2}(\\Gamma _2[2]) \\rightarrow \\oplus _{i=1}^{10}\\, {\\rm Sym}^2 S_6(\\Gamma _1[2]) \\, ,$ where the action on the right is the induced representation from the extension of $\\mathfrak {S}_3 \\times \\mathfrak {S}_3$ by ${\\mathbb {Z}}/2{\\mathbb {Z}}$ to $\\mathfrak {S}_6$ .",
"Now $S_6(\\Gamma _1[2])$ is 1-dimensional and of type $s[1^3]$ and we check that the representation of $\\mathfrak {S}_6$ on the 10-dimensional space $\\oplus _{i=1}^{10} \\rm Sym^2 S_6(\\Gamma _1[2])$ is of type s[6]+s[4,2].",
"Since $S_{8,2}(\\Gamma _2)=(0)$ , we conclude that only $S_{8,2}(\\Gamma _2[2])^{s[4,2]}$ can be non-zero.",
"If $S_{8,2}(\\Gamma _2[2])^{s[4,2]}$ is non-zero, then $S_{8,2}(\\Gamma _0[2])$ is non-zero (see [10]).",
"By restricting we get an element in a similar decomposition as before but with $\\Gamma _1[2]$ replaced by $\\Gamma _0[2]$ .",
"As we know that all theses spaces are zero, we can divide by $\\chi _5$ .",
"This contradiction concludes our claim.",
"More generally we have Proposition 7.2 For $j<12$ we have $S_{j,2}(\\Gamma _2[2])=(0)$ .",
"The space $S_{j,2}(\\Gamma _2[2])$ is defined over ${\\mathbb {Q}}$ .",
"All the cusps of $\\Gamma _2[2]$ are defined over ${\\mathbb {Q}}$ and the action of $\\mathfrak {S}_6$ is defined over ${\\mathbb {Q}}$ .",
"The q-expansion principle says that a modular form in $S_{j,k}(\\Gamma _2[2])$ with $k\\ge 3$ is defined over ${\\mathbb {Q}}$ if its Fourier coefficients at all cusps are defined over ${\\mathbb {Q}}$ , see [19] and [13].",
"We apply this to $f\\chi _{10}$ with $f \\in S_{j,2}(\\Gamma _2[2])$ defined over ${\\mathbb {Q}}$ and we conclude that the Fourier coefficients of $\\sigma (f\\chi _{10})$ with $\\sigma \\in \\mathfrak {S}_6$ are real, hence also those of $\\sigma (f)$ and if $f\\ne 0$ we find by looking at the `first' non-zero term in a Fourier expansion that $\\sum _{\\sigma \\in \\mathfrak {S}_6} \\sigma (f)^2$ is non-zero and because of $\\sigma (f^2)=\\sigma (f)^2$ also invariant under $\\mathfrak {S}_6$ .",
"Thus it defines a non-zero element of $S_{2j,4}(\\Gamma _2)$ .",
"So $S_{j,2}(\\Gamma _2[2])$ implies $S_{2j,4}(\\Gamma _2)\\ne (0)$ .",
"But we know that $S_{2j,4}(\\Gamma _2)=(0)$ for $j<12$ .",
"Remark 7.3 Note that the argument of the proof shows that our conjecture on the vanishing of $S_{j,2}(\\Gamma _2)$ for all $j$ implies the vanishing of $S_{j,1}(\\Gamma _2[2])$ for all $j$ .",
"In order to put our evidence for the vanishing of $S_{j,2}(\\Gamma _2)$ in perspective we show a small table that gives for each value of $k$ the smallest $j_0$ such that $\\dim S_{j_0,k}(\\Gamma _2)\\ne 0$ .",
"height2pt $k$ 3 4 5 6 7 8 height2pt $j_0$ 36 24 18 12 12 6 We can easily construct the generators of the corresponding spaces.",
"In [9] we constructed a generator $\\chi _{6,3}$ of $S_{6,3}(\\Gamma _2,\\epsilon )$ and above we gave the generator $\\chi _{12,2}$ of $S_{12,2}(\\Gamma _2,\\epsilon )$ .",
"The modular forms $\\chi _{12,2}^2$ , $\\chi _{6,3}\\chi _{12,2}$ , $\\chi _{6,3}^2$ , $\\chi _5\\chi _{12,2}$ and $\\chi _5\\chi _{6,3}$ give the generators for $k=4,\\ldots ,8$ .",
"We end by constructing a generator of $S_{36,3}(\\Gamma _2)$ ; the non-vanishing of this space plays a role in the appendix.",
"We look in ${\\rm Sym}^{11}({\\rm Sym}^6(V))$ and at $A[51,15]$ occuring with multiplicity 17 there.",
"We find the covariant $297 \\, C_{1,6}^2C_{3,8}^3 -8316 \\, C_{1,6}C_{3,8}C_{3,12}C_{4,10}+4116 \\, C_{1,6}C_{3,12}^2C_{4,6} -5488 \\, C_{2,0}C_{3,12}^3+9030 \\, C_{2,4}C_{3,8}C_{3,12}^2$ giving a form $f$ of weight $(36,48)$ that is divisible by $\\chi _5^{9}$ and $f/\\chi _5^9$ generates $S_{36,3}(\\Gamma _2)$ .",
"As a check we note that the Fourier coefficient at $n=[1,1,1]$ is of the form $[0, 0, 0, 0, 0, 0, 0, 32/6089428125, 464/6089428125, \\ldots ]$ and the coefficient at $n=[5,5,5]$ is $[0, 0, 0, 0, 0, 0, 0, -8687121398144/81192375, -125963260273088/81192375,\\ldots ]$ giving the eigenvalue for the Hecke operator $T_5$ as their ratio $ -20360440776900$ , in agreement with the value given in [2]."
],
[
"by Gaëtan Chenevier",
"Let $j$ and $k$ be integers with $j\\ge 0$ , and $\\Gamma \\subset {\\rm Sp}_4(\\mathbb {Z})$ a congruence subgroup.",
"Recall that $S_{j,k}(\\Gamma )$ denotes the space of cuspidal Siegel modular forms for the subgroup $\\Gamma $ with values in the representation ${\\rm Sym}^j\\, \\otimes \\,\\det ^k$ of ${\\rm GL}_2(\\mathbb {C})$ .",
"We first consider the full Siegel modular group $\\Gamma _2 = {\\rm Sp}_4(\\mathbb {Z})$ and provide alternative proofs of the following results : Proposition 1.1 We have ${S}_{j,1}(\\Gamma _2)=0$ for any $j$ , and ${S}_{j,2}(\\Gamma _2)=0$ for $j \\le 38$ .",
"The vanishing of ${S}_{j,2}(\\Gamma _2)$ for all $j\\le 52$ is also proved in this paper by Cléry and van der Geer (Theorem REF ).",
"The vanishing of ${S}_{j,1}(\\Gamma _2)$ was at least known to Ibukiyama, who asserts in [16] that it is a consequence of the vanishing of all Jacobi forms of weight 1 for ${\\rm SL}_2(\\mathbb {Z})$ proven by Skoruppa [23].",
"Here we shall rather use automorphic representation theoretic methods.",
"First we need to fix some notations and make some preliminary remarks.",
"We denote by $r : {\\rm Sp}_4(\\mathbb {C}) \\rightarrow {\\rm GL}_4(\\mathbb {C})$ the tautological inclusion.",
"For a real reductive Lie group $H$ we shall denote the infinitesimal character of the Harish-Chandra module $U$ by ${\\rm inf}\\, U$ .",
"In the case $H={\\rm GL}_n(\\mathbb {R})$ (resp.",
"$H={\\rm PGSp}_4(\\mathbb {R})$ ), and following Harish-Chandra and Langlands, ${\\rm inf} \\, U$ may be viewed in a canonical way as a semisimple conjugacy class in the Lie algebra $\\mathfrak {h}=\\mathfrak {gl}_n(\\mathbb {C})$ (resp.",
"$\\mathfrak {h}=\\mathfrak {sp}_4(\\mathbb {C})$ ).",
"In both cases we may and shall identify this conjugacy class with the multiset of its eigenvalues in the natural representation of $\\mathfrak {h}$ .",
"We denote by ${\\rm W}_\\mathbb {R}$ the Weil group of $\\mathbb {R}$ (a certain extension of $\\mathbb {Z}/2\\mathbb {Z}$ by $\\mathbb {C}^\\times $ ), and for any integer $w$ we define ${\\rm I}_{w}$ as the 2-dimensional representation of ${\\rm W}_{\\mathbb {R}}$ induced from the unitary character $z \\mapsto (z/|z|)^{w}$ of $\\mathbb {C}^\\times $ .",
"(a) As we have ${S}_{j,k}(\\Gamma _2)=0$ for any odd $j$ or for $k\\le 0$ , we may once and for all assume $j \\equiv 0 \\bmod 2$ and $k>0$ .",
"As is well-known, for any such $(j,k)$ there is an irreducible unitary Harish-Chandra module for ${\\rm PGSp}_4(\\mathbb {R})$ , unique up to isomorphism, generated by a highest-weight vector of $K$ -type ${\\rm Sym}^j \\,\\otimes \\det ^k$ .",
"This module, that we shall denote by ${\\rm U}_{j,k}$ , is a holomorphic discrete series for $k \\ge 3$ , a limit of holomorphic discrete series for $k=2$ , and non-tempered for $k=1$ ; it is non-generic in all cases.",
"We have ${\\rm inf}\\, {\\rm U}_{j,k} =\\lbrace \\frac{j+2k-3}{2}, \\,\\frac{j+1}{2}, \\,-\\frac{j+1}{2}, \\,-\\frac{j+2k-3}{2}\\rbrace $ .",
"More precisely, if $\\varphi : {\\rm W}_\\mathbb {R}\\rightarrow {\\rm Sp}_4(\\mathbb {C})$ denotes the Langlands parameter of ${\\rm U}_{j,k}$ , then we have $r \\circ \\varphi \\simeq {\\rm I}_{j+2k-3} \\oplus {\\rm I}_{j+1}$ for $k>1$ , and $r \\circ \\varphi \\simeq {\\rm I}_{j} \\otimes |.|^{1/2}\\oplus {\\rm I}_{j} \\otimes |.|^{-1/2}$ for $k=1$ (see e.g.",
"[22] for a survey of those properties, and the references therein).",
"The relevance of ${\\rm U}_{j,k}$ here is that if $\\pi $ is a cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by an element of ${S}_{j,k}(\\Gamma _2)$ , then the Archimedean component $\\pi _\\infty $ of $\\pi $ is isomorphic to ${\\rm U}_{j,k}$ .",
"The other important property of $\\pi $ is that $\\pi _p$ is unramified for each prime $p$ (i.e.",
"admits non-zero invariants under ${\\rm PGSp}_4(\\mathbb {Z}_p)$ ).",
"As ${\\rm PGSp}_4$ is isomorphic to the split classical group ${\\rm SO}_5$ over $\\mathbb {Z}$ , we may apply Arthur's theory [1] to such a $\\pi $ .",
"(b) One of the main results of Arthur [1] associates to any discrete automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ a unique isobaric automorphic representation $\\pi ^{\\rm GL}$ of ${\\rm GL}_4$ over $\\mathbb {Q}$ , characterized by the following property : for any prime $p$ such that $\\pi _p$ is unramified, then $(\\pi ^{\\rm GL})_p$ is unramified as well and its Satake parameter is the image of the one of $\\pi _p$ under the map $r$ .",
"The infinitesimal character of $(\\pi ^{\\rm GL})_\\infty $ is the image of ${\\rm inf}\\, \\pi _\\infty $ under the derivative of $r$ , namely $\\mathfrak {sp}_4(\\mathbb {C}) \\rightarrow \\mathfrak {gl}_4(\\mathbb {C})$ .",
"Moreover, there is a unique collection of distinct triplets $(d_i,n_i,\\pi _i)_{i \\in I}$ , with integers $d_i,n_i \\ge 1$ and cuspidal selfdual automorphic representations $\\pi _i$ of ${\\rm GL}_{n_i}$ with $\\pi ^{\\rm GL} \\,\\simeq \\,\\boxplus _{i \\in I} \\,(\\,\\boxplus _{l=0}^{d_i-1}\\,\\, \\,\\pi _i \\,\\otimes |.|^{\\frac{d_i-1}{2}-l}\\,)\\, \\, \\, \\, \\, {\\rm and} \\,\\,\\,\\,\\,4 = \\sum _{i \\in I} n_i d_i.$ The selfdual representation $\\pi _i$ is symplectic in Arthur's sense if, and only if, $d_i$ is odd.",
"All of this is included in [1].",
"(c) For $\\pi $ as in (b), then ${\\rm inf}\\, \\pi _\\infty $ is the union, over all $i \\in I$ and all $0 \\le l< d_i$ , of the multisets $\\frac{d_i-1}{2}-l\\,+\\,{\\rm inf}\\, (\\pi _i)_\\infty $ .",
"In particular, if we have $\\lambda \\in \\frac{1}{2}\\mathbb {Z}$ and $\\lambda -\\mu \\in \\mathbb {Z}$ for all $\\lambda ,\\mu \\in \\,{\\rm inf}\\, \\pi _\\infty $ , then ${\\rm inf}\\, (\\pi _i)_\\infty $ has the same property for each $i$ : such a $\\pi _i$ is called algebraic.",
"If $\\omega $ is a cuspidal selfdual algebraic automorphic representation of ${\\rm GL}_m$ over $\\mathbb {Q}$ , then $\\omega _\\infty $ is tempered by the Jaquet-Shalika estimates, and its Langlands parameter is trivial on the central subgroup $\\mathbb {R}_{>0}$ of ${\\rm W}_\\mathbb {R}$ (this is the so-called Clozel purity lemma, see e.g.",
"[6]).",
"(d) The only selfdual cuspidal automorphic representation $\\pi $ of ${\\rm GL}_1$ over $\\mathbb {Q}$ such that $\\pi _p$ is unramified for each prime $p$ is the trivial Hecke character 1 (which is of course selfdual orthogonal).",
"Moreover, for any integer $k\\ge 1$ , the number of cuspidal automorphic representations $\\pi $ of ${\\rm GL}_2$ over $\\mathbb {Q}$ such that $\\pi _p$ is unramified for each prime $p$ , and with ${\\rm inf}\\, \\pi _\\infty =\\lbrace -\\frac{k-1}{2},\\frac{k-1}{2}\\rbrace $ , is the dimension of the space ${S}_k(\\Gamma _1)$ of cuspidal modular forms of weight $k$ for $\\Gamma _1={\\rm SL}_2(\\mathbb {Z})$ .",
"Indeed, this is well-known for $k>1$ , and for $k=1$ it follows from the fact that there is no Maass form of eigenvalue $1/4$ for ${\\rm SL}_2(\\mathbb {Z})$ (a fact due to Selberg, see also [6] for a short proof).",
"(of the vanishing of ${S}_{j,1}(\\Gamma _2)$ for any $j$ ).",
"It is enough to show that there is no discrete automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ which is unramified at every prime and with $\\pi _\\infty \\simeq {\\rm U}_{j,1}$ .",
"For that we study $\\pi ^{\\rm GL}$ , and the associated collection $(d_i,n_i,\\pi _i)_{i \\in I}$ given by (b) above.",
"By (a), the infinitesimal character of $(\\pi ^{\\rm GL})_\\infty $ is ${\\rm inf}\\, {\\rm U}_{j,1} \\, =\\,\\lbrace \\,\\frac{j+1}{2}, \\,\\frac{j-1}{2}, \\,-\\frac{j-1}{2}, \\,-\\frac{j+1}{2}\\rbrace .$ If we have $d_i=1$ for each $i$ , then $(\\pi ^{\\rm GL})_\\infty $ is tempered by (c), hence so is $\\pi _\\infty $ by Arthur's local-global compatibility [1], a contradiction as ${\\rm U}_{j,1}$ is non-tempered by (a).",
"Fix $i \\in I$ with $d_i\\ge 2$ .",
"If we have $n_i \\ge 2$ then we must have $I=\\lbrace i\\rbrace $ and $n_i=d_i=2$ by the equality $4 = \\sum _i n_i d_i$ .",
"By (c) and the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ above, we necessarily have ${\\rm inf} \\, (\\pi _i)_\\infty = \\lbrace j/2,-j/2\\rbrace $ .",
"But this is absurd by (d) and the vanishing ${S}_{j+1}(\\Gamma _1)=0$ for any even integer $j \\ge 0$ .",
"We have thus $(d_i,n_i,\\pi _i)=(2,1,1)$ .",
"Choose $i^{\\prime } \\in I-\\lbrace i\\rbrace $ .",
"As we have $(d_{i^{\\prime }},n_{i^{\\prime }},\\pi _{i^{\\prime }}) \\ne (d_i,n_i,\\pi _i)$ , the previous argument shows $d_{i^{\\prime }}=1$ , so $\\pi _{i^{\\prime }}$ is symplectic by (b), which forces $n_{i^{\\prime }}$ to be even, hence the only possibility is $n_{i^{\\prime }}=2$ and $I=\\lbrace i,i^{\\prime }\\rbrace $ .",
"But then the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ and (c) show that we have either $j=0$ and ${\\rm inf} (\\pi _{i^{\\prime }})_\\infty = \\lbrace 1/2, -1/2 \\rbrace $ or $j=1$ and ${\\rm inf} (\\pi _{i^{\\prime }})_\\infty = \\lbrace 3/2, -3/2 \\rbrace $ .",
"Both cases are absurd by (d) as we have ${S}_4(\\Gamma _1)={S}_2(\\Gamma _1)=0$ , and we are done.",
"The second assertion of the proposition will be a consequence of the following two lemmas.",
"Lemma 1.2 Let $j \\ge 0$ be an even integer.",
"The integer $\\dim {S}_{j,2}(\\Gamma _2)$ is the number of cuspidal, selfdual symplectic, automorphic representations $\\Pi $ of ${\\rm GL_4}$ over $\\mathbb {Q}$ whose local components $\\Pi _p$ are unramified for each prime $p$ , and with ${\\rm inf} \\, \\Pi _\\infty \\, = \\,\\lbrace \\,\\frac{j+1}{2}, \\,\\frac{j+1}{2}, \\,-\\frac{j+1}{2}, \\,-\\frac{j+1}{2}\\rbrace $ .",
"Let $\\pi $ be a cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by an arbitrary Hecke eigenform $F$ in ${S}_{j,2}(\\Gamma _2)$ .",
"Consider its associated automorphic representation $\\pi ^{\\rm GL}$ of ${\\rm GL}_4$ and collection of $(d_i,n_i,\\pi _i)$ 's as in (b).",
"We claim that $\\pi ^{\\rm GL}$ is necessarily cuspidal, i.e.",
"$I=\\lbrace i\\rbrace $ is a singleton and $d_i=1$ , so that $\\Pi =\\pi ^{\\rm GL}$ satisfies all the assumptions of the statement by (a) and (b).",
"Let us show first that we have $d_i=1$ for each $i \\in I$ .",
"Otherwise, (c) shows that two elements of ${\\rm inf}\\, {\\rm U}_{j,2}$ must differ by 1, which only happens for $j=0$ .",
"But for $j=0$ an argument similar to the one in the previous proof shows that if $d_i>1$ then we have $I=\\lbrace i,i^{\\prime }\\rbrace $ with $(d_i,n_i,\\pi _i)=(2,1,1)$ , $n^{\\prime }_i=2$ and ${\\rm inf}\\, (\\pi _i)_{\\infty } = \\lbrace 1/2,-1/2\\rbrace $ , which is absurd by the vanishing ${S}_2(\\Gamma _1)=0$ and (d), and we are done.",
"As a consequence, the Langlands parameter of $(\\pi ^{\\rm GL})_\\infty $ is ${\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ by (c).",
"Let us denote by $\\psi $ Arthur's substitute for the global parameter of the representation $\\pi $ of ${\\rm PGSp}_{4}$ defined in [1].",
"We have just proved that $\\psi _\\infty $ is the tempered Langlands parameter of ${\\rm PGSp}_4(\\mathbb {R})$ with $r \\,\\circ \\,\\psi _\\infty \\simeq {\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ , i.e.",
"the Langlands parameter of ${\\rm U}_{j,2}$ by (a).",
"We have $\\mathcal {S}_{\\psi _\\infty } = \\mathbb {Z}/2\\mathbb {Z}$ : the corresponding $L$ -packet of ${\\rm PGSp}_4(\\mathbb {R})$ (limit of discrete series) has two elements, namely ${\\rm U}_{j,2}$ and a generic limit of discrete series with same infinitesimal character.",
"We now apply Arthur's multiplicity formula to the element $\\pi $ of the global packet $\\Pi _\\psi $ defined by Arthur.",
"As we have $d_i=1$ for all $i$ , either $\\pi ^{\\rm GL}$ is cuspidal or we have $I=\\lbrace i,i^{\\prime }\\rbrace $ with $n_i=n_{i^{\\prime }}=2$ and ${\\rm inf}\\, (\\pi _i)_\\infty ={\\rm inf}\\, (\\pi _{i^{\\prime }})_\\infty = \\lbrace \\frac{j+1}{2},-\\frac{j+1}{2}\\rbrace $ .",
"If $\\pi ^{\\rm GL}$ is not cuspidal, then according to Arthur's definitions the natural map $\\mathcal {S}_\\psi \\rightarrow \\mathcal {S}_{\\psi _\\infty }$ is an isomorphism of groups of order 2.",
"But then his multiplicity formula shows that $\\pi _\\infty $ has to be generic since $\\pi _p$ is unramified for each prime $p$ , a contradiction as ${\\rm U}_{j,2}$ is not generic.",
"(We have shown that $\\pi $ is not of “Yoshida type”.)",
"We have thus proved that $\\pi ^{\\rm GL}$ is cuspidal.",
"Note that in this case we have $\\mathcal {S}_\\psi =1$ , thus by the multiplicity formula again, the multiplicity of $\\pi $ in the automorphic discrete spectrum of ${\\rm PGSp}_4$ is 1; in particular, the Hecke eigenspace of the eigenform $F$ we started from has dimension 1.",
"It thus only remains to show that any $\\Pi $ as in the statement is in the image of the construction of the first paragraph above.",
"Let $\\Pi $ be as in the statement.",
"The Langlands parameter of $\\Pi _\\infty $ is the image under $r$ of the one of ${\\rm U}_{j,2}$ by (a) and (c).",
"A trivial application of Arthur's multiplicity formula shows the existence of a discrete automorphic $\\pi $ for ${\\rm PGSp}_4$ with $\\pi _\\infty \\simeq {\\rm U}_{j,2}$ , which is unramified at every prime, and satisfying $\\pi ^{\\rm GL} \\simeq \\Pi $ .",
"As ${\\rm U}_{j,2}$ is tempered, a classical result of Wallach ensures that $\\pi $ is actually cuspidal, hence generated by an element of ${S}_{j,2}(\\Gamma _2)$ : this concludes the proof.",
"Lemma 1.3 For any even integer $0 \\le j \\le 38$ there is no $\\Pi $ as in Lemma REF .",
"In order to contradict the existence of a $\\Pi $ as in Lemma REF for small $j$ , and following work of Odlyzko, Mestre, Fermigier, Miller and Chenevier-Lannes, we shall apply the so-called explicit formula “à la Riemann-Weil” to a suitable test function $F$ and to the complete Rankin-Selberg $L$ -function ${\\rm L}(s,\\Pi \\times \\Pi ^{\\prime })$ , first to $\\Pi ^{\\prime } = \\Pi ^{\\vee }$ (the contragredient of $\\Pi $ ) and then to some other well-chosen cuspidal automorphic representations $\\Pi ^{\\prime }$ .",
"Let us stress that the analytic properties of those Rankin-Selberg $L$ -functions (meromorphic continuation to $\\mathbb {C}$ , functional equation, determination of the poles, and boundedness in vertical strips away from the poles) which have been established by Gelbart, Jacquet, Shalika and Shahidi, will play a crucial role in the argument.",
"It will be convenient to follow the exposition of the explicit formula given in [6], which is designed for this kind of applications, and from which we shall borrow our notations.",
"In particular, we choose for the test function $F$ the scaling of Odlyzko's function which is denoted by ${\\rm F}_{\\lambda }$ in [6], and we denote by ${\\rm K}_{\\infty }$ the Grothendieck ring of finite dimensional complex representations of the quotient of the compact group ${\\rm W}_{\\mathbb {R}}$ by its central subgroup $\\mathbb {R}_{>0}$ , and by ${\\rm J}_{F} : {\\rm K}_{\\infty } \\rightarrow \\mathbb {R}$ the concrete linear form associated to $F$ defined in Proposition-Def.",
"3.7 loc.",
"cit.",
"Let $j\\ge 0$ be an even integer and let $\\Pi $ be as in Lemma REF .",
"As already explained, it follows from (c) that the Langlands parameter of $\\Pi _\\infty $ is ${\\rm I}_{j+1} \\oplus {\\rm I}_{j+1}$ , whose square in the ring ${\\rm K}_\\infty $ is $4\\, ( {\\rm I}_{2j+2}+ {\\rm I}_{0})$ .",
"The explicit formula applied to $\\Pi \\times \\Pi ^\\vee $ leads to the inequality [6] : $ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{2j+2})+{\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0}) \\le \\frac{2}{\\pi ^{2}} \\lambda $ for all $\\lambda >0$ .",
"As ${\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{w})$ is a non-increasing function of $w$ , the truth of the proposition for $j \\le 34$ is a consequence of the following numerical computation for $\\lambda = 3.3$ ${\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{70})+ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0}) \\simeq 0.679 \\, \\, \\, \\, \\, \\, {\\rm and}\\, \\, \\, \\, \\, \\frac{2}{\\pi ^{2}}\\lambda \\simeq 0.669$ (the values are given up to $10^{-3}$ , and the left-hand side has been computed using the formula for ${\\rm J}_{F_{\\lambda }}$ given in [6]).",
"We now explain how to deal with the cases $j=36$ and 38.",
"By Tsushima's formula (proved for $k=3$ independently by Petersen and Taïbi), we know that the first value of $j$ such that ${S}_{j,3}(\\Gamma _2)$ is non-zero is $j=36$ , in which case it has dimension 1.",
"Let $\\pi $ be the cuspidal automorphic representation of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ generated by ${S}_{36,3}(\\Gamma _2)$ and set $\\Pi ^{\\prime } = \\pi ^{\\rm GL}$ .",
"This $\\Pi ^{\\prime }$ is a selfdual cuspidal representation by [6], and the Langlands parameter of $\\Pi ^{\\prime }_\\infty $ is ${\\rm I}_{39} \\oplus {\\rm I}_{37}$ (the existence of such a $\\Pi ^{\\prime }$ actually “explains” why the argument above breaks down at $j=36$ , as the Langlands parameter of $\\Pi ^{\\prime }_\\infty $ is “close” to ${\\rm I}_{37}\\oplus {\\rm I}_{37}$ ).",
"We now apply the explicit formula to $\\Pi \\times \\Pi ^\\vee $ , $\\Pi ^{\\prime } \\times {\\Pi ^{\\prime }}^\\vee $ and $\\Pi \\times {\\Pi ^{\\prime }}^\\vee $ .",
"It leads to a simple criterion, given in [6], for $\\Pi $ not to exist : the explicit quantity denoted there by ${\\rm t}(V,V^{\\prime },\\lambda )$ has to be $\\ge 0$ for all $\\lambda $ , where $V$ and $V^{\\prime }$ are the respective Langlands parameter of $\\Pi _\\infty $ and $\\Pi ^{\\prime }_\\infty $ .",
"But a computation gives ${\\rm t}({\\rm I}_{37} + {\\rm I}_{37},{\\rm I}_{39} + {\\rm I}_{37},4) \\simeq -0.429\\,\\,\\,\\,\\, {\\rm and} \\,\\,\\, \\, \\, {\\rm t}({\\rm I}_{39}+ {\\rm I}_{39},{\\rm I}_{39} + {\\rm I}_{37},4) \\simeq -0.039$ (these values are given up to $10^{-3}$ ) which are both $<0$ .",
"This concludes the proof.",
"Remark 1.4 (i) As we have ${S}_{j,3}(\\Gamma _2)=0$ for $j<36$ by Tsushima's formula, the vanishing of ${S}_{j,2}(\\Gamma _2)$ for $j \\le 38$ is a very mild evidence toward Conjecture 1.1 of Cléry and van der Geer.",
"(ii) Lemma REF and the explicit formula can also be used to obtain upper bounds on $\\dim {S}_{j,2}(\\Gamma _2)$ .",
"Indeed, keeping the notations in the above proof, and applying [6] (due to Taïbi), we get that the inequality $({\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{2j+2})+ {\\rm J}_{{\\rm F}_{\\lambda }}({\\rm I}_{0})) \\dim {S}_{j,2}(\\Gamma _2) \\le \\frac{2}{\\pi ^{2}} \\lambda $ holds for all $\\lambda >0$ .",
"When the parenthesis on left hand side is $> 0$ , which happens (for big enough $\\lambda $ ) for all $j \\le 138$ , we obtain an explicit upper bound for $\\dim {S}_{j,2}(\\Gamma _2)$ .",
"For instance, we get $\\dim {S}_{j,2}(\\Gamma _2) \\le 1$ for all $j < 54$ and $\\dim {S}_{j,2}(\\Gamma _2) \\le 2$ for all $j < 66$ (choose respectively $\\lambda =5$ and $\\lambda =6$ ).",
"Our last and main result concerns the kernel $\\Gamma _2[2]$ of the reduction ${\\rm Sp}_4(\\mathbb {Z}) \\rightarrow {\\rm Sp}_4(\\mathbb {Z}/2\\mathbb {Z})$ .",
"Theorem 1.5 We have ${S}_{j,1}(\\Gamma _2[2])=0$ for any $j$ .",
"Our proof will be an elaboration of the one of Proposition REF .",
"We shall also use the vanishing ${S}_{j,1}(\\Gamma _2[2])=0$ for $j\\le 8$ , proved by Cléry and van der Geer in this paper (Proposition REF ).",
"As we have $-1 \\in \\Gamma _2[2]$ we may also assume $j$ is even.",
"Let us denote by $J$ the principal congruence subgroup of ${\\rm PGSp}_4(\\mathbb {Z}_2)$ and by $\\mathbb {A}$ the adele ring of $\\mathbb {Q}$ ; we easily check ${\\rm PGSp}_4(\\mathbb {A}) = {\\rm PGSp}_4(\\mathbb {Q}) \\cdot ({\\rm PGSp}_4(\\mathbb {R})^0 \\times J \\times \\prod _{p\\ne 2} {\\rm PGSp}_4(\\mathbb {Z}_p)).$ Moreover, classical arguments show that we have $S_{j,1}(\\Gamma _2[2])=0$ if, and only if, there is no cuspidal automorphic representation $\\pi $ of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ which is unramified at every odd prime, such that $\\pi _2$ has a non-zero invariant under $J$ , and with $\\pi _\\infty \\simeq {\\rm U}_{j,1}$ .",
"Thus we fix such a $\\pi $ and consider $\\pi ^{\\rm GL}$ , as well as the associated collection $(d_i,n_i,\\pi _i)_{i \\in I}$ , given by (b) above.",
"By the same argument as in the case $\\Gamma =\\Gamma _2$ , there exists $i \\in I$ with $d_i \\ge 2$ .",
"If we have $n_i=1$ , which forces ${\\rm inf}\\, \\pi _i = \\lbrace 0\\rbrace $ , we must have $d_i=2$ and $j \\in \\lbrace 0,2\\rbrace $ by the shape of ${\\rm inf}\\, {\\rm U}_{j,1}$ .",
"But this is a contradiction as ${S}_{j,1}(\\Gamma _2[2])=0$ for $j=0,2$ .",
"So we must have $n_i=d_i=2$ , $I=\\lbrace i\\rbrace $ , and $\\pi _i$ is orthogonal with ${\\rm inf} (\\pi _i)_\\infty = \\lbrace -j/2,j/2\\rbrace $ .",
"The classification of orthogonal cuspidal automorphic representations of ${\\rm GL}_2$ over $\\mathbb {Q}$ , a very special case of Arthur's results, is well-known.",
"First of all, the central character of such a representation has order 2, hence corresponds to some uniquely defined quadratic extension $K$ of $\\mathbb {Q}$ .",
"Moreover, for any Hecke character $\\chi $ of $K$ which is trivial on the idele group of $\\mathbb {Q}$ , and with $\\chi ^2 \\ne 1$ , the automorphic induction of $\\chi $ to $\\mathbb {Q}$ is an orthogonal cuspidal automorphic representation of ${\\rm GL}_2$ over $\\mathbb {Q}$ that we shall denote by ${\\rm ind}(\\chi )$ .",
"It turns out that they all have this form, and that we have moreover ${\\rm ind}(\\chi ) \\simeq {\\rm ind}(\\chi ^{\\prime })$ if, and only if, we have $\\chi =\\chi ^{\\prime }$ or $\\chi ^{-1} = \\chi ^{\\prime }$ .",
"Last but not least, to any $\\chi $ as above Arthur associates a global packet $\\Pi (\\chi )=\\bigotimes ^{\\prime }_v \\Pi _v(\\chi _v)$ of irreducible admissible representations of ${\\rm PGSp}_4$ over $\\mathbb {Q}$ , whose elements are exactly the discrete automorphic representations $\\omega $ which satisfy $\\omega ^{\\rm GL} = {\\rm ind}(\\chi ) \\otimes |.|^{1/2} \\boxplus {\\rm ind}(\\chi ) \\otimes |.|^{-1/2}$ (Soudry type); in this “stable case” any element of $\\Pi (\\chi )$ is automorphic by Arthur's multiplicity formula.",
"Going back to our specific situation, let $K$ and $\\chi $ be such that $\\pi _i \\simeq {\\rm ind}(\\chi )$ .",
"Since $\\pi _i$ is unramified outside 2, then so is $K$ and we necessarily have $K=\\mathbb {Q}(\\sqrt{d})\\, \\, \\, {\\rm with}\\, \\, \\, d\\,\\in \\lbrace -2,\\,-1,\\,2\\rbrace .$ We first claim $d \\ne 2$ .",
"Indeed, a Hecke character of real quadratic field has the form $|.|^{s_0} \\chi _0$ with $\\chi _0$ a finite order character and $s_0 \\in \\mathbb {C}$ .",
"We would thus have $\\lbrace s_0,s_0\\rbrace = {\\rm inf} (\\pi _i)_\\infty =\\lbrace j/2,-j/2\\rbrace $ , which implies $s_0=j=0$ , which is again absurd.",
"So $K$ is imaginary quadratic.",
"As $\\chi _\\infty $ is trivial on $\\mathbb {R}^\\times $ by assumption on $\\chi $ , and up to replacing $\\chi $ by $\\chi ^{-1}$ if necessary, the shape of ${\\rm inf}\\, (\\pi _i)_\\infty $ implies then $\\chi _\\infty (z) = (z/\\overline{z})^{j/2}$ .",
"Here comes the main trick.",
"Let $\\eta $ be the Hecke character of the statement of Lemma REF below, and set $w_K=|\\mathcal {O}_K^\\times |$ .",
"We may assume $j\\ge 2$ , so there are unique integers $r$ and $j^{\\prime }/2$ , with $r\\ge 0$ and $1 \\le j^{\\prime }/2 \\le w_K$ , such that $j/2 \\,=\\,r \\,w_K \\, +\\, j^{\\prime }/2$ .",
"Consider the Hecke character $\\chi ^{\\prime } = \\chi \\eta ^{-r}$ of $K$ .",
"It is obviously trivial on the idele group of $\\mathbb {Q}$ , and it satisfies $(\\chi ^{\\prime })^2 \\ne 1$ as we have $\\chi ^{\\prime }_\\infty (z)= (z/\\overline{z})^{j^{\\prime }/2}$ with $j^{\\prime }>0$ .",
"Consider now the packet $\\Pi (\\chi ^{\\prime })$ .",
"As we have $\\chi _2=\\chi ^{\\prime }_2$ , the local Arthur packets $\\Pi _2(\\chi _2)$ and $\\Pi _2(\\chi ^{\\prime }_2)$ do coincide.",
"As $\\pi $ belongs to $\\Pi (\\chi )$ , its local component $\\pi _2$ also belongs to $\\Pi _2(\\chi _2)$ .",
"We may thus consider a representation $\\pi ^{\\prime }=\\bigotimes ^{\\prime }_v \\pi ^{\\prime }_v$ in $\\Pi (\\chi ^{\\prime })$ with $\\pi ^{\\prime }_2\\simeq \\pi _2$ , with $\\pi ^{\\prime }_p$ unramified for all odd prime $p$ (since $\\chi ^{\\prime }_p$ is unramified for such a $p$ ), and with $\\pi ^{\\prime }_\\infty \\simeq {\\rm U}_{j^{\\prime },1}$ .",
"This last property holds because the Langlands packet associated to the Arthur packet $\\Pi _\\infty (\\chi ^{\\prime }_\\infty )$ , which is included in $\\Pi _\\infty (\\chi ^{\\prime }_\\infty )$ by [1], is the one of ${\\rm U}_{j^{\\prime },1}$ by Remark (a) above.",
"As already explained, the representation $\\pi ^{\\prime }$ is discrete automorphic by Arthur, and even cuspidal as its Archimedean component is tempered.",
"As $\\pi ^{\\prime }_2 \\simeq \\pi _2$ has non-zero invariants under the principal congruence subgroup $J$ of ${\\rm PGSp}_4(\\mathbb {Z}_2)$ , it follows that $\\pi ^{\\prime }$ is generated by an element in ${S}_{j^{\\prime },1}(\\Gamma _2[2])$ by the first paragraph above.",
"But now we have the inequality $j^{\\prime } \\,\\le \\,2 \\,w_K \\,\\le \\,8$ , a contradiction by the vanishing ${S}_{j^{\\prime },1}(\\Gamma _2[2])=0$ for $j^{\\prime }\\le 8$ .",
"Lemma 1.6 Let $K=\\mathbb {Q}(\\sqrt{d}) \\subset \\mathbb {C}$ with $d=-1,-2$ and set $w_K=|\\mathcal {O}_K^\\times |$ .",
"There is a Hecke character $\\eta $ of $K$ which is unramified outside $\\lbrace 2,\\infty \\rbrace $ and which satisfies $\\eta _2=1$ and $\\eta _\\infty (z)=(z/\\overline{z})^{w_K}$ for all $z \\in K_\\infty ^\\times $ .",
"Moreover, $\\eta $ is trivial on the idele group of $\\mathbb {Q}$ .",
"Denote by $\\mathbb {A}_F$ the adele ring of the number field $F$ .",
"As $\\mathcal {O}_K$ has class number 1 we have the decomposition $\\mathbb {A}_K^\\times = K^\\times \\cdot (K_\\infty ^\\times \\times K_2^\\times \\times \\prod _{v\\ne 2,\\infty } {\\mathcal {O}^\\times _{K_v}})$ .",
"This implies first the (unrequired) uniqueness of $\\eta $ , and shows that its existence is equivalent to the fact that the morphism $z \\mapsto (z/\\overline{z})^{w_K}, \\mathbb {C}^\\times \\rightarrow \\mathbb {C}^\\times $ , is trivial on the subgroup $(\\mathcal {O}_K[1/2])^\\times $ .",
"This is indeed the case as this latter group is generated by $\\mathcal {O}_K^\\times $ and by some element $\\pi \\in \\mathcal {O}_K$ with norm 2 and which satisfies $\\pi /\\overline{\\pi } \\in \\mathcal {O}_K^\\times $ .",
"The last assertion follows from the equality $\\mathbb {A}_\\mathbb {Q}^\\times = \\mathbb {Q}^\\times \\cdot (\\mathbb {R}^\\times \\times \\prod _p \\mathbb {Z}_p^\\times )$ and the properties of $\\eta $ ."
],
[
"Proof of Theorem ",
"In this second appendix, we explain how to deduce Theorem REF stated in the paper of Cléry and van der Geer from the works of Rösner [20] and Weissauer [26], for the convenience of the reader.",
"We first recall some results on Yoshida lifts taken from Weissauer's work [26].",
"Let us fix $f$ and $g$ two non-proportional elliptic eigen newforms of same even weight $j+2$ , and let us denote by $\\pi $ and $\\pi ^{\\prime }$ the (distinct) cuspidal automorphic representations of ${\\rm GL_2}(\\mathbb {A})$ that they generate, $\\mathbb {A}$ being the adele ring of $\\mathbb {Q}$ .",
"In particular, $\\pi _\\infty $ and $\\pi ^{\\prime }_\\infty $ are isomorphic discrete series, and we may assume that $\\pi $ and $\\pi ^{\\prime }$ are normalized such that this discrete series has trivial central character (as $j$ is even).",
"For Yoshida lifts $Y(f,g)$ of $f$ and $g$ to exist, we also need to assume that $f$ and $g$ have the same nebentypus, i.e.",
"that $\\pi $ and $\\pi ^{\\prime }$ have the same central character.",
"Let $\\Pi (\\pi ,\\pi ^{\\prime })=\\bigotimes ^{\\prime }_v \\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ be the restricted tensor product, over all the places $v$ of $\\mathbb {Q}$ , of the local $L$ -packet $\\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ of irreducible admissible representations of ${\\rm GSp}_4(\\mathbb {Q}_v)$ associated to the pair $\\lbrace \\pi _v,\\pi ^{\\prime }_v\\rbrace $ by Weissauer.",
"For each place $v$ of $\\mathbb {Q}$ , this local $L$ -packet has either 1 or 2 elements, including a unique generic element; it has another element if, and only if, both $\\pi _v$ and $\\pi ^{\\prime }_v$ are discrete series [26] [20].",
"In particular, the (Langlands) Archimedean packet $\\Pi _\\infty (\\pi _\\infty ,\\pi ^{\\prime }_\\infty )$ has two elements, the non-generic one being the holomorphic limit of discrete series ${\\rm U}_{j,2}$ recalled in appendix .",
"Also, if both $\\pi $ and $\\pi ^{\\prime }$ are unramified at the finite place $v$ then $\\Pi _v(\\pi _v,\\pi ^{\\prime }_v)$ is a singleton (thus $\\Pi (\\pi ,\\pi ^{\\prime })$ is finite).",
"The multiplicity formula proved by Weissauer [26] states that an element $\\varpi $ of $\\Pi (\\pi ,\\pi ^{\\prime })$ is discrete automorphic if, and only if, there is an even number of places $v$ such that $\\varpi _v$ is non-generic.",
"He also shows that such an element has multiplicity one in the discrete spectrum of ${\\rm GSp}_4$ ; it is necessarily cuspidal as the two elements of $\\Pi _\\infty (\\pi _\\infty ,\\pi ^{\\prime }_\\infty )$ are tempered.",
"Let us denote by $J \\subset {\\rm GSp}_4(\\mathbb {Z}_2)$ the principal congruence subgroup.",
"Some classical arguments show that the Yoshida lifts $Y(f,g)$ which belong to the space $YS_{j,2}^{s[w]}$ of the statement are in natural bijection with certain vectors of the finite part of the cuspidal automorphic representations $\\varpi $ in $\\Pi (\\pi ,\\pi ^{\\prime })$ having the following properties : (i) $\\varpi _\\infty \\simeq {\\rm U}_{j,2}$ , (ii) $\\varpi _p^{{\\rm GSp}_4(\\mathbb {Z}_p)} \\ne 0$ (in which case we have $\\, \\dim \\, \\varpi _p^{{\\rm GSp}_4(\\mathbb {Z}_p)}=1$ ), (iii) the $s[w]$ -isotypic component $\\varpi _2^{J}$ is non-zero.",
"More precisely, the Yoshida lifts $Y(f,g)$ corresponding to such a $\\varpi $ form a linear subspace $YS_{j,2}^{s[w]}[\\varpi ] \\subset YS_{j,2}^{s[w]}$ isomorphic to the $s[w]$ -isotypic component of $\\varpi _2^{J}$ as an $S_6$ -representation.",
"The space $YS_{j,2}^{s[w]}$ of the statement is then the direct sum of its subspaces $YS_{j,2}^{s[w]}[\\varpi ]$ where $f,g$ and $\\varpi $ vary, with $\\varpi $ cuspidal automorphic satisfying (i), (ii) and (iii).",
"We still fix elliptic newforms $f$ and $g$ as above, hence $\\pi $ and $\\pi ^{\\prime }$ as well.",
"By [20], we know first that $\\Pi _p(\\pi _p,\\pi ^{\\prime }_p)$ has an element with non-zero invariants under ${\\rm GSp}_4(\\mathbb {Z}_p)$ if, and only if, both $\\pi _p$ and $\\pi ^{\\prime }_p$ are unramified.",
"Moreover, the same corollary asserts that if $\\Pi _2(\\pi _2,\\pi ^{\\prime }_2)$ has an element with non-zero $J$ -invariants, then both $\\pi _2$ and $\\pi ^{\\prime }_2$ have non-zero invariants under the principal congruence subgroup of ${\\rm GL}_2(\\mathbb {Z}_2)$ .",
"As a first consequence, if $\\varpi \\in \\Pi (\\pi ,\\pi ^{\\prime })$ does satisfy (ii) and (iii) then $f$ and $g$ are newforms on $\\Gamma _0(N)$ with $N|4$ (and both $\\pi $ and $\\pi ^{\\prime }$ have a trivial central character).",
"Moreover, by the statement recalled above concerning the multiplicity formula (\"even parity of the number of non-generic places\"), there is at most one cuspidal automorphic $\\varpi \\in \\Pi (\\pi ,\\pi ^{\\prime })$ satisfying (i), (ii) and (iii) above, and it has the property that $\\varpi _2$ is the non-generic element of $\\Pi _2(\\pi _2,\\pi ^{\\prime }_2)$ .",
"In particular, this latter $L$ -packet has two elements and both $\\pi _2$ and $\\pi ^{\\prime }_2$ are discrete series of ${\\rm GL}_2(\\mathbb {Q}_2)$ (thus neither $f$ nor $g$ can have level 1).",
"By Rösner [20], there are only three possibilities for the isomorphism class of a representation of ${\\rm PGL}_2(\\mathbb {Q}_2)$ generated by an elliptic newform of level $\\Gamma _0(2)$ or $\\Gamma _0(4)$ , namely the Steinberg representation St and its unramified quadratic twist ${\\rm St}^{\\prime }$ in level $\\Gamma _0(2)$ , and a certain supercuspidal representation ${\\rm Sc}$ in level $\\Gamma _0(4)$ .",
"In particular, there are all discrete series.",
"Assume now that $\\sigma $ and $\\sigma ^{\\prime }$ are (possibly equal) elements of the set $\\lbrace {\\rm St}, {\\rm St^{\\prime }}, {\\rm Sc}\\rbrace $ and let $\\tau $ be the non-generic element of $\\Pi _2(\\sigma ,\\sigma ^{\\prime })$ .",
"View the finite dimensional vector space $\\tau ^J$ as a representation of $S_6$ .",
"In order to prove the theorem it only remains to show that either $\\tau ^J$ is 0 or we are in exactly one of the following situations : (a) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm St},{\\rm St^{\\prime }}\\rbrace $ and $\\tau ^J \\simeq s[1^6]$ , (b) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm Sc}\\rbrace $ and $\\tau ^J \\simeq s[2,1^4]$ , (c) $\\lbrace \\sigma ,\\sigma ^{\\prime }\\rbrace =\\lbrace {\\rm St}\\rbrace $ or $\\lbrace {\\rm St^{\\prime }}\\rbrace $ and $\\tau ^J \\simeq s[2^3]$ .",
"This is a delicate analysis which fortunately has been carried out by Rösner.",
"Indeed, this is exactly the content of the left-bottom part of [20] with $q=2$ .",
"This table shows that there are just three non-zero possible representations for $\\tau ^J$ , denoted by $\\theta _2$ , $\\theta _5$ and $\\chi _9(1)$ there but which correspond respectively to the representations $s[2^3]$ , $s[1^6]$ and $s[2,1^4]$ by Rösner's other table [20], exactly according to the three cases above (read the table with $\\Pi _1={\\rm Sc}$ , $\\xi _\\mu {\\rm St} = {\\rm St}^{\\prime }$ , and take for $\\mu $ either the trivial character of $\\mathbb {Q}_2^\\times $ or its unramified quadratic character).",
"$\\square $ Acknowledgement.",
"Gaëtan Chenevier is supported by the C.N.R.S.",
"and by the project ANR-14-CE25 (PerCoLaTor).",
"He would like to thank Fabien Cléry and Gerard van der Geer for inviting him to include these two appendices to their paper.",
"Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France.",
"E-mail address: [email protected]"
]
] | 1709.01748 |
[
[
"Index of Dirac operators and classification of topological insulators"
],
[
"Abstract Real and complex Clifford bundles and Dirac operators defined on them are considered.",
"By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard.",
"Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained.",
"This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems."
],
[
"Introduction",
"The recent discovery of new topological phases of matter has extended the connections between condensed matter physics and topology , , , .",
"The prominent examples of topological phases are topological insulators and they correspond to bulk insulating and edge conducting materials.",
"Topology of the bulk implies the edge conductivity and this differentiates topological insulators from trivial insulators.",
"Besides the classical integer quantum Hall effect, Chern insulators and quantum spin Hall effect are also realized experimentally as examples of topological insulators , .",
"Moreover, topological superconductors can also be described as bulk insulating and edge conducting materials that exemplifies the topological phases of matter .",
"This bulk/edge correspondence in topological materials is a manifestation of the holography principle in condensed matter physics.",
"The existence of topological phases depends on the symmetries of the Hamiltonian and the time-reversal (T) and charge conjugation (C) symmetries play prominent roles in the classification of topological materials.",
"Topological insulators and superconductors can be characterized by the relevant topological invariants defined from the eigenstates of the corresponding Hamiltonians.",
"There are two types of topological materials depending on the group that the topological invariants take values which are $\\mathbb {Z}$ -insulators (Chern insulators) and $\\mathbb {Z}_2$ -insulators.",
"$\\mathbb {Z}$ -invariants are generally characterized by Chern numbers or winding numbers defined from the Berry curvature of the system and $\\mathbb {Z}_2$ -invariants correspond to Kane-Mele invariants or Chern-Simons invariants , .",
"Classification of topological phases in different dimensions and symmetry classes has been obtained by using Clifford algebras and K-theory , , , , , , .",
"Because of the Bott periodicity of K-groups, the topological phases can be described by a finite periodic table.",
"The rows of the table correspond to the ten different symmetry classes of Hamiltonians which are classified in and the columns describe different dimensions.",
"The periodic table of topological insulators and superconductors shows that there are five different topological symmetry classes in each dimension and three of them are $\\mathbb {Z}$ -insulators and two of them are $\\mathbb {Z}_2$ -insulators.",
"Since the symmetry classes of Hamiltonians can be related to Cartan symmetric spaces, the periodic table can be divided into two parts that characterize the complex and real classes separately.",
"Besides Clifford algebras and K-theory, the periodic table can also be obtained by analyzing the stability of gapless boundary states against perturbations or by the dimensional reduction method of massive Dirac Hamiltonians .",
"Although the relations between Clifford algebras, K-groups and topological phases are known, the origin of the periodic table in mathematical terms is not discussed extensively in the literature.",
"In this paper, we show that the periodic table of topological insulators and superconductors can be obtained directly from the Clifford chessboard of Clifford algebras and the index theorems of Dirac operators.",
"We start by turning the Clifford chessboard to the table of Clifford bundles and defining relevant Dirac operators on these bundles.",
"Index theorems for Dirac operators relates the analytic index to the topological index , , and we find the table of indices of Dirac operators in that way.",
"From the relations between index theorems, K-theory groups, Grothendieck groups of Clifford algebra representations and symmetric spaces, we obtain the periodic table of topological phases for all dimensions and symmetry classes.",
"These steps are applied both to real and complex classes separately.",
"Instead of starting with the symmetry classes of free-fermion Hamiltonians and obtaining the topological classes by using Clifford algebras and K-theory which is done in the literature , , , , we start from the Clifford chessboard and find the symmetry classes of Hamiltonians and topological phases by using the index theorems of Dirac operators.",
"This approach gives the different topological classes in all dimensions and relates the topological invariants in these classes with the indices of Dirac operators.",
"The paper is organized as follows.",
"In Section 2, the mathematical concepts discussed in the subsequent sections are connected to the properties of the physical systems that describe topological materials.",
"Section 3 includes the periodicity relations of Clifford algebras and the construction of the Clifford chessboard.",
"In Section 4, we obtain the periodic table of real topological phases by starting with the table of Clifford bundles and by using the index theorems of Dirac operators and relations with K-groups, Grothendieck groups and symmetric spaces.",
"Section 5 deals with the problem of obtaining the periodic table of complex topological phases with the methods discussed in Section 4.",
"Section 6 concludes the paper."
],
[
"Dirac Hamiltonians of Topological Materials",
"In this section, the mathematical concepts used in the subsequent sections to obtain the classification table of topological insulators are connected to the properties of real physical systems via their Hamiltonians and eigenstates.",
"The free-fermion Hamiltonians are classified in ten different symmetry classes depending on the existence or non-existence of anti-unitary symmetries such as time-reversal and charge-conjugation .",
"These so-called Cartan classes represent the defining Hamiltonians of free-fermion systems.",
"The low energy effective Hamiltonians of topological insulators and superconductors in all dimensions and symmetry classes are characterized by Dirac-like Hamiltonians.",
"They can be written generally in terms of the momentum variables $\\bf {k}$ in the following form , $H(\\bf {k})=\\bf {d}(\\bf {k}).\\bf {\\sigma }$ where $\\bf {d}(\\bf {k})$ denotes the functions written in terms of $\\bf {k}$ and $\\bf {\\sigma }$ are Clifford algebra generators in relevant dimensions.",
"For example, the two dimensional Haldane model of graphene , which is a Chern insulator characterized by a $\\mathbb {Z}$ topological invariant, has the following Bloch Hamiltonian $H({\\bf {k}})=\\sum _{i=1}^3\\left\\lbrace 2t_2\\cos (\\phi )\\cos ({\\bf {k}}.",
"{\\bf {b}}_i)\\sigma _0+t_1\\left[\\cos ({\\bf {k}}.",
"{\\bf {a_i}})\\sigma _1+\\sin ({\\bf {k}}.",
"{\\bf {a}}_i)\\sigma _2\\right]+\\left[\\frac{M}{3}-2t_2\\sin (\\phi )\\sin ({\\bf {k}}.",
"{\\bf {b}}_i)\\right]\\sigma _3\\right\\rbrace $ where $t_1$ and $t_2$ are nearest neighbour and next nearest neighbour hopping parameters, $\\phi $ is the Haldane phase and ${\\bf {a}}_i$ and ${\\bf {b}}_i$ are nearest neighbour and next nearest neighbour displacement vectors.",
"$M$ is the on-site energy and $\\sigma _i$ are Pauli matrices that satisfy the Clifford algebra relations.",
"At the low energy limit around the ${\\bf {K}}$ point, this Hamiltonian reduces to the following continuum limit $H({\\bf {K}})=-3t_2\\cos (\\phi )\\sigma _0+\\frac{3}{2}t_1\\left(\\kappa _2\\sigma _1-\\kappa _1\\sigma _2\\right)+\\left(M-3\\sqrt{3}t_2\\sin (\\phi )\\right)\\sigma _3$ where $\\kappa _i$ are the momentum space Pauli matrices.",
"This Hamiltonian is in the form of a Dirac Hamiltonian which corresponds to a Dirac operator written in terms of the Clifford algebra basis $\\sigma _i$ .",
"Similarly, the two-dimensional time-reversal invariant $\\mathbb {Z}_2$ insulator which is characterized by the Kane-Mele model has the following form of Bloch Hamiltonian $H({\\bf {k}})=\\sum _{i=1}^5d_i({\\bf {k}})\\Gamma _i+\\sum _{i<j=1}^5d_{ij}({\\bf {k}})\\Gamma _{ij}$ where $\\Gamma _i$ and $\\Gamma _{ij}=\\frac{1}{2i}[\\Gamma _i,\\Gamma _j]$ are Clifford algebra generators which are defined in terms of sublattice Pauli matrices $\\sigma _i$ and spin Pauli matrices $s_i$ as follows $\\Gamma _{1,2,3,4,5}=\\left(\\sigma _1\\otimes s_0, \\sigma _3\\otimes s_0, \\sigma _2\\otimes s_1, \\sigma _2\\otimes s_2, \\sigma _2\\otimes s_3\\right).$ The functions $d_i({\\bf {k}})$ and $d_{ij}({\\bf {k}})$ are given by $d_1({\\bf {k}})&=&t\\left[1+2\\cos \\left(\\frac{k_x}{2}\\right)\\cos \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\right]\\nonumber \\\\d_2({\\bf {k}})&=&M\\nonumber \\\\d_3({\\bf {k}})&=&\\lambda _R\\left[1-\\cos \\left(\\frac{k_x}{2}\\right)\\cos \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\right]\\nonumber \\\\d_4({\\bf {k}})&=&-\\sqrt{3}\\lambda _R\\sin \\left(\\frac{k_x}{2}\\right)\\sin \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\\\d_{12}({\\bf {k}})&=&-2t\\cos \\left(\\frac{k_x}{2}\\right)\\sin \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\nonumber \\\\d_{15}({\\bf {k}})&=&\\lambda _{SO}\\left[2\\sin (k_x)-4\\sin \\left(\\frac{k_x}{2}\\right)\\cos \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\right]\\nonumber \\\\d_{23}({\\bf {k}})&=&-\\lambda _R\\cos \\left(\\frac{k_x}{2}\\right)\\sin \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\nonumber \\\\d_{24}({\\bf {k}})&=&\\sqrt{3}\\lambda _R\\sin \\left(\\frac{k_x}{2}\\right)\\cos \\left(\\frac{\\sqrt{3}k_y}{2}\\right)\\nonumber $ where $t$ is the nearest neighbour hopping parameter and $\\lambda _{SO}$ and $\\lambda _R$ are spin-orbit and Rashba coupling parameters, respectively.",
"One can easily see that the continuum limit of Hamiltonian (4) corresponds to a sum of two Haldane model Dirac Hamiltonians with different spins.",
"So, we again have a corresponding Dirac operator written in terms of the Clifford algebra basis given in (5).",
"It is also known that the three-dimensional models that describe topological materials again have effective Dirac Hamiltonians written in terms of relevant Clifford algebra basis.",
"Indeed, effective Dirac Hamiltonians are characteristic properties of physical systems that correspond to topological insulators and superconductors in all dimensions .",
"Although the real physical systems arise in two and three dimensions, the mathematical structures describing Dirac Hamiltonians and topological materials can be generalized to all higher dimensional systems.",
"Depending on the symmetry classes of physical models determined by the existence or non-existence of anti-unitary symmetries such as time-reversal, charge conjugation and chiral symmetries, the Clifford algebra generators appearing in the effective Dirac Hamiltonians can be positive or negative generators of the algebra.",
"So, the symmetry properties of the Hamiltonians determine the properties of the Clifford algebra generators.",
"On the other hand, the eight-fold periodicity of real Clifford algebras and the two-fold periodicity of complex Clifford algebras which are given in (16) simplify the dimensional pattern of Dirac Hamiltonians for higher dimensional topological materials.",
"This means that in real class topological materials, the symmetry properties of Dirac Hamiltonians for different dimensions and symmetry classes repeat itself after eight dimensions.",
"Similarly, for complex class topological materials, the properties of Dirac Hamiltonians repeat itself after two dimensions.",
"Namely, the properties of all higher dimensional models of topological materials can be deduced from the lower dimensional systems.",
"In condensed matter systems, the electronic states are described by Bloch wave functions.",
"So, the eigenvalue equation for the above Hamiltonians can be written as $H({\\bf {k}})u_n({\\bf {k}})=E_n({\\bf {k}})u_n({\\bf {k}})$ where $E_n(\\bf {k})$ are $n$ eigenvalues corresponding to $n$ eigenstates $u_n(\\bf {k})$ .",
"Indeed, $u_n(\\bf {k})$ correspond to the sections of a rank $n$ Hilbert bundle $E\\rightarrow B$ which is called Bloch bundle over the Brillouin zone $B$ of the system.",
"Since the Hamiltonians are constructed from the Clifford algebra generators, $u_n(\\bf {k})$ are elements of the representation space of Clifford algebras.",
"Thus, the bundle $E$ corresponding to the bundle of eigenstates of the Hamiltonian is in fact a Dirac bundle which is a bundle corresponding to a left module of the Clifford algebra as defined in Section 4.A and the Dirac Hamiltonians of topological materials correspond to the Dirac operators on this bundle.",
"Moreover, the anti-unitary symmetries that define the symmetry classes of Dirac Hamiltonians play the role of $Cl_k$ -actions that are defined in Section 4.A.",
"So, the Dirac bundles and Dirac operators defined in sections 4.A and 4.B correspond to these physical structures defined for topological materials in real classes and similarly the Dirac operators defined in section 5 correspond to complex class topological materials.",
"Index structure of Dirac-like Hamiltonians that appear in topological materials is compatible with the index theorems of Dirac operators.",
"For some special cases in low dimensions, it is shown that indices of Dirac-like Hamiltonians correspond to topological invariants that characterize non-trivial topological classes of topological insulators and superconductors .",
"The analysis in section 4.C gives a general relation between the indices of Dirac operators corresponding to Dirac Hamiltonians of topological materials and the topological invariants of topological insulators and superconductors in all dimensions and symmetry classes.",
"$\\mathbb {Z}$ -class topological invariants which are Chern and winding numbers and $\\mathbb {Z}_2$ -class topological invariants that are Kane-Mele and Chern-Simons invariants naturally arise from the indices of Dirac operators defined in (22) and (29).",
"So, the analysis gives correct periodic table of topological materials and shows that the topological invariants characterizing the topological classes come from the indices of the representing Dirac operators in all cases.",
"In addition, it also shows that the classification and periodic table of topological materials are originated from the Clifford algebras that are used in the construction of Dirac Hamiltonians of these materials.",
"Adding energy levels which do not cross the Fermi energy to the system defined in (1) does not give a non-trivial topological structure.",
"Namely, adding topologically trivial bands to the Bloch bundle does not change the topological class of the system.",
"So, for the Bloch bundle $E$ , we have an equivalence of bundles $E\\oplus F^n\\cong E\\oplus G^m$ where $F^n$ and $G^m$ are $n$ -dimensional and $m$ -dimensional trivial bundles, respectively.",
"This is the definition of the stable equivalence of bundles and stably equivalent bundles are classified by K-theory.",
"So, $KO$ -groups and $K$ -groups appearing in the analysis of sections 4.D and 5 are related to the Bloch bundles defined on the Brillouin zones of topological materials.",
"They show the stable equivalence classes of these Bloch bundles in different dimensions and symmetry classes.",
"However, there are two different cases for the Brillouin zones of topological materials; periodic lattices and continuum models.",
"The Brillouin zones of periodic lattices are described by $d$ -dimensional torus $T^d$ and for the continuum models they correspond to $d$ -dimensional spheres $S^d$ .",
"Continuum models are effective low energy large distance theories and as is stated above that the Dirac-like Hamiltonians are low energy effective Hamiltonians of topological materials.",
"For large $\\bf {k}$ , the energy bands generally have a trivial topological structure and by taking a one-point compactification at $\\bf {k}\\rightarrow \\infty $ , the Brillouin zone of the model has the topology of $S^d$ .",
"Since the analysis of K-theory groups in the paper is based on the bundles over $S^d$ , they only correspond to the continuum models of topological materials.",
"Classification scheme for periodic lattices is more complicated and may give a different periodic table from the one in the paper since the calculation of K-theory groups of $T^d$ can give different results than for $S^d$ .",
"Indeed, the K-groups of $T^d$ can be written as direct sums of K-groups of $S^d$ as is given in and they characterize the so-called weak topological insulators.",
"On the other hand, the continuum models and K-groups of $S^d$ determine the topological classes of strong topological insulators."
],
[
"Clifford algebras and periodicity relations",
"On a vector space $V$ with a quadratic form $q$ , the Clifford algebra $Cl(V,q)$ is generated by $V$ with the multiplication rule $x.y+y.x=-2q(x,y)$ for $x,y\\in V$ .",
"If we take the vector space $V=\\mathbb {R}^{p+q}$ with the quadratic form $Q(x)=q(x,x)$ which is given by $Q(x)=x_1^2+...+x_p^2-x_{p+1}^2-...-x_{p+q}^2$ then we denote the Clifford algebra on $\\mathbb {R}^{p+q}$ as $Cl_{p,q}\\equiv Cl(\\mathbb {R}^{p+q},Q)$ and it is generated by $p$ negative and $q$ positive generators.",
"In particular, if we consider any $Q$ -orthonormal bases $\\lbrace e_1,...e_{p+q}\\rbrace $ of $\\mathbb {R}^{p+q}$ , then $Cl_{p,q}$ is generated by $e_1,..., e_{p+q}$ that satisfy the following relation $e_i.e_j+e_j.e_i&=&\\left\\lbrace \\begin{array}{ll}-2\\delta _{ij}, & \\hbox{ for $i\\le p$} \\\\+2\\delta _{ij}, & \\hbox{ for $i>p$}\\end{array}\\right.$ The dimension of the Clifford algebra $Cl_{p,q}$ for $p+q=n$ is equal to $2^n$ .",
"As special cases, we denote $Cl_n\\equiv Cl_{n,0}\\nonumber \\\\Cl^*_n\\equiv Cl_{0,n}$ for the Clifford algebras with only negative and only positive generators, respectively.",
"An automorphism $\\eta :Cl_{p,q}\\rightarrow Cl_{p,q}$ can be defined on $Cl_{p,q}$ that gives the $\\mathbb {Z}_2$ -grading of the Clifford algebra; $Cl_{p,q}=Cl^0_{p,q}\\oplus Cl^1_{p,q}$ .",
"This corresponds to the superalgebra structure of the Clifford algebras and the even part $Cl^0_{p,q}$ constitute a subalgebra structure.",
"There is an algebra isomorphism between Clifford algebras and their even subalgebras which is written as $Cl_{p,q}\\cong Cl^0_{p+1,q}\\nonumber \\\\Cl_n\\cong Cl^0_{n+1}$ Other isomorphisms for Clifford algebras are $Cl_{p,q}\\cong Cl_{p-4,q+4}\\nonumber \\\\Cl_{p,q+1}\\cong Cl_{q,p+1}$ Moreover, some tensor product isomorphisms of Clifford algebras can also be stated as follows $Cl_{p,q+2}\\cong Cl_{q,p}\\otimes Cl_{0,2}\\nonumber \\\\Cl_{p+2,q}\\cong Cl_{q,p}\\otimes Cl_{2,0}\\\\Cl_{p+1,q+1}\\cong Cl_{p,q}\\otimes Cl_{1,1}\\nonumber $ On the other hand, if we choose $V=\\mathbb {C}^{p+q}$ , then we can define the complex Clifford algebras as the complexification of real Clifford algebras with complex quadratic form; $\\mathbb {C}l_{p,q}\\equiv Cl_{p,q}\\otimes _{\\mathbb {R}}\\mathbb {C}\\cong Cl(\\mathbb {C}^{p+q},Q\\otimes \\mathbb {C})$ .",
"Since there is no distinction between positive and negative generators in the complex case, we can simply denote the complex Clifford algebras as $\\mathbb {C}l_n$ .",
"The structure of real and complex Clifford algebras can be obtained from the following periodicity relations , $Cl_{p+8,q}&\\cong & Cl_{p,q}\\otimes Cl_{8,0}\\nonumber \\\\Cl_{p,q+8}&\\cong & Cl_{p,q}\\otimes Cl_{0,8}\\\\\\mathbb {C}l_{n+2}&\\cong & \\mathbb {C}l_n\\otimes _{\\mathbb {C}}\\mathbb {C}l_2\\nonumber $ As can be seen from the above isomorphisms, while the real Clifford algebras have eight-fold periodicity, the complex Clifford algebras have two-fold periodicity.",
"Indeed, Clifford algebras correspond to simple or semi-simple matrix algebras constructed from the division algebras $\\mathbb {R}$ , $\\mathbb {C}$ and $\\mathbb {H}$ .",
"From a simple observation of the Clifford algebra relation (11) satisfied by the generators of the Clifford algebra, the lower dimensional Clifford algebras can be obtained in terms of division algebras as follows $Cl_{1,0}&\\cong & \\mathbb {C}\\nonumber \\\\Cl_{0,1}&\\cong & \\mathbb {R}\\oplus \\mathbb {R}\\nonumber \\\\Cl_{1,1}&\\cong & \\mathbb {R}(2)\\nonumber \\\\Cl_{2,0}&\\cong & \\mathbb {H}\\nonumber \\\\Cl_{0,2}&\\cong & \\mathbb {R}(2)\\nonumber $ where $\\mathbb {R}(2)$ denotes the 2$\\times $ 2 matrix algebra with real elements.",
"These basic Clifford algebras and the periodicity relations in (14), (15) and (16) determine the corresponding division algebras of higher dimensional Clifford algebras as in the following table Table: NO_CAPTION Table: NO_CAPTION where $\\mathbb {R}(2^n)$ , $\\mathbb {C}(2^n)$ and $\\mathbb {H}(2^n)$ denotes the $2^n\\times 2^n$ matrix algebras that take values in $\\mathbb {R}$ , $\\mathbb {C}$ and $\\mathbb {H}$ , respectively.",
"Especially, we can write the Clifford algebras $Cl^*_n$ and $\\mathbb {C}l_n$ as follows Table: NO_CAPTIONFrom the considerations given above, one can construct a table of Clifford algebras which is called the Clifford chessboard in the following way Table: NO_CAPTION"
],
[
"Real Clifford bundles and real classes of periodic table",
"In this section, we consider vector bundles whose fibres correspond to the real Clifford algebras.",
"By considering Dirac operators and index theorems, we will obtain the periodic table of real classes in the classification of topological insulators and superconductors in several steps."
],
[
"$Cl_k$ -bundles and table of {{formula:7c30e545-f042-4f49-9a92-1549739b9d2f}}",
"On a manifold $M$ , the Clifford bundle $Cl(E)$ is defined as the bundle of Clifford algebras over $M$ .",
"If the fibers correspond to the real Clifford algebras, then we call it the real Clifford bundle.",
"The algebra structure of the space of sections of $Cl(E)$ arises from the fibrewise multiplication in $Cl(E)$ .",
"Moreover, a Dirac bundle over $M$ is a bundle $S$ of left modules of $Cl(M)$ with a Riemannian metric and a connection on it where $Cl(M)$ is the Clifford algebra defined on $M$ .",
"A special case of a Dirac bundle is the spinor bundle over $M$ whose fibres correspond to the spinor spaces as left modules of $Cl(M)$ .",
"The Clifford algebra $Cl_k$ denotes the real Clifford algebra with $k$ negative generators as defined in Section 3.",
"A $Cl_k$ -Dirac bundle over $M$ is a real Dirac bundle $S$ over $M$ with a right action $Cl_k\\hookrightarrow \\text{Aut}(S)$ which is parallel and commutes with multiplication by the elements of $Cl(M)$ .",
"So, a $Cl_k$ -Dirac bundle is a Dirac bundle with an extra multiplication by $Cl_k$ .",
"Because of the $\\mathbb {Z}_2$ -grading property of $Cl_k$ , the $Cl_k$ -Dirac bundle also carry a $\\mathbb {Z}_2$ -grading $S=S^0\\oplus S^1$ with a $\\mathbb {Z}_2$ -grading for the $Cl_k$ -action.",
"Similarly, one can define $Cl^*_k$ -Dirac bundles from the definition (4) whose fibres correspond to the left modules of Clifford algebras with $k$ positive generators.",
"Let us consider $Cl^*_k$ -Dirac bundles.",
"Since the real Clifford algebras have eight-fold periodicity as stated in (16), it is enough to consider $Cl^*_{k(\\text{mod }8)}$ .",
"We can construct a square table of $8\\times 8$ entries whose elements correspond to $Cl^*_{s-n (\\text{mod }8)}$ -Dirac bundles where $s,n=0,1,...,7$ Table: NO_CAPTIONThe table can be extended to bigger values of $s$ and $n$ , however, it repeats the pattern because of the eight-fold periodicity.",
"Moreover, it is equivalent to the table of Clifford chessboard because of the following isomorphism $Cl_{n,s}\\cong Cl^*_{s-n(\\text{mod }8)}.$ So, we obtain the table of $Cl^*_{s-n (\\text{mod }8)}$ -Dirac bundles from the Clifford chessboard of Clifford algebras."
],
[
"$Cl_k$ -Dirac operators",
"On a Dirac bundle $S$ on $M$ with sections $\\Gamma (S)$ , a first-order differential operator $D:\\Gamma (S)\\longrightarrow \\Gamma (S)$ called Dirac operator can be defined.",
"For the frame basis $\\lbrace X_a\\rbrace $ and co-frame basis $\\lbrace e^a\\rbrace $ , the Dirac operator can be written in terms of the connection $\\nabla $ and local coordinates as $D=e^a.\\nabla _{X_a}$ where $.$ denotes the Clifford product.",
"On a Riemannian manifold $M$ , the Dirac operator of any Dirac bundle is self-adjoint.",
"If the Dirac bundle is $\\mathbb {Z}_2$ -graded $S=S^0\\oplus S^1$ , then the Dirac operator can be written in the form $D=\\left(\\begin{array}{cc}0 & D^1 \\\\D^0 & 0 \\\\\\end{array}\\right)$ where $D^0:\\Gamma (S^0)\\longrightarrow \\Gamma (S^1)$ and $D^1:\\Gamma (S^1)\\longrightarrow \\Gamma (S^0)$ .",
"Since $D$ is self-adjoint, $D^0$ and $D^1$ are adjoints of each other and the index of $D^0$ is written as $\\text{ind}(D^0)=\\text{dim}(\\text{ker } D^0)-\\text{dim}(\\text{ker } D^1)$ where $\\text{ker }D^0$ denotes the kernel of $D^0$ .",
"For $Cl_k$ -Dirac bundles, we denote the Dirac operator as $\\displaystyle {\\lnot }D_k$ and it commutes with the $Cl_k$ -action which means that it is a $Cl_k$ -linear operator.",
"If the bundle is $\\mathbb {Z}_2$ -graded, then the $Cl_k$ -Dirac operator is $\\displaystyle {\\lnot }D_k=\\left(\\begin{array}{cc}0 & \\displaystyle {\\lnot }D^1_k \\\\\\displaystyle {\\lnot }D^0_k & 0 \\\\\\end{array}\\right)$ The index of the $Cl_k$ -Dirac operator is given by $\\text{ind}\\displaystyle {\\lnot }D^0_k=\\text{dim}(\\text{ker }\\displaystyle {\\lnot }D^0_k)-\\text{dim}(\\text{ker }\\displaystyle {\\lnot }D^1_k)$ The constructions for $Cl^*_k$ -Dirac bundles are similar.",
"By defining $Cl^*_k$ -Dirac operators on $Cl^*_k$ -Dirac bundles, we can construct the table of $Cl^*_{s-n}$ -Dirac operators from the table of $Cl^*_{s-n}$ -bundles defined in subsection 4.A as follows Table: NO_CAPTION"
],
[
"Index theorem for $Cl_k$ -Dirac operators",
"If the Dirac bundle $S$ is the Clifford bundle on $M$ , then the Dirac operator $D$ corresponds to the Hodge-de Rham operator and its square $D^2=\\Delta $ is the Hodge Laplacian.",
"The differential forms that are in the kernel of the Hodge Laplacian are called harmonic forms.",
"Similarly, if $S$ is the spinor bundle on $M$ , then $D$ is the usual Dirac operator on spinors and the spinors that are in the kernel of $D$ , namely the spinors that satisfy $D\\psi =0$ , are called harmonic spinors since $\\text{ker }D=\\text{ker }D^2$ for any compact Riemannian manifold $M$ .",
"These definitions can be extended to the case of $Cl_k$ -Dirac operators and we denote the kernel of $\\displaystyle {\\lnot }D_k$ , that is the harmonic space, as $\\textbf {H}_k=\\text{ker }\\displaystyle {\\lnot }D_k$ .",
"Now, we consider the index theorem for $Cl_k$ -Dirac operators.",
"The analytic index of $\\displaystyle {\\lnot }D_k$ defined in (21) can be written in terms of the dimension of the harmonic space $\\textbf {H}_k$ and characteristic classes of the bundle .",
"The Atiyah-Singer index theorem for $\\displaystyle {\\lnot }D_k$ gives the following equality $\\text{ind}(\\displaystyle {\\lnot }D_k)&=&\\left\\lbrace \\begin{array}{ll}\\text{dim}_{\\mathbb {C}}\\textbf {H}_k (\\text{mod }2), & \\hbox{ for $k\\equiv 1$ $(\\text{mod }8)$} \\\\\\text{dim}_{\\mathbb {H}}\\textbf {H}_k (\\text{mod }2), & \\hbox{ for $k\\equiv 2$ $(\\text{mod }8)$} \\\\\\frac{1}{2}\\widehat{A}(M), & \\hbox{ for $k\\equiv 4$ $(\\text{mod }8)$} \\\\\\widehat{A}(M), & \\hbox{ for $k\\equiv 0$ $(\\text{mod }8)$}\\end{array}\\right.$ where $\\text{dim}_{\\mathbb {C}}$ and $\\text{dim}_{\\mathbb {H}}$ denote the complex and quaternionic dimensions, respectively.",
"$\\widehat{A}(M)$ is the $\\widehat{A}$ -genus of the manifold $M$ and it is defined as a power series expansion $\\widehat{A}(M)=\\prod _{i=1}^n \\frac{x_i/2}{\\text{sinh}(x_i/2)}.$ It can be written in terms of the Pontrjagin classes $p_i$ as follows $\\widehat{A}(M)&=&1-\\frac{1}{24}p_1+\\frac{1}{5760}\\left(7p_1^2-4p_2\\right)+\\frac{1}{967680}\\left(-31p_1^3+44p_1p_2-16p_3\\right)+...$ $\\widehat{A}$ -genus is an integer number for compact manifolds and it is an even integer for the dimensions $4 (\\text{mod }8)$ .",
"So, the index of $\\displaystyle {\\lnot }D_k$ takes values in $\\mathbb {Z}_2$ for $k\\equiv 1 \\text{ and }2 (\\text{mod }8)$ and in $\\mathbb {Z}$ for $k\\equiv 0 \\text{ and }4 (\\text{mod}8)$ .",
"Then, we can write the table of $Cl^*_{s-n}$ -Dirac operators in subsection 4.B in terms of the index of $Cl^*_{s-n}$ -Dirac operators in the following way Table: NO_CAPTIONThe Atiyah-Singer index theorem in (22) attaches different topological invariants to $Cl^*_k$ -Dirac operators $\\displaystyle {\\lnot }D_k$ for different values of $k$ .",
"This means that we have non-trivial topological classes for different values of $s$ and $n$ which have non-zero index.",
"For the cases of zero index, there are only trivial classes.",
"In that way, we obtain a table of topological equivalence classes of $Cl^*_k$ -Dirac bundles from the Clifford chessboard by using the index theorems."
],
[
"Relation with $KO$ -groups",
"The Dirac operator $D$ on a real Dirac bundle is a self-adjoint operator with finite dimensional kernel and cokernel.",
"So, it is an element of the set of real Fredholm operators $\\mathfrak {F}_{\\mathbb {R}}$ .",
"The index of Fredholm operators is constant on connected components of $\\mathfrak {F}_{\\mathbb {R}}$ and we have the following isomorphism $\\text{ind}:\\pi _0(\\mathfrak {F}_{\\mathbb {R}})\\stackrel{\\simeq }{\\longrightarrow }\\mathbb {Z}$ Moreover, $\\mathfrak {F}_{\\mathbb {R}}$ is a classifying space for $KO$ -groups.",
"The stable equivalence classes of real vector bundles on a manifold $M$ are characterized by $KO$ -groups and they are real equivalences of the $K$ -groups of complex vector bundles.",
"For a Haussdorf space $X$ , index map defines an isomorphism $\\text{ind}:[X,\\mathfrak {F}_{\\mathbb {R}}]\\longrightarrow KO(X)$ where $[X,\\mathfrak {F}_{\\mathbb {R}}]$ denotes the homotopy classes of maps between $X$ and $\\mathfrak {F}_{\\mathbb {R}}$ and $KO(X)$ is the $KO$ -group on $X$ that classifies the stably equivalent real vector bundles on $X$ .",
"So, this gives an isomorphism between the homotopy groups of $\\mathfrak {F}_{\\mathbb {R}}$ and the $KO$ -groups at the point $\\text{ind}:\\pi _k(\\mathfrak {F}_{\\mathbb {R}})\\stackrel{\\simeq }{\\longrightarrow }KO^{-k}(\\text{pt})\\equiv \\widetilde{KO}(S^k)$ where $KO^{-k}(X)$ is equivalent to the reduced $KO$ -group of $k$ -fold suspension $\\Sigma ^kX$ of $X$ ; $KO^{-k}(X)\\equiv \\widetilde{KO}(\\Sigma ^kX)$ .",
"The reduced $KO$ -group $\\widetilde{KO}(X)$ is defined as the kernel of the map $KO(X)\\longrightarrow KO(\\text{pt})\\cong \\mathbb {Z}$ and $KO(\\text{pt})$ is the $KO$ -group at the point.",
"Since the $k$ -fold suspension of the point is equivalent to the $k$ -sphere $\\Sigma ^k(\\text{pt})\\equiv S^k$ , we have $KO^{-k}(\\text{pt})\\equiv \\widetilde{KO}(S^k)$ .",
"Let us denote the subset of Fredholm operators which are $\\mathbb {Z}_2$ -graded, $Cl_k$ -linear and self-adjoint as $\\mathfrak {F}_k$ .",
"Then, the index of the operators in $\\mathfrak {F}_k$ is constant on connected components of $\\mathfrak {F}_k$ $\\text{ind}:\\pi _0(\\mathfrak {F}_k)\\longrightarrow KO^{-k}(\\text{pt})$ So, $\\mathfrak {F}_k$ is a classifying space for $KO^{-k}$ and for a Haussdorf space $X$ and for any $k$ , there is an isomorphism $\\text{ind}:[X,\\mathfrak {F}_k]\\longrightarrow KO^{-k}(X)$ This means that the index of the $Cl_k$ -Dirac operator $\\displaystyle {\\lnot }D_k$ takes values in the group $KO^{-k}(\\text{pt})$ and we can write the table of the index of $Cl^*_{s-n}$ -Dirac operators in subsection 4.C as the table of $KO^{-(s-n)}(\\text{pt})$ groups Table: NO_CAPTIONSince the $KO$ -groups of the point is given as in the following table Table: NO_CAPTION we obtain the table of $KO^{-(s-n)}(\\text{pt})$ groups as Table: NO_CAPTIONand this is compatible with the fact that the index of $\\displaystyle {\\lnot }D_k$ takes values in $\\mathbb {Z}$ and $\\mathbb {Z}_2$ as stated in subsection 4.C."
],
[
"Isomorphism between $KO$ -groups and Grothendieck groups",
"Now, we consider the Atiyah-Bott-Shapiro (ABS) isomorphism which relates $KO$ -groups with the real representations of Clifford algebras .",
"Let us denote the group of equivalence classes of irreducible real representations of $Cl_k$ as $\\mathfrak {M}_k$ which is called the Grothendieck group.",
"This is the free abelian group generated by the distinct irreducible representations over $\\mathbb {R}$ .",
"For different values of $k$ , we have the Grothendieck groups as in the following table Table: NO_CAPTIONThe inclusion $i:\\mathbb {R}^k\\hookrightarrow \\mathbb {R}^{k+1}$ given by $i(x_1,...,x_k)=(x_1,...,x_k,0)$ induces an algebra homomorphism $i_*:Cl_k\\longrightarrow Cl_{k+1}$ .",
"By restricting the action from $Cl_{k+1}$ to $Cl_k$ , we obtain the homomorphism $i^*:\\mathfrak {M}_{k+1}\\longrightarrow \\mathfrak {M}_k$ .",
"So, we can define the groups $\\mathfrak {M}_k/i^*\\mathfrak {M}_{k+1}$ .",
"In this quotient group, a representation that can be obtained by restricting a representation of $Cl_{k+1}$ to $Cl_k$ is equivalent to zero.",
"Similarly, the Grothendieck group of real $\\mathbb {Z}_2$ -graded modules over $Cl_k$ is denoted by $\\widehat{\\mathfrak {M}}_k$ and there is the natural isomorphism $\\widehat{\\mathfrak {M}}_k=\\mathfrak {M}_{k-1}$ that comes from (13).",
"This implies the following isomorphism $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}\\cong \\mathfrak {M}_{k-1}/i^*\\mathfrak {M}_k$ From the groups $\\mathfrak {M}_k$ defined in the above table, we obtain $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}&\\cong &\\left\\lbrace \\begin{array}{ll}\\mathbb {Z}, & \\hbox{ for $k\\equiv 0$ \\text{or} $4(\\text{mod }8)$} \\\\\\mathbb {Z}_2, & \\hbox{ for $k\\equiv 1$ \\text{or} $2(\\text{mod }8)$} \\\\0, & \\hbox{ otherwise}\\end{array}\\right.$ where the $\\mathbb {Z}$ groups arise when $\\mathfrak {M}_{k-1}$ correspond to $\\mathbb {Z}\\oplus \\mathbb {Z}$ and the $\\mathbb {Z}_2$ groups arise when the dimension of the representation of $Cl_k$ is double of the dimension of the representation of $Cl_{k-1}$ .",
"The graded tensor product of modules gives a multiplication in $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}$ and this defines the graded ring $\\widehat{\\mathfrak {M}}_*/i^*\\widehat{\\mathfrak {M}}_{*+1}\\equiv \\bigoplus _{k\\ge 0}\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}$ .",
"ABS isomorphism defines an isomorphism between the $KO$ -groups at the point and the quotients of Grothendieck groups in the following way $\\phi :\\widehat{\\mathfrak {M}}_*/i^*\\widehat{\\mathfrak {M}}_{*+1}\\stackrel{\\simeq }{\\longrightarrow }KO^{-*}(\\text{pt})$ This can easily be seen from the definitions in this and previous subsections.",
"So, we can write the table of $KO^{-(s-n)}(\\text{pt})$ groups in subsection 4.D as the table of $\\widehat{\\mathfrak {M}}_{s-n}/i^*\\widehat{\\mathfrak {M}}_{s-n+1}$ groups as follows Table: NO_CAPTIONand the Clifford chessboard in section 3 turns into the table of quotients of Grothendieck groups."
],
[
"From Grothendieck groups to symmetric spaces",
"$KO$ -groups and Grothendieck groups are in relation with Cartan symmetric spaces through their connected components.",
"If we consider the following sequence of groups $O(16n)\\longrightarrow U(8n)\\longrightarrow Sp(4n)\\longrightarrow Sp(2n)\\times Sp(2n)\\longrightarrow Sp(2n)\\longrightarrow U(2n)\\longrightarrow O(2n)\\longrightarrow O(n)\\times O(n)\\longrightarrow O(n)$ and denote every element in the sequence by $G_i$ , then the coset spaces $R_i\\equiv G_i/G_{i+1}$ correspond to the eight of ten symmetric spaces of Cartan.",
"These symmetric spaces and the groups that give their connected components are given as follows Table: NO_CAPTIONAs can be seen from the last column of the table, the connected components of the symmetric spaces are in one-to-one correspondence with $KO$ -groups at the point and hence to the Grothendieck groups.",
"Indeed, Cartan symmetric spaces correspond to the classifying spaces of vector bundles and we have the following isomorphism for a Haussdorf space $X$ $KO^{-k}(X)\\cong [X,R_k]$ and so $KO^{-k}(\\text{pt})\\cong \\pi _0(R_k).$ From the ABS isomorphism in subsection 4.E, we also have $\\widehat{\\mathfrak {M}}_k/i^*\\widehat{\\mathfrak {M}}_{k+1}\\cong \\pi _0(R_k).$ Hence, we can write the table of the quotients of Grothendieck groups in subsection 4.E in terms of the connected components of the symmetric spaces as follows Table: NO_CAPTIONHowever, the homotopy groups of symmetric spaces have the property $\\pi _n(R_k)=\\pi _{n+1}(R_{k-1})$ , so we have $\\pi _0(R_{s-n(\\text{mod }8)})=\\pi _{8-n(\\text{mod }8)}(R_s)$ and the table turns into Table: NO_CAPTIONThis shows that every row in the table denoted by $s$ corresponds to a Cartan symmetric space $R_s$ and every column denoted by $n$ corresponds to the $(8-n)$ th homotopy group of the symmetric space."
],
[
"From symmetric spaces to periodic table",
"Eight symmetric spaces defined in the previous subsection are in one-to-one correspondence with the extensions of some real Clifford algebras.",
"By starting with a real Clifford algebra and answering the question how many different types of generators can be added to the existing ones gives the symmetric spaces .",
"The symmetric spaces and corresponding Clifford algebra extensions are given as follows Table: Conclusion"
]
] | 1709.01778 |
[
[
"Ergodic Exploration using Binary Sensing for Non-Parametric Shape\n Estimation"
],
[
"Abstract Current methods to estimate object shape---using either vision or touch---generally depend on high-resolution sensing.",
"Here, we exploit ergodic exploration to demonstrate successful shape estimation when using a low-resolution binary contact sensor.",
"The measurement model is posed as a collision-based tactile measurement, and classification methods are used to discriminate between shape boundary regions in the search space.",
"Posterior likelihood estimates of the measurement model help the system actively seek out regions where the binary sensor is most likely to return informative measurements.",
"Results show successful shape estimation of various objects as well as the ability to identify multiple objects in an environment.",
"Interestingly, it is shown that ergodic exploration utilizes non-contact motion to gather significant information about shape.",
"The algorithm is extended in three dimensions in simulation and we present two dimensional experimental results using the Rethink Baxter robot."
],
[
"Introduction",
"Tactile sensing is often associated with shape estimation [1], [2], [3], [4] and mapping problems [5] in conditions where visual sensing may be limited.",
"In some instances, tactile sensing is used to supplement vision-based sensing to improve shape estimates [6], [7].",
"The richness of touch as a sensing modality is underscored by the development of novel tactile sensors [8], [9], [10], [11] for use in a myriad of applications ranging from robot-assisted tumor detection [7], to texture recognition, and feature localization [12], [13].",
"These advances in tactile sensor technology require corresponding advances in active exploration algorithms and the interpretation of tactile-based sensor data.",
"Recent approaches for active exploration use random sampling-based search algorithms for sensor motion planning that require a separate controller for path following [14].",
"Other methods use task-specific probabilistic spatial methods for shape estimation that are updated based on sensor poses that minimize measurement uncertainty [2], [15], [16], [17], [18].",
"A common approach in the literature is motion planning for sensing followed by feedback regulation of the generated plan [19], [20], [18], [17].",
"Moreover, most methods focus on one object at a time.",
"In contrast, the presented work integrates planning and control into a feedback law.",
"As a result, the method uses sensor motion to actively sense for time-varying spatial information.",
"We take the framework in [21] and use the feedback law developed here to enable real-time execution during shape estimation.",
"Lastly, the feedback in the active sensing algorithm compensates for low resolution sensors and the specification of the algorithm is independent of the number of objects.",
"We show that ergodic exploration with Sequential Action Control (eSAC) can be used for active exploration with respect to time-varying tactile-based distributed information [22], [23].",
"Previous applications of ergodic theory utilize parametric measurement models for localization tasks [24], [21].",
"The current work demonstrates localization and estimation of non-parametric shapes using a binary sensor model with classification methods.",
"In contrast to other methods of active sensing for shape estimation [19], [25], [26], [16], [18], the proposed algorithm automatically encodes dynamical constraints without any overhead spatial discretization or motion planning.",
"In addition, the algorithm incorporates sensor measurement information to actively adjust shape estimates and synthesize tactile-information based control actions.",
"As a result, the algorithm automatically adjusts the control synthesis for multiple objects in an environment.",
"Notably, ergodic exploration uses non-contact motion data (sensor motion not in contact with an object) [27], [28] to improve the shape estimate.",
"The idea of utilizing free space is often found in other related works of pose estimation and tracking [27], [28], [29], [30] and is emphasized in our work.",
"As a final contribution, we show the algorithm is modular with respect to the choice of shape representation and tactile information distribution.",
"The paper outline is as follows: Section motivates the use of a binary contact sensor and outlines the implications for tactile sensing.",
"Section describes the robot control algorithm for active exploration and motivates ergodicity as a measure for exploration.",
"Section explains how shape estimation is achieved and how measurements update the shape estimate and the control policy.",
"Experimental and simulated results and conclusion are shown in sections and respectively."
],
[
"Tactile Information",
"In biological systems, tactile sensing is generally an active process that incorporates feedback from multiple environmental cues [31], [32].",
"Humans, for instance, use fingers to grasp objects for manipulation and feature detection.",
"Many rodents use hair-like appendages (whiskers) for tactile sensing [33], [34], [35].",
"These biological sensors have a large set of actuators and sensor channels, thus the process control for active sensing is complex.",
"We investigate a lower resolution version of tactile sensing that is simple to control for active sensing.",
"Specifically, we show that a binary form of tactile sensing (i.e., collision detection) [10], [8], has enough information for shape estimation, when combined with an active exploration algorithm that automatically takes into account regions of shape information.",
"A binary tactile measurement model consists of a transition state from “no collision” to “collision” and vice versa.",
"We denote this measurement model as $\\Upsilon (x) ={\\left\\lbrace \\begin{array}{ll}\\hfill 1, \\hfill & \\phi (x) \\le 0 \\\\\\hfill 0, \\hfill & \\phi (x) > 0 \\\\\\end{array}\\right.",
"}$ where $x$ is the sensor state and $\\phi (x)$ is a boundary function that determines a transition state if $\\phi (x) \\le 0$ (output of 1).",
"The goal of shape estimation is to determine $\\phi (x)$ ."
],
[
"Motivation",
"Typical algorithms used in active exploration and information maximization cast the problem of information acquisition as “exploratory” (wide spread search for diffuse information such as localization) or “exploitative” (direct search for highly structured information such as shape contours) [36], [37].",
"Ergodic exploration [22], [38], [23], [24], [21] is responsive to both diffuse information densities and highly focused information.",
"Thus, ergodic control seamlessly encodes wide spread coverage for diffuse information and localized search for focused information for both needs."
],
[
"Ergodic Metric",
"A trajectory $x(t)$ of a sensor is ergodic with respect to a probability (“target”) distribution $\\Phi (x)$ when the fraction of time the trajectory spends in an area within the search space is equal to the spatial statistics across the search space [22], [39].",
"The ergodic metric that measures this characteristic is given as [22], [23], $ \\mathcal {E} (x(t)) = \\sum _{k \\in {\\mathbb {Z}}^n} \\Lambda _{k} [c_{k}(x(t)) - \\phi _{k}]^2$ where $c_{k} (x(t)) & = & \\frac{1}{t_f - t_0}\\int _{t_0}^{t_f} F_k (x(\\tau )) d\\tau , \\\\\\phi _{k} & = & \\int _{\\mathcal {X}} \\Phi ({\\bf x}) F_k ({\\bf x}) d{\\bf x} .$ Here, $F_k (x)$ is a Fourier basis function, and $\\Lambda _{k}$ are weights described in [22].",
"The target distribution $\\Phi (x)$ is what drives the control synthesis for active-exploration.",
"Comparison between robot trajectory and distribution is done with equation (REF ) which takes the Fourier power series decomposition using (3) and (4) and directly compares the statistics of a trajectory with that of the spatial statistics $\\Phi (x)$ .",
"As the ergodic metric is convex [22], convex objectives subject to linear affine constraints are still convex.",
"As a consequence, for linear affine dynamical systems, there are no local minimizers and must eventually search the whole space.",
"A path is “ergodic” with respect to a distribution if the fraction of time the trajectory spends in a region in space is equivalent to the measure associated with that region."
],
[
"Ergodic Sequential Action Control",
"Ergodic control determines a discrete set of control inputs that aims to reduce the ergodic metric over time.",
"Ergodic Sequential Action Control (eSAC) is a control scheme that generates ergodic control actions.",
"Although trajectory optimization has been used to generate control synthesis for ergodic exploration [23], [21], sequential action control provides a closed-form control that can immediately respond to changes in the ergodic metric in an infinite horizon setting [40].",
"In comparison with sample-based planners, SAC incorporates dynamical constraints.",
"In particular, non-linear feedback control is desired when dealing with collisions and contact information that must be incorporated at the time of measurement acquisition.",
"Thus, the motivating factor for using eSAC is a combination of compactness in its formulation and the ability to quickly react to measurements.",
"Assuming control-affine dynamics of the form $\\dot{x} = f(x(t), u) = g(x) + h(x) u$ where $x \\in \\mathbb {R}^n$ and $u \\in \\mathbb {R}^m$ , the trajectory cost is $J(x(t)) = q \\mathcal {E} (x(t)) + \\int _{t_0}^{t_f} \\frac{1}{2} u(\\tau )^T R u(\\tau ) d\\tau $ where $q \\in \\mathbb {R}$ weights the ergodic metric and $R \\in \\mathbb {R}^{m \\times m}$ is a positive definite matrix.",
"During each sampling instance, eSAC computes the optimal action to reduce the cost function $J(x(t))$ in the following steps:"
],
[
"Prediction",
"eSAC forward simulates the system dynamics starting from the sampled sensor state $x_0$ at time $t_0$ to a finite time horizon $t_0 + T$ assuming a default control $u_0$ similar to NMPC methods [41], [42].",
"Cost sensitivity is computed by backwards simulating an adjoint variable, $\\rho \\in \\mathbb {R}^n$ that satisfies the following differential equation $\\dot{\\rho } &=& 2 \\frac{q}{T} \\sum _{k \\in \\mathbb {Z}^n} \\Lambda _k (c_k (x(t)) - \\phi _k) \\frac{\\partial }{\\partial x} F_k (x(t)) \\nonumber \\\\& &\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ - \\frac{\\partial }{\\partial x} f(x, u_0)^T \\rho \\\\&&\\text{subject to } \\rho (t_0 + T) = \\vec{0} \\nonumber .$ The pair $x(t)$ and $\\rho (t)$ are used to compute the optimal control action.",
"Figure: (A) Simulated time sequence of tactile-based eSAC estimating a square (green) in 𝒳⊂ℝ 2 ⊂0,1×0,1\\mathcal {X} \\subset \\mathbb {R}^2 \\subset \\left[0,1 \\right] \\times \\left[0,1 \\right] .",
"Robot trajectory (magenta) translates in time from the previous final state (shown as a circle) to the current initial condition (shown as a magenta square).",
"The boundary of the surface estimate is shown as a black line and the posterior likelihood is shown in the density plot.",
"High likelihood of acquiring a collision is shown in red and low likelihood is shown in blue.",
"Bottom Row: Γ\\Gamma metric is defined as Γ=∫ x φ emp (x)-φ actual (x) 2 dx\\Gamma = \\int _x \\left(\\phi _{\\text{emp}}(x) - \\phi _{\\text{actual}} (x) \\right) ^2 dx the integrated difference of the shape and the estimated shape squared.",
"The Γ\\Gamma measure drops quickly after first contact and then remains at an equilibrium as the shape estimate is updated.",
"(B) Note the posterior at the end of the simulation (top subplot) resembles the Expected Information Density (EID) (see for derivation of EID) of the parametric model of the square shape (bottom subplot)."
],
[
"Calculate Optimal Control Actions",
"eSAC produces a schedule of control actions $u^* (t) \\in [t_0, t_0 + T]$ that optimizes $J_u = \\frac{1}{2} \\int _{t_0}^{t_0 + T} \\left[ \\frac{dJ_{\\text{t}rack}}{d \\lambda } - \\alpha _d \\right]^2 + \\Vert u(t)\\Vert _R^2 dt \\\\\\text{where } \\frac{dJ_{\\text{t}rack}}{d \\lambda } = \\rho (t)^T (f(x(t), u) - f(x(t), u_0)).$ The term $\\frac{dJ_{\\text{t}rack}}{d \\lambda }$ computes the rate of change of the cost with respect to a switch of infinitesimal duration $\\lambda $ [40], [43].",
"The value $\\alpha _d \\in \\mathbb {R}^-$ dictates the aggressiveness of the control (more negative values tend to saturate the control).",
"In this work a value for $\\alpha _d = -555$ was used.",
"The closed form solution of $u^*$ is then given as $u^* = (\\Lambda + R^T)^{-1} [\\gamma u_0 + h(x)^T \\rho \\alpha _d]$ where $\\gamma \\triangleq h(x)^T \\rho \\rho ^T h(x)$ .",
"Additional information on the derivation of SAC can be found in [40].",
"In this work, we used values of $q=30$ , $R=\\text{diag} ( [ 0.01, 0.01 ] )$ , $T=0.8$ for eSAC.",
"eSAC [1] given $x_0, c_{k,0}, \\phi _{k,0}, t_{curr}, T, t_s, i=0$ $(x(t), \\rho (t)) \\leftarrow $ prediction($x_0, t_{curr}, T$ ) $u^* (t) \\leftarrow $ calcControl($x(t), \\rho (t)$ ) return $u^*(t) \\in \\left[t_{curr}, t_{cuss}+t_s \\right]$"
],
[
"Shape Estimation",
"The process for shape estimation is described in Fig.",
"REF .",
"Shape estimates are obtained through repeated samples of the search space using directed motion of the sensor, as calculated by eSAC.",
"The samples are processed using kernel basis functions in order to create decision boundaries.",
"The decision boundaries are then processed into a posterior likelihood estimate using a common method in machine learning to generate statistics on a classification fit known as Platt Scaling [44], [45], [46], [47].",
"The returned spatial statistics on the fit is used for active sampling of the search space, thus updating the shape estimates.",
"The process iterates at a user-specified sampling frequency.",
"We further explain the process in the following sections.",
"Figure: Block diagram for shape estimation.",
"The measurements are processed by a Gaussian Kernel and made into a shape estimate.",
"With Platt Scaling, the shape estimate is transformed into a posterior estimate of the likelihood of acquiring a specific binary sensor measurement, which then is converted into control using eSAC."
],
[
"Binary Measurements for Shape Estimation",
"Given a set of measurements $y_k \\in \\left[0, 1 \\right]$ at time indexed by $k$ sampled at the corresponding set of sensor state $x_k$ , shape estimation is accomplished by probabilistic classification methods [18], [16], [25], [26].",
"Using the set of measurements, a decision boundary for the object is approximated as $\\phi (x) \\approx \\sum _{k} \\alpha _k y_k K(x_k, x) + b$ where $K(x_k, x)$ is the kernel basis function that determines the basis shape of the decision boundary and the parameters $\\alpha _k$ and $b$ are optimized parameters based on the set of $y_k$ and $x_k$ .",
"The choice of kernel basis function determines the shape of the decision function [48].",
"Arbitrary shape estimation is desired so a Gaussian kernel basis (also known as a radial basis function) is chosen [49].",
"However, any kernel basis function can be used as a design choice.",
"The Gaussian Kernel is given as $K(x_k,x) = e^{-\\frac{(x_k - x)^2}{\\sigma ^2}} .$ A kernel of this form maps data into infinite dimensional feature space which provides flexibility for decision boundaries [50].",
"(In this work, the python package Scikit-Learn [51] is used for optimizing the boundary fit.)",
"Figure: Time series shape estimation of diamond and clover shape.",
"The posterior likelihood estimate is shown at 0.1,1,2,6,110.1, 1, 2, 6, 11, and 30 seconds.",
"Final shape estimates at 30s30s is shown in the enclosed box.",
"Likelihood of measuring a collision measurement is denoted by the contours.",
"The sensor trajectory (shown in magenta) traverses from the previous time window onto the current posterior.",
"Shape estimates are depicted in the black line.",
"Actual continuous contact motion does not begin until around 6 seconds into the simulation.",
"Nonetheless, through non-contact motion, the algorithm is able to estimate the shapes.",
"After continuous contact occurs, the algorithm refines the shape estimate.",
"The final shape estimates are highlighted in the enclosed rectangular box.",
"Note that the shape estimate (denoted as the black line) for the square is underneath the actual shape.",
"With the clover, part of the shape estimate is outside the shape boundary."
],
[
"Posterior Likelihood Estimate (Platt Scaling)",
"As the measurement model is unknown, we design the controller to be ergodic with respect to the likelihood distribution of acquiring a collision measurement.",
"The distribution is updated for each measurement via Platt Scaling, defined as $P( y_k = 1 | x) = \\frac{1}{1 + e^{A \\phi (x) + B}} ,$ where $A$ and $B$ are solved through a regression fit and $\\phi (x)$ is the current shape estimate.",
"Figure: Time series shape estimation of a spatial sine wave.",
"High likelihood probabilities of a collision are shown in red and low probabilities in blue.",
"Start point of sensor shown as the magenta square.",
"By following the trajectory from 4.5s4.5s to 5.5s5.5s, the shape estimate depicted by the black line is updated by the sensor even without a collision.",
"At 6s6s the middle peak of the sine wave has been estimated and the sensor trajectory has not collided with the sine wave boundary.Figure: Modularity in the eSAC algorithm is shown with the example shapes used in and their approach for generating shape estimate and a target distribution.",
"(A) A Gaussian Process (GP) is used for shape representation and the uncertainty in the GP fit as the target distribution (this implies the robot will search near regions of high uncertainty in the fit).",
"(B) Support Vector Machine (SVM) approach for shape representation using the posterior likelihood as the target distribution.",
"Both methods using eSAC provide similar results in final shape estimates."
],
[
"Simulation Results",
"Active shape estimation in $\\mathbb {R}^2$ is done with double integrator dynamics given as $\\dot{{\\bf x}} = f({\\bf x},u) =\\begin{bmatrix}\\dot{x}, &\\dot{y}, &u_1, &u_2\\end{bmatrix}^T .$ The binary measurements are collisions detected during the crossing of the boundary function $\\phi (x) = 0$ .",
"Estimation in $\\mathbb {R}^3$ is done with a similar double integrator model with dynamics $\\dot{{\\bf x}} = f({\\bf x},u) =\\begin{bmatrix}\\dot{x}, &\\dot{y}, &\\dot{z}, &u_1, &u_2, &u_3\\end{bmatrix}^T .$ No assumptions are needed on the location of the object and we initialize with a uniform distribution as the target distribution.",
"Figure: Time series of eSAC exploring multiple objects.",
"The posterior is viewed at times 0.1,1,2,6,11,300.1, 1, 2, 6, 11, 30.",
"Trajectories start at the magenta square.",
"In each subplot, the sensor trajectory (magenta) starts from the endpoint of the previous time instant.",
"Although the robot has no prior knowledge of the number of objects in the sensor state, the algorithm is still able to correctly estimate all the objects.",
"Therefore, regardless of the number of shapes, the algorithm is still able to estimate all the shapes in the environment given a large enough time horizon.Figure: Baxter Robot used for experiments.",
"An object is pinned down to the table for shape estimation using the end-effector probe.",
"The probe itself has no sensors.",
"Thus, the measured joint torques are used to detect collisions based on rapidly changing measurements at specific joint locations near the end-effector.First, we demonstrate that the proposed algorithm's shape estimate converges to the actual shape.",
"Figure REF shows time evolution of the posterior as well as the resulting shape estimate.",
"The estimate depicted by the black line in Fig.",
"REF eventually converges to the shape, although much of the shape restructuring is accomplished from non-contact motion up until $6s$ .",
"Moreover, by comparing the posterior and the windowed sensor path, it is shown in Fig.",
"REF that the sensor path is drawn to high likelihood densities.",
"This ensures that the posterior estimate is verified and updated as new sensor information is acquired.",
"Note that the sensor is unable to access the interior of the shape, resulting in high likelihood probabilities within the shape.",
"The posterior likelihood of the square shape estimate shows high likelihood probabilities near the corners of the square.",
"As an aside, we take note of the similarities that the resultant likelihood has with the expected information density (EID) of the known shape measurement model (see [21] for EID derivation).",
"Specifically, high likelihood near the corners correspond to the similar large expectation of information for a square shape.",
"If we define the measurement model of the known square as $\\Upsilon ({x}, x)$ where $x$ is search space and ${x}$ is the robot's state space, then a measure of information is the Fisher Information Matrix [52] defined by $I ({x}, x) = \\frac{\\partial \\Upsilon }{\\partial x} ^T \\Sigma ^{-1} \\frac{\\partial \\Upsilon }{\\partial x},$ where $\\Sigma $ is the measurement covariance.",
"Assuming the measurement model is of the same form as in equation (REF ) then the region with the largest information is at the corners where the slopes of the edges collide.",
"Thus, large likelihood estimates should exist near corners if they have not been previously searched.",
"Figure: eSAC trajectory for a Torus 𝕊 2 \\mathbb {S}^2 in 𝒳⊂ℝ 3 ⊂0,1×0,1×0,1\\mathcal {X} \\subset \\mathbb {R}^3 \\subset \\left[0,1 \\right] \\times \\left[0,1 \\right] \\times \\left[0,1 \\right] search space.",
"Top: In magenta, the robot trajectory on a Torus.",
"Bottom: Density plot of the shape estimate.",
"Although the simulation was run for 40s40s, the use of eSAC is shown to capture the doughnut shape inner circle feature which defines the torus shape.",
"Utilizing an array of tactile sensors would reduce the error on the outer ring of the torus as larger coverage is needed for ℝ 3 \\mathbb {R}^3.Figure: Comparison of gEER and eSAC for detecting and refining multiple shape estimates.",
"We randomly selected 20 initial conditions with uniform distribution in the search space for both gEER and eSAC methods.",
"(A) Trial samples shown from both eSAC and gEER.",
"eSAC consistently detects all objects in the search space.",
"(B) Refined estimates of the shape after detection.",
"eSAC is shown to coarsely explore the whole region while spending a larger fraction of time around the shape estimates.",
"gEER is driven by larger values which reduce uncertainty which is susceptible to local minima.",
"(C) eSAC is shown to detect all shapes in the environment within 40s40s of simulation.",
"In all 20 trials, gEER does not successfully detect all shapes in the environment.Figure: Baxter experimental results for shape estimation.",
"(A) shows the shapes used for estimation.",
"(B) shows the location of Baxter's probe at the end of the estimation as well as the estimate of the shape as the dotted black line.",
"The posterior at the end of the experiment is shown in the contour map.",
"Note the prominent posterior likelihood in the triangle estimation around the edges are colored with high likelihood estimate, indicating that the estimated measurement model retains information about corners that is useful for localization.",
"Error in shape area is within 2.5%,7.5%,2.5 \\%, 7.5 \\%, and 10.0%10.0 \\% for the circle, triangle, and square respectively.",
"(Coarse discretization of the distribution was created to generate the figure to prevent run-time degradation on Baxter which results in an off-centered visual error of 1cm1cm in the distribution.",
")The non-contact motion shown in Fig.",
"REF is beneficial for tactile-based exploration.",
"The sensor trajectory from time $4.5s$ to $6s$ updates the shape estimation of the sine wave boundary without measuring a collision for $t \\in [4.5,6s]$ .",
"Knowing that no contact has occurred indicates that the shape is not in the current sensor state, but likely in other unexplored regions of sensor state.",
"(Interestingly, tactile sensor arrays appear often in biological systems such as the rat whisker system [53]).",
"The algorithm is shown in Fig.",
"REF to estimate both a clover shape and a diamond shape.",
"Between times $2s$ to $6s$ in Fig.",
"REF , non-contact motion is shown to drive the sensor trajectory towards an enclosing path that estimates the location of the center of the shape.",
"After a few seconds, the sensor regularly is in contact with the shape as the estimate converges.",
"The final estimates of the diamond and clover are shown by black lines in the rectangular box in Fig.",
"REF .",
"The final time posterior shown in Fig.",
"REF and Fig.",
"REF show a resemblance in the posterior likelihood near the edges of the shape.",
"This similarity in posterior estimates shows the algorithm approximates regions where there is high expected tactile information.",
"Shown in Fig.",
"REF , multiple objects are allocated randomly in the sensor state.",
"The sensor has no prior knowledge of where the shapes are, nor the number of shapes.",
"The sensor contacts the first shape around $1s$ (Fig.",
"REF ), then between 2 and $6s$ the sensor comes into contact with the remaining two shapes.",
"Following the trajectory (magenta) traversed from time $11-30s$ shown in Fig.",
"REF , the sensor distributes the time spent amongst the three shapes.",
"Because the ergodic metric is convex with respect to the information distribution and thus convex with respect to the shape, unexplored regions of sensor state must be explored [22].",
"We further expand upon multiple shape detection with a comparison with a version of Greedy expected entropy reduction (gEER) exploration algorithm used in [21], [54], [55], [56], [57].",
"The algorithm samples nearby states centered at the robot probe's current location for regions with the largest probability of acquiring a collision measurement.",
"The algorithm then moves the probe to that location, sampling along the way to adjust the shape estimate.",
"In Fig.",
"REF , we run 20 trials of uniformly sampled initial conditions for both eSAC and gEER.",
"Figure REF (A) shows sample shape estimates for $80s$ simulations of both algorithms.",
"eSAC is shown to detect all the shapes and begin shape refinement while gEER algorithm at most detects two shapes, but refines only one.",
"We can see in (C) that eSAC detects all shapes within $40s$ of simulation time whereas the gEER detects two shapes at most.",
"The time it takes to detect objects does depend on the distance of the shape and the control saturation of the robot probe which we maintained constant with both algorithms.",
"In (B), we see the difference in the algorithm's area coverage.",
"In particular, eSAC covers most of the search space coarsely with densely collected collision measurements around the shape.",
"Modularity with respect to shape representation is demonstrated by comparing with the shapes used in [19] with the Gaussian Process (GP) for shape estimation (see Fig.",
"REF ).",
"Here, eSAC is shown to work with both a GP and SVM for shape estimates.",
"In particular, in Fig.",
"REF (A), eSAC uses the touch point selection metric used in [19] for control synthesis which is based on the covariance of the GP fit.",
"In comparison with eSAC, the work done in [19] uses an Markov Decision Process (MDP) in order to control a robot manipulator towards acquiring touch data for shape estimation.",
"In [19], a grid is defined and a path planner is used to traverse the grid.",
"The large overhead state discretization and motion planning code that is necessary for the MDP formulation is not required for eSAC.",
"Moreover, eSAC automatically encodes robot dynamic constraints.",
"Notably, eSAC can be used with other visual-based estimators that feed initial shape estimates such that tactile data is used to refine the estimate.",
"This algorithm can be trivially extended to $\\mathbb {R}^3$ as shown in simulation.",
"Figure REF shows the sensor trajectory in the top plot (magenta) actively sensing the torus shape.",
"Shape estimation of the torus is shown to be successful in the lower sub-plot of Fig.",
"REF resulting in scalability to objects in $\\mathbb {R}^3$ .",
"Performance for estimation in $\\mathbb {R}^3$ depends on the robot dynamical constraints as well as the sensor area coverage."
],
[
"Experimental Results",
"Experiments are done using the Baxter robot (Fig.",
"REF ).",
"Binary measurements are taken by the end-effector probe during a large change of joint torques, indicating a collision has occurred.",
"During collision, controller weight $R$ prevents the robot from dragging along the object.",
"In the case that the control weight $R$ is not significant, heuristics are used to prevent overexertion.",
"Baxter's end-effector is used as a probe for the $\\mathbb {R}^2$ search space.",
"No assumptions are needed on the location of the object or the height of the object.",
"Experimental results show that shape estimation can be accomplished with a robotic system with multiple sensors (Fig.",
"REF ).",
"Although there is odometry error within the robot's joint states, shape estimations are still visually seen to match.",
"Within the posterior regime seen in Fig.",
"REF , the probability densities for the shapes with corners are shown to have unique features.",
"In particular, the posterior for the triangle in Fig.",
"REF extends outwards along the three corners indicating high probability of obtaining a collision measurement.",
"Further measurements surrounding the edge regions reduce the likelihood of acquiring estimate which can be seen in with results from estimating the square shape."
],
[
"Conclusions and Future Work",
"In this paper, an algorithm is presented for shape estimation using a binary sensor combined with ergodic exploration.",
"It is shown that a binary measurement sensor can be treated as a lower resolution tactile sensor for shape estimation with active sensing.",
"In addition, using ergodic exploration as the principle for active sensing uses sensor motion to capture shape features during estimation.",
"The algorithm is shown to estimate shapes in $\\mathbb {R}^2$ and $\\mathbb {R}^3$ as well as an any number of shapes in the sensor domain.",
"The algorithm is shown to be modular with respect to choice of shape representation.",
"The algorithm is shown to have potential applications for active sensing with low resolution sensors with respect to shape estimation as well as applications in localization.",
"Future research directions include the use of multiple sensors for estimation.",
"Another direction is to augment the current algorithm to include localization using the estimated measurement model.",
"Thus, simultaneous localization and shape estimation should be possible."
]
] | 1709.01560 |
[
[
"Pure radiation in space-time models that admit integration of the\n eikonal equation by the separation of variables method"
],
[
"Abstract We consider space-time models with pure radiation, which admit integration of the eikonal equation by the method of separation of variables.",
"For all types of these models, the equations of the energy-momentum conservation law were integrated.",
"The resulting form of metric, energy density and wave vectors of radiation as functions of metric for all types of spaces under consideration is presented.",
"The solutions obtained can be used for any metric theories of gravitation."
],
[
"Introduction",
"At present, activity has increased in the field of studying realistic cosmological models, which is due to observational data on the accelerated expansion of the universe, the presence of isotropy disturbances in background cosmological radiation, and so on.",
"If Einstein's theory of gravity is correct, then there must exist \"dark\" energy and \"dark\" matter, which interact gravitationally, but do not interact electromagnetically.",
"The study of the properties of these exotic substances is based on astrophysical observations and theoretical modeling sometimes leads to exotic equations of state.",
"There is another option implemented, in which GR is an approximation to a more realistic theory of gravity, the search for which is also actively pursued.",
"In any case, the situation needs new ideas, methods and tools for studying problems of cosmology and astrophysics.",
"In this research, we propose an approach to the modeling of space-time with pure radiation based on a combined approach with the following requirements: realistic theory of gravity is a metric theory, i.e.",
"Gravity is modeled by a metric tensor, and free bodies and radiation move along the geodesic lines of space-time; the law of conservation of energy-momentum of matter is satisfied; to construct integrable models, we will use spaces that allow the integration of the eikonal equation by the method of separation of variables.",
"As will be shown later, for the space-time models that meet these requirements, one can integrate the equations of the energy-momentum conservation law and obtain expressions for the energy density and the wave vector of pure radiation in the form of functional expressions in terms of functions included in the space-time metric.",
"The explicit form of the field equations of any gravitation theory is not required.",
"Solutions are written in coordinate systems that allow separation of variables in the eikonal equation.",
"At the same time, part of the solutions obtained includes the use of an isotropic (wave-like) variable, which is used in modeling wave processes similar to electromagnetic or gravitational radiation (massless fields).",
"In this paper, we find and enumerate all types of models considered.",
"The results obtained can be used when comparing similar models in Einstein's theory of gravity and in various modified theories of gravitation.",
"When considering metric theories of gravity, then the motion of the test particles and radiation along geodesic lines, just as like the law of conservation of energy-momentum of matter, remain unchanged theory fundamentals: $\\nabla ^iT_{ij}=0,\\qquad i,j,k=0...3,$ where $T_{ij}$ is the energy-momentum tensor of matter.",
"Therefore, to study and compare modified theories of gravity, the use of space-time models that allow the existence of coordinate systems, where the eikonal equation for massless test particles (null geodesic line for the metric $g^{ij}$ ) $g^{ij}S_{,i}S_{,j}=0,$ or the Hamilton-Jacobi equation for test uncharged mass test particles (an analogue of the geodesic equations) $g^{ij}S_{,i}S_{,j}=m^2,$ or the Hamilton-Jacobi equation for a charged test particle in an electromagnetic field (with a potential $A_i$ ) $g^{ij}(S_{,i}+A_i)(S_{,j}+A_j)=m^2,$ allowing integration by the method of complete separation of variables is of great interest (S is the action of the test particle in the Hamilton-Jacobi formalism, $S_{,i}=\\partial S/\\partial x^i$ ).",
"For these models, as it turns out, it is possible to integrate the differential equations of energy-momentum conservation law (REF ) for dust matter with the energy-momentum tensor of the form $&T_{ij}=\\rho \\, u_iu_j, &\\\\& u_iu^i=1,&$ where $\\rho $ – energy density of dust matter and $u_i$ – field of the four-velocity of dust matter.",
"The same situation and for pure radiation (in the high-frequency approximation of geometric optics) with the energy-momentum tensor of the form $& T_{ij}=\\varepsilon \\, {L}_i{L}_j, &\\\\&g^{ij} {L}_i{L}_i=0,&$ where $\\varepsilon $ – energy density and $L_i$ – wave vector of radiation.",
"As a result, for the energy density of dust matter $\\rho $ and radiation $\\varepsilon $ , for the field of four-velocity of matter $u_i$ and for the wave vector of radiation $L_i$ functional expressions can be obtained only through the space-time metric.",
"Thus, in these models, the use of field equations of specific theories of gravity can be reduced only to equations for the metric of space-time.",
"The spaces in which integration of the eikonal equation (REF ) is possible by the method of complete separation of variables are called conformally Stackel spaces (CSS), in contrast to the Stackel spaces, which admit complete separation of variables in the Hamilton-Jacobi equation for test uncharged masses (REF ).",
"In contrast to the case of the Hamilton-Jacobi equation for test mass particles, in the case of integration of the eikonal equation, the space-time metric admits an arbitrary conformal factor.",
"The task of this paper is to obtain a classification of functional expressions for the energy density of radiation and the wave vector by integrating the differential energy-momentum conservation law (REF ) for pure radiation for all possible space-time models, where the eikonal equation (REF ) admits integration by the method of separation of variables.",
"An analogous problem for the case of Stackel spaces and dust matter was considered earlier (see ref. 00).",
"The theory of Stackel spaces for the case of the Lorentz signature was considered in the works of various authors (see, for example, refs.",
"0-1) and was formulated in the final form in the works of V.N.",
"Shapovalov (see refs. 2-3).",
"We are based on the formulations of the theory and notation in (ref. 4).",
"The covariant condition of Stackel space is the presence of the so-called ”complete set” of a commuting Killing vector and tensor fields, which satisfy some additional algebraic relations (see details in ref. 4).",
"The Stackel space metric tensor in privileged coordinate systems (where separation of variables is allowed) is determined by a set of arbitrary functions where each function depends only on one variable.",
"Stackel space-times (SST) application in gravitation theories (refs.",
"5–13) is based on the fact that exact integrable models can be developed for these spaces.",
"The majority of well-known exact solutions is classified as SST (Schwarzschild, Kerr, Friedman-Robertson-Walker, NUT, etc.).",
"It is important to note that the other single-particle equations of motion - Klein-Gordon-Fock, Dirac, Weyl - admit a separation of variables in SST.",
"The same method can be used to obtain solutions in the theories of modified gravity (refs.",
"14–16)."
],
[
"Pure radiation in conformally Stackel space-times",
"The Stackel spaces type is defined by a set of two numbers $(N.N_0)$ , where $N$ – the number of commuting Killing vectors $Y^i_{(p)}$ ($p=1,N$ ) admitted by the ”complete set” (the dimension of the Abelian group of space-time motions), and $N_0=N-rank|Y^i_{(p)}g_{ij}Y^j_{(q)}|$ – is the number of null (\"wave-like\") variables in privileged coordinate systems.",
"For 4-dimensional Stackel spaces of Lorentz signature $N=0...3$ , $N_0=0,1$ .",
"Stackel spaces metrics are conveniently written in privileged coordinate systems in a contravariant form.",
"The coordinates of space-time will be numbered from 0 to 3 by indices $i$ , $ j$ , $k$ .",
"Ignored variables (on which the metric does not depend) will be numbered in indices $ p$ , $q$ , $r$ and non-ignored by Greek letters (indices) $ \\mu $ , $ \\nu $ , $ \\sigma $ .",
"The following notation will be used in the paper: $P=\\ln \\left|\\frac{{\\varepsilon }^2}{\\Delta ^2 \\det g^{ij}}\\right|,$ where $g^{ij}$ is the space-time metric, $\\varepsilon $ is the energy density of radiation and $\\Delta $ is the conformal factor of metric $g_{ij}$ .",
"For conformally Stackel spaces the conformal factor of metric is an arbitrary function of all variables.",
"In the privlieged coordinate system, the wave vector of the radiation has a \"separated\" form, i.e.",
"the corresponding covariant components of the wave vector depend only on one variable ${L}_i={L}_i(x^i)$ : $& L_0=L_0(x^0),\\quad L_1=L_1(x^1),&\\\\&L_2=L_2(x^2),\\quad L_3=L_3(x^3).&\\nonumber $ Later in the paper we obtain a functional form of the energy density and the wave vector of radiation for all types of conformally Stackel space-times in privileged coordinate systems (where the separation of variables in the eikonal equation (REF ) is allowed)."
],
[
"Conformally Stackel space-times (3.0) type",
"Conformally Stackel space-times (3.0) type admit three commuting Killing vectors.",
"In the privileged coordinate system, the metric of a conformally Stackel space of type (3.0) can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&a_0&b_0&c_0\\cr 0&b_0&d_0&e_0\\cr 0&c_0&e_0&f_0},$ where $\\Delta =\\Delta (x^0,x^1,x^2,x^3)$ and $ a_0, b_0, c_0, d_0, e_0, f_0$ are functions of a variable $x^0$ .",
"The wave vector of the radiation $L_i$ in the privileged coordinate system has the following \"separated\" form: $&{L}_0={L}_0(x^0),\\quad {L}_1=\\alpha ,\\quad {L}_2=\\beta ,\\quad {L}_3=\\gamma ,&\\\\&\\alpha , \\beta , \\gamma - const.&\\nonumber $ Wave vector norm condition () takes the form: $\\alpha ^2a_0+2\\alpha \\beta b_0+\\beta ^2d_0+2\\alpha \\gamma c_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=-{L}_0{}^2.$ From the equations of the energy-momentum conservation law (REF ) we obtain: $& {L}_0P_{,0}+(\\alpha a_0+\\beta b_0+\\gamma c_0)P_{,1}+(\\alpha b_0+\\beta d_0+\\gamma e_0)P_{,2} &\\nonumber \\\\&+\\mbox{} (\\alpha c_0+\\beta e_0+\\gamma f_0)P_{,3}+2{L}_0{}^{\\prime }=0,&$ here and in the sequel, a comma means a partial differentiation, and a prime means differentiation with respect to a single variable on which the function depends.",
"Using condition (REF ) and finding the integrals of equation (REF ), one can obtain expressions for the wave vector and the energy density of radiation.",
"Below are listed all the solutions we have obtained for conformally Stackel space-times (3.0) type.",
"The wave vector of the radiation has the form: $&{L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right),\\quad \\alpha , \\beta , \\gamma - const,&\\\\[1ex]&{L}_0=\\sqrt{\\alpha ^2a_0+2\\alpha \\beta b_0+\\beta ^2d_0+2\\alpha \\gamma c_0+2\\beta \\gamma e_0+\\gamma ^2 f_0}.&\\nonumber $ For the energy density of radiation we obtain the expression: $&{\\varepsilon }=F(X,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}/{{L}_0},&\\\\[1ex]&X=x^1-\\int {(\\alpha a_0+\\beta b_0+\\gamma c_0)}/{{L}_0}\\,dx^0,&\\nonumber \\\\&Y=x^2-\\int {(\\alpha b_0+\\beta d_0+\\gamma e_0)}/{{L}_0}\\,dx^0,&\\nonumber \\\\&Z=x^3-\\int {(\\alpha c_0+\\beta e_0+\\gamma f_0)}/{{L}_0}\\,dx^0,& \\nonumber $ where $ F (X, Y, Z) $ is an arbitrary function of its variables."
],
[
"CSS (3.0) type, case 2. (${L}_0= 0$ ).",
"For this degenerate case there is an additional condition for the functions of the metric (REF ): $&\\alpha ^2a_0+2\\alpha \\beta b_0+2\\alpha \\gamma c_0+\\beta ^2d_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=0,& \\nonumber \\\\&\\alpha , \\beta , \\gamma - const.&$ The wave vector takes the form: ${L}_i=(0,\\alpha , \\beta , \\gamma ).$ For the energy density of radiation we obtain the expression: $&{\\varepsilon }=F(x^0,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}&\\\\[1ex]&Y={x^1}/{(\\alpha a_0+\\beta b_0+\\gamma c_0)}-{x^2}/{(\\alpha b_0+\\beta d_0+\\gamma e_0)},& \\nonumber \\\\&Z={x^1}/{(\\alpha a_0+\\beta b_0+\\gamma c_0)}-{x^3}/{(\\alpha c_0+\\beta e_0+\\gamma f_0)},& \\nonumber $ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (3.0) type, case 3. (${L}_0=0$ , {{formula:70047d94-5e5d-40db-85a6-32afca0e0cfd}} ).",
"In this degenerate case, when one of the terms in (REF ) additionally vanishes together with $ {L}_0 $ (for example, the term with $ P_{, 1} $ ), we obtain additional conditions for the metric (REF ): $\\alpha a_0+\\beta b_0+\\gamma c_0=0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\alpha ^2a_0+2\\alpha \\beta b_0+2\\alpha \\gamma c_0+\\beta ^2d_0+2\\beta \\gamma e_0+\\gamma ^2 f_0=0.$ The wave vector of the radiation has the form: ${L}_i=(0,\\alpha , \\beta , \\gamma ).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,x^1,Z){\\Delta \\sqrt{- \\det g^{ij}}}$ $Z=x^2 (\\alpha c_0+\\beta e_0+\\gamma f_0)- x^3 (\\alpha b_0+\\beta d_0+\\gamma e_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables."
],
[
"Conformally Stackel space-times (3.1) type",
"Conformally Stackel space-times (3.1) type admits 3 commuting Killing vectors $Y^i_{(p)}$ ($p=1,3$ ), but $rank\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=2$ .",
"In a privileged coordinate system the metric of a conformally Stackel space-times (3.1) type can be written in the following form, where the variable $ x^0 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{0&1&a_0&b_0\\cr 1&0&0&0\\cr a_0&0&c_0&f_0\\cr b_0&0&f_0&d_0},$ where $\\Delta =\\Delta (x^0,x^1,x^2,x^3)$ and $a_0, b_0, c_0, d_0, f_0$ are functions of a variable $ x^0$ .",
"The wave vector of the radiation has the form: $&{L}_0={L}_0(x^0),\\quad {L}_1=\\alpha ,\\quad {L}_2=\\beta ,\\quad {L}_3=\\gamma ,&\\\\&\\alpha , \\beta , \\gamma - const.\\nonumber $ The system of equations (REF ), () takes the form: $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0+2(\\alpha +\\beta a_0+\\gamma b_0){L}_0=0,$ $&(\\alpha +\\beta a_0+\\gamma b_0)P_{,0}+{L}_0P_{,1}+(a_0{L}_0+\\beta c_0+\\gamma f_0)P_{,2}&\\nonumber \\\\&\\mbox{}+(b_0{L}_0+\\beta f_0+\\gamma d_0)P_{,3}+2\\beta a_0^{\\prime }+2\\gamma b_0^{\\prime }=0.&$ From equations (REF )-(REF ) we can obtain functional expressions for the wave vector and the energy density of radiation.",
"Below all the solutions for conformally Stackel space-times (3.1) are listed."
],
[
"CSS (3.1) type, case 1. ($\\alpha +\\beta a_0+\\gamma b_0\\ne 0$ ).",
"The wave vector of the radiation has the form: $ {L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right),\\quad \\alpha , \\beta , \\gamma - const,$ $ {L}_0=\\frac{-(\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0)}{2\\,(\\alpha +\\beta a_0+\\gamma b_0)}.$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(X,Y,Z){\\Delta \\sqrt{- \\det g^{ij}}}/{(\\alpha +\\beta a_0+\\gamma b_0)},$ $X&=&x^1-\\int \\frac{{L}_0}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber \\\\Y&=&x^2-\\int \\frac{(a_0{L}_0+\\beta c_0+\\gamma f_0)}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber \\\\Z&=&x^3-\\int \\frac{(b_0{L}_0+\\beta f_0+\\gamma d_0)}{(\\alpha +\\beta a_0+\\gamma b_0)}\\,dx^0,\\nonumber $ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (3.1) type, case 2. ($L_0\\ne 0$ , {{formula:06fa1d48-9bb8-49b0-b333-846375975dad}} ). ",
"In this degenerate case there are additional conditions for the metric (REF ): $\\alpha +\\beta a_0+\\gamma b_0= 0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: $ {L}_i=\\left( L_0(x^0), \\alpha , \\beta , \\gamma \\right).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,Y,Z) \\Delta \\sqrt{- \\det g^{ij}}.$ $Y&=& x^1-\\frac{x^2(a_0{L}_0+\\beta c_0+\\gamma f_0) }{L_0},\\nonumber \\\\Z&=& x^1-\\frac{x^3(b_0{L}_0+\\beta f_0+\\gamma d_0) }{L_0},\\nonumber $ where $L_0(x^0)$ and $F(x^0,Y,Z)$ are arbitrary functions of their variables."
],
[
"CSS (3.1) type, case 3. ($L_0= 0$ , {{formula:88ed6080-40a1-42aa-a8a8-2b68967c0299}} ).",
"In this degenerate case there are additional conditions for the metric (REF ): $\\alpha +\\beta a_0+\\gamma b_0= 0,\\quad \\alpha , \\beta , \\gamma - const,$ $\\beta ^2 c_0+2\\beta \\gamma f_0+\\gamma ^2 d_0=0.$ The wave vector of the radiation has the form: ${L}_i=\\left( 0, \\alpha , \\beta , \\gamma \\right).$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,x^1,Z) \\Delta \\sqrt{- \\det g^{ij}}.$ $Z=x^2(\\beta f_0+\\gamma d_0)-x^3(\\beta c_0+\\gamma f_0),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables.",
"Conformally Stackel space-times (2.0) type admit two commuting Killing vectors.",
"In the privileged coordinate system, the metric can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&{{\\epsilon }} &0&0\\cr 0&0&A&B\\cr 0&0&B&C},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),\\quad {\\epsilon } =\\pm 1,\\quad A=a_0(x^0)+a_1(x^1),&\\\\&B=b_0(x^0)+b_1(x^1),\\qquad C=c_0(x^0)+c_1(x^1) .&$ The wave vector of the radiation ${L}_i$ has the form: $&{L}_0={L}_0(x^0),\\qquad {L}_1={L}_1(x^1),&\\\\&{L}_2=\\alpha ,\\qquad {L}_3=\\beta , \\qquad \\alpha , \\beta - const.&\\nonumber $ From the norm condition () one can obtain: $&{{L}_0}^2+\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0-\\gamma =0,&\\\\&{\\epsilon } {{L}_1}^2+\\alpha ^2 a_1+2\\alpha \\beta b_1+\\beta ^2 c_1+\\gamma =0.&$ From the conservation law (REF ) one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+{\\epsilon } {L}_1P_{,1}+(\\alpha A+\\beta B)P_{,2}+(\\alpha B+\\beta C)P_{,3}&\\nonumber \\\\&\\mbox{}+2{L}_0^{\\prime }+2{\\epsilon } {L}_1^{\\prime }=0.&$ Below are listed all the solutions for conformally Stackel space-times (2.0) type."
],
[
"CSS (2.0) type, case 1. (${L}_0{L}_1\\ne 0$ ):",
"The wave vector of the radiation has the form: $&{L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta ,\\gamma - const,&\\nonumber \\\\[1ex]&{L}_0=\\sqrt{\\gamma -\\alpha ^2 a_0-2\\alpha \\beta b_0-\\beta ^2 c_0},&\\nonumber \\\\&{L}_1=\\sqrt{{\\epsilon } (-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}.&$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/({{L}_0\\,{L}_1}),$ $&X=\\int 1/{{L}_0}\\, dx^0- \\int {\\epsilon }/{{L}_1}\\, dx^1,&\\\\&Y=x^2-\\int {(\\alpha a_0+\\beta b_0)}/{{L}_0}\\, dx^0 - {\\epsilon } \\int {(\\alpha a_1+\\beta b_1)}/{{L}_1}\\, dx^1,&\\\\&Z=x^3-\\int {(\\alpha b_0+\\beta c_0)}/{{L}_0}\\, dx^0 - {\\epsilon } \\int {(\\alpha b_1+\\beta c_1)}/{{L}_1}\\, dx^1,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (2.0) type, case 2. (${L}_0=0, {L}_1\\ne 0$ ):",
"Let us consider the degenerate case when $ {L}_0=0$ or ${L}_1 = 0 $ and fo rcertainty we take $ {L}_0 = 0 $ and $ {L}_1 \\ne 0 $ , then the wave vector will take the following form: ${L}_i=\\left( 0, {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta , \\gamma - const,$ ${L}_1=\\sqrt{{\\epsilon } (-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}.$ In this degenerate case there are additional conditions for the metric (REF ): $\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0-\\gamma =0.$ For the energy density of radiation we obtain the expression: ${\\varepsilon }=F(x^0,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_1},$ $Y=x^2-\\int {(\\alpha a_1+\\beta b_1)}/{{L}_1}\\, dx^1,$ $Z=x^3- {\\epsilon } \\int {(\\alpha b_1+\\beta c_1)}/{{L}_1}\\, dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (2.0) type, case 3. (${L}_0={L}_1= 0$ ):",
"In this degenerate case, there are additional conditions for the metric (REF ): $\\alpha ^2 A+2\\alpha \\beta B+\\beta ^2 C=0,\\qquad \\alpha , \\beta - const.$ The wave vector of the radiation has the form: ${L}_i=\\left( 0, 0, \\alpha , \\beta \\right).$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,x^1,Z)\\,\\Delta \\sqrt{-\\det g^{ij}},$ $Z=x^2 (\\alpha B+\\beta C) -x^3(\\alpha A+\\beta B),$ where $F(x^0,x^1,Z)$ is an arbitrary function of its variables."
],
[
"Conformally Stackel space-times (2.1) type",
"Conformally Stackel space-times (2.1) type admits two commuting Killing vectors $Y^i_{(p)}$ ($p=1,2$ ), but $rank\\,|Y^i_{(p)}g_{ij}Y^j_{(q)}|=1$ .",
"In a privileged coordinate system the metric of a conformally Stackel space-times (2.1) type can be written in the following form, where $ x^1 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{1&0&0&0\\cr 0&0&f_1(x^1)&1\\cr 0&f_1(x^1)&A&B\\cr 0&1&B&C},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3), \\quad A=a_0(x^0)+a_1(x^1),&\\\\&B=b_0(x^0)+b_1(x^1),\\quad C=c_0(x^0)+c_1(x^1).&$ The wave vector of radiation $L_i$ has the form: $&{L}_0={L}_0(x^0),\\quad {L}_1={L}_1(x^1),&\\nonumber \\\\&{L}_2=\\alpha ,\\quad {L}_3=\\beta , \\quad \\alpha , \\beta , \\gamma - const.&$ From the norm condition () one can obtain: $&\\gamma ={{L}_0}^2+\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0,&\\\\&-\\gamma =2(\\alpha f_1+\\beta ){L}_1+\\alpha ^2 a_1+2\\alpha \\beta b_1+\\beta ^2 c_1.&$ From the conservation law (REF ), one can obtain the equation for the energy density of radiation: $&{L}_0P_{,0}+(\\alpha f_1+\\beta ) P_{,1}+(\\alpha A+\\beta B+f_1 {L}_1)P_{,2}&\\nonumber \\\\&\\mbox{}+(\\alpha B+\\beta C+{L}_1)P_{,3}+2\\alpha f_1^{\\prime }+2{{L}_0}^{\\prime }=0.&$ From the system of equations (REF )–(REF ) we can obtain functional expressions through the metric for the wave vector and the radiation energy density.",
"Below all the solutions for conformally Stackel space-times (2.1) type are listed."
],
[
"CSS (2.1) type, case 1. ${L}_0(\\alpha f_1+\\beta )\\ne 0$ :",
"The wave vector of radiation has the form: $&{L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , \\beta \\right), \\qquad \\alpha , \\beta ,\\gamma - const,&\\nonumber \\\\[1ex]&{L}_0=\\sqrt{\\gamma -\\alpha ^2 a_0-2\\alpha \\beta b_0-\\beta ^2 c_0},&\\\\&{L}_1={(-\\gamma -\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1)}/\\big ({2\\,(\\alpha f_1+\\beta )}\\big ).&\\nonumber $ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{\\big ({L}_0(\\alpha f_1+\\beta )\\big )},$ $X=\\int \\frac{dx^0}{{L}_0}- \\int \\frac{dx^1}{(\\alpha f_1+\\beta )},$ $Y=x^2-\\int \\frac{(\\alpha a_0+\\beta b_0)}{{L}_0}\\,dx^0 -\\int \\frac{(\\alpha a_1+\\beta b_1+f_1L_1)}{\\alpha f_1+\\beta }\\, dx^1,$ $Z=x^3-\\int \\frac{(\\alpha b_0+\\beta c_0)}{{L}_0}\\,dx^0 -\\int \\frac{(\\alpha b_1+\\beta c_1+L_1)}{\\alpha f_1+\\beta }\\, dx^1,$ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (2.1) type, case 2. $(\\alpha f_1+\\beta )= 0$ , {{formula:89e7bb2c-cabe-4b63-969f-80c8a732525b}} .",
"In this degenerate case, there are additional conditions for the metric (REF ): $a_1=f_1=0.$ The wave vector of radiation has the form ($\\beta =\\gamma =0$ ): ${L}_i=\\left( L_0(x^0), {L}_1(x^1), \\alpha , 0\\right),$ ${L}_0=\\alpha \\sqrt{- a_0},\\quad L_1=\\sigma -\\alpha b_1,\\quad \\alpha , \\sigma - const.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{L_0(x^0)},$ $Y=\\alpha x^2+\\int {L_0}\\,dx^0,\\quad Z= x^3 -\\int {(\\sigma +\\alpha b_0)}/{L_0}\\,dx^0,$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (2.1) type, case 3. ${L}_0= 0$ , {{formula:b7686906-83ba-49e5-96c2-9923fa201a29}} .",
"In this degenerate case, there are additional conditions for the metric (REF ): $&\\alpha ^2 a_0+2\\alpha \\beta b_0+\\beta ^2 c_0=\\gamma ,&\\\\&p(\\alpha a_0+\\beta b_0)+q (\\alpha b_0+\\beta c_0)=r,&\\\\&\\alpha , \\beta , \\gamma , p, q, r \\mbox{ -- const}.&\\nonumber $ The wave vector of radiation has the form: $&{L}_i=\\left( 0, {L}_1(x^1), \\alpha , \\beta \\right),&\\\\&{L}_1= {(-\\alpha ^2 a_1-2\\alpha \\beta b_1-\\beta ^2 c_1-\\gamma )}/{(2\\,(\\alpha f_1+\\beta ))}.&\\nonumber $ The energy density of the radiation has the form: $ {\\varepsilon }=F(x^0,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{(\\alpha f_1+\\beta )},$ $Y=\\alpha x^2+\\beta x^3 +\\int L_1(x^1)\\,dx^1,$ $Z=px^2+qx^3$ $\\mbox{}-\\int \\frac{r+ p(\\alpha a_1+\\beta b_1+f_1L_1)+q(\\alpha b_1+\\beta c_1+L_1)}{(\\alpha f_1+\\beta )}\\,dx^1,$ where $F(x^0,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (2.1) type, case 4. ${L}_0= 0$ , {{formula:468ec54a-5663-4dcf-914e-e423945394f8}} , {{formula:fdcb860c-4b2a-4abd-a024-0e7e81517f01}} .",
"The wave vector of radiation has the form: ${L}_i=\\left( 0, {L}_1(x^1), 0, 0\\right),\\qquad \\alpha = \\beta =\\gamma =0.$ The energy density of the radiation has the form: $ {\\varepsilon }=F(x^0,x^1,Z)\\,\\Delta \\sqrt{-\\det g^{ij}},$ $Z=x^2 -x^3f_1(x^1),$ where $L_1(x^1)$ and $F(x^0,x^1,Z)$ are arbitrary functions of their variables."
],
[
"Conformally Stackel space-times (1.0) type",
"Conformally Stackel space-times (1.0) type admits one Killing vector, and in a privileged coordinate system the metric of this space-time can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{\\Omega &0&0&0\\cr 0&V^1&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^1=t_2(x^2)-t_3(x^3),\\qquad V^2=t_3(x^3)-t_1(x^1),&\\\\&V^3=t_1(x^1)-t_2(x^2),\\quad \\Omega =\\omega _\\nu (x^\\nu ) V^\\nu ,\\quad \\mu ,\\,\\nu =1...3.&$ The wave vector of radiation has the form: $ {L}_i=\\Big (\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ),\\qquad \\alpha =const.$ From equations (REF ), () we have: $ \\Omega \\alpha ^2 + V^\\mu {{L}_\\mu }^2=0,$ $ \\alpha \\Omega P_{,0}+ V^\\mu ({L}_\\mu P_{,\\mu }+2{L}_\\mu ^{\\prime })=0.",
"$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.",
"Below are listed all the solutions for conformally Stackel space-times (1.0) type."
],
[
"CSS (1.0) type, case 1. ${L}_1{L}_2{L}_3\\ne 0$ .",
"The wave vector of radiation has the form: $&{L}_i=\\left(\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\right),\\quad \\alpha ,\\beta ,\\gamma \\mbox{ -- const}.&\\nonumber \\\\[1ex]&{L}_\\mu =\\sqrt{-\\alpha ^2\\omega _\\mu +\\beta t_\\mu +\\gamma },\\quad \\mu , \\nu =1...3.&$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_1\\,{L}_2\\,{L}_3)},$ $&X=x^0 -\\alpha \\sum _\\mu \\int ({\\omega _\\mu }/{{L}_\\mu })\\, dx^\\mu ,&\\\\&Y=\\sum _\\mu \\int ({t_\\mu }/{{L}_\\mu })\\, dx^\\mu ,\\qquad Z= \\sum _\\mu \\int (1/{{L}_\\mu }){\\,dx^\\mu } ,&$ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (1.0) type, case 2. ${L}_1=0, {L}_2{L}_3 \\ne 0$ .",
"In this degenerate case, when one of the components of the wave vector $ {L}_\\mu $ becomes zero (for definiteness, let $ L_1 = 0 $ , $ {L}_2 {L}_3 \\ne 0 $ ) we have an additional condition for the metric (REF ): $\\alpha ^2\\omega _1=\\beta t_1+\\gamma ,\\qquad \\mbox{$\\alpha ,\\beta ,\\gamma $ -- const}.$ The wave vector of radiation has the form: ${L}_i=\\Big (\\alpha ,0,{L}_2(x^2),{L}_3(x^3)\\Big ),$ ${L}_2=\\sqrt{-\\alpha ^2\\omega _2+\\beta t_2+\\gamma },\\quad {L}_3=\\sqrt{-\\alpha ^2\\omega _3+\\beta t_3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,\\Delta \\sqrt{-\\det g^{ij}}/\\prod _{\\mu \\ne 1}{L}_\\mu ,$ $Y=\\alpha x^0 -\\alpha ^2\\sum _{\\mu \\ne 1} \\int ({\\omega _\\mu }/{{L}_\\mu })\\, dx^\\mu +\\beta \\sum _{\\mu \\ne 1} \\int ({t_\\mu }/{{L}_\\mu })\\, dx^\\mu ,$ $Z=\\sum _{\\mu \\ne 1} \\int (1/{{L}_\\mu }){\\,dx^\\mu },$ where $F(x^1,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (1.0) type, case 3. $L_1\\ne 0, {L}_2=0, {L}_3= 0$ .",
"In this degenerate case, when two of the components of the wave vector $ {L}_\\mu $ becomes zero (for definiteness, let $ L_1 \\ne 0 $ , $ {L}_2 ={L}_3= 0 $ ) we have additional conditions for the metric (REF ): $\\alpha ^2\\omega _2=\\beta t_2+\\gamma ,\\quad \\alpha ^2\\omega _3=\\beta t_3+\\gamma ,$ where $\\alpha $ , $\\beta $ , $\\gamma $ – const.",
"The wave vector of radiation has the form: ${L}_i=\\left(\\alpha ,{L}_1(x^1),0,0\\right),\\quad {L}_1=\\sqrt{\\beta t_1-\\alpha ^2\\omega _1+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^2,x^3, Y)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_1(x^1)},$ $Y=\\alpha x^0+\\int L_1(x^1) \\,dx^1,$ where $F(x^2,x^3,Y)$ is an arbitrary function of its variables."
],
[
"Conformally Stackel space-times (1.1) type",
"Conformally Stackel space-times (1.1) type admits one Killing vector.",
"In a privileged coordinate system the metric of a conformally Stackel space-times (1.1) type can be written in the following form, where $ x^1 $ is a null (\"wave-like\") variable: $g^{ij}=\\frac{1}{\\Delta }{\\Omega &V^1&0&0\\cr V^1&0&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^1=t_2(x^2)-t_3(x^3),\\quad V^2=t_3(x^3)-t_1(x^1),&\\\\&V^3=t_1(x^1)-t_2(x^2),\\quad \\Omega =\\omega _\\mu (x^\\mu ) V^\\mu ,\\quad \\mu ,\\nu =1...3.$ The wave vector of radiation has the following \"separated\" form: $ {L}_i=\\Big (\\alpha ,{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ),\\quad \\alpha - const.$ From equations (REF ), () we have: $ \\alpha V^1(2 {L}_1+\\alpha \\omega _1)+V^2({L}_2{}^2+\\alpha ^2\\omega _2)+V^3({L}_3{}^2+\\alpha ^2\\omega _3)=0,$ $(V^1{L}_1+\\alpha \\Omega )P_{,0}+ \\alpha V^1P_{,1}+ {L}_2V^2P_{,2}+ {L}_3V^3P_{,3}$ $\\mbox{}+ 2V^2{L}_2^{\\prime }+ 2V^3{L}_3^{\\prime } = 0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.",
"Below are listed all the solutions for conformally Stackel space-times (1.1) type."
],
[
"CSS (1.1) type, case 1. $\\alpha L_2 L_3\\ne 0$ .",
"The wave vector of radiation has the form: ${L}_0=\\alpha ,\\quad {L}_1=\\frac{1}{2\\alpha }(\\beta t_1 -\\alpha ^2\\omega _1+\\gamma ),\\quad \\alpha ,\\beta , \\gamma - const,$ ${L}_2=\\sqrt{\\beta t_2-\\alpha ^2\\omega _2+\\gamma },\\quad {L}_3=\\sqrt{\\beta t_3-\\alpha ^2\\omega _3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_2\\,{L}_3)},$ $X=x^0 -\\frac{1}{\\alpha } \\int ({L}_1+\\alpha \\omega _1)\\,dx^1 -\\alpha \\left( \\int \\frac{\\omega _2}{{L}_2}\\,dx^2 +\\int \\frac{\\omega _3}{{L}_3}\\,dx^3 \\right),$ $Y=-\\frac{1}{\\alpha }\\int t_1\\, dx^1 +\\int \\frac{t_2}{{L}_2}\\,dx^2+\\int \\frac{t_3}{{L}_3}\\,dx^3 ,$ $Z= \\frac{x^1}{\\alpha }+\\int \\frac{dx^2}{{L}_2}+\\int \\frac{dx^3}{{L}_3},$ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (1.1) type, case 2. $\\alpha = 0$ .",
"In this degenerate case, when $ {L}_0 = 0 $ , we obtain an additional condition for the metric (REF ): $t_1=0.$ The wave vector of the radiation $L_i$ has the following form ($\\gamma =0$ ): ${L}_0=0,\\quad {L}_1=L_1(x^1),\\quad {L}_2=\\sqrt{\\beta t_2},\\quad {L}_3=\\sqrt{\\beta t_3}.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_2\\,{L}_3)},$ $Y=\\int \\frac{t_2}{{L}_2}\\,dx^2+\\int \\frac{t_3}{{L}_3}\\,dx^3 ,\\quad Z=\\frac{x^0}{L_1}+ \\int \\frac{dx^2}{{L}_2}+\\int \\frac{dx^3}{{L}_3},$ where $F(x^1,Y,Z)$ and $L_1(x^1)$ are arbitrary functions of its variables."
],
[
"CSS (1.1) type, case 3. $\\alpha L_2\\ne 0, L_3=0$ .",
"In the degenerate case, when one of the components of the wave vector $L_3$ becomes zero (similarly for $ L_2 $ with the replacement of the indices 3 by 2) we have additional conditions for the metric (REF ): $\\alpha ^2\\omega _3=\\beta t_3+\\gamma , \\qquad \\alpha , \\beta , \\gamma - const,$ $(p\\alpha -q\\beta )\\,t_3=0,\\qquad p,q - const.$ The wave vector of the radiation $L_i$ has the following form: ${L}_0=\\alpha ,\\qquad {L}_1=\\frac{1}{2\\alpha }(\\beta t_1 -\\alpha ^2\\omega _1+\\gamma ),$ ${L}_2=\\sqrt{\\beta t_2-\\alpha ^2\\omega _2+\\gamma },\\qquad {L}_3=0.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^1,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{{L}_2},$ $Y=\\alpha x^0+\\int L_1\\,dx^1+\\int L_2\\,dx^2,$ $Z=qx^0+\\frac{1}{\\alpha }\\int \\Big ( q(\\gamma /\\alpha -\\alpha \\omega _1 -L_1)+pt_1\\Big )dx^1$ $\\mbox{}+\\int \\frac{q(\\gamma /\\alpha -\\alpha \\omega _2)+pt_2}{L_2}\\,dx^2,$ where $F(x^1,Y,Z)$ is an arbitrary function of their variables."
],
[
"Conformally Stackel space-times (0.0) type",
"The metric of conformally Stackel space-times (0.0) type in a privileged coordinate system can be written in the following form: $g^{ij}=\\frac{1}{\\Delta }{V^0 &0&0&0\\cr 0&V^1&0&0\\cr 0&0&V^2&0\\cr 0&0&0&V^3},$ $&\\Delta =\\Delta (x^0,x^1,x^2,x^3),&\\\\&V^0=a_1(b_2-b_3)+a_2(-b_1+b_3)+a_3(b_1-b_2),&\\\\&V^1= a_0(-b_2+b_3)+a_2(b_0-b_3)+a_3(-b_0+b_2),&\\\\&V^2=a_0(b_1-b_3)+a_1(-b_0+b_3)+a_3(b_0-b_1),&\\\\&V^3=a_0(-b_1+b_2)+a_1(b_0-b_2)+a_2(-b_0+b_1),&$ where functions $ a $ , $ b $ , $ c $ are functions of only one variable whose index corresponds to the lower index of the function.",
"For example $ a_0 = a_0 (x ^ 0) $ , $ b_1 = b_1 (x ^ 1) $ , etc.",
"The wave vector of the radiation in a privileged coordinate system has the following \"separated\" form: $ {L}_i=\\Big ({L}_0(x^0),{L}_1(x^1),{L}_2(x^2),{L}_3(x^3)\\Big ).$ From the law of energy-momentum conservation (REF ) and condition () we have: $V^i{L}_i{}^2=0,\\qquad i=0...3,$ $ V^i({L}_iP_{,i}+2{L}_i^{\\prime })=0.$ From this system of equations we obtain functional expressions for the wave vector and the radiation energy density through the metric.",
"Below are listed all the solutions for conformally Stackel space-times (0.0) type."
],
[
"CSS (0.0) type, case 1. ${L}_0{L}_1{L}_2{L}_3\\ne 0$ .",
"The wave vector of the radiation $L_i$ has the following form ($\\alpha $ , $\\beta $ , $\\gamma $ – const): ${L}_i=\\sqrt{\\alpha a_i+\\beta b_i+\\gamma },\\qquad i,j \\mbox{ = 0...3.",
"}$ The energy density of the radiation has the form: ${\\varepsilon }=F(X,Y,Z)\\,{\\Delta \\sqrt{-\\det g^{ij}}}/{({L}_0\\,{L}_1\\,{L}_2\\,{L}_3)},$ $X=\\sum _i \\int \\frac{dx^i}{{L}_i},\\quad Y=\\sum _i \\int \\frac{a_i}{{L}_i}\\,dx^i,\\quad Z=\\sum _i \\int \\frac{b_i}{{L}_i}\\,dx^i,$ where $F(X,Y,Z)$ is an arbitrary function of its variables."
],
[
"CSS (0.0) type, case 2. ${L}_0=0, \\quad {L}_1{L}_2{L}_3\\ne 0$ .",
"In the degenerate case, when one of the components of the wave vector turns to zero (for definiteness a component with the number $ i=0 $ , that is, $ {L}_0 = 0 $ ), then we obtain an additional condition for the metric (REF ): $\\alpha a_0+\\beta b_0+\\gamma =0,\\qquad \\mbox{$\\alpha $, $\\beta $, $\\gamma $ -- const.",
"}$ For the wave vector of radiation ${L}_i$ we obtain: ${L}_{0}=0,\\qquad {L}_i=\\sqrt{\\alpha a_i+\\beta b_i+\\gamma },\\qquad i\\ne 0.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,X,Y)\\,\\Delta \\sqrt{-\\det g^{ij}}/{\\prod _{i\\ne 0}{L}_i},$ $X=\\sum _{i\\ne 0} \\int \\frac{dx^i}{{L}_i},\\qquad Y=\\sum _{i\\ne 0} \\int {L}_i\\,dx^i,$ where $F(x^0,X,Y)$ is an arbitrary function of its variables."
],
[
"CSS (0.0) type, case 3. ${L}_0={L}_1= 0, \\quad {L}_2{L}_3\\ne 0$ .",
"In the degenerate case, when the two components of the wave vector vanish (let it be the components $ {L}_0 =0$ , $ {L}_1 = 0 $ ), then we obtain additional conditions for the metric ($\\alpha $ , $\\beta $ , $\\gamma $ – const): $\\alpha a_0+\\beta b_0+\\gamma =0,\\quad \\alpha a_1+\\beta b_1+\\gamma =0.$ For the wave vector of radiation we have: ${L}_i=\\Big (0,0,L_2(x^2), L_3(x^3)\\Big ),$ ${L}_2=\\sqrt{\\alpha a_2+\\beta b_2+\\gamma },\\quad {L}_3=\\sqrt{\\alpha a_3+\\beta b_3+\\gamma }.$ The energy density of the radiation has the form: ${\\varepsilon }=F(x^0,x^1,Y)\\,\\frac{\\Delta \\sqrt{-\\det g^{ij}}}{{L}_2{L}_3},$ $Y= \\int {L}_2\\,dx^2 + \\int {L}_3\\,dx^3,$ where $F(x^0,x^1,Y)$ is an arbitrary function of its variables.",
"Note that for Stackel space-times (0.0) type the three components of the wave vector of the radiation can not be zero, since this leads to a violation of the norm condition.",
"In the paper we obtain and enumerate all solutions of the equations of the energy-momentum conservation law of pure radiation for models of space-times that admit separation of variables in the eikonal equation.",
"The forms of the energy-momentum tensor of pure radiation (energy density and wave vector of radiation) for all types of space-times under consideration are obtained.",
"In privileged coordinate systems (where the separation of variables is admitted), the energy density and the wave vector of the radiation are determined through functions of the space-time metric.",
"The results obtained can be used to construct integrable models for various metric theories of gravitation, including for comparing similar models in the theory of gravity of Einstein and in modified theories of gravity.",
"Research partially supported by the Ministry of Education and Science of Russia under contract 3.1386.2017/4.6."
]
] | 1709.01718 |
[
[
"Distributed Deep Neural Networks over the Cloud, the Edge and End\n Devices"
],
[
"Abstract We propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and end devices.",
"While being able to accommodate inference of a deep neural network (DNN) in the cloud, a DDNN also allows fast and localized inference using shallow portions of the neural network at the edge and end devices.",
"When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.",
"Due to its distributed nature, DDNNs enhance sensor fusion, system fault tolerance and data privacy for DNN applications.",
"In implementing a DDNN, we map sections of a DNN onto a distributed computing hierarchy.",
"By jointly training these sections, we minimize communication and resource usage for devices and maximize usefulness of extracted features which are utilized in the cloud.",
"The resulting system has built-in support for automatic sensor fusion and fault tolerance.",
"As a proof of concept, we show a DDNN can exploit geographical diversity of sensors to improve object recognition accuracy and reduce communication cost.",
"In our experiment, compared with the traditional method of offloading raw sensor data to be processed in the cloud, DDNN locally processes most sensor data on end devices while achieving high accuracy and is able to reduce the communication cost by a factor of over 20x."
],
[
"Introduction",
"Neural networks (NNs), and deep neural networks (DNNs) in particular, have achieved great success in numerous applications in recent years.",
"For example, deep Convolutional Neural Networks (CNNs) continuously achieve state-of-the-art performances on various tasks in computer vision as shown in Figure REF .",
"At the same time, the number of end devices, including Internet of Things (IoT) devices, has increased dramatically.",
"These devices are appealing targets for machine learning applications as they are often directly connected to sensors (e.g., cameras, microphones, gyroscopes) that capture a large quantity of input data in a streaming fashion.",
"However, the current state of machine learning systems on end devices leaves an unsatisfactory choice: either (1) offload input sensor data to large NN models (e.g., DNNs) in the cloud, with the associated communication costs, latency issues and privacy concerns, or (2) perform classification directly on the end device using simple Machine Learning (ML) models e.g., linear Support Vector Machine (SVM), leading to reduced system accuracy.",
"To address these shortcomings, it is natural to consider the use of a distributed computing approach.",
"Hierarchically distributed computing structures consisting of the cloud, the edge and devices (see, e.g., [1], [2]) have inherent advantages, such as supporting coordinated central and local decisions, and providing system scalability, for large-scale intelligent tasks based on geographically distributed IoT devices.",
"An example of one such distributed approach is to combine a small NNThe term network layer may refer to either a layer in a NN or a layer in the distributed computing hierarchy (e.g., edge or cloud).",
"In order to remove ambiguity, when we refer to network layers for NN we explicitly use the term NN layers.",
"model (less number of parameters) on end devices and a larger NN model (more number of parameters) in the cloud.",
"The small model at an end device can quickly perform initial feature extraction, and also classification if the model is confident.",
"Otherwise, the end device can fall back to the large NN model in the cloud, which performs further processing and final classification.",
"This approach has the benefit of low communication costs compared to always offloading NN input to the cloud and can achieve higher accuracy compared to a simple model on device.",
"Additionally, since a summary based on extracted features from the end device model are sent instead of raw sensor data, the system could provide better privacy protection.",
"However, this kind of distributed approach over a computing hierarchy is challenging for a number of reasons, including: End devices such as embedded sensor nodes often have limited memory and battery budgets.",
"This makes it an issue to fit models on the devices that meet the required accuracy and energy constraints.",
"A straightforward partitioning of NN models over a computing hierarchy may incur prohibitively large communication costs in transferring intermediate results between computation nodes.",
"Incorporating geographically distributed end devices is generally beyond the scope of DNN literature.",
"When multiple sensor inputs on different end devices are used, they need to be aggregated together for a single classification objective.",
"A trained NN will need to support such sensor fusion.",
"Multiple models at the cloud, the edge and the device need to be learned jointly to allow coordinated decision making.",
"Computation already performed on end device models should be useful for further processing on edge or cloud models.",
"Usual layer-by-layer processing of a DNN from the NN's input layer all the way to the NN's output layer does not directly provide a mechanism for local and fast inference at earlier points in the neural networks (e.g., end devices).",
"A balance is needed between the accuracy of a model (with the associated model size) at a given distributed computing layer and the cost of communicating to the layer above it.",
"The solution must have reasonably good lower NN layers on the end devices capable of accurate local classification for some input while also providing useful features for classification in the cloud for other input.",
"To address these concerns under the same optimization framework, it is desirable that a system could train a single end-to-end model, such as a DNN, and partition it between end devices and the cloudFor presentation simplicity, we often just consider the device-cloud scenario.",
"Our methodology can similarly apply to general device-edge (fog)-cloud scenarios., in order to provide a simpler and more principled approach.",
"To this end, we propose distributed deep neural networks (DDNNs) over distributed computing hierarchies, consisting of the cloud, the edge (fog) and geographically distributed end devices.",
"In implementing a DDNN, we map sections of a single DNN onto a distributed computing hierarchy.",
"By jointly training these sections, we show that DDNNs can effectively address the aforementioned challenges.",
"Specifically, while being able to accommodate inference of a DNN in the cloud, a DDNN allows fast and localized inference using some shallow portions of the DNN at the edge and end devices.",
"Moreover, via distributed computing, DDNNs naturally enhance sensor fusion, data privacy and system fault tolerance for DNN applications.",
"When supported by a scalable distributed computing hierarchy, a DDNN can scale up in neural network size and scale out in geographical span.",
"DDNN leverages our earlier work on BranchyNet [3] which allows early exit points to be placed in a DNN.",
"Samples can be classified and exited locally when the system is confident and offloaded to the edge and the cloud when additional processing is required.",
"In addition, DDNN leverages the recent work of binary neural networks (BNNs) [4], which greatly reduce the required memory cost of neural network layers and enables multi-layer NNs to run on end devices with small memory footprints [5].",
"By training DDNN end-to-end, the network optimally configures lower NN layers to support local inference at end devices, and higher NN layers in the cloud to improve overall classification accuracy of the system.",
"As a proof of concept, we show a DDNN can exploit geographical diversity of sensors (on a multi-view multi-camera dataset) in sensor fusion to improve recognition accuracy.",
"The contributions of this paper include A novel DDNN framework and its implementation that maps sections of a DNN onto a distributed computing hierarchy.",
"A joint training method that minimizes communication and resource usage for devices and maximizes usefulness of extracted features which are utilized in the cloud, while allowing low-latency classification via early exit for a high percentage of input samples.",
"Aggregation schemes that allows automatic sensor fusion of multiple sensor inputs to improve the overall performance (accuracy and fault tolerance) of the system.",
"The DDNN codebase is open source and can be found here: https://github.com/kunglab/ddnn.",
"Figure: Progression towards deeper neural network structures in recent years (see, e.g., , , , , )."
],
[
"Related Work",
"In this section, we briefly review related work in distributed computing hierarchies and recent deep learning algorithms that enable our proposed method to run in a distributed fashion.",
"We then discuss other approaches involving distributed deep networks."
],
[
"Distributed Computing Hierarchy",
"The framework of a large-scale distributed computing hierarchy has assumed new significance in the emerging era of IoT.",
"It is widely expected that most of data generated by the massive number of IoT devices must be processed locally at the devices or at the edge, for otherwise the total amount of sensor data for a centralized cloud would overwhelm the communication network bandwidth.",
"In addition, a distributed computing hierarchy offers opportunities for system scalability, data security and privacy, as well as shorter response times (see, e.g., [2], [11]).",
"For example, in [11], a face recognition application shows a reduced response time is achieved when a smartphone's photos are proceeded by the edge (fog) as opposed to the cloud.",
"In this paper, we show that DDNN can systematically exploit the inherent advantages of a distributed computing hierarchy for DNN applications and achieve similar benefits."
],
[
"Deep Neural Network Extensions",
"Binarized neural networks (BNNs) are a recent type of neural networks, where the weights in linear and convolutional layers are constrained to $\\lbrace -1, 1\\rbrace $ (stored as 0 and 1 respectively).",
"This representation has been shown to achieve similar classification accuracy for some datasets such as MNIST and CIFAR-10 [12] when compared to a standard floating-point neural network while using less memory and reduced computation due to the binary format [4].",
"Embedded binarized neural networks (eBNNs) extends BNNs to allow the network to fit on embedded devices by reducing floating-point temporaries through reordering the operations in inference [5].",
"These compact models are especially attractive in end device settings, where memory can be a limiting factor and low power consumption is required.",
"In DDNN, we use BNNs, eBNNs and the alike to accommodate the end devices, so that they can be jointly trained with the NN layers in the edge and cloud.",
"BranchyNet proposed a solution of classifying samples at earlier points in a neural network, called early exit points, through the use of an entropy-based confidence criteria [3].",
"If at an early exit point a sample is deemed confident based on the entropy of the computed probability vector for target classes, then it is classified and no further computation is performed by the higher NN layers.",
"In DDNN, exit points are placed at physical boundaries (e.g., between the last NN layer on an end device and the first NN layer in the next higher layer of the distributed computing hierarchy such as the edge or the cloud).",
"Input samples that can already be classified early will exit locally, thereby achieving a lowered response latency and saving communication to the next physical boundary.",
"With similar objectives, SACT [13] allocates computation on a per region basis in an image, and exits each region independently when it is deemed to be of sufficient quality."
],
[
"Distributed Training of Deep Networks",
"Current research on distributing deep networks is mainly focused on improving the runtime of training the neural network.",
"In 2012, Dean et al.",
"proposed DistBelief, which maps large DNNs over thousands of CPU cores during training [14].",
"More recently, several methods have been proposed to scale up DNN training across GPU clusters [15], [16], which further reduces the runtime of network training.",
"Note that this form of distributing DNNs (over homogeneous computing units) is fundamentally different from the notion presented in this paper.",
"We proposes a way to train and perform feedforward inference over deep networks that can be deployed over a distributed computing hierarchy, rather than processed in parallel over bus- or switch-connected CPUs or GPUs in the cloud.",
"In this section we give an overview of the proposed distributed deep neural network (DDNN) architecture and describe how training and inference in DDNN is performed."
],
[
"DDNN Architecture",
"DDNN maps a trained DNN onto heterogeneous physical devices distributed locally, at the edge, and in the cloud.",
"Since DDNN relies on a jointly trained DNN framework at all parts in the neural network, for both training and inference, many of the difficult engineering decisions are greatly simplified.",
"Figure REF provides an overview of the DDNN architecture.",
"The configurations presented show how DDNN can scale the inference computation across different physical devices.",
"The cloud-based DDNN in (a) can be viewed as the standard DNN running in the cloud as described in the introduction.",
"In this case, sensor input captured on end devices is sent to the cloud in original format (raw input format), where all layers of DNN inference is performed.",
"We can extend this model to include a single end device, as shown in (b), by performing a portion of the DNN inference computation on the device rather than sending the raw input to the cloud.",
"Using an exit point after device inference, we may classify those samples which the local network is confident about, without sending any information to the cloud.",
"For more difficult cases, the intermediate DNN output (up to the local exit) is sent to the cloud, where further inference is performed using additional NN layers and a final classification decision is made.",
"Note that the intermediate output can be designed to be much smaller than the sensor input (e.g., a raw image from a video camera), and therefore drastically reduce the network communication required between the end device and the cloud.",
"The details of how communication is considered in the network is discussed in section REF .",
"DDNN can also be extended to multiple end devices which may be geographically distributed, shown in (c), that work together to make a classification decision.",
"Here, each end device performs local computation as in (b), but their output is aggregated together before the local exit point.",
"Since the entire DDNN is jointly trained across all end devices and exit points, the network automatically aggregates the input with the objective of achieving maximum classification accuracy.",
"This automatic data fusion (sensor fusion) simplifies runtime inference by avoiding the necessity of manually combining output from multiple end devices.",
"We will discuss the design of feature aggregation in detail in section REF .",
"As before, if the local exit point is not confident about the sample, each end devices sends intermediate output to the cloud, where another round of feature aggregation is performed before making a final classification decision.",
"DDNN scales vertically as well, by using an edge layer in the distributed computing hierarchy between the end devices and cloud, shown in (d) and (e).",
"The edge acts similarly to the cloud, by taking output from the end devices, performing aggregation and classification if possible, and forwarding its own intermediate output to the cloud if more processing is needed.",
"In this way, DDNN naturally adjusts the network communication and response time of the system on a per sample basis.",
"Samples that can be correctly classified locally are exiting without any communication to the edge or cloud.",
"Samples that require more feature extraction than can be provided locally are sent to the edge, and eventually the cloud if necessary.",
"Finally, DDNNs can also scale geographically across the edge layer as well, which is shown in (f).",
"Figure: Overview of the DDNN architecture.",
"The vertical lines represent the DNN pipeline, which connects the horizontal bars (NN layers).",
"(a) is the standard DNN (processed entirely in the cloud), (b) introduces end devices and a local exit point that may classify samples before the cloud, (c) extends (b) by adding multiple end devices which are aggregated together for classification, (d) and (e) extend (b) and (c) by adding edge layers between the cloud and end devices, and (f) shows how the edge can also be distributed like the end devices."
],
[
"DDNN Aggregation Methods",
"In DDNN configurations with multiple end devices (e.g., (c), (e), and (f) in Figure REF ), the output from each end device must be aggregated in order to perform classification.",
"We present several different schemes for aggregating the output.",
"Each aggregation method makes different assumptions about how the device output should be combined and therefore can result in different system accuracy.",
"We present three approaches: Max pooling (MP).",
"MP aggregates the input vectors by taking the max of each component.",
"Mathematically, max pooling can be written as $ {\\hat{v}}_{j} = \\max _{1 \\le i \\le n} v_{ij}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.",
"Average pooling (AP).",
"AP aggregates the input vectors by taking the average of each component.",
"This is written as $ {\\hat{v}}_{j} = \\sum _{i=1}^n \\frac{v_{ij}}{n}, $ where $n$ is the number of inputs and $v_{ij}$ is the $j$ -th component of the input vector and ${\\hat{v}}_{j}$ is the $j$ -th component of the resulting output vector.",
"Averaging may reduce noisy input presented in some end devices.",
"Concatenation (CC).",
"CC simply concatenates the input vectors together.",
"CC retains all information which is useful for higher layers (e.g., the cloud) that can use the full information to extract higher level features.",
"Note that this expands the dimension of the resulting vector.",
"To map this vector back to the same number of dimensions as input vectors, we add an additional linear layer.",
"We analyzes these aggregation methods in Section REF ."
],
[
"DDNN Training",
"While DDNN inference is distributed over the distributed computing hierarchy, the DDNN system can be trained on a single powerful server or in the cloud.",
"One aspect of DDNN that is different from most conventional DNN pipelines is the use of multiple exit points as shown in Figure REF .",
"At training time, the loss from each exit is combined during back-propagation so that the entire network can be jointly trained, and each exit point achieves good accuracy relative to its depth.",
"For this work, we follow joint training as described in GoogleNet [9] and BranchyNet [3].",
"For the system evaluation discussed in Section , we apply DDNNs to a classification task.",
"We use the softmax cross entropy loss function as the optimization objective.",
"We now describe formally how we train DDNNs.",
"Let $$ be a one-hot ground-truth label vector, $$ be an input sample and $\\mathcal {C}$ be the set of all possible labels.",
"For each exit, the softmax cross entropy objective function can be written as L(, ;) = -1|C| cC yc yc, where = softmax() = ()cC (zc), and = fexitn(;), where $f_{\\text{exit}_n}$ is a function representing the computation of the neural network layers from an entry point to the $n$ -th exit branch and $\\theta $ represents the network parameters such as weights and biases of those layers.",
"To train the DDNN we form a joint optimization problem as minimizing a weighted sum of the loss functions of each exit: L(, ;) = n=1N wn L(exitn, ;), where $N$ is the total number of exit points and $w_n$ is the associated weight of each exit.",
"Equal weights are used for the experimental results of this paper."
],
[
"DDNN Inference",
"Inference in DDNN is performed in several stages using multiple preconfigured exit thresholds $$ (one element $T$ at each exit point) as a measure of confidence in the prediction of the sample.",
"One way to define $$ is by searching over the ranges of $$ on a validation set and pick the one with the best accuracy.",
"We use a normalized entropy threshold as the confidence criteria (instead of unnormalized entropy as used in [3]) that determines whether to classify (exit) a sample at a particular exit point.",
"The normalized entropy is defined as $\\eta (\\mathbf {x}) =-\\sum _{i=1}^{|\\mathcal {C}|}{\\frac{x_{i}\\log x_{i}}{\\log |\\mathcal {C}|}},$ where $\\mathcal {C}$ is the set of all possible labels and $\\textbf {x}$ is a probability vector.",
"This normalized entropy $\\eta $ has values between 0 and 1 which allows easier interpretation and searching of its corresponding threshold $T$ .",
"For example, $\\eta $ close to 0 means that the DDNN is confident about the prediction of the sample; $\\eta $ close to 1 means it is not confident.",
"At each exit point, $\\eta $ is computed and compared against $T$ in order to determine if the sample should exit at that point.",
"At a given exit point, if the predictor is not confident in the result (i.e., $\\eta > T$ ), the system falls back to a higher exit point in the hierarchy until the last exit is reached which always performs classification.",
"We now provide an example of the inference procedure for a DDNN which has multiple end devices and three exit points (configuration (e) in Figure REF ): Each end device first sends summary information to local aggregator.",
"The local aggregator determines if the combined summary information is sufficient for accurate classification.",
"If so, the sample is classified (exited).",
"If not, each device sends more detailed information to the edge in order to perform further processing for classification.",
"If the edge believes it can correctly classify the sample it does so and no information is sent to the cloud.",
"Otherwise, the edge forwards intermediate computation to the cloud which makes the final classification."
],
[
"Communication Cost of DDNN Inference",
"The total communication cost for an end device with the local and cloud aggregator is calculated as $c = 4\\times |\\mathcal {C}| + (1-l)\\frac{f\\times o}{8}$ where $l$ is the percentage of samples exited locally, $\\mathcal {C}$ is the set of all possible labels (3 in our experiments), $f$ is the number of filters, and $o$ is the output size of a single filter for the final NN layer on the end-device.",
"The constant 4 corresponds to 4 bytes which are used to represent a floating-point number and the constant 8 corresponds to bits used to express a byte output.",
"The first term assumes a single floating-point per class, which conveys the probability that the sample to be transmitted from the end device to the local aggregator belongs to this class.",
"This step happens regardless of whether the sample is exited locally or at a later exit point.",
"The second term is the communication between end device and cloud which happens $(1-l)$ fraction of the time, when the sample is exited in the cloud rather than locally."
],
[
"Accuracy Measures",
"Throughout the evaluation in Section , we use different accuracy measures for the various exit points in a DDNN as follows: Local Accuracy is the accuracy when exiting 100% of samples at the local exit of a DDNN.",
"Edge Accuracy is the accuracy when exiting 100% of samples at the edge exit of a DDNN.",
"Cloud Accuracy is the accuracy when exiting 100% of samples at the cloud exit of a DDNN.",
"Overall Accuracy is the accuracy when exiting some percentage of samples at each exit point in the hierarchy.",
"The samples classified at each exit point are determined by the entropy threshold $T$ for that exit.",
"The impact of $T$ on classification accuracy and communication cost is discussed in Section REF .",
"Individual Accuracy is the accuracy of an end device NN model trained separately from DDNN.",
"The NN model for each end device consists of a ConvP block followed by a FC block (a single end device portion as shown in Figure REF ).",
"In the evaluation, individual accuracy for each device is computed by classifying all samples using the individual NN model and not relying on the local or cloud exit points of a DDNN."
],
[
"DDNN System Evaluation",
"In this section, we evaluate DDNN on a scenario with multiple end devices and demonstrate the following characteristics of the approach: DDNNs allow multiple end devices to work collaboratively in order to improve accuracy at both the local and cloud exit points.",
"DDNNs seamlessly extend the capability of end devices by offloading difficult samples to the cloud.",
"DDNNs have built-in fault tolerance.",
"We illustrate that missing any single end device does not dramatically affect the accuracy of the system.",
"Additionally, we show how performance gradually degrades as more end devices are lost.",
"DDNNs reduce communication costs for end devices compared to traditional system that offloads all input sensor data to the cloud.",
"We first introduce the DDNN architecture and dataset used in our evaluation."
],
[
"DDNN Evaluation Architecture",
"To accommodate the small memory size of the end devices, we use Binary Neural Network [4] blocksA block consists of one or more conventional NN layers.",
"We make use of two types of blocks in [5]: the fused binary fully connected (FC) block and fused binary convolution-pool (ConvP) block as shown in Figure REF .",
"FC blocks each consist of a fully connected layer with $m$ nodes for some $m$ , batch normalization and binary activation.",
"ConvP blocks each consist of a convolutional layer with $f$ filters for some $f$ , a pooling layer and batch normalization and binary activation.",
"A convolution layer has a kernel of size 3x3 with stride 1 and padding 1.",
"A pooling layer has a kernel of size 3x3 with stride 2 and padding 1.",
"For our experiments, we use version (c) from Figure REF , with six end devices.",
"The system presented can be generalized to a more elaborated structure which includes an edge layer, as shown in (d), (e) or (f) of Figure REF .",
"Figure REF depicts a detailed view of the DDNN system used in our experiments.",
"In this system, we have six end devices shown in red, a local aggregator, and a cloud aggregator.",
"During training, output from each device is aggregated together at each exit point using one of the aggregation schemes described in Section REF .",
"We provide detailed analysis on the impact of aggregation schemes at both the local and cloud exit points in Section REF .",
"All DDNNs in our experiments are trained with Adam [17] using the following hyper-parameter settings: $\\alpha $ of 0.001, $\\beta _1$ of 0.9, $\\beta _2$ of 0.999, and $\\epsilon $ of 1e-8.",
"We train each DDNN for 100 epochs.",
"When training the DDNN, we use equal weights for the local and cloud exit points.",
"We explored heavily weighting both the local exit and the cloud exit, but neither weighting scheme significantly changed the accuracy of the system.",
"This indicates that this solution to the dataset and the problem we are exploring is not sensitive to the weights, but this may not be true for other datasets and problemsIn GoogleNet [9], a less than 1% difference in accuracy was observed based on the values of the weight parameters.",
"Figure: Fused binary blocks consisting of one or more standard NN layers.",
"The fused binary fully connected (FC) block is a fully connected layer with nn nodes, batch normalization and binary activation.",
"The fused binary convolution-pool (ConvP) block consists of a convolutional layer with ff filters, a pooling layer, batch normalization and binary activation.",
"The convolution layer has a kernel of size 3x3 with stride 1 and padding 1.",
"The pooling layer has a kernel of size 3x3 with stride 2 and padding 1.",
"These blocks are used as they are presented in .Figure: The DDNN architecture used in the system evaluation.",
"The FC and ConvP blocks in red and blue correspond to layers run on end devices and the cloud respectively.",
"The dashed orange boxes represent the end devices and show which blocks of the DDNN are mapped onto each device.",
"The local aggregator shown in red combines the exit output (a short vector with length equal to the number of classes) from each end device in order to determine if local classification for the given input sample can be performed accurately.",
"If the local exit is not confident (i.e.",
"η(x)>T\\eta (x) > T), the activation output after the last convolutional layer from each end device is sent to the cloud aggregator (shown in blue), which aggregates the input from each device, performs further NN layer processing, and outputs a final classification result.",
"The aggregation of input for multiple end devices is discussed in Section ."
],
[
"Multi-view Multi-camera Dataset",
"We evaluate the proposed DDNN framework on a multi-view multi-camera dataset [18].",
"This dataset consists of images acquired at the same time from six cameras placed at different locations facing the same general area.",
"For the purpose of our evaluation, we assume that each camera is attached to an end device, which can transmit the captured images over a bandwidth-constraint wireless network to a physical endpoint connected to the cloud.",
"The dataset provides object bounding box annotations.",
"Multiple bounding boxes may exist in a single image, each of which corresponds to a different object in the frame.",
"In preparing the dataset, for each bounding box, we extract an image, and manually synchronizeIn practical object tracking systems, this synchronization step is typically automated [19].",
"the same object across the multiple devices that the object appears in for the given frame.",
"Examples of the extracted images are shown in Figure REF .",
"Each row corresponds to a single sample used for classification.",
"We resize each extracted sample to a 32x32 RGB pixel image.",
"For each device that a given object does not appear in, we use a blank image and assign a label of -1, meaning that the object is not present in the frame.",
"Labels 0, 1, and 2 correspond to car, bus and person, respectively.",
"Objects that are not present in a frame (i.e., label of -1) are not used during training.",
"We split the dataset into 680 training samples and 171 testing samples.",
"Figure REF shows the distribution of samples at each device.",
"Due to the imbalanced number of class samples in the dataset, the individual accuracy of each end device differs widely, as shown by the “Individual\" curve of Figure REF .",
"A full description of the training process for the individual NN models is provided in Section REF .",
"The processed dataset used in this paper is available at [20].",
"Figure: Example images of three objects (person, bus, car) from the multi-view multi-camera dataset.",
"The six devices (each with their own camera) capture the same object from different orientations.",
"An all grey image denotes that the object is not present in the frame.Figure: The distribution of class samples for each end device in the multi-view multi-camera dataset."
],
[
"Impact of Aggregation Schemes",
"In order to perform classification on the input from multiple end devices, we must aggregate the information from each end device.",
"We consider three aggregation methods (max pooling, average pooling, and concatenation) outlined in Section REF , at both the local and cloud exit points.",
"The accuracy of different aggregation schemes are shown in Table REF .",
"The first two letters identify the local aggregation scheme and the last two letters identify the scheme used by the cloud aggregator.",
"For example, MP-CC means the local aggregator uses max-pooling and the cloud uses concatenation.",
"Recall that each input to the local aggregator is a floating-point vector of length equal to the number of classes (corresponding to the output from the final FC block for a single device as shown in Figure REF ) and the device output sent to the cloud aggregator is the output from the final ConvP block.",
"Table: Accuracy of aggregation schemes.",
"The first two letters identify the local aggregation scheme, and the last two letters identify the cloud aggregation scheme.",
"For example, MP-CC means the local aggregator uses max-pooling and the cloud aggregator uses concatenation.",
"The accuracy of each exit point (either local or cloud) is computed using the entire test set.",
"In practice, we will exit a portion of samples locally based on the entropy threshold TT and send the remaining samples in the cloud.",
"Due to its high performance, MP-CC is used in the remaining experiments of this paper.The MP-MP scheme has good classification accuracy for the local aggregator but poor performance in the cloud.",
"The elements in the vectors at the local aggregator correspond to the same features (e.g., the first item is the likelihood that the input corresponds to that class).",
"Therefore, max pooling corresponds to taking the max response for each class over all end devices, and shows good performance.",
"On the other hand, since the information sent from the end devices to the cloud is the activation output from the filters at each device, which corresponds to different visual features in the input from the viewpoint of each individual end device, max pooling these features does not perform well.",
"Comparing MP-MP and MP-CC schemes, though both use MP for local aggregators, MP-CC increases the accuracy of the local classifier.",
"In the training phrase, during backpropagation the MP-MP scheme only passes gradients through a device that gives the highest response while MP-CC scheme passes gradients through all devices.",
"Therefore, using CC aggregator in the cloud allows all devices to learn better filters (filter weights) that give a stronger response for the local MP aggregator, resulting in a better classification accuracy.",
"The CC-CC scheme shows an opposite trend where the local accuracy is poor while the cloud accuracy is high.",
"Concatenating the local information (instead of a pooling scheme), does not enforce any relationship between output for the same class on multiple devices and therefore performs worse.",
"Concatenating the output for the cloud aggregator maintains the most information for NN layer processing in the cloud and therefore performs well.",
"Generally, for the local aggregator, average pooling performs worse than max pooling.",
"This is because some of the end devices do not have the object present in the given frame.",
"Average pooling take average of all outputs from end devices; this compromises the strong outputs from end devices in which the object is present.",
"Based on these results, we use the MP-CC aggregation scheme throughout the paper."
],
[
"Entropy Threshold",
"The entropy threshold for an exit point, $T$ , corresponds to the level of confidence that is required in order to exit a sample.",
"A threshold value of 0 would mean that no samples will exit and a value of 1 would mean that all samples exit at that point.",
"Figure REF shows the relationship between $T$ at the local aggregator and the overall accuracy of the DDNN.",
"We observe that as more samples are exited at the local exit, the overall accuracy decreases.",
"This is expected, as the accuracy of the local exit is typically lower than that of the cloud exit.",
"We need to set the threshold appropriately to achieve a balance between the communication cost, as defined in Section REF , latency and accuracy of the system.",
"In this case, we see that setting the threshold to $0.8$ results in the best overall accuracy with significantly reduced communication, i.e., 97% accuracy while exiting 60.82% of samples locally as shown in Table REF where in addition to local exit (%) and overall accuracy (%), communication cost in bytes is given.",
"We set $T=0.8$ for the remaining experiments in the system evaluation, unless noted otherwise.",
"The local classifier may do better than cloud for certain samples where low-level features are more robust in classification than higher-level features.",
"By setting an appropriate threshold $T$ , we can improve overall accuracy.",
"In this experiment, $T=0.8$ corresponds to that sweet spot where some samples which are incorrectly classified by the cloud classifier can actually be correctly classified by the local classifier.",
"Such a threshold indicates the optimal point where both local and cloud classifier work best together.",
"Table: Effects of different exit threshold (TT) settings for the local exit.",
"T=0.8T=0.8 is used in the remaining experiments.Figure: Overall accuracy of the system as the entropy threshold for the local exit is varied from 0 to 1.",
"For this experiment, 4 filters are used in the ConvP blocks on the end devices."
],
[
"Impact of Scaling Across End Devices",
"In order to scale DDNNs across multiple end devices, we distribute the lower sections of Figure REF , shown in red, over the corresponding devices, outlined in orange.",
"Figure REF shows how the accuracy of the system improves as additional end devices (each with its attached input cameras) are added.",
"The devices are added in order sorted by their individual accuracy from worst to best (i.e., the device with the lowest accuracy first and the device with the highest accuracy last).",
"The first observation is the large variation in the individual accuracy of the end devices, as noted earlier.",
"Due to the nature of the dataset, some devices are naturally better positioned and generally have clearer observations of the objects.",
"Looking at the viewpoints of each camera in Figure REF , we see that the selected examples for Device 6 have clear frontal views of each object.",
"This viewpoint gives Device 6 the highest individual accuracy at over 70%.",
"By comparison, Device 2 has the lowest individual accuracy at under 40%.",
"The “Local” and “Cloud” curves show the accuracy of the system at each exit point when all samples are exited at that point.",
"We observe that the cloud exit point outperforms the local exit point at all numbers of end devices.",
"The gap is widest when there are fewer devices.",
"This suggests that the additional NN layers in the cloud significantly improve the final classification result when the problem is more difficult due to limited labeled training data for an end device.",
"Once all six end devices are added, both the local and cloud aggregators have high accuracy.",
"The “Overall” curve represents the overall accuracy of the system when the threshold for the local exit point is set to $0.8$ .",
"We see that this curve is roughly equivalent to exiting all samples at the cloud (but at a much reduced communication cost as 60.82% of samples are exited locally).",
"Generally, these results show that by combining multiple viewpoints we can increase the classification accuracy at both the local and cloud level by a substantial margin when compared to the individual accuracy of any device.",
"The resulting accuracy of the DDNN system is superior to any individual device accuracy by over 20%.",
"Moreover, we note that the 60.82% of samples which exit locally enjoy lowered latency in response time.",
"Figure: Accuracy of the DDNN system as additional end devices are added.",
"The accuracy of “Overall” is obtained by exiting a percentage of the samples locally and the rest in the cloud.",
"The accuracy of “Cloud” and “Local” are computed by exiting all samples at each point, respectively.",
"The end devices are ordered by their “Individual” classification accuracy, sorted from worst to best."
],
[
"Impact of Cloud Offloading on Accuracy Improvements",
"DDNNs improve the overall accuracy of the system by offloading difficult samples to the cloud, which perform further NN layer processing and final classification.",
"Figure REF shows the accuracy and communication costs of DDNN as the number of filters on the end devices increases.",
"For all settings, the NN layers stored on an end device require under 2 KB of memory.",
"In this experiment, we configure the local exit threshold $T$ such that around $75\\%$ of samples are exited locally and around $25\\%$ of samples are offloaded to the cloud.",
"We see that DDNNs achieve around a $5\\%$ improvement in accuracy compared to using just the local aggregator.",
"This demonstrates the advantage for offloading to the cloud even when larger models (more filters) with improved local accuracy are used on the end devices.",
"Figure: Accuracy and communication cost (in bytes) for increasingly larger end device memory sizes that accommodate additional filters.",
"We notice that cloud offloading leads to improved accuracy."
],
[
"Fault Tolerance of DDNNs",
"A key motivation for distributed systems is fault tolerance.",
"Fault tolerance implies that the system still works well even when some parts are broken.",
"In order to test the fault tolerance of DDNN, we simulate end device failures and look at the resulting accuracy of the system.",
"Figure REF shows the accuracy of the system under the presence of individual device failures.",
"Regardless of the device that is missing, the system still achieves over a $95\\%$ overall classification accuracy.",
"Specifically, even when the device with the highest individual accuracy has failed, which is Device 6, the overall accuracy is reduced by only $3\\%$ .",
"This suggests that for this dataset, the automatic fault tolerance provided by DDNN makes the system reliable even in the presence of device failure.",
"We can also view figure REF from the perspective of providing fault tolerance for the system.",
"As we decrease the number of end devices from 6 to 4, we observe that the overall accuracy of the system drops only $4\\%$ .",
"This suggests that the system can also be robust to mutliple failing end devices.",
"Figure: The impact on DDNN system accuracy when any single end device has failed."
],
[
"Reducing Communication Costs",
"DDNNs significantly reduces the communication cost of inference compared to the standard method of offloading raw sensor input to the cloud.",
"Sending a 32x32 RGB pixel image (the input size of our dataset) to the cloud costs 3072 bytes per image sample.",
"By comparison, as shown in Table REF , the largest DDNN model used in our evaluation section requires only 140 bytes of communication per sample on average (an over 20x reduction in communication costs).",
"This communication reduction for an end device results from transmitting class-label related intermediate results to the local aggregator for all samples and binarized communication with the cloud when additional NN layer processing is required for classification with improved accuracy."
],
[
"DDNN Provision for Horizontal and Vertical Scaling",
"The evaluation in the previous section shows that DDNN is able to achieve high overall accuracy through provisioning the network to scale both horizontally, across end devices, and vertically, over the network hierarchy.",
"Specifically, we show that DDNN scales vertically, by exiting easier input samples locally for low-latency response and offloading difficult samples to the cloud for high overall recognition accuracy, while maintaining a small memory footprint on the end devices and incurring a low communication cost.",
"This result is not obvious, as we need sufficiently good feature representations from the lower parts of the DNN (running on the end devices with limited resources) in order for the upper parts of the neural network (running in the cloud) to achieve high accuracy under the low communication cost constraint.",
"Therefore, we show in a positive way that the proposed method of jointly training a single DNN with multiple exit points at each part of the distributed hierarchy allows us to meet this goal.",
"That is, DDNN optimizes the lower parts of the DNN to create a sufficiently good feature representations to support both samples exited locally and those processed further in the cloud.",
"To meet the goal of horizontal scaling, we provide a principled way of jointly training a DNN with inputs from multiple devices through feature pooling via local and cloud aggregators and demonstrate that by aggregating features from each device we can dramatically improve the accuracy of the system both at the local and cloud level.",
"Filters on each device are automatically tuned to process the geographically unique inputs and work together toward to the same overall objective leading to high overall accuracy.",
"Additionally, we show that DDNN provides built-in fault tolerance across the end devices and is still able to achieve high accuracy in the presence of failed devices."
],
[
"Conclusion",
"In this paper, we propose a novel distributed deep neural network architecture (DDNN) that is distributed across computing hierarchies, consisting of the cloud, the edge and end devices.",
"We demonstrate for a multi-view, multi-camera dataset that DDNN scales vertically from a few NN layers on end devices or the edge to many NN layers in the cloud and scales horizontally across multiple end devices.",
"The aggregation of information communicated from different devices is built into the joint training of DDNN and is handled automatically at inference time.",
"This approach simplifies the implementation and deployment of distributed cloud offloading and automates sensor fusion and system fault tolerance.",
"The experimental results suggest that with our DDNN framework, a single DNN properly trained can be mapped onto a distributed computing hierarchy to meet the accuracy, communication and latency requirements of a target application while gaining inherent benefits associated with distributed computing such as fault tolerance and privacy.",
"DDNNs reduce the required communication compared to a standard cloud offloading approach by exiting many samples at the local aggregator and sending a compact binary feature representation to the cloud when additional processing is required.",
"For our evaluation dataset, the communication cost of DDNN is reduced by a factor of over 20x compared to offloading raw sensor input to a DNN in the cloud which performs all of the inference computation.",
"DDNN provides a framework for further research in mapping DNN into a distributed computing hierarchy.",
"For future work, we will investigate the performance of DDNNs on applications with a larger dataset with multiple types of input modalities [21] and more end devices.",
"Currently, all layers in DDNN are binary.",
"While binary layers are a requirement for end devices due to the limited space on devices, it is not necessary in the cloud.",
"We will explore other types of aggregation schemes and mixed precisions schemes where the end devices use binary NN layers and the cloud uses mixed-precision or floating-point NN layers."
],
[
"Acknowledgment",
"This work is supported in part by gifts from the Intel Corporation and in part by the Naval Supply Systems Command award under the Naval Postgraduate School Agreements No.",
"N00244-15-0050 and No.",
"N00244-16-1-0018."
]
] | 1709.01921 |
[
[
"A Stochastic Lagrangian particle system for the Navier-Stokes equations"
],
[
"Abstract This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process.",
"If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\to \\infty$.",
"In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \\to \\infty$.",
"The objective of this short note is to describe a resetting procedure that removes this deficiency.",
"We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy."
],
[
"Introduction",
"The Navier-Stokes equations $ \\partial _t u + (u \\cdot \\nabla )&u - \\nu \\Delta u+ \\nabla p= 0,\\\\ \\nabla &\\cdot u=0$ describe the evolution of a velocity field $u$ of an incompressible fluid with kinematic viscosity $ \\nu >0 $ , and where $p$ denotes the pressure.",
"When the viscosity vanishes, we end up with the incompressible Euler equation: $\\partial _t u + (u \\cdot \\nabla )&u+ \\nabla p= 0,\\\\\\nabla &\\cdot u=0$ which describe the motion of an ideal incompressible fluid.",
"The mathematical theory of these equations have been extensively studied and the existence of regular solutions is still an open problem in PDE theory C,F.",
"We are interested in developing probabilistic techniques, that could help solve this problem.",
"Probabilistic representations of solutions of partial differential equations as the expected value of functionals of stochastic processes date back to the work of Einstein, Feynman, Kac, and Kolmogorov in physics and mathematics.",
"The Feynman-Kac formula is the most well-known example, which has provided a link between linear parabolic partial differential equations and probability theory [13], [15].",
"These stochastic representation methods have provided in some cases tools to show existence and uniqueness of solutions to partial differential equations.",
"For nonlinear partial differential equations the earliest work was done by McKean [14], where a probabilistic representation of the solution for the nonlinear Kolmogorov-Petrovsky-Piskunov equation was given.",
"The theory for nonlinear partial differential equations, however, is far less understood.",
"The questions studied in this work are motivated by the development of a probabilistic formulation of (REF )-() proposed by P.Constantin and G.Iyer in [6].",
"There, the Navier-Stokes equation is interpreted as the average of a stochastic perturbation of the Euler equation.",
"More specifically, a Weber formula is used to represent the velocity of the inviscid equation in terms of the particle trajectories of the inviscid equation without including time derivatives, then the classical Lagrangian trajectories are replaced by stochastic flows.",
"Averaging these stochastic trajectories gives us the solution of (REF )-().",
"In [11] G.Iyer and J.Mattingly used a Monte-Carlo method to approximate the described probabilistic formulation.",
"They took $N$ independent copies of the Wiener process and replaced the expected value in the above formalism with the empirical mean, $\\frac{1}{N}$ times the sum over these $N$ independent copies (we review the details of this method in Section ).",
"By the law of the large numbers it is natural to expect that any average could be replaced by its empirical mean: $1/N\\sum _{i=1}^N X_i \\approx \\mathbb {E}(X)$ , where $X$ and $X_i$ , $i=1,\\dots , N$ are i.i.d.",
"It turns out that a straightforward approximation of this average by its empirical mean is not adequate here.",
"It was shown in [11] that in two dimensions the $N$ -particle system for the Navier-Stokes equations does not dissipate all its energy as $t \\rightarrow \\infty $ .",
"In contrast, the solution of the corresponding Navier-Stokes equations does dissipate all of its energy as $t \\rightarrow \\infty $ .",
"The goal of this paper is to alleviate this deficiency of the particle system developed in [11] by modifying it, so that the modified particle system dissipates all of its energy.",
"Our modification is inspired by [12], where G.Iyer and A.Novikov studied a particle system formulation for the Burgers equation.",
"There we can find another example, where the particle system does not fully mimic properties of the corresponding PDE.",
"Namely, the viscous Burgers equation does not develop shocks, but the corresponding $N$ -particle system shocks almost surely in finite time.",
"In order to remove these shocks, the authors [12] considered a resetting procedure that prevented their formation.",
"In this paper we propose another resetting procedure, such that the particle system for the Navier-Stokes equations dissipate all its energy.",
"The particle system in [11] does not completely dissipate its energy, because, roughly speaking, the gradients of the velocities for the $N$ particles become decorrelated with time.",
"We reinforce correlation of these velocities and their gradients by resetting, and this allows complete dissipation of energy.",
"When the resetting condition holds, then the particle system dissipates its energy exponentially.",
"Once the resetting condition is not satisfied, we reset the particle system, and restart the procedure again.",
"In addition to the exponential dissipation of energy, the resetting procedure itself adds more dissipation each time we average our data.",
"Our theorem states that if we keep repeating the resetting procedure, the particle system will dissipate all its energy.",
"We now highlight briefly some other probabilistic formulations for the Navier-Stokes equations and related work.",
"The initial work on probabilistic representation of the Navier-Stokes equations was done by A.Chorin.",
"It was shown in [1] that in two dimensions vorticity evolves according to Fokker-Planck type equation.",
"Using this A.Chorin gave a probabilistic representation for the vorticity using random walks and a particle limit, and then related it to the velocity vector using the Biot-Savart Law.",
"Using a different approach Y.Le Jan and A.Sznitman in [17] developed a probabilistic representation of Navier-Stokes equations, where they used a backward-in-time branching process in Fourier space to express the velocity of the three dimensional viscous fluid as the average of a stochastic process.",
"These works do not allow to develop a self contained well-posedness theory.",
"Here our motivation is to develop new probabilistic techniques that may help us establish regularity theory for partial differential equations of Fluid Dynamics.",
"The stochastic representation in [6] allows such theory, and we, therefore, chose to work with it.",
"Building on the work done by P.Constantin and G.Iyer in [6], X.Zhang in [18] gave a stochastic representation for the backward incompressible Navier-Stokes equation using stochastic Lagrangian path.",
"Later, linking the work of X.Zhang in [18] with P.Constantin and G.Iyer in [6], F.Delbaen, J.Qiu, and S.Tang in [7] developed a coupled forward-backward stochastic differential system in the space of fields for the incompressible Navier-Stokes equation in the whole space.",
"Using probabilistic tools, they were able to obtain local uniqueness results for the forward-backward stochastic differential system.",
"In addition, they were able to show the existence of global solutions for the case with small Reynolds number or when the dimension is two.",
"We also mention [2], where F.Cipriano and A.Cruzeiro using the Brownian motions on the group of homeomorphisms on the torus, established a stochastic variational principle for the two dimensional Navier-Stokes equations.",
"Moreover, A.Cruzeiro and E.Shamarova in [3] formulated a connection between the Navier-Stokes equations and a system of forward-backward stochastic differential equations on the group of volume-preserving diffeomorphisms of a flat torus.",
"This paper is organized as follows.",
"In Section we describe the stochastic-Lagrangian representation of the Navier-Stokes equations and construct the particle system.",
"In addition, we explain the resetting scheme which will then be used in Section .",
"In Section , the main section of this paper, we study the energy of the Navier-Stokes's particle system and we use the resetting procedure and show that by repeating it often enough, the particle system for the Navier-Stokes equations dissipates all its energy."
],
[
"The particle system and the resetting",
"In this section we construct the particle system for the Navier-Stokes equations based on stochastic Lagrangian trajectories.",
"We begin by describing a stochastic Lagrangian formulation of the Navier-Stokes equations [11].",
"Let $B_t$ be a standard 2 or 3-dimensional Brownian motion on a torus $\\mathbb {T}$ , and let $u_0$ be some given periodic, and divergence free $C^{{k,\\alpha }}$ initial data.",
"Let $\\mathbb {E}$ denote the expected value with respect to the Wiener measure and $\\mathrm {P}$ be the Leray-Hodge projection onto divergence free vector fields.",
"Consider the system of equations $\\ dX_t(x) &= u_t ( X_t(x) ) \\, dt + \\sqrt{2\\nu }\\,dB_t, \\qquad X_0(x) = x, \\\\u_t &= \\mathbb {E}\\, \\mathrm {P}\\left[ (\\nabla ^*Y_t) (u_0 \\circ Y_t)\\right],\\qquad Y_t = X_t^{-1}.",
"$ Above $X_t$ is the stochastic flow of diffeomorphisms on $\\mathbb {T}$ , and we denote $Y_t = X_t^{-1}$ to be the spatial inverse of $X_t$ for any given time $t \\geqslant 0$ .",
"We denote $\\nabla ^*Y_t$ to be the transpose of the Jacobian of $Y_t$ .",
"It was shown in [6] that if the initial data $u_0 \\in C^{{k,\\alpha }}$ is a deterministic divergence free vector field with $k \\geqslant 2$ and if we impose periodic boundary conditions on $u_t$ and $X_t - \\mathbb {I}$ , then for a short time the system (REF )-() is equivalent to the incompressible Navier-Stokes equations, that is, $u_t$ satisfies $\\partial _t u_t + (u_t \\cdot \\nabla ) u_t - \\nu \\Delta u_t+ \\nabla p= 0, \\hspace{2.84544pt}\\nabla \\cdot u_t=0.$ In the case when the viscosity is zero $\\nu = 0$ , the equations (REF )-() are the Lagrangian formulation for the incompressible Euler equation developed in [4].",
"Note that we need the law of the entire flow $X$ in order to compute $u$ , this is due to the fact that the term $\\nabla ^{*}Y$ is present in ().",
"In order to approximate the system (REF )-(), we replace the flow $X_t$ with $N$ different copies $X^{i}_t$ where each one is driven independently by a Wiener process $B^i_t$ , $i=1,2, \\dots , N$ .",
"Fix a (sufficiently large) $N$ , and we end up with the following approximate system $dX^{i}_t &= u_t \\left( X^{i}_t \\right) \\, dt + \\sqrt{2\\nu }\\,dB^i_t,\\quad Y^{i}_t = \\left( X^{i}_t \\right)^{-1},\\\\u_t &= \\frac{1}{N} \\sum _{i=1}^N u^{i}_t, \\hspace{8.5359pt}u^{i}_t = \\mathrm {P}[ (\\nabla ^*Y^{i}_t) u_0\\circ Y^{i}_t], $ with initial data $X_0(x) = x$ .",
"We impose periodic boundary conditions on the initial data $u_0$ , and the displacement $\\lambda ^{i}_t(x) =X^{I}_t(x) - x$ .",
"The following Lemma describes the evolution of the velocity of the particle system () as SPDE.",
"Lemma 2.1 (Lemma 4.2 in [11]) Let $u^{i}_t = \\mathrm {P}[ (\\nabla ^*Y^{i}_t) u_{0} \\circ Y^{i}_t]$ be the $i^\\text{th}$ summand in ().",
"Then $u^{i}_t$ satisfies the SPDE $\\ d u^{i}_t + \\left[ (u_t \\cdot \\nabla ) u^{i}_t - \\nu \\triangle u^{i}_t + (\\nabla ^*u_t) u^{i}_t + \\nabla p^{i}_t\\right]dt + \\sqrt{2\\nu } \\nabla u^{i}_t dB^i_t = 0,$ and $u_t$ satisfies the SPDE $d u_t + \\left[ (u_t \\cdot \\nabla ) u_t - \\nu \\triangle u_t + \\nabla p_t \\right] \\, dt + \\frac{\\sqrt{2\\nu }}{N} \\sum _{i=1}^N \\nabla u^{i}_t dB^i_t = 0.$ In contrast to the true Navier-Stokes equations (REF )-() the particle system (REF )-(), for any finite $N$ , may not dissipate all of its energy as $t \\rightarrow \\infty $ .",
"In two dimensions this was proven in [11].",
"In this work, we propose to alleviate this deficiency by considering a resetting scheme, in which we start by solving the system (REF )-() on the interval $(t_0, t_1] $ , where the resetting time $t_1$ is going to be specified later according to our proposed resetting condition $(\\ref {times})$ below.",
"Next, we average our data, replace the original initial data with $u^{N}_{t_1}$ , and we restart the system (REF )-() for the next time interval using this new initial data.",
"We keep repeating this procedure on each interval $(t_m, t_{m+1}]$ for $m \\in \\mathbb {N}$ .",
"The resetting criterion comes from comparison of the rate of change of the energies of the true Navier-Stokes equation and the particle system.",
"Namely, the rate of change of the energy of our particle system (REF )-() is (see Theorem REF below) as follows.",
"$ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\frac{1}{N^2} \\hspace{4.26773pt} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^{j}_t, \\nabla u^{i}_t \\rangle \\right].$ Observe that the rate of change of energy depends on the average of inner products of the gradients of $N$ velocities.",
"In contrast, for the true Navier-Stokes equation the rate of change of the energy $\\frac{1}{2\\nu } \\partial _t \\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2.$ For large $N$ the right-hand sides of (REF ) and (REF ) are essentiallyIf $\\nabla u^{j}_t=\\nabla u^{i}_t$ , then the right-hand side of (REF ) is $ (N-1) N^{-1} \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\rightarrow \\Vert \\nabla u_t \\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ , as $N \\rightarrow \\infty $ .",
"the same, if $\\nabla u^{j}_t=\\nabla u^{i}_t$ for all $i$ and $j$ .",
"This observation motivates our approach.",
"We will use resetting to keep the sum of the expected value of the inner products $ \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^{j}_t, \\nabla u^{i}_t \\rangle \\right] \\geqslant \\hspace{1.42271pt} c N^2 \\hspace{1.42271pt} \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ on each interval $(t_m, t_{m+1}]$ where $m \\in \\mathbb {N}$ and for some constant $c >0$ that does not depend on $N$ .",
"This will make the inner products in (REF ) to be positive and thus the rate of energy dissipation will be negative on each interval $(t_m, t_{m+1}]$ .",
"Therefore we consider the following resetting system $dX^{i}_{t } &= u_t \\left( X^{i}_{t } \\right) \\, dt + \\sqrt{2\\nu }\\,dB^i_t, \\hspace{5.69046pt} X^{i}_{t_m }(x_0)=x_0,~Y^{i}_{t }= \\left( X^{i}_{t } \\right)^{-1},\\\\u_{t } &= \\frac{1}{N} \\sum _{i=1}^N u^{i}_t, \\hspace{5.69046pt} u^{i}_t = \\mathrm {P}\\left[ (\\nabla ^*Y^{i}_{t }) (u_{t_m} \\circ Y^{i}_{ t }) \\right], \\hspace{5.69046pt} \\text{for} \\hspace{5.69046pt}t \\in (t_m, t_{m+1}], $ where $m \\in \\mathbb {N}$ , the set of non-negative integers, $t_0=0$ , and the resetting times are defined recursively $ t_{m} = \\inf \\Big \\lbrace t > t_{m-1}: \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\nabla u^i_t, \\nabla u^j_t \\rangle \\right] < (1- \\varepsilon ) N(N-1) \\left\\Vert \\nabla u_{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\Big \\rbrace .$ for some positive fixed $\\varepsilon <1$ .",
"We say we reset the system at every $t = t_m$ , because we treat $u_{ t_m}$ as initial conditions for each of the intervals $t \\in [ t_m, t_{m+1} )$ .",
"Theorem 2.2 Suppose we are in two dimensions.",
"Let the initial condition $u_{0}$ be a $\\mathcal {F}_{0}$ -measurable, periodic mean zero function such that the norm $\\Vert u_{0}\\Vert _{{1,\\alpha }}$ , $\\alpha >0$ is almost surely bounded.",
"If we let $\\lbrace t_m\\rbrace _{m=0}^{\\infty }$ to be the sequence of resetting times defined in (REF ), then the particle system with resetting (REF )-() dissipates all its energy.",
"We remark that the particle system with resetting (REF )-() dissipates its energy using two mechanisms.",
"It dissipates energy exponentially anytime the inequality $(\\ref {EnergyCondition})$ holds, and it dissipates energy when we average our data each time we reset our system.",
"We also want to highlight that the manner in which we defined our resetting times in (REF ) causes the length of time increments $\\delta _m= t_m-t_{m-1}$ to vary among resetting intervals.",
"This gives rise to the case that if the sequence of time increments $\\delta _m $ decays to zero too fast, then the limit of the sequence of resetting times $t_m \\rightarrow T$ , for some constant time $T$ .",
"Thus, we have to consider two cases.",
"First case is when the limit of sequence of resetting times $t_m \\rightarrow \\infty $ , as $m \\rightarrow \\infty $ .",
"In this case we show that the energy of the particle system with resetting (REF )-() dissipates its energy mainly exponentially.",
"The second case is when the limit of the sequence of resetting times $t_m \\rightarrow T$ , for some finite time $T$ .",
"In this case, we show that the system (REF )-() dissipates its energy mainly by averaging each time we reset."
],
[
" Energy decay by resetting",
"By Lemma REF , $u^{i}_t$ satisfies the SPDE $d u^{i}_t + \\left[ (u_t \\cdot \\nabla ) u^{i}_t - \\nu \\triangle u^{i}_t + (\\nabla ^*u_t) u^{i}_t + \\nabla p^{i}_t\\right]dt + \\sqrt{2\\nu } \\nabla u^{i}_t dB^i_t = 0,$ on each interval $t \\in § (t_m, t_{m+1}]$ .",
"In two dimensions, the vorticity $\\omega ^i_t = \\nabla \\times u^i_t$ solves $d \\omega ^i_t+ [ (u_t \\cdot \\nabla ) \\omega ^i -\\nu \\Delta \\omega ^i_t] dt +\\sqrt{2 \\nu } \\nabla \\omega ^i_t dB^i_t=0.$ By Itô's formula, we obtain $\\frac{1}{2} d \\left|\\omega ^i_t\\right|^2 + \\omega ^i_t \\cdot \\left[ (u_t \\cdot \\nabla ) \\omega ^i_t - \\nu \\triangle \\omega ^i_t \\right] \\, dt+ \\sqrt{2 \\nu } \\omega ^i_t \\cdot (\\nabla \\omega ^i_t\\, dB^i_t) -\\nu \\left|\\nabla \\omega ^i_t\\right|^2 \\, dt=0$ Integrating in space and using the fact that $u^i_t$ and $u_t$ are divergences free, we have $ d \\Vert \\omega ^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=0$ for all $ t \\in [t_m, t_{m+1})$ .",
"Thus the norm of all the vorticities is preserved on such time-intervals.",
"Since $\\Vert \\omega ^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 =\\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ we have $\\left\\Vert \\nabla u_{t_m}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 =\\left\\Vert \\nabla u^1_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=\\left\\Vert \\nabla u^2_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2=\\dots =\\left\\Vert \\nabla u^N_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for $t \\in § (t_m, t_{m+1}]$ .",
"Using Lemma REF and Itô's formula, we also have $\\frac{1}{2} d \\left|u_t\\right|^2 + u_t \\cdot \\left[ (u_t \\cdot \\nabla ) u_t - \\nu \\triangle u_t + \\nabla p \\right] \\, dt+ \\frac{\\sqrt{2 \\nu }}{N} \\sum _{i=1}^N u_t \\cdot (\\nabla u^i_t\\, dB^i_t) \\\\=\\frac{\\nu }{N^2} \\sum _{i=1}^N \\vert \\nabla u^i_t\\vert ^2 \\, dt$ Integrating in space and taking expected values, we obtain $ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= \\mathbb {E}\\left[\\frac{1}{N^2} \\sum _{i=1}^N \\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 -\\Vert \\nabla u_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\right]\\\\ &= \\mathbb {E}\\left[ \\frac{1}{N^2} \\sum _{i=1}^N \\Vert \\nabla u^i_t\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 - \\frac{1}{N^2} \\bigg [ \\sum _{i=1}^N \\Vert \\nabla u^i_t \\Vert ^2_{L^2}+ \\sum _{i \\ne j }^N \\langle \\nabla u^j_t, \\nabla u^i_t \\rangle \\bigg ] \\right]$ This simplifies to $ \\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 &= - \\frac{1}{N^2} \\hspace{4.26773pt} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\nabla u^j_t, \\nabla u^i_t \\rangle \\right]$ Case I: $\\lim _{m \\rightarrow \\infty } t_m \\rightarrow \\infty $ .",
"In this case the sequence of resetting times goes to infinity.",
"Using resetting, we have $ \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\nabla u^i_t, \\nabla u^j_t \\rangle \\right] \\geqslant (1- \\varepsilon ) N(N-1) \\left\\Vert \\nabla u_{t_{m} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for all $t \\in [t_m, t_{m+1})$ .",
"Thus, using $(\\ref {ResettingCondtion}) $ we can obtain the following estimate on the rate of change of energy $(\\ref {RateEnergy3})$ $\\frac{1}{2\\nu } \\partial _t \\mathbb {E}\\left\\Vert u_t\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant - (1- \\varepsilon )\\frac{N-1}{N} \\left\\Vert \\nabla u_{t_m }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ $\\leqslant -C(1- \\varepsilon )\\frac{N-1}{N} \\mathbb {E}\\left\\Vert u_{t_m }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant -C(1- \\varepsilon )\\frac{N-1}{N} \\mathbb {E}\\left\\Vert u_{t }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ for some constant $C>0$ that arises from using Poincaré's inequality.",
"Using the Gronwall's inequality, we obtain exponential dissipation of energy.",
"Case II: $ \\lim _{m \\rightarrow \\infty } t_m \\rightarrow T $ .",
"In this case the sequence of resetting times converges to a finite time $T$ .",
"At every resetting time we have $\\omega _{t_{m} }= \\frac{1}{N} \\hspace{4.26773pt} \\sum _{i=1 }^N \\omega ^i_{t_{m}}.$ Thus, $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2= \\frac{1}{N^2} \\sum _{i=1 }^N \\mathbb {E}\\left\\Vert \\omega ^i_{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 + \\frac{1}{N^2} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}} , \\omega ^j_{t_{m}} \\rangle \\right]$ $= \\frac{1}{N} \\left\\Vert \\omega _{t_{m-1}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 + \\frac{1}{N^2} \\sum _{i \\ne j }^N \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}} , \\omega ^j_{t_{m}} \\rangle \\right].",
"$ Since $\\langle \\nabla u^i_{t_{m}}, \\nabla u^j_{t_{m}} \\rangle =\\langle \\omega ^i_{t_{m}}, \\omega ^j_{t_{m}} \\rangle $ , we have $ \\sum _{i \\ne j }^N \\hspace{4.26773pt} \\mathbb {E}\\left[ \\langle \\omega ^i_{t_{m}}, \\omega ^j_{t_{m}} \\rangle \\right] \\leqslant (1- \\varepsilon ) N(N-1) \\left\\Vert \\omega _{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2$ $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big ) \\left\\Vert \\omega _{t_{m-1} }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2, \\alpha =1 -\\varepsilon +\\frac{\\varepsilon }{N} <1.$ Iterating over $m$ , we have $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big )^m \\left\\Vert \\omega _{0 }\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2, \\hspace{2.84544pt} \\text{where} \\hspace{2.84544pt} \\omega _0 = \\nabla \\times u_0.$ Thus, if $\\lim _{m \\rightarrow \\infty } t_m \\rightarrow T$ , for some finite time $T$ , this means we are going to reset our particle system a countable number of times.",
"Hence, $\\mathbb {E}\\left\\Vert \\omega _{t_{m}}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\leqslant \\big ( 1 -\\alpha \\big )^m \\left\\Vert \\omega _{0}\\right\\Vert _{\\smash{L^{\\!2}_{\\vphantom{h}}}\\vphantom{L^{\\!2}}}^2 \\rightarrow 0\\hspace{5.69046pt} \\text{as} \\hspace{5.69046pt} m \\rightarrow \\infty .$ Thus, the particle system dissipates all its energy in a finite time $T$ ."
]
] | 1709.01536 |
[
[
"Contact forms with large systolic ratio in dimension three"
],
[
"Abstract The systolic ratio of a contact form on a closed three-manifold is the quotient of the square of the shortest period of closed Reeb orbits by the contact volume.",
"We show that every co-orientable contact structure on any closed three-manifold is defined by a contact form with arbitrarily large systolic ratio.",
"This shows that the many existing systolic inequalities in Finsler and Riemannian geometry are not purely contact-topological phenomena."
],
[
"Introduction and main result",
"Given a Riemannian metric $g$ on a closed surface $S$ , we denote the length of a shortest non-constant closed geodesic by $\\ell _{\\min }(S,g)$ and consider the scaling invariant ratio $\\rho (S,g) := \\frac{\\ell _{\\min }(S,g)^2}{\\mathrm {area}(S,g)},$ where $\\mathrm {area}(S,g)$ denotes the Riemannian area.",
"It is a classical question in systolic geometry to find upper bounds, and possibly optimal ones, for this ratio.",
"When $S$ is not the 2-sphere, the fact that $S$ is not simply connected allows one to bound $\\rho (S,g)$ from above by the ratio $\\rho _{\\rm nc}(S,g) :=\\frac{\\mathrm {sys}_1(S,g)^2}{\\mathrm {area}(S^2,g)},$ where $\\mathrm {sys}_1(S,g)$ denotes the length of a shortest non-contractible curve on $(S,g)$ , which is of course a closed geodesic.",
"When $S$ is the 2-torus 2, a classical result of Loewner from the end of the 1940s states that $\\rho _{\\rm nc}(2,\\cdot )$ is maximised by the flat metric corresponding to the lattice generated by two sides of an equilateral triangle in $\\mathbb {R}^2$ .",
"The corresponding maximal value of $\\rho _{\\rm nc}$ is $2/\\sqrt{3}$ .",
"Shortly after, Pu [15] considered the case of the projective plane $\\mathbb {R}\\mathbb {P}^2$ and showed that $\\rho _{\\rm nc}(\\mathbb {R}\\mathbb {P}^2,\\cdot )$ is maximized by the standard constant curvature metric, with maximal value $\\pi /2$ .",
"In both cases, the metrics maximizing $\\rho _{\\rm nc}$ have no contractible closed geodesics, so these metrics maximize also the ratio $\\rho $ .",
"For an arbitrary closed non-simply connected surface $S$ , the ratio $\\rho _{\\rm nc}(S,\\cdot )$ is bounded from above by the non-optimal universal constant 2, as shown by Gromov in [12], but the supremum is in general not achieved.",
"Actually, in [12] Gromov proved a far reaching generalization of this result by showing that for any essential $n$ -dimensional closed Riemannian manifold $(M,g)$ , the quotient $\\rho _{\\rm nc}(M,g) := \\frac{\\mathrm {sys}_1(M,g)^n}{\\mathrm {vol}(M,g)}$ has an upper bound which depends only on the dimension $n$ .",
"Here, a non-simply connected closed oriented manifold $M$ is said to be essential if its fundamental class is non-zero in the homology of the Eilenberg-MacLane space $K(\\pi _1(M),1)$ .",
"This condition generalizes asphericity, that is the fact that all homotopy groups of degree larger than one vanish, and characterizes manifolds for which the ratio $\\rho _{\\rm nc}$ is bounded from above, see [6].",
"Going back to surfaces, from the fact mentioned above we deduce that the number 2 is an upper bound also for $\\rho (S,\\cdot )$ , when $S\\ne S^2$ .",
"In the case of the two-sphere $S^2$ , the first upper bound for $\\rho (S^2,\\cdot )$ was found by Croke in [7] and later improved by several authors, the best one so far being the bound 32 found by Rotman in [16].",
"We conclude that $\\rho (S,\\cdot )$ is always bounded from above when $S$ is a closed surface, and there is an upper bound which is independent of $S$ .",
"The ratios $\\rho $ and $\\rho _{\\rm nc}$ generalize to Finsler metrics, by replacing the Riemannian area by the Holmes-Thompson area.",
"Recent results of Àlvarez Paiva, Balacheff and Tzanev allow to extend the Riemannian bounds to the Finsler setting: The value of the supremum might get larger (and sometimes it does get larger, as in the case of 2, for which the Loewner metric does not maximize $\\rho $ and $\\rho _{\\rm nc}$ , even locally, among Finsler metrics) but is in any case finite.",
"See [5].",
"The next natural generalization of the ratio $\\rho $ is to the setting of contact geometry.",
"We recall that a contact form $\\alpha $ on a 3-manifold $M$ is a smooth 1-form such that $\\alpha \\wedge d\\alpha $ is a volume form.",
"The kernel of $\\alpha $ is a plane distribution and is called the contact structure defined by $\\alpha $ .",
"In this paper, contact structures are always assumed to be co-orientable, that is, defined by a global contact form.",
"If two contact forms $\\alpha _1,\\alpha _2$ on $M$ define the same contact structure then the volume forms $\\alpha _1\\wedge d\\alpha _1$ and $\\alpha _2\\wedge d\\alpha _2$ define the same orientation.",
"This orientation is said to be induced by the contact structure, and in this paper all contact 3-manifolds are oriented by the contact structure.",
"A contact form $\\alpha $ induces a nowhere vanishing vector field $R_{\\alpha }$ on $M$ , which is defined by the conditions $\\imath _{R_{\\alpha }} d\\alpha = 0 , \\qquad \\alpha (R_{\\alpha })=1,$ and is called the Reeb vector field for $\\alpha $ .",
"Any Reeb vector field on a closed 3-manifold $M$ admits periodic orbits, as proved by Taubes in [17], and we denote by $T_{\\min }(M,\\alpha )$ the minimum among the periods of all periodic orbits of $R_{\\alpha }$ .",
"Then it is natural to define $\\rho (M,\\alpha ):= \\frac{T_{\\min }(M,\\alpha )^2}{\\mathrm {vol}(M,\\alpha \\wedge d\\alpha )}.$ We will refer to this quantity as the systolic ratio of $(M,\\alpha )$ .",
"It is scaling invariant, because if we multiply $\\alpha $ by a non-zero real number $c$ then both $T_{\\min }^2$ and the volume get multiplied by $c^2$ .",
"The inverse of this quantity appears in [4] under the name of systolic volume of $(M,\\alpha )$ .",
"Let $F$ be a smooth Finsler metric on the closed surface $S$ .",
"The unit cotangent sphere bundle $S^*_FS$ is the space of cotangent vectors having norm 1 with respect to the dual Finsler metric $F^*$ .",
"The canonical Liouville 1-form $p\\, dq$ of $T^* S$ restricts to a contact form $\\alpha _F$ on $S_F^*S$ .",
"Two different Finsler metrics $F$ and $F^{\\prime }$ induce contact forms $\\alpha _F$ and $\\alpha _{F^{\\prime }}$ which correspond to the same contact structure, once $S_F^*S$ and $S_{F^{\\prime }}^*S$ are identified by means of the radial projection.",
"The flow of the corresponding Reeb vector field $R_{\\alpha _F}$ is precisely the geodesic flow of $F$ , once this is read on the cotangent bundle by the Legendre transform.",
"Therefore, $\\ell _{\\min }(S,F)=T_{\\min }(\\alpha _F)$ .",
"Moreover, the volume of $S^*_FS$ with respect to $\\alpha _F \\wedge d\\alpha _F$ is, essentially by definition, $2\\pi $ times the Holmes-Thompson area of $(S,F)$ .",
"We conclude that $\\rho (S^*_FS,\\alpha _F) = \\frac{\\rho (S,F)}{2\\pi },$ and the systolic ratio of a contact form is a genuine generalization of the corresponding notion from Riemannian and Finsler geometry.",
"All of this generalizes to higher dimensions, but here we restrict our attention to three-dimensional contact manifolds.",
"Seeing Riemannian and Finsler geometry in the larger contact setting is often fruitful.",
"On the one hand, results about existence and multiplicity of Riemannian and Finsler closed geodesics have often natural generalizations to Reeb flows, see e.g.",
"[13], and the same holds for some statements about the topological entropy of geodesic flows, see e.g.",
"[14], [9], [1].",
"On the other hand, techniques from contact geometry have been recently found to be useful to address systolic questions in Riemannian and Finlser geometry, such as the local systolic maximality of Zoll metrics, see [4], [2], [3].",
"It is therefore a natural question to ask whether the systolic ratio of contact forms inducing a given contact structure on a closed three-manifold also has uniform upper bounds.",
"The purpose of this paper is to give a negative answer to this question.",
"More precisely, we shall prove the following: Theorem Let $\\xi $ be a contact structure on a closed 3-manifold $M$ .",
"For every $c>0$ there exists a contact form $\\alpha $ satisfying $\\ker \\alpha =\\xi $ and $\\rho (M,\\alpha )\\ge c.$ When applied to the cotangent sphere bundle of an arbitrary closed surface $S$ , the above theorem has the following consequence: If $T$ is any positive number, then there exists a fiberwise starshaped domain $A\\subset T^*S$ (i.e.",
"a connected open neighborhood of the zero-section with smooth boundary $\\partial A$ such that for every $q\\in S$ the set $\\partial A \\cap T_q^* S$ is a closed curve in $T^*_q S\\setminus \\lbrace 0\\rbrace $ which is transverse to the radial direction) with volume 1 (with respect to the standard symplectic form $dp\\wedge dq$ of $T^*S$ ) and such that every closed orbit of the Reeb vector field on $\\partial A$ determined by the restriction of the Liouville form $p\\, dq$ has period larger than $T$ .",
"When $A$ is fiberwise convex then $\\partial A$ is the unit sphere bundle of some Finsler metric and the results of Àlvarez Paiva, Balacheff and Tzanev mentioned above prevent this phenomenon.",
"This shows that convexity plays an essential role in systolic inequalities.",
"In the special case of the tight three-sphere $(S^3,\\xi _{\\rm std})$ , that is the standard unit sphere in $\\mathbb {R}^4$ endowed with coordinaten $(x_1,y_1,x_2,y_2)$ and with the contact structure $\\xi _{\\rm std}$ which is given by the kernel of the contact form $\\alpha _0 = \\frac{1}{2} \\sum _{j=1}^2 (x_j \\, dy_j - y_j\\, dx_j) \\Big |_{S^3}$ the unboundedness of the systolic ratio was proved in [3].",
"The argument in [3] starts with the construction of a special symplectomorphism of the disk with a good lower bound on actions of periodic points and with a suitable negative value of the Calabi invariant (see Section and references therein for the definition of these notions).",
"This disk-map is then embedded as a return map to a disk-like global surface of section of some Reeb flow on $(S^3,\\xi _{\\rm std})$ in a way that there is a precise “dictionary”: Actions of periodic points correspond to periods of closed Reeb orbits and the Calabi invariant corresponds to the contact volume, up to explicit additive constants.",
"The tight three-sphere is very simple from a contact-topological point of view, since it admits the trivial supporting open book decomposition with unknotted binding, disk-like pages and trivial monodromy (see Section below and references therein for the definition of these notions).",
"Moreover, the construction of the disk-map in [3] uses the special symmetries of the disk.",
"In order to deal with a general contact structure $\\xi $ on a general orientable closed three-manifold $M$ we argue in the following way.",
"First we use the fact that by a theorem of Giroux [11] $\\xi $ is supported by an open book decomposition of $M$ to construct a contact form $\\alpha $ which is adapted to the open book decomposition and is such that the supports of the monodromy and return maps are contained in a region of small contact area, as visualized in Figure 1.",
"The construction of this special contact form is explained in Section , where the necessary background about open book decompositions is also recalled.",
"The next step is to construct suitable contact forms on a solid torus which are standard near the boundary, have a small contact volume and no closed Reeb orbits with small period.",
"These are constructed in Section , by building on results from [3].",
"Finally, the latter solid tori are used as plugs to modify the contact form $\\alpha $ constructed in Section .",
"Indeed, these plugs can be inserted in the region spanned by the large portions of the pages on which the monodromy and first return map are the identity: Their effect is to eat up volume without creating short periodic orbits.",
"In this way we can reduce the contact volume as much as we wish, while keeping $T_{\\min }$ bounded away from zero, and hence making the systolic ratio arbitrarily large.",
"This final step is performed in Section ."
],
[
"The research of A. Abbondandolo and B. Bramham is supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the Deutsche Forschungsgemeinschaft.",
"U. Hryniewicz thanks the Floer Center for Geometry (Bochum) for its warm hospitality, and acknowledges the generous support of the Alexander von Humboldt Foundation.",
"U. Hryniewicz is also supported by CNPq grant 309966/2016-7.",
"P. Salomão is supported by FAPESP grant 2016/25053-8 and CNPq grant 306106/2016-7.",
"Figure: On the left a typical page as one would picture it in an open book decomposition.",
"On the right, the same page on an open book supporting a contact structure where the topology, and the supports of the monodromy and return maps are squeezed in a region with small contact area.",
"This leaves lots of space to embed solid tori with high systolic ratio."
],
[
"A special contact form",
"We recall that a global surface of section for a smooth flow $\\phi ^t$ on a 3-dimensional manifold $M$ is a compact surface $S$ smoothly embedded in $M$ whose boundary $\\partial S$ is $\\phi ^t$ -invariant and such that all orbits which are not contained in $\\partial S$ meet the surface transversally infinitely many times in the future and in the past.",
"The corresponding first return time function is the smooth function $\\tau : S \\setminus \\partial S \\rightarrow (0,+\\infty ), \\qquad \\tau (p) := \\inf \\lbrace t>0 \\mid \\phi ^t(p)\\in S\\rbrace ,$ and the corresponding first return map is the smooth diffeomorphism $\\varphi : S \\setminus \\partial S \\rightarrow S \\setminus \\partial S, \\qquad \\varphi (p) := \\phi ^{\\tau (p)}(p).$ When $\\phi ^t$ is the Reeb flow of a contact form $\\alpha $ , the transversality of the flow to $S\\setminus \\partial S$ implies that the restriction of $d\\alpha $ to $S\\setminus \\partial S$ is an area form.",
"The exactness of $d\\alpha $ implies that in the Reeb case the boundary of $S$ is necessarily non-empty.",
"Since Reeb vector fields are non-singular, the finitely many circles forming the boundary of $S$ are periodic orbits.",
"Moreover, it is well known that the contact volume of $M$ coincides with the integral of $\\tau $ on $S$ , once this surface is equipped with the area form given by the restriction of $\\alpha $ : $\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) = \\int _S \\tau \\, d\\alpha .$ See e.g.",
"[3] for the easy proof.",
"Proposition 1 Let $M$ be a closed connected 3-manifold with a contact structure $\\xi $ .",
"Then there is an embedded compact surface $S\\subset M$ with the following property: For every $\\epsilon >0$ there exists a contact form $\\alpha $ on $M$ satisfying $\\xi =\\ker \\alpha $ such that $S$ is a global surface of section for the Reeb flow of $\\alpha $ and: If we orient $S$ by $d\\alpha $ and give $\\partial S$ the boundary orientation, we have that $\\int _C \\alpha = 1$ for every connected component $C$ of $\\partial S$ .",
"In particular, $\\int _S d\\alpha = \\int _{\\partial S} \\alpha = \\ell ,$ where $\\ell \\ge 1$ denotes the number of circles forming the boundary of $S$ .",
"The first return time function $\\tau $ of the Reeb flow of $\\alpha $ extends to a smooth function on $S$ and the corresponding first return map $\\varphi $ extends to a smooth diffeomorphism of $S$ onto itself.",
"There exists an open tubular neighborhood $U$ of $\\partial S$ in $S$ such that the support of $\\varphi $ is contained in $S\\setminus U$ and $\\int _{S\\setminus U} d\\alpha < \\epsilon .$ $\\Vert \\tau -1\\Vert _{\\infty } < \\epsilon $ and $\\tau $ is constantly equal to 1 on the neighborhood $U$ from (iii).",
"In order to prove this proposition, we need to recall some facts about open book decompositions.",
"See [8] for more details.",
"An open book decomposition of an oriented closed 3-manifold $M$ is a pair $(\\Pi ,L)$ , where $L\\subset M$ is a smooth link and $\\Pi :M\\setminus L \\rightarrow \\mathbb {R}/2\\pi \\mathbb {Z}$ is a smooth (locally trivial) fibration which near $L$ is a normal angular coordinate.",
"More precisely, there exists a compact neighborhood $N$ of $L$ and a diffeomorphism $N \\simeq L \\times such that $ L L {0}$ and $ |N L$ is represented as$$(\\lambda ,r e^{ix})\\mapsto x,$$where $$ belongs to $ L$ and $ (r,x)(0,1]R/2Z$ are polar coordinates on the punctured closed unit disc $ {0}.",
"The link $L$ is called the binding and the fibers of $\\Pi $ are called the pages of the open book decomposition.",
"The closure of each page is a Seifert surface for $L$ , that is, a compact smoothly embedded surface with boundary $L$ .",
"The co-orientation of the pages given by the map $\\Pi $ and the ambient orientation induce an orientation of the pages.",
"The link $L$ inherits the boundary orientation.",
"Let $S$ be the closure in $M$ of the page $\\Pi ^{-1}(0)$ .",
"The coordinates introduced above induce coordinates $(\\lambda ,r) \\in \\partial S\\times [0,1]$ on the compact neighborhood $N\\cap S$ of $\\partial S$ in $S$ , where $[0,1]$ is seen as the intersection of $ with the positive real half-axis.$ The vector field $\\partial _x$ on $N \\simeq L \\times is smooth, vanishes at $ LL {0}$ and generates the group of rotations\\begin{equation}\\bigl (t,(\\lambda ,\\rho e^{ix}) \\bigr ) \\mapsto (\\lambda ,\\rho e^{i(x+t)}).\\end{equation}Let $ Z$ be a smooth vector field on $ M$ which coincides with $ x$ on $ N$ and is positively transverse to all the pages, meaning that $ dZ >0$ on $ ML$.",
"The set of such vector fields is obviously convex and in particular path connected.$ The surface $S$ is a global surface of section for the flow of $Z$ .",
"The form () of the flow of $Z$ on $N$ implies that the corresponding first return map is the identity on the set $N \\cap (S\\setminus \\partial S)$ , and hence it extends smoothly to the boundary and gives us an orientation preserving diffeomorphism $h: S \\rightarrow S,$ which is the identity on $N \\cap S$ .",
"The diffeomorphism $h$ is called the monodromy map of the open book decomposition which depends on the choice of the vector field $Z$ .",
"However as the set of such vector fields is path connected the isotopy class of $h$ within the space of diffeomorphisms of $S$ that are the identity in a neighborhood of $\\partial S$ is uniquely determined.",
"The mapping torus of $h$ is the smooth 3-manifold with boundary $S(h) := \\left([0,2\\pi ]\\times S\\right)/ \\sim \\qquad (2\\pi ,q) \\sim (0,h(q)).$ It is equipped with the fibration $x: S(h) \\rightarrow \\mathbb {R}/2\\pi \\mathbb {Z},$ given by the projection onto the first component.",
"The piece $([0,2\\pi ]\\times (S\\cap N))/\\sim $ is diffeomorphic to $\\mathbb {R}/2\\pi \\mathbb {Z}\\times (S\\cap N)$ , because $h$ is the identity on $S\\cap N$ , and is equipped with coordinates $(x,\\lambda ,r) \\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\partial S \\times [0,1].$ In particular, the boundary of $S(h)$ is the product $\\mathbb {R}/2\\pi \\mathbb {Z}\\times \\partial S$ , which is a union of finitely many 2-tori.",
"The flow of $Z$ preserves the family of leaves in the neighborhood $N$ and is transverse to the interiors of the leaves.",
"Thus we may renormalise $Z$ outside of $N$ so that its flow preserves the leaves also in $M\\backslash N$ .",
"Then the flow of $Z$ induces a diffeomorphism between the interior of $S(h)$ and $M\\setminus L$ , which identifies the fibrations $x$ and $\\Pi $ .",
"Moreover, $M$ is obtained from $S(h)$ by collapsing each boundary torus onto a circle.",
"More precisely, we can identify $M$ with the quotient $\\bigl (S(h) \\sqcup (\\partial S\\times \\bigr )/\\sim $ where $(x,\\lambda ,r) \\sim (\\lambda ,r e^{ix})$ .",
"Here, the smooth structure near the link is induced by the identification $\\partial S \\times N$ .",
"By a theorem of Giroux [11] (see also [8] for a detailed proof), the contact structure $\\xi $ is supported by some open book decomposition $(\\Pi ,L)$ of $M$ : This means that $\\xi =\\ker \\alpha $ , where $\\alpha $ is a contact form such that: the restriction of $d\\alpha $ to each page is a positive area form; $\\alpha $ is positive on $L$ .",
"As smoothly embedded compact surface $S$ we choose the closure of the page $\\Pi ^{-1}(0)$ .",
"We freely draw from the notation about open book decompositions established above; in particular, $N$ is the closed tubular neighborhood of $L=\\partial S$ with diffeomorphism $N\\simeq L\\times and $ h:S S$ is a choice of monodromy map which is supported in $ SN$.$ If $\\alpha $ and $\\alpha ^{\\prime }$ are contact forms satisfying (A1) and (A2), then $\\ker \\alpha $ and $\\ker \\alpha ^{\\prime }$ are isotopic (see [11] and [8]).",
"In this case, Gray's stability theorem (see [10]) implies the existence of a diffeomorphism bringing $\\ker \\alpha $ into $\\ker \\alpha ^{\\prime }$ .",
"Since the properties (i), (ii), (iii) and (iv) that we wish our contact form to have are preserved by the action of a diffeomorphism, it is enough to find a contact form $\\alpha $ which satisfies them together with the conditions (A1) and (A2) above, i.e.",
"without explicitly assuming that $\\ker \\alpha $ is the given contact structure.",
"Denote by $C_1,\\dots ,C_{\\ell }$ the connected components of $\\partial S$ .",
"The tubular neighborhood $N\\cap S$ of $\\partial S$ in $S$ is the union of $\\ell $ closed annuli $A_j$ carrying coordinates $(\\theta ,r)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,1]$ , where $\\theta $ is an orientation preserving parametrization of $C_j$ .",
"Here the boundary component $C_j$ corresponds to $r=0$ .",
"The fact that $C_j$ is given the boundary orientation from $S$ implies that $2r\\,d\\theta \\wedge dr$ is a positive 2-form on each half-open annulus $A_j\\setminus \\partial S$ .",
"Choose a 2-form $\\Omega $ on $S$ such that $\\Omega >0$ on $S\\setminus L$ , $\\int _S\\Omega = 2\\pi \\ell $ and $\\Omega = 2r\\, d\\theta \\wedge dr \\qquad \\mbox{on each closed annulus } A_j^{\\prime }:= \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,\\rho ]\\subset A_j,$ for a suitably small number $\\rho \\in (0,1]$ .",
"Perhaps after shrinking $\\rho $ , we can assume that there exists a primitive $\\eta $ of $\\Omega $ agreeing with $(1-r^2)d\\theta $ on each $A_j^{\\prime }$ .",
"To see this, choose any 1-form $\\sigma _0$ on $S$ which agrees with $(1-r^2)d\\theta $ near the $C_j$ 's.",
"Then $\\Omega -d\\sigma _0$ is a 2-form with support contained in $S\\setminus \\partial S$ .",
"Since $\\int _S \\bigl ( \\Omega - d\\sigma _0 \\bigr ) = 2\\pi \\ell - \\int _{\\partial S} \\sigma _0 = 2\\pi \\ell - 2\\pi \\ell = 0,$ we can invoke de Rham's theorem to find a 1-form $\\sigma _1$ supported in $S\\setminus \\partial S$ and such that $\\Omega -d\\sigma _0=d\\sigma _1$ .",
"Then $\\eta = \\sigma _0+\\sigma _1$ has the desired property.",
"Let $\\beta :\\mathbb {R}\\rightarrow [0,1]$ be a smooth function satisfying $\\beta (0)=0$ , $\\beta (2\\pi )=1$ and $\\mathrm {supp\\,}(\\beta ^{\\prime })\\subset (0,2\\pi )$ .",
"The family of 1-forms $\\tilde{\\alpha }_s := dx + s \\bigl ( (1-\\beta (x)) \\eta + \\beta (x) h^*\\eta \\bigr ), \\qquad s\\in (0,+\\infty ),$ is well defined on the mapping torus $S(h)$ of $h$ which is defined in (REF ).",
"Here $x$ denotes the fibration (REF ), which corresponds to the fibration $\\Pi $ under the identification of the interior of $S(h)$ with $M\\setminus L$ .",
"From now on, we shall identify $M$ with the manifold $\\left( S(h) \\sqcup \\bigsqcup _{j=1}^{\\ell } \\bigl ( \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho ) \\right) / \\sim $ as in (REF ), where a point in $\\mathbb {R}/2\\pi \\mathbb {Z}\\times A_j^{\\prime }\\subset S(h)$ with coordinates $(x,\\theta ,r)$ , is identified with the point $(\\theta ,r e^{ix})$ in the corresponding copy of $\\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho .",
"With this identification, the set$$V := \\bigsqcup _{j=1}^{\\ell } \\bigl ( \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho )$$is a compact tubular neighborhood of $ L$ in $ M$.$ We will think of $\\tilde{\\alpha }_s$ as defined in $M\\setminus L \\simeq S(h)\\setminus \\partial S(h)$ .",
"The fact that $h$ is the identity on $A_j$ and the form of $\\eta $ on $A_j^{\\prime }$ imply that $\\tilde{\\alpha }_s=dx + s (1-r^2)\\, d\\theta \\qquad \\mbox{on }V\\setminus L,$ with respect to the standard coordinates $(\\theta ,r,x)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times (0,\\rho ] \\times \\mathbb {R}/2\\pi \\mathbb {Z}$ on $V \\setminus L$ .",
"Fix $\\delta >0$ .",
"There exists $s_1>0$ , depending only on $\\rho $ and $\\delta $ , with the following properties: If $s\\in (0,s_1)$ then there are smooth functions $f,g:[0,\\rho ] \\rightarrow [0,+\\infty )$ defining a curve in the complex plane $\\gamma : [0,\\rho ] \\rightarrow \\qquad \\gamma (r) = f(r) + i g(r),$ satisfying: $\\gamma (r) = 1 + is(1-r^2)$ on $[r_1,\\rho ]$ , for some $r_1 \\in (0,\\rho )$ ; $g^{\\prime }<0$ on $(0,\\rho ]$ .",
"$\\gamma ([0,r_0]) \\subset \\lbrace x+iy \\mid x+y=1+\\delta \\rbrace $ for some $r_0\\in (0,r_1)$ such that $g(r_0)-g(\\rho )=g(r_0)-s(1-\\rho ^2) \\le 2\\delta .$ Moreover, $\\gamma (r) = r^2 + i (1+\\delta -r^2)$ when $r$ is close enough to 0.",
"The derivative of the argument of $\\gamma $ , that is the function $ \\frac{g^{\\prime }f-f^{\\prime }g}{f^2+g^2} $ is negative on $(0,\\rho ]$ .",
"The derivative of the argument of $\\gamma ^{\\prime }$ , that is the function $ \\frac{g^{\\prime \\prime }f^{\\prime }-f^{\\prime \\prime }g^{\\prime }}{(f^{\\prime })^2+(g^{\\prime })^2} $ is non-positive on $[0,\\rho ]$ .",
"Figure: Functions f,gf,g with the above properties exist.Finally, we define a smooth 1-form $\\alpha _s$ on $M$ by $\\alpha _s = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\frac{\\tilde{\\alpha }_s}{2\\pi (1+\\delta )}} & \\mbox{on } M\\setminus V, \\\\ \\\\\\displaystyle {\\frac{f(r)\\,dx + g(r)\\,d\\theta }{2\\pi (1+\\delta )}} & \\mbox{on } V. \\end{array} \\right.$ The smoothness of $\\alpha _{s}$ near the boundary of $V$ follows from (REF ) and (B1).",
"By (B3) we have the formula $\\alpha _{s} = \\frac{r^2 \\, dx + (1+\\delta -r^2)\\, d\\theta }{2\\pi (1+\\delta )} \\qquad \\mbox{near $L$},$ which shows that $\\alpha _{s}$ is smooth in a neighborhood of $L$ .",
"Therefore, $\\alpha _{s}$ is a smooth 1-form on $M$ for every $s\\in (0,s_1)$ .",
"We claim that there exists $s_2 \\in (0,s_1)$ , depending on $\\delta ,\\rho ,h,\\eta ,\\beta $ , such that if $s \\in (0,s_2)$ then $\\alpha _s$ is a contact form and satisfies conditions (A1) and (A2).",
"Let us prove this claim.",
"By (B3) the 1-form $\\alpha _{s}$ has the following expression on $L$ : $\\alpha _{s} = \\frac{1}{2\\pi } \\, d\\theta \\qquad \\mbox{on } L \\simeq \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\lbrace 0\\rbrace \\subset \\mathbb {R}/2\\pi \\mathbb {Z}\\times \\rho $ which shows that condition (A2) holds for every $s\\in (0,s_1)$ .",
"On $V$ we have $d\\alpha _s = \\frac{f^{\\prime }dr \\wedge dx-g^{\\prime }d\\theta \\wedge dr}{2\\pi (1+\\delta )}.$ The restriction of $d\\alpha _{s}$ on the page $\\Pi ^{-1}(x_0) = x^{-1}(x_0)$ intersected with $V$ is $-\\frac{g^{\\prime }d\\theta \\wedge dr}{2\\pi (1+\\delta )},$ which is a positive area form thanks to (B2) and the positivity of $d\\theta \\wedge dr$ .",
"So the condition (A1) holds on $V$ for every $s\\in (0,s_1)$ .",
"Moreover, $\\alpha _s \\wedge d\\alpha _s = \\frac{f^{\\prime }g-fg^{\\prime }}{4\\pi ^2 (1+\\delta )^2} \\ dx\\wedge d\\theta \\wedge dr = \\frac{f^{\\prime }g-fg^{\\prime }}{4\\pi ^2 (1+\\delta )^2 r} \\ d\\theta \\wedge (r\\, dr \\wedge dx).$ By the last part of (B3), $\\frac{f^{\\prime }g-fg^{\\prime }}{r} =2(1+\\delta )$ when $r$ is close enough to 0.",
"Together with (B4), this implies that the smooth function $(f^{\\prime }g-fg^{\\prime })/r$ is strictly positive on $V$ .",
"This implies that $\\alpha _{s}$ is a contact form on $V$ for every $s\\in (0,s_1)$ .",
"Now we analyze $\\alpha _s$ on $M\\setminus V$ .",
"The formula $\\alpha _s=\\frac{dx+s \\bigl ((1-\\beta )\\eta +\\beta h^*\\eta \\bigr )}{2\\pi (1+\\delta )} \\qquad \\mbox{on }M\\setminus V$ gives $d\\alpha _s = s \\, \\frac{\\beta ^{\\prime }dx\\wedge (h^*\\eta -\\eta ) + \\omega }{2\\pi (1+\\delta )} \\qquad \\mbox{on }M\\setminus V,$ where $\\omega $ is the smooth 2-form on $M\\setminus L$ with kernel $\\partial _x$ and whose restriction to the page $\\Pi ^{-1}(x) \\simeq S\\setminus \\partial S$ is the positive area form $\\omega _x := (1-\\beta (x))d\\eta +\\beta (x) h^*d\\eta .$ The restriction of $d\\alpha _{s}$ to the page $\\Pi ^{-1}(x_0)=x^{-1}(x_0)\\simeq S\\setminus S$ intersected with $M\\setminus V$ is a positive multiple of the positive area form $\\omega _{x_0}$ , so $\\alpha _{s}$ satisfies (A1) on $M\\setminus V$ for every $s \\in (0,s_1)$ .",
"Moreover, $\\frac{4\\pi ^2 (1+\\delta )^2}{s} \\, \\alpha _s \\wedge d\\alpha _s = dx\\wedge \\omega + s \\ \\beta ^{\\prime } \\eta \\wedge dx \\wedge h^*\\eta \\qquad \\mbox{on } M\\setminus V.$ Since $dx\\wedge \\omega $ is a volume form on $M\\setminus V$ , the above formula shows that we can choose $s_2\\in (0,s_1]$ , depending on the data $h,\\eta ,\\beta $ , such that $\\alpha _s$ is a contact form on $M\\setminus V$ whenever $s\\in (0,s_2)$ .",
"We conclude that $\\alpha _s$ is a contact form satisfying conditions (A1) and (A2), for every $s\\in (0,s_2)$ .",
"The next task is to understand the Reeb flow of $\\alpha _s$ and to show it fullfills the requirements (i), (ii), (iii) and (iv) with respect to the surface $S$ , when $s$ is small enough.",
"Condition (i) is actually fulfilled for any $s\\in (0,s_1)$ , due to (REF ), so we need to focus only on (ii), (iii) and (iv).",
"Denote by $R_{s}$ the Reeb vector field of $\\alpha _{s}$ .",
"We start by analysing the flow of $R_{s}$ on $M\\setminus V$ .",
"Since $\\beta ^{\\prime }$ is supported in $(0,2\\pi )$ , $h^* \\eta -\\eta $ is compactly supported in $S\\setminus V$ and $\\omega $ restricts to an area form on each page, there is a unique smooth vector field $Y$ on $M$ which is supported in $M\\setminus V$ , is tangent to the pages and satisfies $\\omega (Y,\\cdot ) = -\\beta ^{\\prime }(x) (h^*\\eta -\\eta ) \\qquad \\mbox{on } \\Pi ^{-1}(x).$ By (REF ), we have the identity $\\begin{split}\\frac{2\\pi (1+\\delta )}{s} \\ d\\alpha _s(\\partial _x+Y,\\cdot ) &= \\beta ^{\\prime }(h^*\\eta -\\eta ) - \\beta ^{\\prime }(h^*\\eta -\\eta )(Y)\\, dx + \\omega (Y,\\cdot ) \\\\ &=\\beta ^{\\prime }(h^*\\eta -\\eta ) + \\omega (Y,Y)\\, dx - \\beta ^{\\prime }(h^*\\eta -\\eta ) = 0,\\end{split}$ which shows that the non-vanishing vector field $\\partial _x+Y$ is in the kernel of $d\\alpha _{s}$ .",
"Therefore, the Reeb vector field $R_{s}$ of $\\alpha _{s}$ has the form $R_s = \\frac{\\partial _x+Y}{\\alpha _s(\\partial _x+Y)} \\qquad \\mbox{on } M\\setminus V,$ for every $s\\in (0,s_2)$ .",
"The fact that $Y$ is supported in $M\\setminus V$ and the form of $V$ imply that $M\\setminus V$ is invariant for the Reeb flow, and hence the same is true for its complement $V$ .",
"Moreover, the above formula implies that $R_{s}$ is transverse to the portion of the pages $x^{-1}(x_0)$ lying in $M\\setminus V$ , and in particular to $S\\setminus V$ .",
"Next we study the Reeb flow on $V$ .",
"By the form of $\\alpha _{s}$ on $V$ , $R_{s}$ has the form $R_{s} = 2\\pi (1+\\delta ) \\frac{f^{\\prime } \\partial _{\\theta } - g^{\\prime } \\partial _x}{f^{\\prime }g-g^{\\prime }f} \\qquad \\mbox{on } V.$ The condition (B2) implies that $R_{s}$ is transverse the the portions of the pages $x^{-1}(x_0)$ which lie in $V$ , and in particular to $(S\\setminus \\partial S)\\cap V$ .",
"The expression for $R_{s}$ near $L$ becomes, thanks to (B3), $R_{s} = 2\\pi (\\partial _{\\theta } + \\partial _x) \\qquad \\mbox{for } (\\theta ,r,x)\\in \\mathbb {R}/2\\pi \\mathbb {Z}\\times [0,r_0] \\times \\mathbb {R}/2\\pi \\mathbb {Z}.$ In particular, since $\\partial _x$ vanishes on $L$ , $R_{s} = 2\\pi \\,\\partial _{\\theta }\\qquad \\mbox{on } L,$ and the link $L$ consists of periodic orbits of $R_{s}$ of period 1.",
"We conclude that the vector field $R_{s}$ is transverse to the pages $x^{-1}(x_0)$ and leaves the binding $L$ invariant.",
"In particular, in view of the form of $R_s$ near $L$ described above, $S$ is a global surface of section for the Reeb flow, and the first return time function and first return map $\\tau : S \\setminus \\partial S \\rightarrow (0,+\\infty ), \\qquad \\varphi : S \\setminus \\partial S \\rightarrow S \\setminus \\partial S$ are well defined.",
"By (REF ), on $(S\\setminus \\partial S)\\cap V$ the first return time function is explicitly given by the formula $\\tau = \\frac{g^{\\prime }f-f^{\\prime }g}{(1+\\delta )g^{\\prime }} \\qquad \\mbox{on } (S\\setminus \\partial S)\\cap V,$ and the first return time map by $\\varphi : (r,\\theta ) \\mapsto \\left( r,\\theta -2\\pi \\frac{f^{\\prime }(r)}{g^{\\prime }(r)} \\right) \\qquad \\mbox{on } (S\\setminus \\partial S) \\cap V.$ By (B3), we have $\\tau (r,\\theta ) = 1 \\qquad \\mbox{and} \\qquad \\varphi (r,\\theta ) = (r,\\theta -2\\pi ) \\qquad \\forall (r,\\theta )\\in (0,r_0]\\times \\mathbb {R}/2\\pi \\mathbb {Z}.$ This implies that $\\tau $ and $\\varphi $ extend smoothly to the boundary of $S$ and proves (ii).",
"By (REF ), the restriction of the first return time function $\\tau $ to $S\\cap V$ depends only on $r$ and we have $\\partial _r \\tau = \\frac{g(f^{\\prime }g^{\\prime \\prime }-g^{\\prime }f^{\\prime \\prime })}{(1+\\delta )(g^{\\prime })^2}.$ By (B5), this function is non-positive on $[0,\\rho ]$ .",
"Together with (REF ) and $\\tau (\\rho ,\\theta ) = \\frac{1}{1+\\delta },$ where we have used (REF ) and (B1), we find the bounds $\\frac{1}{1+\\delta } \\le \\tau \\le 1 \\qquad \\mbox{on } S\\cap V,$ from which $\\sup _{S\\cap V} |\\tau -1| \\le 1 - \\frac{1}{1+\\delta } = \\frac{\\delta }{1+\\delta } < \\delta .$ From (REF ) we have $dx(R_{s}) = \\frac{1}{\\alpha _{s}(\\partial _x + Y)} \\qquad \\mbox{on } M\\setminus V.$ Since the function $\\alpha _{s}(\\partial _x + Y) = \\frac{1}{2\\pi (1+\\delta )} \\tilde{\\alpha }_{s}(\\partial _x + Y) = \\frac{1}{2\\pi (1+\\delta )} \\bigl ( 1 + s ((1-\\beta ) \\eta (Y) + \\beta h^* \\eta (Y))\\bigr )$ converges to $1/(2\\pi (1+\\delta ))$ for $s\\rightarrow 0$ uniformly on $M\\setminus V$ , by (REF ) the function $dx(R_{s})$ converges to $2\\pi (1+\\delta )$ for $s\\rightarrow 0$ uniformly on $M\\setminus V$ .",
"This implies that $\\tau $ converges uniformly to $1/(1+\\delta )$ on $S\\setminus V$ .",
"Together with (REF ), this implies that there exists $s_3\\in (0,s_2]$ such that for every $s\\in (0,s_3)$ we have $\\Vert \\tau -1\\Vert _{\\infty } < 2\\delta .$ Define $U$ to be the open tubular neighborhood of $L=\\partial S$ in $S$ consisting of those $(r,\\theta )$ in $S\\cap V$ with $0\\le r < r_0$ .",
"By (REF ), $\\tau =1$ and $\\varphi =\\mathrm {id}$ on $U$ .",
"Together with (REF ), this proves that (iv) holds, by choosing $\\delta \\le \\epsilon /2$ .",
"The identity (REF ) gives us a constant $C$ , depending only on $\\beta $ , $\\eta $ and $h$ , such that $\\int _{S\\setminus V} d\\alpha _{s} \\le C s \\qquad \\forall s\\in (0,s_1).$ We now fix a $s_4\\in (0,s_3]$ such that $C s_4<\\delta $ .",
"By (REF ), the $d\\alpha _{s}$ -area of the annulus in $A_j^{\\prime }$ corresponding to the values of $r$ in the interval $[r_0,\\rho ]$ is $\\frac{1}{2\\pi (1+\\delta )} \\int _{\\mathbb {R}/2\\pi \\mathbb {Z}\\times [r_0,\\rho ]} (-g^{\\prime }) \\, d\\theta \\wedge dr = \\frac{1}{1+\\delta } \\bigl ( g(r_0) - g(\\rho ) \\bigr ) \\le \\frac{2\\delta }{1+\\delta }< 2\\delta ,$ where the first upper bound follows from (B3).",
"If $s$ is in the interval $(0,s_4)$ , the above two inequalities imply that $\\int _{S\\setminus U} d\\alpha _{s} = \\int _{S\\setminus V} d\\alpha _{s} + \\frac{1}{2\\pi (1+\\delta )} \\int _{\\mathbb {R}/2\\pi \\mathbb {Z}\\times [r_0,\\rho ]} (-g^{\\prime }) \\, d\\theta \\wedge dr < 3\\delta ,$ and the conclusion (iii) follows by choosing $\\delta =\\epsilon /3$ ."
],
[
"Construction of the plug",
"The aim of this section is to show how some results from [3] can be used in order to build a contact form on a solid torus such that the contact volume is small and all closed orbits of the corresponding Reeb flow have large period.",
"In the next section, this solid torus will be used as a plug to modify the special contact form which we constructed in the previous section.",
"Here is the statement which summarizes the properties of the plug.",
"Proposition 2 Fix positive numbers $r$ and $\\epsilon $ .",
"Let $\\lambda $ be a primitive of the standard area form $dx\\wedge dy$ on the closed disc $r\\mathbb {D}$ .",
"Then there exists a smooth contact form $\\beta $ on the solid torus $r\\mathbb {D} \\times \\mathbb {R}/\\mathbb {Z}$ with the following properties: $\\beta = \\lambda + ds$ in a neighborhood of $\\partial (r \\times \\mathbb {R}/\\mathbb {Z}$ , where $s$ denotes the coordinate on $\\mathbb {R}/ \\mathbb {Z}$ ; in particular, the Reeb vector field $R_{\\beta }$ of $\\beta $ coincides with $\\partial _s$ near the boundary of $r\\mathbb {R}/\\mathbb {Z}$ , and its flow is globally well-defined; the contact form $\\beta $ is smoothly isotopic to the contact form $\\lambda + ds$ on $r\\mathbb {R}/ \\mathbb {Z}$ through a path of contact forms which agree with $\\lambda + ds$ in a neighborhood of $\\partial (r \\times \\mathbb {R}/ \\mathbb {Z}$ ; all the closed orbits of $R_{\\beta }$ have period at least 1; $\\mathrm {vol}\\,(r\\mathbb {R}/ \\mathbb {Z}, \\beta \\wedge d\\beta ) < \\epsilon $ .",
"Before discussing the proof of the above proposition, we recall the definition of the Calabi invariant for compactly supported area-preserving diffeomorphisms of the plane.",
"Endow the plane $\\mathbb {R}^2$ with the standard area form $dx\\wedge dy$ and let $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ be the group of compactly supported area-preserving diffeomorphisms of $\\mathbb {R}^2$ .",
"Let $\\varphi $ be an element of $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ and let $\\lambda $ be a primitive of $dx\\wedge dy$ .",
"To $\\varphi $ and $\\lambda $ we can associate the unique compactly supported smooth function $\\sigma _{\\varphi ,\\lambda }: \\mathbb {R}^2 \\rightarrow \\mathbb {R}$ satisfying $\\varphi ^* \\lambda - \\lambda = d\\sigma _{\\varphi ,\\lambda }.$ This function is called the action of $\\varphi $ with respect to $\\lambda $ .",
"Its value at fixed points of $\\varphi $ is independent of the choice of the primitive $\\lambda $ , and so is its integral $\\mathrm {CAL}(\\varphi ) = \\int _{\\mathbb {R}^2} \\sigma _{\\varphi ,\\lambda } \\, dx\\wedge dy,$ which is called the Calabi invariant of $\\varphi $ .",
"The Calabi invariant defines a homomorphism from $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ onto $\\mathbb {R}$ .",
"Let $\\lambda _0$ be the following radially symmetric primitive of $dx\\wedge dy$ $\\lambda _0 := \\frac{1}{2} ( x\\, dy - y \\, dx).$ We shall make use of the following result: Proposition 3 Fix positive numbers $r$ and $L$ .",
"For every $\\epsilon >0$ there exists a positive integer $n$ and an area-preserving diffeomorphism $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ such that the following properties hold: $\\sigma _{\\varphi ,\\lambda _0} \\ge - L + L/n $ ; $\\mathrm {CAL}(\\varphi )< - L\\pi r^2 + \\epsilon $ ; all the fixed points of $\\varphi $ have non-negative action; all the periodic points of $\\varphi $ which are not fixed points have period at least $n$ .",
"This result is proved in [3] for $r=1$ .",
"The general case follows by a simple rescaling argument, using that if $\\varphi $ is in $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ then the rescaled map $\\varphi _r(z) := r \\varphi \\left( \\frac{z}{r} \\right)$ is also in $\\mathrm {Diff}_c(\\mathbb {R}^2,dx\\wedge dy)$ and $\\sigma _{\\varphi _r,\\lambda _0}(z) = r^2 \\sigma _{\\varphi ,\\lambda _0}\\left( \\frac{z}{r} \\right), \\qquad \\mathrm {CAL}(\\varphi _r)= r^4 \\, \\mathrm {CAL}(\\varphi ).$ The last ingredient which we need is the following result from [3][Proposition 3.1], which allows us to lift an area preserving diffeomorphism of a disc to a Reeb flow on a solid torus.",
"Proposition 4 Fix positive numbers $r$ and $L$ .",
"Let $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ and assume that the function $\\tau := \\sigma _{\\varphi ,\\lambda _0} + L$ is positive on $r.",
"Then there exists a smooth contact form $$ on the solid torus $ rR/L Z$ with the following properties:\\begin{enumerate}[({c}1)]\\item \\beta = \\lambda _0 + ds in a neighborhood of \\partial (r \\times \\mathbb {R}/L \\mathbb {Z}, where s denotes the coordinate on \\mathbb {R}/L \\mathbb {Z}; in particular, the Reeb vector field R_{\\beta } of \\beta coincides with \\partial _s near the boundary of r\\mathbb {R}/L \\mathbb {Z}, and its flow is globally well-defined;\\item the surface r\\lbrace 0\\rbrace is transverse to the flow of R_{\\beta }, and the orbit of every point in r\\mathbb {R}/L \\mathbb {Z} intersects r\\lbrace 0\\rbrace both in the future and in the past;\\item the first return map and the first return time of the flow of R_{\\beta } associated to the surface r\\lbrace 0\\rbrace \\cong r are the map \\varphi and the function \\tau ;\\item \\mathrm {vol}\\,(r\\mathbb {R}/L \\mathbb {Z}, \\beta \\wedge d\\beta ) = L \\,\\pi r^2+ \\mathrm {CAL}(\\varphi );\\item the contact form \\beta is smoothly isotopic to \\lambda _0 + ds on r\\mathbb {R}/L \\mathbb {Z} through a path of contact forms which agree with \\lambda _0 + ds in a neighborhood of \\partial (r \\times \\mathbb {R}/L \\mathbb {Z}.\\end{enumerate}$ Again, Proposition 3.1 in [3] is stated for $r=1$ , but the general case follows by rescaling.",
"In fact consider $r,L$ and $\\varphi $ as in the statement of Proposition REF .",
"Then, as explained above, we have $\\sigma _{\\varphi _{r^{-1}},\\lambda _0}(z) = r^{-2} \\sigma _{\\varphi ,\\lambda _0}\\left( rz \\right)$ .",
"Applying [3] to the map $\\varphi _{r^{-1}}$ with $Lr^{-2}$ in the place of $L$ we get a contact form $\\beta _{r^{-1}}$ on $\\mathbb {R}/Lr^{-2}\\mathbb {Z}$ satisfying the desired conclusions.",
"Consider now the diffeomorphism $\\Phi _{r^{-1}}:r\\mathbb {R}/L\\mathbb {Z}\\rightarrow \\mathbb {R}/Lr^{-2}\\mathbb {Z}$ defined as $(z,s) \\mapsto (r^{-1}z,r^{-2}s)$ .",
"Direct calculations reveal that $\\beta = r^2\\Phi _{r^{-1}}^*\\beta _{r^{-1}}$ is the desired contact form.",
"The statement of [3] is actually slightly more general, since it allows $\\lambda _0$ to be replaced by a more general primitive of $dx\\wedge dy$ , and gives more properties of the contact form $\\beta $ .",
"Building on the above two propositions, it is now easy to prove Proposition REF .",
"The first step is to prove the existence of a contact form $\\beta _0$ on $r\\mathbb {R}/\\mathbb {Z}$ which satisfies the required conditions (a1)-(a4), but where in (a1) and (a2) the primitive $\\lambda $ of $dx\\wedge dy$ is the radially symmetric primitive $\\lambda _0$ .",
"Let $n\\in \\mathbb {N}$ , $\\varphi \\in \\mathrm {Diff}_c(\\mathrm {int}(r\\mathbb {D}),dx\\wedge dy)$ be given by an application of Proposition REF with $L=1$ .",
"By statement (b1) in this proposition, we have the lower bound $1 + \\sigma _{\\varphi ,\\lambda _0} \\ge \\frac{1}{n}.$ In particular, the function $\\tau := 1 + \\sigma _{\\varphi ,\\lambda _0}$ is everywhere positive.",
"Then Proposition REF implies the existence of a contact form $\\beta _0$ on the solid torus $r\\mathbb {D} \\times \\mathbb {R}/\\mathbb {Z}$ which satisfies the conditions (c1)-(c5) with $L=1$ .",
"We just need to check that $\\beta _0$ satisfies (a1)-(a4) with respect to $\\lambda _0$ .",
"Conditions (a1) and (a2) are precisely (c1) and (c5).",
"Condition (a4) follows from (c4) and (b2): $\\mathrm {vol}\\,(r\\mathbb {R}/ \\mathbb {Z}, \\beta _0\\wedge d\\beta _0) \\stackrel{(c4)}{=} \\pi r^2 + \\mathrm {CAL}(\\varphi ) \\stackrel{(b2)}{<} \\epsilon .$ There remains to prove that all closed orbits of the Reeb flow of $\\beta _0$ have period at least 1.",
"By (c2) and (c3), closed orbits of $R_{\\beta _0}$ are in one-to-one correspondence with periodic points of $\\varphi $ , and if $z\\in r\\mathbb {D}$ is a periodic point of $\\varphi $ with minimal period $k\\in \\mathbb {N}$ , then the corresponding closed orbit has period $T:= \\sum _{j=0}^{k-1} \\tau (\\varphi ^j(z)).$ When $k=1$ , $z$ is a fixed point of $\\varphi $ and by (b3) we have $T=\\tau (z) = 1 + \\sigma _{\\varphi ,\\lambda _0}(z) \\ge 1.$ By condition (b4), all periodic points of $\\varphi $ which are not fixed points have period $k\\ge n$ .",
"If $z$ is such a point, condition (b1) gives us $T = \\sum _{j=0}^{k-1} \\tau (\\varphi ^j(z)) = \\sum _{j=0}^{k-1} \\bigl (1+\\sigma _{\\varphi ,\\lambda _0}(\\varphi ^j(z))\\bigr ) \\ge \\sum _{j=0}^{k-1} \\frac{1}{n} = \\frac{k}{n} \\ge 1.$ This proves (a3) and concludes the first step.",
"Now we wish to modify $\\beta _0$ in a neighborhood of the boundary in order to obtain (a1) and (a2) with respect to the given primitive $\\lambda $ of $dx\\wedge dy$ , while keeping conditions (a3) and (a4).",
"The 1-form $\\lambda -\\lambda _0$ is closed and hence exact on $r.",
"Let $ u$ be a smooth function on $ r such that $\\lambda - \\lambda _0 = du.$ Let $\\lbrace \\beta _0^t\\rbrace _{t\\in [0,1]}$ be a smooth path of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ going from $\\beta _0$ to $\\lambda _0+ds$ and agreeing with $\\lambda _0+ds$ in a neighborhood $(rr^{\\prime }\\times \\mathbb {R}/\\mathbb {Z}$ of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ , where $r^{\\prime }\\in (0,r)$ .",
"Let $\\chi $ be a smooth function on $r such that $ =0$ on $ r' and $\\chi =1$ on a neighborhood of $\\partial (r$ .",
"Consider the following smooth contact form on $r\\mathbb {R}/\\mathbb {Z}$ : $\\beta := \\beta _0 + d(\\chi u).$ This contact form satisfies (a1) with respect to $\\lambda $ .",
"The formula $\\beta ^t := \\beta _0^t + d(\\chi u)$ defines a smooth path of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ going from $\\beta $ to the 1-form $\\lambda _0 + ds+d(\\chi u),$ and agreeing with $\\lambda +ds$ in a neighborhood of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ .",
"The latter contact form can be joined to $\\lambda + ds$ by the smooth isotopy $\\lambda _0 + t\\, du + (1-t) d(\\chi u) + ds,$ which consists of contact forms on $r\\mathbb {R}/\\mathbb {Z}$ agreeing with $\\lambda +ds$ in a neighborhood of the boundary of $r\\mathbb {R}/\\mathbb {Z}$ .",
"This proves that $\\beta $ satisfies (a2) with respect to $\\lambda $ .",
"The volume form induced by $\\beta $ is $\\beta \\wedge d\\beta = \\beta _0 \\wedge d\\beta _0 + d(\\chi u) \\wedge d\\beta _0.$ The 3-form $d(\\chi u) \\wedge d\\beta _0$ vanishes identically, because $d(\\chi u)$ is supported in the region in which $d\\beta _0=d\\lambda _0$ and the contractions of this 1-form and this 2-form along $\\partial _s$ are both zero.",
"We deduce that $\\beta \\wedge d\\beta = \\beta _0 \\wedge d\\beta _0$ on the whole $r\\mathbb {R}/\\mathbb {Z}$ , and the fact that $\\beta _0$ satisfies (a4) implies that also $\\beta $ does.",
"The fact that $\\beta $ differs from $\\beta _0$ by an exact 1-form which vanishes along the direction of $R_{\\beta _0}$ implies that the Reeb vector field of $\\beta $ coincides with the one of $\\beta _0$ .",
"Therefore, the fact that $\\beta _0$ satisfies (a3) implies that also $\\beta $ does."
],
[
"Proof of the main theorem",
"We are now ready to prove the theorem stated in the introduction.",
"Let $\\xi $ be a contact structure on the closed three-manifold $M$ .",
"Let $S$ be the smoothly embedded compact surface $S\\subset M$ given by Proposition REF and let $\\epsilon <1$ be a fixed positive number.",
"By Proposition REF we can find a contact form $\\tilde{\\alpha }$ on $M$ such that $\\ker \\tilde{\\alpha } = \\xi $ , $S$ is a global surface of section for the Reeb flow of $\\tilde{\\alpha }$ and statements (i)-(iv) hold, where $\\alpha $ is renamed as $\\tilde{\\alpha }$ .",
"In the following, $U$ , $\\varphi $ and $\\tau $ are the objects appearing in this proposition.",
"Statements labeled by a roman number refer to the statements of this proposition.",
"Denote by $C_1,\\dots ,C_{\\ell }$ the circles forming the boundary of $S$ .",
"Each connected component of $U$ is a half-open annulus $A_j$ and $d\\tilde{\\alpha }$ restricts to an area form on the interior of each $A_j$ .",
"Let $a_j>0$ be the $d\\tilde{\\alpha }$ -area of each $A_j$ .",
"By (i) and (iii) we have $\\ell = \\int _S d\\tilde{\\alpha } = \\int _{S\\setminus U} d\\tilde{\\alpha }+ \\sum _{j=1}^{\\ell } \\int _{A_j} d\\tilde{\\alpha } < \\epsilon + \\sum _{j=1}^{\\ell } a_j.$ For any $j\\in \\lbrace 1,\\dots ,\\ell \\rbrace $ , let $r_j>0$ be such that $\\pi r_j^2 = (1-\\epsilon ) a_j,$ and let $\\psi _j: r_j \\mathbb {D} \\hookrightarrow A_j \\setminus \\partial S$ be a smooth embedding such that $\\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.",
"Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\Psi _j: r_j \\mathbb {R}/\\mathbb {Z}\\hookrightarrow M$$such that\\begin{equation}\\Psi _j(\\cdot ,0) = \\psi _j \\qquad \\mbox{and} \\qquad \\Psi _j^*(R_{\\tilde{\\alpha }}) = \\partial _s,\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\begin{equation}\\Psi _j^*(\\tilde{\\alpha }) = \\lambda _j + ds\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.",
"Indeed, set for simplicity $\\tilde{\\alpha }_j:= \\Psi _j^*(\\tilde{\\alpha })$ .",
"The the second identity in () implies that the Reeb vector field of $\\tilde{\\alpha }_j$ is $\\partial _s$ .",
"Since any contact form is invariant by its Reeb flow, $\\tilde{\\alpha }_j$ is invariant under the translations $(x,y,s) \\mapsto (x,y,s+t)$ and hence has the form $\\tilde{\\alpha }_j = \\lambda _j + u_j(x,y)\\, ds,$ where $\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.",
"Then the condition $\\tilde{\\alpha }_j(\\partial _s)=1$ implies that $u_j=1$ .",
"Finally, the first identity in () implies that $d\\lambda _j = d\\tilde{\\alpha }_j|_{r_j \\lbrace 0\\rbrace } = \\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy,$ which concludes the proof of the claim.",
"Denote by $W_j$ the image of the embedding $\\Psi _j$ .",
"Its volume with respect to $\\tilde{\\alpha } \\wedge d\\tilde{\\alpha }$ is $\\begin{split}\\mathrm {vol}\\,( W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) &=\\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, \\tilde{\\alpha }_j \\wedge d\\tilde{\\alpha }_j ) = \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, d\\lambda _j\\wedge ds) \\\\ &= \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z},dx\\wedge dy\\wedge ds) = \\pi r_j^2 = (1-\\epsilon )a_j.\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }$ has the upper bound $\\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) = \\int _S \\tau \\, d\\tilde{\\alpha } \\le (1+\\epsilon ) \\int _S d\\tilde{\\alpha } = (1+\\epsilon ) \\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\begin{split}\\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } &W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) = \\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) - \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(W_j,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) \\\\&\\le (1+\\epsilon ) \\ell - (1-\\epsilon ) \\sum _{j=1}^{\\ell } a_j < (1+\\epsilon ) \\ell - (1-\\epsilon )(\\ell - \\epsilon ) \\\\ &= \\epsilon (2\\ell +1) - \\epsilon ^2 < \\epsilon (2\\ell +1),\\end{split}$ where the second inequality follows from (REF ).",
"Thanks to Proposition REF , on every solid torus $r_j \\mathbb {R}/\\mathbb {Z}$ there is a contact form $\\beta _j$ which agrees with $\\lambda _j+ds$ near the boundary, has total volume less than $\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.",
"Moreover, $\\beta _j$ is smoothly isotopic to $\\lambda _j+ds$ through contact forms which agree with $\\lambda _j+ds$ near the boundary.",
"Let $\\alpha $ be the contact form on $M$ which coincides with $\\tilde{\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\beta _j$ by the embedding $\\Psi _j$ .",
"This form is indeed smooth due to ().",
"It is smoothly isotopic to the contact form $\\tilde{\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\xi =\\ker \\tilde{\\alpha }$ .",
"By (REF ) and the fact that the volume of each $r_j \\mathbb {R}/\\mathbb {Z}$ with respect to $\\beta _j\\wedge d\\beta _j$ is smaller than $\\epsilon $ , the volume of $M$ with respect to $\\alpha \\wedge d\\alpha $ has the upper bound $\\begin{split}\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) &= \\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) + \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(r_j \\mathbb {R}/\\mathbb {Z},\\beta _j\\wedge d\\beta _j) \\\\ &< \\epsilon (2\\ell +1) + \\epsilon \\ell = \\epsilon (3\\ell +1).\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\alpha $ .",
"The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\beta _j$ 's.",
"The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .",
"Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\alpha $ have period larger than $1-\\epsilon $ .",
"We conclude that $T_{\\min }(\\alpha ) > 1-\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\alpha $ has the lower bound $\\rho _{\\mathrm {sys}}(\\alpha ) = \\frac{T_{\\min }(\\alpha )^2}{\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha )} > \\frac{(1-\\epsilon )^2}{\\epsilon (3\\ell +1)}.$ Since the latter quantity tends to $+\\infty $ for $\\epsilon \\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\xi $ on $M$ can be made arbitrarily large.",
"This concludes the proof of the theorem stated in the introduction.",
"We are now ready to prove the theorem stated in the introduction.",
"Let $\\xi $ be a contact structure on the closed three-manifold $M$ .",
"Let $S$ be the smoothly embedded compact surface $S\\subset M$ given by Proposition REF and let $\\epsilon <1$ be a fixed positive number.",
"By Proposition REF we can find a contact form $\\tilde{\\alpha }$ on $M$ such that $\\ker \\tilde{\\alpha } = \\xi $ , $S$ is a global surface of section for the Reeb flow of $\\tilde{\\alpha }$ and statements (i)-(iv) hold, where $\\alpha $ is renamed as $\\tilde{\\alpha }$ .",
"In the following, $U$ , $\\varphi $ and $\\tau $ are the objects appearing in this proposition.",
"Statements labeled by a roman number refer to the statements of this proposition.",
"Denote by $C_1,\\dots ,C_{\\ell }$ the circles forming the boundary of $S$ .",
"Each connected component of $U$ is a half-open annulus $A_j$ and $d\\tilde{\\alpha }$ restricts to an area form on the interior of each $A_j$ .",
"Let $a_j>0$ be the $d\\tilde{\\alpha }$ -area of each $A_j$ .",
"By (i) and (iii) we have $\\ell = \\int _S d\\tilde{\\alpha } = \\int _{S\\setminus U} d\\tilde{\\alpha }+ \\sum _{j=1}^{\\ell } \\int _{A_j} d\\tilde{\\alpha } < \\epsilon + \\sum _{j=1}^{\\ell } a_j.$ For any $j\\in \\lbrace 1,\\dots ,\\ell \\rbrace $ , let $r_j>0$ be such that $\\pi r_j^2 = (1-\\epsilon ) a_j,$ and let $\\psi _j: r_j \\mathbb {D} \\hookrightarrow A_j \\setminus \\partial S$ be a smooth embedding such that $\\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy.$ Such an area-preserving embedding exists because the Euclidean area of $r_j is smaller than the $ d$-area of $ Aj$.",
"Using the fact that $$ is the identity and $ =1$ on each $ Aj$, we can use the Reeb flow of $$ to lift each $ j$ to an embedding$$\\Psi _j: r_j \\mathbb {R}/\\mathbb {Z}\\hookrightarrow M$$such that\\begin{equation}\\Psi _j(\\cdot ,0) = \\psi _j \\qquad \\mbox{and} \\qquad \\Psi _j^*(R_{\\tilde{\\alpha }}) = \\partial _s,\\end{equation}where $ s$ denoted the coordinate on $ R/Z$.We claim that\\begin{equation}\\Psi _j^*(\\tilde{\\alpha }) = \\lambda _j + ds\\end{equation}for some primitive $ j$ of $ dxdy$ on $ rj.",
"Indeed, set for simplicity $\\tilde{\\alpha }_j:= \\Psi _j^*(\\tilde{\\alpha })$ .",
"The the second identity in () implies that the Reeb vector field of $\\tilde{\\alpha }_j$ is $\\partial _s$ .",
"Since any contact form is invariant by its Reeb flow, $\\tilde{\\alpha }_j$ is invariant under the translations $(x,y,s) \\mapsto (x,y,s+t)$ and hence has the form $\\tilde{\\alpha }_j = \\lambda _j + u_j(x,y)\\, ds,$ where $\\lambda _j$ is a 1-form on $r_j and $ uj$ is a function on $ rj.",
"Then the condition $\\tilde{\\alpha }_j(\\partial _s)=1$ implies that $u_j=1$ .",
"Finally, the first identity in () implies that $d\\lambda _j = d\\tilde{\\alpha }_j|_{r_j \\lbrace 0\\rbrace } = \\psi _j^*(d\\tilde{\\alpha }) = dx\\wedge dy,$ which concludes the proof of the claim.",
"Denote by $W_j$ the image of the embedding $\\Psi _j$ .",
"Its volume with respect to $\\tilde{\\alpha } \\wedge d\\tilde{\\alpha }$ is $\\begin{split}\\mathrm {vol}\\,( W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) &=\\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, \\tilde{\\alpha }_j \\wedge d\\tilde{\\alpha }_j ) = \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z}, d\\lambda _j\\wedge ds) \\\\ &= \\mathrm {vol}\\,( r_j \\mathbb {R}/\\mathbb {Z},dx\\wedge dy\\wedge ds) = \\pi r_j^2 = (1-\\epsilon )a_j.\\end{split}$ By (REF ), (i) and (iv), the total volume of $M$ with respect to $\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }$ has the upper bound $\\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) = \\int _S \\tau \\, d\\tilde{\\alpha } \\le (1+\\epsilon ) \\int _S d\\tilde{\\alpha } = (1+\\epsilon ) \\ell .$ Since the sets $W_j$ are pairwise disjoint, being the saturations by the flow of pairwise disjoint sets on a global surface of section, the volume of the complement of their union has the upper bound $\\begin{split}\\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } &W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) = \\mathrm {vol}\\,(M,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) - \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(W_j,\\tilde{\\alpha }\\wedge d\\tilde{\\alpha }) \\\\&\\le (1+\\epsilon ) \\ell - (1-\\epsilon ) \\sum _{j=1}^{\\ell } a_j < (1+\\epsilon ) \\ell - (1-\\epsilon )(\\ell - \\epsilon ) \\\\ &= \\epsilon (2\\ell +1) - \\epsilon ^2 < \\epsilon (2\\ell +1),\\end{split}$ where the second inequality follows from (REF ).",
"Thanks to Proposition REF , on every solid torus $r_j \\mathbb {R}/\\mathbb {Z}$ there is a contact form $\\beta _j$ which agrees with $\\lambda _j+ds$ near the boundary, has total volume less than $\\epsilon $ and is such that all its closed Reeb orbits have period at least 1.",
"Moreover, $\\beta _j$ is smoothly isotopic to $\\lambda _j+ds$ through contact forms which agree with $\\lambda _j+ds$ near the boundary.",
"Let $\\alpha $ be the contact form on $M$ which coincides with $\\tilde{\\alpha }$ outside of the union of the $W_j$ and on each $W_j$ is given by the push-forward of $\\beta _j$ by the embedding $\\Psi _j$ .",
"This form is indeed smooth due to ().",
"It is smoothly isotopic to the contact form $\\tilde{\\alpha }$ , and hence by Gray stability its kernel is diffeomorphic to the given structure $\\xi =\\ker \\tilde{\\alpha }$ .",
"By (REF ) and the fact that the volume of each $r_j \\mathbb {R}/\\mathbb {Z}$ with respect to $\\beta _j\\wedge d\\beta _j$ is smaller than $\\epsilon $ , the volume of $M$ with respect to $\\alpha \\wedge d\\alpha $ has the upper bound $\\begin{split}\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha ) &= \\mathrm {vol}\\,\\Bigl ( M \\setminus \\bigcup _{j=1}^{\\ell } W_j , \\tilde{\\alpha }\\wedge d\\tilde{\\alpha }\\Bigr ) + \\sum _{j=1}^{\\ell } \\mathrm {vol}\\,(r_j \\mathbb {R}/\\mathbb {Z},\\beta _j\\wedge d\\beta _j) \\\\ &< \\epsilon (2\\ell +1) + \\epsilon \\ell = \\epsilon (3\\ell +1).\\end{split}$ The sets $W_j$ are invariant for the Reeb flow of $\\alpha $ .",
"The closed orbits which are contained in the $W_j$ 's have period at least 1, thanks to the corresponding property of the $\\beta _j$ 's.",
"The components of the boundary of $S$ are closed orbits of period 1, thanks to statement (i) in Proposition REF .",
"Statement (iv) in the same proposition tells us that all the other closed orbits of the Reeb flow of $\\alpha $ have period larger than $1-\\epsilon $ .",
"We conclude that $T_{\\min }(\\alpha ) > 1-\\epsilon .$ Together with (REF ), we deduce that the systolic ratio of $\\alpha $ has the lower bound $\\rho _{\\mathrm {sys}}(\\alpha ) = \\frac{T_{\\min }(\\alpha )^2}{\\mathrm {vol}\\,(M,\\alpha \\wedge d\\alpha )} > \\frac{(1-\\epsilon )^2}{\\epsilon (3\\ell +1)}.$ Since the latter quantity tends to $+\\infty $ for $\\epsilon \\rightarrow 0$ , the systolic ratio of a contact form inducing the contact structure $\\xi $ on $M$ can be made arbitrarily large.",
"This concludes the proof of the theorem stated in the introduction."
]
] | 1709.01621 |
[
[
"On Gauge Invariant Cosmological Perturbations in UV-modified Horava\n Gravity: A Brief Introduction"
],
[
"Abstract We revisit gauge invariant cosmological perturbations in UV-modified, z = 3 Horava gravity with one scalar matter field, which has been proposed as a renormalizable gravity theory without the ghost problem in four dimensions.",
"We confirm that there is no extra graviton modes and general relativity is recovered in IR, which achieves the consistency of the model.",
"From the UV-modification terms which break the detailed balance condition in UV, we obtain scale-invariant power spectrums for non-inflationary backgrounds, like the power-law expansions, without knowing the details of early expansion history of Universe.",
"This could provide a new framework for the Big Bang cosmology."
],
[
"Introduction",
"(1).",
"As a particle physicist's point of view, a surprise of cosmology is as follows.",
"In particle physics, we need (fundamental) scalar fields (called Higgs fields) for “theoretical\" reasons (i.e., to give masses to fundamental particles with renormalizability), but we have waited for a long time (about 40 years, after 1964 papers) for an experimental confirmation at LHC (4 July, 2012).",
"This is (thought to be) the first elementary scalar particle discovered in Nature.",
"However, in cosmology, it has been known for a long time that there is scalar (cosmological fluctuation mode) in the sky after the discovery of CMB (1964, the same year that Higgs field has been proposed !",
").",
"But, the existence of a scalar mode in the sky is a big mystery from the following reasons.",
"The small fluctuations in CMB are thought to be due to (space-time) fluctuations in the early epoch of our (Big Bang) Universe, which should be described by General Relativity (GR).",
"However, GR alone can not have scalar (fluctuation) mode but only the (usual) tensor mode (2 polarizations, called gravitational waves) !",
"The cosmological/primordial tensor modes have not been discovered yet, though astrophysical tensor modes (gravitational waves) from black hole mergers have been detected in LIGO.",
"So, it would not be quite strange (at least to me) even though we discover the cosmological/primordial tensor modes (called B-mode) in the near future.",
"Rather, the existence of the scalar mode is much more surprising !",
"The simplest way to explain the data is introduce a “cosmic scalar\" field and “assume\" some peculiar behaviors in the early epoch (known as the inflationary epoch).",
"But, there are huge numbers of explicit models and we still do not know what is the right one and its origin–Is Higgs the remnant of the primordial scalar ?",
"This situation is opposite to that of particle physics: In cosmology, we have discovered scalar mode (CMB) long ago but we do not have its theory yet !",
"Another way to explain the scalar mode is to modify GR, like $f(R)$ , massive gravity, etc., so that the gravity itself has additional scalar modes.",
"But, in that case the scalar (gravitation) could be disaster since it could affect the known (or, well-established) GR test in solar system (,i.e., low energy (IR) test of GR), unless we decouple it from low energy GR sector.",
"And also, this could affect the dark matter and dark energy problems as well.",
"Sometimes, the scalar could be ghost as well, which should be avoided.",
"Today, I will consider another modified gravity theory but without the scalar gravitation mode so that the usual scalar matters are needed in explaining the cosmology data.",
"(2).",
"One the other hand, our universe is considered to be created from “quantum (vacuum) fluctuations\".",
"Actually, since the scalar power spectrum in inflationary cosmology contains Planck constant $\\hbar $ as well as Newton's constant $G$ ($H$ is the Hubble parameter), $\\Delta ^2(k)= 8 \\pi G \\hbar ( {2 H^2}/{\\pi ^2}),$ one can consider the power spectrum as a “quantum gravity effect\" and so does the inflationary cosmology as a “quantum cosmology\" !",
"Moreover, recent LIGO's detections of gravitational waves from merging black holes seems to imply that we need to consider quantum gravity more seriously.",
"Actually, we open the strong gravity test era of GR, beyond the weak gravity test in solar system.",
"(3).",
"Do we have quantum gravity then ?",
"For quantum theory of particle's interactions, “renormalization\" has been a powerful constraint and Higgs particle is its natural consequence.",
"Now, what if we require renormalizability in quantum gravity ?",
"But, it is well-known that the renormalizable quantum gravity can not be realized in Einstein's gravity or its (relativistic) higher-derivative generalizations: There are “ghosts\" which have kinetic terms with a wrong sign, in addition to massless gravitons.",
"Recently, Hořava (or Hořava-Lifshitz (HL)) gravity has been proposed as a renormalzable gravity [1].",
"There is no ghosts (in the tensor modes) by abandoning the equal-footing treatment of space and time (i.e., Lorentz symmetry) in the higher energy regime (UV).",
"It is power-counting renormalizable but no proof of renormalizability yet.",
"This resembles the situations of Yang-Mills theory when it first appeared.",
"In cosmology, this theory may provide inflationary effect without “inflationary phase (i.e., early de Sitter or accelerating phase)\" so that scale-invariant (scalar/tensor) spectrum can be also produced.",
"This has been first argued by Mukohyama [2] but there is no rigorous analysis about this yet.",
"On the contrary, in the works of Gao et.",
"al.",
"[3] and Gong et.",
"al.",
"[4], scalar spectrum are not scale-invariant though tensor spectrum does.",
"This is today's topic and it provides a natural formulation of cosmology, known as “effective field theory (EFT)\", by construction."
],
[
"Hořava (-Lifshitz) Gravity: Basic Idea",
"In 2009, Hořava proposed a renormalizable, higher-derivative gravity theory, without ghost problems, by abandoning Lorentz symmetry in UV but keeping, so called, “foliation preserving\" diffemorphism ($FPDiff$ ) [1].",
"The basic ingredients of the Hořava are (1) Einstein gravity (with a Lorentz deformation parameter $\\lambda $ ), (2) non-covariant deformations with higher-spatial derivatives (up to six orders), and (3) “detailed balance\" in the coefficients (with five constant parameters, $\\kappa , {\\lambda }, \\nu , \\mu , {\\Lambda }_W$ ), in contrast to the Einstein gravity case with the three physical parameters $c, G, {\\Lambda }$ and ${\\lambda }=1$ .",
"Here, six orders of spatial derivatives $(z=3)$ came from the power counting renormalizability in $3+1$ dimensions.",
"(For the limitation of space and time, I can not explain all the details of the construction.",
"So, please read the original work of Hořava [1] about this.)",
"From the Hořava's construction, some improved UV behaviors, without ghosts, are expected so that the gravity theory “could\" be renormalizable and one might have predictable quantum gravity theory.",
"But, in cosmology applications, it seems that the detailed-balance condition is too strong to get the required, scale invariant cosmological perturbations.",
"For example, tensor spectrum is scale invariant but scalar spectrum is not !",
"[3], [4].",
"So, we need to break the detailed balancing in some way but without altering the improved UV behaviors of scale invariant tensor modes in the previous works [3], [4] and we called it as “UV modifications\" [5]."
],
[
"Cosmological Perturbations in Hořava gravity",
"We start by considering the four-dimensional, UV-modified Hořava gravity action with $z=3$ , which is power-counting renormalizable [1], $S_\\mathrm {g} &=& \\int d \\eta d^3x \\sqrt{g} N \\left[ \\frac{2}{\\kappa ^2}\\left(K_{ij}K^{ij} - \\lambda K^2 \\right) - {\\cal V} \\right], \\\\-{\\cal V}&=& \\sigma + \\xi R + \\alpha _1 R^2+ \\alpha _2 R_{ij}R^{ij}+\\alpha _3 \\frac{\\epsilon ^{ijk}}{\\sqrt{g}}R_{il}\\nabla _jR^l{}_k+ \\alpha _4 \\nabla _{i}R_{jk} \\nabla ^{i}{R}^{jk}+{\\alpha }_5 \\nabla _{i}R_{jk}\\nabla ^{j} {R}^{ik}+{\\alpha }_6 \\nabla _{i}R\\nabla ^{i}R , \\numero $ where (the prime $(^{\\prime })$ denotes the derivative with respect to $\\eta $ ) $K_{ij}=\\frac{1}{2N}\\left({g_{ij}}^{\\prime }-\\nabla _i N_j-\\nabla _jN_i\\right)\\, ,~ds^2 =-N^2 c^2 d\\eta ^2 +g_{ij}\\left(dx^i+N^i d\\eta \\right) \\left(dx^j+N^j d\\eta \\right).",
"\\numero $ With the detailed balance condition, the number of independent coupling constants can be reduced to six, i.e., $\\kappa ,\\lambda ,\\mu ,\\nu ,{\\Lambda }_W,{\\omega }$ for a viable model in IR [8], [9], $\\sigma = {8 (3 {\\lambda }-1)},~\\xi ={8 (3 {\\lambda }-1)},~{\\alpha }_1={32 (3 {\\lambda }-1)}, ~{\\alpha }_2=-{8}, ~{\\alpha }_3 = {2 \\nu ^2},~{\\alpha }_4=-{8 2 \\nu ^4}=-{\\alpha }_5=-8 {\\alpha }_6,$ But, we do not restrict to this case only, at least for the UV couplings ${\\alpha }_4,{\\alpha }_5,{\\alpha }_6$ so that the power-counting renormalizable and scale-invariant cosmological scalar fluctuations can be obtained.",
"For the power-counting renormalizable matter action, we consider $z=3$ scalar field action [6], [7], $S_\\mathrm {m} = \\int d\\eta d^3x \\sqrt{g} N \\left[ \\frac{1}{2N^2}\\left( \\phi ^{\\prime } - N^i {\\partial }_i \\phi \\right)^2 - V(\\phi )- Z({\\partial }_i\\phi ) \\right] \\, ,$ where $Z({\\partial }_i\\phi ) = \\sum _{n=1}^3 \\xi _n \\partial _i^{(n)}\\phi \\partial ^{i(n)}\\phi \\,$ with the superscript $(n)$ denoting $n$ -th spatial derivatives, and $V(\\phi )$ is the matter's potential without derivatives.",
"The actions, $S_\\mathrm {g}$ and $S_\\mathrm {m}$ are invariant under the foliation preserving FPDiff [1], $\\delta x^i &=&-\\zeta ^i (\\eta , {\\bf x}), ~\\delta \\eta =-f(\\eta ),~\\delta g_{ij}={\\partial }_i\\zeta ^k g_{jk}+{\\partial }_j \\zeta ^k g_{ik}+\\zeta ^k{\\partial }_k g_{ij}+f g^{\\prime }_{ij},\\\\\\delta N_i &=& {\\partial }_i \\zeta ^j N_j+\\zeta ^j {\\partial }_j N_i+\\zeta ^{^{\\prime }j}g_{ij}+f {N^{\\prime }}_i+f^{\\prime } N_i,~\\delta N= \\zeta ^j {\\partial }_j N+f N^{\\prime }+f^{\\prime } N, ~\\delta \\phi =\\zeta ^j {\\partial }_j \\phi +f \\phi ^{\\prime }.",
"\\numero $ In order to study the cosmological perturbations around the homogeneous and isotropic backgrounds (as seen in CMB), we expand the metric and the scalar field as, $N = a(\\eta )[1+{\\cal A}(\\eta ,{\\bf x})] \\, ,~N_i = a^2(\\eta ){{\\cal B}(\\eta ,{\\bf x})}_i \\, , ~g_{ij} = a^2(\\eta ) [{\\delta }_{ij}+h_{ij} (\\eta ,{\\bf x})],~\\phi = \\phi _0(\\eta ) + \\delta \\phi (\\eta ,{\\bf x}) \\, ,$ by considering spatially flat ($k=0$ ) backgrounds and the conformal (or comoving) time $\\eta $ , for simplicity.",
"By substituting the metric and scalar field of (REF ) into the actions one can obtain the linear-order perturbation part of the total action $S=S_g +S_m$ which gives the Friedman's equations for the background, $&&{\\cal H}^2 =-\\frac{\\kappa ^2}{6(1-3\\lambda )} \\left( {2}{\\phi _0^{\\prime }}^2+ a^2 \\left(V_0 -\\sigma \\right) \\right) \\, ,~{\\cal H}^2 +2{\\cal H}^{\\prime } =\\frac{\\kappa ^2}{2(1-3\\lambda )} \\left( {2}{\\phi _0^{\\prime }}^2- a^2 \\left(V_0 -\\sigma \\right) \\right) \\, , \\numero \\\\&&\\phi _0^{\\prime \\prime } + 2{\\cal H}\\phi _0^{\\prime } + a^2V_{\\phi _0} = 0 \\, ,$ with the comoving Hubble parameter ${\\cal H} \\equiv a^{\\prime }/a$ , $V_0 \\equiv V(\\phi _0), V_{\\phi _0}\\equiv ({\\partial }V/{\\partial }\\phi )_{\\phi _0}$ , and $h\\equiv h^i_i$ .",
"Here, it important to note that there is no higher-derivative corrections to the Friedman's equations for spatially flat case and so the background equations are the same as those of GR [8], [9].",
"However, even in this case, the higher-derivative effects can reappear in the perturbed parts.",
"The quadratic part of the total perturbed action is given by ${\\delta }_2 S&=& \\int d \\eta d^3x \\left\\lbrace {{\\kappa }^2} \\left[ (1-3 {\\lambda }) {\\cal H} \\left(3 {\\cal H} {\\cal A}^2 + {\\cal A} (2 {\\partial }{\\cal B}^i-h^{\\prime }) \\right)+ (1-{\\lambda }) ({\\partial }_i {\\cal B}^i)^2+{2} {\\partial }_i {\\cal B}_j {\\partial }^i {\\cal B}^j \\right.",
"\\right.",
"\\numero \\\\&&\\left.",
"\\left.",
"-{\\partial }_i {\\cal B}_j {h^{ij}}^{\\prime } +{4} {h_{ij}}^{\\prime } {h^{ij}}^{\\prime }+{\\lambda }\\left({\\partial }_i {\\cal B}^i h^{\\prime }-{4}h^{\\prime 2} \\right) \\right]+a^2 \\xi \\left({\\cal A} +{2} h \\right)\\left( {\\partial }_i {\\partial }_j h^{ij} -\\Delta h \\right) \\right.",
"\\numero \\\\&&\\left.+a^2 \\left[ {2} {\\delta }\\phi ^{\\prime 2}-{\\cal A} \\phi _0^{\\prime } {\\delta }\\phi ^{\\prime } +{2} {\\cal A}^2 \\phi _0^{\\prime 2} +{\\partial }_i {\\cal B}^i \\phi _0^{\\prime } {\\delta }\\phi -{2} V_{\\phi _0 \\phi _0} {\\delta }\\phi ^2 -a^2 V_{\\phi _0} {\\delta }\\phi {\\cal A}-{2} \\phi _0^{\\prime } {\\delta }\\phi h^{\\prime }\\right] \\right.\\numero \\\\&&\\left.",
"-a^4 \\left({\\cal V}^{(2)}+{\\delta }Z \\right)\\right\\rbrace ,$ where ${\\cal V}^{(2)}$ is the quadratic part of the potential ${\\cal V}$ and $\\Delta \\equiv {\\delta }^{ij} {\\partial }_i {\\partial }_j$ , ${\\delta }Z=\\sum _{n=1}^3 \\xi _n~ \\partial _i^{(n)}{\\delta }\\phi \\partial ^{i(n)}{\\delta }\\phi $ .",
"Now, in order to separate the scalar, vector, and tensor contributions, we consider the most general (SVT) decompositions $({\\partial }_i S^i = {\\partial }_i F^i = \\tilde{H} = {\\partial }_i \\tilde{H}^i_j = 0 )$ , ${\\cal B}_i = {\\partial }_i {\\cal B} + S_i \\, ,~h_{ij} = 2 {\\cal R} {\\delta }_{ij}+ {\\partial }_i {\\partial }_j {\\cal E} + {\\partial }_{(i} F_{j)} + \\tilde{H}_{ij} \\, .$ Then, the pure tensor, vector, and scalar parts of the total action are given by, respectively, $\\delta _2 S^{(t)} &=& \\int d \\eta d^3x~ a^2\\left[ \\frac{2}{\\kappa ^2} {\\tilde{H}_{ij}}^{\\prime } {\\tilde{H}}^{ij^{\\prime }}+\\xi \\tilde{H}_{ij}\\Delta \\tilde{H}^{ij}+ \\frac{\\alpha _2}{a^2}\\Delta \\tilde{H}_{ij}\\Delta \\tilde{H}^{ij}+ \\frac{\\alpha _3}{a^3} \\epsilon ^{ijk} \\Delta \\tilde{H}_{il} \\Delta {\\partial }_j \\tilde{H}^l_{k} \\right.",
"\\numero \\\\&&~~~~~~~~~~~~~~~~~~~~\\left.- \\frac{\\alpha _4}{a^4} \\Delta \\tilde{H}_{ij}\\Delta ^2 \\tilde{H}^{ij} \\right],\\\\\\delta _2 S^{(v)} &=& \\frac{1}{\\kappa ^2} \\int d \\eta d^3x~ a^2 {\\partial }_i\\left( S^j-{F^j}^{\\prime } \\right) {\\partial }_i \\left( S_j-F_j^{\\prime } \\right) \\, ,\\\\\\delta _2 S^{(s)}&=& \\int d \\eta d^3x ~a^2 \\left\\lbrace \\frac{2(1-3\\lambda )}{\\kappa ^2}\\left[ 3{{\\cal R}^{\\prime }}^2 - 6{\\cal H}{\\cal A}{\\cal R}^{\\prime } +3{\\cal H}^2 {\\cal A}^2- 2\\left( {\\cal R}^{\\prime } - {\\cal H}{\\cal A}\\right) \\Delta ({\\cal B}-{\\cal E}^{\\prime })\\right] \\right.\\nonumber \\\\&&+ \\frac{2(1-\\lambda )}{\\kappa ^2} \\left[ \\Delta \\left({\\cal B}-{\\cal E}^{\\prime }\\right) \\right]^2- 2\\xi ({\\cal R}+2{\\cal A})\\Delta {\\cal R}+ \\frac{2}{a^2} \\left( 8\\alpha _1+3\\alpha _2 \\right)(\\Delta {\\cal R})^2\\nonumber \\\\&& \\left.-\\frac{2}{a^4} \\left( 3\\alpha _4+2\\alpha _5+8 {\\alpha }_6 \\right)\\Delta {\\cal R}\\Delta ^2 {\\cal R}-a^2V_{\\phi _0} {\\cal A}\\delta \\phi -\\frac{1}{2a^2}V_{\\phi _0\\phi _0}\\delta \\phi ^2 - \\delta {Z} \\right.\\numero \\\\&& \\left.+ \\frac{1}{2}{\\delta \\phi ^{\\prime }}^2 - \\phi _0^{\\prime }\\delta \\phi ^{\\prime } {\\cal A}+{2} {\\phi _0^{\\prime }}^2 {\\cal A}^2+ [\\Delta ({\\cal B}-{\\cal E}^{\\prime })-3 {\\cal R}^{\\prime }]\\phi _0^{\\prime }\\delta \\phi \\right\\rbrace \\, .$ Here, it is important to note that sixth-order-derivative terms in the gravity action contribute to scalar as well as tensor perturbations, through the specific combination of `$3\\alpha _4+2\\alpha _5+8 {\\alpha }_6$ ' for the former but through only `${\\alpha }_4$ ' for the latter.",
"This is what we need for renormalizability and scale-invariant spectrums (as we can see shortly), for both scalar and tensor.",
"In the detailed-balanced case, the combination `$3\\alpha _4+2\\alpha _5+8 {\\alpha }_6$ ' vanishes though ${\\alpha }_4$ does not.",
"So in that case, the theory would not be renormalizable and nor scale invariant for scalar part, which is in contradict to observational data !",
"Now, in order to exhibit the true dynamical degrees of freedom we consider the Hamiltonian reduction method [10], for the cosmologically perturbed actions [11], [4] and, after some computation, the action reduces to $\\delta _2S_\\star ^{(s)}= \\int d\\eta d^3x \\frac{1}{2}\\left\\lbrace {u^{\\prime }}^2- u \\left[ {2}\\left( {\\cal G}_1^{\\prime }{\\cal G}_1^{-1} \\right)^{\\prime }- {4}\\left({\\cal G}_1^{\\prime }{\\cal G}_1^{-1}\\right)^2- {\\cal G}_1 \\left( {\\cal G}_2 {\\cal G}_1^{-1} \\right)^{\\prime }- {\\cal G}_2^2 + 4{\\cal G}_1{\\cal G}_3 \\right]u \\right\\rbrace \\,$ for a true scalar degree of freedom $u$ .",
"In UV limit ($3 \\tilde{{\\alpha }_4} \\equiv 3 {\\alpha }_4 +2 {\\alpha }_5+8 {\\alpha }_6, ~z \\equiv a \\phi _0^{\\prime }/{\\cal H}$ ), its equations of motion reduce to $u^{\\prime \\prime } =- {\\omega }_{u(UV)}^2 u \\, , ~~{\\omega }_{u(UV)}^2= {a^2 z^2 } \\left[ 2+{4 (1-3 {\\lambda })}{a^2}\\right] \\Delta ^3.$ Here, it is important to note that there are sixth-spatial derivatives, as required by the scale invariance of the observed power spectrum [2] as well as the (power-counting) renormalizability [1].",
"This occurs only when there are sixth-derivative terms in the starting scalar action as well as some breaking of the detailed balance condition in sixth-derivative terms for the gravity action (i.e., $\\tilde{{\\alpha }_4} \\ne 0$ ) Regarding the scale invariance of the power spectrums, it has been noted that Hořava gravity could provide an alternative mechanism for the early Universe without introducing the hypothetical inflationary epoch [2]: The basic reason of the alternative mechanism comes from the momentum-dependent speeds of gravitational perturbations which could be much larger than the current, low energy (i.e., IR) speed $c$ so that the exponentially expanding early space-time could be mimicked.",
"In order to see this explicitly in our case, we consider the power-law expansions [12] ($t$ is the physical time, defined by $dt=a d \\eta $ ), $a =a_0 t^p, ~(1/3 < p <1).$ Then we can produce the scale-invariant power spectrums for the quantum field $\\hat{\\zeta }$ of the $\\zeta $ (scalar) perturbation, as follows, $\\left<0|\\hat{\\zeta }_{\\bf k} (\\eta ) \\hat{\\zeta }_{{\\bf k}^{\\prime }} (\\eta )|0\\right>=(2 \\pi )^3 {\\delta }({\\bf k}+{\\bf k}^{\\prime }) {k^3} \\Delta ^2_{\\zeta } (k),~~\\Delta ^2_{\\zeta }={2 \\pi ^2} | \\zeta _{\\bf k} |^2={8 \\pi ^2} \\sqrt{{6 {\\kappa }^2 \\xi _3 \\tilde{{\\alpha }}_4}}~,$ without knowing the details of the history of the early Universe and the form of the (non-derivative) potential $V (\\phi )$ .",
"This result shows that one can achieve the \"inflation without inflation\" picture, as argued by Mukohyama [2].",
"The basic reason of this is that, in Hořava gravity, due to the momentum-dependent, superluminal speeds of fluctuations is possible in the early Universe, which is assumed to be UV region, so that one can mimic \"the inflationary scenario without inflationary epoch\" !",
"(See Fig.",
"1 in [5].)"
],
[
"Acknowledgments",
"This was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1A2B401304)."
]
] | 1709.01671 |
[
[
"Inner amenability and approximation properties of locally compact\n quantum groups"
],
[
"Abstract We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction.",
"We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\\mathbb{G}$.",
"Similar homological techniques are used to prove that $\\ell^1(\\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\\mathbb{G}$; a discrete Kac algebra $\\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\\widehat{\\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\\mathbb{G}$ implies $C_u(\\widehat{\\mathbb{G}})=L^1(\\widehat{\\mathbb{G}})\\widehat{\\otimes}_{L^1(\\widehat{\\mathbb{G}})}C_0(\\widehat{\\mathbb{G}})$ completely isometrically.",
"The latter result allows us to partially answer a conjecture of Voiculescu when $\\mathbb{G}$ has the approximation property."
],
[
"Introduction",
"The class of inner amenable locally compact groups has played an important role in the development of abstract harmonic analysis and its connection to operator algebras [46], [48], [58].",
"In particular, it provides a sufficiently general class of locally compact groups for which approximation properties of $G$ can be characterized through approximation properties of its reduced $C^*$ -algebra $C^*_\\lambda (G)$ and von Neumann algebra $VN(G)$ .",
"Indeed, within the class of inner amenable groups, amenability of $G$ is equivalent to nuclearity of $C^*_\\lambda (G)$ as well as injectivity of $VN(G)$ [46]; exactness of $G$ is equivalent to exactness of $C^*_\\lambda (G)$ [1], and the Haagerup property of $G$ is equivalent to the Haagerup property of $VN(G)$ [52].",
"In [33], a notion of inner amenability was introduced for an arbitrary locally compact quantum group $\\mathbb {G}$ by means of the existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,x\\star f\\rangle =\\langle m,f\\star x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Although such a condition may be interesting in its own right, if $\\mathbb {G}$ is co-amenable, then any cluster point in $L^{\\infty }(\\mathbb {G})^*$ of a bounded approximate identity in $L^1(\\mathbb {G})$ will give rise to such a state.",
"In particular, any locally compact group is an “inner amenable” locally compact quantum group in the sense of [33].",
"In this paper we introduce a bona fide generalization of inner amenability to locally compact quantum groups.",
"We study its basic properties, examples, and related notions of strong and topological inner amenability.",
"Generalizing a well-known result of Lau–Paterson [46], we show that $\\mathbb {G}$ is amenable if and only if $L^{\\infty }(\\widehat{\\mathbb {G}})$ is injective and $\\mathbb {G}$ is inner amenable.",
"We also derive sufficient conditions under which the bicrossed product of a matched pair of locally compact groups is inner amenable.",
"We pursue homological manifestations of this concept in section 4, showing that inner amenability of $\\mathbb {G}$ entails the relative 1-flatness of $L^1(\\widehat{\\mathbb {G}})$ .",
"As an application of our techniques, we prove that $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.",
"This resolves a recent conjecture of Ng–Viselter [51], and generalizes a recent result of Ng [50] from locally compact groups to arbitrary Kac algebras, wherein $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.",
"The method of proof shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.",
"Indeed, we show that for strongly inner amenable quantum groups, weak amenability and the approximation property follow respectively from the w*CBAP and w*OAP of $L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"In particular, for inner amenable locally compact groups $G$ , the w*CBAP of $VN(G)$ implies weak amenability of $G$ and the approximation property is equivalent to the w*OAP of $VN(G)$ .",
"Similar results are proved in the setting of topological inner amenability and approximation properties of $C_0()$ .",
"These techniques may be viewed as new tools for the development of harmonic analysis on quantum groups beyond the unimodular discrete setting (for which the above results greatly simplify).",
"In section 5 we establish the self-duality of (non-relative) 1-biflatness, that is, $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.",
"This shows, in particular, that $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group $\\mathbb {G}$ .",
"Section 6 is devoted to the question of operator amenability of $L^1_{cb}(\\mathbb {G})$ , the $cb$ -multiplier closure of $L^1(\\mathbb {G})$ .",
"For unimodular discrete quantum groups $\\mathbb {G}$ with Kirchberg's factorization property, we show that $\\mathbb {G}$ is weakly amenable if and only if $L^1_{cb}()$ is operator amenable.",
"This result is new even for the class of weakly amenable residually finite discrete groups such that $C^*(G)$ is not residually finite-dimensional (see [6]).",
"We finish in section 7 with a strengthening of [14], by showing that $C_u(\\mathbb {G})^*\\cong M_{cb}^l(L^1(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically when $$ is amenable.",
"As a corollary, when $$ has the approximation property, we prove that $\\mathbb {G}$ is co-amenable if and only if $$ is amenable."
],
[
"Operator Modules",
"Let $\\mathcal {A}$ be a completely contractive Banach algebra.",
"We say that an operator space $X$ is a right operator $\\mathcal {A}$ -module if it is a right Banach $\\mathcal {A}$ -module such that the module map $m_X:X\\widehat{\\otimes }\\mathcal {A}\\rightarrow X$ is completely contractive, where $\\widehat{\\otimes }$ denotes the operator space projective tensor product.",
"We say that $X$ is faithful if for every non-zero $x\\in X$ , there is $a\\in \\mathcal {A}$ such that $x\\cdot a\\ne 0$ , and we say that $X$ is essential if $\\langle X\\cdot \\mathcal {A}\\rangle =X$ , where $\\langle \\cdot \\rangle $ denotes the closed linear span.",
"Note that our definition of faithfulness, the standard notion in operator modules, is opposite in nature to the usual definition in algebra.",
"We denote by $\\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ the category of right operator $\\mathcal {A}$ -modules with morphisms given by completely bounded module homomorphisms.",
"Left operator $\\mathcal {A}$ -modules and operator $\\mathcal {A}$ -bimodules are defined similarly, and we denote the respective categories by $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ and $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"Regarding terminology, in what follows we will often omit the term “operator” when discussing homological properties of operator modules as we will be working exclusively in the operator space category.",
"Let $\\mathcal {A}$ be a completely contractive Banach algebra, $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ and $Y\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ .",
"The $\\mathcal {A}$ -module tensor product of $X$ and $Y$ is the quotient space $X\\widehat{\\otimes }_{\\mathcal {A}}Y:=X\\widehat{\\otimes }Y/N$ , where $N=\\langle x\\cdot a\\otimes y-x\\otimes a\\cdot y\\mid x\\in X, \\ y\\in Y, \\ a\\in \\mathcal {A}\\rangle ,$ and, again, $\\langle \\cdot \\rangle $ denotes the closed linear span.",
"It follows that (see [9]) $\\mathcal {CB}_{\\mathcal {A}}(X,Y^*)\\cong N^{\\perp }\\cong (X\\widehat{\\otimes }_{\\mathcal {A}} Y)^*,$ where $\\mathcal {CB}_{\\mathcal {A}}(X,Y^*)$ is the space of completely bounded right $\\mathcal {A}$ -module maps $\\Phi :X\\rightarrow Y^*$ .",
"If $Y=\\mathcal {A}$ , then clearly $N\\subseteq \\mathrm {Ker}(m_X)$ where $m_X:X\\widehat{\\otimes }\\mathcal {A}\\rightarrow X$ is the module map.",
"If the induced mapping $\\widetilde{m}_X:X\\widehat{\\otimes }_{\\mathcal {A}}\\mathcal {A}\\rightarrow X$ is a completely isometric isomorphism we say that $X$ is an induced $\\mathcal {A}$ -module.",
"A similar definition applies for left modules.",
"In particular, we say that $\\mathcal {A}$ is self-induced if $\\widetilde{m}_\\mathcal {A}:\\mathcal {A}\\widehat{\\otimes }_{\\mathcal {A}}\\mathcal {A}\\cong \\mathcal {A}$ completely isometrically.",
"Let $\\mathcal {A}$ be a completely contractive Banach algebra and $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"The identification $\\mathcal {A}^+=\\mathcal {A}\\oplus _1 turns the unitization of $ A$ into a unital completely contractive Banach algebra, and it follows that $ X$ becomes a right operator $ A+$-module via the extended action\\begin{equation*}x\\cdot (a+\\lambda e)=x\\cdot a+\\lambda x, \\ \\ \\ a\\in \\mathcal {A}^+, \\ \\lambda \\in \\ x\\in X.\\end{equation*}Let $ C1$.",
"Then $ X$ is \\emph {relatively $ C$-projective} if there exists a morphism $ +:XXA+$ which is a right inverse to the extended module map $ mX+:XA+X$ such that $ +cbC$.",
"When $ X$ is essential, then $ X$ is relatively $ C$-projective if and only if there exists a morphism $ :XXA$ satisfying $ cbC$ and $ mX=idX$ by the operator analogue of \\cite [Proposition 1.2]{DP}.",
"We say that $ X$ is \\emph {$ C$-projective} if for every $ Y,Zmod A$, every complete quotient morphism $ :YZ$, every morphism $ :XZ$, and every $ >0$, there exists a morphism $ :XY$ such that $ cb< Ccb+$ and $ =$, i.e., the following diagram commutes:\\begin{equation*}\\begin{tikzcd}&Y [d, two heads, \"\\Psi \"]\\\\X [ru, dotted, \"\\widetilde{\\Phi }_\\varepsilon \"] [r, \"\\Phi \"] &Z\\end{tikzcd}\\end{equation*}$ Given a completely contractive Banach algebra $\\mathcal {A}$ and $X\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ , there is a canonical completely contractive morphism $\\Delta ^+:X\\rightarrow \\mathcal {CB}(\\mathcal {A}^+,X)$ given by $\\Delta ^+(x)(a)=x\\cdot a, \\ \\ \\ x\\in X, \\ a\\in \\mathcal {A}^+,$ where the right $\\mathcal {A}$ -module structure on $\\mathcal {CB}(\\mathcal {A}^+,X)$ is defined by $(\\Psi \\cdot a)(b)=\\Psi (ab), \\ \\ \\ a\\in \\mathcal {A}, \\ \\Psi \\in \\mathcal {CB}(\\mathcal {A}^+,X), \\ b\\in \\mathcal {A}^+.$ An analogous construction exists for objects in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod}$ .",
"Let $C\\ge 1$ .",
"Then $X$ is relatively $C$ -injective if there exists a morphism $\\Phi ^+:\\mathcal {CB}(\\mathcal {A}^+,X)\\rightarrow X$ such that $\\Phi ^+\\circ \\Delta ^+=\\textnormal {id}_{X}$ and $\\Vert \\Phi ^+\\Vert _{cb}\\le C$ .",
"When $X$ is faithful, then $X$ is relatively $C$ -injective if and only if there exists a morphism $\\Phi :\\mathcal {CB}(\\mathcal {A},X)\\rightarrow X$ such that $\\Phi \\circ \\Delta =\\textnormal {id}_{X}$ and $\\Vert \\Phi \\Vert _{cb}\\le C$ by the operator analogue of [17], where $\\Delta (x)(a):=\\Delta ^+(x)(a)$ for $x\\in X$ and $a\\in \\mathcal {A}$ .",
"We say that $X$ is $C$ -injective if for every $Y,Z\\in \\mathbf {mod\\hspace{2.0pt}\\mathcal {A}}$ , every completely isometric morphism $\\Psi :Y\\hookrightarrow Z$ , and every morphism $\\Phi :Y\\rightarrow X$ , there exists a morphism $\\widetilde{\\Phi }:Z\\rightarrow X$ such that $\\Vert \\widetilde{\\Phi }\\Vert _{cb}\\le C\\Vert \\Phi \\Vert _{cb}$ and $\\widetilde{\\Phi }\\circ \\Psi =\\Phi $ , that is, the following diagram commutes: $\\begin{tikzcd}Z [rd, dotted, \"\\widetilde{\\Phi }\"]\\\\Y [u, hook, \"\\Psi \"] [r, \"\\Phi \"] &X\\end{tikzcd}$ There is a natural categorical equivalence between $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ and $\\mathbf {mod\\hspace{2.0pt}\\mathcal {A}^{\\mathrm {op}}\\widehat{\\otimes }\\mathcal {A}}$ given by $axb=x\\cdot (a\\otimes b), \\ \\ \\ a,b\\in \\mathcal {A}, \\ x\\in X, \\ X\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}.$ With this identification, we obtain the following bimodule analogue of [14].",
"Proposition 2.1 Let $\\mathcal {A}$ be a completely contractive Banach algebra and $X\\in \\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"If $X$ is $C_1$ -injective in $\\mathbf {mod}- and is relatively $ C2$-injective in $ A mod A$, then $ X$ is $ C1C2$-injective in $ A mod A$.$"
],
[
"Locally Compact Quantum Groups",
"A locally compact quantum group is a quadruple $\\mathbb {G}=(L^{\\infty }(\\mathbb {G}),\\Gamma ,\\varphi ,\\psi )$ , where $L^{\\infty }(\\mathbb {G})$ is a Hopf-von Neumann algebra with co-multiplication $\\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , and $\\varphi $ and $\\psi $ are fixed left and right Haar weights on $L^{\\infty }(\\mathbb {G})$ , respectively [43], [44].",
"For every locally compact quantum group $\\mathbb {G}$ , there exists a left fundamental unitary operator $W$ on $L^2(\\mathbb {G},\\varphi )\\otimes L^2(\\mathbb {G},\\varphi )$ and a right fundamental unitary operator $V$ on $L^2(\\mathbb {G},\\psi )\\otimes L^2(\\mathbb {G},\\psi )$ implementing the co-multiplication $\\Gamma $ via $\\Gamma (x)=W^*(1\\otimes x)W=V(x\\otimes 1)V^*, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Both unitaries satisfy the pentagonal relation; that is, $W_{12}W_{13}W_{23}=W_{23}W_{12}\\hspace{10.0pt}\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}\\hspace{10.0pt}V_{12}V_{13}V_{23}=V_{23}V_{12}.$ By [44], we may identify $L^2(\\mathbb {G},\\varphi )$ and $L^2(\\mathbb {G},\\psi )$ , so we will simply use $L^2(\\mathbb {G})$ for this Hilbert space throughout the paper.",
"We denote by $R$ the unitary antipode of $\\mathbb {G}$ .",
"Let $L^1(\\mathbb {G})$ denote the predual of $L^{\\infty }(\\mathbb {G})$ .",
"Then the pre-adjoint of $\\Gamma $ induces an associative completely contractive multiplication on $L^1(\\mathbb {G})$ , defined by $\\star :L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})\\ni f\\otimes g\\mapsto f\\star g=\\Gamma _*(f\\otimes g)\\in L^1(\\mathbb {G}).$ The multiplication $\\star $ is a complete quotient map from $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ onto $L^1(\\mathbb {G})$ , implying $\\langle L^1(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle =L^1(\\mathbb {G}).$ Moreover, $L^1(\\mathbb {G})$ is always self-induced.",
"The proof follows from [63] (see [14] for details).",
"The canonical $L^1(\\mathbb {G})$ -bimodule structure on $L^{\\infty }(\\mathbb {G})$ is given by $f\\star x=(\\textnormal {id}\\otimes f)\\Gamma (x) \\ \\ \\ \\ x\\star f=(f\\otimes \\textnormal {id})\\Gamma (x), \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}), \\ f\\in L^1(\\mathbb {G}).$ A left invariant mean on $L^{\\infty }(\\mathbb {G})$ is a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,x\\star f \\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}), \\ f\\in L^1(\\mathbb {G}).$ Right and two-sided invariant means are defined similarly.",
"A locally compact quantum group $\\mathbb {G}$ is said to be amenable if there exists a left invariant mean on $L^{\\infty }(\\mathbb {G})$ .",
"It is known that $\\mathbb {G}$ is amenable if and only if there exists a right (equivalently, two-sided) invariant mean (cf.",
"[26]).",
"We say that $\\mathbb {G}$ is co-amenable if $L^1(\\mathbb {G})$ has a bounded left (equivalently, right or two-sided) approximate identity (cf.",
"[5]).",
"The left regular representation $\\lambda :L^1(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ of $\\mathbb {G}$ is defined by $\\lambda (f)=(f\\otimes \\textnormal {id})(W), \\ \\ \\ f\\in L^1(\\mathbb {G}),$ and is an injective, completely contractive homomorphism from $L^1(\\mathbb {G})$ into $\\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Then $L^{\\infty }(\\widehat{\\mathbb {G}}):=\\lbrace \\lambda (f) : f\\in L^1(\\mathbb {G})\\rbrace ^{\\prime \\prime }$ is the von Neumann algebra associated with the dual quantum group $$ .",
"Analogously, we have the right regular representation $\\rho :L^1(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\rho (f)=(\\textnormal {id}\\otimes f)(V), \\ \\ \\ f\\in L^1(\\mathbb {G}),$ which is also an injective, completely contractive homomorphism from $L^1(\\mathbb {G})$ into $\\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Then $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime }):=\\lbrace \\rho (f) : f\\in L^1(\\mathbb {G})\\rbrace ^{\\prime \\prime }$ is the von Neumann algebra associated to the quantum group $^{\\prime }$ .",
"It follows that $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })=L^{\\infty }(\\widehat{\\mathbb {G}})^{\\prime }$ , and the left and right fundamental unitaries satisfy $W\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})$ and $V\\in L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ [44].",
"Moreover, dual quantum groups always satisfy $L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })=$ [66].",
"The modular conjugations of the dual Haar weights give rise to conjugate linear isometries $J,\\widehat{J}:L^2(\\mathbb {G})\\rightarrow L^2(\\mathbb {G})$ satisfying $JL^{\\infty }(\\mathbb {G})J=L^{\\infty }(\\mathbb {G})^{\\prime }\\hspace{10.0pt}\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}\\hspace{10.0pt}\\widehat{J}L^{\\infty }(\\widehat{\\mathbb {G}})\\widehat{J}=L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime }).$ Moreover, the unitary $U:=\\widehat{J}J$ intertwines the left and right regular representations via $\\rho (f)=U\\lambda (f)U^*$ , $f\\in L^1(\\mathbb {G})$ .",
"At the level of the fundamental unitaries, this relation becomes $V=\\sigma (1\\otimes U)W(1\\otimes U^*)\\sigma .$ We also record the adjoint formulas $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*$ and $(J\\otimes \\widehat{J})V(J\\otimes \\widehat{J})=V^*$ .",
"If $G$ is a locally compact group, we let $\\mathbb {G}_a=(L^{\\infty }(G),\\Gamma _a,\\varphi _a,\\psi _a)$ denote the commutative quantum group associated with the commutative von Neumann algebra $L^{\\infty }(G)$ , where the co-multiplication is given by $\\Gamma _a(f)(s,t)=f(st)$ , and $\\varphi _a$ and $\\psi _a$ are integration with respect to a left and right Haar measure, respectively.",
"The dual $_a$ of $\\mathbb {G}_a$ is the co-commutative quantum group $\\mathbb {G}_s=(VN(G),\\Gamma _s,\\varphi _s,\\psi _s)$ , where $VN(G)$ is the left group von Neumann algebra with co-multiplication $\\Gamma _s(\\lambda (t))=\\lambda (t)\\otimes \\lambda (t)$ , and $\\varphi _s=\\psi _s$ is Haagerup's Plancherel weight (cf.",
"[62]).",
"Then $L^1(\\mathbb {G}_a)$ is the usual group convolution algebra $L^1(G)$ , and $L^1(\\mathbb {G}_s)$ is the Fourier algebra $A(G)$ .",
"It is known that every commutative locally compact quantum group is of the form $\\mathbb {G}_a$ [60].",
"Therefore, every commutative locally compact quantum group is co-amenable, and is amenable if and only if the underlying locally compact group is amenable.",
"By duality, every co-commutative locally compact quantum group is of the form $\\mathbb {G}_s$ , which is always amenable, and is co-amenable if and only if the underlying locally compact group is amenable, by Leptin's theorem [47].",
"For a locally compact quantum group $\\mathbb {G}$ , we let $C_0(\\mathbb {G}):=\\overline{\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))}^{\\Vert \\cdot \\Vert }$ denote the reduced quantum group $C^*$ -algebra of $\\mathbb {G}$ .",
"We say that $\\mathbb {G}$ is compact if $C_0(\\mathbb {G})$ is a unital $C^*$ -algebra, in which case we denote $C_0(\\mathbb {G})$ by $C(\\mathbb {G})$ .",
"We say that $\\mathbb {G}$ is discrete if $L^1(\\mathbb {G})$ is unital, in which case we denote $L^1(\\mathbb {G})$ by $\\ell ^1(\\mathbb {G})$ .",
"It is well-known that $\\mathbb {G}$ is compact if and only if $$ is discrete, and in that case, $\\ell ^1()\\cong \\bigoplus _{1} \\lbrace T_{n_\\alpha }(\\mid \\alpha \\in \\mathrm {Irr}(\\mathbb {G})\\rbrace $ , where $T_{n_\\alpha }($ is the space of $n_\\alpha \\times n_\\alpha $ trace class operators, and $\\mathrm {Irr}(\\mathbb {G})$ denotes the set of (equivalence classes of) irreducible co-representations of the compact quantum group $\\mathbb {G}$ [68].",
"For general $\\mathbb {G}$ , the operator dual $M(\\mathbb {G}):=C_0(\\mathbb {G})^*$ is a completely contractive Banach algebra containing $L^1(\\mathbb {G})$ as a norm closed two-sided ideal via the map $L^1(\\mathbb {G})\\ni f\\mapsto f|_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ .",
"The co-multiplication satisfies $\\Gamma (C_0(\\mathbb {G}))\\subseteq M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))$ , where $M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))$ is the multiplier algebra of the $C^*$ -algebra $C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})$ .",
"We let $C_u(\\mathbb {G})$ be the universal quantum group $C^*$ -algebra of $\\mathbb {G}$ , and denote the canonical surjective *-homomorphism onto $C_0(\\mathbb {G})$ by $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ [42].",
"The space $C_u(\\mathbb {G})^*$ then has the structure of a unital completely contractive Banach algebra such that the map $L^1(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})^*$ given by the composition of the inclusion $L^1(\\mathbb {G})\\subseteq M(\\mathbb {G})$ and $\\pi _{\\mathbb {G}}^*:M(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})^*$ is a completely isometric homomorphism, and it follows that $L^1(\\mathbb {G})$ is a norm closed two-sided ideal in $C_u(\\mathbb {G})^*$ [42].",
"An element $\\hat{b}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ is said to be a completely bounded left multiplier of $L^1(\\mathbb {G})$ if $\\hat{b}\\lambda (f)\\in \\lambda (L^1(\\mathbb {G}))$ for all $f\\in L^1(\\mathbb {G})$ and the induced map $m_{\\hat{b}}^l:L^1(\\mathbb {G})\\ni f\\mapsto \\lambda ^{-1}(\\hat{b}\\lambda (f))\\in L^1(\\mathbb {G})$ is completely bounded on $L^1(\\mathbb {G})$ .",
"We let $M_{cb}^l(L^1(\\mathbb {G}))$ denote the space of all completely bounded left multipliers of $L^1(\\mathbb {G})$ , which is a completely contractive Banach algebra with respect to the norm $\\Vert [\\hat{b}_{ij}]\\Vert _{M_n(M_{cb}^l(L^1(\\mathbb {G})))}=\\Vert [m^l_{\\hat{b}_{ij}}]\\Vert _{cb}.$ Completely bounded right multipliers are defined analogously and we denote by $M_{cb}^r(L^1(\\mathbb {G}))$ the corresponding completely contractive Banach algebra.",
"There is a canonical, injective completely contractive homomorphism $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ , extending $\\lambda $ , which maps $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .",
"In general, given $\\hat{b}\\in M_{cb}^l(L^1(\\mathbb {G}))$ , the adjoint $\\Theta ^l(\\hat{b}):=(m_{\\hat{b}}^l)^*$ defines a normal completely bounded left $L^1(\\mathbb {G})$ -module map on $L^{\\infty }(\\mathbb {G})$ , and by [37] have the completely isometric identification $\\Theta ^l:M_{cb}^l(L^1(\\mathbb {G}))\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G})).$ Moreover, it follows from [37] that $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ .",
"It is known that $M_{cb}^l(L^1(\\mathbb {G}))$ is a dual operator space [35], with predual $Q_{cb}^l(L^1(\\mathbb {G}))$ .",
"By the general result [34], it follows from [41] that for arbitrary locally compact quantum groups $Q_{cb}^l(L^1(\\mathbb {G}))=\\lbrace \\Omega _{A,\\rho }\\mid A\\in C_0(\\mathbb {G})\\otimes _{\\min }\\mathcal {K}(\\ell ^2), \\ \\rho \\in L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(\\ell ^2)\\rbrace ,$ where $\\langle \\hat{b},\\Omega _{A,\\rho }\\rangle =\\langle (\\Theta ^l(\\hat{b})\\otimes \\textnormal {id}_{K_{\\infty }})(A),\\rho \\rangle $ , $\\hat{b}\\in M_{cb}^l(L^1(\\mathbb {G}))$ .",
"We say that $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .",
"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .",
"We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.",
"Let $\\mathbb {G}$ be a locally compact quantum group.",
"The right fundamental unitary $V$ of $\\mathbb {G}$ induces a co-associative co-multiplication $\\Gamma ^r:\\mathcal {B}(L^2(\\mathbb {G}))\\ni T\\mapsto V(T\\otimes 1)V^*\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})),$ and the restriction of $\\Gamma ^r$ to $L^{\\infty }(\\mathbb {G})$ yields the original co-multiplication $\\Gamma $ on $L^{\\infty }(\\mathbb {G})$ .",
"The pre-adjoint of $\\Gamma ^r$ induces an associative completely contractive multiplication on the space of trace class operators $\\mathcal {T}(L^2(\\mathbb {G}))$ , defined by $\\rhd :\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\otimes \\tau \\mapsto \\omega \\rhd \\tau =\\Gamma ^r_*(\\omega \\otimes \\tau )\\in \\mathcal {T}(L^2(\\mathbb {G})).$ Analogously, the left fundamental unitary $W$ of $\\mathbb {G}$ induces a co-associative co-multiplication $\\Gamma ^l:\\mathcal {B}(L^2(\\mathbb {G}))\\ni T\\mapsto W^*(1\\otimes T)W\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})),$ and the restriction of $\\Gamma ^l$ to $L^{\\infty }(\\mathbb {G})$ is also equal to $\\Gamma $ .",
"The pre-adjoint of $\\Gamma ^l$ induces another associative completely contractive multiplication $\\lhd :\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\otimes \\tau \\mapsto \\omega \\lhd \\tau =\\Gamma ^l_*(\\omega \\otimes \\tau )\\in \\mathcal {T}(L^2(\\mathbb {G})).$ The algebra $\\mathcal {B}(L^2(\\mathbb {G}))$ inherits a canonical $\\mathcal {T}(L^2(\\mathbb {G}))$ -bimodule structure with respect to both the left $\\lhd $ and right $\\rhd $ products.",
"It was shown in [35] that the pre-annihilator $L^{\\infty }(\\mathbb {G})_{\\perp }$ of $L^{\\infty }(\\mathbb {G})$ in $\\mathcal {T}(L^2(\\mathbb {G}))$ is a norm closed two sided ideal in $(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ and $(\\mathcal {T}(L^2(\\mathbb {G})),\\lhd )$ , respectively, and the complete quotient map $\\pi :\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\omega \\mapsto f=\\omega |_{L^{\\infty }(\\mathbb {G})}\\in L^1(\\mathbb {G})$ is an algebra homomorphism from $\\mathcal {T}_{\\rhd }:=(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ , respectively, $\\mathcal {T}_{\\lhd }:=(\\mathcal {T}(L^2(\\mathbb {G})),\\lhd )$ , onto $L^1(\\mathbb {G})$ .",
"It follows that the right $\\lhd $ -module and left $\\rhd $ -module structures degenerate to an $L^1(\\mathbb {G})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ : $ f\\rhd T = (\\textnormal {id}\\otimes f)\\Gamma ^r(T), \\ \\ \\ T\\lhd f = (f\\otimes \\textnormal {id})\\Gamma ^l(T), \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})).$ By [44] the unitary antipode $R$ satisfies $R(x)=\\widehat{J}x^*\\widehat{J}$ , for $x\\in L^{\\infty }(\\mathbb {G})$ .",
"It therefore extends to a *-anti-automorphism (still denoted) $R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ , via $R(T)=J̉T^*J̉$ , $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"The extended antipode maps $L^{\\infty }(\\mathbb {G})$ and $L^{\\infty }(\\widehat{\\mathbb {G}})$ onto $L^{\\infty }(\\mathbb {G})$ and $L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ , respectively, and satisfies the generalized antipode relations; that is, $( R\\otimes R)\\circ \\Gamma ^r=\\Sigma \\circ \\Gamma ^l\\circ R\\hspace{10.0pt}\\mathrm {and}\\hspace{10.0pt}( R\\otimes R)\\circ \\Gamma ^l=\\Sigma \\circ \\Gamma ^r\\circ R,$ where $\\Sigma $ is the flip map on $\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Let $\\mathbb {G}$ and $\\mathbb {H}$ be two locally compact quantum groups.",
"Then $\\mathbb {H}$ is said to be a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes if there exists a normal, unital, injective *-homomorphism $\\gamma :L^{\\infty }(\\widehat{\\mathbb {H}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ satisfying $(\\gamma \\otimes \\gamma )\\circ \\Gamma _{}=\\Gamma _{}\\circ \\gamma $ .",
"This is not the original definition of Vaes [64], but was shown to be equivalent in [24]."
],
[
"Inner Amenability",
"Given a locally compact quantum group $\\mathbb {G}$ the composition $W\\sigma V\\sigma \\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ defines a unitary co-representation of $\\mathbb {G}$ called the conjugation co-representation, where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\sigma V\\sigma \\xi (s,t)=\\xi (s,s^{-1}ts)\\Delta (s)^{1/2}, \\ \\ \\ \\xi \\in L^2(G\\times G), \\ s,t\\in G.$ Thus, $W\\sigma V\\sigma $ is the unitary generator of the conjugation representation $\\beta _2:G\\rightarrow \\mathcal {B}(L^2(G))$ , where $\\beta _2(s)\\xi (t)=\\lambda (s)\\rho (s)\\xi (t)=\\xi (s^{-1}ts)\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\in L^{\\infty }(G)^*$ satisfying $\\langle m,\\beta _{\\infty }(s)f\\rangle =\\langle m,f\\rangle \\ \\ \\ s\\in G, \\ f\\in L^{\\infty }(G),$ where $\\beta _\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\in G$ , $f\\in L^{\\infty }(G)$ , is the conjugation action on $L^{\\infty }(G)$ .",
"Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\in \\ell ^{\\infty }(G)^*$ such that $m\\ne \\delta _e$ .",
"In what follows, inner amenability will always refer to the definition (REF ) given above.",
"In particular, every discrete group is inner amenable.",
"Definition 3.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"We say that $\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\xi _i)$ of unit vectors such that $\\Vert W\\sigma V\\sigma (\\eta \\otimes \\xi _i)-\\eta \\otimes \\xi _i\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ $\\mathbb {G}$ is inner amenable if there exists a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ satisfying $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}}).$ Such a state is said to be inner invariant.",
"$\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\in C_0()^*$ such that $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in C_0().$ Such a state is said to be topologically inner invariant.",
"Examples 3.3 The following known examples are worth mentioning.",
"Any co-commutative quantum group $\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\sigma V_s\\sigma =W_sW_s^*=1$ .",
"Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\xi :=\\Lambda _\\varphi (1)$ being conjugation invariant, where $\\varphi $ is the Haar trace on the compact dual.",
"It was shown in [51] that if $\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\mathbb {G}$ is topologically inner amenable.",
"Further examples, including the bicrossed product construction will be studied below.",
"The following proposition is standard.",
"We include the proof for completeness.",
"Proposition 3.4 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $\\mathbb {G}$ is strongly inner amenability then it is inner amenable.",
"Let $(\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\sigma V\\sigma $ .",
"Passing to a subnet, we may assume that $(\\omega _{J\\xi _i}|_{L^{\\infty }(\\widehat{\\mathbb {G}})})$ converges weak* to a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ .",
"For any $f=\\omega _{\\xi ,\\eta }\\in L^1(\\mathbb {G})$ and $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*, \\ \\ \\ (\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\otimes J)=\\sigma V^*\\sigma ,$ we have $\\langle m,\\hat{x}\\lhd f\\rangle &=\\lim _i\\langle \\omega _{J\\xi _i},\\hat{x}\\lhd f\\rangle \\\\&=\\lim _i\\langle W^*(1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),\\eta \\otimes J\\xi _i\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),W(\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})\\sigma V^*\\sigma (\\xi \\otimes J\\xi _i),\\sigma V^*\\sigma (\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\langle m,\\hat{x}\\rangle \\langle f,1\\rangle .$ In the commutative case, the converse of Proposition REF holds.",
"Proposition 3.5 A commutative quantum group $\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.",
"If $\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.",
"Let $m\\in VN(G)^*$ satisfy $\\langle m,x\\rangle =\\langle m,x \\lhd f\\rangle $ for all $x\\in VN(G)$ and all states $f\\in L^1(G)$ .",
"Then for $t\\in G$ $\\lambda (t)x\\lambda (t)^* \\lhd f=\\int _G \\lambda (s^{-1}t)x\\lambda (t^{-1}s) f(s)ds=\\int _G\\lambda (r)^*x\\lambda (r)f(tr)dr=x \\lhd f_t,$ so that $\\langle m,\\lambda (t)x\\lambda (t)^*\\rangle =\\langle m,\\lambda (t)x\\lambda (t)^* \\lhd f\\rangle =\\langle m,x \\lhd f_t\\rangle =\\langle m,x\\rangle $ for all $x\\in VN(G)$ , $t\\in G$ and states $f\\in L^1(G)$ .",
"Thus, $G$ is inner amenable by [16].",
"Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\xi _i)$ in $L^2(G)$ satisfying $\\Vert \\lambda (s)\\xi _i-\\rho (s)^*\\xi _i\\Vert _{L^2(G)}\\rightarrow 0$ uniformly on compact subsets of $G$ .",
"For any $\\eta \\in C_c(G)$ we then have $\\langle W\\sigma V\\sigma (\\eta \\otimes \\xi _i),\\eta \\otimes \\xi _i\\rangle =\\iint \\xi _i(ts)\\overline{\\xi _i}(st)\\Delta (s)^{1/2}|\\eta (s)|^2 \\ ds \\ dt\\rightarrow 1.$ It follows that $(\\xi _i)$ is strongly inner invariant.",
"Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].",
"Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\mathcal {B}(L^2(G))$ , via $\\langle m,f \\rhd T\\rangle =\\langle m, T \\lhd f\\rangle , \\ \\ \\ f\\in L^1(G), \\ T\\in \\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.",
"For details, see [13].",
"Proposition 3.6 A commutative quantum group $\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.",
"The existence of a tracial state on $C_\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.",
"By [51] $\\mathbb {G}_a$ is topologically inner amenable if $C_\\lambda ^*(G)$ has a tracial state.",
"Conversely, if $m$ is an inner invariant state on $C_\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.",
"Viewing $m$ as a positive definite function in $B_\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\in G$ .",
"A simple integral calculation then shows that $m$ is a tracial state on $C_\\lambda ^*(G)$ .",
"Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\infty }(\\widehat{\\mathbb {G}})$ to $C_0()$ .",
"This is not the case however.",
"In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\bigoplus _{n\\in \\mathbb {N}} \\Gamma _n\\rtimes \\prod _{n\\in \\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\Gamma _n$ for which $C^*_\\lambda (\\Gamma _n\\rtimes F_n)$ admits a unique tracial state.",
"In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .",
"Thus, $G$ is inner amenable.",
"However, in [30], the authors also show that the continuous function $m$ in $B_\\lambda (G)$ associated to a tracial state on $C^*_\\lambda (G)$ must be discontinuous.",
"Whence, there is no tracial state on $C^*_\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .",
"Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\lambda (G)$ .",
"In the other direction, in general it is not possible to lift a tracial state on $C^*_\\lambda (G)$ to an inner invariant mean on $VN(G)$ .",
"In [30] it is shown that $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\mathbb {R})$ .",
"Proposition 3.8 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is co-amenable then $\\mathbb {G}$ is strongly inner amenable.",
"If $\\mathbb {G}$ is amenable then it is inner amenable.",
"If $$ is co-amenable, let $E\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\omega _{\\xi _i}$ with $J̉\\xi _i=\\xi _i$ (since $L^{\\infty }(\\widehat{\\mathbb {G}})\\curvearrowright L^2(\\mathbb {G})$ is standard).",
"Then $1=(\\textnormal {id}\\otimes E)(W)=\\lim _i(\\textnormal {id}\\otimes \\omega _{\\xi _i})(W)$ strongly, from which it follows that $\\Vert W(\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Since $\\sigma V\\sigma =(J̉\\otimes J̉)W^*(J̉\\otimes J̉)$ , and $J̉\\xi _i=\\xi _i$ , we also have $\\Vert \\sigma V\\sigma (\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Whence, $\\mathbb {G}$ is strongly inner amenable.",
"Now, suppose $\\mathbb {G}$ is amenable and let $m\\in L^{\\infty }(\\mathbb {G})^*$ be a two-sided invariant mean.",
"We will show a stronger statement by providing a state $M\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ such that $\\langle M,\\rho \\rhd T\\rangle =\\langle M,T\\lhd \\rho \\rangle , \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})),$ upon which restriction to $L^{\\infty }(\\widehat{\\mathbb {G}})$ is the desired state.",
"Letting $m$ also denote its restriction to $\\mathrm {LUC}(\\mathbb {G}):=\\langle L^{\\infty }(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle $ , let $m_0:=\\rho _0\\circ \\Theta ^r(m)\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ , where $\\rho _0\\in \\mathcal {T}(L^2(\\mathbb {G}))$ is a fixed normal state, and $\\Theta ^r:\\mathrm {LUC}(\\mathbb {G})^*\\ni n\\mapsto (T\\mapsto (\\textnormal {id}\\otimes n)V^*(T\\otimes 1)V)\\in \\mathcal {CB}_{\\mathcal {T}_{\\rhd }}(\\mathcal {B}(L^2(\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).",
"Since $\\mathrm {LUC}(\\mathbb {G})=\\langle \\mathcal {B}(L^2(\\mathbb {G}))\\rhd \\mathcal {T}(L^2(\\mathbb {G}))\\rangle $ [35], one may view the map $\\Theta ^r$ as follows: $\\langle \\Theta ^r(n)(T),\\rho \\rangle =\\langle n,T\\rhd \\rho \\rangle , \\ \\ \\ n\\in \\mathrm {LUC}(\\mathbb {G})^*, \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Consider the state $R^*(m_0)\\square m_0\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\square $ is the left Arens product on $\\mathcal {B}(L^2(\\mathbb {G}))^*$ extending the multiplication in $\\mathcal {T}_\\rhd =(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ .",
"Fix $\\rho ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Firstly, since $m\\in \\mathrm {LUC}(\\mathbb {G})^*$ is an invariant mean $m\\square \\pi (\\rho )=\\langle \\pi (\\rho ),1\\rangle m=\\langle \\rho ,1\\rangle m$ .",
"Thus, $\\langle m_0\\square \\rho ,T\\rangle &=\\langle m_0,\\rho \\rhd T\\rangle =\\langle \\rho _0,\\Theta ^r(m)\\circ \\Theta ^r(\\pi (\\rho ))(T)\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m\\square \\pi (\\rho ))(T)\\rangle =\\langle \\rho ,1\\rangle \\langle \\rho _0,\\Theta ^r(m)(T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle m_0,T\\rangle .$ Hence, $m_0\\square \\rho =\\langle \\rho ,1\\rangle m_0$ .",
"Secondly, $\\Theta ^r(m)$ is a conditional expectation onto $L^{\\infty }(\\widehat{\\mathbb {G}})$ commuting with both the right $\\mathcal {T}_\\rhd $ - and right $\\mathcal {T}_\\lhd $ -actions [15].",
"Since $\\hat{x}\\rhd \\omega =\\langle \\omega ,\\hat{x}\\rangle 1$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ and $\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we also have $\\langle m_0\\square (T\\lhd \\rho ),\\omega \\rangle &=\\langle m_0,(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,\\Theta ^r(m)((T\\lhd \\rho )\\rhd \\omega )\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T\\lhd \\rho ),\\omega \\rangle \\\\&=\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T)\\lhd \\rho ,\\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T),\\rho \\lhd \\omega \\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T)\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle \\rho _0,\\Theta ^r(m)(T\\rhd (\\rho \\lhd \\omega ))\\rangle \\\\&=\\langle m_0,T\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle m_0\\square T,\\rho \\lhd \\omega \\rangle \\\\&=\\langle (m_0\\square T)\\lhd \\rho ,\\omega \\rangle .\\\\$ Thus, $m_0\\square (T\\lhd \\rho )=(m_0\\square T)\\lhd \\rho $ .",
"Putting things together, on the one hand we obtain $\\langle R^*(m_0)\\square m_0,\\rho \\rhd T\\rangle =\\langle R^*(m_0),(m_0\\square \\rho )\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle ,$ and on the other, $\\langle R^*(m_0)\\square m_0,T\\lhd \\rho \\rangle &=\\langle R^*(m_0),m_0\\square (T\\lhd \\rho )\\rangle =\\langle R^*(m_0),(m_0\\square T)\\lhd \\rho \\rangle \\\\&=\\langle m_0,R((m_0\\square T)\\lhd \\rho )\\rangle =\\langle m_0,R_*(\\rho )\\rhd R(m_0\\square T)\\rangle \\\\&=\\langle m_0\\square R_*(\\rho ),R(m_0\\square T)\\rangle =\\langle R_*(\\rho ),1\\rangle \\langle m_0, R(m_0\\square T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0)\\square m_0, T\\rangle .$ Therefore, $M:=R^*(m_0)\\square m_0$ is the required state.",
"Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].",
"Corollary 3.9 A locally compact quantum group $\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $2pt\\mathbf {mod}$ .",
"Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.",
"Proposition 3.10 Let $\\mathbb {G}$ and $\\mathbb {H}$ be locally compact quantum groups such that $\\mathbb {H}$ is a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes.",
"If $\\mathbb {G}$ is inner amenable then $\\mathbb {H}$ is inner amenable.",
"Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\circ \\gamma \\in L^{\\infty }(\\widehat{\\mathbb {H}})^*$ , and let $W_{\\mathbb {G}}$ and $W_{\\mathbb {H}}$ denote the left fundamental unitaries of $\\mathbb {G}$ and $\\mathbb {H}$ , respectively.",
"We also denote by $\\mathbb {W}_{\\mathbb {G}}\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_u())$ and $\\mathbb {W}_{\\mathbb {H}}\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\pi _{\\mathbb {G}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})=W_{\\mathbb {G}}$ and $(\\pi _{\\mathbb {H}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})=W_{\\mathbb {H}}$ .",
"Finally, we define $\\W _{\\mathbb {G}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0()), \\ \\ \\ \\W _{\\mathbb {H}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_0()).$ Let $\\pi _{\\mathbb {G},\\mathbb {H}}:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {H})$ be the surjection from [24].",
"Its dual morphism is a non-degenerate $*$ -homomorphism $\\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}:C_u()\\rightarrow M(C_u())$ satisfying $(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})=(\\textnormal {id}\\otimes \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}}).$ From the relation $\\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}}=\\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}$ [24], we have $(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})&=(\\textnormal {id}\\otimes \\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}})(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})\\\\&=(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}).$ Thus, for any $x̉\\in L^{\\infty }(\\widehat{\\mathbb {H}})$ and $g\\in L^{1}(\\mathbb {H})$ we have $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\gamma ((g\\otimes \\textnormal {id})W_{\\mathbb {H}}^*(1\\otimes x̉)W_{\\mathbb {H}})\\rangle \\\\&=\\langle m̉,\\gamma ((\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\W _{\\mathbb {H}}^*(1\\otimes x̉)\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})^*(1\\otimes \\gamma (x̉))(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}^*(1\\otimes \\gamma (x̉))\\W _{\\mathbb {G}})\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g)))(\\gamma (\\hat{x}))\\rangle ,$ where the last equality follows from Lemma REF .",
"By invariance of $m̉$ , we may convolve with any state $f\\in L^1(\\mathbb {G})$ to obtain $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(g))(\\gamma (\\hat{x}))\\lhd _{\\mathbb {G}}f\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)(\\gamma (\\hat{x}))\\rangle \\\\&=\\langle m̉,\\gamma (\\hat{x})\\lhd _{\\mathbb {G}}(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)\\rangle \\\\&=\\langle \\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f,1\\rangle \\langle m̉,\\gamma (\\hat{x})\\rangle \\\\&=\\langle g,1\\rangle \\langle n̉,\\hat{x}\\rangle .$"
],
[
"Examples arising from the bicrossed product construction",
"Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\rightarrow G$ and an anti-homomorphism $j:G_2\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.",
"Suppose further that $G_1\\times G_2\\ni (g,s)\\mapsto i(g)j(s)\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.",
"Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].",
"Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\alpha :G_1\\times G_2\\rightarrow G_2$ and $\\beta :G_1\\times G_2\\rightarrow G_1$ satisfying mutual co-cycle relations [65].",
"It is known that the von Neumann crossed product $G_1\\rtimes _\\alpha L^\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .",
"The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\infty }(G_1)^\\beta \\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .",
"Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.",
"In preparation we collect some useful formulae from [65], to which we refer the reader for details.",
"To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .",
"The fundamental unitary $W$ satisfies $W^*\\xi (g,s,h,t)=\\xi (\\beta _{t}(h)^{-1}g,s,h,\\alpha _{\\beta _t(h)^{-1}g}(s)t), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2).$ Letting $\\Delta $ , $\\Delta _1$ , and $\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\xi (g,s)=\\Delta (\\alpha _g(s))^{1/2}\\Delta _1(\\beta _s(g)g^{-1})^{1/2}\\Delta _2(\\alpha _g(s)s^{-1})^{1/2}\\overline{\\xi }(\\beta _s(g),s^{-1}), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2).$ Let $\\Psi :G_2\\times G_1\\rightarrow (0,\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\Psi (s,g):=\\frac{d\\beta _s(g)}{dg}$ .",
"It follows that $\\Psi (s,g)=\\Delta (\\alpha _g(s))\\Delta _1(\\beta _s(g)g^{-1})\\Delta _2(\\alpha _g(s))$ .",
"The action $\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )^{1/2}\\xi (\\beta _{s^{-1}}(g)), \\ \\ \\ \\xi \\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )f(\\beta _{s^{-1}}(g)), \\ \\ \\ f\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\beta $ preserves $\\Delta _1$ , there exists an asymptotically $\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\Vert \\cdot \\Vert _1=1}$ with $\\mathrm {supp}(f_i)\\rightarrow \\lbrace e\\rbrace $ satisfying $\\Vert v^1_sf_i-f_i\\Vert _1\\rightarrow 0$ uniformly on compacta.",
"Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.",
"Using the co-cycle properties of $\\alpha $ and $\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\otimes J̉)W^*(J̉\\otimes J̉)\\xi (g,s,h,t)\\\\&=\\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2}\\xi (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\alpha _{h^{-1}\\beta _s(g)}(s^{-1})^{-1})$ for all $\\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2)$ .",
"Let $\\eta _i:=\\sqrt{f_i}$ .",
"By hypothesis $(iii)$ we have $\\Vert v_s\\eta _i - \\eta _i\\Vert _{L^2(G_1)}^2\\le \\Vert v_s^1f_i-f_i\\Vert _{L^1(G_1)}\\rightarrow 0$ uniformly on compacta.",
"Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\times G_2\\times G_1\\rightarrow it follows that\\begin{equation} \\int f(g,s,h) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dh \\rightarrow f(g,s,e)\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\xi _j)$ in $C_c(G_2)_{\\Vert \\cdot \\Vert _2=1}$ satisfying $\\Vert \\rho (s)\\xi _j-\\lambda (s)^*\\xi _j\\Vert _{L^2(G_2)}\\rightarrow 0$ uniformly on compacta.",
"Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\xi (g)=\\xi (g^{-1})\\Delta _1(g^{-1})$ .",
"Then for any $\\eta \\in C_c(G_1\\times G_2)$ , and any $j$ , we have $&\\langle W\\sigma V\\sigma (\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),\\eta \\otimes U_1\\eta _i\\otimes \\xi _j\\rangle =\\langle (J̉\\otimes J̉)W^*(J̉\\otimes J̉)(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),W^*(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j)\\rangle \\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ U_1\\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{U_1\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ \\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\ \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\ \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ \\overline{\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiint \\bigg (\\int \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ dt\\bigg )\\\\&\\times \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dg \\ ds \\ dh\\\\&\\rightarrow \\iiint \\xi _j(t(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1})\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2} \\ dg \\ ds \\ dt\\\\$ by ().",
"But $\\alpha _{\\beta _s(g)}(s^{-1})^{-1}=\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2}\\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\Delta _2(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\\\&=\\Delta _2(\\alpha _g(s))^{1/2}, \\ \\ \\ a.e.$ The final integral in the above calculation therefore reduces to $\\iiint \\Delta _2(\\alpha _g(s))^{1/2} \\ \\xi _j(t\\alpha _{g}(s))\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ dg \\ ds \\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\iint |\\eta (g,s)|^2 \\ dg \\ ds = \\Vert \\eta \\Vert _{L^2(G_1\\times G_2)}^2.$ Denoting the index sets of $(\\eta _i)$ and $(\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\mathcal {I}:=J\\times I^{J}$ and for $I=(j,(i_j)_{j\\in J})\\in \\mathcal {I}$ we let $\\xi _I:=U_1\\eta _{i_j}\\otimes \\xi _j\\in L^2(G_1\\times G_2)$ .",
"By [39] and the above analysis, the resulting net $(\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .",
"Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.",
"Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.",
"Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.",
"For example, $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\mathbb {R}^2$ and $\\mathbb {F}_6$ are.",
"Example 3.14 We consider a discretized version of [26].",
"Let $G=\\bigg \\lbrace \\begin{pmatrix} a & b & x\\\\ c & d & y\\\\ 0 & 0 & 1\\end{pmatrix}\\mid \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\in SL(2,\\mathbb {Q}), \\ x,y\\in \\mathbb {Q}\\bigg \\rbrace ,$ $G_1=(\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\mathbb {Q})$ , viewed as discrete groups.",
"The embeddings $G_1\\ni (x,y)\\mapsto \\begin{pmatrix} 1 & 0 & -x\\\\ -x & 1 & -y+\\frac{1}{2}x^2\\\\ 0 & 0 & 1\\end{pmatrix}\\in G, \\ \\ G_2\\ni \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\mapsto \\begin{pmatrix}d & -b & 0\\\\ -c & a & 0\\\\ 0 & 0 & 1\\end{pmatrix}\\in G$ determine a matched pair structure for $(G_1,G_2)$ .",
"The corresponding actions are given by $\\alpha _{(x,y)}\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}&=\\begin{pmatrix}a+bx & b\\\\ c+dx-(a+bx)(ax+b(y+\\frac{1}{2}x^2)) & d-b(ax+b(y+\\frac{1}{2}x^2))\\end{pmatrix} \\\\\\beta _{\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}}(x,y)&=\\bigg (ax+by+\\frac{b}{2}x^2,cx+d\\bigg (y+\\frac{1}{2}x^2\\bigg )-\\frac{1}{2}\\bigg (ax+b\\bigg (y+\\frac{1}{2}x^2\\bigg )\\bigg )^2\\bigg ).$ By Corollary REF , the bicrossed product $\\mathbb {G}:=VN(\\mathbb {Q}^2)^\\beta \\bowtie _\\alpha L^\\infty (SL(2,\\mathbb {Q}))$ is strongly inner amenable.",
"Since $SL(2,\\mathbb {Q})$ is not amenable, it follows from [26] that $\\mathbb {G}$ is not amenable.",
"Moreover, since $\\mathbb {Q}^2$ is not compact, $\\mathbb {G}$ is not discrete by [65].",
"Thus, $\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group."
],
[
"Relative Injectivity and the Averaging Technique",
"In [56] Ruan and Xu showed that the dual $L^{\\infty }(\\widehat{\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\mathbb {G}$ is relatively 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}$ .",
"We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.",
"Below we let $\\widetilde{\\mathbb {G}}:=\\mathrm {Gr}()$ denote the intrinsic group of $$ .",
"Proposition 4.1 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Consider the following conditions: $$ is inner amenable; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ ; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ ; $\\widetilde{}=\\mathrm {Gr}(\\mathbb {G})$ is inner amenable.",
"Then $(i)\\Rightarrow (ii)\\Leftrightarrow (iii)\\Rightarrow (iv)$ .",
"When $\\mathbb {G}$ is co-commutative, the conditions are equivalent.",
"$(i)\\Rightarrow (ii)$ : Given a state $n\\in L^{\\infty }(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $m:=n\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant.",
"It suffices to provide a completely contractive morphism which is a left inverse to the map $\\Delta :L^{\\infty }(\\mathbb {G})\\rightarrow \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ given by $\\Delta (x)(f)=x\\rhd f, \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Identifying $\\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))\\cong L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ via $\\langle \\Phi ,f\\otimes g\\rangle =\\langle \\Phi (f),g\\rangle , \\ \\ \\ \\Phi \\in \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G})), \\ f,g\\in L^1(\\mathbb {G}),$ we have $\\Delta =\\Gamma $ , and that the corresponding $L^1(\\mathbb {G})$ -module structure on $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ is defined by $X\\unrhd f=(f\\otimes \\textnormal {id}\\otimes \\textnormal {id})(\\Gamma ^r\\otimes \\textnormal {id})(X)$ for $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $f\\in L^1(\\mathbb {G})$ .",
"First, consider the map $\\Phi :\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni A\\mapsto (\\textnormal {id}\\otimes m)(V^*AV)\\in \\mathcal {B}(L^2(\\mathbb {G})).$ Clearly, $\\Phi $ is a completely contractive left inverse to $\\Gamma $ .",
"We show that $\\Phi $ is a right $\\mathcal {T}_\\rhd $ -module map.",
"This will complete the proof since [14] will entail the invariance $\\Phi (L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ , and the restricted module action $\\mathcal {T}_\\rhd \\curvearrowright L^{\\infty }(\\mathbb {G})$ is the pertinent $L^1(\\mathbb {G})$ -module action.",
"To this end, fix $A\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $\\rho \\in \\mathcal {T}(L^2(\\mathbb {G}))$ .",
"Then $\\Phi (A\\unrhd \\rho )&=\\Phi ((\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\prime }=\\sigma V^*\\sigma $ , where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ , for any $\\tau ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we have $&\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*(\\sigma \\otimes 1)V_{23}^*A_{23}V_{23}(\\sigma \\otimes 1)V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\omega \\otimes \\tau \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m)(V^*(1\\otimes (\\tau \\otimes \\textnormal {id})(V^*AV))V),\\omega \\rangle \\\\&=\\langle ( m\\otimes \\textnormal {id})(V̉^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV)\\otimes 1)V̉^{\\prime *}),\\omega \\rangle \\\\&=\\langle m,\\omega \\widehat{\\rhd }^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV))\\rangle \\\\&=\\langle m,(\\tau \\otimes \\textnormal {id})(V^*AV)\\rangle \\langle \\omega ,1\\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m\\otimes \\textnormal {id})(V^*AV\\otimes 1),\\tau \\otimes \\omega \\rangle \\\\&=\\langle \\Phi (A)\\otimes 1,\\tau \\otimes \\omega \\rangle .$ As $\\tau $ and $\\omega $ were arbitrary, we have $\\Phi (A\\unrhd \\rho )&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\Phi (A)\\otimes 1)V^*)\\\\&=\\Phi (A)\\rhd \\rho .$ $(ii)\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\mathbb {G})$ -module left inverse $\\Phi $ to $\\Gamma $ , it follows that $R\\circ \\Phi \\circ \\Sigma \\circ (R\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ .",
"$(iii)\\Rightarrow (iv)$ : Recall that $\\mathrm {Gr}(\\mathbb {G})$ is a group of unitaries in $L^{\\infty }(\\mathbb {G})$ , so it acts naturally on $L^{\\infty }(\\mathbb {G})$ by conjugation.",
"The existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ which is $\\mathrm {Gr}(\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\Gamma (L^{\\infty }(\\mathbb {G}))-L^{\\infty }(\\mathbb {G})$ -bimodule property of $\\Phi $ .",
"Since $\\widetilde{\\widehat{\\mathbb {G}}}$ is a closed quantum subgroup of $\\widehat{\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\gamma :L^{\\infty }(\\widehat{\\widetilde{}})\\rightarrow L^{\\infty }(\\mathbb {G})$ intertwining the co-multiplications.",
"As $L^{\\infty }(\\widehat{\\widetilde{}})=VN(\\widetilde{\\widehat{\\mathbb {G}}})=VN(\\mathrm {Gr}(\\mathbb {G}))$ , the state $m\\circ \\gamma \\in VN(\\mathrm {Gr}(\\mathbb {G}))^*$ is $\\mathrm {Gr}(\\mathbb {G})$ -invariant, making $\\mathrm {Gr}(\\mathbb {G})$ inner amenable by [16].",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, then $\\mathrm {Gr}(\\mathbb {G}_s)=G$ and the implication $(iv)\\Rightarrow (i)$ follows immediately from [16].",
"Corollary 4.2 Let $\\mathbb {G}$ be a locally compact quantum group for which $L^{\\infty }(\\widehat{\\mathbb {G}})$ is an injective von Neumann algebra.",
"Then the following are equivalent: $\\mathbb {G}$ is amenable; $\\mathbb {G}$ is inner amenable; $L^{\\infty }(\\widehat{\\mathbb {G}})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"Propositions REF and REF yield the implications $(i)\\Rightarrow (ii)\\Rightarrow (iii)$ .",
"Assume $(iii)$ .",
"Since $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $\\mathbf {mod}\\hspace{2.0pt}it follows from \\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.",
"Hence, $ G$ is amenable by \\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\mathbb {G}$ and amenability of $$ .",
"One of their main results is the following.",
"Theorem 4.3 (Ng–Viselter) Let $\\mathbb {G}$ be a locally compact quantum group.",
"Consider the following conditions: $\\mathbb {G}$ is co-amenable; $C_0(\\mathbb {G})$ is nuclear and there exists a state $\\rho \\in C_0(\\mathbb {G})^*$ such that $\\langle \\rho ,x\\widehat{\\lhd } \\hat{f}\\rangle =\\langle \\rho ,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ $$ is amenable.",
"Then $(a)\\Rightarrow (b)\\Rightarrow (c)$ .",
"Moreover, if $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra), then $(a)\\Rightarrow (b^{\\prime })\\Rightarrow (b)$ , where $C_0(\\mathbb {G})$ is nuclear and has a tracial state.",
"They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .",
"We now show that $(b)$ is indeed equivalent to $(a)$ .",
"Theorem 4.4 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.",
"Suppose that $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.",
"Given a state $m\\in C_0(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $n:=m\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant, as in the proof of Proposition REF .",
"Let $n\\in M(C_0(\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\textnormal {id}\\otimes n):M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ denote the unique extension of the slice map $(\\textnormal {id}\\otimes n):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))$ which is strictly continuous on the unit ball.",
"By strict density of $C_0(\\mathbb {G})$ in $M(C_0(\\mathbb {G}))$ , it follows that $\\langle n,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle n,x\\rangle \\langle \\hat{f}^{\\prime },1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in M(C_0(\\mathbb {G})).$ Since $V\\in M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))$ , the map $\\Phi :C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\Phi (A)=(\\textnormal {id}\\otimes n)(V^*AV), \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.",
"Using the extended $\\widehat{\\rhd }^{\\prime }$ -invariance on $M(C_0(\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\Phi (A\\unrhd \\rho )=\\Phi (A)\\rhd \\rho , \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Since $\\mathrm {Ad}(V^*):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})$ and $\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})=(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))^*,$ we have $\\mathrm {Ad}(V^*)^*:\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})\\rightarrow \\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G}).$ Letting $r:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto \\rho |_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\Phi ^*|_{\\mathcal {T}(L^2(\\mathbb {G}))}:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto (r\\otimes \\textnormal {id})(\\mathrm {Ad}(V^*)^*(\\rho \\otimes n))\\in M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ The proof of [14] entails the inclusion $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ .",
"In fact, since $\\Phi $ is a right $L^1(\\mathbb {G})$ -module map and $C_0(\\mathbb {G})$ is essential, more is true: $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq \\mathrm {LUC}(\\mathbb {G})\\subseteq M(C_0(\\mathbb {G})).$ The unique strict extension $\\widetilde{\\Phi }:M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow M(C_0(\\mathbb {G}))$ , which exists by [45], satisfies $\\widetilde{\\Phi }\\circ \\Gamma |_{C_0(\\mathbb {G})}=\\textnormal {id}_{C_0(\\mathbb {G})}$ .",
"If $\\rho \\in L^{\\infty }(\\mathbb {G})_{\\perp }$ then the invariance $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ implies $\\Phi ^*(\\rho )=0$ .",
"Thus, $\\Phi ^*$ induces a completely positive left $L^1(\\mathbb {G})$ -module map $\\Phi ^*:L^1(\\mathbb {G})=(\\mathcal {T}(L^2(\\mathbb {G}))/L^{\\infty }(\\mathbb {G})_{\\perp })\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\mathbb {G})$ , for $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle m_{M(\\mathbb {G})}(\\Phi ^*(f)),x\\rangle =\\langle \\Phi ^*(f),\\Gamma (x)\\rangle =\\langle f,\\widetilde{\\Phi }(\\Gamma (x))\\rangle =\\langle f,x\\rangle .$ Hence, $m_{M(\\mathbb {G})}\\circ \\Phi ^*$ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .",
"Now, by nuclearity of $C_0(\\mathbb {G})$ , there exists a net $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.",
"Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*.$ Then $\\psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}),M(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}))$ .",
"By [20] there exist contractive positive functionals $\\nu _i\\in C_u(\\mathbb {G})^*$ such that $\\psi _i=m^r_{\\nu _i}$ .",
"Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.",
"Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .",
"Since $(b)\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.",
"Hence, by [14] $M^r_{cb}(L^1(\\mathbb {G}))= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))=C_u(\\mathbb {G})^*$ .",
"For each $i$ and $k$ the map $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*\\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ , so there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*=m^r_{\\nu _k^i}$ .",
"Thus, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*(f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi ^*(f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i.$ Hence, $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Passing to a subnet we may assume $\\nu _i\\rightarrow \\nu $ weak* in $M(\\mathbb {G})$ .",
"But then for any $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle f\\star \\nu , x\\rangle =\\langle \\nu , x\\star f\\rangle =\\lim _i\\langle \\nu _i,x\\star f\\rangle =\\lim _i\\langle f\\star \\nu _i,x\\rangle =\\langle f,x\\rangle .$ It follows that $\\nu $ is a right identity for $M(\\mathbb {G})$ , which implies that $M(\\mathbb {G})$ is unital, whence $\\mathbb {G}$ is co-amenable (cf.",
"[5]).",
"Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\lambda (G)$ is nuclear and has a tracial state.",
"Corollary 4.5 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra).",
"Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.",
"The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in L^{\\infty }(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ while $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in C_0(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\mathbb {G})$ is 1-flat [14], while $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective, as we now prove.",
"Theorem 4.6 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective in $M(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ .",
"If $\\mathbb {G}$ is co-amenable then $M(\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.",
"Any unital completely contractive Banach algebra $\\mathcal {A}$ is $\\Vert e_{\\mathcal {A}}\\Vert $ -projective, so the claim follows.",
"Conversely, if $M(\\mathbb {G})$ is 1-projective, then $C_0(\\mathbb {G})^{**}=M(\\mathbb {G})^*$ is 1-injective in $\\mathbf {mod}-M(\\mathbb {G})$ .",
"It follows that the inclusion morphism $M(C_0(\\mathbb {G}))\\hookrightarrow C_0(\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\mathbb {G})$ -module map $\\Phi : \\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}$ .",
"Applying the argument of [51] we see that $(\\textnormal {id}\\otimes \\Phi ):C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}\\overline{\\otimes }C_0(\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\textnormal {id}\\otimes \\Phi )(W̉)=W̉$ .",
"By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\textnormal {id}\\otimes \\Phi )$ , and hence $(\\textnormal {id}\\otimes \\Phi )(W̉^*XW̉)=W̉^*(\\textnormal {id}\\otimes \\Phi )(X)W̉, \\ \\ \\ X\\in C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})).$ In particular, for every $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ and $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ we have $\\Phi (T\\hat{f})&=\\Phi ((\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes T)W̉))=(\\hat{f}\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\Phi )((W̉^*(1\\otimes T)W̉))\\\\&=(\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes \\Phi (T))W̉)=\\Phi (T)\\hat{f}.$ Thus, $\\Phi $ is a morphism with respect to the canonical $L^1(\\widehat{\\mathbb {G}})$ -module structure on $C_0(\\mathbb {G})^{**}$ .",
"Moreover, the $M(\\mathbb {G})$ -module property entails $\\pi \\circ \\Phi (L^{\\infty }(\\widehat{\\mathbb {G}}))=$ , where $\\pi :C_0(\\mathbb {G})^{**}\\rightarrow L^{\\infty }(\\mathbb {G})$ is the canonical surjection.",
"Since $\\pi $ is clearly an $L^1(\\widehat{\\mathbb {G}})$ -module map, it follows that $\\pi \\circ \\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\mathbb {G})$ .",
"Now, by 1-projectivity of $M(\\mathbb {G})$ , for every $\\varepsilon >0$ there exists a morphism $\\Phi _\\varepsilon :M(\\mathbb {G})\\rightarrow M(\\mathbb {G})^+\\widehat{\\otimes }M(\\mathbb {G})$ such that $m^+\\circ \\Phi _\\varepsilon =\\textnormal {id}_{M(\\mathbb {G})}$ , and $\\Vert \\Phi _\\varepsilon \\Vert _{cb}<1+\\varepsilon $ .",
"By amenability of $$ , we know $C_u(\\mathbb {G})^*= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\Phi _\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\mu _i)$ in $M(\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\mathbb {G}$ .",
"The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.",
"In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to the co-multiplication that is an $L^1(\\mathbb {G})$ -module map.",
"Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow \\Gamma (L^{\\infty }(\\mathbb {G}))$ onto the image of $\\Gamma $ (see [23] and [41]).",
"It is the combination of the $L^1(\\mathbb {G})$ -module property of $\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\infty }(\\mathbb {G})$ or $C_0(\\mathbb {G})$ to approximation properties of $$ .",
"Thus, provided one has a suitably nice $L^1(\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.",
"This is where inner amenability enters the picture.",
"Recall that a locally compact quantum group $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .",
"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .",
"We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.",
"Proposition 4.7 Let $\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.",
"If $L^{\\infty }(\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .",
"$L^{\\infty }(\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.",
"Let $(\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\mathbb {G})$ .",
"It follows verbatim from [56] that $\\Phi _i:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni X\\mapsto (\\omega _{\\xi _i}\\otimes \\textnormal {id})(W^*(U^*\\otimes 1)X(U\\otimes 1)W)\\in L^{\\infty }(\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\mathbb {G})$ -module maps, which cluster weak* in $\\mathcal {CB}(L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=((L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\widehat{\\otimes }L^1(\\mathbb {G}))^*$ to a module left inverse to $\\Gamma $ .",
"Passing to a subnet we may assume convergence.",
"$(i)$ Let $(\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\Vert \\varphi _j\\Vert _{cb}\\le C$ .",
"Since $\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\in L^1(\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\in L^{\\infty }(\\mathbb {G})$ such that $\\varphi _j=\\sum _{k=1}^{n_j} x^j_k\\otimes f^j_k$ .",
"Put $\\Phi _{ij}=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G}).$ Then $\\Phi _{ij}$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map with $\\Vert \\Phi _{ij}\\Vert _{cb}\\le C$ .",
"Also, for each $i,j$ and $1\\le k\\le n_j$ , the map $L^{\\infty }(\\mathbb {G})\\ni x\\mapsto \\Phi _i(x^j_k\\otimes x)\\in L^{\\infty }(\\mathbb {G})$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map.",
"Since $x^j_k$ is a linear combination of positive elements in $L^{\\infty }(\\mathbb {G})$ , and $\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\mathbb {G})$ -module maps $L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ .",
"By [20], there exist $\\mu ^{ij}_k\\in C_u(\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\mu ^{ij}_k$ .",
"Hence, for each $x\\in L^{\\infty }(\\mathbb {G})$ $\\Phi _{ij}(x)&=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma (x)\\\\&=\\sum _{k=1}^{n_j}\\Phi _i(x^j_k \\otimes x\\star f^j_k)\\\\&=\\sum _{k=1}^{n_j}x\\star f^j_k\\star \\mu ^{ij}_k\\\\&=x\\star f_{ij},$ where $f_{ij}=\\sum _{k=1}^{n_i} f^j_k\\star \\mu ^{ij}_k\\in L^1(\\mathbb {G})$ as $L^1(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Thus, $\\Vert f_{ij}\\Vert _{cb}\\le C$ , and it follows that $\\lim _i\\lim _jx\\star f_{ij}=\\lim _i\\Phi _i(\\Gamma (x))=x,$ point-weak* in $L^{\\infty }(\\mathbb {G})$ .",
"Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .",
"$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.",
"[34]).",
"The converse was shown for Kac algebras in [41].",
"Their proof extends verbatim to arbitrary locally compact quantum groups.",
"Corollary 4.8 Let $G$ be an inner amenable locally compact group.",
"If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.",
"$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.",
"Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ that can be approximated by normal $L^1(\\mathbb {G})$ -module maps.",
"It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .",
"Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.",
"That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?",
"By [34] it is known that in this case $\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\mathcal {CB}^\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\mathcal {CB}^\\sigma (VN(G))$ .",
"Proposition 4.11 Let $\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.",
"If $C_0(\\mathbb {G})$ has the CBAP, then there exists a net $(\\nu _i)$ in $M(\\mathbb {G})$ such that $\\Vert \\nu _i\\Vert _{cb}\\le \\Lambda _{cb}(C_0(\\mathbb {G}))$ and $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .",
"If $C_0(\\mathbb {G})$ has the OAP then $M(\\mathbb {G})$ is $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"Let $\\Phi :L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G})$ be the completely positive left $L^1(\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.",
"Recall that $m_{M(\\mathbb {G})}\\circ \\Phi $ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .",
"$(i)$ Let $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\Vert \\varphi _i\\Vert _{cb}\\le C$ .",
"Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .",
"For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\mathbb {G})$ , and hence the map $(\\textnormal {id}\\otimes x_k^i)\\Phi \\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\mathbb {G})$ -module maps on $L^1(\\mathbb {G})$ , so by [20] there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi =m^r_{\\nu _k^i}$ .",
"Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi $ .",
"Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.",
"Moreover, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi (f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi (f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i\\\\&=f\\star \\nu _i,$ where $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Then $\\Vert \\nu _i\\Vert _{cb}\\le C$ , and it follows that $\\nu _i\\star x\\rightarrow x$ weak* in $L^{\\infty }(\\mathbb {G})$ for all $x\\in C_0(\\mathbb {G})$ .",
"Since $(\\nu _i)$ is bounded in $\\mathcal {CB}_{L^1(\\mathbb {G})}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ , we have $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .",
"$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\mathbb {G}))$ .",
"Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .",
"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).",
"We now obtain the same conclusion under a priori weaker hypotheses.",
"Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.",
"It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .",
"Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .",
"However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.",
"Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .",
"Hence, $m$ satisfies (REF ).",
"A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.",
"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .",
"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.",
"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.",
"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.",
"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.",
"Proposition 5.2 Let $G$ be a locally compact group.",
"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.",
"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.",
"Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .",
"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].",
"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.",
"Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].",
"We now show that 1-biflatness of quantum convolution algebras is a self dual property.",
"Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.",
"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .",
"Also, [15] implies that $E$ is a right $\\lhd $ -module map.",
"Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .",
"Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .",
"Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .",
"Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.",
"Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).",
"Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .",
"By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .",
"It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.",
"Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].",
"Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.",
"Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .",
"Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.",
"The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].",
"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.",
"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].",
"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].",
"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .",
"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .",
"Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.",
"$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].",
"$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.",
"The bimodule analogue of [14] then yields $(iii)$ .",
"$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .",
"As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.",
"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.",
"Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .",
"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.",
"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.",
"In [16] the authors gave examples of weakly amenable connected groups (e.g.",
"$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.",
"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].",
"Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .",
"As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .",
"Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.",
"There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .",
"Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .",
"Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).",
"Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.",
"Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .",
"Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.",
"Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.",
"Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.",
"Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .",
"It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].",
"Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .",
"By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .",
"Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.",
"Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .",
"By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .",
"Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .",
"It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .",
"Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .",
"By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .",
"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .",
"The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .",
"By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.",
"Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .",
"Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .",
"Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .",
"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .",
"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.",
"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.",
"There are examples of weakly amenable residually finite groups (e.g.",
"$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].",
"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.",
"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].",
"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .",
"Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .",
"If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .",
"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.",
"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].",
"We now generalize this implication to arbitrary locally compact quantum groups.",
"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.",
"By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].",
"Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].",
"We present the proof for the convenience of the reader.",
"Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .",
"Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.",
"Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .",
"Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.",
"Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].",
"For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .",
"In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.",
"We show that it is a complete order bijection.",
"To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .",
"On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .",
"By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"We now show that $\\tilde{\\lambda }$ is a complete isometry.",
"To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].",
"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.",
"Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].",
"By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .",
"Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .",
"Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].",
"The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.",
"To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.",
"Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .",
"Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.",
"Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.",
"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].",
"Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.",
"Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.",
"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .",
"Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).",
"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .",
"Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.",
"Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .",
"Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.",
"It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.",
"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.",
"By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .",
"Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .",
"Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.",
"The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.",
"Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .",
"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.",
"The author was partially supported by the NSERC Discovery Grant 1304873."
],
[
"Inner Amenability",
"Given a locally compact quantum group $\\mathbb {G}$ the composition $W\\sigma V\\sigma \\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))$ defines a unitary co-representation of $\\mathbb {G}$ called the conjugation co-representation, where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_a$ for some locally compact group $G$ , it follows that $W\\sigma V\\sigma \\xi (s,t)=\\xi (s,s^{-1}ts)\\Delta (s)^{1/2}, \\ \\ \\ \\xi \\in L^2(G\\times G), \\ s,t\\in G.$ Thus, $W\\sigma V\\sigma $ is the unitary generator of the conjugation representation $\\beta _2:G\\rightarrow \\mathcal {B}(L^2(G))$ , where $\\beta _2(s)\\xi (t)=\\lambda (s)\\rho (s)\\xi (t)=\\xi (s^{-1}ts)\\Delta (s)^{1/2}.$ Following Paterson [53], we say that a locally compact group $G$ is inner amenable if there exists a state $m\\in L^{\\infty }(G)^*$ satisfying $\\langle m,\\beta _{\\infty }(s)f\\rangle =\\langle m,f\\rangle \\ \\ \\ s\\in G, \\ f\\in L^{\\infty }(G),$ where $\\beta _\\infty (s)f(t)=f(s^{-1}ts)$ , $s,t\\in G$ , $f\\in L^{\\infty }(G)$ , is the conjugation action on $L^{\\infty }(G)$ .",
"Remark 3.1 In [27], Effros defined a discrete group $G$ to be “inner amenable” if there exists a conjugation invariant mean $m\\in \\ell ^{\\infty }(G)^*$ such that $m\\ne \\delta _e$ .",
"In what follows, inner amenability will always refer to the definition (REF ) given above.",
"In particular, every discrete group is inner amenable.",
"Definition 3.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"We say that $\\mathbb {G}$ is strongly inner amenable (see [52]) if there exists a net $(\\xi _i)$ of unit vectors such that $\\Vert W\\sigma V\\sigma (\\eta \\otimes \\xi _i)-\\eta \\otimes \\xi _i\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ $\\mathbb {G}$ is inner amenable if there exists a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ satisfying $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}}).$ Such a state is said to be inner invariant.",
"$\\mathbb {G}$ is topologically inner amenable if there exists a state $m\\in C_0()^*$ such that $\\langle m,\\hat{x}\\lhd f\\rangle =\\langle f,1\\rangle \\langle m,\\hat{x}\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\hat{x}\\in C_0().$ Such a state is said to be topologically inner invariant.",
"Examples 3.3 The following known examples are worth mentioning.",
"Any co-commutative quantum group $\\mathbb {G}_s$ is trivially strongly inner amenable, as $V_s=W_a$ so that $W_s\\sigma V_s\\sigma =W_sW_s^*=1$ .",
"Any unimodular discrete quantum group is strongly inner amenable; the unit vector $\\xi :=\\Lambda _\\varphi (1)$ being conjugation invariant, where $\\varphi $ is the Haar trace on the compact dual.",
"It was shown in [51] that if $\\mathbb {G}$ has trivial scaling group, and $C_0()$ possesses a tracial state, then $\\mathbb {G}$ is topologically inner amenable.",
"Further examples, including the bicrossed product construction will be studied below.",
"The following proposition is standard.",
"We include the proof for completeness.",
"Proposition 3.4 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $\\mathbb {G}$ is strongly inner amenability then it is inner amenable.",
"Let $(\\xi _i)$ be a net of unit vectors asymptotically invariant under the conjugation co-representation $W\\sigma V\\sigma $ .",
"Passing to a subnet, we may assume that $(\\omega _{J\\xi _i}|_{L^{\\infty }(\\widehat{\\mathbb {G}})})$ converges weak* to a state $m\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ .",
"For any $f=\\omega _{\\xi ,\\eta }\\in L^1(\\mathbb {G})$ and $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ , using strong inner amenability together with the adjoint relations $(\\widehat{J}\\otimes J)W(\\widehat{J}\\otimes J)=W^*, \\ \\ \\ (\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\otimes J)=\\sigma V^*\\sigma ,$ we have $\\langle m,\\hat{x}\\lhd f\\rangle &=\\lim _i\\langle \\omega _{J\\xi _i},\\hat{x}\\lhd f\\rangle \\\\&=\\lim _i\\langle W^*(1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),\\eta \\otimes J\\xi _i\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})W(\\xi \\otimes J\\xi _i),W(\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)W^*(\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\xi \\otimes \\xi _i),(\\widehat{J}\\otimes J)\\sigma V\\sigma (\\widehat{J}\\eta \\otimes \\xi _i)\\rangle \\\\&=\\lim _i\\langle (1\\otimes \\hat{x})\\sigma V^*\\sigma (\\xi \\otimes J\\xi _i),\\sigma V^*\\sigma (\\eta \\otimes J\\xi _i)\\rangle \\\\&=\\langle m,\\hat{x}\\rangle \\langle f,1\\rangle .$ In the commutative case, the converse of Proposition REF holds.",
"Proposition 3.5 A commutative quantum group $\\mathbb {G}_a$ is strongly inner amenable if and only if it is inner amenable if and only if its underlying group $G$ is inner amenable.",
"If $\\mathbb {G}_a$ is strongly inner amenable then it is inner amenable.",
"Let $m\\in VN(G)^*$ satisfy $\\langle m,x\\rangle =\\langle m,x \\lhd f\\rangle $ for all $x\\in VN(G)$ and all states $f\\in L^1(G)$ .",
"Then for $t\\in G$ $\\lambda (t)x\\lambda (t)^* \\lhd f=\\int _G \\lambda (s^{-1}t)x\\lambda (t^{-1}s) f(s)ds=\\int _G\\lambda (r)^*x\\lambda (r)f(tr)dr=x \\lhd f_t,$ so that $\\langle m,\\lambda (t)x\\lambda (t)^*\\rangle =\\langle m,\\lambda (t)x\\lambda (t)^* \\lhd f\\rangle =\\langle m,x \\lhd f_t\\rangle =\\langle m,x\\rangle $ for all $x\\in VN(G)$ , $t\\in G$ and states $f\\in L^1(G)$ .",
"Thus, $G$ is inner amenable by [16].",
"Conversely, if $G$ is inner amenable then there exists a net of unit vectors $(\\xi _i)$ in $L^2(G)$ satisfying $\\Vert \\lambda (s)\\xi _i-\\rho (s)^*\\xi _i\\Vert _{L^2(G)}\\rightarrow 0$ uniformly on compact subsets of $G$ .",
"For any $\\eta \\in C_c(G)$ we then have $\\langle W\\sigma V\\sigma (\\eta \\otimes \\xi _i),\\eta \\otimes \\xi _i\\rangle =\\iint \\xi _i(ts)\\overline{\\xi _i}(st)\\Delta (s)^{1/2}|\\eta (s)|^2 \\ ds \\ dt\\rightarrow 1.$ It follows that $(\\xi _i)$ is strongly inner invariant.",
"Definition REF (ii) is therefore a bona fide generalization of inner amenability to quantum groups, contrary to the definition proposed in [33].",
"Curiously, if one “lifts” the relation (REF ) proposed in [33] to the level of $\\mathcal {B}(L^2(G))$ , via $\\langle m,f \\rhd T\\rangle =\\langle m, T \\lhd f\\rangle , \\ \\ \\ f\\in L^1(G), \\ T\\in \\mathcal {B}(L^2(G)),$ then a similar argument as in Proposition REF shows that one obtains a proper generalization of inner amenability.",
"For details, see [13].",
"Proposition 3.6 A commutative quantum group $\\mathbb {G}_a$ is topologically inner amenable if and only if $C^*_\\lambda (G)$ possesses a tracial state if and only if the amenable radical of $G$ is open.",
"The existence of a tracial state on $C_\\lambda ^*(G)$ was recently investigated by Forrest–Spronk–Wiersma [30], and Kennedy–Raum [40], where it was shown to be equivalent to the openness of the amenable radical of $G$ , that is, the largest amenable normal subgroup.",
"By [51] $\\mathbb {G}_a$ is topologically inner amenable if $C_\\lambda ^*(G)$ has a tracial state.",
"Conversely, if $m$ is an inner invariant state on $C_\\lambda ^*(G)$ , it follows as in the proof of Proposition REF that $m$ is $G$ -invariant under the canonical conjugation action.",
"Viewing $m$ as a positive definite function in $B_\\lambda (G)$ , it follows that $m(s)=m(tst^{-1})$ , $s,t\\in G$ .",
"A simple integral calculation then shows that $m$ is a tracial state on $C_\\lambda ^*(G)$ .",
"Remark 3.7 At first glance one might think that inner amenability implies topological inner amenability via the restriction of an inner invariant state $m$ on $L^{\\infty }(\\widehat{\\mathbb {G}})$ to $C_0()$ .",
"This is not the case however.",
"In [59] Suzuki provided elementary constructions of non-discrete $C^*$ -simple groups of the form $G=\\bigoplus _{n\\in \\mathbb {N}} \\Gamma _n\\rtimes \\prod _{n\\in \\mathbb {N}} F_n$ where $F_n$ is a finite group acting on the discrete group $\\Gamma _n$ for which $C^*_\\lambda (\\Gamma _n\\rtimes F_n)$ admits a unique tracial state.",
"In [30], it was shown that each compactly generated open subgroup $H$ of $G$ is IN (meaning $H$ has a compact conjugation invariant neighbourhood of the identity), and that one may build a resulting net of normalized characteristic functions whose vector functionals cluster to an inner invariant state on $VN(G)$ .",
"Thus, $G$ is inner amenable.",
"However, in [30], the authors also show that the continuous function $m$ in $B_\\lambda (G)$ associated to a tracial state on $C^*_\\lambda (G)$ must be discontinuous.",
"Whence, there is no tracial state on $C^*_\\lambda (G)$ and $G$ is not topologically inner amenable by Proposition REF .",
"Therefore, there are “continuity restrictions” which forbid the restriction of an inner invariant state on $VN(G)$ to a tracial state on $C^*_\\lambda (G)$ .",
"In the other direction, in general it is not possible to lift a tracial state on $C^*_\\lambda (G)$ to an inner invariant mean on $VN(G)$ .",
"In [30] it is shown that $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is topologically inner amenable but not inner amenable, where $\\mathbb {F}_6$ is viewed as a closed subgroup of $SL(2,\\mathbb {R})$ .",
"Proposition 3.8 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is co-amenable then $\\mathbb {G}$ is strongly inner amenable.",
"If $\\mathbb {G}$ is amenable then it is inner amenable.",
"If $$ is co-amenable, let $E\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ be a co-unit and approximate $E$ in the weak* topology by vector states $\\omega _{\\xi _i}$ with $J̉\\xi _i=\\xi _i$ (since $L^{\\infty }(\\widehat{\\mathbb {G}})\\curvearrowright L^2(\\mathbb {G})$ is standard).",
"Then $1=(\\textnormal {id}\\otimes E)(W)=\\lim _i(\\textnormal {id}\\otimes \\omega _{\\xi _i})(W)$ strongly, from which it follows that $\\Vert W(\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Since $\\sigma V\\sigma =(J̉\\otimes J̉)W^*(J̉\\otimes J̉)$ , and $J̉\\xi _i=\\xi _i$ , we also have $\\Vert \\sigma V\\sigma (\\eta \\otimes \\xi _i)-(\\eta \\otimes \\xi _i)\\Vert \\rightarrow 0, \\ \\ \\ \\eta \\in L^2(\\mathbb {G}).$ Whence, $\\mathbb {G}$ is strongly inner amenable.",
"Now, suppose $\\mathbb {G}$ is amenable and let $m\\in L^{\\infty }(\\mathbb {G})^*$ be a two-sided invariant mean.",
"We will show a stronger statement by providing a state $M\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ such that $\\langle M,\\rho \\rhd T\\rangle =\\langle M,T\\lhd \\rho \\rangle , \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})),$ upon which restriction to $L^{\\infty }(\\widehat{\\mathbb {G}})$ is the desired state.",
"Letting $m$ also denote its restriction to $\\mathrm {LUC}(\\mathbb {G}):=\\langle L^{\\infty }(\\mathbb {G})\\star L^1(\\mathbb {G})\\rangle $ , let $m_0:=\\rho _0\\circ \\Theta ^r(m)\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ , where $\\rho _0\\in \\mathcal {T}(L^2(\\mathbb {G}))$ is a fixed normal state, and $\\Theta ^r:\\mathrm {LUC}(\\mathbb {G})^*\\ni n\\mapsto (T\\mapsto (\\textnormal {id}\\otimes n)V^*(T\\otimes 1)V)\\in \\mathcal {CB}_{\\mathcal {T}_{\\rhd }}(\\mathcal {B}(L^2(\\mathbb {G})))$ is the canonical completely contractive homomorphism (see [15] or [35]).",
"Since $\\mathrm {LUC}(\\mathbb {G})=\\langle \\mathcal {B}(L^2(\\mathbb {G}))\\rhd \\mathcal {T}(L^2(\\mathbb {G}))\\rangle $ [35], one may view the map $\\Theta ^r$ as follows: $\\langle \\Theta ^r(n)(T),\\rho \\rangle =\\langle n,T\\rhd \\rho \\rangle , \\ \\ \\ n\\in \\mathrm {LUC}(\\mathbb {G})^*, \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Consider the state $R^*(m_0)\\square m_0\\in \\mathcal {B}(L^2(\\mathbb {G}))^*$ where $R$ is the extended unitary antipode and $\\square $ is the left Arens product on $\\mathcal {B}(L^2(\\mathbb {G}))^*$ extending the multiplication in $\\mathcal {T}_\\rhd =(\\mathcal {T}(L^2(\\mathbb {G})),\\rhd )$ .",
"Fix $\\rho ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Firstly, since $m\\in \\mathrm {LUC}(\\mathbb {G})^*$ is an invariant mean $m\\square \\pi (\\rho )=\\langle \\pi (\\rho ),1\\rangle m=\\langle \\rho ,1\\rangle m$ .",
"Thus, $\\langle m_0\\square \\rho ,T\\rangle &=\\langle m_0,\\rho \\rhd T\\rangle =\\langle \\rho _0,\\Theta ^r(m)\\circ \\Theta ^r(\\pi (\\rho ))(T)\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m\\square \\pi (\\rho ))(T)\\rangle =\\langle \\rho ,1\\rangle \\langle \\rho _0,\\Theta ^r(m)(T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle m_0,T\\rangle .$ Hence, $m_0\\square \\rho =\\langle \\rho ,1\\rangle m_0$ .",
"Secondly, $\\Theta ^r(m)$ is a conditional expectation onto $L^{\\infty }(\\widehat{\\mathbb {G}})$ commuting with both the right $\\mathcal {T}_\\rhd $ - and right $\\mathcal {T}_\\lhd $ -actions [15].",
"Since $\\hat{x}\\rhd \\omega =\\langle \\omega ,\\hat{x}\\rangle 1$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ and $\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we also have $\\langle m_0\\square (T\\lhd \\rho ),\\omega \\rangle &=\\langle m_0,(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,\\Theta ^r(m)((T\\lhd \\rho )\\rhd \\omega )\\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T\\lhd \\rho )\\rhd \\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T\\lhd \\rho ),\\omega \\rangle \\\\&=\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T)\\lhd \\rho ,\\omega \\rangle =\\langle \\rho _0,1\\rangle \\langle \\Theta ^r(m)(T),\\rho \\lhd \\omega \\rangle \\\\&=\\langle \\rho _0,\\Theta ^r(m)(T)\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle \\rho _0,\\Theta ^r(m)(T\\rhd (\\rho \\lhd \\omega ))\\rangle \\\\&=\\langle m_0,T\\rhd (\\rho \\lhd \\omega )\\rangle =\\langle m_0\\square T,\\rho \\lhd \\omega \\rangle \\\\&=\\langle (m_0\\square T)\\lhd \\rho ,\\omega \\rangle .\\\\$ Thus, $m_0\\square (T\\lhd \\rho )=(m_0\\square T)\\lhd \\rho $ .",
"Putting things together, on the one hand we obtain $\\langle R^*(m_0)\\square m_0,\\rho \\rhd T\\rangle =\\langle R^*(m_0),(m_0\\square \\rho )\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle ,$ and on the other, $\\langle R^*(m_0)\\square m_0,T\\lhd \\rho \\rangle &=\\langle R^*(m_0),m_0\\square (T\\lhd \\rho )\\rangle =\\langle R^*(m_0),(m_0\\square T)\\lhd \\rho \\rangle \\\\&=\\langle m_0,R((m_0\\square T)\\lhd \\rho )\\rangle =\\langle m_0,R_*(\\rho )\\rhd R(m_0\\square T)\\rangle \\\\&=\\langle m_0\\square R_*(\\rho ),R(m_0\\square T)\\rangle =\\langle R_*(\\rho ),1\\rangle \\langle m_0, R(m_0\\square T)\\rangle \\\\&=\\langle \\rho ,1\\rangle \\langle R^*(m_0),m_0\\square T\\rangle =\\langle \\rho ,1\\rangle \\langle R^*(m_0)\\square m_0, T\\rangle .$ Therefore, $M:=R^*(m_0)\\square m_0$ is the required state.",
"Combining [15] with (the proof of) Proposition REF , we obtain a quantum group analogue of a well-known result of Lau–Paterson [46].",
"Corollary 3.9 A locally compact quantum group $\\mathbb {G}$ is amenable if and only if it is inner amenable and $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $2pt\\mathbf {mod}$ .",
"Generalizing the hereditary property in the group setting, we show that inner amenability passes to closed quantum subgroups.",
"Proposition 3.10 Let $\\mathbb {G}$ and $\\mathbb {H}$ be locally compact quantum groups such that $\\mathbb {H}$ is a closed quantum subgroup of $\\mathbb {G}$ in the sense of Vaes.",
"If $\\mathbb {G}$ is inner amenable then $\\mathbb {H}$ is inner amenable.",
"Let $m̉$ be an inner invariant state in the sense of Definition REF (ii), let $n̉:=m̉\\circ \\gamma \\in L^{\\infty }(\\widehat{\\mathbb {H}})^*$ , and let $W_{\\mathbb {G}}$ and $W_{\\mathbb {H}}$ denote the left fundamental unitaries of $\\mathbb {G}$ and $\\mathbb {H}$ , respectively.",
"We also denote by $\\mathbb {W}_{\\mathbb {G}}\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_u())$ and $\\mathbb {W}_{\\mathbb {H}}\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_u())$ the universal multiplicative unitaries satisfying $(\\pi _{\\mathbb {G}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})=W_{\\mathbb {G}}$ and $(\\pi _{\\mathbb {H}}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})=W_{\\mathbb {H}}$ .",
"Finally, we define $\\W _{\\mathbb {G}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0()), \\ \\ \\ \\W _{\\mathbb {H}}:=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {H}})\\in M(C_u(\\mathbb {H})\\otimes _{\\min } C_0()).$ Let $\\pi _{\\mathbb {G},\\mathbb {H}}:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {H})$ be the surjection from [24].",
"Its dual morphism is a non-degenerate $*$ -homomorphism $\\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}:C_u()\\rightarrow M(C_u())$ satisfying $(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})=(\\textnormal {id}\\otimes \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}}).$ From the relation $\\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}}=\\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}}$ [24], we have $(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})&=(\\textnormal {id}\\otimes \\gamma \\circ \\pi _{\\widehat{\\mathbb {H}}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}}\\circ \\hat{\\pi }_{\\mathbb {G},\\mathbb {H}})(\\mathbb {W}_{\\mathbb {H}})\\\\&=(\\textnormal {id}\\otimes \\pi _{\\widehat{\\mathbb {G}}})(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\mathbb {W}_{\\mathbb {G}})\\\\&=(\\pi _{\\mathbb {G},\\mathbb {H}}\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}).$ Thus, for any $x̉\\in L^{\\infty }(\\widehat{\\mathbb {H}})$ and $g\\in L^{1}(\\mathbb {H})$ we have $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\gamma ((g\\otimes \\textnormal {id})W_{\\mathbb {H}}^*(1\\otimes x̉)W_{\\mathbb {H}})\\rangle \\\\&=\\langle m̉,\\gamma ((\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\W _{\\mathbb {H}}^*(1\\otimes x̉)\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {H}}^*(g)\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}})^*(1\\otimes \\gamma (x̉))(\\textnormal {id}\\otimes \\gamma )(\\W _{\\mathbb {H}}))\\rangle \\\\&=\\langle m̉,(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\otimes \\textnormal {id})(\\W _{\\mathbb {G}}^*(1\\otimes \\gamma (x̉))\\W _{\\mathbb {G}})\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g)))(\\gamma (\\hat{x}))\\rangle ,$ where the last equality follows from Lemma REF .",
"By invariance of $m̉$ , we may convolve with any state $f\\in L^1(\\mathbb {G})$ to obtain $\\langle n̉,x̉\\lhd _{\\mathbb {H}}g\\rangle &=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(g))(\\gamma (\\hat{x}))\\lhd _{\\mathbb {G}}f\\rangle \\\\&=\\langle m̉,\\Theta ^l(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)(\\gamma (\\hat{x}))\\rangle \\\\&=\\langle m̉,\\gamma (\\hat{x})\\lhd _{\\mathbb {G}}(\\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f)\\rangle \\\\&=\\langle \\pi _{\\mathbb {G},\\mathbb {H}}^*(\\pi _{\\mathbb {H}}^*(g))\\star _{\\mathbb {G}}f,1\\rangle \\langle m̉,\\gamma (\\hat{x})\\rangle \\\\&=\\langle g,1\\rangle \\langle n̉,\\hat{x}\\rangle .$"
],
[
"Examples arising from the bicrossed product construction",
"Let $G, G_1$ and $G_2$ be locally compact groups with fixed left Haar measures for which there exists a homomorphism $i:G_1\\rightarrow G$ and an anti-homomorphism $j:G_2\\rightarrow G$ which have closed ranges and are homeomorphisms onto these ranges.",
"Suppose further that $G_1\\times G_2\\ni (g,s)\\mapsto i(g)j(s)\\in G$ is a homeomorphism onto an open subset of $G$ having complement of measure zero.",
"Then $(G_1,G_2)$ is said to be a matched pair of locally compact groups [65].",
"Any matched pair $(G_1,G_2)$ determines a matched pair of actions $\\alpha :G_1\\times G_2\\rightarrow G_2$ and $\\beta :G_1\\times G_2\\rightarrow G_1$ satisfying mutual co-cycle relations [65].",
"It is known that the von Neumann crossed product $G_1\\rtimes _\\alpha L^\\infty (G_2)$ admits a quantum group structure, called the bicrossed product of the matched pair $(G_1,G_2)$ .",
"The von Neumann algebra of the dual quantum group is given by the crossed product $L^{\\infty }(G_1)^\\beta \\ltimes G_2$ , and therefore, following [49], we denote the bicrossed product quantum group by $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .",
"Below we present sufficient conditions on the matched pair $(G_1,G_2)$ under which the bicrossed product is (strongly) inner amenable.",
"In preparation we collect some useful formulae from [65], to which we refer the reader for details.",
"To ease the presentation we suppress the notations $i$ and $j$ for the embeddings into $G$ .",
"The fundamental unitary $W$ satisfies $W^*\\xi (g,s,h,t)=\\xi (\\beta _{t}(h)^{-1}g,s,h,\\alpha _{\\beta _t(h)^{-1}g}(s)t), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2).$ Letting $\\Delta $ , $\\Delta _1$ , and $\\Delta _2$ denote the modular functions for the groups $G$ , $G_1$ , and $G_2$ , respectively, the modular conjugation $J̉$ of the dual Haar weight satisfies $J̉\\xi (g,s)=\\Delta (\\alpha _g(s))^{1/2}\\Delta _1(\\beta _s(g)g^{-1})^{1/2}\\Delta _2(\\alpha _g(s)s^{-1})^{1/2}\\overline{\\xi }(\\beta _s(g),s^{-1}), \\ \\ \\ \\xi \\in L^2(G_1\\times G_2).$ Let $\\Psi :G_2\\times G_1\\rightarrow (0,\\infty )$ be the (continuous) function determined by the Radon-Nikodym derivatives $\\Psi (s,g):=\\frac{d\\beta _s(g)}{dg}$ .",
"It follows that $\\Psi (s,g)=\\Delta (\\alpha _g(s))\\Delta _1(\\beta _s(g)g^{-1})\\Delta _2(\\alpha _g(s))$ .",
"The action $\\beta $ determines a unitary representation of $G_2$ on $L^2(G_1)$ given by $v_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )^{1/2}\\xi (\\beta _{s^{-1}}(g)), \\ \\ \\ \\xi \\in L^2(G_1).$ We denote by $v^1$ the corresponding action of $G_2$ on $L^1(G_1)$ , given by $v^1_s\\xi (g)=\\bigg (\\frac{d\\beta _{s^{-1}}(g)}{dg}\\bigg )f(\\beta _{s^{-1}}(g)), \\ \\ \\ f\\in L^1(G_1).$ Proposition 3.11 Let $(G_1,G_2)$ be a matched pair of locally compact groups such that $G_2$ is inner amenable, $\\beta $ preserves $\\Delta _1$ , there exists an asymptotically $\\beta $ -invariant BAI for $L^1(G_1)$ , i.e., a net $(f_i)$ of non-negative functions in $C_c(G_1)_{\\Vert \\cdot \\Vert _1=1}$ with $\\mathrm {supp}(f_i)\\rightarrow \\lbrace e\\rbrace $ satisfying $\\Vert v^1_sf_i-f_i\\Vert _1\\rightarrow 0$ uniformly on compacta.",
"Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.",
"Using the co-cycle properties of $\\alpha $ and $\\beta $ [65] together with the definitions of $W$ and $J̉$ , one sees that $&(J̉\\otimes J̉)W^*(J̉\\otimes J̉)\\xi (g,s,h,t)\\\\&=\\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2}\\xi (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s,\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1},t\\alpha _{h^{-1}\\beta _s(g)}(s^{-1})^{-1})$ for all $\\xi \\in L^2(G_1\\times G_2\\times G_1\\times G_2)$ .",
"Let $\\eta _i:=\\sqrt{f_i}$ .",
"By hypothesis $(iii)$ we have $\\Vert v_s\\eta _i - \\eta _i\\Vert _{L^2(G_1)}^2\\le \\Vert v_s^1f_i-f_i\\Vert _{L^1(G_1)}\\rightarrow 0$ uniformly on compacta.",
"Combining this with the support condition in $(iii)$ , for any uniformly continuous function $f:G_1\\times G_2\\times G_1\\rightarrow it follows that\\begin{equation} \\int f(g,s,h) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dh \\rightarrow f(g,s,e)\\end{equation}uniformly for $ (g,s)$ in compact subsets $ KG1G2$.$ By $(i)$ there exits a net $(\\xi _j)$ in $C_c(G_2)_{\\Vert \\cdot \\Vert _2=1}$ satisfying $\\Vert \\rho (s)\\xi _j-\\lambda (s)^*\\xi _j\\Vert _{L^2(G_2)}\\rightarrow 0$ uniformly on compacta.",
"Let $U_1$ be the self-adjoint unitary on $L^2(G_1)$ satisfying $U_1\\xi (g)=\\xi (g^{-1})\\Delta _1(g^{-1})$ .",
"Then for any $\\eta \\in C_c(G_1\\times G_2)$ , and any $j$ , we have $&\\langle W\\sigma V\\sigma (\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),\\eta \\otimes U_1\\eta _i\\otimes \\xi _j\\rangle =\\langle (J̉\\otimes J̉)W^*(J̉\\otimes J̉)(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j),W^*(\\eta \\otimes U_1\\eta _i\\otimes \\xi _j)\\rangle \\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ U_1\\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{U_1\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ \\eta _i(\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h^{-1})g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h^{-1}) \\ \\xi _j(t(\\alpha _{h^{-1}\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h^{-1})^{-1/2} \\ \\overline{\\eta }(\\beta _t(h)^{-1}g,s) \\ \\overline{\\eta _i}(h^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h)^{-1}g}(s)t) \\ \\Delta _1(h^{-1}) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiiint \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\ \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1})\\\\&\\times \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ \\overline{\\eta _i}(h) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ dg \\ ds \\ dh \\ dt\\\\&=\\iiint \\bigg (\\int \\xi _j(t(\\alpha _{h\\beta _s(g)}(s^{-1}))^{-1}) \\ \\overline{\\xi _j}(\\alpha _{\\beta _t(h^{-1})^{-1}g}(s)t) \\ \\overline{\\eta }(\\beta _t(h^{-1})^{-1}g,s) \\ dt\\bigg )\\\\&\\times \\Delta (\\alpha _{h\\beta _s(g)}(s^{-1}))^{1/2} \\ \\eta (\\beta _{\\alpha _{\\beta _s(g)}(s^{-1})}(h)g,s) \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),h)^{-1/2} \\ (v_{\\alpha _{\\beta _s(g)}(s^{-1})^{-1}}\\eta _i)(h) \\overline{\\eta _i}(h) \\ dg \\ ds \\ dh\\\\&\\rightarrow \\iiint \\xi _j(t(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1})\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\ \\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2} \\ dg \\ ds \\ dt\\\\$ by ().",
"But $\\alpha _{\\beta _s(g)}(s^{-1})^{-1}=\\alpha _g(s)$ almost everywhere in $(g,s)$ , so that $\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2}\\Psi (\\alpha _{\\beta _s(g)}(s^{-1}),e)^{-1/2}&=\\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{1/2} \\Delta (\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\Delta _2(\\alpha _{\\beta _s(g)}(s^{-1}))^{-1/2}\\\\&=\\Delta _2(\\alpha _g(s))^{1/2}, \\ \\ \\ a.e.$ The final integral in the above calculation therefore reduces to $\\iiint \\Delta _2(\\alpha _g(s))^{1/2} \\ \\xi _j(t\\alpha _{g}(s))\\ \\overline{\\xi _j}(\\alpha _{g}(s)t) \\ |\\eta (g,s)|^2 \\ dg \\ ds \\ dt.$ By the compact convergence (REF ) the above expression converges in $j$ to $\\iint |\\eta (g,s)|^2 \\ dg \\ ds = \\Vert \\eta \\Vert _{L^2(G_1\\times G_2)}^2.$ Denoting the index sets of $(\\eta _i)$ and $(\\xi _j)$ by $I$ and $J$ , respectively, we form the product $\\mathcal {I}:=J\\times I^{J}$ and for $I=(j,(i_j)_{j\\in J})\\in \\mathcal {I}$ we let $\\xi _I:=U_1\\eta _{i_j}\\otimes \\xi _j\\in L^2(G_1\\times G_2)$ .",
"By [39] and the above analysis, the resulting net $(\\xi _I)$ is asymptotically conjugation invariant for $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ .",
"Corollary 3.12 Let $(G_1,G_2)$ be matched pair of discrete groups.",
"Then $VN(G_1)^\\beta \\bowtie _\\alpha L^\\infty (G_2)$ is strongly inner amenable.",
"Remark 3.13 Unlike amenability, inner amenability for locally compact groups does not pass to extensions.",
"For example, $\\mathbb {R}^2\\rtimes \\mathbb {F}_6$ is not inner amenable (see Remark REF ), while both $\\mathbb {R}^2$ and $\\mathbb {F}_6$ are.",
"Example 3.14 We consider a discretized version of [26].",
"Let $G=\\bigg \\lbrace \\begin{pmatrix} a & b & x\\\\ c & d & y\\\\ 0 & 0 & 1\\end{pmatrix}\\mid \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\in SL(2,\\mathbb {Q}), \\ x,y\\in \\mathbb {Q}\\bigg \\rbrace ,$ $G_1=(\\mathbb {Q}^2,+)$ , and $G_2=SL(2,\\mathbb {Q})$ , viewed as discrete groups.",
"The embeddings $G_1\\ni (x,y)\\mapsto \\begin{pmatrix} 1 & 0 & -x\\\\ -x & 1 & -y+\\frac{1}{2}x^2\\\\ 0 & 0 & 1\\end{pmatrix}\\in G, \\ \\ G_2\\ni \\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}\\mapsto \\begin{pmatrix}d & -b & 0\\\\ -c & a & 0\\\\ 0 & 0 & 1\\end{pmatrix}\\in G$ determine a matched pair structure for $(G_1,G_2)$ .",
"The corresponding actions are given by $\\alpha _{(x,y)}\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}&=\\begin{pmatrix}a+bx & b\\\\ c+dx-(a+bx)(ax+b(y+\\frac{1}{2}x^2)) & d-b(ax+b(y+\\frac{1}{2}x^2))\\end{pmatrix} \\\\\\beta _{\\begin{pmatrix}a & b\\\\ c & d\\end{pmatrix}}(x,y)&=\\bigg (ax+by+\\frac{b}{2}x^2,cx+d\\bigg (y+\\frac{1}{2}x^2\\bigg )-\\frac{1}{2}\\bigg (ax+b\\bigg (y+\\frac{1}{2}x^2\\bigg )\\bigg )^2\\bigg ).$ By Corollary REF , the bicrossed product $\\mathbb {G}:=VN(\\mathbb {Q}^2)^\\beta \\bowtie _\\alpha L^\\infty (SL(2,\\mathbb {Q}))$ is strongly inner amenable.",
"Since $SL(2,\\mathbb {Q})$ is not amenable, it follows from [26] that $\\mathbb {G}$ is not amenable.",
"Moreover, since $\\mathbb {Q}^2$ is not compact, $\\mathbb {G}$ is not discrete by [65].",
"Thus, $\\mathbb {G}$ is an example of a non-discrete, non-amenable, inner amenable quantum group."
],
[
"Relative Injectivity and the Averaging Technique",
"In [56] Ruan and Xu showed that the dual $L^{\\infty }(\\widehat{\\mathbb {G}})$ of a strongly inner amenable Kac algebra $\\mathbb {G}$ is relatively 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}$ .",
"We will now show that relative 1-injectivity follows from the a priori weaker notion of inner amenability.",
"Below we let $\\widetilde{\\mathbb {G}}:=\\mathrm {Gr}()$ denote the intrinsic group of $$ .",
"Proposition 4.1 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Consider the following conditions: $$ is inner amenable; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ ; $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ ; $\\widetilde{}=\\mathrm {Gr}(\\mathbb {G})$ is inner amenable.",
"Then $(i)\\Rightarrow (ii)\\Leftrightarrow (iii)\\Rightarrow (iv)$ .",
"When $\\mathbb {G}$ is co-commutative, the conditions are equivalent.",
"$(i)\\Rightarrow (ii)$ : Given a state $n\\in L^{\\infty }(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $m:=n\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant.",
"It suffices to provide a completely contractive morphism which is a left inverse to the map $\\Delta :L^{\\infty }(\\mathbb {G})\\rightarrow \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ given by $\\Delta (x)(f)=x\\rhd f, \\ \\ \\ T\\in \\mathcal {B}(L^2(\\mathbb {G})), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Identifying $\\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))\\cong L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ via $\\langle \\Phi ,f\\otimes g\\rangle =\\langle \\Phi (f),g\\rangle , \\ \\ \\ \\Phi \\in \\mathcal {CB}(L^1(\\mathbb {G}),L^{\\infty }(\\mathbb {G})), \\ f,g\\in L^1(\\mathbb {G}),$ we have $\\Delta =\\Gamma $ , and that the corresponding $L^1(\\mathbb {G})$ -module structure on $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ is defined by $X\\unrhd f=(f\\otimes \\textnormal {id}\\otimes \\textnormal {id})(\\Gamma ^r\\otimes \\textnormal {id})(X)$ for $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $f\\in L^1(\\mathbb {G})$ .",
"First, consider the map $\\Phi :\\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni A\\mapsto (\\textnormal {id}\\otimes m)(V^*AV)\\in \\mathcal {B}(L^2(\\mathbb {G})).$ Clearly, $\\Phi $ is a completely contractive left inverse to $\\Gamma $ .",
"We show that $\\Phi $ is a right $\\mathcal {T}_\\rhd $ -module map.",
"This will complete the proof since [14] will entail the invariance $\\Phi (L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ , and the restricted module action $\\mathcal {T}_\\rhd \\curvearrowright L^{\\infty }(\\mathbb {G})$ is the pertinent $L^1(\\mathbb {G})$ -module action.",
"To this end, fix $A\\in \\mathcal {B}(L^2(\\mathbb {G}))\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ and $\\rho \\in \\mathcal {T}(L^2(\\mathbb {G}))$ .",
"Then $\\Phi (A\\unrhd \\rho )&=\\Phi ((\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}A_{13}V_{12}^*))\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{23}^*V_{12}A_{13}V_{12}^*V_{23})\\\\&=(\\textnormal {id}\\otimes m)(\\rho \\otimes \\textnormal {id}\\otimes \\textnormal {id})(V_{12}V_{23}^*V_{13}^*A_{13}V_{13}V_{23}V_{12}^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*).$ Now, using the fact that $V̉^{\\prime }=\\sigma V^*\\sigma $ , where $\\sigma $ is the flip map on $L^2(\\mathbb {G})\\otimes L^2(\\mathbb {G})$ , for any $\\tau ,\\omega \\in \\mathcal {T}(L^2(\\mathbb {G}))$ , we have $&\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*(\\sigma \\otimes 1)V_{23}^*A_{23}V_{23}(\\sigma \\otimes 1)V_{23}),\\tau \\otimes \\omega \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{13}^*V_{23}^*A_{23}V_{23}V_{13}),\\omega \\otimes \\tau \\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m)(V^*(1\\otimes (\\tau \\otimes \\textnormal {id})(V^*AV))V),\\omega \\rangle \\\\&=\\langle ( m\\otimes \\textnormal {id})(V̉^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV)\\otimes 1)V̉^{\\prime *}),\\omega \\rangle \\\\&=\\langle m,\\omega \\widehat{\\rhd }^{\\prime }((\\tau \\otimes \\textnormal {id})(V^*AV))\\rangle \\\\&=\\langle m,(\\tau \\otimes \\textnormal {id})(V^*AV)\\rangle \\langle \\omega ,1\\rangle \\\\&=\\langle (\\textnormal {id}\\otimes m\\otimes \\textnormal {id})(V^*AV\\otimes 1),\\tau \\otimes \\omega \\rangle \\\\&=\\langle \\Phi (A)\\otimes 1,\\tau \\otimes \\omega \\rangle .$ As $\\tau $ and $\\omega $ were arbitrary, we have $\\Phi (A\\unrhd \\rho )&=(\\rho \\otimes \\textnormal {id})(V(\\textnormal {id}\\otimes \\textnormal {id}\\otimes m)(V_{23}^*V_{13}^*A_{13}V_{13}V_{23})V^*)\\\\&=(\\rho \\otimes \\textnormal {id})(V(\\Phi (A)\\otimes 1)V^*)\\\\&=\\Phi (A)\\rhd \\rho .$ $(ii)\\Leftrightarrow (iii)$ Given a completely contractive left (respectively, right) $L^1(\\mathbb {G})$ -module left inverse $\\Phi $ to $\\Gamma $ , it follows that $R\\circ \\Phi \\circ \\Sigma \\circ (R\\otimes R)$ is a completely contractive right (respectively, left) $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ .",
"$(iii)\\Rightarrow (iv)$ : Recall that $\\mathrm {Gr}(\\mathbb {G})$ is a group of unitaries in $L^{\\infty }(\\mathbb {G})$ , so it acts naturally on $L^{\\infty }(\\mathbb {G})$ by conjugation.",
"The existence of a state $m\\in L^{\\infty }(\\mathbb {G})^*$ which is $\\mathrm {Gr}(\\mathbb {G})$ -invariant follows directly from the argument of [16], using the $\\Gamma (L^{\\infty }(\\mathbb {G}))-L^{\\infty }(\\mathbb {G})$ -bimodule property of $\\Phi $ .",
"Since $\\widetilde{\\widehat{\\mathbb {G}}}$ is a closed quantum subgroup of $\\widehat{\\mathbb {G}}$ in the sense of Vaes [22], there exists a normal $*$ -homomorphism $\\gamma :L^{\\infty }(\\widehat{\\widetilde{}})\\rightarrow L^{\\infty }(\\mathbb {G})$ intertwining the co-multiplications.",
"As $L^{\\infty }(\\widehat{\\widetilde{}})=VN(\\widetilde{\\widehat{\\mathbb {G}}})=VN(\\mathrm {Gr}(\\mathbb {G}))$ , the state $m\\circ \\gamma \\in VN(\\mathrm {Gr}(\\mathbb {G}))^*$ is $\\mathrm {Gr}(\\mathbb {G})$ -invariant, making $\\mathrm {Gr}(\\mathbb {G})$ inner amenable by [16].",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, then $\\mathrm {Gr}(\\mathbb {G}_s)=G$ and the implication $(iv)\\Rightarrow (i)$ follows immediately from [16].",
"Corollary 4.2 Let $\\mathbb {G}$ be a locally compact quantum group for which $L^{\\infty }(\\widehat{\\mathbb {G}})$ is an injective von Neumann algebra.",
"Then the following are equivalent: $\\mathbb {G}$ is amenable; $\\mathbb {G}$ is inner amenable; $L^{\\infty }(\\widehat{\\mathbb {G}})$ is relatively 1-injective in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"Propositions REF and REF yield the implications $(i)\\Rightarrow (ii)\\Rightarrow (iii)$ .",
"Assume $(iii)$ .",
"Since $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $\\mathbf {mod}\\hspace{2.0pt}it follows from \\cite [Proposition 2.3]{C} that $ L(G)$ is 1-injective in $ mod L1(G)$.",
"Hence, $ G$ is amenable by \\cite [Theorem 5.1]{C}.$ In the recent article [51], Ng and Viselter utilized topological inner amenability to elucidate the connection between co-amenability of $\\mathbb {G}$ and amenability of $$ .",
"One of their main results is the following.",
"Theorem 4.3 (Ng–Viselter) Let $\\mathbb {G}$ be a locally compact quantum group.",
"Consider the following conditions: $\\mathbb {G}$ is co-amenable; $C_0(\\mathbb {G})$ is nuclear and there exists a state $\\rho \\in C_0(\\mathbb {G})^*$ such that $\\langle \\rho ,x\\widehat{\\lhd } \\hat{f}\\rangle =\\langle \\rho ,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ $$ is amenable.",
"Then $(a)\\Rightarrow (b)\\Rightarrow (c)$ .",
"Moreover, if $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra), then $(a)\\Rightarrow (b^{\\prime })\\Rightarrow (b)$ , where $C_0(\\mathbb {G})$ is nuclear and has a tracial state.",
"They conjectured that condition $(b)$ above is equivalent to either condition $(a)$ or condition $(c)$ .",
"We now show that $(b)$ is indeed equivalent to $(a)$ .",
"Theorem 4.4 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.",
"Suppose that $C_0(\\mathbb {G})$ is nuclear and $$ is topologically inner amenable.",
"Given a state $m\\in C_0(\\mathbb {G})^*$ which is right $\\widehat{\\lhd }$ invariant, it follows that $n:=m\\circ R$ is left $\\widehat{\\rhd }^{\\prime }$ invariant, as in the proof of Proposition REF .",
"Let $n\\in M(C_0(\\mathbb {G}))^*$ also denote the unique extension to $M(C_0(\\mathbb {G}))$ which is strictly continuous on the unit ball, and $(\\textnormal {id}\\otimes n):M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ denote the unique extension of the slice map $(\\textnormal {id}\\otimes n):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))$ which is strictly continuous on the unit ball.",
"By strict density of $C_0(\\mathbb {G})$ in $M(C_0(\\mathbb {G}))$ , it follows that $\\langle n,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle n,x\\rangle \\langle \\hat{f}^{\\prime },1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in M(C_0(\\mathbb {G})).$ Since $V\\in M(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))$ , the map $\\Phi :C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ defined by $\\Phi (A)=(\\textnormal {id}\\otimes n)(V^*AV), \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}),$ is a non-zero, strict, completely positive contraction.",
"Using the extended $\\widehat{\\rhd }^{\\prime }$ -invariance on $M(C_0(\\mathbb {G}))$ , it follows verbatim from the proof of Proposition REF that $\\Phi (A\\unrhd \\rho )=\\Phi (A)\\rhd \\rho , \\ \\ \\ A\\in C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}), \\ \\rho \\in \\mathcal {T}(L^2(\\mathbb {G})).$ Since $\\mathrm {Ad}(V^*):\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})\\rightarrow \\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G})$ and $\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})=(\\mathcal {K}(L^2(\\mathbb {G}))\\otimes _{\\min } C_0(\\mathbb {G}))^*,$ we have $\\mathrm {Ad}(V^*)^*:\\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G})\\rightarrow \\mathcal {T}(L^2(\\mathbb {G}))\\widehat{\\otimes }M(\\mathbb {G}).$ Letting $r:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto \\rho |_{C_0(\\mathbb {G})}\\in M(\\mathbb {G})$ be the (completely positive) restriction map, it follows that $\\Phi ^*|_{\\mathcal {T}(L^2(\\mathbb {G}))}:\\mathcal {T}(L^2(\\mathbb {G}))\\ni \\rho \\mapsto (r\\otimes \\textnormal {id})(\\mathrm {Ad}(V^*)^*(\\rho \\otimes n))\\in M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ The proof of [14] entails the inclusion $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ .",
"In fact, since $\\Phi $ is a right $L^1(\\mathbb {G})$ -module map and $C_0(\\mathbb {G})$ is essential, more is true: $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq \\mathrm {LUC}(\\mathbb {G})\\subseteq M(C_0(\\mathbb {G})).$ The unique strict extension $\\widetilde{\\Phi }:M(C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\rightarrow M(C_0(\\mathbb {G}))$ , which exists by [45], satisfies $\\widetilde{\\Phi }\\circ \\Gamma |_{C_0(\\mathbb {G})}=\\textnormal {id}_{C_0(\\mathbb {G})}$ .",
"If $\\rho \\in L^{\\infty }(\\mathbb {G})_{\\perp }$ then the invariance $\\Phi (C_0(\\mathbb {G})\\otimes _{\\min } C_0(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})$ implies $\\Phi ^*(\\rho )=0$ .",
"Thus, $\\Phi ^*$ induces a completely positive left $L^1(\\mathbb {G})$ -module map $\\Phi ^*:L^1(\\mathbb {G})=(\\mathcal {T}(L^2(\\mathbb {G}))/L^{\\infty }(\\mathbb {G})_{\\perp })\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G}).$ By strict continuity and the definition of the multiplication on $M(\\mathbb {G})$ , for $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle m_{M(\\mathbb {G})}(\\Phi ^*(f)),x\\rangle =\\langle \\Phi ^*(f),\\Gamma (x)\\rangle =\\langle f,\\widetilde{\\Phi }(\\Gamma (x))\\rangle =\\langle f,x\\rangle .$ Hence, $m_{M(\\mathbb {G})}\\circ \\Phi ^*$ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .",
"Now, by nuclearity of $C_0(\\mathbb {G})$ , there exists a net $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ of finite-rank completely positive contractions converging to the identity in the point-norm topology.",
"Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*.$ Then $\\psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}),M(\\mathbb {G}))=_{L^1(\\mathbb {G})}\\mathcal {CP}(L^1(\\mathbb {G}))$ .",
"By [20] there exist contractive positive functionals $\\nu _i\\in C_u(\\mathbb {G})^*$ such that $\\psi _i=m^r_{\\nu _i}$ .",
"Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.",
"Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .",
"Since $(b)\\Rightarrow (c)$ in Theorem REF we know that $$ is amenable.",
"Hence, by [14] $M^r_{cb}(L^1(\\mathbb {G}))= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))=C_u(\\mathbb {G})^*$ .",
"For each $i$ and $k$ the map $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*\\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ , so there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi ^*=m^r_{\\nu _k^i}$ .",
"Thus, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi ^*(f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi ^*(f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i.$ Hence, $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Passing to a subnet we may assume $\\nu _i\\rightarrow \\nu $ weak* in $M(\\mathbb {G})$ .",
"But then for any $f\\in L^1(\\mathbb {G})$ and $x\\in C_0(\\mathbb {G})$ we have $\\langle f\\star \\nu , x\\rangle =\\langle \\nu , x\\star f\\rangle =\\lim _i\\langle \\nu _i,x\\star f\\rangle =\\lim _i\\langle f\\star \\nu _i,x\\rangle =\\langle f,x\\rangle .$ It follows that $\\nu $ is a right identity for $M(\\mathbb {G})$ , which implies that $M(\\mathbb {G})$ is unital, whence $\\mathbb {G}$ is co-amenable (cf.",
"[5]).",
"Theorem REF generalizes, and provides a different proof of, the main result in [50], which says that a locally compact group $G$ is amenable if and only if $C^*_\\lambda (G)$ is nuclear and has a tracial state.",
"Corollary 4.5 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has trivial scaling group (for instance, if $\\mathbb {G}$ is a Kac algebra).",
"Then $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and has a tracial state.",
"The combination of Theorem REF with Corollary REF elucidates the relationship between co-amenability and amenability of the dual: $$ is amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in L^{\\infty }(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ while $\\mathbb {G}$ is co-amenable if and only if $C_0(\\mathbb {G})$ is nuclear and there exists a state $m\\in C_0(\\mathbb {G})^*$ such that $\\langle m,x\\widehat{\\lhd } f̉\\rangle =\\langle m,x\\rangle \\langle \\hat{f},1\\rangle , \\ \\ \\ \\hat{f}\\in L^1(\\widehat{\\mathbb {G}}), \\ x\\in C_0(\\mathbb {G}).$ This subtle difference can also be phrased in terms of homology: $$ is amenable if and only if $L^1(\\mathbb {G})$ is 1-flat [14], while $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective, as we now prove.",
"Theorem 4.6 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $\\mathbb {G}$ is co-amenable if and only if $M(\\mathbb {G})$ is 1-projective in $M(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ .",
"If $\\mathbb {G}$ is co-amenable then $M(\\mathbb {G})$ is unital by [5], and moreover the unit has norm one.",
"Any unital completely contractive Banach algebra $\\mathcal {A}$ is $\\Vert e_{\\mathcal {A}}\\Vert $ -projective, so the claim follows.",
"Conversely, if $M(\\mathbb {G})$ is 1-projective, then $C_0(\\mathbb {G})^{**}=M(\\mathbb {G})^*$ is 1-injective in $\\mathbf {mod}-M(\\mathbb {G})$ .",
"It follows that the inclusion morphism $M(C_0(\\mathbb {G}))\\hookrightarrow C_0(\\mathbb {G})^{**}$ extends to a unital completely contractive, hence completely positive $M(\\mathbb {G})$ -module map $\\Phi : \\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}$ .",
"Applying the argument of [51] we see that $(\\textnormal {id}\\otimes \\Phi ):C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow C_0(\\mathbb {G})^{**}\\overline{\\otimes }C_0(\\mathbb {G})^{**}$ is a unital completely positive morphism satisfying $(\\textnormal {id}\\otimes \\Phi )(W̉)=W̉$ .",
"By unitarity it follows that $W̉$ is in the multiplicative domain of $(\\textnormal {id}\\otimes \\Phi )$ , and hence $(\\textnormal {id}\\otimes \\Phi )(W̉^*XW̉)=W̉^*(\\textnormal {id}\\otimes \\Phi )(X)W̉, \\ \\ \\ X\\in C_0(\\mathbb {G})^{**}\\overline{\\otimes }\\mathcal {B}(L^2(\\mathbb {G})).$ In particular, for every $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ and $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ we have $\\Phi (T\\hat{f})&=\\Phi ((\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes T)W̉))=(\\hat{f}\\otimes \\textnormal {id})(\\textnormal {id}\\otimes \\Phi )((W̉^*(1\\otimes T)W̉))\\\\&=(\\hat{f}\\otimes \\textnormal {id})(W̉^*(1\\otimes \\Phi (T))W̉)=\\Phi (T)\\hat{f}.$ Thus, $\\Phi $ is a morphism with respect to the canonical $L^1(\\widehat{\\mathbb {G}})$ -module structure on $C_0(\\mathbb {G})^{**}$ .",
"Moreover, the $M(\\mathbb {G})$ -module property entails $\\pi \\circ \\Phi (L^{\\infty }(\\widehat{\\mathbb {G}}))=$ , where $\\pi :C_0(\\mathbb {G})^{**}\\rightarrow L^{\\infty }(\\mathbb {G})$ is the canonical surjection.",
"Since $\\pi $ is clearly an $L^1(\\widehat{\\mathbb {G}})$ -module map, it follows that $\\pi \\circ \\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ is an invariant mean, entailing the amenability of $$ and therefore the nuclearity of $C_0(\\mathbb {G})$ .",
"Now, by 1-projectivity of $M(\\mathbb {G})$ , for every $\\varepsilon >0$ there exists a morphism $\\Phi _\\varepsilon :M(\\mathbb {G})\\rightarrow M(\\mathbb {G})^+\\widehat{\\otimes }M(\\mathbb {G})$ such that $m^+\\circ \\Phi _\\varepsilon =\\textnormal {id}_{M(\\mathbb {G})}$ , and $\\Vert \\Phi _\\varepsilon \\Vert _{cb}<1+\\varepsilon $ .",
"By amenability of $$ , we know $C_u(\\mathbb {G})^*= _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ [14], so one does not require the complete positivity of $\\Phi _\\varepsilon $ to perform the averaging argument from Theorem REF , which yields a bounded net $(\\mu _i)$ in $M(\\mathbb {G})$ that clusters to a right identity, entailing the co-amenability of $\\mathbb {G}$ .",
"The averaging argument used above, together with its variants used in [3], [14], shows that it is inner amenability, as opposed to discreteness, that underlies the original averaging technique of Haagerup.",
"In the setting of unimodular discrete quantum groups $$ , the technique relies on the existence of a normal left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to the co-multiplication that is an $L^1(\\mathbb {G})$ -module map.",
"Such a map is typically built from a trace-preserving normal conditional expectation $E:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow \\Gamma (L^{\\infty }(\\mathbb {G}))$ onto the image of $\\Gamma $ (see [23] and [41]).",
"It is the combination of the $L^1(\\mathbb {G})$ -module property of $\\Phi $ together with a suitable finite-dimensional approximation that allows one to average approximation properties of $L^{\\infty }(\\mathbb {G})$ or $C_0(\\mathbb {G})$ to approximation properties of $$ .",
"Thus, provided one has a suitably nice $L^1(\\mathbb {G})$ -module left inverse to the co-multiplication, the same averaging technique applies.",
"This is where inner amenability enters the picture.",
"Recall that a locally compact quantum group $\\mathbb {G}$ is weakly amenable if there exists an approximate identity $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ which is bounded in $M_{cb}^l(L^1(\\widehat{\\mathbb {G}}))$ .",
"The infimum of bounds for such approximate identities is the Cowling–Haagerup constant of $\\mathbb {G}$ , and is denoted $\\Lambda _{cb}(\\mathbb {G})$ .",
"We say that $\\mathbb {G}$ has the approximation property if there exists a net $(\\hat{f}_i)$ in $L^1(\\widehat{\\mathbb {G}})$ such that $^l(\\hat{\\lambda }(\\hat{f}_i))$ converges to $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ in the stable point-weak* topology.",
"Proposition 4.7 Let $\\mathbb {G}$ be a locally compact quantum group whose dual $$ is strongly inner amenable.",
"If $L^{\\infty }(\\mathbb {G})$ has the w*CBAP then $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .",
"$L^{\\infty }(\\mathbb {G})$ has the w*OAP if and only if $$ has the approximation property.",
"Let $(\\xi _i)$ be a net of asymptotically conjugation invariant unit vectors in $L^2(\\mathbb {G})$ .",
"It follows verbatim from [56] that $\\Phi _i:L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\ni X\\mapsto (\\omega _{\\xi _i}\\otimes \\textnormal {id})(W^*(U^*\\otimes 1)X(U\\otimes 1)W)\\in L^{\\infty }(\\mathbb {G}).$ defines a net of unital completely positive left $L^1(\\mathbb {G})$ -module maps, which cluster weak* in $\\mathcal {CB}(L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=((L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G}))\\widehat{\\otimes }L^1(\\mathbb {G}))^*$ to a module left inverse to $\\Gamma $ .",
"Passing to a subnet we may assume convergence.",
"$(i)$ Let $(\\varphi _j)$ be a net of finite-rank, normal, completely bounded maps converging to the identity point-weak*, with $\\Vert \\varphi _j\\Vert _{cb}\\le C$ .",
"Since $\\varphi _j$ is finite-rank, there exists $f^j_1,...,f^j_{n_j}\\in L^1(\\mathbb {G})$ and $x^j_1,...,x^j_{n_j}\\in L^{\\infty }(\\mathbb {G})$ such that $\\varphi _j=\\sum _{k=1}^{n_j} x^j_k\\otimes f^j_k$ .",
"Put $\\Phi _{ij}=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma :L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G}).$ Then $\\Phi _{ij}$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map with $\\Vert \\Phi _{ij}\\Vert _{cb}\\le C$ .",
"Also, for each $i,j$ and $1\\le k\\le n_j$ , the map $L^{\\infty }(\\mathbb {G})\\ni x\\mapsto \\Phi _i(x^j_k\\otimes x)\\in L^{\\infty }(\\mathbb {G})$ is a normal completely bounded left $L^1(\\mathbb {G})$ -module map.",
"Since $x^j_k$ is a linear combination of positive elements in $L^{\\infty }(\\mathbb {G})$ , and $\\Phi _i$ is completely positive, the map (REF ) is a linear combination of normal completely positive $L^1(\\mathbb {G})$ -module maps $L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ .",
"By [20], there exist $\\mu ^{ij}_k\\in C_u(\\mathbb {G})^*$ such that (REF ) is given by right multiplication by $\\mu ^{ij}_k$ .",
"Hence, for each $x\\in L^{\\infty }(\\mathbb {G})$ $\\Phi _{ij}(x)&=\\Phi _i\\circ (\\varphi _j\\otimes \\textnormal {id})\\circ \\Gamma (x)\\\\&=\\sum _{k=1}^{n_j}\\Phi _i(x^j_k \\otimes x\\star f^j_k)\\\\&=\\sum _{k=1}^{n_j}x\\star f^j_k\\star \\mu ^{ij}_k\\\\&=x\\star f_{ij},$ where $f_{ij}=\\sum _{k=1}^{n_i} f^j_k\\star \\mu ^{ij}_k\\in L^1(\\mathbb {G})$ as $L^1(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Thus, $\\Vert f_{ij}\\Vert _{cb}\\le C$ , and it follows that $\\lim _i\\lim _jx\\star f_{ij}=\\lim _i\\Phi _i(\\Gamma (x))=x,$ point-weak* in $L^{\\infty }(\\mathbb {G})$ .",
"Combining the iterated limit into a single net as in Proposition REF and appealing to the standard convexity argument, it follows that $$ is weakly amenable with $\\Lambda _{cb}()\\le \\Lambda _{cb}(L^{\\infty }(\\mathbb {G}))$ .",
"$(ii)$ The proof that w*OAP implies the approximation property follows by a similar argument to $(i)$ , appealing to well-known properties of the stable point-weak* topology on von Neumann algebras (cf.",
"[34]).",
"The converse was shown for Kac algebras in [41].",
"Their proof extends verbatim to arbitrary locally compact quantum groups.",
"Corollary 4.8 Let $G$ be an inner amenable locally compact group.",
"If $VN(G)$ has the w*CBAP then $G$ is weakly amenable.",
"$VN(G)$ has the w*OAP if and only if $G$ has the approximation property.",
"Remark 4.9 From the homological perspective, the (potential) distinction between strong inner amenability and inner amenability of $$ , is that the former case generates an $L^1(\\mathbb {G})$ -module left inverse to $\\Gamma $ that can be approximated by normal $L^1(\\mathbb {G})$ -module maps.",
"It is not clear if a similar approximation can be achieved in the latter case, whence the strong inner amenability assumption in Proposition REF .",
"Remark 4.10 To the author's knowledge, the converse of Proposition REF $(i)$ is not known to hold even in the co-commutative setting.",
"That is, if $G$ is a weakly amenable locally compact group, does $VN(G)$ have the weak*CBAP?",
"By [34] it is known that in this case $\\textnormal {id}_{VN(G)}$ can be approximated in the point-weak* topology by a bounded net in $\\mathcal {CB}^\\sigma (VN(G))$ , each of whose elements is the limit in the point-weak* topology of bounded net of finite-rank elements in $\\mathcal {CB}^\\sigma (VN(G))$ .",
"Proposition 4.11 Let $\\mathbb {G}$ be a topologically inner amenable locally compact quantum group.",
"If $C_0(\\mathbb {G})$ has the CBAP, then there exists a net $(\\nu _i)$ in $M(\\mathbb {G})$ such that $\\Vert \\nu _i\\Vert _{cb}\\le \\Lambda _{cb}(C_0(\\mathbb {G}))$ and $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .",
"If $C_0(\\mathbb {G})$ has the OAP then $M(\\mathbb {G})$ is $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ -dense in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"Let $\\Phi :L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})\\widehat{\\otimes }M(\\mathbb {G})$ be the completely positive left $L^1(\\mathbb {G})$ -module map constructed in Theorem REF from topological inner amenability.",
"Recall that $m_{M(\\mathbb {G})}\\circ \\Phi $ is the canonical inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ .",
"$(i)$ Let $\\varphi _i:C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ be a net of finite-rank completely bounded maps converging to the identity in the point-norm topology such that $\\Vert \\varphi _i\\Vert _{cb}\\le C$ .",
"Write $\\varphi _i=\\sum _{k=1}^{n_i}x_k^i\\otimes \\mu _k^i$ for some $x_k^i\\in C_0(\\mathbb {G})$ and $\\mu _k^i\\in M(\\mathbb {G})$ .",
"For each $i$ and $k$ , $x_k^i$ is a linear combination of positive elements in $C_0(\\mathbb {G})$ , and hence the map $(\\textnormal {id}\\otimes x_k^i)\\Phi \\in _{L^1(\\mathbb {G})}\\mathcal {CB}(L^1(\\mathbb {G}))$ is a linear combination of completely positive left $L^1(\\mathbb {G})$ -module maps on $L^1(\\mathbb {G})$ , so by [20] there exists $\\nu _k^i$ such that $(\\textnormal {id}\\otimes x_k^i)\\Phi =m^r_{\\nu _k^i}$ .",
"Define $\\psi _i:L^1(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ by $\\psi _i=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi $ .",
"Since $\\varphi ^*_i:M(\\mathbb {G})\\rightarrow M(\\mathbb {G})$ forms a bounded net converging to the identity point-weak*, and $m_{M(\\mathbb {G})}$ is separately weak* continuous, it follows that $\\psi _i$ converges to the inclusion $L^1(\\mathbb {G})\\hookrightarrow M(\\mathbb {G})$ point-weak*.",
"Moreover, for $f\\in L^1(\\mathbb {G})$ we have $\\psi _i(f)&=m_{M(\\mathbb {G})}\\circ (\\textnormal {id}\\otimes \\varphi _i^*)\\circ \\Phi (f)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}((\\textnormal {id}\\otimes x_k^i)\\Phi (f)\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}m_{M(\\mathbb {G})}(f\\star \\nu _k^i\\otimes \\mu _k^i)\\\\&=\\sum _{k=1}^{n_i}f\\star \\nu _k^i\\star \\mu _k^i\\\\&=f\\star \\nu _i,$ where $\\nu _i=\\sum _{k=1}^{n_i}\\nu _k^i\\star \\mu _k^i\\in M(\\mathbb {G})$ as $M(\\mathbb {G})$ is a closed ideal in $C_u(\\mathbb {G})^*$ .",
"Then $\\Vert \\nu _i\\Vert _{cb}\\le C$ , and it follows that $\\nu _i\\star x\\rightarrow x$ weak* in $L^{\\infty }(\\mathbb {G})$ for all $x\\in C_0(\\mathbb {G})$ .",
"Since $(\\nu _i)$ is bounded in $\\mathcal {CB}_{L^1(\\mathbb {G})}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ , we have $\\nu _i\\rightarrow 1$ $\\sigma (M_{cb}^r(L^1(\\mathbb {G})),Q_{cb}^r(L^1(\\mathbb {G})))$ .",
"$(ii)$ The proof follows similarly to $(i)$ , averaging the finite-rank maps arising from the OAP of $C_0(\\mathbb {G})$ and using the stable point-norm topology together with the structure of $Q_{cb}^r(L^1(\\mathbb {G}))$ .",
"Self-duality of Biflatness As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .",
"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).",
"We now obtain the same conclusion under a priori weaker hypotheses.",
"Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.",
"It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .",
"Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .",
"However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.",
"Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .",
"Hence, $m$ satisfies (REF ).",
"A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.",
"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .",
"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.",
"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.",
"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.",
"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.",
"Proposition 5.2 Let $G$ be a locally compact group.",
"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.",
"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.",
"Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .",
"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].",
"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.",
"Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].",
"We now show that 1-biflatness of quantum convolution algebras is a self dual property.",
"Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.",
"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .",
"Also, [15] implies that $E$ is a right $\\lhd $ -module map.",
"Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .",
"Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .",
"Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .",
"Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.",
"Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).",
"Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .",
"By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .",
"It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.",
"Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].",
"Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.",
"Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .",
"Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.",
"The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].",
"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.",
"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].",
"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].",
"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .",
"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .",
"Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.",
"$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].",
"$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.",
"The bimodule analogue of [14] then yields $(iii)$ .",
"$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .",
"As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.",
"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.",
"Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .",
"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.",
"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.",
"In [16] the authors gave examples of weakly amenable connected groups (e.g.",
"$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.",
"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].",
"Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .",
"As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .",
"Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.",
"There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .",
"Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .",
"Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).",
"Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.",
"Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .",
"Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.",
"Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.",
"Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.",
"Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .",
"It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].",
"Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .",
"By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .",
"Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.",
"Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .",
"By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .",
"Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .",
"It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .",
"Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .",
"By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .",
"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .",
"The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .",
"By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.",
"Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .",
"Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .",
"Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .",
"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .",
"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.",
"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.",
"There are examples of weakly amenable residually finite groups (e.g.",
"$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].",
"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.",
"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].",
"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .",
"Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .",
"If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .",
"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.",
"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].",
"We now generalize this implication to arbitrary locally compact quantum groups.",
"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.",
"By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].",
"Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].",
"We present the proof for the convenience of the reader.",
"Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .",
"Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.",
"Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .",
"Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.",
"Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].",
"For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .",
"In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.",
"We show that it is a complete order bijection.",
"To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .",
"On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .",
"By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"We now show that $\\tilde{\\lambda }$ is a complete isometry.",
"To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].",
"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.",
"Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].",
"By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .",
"Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .",
"Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].",
"The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.",
"To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.",
"Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .",
"Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.",
"Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.",
"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].",
"Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.",
"Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.",
"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .",
"Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).",
"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .",
"Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.",
"Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .",
"Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.",
"It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.",
"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.",
"By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .",
"Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .",
"Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.",
"The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.",
"Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .",
"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.",
"The author was partially supported by the NSERC Discovery Grant 1304873."
],
[
"Self-duality of Biflatness",
"As in the one-sided case, for a completely contractive Banach algebra $\\mathcal {A}$ , we say that an operator $\\mathcal {A}$ -bimodule $X$ is $C$ -biflat (respectively, relatively $C$ -biflat) if its dual $X^*$ is $C$ -injective (respectively, relatively $C$ -injective) in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ .",
"Equivalently, $X$ is $C$ -biflat if for any 1-exact sequence $0\\rightarrow Y\\hookrightarrow Z\\twoheadrightarrow Z/Y\\rightarrow 0$ in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ , the sequence $0\\rightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y\\hookrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z\\twoheadrightarrow X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Z/Y\\rightarrow 0$ is $C$ -exact in the category of operator spaces and completely bounded maps, where $X_{\\mathcal {A}}\\widehat{\\otimes }_{\\mathcal {A}}Y$ is the bimodule tensor product of $X$ and $Y$ , defined by $X\\widehat{\\otimes }Y/_{\\mathcal {A}}N_{\\mathcal {A}}$ , where $_{\\mathcal {A}}N_{\\mathcal {A}}=\\langle a\\cdot x\\otimes y - x\\otimes y\\cdot a, \\ x^{\\prime }\\cdot a^{\\prime }\\otimes y^{\\prime } - x^{\\prime }\\otimes a^{\\prime }\\cdot y^{\\prime }\\mid a,a^{\\prime }\\in \\mathcal {A}, \\ x,x^{\\prime }\\in X, \\ y,y^{\\prime }\\in Y\\rangle .$ In [56] Ruan and Xu provided a sufficient condition for relative 1-biflatness of $L^1(\\widehat{\\mathbb {G}})$ for any Kac algebra $\\mathbb {G}$ by means of the existence of a certain net of unit vectors $(\\xi _i)$ which are asymptotically invariant under the conjugate co-representation $W\\sigma V \\sigma $ and for which $\\omega _{\\xi _i}|_{L^{\\infty }(\\mathbb {G})}$ is a bounded approximate identity of $L^1(\\mathbb {G})$ .",
"In the group setting, this is precisely the quasi-SIN, or QSIN condition (see [48], [58]).",
"We now obtain the same conclusion under a priori weaker hypotheses.",
"Proposition 5.1 Let $\\mathbb {G}$ be a locally compact quantum group for which there exists a right invariant mean $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ Then $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse holds.",
"It suffices to provide a completely contractive $L^1(\\mathbb {G})$ -bimodule map $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ which is a left inverse to $\\Gamma $ .",
"Defining $\\Phi (X)=(\\textnormal {id}\\otimes m)(V^*XV)$ , $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})$ , as in Proposition REF , it immediately follows that $\\Phi $ is a completely contractive right $L^1(\\mathbb {G})$ -module map and $\\Phi \\circ \\Gamma =\\textnormal {id}_{L^{\\infty }(\\mathbb {G})}$ .",
"However, since $m$ is also a right invariant mean on $L^{\\infty }(\\mathbb {G})$ , the module argument from [15] shows that $\\Phi $ is also a left $L^1(\\mathbb {G})$ -module map.",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, the converse follows from the proof of [16], wherein the existence of a state $m\\in VN(G)^*$ invariant under both the $A(G)$ -action and the right $L^1(G)$ -action was established.",
"Owing to the fact that $V_s=W_a$ we have $f\\widehat{\\rhd }_{s}^{\\prime }x&=(\\textnormal {id}\\otimes f)(V̉^{\\prime }_s(x\\otimes 1)V̉_s^{\\prime *})=(f\\otimes \\textnormal {id})(V_s^*(1\\otimes x)V_s)\\\\&=(f\\otimes \\textnormal {id})(W_a^*(1\\otimes x)W_a)=x\\lhd _a f$ for all $f\\in L^1(G)$ and $x\\in VN(G)$ .",
"Hence, $m$ satisfies (REF ).",
"A completely contractive Banach algebra $\\mathcal {A}$ is operator amenable if it is relatively $C$ -biflat in $\\mathbf {\\mathcal {A}\\hspace{2.0pt}mod\\hspace{2.0pt}\\mathcal {A}}$ for some $C\\ge 1$ , and has a bounded approximate identity.",
"By [54] this is equivalent to the existence of a bounded approximate diagonal in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ , that is, a bounded net $(A_\\alpha )$ in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ satisfying $a\\cdot A_\\alpha - A_\\alpha \\cdot a, \\ m_{\\mathcal {A}}(A_\\alpha )\\cdot a \\rightarrow 0, \\ \\ \\ a\\in \\mathcal {A}.$ We let $OA(\\mathcal {A})$ denote the operator amenability constant of $\\mathcal {A}$, the infimum of all bounds of approximate diagonals in $\\mathcal {A}\\widehat{\\otimes }\\mathcal {A}$ .",
"This notion is the operator module analogue of the classical concept introduced by Johnson [36], who showed that the group algebra $L^1(G)$ of a locally compact group $G$ is (operator) amenable if and only if $G$ is amenable.",
"In [54], Ruan established the dual result, showing that the Fourier algebra $A(G)$ is operator amenable precisely when $G$ is amenable.",
"Thus, $L^1(G)$ is operator amenable if and only if $A(G)$ is operator amenable.",
"To motivate our next result, we now recast this equivalence at the level of (non-relative) biflatness.",
"Proposition 5.2 Let $G$ be a locally compact group.",
"Then $L^1(G)$ is 1-biflat if and only if $G$ is amenable if and only if $A(G)$ is 1-biflat.",
"By [36] $G$ is amenable if and only if $L^1(G)$ is amenable if and only if $L^1(G)$ is relatively 1-biflat.",
"Since $L^{\\infty }(G)$ is a 1-injective operator space, Proposition REF entails the equivalence with 1-injectivity of $L^{\\infty }(G)$ in $L^1(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(G)$ .",
"Dually, if $G$ were amenable, then $VN(G)$ is a 1-injective operator space, and it is relatively 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ by (the proof of) [54].",
"Thus, Proposition REF entails the 1-injectivity of $VN(G)$ as an operator $A(G)$ -bimodule.",
"Conversely, if $VN(G)$ is 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}A(G)$ , then it is clearly 1-injective in $A(G)\\hspace{2.0pt}\\mathbf {mod}$ , so $G$ is amenable by [14].",
"We now show that 1-biflatness of quantum convolution algebras is a self dual property.",
"Theorem 5.3 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then $L^1(\\mathbb {G})$ is 1-biflat if and only if $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat.",
"Clearly, it suffices to show one direction by Pontrjagin duality, so suppose that $L^1(\\widehat{\\mathbb {G}})$ is 1-biflat, that is, $L^{\\infty }(\\widehat{\\mathbb {G}})$ is 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"Consider the canonical $L^1(\\widehat{\\mathbb {G}})$ -bimodule structure on $\\mathcal {B}(L^2(\\mathbb {G}))$ given by $\\hat{f}\\widehat{\\rhd }T=(\\textnormal {id}\\otimes \\hat{f})\\widehat{V}(T\\otimes 1)V̉^* \\hspace{10.0pt}\\hspace{10.0pt}\\textnormal {and}\\hspace{10.0pt}\\hspace{10.0pt}T\\widehat{\\lhd }\\hat{f}=(\\hat{f}\\otimes \\textnormal {id})W̉^*(1\\otimes T)W̉,$ for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"Then by 1-injectivity, $\\textnormal {id}_{L^{\\infty }(\\widehat{\\mathbb {G}})}$ extends to a completely contractive $L^1(\\widehat{\\mathbb {G}})$ -bimodule projection $E:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"By the left $\\widehat{\\rhd }$ -module property, it follows from the standard argument that $E(L^{\\infty }(\\mathbb {G}))\\subseteq L^{\\infty }(\\mathbb {G})\\cap L^{\\infty }(\\widehat{\\mathbb {G}})=$ .",
"Also, [15] implies that $E$ is a right $\\lhd $ -module map.",
"Let $R$ be the extended unitary antipode of $\\mathbb {G}$ .",
"Then $(R\\otimes R)(V̉^{\\prime })=(R\\otimes R)(\\sigma V^*\\sigma )=\\Sigma (R\\otimes R)(V^*)=\\Sigma (J̉\\otimes J̉)(V)(J̉\\otimes J̉)=\\Sigma W̉,$ where the last equality follows from equation (REF ) and the adjoint relations of $W$ and $V$ .",
"Let $E_R:\\mathcal {B}(L^2(\\mathbb {G}))\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}}^{\\prime })$ be the projection of norm one, $E_R=R\\circ E\\circ R$ .",
"Then for $\\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime })$ and $T\\in \\mathcal {B}(L^2(\\mathbb {G}))$ , we have $E_R(\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }T)&=R(E(R((\\textnormal {id}\\otimes \\hat{f}^{\\prime })V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)(R\\otimes R)(V̉^{\\prime }(T\\otimes 1)V̉^{\\prime *})))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((R\\otimes R)(V̉^{\\prime *})(R(T)\\otimes 1)(R\\otimes R)(V̉^{\\prime }))))\\\\&=R(E((\\textnormal {id}\\otimes \\hat{f}^{\\prime }\\circ R)((\\Sigma W̉^*)(R(T)\\otimes 1)(\\Sigma W̉))))\\\\&=R(E((\\hat{f}^{\\prime }\\circ R\\otimes \\textnormal {id})(W̉^*(1\\otimes R(T))W̉)))\\\\&=R(E(R(T)\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R)))\\\\&=R(E(R(T))\\widehat{\\lhd }(\\hat{f}^{\\prime }\\circ R))\\\\&=\\hat{f}^{\\prime }\\widehat{\\rhd }^{\\prime }E_R(T).$ Thus, $E_R$ is a left $\\widehat{\\rhd }^{\\prime }$ -module map.",
"Since $R(L^{\\infty }(\\mathbb {G}))=L^{\\infty }(\\mathbb {G})$ , the restriction $E_R|_{L^{\\infty }(\\mathbb {G})}$ defines a state $m\\in L^{\\infty }(\\mathbb {G})^*$ satisfying $\\langle m,f̉^{\\prime }\\widehat{\\rhd }^{\\prime }x\\rangle =\\langle \\hat{f}^{\\prime },1\\rangle \\langle m,x\\rangle , \\ \\ \\ \\hat{f}^{\\prime }\\in L^1(\\widehat{\\mathbb {G}}^{\\prime }), \\ x\\in L^{\\infty }(\\mathbb {G}).$ But $E$ was also a right $\\lhd $ -module map, which implies that $E_R$ is a left $\\rhd $ -module map by the generalized antipode relation (REF ).",
"Thus, we also have $\\langle m,f\\rhd x\\rangle =\\langle f,1\\rangle \\langle m,x\\rangle , \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ x\\in L^{\\infty }(\\mathbb {G}),$ meaning that $m$ is a right invariant mean on $L^{\\infty }(\\mathbb {G})$ .",
"By Proposition REF it follows that $L^{\\infty }(\\mathbb {G})$ is relatively 1-injective in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"By 1-injectivity of $L^{\\infty }(\\widehat{\\mathbb {G}})$ in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ , there exists a completely contractive morphism $\\Phi :L^{\\infty }(\\widehat{\\mathbb {G}})\\overline{\\otimes }L^{\\infty }(\\widehat{\\mathbb {G}})\\rightarrow L^{\\infty }(\\widehat{\\mathbb {G}})$ which is a left inverse to $$ .",
"It follows that $\\Phi |_{L^{\\infty }(\\widehat{\\mathbb {G}})\\otimes 1}$ defines a state $m̉\\in L^{\\infty }(\\widehat{\\mathbb {G}})^*$ which is a right $L^1(\\widehat{\\mathbb {G}})$ -module map, i.e., $$ is amenable.",
"Hence, $L^{\\infty }(\\mathbb {G})$ is a 1-injective operator space by [5].",
"Proposition REF then implies the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Corollary 5.4 $\\ell ^1(\\mathbb {G})$ is not relatively 1-biflat for any non-unimodular discrete quantum group.",
"Since $\\ell ^\\infty (\\mathbb {G})$ is always a 1-injective operator space for any disrete quantum group $\\mathbb {G}$ , if $\\ell ^1(\\mathbb {G})$ were relatively 1-biflat, then by Proposition REF , $\\ell ^\\infty (\\mathbb {G})$ would be 1-injective in $\\ell ^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}\\ell ^1(\\mathbb {G})$ .",
"Then by Theorem REF $L^{\\infty }(\\widehat{\\mathbb {G}})$ would be 1-injective in $L^1(\\widehat{\\mathbb {G}})\\hspace{2.0pt}\\mathbf {mod}\\hspace{2.0pt}L^1(\\widehat{\\mathbb {G}})$ .",
"But then, [12] would entail that $$ is a compact Kac algebra, and therefore $\\mathbb {G}$ is unimodular.",
"The relative biprojectivity of $L^1(\\mathbb {G})$ , that is, relative projectivity of $L^1(\\mathbb {G})$ as an operator bimodule over itself, has been completely characterized: $L^1(\\mathbb {G})$ is relatively $C$ -biprojective if and only if $L^1(\\mathbb {G})$ is relatively 1-biprojective if and only if $\\mathbb {G}$ is a compact Kac algebra [2], [20], [12].",
"The corresponding characterization for (relative) $C$ -biflatness remains an interesting open question.",
"In the co-commutative setting, the relative 1-biflatness of $A(G)$ has been studied in [4], [16], [56].",
"It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup $G_\\Delta $ [16].",
"The authors in [16] conjecture that it is equivalent to the QSIN property of $G$ .",
"We finish this subsection with a generalization of [38] beyond co-amenable quantum groups, which at the same time characterizes the (non-relative) 1-biprojectivity of $L^1(\\mathbb {G})$ .",
"Theorem 5.5 Let $\\mathbb {G}$ be a locally compact quantum group.",
"Then the following conditions are equivalent: $\\mathbb {G}$ is finite–dimensional.",
"$\\mathcal {T}_\\rhd $ is relatively 1-biprojective; $L^1(\\mathbb {G})$ is 1-biprojective; $(i)\\Rightarrow (ii)$ follows from [38].",
"$(ii)\\Rightarrow (iii)$ follows similarly to the proof of [14], giving the relative 1-biprojectivity of $L^1(\\mathbb {G})$ together with the 1-projectivity of $L^1(\\mathbb {G})$ as an operator space.",
"The bimodule analogue of [14] then yields $(iii)$ .",
"$(iii)\\Rightarrow (i)$ The 1-biprojectivity of $L^1(\\mathbb {G})$ ensures the existence of a normal completely bounded $L^1(\\mathbb {G})$ -bimodule left inverse $\\Phi :L^{\\infty }(\\mathbb {G})\\overline{\\otimes }L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ to $\\Gamma $ .",
"As usual, the restriction $\\Phi |_{L^{\\infty }(\\mathbb {G})\\otimes 1}:L^{\\infty }(\\mathbb {G})\\rightarrow L^{\\infty }(\\mathbb {G})$ maps into $, and, moreover, it is a right $ L1(G)$-module map, so $ G$ is compact by normality of $$.",
"Since compact quantum groups are regular, we may repeat the proof of $ (iii)(i)$ from \\cite [Theorem 5.14]{C} to deduce the discreteness of $ G$.",
"Thus, $ G$ is finite-dimensional by \\cite [Theorem 4.8]{KN}.$ Operator Amenability of $L^1_{cb}(\\mathbb {G})$ For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .",
"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.",
"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.",
"In [16] the authors gave examples of weakly amenable connected groups (e.g.",
"$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.",
"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].",
"Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .",
"As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .",
"Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.",
"There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .",
"Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .",
"Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).",
"Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.",
"Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .",
"Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.",
"Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.",
"Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.",
"Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .",
"It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].",
"Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .",
"By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .",
"Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.",
"Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .",
"By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .",
"Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .",
"It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .",
"Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .",
"By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .",
"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .",
"The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .",
"By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.",
"Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .",
"Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .",
"Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .",
"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .",
"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.",
"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.",
"There are examples of weakly amenable residually finite groups (e.g.",
"$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].",
"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.",
"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].",
"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .",
"Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .",
"If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .",
"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold.",
"Decomposability For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].",
"We now generalize this implication to arbitrary locally compact quantum groups.",
"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.",
"By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].",
"Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].",
"We present the proof for the convenience of the reader.",
"Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .",
"Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.",
"Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .",
"Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.",
"Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].",
"For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .",
"In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.",
"We show that it is a complete order bijection.",
"To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .",
"On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .",
"By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"We now show that $\\tilde{\\lambda }$ is a complete isometry.",
"To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].",
"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.",
"Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].",
"By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .",
"Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .",
"Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].",
"The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.",
"To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.",
"Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .",
"Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.",
"Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.",
"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].",
"Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.",
"Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.",
"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .",
"Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).",
"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .",
"Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.",
"Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .",
"Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.",
"It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.",
"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.",
"By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .",
"Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .",
"Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.",
"The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.",
"Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ .",
"Acknowledgements The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.",
"The author was partially supported by the NSERC Discovery Grant 1304873."
],
[
"Operator Amenability of $L^1_{cb}(\\mathbb {G})$",
"For a locally compact quantum group $\\mathbb {G}$ , let $L^1_{cb}(\\mathbb {G})$ denote the closure of $L^1(\\mathbb {G})$ inside $M_{cb}^l(L^1(\\mathbb {G}))$ .",
"Recall that $$ is weakly amenable precisely when $L^1_{cb}(\\mathbb {G})$ has a bounded approximate identity.",
"In analogy to Ruan's result – equating amenability of a locally compact group $G$ to operator amenability of $A(G)$ – it was suggested in [29] that $A_{cb}(G)$ may be operator amenable exactly when $G$ is weakly amenable.",
"In [16] the authors gave examples of weakly amenable connected groups (e.g.",
"$G=SL(2,\\mathbb {R})$ ) for which $A_{cb}(G)$ is not operator amenable.",
"We now relate weak amenability of $$ to operator amenability of $L^1_{cb}(\\mathbb {G})$ for unimodular discrete quantum groups with Kirchberg's factorization property in the sense of [7].",
"Let $\\mathbb {G}$ be a compact Kac algebra and let $\\varphi $ and $R$ denote the Haar trace and unitary antipode on $C(\\mathbb {G})$ , as well as their universal extensions to $C_u(\\mathbb {G})$ .",
"As in [7], we define $*$ -homomorphisms $\\lambda ,\\rho :C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ by $\\lambda (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (xy), \\ \\ \\ \\rho (x)\\Lambda _\\varphi (y)=\\Lambda _\\varphi (y R(x)), \\ \\ \\ x,y\\in C_u(\\mathbb {G}).$ Since $\\lambda $ and $\\rho $ have commuting ranges, we obtain a canonical representation $\\lambda \\times \\rho :C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G})\\rightarrow \\mathcal {B}(L^2(\\mathbb {G}))$ .",
"The unimodular discrete dual $$ is said to have Kirchberg's factorization property if $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"When $\\mathbb {G}=\\mathbb {G}_s$ is co-commutative, this notion coincides with Kirchberg's factorization property for the underlying discrete group $G$ .",
"Lemma 6.1 Let $\\mathcal {A}$ be a $C^*$ -algebra.",
"There exists a complete isometry $\\iota :\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ such that $\\iota (\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*_{\\Vert \\cdot \\Vert \\le 1})$ is weak* dense in $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*_{\\Vert \\cdot \\Vert \\le 1}$ .",
"Let $\\pi _\\mathcal {A}:\\mathcal {A}\\rightarrow \\mathcal {A}^{**}$ denote the universal representation of $\\mathcal {A}$ .",
"Then the universal cover of the representation $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}$ is a normal surjective $*$ -homomorphism $\\pi $ of $(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{**}$ onto $(\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A})(\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^{\\prime \\prime }=\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }\\overline{\\otimes }\\pi _\\mathcal {A}(\\mathcal {A})^{\\prime \\prime }=\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^*$ (see [61]).",
"Its pre-adjoint $\\iota :=\\pi _*:\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*\\hookrightarrow (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ is a complete isometry.",
"Now, Let $F\\in (\\mathcal {A}\\otimes _{\\min }\\mathcal {A})^*$ , $\\Vert F\\Vert \\le 1$ .",
"Since $\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}:\\mathcal {A}\\otimes _{\\min }\\mathcal {A}\\rightarrow \\mathcal {A}^{**}\\otimes _{\\min }\\mathcal {A}^{**}\\subseteq \\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**}$ is a complete isometry, we may take a norm preserving Hahn–Banach extension $\\tilde{F}\\in (\\mathcal {A}^{**}\\overline{\\otimes }\\mathcal {A}^{**})^*=(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)^{**}$ satisfying $\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\langle F,A\\rangle , \\ \\ \\ A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}.$ By Goldstine's theorem, there exists a net $(f_j)$ in $(\\mathcal {A}^*\\widehat{\\otimes }\\mathcal {A}^*)_{\\Vert \\cdot \\Vert \\le 1}$ such that $f_j\\rightarrow \\tilde{F}$ weak*.",
"Thus, for all $A\\in \\mathcal {A}\\otimes _{\\min }\\mathcal {A}$ , $\\langle F,A\\rangle =\\langle \\tilde{F},\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle f_j,\\pi _\\mathcal {A}\\otimes \\pi _\\mathcal {A}(A)\\rangle =\\lim _j\\langle \\iota (f_j),A\\rangle .$ Theorem 6.2 Let $$ be a unimodular discrete quantum group with Kirchberg's factorization property.",
"Then $$ is weakly amenable if and only if $L^1_{cb}(\\mathbb {G})$ is operator amenable.",
"Moreover, $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))\\le \\Lambda _{cb}()^2$ .",
"It is clear that $\\Lambda _{cb}()\\le OA(L^1_{cb}(\\mathbb {G}))$ , as any operator amenable Banach algebra admits a bounded approximate identity with pertinent control over the norm [54].",
"Conversely, by Kirchberg's factorization property the representation $\\lambda \\times \\rho $ factors through $C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G})$ .",
"Composing with $\\omega _{\\Lambda _\\varphi (1)}$ , we obtain a state $\\mu :=\\omega _{\\Lambda _\\varphi (1)}\\circ \\lambda \\times \\rho \\in (C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ .",
"By $R$ -invariance of $\\varphi $ , one can easily verify that $\\mu =\\mu \\circ \\Sigma $ .",
"Let $\\Gamma _u:C_u(\\mathbb {G})\\rightarrow C_u(\\mathbb {G})\\otimes _{\\min } C_u(\\mathbb {G})$ denote the universal co-multiplication.",
"Then, similar to the calculations in [55], for all $u\\in \\mathrm {Irr}(\\mathbb {G})$ , $1\\le i,j\\le n_u$ , $\\langle \\Gamma _u^*(\\mu )\\star f,u_{ij}\\rangle &=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\langle \\mu ,u_{ik}\\otimes u_{kl}\\rangle =\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} R(u_{kl}))\\\\&=\\sum _{k,l=1}^{n_u}\\langle f,u_{lj}\\rangle \\varphi (u_{ik} u_{lk}^{*})=\\frac{1}{n_u}\\sum _{k=1}^{n_u}\\langle f,u_{ij}\\rangle \\\\&=\\langle f,u_{ij}\\rangle .$ Hence, $\\Gamma _u^*(\\mu )\\star f = f$ for all $f\\in L^1(\\mathbb {G})$ .",
"By [42], $R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x^*)(1\\otimes y)))=(\\textnormal {id}\\otimes \\varphi )((1\\otimes x^*)\\Gamma _u(y))$ for all $x,y\\in C_u(\\mathbb {G})$ , so that $\\langle (f\\otimes 1)\\star \\mu ,x\\otimes y\\rangle &=\\varphi ((x\\star f) R(y))=(f\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))\\\\&=f\\circ R(R((\\textnormal {id}\\otimes \\varphi )(\\Gamma _u(x)(1\\otimes R(y)))))\\\\&=f\\circ R((\\textnormal {id}\\otimes \\varphi )((1\\otimes x)\\Gamma _u(R(y))))\\\\&=(f\\circ R\\otimes \\varphi )((1\\otimes x)\\Sigma \\circ R\\otimes R\\circ \\Gamma _u(y))\\\\&=(\\varphi \\otimes f)(\\Gamma _u(y)(R(x)\\otimes 1))\\\\&=\\varphi ((f\\star y)R(x))\\\\&=\\langle \\mu \\star (1\\otimes f),x\\otimes y\\rangle .$ Thus, $(f\\otimes 1)\\star \\mu =\\mu \\star (1\\otimes f)$ for all $f\\in L^1(\\mathbb {G})$ .",
"Since $\\mu =\\mu \\circ \\Sigma $ , we also have $(1\\otimes f)\\star \\mu =\\mu \\star (f\\otimes 1)$ , for all $f\\in L^1(\\mathbb {G})$ .",
"It follows that $F\\star \\mu =\\mu \\star \\Sigma F$ for all $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ , where we let $\\Sigma $ also denote the flip homomorphism on $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"The proof of Lemma REF implies the existence of a net $(\\mu _i)$ of states in $C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*$ such that $\\mu _i\\rightarrow \\mu $ weak* in $(C_u(\\mathbb {G})\\otimes _{\\min }C_u(\\mathbb {G}))^*$ , and hence in $(C_u(\\mathbb {G})\\otimes _{\\max }C_u(\\mathbb {G}))^*=C_u(\\mathbb {G}\\times \\mathbb {G})^*$ .",
"Since $m_{C_u(\\mathbb {G})^*}=\\Gamma _u^*|_{C_u(\\mathbb {G})^*\\widehat{\\otimes }C_u(\\mathbb {G})^*}$ , it follows that $m_{C_u(\\mathbb {G})^*}(\\mu _i)\\rightarrow \\Gamma _u^*(\\mu )$ weak* in $C_u(\\mathbb {G})^*$ .",
"By [57] (note that $m_{C_u(\\mathbb {G})^*}(\\mu _i),\\Gamma _u^*(\\mu )$ are states), we have $\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-f\\Vert _{L^1(\\mathbb {G})}=\\Vert m_{C_u(\\mathbb {G})^*}(\\mu _i)\\star f-\\Gamma _u^*(\\mu )\\star f\\Vert _{L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ f\\in L^1(\\mathbb {G}).$ Since $\\mu _i\\rightarrow \\mu $ weak* in $C_u(\\mathbb {G}\\times \\mathbb {G})^*$ , again by [57] we obtain $\\Vert F\\star \\mu _i-\\mu _i\\star \\Sigma F\\Vert _{L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})}\\rightarrow 0, \\ \\ \\ F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G}),$ from which we have $&\\Vert F\\star ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\Vert =\\Vert (F\\star (1\\otimes f))\\star \\mu _i-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-(\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)\\Vert +\\Vert (\\mu _i\\star \\Sigma F)\\star (f\\otimes 1)-F\\star \\mu _i\\star (f\\otimes 1)\\Vert \\\\&\\le \\Vert (F\\star (1\\otimes f))\\star \\mu _i-\\mu _i\\star (\\Sigma (F\\star (1\\otimes f)))\\Vert +\\Vert \\mu _i\\star \\Sigma F-F\\star \\mu _i\\Vert \\Vert f\\otimes 1\\Vert \\\\&\\rightarrow 0$ for every $f\\in L^1(\\mathbb {G})$ , $F\\in L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ .",
"Similarly, $\\Vert ((1\\otimes f)\\star \\mu _i-\\mu _i\\star (f\\otimes 1))\\star F\\Vert \\rightarrow 0$ .",
"Now, if $$ is weakly amenable, then by [41] there exists an approximate identity $(f_j)$ for $L^1(\\mathbb {G})$ in $\\mathcal {Z}(L^1(\\mathbb {G}))$ such that $\\sup _j\\Vert f_j\\Vert _{cb}<\\infty $ .",
"The tensor square of the canonical complete contraction $C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ allows us to view $\\mu _i\\in M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))$ with $\\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1$ for all $i$ .",
"By the universal property of the operator space projective tensor product, we may also view each $\\mu _i$ as an element of $\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))$ by right multiplication, as well as in $\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))$ by left multiplication.",
"Moreover, $\\Vert \\mu _i\\Vert _{\\mathcal {CB}(M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }L^1_{cb}(\\mathbb {G}))},\\Vert \\mu _i\\Vert _{\\mathcal {CB}(L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G})))}\\le \\Vert \\mu _i\\Vert _{ M_{cb}^l(L^1(\\mathbb {G}))\\widehat{\\otimes }M_{cb}^l(L^1(\\mathbb {G}))}\\le 1.$ Define $\\mu _{ij}:=(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)$ .",
"Then $\\mu _{ij}\\in L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _{ij}\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Vert f_j\\Vert _{cb}^2\\le \\Lambda _{cb}()^2.$ Given $f\\in L^1(\\mathbb {G})$ , for each $j$ we have $\\lim _i f\\star \\mu _{ij}-\\mu _{ij}\\star f&=\\lim _i(f\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1)-(1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f)\\\\&=\\lim _i(f\\otimes f_j^2)\\star \\mu _i-\\mu _i\\star (f_j^2\\otimes f)\\\\&=0$ in $L^1(\\mathbb {G})\\widehat{\\otimes }L^1(\\mathbb {G})$ and therefore in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ .",
"Furthermore, $\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}(\\mu _{ij})\\star f&=\\lim _j \\lim _i m_{L^1_{cb}(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes 1))\\star f\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}((1\\otimes f_j)\\star \\mu _i\\star (f_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{L^1(\\mathbb {G})}(\\mu _i\\star (f^2_j\\otimes f))\\\\&=\\lim _j \\lim _i m_{C_u(\\mathbb {G})^*}(\\mu _i)m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=\\lim _j m_{L^1(\\mathbb {G})}(f^2_j\\otimes f)\\\\&=f,$ where the 4th equality follows from the fact that $f_j\\in \\mathcal {Z}(L^1(\\mathbb {G}))$ .",
"Combining the iterated limit into a single net as in Proposition REF , we obtain a bounded approximate diagonal $(\\mu _I)$ in $L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})$ with $\\Vert \\mu _I\\Vert _{L^1_{cb}(\\mathbb {G})\\widehat{\\otimes }L^1_{cb}(\\mathbb {G})}\\le \\Lambda _{cb}()^2$ .",
"Remark 6.3 There is a corresponding statement for the closure of $L^1(\\mathbb {G})$ in $M_{cb}^r(L^1(\\mathbb {G}))$ .",
"It is proved in the exact same way using the fact that $f\\star \\Gamma _u^*(\\mu )=f$ for all $f\\in L^1(\\mathbb {G})$ , which is easily verified.",
"Examples 6.4 ${}$ It was shown in [29] that $A_{cb}(G)$ is operator amenable for any weakly amenable discrete group $G$ such that $C^*(G)$ is residually finite-dimensional.",
"There are examples of weakly amenable residually finite groups (e.g.",
"$G=SL(2,\\mathbb {Z}[\\sqrt{2}])$ ) for which $C^*(G)$ is not residually finite-dimensional [6].",
"Since residually finite groups have Kirchberg's factorization property, Theorem REF is new even for this class of discrete groups.",
"When $$ is an amenable unimodular discrete quantum group, we recover Ruan's result on the operator amenability of $L^1(\\mathbb {G})=L^1_{cb}(\\mathbb {G})$ [55].",
"Using results from [10] and [11], it was shown in [7] that the discrete duals $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ of the free orthogonal and unitary quantum groups have Kirchberg's factorization property for $N\\ne 3$ .",
"Since $\\widehat{O_N^+}$ and $\\widehat{U_N^+}$ are always weakly amenable with Cowling–Haagerup constant 1 [31], we have $OA(L^1_{cb}(O_N^+))=OA(L^1_{cb}(U_N^+))=1$ for all $N\\ne 3$ .",
"If $\\mathbb {G}_1$ and $\\mathbb {G}_2$ are compact quantum groups with Kirchberg's factorization property and $\\Lambda _{cb}()=\\Lambda _{cb}()=1$ , then $\\mathbb {G}=\\mathbb {G}_1\\ast \\mathbb {G}_2$ also has the factorization property [11] and $\\Lambda _{cb}()=1$ [32], so $OA(L^1_{cb}(\\mathbb {G}))=1$ .",
"Remark 6.5 It would be interesting to find an example of a unimodular discrete quantum group $$ with Kirchberg's factorization property for which equality of the constants in Theorem REF does not hold."
],
[
"Decomposability",
"For a locally compact group $G$ , it is well-known that $B(G)=M_{cb}A(G)$ completely isometrically whenever $G$ is amenable [25].",
"We now generalize this implication to arbitrary locally compact quantum groups.",
"Moreover, we show that the corresponding complete isometry is a weak*-weak* homeomorphism.",
"By [42], the universal co-representation $\\W _{\\mathbb {G}}=(\\textnormal {id}\\otimes \\pi _{})(\\mathbb {W}_{\\mathbb {G}})\\in M(C_u(\\mathbb {G})\\otimes _{\\min } C_0())$ from the proof of Proposition REF satisfies $\\hat{\\lambda }_u(f̉)=(\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}^*)$ for all $f̉\\in L^1_*(\\widehat{\\mathbb {G}})$ , $(\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}$ for all $x\\in C_u(\\mathbb {G})$ , where $\\hat{\\lambda }_u$ is the embedding of $L^1_*(\\widehat{\\mathbb {G}})$ into $C_u(\\mathbb {G})$ , and $\\pi _{\\mathbb {G}}:C_u(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ is the (unique) extension of $\\hat{\\lambda }:L^1_*(\\widehat{\\mathbb {G}})\\rightarrow C_0(\\mathbb {G})$ [42].",
"Moreover, $C_u(\\mathbb {G})=\\overline{\\text{span}\\lbrace (\\textnormal {id}\\otimes f̉)(\\W _{\\mathbb {G}}) : f̉\\in L^1(\\widehat{\\mathbb {G}})\\rbrace }^{\\Vert \\cdot \\Vert _u}.$ We will need the following representation of $\\Theta ^l(\\mu )$ for $\\mu \\in C_u(\\mathbb {G})^*$ , which may be found in [19].",
"We present the proof for the convenience of the reader.",
"Lemma 7.1 For $\\mu \\in C_u(\\mathbb {G})^*$ , $\\Theta ^l(\\mu )(x)=(\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ First let $x=\\hat{\\lambda }(\\hat{f})\\in C_0(\\mathbb {G})$ for some $\\hat{f}\\in L^1_*(\\widehat{\\mathbb {G}})$ .",
"Then, for all $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu )(x),f\\rangle &=\\langle x,m^l_\\mu (f)\\rangle =\\langle \\hat{\\lambda }(\\hat{f}),\\mu \\star f\\rangle \\\\&=\\langle \\pi _{\\mathbb {G}}\\circ \\hat{\\lambda }_u(\\hat{f}),\\mu \\star f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\\\&=\\langle \\Gamma _u(\\hat{\\lambda }_u(\\hat{f})),\\mu \\otimes \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle (\\textnormal {id}\\otimes \\pi _{\\mathbb {G}})(\\Gamma _u(\\hat{\\lambda }_u(\\hat{f}))),\\mu \\otimes f\\rangle \\\\&=\\langle \\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(\\hat{\\lambda }_u(\\hat{f})))\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle =\\langle \\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},\\mu \\otimes f\\rangle \\\\&=\\langle (\\mu \\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}},f\\rangle .$ As $\\hat{\\lambda }(L^1_*(\\widehat{\\mathbb {G}}))$ is norm dense in $C_0(\\mathbb {G})$ , and since $C_0(\\mathbb {G})$ is weak* dense in $L^{\\infty }(\\mathbb {G})$ , the result follows.",
"Recall that $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is the map taking $\\mu \\in C_u(\\mathbb {G})^*$ to the operator of left multiplication by $\\mu $ on $L^1(\\mathbb {G})$ .",
"Theorem 7.2 Let $\\mathbb {G}$ be a locally compact quantum group.",
"If $$ is amenable then $\\tilde{\\lambda }:C_u(\\mathbb {G})^*\\rightarrow M_{cb}^l(L^1(\\mathbb {G}))$ is a weak*–weak* homeomorphic completely isometric isomorphism.",
"Amenability of $$ entails the surjectivity of $\\tilde{\\lambda }$ from (the left version of) [14].",
"For simplicity, throughout the proof we denote by $\\Theta ^l(\\mu )$ the map $\\Theta ^l(\\tilde{\\lambda }(\\mu ))$ for $\\mu \\in C_u(\\mathbb {G})^*$ .",
"In [19] Daws shows that $\\Theta ^l:C_u(\\mathbb {G})^*_+\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is an order bijection.",
"We show that it is a complete order bijection.",
"To this end, let $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"By Lemma REF $\\Theta ^l(\\mu _{ij})(x)=(\\textnormal {id}\\otimes \\mu _{ij})\\W _{\\mathbb {G}}^*(1\\otimes x)\\W _{\\mathbb {G}}, \\ \\ \\ x\\in L^{\\infty }(\\mathbb {G}).$ Thus, for any $x_1,..,x_m\\in L^{\\infty }(\\mathbb {G})$ we have $((\\Theta ^l)^n([\\mu _{ij}]))^m([x_k^*x_l])&=[(\\mu _{ij}\\otimes \\textnormal {id})\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}]\\\\&=[(\\mu _{ij}\\otimes \\textnormal {id})]^m([\\W _{\\mathbb {G}}^*(1\\otimes x_k^*x_l)\\W _{\\mathbb {G}}])\\ge 0.$ It follows that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ .",
"On the other hand, suppose $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)$ such that $(\\Theta ^l)^n([\\mu _{ij}])\\in \\mathcal {CP}(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Let $P̉$ denote the positive operator implementing the scaling group $\\hat{\\tau }$ on $$ , via $\\hat{\\tau }_t(\\hat{x})=P̉^{it}\\hat{x}P̉^{-it}$ , $\\hat{x}\\in L^{\\infty }(\\widehat{\\mathbb {G}})$ .",
"Using [20], for $\\xi _1,...\\xi _m\\in \\mathcal {D}(P̉^{1/2})$ , $\\eta _1,...,\\eta _m\\in \\mathcal {D}(P̉^{-1/2})$ , and $[z_{ik}]\\in {nm}$ , $\\langle [\\mu _{ij}]^m([\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})])[z_{ik}],[z_{ik}]\\rangle &=\\sum _{i,j=1}^n\\sum _{k,l=1}^m\\overline{z_{ik}}z_{jl}\\langle \\mu _{ij},\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\rangle \\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\mu _{ij}^*,\\lambda _u(\\omega _{\\xi _l,\\eta _l})\\lambda _u(\\omega _{\\xi _k,\\eta _k})^*\\rangle }\\\\&=\\overline{\\sum _{i,j=1}^n\\sum _{k,l=1}^mz_{ik}\\overline{z_{jl}}\\langle \\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)\\eta _k,\\eta _l\\rangle }\\\\&=\\overline{\\sum _{k,l=1}^m \\langle [\\Theta ^l(\\mu _{ij})(\\xi _l\\xi _k^*)]y_k,y_l\\rangle }\\ge 0,$ where $y_k=[z_{1k}\\eta _k \\ \\cdots \\ z_{nk}\\eta _k]^T\\in L^2(\\mathbb {G})^n$ for $1\\le k\\le m$ .",
"By density of $\\lbrace \\omega _{\\xi ,\\eta }\\mid \\xi \\in \\mathcal {D}(P̉^{1/2}), \\eta \\in \\mathcal {D}(P̉^{-1/2})\\rbrace $ in $L^1_*(\\widehat{\\mathbb {G}})$ [21], it follows that $[\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+$ .",
"We now show that $\\tilde{\\lambda }$ is a complete isometry.",
"To do so we introduce a decomposability norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , given by $\\Vert \\Phi \\Vert _{L^1dec}:=\\inf \\bigg \\lbrace \\max \\lbrace \\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}\\rbrace \\mid \\begin{bmatrix}\\Psi _1 & \\Phi \\\\ \\Phi ^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0\\bigg \\rbrace ,$ where $\\Psi _i\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"It is evident that $\\Vert \\cdot \\Vert _{L^1dec}$ is a norm on $_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ .",
"That $\\Vert \\Phi \\Vert _{cb}\\le \\Vert \\Phi \\Vert _{L^1dec}$ for all $\\Phi \\in \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ follows verbatim from the first part of [28].",
"In a similar fashion we obtain a decomposable norm on $M_n(_{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}))= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G}))).$ Since $\\Theta ^l$ is a completely positive contraction from $(C_u(\\mathbb {G})^*)^+$ onto $_{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}))$ , one easily sees that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}\\le \\Vert [\\mu _{ij}]\\Vert _{dec}, \\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*),$ where $\\Vert \\cdot \\Vert _{dec}$ is the standard decomposable norm for maps between $C^*$ -algebras.",
"Conversely, if $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{dec}<1$ then there exist $\\Psi _1,\\Psi _2\\in \\ _{L^1(\\mathbb {G})}\\mathcal {CP}^{\\sigma }(L^{\\infty }(\\mathbb {G}),M_n(L^{\\infty }(\\mathbb {G})))$ such that $\\Vert \\Psi _1\\Vert _{cb},\\Vert \\Psi _2\\Vert _{cb}<1$ , and $\\begin{bmatrix}\\Psi _1 & (\\Theta ^l)^n([\\mu _{ij}])\\\\ (\\Theta ^l)^n([\\mu _{ij}])^* & \\Psi _2\\end{bmatrix}\\ge _{cp}0.$ Since $(\\Theta ^l)^n$ is a complete order bijection there exist $[\\nu ^k_{ij}]\\in M_n(C_u(\\mathbb {G})^*)^+=\\mathcal {CP}(C_u(\\mathbb {G}),M_n)$ such that $\\Psi _k=(\\Theta ^l)^n([\\nu ^k_{ij}])$ , $k=1,2$ , and $\\begin{bmatrix}[\\nu ^1_{ij}] & [\\mu _{ij}]\\\\ [\\mu _{ij}]^* & [\\nu ^2_{ij}]\\end{bmatrix}\\ge _{cp}0.$ It follows that $[\\nu ^k_{ij}]$ is a strictly continuous completely positive map $C_u(\\mathbb {G})\\rightarrow M_n$ , and therefore admits a unique extension to a completely positive map $\\widetilde{[\\nu ^k_{ij}]}:M(C_u(\\mathbb {G}))\\rightarrow M_n$ which is strictly continuous on the unit ball [45].",
"By uniqueness, $\\widetilde{[\\nu ^k_{ij}]}=[\\tilde{\\nu }^k_{ij}]$ , where $\\tilde{\\nu }^k_{ij}$ is the unique strict extension of the functional $\\nu ^k_{ij}$ .",
"Thus, by completely positivity $\\Vert [\\nu ^k_{ij}] \\Vert _{cb}=\\Vert \\widetilde{[\\nu ^k_{ij}]}(1_{M(C_u(\\mathbb {G}))})\\Vert =\\Vert [\\tilde{\\nu }^k_{ij}(1_{M(C_u(\\mathbb {G}))})]\\Vert =\\Vert \\Psi _k(1)\\Vert =\\Vert \\Psi _k\\Vert _{cb}<1,$ so that $\\Vert [\\mu _{ij}]\\Vert _{dec}<1$ .",
"Therefore $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{dec}.$ However, $\\Vert [\\mu _{ij}]\\Vert _{dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}$ by injectivity of $M_n$ (see [28]), so that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}, ,\\ \\ \\ [\\mu _{ij}]\\in M_n(C_u(\\mathbb {G})^*).$ Now, amenability of $$ entails the the 1-injectivity of $L^{\\infty }(\\mathbb {G})$ in $L^1(\\mathbb {G})\\hspace{2.0pt}\\mathbf {mod}$ by the left version of [14].",
"The matricial analogues of the proofs of [14] show that $\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{cb}=\\Vert (\\Theta ^l)^n([\\mu _{ij}])\\Vert _{L^1dec}=\\Vert [\\mu _{ij}]\\Vert _{cb}.$ Hence, $\\Theta ^l:C_u(\\mathbb {G})^*\\rightarrow \\ _{L^1(\\mathbb {G})}\\mathcal {CB}^\\sigma (L^{\\infty }(\\mathbb {G}))$ is a completely isometric isomorphism.",
"To prove that $\\Theta ^l$ is a weak*-weak* homeomorphism, it suffices to show that it is weak* continuous on bounded sets.",
"Let $(\\mu _i)$ be a bounded net in $C_u(\\mathbb {G})^*$ converging weak* to $\\mu $ .",
"Since $C_u(\\mathbb {G})^*$ is a dual Banach algebra [18], multiplication is separately weak* continuous.",
"Hence, for $\\hat{f}\\in L^1(\\widehat{\\mathbb {G}})$ and $f\\in L^1(\\mathbb {G})$ , $\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle =\\langle \\hat{\\lambda }_u(\\hat{f}),\\mu _i\\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle \\rightarrow \\langle \\hat{\\lambda }_u(\\hat{f}),\\mu \\star _u \\pi _{\\mathbb {G}}^*(f)\\rangle =\\langle \\Theta ^l(\\mu _i)(\\hat{\\lambda }(\\hat{f})),f\\rangle .$ The density of $\\hat{\\lambda }(L^1(\\widehat{\\mathbb {G}}))$ in $C_0(\\mathbb {G})$ and boundedness of $\\Theta ^l(\\mu _i)$ in $_{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))=(Q_{cb}^l(\\mathbb {G}))^*$ (see [14]) establish the claim.",
"Remark 7.3 We note that the conclusion of Theorem REF was obtained under the a priori stronger assumption that $\\mathbb {G}$ is co-amenable [35].",
"Corollary 7.4 Let $\\mathbb {G}$ be a locally compact quantum group such that $$ has the approximation property.",
"Then $\\mathbb {G}$ is co-amenable if and only if $$ is amenable.",
"Assuming $$ has the AP, there exists a stable approximate identity $(f_i)$ for $L^1(\\mathbb {G})$ .",
"Moreover, as noted in the proof of Proposition REF , $L^{\\infty }(\\mathbb {G})$ has the w*OAP, and therefore the dual slice map property (see [28]).",
"Any operator space is a complete quotient of the space of trace class operators for some Hilbert space [8], so let $H$ be a Hilbert space such that $\\mathcal {T}(H)\\twoheadrightarrow C_0(\\mathbb {G})$ .",
"Then $L^1(\\mathbb {G})\\widehat{\\otimes }\\mathcal {T}(H)\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ by projectivity of $\\widehat{\\otimes }$ , and $L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})\\overline{\\otimes }\\mathcal {B}(H)$ is a weak*-weak* continuous complete isometry.",
"Hence, $\\Theta ^l(f_i)\\otimes \\textnormal {id}_{M(\\mathbb {G})}(X)\\rightarrow X,$ weak* for all $X\\in L^{\\infty }(\\mathbb {G})\\overline{\\otimes }M(\\mathbb {G})$ , so that $\\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A)\\rightarrow A$ weakly for all $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"By the standard convexity argument, we may assume that the net $(f_i)$ satisfies $\\Vert \\Theta ^l(f_i)_*\\otimes \\textnormal {id}_{C_0(\\mathbb {G})}(A) - A\\Vert \\rightarrow 0, \\ \\ \\ A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G}).$ Consider the multiplication map $m:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ .",
"Let $\\tilde{m}:L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow C_0(\\mathbb {G})$ denote the induced map and $q:L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ denote the quotient map.",
"It follows that $q(f\\star A)=q(f\\otimes m(A))$ for all $f\\in L^1(\\mathbb {G})$ and $A\\in L^1(\\mathbb {G})\\widehat{\\otimes }C_0(\\mathbb {G})$ .",
"Thus, if $m(A)=0$ , then $q(A)=\\lim _i q(f_i\\star A)=\\lim _iq(f_i\\otimes m(A))=0,$ so that the induced multiplication $\\tilde{m}$ is injective.",
"Now, assuming $$ is amenable, Theorem REF implies that $C_u(\\mathbb {G})^*\\cong \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G}))$ completely isometrically and weak*-weak* homeomorphically, that is, $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ completely isometrically.",
"By the left version of [14] $L^1(\\mathbb {G})$ is 1-flat in $\\mathbf {mod}\\hspace{2.0pt}L^1(\\mathbb {G})$ .",
"Thus, the following sequence is 1-exact $0\\rightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})\\hookrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\twoheadrightarrow L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})\\rightarrow 0.$ Since $f\\star x=(\\textnormal {id}\\otimes f\\circ \\pi _{\\mathbb {G}})\\circ \\Gamma _u(x)=(\\textnormal {id}\\otimes f)\\W _{\\mathbb {G}}^*(1\\otimes \\pi _{\\mathbb {G}}(x))\\W _{\\mathbb {G}}=0$ for all $x\\in \\mathrm {Ker}(\\pi _{\\mathbb {G}})$ , it follows that $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}\\mathrm {Ker}(\\pi _{\\mathbb {G}})=0$ .",
"Hence, $L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})$ .",
"Moreover, as $\\Theta ^l(\\mu \\star f)(x)=\\Theta ^l(f)(\\Theta ^l(\\mu )(x))=(\\Theta ^l(\\mu )(x))\\star f, \\ \\ \\ f\\in L^1(\\mathbb {G}), \\ \\mu \\in C_u(\\mathbb {G})^*, \\ x\\in C_0(\\mathbb {G}),$ it follows that $C_u(\\mathbb {G})\\cong L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})} C_0(\\mathbb {G})$ is an isomorphism of left $L^1(\\mathbb {G})$ -modules, i.e., $C_u(\\mathbb {G})$ is induced.",
"The commutative diagram $\\begin{tikzcd}L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_u(\\mathbb {G}) [r, equal][d, equal] &L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G})[d, \"\\tilde{m}\"]\\\\C_u(\\mathbb {G}) [r, two heads] &C_0(\\mathbb {G})\\end{tikzcd}$ then implies that $\\tilde{m}$ is a complete quotient map.",
"Thus, $C_0(\\mathbb {G})$ is an induced $L^1(\\mathbb {G})$ -module and $M(\\mathbb {G})\\cong (L^1(\\mathbb {G})\\widehat{\\otimes }_{L^1(\\mathbb {G})}C_0(\\mathbb {G}))^*= \\ _{L^1(\\mathbb {G})}\\mathcal {CB}(C_0(\\mathbb {G}),L^{\\infty }(\\mathbb {G})).$ The measure corresponding to the inclusion $C_0(\\mathbb {G})\\hookrightarrow L^{\\infty }(\\mathbb {G})$ is necessarily a left unit for $M(\\mathbb {G})$ , which entails the co-amenability of $\\mathbb {G}$ ."
],
[
"Acknowledgements",
"The author would like to thank Michael Brannan and Ami Viselter for helpful discussions at various points during this project, as well as the anonymous referee whose valuable comments significantly improved the presentation of the paper.",
"The author was partially supported by the NSERC Discovery Grant 1304873."
]
] | 1709.01770 |
[
[
"Puiseux monoids and transfer homomorphisms"
],
[
"Abstract There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades.",
"The factorization theory of finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most systematically studied.",
"Puiseux monoids, which are additive submonoids of $\\mathbb{Q}_{\\ge 0}$ consisting of nonnegative rational numbers, have only been studied recently.",
"In this paper, we provide evidence that this family comprises plenty of monoids with a basically unexplored atomic structure.",
"We do this by showing that the arithmetical invariants of the well-studied atomic monoids mentioned earlier cannot be transferred to most Puiseux monoids via homomorphisms that preserve atomic configurations, i.e., transfer homomorphisms.",
"Specifically, we show that transfer homomorphisms from a non-finitely generated atomic Puiseux monoid to a finitely generated monoid do not exist.",
"We also find a large family of Puiseux monoids that fail to be strongly primary.",
"In addition, we prove that the only nontrivial Puiseux monoid that accepts a transfer homomorphism to a Krull monoid is $\\mathbb{N}_0$.",
"Finally, we classify the Puiseux monoids that happen to be C-monoids."
],
[
"Introduction",
"The study of the phenomenon of non-unique factorizations in the ring of integers $\\mathcal {O}_K$ of an algebraic number field $K$ was initiated by L. Carlitz in the 1950's, and it was later carried out on more general integral domains.",
"As a result, many techniques to measure the non-uniqueness of factorizations in several families of integral domains were systematically developed during the second half of the last century (see [2] and references therein).",
"However, it was not until recently that questions about the non-uniqueness of factorizations were abstractly formulated in the context of commutative cancellative monoids.",
"This was possible because most of the factorization-related questions inside an integral domain are purely multiplicative in essence.",
"The fundamental goal of abstract (or modern) factorization theory is to measure how far is a commutative cancellative monoid from being factorial by using different arithmetical invariants.",
"At this point, the arithmetical invariants of several families of atomic monoids have been intensively studied.",
"Finitely generated monoids, strongly primary monoids, Krull monoids, and C-monoids are among the most studied.",
"These families of monoids not only have very diverse arithmetical properties, but also have proved to be useful in the study of the factorization theory of less-understood atomic monoids via transfer homomorphisms.",
"A monoid homomorphism is said to be transfer if somehow it allows to shift the atomic structure of its codomain back to its domain (see Definition REF ).",
"Therefore if one is willing to know the factorization invariants of a given monoid, it suffices to find a transfer homomorphism from such a monoid to a better-understood monoid and carry over the desired factorization properties.",
"Puiseux monoids were recently introduced as a rational generalization of numerical monoids.",
"Many families of atomic Puiseux monoids were explored in [16], [17], [19], and their elasticity was studied in [20].",
"However, it is still unanswered whether the non-unique factorization behavior in Puiseux monoids is somehow similar to that of some of the monoids whose factorization properties are already well-understood.",
"To give a partial answer to this, we will determine which atomic Puiseux monoids can be the domain of a transfer homomorphism to some of the monoids whose arithmetical invariants have already been studied.",
"In particular, we consider finitely generated monoids, Krull monoids, and C-monoids as our transfer codomains.",
"The content of this paper is organized as follows.",
"In Section , we establish the notation we shall be using later, and we formally present most of the fundamental concepts needed in this paper.",
"Then, in Section , we show that homomorphisms between Puiseux monoids can only be given by rational multiplication, which will allow us to characterize the transfer homomorphisms between Puiseux monoids.",
"We also present a family of Puiseux monoids whose members have $\\mathbb {Z}$ as their group of automorphisms.",
"Section is devoted to characterize the Puiseux monoids admitting a transfer homomorphism to some finitely generated monoid.",
"Then, in Section we investigate which Puiseux monoids are strongly primary.",
"Finally, in Section , we prove that the only Puiseux monoid that is transfer Krull is the additive monoid $\\mathbb {N}_0$ .",
"We use this information to classify the Puiseux monoids which happen to be C-monoids."
],
[
"Background",
"To begin with let us introduce the fundamental concepts related to our exposition as an excuse to establish the notation we need.",
"The reader can consult Grillet [21] for information on commutative semigroups and Geroldinger and Halter-Koch [10] for extensive background in non-unique factorization theory of atomic monoids.",
"Throughout this sequel, we let $\\mathbb {N}$ denote the set of positive integers, and we set $\\mathbb {N}_0 := \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .",
"For $X \\subseteq \\mathbb {R}$ and $r \\in \\mathbb {R}$ , we set $X_{\\le r} := \\lbrace x \\in X \\mid x \\le r\\rbrace $ ; with a similar spirit we use the symbols $X_{\\ge r}$ , $X_{< r}$ , and $X_{> r}$ .",
"If $q \\in \\mathbb {Q}_{> 0}$ , then we call the unique $a,b \\in \\mathbb {N}$ such that $q = a/b$ and $\\gcd (a,b)=1$ the numerator and denominator of $q$ and denote them by $\\mathsf {n}(q)$ and $\\mathsf {d}(q)$ , respectively.",
"For each subset $Q$ of $\\mathbb {Q}_{>0}$ , we call the sets $\\mathsf {n}(Q) = \\lbrace \\mathsf {n}(q) \\mid q \\in Q\\rbrace $ and $\\mathsf {d}(Q) = \\lbrace \\mathsf {d}(q) \\mid q \\in Q\\rbrace $ the numerator set and denominator set of $Q$ , respectively.",
"As usual, a semigroup is a pair $(S, *)$ , where $S$ is a set and $*$ is an associative binary operation in $S$ ; we write $S$ instead of $(S,*)$ provided that $*$ is clear from the context.",
"However, inside the scope of this paper, a monoid is a commutative cancellative semigroup with identity (cf.",
"the standard definition of monoid).",
"To comply with established conventions, we will be using simultaneously additive and multiplicative notations; however, the context will always save us from the risk of ambiguity.",
"Let $M$ be a monoid written additively.",
"We set $M^\\bullet := M \\!",
"\\setminus \\!",
"\\lbrace 0\\rbrace $ and, as usual, we let $M^\\times $ denote the set of units (i.e., invertible elements) of $M$ .",
"The monoid $M$ is reduced if $M^\\times = \\lbrace 0\\rbrace $ .",
"For $a, b \\in M$ , we say that $a$ divides $b$ in $M$ if there exists $c \\in M$ such that $b = a + c$ ; in this case we write $a \\mid _M b$ .",
"An element $a \\in M \\!",
"\\setminus \\!",
"M^\\times $ is an atom if whenever $a = u + v$ for some $u,v \\in M$ , either $u \\in M^\\times $ or $v \\in M^\\times $ .",
"Atoms are the building blocks in factorization theory; this motivates the especial notation $\\mathcal {A}(M) := \\lbrace a \\in M \\mid a \\text{ is an atom of } M\\rbrace .$ For $S \\subseteq M$ , we let $\\langle S \\rangle $ denote the smallest submonoid of $M$ containing $S$ , and we say that $S$ generates $M$ if $M = \\langle S \\rangle $ .",
"The monoid $M$ is said to be finitely generated if it can be generated by a finite set.",
"On the other hand, we say that $M$ is atomic if $M = \\langle \\mathcal {A}(M) \\rangle $ .",
"A monoid is factorial if every element can be written as a sum of primes.",
"As every prime is an atom, every factorial monoid is atomic.",
"Let $\\rho \\subseteq M \\times M$ be an equivalence relation on $M$ , and let $[a]_\\rho $ denote the equivalence class of $a \\in M$ .",
"We say that $\\rho $ is a congruence if for all $a,b,c \\in M$ such that $(a,b) \\in \\rho $ it follows that $(ca,cb) \\in \\rho $ .",
"Congruences are precisely the equivalence relations that are compatible with the operation of $M$ , meaning that $M/\\rho := \\lbrace [a]_\\rho \\mid a \\in M\\rbrace $ is a commutative semigroup with identity (no necessarily cancellative).",
"Two elements $a,b \\in M$ are associates, and we write $a \\simeq b$ , if $a = ub$ for some $u \\in M^\\times $ .",
"Being associates defines a congruence relation $\\simeq $ on $M$ , and $M_{\\text{red}} := M/\\!\\!\\simeq $ is called the associated reduced semigroup of $M$ .",
"We say that a multiplicative monoid $F$ is free abelian with basis $P \\subset F$ if every element $a \\in F$ can be written uniquely in the form $a = \\prod _{p \\in P} p^{\\mathsf {v}_p(a)},$ where $\\mathsf {v}_p(a) \\in \\mathbb {N}_0$ and $\\mathsf {v}_p(a) > 0$ only for finitely many elements $p \\in P$ .",
"The monoid $F$ is determined by $P$ up to canonical isomorphism, so we shall also denote $F$ by $\\mathcal {F}(P)$ .",
"By the fundamental theorem of arithmetic, the multiplicative monoid $\\mathbb {N}$ is free on the set of prime numbers.",
"In this case, we can extend $\\mathsf {v}_p$ to $\\mathbb {Q}_{\\ge 0}$ as follows.",
"For $r \\in \\mathbb {Q}_{> 0}$ let $\\mathsf {v}_p(r) := \\mathsf {v}_p(\\mathsf {n}(r)) - \\mathsf {v}_p(\\mathsf {d}(r))$ and set $\\mathsf {v}_p(0) = \\infty $ .",
"The free abelian monoid on $\\mathcal {A}(M)$ , denoted by $\\mathsf {Z}(M)$ , is called the factorization monoid of $M$ , and the elements of $\\mathsf {Z}(M)$ are called factorizations.",
"If $z = a_1 \\dots a_n \\in \\mathsf {Z}(M)$ for some $n \\in \\mathbb {N}_0$ and $a_1, \\dots , a_n \\in \\mathcal {A}(M)$ , then $n$ is the length of the factorization $z$ ; the length of $z$ is denoted by $|z|$ .",
"The unique homomorphism $\\phi \\colon \\mathsf {Z}(M) \\rightarrow M \\ \\ \\text{satisfying} \\ \\ \\phi (a) = a \\ \\ \\text{for all} \\ \\ a \\in \\mathcal {A}(M)$ is called the factorization homomorphism of $M$ .",
"Additionally, for $x \\in M^\\bullet $ , $\\mathsf {Z}(x) := \\phi ^{-1}(x) \\subseteq \\mathsf {Z}(M)$ is the set of factorizations of $x$ .",
"By definition, we set $\\mathsf {Z}(0) = \\lbrace 0\\rbrace $ .",
"Note that the monoid $M$ is atomic if and only if $\\mathsf {Z}(x)$ is not empty for all $x \\in M$ .",
"For each $x \\in M$ , the set of lengths of $x$ is defined by $\\mathsf {L}(x) := \\lbrace |z| : z \\in \\mathsf {Z}(x)\\rbrace .$ We say that the monoid $M$ is half-factorial if $|\\mathsf {L}(x)| = 1$ for all $x \\in M$ .",
"On the other hand, if $\\mathsf {L}(x)$ is a finite set for all $x \\in M$ , then we say that $M$ is a BF-monoid.",
"The system of sets of lengths of $M$ is defined by $\\mathcal {L}(M) := \\lbrace \\mathsf {L}(x) \\mid x \\in M\\rbrace .$ The system of sets of lengths is an arithmetical invariant of atomic monoids that has received significant attention in recent years (see [1], [4] and the literature cited there).",
"A very special family of atomic monoids is that one comprising all numerical monoids, cofinite submonoids of the additive monoid $\\mathbb {N}_0$ .",
"We say that a numerical monoid is proper if it is strictly contained in $\\mathbb {N}_0$ .",
"Each numerical monoid has a unique minimal set of generators, which is finite.",
"Moreover, if $\\lbrace a_1, \\dots , a_n\\rbrace $ is the minimal set of generators for a numerical monoid $N$ , then $\\mathcal {A}(N) = \\lbrace a_1, \\dots , a_n\\rbrace $ and $\\gcd (a_1, \\dots , a_n) = 1$ .",
"As a result, every numerical monoid is atomic and contains only finitely many atoms.",
"The Frobenius number of $N$ , denoted by $F(N)$ , is the minimum $n \\in \\mathbb {N}$ such that $\\mathbb {Z}_{> n} \\subset N$ .",
"An introduction to numerical monoids can be found in [7].",
"An additive submonoid of $\\mathbb {Q}_{\\ge 0}$ is called a Puiseux monoid.",
"Puiseux monoids are a natural generalization of numerical monoids.",
"However, the general atomic structure of Puiseux monoids drastically differs from that one of numerical monoids.",
"Puiseux monoids are not always atomic; for instance, consider $\\langle 1/2^n \\mid n \\in \\mathbb {N}\\rangle $ .",
"On the other hand, if an atomic Puiseux monoid $M$ is not isomorphic to a numerical monoid, then $\\mathcal {A}(M)$ is infinite.",
"The atomic structure of Puiseux monoids has been studied in [17] and [19], where several families of atomic Puiseux monoids were described."
],
[
"Homomorphisms Between Puiseux Monoids",
"In this section we present characterizations of homomorphisms and transfer homomorphisms between Puiseux monoids.",
"Let us start by introducing the concept of a transfer homomorphism, which is going to play a central role in this paper.",
"Definition 3.1 A monoid homomorphism $\\theta \\colon M \\rightarrow N$ is said to be a transfer homomorphism if the following conditions hold: (T1) $N = \\theta (M) N^\\times $ and $\\theta ^{-1}(N^\\times ) = M^\\times $ ; (T2) if $\\theta (a) = b_1 b_2$ for $a \\in M$ and $b_1,b_2 \\in N$ , then there exist $a_1, a_2 \\in M$ such that $a = a_1 a_2$ and $\\theta (a_i) = b_i$ for $i \\in \\lbrace 1,2\\rbrace $ .",
"We proceed to characterize the homomorphisms between Puiseux monoids.",
"This will immediately yield a characterization of those homomorphisms between Puiseux monoids that happen to be transfer homomorphisms.",
"Proposition 3.2 If $\\phi \\colon M \\rightarrow N$ be a homomorphism between Puiseux monoids, then the following conditions hold.",
"There exists $q \\in \\mathbb {Q}_{\\ge 0}$ such that $\\phi (x) = qx$ for all $x \\in M$ , i.e., $\\phi $ is given by rational multiplication.",
"The homomorphism $\\phi $ is a transfer homomorphism if and only if it is surjective.",
"Let us argue first that $\\phi $ is given by rational multiplication.",
"It is clear that if a map $P \\rightarrow P^{\\prime }$ between two Puiseux monoids is multiplication by a rational number, then it is a monoid homomorphism.",
"Thus, it suffices to verify that the only homomorphisms of Puiseux monoids are those given by rational multiplication.",
"To do this, consider the Puiseux monoid homomorphism $\\varphi \\colon P \\rightarrow P^{\\prime }$ .",
"Because the trivial homomorphism is multiplication by 0, there is no loss in assuming that $P \\ne \\lbrace 0\\rbrace $ .",
"Let $\\lbrace n_1, \\dots , n_k\\rbrace $ be a minimal set of generators for the additive monoid $N = P \\cap \\mathbb {N}_0$ .",
"Notice that $N \\ne \\lbrace 0\\rbrace $ and, therefore, $k \\ge 1$ .",
"The fact that $\\varphi $ is nontrivial implies that $\\varphi (n_j) \\ne 0$ for some $j \\in \\lbrace 1,\\dots ,k\\rbrace $ .",
"Set $q = \\varphi (n_j)/n_j$ , and take $r \\in P^\\bullet $ and $c_1, \\dots , c_k \\in \\mathbb {N}_0$ satisfying that $\\mathsf {n}(r) = c_1 n_1 + \\dots + c_k n_k$ .",
"Since $n_i \\varphi (n_j) = \\varphi (n_i n_j) = n_j \\varphi (n_i)$ for each $i \\in \\lbrace 1,\\dots ,k\\rbrace $ , one obtains $\\varphi (r) = \\frac{1}{\\mathsf {d}(r)} \\varphi (\\mathsf {n}(r)) = \\frac{1}{\\mathsf {d}(r)} \\sum _{i=1}^k c_i \\varphi (n_i) = \\frac{1}{\\mathsf {d}(r)} \\sum _{i=1}^k c_i n_i \\frac{\\varphi (n_j)}{n_j} = rq.$ As a result, the homomorphism $\\varphi $ is just multiplication by $q \\in \\mathbb {Q}_{>0}$ .",
"It is easy to see that condition (2) is a direct consequence of condition (1), which completes the proof.",
"Remark 3.3 A Puiseux monoid $M$ is said to be increasing (resp., decreasing) if $M$ can be generated by an increasing (resp., decreasing) sequence of rational numbers.",
"Also, we say that $M$ is bounded if $M$ can be generated by a bounded sequence of rational numbers, and it is said to be strongly bounded if it can be generated by a sequence of rational numbers whose numerator set is bounded.",
"Finally, $M$ is called dense if it contains 0 as a limit point.",
"Although the definitions just given are not algebraic in nature, we should notice that they are all preserved by Puiseux monoid isomorphisms.",
"This explains why all of them have been useful in the study of the atomic structure of Puiseux monoids (see [17] and [19]).",
"With notation as in Definition REF , when $M$ and $N$ are reduced, we can restate the first condition above as (T1') $\\theta $ is surjective and $\\theta ^{-1}(1) = 1$ .",
"We have already mentioned that a transfer homomorphism allows us to shift the atomic structure and the arithmetic of length of factorizations from its codomain to its domain.",
"This property is formally described in the following proposition.",
"Proposition 3.4 [9] If $\\theta \\colon M \\rightarrow N$ is a transfer homomorphism of atomic monoids, then the following conditions hold: $a \\in \\mathcal {A}(M)$ if and only if $\\theta (a) \\in \\mathcal {A}(N)$ ; $M$ is atomic if and only if $N$ is atomic; $\\mathsf {L}_M(x) = \\mathsf {L}_N(\\theta (x))$ for all $x \\in M$ ; $\\mathcal {L}(M) = \\mathcal {L}(N)$ , and so $M$ is a BF-monoid if and only if $N$ is a BF-monoid.",
"For a Puiseux monoid $M$ , let $\\text{Aut}(M)$ denote the group of automorphisms of $M$ .",
"As we have seen in Proposition REF , the set of homomorphisms between Puiseux monoids is very exclusive.",
"In particular, we might wonder whether $\\text{Aut}(M)$ is always trivial.",
"However, it is not hard to verify, for instance, that when $M_1 = \\langle 1/2^n \\mid n \\in \\mathbb {N}\\rangle $ , multiplication by $1/2$ is in $\\text{Aut}(M_1)$ .",
"This example might not be the most desirable because $M_1$ fails to be atomic; in fact, $M_1$ does not contain any atoms.",
"The next proposition exhibits a family of atomic monoids whose groups of automorphisms are nontrivial.",
"First, let us introduce a family of atomic Puiseux monoids whose atomicity is used in the proof.",
"For $r \\in \\mathbb {Q}_{>0}$ , the monoid $M_r = \\langle r^n \\mid n \\in \\mathbb {N}\\rangle $ is the multiplicatively $r$ -cyclic Puiseux monoid.",
"If $\\mathsf {n}(r), \\mathsf {d}(r) > 1$ , then $M_r$ is atomic with $\\mathcal {A}(M_r) = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ (see [19]).",
"Proposition 3.5 Let $r \\in \\mathbb {Q}_{>0}$ such that $\\mathsf {n}(r), \\mathsf {d}(r) > 1$ .",
"If $M = \\langle r^n \\mid n \\in \\mathbb {Z}\\rangle $ , then $\\emph {Aut}(M) \\cong \\mathbb {Z}$ .",
"Set $A = \\lbrace r^n \\mid n \\in \\mathbb {Z}\\rbrace $ .",
"For $n \\in \\mathbb {Z}$ , the fact that $r^n A = A$ implies that multiplication by $r^n$ is an endomorphism of $M$ whose inverse is given by multiplication by $r^{-n}$ .",
"Thus, multiplication by any integer power of $r$ is an automorphism of $M$ .",
"To prove that these are the only elements of $\\text{Aut}(M)$ , let us first argue that $M$ is atomic with $\\mathcal {A}(M) = A$ .",
"Assume first that $r < 1$ .",
"Fix $k \\in \\mathbb {Z}$ , and let us check that $r^k \\in \\mathcal {A}(M)$ .",
"To do this notice that the monoid $\\langle r^n \\mid n \\ge k \\rangle $ is the isomorphic image (under multiplication by $r^{k-1}$ ) of the multiplicatively $r$ -cyclic Puiseux monoid $M_r$ , which is atomic with set of atoms $A = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ .",
"Since $r \\in \\mathcal {A}(M_r)$ , it follows that $r \\notin \\langle r^n \\mid n > 1 \\rangle $ .",
"Then $r^k \\notin \\langle r^n \\mid n > k \\rangle $ .",
"As $r < 1$ , no atom in $\\lbrace r^n \\mid n < k\\rbrace $ divides $r^k$ .",
"Hence $r^k \\notin \\langle A \\setminus \\lbrace r^k\\rbrace \\rangle $ and, therefore, $r^k \\in \\mathcal {A}(M)$ .",
"As a result, $\\mathcal {A}(M) = A$ .",
"Now suppose that $r > 1$ .",
"As before, fix $k \\in \\mathbb {Z}$ .",
"Because $r > 1$ , proving that $r^k \\in \\mathcal {A}(M)$ amounts to showing that $r^k \\notin \\langle r^n \\mid n < k \\rangle $ .",
"Let us assume, by way of contradiction, that this is not the case.",
"Then $r^k = a_1 r^{n_1} + \\dots + a_t r^{n_t}$ for some $a_1, \\dots , a_t \\in \\mathbb {N}$ and $n_1, \\dots , n_t \\in \\mathbb {N}$ with $k > n_1 > \\dots > n_t$ .",
"As a consequence, $r^{k - n_t + 1} \\in \\langle r^{n_1 - n_t + 1}, r^{n_2 - n_t + 1}, \\dots , r \\rangle $ , which contradicts the fact that $r^{k - n_t + 1} \\in \\mathcal {A}(M_r)$ .",
"As in the previous case, we conclude that $\\mathcal {A}(M) = A$ .",
"By Proposition REF , any automorphism of $M$ is given by rational multiplication.",
"Take $s \\in \\mathbb {Q}_{>0}$ such that $\\phi _s \\in \\text{Aut}(M)$ , where $\\phi _s$ consists in left multiplication by $s$ .",
"Because $\\phi _s$ must send atoms to atoms, it follows that $sr = \\phi _s(r) \\in A$ .",
"Therefore $s$ must be an integer power of $r$ .",
"Hence $\\text{Aut}(M)$ is precisely $A$ when seen as a multiplicative subgroup of $\\mathbb {Q}$ .",
"As $A$ is the infinite cyclic group, the proof follows."
],
[
"Finite Transfer Puiseux Monoids",
"Now we turn to characterize the transfer homomorphisms from Puiseux monoids to finitely generated monoids.",
"Definition 4.1 We say that a Puiseux monoid $M$ is transfer finite if there exists a transfer homomorphism from $M$ to a finitely generated monoid.",
"By the fundamental structure theorem of finitely generated abelian groups, it immediately follows that every finitely generated monoid $F$ is a submonoid of a group $T \\times \\mathbb {Z}^\\beta $ for some finite abelian group $T$ and $\\beta \\in \\mathbb {N}_0$ .",
"In case of $F$ being reduced, it can be thought of as a submonoid of $T \\times \\mathbb {N}_0^\\beta $ .",
"Condition (T2) in the definition of a transfer homomorphism $\\theta \\colon M \\rightarrow F$ is crucial to transfer the factorization behavior of $F$ to $M$ .",
"However, the reader might wonder how much the set $\\text{Hom}(M,F)$ will increase if we drop condition (T2).",
"Surprisingly, the set of homomorphisms will remain the same as long as we impose $F$ to be reduced.",
"This fact facilitates to classify the Puiseux monoids that happen to be transfer finite, as we will prove in the next theorem.",
"First, notice that if $\\phi \\colon M \\rightarrow N$ is a monoid homomorphism, then the map $\\phi _{\\text{red}} \\colon M_{\\text{red}} \\rightarrow N_{\\text{red}}$ defined by $\\phi _{\\text{red}}(aM^\\times ) = \\phi (a)N^\\times $ is also a monoid homomorphism.",
"Theorem 4.2 Let $M$ be a nontrivial Puiseux monoid, and let $F$ be a finitely generated (additive) monoid.",
"If $\\theta \\colon M \\rightarrow F$ is a homomorphism satisfying $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ , then $M$ is isomorphic to a numerical monoid.",
"The Puiseux monoid $M$ is transfer finite if and only if it is isomorphic to a numerical monoid.",
"We argue first part (1).",
"It is easy to see that $\\theta _{\\text{red}} \\colon M_{\\text{red}} = M \\rightarrow F_{\\text{red}}$ is also a transfer homomorphism.",
"So we can assume, without loss of generality, that $F$ is reduced.",
"Suppose that $F$ is a submonoid of $T \\times \\mathbb {N}_0^\\beta $ , where $T$ is a finite abelian group and $\\beta \\in \\mathbb {N}_0$ .",
"First, assume, by way of contradiction, that $\\beta = 0$ .",
"In this case, it is not hard to verify that $\\theta (M)$ must be a subgroup of $T$ .",
"If $\\alpha = |\\theta (M)|$ and $r \\in M^\\bullet $ , then $\\theta (\\alpha r) = \\alpha \\theta (r) = 0$ .",
"This contradicts that $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ .",
"Thus, $\\beta \\ge 1$ .",
"Define $\\pi \\colon T \\times \\mathbb {N}_0^\\beta \\rightarrow \\mathbb {N}_0^\\beta $ by $\\pi (t,\\mathbf {v}) = \\mathbf {v}$ for all $t \\in T$ and $\\mathbf {v} \\in \\mathbb {N}^\\beta $ .",
"Let us verify that $\\pi (\\theta (M))$ is finitely generated.",
"Take $\\textbf {x} = (x_1, \\dots , x_\\beta ) \\in \\pi (\\theta (M))^\\bullet $ , and let $d = \\gcd (x_1, \\dots , x_\\beta )$ .",
"We shall verify that $\\pi (\\theta (M)) \\subseteq \\langle \\textbf {x}/d \\rangle $ .",
"To do so, consider $\\textbf {y} = (y_1, \\dots , y_\\beta ) \\in \\pi (\\theta (M))^\\bullet $ .",
"Now take $r,s \\in M^\\bullet $ such that $\\pi (\\theta (r)) = \\textbf {x}$ and $\\pi (\\theta (s)) = \\textbf {y}$ , and take $m,n \\in \\mathbb {N}$ satisfying that $\\gcd (m,n) = 1$ and $mr = ns$ .",
"Because $m \\textbf {x} = \\pi (\\theta (mr)) = \\pi (\\theta (ns)) = n \\textbf {y},$ one finds that $mx_i = ny_i$ for $i = 1, \\dots , \\beta $ .",
"As $\\gcd (m,n) = 1$ , it follows that $n$ divides each $x_i$ , i.e., $d/n \\in \\mathbb {N}$ .",
"As a result, $\\mathbf {y} = \\frac{m}{n} \\textbf {x} = \\bigg (\\frac{md}{n} \\bigg ) \\frac{\\mathbf {x}}{d} \\in \\bigg \\langle \\frac{\\mathbf {x}}{d} \\bigg \\rangle .$ Hence $\\pi (\\theta (M)) \\subseteq \\langle \\textbf {x}/d \\rangle $ .",
"Because $\\langle \\mathbf {x}/d \\rangle $ is isomorphic to $\\mathbb {N}_0$ , it follows that $\\pi (\\theta (M))$ is finitely generated.",
"We show now that $M$ is also finitely generated, which amounts to proving that $\\pi \\circ \\theta \\colon M \\rightarrow \\mathbb {N}_0^\\beta $ is injective.",
"First, let us verify that $\\pi $ is injective when restricted to $\\theta (M)$ .",
"As $\\theta (M)$ is a submonoid of the reduced monoid $F$ , it is also reduced.",
"Suppose that $(t_1, \\mathbf {v}), (t_2, \\mathbf {v}) \\in \\theta (M)$ , and let us check that $t_1 = t_2$ .",
"If $\\mathbf {v} = \\mathbf {0}$ , then $t_1 = t_2 = 0$ because $\\theta (M)^\\times $ is trivial.",
"Otherwise, there exist $r,s \\in M^\\bullet $ such that $\\theta (r) = (t_1, \\mathbf {v})$ and $\\theta (s) = (t_2, \\mathbf {v})$ .",
"Take $m,n \\in \\mathbb {N}$ such that $mr = ns$ .",
"Since $m (t_1, \\mathbf {v}) = m \\theta (r) = n \\theta (s) = n (t_2, \\mathbf {v})$ and $\\mathbf {v} \\ne \\mathbf {0}$ , one finds that $m = n$ and, therefore, $r = s$ .",
"This, in turn, implies that $t_1 = t_2$ .",
"Hence the restriction of $\\pi $ to $\\theta (M)$ is injective.",
"To conclude the proof of part (1), we show that $\\theta $ is also injective.",
"Let $r,s \\in M$ such that $\\theta (r) = \\theta (s) \\ne 0$ .",
"Taking $m,n \\in \\mathbb {N}$ satisfying $mr = ns$ , we have $m \\theta (r) = \\theta (mr) = \\theta (ns) = n \\theta (s).$ Since $\\theta (M)$ is reduced, the element $\\theta (r)$ must be torsion-free in $T \\times \\mathbb {N}_0^\\beta $ .",
"Thus, $m = n$ , which implies that $r = s$ .",
"As $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ , it follows that $|\\theta ^{-1}(a)| = 1$ for all $a \\in \\theta (M)$ .",
"Therefore $\\theta $ is injective, leading us to the injectivity of $\\pi \\circ \\theta $ .",
"Now that fact that $\\pi (\\theta (M))$ is finitely generated implies that $M$ is also finitely generated.",
"Hence $M$ must be isomorphic to a numerical monoid.",
"Finally, let us argue part (2) of the theorem.",
"For the direct implication, assume that the homomorphism $\\theta \\colon M \\rightarrow F$ is a transfer homomorphism.",
"As we did in the proof of part (1), we can assume that $F$ is a reduced.",
"As both $M$ and $F$ are reduced, condition (T1') yields $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ .",
"Now it follows by part (1) that the Puiseux monoid $M$ is isomorphic to a numerical monoid.",
"For the reverse implication, just take $\\theta $ to be the identity map.",
"Imposing the homomorphism $\\theta \\colon M \\rightarrow F$ in Theorem REF to satisfy $\\theta ^{-1}(0) = \\lbrace 0\\rbrace $ is not superfluous even if $M$ is atomic.",
"Then next example sheds some light upon this observation.",
"Example 4.3 Let $p_1, p_2, \\dots , $ be an enumeration of the odd prime numbers, and let $M = \\langle 1/p_n \\mid n \\in \\mathbb {N}\\rangle $ .",
"It is not hard to verify that $\\mathcal {A}(M) = \\lbrace 1/p_n \\mid n \\in \\mathbb {N}\\rbrace $ .",
"This implies that $M$ is atomic.",
"Now define $\\theta \\colon M \\rightarrow \\mathbb {Z}_2$ by setting $\\theta (0) = 0$ , $\\theta (r) = 0$ if $\\mathsf {n}(r)$ is even, and $\\theta (r) = 1$ if $\\mathsf {n}(r)$ is odd.",
"It follows immediately that $\\theta $ is a surjective monoid homomorphism.",
"However, $M$ is not isomorphic to any numerical monoid because it contains infinitely many atoms."
],
[
"Strongly Primary Puiseux Monoids",
"In this section we investigate which Puiseux monoids are strongly primary.",
"In the case of Puiseux monoids, being strongly primary is equivalent to being finitary.",
"In general, finitary monoids provide a common algebraic framework to study not only the arithmetic of strongly primary monoids but also that one of $v$ -noetherian $G$ -monoids (see [10]).",
"The structure of strongly primary monoids was first studied by Satyanarayana in [23] and has received substantial attention in the literature since then (see [15] and references therein).",
"In particular, the class of strongly primary monoids yields multiplicative models for a large class of one-dimensional local domains (see [10]).",
"All monoids mentioned in this section are assumed to be reduced.",
"Definition 5.1 Let $M$ be a monoid.",
"A submonoid $S$ of $M$ is called divisor-closed provided that for all $x \\in M$ and $s \\in S$ the fact that $x \\mid _M s$ implies that $x \\in S$ .",
"The monoid $M$ is called primary if it is nontrivial and its only divisor-closed submonoids are $\\lbrace 0\\rbrace $ and $M$ .",
"The monoid $M$ is called finitary if $M$ is a BF-monoid and there exist $n \\in \\mathbb {N}$ and a finite subset $S \\subseteq M^\\bullet $ such that $n M^\\bullet \\subseteq S + M$ .",
"The monoid $M$ is called strongly primary provided that $M$ is both primary and finitary.",
"Let $M$ be a Puiseux monoid, and let $M^{\\prime }$ be a nontrivial proper submonoid of $M$ .",
"Notice that for all $x \\in M \\setminus M^{\\prime }$ and $y \\in M^{\\prime }$ satisfying that $x \\mid _M y$ , the fact that $x \\mid _M \\mathsf {n}(x) \\mathsf {d}(y) y \\in M^{\\prime }$ immediately implies that $M^{\\prime }$ is not a divisor-closed submonoid of $M$ .",
"Thus, $M$ is primary.",
"On the other hand, suppose that $F$ is a finitely generated monoid, say $F = \\langle S \\rangle $ for some finite subset $S$ of $F^\\bullet $ .",
"Then the fact that $F^\\bullet = S + F$ immediately implies that $F$ is finitary.",
"In particular, every nontrivial finitely generated Puiseux monoid is finitary and, therefore, strongly primary.",
"The next proposition summarizes the observations made in this paragraph.",
"Proposition 5.2 Every nontrivial Puiseux monoid is primary.",
"Every finitely generated Puiseux monoid is strongly primary.",
"A Puiseux monoid that is not finitely generated may fail to be strongly primary (see, for example, Proposition REF ).",
"However, in Proposition REF and Proposition REF we exhibit two infinite families of non-finitely generated Puiseux monoids that are strongly primary.",
"To argue Proposition REF , we will use the following result, which is a weaker version of [16].",
"Proposition 5.3 If $M$ is a Puiseux monoid satisfying that 0 is not a limit point of $M^\\bullet $ , then $M$ is a BF-monoid.",
"The next two propositions introduce two families of (non-finitely generated) strongly primary Puiseux monoids.",
"Proposition 5.4 Let $p, q \\in \\mathbb {N}$ such that $\\gcd (p,q) = 1$ , and let $\\lbrace S_n\\rbrace $ be an inclusion-decreasing sequence of numerical monoids.",
"If a function $f \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ satisfies that $f(1) = 1$ and $q^{f(n+1) - f(n)} - p^n > p \\max \\lbrace F(S_n), a \\mid a \\in \\mathcal {A}(S_n) \\rbrace $ for every $n \\in \\mathbb {N}$ , then the Puiseux monoid $\\bigg \\langle \\frac{q^{f(n)}}{p^n} s \\ \\bigg {|} \\ n \\in \\mathbb {N}\\ \\text{ and} \\ s \\in S_n \\bigg \\rangle $ is strongly primary.",
"Set $M = \\big \\langle q^{f(n)}s/p^n \\mid n \\in \\mathbb {N}\\ \\text{and} \\ s \\in S_n \\big \\rangle $ , and for each $n \\in \\mathbb {N}$ set $A_n = \\mathcal {A}(S_n)$ .",
"First, we argue that $M$ is a BF-monoid.",
"To do so, observe that for each $n \\in \\mathbb {N}$ the fact that $q^{f(n+1) - f(n)} > p \\max A_n$ implies that $ \\min \\frac{q^{f(n+1)}}{p^{n+1}} A_{n+1} \\ge \\frac{q^{f(n+1)}}{p^{n+1}} > \\frac{q^{f(n)}}{p^n} \\max A_n.$ Therefore the generating set $\\cup _{n \\in \\mathbb {N}} (q^{f(n)}/p^n) A_n$ of $M$ can be listed as an increasing sequence of rational numbers.",
"As $M$ is generated by an increasing sequence of rational numbers, 0 cannot be a limit point of $M^\\bullet $ .",
"Hence $M$ is a BF-monoid by Proposition REF .",
"We proceed to prove that $M$ is finitary.",
"Since $M \\cap \\mathbb {N}_0$ is a submonoid of $(\\mathbb {N}_0,+)$ , it is atomic and $\\mathcal {A}(M \\cap \\mathbb {N}_0)$ is finite.",
"Take $S$ to be the finite set $\\mathcal {A}(M \\cap \\mathbb {N}_0) \\cup q A_1 \\subset M$ .",
"We are done once we verify the inclusion $p M^\\bullet \\subseteq S + M$ .",
"Fix $n \\in \\mathbb {N}$ and $a \\in A_{n+1}$ .",
"Because $q^{f(n)}a \\in q^{f(n)}S_{n+1} \\subseteq q^{f(n)}S_n \\in M \\cap \\mathbb {N}_0$ , the element $q^{f(n)}a$ is divisible in $M$ by an element of $S$ .",
"On the other hand, $(q^{f(n+1) - f(n)} - p^n)a \\ge q^{f(n+1) - f(n)} - p^n > F(S_n)$ , which implies that $(q^{f(n+1) - f(n)} - p^n)a \\, \\frac{q^{f(n)}}{p^n} \\in M$ .",
"Thus, $p\\bigg ( \\frac{q^{f(n+1)}}{p^{n+1}} a \\bigg ) = q^{f(n)}a + \\big (q^{f(n+1) - f(n)} - p^n\\big ) \\frac{q^{f(n)}}{p^n} a \\in S + M.$ In addition, the fact that $qa \\in qS_{n+1} \\subseteq qS_1$ guarantees $qa$ is divisible in $M$ by some element of $S$ .",
"This, in turn, implies that $p \\big (q^{f(1)}/p\\big )a = qa \\in S + M$ .",
"As a result, for each $x \\in M^\\bullet $ it follows that $px \\in m(S+M) \\subseteq S + M$ for some $m \\in \\mathbb {N}$ .",
"Hence $pM^\\bullet \\subseteq S + M$ , as desired.",
"Example 5.5 Consider the Puiseux monoid $M = \\bigg \\langle \\frac{3^{n^2 + 1}}{2^n}, \\frac{5 \\cdot 3^{n^2}}{2^n} \\ \\bigg {|} \\ n \\in \\mathbb {N}\\bigg \\rangle .$ Taking $S_n$ to be the numerical monoid $\\langle 3, 5 \\rangle $ for each $n \\in \\mathbb {N}$ and defining the function $f \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ by $f(n) = n^2$ , we can rewrite the Puiseux monoid $M$ as follows: $M = \\bigg \\langle \\frac{3^{f(n)}}{2^n} s \\ \\bigg {|} \\ n \\in \\mathbb {N}\\ \\text{ and} \\ s \\in S_n \\bigg \\rangle .$ Since $F(S_n) = 7$ for each $n \\in \\mathbb {N}$ , it follows that $3^{f(n+1) - f(n)} - 2^n = 3^{2n+1} - 2^n > 14 = 2 \\max \\lbrace 3,5, F(S_n)\\rbrace .$ As $f(1) = 1$ , Proposition REF guarantees that $M$ is a strongly primary Puiseux monoid.",
"As in Section , for $r \\in \\mathbb {Q}_{> 0}$ we let $M_r$ denote the multiplicatively $r$ -cyclic Puiseux monoid $\\langle r^n \\mid n \\in \\mathbb {N}\\rangle $ .",
"Proposition 5.6 For each $r \\in \\mathbb {Q}_{> 1}$ , the Puiseux monoid $M_r$ is strongly primary.",
"Since $r > 1$ , it follows that 0 is not a limit point of $M_r^\\bullet $ .",
"Thus, Proposition REF ensures that $M_r$ is a BF-monoid.",
"On the other hand, it was proved in [19] that $\\mathcal {A}(M_r) = \\lbrace r^n \\mid n \\in \\mathbb {N}\\rbrace $ .",
"Now we check that $\\mathsf {n}(r)$ divides $\\mathsf {n}(r)r^j$ in $M_r$ for every $j \\in \\mathbb {N}_0$ .",
"If $j=0$ , then $\\mathsf {n}(r) \\mid _{M_r} \\mathsf {n}(r)r^j$ follows trivially.",
"Hence it suffices to assume that $j \\in \\mathbb {N}$ .",
"In this case, the fact that $\\mathsf {n}(r)(r-1) = r( \\mathsf {n}(r) - \\mathsf {d}(r))$ implies that $\\mathsf {n}(r) r^j - \\mathsf {n}(r) = \\mathsf {n}(r) (r-1) \\sum _{i=0}^{j-1} r^i = r \\big (\\mathsf {n}(r) - \\mathsf {d}(r)\\big ) \\sum _{i=0}^{j-1} r^i = \\big (\\mathsf {n}(r) - \\mathsf {d}(r)\\big ) \\sum _{i=0}^{j-1} r^{i+1} \\in M_r.$ Therefore $\\mathsf {n}(r)$ divides $\\mathsf {n}(r) r^j$ in $M_r$ .",
"To show now that $M_r$ is finitary, we take $n = \\mathsf {d}(r)$ and $S = \\lbrace \\mathsf {n}(r)\\rbrace $ and verify that $n M_r^\\bullet \\subseteq S + M_r$ .",
"For $q \\in M_r^\\bullet $ , take $\\alpha _1, \\dots , \\alpha _t \\in \\mathbb {N}_0$ such that $q = \\sum \\alpha _i r^i$ , and fix $k \\in \\lbrace 1, \\dots , t\\rbrace $ such that $\\alpha _k > 0$ .",
"Because $\\mathsf {n}(r)$ divides $\\mathsf {n}(r) r^{k-1}$ in $M_r$ , there exists $s \\in M_r$ satisfying that $\\mathsf {n}(r) r^{k-1} = \\mathsf {n}(r) + s$ .",
"Thus, $n q &= \\mathsf {d}(r) \\alpha _k r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\\\&= \\mathsf {n}(r) r^{k-1} + (\\alpha _k - 1) \\mathsf {d}(r) r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\\\&= \\mathsf {n}(r) + \\big ( s + (\\alpha _k - 1) \\mathsf {d}(r) r^k + \\sum _{i \\ne k} \\mathsf {d}(r) \\alpha _i r^i \\big ),$ which implies that $nq \\in S + M_r$ .",
"Since $n M_r^\\bullet \\subseteq S + M_r$ , one obtains that $M_r$ is finitary and, therefore, strongly primary.",
"We conclude this section providing a family of Puiseux monoids that fail to be strongly primary.",
"Proposition 5.7 Let $\\lbrace a_n\\rbrace $ be a sequence of positive rational numbers satisfying that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ for any $i \\ne j$ .",
"Then the Puiseux monoid $\\langle a_n \\mid n \\in \\mathbb {N}\\rangle $ is atomic but not strongly primary.",
"Set $M = \\langle a_n \\mid n \\in \\mathbb {N}\\rangle $ .",
"It is not difficult to verify that $\\mathcal {A}(M) = \\lbrace a_n \\mid n \\in \\mathbb {N}\\rbrace $ from the fact that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ for any $i \\ne j$ ; we leave the details to the reader.",
"This implies, in particular, that $M$ is atomic.",
"Suppose, by way of contradiction, that $M$ is finitary.",
"Choose $n \\in \\mathbb {N}$ and a finite subset $S$ of $M^\\bullet $ such that $n M^\\bullet \\subseteq S + M$ .",
"Let $S^{\\prime }$ be a finite subset of $\\mathcal {A}(M)$ such that for each $s \\in S$ there is at least one atom in $S^{\\prime }$ dividing $s$ in $M$ .",
"After substituting $S$ by $S^{\\prime }$ , we can assume that $S \\subset \\mathcal {A}(M)$ .",
"Since $n M^\\bullet \\subseteq S + M$ and $\\inf (S+M) \\ge \\min S$ , it follows that 0 cannot be a limit point of $M^\\bullet $ .",
"Fix $\\epsilon > 0$ such that $\\epsilon < \\inf M^\\bullet $ .",
"Since $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ , there exists $j \\in \\mathbb {N}$ such that $\\mathsf {d}(a_j) > \\max \\lbrace n, \\max \\mathsf {d}(S)\\rbrace $ .",
"Because $n M^\\bullet \\subseteq S + M$ , one can write $ n a_j = a_s + \\sum _{i=1}^k \\alpha _i a_i$ for some $k \\in \\mathbb {N}$ , $a_s \\in S$ , and $\\alpha _i \\in \\mathbb {N}_0$ .",
"Applying the $\\mathsf {d}(a_j)$ -valuation to both sides of (REF ) and using the fact that $\\gcd (\\mathsf {d}(a_i), \\mathsf {d}(a_j)) = 1$ , it is not hard to find that $\\mathsf {d}(a_j)$ divides $n - \\alpha _j$ .",
"Now $\\mathsf {d}(a_j) > n$ yields $n = \\alpha _j$ .",
"This, along with (REF ), would force $a_s = 0$ , which contradicts that $S \\subset \\mathcal {A}(M)$ .",
"Thus, $M$ is not strongly primary.",
"Remark 5.8 The previous proposition not only shows that Puiseux monoids are not, in general, strongly primary, but also illustrates that a natural bounding, ordering, or topological restriction under which a Puiseux monoid is guaranteed to be strongly primary is rather unlikely.",
"For example, consider the Puiseux monoids $M_1 = \\bigg \\langle \\frac{1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle , \\ M_2 = \\bigg \\langle \\frac{p-1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle , \\ \\text{and} \\ M_3 = \\bigg \\langle \\frac{p^2+1}{p} \\ \\bigg {|} \\ p \\ \\text{is prime} \\bigg \\rangle .$ It is not hard to check that the sets of atoms of $M_1$ , $M_2$ , and $M_3$ are precisely the generating sets displayed.",
"Therefore $M_1$ is strongly bounded, $M_2$ is bounded, and $M_3$ is not bounded.",
"We can also see that $M_1$ is a decreasing Puiseux monoid, while $M_2$ is increasing.",
"Furthermore, notice that 0 is a limit point of $M_1^\\bullet $ , but 0 is not a limit point of $M_2^\\bullet $ .",
"Finally, Proposition REF ensures that $M_1$ , $M_2$ , and $M_3$ are all strongly primary."
],
[
"Puiseux Monoids Are Almost Never Transfer Krull",
"We dedicate this section to show that the atomic structure of Puiseux monoids almost never can be obtained by transferring back that one of Krull monoids; specifically we shall prove that the existence of a transfer homomorphism from a nontrivial Puiseux monoid to a Krull monoid forces the domain to be isomorphic to $(\\mathbb {N}_0,+)$ .",
"The we use this information to show that only finitely generated Puiseux monoids admit transfer homomorphisms to C-monoids.",
"Let us start by giving the definition of a Krull monoid.",
"Definition 6.1 A monoid $K$ is called a Krull monoid if there is a monoid homomorphism $\\varphi \\colon K \\rightarrow D$ , where $D$ is a free abelian monoid and $\\varphi $ satisfies the following two conditions: if $a, b \\in K$ and $\\varphi (a) \\mid _D \\varphi (b)$ , then $a \\mid _K b$ ; for every $d \\in D$ there exist $a_1, \\dots , a_n \\in K$ with $d = \\gcd \\lbrace \\varphi (a_1), \\dots , \\varphi (a_n)\\rbrace $ .",
"With notation as in Definition REF , it is easy to see that $K$ is a Krull monoid if and only if $K_{\\text{red}}$ is a Krull monoid.",
"The basis elements of $D$ are called the prime divisors of $K$ .",
"The abelian group Cl$(K) := D/\\varphi (K)$ is called the class group of $K$ (see [10]).",
"As Krull monoids are isomorphic to submonoids of free abelian monoids, Krull monoids are atomic.",
"The factorization theory of Krull monoids has been significantly studied (see [5], [13] and references therein).",
"The class of Krull monoids contains many well-studied types of monoids, including the multiplicative monoid of the ring of integers of an algebraic number, the Hilbert monoids, and the regular congruence monoids.",
"These and further examples of Krull monoids are presented in [9] and [10].",
"From the point of view of factorization theory, perhaps the most important family of Krull monoids is that one consisting of block monoids, which we are about to introduce.",
"This is because block monoids capture the essence of the arithmetic of lengths of factorizations in Krull monoids.",
"Let $G$ be an abelian group and $\\mathcal {F}(G)$ the free abelian monoid on $G$ .",
"An element $X = g_1 \\dots g_l \\in \\mathcal {F}(G)$ is called a sequence over $G$ .",
"The length of $X$ is defined as $|X| = l =\\sum _{g \\in G} \\mathsf {v}_g(X).$ For every $I \\subseteq [1, l]$ , the sequence $Y = \\prod _{i \\in I} g_i$ is called a subsequence of $X$ .",
"The subsequences are precisely the divisors of $X$ in the free abelian monoid $\\mathcal {F}(G)$ .",
"The submonoid $\\mathcal {B}(G) := \\bigg \\lbrace X \\in \\mathcal {F}(G) \\ \\bigg {|} \\ \\sum _{g \\in G} \\mathsf {v}_g(X) g = 0 \\bigg \\rbrace $ of $\\mathcal {F}(G)$ is called the block monoid on $G$ , and its elements are referred to as zero-sum sequences or blocks over $G$ ([10] is a good general reference on block monoids).",
"Furthermore, if $G_0$ is a subset of $G$ , then the submonoid $\\mathcal {B}(G_0) := \\lbrace X \\in \\mathcal {B}(G) \\mid \\mathsf {v}_g(X) = 0 \\ \\emph { if } \\ g \\notin G_0 \\rbrace $ of $\\mathcal {B}(G)$ is called the restriction of the block monoid $\\mathcal {B}(G)$ to $G_0$ .",
"For $X \\in \\mathcal {B}(G_0)$ , the support of $X$ in $G_0$ is defined to be $\\text{supp}_{G_0}(X) := \\lbrace g \\in G_0 \\mid \\mathsf {v}_g(X) > 0\\rbrace .$ As mentioned before, the relevance of block monoids in the theory of non-unique factorizations lies in the next result.",
"Proposition 6.2 [10] Let $K$ be a Krull monoid with class group $G$ and let $G_0$ be the set of classes of $G$ which contain prime divisors.",
"Then $\\mathcal {L}(K) = \\mathcal {L}(\\mathcal {B}(G_0)).$ As a consequence, understanding the arithmetic of lengths of factorizations in Krull monoids amounts to understanding the same in block monoids.",
"Definition 6.3 A Puiseux monoid $M$ is transfer Krull if there exist an abelian group $G$ , a subset $G_0$ of $G$ , and a transfer homomorphism $\\theta \\colon M \\rightarrow \\mathcal {B}(G_0)$ .",
"Remark: Our definition of a transfer Krull monoid coincides with the definition given in [8]; this is because in the present setting the concepts of a transfer homomorphism and the concept of a weak transfer homomorphism coincide by [3].",
"We denote the field of fractions of an integral domain $R$ by $\\mathsf {q}(R)$ .",
"For subsets $X,Y$ of $\\mathsf {q}(R)$ we set $(X : Y) := \\lbrace x \\in \\mathsf {q}(R) \\mid xY \\subseteq X\\rbrace $ .",
"In addition, $R$ is called a Krull domain if $R^\\bullet $ is a Krull monoid.",
"In this case, the divisor class group of $R$ , denoted by $\\mathcal {C}(R)$ , measures the extent to which factorizations in $R$ fail to be unique (see [10]).",
"Unlike Krull domains/monoids, which have been central objects in commutative algebra since mid-nineteenth century, transfer Krull monoids (which generalize the concept of Krull monoids) were introduced more recently.",
"Let us proceed to present a few examples of transfer Krull monoids.",
"Examples of transfer Krull monoids: Let $H$ be a half-factorial monoid, and let $\\theta \\colon H \\rightarrow \\mathcal {B}(\\lbrace 0\\rbrace )$ be the map defined by $\\theta (h) = 0$ if $h \\in \\mathcal {A}(H)$ and $\\theta (h) = 1$ if $h \\in H^\\times $ .",
"As the map $\\theta $ is a transfer homomorphism, it follows that $H$ is a transfer Krull monoid.",
"Let $R$ be a Krull domain, and let $K$ be a subring of $R$ with the same field of fractions.",
"Suppose, in addition, that the following three conditions hold: $R = KR^\\times $ ; $K \\cap R^\\times = K^\\times $ ; $(K : R)$ is a maximal ideal of $K$ (see [10]).",
"Then the inclusion map $K^\\bullet \\hookrightarrow R^\\bullet $ is a transfer homomorphism and, therefore, $K^\\bullet $ is a transfer Krull monoid.",
"Transfer Krull monoids can also be defined in a non-commutative context (see, for instance, [3]).",
"Let $R$ be a bounded HNP (hereditary Noetherian prime) ring.",
"If every stably free left $R$ -ideal is free, then $R^\\bullet $ is a transfer Krull monoid (see [24] for details).",
"There are also many monoids that fail to be transfer Krull.",
"Examples of non-transfer Krull monoids in a non-commutative setting are provided by [6], [12], and [25].",
"On the other hand, Theorem REF and the next proposition (which follows from [14]) yield examples of non-transfer Krull monoids in a commutative context.",
"Proposition 6.4 Every proper numerical monoid fails to be transfer Krull.",
"The next lemma will be used in the proof of Theorem REF .",
"Lemma 6.5 If $\\lbrace a_n\\rbrace $ is an infinite sequence of positive integers, then there exists $m \\in \\mathbb {N}$ such that $a_{m+1} \\in \\langle a_1, \\dots , a_m \\rangle $ .",
"If $\\lbrace a_n\\rbrace $ is bounded there is a term that repeats infinitely many times, making the conclusion of the lemma obvious.",
"Thus, suppose that $\\lbrace a_n\\rbrace $ is not bounded.",
"Let $\\lbrace a_{n_j}\\rbrace $ be a subsequence of $\\lbrace a_n\\rbrace $ satisfying that $ a_{n_{j+1}} > \\prod _{i=1}^{j} a_{n_i}$ for every $j \\in \\mathbb {N}$ .",
"Now, for each natural number $j$ , set $d_j = \\gcd (a_{n_1}, \\dots , a_{n_j})$ , and notice that $d_{j+1} \\mid d_j$ for every $j \\in \\mathbb {N}$ .",
"Therefore $d_{k+1} = d_k$ must hold for some $k$ .",
"In particular, $d_k \\mid a_{n_{k+1}}$ .",
"On the other hand, condition (REF ) ensures that $a_{n_{k+1}}/d_k$ is greater than the Frobenius number of the numerical monoid $\\langle a_{n_1}/d_k, \\dots , a_{n_k}/d_k \\rangle $ .",
"This implies that $a_{n_{k+1}} \\in \\langle a_{n_1}, \\dots , a_{n_k} \\rangle $ .",
"The lemma follows by taking $m = n_{k+1}-1$ .",
"Now we are in a position to prove that atomic Puiseux monoids are almost never transfer Krull.",
"Theorem 6.6 If a nontrivial Puiseux monoid is transfer Krull, then it must be isomorphic to $(\\mathbb {N}_0,+)$ .",
"Let $M$ be a nontrivial Puiseux monoid that happens to be transfer Krull.",
"As Krull monoids are atomic, $M$ is atomic by Proposition REF .",
"Let $G$ be an abelian group, and let $\\theta \\colon M \\rightarrow \\mathcal {B}(G_0)$ be a transfer homomorphism, where $G_0$ is a subset of $G$ .",
"Because both $M$ and $\\mathcal {B}(G_0)$ are reduced, $\\theta ^{-1}(\\emptyset ) = \\lbrace 0\\rbrace $ .",
"Assume, by way of contradiction, that $M$ is not isomorphic to a numerical monoid.",
"Take $X \\in \\mathcal {B}(G_0)^\\bullet $ and $r,s \\in M^\\bullet $ such that $\\theta (r) = \\theta (s) = X$ .",
"Taking $m,n \\in \\mathbb {N}$ such that $mr = ns$ , one obtains $ \\prod _{g \\in G_0} g^{m \\mathsf {v}_g(X)} = \\theta (r)^m = \\theta (s)^n = \\prod _{g \\in G_0} g^{n \\mathsf {v}_g(X)}.$ Since $|X| \\ge 1$ and $m \\mathsf {v}_g(X) = n \\mathsf {v}_g(X)$ for every $g \\in G_0$ , it follows that $m=n$ , which yields $r = s$ .",
"Hence the preimage under $\\theta $ of each element of $\\mathcal {B}(G_0)^\\bullet $ is a singleton.",
"This, along with the fact that $\\theta ^{-1}(\\emptyset ) = \\lbrace 0\\rbrace $ , implies that $\\theta $ is injective.",
"In addition, the same equality (REF ) implies that $\\text{supp}_{G_0}(\\theta (a)) = \\text{supp}_{G_0}(\\theta (a^{\\prime }))$ for all $a,a^{\\prime } \\in \\mathcal {A}(M)$ .",
"As a consequence, any two elements of $\\theta (M^\\bullet )$ have the same support, and we can assume, without loss of generality, that $G_0$ is finite.",
"Let $G_0 =: \\lbrace g_1, \\dots , g_t\\rbrace $ be the common support.",
"List the set $\\mathcal {A}(M)$ as a sequence $\\lbrace a_n\\rbrace $ , and let $A_n = \\theta (a_n)$ for each $n \\in \\mathbb {N}$ .",
"Because $\\theta $ is injective, $A_i \\ne A_j$ when $i \\ne j$ .",
"Now, for any pair $(i,j) \\in \\mathbb {N}^2$ , there exist $c_i,c_j \\in \\mathbb {N}$ such that $c_i a_i = c_j a_j$ .",
"For each $n \\in \\lbrace 1, \\dots , t\\rbrace $ , we can apply $\\mathsf {v}_{g_n} \\circ \\theta $ to the equality $c_i a_i = c_j a_j$ to get $c_i \\mathsf {v}_{g_n}(A_i) = c_j \\mathsf {v}_{g_n}(A_j)$ .",
"After rewriting this equality, one obtains that $ \\frac{\\mathsf {v}_{g_n}(A_i)}{\\mathsf {v}_{g_n}(A_j)} = \\frac{c_j}{c_i} = \\frac{\\mathsf {v}_{g_1}(A_i)}{\\mathsf {v}_{g_1}(A_j)}$ for each $n \\in \\lbrace 1, \\dots , t\\rbrace $ .",
"On the other hand, notice that Lemma REF guarantees the existence of $m \\in \\mathbb {N}$ and $\\alpha _1, \\dots , \\alpha _m \\in \\mathbb {N}_0$ such that $ \\mathsf {v}_{g_1}(A_{m+1}) = \\sum _{i=1}^m \\alpha _i \\mathsf {v}_{g_1}(A_i).$ By (REF ), it follows that the equality (REF ) holds when we replace $g_1$ by any other element of $G_0$ (exactly with the same $\\alpha _i$ 's).",
"As a result, we obtain $A_{m+1} = \\prod _{j=1}^{|G_0|} g_j^{\\mathsf {v}_{g_j}(A_{m+1})} = \\prod _{j=1}^{|G_0|} \\prod _{i=1}^m g_j^{\\alpha _i \\mathsf {v}_{g_j}(A_i)} = \\prod _{i=1}^m \\bigg (\\prod _{j=1}^{|G_0|} g_j^{\\mathsf {v}_{g_j}(A_i)}\\bigg )^{\\alpha _i} = \\prod _{i=1}^m A_i^{\\alpha _i}.$ This contradicts the fact that $A_{m+1}$ is an atom of the block monoid $\\mathcal {B}(G_0)$ .",
"Therefore $M$ must be isomorphic to a numerical monoid.",
"Now the direct implication of the proof follows by Proposition REF .",
"For the reverse implication, it suffices to notice that $\\theta \\colon 1 \\mapsto [1_G]$ is an isomorphism from $(\\mathbb {N}_0,+)$ to the block monoid $\\mathcal {B}(G)$ , where $G$ is the trivial group.",
"Corollary 6.7 A Puiseux monoid is a Krull monoid if and only if it can be generated by one element.",
"Perhaps the second most-systematically studied family of atomic monoids is that one comprising the C-monoids.",
"We would like to know under which conditions a Puiseux monoid happens to be a C-monoid.",
"Any monoid $M$ can be embedded into a quotient group $\\mathsf {g}(M)$ , which is unique up to canonical isomorphism.",
"Let $D$ be a multiplicative monoid with quotient group $\\mathsf {g}(D)$ , and let $M$ be a submonoid of $D$ .",
"Two elements $x,y \\in D$ are said to be $M$ -equivalent provided that $x^{-1}M \\cap D = y^{-1}M \\cap D$ .",
"It can be easily checked that being $M$ -equivalent defines a congruence relation on $D$ .",
"For each $x \\in D$ , let $[x]^D_M$ denote the congruence class of $x$ .",
"The set $\\mathcal {C}^*(M,D) := \\big \\lbrace [x]^D_M \\mid x \\in (D \\setminus D^\\times ) \\cup \\lbrace 1\\rbrace \\big \\rbrace $ is a commutative semigroup with identity, which is called the reduced class semigroup of $M$ in $D$ .",
"Definition 6.8 A monoid $M$ is called a C-monoid if it is a submonoid of a factorial monoid $F$ such that $F^\\times \\cap M = M^\\times $ and $\\mathcal {C}^*(M,F)$ is finite.",
"With notation as in Definition REF , we say that $M$ is a C-monoid defined in $F$ .",
"A C-monoid can be defined in more than one factorial monoid $F$ ; however, there is a canonical way of choosing $F$ (see [9]).",
"Because C-monoids are submonoids of factorial monoids, they are atomic.",
"The family of C-monoids allows us to study the arithmetic of non-integrally closed Noetherian domains.",
"Given a multiplicative monoid $M$ with quotient group $\\mathsf {g}(M)$ , we say that $x \\in \\mathsf {g}(M)$ is almost integral over $M$ if there exists $c \\in M$ such that $cx^n \\in M$ for every $n \\in \\mathbb {N}$ .",
"The subset of $\\mathsf {g}(M)$ consisting of all almost integral elements over $M$ is denoted by $\\widehat{M}$ and called the complete integral closure of $M$ .",
"Let $R$ be an integral domain with field of fractions $\\mathsf {q}(R)$ .",
"An ideal $I$ of $R$ is divisorial if $(R : (R : I)) = I$ .",
"The domain $R$ is called a Mori domain if it satisfies the ascending chain condition on divisorial ideals.",
"Finally, for the domain $R$ we set $\\widehat{R} = \\widehat{R^\\bullet } \\cup \\lbrace 0\\rbrace $ , where $R^\\bullet $ is the multiplicative monoid of $R$ .",
"Example 6.9 If $A$ is a Mori domain, then $R = \\widehat{A}$ is a Krull domain.",
"Moreover, if $\\mathfrak {f} = (A : R)$ is nonzero and both the quotient ring $R/\\mathfrak {f}$ and the class group $\\mathcal {C} (R)$ are finite, then $A^\\bullet $ is a C-monoid (see [10]).",
"More examples of C-monoids can be found in [11] and [22].",
"The next theorem is used in the proof of Proposition REF .",
"Theorem 6.10 [10] The complete integral closure of a C-monoid is a Krull monoid.",
"Proposition 6.11 A nontrivial Puiseux monoid is a C-monoid if and only if it is isomorphic to a numerical monoid.",
"Let $M$ be a nontrivial Puiseux monoid that is also a C-monoid.",
"Let $\\widehat{M}$ be the complete integral closure of $M$ .",
"Observe first that if $x \\in \\mathsf {g}(M) \\cap \\mathbb {Q}_{<0}$ , then $S_{x,r} := \\lbrace r + nx \\mid n \\in \\mathbb {N}\\rbrace $ contains only finitely many positive rational numbers for all $r \\in M$ .",
"As a result, $|S_{x,r} \\cap M| < \\infty $ for all $r \\in M$ , which implies that no negative element of $\\mathsf {g}(M)$ is almost integral over $M$ .",
"Because $\\widehat{M}$ is a monoid and it is contained in $\\mathbb {Q}_{\\ge 0}$ , it must be a Puiseux monoid.",
"By Theorem REF , the monoid $\\widehat{M}$ is a Krull monoid; in particular, it is transfer Krull.",
"Now Theorem REF ensures that $\\widehat{M}$ is isomorphic to $(\\mathbb {N}_0,+)$ .",
"Finally, the fact that $M$ is a submonoid of $\\widehat{M}$ forces $M$ to be isomorphic to a numerical monoid.",
"For the reverse implication, it suffices to note that for every proper numerical monoid $N$ , any two natural numbers greater than the Frobenius number of $N$ are $N$ -equivalent, which implies that $\\mathcal {C}^*(N, \\mathbb {N}_0)$ is finite.",
"As $\\mathbb {N}_0$ is also a C-monoid, the proof follows."
],
[
"Acknowledgements",
"While working on this paper, I was supported by the NSF-AGEP fellowship.",
"I am grateful to Alfred Geroldinger not only for proposing the main questions motivating this paper but also for his enlightening guidance through early drafts.",
"Also, I would like to thank Salvatore Tringali and the anonymous referee, whose helpful suggestions help me improve the final version of this paper."
]
] | 1709.01693 |
[
[
"Nanoscale chemical mapping of laser-solubilized silk"
],
[
"Abstract A water soluble amorphous form of silk was made by ultra-short laser pulse irradiation and detected by nanoscale IR mapping.",
"An optical absorption-induced nanoscale surface expansion was probed to yield the spectral response of silk at IR molecular fingerprinting wavelengths with a high ~20 nm spatial resolution defined by the tip of the probe.",
"Silk microtomed sections of 1-5 micrometers in thickness were prepared for nanoscale spectroscopy and a laser was used to induce amorphisation.",
"Comparison of silk absorbance measurements carried out by table-top and synchrotron Fourier transform IR spectroscopy proved that chemical imaging obtained at high spatial resolution and specificity (able to discriminate between amorphous and crystalline silk) is reliably achieved by nanoscale IR.",
"A nanoscale material characterization using synchrotron IR radiation is discussed."
],
[
"Introduction",
"In analytical material science, absorption of IR light is used for fingerprinting (chemical imaging) of particular molecules, specific compounds, and provides insight into interactions in their immediate vicinity.",
"However, challenges arise when this information needs to be obtained from sub-wavelength and sub-cellular dimensions, in particular at IR and terahertz spectral bands of absorption [1] or scattering [2].",
"Optical properties of sub-wavelength structures and patterns have opened an entirely new direction in photonics and design of highly efficient optical elements for control of intensity, phase, polarisation, spin and orbital momenta of light based on flat planar geometries, yet rich in nanoscale features [3], [4].",
"We aim at reaching that level of control in the domain of chemical spectral imaging.",
"There, interpretation of data from near-field requires further knowledge of probe interaction with substrate, phase information of the reflected/transmitted light from sub-wavelength structures to reveal complex peculiarities of light-matter interactions at nanoscale and is now advancing with strongly concentrated efforts [5], [6], [7], [8], [9], [10].",
"Recently, an electron tunneling control by a single-cycle terahertz pulse illuminated onto a tip of a scanning transmission microscope (STM) needle was demonstrated at 10 V/nm fields [11].",
"STM reaches an atomic precision in surface probing and its spectroscopic characterisation and can be carried out on a water surface [12].",
"Absorbance spectra quantify and identify the resonant molecules, chemical structures through their individual or collective excitations as detected in transmission (or inferred from reflection).",
"By sweeping excitation wavelength through the finger printing region of a particular material, the usual optical excitation relaxation pathway ending with a thermal energy deposition into host material can be sensitively measured using an atomic force microscopy (AFM) approach [13], [14] which opened up a rapidly growing AFM-IR field [15] (also known as nano-IR; Fig.",
"REF (a)).",
"How reliably one can determine IR properties with an AFM nano-tip based on the thermal expansion is currently still under debate due to a lack of knowledge of the actual anisotropy of thermal and mechanical properties at the nanoscale, 3D molecular conformation, alignment, and the interaction volume.",
"To establish the correspondence between nano-IR and spectroscopy it is necessary to compare IR spectral imaging in near-field and far-field modes with nano-IR.",
"Another challenge of the nano-IR technique is that a very thin sub-micrometer film has to be prepared and mounted on a thermally conductive substrate.",
"Microtomed thin sections usually have thicknesses above the optimum and need to be embedded into an epoxy host which interferes with the nano-IR signal from micro-specimens.",
"In order to demonstrate that microtomed slices of samples with features at or below micrometers in lateral cross-section can be measured using nano-IR and provide reliable IR spectral information we acquired absorbance spectra from areas down to single pixel hyper-spectral resolution accessible on table-top FT-IR spectrometers, and at high-resolution which was achieved with a solid immersion lens using high-brightness synchrotron FT-IR microscopy.",
"Such comparison of IR properties read out from nanoscale and far-field was carried out in this study using silk - a bio-polymer complex in its structure comprised of crystalline and amorphous building blocks [16], [17], [18], [19].",
"Here, micrometer-thick slices of silk were used to measure thermal expansion under a specific wavelength of excitation with $\\sim 20$ -nm-sharp AFM tips and to compare with high-resolution spectra measured with ATR FT-IR method using synchrotron radiation at the Infrared Microspectroscopy (IRM) Beamline (Australian Synchrotron) as well as a table-top FT-IR spectrometer.",
"Amorphisation of silk induced by single ultra-short laser pulses [19] has been spectroscopically recognised using nano-IR.",
"Figure: Principle of nano-IR: an AFM tip follows height changesinduced by a spectral sweep of excitation at IR wavelengths (∼6μ\\sim 6~\\mu m).",
"IR laser beam is p-polarised and incident atΘ≃60 ∘ \\Theta \\simeq 60^\\circ angle to minimise reflective losses (closeto the Brewster angle) and to illuminate a much larger sample areawhile AFM readout occurs from a AFM needle contact (∼20\\sim 20 nm)continuously scanned with 0.1 Hz frequency and digitised to obtainmap with x×y=25×100x\\times y = 25\\times 100 nm 2 ^2 pixels.",
"AFM and nano-IRmapping was carried out at a selected excitation wavelength.",
"(b)SEM images of silk fiber melted/amorphised by a single laser pulseas-fabricated and after immersion into water; laser wavelength515 nm, pulse duration 230 fs, focused with objective lens withnumerical aperture NA=0.5NA = 0.5, pulse energy 425 nJ, linearpolarisation was along the fiber (different fibers exposed at thesame conditions were used for the two images).Figure: AFM (a) and chemical map at Amide I 1660 cm -1 ^{-1} band(b) measured with nano-IR; inset in (b) shows the ablation pitregion imaged at crystalline band at 1700 cm -1 ^{-1}.",
"(c) Pointspectra measured on epoxy, amorphous-rich, and predominantlycrystalline regions of the T-cross-section of a silk fiber; insetshows AFM image of a microtome slice of a silk fiber in the epoxymatrix and laser irradiated crater region."
],
[
"Samples and methods",
"Domestic (Bombyx mori) silk fibers, stripped of their sericin rich cladding [20], were probed during the spectroscopic imaging experiments.",
"Pulsed laser radiation was used to induce local structural modifications of silk.",
"The AFM-based nano-IR experiments with $\\sim $ 25-nm-diameter tips were benchmarked against more conventional methods, such as attenuated total reflectance (ATR) at the Australian Synchrotron IRM Beamline ($\\sim 1.9~\\mu $ m resolution), and a table-top FT-IR spectrometer ($\\sim 6~\\mu $ m resolution).",
"For this set of experiments method-agnostic silk samples, in the form of thin slices, had to be prepared.",
"For cross-sectional observation, the natural silk fibers were aligned and embedded into an epoxy adhesive (jER 828, Mitsubishi Chemical Co., Ltd.).",
"Fibers fixed in the epoxy matrix were microtomed into 1-5 $\\mu $ m-thick slices which are mechanically robust enough to be measured using standard FT-IR setups without any supporting substrate.",
"This was particularly important to increase sensitivity of the far-field absorbance measurements and to diminish reflective losses.",
"Both longitudinal (L) and transverse (T) slices of the silk fibers were prepared by microtome (RV-240, Yamato Khoki Industrial Co., Ltd).",
"The slices were thinner than the original silk fibers.",
"For synchrotron ATR FT-IR, an aluminum disk was used to support the thin silk sections when they were brought into contact with a 100 $\\mu $ m diameter facet tip of the Ge ATR hemisphere (refractive index $n = 4$ ).",
"Synchrotron ATR-FTIR mapping measurement was performed using a in-house developed ATR-FTIR device at the IR Microspectroscopy Beamline, which has a high-speed, high-resolution surface characterisation capabilities with spatial resolution down to $~1.9~\\mu $ m [21].",
"A 100 $\\mu $ m tip ATR accessory for a FT-IR spectrometer (Hyperion 3000, Bruker) with Ge contact lens of $NA = n\\sin \\varphi \\simeq 2.4$ of refractive index $n = 4$ and $\\varphi = 36.9^\\circ $ a half-angle of the focusing cone was used.",
"A deep sub-wavelength resolution $r = 0.61\\lambda _{IR}/NA \\simeq 1.5~\\mu $ m is achievable for the IR wavelengths of interest at the Amide band of $\\lambda _{IR} = 1600-1700$ cm$^{-1}$ or 6.25 - 5.9 $\\mu $ m. The nano-IR experiments, i.e., an AFM readout of the height changes in response to IR sample excitation, were carried out with nano-IR2 (Anasys Instruments, Santa Barbara, CA) tool with a $\\sim $ 20 nm diameter tip.",
"During continuous scan over the selected region with 0.1 Hz frequency, digitisation was carried out resulting in $25\\times 100$ nm$^2$ pixels in $x\\times y$ .",
"The oscillation frequency of the AFM needle was 190 kHz.",
"Table-top FT-IR spectrometer (Spotlight, PerkinElmer) was used with a detector array with $\\sim 6~\\mu $ m pixel resolution.",
"Localized modification of silk was carried out via exposure to 515 nm wavelength and 230 fs duration pulses (Pharos, Light Conversion Ltd.) in an integrated industrial laser fabrication setup (Workshop of Photonics, Ltd.).",
"Fibers were imaged and laser radiation was focused using an objective lens of numerical aperture $NA = 0.5$ (Mitutoyo).",
"Single pulse exposures were carried out with pulse energy, $E_p$ .",
"Optical and electron scanning microscopy (SEM) were used for structural characterisation of the laser modified regions.",
"Figure: Point spectra and chemical maps of T cross sections ofsilk (red) in epoxy (blue).",
"(a) Synchrotron ATR-FTIR spectra andits chemical map based on Amide A band.",
"(b) Table-top FT-IRspectra and the corresponding Amide A chemical map.",
"Note that theCCD array has the pixel size of 6.25×6.25μ6.25\\times 6.25~\\mu m 2 ^2comparable with the T-cross-section of the silk fiber.",
"Chemicalmaps (on the right-side) are presented as measured (withoutsmoothing)."
],
[
"Results and Discussion",
"A highly crystalline natural silk fiber can be thermally amorphised only through rapid $2\\times 10^3$ K/s thermal quenching from melt [22] as demonstrated for a very tiny amounts of silk measured in nanograms (1 ng occupies a sphere of 5.7 $\\mu $ m diameter).",
"Amorphous silk is water soluble and can be used as 3D printing material for scaffolds, desorbable implants [23], [24], [25], bio-resists [26], and even be utilised as an electron beam resist [27], [28].",
"Amorphous-to-crystalline transition of silk fibroin is usually achieved via a simple water and alcohol bath processing at moderately elevated temperatures $\\sim 80^\\circ $ [28].",
"An UV 266 nm wavelength nanosecond pulsed laser irradiation was also used to enrich amorphous silk fibroin with crystalline $\\beta $ -sheets [29].",
"To realise a fast thermal quenching and to retrieve the amorphous silk phase [19], we used ultrashort 230 fs duration and 515 nm wavelength laser pulses tightly focused into focal spot of $d = 1.22\\lambda /NA \\simeq 1.3~\\mu $ m; the numerical aperture of the objective lens was $NA = 0.5$ and the wavelength was $\\lambda = 515$ nm.",
"Single pulse irradiation was carried out to ablate nanograms of silk and create a molten phase which is thermally quenched fast enough [22] to prevent crystallisation (Fig.",
"REF (b)).",
"Threshold of optically recognisable modification of silk fiber during laser irradiation was at 8 nJ which corresponds to 2.8 TW/cm$^2$ average power (0.6 J/cm$^2$ fluence) per pulse and is typical for polymer glasses [30].",
"Nano-IR spectrum, the low-frequency band of the AFM detected height changes in response to a spectral sweep of excitation at the IR absorbance bands at 1 kHz frequency was measured on a transverse (T) microtome cross section of silk fiber embedded into a micro-thin epoxy (Fig.",
"REF (a)).",
"Single wavelength chemical map measured at the specific Amide I band of 1660 cm$^{-1}$ associated with the amorphous silk components is shown in (b) with clearly discernable amorphous rim of the ablation crater, which, in contrast, is not recognisable at the crystalline $\\beta $ -sheet region of 1700 cm$^{-1}$ (inset of (b)).",
"Typical single point measurement spectra are shown in Fig.",
"REF (c) for the Amide I region with clear distinction between epoxy matrix, amorphous, and crystalline counterparts of silk.",
"Amorphous silk of the molten quenched phase has only 200 nm thickness as observed by SEM and AFM, however, it was distinguished using illumination at the corresponding absorption bands ($\\lambda _{ex} \\simeq 6~\\mu $ m).",
"Thin microtomed slices placed on a high thermal conductivity substrate have been shown to enhance speed of nano-IR imaging since a thermalisation time scale is $t_{th}\\simeq \\rho c_ph^2/\\eta $ [31], where $h$ is the thickness of the film, $\\eta $ is thermal conductivity, $\\rho $ is the mass density, and $c_p$ is the specific heat capacity."
],
[
"Conclusions and outlook",
"It was shown that by using micro-thin slices which are sub-wavelength at the IR spectral range of interest, it was possible to obtain spectral band readout using nano-IR method - the surface height changes due to thermal expansion following the absorbance spectrum of silk.",
"Amorphous silk created by ultra-fast thermal quenching at the irradiation location of fs-laser pulse was distinguished on the chemical map and imaged with lateral resolution defined by digitisation $x\\times y \\equiv 25\\times 100$ nm$^2$ and can potentially reach the limit defined by tip [32] which was $20\\times 20$ nm$^2$ in this study.",
"Chemical mapping result acquired using the nano-IR method is consistent with far-field spectroscopy of silk carried out with table-top and synchrotron FT-IR (see Fig.",
"REF (b)).",
"The far-field FT-IR detectors have a typical pixel size of $\\sim 6~\\mu $ m which limits the resolution to the entire T-cross-section of the silk fiber, however, a good correspondence with nanoscale IR spectral mapping is confirmed between different spectroscopic methods.",
"Laser-induced amorphisation of crystalline silk has been spectroscopically resolved with high spatial resolution.",
"We can envisage, that a simple principle of the nano-IR technique allows for a coupling with synchrotron light sources and to complement it with a phase and amplitude mapping using scanning near-field microscopy.",
"The high brightness of synchrotron radiation would also enable a fast mapping required for temporally resolved evolution of photo or thermally excited processes, and even a molecular alignment mapping could be realised by using the four polarisation method [33].",
"These functionalities will bring new developments into a cutting edge nanoscale molecular characterisation."
],
[
"Acknowledgments",
"J.M.",
"acknowledges a partial support by a JSPS KAKENHI Grant No.16K06768.",
"We acknowledge the Swinburne's startup grant for Nanotechnology facility and partial support via ARC Discovery DP130101205 and DP170100131 grants.",
"Experiments were carried out via beamtime project No.",
"11119 at the Australian Synchrotron IRM Beamline.",
"Window-on-Photonics R&D, Ltd. is acknowledged for joint development grant and laser fabrication facility."
]
] | 1709.01799 |
[
[
"The Cosmological Memory Effect"
],
[
"Abstract The \"memory effect\" is the permanent change in the relative separation of test particles resulting from the passage of gravitational radiation.",
"We investigate the memory effect for a general, spatially flat FLRW cosmology by considering the radiation associated with emission events involving particle-like sources.",
"We find that if the resulting perturbation is decomposed into scalar, vector, and tensor parts, only the tensor part contributes to memory.",
"Furthermore, the tensor contribution to memory depends only on the cosmological scale factor at the source and observation events, not on the detailed expansion history of the universe.",
"In particular, for sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of $(1 + z)$."
],
[
"Introduction",
"The passage of gravitational radiation through a configuration of test particles that make up a gravitational wave detector can induce a permanent change in the relative separation of the test particles, a phenomenon which has come to be known as the gravitational wave memory effect.",
"The memory effect on a flat background was first recognized in the linear regime for sources in non-relativistic motion by Zel'dovich and Ponarev [1].",
"Afterwards, Christodoulou [2] discovered that there could be additional contributions to memory arising from the nonlinearity of the Einstein equation and associated with the Bondi flux of the gravitational waves to null infinity.",
"Shortly thereafter, it was argued that the nonlinear memory of Christodoulou could be interpreted as corresponding to a linear memory caused by the effective stress-energy associated with the primary gravitational radiation [3], [4].",
"More recently, it has been found to be useful to make a distinction between ordinary and null memory in the linearized gravity context [5],[6]: Ordinary memory is caused by massive matter sources and null memory is caused by null matter sources.",
"The gravitational radiation emitted from a source event involving both massive and null matter will induce both ordinary and null memory [5],[6],[7].",
"Neither the ordinary nor null memory should be viewed as a tidal effect with a Newtonian analogue but rather as a byproduct of the gravitational radiation emitted from a burst-type event [7],[8].",
"The above work has concerned itself with memory on an asymptotically flat spacetime.",
"If a spacetime is not asymptotically flat, it is not clear what “memory” should mean, even if we treat the gravitational radiation as a perturbation off of a background metric.",
"Memory has been defined in terms of the net change in the separation of test particles, but this could include motion due to the background curvature rather than the radiation.",
"For example, in cosmological Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, the proper separation of FLRW observers will change with time.",
"Furthermore, if there is no notion of null infinity, it is not clear where we should put our detector so that it will be exposed to radiation but isolated from other non-radiative gravitational forces.",
"Recently, Bieri, Garfinkle and Yau [9], Kehagias and Riotto [10], and Chu [11],[12] have considered the memory effect in FLRW spacetimes.",
"However, their choice of methods for resolving the above difficulties have so far limited the applicability of their results.",
"Bieri and Garfinkle consider Weyl tensor perturbations, so their analysis is greatly simplified by working in vacuo, and, consequently, they have considered only memory in a background vacuum de Sitter universe.",
"Meanwhile, Kehagias and Riotto's use of BMS transformations depends on the existence of a null infinity for the FLRW spacetime, so they restrict consideration to a decelerating universe without a cosmological constant.",
"Furthermore, both groups consider only null memory.",
"While Chu does not make either of these specific restrictions, his preferred definition of memory does not distinguish test particle motion due to radiative and non-radiative gravity.",
"In this paper, we will investigate the total memory effect—both ordinary and null—in a general, spatially flat, FLRW spacetime.",
"We will simplify the problem not by limiting the cosmological models under consideration, but rather the kinds of radiation sources.",
"Specifically, we will consider—in the context of linearParticle-like sources in Einsten's equation do not make sense outside of the context of linear perturbation theory [13].",
"perturbation theory off of an FLRW background—only point-particle sources, i.e., sources whose stress-energy is confined by Dirac delta functions to worldlines that meet at a vertex, which we will call the source event.",
"We previously considered such sources in Minkowski spacetime [7], [8], and found that the retarded solution gives rise to a contribution to the curvature tensor of the form of the derivative of a delta function in retarded time.",
"Upon integration of the geodesic deviation equation, this derivative of a delta function in curvature gives rise to a well defined memory effect—i.e., a change in the separation of test particles coincident in time with the passage of the radiation from the source event—that includes both the ordinary and null memory [7], [8].",
"Thus, for the idealized sources that we consider, the memory effect can be associated with the presence of a derivative of a delta function in the Riemann curvature of the retarded solution arising from the source event.",
"This enables us to distinguish the memory effect induced by the passage of radiation from non-radiative gravitational effects, and it does not require us to take limits to null infinity.",
"It thereby yields a well defined notion of the memory effect that is applicable to linearized perturbations about arbitrary background spacetimes.",
"Another advantage to considering the above idealized particle sources, where the radiation emerges from a single “source event” in the background spacetime, is that—since all spacetimes are “locally flat” on sufficiently small scales—there is a well defined notion of having the “same source” in different spacetimes.",
"Similarly, there is a well defined notion of having the “same detector”—i.e., geodesic test particles initially at rest and with small separation—in different spacetimes.",
"Thus, we can compare the memory effect in two different spacetimes provided only that we specify the location and “rest frame” of both the source and the detector in the two spacetimes.",
"Since any spatially flat FLRW spacetime is conformal to Minkowski spacetime, it is particularly useful to state our results by comparing the memory effect in FLRW spacetime to that in Minkowski spacetime.",
"Stated in this manner, the main result of our paper is as follows: Consider a spatially flat FLRW solution to Einstein's equation with arbitrary fluid matter, and with a given particle source perturbation, as described above.",
"Now place the same source and detector in Minkowski spacetime such that the source and detector are at rest with respect to each other and the source is at a distance, $d$ , in Minkowski spacetime equal to the luminosity distance, $d_L$ , in the FLRW spacetime.",
"Then the memory effect in the FLRW spacetime is enhanced over the Minkowski value by a factor of $(1+z)$ , where $z$ denotes the redshift factor between the source and observer/detector.",
"This result applies to both ordinary and null memory.",
"It is in agreement with the results obtained for null memory in the special cases considered in [9] and [10].",
"Note that in a spatially flat FLRW spacetime, the luminosity distance $d_L$ and the angular diameter distance, $d_A$ , are related by $d_A = d_L/(1+z)^2 $ It follows that for a Minkowski source at $d = d_A$ , the memory effect in the FLRW spacetime will be decreased from the Minkowski value by a factor of $(1+z)^{-1}$ .",
"Finally, it should be noted that although the memory effect in a spatially flat FLRW spacetime is related to the Minkowski memory effect in this simple way, the waveforms will be different; in particular, there will be “tail effects” in the FLRW spacetime.",
"Although our analysis is restricted to the context of linear perturbation theory with idealized particle sources, we expect that our main result stated above should be valid completely generally for any sources whose spatial and time variation scales are small compared with the Hubble scale.",
"Indeed, the main difficulty in generalizing our results to non-particle-like sources and to the nonlinear regime would be to give a precise definition of “memory” outside of the context we consider.",
"Thus, if one wishes to compute the memory effect resulting from, say, the coalescence of two black holes in a distant galaxy in a spatially flat FLRW spacetime, it should suffice to compute the memory effect arising from a similar coalescence in an asymptotically flat spacetime and then use the above correspondence.",
"However, we shall not attempt to formulate or prove such a generalization here.",
"We shall begin in section by describing the particle sources that we shall consider and characterizing the memory effect for perturbations of arbitrary curved spacetimes.",
"In section , we analyze linearized perturbations off of spatially flat FLRW spacetimes with such particle sources.",
"In section , we consider the tensor mode contribution to memory.",
"We show that only the “light cone portion” of the retarded Green's function will contribute to memory, and that its contribution can be related in a simple way to the memory caused by similar sources in a flat spacetime.",
"In the Appendix, we show that the scalar modes do not contribute to memory.",
"Latin indices from the early alphabet ($a,b, \\dots $ ) denote abstract spacetime indices.",
"Greek indices ($\\mu , \\nu , \\dots $ ) denote spacetime components of tensors, whereas Latin indices from the mid-alphabet ($i,j, \\dots $ ) denote spatial components."
],
[
"Particle Sources",
"In asymptotically flat spacetimes, the memory effect can be characterized in a precise manner by considering a detector composed of test particles near null infinity.",
"Radiation effects fall off as $1/r$ , whereas Newtonian tidal effects fall off as $1/r^3$ , so by considering only the $O(1/r)$ effects on the test particles, we can distinguish between effects produced by gravitational radiation and all other gravitational effects.",
"However, in a non-asymptotically-flat spacetime, it is not clear how even to define memory, since there is no obvious way to make a clean distinction between effects due to “radiation” as compared with other tidal gravitational effects.",
"In our previous investigation of the memory effect in linearized gravity [7],[8], we considered the idealized process of the instantaneous decay of a massive particle into two other particles.",
"We found that the retarded solution to the linearized Einstein equation with such a source has the property that the $O(1/r)$ part of the curvature has the form of a derivative of a delta-function of retarded time at the retarded time of the decay event.",
"Integration of the geodesic deviation equation then shows that the $O(1/r)$ effect on test particles is to produce a sharp step function in their relative separation.",
"Thus, for this kind of idealized source, the memory effect can be characterized by the presence of a derivative of a delta-function in the linearized curvature and a corresponding step function behavior in the relative separation of test particlesOf course, if one were to consider a less idealized source with a smoothed out energy-momentum tensor, then the Riemann tensor also will be smoothed out, and the relative separation of the test particles will not undergo a sharp, sudden change in separation; rather, separation of the particles that results in a memory effect would occur continuously on the same timescale as that of the event itself.. For our present purposes, the main advantage of considering sources consisting of particles undergoing instantaneous interactions is that the characterization of the memory effect in terms of derivative of a delta-function behavior of the curvature holds at all distances from the interaction event, i.e., one does not need to go to null infinity to extract this characterization of the memory effect.",
"This characterization may therefore be imported straightforwardly to other spacetimes.",
"Thus, in this paper, we shall restrict consideration to linearized gravity off of a smooth background spacetime, with a linearized perturbation sourced by (massive or massless) point particles.",
"The interactions of the particles will be modeled by having their worldlines intersect (and, possibly, begin or end) at a single event, $q$ , in spacetime as illustrated in figure 1.",
"Conservation of stress-energy then requires that (i) the particle worldlines are geodesics [14] away from $q$ , and (ii) total 4-momentum is conserved at $q$ .",
"“Memory” will then be characterized by the presence of a derivative of a delta-function in the curvature of the retarded solution with this source.",
"This characterization does not require that the detector be placed near “infinity.” Figure: A spacetime diagram of the sort of gravitational wave source we will consider.",
"Here 5 point particles enter a single “source event” qq, and 3 emerge.",
"The worldlines of the incoming and outgoing particles must be timelike or null geodesics.To specify more precisely the type of source we consider, we assume that local coordinates $(t,\\mathbf {x})$ have been introduced in a neighborhood of $q$ so that $\\nabla t$ is past-directed timelike and so that the event $q$ is labeled by $t=\\mathbf {x} = 0$ .",
"The worldline, $\\gamma $ , of each incoming massive particle must be a timelike geodesic [14] with endpoint at $q$ .",
"We can parametrize $\\gamma $ by $t$ and specify it by giving $\\mathbf {x}(t) = \\mathbf {z}(t)$ , where $\\mathbf {z}(0) = \\mathbf {0}$ .",
"The stress-energy of each incoming massive particle then takes the form $T^{(M, {\\rm in})}_{ab} = m u_a u_b \\, \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(t)\\right) \\frac{1}{\\sqrt{-g}} \\frac{d\\tau }{dt} \\Theta (-t) \\, .$ Here $u^{a}$ is the unit tangent (4-velocity) to $\\gamma $ , $\\tau $ is the proper time along $\\gamma $ , $\\Theta $ is the Heaviside step function, and $\\delta ^{(3)}$ is the “coordinate delta-function,” i.e., $\\int \\delta ^{(3)}(\\mathbf {x} - \\mathbf {z}(t)) d^3 \\mathbf {x} = 1$ .",
"Each incoming massless particle moves on a null geodesic, $\\alpha $ , given by $\\mathbf {x}(t) = \\mathbf {y}(t)$ , with $\\mathbf {y}(0) = \\mathbf {0}$ .",
"The stress-energy of each incoming massless particle takes the form $T^{(N, {\\rm in})}_{ab} = k_a k_b \\, \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {y}(t)\\right) \\frac{1}{\\sqrt{-g}} \\frac{d\\lambda }{dt} \\Theta (-t) \\, .$ Here, $\\lambda $ is an affine parameter of $\\alpha $ and $k^a$ is the corresponding tangent, with the scaling of $\\lambda $ chosen so that (REF ) holds.",
"The stress-energy of each of the outgoing massive and massless particles takes the form of (REF ) and (REF ) except that $\\Theta (-t)$ is replaced by $\\Theta (t)$ .",
"The total stress-energy of the particle sources we consider takes the form $T^{(P)}_{ab} = \\sum _{l, {\\rm in}} T^{(M,l)}_{ab} + \\sum _{n, {\\rm in}} T^{(N,n)}_{ab} + \\sum _{l^{\\prime }, {\\rm out}} T^{(M, l^{\\prime })}_{ab} + \\sum _{n^{\\prime }, {\\rm out}} T^{(N, n^{\\prime })}_{ab} \\, ,$ where each $T^{(M,l)}_{ab}$ takes the form of (REF ), each $T^{(N,n)}_{ab}$ takes the form of (REF ), and each $T^{(M, l^{\\prime })}_{ab}$ and $T^{(N,n^{\\prime })}_{ab}$ also take these forms with $\\Theta (t)$ replaced by $\\Theta (-t)$ .",
"Conservation of stress-energy, $\\nabla ^a T_{ab} = 0$ , holds in the distributional sense away from $q$ by virtue of the fact that each particle moves on a geodesic [14].",
"Conservation of stress-energy will hold at $q$ if and only if we have at $q$ $\\sum _{l, {\\rm in}} m^{(l)}u^{(l)}_a+ \\sum _{n, {\\rm in}} k^{(n)}_a = \\sum _{l^{\\prime }, {\\rm out}} m^{(l^{\\prime })}u^{(l^{\\prime })}_a + \\sum _{n^{\\prime }, {\\rm out}} k^{(n^{\\prime })}_a \\, ,$ Equation (REF ) with condition (REF ) defines the sources that we consider in this paper.",
"We are interested in the solution to the linearized Einstein equation “produced by such a source” in an arbitrary background spacetime.",
"For hyperbolic equations, what we mean by “produced by a source” is the solution obtained by convolving the source with the retarded Green's function.",
"In general spacetimes, issues of convergence of the retarded solution will arise from the contribution of the sources at arbitrarily early timesEven in Minkowski spacetime, the contribution of null sources at arbitrarily early times does not converge to a distribution [8]..",
"However, we are not interested in such issues of convergence here, but rather the effects arising from near the source event $q$ .",
"Thus, we shall simply consider the contributions to the retarded Green's function integral arising from the above particle sources in a small neighborhood of $q$ .",
"The memory effect will then be identified with the presence of a derivative of a delta-function in the curvature “produced” by these particle sources near event $q$ ."
],
[
"Perturbation Theory in Cosmological Spacetimes",
"We now wish to consider linearized perturbations off of a spatially flat FLRW background, $ds^2= - d \\tau ^2 +a^2(\\tau )(dx^2 + dy^2 + dz^2).$ As usual, it is convenient to introduce conformal time $d\\eta =d\\tau /a$ so that the background FLRW metric takes the manifestly conformally flat form $ds^2 =a^2(\\eta )\\left(- d \\eta ^2 + dx^2 + dy^2 + dz^2\\right).$ Throughout the rest of the paper, “0” and “$i$ ” (i.e., spatial) indices will denote components of tenors with respect to these coordinates, and an overdot will denote a derivative with respect to $\\eta $ .",
"We will write $\\partial ^i=\\delta ^{ij}\\partial _j$ and $\\nabla ^2 = \\partial ^i\\partial _i = \\delta ^{ij} \\partial _i \\partial _j$ , i.e., $\\nabla ^2$ is the Laplacian with respect to the spatial metric $\\delta _{ij}$ given by $dx^2 + dy^2 +dz^2$ .",
"We assume that Einstein's equation holds (possibly with a cosmological constant $\\Lambda $ ) and that the matter stress-energy—apart from the particle matter that we will add as a perturbation—is that of a perfect fluid $T^{(F)}_{ab}=(\\rho +p)u_a u_b+pg_{ab} \\, ,$ with 4-velocity $u^a$ , density $\\rho $ and pressure $p$ .",
"The fluid is assumed to be described by a one-parameter (“barotropic”) equation of state $p=p(\\rho )$ .",
"The density and pressure are perturbations away from homogeneous background values $\\bar{\\rho }$ and $\\bar{p}$ which satisfy the Friedmann equations $\\left(\\frac{1}{a}\\frac{da}{d\\tau }\\right)^2=\\frac{1}{3}\\left(8\\pi \\bar{\\rho }+\\Lambda \\right) \\, ,\\\\\\frac{1}{a}\\frac{d^2a}{d\\tau ^2}=\\frac{1}{3}\\left(-4\\pi (\\bar{\\rho }+3\\bar{p})+\\Lambda \\right) \\, .",
"$ We write the perturbed metric as $g_{ab}= \\bar{g}_{ab}+a^2h_{ab}$ where $\\bar{g}_{ab}$ denotes the background FLRW metric.",
"The perturbed fluid is described by $\\delta u^\\mu $ , $\\delta \\rho $ , and $\\delta p = c_s^2 \\delta \\rho $ , where $c_s^2=\\frac{d p}{d\\rho } \\, .$ We wish to consider the metric perturbation resulting from the presence of a particle stress-energy of the form (REF ).",
"The particle sources are assumed to have no direct interaction with the fluid present in the FLRW background; the particle stress-energy is separately conserved.",
"However, since the particles affect the perturbed metric, they automatically affect the fluid (even at the linearized level), so the fluid perturbations cannot be ignored.",
"Analysis of the perturbations is most easily done using the gauge-invariant methods of Bardeen [15] with modifications by Durrer [16],[17] allowing for additional forms of matter perturbationsBoth Bardeen and Durrer allow for general stress-energies with non-fluid properties like anisotropic pressures.",
"However, as we have discussed in section , we want our perturbed fluid and particles to interact only gravitationally, which means that the stress-energies of the fluid and the particles must be conserved independently.",
"Bardeen does not discuss this scenario, but it corresponds to Durrer's notion of cosmological seeds..",
"These methods rely on decomposing the metric, fluid stress-energy, and particle stress-energy perturbations into scalar, vector, and tensor parts, and working with gauge invariant quantities in each sector.",
"We can decompose a general symmetric tensor field $X_{ab}$ on spacetime into its scalar, vector, and tensor parts by writing its coordinate components as $X_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi &\\partial _i\\chi \\;\\;+\\;\\;\\xi _i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi \\\\+\\\\ \\xi _i\\end{array}}&{\\begin{array}{c}\\psi \\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega \\\\+\\partial _{(i}\\zeta _{j)}+{X}_{ij}\\end{array}}\\end{array}\\right) \\, ,$ where the scalar parts are given byIf the spatial slices have topology $\\bf {R}^3$ , we need to impose boundary conditions at infinity in order to get a unique solution to the Poisson equations for $\\chi $ and $\\omega $ (and $\\zeta _i$ below), which, in turn, may put restrictions on the asymptotic behavior of $X_{ab}$ .",
"However, as we are ultimately interested in singular behavior of the perturbations, it does not matter what solutions of the Poisson equations we choose.",
"For convenience, we shall assume that the spatial slices have the topology of three-tori, with the dimensions of the tori being much larger than the dimensions of the physical problem.",
"The solutions are then unique up to the addition of constants, which do not affect the decomposition.",
"=X00 2=iX0i =13ijXij 22=32(ij-13ij2)Xij , the vector parts are given by i=X0i-i 2i=2(jXij-i-232i) , and the tensor part is ${X}_{ij} =X_{ij}-\\psi \\delta _{ij}-\\left(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2\\right)\\omega -\\partial _{(i}\\zeta _{j)} \\, .",
"$ If the metric perturbation is written in this way, $h_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi ^{(h)}&\\partial _i\\chi ^{(h)}\\;\\;+\\;\\;\\xi ^{(h)}_i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi ^{(h)}\\\\+\\\\ \\xi ^{(h)}_i\\end{array}}&{\\begin{array}{c}\\psi ^{(h)}\\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega ^{(h)}\\\\+\\partial _{(i}\\zeta ^{(h)}_{j)}+{h}_{ij}\\end{array}}\\end{array}\\right),$ then =(h)+2(h)+2aa(h)+(h)+aa(h) =(h)+2aa(h)-132(h)-aa(h) are gauge-invariant scalar quantities, whereas $\\Xi _i=\\xi ^{(h)}_i-\\dot{\\zeta }^{(h)}_i$ is a gauge-invariant vector quantity, and ${h}_{ij}$ is a gauge-invariant tensor quantity.",
"The above two scalar fields, $\\Phi $ and $\\Psi $ , one transverse three-vector field, $\\Xi _i$ , and one transverse-traceless three-tensor field, ${h}_{ij}$ , contain all of the physical (non-gauge) information concerning the metric perturbation.",
"The stress-energy tensor of the particles (REF ) can also be decomposed in this way: $T^{(P)}_{\\mu \\nu }=\\left(\\begin{array}{c|c}\\varphi ^{(P)}&\\partial _i\\chi ^{(P)}\\;\\;+\\;\\;\\xi ^{(P)}_i\\\\\\hline {\\begin{array}{c}\\partial _i\\chi ^{(P)}\\\\+\\\\ \\xi ^{(P)}_i\\end{array}}&{\\begin{array}{c}\\psi ^{(P)}\\delta _{ij}+(\\partial _i\\partial _j-\\frac{1}{3}\\delta _{ij}\\nabla ^2)\\omega ^{(P)}\\\\+\\partial _{(i}\\zeta ^{(P)}_{j)}+{T}_{ij}\\end{array}}\\end{array}\\right).$ Because there is no “background” particle stress-energy, each of the individual component fields $\\varphi ^{(P)},\\chi ^{(P)},$ etc.",
"are already gauge invariant to first order.",
"These quantities are related to $T^{(P)}_{\\mu \\nu }$ by eqs.",
"()-(REF ).",
"Since $T^{(P)}_{\\mu \\nu }$ is distributional, these quantities will also be distributional.",
"We can also find gauge-invariant combinations of the perturbed stress-energy $\\delta T^{(F)}_{\\mu \\nu }$ of the fluid, (REF ).",
"We define $\\delta _\\rho = \\frac{\\rho -\\bar{\\rho }}{\\bar{\\rho }} \\, .$ We decompose the perturbed 4-velocity as $\\delta u^\\mu =\\frac{1}{a}\\left(\\begin{array}{c}\\delta u^0\\\\\\partial ^i v+v^i\\end{array}\\right)$ with $\\partial _i v^i = 0$ , and we remind the reader that $\\partial ^i v = \\delta ^{ij} \\partial _j v$ .",
"Note that the quantity $\\delta u^0$ is not independent, since it is fixed by the normalization condition $g_{ab}u^au^b=-1$ .",
"In terms of these quantities and the perturbed metric, we can obtain the following gauge-invariant fluid variables: $V=v+\\frac{1}{2}\\dot{\\omega }^{(h)}\\\\A=\\delta _\\rho +3\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\frac{1}{2}\\left(\\psi ^{(h)}-\\frac{1}{3}\\nabla ^2\\omega ^{(h)}\\right)-\\frac{\\dot{a}}{a}V-\\Phi \\right)\\\\W_i=\\delta _{ij}v^j+\\frac{1}{2}\\dot{\\xi }^{(h)}_i.$ The fields $V$ , $A$ , and $W_i$ thus provide us, respectively, with gauge-invariant measures of fluid's peculiar velocity with respect to the Hubble flow, its perturbed density, and its vorticity.",
"The linearized Einstein equation decomposes into decoupled sets of equations involving the scalar, vector, and tensor parts of the perturbations.",
"These equations can be written entirely in terms of the gauge-invariant quantities introduced above.",
"The scalar equations are $\\nabla ^2\\Psi -3\\frac{\\dot{a}}{a}\\left(\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi \\right) = -8\\pi \\left(a^2\\bar{\\rho }A-3a\\dot{a}(\\bar{\\rho }+\\bar{p})V+\\varphi ^{(P)}\\right)\\\\\\partial _i\\left(\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi \\right) = -8\\pi \\left(a^2(\\bar{\\rho }+\\bar{p})\\partial _iV-\\partial _i\\chi ^{(P)}\\right)\\\\\\partial _i\\partial _j\\left(\\Psi -\\Phi \\right) = -16\\pi \\partial _i\\partial _j\\omega ^{(P)}$ $\\ddot{\\Psi }+2\\frac{\\dot{a}}{a}\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\dot{\\Phi }+\\left(2\\frac{\\ddot{a}}{a}-\\left(\\frac{\\dot{a}}{a}\\right)^2\\right)\\Phi +\\frac{2}{3}\\nabla ^2\\left(\\Psi -\\Phi \\right)\\\\=-16\\pi \\left(a^2c_s^2(\\bar{\\rho })A-3(\\bar{\\rho }+\\bar{\\phi })\\frac{\\dot{a}}{a}V+\\psi ^{(P)}\\right) \\, .$ The vector equations are $\\nabla ^2\\Xi _i=-8\\pi \\left(2a^2(\\bar{\\rho }+\\bar{p})W_i-\\xi ^{(P)}_i\\right)\\\\\\partial _{(i}\\dot{\\Xi }_{j)}+2\\frac{\\dot{a}}{a}\\partial _{(i}\\Xi _{j)}=-8\\pi \\partial _{(i}\\zeta ^{(P)}_{j)} \\, .$ Finally, the tensor equation is $-\\ddot{{h}}_{ij}-2\\frac{\\dot{a}}{a}\\dot{{h}}_{ij}+\\nabla ^2{h}_{ij} =-16\\pi {T}_{ij} \\, .$ If the various perturbation fields do not grow in an unbounded fashion at large distances, the unique solutions to () and () are $\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\Phi =-8\\pi \\left(a^2(\\bar{\\rho }+\\bar{p})V-\\chi ^{(P)}\\right)\\\\\\Psi -\\Phi =-16\\pi \\omega ^{(P)} \\, .$ Equations (REF ) and (REF ) then simplify to $\\nabla ^2\\Psi =-8\\pi \\left(a^2\\bar{\\rho }A+\\varphi ^{(P)}+3\\chi ^{(P)}\\right)$ $\\ddot{\\Psi }+2\\frac{\\dot{a}}{a}\\dot{\\Psi }+\\frac{\\dot{a}}{a}\\dot{\\Phi }+\\left(2\\frac{\\ddot{a}}{a}-\\left(\\frac{\\dot{a}}{a}\\right)^2\\right)\\Phi \\\\=-16\\pi \\left(a^2c_s^2\\left(\\bar{\\rho }A-3(\\bar{\\rho }+\\bar{p})\\frac{\\dot{a}}{a}V\\right)+\\psi ^{(P)}-\\frac{2}{3}\\nabla ^2\\omega ^{(P)}\\right) \\, .$ Similarly, () implies that $\\dot{\\Xi }_{i}+2\\frac{\\dot{a}}{a}\\Xi _{i}=-8\\pi \\zeta ^{(P)}_{i} \\, .$ The full set of Einstein's equation thus reduces to (REF )-(REF ) together with (REF ) and (REF ).",
"The perturbed conservation of stress energy for the fluid, $\\delta [\\nabla ^\\mu T^{(F)}_{\\mu \\nu }] = 0$ , yields the scalar equations $\\dot{V}+\\frac{\\dot{a}}{a}V=-\\frac{c_s^2\\bar{\\rho }E}{\\bar{\\rho }+\\bar{p}}-\\frac{1}{2}\\Phi \\\\\\dot{A}-3\\frac{\\bar{p}}{\\bar{\\rho }}\\frac{\\dot{a}}{a}A=-\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\nabla ^2V-3\\chi ^{(P)}\\right)$ as well as the vector equation $\\dot{W}_i-3\\frac{\\dot{a}}{a}c_s^2W_i=0 \\, .$ Similarly, conservation of stress-energy for the particles can also be expressed in terms of the fields (REF ) as $\\dot{\\varphi }^{(P)}+\\frac{\\dot{a}}{a}\\varphi ^{(P)}-\\nabla ^2\\chi ^{(P)}+3\\frac{\\dot{a}}{a}\\psi ^{(P)}=0\\\\\\dot{\\chi }^{(P)}+2\\frac{\\dot{a}}{a}\\chi ^{(P)}-\\psi ^{(P)}-\\frac{2}{3}\\nabla ^2\\omega ^{(P)}=0\\\\\\dot{\\xi }^{(P)}_i+2\\frac{\\dot{a}}{a}\\xi ^{(P)}_i-\\frac{1}{2}\\nabla ^2\\zeta ^{(P)}_i=0.$ A very useful equation for $A$ can be derived as follows [16]: We differentiate () with respect to $\\eta $ , and substitute from (REF ) to eliminate $\\dot{V}$ .",
"Then we use () to eliminate $\\nabla ^2 V$ , and we use () and (REF ) to eliminate $\\nabla ^2\\Phi $ .",
"Finally, we use () to eliminate $\\dot{\\chi }^{(P)}$ .",
"We thereby obtain a wave equation for $A$ with particle sources, $-\\ddot{A}-\\frac{\\dot{a}}{a}\\left(1+3c_s^2-6\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\dot{A}+c_s^2\\nabla ^2 A+3\\left(\\frac{\\ddot{a}}{a}\\frac{\\bar{p}}{\\bar{\\rho }}-3\\left(\\frac{\\dot{a}}{a}\\right)^2\\left(c_s^2-\\frac{\\bar{p}}{\\bar{\\rho }}\\right)+a^2\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\frac{4\\pi }{3}\\bar{\\rho }\\right)A\\\\=-4\\pi a^2\\left(1+\\frac{\\bar{p}}{\\bar{\\rho }}\\right)\\left(\\varphi ^{(P)}+3\\psi ^{(P)}\\right).$ Physically, this equation describes the propagation of sound waves in the fluid.",
"Although there is no direct coupling between the particles and the fluid, there are “particle source terms” in (REF ) resulting from the gravitational interactions between the particles and the fluid.",
"As previously explained, the memory effect will be identified with the presence of a derivative of a delta function in the curvature.",
"The curvature is given by an expression involving at most 2 derivatives of the metric variables.",
"In particular, in the Newtonian gauge, the perturbation to the electric components of the Riemann tensor is [18] $\\delta R_{i00}^{\\quad j}=-\\frac{1}{2}\\Bigg (\\left(\\partial _i\\partial _k-\\frac{1}{3}\\delta _{ik}\\nabla ^2\\right)\\Phi +\\left(\\ddot{\\Psi }+\\frac{\\dot{a}}{a}(\\dot{\\Psi }-\\dot{\\Phi })\\right)\\delta _{ik}\\\\-\\partial _{(i}\\left(\\dot{\\Xi }_{k)}+\\frac{\\dot{a}}{a}\\Xi _{k)}\\right)+\\left(\\ddot{{h}}_{ik}+\\frac{\\dot{a}}{a}\\dot{{h}}_{ik}\\right)\\Bigg )\\delta ^{jk}.$ Thus, a derivative of a delta function in the curvature requires a step function (or worse) discontinuity in the gauge invariant metric variables.",
"We are now in a position to analyze how discontinuities could arise.",
"First, it is important to note that the equations ()-(REF ) giving the scalar, vector, and tensor parts of a tensor $X_{\\mu \\nu }$ involve solving elliptic and/or algebraic equations, with “source” given by components of $X_{\\mu \\nu }$ and their derivatives.",
"It follows immediately that the scalar, vector, and tensor parts of $X_{\\mu \\nu }$ are smooth wherever $X_{\\mu \\nu }$ itself is smooth.",
"In particular, the scalar, vector, and tensor parts of the particle stress-energy (REF ) are smooth away from the worldlines of the particles.",
"Consider, now, the scalar perturbations.",
"Eqs.",
"(REF ) and (REF ) are elliptic in $\\Phi $ and $\\Psi $ , so these quantities—which fully characterize the scalar part of the metric perturbation—can be singular only where the source terms in these equations are singular.",
"These source terms involve the scalar part of the particle source and the quantity $A$ .",
"The scalar part of the particle source is smooth away from the particle world lines.",
"The quantity $A$ satisfies the hyperbolic equation (REF ), which, in turn is sourced by the scalar parts of the particle stress-energy.",
"We shall analyze the possible singular behavior of $A$ in the Appendix.",
"We shall show there that, although $A$ can be discontinuous along the sound cone of the source event $q$ , there cannot be any discontinuities in $\\Phi $ or $\\Psi $ .",
"Thus, no memory effect can occur in the scalar sector.",
"Consider, now, the vector perturbations.",
"The quantity $\\Xi _i$ satisfies the elliptic equation (REF ) and thus is smooth wherever the sources are smooth.",
"However, the particle source term $\\xi _i^{(P)}$ is smooth away from the worldlines of the particles and the fluid source term $W_i$ satisfies the source free evolution equation (REF ) and is thus nonsingular everywhere.",
"Thus, $\\Xi _i$ is smooth away from the particle worldlines and no memory effect can occur in the vector sector.",
"In the next section, we calculate the memory effect occurring in the tensor sector."
],
[
"The Retarded Gravitational Field and the Memory Effect",
"The tensor perturbations are described by the quantity ${h}_{ij}$ , which satisfies (see (REF )) $-\\ddot{{h}}_{ij}-2\\frac{\\dot{a}}{a}\\dot{{h}}_{ij}+\\nabla ^2{h}_{ij} =-16\\pi {T}_{ij} \\, , $ where ${T}_{ij}$ is the tensor part of the particle stress energy.",
"Thus, each component of ${h}_{ij}$ in the coordinates (REF ) satisfies a decoupled scalar wave equation and it suffices to analyze the behavior of solutions to the scalar wave equation.",
"We are interested in the contribution to the retarded integral ${h}_{ij}(x)=16\\pi \\int \\sqrt{-g(x^{\\prime })}d^4x^{\\prime }G^{\\rm ret}(x,x^{\\prime }){T}_{ij}(x^{\\prime })$ arising from a small neighborhood of the source event $q$ (see section ), where $G^{\\rm ret}(x,x^{\\prime })$ denotes the retarded Green's function for the scalar wave equation $- \\ddot{\\phi } - 2\\frac{\\dot{a}}{a} \\dot{\\phi } + \\nabla ^2 \\phi = -16\\pi T \\, .$ Specifically, we seek to determine whether a discontinuity can arise in ${h}_{ij}$ and, if so, to determine its magnitude.",
"Such discontinuities will give rise to derivative of delta function contributions to the curvature, which, in turn, will produce a memory effect.",
"To proceed, we need to know the form of $G^{\\rm ret}(x,x^{\\prime })$ .",
"Consider an equation of the general form $L[\\phi ]=g^{\\mu \\nu }\\partial _\\mu \\partial _\\nu \\phi +b^\\mu \\partial _\\mu \\phi +c\\phi =-16\\pi T \\, ,$ where $g_{\\mu \\nu }$ is a metric of Lorentz signature.",
"It is well known [19], [20] that, in 4 spacetime dimensions, the retarded Green's function for this equation takes the form $G^{\\rm ret}(x,x^{\\prime })=\\left[U(x,x^{\\prime })\\delta (\\sigma )+V(x,x^{\\prime })\\Theta (-\\sigma )\\right] \\Theta (t-t^{\\prime }) \\, , $ where $\\sigma $ denotes the squared geodesic distance between $x$ and $x^{\\prime }$ in the metric $g_{\\mu \\nu }$ and $t$ is a global time function.",
"The quantities $U$ (the Van Vleck-Morette determinant) and $V$ are smooth functions; we refer to $U(x,x^{\\prime })\\delta (\\sigma )$ as the “direct part” and $V(x,x^{\\prime })\\Theta (-\\sigma )$ as the “tail part” of $G^{\\rm ret}(x,x^{\\prime })$ .",
"In general, the form (REF ) for $G^{\\rm ret}(x,x^{\\prime })$ will hold only locally in a convex normal neighborhood, but in the case of (REF ), the spacetime metric corresponding to (REF ) is flat, and this form of $G^{\\rm ret}(x,x^{\\prime })$ holds globally, with $\\sigma (x,x^{\\prime })=-(\\eta -\\eta ^{\\prime })^2+(x-x^{\\prime })^2+(y-y^{\\prime })^2+(z-z^{\\prime })^2 \\, .$ The Van Vleck-Morette $U$ is determined by integrating an ODE along a geodesic connecting $x$ and $x^{\\prime }$ [19], [20].",
"The quantities appearing in this ODE depend on $g^{\\mu \\nu }$ and $b^\\mu $ but do not depend on $c$ .",
"We could integrate this ODE directly, but we can greatly simplify the calculation of $U$ by working with the rescaled variable ${\\tilde{\\phi }} = a\\phi $ (where $\\phi $ denotes a component of ${h}_{ij}$ ), which satisfies the equation $- \\ddot{\\tilde{\\phi }} + \\nabla ^2 {\\tilde{\\phi }} + \\frac{\\ddot{a}}{a}{\\tilde{\\phi }} =-16\\pi a T \\, .$ This change of variables eliminates the term involving $b^\\mu $ in (REF ), so $\\tilde{U}$ is determined by exactly the same equation for the wave equation in flat spacetime.",
"Thus we obtain the unique solution $\\tilde{U}(x,x^{\\prime }) = (4\\pi )^{-1}$ .",
"However, the retarded Green's function for $\\phi $ is related to the retarded Green's function for $\\tilde{\\phi }$ by [21]-[23] $G^{\\rm ret}(x,x^{\\prime })=\\frac{a(\\eta ^{\\prime })}{a(\\eta )}\\tilde{G}^{\\rm ret}(x,x^{\\prime }) \\, .$ Thus, we obtain $U(x,x^{\\prime })=\\frac{1}{4\\pi }\\frac{a(\\eta ^{\\prime })}{a(\\eta )} \\, .$ This holds for any FLRW universe, i.e., we have not assumed any particular equation of state $\\bar{P}= \\bar{P}(\\bar{\\rho })$ (and, thus, we have not assumed any particular expansion law) in the background spacetime.",
"By contrast, $V(x,x^{\\prime })$ will depend on the expansion history of the FLRW universe between $\\eta ^{\\prime }$ and $\\eta $ .",
"Note, however, that by spatial Euclidean invariance, $V$ depends on $(x,x^{\\prime })$ only via $\\eta $ , $\\eta ^{\\prime }$ , and $|\\mathbf {x} - \\mathbf {x}^{\\prime }|$ .",
"As previously stated, we are interested in the possible discontinuities in ${h}_{ij}$ resulting from the source behavior near $q$ , where we take $q$ to have coordinates $\\mathbf {x} = t =0$ .",
"To analyze this, let us first introduce a new toy mathematical problem, wherein we consider retarded solutions to the equation $-\\ddot{H}_{ij}-2\\frac{\\dot{a}}{a}\\dot{H}_{ij}+\\nabla ^2H_{ij} =-16\\pi T^{(P)}_{ij} \\, .$ Eq.",
"(REF ) differs from the equation of interest (REF ) in that we have not taken the tensor part of the particle source and, correspondingly, we do not require $H_{ij}$ to be transverse or traceless.",
"By (REF ), $T^{(P)}_{ij}$ consists of a sum of terms, each one of which has the form $ \\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(t)\\right) \\Theta (\\pm t)$ , where $\\mathbf {z}(t)$ describes a timelike or null geodesic.",
"The contribution of the tail part, $V(x,x^{\\prime })\\Theta (-\\sigma )$ , of $G^{\\rm ret}(x,x^{\\prime })$ to the retarded integral is thus a sum of terms of the form $H^{\\rm tail}_{ij} (x) =\\int d^4x^{\\prime } f_{ij}(x^{\\prime }) V(x,x^{\\prime })\\Theta \\left(-\\sigma (x,x^{\\prime })\\right) \\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right) \\Theta (\\pm t^{\\prime }) \\, ,$ where $f_{ij}$ is smooth.",
"It is not difficult to see that $H^{\\rm tail}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ .",
"On the other hand, if $\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic—i.e., if one of the ingoing or outgoing null particles is “aimed” directly at an observer at $x$ —then the singularities of $\\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right)$ and $\\Theta \\left(-\\sigma (x,x^{\\prime })\\right)$ will coincide and $H^{\\rm tail}_{ij}$ will, in general, be “highly singular” at $x$ in the sense that, in general, it will be defined only distributionally in a neighborhood of $x$ .",
"We exclude such special points from consideration.",
"The case of main interest is one where $x$ lies near the future light cone of $q$ but does not lie on the special direction defined by $\\mathbf {z}(t)$ (if null).",
"Then the $\\delta ^{(3)}\\left(\\mathbf {x}^{\\prime } - \\mathbf {z}(t^{\\prime })\\right)$ singularity in the integral will be transverse to the step function singularity of $\\Theta \\left(-\\sigma (x,x^{\\prime })\\right)$ as well as to that of $\\Theta (\\pm t)$ .",
"One can then integrate over $\\mathbf {x}^{\\prime }$ , leaving one with an integral only over $t^{\\prime }$ .",
"The integrand will be proportional to $V(x; \\mathbf {z}(t^{\\prime }), t^{\\prime }) \\Theta (u-t^{\\prime }) \\Theta (\\pm t^{\\prime })$ , where $u = t-|\\mathbf {x}|$ denotes the retarded time of $x$ .",
"The integral over $t^{\\prime }$ thus yields a result of the form $F_{ij}(x) u \\Theta (u)$ , where $F_{ij}$ is smooth.",
"Thus, we see that $H^{\\rm tail}_{ij}$ is continuous (although not continuously differentiable) at $x$ .",
"The analysis of the contribution, $H^{\\rm dir}_{ij}$ , of the “direct part,” $U(x,x^{\\prime }) \\delta (\\sigma )$ , of $G^{\\rm ret}(x,x^{\\prime })$ to $H_{ij}$ is similar, with $U(x,x^{\\prime }) \\delta (\\sigma )$ replacing $V(x,x^{\\prime })\\Theta (-\\sigma )$ in (REF ).",
"Again, $H^{\\rm dir}_{ij}$ is smooth whenever $x$ does not lie on the future light cone of $q$ , and is highly singular if $\\mathbf {z}(t)$ is a null geodesic and if $x$ lies on (the continuation of) this null geodesic.",
"If we exclude such special points, then the integral over $\\mathbf {x}^{\\prime }$ can again be done, and we are again left with an integral over $t^{\\prime }$ .",
"However, now the integrand is proportional to $\\delta (u-t^{\\prime }) \\Theta (\\pm t^{\\prime })$ , where $u$ is the retarded time of $x$ .",
"Consequently, $H^{\\rm dir}_{ij}$ has the form $\\tilde{F}_{ij}(x) \\Theta (u)$ for some smooth $\\tilde{F}_{ij}$ , and thus it has a discontinuity along the future light cone of $q$ .",
"The above analysis is for the toy problem (REF ), where we did not take the tensor part, ${T}_{ij}$ , of the source $T^{(P)}_{ij}$ .",
"It would be cumbersome to compute ${T}_{ij}$ and then perform a similar direct analysis of the behavior of the retarded Green's function integral involving ${T}_{ij}$ .",
"Fortunately, we can bypass this by noting that the operation of “taking the tensor part\" commutes with the wave operator appearing in (REF ) and (REF ).",
"It follows that the desired quantity, ${h}_{ij}$ , given by (REF ), is related to $H_{ij}$ by ${h}_{ij}=[H_{ij}]^T \\, ,$ where “$[X_{ij}]^T$ ” denotes the operation of taking the “tensor part” of $X_{ij}$ , as given by (REF ).",
"Thus, our analysis reduces to extracting information about how the operation of taking the tensor part of a quantity affects the nature of its singularities.",
"To analyze this, we note first that since “taking the tensor part” consists of algebraic operations involving differentiations and inversions of Laplacians (see (REF )-(REF )), the tensor part, ${X}_{ij}$ , of a distribution $X_{ij}$ must be smooth wherever $X_{ij}$ is smooth.",
"It then follows that the singular behavior of the tensor part of $X_{ij}$ at $x$ is the same as that of the tensor part of $\\psi X_{ij}$ , where $\\psi $ is any smooth function of compact support with $\\psi = 1$ in a neighborhood of $x$ .",
"(Proof: $X_{ij} - \\psi X_{ij}$ vanishes in a neighborhood of $x$ and hence is smooth there, so the tensor part of this difference is smooth in a neighborhood of $x$ .)",
"However, the singular behavior of $\\psi X_{ij}$ is characterized by the decay (or lack thereof) of its Fourier transform at large $k_\\mu $ .",
"The key point is that in the operation of “taking the tensor part,” there are exactly as many total “inverse derivatives” from Laplacian inversions in (REF ) as there are differentiations.",
"It follows that the Fourier transform of the tensor part of $\\psi X_{ij}$ is related to the Fourier transform of $\\psi X_{ij}$ by a function that is everywhere bounded in $k_\\mu $ .",
"In particular, the decay of the Fourier transform of the tensor part of $\\psi X_{ij}$ at large $k_\\mu $ cannot be slower than that of $\\psi X_{ij}$ .",
"The above argument can be applied to the present case as follows to get the key conclusion that we need.",
"It is easily seen from the explicit behavior of $H^{\\rm tail}_{ij}$ found above that the Fourier transform of $\\psi H^{\\rm tail}_{ij}$ lies in $L^1$ for any smooth function $\\psi $ of compact support.",
"Therefore, the Fourier transform of the tensor part of $\\psi H^{\\rm tail}_{ij}$ —which differs from the Fourier transform of $\\psi H^{\\rm tail}_{ij}$ by a bounded function of $k_\\mu $ —also lies in $L^1$ .",
"But that implies that the tensor part of $\\psi H^{\\rm tail}_{ij}$ is continuous for all $\\psi $ , which implies that the tensor part of $H^{\\rm tail}_{ij}$ is continuous.",
"Thus, we have shown that the tail contribution to ${h}_{ij}$ is continuous and thus cannot contribute to the memory effect.",
"The above conclusion is all that is needed to derive our results on the memory effect, because, as we have seen above, the direct contribution to the retarded solution is universal, and does not depend on the expansion history.",
"Furthermore, Minkowski spacetime lies within the class of $k=0$ FLRW spacetimes to which our analysis applies.",
"Thus, we can relate the memory effect in an arbitrary FLRW spacetime to that in Minkowski spacetime as follows.",
"Consider a source event at $q$ in the FLRW spacetime that is observed at event $p$ .",
"Let $\\eta _s$ and $\\eta _o$ denote the conformal times of the events $q$ and $p$ respectively.",
"For convenience, rescale the coordinates, if necessary, so that $a(\\eta _s) = 1$ .",
"This corresponds to choosing the comoving coordinates to corrspond to proper distances at $\\eta = \\eta _s$ .",
"Now, identify the FLRW spacetime with Minkowski spacetime by identifying the coordinates (REF ) of the FLRW spacetime with global inertial coordinates of Minkowski spacetime.",
"Place a source and observer at the events ${\\bar{q}}$ and ${\\bar{p}}$ of Minkowski spacetime that are identified in this manner with events $q$ and $p$ in the FLRW spacetime.",
"Since $a(\\eta _s) =1$ , the Minkowski source will physically correspond to the FLRW source provided that the masses and 4-velocities of each of the particles agree (under this identification) at $q$ .",
"It follows immediately from (REF ) that the direct part, ${h}_{ij}^{\\rm dir}$ , of ${h}_{ij}(x)$ near $p$ will be a factor of $1/a(\\eta _o)$ times the same function of $x^\\mu $ as it is in the Minkowski case, i.e., near $p$ , ${h}_{ij}^{\\rm dir} (x^\\mu ) = \\frac{1}{a(\\eta _o)} {\\bar{h}}_{ij}^{\\rm dir} (x^\\mu ) \\, .$ It then follows immediately from (REF ) that the direct parts of the linearized Riemann curvature tensor are similarly related, i.e., near $p$ $\\delta {R^{\\rm dir}_{i00}}^{j} (x^\\mu ) = \\frac{1}{a(\\eta _o)} \\delta {\\bar{R}}^{\\rm dir}_{i00}{}^j (x^\\mu ) \\, .$ Suppose, now, that we place a gravitational wave detector at $p$ , composed of two nearby particles initially at rest in the cosmic reference frame.",
"By the geodesic deviation equation, the deviation vector, $D^i$ , describing the displacement of the particles will satisfy $v^b \\nabla _b (v^c\\nabla _c {D}^a) = R_{def}^{\\quad a}{D}^d v^e v^f \\, ,$ where $v^a$ is the unit tangent to the geodesic.",
"Since $v^\\mu \\approx 1/a(\\eta _o) (\\partial /\\partial \\eta )^\\mu $ and the Hubble expansion is negligible over the relevant timescale, we can rewrite this equation as $\\frac{d^2}{d \\eta ^2} D^j = R_{i00}{}^j D^i \\, .$ Let $\\Delta D^i$ denote the coordinate components of the “memory displacement,” obtained by integrating (REF ) twice with respect to $\\eta $ .",
"In view of (REF ) and the fact, proven above, that the “direct part” of the Riemann tensor contains the full memory effect, we see that $\\Delta D^i = \\frac{1}{a(\\eta _o)} \\overline{\\Delta D}^i \\, ,$ where $\\overline{\\Delta D}^i$ denotes the corresponding memory displacement in Minkowski spacetime, assuming that the initial displacement was $\\overline{D}^i = D^i$ .",
"Thus, the relationship between $\\Delta D^i$ and $D^i$ in an arbitrary FLRW spacetime differs from the corresponding Minkowski result by a factor of $1/a(\\eta _o)$ .",
"Thus, we have shown that if we identify the FLRW spacetime with Minkowski spacetime via the coordinates (REF ) in such a way that $a(\\eta _s) = 1$ , and we place the same physical source at $q$ and the same physical detector at $p$ in both spacetimes, then the memory effect in the FLRW spacetime will be a factor of $1/a(\\eta _o) = 1/(1 + z)$ smaller than the corresponding memory effect in Minkowski spacetime.",
"Note that placing the source at the same proper distance at the time of emission corresponds to placing the source at the same angular diameter distance in both spacetimes.",
"The above result compares the memory effect in FLRW and Minkowski spacetime when the source and detector are at the same proper distance at the source emission time, i.e., when they are at the same location with $a(\\eta _s) = 1$ .",
"Since the memory effect in Minkowski spacetime falls off as $1/r$ , this result may be reformulated in numerous equivalent ways.",
"In particular, we have If the source and detector are placed so that they are at the same proper distance at the time of detection (rather than emission), then the memory effect in the FLRW spacetime is identical to the corresponding memory effect in Minkowski spacetime.",
"If the source and detector are placed so that the source is at the same luminosity distance in both cases, then the memory effect in the FLRW spacetime is larger by a factor of $(1+z)$ as compared with the corresponding memory effect in Minkowski spacetime.",
"The above result is in agreement with the results of [9], [10], [11], and [12] in the cases where the results of those references apply.",
"Acknowledgements We wish to thank Lydia Bieri and David Garfinkle for helpful discussions.",
"This research was supported in part by NSF grants PHY 12-02718 and PHY 15-05124 to the University of Chicago."
],
[
"Acoustic Shock Waves and Metric Continuity",
"In this Appendix, we show that the the scalar sector does not contribute to the memory effect.",
"The gauge-invariant density perturbation $A$ satisfies (REF ).",
"Equation (REF ) is a hyperbolic wave equation of the general form (REF ), with the Lorentz metric $g_{\\mu \\nu }$ now being the “acoustic metric,” $ds^2= -d \\eta ^2 + \\frac{1}{c_s^2}[dx^2 + dy^2 + dz^2] \\, ,$ and with the source term $T$ proportional to the scalar particle fields $\\varphi ^{(P)}$ and $\\psi ^{(P)}$ .",
"Thus, the retarded Green's function for (REF ) takes the general Hadamard form (REF ), with $\\sigma $ replaced by the squared geodesic distance, $\\sigma _s$ , in the acoustic metric (REF ), i.e., we have $G_s(x,x^{\\prime })=\\left[U_s(x,x^{\\prime })\\delta (\\sigma _s)+V_s(x,x^{\\prime })\\Theta (-\\sigma _s) \\right] \\Theta (\\eta -\\eta ^{\\prime }) \\, ,$ where $U_s$ and $V_s$ are again smooth functions in both $x$ and $x^{\\prime }$ Furthermore, it can be seen from (REF ) that both $\\varphi ^{(P)}$ and $\\psi ^{(P)}$ are obtained from $T^{(P)}_{\\mu \\nu }$ by algebraic operations (i.e., no differentiations or Laplace inversions).",
"It follows immediately that the source term appearing in (REF ) takes the same form (namely, proportional to $\\delta ^{(3)}\\left(\\mathbf {x} - \\mathbf {z}(\\eta )\\right) \\Theta (\\pm \\eta )$ ), as considered in Section .",
"We may therefore repeat the analysis of Section to draw the following conclusion: Suppose that all of the particles in $T^{(P)}_{\\mu \\nu }$ are moving with velocity smaller than the speed of soundIf any of the particles are moving with velocity greater than the speed of sound, there will be additional “Cherenkov radiation” singularities occurring at points $x$ where the past sound cone of $x$ intersects a particle world line orthogonally (in the sound metric).",
"These additional singularities are not of interest for the memory effect.. Then $A$ is smooth except on the future sound cone of $q$ .",
"Furthermore, on the sound cone, $A$ will, in general, be discontinuous, but it cannot have “worse” singular behavior.",
"The metric perturbation variables $\\Phi $ and $\\Psi $ satisfy elliptic equations, with source terms given by $A$ and the scalar parts of the particle sources.",
"It follows that $\\Phi $ and $\\Psi $ must be smooth everywhere apart from the worldlines of the particles and the points at which $A$ fails to be smooth, i.e., the future sound cone of $q$ .",
"We are not interested in the singularities at the particle worldlines.",
"However, on the future sound cone of $q$ , $\\nabla ^2 \\Phi $ and $\\nabla ^2 \\Psi $ are at worst discontinuous, so $\\Phi $ and $\\Psi $ themselves are at least $C^1$ .",
"Thus, they cannot contribute a derivative of delta-function to the Riemann curvature (REF ), and thus do not contribute to any memory effect."
]
] | 1606.04894 |
[
[
"The stochastic heat equation, 2D Toda equations and dynamics for the\n multilayer process"
],
[
"Abstract We show that solutions of the stochastic heat equation driven by space-time white noise, although not smooth, meaningfully solve the two-dimensional Toda equations.",
"Then by extending our arguments we show the time evolution of the multilayer process introduced by O'Connell and Warren is conjugate to a flow induced by the stochastic heat equation.",
"In particular this establishes a Markov property conjectured by O'Connell and Warren.",
"It also defines, for the first time, the multilayer process started from a general initial condition."
],
[
"Introduction",
"In [23], O'Connell and Warren introduced the following: for each $n = 1,2,\\ldots $ , $t>0$ and $x$ , $y\\in \\mathbf {R}$ define $Z_n(t,x,y) = p_t(x-y)^n \\bigg (1 + \\sum _{k=1}^\\infty \\int _{\\Delta _k(t)} \\int _{\\mathbf {R}^k} R_k(\\mathbf {s},\\mathbf {y^\\prime }; t,x,y) \\;W^{\\otimes k}(\\mathrm {d}\\mathbf {s},\\mathrm {d}\\mathbf {y}^\\prime ) \\bigg ),$ where $\\Delta _k(t) = \\lbrace 0 < s_1 < s_2 < \\cdots < s_k < t\\rbrace $ .",
"$\\mathbf {s} = (s_1,\\ldots ,s_k)$ , $\\mathbf {y}^\\prime = (y_1^\\prime ,\\ldots ,y_k^\\prime )$ and $R_k(\\mathbf {s}, \\mathbf {y}^\\prime ; t,x,y)$ is the $k$ -point correlation function for a collection of $n$ non-intersecting Brownian bridges each of which starts at $x$ at time 0 and ends at $y$ at time $t$ .",
"$p_t(x-y)= (2\\pi t)^{-1/2} e^{-(x-y)^2/2t}$ is the transition density of Brownian motion.",
"The integral is a multiple stochastic integral with respect to space-time white noise.",
"It was shown in [23] by considering local times of non-intersecting Brownian bridges that the infinite sum in the definition is convergent in $L^2$ with respect to the white noise.",
"Observe that $u = Z_1$ is the solution to the (multiplicative) stochastic heat equation (SHE) with delta initial data: ${\\left\\lbrace \\begin{array}{ll}\\partial _t u(t,x,y) = \\Big ( \\frac{1}{2} \\Delta _y + \\dot{W}(t,y) \\Big ) u(t,x,y), \\quad t\\in (0,\\infty ), y\\in \\mathbf {R}, \\\\u(0,x,y) = \\delta (x-y), \\quad x\\in \\mathbf {R}.\\end{array}\\right.",
"}$ By a solution to the above we mean a random field $u$ which satisfies almost surely the mild form of the equation: $u(t,x,y) = p_t(x-y) + \\int _0^t \\int _\\mathbf {R}p_{t-s}(y-y^\\prime ) u(s,x,y^\\prime ) \\;W(\\mathrm {d}s,\\mathrm {d}y^\\prime ).$ Iterating equation (REF ) multiple times gives the chaos expansion (REF ) for $n=1$ .",
"One can express $Z_n(t,x,y)$ in a more suggestive notation: $Z_n(t,x,y) = p_t(x-y)^n \\mathbf {E}_{x,y;t}^X \\bigg [ {E}\\mathrm {xp} \\bigg ( \\sum _{i=1}^n \\int _0^t W(s,X_s^i) \\;\\mathrm {d}s \\bigg ) \\bigg ],$ where $(X_s^1,\\ldots ,X_s^n, 0\\le s\\le t)$ denotes the trajectories of the above mentioned collection of $n$ non-intersecting Brownian bridges and $\\mathbf {E}_{x,y;t}^X$ is the corresponding expectation.",
"${E}\\mathrm {xp}$ is the Wick exponential defined by ${E}\\mathrm {xp}(M_t) := \\exp \\big ( M_t - \\frac{1}{2}\\langle M,M\\rangle _t \\big )$ for a martingale $M$ .",
"The Feynman–Kac formula (REF ) is not rigorous as it is unclear how one would define the integral of the white noise along a Brownian path and moreover to exponentiate such an expression.",
"However, one can obtain an rigorous expression by replacing $W$ in (REF ) with a smoothed version of the space-time white noise.",
"Indeed, Bertini and Cancrini showed in [2] that such expression (in the case $n=1$ ) has a meaningful limit as one takes away the smoothing and that the limit solves the SHE.",
"Bymeans of the Feynman–Kac formula, one can interpret the solution to the stochastic heat equation as the partition function (up to a multiplication by the heat kernel) of the continuum directed random polymer, and then similarly, $Z_n$ is the partition function of a natural extension of the continuum polymer involving multiple Brownian paths.",
"The Cole–Hopf solution $h = \\log u$ to the KPZ equation with narrow wedge initial data corresponds via the Feynman–Kac formula to the free energy of the continuum directed random polymer.",
"With this interpretation $h$ can be regarded as the continuum analogue of the longest increasing subsequence of a random permutation, length of the first row of a random Young diagram, directed last passage percolation and free energy of a discrete/semi-discrete polymer in random media etc., see [3], [4], [5], [16], [17], [24], [18], [10] and the references therein.",
"In each of these discrete models, there is further structure provided either by multiple non-intersecting up-right paths on lattices, multi-layer growth dynamics or Young diagrams constructed from the RSK correspondence.",
"The work in the above mentioned references have shown that in some cases, utilisation of this additional structure have lead to derivations of exact formulae for the distribution of quantities of interest.",
"The above mentioned discrete models provide examples of what is called integrability or exact solvability.",
"The motivation for introducing the partition functions $Z_n$ , which are the continuum analogue of the structures mentioned above, is that they should provide insight to the integrable structure in the continuum setting.",
"There has been other recent work on multiple polymer paths and the multilayer process in the stochastic heat equation setting.",
"In [11] and [12], in a manifestation of the exact solvability, the Bethe Ansatz is used to make exact and asymptotic distributional statements.",
"In [8], the KPZ line ensemble, which is expected to be given by the logarithm of the multilayer process we are considering here is shown to have a remarkable Gibbs property which generalises that of the Airy line ensemble, [7].",
"Very recent results in [9] show the multilayer process arises as a scaling limit of discrete polymer models.",
"It was shown in [23] by considering a smooth space-time potential that $(Z_n, n\\ge 1)$ should satisfy a system of coupled SPDEs, however unfortunately it is not immediately obvious that such SPDEs make sense in the white noise setting.",
"Nevertheless, it does suggests that the process should have a Markovian evolution.",
"Indeed, we have the following theorem which is the main result of this paper.",
"Theorem 1.1 For each $n\\ge 1$ and $x\\in \\mathbf {R}$ , $\\big (Z_1(t,x,\\cdot ),\\ldots ,Z_n(t,x,\\cdot ) ; t\\ge 0\\big ),$ is a Markov process with respect to the filtration generated by the space-time white noise taking values in the space ${\\mathfrak {L}}_n:= C(\\mathbf {R})\\times \\cdots \\times C(\\mathbf {R})$ , where $C(\\mathbf {R}) := C(\\mathbf {R},\\mathbf {R}_+)$ is the space of continuous functions from $\\mathbf {R}$ to $\\mathbf {R}_+ = (0,\\infty )$ .",
"In the case of the stochastic heat equation ($n=1$ ), the Markov property can be seen from the Feynman–Kac formula since $u(s+t,x,y)$ can be written in the form $\\mathbf {E}_{x,y;t}^X\\Big [ {E}\\mathrm {xp}\\big ( F_X(0,s)\\big ) {E}\\mathrm {xp}\\big ( F_X(s,t)\\big )\\Big ],$ where $F_X(s,t)$ is a function of the Brownian bridge $X$ starting from $(s,X_s)$ and ending at $(t,y)$ and the white noise over the time interval $[s,t]$ , which is independent of the white noise over $[0,s]$ and the bridge from $(0,x)$ to $(s,X_s)$ .",
"From this one obtains the flow property of $u$ : $u(t,x,y) = \\int _\\mathbf {R}u(s,x,z) u(s,t,z,y) \\;\\mathrm {d}z,$ where $u(s,s+\\cdot ,z,\\cdot )$ is the solution to the SHE driven by the shifted white noise $\\dot{W}(s+\\cdot ,\\cdot )$ .",
"However, this argument does not apply for $n\\ge 2$ since the definition of $Z_n$ involves non-intersecting Brownian bridges with common starting and ending points but at any intermediate time each of the bridges are at distinct locations.",
"Nevertheless, Theorem REF is true and we shall prove it by considering a natural extension $M_n$ of $Z_n$ which corresponds to allowing the multiple polymer paths to have differing starting and ending points from one another.",
"This extended process can easily be seen to have the Markov property, and the key to understanding Theorem REF is that the extension $M_n$ turns out to be able to be recovered from the values of $(Z_1, Z_2, \\ldots , Z_n)$ .",
"In fact we establish in Theorem REF what amounts to a conjugacy between the random dynamical systems that describe the evolution of $(M_1, M_2, \\ldots , M_n)$ and $(Z_1, Z_2, \\ldots , Z_n)$ as shown in Figure REF .",
"This develops an idea that was suggested in [23], but only rigorously established when $n=2$ .",
"Figure: The evolution of the multilayer process is described by a conjugacyLet $W_n = \\lbrace x\\in \\mathbf {R}^n : x_1\\ge \\cdots \\ge x_n\\rbrace $ be the Weyl chamber in $\\mathbf {R}^n$ , then define for $n\\ge 1$ , $(t,\\mathbf {x},\\mathbf {y})\\in (0,\\infty )\\times W_n\\times W_n$ , $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{p_n^*(t,\\mathbf {x},\\mathbf {y})}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})} \\bigg (1 + \\sum _{k=1}^\\infty \\int _{\\Delta _{k}(t)} \\int _{\\mathbf {R}^k} R_k(\\mathbf {s},\\mathbf {y}^\\prime ; t,\\mathbf {x},\\mathbf {y}) \\;W^{\\otimes k}(\\mathrm {d}\\mathbf {s},\\mathrm {d}\\mathbf {y}^\\prime ) \\bigg ),$ where $R_k$ is the $k$ -point correlation function of a collection of $n$ non-intersection Brownian bridges which starts at $\\mathbf {x}$ at time 0 and ends at $\\mathbf {y}$ at time $t$ .",
"$p_n^*(t,\\mathbf {x},\\mathbf {y}) = \\det [p_t(x_i-x_j)]_{i,j=1}^n$ is by the Karlin–McGregor formula [21] the transition density of Brownian motion killed at the boundary of $W_n$ and $\\Delta _n(\\mathbf {x}) = \\prod _{1\\le i<j\\le n} (x_i-x_j)$ is the Vandermonde determinant.",
"Notice that $M_n(t,\\mathbf {x},\\mathbf {y})$ is well defined for $\\mathbf {x}$ , $\\mathbf {y}$ at the boundary of the Weyl chamber since $p_n^*(t,\\mathbf {x},\\mathbf {y})/\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})$ is a smooth function of $(\\mathbf {x},\\mathbf {y})$ over $\\mathbf {R}^n\\times \\mathbf {R}^n$ by [1] and the $k$ -point correlation function $R_k$ extends continuously to the boundary.",
"Moreover we can extend $M_n$ by symmetry to a function on $\\mathbf {R}^n\\times \\mathbf {R}^n$ , $M_n(t,\\mathbf {x},\\mathbf {y})$ satisfying $M_n(t, \\pi \\mathbf {x},\\sigma \\mathbf {y}) = M_n(t,\\mathbf {x},\\mathbf {y})$ for any permutations $\\pi $ , $\\sigma $ of $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"We have the following which is the main result of [22].",
"Theorem 1.2 For all $n\\ge 1$ , $M_n$ has a version that is continuous over $(0,\\infty )\\times {\\mathbf {R}}^n \\times {\\mathbf {R}}^n$ and $\\mathbb {P}[M_n(t,\\mathbf {x},\\mathbf {y}) > 0 \\text{ for all } t>0 \\text{ and } \\mathbf {x},\\mathbf {y}\\in {\\mathbf {R}}^n] = 1$ .",
"Moreover, when all the coordinates of $\\mathbf {x}$ and $\\mathbf {y}$ are equal, $M_n$ agrees almost surely up to a strictly positive multiplicative constant with $Z_n$ , that is $M_n(t,a\\mathbf {1}_n,b\\mathbf {1}_n) = c_{n,t} Z_n(t,a,b), \\quad c_{n,t} = c_n t^{-n(n-1)/2}, c_n = \\left(\\prod _{i=1}^{n-1} i!\\right)^{-1},$ where $\\mathbf {1}_n= (1,\\ldots ,1) \\in {\\mathbf {R}}^n$ .",
"This implies the almost sure continuity and everywhere strict positivity of $Z_n$ for all $n$ .",
"It was shown in [14] that the difference of two solutions to the KPZ equation (with the same white noise) starting from two different Hölder continuous initial data is in $C^{\\frac{3}{2}-\\varepsilon }$ for every $\\varepsilon >0$ .",
"Since $M_2(t,\\mathbf {x},\\cdot )$ and $u(t,x,\\cdot )$ are Hölder continuous of order up to 1/2, Theorem REF and equation (REF ) imply that the ratio of $u(t,x_1,\\cdot )$ and $u(t,x_2,\\cdot )$ is in $C^{\\frac{3}{2}-\\varepsilon }$ for every $\\varepsilon >0$ and hence generalises the result of [14] to delta initial data.",
"In fact Theorem REF implies that there is much more regularity present when considering multiple solutions to the stochastic heat equation driven by the same white noise than might be initially expected.",
"We have the following determinantal expression for $M_n$ (see [23], and also proved in the Appendix to this paper) $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{\\det [u(t,x_i,y_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {x})}\\Delta _n{(\\mathbf {y})}}, \\quad t>0, \\mathbf {x},\\mathbf {y}\\in {\\mathbf {R}}^n_{\\ne },$ where the entries in the determinant are solutions to (REF ) each driven by the same white noise.",
"${\\mathbf {R}}^n_{\\ne }$ denotes the subset of points in ${\\mathbf {R}}^n$ whose coordinates are all distinct.",
"In view of this representation Theorem REF implies we can meaningfully assign values to Wronskians whose entries are formally given as derivatives of solutions of the stochastic heat equation.",
"Suppose that we replace the space-time white noise with a smooth space-time potential in the definition of $Z_n$ then it can be shown that $Z_n$ is given by the bi-directional Wronskian $Z_n(t,x,y) = c_n t^{n(n-1)/2} \\det [\\partial _x^{i-1} \\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n,$ where $u(t,x,y)$ is the solution to (REF ) driven by the smooth potential.",
"Now let $\\tau _n(x,y) = \\det [\\partial _x^{i-1} \\partial _y^{j-1} u(t,x,y) ]_{i,j=1}^n$ then $\\tau _n$ satisfies the two-dimensional Toda equation (2DTE) $\\partial _{xy} q_n = e^{q_{n+1}-q_n} - e^{q_n-q_{n-1}}, \\quad n\\ge 1,$ where $q_n = \\log (n/{n-1})$ or equivalently, $\\partial _{xy} \\log n = \\frac{{n-1}{n+1}}{n^2},$ with the convention that $0 \\equiv 1$ .",
"Evaluating the derivative and rearranging we obtain $n \\partial _{xy} n - (\\partial _xn) (\\partial _yn) = {n-1}{n+1}.$ We introduce the following notation.",
"For an $(n+1)\\times (n+1)$ determinant $D$ , let $D\\begin{bmatrix} i \\\\ j\\end{bmatrix}$ be the $n\\times n$ determinant obtained from $D$ by removing the $i$ th row and the $j$ th column and similarly let $D\\begin{bmatrix} i & j \\\\ k & l\\end{bmatrix}$ be the $(n-1)\\times (n-1)$ determinant obtained from $D$ by removing the $i$ th and $j$ th rows and the $k$ th and $l$ th columns.",
"Then, by properties of Wronskians (REF ) can be written as ${n+1}\\begin{bmatrix} n+1 \\\\ n+1\\end{bmatrix} {n+1}\\begin{bmatrix} n \\\\ n\\end{bmatrix} - {n+1} \\begin{bmatrix} n \\\\ n+1\\end{bmatrix} {n+1} \\begin{bmatrix} n+1 \\\\ n\\end{bmatrix} = {n+1} \\begin{bmatrix} n & n+1 \\\\ n & n+1\\end{bmatrix} {n+1},$ which is nothing but the Jacobi identity for determinants [15].",
"In Section we study Wronskians defined as the continuous extensions of determinantal expressions such as (REF ) where $u$ is not necessarily differentiable.",
"With the help of the Jacobi idenitity we show that such generalised Wronksians satisfy the 2D Toda equations in an integrated form.",
"Then in Section , using a key lemma from the preceeding section, we construct the conjugacy that describes the evolution of the multilayer process.",
"A fuller description of this conjugacy is the content of Section which follows now."
],
[
"Acknowledgements",
"The research of C.H.L.",
"was supported by EPSRC grant number EP/H023364/1 through the MASDOC DTC.",
"This research of J.W.",
"was supported in part by the National Science Foundation under Grant No.",
"NSF PHY11-25915."
],
[
"Description of the dynamics of the multilayer process",
"The multilayer process is, for a fixed $x\\in {\\mathbf {R}}$ , and $n\\ge 1$ the finite sequence of “lines” $\\big (Z_1(t,x,\\cdot ),\\ldots ,Z_n(t,x,\\cdot ) )$ , which we consider to be evolving in the time variable $t>0$ .",
"Actually it seems more natural to work with a slightly different normalisation than that originally used by [23], and consider instead the process ${\\tau }_{t}(x,\\cdot )=\\big (\\tau _1(t,x,\\cdot ),\\ldots , \\tau _n(t,x,\\cdot ) )$ , where $\\tau _n(t,x,y)=c_{n}^{-1} t ^{-n(n-1)/2} Z_n(t,x,y)= c_n^{-2} M_n( t,x{\\mathbf {1}}_n, y{\\mathbf {1}}_n).$ As discussed in the Introduction, we describe the evolution of the multilayer process with the aid of the extended process $M_n$ .",
"It was shown in [22] that $M_n$ , defined by the chaos expansion (REF ), satisfies an evolution equation which can be regarded as a type of multi-dimensional SHE.",
"$M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{p_n^*(t,\\mathbf {x},\\mathbf {y})}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})} + \\frac{1}{(n-1)!}",
"\\int _0^t \\int _{\\mathbf {R}^n} Q_{t-s}(\\mathbf {y},\\mathbf {y}^\\prime ) M_n(s,\\mathbf {x},\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}_*^\\prime \\;W(\\mathrm {d}s,\\mathrm {d}y_1^\\prime ),$ almost surely for all $(t,x,y)\\in (0,\\infty )\\times \\mathbf {R}^n\\times \\mathbf {R}^n$ where $\\mathrm {d}\\mathbf {y}_*^\\prime = \\mathrm {d}y_2^\\prime \\ldots \\mathrm {d}y_n^\\prime $ .",
"The function $Q_t(\\mathbf {x},\\mathbf {y}) = \\frac{\\Delta _n(\\mathbf {y})}{\\Delta _n(\\mathbf {x})} p_n^*(t,\\mathbf {x},\\mathbf {y})$ is the transition density of Dyson Brownian motion [13] starting from $\\mathbf {x}\\in W_n$ and ending at $\\mathbf {y}\\in W_n$ .",
"Note that in (REF ) we have extended $Q_t$ by symmetry to a function on $\\mathbf {R}^n\\times \\mathbf {R}^n$ so the integral over $\\mathbf {R}^n$ is defined.",
"We can consider this same evolution for initial data $g$ , which is assumed to be a symmetric function on ${\\mathbf {R}}^n$ , $M^g_n(t,\\mathbf {y}) =\\frac{1}{n!}",
"\\int _{{\\mathbf {R}}^n} g(\\mathbf {y}^\\prime )Q_{t}(\\mathbf {y},\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}^\\prime + \\\\\\frac{1}{(n-1)!}",
"\\int _0^t \\int _{\\mathbf {R}^n} Q_{t-s}(\\mathbf {y},\\mathbf {y}^\\prime ) M^g_n(s,\\mathbf {y}^\\prime ) \\;\\mathrm {d}\\mathbf {y}_*^\\prime \\;W(\\mathrm {d}s,\\mathrm {d}y_1^\\prime ),$ For bounded $g$ , existence and uniqueness for this equation were established in [22].",
"If $g$ takes the form $g(\\mathbf {x})= \\frac{\\det [f_i(x_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {x})}}, \\quad \\mathbf {x}\\in {\\mathbf {R}}^n_{\\ne },$ then we have the representation $M_n^g(t,\\mathbf {y})=\\frac{\\det [u^{f_i}(t,y_j)]_{i,j=1}^n}{\\Delta _n{(\\mathbf {y})}}, \\quad t>0, \\mathbf {y}\\in {\\mathbf {R}}^n_{\\ne },$ where $u^{f_i}$ denotes the solution to the solution to the SHE with initial condition at time 0 given by $f_i$ .",
"This is a consequence of equation (REF ).",
"From now on we use the notation $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n (\\mathbf {x})= \\det [f_i(x_j)]_{i,j=1}^n$ We let ${\\mathfrak {F}}_n$ be the set of sequences $({\\mathfrak {f}}_1, {\\mathfrak {f}}_2, \\ldots , {\\mathfrak {f}}_n)$ with each ${\\mathfrak {f}}_k:{\\mathbf {R}}^k \\rightarrow {\\mathbf {R}}$ being a continuous, strictly positive symmetric function, and such that there exists a sequence of continuous functions $f_1, f_2, \\ldots , f_k, \\ldots $ each defined on ${\\mathbf {R}}$ so that, for every $k$ , ${\\mathfrak {f}_k}(\\mathbf {x})= \\frac{f_1\\wedge f_2 \\wedge \\ldots \\wedge f_k(\\mathbf {x})}{\\Delta _k(\\mathbf {x})}$ for $\\mathbf {x}\\in {\\mathbf {R}}^k$ with all coordinates distinct.",
"In view of equation (REF ), for $\\mathfrak {f} \\in {\\mathfrak {F}}_n$ with bounded components, we can define an ${\\mathfrak {F}}_n$ -valued process $(\\mathfrak {f}_t)_{t \\ge 0}$ with initial value $\\mathfrak {f}$ by $\\mathfrak {f}_t= \\bigl ( M_1^{{\\mathfrak {f}}_1}(t,\\cdot ), M_2^{{\\mathfrak {f}}_2}(t,\\cdot ), \\ldots , M_n^{{\\mathfrak {f}}_n}(t,\\cdot ) \\bigr )$ We will denote, for any $t>0$ , the ${\\mathfrak {F}}_n$ -valued random variable $\\mathfrak {f}_t$ by ${\\mathcal {M}}_t \\mathfrak {f}$ .",
"For $0<s<t$ , replacing the white noise in (REF ) by its shift $\\dot{W}^{(s)}(\\cdot ,\\cdot )= \\dot{W}(s+\\cdot ,\\cdot )$ we can define analogously ${\\mathcal {M}}_{s,t} \\mathfrak {f}$ where now $\\mathfrak {f}$ might be a random element of $\\mathfrak {F}_n$ determined by the white noise prior to time $s$ , and whose $p$ th-moments are uniformly bounded.",
"Then, for bounded ${\\mathfrak {f}}$ , the existence and uniqueness results proved in [22] imply that ${\\mathcal {M}}_t \\mathfrak {f} = {\\mathcal {M}}_{s,t} {\\mathcal {M}}_s \\mathfrak {f}.$ Recall the definition of ${\\mathfrak {L}}_n$ from Theorem REF .",
"We are able to define a bijective correspondence between ${\\mathfrak {F}}_n$ and ${\\mathfrak {L}}_n$ .",
"Setting $\\ell _k( x) = c_n^{-1} {\\mathfrak {f}}_k( x {\\mathbf {1}}_k)$ defines a map ${\\mathcal {R}}: ({\\mathfrak {f}}_1, {\\mathfrak {f}}_2, \\ldots , {\\mathfrak {f}}_n) \\mapsto (\\ell _1, \\ell _2, \\ldots , \\ell _n )$ .",
"Somewhat suprisingly this is invertible, with an inverse which takes an explicit form - see Propositions REF and REF .",
"In fact it's easy to see from the form of ${\\mathcal {R}}^{-1}$ that if ${\\mathfrak {F}}_n$ and ${\\mathfrak {L}}_n$ are each equipped with the topology of locally uniform convergence, then ${\\mathcal {R}}$ beceomes a homeomorphism between these spaces.",
"We use $ {\\mathcal {R}}$ to induce a map $\\langle \\cdot , \\cdot \\rangle _{{\\mathfrak {L}}_n}: {\\mathfrak {L}}_n \\times {\\mathfrak {L}}_n\\rightarrow (0,\\infty ]^n,$ This is defined by setting $\\langle {\\ell }, {\\ell }^\\prime \\rangle _{{\\mathfrak {L}}_n}= (\\beta _1, \\beta _2, \\ldots , \\beta _n)$ , where $\\beta _k= \\frac{1}{k!",
"}\\int _{{\\mathbf {R}}^k} {\\mathfrak {f}}_k(\\mathbf {x}) {\\mathfrak {f}}^\\prime _k(\\mathbf {x}) \\Delta _k(\\mathbf {x})^2 \\;\\mathrm {d}\\mathbf {x},$ with ${\\ell }={\\mathcal {R}}({\\mathfrak {f}})$ and ${\\ell }^\\prime ={\\mathcal {R}}({\\mathfrak {f}}^\\prime )$ .",
"Theorem 2.1 For each $x\\in {\\mathbf {R}}$ , the multiline line process $({\\tau }_{t})_{t>0}=({\\tau }(t,x,\\cdot ))_{t>0}$ is an ${\\mathfrak {L}}_n$ -valued process whose evolution is given by ${\\tau }_{t}={\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1} {\\tau }_{s} \\quad \\text{ for $0<s<t$}.$ Moreover, we have the following flow property, for any $0<s<t$ , and for any $x,y\\in {\\mathbf {R}}$ , with probability one, ${\\tau }(t,x,y)= \\langle {\\tau }(s, x,\\cdot ), {\\tau }(s,t,\\cdot ,y) \\rangle _{{\\mathfrak {L}}_n},$ where ${\\tau }(s,t, \\cdot , \\cdot )$ is defined analogously to ${\\tau }(t-s, \\cdot , \\cdot )$ but replacing the white noise with the shifted noise $\\dot{W}^{(s)}$ .",
"In the light of this theorem, it is clear how the multiline process should be defined starting from an initial condition ${\\ell }\\in {\\mathfrak {L}}_n$ .",
"Provided ${\\mathcal {R}}^{-1}({\\ell }) $ is bounded, then ${\\tau }_t^{\\ell }= {\\mathcal {R}} {\\mathcal {M}}_t {\\mathcal {R}}^{-1}({\\ell })$ defines an ${\\mathfrak {L}}_n$ -valued process, which starts from ${\\ell }$ .",
"We also have the representation ${\\tau }_t^{\\ell } (y)= \\langle {\\ell } , {\\tau }_t(\\cdot ,y) \\rangle _{{\\mathfrak {L}}_n}.$ In essence there is a conjugacy between the random dynamical system on ${\\mathfrak {L}}_n$ which drives the evolution of the multilayer process, and the random dynamical system on ${\\mathfrak {F}}_n$ induced by the stochastic heat equation.",
"More precisely, if ${\\mathfrak {F}}^0_n\\subset {\\mathfrak {F}}_n$ is invariant for (some extension of) ${\\mathcal {M}}_{s,t}$ then we can define a conjugate dynamical system on ${\\mathcal {R}}\\bigl ( {\\mathfrak {F}}^0_n \\bigr )\\subset {\\mathfrak {L}}_n$ via ${\\mathcal {T}}_{s,t} = {\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1}.$"
],
[
"Wronskians for non-smooth functions and the Toda equations",
"Suppose $f_1, f_2, \\ldots , f_n:{\\mathbf {R}}\\rightarrow {\\mathbf {R}} $ are continuous functions such that $\\frac{f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n (\\mathbf {x})}{\\Delta _n(\\mathbf {x})}$ extends continuously to ${\\mathbf {R}}^n$ .",
"Then we define the Wronskian $W(f_1, f_2, \\ldots f_n) (x)$ to be the product of the constant $c_n^{-1}$ and the value of the extension at $\\mathbf {x}=x{\\bf 1}_n$ .",
"If the functions $f_1, f_2,\\ldots , f_n$ are smooth enough, then this agrees with the usual definition.",
"Hirota's bilinear derivative, which we denote as $D(f_1,f_2)$ is the Wronskian $W(f_2,f_1)=-W(f_1,f_2)$ .",
"Suppose that $f_2$ is strictly positive.",
"Then we can write $\\frac{f_1 \\wedge f_2 (x_1,x_2)}{(x_2-x_1)}= \\frac{ f_2(x_2) f_2(x_1)}{(x_2-x_1)} \\left( \\frac{f_1(x_1)}{f_2(x_1)} -\\frac{ f_1(x_2)}{f_2(x_2)}\\right),$ and so we see a continuous extension to $\\lbrace x_1=x_2\\rbrace $ existing is equivalent to the ratio $f_1/f_2$ being continuously differentiable, and then we have $\\frac{f_1(b)}{f_2(b)}-\\frac{f_1(a)}{f_2(a)}=\\int _a^b \\frac{D(f_1,f_2)(x)}{f_2(x)^2} \\;\\mathrm {d}x.$ The following key lemma is the basis for all the results of this paper.",
"It extends to our generalised Wronskians an easy but important consequence of the Jacobi identity for classical Wronskians.",
"Lemma 3.1 Suppose that $f_1 \\wedge \\ldots \\wedge f_n/\\Delta _n$ , $f_1 \\wedge \\ldots \\wedge f_n \\wedge g/\\Delta _{n+1}$ , $f_1 \\wedge \\ldots \\wedge f_n \\wedge h/\\Delta _{n+1} $ and $f_1 \\wedge \\ldots \\wedge f_n \\wedge g \\wedge h/\\Delta _{n+2}$ extend to continuous functions on ${\\mathbf {R}}^{n}$ , ${\\mathbf {R}}^{n+1}$ , ${\\mathbf {R}}^{n+1}$ and ${\\mathbf {R}}^{n+2}$ respectively.",
"Further assume $f_1 \\wedge \\ldots \\wedge f_n \\wedge g/\\Delta _{n+1}$ extends as a strictly positive function.",
"Then $\\frac{W(f_1, \\ldots , f_n,h)}{ W(f_1, \\ldots , f_n,g) } \\text{ is differentiable}$ and $D( W(f_1, \\ldots , f_n,h), W(f_1, \\ldots , f_n,g)) = W(f_1, \\ldots , f_n)W(f_1, \\ldots , f_n,g,h).$ Denote ${f_1 \\wedge f_2 \\ldots \\wedge f_n }/{\\Delta _n}$ ,${f_1 \\wedge f_2 \\ldots \\wedge f_n \\wedge g}/{\\Delta _{n+1}}$ , ${f_1 \\wedge f_2 \\ldots \\wedge f_n \\wedge h }/{\\Delta _{n+1}}$ and ${f_1 \\wedge f_2 \\ldots \\wedge f_n \\wedge g \\wedge h}/{\\Delta _{n+2}}$ by $F$ , $G$ ,$H$ and $K$ respectively.",
"Let ${\\bf x}= (x_1, x_2, \\ldots , x_n)$ be a point in $ {\\mathbf {R}}^n$ , and for $ y,z \\in {\\mathbf {R}}$ consider points $({\\bf x},y),({\\bf x},y)\\in {\\mathbf {R}}^{n+1}$ and $({\\bf x},y,z)\\in {\\mathbf {R}}^{n+2}$ .",
"The Jacobi identity gives $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n \\wedge g( {\\bf x},y) f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n \\wedge h ({\\bf x},z)-\\\\f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n \\wedge h( {\\bf x},y) f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n \\wedge g ({\\bf x},z)=\\\\f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n( {\\bf x}) f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n \\wedge g \\wedge h ({\\bf x},y,z)$ Now, as can be easily be checked directly, $\\frac{\\Delta _{n+1}({\\bf x},y) \\Delta _{n+1}({\\bf x},z)}{\\Delta _{n}({\\bf x}) \\Delta _{n+2}({\\bf x},y,z)}=\\frac{1}{z-y},$ and so we deduce that $\\frac{1}{z-y} \\bigl ( G( {\\bf x},y) H ({\\bf x},z)-H( {\\bf x},y)G ({\\bf x},z)\\bigr )=F({\\bf x}) K({\\bf x},y,z).$ Notice that the continuity of $F$ , $G$ , $H$ and $K$ implies that we no longer need to assume that $x_1, x_2, \\ldots x_n,y,z$ are all distinct.",
"Moreover dividing through by $ G( {\\bf x},y) G( {\\bf x},z)$ which by hypothesis is strictly positive, the same continuity implies that we can take a limit as $y-z$ tends to 0 and thereby deduce that $y \\mapsto \\frac{H({\\bf x},y)}{G({\\bf x},y)} \\text{ is continuously differentiable}$ and that its derivative is ${F({\\bf x}) K({\\bf x},y,y)}/{G({\\bf x},y)^2}.$ But $G$ and $H$ , and hence the ratio $H/G$ are symmetric functions on ${\\mathbf {R}}^{n+1}$ , and so in fact the partial derivative in each variable of $H/G$ exists and is continuous.",
"By a standard result from calculus this implies that $H/G$ is a differentiable function on ${\\mathbf {R}}^{n+1}$ .",
"In particular, we have $x \\mapsto \\frac{H(x {\\bf 1}_n)}{G(x {\\bf 1}_n)} \\text{ is continuously differentiable}$ and its derivative, given by summing the partial derivatives, is $(n+1)\\frac{F({x\\bf 1_n}) K(x{\\bf 1}_{n+2})}{G(x{\\bf 1}_{n+1})^2}.$ In view of the definition of the Wronskian $W(f_1, \\ldots f_n)= c_n^{-1} F({x\\bf 1_n})$ and similarly for the other Wronskians appearing in the statement, the lemma is proved, noting that$ c_n c_{n+2}/c_n^2=1/(n+1)$ .",
"We turn now to determinants constructed from a kernel function of two real variables.",
"When a continuous extension exists in this setting we can define a quantity that generalizes the notion of bi-directional Wronskians, and as in the case of smooth functions these give rise to solutions of the two dimensional Toda equations.",
"This is the content of the following proposition, the proof of which is to first apply the preceeding lemma in one of the variables of the kernel, and then repeat the arguments used in the proof of the lemma in the second variable.",
"Proposition 3.2 Suppose that $u$ is a continuous function defined on ${\\mathbf {R}} \\times {\\mathbf {R}}$ such that, for every $n \\ge 1$ , $( \\mathbf {x},\\mathbf {y}) \\mapsto \\frac{ \\det [ u(x_i,y_j) ]_{n \\times n}}{ \\Delta _n(\\mathbf {x}) \\Delta _n(\\mathbf {y})}$ extends to a strictly postive, continuous function $M_n$ on ${\\mathbf {R}}^n\\times {\\mathbf {R}}^n$ .",
"Define, for $n\\ge 1$ , $\\tau _n$ via $\\tau _n(x,y)=c_n^{-2} M_n( x{\\bf 1}_n, y{\\bf 1}_n),$ and $\\tau _0(x,y)=1$ .",
"Then $(\\tau _n; n \\ge 1)$ satisfy the two dimensional Toda equations in integrated form $\\log \\left( \\frac{\\tau _n(a,c)\\tau _n(b,d)}{\\tau _n(a,d)\\tau _n(b,c)}\\right)= \\int _a^b \\int _c^d \\frac{\\tau _{n-1}(x,y) \\tau _{n+1}(x,y)}{\\tau _n(x,y)^2} \\;\\mathrm {d}y\\mathrm {d}x.$ For any choice of distinct $x_1,x_2,\\ldots ,x_n, y,z\\in {\\mathbf {R}}$ we can apply the preceeding lemma to the functions $f_i(\\cdot )=u(x_i, \\cdot )$ for $i=1,2,\\ldots ,n$ , $g(\\cdot )= u(y, \\cdot ) $ , and $h(\\cdot )= u(z, \\cdot ) $ the necessary extensions existing by virtue of the existence of $M_n,M_{n+1}$ and $M_{n+2}$ .",
"Noting that $M_n( \\mathbf {x}, c{\\bf 1}_n)= c_n^{-1} \\frac{ W(f_1,f_2, \\ldots , f_n)(c)}{\\Delta _n(\\mathbf {x})},$ with similar expressions for the Wronskians $ W(f_1,f_2, \\ldots , f_n,g)$ , $ W(f_1,f_2, \\ldots , f_n,h)$ and $W(f_1,f_2, \\ldots , f_n,g,h)$ and using $\\frac{\\Delta _{n+1}({\\bf x},y) \\Delta _{n+1}({\\bf x},z)}{\\Delta _{n}({\\bf x}) \\Delta _{n+2}({\\bf x},y,z)}=\\frac{1}{z-y},$ we can write the conclusion of applying the lemma as $\\frac{1}{z-y}\\left( \\frac{M_{n+1}( (\\mathbf {x},z), d{\\bf 1}_{n+1})}{M_{n+1}( (\\mathbf {x},z), c{\\bf 1}_{n+1})}- \\frac{M_{n+1}( (\\mathbf {x},y), d{\\bf 1}_{n+1})}{M_{n+1}( (\\mathbf {x},y), c{\\bf 1}_{n+1})} \\right)=\\\\ \\frac{M_{n+1}( (\\mathbf {x},y), d{\\bf 1}_{n+1})}{M_{n+1}( (\\mathbf {x},z), c{\\bf 1}_{n+1})}\\int _c^d \\frac{ M_n(\\mathbf {x},v{\\bf 1}_n) M_{n+2}((\\mathbf {x},y,z),v{\\bf 1}_{n+2})}{M_{n+1}((\\mathbf {x},y),v{\\bf 1}_{n+1})^2} \\mathrm {d}v.$ Since the right hand side is a continuous function of $\\mathbf {x}$ , $y$ and $z$ we can both drop the assumption that $x_1, x_2, \\ldots , x_n,y$ are all distinct, and then let $z-y$ tend to 0 and deduce that $y \\mapsto \\frac{M_{n+1}( (\\mathbf {x},y), d{\\bf 1}_{n+1})}{M_{n+1}( (\\mathbf {x},y), c{\\bf 1}_{n+1})}$ is continuously differentiable with derivative $\\frac{M_{n+1}( (\\mathbf {x},y), d{\\bf 1}_{n+1})}{M_{n+1}( (\\mathbf {x},y), c{\\bf 1}_{n+1})}\\int _c^d \\frac{ M_n(\\mathbf {x},v{\\bf 1}_n) M_{n+2}((\\mathbf {x},y,y),v{\\bf 1}_{n+2})}{M_{n+1}((\\mathbf {x},y),v{\\bf 1}_{n+1})^2} \\;\\mathrm {d}v.$ $M_{n+1}( \\cdot , d{\\bf 1}_{n+1})/ M_{n+1}( \\cdot , c{\\bf 1}_{n+1})$ being a symmetric function on ${\\mathbf {R}}^{n+1}$ , this shows that all its partial deriavtives exist and are continuous, and hence, that it is in fact continuously differentiable.",
"Then we calculate $\\frac{\\mathrm {d}}{\\mathrm {d}x}\\left(\\frac{M_{n+1}( x{\\bf 1}_{n+1}, d{\\bf 1}_{n+1})}{ M_{n+1}( x{\\mathbf {1}}_{n+1}, c{\\bf 1}_{n+1})}\\right)= \\\\(n+1)\\frac{M_{n+1}( (x{\\bf 1}_{n+1}, d{\\bf 1}_{n+1})}{M_{n+1}( ((x{\\bf 1}_{n+1}, c{\\bf 1}_{n+1})}\\int _c^d \\frac{ M_n( (x{\\bf 1}_{n},v{\\bf 1}_n) M_{n+2}((x{\\bf 1}_{n+2},v{\\bf 1}_{n+2})}{M_{n+1}((x{\\bf 1}_{n+1},v{\\bf 1}_{n+1})^2} \\;\\mathrm {d}v$ from which the Toda equation follows easily.",
"In view of Theorem REF , and the representation (REF ), the above proposition applies directly to $u(x,y)=u(t,x,y)$ , the jointly continuous version of the solution to the stochastic heat equation."
],
[
"Construction of the conjugacy and proof of theorems",
"We begin with a lemma that ensures all the Wronskians we need exist.",
"Lemma 4.1 Suppose $f_1, f_2, \\ldots , f_n:{\\mathbf {R}}\\rightarrow {\\mathbf {R}} $ are continuous functions such that $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n /\\Delta _n$ extends continuously to ${\\mathbf {R}}^n$ .",
"Suppose also that $f_1,f_2, \\ldots , f_n$ are linearly independent.",
"Then, $f_{i_1} \\wedge f_{i_2} \\wedge \\ldots \\wedge f_{i_k} /\\Delta _k$ extends continuously to ${\\mathbf {R}}^k $ for any subset $\\lbrace i_1,i_2, \\ldots , i_k\\rbrace \\subseteq \\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"According to Laplace's expansion for determinants, $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n (x_1,x_2,\\ldots ,x_n)= \\\\\\sum _{k=1}^n (-1)^{k+1} f_k(x_1) f_1\\wedge \\ldots \\wedge f_{k-1}\\wedge f_{k+1} \\wedge \\ldots \\wedge f_n ( x_2,\\ldots ,x_n).$ Now the linear independence of $f_1, f_2, \\ldots , f_n$ ensures $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n$ is not identically zero, and so we can find $z_1, z_2, \\ldots , z_n$ so that the matrix $A$ with $(j,k)$ th entry $f_{k}(z_j)$ is invertible.",
"So the $n$ linear equations arising from Laplace's expansion by choosing $ x_1=z_1, z_2, \\ldots ,z_n$ successfully can be solved to give, for each $k$ , $f_1\\wedge \\ldots \\wedge f_{k-1}\\wedge f_{k+1} \\wedge \\ldots \\wedge f_n ( x_2,\\ldots ,x_n)= \\\\ (-1)^{k+1} \\sum _{j=1}^n A^{-1}_{kj} f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_n (z_j,x_2,\\ldots ,x_n).$ Now dividing this equation through by $\\Delta _{n-1}(x_2,\\ldots x_n)$ the right hand side is a linear combination of continuous functions on ${\\mathbf {R}}^{n-1}$ which proves the assertion of the lemma for $\\lbrace i_1,i_2, \\ldots , i_{n-1}\\rbrace = \\lbrace 1,2,\\ldots ,n\\rbrace \\setminus \\lbrace k\\rbrace $ .",
"The general case follows by iterating the argument.",
"A sequence of continuous functions $f_1, f_2, \\ldots , f_n$ with the property that for each $k \\in \\lbrace 1,2,\\ldots , n\\rbrace $ , $f_1\\wedge f_2\\wedge \\ldots \\wedge f_k(\\mathbf {x})/\\Delta _k(\\mathbf {x})$ extends to a continuous, strictly positive function on ${\\mathbf {R}}^k$ form an extended, complete Tchebysheff system, as defined in [19], where $f_1,\\ldots , f_n$ are assumed to be $(n-1)$ -times continuously differentiable so that Wronskians exist in the classical sense.",
"The following proposition is essentially a generalisation to the non-differentiable case of a representation for such a system in terms of its successive Wronskians, see [19].",
"In the statement of the proposition the integrals denote any choice of indefnite integral.",
"Proposition 4.2 Suppose $f_1, f_2, \\ldots , f_n$ are continuous functions with the property that that, for each $k \\in \\lbrace 1,2,\\ldots , n\\rbrace $ , $f_1\\wedge f_2\\wedge \\ldots \\wedge f_k(\\mathbf {x})/\\Delta _k(\\mathbf {x})$ extends to a continuous, strictly positive function on ${\\mathbf {R}}^k$ .",
"Denote $W(f_1, f_2, \\ldots , f_k)$ by $\\ell _k$ for $ 1\\le k\\le n$ .",
"Then $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_k$ for $k=1, 2, \\ldots , n$ can be recovered from $\\ell _1, \\ell _2, \\ldots , \\ell _n$ : indeed if we define $\\tilde{f}_1&= \\ell _1, \\\\\\tilde{f}_2&= \\ell _1\\int \\frac{\\ell _2}{\\ell _1^2}, \\\\\\tilde{f}_3&= \\ell _1\\int \\frac{\\ell _2}{\\ell _1^2} \\int \\frac{\\ell _1 \\ell _3}{\\ell ^2_2}, \\\\\\tilde{f}_n&= \\ell _1\\int \\frac{\\ell _2}{\\ell _1^2} \\int \\frac{\\ell _1 \\ell _3}{\\ell ^2_2} \\int \\cdots \\int \\frac{\\ell _{n-2}\\ell _n}{\\ell _{n-1}^2}, \\\\$ then $f_1 \\wedge f_2 \\wedge \\ldots \\wedge f_k= \\tilde{f}_1 \\wedge \\tilde{f}_2 \\wedge \\ldots \\wedge \\tilde{f}_k.$ First note that since $W(f_1,f_k)$ exists by the previous lemma, and $f_1$ is positive, $f_k= f_1 \\int \\frac{W(f_1,f_k)}{f_1^2},$ as in equation (REF ), with some unknown constant of integration.",
"Then since for $2 \\le r \\le k$ , all the necessary continuous extensions exist by the preceeding lemma, and $W(f_1, f_2, \\ldots ,f_{r})$ being strictly positive by hypothesis we can apply Lemma REF to obtain $W( f_1, f_2,\\ldots , f_{r-1},f_k)= W( f_1, f_2,\\ldots , f_r) \\int \\frac{ W( f_1, \\ldots , f_{r-1}) W( f_1, \\ldots , f_{r-1} f_{r},f_k) }{W( f_1, f_2,\\ldots , f_r)^2}.$ Applying this successively for $r=2, \\ldots , k$ , and substituting into our expression for $f_k$ from the previous step of the iteration, gives the result, noting the unknown constants of integration don't affect the value of the determinants.",
"Recalling from Section the definition of the map ${\\mathcal {R}}$ the preceeding proposition implies that ${\\mathcal {R}}: {\\mathfrak {F}}_n \\rightarrow {\\mathfrak {L}}_n$ is injective and gives an explicit form for its left inverse.",
"To show that ${\\mathcal {R}}$ is surjective, we need to consider whether the extension of $\\tilde{f}_1 \\wedge \\tilde{f}_2 \\wedge \\ldots \\wedge \\tilde{f}_k/ \\Delta _k$ exists when $\\tilde{f}_1, \\ldots , \\tilde{f}_k$ are constructed as in the proposition but $\\ell _1, \\ldots , \\ell _k$ are arbitrary strictly positive continuous functions.",
"This follows easily from the following proposition.",
"Proposition 4.3 Suppose $\\rho _1,\\rho _2, \\ldots , \\rho _n$ are continuous functions on ${\\mathbf {R}}$ , and that $ g_1=\\rho _1, g_2= \\rho _1 \\int \\rho _2, \\ldots , g_n= \\rho _1 \\int \\rho _{2} \\int \\cdots \\int \\rho _n.$ Then $(g_1\\wedge g_2 \\wedge \\ldots \\wedge g_n)/ \\Delta _n$ has a continuous extension to ${\\mathbf {R}}^{n}$ and $W(g_1,g_2, \\ldots , g_n)= \\rho _1^n W(h_1, h_2, \\ldots , h_{n-1}),$ where $ h_1=\\rho _2, h_2= \\rho _2 \\int \\rho _3, \\ldots h_{n-1}= \\rho _2 \\int \\rho _{3} \\int \\cdots \\int \\rho _n.$ If $\\rho _1,\\rho _2, \\ldots , \\rho _n$ are all strictly positive then so too is the extension of $(g_1\\wedge g_2\\wedge \\ldots \\wedge g_n)/ \\Delta _n$ .",
"This is by induction on $n$ .",
"Let $H$ denote the continuous extension of $(h_1\\wedge h_2 \\wedge \\ldots \\wedge h_{n-1})/ \\Delta _{n-1}$ which we assume exists by the inductive hypothesis.",
"Denote $(g_1\\wedge g_2 \\wedge \\ldots \\wedge g_n)/ \\Delta _n$ by $G$ .",
"Since this is symmetric, it is enough to show that this extends from a function on $W_n= \\lbrace \\mathbf {x}\\in {\\mathbf {R}}^n: x_1<x_2 <\\cdots <x_n\\rbrace $ to a continuous function on $ \\lbrace \\mathbf {x}\\in {\\mathbf {R}}^n: x_1\\le x_2 \\le \\cdots \\le x_n\\rbrace $ .",
"Elementary operations on determinants verify that $g_1\\wedge g_2 \\wedge \\ldots \\wedge g_n (\\mathbf {x})=\\left\\lbrace \\prod _{k=1}^n g_1(x_k) \\right\\rbrace \\int _{\\mathbf {y}\\preceq \\mathbf {x}} h_1\\wedge h_2 \\wedge \\ldots \\wedge h_{n-1}(\\mathbf {y}) \\;\\mathrm {d}\\mathbf {y},$ where $\\mathbf {y}\\preceq \\mathbf {x}$ means $\\mathbf {y}$ is interlaced with $\\mathbf {x}$ , that is, $x_1\\le y_1 \\le x_2 \\le y_2 \\le \\cdots \\le x_{n-1} \\le y_{n-1} \\le x_n.$ Consequently, $G(\\mathbf {x})= \\int H(\\mathbf {y}) \\frac{\\Delta _{n-1} (\\mathbf {y})}{\\Delta _n(\\mathbf {x}) }{\\mathbf {1}}( \\mathbf {y}\\preceq \\mathbf {x}) \\;\\mathrm {d}\\mathbf {y}.$ Now the kernel $(n-1)!\\frac{\\Delta _{n-1} (\\mathbf {y})}{\\Delta _n(\\mathbf {x}) }{\\mathbf {1}}( \\mathbf {y}\\preceq \\mathbf {x}) $ is well known to have the following random matrix interpretation.",
"If ${\\mathcal {M}}$ is an $n\\times n$ random Hermitian matrix having the GUE distribution, and $\\Pi {\\mathcal {M}}$ denotes its principal $(n-1) \\times (n-1)$ minor then the conditional distribution of the eigenvalues of $\\Pi {\\mathcal {M}}$ given that the eigenvalues of ${\\mathcal {M}}$ equal $\\mathbf {x}\\in W_n$ has density given by this kernel.",
"Consequently we can write $G(\\mathbf {x}) = \\frac{1}{(n-1)!",
"}{\\mathbf {E}} \\bigl [ H( \\Pi {\\mathcal {M}} ) | \\text{sp}({\\mathcal {M}})=\\mathbf {x}\\bigr ]= \\frac{1}{(n-1)!}",
"{\\mathbf {E}} \\bigl [ H( \\Pi ( {\\mathcal {U}} {\\mathcal {D}}(\\mathbf {x}) {\\mathcal {U}}^*))\\bigr ]$ where we interpret $H$ as a continuous function of $(n-1) \\times (n-1)$ Hermitian matrices that is invariant under the action of the unitary group, ${\\mathcal {U}}$ denotes an $n\\times n$ random unitary matrix distributed according to Haar measure, and $ {\\mathcal {D}}(\\mathbf {x})$ is a diagonal matrix with diagonal entries $(x_1,x_2, \\ldots , x_n)$ .",
"But this representation naturally defines $G$ on the boundary of the Weyl chamber $W_n$ , and moreover since $\\Pi $ and $H$ are continuous, the dominated convergence theorem implies that $G$ is continuous too.",
"Moreover, if $H$ and $g_1$ are both strictly positive then so too must be $G$ .",
"Finally, the relationship between Wronskians is simply the fact that $G(x{\\mathbf {1}}_n)=\\frac{c_n}{c_{n-1}} g_1(x)^n H(x{\\mathbf {1}}_{n-1}).$ To prove Theorem REF we must verify that the “fundamental solution” ${\\tau }(t,x, y)$ evolves according to ${\\tau }_{t}={\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1} {\\tau }_{s},$ and satisfies the flow property stated in the theorem.",
"Take a sequence of points $\\mathbf {x}(k) \\in {\\mathbf {R}}^n_{\\ne }$ with $\\mathbf {x}(k) \\rightarrow x{\\mathbf {1}}_n$ as $k \\rightarrow \\infty $ .",
"Then by virtue of Theorem REF , with probability one, $M_n( t, \\mathbf {x}(k), \\mathbf {y}) \\rightarrow M_n(t, x{\\mathbf {1}}_n, \\mathbf {y}) \\text{ locally uniformly in $\\mathbf {y}\\in {\\mathbf {R}}^n$}.$ Consequently, $c_n^{-1}M_n( t, \\mathbf {x}(k), y{\\mathbf {1}}_n) \\rightarrow \\tau _n(t,x,y) \\text{ locally uniformly in $y \\in {\\mathbf {R}}$.",
"}$ Now $M_n( t, \\mathbf {x}(k), \\cdot ) \\in C(\\mathbf {R}^n, \\mathbf {R})$ for each $k$ , and ${\\mathcal {R}}$ being a homeomorphism, we deduce that $(M_k( t, x{\\mathbf {1}}_k, \\cdot ))_{1\\le k\\le n} \\in {\\mathfrak {F}}_n$ and ${\\mathcal {R}} \\bigl (M_k(t, x{\\mathbf {1}}_k, \\cdot ), 1\\le k\\le n \\bigr )= {\\tau }(t, x, \\cdot ).$ Now it was proved in [22] that, for a fixed $s>0$ and $k\\ge 1$ , $M_k(s, x{\\mathbf {1}}_k,\\mathbf {y})$ has uniformly bounded $p$ th moments as $y\\in {\\mathbf {R}}^k$ varies, and consequently we can solve the evolution equation for $M_k$ with shifted white noise and initial data $\\bigl (M_k(s, x{\\mathbf {1}}_k,\\mathbf {y}); \\mathbf {y}\\in {\\mathbf {R}}\\bigr )$ .",
"Thus, as at equation as a (REF ), it is meaningful to apply ${\\mathcal {M}}_{s,t}$ to $(M_k( s, x{\\mathbf {1}}_k, \\cdot ))_{1\\le k\\le n}$ , giving $(M_k( t, x{\\mathbf {1}}_k, \\cdot ))_{1\\le k\\le n}$ .",
"It now follows from (REF ), with the time $t$ replaced by the time $s$ , that ${\\tau }_t(x, \\cdot )$ satisfies ${\\tau }_{t}={\\mathcal {R}} {\\mathcal {M}}_{s,t} {\\mathcal {R}}^{-1} {\\tau }_{s}.$ Corollary 6.2 of [23], states that, for given $\\mathbf {x}, \\mathbf {y}\\in {\\mathbf {R}}^n$ and $0<s<t$ , with probability one, $M_n(t,\\mathbf {x},\\mathbf {y})= \\frac{1}{n!}",
"\\int _{\\mathbf {R}^n} M_n(s,\\mathbf {x}, \\mathbf {z})M_n(s,t,\\mathbf {z},\\mathbf {y}) \\Delta _n(\\mathbf {z})^2 \\;\\mathrm {d}\\mathbf {z}$ where $M_n(s,t, \\cdot ,\\cdot )$ is defined as $M_n(t-s,\\cdot ,\\cdot )$ by the chaos expansion but with the shifted white noise $\\dot{W}^{(t-s)}(\\cdot ,\\cdot )= \\dot{W}((t-s)+\\cdot ,\\cdot )$ .",
"Notice that the argument that gave (REF ) is easily adapted to give ${\\mathcal {R}} \\bigl (M_k(t, \\cdot , y{\\mathbf {1}}_k ), 1\\le k\\le n \\bigr )= {\\tau }_t( \\cdot ,y).$ And now combining (REF ), (REF ), and the definition of the pairing $\\langle \\cdot ,\\cdot \\rangle _{{\\mathfrak {L}}_n}$ when we take $\\mathbf {x}=x{\\mathbf {1}}_n$ and $\\mathbf {y}=y{\\mathbf {1}}_n$ , we obtain the desired equation for ${\\tau }(t,x, y)$ from (REF ).",
"Theorem REF is a fairly immediate consequence of the construction in Theorem REF .",
"Let $0\\le s < t$ and fix $x\\in \\mathbf {R}$ .",
"Let $C\\in {B}({\\mathfrak {L} }_n)$ be a Borel set.",
"Denote $\\mathbf {Z}_t = \\big (Z_1(t,x,\\cdot ),\\ldots ,Z_n(t,x,\\cdot )\\big )$ .",
"Let $\\bigl ({F}_t\\bigr )_{t \\ge 0}$ denote the filtration generated by the white noise $\\big ( \\dot{W}(s,A); 0\\le s\\le t, A\\in {B}({\\mathbf {R}}) \\bigr )$ .",
"We will show that the conditional probability that $\\mathbf {Z}_t\\in C$ given ${F}_s$ is measurable with respect to $\\sigma \\big (\\mathbf {Z}_s\\big )$ .",
"By (REF ), the process $\\mathbf {Z}_t$ is proportional to $\\big (M_1(t,x\\mathbf {1}_1,\\cdot \\mathbf {1}_1),\\ldots ,M_n(t,x\\mathbf {1}_n,\\cdot \\mathbf {1}_n)\\big )$ and since $M_k(t,x\\mathbf {1}_k,\\cdot ) $ can almost surely be identified as the solution of the evolution equation (REF ) with initial condition $M_k(s,x\\mathbf {1}_k,\\cdot ) $ and driven by the shifted white noise which is independent of ${F}_s$ , we have, $\\mathbb {P}[\\mathbf {Z}_t\\in C | {F}_s] \\text{ is measurable with respect to $\\big (M_1(s,x\\mathbf {1}_1,\\cdot ),\\ldots ,M_n(s,x\\mathbf {1}_n,\\cdot )\\big )$.",
"}$ However, for each $1 \\le k \\le n$ , $M_k(s,x\\mathbf {1}_k,\\cdot )$ is a function of $\\mathbf {Z}_s$ and the result follows."
],
[
"Appendix",
"We show that $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{\\det [u(t,x_i,y_j)]_{i,j=1}^n}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})}$ solves $M_n(t,\\mathbf {x},\\mathbf {y}) = \\frac{p_n^*(t,\\mathbf {x},\\mathbf {y})}{\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})} + A_n \\int _0^t \\int _{\\mathbf {R}^n} Q_{t-s}(\\mathbf {y},\\mathbf {z}) M_n(s,\\mathbf {x},\\mathbf {z}) \\;\\mathrm {d}\\mathbf {z}_* W(\\mathrm {d}s,\\mathrm {d}z_1),$ where $Q_t(\\mathbf {y},\\mathbf {z}) = \\frac{\\Delta _n(\\mathbf {z})}{\\Delta _n(\\mathbf {y})}p_n^*(t,\\mathbf {y},\\mathbf {z})$ , $\\mathrm {d}\\mathbf {z}_* = \\mathrm {d}z_2\\ldots \\mathrm {d}z_n$ and $A_n = \\frac{1}{(n-1)!",
"}$ .",
"It suffices to prove that $K_n(t,\\mathbf {x},\\mathbf {y}) := \\det [u(t,x_i,y_j)]_{i,j=1}^n$ satisfies $K_n(t,\\mathbf {x},\\mathbf {y}) = p_n^*(t,\\mathbf {x},\\mathbf {y}) + A_n \\int _0^t \\int _{\\mathbf {R}^n} p_n^*(t-s,\\mathbf {y},\\mathbf {z}) K_n(s,\\mathbf {x},\\mathbf {z}) \\;\\mathrm {d}\\mathbf {z}_* W(\\mathrm {d}s,\\mathrm {d}z_1),$ then the result follows upon dividing by $\\Delta _n(\\mathbf {x})\\Delta _n(\\mathbf {y})$ .",
"First note that $u$ is the solution to the following mild equation $u(t,x,y)&= p_t(x-y) + \\int _0^t \\int _\\mathbf {R}p_{t-s}(y-z) u(s,x,z) \\;W(\\mathrm {d}s,\\mathrm {d}z) \\\\&= p_t(x-y) + I(t,x,y).$ Using this and expanding the determinant we have $K_n(t,\\mathbf {x},\\mathbf {y})&= \\sum _{\\sigma } (-1)^\\sigma \\prod _{i=1}^n \\big ( p_t(x_{\\sigma i}-y_i) + I(t,x_{\\sigma i},y_i) \\big ) \\\\&= \\sum _{m=0}^n \\sum _\\sigma \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m} (-1)^\\sigma \\prod _{r=1}^m I(t,x_{\\sigma i_r},y_{i_r}) \\prod _{r=m+1}^n p_t(x_{\\sigma j_r}-y_{j_r}),$ where $D_m := \\lbrace (\\mathbf {i}, \\mathbf {j}) : 1\\le i_1<\\cdots <i_m\\le n, 1\\le j_{m+1}<\\cdots <j_n\\le n, i_k \\ne j_l \\;\\forall k,l \\rbrace $ and empty products are defined to be equal to 1.",
"For martingales $X^1,\\ldots , X^n$ we have the following product formula $\\prod _{i=1}^n X_t^i = \\prod _{i=1}^n X_0^i + \\sum _{i=1}^n \\int _0^t \\prod _{j\\ne i} X_s^j \\;\\mathrm {d}X_s^i + \\sum _{i\\ne j} \\int _0^t \\prod _{k\\ne i,j} X_s^k \\;\\mathrm {d}\\langle X^i,X^j\\rangle _s,$ which follows from the familiar formula for the $n=2$ case and induction.",
"Using this we have $K_n(t,\\mathbf {x},\\mathbf {y})&= \\sum _{m=0}^n \\sum _\\sigma \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m} (-1)^\\sigma \\prod _{r=m+1}^n p_t(x_{\\sigma j_r}-y_{j_r}) \\\\&\\qquad \\times \\bigg ( \\sum _{k=1}^m \\int _0^t \\prod _{r\\ne k} I_s^{\\sigma i_r} \\;\\mathrm {d}I_s^{\\sigma i_k} - \\sum _{k\\ne l} \\int _0^t \\prod _{r\\ne k,l} I_s^{\\sigma i_r} \\;\\mathrm {d}\\langle I^{\\sigma i_k},I^{\\sigma i_l}\\rangle _s \\bigg ),$ where $I_t^{\\sigma i} := I(s,x_{\\sigma i},y_i)$ and by [20] $\\langle I^{\\sigma i}, I^{\\sigma j} \\rangle _t = \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_i-z)p_{t-s}(y_j-z)u(s,x_{\\sigma i},z)u(s,x_{\\sigma j},z) \\;\\mathrm {d}z\\mathrm {d}s.$ Consider $\\sigma ^\\prime = \\sigma \\circ (i,j)$ then $\\langle I^{\\sigma ^\\prime i}, I^{\\sigma ^\\prime j} \\rangle _t = \\langle I^{\\sigma i}, I^{\\sigma j} \\rangle _t$ .",
"Moreover since $(-1)^{\\sigma ^\\prime } = -(-1)^\\sigma $ we have, by considering such pairs of permutations, that for each $m = 2,\\ldots ,n$ $\\sum _{\\sigma } \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m} \\sum _{k\\ne l} (-1)^\\sigma \\int _0^t \\prod _{r\\ne k,l} I_s^{\\sigma i_r} \\;\\mathrm {d}\\langle I^{\\sigma i_k},I^{\\sigma i_l}\\rangle _s = 0.$ Therefore, $K_n(t,\\mathbf {x},\\mathbf {y}) = \\sum _{m=0}^n \\sum _\\sigma \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m} \\sum _{k=1}^m (-1)^\\sigma \\prod _{r=m+1}^n p_t(x_{\\sigma j_r}-y_{j_r}) \\int _0^t \\prod _{r\\ne k} I_s^{\\sigma i_r} \\;\\mathrm {d}I_s^{\\sigma i_k}.$ On the other hand, $\\int _0^t \\int _{\\mathbf {R}^n} & p_n^*(t-s,\\mathbf {y},\\mathbf {z}) K_n(s,\\mathbf {x},\\mathbf {z}) \\;\\mathrm {d}\\mathbf {z}_* W(\\mathrm {d}s,\\mathrm {d}z_1) \\\\&= \\sum _\\sigma \\sum _\\pi (-1)^\\sigma (-1)^\\pi \\int _0^t \\int _{\\mathbf {R}^n} \\prod _{i=1}^n p_{t-s}(y_{\\pi i}-z_i) u(s,x_{\\sigma i}, z_i) \\;\\mathrm {d}\\mathbf {z}_*W(\\mathrm {d}s,\\mathrm {d}z_1).$ Observe that $u(t,x,y)&= p_t(x-y) + I(t,x,y) \\\\&= \\int _\\mathbf {R}p_{t-s}(y-z) u(s,x,z) \\;\\mathrm {d}z + \\int _s^t \\int _\\mathbf {R}p_{t-r}(y-z) u(r,x,z) \\;W(\\mathrm {d}r,\\mathrm {d}z),$ and so $\\int _\\mathbf {R}p_{t-s}(y-z) u(s,x,z) \\;\\mathrm {d}z = p_t(x-y) + I(s,x,y).$ Using this the right hand side of (REF ) is equal to $\\sum _\\sigma \\sum _\\pi & (-1)^\\sigma (-1)^\\pi \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_{\\pi 1}-z) u(s,x_{\\sigma 1},z) \\\\&\\qquad \\qquad \\times \\prod _{i=2}^n \\big ( p_t(x_{\\sigma i}-y_{\\pi i}) + I(s,x_{\\sigma i},y_{\\pi i}) \\big ) \\;W(\\mathrm {d}s,\\mathrm {d}z) \\\\&= \\sum _{m=1}^n \\sum _\\sigma \\sum _\\pi \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m^\\prime } (-1)^\\sigma (-1)^\\pi \\prod _{r=m+1}^n p_t(x_{\\sigma j_r}-y_{\\pi j_r}) \\\\&\\qquad \\qquad \\times \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_{\\pi 1}-z) u(s,x_{\\sigma 1},z) \\prod _{r=2}^m I(s,x_{\\sigma i_r},y_{\\pi i_r}) \\;W(\\mathrm {d}s,\\mathrm {d}z),$ where $D_m^\\prime = \\lbrace (\\mathbf {i},\\mathbf {j}) : 2\\le i_2<\\cdots <i_m\\le n, 2\\le j_{m+1}<\\cdots <j_n\\le n, i_k\\ne j_l \\;\\forall k,l\\rbrace $ .",
"Note that there are $\\binom{n-1}{m-1}$ terms in the sum over $D_m^\\prime $ .",
"Let $(\\mathbf {a},\\mathbf {b})\\in D_m$ where $D_m$ was defined above then for $1\\le k\\le m$ , split the sum over $\\pi \\in S_n$ into groups $G_k(\\mathbf {a},\\mathbf {b})$ of permutations such that $\\pi 1 = a_k$ , $\\pi i_r \\in \\lbrace a_i : 1\\le i \\le m, i\\ne k\\rbrace $ and for $r = 2,\\ldots ,m$ and $\\pi i_r \\in \\lbrace b_{m+1},\\ldots ,b_n\\rbrace $ for $r = m+1,\\ldots ,n$ .",
"Then the right hand side of the previous display is equal to $&\\sum _{m=1}^n \\sum _{(\\mathbf {a},\\mathbf {b})\\in D_m} \\sum _{k=1}^m \\sum _{\\pi \\in G_k(\\mathbf {a},\\mathbf {b})} \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m^\\prime } \\sum _\\sigma (-1)^\\sigma (-1)^\\pi \\prod _{r=m+1}^n p_t(x_{\\sigma j_r}-y_{\\pi j_r}) \\\\&\\qquad \\qquad \\times \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_{\\pi 1}-z) u(s,x_{\\sigma 1},z) \\prod _{r=2}^m I(s,x_{\\sigma i_r},y_{\\pi i_r}) \\;W(\\mathrm {d}s,\\mathrm {d}z) \\\\&= \\sum _{m=1}^n \\sum _{(\\mathbf {a},\\mathbf {b})\\in D_m} \\sum _{k=1}^m \\sum _\\sigma A_n^{-1} (-1)^\\sigma \\prod _{r=m+1}^n p_t(x_{\\sigma b_r}-y_{b_r}) \\\\&\\qquad \\qquad \\times \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_{a_k}-z) u(s,x_{\\sigma a_k},z) \\prod _{r\\ne k} I(s,x_{\\sigma a_r},y_{a_r}) \\;W(\\mathrm {d}s,\\mathrm {d}z),$ where the last equality is due to the fact that for each $\\mathbf {a}$ , $\\mathbf {b}$ and $k$ each term in the sum $\\sum _{\\pi \\in G_k(\\mathbf {a},\\mathbf {b})} \\sum _{(\\mathbf {i},\\mathbf {j})\\in D_m}$ are equal and there are in total $\\binom{n-1}{m-1}(m-1)!(n-m)!",
"= A_n^{-1}$ terms in the sum.",
"Finally, observing that the $m=0$ term in (REF ) is equal to $p_n^*(t,\\mathbf {x},\\mathbf {y})$ shows that $\\det [u(t,x_i,y_j)]_{i,j=1}^n$ satisfies equation (REF ).",
"In the calculations above we saw stochastic integrals of the form $\\int _0^t \\int _\\mathbf {R}p_{t-s}(y_1-z) u(s,x_1,z) \\prod _{i=2}^m I(s,x_i,y_i) \\;W(\\mathrm {d}s,\\mathrm {d}z).$ This integral is well defined since the integrand is adapted and continuous and by Hölder's inequality $\\int _0^t \\int _\\mathbf {R}& p_{t-s}(y_1-z)^2 \\bigg ṷ(s,x_1,z) \\prod _{i=2}^m I(s,x_i,y_i) \\bigg 2^2 \\;\\mathrm {d}z\\mathrm {d}s \\\\&\\le \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_1-z)^2 ṷ(s,x_1,z) {2m}^2 \\prod _{i=2}^m I̭(s,x_i,y_i) {2m}^2 \\;\\mathrm {d}z\\mathrm {d}s,$ where $p := (\\mathbf {E}[|\\cdot |^p])^{1/p}$ .",
"By the Burkholder–Davis–Gundy inequality, there is a constant $c = c(m)$ such that $I̭(s,x,y) {2m}^2 \\le c \\int _0^s \\int _\\mathbf {R}p_{t-r}(y-z)^2 ṷ(r,x,z){2m}^2 \\;\\mathrm {d}z\\mathrm {d}r.$ Since $ṷ(r,x,z) p \\le c p_r(x-z)$ for a constant $c$ depending on $p$ , see for example [2], we have for constants depending only on $m$ $I̭(s,x,y) {2m}^2&\\le c^\\prime \\int _0^s \\int _\\mathbf {R}p_{t-r}(y-z)^2 p_r(x-y)^2 \\;\\mathrm {d}z\\mathrm {d}r \\\\&\\le c^{\\prime \\prime } p_{t/2}(x-y) \\int _0^s \\frac{1}{\\sqrt{t-r}} \\frac{1}{\\sqrt{r}} \\;\\mathrm {d}r \\\\&\\le c^{\\prime \\prime } \\pi p_{t/2}(x-y),$ where in the last line we used the fact that $\\int _0^s \\frac{1}{\\sqrt{t-r}} \\frac{1}{\\sqrt{r}} \\;\\mathrm {d}r \\le \\int _0^t \\frac{1}{\\sqrt{t-r}} \\frac{1}{\\sqrt{r}} \\;\\mathrm {d}r = \\pi .$ Therefore, the right hand side of (REF ) is less than $c^{\\prime \\prime \\prime } \\prod _{i=2}^m p_{t/2}(x_i-y_i) \\int _0^t \\int _\\mathbf {R}p_{t-s}(y_1-z)^2 p_s(x_1-z)^2 \\;\\mathrm {d}z{s} \\le c^{\\prime \\prime \\prime \\prime } \\prod _{i=1}^m p_{t/2}(x_i-y_i) < \\infty .$ Consequently, by [6], the integral (REF ) is well defined."
]
] | 1606.05139 |
[
[
"Astrophysical applications of the post-Tolman-Oppenheimer-Volkoff\n formalism"
],
[
"Abstract The bulk properties of spherically symmetric stars in general relativity can be obtained by integrating the Tolman-Oppenheimer-Volkoff (TOV) equations.",
"In previous work we developed a \"post-TOV\" formalism - inspired by parametrized post-Newtonian theory - which allows us to classify in a parametrized, phenomenological form all possible perturbative deviations from the structure of compact stars in general relativity that may be induced by modified gravity at second post-Newtonian order.",
"In this paper we extend the formalism to deal with the stellar exterior, and we compute several potential astrophysical observables within the post-TOV formalism: the surface redshift $z_s$, the apparent radius $R_{\\rm app}$, the Eddington luminosity at infinity $L_{\\rm E}^\\infty$ and the orbital frequencies.",
"We show that, at leading order, all of these quantities depend on just two post-TOV parameters $\\mu_1$ and $\\chi$, and we discuss the possibility to measure (or set upper bounds on) these parameters."
],
[
"Introduction",
"Compact stars are ideal astrophysical environments to probe the coupling between matter and gravity in a high-density, strong gravity regime not accessible in the laboratory.",
"Cosmological observations and high-energy physics considerations have spurred extensive research on the properties of neutron stars, whether isolated or in binary systems, in modified theories of gravity (see e.g.",
"[2], [3] for reviews).",
"Different extensions of general relativity (GR) affect the bulk properties of the star (such as the mass $M$ and radius $R$ ) in similar ways for given assumptions on the equation of state (EOS) of high-density matter.",
"Therefore it is interesting to understand whether these deviations from the predictions of GR can be understood within a simple parametrized formalism.",
"The development of such a generic framework to understand how neutron star properties are affected in modified gravity is even more pressing now that gravitational-wave observations are finally a reality [4], since the observation of neutron star mergers could allow us to probe the dynamical behavior of these objects in extreme environments.",
"In previous work we developed a post-Tolman-Oppenheimer-Volkoff (henceforth, post-TOV) formalism valid for spherical stars [1].",
"The basic idea is quite simple.",
"The structure of nonrotating, relativistic stars can be determined by integrating two ordinary differential equations: one of these equations gives the “mass function” and the other equation – a generalization of the hydrostatic equilibrium condition in Newtonian gravity – determines the pressure profile and the stellar radius, defined as the point where the pressure vanishes.",
"The post-TOV formalism, reviewed in Section below, adds a relatively small number of parametrized corrections with parameters $\\mu _i$ ($i=1, \\dots \\,, 5$ ) and $\\pi _i$ ($i=1, \\dots \\,, 4$ ) to the mass and pressure equations.",
"These corrections have two properties: (i) they are of second post-Newtonian (PN) order, because first-order deviations are already tightly constrained by observations; and (ii) they are general enough to capture in a phenomenological way all possible deviations from the mass-radius relation in GR.",
"Other parametrizations were explored in [5], [6] by modifying ad hoc the TOV equations.",
"In this paper we turn to the investigation of astrophysical applications of the formalism.",
"Part of our analysis is inspired by previous work by Psaltis [7], who showed that, under the assumption of spherical symmetry, many properties of neutron stars in metric theories of gravity can be calculated using only conservation laws, Killing symmetries, and the Einstein equivalence principle, without requiring the validity of the GR field equations.",
"Psaltis computed the gravitational redshift of a surface atomic line $z_s$ , the Eddington luminosity at infinity $L_{\\rm E}^\\infty $ (thought to be equal to the touchdown luminosity of a radius-expansion burst), and the apparent surface area of a neutron star (which is potentially measurable during the cooling tails of bursts).",
"We first extend our previous work to study the exterior of neutron stars.",
"Then we compute the surface redshift $z_s$ , the apparent radius $R_{\\rm app}$ and the Eddington luminosity at infinity $L_{\\rm E}^\\infty $ .",
"In addition, we study geodesic motion in the neutron star spacetime within the post-TOV formalism.",
"We focus on the orbital and epicyclic frequencies, that according to some models – such as the relativistic precession model [8], [9] and the epicyclic resonance model [10] – may be related with the quasi-periodic oscillations (QPOs) observed in the X-ray spectra of accreting neutron stars.",
"Our main result is that, at leading order, all of these quantities depend on just two post-TOV parameters: $\\mu _1$ and the combination $\\chi \\equiv \\pi _2 - \\mu _2 - 2\\pi \\mu _1\\,.$ We also express the leading multipoles in a multipolar expansion of the neutron star spacetime in terms of $\\mu _1$ and $\\chi $ , and we discuss the possibility to measure (or set upper bounds on) these parameters with astrophysical observations.",
"The plan of the paper is as follows.",
"In Section we present a short review of the post-TOV formalism developed in [1].",
"In Section we extend the formalism to deal with the stellar exterior, computing a “post-Schwarzschild” exterior metric.",
"In Section we compute the surface redshift $z_s$ and relate it to the stellar compactness $M/R$ .",
"In Section , following [7], we study the properties of bursting neutron stars in the post-TOV framework.",
"In Section we calculate the orbital frequencies.",
"In Section we look at the leading-order multipoles of post-TOV stars.",
"Then we present some conclusions and possible directions for future work."
],
[
"Overview of the post-TOV formalism",
"Let us begin with a review of the post-TOV formalism introduced in [1].",
"The core of this formalism consists of the following set of “post-TOV\" structure equations for static spherically symmetric stars (we use geometrical units $G=c=1$ ): $\\frac{dp}{dr} &= \\left(\\frac{dp}{dr} \\right)_{\\rm GR} -\\frac{\\rho m}{r^2} \\left(\\, {\\cal P}_1 + {\\cal P}_2\\, \\right),\\\\\\nonumber \\\\\\frac{dm}{dr} & = \\left( \\frac{dm}{dr} \\right)_{\\rm GR} + 4\\pi r^2\\rho \\left( {\\cal M}_1 + {\\cal M}_2\\right),$ where ${\\cal P}_1 &\\equiv \\delta _1 \\frac{m}{r} + 4\\pi \\delta _2 \\frac{r^3 p}{m},\\quad {\\cal M}_1 \\equiv \\delta _3 \\frac{m}{r} + \\delta _4 \\Pi ,\\\\\\nonumber \\\\{\\cal P}_2 &\\equiv \\pi _1 \\frac{m^3}{r^5\\rho } + \\pi _2 \\frac{m^2}{r^2}+ \\pi _3 r^2 p + \\pi _4 \\frac{\\Pi p}{\\rho },\\\\\\nonumber \\\\{\\cal M}_2 &\\equiv \\mu _1 \\frac{m^3}{r^5\\rho } + \\mu _2 \\frac{m^2}{r^2}+ \\mu _3 r^2 p+ \\mu _4 \\frac{\\Pi p}{\\rho } + \\mu _5 \\Pi ^3 \\frac{r}{m} .",
"\\nonumber \\\\$ Here $r$ is the circumferential radius, $m$ is the mass function, $p$ is the fluid pressure, $\\rho $ is the baryonic rest mass density, $\\epsilon $ is the total energy density and $\\Pi \\equiv (\\epsilon -\\rho )/\\rho $ is the internal energy per unit baryonic mass.",
"A “GR” subscript denotes the standard TOV equations in GR, i.e.",
"$& \\left(\\frac{dp}{dr} \\right)_{\\rm GR} = -\\frac{(\\epsilon + p)}{r^2} \\frac{ (m_{\\rm T} + 4\\pi r^3 p )}{ ( 1-2m_{\\rm T} /r )},\\\\\\nonumber \\\\& \\left( \\frac{dm}{dr} \\right)_{\\rm GR} = \\frac{d m_{\\rm T}}{dr} = 4\\pi r^2 \\epsilon ,$ where $m_{\\rm T}$ is the GR mass function.",
"The dimensionless combinations ${\\cal P}_1,{\\cal M}_1$ and ${\\cal P}_2, {\\cal M}_2$ represent a parametrized departure from the GR stellar structure and are linear combinations of 1PN- and 2PN-order terms, respectively.",
"These terms feature the phenomenological post-TOV parameters $\\delta _i$ ($i=1,\\,\\dots \\,,4$ ), $\\pi _i$ ($i=1,\\,\\dots \\,,4$ ) and $\\mu _i$ ($i=1, \\dots \\,, 5$ ).",
"In particular, the coefficients $\\delta _i$ attached to the 1PN terms are simple algebraic combinations of the traditional PPN parameters $\\delta _1 \\equiv 3 (1+ \\gamma ) -6\\beta + \\zeta _2$ , $\\delta _2 \\equiv \\gamma -1+\\zeta _4$ , $\\delta _3 \\equiv -\\frac{1}{2} \\left( 11 + \\gamma -12\\beta + \\zeta _2 -2\\zeta _4 \\right)$ , $\\delta _4\\equiv \\zeta _3.$ As such, they are constrained to be very close to zero by existing Solar System and binary pulsar observationsUsing the latest constraints on the PPN parameters [11] we obtain the following upper limits: $| \\delta _1| \\lesssim 6\\times 10^{-4}, |\\delta _2| \\lesssim 7\\times 10^{-3}, | \\delta _3 | \\lesssim 7\\times 10^{-3}, |\\delta _4| \\lesssim 10^{-8}$ .",
": $|\\delta _i| \\ll 1$ .",
"This result translates to negligibly small 1PN terms in Eq.",
"(): ${\\cal P}_{1}\\ll 1$ , ${\\cal M}_1 \\ll 1$ .",
"On the other hand, $\\pi _i$ and $\\mu _i$ are presently unconstrained, and consequently ${\\cal P}_2, {\\cal M}_2$ should be viewed as describing the dominant (significant) departure from GR.",
"The GR limit of the formalism corresponds to setting all of these parameters to zero, i.e.",
"$\\delta _i, \\pi _i, \\mu _i \\rightarrow 0$ .",
"Alternatively, the stellar structure equations () can be formally derived – if we neglect the small terms ${\\cal P}_1, {\\cal M}_1$ – from a covariantly conserved perfect fluid stress energy tensor [1]: $\\nabla _\\nu T^{\\mu \\nu } = 0, \\qquad T^{\\mu \\nu } = (\\epsilon _{\\rm eff} + p) u^\\mu u^\\nu + p g^{\\mu \\nu },$ where the effective, gravity-modified energy density is $\\epsilon _{\\rm eff} = \\epsilon + \\rho {\\cal M}_2,$ and the covariant derivative is compatible with the effective post-TOV metric $g_{\\mu \\nu } = \\mbox{diag} [\\, -e^{\\nu (r)}, (1-2m(r)/r)^{-1}, r^2, r^2 \\sin ^2\\theta \\,],$ with $\\frac{d\\nu }{dr} = \\frac{2}{r^2\\\\} \\left[\\, (1-{\\cal M}_2) \\frac{m+4\\pi r^3 p}{1-2m/r} + m{\\cal P}_2 \\, \\right].$ This post-TOV metric is valid in the interior of the star.",
"In the following section we discuss how an exterior post-TOV metric can be constructed within our framework."
],
[
"The exterior “post-Schwarzschild” metric",
"For the applications of the post-TOV formalism considered in this work, we must specify how the $g_{tt}$ and $g_{rr}$ metric elements are calculated in the interior and exterior regions of the fluid distribution.",
"In this section we will construct an exterior spacetime in a “post-Schwarzschild” form.",
"From the effective post-TOV metric, we have that inside the fluid body $g_{tt} = -\\exp [\\nu (r)]$ , where $\\nu (r)$ is determined in terms of the fluid variables and post-TOV parameters from Eq.",
"(REF ).",
"We will assume that outside the fluid distribution the same effective metric expression holdsThis assumption is based on simplicity.",
"While we keep an agnostic view on the validity of Birkhoff's theorem within our formalism (and in modified gravity theories in general), the interior post-TOV metric is arguably the best guide towards the construction of the exterior metric.. Then we get the equations $&& \\frac{d\\nu }{dr} = \\left(\\frac{d\\nu }{dr} \\right)_{\\rm GR} + \\frac{2}{r^2} \\left[\\, - \\mu _2 \\frac{m^2}{r^2} \\frac{m}{1-2m/r}+ \\pi _2 \\frac{m^3}{r^2} \\, \\right],\\nonumber \\\\\\\\\\nonumber \\\\&& \\frac{dm}{dr} = 4\\pi \\mu _1 \\frac{m^3}{r^3},$ where $\\left(\\frac{d\\nu }{dr} \\right)_{\\rm GR} \\equiv \\frac{2}{r^2}\\frac{m}{1-2m/r}.$ These equations originate from the general expressions (REF ) and () after setting all fluid parameters to zero, i.e $p=\\epsilon =\\rho =\\Pi =0$ , and keeping the surviving terms in ${\\cal M}_2$ and ${\\cal P}_2$ .",
"It is not difficult to see that, in the nomenclature of [1], the only 2PN post-TOV terms that can appear in the exterior equations are those of “family F1” and “family F2”.",
"The F1 term (coefficient $\\pi _1$ ) should not appear in the ${\\cal P}_2$ correction of the interior $d\\nu /dr$ equation because it is divergent at the surface.",
"This implies that the F1 term should not appear in the $dp/dr$ equation either.",
"As it stands, Eq.",
"(REF ) contains higher than 2PN order terms.",
"It should therefore be PN-expanded with respect to the post-TOV terms: $\\frac{d\\nu }{dr} = \\left(\\frac{d\\nu }{dr} \\right)_{\\rm GR} + 2( \\pi _2 - \\mu _2) \\frac{m^3}{r^4}.$ Thus (REF ) and () are our “final” post-TOV equations for the stellar exterior.",
"The mass equation is decoupled and can be directly integrated.",
"The result is $m(r) = \\frac{r}{\\sqrt{4\\pi \\mu _1 + K r^2}},$ where $K$ is an integration constant.",
"The fact that $dm/dr \\ne 0$ outside the star implies the presence of an “atmosphere” due to the non-GR degree of freedom.",
"This is reminiscent of the exterior structure of neutron stars in scalar-tensor theories [12].",
"The constant $K$ is fixed by setting $m(r\\rightarrow \\infty ) $ equal to the system's ADM mass $M_{\\infty }$ .",
"Then, $m(r) = M_\\infty \\left( 1+ 4\\pi \\mu _1 \\frac{M^2_\\infty }{r^2} \\right)^{-1/2}.$ Thus the ADM mass is related to the Schwarzschild mass $M \\equiv m(R)$ by $M = M_\\infty \\left( 1+ 4\\pi \\mu _1 \\frac{M^2_\\infty }{R^2} \\right)^{-1/2}.$ As expected, in the GR limit the two masses coincide $m(r> R) = M_\\infty = M.$ Assuming a post-TOV correction ${\\cal F} \\equiv 4\\pi |\\mu _1 | (M_\\infty /R)^2 \\ll 1$ we can re-expand our result (REF ), $M = M_\\infty \\left( 1- 2\\pi \\mu _1 \\frac{M^2_\\infty }{R^2} \\right).$ The inverse relation $M_\\infty = M_\\infty (M)$ readsAt first glance, obtaining $M_\\infty (M)$ entails solving a cubic equation.",
"However, the procedure is greatly simplified if we recall that the post-TOV formalism must reduce to GR for $\\lbrace \\mu _i, \\pi _i\\rbrace \\rightarrow 0$ .",
"Having that in mind we can treat $\\mu _1$ as a small parameter and solve (REF ) perturbatively.",
"The only regular solution in the $\\mu _1 \\rightarrow 0$ limit is Eq.",
"(REF ).",
"$M_\\infty = M \\left( 1+ 2\\pi \\mu _1 \\frac{M^2}{R^2} \\right).$ The three mass relations (REF ), (REF ) and (REF ) are equivalent in the ${\\cal F} \\ll 1$ limit.",
"Equations (REF ) and (REF ) are “exact\" post-TOV results and do not require ${\\cal F}$ or $\\mu _1$ to be much smaller than unity, although Eq.",
"(REF ) does place a lower limit on $\\mu _1$ because the argument of the square root must be nonnegative.",
"Unfortunately, the use of (REF ) in the calculation of the metric components leads to very cumbersome expressions.",
"To make progress (while also keeping up with the post-TOV spirit), we hereafter use the ${\\cal F} \\ll 1$ approximations (REF )-(REF ).",
"This step, however, introduces a certain degree of error.",
"This is quantified in Fig.",
"REF (left panel), where we show the relative percent error in calculating $M_{\\infty }$ using the post-TOV expanded Eqs.",
"(REF ) and (REF ) rather than Eq.",
"(REF ).",
"Using EOS Sly4 [13], we considered values of $\\mu _1$ for which Eqs.",
"(REF ) and (REF ) admit a positive solution for $M_{\\infty }$ .",
"As test beds, we consider neutron stars with central energy densities which result in a canonical $1.4\\,M_{\\odot }$ and the maximum allowed mass in GR, i.e.",
"$2.05\\, M_{\\odot }$ .",
"As evident from Fig.",
"REF , the error can become significant as we increase the value of $|\\mu _1|$ .",
"By demanding that the errors remain within $5\\%$ we can narrow down the admissible values of $\\mu _1$ to $[-1.0, 0.1]$ .",
"We observe that while $M_{\\infty }$ can deviate greatly from the GR value (e.g.",
"$M_{\\infty }$ reduces by $\\approx 21\\%$ when $\\mu _1 = -1.0$ with respect to a $1.4 \\, M_{\\odot }$ neutron star), ${\\cal F}$ remains below unity (see right panel of Fig.",
"REF ).",
"This is because large negative values of the parameter $\\mu _1$ make the star less compact (i.e.",
"Newtonian), as evidenced in Fig.",
"1 of [1].",
"We emphasize that the larger errors for some values of $\\mu _1$ are not an issue with the post-TOV formalism itself, but serve to constrain the values of $\\mu _1$ for which the perturbative expansion is valid.",
"From a practical point of view, excluding large values of $|\\mu _1|$ is a sensible strategy, since the resulting stellar parameters are so different with respect to their GR values that these cannot be considered as meaningful post-TOV corrections.",
"Hereafter, whenever we refer to $M_{\\infty }$ we mean the mass calculated using Eq.",
"(REF ) with $\\mu _1 \\in [-1.0, 0.1]$ .",
"Figure: Errors in M ∞ M_{\\infty }.We show the percent error [% error ≡100×(x value -x ref )/x ref \\%\\, {\\rm error} \\equiv 100 \\times (x_{\\rm value} - x_{\\rm ref})/x_{\\rm ref}]in calculating M ∞ M_{\\infty } using Eqs.",
"() and ()with respect to () for various values of μ 1 \\mu _1 using EOS SLy4.The range of μ 1 \\mu _1 is chosen such that using any of Eqs.",
"(), ()or () one can obtain a real root corresponding to M ∞ M_{\\infty }.The post-TOV models are constructed using a fixed central value of the energy density,which results in either a canonical (1.4M ⊙ 1.4\\, M_{\\odot }) or a maximum-mass(2.05M ⊙ 2.05\\, M_{\\odot }) neutron star in GR.Top panel: errors for a maximum-mass GR star.Bottom panel: errors for a canonical-mass GR star.Right panel: the absolute value of the post-TOV correctionℱ=4πμ 1 (M ∞ /R) 2 {\\cal F} =4 \\pi \\mu _1 (M_{\\infty }/R)^2 as a function of μ 1 \\mu _1.",
"Thecondition ℱ≪1{\\cal F} \\ll 1 boundsthe range of acceptable values of μ 1 \\mu _1 for which the expansionsleading toEqs.",
"() and () are valid.Errors are below 5 %\\% when μ 1 ∈[-1.0,0.1]\\mu _1 \\in [-1.0, 0.1].Within this approximation we are free to use the Taylor-expanded form of (REF ), i.e.",
"$m(r) = M_\\infty \\left( 1 - 2\\pi \\mu _1 M^2_\\infty /r^2 \\right)\\,.$ This expression leads to the exterior $g_{rr}$ metric $g_{rr} (r) = \\left( 1 - \\frac{2 M_\\infty }{r} \\right)^{-1} -4\\pi \\mu _1 \\frac{M^3_\\infty }{r^3}+ {\\cal O} \\left(\\frac{\\mu _1 M_\\infty ^4}{ r^4} \\right).",
"\\nonumber \\\\$ This expression allows us to identify $M_\\infty $ as the spacetime's gravitating mass (see also the result for $g_{tt}$ below).",
"The next step is to use our result for $m(r)$ in (REF ) and integrate to obtain $\\nu (r)$ .",
"After expanding to 2PN post-TOV order we obtain: $\\frac{d\\nu }{dr} = \\frac{2M_\\infty }{r^2} \\left( 1-\\frac{2M_\\infty }{r} \\right)^{-1}+ 2 \\chi \\frac{M^3_\\infty }{r^4}.$ where the parameter $\\chi $ , defined in Eq.",
"(REF ), quantifies the departure from the Schwarzschild metric.",
"Integrating, $\\nu (r) = \\log \\left( 1-\\frac{2M_\\infty }{r} \\right) - \\frac{2\\chi }{3}\\frac{M^3_\\infty }{r^3},$ where the integration constant has been eliminated by requiring asymptotic flatness.",
"The resulting exterior $g_{tt}$ metric component is $g_{tt} (r) = -\\left( 1-\\frac{2M_\\infty }{r} \\right) + \\frac{2\\chi }{3} \\frac{M^3_\\infty }{r^3}+{\\cal O} \\left( \\frac{\\chi M_\\infty ^4}{r^4} \\right)\\,.$ Eqs.",
"(REF ) and (REF ) represent our final results for the 2PN-accurate exterior post-Schwarzschild metric.",
"From this construction it follows that post-TOV stars for which $\\mu _1 = \\mu _2 = \\pi _2 = 0$ have the Schwarzschild metric as the exterior spacetime.",
"The following sections describe how the exterior metric can be used to compute observables of relevance for neutron star astrophysics."
],
[
"Surface redshift & stellar compactness",
"The surface redshift is among the most basic neutron star observables that could be affected by a change in the gravity theory.",
"The surface redshift is defined in the usual way as $z_s \\equiv \\frac{\\lambda _{\\infty } - \\lambda _{\\rm s}}{\\lambda _{\\rm s}}= \\frac{f_{\\rm s}}{f_{\\infty }} - 1,$ where $\\lambda $ and $f$ are the wavelength and frequency of a photon, respectively.",
"Here and below, the subscripts $s$ and $\\infty $ will denote the value of various quantities at the stellar surface $r = R$ and at spatial infinity.",
"The familiar redshift formula $\\frac{f_\\infty }{f_{\\rm s}} = \\left[\\frac{g_{tt} (R)}{g_{tt} (\\infty )}\\right]^{1/2}$ is valid for any static spacetime, regardless of the form of the field equations.",
"Using the metric (REF ), we easily obtain (at first post-TOV order) $\\frac{f_{\\rm s}}{f_{\\infty }} = \\left( 1 - \\frac{2 M_{\\infty }}{R}\\right)^{-1/2} + \\frac{\\chi }{3} \\frac{M_{\\infty }^3}{R^3}.$ Given that the frequency shift depends only on the ratio $M_\\infty /R$ , it is more convenient to work in terms of the stellar compactness $C = {M_\\infty }/{R}.$ Then from the definition of the surface redshift we obtain $z_{\\rm s} = z_{\\rm GR} + \\frac{\\chi }{3} C^3,$ where $z_{\\rm GR} \\equiv \\left(\\, 1 - 2C \\, \\right)^{-1/2} - 1,$ is the standard redshift formula in GR, while the second term represents the post-TOV correction.",
"Observe that $z_{\\rm s}$ can be smaller or larger than $z_{\\rm GR}$ depending on the sign of the parameter $\\chi $ .",
"This is shown in Fig.",
"REF (left panel), where we plot the percent difference $\\delta z_{\\rm s}/z_{\\rm s}\\equiv 100 \\times (z_s - z_{\\rm GR})/z_{\\rm GR} $ as a function of $C$ for two representative cases ($\\chi = \\pm 0.1$ ).",
"A characteristic property of the redshift is that it is a function of $C$ , and as such it cannot be used to disentangle mass and radius individually.",
"A given observed surface redshift $z_{\\rm obs}$ can be experimentally interpreted either as $z_{\\rm obs} = z_{\\rm GR} (C)$ or $z_{\\rm obs} =z_s (C,\\chi )$ , and therefore lead to different estimates for $C$ (for a given $\\chi $ ).",
"Fig.",
"REF (right panel), where we plot the percent difference $\\delta C / C \\equiv 100 \\times (C-C_{\\rm GR})/C_{\\rm GR}$ as a function of $z_{\\rm s}$ , shows how much the inferred compactness would differ in the two cases where $\\chi = \\pm 0.1$ .",
"A positive (negative) $\\chi $ leads to a lower (higher) inferred compactness with respect to GR.",
"The figure suggests that the “error” in $C$ becomes significant for redshifts $z_{\\rm s} \\gtrsim 1$ .",
"Figure: Surface redshift and stellar compactness.",
"Relative percent changeswith respect to GR for both z s z_{\\rm s} and CC for different values of χ\\chi .It is a straightforward exercise to invert the redshift formula and obtain a post-TOV expression $C= C(z_{\\rm s})$ .",
"We first write $1 + z_{\\rm s} = \\frac{1}{\\sqrt{-g_{tt} (R)}} \\,\\Rightarrow \\,g_{tt} (R) = -\\frac{1}{(1+z_{\\rm s})^2}.$ Upon inserting the post-Schwarzschild metric (REF ) we get a cubic equation for the compactness, $-1 + \\frac{1}{(1+z_{\\rm s})^2} + 2 C + \\frac{2}{3} \\chi C^3 = 0.$ In solving this equation we take into account that the small parameters are $C$ and $z_{\\rm s} \\sim {\\cal O}(C)$ and that the $\\chi \\rightarrow 0$ limit should be smooth.",
"We find, $C = C_{\\rm GR} \\left(\\, 1 - \\frac{1}{3} \\chi z_{\\rm s}^2 \\, \\right),$ where $C_{\\rm GR} = \\frac{1}{2} \\left[ 1 - (1+z_{\\rm s})^{-2} \\right],$ is the corresponding solution in GR.",
"As was the case for the post-TOV redshift formula, the compactness of a post-TOV star can be pushed above (below) the GR value by choosing a negative (positive) parameter $\\chi $ .",
"The two main results of this section, Eqs.",
"(REF ) and (REF ), are also interesting from a different prespective, namely, their dependence on the single post-TOV parameter $\\chi $ .",
"This dependence entails a degeneracy with respect to the coefficient triad $\\lbrace \\mu _1, \\mu _2, \\pi _2 \\rbrace $ when (for example) a neutron star redshift observation is used as a gravity theory discriminator.",
"The redshift/compactness $\\chi $ -degeneracy is another reminder of the intrinsic difficulty in distinguishing non-GR theories of gravity from neutron star physics (see e.g.",
"discussion around Fig.",
"1 in [1])."
],
[
"Bursting neutron stars",
"A potential testbed for measuring deviations from GR with a parametrized scheme like our post-TOV formalism is provided by accreting neutron stars exhibiting the so-called type I bursts.",
"These are X-ray flashes powered by the nuclear detonation of accreted matter on the stellar surface layers [14].",
"The luminosity associated with these events can reach the Eddington limit and may cause a photospheric radius expansion (see e.g.",
"[15], [16]), thus offering a number of observational “handles\" to the system (see below for more details).",
"A paper by Psaltis [7] proposed type I bursting neutron stars as a means to constrain possible deviations from GR.",
"Psaltis' analysis, based on a static and spherically symmetric model for describing the spacetime outside a non-rotating neutron star, is general enough to allow a direct adaptation to the post-TOV scheme.",
"For that reason we can omit most of the technical details discussed in [7] and instead focus on the key results derived in that paper.",
"There is a number of observable quantities associated with type I bursting neutron stars that can be used to set up a test of GR.",
"The first one is the surface redshift $z_{\\rm s}$ ; in Section we have derived post-TOV formulae for $z_{\\rm s}$ and the stellar compactness $C$ , which are used below in the derivation of a constraint equation between the post-Schwarzschild metric and the various observables.",
"The luminosity (as measured at infinity) of a source located at a (luminosity) distance $D$ is $L_\\infty = 4\\pi D^2 F_\\infty ,$ where $F_\\infty $ is the (observable) flux.",
"This luminosity can be written in a black-body form with the help of an apparent surface area $S_{\\rm app}$ and a color temperature (as measured at infinity) $\\bar{T}_{\\infty }$ : $4\\pi D^2 F_\\infty = \\sigma _{\\rm SB} S_{\\rm app} \\bar{T}^4_\\infty ,$ where $\\sigma _{\\rm SB}$ is the Stefan-Boltzmann constant.",
"We then define the second observable parameter used in this analysis, i.e.",
"the apparent radius $R_{\\rm app} \\equiv \\left( \\frac{S_{\\rm app}}{4\\pi } \\right)^{1/2} = D \\left( \\frac{F_\\infty }{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4} \\right)^{1/2}.$ As evident from its form, $R_{\\rm app}$ is independent of the underpinning gravitational theory, at least to the extent that the theory does not appreciably modify the (luminosity) distance to the source.",
"The surface color temperature is related to the intrinsic effective temperature $T_{\\rm eff}$ via the standard color correction factor $f_{\\rm c}$ [17], [18], $\\bar{T}_{\\rm s} = f_{\\rm c} T_{\\rm eff}.$ The observed temperature at infinity picks up a redshift factor with respect to its local surface value, that is, $\\bar{T}_\\infty = f_{\\rm c} \\sqrt{-g_{tt} (R)} T_{\\rm eff}.$ The effective temperature is the one related to the source's intrinsic luminosity, $L_{\\rm s} = 4\\pi R^2 \\sigma _{\\rm SB} T_{\\rm eff}^4.$ As shown in [7], $L_\\infty = - g_{tt} (R) L_{\\rm s} = 4\\pi R^2 \\sigma _{\\rm SB} \\left( \\frac{\\bar{T}_\\infty }{f_{\\rm c}} \\right)^4 [-g_{tt} (R)]^{-1}.$ Combining this with the preceding formulae leads to, $\\frac{R_{\\rm app}}{R} = \\frac{1+z_{\\rm s}}{f_{\\rm c}^2}.$ The third relevant observable is the Eddington luminosity/flux at infinity.",
"This is given by [7], $L_{\\rm E}^\\infty = 4\\pi D^2 F^\\infty _{\\rm E} = \\frac{4\\pi }{\\kappa } \\frac{R^2}{(1+z_{\\rm s})^2} g_{\\rm eff},$ where $\\kappa $ is the opacity of the matter interacting with the radiation fieldTypically, this interaction manifests itself as Thomson scattering in a hydrogen-helium plasma, in which case the opacity is $\\kappa \\approx 0.2 (1+ X)\\,\\mbox{cm}^2/\\mbox{gr}$ where $X$ is the hydrogen mass fraction [16].",
"and $g_{\\rm eff}$ is an effective surface gravitational acceleration, defined as: $g_{\\rm eff} =\\frac{1}{2\\sqrt{g_{rr} (R)}} \\frac{g^\\prime _{tt} (R)}{g_{tt} (R)}.$ This parameter is key to the present analysis as it encodes the departure from the general relativistic Schwarzschild metric.",
"Figure: Bursting neutron star constraints.",
"The surfaces arecontours of constant1+(2/3)χz s 2 z s (2+z s )/(1+z s ) 4 \\left[\\, 1 + (2/3) \\chi z_{\\rm s}^2 \\, \\right] z_{\\rm s}(2+z_{\\rm s})/(1+z_{\\rm s})^4 in the (z s ,χ)(z_{\\rm s},\\,\\chi ) plane.",
"This quantity is a combinationof observables – cf.",
"the right-hand side of Eq.",
"()– and therefore it is potentially measurable; a measurement willsingle out a specific contour in this plot.",
"A further measurement of(say) the redshift z s z_{\\rm s} corresponds to the intersectionbetween one such contour and a line with z s = const .z_{\\rm s}={\\rm const.",
"},so it can lead to a determination of χ\\chi .Having at our disposal the above three observable combinations, the strategy is to combine them and derive a constraint equation between the observables and the spacetime metric.",
"To this end, we first need to eliminate the not directly observable stellar radius $R$ between (REF ) and (REF ) and subsequently solve with respect to $g_{\\rm eff}$ .",
"We obtain $g_{\\rm eff} = \\kappa \\sigma _{\\rm SB} \\frac{F_{\\rm E}^\\infty }{F_\\infty } \\left( \\frac{\\bar{T}_\\infty }{f_{\\rm c}} \\right)^4 (1+z_{\\rm s})^4.$ The remaining task is to express $ g_{\\rm eff} $ in terms of $z_{\\rm s}$ .",
"Using the post-Schwarzschild metric, Eqs.",
"(REF ) and (REF ), in (REF ) we obtain the post-TOV result $g_{\\rm eff} = \\frac{C}{R} (1+z_{\\rm s} ) \\left(\\, 1 + \\chi C^2 \\, \\right).$ Making use of Eq.",
"(REF ) for the compactness leads to the desired result [cf.",
"Eq.",
"(39) in [7]]: $g_{\\rm eff} = \\frac{z_{\\rm s}}{2R} \\frac{(2+z_{\\rm s})}{(1+z_{\\rm s})} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right),$ where, as evident, the prefactor represents the GR result.",
"Finally, after eliminating $R$ with the help of (REF ) and (REF ), we obtain the “observable\" effective gravity: $g_{\\rm eff} = \\frac{z_{\\rm s}(2+z_{\\rm s})}{2 D f_{\\rm c}^2} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right)\\left( \\frac{\\sigma _{\\rm SB} \\bar{T}^4_\\infty }{F_\\infty } \\right)^{1/2}.$ This can then be combined with (REF ) to give $\\frac{z_{\\rm s}(2+z_{\\rm s})}{(1+z_{\\rm s})^4} \\left(\\, 1 + \\frac{2}{3} \\chi z_{\\rm s}^2 \\, \\right)= 2D \\kappa \\frac{F_{\\rm E}^\\infty }{ f_{\\rm c}^2} \\left( \\frac{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4 }{F_\\infty } \\right)^{1/2},$ and consequently $\\chi = \\frac{3}{2 z^2_{\\rm s}} \\left[\\, 2D \\kappa \\frac{ (1+z_{\\rm s})^4}{z_{\\rm s}(2+z_{\\rm s})} \\frac{F_{\\rm E}^\\infty }{ f_{\\rm c}^2}\\left( \\frac{\\sigma _{\\rm SB} \\bar{T}_\\infty ^4 }{F_\\infty } \\right)^{1/2} - 1 \\, \\right].$ This equation is the main result of this section and provides, at least as a proof of principle, a quantitative connection between the post-Schwarzschild correction to the exterior metric [in the form of the $\\chi $ coefficient defined in Eq.",
"(REF )] and observable quantities in a type I bursting neutron star.",
"Ref.",
"[7] arrives at a similar result [their Eq.",
"(49)] which has the same physical meaning, but is not identical to Eq.",
"(REF ) due to the different assumed form of the exterior metric.",
"Our results are illustrated in Fig.",
"REF , where we show the left-hand side of Eq.",
"(REF ) as a contour plot in the $(z_{\\rm s},\\,\\chi )$ plane.",
"Each contour represents a specific measurement of this observable quantity.",
"An additional surface redshift measurement can lead, at least in principle, to the determination of the post-TOV parameter $\\chi $ , as given by Eq.",
"(REF )."
],
[
"Quasi-periodic oscillations",
"The post-Schwarzschild metric allows us to compute the geodesic motion of particles in the exterior spacetime of post-TOV neutron stars.",
"Geodesics in neutron star spacetimes play a key role in the theoretical modelling of the QPOs observed in the X-ray spectra of accreting neutron stars.",
"The detailed physical mechanism(s) responsible for the QPO-like time variability in the flux of these systems is still a matter of debate, but some of the most popular models are based on the notion of a radiating hot “blob” of matter moving in nearly circular geodesic orbits.",
"The QPO frequencies are identified either with the orbital frequencies, or with simple combinations of the orbital frequencies.",
"The most popular models are variants of the relativistic precession [8], [9] and epicyclic resonance [10] models.",
"In this section we discuss the relevant orbital frequencies within the post-TOV formalism and derive formulae that could easily be used in the aforementioned QPO models.",
"In principle, matching the orbital frequencies to the QPO data would allow one to extract post-TOV parameters such as $\\chi $ and $\\mu _1$ (see [19], [20], [21] for a similar exercise in the context of GR and scalar-tensor theory).",
"For nearly circular orbits in a spherically symmetric spacetime, the only perturbations of interest are the radial ones (i.e., there is periastron precession but no Lense-Thirring nodal precession) and therefore we can associate two frequencies to every circular orbit: the orbital azimuthal frequency of the circular orbit $\\Omega _\\varphi $ and the radial epicyclic frequency $\\Omega _r$ .",
"Geodesics in a static, spherically symmetric spacetime are characterized by the two usual conserved quantities, the energy $E =-g_{tt} \\dot{t}$ and the angular momentum $ L= g_{\\phi \\phi } \\dot{\\phi }$ .",
"Here both constants are defined per unit particle mass, and the dots stand for differentiation with respect to proper time.",
"The four-velocity normalization condition $u^au_a=-1$ yields an effective potential equation for the particle's radial motion, $g_{rr} \\dot{r}^2 = -\\frac{E^2}{g_{tt}}-\\frac{L^2}{g_{\\phi \\phi }}-1\\equiv V_{\\rm eff}(r).$ The conditions for circular orbits are $V_{\\rm eff}(r)=V^{\\prime }_{\\rm eff}(r)=0$ , where the prime denotes differentiation with respect to the radial coordinate.",
"Hereafter $r$ will denote the circular orbit radius.",
"From these conditions we can determine the orbital frequency $\\Omega _\\varphi \\equiv \\dot{\\phi }/\\dot{t}$ measured by an observer at infinity.Apart from its implications for the QPOs, the post-TOV corrected orbital frequency would imply a shift in the corotation radius $r_{\\rm co}$ in an accreting system.",
"This radius is defined as $\\Omega _*= \\Omega _\\varphi (r_{\\rm co})$ , where $\\Omega _*$ is the stellar angular frequency, and plays a key role in determining the torque-spin equilibrium in magnetic field-disk coupling models.",
"Using the above definition we find the following result for the post-TOV corotation radius: $r_{\\rm co}= M_{\\infty }(M_{\\infty } \\Omega _*)^{-2/3} \\left[ \\, 1 + (\\chi /3) (M_{\\infty } \\Omega _*)^{4/3} \\, \\right]$ .",
"The square of the orbital frequency is then given as $\\Omega _\\varphi ^2 &= -\\frac{g^{\\prime }_{tt}}{g^{\\prime }_{\\phi \\phi }} =\\frac{M_{\\infty }}{r^3}\\left(1+\\chi \\frac{M_{\\infty }^2}{r^2}\\right).$ The Schwarzschild frequency is recovered for $\\chi =0$ .",
"The radial epicyclic frequency can be calculated from the equation for radially perturbed circular orbits, which follows from Eq.",
"(REF ): $\\Omega _r^2 &= -\\frac{g^{rr}}{2 \\dot{t}^2} V^{\\prime \\prime }_{\\rm eff}(r)\\approx \\frac{M_{\\infty }}{r^3}\\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{\\chi M_{\\infty }^2}{r^2} + {\\cal O} (r^{-3})\\right] \\\\&= \\Omega _\\varphi ^2 \\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{2 \\chi M_{\\infty }^2}{r^2} + {\\cal O} (r^{-3}) \\right], $ where again the frequency is calculated with respect to observers at infinity.",
"From the post-TOV expanded result we can see that the first two terms correspond to the Schwarzschild epicyclic frequency.",
"The additional post-TOV terms in these formulae produce a shift in the frequency and radius of the innermost stable circular orbit (ISCO) with respect to their GR values – the latter quantity is determined by the condition $\\Omega _r^2=0$ , which in GR leads to the well-known result $r_{\\rm isco}=6M_{\\infty }$ .",
"The corresponding post-TOV ISCO is obtained from (REF ), up to linear order in $\\chi $ , as: $r_{\\rm isco} \\approx 6 M_{\\infty } \\left(1+\\frac{19}{324}\\chi \\right).$ The post-TOV ISCO parameters $r_{\\rm isco}$ and $(\\Omega _\\varphi )_{\\rm isco}$ are plotted in Fig.",
"REF as functions of the parameter $\\chi $ .",
"As evident from Eq.",
"(REF ), a positive (negative) $\\chi $ implies $r_{\\rm isco} > 6M_{\\infty }$ ($r_{\\rm isco} < 6 M_{\\infty }$ ).",
"If one takes Eq.",
"(REF ) at face value for the given post-Schwarzschild metric, for negative enough values of $\\chi $ there is no ISCO solution, but this occurs well beyond the point where it is safe to use our perturbative formalism.",
"The orbital frequency profile remains rather simple, with $(\\Omega _\\varphi )_{\\rm isco}$ exceeding the GR value when $r_{\\rm isco} < 6M_{\\infty }$ (and vice versa).",
"Figure: ISCO quantities.",
"ISCO quantities as functions ofχ\\chi .",
"The solid curve corresponds to the relative difference (inpercent) of r isco r_{\\rm isco} with respect to GR, while the dashedcurve corresponds to the relative difference of the orbitalfrequency at the ISCO, (Ω ϕ ) isco ( \\Omega _\\varphi )_{\\rm isco}.Besides the frequency pair $\\lbrace \\Omega _\\varphi , \\Omega _r \\rbrace $ , a third prominent quantity in the QPO models is the frequency $\\Omega _{\\rm per}=\\Omega _\\varphi -\\Omega _r,$ associated with the orbital periastron precession (for example, in the relativistic precession model [8], [9] this frequency is typically associated with the low-frequency QPO) .",
"Given our earlier results, it is straightforward to derive a series expansion in powers of $1/r$ for $\\Omega _{\\rm per}$ .",
"However, it is usually more desirable to produce a series expansion with respect to an observable quantity, such as the circular orbital velocity $U_{\\infty }=(M_{\\infty }\\Omega _\\varphi )^{1/3}$ .",
"This can be done by first expanding $U_\\infty $ with respect to $1/r$ and then inverting the expansion, thus producing a series in $U_\\infty $ .",
"The outcome of this recipe is $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 1-\\frac{\\Omega _r}{\\Omega _\\varphi }= 3 U_{\\infty }^2 +\\left(\\frac{9}{2} + \\chi \\right) U_{\\infty }^4 + {\\cal O} (U_{\\infty }^{6}).$ A similar “Keplerian\" version of this expression can be produced if we opt for using the velocity $U_{\\rm K}$ and mass $M_{\\rm K}$ that an observer would infer from the motion of (say) a binary system under the assumption of exactly Keplerian orbits.",
"These are $U_{\\rm K} =(M_{\\rm K}\\Omega _\\varphi )^{1/3}$ and $M_{\\rm K}=r^3\\Omega ^2_\\varphi $ , so that $M_{\\rm K} =M_{\\infty }\\left(1+\\chi M_{\\infty }^2/r^2\\right)$ .",
"The resulting series is identical to Eq.",
"(REF ) when truncated to $U_{\\rm K}^4$ order.",
"Higher-order terms, however, are different (see the following section).",
"The above results for the frequencies $\\lbrace \\Omega _\\varphi , \\Omega _r, \\Omega _{\\rm per}\\rbrace $ suggest that a QPO-based test of GR within the post-TOV formalism could in principle allow the extraction of the post-TOV parameter $\\chi $ .",
"In this sense these frequencies probe the same kind of deviation from GR (and suffer from the same degree of degeneracy) as the observations of bursting neutron stars discussed in Section .",
"Figure: Orbital frequencies.Plots of Ω r \\Omega _r against Ω ϕ \\Omega _{\\varphi } fordifferent values of χ=±0.1,±0.5\\chi =\\pm 0.1,\\, \\pm 0.5 and ±1\\pm 1.",
"The black solid curve corresponds to the GR case.The dashed curves correspond to positive values of χ\\chi (and r isco >6M ∞ r_{\\rm isco}>6M_{\\infty })while the dashed-dotted curves correspond to negative values of χ\\chi (andr isco <6M ∞ r_{\\rm isco}<6M_{\\infty }).We conclude this section by sketching how this procedure works in practice in the context of the relativistic precession model.",
"The twin $\\mbox{kHz}$ QPO frequencies $\\lbrace \\nu _1, \\nu _2 \\rbrace $ seen in the flux of bright low-mass X-ray binaries are identified with the azimuthal and periastron precession orbital frequencies.",
"More specifically, the high-frequency member of the pair is identified with the azimuthal frequency ($\\nu _2 = \\nu _\\varphi = \\Omega _\\varphi /2\\pi $ ), while the low-frequency member is identified with the periastron precession ($\\nu _1 = \\nu _{\\rm per} = \\Omega _{\\rm per}/2\\pi $ ).",
"With this interpretation, the QPO separation is equal to the radial epicyclic frequency: $\\Delta \\nu = \\nu _2 - \\nu _1 = \\Omega _r/2\\pi $ .",
"We use our previous results [Eqs.",
"(REF ), (), (REF )] to plot these orbital frequencies (clearly, $\\nu _1/\\nu _2 = \\Omega _{\\rm per}/\\Omega _\\varphi $ and $\\Delta \\nu /\\nu _2 = \\Omega _r/\\Omega _\\varphi $ ) as functions of each other and for varying $\\chi $ .",
"As it turns out, deviations from GR are best illustrated by plotting $\\Omega _r (\\Omega _\\varphi )$ (or equivalenty $\\Delta \\nu (\\nu _2)$ ).",
"In Fig.",
"REF we plot the dimensionless combinations $M_\\infty \\Omega _r,~M_\\infty \\Omega _\\varphi $ (in units of kHz for the frequencies and solar masses for $M_{\\infty }$ ).",
"As we can see, the post-TOV models considered here ($ -1 <\\chi < 1 $ ) are qualitatively similar to the GR result (black solid curve), all cases showing the characteristic hump in $\\Omega _r$ as $\\Omega _\\varphi $ increases (so that the orbital radius decreases).",
"This feature is evidently associated with the existence of an ISCO (where $\\Omega _r \\rightarrow 0$ ) and is consistent with a similar trend seen in observations [9]."
],
[
"Multipolar structure of the spacetime",
"Expansions like Eq.",
"(REF ) contain information about the multipolar structure of the background spacetime.",
"That expansion can be directly compared against a similar expansion derived by Ryan [19] for an axisymmetric, stationary spacetime in GR with an arbitrary set of mass ($M_0=M_{\\infty }, M_2, M_4, ...$ ) and current ($S_1, S_3, S_5, ...$ ) Geroch-Hansen multipole moments [22], [23], [24]:A multipolar expansion in scalar-tensor theory can be found in [25].",
"Specific calculations were also carried out in other theories: for example, the quadrupole moment was computed in Einstein-dilaton-Gauss-Bonnet gravity [26].",
"$\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U^2 - 4\\frac{S_1}{M_{\\infty }^2} U^3+ \\left(\\, \\frac{9}{2} - \\frac{3}{2} \\frac{M_2}{M_{\\infty }^3} \\,\\right) U^4 -10 \\frac{S_1}{M_{\\infty }^2} U^5\\nonumber \\\\& + \\left(\\, \\frac{27}{2} -2\\frac{S_1^2}{M_{\\infty }^4} -\\frac{21}{2} \\frac{M_2}{M_{\\infty }^3} \\, \\right) U^6 + {\\cal O} (U^{7} ).$ where $U = (M_{\\infty } \\Omega _\\varphi )^{1/3} $ denotes the orbital velocity.",
"To understand the PN accuracy of the post-TOV expansion in this context, it is useful to consider the effect of higher PN order terms in the metric.",
"Imagine that the $g_{tt}$ and $g_{rr}$ metric components [see Eqs.",
"(REF ), (REF )] included 3PN corrections of the schematic form, $g_{tt}(r)&=g_{tt}^{\\rm 2PN}+\\alpha _{tt} \\frac{M_{\\infty }^3}{r^3},\\\\g_{rr}(r)&=g_{rr}^{\\rm 2PN}+\\alpha _{rr} \\frac{M_{\\infty }^3}{r^3}.$ We can use the coefficients $\\alpha _{tt}$ and $\\alpha _{rr}$ as bookeeping parameters in order to understand how these omitted higher-order contributions affect the results of the previous section.",
"The recalculation of the various expressions reveals that the orbital frequency remains unchanged to 2PN order; the 3PN term of Eq.",
"(REF ) contributes at the next order, as expected.",
"The same applies to the epicyclic frequency, as we can see for example from the modified Eq.",
"(), where the next-order correction is a mixture of $g_{rr}^{\\rm 2PN}$ and the 3PN term in $g_{tt}$ : $\\Omega _r^2&=\\Omega _\\varphi ^2 \\left[1-\\frac{6 M_{\\infty }}{r} -\\frac{2 \\chi M_{\\infty }^2}{r^2} \\right.\\nonumber \\\\&\\left.+ \\frac{(4 \\pi \\mu _1 -6\\alpha _{tt})M_{\\infty }^3}{r^3} + {\\cal O} (r^{-4}) \\right].$ Proceeding in a similar way we find the next-order correction to the Ryan-like expansion (REF ): $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U_{\\infty }^2 +\\left(\\frac{9}{2} + \\chi \\right) U_{\\infty }^4\\nonumber \\\\&+\\left[\\frac{27}{2} + 2 (\\chi -\\pi \\mu _1 )+3\\alpha _{tt}\\right] U_{\\infty }^6+ {\\cal O} (U_{\\infty }^{8}).$ We can see that the 3PN term “contaminates\" the PN correction that was omitted in Eq.",
"(REF ).",
"Repeating the same exercise for the Keplerian version of the multipole expansion (i.e.",
"where the orbital velocity $U_\\infty $ is replaced by $U_{\\rm K}$ ) we find: $\\frac{\\Omega _{\\rm per}}{\\Omega _\\varphi } &= 3 U^2_{\\rm K} +\\left(\\frac{9}{2} + \\chi \\right) U^4_{\\rm K} \\nonumber \\\\&+\\left(\\frac{27}{2} - 2 \\pi \\mu _1+3\\alpha _{tt}\\right) U^6_{\\rm K} + {\\cal O} (U^{8}_{\\rm K} ).$ At the PN order considered in the previous section the two expressions were identical but, as we can see, they differ at the next order.",
"We now have Ryan-type multipole expansions of the post-Schwarzschild spacetime up to 3PN in the circular orbital velocity, which we can compare against Eq.",
"(REF ) to draw (with some caution) analogies and differences between GR and modified theories of gravity.",
"For instance, odd powers of $U_\\infty $ are missing in Eq.",
"(REF ) because the nonrotating post-Schwarzschild spacetime has vanishing current multipole moments.",
"Furthermore, we can see that the quadrupole moment $M_2$ , first appearing in the coefficient of $U^4_\\infty $ in Eq.",
"(REF ), can be associated with $\\chi $ .",
"The parameter $\\chi $ is an effective quadrupole moment in the sense that $M_2^{\\rm eff} = -\\frac{2}{3} \\chi M_{\\infty }^3.$ Indeed, this relation implies that a positive (negative) $\\chi $ could be associated with an oblate (prolate) source of the gravitational field.",
"The identification (REF ) holds at $ {\\cal O} (U_{\\infty }^{4})$ .",
"The next-order term $U^6_\\infty $ would, in general, lead to a different effective $M_2$ .",
"Hence, the comparison between the $U_{\\infty }^4$ and $U_{\\infty }^6$ terms could provide a null test for the GR-predicted quadrupole.",
"However, there is a special case where these two terms could be consistent with the same effective quadrupole (REF ): this occurs when the post-TOV parameters satisfy the condition $5\\chi = -2\\pi \\mu _1+3\\alpha _{tt}$ , in which case the expansion (REF ) behaves as a “GR mimicker\".",
"A different kind of “multipole” expansion in powers of $1/r$ can be applied to the metric functions $\\nu (r), m(r)$ [see Eqs.",
"(REF )–(REF )], leading to an alternative calculation of the ADM mass $M_\\infty $ of a post-TOV star.",
"We consider the expansions $\\nu (r) = \\sum _{n = 0}^{\\infty } \\frac{\\nu _{n}}{r^n}, \\qquad m(r) = \\sum _{n = 0}^{\\infty } \\frac{m_{n}}{r^n},$ where $\\nu _n$ and $m_n$ are constant coefficients.",
"In addition, we impose that $\\nu _0 = 0$ and $m_0 = M_{\\infty }$ .",
"We subsequently substitute these expansions into Eqs.",
"(REF )–(), expand for $r/R \\gg 1$ , and then solve for the coefficients.",
"The outcome of this exercise in the vacuum exterior spacetime is $m(r) &= M_{\\infty } - 2 \\pi \\mu _1 \\frac{M_{\\infty }^3}{r^2} + {\\cal O}(r^{-4})\\,,\\\\\\nonumber \\\\\\nu (r) &= - \\frac{2 M_{\\infty }}{r} - \\frac{2 M_{\\infty }^2}{r^2}- \\frac{2}{3} \\frac{M_{\\infty }^3}{r^3}\\left( 4 + \\chi \\right) + {\\cal O}(r^{-4})\\,.$ As expected, the top equation is consistent with our earlier result, Eq.",
"(REF ).",
"To get an agreement between Eqs.",
"(REF ) and () we must expand the logarithm appearing in the former equation in powers of $M_{\\infty }/r$ , thus recovering Eq.",
"()."
],
[
"Conclusions",
"In this paper we have demonstrated the applicability of the post-TOV formalism to a number of facets of neutron star astrophysics.",
"Let us summarize our main results.",
"The exterior post-Schwarzschild metric [Eqs.",
"(REF ) and (REF )] depends only on the ADM mass $M_\\infty $ (given by Eq.",
"(REF )) and on just two post-TOV parameters $\\mu _1$ and $\\chi $ .",
"These are subsequently used to produce a post-TOV formula for the surface redshift, Eq.",
"(REF ), which is a function of the stellar compactness and $\\chi $ .",
"Next, we have shown how a basic post-TOV model for type I bursting neutron stars can be constructed.",
"The key equation here is (REF ), which gives $\\chi $ (the only post-TOV parameter appearing in the model) in terms of observable quantities.",
"We also computed geodesic motion in the post-Schwarzschild exterior of post-TOV neutron star models, finding expressions for the orbital, epicyclic and periastron precession frequencies of nearly circular orbits [Eqs.",
"(REF ), (), (REF )] and for the ISCO radius [Eq.",
"(REF )].",
"These results can be fed into models for QPOs from accreting neutron stars, such as the relativistic precession model.",
"Finally, on a more theoretical level, we have sketched how the post-TOV parameters enter in the spacetime's multipolar structure [Eq.",
"(REF )].",
"The meticulous reader may have noticed that, in spite of the exterior metric being a function $g_{tt} (\\chi )$ and $g_{rr} (\\mu _1)$ , all other post-TOV results feature only $\\chi $ , while $\\mu _1$ is either not present or enters at higher order.",
"This is not a coincidence: these quantities either depend solely on $g_{tt}$ (e.g.",
"the redshift) or receive their leading-order contributions from $g_{tt}$ (e.g.",
"the orbital frequencies).",
"The post-TOV formalism developed in [1] and in this paper can be viewed as a basic “stage-one” version of a more general framework.",
"There are several directions one can follow for taking the formalism to a more sophisticated level, and here we discuss just a couple of possibilities.",
"An obvious improvement is the addition of stellar rotation.",
"This is necessary because all astrophysical compact stars rotate, some of them quite rapidly, and the influence of rotation is ubiquitous, affecting to some extent all of the effects discussed in this paper.",
"As a first stab at the problem, it would make sense to work in the Hartle-Thorne slow rotation approximation [27], [28], which should be accurate enough for all but the fastest spinning neutron stars [29].",
"There are equally important possibilities for improvement on the modified gravity sector of the formalism.",
"The present post-TOV theory is oblivious to the existence of dimensionful coupling constants, such as the ones appearing in many modified theories of gravity (e.g.",
"$f(R)$ theories or theories quadratic in the curvature).",
"These coupling parameters should be added to the existing set of fluid parameters, and participate in the algorithmic generation of “families\" of post-TOV terms (see [1] for details).",
"The extended set of parameters will most likely lead to a proliferation of post-TOV terms, and result in more complicated stellar structure equations than the ones used so far [i.e.",
"Eqs. ()].",
"Among other things, this enhancement may allow one to study in more generality to what extent other theories of gravity are mapped onto the post-TOV formalism.",
"Another limitation of the formalism is that it is intrinsically perturbative with respect to GR solutions.",
"It is important to generalize to theories of gravity that present screening mechanisms; the viability of perturbative expansions in these theories is a topic of active research (see e.g.",
"[30], [31], [32], [33]).",
"We hope that the astrophysical applications outlined in this work will stimulate more research to address these issues.",
"K.G.",
"is supported by the Ramón y Cajal Programme of the Spanish Ministerio de Ciencia e Innovación and by NewCompStar (a COST-funded Research Networking Programme).",
"H.O.S., G.P.",
"and E.B.",
"are supported by NSF CAREER Grant No.",
"PHY-1055103.",
"E.B.",
"is supported by FCT contract IF/00797/2014/CP1214/CT0012 under the IF2014 Programme.",
"This work was supported by the H2020-MSCA-RISE-2015 Grant No.",
"StronGrHEP-690904.",
"H.O.S.",
"thanks the Instituto Superior Técnico for hospitality, and the Department of Physics and Astronomy of the University of Mississippi for financial support."
]
] | 1606.05106 |
[
[
"On logarithmic coefficients of some close-to-convex functions"
],
[
"Abstract The logarithmic coefficients $\\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\\mathbb{D}=\\{z\\in\\mathbb{C}:|z|<1\\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\\log \\frac{f(z)}{z}= 2\\sum_{n=1}^{\\infty} \\gamma_n z^n$.",
"Recently, D.K.",
"Thomas [On the logarithmic coefficients of close to convex functions, {\\it Proc.",
"Amer.",
"Math.",
"Soc.}",
"{\\bf 144} (2016), 1681--1687] proved that $|\\gamma_3|\\le \\frac{7}{12}$ for functions in a subclass of close-to-convex functions (with argument $0$) and claimed that the estimate is sharp by providing a form of a extremal function.",
"In the present paper, we pointed out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument $0$).",
"We also determine a sharp upper bound of $|\\gamma_3|$ for close-to-convex functions (with argument $0$) with respect to the Koebe function."
],
[
"Introduction",
"Let $\\mathcal {A}$ denote the class of analytic functions $f$ in the unit disk $\\mathbb {D}=\\lbrace z\\in \\mathbb {C}:|z|<1\\rbrace $ normalized by $f(0)=0=f^{\\prime }(0)-1$ .",
"If $f\\in \\mathcal {A}$ then $f(z)$ has the following representation $f(z)= z+\\sum _{n=2}^{\\infty }a_n(f) z^n.$ We will simply write $a_n:=a_n(f)$ when there is no confusion.",
"Let $\\mathcal {S}$ denote the class of all univalent (i.e.",
"one-to-one) functions in $\\mathcal {A}$ .",
"A function $f\\in \\mathcal {A}$ is called starlike (convex respectively) if $f(\\mathbb {D})$ is starlike with respect to the origin (convex respectively).",
"Let $\\mathcal {S}^*$ and $\\mathcal {C}$ denote the class of starlike and convex functions in $\\mathcal {S}$ respectively.",
"It is well-known that a function $f\\in \\mathcal {A}$ is in $\\mathcal {S}^*$ if and only if ${\\rm Re\\,}\\left(zf^{\\prime }(z)/f(z)\\right)>0$ for $z\\in \\mathbb {D}$ .",
"Similarly, a function $f\\in \\mathcal {A}$ is in $\\mathcal {C}$ if and only if ${\\rm Re\\,}\\left(1+(zf^{\\prime \\prime }(z)/f^{\\prime }(z))\\right)>0$ for $z\\in \\mathbb {D}$ .",
"From the above it is easy to see that $f\\in \\mathcal {C}$ if and only if $zf^{\\prime }\\in \\mathcal {S}^*$ .",
"Given $\\alpha \\in (-\\pi /2,\\pi /2)$ and $g\\in \\mathcal {S}^*$ , a function $f\\in \\mathcal {A}$ is said to be close-to-convex with argument $\\alpha $ and with respect to $g$ if ${\\rm Re\\,} \\left(e^{i\\alpha }\\frac{zf^{\\prime }(z)}{g(z)}\\right)>0 \\quad z\\in \\mathbb {D}.$ Let $\\mathcal {K}_{\\alpha }(g)$ denote the class of all such functions.",
"Let $\\mathcal {K}(g):= \\bigcup _{\\alpha \\in (-\\pi /2,\\pi /2)} \\mathcal {K}_{\\alpha }(g) \\quad \\mbox{ and }\\quad \\mathcal {K}_{\\alpha }:= \\bigcup _{g\\in \\mathcal {S}^*} \\mathcal {K}_{\\alpha }(g)$ be the classes of functions called close-to-convex functions with respect to $g$ and close-to-convex functions with argument $\\alpha $ , respectively.",
"The class $\\mathcal {K}:= \\bigcup _{\\alpha \\in (-\\pi /2,\\pi /2)} \\mathcal {K}_{\\alpha }= \\bigcup _{g\\in \\mathcal {S}^*} \\mathcal {K}(g)$ is the class of all close-to-convex functions.",
"It is well-known that every close-to-convex function is univalent in $\\mathbb {D}$ (see [2]).",
"Geometrically, $f\\in \\mathcal {K}$ means that the complement of the image-domain $f(\\mathbb {D})$ is the union of non-intersecting half-lines.",
"The logarithmic coefficients of $f\\in \\mathcal {S}$ are defined by $\\log \\frac{f(z)}{z}= 2\\sum _{n=1}^{\\infty } \\gamma _n z^n$ where $\\gamma _n$ are known as the logarithmic coefficients.",
"The logarithmic coefficients $\\gamma _n$ play a central role in the theory of univalent functions.",
"Very few exact upper bounds for $\\gamma _n$ seem have been established.",
"The significance of this problem in the context of Bieberbach conjecture was pointed out by Milin in his conjecture.",
"Milin conjectured that for $f\\in \\mathcal {S}$ and $n\\ge 2$ , $\\sum _{m=1}^{n}\\sum _{k=1}^{m} \\left(k|\\gamma _k|^2-\\frac{1}{k}\\right)\\le 0,$ which led De Branges, by proving this conjecture, to the proof of the Bieberbach conjecture [1].",
"More attention has been given to the results of an average sense (see [2], [3]) than the exact upper bounds for $|\\gamma _n|$ .",
"For the Koebe function $k(z)=z/(1-z)^2$ , the logarithmic coefficients are $\\gamma _n=1/n$ .",
"Since the Koebe function $k(z)$ plays the role of extremal function for most of the extremal problems in the class $\\mathcal {S}$ , it is expected that $|\\gamma _n|\\le \\frac{1}{n}$ holds for functions in $\\mathcal {S}$ .",
"But this is not true in general, even in order of magnitude [2].",
"Indeed, there exists a bounded function $f$ in the class $\\mathcal {S}$ with logarithmic coefficients $\\gamma _n\\ne O(n^{-0.83})$ (see [2]).",
"By differentiating (REF ) and equating coefficients we obtain $\\gamma _1=\\frac{1}{2} a_2$ $\\gamma _2=\\frac{1}{2}(a_3-\\frac{1}{2}a_2^2)$ $\\gamma _3=\\frac{1}{2}(a_4-a_2a_3+\\frac{1}{3}a_2^3).$ If $f\\in \\mathcal {S}$ then $|\\gamma _1|\\le 1$ follows at once from (REF ).",
"Using Fekete-Szegö inequality [2] in (REF ), we can obtain the sharp estimate $|\\gamma _2|\\le \\frac{1}{2}(1+2e^{-2})=0.635\\ldots .$ For $n\\ge 3$ , the problem seems much harder, and no significant upper bound for $|\\gamma _n|$ when $f\\in \\mathcal {S}$ appear to be known.",
"If $f\\in \\mathcal {S}^*$ then it is not very difficult to prove that $|\\gamma _n|\\le \\frac{1}{n}$ for $n\\ge 1$ and equality holds for the Koebe function $k(z)=z/(1-z)^2$ .",
"The inequality $|\\gamma _n|\\le \\frac{1}{n}$ for $n\\ge 2$ extends to the class $\\mathcal {K}$ was claimed in a paper of Elhosh [4].",
"However, Girela [6] pointed out some error in the proof of Elhosh [4] and, hence, the result is not substantiated.",
"Indeed, Girela proved that for each $n\\ge 2$ , there exists a function $f\\in \\mathcal {K}$ such that $|\\gamma _n|> \\frac{1}{n}$ .",
"In the same paper it has been shown that $|\\gamma _n|\\le \\frac{3}{2n}$ holds for $n\\ge 1$ whenever $f$ belongs to the set of extreme points of the closed convex hull of the class $\\mathcal {K}$ .",
"Recently, Thomas [12] proved that $|\\gamma _3|\\le \\frac{7}{12}$ for functions in $\\mathcal {K}_0$ (close-to-convex functions with argument 0) with the additional assumption that the second coefficient of the corresponding starlike function $g$ is real.",
"Thomas claimed that this estimate is sharp and has given a form of the extremal function.",
"But after rigorous reading of the paper [12], we observed that such functions do not belong to the class $\\mathcal {K}_0$ (more details will be given in Section ).",
"By fixing a starlike function $g$ in the class $\\mathcal {S}^*$ , the inequality (REF ) assertions a specific subclass of close-to-convex functions.",
"One of such important subclass is the class of close-to-convex functions with respect to the Koebe function $k(z)=z/(1-z)^2$ .",
"In this case, the inequality (REF ) becomes ${\\rm Re\\,} \\left(e^{i\\alpha }(1-z)^2f^{\\prime }(z)\\right)>0,\\quad z\\in \\mathbb {D}$ and defines the subclass $\\mathcal {K}_{\\alpha }(k)$ .",
"Several authors have been extensively studied the class of functions $f\\in \\mathcal {S}$ that satisfies the condition (REF ) (see [5], [7], [9], [11]).",
"Geometrically (REF ) says that the function $h:=e^{i\\delta }f$ has the boundary normalization $\\lim _{t\\rightarrow \\infty } h^{-1}(h(z)+t)=1$ and $h(\\mathbb {D})$ is a domain such that $\\lbrace w + t : t\\ge 0\\rbrace \\subseteq h(\\mathbb {D})$ for every $w\\in h(\\mathbb {D})$ .",
"Clearly, the image domain $h(\\mathbb {D})$ is convex in the positive direction of the real axis.",
"Denote by $\\mathcal {CR}^+:=\\mathcal {K}_{0}(k)$ the class of close-to-convex functions with argument 0 and with respect to Koebe function $k(z)$ .",
"That is $\\mathcal {CR}^+=\\left\\lbrace f\\in \\mathcal {A}: {\\rm Re\\,} (1-z)^2f^{\\prime }(z)>0, ~~ z\\in \\mathbb {D} \\right\\rbrace .$ Then clearly functions in $\\mathcal {CR}^+$ are convex in the positive direction of the real axis.",
"In the present article, we determine the upper bound of $|\\gamma _3|$ for functions in $\\mathcal {K}_0$ and $\\mathcal {CR}^+$ ."
],
[
"Main Results",
"Let $\\mathcal {P}$ denote the class of analytic functions $P$ with positive real part on $\\mathbb {D}$ which has the form $P(z)= 1+\\sum _{n=1}^{\\infty }c_n z^n.$ Functions in $\\mathcal {P}$ are sometimes called Carathéodory function.",
"To prove our main results, we need some preliminary lemmas.",
"The first one is known as Carathéodory's lemma (see [2] for example) and the second one is due to Libera and Złotkiewicz [10].",
"Lemma 2.1 [2] For a function $P\\in \\mathcal {P}$ of the form (REF ), the sharp inequality $|c_n|\\le 2$ holds for each $n\\ge 1$ .",
"Equality holds for the function $P(z)=(1+z)/(1-z)$ .",
"Lemma 2.2 [10] Let $P\\in \\mathcal {P}$ be of the form (REF ).",
"Then there exist $x, t\\in \\mathbb {C}$ with $|x|\\le 1$ and $|t|\\le 1$ such that $2c_2 = c_1^2 + x(4 - c_1^2)$ and $4c_3= c_1^3+ 2(4-c_1^2)c_1x-c_1(4-c_1^2)x^2+2(4-c_1^2)(1-|x|^2)t.$ In [12], Thomas claimed that his result (i.e.",
"$|\\gamma _3|\\le 7/12$ ) is sharp for functions in the class $\\mathcal {K}_0$ by ascertaining the equality holds for a function $f$ defined by $zf^{\\prime }(z)=g(z)P(z)$ where $g\\in \\mathcal {S}^*$ with $b_2(g)=b_3(g)=b_4(g)=2$ and $P\\in \\mathcal {P}$ with $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .",
"But in view of Lemma REF , it is easy to see that there does not exist a function $P\\in \\mathcal {P}$ with the property $c_1(P)=0$ , $c_2(P)=c_3(P)=2$ .",
"Thus we can conclude that the result obtained by Thomas is not sharp.",
"The main aim of the present paper is to obtain a better upper bound for $|\\gamma _3|$ for functions in the class $\\mathcal {K}_0$ than that of obtained by Thomas [12].",
"To prove our main results we also need the following Fekete-Szegö inequality for functions in the class $\\mathcal {S}^*$ .",
"Lemma 2.3 [8] Let $g\\in \\mathcal {S}^*$ be of the form $g(z)=z+\\sum _{n=2}^{\\infty }b_n z^n$ .",
"Then for any $\\lambda \\in \\mathbb {C}$ , $|b_3-\\lambda b_2^2|\\le \\max \\lbrace 1,|3-4\\lambda |\\rbrace .$ The inequality is sharp for $k(z)=z/(1-z)^2$ if $|3-4\\lambda |\\ge 1$ and for $(k(z^2))^{1/2}$ if $|3-4\\lambda |<1$ .",
"For $f\\in \\mathcal {K}_0$ (close-to-convex functions with argument 0), we obtained the following improved result for $|\\gamma _3|$ (compare [12]).",
"Theorem 2.1 If $f\\in \\mathcal {K}_0$ then $|\\gamma _3|\\le \\frac{1}{18} (3+4 \\sqrt{2})=0.4809$ .",
"Let $f\\in \\mathcal {K}_0$ be of the form (REF ).",
"Then there exists a starlike function $g(z)=z+\\sum _{n=2}^{\\infty }b_n z^n$ and a Carathéodory function $P\\in \\mathcal {P}$ of the form (REF ) such that $zf^{\\prime }(z)=g(z)P(z).$ A comparison of the coefficients on the both sides of (REF ) yields $a_2&=\\frac{1}{2}(b_2+c_1)\\\\a_3&=\\frac{1}{3}(b_3+b_2c_1+c_2)\\\\a_4&=\\frac{1}{4}(b_4+b_3c_1+b_2c_2+c_3).$ By substituting the above $a_2, a_3$ and $a_4$ in (REF ) and then further simplification gives $2\\gamma _3&= a_4-a_2a_3+\\frac{1}{3}a_2^3\\\\&=\\frac{1}{24}\\left((6b_4-4b_2b_3+b_2^3)+\\frac{c_1}{2}\\left(b_3-\\frac{1}{2}b_2^2\\right)+b_2(2c_2-c_1^2)+c_1^3-4c_1c_2+6c_3\\right)\\nonumber .$ In view of Lemma REF and writing $c_2$ and $c_3$ in terms of $c_1$ we obtain $48\\gamma _3&= (6b_4-4b_2b_3+b_2^3)+ 2c_1\\left(b_3-\\frac{1}{2}b_2^2\\right)+ b_2x(4-c_1^2)\\\\&\\quad +\\frac{1}{2}c_1^3+c_1x(4-c_1^2)-\\frac{3}{2}c_1x^2(4-c_1^2)+3(4-c_1^2)(1-|x|^2)t,\\nonumber $ where $|x|\\le 1$ and $|t|\\le 1$ .",
"Note that if $\\gamma _3(g)$ denote the third logarithmic coefficient of $g\\in \\mathcal {S}^*$ then $|\\gamma _3(g)|=\\frac{1}{2}|b_4-b_2b_3+\\frac{1}{3}b_2^3|\\le \\frac{1}{3}$ .",
"Since $g\\in \\mathcal {S}^*$ , in view of Lemma REF we obtain $|6b_4-4b_2b_3+b_2^3|\\le 6|b_4-b_2b_3+\\frac{1}{3}b_2^3|+2|b_2||b_3-\\frac{1}{2}b_2^2|\\le 8.$ Since the class $\\mathcal {K}_0$ is invariant under rotation, without loss of generality we can assume that $c_1=c$ , where $0\\le c\\le 2$ .",
"Taking modulus on both the sides of (REF ) and then applying triangle inequality and further using the inequality (REF ) and Lemma REF , it follows that $48|\\gamma _3|\\le 8+ 2c+ 2|x|(4-c^2)+\\left|\\frac{1}{2}c^3+cx(4-c^2)-\\frac{3}{2}cx^2(4-c^2)\\right|+3(4-c^2)(1-|x|^2),$ where we have also used the fact $|t|\\le 1$ .",
"Let $x=re^{i\\theta }$ where $0\\le r\\le 1$ and $0\\le \\theta \\le 2\\pi $ .",
"For simplicity, by writing $\\cos \\theta =p$ we obtain $48|\\gamma _3|\\le \\psi (c,r)+\\left|\\phi (c,r,p)\\right|=:F(c,r,p)$ where $\\psi (c,r)=8+ 2c+ 2r(4-c^2)+3(4-c^2)(1-r^2)$ and $\\phi (c,r,p)&=\\left(\\frac{1}{4}c^6+c^2r^2(4-c^2)^2+\\frac{9}{4}c^2r^4(4-c^2)^2+c^4(4-c^2)rp\\right.\\\\&\\qquad \\quad \\left.-\\frac{3}{2}c^4r^2(4-c^2)(2p^2-1)-3c^2(4-c^2)r^3p \\right)^{1/2}.$ Thus we need to find the maximum value of $F(c,r,p)$ over the rectangular cube $R:=[0,2]\\times [0,1]\\times [-1,1]$ .",
"By elementary calculus one can verify the followings: $&\\max _{0\\le r\\le 1} \\psi (0,r)=\\psi \\left(0,\\frac{1}{3}\\right)=\\frac{64}{3},\\quad \\max _{0\\le r\\le 1} \\psi (2,r)=12,\\\\[2mm]&\\max _{0\\le c\\le 2} \\psi (c,0)=\\psi \\left(\\frac{1}{3},0\\right)=\\frac{61}{3},\\quad \\max _{0\\le c\\le 2} \\psi (c,1)=\\psi (0,1)=16 \\quad \\mbox{ and }\\\\[2mm]&\\max _{(c,r)\\in [0,2]\\times [0,1]} \\psi (c,r)=\\psi \\left(\\frac{3}{10},\\frac{1}{3}\\right)=\\frac{649}{30}=21.6333.$ We first find the maximum value of $F(c,r,p)$ on the boundary of $R$ , i.e on the six faces of the rectangular cube $R$ .",
"On the face $c=0$ , we have $F(0,r,p)=\\psi (0,r)$ , where $(r,p)\\in R_1:=[0,1]\\times [-1,1]$ .",
"Thus $\\max _{(r,p)\\in R_1} F(0,r,p)= \\max _{0\\le r\\le 1} \\psi (0,r)=\\psi \\left(0,\\frac{1}{3}\\right)=\\frac{64}{3}=21.33.$ On the face $c=2$ , we have $F(2,r,p)= 16$ , where $(r,p)\\in R_1$ .",
"On the face $r=0$ , we have $F(c,0,p)=8+ 2c+3(4-c^2)+\\frac{1}{2}c^3$ , where $(c,p)\\in R_2:=[0,2]\\times [-1,1]$ .",
"By using elementary calculus it is easy to see that $\\max _{(c,p)\\in R_2} F(c,0,p)= F\\left(\\frac{2}{3} (3-\\sqrt{6}),0,p\\right)=\\frac{16}{9} \\left(9+\\sqrt{6}\\right)=20.3546.$ On the face $r=1$ , we have $F(c,1,p)=\\psi (c,1)+ |\\phi (c,1,p)|$ , where $(c,p)\\in R_2$ .",
"We first prove that $\\phi (c,1,p)\\ne 0$ in the interior of $R_2$ .",
"On the contrary, if $\\phi (c,1,p)=0$ in the interior of $R_2$ then $|\\phi (c,1,p)|^2=\\left|\\frac{1}{2}c^3+ce^{i\\theta }(4-c^2)-\\frac{3}{2}ce^{2i\\theta }(4-c^2)\\right|^2=0$ and hence $\\frac{1}{2}c^3+cp(4-c^2)-\\frac{3}{2}c(4-c^2)(2p^2-1)=0 ~\\mbox{ and } c(4-c^2)\\sin \\theta -\\frac{3}{2}c(4-c^2)\\sin 2\\theta =0.$ Further, (REF ) reduces to $\\frac{1}{2}c^2+p(4-c^2)-\\frac{3}{2}(4-c^2)(2p^2-1)=0 \\quad \\mbox{and}\\quad 1-3p=0,$ which is equivalent to $p=1/3$ and $c^2=6$ .",
"This contradicts the range of $c\\in (0,2)$ .",
"Thus $\\phi (c,1,p)\\ne 0$ in the interior of $R_2$ .",
"Next, we prove that $F(c,1,p)$ has no maximum at any interior point of $R_2$ .",
"Suppose that $F(c,1,p)$ has the maximum at an interior point of $R_2$ .",
"Then at such point $\\frac{\\partial F(c,1,p)}{\\partial c}=0$ and $\\frac{\\partial F(c,1,p)}{\\partial p}=0$ .",
"From $\\frac{\\partial F(c,1,p)}{\\partial p}=0$ , (for points in the interior of $R_2$ ), a straight forward calculation gives $p=\\frac{2 \\left(c^2-3\\right)}{3 c^2}.$ Substituting the value of $p$ as given in (REF ) in the relation $\\frac{\\partial F(c,1,p)}{\\partial c}=0$ and further simplification gives $3c^3-2c+(2c-1) \\sqrt{6(c^2+2)}=0.$ It is easy to show that the function $\\rho (c)=3c^3-2c+(2c-1) \\sqrt{6(c^2+2)}$ is strictly increasing in $(0,2)$ .",
"Since $\\rho (0)<0$ and $\\rho (2)>0$ , the equation (REF ) has exactly one solution in $(0,2)$ .",
"By solving the equation (REF ) numerically, we obtain the approximate root in $(0,2)$ as $0.5772$ .",
"But the corresponding value of $p$ obtained by (REF ) is $-5.3365$ which does not belong to $(-1,1)$ .",
"Thus $F(c,1,p)$ has no maximum at any interior point of $R_2$ .",
"Thus we find the maximum value of $F(c,1,p)$ on the boundary of $R_2$ .",
"Clearly, $F(0,1,p)=F(2,1,p)=16$ , $F(c,1,-1)={\\left\\lbrace \\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(10-3c^2)& \\mbox{for}\\quad 0\\le c\\le \\sqrt{\\frac{10}{3}}\\\\[2mm]8+ 2c+ 2(4-c^2)-c(10-3c^2)& \\mbox{for}\\quad \\sqrt{\\frac{10}{3}}< c\\le 2\\end{array}\\right.",
"}$ and $F(c,1,1)={\\left\\lbrace \\begin{array}{ll}8+ 2c+ 2(4-c^2)+c(2-c^2)& \\mbox{for}\\quad 0\\le c\\le \\sqrt{2}\\\\[2mm]8+ 2c+ 2(4-c^2)-c(2-c^2)& \\mbox{for}\\quad \\sqrt{2}< c\\le 2.\\end{array}\\right.",
"}$ By using elementary calculus we find that $\\max _{0\\le c\\le 2}F(c,1,-1)= F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023\\quad \\mbox{ and }$ $\\max _{0\\le c\\le 2}F(c,1,1)= F\\left(\\frac{2}{3},1,1\\right)=\\frac{427}{27} =17.48.$ Hence, $\\max _{(c,p)\\in R_2} F(c,1,p)= F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023.$ On the face $p=-1$ , $F(c,r,-1)={\\left\\lbrace \\begin{array}{ll}\\psi (c,r)+\\eta _1(c,r) & \\mbox{ for }\\quad \\eta _1(c,r)\\ge 0\\\\[2mm]\\psi (c,r)-\\eta _1(c,r)& \\mbox{ for }\\quad \\eta _1(c,r)< 0,\\end{array}\\right.",
"}$ where $\\eta _1(c,r)=c^3 (3r^2+2r+1)-4cr(3r+2)$ and $(c,r)\\in R_3:=[0,2]\\times [0,1]$ .",
"Differentiating partially $F(c,r,-1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\max _{(c,r)\\in {\\rm int \\,}R_3\\setminus S_1} F(c,r,-1)= F\\left(2(\\sqrt{2}-1),\\frac{1}{3}(1+\\sqrt{2}),-1\\right)=\\frac{8}{3} (3+4 \\sqrt{2})=23.0849,$ where $S_1=\\lbrace (c,r)\\in R_3: \\eta _1(c,r)=0\\rbrace $ .",
"Now we find the maximum value of $F(c,r,-1)$ on the boundary of $R_3$ and on the set $S_1$ .",
"Note that $\\max _{(c,r)\\in S_1} F(c,r,-1)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\frac{649}{30}=21.6333.$ On the other hand by using elementary calculus, as before, we find that $\\max _{(c,r)\\in \\partial R_3} F(c,r,-1)=F\\left(\\frac{2}{9} (2 \\sqrt{7}-1),1,-1\\right)=\\frac{8}{243} \\left(403+112 \\sqrt{7}\\right)=23.023,$ where $\\partial R_3$ denotes the boundary of $R_3$ .",
"Hence, by combining the above cases we obtain $\\max _{(c,r)\\in R_3} F(c,r,-1)=F\\left(2(\\sqrt{2}-1),\\frac{1}{3}(1+\\sqrt{2}),-1\\right)=\\frac{8}{3} (3+4 \\sqrt{2})=23.0849.$ On the face $p=1$ , $F(c,r,1)={\\left\\lbrace \\begin{array}{ll}\\psi (c,r)+\\eta _2(c,r) & \\mbox{ for }\\quad \\eta _2(c,r)\\ge 0\\\\[2mm]\\psi (c,r)-\\eta _2(c,r)& \\mbox{ for }\\quad \\eta _2(c,r)< 0,\\end{array}\\right.",
"}$ where $\\eta _2(c,r)=c^3 (3r^2-2r+1)+4cr(3r-2)$ and $(c,r)\\in R_3$ .",
"Differentiating partially $F(c,r,1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\max _{(c,r)\\in {\\rm int \\,}R_3\\setminus S_2} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.89,$ where $S_2=\\lbrace (c,r)\\in R_3: \\eta _2(c,r)=0\\rbrace $ .",
"Now, we find the maximum value of $F(c,r,1)$ on the boundary of $R_3$ and on the set $S_2$ .",
"By noting that $\\max _{(c,r)\\in S_2} F(c,r,1)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\frac{649}{30}=21.6333$ and proceeding similarly as in the previous case, we find that $\\max _{(c,r)\\in R_3} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.89.$ Let $S^{\\prime }=\\lbrace (c,r,p)\\in R: \\phi (c,r,p)=0\\rbrace $ .",
"Then $\\max _{(c,r,p)\\in S^{\\prime }} F(c,r,p)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=\\psi \\left(\\frac{3}{10},\\frac{1}{3}\\right)=\\frac{649}{30}=21.6333.$ We prove that $F(c,r,p)$ has no maximum value at any interior point of $R\\setminus S^{\\prime }$ .",
"Suppose that $F(c,r,p)$ has a maximum value at an interior point of $R\\setminus S^{\\prime }$ .",
"Then at such point $\\frac{\\partial F}{\\partial c}=0$ , $\\frac{\\partial F}{\\partial r}=0$ and $\\frac{\\partial F}{\\partial p}=0$ .",
"Note that $\\frac{\\partial F}{\\partial c}$ , $\\frac{\\partial F}{\\partial r}$ and $\\frac{\\partial F}{\\partial p}$ may not exist at points in $S^{\\prime }$ .",
"In view of $\\frac{\\partial F}{\\partial p}=0$ (for points in the interior of $R\\setminus S^{\\prime }$ ), a straight forward but laborious calculation gives $p= \\frac{3 c^2 r^2+c^2-12 r^2}{6 c^2 r}.$ Substituting the value of $p$ as given in (REF ) in the relations $\\frac{\\partial F}{\\partial c}=0$ and $\\frac{\\partial F}{\\partial r}=0$ and simplifying (again, a long and laborious calculation), we obtain $\\frac{3 \\sqrt{6} c^3 (1-3 r^2)+12 (c(3r^2-2r-3)+1)\\sqrt{c^2+2} )+4\\sqrt{6}c}{6\\sqrt{c^2+2}}=0$ and $(4-c^2) \\left(( \\sqrt{6(c^2+2)}-6) r+2\\right)=0.$ Since $0<c<2$ , solving the equation (REF ) for $r$ , we obtain $r=\\frac{2}{6-\\sqrt{6(c^2+2)}}.$ Substituting the value of $r$ in (REF ) and then further simplification gives $3 c^3+6 c -(6 c-2) \\sqrt{6 \\left(c^2+2\\right)}=0.$ Taking the last term on the right hand side and squaring on both sides yields $3 \\left(c^2+2\\right) \\left(3 c^4-66 c^2+48 c-8\\right)=0.$ Clearly $c^2+2\\ne 0$ in $0<c<2$ .",
"On the other hand the polynomial $q(c)=3 c^4-66 c^2+48 c-8$ has exactly two roots in $(0,2)$ , one lies in $(0,1/3)$ and another lies in $(1/3,1/2)$ .",
"This can be seen using the well-known Strum theorem for isolating real roots and hence for the sake of brevity we omit the details.",
"By solving the equation $q(c)=0$ numerically, we obtain two approximate roots $0.2577$ and $0.4795$ in $(0,2)$ .",
"But the corresponding value of $p$ obtained from (REF ) and (REF ) are $-23.6862$ and $-6.80595$ which do not belong to $(-1,1)$ .",
"This proves that $F(c,r,p)$ has no maximum in the interior of $R\\setminus S^{\\prime }$ Thus combining all the above cases we find that $\\max _{(c,r,p)\\in R} F(c,r,p)=F\\left(2(\\sqrt{2}-1),\\frac{1}{3}(1+\\sqrt{2}),-1\\right)=\\frac{8}{3} (3+4 \\sqrt{2})=23.0849,$ and hence from (REF ) we obtain $|\\gamma _3|\\le \\frac{1}{18} (3+4 \\sqrt{2})=0.4809.$ We obtained the following sharp upper bound for $|\\gamma _3|$ for functions in the class $\\mathcal {CR}^+$ .",
"Theorem 2.2 Let $f\\in \\mathcal {CR}^+$ be of the form (REF ) with $1\\le a_2 \\le 2$ .",
"Then $|\\gamma _3|\\le \\frac{1}{243} (28+19 \\sqrt{19})=0.4560.$ The inequality is sharp.",
"If $f\\in \\mathcal {CR}^+$ then there exists a Carathéodory function $P\\in \\mathcal {P}$ of the form (REF ) such that $zf^{\\prime }(z)=g(z)P(z)$ , where $g(z):=k(z)=z/(1-z)^2$ .",
"Following the same method as used in Theorem REF and noting that $g(z):=k(z)=z+2z^2+3z^3+4z^4+\\cdots $ , a simple computation in (REF ) shows that $48\\gamma _3= 8+ 2c_1+\\frac{1}{2}c_1^3+ (4-c_1^2)(2x +c_1x-\\frac{3}{2}c_1x^2)+3(4-c_1^2)(1-|x|^2)t,$ where $|x|\\le 1$ and $|t|\\le 1$ .",
"Since $1\\le a_2 \\le 2$ and $2a_2=2+c_1$ , then $0\\le c_1\\le 2$ .",
"Taking modulus on the both sides of (REF ) and then applying triangle inequality and writing $c=c_1$ , it follows that $48|\\gamma _3|\\le \\left|8+ 2c_1+\\frac{1}{2}c_1^3+ (4-c_1^2)(2x +c_1x-\\frac{3}{2}c_1x^2)\\right|+3(4-c^2)(1-|x|^2),$ where we have also used the fact $|t|\\le 1$ .",
"Let $x=re^{i\\theta }$ where $0\\le r\\le 1$ and $0\\le \\theta \\le 2\\pi $ .",
"For simplicity, by writing $\\cos \\theta =p$ we obtain $48|\\gamma _3|\\le \\psi (c,r)+\\left|\\phi (c,r,p)\\right|=:F(c,r,p)$ where $\\psi (c,r)=3(4-c^2)(1-r^2)$ and $\\phi (c,r,p)&=\\left( (8+2c+\\frac{1}{2}c^3)^2 +r^2(4-c^2)^2(4+c^2+\\frac{9}{4}c^2r^2+4c-6crp-3c^2rp)\\right.\\\\&\\qquad \\quad \\left.",
"+2(4-c^2)(8+2c+\\frac{1}{2}c^3)(2rp+crp-\\frac{3}{2}cr^2(2p^2-1)) \\right)^{1/2}.$ Thus we need to find the maximum value of $F(c,r,p)$ over the rectangular cube $R=[0,2]\\times [0,1]\\times [-1,1]$ .",
"We first find the maximum value of $F(c,r,p)$ on the boundary of $R$ , i.e on the six faces of the rectangular cube $R$ .",
"As before, let $R_1=[0,1]\\times [-1,1], R_2=[0,2]\\times [-1,1]$ and $R_3=[0,2]\\times [0,1]$ .",
"By elementary calculus it is not very difficult to prove that $\\max _{(r,p)\\in R_1} F(0,r,p)&= F(0,\\frac{1}{3},1)=\\frac{64}{3}=21.33,\\\\\\max _{(r,p)\\in R_1} F(2,r,p)&= F(2,r,p)= 16,\\\\\\max _{(c,p)\\in R_2} F(c,0,p)&= F\\left(\\frac{2}{3} (3-\\sqrt{6}),0,p\\right)=\\frac{16}{9} \\left(9+\\sqrt{6}\\right)=20.3546.$ On the face $r=1$ , we have $F(c,1,p)=|\\phi (c,1,p)|$ where $(c,p)\\in R_2$ .",
"As in the proof of Theorem REF , one can verify that $\\phi (c,1,p)\\ne 0$ in the interior of $R_2$ (otherwise, one can simply proceed to find maximum value $F(c,1,p)$ at an interior point of $R_2\\setminus T$ , where $T=\\lbrace (c,p)\\in R_2: \\phi _1(c,1,p)=0\\rbrace $ , as $F(c,1,p)=0$ in $T$ ).",
"Suppose that $F(c,1,p)$ has the maximum value at an interior point of $R_2$ .",
"Then at such point $\\frac{\\partial F}{\\partial c}=0$ and $\\frac{\\partial F}{\\partial p}=0$ .",
"From $\\frac{\\partial F}{\\partial p}=0$ (for points in the interior of $R_2$ ), it follows that $p=\\frac{2 \\left(c^3-2 c+4\\right)}{3 c \\left(c^2-2 c+8\\right)}.$ By substituting the above value of $p$ given in (REF ) in the relation $\\frac{\\partial F}{\\partial c}=0$ and further computation (a long and laborious calculation) gives $3 c^8-17 c^7+76 c^6-136 c^5+120 c^4+640 c^3-832 c^2-192 c+128=0.$ This equation has exactly two real roots in $(0,2)$ , one lies in $(0,1)$ and another lies in $(1,2)$ .",
"This can be seen using the well-known Strum theorem for isolating real roots therefore for the sake of brevity we omit the details.",
"Solving this equation numerically we obtain two approximate roots $0.3261$ and $1.2994$ in $(0,2)$ and the corresponding values of $p$ are $0.9274$ and $0.2602$ respectively.",
"Thus the extremum points of $F(c,1,p)$ in the interior of $R_2$ lie in a small neighborhood of the points $A_1=(0.3261,1,0.9274)$ and $A_2=(1.2994,1,0.2602)$ (on the plane $r=1$ ).",
"Now $F(A_1)=15.8329$ and $F(A_2)=18.6303$ .",
"Since the function $F(c,1,p)$ is uniformly continuous on $R_2$ , the value of $F(c,1,p)$ would not vary too much in the neighborhood of the points $A_1$ and $A_2$ .",
"Again, proceeding similarly as in the proof of Theorem REF , we find that $\\max _{(c,p)\\in \\partial R_2} F(c,1,p)= F(2,1,p)=16$ and hence $\\max _{(c,p)\\in R_2} F(c,1,p) \\approx 18.6306< \\frac{64}{3}.$ On the face $p=-1$ , $F(c,r,-1)={\\left\\lbrace \\begin{array}{ll}\\psi (c,r)+\\eta _1(c,r) & \\mbox{ for }\\quad \\eta _1(c,r)\\ge 0\\\\[2mm]\\psi (c,r)-\\eta _1(c,r)& \\mbox{ for }\\quad \\eta _1(c,r)\\le 0,\\end{array}\\right.",
"}$ where $\\eta _1(c,r)=c^3-3cr^2 (4-c^2)+2 (c-2) (c+2)^2 r+4 c+16$ and $(c,r)\\in R_3$ .",
"Again, proceeding similarly as in the proof of Theorem REF , we can show that $F(c,r,-1)$ has no maximum in the interior of $R_3\\setminus S_1$ , where $S_1=\\lbrace (c,r)\\in R_3: \\eta _1(c,r)=0\\rbrace $ .",
"Computing the maximum value on the boundary of $R_3$ and on the set $S_1$ we conclude that $\\max _{(c,r)\\in R_3} F(c,r,-1)= F(0,0,-1)=20.$ On the face $p=1$ , we have $F(c,r,1)=\\psi (c,r)+\\eta _2(c,r)$ , where $\\eta _2(c,r)&=(c+2) (8-2c+c^2+8r-2c^2r-6cr^2+3c^2r^2)\\\\& \\ge (c+2)\\left(3+(1-c)^2+r(8-2c^2)+r^2(3c^2-6c+4)\\right)\\\\&\\ge 0$ for $(c,r)\\in R_3$ .",
"Differentiating partially $F(c,r,1)$ with respect to $c$ and $r$ and a routine calculation shows that $\\max _{(c,r)\\in {\\rm int \\,}R_3} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.8902,$ and on the boundary of $R_3$ we have $\\max _{(c,r)\\in \\partial R_3} F(c,r,1)=F(0,\\frac{1}{3},1)=\\frac{64}{3}=21.33.$ Thus, $\\max _{(c,r)\\in R_3} F(c,r,1)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.8902.$ Let $S^{\\prime }=\\lbrace (c,r,p)\\in R: \\phi (c,r,p)=0\\rbrace $ .",
"Then $\\max _{(c,r,p)\\in S^{\\prime }} F(c,r,p)\\le \\max _{(c,r)\\in R_3} \\psi (c,r)=12.$ We now prove that $F(c,r,p)$ has no maximum at an interior point of $R\\setminus S^{\\prime }$ .",
"Suppose that $F(c,r,p)$ has a maximum at an interior point of $R\\setminus S^{\\prime }$ .",
"Then at such point $\\frac{\\partial F}{\\partial c}=0$ , $\\frac{\\partial F}{\\partial r}=0$ and $\\frac{\\partial F}{\\partial p}=0$ .",
"Note that $\\frac{\\partial F}{\\partial c}$ , $\\frac{\\partial F}{\\partial r}$ and $\\frac{\\partial F}{\\partial p}$ may not exist at points in $S^{\\prime }$ .",
"In view of $\\frac{\\partial F}{\\partial p}=0$ (for points in the interior of $R\\setminus S^{\\prime }$ ), a straight forward but laborious calculation gives $p= \\frac{3 c^3 r^2+c^3-12 c r^2+4 c+16}{6cr (c^2-2 c+8)}.$ Substituting the value of $p$ given in (REF ) in the relation $\\frac{\\partial F}{\\partial r}=0$ and then further simplifying (again, a long and laborious calculation), we obtain $r(4-c^2) \\left( c \\sqrt{\\frac{6(c^3-4 c^2+14 c+4)}{c(c^2-2 c+8)}}-6\\right)=0.$ Since $0<c<2$ and $0<r<1$ , we can divide by $r(4-c^2)$ on both the sides of (REF ).",
"Further, a simple computation shows that $\\frac{6 (4-c^2) (c^2-4 c+12)}{c^2-2 c+8}=0.$ But this equation has no real roots in $(0,2)$ .",
"Therefore, $F(c,r,p)$ has no maximum at an interior point of $R\\setminus S^{\\prime }$ .",
"Thus combining all the cases we find that $\\max _{(c,r,p)\\in R} F(c,r,p)=F\\left(\\frac{1}{3} (10-2 \\sqrt{19}),\\frac{1}{3},1\\right)=\\frac{16}{81} \\left(28+19 \\sqrt{19}\\right)=21.8902,$ and hence, from (REF ) we obtain $|\\gamma _3|\\le \\frac{1}{243} (28+19 \\sqrt{19})=0.4560.$ We now show that the inequality (REF ) is sharp.",
"It is pertinent to note that equality holds in (REF ) if we choose $c_1=c=\\frac{1}{3} (10-2 \\sqrt{19})$ , $x=1$ and $t=1$ in (REF ).",
"For such values of $c_1, x$ and $t$ , Lemma REF elicit $c_2=\\frac{2}{27} (97-20 \\sqrt{19})$ and $c_3=\\frac{1}{243} (2050-362 \\sqrt{19})$ .",
"A function $P\\in \\mathcal {P}$ having the first three coefficients $c_1, c_2$ and $c_3$ as above is given by $P(z) &=(1-2\\lambda ) \\frac{1+z}{1-z}+\\lambda \\frac{1+uz}{1-uz}+\\lambda \\frac{1+\\overline{u}z}{1-\\overline{u}z}\\\\& =1+\\frac{1}{3} (10-2 \\sqrt{19})z +\\frac{2}{27} (97-20 \\sqrt{19})z^2 +\\frac{1}{243} (2050-362 \\sqrt{19})z^3+\\cdots ,\\nonumber $ where $\\lambda =\\frac{1}{9} (-1-\\sqrt{19})$ and $u=\\alpha +i\\sqrt{1-\\alpha ^2}$ with $\\alpha =\\frac{1}{18} (-13+4 \\sqrt{19})$ .",
"Hence the inequality (REF ) is sharp for a function $f$ defined by $(1-z)^2f^{\\prime }(z)=P(z)$ , where $P(z)$ is given by (REF ).",
"This completes the proof.",
"Remark 2.1 In [12], Thomas proved that $|\\gamma _3|\\le \\frac{7}{12}=0.5833$ for functions in the class $\\mathcal {K}_0$ with an additional condition that the second coefficient $b_2$ of the corresponding starlike function $g$ is real.",
"However, in Theorem REF we obtained a much improved bound $|\\gamma _3|\\le \\frac{1}{18} (3+4 \\sqrt{2})=0.4809$ for functions in the whole class $\\mathcal {K}_0$ without assuming any additional condition on functions in the class $\\mathcal {K}_0$ .",
"While for functions in the class $\\mathcal {CR}^+$ (with $1\\le a_2\\le 2$ ) we obtained the sharp bound $|\\gamma _3|\\le \\frac{1}{243} (28+19 \\sqrt{19})=0.4560$ .",
"We conjecture that for the whole class $\\mathcal {K}_0$ the sharp upper bound for $|\\gamma _3|$ is $|\\gamma _3|\\le \\frac{1}{243} (28+19 \\sqrt{19})=0.4560$ .",
"Acknowledgement: The first author thank University Grants Commission for the financial support through UGC-SRF Fellowship.",
"The second author thank SERB (DST) for financial support."
]
] | 1606.05162 |
[
[
"Haldane linearisation done right: Solving the nonlinear recombination\n equation the easy way"
],
[
"Abstract The nonlinear recombination equation from population genetics has a long history and is notoriously difficult to solve, both in continuous and in discrete time.",
"This is particularly so if one aims at full generality, thus also including degenerate parameter cases.",
"Due to recent progress for the continuous time case via the identification of an underlying stochastic fragmentation process, it became clear that a direct general solution at the level of the corresponding ODE itself should also be possible.",
"This paper shows how to do it, and how to extend the approach to the discrete-time case as well."
],
[
"Introduction",
"The recombination equation is a well-known dynamical system from mathematical population genetics [15], [10], [9], [3], which describes the evolution of the genetic composition of a population that evolves under recombination.",
"The genetic composition is described via a probability distribution (or measure) on a space of sequences of finite length, and recombination is the genetic mechanism in which two parent individuals are involved in creating the mixed sequence of their offspring during sexual reproduction.",
"The model comes in a continuous-time and a discrete-time version.",
"It can accommodate a variety of different mechanisms by which the genetic material of the offspring is partitioned across its parents.",
"In all cases, the resulting equations are nonlinear and notoriously difficult to solve.",
"Elucidating the underlying structure and finding solutions has been a challenge to theoretical population geneticists for nearly a century now.",
"The first studies go back to Jennings in 1917 [14] and Robbins in 1918 [20].",
"Geiringer in 1944 [13] and Bennett in 1954 [8] were the first to state the generic general form of the solution in terms of a convex combination of certain basis functions, and developed methods for the recursive evaluation of the corresponding coefficients to obtain the solution itself, at least in principle.",
"The approach was later continued within the systematic framework of genetic algebras; compare [15], [17].",
"It could be shown that, despite the nonlinearity, the dynamical system may be (exactly) transformed into a linear one by embedding it into a higher-dimensional space.",
"More explicitly, a large number of further components are added that correspond to multilinear transformations of the original measure.",
"This method is known as Haldane linearisation [17].",
"However, this line of research led to astonishingly few concrete or applicable results.",
"This raises the question whether the program may be completed outside the abstract framework, and what kinds of results can be obtained via different approaches.",
"A first step forward, for an important special case, was achieved in [6], via a rather powerful use of the inclusion-exclusion principle in the form of the Möbius inversion formula.",
"At the same time, a general formalism via nonlinear operators, called recombinators, was introduced that also allowed for an alternative consideration starting from the nonlinear equation with a single such operator and extending this to the same solution [5].",
"After various intermediate steps of gradual generalisations, the complete equation, in the setting of general partitions, was analysed and solved in [4].",
"In the generic parameter case, the general solution was given in recursive form.",
"Also, the principal form of the solution in degenerate cases was analysed, but no general formula was given.",
"The most important insight, however, was the identification of an underlying stochastic fragmentation process in [4].",
"This means that the solution of the nonlinear recombination equation has a representation in terms of the solution of the Kolmogorov forward equation for this very process, which is a linear ODE.",
"In [7], the slightly simpler setting of ordered or interval partitions was analysed, with focus on explicit solution formulas for all parameter values (thus including the degenerate cases) for sequences of length up to five.",
"Here, the above-mentioned Kolmogorov equation was investigated further, and the Markov generator from [4] was derived explicitly.",
"This demonstrated two important things: The solvability of the nonlinear recombination ODE ultimately rests upon the fact that this solution essentially also solves a system of linear equations; The degenerate cases are in one-to-one correspondence to the cases where the Markov generator fails to be diagonalisable, and the appearance of Jordan blocks, well known from classic ODE theory, determines the solutions then.",
"Now, with hindsight, one can ask whether one can treat the original nonlinear ODE in such a way that this becomes immediately transparent, without resorting to the underlying stochastic process.",
"The answer is affirmative, and this paper explains how to do it.",
"Effectively, this new approach means to re-interpret the original Haldane linearisation in a suitable way, without any need for genetic algebras.",
"Moreover, as we shall see, a completely analogous approach also works for the discrete-time recombination equation.",
"This paper builds on previous work, most importantly on [4], [7].",
"Some of the results from these papers will be freely used below, and not re-derived here (though we will always provide precise references).",
"Also, the biological background is explained in [4].",
"After recalling the preliminaries and our notation in Section , the general recombination equation in continuous time, together with its reduction to subsystems via marginalisation, is discussed in Section .",
"This is followed by its general solution via our new and simplified strategy (Section ), which leads to the first main result in Theorem REF .",
"A stratified interpretation in terms of the underlying partitioning process is offered in Section , which gives our second main result (Theorem REF ).",
"The discrete-time version of the recombination equation is then discussed and solved in Section , by the same method, which leads to our third main result in Theorem REF .",
"The corresponding stochastic process is also identified and briefly summarised."
],
[
"Partitions, product spaces, measures and\nrecombinators",
"Let $S$ be a finite set, and consider the lattice $\\mathbb {P}(S)$ of partitions of $S$ ; see [1] for general background on lattice theory and [4] for details of the present setting.",
"Here, we write a partition of $S$ as $\\mathcal {A}= \\lbrace A_{1}, \\dots , A_{m} \\rbrace $ , where $m = |\\mathcal {A}|$ is the number of its (non-empty) parts (also called blocks), and one has $A_{i} \\cap A_{j} = \\varnothing $ for all $i\\ne j$ together with $A_{1} \\cup \\dots \\cup A_{m} = S$ .",
"The natural ordering relation is denoted by $\\preccurlyeq $ , where $\\mathcal {A}\\preccurlyeq \\mathcal {B}$ means that $\\mathcal {A}$ is finer than $\\mathcal {B}$ , or that $\\mathcal {B}$ is coarser than $\\mathcal {A}$ .",
"The conditions $\\mathcal {A}\\preccurlyeq \\mathcal {B}$ and $\\mathcal {B}\\succcurlyeq \\mathcal {A}$ are synonymous, while $\\mathcal {A}\\prec \\mathcal {B}$ means $\\mathcal {A}\\preccurlyeq \\mathcal {B}$ together with $\\mathcal {A}\\ne \\mathcal {B}$ , so $\\mathcal {A}$ is strictly finer than $\\mathcal {B}$ .",
"The joint refinement of two partitions $\\mathcal {A}$ and $\\mathcal {B}$ is written as $\\mathcal {A}\\wedge \\mathcal {B}$ , and is the coarsest partition below $\\mathcal {A}$ and $\\mathcal {B}$ .",
"The unique minimal partition within the lattice $\\mathbb {P}(S)$ is denoted as $\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}0\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}= \\big \\lbrace \\lbrace x\\rbrace \\mid x \\in S \\big \\rbrace $ , while the unique maximal one is $\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}= \\lbrace S \\rbrace $ .",
"When $U$ and $V$ are disjoint (finite) sets, two partitions $\\mathcal {A}\\in \\hspace{0.5pt}\\mathbb {P}(U)$ and $\\mathcal {B}\\in \\hspace{0.5pt}\\mathbb {P}(V)$ can be joined to form an element of $\\mathbb {P}(U\\hspace{-0.5pt}\\cup V)$ .",
"We denote such a joining by $\\mathcal {A}\\sqcup \\mathcal {B}$ , and similarly for multiple joinings.",
"Conversely, if $U\\hspace{-0.5pt}\\subseteq S$ , a partition $\\mathcal {A}\\in \\hspace{0.5pt}\\mathbb {P}(S)$ , with $\\mathcal {A}= \\lbrace A_{1}, \\dots , A_{m} \\rbrace $ say, defines a unique partition of $U$ by restriction.",
"The latter is denoted by $\\mathcal {A}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}$ , and its parts are precisely all non-empty sets of the form $A_{i} \\cap U$ with $1\\leqslant i \\leqslant m$ .",
"Fix now $S=\\lbrace 1,2,\\dots ,n\\rbrace $ and define $X := X_{1} \\times \\dots \\times X_{n}$ , where each $X_{i}$ is a locally compact space (which we mean to include the Hausdorff property).",
"The natural projection of $X$ to its $i$ th component is denoted by $\\pi _{i}$ , so $\\pi _{i} (X)= X_{i}$ .",
"For an arbitrary non-empty subset $U\\hspace{-0.5pt}\\subseteq S$ , we use the notation $\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} \\!",
": \\; X \\longrightarrow X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}:=\\mbox{\\LARGE $\\times $}_{i\\in U} X_{i}$ for the projection to $X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}$ .",
"Let $\\mathcal {M}(X)$ denote the space of signed, finite and regular Borel measures on $X$ , equipped with the usual total variation norm $\\Vert .",
"\\Vert $ , which makes it into a Banach space.",
"Also, we need the closed subset (or cone) $\\mathcal {M}_{+} (X)$ of positive measures, which includes the zero measure.",
"Within $\\mathcal {M}_{+} (X)$ , we denote the closed subset of probability measures by $\\mathcal {P}(X)$ .",
"Note that $\\mathcal {M}_{+} (X)$ and $\\mathcal {P}(X)$ are convex sets.",
"The restriction of a measure $\\mu \\in \\mathcal {M}(X)$ to a subspace $X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}$ is written as $\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\mu :=\\mu \\circ \\pi ^{-1}_{U}$ , which is consistent with marginalisation of measures.",
"When the context is clear, we will use the abbreviation $\\mu ^{U}\\hspace{-0.5pt}:= \\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}.\\mu $ .",
"For any Borel set $A\\subseteq X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}$ , one thus has the relation $\\mu ^{U}\\!",
"(A) =\\bigl (\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\mu \\bigr ) (A) = \\mu \\bigl ( \\pi ^{-1}_{U}(A)\\bigr )$ .",
"Given a measure $\\mu \\in \\mathcal {M}(X)$ and a partition $\\mathcal {A}= \\lbrace A_{1},\\dots ,A_{m} \\rbrace \\in \\mathbb {P}(S)$ , we define the mapping $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} \\!",
": \\, \\mathcal {M}(X)\\longrightarrow \\mathcal {M}(X)$ by $\\mu \\mapsto R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}}(\\mu )$ with $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} (0) := 0$ and, for $\\mu \\ne 0$ , $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} (\\mu ) \\, := \\, \\frac{1}{\\Vert \\mu \\Vert ^{m-1}}\\bigotimes _{i=1}^{m} \\bigl (\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"A_{i}} .",
"\\hspace{0.5pt}\\mu \\bigr )\\, = \\, \\frac{\\hspace{0.5pt}\\mu ^{A^{}_{1}} \\otimes \\dots \\otimes \\mu ^{A^{}_{m}}}{ \\Vert \\mu \\Vert ^{m-1}} \\hspace{0.5pt}.$ Note that the product is (implicitly) `site ordered', which means that it matches the ordering of the sites as specified by the set $S$ .",
"We shall use (implicit) site ordering also for product sets.",
"We call a mapping of type $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!\\mathcal {A}}$ a recombinator.",
"Note that recombinators are nonlinear whenever $\\mathcal {A}\\ne \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}$ .",
"Let us recall some results from [4] as follows.",
"Proposition 1 Let $S=\\lbrace 1,2,\\dots ,n \\rbrace $ and $X=X_{1} \\times \\dots \\times X_{n}$ as above.",
"Now, let $\\mathcal {A}\\in \\mathbb {P}(S)$ be arbitrary, and consider the corresponding recombinator $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}}$ as defined by Eq.",
"(REF ).",
"Then, the following assertions are true.",
"$R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}$ is positive homogeneous of degree 1, which means that $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}} (a \\mu ) = a R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}(\\mu )$ holds for all $\\mu \\in \\mathcal {M}(X)$ and all $a\\geqslant 0$ .",
"$R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}\\vphantom{I}$ is globally Lipschitz on $\\mathcal {M}(X)$ , with Lipschitz constant $L\\leqslant 2\\hspace{0.5pt}\\vert \\mathcal {A}\\vert + 1$ .",
"On $\\mathcal {P}(X)$ , the recombinator $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}$ is Lipschitz with $L\\leqslant \\vert \\mathcal {A}\\vert $ .",
"$\\Vert R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}} (\\mu )\\Vert \\leqslant \\Vert \\mu \\Vert \\vphantom{I}$ holds for all $\\mu \\in \\mathcal {M}(X)$ .",
"$R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}\\vphantom{I}$ maps $\\mathcal {M}_{+} (X)$ into itself.",
"$R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\!",
"\\mathcal {A}}\\vphantom{I}$ preserves the norm of positive measures, and hence also maps $\\mathcal {P}(X)$ into itself.",
"On $\\mathcal {M}_{+} (X)$ , the recombinators satisfy $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} = R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\wedge \\mathcal {B}} $ .",
"In particular, each recombinator is an idempotent and any two recombinators commute.",
"$\\Box $ Several of these properties will be used below without further mentioning, some in results that we simply recall from previous work.",
"Let us mention (without proof) that the Lipschitz constant in claim 2 can be improved to $L \\leqslant 2\\hspace{0.5pt}\\vert \\mathcal {A}\\vert - 1$ .",
"We are now set to define and analyse the recombination ODE."
],
[
"The general recombination equation and\nmarginalisation",
"The general recombination equation in continuous time is formulated within the Banach space $(\\mathcal {M}(X), \\Vert .\\Vert )$ , as the nonlinear ODE $\\dot{\\omega }^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\!\\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} \\!",
"\\!",
"\\varrho (\\mathcal {A}) \\hspace{0.5pt}\\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} - \\mathbb {1}\\bigr ) (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ with non-negative numbers $\\varrho (\\mathcal {A})$ that have the meaning of recombination rates in our context.",
"We will usually assume that an initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} \\in \\mathcal {M}(X)$ for $t=0$ is given for the ODE (REF ), and then speak of the corresponding Cauchy problem (or initial value problem).",
"With $:= \\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} \\varrho (\\mathcal {A}) \\hspace{0.5pt}\\bigl (R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} - \\mathbb {1}\\bigr )$ , we can now simply write $\\dot{\\omega }^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\hspace{0.5pt},$ but we must keep in mind that $$ is a nonlinear operator.",
"Nevertheless, one has the following basic result [6], [4]; see [2] for general background on ODEs on Banach spaces.",
"Proposition 2 Let $S$ be a finite set and $X$ the corresponding locally compact product space as introduced above.",
"Then, the Cauchy problem of Eq.",
"(REF ) with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\in \\mathcal {M}(X)$ has a unique solution.",
"Moreover, the cone $\\mathcal {M}_{+}(X)$ is forward invariant, and the flow is norm-preserving on $\\mathcal {M}_{+} (X)$ .",
"In particular, $ \\mathcal {P}(X)$ is forward invariant under the flow.",
"$\\Box $ Without loss of generality, when we start with a positive measure, we may thus restrict our attention to the investigation of the recombination equation on the cone $\\mathcal {M}_{+} (X)$ , and on $\\mathcal {P}(X)$ in particular.",
"So, let us assume that we consider the Cauchy problem with $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} \\in \\mathcal {P}(X)$ .",
"The way to a solution of the recombination equation in previous papers started with the ansatz $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} \\!",
"a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A})\\, R^{}_{\\hspace{-0.5pt}\\mathcal {A}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0})$ which effectively means a complete separation of the time evolution and the recombination of the initial condition.",
"This led to a nonlinear ODE system for the coefficient functions $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A})$ that could be solved recursively in [4], for generic recombination rates.",
"Via the identification of an underlying Markov partitioning process in [4], the further analysis of [7] showed that these coefficient functions also solve a linear ODE system with constant coefficient matrix, $Q$ say, which makes the entire solvability understandable in retrospect.",
"As mentioned in the Introduction, the degenerate cases then correspond to $Q$ not being diagonalisable.",
"In this approach, which meant a significant progress and simplification in comparison to earlier attempts [15], [11], [12] while being more general at the same time, the number of steps were still formidable, and another simplification was suggestive.",
"This is precisely what we want to describe now.",
"Here, the golden key emerges from also considering the time evolution of $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}}(\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ for an arbitrary $\\mathcal {B}\\in \\mathbb {P}(S)$ , where $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ is a solution of the recombination equation (REF ).",
"In view of Propositions REF and REF , it suffices to look at probability measures, so that we get $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t}\\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\, = \\,\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t} \\left( \\omega ^{B_{1}}_{t}\\otimes \\cdots \\otimes \\omega ^{B_{\\vert \\mathcal {B}\\vert }}_{t} \\right)\\, = \\, \\sum _{i=1}^{\\vert \\mathcal {B}\\vert }\\hspace{0.5pt}\\Bigl ((\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}B_{i}} \\hspace{-0.5pt}.",
"\\hspace{0.5pt}\\dot{\\omega }^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\otimes \\bigotimes _{j\\ne i} \\omega ^{B_{j}}_{t} \\Bigr ) .$ To proceed, it will be instrumental to understand the behaviour of recombination on subsystems defined by a set $\\varnothing \\ne U \\hspace{-0.5pt}\\subseteq S$ , where we begin by recalling [4].",
"Note that this result effectively underlies assertion (7) of Proposition REF .",
"Lemma 1 Let $S$ be a finite set as above, and $\\mathcal {A}= \\lbrace A_{1}, \\dots ,A_{\\vert \\mathcal {A}\\vert } \\rbrace \\in \\mathbb {P}(S)$ an arbitrary partition.",
"If $U\\hspace{-0.5pt}\\subseteq S$ is non-empty and $\\omega \\in \\mathcal {M}_{+} (X)$ , one has $\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\bigl ( R^{\\hspace{0.5pt}S}_{\\hspace{-0.5pt}\\mathcal {A}} (\\omega ) \\bigr ) \\, = \\,R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}} (\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\hspace{0.5pt}\\omega ) \\hspace{0.5pt},$ where $\\mathcal {A}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} \\in \\mathbb {P}(U)$ and the upper index of a recombinator indicates on which measure space it acts, with $R^{S}_{\\hspace{-0.5pt}\\mathcal {A}} = R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}}$ .",
"$\\Box $ To continue, it is clear that we will need the recombination rates on subsystems defined by some $\\varnothing \\ne U \\subseteq S$ , as induced by the marginalisation $\\varrho ^{U} \\!",
"(\\mathcal {A}) \\, =\\sum _{\\begin{array}{c}\\mathcal {B}\\in \\mathbb {P}(S) \\\\ \\mathcal {B}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} = \\hspace{0.5pt}\\mathcal {A}\\end{array}}\\!",
"\\varrho ^{S} (\\mathcal {B}) \\hspace{0.5pt},$ where $\\varrho ^{S} (\\mathcal {B}) = \\varrho (\\mathcal {B})$ .",
"Now, Lemma REF implies the following marginalisation consistency on the level of probability measures, where we use the notation $\\omega ^{U}_{t} = \\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} \\hspace{-0.5pt}.",
"\\hspace{0.5pt}\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ as introduced earlier.",
"The version we state here is a special case of [4]; see also Lemma REF below.",
"Proposition 3 Let $\\varnothing \\ne U \\subseteq S$ .",
"If $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ is a solution of the Cauchy problem of Eq.",
"(REF ) with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} \\in \\mathcal {P}(X)$ , the marginal measures $(\\omega ^{U}_{t})^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t\\geqslant 0}$ on $X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}$ solve the ODE $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t} \\, \\omega ^{U}_{t} \\,= \\!",
"\\sum _{\\mathcal {A}\\in \\mathbb {P}(U)}\\!\\varrho ^{U} \\!",
"(\\mathcal {A}) \\, \\bigl ( R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} - \\mathbb {1}\\bigr )(\\omega ^{U}_{t})$ with initial condition $\\omega ^{U}_{0} = \\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U}\\hspace{-0.5pt}.",
"\\hspace{0.5pt}\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}$ and marginalised rates $\\varrho ^{U} \\!",
"( \\mathcal {A})$ according to Eq.",
"(REF ).",
"In particular, one has $\\omega ^{U}_{t}\\!",
"\\in \\mathcal {P}(X^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U})$ for all $t\\geqslant 0$ .",
"$\\Box $ In this context, it is helpful to also note a factorisation property of the recombinators on $\\mathcal {M}_{+} (X)$ .",
"Lemma 2 Let $\\lbrace U , V \\rbrace $ be a partition of $S$ , and assume that two partitions $\\mathcal {A}\\in \\mathbb {P}(U)$ and $\\mathcal {B}\\in \\mathbb {P}(V)$ are given.",
"Then, for any $0 \\ne \\mu \\in \\mathcal {M}_{+} (X)$ , one has $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\sqcup \\mathcal {B}} \\hspace{0.5pt}(\\mu ) \\; = \\;\\frac{1}{\\Vert \\mu \\Vert } \\,R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} (\\mu ^{U}) \\otimes R^{V}_{\\mathcal {B}} (\\mu ^{\\hspace{-0.5pt}V}) \\hspace{0.5pt},$ which simplifies to $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\sqcup \\mathcal {B}} \\hspace{0.5pt}(\\mu ) =R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} (\\mu ^{U}) \\otimes R^{V}_{\\mathcal {B}} (\\mu ^{\\hspace{-0.5pt}V})$ for $\\mu \\in \\mathcal {P}(X)$ .",
"Observe first that $\\lbrace U,V\\rbrace \\wedge (\\mathcal {A}\\sqcup \\mathcal {B}) = \\mathcal {A}\\sqcup \\mathcal {B}$ due to our assumptions.",
"By assertions (1) and (7) from Proposition REF , we then know that $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\sqcup \\mathcal {B}} \\hspace{0.5pt}(\\mu ) \\, = \\, \\bigl (R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\sqcup \\mathcal {B}} \\, R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\lbrace U,V\\rbrace } \\bigr )(\\mu ) \\, = \\, R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}\\sqcup \\mathcal {B}} \\bigl (\\tfrac{1}{\\Vert \\mu \\Vert } \\, \\mu ^{U} \\!",
"\\otimes \\mu ^{V} \\bigr )\\, = \\, \\frac{1}{\\Vert \\mu \\Vert }R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} (\\mu ^{U}) \\otimes R^{V}_{\\mathcal {B}}(\\mu ^{\\hspace{-0.5pt}V}) \\hspace{0.5pt},$ from which the second claim is immediate.",
"We are now set to proceed with solving Eq.",
"(REF )."
],
[
"Solution of the recombination\nequation in continuous time",
"Let us consider the time evolution of $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}}(\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ for an arbitrary partition $\\mathcal {B}\\in \\mathbb {P}(S)$ , here written as $\\mathcal {B}= \\lbrace B^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{1}, \\ldots , B^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{m}\\rbrace $ .",
"As before, we assume $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ to be a solution of the recombination equation (REF ) with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\in \\mathcal {P}(X)$ .",
"Using the product rule as above, and employing Proposition REF , we obtain $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t} \\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} ( \\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} )& = & \\sum _{i=1}^{m} \\Bigl ( \\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t} \\,\\omega ^{B_{i}}_{t} \\Bigr ) \\otimes \\bigotimes _{j\\ne i} \\omega ^{B_{j}}_{t} \\nonumber \\\\[1mm]& = & \\sum _{i=1}^{m} \\, \\sum _{\\mathcal {A}_{i} \\in \\mathbb {P}(B_{i})} \\!\\varrho ^{B_{i}} (\\mathcal {A}_{i}) \\, \\bigl ( R^{B_{i}}_{\\!",
"\\mathcal {A}_{i}}- \\mathbb {1}\\bigr ) (\\omega ^{B_{i}}_{t}) \\otimes \\bigotimes _{j\\ne i} \\omega ^{B_{j}}_{t} \\nonumber \\\\[1mm]& = & \\sum _{i=1}^{m} \\, \\sum _{\\mathcal {A}_{i} \\in \\mathbb {P}(B_i)} \\!\\varrho ^{B_{i}} (\\mathcal {A}_{i})\\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{(\\mathcal {B}\\setminus B_i) \\sqcup \\hspace{0.5pt}\\mathcal {A}_i}- R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} \\bigr ) (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\nonumber \\\\[1mm]& = & \\sum _{i=1}^{m} \\,\\sum _{\\begin{array}{c}\\mathcal {A}_{i} \\in \\mathbb {P}(B_i) \\\\ \\mathcal {A}_{i} \\ne \\lbrace B_{i}\\rbrace \\end{array} }\\!",
"\\varrho ^{B_{i}} (\\mathcal {A}_{i})\\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{(\\mathcal {B}\\setminus B_i) \\sqcup \\hspace{0.5pt}\\mathcal {A}_i}- R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} \\bigr ) (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\hspace{0.5pt},$ where $\\mathcal {B}\\setminus B_{i}$ denotes the partition of $S \\setminus B_{i}$ that emerges from $\\mathcal {B}$ by removing $B_{i}$ .",
"Note that the crucial third step follows from Lemma REF used backwards.",
"Clearly, we may restrict the inner summation to $\\mathcal {A}_{i}\\ne \\lbrace B_{i}\\rbrace $ , as the then omitted term vanishes anyhow, which gives the last line.",
"We can now state our main result as follows.",
"Theorem 1 Let $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ be a solution of the recombination equation (REF ), with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} \\in \\mathcal {P}(X)$ .",
"Then, for any partition $\\mathcal {B}\\in \\mathbb {P}(S)$ , the measure $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ satisfies an ODE of the linear form $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t}\\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} ( \\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} )\\, = \\sum _{\\mathcal {C}\\in \\mathbb {P}(S)} Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}\\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\hspace{0.5pt},$ where the coefficients are explicitly given by $Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} := {\\left\\lbrace \\begin{array}{ll}\\varrho ^{B_i} (\\mathcal {A}^{}_i), &\\text{if } \\hspace{0.5pt}\\mathcal {C}= (\\mathcal {B}\\setminus B^{}_i) \\sqcup \\mathcal {A}_{i}\\text{ for some } \\lbrace B_{i} \\rbrace \\ne \\mathcal {A}_i \\in \\mathbb {P}(B_i) \\\\ &\\text{and precisely one index } 1 \\leqslant i \\leqslant \\vert \\mathcal {B}\\vert \\hspace{0.5pt}, \\\\- \\sum \\limits _{i=1}^{\\vert \\mathcal {B}\\vert }\\sum \\limits _{\\begin{array}{c}\\mathcal {A}_{i} \\ne \\lbrace B_{i} \\rbrace \\\\ \\mathcal {A}_{i} \\in \\mathbb {P}(B_{i})\\end{array} }\\!",
"\\!",
"\\varrho ^{B_{i}} (\\mathcal {A}_{i}), & \\text{if } \\mathcal {C}= \\mathcal {B}\\hspace{0.5pt}, \\\\0, & \\text{otherwise}\\hspace{0.5pt}.",
"\\end{array}\\right.",
"}$ In particular, the matrix $Q = \\bigl ( Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}\\bigr )_{\\mathcal {B},\\mathcal {C}\\in \\mathbb {P}(S)}$ is a Markov generator with triangular structure, and $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ is a probability measure for all $t \\geqslant 0$ .",
"It is clear from Eq.",
"(REF ) that the derivative of $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ can indeed be written as a linear combination, namely $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t}\\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} ( \\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\,= \\sum _{\\mathcal {C}\\prec \\mathcal {B}} Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}\\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} - R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} \\bigr )(\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\,= \\sum _{\\mathcal {C}\\preccurlyeq \\mathcal {B}} Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\,= \\!",
"\\sum _{\\mathcal {C}\\in \\mathbb {P}(S)} \\!",
"\\!",
"Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\hspace{0.5pt},$ where the second step follows from a simple change of summation, while the third just reflects the fact that $Q$ has a triangular structure.",
"It remains to show that the coefficients $Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}$ are those given by Eq.",
"(REF ).",
"If $\\mathcal {C}\\prec \\mathcal {B}$ , Eq.",
"(REF ) tells us that this coefficient must vanish unless $\\mathcal {C}$ refines precisely one part of $\\mathcal {B}$ , in which case its value is as claimed, and non-negative due to our general assumption on the recombination coefficients.",
"When $\\mathcal {C}=\\mathcal {B}$ , we read from our change of summation that all non-diagonal coefficients of the row of $Q$ defined by $\\mathcal {B}$ must occur on the diagonal once, with negative sign.",
"All other coefficients clearly vanish.",
"The Markov generator property is then clear, and the last claim follows from our general properties of the recombinators in conjunction with Proposition REF .",
"The meaning of the Markov generator $Q$ will become clear in the next section.",
"Let us now define the (column) vector $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_t :=\\bigl ( \\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {B}) \\bigr )_{\\mathcal {B}\\in \\mathbb {P}(S)}$ with $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {B}) := R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ .",
"With this abbreviation, the ODEs from Theorem REF now turn into the linear ODE system $\\frac{\\,\\mathrm {d}}{\\,\\mathrm {d}t} \\,\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, Q \\hspace{0.5pt}\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ with initial condition $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}= \\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}}(\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}) \\bigr )_{\\mathcal {B}\\in \\mathbb {P}(S)}$ and solution $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, \\hspace{0.5pt}\\mathrm {e}^{t \\hspace{0.5pt}Q} \\hspace{0.5pt}\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} \\hspace{0.5pt}.$ Note that $\\lbrace \\hspace{0.5pt}\\mathrm {e}^{t\\hspace{0.5pt}Q} \\mid t \\geqslant 0 \\rbrace $ is the Markov semigroup generated by $Q$ .",
"In particular, for the first component of $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ , we now get $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, \\varphi ^{}_t (\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}) \\, =\\!",
"\\sum _{\\mathcal {A}\\in \\hspace{0.5pt}\\mathbb {P}(S)}\\!",
"\\!",
"a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}(\\mathcal {A}) \\, R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {A}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}),$ with $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) = (\\hspace{0.5pt}\\mathrm {e}^{t \\hspace{0.5pt}Q})^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}}$ , which leads us back to Eq.",
"(REF ).",
"Clearly, $\\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) =1$ since each $\\hspace{0.5pt}\\mathrm {e}^{t \\hspace{0.5pt}Q}$ with $t\\geqslant 0$ is a Markov matrix, so all row sums are 1.",
"Via the Kolmogorov forward equation for the Markov semigroup $\\lbrace e^{tQ} \\mid t \\geqslant 0 \\rbrace $ , compare [19], the following consequence is now immediate; see [4], [7] for the original (but much longer) derivation.",
"Corollary 1 Under the assumptions of Theorem $\\ref {thm:main}$ , the coefficient functions of Eq.",
"(REF ) are $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) =( e^{t Q})^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}}$ , with the Markov generator $Q$ as in Eq.",
"(REF ), and satisfy the ODEs $\\dot{a}^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) \\, = \\sum _{\\mathcal {B}\\succcurlyeq \\mathcal {A}}a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {B}) \\, Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {A}}$ with initial conditions $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} (\\mathcal {A}) =\\delta ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}}$ .",
"In particular, $\\bigl (a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}(\\mathcal {A})\\bigr )_{\\mathcal {A}\\in \\mathbb {P}(S)}$ is a probability vector for all $t\\geqslant 0$ .",
"$\\Box $ Let us pause to discuss what Eq.",
"(REF ) tells us.",
"First, the solution of the recombination equation may be expressed in terms of a convex combination of the initial measure recombined in all possible ways.",
"This is quite plausible, given that the differential equation means a continuous replacement of the current measure by its recombined versions.",
"Second, the procedure just described has uncovered a linear structure that underlies the nonlinear recombination equation, and thus reduced the problem to a linear one.",
"With hindsight, we recognise a streamlined version of Haldane linearisation: The $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ with $\\mathcal {B}\\ne \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}$ can be viewed as additional components that are used to enlarge the system in order to unravel its intrinsic linear structure.",
"The latter is conveyed by the Markov generator $Q$ , whose meaning still remains to be elucidated, as will be done next."
],
[
"The backward point of view: Partitioning\nprocess",
"Now that we have understood the structure of the ODE system for the coefficients $a^{}_t$ in the usual (forward) direction of time, let us consider a related (stochastic) process that will provide an additional meaning for $a^{}_t$ .",
"Let $\\lbrace ^{}_t\\rbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t\\geqslant 0}$ be a Markov chain in continuous time with values in $\\mathbb {P}(S)$ that is constructed as follows.",
"Start with $^{}_0= \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}$ .",
"If the current state is $^{}_t=\\mathcal {B}$ , then part $B_{i}$ of $\\mathcal {B}$ , with $1 \\leqslant i\\leqslant |\\mathcal {B}|$ , is replaced by $\\lbrace B_{i} \\rbrace \\ne \\mathcal {A}_i \\in \\mathbb {P}(B_i)$ at rate $\\varrho ^{B_i} (\\mathcal {A}_i)$ , independently of all other parts.",
"That is, the transition from $\\mathcal {B}$ to $(\\mathcal {B}\\setminus B_{i})\\sqcup \\mathcal {A}_{i}$ occurs at rate $\\varrho ^{B_{i}} (\\mathcal {A}_{i})$ for all $\\lbrace B_{i} \\rbrace \\ne \\mathcal {A}_{i} \\in \\mathbb {P}(B_{i}) $ and $1 \\leqslant i \\leqslant |\\mathcal {B}|$ .",
"Put differently, the transition from $\\mathcal {B}$ to $\\mathcal {C}$ happens at rate $Q^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}$ of Eq.",
"(REF ).",
"This way, we have given a meaning to the generator $Q$ of Section : It holds the transition rates of the process of progressive refinements $\\lbrace _{t} \\rbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t \\geqslant 0}$ , which we have just described, and which we call the underlying partitioning process.",
"The argument is illustrated in Figure REF .",
"Since $Q$ is the Markov generator of $\\lbrace _t\\rbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t\\geqslant 0}$ , we can further conclude that $(\\hspace{0.5pt}\\mathrm {e}^{t \\hspace{0.5pt}Q})^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\, =\\, \\mathbf {P} \\bigl ( _{t} =\\mathcal {C}\\mid _0 = \\mathcal {B}\\bigr )$ (where $\\mathbf {P}$ denotes probability), that is, the transition probability from `state' $\\mathcal {B}$ to `state' $\\mathcal {C}$ during a time interval of length $t$ .",
"In particular, $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) = \\bigl (\\hspace{0.5pt}\\mathrm {e}^{t \\hspace{0.5pt}Q} \\bigr )_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}} = \\mathbf {P} \\bigl ( ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} = \\mathcal {A}\\mid ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} = \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\bigr )$ .",
"We have therefore shown our second main result, which can now be stated as follows.",
"Theorem 2 The probability vector $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ from Eqs.",
"(REF ) and (REF ) agrees with the distribution of the partitioning process $\\lbrace _{t} \\rbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t \\geqslant 0}$ .",
"Explicitly, we have $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) \\, = \\,\\mathbf {P} \\bigl (^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} = \\mathcal {A}\\mid ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} = \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\bigr )$ for any $\\mathcal {A}\\in \\mathbb {P}(S)$ and all $t \\geqslant 0$ .",
"$\\Box $ Figure: Illustrative realisation of the partitioningprocess."
],
[
"Recombination equation in discrete\ntime",
"Let us turn our attention to the discrete-time analogue of Eq.",
"(REF ), which is often considered in population genetics [9], [12], [15], [16].",
"We can use the same general setting of Section , in particular the space $\\mathcal {M}(X)$ of measures and the action of the recombinators on it.",
"Then, one has to consider the nonlinear iteration $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t+1} \\, = \\!",
"\\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} \\!r (\\mathcal {A}) \\, R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {A}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\hspace{0.5pt},$ where the parameters are now recombination probabilities, so $r(\\mathcal {A}) \\geqslant 0$ for $\\mathcal {A}\\in \\mathbb {P}(S)$ together with $\\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} r (\\mathcal {A}) = 1$ .",
"Moreover, $t\\in \\mathbb {N}_{0}$ denotes discrete time (counting generations, say) with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}$ .",
"Clearly, the positive cone $\\mathcal {M}_{+} (X)$ is preserved under the iteration, as is the norm of a positive measure.",
"In view of the general properties of the recombinators from Proposition REF , we can further confine our discussion to $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\in \\mathcal {P}(X)$ , which immediately implies that $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\in \\mathcal {P}(X)$ for all $t\\in \\mathbb {N}$ as well.",
"As above in Proposition REF , we need marginalisation consistency for subsystems.",
"A generalisation of [10] to our setting leads to the following result.",
"Lemma 3 Let $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ with $t\\in \\mathbb {N}_{0}$ be a solution of the discrete recombination equation (REF ), with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\in \\mathcal {P}(X)$ .",
"Then, for any $\\varnothing \\ne \\hspace{-0.5pt}U \\subseteq S$ , the marginal measures $\\omega ^{U}_{t} \\!",
"$ satisfy the induced recombination equation $\\omega ^{\\hspace{0.5pt}U}_{t+1} \\, = \\!",
"\\sum _{\\mathcal {A}\\in \\mathbb {P}(U)} \\!\\!r^{U} \\!",
"(\\mathcal {A}) \\, R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} (\\omega ^{U}_{t}) \\hspace{0.5pt},$ for all $t\\in \\mathbb {N}_{0}$ , with $R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}}$ as before.",
"Here, $r^{U}\\!",
"(\\mathcal {A}) \\; := \\!",
"\\sum _{\\begin{array}{c}\\mathcal {B}\\in \\mathbb {P}(S) \\\\\\mathcal {B}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} = \\mathcal {A}\\end{array}} \\!",
"\\!",
"r (\\mathcal {B}) \\, \\geqslant \\, 0$ are the induced recombination probabilities for the subsystem, with $\\sum _{\\mathcal {A}\\in \\mathbb {P}(U)} r^{U}\\!",
"(\\mathcal {A}) = 1$ .",
"The claim follows from a simple calculation, $\\begin{split}\\omega ^{U}_{t+1} \\; & = \\; \\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\hspace{0.5pt}\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t+1}\\; = \\; \\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} \\hspace{0.5pt}.",
"\\!",
"\\sum _{\\mathcal {B}\\in \\mathbb {P}(S)} \\!",
"r (\\mathcal {B}) \\,R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\; = \\sum _{\\mathcal {B}\\in \\mathbb {P}(S)} \\!",
"r(\\mathcal {B}) \\,\\bigl (\\pi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{U} .",
"\\hspace{0.5pt}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\bigr ) \\\\[1mm]& = \\sum _{\\mathcal {B}\\in \\mathbb {P}(S)} \\!",
"r(\\mathcal {B}) \\, R^{\\hspace{0.5pt}U}_{\\mathcal {B}|^{}_{U}}\\hspace{-0.5pt}(\\omega ^{U}_{t}) \\; = \\!",
"\\sum _{\\mathcal {A}\\in \\mathbb {P}(U)} \\,\\sum _{\\begin{array}{c}\\mathcal {B}\\in \\mathbb {P}(S) \\\\ \\mathcal {B}|^{}_{U} = \\mathcal {A}\\end{array}}\\!",
"r(\\mathcal {B}) \\, R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}} (\\omega ^{U}_{t}) \\; = \\!\\sum _{\\mathcal {A}\\in \\mathbb {P}(U)} \\!",
"r^{U} \\!",
"(\\mathcal {A}) \\, R^{\\hspace{0.5pt}U}_{\\hspace{-0.5pt}\\mathcal {A}}(\\omega ^{U}_{t}) \\hspace{0.5pt},\\end{split}$ where the first step in the second line is a consequence of Lemma REF .",
"Let now $\\mathcal {B}= \\lbrace B^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{1}, \\ldots , B^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{m}\\rbrace $ be an arbitrary partition of $S$ , and consider $\\begin{split}R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t+1}) \\, & = \\,\\omega ^{B_{1}}_{t+1} \\otimes \\dots \\otimes \\omega ^{B_{m}}_{t+1} \\, = \\; \\bigotimes _{i=1}^{m} \\,\\sum _{\\mathcal {A}_{i}\\in \\mathbb {P}(B_{i})} \\!\\!",
"r^{B_{i}} (\\mathcal {A}_{i}) \\,R^{B_{i}}_{\\hspace{-0.5pt}\\mathcal {A}_{i}} (\\omega ^{B_{i}}_{t}) \\\\[1mm]& = \\sum _{\\mathcal {A}_{1} \\in \\mathbb {P}(B_{1})} \\!",
"\\cdots \\!\\!\\sum _{\\mathcal {A}_{m}\\in \\mathbb {P}(B_{m})} \\biggl (\\,\\prod _{i=1}^{m} r^{B_{i}} (\\hspace{-0.5pt}\\mathcal {A}_{i}) \\biggr )\\, R^{B_{1}}_{\\hspace{-0.5pt}\\mathcal {A}_{1}} (\\omega ^{B_{1}}_{t}) \\otimes \\dots \\otimes R^{B_{m}}_{\\hspace{-0.5pt}\\mathcal {A}_{m}} (\\omega ^{B_{m}}_{t}) \\hspace{0.5pt},\\end{split}$ where we have used Lemma REF .",
"Invoking the factorisation property from Lemma REF , we thus see that we can rewrite the last expression as a linear combination of terms of the form $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ with $\\mathcal {C}\\in \\mathbb {P}(S)$ being a refinement of $\\mathcal {A}_{1} \\sqcup \\dots \\sqcup \\mathcal {A}_{m}$ .",
"We can now formulate the following general result, which also resembles some recent findings from [16].",
"Theorem 3 Let $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ with $t\\in \\mathbb {N}_{0}$ be a solution of the discrete recombination equation (REF ), with initial condition $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\in \\mathcal {P}(X)$ .",
"Then, for any $\\mathcal {B}\\in \\mathbb {P}(S)$ and any $t\\in \\mathbb {N}_{0}$ , the measure $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ is a probability measure and satisfies the linear recursion $R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t+1}) \\; = \\,\\sum _{\\mathcal {C}\\preccurlyeq \\mathcal {B}} M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\,R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}) \\; =\\sum _{\\mathcal {C}\\in \\mathbb {P}(S)} \\!",
"M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\,R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {C}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})$ with the coefficients $M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\, = \\, {\\left\\lbrace \\begin{array}{ll}\\prod _{i=1}^{\\vert \\mathcal {B}\\vert } r^{B_{i}} (\\mathcal {C}|^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{B_{i}}) ,& \\text{if } \\mathcal {C}\\preccurlyeq \\mathcal {B}, \\\\0 , & \\text{otherwise}.",
"\\end{array}\\right.",
"}$ In particular, $M = \\bigl (M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}\\bigr )_{\\mathcal {B},\\mathcal {C}\\in \\mathbb {P}(S)}$ is a triangular Markov matrix.",
"The first claim, as mentioned earlier, is a direct consequence of part (6) of Proposition REF .",
"Our above calculation, in conjunction with Lemma REF , proves the second claim, where the determination of the coefficients $M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}}$ is a straight-forward exercise.",
"Clearly, we have $M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\geqslant 0$ for all $\\mathcal {B},\\mathcal {C}\\in \\mathbb {P}(S)$ because $r (\\mathcal {A}) \\geqslant 0$ by assumption, hence also $r^{U} \\!",
"(\\mathcal {B}) \\geqslant 0$ for all $\\varnothing \\ne \\hspace{-0.5pt}U\\subseteq S$ by the formula in Lemma REF .",
"It remains to show that each row of $M$ sums to 1.",
"Indeed, given any $\\mathcal {B}\\in \\mathbb {P}(S)$ , one has $\\sum _{\\mathcal {C}\\in \\mathbb {P}(S)} \\!",
"M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\, = \\sum _{\\mathcal {C}\\preccurlyeq \\mathcal {B}}M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {C}} \\, = \\, \\prod _{i=1}^{\\vert \\mathcal {B}\\vert } \\,\\sum _{\\mathcal {A}_{i}\\in \\mathbb {P}(B_{i})} \\!\\!",
"r^{B_{i}} (\\mathcal {A}_{i}) \\, = \\, 1$ because $\\bigl ( r^{U} \\!",
"(\\mathcal {A}) \\bigr )_{\\mathcal {A}\\in \\mathbb {P}(U)}$ is a probability vector for all $\\varnothing \\ne \\hspace{-0.5pt}U \\subseteq S$ by Lemma REF .",
"Note that the matrix entry $M^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {B}\\mathcal {B}}$ is the probability that `nothing happens' to partition $\\mathcal {B}$ in one step.",
"These diagonal entries are, due to the triangular structure, the eigenvalues of $M$ .",
"For a special case of our setting, they have been determined earlier, by rather different methods and without reference to a linear structure, in [15]; see also [18], [21].",
"Theorem REF now gives them a clearer meaning in a more general setting.",
"Also, the degenerate cases that had to be excluded in previous attempts [15], [12] precisely correspond to the cases where $M$ fails to be diagonalisable.",
"They pose no problem in the above approach.",
"At this point, we can repeat our previous interpretation.",
"If $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} = \\bigl ( R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{-0.5pt}\\mathcal {A}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t})\\bigr )_{\\mathcal {A}\\in \\mathbb {P}(S)}$ is considered as a column vector, with $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ a solution of the recombination equation, we get $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t+1} = M \\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t}$ , and hence $\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, M^{t} \\hspace{0.5pt}\\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}$ for all $t \\in \\mathbb {N}_{0}$ .",
"The first component of the vector is $\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\, = \\, \\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt})\\, = \\, \\bigl ( M^{t} \\varphi ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}\\bigr ) (\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt})\\, = \\!",
"\\sum _{\\mathcal {A}\\in \\mathbb {P}(S)} \\!",
"a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) \\,R^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\mathcal {A}} (\\omega ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0}) \\hspace{0.5pt},$ with $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} (\\mathcal {A}) = (M^{t})^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}}$ .",
"In particular, $a^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} (\\mathcal {A}) = \\delta ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{\\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}\\mathcal {A}}$ .",
"There is again an underlying stochastic process, in analogy to the continuous-time case in Section .",
"Here, $\\lbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} \\rbrace ^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t \\in \\mathbb {N}_{0}}$ is a Markov chain in discrete time with values in $\\mathbb {P}(S)$ , starting at $^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{0} = \\hspace{0.5pt}\\hspace{0.5pt}\\underline{\\hspace{-0.5pt}\\hspace{-0.5pt}1\\hspace{-0.5pt}\\hspace{-0.5pt}}\\hspace{0.5pt}\\hspace{0.5pt}$ .",
"When $^{\\hphantom{g}\\hspace{-0.5pt}\\hspace{-0.5pt}}_{t} = \\mathcal {B}$ , in the time step from $t$ to $t+1$ , part $B_{i}$ of $\\mathcal {B}$ is replaced by $\\mathcal {A}_{i} \\in \\mathbb {P}(B_{i})$ with probability $r^{B_{i}}_{\\mathcal {A}_{i}}$ , independently for each $1 \\leqslant i \\leqslant \\vert \\mathcal {B}\\vert $ .",
"Note that, unlike in the continuous-time case, several parts can be refined in one step, which makes the discrete-time case actually more complicated.",
"Of course, $\\mathcal {A}_{i} = \\lbrace B_{i}\\rbrace $ , which means no action on this part, is also possible.",
"Put together, it is not difficult to verify that one ends up precisely with the transition matrix $M$ from Eq.",
"(REF )."
],
[
"Acknowledgements",
"This work was supported by the German Research Foundation (DFG), within the SPP 1590."
]
] | 1606.05175 |
[
[
"Exterior Navier-Stokes flows for bounded data"
],
[
"Abstract We prove unique existence of mild solutions on $L^{\\infty}_{\\sigma}$ for the Navier-Stokes equations in an exterior domain in $ \\mathbb{R}^{n}$, $n\\geq 2$, subject to the non-slip boundary condition."
],
[
"Introduction",
"We consider the initial-boundary value problem of the Navier-Stokes equations in an exterior domain $\\Omega \\subset \\mathbb {R}^{n}$ , $n\\ge 2$ : $\\begin{aligned}\\partial _t u-\\Delta {u}+u\\cdot \\nabla u+\\nabla {p}&= 0 \\quad \\textrm {in}\\quad \\Omega \\times (0,T), \\\\u&=0\\quad \\textrm {in}\\quad \\Omega \\times (0,T), \\\\u&=0\\quad \\textrm {on}\\quad \\partial \\Omega \\times (0,T), \\\\u&=u_0\\quad \\hspace{-4.0pt} \\textrm {on}\\quad \\Omega \\times \\lbrace t=0\\rbrace .\\end{aligned}\\qquad \\mathrm {(1.1)}$ There is a large literature on the solvability of the exterior problem for initial data decaying at space infinity.",
"However, a few results are available for non-decaying data.",
"A typical example of non-decaying flow is a stationary solution of (1.1) having a finite Dirichlet integral, called $D$ -solution .",
"It is known that $D$ -solutions are bounded in $\\Omega $ and asymptotically constant as $|x|\\rightarrow \\infty $ ; see Remarks 1.2 (ii).",
"In this paper, we do not impose on $u_0$ conditions at space infinity.",
"The purpose of this paper is to establish a solvability of (1.1) for merely bounded initial data.",
"We set the solenoidal $L^{\\infty }$ -space, $L^{\\infty }_{\\sigma }(\\Omega )=\\left\\lbrace f\\in L^{\\infty }(\\Omega )\\ \\Bigg |\\ \\int _{\\Omega }f\\cdot \\nabla \\varphi \\textrm {d}x=0 \\quad \\textrm {for}\\ \\varphi \\in \\hat{W}^{1,1}(\\Omega ) \\right\\rbrace ,$ by the homogeneous Sobolev space $\\hat{W}^{1,1}(\\Omega )=\\lbrace \\varphi \\in L^{1}_{\\textrm {loc}}(\\Omega )\\ |\\ \\nabla \\varphi \\in L^{1}(\\Omega )\\ \\rbrace $ .",
"For exterior domains, the space $L^{\\infty }_{\\sigma }$ agrees with the space of all bounded divergence-free vector fields, whose normal trace is vanishing on $\\partial \\Omega $ .",
"The $L^{\\infty }$ -type solvability for (1.1) is recently established on $C_{0,\\sigma }$ in the previous work of the author , where $C_{0,\\sigma }$ is the $L^{\\infty }$ -closure of $C_{c,\\sigma }^{\\infty }$ , the space of all smooth solenoidal vector fields with compact support in $\\Omega $ .",
"Since the condition $u_0\\in C_{0,\\sigma }$ imposes the decay $u_0\\rightarrow 0$ as $|x|\\rightarrow \\infty $ , we develop an existence theorem for non-decaying space $L^{\\infty }_{\\sigma }$ , which in particular includes asymptotically constant vector fields.",
"Moreover, the space $L^{\\infty }_{\\sigma }$ includes vector fields rotating at space infinity; see Remarks 1.2 (iv).",
"When $\\Omega $ is the whole space or a half space , , the existence of mild solutions of (1.1) on $L^{\\infty }_{\\sigma }$ is proved by explicit formulas of the Stokes semigroup.",
"In this paper, we prove unique existence of mild solutions on $L^{\\infty }_{\\sigma }$ for exterior domains based on $L^{\\infty }$ -estimates of the Stokes semigroup , .",
"To state a result, let $S(t)$ denote the Stokes semigroup.",
"It is proved in that $S(t)$ is an analytic semigroup on $L^{\\infty }_{\\sigma }$ for exterior domains of class $C^{3}$ .",
"Let $\\mathbb {P}$ denote the Helmholtz projection.",
"We write $F=(\\sum _{i=1}^{n}\\partial _{i}F_{ij})$ for matrix-valued functions $F=(F_{ij})$ .",
"It is proved in that the composition operator $S(t)\\mathbb {P} satisfies an estimate of the form\\begin{equation*}\\big \\Vert S(t)\\mathbb {P}F̥\\big \\Vert _{L^{\\infty }(\\Omega )}\\le \\frac{C_{\\alpha }}{t^{\\frac{1-\\alpha }{2}}}\\big \\Vert F\\big \\Vert _{L^{\\infty }(\\Omega )}^{1-\\alpha }\\big \\Vert \\nabla F\\big \\Vert _{L^{\\infty }(\\Omega )}^{\\alpha }, \\end{equation*}for \\qquad \\mathrm {(1.2)}$ FC10W1,2()$, $ tT0$ and $ (0,1)$.",
"Here, $ W1,2()$ denotes the Sobolev space and $ C01()$ denotes the $ W1,$-closure of $ Cc()$, the space of all smooth functions with compact support in $$.",
"Although the projection $ P$ may not act as a bounded operator on $ L$, the $ L$-estimate (1.2) implies that the composition $ S(t)P is uniquely extendable to a bounded operator from $C_{0}^{1}$ to $C_{0,\\sigma }$ .",
"Note that $F\\in C^{1}_{0}$ imposes a decay condition at space infinity.",
"Thus the extension to $C_{0}^{1}$ is not sufficient for studying non-decaying solutions.",
"In this paper, we prove that the composition $S(t)\\mathbb {P} is uniquely extendable to a bounded operator $S(t)P$ from the non-decaying space $ W1,0$ to $ L$, where $ W1,0$ is the space of all functions in $ W1,$ vanishing on $$.$ By means of the new extension, we study the integral equation on $L^{\\infty }_{\\sigma }$ of the form $u(t)=S(t)u_{0}-\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P} (uu)(s)ds.",
"\\qquad \\mathrm {(1.3)}}Here, uu=(u_iu_j) is the tensor product.",
"We call solutions of (1.3) mild solution on L^{\\infty }_{\\sigma }.",
"Since the projection \\mathbb {P} may not be bounded on L^{\\infty }, the extension \\overline{S(t)\\mathbb {P} is not expressed by the individual operators.",
"We thus prove that mild solutions satisfy (1.1) by using a weak form.",
"Let C^{\\infty }_{c,\\sigma }(\\Omega \\times [0,T)) denote the space of all smooth solenoidal vector fields with compact support in \\Omega \\times [0,T).",
"Let C([0,T]; X) (resp.",
"C_{w}([0,T]; X)) denote the space of all (resp.",
"weakly-star) continuous functions from [0,T] to a Banach space X.",
"Let BUC_{\\sigma }(\\Omega ) denote the space of all solenoidal vector fields in BUC(\\Omega ) vanishing on \\partial \\Omega , where BUC(\\Omega ) is the space of all bounded uniformly continuous functions in \\overline{\\Omega }.",
"Let [\\cdot ]_{\\Omega }^{(\\beta )} denote the \\beta -th Hölder semi-norm in \\overline{\\Omega }.",
"The main result of this paper is the following:}\\vspace{10.0pt}$ Theorem 1.1 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .",
"For $u_0\\in L^{\\infty }_{\\sigma }$ , there exist $T\\ge \\varepsilon /||u_0||_{\\infty }^{2}$ and a unique mild solution $u\\in C_{w}([0,T]; L^{\\infty })$ such that $\\int _{0}^{T}\\int _{\\Omega }\\big (u\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )+u u: \\nabla \\varphi \\big )\\textrm {d}x\\textrm {d}t=-\\int _{\\Omega }u_0(x)\\cdot \\varphi (x,0)\\textrm {d}x \\qquad \\mathrm {(1.4)}$ for all $\\varphi \\in C^{\\infty }_{c,\\sigma }(\\Omega \\times [0,T))$ , with some constant $\\varepsilon =\\varepsilon _{\\Omega }$ .",
"The solution $u$ satisfies $\\sup _{0< t\\le T}\\left\\lbrace \\big \\Vert u\\big \\Vert _{L^{\\infty }(\\Omega )}(t)+t^{\\frac{1}{2}}\\big \\Vert \\nabla u\\big \\Vert _{L^{\\infty }(\\Omega )}(t)+t^{\\frac{1+\\beta }{2}}\\Big [\\nabla u\\Big ]^{(\\beta )}_{\\Omega }(t) \\right\\rbrace \\le C_1\\big \\Vert u_0\\big \\Vert _{L^{\\infty }(\\Omega )}, \\qquad \\mathrm {(1.5)}$ $\\sup _{x\\in \\Omega }\\left\\lbrace \\Big [u\\Big ]^{(\\gamma )}_{[\\delta ,T]}(x)+\\Big [\\nabla u\\Big ]^{(\\frac{\\gamma }{2})}_{[\\delta ,T]}(x) \\right\\rbrace \\le C_2\\big \\Vert u_0\\big \\Vert _{L^{\\infty }(\\Omega )}, \\qquad \\mathrm {(1.6)}$ for $\\beta , \\gamma \\in (0,1)$ and $\\delta \\in (0,T)$ with the constant $C_1$ , independent of $u_0$ and $T$ .",
"The constant $C_2$ depends on $\\gamma $ , $\\delta $ and $T$ .",
"If $u_0\\in BUC_{\\sigma }$ , $u$ , $t^{1/2}\\nabla u\\in C([0,T]; BUC)$ and $t^{1/2}\\nabla u$ vanishes at time zero.",
"Remarks 1.2 (i) (Blow-up rate) By the estimate of the existence time in Theorem 1.1, we obtain a blow-up rate of mild solutions $u\\in C_{w}([0,T_*); L^{\\infty })$ of the form $\\Vert u\\Vert _{L^{\\infty }(\\Omega )}\\ge \\frac{\\varepsilon ^{\\prime }}{\\sqrt{T_*-t}}\\quad \\textrm {for}\\ t<T_{*},$ with $\\varepsilon ^{\\prime }=\\varepsilon ^{1/2}$ , where $t=T_{*}$ is the blow-up time.",
"The above blow-up estimate was first proved by Leray for $\\Omega =\\mathbb {R}^{3}$ .",
"See for $n\\ge 3$ and (, ) for a half space.",
"The statement of Theorem 1.1 is valid also for a half space and improves regularity properties of mild solutions on $L^{\\infty }_{\\sigma }$ proved in , .",
"(ii)($D$ -solutions) In , Leray proved the existence of $D$ -solutions $u$ satisfying $u-u_{\\infty }\\in L^{6}(\\Omega )$ for $u_{\\infty }\\in \\mathbb {R}^{3}$ in the exterior domain $\\Omega \\subset \\mathbb {R}^{3}$ .",
"His construction is based on an approximation for $R\\rightarrow \\infty $ of the problem $-\\Delta u_{R}+u_{R}\\cdot \\nabla u_{R}+\\nabla p_{R}&=0\\quad \\textrm {in}\\ \\Omega _R,\\\\u_R&=0\\quad \\textrm {in}\\ \\Omega _R,\\\\u_{R}&=0\\quad \\textrm {on}\\ \\partial \\Omega ,\\\\u_{R}&=u_{\\infty }\\hspace{4.0pt} \\textrm {on}\\ \\lbrace |x|=R\\rbrace ,$ for $\\Omega _{R}=\\Omega \\cap \\lbrace |x|<R\\rbrace $ ().",
"See also () for a different construction.",
"If the Dirichlet integral is finite, stationary solutions of (1.1) are locally bounded in $\\overline{\\Omega }$ (e.g., ).",
"Moreover, $D$ -solutions are bounded as $|x|\\rightarrow \\infty $ by $u-u_{\\infty }\\in L^{6}(\\Omega )$ .",
"Thus, $D$ -solutions are elements of $L^{\\infty }_{\\sigma }$ for $n=3$ .",
"When $n=2$ , more analysis is needed for information about the behavior as $|x|\\rightarrow \\infty $ since a finite Dirichlet integral does not imply decays at space infinity (e.g., $u=(\\log |x|)^{\\alpha }$ for $0<\\alpha <1/2$ ).",
"Leray's construction gives $D$ -solutions also in $\\Omega \\subset \\mathbb {R}^{2}$ .",
"It is proved in () that Leray's solutions are bounded in $\\overline{\\Omega }$ and converge to some constant $\\overline{u}_{\\infty }$ in the sense that $\\int _{0}^{2\\pi }|u(re_{r})-\\overline{u}_{\\infty }|\\textrm {d}\\theta \\rightarrow 0$ as $r\\rightarrow \\infty $ , where $(r,\\theta )$ is the polar coordinate and $e_{r}=(\\cos \\theta ,\\sin \\theta )$ .",
"Moreover, every $D$ -solutions are bounded and asymptotically constant in the above sense .",
"Thus, $D$ -solutions are elements of $L^{\\infty }_{\\sigma }$ also for $n=2$ .",
"Theorem 1.1 yields a local solvability of (1.1) around $D$ -solutions without imposing decay conditions for initial disturbance.",
"(iii) (Global well-posedness for $n=2$ ) It is well known that the exterior problem (1.1) for $n=2$ is globally well-posed for initial data having finite energy, e.g., .",
"However, global well-posedness is unknown for non-decaying data $u_0\\in L^{\\infty }_{\\sigma }$ .",
"For the whole space, the vorticity $\\omega =\\partial _{1}u^{2}-\\partial _{2}u^{1}$ satisfies the a priori estimate $||\\omega ||_{L^{\\infty }(\\mathbb {R}^{2})}\\le ||\\omega _0||_{L^{\\infty }(\\mathbb {R}^{2})}\\quad t>0.$ It is proved in that the Cauchy problem of (1.1) for $n=2$ is globally well-posed for $u_0\\in L^{\\infty }_{\\sigma }$ based on the local solvability result in .",
"We proved a local solvability on $L^{\\infty }_{\\sigma }$ for exterior domains.",
"Note that global solutions exist for rotationally symmetric initial data $u_0\\in L^{\\infty }_{\\sigma }$ ; see below (iv).",
"(iv) (Rotating flows) An example of $u_0\\in L^{\\infty }_{\\sigma }$ which is not asymptotically constant is a vector field rotating at space infinity.",
"For example, we consider the two-dimensional unit disk $\\Omega ^{c}$ centered at the origin and a rotationally symmetric initial data $u_{0}=u_{0}^{\\theta }(r)e_{\\theta }(\\theta )$ for $e_{\\theta }(\\theta )=(-\\sin {\\theta },\\cos {\\theta })$ .",
"Observe that $u_0$ is a solenoidal vector field in $\\Omega $ and a direction of $u_0$ varies for $\\theta \\in [0,2\\pi ]$ and $u_{0}^{\\theta }\\in L^{\\infty }(1,\\infty )$ .",
"Solutions of (1.1) for $u_0$ are rotationally symmetric and given by $u=e^{t\\Delta _{D}}u_0\\qquad p=\\int _{1}^{|x|}\\frac{|u|^{2}}{r}\\textrm {d}r,$ where $\\Delta _{D}$ denotes the Laplace operator subject to the Dirichlet boundary condition.",
"The solution $u$ is bounded in $\\Omega \\times (0,\\infty )$ and non-decaying as $|x|\\rightarrow \\infty $ .",
"(v) (Associated pressure) We invoke that the associated pressure of mild solutions on $L^{p}$ ($p\\ge n$ ) is determined by the projection operator $\\mathbb {Q}=I-\\mathbb {P}$ and $\\nabla p=\\mathbb {Q}\\Delta u-\\mathbb {Q}(u\\cdot \\nabla u).$ Since the projection $\\mathbb {Q}$ may not be bounded on $L^{\\infty }$ , this representation is no longer available for mild solutions on $L^{\\infty }_{\\sigma }$ .",
"When $\\Omega =\\mathbb {R}^{n}$ or $\\mathbb {R}^{n}_{+}$ , the projection $\\mathbb {Q}$ has explicit kernels and we are able to find associated pressure of mild solutions on $L^{\\infty }$ ; see for $\\Omega =\\mathbb {R}^{n}$ and , , for $\\Omega =\\mathbb {R}^{n}_{+}$ .",
"Although explicit kernels are not available for exterior domains, we are able to find the associated pressure of mild solutions on $L^{\\infty }$ .",
"We set $\\nabla p=\\mathbb {K}W-\\mathbb {Q}F̥ $ for $W=-(\\nabla u-\\nabla ^{T} u)n_{\\Omega }$ and $F=uu$ , where $n_{\\Omega }$ is the unit outward normal on $\\partial \\Omega $ and $\\mathbb {K}$ is a solution operator of the homogeneous Neumann problem (harmonic-pressure operator) .",
"Note that $W=-\\textrm {curl}\\ u\\times n_{\\Omega }$ for $n=3$ .",
"The operators $\\mathbb {K}$ and $\\mathbb {Q} act for bounded functions and the associated pressure on $ L$ is uniquely determined by (1.7) in the sense of distribution; see Remark 3.5 for a detailed discussion.$ For asymptotically constant initial data $u_0$ (i.e., $u_{0}\\rightarrow u_{\\infty }$ as $|x|\\rightarrow \\infty $ ), local solvability of (1.1) for $n=3$ is proved in by means of the Oseen semigroup.",
"In the paper, the problem (1.1) is reduced to an initial-boundary problem for decaying data by shifting $u$ by a constant $u_{\\infty }$ .",
"Our analysis is based on the $L^{\\infty }$ -estimates of the Stokes semigroup which yields a local-in-time solvability of (1.1) without conditions for $u_0$ at space infinity.",
"The $L^{\\infty }$ -theory for the Cauchy problem of the Navier-Stokes equations is developed by Knightly , , Cannon and Knightly , Cannone () and Giga et al.",
".",
"For the whole space, mild solutions on $L^{\\infty }$ are smooth and satisfy (1.1) in a classical sense .",
"For a half space, mild solutions on $L^{\\infty }$ are constructed in (see also , ).",
"There are a few results on solvability of the exterior problem for non-decaying data.",
"In , unique existence of continuous solutions of (1.1) for $n\\ge 3$ is proved for non-decaying and Hölder continuous initial data.",
"The result is extended in for merely bounded $u_0\\in L^{\\infty }_{\\sigma }$ and $n\\ge 3$ by using $L^{\\infty }$ -estimates of the Stokes semigroup , .",
"Note that mild solutions on $L^{\\infty }_{\\sigma }$ are not constructed without the composition operator $\\overline{S(t)\\mathbb {P}.",
"We proved the unique existence of mild solutions on L^{\\infty }_{\\sigma }, which in particular yields a local-in-time solvability for n=2.",
"The integral form (1.3) is fundamental for studying solutions of (1.1).",
"We expect that mild solutions on L^{\\infty } are sufficiently smooth and satisfy (1.1) in a classical sense.\\\\}$ The article is organized as follows.",
"In Section 2, we extend the composition operator $S(t)\\mathbb {P} to a bounded operator from $ W1,0$ to $ L$ by approximation as we did the Stokes semigroup in \\cite {AG2}.",
"We extend $ S(t)P as a solution operator $F\\longmapsto v(\\cdot ,t)$ for solutions $(v,q)$ of the Stokes equations for $v_0=\\mathbb {P}F$ .",
"Note that $v_0=\\mathbb {P}F$ for $F\\in W^{1,\\infty }_{0}$ is not an element of $L^{\\infty }$ in general since the projection $\\mathbb {P}$ is not bounded on $L^{\\infty }$ .",
"We understand $v_0=\\mathbb {P}F$ as distribution by using the fact that $\\nabla \\mathbb {P}\\varphi \\in L^{1}$ for $\\varphi \\in C_{c}^{\\infty }$ (Lemma A.1).",
"We approximate $F\\in W^{1,\\infty }_{0}$ by a sequence $\\lbrace F_m\\rbrace \\subset C_{c}^{\\infty }$ locally uniformly in $\\overline{\\Omega }$ and obtain a unique extension $\\overline{S(t)\\mathbb {P}: F\\longmapsto v(\\cdot ,t) by a limit v of the sequence v_m=S(t)\\mathbb {P}F_m.",
"}In Section 3, we prove Theorem 1.1.",
"We approximate initial data $ u0L$ by a sequence $ {u0,m}Cc,$ satisfying $ u0,mu0$ a.e.",
"in $$ and $ ||u0,m||C||u0||$.",
"Since the property of mild solutions (1.4) may not follow from a direct iteration argument on $ L$, we construct mild solutions by approximation.",
"We apply an existence theorem on $ C0,$ \\cite {A4} and construct a sequence of mild solutions $ umC([0,T]; C0,)$ satisfying (1.4)-(1.6) for $ u0,mC0,$.",
"We prove that $ um$ subsequently converges to a mild solution $ u$ for $ u0L$ locally uniformly in $(0,T]$.$ In Appendix A, we show that $\\nabla \\mathbb {P}\\varphi \\in L^{1}$ for $\\varphi \\in C_{c}^{\\infty }$ by means of the layer potential."
],
[
"An extension of the composition operator",
"In this section, we prove that the composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\infty }_{0}$ to $L^{\\infty }_{\\sigma }$ .",
"We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\mathbb {P}\\partial :f\\longmapsto v(\\cdot ,t)$ .",
"In what follows, $\\partial =\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots ,n$ ."
],
[
"The Stokes system",
"We consider the Stokes equations, $\\begin{aligned}\\partial _{t}v-\\Delta v+\\nabla q&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {on}\\ \\partial \\Omega \\times (0,T), \\\\v&=v_0\\quad \\hspace{-4.0pt} \\textrm {on}\\ \\Omega \\times \\lbrace t=0\\rbrace .\\end{aligned}\\qquad \\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\bigl |v(x,t)\\bigr |+t^{\\frac{1}{2}}\\bigl |\\nabla v(x,t)\\bigr |+t\\bigl |\\nabla ^{2}v(x,t)\\bigr |+t\\bigl |\\partial _{t}v(x,t)\\bigr |+t\\bigl |\\nabla q(x,t)\\bigr |.$ Let $d(x)$ denote the distance from $x\\in \\Omega $ to $\\partial \\Omega $ .",
"Let $(v, \\nabla q)\\in C^{2+\\mu ,1+\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T])\\times C^{\\mu ,\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T]),\\ \\mu \\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .",
"We say that $(v,q)$ is a solution of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , if $\\sup _{0<t\\le T}\\Bigg \\lbrace t^{\\gamma }\\big \\Vert N(v,q)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d \\nabla q\\big \\Vert _{\\infty }(t)\\Bigg \\rbrace <\\infty , \\qquad \\mathrm {(2.2)}$ for some $\\gamma \\in [0,1/2)$ and $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =\\int _{\\Omega }f\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x \\qquad \\mathrm {(2.3)}$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ with $\\varphi _{0}(x)=\\varphi (x,0)$ .",
"The left-hand side is finite since $\\varphi (\\cdot ,t)$ is supported in $\\Omega $ and $\\gamma <1/2$ .",
"The right-hand side is finite since $\\partial \\mathbb {P}\\varphi _0$ is integrable in $\\Omega $ for $\\varphi _0\\in C_{c}^{\\infty }(\\Omega )$ by Lemma A.1.",
"As explained later in Remarks 2.9 (i), the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ and we are able to define $\\mathbb {P}\\partial f$ in the sense of distribution.",
"The goal of this section is to prove: Theorem 2.1 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.",
"Let $T>0$ .",
"For $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\sup _{0<t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v,q)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d \\nabla q\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }, \\qquad \\mathrm {(2.4)}$ for $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ and some constant C, depending on $\\alpha $ , $T$ and $\\Omega $ .",
"Theorem 2.1 implies the following: Theorem 2.2 The composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator $\\overline{S(t)\\mathbb {P}\\partial }$ from $W^{1,\\infty }_{0}(\\Omega )$ to $L^{\\infty }_{\\sigma }(\\Omega )$ together with the estimate $\\sup _{0< t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}\\overline{S(t)\\mathbb {P}\\partial } f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }, \\qquad \\mathrm {(2.5)}$ for $f\\in W^{1,\\infty }_{0}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ ."
],
[
"Hölder estimates and uniqueness",
"In order to prove Theorem 2.1, we recall local Hölder estimates and a uniqueness result for the Stokes equations.",
"In the subsequent section, we give a proof for Theorem 2.1 by approximation.",
"We set the Hölder semi-norm $\\Big [f\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q}=\\sup _{t\\in (0,T]}\\Big [f\\Big ]^{(\\mu )}_{\\Omega }(t)+\\sup _{x\\in \\Omega }\\Big [f\\Big ]^{(\\frac{\\mu }{2})}_{(0,T]}(x),\\quad \\mu \\in (0,1),$ for $Q=\\Omega \\times (0,T]$ .",
"We set $N=\\sup _{\\delta \\le t\\le T}\\big \\Vert N(v,q)\\big \\Vert _{L^{\\infty }(\\Omega )}(t)$ for solutions $(v,q)$ of (2.1).",
"The following local Hölder estimate is proved in based on the Schauder estimates for the Stokes equations (, ).",
"Proposition 2.3 Let $\\Omega $ be an exterior domain with $C^3$ -boundary.",
"(i) (Interior estimates) For $\\mu \\in (0,1)$ , $\\delta >0$ , $T>0$ , $R>0$ , there exists a constant $C=C\\bigl (\\mu ,\\delta ,T, R,d\\bigr )$ such that $\\Big [\\nabla ^{2}v\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [v_t\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [\\nabla q\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}\\le CN \\qquad \\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\prime }=B_{x_0}(R)\\times (2\\delta ,T]$ and $x_0\\in \\Omega $ satisfying $\\overline{B_{x_0}(R)}\\subset \\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\partial \\Omega $ .",
"(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\mu \\in (0,1),\\ \\delta >0$ , $T>0$ and $R\\le R_{0}$ , there exists a constant $C$ depending on $\\mu $ , $\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\partial \\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\prime }=\\Omega _{x_0,R}\\times (2\\delta ,T]$ and $\\Omega _{x_0,R}=B_{x_0}(R)\\cap \\Omega $ , $x_0\\in \\partial \\Omega $ .",
"We observe the uniqueness of solutions for (2.1).",
"The uniqueness of the Stokes equations (2.1) for $v_0\\in L^{\\infty }_{\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .",
"In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , may not be bounded at $t=0$ .",
"The corresponding uniqueness result for a half space is recently proved in .",
"We deduce the result for exterior domains by the same blow-up argument as we did in .",
"Proposition 2.4 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.",
"Let $(v,\\nabla q)\\in C^{2,1}(\\overline{\\Omega }\\times (0,T])\\times C(\\overline{\\Omega }\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\gamma \\in [0,1/2)$ .",
"Assume that $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =0,$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ .",
"Then, $v\\equiv 0$ and $\\nabla q\\equiv 0$ ."
],
[
"Approximation",
"We prove Theorem 2.1.",
"We show existence of solutions for the Stokes equations (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , by approximation.",
"We approximate $f\\in W^{1,\\infty }_{0}$ by elements of $C^{\\infty }_{c}$ locally uniformly in $\\overline{\\Omega }$ .",
"Lemma 2.5 Let $\\Omega $ be an exterior domain with Lipschitz boundary.",
"There exist constants $C_1$ , $C_2$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C_{c}^{\\infty }(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert f_{m}\\big \\Vert _{\\infty }\\le C_1\\big \\Vert f\\big \\Vert _{\\infty }, \\\\&\\big \\Vert \\nabla f_{m}\\big \\Vert _{\\infty }\\le C_2\\big \\Vert f\\big \\Vert _{1,\\infty }, \\\\&f_m\\rightarrow f\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.",
"Proposition 2.6 The statement of Lemma 2.5 holds when $\\Omega =\\mathbb {R}^{n}$ with $C_1=1$ .",
"We cutoff the function $f\\in W^{1,\\infty }(\\mathbb {R}^{n})$ .",
"Let $\\theta \\in C^{\\infty }_{c}[0,\\infty )$ be a cut-off function satisfying $\\theta \\equiv 1$ in $[0,1]$ , $\\theta \\equiv 0$ in $[2,\\infty )$ and $0\\le \\theta \\le 1$ .",
"We set $\\theta _{m}(x)=\\theta (|x|/m)$ for $m\\ge 1$ so that $\\theta _{m}\\equiv 1$ for $|x|\\le m$ and $\\theta _{m}\\equiv 0$ for $|x|\\ge 2m$ .",
"Then, $f_m=f\\theta _{m}$ satisfies (2.7).",
"Proposition 2.7 Let $\\Omega $ be a bounded domain with Lipschitz boundary.",
"There exists a constant $C_3$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C^{\\infty }_{c}(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert \\nabla f_m\\big \\Vert _{\\infty }\\le C_3\\big \\Vert \\nabla f\\big \\Vert _{\\infty }\\\\f_m&\\rightarrow f\\quad \\textrm {uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.8)}$ We begin with the case when $\\Omega $ is star-shaped, i.e., $\\lambda \\Omega _{x_0}\\subset \\overline{\\Omega }$ for some $x_0\\in \\Omega $ and all $\\lambda <1$ , where $\\lambda \\Omega _{x_0}=\\lbrace x_0+\\lambda (x-x_0)\\ |\\ x\\in \\Omega \\rbrace $ .",
"We may assume $x_0=0\\in \\Omega $ and $\\lambda \\Omega \\subset \\overline{\\Omega }$ by translation.",
"For $f\\in W^{1,\\infty }_{0}(\\Omega )$ , we set $f_{\\lambda }(x)={\\left\\lbrace \\begin{array}{ll}&f\\big (x/\\lambda \\big )\\quad x\\in \\lambda \\Omega ,\\\\&0\\qquad \\hspace{15.0pt} x\\in \\Omega \\backslash \\overline{\\lambda \\Omega } .\\end{array}\\right.",
"}$ Then, $f_{\\lambda }$ is in $ W^{1,\\infty }(\\Omega )$ since $f$ is vanishing on $\\partial \\Omega $ .",
"It follows that $\\big \\Vert \\nabla f_{\\lambda }\\big \\Vert _{\\infty }\\le \\frac{1}{\\lambda }\\big \\Vert \\nabla f\\big \\Vert _{\\infty },$ and $f_{\\lambda }\\rightarrow f$ uniformly in $\\overline{\\Omega }$ as $\\lambda \\rightarrow 1$ .",
"By a mollification of $f_{\\lambda }$ , we obtain a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfying (2.8) with $C_3=2$ .",
"For general $\\Omega $ , we take an open covering $\\lbrace D_j\\rbrace _{j=1}^{N}$ so that $\\overline{\\Omega }\\subset \\cup _{j=1}^{N}D_j$ and $\\Omega _{j}=\\Omega \\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\in \\Omega _{j}$ .",
"We take a partition of unity $\\lbrace \\xi _{j}\\rbrace _{j=1}^{N}\\subset C_{c}^{\\infty }(\\mathbb {R}^{n})$ such that $\\sum _{j=1}^{N}\\xi _{j}=1$ , $0\\le \\xi _{j}\\le 1$ , spt $\\xi _{j}\\subset \\overline{D_j}$ and set $f=\\sum _{j=1}^{N}f_j,\\quad f_j=f\\xi _j.$ Since spt $f_j\\subset \\overline{\\Omega }_{j}$ , $\\xi _{j}=0$ on $\\partial D_{j}$ and $f=0$ on $\\partial \\Omega $ , $f_{j}$ is in $W^{1,\\infty }_{0}(\\Omega _{j})$ .",
"Since $\\Omega _j$ is star-shaped for some $x_j\\in \\Omega _{j}$ , there exists $\\lbrace f_{j,m}\\rbrace \\subset C_{c}^{\\infty }(\\Omega _j)$ satisfying (2.8) in $\\Omega _j$ with $C_3=2$ .",
"We extend $f_{j,m}\\in C_{c}^{\\infty }(\\Omega _j)$ to $\\Omega \\backslash \\overline{\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\sum _{j=1}^{N}f_{j,m}$ .",
"Then, $f_m\\in C_{c}^{\\infty }(\\Omega )$ converges to $f$ uniformly in $\\overline{\\Omega }$ .",
"We estimate $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le \\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j,m}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}\\le 2\\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}.$ Since $\\nabla f_j=\\nabla f\\xi _j+f\\nabla \\xi _j$ and $\\big \\Vert f\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C_{p}\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )}.$ Thus, $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfies (2.8).",
"The proof is complete.",
"The assertion follows from Propositions 2.6 and 2.7.",
"We recall the a priori estimate of $S(t)\\mathbb {P}\\partial $ for $f\\in C_{c}^{\\infty }(\\Omega )$ .",
"Proposition 2.8 There exists a constant $C$ such that $\\sup _{0\\le t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}S(t)\\mathbb {P}\\partial f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert \\nabla f\\big \\Vert _{\\infty }^{\\alpha } \\qquad \\mathrm {(2.9)}$ for $f\\in C^{\\infty }_{c}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ .",
"For $f\\in W^{1,\\infty }_{0}$ , we take a sequence $\\lbrace f_m\\rbrace \\subset C^{\\infty }_{c}$ satisfying (2.7).",
"For $v_{0,m}=\\mathbb {P}\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\int _{0}^{T}\\int _{\\Omega }\\big ( v_{m}\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q_m\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t=\\int _{\\Omega }f_m\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x,$ for $\\varphi \\in C^{\\infty }_{c}\\big (\\Omega \\times [0,T)\\big )$ .",
"By (2.9) and (2.7), there exists a constant $C$ independent of $m\\ge 1$ such that $\\sup _{0\\le t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v_m,q_m)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d\\nabla q_{m}\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla v_m$ , $\\nabla ^{2}v_m$ , $\\partial _{t}v_m$ and $\\nabla q_{m}$ .",
"By sending $m\\rightarrow \\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\mathbb {P}\\partial f$ .",
"By Proposition 2.4, the limit $(v,q)$ is unique.",
"We proved the unique existence of solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ and $f\\in W^{1,\\infty }_{0}$ satisfying (2.4).",
"The proof is now complete.",
"Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\mathbb {P}\\partial $ for $f\\in W^{1,\\infty }_{0}$ .",
"We take a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\mathbb {P}\\partial f_m$ satisfies $(v_{0,m}, \\varphi )=-(f_m, \\partial \\mathbb {P}\\varphi )\\quad \\textrm {for}\\ \\varphi \\in C^{\\infty }_{c}(\\Omega ).$ Since $\\partial \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ by Lemma A.1, the sequence $\\lbrace v_{0,m}\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\varphi )=-(f,\\partial \\mathbb {P}\\varphi )$ .",
"Since the limit $v_0$ is unique, the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ .",
"(ii) We recall that for a sequence $\\lbrace v_{0,m}\\rbrace _{m=1}^{\\infty }\\subset L^{\\infty }_{\\sigma }$ satisfying $||v_{0,m}||_{\\infty }&\\le K_1,\\\\v_{0,m}\\rightarrow v_{0}&\\quad \\textrm {a.e.",
"}\\ \\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .",
"From the proof of Theorem 2.1, we observe that for a sequence $\\lbrace f_m\\rbrace \\subset W^{1,\\infty }_{0}$ satisfying $||f_m||_{1,\\infty }&\\le K_2,\\\\f_m\\rightarrow f\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega },$ $\\overline{S(t)\\mathbb {P}\\partial } f_m$ subsequently converges to $\\overline{S(t)\\mathbb {P}\\partial } f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .",
"(iii) The extension $\\overline{S(t)\\mathbb {P}\\partial }$ satisfies the property $S(t)\\overline{S(s)\\mathbb {P}\\partial } f=\\overline{S(t+s)\\mathbb {P}\\partial } f$ for $t\\ge 0$ , $s>0$ and $f\\in W^{1,\\infty }_{0}$ .",
"In fact, this property holds for $f_m\\in C_{c}^{\\infty }$ satisfying (2.7).",
"By choosing a subsequence, $v_m(\\cdot ,t)=S(t)\\mathbb {P}\\partial f_m$ converges to $v(\\cdot ,t)=S(t)\\mathbb {P}\\partial f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ as in the proof of Theorem 2.1.",
"For fixed $s>0$ , sending $m\\rightarrow \\infty $ implies $S(t)S(s)\\mathbb {P}\\partial f_m=S(t)v_m(s)&\\rightarrow S(t)v(s) \\\\&=S(t)\\overline{S(s)\\mathbb {P}\\partial }f\\quad \\textrm {locally uniformly in \\overline{\\Omega }\\times (0,\\infty )}.$ Thus the property is inherited to $\\overline{S(t)\\mathbb {P}\\partial } f$ ."
],
[
"Mild solutions on $L^{\\infty }_{\\sigma }$",
"We prove Theorem 1.1 by approximation.",
"We show that a sequence of mild solutions $\\lbrace u_m\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by the $L^{\\infty }$ -estimates (1.5) and (1.6).",
"Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).",
"We first recall the existence of mild solutions on $C_{0,\\sigma }$ Proposition 3.1 For $u_0\\in C_{0,\\sigma }$ , there exist $T\\ge \\varepsilon _0/||u_0||_{\\infty }^{2}$ and a unique mild solution $u\\in C([0,T]; C_{0,\\sigma })$ satisfying (1.3)-(1.6).",
"We approximate $u_0\\in L^{\\infty }_{\\sigma }$ by elements of $C_{c,\\sigma }^{\\infty }\\subset C_{0,\\sigma }$ .",
"We take a sequence $\\lbrace u_{0,m}\\rbrace _{m=1}^{\\infty }\\subset C_{c,\\sigma }^{\\infty }(\\Omega )$ satisfying $\\begin{aligned}&\\Vert u_{0,m}\\Vert _{\\infty }\\le C\\Vert u_{0}\\Vert _{\\infty }\\\\&u_{0,m}\\rightarrow u_{0}\\quad \\textrm {a.e.",
"in}\\ \\Omega ,\\end{aligned}\\qquad \\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\ge 1$ .",
"We apply Proposition 3.1 and observe that there exists $T_m\\ge \\varepsilon _0/||u_{0,m}||_{\\infty }^{2}$ and a unique mild solution $u_m\\in C([0,T_m]; C_{0,\\sigma })$ satisfying $\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P}F_m(s)ds,\\\\F_m&=u_mu_m.",
"}\\qquad \\mathrm {(3.2)}\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\ge \\varepsilon /||u_0||_{\\infty }^{2} for \\varepsilon =\\varepsilon _0 C^{-2}/2 so that T_m\\ge T and u_m\\in C([0,T]; C_{0,\\sigma }) for m\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla u_m$ .",
"It follows from (1.5), (1.6) and (3.1) that $\\sup _{0\\le t\\le T}\\Big \\lbrace \\Vert u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1}{2}}\\Vert \\nabla u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1+\\beta }{2}}\\big [\\nabla u_{m}\\big ]_{\\Omega }^{(\\beta )}(t)\\Big \\rbrace &\\le C_1^{\\prime }\\Vert u_0\\Vert _{\\infty }, \\\\\\sup _{x\\in \\Omega }\\Big \\lbrace \\big [u_{m}\\big ]_{[\\delta ,T]}^{(\\gamma )}(x)+\\big [\\nabla u_{m}\\big ]_{[\\delta ,T]}^{(\\frac{\\gamma }{2})}(x)\\Big \\rbrace &\\le C_2^{\\prime }\\Vert u_0\\Vert _{\\infty }, $ for $\\beta ,\\gamma \\in (0,1)$ and $\\delta \\in (0,T]$ with some constants $C_1^{\\prime }$ and $C_2^{\\prime }$ , independent of $m\\ge 1$ .",
"Since $u_m$ and $\\nabla u_m$ are uniformly bounded and equi-continuous in $\\overline{\\Omega }\\times [\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.",
"Proposition 3.3 The limit $u\\in C_{w}([0,T]; L^{\\infty })$ is a mild solution for $u_0\\in L^{\\infty }_{\\sigma }$ .",
"We observe that the limit $u$ satisfies (1.4) by sending $m\\rightarrow \\infty $ .",
"The estimates (3.3) and (3.4) are inherited to $u$ .",
"We prove that $u$ satisfies the integral equation (1.3).",
"By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by Remarks 2.9 (ii).",
"It follows from (3.3) and Proposition 3.2 that $\\begin{aligned}||F_m||_{\\infty }&\\le K,\\\\||\\nabla F_{m}||_{\\infty }&\\le \\frac{2}{s^{\\frac{1}{2}}}K,\\\\F_m\\rightarrow F\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T]\\ \\textrm {as}\\ m\\rightarrow \\infty ,\\end{aligned}\\qquad \\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\prime }||u_0||_{\\infty }$ .",
"By choosing a subsequence, we have $\\overline{S(\\eta )\\mathbb {P} F_m\\rightarrow \\overline{S(\\eta )\\mathbb {P} F\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T],}for each s\\in (0,t) as in Remarks 2.9 (ii).",
"It follows from (3.5) and (2.5) that\\begin{equation*}\\big \\Vert \\overline{S(t-s)\\mathbb {P} F_m\\big \\Vert _{\\infty }\\le \\frac{C}{(t-s)^{\\frac{1-\\alpha }{2}}}\\Bigg (1+\\frac{2}{s^{\\frac{\\alpha }{2}}}\\Bigg )K^{2}}for 0<s< t and \\alpha \\in (0,1).",
"By the dominated convergence theorem, we have{\\begin{@align*}{1}{-1}\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P} F_m\\textrm {d}s\\rightarrow \\int _{0}^{t}\\overline{S(t-s)\\mathbb {P} F\\textrm {d}s\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times [0,T].",
"}}Thus sending m\\rightarrow \\infty implies that the limit u is a mild solution for u_0\\in L^{\\infty }_{\\sigma }.",
"Since S(t)u_0 is weakly-star continuous on L^{\\infty } at t=0 \\cite {AG2}, so is u.\\end{@align*}}\\vspace{5.0pt}\\end{equation*}}It remains to show continuity at t=0 for u_0\\in BUC_{\\sigma }.$ Proposition 3.4 For $u_0\\in BUC_{\\sigma }$ , $S(t)u_0$ , $t^{1/2}\\nabla S(t)u_0\\in C([0,T]; BUC)$ and $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"Since $S(t)$ is a $C_0$ -analytic semigroup on $BUC_{\\sigma }$ , $S(t)u_0\\in C([0,T]; BUC_{\\sigma })$ .",
"Moreover, $t^{1/2}\\nabla S(t)u_0$ is continuous and bounded for $t\\in (0,T]$ in $BUC$ .",
"We show that $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"We divide $u_0$ into two terms by using the Bogovskiĭ operator.",
"For $u_0\\in BUC_{\\sigma }$ , there exists $u_{0}^{1}\\in C_{0,\\sigma }$ with compact support in $\\overline{\\Omega }$ and $u_{0}^{2}\\in BUC_{\\sigma }$ supported away from $\\partial \\Omega $ such that $u_0=u_0^{1}+u_0^{2}$ (see ).",
"Let $A$ denote the Stokes operator and $D(A)$ denote the domain of $A$ in $BUC_{\\sigma }$ .",
"Since $S(t)$ is a $C_0$ -semigroup on $BUC_{\\sigma }$ , $D(A)$ is dense in $BUC_{\\sigma }$ .",
"It follows from the resolvent estimate that $||\\nabla v||_{\\infty }\\le C(||v||_{\\infty }+||Av||_{\\infty })\\quad \\textrm {for}\\ v\\in D(A).",
"$ We take an arbitrary $\\epsilon >0$ .",
"For $u_{0}^{1}\\in C_{0,\\sigma }$ , there exists $\\lbrace u_{0,m}^{1}\\rbrace \\subset C_{c,\\sigma }^{\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\infty }\\le \\epsilon $ for $m\\ge N^{1}_{\\epsilon }$ .",
"We apply (3.5) and observe that $t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }&\\le t^{\\frac{1}{2}}C\\big (||S(t)u_{0,m}^{1}||_{\\infty }+||S(t)Au_{0,m}^{1}||_{\\infty }\\big )\\\\&\\le t^{\\frac{1}{2}}C^{\\prime }\\big (||u_{0,m}^{1}||_{\\infty }+||Au_{0,m}^{1}||_{\\infty }\\big )\\rightarrow 0\\quad \\textrm {as}\\ t\\rightarrow 0.$ We estimate $\\overline{\\lim _{t\\rightarrow 0}}t^{\\frac{1}{2}}||\\nabla S(t)u_0^{1}||_{\\infty }&\\le \\overline{\\lim _{t\\rightarrow 0}}\\big (t^{\\frac{1}{2}}||\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\infty }+t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }\\big ) \\\\&\\le C^{\\prime \\prime }\\epsilon .$ We set $u_{0,m}^{2}=\\eta _{\\delta _m}*u_{0}^{2}$ by the mollifier $\\eta _{\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\overline{\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\infty }\\le \\epsilon $ for $m\\ge N_{\\epsilon }^{2}$ .",
"Since $u_{0,m}^{2}$ is supported away from $\\partial \\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\Delta u_{0,m}^{2}$ (see ).",
"By a similar way as for $u_{0}^{1}$ , we estimate $\\overline{\\lim }_{t\\rightarrow 0}t^{1/2}||\\nabla S(t)u_0^{2}||_{\\infty }\\le C^{\\prime \\prime }\\epsilon $ .",
"We proved $\\overline{\\lim }_{t\\rightarrow 0}t^{\\frac{1}{2}}||\\nabla S(t)u_0||_{\\infty }\\le 2C^{\\prime \\prime }\\epsilon .$ Since $\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"The assertion follows from Propositions 3.1-3.4.",
"The proof is now complete.",
"Remark 3.5 We set the associated pressure of mild solutions on $L^{\\infty }$ by (1.7) and the harmonic-pressure operator $\\mathbb {K}: L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )\\longrightarrow L^{\\infty }_{d}(\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\Delta q&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial q}{\\partial n}&={\\partial \\Omega }W\\quad \\textrm {on}\\ \\partial \\Omega .$ Note that $\\Delta u\\cdot n={\\partial \\Omega }W$ by the divergence-free condition of $u$ .",
"Here, $L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )$ denotes the space of all bounded tangential vector fields on $\\partial \\Omega $ and $L^{\\infty }_{d}(\\Omega )$ is the space of all functions $f\\in L^{1}_{\\textrm {loc}}(\\Omega )$ such that $df$ is bounded in $\\Omega $ for $d(x)=\\inf _{y\\in \\partial \\Omega }|x-y|$ , $x\\in \\Omega $ .",
"Since $W=-(\\nabla u-\\nabla ^{T}u)n$ is bounded on $\\partial \\Omega $ for mild solutions on $L^{\\infty }$ , $\\nabla q=\\mathbb {K}W$ is defined as an element of $L^{\\infty }_{d}$ .",
"Moreover, $\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\in W^{1,\\infty }_{0}$ as a distribution by Remarks 2.9 (i).",
"Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\infty }$ .",
"acknowledgements The author is grateful to the anonymous referees for their valuable comments.",
"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.",
"appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .",
"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.",
"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.",
"Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .",
"Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .",
"We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .",
"It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .",
"We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .",
"We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .",
"We may assume that $0\\in \\Omega ^{c}$ by translation.",
"We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .",
"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.",
"We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .",
"In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .",
"Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .",
"Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .",
"In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .",
"By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .",
"We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .",
"We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .",
"Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .",
"Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .",
"The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .",
"We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .",
"Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.",
"By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .",
"Thus, $g$ is continuous on $\\partial \\Omega $ .",
"Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .",
"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.",
"The proof is complete.",
"We estimate $\\Phi _2$ by means of the layer potential.",
"Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .",
"The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .",
"The assertion (i) is based on the Fredholm's theorem.",
"See .",
"Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .",
"Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).",
"When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .",
"Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.",
"Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.",
"Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.",
"We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.",
"The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .",
"By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .",
"We shall show that $\\Psi \\equiv 0$ .",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .",
"Suppose that $x_0\\in \\partial \\Omega $ .",
"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .",
"Thus $x_0\\in \\Omega $ .",
"We apply the strong maximum principle and conclude that $\\Psi $ is constant.",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .",
"The proof is complete.",
"Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .",
"Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .",
"Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .",
"The proof is complete.",
"By Propositions A.2 and A.6, the assertion follows.",
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[
"An extension of the composition operator",
"In this section, we prove that the composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator from $W^{1,\\infty }_{0}$ to $L^{\\infty }_{\\sigma }$ .",
"We prove unique existence of solutions of the Stokes equations for initial data $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , and extend the composition as a solution operator $S(t)\\mathbb {P}\\partial :f\\longmapsto v(\\cdot ,t)$ .",
"In what follows, $\\partial =\\partial _{j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots ,n$ ."
],
[
"The Stokes system",
"We consider the Stokes equations, $\\begin{aligned}\\partial _{t}v-\\Delta v+\\nabla q&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {in}\\ \\Omega \\times (0,T), \\\\v&=0\\quad \\textrm {on}\\ \\partial \\Omega \\times (0,T), \\\\v&=v_0\\quad \\hspace{-4.0pt} \\textrm {on}\\ \\Omega \\times \\lbrace t=0\\rbrace .\\end{aligned}\\qquad \\mathrm {(2.1)}$ We set the norm $N(v,q)(x,t)=\\bigl |v(x,t)\\bigr |+t^{\\frac{1}{2}}\\bigl |\\nabla v(x,t)\\bigr |+t\\bigl |\\nabla ^{2}v(x,t)\\bigr |+t\\bigl |\\partial _{t}v(x,t)\\bigr |+t\\bigl |\\nabla q(x,t)\\bigr |.$ Let $d(x)$ denote the distance from $x\\in \\Omega $ to $\\partial \\Omega $ .",
"Let $(v, \\nabla q)\\in C^{2+\\mu ,1+\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T])\\times C^{\\mu ,\\frac{\\mu }{2}}(\\overline{\\Omega }\\times (0,T]),\\ \\mu \\in (0,1)$ , satisfy the equations and the boundary condition of $(2.1)$ .",
"We say that $(v,q)$ is a solution of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , if $\\sup _{0<t\\le T}\\Bigg \\lbrace t^{\\gamma }\\big \\Vert N(v,q)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d \\nabla q\\big \\Vert _{\\infty }(t)\\Bigg \\rbrace <\\infty , \\qquad \\mathrm {(2.2)}$ for some $\\gamma \\in [0,1/2)$ and $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =\\int _{\\Omega }f\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x \\qquad \\mathrm {(2.3)}$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ with $\\varphi _{0}(x)=\\varphi (x,0)$ .",
"The left-hand side is finite since $\\varphi (\\cdot ,t)$ is supported in $\\Omega $ and $\\gamma <1/2$ .",
"The right-hand side is finite since $\\partial \\mathbb {P}\\varphi _0$ is integrable in $\\Omega $ for $\\varphi _0\\in C_{c}^{\\infty }(\\Omega )$ by Lemma A.1.",
"As explained later in Remarks 2.9 (i), the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ and we are able to define $\\mathbb {P}\\partial f$ in the sense of distribution.",
"The goal of this section is to prove: Theorem 2.1 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.",
"Let $T>0$ .",
"For $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}(\\Omega )$ , there exists a unique solution $(v,q)$ of (2.1) satisfying $\\sup _{0<t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v,q)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d \\nabla q\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }, \\qquad \\mathrm {(2.4)}$ for $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ and some constant C, depending on $\\alpha $ , $T$ and $\\Omega $ .",
"Theorem 2.1 implies the following: Theorem 2.2 The composition operator $S(t)\\mathbb {P}\\partial $ is uniquely extendable to a bounded operator $\\overline{S(t)\\mathbb {P}\\partial }$ from $W^{1,\\infty }_{0}(\\Omega )$ to $L^{\\infty }_{\\sigma }(\\Omega )$ together with the estimate $\\sup _{0< t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}\\overline{S(t)\\mathbb {P}\\partial } f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }, \\qquad \\mathrm {(2.5)}$ for $f\\in W^{1,\\infty }_{0}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ ."
],
[
"Hölder estimates and uniqueness",
"In order to prove Theorem 2.1, we recall local Hölder estimates and a uniqueness result for the Stokes equations.",
"In the subsequent section, we give a proof for Theorem 2.1 by approximation.",
"We set the Hölder semi-norm $\\Big [f\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q}=\\sup _{t\\in (0,T]}\\Big [f\\Big ]^{(\\mu )}_{\\Omega }(t)+\\sup _{x\\in \\Omega }\\Big [f\\Big ]^{(\\frac{\\mu }{2})}_{(0,T]}(x),\\quad \\mu \\in (0,1),$ for $Q=\\Omega \\times (0,T]$ .",
"We set $N=\\sup _{\\delta \\le t\\le T}\\big \\Vert N(v,q)\\big \\Vert _{L^{\\infty }(\\Omega )}(t)$ for solutions $(v,q)$ of (2.1).",
"The following local Hölder estimate is proved in based on the Schauder estimates for the Stokes equations (, ).",
"Proposition 2.3 Let $\\Omega $ be an exterior domain with $C^3$ -boundary.",
"(i) (Interior estimates) For $\\mu \\in (0,1)$ , $\\delta >0$ , $T>0$ , $R>0$ , there exists a constant $C=C\\bigl (\\mu ,\\delta ,T, R,d\\bigr )$ such that $\\Big [\\nabla ^{2}v\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [v_t\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}+\\Big [\\nabla q\\Big ]^{(\\mu ,\\frac{\\mu }{2})}_{Q^{\\prime }}\\le CN \\qquad \\mathrm {(2.6)}$ holds for all solutions $(v,q)$ of (2.1) for $Q^{\\prime }=B_{x_0}(R)\\times (2\\delta ,T]$ and $x_0\\in \\Omega $ satisfying $\\overline{B_{x_0}(R)}\\subset \\Omega $ , where $d$ denotes the distance from $B_{x_0}(R)$ to $\\partial \\Omega $ .",
"(ii) (Estimates up to the boundary) There exists $R_0>0$ such that for $\\mu \\in (0,1),\\ \\delta >0$ , $T>0$ and $R\\le R_{0}$ , there exists a constant $C$ depending on $\\mu $ , $\\delta $ , $T$ , $R$ and $C^{3}$ -regularity of $\\partial \\Omega $ such that (2.6) holds for all solutions $(v, q)$ of (2.1) for $Q^{\\prime }=\\Omega _{x_0,R}\\times (2\\delta ,T]$ and $\\Omega _{x_0,R}=B_{x_0}(R)\\cap \\Omega $ , $x_0\\in \\partial \\Omega $ .",
"We observe the uniqueness of solutions for (2.1).",
"The uniqueness of the Stokes equations (2.1) for $v_0\\in L^{\\infty }_{\\sigma }$ in an exterior domain is proved based on the uniqueness result in a half space by a blow-up argument; see .",
"In order to prove Theorem 2.1, we need a stronger uniqueness result since solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , may not be bounded at $t=0$ .",
"The corresponding uniqueness result for a half space is recently proved in .",
"We deduce the result for exterior domains by the same blow-up argument as we did in .",
"Proposition 2.4 Let $\\Omega $ be an exterior domain with $C^{3}$ -boundary.",
"Let $(v,\\nabla q)\\in C^{2,1}(\\overline{\\Omega }\\times (0,T])\\times C(\\overline{\\Omega }\\times (0,T])$ satisfy the equations and the boundary condition of (2.1), and (2.2) for some $\\gamma \\in [0,1/2)$ .",
"Assume that $\\int _{0}^{T}\\int _{\\Omega }\\big (v\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t =0,$ for all $\\varphi \\in C^{\\infty }_{c}(\\Omega \\times [0,T))$ .",
"Then, $v\\equiv 0$ and $\\nabla q\\equiv 0$ ."
],
[
"Approximation",
"We prove Theorem 2.1.",
"We show existence of solutions for the Stokes equations (2.1) for $v_0=\\mathbb {P}\\partial f$ , $f\\in W^{1,\\infty }_{0}$ , by approximation.",
"We approximate $f\\in W^{1,\\infty }_{0}$ by elements of $C^{\\infty }_{c}$ locally uniformly in $\\overline{\\Omega }$ .",
"Lemma 2.5 Let $\\Omega $ be an exterior domain with Lipschitz boundary.",
"There exist constants $C_1$ , $C_2$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C_{c}^{\\infty }(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert f_{m}\\big \\Vert _{\\infty }\\le C_1\\big \\Vert f\\big \\Vert _{\\infty }, \\\\&\\big \\Vert \\nabla f_{m}\\big \\Vert _{\\infty }\\le C_2\\big \\Vert f\\big \\Vert _{1,\\infty }, \\\\&f_m\\rightarrow f\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.7)}$ The proof of Lemma 2.5 is reduced to the whole space and bounded domains.",
"Proposition 2.6 The statement of Lemma 2.5 holds when $\\Omega =\\mathbb {R}^{n}$ with $C_1=1$ .",
"We cutoff the function $f\\in W^{1,\\infty }(\\mathbb {R}^{n})$ .",
"Let $\\theta \\in C^{\\infty }_{c}[0,\\infty )$ be a cut-off function satisfying $\\theta \\equiv 1$ in $[0,1]$ , $\\theta \\equiv 0$ in $[2,\\infty )$ and $0\\le \\theta \\le 1$ .",
"We set $\\theta _{m}(x)=\\theta (|x|/m)$ for $m\\ge 1$ so that $\\theta _{m}\\equiv 1$ for $|x|\\le m$ and $\\theta _{m}\\equiv 0$ for $|x|\\ge 2m$ .",
"Then, $f_m=f\\theta _{m}$ satisfies (2.7).",
"Proposition 2.7 Let $\\Omega $ be a bounded domain with Lipschitz boundary.",
"There exists a constant $C_3$ such that for $f\\in W^{1,\\infty }_{0}(\\Omega )$ there exists a sequence of functions $\\lbrace f_m\\rbrace _{m=1}^{\\infty }\\subset C^{\\infty }_{c}(\\Omega )$ such that $\\begin{aligned}&\\big \\Vert \\nabla f_m\\big \\Vert _{\\infty }\\le C_3\\big \\Vert \\nabla f\\big \\Vert _{\\infty }\\\\f_m&\\rightarrow f\\quad \\textrm {uniformly in}\\ \\overline{\\Omega }\\quad \\textrm {as}\\ m\\rightarrow \\infty .\\end{aligned}\\qquad \\mathrm {(2.8)}$ We begin with the case when $\\Omega $ is star-shaped, i.e., $\\lambda \\Omega _{x_0}\\subset \\overline{\\Omega }$ for some $x_0\\in \\Omega $ and all $\\lambda <1$ , where $\\lambda \\Omega _{x_0}=\\lbrace x_0+\\lambda (x-x_0)\\ |\\ x\\in \\Omega \\rbrace $ .",
"We may assume $x_0=0\\in \\Omega $ and $\\lambda \\Omega \\subset \\overline{\\Omega }$ by translation.",
"For $f\\in W^{1,\\infty }_{0}(\\Omega )$ , we set $f_{\\lambda }(x)={\\left\\lbrace \\begin{array}{ll}&f\\big (x/\\lambda \\big )\\quad x\\in \\lambda \\Omega ,\\\\&0\\qquad \\hspace{15.0pt} x\\in \\Omega \\backslash \\overline{\\lambda \\Omega } .\\end{array}\\right.",
"}$ Then, $f_{\\lambda }$ is in $ W^{1,\\infty }(\\Omega )$ since $f$ is vanishing on $\\partial \\Omega $ .",
"It follows that $\\big \\Vert \\nabla f_{\\lambda }\\big \\Vert _{\\infty }\\le \\frac{1}{\\lambda }\\big \\Vert \\nabla f\\big \\Vert _{\\infty },$ and $f_{\\lambda }\\rightarrow f$ uniformly in $\\overline{\\Omega }$ as $\\lambda \\rightarrow 1$ .",
"By a mollification of $f_{\\lambda }$ , we obtain a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfying (2.8) with $C_3=2$ .",
"For general $\\Omega $ , we take an open covering $\\lbrace D_j\\rbrace _{j=1}^{N}$ so that $\\overline{\\Omega }\\subset \\cup _{j=1}^{N}D_j$ and $\\Omega _{j}=\\Omega \\cap D_{j}$ is Lipschitz and star-shaped for some $x_j\\in \\Omega _{j}$ .",
"We take a partition of unity $\\lbrace \\xi _{j}\\rbrace _{j=1}^{N}\\subset C_{c}^{\\infty }(\\mathbb {R}^{n})$ such that $\\sum _{j=1}^{N}\\xi _{j}=1$ , $0\\le \\xi _{j}\\le 1$ , spt $\\xi _{j}\\subset \\overline{D_j}$ and set $f=\\sum _{j=1}^{N}f_j,\\quad f_j=f\\xi _j.$ Since spt $f_j\\subset \\overline{\\Omega }_{j}$ , $\\xi _{j}=0$ on $\\partial D_{j}$ and $f=0$ on $\\partial \\Omega $ , $f_{j}$ is in $W^{1,\\infty }_{0}(\\Omega _{j})$ .",
"Since $\\Omega _j$ is star-shaped for some $x_j\\in \\Omega _{j}$ , there exists $\\lbrace f_{j,m}\\rbrace \\subset C_{c}^{\\infty }(\\Omega _j)$ satisfying (2.8) in $\\Omega _j$ with $C_3=2$ .",
"We extend $f_{j,m}\\in C_{c}^{\\infty }(\\Omega _j)$ to $\\Omega \\backslash \\overline{\\Omega _j}$ by the zero extension (still denoted by $f_{j,m}$ ) and set $f_m=\\sum _{j=1}^{N}f_{j,m}$ .",
"Then, $f_m\\in C_{c}^{\\infty }(\\Omega )$ converges to $f$ uniformly in $\\overline{\\Omega }$ .",
"We estimate $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le \\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j,m}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}\\le 2\\sum _{j=1}^{N}\\big \\Vert \\nabla f_{j}\\big \\Vert _{L^{\\infty }(\\Omega _{j})}.$ Since $\\nabla f_j=\\nabla f\\xi _j+f\\nabla \\xi _j$ and $\\big \\Vert f\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C_{p}\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )},$ by the Poincaré inequality (e.g., ), we obtain $\\big \\Vert \\nabla f_m\\big \\Vert _{L^{\\infty }(\\Omega )}\\le C\\big \\Vert \\nabla f\\big \\Vert _{L^{\\infty }(\\Omega )}.$ Thus, $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }(\\Omega )$ satisfies (2.8).",
"The proof is complete.",
"The assertion follows from Propositions 2.6 and 2.7.",
"We recall the a priori estimate of $S(t)\\mathbb {P}\\partial $ for $f\\in C_{c}^{\\infty }(\\Omega )$ .",
"Proposition 2.8 There exists a constant $C$ such that $\\sup _{0\\le t\\le T}t^{\\gamma +\\frac{|k|}{2}+s}\\Big \\Vert \\partial _{t}^{s}\\partial _{x}^{k}S(t)\\mathbb {P}\\partial f\\Big \\Vert _{\\infty }\\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert \\nabla f\\big \\Vert _{\\infty }^{\\alpha } \\qquad \\mathrm {(2.9)}$ for $f\\in C^{\\infty }_{c}(\\Omega )$ , $0\\le 2s+|k|\\le 2$ and $\\alpha \\in (0,1)$ with $\\gamma =(1-\\alpha )/2$ .",
"For $f\\in W^{1,\\infty }_{0}$ , we take a sequence $\\lbrace f_m\\rbrace \\subset C^{\\infty }_{c}$ satisfying (2.7).",
"For $v_{0,m}=\\mathbb {P}\\partial f_m$ , there exists a solution of the Stokes equations $(v_m,q_m)$ satisfying $\\int _{0}^{T}\\int _{\\Omega }\\big ( v_{m}\\cdot (\\partial _{t}\\varphi +\\Delta \\varphi )-\\nabla q_m\\cdot \\varphi \\big )\\textrm {d}x\\textrm {d}t=\\int _{\\Omega }f_m\\cdot \\partial \\mathbb {P}\\varphi _{0}\\textrm {d}x,$ for $\\varphi \\in C^{\\infty }_{c}\\big (\\Omega \\times [0,T)\\big )$ .",
"By (2.9) and (2.7), there exists a constant $C$ independent of $m\\ge 1$ such that $\\sup _{0\\le t\\le T}\\Big \\lbrace t^{\\gamma }\\big \\Vert N(v_m,q_m)\\big \\Vert _{\\infty }(t)+t^{\\gamma +\\frac{1}{2}}\\big \\Vert d\\nabla q_{m}\\big \\Vert _{\\infty }(t)\\Big \\rbrace \\le C\\big \\Vert f\\big \\Vert _{\\infty }^{1-\\alpha }\\big \\Vert f\\big \\Vert _{1,\\infty }^{\\alpha }.$ We apply Proposition 2.3 and observe that there exists a subsequence of $(v_m,q_m)$ such that $(v_m,q_m)$ converges to a limit $(v,q)$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla v_m$ , $\\nabla ^{2}v_m$ , $\\partial _{t}v_m$ and $\\nabla q_{m}$ .",
"By sending $m\\rightarrow \\infty $ , we obtain a solution $(v,q)$ of (2.1) for $v_0=\\mathbb {P}\\partial f$ .",
"By Proposition 2.4, the limit $(v,q)$ is unique.",
"We proved the unique existence of solutions of (2.1) for $v_0=\\mathbb {P}\\partial f$ and $f\\in W^{1,\\infty }_{0}$ satisfying (2.4).",
"The proof is now complete.",
"Remarks 2.9 (i) By the approximation (2.7) we are able to extend the operator $\\mathbb {P}\\partial $ for $f\\in W^{1,\\infty }_{0}$ .",
"We take a sequence $\\lbrace f_m\\rbrace \\subset C_{c}^{\\infty }$ satisfying (2.7) by Lemma 2.5 and observe that $v_{0,m}=\\mathbb {P}\\partial f_m$ satisfies $(v_{0,m}, \\varphi )=-(f_m, \\partial \\mathbb {P}\\varphi )\\quad \\textrm {for}\\ \\varphi \\in C^{\\infty }_{c}(\\Omega ).$ Since $\\partial \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ by Lemma A.1, the sequence $\\lbrace v_{0,m}\\rbrace $ converges to a limit $v_0$ in the distributional sense and the limit $v_0$ satisfies $(v_0,\\varphi )=-(f,\\partial \\mathbb {P}\\varphi )$ .",
"Since the limit $v_0$ is unique, the operator $\\mathbb {P}\\partial $ is uniquely extendable for $f\\in W^{1,\\infty }_{0}$ .",
"(ii) We recall that for a sequence $\\lbrace v_{0,m}\\rbrace _{m=1}^{\\infty }\\subset L^{\\infty }_{\\sigma }$ satisfying $||v_{0,m}||_{\\infty }&\\le K_1,\\\\v_{0,m}\\rightarrow v_{0}&\\quad \\textrm {a.e.",
"}\\ \\Omega ,$ with some constant $K_1$ , there exists a subsequence such that $S(t)v_{0,m}$ converges to $S(t)v_0$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .",
"From the proof of Theorem 2.1, we observe that for a sequence $\\lbrace f_m\\rbrace \\subset W^{1,\\infty }_{0}$ satisfying $||f_m||_{1,\\infty }&\\le K_2,\\\\f_m\\rightarrow f\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega },$ $\\overline{S(t)\\mathbb {P}\\partial } f_m$ subsequently converges to $\\overline{S(t)\\mathbb {P}\\partial } f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ .",
"(iii) The extension $\\overline{S(t)\\mathbb {P}\\partial }$ satisfies the property $S(t)\\overline{S(s)\\mathbb {P}\\partial } f=\\overline{S(t+s)\\mathbb {P}\\partial } f$ for $t\\ge 0$ , $s>0$ and $f\\in W^{1,\\infty }_{0}$ .",
"In fact, this property holds for $f_m\\in C_{c}^{\\infty }$ satisfying (2.7).",
"By choosing a subsequence, $v_m(\\cdot ,t)=S(t)\\mathbb {P}\\partial f_m$ converges to $v(\\cdot ,t)=S(t)\\mathbb {P}\\partial f$ locally uniformly in $\\overline{\\Omega }\\times (0,\\infty )$ as in the proof of Theorem 2.1.",
"For fixed $s>0$ , sending $m\\rightarrow \\infty $ implies $S(t)S(s)\\mathbb {P}\\partial f_m=S(t)v_m(s)&\\rightarrow S(t)v(s) \\\\&=S(t)\\overline{S(s)\\mathbb {P}\\partial }f\\quad \\textrm {locally uniformly in \\overline{\\Omega }\\times (0,\\infty )}.$ Thus the property is inherited to $\\overline{S(t)\\mathbb {P}\\partial } f$ ."
],
[
"Mild solutions on $L^{\\infty }_{\\sigma }$",
"We prove Theorem 1.1 by approximation.",
"We show that a sequence of mild solutions $\\lbrace u_m\\rbrace $ subsequently converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by the $L^{\\infty }$ -estimates (1.5) and (1.6).",
"Then, by an approximation argument for linear operators, we show that the limit $u$ satisfies the integral equation (1.3).",
"We first recall the existence of mild solutions on $C_{0,\\sigma }$ Proposition 3.1 For $u_0\\in C_{0,\\sigma }$ , there exist $T\\ge \\varepsilon _0/||u_0||_{\\infty }^{2}$ and a unique mild solution $u\\in C([0,T]; C_{0,\\sigma })$ satisfying (1.3)-(1.6).",
"We approximate $u_0\\in L^{\\infty }_{\\sigma }$ by elements of $C_{c,\\sigma }^{\\infty }\\subset C_{0,\\sigma }$ .",
"We take a sequence $\\lbrace u_{0,m}\\rbrace _{m=1}^{\\infty }\\subset C_{c,\\sigma }^{\\infty }(\\Omega )$ satisfying $\\begin{aligned}&\\Vert u_{0,m}\\Vert _{\\infty }\\le C\\Vert u_{0}\\Vert _{\\infty }\\\\&u_{0,m}\\rightarrow u_{0}\\quad \\textrm {a.e.",
"in}\\ \\Omega ,\\end{aligned}\\qquad \\mathrm {(3.1)}$ with some constant $C$ , independent of $m\\ge 1$ .",
"We apply Proposition 3.1 and observe that there exists $T_m\\ge \\varepsilon _0/||u_{0,m}||_{\\infty }^{2}$ and a unique mild solution $u_m\\in C([0,T_m]; C_{0,\\sigma })$ satisfying $\\begin{aligned}u_m(t)&=S(t)u_{0,_m}-\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P}F_m(s)ds,\\\\F_m&=u_mu_m.",
"}\\qquad \\mathrm {(3.2)}\\end{aligned}Since T_m is estimated from below by (3.1), we take T\\ge \\varepsilon /||u_0||_{\\infty }^{2} for \\varepsilon =\\varepsilon _0 C^{-2}/2 so that T_m\\ge T and u_m\\in C([0,T]; C_{0,\\sigma }) for m\\ge 1.$ Proposition 3.2 There exists a subsequence such that $u_m$ converges to a limit $u$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ together with $\\nabla u_m$ .",
"It follows from (1.5), (1.6) and (3.1) that $\\sup _{0\\le t\\le T}\\Big \\lbrace \\Vert u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1}{2}}\\Vert \\nabla u_{m}\\Vert _{\\infty }(t)+t^{\\frac{1+\\beta }{2}}\\big [\\nabla u_{m}\\big ]_{\\Omega }^{(\\beta )}(t)\\Big \\rbrace &\\le C_1^{\\prime }\\Vert u_0\\Vert _{\\infty }, \\\\\\sup _{x\\in \\Omega }\\Big \\lbrace \\big [u_{m}\\big ]_{[\\delta ,T]}^{(\\gamma )}(x)+\\big [\\nabla u_{m}\\big ]_{[\\delta ,T]}^{(\\frac{\\gamma }{2})}(x)\\Big \\rbrace &\\le C_2^{\\prime }\\Vert u_0\\Vert _{\\infty }, $ for $\\beta ,\\gamma \\in (0,1)$ and $\\delta \\in (0,T]$ with some constants $C_1^{\\prime }$ and $C_2^{\\prime }$ , independent of $m\\ge 1$ .",
"Since $u_m$ and $\\nabla u_m$ are uniformly bounded and equi-continuous in $\\overline{\\Omega }\\times [\\delta ,T]$ , the assertion follows from the Ascoli-Arzelà theorem.",
"Proposition 3.3 The limit $u\\in C_{w}([0,T]; L^{\\infty })$ is a mild solution for $u_0\\in L^{\\infty }_{\\sigma }$ .",
"We observe that the limit $u$ satisfies (1.4) by sending $m\\rightarrow \\infty $ .",
"The estimates (3.3) and (3.4) are inherited to $u$ .",
"We prove that $u$ satisfies the integral equation (1.3).",
"By (3.1) and choosing a subsequence, $S(t)u_{0,m}$ converges to $S(t)u_0$ locally uniformly in $\\overline{\\Omega }\\times (0,T]$ by Remarks 2.9 (ii).",
"It follows from (3.3) and Proposition 3.2 that $\\begin{aligned}||F_m||_{\\infty }&\\le K,\\\\||\\nabla F_{m}||_{\\infty }&\\le \\frac{2}{s^{\\frac{1}{2}}}K,\\\\F_m\\rightarrow F\\quad &\\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T]\\ \\textrm {as}\\ m\\rightarrow \\infty ,\\end{aligned}\\qquad \\mathrm {(3.5)}$ for $F=uu$ and $K=C_1^{\\prime }||u_0||_{\\infty }$ .",
"By choosing a subsequence, we have $\\overline{S(\\eta )\\mathbb {P} F_m\\rightarrow \\overline{S(\\eta )\\mathbb {P} F\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times (0,T],}for each s\\in (0,t) as in Remarks 2.9 (ii).",
"It follows from (3.5) and (2.5) that\\begin{equation*}\\big \\Vert \\overline{S(t-s)\\mathbb {P} F_m\\big \\Vert _{\\infty }\\le \\frac{C}{(t-s)^{\\frac{1-\\alpha }{2}}}\\Bigg (1+\\frac{2}{s^{\\frac{\\alpha }{2}}}\\Bigg )K^{2}}for 0<s< t and \\alpha \\in (0,1).",
"By the dominated convergence theorem, we have{\\begin{@align*}{1}{-1}\\int _{0}^{t}\\overline{S(t-s)\\mathbb {P} F_m\\textrm {d}s\\rightarrow \\int _{0}^{t}\\overline{S(t-s)\\mathbb {P} F\\textrm {d}s\\quad \\textrm {locally uniformly in}\\ \\overline{\\Omega }\\times [0,T].",
"}}Thus sending m\\rightarrow \\infty implies that the limit u is a mild solution for u_0\\in L^{\\infty }_{\\sigma }.",
"Since S(t)u_0 is weakly-star continuous on L^{\\infty } at t=0 \\cite {AG2}, so is u.\\end{@align*}}\\vspace{5.0pt}\\end{equation*}}It remains to show continuity at t=0 for u_0\\in BUC_{\\sigma }.$ Proposition 3.4 For $u_0\\in BUC_{\\sigma }$ , $S(t)u_0$ , $t^{1/2}\\nabla S(t)u_0\\in C([0,T]; BUC)$ and $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"Since $S(t)$ is a $C_0$ -analytic semigroup on $BUC_{\\sigma }$ , $S(t)u_0\\in C([0,T]; BUC_{\\sigma })$ .",
"Moreover, $t^{1/2}\\nabla S(t)u_0$ is continuous and bounded for $t\\in (0,T]$ in $BUC$ .",
"We show that $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"We divide $u_0$ into two terms by using the Bogovskiĭ operator.",
"For $u_0\\in BUC_{\\sigma }$ , there exists $u_{0}^{1}\\in C_{0,\\sigma }$ with compact support in $\\overline{\\Omega }$ and $u_{0}^{2}\\in BUC_{\\sigma }$ supported away from $\\partial \\Omega $ such that $u_0=u_0^{1}+u_0^{2}$ (see ).",
"Let $A$ denote the Stokes operator and $D(A)$ denote the domain of $A$ in $BUC_{\\sigma }$ .",
"Since $S(t)$ is a $C_0$ -semigroup on $BUC_{\\sigma }$ , $D(A)$ is dense in $BUC_{\\sigma }$ .",
"It follows from the resolvent estimate that $||\\nabla v||_{\\infty }\\le C(||v||_{\\infty }+||Av||_{\\infty })\\quad \\textrm {for}\\ v\\in D(A).",
"$ We take an arbitrary $\\epsilon >0$ .",
"For $u_{0}^{1}\\in C_{0,\\sigma }$ , there exists $\\lbrace u_{0,m}^{1}\\rbrace \\subset C_{c,\\sigma }^{\\infty }$ such that $||u_{0}^{1}-u_{0,m}^{1}||_{\\infty }\\le \\epsilon $ for $m\\ge N^{1}_{\\epsilon }$ .",
"We apply (3.5) and observe that $t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }&\\le t^{\\frac{1}{2}}C\\big (||S(t)u_{0,m}^{1}||_{\\infty }+||S(t)Au_{0,m}^{1}||_{\\infty }\\big )\\\\&\\le t^{\\frac{1}{2}}C^{\\prime }\\big (||u_{0,m}^{1}||_{\\infty }+||Au_{0,m}^{1}||_{\\infty }\\big )\\rightarrow 0\\quad \\textrm {as}\\ t\\rightarrow 0.$ We estimate $\\overline{\\lim _{t\\rightarrow 0}}t^{\\frac{1}{2}}||\\nabla S(t)u_0^{1}||_{\\infty }&\\le \\overline{\\lim _{t\\rightarrow 0}}\\big (t^{\\frac{1}{2}}||\\nabla S(t)(u_0^{1}-u_{0,m}^{1})||_{\\infty }+t^{\\frac{1}{2}}||\\nabla S(t)u_{0,m}^{1}||_{\\infty }\\big ) \\\\&\\le C^{\\prime \\prime }\\epsilon .$ We set $u_{0,m}^{2}=\\eta _{\\delta _m}*u_{0}^{2}$ by the mollifier $\\eta _{\\delta _{m}}$ so that $u_{0,m}^{2}$ is smooth in $\\overline{\\Omega }$ and $||u_{0}^{2}-u_{0,m}^{2}||_{\\infty }\\le \\epsilon $ for $m\\ge N_{\\epsilon }^{2}$ .",
"Since $u_{0,m}^{2}$ is supported away from $\\partial \\Omega $ , we have $AS(t)u_{0,m}^{2}=S(t)\\Delta u_{0,m}^{2}$ (see ).",
"By a similar way as for $u_{0}^{1}$ , we estimate $\\overline{\\lim }_{t\\rightarrow 0}t^{1/2}||\\nabla S(t)u_0^{2}||_{\\infty }\\le C^{\\prime \\prime }\\epsilon $ .",
"We proved $\\overline{\\lim }_{t\\rightarrow 0}t^{\\frac{1}{2}}||\\nabla S(t)u_0||_{\\infty }\\le 2C^{\\prime \\prime }\\epsilon .$ Since $\\epsilon >0$ is arbitrary, we proved $t^{1/2}||\\nabla S(t)u_0||_{\\infty }\\rightarrow 0$ as $t\\rightarrow 0$ .",
"The assertion follows from Propositions 3.1-3.4.",
"The proof is now complete.",
"Remark 3.5 We set the associated pressure of mild solutions on $L^{\\infty }$ by (1.7) and the harmonic-pressure operator $\\mathbb {K}: L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )\\longrightarrow L^{\\infty }_{d}(\\Omega )$ , which is a solution operator of the homogeneous Neumann problem, $\\Delta q&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial q}{\\partial n}&={\\partial \\Omega }W\\quad \\textrm {on}\\ \\partial \\Omega .$ Note that $\\Delta u\\cdot n={\\partial \\Omega }W$ by the divergence-free condition of $u$ .",
"Here, $L^{\\infty }_{\\textrm {tan}}(\\partial \\Omega )$ denotes the space of all bounded tangential vector fields on $\\partial \\Omega $ and $L^{\\infty }_{d}(\\Omega )$ is the space of all functions $f\\in L^{1}_{\\textrm {loc}}(\\Omega )$ such that $df$ is bounded in $\\Omega $ for $d(x)=\\inf _{y\\in \\partial \\Omega }|x-y|$ , $x\\in \\Omega $ .",
"Since $W=-(\\nabla u-\\nabla ^{T}u)n$ is bounded on $\\partial \\Omega $ for mild solutions on $L^{\\infty }$ , $\\nabla q=\\mathbb {K}W$ is defined as an element of $L^{\\infty }_{d}$ .",
"Moreover, $\\mathbb {Q}F̥$ is uniquely defined for $F=uu\\in W^{1,\\infty }_{0}$ as a distribution by Remarks 2.9 (i).",
"Thus the associated pressure is defined by (1.7) for mild solutions on $L^{\\infty }$ .",
"acknowledgements The author is grateful to the anonymous referees for their valuable comments.",
"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015.",
"appendix $L^{1}$ -estimates for the Neumann problem In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .",
"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.",
"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.",
"Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .",
"Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .",
"We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .",
"It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .",
"We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .",
"We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .",
"We may assume that $0\\in \\Omega ^{c}$ by translation.",
"We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .",
"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.",
"We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .",
"In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .",
"Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .",
"Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .",
"In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .",
"By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .",
"We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .",
"We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .",
"Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .",
"Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .",
"The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .",
"We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .",
"Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.",
"By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .",
"Thus, $g$ is continuous on $\\partial \\Omega $ .",
"Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .",
"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.",
"The proof is complete.",
"We estimate $\\Phi _2$ by means of the layer potential.",
"Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .",
"The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .",
"The assertion (i) is based on the Fredholm's theorem.",
"See .",
"Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .",
"Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).",
"When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .",
"Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.",
"Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.",
"Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.",
"We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.",
"The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .",
"By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .",
"We shall show that $\\Psi \\equiv 0$ .",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .",
"Suppose that $x_0\\in \\partial \\Omega $ .",
"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .",
"Thus $x_0\\in \\Omega $ .",
"We apply the strong maximum principle and conclude that $\\Psi $ is constant.",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .",
"The proof is complete.",
"Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .",
"Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .",
"Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .",
"The proof is complete.",
"By Propositions A.2 and A.6, the assertion follows.",
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[
"acknowledgements",
"The author is grateful to the anonymous referees for their valuable comments.",
"This work was partially supported by JSPS through the Grant-in-aid for Research Activity Start-up 15H06312 and Kyoto University Research Founds for Young Scientists (Start-up) FY2015."
],
[
"$L^{1}$ -estimates for the Neumann problem",
" In Appendix A, we prove that $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ , $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ , for an exterior domain $\\Omega $ .",
"We first estimate $L^{1}$ -norms of solutions for the Poisson equation in $\\mathbb {R}^{n}$ by using the heat semigroup.",
"Then, we reduce the problem to the homogeneous Neumann problem and estimate solutions by a layer potential.",
"Lemma 1.1 Let $\\Omega $ be an exterior domain with $C^{2}$ -boundary in $\\mathbb {R}^{n}$ , $n\\ge 2$ .",
"Then, $\\nabla \\mathbb {P}\\varphi \\in L^{1}(\\Omega )$ for $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ .",
"We set $\\nabla \\Phi =\\varphi $ for $=I-\\mathbb {P}$ .",
"It suffices to show that $\\nabla ^{2}\\Phi $ is integrable in $\\Omega $ .",
"We recall that the $\\Phi $ solves the Neumann problem $\\begin{aligned}\\Delta \\Phi =\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\varphi }{\\partial n}=0\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.1)}$ See .",
"We observe that $\\Phi \\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ by the elliptic regularity theory (e.g., ) since $\\varphi $ is smooth in $\\Omega $ and the boundary is $C^{2}$ .",
"We may assume that $0\\in \\Omega ^{c}$ by translation.",
"We take $R>0$ such that $\\Omega ^{c}\\subset B_{0}(R)$ .",
"Let $E$ denote the fundamental solution of the Laplace equation, i.e., $E(x)=C_{n}|x|^{-(n-2)} $ for $n\\ge 3$ and $E(x)=-(2\\pi )^{-1}\\log {|x|}$ for $n=2$ , where $C_{n}=(an(n-2))^{-1}$ and $a$ denotes the volume of $n$ -dimensional unit ball.",
"We first show that the statement of Lemma A.1 is valid for $\\Omega =\\mathbb {R}^{n}$ .",
"In the sequel, we do not distinguish $\\varphi \\in C_{c}^{\\infty }(\\Omega )$ and its zero extension to $\\mathbb {R}^{n}\\backslash \\Omega $ .",
"Proposition 1.2 Set $h=E*\\varphi $ and $\\Phi _1=-\\textrm {div}\\ h$ .",
"Then, $\\nabla ^{3}h$ is integrable in $\\mathbb {R}^{n}$ .",
"In particular, $\\nabla ^{2}\\Phi _1\\in L^{1}(\\mathbb {R}^{n})$ .",
"By using the heat semigroup, we transform $h$ into $h=\\int _{0}^{\\infty }e^{t\\Delta }\\varphi \\textrm {d}t.$ We divide $h$ into two terms and observe that $\\partial ^{3}_{x}h=\\int _{0}^{1}\\partial _{x} e^{t\\Delta }\\partial ^{2}_{x}\\varphi \\textrm {d}t+\\int _{1}^{\\infty }\\partial ^{3}_{x}e^{t\\Delta }\\varphi \\textrm {d}t,$ where $\\partial _{x}=\\partial _{x_j}$ indiscriminately denotes the spatial derivatives for $j=1,\\cdots n$ .",
"We estimate $||\\partial ^{3}_{x} h||_{L^{1}(\\mathbb {R}^{n})}&\\lesssim \\int _{0}^{1}\\frac{1}{t^{1/2}}||\\partial ^{2}_{x} \\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t+\\int _{1}^{\\infty }\\frac{1}{t^{3/2}}||\\varphi ||_{L^{1}(\\mathbb {R}^{n})} \\textrm {d}t\\\\&\\lesssim ||\\partial ^{2}_{x}\\varphi ||_{L^{1}(\\mathbb {R}^{n})}+||\\varphi ||_{L^{1}(\\mathbb {R}^{n})}.$ We proved $\\nabla ^{3}h\\in L^{1}(\\mathbb {R}^{n})$ .",
"We reduce (A.1) to the homogeneous Neumann problem $\\begin{aligned}-\\Delta \\Phi _2&=0\\quad \\textrm {in}\\ \\Omega ,\\\\\\frac{\\partial \\Phi _2}{\\partial n}&=g\\quad \\textrm {on}\\ \\partial \\Omega .\\end{aligned}\\qquad \\mathrm {(A.2)}$ We write connected components of $\\Omega $ by unbounded $\\Omega _0$ and bounded $\\Omega _1$ , $\\cdots $ , $\\Omega _N$ , i.e., $\\Omega =\\Omega _{0}\\cup (\\cup _{j=1}^{N}\\Omega _{j})$ .",
"Proposition 1.3 Set $\\Phi _2=\\Phi -\\Phi _1$ .",
"Then, $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ solves (A.2) for $g=\\textrm {div}_{\\partial \\Omega }(An)$ and $A=\\nabla h-\\nabla ^{T} h$ .",
"The function $g\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega _{j}}g \\textrm {d}{\\mathcal {H}}=0\\quad \\textrm {for}\\ j=0,1,\\cdots ,N. $ We observe that $\\Phi _2\\in C^{2}(\\Omega )\\cap C^{1}(\\overline{\\Omega })$ satisfies $-\\Delta \\Phi _2=0$ in $\\Omega $ and $\\partial \\Phi _2/\\partial n=\\partial (h̥)/\\partial n$ on $\\partial \\Omega $ .",
"We take an arbitrary $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ .",
"Since $An=(\\sum _{1\\le j\\le n}(\\partial _jh^{i}-\\partial _ih^{j})n^{j})_{1\\le i\\le n}$ is a tangential vector field on $\\partial \\Omega $ (i.e., $An\\cdot n=0$ on $\\partial \\Omega $ ), applying integration by parts yields $\\int _{\\partial \\Omega } g\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }\\textrm {div}_{\\partial \\Omega }(An)\\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(An)\\cdot \\nabla \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }(\\partial _jh^{i}-\\partial _ih^{j})n^{j}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}\\\\&=-\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _i \\rho \\textrm {d}{\\mathcal {H}}+\\int _{\\partial \\Omega }\\partial _j h^{i}n^{i}\\partial _j \\rho \\textrm {d}{\\mathcal {H}},$ where the symbol of summation is suppressed.",
"By integration by parts, we have $\\int _{\\partial \\Omega }\\partial _jh^{i}n^{j}\\partial _i\\rho \\textrm {d}{\\mathcal {H}}&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho +\\nabla h^{i}\\cdot \\nabla \\partial _i \\rho ) \\textrm {d}x\\\\&=\\int _{\\partial \\Omega }(\\Delta h^{i}\\partial _i\\rho -\\nabla h̥\\cdot \\nabla \\rho ) \\textrm {d}x+\\int _{\\partial \\Omega }\\nabla h^{i}\\cdot \\nabla \\rho n^{i}\\textrm {d}{\\mathcal {H}}.$ Since $-\\Delta h=\\varphi $ is supported in $\\Omega $ , it follows that $\\int _{\\partial \\Omega }g\\rho {\\mathcal {H}}&=-\\int _{\\Omega }(\\Delta h-\\nabla h)\\cdot \\nabla \\rho \\textrm {d}x \\\\&=-\\int _{\\Omega }(\\Delta h\\cdot \\nabla \\rho +\\Delta h\\rho )\\textrm {d}x+\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}\\\\&=\\int _{\\partial \\Omega }\\frac{\\partial }{\\partial n}h\\rho \\textrm {d}{\\mathcal {H}}.$ Since $\\partial \\Omega $ is $C^{2}$ , $n$ is extendable to a $C^{1}$ -function in a tubular neighborhood of $\\partial \\Omega $ .",
"Thus, $g$ is continuous on $\\partial \\Omega $ .",
"Since $\\rho \\in C_{c}^{\\infty }(\\mathbb {R}^{n})$ is arbitrary, we proved $\\partial (h)/\\partial n=g$ on $\\partial \\Omega $ .",
"Since $g$ is a surface-divergence form, by integration by parts, (A.3) follows.",
"The proof is complete.",
"We estimate $\\Phi _2$ by means of the layer potential.",
"Proposition 1.4 (i) For $g\\in C(\\partial \\Omega )$ satisfying (A.3), there exists a moment $h\\in C(\\partial \\Omega )$ satisfying $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ and $-g(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla _x E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ (ii) Set the single layer potential $\\tilde{\\Phi }_{2}(x)=-\\int _{\\partial \\Omega }E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Then, $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative $\\partial _n \\tilde{\\Phi }_{2}$ exits and is continuous on $\\partial \\Omega $ .",
"The function $\\tilde{\\Phi }_{2}$ satisfies (A.2) and decays as $|x|\\rightarrow \\infty $ .",
"The assertion (i) is based on the Fredholm's theorem.",
"See .",
"Since $h$ is bounded on $\\partial \\Omega $ , $\\tilde{\\Phi }_{2}$ is continuous in $\\overline{\\Omega }$ .",
"Moreover, we have $-\\frac{\\partial \\tilde{\\Phi }_{2}}{\\partial n}(x)=\\frac{1}{2}h(x)+\\int _{\\partial \\Omega }n(x)\\cdot \\nabla E(x-y)h(y)\\textrm {d}{\\mathcal {H}}(y)\\quad x\\in \\partial \\Omega .$ See .",
"Thus $\\tilde{\\Phi }_{2}$ satisfies (A.2) by the assertion (i).",
"When $n\\ge 3$ , $\\tilde{\\Phi }_{2}(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ since the fundamental solution decays as $|x|\\rightarrow \\infty $ .",
"Moreover, when $n=2$ , the average of $h$ on $\\partial \\Omega $ is zero and we have $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\frac{1}{2\\pi }\\int _{\\partial \\Omega }\\log {\\Bigg (\\frac{|x-y|}{|x|}\\Bigg )}h(y)\\textrm {d}\\mathcal {H}(y)\\rightarrow 0\\quad \\textrm {as}\\ |x|\\rightarrow \\infty .$ The proof is complete.",
"Proposition 1.5 The function $\\tilde{\\Phi }_2$ agrees with ${\\Phi }_2$ up to constant.",
"Since $\\nabla \\Phi _2=\\nabla \\Phi -\\nabla \\Phi _1$ is $L^{p}$ -integrable in $\\Omega $ for all $p\\in (1,\\infty )$ (e.g., ), we may assume that $\\Phi _2\\rightarrow 0$ as $|x|\\rightarrow \\infty $ by shifting $\\Phi _2$ by a constant.",
"We set $\\Psi =\\Phi _2-\\tilde{\\Phi }_2 $ and observe that $\\Psi $ is continuous in $\\overline{\\Omega }$ .",
"Moreover, the normal derivative exists and is continuous on $\\partial \\Omega $ by Proposition A.4.",
"The function $\\Psi $ satisfies $-\\Delta \\Psi =0$ in $\\Omega $ , $\\partial \\Psi /\\partial n=0$ on $\\partial \\Omega $ and $\\Psi \\rightarrow 0$ as $|x|\\rightarrow \\infty $ .",
"By the elliptic regularity theory , $\\Psi $ is smooth in $\\Omega $ and continuously differentiable in $\\overline{\\Omega }$ .",
"We shall show that $\\Psi \\equiv 0$ .",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , there exits a point $x_0\\in \\overline{\\Omega }$ such that $\\sup _{x\\in \\Omega }\\Psi (x)=\\Psi (x_0)$ .",
"Suppose that $x_0\\in \\partial \\Omega $ .",
"Since the boundary of class $C^{2}$ satisfies the interior sphere condition, the Hopf's lemma implies that $\\partial \\Psi (x_0)/\\partial n>0$ .",
"Thus $x_0\\in \\Omega $ .",
"We apply the strong maximum principle and conclude that $\\Psi $ is constant.",
"Since $\\Psi $ decays as $|x|\\rightarrow \\infty $ , we have $\\Psi \\equiv 0$ .",
"The proof is complete.",
"Proposition 1.6 $\\nabla ^{2}\\Phi _2$ is integrable in $\\Omega $ .",
"Since $\\nabla ^{2}\\Phi _2$ is integrable near the boundary $\\partial \\Omega $ , it suffices to show that $\\nabla ^{2}\\Phi _2\\in L^{1}(\\lbrace |x|\\ge 2R\\rbrace )$ .",
"Since $h\\in C(\\partial \\Omega )$ satisfies $\\int _{\\partial \\Omega }h\\textrm {d}{\\mathcal {H}}=0$ , we observe that $\\tilde{\\Phi }_{2}(x)&=-\\int _{\\partial \\Omega }(E(x-y)-E(x))h(y)\\textrm {d}{\\mathcal {H}}(y)\\\\&=\\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }y\\cdot (\\nabla E)(x-ty)h(y)\\textrm {d}{\\mathcal {H}}(y).$ Since $\\Omega ^{c}\\subset B_{0}(R)$ , for $|x|\\ge 2R$ we observe that $|x-ty|&\\ge \\big |\\ |x|-|ty|\\ \\big | \\\\&\\ge |x|-R \\\\&\\ge \\frac{|x|}{2}.$ Since $\\tilde{\\Phi }_{2}$ agrees with $\\Phi _2$ up to constant, we estimate $|\\nabla ^{2}\\Phi _2(x)|&\\lesssim \\int _{0}^{1}\\textrm {d}t\\int _{\\partial \\Omega }\\frac{|h(y)|}{|x-ty|^{n+1}}\\textrm {d}{\\mathcal {H}}(y)\\\\&\\lesssim \\frac{1}{|x|^{n+1}}||h||_{L^{1}(\\partial \\Omega )}.$ Thus, $\\nabla ^{2}\\Phi _2$ is integrable in $\\lbrace |x|\\ge 2R\\rbrace $ .",
"The proof is complete.",
"By Propositions A.2 and A.6, the assertion follows.",
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]
] | 1606.05040 |
[
[
"LOLAS-2 : redesign of an optical turbulence profiler"
],
[
"Abstract We present the development, tests and first results of the second generation Low Layer Scidar (LOLAS-2).",
"This instrument constitutes a strongly improved version of the prototype Low Layer Scidar, which is aimed at the measurement of optical turbulence profiles close to the ground, with high altitude-resolution.",
"The method is based on the Generalised Scidar principle which consists in taking double-star scintillation images on a defocused pupil plane and calculating in real time the autocovariance of the scintillation.",
"The main components are an open-truss 40-cm Ritchey-Chr\\'etien telescope, a german-type equatorial mount, an Electron Multiplying CCD camera and a dedicated acquisition and real-time data processing software.",
"The new optical design of LOLAS-2 is significantly simplified compared with the prototype.",
"The experiments carried out to test the permanence of the image within the useful zone of the detector and the stability of the telescope focus show that LOLAS-2 can function without the use of the autoguiding and autofocus algorithms that were developed for the prototype version.",
"Optical turbulence profiles obtained with the new Low Layer Scidar have the best altitude-resolution ever achieved with Scidar-like techniques (6.3 m).",
"The simplification of the optical layout and the improved mechanical properties of the telescope and mount make of LOLAS-2 a more robust instrument."
],
[
"Introduction",
"Turbulent flows in the atmosphere combined with stratified temperatures provoke turbulent fluctuations of the refractive index of air, commonly known as optical turbulence.",
"The intensity of such fluctuations is determined by the second order structure constant $C_{\\mathrm {N}}^2$ .",
"The measurement of $C_{\\mathrm {N}}^2(h)$ , $h$ being the altitude, is of major importance for the development of novel adaptive optical (AO) systems that overcome the angular-resolution degradation introduced by optical turbulence.",
"The statistical study of optical turbulence profiles is also crucial for the characterization of sites where next generation of optical telescopes are to be installed (e.g., [16]).",
"One important limitation of AO systems that are designed to account for phase fluctuations generated all along the atmosphere is the tiny field of view over which the wavefront is corrected.",
"One way to improve image quality over a wide field of view is to correct only the wavefront perturbations that come from turbulent layers close to the ground.",
"This method is known as ground-layer adaptive optics (GLAO) [14], [18].",
"Indeed, the compensation of lower-altitude turbulent-layers provides wider corrected fields of view [7] and turbulence close to the ground is generally the most intense (e.g., [3]).",
"To develop a GLAO system for a given site, it is required to have as much knowledge as possible about the vertical distribution of optical turbulence in the ground layer (e.g, [11]).",
"This requires measurements of $C_{\\mathrm {N}}^2(h)$ with very high altitude-resolution close to the ground.",
"Scintillation Detection and Ranging (SCIDAR) has extensively been used for $C_{\\mathrm {N}}^2$ profiling since its invention by [15].",
"The generalised version of the SCIDAR made it possible to detect turbulence near the ground [10], [4].",
"A more recent implementation of the Generalized SCIDAR on a 40-cm telescope that used a widely-separated double star as a light source and an Electron Multiplying Charge Coupled Device (EMCCD) as the detector, gave place to the Low Layer SCIDAR (LOLAS) [1].",
"The LOLAS prototype was used to characterize the ground-layer turbulence at Mauna Kea, together with a Slope Detection and Ranging instrument [8].",
"Although this several-years campaign gave definitive results, the experience showed that a number of aspects on the prototype LOLAS could be improved in order to optimize data acquisition.",
"In this paper we describe the development and tests of the second generation Low Layer Scidar (LOLAS-2).",
"A few instruments based on optical methods have been developed to measure $C_{\\mathrm {N}}^2(h)$ profiles with the required vertical resolution close to the ground.",
"For example, the surface-layer Slope Detection and Ranging (SL-SLODAR) [12] makes use of two Shack-Hartmann wavefront sensors, each looking at one component of a widely-separated double star.",
"The slope of the wavefronts coming from each star is measured on 5-cm square subapertures whereas LOLAS-2 measures scintillation on square elements of 1-cm side approximately, which makes the SL-SLODAR five times more sensitive than the LOLAS-2 for equal integration times or five times faster for equal number of photons per element.",
"On the other hand, if the instruments were using the same double star separation, LOLAS-2 provides 5 times better altitude resolution.",
"A similar concept that also uses a double star but measures the scintillation from each component on two cameras was implemented in the Stereo-Scidar by [17].",
"The Lunar Scintillometer, which consists of an array of photodiodes that measure scintillation from the moon, has been used to obtain high altitude-resolution turbulence profiles near the ground [19].",
"[9] use a Generalized SCIDAR to distinguish layers at very similar altitudes but moving at different velocities.",
"The present paper is organized as follows: § gives an overview of the LOLAS method.",
"In § REF we briefly describe the prototype version of the instrument and in § REF the second generation of the instrument is presented.",
"Tests on the instrument performances and some measurements are shown in § and § .",
"Conclusions are given in § ."
],
[
"Method",
"The Low Layer Scidar is based on the principle of the Generalized Scintillation Detection and Ranging (G-SCIDAR) technique [15], [10], [4].",
"In this section, we present a brief overview of the LOLAS method, since a complete description can be found in [1].",
"The G-SCIDAR principle can be summarized as follows: double-star scintillation patterns are recorded on short exposure-time images on a virtual plane located a distance $h_\\mathrm {gs}$ from the telescope pupil.",
"In G-SCIDAR experiments this virtual plane is located below the telescope pupil, thus $h_\\mathrm {gs}<0$ .",
"A schematic view of the optical setup is shown in Fig.",
"REF .",
"Each image consists of a randomly distributed intensity pattern.",
"The autocovariance of this stochastic illumination is obtained by computing the spatial normalized autocorrelation of each image and averaging those autocorrelations over thousands of statistically independent image samples.",
"Each layer of optical turbulence contributes to the resulting scintillation autocovariance with three covariance peaks: one centred at the autocovariance origin and two others, identical to each other, separated from the origin by $\\mathbf {d}_\\mathrm {l}=-\\rho \\vert h-h_\\mathrm {gs}\\vert $ and $\\mathbf {d}_\\mathrm {r}=\\rho \\vert h-h_\\mathrm {gs} \\vert $ , respectively, where $\\rho $ denotes the angular separation of the double star and $h$ the layer altitude above the ground.",
"Knowing $\\rho $ and the conjugation altitude $h_\\mathrm {gs}$ , the experimental determination of $\\mathbf {d}_\\mathrm {l}$ (or $\\mathbf {d}_\\mathrm {r}$ ) leads to an estimate of the layer altitude $h$ .",
"The measured autocovariance peaks $C_\\mathrm {gs}(\\mathrm {\\mathbf {r}})$ are proportional to the optical turbulence strength at altitude $h$ , $C_\\mathrm {N}^2(h)$ , and to the scintillation autocovariance function $K(\\mathbf {r}, \\vert h-h_\\mathrm {gs} \\vert )$ .",
"In the realistic case of multiple layers, the response of each layer adds up to the measured autocovariance, resulting in Eq.",
"1 of [1].",
"This equation consists of an integral with respect the altitude $h$ that is similar to a convolution, except that the kernel $K(\\mathbf {r}, \\vert h-h_\\mathrm {gs} \\vert )$ depends on the integration variable.",
"One needs to invert this integral equation to retrieve $C_\\mathrm {N}^2(h)$ .",
"Due to its similarity with a convolution integral, we developed an algorithm based on the CLEAN method, but in which the kernel is recalculated for each different value of $h$ .",
"Our modified CLEAN algorithm is based on that reported by [13].",
"As explained by [1], the achievable altitude resolution using the modified CLEAN method, when the target is at the zenith is $\\Delta h=0.52\\sqrt{\\lambda (\\vert h-h_\\mathrm {gs} \\vert )}/\\rho $ , where $\\lambda $ is the wavelength and $\\rho =\\vert \\rho \\vert $ .",
"When the star is located at an elevation angle $\\theta $ , $\\Delta h$ is decreased by a factor $\\cos \\theta $ .",
"We take $\\lambda =0.5 \\mu \\mathrm {m}$ , which corresponds to the maximum sensitivity of our detector.",
"For a telescope having an aperture $D$ , the maximum altitude for which the $C_{\\mathrm {N}}^2$ value can be measured is given by $h_\\mathrm {max}=D/\\rho $ [1].",
"The Low Layer Scidar concept consists of putting into practice a G-SCIDAR on a dedicated portable telescope, using widely separated double stars as light sources.",
"As can be seen from the above expressions for $\\Delta h$ and $h_\\mathrm {max}$ , the wider the separation, the better the altitude resolution but the shorter the maximum altitude.",
"LOLAS was designed to use a 40-cm telescope, an EMCCD to improve sensitivity and a real-time computation of the scintillation autocovariance.",
"Sections REF and REF describe the prototype LOLAS version and the second generation instrument, respectively."
],
[
"Prototype version",
"The instrumental setup of LOLAS has been widely explained elsewhere [1], [8], [6].",
"We present a summary of the most important instrumental characteristics.",
"Figure REF shows a schematic view of the prototype LOLAS.",
"The scintillation images are obtained with a Schmidt-Cassegrain telescope of focal ratio f/10 and diameter $D = 40.64$ cm and installed on an equatorial mount, manufactured by Meade.",
"The optics consists of two achromatic lenses of 50 mm focal-length.",
"With this optical arrangement, the virtual analysis plane is located 1.94 km below the pupil.",
"The diameter of the pupil image on the detector is $D^\\prime = 24.5$ mm.",
"The scintillation images are captured by an EMCCD camera (Andor iXon) with $512\\times 512$ square pixels of $16~\\mu \\mathrm {m}$ .",
"The frames are binned 2 $\\times $ 2, and the active zone is limited to an array of $256 \\times 80$ binned pixels.",
"The exposure time of each frame ranges from 3 to 10 ms, depending on the wind conditions.",
"The typical number of images to obtain one autocovariance is set to 30000.",
"The EMCCD camera is mounted on a base that is attached to the rear of the telescope (see Fig.",
"REF ).",
"The same equipment, but with different optics, is used to form the SLODAR instrument.",
"To switch between each instrument with the required positioning accuracy, a manual exchange mechanism was installed in front of the camera."
],
[
"Second generation",
"As seen in Fig.",
"REF , LOLAS-2 uses a Ritchey-Chrétien open telescope of focal ratio f/9 and diameter $D = 40.64$ cm, manufactured by RC Optical Systems, a German equatorial mount (1200GTO) manufactured by Astro-Physics, an EMCCD camera (Andor iXon) to acquire scintillation images and a Sbig St-402ME camera for the finder telescope.",
"One advantage of using a Ritchey-Chrétien telescope is that when the position of the focal plane is changed, the effective focal length of the telescope remains unchanged.",
"Our RC Optical Systems telescope is equipped with a system that maintains the focus by monitoring the secondary mirror position in closed loop at a frequency of 6 kHz, reaching an accuracy in the secondary mirror and the focus positions of 0.6 and 25.4 $\\mu \\mathrm {m}$ , respectively.",
"This control system is part of the telescope.",
"In addition, the fact that the telescope is open prevents air at different temperatures to get trapped inside the tube in a turbulent convective flow, which would add an instrumental bias to the $C_{\\mathrm {N}}^2$ measurements at ground level.",
"This was the case with the prototype LOLAS.",
"Removal of the spurious turbulence from the measurements was performed in a post-processing procedure using the method described by [5], as reported by [8].",
"In LOLAS-2, this post-processing step is avoided, making the data reduction faster and simpler.",
"Concerning the mount, LOLAS-2 incorporates a German-type mount that reduces considerably the lever arm between the equatorial and declination axis.",
"Moreover, the Astro-Physics mount has a higher stiffness and the worm gear accuracy is significantly better, compared to those of the Meade mount.",
"The prototype version uses the optics of the telescope and achromatic doublets to define the spatial sampling and conjugation distance $h_{gs}$ below the pupil.",
"The second generation instrument was developed so as to dispense the use of the achromatic doublets and the exchange mechanism.",
"It only uses the telescope optics, making it a simpler and more robust instrument.",
"The beam is no longer collimated, like in the prototype version.",
"The EMCDD is placed directly a distance $L$ before the telescope focal plane.",
"Distance $L$ is chosen such that the detector plane is made the conjugate of the a virtual plane located a distance $h_\\mathrm {gs}$ below the telescope pupil (see Fig.",
"REFa) and the spatial sampling on this plane is well-suited to sample the scintillation speckles (see Fig.",
"REFb).",
"Using the thin lens equation, it can easily be shown that $h_\\mathrm {gs}$ is related to $L$ by the following expression: $h_\\mathrm {gs} = -\\frac{{F_\\mathrm {tel}}^2-F_\\mathrm {tel}L}{L},$ where $F_\\mathrm {tel}$ is the focal length of the telescope.",
"For the spatial sampling, Fig.",
"REFb illustrates the demagnification relation: $\\frac{\\mathcal {L}_{D}}{L}= \\frac{\\mathcal {L}_\\mathrm {min}}{F_\\mathrm {tel}}$ where $\\mathcal {L}_\\mathrm {min}$ represents the typical size of the smallest scintillation speckles on the pupil and $\\mathcal {L}_{D}$ is its corresponding size on the detector plane.",
"[13] showed that the full width at half maximum of the autocovariance of the scintillation produced at altitude $h$ is given by $\\mathcal {L}(h)=0.78\\sqrt{\\lambda \\vert h-h_\\mathrm {gs} \\vert )}.$ This is, the typical size of the smallest speckle is $\\mathcal {L}_\\mathrm {min}\\equiv \\mathcal {L}(0)=0.78\\sqrt{\\lambda \\vert h_\\mathrm {gs}\\vert }$ .",
"Solving Eq.",
"REF for $\\mathcal {L}_{D}$ and replacing the above expression for $\\mathcal {L}_\\mathrm {min}$ gives: $\\mathcal {L}_{D}= L\\frac{0.78\\sqrt{\\lambda \\vert h_\\mathrm {gs}\\vert }}{F_\\mathrm {tel}}.$ For ground-level turbulence to be detectable, $h_\\mathrm {gs}$ must be smaller than $-1000$ m. A good spatial sampling of the scintillation speckles is obtained when $\\mathcal {L}_{D}\\simeq 2p$ , where $p$ is the size of the elementary sampling element.",
"We chose to acquire images with the camera pixels binned $2\\times 2$ .",
"In that case, $p=2d_\\mathrm {pix}$ .",
"The camera pixel size is $d_\\mathrm {pix}=16\\;\\mu \\mathrm {m}$ .",
"Table gives values of different parameters of interest for different values of $L$ .",
"It can be seen that a good compromise is obtained when the detector is located a distance $L=11$ mm before the telescope focal plane, as the conjugation distance is large enough ($h_\\mathrm {gs} = -1212$ m) and the number of spatial samples per smallest speckle width is 1.8, which is very close to 2, the Nyquist criterion.",
"Even though the undersampling is small, it is taken into account in the kernel of the inversion process that calculates $C_{\\mathrm {N}}^2$ profiles from the measured autocovariances.",
"As a consequence of this analysis, the value for $L$ is set to 11 mm in LOLAS-2.",
"For this value of $L$ , the distance separating two contiguous binned pixels corresponds to an angular separation of $1.8^{\\prime \\prime }$ ."
],
[
"Instrument performance",
"In this section we present results concerning the focus stability and guiding performance obtained the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM), Baja California, Mexico.",
"The data was obtained on the nights of 2013 June 15, 16 and 17.",
"Table summarises the pertinent characteristics of the double-star targets used."
],
[
"Focus stability",
"Focus stability is extremely important to maintain the spatial sampling and conjugation altitude along data acquisition.",
"A variation of the telescope focus position translates into a variation of the pupil image diameter $d$ on the detector.",
"This diameter is continuously monitored during the standard data acquisition of the instrument.",
"Images are sent by the EMCCD to the computer in packets of 200 consecutive 256x80 pixels frames.",
"In each frame, the image is centred (as explained in §REF ) and then co-added to form a mean image made of 200 frames.",
"The pupil diameters are calculated from this mean image as follows (see Fig.",
"REF ): the mean image is integrated along columns to form a row of the accumulated values.",
"The width at half maximum of the left and right pupils on the accumulated-values row determines their diameter in number of pixels $N_{d,l}$ and $N_{d,r}$ , respectively.",
"To estimate the focus stability, the diameter of each pupil was monitored during several hours on three nights, using the same double star as source.",
"The total number of pupil size measurements was 4566.",
"The mean and standard deviation of the measured diameters are $\\left<N_{d}\\right>=38.13$ and $\\sigma _{N_{d}}=0.40$ pixels.",
"The average of the temperature and its variation for each night was $13.5\\pm 0.22$ , $14.6\\pm 0.23$ and $14.1\\pm 0.07$ Celcius degrees, according to the weather station at SPM.",
"The obtained standard deviations can be due to uncertainties in the estimation procedure and/or to actual physical variation in the focus or camera positions.",
"If we consider the latter to be the cause of the measured diameter fluctuations, the consequence of the deviation of 0.40 pixels in pupil diameter implies a variation of 13 m in altitude conjugation and 0.32 $\\mu $ m in the sampling element size on the detector, approximately, which are tolerable values.",
"This result suggests that the use of an auto-focus algorithm, which was developed for the prototype LOLAS, could be avoided in the second generation instrument, although more tests under varying temperature conditions should be performed."
],
[
"Guiding test",
"Maintaining a constant position of the pupil images on the EMCCD is important to correctly compute the mean image, the autocorrelation of which is used to normalize the mean autocorrelation of scintillation images, so obtaining the scintillation autocovariance.",
"On wind conditions commonly encountered in astronomical observatories, telescope shake may cause pupil images to move on the detector.",
"This eventual image wander is corrected for by centering the image in every single frame captured by the detector.",
"The guiding performance of the telescope mount is important to keep pupil images within the useful window of the EMCCD during data acquisition that may last hours.",
"The image position on the active area of the EMCCD is determined as follows: we construct an artificial reference image $I_\\mathrm {r}(\\mathbf {r})$ formed by two disks of 38 pixels in diameter each and separated from each other by the same distance as the observed double star.",
"We compute the cross-correlation $C_\\mathrm {c}(\\mathbf {r})$ of the current image $I_i(\\mathbf {r})$ with $I_\\mathrm {r}(\\mathbf {r})$ .",
"The position of the pupil images $(X,Y)$ in $I_i(\\mathbf {r})$ is set as the position of the maximum value of $C_\\mathrm {c}(\\mathbf {r})$ with respect the frame centre.",
"In standard operation of LOLAS-2, image positions are not saved on disk.",
"To test the guiding performance and image stability we recorded image positions $(X,Y)$ during several observations.",
"The $(X,Y)$ coordinates on the EMCCD were rotated according to the double-star position angle to obtain image positions in right ascension ($\\alpha $ ) and declination ($\\delta $ ) coordinates system.",
"This allows us to investigate separately the guiding and stability behaviour on the two rotation axis of the mount.",
"Figures REF and REF show two examples.",
"For the results obtained on 2013 June 17 UT (Fig.",
"REF ) wind was blowing from the South-West with mean speed of 15.5 km h$^{-1}$ and a maximum value of 18.0 km h$^{-1}$ .",
"The image positions remained very stable apart from a slow continuous drift on $\\delta $ and a slow oscillation in $\\alpha $ , presumably due to an inexact alignment of the mount and a periodic error on the right-ascension gear-mechanism, respectively.",
"On 2013 June 15 UT wind was less benevolent, blowing from the West at a mean speed of 26.6 km h$^{-1}$ with gusts reaching 40 km h$^{-1}$ .",
"Figure REF shows an example of that night.",
"Even though images were moving significantly, the mean position remained constant and pupil images remained within the EMCCD working window, which enabled turbulence profiles to be measured in these conditions.",
"To investigate further the dependence of image jitter on prevailing wind speed, in Fig.",
"REF it is shown a plot of the standard deviation of image position as a function of wind speed.",
"For each of the available 4566 frame-packets, the standard deviation $\\sigma _\\eta $ of the image position was calculated.",
"The frame rate within each packet is 13-ms per frame.",
"The wind conditions that prevailed at the time of each measurement were obtained from the database of the OAN-SPM weather stationwww.astrossp.unam.mx/weather15/.",
"Bar extremities on Fig.",
"REF indicate the 25 and 75 percentiles of the $\\sigma _\\eta $ values for a given wind speed.",
"As expected, strong winds produce large image jitter, but surprisingly, $\\sigma _\\eta $ values remain lower than $5^{\\prime \\prime }$ for wind speeds lower than 30 $\\mathrm {ms^{-1}}$ .",
"Even though the force exerted by the wind on the telescope is proportional to the wind speed, the mount and telescope structures move significantly only if the wind speed exceeds a certain value that seems to lie between 30 and 35 $\\mathrm {ms^{-1}}$ .",
"The telescope and mount stiffness, together with the guiding performance have proven to suffice for conducting observations in moderate to strong wind conditions, without the need of an auto guiding procedure like in the prototype LOLAS."
],
[
" $C_{\\mathrm {N}}^2$ Measurement examples",
"In this section we present a few examples of measured scintillation autocorrelations and corresponding turbulence profiles.",
"The first results of LOLAS-2 were obtained in 2013 November at the OAN-SPM.",
"The instrument was installed on a concrete pillar (see Fig.",
"REF ) to reduce vibrations.",
"Figure REF shows examples of autocovariance maps obtained from 30 000 scintillation images, during the nights 2013 November 16 and 17.",
"In the central peaks of Figs.",
"REFa and REFc , the contribution of all turbulent layers in the atmosphere are added up.",
"The correlated speckles produced by turbulent layers above $h_{\\mathrm {max}}$ are separated by a distance longer than the pupil diameter, thereby not giving rise to lateral peaks inside the autocovariance-map boundaries and avoiding the corresponding ${C_{\\mathrm {N}}^2}$ estimate.",
"Only the layers below $h_{\\mathrm {max}}$ form visible lateral peaks from which ${C_{\\mathrm {N}}^2}$ values are retrieved.",
"A white rectangle frames the right-hand side lateral peaks in Figs.",
"REFa and REFc.",
"Enlarged views are shown in Fig.",
"REFb and REFd.",
"The most intense central peak inside each of those frames corresponds to the turbulence at ground level.",
"Weaker peaks can clearly be seen in Fig.",
"REFa, which correspond to turbulence above the ground.",
"Fig.",
"REFd only exhibits ground-level turbulence.",
"The ${C_{\\mathrm {N}}^2(h)}$ profiles are obtained using a modified CLEAN algorithm [1] based on the one presented by [13].",
"In Fig.",
"REF we present a few profiles obtained on 2013 November 16 and 17 using autocovariance maps whose examples are shown in Fig REF .",
"The purpose of Fig.",
"REF is only to display the ability of the instrument in measuring ${C_{\\mathrm {N}}^2}$ profiles and their characteristics.",
"It is not intended for studying the optical turbulence at the site.",
"Altitude resolution, maximum sensed altitude and noise level values are summarised in Table .",
"Note that the altitude resolution obtained when using 12 Cam as a target is the best ever achieved with Scidar-like techniques.",
"The noise levels indicated in Table are the $C_{\\mathrm {N}}^2$ values that correspond to the $3\\sigma $ , i.e.",
"three times the standard deviation on the autocovariance-map background [1].",
"The profiles corresponding to November 16 show three turbulent layers: one at ground level, the second at 81 m and the weakest at 390 m. On November 17, almost all turbulence below 400 m was concentrated at ground level.",
"Only a weak turbulent layer is detected at 170 m above the ground at 11:27 UT that night.",
"It is worth recalling that $C_{\\mathrm {N}}^2$ values obtained with Scidar techniques in which the pupil images are not superimposed on the detector, like LOLAS, do not need to be corrected for the normalization error pointed out by [2]."
],
[
"Conclusions",
"The second generation of the Low Layer Scidar incorporates a simplified optical layout, a stiff and precise telescope mount and an open Ritchey-Chrétien telescope with excellent focus stability.",
"The analysis of the image positions showed that the guiding quality of the mount permits to avoid the autoguiding algorithm that was used in the prototype LOLAS.",
"Similarly, the excellent focus stability of the telescope allows for observations without the autofocus algorithm employed in the former version of the instrument.",
"Examples of measurements obtained with LOLAS-2 illustrate the capability of the instrument for very high altitude-resolution profiling of the optical turbulence.",
"A statistical analysis of $C_{\\mathrm {N}}^2(h)$ profiles obtained so far at the OAN-SPM will be presented in a forthcoming paper.",
"We are deeply grateful to the staff of the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM) for their kind help in all the logistics for the observations.",
"Wind data used in the work was provided by the OAN-SPM wheather station (http://tango.astrosen.unam.mx).",
"Financial support was provided by DGAPA–UNAM through grants IN103913 and IN115013.",
"1 lcccc Instrumental parameters for different values of $L$ 0pt Parameter $L_1$ =15 mm $\\mathbf {L_2=}$11 mm $L_3$ =7.5 mm $L_4$ =5 mm $h_\\mathrm {gs}$ [m] -886 -1212 -1777 -2668 $d$ [mm]a 1.66 1.22 0.83 0.55 $N_{d}$ [pixels]b 52 38 26 17 $\\mathcal {L}_{D}/p$ c 2.58 1.80 1.29 0.86 aDiameter of the pupil image on the detector plane.",
"bNumber of binned pixels along a pupil image diameter (Fig.",
"REF ).",
"cNumber of binned pixels along a speckle (for $h=0$ ).",
"lccccc Targets 0pt Name ${\\alpha _{2000}}$ a ${\\delta _{2000}}$ a ${m_{1}}$ b ${m_{2}}$ b $\\rho $ c [$^{\\prime \\prime }$ ] 15 Tri 2$^{\\mathrm {h}}$ :35 34$^{\\circ }$ :41 5.5 6.7 138.9 12 Cam 5$^{\\mathrm {h}}$ :06 58$^{\\circ }$ :58 5.2 6.2 181.3 a Right ascension ($\\alpha _{2000}$ ) and Declination ($\\delta _{2000}$ ).",
"b Visible magnitude of each star.",
"c Angular separation.",
"lccccc Observational parameters 0pt Date Target $\\Delta h$ a ${h_{\\mathrm {max}}}$ b $\\tau $ c N. L.d 2013/11/16 15 Tri 11.7 603 3 9.5$\\times 10^{-16}$ 2013/11/17 12 Cam 6.3 462 2 1.3$\\times 10^{-15}$ a Altitude resolution [m].",
"b Maximum altitude [m].",
"c Exposure time [ms].",
"d Noise level [$\\mathrm {m}^{-2/3}$ ]."
]
] | 1606.05217 |
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